Temperature Viscosities of liquids Viscosity (mN s/m2) or (cP) 100 60 50 40 30 20 475 450 425 400 375 350 325 300 275 250 390 380 330 320 290. Coulson & Richardson's Chemical Engineering. Chemical Engineering, Volume 1, Sixth edition. Fluid Flow, Heat Transfer and Mass Transfer.
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CHEMICAL ENGINEERING Solutions to the Problems in Chemical Engineering Volume 1
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Coulson & Richardson’s Chemical Engineering Chemical Engineering, Volume 1, Sixth edition Fluid Flow, Heat Transfer and Mass Transfer J. M. Coulson and J. F. Richardson with J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 2, Fourth edition Particle Technology and Separation Processes J. M. Coulson and J. F. Richardson with J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 3, Third edition Chemical & Biochemical Reactors & Process Control Edited by J. F. Richardson and D. G. Peacock Solutions to the Problems in Volume 1, First edition J. R. Backhurst and J. H. Harker with J. F. Richardson Chemical Engineering, Volume 5, Second edition Solutions to the Problems in Volumes 2 and 3 J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 6, Third edition Chemical Engineering Design R. K. Sinnott
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Coulson & Richardson’s
CHEMICAL ENGINEERING J. M. COULSON and J. F. RICHARDSON
Solutions to the Problems in Chemical Engineering Volume 1 By
J. R. BACKHURST and J. H. HARKER University of Newcastle upon Tyne
With
J. F. RICHARDSON University of Wales Swansea
OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI
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Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd
First published 2001 J. F. Richardson, J. R. Backhurst and J. H. Harker 2001
All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 4950 X Typeset by Laser Words, Madras, India
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Contents Preface 1. Units and dimensions
iv 1
2. Flow of fluids — energy and momentum relationships
16
3. Flow in pipes and channels
19
4. Flow of compressible fluids
60
5. Flow of multiphase mixtures
74
6. Flow and pressure measurement
77
7. Liquid mixing
103
8. Pumping of fluids
109
9. Heat transfer
125
10. Mass transfer
217
11. The boundary layer
285
12. Momentum, heat and mass transfer
298
13. Humidification and water cooling
318
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Preface Each of the volumes of the Chemical Engineering Series includes numerical examples to illustrate the application of the theory presented in the text. In addition, at the end of each volume, there is a selection of problems which the reader is invited to solve in order to consolidate his (or her) understanding of the principles and to gain a better appreciation of the order of magnitude of the quantities involved. Many readers who do not have ready access to assistance have expressed the desire for solutions manuals to be available. This book, which is a successor to the old Volume 4, is an attempt to satisfy this demand as far as the problems in Volume 1 are concerned. It should be appreciated that most engineering problems do not have unique solutions, and they can also often be solved using a variety of different approaches. If therefore the reader arrives at a different answer from that in the book, it does not necessarily mean that it is wrong. This edition of the solutions manual relates to the sixth edition of Volume 1 and incorporates many new problems. There may therefore be some mismatch with earlier editions and, as the volumes are being continually revised, they can easily get out-of-step with each other. None of the authors claims to be infallible, and it is inevitable that errors will occur from time to time. These will become apparent to readers who use the book. We have been very grateful in the past to those who have pointed out mistakes which have then been corrected in later editions. It is hoped that the present generation of readers will prove to be equally helpful! J. F. R.
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SECTION 1
Units and Dimensions PROBLEM 1.1 98% sulphuric acid of viscosity 0.025 N s/m2 and density 1840 kg/m3 is pumped at 685 cm3 /s through a 25 mm line. Calculate the value of the Reynolds number.
Solution Cross-sectional area of line D 00030004/400050.0252 D 0.00049 m2 . Mean velocity of acid, u D 0003685 ð 1000036 0005/0.00049 D 1.398 m/s. ∴ Reynolds number, Re D du / D 00030.025 ð 1.398 ð 18400005/0.025 D 2572
PROBLEM 1.2 Compare the costs of electricity at 1 p per kWh and gas at 15 p per therm.
Solution Each cost is calculated in p/MJ. 1 kWh D 1 kW ð 1 h D 00031000 J/s000500033600 s0005 D 3,600,000 J or 3.6 MJ 1 therm D 105.5 MJ ∴ cost of electricity D 1 p/3.6 MJ or 00031/3.60005 D 0.28 p/MJ cost of gas D 15 p/105.5 MJ or 000315/105.50005 D 0.14 p/MJ
PROBLEM 1.3 A boiler plant raises 5.2 kg/s of steam at 1825 kN/m2 pressure, using coal of calorific value 27.2 MJ/kg. If the boiler efficiency is 75%, how much coal is consumed per day? If the steam is used to generate electricity, what is the power generation in kilowatts assuming a 20% conversion efficiency of the turbines and generators? 1
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution From the steam tables, in Appendix A2, Volume 1, total enthalpy of steam at 1825 kN/m2 D 2798 kJ/kg. ∴
enthalpy of steam D 00035.2 ð 27980005 D 14,550 kW
Neglecting the enthalpy of the feed water, this must be derived from the coal. With an efficiency of 75%, the heat provided by the coal D 000314,550 ð 1000005/75 D 19,400 kW. For a calorific value of 27,200 kJ/kg, rate of coal consumption D 000319,400/27,2000005 D 0.713 kg/s or:
00030.713 ð 3600 ð 240005/1000 D 61.6 Mg/day
20% of the enthalpy in the steam is converted to power or: 000314,550 ð 200005/100 D 2910 kW or 2.91 MW say 3 MW
PROBLEM 1.4 The power required by an agitator in a tank is a function of the following four variables: (a) (b) (c) (d)
diameter of impeller, number of rotations of the impeller per unit time, viscosity of liquid, density of liquid.
From a dimensional analysis, obtain a relation between the power and the four variables. The power consumption is found, experimentally, to be proportional to the square of the speed of rotation. By what factor would the power be expected to increase if the impeller diameter were doubled?
Solution If the power P D f0003DN 0005, then a typical form of the function is P D kDa Nb c d , where k is a constant. The dimensions of each parameter in terms of M, L, and T are: power, P D ML2 /T3 , density, D M/L3 , diameter, D D L, viscosity, D M/LT, and speed of rotation, N D T00031 Equating dimensions: M: 1 DcCd L : 2 D a 0003 3c 0003 d T : 00033 D 0003b 0003 d Solving in terms of d : a D 00035 0003 2d0005, b D 00033 0003 d0005, c D 00031 0003 d0005 0001 5 3 0002 D N d ∴ PDk D2d Nd d or: that is:
P/D5 N3 D k0003D2 N /000b00050003d NP D k Rem
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UNITS AND DIMENSIONS
Thus the power number is a function of the Reynolds number to the power m. In fact NP is also a function of the Froude number, DN2 /g. The previous equation may be written as: P/D5 N3 D k0003D2 N /000b0005m P / N2
Experimentally: From the equation,
P / Nm N3 , that is m C 3 D 2 and m D 00031
Thus for the same fluid, that is the same viscosity and density: 0003P2 /P1 00050003D15 N31 /D25 N32 0005 D 0003D12 N1 /D22 N2 000500031 or: 0003P2 /P1 0005 D 0003N22 D23 0005/0003N21 D13 0005 In this case, N1 D N2 and D2 D 2D1 . ∴
0003P2 /P1 0005 D 8D13 /D13 D 8
A similar solution may be obtained using the Recurring Set method as follows: P D f0003D, N, , 0005, f0003P, D, N, , 0005 D 0 Using M, L and T as fundamentals, there are five variables and three fundamentals and therefore by Buckingham’s 0004 theorem, there will be two dimensionless groups. Choosing D, N and as the recurring set, dimensionally: 0003 0004 D0006L L0006D 00031 N0006T T 0006 N00031 Thus: 00033
0006 ML M 0006 L3 D D3 First group, 00041 , is P0003ML2 T00033 000500031 0006 P0003 D3 D2 N3 000500031 0006
P
D5 N3
Second group, 00042 , is 0003ML00031 T00031 000500031 0006 0003 D3 D00031 N000500031 0006 0001
Thus:
f
P ,
D5 N3 D2 N
0002
D2 N
D0
Although there is little to be gained by using this method for simple problems, there is considerable advantage when a large number of groups is involved.
PROBLEM 1.5 It is found experimentally that the terminal settling velocity u0 of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle 0003 weight of displaced fluid), W; fluid density, , and fluid viscosity, . Obtain a relationship for u0 using dimensional analysis. Stokes established, from theoretical considerations, that for small particles which settle at very low velocities, the settling velocity is independent of the density of the fluid
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
except in so far as this affects the buoyancy. Show that the settling velocity must then be inversely proportional to the viscosity of the fluid.
Solution u0 D kda Wb c d , then working in dimensions of M, L and T:
If:
0003L/T0005 D k0003La 0003ML/T2 0005b 0003M/L3 0005c 0003M/LT0005d 0005 Equating dimensions: M: 0 DbCcCd L : 1 D a C b 0003 3c 0003 d T : 00031 D 00032b 0003 d Solving in terms of b: a D 00031, c D 0003b 0003 10005, and d D 00031 0003 2b0005 ∴
u0 D k00031/d00050003Wb 00050003 b / 00050003000b/000b2b 0005 where k is a constant,
or:
u0 D k0003000b/d 00050003W /000b2 0005b
Rearranging: 0003du0 /000b0005 D k0003W /000b2 0005b where (W /000b2 ) is a function of a form of the Reynolds number. For u0 to be independent of , b must equal unity and u0 D kW/d Thus, for constant diameter and hence buoyant weight, the settling velocity is inversely proportional to the fluid viscosity.
PROBLEM 1.6 A drop of liquid spreads over a horizontal surface. What are the factors which will influence: (a) the rate at which the liquid spreads, and (b) the final shape of the drop? Obtain dimensionless groups involving the physical variables in the two cases.
Solution (a) The rate at which a drop spreads, say R m/s, will be influenced by: viscosity of the liquid, ; volume of the drop, V expressed in terms of d, the drop diameter; density of the liquid, ; acceleration due to gravity, g and possibly, surface tension of the liquid,
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UNITS AND DIMENSIONS
5
0019. In this event: R D f0003000b, d, , g, 00190005. The dimensions of each variable are: R D L/T, D M/LT, d D L, D M/L3 , g D L/T2 , and 0019 D M/T2 . There are 6 variables and 3 fundamentals and hence 00036 0003 30005 D 3 dimensionless groups. Taking as the recurring set, d, and g, then: d 0006 L,
0006 M/L3 g 0006 L/T2
LDd ∴ M D L3 D d3 ∴ T2 D L/g D d/g and T D d0.5 /g0.5
Thus, dimensionless group 1: RT/L D Rd0.5 /dg0.5 D R/0003dg00050.5 dimensionless group 2: LT/M D d0003d0.5 0005/0003g0.5 d3 0005 D /0003g0.5 d1.5 0005
∴
or:
dimensionless group 3: 0019T2 /M D 0019d/0003g d3 0005 D 0019/0003g d2 0005 0001 0002 0019 0.5 R/0003dg0005 D f 0.5 1.5 , g d g d2 0001 0002 2 R2 0019 Df , dg g 2 d3 g d2
(b) The final shape of the drop as indicated by its diameter, d, may be obtained by using the argument in (a) and putting R D 0. An alternative approach is to assume the final shape of the drop, that is the final diameter attained when the force due to surface tension is equal to that attributable to gravitational force. The variables involved here will be: volume of the drop, V; density of the liquid, ; acceleration due to gravity, g, and the surface tension of the liquid, 0019. In this case: d D f0003V, , g, 00190005. The dimensions of each variable are: d D L, V D L3 , D M/L3 , g D L/T2 , 0019 D M/T2 . There are 5 variables and 3 fundamentals and hence 00035 0003 30005 D 2 dimensionless groups. Taking, as before, d,
and g as the recurring set, then: d 0006 L,
0006 M/L3 g 0006 L/T2
LDd ∴ M D L3 D d3 ∴ T2 D L/g D d/g and T D d0.5 /g0.5
Dimensionless group 1: V/L3 D V/d3 Dimensionless group 2: 0019T2 /M D 0019d/0003g d3 0005 D 0019/0003g d2 0005 0001 0002 0019 and hence: 0003d3 /V0005 D f g d2
PROBLEM 1.7 Liquid is flowing at a volumetric flowrate of Q per unit width down a vertical surface. Obtain from dimensional analysis the form of the relationship between flowrate and film thickness. If the flow is streamline, show that the volumetric flowrate is directly proportional to the density of the liquid.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution The flowrate, Q, will be a function of the fluid density, , and viscosity, , the film thickness, d, and the acceleration due to gravity, g, or:
Q D f0003 , g, , d0005, or: Q D K a gb c dd where K is a constant.
The dimensions of each variable are: Q D L2 /T, D M/L3 , g D L/T2 , D M/LT and d D L. Equating dimensions: M: 0 DaCc L : 2 D 00033a C b 0003 c C d T : 00031 D 00032b 0003 c from which, c D 1 0003 2b, a D 0003c D 2b 0003 1, and d D 2 C 3a 0003 b C c D 2 C 6b 0003 3 0003 b C 1 0003 2b D 3b ∴
or:
Q D K0003 2b00031 gb 100032b d3b 0005 Q
D K0003 2 gd3 /000b2 0005b and Q / 100032b .
For streamline flow, Q / 00031 and:
00031 D 1 0003 2b and b D 1
∴
Q / D K0003 2 gd3 /000b2 0005, Q D K0003 gd3 /000b0005
and:
Q is directly proportional to the density,
PROBLEM 1.8 Obtain, by dimensional analysis, a functional relationship for the heat transfer coefficient for forced convection at the inner wall of an annulus through which a cooling liquid is flowing.
Solution Taking the heat transfer coefficient, h, as a function of the fluid velocity, density, viscosity, specific heat and thermal conductivity, u, , , Cp and k, respectively, and of the inside and outside diameters of the annulus, di and d0 respectively, then: h D f0003u, di , d0 , , , Cp , k0005 The dimensions of each variable are: h D H/L2 Tq, u D L/T, di D L, d0 D L, D M/L3 , D M/LT, Cp D H/Mq, k D H/LTq. There are 8 variables and 5 fundamental dimensions and hence there will be 00038 0003 50005 D 3 groups. H and q always appear however as
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UNITS AND DIMENSIONS
7
the group H/q and in effect the fundamental dimensions are 4 (M, L, T and H/q) and there will be 00038 0003 40005 D 4 groups. For the recurring set, the variables di , , k and will be chosen. Thus: di
k
0006 L, L 0006 M/L3 M 0006 M/LT, T 0006 0003H/q0005/LT, 0003H/q0005
D di D L3 D d3i D M/L D d3i /di D d2i / D kLT D kdi d2i / D k d3i /000b
Dimensionless group 1: hL2 T/0003H/q0005 D hd2i d2i /000b0003k d3i /000b0005 D hdi /k Dimensionless group 2: uT/L D u d2i /000bdi D di u / Dimensionless group 3: d0 /L D d0 /di Dimensionless group 4: Cp M/0003H/q0005 D Cp d3i /k0003 d3i /000b0005 D Cp /k ∴
hdi /k D f0003di u /000b, Cp /k, d0 /di 0005 which is a form of equation 9.94.
PROBLEM 1.9 Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection. It is found by experiment that, when the flow is turbulent, increasing the flowrate by a factor of 2 always results in a 50% increase in the coefficient. How would a 50% increase in density of the fluid be expected to affect the coefficient, all other variables remaining constant?
Solution For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as: hd/k D f0003du /000b, Cp /k0005 or:
hd/k D K0003du /000b0005n 0003Cp /k0005m
∴
h2 /h1 D 0003u2 /u1 0005n .
Increasing the flowrate by a factor of 2 results in a 50% increase in the coefficient, or: 1.5 D 2.0n and n D 0003ln 1.5/ ln 2.00005 D 0.585. Also:
h2 /h1 D 0003 2 / 1 00050.585
When 0003 2 / 1 0005 D 1.50, h2 /h1 D 00031.5000050.585 D 1.27 and the coefficient is increased by 27%
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 1.10 A stream of droplets of liquid is formed rapidly at an orifice submerged in a second, immiscible liquid. What physical properties would be expected to influence the mean size of droplet formed? Using dimensional analysis obtain a functional relation between the variables.
Solution The mean droplet size, dp , will be influenced by: diameter of the orifice, d; velocity of the liquid, u; interfacial tension, 0019; viscosity of the dispersed phase, ; density of the dispersed phase, d ; density of the continuous phase, c , and acceleration due to gravity, g. It would also be acceptable to use the term 0003 d 0003 c 0005g to take account of gravitational forces and there may be some justification in also taking into account the viscosity of the continuous phase. On this basis:
dp D f0003d, u, 0019, , d , c , g0005
The dimensions of each variable are: dp D L, d D L, u D L/T, 0019 D M/T2 , D M/LT,
d D M/L3 , c D M/L3 , and g D L/T2 . There are 7 variables and hence with 3 fundamental dimensions, there will be 00037 0003 30005 D 4 dimensionless groups. The variables d, u and 0019 will be chosen as the recurring set and hence: d 0006 L, LDd u 0006 L/T, T D L/u D d/u 0019 0006 M/T2 , M D 0019T2 D 0019d2 /u2 Thus, dimensionless group 1: LT/M D d0003d/u0005/00030019d2 /u2 0005 D u/0019 dimensionless group 2: d L3 /M D d d3 /00030019d2 /u2 0005 D d du2 /0019 dimensionless group 3: c L3 /M D c d3 /00030019d2 /u2 0005 D c du2 /0019 dimensionless group 4: gT2 /L D g0003d2 /u2 0005/d D gd/u2 and the function becomes: dp D f0003000bu/0019, d du2 /0019, c du2 /0019, gd/u2 0005
PROBLEM 1.11 Liquid flows under steady-state conditions along an open channel of fixed inclination to the horizontal. On what factors will the depth of liquid in the channel depend? Obtain a relationship between the variables using dimensional analysis.
Solution The depth of liquid, d, will probably depend on: density and viscosity of the liquid,
and ; acceleration due to gravity, g; volumetric flowrate per unit width of channel, Q,
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UNITS AND DIMENSIONS
and the angle of inclination, ', or:
d D f0003 , , g, Q, '0005
Excluding ' at this stage, there are 5 variables and with 3 fundamental dimensions there will be 00035 0003 30005 D 2 dimensionless groups. The dimensions of each variable are: d D L,
D M/L3 , D M/LT, g D L/T2 , Q D L2 /T, and, choosing Q, and g as the recurring set, then: Q D L2 /T g D L/T2
D M/L3
T D L2 /Q L D gT2 D gL4 /Q2 , L3 D Q2 /g, L D Q2/3 /g1/3 and T D Q4/3 /Qg2/3 D Q1/3 /g2/3 M D L3 D 0003Q2 /g0005 D Q2 /g
Thus, dimensionless group 1: d/L D dg1/3 /Q2/3 or d3 g/Q2 dimensionless group 2: LT/M D 0003Q2/3 /g1/3 00050003Q1/3 /g2/3 0005/Q2 g D /Q
and the function becomes: d3 g/Q2 D f0003000b/Q , '0005
PROBLEM 1.12 Liquid flows down an inclined surface as a film. On what variables will the thickness of the liquid film depend? Obtain the relevant dimensionless groups. It may be assumed that the surface is sufficiently wide for edge effects to be negligible.
Solution This is essentially the same as Problem 1.11, though here the approach used is that of equating indices. d D K0003 a , b , gc , Qd , ' e 0005
If, as before:
then, excluding ' at this stage, the dimensions of each variable are: d D L, D M/L3 , D M/LT, g D L/T2 , Q D L2 /T. Equating dimensions: M: 0DaCb
0003i0005
L : 1 D 00033a 0003 b C c C 2d
0003ii0005
T : 0 D 0003b 0003 2c 0003 d
0003iii0005
Solving in terms of b and c then: from (i)
a D 0003b
from (iii)
d D 0003b 0003 2c
and in (ii)
1 D 3b 0003 b C c 0003 2b 0003 4c or: c D 00031/3 ∴
d D 2/3 0003 b
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
d D K0003 0003b Ð b Ð g00031/3 Ð Q2/30003b 0005
Thus:
dg1/3 /Q2/3 D K0003000b/ Q0005b and:
d3 g/Q2 D K0003000b/ Q0005b 0003'0005e as before.
PROBLEM 1.13 A glass particle settles under the action of gravity in a liquid. Upon what variables would the terminal velocity of the particle be expected to depend? Obtain a relevant dimensionless grouping of the variables. The falling velocity is found to be proportional to the square of the particle diameter when other variables are kept constant. What will be the effect of doubling the viscosity of the liquid? What does this suggest regarding the nature of the flow?
Solution See Volume 1, Example 1.3
PROBLEM 1.14 Heat is transferred from condensing steam to a vertical surface and the resistance to heat transfer is attributable to the thermal resistance of the condensate layer on the surface. What variables are expected to affect the film thickness at a point? Obtain the relevant dimensionless groups. For streamline flow it is found that the film thickness is proportional to the one third power of the volumetric flowrate per unit width. Show that the heat transfer coefficient is expected to be inversely proportional to the one third power of viscosity.
Solution For a film of liquid flowing down a vertical surface, the variables influencing the film thickness υ, include: viscosity of the liquid (water), ; density of the liquid, ; the flow per unit width of surface, Q, and the acceleration due to gravity, g. Thus: υ D f0003000b, , Q, g0005. The dimensions of each variable are: υ D L, D M/LT, D M/L3 , Q D L2 /T, and g D L/T2 . Thus, with 5 variables and 3 fundamental dimensions, 00035 0003 30005 D 2 dimensionless groups are expected. Taking , and g as the recurring set, then: 0006 M/LT, M D LT ∴ L3 D LT, T D L2 /000b
0006 M/L3 , M D L3 2 g 0006 L/T D 2 L/ 2 L4 D 2 / 2 L3 ∴ L3 D 2 / 2 g and L D 2/3 /0003 2/3 g1/3 0005 ∴
T D 0003000b2 / 2 g00052/3 / D 1/3 /0003 1/3 g2/3 0005
and:
M D 0003000b2 / 2 g00051/3 0003000b1/3 /0003 1/3 g2/3 00050005 D 2 /0003 g0005
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UNITS AND DIMENSIONS
11
Thus, dimensionless group 1: QT/L2 D Q0003000b1/3 /0003 1/3 g2/3 00050005/0003000b4/3 /0003 4/3 g2/3 00050005 D Q / dimensionless group 2: υL D υ000b2/3 /0003 2/3 g1/3 0005 or, cubing D υ3 2 g/000b2 0003υ3 2 g/000b2 0005 D f0003Q /000b0005
and: This may be written as:
0003υ3 2 g/000b2 0005 D K0003Q /000b0005n
For streamline flow, υ / Q1/3 or n D 1 and hence: 0003υ3 2 g/000b2 0005 D KQ /000b, υ3 D KQ000b/0003 g0005 and υ D 0003KQ000b/ g00051/3 As the resistance to heat transfer is attributable to the thermal resistance of the condensate layer which in turn is a function of the film thickness, then: h / k/υ where k is the thermal conductivity of the film and since υ / 1/3 , h / k/000b1/3 , that is the coefficient is inversely proportional to the one third power of the liquid viscosity.
PROBLEM 1.15 A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables would you expect the falling velocity of the particle to depend? Obtain the relevant dimensionless groups. For particles of a given density settling in a vessel of large diameter, the settling velocity is found to be inversely proportional to the viscosity of the liquid. How would this depend on particle size?
Solution This problem is very similar to Problem 1.13, although, in this case, the liquid through which the particle settles is contained in a narrow vessel. This introduces another variable, D, the vessel diameter and hence the settling velocity of the particle is given by: u D f0003d, , , D, s , g0005. The dimensions of each variable are: u D L/T, d D L, D M/L3 , D M/LT, D D L, s D M/L3 , and g D L/T2 . With 7 variables and 3 fundamental dimensions, there will be 00037 0003 30005 D 4 dimensionless groups. Taking d, and as the recurring set, then: d 0006 L, LDd 3
0006 M/L , M D L3 D d3 0006 M/LT, T D M/L D d3 /d D d2 / Thus: dimensionless group 1: uT/L D u d2 /0003000bd0005 D du / dimensionless group 2: D/L D D/d dimensionless group 3: s L3 /M D s d3 /0003 d3 0005 D s /
and dimensionless group 4: gT2 /L D g 2 d4 /0003000b2 d0005 D g 2 d3 /000b2 Thus:
0003du /000b0005 D f00030003D/d00050003 s / 00050003g 2 d3 /000b2 00050005
In particular, 0003du /000b0005 D K0003g 2 d3 /000b2 0005n where K is a constant.
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12
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
For particles settling in a vessel of large diameter, u / 1/000b. But 0003u/000b0005 / 00031/000b2 0005n and, when n D 1, n / 1/000b. In this case: 0003du /000b0005 D K0003g 2 d3 /000b2 0005 du / d3 and u / d2
or:
Thus the settling velocity is proportional to the square of the particle size.
PROBLEM 1.16 A liquid is in steady state flow in an open trough of rectangular cross-section inclined at an angle ' to the horizontal. On what variables would you expect the mass flow per unit time to depend? Obtain the dimensionless groups which are applicable to this problem.
Solution This problem is similar to Problems 1.11 and 1.12 although, here, the width of the trough and the depth of liquid are to be taken into account. In this case, the mass flow of liquid per unit time, G will depend on: fluid density, ; fluid viscosity, ; depth of liquid, h; width of the trough, a; acceleration due to gravity, g and the angle to the horizontal, '. Thus: G D f0003 , , h, a, g, '0005. The dimensions of each variable are: G D M/T, D M/L3 , D M/LT, h D L, a D L, g D L/T2 and neglecting ' at this stage, with 6 variables with dimensions and 3 fundamental dimensions, there will be 00036 0003 30005 D 3 dimensionless groups. Taking h, and as the recurring set then: h 0006 L, LDh 3
0006 M/L , M D L3 D h3 0006 M/LT, T D M/L D h3 /0003h000b0005 D h2 / Thus: dimensionless group 1: GT/M D G h2 /0003 h3 0005 D G/000bh dimensionless group 2: a/L D a/h dimensionless group 3: gT2 /L D g 2 h4 /0003000b2 h0005 D g 2 h3 /000b2 and:
0003G/000bh0005 D f00030003a/h00050003g 2 h3 /000b2 00050005
PROBLEM 1.17 The resistance force on a spherical particle settling in a fluid is given by Stokes’ Law. Obtain an expression for the terminal falling velocity of the particle. It is convenient to express experimental results in the form of a dimensionless group which may be plotted against a Reynolds group with respect to the particle. Suggest a suitable form for this dimensionless group. Force on particle from Stokes’ Law D 30004000bdu; where is the fluid viscosity, d is the particle diameter and u is the velocity of the particle relative to the fluid.
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UNITS AND DIMENSIONS
13
What will be the terminal falling velocity of a particle of diameter 10 µm and of density 1600 kg/m3 settling in a liquid of density 1000 kg/m3 and of viscosity 0.001 Ns/m2 ? If Stokes’ Law applies for particle Reynolds numbers up to 0.2, what is the diameter of the largest particle whose behaviour is governed by Stokes’ Law for this solid and liquid?
Solution The accelerating force due to gravity D 0003mass of particle 0003 mass of liquid displaced0005g. For a particle of radius r, volume D 40004r 3 /3, or, in terms of diameter, d, volume D 400040003d3 /23 0005/3 D 0004d3 /6. Mass of particle D 0004d3 s /6, where s is the density of the solid. Mass of liquid displaced D 0004d3 /6, where is the density of the liquid, and accelerating force due to gravity D 00030004d3 s /6 0003 0004d3 /60005g D 00030004d3 /600050003 s 0003 0005g. At steady state, that is when the terminal velocity is attained, the accelerating force due to gravity must equal the drag force on the particle F, or: 00030004d3 /600050003 s 0003 0005g D 30004000bdu0 where u0 is the terminal velocity of the particle. u0 D 0003d2 g/18000b00050003 s 0003 0005
Thus:
(i)
It is assumed that the resistance per unit projected area of the particle, R0 , is a function of particle diameter, d; liquid density, ; liquid viscosity, , and particle velocity, u or R0 D f0003d, , , u0005. The dimensions of each variable are R0 D M/LT2 , d D L, D M/L3 , D M/LT and u D L/T. With 5 variables and 3 fundamental dimensions, there will be 00035 0003 30005 D 2 dimensionless groups. Taking d, and u as the recurring set, then: d 0006 L, LDd 3
0006 M/L , M D L3 D d3 u 0006 L/T, T D L/u D d/u Thus: dimensionless group 1: R0 LT2 /M D R0 d0003d2 /u2 0005/0003 d3 0005 D R0 / u2 dimensionless group 2: LT/M D d0003d/u0005/0003 d3 0005 D /0003du 0005 R0 / u2 D f0003000b/du 0005
and:
R0 / u2 D K0003du /000b0005n D K Ren
or:
(ii)
In this way the experimental data should be plotted as the group (R/ u2 ) against Re . For this particular example, d D 10 µm D 000310 ð 1000036 0005 D 1000035 m; s D 1600 kg/m3 ;
D 1000 kg/m3 and D 0.001 Ns/m2 . Thus, in equation (i):
u0 D 000300031000035 00052 ð 9.81/000318 ð 0.0010005000500031600 0003 10000005 D 3.27 ð 1000035 m/s or 0.033 mm/s
When Re D 0.2, du / D 0.2 or when the terminal velocity is reached: du0 D 0.2000b/ D 00030.2 ð 0.0010005/1000 D 2 ð 1000037 or:
u0 D 00032 ð 1000037 0005/d
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
u0 D 0003d2 g/18000b00050003 s 0003 0005
In equation (i):
00032 ð 1000037 0005/d D 0003d2 ð 9.81/000318 ð 0.0010005000500031600 0003 10000005 ∴
d3 D 6.12 ð 10000313 d D 8.5 ð 1000035 m or 85 µm
and:
PROBLEM 1.18 A sphere, initially at a constant temperature, is immersed in a liquid whose temperature is maintained constant. The time t taken for the temperature of the centre of the sphere to reach a given temperature 'c is a function of the following variables: Diameter of sphere, d Thermal conductivity of sphere, k Density of sphere,
Specific heat capacity of sphere, Cp Temperature of fluid in which it is immersed, 's . Obtain relevant dimensionless groups for this problem.
Solution In this case, t D f0003d, k, , Cp , 'c , 's 0005. The dimensions of each variable are: t D T, d D L, k D ML/Tq, Cp D L2 /T2 q, 'c D q, 's D q. There are 7 variables and hence with 4 fundamental dimensions, there will be 00037 0003 40005 D 3 dimensionless groups. Taking d, , Cp and 'c as the recurring set, then: d
'c Cp
0006 L, L 3 0006 M/L , M 0006 q, q 0006 L2 /T2 q Cp
D d, D L3 D d3 D 'c D d2 /T2 'c and T2 D d2 /Cp 'c or: T D d/Cp0.5 'c0.5
Thus: dimensionless group 1: t/T D tCp0.5 'c0.5 /d dimensionless group 2: kTq/ML D k0003d/Cp0.5 'c0.5 0005'c /0003 d3 0005d D k'c0.5 /Cp0.5 d3 dimensionless group 3: 's /q D 's /'c
PROBLEM 1.19 Upon what variables would the rate of filtration of a suspension of fine solid particles be expected to depend? Consider the flow through unit area of filter medium and express the variables in the form of dimensionless groups. It is found that the filtration rate is doubled if the pressure difference is doubled. What would be the effect of raising the temperature of filtration from 293 to 313 K?
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UNITS AND DIMENSIONS
15
The viscosity of the liquid is given by: D 0 00031 0003 0.0150003T 0003 27300050005 where is the viscosity at a temperature T K and 0 is the viscosity at 273 K.
Solution The volume flow of filtrate per unit area, u m3 /m2 s, will depend on the fluid density, ; fluid viscosity, ; particle size, d; pressure difference across the bed, P, and the voidage of the cake, e or: v D f0003 , , d, P, e0005. The dimensions of each of these variables are u D L/T, D M/L3 , D M/LT, d D L, P D M/LT2 and e D dimensionless. There are 6 variables and 3 fundamental dimensions and hence 00036 0003 30005 D 3 dimensionless groups. Taking, d, and as the recurring set, then: d 0006 L, LDd
0006 M/L3 , M D L3 D d3 0006 M/LT, T D M/L D d3 /0003d000b0005 D d2 / Thus: dimensionless group 1: uT/L D u d2 /0003000bd0005 D du / dimensionless group 2: PLT2 /M D Pd0003 d2 /000b00052 / d3 D P d2 /000b2 and the function is:
P d2 /000b2 D f0003du /000b0005
This may be written as:
P d2 /000b2 D K0003du /000b0005n
Since the filtration rate is doubled when the pressure difference is doubled, then: u / P and n D 1, P d2 /000b2 D Kdu / and:
u D 00031/K0005Pd/000b, or u / 1/000b
0003000b293 /000b313 0005 D 0 00031 0003 0.0150003293 0003 27300050005/000b0 00031 0003 0.0150003313 0003 27300050005 D 00030.7/0.40005 D 1.75 ∴
0003v313 /v293 0005 D 0003000b293 /000b313 0005 D 1.75
and the filtration rate will increase by 75%.
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SECTION 2
Flow of Fluids — Energy and Momentum Relationships PROBLEM 2.1 Calculate the ideal available energy produced by the discharge to atmosphere through a nozzle of air stored in a cylinder of capacity 0.1 m3 at a pressure of 5 MN/m2 . The initial temperature of the air is 290 K and the ratio of the specific heats is 1.4.
Solution From equation 2.1: dU D υq 0002 υW. For an adiabatic process: υq D 0 and dU D 0002υW, and for an isentropic process: dU D Cv dT D 0002υW from equation 2.25. As 0007 D Cp /Cv and Cp D Cv C R (from equation 2.27), Cv D R/ 0007 0002 1 ∴
W D 0002Cv T D 0002RT/ 0007 0002 1 D RT1 0002 RT2 / 0007 0002 1000b
and: 0007
RT1 D P1 v1 and RT2 D P2 v2 and hence: W D P1 v1 0002 P2 v2 / 0007 0002 1 0007
P1 v1 D P2 v2 and substituting for v2 gives: W D [ P1 v1 / 0007 0002 1000b] 1 0002 P2 /P1 000700021000b/0007 U D 0002W D [ P1 v1 / 0007 0002 1000b][ P2 /P1 000700021000b/0007 0002 1]
and: In this problem:
P1 D 5 MN/m2 , P2 D 0.1013 MN/m2 , T1 D 290 K, and 0007 D 1.4. The specific volume, v1 D 22.4/29 290/273 0.1013/5 D 0.0166 m3 /kg. ∴
0002W D [ 5 ð 106 ð 0.0166000b/0.4][ 0.1013/5000b0.4/1.4 0002 1] D 00020.139 ð 106 J/kg
Mass of gas D 0.1/0.0166 D 6.02 kg ∴
U D 0002 0.139 ð 106 ð 6.20 D 00020.84 ð 106 J or 0002 840 kJ
PROBLEM 2.2 Obtain expressions for the variation of: (a) internal energy with change of volume, (b) internal energy with change of pressure, and (c) enthalpy with change of pressure, all at constant temperature, for a gas whose equation of state is given by van der Waals’ law. 16
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FLOW OF FLUIDS — ENERGY AND MOMENTUM RELATIONSHIPS
17
Solution See Volume 1, Example 2.2.
PROBLEM 2.3 Calculate the energy stored in 1000 cm3 of gas at 80 MN/m2 at 290 K using STP as the datum.
Solution The key to this solution lies in the fact that the operation involved is an irreversible expansion. Taking Cv as constant between T1 and T2 , U D 0002W D nCv T2 0002 T1 where n is the kmol of gas and T2 and T1 are the final and initial temperatures, then for a constant pressure process, the work done, assuming the ideal gas laws apply, is given by: W D P2 V2 0002 V1 D P2 [ nRT2 /P2 0002 nRT1 /P1 ] 0001 0002 RT2 RT1 Equating these expressions for W gives: 0002Cv T2 0002 T1 D P2 0002 P2 P1 In this example: P1 D 80000 kN/m2 , P2 D 101.3 kN/m2 , V1 D 1 ð 1000023 m3 , R D 8.314 kJ/kmol K, and T1 D 290 K Hence: 0002Cv T2 0002 290 D 101.3R[ T2 /101.3 0002 290/80,000000b] By definition, 0007 D Cp /Cv and Cp D 0012v C R (from equation 2.27) or: Cv D R/ 0007 0002 1 Substituting:
T2 D 174.15 K.
PV D nRT and n D 80000 ð 1000023 / 8.314 ð 290 D 0.033 kmol ∴ U D 0002W D Cv n T2 0002 T1 D 1.5 ð 8.314 ð 0.033 174.15 0002 290 D 000247.7 kJ
PROBLEM 2.4 Compressed gas is distributed from a works in cylinders which are filled to a pressure P by connecting them to a large reservoir of gas which remains at a steady pressure P and temperature T. If the small cylinders are initially at a temperature T and pressure P0 , what is the final temperature of the gas in the cylinders if heat losses can be neglected and if the compression can be regarded as reversible? Assume that the ideal gas laws are applicable.
Solution From equation 2.1, dU D υq 0002 υW. For an adiabatic operation, q D 0 and υq D 0 and υW D Pdv or dU D 0002Pdv. The change in internal energy for any process involving an
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
ideal gas is given by equation 2.25: Cv dT D 0002Pdv D dU. P D RT/Mv and hence: dT/T D 0002R/MCv dv/v By definition: ∴
0007 D Cp /Cv and Cp D Cv C R/M (from equation 2.27) R/MCv D 0007 0002 1 and dT/T D 0002 0007 0002 1 dv/v000b
Integrating between conditions 1 and 2 gives: ln T2 /T1 D 0002 0007 0002 1 ln v2 /v1 or T2 /T1 D v2 /v1 000700021 P1 v1 /T1 D P2 v2 /T2 and hence v1 /v2 D P2 /P1 T1 /T2 and:
T2 /T1 D P2 /P1 000700021000b/0007
Using the symbols given, the final temperature, T2 D T P/P0 000700021000b/0007
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SECTION 3
Flow in Pipes and Channels PROBLEM 3.1 Calculate the hydraulic mean diameter of the annular space between a 40 mm and a 50 mm tube.
Solution The hydraulic mean diameter, dm , is defined as four times the cross-sectional area divided by the wetted perimeter. Equation 3.69 gives the value dm for an annulus of outer radius r and inner radius ri as: dm D 400050006r 2 0002 ri2 0007/200050006r C ri 0007 D 20006r 0002 ri 0007 D 0006d 0002 di 0007 If r D 25 mm and ri D 20 mm, then: dm D 2000625 0002 200007 D 10 mm
PROBLEM 3.2 3
0.015 m /s of acetic acid is pumped through a 75 mm diameter horizontal pipe 70 m long. What is the pressure drop in the pipe? Viscosity of acid D 2.5 mNs/m2 , density of acid D 1060 kg/m3 , and roughness of pipe surface D 6 ð 1000025 m.
Solution Cross-sectional area of pipe D 00060005/4000700060.07500072 D 0.0044 m2 . Velocity of acid in the pipe, u D 00060.015/0.00440007 D 3.4 m/s. Reynolds number D ud/ D 00061060 ð 3.4 ð 0.070007/00062.5 ð 1000023 0007 D 1.08 ð 105 Pipe roughness e D 6 ð 1000025 m and e/d D 00066 ð 1000025 0007/0.075 D 0.0008 The pressure drop is calculated from equation 3.18 as: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 From Fig. 3.7, when Re D 1.08 ð 105 and e/d D 0.0008, R/000bu2 D 0.0025. Substituting: 0002Pf D 00064 ð 0.00250007000670/0.075000700061060 ð 3.42 0007 D 114,367 N/m2 or: 114.4 kN/m2 19
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.3 A cylindrical tank, 5 m in diameter, discharges through a mild steel pipe 90 m long and 230 mm diameter connected to the base of the tank. Find the time taken for the water level in the tank to drop from 3 m to 1 m above the bottom. The viscosity of water is 1 mNs/m2 .
Solution If at any time the depth of water in the tank is h and levels 1 and 2 are the liquid levels in the tank and the pipe outlet respectively, then the energy balance equation states that: 0006u2 /20007 C gz C v0006P2 0002 P1 0007 C F D 0 In this example, P1 D P2 D atmospheric pressure and v0006P2 0002 P1 0007 D 0. Also u1 /u2 D 00060.23/500072 D 0.0021 so that u1 may be neglected. The energy balance equation then becomes: u2 /2 0002 hg C 40006R/000bu2 00070006l/d0007u2 D 0 The last term is obtained from equation 3.19 and z D 0002h. Substituting the known data: u2 /2 0002 9.81h C 40006R/000bu2 0007000690/0.230007u2 D 0 or: from which:
u2 0002 19.62h C 31300006R/000bu2 0007u2 D 0 p 0001 u D 4.43 h/ [1 C 31300006R/000bu2 0007]
In falling from a height h to h 0002 dh, the quantity of water discharged D 00060005/4000752 00060002dh0007 D 19.63dh m3 . p 0001 Volumetric flow rate D 00060005/4000700060.2300072 u D 0.0415u D 0.184 h/ [1 C 31300006R/000bu2 0007], and the time taken for the level to fall from h to h 0002 dh is: 0002 0002 000219.63 dh p [1 C 31300006R/000bu2 0007] D 0002106.7h00020.5 [1 C 31300006R/000bu2 0007] dh 0.184 h ∴ the time taken for the level to fall from 3 m to 1 m is: 0003 1 0002 t D 0002106.7 [1 C 31300006R/000bu2 0007] h00020.5 dh 3 2
R/000bu depends upon the Reynolds number which will fall as the level in the tank falls and upon the roughness of the pipe e which is not specified in this example. The pressure drop along the pipe D h000bg D 4Rl/d N/m2 and R D h000bgd/4l. From equation 3.23: 0006R/000bu2 0007 Re2 D Rd2 / 2 D h000b2 gd3 /4l 2 D 0006h ð 10002 ð 9.81 ð 0.233 0007/00064 ð 90 ð 1000026 0007 D 3.315 ð 108 h Thus as h varies from 3 m to 1 m, 0006R/000bu2 00070006Re00072 varies from (9.95 ð 108 ) to (3.315 ð 108 .)
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FLOW IN PIPES AND CHANNELS
21
If R/000bu2 is taken as 0.002, Re will vary from (7.05 ð 105 ) to (4.07 ð 105 ). From Fig. 3.7 this corresponds to a range of e/d of between 0.004 and 0.005 or a roughness of between 0.92 and 1.15 mm, which is too high for a commercial pipe. If e is taken as 0.05 mm, e/d D 0.0002, and, for Reynolds numbers near 106 , R/000bu2 D 0.00175. Substituting R/000bu2 D 0.00175 and integrating gives a time of 398 s for the level to fall from 3 m to 1 m. If R/000bu2 D 0.00175, Re varies from (7.5 ð 105 ) to (4.35 ð 105 ), and from Fig. 3.7, e/d D 0.00015, which is near enough to the assumed value. Thus the time for the level to fall is approximately 400 s.
PROBLEM 3.4 Two storage tanks A and B containing a petroleum product discharge through pipes each 0.3 m in diameter and 1.5 km long to a junction at D. From D the product is carried by a 0.5 m diameter pipe to a third storage tank C, 0.8 km away. The surface of the liquid in A is initially 10 m above that in C and the liquid level in B is 7 m higher than that in A. Calculate the initial rate of discharge of the liquid if the pipes are of mild steel. The density of the petroleum product is 870 kg/m3 and the viscosity is 0.7 mNs/m2 .
Solution See Volume 1, Example 3.4
PROBLEM 3.5 Find the drop in pressure due to friction in a glazed porcelain pipe 300 m long and 150 mm diameter when water is flowing at the rate of 0.05 m3 /s.
Solution For a glazed porcelain pipe, e D 0.0015 mm, e/d D 00060.0015/1500007 D 0.00001. Cross-sectional area of pipe D 00060005/4000700060.1500072 D 0.0176 m2 . Velocity of water in pipe, u D 00060.05/0.01760007 D 2.83 m/s. Reynolds number D ud/ D 00061000 ð 2.83 ð 0.150007/1000023 D 4.25 ð 105 From Fig. 3.7, R/000bu2 D 0.0017. The pressure drop is given by equation 3.18: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 or:
4 ð 0.00170006300/0.15000700061000 ð 2.832 0007 D 108,900 N/m2 or 1 MN/m2
PROBLEM 3.6 Two tanks, the bottoms of which are at the same level, are connected with one another by a horizontal pipe 75 mm diameter and 300 m long. The pipe is bell-mouthed at each
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
end so that losses on entry and exit are negligible. One tank is 7 m diameter and contains water to a depth of 7 m. The other tank is 5 m diameter and contains water to a depth of 3 m. If the tanks are connected to each other by means of the pipe, how long will it take before the water level in the larger tank has fallen to 6 m? Assume the pipe to be of aged mild steel.
Solution The system is shown in Fig. 3a. If at any time t the depth of water in the larger tank is h and the depth in the smaller tank is H, a relationship between h and H may be found. Area of larger tank D 00060005/4000772 D 38.48 m2 , area of smaller tank D 00060005/4000752 D 19.63 m2 .
dh 7m h
H 75 mm bore
7m
300 m
x
5m
5m
Figure 3a.
When the level in the large tank falls to h, the volume discharged D 00067 0002 h0007 ð 38.48 m3 . The level in the small tank will rise by a height x, given by: x D 38.4800067 0002 h0007/19.63 D 000613.72 0002 1.95h0007 H D 0006x C 30007 D 000616.72 0002 1.95h0007 The energy balance equation is: u2 /2 C gz C v0006P1 0002 P2 0007 D F u2 /2 may be neglected, and P1 D P2 D atmospheric pressure, so that: gz D F D gz C 40006R/000bu2 00070006l/d0007u2 , z D 0006h 0002 H0007 D 00062.95h 0002 16.720007 or: and:
00062.95h 0002 16.720007g D 40006R/000bu2 00070006l/d0007u2 0001 u D [00062.95h 0002 16.720007g/40006R/000bu2 00070006l/d0007]
As the level falls from h to h 0002 dh in time dt, the volume discharged D 38.480006dh0007 m3 . Hence:
time, dt D
00060005/4000700060.07500072 0001
or:
dt D
0001
000238.48 dh [00062.95h 0002 16.720007g/40006R/000bu2 00070006l/d0007]
00022780 dh [40006R/000bu2 00070006l/d0007] p 00062.95h 0002 16.720007
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FLOW IN PIPES AND CHANNELS
If R/000bu2 is taken as 0.002, then: 0003 0003 dt D 000215,740
6
p 7
dh 00062.95h 0002 16.720007
23
and t D 10590 s
Average volumetric flowrate D 38.4800067 0002 60007/10590 D 0.00364 m3 /s Cross-sectional area of pipe D 0.00442 m2 . Average velocity in the pipe D 00060.00364/0.004420007 D 0.82 m/s. Reynolds number D 00061000 ð 0.82 ð 0.750007/1000023 D 6.2 ð 104 . From Fig. 3.7, if e D 0.05 mm, e/d D 0.00067 and R/000bu2 D 0.0025, which is near enough to the assumed value of 0.002 for a first estimate. Thus the time for the level to fall is approximately 10590 s (2.94 h).
PROBLEM 3.7 Two immiscible fluids A and B, of viscosities A and B , flow under streamline conditions between two horizontal parallel planes of width b, situated a distance 2a apart (where a is much less than b), as two distinct parallel layers one above the other, each of depth a. Show that the volumetric rate of flow of A is: 0004 00050004 0005 0002Pa3 b 7 A C B 12 A l A C B where, 0002P is the pressure drop over a length l in the direction of flow. l Fluid B
RA = Shear stress at centre-plane on A CL RB = Shear stress at centre-plane on B
s Fluid A
Figure .
Solution Considering a force balance on the fluid lying within a distance s from the centre plane, then: 0004 0005 dus For A: 0002P0006sb0007 D bl A C RA l ds A where RA is the shear stress at the centre plane, RA 0002P sds 0002 ds or: 0002d s D A l A Integrating:
00060002us 0007A D
RA 0002P s2 0002 s C k1 A l 2 A
RB 0002P s2 0002 s C k2 A l 2 B where RB is the shear stress at the centre plane on B. Similarly for B:
00060002us 0007B D
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Noting that:
RA D 0002RB
At s D a:
0006us 0007A D 0 0006us 0007B D 0
On the centre plane: At s D 0 0006us 0007A D 0006us 0007B AD0002
RA 00060002P0007 a2 C a A l 2 A
BD0002
RB 00060002P0007 a2 C a B l 2 B
Thus:
usA D
0002P 2 RA 0006a 0002 s0007 fa 0002 s2 g 0002 2 A l A
and:
usB D
0002P 2 RA fa 0002 s2 g C 0006a 0002 s0007 0006since RA D 0002RB 0007 2 B l B
Centre line velocity, and: Equating: ∴
0006uA 0007CL D
RA a 0002Pa2 C 2 A l A
RA a 0002Pa2 C 2 B l B 0007 0006 0007 2 0006 1 1 0002Pa 1 1 0002 D aRA C 2l A B A B 0006 0007 0002Pa B 0002 A RA D 2l B C A 0006uB 0007CL D
Substituting: B 0002 A 0002P 2 0002Pa 0006a 0002 s2 0007 0002 0006a 0002 s0007 2 A l 2 A l B C A 0006 0007 0002P A 0002 B 2 2 D 0006a 0002 s 0007 C a0006a 0002 s0007 2 A l A C B
0006us 0007A D
Total flowrate of A D QA is given by: 00060003 a 0004 0005 0007 0003 a A 0002 B 0002Pb 2 2 2 QA D bus ds D 0006a 0002 s 0007 C 0006a 0002 as0007 ds 2 A l A C B 0 0 0004 0005 0004 0005 a A 0002 B 0002Pb s3 s2 D a2 s 0002 C a2 s 0002 a 2 A l 3 A C B 2 0 3 3 A 0002 B a 0002Pb 2a C D 2 A l 3 A C B 2 A 0002 B 0002Pba3 2 C D 2 A l 3 20006 A C B 0007
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FLOW IN PIPES AND CHANNELS
25
0002Pba3 4 A C 4 B C 3 A 0002 3 B 2 A l 60006 A C B 0007 3 0002Pba 7 A C B D 12 A l A C B
D
PROBLEM 3.8 Glycerol is pumped from storage tanks to rail cars through a single 50 mm diameter main 10 m long, which must be used for all grades of glycerol. After the line has been used for commercial material, how much pure glycerol must be pumped before the issuing liquid contains not more than 1% of the commercial material? The flow in the pipeline is streamline and the two grades of glycerol have identical densities and viscosities.
Solution Making a force balance over an element distance r from the axis of a pipe whose radius is a, then: 0002P0005r 2 D 0002 0006du/dr000720005rl where u is the velocity at distance r and l is the length of the pipe. Hence: and:
du D 000200060002P/2 l0007r dr u D 000200060002P/4 l0007r 2 C constant
When r D a, u D 0, the constant D 00060002P/4 l0007a2 and hence u D 00060002P/4 l00070006a2 0002 r 2 0007. At a distance r from the axis, the time taken for the fluid to flow through a length l is given by 4 l2 /0002P0006a2 0002 r 2 0007. The volumetric rate of flow from r D 0 to r D r is: 0003 r D 00060002P/4 l00070006a2 0002 r 2 000720005rdr 0
D 00060002P0005/8 l000700062a2 r 2 0002 r 4 0007 Volumetric flowrate over the whole pipe D 0002P0005a4 /8 l and the mean velocity D 0002Pa2 /8 l. The required condition at the pipe outlet is: 00060002P0005/8 l000700062a2 r 2 0002 r 4 0007 D 0.99 0002P0005a4 /8 l from which r D 0.95a. The time for fluid at this radius to flow through length l D 4 l2 /0002Pa2 00061 0002 0.952 0007 D 000641 l2 0007/00060002Pa2 ) Hence, the volume to be pumped D 000641 l2 /00060002P0007a2 00070006000500060002P0007a4 /8 l0007 D 410005a2 /8 D 41000500060.2500072 /8 D 0.10 m3
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.9 A viscous fluid flows through a pipe with slightly porous walls so that there is a leakage of kP, where P is the local pressure measured above the discharge pressure and k is a constant. After a length l, the liquid is discharged into a tank. If the internal diameter of the pipe is d and the volumetric rate of flow at the inlet is Qo , show that the pressure drop in the pipe is given by 0002P D 0006Qo /0005kd0007a tanh al, a D 0006128k /d3 00070.5
where:
Assume a fully developed flow with 0006R/000bu2 0007 D 8 Re00021 .
Solution Across a small element of the pipe, the change in liquid flow is: 0002dQ D kP0005ddl and the change in velocity is: du D 00024kPdl/d R/000bu2 D 8 /ud000b
Also:
and: R D 8 u/d
0006i0007
Making a force balance over the element: 0002dP00060005/40007d2 D R0005ddl D 8 u0005dl 0002dP D 32 u dl/d2
and:
0006ii0007
From equations (i) and (ii): 0002dP/du D 00028 u/kPd and:
PdP D 8 u du/kd
Over the whole pipe: 0006Po2 /2 0002 P2 /20007 D 00068 /kd0007[0006uo2 /20007 0002 0006u2 /20007] u2 D uo2 C 0006P2 0002 Po2 00070006kd/8 0007
and:
Assuming zero outlet pressure as a datum, then substituting for u in equation (ii):
0002 0003 0 0002dp [uo2 C 0006P2 0002 Po2 00070006kd/8 0007] D 32 l/d2 Po
Thus:
0001
00068 /kd0007 sinh
00021
0002
Po
00068 uo2 /kd0007
0002
Po2
D 32 l/d2
0002
and:
Po
0002
[8 uo2 /kd0007 0002 Po2 ] D sinh
0006128 k/d3 0007l
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FLOW IN PIPES AND CHANNELS
Writing
0001
27
0006128 k/d3 0007 D a:
00068 uo2 /kd0007 D [a2 0006d3 /128 k0007][8 Qo2 ð 160007/0006kd00052 d4 0007] D Qo2 /0006k 2 d2 00052 a2 0007 and:
Po2 D [0006Qo2 /0006k 2 d2 00052 a2 0007 sinh2 al]/00061 C sinh2 al0007 0002P D Po D 0006Qo /kd00050007a tanh al
or:
PROBLEM 3.10 A petroleum product of viscosity 0.5 m Ns/m2 and density 700 kg/m3 is pumped through a pipe of 0.15 m diameter to storage tanks situated 100 m away. The pressure drop along the pipe is 70 kN/m2 . The pipeline has to be repaired and it is necessary to pump the liquid by an alternative route consisting of 70 m of 200 mm pipe followed by 50 m of 100 mm pipe. If the existing pump is capable of developing a pressure of 300 kN/m2 , will it be suitable for use during the period required for the repairs? Take the roughness of the pipe surface as 0.05 mm.
Solution This problem may be solved by using equation 3.23 and Fig. 3.8 to find the volumetric flowrate and then calculating the pressure drop through the alternative pipe system. From equation 3.23: 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 000670,000 ð 0.153 ð 7000007/00064 ð 100 ð 0.52 ð 1000026 0007 D 1.65 ð 109 From Fig. 3.8, Re D 8.8 ð 105 D 0006700 ð 10.15u0007/00060.5 ð 1000023 0007 and the velocity u D 4.19 m/s. Cross-sectional area D 00060005/400070.152 D 0.0177 m2 . Volumetric flowrate D 00064.19 ð 0.01770007 D 0.074 m3 /s. The velocity in the 0.2 m diameter pipe D 0.074/00060005/400070.22 D 2.36 m/s. The velocity in the 0.1 m diameter pipe D 9.44 m/s. Reynolds number in the 0.2 m pipe D 0006700 ð 2.36 ð 0.2/0.5 ð 1000023 0007 D 6.6 ð 105 . Reynolds number in the 0.1 m pipe D 0006700 ð 9.44 ð 0.1/0.5 ð 1000023 0007 D 1.32 ð 106 . The values of e/d for the 0.2 m and the 0.1 m pipes are 0.00025 and 0.0005 respectively. From Fig. 3.7, R/000bu2 D 0.0018 and 0.002 respectively, and from equation 3.18: 0002Pf D [4 ð 0.0018000670/0.200070006700 ð 2.362 0007] C [4 ð 0.002000650/0.100070006700 ð 9.442 0007] D 255,600 N/m2 D 255.6 kN/m2 In addition, there will be a small pressure drop at the junction of the two pipes although this has been neglected in this solution. Thus the existing pump is satisfactory for this duty.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.11 Explain the phenomenon of hydraulic jump which occurs during the flow of a liquid in an open channel. A liquid discharges from a tank into an open channel under a gate so that the liquid is initially travelling at a velocity of 1.5 m/s and a depth of 75 mm. Calculate, from first principles, the corresponding velocity and depth after the jump.
Solution See Volume 1, Example 3.9.
PROBLEM 3.12 What is a non-Newtonian fluid? Describe the principal types of behaviour exhibited by these fluids. The viscosity of a non-Newtonian fluid changes with the rate of shear according to the approximate relationship: 0004 0005 dux 00020.5 a D k 0002 dr where a is the viscosity, and du/dr is the velocity gradient normal to the direction of motion. Show that the volumetric rate of streamline flow through a horizontal tube of radius a is: 0004 0005 0005 5 0002P 2 a 5 2kl where 0002P is the pressure drop over a length l of the tube.
Solution For a power-law fluid, the apparent viscosity is given by equation 3.123. Noting that the velocity gradient dux /dr is negative then: 0005 0004 0004 0005 dux n00021 dux n00021 D k 0002 a D k dy dr Thus, in this problem: n 0002 1 D 00020.5 and: n 0002 1 D 00020.5, giving n D 0.5 The fluid is therefore shear-thinning. Equation 3.136 gives the mean velocity in a pipe: 0004
uD
0002P 4kl
00051 0004 n
0005 nC1 n d n 6n C 2
(equation 3.136)
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Putting n D 0.5, the volumetric flowrate Q is given by: 0004 0005 0004 0005 0002P 2 1 0005 2 QD d d3 4kl 10 4 Putting d D 2a, then: QD
0005
5
0004
a
5
0002P 2kl
00052
PROBLEM 3.13 Calculate the pressure drop when 3 kg/s of sulphuric acid flows through 60 m of 25 mm pipe ( D 1840 kg/m3 , D 0.025 N s/m2 ).
Solution Reynolds number D ud/ D 4G/0005 d D 4.30/00060005 ð 0.025 ð 0.0250007 D 6110 If e is taken as 0.05 mm, then: e/d D 00060.05/250007 D 0.002. From Fig. 3.7, R/000bu2 D 0.0046. Acid velocity in pipe D 3.0/[1840 ð 00060005/4000700060.02500072 ] D 3.32 m/s. From equation 3.18, the pressure drop due to friction is given by: 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 4 ð 0.0046000660/0.02500071840 ð 3.322 D 895,620 N/m2 or 900 kN/m2
PROBLEM 3.14 The relation between cost per unit length C of a pipeline installation and its diameter d is given by: C D a C bd where a and b are independent of pipe size. Annual charges are a fraction ˇ of the capital cost. Obtain an expression for the optimum pipe diameter on a minimum cost basis for a fluid of density and viscosity flowing at a mass rate of G. Assume that the fluid is in turbulent flow and that the Blasius equation is applicable, that is the friction factor is proportional to the Reynolds number to the power of minus one quarter. Indicate clearly how the optimum diameter depends on flowrate and fluid properties.
Solution The total annual cost of a pipeline consists of a capital charge plus the running costs. The chief element of the running cost is the power required to overcome the head loss which is given by: hf D 80006R/000bu2 00070006l/d00070006u2 /2g0007 (equation 3.20)
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If R/000bu2 D 0.04/ Re0.25 , the head loss per unit length l is: hf /l D 800060.04/ Re0.25 00070006l/d00070006u2 /2g0007 D 0.0160006u2 /d00070006 /000bud00070.25 D 0.016u1.75 0.25 /0006000b0.25 d1.25 0007 The velocity u D G/000bA D G/000b00060005/40007d2 D 1.27G/000bd2 ∴
hf /l D 0.01600061.27G/000bd2 00071.75 0.25 /0006000b0.25 d1.25 0007 D 0.024G1.75 0.25 /0006000b2 d4.75 0007
The power required for pumping if the pump efficiency is * is: P D Gg00060.024G1.75 0.25 /000b2 d4.75 0007/* If * D 0.5, P D 0.47G2.75 0.25 /0006000b2 d4.75 0007 W If c D power cost/W, the cost of pumping is given by: 0.47cG2.75 0.25 /000b2 d4.75 The total annual cost is then D 0006ˇa C ˇbd0007 C 0006,G2.75 0.25 0.25 /000b2 d4.75 0007 where , D 0.47c Differentiating the total cost with respect to the diameter gives: dC/dd D ˇb 0002 4.75,G2.75 0.25 /000b2 d5.75 For minimum cost, dC/dd D 0, d5.75 D 4.75,G2.75 0.25 /000b2 ˇb and d D KG0.48 0.43 /000b0.35 where:
K D 00064.75,ˇb00070.174
PROBLEM 3.15 A heat exchanger is to consist of a number of tubes each 25 mm diameter and 5 m long arranged in parallel. The exchanger is to be used as a cooler with a rating of 4 MW and the temperature rise in the water feed to the tubes is to be 20 deg K. If the pressure drop over the tubes is not to exceed 2 kN/m2 , calculate the minimum number of tubes that are required. Assume that the tube walls are smooth and that entrance and exit effects can be neglected. Viscosity of water D 1 mNs/m2 .
Solution Heat load D 0006mass flow ð specific heat ð temperature rise0007, or: 4000 D 0006m ð 4.18 ð 200007 and:
m D 47.8 kg/s
Pressure drop D 2 kN/m2 D 2000/00061000 ð 9.810007 D 0.204 m of water. From equation 3.23, 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 00062000 ð 0.253 ð 10000007/00064 ð 5 ð 1000026 0007 D 1.56 ð 106 If the tubes are smooth, then from Fig. 3.8: Re D 2.1 ð 104 . ∴ water velocity D 00062.1 ð 104 ð 1000023 0007/00061000 ð 0.0250007 D 0.84 m/s.
Cross-sectional area of each tube D 00060005/400070.252 D 0.00049 m2 .
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Mass flow rate per tube D 00060.84 ð 0.000490007 D 0.000412 m3 /s D 0.412 kg/s Hence the number of tubes required D 000647.8/0.4120007 D 116 tubes
PROBLEM 3.16 Sulphuric acid is pumped at 3 kg/s through a 60 m length of smooth 25 mm pipe. Calculate the drop in pressure. If the pressure drop falls by one half, what will be the new flowrate? Density of acid D 1840 kg/m3 . Viscosity of acid D 25 mN s/m2 .
Solution Cross-sectional area of pipe D 00060005/400070.0252 D 0.00049 m2 . Volumetric flowrate of acid D 00063.0/18400007 D 0.00163 m3 /s. Velocity of acid in the pipe D 00060.00163/0.000490007 D 3.32 m/s. Reynolds number, ud/ D 00061840 ð 3.32 ð 0.025/25 ð 1000023 0007 D 6120 From Fig. 3.7 for a smooth pipe and Re D 6120, R/000bu2 D 0.0043. The pressure drop is calculated from equation 3.18: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 D 4 ð 0.0043000660/0.025000700061840 ð 3.322 0007 D 837,200 N/m2 or 840 kN/m2 If the pressure drop falls to 418,600 N/m2 , equation 3.23 and Fig. 3.8 may be used to calculate the new flow. From equation 3.23 : 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 0006418,600 ð 0.0253 ð 18400007/00064 ð 60 ð 252 ð 1000026 0007 D 8.02 ð 104 From Fig. 3.8: Re D 3800 and the new velocity is: u0 D 00063800 ð 25 ð 1000023 0007/00061840 ð 0.0250007 D 2.06 m/s and the mass flowrate D 00062.06 ð 0.00049 ð 18400007 D 1.86 kg/s
PROBLEM 3.17 A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is flat? The shear stress acting within the fluid, Ry , varies with velocity gradient dux /dy according to the relation: Ry 0002 Rc D 0002k0006dux /dy0007 where Rc and k are constants for the material.
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Solution The shearing characteristics of non-Newtonian fluids are shown in Fig. 3.24 of Volume 1. This type of fluid remains rigid when the shear stress is less than the yield stress RY and flows like a Newtonian fluid when the shear stress exceeds RY . Examples of Bingham plastics are many fine suspensions and pastes including sewage sludge and toothpaste. The velocity profile in laminar flow is shown in Fig. 3c. Pipe wall Rw Ry
r
Rc
y
ro
R=o
Plug flow region
Velocity distribution
Figure 3c.
A force balance over the pipe assuming no slip at the walls gives: 0002P0005r 2 D Rw 20005rL, and 0006i0007 0002P/L D 2Rw /r where Rw D shear stress at the wall. A force balance over the annular core where y > r0 gives: 0002P0005y 2 D 20005yLRy Hence:
Ry D yRw /r and y D rRy /Rw
(ii)
when:
Ry D RY and r0 D rRY /Rw
(iii)
∴ from equation (ii):
Integrating:
Ry 0002 RY D 0002k0006dux /dy0007 0004 0005 Ry 0002 RY 1 yRw dux D D 0002 RY 0002 dy k k r
(iv)
0002kux D 0006y 2 Rw /2r0007 0002 RY y C C
When y D r, ux D 0, C D 00060002rRw /20007 C RY r. 0004 0005 r y2 0002 ∴ kux D Rw 0002 RY 0006r 0002 y0007 2 2r Substituting for y from equation (iii) gives: 0004 0005 0004 0005 r RY r RY r 0002 0002 RY r 0002 r and ku0 D 0006Rw 0002 RY 00072 ku0 D Rw 2 Rw 2 Rw 2Rw
(v)
(vi)
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The total volumetric flowrate Q is obtained by integrating the equation for the velocity profile to give: 0003 r Qtotal D 0005y 2 00060002dux /dy0007 dy 0
From equation (iv):
Qtotal
1 D k
0003
0004
r
0005y 0
2
yRw 0002 RY r
0005
0005r 3 dy D k
0004
Rw RY 0002 4 3
0005
m3 /s
Over the central core, the volumetric flowrate Qcore is: Qcore D 0005r02 u0 D 00050006rRY /Rw 00072 u0 (from 0006300070007 From equation (vi): Qcore D 00050006rRY /Rw 00072 0006r/2kRw 00070006Rw 0002 RY 00072 D 00060005r 3 RY2 /2kRw3 00070006Rw 0002 RY 00072 If half the total flow is to be within the central core, then: Qcore D Qtotal /2
and:
00060005r 3 RY2 /2kRw3 00070006Rw 0002 RY 00072 D 00060005r 3 /2k00070006Rw /4 0002 RY /30007 0004 0005 Rw RY 2 2 3 0002 RY 0006Rw 0002 RY 0007 D Rw 4 3
PROBLEM 3.18 2
Oil of viscosity 10 mN s/m and density 950 kg/m3 is pumped 8 km from an oil refinery to a distribution depot through a 75 mm diameter pipeline and is then despatched to customers at a rate of 500 tonne/day. Allowance must be made for periods of maintenance which may interrupt the supply from the refinery for up to 72 hours. If the maximum permissible pressure drop over the pipeline is 3450 kN/m2 , what is the shortest time in which the storage tanks can be completely recharged after a 72 hour shutdown? The roughness of the pipe surface is 0.05 mm.
Solution From equation 3.23:
R 0002Pf d3 2 Re D u2 4l 2
0002Pf D 3450 kN/m2 D 3.45 ð 106 N/m2 , d D 0.075 m, D 950 kg/m3 , l D 8000 m and D 10 m Ns/m2 0.01 Ns/m2 . R ∴ Re2 D 00063.45 ð 106 ð 0.0753 ð 9500007/00064 ð 8000 ð 0.012 0007 D 4.32 ð 105 u2 e/d D 00060.05/750007 D 0.0007 From Fig. 3.8 with 0006R/000bu2 0007 Re2 D 00064.32 ð 105 0007, e/d D 0.0007. Re D 1.1 ð 104 D du/
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∴
u D Re /d D 00061.1 ð 104 ð 0.01/0.0750007 D 1.47 ð 103 kg/m2 s
∴ mass flowrate D 00061.47 ð 103 ð 1000023 ð 3600 ð 24 ð 00060005/400070.0752 0007 D 561 tonne/day
Depletion of storage in 72 h D 0006561 ð 72/240007 D 1683 tonne Maximum net gain in capacity in the system D 0006561 0002 5000007 D 61 tonne/day and the time to recharge the tanks D 00061683/610007 D 27.6 days
PROBLEM 3.19 Water is pumped at 1.4 m3 /s from a tank at a treatment plant to a tank at a local works through two parallel pipes, 0.3 m and 0.6 m diameter respectively. What is the velocity in each pipe and, if a single pipe is used, what diameter will be needed if this flow of water is to be transported, the pressure drop being the same? Assume turbulent flow with the friction factor inversely proportional to the one quarter power of the Reynolds number.
Solution The pressure drop through a pipe is given by equation 3.18: 0004 0005 R l 2 0002P D 4 u u2 d In this case, R/000bu2 D K Re00021/4 where K is a constant. 0004 0005 ud 00021/4 l 2 u Hence: 0002P D 4K d DK
u1.75 l000b0.75 D K0 u1.75 /d1.25 d1.25 0.25
For pipe 1 in which the velocity is u1 , 0002P D K0 u11.75 /0.31.25 and the diameter is 0.3 m. Similarly for pipe 2, 0002P D K0 u11.75 /0.61.25 Hence 0006u2 /u1 00071.75 D 00060.6/0.300071.25 D 2.38 and u2 /u1 D 1.64 The total volumetric flowrate D 1.4 m3 /s D 0005/40006d21 u1 C d22 u2 0007 and substituting for d1 and d2 and u2 D 1.64u, u1 D 2.62 m/s and u2 D 4.30 m/s If a single pipe of diameter d3 is used for the same duty at the same pressure drop and the velocity is u3 , then: 00060005/40007d23 u3 D 1.4 and d23 u3 D 1.78 and: and: Since u1 D 2.62 m/s, then:
000610007
0002P D K1 u31.75 /d31.25 0006u3 /u1 00071.75 D 0006d3 /0.300071.25 0.185u31.75 D 4.5d31.25
000620007
From equations (1) and (2), the required diameter, d3 D 0.63 m
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PROBLEM 3.20 Oil of viscosity 10 mNs/m2 and specific gravity 0.90, flows through 60 m of 100 mm diameter pipe and the pressure drop is 13.8 kN/m2 . What will be the pressure drop for a second oil of viscosity 30 mNs/m2 and specific gravity 0.95 flowing at the same rate through the pipe? Assume the pipe wall to be smooth.
Solution For the first oil, with a velocity in the pipe of u m/s then: Re D u ð 00060.90 ð 10000007 ð 0006100/10000007/000610 ð 1000023 0007 D 9000u R 0002Pd3 2 Re D u2 4l 2 D 000613.8 ð 10000007 ð 0.103 ð 900/00064 ð 60 ð 0.012 0007 D 5.2 ð 105 From Fig. 3.8, when 0006R/000bu2 0007 Re2 D 5.2 ð 105 for a smooth pipe, Re D 12000. Hence, velocity u D 000612,000/90000007 D 1.33 m/s. For the second oil, the same velocity is used although the density and viscosity are now 950 kg/m3 and 0.03 Ns/m2 . Hence:
Re D 00061.33 ð 0.10 ð 950/0.030007 D 4220
For a smooth pipe, Fig. 3.7 gives a friction factor, R/000bu2 D 0.0048 for this value of Re. From Equation 3.18: 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 4 ð 0.0048 ð 000660/0.100007 ð 950 ð 1.332 D 1.94 ð 104 N/m2 19.4 kN/m2
PROBLEM 3.21 Crude oil is pumped from a terminal to a refinery through a 0.3 m diameter pipeline. As a result of frictional heating, the temperature of the oil is 20 deg K higher at the refinery end than at the terminal end of the pipe and the viscosity has fallen to one half its original value. What is the ratio of the pressure gradient in the pipeline at the refinery end to that at the terminal end? Viscosity of oil at terminal D 90 mNs/m2 . Density of oil (approximately constant) D 960 kg/m3 . Flowrate of oil D 20,000 tonne/day. Outline a method for calculating the temperature of the oil as a function of distance from the inlet for a given value of the heat transfer coefficient between the pipeline and the surroundings.
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Solution Oil flowrate D 000620,000 ð 10000007000624 ð 36000007 D 231.5 kg/s or 0006231/9600007 D 0.241 m3 /s Cross section area of pipe D 00060005/40007 ð 0.32 D 0.0707 m2 Oil velocity in pipe D 00060.241/0.07070007 D 3.40 m/s Reynolds number at terminal D 00063.40 ð 0.3 ð 960/0.090007 D 10,880 Reynolds number at the refinery is twice this value or 21,760. 0002P D 40006R/000bu2 00070006l/d00070006000bu2 0007
From equation 3.18:
(equation 3.18)
2
00020006P/l0007refinery 0006R/000bu 0007refinery D 00020006P/l0007terminal 0006R/000bu2 0007terminal
and: which, from Fig. 3.7:
D 00060.0030/0.003750007 D 0.80
In a length of pipe dl: 0002dP D 40006R/000bu2 00070006dl/d0007000bu2 N/m2 Energy dissipated D 0002dPQ D 00060005/40007d2 u40006R/000bu2 00070006dl/d00070006000bu2 0007 W where u is the velocity in the pipe. The heat loss to the surroundings at a distance l from the inlet is h0006T 0002 TS 00070005dl W where TS is the temperature of the surroundings and T is the temperature of the fluid. Heat gained by the fluid D 00060005/40007d2 u000bCp dT W where Cp (J/kg K) is the specific heat capacity of the fluid. Thus an energy balance over the length of pipe dl gives: 0006R/000bu2 0007d000bu3 dl D h0006T 0002 Ts 00070005d dl C 00060005/40007d2 u000bCp dT 0006R/000bu2 0007 varies with temperature as illustrated in the first part of this problem, and hence this equation may be written as: A dl D B dl C C dT CdT D dl or: A0002B (where A and B are both functions of temperature and C is a constant). Integrating between l1 and l2 , T1 and T2 gives: 0003 l2 0003 T2 CdT dl D A0002B l1 T1 If T1 , Ts , h and T are known (20 deg K in this problem), the integral may then be evaluated.
PROBLEM 3.22 Oil with a viscosity of 10 mNs/m2 and density 900 kg/m3 is flowing through a 500 mm diameter pipe 10 km long. The pressure difference between the two ends of the pipe is
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106 N/m2 . What will the pressure drop be at the same flowrate if it is necessary to replace the pipe by one only 300 mm in diameter? Assume the pipe surface to be smooth.
Solution D 0.01 Ns/m2 , D 900 kg/m3 , d D 0.50 m, l D 10000 m and 0002P D 1 ð 106 N/m2 . 0006R/000bu2 0007 Re2 D 00060002Pd3 0007/00064l 2 0007
(equation 3.23)
D 00061.106 ð 0.503 ð 900/00064 ð 10000 ð 0.012 00070007 D 2.81 ð 107 From Fig. 3.8, Re D ud000b/ D 1.2 ð 105 u D Re /000bd D 00061.2 ð 105 ð 0.01/900 ð 0.500007 D 2.67 m/s If the diameter of the new pipe is 300 mm, the velocity is then: D 2.67 ð 00060.5/0.300072 D 7.42 m/s Reynolds number D 00067.42 ð 0.30 ð 900/0.010007 D 2.0 ð 105 From Fig. 3.7, R/000bu2 D 0.0018 and from equation 3.18: 0002P D 00064 ð 0.0018 ð 000610000/0.30007 ð 900 ð 7.422 0007 D 1.19 ð 107 N/m2
PROBLEM 3.23 Oil of density 950 kg/m3 and viscosity 1000022 Ns/m2 is to be pumped 10 km through a pipeline and the pressure drop must not exceed 2 ð 105 N/m2 . What is the minimum diameter of pipe which will be suitable, if a flowrate of 50 tonne/h is to be maintained? Assume the pipe wall to be smooth. Use either a pipe friction chart or the Blasius equation 0006R/000bu2 D 0.0396 Re00021/4 0007.
Solution From equation 3.6 a force balance on the fluid in the pipe gives: R D 0002P0006d/4l0007 or:
D 2 ð 105 0006d/4 ð 104 0007 D 5d
Velocity in the pipe D G/000bA D 000650 ð 1000/36000007/0006950 ð 00060005/40007d2 0007 D 0.186/d2 Hence:
R/000bu2 D 5d/0006950 ð 00060.186/d2 00072 0007 D 15.21d5 Re D ud000b/ D 00060.186 ð d2 0007 ð d ð 950/00061 ð 1000022 0007 D 1.77 ð 103 /d
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The Blasius equation is: R/000bu2 D 0.0396 Re00020.25 and hence:
(equation 3.11)
15.21d5 D 0.03960006d/1.77 ð 103 00070.25
and:
d D 0.193 m
In order to use the friction chart, Fig. 3.7, it is necessary to assume a value of R/000bu2 , calculate d as above, check the resultant value of Re and calculate R/000bu2 and compare its value with the assumed value. If R/000bu2 is assumed to be D 0.0030, then 15.21d5 D 0.0030 and d D 0.182 m ∴
Re D 00061.77 ð 103 0007/0.182 D 9750
From Fig. 3.7, R/000bu2 D 0.0037 which does not agree with the original assumption. If R/000bu2 is taken as 0.0024, d is calculated D 0.175 m, Re is 1.0 ð 105 and R/000bu2 D 0.0022. This is near enough giving the minimum pipe diameter D 0.175 m.
PROBLEM 3.24 On the assumption that the velocity profile in a fluid in turbulent flow is given by the Prandtl one-seventh power law, calculate the radius at which the flow between it and the centre is equal to that between it and the wall, for a pipe 100 mm in diameter.
Solution See Volume 1, Example 3.5.
PROBLEM 3.25 A pipeline 0.5 m diameter and 1200 m long is used for transporting an oil of density 950 kg/m3 and of viscosity 0.01 Ns/m2 at 0.4 m3 /s. If the roughness of the pipe surface is 0.5 mm, what is the pressure drop? With the same pressure drop, what will be the flowrate of a second oil of density 980 kg/m3 and of viscosity 0.02 Ns/m2 ?
Solution D 0.01 Ns/m2 , d D 0.5 m and A D 00060005/400070.52 D 0.196 m2 , l D 1200 m, D 950 kg/m3 and: u D 00060.4/0.1960007 D 2.04 m/s. Reynolds number D ud/ D 0006950 ð 2.04 ð 0.50007/0.01 D 9.7 ð 104 e/d D 00060.5/5000007 D 0.001 and from Fig. 3.7, R/000bu2 D 0.0027
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From equation 3.8, 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 00064 ð 0.0027 ð 1200 ð 950 ð 2.042 /0.50007 D 1.03 ð 105 N/m2 0006R/000bu2 0007 Re2 D 0002Pd3 /4l 2 5
(equation 3.23) 3
2
D 00061.03 ð 10 ð 0.5 ð 980/4 ð 1200 ð 0.02 0007 D 6.6 ð 106 From Fig. 3.8, Re D 4.2 ð 104 D 0006980u ð 0.5/0.020007 and:
u D 1.71 m/s
∴
volumetric flowrate D 00061.71 ð 0.1960007 D 0.34 m3 /s
PROBLEM 3.26 Water (density 1000 kg/m3 , viscosity 1 mNs/m2 ) is pumped through a 50 mm diameter pipeline at 4 kg/s and the pressure drop is 1 MN/m2 . What will be the pressure drop for a solution of glycerol in water (density 1050 kg/m3 , viscosity 10 mNs/m2 ) when pumped at the same rate? Assume the pipe to be smooth.
Solution Cross-sectional area of pipe D 000600060005/40007 ð 0.052 0007 D 0.00196 m2 Water velocity, u D 4/00061000 ð 0.001960007 D 2.04 m/s. Reynolds number, Re D 00062.04 ð 1000 ð 0.05/1 ð 1000023 0007 D 102,000 From Fig. 3.7, R/000bu2 D 0.0022 From equation 3.18, 0002P D 00064 ð 0.0022 ð 0006l/0.050007 ð 1000 ð 2.042 0007 D 732l For glycerol/water flowing at the same velocity: Re D 00062.4 ð 1050 ð 0.05/1 ð 1000022 0007 D 10,700 From Fig. 3.7, and: ∴
and
R/000bu2 D 0.0037 P D 00064 ð 0.0037 ð 0006l/0.050007 ð 1050 ð 2.042 0007 D 1293l.
0002Pglycerol / 0002 Pwater D 00061293l/732/l0007 D 1.77 0002Pglycerol D 00061.77 ð 1 ð 106 0007 D 1.77 ð 106 N/m2
PROBLEM 3.27 A liquid is pumped in streamline flow through a pipe of diameter d. At what distance from the centre of the pipe will the fluid be flowing at the average velocity?
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Solution A force balance on an element of fluid of radius r gives: du 0002P0005r 2 D 000220005rl dr Pr Pr 2 du D0002 or 0002 u D C constant dr 2 l 4 l 0004 0005 Pa2 When r D d/2 D a, u D 0 and the constant D 0002 0002 4 l 0002P 2 ∴ 0006a 0002 r 2 0007 uD 4 l
or:
0002
The flow, dQ, through an annulus, radius r and thickness dr is given by: dQ D 20005r dr ur 0002P 2 2 0006a 0002 r 0007 D 20005r dr 4 l 0004 0005 4 0002P Q D 0005a 8 l
and: The average velocity is:
uav D Q/0005a2 D
0002Pa2 8 l
The radius at which u D uav is: 0002Pa2 0002P 2 D 0006a 0002 r 2 0007 8 l 4 l from which: r 2 D a2 /2 D d2 /8 and r D 0.35d
PROBLEM 3.28 Cooling water supplied to a heat exchanger flows through 25 mm diameter tubes each 5 m long arranged in parallel. If the pressure drop over the heat exchanger is not to exceed 8000 N/m2 , how many tubes must be included for a total flowrate of water of 110 tonne/h? Density of water D 1000 kg/m3 . Viscosity of water D 1 mNs/m2 . Assume pipes to be smooth-walled. If ten per cent of the tubes became blocked, what would the new pressure drop be?
Solution R 0002Pd3 ð Re2 D 2 u 4l 2
(equation 3.23)
D 00068000 ð 00060.02500073 ð 10000007/00064 ð 5 ð 00061 ð 1000023 00072 0007 D 6.25 ð 106
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From Fig. 3.8: Re D ud/ D 5 ð 104 ∴ u D 00065 ð 104 ð 1 ð 1000023 0007/00061000 ð 0.0250007 D 2.0 m/s
Flowrate per tube D 00062.0 ð 00060005/40007 ð 0.0252 0007 D 0.982 ð 104 m3 /s Total flowrate D 0006110 ð 10000007/00061000 ð 36000007 D 0.3056 m3 /s ∴ Number of tubes required D 0.3056/00060.982 ð 1000024 0007 D 31.1 or 32 tubes
If 10 per cent of the tubes are blocked, velocity of fluid D 00062.0/0.90007 D 2.22 m/s Re D 00065 ð 104 0007/0.9 D 5.5 ð 104 and, from Fig. 3.7, R/000bu2 D 0.00245. From equation 3.18, pressure drop is: 0002P D 00064 ð 0.00245 ð 00065/0.0250007 ð 1000 ð 2.222 0007 D 9650 N/m2 , an increase of 20.6%
PROBLEM 3.29 The effective viscosity of a non-Newtonian fluid may be expressed by the relationship: 0004 0005 dux 00 a D k 0002 dr where k 00 is constant. Show that the volumetric flowrate of this fluid in a horizontal pipe of radius a under isothermal laminar flow conditions with a pressure drop 0002P/l per unit length is: 0004 0005 20005 7/2 0002P 1/2 QD a 7 2k 00 l
Solution In Section 3.4.1 of Volume 1 it is shown that for any fluid, the shear stress, Rr , at a distance r from the centre of the pipe may be found from a force balance for an element of fluid of length l across which the pressure drop is 0002P by: 0004 0005 r 0002P 0002P0005r 2 D 20005rl00060002Rr 0007 or 0002 Rr D (equation 3.7) 2 l The viscosity is related to the velocity of the fluid, ux , and the shear stress, Rr , by: (from equation 3.4)
Rr D a 00060002dux /dr0007 If, for the non-Newtonian fluid, a D k 00 00060002dux /dr0007 then:
Rr D k 00 00060002dux /dr000700060002dux /dr0007 D k 00 00060002dux /dr00072
Combining the two equations for Rr : k 00 00060002dux /dr00072 D
r 2
0004
0002P l
0005
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0005 0002P 1/2 1/2 r dr 2k 00 l 0004 0005 2 0002P 1/2 3/2 ∴ 0002ux D r C constant. 3 2k 00 l 000e 0004 000f 0005 2 0002P 1/2 3/2 a When r D a (at the wall), ux D 0 and the constant D 0002 3 2k 00 l 0004 0005 0002P 1/2 2 3/2 0006a 0002 r 3/2 0007r and: ux D 2k 00 l 3
or:
0002
dux D dr
0004
The volumetric flowrate Q is: 0004 0005 0003 Q 0003 a 0003 0002P 1/2 2 a 3/2 dQ D 20005rux dr D 20005 0006ra 0002 r 5/2 0007 dr 2k 00 l 3 0 0 0 0004 0005 0004 0005 0004 0005 0002P 1/2 2 3 7/12 20005 0002P 1/2 7/2 a D a D 20005 2k 00 l 3 14 7 2k 00 l
PROBLEM 3.30 Determine the yield stress of a Bingham fluid of density 2000 kg/m3 which will just flow out of an open-ended vertical tube of diameter 300 mm under the influence of its own weight.
Solution The shear stress at the pipe wall, R0 , in a pipe of diameter d, is found by a force balance as given Volume 1, Section 3.4.1: 00060002R0 00070005 dl D 00060002P000700060005/40007d2 or:
0002R0 D 00060002P00070006d/4l0007
(equation 3.6)
If the fluid just flows from the vertical tube, then: 0002P/l D g and:
0002R0 D g0006d/40007 D 00062000 ð 981 ð 0.30007/4 D 1472 N/m2
PROBLEM 3.31 A fluid of density 1200 kg/m3 flows down an inclined plane at 15° to the horizontal. If the viscous behaviour is described by the relationship: 0004 0005 dux n Ryx D 0002k dy
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where k D 4.0 Ns0.4 /m2 and n D 0.4, calculate the volumetric flowrate per unit width if the fluid film is 10 mm thick.
Solution Flow with a free surface is discussed in Section 3.6 and the particular case of laminar flow down an inclined surface in Section 3.6.1. For a flow of liquid of depth υ, width w and density down a surface inclined at an angle 4 to the horizontal, a force balance in the x direction (parallel to the surface) may be written. The weight of fluid flowing down the plane at a distance y from the free surface is balanced by the shear stress at the plane. For unit width and unit height: 0002Ryx D g sin 4y Ryx D 0002k0006dx /dy 0007n and k0006dux /dy0007n D g sin 4y Substituting k D 4.0 Ns0.4 /m2 , n D 0.4, D 1200 kg/m3 and 4 D 15° : 4.00006dux /dy00070.4 D 00061200 ð 9.81 ð sin 15° 0007y or: 0006dux /dy D 762y0007 and dux /dy D 1.60 ð 107 y 2.5 ux D 4.57 ð 106 y 3.5 C constant When the film thickness y D υ D 0.01 m, ux D 0. Hence 0 D 0.457 C c and c D 00020.457. ∴
ux D 4.57 ð 106 y 3.5 0002 0.457 The volumetric flowrate down the surface is then: 0003 Q 0003 w0003 0 dQ D ux dw dy 0
or, for unit width: Q/W D
00100
0
0.01 00064.57
0.01
ð 106 y 3.5 0002 0.4570007 dy D 0.00357 0006m3 /s0007/m
PROBLEM 3.32 A fluid with a finite yield stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion is 0.01 Nm. Calculate the torque needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/m2 .
Solution Concentric-cylinder viscometers are in widespread use. Figure 3d represents a partial section through such an instrument in which liquid is contained and sheared between the stationary inner and rotating outer cylinders. Either may be driven, but the flow regime
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Pointer Retaining spring
ro ri
h Rotating outer cylinder
Stationary inner cylinder Figure 3d.
Partial section of concentric-cylinder viscometer
which is established with the outer rotating and the inner stationary is less disturbed by centrifugal forces. The couple transmitted through the fluid to the suspended stationary inner cylinder is resisted by a calibrated spring, the deflection of which allows calculation of the torque, T, and hence the inner wall shearing stress Ri is given by: T D 0002Ri ri 20005ri h This torque T originates from the outer cylinder which is driven at a uniform speed. On the inner surface of the outer cylinder the shear stress is Ro and:
∴
and:
T D 0002Ro ro 20005ro h 0002T Ro D 20005ro2 h Ri D
0002T 20005ri2 h
For any intermediate radius r, the local shear stress is: 0004 20005 0004 20005 ro r 0002T Rr D D Ro 2 D Ri i2 20005r 2 h r r In this example, ri D 0.015 m, r2 D 0.035 m, h D 0.05 m and T D 0.01 Nm which just produces motion at the surface of the inner cylinder. Using these equations: Ri D T/000620005ri2 h0007 D [0.01/000620005 ð 0.0152 ð 0.050007] D 141.5 N/m2
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As motion just initiates under the action of this torque, this shear stress must equal the yield stress and: RY D 141.5 N/m2 If all the fluid is to be in motion, the shear stress at the surface of the outer cylinder must be at least this value and the shear stress at the inner cylinder will be higher, and will be given by: Ri D Ro 0006ro /ri 00072 D [141.500060.035/0.01500072 ] D 770 N/m2 The required torque is then: T D Ri ð 20005ri2 h D 0006770 ð 20005 ð 0.0152 ð 0.050007 D 0.054 Nm
PROBLEM 3.33 Experiments with a capillary viscometer of length 100 mm and diameter 2 mm gave the following results: Applied pressure (N/m2 ) 1 ð 103 2 ð 103 5 ð 103 1 ð 104 2 ð 104 5 ð 104 1 ð 105
Volumetric flowrate (m3 /s) 1 ð 1000027 2.8 ð 1000027 1.1 ð 1000026 3 ð 1000026 9 ð 1000026 3.5 ð 1000025 1 ð 1000024
Suggest a suitable model to describe the fluid properties.
Solution Inspection of the data shows that the pressure difference increases less rapidly than the flowrate. Taking the first and the last entries in the table, it is seen that when the flowrate increases from 1 ð 1000027 to 1 ð 1000024 m3 /s, that is by a factor of 1000, the pressure difference increases from 1 ð 103 to 1 ð 105 N/m2 that is by a factor of only 100. In this way, the fluid appears to be shear-thinning and the simplest model, the power-law model, will be tried. From equation 3.136: Q D 00060005/40007d2 u D 00060002P/4kl00071/n [n/00066n C 20007]00060005/40007d00063nC10007/n Using the last set of data: 1.0 ð 1000024 D [00061 ð 105 0007/00064k ð 0.10007]1/n 00060005/800070006n/00063n C 10007000700062 ð 1000023 000700063nC10007/n or:
Q D K00060002P00071/n
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A plot of Q against 0002P on logarithmic axes, shown in Figure 3e, gives a slope, 00061/n0007 D 1.5 which is constant over the entire range of the experimental data. This confirms the validity of the power-law model and, for this system: n D 0.67 10−4
Slope, 1/n = 1.5
Q (m3/s)
10−5
10−6
10−7 103
104
105
106
−∆P (N/m ) 2
Figure 3e.
The value of the consistency coefficient k may be obtained by substituting n D 0.67 and the experimental data for any one set of data and, if desired, the constancy of this value may be confirmed by repeating this procedure for each set of the data. For the last set of data: Q D 00060005/40007d2 u D 00060002P/4kl00071/n [n/00066n C 20007]00060005/40007d00063nC10007/n 5
1.5
(from equation 3.136)
4.5
Thus: 1 ð 1000024 D [00061 ð 10 0007/00064 ð 0.1k0007] 00061/9000700060005/4000700060020007 and:
k D 0.183 Nsn m2
In S.I. units, the power-law equation is therefore: R D 0.1830006dux /dy00070.67 or:
7 D 0.183,P 0.67
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PROBLEM 3.34 Data obtained with a cone and plate viscometer, cone half-angle 89° cone radius 50 mm, were: cone speed (Hz) measured torque (Nm) 4.6 ð 1000021 7 ð 1000021 1.0 3.4 6.4 3.0 ð 10
0.1 0.5 1 5 10 50
Suggest a suitable model to describe the fluid properties.
ω r0 r θ
r0 r
dr
Figure 3f. Cone and plate viscometer
Solution A cone and plate viscometer, such as the Ferranti–Shirley or the Weissenberg instruments, shears a fluid sample in the small angle (usually 4° or less) between a flat surface and a rotating cone whose apex just touches the surface. Figure 3f illustrates one such arrangement. This geometry has the advantage that the shear rate is everywhere uniform and equal to ω/ sin 4, since the local cone velocity is ωr and the separation between the
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solid surfaces at that radius is r sin 4 where ω is the angular velocity of rotation. The shear stress Rr acting on a small element or area dr wide will produce a couple 000220005r dr Rr r about the axis of rotation. With a uniform velocity gradient at all points in contact with the cone surface, the surface stress R will also be uniform, so that the suffix can be omitted and the total couple about the axis is: 0003 r0 2 0002C D 20005r 2 R dr D 0005r03 R 3 0 The shear stress within the fluid can therefore be evaluated from this equation. In this problem, 4 D 1° D 0.0175 rad and r0 D 0.05 m When the cone speed is 0.1 Hz, ω D 20005 ð 0.1 D 0.628 Hence the shear rate, ω/ sin 4 D 00060.628/0.01750007 D 36 s00021 3c The shear stress is given by: R D 20005703 When c D 4.6 ð 1000022 Nm, R D 00063 ð 4.6 ð 1000022 /000620005 ð 0.053 00070007 D 176 N/m2 The remaining data may be treated in the same way to give: Cone speed (Hz)
Shear rate (s00021 )
Torque (Nm)
Shear stress (N/m2 )
0.1 0.5 1 5 10 50
36 180 360 1800 3600 18000
0.46 0.70 1.0 3.4 6.4 30.0
1760 2670 3820 13000 24500 114600
These data may be plotted on linear axes as shown in Fig. 3.24 or on logarithmic axes as in Fig. 3.26 given here as Figs 3g and 3h, respectively. It will be seen from Fig. 3g that linear axes produce an excellent straight line with an intercept of 1500 N/m2 and this indicates a Bingham plastic type of material whose characteristics are described by equation 3.122 dux jRy j 0002 Ry D p (equation 3.122) dy From Fig. 3g, the slope is p D 6.4 Ns/m2 and the graph confirms Bingham plastic behaviour.
PROBLEM 3.35 Tomato pur´ee of density 1300 kg/m3 is pumped through a 50 mm diameter factory pipeline at a flowrate of 0.00028 m3 /s. It is suggested that in order to double production: (a) a similar line with pump should be put in parallel to the existing one, or
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25000
Shear stress (N/m2)
20000
15000
10000
Slope = 6.4 Ns/m2
5000 Intercept = 1500 N/m2 0
500 1000 1500 2000
3000
Shear rate (s−1)
Figure 3g.
Shear stress (N /m2)
100,000
10,000
1000 10
100
1000
10,000
100,000
Shear rate (s−1)
Figure 3h.
(b) a large pump should force the material through the present line, or (c) a large pump should supply the liquid through a line of twice the cross-sectional area. Given that the flow properties of the pur´ee can be described by the Casson equation: 0004
1/2
00060002Ry 0007
1/2
D 00060002RY 0007
dux C 0002 c dy
00051/2
where RY is a yield stress, here 20 N/m2 , c is a characteristic Casson plastic viscosity, 5 Ns/m2 , and dux /dy is the velocity gradient, evaluate the relative pressure drops of the three suggestions, assuming laminar flow throughout.
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Solution The Casson equation is a particular form of equation 3.122 which applies to a number of foodstuffs as well as tomato puree.
or: and:
0004 0005 dux 1/2 00060002Ry 00071/2 D 00060002RY 00071/2 C 0002 c dy 0004 00051/2 dux 0002 c D 00060002Ry 00071/2 0002 00060002RY 00071/2 dy
dux D 00060002RY 0007 C 00060002Ry 0007 0002 20006Ry RY 00071/2 dy 0012 1 0011 dux D 0002 00060002Ry 0007 C 00060002RY 0007 0002 00062Ry RY 00071/2 dy c
0002 c
The Rabinowitch–Mooney equation gives the total volumetric flowrate Q through the pipe as: 0003 Rw Q 1 D0002 0006Ry 00072 f0006Ry 0007 dRy (equation 3.149) 0005a3 c Rw3 0 where a is the pipe radius and Rw is the stress at the wall. Substituting for f0006Ry 0007: 1 Q D 3 0005a c Rw3
0003 0
Rw
1/2
0006Ry3 C Ry2 RY C 2Ry5/2 RY 0007 dRy
000e 000fR w Ry4 Ry3 RY 1 4 7/2 1/2 D C C Ry RY c Rw3 4 3 7 0 RY 4 1 Rw 1/2 C C Rw1/2 RY D c 4 3 7
In this problem, RY D 20 N/m2 , c D 5 Ns/m2 , Q D 2.8 ð 1000024 m3 /s, a D 0.025 m and substituting these values, Rw D 030.24 N/m2 . From equation 3.138, Rw D 0006D/4000700060002P/l0007 D 30.24 N/m2 and:
00060002P/l0007 D 2420 0006N/m2 0007/m
For case (a), the pressure drop will remain unchanged. For case (b), the flowrate D 2Q and substituting 2Q for Q enables Rw to be recalculated as 98.0 N/m2 and (0002P/l) to be determined as 7860 0006N/m2 0007/m. p For case (c), the flowrate D 2Q and the pipe diameter D a 2. Again recalculation of Rw gives a value of 14.52 N/m2 and 00060002P/l0007 D 821 0006N/m2 0007/m.
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PROBLEM 3.36 The rheological properties of a particular suspension can be approximated reasonably well by either a “power law” or a “Bingham plastic” model over the shear rate range of 10 to 50 s00021 . If the consistency k is 10 Nsn /m2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham plastic model? What will be the pressure drop when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power law model? Calculate the centre line velocity for this pressure drop for the Bingham plastic model and comment on the result.
Solution See Volume 1, Example 3.10.
PROBLEM 3.37 Show how, by suitable selection of the index n, the power-law may be used to describe the behaviour of both shear-thinning and shear-thickening non-Newtonian fluids over a limited range of shear rates. What are the main objections to the use of the power law? Give some examples of different types of shear-thinning fluids. A power-law fluid is flowing under laminar conditions through a pipe of circular crosssection. At what radial position is the fluid velocity equal to the mean velocity in the pipe? Where does this occur for a fluid with an n-value of 0.2?
Solution Steady state shear-dependent behaviour is discussed in Volume 1, Section 3.7.1. du (equation 3.4) dy 0004 0005n du (equation 3.119) For a non-Newtonian power law fluid, R D k dy 0004 0005n00021 0004 0005 du du du D a Dk dy dy dy For a Newtonian fluid,
RD
0004
where the apparent viscosity a D k
du dy
0005n00021
When n < 1, shear-thinning behaviour is represented n > 1, shear-thickening behaviour is represented n D 1, the behaviour is Newtonian.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
For shear-thinning fluids, a ! 1 at zero shear stress and a ! 0 at infinite shear stress. Paint often exhibits shear thinning behaviour as its apparent viscosity is very high while in the can and when just applied to a wall but its apparent viscosity is very low as the brush applies it to the surface when it flows readily to give an even film. Toothpaste remains in its tube and on the brush when not subjected to shear but when sheared, as it is when the tube is squeezed, it flows readily through the nozzle to the brush. For a fluid flowing in a pipe of radius a, length l with a central core of radius r, a force balance gives: dur n 2 0002P0005r D k 0002 20005rl dr 0002Pr dur n Dk 0002 2l dr 0002P 1/n 1/n dur or: 0002 r D dr 2kl nC1 0002P 1/n n r n CC Integrating: 0002ur D 2kl nC1 1/n nC1 0002P n a n When r D a, ur D 0 and C D 0002 2kl nC1 1/n nC1 nC1 0002P n ∴ ur D a n 0002r n 2kl nC1 The mean velocity is u given by the volumetric flow/area 0003 Q 0003 a 1 1 dQ D 2 20005r dr ux or: uD 2 0005a 0 0005a 0 0004 0005 0003 a nC1 2nC1 0002P 1/n n 1 n r0002r n 20005 ∴ uD a dr 0005a2 2kl nC1 0 0004 0005 nC1 0002P 1/n n ∴ uD a n 2kl 3n C 1 When the mean velocity D average velocity, then: 0004 0005 0004 0005 nC1 nC1 nC1 0002P 1/n n 0002P 1/n n a n D a n 0002r n 2kl 3n C 1 2kl nC1
or:
When n D 0.2 then:
r nC1
nC1 n D a 3n C 1 0004 0005 n nC1 r 2n D a 3n C 1
10002
r D a
0004
0.4 1.6
0005 0.2 1.2
D 0.794
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PROBLEM 3.38 A liquid whose rheology can be represented by the power-law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow is 1 m/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n? Water, of viscosity 1 mNs/m2 flowing through the pipe at the same mean velocity gives rise to a pressure drop of 104 N/m2 compared with 105 N/m2 for the non-Newtonian fluid. What is the consistency (“k” value) of the non-Newtonian fluid?
Solution In problem 3.37, the mean velocity, u, is shown to be: 0004
uD
0002P 2kl
00051/n
a
nC1 n
n 3n C 1
and the velocity at any distance y from the pipe axis is: 0004
ur D
0002P 2kl
00051/n
nC1 nC1 n a n 0002r n nC1
The maximum velocity, umax , will occur when y D 0 and: 0004 0005 nC1 0002P 1/n n a n umax D 2kl nC1 ∴
As shown previously:
umax 1.2 3n C 1 D D and n D 0.111 u 1.0 nC1 0004 0005 0002P 1/n nC1 n uD a n 2kl 3n C 1
When n D 0.111 for the non-Newtonian fluid, 0002P D 105 N/m2 and u D 1 m/s 0004 5 00059 10 ∴ 1D a10 ð 0.083 2kl When n D 1 for water, 0002P D 104 N/m2 and u D 1 m/s and k D . 0004 40005 10 ∴ 1D a2 ð 0.25 2 l 0004 4 00059 10 a18 ð 3.81 ð 1000026 or: 1D 2 l 0004 4 00059 0004 5 00059 10 10 ∴ a10 ð 0.083 D a18 ð 3.81 ð 1000026 2kl 2 l
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From which, when a D 0.0025 m and D 1 ð 1000023 Ns/m2 , k D 6.24 Nsn /m2
PROBLEM 3.39 Two liquids of equal densities, the one Newtonian and the other a non-Newtonian “powerlaw” fluid, flow at equal volumetric rates down two wide vertical surfaces of the same widths. The non-Newtonian fluid has a power-law index of 0.5 and has the same apparent viscosity, in SI units, as the Newtonian fluid when its shear rate is 0.01 s00021 . Show that, for equal surface velocities of the two fluids, the film thickness for the Newtonian fluid is 1.125 times that of the non-Newtonian fluid.
Solution For a power-law fluid:
0004
RDk 0004
Dk
du dy du dy
0005n
(equation 3.121) 0005n00021 0004
dux dy
0005
D a 0006dux /dy0007
where a is the apparent velocity D k0006dux /dy0007n00021 For a Newtonian fluid: 0004 0005 dux RD dy
(equation 3.122)
(equation 3.3)
When n D 0.5 and 0006dux /dy0007 D 0.01, D a and: a D D k0006dux /dy0007n00021 D k00060.01000700020.5 D 10 k D and k D 0.1 . The equation of state of the power-law fluid is therefore: R D 0.1 0006dux /dy00070.5 For a fluid flowing down a vertical surface, length l and width w and film thickness S, at a distance y from the solid surface, a force balance gives: 0006S 0002 y0007wl000bg D Rwl D k0006dux /dy0007n wl g 1/n dux D 0006S 0002 y00071/n or: dy k 0004 0005 g 1/n nC1 n and: ux D 0006S 0002 y0007 n 0002 C const. k nC1 g 1/n nC1 n When y D 0, ux D 0 and the constant D S n K nC1 g 1/n n nC1 nC1 S n 0002 0006S 0002 y0007 n and: ux D k nC1
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55
At the free surface where y D S: us D
g 1/n 0004
k
n nC1
0005
S
nC1 n
The volumetric flowrate Q is given by: 0005 0003 s g 1/n 0004 n 0005 0004 nC1 nC1 QD w dy S n 0002s0002y n k nC1 0 g 1/n 0004 n 0005 2nC1 Dw S n k 2n C 1
(i)
(ii)
For the non-Newtonian fluid, k D 0.1 , n D 0.5 and equation (ii), when expressed in S.I. units, becomes: 0004 0005 0004 00052 g 2 gs2 4 Dw ð 0.25 s D 25 w (iii) 0.1 For the Newtonian fluid, n D 1 and k D and substituting in equation (ii): 0004 0005 g 3 ð 0.33SN QDw
(iv)
where SN is the thickness of the Newtonian film. For equal flowrates, from equations (iii) and (iv): 0004
25w
000bgs2
or:
00052
0004
0005 g 3 SN 0004 0005 g 3 SN D 75 S4
D 0.33w
(v)
For equal surface velocities, the term 0006000bg/K0007 in equation (i) can be substituted from equation (v) and: 0004
0005 g 2 For the non-Newtonian fluid: us D ð 0.33S3 0.1 0004 3 00052 SN D 100 ð 0.33S3 75S4 6 D 0.00592SN /S5 0004
0005 g 2 ð 0.5SN 0004 3 0005 SN 2 5 D ð 0.5SN D 0.0067SN /S4 75S4
For the Newtonian fluid: us D
and:
SN /S D 00060.00667/0.005920007 D 1.126
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.40 A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 ð 104 N/m2 . When the flowrate is doubled, the pressure drop increases by a factor of 1.5. A pitot tube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Any equations used must be derived from the basic relation between shear stress R and shear rate: R D k0006,0007 P n
radial distance from centre of pipe (s mm)
velocity (m/s)
0 10 20 30
0.80 0.77 0.62 0.27
Solution For a power-law fluid: At the initial flowrate:
4 ð 104 D kun
With a flow of:
6 ð 104 D k00062u0007n 1.5 D 2n
Dividing: and hence:
n D 0.585
For the power-law fluid:
dux n R D k dy
A force balance on a fluid core of radius s in pipe of radius r gives:
or:
Integrating:
Rs 20005sl D 0002P0005s2 0004 0005 dux n Ps Rs D k 0002 D0002 ds 2l 0004 0005 P 1/n 1/n dux D 0002 s 0002 ds 2kl 0004 0005 0004 0005 nC1 n P 1/n 0002ux D 0002 s n C constant 2ks nC1
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When s D r, ux D 0 (the no-slip condition): 0004
00051 0004 0005 nC1 0002P n n r n and hence the constant D 0002 2kl nC1 0004 00051/n 0004 0005 nC1 nC1 n P Substituting: ux D 0002 r n 0002s n 2kl nC1 0004 0005 0004 0005 nC1 0002P 1/n n On the centre line: uCL D r n 2kl nC1 s ux n and hence: D10002 uCL r s 2.71 at when n D 0.585 : D10002 r
nC1
The following data are obtained for a pipe radius of r D 35 mm: experimental
radius s (mm)
s 2.71 ux D10002 uCL r
ux (m/s)
ux /uCL
0 10 20 30
1 0.966 0.781 0.341
0.80 0.77 0.62 0.27
1 0.96 0.77 0.34
Thus, the calculated and experimental values of ux /uCL agree within reasonable limits of experimental accuracy.
PROBLEM 3.41 A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2 ) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient 0002P/l and volumetric flowrate Q is: 4 1 000500060002P0007r 4 1 0002 X C X4 QD 8l p 3 3 where l is the pipe radius, p is the plastic viscosity, and X is the ratio of the yield stress to the shear stress at the pipe wall. Calculate the flowrate for this pipeline when the pressure drop is 600 kN/m2 . It may be assumed that the flow is laminar.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution For a Bingham-plastic material, the shear stress Rs at radius s is given by: 0004 0005 dux Rs 0002 RY D C p 0002 0006Rs 0002 RY 0007 ds dux D 0 0006Rs 0003 RY 0007 ds The central unsheared core has radius rc D r0006RY /R0007 (where r D pipe radius and R D wall shear stress) since the shear stress is proportional to the radius s. In the annular region:
1 dux 1 s D 0002 0006R 0002 RY 0007 D 0002P 0002 RY 0006from a force balance0007 ds p p 2l 0004 0005 0003 1 s2 0002P 0002 RY s C constant 0002ux D 0002 dux D p 4l For the no-slip condition: ux D 0, when s D r 0004 0005 r2 1 0002P 0002 RY s C constant Thus: 0D p 4l 0006 0007 1 0002P 2 2 and: us D 0006r 0002 s 0007 0002 RY 0006r 0002 s0007 p 4l Substituting:
0002P D
2R l/r
us D
1 p
0006
0007 R 2 0006r 0002 s2 0007 0002 RY 0006r 0002 s0007 2r
(i)
The volumetric flowrate through elemental annulus, dQA D us 20005sds 0006 0007 0003 r 1 R 2 Thus: QA D 0006r 0002 s2 0007 0002 RY 0006r 0002 s0007 20005sds 2r rc p 0004 2 2 0005 0004 0005 r 1 r s s4 s3 RY rs2 20005 0002 0002 R 0002 D p 2r 2 4 R 2 3 rc Writing
RY RY D X and rc D r , then : R R 0006 0004 4 0005 0004 3 0005 0004 0005 1 r r r4 r3 X4 r 4 20005 1 X2 r 4 0002 0002 0002 R 0002X 0002 QA D p 2r 2 4 2 3 2r 2 4 0004 3 2 00050007 r X r 3 X3 0002 CX 2 3 0006 0007 20005R 3 1 1 1 1 1 1 D r 0002 X 0002 X2 C X4 C X3 0002 X4 p 8 6 4 8 2 3
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FLOW IN PIPES AND CHANNELS
D
0005Rr 3 4 p
0006
5 1 0002 4/3X 0002 2X2 C 4X3 0002 X4 3
59
0007
(ii)
In the core region Substituting: s D rc D 0006RY /R0007r D Xr in equation (i) for the core velocity uc gives: 0006 0007 R 2 1 0006r 0002 X2 r 2 0007 0002 RY 0006r 0002 Xr uc D p 2r 0006 0007 Rr 1 Rr 2 00061 0002 X 0007 0002 X00061 0002 X0007 0002 D f200061 0002 X2 0007 0002 4X C 4X2 0007g p 2 4 p D
0014 Rr 0013 2 0002 4X C 2X2 4 p
The flowrate through the core is: uc 0005rc2 D uc 0005X2 r 2 D Qc Thus:
Qc D D
Rr 0005X2 r 2 f2 0002 4X C 2X2 g 4 p Rr 3 0005 f2X2 0002 4X3 C 2X4 g 4 p
The total flowrate is: 0006QA C Qc 0007 D Q and:
QD
0014 0005Rr 3 0013 1 0002 43 X C 13 X4 4 p
Putting
RD
0002Pr then : 2l QD
When: Then:
000500060002P0007r 4 f1 0002 43 X C 13 X4 g 8l p
0002P D 6 ð 105 N/m2 , l D 200 m d D 40 mm and r D 0.02 m. R D 0002P
r 0.02 D6ð ð 105 D 30 N/m2 2l 400
p D 0.150 Ns/m2 RY D 14.35 N/m2 and: Thus:
RY 14.35 D D 0.478 R 30 0006 0007 1 4 00060005000700066 ð 105 000700060.0200074 3 1 0002 ð 0.478 C 00060.4780007 QD 8 ð 200 ð 0.150 3 3 XD
D 0.000503 m3 /s
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SECTION 4
Flow of Compressible Fluids PROBLEM 4.1 A gas, having a molecular weight of 13 kg/kmol and a kinematic viscosity of 0.25 cm2 /s, flows through a pipe 0.25 m internal diameter and 5 km long at the rate of 0.4 m3 /s and is delivered at atmospheric pressure. Calculate the pressure required to maintain this rate of flow under isothermal conditions. The volume occupied by 1 kmol at 273 K and 101.3 kN/m2 is 22.4 m3 . What would be the effect on the required pressure if the gas were to be delivered at a height of 150 m (i) above, and (ii) below its point of entry into the pipe?
Solution From equation 4.57 and, as a first approximation, omitting the kinetic energy term: 0002P2 0001 P1 0004/vm C 40002R/u2 00040002l/d00040002G/A00042 D 0 At atmospheric pressure and 289 K, the density D 000213/22.400040002273/2890004 D 0.542 kg/m3 Mass flowrate of gas, G D 00020.4 ð 0.5420004 D 0.217 kg/s. Cross-sectional area, A D 0002000e/4000400020.2500042 D 0.0491 m2 . Gas velocity, u D 00020.4/0.04910004 D 8.146 m/s ∴
G/A D 00020.217/0.04910004 D 4.413 kg/m2 s
Reynolds number,
Re D du/0010 D 00020.25 ð 8.146/0.25 ð 1000014 0004 D 8.146 ð 104
From Fig. 3.7, for e/d D 0.002, R/u2 D 0.0031 v2 D 00021/0.5420004 D 1.845 m3 /kg v1 D 000222.4/1300040002298/27300040002101.3/P1 0004 D 190.5/P1 m3 /kg
and:
vm D 00020.923P1 C 95.250004/P1 m3 /kg
Substituting in equation 4.57: P1 0002P1 0001 101.30004103 /00020.923P1 C 95.250004 D 400020.0031000400025000/0.25000400024.72600042 and:
P1 D 111.1 kN/m2 60
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The kinetic energy term D 0002G/A00042 ln0002P1 /P2 0004 D 00024.41300042 ln0002111.1/101.30004 D 1.81 kg2 /m4 s2 This is negligible in comparison with the other terms which equal 5539 kg2 /m4 s2 so that the initial approximation is justified. If the pipe is not horizontal, the term g dz in equation 4.49 must be included in the calculation. If equation 4.49 is divided by v2 , this term on integration becomes gz/v2m . ∴
vm D 00020.923 ð 111.1 C 95.250004/111.1 D 1.781 m3 /kg vair D 000224.0/290004 D 0.827 m3 /kg
As the gas is less dense than air, vm is replaced by 0002vair 0001 vm 0004 D 00010.954 m3 /kg. ∴
gz/v2m D 00029.81 ð 150/0.9542 0004 D 1616 N/m2 or 0.16 kN/m2
(i) If the delivery point is 150 m above the entry level, then since gas is less dense, P1 D 0002111.1 0001 0.160004 D 110.94 kN/m2 (ii) If the delivery point is 150 m below the entry level then, P1 D 0002111.1 C 0.160004 D 111.26 kN/m2
PROBLEM 4.2 Nitrogen at 12 MN/m2 pressure is fed through a 25 mm diameter mild steel pipe to a synthetic ammonia plant at the rate of 1.25 kg/s. What will be the pressure drop over a 30 m length of pipe for isothermal flow of the gas at 298 K? Absolute roughness of the pipe surface D 0.005 mm. Kilogram molecular volume D 22.4 m3 . Viscosity of nitrogen D 0.02 mN s/m2 .
Solution Molecular weight of nitrogen D 28 kg/kmol. Assuming a mean pressure in the pipe of 10 MN/m2 , the specific volume, vm at 10 MN/m2 and 298 K is: vm D 000222.4/2800040002101.3/10 ð 103 00040002298/2730004 D 0.00885 m3 /kmol
Reynolds number, ud/0010 D d0002G/A0004/00100004. A D 0002000e/4000400020.02500042 D 4.91 ð 1000013 m2 . ∴ 0002G/A0004 D 00021.25/4.91 ð 1000013 0004 D 2540 kg/m2 s
and:
Re D 00020.025 ð 2540/0.02 ð 1000013 0004 D 3.18 ð 106
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
From Fig. 3.7, for Re D 3.18 ð 106 and e/d D 00020.005/250004 D 0.0002, R/u2 D 0.0017 In equation 4.57 and neglecting the first term: 0002P2 0001 P1 0004/vm C 40002R/u2 00040002l/d00040002G/A00042 D 0 or:
P1 0001 P2 D 4vm 0002R/u2 00040002l/d00040002G/A00042 D 4 ð 0.0088500020.00170004000230/0.02500040002254000042 D 466,000 N/m2 or 0.466 MN/m2
This is small in comparison with P1 D 12 MN/m2 , and the average pressure of 10 MN/m2 is seen to be too low. A mean pressure of 11.75 kN/m2 is therefore selected and the calculation repeated to give a pressure drop of 0.39 MN/m2 . The mean pressure is then 000212 C 11.610004/2 D 11.8 MN/m2 which is close enough to the assumed value. It remains to check if the assumption that the kinetic energy term is negligible is justified. Kinetic energy term D 0002G/A00042 ln0002P1 /P2 0004 D 0002254000042 ln000212/11.610004 D 2.13 ð 105 kg2/m4s2 The term 0002P1 0001 P2 0004/vm , where vm is the specific volume at the mean pressure of 11.75 MN/m2 D 00020.39 ð 106 0004/0.00753 D 5.18 ð 107 kg2 /m4 s. Hence the omission of the kinetic energy term is justified and the pressure drop D 0.39 MN/m2
PROBLEM 4.3 Hydrogen is pumped from a reservoir at 2 MN/m2 pressure through a clean horizontal mild steel pipe 50 mm diameter and 500 m long. The downstream pressure is also 2 MN/m2 and the pressure of this gas is raised to 2.6 MN/m2 by a pump at the upstream end of the pipe. The conditions of flow are isothermal and the temperature of the gas is 293 K. What is the flowrate and what is the effective rate of working of the pump? Viscosity of hydrogen D 0.009 mN s/m2 at 293 K.
Solution Neglecting the kinetic energy term in equation 4.55, then: 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 where P1 D 2.6 MN/m2 and P2 D 2.0 MN/m2 . Thus:
v1 D 000222.4/200040002293/273000400020.1013/2.60004 D 0.468 m3 /kg
If Re D 107 and e/d D 0.001, from Fig. 3.7, R/u2 D 0.0023.
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Substituting: 00022.02 0001 2.62 00041012 /00022 ð 2.6 ð 106 ð 0.4680004 C 400020.002300040002500/0.0500040002G/A00042 D 0 from which G/A D 111 kg/m2 s. ∴
Re D d0002G/A0004/0010 D 00020.05 ð 111/00020.009 ð 1000013 0004 D 6.2 ð 105
Thus the chosen value of Re was too high. If Re is taken as 6.0 ð 105 and the problem reworked, G/A D 108 kg/m2 s and Re D 6.03 ð 105 which is in good agreement. A D 0002000e/4000400020.0500042 D 0.00197 m2 and:
G D 108 ð 0.00197 D 0.213 kg/s
The power requirement is given by equation 8.71 as 00021/00150004GP1 v1 ln0002P1 /P2 0004 If a 60% efficiency is assumed, then the power requirement is: D 00021/0.60004 ð 0.213 ð 2.6 ð 106 ð 0.468 ln00022.6/20004 D 00021.13 ð 105 0004 W or 113 kW
PROBLEM 4.4 In a synthetic ammonia plant the hydrogen is fed through a 50 mm steel pipe to the converters. The pressure drop over the 30 m length of pipe is 500 kN/m2 , the pressure at the downstream end being 7.5 MN/m2 . What power is required in order to overcome friction losses in the pipe? Assume isothermal expansion of the gas at 298 K. What error is introduced by assuming the gas to be an incompressible fluid of density equal to that at the mean pressure in the pipe? 0010 D 0.02 mNs/m2 .
Solution If the downstream pressure D 7.5 MN/m2 and the pressure drop due to friction D 500 kN/m2 , the upstream pressure D 8.0 MN/m2 and the mean pressure D 7.75 MN/m2 . The mean specific volume is: vm D 000222.4/200040002298/273000400020.1013/7.750004 D 0.16 m3 /kg and:
v1 D 000222.4/200040002298/273000400020.1013/8.00004 D 0.15 m3 /kg
It is necessary to assume a value of R/u2 , calculate G/A and the Reynolds number and check that the value of e/d is reasonable. If the gas is assumed to be an incompressible fluid of density equal to the mean pressure in the pipe and R/u2 D 0.003, the pressure drop due to friction D 500 kN/m2 is: ∴
and
0002500 ð 103 /0.160004 D 400020.0030004000230/0.0500040002G/A00042 G/A D 658 kg/m2 s. Re D d0002G/A0004/0010 D 00020.05 ð 658/0.02 ð 1000013 0004 D 1.65 ð 106
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
From Fig. 3.7 this corresponds to a value of e/d of approximately 0.002, which is reasonable for a steel pipe. For compressible flow: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
Substituting: 0002G/A00042 ln00028.0/7.50004 C 00027.52 0001 8.02 00041012 /00022 ð 8.0 ð 106 ð 0.150004 C 400020.0030004000230/0.0500040002G/A00042 D 0 from which: G/A D 667 kg/m2 s and G D 667 ð 0002000e/4000400020.0500042 D 1.31 kg/s Little error is made by the simplifying assumption in this particular case. The power requirement is given by equation 8.71: D 00021/00150004GP1 v1 ln0002P1 /P2 0004 If the compressor efficiency D 60%, power requirement D 00021/0.60004 ð 1.31 ð 8.0 ð 106 ð 0.15 ln00028/7.50004 D 00021.69 ð 105 0004 W or 169 kW
PROBLEM 4.5 A vacuum distillation plant operating at 7 kN/m2 pressure at the top has a boil-up rate of 0.125 kg/s of xylene. Calculate the pressure drop along a 150 mm bore vapour pipe used to connect the column to the condenser. The pipe length may be taken as equivalent to 6 m, e/d D 0.002 and 0010 D 0.01 mN s/m2 .
Solution From vapour pressure data, the vapour temperature D 338 K and the molecular weight of xylene D 106 kg/kmol. In equation 4.55: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 Cross-sectional area of pipe, A D 0002000e/4000400020.1500042 D 1.76 ð 1000012 m2 G/A D 00020.125/1.76 ð 1000012 0004 D 7.07 kg/m2 s The Reynolds number, is ud/0010 D d0002G/A0004/0010 D 00020.15 ð 7.07/00020.01 ð 1000013 0004 D 1.06 ð 105 From Fig. 3.7, with e/d D 0.002 and Re D 1.06 ð 105 , 0002R/u2 0004 D 0.003. Specific volume, v1 D 000222.4/10600040002338/27300040002101.3/7.00004 D 3.79 m3 /kg.
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FLOW OF COMPRESSIBLE FLUIDS
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Substituting in equation 4.55: 00027.000042 ln00027/P2 0004 C 0002P22 0001 72 0004 ð 106 /2 ð 7 ð 103 ð 3.79 C 4 ð 0.00300026/0.15000400027.0700042 D 0 where P2 is the pressure at the condenser 0002kN/m2 0004. Solving by trial and error: P2 D 6.91 kN/m2 ∴
0002P1 0001 P2 0004 D 00027.0 0001 6.910004 D 0.09 kN/m2 or 90 N/m2
PROBLEM 4.6 Nitrogen at 12 MN/m2 pressure is fed through a 25 mm diameter mild steel pipe to a synthetic ammonia plant at the rate of 0.4 kg/s. What will be the drop in pressure over a 30 m length of pipe assuming isothermal expansion of the gas at 300 K? What is the average quantity of heat per unit area of pipe surface that must pass through the walls in order to maintain isothermal conditions? What would be the pressure drop in the pipe if it were perfectly lagged? 0010 D 0.02 mNs/m2 .
Solution At high pressure, the kinetic energy term in equation 4.55 may be neglected to give: 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 Specific volume at entry of pipe, v1 D 000222.4/2800040002300/273000400020.1013/120004 D 0.00742 m3 /kg Cross-sectional area of pipe, A D 0002000e/4000400020.02500042 D 0.00049 m2 ∴ G/A D 00020.4/0.000490004 D 816 kg/m2 s.
Reynolds number, d0002G/A0004/0010 D 0.025 ð 816/00020.02 ð 1000013 0004 D 1.02 ð 106 If e/d D 0.002 and Re D 1.02 ð 106 , R/u2 D 0.0028 from Fig. 3.7. Substituting: 0002122 0001 P22 00041012 /00022 ð 12 ð 106 ð 0.007420004 D 400020.00280004000230/0.0250004000281600042 and: P2 D 11.93 MN/m2 and: pressure drop D 000212.0 0001 11.930004 D 0.07 MN/m2 0005 70 kN/m2 The heat required to maintain isothermal flow is given in Section 4.5.2 as Gu2 /2. The velocity at the high pressure end of the pipe D volumetric flow/area D 0002G/A0004v1 D 0002816 ð 0.00720004 D 6.06 m/s and the velocity in the plant is taken as zero. Thus:
Gu2 /2 D 0.4 ð 00026.0600042 /2 D 7.34 W
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Outside area of pipe D 000230 ð 000e ð 0.0250004 D 2.36 m2 . Heat required D 00027.34/2.360004 D 3.12 W/m2 This low value of the heat required stems from the fact that the change in kinetic energy is small and conditions are almost adiabatic. If the pipe were perfectly lagged, the flow would be adiabatic and the pressure drop would then be calculated from equations 4.77 and 4.72. The specific volume at the low pressure end v2 to be calculated from: 0001 0002 0003 00040001 0002 00032 0004 0002 0003 0016 0001 1 P1 A 2 0016 C1 v1 v2 2 80002R/u 00040002l/d0004 D C ln 10001 0001 20016 v1 G v2 0016 v1 (equation 4.77) For nitrogen, 0016 D 1.4 and hence: 0001 0002 0003 00040001 0002 0003 0004 1.4 0001 1 12 ð 106 1 2 0.00742 2 800020.00280004000230/0.0250004 D C 10001 2 ð 1.4 0.00742 816 v2 1.4 C 1 0005 v2 0006 ln 0001 1.4 0.00742 Solving by trial and error, v2 D 0.00746 m3 /kg. Thus: 0002 0003 0002 0003 0002 0003 0002 0003 1 G 2 2 0016 0016 1 G 2 2 v1 C P1 v1 D v2 C P2 v2 2 A 0016 00011 2 A 0016 00011
(equation 4.72)
Substitution gives: 000281600042 00020.0074200042 /2 C [1.4/00021.4 0001 10004]12 ð 106 ð 0.00742 D 000281600042 00020.0074600042 /2 C [1.4/00021.4 0001 10004]P2 ð 106 ð 0.00746 and: P1 D 11.94 MN/m2 The pressure drop for adiabatic flow D 000212.0 0001 11.940004 D 0.06 MN/m2 or 60 kN/m2
PROBLEM 4.7 Air, at a pressure of 10 MN/m2 and a temperature of 290 K, flows from a reservoir through a mild steel pipe of 10 mm diameter and 30 m long into a second reservoir at a pressure P2 . Plot the mass rate of flow of the air as a function of the pressure P2 . Neglect any effects attributable to differences in level and assume an adiabatic expansion of the air. 0010 D 0.018 mN s/m2 , 0016 D 1.36.
Solution G/A is required as a function of P2 . v2 cannot be found directly since the downstream temperature T2 is unknown and varies as a function of the flowrate. For adiabatic flow,
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FLOW OF COMPRESSIBLE FLUIDS
v2 may be calculated from equation 4.77 using specified values of G/A and substituted in equation 4.72 to obtain the value of P2 . In this way the required data may be calculated. 0001 0002 0003 00040001 0002 00032 0004 0002 0003 0016 0001 1 P1 A 2 0016 C1 v1 v2 2 C ln 10001 0001 80002R/u 00040002l/d0004 D 20016 v1 G v2 0016 v1
(equation 4.77) 0.50002G/A00042 v21 C [0016/00020016 0001 10004]P1 v1 D 0.50002G/A00042 v22 C [0016/00020016 0001 10004]P2 v2 (equation 4.72) or:
0.50002G/A00042 0002v21 0001 v22 0004 C [0016/00020016 0001 10004]P1 v1 D P2 [0016/00020016 0001 10004]v2
When P2 D P1 D 10 MN/m2 , G/A D 0. If G/A is 2000 kg/m2 s, then: Re D 00020.01 ð 2000/0.018 ð 1000013 0004 D 1.11 ð 106 When e/d D 0.0002, R/u2 D 0.0028 from Fig. 3.7 and: v1 D 000222.4/2900040002290/273000400020.1013/100004 D 0.0083 m3 /kg
Substituting in equation 4.77: 0001
0002 00032 0004 0.36 1 10 ð 106 C 800020.00280004000230/0.010004 D 2 ð 1.36 0.0083 2000 0001 0004 0002 0003 0.0083 2 2.36 0005 v2 0006 ð 10001 0001 ln v2 1.36 0.0083
and: v2 D 0.00942 m3 /kg. Substituting for v2 in equation 4.72 gives: P2 D [0.50002200000042 00020.00832 0001 0.009422 0004 C 00021.36/0.36000410 ð 106ð0.0083 ]/00021.36/0.360004 ð 0.00942 and: P2 D 8.75 MN/m2 . In a similar way the following table may be produced. G/A0002kg/m2 s0004
v2 0002m3 /kg0004
P2 0002MN/m2 0004
0 2000 3000 3500 4000 4238
0.0083 0.00942 0.012 0.0165 0.025 0.039
10.0 8.75 6.76 5.01 3.37 2.04
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
5000
G /A (kg / m2s)
4000
3000
2000
1000
0 10
9
8
7
4 3 6 5 Pressure P2 (MN/m2)
2
1
0
Figure 4a.
These data are plotted in Fig. 4a. It is shown in Section 4.5.4, Volume 1, that the maximum velocity which can p occur in a pipe under adiabatic flow conditions is the sonic velocity which is equal to p 0016P2 v2 . From the above table 0016P2 v2 at maximum flow is: 0007 1.36 ð 2.04 ð 106 ð 0.039 D 329 m/s The temperature at this condition is given by P2 v2 D RT/M, and: T2 D 000229 ð 0.039 ð 2.04 ð 106 /83140004 D 227 K The velocity of sound in air at 227 K D 334 m/s, which serves as a check on the calculated data.
PROBLEM 4.8 Over a 30 m length of 150 mm vacuum line carrying air at 293 K, the pressure falls from 1 kN/m2 to 0.1 kN/m2 . If the relative roughness e/d is 0.002, what is approximate flowrate?
Solution The specific volume of air at 293 K and 1 kN/m2 is: v1 D 000222.4/2900040002293/27300040002101.3/1.00004 D 83.98 m3 /kg
It is necessary to assume a Reynolds number to determine R/u2 and then calculate a value of G/A which should correspond to the original assumed value. Assume a Reynolds number of 1 ð 105 .
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When e/d D 0.002 and Re D 105 , R/u2 D 0.003 from Fig. 3.7. 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
Substituting: 0002G/A00042 ln00021.0/0.10004 C 00020.12 0001 12 0004 ð 106 /00022 ð 1 ð 103 ð 83.980004 C 400020.0030004000230/0.1500040002G/A00042 D 0 and: 0002G/A0004 D 1.37 kg/m2 s. The viscosity of air is 0.018 mN s/m2 . ∴ Re D 00020.15 ð 1.370004/00020.018 ð 1000013 0004 D 1.14 ð 104
Thus the chosen value of Re is too high. When Re D 1 ð 104 , R/u2 D 0.0041 and G/A D 1.26 kg/m2 s. Re now equals 1.04 ð 104 which agrees well with the assumed value. G D 1.26 ð 0002000e/40004 ð 00020.1500042 D 0.022 kg/s
Thus:
PROBLEM 4.9 A vacuum system is required to handle 10 g/s of vapour (molecular weight 56 kg/kmol) so as to maintain a pressure of 1.5 kN/m2 in a vessel situated 30 m from the vacuum pump. If the pump is able to maintain a pressure of 0.15 kN/m2 at its suction point, what diameter of pipe is required? The temperature is 290 K, and isothermal conditions may be assumed in the pipe, whose surface can be taken as smooth. The ideal gas law is followed. Gas viscosity D 0.01 mN s/m2 .
Solution Use is made of equation 4.55 to solve this problem. It is necessary to assume a value of the pipe diameter d in order to calculate values of G/A, the Reynolds number and R/u2 . If d D 0.10 m, A D 0002000e/4000400020.1000042 D 0.00785 m2 ∴
G/A D 000210 ð 1000013 /0.007850004 D 1.274 kg/m2 s
and
Re D d0002G/A0004/0010 D 0.10 ð 1.274/00020.01 ð 1000013 0004 D 1.274 ð 104
For a smooth pipe, R/u2 D 0.0035, from Fig. 3.7. Specific volume at inlet, v1 D 000222.4/5600040002290/27300040002101.3/1.50004 D 28.7 m3 /kg 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 (equation 4.55) Substituting gives: 00021.27400042 ln00021.5/0.150004 C 00020.152 0001 1.52 0004 ð 106 /00022 ð 1.5 ð 103 ð 28.70004 C 00020.00350004000230/0.10000400021.27400042 D 000116.3 and the chosen value of d is too large.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
A further assumed value of d D 0.05 m gives a value of the right hand side of equation 4.55 of 25.9 and the procedure is repeated until this value is zero. This occurs when d D 0.08 m or 80 mm.
PROBLEM 4.10 In a vacuum system, air is flowing isothermally at 290 K through a 150 mm diameter pipeline 30 m long. If the relative roughness of the pipewall e/d is 0.002 and the downstream pressure is 130 N/m2 , what will the upstream pressure be if the flowrate of air is 0.025 kg/s? Assume that the ideal gas law applies and that the viscosity of air is constant at 0.018 mN s/m2 . What error would be introduced if the change in kinetic energy of the gas as a result of expansion were neglected?
Solution As the upstream and mean specific volumes v1 and vm are required in equations 4.55 and 4.56 respectively, use is made of equation 4.57: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/00022RT/M0004 C 40002R/u2 00040002l/d00040002G/A00042 D 0 R D 8.314 kJ/kmol K and hence: 2RT/M D 00022 ð 8.314 ð 103 ð 2900004/29 D 1.66 ð 105 J/kg The second term has units of 0002N/m2 00042 /0002J/kg0004 D kg2 /s2 m4 which is consistent with the other terms. A D 0002000e/4000400020.1500042 D 0.0176 m2 ∴
G/A D 00020.025/0.01760004 D 1.414
and
Re D d0002G/A0004/0010 D 00020.15 ð 1.4140004/00020.018 ð 1000013 0004 D 1.18 ð 104
For smooth pipes and Re D 1.18 ð 104 , R/u2 D 0.0040 from Fig. 3.7. Substituting in equation 4.57 gives: 00021.41400042 ln0002P1 /1300004 C 00021302 0001 P12 0004/1.66 ð 105 C 4 ð 0.0040000230/0.15000400021.41400042 D 0 Solving by trial and error, the upstream pressure, P1 D 1.36 kN/m2 If the kinetic energy term is neglected, equation 4.57 becomes: 0002P22 0001 P12 0004/00022RT/M0004 C 40002R/u2 00040002l/d00040002G/A00042 D 0 and P1 D 1.04 kN/m2 Thus a considerable error would be introduced by this simplifying assumption.
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FLOW OF COMPRESSIBLE FLUIDS
PROBLEM 4.11 Air is flowing at the rate of 30 kg/m2 s through a smooth pipe of 50 mm diameter and 300 m long. If the upstream pressure is 800 kN/m2 , what will the downstream pressure be if the flow is isothermal at 273 K? Take the viscosity of air as 0.015 mN s/m2 and assume that volume occupies 22.4 m3 . What is the significance of the change in kinetic energy of the fluid?
Solution 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
The specific volume at the upstream condition is: v1 D 000222.4/2900040002273/27300040002101.3/8000004 D 0.098 m3 /kg
G/A D 30 kg/m2 s ∴
Re D 00020.05 ð 300004/00020.015 ð 1000013 0004 D 1.0 ð 105
For a smooth pipe, R/u2 D 0.0032 from Fig. 3.7. Substituting gives: 00023000042 ln0002800/P2 0004 C 0002P22 0001 8002 0004 ð 106 /00022 ð 800 ð 103 ð 0.0980004 C 400020.003200040002300/0.05000400023000042 D 0 and the downstream pressure, P2 D 793 kN/m2 The kinetic energy term D 0002G/A00042 ln0002800/7930004 D 7.91 kg2 /m4 s2 This is insignificant in comparison with 69,120 kg2 /m4 s2 which is the value of the other terms in equation 4.55.
PROBLEM 4.12 If temperature does not change with height, estimate the boiling point of water at a height of 3000 m above sea-level. The barometer reading at sea-level is 98.4 kN/m2 and the temperature is 288.7 K. The vapour pressure of water at 288.7 K is 1.77 kN/m2 . The effective molecular weight of air is 29 kg/kmol.
Solution The air pressure at 3000 m is P2 and the pressure at sea level, P1 D 98.4 kN/m2 . v dP C g dz D 0 v D v1 0002P/P1 0004
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
dP C g dz D 0 P1 v1 P P1 v1 ln0002P2 /P1 0004 C g0002z2 0001 z1 0004 D 0
and: and:
v1 D 000222.4/2900040002288.7/27300040002101.3/98.40004 D 0.841 m3 /kg. ∴
98,400 ð 0.841 ln0002P2 /98.40004 C 9.8100023000 0001 00004 D 0 P2 D 68.95 kN/m2
and:
The relationship between vapour pressure and temperature may be expressed as: log P D a C bT When,
T D 288.7, P D 1.77 kN/m2
and when,
T D 373, P D 101.3 kN/m2
∴
log P D 00015.773 C 0.0209 T When P2 D 68.95, T D 364 K.
PROBLEM 4.13 A 150 mm gas main is used for transferring gas (molecular weight 13 kg/kmol and kinematic viscosity 0.25 cm2 /s) at 295 K from a plant to a storage station 100 m away, at a rate of 1 m3 /s. Calculate the pressure drop if the pipe can be considered to be smooth. If the maximum permissible pressure drop is 10 kN/m2 , is it possible to increase the flowrate by 25%?
Solution If the flow of 1 m3 /s is at STP, the specific volume of the gas is: 000222.4/130004 D 1.723 m3 /kg. The mass flowrate, G D 00021.0/1.7230004 D 0.58 kg/s. Cross-sectional area, A D 0002000e/4000400020.1500042 D 0.0176 m2 ∴
G/A D 32.82 kg/m2 s 0010/ D 0.25 cm2 /s D 0.25 ð 1000014 m2 /s
and ∴
0010 D 00020.25 ð 1000014 000400021/1.7230004 D 1.45 ð 1000015 N s/m2 Re D 00020.15 ð 32.82/1.45 ð 1000015 0004 D 3.4 ð 105
For smooth pipes, R/u2 D 0.0017, from Fig. 3.7. The pressure drop due to friction is: 40002R/u2 00040002l/d00040002G/A00042 D 400020.001700040002100/0.150004000232.8200042 D 4883 kg2 /m4 s2 and: 0001P D 00024883/1.7230004 D 2834 N/m2 or 2.83 kN/m2 .
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FLOW OF COMPRESSIBLE FLUIDS
73
If the flow is increased by 25%, G D 00021.25 ð 0.580004 D 0.725 kg/s G/A D 41.19 kg/m2 s and:
Re D 00020.15 ð 41.90004/00021.45 ð 105 0004 D 4.3 ð 105
and, from Fig. 3.7, R/u2 D 0.00165 The pressure drop D 400020.0016500040002100/0.150004000241.1900042 1.723 D 4.33 kN/m2 (which is less than 10 kN/m2 ) It is therefore possible to increase the flowrate by 25%.
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SECTION 5
Flow of Multiphase Mixtures PROBLEM 5.1 It is required to transport sand of particle size 1.25 mm and density 2600 kg/m3 at the rate of 1 kg/s through a horizontal pipe 200 m long. Estimate the air flowrate required, the pipe diameter, and the pressure drop in the pipe-line.
Solution For conventional pneumatic transport in pipelines, a solids-gas mass ratio of about 5 is employed. Mass flow of air D 00011/50003 D 0.20 kg/s and, taking the density of air as 1.0 kg/m3 , volumetric flowrate of air D 00011.0 ð 0.200003 D 0.20 m3 /s In order to avoid excessive pressure drops, an air velocity of 30 m/s seems reasonable. Ignoring the volume occupied by the sand (which is about 0.2% of that occupied by the 2 air), the cross-sectional area of pipe p required D 00010.20/300003 D 0.0067 m , equivalent to a pipe diameter of 00014 ð 0.0067/00050003 D 0.092 m or 92 mm. Thus a pipe diameter of 101.6 mm (100 mm) would be specified. From Table 5.3 for sand of particle size 1.25 mm and density 2600 kg/m3 , the freefalling velocity is: u0 D 4.7 m/s p In equation 5.37, 0001uG 0004 us 0003 D 4.7/[0.468 C 7.25 00014.7/26000003] D 6.05 m/s The cross-sectional area of a 101.6 mm i.d. pipe D 00010005 ð 0.10162 /40003 D 0.0081 m2 . ∴
and:
air velocity, uG D 00010.20/0.00810003 D 24.7 m/s us D 000124.7 0004 6.050003 D 18.65 m/s
Taking the viscosity and density of air as 1.7 ð 1000045 N s/m2 and 1.0 kg/m3 respectively, the Reynolds number for the air flow alone is: Re D 00010.102 ð 24.7 ð 1.00003/00011.7 ð 1000045 0003 D 148,000 and from Fig. 3.7, the friction factor D 0.002.
74
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FLOW OF MULTIPHASE MIXTURES
0004Pair D 4 0001l/d00030011u2
(equation 3.18)
D 00014 ð 0.0020001200/0.1020003 ð 1.0 ð 24.72 0003 D 9570 N/m2 or 9.57 kN/m2 assuming isothermal conditions and incompressible flow. 00010004Px / 0004 Pair 00030001us2 /F0003 D 00012805/u0 0003 ∴
(equation 5.38)
0004Px D 0001280500010004Pair 0003F0003/0001u0 us2 0003 D 00012805 ð 9.57 ð 1.00003/00014.7 ð 18.652 0003 D 16.4 kN/m2
PROBLEM 5.2 Sand of a mean diameter 0.2 mm is to be conveyed in water flowing at 0.5 kg/s in a 25 mm ID horizontal pipe 100 m long. What is the maximum amount of sand which may be transported in this way if the head developed by the pump is limited to 300 kN/m2 ? Assume fully suspended heterogeneous flow.
Solution See Volume 1, Example 5.2.
PROBLEM 5.3 Explain the various mechanisms by which particles may be maintained in suspension during hydraulic transport in a horizontal pipeline and indicate when each is likely to be important. A highly concentrated suspension of flocculated kaolin in water behaves as a pseudohomogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald–de Waele power-law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced, even though the flowrate of suspension is kept constant. Explain how this is possible in “slug” flow, and estimate the possible reduction in pressure gradient for equal volumetric, flowrates of suspension and air.
Solution If u is the superficial velocity of slurry, then: For slurry alone: The pressure drop in a pipe of length l is: Kun l. If the air: slurry volumetric ratio is R, there is no slip between the slurry and the air and the system consists of alternate slugs of air and slurry, then: The linear velocity of slurry is 0001R C 10003u
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
1 RC1 Assuming that the pressure drop is the sum of the pressure drops along the slugs, then: 0001 0002 l n the new pressure drop is: Kf0001R C 10003ug D Kun l0001R C 10003n00041 RC1 Fraction of pipe occupied by slurry slugs is
Kun l0001R C 10003n00041 pressure gradient with air D D 0001R C 10003n00041 pressure gradient without air Kun l
Then:
rD
For
n D 0.15 and: R D 1 r D 200040.85 D 0.55
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SECTION 6
Flow and Pressure Measurement PROBLEM 6.1 Sulphuric acid of density 1300 kg/m3 is flowing through a pipe of 50 mm internal diameter. A thin-lipped orifice, 10 mm diameter, is fitted in the pipe and the differential pressure shown by a mercury manometer is 10 cm. Assuming that the leads to the manometer are filled with the acid, calculate (a) the mass of acid flowing per second, and (b) the approximate loss of pressure caused by the orifice. The coefficient of discharge of the orifice may be taken as 0.61, the density of mercury as 13,550 kg/m3 , and the density of water as 1000 kg/m3 .
Solution See Volume 1, Example 6.2.
PROBLEM 6.2 The rate of discharge of water from a tank is measured by means of a notch, for which the flowrate is directly proportional to the height of liquid above the bottom of the notch. Calculate and plot the profile of the notch if the flowrate is 0.1 m3 /s when the liquid level is 150 mm above the bottom of the notch.
Solution The velocity of fluid discharged as a height h above the bottom of the notch is: 0001 u D 00052gh0007 The velocity therefore varies from zero at the bottom of the notch to a maximum value at the free surface. For a horizontal element of fluid of width 2w and depth dh at a height h above the bottom of the notch, the discharge rate of fluid is given by: 0001 dQ D 00052gh00072wdh If the discharge rate is linearly related to the height of the liquid over the notch, H, w will be a function of h and it may be supposed that: w D khn where k is a constant. 77
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Substituting for w in the equation for dQ and integrating to give the discharge rate over the notch Q then: 0002 H 0001 Q D 2 00052g0007 k hn h0.5 dh 0
0002 0001 D 2 00052g0007 k
H
hnC0.5 dh
0
0001 D 2 00052g0007 k[1/0005n C 1.50007]H0005nC1.50007
Since it is required that Q / H: n C 1.5 D 1 and:
n D 00040.5 0001 Q D 2 00052g0007 kH
Thus:
Since Q D 0.1 m3 /s when H D 0.15 m: 0001 k D 00050.1/0.150007[1/00052 00052g0007] D 0.0753 m1.5
Thus, with w and h in m:
w D 0.0753h00040.5
and, with w and h in mm:
w D 2374h00040.5
and using this equation, the profile is plotted as shown in Figure 6a.
500
400
h (mm)
300
200
100
300
200
100
0
100
200
300
Distance from centre line, w (mm)
Figure 6a.
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FLOW AND PRESSURE MEASUREMENT
79
PROBLEM 6.3 Water flows at between 3 l and 4 l/s through a 50 mm pipe and is metered by means of an orifice. Suggest a suitable size of orifice if the pressure difference is to be measured with a simple water manometer. What is the approximate pressure difference recorded at the maximum flowrate?
Solution Equations 6.19 and 6.21 relate the pressurepdrop to the mass flowrate. If equation 6.21 is used as a first approximation, G D CD A0 0011 00052gh0007. For the maximum flow of 4 l/s, G D 4 kg/s. The largest practicable height of a water manometer will be taken as 1 m and equation 6.21 is then used to calculate the orifice area A0 . If the coefficient of discharge CD is taken as 0.6, then: 0001 4.0 D 0.6A0 ð 1000 00052 ð 9.81 ð 1.00007, A0 D 0.0015 m2 and d0 D 0.0438 m The diameter, d0 , is comparable with the pipe diameter and hence the area correction term must be included and: [1 0004 0005A0 /A1 00072 ] D [1 0004 000543.82 /502 00072 ] D 0.641. Therefore the value of A0 must be recalculated as: 0003 4.0 D 0.6A0 ð 1000 00052 ð 9.8 ð 1.00007/[1 0004 0005A0 /A1 00072 ] from which A0 D 0.00195 m2 and d D 0.039 m or 39 mm 0003
0003
[1 0004 0005A0 /A1 00072 ] D
[1 0004 0005392 /502 00072 ] D 0.793
Substituting in equation 6.19: 0001 4.0 D 00050.6 ð 0.001950007 ð 1000 00052 ð 0.00100050004P0007/0.7930007
and:
0004P D 12320 N/m2 or 12.3 kN/m2
PROBLEM 6.4 The rate of flow of water in a 150 mm diameter pipe is measured by means of a venturi meter with a 50 mm diameter throat. When the drop in head over the converging section is 100 mm of water, the flowrate is 2.7 kg/s. What is the coefficient for the converging cone of the meter at that flowrate and what is the head lost due to friction? If the total loss of head over the meter is 15 mm water, what is the coefficient for the diverging cone?
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution The equation relating the mass flowrate G and the head loss across a venturi meter is given by: 0004 CD A0 2v0005P1 0004 P2 0007 GD (equation 6.19) v 1 0004 0005A0 /A1 00072 G D CD 0011 0003
A 1 A2
0005A21 0004 A21 0007 0001 G D CD 0011C0 00052ghv 0007
0001
00052v0005P1 0004 P2 00070007
(equation 6.32) (equation 6.33)
where C0 is a constant for the meter and hv is the loss in head over the converging cone expressed as height of fluid. A1 D 00050016/4000700050.1500072 D 0.0176 m2 A2 D 00050016/4000700050.0500072 D 0.00196 m2 0001 C0 D 00050.0176 ð 0.00196/ 00050.01762 0004 0.001962 00070007 D 0.00197 m2 ∴
hv D 0.1 m p 2.7 D 0005CD ð 1000 ð 0.001970007 00052 ð 9.81 ð 0.100007 and CD D 0.978
In equation 6.33, if there were no losses, the coefficient of discharge of the meter would be unity, and for a flowrate G the loss in head would be 0005hv 0004 hf 0007 where hf is the head loss due to friction. 0001 Thus: G D 0011C0 [2g0005hv 0004 hf 0007] Dividing this equation by equation 6.33 and squaring gives: 1 0004 0005hf /hv 0007 D C2D and hf D hv 00051 0004 C2D 0007 ∴
hf D 10000051 0004 0.9782 0007 D 4.35 mm
0 If the head recovered over the diverging cone 0001 is hv and the coefficient of discharge for 0 0 0 0 the converging cone is CD , then G D CD 0011C 00052ghv 0007 If the whole of the excess kinetic energy is recovered as pressure energy, the coefficient 0 C0D will equal unity and 0003 G will be obtained with a recovery of head equal to hv plus some quantity hf0 , G D 0011C0 [2g0005hv0 C hf0 0007 Equating these two equations and squaring gives: 0
0
CD2 D 1 C 0005hf0 /hv0 0007 and hf0 D hv0 0005CD2 0004 10007 Thus the coefficient of the diverging cone is greater than unity and the total loss of head D hf C hf0 . Head loss over diverging cone D 000515.0 0004 4.350007 D 10.65 mm
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The coefficient of the diverging cone C0D is given by: 0001 G D C0D 0011C0 00052ghv0 0007 and:
hv0 D 0005100 0004 150007 D 85 mm p 2.7 D 0005C0D ð 1000 ð 0.001970007 00052 ð 9.81 ð 0.0850007 or C0D D 1.06
PROBLEM 6.5 A venturi meter with a 50 mm throat is used to measure a flow of slightly salt water in a pipe of inside diameter 100 mm. The meter is checked by adding 20 cm3 /s of normal sodium chloride solution above the meter and analysing a sample of water downstream from the meter. Before addition of the salt, 1000 cm3 of water requires 10 cm3 of 0.1 M silver nitrate solution in a titration. 1000 cm3 of the downstream sample required 23.5 cm3 of 0.1 M silver nitrate. If a mercury-under-water manometer connected to the meter gives a reading of 221 mm, what is the discharge coefficient of the meter? Assume that the density of the liquid is not appreciably affected by the salt.
Solution If the flow of the solution is x m3 /s, then a mass balance in terms of sodium chloride gives: 0005x ð 0.05850007 C 000520 ð 1000046 ð 58.50007 D 0.1375000520 ð 1000046 C x0007 x D 0.0148 m3 /s
and:
or, assuming the density of the solution is 1000 kg/m3 , the mass flowrate is: 00050.0148 ð 10000007 D 14.8 kg/s For the venturi meter, the area of the throat is given by: A1 D 00050016/40007000550/100000072 D 0.00196 m2 and the area of the pipe is: A2 D 00050016/400070005100/100000072 D 0.00785 m2 From equations 6.32 and 6.33: 0003 C0 D A1 A2 / 0005A21 0004 A22 0007 D 0.00204 m2 h D 221 mm Hgunder-water D 0.221000513500 0004 10000007/1000 D 2.78 m water 0001 and hence: 14.8 D 0005CD ð 1000 ð 0.002040007 00052 ð 9.81 ð 2.780007 and:
CD D 0.982
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PROBLEM 6.6 A gas cylinder containing 30 m3 of air at 6 MN/m2 pressure discharges to the atmosphere through a valve which may be taken as equivalent to a sharp edged orifice of 6 mm diameter (coefficient of discharge D 0.6). Plot the rate of discharge against the pressure in the cylinder. How long will it take for the pressure in the cylinder to fall to (a) 1 MN/m2 , and (b) 150 kN/m2 ? Assume an adiabatic expansion of the gas through the valve and that the contents of the cylinder remain at 273 K.
Solution Area of orifice D 00050016/4000700050.00600072 D 2.828 ð 1000045 m2 . The critical pressure ratio wc is: (equation 4.43) wc D [2/0005k C 10007]k/0005k000410007 Taking k D 001a D 1.4 for air, wc D 0.527. Thus sonic velocity will occur until the cylinder pressure falls to a pressure of: P2 D 0005101.3/0.5270007 D 192.2 kN/m2 . For pressures in excess of 192.2 kN/m2 , the rate of discharge is given by: 0003 G D CD A0 0005kP1 /v1 000700052/0005k C 1000700070005kC10007/0005k000410007 (equation 6.29) p If k D 1.4, G D 1.162 ð 1000045 0005P1 /v1 0007 If Pa and va are atmospheric pressure and the specific volume at atmospheric pressure respectively, Pa va D P1 v1 and v1 D Pa va /P1 Pa D 101,300 N/m2 and va D 000522.4/290007 D 0.773 m3 /kg ∴
and:
v1 D 0005101,300 ð 0.773/P1 0007 D 000578,246/P1 0007 0003 G D 1.162 ð 1000045 0005P12 /78,2460007 D 4.15 ð 1000048 P1 kg/s
If P1 is expressed in MN/m2 , then: G D 0.0415 P1 kg/s. For pressures lower than 192.2 kN/m2 : G2 D 0005A0 CD /v2 00072 2P1 v1 0005k/k 0004 10007[1 0004 0005P2 /P1 00070005k000410007/k ]
(equation 6.26)
3
v2 D va D 0.773 m /kg
P2 D Pa D 101,300 N/m2 v1 D Pa va /P1
Substituting gives:
2
G D 2.64 ð 1000044 [1 0004 0005Pa /P1 00070.286 ]
Thus a table of G as a function of pressure may be produced as follows:
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P < 192.2 kN/m2
P > 192.2 kN/m2
P (MN/m2 )
G (kg/s)
P (MN/m2 )
G (kg/s)
0.1013 0.110 0.125 0.150 0.175
0 0.0024 0.0039 0.0053 0.0062
0.2 0.5 1.0 2.0 6.0
0.0083 0.0208 0.0416 0.0830 0.249
These data are plotted in Fig. 6b, from which discharge rate is seen to be linear until the cylinder pressure falls to 0.125 MN/m2 . 0.25
G (kg /s)
0.20
0.15 0.010
0.10
G (kg /s)
Low pressure range 0.005
0.05 0
0.1 0.2
0.3 0.4 0.5
P1 (MN/m2) 0
1
2
3
4
5
6
P1 (MN/m2)
Figure 6b.
If m is the mass of air in the cylinder at any pressure P1 over the linear part of the curve, G D dm/dt D 0.0415P1 . ∴
dt D dm/0.0415P1 m D 000529/22.400070005P1 /0.10130007 ð 30 D 383.4P1 kg
∴
and
dt D 383.4dm/0.0415m D 9240d m/m t D 9240 ln0005m1 /m2 0007
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At 6 MN/m2 and 1 MN/m2 , the masses of air in the cylinder are 2308 and 383.4 kg respectively. ∴
The time for the pressure to fall to 1 MN/m2 D 9240 ln00052308/3834.40007 D 16,600 s 00054.61 h0007
As 0.15 MN/m2 is still within the linear region, the time for the pressure to fall to this value is 34,100 s .00059.47 h0007
PROBLEM 6.7 2
Air, at 1500 kN/m and 370 K, flows through an orifice of 30 mm2 to atmospheric pressure. If the coefficient of discharge is 0.65, the critical pressure ratio 0.527, and the ratio of the specific heats is 1.4, calculate the mass flowrate.
Solution If the critical pressure ratio wc is 0.527 (from Problem 6.6), sonic velocity will occur until the pressure falls to 0005101.3/0.5270007 D 192.2 kN/m2 . For pressures above this value, the mass flowrate is given by: 0001 G D CD A0 0005kP1 /v1 0007[2/0005k C 10007]0005kC10007/0005k000410007 (equation 6.29) 0001 p If k D 1.4, G D CD A0 00051.4P1 /v1 000700052/2.400072.4/0.4 D CD A0 00050.468P1 /v1 0007 P1 D 1,500,000 N/m2 v1 D 000522.4/2900070005370/27300070005101.3/15000007 D 0.0707 m3 /kg p Substituting gives: G D 00050.65 ð 30 ð 1000046 0007 00050.486 ð 1,500,000/0.07070007 D 0.061 kg/s
and:
PROBLEM 6.8 Water flows through an orifice of 25 mm diameter situated in a 75 mm pipe at the rate of 300 cm3 /s. What will be the difference in level on a water manometer connected across the meter? Viscosity of water is 1 mN s/m2 .
Solution See Volume 1, Example 6.1.
PROBLEM 6.9 Water flowing at 1.5 l/s in a 50 mm diameter pipe is metered by means of a simple orifice of diameter 25 mm. If the coefficient of discharge of the meter is 0.62, what will
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be the reading on a mercury-under-water manometer connected to the meter? What is the Reynolds number for the flow in the pipe? Density of water D 1000 kg/m3 . Viscosity of water D 1 mN s/m2 .
Solution Mass flowrate, G D 00051500 ð 1000046 ð 10000007 D 1.5 kg/s. Area of orifice, A0 D 00050016/4000700050.02500072 D 0.00049 m2 . Area of pipe, A1 D 00050016/4000700050.05000072 D 0.00196 m2 . Reynolds number D 0011ud/
D d0005G/A1 0007/ D 0.0500051.5/0.001960007/00051 ð 1000043 0007 D 3.83 ð 104
0001 The orifice meter equations are 6.19 and 6.21; the latter being used when [1 0004 0005A0 /A1 00072 ] approaches unity. 0001 0001 [1 0004 0005A0 /A1 00072 ] D [1 0004 0005252 /502 00072 ] D 0.968 Thus: p Using equation 6.21, G D CD A0 0011 00052gh0007 gives: 0001 1.5 D 0.62 ð 0.00049 ð 1000 00052 ð 9.81h0007, and h D 1.24 m of water
Using equation 6.19 in terms of h gives: 0001 1.5 D 00050.62 ð 0.00049 ð 1000/0.9680007 00052gh0007 and h D 1.16 m of water
This latter value of h should be used. The height of a mercury-under-water manometer would then be 1.16/0005000513.55 0004 1.000007/1.000007 D 0.092 m or 92 mm Hg.
PROBLEM 6.10 What size of orifice would give a pressure difference of 0.3 m water gauge for the flow of a petroleum product of density 900 kg/m3 at 0.05 m3 /s in a 150 mm diameter pipe?
Solution As in previous problems, equations 6.19 and 6.21 may be used to calculate the flow through an orifice. In this problem the size of the orifice is to be found so that the simpler equation will be used in the first instance. 0001 G D CD A0 0011 00052gh0007 (equation 6.21) G D 00050.05 ð 9000007 D 45.0 kg/s 0011 D 900 kg/m3
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h D 0.3 m of water or 00050.3/0.90007 D 0.333 m of petroleum product CD D 0.62 (assumed) p 45.0 D 00050.62 ð A0 ð 9000007 00052 ð 9.81 ð 0.3330007
∴
A0 D 0.3155 m2 and d0 D 0.2 m.
Thus:
This orifice diameter is larger than the pipe size so that it was clearly wrong to use the simpler equation. 0003 Thus: G D CD A0 0011 [2gh/00051 0004 0005A0 /A1 00072 0007] (equation 6.19) A1 D 00050016/4000700050.1500072 D 0.0177 m2 0003 45.0 D 00050.62 ð A0 ð 9000007 [2 ð 9.81 ð 0.33/00051 0004 0005A0 /0.017700072 0007]
∴
A0 D 0.154 m2 and d0 D 0.14 m
Thus:
PROBLEM 6.11 The flow of water through a 50 mm pipe is measured by means of an orifice meter with a 40 mm aperture. The pressure drop recorded is 150 mm on a mercury-underwater manometer and the coefficient of discharge of the meter is 0.6. What is the Reynolds number in the pipe and what would the pressure drop over a 30 m length of the pipe be expected to be? Friction factor, ! D R/0011u2 D 0.0025. Density of mercury D 13,600 kg/m3 . Viscosity of water D 1 mN s/m2 . What type of pump would be used, how would it be driven and what material of construction would be suitable?
Solution Area of pipe, A1 D 00050016/4000700050.0500072 D 0.00197 m2 . Area of orifice, A0 D 00050016/4000700050.0400072 D 0.00126 m2 . h D 150 mmHg under water D 0.15 ð 000513600 0004 10000007/1000 1.88 m of water. 1 0004 0005A0 /A00072 D 0.591, and hence: 0003 G D CD A0 0011 [2gh/00051 0004 0005A0 /A00072 0007] (equation 6.19) 0001 D 00050.6 ð 0.00126 ð 10000007 2 ð 9.81 ð 1.88/0.591 D 5.97 kg/s Reynolds number, 0011ud/ D d0005G/A1 0007/ D 0.0500056.22/0.001970007/00051 ð 1000043 0007 D 1.52 ð 105 The pressure drop is given by: 0004P/v D 40005R/0011u2 00070005l/d00070005G/A00072 D 400050.00250007000530/0.05000700055.97/0.0019700072 D 5.74 ð 107 kg2 /m4 s2
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0004P D 5.74 ð 107 ð 00051/10000007 D 00055.74 ð 104 0007 N/m2 or 57.4 kN/m2 Power required D head loss (m) ð G ð g D 00055.74 ð 104 /1000 ð 9.81000700055.97 ð 9.810007 D 343 W For a pump efficiency of 60%, the actual power requirement D 0005343/0.60007 D 571 W. Water velocity D 5.97/00050.00197 ð 10000007 D 3.03 m/s. For this low-power requirement at a low head and comparatively low flowrate, a centrifugal pump, electrically driven and made of stainless steel, would be suitable.
PROBLEM 6.12 A rotameter has a tube 0.3 m long which has an internal diameter of 25 mm at the top and 20 mm at the bottom. The diameter of the float is 20 mm, its effective density is 4800 kg/m3 , and its volume 6.6 cm3 . If the coefficient of discharge is 0.72, at what height will the float be when metering water at 100 cm3 /s?
Solution See Volume 1, Example 6.4.
PROBLEM 6.13 Explain why there is a critical pressure ratio across a nozzle at which, for a given upstream pressure, the flowrate is a maximum. Obtain an expression for the maximum flow for a given upstream pressure for isentropic flow through a horizontal nozzle. Show that for air (ratio of specific heats, 001a D 1.4) the critical pressure ratio is 0.53 and calculate the maximum flow through an orifice of area 30 mm2 and coefficient of discharge 0.65 when the upstream pressure is 1.5 MN/m2 and the upstream temperature 293 K. Kilogram molecular volume D 22.4 m3 .
Solution The reasons for critical pressure ratios are discussed in Section 4.4.1. The maximum rate of discharge is given by: 0003 (equation 6.29) Gmax D CD A0 0005kP1 /v1 000700052/0005k C 1000700070005kC10007/0005k000410007 For an isentropic process, k D 001a D 1.4 for air. The critical pressure ratio,
wc D 00052/k C 10007k/0005k000410007
(equation 4.430007
Substituting for k D 001a D 1.4, wc D 00052/2.400071.4/0.4 D 0.523
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The maximum rate of discharge is given by equation 6.29. P1 D 1.5 ð 106 N/m2 A0 D 30 ð 1000046 m2 k D 1.4 and CD D 0.65 At P1 D 1.5 MN/m2 and T1 D 293 K, the specific volume v1 is: v1 D 000522.4/2900070005293/273000700050.1013/1.50007 D 0.056 m3 /kg 0001 Substituting, Gmax D 0.65 ð 30 ð 1000046 00051.4 ð 1.5 ð 106 /0.056000700052/2.400072.4/0.4 D 0.069 kg/s
PROBLEM 6.14 A gas cylinder containing air discharges to atmosphere through a valve whose characteristics may be considered similar to those of a sharp-edged orifice. If the pressure in the cylinder is initially 350 kN/m2 , by how much will the pressure have fallen when the flowrate has decreased to one-quarter of its initial value? The flow through the valve may be taken as isentropic and the expansion in the cylinder as isothermal. The ratio of the specific heats at constant pressure and constant volume is 1.4.
Solution From equation 4.43: the critical pressure ratio, wc D [2/0005k C 10007]k/0005k000410007 D 00052/2.400071.4/0.4 D 0.528 If the cylinder is discharging to atmospheric pressure, sonic velocity will occur until the cylinder pressure has fallen to 0005101.3/0.5280007 D 192 kN/m2 The maximum discharge when the cylinder pressure exceeds 192 kN/m2 is given by: 0004 0005 00060005kC10007/0005k000410007 kP1 2 Gmax D CD A0 (equation 6.29) v1 0005k C 10007 If Pa and va are the pressure and specific volume at atmospheric pressure, then:
and:
Gmax
1/v1 D P1 /Pa va 0004 0005 00060005kC10007/0005k000410007 kP12 2 D C D A0 Pa va k C 1 0003 D CD A0 P1 [0005k/Pa va 000700052/k C 10007]0005kC10007/0005k000410007
If G350 and G192 are the rates of discharge at 350 and 192 kN/m2 respectively, then: G350 /G192 D 0005350/1920007 D 1.82 or:
G192 D 0.55G350
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For pressures below 192 kN/m2 : 0007 0005 0006
0005 00060005k000410007/k k P2 C D A0 2P1 v1 10004 GD v2 k00041 P1
(equation 6.26)
Substituting for 1/v1 D P1 /Pa va and v2 D va gives: 0007 0005 0006
0005 00060005k000410007/k k P2 C D A0 GD 2Pa va 10004 va k00041 P1 and:
G2 D 0005CD A0 /va 00072 2Pa va [k/0005k 0004 10007][1 0004 0005P2 /P1 00070005k000410007/k ] D 0005CD A0 /va 00072 2Pa va ð 3.5[1 0004 0005P2 /P1 00070.286 ]
When P1 D 192 kN/m2 , G192 D 0.55G350 , P2 , atmospheric pressure, 101.3 kN/m2 and: 00050.55G350 00072 D 0005CD A0 /va 00072 2Pa va ð 3.5[1 0004 0005101.3/19200070.286 ] When the final pressure P1 is reached, the flowrate is 0.25G350 . ∴
00050.25G350 00072 D 0005CD A0 /va 00072 2Pa va ð 3.500051 0004 0005101.3/P1 00070.286 0007
Dividing these two equations gives: 0005 0006 0.55 2 1 0004 0005101.3/19200070.286 D 0.25 1 0004 0005101.3/P1 00070.286 and:
P1 D 102.3 kN/m2
PROBLEM 6.15 Water discharges from the bottom outlet of an open tank 1.5 m by 1 m in cross-section. The outlet is equivalent to an orifice 40 mm diameter with a coefficient of discharge of 0.6. The water level in the tank is regulated by a float valve on the feed supply which shuts off completely when the height of water above the bottom of the tank is 1 m and which gives a flowrate which is directly proportional to the distance of the water surface below this maximum level. When the depth of water in the tank is 0.5 m the inflow and outflow are directly balanced. As a result of a short interruption in the supply, the water level in the tank falls to 0.25 m above the bottom but is then restored again. How long will it take the level to rise to 0.45 m above the bottom?
Solution The mass flowrate G is related to the head h for the flow through an orifice when the area of the orifice is small in comparison with the area of the pipe by: 0001 G D CD A0 0011 00052gh0007 (equation 6.21)
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If h is the distance of the water level below the maximum depth of 1 m, then the head above the orifice is equal to 00051 0004 h0007 and: 0001 G D CD A0 0011 [2g00051 0004 h0007] When the tank contains 0.5 m of water, the flowrate is given by: 0001 G D 00050.6 ð 00050016/4000700050.0400072 ð 1000 00052 ð 9.81 ð 0.500070007 D 2.36 kg/s The input to the tank is stated to be proportional to h, and when the tank is half full the inflow is equal to the outflow, or:
2.36 D 0005K ð 0.50007 and K D 4.72 kg/ms
p p Thus the inflow D 4.72h kg/s and the outflow D CD A0 0011 2g 00051 0004 h0007 kg/s. p p p The net rate of filling D 4.72h 0004 CD A0 0011 2g 00051 0004 h0007 D 4.72h 0004 3.34 00051 0004 h0007 Time to fill the tank D (mass of water/rate of filling) D 1 ð 1.5 ð 00050.45 0004 0.250007 ð 1000/rate D 300/rate The time to fill from 0.25 to 0.45 m above the bottom of the tank is then: 0002 0.75 300dh p time D 0.55 4.72h 0004 3.34 00051 0004 h0007 This integral is most easily solved graphically as shown in Fig. 6c, where the area under the curve D 0.233 s/m and the time D 0005300 ð 0.2330007 D 70 s.
3.0
1/(4.72−3.34√ 1−h)
2.5
2.0
1.5
Area under curve = 0.233 s/m
1.0
0.5
0 0.55
0.55
0.65 h (m)
0.70
0.70
Figure 6c.
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PROBLEM 6.16 The flowrate of air at 298 K in a 0.3 m diameter duct is measured with a pitot tube which is used to traverse the cross-section. Readings of the differential pressure recorded on a water manometer are taken with the pitot tube at ten different positions in the crosssection. These positions are so chosen as to be the mid-points of ten concentric annuli each of the same cross-sectional area. The readings are: Position Manometer reading (mm water) Position Manometer reading (mm water)
1 18.5 6 14.7
2 18.0 7 13.7
3 17.5 8 12.7
4 16.8 9 11.4
5 15.7 10 10.2
The flow is also metered using a 150 mm orifice plate across which the pressure differential is 50 mm on a mercury-under-water manometer. What is the coefficient of discharge of the orifice meter?
Solution Cross-sectional area of duct D 00050016/4000700050.300072 D 0.0707 m2 . Area of each concentric annulus D 0.00707 m2 . If the diameters of the annuli are designated d1 , d2 etc., then: 0.00707 D 00050016/4000700050.32 0004 d21 0007 0.00707 D 00050016/400070005d2 0004 d22 0007 0.00707 D 00050016/400070005d22 0004 d23 0007 and so on, and the mid-points of each annulus may be calculated across the duct. For a pitot tube, the velocity may be calculated from the head h as u D For position 1, h D 18.5 mm of water.
p
00052gh0007
The density of the air D 000529/22.400070005273/2980007 D 1.186 kg/m3 .
and:
h D 000518.5 ð 1000043 ð 1000/1.1860007 D 15.6 m of air 0001 u D 00052 ð 9.81 ð 15.60007 D 17.49 m/s
In the same way, the velocity distribution across the tube may be found as shown in the following table. Mass flowrate, G D 00051.107 ð 1.1860007 D 1.313 kg/s For the orifice, [1 0004 0005A0 /A1 00072 ] D [1 0004 00050.15/0.300072 ] D 0.938 h D 50 mm Hg-under-water D 00050.05 ð 000513.55 0004 10007 ð 1000/1.1860007 D 529 m of air p and: 1.313 D CD 00050016/4000700050.1500072 ð 1.186 00052 ð 9.81 ð 529/0.9380007 and CD D 0.61
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Position
1 2 3 4 5 6 7 8 9 10
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Distance from axis of duct (mm)
Manometer reading Water (mm)
Air (m)
24 57 75 89 101 111 121 130 139 147
18.5 18.0 17.5 16.8 15.7 14.7 13.7 12.7 11.4 10.2
15.6 15.17 14.75 14.16 13.23 12.39 11.55 10.71 9.61 8.60
Air velocity (u m/s)
Velocity ð area of annulus 0005m3 /s0007
17.5 17.3 17.0 16.7 16.1 15.6 15.1 14.5 13.7 13.0
0.124 0.122 0.120 0.118 0.114 0.110 0.107 0.103 0.097 0.092 Total D 1.107
The velocity profile across the duct is plotted in Fig. 6d. Centre line of duct
Duct wall
20
Velocity (m/s)
15
10
5
0
20
40
60
80 100 120 140
Distance from duct axis (mm)
Figure 6d.
PROBLEM 6.17 Explain the principle of operation of the pitot tube and indicate how it can be used in order to measure the total flowrate of fluid in a duct. If a pitot tube is inserted in a circular
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cross-section pipe in which a fluid is in streamline flow, calculate at what point in the cross-section it should be situated so as to give a direct reading representative of the mean velocity of flow of the fluid.
Solution The principle of operation of a pitot tube is discussed in Section 6.3.1. It should be emphasised that the pitot tube measures the point velocity of a flowing fluid and not the average velocity so that in order to find the average velocity, a traverse across the duct is necessary. Treatment of typical results is illustrated in Problem 6.16. The point velocity is p given by u D 00052gh0007 where h is the difference of head expressed in terms of the flowing fluid. For streamline flow, the velocity distribution is discussed in Section 3.3.4 and: us /uCL D 1 0004 0005s2 /r 2 0007
(equation 3.32)
where us and uCL are the point velocities at a distance s from the wall and at the axis respectively and r is the radius of the pipe. The average velocity is: uav D umax /2
(equation 3.36)
When us D uav D umax /2, us /umax D 0005umax /20007/umax D 1 0004 0005s2 /r 2 0007 and:
0.5 D s2 /r 2 from which s D 0.707 r
PROBLEM 6.18 The flowrate of a fluid in a pipe is measured using a pitot tube, which gives a pressure differential equivalent to 40 mm of water when situated at the centre line of the pipe and 22.5 mm of water when midway between the axis and the wall. Show that these readings are consistent with streamline flow in the pipe.
Solution For streamline flow in a pipe, a force balance gives: 0004P0016r 2 D 0004 and:
0004u D
du 20016rl dr
0004P r 2 0004P r and 0004 u D C constant. 2 l 2 l 2
When r D a (at the wall), u D 0, the constant D 0004Pa2 /4 l and:
uD0004
P 2 0005a 0004 r 2 0007 4 l
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The maximum velocity, umax D
0004Pa2 4 l u
and:
umax
D10004
r 2
a
When r D a/2, u/umax D 0.75. The pitot tube is discussed in Section 6.3.1 and: p uDk h
(from equation 6.10)
At the centre-line, u D umax and h D 40 mm. p ∴ umax D K 40 D 6.32 K At a point midway between the axis and the wall, u D u1/2 and h D 22.5 mm. p ∴ u1/2 D K 22.5 D 4.74 K u1/2 /umax D 00054.74 K/632 K0007 D 0.75 and hence the flow is streamline.
PROBLEM 6.19 Derive a relationship between the pressure difference recorded by a pitot tube and the velocity of flow of an incompressible fluid. A pitot tube is to be situated in a large circular duct in which fluid is in turbulent flow so that it gives a direct reading of the mean velocity in the duct. At what radius in the duct should it be located, if the radius of the duct is r? The point velocity in the duct can be assumed to be proportional to the one-seventh power of the distance from the wall.
Solution An energy balance for an incompressible fluid in turbulent flow is given by: u2 /2 C gz C vP C F D 0
(equation 2.55)
Ignoring functional losses and assuming the pitot tube to be horizontal, 0005u22 0004 u12 0007/2 D 0004v0005P2 0004 P1 0007 If the fluid is brought to rest of plane 2, then:
and:
0004u12 /2 D 0004v0005P2 0004 P1 0007 0001 p u1 D 2v0005P2 0004 P1 0007 D 2gh
(equation 6.10)
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95
If the duct radius is r, the velocity uy at a distance y from the wall (and s from the centreline) is given by the one-seventh power law as: y 1/7 uy D us (equation 3.59) r where us is the velocity at the centreline. The flow, dQ, through an annulus of thickness dy1 distance y from the axis is: y 1/7 dQ D 20016sdyus r Multiplying and dividing through by r 2 gives: s y 1/7 y d dQ D 20016r 2 us r r r
y 1/7 y y or, since s D 0005r 0004 y0007: D 20016r 2 us 1 0004 d r r r 0002 1 000e 1/7 8/7 000f y y y The total flow is: Q D 20016r 2 us 0004 d r r r 0 000e 000f 7 y 8/7 7 y 15/7 1 D 20016r 2 us 0004 D 0.8170016r 2 us 8 r 15 r 0 The average velocity, uav D Q/0016r 2 D 0.817us uy D uav , 0.817us D us 0005y/r00071/7
Thus: ∴
0005y/r0007 D 0.243 and s/r D 0.757
PROBLEM 6.20 A gas of molecular weight 44 kg/kmol, temperature 373 K and pressure 202.6 kN/m2 is flowing in a duct. A pitot tube is located at the centre of the duct and is connected to a differential manometer containing water. If the differential reading is 38.1 mm water, what is the velocity at the centre of the duct? The volume occupied by 1 kmol at 273 K and 101.3 kN/m2 is 22.4 m3 .
Solution As shown in section 6.2.5, for a pitot tube: u12 /2 C P1 v D u22 /2 C P2 v u2 D 0, and hence, u1 D
p 20005P2 0004 P1 0007v
Difference in head D 38.1 mm of water ∴ P2 0004 P1 D 0005000538.1/10000007 ð 1000 ð 9.810007 D 373.8 N/m2
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The specific volume, v D 000522.4/4400070005373/27300070005101.3/202.60007 D 0.348 m3 /kg p ∴ u1 D 2 ð 373.8 ð 0.348 D 16.1 m/s
PROBLEM 6.21 Glycerol, of density 1260 kg/m3 and viscosity 50 mNs/m2 , is flowing through a 50 mm pipe and the flowrate is measured using an orifice meter with a 38 mm orifice. The pressure differential is 150 mm as indicated on a manometer filled with a liquid of the same density as the glycerol. There is reason to suppose that the orifice meter may have become partially blocked and that the meter is giving an erroneous reading. A check is therefore made by inserting a pitot tube at the centre of the pipe. It gives a reading of 100 mm on a water manometer. What does this suggest?
Solution From the reading taken from the pitot tube, the velocity in the pipe, and hence the mass flowrate, can be calculated. From the orifice meter, the mass flowrate can also be calculated and compared with the accurate value. 0001 For the pitot tube, u D 2gh (equation 6.10) where u D umax at the pipe axis, and the head loss h is in m of the liquid flowing. Now: ∴
h D 0005100/10000007 ð 00051000/12600007 D 0.0794 m of glycerol p umax D 2 ð 9.81 ð 0.0794 D 1.25 m/s
Reynolds number D 00051260 ð 1.25 ð 0.05/0.050007 D 1575 ∴
uav D 0.5umax D 0.63 m/s
(equation 3.36)
Mass flowrate D 00050.63 ð 1260 ð 00050016/40007 ð 0.052 0007 D 1.56 kg/s For the orifice meter: 0004 A0 2v0005P1 0004 P2 0007 (equation 6.190007 mass flowrate, G D CD v 1 0004 0005A0 /A1 00072 0004 2gh D C D A0 0011 1 0004 0005d0 /d00074 0004 2 ð 9.81 ð 0005150/10000007 2 D 0005CD ð 00050016/40007 ð 0.038 ð 12600007 D 2.99CD 1 0004 00050.038/0.0500074 ∴ CD D 00051.56/2.990007 D 0.53 which confirms that the meter is faulty.
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PROBLEM 6.22 The flowrate of air in a 305 mm diameter duct is measured with a pitot tube which is used to traverse the cross-section. Readings of the differential pressure recorded on a water manometer are taken with the pitot tube at ten different positions in the cross-section. These positions are so chosen as to be the mid-points of ten concentric annuli each of the same cross-sectional area. The readings are as follows: Position Manometer reading (mm water) Position Manometer reading (mm water)
1 18.5 6 14.7
2 18.0 7 13.7
3 17.5 8 12.7
4 16.8 9 11.4
5 15.8 10 10.2
The flow is also metered using a 50 mm orifice plate across which the pressure differential is 150 mm on a mercury-under-water manometer. What is the coefficient of discharge of the orifice meter?
Solution For a pitot tube, the velocity at any point in the duct is: 0001 u D 2gh
(equation 6.10)
where h is the manometer reading in m of the fluid which flows in the duct. ∴
h D 0005reading in mm water/10000007 ð 00050011w /0011air 0007 m
The total volumetric air flowrate is given by: Q D 0005area of duct ð average velocity0007 00100001 2gh D area ð 00051/100007 00100001 0001 manometer reading D 00050016/40007 ð 0.3052 ð 0.1 ð 00050011w /0011a 0007/1000 ð p p p p 0001 0001 D 0.00102 00050011w /0011a 00070005 18.5 C 18.0 C 17.5 C Ð Ð Ð 10.20007 D 0.0394 00050011w /0011a 0007 For the orifice meter, the volumetric flowrate is given by: 0001 A0 2gh Q D CD 0001 1 0004 0005A0 /A1 00072 A0 D 00050016/40007 ð 0.152 D 0.0177 m2 , A1 D 00050016/40007 ð 0.3052 D 0.0731 m2 , h D 50 mm Hg under water
∴
∴
D 00050.05000513.6 0004 1.00007/1.00007 D 0.63 m of water D 0.6300050011w /0011a 0007 m of air. 0001 0.0177 Q D CD ð 0001 2 ð 9.81 ð 0.6300050011w /0011a 0007 1 0004 00050.0177/0.073100072 0001 D 0.066 00050011w /0011a 0007 ð CD p p 0.0394 00050011w /0011a 0007 D 0.066 00050011w /0011a 0007 ð CD and CD D 0.60
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PROBLEM 6.23 The flow of liquid in a 25 mm diameter pipe is metered with an orifice meter in which the orifice has a diameter of 19 mm. The aperture becomes partially blocked with dirt from the liquid. What fraction of the area can become blocked before the error in flowrate at a given pressure differential exceeds 15 per cent? Assume that the coefficient of discharge of the meter remains constant when calculated on the basis of the actual free area of the orifice.
Solution If two sections in the pipe are chosen, 1 being upstream and 2 at the orifice, then from an energy balance: u12 /2 C P1 v D u22 /2 C P2 v
(from equation 2.55)
and G, the mass flowrate D u2 A2 /v D u1 A1 /v ∴
or:
0005u22 /200070005A2 /A1 00072 C P1 v D u22 /2 C P2 v u22 D
20005P1 0004 P2 0007v 1 0004 0005A2 /A1 00072
The volumetric flowrate, Q D CD A2 u2 ∴
Q2 D C2D A22 ð
20005P1 0004 P2 0007v 1 0004 0005A2 /A1 00072
D 2C2D 0005P1 0004 P2 0007vA21 A22 /0005A21 0004 A22 0007 or:
Q D K0003
A1 A2
000510007
A21 0004 A22
If the area of the orifice is reduced by partial blocking, the new orifice area D rA2 where f is the fraction available for flow. The new flowrate D 0.85 Q when the error is 15 per cent and: KA1 fA2 0.85Q D 0003 A21 0004 f2 A22
000520007
A1 D 00050016/40007 ð 252 D 491 mm2 A2 D 00050016/40007 ð 192 D 284 mm2 ∴ Dividing equation (2) by equation (1) and substituting gives: 0001 f 00054912 0004 2842 0007 0.85 D 0001 00054912 0004 r 2 2842 0007
from which f D 0.89 or 11 per cent of the area is blocked.
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99
PROBLEM 6.24 Water is flowing through a 100 mm diameter pipe and its flowrate is metered by means of a 50 mm diameter orifice across which the pressure drop is 13.8 kN/m2 . A second stream, flowing through a 75 mm diameter pipe, is also metered using a 50 mm diameter orifice across which the pressure differential is 150 mm measured on a mercury-underwater manometer. The two streams join and flow through a 150 mm diameter pipe. What would you expect the reading to be on a mercury-under-water manometer connected across a 75 mm diameter orifice plate inserted in this pipe? The coefficients of discharge for all the orifice meters are equal. Density of mercury D 13600 kg/m3 .
Solution As in Problem 6.23: u22
p 20005P1 0004 P2 0007v 20005P1 0004 P2 0007v D and Q D CD A2 2 1 0004 0005A1 /A2 0007 1 0004 0005A2 /A1 00072
For pipe 1, A2 D 00050016/40007 ð 0.052 D 0.00196 m2 , A1 D 00050016/40007 ð 0.102 D 0.00785 m2 , 0005P1 0004 P2 0007 D 13,800 N/m2
∴
For pipe 2,
v D 00051/10000007 D 0.001 m3 /kg 0004 2 ð 13,800 ð 0.001 Q1 D CD ð 0.00196 D 0.011CD 1 0004 00050.00196/0.0078900072 p p 20005P1 0004 P2 0007v D 2gh (equation 6.10)
A2 D 0.00196 m2 , A1 D 00050016/40007 ð 0.0752 D 0.0044 m2 Head loss, h D 150 mm Hg-under-water or 0005150/10000007 ð [000513,600 0004 10000007/1000] D 1.89 m water. 0004 2 ð 9.81 ð 1.89 ∴ Q2 D CD ð 0.00196 D 0.0133CD 1 0004 00050.00196/0.004400072 Total flow in pipe 3, Q3 D 0005Q1 C Q2 0007 D 00050.011CD C 0.0133CD 0007 D 0.0243CD For pipe 3, A2 D 00050016/40007 ð 0.0752 D 0.0044 m2 , A1 D 00050016/40007 ð 0.152 D 0.0176 m2 0004 p 2 ð 9.81 ð h and: Q3 D CD ð 0.0044 D 0.020CD h 2 00051 0004 00050.0044/0.01760007 0007 p ∴ 0.0243CD D 0.020CD h and: or
h D 1.476 m of water 00051.476 ð 10000007/0005000513600 0004 10000007/10000007 D 117 mm of Hg-under-water.
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PROBLEM 6.25 Water is flowing through a 150 mm diameter pipe and its flowrate is measured by means of a 50 mm diameter orifice, across which the pressure differential is 2.27 ð 104 N/m2 . The coefficient of discharge of the orifice meter is independently checked by means of a pitot tube which, when situated at the axis of the pipe, gave a reading of 100 mm on a mercury-under-water manometer. On the assumption that the flow in the pipe is turbulent and that the velocity distribution over the cross-section is given by the Prandtl one-seventh power law, calculate the coefficient of discharge of the orifice meter.
Solution For the pitot tube: uD
0001
2gh
(equation 6.10)
where h is the manometer reading in m of the same fluid which flows in the pipe. ∴
h D 0005100/10000007 ð 0005000513.6 0004 1.00007/1.00007 D 1.26 m of water p The velocity at the pipe axis, u D 00052 ð 9.81 ð 1.260007 D 4.97 m/s For turbulent flow, the Prandtl one-seventh power law can be used to give: uav D 0.82 ð uaxis ∴
(equation 3.60)
uav D 0.82 ð 4.97 D 4.08 m/s
For the orifice meter, the average velocity is: 0004 20005P1 0004 P2 0007v uD 1 0004 0005A2 /A1 00072 A2 D 00050016/40007 ð 0.052 D 0.00196 m2 , A1 D 00050016/40007 ð 0.152 D 0.0177 m2 , v D 0.001 m3 /kg ∴
uav D 6.78 m/s
The coefficient of discharge D 0005uav from pitot0007/0005uav from orifice meter0007. D 00054.08/6.780007 D 0.60
PROBLEM 6.26 Air at 323 K and 152 kN/m2 flows through a duct of circular cross-section, diameter 0.5 m. In order to measure the flowrate of air, the velocity profile across a diameter of the duct is measured using a pitot-static tube connected to a water manometer inclined at an angle of cos00041 0.1 to the vertical. The following results are obtained:
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Distance from duct centre-line (m)
Manometer Reading hm (mm)
0 0.05 0.10 0.15 0.175 0.20 0.225
104 100 96 86 79 68 50
101
Calculate the mass flowrate of air through the duct, the average velocity, the ratio of the average to the maximum velocity and the Reynolds number. Comment on these results. Discuss the application of this method of measuring gas flowrates, with particular emphasis on the best distribution of experimental points across the duct and on the accuracy of the results. Take the viscosity of air as 1.9 ð 1000042 mN s/m2 and the molecular weight of air as 29 kg/kmol.
Solution If hm is the manometer reading, the vertical manometer height will be 0.1hm (mm of water). For a pitot tube, the velocity at any point is: 0001 u D 2gh (equation 6.10) where h is the manometer reading in terms of the fluid flowing in the duct. Thus:
h D 00050.1hm /10000007 ð 00050011w /0011air 0007 0011air D 000529/22.400070005152/101.300070005273/3230007 D 1.64 kg/m3
∴
and:
h D 00050.1hm /1000000700051000/1.640007 D 0.061hm 0001 0001 u D 2 ð 9.81 ð 0.061hm D 1.09 hm (m/s)
If the duct is divided into a series of elements with the measured radius at the centre-line of the element, the velocity of the element can be found from the previous equation and the volumetric flowrate calculated. By adopting this procedure across the whole section, the required values may be determined. For example, at 0.05 m, where hm D 10 mm, Inner radius of element D 0.025 m Outer radius of element D 0.075 m Area of element ∴
D 001600050.0752 0004 0.0252 0007 D 0.0157 m2 0001 p u D 00051.09 hm 0007 D 1.09 100 D 10.9 m/s
Volumetric flowrate in the element D 000510.9 ð 0.01570007 D 0.171 m3 /s
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The following table is constructed in the same way. Distance Outer Inner from duct radius of radius of centre line element element (m) (m) (m) 0 0.05 0.10 0.15 0.175 0.20 0.225 0.25
0.025 0.075 0.125 0.1625 0.1875 0.2125 0.2375 0.25
Area of element 0005m00072
0 0.025 0.075 0.125 0.1625 0.1875 0.2125 0.2375
0.00196 0.0157 0.0314 0.0339 0.0275 0.0314 0.0353 0.0192 0011
hm Velocity (mm) u (m/s) 104 100 96 86 79 68 50 0
D 0.1964 m2
Volumetric flowrate Q m3 /s
11.1 10.9 10.7 10.1 9.7 8.9 7.7 0
0.0218 0.171 0.336 0.342 0.293 0.279 0.272 0 0011
D 1.715 m3 /s
Average velocity D 00051.715/0.19640007 D 8.73 m/s Mass flowrate D 00051.715 ð 1.640007 D 2.81 kg/s uav /umax D 00058.73/11.10007 D 0.79 Re D 00058.73 ð 1.64 ð 0.050007/00051.9 ð 1000045 0007 D 3.77 ð 104 The velocity distribution in turbulent flow is discussed in Section 3.3.6 where the Prandtl one-seventh power law is used to give: uav D 0.82umax
(equation 3.63)
This is close to that measured in this duct though strictly it only appears at very high values of Re. Reference to Fig. 3.14 shows that, at Re D 3.8 ð 104 , the velocity ratio is about 0.80 which shows remarkably good agreement.
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SECTION 7
Liquid Mixing PROBLEM 7.1 A reaction is to be carried out in an agitated vessel. Pilot plant experiments were performed under fully turbulent conditions in a tank 0.6 m in diameter, fitted with baffles and provided with a flat-bladed turbine. It was found that satisfactory mixing was obtained at a rotor speed of 4 Hz, when the power consumption was 0.15 kW and the Reynolds number was 160,000. What should be the rotor speed in order to retain the same mixing performance if the linear scale of the equipment is increased 6 times? What will be the power consumption and the Reynolds number?
Solution See Volume 1, Example 7.3.
PROBLEM 7.2 A three-bladed propeller is used to mix a fluid in the laminar region. The stirrer is 0.3 m in diameter and is rotated at 1.5 Hz. Due to corrosion, the propeller has to be replaced by a flat two-bladed paddle, 0.75 m in diameter. If the same motor is used, at what speed should the paddle rotate?
Solution For mixing in the laminar region, the power requirement is: P D k 0 N2 D3
(equation 7.17)
where k 0 D 1964 for a propeller and 1748 for a flat paddle. Thus, for a propeller 0.3 m in diameter rotating at 1.5 Hz: P D 00041964 ð 1.52 ð 0.32 0006 D 119.3 W and for a paddle, 0.75 m in diameter using the same motor: 119.3 D 00041748N2 ð 0.753 0006 and N D 0.403 Hz (24 rpm) 103
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PROBLEM 7.3 Compare the capital and operating costs of a three-bladed propeller with those of a constant speed six-bladed turbine, both constructed from mild steel. The impeller diameters are 0.3 and 0.45 m respectively and both stirrers are driven by a 1 kW motor. What is the recommended speed of rotation in each case? Assume operation for 8000 h/year, power costs of £0.01/kWh and interest and depreciation at 15%/year.
Solution The capital cost of an impeller, C D FM CB Pn where FM is a factor for the material of construction which for mild steel D 1.0, CB is a base cost 0004£0006, P is the power (kW) and n is an index (0004). For the propeller: CB D 960£000419900006 and n D 0.34 ∴
C D 00041.0 ð 960 ð 10.34 0006 D £960
Interest and depreciation D 0004960 ð 15/1000006 D £144/year Operating costs D 00041 ð 0.01 ð 80000006 D £80/year
0001
a total of £224/year
For the turbine: CB D 3160£000419900006 and n D 0.10 ∴
C D 00041.0 ð 3160 ð 10.10 0006 D £3160
Interest and depreciation D 00043160 ð 15/1000006 D £474/year Operating costs D 00041 ð 0.01 ð 80000006 D £80/year
0001
a total of £554/year.
In equation 7.13, k 0 D 165 for a propeller and 3245 for a turbine. For the propeller: P D 165N3 D5 or:
1000 D 165N3 0.35 and N D 13.5 Hz (810 rpm)
For the turbine: P D 3245N2 D5 or:
1000 D 00043245N3 0.455 0006 and N D 2.54 Hz (152 rpm)
PROBLEM 7.4 In a leaching operation, the rate at which solute goes into solution is given by: dM/dt D k0004cs 0004 c0006 kg/s where M kg is the amount of solute dissolving in t s, k 0004m3 /s0006 is a constant and cs and c are the saturation and bulk concentrations of the solute respectively in kg/m3 . In a pilot test on a vessel 1 m3 in volume, 75% saturation was attained in 10 s. If 300 kg of a solid
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containing 28% by mass of a water soluble solid is agitated with 100 m3 of water, how long will it take for all the solute to dissolve assuming conditions are the same as in the pilot unit? Water is saturated with the solute at a concentration of 2.5 kg/m3 .
Solution The mass of solute M, dissolving in t s is: dM/dt D k0004cs 0004 c0006 kg/s
(i)
For a batch of solution, V m3 in volume: dM D Vdc and substituting for dM in (i): Integrating:
V dc/dt D k0004cs 0004 c0006/V (ii)
ln00040004cs 0004 c0 0006/0004cs 0004 c00060006 D kt/V
where c0 is the concentration of the solute when t D 0. For pure water, c0 D 0 kg/m3 when t D 0 and hence equation (ii) becomes: c D cs 00041 0004 e0004kt/V 0006 kg/m3
(iii)
For the pilot test, the batch volume, V D 1 m3 , and cs D 2.5 kg/m3 at saturation. When t D 10 s, 75% saturation is achieved or: c D 00042.5 ð 75/1000006 D 1.875 kg/m3 Therefore, in equation (iii): 1.875 D 2.500041 0004 e000410k/1 0006 and k D 0.138 For the full-scale unit, the batch volume, V D 100 m3 . Mass of solute present D 0004300 ð 28/1000006 D 84 kg and c D 000484/1000006 D 0.84 kg/m3 . Therefore, in equation (iii): 0.84 D 2.500041 0004 e00040.138t/100 0006 and t D 297 s
PROBLEM 7.5 For producing an oil-water emulsion, two portable three-bladed propeller mixers are available; a 0.5 m diameter impeller rotating at 1 Hz and a 0.35 m impeller rotating at 2 Hz. Assuming turbulent conditions prevail, which unit will have the lower power consumption?
Solution Under turbulent conditions, the power requirements for mixing are given by: P D kN3 D5
(equation 7.13)
In this case: P1 D 0004k13 ð 0.55 0006 D 0.03125k and P2 D 0004k23 ð 0.355 0006 D 0.0420k ∴
P1 /P2 D 00040.03125k/0.0420k0006 D 0.743
Thus the 0.5 m diameter impeller will have the lower power consumption; some 75% of that of the 0.35 m diameter impeller.
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PROBLEM 7.6 A reaction is to be carried out in an agitated vessel. Pilot-plant experiments were performed under fully turbulent conditions in a tank 0.6 m in diameter, fitted with baffles and provided with a flat-bladed turbine. It was found that satisfactory mixing was obtained at a rotor speed of 4 Hz, when the power consumption was 0.15 kW and the Reynolds number 160,000. What should be the rotor speed in order to retain the same mixing performance if the linear scale of the equipment is increased 6 times? What will be the power consumption and the Reynolds number?
Solution See Volume 1, Example 7.3.
PROBLEM 7.7 Tests on a small scale tank 0.3 m diameter (Rushton impeller, diameter 0.1 m) have shown that a blending process between two miscible liquids (aqueous solutions, properties approximately the same as water, i.e. viscosity 1 mN s/m2 , density 1000 kg/m3 ) is satisfactorily completed after 1 minute using an impeller speed of 250 rev/min. It is decided to scale up the process to a tank of 2.5 m diameter using the criterion of constant tip-speed. (a) What speed should be chosen for the larger impeller? (b) What power will be required? (c) What will be the blend time in the large tank?
Solution a) In the small scale tank, the 0.1 m diameter impeller is rotated at 250 rev/min or: 0004250/600006 D 4.17 Hz. The tip speed is then: 0013DN D 00040013 ð 0.1 ð 4.170006 D 1.31 m/s If this is the same in the large scale tank, where D D 00042.5/30006 D 0.83 m, then: 1.31 D 00040013 ð 0.83 ð N0006 from which the speed of rotation to the larger impeller, N D 0.346 Hz or 20.8 rev/min b) In the large scale tank: and 0015 D 1 ð 1000043 Ns/m2 . Thus,
N D 0.346 Hz,
D D 0.83 m,
0014 D 1000 kg/m3
Re D D2 N0014/0015 D 00040.832 ð 0.346 ð 10000006/00041 ð 1000043 0006 D 238,360.
From Fig. 7.6, for a propeller mixer, the Power number, Np D 0.6. Thus:
0.6 D P/0014N3 D5
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and:
107
P D 0.60014N3 D5 D 00040.6 ð 1000 ð 0.3463 ð 0.835 0006 D 9.8 W
c) In the smaller tank: Re D D2 N0014/0015 D 00040.12 ð 4.17 ð 10000006/00041 ð 1000043 0006 D 41700 In Equation 7.22, the dimensionless mixing time is: 0018m D Ntm D kRe or for tm D 60s: 00044.17 ð 100006 D k ð 41700 and:
k D 0.0060
Thus in the larger tank: Ntm D 0.0060 Re or: and:
0.346 tm D 00040.0060 ð 238,3600006 tm D 4140 s or 1.15 min
PROBLEM 7.8 An agitated tank with a standard Rushton impeller is required to disperse gas in a solution of properties similar to those of water. The tank will be 3 m diameter (1 m diameter impeller). A power level of 0.8 kW/m3 is chosen. Assuming fully turbulent conditions and that the presence of the gas does not significantly affect the relation between the Power and Reynolds numbers: (a) What power will be required by the impeller? (b) At what speed should the impeller be driven? (c) If a small pilot scale tank 0.3 m diameter is to be constructed to test the process, at what speed should the impeller be driven?
Solution (a) Assuming that the depth of liquid D tank diameter, then: volume of liquid D 00040013D2 /40006 H D 00040013 ð 32 ð 30006/4 D 21.2 m3 With a power input of 0.8 kW/m3 , the power required be the impeller is: P D 00040.8 ð 21.20006 D 17.0 kW (b) For fully turbulent conditions and 0015 D 1 mN s/m2 , and the power number, from Fig. 7.6 is approximately 0.7. On this basis: P/0014N3 D5 D 0.7 or: from which:
000417.0 ð 103 0006/00041000N3 ð 15 0006 D 0.7 N D 2.90 Hz or 173 rev/ min
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(c) For the large tank, from Equation 7.13: P D kN3 D5 or:
000417.0 ð 103 0006 D k ð 2.903 ð 15
from which:
k D 697
Thus, for the smaller tank, assuming power/unit volume is constant: volume of fluid D 00040013/400060.32 ð 0.3 D 0.021 m3 and:
power supplied, P D 00040.021 ð 0.8 ð 103 0006 D 17 W
Thus, for the smaller tank: 17 D 697 N3 ð 0.15 and:
N D 13.5 Hz or 807 rev/ min .
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SECTION 8
Pumping of Fluids PROBLEM 8.1 A three-stage compressor is required to compress air from 140 kN/m2 and 283 K to 4000 kN/m2 . Calculate the ideal intermediate pressures, the work required per kilogram of gas, and the isothermal efficiency of the process. It may be assumed that the compression is adiabatic and interstage cooling is provided to cool the air to the initial temperature. Show qualitatively, by means of temperature–entropy diagrams, the effect of unequal work distribution and imperfect intercooling, on the performance of the compressor.
Solution It is shown in Section 8.3.4 that the work done is a minimum when the intermediate pressures Pi1 and Pi2 are related to the initial and final pressures P1 and P2 by: Pi1 /P1 D Pi2 /Pi1 D P2 /Pi2
(equation 8.45)
P1 D 140 kN/m2 and P2 D 4000 kN/m2 . ∴
P2 /P1 D 28.57
∴
Pi2 /Pi1 D P2 /Pi2 D
p 3
28.57 D 3.057,
2
Pi1 D 428 kN/m , and:
Pi2 D 1308 kN/m2
The specific volume of the air at the inlet is: v1 D 000622.4/2900070006283/27300070006101.3/1400007 D 0.579 m3 /kg
Hence, for 1 kg of air, the minimum work of compression in a compressor of n stages is: 0004 0001 0002 00030001 00020006 000310007/n
P2 W D nP1 v1 00031 (equation 8.46)
00031 P1 Thus: W D 00063 ð 140,000 ð 0.579000700061.4/0.40007[000628.5700070.4/3ð1.4 0003 1] D 319,170 J/kg The isothermal work of compression is: Wiso D P1 V1 ln0006P2 /P1 0007
(equation 8.36)
D 0006140,000 ð 0.579 ln 28.570007 D 271,740 J/kg The isothermal efficiency D 0006100 ð 271,7400007/319,170 D 85.1% 109
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Compression cycles are shown in Figs 8a and 8b. The former indicates the effect of various values of n in PVn D constant and it is seen that the work done is the area under the temperature–entropy curve. Figure 8b illustrates the three-stage compressor of this problem. The final temperature T2 , found from T2 /T1 D 0006P2 /P1 00070006 000310007/ , is 390 K. The dotted lines illustrate the effect of imperfect interstage cooling.
Adiabatic temperature
Rise Temperature, T
P2
n>γ P1
T2
n=γ n<γ n=1
T1
Line of constant pressure Work done where PV n = constant n<γ 0
Entropy, S
Figure 8a.
4000 kN/m2
1308 428
Indicates imperfect cooling Temperature (K)
140 390
Delivery for imperfect cooling
283 Inlet Delivery Work done stage 3
0
Work done stage 2
Work done stage 1
Entropy (kJ/kg K)
Figure 8b.
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PROBLEM 8.2 A twin-cylinder, single-acting compressor, working at 5 Hz, delivers air at 515 kN/m2 at the rate of 0.2 m3 /s. If the diameter of the cylinder is 20 cm, the cylinder clearance ratio 5%, and the temperature of the inlet air 283 K, calculate the length of stroke of the piston and the delivery temperature.
Solution For adiabatic conditions, PV D constant P2 /P1 D 0006T2 /T1 0007 /0006 000310007 or T2 D T1 0006P2 /P1 00070006 000310007/
and:
Thus the delivery temperature D 2830006515/101.300070.4/1.4 D 500 K The volume handled per cylinder D 00060.2/20007 D 0.1 m3 /s. Volume per stroke D 00060.1/50007 D 0.02 m3 /s at 515 kN/m2 . Volume at the inlet condition D 00060.02 ð 2830007/500 D 0.0126 m3 /s. From equation 8.42, 0.0126 D Vs [1 C c 0003 c0006P2 /P1 00071/ ] where c is the clearance and Vs the swept volume. Thus: ∴
0.0126 D Vs [1 ð 0.05 0003 0.050006515/101.300071/1.4 ] and Vs D 0.0142 m3 0006000f/4000700060.200072 ð stroke D 0.0142 and the stroke D 0.45 m
PROBLEM 8.3 A single-stage double-acting compressor running at 3 Hz is used to compress air from 110 kN/m2 and 282 K to 1150 kN/m2 . If the internal diameter of the cylinder is 20 cm, the length of stroke 25 cm, and the piston clearance 5%, calculate: (a) the maximum capacity of the machine, referred to air at the initial temperature and pressure, and (b) the theoretical power requirements under isentropic conditions.
Solution The volume per stroke D 00062 ð 0006000f/4000700060.200072 ð 0.250007 D 0.0157 m3 The compression ratio D 00061150/1100007 D 10.45. The swept volume Vs is given by: 0.0157 D Vs [1 C 0.05 0003 0.05000610.4500071/1.4 ] and Vs D 0.0217 m3
(equation 8.42)
The work of compression/cycle is: W D P1 0006V1 0003 V4 00070006 / 0003 10007[0006P2 /P1 00070006 000310007/ 0003 1]
(equation 8.41)
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and substituting for 0006V1 0003 V4 0007, gives: W D P1 Vs [1 C c 0003 c0006P2 /P1 00071/ ][ / 0003 10007][0006P2 /P1 00070006 000310007/ 0003 1] 0.286
D 0006110,000 ð 0.0157000700061.4/0.40007[000610.450007
(equation 8.43)
0003 1] D 5781 J
The theoretical power requirement D 00063 ð 57810007 D 17,340 W or 17.3 kW The capacity D 00063 ð 0.01570007 D 0.047 m3 /s
PROBLEM 8.4 Methane is to be compressed from atmospheric pressure to 30 MN/m2 in four stages. Calculate the ideal intermediate pressures and the work required per kilogram of gas. Assume compression to be isentropic and the gas to behave as an ideal gas. Indicate on a temperature–entropy diagram the effect of imperfect intercooling on the work done at each stage.
Solution The ideal intermediate pressures are obtained when the compression ratios in each stage are equal. If the initial, intermediate, and final pressures from this compressor are P1 , P2 , P3 , P4 , and P5 , then: P2 /P1 D P3 /P2 D P4 /P3 D P5 /P4 D P5 /P1 as in problem 8.1. P5 /P1 D 000630,000/101.30007 D 296.2 and: Hence:
0006P5 /P1 00070.25 D 4.148 P2 D 4.148P1 D 00064.148 ð 101.30007 D 420 kN/m2 P3 D 4.148P2 D 1.74 MN/m2 P4 D 4.148P3 D 7.23 MN/m2
The work required per kilogram of gas is: 00030001 0002 0004
P5 0006 000310007/n
00031 W D nP1 V1
00031 P1
(equation 8.46)
For methane, the molecular mass D 16 kg/kmol and the specific volume at STP D 000622.4/160007 D 1.40 m3 /kg. If D 1.4, the work per kilogram is: W D 00064 ð 101,300 ð 1.40000700061.4/0.40007[0006296.200070.4/00064ð1.40007 0003 1] D 710,940 J/kg or 711 kJ/kg The effect of imperfect cooling is shown in Figs 8a and 8b.
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PROBLEM 8.5 An air-lift raises 0.01 m3 /s of water from a well 100 m deep through a 100 mm diameter pipe. The level of the water is 40 m below the surface. The air consumed is 0.1 m3 /s of free air compressed to 800 kN/m2 . Calculate the efficiency of the pump and the mean velocity of the mixture in the pipe.
Solution See Volume 1, Example 8.6
PROBLEM 8.6 In a single-stage compressor: Suction pressure D 101.3 kN/m2 . Suction temperature D 283 K. Final pressure D 380 kN/m2 . If each new charge is heated 18 deg K by contact with the clearance gases, calculate the maximum temperature attained in the cylinder.
Solution The compression ratio D 0006380/101.30007 D 3.75. On the first stroke, the air enters at 283 K and is compressed adiabatically to 380 kN/m2 . Thus:
T2 /T1 D 0006P2 /P1 00070006 000310007/
D 3.750.286 D 1.459
Hence, the exit temperature is: T2 D 00061.459 ð 2830007 D 413 K The clearance volume gases which remain in the cylinder are able to raise the temperature of the next cylinder full of air by 18 deg K leaving the cylinder and its contents at 0006283 C 180007 D 301 K. After compression, the exit temperature is: T D 0006301 ð 3.750.286 0007 D 439.2 K On each subsequent stroke, the inlet temperature is always 301 K and hence the maximum temperature attained is 439.2 K.
PROBLEM 8.7 A single-acting reciprocating pump has a cylinder diameter of 115 mm and a stroke of 230 mm. The suction line is 6 m long and 50 mm diameter and the level of the water in the suction tank is 3 m below the cylinder of the pump. What is the maximum speed at which the pump can run without an air vessel if separation is not to occur in the suction line? The piston undergoes approximately simple harmonic motion. Atmospheric pressure is equivalent to a head of 10.4 m of water and separation occurs at pressure corresponding to a head of 1.22 m of water.
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Solution The tendency for separation to occur will be greatest at the inlet to the cylinder and at the beginning of the suction stroke. If the maximum speed of the pump is N Hz, the angular velocity of the driving mechanism is 2000fN radian/s. The acceleration of the piston D 00060.5 ð 0.2300070006000fN00072 cos 00062000fN0007 m/s2 . The maximum acceleration, when t D 0, is 4.54 N2 m/s2 . Maximum acceleration of the liquid in the suction pipe is: 00060.115/0.0500072 00064.54N2 0007 D 24.02 N2 m/s Accelerating force on the liquid D 000624.02N2 0006000f/2000700060.0500072 ð 6 ð 10000007. Pressure drop in suction line due to acceleration D 000624.02N2 ð 6 ð 10000007 D 1.44 ð 105 N2 N/m2 D 00061.44 ð 105 N2 /1000 ð 9.810007 D 14.69 N2 m of water Pressure head at the cylinder when separation is about to occur: 1.22 D 000610.4 0003 3.0 0003 14.69 N2 0007 m of water and : N D 0.65 Hz
PROBLEM 8.8 An air-lift pump is used for raising 0.8 l/s of a liquid of density 1200 kg/m3 to a height of 20 m. Air is available at 450 kN/m2 . If the efficiency of the pump is 30%, calculate the power requirement, assuming isentropic compression of the air 0006 D 1.40007.
Solution Volume flow of liquid D 800 cm3 /s or 800 ð 1000036 m3 /s Mass of flowrate of liquid D 0006800 ð 1000036 ð 1200 D 0.96 kg/s Work done per second D 00060.96 ð 20 ð 9.810007 D 188.4 W Actual work of expansion of air D 0006188.4/0.30007 D 627.8 W. The mass of air required per unit time is: W D Pa va m ln0006P/Pa 0007 D Pa Va ln0006P/Pa 0007
(equation 8.49)
where Va is the volume of air at STP, Thus:
627.8 D 101,300Va ln0006450/101.30007 and Va D 0.0042 m3
The work done in the isentropic compression of this air is: P1 V1 [ /0006 0003 10007][0006P2 /P1 00070006 000310007/ 0003 1] D 0006101,300 ð 0.0042000700061.4/0.40007[0006450/101.30007
(equation 8.37) 0.286
0003 1] D 792 J
Power required D 792 J/s D 792 W or 0.79 kW.
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PROBLEM 8.9 A single-acting air compressor supplies 0.1 m3 /s of air (at STP) compressed to 380 kN/m2 from 101.3 kN/m2 pressure. If the suction temperature is 288.5 K, the stroke is 250 mm, and the speed is 4 Hz, find the cylinder diameter. Assume the cylinder clearance is 4% and compression and re-expansion are isentropic 0006 D 1.40007. What is the theoretical power required for the compression?
Solution See Volume 1, Example 8.3.
PROBLEM 8.10 Air at 290 K is compressed from 101.3 to 2000 kN/m2 pressure in a two-stage compressor operating with a mechanical efficiency of 85%. The relation between pressure and volume during the compression stroke and expansion of the clearance gas is PV1.25 D constant. The compression ratio in each of the two cylinders is the same and the interstage cooler may be taken as perfectly efficient. If the clearances in the two cylinders are 4% and 5% respectively, calculate: (a) (b) (c) (d)
the the the the
work of compression per unit mass of gas compressed; isothermal efficiency; isentropic efficiency 0006 D 1.40007; ratio of the swept volumes in the two cylinders.
Solution See Volume 1, Example 8.4.
PROBLEM 8.11 Explain briefly the significance of the “specific speed” of a centrifugal or axial-flow pump. A pump is designed to be driven at 10 Hz and to operate at a maximum efficiency when delivering 0.4 m3 /s of water against a head of 20 m. Calculate the specific speed. What type of pump does this value suggest? A pump built for these operating conditions has a measured overall efficiency of 70%. The same pump is now required to deliver water at 30 m head. At what speed should the pump be driven if it is to operate at maximum efficiency? What will be the new rate of delivery and the power required?
Solution Specific speed is discussed in Section 8.2.3 of Volume 1, where it is shown to be Ns D NQ1/2 /0006gh00073/4 . This expression is dimensionless providing that the pump speed, throughput, and head are expressed in consistent units.
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In this problem, N D 10 Hz, Q D 0.4 m3 /s, and h D 20 m. Thus:
Ns D 000610 ð 00060.400070.5 /00069.81 ð 2000070.75 0007 D 0.121
Reference should be made to specialist texts on pumps where classifications of pump types as a function of specific speed are presented. A centrifugal pump is suggested here. Q / N and Q1 /Q2 D N1 /N2 and: Thus: and:
2
(equation 8.15) 2
h / N and h1 /h2 D 0006N1 /N2 0007
(equation 8.16)
000620/300007 D 000610/N2 00072 from which N2 D 12.24 Hz 0.4/Q2 D 000610/12.240007 from which Q2 D 0.49 m3 /s
Power required D 00061/n00070006mass flow ð head ð g0007 D 00061/0.7000700060.49 ð 1000 ð 30 ð 9.810007 D 206 W
PROBLEM 8.12 A centrifugal pump is to be used to extract water from a condenser in which the vacuum is 640 mm of mercury. At the rated discharge, the net positive suction head must be at least 3 m above the cavitation vapour pressure of 710 mm mercury vacuum. If losses in the suction pipe account for a head of 1.5 m, what must be the least height of the liquid level in the condenser above the pump inlet?
Solution The system is illustrated in Fig. 8c. From an energy balance, the head at the suction point of the pump is: hi D 0006P0 /0017h0007 C x 0003 0006ui2 /2g0007 0003 hf
Condenser pressure = P0 kN/m2 Liquid level
xm
Pump Velocity = ui
Figure 8c.
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The losses in the suction pipe D 1.5 m, and 0006ui 2 /2g0007 D hf D 1.5 The net positive suction head (NPSH) is discussed in Section 8.2.3 where it is shown that: NPSH D hi 0003 0006Pv /0017g0007 where Pv is the vapour pressure of the liquid being pumped. The minimum height x is then obtained from: 3 D 0006P0 /0017g0007 C x 0003 1.5 0003 0006Pv /0017g0007 P0 D 0006760 0003 6400007 D 120 mm Hg D 16,000 N/m2 Pv D 0006760 0003 7100007 D 50 mm Hg D 6670 N/m2 0017 D 1000 kg/m3 , g D 9.81 m/s2 ∴
x D 3 C 1.5 0003 000616,000 C 66700007/00061000 ð 9.810007 D 3.55 m
PROBLEM 8.13 What is meant by the Net Positive Suction Head (NPSH) required by a pump? Explain why it exists and how it can be made as low as possible. What happens if the necessary NPSH is not provided? A centrifugal pump is to be used to circulate liquid of density 800 kg/m3 and viscosity 0.5 mN s/m2 from the reboiler of a distillation column through a vaporiser at the rate of 400 cm3 /s, and to introduce the superheated liquid above the vapour space in the reboiler which contains liquid to a depth of 0.7 m. Suggest a suitable layout if a smooth-bore 25 mm pipe is to be used. The pressure of the vapour in the reboiler is 1 kN/m2 and the NPSH required by the pump is 2 m of liquid.
Reboiler P = l kN / m2 0.7 m
h0
Vaporiser
Figure 8d.
Solution See Volume 1, Example 8.2
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PROBLEM 8.14 1250 cm3 /s of water is to be pumped through a steel pipe, 25 mm diameter and 30 m long, to a tank 12 m higher than its reservoir. Calculate the approximate power required. What type of pump would you install for the purpose and what power motor (in kW) would you provide? Viscosity of water D 1.30 mN s/m2 . Density of water D 1000 kg/m3 .
Solution For a 25 mm bore pipe, cross-sectional area D 0006000f/4000700060.02500072 D 0.00049 m2 . Viscosity, u D 00061250 ð 1000036 0007/0.00049 D 2.54 m/s. Re D 0017ud/001e D 00061000 ð 2.54 ð 0.0250007/00061.3 ð 1000033 0007 D 48,900
and
From Table 3.1 the roughness e of a steel pipe will be taken as 0.045 mm. Hence e/d D 00060.046/250007 D 0.0018. When e/d D 0.0018 and Re D 4.89 ð 104 , from Fig. 3.7 R/0017u2 D 0.0032 The pressure drop is then calculated from the energy balance equation and equation 3.19. For turbulent flow of an incompressible fluid: u2 /2 C gz C v0006P2 0003 P1 0007 C 40006R/0017u2 00070006l/d0007u2 D 0 The pressure drop is: 0006P1 0003 P2 0007 D 0017[u2 /2 C gz C 40006R/0017u2 00070006l/d0007u2 ] D 0017f[0.5 C 40006R/0017u2 00070006l/d0007]u2 C gzg since the velocity in the tank is equal to zero. Substituting: 0006P1 0003 P2 0007 D 1000f[0.5 C 400060.0032000630/0.0250007]2.542 C 00069.81 ð 120007g D 219,500 N/m2 or 219.5 kN/m2 Power D G[0006u2 /20007 C gz C F] D 0006kg/s00070006m2 /s2 0007 D 0006m2 /s00070006N/m2 0007 00033
(equation 8.60)
5
D 00061.25 ð 10 000700062.195 ð 10 0007 D 275 W If a pump efficiency of 60% is assumed, the pump motor should be rated at 0006275/0.60007 D 458 W. A single stage centrifugal pump would be suitable for this duty.
PROBLEM 8.15 Calculate the pressure drop in, and the power required to operate, a condenser consisting of 400 tubes, 4.5 m long and 10 mm internal diameter. The coefficient of contraction at the entrance of the tubes is 0.6, and 0.04 m3 /s of water is to be pumped through the condenser.
Solution Flow of water through each tube D 00060.04/4000007 D 0.0001 m3 /s. Cross-sectional area of each tube D 0006000f/4000700060.0100072 D 0.0000785 m2 .
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Water velocity D 00060.0001/0.00007850007 D 1.273 m/s. Entry pressure drop is: 0001 00022 0017u2 1 Pf D 0003 00031 2 Cc
(from equation 3.78)
D 00061000 ð 1.2732 /20007[00061/0.60007 0003 1]2 D 360 N/m2 Re D 0017ud/001e D 00061000 ð 1.273 ð 0.010007/00061.0 ð 1000033 0007 D 1.273 ð 104 If e is taken as 0.046 mm from Table 3.1, then e/d D 0.0046 and from Fig. 3.7: R/0017u2 D 0.0043. The pressure drop due to friction is: Pf D 40006R/0017u2 00070006l/d000700060017u2 0007
(equation 3.18)
D 00064 ð 0.004300064.5/0.01000700061000 ð 1.2732 00070007 D 12,540 N/m2 Total pressure drop across one tube D 000612,540 C 3600007 D 12,900 N/m2 or 12.9 kN/m2 0001 0002 0001 0002 pressure volumetric If tubes are connected in parallel, power required D ð drop flowrate D 000612,900 ð 0.040007 D 516 W
PROBLEM 8.16 75% sulphuric acid, of density 1650 kg/m3 and viscosity 8.6 mN s/m2 , is to be pumped for 0.8 km along a 50 mm internal diameter pipe at the rate of 3.0 kg/s, and then raised vertically 15 m by the pump. If the pump is electrically driven and has an efficiency of 50%, what power will be required? What type of pump would you use and of what material would you construct the pump and pipe?
Solution Cross-sectional area of pipe D 0006000f/4000700060.0500072 D 0.00196 m2 . Velocity, u D 3.0/00061650 ð 0.001960007 D 0.93 m/s. Re D 0017ud/001e D 00061650 ð 0.93 ð 0.050007/8.6 ð 1000033 D 8900 If e is taken as 0.046 mm from Table 3.1, e/d D 0.00092. From Fig. 3.7, R/0017u2 D 0.0040. Head loss due to friction is: hf D 0003Pf /0017g D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
2
D 00064 ð 0.00400070006800/0.05000700060.93 /9.810007 D 22.6 m Total head D 000622.6 C 150007 D 37.6 m.
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Power D 0006mass flowrate ð head ð g0007
(equation 8.61)
D 00063.0 ð 37.6 ð 9.810007 D 1105 W If the pump is 50% efficient, power required D 00061105/0.50007 D 2210 W or 2.2 kW. For this duty a PTFE lined pump and lead piping would be suitable.
PROBLEM 8.17 60% sulphuric acid is to be pumped at the rate of 4000 cm3 /s through a lead pipe 25 mm diameter and raised to a height of 25 m. The pipe is 30 m long and includes two right-angled bends. Calculate the theoretical power required. The density of the acid is 1531 kg/m3 and its kinematic viscosity is 4.25 ð 1000035 m2 /s. The density of water may be taken as 1000 kg/m3 .
Solution Cross-sectional area of pipe D 0006000f/4000700060.02500072 D 0.00049 m2 Velocity, u D 00064000 ð 1000036 /0.000490007 D 8.15 m/s. Re D 0017ud/001e D ud/0006001e/00170007 D 00068.15 ð 0.0250007/00064.25 ð 1000035 0007 D 4794 If e is taken as 0.05 mm from Table 3.1, e/d D 0.002 and from Fig. 3.7, R/0017u2 D 0.0047. Head loss due to friction is given by: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.00470007000630/0.025000700068.152 /9.810007 D 152.8 m and z D 25.0 m From Table 3.2, 0.8 velocity heads 0006u2 /2g0007 are lost through each 90° bend so that the loss through two bends is 1.6 velocity heads or 00061.6 ð 8.152 0007/00062 ð 9.810007 D 5.4 m. Total head loss D 0006152.8 C 25 C 5.40007 D 183.2 m. Mass flowrate D 00064000 ð 1000036 ð 1.531 ð 10000007 D 6.12 kg/s. From equation 8.61 the theoretical power requirement D 00066.12 ð 183.2 ð 9.810007 D 11,000 W or 11.0 kW.
PROBLEM 8.18 1.3 kg/s of 98% sulphuric acid is to be pumped through a 25 mm diameter pipe, 30 m long, to a tank 12 m higher than its reservoir. Calculate the power required and indicate the type of pump and material of construction of the line that you would choose. Viscosity of acid D 0.025 N s/m2 . Density D 1840 kg/m3 .
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Solution Cross-sectional area of pipe D 0006000f/4000700060.002500072 D 0.00049 m2 . Volumetric flowrate D 1.3/00061.84 ð 10000007 D 0.00071 m3 /s. Velocity in the pipe, u D 00060.00071/0.000490007 D 1.45 m/s Re D 0017ud/001e D 000600061.84 ð 10000007 ð 1.45 ð 0.0250007/0.025 D 2670 This value of the Reynolds number lies within the critical zone. If the flow were laminar, the value of R/0017u2 from Fig. 3.7 would be 0.003. If the flow were turbulent, the value of R/0017u2 would be considerably higher, and this higher value should be used in subsequent calculation to provide a margin of safety. If the roughness is taken as 0.05 mm, e/d D 00060.05/250007 D 0.002 and, from Fig 3.7, R/0017u2 D 0.0057. The head loss due to friction, hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.0057000630/0.025000700061.452 /9.810007 D 5.87 m z D 12 m so that the total head D 17.87 m. The theoretical power requirement, from equation 8.61, is: power D 000617.87 ð 1.3 ð 9.810007 D 227 W If the pump is 50% efficient, actual power D 0006227/0.50007 D 454 W A PTFE lined centrifugal pump and lead or high silicon iron pipe would be suitable for this duty.
PROBLEM 8.19 A petroleum fraction is pumped 2 km from a distillation plant to storage tanks through a mild steel pipeline, 150 mm in diameter, at the rate of 0.04 m3 /s. What is the pressure drop along the pipe and the power supplied to the pumping unit if it has an efficiency of 50%? The pump impeller is eroded and the pressure at its delivery falls to one half. By how much is the flowrate reduced? Density of the liquid D 705 kg/m3 . Viscosity of the liquid D 0.5 mN s/m2 . Roughness of pipe surface D 0.004 mm.
Solution Cross-sectional area of pipe D 0006000f/400070.152 D 0.0177 m2 . Velocity in the pipe D 00060.04/0.01770007 D 2.26 m/s. Reynolds number D 00060.705 ð 1000 ð 2.26 ð 0.150007/00060.5 ð 1000033 0007 D 4.78 ð 105 e D 0.004 mm, e/d D 00060.004/1500007 D 0.000027 and from Fig. 3.7, R/0017u2 D 0.00165 The pressure drop is: 0003Pf D 40006R/0017u2 00070006l/d000700060017u2 0007
(equation 3.18)
0003Pf D 00064 ð 0.00165000700062000/0.1500070006705 ð 2.262 0007 D 316,900 N/m2 or 320 kN/m2
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If the pump efficiency is 50%, power D 0006head ð mass flowrate ð g0007/0.5 D pressure drop (N/m2 0007 ð volumetric flowrate (m3 s0007/0.5 D 0006316,900 ð 0.040007/0.5 D 25,350 W or 25.4 W If, due to impeller erosion, the delivery pressure is halved, the new flowrate may be found from: 0006R/0017u2 0007 Re2 D 0003Pf d3 0017/4l001e2 (equation 3.23) The new pressure drop D 0006316,900/20007 D 158,450 N/m2 and: 0006R/0017u2 0007 Re2 D 0006158,450 ð 0.153 ð 7050007/00064 ð 2000 ð 0.52 ð 1000036 0007 D 1.885 ð 108 From Fig. 3.8, when 0006R/0017u2 0007 Re2 D 1.9 ð 108 and e/d D 0.000027, Re D 3.0 ð 105 and:
00063.0 ð 105 0007 D 0006705 ð 0.15 ð u0007/00060.5 ð 1000033 0007 and: u D 1.418 m/s
The volumetric flowrate is now: 00061.418 ð 0.01770007 D 0.025 m3 /s
PROBLEM 8.20 Calculate the power required to pump oil of density 850 kg/m3 and viscosity 3 mN s/m2 at 4000 cm3 /s through a 50 mm pipeline 100 m long, the outlet of which is 15 m higher than the inlet. The efficiency of the pump is 50%. What effect does the nature of the surface of the pipe have on the resistance?
Solution Cross-sectional area of pipe D 0006000f/400070.052 D 0.00196 m2 Velocity of oil in the pipe D 00064000 ð 1000036 0007/0.00196 D 2.04 m/s. Re D 0017ud/001e D 00060.85 ð 1000 ð 2.04 ð 0.050007/00063 ð 1000033 0007 D 2.89 ð 104 If the pipe roughness e is taken to be 0.05 mm, e/d D 0.001, and from Fig. 3.7, R/0017u2 D 0.0031. Head loss due to friction is: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.003100070006100/0.05000700062.042 /9.810007 D 10.5 m The total head D 000610.5 C 150007 D 25.5 m The mass flowrate D 00064000 ð 1000036 ð 8500007 D 3.4 kg/s Power required D 000625.5 ð 3.4 ð 9.81/0.50007 D 1700 W or 1.7 kW The roughness of the pipe affects the ratio e/d. The rougher the pipe surface, the higher will be e/d and there will be an increase in R/0017u2 . This will increase the head loss due to friction and will ultimately increase the power required.
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PUMPING OF FLUIDS
123
PROBLEM 8.21 600 litres/s of water at 320 K is pumped in a 40 mm i.d. pipe through a length of 150 m in a horizontal direction and up through a vertical height of 10 m. In the pipe there is a control valve which may be taken as equivalent to 200 pipe diameters and other pipe fittings equivalent to 60 pipe diameters. Also in the line there is a heat exchanger across which there is a loss in head of 1.5 m of water. If the main pipe has a roughness of 0.0002 m, what power must be delivered to the pump if the unit is 60% efficient?
Solution Mass flowrate of water D 0006600 ð 1000036 ð 10000007 D 0.6 kg/s. Cross-sectional area of pipe D 0006000f/400070.042 D 0.00126 m2 . Velocity of water in the pipe D 0006600 ð 1000036 /0.001260007 D 0.476 m/s. Re D 0017ud/001e D 00061000 ð 0.476 ð 0.040007/00061 ð 1000033 0007 D 1.9 ð 104 . If e D 0.0002 m, e/d D 0.005, and from Fig. 3.7, R/0017u2 D 0.0042. The valve and fittings are equivalent to 260 pipe diameters which is equal to 0006260 ð 0.040007 D 10.4 m of pipe. The equivalent length of pipe is therefore 0006150 C 10.40007 D 160.4 m. The head loss due to friction is: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.004200070006160.4/0.04000700060.4762 /9.810007 D 1.56 m ∴
total head D 00061.56 C 1.5 C 100007 D 13.06 m.
and:
power required D 000613.06 ð 0.6 ð 9.810007/0.6 D 128 W
PROBLEM 8.22 A pump developing a pressure of 800 kN/m2 is used to pump water through a 150 mm pipe 300 m long to a reservoir 60 m higher. With the valves fully open, the flowrate obtained is 0.05 m3 /s. As a result of corrosion and scaling the effective absolute roughness of the pipe surface increases by a factor of 10. By what percentage is the flowrate reduced? Viscosity of water D 1 mN s/m2 .
Solution 800 kN/m2 is equivalent to a head of 80,000/00061000 ð 9.810007 D 81.55 m of water. If the pump is required to raise the water through a height of 60 m, then neglecting kinetic energy losses, the head loss due to friction in the pipe D 000681.55 0003 600007 D 21.55 m. The flowrate under these conditions is 0.05 m3 /s.
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The cross-sectional area of the pipe D 0006000f/400070.152 D 0.0177 m2 . Velocity of the water D 00060.05/0.01770007 D 2.82 m/s. Head loss due to friction is: hf D 80006R/0017u2 00070006l/d00070006u2 /2g0007 Thus:
21.55 D 80006R/0017u2 00070006300/0.15000700062.822 /00062 ð 9.8100070007
and:
R/0017u2 D 0.0033
(equation 3.20)
Re D 0017ud/001e D 00061000 ð 2.82 ð 0.150007/1000033 D 4.23 ð 105 From Fig. 3.7, e/d is 0.003. If, as a result of scaling and fouling, the roughness increases by a factor of 10, the new value of e/d D 0.03. Fig. 3.7 can no longer be used since the new velocity, and hence the Reynolds number, is unknown. Use is made of equation 3.23 and Fig. 3.8 to find the new velocity. The maximum head loss due to friction is still equal to 21.55 m as the pump head is unchanged. Thus:
21.55 m D 000621.55 ð 1000 ð 9.810007 D 211,410 N/m2 0006R/0017u2 0007 Re2 D 0003Pf d3 0017/4l001e2
(equation 3.23)
D 0006211,410 ð 0.153 ð 1000/4 ð 300 ð 1000036 0007 D 6.0 ð 108 From Fig. 3.8, Re D 2.95 ð 105 when e/d D 0.03. Hence the new velocity D 00062.95 ð 105 ð 1000033 0007/00061000 ð 0.150007 D 1.97 m/s Reduction in flow D 000610000062.82 0003 1.970007/2.820007 D 30.1 per cent.
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SECTION 9
Heat Transfer PROBLEM 9.1 Calculate the time taken for the distant face of a brick wall, of thermal diffusivity, DH D 0.0042 cm2 /s and thickness l D 0.45 m, initially at 290 K, to rise to 470 K if the near face is suddenly raised to a temperature of 0006 0 D 870 K and maintained at that temperature. Assume that all the heat flow is perpendicular to the faces of the wall and that the distant face is perfectly insulated.
Solution The temperature at any distance x from the near face at time t is given by: 0006D
ND1 0001
p p
00041000bN 0006 0 ferfc[ 2lN C x000b/ 2 DH t000b] C erfc[2 N C 1000bl 0004 x/ 2 DH t000b]g
ND0
(equation 9.37) and the temperature at the distant face is: 0006D
ND1 0001
p
00041000bN 0006 0 f2 erfc[ 2N C 1000bl]/ 2 DH t000bg
ND0
Choosing the temperature scale such that the initial temperature is everywhere zero, 0006/20006 0 D 470 0004 290000b/2 870 0004 290 D 0.155 p DH D 0.0042 cm2 /s or 4.2 ð 1000047 m2 /s, DH D 6.481 ð 104 Thus: 0.155 D
ND1 0001
and l D 0.45 m
00041 erfc 347 2N C 1000b/t0.5
ND0
D erfc 347t00040.5 0004 erfc 1042t00040.5 C erfc 1736t00040.5 Considering the first term only, 347t00040.5 D 1.0 and t D 1.204 ð 105 s The second and higher terms are negligible compared with the first term at this value of t and hence: t D 0.120 Ms (33.5 h)
125
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 9.2 Calculate the time for the distant face to reach 470 K under the same conditions as Problem 9.1, except that the distant face is not perfectly lagged but a very large thickness of material of the same thermal properties as the brickwork is stacked against it.
Solution This problem involves the conduction of heat in an infinite medium where it is required to determine the time at which a point 0.45 m from the heated face reaches 470 K. The boundary conditions are therefore: 0006 D 0,
t D 0;
0006 D 00060 , t > 0
0006 D 870 0004 290 D 580 deg K, 0006 D 0,
x D 1,
for all values of x x D 0, t > 0
t>0
0006 D 0,
x D 0, t D 0 0002 2 0003 ∂ 0006 ∂2 0006 ∂2 0006 ∂0006 D DH C C ∂t ∂x 2 ∂y 2 ∂z2 D DH
∂2 0006 ∂x 2
The Laplace transform of:
(for unidirectional heat transfer) 0004 1 0006e0004pt dt 0006 D 0006N D
(equation 9.29) (i)
0
d2 0006N p N 0006N tD0 00060004 D 2 dx DH DH p p Integrating equation (ii): 0006N D B1 ex p/DH C B2 e0004x p/DH C 0006tD0 /p p p p p d0006N and: D B1 p/DH ex p/DH 0004 B2 p/DH e0004x p/DH dx 0004 1 N 0006 0 e0004pt dt D 0006 0 /p In this case, 0006 t>0 D
and hence:
0005
and:
xD0
∂0006 ∂t
0006
0
0004
1
D t>0 xD0
0
0002
∂0006 ∂t
(ii) (iii) (iv)
0003
e0004pt dt D 0
Substituting the boundary conditions in equations (iii) and (iv): 0006N t>0 D 0006 0t>0 /p D B1 C B2 C 0006tD0 /p or xD0
and: ∴
0005
B1 C B2 D 0006 0t>0 /p
xD0
∂0006N ∂t
0006
xD0
p p D 0 D B1 p/DH e1 0004 B2 p/DH e00041 t>0 xD0
p B1 p/DH D 0 and B1 D 0,
B2 D 0006 0t>0 /p xD0
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HEAT TRANSFER
p
p p D 0006 0 p00041 e0004k p where k D x/ DH p p The Laplace transform of p00041 e0004k p D erfc k/2 t (from Volume 1, Appendix). 0007 x 0 and: 0006 D 0006 t>0 erfc p 2 DH t xD0
0006N D B2 e0004x
From (iii),
p/DH
(v)
When x D 0.45 m, 0006 D 470 0004 290 D 180 deg K, and hence in (v), with DH D 4.2 ð 1000047 m2 /s, p
180/580 D erfcf[0.45/ 6.481 ð 1000044 ][1/ 2 t000b]g D 0.31 p ∴
0.45/6.481 ð 1000044 /2 t D 0.73 t D 2.26 ð 105 s or
and:
0.226 Ms 62.8 h000b
As an alternative method of solution, Schmidt’s method is used with the construction shown in Fig. 9a. In this case x D 0.1 m and it is seen that at x D 0.45 m, the temperature is 470 K after a time 20 t. In equation 9.43: t D 0.1000b2 / 2 ð 4.2 ð 1000047 D 1.191 ð 104 s and hence the required time, t D 20 ð 1.191 ð 104 D 2.38 ð 105 s D 0.238 Ms 66.1 h The difference here is due to inaccuracies resulting from the coarse increments of x. 900
870
800 9 7
Temperature (K)
16
700 19 17
14 12 10
5
8
3
6
15
600
13 11 20 18 16
500
4
1
9 7
14
470
19
2
12
17 10 20 18
400
16 19
14
17
12
5
15 13
8
11 6
3
9
15 13
300
290
1.5
10
1.4
1.3
1.2
1.1
9
10
11
1.0
0.9
0.8
0.7
11 9 7
0.6
10
7
4
8 5 6
0.5
0.4
0.3
0.2
0.1
0
Distance from hot face (m)
Figure 9a.
PROBLEM 9.3 Benzene vapour, at atmospheric pressure, condenses on a plane surface 2 m long and 1 m wide maintained at 300 K and inclined at an angle of 45° to the horizontal. Plot the thickness of the condensate film and the point heat transfer coefficient against distance from the top of the surface.
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Solution At 101.3 kN/m2 , benzene condenses at Ts D 353 K. With a wall temperature of Tw D 300 K, the film properties at a mean temperature of 327 K are: 0018 D 4.3 ð 1000044 N s/m2 , 0019 D 860 kg/m3 , k D 0.151 W/m K and 001a D 423 kJ/kg D 4.23 ð 105 J/kg Thus: s D f[40018k Ts 0004 Tw x]/ g sin 001c001a00192 g0.25 00044
D f[4 ð 4.3 ð 10
(equation 9.168)
ð 0.151 353 0004 300000bx]/ 9.81 sin 45° ð 4.23 ð 105 ð 8602 g0.25
D 2.82 ð 1000044 x 0.25 m Similarly: h D f 00192 g sin 001c001ak 3 /[40018 Ts 0004 Tw x]g0.25
(equation 9.169)
D f 8602 ð 9.81 sin 45° ð 4.23 ð 105 ð 0.1513 /[4 ð 4.3 ð 1000044 353 0004 300000bx]g0.25 D 535x 00040.25 W/m2 K Values of x between 0 and 2.0 m in increments of 0.20 m are now substituted in these equations with the following results, which are plotted in Fig. 9b.
Thickness of film (s mm)
750 700
0.25
650
0.20
600
0.15
550
0.10
500
0.05
Heat transfer coefficient (h W/m2 K)
800
450 0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Distance from top of surface (x m) Figure 9b.
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HEAT TRANSFER
x (m)
x 0.25
x 00040.25
s (m)
h
W/m2 K000b
0 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0 0.562 0.669 0.795 0.880 0.946 1.000 1.047 1.088 1.125 1.158 1.189
1 1.778 1.495 1.258 1.136 1.057 1.000 0.956 0.919 0.889 0.863 0.841
0 1.58 ð 1000044 1.89 ð 1000044 2.24 ð 1000044 2.48 ð 1000044 2.67 ð 1000044 2.82 ð 1000044 2.95 ð 1000044 3.07 ð 1000044 3.17 ð 1000044 3.27 ð 1000044 3.35 ð 1000044
1 951 800 673 608 566 535 512 492 476 462 450
PROBLEM 9.4 It is desired to warm 0.9 kg/s of air from 283 to 366 K by passing it through the pipes of a bank consisting of 20 rows with 20 pipes in each row. The arrangement is in-line with centre to centre spacing, in both directions, equal to twice the pipe diameter. Flue gas, entering at 700 K and leaving at 366 K, with a free flow mass velocity of 10 kg/m2 s, is passed across the outside of the pipes. Neglecting gas radiation, how long should the pipes be? For simplicity, outer and inner pipe diameters may be taken as 12 mm. Values of k and 0018, which may be used for both air and flue gases, are given below. The specific heat capacity of air and flue gases is 1.0 kJ/kg K. Temperature (K)
Thermal conductivity k(W/m K)
Viscosity 0018(mN s/m2 )
250 500 800
0.022 0.044 0.055
0.0165 0.0276 0.0367
Solution Heat load, Q D 0.9 ð 1.0 366 0004 283 D 74.7 kW Temperature driving force, 00061 D 700 0004 366 D 334 deg K, 00062 D 366 0004 283 D 83 deg K and in equation 9.9, 0006m D 334 0004 83000b/ ln 334/83 D 180 deg K
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Film coefficients
Inside: hi di /k D 0.023 dG0 /0018000b0.8 Cp 0018/k000b0.4 00042
di D 12 mm or 1.2 ð 10
(equation 9.61)
m.
The mean air temperature D 0.5 366 C 283 D 325 K and k D 0.029 W/m K. Cross-sectional area of one tube D $/4 1.2 ð 1000042 2 D 1.131 ð 1000044 m2 Area for flow D 20 ð 20000b1.131 ð 1000044 D 4.52 ð 1000042 m2 . Thus, mass velocity G D 0.9/ 4.52 ð 1000042 D 19.9 kg/m2 s. At 325 K, 0018 D 0.0198 mN s/m2 or 1.98 ð 1000045 N s/m2 Cp D 1.0 ð 103 J/kg K Thus: hi ð 1.2 ð 1000042 / 2.9 ð 1000042 D 0.023 1.2 ð 1000042 ð 19.9/1.98 ð 1000045 0.8 ð 1.0 ð 103 ð 1.98 ð 1000045 /0.029000b0.4 0.4138hi D 0.023 1.206 ð 104 0.8 0.683000b0.4
and hi D 87.85 W/m2 K
Outside: 0.6
Cp 0018/k000b0.3 ho do /k D 0.33Ch do G0 /0018000bmax
do D 12.0 mm
or
(equation 9.90)
1.2 ð 1000042 m
G0 D 10 kg/m2 s for free flow G0max D YG0 / Y 0004 do where Y, the distance between tube centres D 2do D 2.4 ð 1000042 m. ∴
G0max D 2.4 ð 1000042 ð 10.0000b/ 2.4 ð 1000042 0004 1.2 ð 1000042 D 20 kg/m2 s At a mean flue gas temperature of 0.5 700 C 366 D 533 K,
0018 D 0.0286 mN s/m2 or 2.86 ð 1000045 N s/m2 , k D 0.045 W/m K and Cp D 1.0 ð 103 J/kg K ∴
Remax D 1.2 ð 1000042 ð 20.0000b/ 2.86 ð 1000045 D 8.39 ð 103
From Table 9.3, when Remax D 8.39 ð 103 , X D 2do , and Y D 2do , Ch D 0.95. Thus: ho ð 1.2 ð 1000042 / 4.5 ð 1000042 D 0.33 ð 0.95 8.39 ð 103 0.6 ð 1.0 ð 103 ð 2.86 ð 1000045 /0.045000b0.3 or:
0.267ho D 0.314 8.39 ð 103 0.6 0.836000b0.3
and ho D 232 W/m2 K
Overall: Ignoring wall and scale resistances, then: 1/U D 1/ho C 1/hi D 0.0114 C 0.0043 D 0.0157 and:
U D 63.7 W/m2 K
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Area required
In equation 9.1, A D Q/U0006m D 74.7 ð 103 / 63.7 ð 180 D 6.52 m2 . Area/unit length of tube D $/4 12 ð 1000042 D 9.43 ð 1000043 m2 /m and hence: total length of tubing required D 6.52/ 9.43 ð 1000043 D 6.92 ð 102 m. The length of each tube is therefore D 6.92 ð 102 / 20 ð 20 D 1.73 m
PROBLEM 9.5 A cooling coil, consisting of a single length of tubing through which water is circulated, is provided in a reaction vessel, the contents of which are kept uniformly at 360 K by means of a stirrer. The inlet and outlet temperatures of the cooling water are 280 K and 320 K respectively. What would be the outlet water temperature if the length of the cooling coil were increased by 5 times? Assume the overall heat transfer coefficient to be constant over the length of the tube and independent of the water temperature.
Solution (equation 9.1)
Q D UATm
where Tm is the logarithmic mean temperature difference. For the initial conditions: Q1 D m1 ð 4.18 320 0004 280 D U1 A1 [ 360 0004 280 0004 360 0004 320000b]/ [ln 360 0004 280000b/ 360 0004 320000b] or: and:
167.2m1 D U1 A1 80 0004 40000b/ ln 80/40 D 57.7U1 A1
m1 /U1 A1 D 0.345
In the second case, m2 D m1 , U2 D U1 , and A2 D 5A1 . ∴
Q2 D m1 ð 4.18 T 0004 280 D 5U1 A1 [ 360 0004 280 0004 360 0004 T000b]/ ln 360 0004 280000b/ 360 0004 T000b
or:
4.18 m1 /U1 A1 T 0004 280000b/5 D 80 0004 360 C T000b/[ln[80/360 0004 T000b]
Substituting for m1 /U1 A1 , 0.289 T 0004 280 D T 0004 280000b/[ln 80/ 360 0004 T000b] or:
ln[80/ 360 0004 T000b] D 3.467 and
T D 357.5 K
PROBLEM 9.6 In an oil cooler, 216 kg/h of hot oil enters a thin metal pipe of diameter 25 mm. An equal mass of cooling water flows through the annular space between the pipe and a
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larger concentric pipe; the oil and water moving in opposite directions. The oil enters at 420 K and is to be cooled to 320 K. If the water enters at 290 K, what length of pipe will be required? Take coefficients of 1.6 kW/m2 K on the oil side and 3.6 kW/m2 K on the water side and 2.0 kJ/kg K for the specific heat of the oil.
Solution Heat load
Mass flow of oil D 6.0 ð 1000042 kg/s. and hence, Q D 6.0 ð 1000042 ð 2.0 420 0004 320 D 12 kW Thus the water outlet temperature is given by: 12 D 6.0 ð 1000042 ð 4.18 T 0004 290 or T D 338 K Logarithmic mean temperature driving force
In equation 9.9: 00061 D 420 0004 338 D 82 deg K, and:
00062 D 320 0004 290 D 30 deg K
0006m D 82 0004 30000b/ ln 82/30 D 51.7 deg K
Overal coefficient
The pipe wall is thin and hence its thermal resistance may be neglected. Thus in equation 9.8: 1/U D 1/ho C 1/hi D 1/1.6 C 1/3.6000b
and U D 1.108 kW/m2 K
Area
In equation 9.1, A D Q/U0006m D 12/ 1.108 ð 51.7 D 0.210 m2 Tube diameter D 25 ð 1000043 m (assuming a mean value) area/unit length D $ ð 25 ð 1000043 ð 1.0 D 7.85 ð 1000042 m2 /m and the tube length required D 0.210/ 7.85 ð 1000042 D 2.67 m
PROBLEM 9.7 The walls of a furnace are built of a 150 mm thickness of a refractory of thermal conductivity 1.5 W/m K. The surface temperatures of the inner and outer faces of the refractory are 1400 K and 540 K respectively. If a layer of insulating material 25 mm thick of
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HEAT TRANSFER
thermal conductivity 0.3 W/m K is added, what temperatures will its surfaces attain assuming the inner surface of the furnace to remain at 1400 K? The coefficient of heat transfer from the outer surface of the insulation to the surroundings, which are at 290 K, may be taken as 4.2, 5.0, 6.1, and 7.1 W/m2 K for surface temperatures of 370, 420, 470, and 520 K respectively. What will be the reduction in heat loss?
Solution Heat flow through the refractory,
Q D kA T1 0004 T2 /x
(equation 9.12)
Thus for unit area,
Q D 1.5 ð 1.0 1400 0004 T2 / 150 ð 1000043 D 14,000 0004 10T2 W/m2
(i)
where T2 is the temperature at the refractory–insulation interface. Similarly, the heat flow through the insulation is: Q D 0.3 ð 1.0 T2 0004 T3 / 25 ð 1000043 D 12T2 0004 12T3 W/m2
(ii)
The flow of heat from the insulation surface at T3 K to the surroundings at 290 K, is: Q D hA T3 0004 290 or T3 0004 290 hW/m2
(iii)
where h is the coefficient of heat transfer from the outer surface. The solution is now made by trial and error. A value of T3 is selected and h obtained by interpolation of the given data. This is substituted in equation (iii) to give Q. T2 is then obtained from equation (ii) and a second value of Q is then obtained from equation (i). The correct value of T3 is then given when these two values of Q coincide. The working is as follows and the results are plotted in Fig. 9c. Q = 14000 − 10 T2
Q (W/m 2)
10000 8000 6000 4000
4050 W/m 2
Q = h (T3 − 290)
2000
662 K 0
300
350
400
450 500 T3 (K)
550
600
650
700
750
Figure 9c.
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T3 (K)
h (W/m2 K)
Q D h T3 0004 290 (W/m2 )
T2 D T3 C Q/12 (K)
Q D 14,000 0004 10T2 (W/m2 )
300 350 400 450 500 550 600 650 700 750
3.2 3.9 4.7 5.6 6.5 7.8 9.1 10.4 11.5 12.7
32 234 517 896 1355 2028 2821 3744 4715 5842
302.7 369.5 443.1 524.7 612.9 719.0 835.1 962.0 1092.9 1236.8
10,973 10,305 9569 8753 7871 6810 5649 4380 3071 1632
A balance is obtained when T3 D 662 K, at which Q D 4050 W/m2 . In equation (i):
4050 D 14,000 0004 10T2 or T2 D 995 K
Thus the temperatures at the inner and outer surfaces of the insulation are 995 K and 662 K respectively With no insulation, Q D 1.5 ð 1.0 1400 0004 540000b/ 150 ð 1000043 D 8600 W/m2 and hence the reduction in heat loss is 8600 0004 4050 D 4550 W/m2 or:
4540 ð 100000b/8600 D 52.9%
PROBLEM 9.8 A pipe of outer diameter 50 mm, maintained at 1100 K, is covered with 50 mm of insulation of thermal conductivity 0.17 W/m K. Would it be feasible to use a magnesia insulation, which will not stand temperatures above 615 K and has a thermal conductivity of 0.09 W/m K, for an additional layer thick enough to reduce the outer surface temperature to 370 K in surroundings at 280 K? Take the surface coefficient of heat transfer by radiation and convection as 10 W/m2 K.
Solution For convection to the surroundings
Q D hA3 T3 0004 T4 W/m where A3 is area for heat transfer per unit length of pipe, m2 /m000b. The radius of the pipe, r1 D 50/2 D 25 mm or 0.025 m. The radius of the insulation, r2 D 25 C 50 D 75 mm or 0.075 m.
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The radius of the magnesia, r3 D 75 C x D 0.075 C 0.001x m where x mm is the thickness of the magnesia. Hence the area at the surface of the magnesia, A3 D 2$ 0.075 C 0.001x m2 /m and Q D 10[2$ 0.075 C 0.001x000b] 370 C 280 D 424.1 C 5.66x W/m
(i)
For conduction through the insulation
Q D k 2$rm l T1 0004 T2 / r2 0004 r1
(equation 9.22)
where rm D r2 0004 r1 / ln r2 /r1 . ∴
Q D 0.17[2$ ð 1.0 r2 0004 r1 ] 1100 0004 T2 /[ r2 0004 r1 ln 0.075/0.025000b] D 0.972 1100 0004 T2 W/m
(ii)
For conduction through the magnesia
In equation 9.22: Q D 0.09[2$ ð 1.0 r3 0004 r2 ] T2 0004 370000b/[ r3 0004 r2 ln 0.075 C 0.001x000b/0.075] D 0.566 T2 0004 370000b/ ln 1 C 0.013x000b
(iii)
For a value of x, Q is found from (i) and hence T2 from (ii). These values are substituted in (iii) to give a second value of Q, with the following results: x (mm)
Q D 424.1 C 5.66x (W/m)
T2 D 1100 0004 1.028Q (K)
Q D 0.566 T2 0004 370000b/ ln 1 C 0.013x (W/m)
5.0 7.5 10.0 12.5 15.0 17.5 20.0
452.4 466.6 480.7 494.9 509.0 523.2 537.3
635 620 606 591 577 562 548
2380 1523 1092 832 657 531 435
From a plot of the two values of Q, a balance is attained when x D 17.5 mm. With this thickness, T2 D 560 K which is below the maximum permitted and hence the use of the magnesia would be feasible.
PROBLEM 9.9 In order to heat 0.5 kg/s of a heavy oil from 311 K to 327 K, it is passed through tubes of inside diameter 19 mm and length 1.5 m forming a bank, on the outside of which steam is condensing at 373 K. How many tubes will be needed?
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In calculating Nu, Pr, and Re, the thermal conductivity of the oil may be taken as 0.14 W/m K and the specific heat as 2.1 kJ/kg K, irrespective of temperature. The viscosity is to be taken at the mean oil temperature. Viscosity of the oil at 319 and 373 K is 154 and 19.2 mN s/m2 respectively.
Solution Heat load
Q D 0.5 ð 2.1 327 0004 311 D 16.8 kW Logarithmic mean driving force
00061 D 373 0004 311 D 62 deg K, 00062 D 373 0004 327 D 46 deg K ∴ in equation 9.9,
0006m D 62 0004 46000b/ ln 62/46 D 53.6 deg K
A preliminary estimate of the overall heat transfer coefficient may now be obtained from Table 9.18. For condensing steam, ho D 10,000 W/m2 K and for oil, hi D 250 W/m2 K (say). Thus 1/U D 1/ho C 1/hi D 0.0041, U D 244 W/m2 K and from equation 9.1, the preliminary area: A D 16.8 ð 103 / 244 ð 53.6 D 1.29 m2 The area/unit length of tube is $ ð 19.0 ð 1000043 ð 1.0 D 5.97 ð 1000042 m2 /m and:
total length of tubing D 1.29/ 5.97 ð 1000042 D 21.5 m
Thus:
number of tubes D 21.5/1.5 D 14.3, say 14 tubes
Film coefficients
The inside coefficient is controlling and hence this must be checked to ascertain if the preliminary estimate is valid. The Reynolds number, Re D di G0 /0018 D 19.0 ð 1000043 G0 /0018 At a mean oil temperature of 0.5 327 C 311 D 319 K, 0018 D 154 ð 1000043 N s/m2 . Area for flow per tube D $/4 19.0 ð 1000043 2 D 2.835 ð 1000044 m2 . ∴
total area for flow D 14 ð 2.835 ð 1000044 D 3.969 ð 1000043 m2
and hence: Thus:
G0 D 0.5/ 3.969 ð 1000043 D 1.260 ð 102 kg/m2 s Re D 19.0 ð 1000043 ð 1.260 ð 102 / 154 ð 1000043 D 15.5
That is, the flow is streamline and hence:
hi di /k 0018s /0018000b0.14 D 2.01 GCp /kl000b0.33
(equation 9.85)
At a mean wall temperature of 0.5 373 C 319 D 346 K, 0018s D 87.0 ð 1000043 N s/m2 .
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The mass flow, G D 0.5 kg/s. ∴
hi ð 19.0 ð 1000043 /0.14 87.0 ð 1000043 /154 ð 1000043 0.14 D 2.01
0.5 ð 2.1 ð 103 / 0.14 ð 1.5 0.33
or:
0.13hi ð 0.923 D 2.01 ð 16.6 and hi D 266 W/m2 K
This is sufficiently close to the assumed value and hence 14 tubes would be specified.
PROBLEM 9.10 A metal pipe of 12 mm outer diameter is maintained at 420 K. Calculate the rate of heat loss per metre run in surroundings uniformly at 290 K, (a) when the pipe is covered with 12 mm thickness of a material of thermal conductivity 0.35 W/mK and surface emissivity 0.95, and (b) when the thickness of the covering material is reduced to 6 mm, but the outer surface is treated so as to reduce its emissivity to 0.10. The coefficients of radiation from a perfectly black surface in surroundings at 290 K are 6.25, 8.18, and 10.68 W/m2 K at 310 K, 370 K, and 420 K respectively. The coefficients of convection may be taken as 1.22 0006/d000b0.25 W/m2 K, where 0006(K) is the temperature difference between the surface and the surrounding air and d(m) is the outer diameter.
Solution Case (a)
Assuming that the heat loss is q W/m and the surface temperature is T K, for conduction through the insulation, from equation 9.12, q D kAm 420 0004 T000b/x The mean diameter is 18 mm or 0.018 m, and hence: Am D $ ð 0.018 D 0.0566 m2 /m x D 0.012 m ∴
q D 0.35 ð 0.0566 420 0004 T000b/0.012 D 693.3 0004 1.67T000bW/m
(i)
For convection and radiation from the surface, from equation 9.119: q D hr C hc A2 T 0004 290 W/m where hr is the film coefficient equivalent to the radiation and hc the coefficient due to convection given by: hc D 1.22[ T 0004 290000b/d]0.25 where d D 36 mm or 0.036 m ∴
hc D 2.80 T 0004 290000b0.25 W/m2 K
If hb is the coefficient equivalent to radiation from a black body, hr D 0.95hb W/m2 K The outer diameter is 0.036 m and hence: A2 D $ ð 0.036 ð 1.0 D 0.1131 m2 /m ∴
q D [0.95hb C 2.80 T 0004 290000b0.25 ]0.1131 T 0004 290 D 0.1074hb T 0004 290 C 0.317 T 0004 290000b1.25 W/m
(ii)
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Values of T are now assumed and together with values of hb from the given data substituted into (i) and (ii) until equal values of q are obtained as follows: T q D 693.3 0004 1.67T hb 0.1074hb T 0004 290 0.317 T 0004 290000b1.25 q 2 (K) (W/m)
W/m K (W/m) (W/m) (W/m) 300 320 340 360 380 400
193.3 160.0 126.7 93.3 60.0 26.7
6.0 6.5 7.1 7.8 8.55 9.55
6.5 20.9 38.1 58.7 82.7 112.8
5.7 22.2 42.1 64.2 87.9 113.0
12.2 43.1 80.2 122.9 170.6 225.8
A balance is obtained when T D 350 K and q D 106 W/m.
Case (b)
For conduction through the insulation, x D 0.006 m and the mean diameter is 15 mm or 0.015 m. ∴
Am D $ ð 0.015 ð 1.0 D 0.0471 m2 /m
∴
q D 0.35 ð 0.0471 420 0004 T000b/0.006 D 1154 0004 2.75T W/m
(i)
The outer diameter is now 0.024 m and A2 D $ ð 0.024 ð 1.0 D 0.0754 m2 /m The coefficient due to convection is: hc D 1.22[ T 0004 290000b/0.024]0.25 D 3.10 T 0004 290000b0.25 W/m2 K The emissivity is 0.10 and hence hr D 0.10hb W/m2 K ∴
q D [0.10hb C 3.10 T 0004 290000b0.25 ]0.0754 T 0004 290 D 0.00754hb T 0004 290 C 0.234 T 0004 290000b1.25 W/m
(ii)
Making the calculation as before: T q D 1154 0004 2.75T hb2 0.0075hb T 0004 290 0.234 T 0004 290000b1.25 q 2 (W/m) (W/m) (W/m) (K) (W/m)
W/m K 300 320 340 360 380 400
329.0 274.0 219.0 164.0 109.0 54.0
6.0 6.5 7.1 7.8 8.55 9.55
0.5 1.5 2.7 4.2 5.8 7.9
4.2 16.4 31.1 47.4 64.9 83.4
4.7 17.9 33.8 51.6 70.7 91.3
A balance is obtained when T D 390 K and q D 81 W/m.
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PROBLEM 9.11 A condenser consists of 30 rows of parallel pipes of outer diameter 230 mm and thickness 1.3 mm with 40 pipes, each 2 m long in each row. Water, at an inlet temperature of 283 K, flows through the pipes at 1 m/s and steam at 372 K condenses on the outside of the pipes. There is a layer of scale 0.25 mm thick of thermal conductivity 2.1 W/m K on the inside of the pipes. Taking the coefficients of heat transfer on the water side as 4.0 and on the steam side as 8.5 kW/m2 K, calculate the water outlet temperature and the total mass flow of steam condensed. The latent heat of steam at 372 K is 2250 kJ/kg. The density of water is 1000 kg/m3 .
Solution Overall coefficient
1 1 1 xw xr D C C C U hi ho kw kr
(equation 9.201)
where xr and kr are the thickness and thermal conductivity of the scale respectively. Considering these in turn, hi D 4000 W/m2 K. The inside diameter, di D 230 0004 2 ð 1.3 D 227.4 mm or 0.2274m. Therefore basing the coefficient on the outside diameter: hio D 4000 ð 0.2274/0.230 D 3955W/m3 K For conduction through the wall, xw D 1.3 mm, and from Table 9.1, kw D 45 W/m K for steel and kw /xw D 45/0.0013 D 34615 W/m2 K The mean wall diameter D 0.230 C 0.2274000b/2 D 0.2287 m and hence the coefficient equivalent to the wall resistance based on the tube o.d. is: 34615 ð 0.2287/0.230 D 34419 W/m2 /K For conduction through the scale, xr D 0.25 ð 1000043 m, kr D 2.1 W/m K and hence: kr /xr D 2.1/0.25 ð 1000043 D 8400 W/m2 K The mean scale diameter D 227.4 0004 0.25 D 227.15 mm or 0.2272 m and hence the coefficient equivalent to the scale resistance based on the tube o.d. is:
8400 ð 0.2272/0.230 D 8298 W/m2 K ∴
and:
1/U D 1/3955 C 1/8500 C 1/34419 C 1/8298 D 5.201 ð 1000044 U D 1923 W/m2 K
Temperature driving force
If water leaves the unit at T K: 00061 D 372 0004 283 D 89 deg K, 00062 D 372 0004 T000b
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and in equation 9.9: 0006m D [89 0004 372 0004 T000b]/ ln[89/ 372 0004 T000b] D T 0004 283000b/ ln[89/ 372 0004 T000b] Area
For 230 mm o.d. tubes, outside area per unit length D $ ð 0.230 ð 1.0 D 0.723 m2 /m. Total length of tubes D 30 ð 40 ð 2 D 2400 m and hence heat transfer area, A D
2400 ð 0.723 D 1735.2 m2 .
Heat load
The cross-sectional area for flow/tube D $/4 0.230000b2 D 0.0416 m2 /tube. Assuming a single-pass arrangement, there are 1200 tubes per pass and hence area for flow D 1200 ð 0.0416 D 49.86 m2 . For a velocity of 1.0 m/s, the volumetric flow D 0.1 ð 49.86 m3 /s and the mass flow D 1000 ð 4.986 D 4986 kg/s. Thus the heat load, Q D 4986 ð 4.18 T 0004 283 D 20,840 T 0004 283 kW or 2.084 ð 107 T 0004 283 W. Substituting for Q, U, A, and 0006m in equation 9.1:
2.084 ð 107 T 0004 283 D 1923 ð 1735.2 T 0004 283000b/ ln[89/ 372 0004 T000b] or:
ln[89/ 372 0004 T000b] D 0.1601 and T D 296 K
The total heat load is, therefore, Q D 20,840 296 0004 283 D 2.71 ð 105 kW and the mass of steam condensed D 2.71 ð 105 /2250 D 120.4 kg/s.
PROBLEM 9.12 In an oil cooler, water flows at the rate of 360 kg/h per tube through metal tubes of outer diameter 19 mm and thickness 1.3 mm, along the outside of which oil flows in the opposite direction at the rate of 6.675 kg/s per tube. If the tubes are 2 m long and the inlettemperatures of the oil and water are 370 K and 280 K respectively, what will be the outlet oil temperature? The coefficient of heat transfer on the oil side is 1.7 kw/m2 K and on the water side 2.5 kW/m2 K and the specific heat of the oil is 1.9 kJ/kg K.
Solution In the absence of information as to the geometry of the unit, the solution will be worked on the basis of one tube — a valid approach as the number of tubes effectively appears on both sides of equation 9.1. If Tw and To are the outlet temperature of the water and the oil respectively, then:
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Heat load
Q D
360/3600 ð 4.18 Tw 0004 280 D 0.418 Tw 0004 280 kW for water and:
Q D
75/1000 ð 1.9 370 0004 To D 0.143 370 0004 To kW for the oil.
From these two equations, Tw D 406.5 0004 0.342To K Area
For 19.0 mm o.d. tubes, surface area D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and for one tube, surface area D 2.0 ð 0.0597 D 0.1194 m2 Temperature driving force
00061 D 370 0004 Tw ,
00062 D To 0004 280000b
and in equation 9.9: 0006m D [ 370 0004 Tw 0004 To 0004 280000b]/[ln 370 0004 Tw / To 0004 280000b] D 650 0004 Tw 0004 To / ln 370 0004 Tw / To 0004 280 Substituting for Tw : 0006m D 243.5 0004 0.658To / ln 0.342To 0004 36.5000b/ To 0004 280 K Overall coefficient
hi D 2.5 kW/m2 K di D 19.0 0004 2 ð 1.3 D 16.4 mm Therefore the inside coefficient, based on the outside diameter is: hio D 2.5 ð 16.4/19.0 D 2.16 kW/m2 K Neglecting the scale and wall resistances then: 1/U D 1/2.16 C 1/1.7 D 1.052 m2 K/kW and:
U D 0.951 kW/m2 K
Substituting in equation 9.1 gives: 0.143 370 0004 To D 0.951 ð 0.1194 243.5 0004 0.658To / ln 0.342To 0004 36.5000b/ To 0004 280 ∴
ln 0.342To 0004 36.5000b/ To 0004 280 D 0.523 and To D 324 K
PROBLEM 9.13 Waste gases flowing across the outside of a bank of pipes are being used to warm air which flows through the pipes. The bank consists of 12 rows of pipes with 20 pipes, each
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0.7 m long, per row. They are arranged in-line, with centre-to-centre spacing equal, in both directions, to one-and-a-half times the pipe diameter. Both inner and outer diameter may be taken as 12 mm. Air with a mass velocity of 8 kg/m2 s enters the pipes at 290 K. The initial gas temperature is 480 K and the total mass flow of the gases crossing the pipes is the same as the total mass flow of the air through them. Neglecting gas radiation, estimate the outlet temperature of the air. The physical constants for the waste gases, assumed the same as for air, are: Temperature (K)
Thermal conductivity (W/m K)
Viscosity (mN s/m2 )
250 310 370 420 480
0.022 0.027 0.030 0.033 0.037
0.0165 0.0189 0.0214 0.0239 0.0260
Specific heat D 1.00 kJ/kg K.
Solution Heat load
The cross area for flow per pipe D $/4 0.012000b2 D 0.000113 m2 and therefore for 12 ð 20 D 240 pipes, the total flow area D 240 ð 0.000113 D 0.027 m2 . Thus:
flow of air D 8.0 ð 0.271 D 0.217 kg/s
which is also equal to the flow of waste gas. If the outlet temperatures of the air and waste gas are Ta and Tw K respectively, then: Q D 0.217 ð 1.0 Ta 0004 290 kW or 217 Ta 0004 290 W and: from which:
Q D 0.217 ð 1.0 480 0004 Tw kW Tw D 770 0004 Ta K
Area
Surface area/unit length of pipe D $ ð 0.012 ð 1.0 D 0.0377 m2 /m. Total length of pipe D 240 ð 0.7 D 168 m and hence the heat transfer area, A D 168 ð 0.0377 D 6.34 m2 . Temperature driving force
00061 D 480 0004 Ta 00062 D Tw 0004 1290000b
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or, substituting for Tw :
00062 D 480 0004 Ta D 00061
Thus in equation 9.9:
0006m D 480 0004 Ta
Overall coefficient
The solution is now one of trial and error in that mean temperatures of both streams must be assumed in order to evaluate the physical properties. Inside the tubes: a mean temperature of 320 K, will be assumed at which, k D 0.028 W/m K, 0018 D 0.0193 ð 1000043 N s/m2 , and Cp D 1.0 ð 103 J/kg K Therefore: hi di /k D 0.023 di G/0018000b0.8 Cp 0018/k000b0.4
(equation 9.61)
hi ð 0.012/0.028 D 0.023 0.012 ð 8.0/0.0193 ð 1000043 0.8 ð 1 ð 103 ð 0.0193 ð 1000043 /0.028000b0.4 ∴
hi D 0.0537 4.974 ð 103 0.8 0.689000b0.4 D 41.94 W/m2 K
Outside the tubes: The cross-sectional area of the tube bundle D 0.7 ð 20 1.5 ð 0.012 D 0.252 m2 and hence the free flow mass velocity, G0 D 0.217/0.252 D 0.861 kg/m2 s. From Fig. 9.27: Y D 1.5 ð 0.012 D 0.018 m and therefore:
G0max D 0.861 ð 0.018000b/ 0.081 0004 0.012 D 2.583 kg/m2 s
At an assumed mean temperature of 450 K, 0018 D 0.0250 ð 1000043 N s/m2 and k D 0.035 W/m K. ∴
Remax D 0.012 ð 2.583000b/ 0.0250 ð 1000043 D 1.24 ð 103 From Table 9.3: for X D 1.5do and Y D 1.5do , Ch D 0.95. In equation 9.90:
ho 0.012/0.035 D 0.33 ð 0.95 1.24 ð 103 0.6 1.0 ð 103 ð 0.0250 ð 1000043 /0.035000b0.3 ∴
ho D 0.914 ð 71.8 ð 0.7140.3 D 59.3 W/m2 K Hence, ignoring wall and scale resistances: 1/U D 1/41.91 C 1/59.3 D 4.07 ð 1000042 U D 24.57 W/m2 K
and: Thus, in equation 9.1:
217 Ta 0004 290 D 24.57 ð 6.34 480 0004 Ta from which:
Ta D 369.4 K
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With this value, the mean air and waste gas temperatures are 330 K and 440 K respectively. These are within 10 deg K of the assumed values in each case. Such a difference would have a negligible effect on the film properties and recalculation is unnecessary.
PROBLEM 9.14 Oil is to be heated from 300 K to 344 K by passing it at 1 m/s through the pipes of a shelland-tube heat exchanger. Steam at 377 K condenses on the outside of the pipes, which have outer and inner diameters of 48 and 41 mm respectively, though due to fouling, the inside diameter has been reduced to 38 mm, and the resistance to heat transfer of the pipe wall and dirt together, based on this diameter, is 0.0009 m2 K/W. It is known from previous measurements under similar conditions that the oil side coefficients of heat transfer for a velocity of 1 m/s, based on a diameter of 38 mm, vary with the temperature of the oil as follows: Oil temperature (K) Oil side coefficient of heat transfer (W/m2 K)
300 74
311 80
322 97
333 136
344 244
The specific heat and density of the oil may be assumed constant at 1.9 kJ/kg K and 900 kg/m3 respectively and any resistance to heat transfer on the steam side may be neglected. Find the length of tube bundle required?
Solution In the absence of further data, this problem will be worked on the basis of one tube. Heat load
Cross-sectional area at the inside diameter of the scale D $/4 0.038000b2 D 0.00113 m2 . ∴
volumetric flow D 0.00113 ð 1.0 D 0.00113 m3 /s
and: ∴
mass flow D 0.00113 ð 900 D 1.021 kg/s heat load, Q D 1.021 ð 1.9 344 0004 300 D 85.33 kW
Temperature driving force
00061 D 377 0004 300 D 77 deg K, 00062 D 377 0004 344 D 33 deg K and, in equation 9.9:
0006m D 77 0004 33000b/ ln 77/33 D 52 deg K
Overall coefficient
Inside: The mean oil temperature D 0.5 344 C 300 D 322 K at which hi based on di D 0.038 m D 97 W/m2 K.
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Basing this value on the outside diameter of the pipe: hio D 97 ð 0.038/0.048 D 76.8 W/m2 K Outside: From Table 9.17, ho for condensing steam will be taken as 10,000 W/m2 K. Wall and scale: The scale resistance based on d D 0.038 m is 0.0009 m2 K/W or: k/x D 1/0.0009 D 1111.1 W/m2 K. Basing this on the tube o.d., k/x D 1111.1 ð 0.038/0.048 D 879.6 W/m2 K. 1/U D 1/hio C 1/ho C x/k
(equation 9.201)
D 0.0130 C 0.0001 C 0.00114 or:
U D 70.2 W/m2 K
Area
A D Q/U0006m D 85.33 ð 103 / 70.2 ð 52 D 23.4 m2 Area per unit length of pipe D $ ð 0.048 ð 1.0 D 0.151 m2 /m and length of tube bundle D 23.4/0.151 D 154.9 m A very large tube length is required because of the very low inside film coefficient and several passes or indeed a multistage unit would be specified. A better approach would be to increase the tube side velocity by decreasing the number of tubes in each pass, though any pressure drop limitations would have to be taken into account. The use of a smaller tube diameter might also be considered.
PROBLEM 9.15 It is proposed to construct a heat exchanger to condense 7.5 kg/s of n-hexane at a pressure of 150 kN/m2 , involving a heat load of 4.5 MW. The hexane is to reach the condenser from the top of a fractionating column at its condensing temperature of 356 K. From experience it is anticipated that the overall heat transfer coefficient will be 450 W/m2 K. Cooling water is available at 289 K. Outline the proposals that you would make for the type and size of the exchanger, and explain the details of the mechanical construction that you consider require special attention.
Solution A shell-and-tube unit is suitable with hexane on the shell side. For a heat load of 4.5 MW D 4.5 ð 103 kW, the outlet water temperature is: 4.5 ð 103 D m ð 4.18 T 0004 289000b
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In order to avoid severe scaling in such a case, the maximum allowable water temperature is 320 K and hence 310 K will be chosen as a suitable value for T. Thus:
4.5 ð 103 D 4.18m 310 0004 289 and m D 51.3 kg/s
The next stage is to estimate the heat transfer area required. (equation 9.1)
Q D UA0006m where the heat load Q D 4.5 ð 106 W U D 450 W/m2 K 00061 D 356 0004 289 D 67 deg K,
00062 D 356 0004 310 D 46 deg K
and from equation 9.9, 0006m D 67 0004 46000b/ ln 67/46 D 55.8 deg K No correction factor is necessary as the shell side fluid temperature is constant. ∴
A D 4.5 ð 106 / 450 ð 55.8 D 179.2 m2 A reasonable tube size must now be selected, say 25.4 mm, 14 BWG.
The outside surface area is therefore $ ð 0.0254 ð 1.0 D 0.0798 m2 /m and hence the total length of tubing required D 179.2/0.0798 D 2246 m. A standard tube length is now selected, say 4.87 m and hence the total number of tubes required D 2246/4.87 D 460. It now remains to decide the number of tubes per pass, and this is obtained from a consideration of the water velocity. For shell and tube units, u D 1.0 0004 1.5 m/s and a value of 1.25 m/s will be selected. The water flow, 51.3 kg/s D 51.3/1000 D 0.0513 m3 /s. The tube i.d. is 21.2 mm and hence the cross-sectional area for flow/tube D
$/4 0.0212000b2 D 0.000353 m2 . Area required to give a velocity of 1.25 m/s D 0.0513/1.25 D 0.0410 m2 and hence number of tubes/pass D 0.0410/0.000353 D 116 and number of passes D 460/116 ³ 4. As the shell side fluid is clean, triangular pitch might be suitable and 460 ð 25 mm o.d. tubes on 32 mm triangular pitch with 4 tube side passes can be accommodated in a 0.838 m i.d. shell and still allow room for impingement plates. The proposed unit will therefore consist of: 460, 25.4 mm o.d. tubes ð 14 BWG, 4.87 m long arranged in 4 tube side passes on 32 mm triangular pitch in a 0.838 m i.d. shell. The general mechanical details of the unit are described in Section 9.9.1 of Volume 1 and points of detail are: (i) impingement baffles should be fitted under each inlet nozzle; (ii) segmental baffles are not usually fitted to a condenser of this type; (iii) the unit should be installed on saddles at say 5° to the horizontal to facilitate drainage of the condensate.
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PROBLEM 9.16 A heat exchanger is to be mounted at the top of a fractionating column about 15 m high to condense 4 kg/s of n-pentane at 205 kN/m2 , corresponding to a condensing temperature of 333 K. Give an outline of the calculations you would make to obtain an approximate idea of the size and construction of the exchanger required. For purposes of standardisation, 19 mm outside diameter tubes of 1.65 mm wall thickness will be used and these may be 2.5, 3.6, or 5 m in length. The film coefficient for condensing pentane on the outside of a horizontal tube bundle may be taken as 1.1 kW/m2 K. The condensation is effected by pumping water through the tubes, the initial water temperature being 288 K. The latent heat of condensation of pentane is 335 kJ/kg. For these 19 mm tubes, a water velocity of 1 m/s corresponds to a flowrate of 0.2 kg/s of water.
Solution The calculations follow the sequence of earlier problems in that heat load, temperature driving force, and overall coefficient are obtained and hence the area evaluated. It then remains to consider the geometry of the unit bearing in mind the need to maintain a reasonable cooling water velocity. As in the previous example, the n-pentane will be passed through the shell and cooling water through the tubes. Heat load
Q D 4.0 ð 335 D 1340 kW assuming there is no sub-cooling of the condensate. As in Problem 9.15, the outlet temperature of the cooling water will be taken as 310 K, and for a flow of G kg/s: 1340 D G ð 4.18 310 0004 288 or G D 14.57 kg/s Temperature driving force
00061 D 333 0004 288 D 45 deg K, and:
00062 D 333 0004 310 D 23 deg K
0006m D 45 0004 23000b/ ln 45/23 D 32.8 deg K
Overall coefficient
Inside: For forced convection to water in tubes: hi D 4280 0.00488T 0004 1000bu0.8 /di0.2 W/m2 K
(equation 9.221)
where T, the mean water temperature D 0.5 310 C 288 D 299 K; u, the water velocity will be taken as 1 m/s — a realistic optimum value, bearing in mind the need to limit the
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pressure drop, and di D 19.0 0004 2 ð 1.65 D 15.7 mm or 0.0157 m. ∴
hi D 4280 0.00488 ð 299 0004 1000b1.00.8 /0.01570.2 D 4280 ð 0.459 ð 1.0000b/0.436 D 4506 W/m2 K
or, based on the outer diameter, hio D 4.506 ð 0.0157000b/0.019 D 3.72 kW/m2 K Wall: For steel, k D 45 W/m K and x D 0.00163 m and hence: x/k D 0.00163/45 D 0.0000362 m2 K/W or 0.0362 m2 K/kW Outside: ho D 1.1 kW/m2 K Ignoring scale resistance: 1/U D 1/ho C x/k C 1/hio D 0.9091 C 0.0362 C 0.2686 and:
U D 0.823 kW/m2 K
Area
Q D UA0006m and hence:
A D 1340/ 0.823 ð 32.8 D 49.6 m2
Outer area of 0.019 m diameter tube D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and hence total length of tubing required D 49.6/0.0597 D 830.8 m. Thus with 2.5, 3.6, and 5.0 m tubes, the number of tubes will be 332, 231 or 166. The total cooling water flow D 14.57 kg/s and for u D 1 m/s, the flow through 1 tube is 0.20 kg/s ∴
the number of tubes/pass D 14.57/0.20 D 73
Clearly 3 passes are usually to be avoided, and hence 2 or 4 are suitable, that is 146 or 292 tubes, 5.0 or 2.5 m long. The former is closer to a standard shell size and 166 ð 19 mm tubes on 25.4 mm square pitch with two tube side passes can be fitted within a 438 mm i.d. shell. In this event, the water velocity would be slightly less than 1 m/s in fact 1 ð 146/166 D 0.88 m/s, though this would not affect the overall coefficient to any significant extent. The proposed unit is therefore 166 ð 19 mm o.d. tubes on 25.4 mm square pitch 5.0 m long with a 438 mm i.d. shell. In making such calculations it is good practice to add an overload factor to the heat load, say 10%, to allow for errors in predicting film coefficients, although this is often taken into account in allowing for extra tubes within the shell. In this particular example, the fact that the unit is to be installed 15 m above ground level is of significance in limiting the pressure drop and it may be that in an actual situation space limitations would immediately specify the tube length.
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PROBLEM 9.17 An organic liquid is boiling at 340 K on the inside of a metal surface of thermal conductivity 42 W/m K and thickness 3 mm. The outside of the surface is heated by condensing steam. Assuming that the heat transfer coefficient from steam to the outer metal surface is constant at 11 kW/m2 K, irrespective of the steam temperature, find the value of the steam temperature would give a maximum rate of evaporation. The coefficients of heat transfer from the inner metal surface to the boiling liquid which depend upon the temperature difference are: Temperature difference between metal Heat transfer coefficient metal surface surface and boiling liquid (deg K) to boiling liquid kW/m2 K 22.2 27.8 33.3 36.1 38.9 41.7 44.4 50.0
4.43 5.91 7.38 7.30 6.81 6.36 5.73 4.54
Solution For a steam temperature Ts K, the heat conducted through the film of condensing steam, Q D hc A Ts 0004 T1 , or: Q D 11 ð 1.0 Ts 0004 T1 D 11.0 Ts 0004 T1 kW/m2
(i)
where T1 is the temperature at the outer surface of the metal. For conduction through the metal, Q D kA T1 0004 T2 /x D 42 ð 1000043 ð 1.0 T1 0004 T2 /0.003 D 14.0 T1 0004 T2 kW/m2
(ii)
where T2 is the temperature at the inner surface of the metal. For conduction through the boiling film: Q D hb T2 0004 340 D hb T2 0004 340 kW/m2
(iii)
where hb kW/m2 K is the film coefficient to the boiling liquid. Thus for an assumed value of T2 the temperature difference T2 0004 340 is obtained and hb from the table of data. Q is then obtained from (iii), T1 from (ii), and hence Ts from (i) as follows:
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T2 (K)
T2 0004 340 (K)
hb
kW/m2 K000b
Q
kW/m2
T1 (K)
Ts (K)
362.2 367.8 373.3 376.1 378.9 381.7 384.4 390.0
22.2 27.8 33.3 36.1 38.9 41.7 44.4 50.0
4.43 5.91 7.38 7.30 6.81 6.36 5.73 4.54
98.4 164.3 245.8 263.5 264.9 265.2 254.4 227.0
369.2 379.5 390.9 394.9 397.8 400.7 402.6 406.2
378.1 394.4 413.3 418.9 421.9 424.8 425.7 426.8
It is fairly obvious that the rate of evaporation will be highest when the heat flux is a maximum. On inspection this occurs when Ts D 425 K.
PROBLEM 9.18 It is desired to warm an oil of specific heat 2.0 kJ/kg K from 300 to 325 K by passing it through a tubular heat exchanger containing metal tubes of inner diameter 10 mm. Along the outside of the tubes flows water, inlet temperature 372 K, and outlet temperature 361 K. The overall heat transfer coefficient from water to oil, based on the inside area of the tubes, may be assumed constant at 230 W/m2 K, and 0.075 kg/s of oil is to be passed through each tube. The oil is to make two passes through the heater and the water makes one pass along the outside of the tubes. Calculate the length of the tubes required.
Solution Heat load
If the total number of tubes is n, there are n/2 tubes in one pass on the oil side, that is the oil passes through 2 tubes in traversing the exchanger. The mass flow of oil is therefore D 0.075 ð n/2 D 0.0375n kg/s and the heat load: Q D 0.0375n ð 2.0 325 0004 300 D 1.875n kW Temperature driving force
00061 D 361 0004 300 D 61 deg K, 00062 D 372 0004 325 D 47 deg K and, in equation 9.9: 0006m D 61 0004 47000b/ ln 61/47 D 53.7 deg K In equation 9.213: X D 00062 0004 00061 / T1 0004 00061 and Y D T1 0004 T2 / 00062 0004 00061 where T1 and T2 are the inlet and outlet temperatures on the shell side and 00061 and 00062 are the inlet and outlet temperatures on the tube side.
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∴
X D 325 0004 300000b/ 372 0004 300 D 0.347
and:
Y D 372 0004 361000b/ 325 0004 300 D 0.44
151
For one shell side pass, two tube side passes, Fig. 9.71 applies and F D 0.98. Area
In equation 9.212, A D Q/UF0006m D 1.875n/ 0.230 ð 0.98 ð 53.7 D 0.155n m2 . The area per unit length based on 10 mm i.d. D $ ð 0.010 ð 1.0 D 0.0314 m2 /m and total length of tubing D 0.155n/0.0314 D 4.94n m. Thus the length of tubes required D 4.94n/n D 4.94 m.
PROBLEM 9.19 A condenser consists of a number of metal pipes of outer diameter 25 mm and thickness 2.5 mm. Water, flowing at 0.6 m/s, enters the pipes at 290 K, and it should be discharged at a temperature not exceeding 310 K. If 1.25 kg/s of a hydrocarbon vapour is to be condensed at 345 K on the outside of the pipes, how long should each pipe be and how many pipes would be needed? Take the coefficient of heat transfer on the water side as 2.5, and on the vapour side as 0.8 kW/m2 K and assume that the overall coefficient of heat transfer from vapour to water, based upon these figures, is reduced 20% by the effects of the pipe walls, dirt and scale. The latent heat of the hydrocarbon vapour at 345 K is 315 kJ/kg.
Solution Heat load
For condensing the organic at 345 K, Q D 1.25 ð 315 D 393.8 kW If the water outlet temperature is limited to 310 K, then the mass flow of water is given by: 393.8 D G ð 4.18 310 0004 290 or G D 4.71 kg/s Temperature driving force
00061 D 345 0004 290 D 55 deg K, 00062 D 345 0004 310 D 35 deg K Therefore in equation 9.9, 0006m D 55 0004 35000b/ ln 55/35 D 44.3 deg K. No correction factor is necessary with isothermal conditions in the shell. Overall coefficient
Inside: hi D 2.5 kW/m2 K. The outside diameter D 0.025 m and di D 25 0004 2 ð 2.5000b/103 D 0.020 m. Basing the inside coefficient on the outer diameter: hio D 2.5 ð 0.020/0.025 D 2.0 kW/m3 K
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Outside: ho D 0.8 kW/m2 K and hence the clean overall coefficient is given by: 1/Uc D 1/hio C 1/ho D 1.75 m2 K/kW or Uc D 0.572 kW/m2 K Thus allowing for scale and the wall: UD D 0.572 100 0004 20000b/100 D 0.457 kW/m2 K Area
In equation 9.1:
A D Q/U0006m D 393.8/ 0.457 ð 44.3 D 19.45 m2
Outside area D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and hence total length of piping D 19.45/0.0785 D 247.6 m. 4.71 kg/s water 4.71/1000 D 0.00471 m3 /s and hence cross-sectional area/pass to give a velocity of 0.6 m/s D 0.00471/0.6 D 0.00785 m2 Cross-sectional area of one tube D $/ 0.020000b2 D 0.000314 m2 . Therefore number of tubes/pass D 0.00785/0.000314 D 25. Thus: with 1 tube pass, total tubes D 25 and tube length D 247.6/25 D 9.90 m with 2 tube passes, total tubes D 50 and tube length D 247.6/50 D 4.95 m with 4 tube passes, total tubes D 100 and tube length D 247.6/100 D 2.48 m A tube length of 2.48 m is perhaps the most practical proposition.
PROBLEM 9.20 An organic vapour is being condensed at 350 K on the outside of a bundle of pipes through which water flows at 0.6 m/s; its inlet temperature being 290 K. The outer and inner diameters of the pipes are 19 mm and 15 mm respectively, although a layer of scale, 0.25 mm thick and of thermal conductivity 2.0 W/m K, has formed on the inside of the pipes. If the coefficients of heat transfer on the vapour and water sides are 1.7 and 3.2 kW/m2 K respectively and it is required to condense 0.025 kg/s of vapour on each of the pipes, how long should these be, and what will be the outlet temperature of water? The latent heat of condensation is 330 kJ/kg. Neglect any resistance to heat transfer in the pipe walls.
Solution For a total of n pipes, mass flow of vapour condensed D 25n ð 1000043 kg/s and hence load, Q D 0.025n ð 330 D 8.25n kW.
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153
For a water outlet temperature of T K and a mass flow of G kg/s: 8.25n D G ð 4.18 T 0004 290 kW or
G D 1.974n/ T 0004 290 kg/s
(i)
00061 D 350 0004 290000b, 00062 D 350 0004 T and hence in equation 9.9: 0006m D [ 350 0004 290 0004 350 0004 T000b]/ ln[ 350 0004 290000b/ 350 0004 T000b] D T 0004 290000b/ ln[60/ 350 0004 T000b] deg K. Considering the film coefficients: hi D 3.2 kW/m2 K, ho D 1.7 kW/m2 K and hence: hio D 3.2 ð 0.015000b/0.019 D 2.526 kW/m2 K. The scale resistance is:
x/k D 0.25 ð 1000043 /2.0 D 0.000125 m2 K/W or 0.125 m2 K/kW Therefore the overall coefficient, neglecting the wall resistance is given by: 1/U D 1/hio C x/k C 1/ho D 0.5882 C 0.125 C 0.396 D 1.109 m2 K/kW or U D 0.902 kW/m2 K Therefore in equation 9.1: A D Q/U0006m D 8.25n/f0.902 T 0004 290000b/ ln[60/ 350 0004 T000b]gm2 D
4.18G T 0004 290 ln[60/ 350 0004 T000b] D 4.634G ln[60/ 350 0004 T000b]m2 0.902 T 0004 290000b
(ii)
The cross-sectional area for flow D $/4 0.015000b2 D 0.000177 m2 /tube. G kg/s G/1000 D 0.001G m3 /s and area/pass to give a velocity of 0.6 m/s D 0.001G/0.6 D 0.00167G m2 . ∴
number of tubes/pass D 0.00167G/0.000177 D 9.42G
(iii)
Area per unit length of tube D $ ð 0.019 ð 1.0 D 0.0597 m2 /m. ∴ total length of tubes D 4.634G ln[60/350 0004 T000b]/0.0597 D 77.6G ln[60/350 0004 T000b]m
length of each tube D 77.6G ln[60/350 0004 T000b]/n m and, substituting from (i), tube length D 77.6 ð 1.974n ln[60/ 350 0004 T000b]/ n T 0004 290000b] D 153.3 ln[60/ 350 0004 T000b]/ T 0004 290 m
(iv)
The procedure is now to select a number of tube passes N and hence m in terms of n from (iii). T is then obtained from (i) and hence the tube length from (iv). The following results are obtained:
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No. of tube passes N 1 2 4 6
Total tubes n
Outlet water temperature T (K)
Tube length (m)
308.6 327.2 364.4 401.6
3.05 3.99 — --
9.42G 18.84G 37.68G 56.52G
Arrangements with 4 and 6 tube side passes require water outlet temperatures in excess of the condensing temperature and are clearly not possible. With 2 tube side passes, T D 327.2 K at which severe scaling would result and hence the proposed unit would consist of one tube side pass and a tube length of 3.05 m. The outlet water temperature would be 308.6 K.
PROBLEM 9.21 A heat exchanger is required to cool continuously 20 kg/s of water from 360 K to 335 K by means of 25 kg/s of cold water, inlet temperature 300 K. Assuming that the water velocities are such as to give an overall coefficient of heat transfer of 2 kW/m2 K, assumed constant, calculate the total area of surface required (a) in a counterflow heat exchanger, i.e. one in which the hot and cold fluids flow in opposite directions, and (b) in a multipass heat exchanger, with the cold water making two passes through the tubes, and the hot water making one pass along the outside of the tubes. In case (b) assume that the hot-water flows in the same direction as the inlet cold water, and that its temperature over any cross-section is uniform.
Solution The heat load, Q D 20 ð 4.18 360 0004 335 D 2090 kW and the outlet cold water temperature is given by: 2090 D 25 ð 4.18 T2 0004 300 or T2 D 320 K Case (a)
00061 D 360 0004 320 D 40 deg K,
00062 D 335 0004 300 D 35 deg K
and in equation 9.9: 0006m D 40 0004 35000b/ ln 40/35 D 37.4 deg K As the flow is true counter-flow, no correction factor is necessary and F D 1.0. Therefore in equation 9.150: A D Q/UF0006m D 2090/ 2.0 ð 1.0 ð 37.4 D 27.94 m2
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Case (b)
Again, 0006m D 37.4 K. In equation 9.212: X D 00062 0004 00061 / T1 0004 00061 D 320 0004 300000b/ 360 0004 300 D 0.33 Y D T1 0004 T2 / 00062 0004 00061 D 360 0004 335000b/ 320 0004 300 D 1.25 Hence, from Fig. 9.71, F D 0.94 and in equation 9.212: A D 2090/ 2.0 ð 0.94 ð 374 D 29.73 m2
PROBLEM 9.22 Find the heat loss per unit area of surface through a brick wall 0.5 m thick when the inner surface is at 400 K and the outside at 310 K. The thermal conductivity of the brick may be taken as 0.7 W/m K.
Solution Q D kA T1 0004 T2 /x
(equation 9.12)
D 0.7 ð 1.0 400 0004 310000b/0.5 D 126 W/m2
PROBLEM 9.23 A furnace is constructed with 225 mm of firebrick, 120 mm of insulating brick, and 225 mm of building brick. The inside temperature is 1200 K and the outside temperature 330 K. If the thermal conductivities are 1.4, 0.2, and 0.7 W/m K, find the heat loss per unit area and the temperature at the junction of the firebrick and insulating brick.
Solution If T1 K and T2 K are the temperatures at the firebrick/insulating brick and the insulating brick/building brick junctions respectively, then in equation 9.12, for conduction through the firebrick: Q D 1.4 ð 1.0 1200 0004 T1 /0.255 D 6.22 1200 0004 T1 W/m2
(i)
For conduction through the insulating brick: Q D 0.2 ð 1.0 T1 0004 T2 /0.120 D 1.67 T1 0004 T2 W/m2
(ii)
and for conduction through the building brick: Q D 0.7 ð 1.0 T2 0004 330000b/0.225 D 3.11 T2 0004 330 W/m2
(iii)
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The thermal resistances of each material, (x/kA), are: firebrick D 1/6.22 D 0.161; insulating brick D 1/1.67 D 0.60; building brick D 1/3.11 D 0.322 K/Wm2 ; and in equation 9.18:
1200 0004 330 D 0.161 C 0.60 C 0.322000bQ or:
Q D 803.3 W/m2
T firebrick/T D x/kA000bfirebrick / x/kA ∴
1200 0004 T1 / 1200 0004 330 D 0.161/ 0.161 C 0.60 C 0.322 D 0.161/1.083 and:
T1 D 1071 K
Similarly for the insulating brick:
1071 0004 T2 / 1200 0004 330 D 0.60/1.083 and:
T2 D 589 K
PROBLEM 9.24 Calculate the total heat loss by radiation and convection from an unlagged horizontal steam pipe of 50 mm outside diameter at 415 K to air at 290 K.
Solution Outside area per unit length of pipe D $ ð 0.050 ð 1.0 D 0.157 m2 /m. Convection
For natural convection from a horizontal pipe to air, the simplified form of equation 9.102 may be used: hc D 1.18 T/do 0.25 In this case: T D 415 0004 290 D 125 deg K and do D 0.050 m. ∴
hc D 1.18 125/0.050000b0.25 D 8.34 W/m2 K Thus, heat loss by convection: qc D hc A T1 0004 T2 D 8.34 ð 0.157 415 0004 290 D 163.7 W/m
Radiation
An extension of equation 9.118 may be used. Taking the emissivity of the pipe as 0.9: qr D 5.67 ð 1000048 ð 0.9 4154 0004 2904 ð 0.157 D 181.0 W/m and the total loss is 344.7 W/m length of pipe.
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157
PROBLEM 9.25 Toluene is continuously nitrated to mononitrotoluene in a cast-iron vessel 1 m in diameter fitted with a propeller agitator of 0.3 m diameter driven at 2 Hz. The temperature is maintained at 310 K by circulating cooling water at 0.5 kg/s through a stainless steel coil of 25 mm outside diameter and 22 mm inside diameter wound in the form of a helix of 0.81 m diameter. The conditions are such that the reacting material may be considered to have the same physical properties as 75% sulphuric acid. If the mean water temperature is 290 K, what is the overall heat transfer coefficient?
Solution The overall coefficient Uo based on the outside area of the coil is given by equation 9.201: 1/Uo D 1/ho C xw /kw do /dw C 1/hi do /di C Ro C Ri do /di where dw is the mean pipe diameter. Inside
The coefficient on the water side is given by equations 9.202 and 9.203: hi D k/d 1 0004 3.5d/dc 0.023 di u0019/0018000b0.8 Cp 0018/k000b0.4 u0019 D 0.5/[ $/4 ð 0.0222 ] D 1315 kg/m2 s
where:
di D 0.022 m, dc D 0.80 m and for water at 290 K: k D 0.59 W/m K, 0018 D 0.00108 Ns/m2 , and Cp D 4180 J/kg K. ∴
hi D 0.59/0.022 1 C 3.5 ð 0.22/0.80 ð 0.023 0.022 ð 1315/0.00108000b0.8 ð 4180 ð 0.00108/0.59000b0.4 D 0.680 26,780000b0.8 7.65000b0.4 D 5490 W/m2 K
Outside
In equation 9.204:
ho dv /k 0018s /0018000b0.14 D 0.87 Cp 0018/k000b0.33 L 2 N0019/0018000b0.62 For 75% sulphuric acid: k D 0.40 W/m K, 0018s D 0.0086 N s/m2 at 300 K, 0018 D 0.0065 N s/m2 at 310 K, Cp D 1880 J/kg K and 0019 D 1666 kg/m3 ∴ ho ð 1.0/0.40 0.0086/0.0065000b0.14 D 0.87 1880 ð 0.0065/0.40000b0.33
ð 0.32 ð 2.0 ð 1665/0.0065000b0.62 ∴
and:
2.5ho 1.323000b0.14 D 0.87 30.55000b0.33 46,108000b0.62 ho D 0.348 ð 3.09 ð 779000b/1.04 D 805.5 W/m2 K
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Overall
Taking kw D 15.9 W/m K and Ro and Ri as 0.0004 and 0.0002 m2 K/W respectively, then in equation 9.201: 1/Uo D 1/805.5 C 0.0015/15.9 0.025/0.0235 C 1/5490 0.025/0.022 C 0.0004 C 0.0002 0.025/0.022 D 0.00124 C 0.00010 C 0.00021 C 0.00040 C 0.00023 D 0.00218 m2 K/W Uo D 458.7 W/m2 K
and:
PROBLEM 9.26 7.5 kg/s of pure iso-butane is to be condensed at a temperature of 331.7 K in a horizontal tubular exchanger using a water inlet temperature of 301 K. It is proposed to use 19 mm outside diameter tubes of 1.6 mm wall arranged on a 25 mm triangular pitch. Under these conditions the resistance of the scale may be taken as 0.0005 m2 K/W. Determine the number and arrangement of the tubes in the shell.
Solution The latent heat of vaporisation of isobutane is 286 kJ/kg and hence the heat load: Q D 7.5 ð 286 D 2145 kW The cooling water outlet should not exceed 320 K and a value of 315 K will be used. The mass flow of water is then: 2145/[4.18 315 0004 301000b] D 36.7 kg/s In order to obtain an approximate size of the unit, a value of 500 W/m2 K will be assumed for the overall coefficient based on the outside area of the tubes. 00061 D 331.7 0004 301 D 30.7 deg K,
00062 D 331.7 0004 315 D 16.7 deg K
and from equation 9.9: 0006m D 30.7 0004 16.7000b/ ln 30.7/16.7 D 23.0 deg K. Thus, the approximate area D 2145 ð 103 / 500 ð 23.0 D 186.5 m2 . The outside area of 0.019 m diameter tubes D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and hence the total length of tubing D 186.5/0.0597 D 3125 m. Adopting a standard tube length of 4.88 m, number of tubes D 3125/4.88 D 640. With the large flow of water involved, a four tube-side pass unit is proposed, and for this arrangement 678 tubes can be accommodated on 25 mm triangular pitch in a 0.78 m i.d. shell. Using this layout, the film coefficients are now estimated and the assumed value of U is checked. Inside
Water flow through each tube D 36.7 678/4 D 0.217 kg/s.
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The tube i.d. D 19.0 0004 2 ð 1.67 D 15.7 mm the cross-sectional area for flow D $/4 0.0157000b2 D 0.000194 m2 and hence the water velocity: u D 0.217/ 1000 ð 0.000194 D 1.12 m/s. From equation 9.221 : hi D 4280[ 0.00488 ð 308 0004 1]1.120.8 /0.01570.2 D 4280 ð 0.503 ð 1.095000b/0.436 D 5407 W/m2 K or, based on outside diameter: hio D 5407 ð 0.0157000b/0.019 D 4468 W/m2 K or 4.47 kW/m2 K Outside
The temperature drop across the condensate film, Tf is given by: (thermal resistance of water film C scale)/ total thermal resistance D 0006m 0004 Tf /0006m or:
1/4.47 C 0.5000b/ 1/0.500 D 23.0 0004 Tf /23.0 Tf D 14.7 deg K
and:
The condensate film is thus at 331.7 0004 14.7 D 317 K. The outside film coefficient is given by: ho D 0.72[ k 3 00192 g001a000b/ jdo 0018Tf ]0.25
(equation 9.177) p At 317 K, k D 0.13 W/m K, 0019 D 508 kg/m3 , 0018 D 0.000136 N s/m2 and j D 678 D 26.0. ∴
ho D 0.72[ 0.133 ð 5082 ð 9.81 ð 286 ð 103 /
26 ð 19.0 ð 1000043 ð 0.000136 ð 14.5000b]0.25 D 814 W/m2 K or 0.814 kW/m2 K
Overall
1/U D 1/4.47 C 1/0.814 C 0.50 D 1.952 U D 0.512 kW/m2 K or 512 W/m2 K
and:
which is sufficiently near the assumed value. For the proposed unit, the heat load: Q D 0.512 ð 678 ð 4.88 ð 0.0597 ð 23.0 D 2328 kW or an overload of:
2328 0004 2145000b100/2145 D 8.5%
PROBLEM 9.27 37.5 kg/s of crude oil is to be heated from 295 to 330 K by heat transferred from the bottom product from a distillation column. The bottom product, flowing at 29.6 kg/s is to
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be cooled from 420 to 380 K. There is available a tubular exchanger with an inside shell diameter of 0.60 m, having one pass on the shell side and two passes on the tube side. It has 324 tubes, 19 mm outside diameter with 2.1 mm wall and 3.65 m long, arranged on a 25 mm square pitch and supported by baffles with a 25% cut, spaced at 230 mm intervals. Would this exchanger be suitable?
Solution Mean temperature of bottom product D 0.5 420 C 380 D 400 K. Mean temperature of crude oil D 0.5 330 C 295 D 313 K. For the crude oil at 313 K: Cp D 1986 J/kg K, 0018 D 0.0029 N s/m2 , k D 0.136 W/m K and 0019 D 824 kg/m3 . For the bottom product at 400 K: Cp D 2200 J/kg K. Heat loads: tube side: Q D 37.5 ð 1.986 330 0004 295 D 2607 kW. shell side: Q D 29.6 ð 2.20 420 0004 380 D 2605 kW.
Outside coefficient
Temperature of wall D 0.5 400 C 313 D 356.5 K and film temperature, Tf D 0.5 400 C 356.5 D 378 K. At 378 K, 0019 D 867 kg/m3 , 0018 D 0.0052 N s/m2 , and k D 0.119 W/m K Cross-sectional area for flow D shell i.d. ð clearance ð baffle spacing000b/pitch D 0.60 ð 0.0064 ð 0.23000b/0.025 D 0.353 m2 (assuming a clearance of 0.0064 m). ∴
G0max D 29.6/0.0353 D 838.5 kg/m2 s Remax D 0.019 ð 838.5000b/0.0052 D 306.4
and:
Therefore in equation 9.90, taking Ch D 1:
ho ð 0.019/0.119 D 0.33 ð 1.0 3064000b0.6 2200 ð 0.0052/0.119000b0.3 or:
ho D 2.07 ð 125 ð 3.94 D 1018 W/m2 K or 1.02 kW/m2 K
Inside coefficient
Number tubes per pass D 324/2 D 162. Inside diameter D [19.0 0004 2 ð 2.1000b]/1000 D 0.0148 m
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and cross-sectional area for flow D $/4 0.0148000b2 D 0.000172 m2 per tube or:
0.000172 ð 162 D 0.0279 m2 per pass. ∴
G0 D 37.5/0.0279 D 1346 kg/m2 s In equation 9.61:
hi ð 0.0148/0.136 D 0.023 0.0148 ð 1346/0.0029000b0.8 1986 ð 0.0029/0.136000b0.4 hi D 0.211 6869000b0.8 42.4000b0.4 D 1110 W/m2 K
or, based on the outside area, hio D 1110 ð 0.0148000b/0.019 D 865 W/m2 K hio D 0.865 kW/m2 K.
or: Overall coefficient
Neglecting the wall and scale resistance, the clean overall coefficient is: 1/Uc D 1/1.02 C 1/0.865 D 2.136 m2 K/kW The area available is A D 324 ð 3.65 ð $ ð 0.019 D 70.7 m2 and hence the minimum value of the design coefficient is: 1/UD D A0006m /Q 00061 D 420 0004 330 D 90 deg K, and: ∴
00062 D 380 0004 295 D 85 deg K
0006m D 90 0004 85000b/ ln 90/85 D 87.5 deg K 1/UD D 70.7 ð 87.5000b/2607 D 2.37 m2 K/kW
The maximum allowable scale resistance is then: R D 1/UD 0004 1/Uc D 2.37 0004 2.136 D 0.234 m2 K/kW This value is very low as seen from Table 9.16, and the exchanger would not give the required temperatures without frequent cleaning.
PROBLEM 9.28 A 150 mm internal diameter steam pipe, carrying steam at 444 K, is lagged with 50 mm of 85% magnesia. What will be the heat loss to the air at 294 K?
Solution In this case: di D 0.150 m, do D 0.168 m and dw D 0.5 0.150 C 0.168 D 0.159 m. ds D 0.168 ð 2 ð 0.050 D 0.268 m and dm (the logarithmic mean of do and ds ) D 0.215 m.
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The coefficient for condensing steam including any scale will be taken as 8500 W/m2 K, kw as 45 W/m K, and kl as 0.073 W/m K. The surface temperature of the lagging will be assumed to be 314 K and hr C hc to be 10 W/m2 K. The thermal resistances are therefore:
1/hi $d D 1/ 8500 ð $ ð 0.150 D 0.00025 mK/W
xw /kw $dw D 0.009/ 45$ ð 0.159 D 0.00040 mK/W
xl /kl $dm D 0.050/ 0.073$ ð 0.215 D 1.0130 mK/W
1/ hr ð hc ds D 1/ 10 ð 0.268 D 0.1190 mK/W Neglecting the first two terms, the total thermal resistance D 1.132 mK/W. From equation 9.261, heat lost per unit length D 444 0004 294000b/1.132 D 132.5 W/m. The surface temperature of the lagging is given by: T lagging000b/T D 1.013/1.132 D 0.895 and:
T lagging D 0.895 444 0004 294 D 134 deg K
Therefore the surface temperature D 444 0004 134 D 310 K which approximates to the assumed value. Assuming an emissivity of 0.9: hr D 5.67 ð 1000048 ð 0.9 3104 0004 2944 / 310 0004 294 D 3.81 W/m2 K. For natural convection: hc D 1.37 T/ds 0.25 D 1.37[ 310 0004 294000b/0.268]0.25 D 3.81 W/m2 K. ∴
hr C hc D 9.45 W/m2 K which again agrees with the assumed value. In practice forced convection currents are usually present and the heat loss would probably be higher than this value. For an unlagged pipe and T D 150 K, hr C hc would be about 20 W/m2 K and the heat loss, Q/l D hr C hc $d0 T D 20$ ð 0.168 ð 150 D 1584 W/m. Thus the heat loss has been reduced by about 90% by the addition of 50 mm of lagging.
PROBLEM 9.29 A refractory material with an emissivity of 0.40 at 1500 K and 0.43 at 1420 K is at a temperature of 1420 K and is exposed to black furnace walls at a temperature of 1500 K. What is the rate of gain of heat by radiation per unit area?
Solution In the absence of further data, the system will be considered as two parallel plates.
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The radiating source is the furnace walls at T1 D 1500 K and for a black surface, e1 D 1.0. The heat sink is the refractory at T2 D 1420 K, at which e2 D 0.43. Putting A1 D A2 in equation 9.150: q D e1 e2 9 T41 0004 T42 / e1 C e2 0004 e1 e2 D 1.0 ð 0.43 ð 5.67 ð 1000048 15004 0004 14204 / 1.0 C 0.43 0004 0.43 ð 1.0 D 2.44 ð 1000048 9.97 ð 1011 /1.0 D 2.43 ð 104 W/m2 or 24.3 kW/m2
PROBLEM 9.30 The total emissivity of clean chromium as a function of surface temperature, T K, is given approximately by: e D 0.38 1 0004 263/T000b. Obtain an expression for the absorptivity of solar radiation as a function of surface temperature, and calculate the values of the absorptivity and emissivity at 300, 400 and 1000 K. Assume that the sun behaves as a black body at 5500 K.
Solution It may be assumed that the absorptivity of the chromium at temperature T1 is the emissivity of the chromium at the geometric mean of T1 and the assumed temperature of the sun, T2 where T2 D 5500 K. Since:
e D 0.38 1 0004 263/T
i 0.5
then, taking the geometric mean temperature as 5500T1 : a D 0.38f1 0004 [263/ 5500T1 0.5 ]g
ii000b
For the given values of T1 , values of e and a are now calculated from (i) and (ii) respectively to give the following data: T1 300 400 1000
T1 T2 0.5 1285 1483 2345
e 0.047 0.130 0.280
a 0.300 0.312 0.337
PROBLEM 9.31 Repeat Problem 9.30 for the case of aluminium, assuming the emissivity to be 1.25 times that for chromium.
Solution In this case:
and:
e D 1.25 ð 0.38 1 0004 263/T1 D 0.475[ 1 0004 263/T1 ]
i000b
a D 0.475 0004 1.66T00040.5 1
ii000b
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The following data are obtained by substituting values for T1 in equations (i) and (ii): T1 300 400 1000
T1 T2 0.5 1285 1483 2345
e 0.059 0.163 0.350
a 0.378 0.391 0.422
PROBLEM 9.32 Calculate the heat transferred by solar radiation on the flat concrete roof of a building, 8 m by 9 m, if the surface temperature of the roof is 330 K. What would be the effect of covering the roof with a highly reflecting surface, such as polished aluminium, separated from the concrete by an efficient layer of insulation? The emissivity of concrete at 330 K is 0.89, whilst the total absorptivity of solar radiation (sun temperature D 5500 K) at this temperature is 0.60. Use the data for aluminium from Problem 9.31 which should be solved first.
Solution The emission from a body with an emissivity, e, at a temperature T is given by: I D e9T4 Thus, for the concrete: I D 0.89 ð 5.67 ð 1000048 ð 3304 D 598.5 W/m2 Taking T D 330 K as the equilibrium temperature, the energy emitted by the concrete must equal the energy absorbed and, since the absorptivity of concrete, a D 0.60, the solar flux is then: Is D 598.5/0.6 D 997.4 W/m2 which approximates to the generally accepted figure of about 1 kW/m2 . With a covering of polished aluminium, then using the data given in Problem 9.31 and an equilibrium surface temperature of T K, the absorptivity is: a D 0.475 0004 1.66T0.5 and, with an area of 8 ð 9 D 72 m2 , the energy absorbed is:
1.0 ð 103 ð 72 0.475 0004 1.66T0.5 D 3.42 ð 104 0004 1.20 ð 105 T0.5 W
(i)
The emissivity is given by: e D 0.475 1 0004 263/T D 0.475 0004 125T00041 and the energy emitted is:
72 ð 5.67 ð 1000048 T4 0.475 0004 125T00041 D 1.94 ð 1000046 T4 0004 5.10 ð 1000044 T3 W
(ii)
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Equating (i) and (ii): 1.94 ð 1000046 T4 0004 5.10 ð 1000044 T3 C 1.20 ð 105 T0.5 D 3.42 ð 104 or:
5.67 ð 10000411 T4 0004 1.49 ð 1000048 T3 C 3.51T0.5 D 1
Solving by trial and error, the equilibrium temperature of the aluminium is: T D 438 K . Substituting T D 438 K in (i), the energy absorbed and emitted is then 2847 W which represents an increase of some 375 per cent compared with the value for the concrete alone.
PROBLEM 9.33 A rectangular iron ingot 15 cm ð 15 cm ð 30 cm is supported at the centre of a reheating furnace. The furnace has walls of silica-brick at 1400 K, and the initial temperature of the ingot is 290 K. How long will it take to heat the ingot to 600 K? It may be assumed that the furnace is large compared with the ingot, and that the ingot is always at uniform temperature throughout its volume. Convection effects are negligible. The total emissivity of the oxidised iron surface is 0.78 and both emissivity and absorptivity may be assumed independent of the surface temperature. (Density of iron D 7.2 Mg/m3 . Specific heat capacity of iron D 0.50 kJ/kg K.)
Solution As there are no temperature gradients within the ingot, the rate of heating is dependent on the rate of radiative heat transfer to the surface. In addition, since the dimensions of the ingot are much smaller than those of the surrounding surfaces, the ingot may be treated as a black body. Volume of ingot D 15 ð 15 ð 30 D 6750 cm3 or 0.00675 m3 . Mass of ingot D 7.2 ð 103 ð 0.00675 D 48.6 kg. For an ingot temperature of T K, the increase in enthalpy D d mCp T000b/dt or mCp dT/dt where t is the time and Cp the specific heat of the ingot. The heat received by radiation D A9a T4f 0004 T4 where the area, A D 4 ð 30 ð 15 C
2 ð 15 ð 15 D 2250 cm2 or 0.225 m2 . The absorptivity a will be taken as the emissivity D 0.78 and the furnace temperature, Tf D 1400 K. Thus: or:
mCp dT/dt D A9a T4f 0004 T4 0004 t 0004 dT mCp 600 dt D 4 4 aA9 0 290 Tf 0004 T
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0002
∴
tD
00030002 0003
48.6 ð 0.50 ð 103 1
0.78 ð 0.225 ð 5.67 ð 1000048
4 ð 14003 0002 0003600 T
1400 C T C 2 tan00041 ð ln D 200 s
1400 0004 T 1400 290
PROBLEM 9.34 A wall is made of brick, of thermal conductivity 1.0 W/m K, 230 mm thick, lined on the inner face with plaster of thermal conductivity 0.4 W/m K and of thickness 10 mm. If a temperature difference of 30 K is maintained between the two outer faces, what is the heat flow per unit area of wall?
Solution For an area of 1 m2 , thermal resistance of the brick:
x1 /k1 A D 0.230/ 1.0 ð 1.0 D 0.230 K/W
thermal resistance of the plaster:
x2 /k2 A D 0.010/ 0.4 ð 1.0 D 0.0025 K/W
and in equation 9.18: 30 D 230 C 0.0025000bQ or Q D 129 W
PROBLEM 9.35 A 50 mm diameter pipe of circular cross-section and with walls 3 mm thick is covered with two concentric layers of lagging, the inner layer having a thickness of 25 mm and a thermal conductivity of 0.08 W/m K, and the outer layer a thickness of 40 mm and a thermal conductivity of 0.04 W/m K. What is the rate of heat loss per metre length of pipe if the temperature inside the pipe is 550 K and the outside surface temperature is 330 K?
Solution From equation 9.22, the thermal resistance of each component is: r2 0004 r1 /k 2$rm l Thus for the wall: r2D 0.050/2 C 0.003 D 0.028 m r1 D 0.050/2 D 0.025 m and:
rm D 0.028 0004 0.025000b/ ln 0.028/0.025 D 0.0265 m.
Taking k D 45 W/m K and l D 1.0 m the thermal resistance is: D 0.028 0004 0.025000b/ 45 ð 2$ ð 0.0265 ð 1.0 D 0.00040 K/W.
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For the inner lagging: r2 D 0.028 C 0.025 D 0.053 m r1 D 0.028 m and:
rm D 0.053 0004 0.028000b/ ln 0.053/0.028 D 0.0392 m.
Therefore the thermal resistance D 0.053 0004 0.028000b/ 0.08 ð 2$ ð 0.0392 ð 1.0 D 1.2688 K/W For the outer lagging: r2 D 0.053 C 0.040 D 0.093 m r1 D 0.053 m and:
rm D 0.093 0004 0.053000b/ ln 0.093/0.053 D 0.0711 m
Therefore the thermal resistance D 0.093 0004 0.053000b/ 0.04 ð 2$ ð 0.0711 ð 1.0 D 2.2385 K/W From equation 9.19: Q D 550 0004 330000b/ 0.0004 C 1.2688 C 2.2385 D 62.7 W/m
PROBLEM 9.36 The temperature of oil leaving a co-current flow cooler is to be reduced from 370 to 350 K by lengthening the cooler. The oil and water flowrates, the inlet temperatures and the other dimensions of the cooler will remain constant. The water enters at 285 K and oil at 420 K. The water leaves the original cooler at 310 K. If the original length is 1 m, what must be the new length?
Solution For the original cooler, for the oil:
Q D Go Cpo 420 0004 370000b
and for the water:
Q D Gw Cpw 310 0004 285000b
∴
Go Cp /Gw Cp D 25/50 D 0.5
where Go and Gw are the mass flows and Cpo and Cpw the specific heat capacities of the oil and water respectively. 00061 D 420 0004 285 D 135 deg K, 00062 D 370 0004 310 D 60 deg K for co-current flow, and from equation 9.9: 0006m D 135 0004 60000b/ ln 135/60 D 92.5 deg K If a is the area per unit length of tube multiplied by the number of tubes, then: A D 1.0 ð a m2 and in equation 9.1: Go Cp 420 0004 370 D Ua 92.5 or Go Cp /Ua D 1.85 For the new cooler, for the oil: Q D Go Cpo 420 0004 350 and for the water,
Q D Gw Cpw T 0004 285000b
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where T is the water outlet temperature. Thus:
T 0004 285 D Go Cp /Gw Cp 420 0004 350 D 0.5 ð 70000b
and:
T D 320 K
∴ 00061 D 420 0004 285 D 135 deg K, 00062 D 350 0004 320 D 30 deg K, again for co-current flow, and from equation 9.9: 0006m D 135 0004 30000b/ ln 135/30 D 69.8 deg K
In equation 9.1: Go so 420 0004 350 D Ual69.8 ∴
l D Go Cp /Ua ð 1.003 D 1.85 ð 1.003 D 1.86 m
PROBLEM 9.37 In a countercurrent-flow heat exchanger, 1.25 kg/s of benzene (specific heat 1.9 kJ/kg K and density 880 kg/m3 ) is to be cooled from 350 K to 300 K with water which is available at 290 K. In the heat exchanger, tubes of 25 mm external and 22 mm internal diameter are employed and the water passes through the tubes. If the film coefficients for the water and benzene are 0.85 and 1.70 kW/m2 K respectively and the scale resistance can be neglected, what total length of tube will be required if the minimum quantity of water is to be used and its temperature is not to be allowed to rise above 320 K?
Solution Heat load:
For the benzene: Q D 1.25 ð 1.9 350 0004 300 D 118.75 kW. In order to use the minimum amount, water must leave the unit at the maximum temperature, 320 K. Thus for G kg/s water: 118.75 D G ð 4.18 320 0004 290 or G D 0.947 kg/s Temperature driving force
00061 D 350 0004 320 D 30 deg K, 00062 D 300 0004 290 D 10 deg K and in equation 9.9: 0006m D 30 0004 10000b/ ln 30/10 D 18.2 deg K. In the absence of further data, it will be assumed that the correction factor is unity. Overall coefficient
Inside: hi D 0.85 kW/m2 K or based on the tube o.d., hio D 0.85 ð 22/25 D 0.748 kW/m2 K.
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Outside: ho D 1.70 kW/m2 K. Wall: Taking ksteel D 45 W/m K, x/k D 0.003/45 D 0.00007 m2 K/W or 0.07 m2 K/kW. Thus neglecting any scale resistance: 1/U D 1/0.748 0004 1/1.70 C 0.07 D 1.995 m2 K/kW U D 0.501 kW/m2 K
and: Area
In equation 9.1: A D Q/U0006m D 118.75/ 0.0501 ð 18.2 D 13.02 m2 . Surface area of a 0.025 m o.d. tube D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and hence total length of tubing required D 1302/0.0785 D 165.8 m
PROBLEM 9.38 Calculate the rate of loss of heat from a 6 m long horizontal steam pipe of 50 mm internal diameter and 60 mm external diameter when carrying steam at 800 kN/m2 . The temperature of the surroundings is 290 K. What would be the cost of steam saved by coating the pipe with a 50 mm thickness of 85% magnesia lagging of thermal conductivity 0.07 W/m K, if steam costs £0.5/100 kg? The emissivity of both the surface of the bare pipe and the lagging may be taken as 0.85, and the coefficient h for the heat loss by natural convection is given by: h D 1.65 T000b0.25 W/m2 K where T is the temperature difference in deg K. The Stefan-Boltzmann constant is 5.67 ð 1000048 W/m2 K4 .
Solution For the bare pipe
Steam is saturated at 800 kN/m2 and 443 K. Neglecting the inside resistance and that of the wall, it may be assumed that the surface temperature of the pipe is 443 K. For radiation from the pipe, the surface area D $ ð 0.060 ð 6.0 D 1.131 m2 and in equation 9.119: qr D 5.67 ð 1000048 ð 0.85 ð 1.131 4434 0004 2904 D 1714 W.
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For convection from the pipe, the heat loss: qc D hc A Ts 0004 T D 1.65 443 0004 290000b0.25 ð 1.131 443 0004 290 D 1.866 443 0004 290000b1.25 D 1004 W and the total loss D 2718 W or 2.71 kW For the insulated pipe
The heat conducted through the lagging ql must equal the heat lost from the surface
qr C qc . Mean diameter of the lagging D [ 0.060 C 2 ð 0.050 C 0.060]/2 D 0.110 m at which the area D $ ð 0.110 ð 6.0 D 2.07 m2 and in equation 9.12: ql D 0.07 ð 2.07 443 0004 Ts /0.050 D 1280 0004 2.90Ts W where Ts is the surface temperature. The outside area D $ 0.060 C 2 ð 0.050 ð 6.0 D 3.016 m2 and from equation 9.119 : qr D 5.67 ð 1000048 ð 0.85 ð 3.016 T4s 0004 2904 D 1.456 ð 1000047 T4s D 1030 W and:
qc D 1.65 Ts 0004 290000b0.25 ð 3.016 Ts 0004 290 D 4.976 Ts 0004 290000b1.25 W
Making a heat balance:
1280 0004 2.90Ts D 1.456 ð 1000047 T4s 0004 1030 C 4.976 Ts 0004 290000b1.25 or:
4.976 Ts 0004 290000b1.25 C 1.456 ð 1000047 T4s C 2.90Ts D 2310
Solving by trial and error: Ts D 305 K and hence the heat lost D 1280 0004 2.90 ð 305 D 396 W. The heat saved by lagging the pipe D 2712 0004 396 D 2317 W or 2.317 kW. At 800 kN/m2 , the latent heat of steam is 2050 kJ/kg and the reduction in the amount of steam condensed D 2.317/2050 D 0.00113 kg/s or:
0.00113 ð 3600 ð 24 ð 365 D 35,643 kg/year
∴
annual saving D 35,643 ð 0.5000b/100 D £178/year
It may be noted that arithmetic mean radius should only be used with thin walled tubes, which is not the case here. If a logarithmic mean radius is used in applying equation 9.8, Ts D 305.7 K and the difference is, in this case, negligible.
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PROBLEM 9.39 A stirred reactor contains a batch of 700 kg reactants of specific heat 3.8 kJ/kg K initially at 290 K, which is heated by dry saturated steam at 170 kN/m2 fed to a helical coil. During the heating period the steam supply rate is constant at 0.1 kg/s and condensate leaves at the temperature of the steam. If heat losses are neglected, calculate the true temperature of the reactants when a thermometer immersed in the material reads 360 K. The bulb of the thermometer is approximately cylindrical and is 100 mm long by 10 mm in diameter with a water equivalent of 15 g, and the overall heat transfer coefficient to the thermometer is 300 W/m2 K. What temperature would a thermometer with a similar bulb of half the length and half the heat capacity indicate under these conditions?
Solution The latent heat of dry saturated steam at 170 kN/m2 and 388 K D 2216 kJ/kg. Therefore heat added to the reactor D 2216 ð 0.1 D 221.6 kJ/s D 221.6 kW which is equal to the increase in enthalpy, dH/dt. The enthalpy of the contents, neglecting the heat capacity of the reactor and losses D mCp dT/dt D 700 ð 3.8 dT/dt or 2660 dT/dt kW ∴
2660 dT/dt D 221.6
and the rate of temperature rise, dT/dt D 0.083 deg K/s. At time t, the temperature of the reactants is: T D 290 C 0.083t K
(i)
The increase in enthalpy of the thermometer is equal to the rate of heat transfer from the fluid, or: (ii)
mCp t dTt /dt D UAt T 0004 Tt where the subscript t refers to the thermometer. ∴
15/1000 ð 4.18 dTt /dt D 0.300 $ ð 0.010 ð 0.100 T 0004 Tt
and:
dTt /dt D 0.0150 T 0004 Tt deg K/s
At time t s, the temperature of the thermometer is therefore: Tt D 290 C [0.0150 T 0004 Tt ]t K
(iii)
When Tt D 360 K, then substituting from equation (i): 360 D 290 C f0.0150[290 C 0.083t 0004 360]gt
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or:
0.00125t2 0004 1.05t 0004 70 D 0 and t D 902 s
Therefore in (i):
T D 290 C 0.083 ð 902 D 364.9 K
With half the length, that is 0.050 m, and half the heat capacity, that is 7.5 g water, then in equation (ii):
7.5/1000 ð 4.18 dTt /dt D 0.300 $ ð 0.010 ð 0.050 T 0004 Tt
dTt /dt D 0.0150 T 0004 Tt
or:
The same result as before and hence the new thermometer would also read 360 K.
PROBLEM 9.40 How long will it take to heat 0.18 m3 of liquid of density 900 kg/m3 and specific heat 2.1 kJ/kg K from 293 to 377 K in a tank fitted with a coil of area 1 m2 ? The coil is fed with steam at 383 K and the overall heat transfer coefficient can be taken as constant at 0.5 kW/m2 K. The vessel has an external surface of 2.5 m2 , and the coefficient for heat transfer to the surroundings at 293 K is 5 W/m2 K. The batch system of heating is to be replaced by a continuous countercurrent heat exchanger in which the heating medium is a liquid entering at 388 K and leaving at 333 K. If the heat transfer coefficient is 250 W/m2 K, what heat exchange area is required? Heat losses may be neglected.
Solution Mass of liquid in the tank D 0.18 ð 900 D 162 kg ∴
mCp D 162 ð 2100 D 340,200 J/deg K
Using the argument given in Problem 9.77: 340,200 dT/dt D 500 ð 1 383 0004 T 0004 5 ð 2.5 T 0004 293 or: or:
D 191,500 0004 500T 0004 12.5T C 3663 D 195,163 0004 512.5T 664 dT/dt D 380.8 0004 T
The time taken to heat the liquid from 293 to 377 K is: 0004 377 dT/ 380.8 0004 T t D 664 293
D 664 ln[ 380.8 0004 293000b/ 380.8 0004 377000b] D 2085 s 0.58 h For the heat exchanger:
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T1 D 388 0004 377 D 11 deg K, T2 D 333 0004 293 D 40 deg K and from equation 9.9: Tm D 40 0004 11000b/ ln 40/11 D 22.5 deg K. Mass flow D 162/2085 D 0.0777 kg/s Heat load: Q D 0.0777 ð 2.1 377 0004 293 D 13.71 kW In equation 9.1: U D 250/1000 D 0.250 kW/m2 K, The area required: A D 13.71/ 0.250 ð 22.5 D 2.44 m2 .
PROBLEM 9.41 The radiation received by the earth’s surface on a clear day with the sun overhead is 1 kW/m2 and an additional 0.3 kW/m2 is absorbed by the earth’s atmosphere. Calculate approximately the temperature of the sun, assuming its radius to be 700,000 km and the distance between the sun and the earth to be 150,000,000 km. The sun may be assumed to behave as a black body.
Solution The total radiation received D 1.3 kW/m2 of the earth’s surface. The equivalent surface area of the sun is obtained by comparing the area of a sphere at the radius of the sun, 7 ð 105 km and the area of a sphere of radius (radius of sun C distance between sun and earth) or: A1 /A2 D 4$ 7 ð 105 2 /4$ 150 ð 106 C 7 ð 105 2 D 2.16 ð 1000045 . Therefore radiation at the sun’s surface D 1.3ð 103 /2.16ð 1000045 D 6.03 ð 107 W/m2 . For a black body, the intensity of radiation is given by equation 9.112: 6.03 ð 107 D 5.67 ð 1000048 T4
and
T D 5710 K
PROBLEM 9.42 A thermometer is immersed in a liquid which is heated at the rate of 0.05 K/s. If the thermometer and the liquid are both initially at 290 K, what rate of passage of liquid over the bulb of the thermometer is required if the error in the thermometer reading after 600 s is to be no more than 1 deg K? Take the water equivalent of the thermometer as 30 g, the heat transfer coefficient to the bulb to be given by U D 735 u0.8 W/m2 K. The area of the bulb is 0.01 m2 where u is the velocity in m/s.
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Solution If T and T0 are the liquid and thermometer temperatures respectively after time t s, then: dT/dt D 0.05 K/s and hence T D 290 C 0.05t When t D 600 s, T 0004 T0 D 1. ∴
T D 290 C 600 ð 0.05 D 320 K and T0 D 319 K Balancing: Gt C0019t dT0 /dt D UA T 0004 T0
∴ ∴ ∴
30/1000000b4.18 dT0 /dt D UA 290 C 0.05t C T0 dT0 /dt C 7.98UAT0 D 2312UA 1 C 0.000173t
0004 e7.98UAt e7.98UAt 0004 0.000173 dt e7.98UAt T0 D 2312UA 1 C 0.000173t 7.98UA 7.98UA D 290 1 C 0.000173t000be7.98UAt 0004 0.050
e7.98UAt Ck 7.98UA
When t D 0, T0 D 290 K and k D 0.00627/UA. ∴
T0 D 290 1 C 0.000173t 0004 0.00627/UA 1 0004 e00047.98UAt
When t D 600 s, T0 D 319 K. ∴ ∴
and:
319 D 320 C 0.00627/UA 1 0004 e00044789UA 00044789UA D ln 1 0004 159.5UA UA D 00040.000209 ln 1 0004 159.5UA000b
Solving by trial and error: UA D 0.00627 kW/K. A D 0.01 m2 and hence: U D 0.627 kW/m2 K or 627 W/m2 K ∴
627 D 735u0.8
and
u D 0.82 m/s
PROBLEM 9.43 In a shell-and-tube type of heat exchanger with horizontal tubes 25 mm external diameter and 22 mm internal diameter, benzene is condensed on the outside of the tubes by means of water flowing through the tubes at the rate of 0.03 m3 /s. If the water enters at 290 K and leaves at 300 K and the heat transfer coefficient on the water side is 850 W/m2 K, what total length of tubing will be required?
Solution Mass flow of water D 0.03 ð 1000 D 30 kg/s
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and hence the heat load D 30 ð 4.18 300 0004 290 D 1254 kW At atmospheric pressure, benzene condenses at 353 K and hence: 00061 D 353 0004 290 D 63 deg K,
00062 D 353 0004 300 D 53 deg K
and from equation 9.9: 0006m D 63 0004 53000b/ ln 63/53 D 57.9 deg K No correction factor is required, because of isothermal conductions on the shell side. For condensing benzene, ho will be taken as 1750 W/m2 K. From Table 9.18: hi D 850 W/m2 K or, based on the outside diameter, hio D 850 ð 22/25 D 748 W/m2 K. Neglecting scale and wall resistances: 1/U D 1/1750 C 1/748 D 0.00191 m2 K/W and:
U D 524 W/m2 K or 0.524 kW/m2 K
Therefore, from equation 9.1: A D 1254/ 0.524 ð 57.9 D 41.3 m2 . Outside area of 0.025 m tubing D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and total length of tubing required D 41.3/0.0785 D 526 m.
PROBLEM 9.44 In a contact sulphuric acid plant, the gases leaving the first convertor are to be cooled from 845 to 675 K by means of the air required for the combustion of the sulphur. The air enters the heat exchanger at 495 K. If the flow of each of the streams is 2 m3 /s at NTP, suggest a suitable design for a shell-and-tube type of heat exchanger employing tubes of 25 mm internal diameter. (a) Assume parallel co-current flow of the gas streams. (b) Assume parallel countercurrent flow. (c) Assume that the heat exchanger is fitted with baffles giving cross-flow outside the tubes.
Solution Heat load
At a mean temperature of 288 K, the density of air D 29/22.4 273/288 D 1.227 kg/m3 , where 29 kg/kmol is taken as the mean molecular mass of air. ∴
mass flow of air D 2.0 ð 1.227 D 2.455 kg/s.
If, as a first approximation, the thermal capacities of the two streams can be assumed equal for equal flowrates, then the outlet air temperature D 495 C 845 0004 675 D 665 K
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and for a mean specific heat of 1.0 kJ/kg K, the heat load is Q D 2.455 ð 1.0 665 0004 495 D 417.4 kW For gas to gas heat transfer, an overall coefficient of 1/ 1/60 C 1/60 D 30 W/m2 K will be taken using the data in Table 9.17. (a) Co-current flow
00061 D 845 0004 495 D 350 deg K,
00062 D 675 0004 665 D 10 deg K
and in equation 9.9: 0006m D 350 0004 10000b/ ln 350/10 D 95.6 deg K. Therefore in equation 9.1: A D 417.4 ð 103 / 30 ð 95.6 D 145.5 m2 . For 25 mm i.d. tubes an o.d. of 32 mm will be assumed for which the outside area D
$ ð 0.032 ð 1.0 D 0.1005 m2 /m and total length of tubing D 145.5/0.1005 D 1447 m. At a mean air temperature of 580 K: 0019 D 29/22.4 273/580 D 0.609 kg/m3 . ∴ volume flow of air D 2.445/0.609 D 4.03 m3 /s. For a reasonable gas velocity of say 15 m/s: area for flow D 4.03/15 D 0.268 m2 . Cross-sectional area of one tube D $/4000b0.0252 D 0.00050 m2 . ∴ number of tubes/pass D 0.268/0.00050 D 545, each of length D 1447/545 D 2.65 m.
In practice, the standard length of 2.44 m would be adopted with 1447/2.44 D 594 tubes in a single pass. (b) Countercurrent flow
In this case, 00061 D 845 0004 665 D 180 deg K, 00062 D 675 0004 495 D 180 deg K, and 0006m D 180 deg K In equation 9.1: A D 417.4 ð 103 / 30 ð 180 D 77.3 m2 and total length of tubing D 77.3/0.1005 D 769 m. With a velocity of 15 m/s, each tube would be 769/545 D 1.41 m long. A better arrangement would be the use of 769/2.44 D 315 tubes, 2.44 m long, though this would give a higher velocity and hence an increased air side pressure drop. With such an arrangement, 315 ð 32 mm o.d. tubes could be accommodated in a 838 mm i.d. shell on 40 mm triangular pitch.
(c) Cross flow
As in (b), 0006m D 180 deg K. From equation 9.213: X D t2 0004 t1 / T1 0004 t1 D 665 0004 495000b/ 845 0004 495 D 0.486 and:
Y D T1 0004 T2 / t2 0004 t1 D 845 0004 675000b/ 665 0004 495 D 1.0
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Thus, assuming one shell pass, two tube-side passes, from Figure 9.71: F D 0.82 and 0006m F D 0.82 ð 180 D 147.6 K Thus, in equation 9.212: A D 417.4 ð 103 / 30 ð 147.6 D 94.3 m2 and: total length of tubing D 94.3/0.1005 D 938 m. Using standard tubes 2.44 m long, number of tubes D 938/2.44 D 384 or 384/2 D 192 tubes/pass. The cross-sectional area for flow would then be 192 ð 0.00050 D 9.61 ð 1000042 m2 and the air velocity D 4.03/9.61 ð 1000042 D 41.9 m/s. This is not excessive providing the minimum acceptable pressure drop is not exceeded. The nearest standard size is 390 ð 32 mm o.d. tubes, 2.44 m in a 940 mm i.d. shell arranged on 40 mm triangular pitch in two passes.
PROBLEM 9.45 A large block of material of thermal diffusivity DH D 0.0042 cm2 /s is initially at a uniform temperature of 290 K and one face is raised suddenly to 875 K and maintained at that temperature. Calculate the time taken for the material at a depth of 0.45 m to reach a temperature of 475 K on the assumption of unidirectional heat transfer and that the material can be considered to be infinite in extent in the direction of transfer.
Solution This problem is identical to Problem 9.2 except for slight variations in temperature, and reference may be made to that solution.
PROBLEM 9.46 A 50% glycerol–water mixture is flowing at a Reynolds number of 1500 through a 25 mm diameter pipe. Plot the mean value of the heat transfer coefficient as a function of pipe length, assuming that: Nu D 1.62 Re Pr d/l000b0.33 . Indicate the conditions under which this is consistent with the predicted value Nu D 4.1 for fully developed flow.
Solution For 50% glycerol–water at, say, 290 K: 0018 D 0.007 N s/m2 , k D 0.415 W/m K and Cp D 3135 J/kg K. ∴ ∴
h ð 0.025000b/0.415 D 1.62[ 1500 ð 3135 ð 0.007/0.415 0.025/l000b]0.33 h D 26.89 1983/l000b0.33 D 330/l0.33 W/m2 K
h is plotted as a function of l over the range l D 0–10 m in Fig. 9d. When Nu D 4.1: h D 4.1k/d D 4.1 ð 0.415000b/0.025 D 68.1 W/m2 K. Taking this as a point value, l D 330/68.1000b3 D 113.8 m
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Mean heat transfer coefficient (h W/m2 K)
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
330
250
200
150 0
1
2
3
4 5 6 7 Pipe length (l m)
8
9
10
Figure 9d.
which would imply that the flow is fully developed at this point. For further discussion on this point reference should be made to the turbulent flow of gases in Section 9.4.3.
PROBLEM 9.47 A liquid is boiled at a temperature of 360 K using steam fed at 380 K to a coil heater. Initially the heat transfer surfaces are clean and an evaporation rate of 0.08 kg/s is obtained from each square metre of heating surface. After a period, a layer of scale of resistance 0.0003 m2 K/W, is deposited by the boiling liquid on the heat transfer surface. On the assumption that the coefficient on the steam side remains unaltered and that the coefficient for the boiling liquid is proportional to its temperature difference raised to the power of 2.5, calculate the new rate of boiling.
Solution When the surface is clean, taking the wall and the inside resistances as negligible, the surface temperature will be 380 K. Thus:
Q D h0 A Ts 0004 T000b
where Q D GL, G kg/s is the rate of evaporation of fluid of latent heat L J/kg, A D 1 m2 , and Ts and T are the surface and fluid temperature respectively. ∴
0.08L D ho ð 1.0 380 0004 360 or ho D 0.004L ho / Ts 0004 T000b2.5
or:
ho D k 0 380 0004 360000b2.5 D 1.79 ð 103 k 0
∴
k 0 D 0.004L/ 1.79 ð 103 D 2.236 ð 1000046 L
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When the scale has formed, the total resistance is: 0.0003 C 1/[2.236 ð 1000046 L Ts 0004 360000b2.5 ] D 0.0003 C 4.472 ð 105 L 00041 Ts 0004 360000b00042.5 For conduction through the scale: GL D 380 0004 Ts /0.0003 D 3.33 ð 103 380 0004 Ts
(i)
For transfer through the outside film: GL D t 0004 360000b/[4.472 ð 105 L 00041 Ts 0004 360000b00042.5 ] 0004 2.236 ð 1000046 L Ts 0004 360000b3.5
(ii)
and for overall transfer: GL D 380 0004 360000b/[0.0003 C 4.472 ð 105 L 00041 Ts 0004 360000b00042.5 ]
(iii)
Inspection of these equations shows that the rate of evaporation G is a function not only of the surface temperature Ts but also of the latent heat of the fluid L. Using equations (i) and (ii) and selecting values of T in the range 360 to 380 K, the following results are obtained: Surface temperature Ts (K)
Mass rate of evaporation G (kg/s)
Latent heat of vaporisation L (kJ/kg)
362 364 366 368 370 372 374 376 378 380
0.000025 0.00029 0.0012 0.0033 0.0071 0.013 0.023 0.036 0.055 0.080
2,400,000 186,000 39,600 12,200 4710 1990 869 364 121 0
At a boiling point of 360 K it is likely that the liquid is organic with a latent heat of, say, 900 kJ/kg. This would indicate a surface temperature of 374 K and an evaporation rate of 0.023 kg/s. A precise result requires more specific data on the latent heat.
PROBLEM 9.48 A batch of reactants of specific heat 3.8 kJ/kg K and of mass 1000 kg is heated by means of a submerged steam coil of area 1 m2 fed with steam at 390 K. If the overall heat transfer coefficient is 600 W/m2 K, calculate the time taken to heat the material from 290 to 360 K if heat losses to the surroundings are neglected. If the external area of the vessel is 10 m2 and the heat transfer coefficient to the surroundings at 290 K is 8.5 W/m2 K, what will be the time taken to heat the reactants over the same temperature range and what is the maximum temperature to which the reactants can be raised? What methods would you suggest for improving the rate of heat transfer?
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Solution Use is made of equation 9.209: ln[ Ts 0004 T1 ]/ Ts 0004 T2 D UAt/GCp ∴
ln[ 390 0004 290000b/ 390 0004 360000b] D 600 ð 1.0t/ 1000 ð 3.8 ð 103 ln 3.33 D 0.000158t and t D 7620 s 2.12 h000b
or:
The heat lost from the vessel: QL D hAv T 0004 Ta , where Ta is the ambient temperature. ∴
QL D 8.5 ð 10.0 T 0004 290 D 85.0T 0004 24650 W
Heat from the steam D heat to the reactants C heat losses ∴
UA Ts 0004 T D GCp dT/dt C 85.0T 0004 24650 600 ð 1.0 390 0004 T D 1000 ð 3.8 ð 103 dT/dt C 85.0T 0004 24650 0004 t 0004 T2 dt D 5548 dT/ 3777.6 0004 T 0
∴
T1
t D 5548 ln[ 377.6 0004 T1 / 337.6T2 ] D 5548 ln[ 377.6 0004 290000b/ 377.6 0004 360000b] D 8904 s 2.47 h000b
The maximum temperature of the reactants is attained when the heat transferred from the steam is equal to the heat losses, or: UA Ts 0004 T D hAv T 0004 Ta Thus:
600 C 1.0 390 0004 T D 8.5 ð 10.0 T 0004 290 and T D 378 K
The heating-up time could be reduced by improving the rate of heat transfer to the fluid, by agitation of the fluid for example, and by reducing heat losses from the vessel by insulation. In the case of a large vessel there is a limit to the degree of agitation and circulation of the fluid through an external heat exchanger is an attractive alternative.
PROBLEM 9.49 What do you understand by the terms “black body” and “grey body” when applied to radiant heat transfer? Two large parallel plates with grey surfaces are situated 75 mm apart; one has an emissivity of 0.8 and is at a temperature of 350 K and the other has an emissivity of 0.4 and is at a temperature of 300 K. Calculate the net rate of heat exchange by radiation per square metre taking the Stefan–Boltzmann constant as 5.67 ð 1000048 W/m2 K4 . Any formula (other than Stefan’s law) which you use must be proved.
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Solution The terms “black body” and “grey body” are discussed in Sections 9.5.2 and 9.5.3. For two large parallel plates with grey surfaces, the heat transfer by radiation between them is given by putting A1 D A2 in equation 150 to give: q D [e1 e2 9/ e1 C e2 0004 e1 e2 ] T41 0004 T42 W/m2 In this case: q D [ 0.8 ð 0.4 ð 5.67 ð 1000048 / 0.8 C 0.4 0004 0.8 ð 0.4000b] 3504 0004 3004 D 0.367 ð 5.67 ð 1000048 ð 6.906 ð 109 D 143.7 W/m2
PROBLEM 9.50 A longitudinal fin on the outside of a circular pipe is 75 mm deep and 3 mm thick. If the pipe surface is at 400 K, calculate the heat dissipated per metre length from the fin to the atmosphere at 290 K if the coefficient of heat transfer from its surface may be assumed constant at 5 W/m2 K. The thermal conductivity of the material of the fin is 50 W/m K and the heat loss from the extreme edge of the fin may be neglected. It should be assumed that the temperature is uniformly 400 K at the base of the fin.
Solution The heat lost from the fin is given by equation 9.254: Qf D hbkA000b00061 tan hmL where h is the coefficient of heat transfer to the surroundings D 5 W/m2 K, b is the fin perimeter D 2 ð 0.075 C 0.003 D 0.153 m, k is the thermal conductivity of the fin D 50 W/mK, A is the cross-sectional area of the fin D 0.003 ð 1.0 D 0.003 m2 , 00061 is pthe temperature p difference at the root D T1 0004 TG D 400 0004 290 D 100 deg K, m D hb/kA D
5 ð 0.153000b/ 50 ð 0.003 D 2.258 and L is the length of the fin D 0.075 m. ∴ Qf D 5 ð 0.153 ð 50 ð 0.003000b[110 tanh 2.258 0004 0.075000b] D 0.339 ð 100 tanh 0.169 D 6.23 W/m
PROBLEM 9.51 Liquid oxygen is distributed by road in large spherical insulated vessels, 2 m internal diameter, well lagged on the outside. What thickness of magnesia lagging, of thermal conductivity 0.07 W/m K, must be used so that not more than 1% of the liquid oxygen evaporates during a journey of 10 ks (2.78 h) if the vessel is initially 80% full? Latent heat of vaporisation of oxygen D 215 kJ/kg. Boiling point of oxygen D 90 K. Density of liquid oxygen D 1140 kg/m3 . Atmospheric temperature D 288 K. Heat transfer coefficient from the outside surface of the lagging surface to atmosphere D 4.5 W/m2 K.
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Solution For conduction through the lagging: Q D 4$k T1 0004 T2 / 1/r1 0004 1/r2
(equation 9.25)
where T1 will be taken as the temperature of boiling oxygen D 90 K and the tank radius, r1 D 1.0 m. In this way, the resistance to heat transfer in the inside film and the walls is neglected. r2 is the outer radius of the lagging. ∴
Q D 4$ ð 0.07 90 0004 T2 / 1/1.0 0004 1/r2 W
(i)
For heat transfer from the outside of the lagging to the surroundings, Q D hA T2 0004 Ta where h D 4.5 W/m2 K, A D 4$r22 and Ta , ambient temperature D 288 K. ∴
Q D 4.5 ð 4$r22 T2 0004 288 D 18$r22 T2 0004 288 W
(ii)
The volume of the tank D 4$r13 /3 D 4$ ð 1.03 /3 D 4.189 m3 . ∴
volume of oxygen D 4.189 ð 80/100 D 3.351 m3 and mass of oxygen D 3.351 ð 1140 D 3820 kg
∴
mass of oxygen which evaporates D 3820 ð 1/100 D 38.2 kg 38.2/ 10 ð 103 D 0.00382 kg/s
or: ∴ heat flow into vessel: ∴ In (ii)
Q D 215 ð 103 ð 0.00382 D 821 W 821 D 18$r22 T2 0004 288 and T2 D 288 0004 14.52/r22
Substituting in (i): 821 D 4$ ð 0.07[90 0004 288 C 14.52/r22 ]/ 1 0004 1/r2 or:
r22 0004 1.27r2 C 0.0198 D 0 and r2 D 1.25 m
Thus the thickness of lagging D r2 0004 r1 D 0.25 m.
PROBLEM 9.52 Benzene is to be condensed at the rate of 1.25 kg/s in a vertical shell and tube type of heat exchanger fitted with tubes of 25 mm outside diameter and 2.5 m long. The vapour condenses on the outside of the tubes and the cooling water enters at 295 K and passes through the tubes at 1.05 m/s. Calculate the number of tubes required if the heat exchanger is arranged for a single pass of the cooling water. The tube wall thickness is 1.6 mm.
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Solution Preliminary calculation
At 101.3 kN/m2 , benzene condenses at 353 K at which the latent heat D 394 kJ/kg. ∴
heat load: Q D 1.25 ð 394 D 492 kW
The maximum water outlet temperature to minimise scaling is 320 K and a value of 300 K will be selected. Thus the water flow is given by: 492 D G ð 4.18 300 0004 295 or: ∴
G D 23.5 kg/s [or 23.5/1000 D 0.0235 m3 /s] area required for a velocity of 1.05 m/s D 0.0235/1.05 D 0.0224 m2
The cross-sectional area of a tube of 25 0004 2 ð 1.6 D 21.8 mm i.d. is:
$/4 ð 0.02182 D 0.000373 m2 and hence number of tubes required D 0.0224/0.000373 D 60 tubes. The outside area D $ ð 0.025 ð 2.5 ð 60 D 11.78 m2 00061 D 353 0004 295 D 58 deg K,
00062 D 353 0004 300 D 53 deg K
and in equation 9.9: 0006m D 58 0004 53000b/ ln 58/53 D 55.5 deg K ∴
U D 492/ 55.5 ð 11.78 D 0.753 kW/m2 K
This is quite reasonable as it falls in the middle of the range for condensing organics as shown in Table 9.17. It remains to check whether the required overall coefficient will be attained with this geometry. Overall coefficient
Inside: The simplified equation for water in tubes may be used: hi D 4280 0.00488T 0004 1000bu0.8 /di0.2 W/m2 K.
(equation 9.221)
where T D 0.5 300 C 295 D 297.5 K u D 105 m/s and di D 0.0218 m ∴
hi D 4280 0.00488 ð 297.5 0004 1000b1.050.8/0.2180.2 D 4322 W/m2 K or 4.32 kW/m2 K Based on the outside diameter: hio D 4.32 ð 0.218/0.025 D 3.77 kW/m2 K
Wall: For steel, k D 45 W/m K, x D 0.0016 m and hence: x/k D 0.0016/45 D 0.000036 m2 K/W
or
0.036 m2 K/kW
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Outside: For condensation on vertical tubes: ho 00182 /k 3 00192 g000b0.33 D 1.47 4M/0018000b00040.33
(equation 9.174)
The wall temperature is approximately 0.5 353 C 297.5 D 325 K, and the benzene film temperature will be taken as 0.5 353 C 325 D 339 K. At 339 K: k D 0.15 W/m K, 0019 D 880 kg/m3 , and 0018 D 0.35 ð 1000043 Ns/m2 . With 60 tubes, the mass flow of benzene per tube, G0 D 1.25/60 D 0.0208 kg/s. For vertical tubes, M D G0 /$do D 0.0208/ $ ð 0.025 D 0.265 kg/ms ∴ ho [ 0.35 ð 1000043 2 /0.152 ð 8802 ð 9.8]0.33 D 1.47[4 ð 0.0208/ 0.35 ð 1000043 ]00040.33 ∴
1.699 ð 1000044 ho D 1.47 ð 1.62 ð 1000041 ho D 1399 W/m2 K or 1.40 kW/m2 K
and: Overall: Neglecting scale resistances:
1/U D 1/hio C x/k C 1/ho D 0.265 C 0.036 C 0.714 D 1.015 m2 K/kW and:
U D 0.985 kW/m2 K
This is in excess of the value required and would allow for a reasonable scale resistance. If this were negligible, the water throughput could be reduced. On the basis of these calculations, 60 tubes are required.
PROBLEM 9.53 One end of a metal bar 25 mm in diameter and 0.3 m long is maintained at 375 K and heat is dissipated from the whole length of the bar to surroundings at 295 K. If the coefficient of heat transfer from the surface is 10 W/m2 K, what is the rate of loss of heat? Take the thermal conductivity of the metal as 85 W/m K.
Solution Use is made of equation 9.254: Qf D
hbkA000b00061 tanh mL
where the coefficient of heat transfer from the surface, h D 10 W/m2 K; the perimeter, b D $ ð 0.025 C 0.0785 m; the cross-sectional area, A D $/4 ð 0.0252 D 0.000491 m2 ; the thermal conductivity of the metal, k D 85 W/m K; the ptemperature difference at the root, 00061 D 375 0004 295 D 80 deg K; the value of m D hb/kA D p [ 10 ð 0.0785000b/ 85 ð 0.000491000b] D 4.337, and the length of the rod, L D 0.3 m. ∴ Qf D 10 ð 0.0785 ð 85 ð 0.000491000b[80 tanh 4.337 ð 0.3000b]
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HEAT TRANSFER
D
185
0.0328 80 tanh 1.3011000b
D 14.49 e1.301 0004 e00041.301 / e1.301 C e00041.301 D 14.49 3.673 0004 0.272000b/ 3.673 C 0.272 D 12.5 W
PROBLEM 9.54 A shell-and-tube heat exchanger consists of 120 tubes of internal diameter 22 mm and length 2.5 m. It is operated as a single-pass condenser with benzene condensing at a temperature of 350 K on the outside of the tubes and water of inlet temperature 290 K passing through the tubes. Initially there is no scale on the walls, and a rate of condensation of 4 kg/s is obtained with a water velocity of 0.7 m/s through the tubes. After prolonged operation, a scale of resistance 0.0002 m2 K/W is formed on the inner surface of the tubes. To what value must the water velocity be increased in order to maintain the same rate of condensation on the assumption that the transfer coefficient on the water side is proportional to the velocity raised to the 0.8 power, and that the coefficient for the condensing vapour is 2.25 kW/m2 K, based on the inside area? The latent heat of vaporisation of benzene is 400 kJ/kg.
Solution Area for heat transfer, based on the tube i.d. D $ ð 0.022 ð 1.0 D 0.0691 m2 /m or:
120 ð 2.5 ð 0.0691 D 20.74 m2 .
With no scale
Heat load: Q D 4 ð 400 D 1600 W. Cross-sectional area of one tube D $/4000b0.0222 D 0.00038 m2 and hence area for flow per pass D 120 ð 0.00038 D 0.0456 m2 . ∴
and:
volume of flow of water D 0.0456 ð 0.7 D 0.0319 m3 /s mass flow of water D 0.0319 ð 1000 D 31.93 kg/s
The water outlet temperature is given by, 1600 D 31.93 ð 4.18 T 0004 290 or T D 302 K 00061 D 350 0004 290 D 60 deg K, 00062 D 350 0004 302 D 48 deg K and in equation 9.9, 0006m D 60 0004 48000b/ ln 60/48 D 53.8 deg K. In equation 9.1, U D Q/A0006m D 1600/ 20.74 ð 53.8 D 1.435 kW/m2 K Neglecting the wall resistance, 1/U D 1/hi C 1/h0i
1/1.435 D 1/hi C 1/2.25 and hi D 3.958 kW/m2 K hi is proportional to u0.8 or 3.958 D k 0.7000b0.8 and k D 5.265
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With scale
hi D 5.265u0.8 kW/m2 K, scale resistance D 0.20 m2 K/kW 1/U D 1/ 5.265u0.8 C 0.20 C 1/2.25000b
and: ∴
U D u0.8 / 0.190 C 0.644u0.8 kW/m2 K
that is:
Q D 1600 kW as before.
The mass flow of water is: u ð 0.0456 ð 1000 D 45.6u kg/s and the outlet water temperature is given by: 1600 D 45.6u ð 4.18 T 0004 290 or:
T D 290 C 8.391/u K 00061 D 350 0004 290 D 60 deg K, 00062 D 350 0004 290 0004 8.391000b/u D 60 0004 8.391000b/u
and: 0006m D 60 0004 60 0004 8.391000b/u000b/ ln[60/ 60 0004 8.391000b/u] D 8.391/fu ln[60u/ 60u 0004 8.391000b]g In equation 9.1: 1600 D [u0.8 / 0.190 C 0.644u0.8 ] ð 20.74 ð 8.391/fu ln[60u/ 60u 0004 8.391000b]g or:
1/fu0.2 0.190 C 0.644u0.8 ln[60u/ 60u 0004 8.391000b]g D 9.194
The left-hand side of this equation is plotted against u in Fig. 9e and the function equals 9.194 when u D 2.06 m/s.
9.2
9.194
9.0
Function
8.8 8.6 8.4 2.06
8.2 8.0 0.8
1.0 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Water velocity (u m/s)
Figure 9e.
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PROBLEM 9.55 Derive an expression for the radiant heat transfer rate per unit area between two large parallel planes of emissivities e1 and e2 and at absolute temperatures T1 and T2 respectively. Two such planes are situated 2.5 mm apart in air. One has an emissivity of 0.1 and is at a temperature of 350 K, and the other has an emissivity of 0.05 and is at a temperature of 300 K. Calculate the percentage change in the total heat transfer rate by coating the first surface so as to reduce its emissivity to 0.025. Stefan–Boltzmann constant D 5.67 ð 1000048 W/m2 K4 . Thermal conductivity of air D 0.026 W/m K.
Solution The theoretical derivation is laid out in Section 9.5.5 and the heat transfer by radiation is given by putting A1 D A2 in equation 9.150 to give: qr D [ e1 e2 9000b/ e1 C e2 0004 e1 e2 ] T41 0004 T42 For conduction between the two planes: (equation 9.12)
qc D kA T1 0004 T2 /x D 0.026 ð 1.0 350 0004 300000b/0.0025 D 520 W/m
2
For radiation between the two planes: qr D [ e1 e2 9000b/ e1 C e2 0004 e1 e2 ] T41 0004 T42 D [ 0.1 ð 0.05 ð 5.67 ð 1000048 / 0.1 C 0.05 0004 0.1 ð 0.05000b] 3504 0004 3004 D 13.5 W/m2 Thus neglecting any convection in the very narrow space, the total heat transferred is 533.5 W/m2 . When e1 D 0.025, the heat transfer by radiation is: qr D [ 0.025 ð 0.05 ð 5.67 ð 1000048 / 0.025 C 0.05 0004 0.025 ð 0.05000b] ð 3504 0004 3004 D 6.64 W/m2 and: qr C qc D 526.64 W/m2 Thus, although the heat transferred by radiation is reduced to 100 ð 6.64000b/13.5 D 49.2% of its initial value, the total heat transferred is reduced to 100 ð 526.64000b/533.5 D 98.7% of the initial value.
PROBLEM 9.56 Water flows at 2 m/s through a 2.5 m length of a 25 mm diameter tube. If the tube is at 320 K and the water enters and leaves at 293 and 295 K respectively, what is the value of the heat transfer coefficient? How would the outlet temperature change if the velocity was increased by 50%?
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Solution The cross-sectional area of 0.025 m tubing D $/4000b0.0252 D 0.000491 m2 . Volume flow of water D 2 ð 0.000491 D 0.000982 m3 /s Mass flow of water D 1000 ð 0.000982 D 0.982 kg/s ∴
Heat load, Q D 0.982 ð 4.18 295 0004 293 D 8.21 kW
Surface area of 0.025 m tubing D $ ð 0.025 ð 1.0 D 0.0785 m2 /m A D 0.0785 ð 2.5 D 0.196 m2
and:
00061 D 320 0004 293 D 27 deg K, 00062 D 320 0004 295 D 25 deg K 0006m D 27 0004 25000b/ ln 27/25 D 25.98 say 26 deg K
and:
In equation 9.1: U D 8.21/ 0.196 ð 26 D 1.612 kW/m2 K An estimate may be made of the inside film coefficient from equation 9.221, where T, the mean water temperature, is 294 K. Thus:
hi D 4280 0.00488 ð 294 0004 1000b2.00.8 /0.0250.2 D 4280 ð 0.435 ð 1.741/0.478 D 6777 W/m2 K or 6.78 kW/m2 K
The scale resistance is therefore given by:
1/1.612 D 1/6.78 C R or: R D 0.473 m2 K/kW With a water velocity of 2.0 ð 150/100 D 3.0 m/s, assuming a mean water temperature of 300 K, then: hi D 4280 0.00488 ð 300 0004 1000b3.00.8 /0.0250.2 D 4280 ð 0.464 ð 2.408/0.478 D 10004 or 10.0 kW/m2 K ∴
1/U D 0.473 C 1/10.0 and U D 1.75 kW/m2 K
For an outlet water temperature of T K: 00061 D 27 deg K, and, taking an arithmetic mean:
00062 D 320 0004 T deg K
0006m D 0.5 27 C 320 0004 T D 173.5 0004 0.5T deg K.
The mass flow of water D 0.982 ð 150000b/100 D 1.473 kg/s, and the heat load, ∴
from which:
Q D 1.473 ð 4.18 T 0004 293 D 6.157T 0004 1804 kW
6.157T 0004 1804 D [1.75 ð 0.196 173.5 0004 0.5T000b] T D 294.5 K
The use of 300 K as a mean water temperature has a minimal effect on the result and recalculation is not necessary.
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PROBLEM 9.57 A liquid hydrocarbon is fed at 295 K to a heat exchanger consisting of a 25 mm diameter tube heated on the outside by condensing steam at atmospheric pressure. The flowrate of hydrocarbon is measured by means of a 19 mm orifice fitted to the 25 mm feed pipe. The reading on a differential manometer containing hydrocarbon-over-water is 450 mm and the coefficient of discharge of the meter is 0.6. Calculate the initial rate of rise of temperature (deg K/s) of the hydrocarbon as it enters the heat exchanger. The outside film coefficient D 6.0 W/m2 K. The inside film coefficient h is given by: hd/k D 0.023 ud0019/0018000b0.8 Cp 0018/k000b0.4 where: u D linear velocity of hydrocarbon (m/s). d D tube diameter (m), 0019 D liquid density
800 kg/m3 , 0018 D liquid viscosity 9 ð 1000044 N s/m2 , Cp D specific heat of liquid (1.7 ð 103 J/kgK), and k D thermal conductivity of liquid (0.17 W/mK).
Solution The effective manometer fluid density, is 200 kg/m3 . The pressure difference across the orifice D 450 mm water or: that is:
450 ð 800/200 D 1800 mm hydrocarbon H D 1.80 m
The area of the orifice D $/4000b0.0192 D 2.835 ð 1000044 m2 In equation 6.21: G D 0.6 ð 2.835 ð 1000044 ð 800 2 ð 9.81 ð 1.80 D 1.36 35.3 D 0.808 kg/s The volume flow D 0.808/800 D 0.00101 m3 /s. Cross-sectional area of a 0.025 m diameter pipe D $/4000b0.0252 D 0.000491 m2 and hence the velocity, u D 0.00101/0.000491 D 2.06 m/s. The inside film coefficient is given by:
hi ð 0.025/0.17 D 0.023
2.06 ð 0.025 ð 800000b/9 ð 1000044 0.8 ð
1.7 ð 103 ð 9 ð 1000044 /0.17000b0.4 or:
hi D 0.1564 4.58 ð 104 0.8 9.0000b0.4 D 2016 W/m2 K or 2.02 kW/m2 K
Neglecting scale and wall resistances: 1/U D 1/6.0 C 1/2.02 and U D 1.511 kW/m2 K For steam at atmospheric pressure, the saturation temperature D 373 K and at the inlet the temperature driving force D 373 0004 295 D 78 deg K.
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The heat flux is: 1.511 ð 78 D 117.9 kW/m2 . For a small length of tube, say 0.001 m, the area for heat transfer D $ ð 0.025 ð 0.001 D 7.854 ð 1000045 m2 and the heat transfer rate D 117.9 ð 7.854 ð 1000045 ð 1000 D 9.27 W. In the small length (0.001 m) of tube, mass of material D 0.000491 ð 0.001 ð 800 D 3.93 ð 1000044 kg and hence temperature rise D [9.27/ 3.93 ð 1000044 ð 1.7 ð 103 ] D 13.9 deg K/s
PROBLEM 9.58 Water passes at a velocity of 1.2 m/s through a series of 25 mm diameter tubes 5 m long maintained at 320 K. If the inlet temperature is 290 K, at what temperature would it leave?
Solution Assuming an outlet water temperature of T K, the mean water temperature is therefore: D 0.5 T C 290 D 0.5T C 145 K. The coefficient may be calculated from: h D 4280 0.00488T 0004 1000bu0.8 /d0.2
(equation 9.221)
D 4280[0.00488 0.5T C 145 0004 1]1.20.8 /0.0250.2 D 25.28T 0004 3028.1 W/m2 K Area for heat transfer D $ ð 0.025 ð 5.0 D 0.393 m2 and the heat load, Q D [1.2 $/4000b0.0252 ð 1000 ð 4.18 ð 103 T 0004 290000b] D 2462T 0004 714,045 W Therefore neglecting any scale resistance:
2462T 0004 714,045 D 25.28T 0004 3028.1000b0.393[320 0004 0.5T C 145000b] from which: and:
T2 C 25.98T 0004 101,851 D 0 T D 306.4 K
[An alternative approach is as follows: The heat transferred per unit time in length dL of pipe, D h ð $ ð 0.025dL 320 0004 Tk W where Tk is the water temperature at L m from the inlet. The rate of increase in the heat content of the water is:
$/4 ð 0.0252 ð 1.2 ð 1000 ð 4.18 ð 103 dT D 2462 dT
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The outlet temperature T0 is then given by: 0004 T0 0004 5 dT D 0.0000319h dL 290 320 0004 T 0 ln 320 0004 T0 D ln 30 0004 0.0001595h D 3.401 0004 0.0001595h
or:
At a mean temperature of say 300 K, in equation 9.221: h D 4280 0.00488 ð 300 0004 1000b1.20.8 /0.0250.2 D 4805 W/m2 K Thus: ln 320 0004 T0 D 3.401 0004 0.0001595 ð 4805 T0 D 306.06 K]
and:
PROBLEM 9.59 Heat is transferred from one fluid stream to a second fluid across a heat transfer surface. If the film coefficients for the two fluids are, respectively, 1.0 and 1.5 kW/m2 K, the metal is 6 mm thick (thermal conductivity 20 W/m K) and the scale coefficient is equivalent to 850 W/m2 K, what is the overall heat transfer coefficient?
Solution From equation 9.201: 1/U D 1/ho C xw /kw C R C 1/hi D 1/1000 C 10.006/20 C 1/850 C 1/1500 D 0.001 C 0.00030 C 0.00118 C 0.00067 D 0.00315 m2 K/W ∴
U D 317.5 W/m2 K or 0.318 kW/m2 K
PROBLEM 9.60 A pipe of outer diameter 50 mm carries hot fluid at 1100 K. It is covered with a 50 mm layer of insulation of thermal conductivity 0.17 W/m K. Would it be feasible to use magnesia insulation, which will not stand temperatures above 615 K and has a thermal conductivity of 0.09 W/m K for an additional layer thick enough to reduce the outer surface temperature to 370 K in surroundings at 280 K? Take the surface coefficient of transfer by radiation and convection as 10 W/m2 K.
Solution The solution is presented as Problem 9.8.
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PROBLEM 9.61 A jacketed reaction vessel containing 0.25 m3 of liquid of density 900 kg/m3 and specific heat 3.3 kJ/kg K is heated by means of steam fed to a jacket on the walls. The contents of the tank are agitated by a stirrer rotating at 3 Hz. The heat transfer area is 2.5 m2 and the steam temperature is 380 K. The outside film heat transfer coefficient is 1.7 kW/m2 K and the 10 mm thick wall of the tank has a thermal conductivity of 6.0 W/m K. The inside film coefficient was 1.1 kW/m2 K for a stirrer speed of 1.5 Hz and proportional to the two-thirds power of the speed of rotation. Neglecting heat losses and the heat capacity of the tank, how long will it take to raise the temperature of the liquid from 295 to 375 K?
Solution For a stirrer speed of 1.5 Hz, hi D 1.1 kW/m2 K. ∴
1.1 D k 0 1.50.67 and k 0 D 0.838 Thus at a stirrer speed of 3 Hz, hi D 0.838 ð 3.00.67 D 1.75 kW/m2 K. The overall coefficient is given by: 1/U D 1/1750 C 0.010/6.0 C 1/1700 C 0.00283 (equation 9.201)
and:
U D 353.8 W/m2 K neglecting scale resistances.
The time for heating the liquid is given by: ln[ Ts 0004 T1 / Ts 0004 T2 ] D UAt/mCp
(equation 9.209)
In this case: m D 0.25 ð 900 D 225 kg and Cp D 3300 J/kg K. ∴
ln[ 380 0004 295000b/ 380 0004 375000b] D 353.8 ð 2.5t/ 225 ð 3300 2.833 D 0.00119t and t D 2381 s 40 min000b
PROBLEM 9.62 By dimensional analysis, derive a relationship for the heat transfer coefficient h for natural convection between a surface and a fluid on the assumption that the coefficient is a function of the following variables: k D thermal conductivity of the fluid, Cp D specific heat of the fluid, 0019 D density of the fluid, 0018 D viscosity of the fluid, ˇg D the product of the coefficient of cubical expansion of the fluid and the acceleration due to gravity, l D a characteristic dimension of the surface, and T D the temperature difference between the fluid and the surface. Indicate why each of these quantities would be expected to influence the heat transfer coefficient and explain how the orientation of the surface affects the process. Under what conditions is heat transfer by natural convection important in Chemical Engineering?
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Solution If the heat transfer coefficient h can be expressed as a product of powers of the variables, then: h D k 0 k a Cpb 0019c 0018d ˇg000be lf Tg where k 0 is a constant. The dimensions of each variable in terms of M, L, T, Q, and q are: heat transfer coefficient, h D Q/L2 Tq thermal conductivity, k D Q/LTq specific heat, Cp D Q/Mq viscosity, 0018 D M/LT density, 0019 D M/L3 the product, ˇg D L/T2 q00041 length, lDL temperature difference, T D q Equating indices: M: L: T: Q: q:
0 00042 00041 1 00041
D 0004b C c C d D 0004a 0004 3c 0004 d C e C f D 0004a 0004 d 0004 2e DaCb D 0004a 0004 b 0004 e C g
Solving in terms of b and c: a D 1 0004 b000b, d D b 0004 c000b, e D c/2000b, f D 3c/2 0004 1000b, g D c/2 and hence: h D k0 or:
0002
0003 0002 00030002 0003 0002 0003c 3c/2 k b c 0018b Cp 0018 b l3/2 0019 ˇg000b1/2 T1/2 c/2 l c/2 0 k C 0019
ˇg D k T k b p 0018c l l k 0018 0002 0003b 0002 3 2 0003c/2 Cp 0018 l 0019 ˇgT hl D k0 k k 00182
where Cp 0018/k is the Prandtl number and l3 00192 ˇgT/00182 the Grashof number. A full discussion of the significance of this result and the importance of free of natural convection is presented in Section 9.4.7.
PROBLEM 9.63 A shell-and-tube heat exchanger is used for preheating the feed to an evaporator. The liquid of specific heat 4.0 kJ/kg K and density 1100 kg/m3 passes through the inside of tubes and is heated by steam condensing at 395 K on the outside. The exchanger heats liquid at 295 K to an outlet temperature of 375 K when the flowrate is 1.75 ð 1000044 m3 /s and to 370 K when the flowrate is 3.25 ð 1000044 m3 /s. What is the heat transfer area and the value of the overall heat transfer coefficient when the flow rate is 1.75 ð 1000044 m3 /s?
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Assume that the film heat transfer coefficient for the liquid in the tubes is proportional to the 0.8 power of the velocity, that the transfer coefficient for the condensing steam remains constant at 3.4 kW/m2 K and that the resistance of the tube wall and scale can be neglected.
Solution i) For a flow of 1.75 ð 1000044 m3 /s: Density of the liquid D 1100 kg/m3 Mass flow
D 1.75 ð 1000044 ð 1100 D 0.1925 kg/s.
Heat load
D 0.1925 ð 4.0 373 0004 295 D 61.6 kW 00061 D 395 0004 295 D 100 deg K,
00062 D 395 0004 375 D 20 deg K
and in equation 9.9: 0006m D 100 0004 20000b/ ln 100/20 D 49.7 deg K Thus, in equation 9.1: U1 A D 61.6/49.7 D 1.239 kW/K ii) For a flow of 3.25 ð 1000044 m3 /s: Mass flow D 3.25 ð 1000044 ð 1100 D 0.3575 kg/s Heat load D 0.3575 ð 4.0 370 0004 295 D 107.3 kW 00061 D 395 0004 295 D 100 deg K,
00062 D 395 0004 370 D 25 deg K
and in equation 9.9: 0006m D 100 0004 25000b/ ln 100/25 D 54.1 deg K Thus in equation 9.1: U2 A D 107.3/54.1 D 1.983 kW/K ∴
U2 /U1 D 1.983/1.239 D 1.60
The velocity in the tubes is proportional to the volumetric flowrate, v cm3 /s and hence hi / v0.8 or hi D k 0 v0.8 , where k 0 is a constant. Neglecting scale and wall resistances: 1/U D 1/h0 C 1/hi D 1/3.4 C 1/k 0 v0.8 and: U D 3.4k 0 v0.8 / 3.4 C k 0 v0.8
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∴
195
U1 D 3.4k 0 ð 1750.8 / 3.4 C k 0 ð 1750.8 D 211.8k 0 / 3.4 C 62.3k 0
and: U2 D 3.4k 0 ð 3250.8 / 3.4 C k 0 ð 3250.8 D 347.5k 0 / 3.4 C 102.2k 0 ∴ ∴ ∴
[347.5k 0 / 3.4 C 102.2k 0 ]/[211.18k 0 / 3.4 C 62.3k 0 ] D 1.60 k 0 D 0.00228 U1 D 3.4 ð 0.00228 ð 1750.8 / 3.4 C 0.00228 ð 1750.8 D 0.136 kW/m2 K
and the heat transfer area, A D 1.239/0.136 D 9.09 m2 .
PROBLEM 9.64 0.1 m3 of liquid of specific heat capacity 3 kJ/kg K and density 950 kg/m3 is heated in an agitated tank fitted with a coil, of heat transfer area 1 m2 , supplied with steam at 383 K. How long will it take to heat the liquid from 293 to 368 K, if the tank, of external area 20 m2 is losing heat to the surroundings at 293 K? To what temperature will the system fall in 1800 s if the steam is turned off? Overall heat transfer coefficient in coil D 2000 W/m2 K. Heat transfer coefficient to surroundings D 10 W/m2 K.
Solution If T K is the temperature of the liquid at time t s, then: heat input from the steam D UA Ts 0004 T D 2000 ð 1 383 0004 T or 2000 383 0004 T W Similarly, heat losses to the surroundings D 10 ð 20 T 0004 293 D 200 T 0004 293 W and, net heat input to the liquid D 2000 383 0004 T 0004 200 T 0004 293 D 824,600 0004 2200T W This is equal to:
Q D mCp dT/dt000b
where m D 0.1 ð 950 D 95 kg and Cp D 3000 J/kg K. ∴
95 ð 3000000bdT/dt D 824,600 0004 2200T000b
or:
129.6 dT/dt D 374.8 0004 T000b
Thus the time taken to heat from 293 to 368 K is: 0004 368 dT/ 374.8 0004 T t D 129.6 293
D 129.6 ln
374.8 0004 293000b/ 374 0004 368 D 1559 s 0.43 h The steam is turned off for 1800s and, during this time, a heat balance gives:
95 ð 3000000bdT/dt D 0004 10 ð 20 T 0004 293 ∴
285,000 dT/dt D 58600 0004 200T or 1425 dT/dt D 293 0004 T000b
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The change in temperature is then given by: 0004 T 0004 dT/ 293 0004 T D 1/1425 368
1800
dt 0
ln
293 0004 368000b/ 293 0004 T D 1800/1425 D 1.263 and T D 311.8 K .
PROBLEM 9.65 The contents of a reaction vessel are heated by means of steam at 393 K supplied to a heating coil which is totally immersed in the liquid. When the vessel has a layer of lagging 50 mm thick on its outer surfaces, it takes one hour to heat the liquid from 293 to 373 K. How long will it take if the thickness of lagging is doubled? Outside temperature D 293 K. Thermal conductivity of lagging D 0.05 W/mK. Coefficient for heat loss by radiation and convection from outside surface of vessel D 10 W/m2 K. Outside area of vessel D 8 m2 . Coil area D 0.2 m2 . Overall heat transfer coefficient for steam coil D 300 W/m2 K.
Solution If T K is the temperature of the liquid at time t s and T1 K the temperature at the outside surface of the vessel, then heat flowing through the insulation is equal to the heat lost by convection and radiation to the surroundings or:
kA/x T 0004 T1 D hc A T1 0004 T0 where hc is the coefficient for heat loss, A the outside surface area of the vessel and T0 the ambient temperature. Thus:
0.05 ð 8/0.050 T 0004 T1 D 10 ð 8 T1 0004 293 T1 D 0.0909T C 266.4 K
and: ∴ Heat loss to the surroundings
D 10 ð 8 0.0909T C 266.4 0004 293000b
D 7.272T 0004 2128 W Heat input from the coil D 300 ð 0.2 393 0004 T D 23580 0004 60T W and net heat input D 23580 0004 60T 0004 7.272T 0004 2128 D 25708 0004 67.3T W which is equal to: Q D mCp dT/dt or:
mCp dT/dt D 25708 0004 67.3T
and:
0.0149mCp dT/dt D 382 0004 T
It takes t D 3600 s to heat the contents from 293 to 373 K, or: 0004 373 3600 D 0.0149mCp dT/ 382 0004 T 293
∴
and:
241610 D mCp ln
382 0004 293000b/ 382 0004 373 D 2.291mCp mCp D 105442 J/K
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If the thickness of the lagging is doubled to 0.100 m, then:
0.05 ð 8/0.100 T 0004 T1 D 10 ð 8 T1 0004 293 T1 D 0.0476T C 279.1 K
and: ∴ Heat loss to the surroundings
D 10 ð 8 0.0476T C 279.1 0004 293000b
D 3.808T 0004 1112 W Heat input from the coil D 300 ð 0.2 393 0004 T D 23580 0004 60T W and net heat input D 23580 0004 60T 0004 3.808T 0004 1112 D 24692 0004 63.808T. ∴
mCp dT/dt D 24,692 0004 63.808T 105442 dT/dt D 24,692 0004 63.808T000b
or:
1652.5 dT/dt D 387 0004 T
Thus, the time taken to heat the contents from 293 to 373 K is: 0004 373 dT/ 387 0004 T t D 1652.5 293
D 1625.5 ln[ 387 0004 293000b/ 387 0004 373000b] D 1652.5 ð 1.904 D 3147 s 0.87 h000b
PROBLEM 9.66 A smooth tube in a condenser which is 25 mm internal diameter and 10 m long is carrying cooling water and the pressure drop over the length of the tube is 2 ð 104 N/m2 . If vapour at a temperature of 353 K is condensing on the outside of the tube and the temperature of the cooling water rises from 293 K at inlet to 333 K at outlet, what is the value of the overall heat transfer coefficient based on the inside area of the tube? If the coefficient for the condensing vapour is 15,000 W/m2 K, what is the film coefficient for the water? If the latent heat of vaporisation is 800 kJ/kg, what is the rate of condensation of vapour?
Solution From equation 3.23: R/0019u2 Re2 D 0004Pf d3 0019/ 4l/00182 Taking the viscosity of water as 1 mN s/m2 0.001 Ns/m2 , then: 0004Pf d3 0019/ 4l/00182 D 20,000 0.025000b3 1000/ 4 ð 10 0.001000b2 D 7,812,500 From Fig. 3.8, for a smooth pipe, Re D 57,000 ∴
and:
du0019/0018 D 0.025u1000000b/0.001 D 57,000 u D 2.28 m/s
∴ Volume flow of water D $ 0.0252 /4000b2.28 D 0.00112 m3 /s
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Mass flow of water D 1000 ð 0.00112 D 1.12 kg/s Heat removed by water D 1.12 ð 4.187 333 0004 293 D 187.6 kW Surface area of tube, based on inside diameter D $ ð 0.025 ð 10 D 0.785 m2 Vapour temperature = 353 K
∴ T1 D 353 0004 293 D 60 deg K
T2 D 353 0004 333 D 20 deg K and from equation 9.9, Tm D 60 0004 20000b/ ln 60/20 D 36.4 deg K From equation 9.1, 187.6 D U ð 0.785 ð 36.4 and the overall coefficient based on the inside diameter is: U D 6.57 kW/m2 K. In equation 9.201, neglecting the wall and scale resistances: 1/U D 1/ho C 1/hi
1/6.57 D 1/15.0 C 1/hi and hi D 11.68 kW/m2 K If the latent heat of condensation is 800 kJ/kg, then assuming the vapour enters and the condensate leaves at the boiling point: rate of condensation D 187.6/800 D 0.235 kg/s
PROBLEM 9.67 A chemical reactor, 1 m in diameter and 5 m long, operates at a temperature of 1073 K. It is covered with a 500 mm thickness of lagging of thermal conductivity 0.1 W/m K. The heat loss from the cylindrical surface to the surroundings is 3.5 kW. What is the heat transfer coefficient from the surface of the lagging to the surroundings at a temperature of 293 K? How would the heat loss be altered if the coefficient were halved?
Solution From equation 9.20, the heat flow at any radius r is given by: Q D 0004k 2$ C 1000bdT/dr W ∴
dr/r D 00042$kl/Q000bdT
Integrating between the limits r1 and r2 at which the temperatures are T1 and T2 respectively: 0004 r2 0004 T2
dr/r D 00042$kl/Q dT r1
∴
T1
ln r2 /r1 D 00042$kl/Q T1 0004 T2
In this case: r1 D 1/2 D 0.50 m, r2 D 0.50 C 500/1000 D 1.0 m, l D 5 m, k D 0.1 W/mK, Q D 3500 W and T1 D 1073 K. ∴
and:
ln 1.0/0.50 D 2$ ð 0.1 ð 5/3500 1073 0004 T2 0.693 D 0.00090 1073 0004 T2 and T2 D 301 K
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The heat flow to the surroundings is: Q D ho Ao T2 0004 To ∴
3500 D ho $ ð 2.0 ð 5.0 301 0004 293 and ho D 14.05 W/m2 K
If this value is halved, that is ho2 D 7.02 W/m2 K, then: Q2 D 7.02 2$ ð 2.0 ð 5.0 T2 0004 293 D 220.5 T2 0004 293 T2 D Q2 /220.5 C 293 K.
and: But:
Q2 /Q1 D T1 0004 T2 2 / T1 0004 T2 1
∴
Q2 /3500 D [1073 0004 Q2 /220.5 0004 293]/ 1073 0004 301 0.000286Q2 D 0.00130 780 0004 0.00454Q2 Q2 D 3473 W — a very slight reduction in the heat loss.
and:
In this case, T2 D 3473/220.5 C 293 D 308.7 K.
PROBLEM 9.68 An open cylindrical tank 500 mm diameter and 1 m deep is three-quarters filled with a liquid of density 980 kg/m3 and of specific heat capacity 3 kJ/kg K. If the heat transfer coefficient from the cylindrical walls and the base of the tank is 10 W/m2 K and from the surface is 20 W/m2 K, what area of heating coil, fed with steam at 383 K, is required to heat the contents from 288 K to 368 K in a half hour? The overall heat transfer coefficient for the coil may be taken as 100 W/m2 K. The surroundings are at 288 K. The heat capacity of the tank itself may be neglected.
Solution The rate of heat transfer from the steam to the liquid is: Uc Ac 383 0004 T D 100Ac 383 0004 T W where Ac is the surface area of the coil. The rate of heat transfer from the tank to the surroundings D UT AT T 0004 288 where UT is the effective overall coefficient and AT the surface area of the tank and liquid surface. In this case: UT AT D 10
$ ð 0.5 ð 1 C $/4000b0.52 C 20 $/4000b0.52 D 21.6 W/K. ∴ rate of heat loss D 21.6 T 0004 288 W ∴ net rate of heat input to the tank D 100Ac 383 0004 T 0004 21.6 T 0004 288 W.
This is equal to mCp dT/dt, where the mean specific heat, Cp D 3000 J/kgK. Volume of liquid D 75/100 $/4000b0.52 ð 1 D 0.147 m3
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Mass of liquid: m D 0.147 ð 980 D 144.3 kg and: ∴
mCp D 144.3 ð 3000 D 432,957 J/K. 432,957 dT/dt D 100Ac 383 0004 T 0004 21.6 T 0004 288 D 38,300Ac C 6221 0004 100Ac C 21.6000bT 0004 ∴
368
dT/
38,300Ac C 6221000b/ 100Ac C 21.6 0004 T 288
0004
1800
D
100Ac C 21.6000b/432,957000b
dt 0
∴
lnf[ 38,300Ac C 6221000b/ 100Ac C 21.6 0004 288]/ [ 38,300Ac C 6221000b/ 100Ac C 21.6 0004 368000b]g D 0.00416 100Ac C 21.6 This equation is solved by trial and error to give: Ac D 5.0 m2 .
PROBLEM 9.69 Liquid oxygen is distributed by road in large spherical vessels, 1.82 m in internal diameter. If the vessels were unlagged and the coefficient for heat transfer from the outside of the vessel to the atmosphere were 5 W/m2 K, what proportion of the contents would evaporate during a journey lasting an hour? Initially the vessels are 80% full. What thickness of lagging would be required to reduce the losses to one tenth? Atmospheric temperature D 288 K. Boiling point of oxygen D 90 K. Density of oxygen D 1140 kg/m3 . Latent heat of vaporisation of oxygen D 214 kJ/kg. Thermal conductivity of lagging D 0.07 W/m K.
Solution Volume of the vessel Volume of liquid oxygen Mass of liquid oxygen Surface area of unlagged vessel Heat leakage into the vessel
∴ ∴
D $d3 /6 D $ ð 1.823 /6 D 3.16 m3 D 80/100000b3.16 D 2.53 m3 D 2.53 ð 1140 D 2879 kg D $ ð 1.822 D 10.41 m2 D hc A T1 0004 T2 D 5.0 ð 10.41 288 0004 90 D 10,302 W or 10.3 kW
Evaporation rate of oxygen D 10.3/214 D 0.048 kg/s Evaporation taking place during 1 h D 0.048 ð 3600 D 173.3 kg
which is 100 ð 173.3/2879 D 6.02% of the contents In order to reduce the losses to one tenth, the heat flow into the vessel must be 1.03 kW and this will be achieved by reducing the temperature driving force to:
288 0004 90000b/10 D 19.8 deg K.
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In this case the outside temperature of the lagging will be 288 0004 19.8 D 268.2 K and the temperature drop through the lagging will be 268.2 0004 90 D 178.2 deg K. Thus, the heat flow through the lagging is: 1030 D kA/x000bTlagging D 0.07 ð 10.41/x000b178.2 from which the thickness of the lagging, x D 0.126 m or 126 mm This calculation does not take into account the increase in the surface area at the lagging surface since it was assumed to be that of the tank, 10.41 m2 . In practice, it will be larger than this and, if this is taken into account, the reasoning is as follows: Radius of the tank D 1.82/2 D 0.91 m ∴ For a lagging thickness of x m, the new radius is 0.91 C x000bm and the surface area is: 4$ 0.91 C x000b2 m2 ∴ convective heat gain D 5.0 ð 4$ 0.91 C x000b2 288 0004 T D 1030 W and the outside temperature of the lagging:
T D 288 0004 16.39/ 0.91 C x000b2 K
(i)
The heat flow through the lagging (taking an arithmetic mean area and neglecting the curvature), 1030 D 0.07/x000b4$ 0.91 C x/2000b2 T 0004 90 Substituting for T from (i) into (ii): 1030 D 0.07/x000b4$ 0.91 C x/2000b2 198 0004 16.39000b/ 0.91 C x000b2 Solving by trial and error: x D 0.151 m or 151 mm
PROBLEM 9.70 Water at 293 K is heated by passing it through a 6.1 m coil of 25 mm internal diameter pipe. The thermal conductivity of the pipe wall is 20 W/m K and the wall thickness is 3.2 mm. The coil is heated by condensing steam at 373 K for which the film coefficient is 8 kW/m2 K. When the water velocity in the pipe is 1 m/s, its outlet temperature is 309 K. What will the outlet temperature be if the velocity is increased to 1.3 m/s, if the coefficient of heat transfer to the water in the tube is proportional to the velocity raised to the 0.8 power?
Solution The surface area of the coil D $do l D $
25 C 2 ð 3.2000b/1000000b6.1 D 0.602 m2 i) When the water velocity is 1 m/s: Area for flow D $d2i /4 D $ 25/1000000b2 /4 D 0.00049 m2 Volume flow D 0.00049 ð 1.0 D 0.00049 m3 /s
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Taking the density as 1000 kg/m3 , Mass flow of water D 1000 ð 0.00049 D 0.491 kg/s and taking the mean specific heat as 4.18 kJ/kg K, Heat load D 0.491 ð 4.18 309 0004 293 D 32.83 kW With steam at 373 K,
T1 D 373 0004 293 D 80 deg K, T2 D 373 0004 309 D 64 deg K
and from equation 9.9, Tm D 80 0004 64000b/ ln 80/64 D 71.7 deg K Therefore from equation 9.1, the overall coefficient, U D 32.83/ 0.602 ð 71.7 D 0.761 kW/m2 K or 761 W/m2 K. From equation 9.201, and neglecting any scale resistance: 1/U D 1/hi C 1/h0 C x/k In this case, ho D 8 kW/m2 K D 8000 W/m2 K, k D 20 W/mK, x D 3.2 mm or 0.0032 m and hi D Ku0.8 where u D 1 m/s and K is a constant. ∴
1/761 D 1/K10.8 C 1/8000 C 0.0032/20 0.00131 D 1/K C 0.000125 C 0.00016 and K D 976
ii) When the velocity is 1.3 m/s Volume flow of water D 0.00049 ð 1.3 D 0.000637 m3 /s Mass flow of water
D 1000 ð 0.000637 D 0.637 kg/s
∴ Heat load
D 0.637 ð 4.18 T 0004 293 D 2.663 T 0004 293 kW or 2663 T 0004 293 W
The inside coefficient, hi D 976 ð 1.30.8 D 1204 W/m2 K and the overall coefficient, U is given by: 1/U D 1/1204 C 1/8000 C 0.0032/20 ∴
U D 896.4 W/m2 K T1 D 373 0004 293 D 80 deg K and T2 D 373 0004 T deg K.
Thus, from equation 9.9: Tm D 80 0004 373 C T000b/ ln[80/ 373 0004 T000b] D T 0004 293000b/ ln[80/ 373 0004 T000b] deg K and in equation 9.1: 2663 T 0004 293 D 896.4 ð 0.602 T 0004 293000b/ ln[80/ 373 0004 T000b] ln 80/ 373 0004 T D 0.2026 and: T D 307.7 K .
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PROBLEM 9.71 Liquid is heated in a vessel by means of steam which is supplied to an internal coil in the vessel. When the vessel contains 1000 kg of liquid it takes half an hour to heat the contents from 293 to 368 K if the coil is supplied with steam at 373 K. The process is modified so that liquid at 293 K is continuously fed to the vessel at the rate of 0.28 kg/s. The total contents of the vessel are always being maintained at 1000 kg. What is the equilibrium temperature which the contents of the vessel will reach, if heat losses to the surroundings are neglected and the overall heat transfer coefficient remains constant?
Solution Use is made of equation 9.209: ln
Ts 0004 T1 / Ts 0004 T2 D UA/mCp t In this case: Ts D 373 K, T1 D 293 K, T2 D 368 K, m D 1000 kg and t D 0.5 h or 1800 s ∴
ln
373 0004 293000b/ 373 0004 368 D UA/Cp 1800/1000 D 2.773 UA/Cp D 1.54 kg/s
and:
For continuous heating, assuming UA/Cp is constant and losses are negligible then: Q D UA Ts 0004 T D mCp T 0004 T1 where T is the temperature of the contents. ∴
UA 373 0004 T D 0.28Cp T 0004 293000b
UA/Cp 373 0004 T D 0.28T 0004 82.04
Substituting for UA/Cp :
1.54 ð 373 0004 1.54T D 0.28T 0004 82.04 and:
T D 360.7 K
PROBLEM 9.72 The heat loss through a firebrick furnace wall 0.2 m thick is to be reduced by addition of a layer of insulating brick to the outside. What is the thickness of insulating brick necessary to reduce the heat loss to 400 W/m2 ? The inside furnace wall temperature is 1573 K, the ambient air adjacent to the furnace exterior is at 293 K and the natural convection heat transfer coefficient at the exterior surface is given by ho D 3.0T0.25 W/m2 K, where T is the temperature difference between the surface and the ambient air. Thermal conductivity of firebrick D 1.5 W/m K. Thermal conductivity of insulating brick D 0.4 W/m K.
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Solution The conduction through the firebrick is given by: Q D 400 D 1.5 ð 1.0/0.2 1573 0004 T2
(equation 9.12)
and T2 , the temperature at the firebrick/insulating brick interface, is: T2 D 1579.7 K. For the natural convection to the surroundings: Q D ho A T3 0004 Ta 400 D 3.0T0.25 ð 1.0 T3 0004 293000b
or: but T3 0004 293 D T and:
400 D 3.0T1.25 ∴
T D 133.30.8 D 50.1 deg K
and the temperature at the outer surface of the insulating brick, T3 D 293 C 50.1 D 343.1 K. Thus, applying equation 9.12 to the insulating brick: 400 D 0.4 ð 1.0/x 1519.7 0004 343.1 and the thickness of the brick, x D 1.18 m
PROBLEM 9.73 2.8 kg/s of organic liquid of specific heat capacity 2.5 kJ/kg K is cooled in a heat exchanger from 363 to 313 K using water whose temperature rises from 293 to 318 K flowing countercurrently. After maintenance, the pipework is wrongly connected so that the two streams, flowing at the same rates as previously, are now in co-current flow. On the assumption that overall heat transfer coefficient is unaffected, show that the new outlet temperatures of the organic liquid and the water will be 320.6 K and 314.5 K, respectively.
Solution i) Countercurrent flow Heat load, Q D 2.8 ð 2.5 363 0004 313 D 350 kW ∴ water flow D 350/4.18 318 0004 293 D 3.35 kg/s
T1 D 363 0004 318 D 35 deg K, T2 D 313 0004 293 D 20 deg K and, from equation 9.9, Tm D 45 0004 20000b/ ln 45/20 D 30.83 deg K.
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From equation 9.1: 350 D UA ð 30.83 and:
UA D 11.35 kW/K.
ii) Co-current flow Heat load, Q D 2.8 ð 2.5 363 0004 T D 7.0 363 0004 T kW for the organic and for the water, Q D 3.35 ð 4.18 T0 0004 293 D 14.0 T0 kW where T and T0 are the outlet temperatures of the organic and water respectively. ∴
and:
363 0004 T D 14.0/7.0 T0 0004 293 T0 D 474.5 0004 0.5T K. T1 D 363 0004 293 D 70 deg K, T2 D T 0004 T0
and from equation 9.9, Tm D 70 0004 T 0004 T0 / ln[70/ T 0004 T0 ] deg K. In equation 9.1: 70 360 0004 T D 11.35 70 0004 T C T0 / ln[70/ T 0004 T0 ] ∴
7.0 360 0004 T D 11.35 70 0004 T C 474.5 0004 0.5T000b/ ln[70/ T 0004 474.5 C 0.5T000b]
or: 0.617 360 0004 T D 544.5 0004 1.5T000b/ ln[70/ 1.5T 0004 474.5000b] Solving by trial and error, T D 319.8 K which is very close to the value suggested, 320.6 K. ∴ The outlet temperature of the water is: T0 D 474.5 0004 0.5 ð 319.8 D 314.6 K which agrees almost exactly with the given value. Thus for co-current flow: Q D 7.0 363 0004 319.8 D 302.4 kW T1 D 70 deg K (as before), T2 D 319.8 0004 314.6 D 5.2 deg K and from equation 9.9, Tm D 70 0004 5.2000b/ ln 70/5.2 D 24.92 deg K ∴ in equation 9.1: 302.4 D UA ð 24.92 and:
UA D 12.10 kW/K
which is in relatively close agreement with the counter-current value.
PROBLEM 9.74 An organic liquid is cooled from 353 to 328 K in a single-pass heat exchanger. When the cooling water of initial temperature 288 K flows countercurrently its outlet temperature is 333 K. With the water flowing co-currently, its feed rate has to be increased in order to give the same outlet temperature for the organic liquid, the new outlet temperature of the water is 313 K. When the cooling water is flowing countercurrently, the film heat transfer coefficient for the water is 600 W/m2 K.
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What is the coefficient for the water when the exchanger is operating with cocurrent flow if its value is proportional to the 0.8 power of the water velocity? Calculate the film coefficient from the organic liquid, on the assumptions that it remains unchanged, and that heat transfer resistances other than those attributable to the two liquids may be neglected.
Solution i) For countercurrent flow: T1 D 353 0004 333 D 20 deg K and T2 D 328 0004 288 D 40 deg K. ∴ From equation 9.9: Tm D 40 0004 20000b/ ln 40/20 D 28.85 deg K. ii) For co-current flow:
T1 D 353 0004 288 D 65 deg K and T2 D 328 0004 313 D 15 deg K. ∴
Tm D 65 0004 15000b/ ln 65/15 D 34.1 deg K.
Taking countercurrent flow as state 1 and co-current flow as state 2, then, in equation 9.1: Q D U1 ATm1 D U2 ATm2 or:
U1 /U2 D 28.85/34.1 D 0.846
(i)
The water velocity, u / 1/T, where T is the rise in temperature of the water, or: ∴
u D K/T where K is a constant. u1 /u2 D T2 /T1 D 313 0004 288000b/ 333 0004 288 D 0.556.
But: hi / u0.8 or: hi D k 0 u0.8 Thus: ∴
hi1 /hi2 D u1 /u2 0.8 D 0.556000b0.8 D 0.625 600/hi2 D 0.625 and hi2 D 960 W/m2 K .
From equation 9.201, ignoring scale and wall resistances: 1/U D 1/ho C 1/hi ∴
1/U1 D 1/ho C 1/600 and U1 D 600ho / 600 C ho
and
1/U2 D 1/ho C 1/960 and U2 D 960ho / 960 C ho
∴ ∴
U1 /U2 D 600/960 960 C ho / 600 C ho D 0.846 (from (i)) 0.625 960 C ho D 0.846 600 C ho and ho D 417 W/m2 K .
PROBLEM 9.75 A reaction vessel is heated by steam at 393 K supplied to a coil immersed in the liquid in the tank. It takes 1800 s to heat the contents from 293 K to 373 K when the outside
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temperature is 293 K. When the outside and initial temperatures are only 278 K, it takes 2700 s to heat the contents to 373 K. The area of the steam coil is 2.5 m2 and of the external surface is 40 m2 . If the overall heat transfer coefficient from the coil to the liquid in the vessel is 400 W/m2 K, show that the overall coefficient for transfer from the vessel to the surroundings is about 5 W/m2 K.
Solution Using the argument in Section 9.8.3, the net rate of heating is given by: mCp dT/dt D Uc Ac Ts 0004 T 0004 Uo Ao T 0004 To where Uc and Uo are the overall coefficients from the coil and the outside of the vessel respectively, Ac and Ao are the areas of the coil and the outside of the vessel and Ts , T and To are the temperature of the steam, the contents and the surroundings respectively. Writing Uc Ac D a and Uo Ao D b: mCp dT/dt D a 393 0004 T 0004 b T 0004 Ta D 393a C bTa 0004 a C b000bT
000e373 Integrating: t D mCp / a C b ln 1/[393a C bTa 0004 a C b000bT000b] Ta s D mCp / a C b ln[ a 393 0004 Ta / 20a 0004 373b C bTa ] s When Ta D 293 K:
1800 D mCp / a C b ln[ 100a000b/ 20a 0004 80b000b]
(i)
When Ta D 278 K:
2700 D mCp / a C b ln[ 115a000b/ 20a 0004 95b000b]
(ii)
Dividing (i) by (ii):
0.667 D ln[5a/ a 0004 4b000b]/ ln[23a/ 4a 0004 19b000b]
(iii)
But Uc D 400 W/m2 K, Ac D 2.5 m2 and hence: a D Uc Ac D 1000 W/K or 1 kW/K. Substituting in (iii): 1.5 D ln[23/ 4 0004 19b000b]/ ln[5/ 1 0004 4b000b] Solving by trial and error: b D 0.2 kW/K or 200 W/K. ∴
Uo ð 40 D 200 and Uo D 200/40 D 5 W/m2 K .
PROBLEM 9.76 Steam at 403 K is supplied through a pipe of 25 mm outside diameter. Calculate the heat loss per metre to surroundings at 293 K, on the assumption that there is a negligible drop in temperature through the wall of the pipe. The heat transfer coefficient h from the outside of the pipe of the surroundings is given by: h D 1.22 T/d000b0.25 W/m2 K where d is the outside diameter of the pipe (m) and T is the temperature difference (deg K) between the surface and surroundings.
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The pipe is then lagged with a 50 mm thickness of lagging of thermal conductivity 0.1 W/m K. If the outside heat transfer coefficient is given by the same equation as for the bare pipe, by what factor is the heat loss reduced?
Solution For 1 m length of pipe: surface area D $dl D $ 25/1000 ð 1.0 D 0.0785 m2 With a negligible temperature drop through the wall, the wall is at the steam temperature, 403 K, and T D 403 0004 293 D 110 deg K. Thus, the coefficient of heat transfer from the pipe to the surroundings is: h D 1.22[110/ 25/1000000b]0.25 D 9.94 W/m2 K. and the heat loss: Q D hA Tw 0004 Ts D 9.94 ð 0.0785 403 0004 293 D 85.8 W/m With the lagging: Q D k 2$rm l T1 0004 T2 / r2 0004 r1
(equation 9.22)
In this case: k D 0.1 W/mK, T1 D 403 K and T2 is the temperature at the surface of the lagging. r1 D 25/1000000b/2 D 0.0125 m r2 D 0.0125 C 50/1000 D 0.0625 m and: rm D 0.0625 0004 0.0125000b/ ln 0.0625/0.0125 D 0.0311 m Thus: Q D 0.1 ð 2$ ð 0.0311 ð 1 403 0004 T2 / 0.0625 0004 0.0125 D 0.391 403 0004 T2 (i) But: Q D 1.22[ T2 0004 293000b/ 2 ð 0.0625000b]0.25 $ ð 2 ð 0.0625 ð 1 T2 0004 293 D 0.806 T2 0004 293000b1.25
(ii)
From (i) and (ii): 0.391 403 0004 T2 D 0.806 T2 0004 293000b1.25 Solving by trial and error: T2 D 313.5 K and hence: Q D 0.39/ 403 0004 313.5 D 35.0 W/m, a reduction of: or:
85.8 0004 35.0 D 50.8 W/m
50.8 ð 100/85.8 D 59.2%
PROBLEM 9.77 A vessel contains 1 tonne of liquid of specific heat capacity 4.0 kJ/kg K. It is heated by steam at 393 K which is fed to a coil immersed in the liquid and heat is lost to the
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surroundings at 293 K from the outside of the vessel. How long does it take to heat the liquid from 293 to 353 K and what is the maximum temperature to which the liquid can be heated? When the liquid temperature has reached 353 K, the steam supply is turned off for two hours and the vessel cools. How long will it take to reheat the material to 353 K? Coil: Area 0.5 m2 . Overall heat transfer coefficient to liquid, 600 W/m2 K. Outside of vessel: Area 6 m2 . Heat transfer coefficient to surroundings, 10 W/m2 K.
Solution If T K is the temperature of the liquid at time t s, then the net rate of heat input to the vessel, Uc Ac Ts 0004 T 0004 Us As T 0004 Ta D mCp dT/dt W where the coefficient at the coil, Uc D 600 W/m2 K, the coefficient at the outside of the vessel, Us D 10 W/m2 K, the areas are: coil, Ac D 0.5 m2 , vessel, As D 6.0 m2 , the temperatures are: steam, Ts D 393 K, ambient, Ta D 293 K, the mass of liquid, m D 1000 kg and the specific heat capacity, Cp D 4.0 kJ/kg K or 4000 J/kg K. Thus: 1000 ð 4000000bdT/dt D 600 ð 0.5 393 0004 T 0004 10 ð 6 T 0004 293 and: ∴
11,111 dT/dt D 376.3 0004 T 0004 T2 t D 11,111 dT/ 376.3 0004 T D 11,111 ln[ 376.3 0004 T1 / 376.3 0004 T2 ]
(i) (ii)
T1
When T1 D 293 K and T2 D 353 K then: t D 11,111 ln 83.3/23.3 D 14,155 s (3.93 h) The maximum temperature to which the liquid can be heated is obtained by putting dT/dt D 0 in (i) to give: T D 376.3 K. During the time the steam is turned off (for a period of 7200 s) a heat balance gives: mCp dT/dt D 0004Us As T 0004 Ta or:
1000 ð 4000000bdT/dt D 0004 10 ð 6 T 0004 293000b
∴
0004
T
Integrating:
66,700 dT/dt D 293 0004 T 0004 dT/ 293 0004 T D 0.000015
353
∴
7200
dt 0
ln
293 0004 353000b/ 293 0004 T D 0.000015 ð 7200 D 0.108 and T D 346.9 K.
The time taken to reheat the liquid to 353 K is then given by (ii): 0004 353 dT/ 376.3 0004 T t D 11,111 346.9
D 11,111 ln[ 376.3 0004 346.9000b/ 376.3 0004 353000b] D 2584 s 0.72 h000b
PROBLEM 9.78 A bare thermocouple is used to measure the temperature of a gas flowing through a hot pipe. The heat transfer coefficient between the gas and the thermocouple is proportional
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
to the 0.8 power of the gas velocity and the heat transfer by radiation from the walls to the thermocouple is proportional to the temperature difference. When the gas is flowing at 5 m/s the thermocouple reads 323 K. When it is flowing at 10 m/s it reads 313 K, and when it is flowing at 15.0 m/s it reads 309 K. Show that the gas temperature is about 298 K and calculate the approximate wall temperature. What temperature will the thermocouple indicate when the gas velocity is 20 m/s?
Solution If the gas and thermocouple temperatures are Tg and Tk respectively, then the rate of heat transfer from the thermocouple to the gas: Q1 D Ku0.8 Tg 0004 T000b
(i)
where K is a constant and u the gas velocity. Similarly, the rate of heat transfer from the walls to the thermocouple is: Q2 D k 0 Tw 0004 T W
(ii)
where k 0 is a constant and Tw is the wall temperature. At equilibrium:
Q1 D Q2 and u0.8 D k 0 /k Tw 0004 T000b/ T 0004 Tg
When u D 5 m/s,
T D 323 K and in (iii): 0.8
5 ∴
(iii)
D k 0 /k Tw 0004 323000b/ 323 0004 Tg D 3.624
k 0 /k D 3.624 323 0004 Tg / Tw 0004 323000b
(iv)
When u D 10 m/s, T D 313 K and in (iii): 100.8 D k 0 /k Tw 0004 313000b/ 313 0004 Tg D 6.31 Substituting for k 0 /k from (iv): 6.31 D 3.624 323 0004 Tg Tw 0004 313000b/[ Tw 0004 323 313 0004 Tg ]
(v)
When u D 15 m/s, T D 309 K and in (iii): 150.8 D k 0 /k Tw 0004 309000b/ 309 0004 Tg D 8.73 Substituting for k 0 /k from (iv): 8.73 D 3.624 323 0004 Tg Tw 0004 309000b/[ Tw 0004 323 309 0004 Tg ]
(vi)
If Tg D 298 K, then in (v): 1.741 D [ 323 0004 298 Tw 0004 313000b]/[ Tw 0004 323 313 0004 298000b] or:
Tw 0004 313000b/ Tw 0004 323 D 1.045 and Tw D 533 K.
If Tg D 298 K, then in (vi): 2.409 D 323 0004 298 Tw 0004 309000b/[ Tw 0004 323 309 0004 298000b] or:
Tw 0004 309000b/ Tw 0004 323 D 1.060 and Tw D 556 K
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211
This result agrees fairly well and a mean value of Tw D 545 K is indicated. In equation (iv):
k 0 /k D 3.624 323 0004 298000b/ 545 0004 323 D 0.408
∴ in equation (iii)
u0.8 D 0.408 545 0004 T000b/ T 0004 298000b
When u D 20 m/s: 10.99 D 0.408 545 0004 T000b/ T 0004 298 and T D 306.8 K.
PROBLEM 9.79 A hydrocarbon oil of density 950 kg/m3 and specific heat capacity 2.5 kJ/kg K is cooled in a heat exchanger from 363 to 313 K by water flowing countercurrently. The temperature of the water rises from 293 to 323 K. If the flowrate of the hydrocarbon is 0.56 kg/s, what is the required flowrate of water? After plant modifications, the heat exchanger is incorrectly connected so that the two streams are in co-current flow. What are the new outlet temperatures of hydrocarbon and water, if the overall heat transfer coefficient is unchanged?
Solution Heat lost by the oil D 0.56 ð 2.5 363 0004 313 D 70.0 kW For a flow of water of G kg/s, heat gained by the water is: 70.0 D G ð 4.18 323 0004 293 and G D 0.56 kg/s i) For countercurrent flow: T1 D 363 0004 323 D 40 deg K, T2 D 313 0004 293 D 20 deg K and from equation 9.9: Tm D 40 0004 20000b/ ln 40/20 D 28.85 deg K In equation 9.1: 70.0 D UA ð 28.85 and UA D 2.43 kW/K. ii) For co-current flow: If To and Tw are the outlet temperature of the oil and water respectively, the heat load is: Q D 0.56 ð 2.5 363 0004 To D 0.56 ð 4.18 Tw 0004 293000b
(i)
508.2 0004 1.4To D 2.34Tw 0004 685.9 and:
Tw D 510.3 0004 0.60To
(ii)
T1 D 363 0004 293 D 70 K T1 D To 0004 Tw D To 0004 510.3 C 0.60To D 1.60To 0004 510.3 K
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
and in equation 9.9: Tm D 70 0004 1.60To C 510.3000b/ ln[70/ 1.60To 0004 510.3000b] D 580.3 0004 1.60To / ln[70/1.60To 0004 510.3000b] deg K and: 508.2 0004 1.4To D 2.43 580.3 0004 1.60To / ln[70/ 1.60To 0004 510.3000b]
508.2 0004 1.4To D 2.126 508.2 0004 1.40To / ln[70/ 1.60To 0004 510.3000b] ∴
70/ 1.60To 0004 510.3 D e2.126 D 8.38 and To D 324.2 K .
and in equationT(ii): w D 510.3 0004 0.60 ð 324.2 D 315.8 K .
PROBLEM 9.80 A reaction mixture is heated in a vessel fitted with an agitator and a steam coil of area 10 m2 fed with steam at 393 K. The heat capacity of the system is equal to that of 500 kg of water. The overall coefficient of heat transfer from the vessel of area 5 m2 is 10 W/m2 K. It takes 1800 s to heat the contents from ambient temperature of 293 to 333 K. How long will it take to heat the system to 363 K and what is the maximum temperature which can be reached? Specific heat capacity of water D 4200 J/kgK.
Solution Following the argument of Problem 9.77 and taking ambient temperature as the initial temperature of the mixture, 293 K, then: net rate of heatingD 500 ð 4200000bdT/dt D Uc ð 10 393 0004 T 0004 10 ð 5 T 0004 293 W ∴
2,100,000 dT/dt D 3930Uc 0004 10Uc T 0004 50T C 14,650 D 3930Uc C 14,650 0004 10Uc C 50000bT.
∴ ∴
2,100,000/ 10Uc C 50 dT/dt D
3930Uc C 14,650000b/ 10Uc C 50 0004 T. 000f 333 t D [2,100,000/ 10Uc C 50000b] 293 dT/[ 3930Uc C 14,650000b/ 10Uc C 50 0004 T]
In heating from 293 to 333 K, the time taken is 1800 s and: 1800 D [2,100,000/ 10Uc C 50000b] lnf[ 3930Uc C 14,650000b/ 10Uc C 50 0004 293]/ [ 3930Uc C 14,650000b/ 10Uc C 50 0004 333]g Solving by trial and error: Uc D 61.0 W/m2 K. Thus, net rate of heating is: 2,100,000 dT/dt D 61.0 ð 10 393 0004 T 0004 10 ð 5 T 0004 293 D 254,380 0004 660 T W or:
3182 dT/dt D 385.4 0004 T W 0004 363 ∴ time for heating, t D 3182 dT/ 385.4 0004 T000b
(i)
293
D 3182 ln[ 385.4 0004 293000b/ 385.4 0004 363000b] D 3182 ln 92.4/22.4 D 4509 s 1.25 h000b
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The maximum temperature which can be attained is obtained by putting dT/dt D 0 in (i) which gives: Tmax D 385.5 K .
PROBLEM 9.81 A pipe, 50 mm outside diameter, is carrying steam at 413 K and the coefficient of heat transfer from its outer surface to the surroundings at 288 K is 10 W/m2 K. What is the heat loss per unit length? It is desired to add lagging of thermal conductivity 0.03 W/m K as a thick layer to the outside of the pipe in order to cut heat losses by 90%. If the heat transfer from the outside surface of the lagging is 5 W/m2 K, what thickness of lagging is required?
Solution Outside area of pipe: $dl D $ 50/1000 ð 1.0 D 0.157 m2 /m. Assuming the pipe wall is at the temperature of the steam, that is the resistance of the wall is negligible, then the heat loss is: Q D hA Tw 0004 Ta D 10 ð 0.157 413 0004 288 D 196 W/m With the addition of lagging of thickness, x mm, the required heat flow is 19.6 W/m The diameter of the lagging D 50 C 2x000b/1000 m and the surface area of the lagging D [$ 50 C 2x000b/1000] ð 1.0 D 0.157 C 0.00628x m2 /m. The heat transferred to the surroundings, 19.6 D 5 0.157 C 0.00628x T2 0004 288 W and:
T2 D 49.14 C 1.809x000b/ 0.157 C 0.00628x K
(i)
For conduction through the lagging: Q D k 2$rm l T1 0004 T2 / r2 0004 r1 W
(equation 9.22)
where Q D 19.6 W/m, k D 0.03 W/mK, l D 1.0 m and the steam temperature, T1 D 413 K. r1 D 25 mm or 0.025 m, r2 D 25 C x mm or 25 C x000b/1000 D 0.025 C 0.001x m ∴
rm D 0.025 C 0.001x 0004 0.025000b/ ln[ 0.025 C 0.001x000b/0.025] D 0.001x/ ln 1 C 0.04x m
Thus, in equation 9.22: 19.6 D 0.03 2$ 0.001x/ ln 1 C 0.04x ð 1.0 413 0004 T2 / 0.025 C 0.001x 0004 0.025 ∴
103,981 D [x/ ln 1 C 0.04x000b] 413 0004 T2 /0.001x
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Substituting for T2 from (i): 104 D [413 0004 49.14 C 1.809x000b/ 0.157 C 0.00628x000b]/ ln 1 C 0.04x Solving by trial and error, x D 52.5 mm.
PROBLEM 9.82 It takes 1800 s (0.5 h) to heat a tank of liquid from 293 to 323 K using steam supplied to an immersed coil when the steam temperature is 383 K. How long will it take when the steam temperature is raised to 393 K? The overall heat transfer coefficient from the steam coil to the tank is 10 times the coefficient from the tank to surroundings at a temperature of 293 K, and the area of the steam coil is equal to the outside area of the tank.
Solution Using the argument in Problem 9.77, mCp dT/dt D Uc Ac Ts 0004 T 0004 Us As T 0004 Ta In this case, Ts D 383 K, Ta D 293 K, Us D Uc /10 and Ac D As D A (say). ∴
mCp dT/dt D Uc Ac 383 0004 T 0004 Uc Ac /10 T 0004 293000b
∴
Uc Ac /mCp dt D dT/ 412.3 0004 1.1T000b
On integration: 0004
323
Uc Ac /mCp t D 293
000e323 dT/ 412.3 0004 1.1T D 1/1.1 ln 1/ 412.3 0004 1.1T 293
Since it takes 1800 s to heat the liquid from 293 to 333 K, then: 1800 Uc Ac /mCp D 0.909 ln[ 412.3 0004 1.1 ð 293000b]/[412.3 0004 1.1 ð 323000b] D 0.909 ln 90/57 and:
Uc Ac /mCp D 0.00023071 s00041
On increasing the steam temperature to 393 K,
Heat transferred from the steam D Uc Ac 393 0004 T W Heat lost to the surroundings D Uc Ac /10 T 0004 293 W and: ∴
mCp dT/dt D Uc Ac 393 0004 T 0004 0.1Uc Ac T 0004 293 W dT/ 422.3 0004 1.1T D Uc Ac /mCp dt D 0.0002307 dt
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On integration: 0004
323
0.0002307t D 293
dT/ 422.3 0004 1.1T D 1/1.1000b[ln 1/ 422.3 0004 1.1T000b]333 293
Thus, on heating from 293 to 323 K: 0.0002307t D 0.909 lnf[422.3 0004 1.1 ð 293000b]/[422.3 0004 1.1 ð 323000b]g D 0.909 ln 100/67 and:
t D 1578 s 0.44h000b
PROBLEM 9.83 A thermometer is situated in a duct in an air stream which is at a constant temperature. The reading varies with the gas flowrate as follows: air velocity (m/s)
thermometer reading (K)
6.1 7.6 12.2
553 543 533
The wall of the duct and the gas stream are at somewhat different temperatures. If the heat transfer coefficient for radiant heat transfer from the wall to the thermometer remains constant, and the heat transfer coefficient between the gas stream and thermometer is proportional to the 0.8 power of the velocity, what is the true temperature of the air stream? Neglect any other forms of heat transfer.
Solution As with Problem 9.78, a heat balance on the thermometer gives: hw Tw 0004 T D hg T 0004 Tg where hw and hg are the coefficients for radiant heat transfer from the wall and for convection to the gas respectively and Tw , T and Tg are the temperatures of the wall, thermometer and gas, respectively, above a datum of 533 K. When u D 12.2 m/s,
hw Tw 0004 0 D hg 0 C Tg
When u D 7.6 m/s, since hg / u0.8 , When u D 6.1 m/s,
(i)
hw Tw 0004 10 D hg 7.6/12.2000b0.8 000410 C Tg 0.8
hw Tw 0004 20 D hg 6.1/12.2 000420 C Tg
(ii) (iii)
Dividing equation (i) by equation (ii): Tw / Tw 0004 10 D 12.2/7.6000b0.8 Tg / Tg 0004 10 D 1.46Tg / Tg 0004 10000b
(iv) 0.8
and dividing equation (i) by equation (iii): Tw / Tw 0004 20 D 12.2/6.1 Tg / Tg 0004 20 D 1.741Tg / Tg 0004 20000b
(v)
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From (v): ∴
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Tw Tg 0004 20Tw D 1.741Tw Tg 0004 34.82Tg Tw D 34.82Tg / 20 C 0.741Tg K
(vi)
Substituting for Tw from (vi) into (iv): 34.82Tg / 27.4Tg 0004 200 D 1.46Tg / Tg 0004 10 and Tg D 000411.20 K and hence, the temperature of the gas is 533 0004 11.2 D 521.8 K.
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SECTION 10
Mass Transfer PROBLEM 10.1 Ammonia gas is diffusing at a constant rate through a layer of stagnant air 1 mm thick. Conditions are fixed so that the gas contains 50% by volume of ammonia at one boundary of the stagnant layer. The ammonia diffusing to the other boundary is quickly absorbed and the concentration is negligible at that plane. The temperature is 295 K and the pressure atmospheric, and under these conditions the diffusivity of ammonia in air is 0.18 cm2 /s. Calculate the rate of diffusion of ammonia through the layer.
Solution See Volume 1, Example 10.1.
PROBLEM 10.2 A simple rectifying column consists of a tube arranged vertically and supplied at the bottom with a mixture of benzene and toluene as vapour. At the top, a condenser returns some of the product as a reflux which flows in a thin film down the inner wall of the tube. The tube is insulated and heat losses can be neglected. At one point in the column, the vapour contains 70 mol% benzene and the adjacent liquid reflux contains 59 mol% benzene. The temperature at this point is 365 K. Assuming the diffusional resistance to vapour transfer to be equivalent to the diffusional resistance of a stagnant vapour layer 0.2 mm thick, calculate the rate of interchange of benzene and toluene between vapour and liquid. The molar latent heats of the two materials can be taken as equal. The vapour pressure of toluene at 365 K is 54.0 kN/m2 and the diffusivity of the vapours is 0.051 cm2 /s.
Solution In this solution, subscripts 1 and 2 refer to the liquid surface and vapour side of the stagnant layer respectively and subscripts B and T refer to benzene and toluene. If the latent heats are equal and there are no heat losses, there is no net change of phase across the stagnant layer. This is an example of equimolecular counter diffusion and: NA D 0002DPA2 0002 PA1 /RTL
(equation 10.23)
where L D thickness of the stagnant layer D 0.2 mm D 0.0002 m. 217
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As the vapour pressure of toluene D 54 kN/m2 , the partial pressure of toluene from Raoult’s law D 1 0002 0.59 ð 54 D 22.14 kN/m2 D PT1 and: PT2 D 1 0002 0.70 ð 101.3 D 30.39 kN/m2 For toluene: NT D 00020.051 ð 1000024 30.39 0002 22.14 /8.314 ð 365 ð 0.0002
D 00026.93 ð 1000025 kmol/m2 s For benzene: PB1 D 101.3 0002 22.14 D 79.16 kN/m2 PB2 D 101.3 0002 30.39 D 70.91 kN/m2 Hence, for benzene: NB D 00020.051 ð 1000024 70.91 0002 79.16 /8.314 ð 365 ð 0.0002
D 6.93 ð 1000025 kmol/m2 s Thus the rate of interchange of benzene and toluene is equal but opposite in direction.
PROBLEM 10.3 By what percentage would the rate of absorption be increased or decreased by increasing the total pressure from 100 to 200 kN/m2 in the following cases? (a) The absorption of ammonia from a mixture of ammonia and air containing 10% of ammonia by volume, using pure water as solvent. Assume that all the resistance to mass transfer lies within the gas phase. (b) The same conditions as (a) but the absorbing solution exerts a partial vapour pressure of ammonia of 5 kN/m2 . The diffusivity can be assumed to be inversely proportional to the absolute pressure.
Solution (a) The rates of diffusion for the two pressures are given by: NA D 0002D/RTL P/PBM PA2 0002 PA1
(equation 10.34)
where subscripts 1 and 2 refer to water and air side of the layer respectively and subscripts A and B refer to ammonia and air. Thus:
PA2 D 0.10 ð 100 D 10 kN/m2 and PA1 D 0 kN/m2 PB2 D 100 0002 10 D 90 kN/m2 and PB1 D 100 kN/m2 PBM D 100 0002 90 / ln100/90 D 94.91 kN/m2
∴
Hence:
P/PBM D 100/94.91 D 1.054 NA D 0002D/RTL 1.05410 0002 0 D 000210.54D/RTL
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219
If the pressure is doubled to 200 kN/m2 , the diffusivity is halved to 0.5D (from equation 10.18) and: PA2 D 0.1 ð 200 D 20 kN/m2 and PA1 D 0 kN/m2 PB2 D 200 0002 20 D 180 kN/m2 and PB1 D 200 kN/m2 ∴
PBM D 200 0002 180 / ln200/180 D 189.82 kN/m2 P/PBM D 200/189.82 D 1.054 i.e. unchanged
Hence: NA D 00020.5D/RTL 1.05420 0002 0 D 000210.54D/RTL, that is the rate is unchanged (b) If the absorbing solution now exerts a partial vapour pressure of ammonia of 5 kN/m2 , then at a total pressure of 100 kN/m2 : PA2 D 10 kN/m2 and PA1 D 5 kN/m2 PB2 D 90 kN/m2 and PB1 D 95 kN/m2 PBM D 95 0002 90 / ln95/90 D 92.48 kN/m2 ∴
P/PBM D 100/92.48 D 1.081 NA D 0002D/RTL ð 1.08110 0002 5 D 00025.406D/RTL
At 200 kN/m2 , the diffusivity D 0.5D and: PA2 D 20 kN/m2 and PA1 D 5 kN/m2 PB2 D 180 kN/m2 and PB1 D 195 kN/m2 ∴
PBM D 195 0002 180 / ln195/180 D 187.4 kN/m2 P/PBM D 1.067 NA D 00020.5D/RTL 1.06720 0002 5 D 00028.0D/RTL Thus the rate of diffusion has been increased by 1008 0002 5.406 /5.406 D 48%.
PROBLEM 10.4 In the Danckwerts’ model of mass transfer it is assumed that the fractional rate of surface renewal s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is se0002st . If the fractional rate of surface renewal were proportional to surface age (say s D bt, where b is a constant), show that the surface age distribution function would then assume the form: 2 2b/0010 1/2 e0002bt /2
Solution From equation 10.117:
f0 t D sft D 0
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In this problem: s D bt and ebt
2 /2
ft D constant D k
∴
ft D ke0002bt
2 /2
The total area of surface considered is unity and: 0001 1 ∴ ft dt D 1 0
0001
1
∴
ke0002bt
2 /2
dt D 1
0
and by substitution as in equation 10.120: k0010/2b 0.5 D 1 k D 2b/0010 0.5 and ft D 2b/0010 1/2 e0002bt
2 /2
PROBLEM 10.5 By consideration of the appropriate element of a sphere show that the general equation for molecular diffusion in a stationary medium and in the absence of a chemical reaction is: 0002 2 0003 ∂CA ∂ 2 CA ∂ CA 1 ∂ 2 CA 1 2 ∂CA cotˇ ∂CA DD C 2 C 2 C 2 2 C ∂t ∂r 2 r ∂ˇ2 r ∂r r ∂ˇ r sin ˇ ∂00162 where CA is the concentration of the diffusing substance, D the molecular diffusivity, t the time, and r, ˇ, 0016 are spherical polar coordinates, ˇ being the latitude angle.
Solution The basic equation for unsteady state mass transfer is: 00040002 0003 0002 2 0003 0002 2 0003 0005 ∂CA ∂ 2 CA ∂ CA ∂ CA DD C C ∂t ∂x 2 yz ∂y 2 zx ∂z2 xy
(equation 10.67) (i)
This equation may be transformed into other systems of orthogonal coordinates, the most useful being the spherical polar system. (Carslaw and Jaeger, Conduction of Heat in Solids, gives details of the transformation.) When the operation is performed: x D r sin ˇ cos 0016 y D r sin ˇ sin 0016 z D r cos ˇ and the equation for CA becomes: 0006 0002 0003 0002 0003 0007 ∂CA D ∂ 1 ∂ ∂CA 1 ∂ 2 CA ∂CA D 2 r2 C sin ˇ C ∂t r ∂r ∂r sin ˇ ∂ˇ ∂ˇ sin2 ˇ ∂00162
ii
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MASS TRANSFER
which may be written as: 0006 2 0002 0003 0007 ∂CA ∂ CA 2 ∂CA 1 1 υ ∂ 2 CA 2 ∂CA C 1 0002 001c
C DD C ∂t ∂r 2 r ∂r r 2 ∂001c ∂001c r 2 1 0002 001c2 ∂00162 where:
001c D cos ˇ.
iii
iv
In this problem ∂CA /∂t is given by: ∂CA DD ∂t
0002
∂ 2 CA ∂ 2 CA 1 ∂ 2 CA 1 2 ∂CA cotˇ ∂CA C 2 C C C 2 2 2 2 2 2 ∂r r ∂ˇ r ∂r r ∂ˇ r sin ˇ ∂0016
0003
v
Comparing equations (iii) and (v) is necessary to prove that: 1 ∂ r 2 ∂001c
0002 0003 ∂ 2 CA ∂ 2 CA 1 1 ∂ 2 CA 1 cot ˇ ∂CA 2 ∂CA 1 0002 001c
C 2 D C C 2 2 2 2 2 2 2 2 ∂001c r 1 0002 001c ∂0016 r ∂ˇ r ∂ˇ r sin ˇ ∂0016
001c D cos ˇ, 1 0002 001c2 D 1 0002 cos2 ˇ D sin2 ˇ ∂ 2 CA 1 ∂ 2 CA 1 D r 2 1 0002 001c2 ∂00162 r 2 sin2 ˇ ∂00162
∴
It now becomes necessary to prove that: 1 ∂ 2 CA cot ˇ ∂CA 1 ∂ C 2 D 2 2 2 r ∂ˇ r ∂ˇ r ∂001c From equation (iv): ∴
0002 0003 2 ∂CA 1 0002 001c
∂001c
001c D cos ˇ ∂001c/∂ˇ D 0002 sin ˇ
and:
vi
∂2 001c/∂ˇ2 D 0002 cos ˇ 0002 0003 0002 0003 ∂ˇ 1 ∂ 1 ∂ 2 ∂CA 2 ∂CA ∂ˇ 1 0002 001c
D 2 1 0002 001c
2 r ∂001c ∂001c r ∂ˇ ∂ˇ ∂001c ∂001c
vii
viii
Substituting from equation (iv) for 001c from equation (vii) for ∂ˇ/∂001c gives: 0002 0003 0002 0003 1 1 1 ∂ ∂CA 1 ∂CA 1 ∂ 2 D 2 1 0002 cos ˇ
0002 sin ˇ D 2 r ∂ˇ ∂ˇ 0002 sin ˇ 0002 sin ˇ r ∂ˇ ∂ˇ 0002 sin ˇ 0006 0007 2 2 1 1 1 ∂ CA ∂ CA ∂CA cot ˇ ∂CA D 2 0002 sin ˇ 0002 cos ˇ
D 2 C C 2 2 2 r ∂ˇ ∂ˇ 0002 sin ˇ r ∂ˇ r ∂ˇ
PROBLEM 10.6 Prove that for equimolecular counter diffusion from a sphere to a surrounding stationary, infinite medium, the Sherwood number based on the diameter of the sphere is equal to 2.
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Solution If the particle has a radius r, and is surrounded by a spherical shell of radius s then, Moles per unit time diffusing through the shell, M is given by: 0002 0003 dCA M D 40010s2 0002D ds At steady state, M is constant and: 0001 s2 0001 CA 2 ds M D 400100002D
dCA 2 s1 s CA1 0002 0003 1 1 M 0002 D 40010DCA1 0002 CA2
s1 s2 If CA1 is the concentration at s1 D r and CA2 is the concentration at s2 D 1, then: M/r D 40010D0002CA
The mass transfer coefficient: hd D
M M D 2 A0002CA
40010r 0002CA
hD 40010r 2 0002CA /r D 40010D0002CA
hD D D/r D 2D/d hD d/D D Sh D 2
PROBLEM 10.7 Show that the concentration profile for unsteady-state diffusion into a bounded medium of thickness L, when the concentration at the interface is suddenly raised to a constant value CAi and kept constant at the initial value of CAo at the other boundary is: 0004nD1 0005 CA 0002 CAo 2 1 z exp0002n2 00102 Dt/L 2 sinnz0010/L . D10002 0002 CAi 0002 CAo L 0010 nD1 n Assume the solution to be the sum of the solution for infinite time (steady-state part) and the solution of a second unsteady-state part, which simplifies the boundary conditions for the second part.
Solution The system is shown in Fig. 10a. The boundary conditions are: At time,
tD0 t>0 t>0
CA D CAo CA D CAi CA D CAo
0< />
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MASS TRANSFER
223
CAi
CAo L y 0
Figure 10a.
Replacing CAi by C0i and CA by C0 where: CA D C0 C CAo and CAi D C0i C CAo , then using these new variables: At:
t D 0 C0 D 0 t > 0 C0 D C0i t > 0 C0 D 0
0< />
The problem states that the solution of the one dimensional diffusion equation is: C0 D steady state solution C
1
exp0002n2 00102 Dt/L 2 An sinn0010y/L
0
where the steady state solution D C0i 0002 C0i y/L. (A derivation of the analogous equation for heat transfer may be found in Conduction of Heat in Solids by H. S. Carslaw and J. C. Jaeger, Oxford, 1960.) 0001 2 L An D initial concentration profile–steady state sinn0010y/L dy L 0 0001 2 L [0 C C0i y/L 0002 C0i ] sinn0010y/L dy D L 0 D 00022C0i /n0010 (this proof is given at the end of this problem).
Hence:
C0 D C0i 0002 C0i y/L 0002 0004
∴
1 2C0i 1 exp0002n2 00102 Dt/L 2 sinn0010y/L
0010 nD0 n
0005 1 21 y 2 2 2 exp0002n 0010 Dt/L sinn0010y/L
C D Co C Ci 0002 Co 1 0002 0002 L 0010 nD0 n
An D
2 L
0001 0
L
[C0i y/L 0002 C0i ] sinn0010y/L dy
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D
2C0i L2
0001
L
y sinn0010y/L dy 0002 0
2C0i L
0001
L
sinn0010y/L dy 0
0001 0001 2C0i L 2C0i L 1 0002 2 0007 0007 D 2 L L 0 0 0006 0007 0001 L 0001 L L n0010y L n0010y Ly 1 D 0002 cos cos dy 0007 C n0010 L 0 L 0 0 n0010
Putting u D y, du D dy dv D sinn0010y/L dy,
and: 0001 ∴ 0
L
vD0002
n0010y L cos n0010 L
0002 0003 0002 2 0003L L Ly n0010y n0010y L 1 D 0002 0007 C sin cos n0010 L 0 n2 0010 2 L 0
L2 L2 L2 cos n0010 C 2 2 sin n0010 D 0002 00021 n n0010 n 0010 n0010 0002 0003L 0001 L n0010y L L L 2 D 0002 cos cos n0010 C 0007 D0002 n0010 L n0010 n0010 0 0 D0002
L L cos n0010 C n0010 n0010 L L D 0002 00021 n C n0010 n0010 D0002
2C0 2C0i 2C0i 2 D 1 0002 0007 An D 2 i 0007 L L L2
0002 0003 0002 0003 L2 2C0i L L n n 0002 00021 0002 0002 00021 C n0010 L n0010 n0010
D 00022C0i /n0010
PROBLEM 10.8 Show that under the conditions specified in Problem 10.7 and assuming the Higbie model of surface renewal, the average mass flux at the interface is given by: 0007
nD1 0006 00102 1 2 2 2 2 2 NA t D CAi 0002 CAo D/L 1 C 2L /0010 Dt
0002 2 exp0002n 0010 Dt/L
6 n nD1 Use the relation
1 1 D 00102 /6. 2 n nD1
Solution The rate of transference across the phase boundary is given by: NA D 0002D∂CA /∂y yD0
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According to the Higbie model, if the element is exposed for a time te , the average rate of transfer is given by: 0001 1 te NA D 0002D∂C/∂z zD0 dt te 0 From Problem 10.7, the concentration C is: 0004 0005 1 21 y 2 2 2 C D CAo C CAi 0002 CAo 1 0002 0002 exp0002n 0010 Dte /L sin n0010y/L L 0010 nD0 n 0004 0005 1 ∂C 20010 1 2 2 2 D CAi 0002 CAo 0002 0002 exp0002n 0010 Dte /L cos n0010y/L ∂y L 0010 nD0 L 0004 0005 0002 0003 1 ∂C 1 20010 2 2 2 D CAi 0002 CAo 0002 0002 exp0002n 0010 Dte /L
∂y yD0 L 0010 0 L 0005 0001 0004 1 20010 DCAi 0002 CAo te 1 2 2 2 exp0002n 0010 Dte /L dt NA D 0002 0002 0002 te L 0010 0 L 0 0004 0005te 0002 0003 1 DCAi 0002 CAo
20010 te L2 2 2 2 D0002 0002 0002 0002 2 2 exp0002n 0010 Dte /L
te L 0010 0 L n 0010 D 0
DCAi 0002 CAo
te 0004 0005 0003 0002 1 0002 1 2 0003 0010 2 te L 2 L 0002 0002 0002 2 exp0002n2 00102 Dte /L 2 C 0002 2 2 L 0010 0 n 0010D 0010 0 L n 0010 D 00041 0005
1 1 D 2L 2 00021 2 2 2 NA D CAi 0002 CAo 1 C 2 exp0002n 0010 Dte /L C L 0010 Dte 0 n2 n2 0 D0002
1 00021 0
D
1 00021 0
n2
2 2
D
1
2
exp0002n 0010 Dte /L C
1 00021 1
D 0002 exp000200102 Dte /L 2 C 1 0006
n2
exp0002n2 00102 Dte /L 2 C
1 00021 1
n2
1 1 n2 0
1 1 1 1 exp0002n 0010 Dte /L C C 2 2 n2 n n 0 1 2 2
2
exp0002n2 00102 Dte /L 2 C 1 C 00102 /6
0007 0010 1 2 2 2 2 2 0002 2 exp0002n 0010 Dte /L C 1 0002 exp00020010 Dte /L
6 n 2
Considering the terms 1 0002 exp000200102 Dte /L 2 and Dte /L 2 to be very small so that 000200102 Dte /L 2 is small and exp000200102 Dte /L 2 ! 1. Therefore, 1 0002 exp000200102 Dte /L 2 is
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approximately zero and:
0007
1 0006 1 D 2L 2 00102 NA D CAi 0002 CAo 1 C 2 0002 2 exp0002n2 00102 Dte /L 2
L 0010 Dte nD1 6 n
PROBLEM 10.9 According to the simple penetration theory the instantaneous mass flux: 0002 00030.5 D NA t D CAi 0002 CAo
0010t What is the equivalent expression for the instantaneous heat flux under analogous conditions? Pure sulphur dioxide is absorbed at 295 K and atmospheric pressure into a laminar water jet. The solubility of SO2 , assumed constant over a small temperature range, is 1.54 kmol/m3 under these conditions and the heat of solution is 28 kJ/kmol. Calculate the resulting jet surface temperature if the Lewis number is 90. Neglect heat transfer between the water and the gas.
Solution The heat flux at any time, f D 0002k∂)/∂x where k is the thermal conductivity, ) the temperature, and y the distance in the direction of transfer. The flux satisfies the same differential equation as ), that is: DH ∂2 f/∂y 2 D ∂f/∂t
y > 0, t > 0
where DH D thermal diffusivity D k/+Cp . This last equation is analogous to the mass transfer equation 10.66: ∂C/∂t D D∂2 C/∂y 2
The solution of the heat transfer equation with f D Fo (constant) at y D 0 when t > 0 is:
y f D Fo erfc p 2 DH t
The temperature rise is due to the heat of solution HS . Heat is liberated at the jet surface at a rate H t D NoA HS , or:
H t D CAi 0002 CAo HS D/0010t 0.5
The temperature rise, T, due to the heat flux H t into the surface is: 0001 L 1 H t 0002 ) d) p p TD +Cp 0010DH 0 ) p CAi 0002 CAo Hs D/DH and: TD +Cp
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The Lewis number D h/Cp +hD D Pr/Sc D DCp +/k. D/DH D DCp +/k D 90 CAi 0002 CAo D 1.54 kmol/m3 HS D 28 kJ/kmol p T D 1.54 ð 28 90 /1000 ð 4.186 D 0.1 deg K
∴
PROBLEM 10.10 In a packed column, operating at approximately atmospheric pressure and 295 K, a 10% ammonia-air mixture is scrubbed with water and the concentration is reduced to 0.1%. If the whole of the resistance to mass transfer may be regarded as lying within a thin laminar film on the gas side of the gas-liquid interface, derive from first principles an expression for the rate of absorption at any position in the column. At some intermediate point where the ammonia concentration in the gas phase has been reduced to 5%, the partial pressure of ammonia in equilibrium with the aqueous solution is 660 N/m2 and the transfer rate is 1000023 kmol/m2 s. What is the thickness of the hypothetical gas film if the diffusivity of ammonia in air is 0.24 cm2 /s?
Solution The equation for the rate of absorption is derived in Section 10.2.2 as: NA D 0002D/RTL PA2 0002 PA1
(equation 10.23)
If subscripts 1 and 2 refer to the water and air side of the stagnant film and subscripts A and B refer to ammonia and air, then: PA1 D 66.0 kN/m2 and PA2 D 0.05 ð 101.3 D 5.065 kN/m2 D D 0.24 ð 1000024 m2 /s, R D 8.314 kJ/kmol K, T D 295 K and NA D 1 ð 1000023 kmol/m2 s ∴
L D 0002D/NA RT PA2 0002 PA1
D 00020.24 ð 1000024 /1000023 ð 8.314 ð 295 66.0 0002 5.065 D 00020.000043 m
The negative sign indicates that the diffusion is taking place in the opposite direction and the thickness of the gas film is 0.043 mm.
PROBLEM 10.11 An open bowl, 0.3 m in diameter, contains water at 350 K evaporating into the atmosphere. If the air currents are sufficiently strong to remove the water vapour as it is formed
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and if the resistance to its mass transfer in air is equivalent to that of a 1 mm layer for conditions of molecular diffusion, what will be the rate of cooling due to evaporation? The water can be considered as well mixed and the water equivalent of the system is equal to 10 kg. The diffusivity of water vapour in air may be taken as 0.20 cm2 /s and the kilogram molecular volume at NTP as 22.4 m3 .
Solution If subscripts 1 and 2 refer to the water and air side of the stagnant layer and subscripts A and B refer to water vapour and air, then the rate of diffusion through a stagnant layer is: NA D 0002D/RTL P/PBM PA2 0002 PA1
(equation 10.34)
where, PA1 is the vapour pressure of water at 350 K D 41.8 kN/m2 . PA2 D 0 (since the air currents remove the vapour as it is formed.) PB1 D 101.3 0002 41.8 D 59.5 kN/m2 and PB2 D 101.3 kN/m2 . ∴
PBM D 101.3 0002 59.5 / ln101.3/59.5 D 78.17 kN/m2 . and: P/PBM D 101.3/78.17 D 1.296.
∴
NA D 00020.2 ð 1000024 /8.314 ð 350 ð 1000023 1.2960 0002 41.8
D 3.72 ð 1000024 kmol/m2 s D 3.72 ð 1000024 ð 18 D 6.70 ð 1000023 kg water/m2 s Area of bowl D 0010/4 0.32 D 0.0707 m2 Therefore the rate of evaporation D 6.70 ð 1000023 ð 0.0707 D 4.74 ð 1000024 kg/s Latent heat of vaporisation D 2318 kJ/kg Specific heat capacity of water D 4.187 kJ/kg K Rate of heat removal D 4.74 ð 1000024 ð 2318 D 1.10 kW If the rate of cooling D d)/dt K/s, then: water equivalent ð specific heat capacity ð d)/dt D 0.0617
or:
10 ð 4.187 ð d)/dt D 1.10 and d)/dt D 0.026 deg K/s
PROBLEM 10.12 Show by substitution that when a gas of solubility CC is absorbed into a stagnant liquid of infinite depth, the concentration at time t and depth y is: y CC erfc p 2 Dt Hence, on the basis of the simple penetration theory, show that the rate of absorption in a packed column will be proportional to the square root of the diffusivity.
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229
Solution The first part of this question is discussed in Section 10.5.2 and the required equation is presented as equation 10.108. In Section 10.5.2 the analysis leads to equation 10.113 which expresses the instantaneous rate of mass transfer when the surface element under consideration has an age t, or: NA t D CAi 0002 CAo D/0010t
The simple penetration theory assumes that each element is exposed for the same time interval te before returning to the bulk solution. The average rate of mass transfer is then: 0001 0001 1 te CAi 0002 CAo te NA t dt D D/0010t 0.5 dt NA D te 0 te 0 D 2CAi 0002 CAo D/0010te p and the rate of absorption is proportional to D.
PROBLEM 10.13 Show that in steady-state diffusion through a film of liquid, accompanied by a firstorder irreversible reaction, the concentration of solute in the film at depth y below the interface is: k L 0002 y
CA Ci D sinh D CAi k L sinh D if CA D 0 at y D L and CA D CAi at y D 0, corresponding to the interface. Hence show that according to the “film theory” of gas-absorption, the rate of absorption per unit area of interface, NA is given by: NA D kL CAi
ˇ tanh ˇ
p where ˇ D Dk /kL , D is the diffusivity of the solute, k the rate constant of the reaction, KL the liquid film mass transfer coefficient for physical absorption, CAi the concentration of solute at the interface, y the distance normal to the interface and yL the liquid film thickness.
Solution The basic equation for diffusion through a film of liquid accompanied by a first-order irreversible reaction is: Dd2 CA /dy 2 D k CA p where a2 D k /D.
or
d2 CA /dy 2 D a2 CA
(equation 10.171) (i)
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The general solution of equation (i) is: CA D A cosh ay C B sinh ay
ii
where A and B are constants. The boundary conditions are: At y D L, CA D 0
iii
At y D 0, CA D CAi
iv
Substituting equation (iii) in equation (ii): 0 D A cosh aL C B sinh aL and substituting equation (iv) in equation (ii): CAi D A C 0 and A D CAi and B D 0002CAi cosh aL/ sinh aL ∴
CA D CAi cosh ay 0002 CAi
cosh aL sinh ay sinh aL
CAi cosh ay sinh aL 0002 cosh aL sinh ay
sinh aL CAi sinh aL 0002 y
D sinh aL p sinh aL 0002 y
sinh k /DL 0002 y
p D CAi D CAi sinh aL sinh k /DL
D
0002
Rate of absorption:
NA D 0002D
dCA dy
0003 yD0
Assuming CA to be small so that bulk flow can be neglected, then: 0002 0003 d sinh aL 0002 y
NA D 0002D dy sinh aL DCAi a cosh aL sinh aL D DCAi a/ tanh aL D DCAi aL/L tanh aL D
kL D D/L ˇ D Dk /kL D k /D L D aL ∴
NA D
kL CAi ˇ tanh ˇ
PROBLEM 10.14 The diffusivity of the vapour of a volatile liquid in air can be conveniently determined by Winkelmann’s method, in which liquid is contained in a narrow diameter vertical tube
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maintained at a constant temperature, and an air stream is passed over the top of the tube sufficiently rapidly to ensure the partial pressure of the vapour there remains approximately zero. On the assumption that the vapour is transferred from the surface of the liquid to the air stream by molecular diffusion, calculate the diffusivity of carbon tetrachloride vapour in air at 321 K and atmospheric pressure from the following experimentally obtained data: Time from commencement of experiment (ks)
Liquid level (cm)
0 1.6 11.1 27.4 80.2 117.5 168.6 199.7 289.3 383.1
0.00 0.25 1.29 2.32 4.39 5.47 6.70 7.38 9.03 10.48
The vapour pressure of carbon tetrachloride at 321 K is 37.6 kN/m2 , and the density of the liquid is 1540 kg/m3 . The kilogram molecular volume is 22.4 m3 .
Solution Equations 10.37 and 10.38 state that: NA D 0002D
CA2 0002 CA1 CT y2 0002 y1 CBm
In this problem, the distance through which the gas is diffusing will be taken as h and CA2 D 0. ∴
NA D DCA /h CT /CBm kmol/m2 s
where CA is the concentration at the interface. If the liquid level falls by a distance dh in time dt, the rate of evaporation is: NA D +L /M dh/dt kmol/m2 s Hence:
+L /M dh/dt D DCA /h CT /CBm
If this equation is integrated, noting that when t D 0, h D h0 , then: h2 0002 h02 D 2MD/+L CA CT /CBm t or: t/h 0002 h0 D +L /2MD CBm /CA CT h 0002 h0 C +L CBm /MDCA CT h0
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Thus a plot of t/h 0002 h0 against h 0002 h0 will be a straight line of slope s where: s D +L CBm /2MDCA CT
or
D D +L CBm /2MCA CT s
The following table may be produced: t (ks) 1.6 11.1 27.4 80.2 117.5 168.6 199.7 289.3 383.1 h 0002 h0 (mm) 2.5 12.9 23.2 43.9 54.7 67.0 73.8 90.3 104.8 t/h 0002 h0
s/m ð 1000026
0.64 0.86 1.18 1.83 2.15 3.52 2.71 3.20 3.66
3.5
t / (h −h0) s/m × 10−6
3.0
2.5
2.0 Slope = 3.04 ×10−7 s/m2
1.5
1.0
0.5
0
20
40
60
80
100
120
(h −h0) mm
Figure 10b.
These data are plotted as Fig. 10b and the slope is: s D 3.54 0002 0.5 1000026 /100 ð 1000023 D 3.04 ð 1000027 s/m2 CT D 1/22.4 273/321 D 0.0380 kmol/m3 M D 154 kg/kmol CA D 37.6/101.3 1/22.4 273/321 D 0.0141 kmol/m3 +L D 1540 kg/m3 CB1 D 0.0380 kmol/m3
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233
CB2 D 0.0380 0002 0.0141 D 0.0239 kmol/m3 ∴
CBm D 0.0380 0002 0.0239 / ln0.0380/0.0239 D 0.0304 kmol/m3
Hence:
D D 1540 ð 0.0304 /2 ð 154 ð 0.0141 ð 0.0380 ð 3.04 ð 1000027
D 9.33 ð 1000026 m2 /s
PROBLEM 10.15 Ammonia is absorbed in water from a mixture with air using a column operating at atmospheric pressure and 295 K. The resistance to transfer can be regarded as lying entirely within the gas phase. At a point in the column the partial pressure of the ammonia is 6.6 kN/m2 . The back pressure at the water interface is negligible and the resistance to transfer can be regarded as lying in a stationary gas film 1 mm thick. If the diffusivity of ammonia in air is 0.236 cm2 /s, what is the transfer rate per unit area at that point in the column? If the gas were compressed to 200 kN/m2 pressure, how would the transfer rate be altered?
Solution See Volume 1, Example 10.3.
PROBLEM 10.16 What are the general principles underlying the two-film penetration and film-penetration theories for mass transfer across a phase boundary? Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. According to the penetration theory, the instantaneous rate of mass transfer per unit area NA t at some time t after the commencement of transfer is given by: D NA t D CA 0010t where CA is the concentration driving force and D is the diffusivity. Obtain expressions for the average rates of transfer on the basis of the Higbie and Danckwerts assumptions.
Solution The various theories for the mechanism of mass transfer across a phase boundary are discussed in Section 10.5. The basic equation for unsteady state equimolecular counter-diffusion is: 00040002 0003 0002 2 0003 0002 2 0003 0005 ∂CA ∂ 2 CA ∂ CA ∂ CA C C (equation 10.67) DD ∂t ∂x 2 yz ∂y z xz ∂z2 xy
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Considering the diffusion of solute A away from the interface in the y-direction this equation becomes: ∂CA ∂ 2 CA DD 2 ∂t ∂y The boundary conditions are: tD0 t>0 t>0
0< /><1 yd0='>1>
CA D CAo CA D CAi CA D CAo
where CAo is the concentration in the bulk of the phase and CAi is the equilibrium concentration at the interface. The instantaneous rate of mass transfer per unit area NA at time t is given by: NA t D CA D/0010t Higbie assumed that every element of surface is exposed to the gas for the same length of time ) before being replaced by liquid of the bulk composition. Amount absorbed in time ): 0001 ) 0001 ) QD NA t d) D CA D/0010) d) D 2CA D)/0010 0
0
000e p p The average rate of absorption: Q/) D 2CA D)/0010 /) D 2CA D/0010)
Danckwerts suggested that each element would not be exposed for the same time but that a random distribution of ages would exist. It is shown in Section 10.5.2 that this age distribution may be expressed ft D se0002st . The average rate of absorption is the value of NA t averaged over all elements of the surface having ages between 0 and 1 is then given by: 0001 1 0001 1 p p 0002s) NA D s NA t e d) D CA D/0010 e0002s) / ) d) D CA Ds 0
0
PROBLEM 10.17 A solute diffuses from a liquid surface at which its molar concentration is CAi into a liquid with which it reacts. The mass transfer rate is given by Fick’s law and the reaction is first order with respect to the solute. In a steady-state process, the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration CA of solute at a depth y from the surface in terms of the molecular diffusivity D and the reaction rate constant k . What is the molar flux at the surface?
Solution 2 2 2 As in Problem p 10.13, the basic equation is: d CA /dy D a CA where a D k /D
Then:
CA D A cosh ay C B sinh ay
(i) ii
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235
The first boundary condition is at y D 0, CA D CAi , and CAi D A. CA D CAi cosh ax C B sinh ax
Hence:
iii
The second boundary condition is that when y D L and: NA D 0002DdCA /dy yD0 D 00022DdCA /dy yDL Differentiating equation (iii): dCA /dy D CAi a sinh ay C Ba cosh ay and:
dCA /dy yD0 D Ba
and:
dCA /dy yDL D aB/2 D CAi a sinh aL C Ba cosh aL
so that:
BD
2CAi sinh aL 1 0002 2 cosh aL
iv
Substituting equation (iv) into equation (iii): 2CAi sinh aL sinh ay 1 0002 2 cosh aL D CAi [cosh ay 0002 2cosh ay cosh aL C sinh aL sinh ay ]/1 0002 2 cosh aL
CA D CAi cosh ay C
D CAi [cosh ay 0002 2 cosh ay C L ] The molar flux at the surface D NA D 0002DdCA /dy yD0 . dCA D CAi [a sinh ay 0002 2a sinh aa C L ] dy dCA /dy yD0 D 00022CAi a2 sinh aL NA D 2DCAi a2 sinh aL a D k /D
NA D 2DCAi k /D sinh L k /D p D 2CAi k sinhL k /D
PROBLEM 10.18 4 cm3 of mixture formed by adding 2 cm3 of acetone to 2 cm3 of dibutyl phthalate is contained in a 6 mm diameter vertical glass tube immersed in a thermostat maintained at 315 K. A stream of air at 315 K and atmospheric pressure is passed over the open top of the tube to maintain a zero partial pressure of acetone vapour at that point. The liquid level is initially 11.5 mm below the top of the tube and the acetone vapour is transferred to the air stream by molecular diffusion alone. The dibutyl phthalate can be regarded as completely non-volatile and the partial pressure of acetone vapour may be calculated from Raoult’s law on the assumption that the density of dibutyl phthalate is sufficiently greater than that of acetone for the liquid to be completely mixed.
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Calculate the time taken for the liquid level to fall to 5 cm below the top of the tube, neglecting the effects of bulk flow in the vapour. 1 kmol occupies 22.4 m3 . Molecular weights of acetone, dibutyl phthalate D 58 and 279 kg/kmol respectively. Liquid densities of acetone, dibutyl phthalate D 764 and 1048 kg/m3 respectively. Vapour pressure of acetone at 315 K D 60.5 kN/m2 . Diffusivity of acetone vapour in air at 315 K D 0.123 cm2 /s.
Solution Considering the situation when the liquid has fallen to a depth h cm below the top of the tube, volume of acetone evaporated D 0010/4 0.6 2 h 0002 1.15 D 0.283h 0002 1.15 cm3 . At this time, the amount of dibutyl phthalate is: 2 ð 1.048/278 D 0.00754 mol and the amount of acetone D [2 0002 0.283h 0002 1.15 ]0.764/58 D 0.0306 0002 0.00372h
∴
8.23 0002 h
0.0306 0002 0.00372h D 0.00754 C 0.0306 0002 0.00372h
10.24 0002 h
0002 0003 8.23 0002 h Partial pressure of acetone D 60.5 kN/m2 10.24 0002 h Molar concentration of acetone vapour at the liquid surface 0002 0003 0002 0003 0002 00030002 0003 60.5 273 1 8.23 0002 h ð ð D 101.3 315 22400 10.24 0002 h 0002 0003 8.23 0002 h D 2.31 ð 1000025 mol/cm3 10.24 0002 h Mole fraction of acetone D
dh 0.764 ð D 0.0132dh/dt mol/cm2 s dt 58 0002 0003 8.23 0002 h 00025 D D/h ð molar concentration at surface D 0.123/h 2.31 ð 10 10.24 0002 h 0002 0003 8.23 0002 h 1 dh D ð 2.84 ð 1000026 0.0132 ∴ dt h 10.24 0002 h 0002 0003 10.24 0002 h dt and: h dh D 8.23 0002 h 4650 Rate of evaporation of acetone: NA D
The time for the liquid level to fall from 1.15 cm to 5 cm below the top of the tube is obtained by integrating this equation: 0003 0003 0001 5 0002 0001 t 0001 5 0002 0001 t 10.24 0002 h 1 16.6 1 h dh D dt D h 0002 2.02 0002 dh D dt 8.23 0002 h 4650 0 h 0002 8.23 4650 0 1.15 1.15 and:
t D 79500 s 79.5 ks ³ 22 h
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PROBLEM 10.19 A crystal is suspended in fresh solvent and 5% of the crystal dissolves in 300 s. How long will it take before 10% of the crystal has dissolved? Assume that the solvent can be regarded as infinite in extent, that the mass transfer in the solvent is governed by Fick’s second law of diffusion and may be represented as a unidirectional process, and that changes in the surface area of the crystal may be neglected. Start your derivations using Fick’s second law.
Solution The mass transfer process is governed by Fick’s second law: ∂CA ∂ 2 CA DD 2 ∂t ∂y
(equation 10.66)
and discussed in Section 10.5.2 The boundary conditions for the crystal dissolving are: When t D 0 t>0 t>0
0< /><1 yd1='>1>
CA D 0 CA D 0 CA D CAs the saturation value
These boundary conditions allow the solution of equation 10.66 using Laplace transforms as the most convenient method: 0001 1 ∂CA ∂CA D dt (equation 10.102) e0002pt dt ∂t 0 0001 1 000f 00101 D e0002pt CA 0 C p e0002pt CA dt D 0 C pCNA (equation 10.103) 0
Taking Laplace transforms of both sides of equation 10.66: ∂2 CNA pCNA D D 2 ∂y ∴
and:
∂2 CNA p 0002 CNA D 0 ∂y 2 D p p CNA D Ae p/D y C Be0002 p/D y
(equation 10.105)
When y D 1, CA D 0 ∴ and A D 0 CNA D 0 When y D 0, CA D CAs and CNA D CAs /po , B D CAs /p p C As 0002 p/D y e ∴ CNA D p p Inverting: CA D CAs erfcy/2 Dt
(See Volume 1, Appendix Table 13)
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0002
Mass transfer rate at the surface D 0002D
0003 yD0
0002 00030012 1 2 y 0002y 2 /4Dt p p e d 0010 y/2pDt
2 Dt 0002 0003 2 1 2 D CAs p 0002 p e0002y /4Dt (equation 10.111) 0010 2 Dt 0003 0002 ∂CA CAs ∴ D 0002p dy yD0 0010Dt 0002 0003 ∂CA D D CAs NA t D 0002D ∂t yD0 0010t 0001 t D Dp The mass transfer in time t D dt D 2 t 0010t 0010 0 p and the mass transfer is proportional to t M1 t1 Thus: D M2 t2
∂CA ∂ D CAs ∂y ∂y
and:
0011
∂CA ∂y
0001
M1 D 5%, M2 D 10%, and t1 D 300 s p 0.5 D 300/t2 and t2 D 1200 s
PROBLEM 10.20 In a continuous steady state reactor, a slightly soluble gas is absorbed into a liquid in which it dissolves and reacts, the reaction being second-order with respect to the dissolved gas. Calculate the reaction rate constant on the assumption that the liquid is semi-infinite in extent and that mass transfer resistance in the gas phase is negligible. The diffusivity of the gas in the liquid is 1000028 m2 /s, the gas concentration in the liquid falls to one half of its value in the liquid over a distance of 1 mm, and the rate of absorption at the interface is 4 ð 1000026 kmol/m2 s.
Solution The equation for mass transfer with chemical reaction is: ∂CA ∂ 2 CA D D 2 0002 k CnA ∂t ∂y
(equation 10.170)
For steady state second order reaction where n D 2: D
d2 CA 0002 k C2A D 0 dy 2
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MASS TRANSFER
Putting dCA /dy D q:
239
dq dCA dq d2 CA dq D Dq D 2 dy dy dCA dy dCA
Thus:
Dq
dq 0002 k C2A D 0 dCA
q dq D k /D C2A dCA q2 /2 D k /D C3A /3 C const. In an infinite system at y D 1, CA D 0 and dCA /dy D 0 and hence the constant D 0. 0003 0002 dCA 2 2 k 3 2k 3/2 dCA CA and D0002 C ∴ D dy 3D dy 3D A noting the negative sign since dCA /dy is negative for all values of CA . 2k 00021/2 Thus: 00022CA D 0002 y C constant 3D At the free surface, y D 0 and CA D CAi D constant. ∴
and: or:
00021/2
constant D 00022CAi 0013 0014 p 00021/2 00021/2 2 CA 0002 CAi D 2k /3D y 00021/2
CA
00021/2
0002 CAi
D
p k /6D y
When y D y1 , CA D CAi /2, and: 00021/2
CAi /2 00021/2 0002 CAi
k /6D y1 p 00021/2 When y1 D 1000023 , substituting gives: CAi D 2.42 ð 1000023 k /6D
p p 00021/2 ∴ CAi 0002 2.42 ð 1000023 k /6D D k /6D y D
The mass transfer rate at the interface, where y D 0, is: 0002 0003 dCA 2k D 3/2 NA t D 0002D CAi D dy yD0 3 0002 00033 2k D 1 p ð D D 8.47 ð 108 D2 /k 3 2.42 ð 1000023 k /6D
When D D 1 ð 1000028 m2 /s and NA D 4 ð 1000026 kmol/m2 s : k D 8.47 ð 108 ð 1 ð 1000028 2 /4 ð 1000026 D 212 m3 /kmol s
PROBLEM 10.21 Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with a
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
constant fractional rate of surface renewal, s, but an upper limit on surface age equal to the life of the jet, 4, show that the surface age frequency distribution function, ft , for this case is given by: ft D s exp0002st/[1 0002 exp0002st ]
for
0< />< />
ft D 0
for
t > 4.
Hence, show that the enhancement, E, for the increase in value of the liquid-phase mass transfer coefficient is: E D [0010s4 1/2 erfs4 1/2 ]/f2[1 0002 exp0002s4 ]g where E is defined as the ratio of the mass transfer coefficient predicted by conditions described above to the mass transfer coefficient obtained from the penetration theory for a jet with an undisturbed surface. Assume that the interfacial concentration of acetone is practically constant.
Solution For the penetration theory: ∂CA ∂ 2 CA DD 2 ∂t ∂y
(equation 10.66)
As shown in Problem 10.19, this equation can be transformed and solved to give: p p N A D Ae p/D y C Be0002 p/D y C The boundary conditions are: When y D 0,
CA D CAi ,
and when y D 1, ∴
B D CAi /p
CA D 0 and A D 0 p
N A D CAi e0002 p/D y C p 0015 NA dC 1 1 0002pp/D y D 0002CAi e dy D p
From Volume 1, Appendix, Table 12, No 84, the inverse: dCA 1 1 0002y 2 /4Dt D 0002CAi e dy D 0010t 0002 0003 dCA D at time t At the surface: NA t D 0002D D CAi dy yD0 0010t The average rate over a time 4 is: 0001 4 1 D dt D p D 2CAi CAi 4 0010 0 00104 t
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241 p 6 0 and NA D 2CAi 0002 CA0 D/00104 for mass transfer without surface D MASS TRANSFER
In general, CA0 renewal. Random surface renewal is discussed in Section 10.5.2 where it is shown that the age distribution function is: D constant e0002st D ke0002st where s is the rate of production of fresh surface per unit total area of surface. If the maximum age of the surface is 4, then: 0001 4 e0002st dt D 1 K 0
k 0002 [e0002st ]40 D 1 s 1 0002 e0002s4 D s/k and K D 0002 ∴ the age distribution function is:
s 1 0002 e0002s4
s 1 0002 e0002s4
0003
e0002st
The mass transfer in time 4 is: 0001 4 0001 4 0002st D D e s s 0002st p dt CAi 0002 CA0
CAi 0002 CA0
e dt D 0002s4 0002s4 0010t 10002e 0010t 10002e t 0 0 p p The integral is conveniently solved by substituting st D ˇ2 and t D ˇ/ s or s dt D 2ˇ dˇ and dt D 2ˇ dˇ/s p 0001 4p 0001 4 p s 0002ˇ 2ˇ dˇ 0010 p 2 erf s4 0002ˇ2 e Dp erf s4 D CAi 0002 CA0 Ds e dˇ D Then: ˇ s s 0 s 1 0002 e0002s4 0 The enhancement factor E is given by: p p erf s4 p p CAi 0002 CA0 Ds 0002s4 0010s4 erf s4 1 0002 e ED D 21 0002 e0002s4
D CAi 0002 CA0
2 00104
PROBLEM 10.22 Solute gas is diffusing into a stationary liquid, virtually free of solvent, and of sufficient depth for it to be regarded as semi-infinite in extent. In what depth of fluid below the surface will 90% of the material which has been transferred across the interface have accumulated in the first minute? Diffusivity of gas in liquid D 1000029 m2 /s.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution As in the previous problem, the basic equation is: ∂ 2 CA ∂CA DD 2 ∂t ∂y
(equation 10.66)
which can be solved using the same boundary conditions to give the rate of mass transfer at depth, y, NA y,t as: D 0002y 2 /4Dt dCA NA y,t D 0002D D CAi e dy 0010t At some other value of y D L, the amount which has been transferred in time t per unit area is: 0001 t D 0002y 2 /4Dt e CAi dt 0010t 0 This integral can be solved by making the substitution:
so that: and:
ˇ2 D y 2 /4Dt p ˇ D y/2 Dt p t D y 2 /4Dˇ2 , t00021/2 D y/2ˇ D dt D 0002y 2 /2D ˇ00023 dˇ
The amount transferred at depth L is then: 0004 p 0005 D y 2 Dt 0002y 2 /4Dt p y p e 0002 0010 erfc p D CAi 0010 D y 2 Dt 0004 0005 Dt 0002y 2 /4Dt y e D CAi 2 0002 y erfc p 0010 2 Dt 0004 0005 Dt 0002y 2 /4Dt y e 2 0002 y erfc p 0010 2 Dt mass transfer at L D and: mass transfer at y D 0 Dt 2 0010 p p y y 2 2 D e0002y /4Dt 0002 p 0010 erfc p D e0002X 0002 X 0010 erfc X 2 Dt 2 Dt p where X D y/2 Dt Under the conditions in this problem, this ratio D 0.1. erfc X D 1 0002 erf X so that erfc X can be calculated from Table 13 in the Appendix of Volume 1. Values of X will be assumed and the right hand side evaluated until a value of X is found such that the right hand side D 0.1.
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MASS TRANSFER
X
e0002X
erf X
erfc X
p X 0010 erfc X
Right hand side
1 0.9 0.97 0.96
0.368 0.445 0.390 0.398
0.843 0.797 0.830 0.825
0.157 0.203 0.170 0.175
0.278 0.324 0.292 0.297
0.0897 0.121 0.098 0.101
2
∴
X D 0.96 D y 2 /4Dt
∴
y D 0.96 ð 4 ð 1000029 ð 60 0.5 D 4.8 ð 1000024 m or 0.48 mm
PROBLEM 10.23 A chamber, of volume 1 m3 , contains air at a temperature of 293 K and a pressure of 101.3 kN/m2 , with a partial pressure of water vapour of 0.8 kN/m2 . A bowl of liquid with a free surface of 0.01 m2 and maintained at a temperature of 303 K is introduced into the chamber. How long will it take for the air to become 90% saturated at 293 K and how much water must be evaporated? The diffusivity of water vapour in air is 2.4 ð 1000025 m2 /s and the mass transfer resistance is equivalent to that of a stagnant gas film of thickness 0.25 mm. Neglect the effects of bulk flow. Saturation vapour pressure of water D 4.3 kN/m2 at 303 K and 2.3 kN/m2 at 293 K.
Solution Moles transferred, n D DA/L CAs 0002 CA
where CA D concentration (kmol/m3 ), CAs is the saturation value of CA at the surface D is the diffusivity and L is the thickness of the stagnant gas film. If the saturated vapour pressure at the interface is 4.3 kN/m2 and if at any time the partial pressure in the air is PA kN/m2 , then the rate of evaporation is given by: 0006 0002 00030007 dn 1 4.3 0.01 ð 2.4 ð 1000025 273 PA D ð 0002 dt 0.25/1000
22.4 303 101.3 101.3 D 3.81 ð 1000027 4.3 0002 PA kmol/s 1 m3 of air at 303 K and 101.3 kN/m2 is equivalent to 1/22.4 273/293 D 0.0416 kmol Initial moisture content D 0.8/101.3 ð 0.0416 D 3.29 ð 1000024 kmol Final moisture content D 0.9 ð 2.3/101.3 ð 0.0416 D 8.50 ð 1000024 kmol ∴ Water evaporated D 8.50 0002 3.29 ð 1000024 D 5.21 ð 1000024 kmol D 5.21 ð 1000024 ð 18 D 9.38 ð 1000023 kg water
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At a pressure P kN/m2 : n D PA /101.3 1/22.4 273/293 D 4.11 ð 1000024 PA kmol/m3 dn/dt D 4.11 ð 1000024 dPA /dt
and: ∴
4.11 ð 1000024 0001
2.3
∴
0.8
dPA D 3.81 ð 1000027 4.3 0002 PA
dt
dPA D 9.27 ð 1000024 dt 4.3 0002 PA
from which t D 604 s 10 min
PROBLEM 10.24 A large deep bath contains molten steel, the surface of which is in contact with air. The oxygen concentration in the bulk of the molten steel is 0.03% by mass and the rate of transfer of oxygen from the air is sufficiently high to maintain the surface layers saturated at a concentration of 0.16% by weight. The surface of the liquid is disrupted by gas bubbles rising to the surface at a frequency of 120 bubbles per m2 of surface per second, each bubble disrupts and mixes about 15 cm2 of the surface layer into the bulk. On the assumption that the oxygen transfer can be represented by a surface renewal model, obtain the appropriate equation for mass transfer by starting with Fick’s second law of diffusion and calculate: (a) The mass transfer coefficient (b) The mean mass flux of oxygen at the surface (c) The corresponding film thickness for a film model, giving the same mass transfer rate. Diffusivity of oxygen in steel D 1.2 ð 1000028 m2 /s. Density of molten steel D 7100 kg/m3 .
Solution If C0 is defined as the concentration above a uniform datum value: ∂ 2 C0 ∂C0 DD 2 ∂t ∂y
(equation 10.100)
The boundary conditions are: when
t D 0, t>0 t>0
0< /><1 yd0='>1>
C0 D 0 C0 D C0i C0 D 0
The equation is most conveniently solved using Laplace transforms. The Laplace transN 0 of C0 is: form C 0001 1 N0 D e0002pt C0 dt (equation 10.101) C 0
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MASS TRANSFER
N0 ∂C D ∂t
0001
∂C0 dt ∂t 0 0001 0002pt 0 1 D [e C ]0 C p
Then:
1
e0002pt
(equation 10.102) 1
N0 e0002pt C0 dt D pC
(equation 10.103)
0
Since the Laplace transform operation is independent of y, ∂ 2 C0 ∂ 2 C0 D 2 ∂y ∂y 2
(equation 10.104)
Taking Laplace transforms of both sides of equation 10.100: N0 D D pC
From which:
N0 ∂2 C ∂y 2
N0 ∂2 C p N0 0002 C D0 2 ∂y D p p N 0 D Ae p/D y C Be0002 p/D y C
(equation 10.105)
N0 D 0 C and N 0 D C0i /p C and p 0 N 0 D Ci e0002 p/D y C p p N0 dC C0 D 0002 p i e0002 p/D y dy pD
When y D 1, When y D 0, ∴
Inverting:
1 C0 ∂C0 2 D 0002 p i ð p e0002y /4Dt ∂y 0010t D
AD0 B D C0i /p
(See Volume 1, Appendix Table 12) 0002
The mass transfer rate at the surface, NA t D 0002D
∂C0 ∂y
0003
D yD0
C0i
D at time t 0010t
The average rate of mass transfer in time t: 0001 D 1 t 0 D 0 dt D 2Ci Ci t 0 0010t 0010t Taking 1 m2 of surface, the area disrupted by the bubbles per second is: 120 ð 15/10000 D 0.18/s ∴ Average surface age duration D 1/0.18 D 5.55 s
C0i D 0.16 0002 0.03 /100 D 0.0013 kg O2 /kg steel D 0.0013/32 ð 7100 D 0.2885 kmol/m3
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Then:
(a) The mass transfer coefficient D 2
D D 21.2 ð 1000028 /0010 ð 5.55 0.5 0010t D 5.25 ð 1000025 m/s
(b) The mean rate of transfer, NA D 2C0i D/0010t 0.5 D 2 ð 0.28851.2 ð 1000028 /0010 ð 5.55 0.5 D 1.51 ð 1000025 kmol/m2 s N 0i (c) The film thickness L is given by: NA D D/L C and:
L D 1.2 ð 1000028 ð 0.2885 /1.51 ð 1000025 D 2.29 ð 1000024 m D 0.23 mm
PROBLEM 10.25 Two large reservoirs of gas are connected by a pipe of length 2L with a full-bore valve at its mid-point. Initially a gas A fills one reservoir and the pipe up to the valve and gas B fills the other reservoir and the remainder of the pipe. The valve is opened rapidly and the gases in the pipe mix by molecular diffusion. Obtain an expression for the concentration of gas A in that half of the pipe in which it is increasing, as a function of distance y from the valve and time t after opening. The whole system is at a constant pressure and the ideal gas law is applicable to both gases. It may be assumed that the rate of mixing in the vessels is high so that the gas concentration at the two ends of the pipe do not change.
Solution The system and nomenclature are shown in Fig. 10c. y = −L
y = +L y =0
CA = CA 0
CA = 0
Figure 10c.
When time t D 0, For gas A: ∂CA ∂ 2 CA DD 2 ∂t ∂y
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MASS TRANSFER
When When When When
t D 0, t D 0, t>0 t>0
0002L < y < 0 0< />
CA CA CA CA
D CA0 D0 D CA0 D0
CB CB CB CB
D0 D CB0 D0 D CB0
For gas B: ∂CB ∂ 2 CB DD 2 ∂t ∂y When When When When
tD0 tD0 t>0 t>0
0002L < y < 0 0< />
and for all values of y: ∂CA ∂CB C D0 ∂y ∂y As in previous problems, these equations may be solved by the use of Laplace transforms. For y > 0: p p N A D Ae p/D y C Be0002 p/D y C and for y < 0:
p
p N A D A0 e C
p/D y
C B 0 e0002
p/D y
C CA0 /p
The boundary conditions may now be used to evaluate the constants thus: p P 00022 p/D L CA0 /p e p AD0002 21 0002 e00022 p/D L
CA0 /p
p 21 0002 e00022 p/D L
p A0 D 0002B0 e2 p/D L p 00022 p/D L C 1
Be p B0 D e2 p/D L C 1
BD
Substituting these values: 0007 nD1 0006 CA0 2nL C y 2n C 1 L 0002 y p erfc p 0002 erfc CA D 2 nD0 2 Dt 2 Dt
This relation can be checked as follows: 0007 1 0006 n C 1 L CA0 nL CA0 (a) When y D 0: CA D erfc p 0002 erfc p D 2 0 2 Dt Dt (b) When y D L: CA D 0
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PROBLEM 10.26 A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Fick’s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, D, the first-order reaction rate constant k , the film thickness L and the concentration CAS of solute in a saturated solution. The reaction is initially carried out at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K? Reaction rate constant at 293 K D 2.5 ð 1000026 s00021 . Energy of activation for reaction (in Arrhenius equation) D 26430 kJ/kmol. Universal gas constant R D 8.314 kJ/kmol K. Molecular diffusivity D D 1000029 m2 /s. Film thickness, L D 10 mm. Solubility of gas at 313 K is 80% of solubility at 293 K.
Solution See Volume 1, Example 10.11
PROBLEM 10.27 Using Maxwell’s law of diffusion obtain an expression for the effective diffusivity for a gas A in a binary mixture of B and C, in terms of the diffusivities of A in the two pure components and the molar concentrations of A, B and C. Carbon dioxide is absorbed in water from a 25 per cent mixture in nitrogen. How will its absorption rate compare with that from a mixture containing 35 per cent carbon dioxide, 40 per cent hydrogen and 25 per cent nitrogen? It may be assumed that the gas-film resistance is controlling, that the partial pressure of carbon dioxide at the gas–liquid interface is negligible and that the two-film theory is applicable, with the gas film thickness the same in the two cases. Diffusivity of CO2 in hydrogen D 3.5 ð 1000025 m2 /s; in nitrogen D 1.6 ð 1000025 m2 /s.
Solution Maxwell’s Law of Diffusion is discussed in Section 10.3.2 where for a two component gaseous mixture: 0002dPA /dy D FAB CA CB uA 0002 uB
(equation 10.77)
For an ideal gas,
PA D CA RT
(equation 10.9a)
and from equation 10.78:
uA D N0A /CA
(equation 10.9b)
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MASS TRANSFER
249
when B is not undergoing mass transfer, or uB D 0, then: 0002RTdCA /dy D FAB CB N0A N0A D 0002
RT dCA RT CT dCA D0002 FAB CB dy FAB CT CB dy
By comparison with Stefan’s Law: N0A D 0002DAB
CT dCA CB dy
Then:
DAB D RT/FAB CT
or:
FAB D
(equation 10.29)
RT DAB CT
Applying to A in a mixture of B and C: 0002dPA /dy D FAB CA CB uA 0002 uB C FAC CA CC uA 0002 uC
For the case where uB D uC D 0: dCA RT RT 0002RT D CB N0A C CC N0A dy DAB CT DAC CT 0002dCA /dy CT CB /DAB C CC /DAC
or:
N0A D
From Stefan’s Law:
N0A D 0002D0
CT dCA dCA /dy CT D0002 CT 0002 CA dy CT 0002 CA /D0
i
ii
where D0 is the effective diffusivity of A in the mixture Comparing equations (i) and (ii): 1 CB CC 1 1 D C D0 DAB CT 0002 CA DAC CT 0002 CA For CO2 in N2 : N0A D D
1 CT CCO2 CT yCO2 DDð (equation 10.33) ð CN2 lm L yN2 lm L
D D 1.6 ð 1000025 m2 /s yN2 1 D 1.0, yN2 2 D 0.75, ∴
yN2 lm D [1 0002 0.75 / ln1/0.75 ] D 0.87 yCO2 D 0.25, CT and L are unknown
CT 1.6 ð 1000025 ð 0.25 D 4.6 ð 1000026 CT /L kmol/m2 s 0.87 L For CO2 in a mixture of H2 and N2 , the effective diffusivity, derived in the first part of the problem, is used to give D0 : Hence:
N0A D
1 1 0.4 0.25 1 C D 2.4 ð 1000025 m2 /s D ð ð D0 3.5 ð 1000025 1 0002 0.35 1.6 ð 1000025 1 0002 0.35
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N0A D D0
CT CCO2 1 CT yCO2 D D0 ð ð ð CN2 CH2 lm L yN2 CH2 lm L
yCO2 D 0.35 yN2 CH2 l D 1.0, yN2 CH2 2 D 0.65 and:
N0A D 2.4 ð 1000025 ð
C ln1/0.65
ð 0.35 D 1.033 ð 1000025 CT /L 1 0002 0.65
L
∴ The ratio of mass transfer rates D 1.033 ð 1000025 /4.6 ð 1000026 D 2.25
PROBLEM 10.28 Given that from the penetration theory for mass transfer across an interface, the instantaneous rate of mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure time in the Higbie model and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be derived from first principles.
Solution Given that the instantaneous mass transfer rate D Kt00021/2 , then for the Higbie model, the average mass transfer rate for an exposure time te is given by: 0001 1 te 00021/2 Kt dt D 2Kte00021/2 te 0 For the Danckwerts model, the random surface renewal analysis, presented in Section 10.5.2, shows that the fraction of the surface with an age between t and t C dt is a function of t D ft dt and that ft D Ke0002st where s is the rate of production of fresh surface per unit total area. For a total surface area of unity: 0006 0002st 00071 0001 1 e Ke0002st dt D 1 D K D K/s 0002s 0 0 ∴
K D s and ft D se0002st dt
The rate of mass transfer for unit area is: 0001 1 0001 00021/2 0002st Kt ð se dt D Ks 0
1
t00021/2 e0002st dt 0
2
Substituting ˇ D st and s dt D 2ˇ dˇ, then: 0001 1p 0001 1 p p p p s 0002ˇ2 2ˇ 2 e dˇ D K s ð 2 Rate D Ks e0002ˇ dˇ D K s ð 2 ð 0010/2 D K 0010s ˇ s 0 0
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MASS TRANSFER
If the rates from each model are equal, then: p 2Kte00021/2 D K 0010s
or
ste D 4/0010
PROBLEM 10.29 Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the mass transfer rate immediately before mixing is only 16 per cent of that calculated from the theory and the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process takes place every second, what thickness of surface film would account for the discrepancy? Diffusivity of ammonia in water D 1.76 ð 1000029 m2 /s.
Solution For the penetration theory:
When t D 0, When t > 0, When t > 0,
∂CA ∂ 2 CA DD 2 (equation 10.66) ∂t ∂y CA D 0 y D 0, CA D CAi D constant yD1 CA D 0
As shown earlier in problems 10.19 and 10.21, this equation may be transformed and solved to give: p p N A D Ae2 p/D y C Be00022 p/D y C (equation 10.105) When y D 0, y D 1,
N A D CAi /p C NA D 0 C
and hence:
A D 0 and B D CAi /p p N A D CAi e0002 p/D y Hence, C p NA p 0002pp/D y ∂C D 0002CAi e and: ∂y D 0002 0003 ∂CA D CAi in time t (as in Problem 10.21). At the surface, NA t D 0002D D ∂y yD0 0010t For the film, the origin is taken at the interface between the film (whose thickness is L) and the mixed fluid. p p N A D Ae p/D y C Be0002 p/D y Again: C (equation 10.105)
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CAx at y D 0 is now a variable y D 1, CA D 0 p NA D C N Ax e0002 p/D y A D 0 and C NA p N 0002pp/D y dC D0002 CAx e dy D 0016 0017 N1 dC pN CAx D0002 dy D
Hence: and: and:
yD0
To maintain mass balance at the film boundary: D CAi 0002 CAx D 0002D L
0002
∂CA ∂y
0003 yD0
Taking Laplace transforms: D L So:
and:
Hence:
Inverting:
0002
CAi N Ax 0002C p
0016
0003
D 0002D
D CAi D L p N Ax C
0002
NA ∂C ∂y
D C L
0017
D
N Ax pD C
yD0
0003
N Ax pD C
D CAi L p D D p C pD L
p p D CAi 1 e0002 p/D y DL p D p C pD L p 1 1/L D 0002CAi p p e0002 p/D y p D p C p L 0016p 0017 Dt CAi y/L Dt/L2 y ∂CA D0002 e e C p erfc ∂y L L 2 Dt 0016p 0017 0002 0003 ∂CA D Dt/L2 Dt NA yD0 D 0002D D CA i e erfc ∂y yD0 L L 0016p 0017 D Dt/L2 Dt CAi e erfc L L D 0.16 D CAi 0010t N Ax dC D0002 dy
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MASS TRANSFER
or: writing
253
Dt Dt/L2 Dt 1 ð 0.16 D 0.0903 e erfc D 2 2 L L 0010 Dt 2 XD then XeX erfc X D 0.0903 2 L
Solving by trial and error: X D 0.101 When t D 1 s, D D 1.76 ð 1000029 m2 /s, and: L D 0.42 mm.
PROBLEM 10.30 A deep pool of ethanol is suddenly exposed to an atmosphere consisting of pure carbon dioxide and unsteady state mass transfer, governed by Fick’s Law, takes place for 100 s. What proportion of the absorbed carbon dioxide will have accumulated in the 1 mm thick layer of ethanol closest to the surface? Diffusivity of carbon dioxide in ethanol D 4 ð 1000029 m2 /s.
Solution See Volume 1, Example 10.6.
PROBLEM 10.31 A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at CAs and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. If the surface concentration is maintained at 0.04 kmol/m3 , the second-order rate constant k2 is 9.5 ð 103 m3 /kmol s and the liquid phase diffusivity D is 1.8 ð 1000029 m2 /s, calculate: (a) The concentration at a depth of 0.1 mm. (b) The molar rate of transfer at the surface kmol/m2 s . (c) The molar rate of transfer at a depth of 0.1 mm. It may be noted that if: dCA d2 CA dq D q, then: Dq 2 dy dy dCA
Solution Considering element of unit area and depth dy, then for a steady state process: RATE IN 0002 RATE OUT D REACTION RATE
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0011 0002 0003 0012 d dCA dCA dCA 0002 0002D C 0002D 0002D dy D k2 C2A dy
dy dy dy dy
or:
D
d2 C A D k2 C2A dy 2
d2 CA k2 0002 C2A D 0 dy 2 D Putting:
qD
dCA d2 C A dq dCA dq dq , then: D Dq D Ð dy dy 2 dy dCA dy dCA
dq k2 0002 C2A D 0 dCA D k2 q dq D C2A D 2 k2 C3A q D CK Integrating: 2 D 3 When y D 1, q D 0, CA D 0 and: K D 0 0002 0003 1 dCA 2 k2 C3A D Thus: 2 dy D 3 dCA 2 k2 3/2 Dš C dy 3D A Substituting:
q
As NA D 0002DdCA /dy is positive, negative root must apply and: 2 k2 00023/2 CA dCA D 0002 dy 3D 2 k2 00021/2 Integrating: 00022CA D 0002 yCK 3D 00021/2
When y D 0, CA D CAs and: K D 00022CAs Thus: or:
1 2 k2 y 0002 D 2 3D 00040002 0005 0003 CA 00021/2 1 k2 00021/2 y 00021 D CAs CAs 6D 0002
0003 CAs 1/2 1 k2 00021/2 y. 00021 D CAs CA 6D 00021/2 CA
(i) For the conditions given:
1 k2 D 6D
00021/2 CAs
001500110002 0003 0002 00030012 1 9.5 ð 103 6 1.8 ð 1000029
D 0.938 ð 106 (m/kmol)0.5
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At depth of 0.1 mm D 1000024 m : 0002
and:
CAs CA
00031/2
00021 0002
CAs CA
1 k2 y D 93.8 6D C1/2
D 93.8 CAs
D 18.8
00031/2
D 19.8 CA D
0.04 D 0.00010 kmol/m3 . 19.82
(ii) The molar transfer rate at surface is: 2 k2 3/2 dCA NA 0002 D D0002 C 0002D . dy 3 D As 2k2 D 3/2 D CAs 3 2 D ð 9.5 ð 103 ð 1.8 ð 1000029 0.04 3/2 3 D 3.38 ð 1000023 ð 0.008 D 2.70 ð 1000025 kmol/m2 s. (iii) The molar transfer rate at depth of 0.1 mm is: 2k2 D 3/2 CA NA D 3 D 3.38 ð 1000023 ð 0.00010 3/2 D 3.38 ð 1000029 kmol/m2 s
PROBLEM 10.32 In calculating the mass transfer rate from the penetration theory, two models for the age distribution of the surface elements are commonly used — those due to Higbie and to Danckwerts. Explain the difference between the two models and give examples of situations in which each of them would be appropriate. (a) In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average mass transfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is p [D/0010t ], where D is the diffusivity. (b) If for unit area of surface the surface renewal rate is s, by how much will the mass transfer coefficient be changed if no surface has an age exceeding 2/s? (c) If the probability of surface renewal is linearly related to age, as opposed to being constant, obtain the corresponding form of the age distribution function.
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It may be noted that:
0001
p
1
e
0002x2
0010 2
dx D
0
Solution (a) If the age distribution function be ft , then the surface in the age group t to t C dt is: ft dt. Then, the surface of age t C dt minus, the surface of age t C dt is the surface destroyed in the dt, or: ft 0002 ft C dt D sft dt 0002f0 t C dt dt D sft dt
or:
f0 t C sft D 0
As dt ! 0, 0018
Using the integrating factor e
sdt
i
D est then: est ft D K (const)
and:
ft D Ke0002st 0006 0002st 00071 0001 1 e K the total surface D 1 D K e0002st dt D K D 0002s 0 s 0
ii
ft D se0002st
and hence:
The mass transfer rate into fraction of surface of age t to t C dt (per unit total area of surface) is: D D 0002st CA se dt D CA e0002st t00021/2 dt 0010t 0010 The mass transfer rate per unit area, NA D Putting st D ˇ2 : and:
Thus:
D CA 0010
0001
1
t00021/2 e0002st dt 0
s dt D 2ˇ dˇ 0001 1 1/2 D s 2 NA D CA e0002ˇ 2ˇ dˇ 0010 ˇ 0 0001 1 sD 2 CA e0002ˇ dˇ D2 0010 0 p sD 0010 p CA D DsCA D2 0010 2 p NA D Ds the mass transfer coefficient D CA
iii
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(b) For an age range of surface from 0 to 2/s, application of equation (ii) gives: 0006 0002st 00072/s 0001 2/s e K 1DK e0002st dt D K D 1 0002 e00022
0002s 0 s 0 s or: KD 1 0002 e00022 s and: ft D e0002st 1 0002 e00022 Thus NA from equation (ii) is multiplied by factor 1/1 0002 e00022 or: 1 p DsCA D 1.16 times the value in equation 3. NA D 1 0002 e00022 The mass transfer rate per unit total area of surface is: 0017 0001 2/s 0016 0001 2/s D D 0002st CA 1.16se dt D 1.16sCA t00021/2 e0002st dt NA D 0010t 0010 0 0 st D ˇ2 p s 00021/2 t D ˇ
Putting: then: and:
s dt D 2ˇ dˇ 0001
p
2
Integrating: 0
p
s 0002ˇ2 2ˇ 2 e dˇ D p ˇ s s
0001
p
2
2
e0002ˇ dˇ 0
p p 2 0010 Dp erf 2 s 2 p 0010 D erf 2 s
p p 0010 CA erf 2 D 1.16 ð 0.954 DsCA s p D 1.107 DsCA p The mass transfer coefficient, hD D 1.107 Ds, an increase of 10.7% NA D
D 1.16s 0010
(c) For probability of surface renewal being linearly related to age, s D kt (where k is a constant) Equation (i) becomes: f0 t C ktft D 0. 0018 2 The integrating factor is: e kt dt D ekt /2 ekt and:
2 /2
ft D K ft D Ke0002kt
2 /2
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0001
1
0002kt2 /2
0001 2 1 0002kt2 /2 p dt D K e d k/2 t k 0
The total surface is: 1 D K e 0 k t D X: Putting p 20001 1 2 2 0010 0010 0002X2 e dX D K DK , 1DK k 0 k 2 2k 2k then: KD 0010 2k 0002kt2 /2 and the age distribution function is: e 0010
PROBLEM 10.33 Explain the basis of the penetration theory for mass transfer across a phase boundary. What are the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed? (Do not present a solution of the differential equation.) Obtain the age distribution function for the surface: (a) On the basis of the Danckwerts’ assumption that the probability of surface renewal is independent of its age. (b) On the assumption that the probability of surface renewal increases linearly with the age of the surface. Using the Danckwerts surface renewal model, estimate: (c) At what age of a surface element is the mass transfer rate equal to the mean value for the whole surface for a surface renewal rate (s) of 0.01 m2 /m2 s? (d) For what proportion of the total mass transfer is surface of an age exceeding 10 seconds responsible?
Solution (a) Danckwerts age distribution function Dividing the total unit surface into elements each of duration dt, then: dt
t − dt
dt
t
dt
t + dt
t + 2dt
If the fraction of surface in age band t to t C dt is ft dt, then: the fraction of surface in age band t 0002 dt to t will be ft 0002 dt dt.
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The surface not going from t 0002 dt/t to t/t C dt D ft 0002 dt dt 0002 ft dt D 0002f0 t 0002 dt dt dt This will be surface destroyed in time dt D destruction rate ð area of surface ð time interval
D s[ft 0002 dt dt]dt 0
Thus: 0002f t 0002 dt dt dt D s[ft 0002 dt dt]dt As dt ! 0 then: f0 t C sft D 0 Using the Danckwerts model: s D const. est f0 t C sest ft D 0 est ft D K ft D Ke0002st There is no upper age limit to surface 0006 0002st 00071 0001 1 e 0002st Thus total surface D 1 D K e dt D K D K/s or: K D s 0002s 0 0 ft D se0002st . (b) s D at where a is a constant. Thus:
eat
2 /2 0
f t C ateat e
2 /2
at2 /2
ft D 0 ft D K0 2/2
ft D K0 e0002at 0001 0002 0003 0001 1 2 1 0002at2/2 a 0002at2/2 0 0 For unit total surface: 1 D K e dt D K e d t 2 0 pa 0 2 0010 D K0 a 2 2a 2a 0002at2/2 0 and ft D e and: K D 0010 0010 (c) regarding surface renewal as random 0001 and:
1
kt00021/2 se0002st dtCA 0001 1 D CA ks t00021/2 e0002st dt.
For unit total surface, mass transfer (mol/area time) D
0
0
where CA is the concentration driving force in moles per unit volume Putting st D x 2 , then: s dt D 2x dx 0001 1 p 1 2 The mass transfer rate D ksCA x 00021 s e0002x 2x dx. s 0 0001 p 1 p p 0010 p 0002x2 D CA 2k s D CA k 0010s e dx D CA 2k s 2 0
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The mass transfer rate at time t is: CA kt00021/2 Thus the age of surface at which rate D average is given by or: p kt00021/2 D k 0010s tD
100 1 D D 31.8 s 0010s 0010
(d) Surface of age less than 10 seconds The mass transfer taking place into surface of age up to 10 s is given by the same expression as for the whole surface but with upper limit of 10 s instead of infinity 0001 k100002t p 2 or: CA 2k s e0002x dx 0
p when t D 10 s: x D st D 0.01 ð 10 D 0.316 The mass transfer into surface up to 10 s age is then: p 0001 0.316 p p p 0010 2 CA 2k s erf 0.316 D CA k 0010s ð 0.345 e0002x dx D CA 2k s 2 0 p
Thus a fraction: 0.345 is contributed by surface of age 0 0002 10 s and: a fraction: 0.655 by surface of age 10 s to infinity .
PROBLEM 10.34 At a particular location in a distillation column, where the temperature is 350 K and the pressure 500 m Hg, the mol fraction of the more volatile component in the vapour is 0.7 at the interface with the liquid and 0.5 in the bulk of the vapour. The molar latent heat of the more volatile component is 1.5 times that of the less volatile. Calculate the mass transfer rates kmol m00022 s00021 of the two components. The resistance to mass transfer in the vapour may be considered to lie in a stagnant film of thickness 0.5 mm at the interface. The diffusivity in the vapour mixture is 2 ð 1000025 m2 s00021 . Calculate the mol fractions and concentration gradients of the two components at the mid-point of the film. Assume that the ideal gas law is applicable and that the Universal Gas Constant R D 8314 J/kmol K.
Solution In this case: T D 350 K, P D 500 mm Hg D
and:
0002
0003 500 ð 101,300 D 0.666 ð 105 N/m2 760
D D 2 ð 1000025 m2 /s 0002 0003 0.666 ð 105 P CT D D D 0.0229 kmol/m3 RT 8314 ð 350
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MASS TRANSFER
If:
A D MVC
B D LVC, then:
9A D 1.59B N0A 9A D 0002N0B 9B N0B D 0002N0A ,
and:
9A D 00021.5N0A 9B
N0A D 0002DCT
0001
dxA N0A C N0B C CA dy CT
D 0002DCT
dxA C N0A 0002 1.5N0A xA dy
5ð1000024
dxA dy 0001 0.5
dy D 0002DCT 0
0.7
N0A ð 5 ð 1000024
(from equations 10.46a and 10.19)
D 0002DCT
N0A 1 C 0.5xA D 0002DCT N0A
dxA C uF CA dy
i
dxA 1 C 0.5xA
0019 001a0.5 D 0002DCT 2 ln1 C 0.5xA
0.7
1 1.25 ð N0A D 00022DCT ln 1.35 5 ð 1000024 D 1.41 ð 1000024 kmol/m2 s N0B D 0002 2.11 ð 1000024 kmol/m2 s
and:
At the mid-point: y D 2.5 ð 1000024 m
0019 001axA N0A ð 2.5 ð 1000024 D 00022 ð 1000025 ð 0.0229 ð 2 ln1 C 0.5xA
0.7
00024
00024
1.41 ð 10 ð 2.5 ð 10 1 C 0.5xA D0002 1.35 2 ð 1000025 ð 0.0229 ð 2 1.35 D 0.0384 ln 1 C 0.5xA ln
and:
xA D 0.598
The concentration gradient is given by equation (i) or: N0 1 C 0.5xA
dxA D A dy 0002DCT When xA is 0.598, then: dxA 1.41 ð 1000024 1 C 0.5 ð 0.598
D dy 00022 ð 1000025 ð 0.0229 D 400 m00021 D 0.4 mm00021
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dxA dCA D CT D 9.16 kmol/m4 . dy dy
and:
PROBLEM 10.35 For the diffusion of carbon dioxide at atmospheric pressure and a temperature of 293 K, at what time will the concentration of solute 1 mm below the surface reach 1 per cent of the value at the surface? At that time, what will the mass transfer rate (kmol m00022 s00021 be: (a) At the free surface? (b) At the depth of 1 mm? The diffusivity of carbon dioxide in water may be taken as 1.5 ð 1000029 m2 s00021 . In the literature, Henry’s law constant K for carbon dioxide at 293 K is given as 1.08 ð 106 where K D P/X, P being the partial pressure of carbon dioxide (mm Hg) and X the corresponding mol fraction in the water.
Solution ∂CA ∂ 2 CA DD 2 ∂t ∂y where CA is concentration of solvent undergoing mass transfer. The boundary conditions are: y D 0 (interface) yD1 tD0
CA D CAs (solution value) CA D 0 CA D 0
t>0 0< />< />
Taking Laplace transforms then: 0003 0001 10002 ∂CA ∂CA 0002pt D e dt ∂t ∂t 0 0001 1 0002pt 1 0002pept CA dt D 0 C pCNA D [CA e ]0 0002 0
∂ CA ∂ CNA D 2 ∂y ∂y 2 2
Thus:
2
pCNA D D
∂2 CNA ∂y 2
∂2 CNA p 0002 CNA D 0 ∂y 2 D p
p CNA D Ae For t > 0; when y D 1 when y D 0
p/Dy
CA D 0, CA D CAs
C Be0002
p/Dy
N A D 0 ∴ A D 0. C 0006 0002pt 00071 0001 1 e CAs 0002pt N . CAs e dt D CAs D CAs D 0002p 0 p 0
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MASS TRANSFER
Thus: and: Inverting:
CAs D B.1 p CAs 0002pp/Dy. CNA D e p CA y D erfc p CAs 2 Dt
See Table in Volume 1, Appendix
Differentiating with respect to y: 0011 0002 0003 0001 1 1 ∂CA 2 y ∂ 0002y 2 /4Dt p p D e d CAs ∂y ∂y 0010 y/2pDt 2 Dt 1 2 1 2 2 D p Ð p 0002e0002y /4Dt D 0002 p e0002y /4Dt 0010 2 Dt 0010Dt 0011 0012 00021 0002y 2 /4Dt The mass transfer rate at t, y, N D 0002D p CAs e 0010Dt 0002 0003 D 0002y 2 /4Dt ∂CA e when: t > 0, then: 0002D D CAs ∂y yDy 0010Dt D At t > 0 and y D 0, then: N0 D CAs 0010t
i
ii
For a concentrated 1% of surface value at y D 1 mm, CA /CAs D 0.01 and: 0011 0012 1000023 p 0.01 D erfc 2 1.5 ð 1000029 t Writing erf x D 1 0002 erfc x, then: 0.99 D erf12.91t00021/2
1.82 D 12.91t00021/2
From tables
t D 50.3 s The mass transfer rate at the interface at t D 50.3 s is given by equation (ii) as: 0015 D 1.5 ð 1000029 D CAs N0 D CAs 0010t 0010 ð 50.3 D 3.08 ð 1000026 CAs kmol/m2 s The mass transfer rate at y D 1 mm. and t D 50.3 s is given by equation (i) as: N D N0 e0002y
2 /4Dt
00026 /4ð1.5ð1000029 ð50.3
D 3.08 ð 1000026 e000210
CAs
D 1.121 ð 1000027 CAs where CAs is in kmol/m3 .
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Henry’s law constant, K D 1.08 ð 106 , where: K D P/X. Also: P D 760 mm Hg X D Mol fraction in liquid X D 760/1.08 ð 106 D 7.037 ð 1000024 kmol CO2 kmol solution (³ per kmol water)
and:
Water molar density D 1000/18 kmol/m3 0002 0003 00024 1000 CAs D 7.037 ð 10 D 0.0391 kmol/m3 18 NA0 D 1.204 ð 1000027 kmol/m2 s.
When y D 0, then: When y D 1 mm, then:
NA D 4.38 ð 1000029 kmol/m2 s.
PROBLEM 10.36 Experiments are carried out at atmospheric pressure on the absorption into water of ammonia from a mixture of hydrogen and nitrogen, both of which may be taken as insoluble in the water. For a constant mole fraction of 0.05 of ammonia, it is found that the absorption rate is 25 per cent higher when the molar ratio of hydrogen to nitrogen is changed from 1 : 1 to 4 : 1. Is this result consistent with the assumption of a steady-state gas-film controlled process and, if not, what suggestions have you to make to account for the discrepancy? Neglect the partial pressure attributable to ammonia in the bulk solution. Diffusivity of ammonia in hydrogen D 52 ð 1000026 m2 /s Diffusivity of ammonia in nitrogen D 23 ð 1000026 m2 /s
Solution Using Maxwell’s Law for mass transfer of A in B then: 0002dPA D FCA CB uA 0002 uB
dy or:
0002RT Ð
(equation 10.77)
dCA D FCA CB N0A /CA 0002 N0B /CB
dy
For an absorption process N0B D 0 and: 0002
FCB dCA D N0A dy RT
i
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265
1 CB dCA D N0A Ð (equation 10.30) dy D CT RT RT or: F D ii
Thus: DD FCT DCT For the three-component system, equation 1 may be written: N0 dCA D A FAB CB C FAC CC
0002 dy RT RT RT and FAC D . From (2): FAB D DAB CT DAC CT 0002 0003 CB 1 dCA CC 0 Substituting: 0002 D NA C iii
dy DAB DAC CT Stefan’s law for 3-components system may be written as: dCA CT where D0 is the effective diffusivity N0A D 0002D0 Ð CT 0002 CA dy From Stefan’s Law: 0002
1 CT 0002 CA dCA D N0A 0 Ð dy D CT Comparing equations (iii) and (iv): 1 1 CB 1 CC D Ð C Ð D0 DAB CT 0002 CA DAC CT 0002 CA or:
0002
iv
xB0 x0 C C DAB DAC where xB0 and xC0 are mole fractions of B, C in the “stationary” gas. Taking A as NH3 , B as H2 and C as N2 , then: Case 1 xB0 D 0.5 xC0 D 0.5 0002 0003 0002 0003 1 0.5 0.5 D C D 0.03135 ð 106 s/m2 D0 52 ð 1000026 23 ð 1000026 D
D0 D 31.9 ð 1000026 m2 /s
and:
xBM D Mass transfer rate,
N0A D D
Case 2
and:
0.05 1 0002 0.95 xT D D 0.975, and D 1.026 00021 ln[1/0.95 ] ln0.95
xBm xT xT 0 D0 1 CA D CA D L xBm L xBm xT 1 CA Ð 31.9 L xBm
xB0 D 0.8 xC0 D 0.2 0002 0003 0002 0003 1 0.8 0.2 D C D 0.0241 ð 106 s/m2 D0 52 ð 1000026 23 ð 1000026 D0 D 41.5 ð 1000026 m2 /s
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xT /xBm , L and CA are as in case 1. ∴
N0A D
xT xT D0 1 CA D CA Ð 41.5 L xBm L xBm
41.5 N0A 2 D 1.30 or 1.3 times greater in the second case. D N0A 1 31.9 The observed ratio of two rates is only 1.25. This may be explained by: 1. Steady-state film conditions do not exist and there is some periodic partial disruption of the film. Penetration model ! N0A / D0.5 c/f. / D for film model. In this problem observed result would be accounted for by: N0A / D0m . where 1.3 m D 1.25 or m D 0.85. 2. The assumption of a gas-film controlled process may not be valid. If there is a liquid-film resistance, the effect of increasing the gas-film diffusivity will be less than predicted for a gas-film controlled process. 3. The value of the film thickness L is not the same because of different hydrodynamic conditions (second mixture having a lower viscosity). In this case, the film thickness would be expected to be reduced giving rise to the reverse effect so this is not a plausible explanation. 4. Experimental inaccuracies!
PROBLEM 10.37 Using a steady-state film model, obtain an expression for the mass transfer rate across a laminar film of thickness L in the vapour phase for the more volatile component in a binary distillation process: (a) where the molar latent heats of two components are equal. (b) where the molar latent heat of the less volatile component (LVC) is f times that of the more volatile component (MVC). For the case where the ratio of the molar latent heats f is 1.5. what is the ratio of the mass transfer rate in case (b) to that in case (a) when the mole fraction of the MVC falls from 0.75 to 0.65 across the laminar film?
Solution Case (a): With equal molar latent heats, equimolecular counter diffusion takes place and there is no bulk flow. Writing Fick’s Law for the MVC gives: NA D 0002D
dCA dy
(equation 10.4)
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MASS TRANSFER
Integrating for the steady state, then: NA D D
xA 0002 xA2
CA 1 0002 CA 2 D DCT 1 y2 0002 y1 L
i
Case (b): The net heat effect at the interface must be zero and hence: N0A 9A C N0B 9B D 0 N0B D 0002
and:
9A 0 1 N D 0002 N0A 9B A f
In this case: Total flux of A D Diffusional flux C Bulk-flow 0002 0003 dxA dCA 1 C uF CA D 0002DCT C xA N0A 0002 N0A or: N0A D 0002D dy dy f 0006 0002 00030007 dxA 1 Thus: N0A 1 0002 xA 1 0002 D 0002DCT f dy 0001 L 0001 CA 2 1 0002 0003 dxA dy D 0002DCT N0A 1 CA1 0 1 0002 xA 1 0002 f 0002 000300070007xA 0006 0006 2 DCT 1 0003 ln 1 0002 xA 1 0002 ∴ N0A L D 0002 0002 1 f xA1 10002 f 0002 0003 1 1 0002 xA2 1 0002 DCT f 0003 ln 0002 0003 D0002 ii
1 1 10002 L 1 0002 xA 1 1 0002 f f From equations (i) and (ii):
RD
N0A case b D NA case a
1
1 1 xA1 0002 xA2 10002 f
0002 0003 1 1 0002 xA 2 1 0002 f 0002 0003 ln 1 1 0002 xA 1 1 0002 f
Substituting f D 1.5 then: 10002
1 1 D f 3
xA1 D 0.75 RD For B:
1 1 3
Ð
xA2 D 0.65
1 0002 0.65 ð 1 ln 0.75 0002 0.65 1 0002 0.75 ð
1 3 1 3
D 1.303
R D 1.303/1.5 D 0.869.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 10.38 Based on the assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration CA of a solute A at a depth y below the interface at a time t after the formation of the interface is given by: 0006 0007 CA y D erfc p CAi 2 Dt
where CAi is the interface concentration, assumed constant and D is the molecular diffusivity of the solute in the solvent. The solvent initially contains no dissolved solute. Obtain an expression for the molar rate of transfer of A per unit area at time t and depth y, and at the free surface (y D 0). In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid. Occasionally, two droplets coalesce forming a single drop of twice the volume. What is the ratio of the mass transfer rate (kmol/s) at a coalesced drop to that at a single droplet when each has fallen the same distance, that is to the bottom of the column? The fluid resistance force acting on the droplet should be taken as that given by Stokes’ law, that is 30010001cdu where 001c is the viscosity of the continuous phase, d the drop diameter and u its velocity relative to the continuous phase. It may be noted that: 0001 1 2 2 erfcx D p ex dx. 0010 x
Solution CA 2 y D erfc p D p CAi 0010 2 Dt
0001
1 p y/2 Dt
e
0002y 2 /4Dt
0002
d
y p 2 Dt
0003
Differentiating with respect to y at constant t gives: 0011 0012 0001 1 1 ∂CA 1 2 ∂ 0002y 2 /4Dt p Dp e dt CAi ∂y 0010 ∂y 2 Dt y/2pDt
Dp
1 0010Dt
0002e0002y
2 /4Dt
1 2 0002e0002y /4Dt D CAi 0010Dt D D bt00021/2 At the interface, when y D 0: NA D CAi 0010t
∂CA D CAi 0002D
Thus: NA t D 0002D ∂y
D 0002y 2 /4Dt e 0010t
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From Stokes’ Law, the terminal falling velocity of the droplet is given by: 0010 30010001cdu0 D d3 +s 0002 + g 6 or:
u0 D
d2 g +s 0002 + D Kd2 18001c
Thus, the time taken for the droplet to travel the depth H of the rising liquid is: H 1 Kd2 2
Since the liquid is rising at a velocity of 12 Kd2 and the relative velocity is Kd2 0002 12 Kd2 D 12 Kd2 , the mass transfer rate (kmol/m2 s) to droplet at end of travel is: K b d 2H The mass transfer rate to the drop is: K K 3 K 1 2 d0010d D 0010b d D 0010b 0002 p d3 b 2H 2H H 2 For coalesced drops, the new diameter is: 21/3 d The terminal falling velocity is: K22/3 d2 Its velocity relative to the liquid is: K22/3 d2 0002 12 Kd2 D Kd2 22/3 0002 12
Thus:
Time of fall of drop D
H Kd2 22/3
0002 12
K 001b 2/3 1 d 2 0002 2 kmol/m2 s H K 001b 2/3 1 d 2 0002 2 001021/3 d 2 kmol/s Mass transfer rate to drop D bd H 001b K D 0010b 22/3 0002 12 22/3 d3 kmol/s H
Mass transfer rate at end of travel D bd
The ratio of the mass transfer rate for the coalesced drop to the mass transfer rate for the single droplet is then: 001b 22/3 0002 12 22/3 p D D 2.34 1/ 2
PROBLEM 10.39 In a drop extractor, a dense organic solvent is introduced in the form of spherical droplets of diameter d and extracts a solute from an aqueous stream which flows upwards at
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a velocity equal to half the terminal falling velocity u0 of the droplets. On increasing the flowrate of the aqueous stream by 50 per cent, whilst maintaining the solvent rate constant, it is found that the average concentration of solute in the outlet stream of organic phase is decreased by 10 per cent. By how much would the effective droplet size have had to change to account for this reduction in concentration? Assume that the penetration theory is applicable with the mass transfer coefficient inversely proportional to the square root of the contact time between the phases and that the continuous phase resistance is small compared with that within the droplets. The drag force F acting on the falling droplets may be calculated from Stokes’ Law, F D 30010001cduo , where 001c is the viscosity of the aqueous phase. Clearly state any assumptions made in your calculation.
Solution For droplets in the Stokes’ law region, the terminal falling velocity is given by: 0010/6 d3 +s 0002 + g D 30010001cdu0 or:
u0 D
d2 g +s 0002 + D kd2 18001c
The mass transfer rate to the droplet is Kte00021/2 moles per unit area per unit time / Kte1/2 moles per unit area in time te / Kte1/2 d2 moles per drop during time of rise The concentration of solute in drop: / Kte1/2 d2 /d3 / Kte1/2 d00021 . Initial case:
u0 D kd2 Rising velocity of liquid D
kd2 2
Velocity of liquid relative to container D kd2 0002
kd2 kd2 D 2 2
H D te . kd2 /2 2H 00021 2H 00022 Thus the concentration in the drop / K d D C1 d / K kd2 k Time of exposure in height H D
Second case:
New drop diameter D d0 Rising velocity of liquid D 1.5 ð
3 kd2 D kd2 2 4
Rising velocity of drop relative to liquid D kd02 Velocity relative to container D kd02 0002 34 kd2 Time of exposure D
kd02
H 0002 34 d2
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271
Concentration of solute in drop D Kte1/2 d0 00021
1/2 H C2 D K d0 00021 kd0 2 0002 34 d2
∴
Given that: C2 /C1 D 0.9, then:
K
1/2 H d000021
kd02 0002 34 d2
D 0.9 2H 00022 d K k
d000022 D 1.62d00024 d02 0002 34 d2
Squaring gives: Writing R D
d0 , then: d
1 D 1.62 R4 0002 34 R2 R4 0002 34 R2 0002 0.6173 D 0 R D 1.11 or 11.1 per cent increase
PROBLEM 10.40 According to the penetration theory for mass transfer across an interface, the ratio of the concentration CA at a depth y and time t to the surface concentration CAs at the liquid is initially free of solute, is given by y CA D erfc p CAs 2 Dt where D is the diffusivity. Obtain a relation for the instantaneous rate of mass transfer at time t both at the surface y D 0 and at a depth y. What proportion of the total solute transferred into the liquid in the first 90 s of exposure will be retained in a 1 mm layer of liquid at the surface, and what proportion will be retained in the next 0.5 mm? The diffusivity is 2 ð 1000029 m2 /s.
Solution For a rectangular particle:
Thus:
Thiele Modulus D 0016 D 9L 0015 k 5 ð 1000024 D D 500 m00021 9D D 2 ð 1000029
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LD
8 2
D 4 mm or 0.004 m
f D 500 ð 0.004 D 2 tanh f tanh 2 0.96 Thus the effectiveness factor, ; D D D D 0.48 f 2 2 For a spherical particle:
(equation 10.202)
Thiele Modulus D f D 9R RD
4 0010r 3 Volume ro D 3 o2 D Surface 40010ro 3
5 ð 1000023 m 3 5 ð 1000023 500 ð D 0.833 3 fD 9
R
RD
Thus the effectiveness factor,
1 1 coth 3f 0002 2 f 3f 1 1 D coth 2.5 0002 0.833 3 ð 0.8332 D 1.217 0002 0.480 D 0.736
;D
(equation 10.215)
PROBLEM 10.41 Obtain an expression for the effective diffusivity of component A in a gaseous mixture of A, B and C in terms of the binary diffusion coefficients DAB for A in B, and DAC for A in C. The gas-phase mass transfer coefficient for the absorption of ammonia into water from a mixture of composition NH3 20%, N2 73%. H2 7% is found experimentally to be 0.030 m/s. What would you expect the transfer coefficient to be for a mixture of composition NH3 5%, N2 60%, H2 35%? All compositions are given on a molar basis. The total pressure and temperature are the same in both cases. The transfer coefficients are based on a steady-state film model and the effective film thickness may be assumed constant. Neglect the solubility of N2 and H2 in water. Diffusivity of NH3 in N2 D 23 ð 1000026 m2 /s. Diffusivity of NH3 in H2 D 52 ð 1000026 m2 /s.
Solution For case 1: 1 The effective diffusivity D is given by: 0 D D 0
0002
73/80 23 ð 1000026
0003
0002
C
7/80 52 ð 1000026
0003
(from equation 10.90)
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MASS TRANSFER
D0 D 24.2 ð 1000026 m2 /s 0003 1 D0 The mass transfer coefficient is: L log mean 1 and 0.8 0016 001700021 0.2 24.2 ð 1000026 1 D D ð 27 ð 1000026 D 0.030 m/s 1 L L ln 0.8 or:
and hence:
0002
L D 0.90 ð 1000023 m
Case 2: 1 The effective diffusivity D is given by: 0 D D 0
0002
60/95 23 ð 1000026
0003
0002
C
35/95 52 ð 1000026
0003
D0 D 28.9 ð 1000026 m2 /s 0002 0003 D0 1 The mean transfer coefficient is: Ð L log mean 1 and 0.95 001700021 0016 0002 0003 28.9 ð 1000026 0.05 D D 0.033 m/s. 1 0.9 ð 1000023 ln 0.95
or:
PROBLEM 10.42 State the assumptions made in the penetration theory for the absorption of a pure gas into a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the far boundary of the liquid. Starting with Fick’s second law of diffusion, obtain an expression for (i) the concentration, and (ii) the mass transfer rate at a time t and a depth y below the surface. After 50 s, at what depth y will the concentration have reached one tenth the value at the surface? What is the mass transfer rate (i) at the surface, and (ii) at the depth y, if the surface concentration has a constant value of 0.1 kmol/m3 ?
Solution CA y D erfc p CAS 2 Dt
(equation 10.108)
Differentiating with respect to y: 0011 0002 0003 0001 1 1 ∂CA 2 y ∂ 0002y 2 /4Dt p p D e d CAS ∂y dy 0010 y/2pDt 2 Dt D 0002p Thus:
NA y,t
1
e0002y
2 /4Dt
0010Dt 0016 0017 0011 0012 D 0002y 2 /4Dt 1 0002y 2 /4Dt D 0002D 0002 p CAS D CAS e e 0010t 0010Dt
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When t D 50 s and CA /CAS D 0.1, then:
or:
y 0.1 D erfc p 00029 2 10 ð 50 y 0.9 D erf p 2 Dt
From Table 13 in the Appendix of Volume 1, the quantity whose error fraction D 0.9 is: y p D 1.16 2 Dt D CAS At the surface: NA yD0,t D 0010t 0015 1000029 ð 0.1 D 0010 ð 50 D 0.252 ð 1000026 kmol/m2 s. At a depth y:
2
NA D 0.252 ð 1000026 ð e00021.16 D 0.0656 ð 1000026 kmol/m2 s.
PROBLEM 10.43 In a drop extractor, liquid droplets of approximately uniform size and spherical shape are formed at a series of nozzles and rise countercurrently through the continuous phase which is flowing downwards at a velocity equal to one half of the terminal rising velocity of the droplets. The flowrates of both phases are then increased by 25 per cent. Because of the greater shear rate at the nozzles, the mean diameter of the droplets is, however, only 90 per cent of the original value. By what factor will the overall mass transfer rate change? It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may be neglected. From the penetration theory, the concentration CA at a depth y below the surface at time t is given by: 0006 0007 0001 1 CA y 2 2 D erfc p where erfc X D p e0002x dx CAS 0010 X 2 Dt
where CAS is the surface concentration for the drops (assumed constant) and D is the diffusivity in the dispersed (droplet) phase. The droplets may be assumed to rise at their terminal velocities and the drag force F on the droplet may be calculated from Stokes’ Law, F D 30010001c du.
Solution Case 1: For a volumetric flowrate Q1 , the numbers of drops per unit time is: Q1 1 6 0010d3
D
6Q1 0010d31
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275
The rising velocity is given by a force balance: 30010001cd1 u D 16 0010d31 +1 0002 +2 g or:
u1 D
d21 g +1 0002 +2 D Kd21 relative to continuous phase 18001c
The downward liquid velocity is 12 Kd21 The upward droplet velocity relative to container is: Kd21 0002 12 Kd21 D 12 Kd21 and the time of contact during rise through height H is: tc D
H 1 2 2 Kd1
.
(i)
The mass transfer rate is: 0002D∂CA /∂y . 0011 0012 0011 0002 00030012 0001 1 2 y y ∂ ∂ 1 ∂CA 0002y 2 /4Dt p p erfc p D e d D Thus: CAS ∂y ∂y ∂y 0010 y/2pDt 2 Dt 2 Dt 0001 1 1 ∂ 2 2 Ð p 0002p D e0002y /4Dt dt. ∂y 2 Dt 0010 y 0002 0003 CAS 0002y 2 /4Dt ∂CA CAS ∂CA D 0002p D 0002p or: e ∂y ∂y yD0 0010Dt 0010Dt The mass transfer rate at the surface is: (moles/area ð time). 0002 0003 D CAS 0002D 0002 p CAS D 0010t 0010Dt The mass transfer in time te1 is: 0001 te1 D D 1/2 1/2 00021/2 CAS te1 CAS D Kte1 t dt D 2 0010 0010 0 Substituting from equation (i): p 2H Mass transfer in moles per unit area of drop / p Kd1 The mass transfer per drop is proportional to: p H 00021 2 p H d1 d1 / 2 d1 2 K K The mass transfer per unit time D Mass transfer per drop ð drops/time p H H Q1 6Q1 d1 ð or: proportional to: 2 / 8.48 3 K K 0010d21 0010d1
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Case 2: diameter D 0.9d D d2 , Q2 D 1.25Q1 The number of drops per unit time is: 7.5Q1 10.29Q1 6Q2 D D 3 3 00100.9d1
0010d2 0010d31 Rising velocity D Kd21 D K0.9d1 2 D 0.81Kd21 Downward liquid velocity D
5 4
ð 12 Kd21 D 0.625 Kd21 relative to continuous phase
Rising velocity relative to container D 0.81 Kd21 0002 0.625 Kd21 D 0.185Kd21 Contact time, te2 D
H 0.185Kd21
Mass transfer per drop Mass transfer per unit time
H 1 / 2.325 0.185K d1
H 00021 d K 1 H 00021 H 2 d1 ð 0.9d1 / 1.883 d1 / 2.325 K K H 10.29Q1 d1 ð / 1.883 K 0010d31 H Q1 / 19.37 K 0010d21
Mass transfer in time te2 per unit area /
kte1/2 2
/
Thus, the factor by which mass transfer rate is increased is: 19.37/8.48 D 2.28
PROBLEM 10.44 According to Maxwell’s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble, and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. Carbon dioxide is absorbed in alkaline water from a mixture consisting of 30% CO2 and 70% N2 , and the mass transfer rate is 0.1 kmol/s. The concentration of CO2 in the gas in contact with the water is effectively zero. The gas is then mixed with an equal molar quantity of a second gas stream of molar composition 20% CO2 , 50%, N2 and 30% H2 . What will be the new mass transfer rate, if the surface area, temperature and pressure remain unchanged? It may be assumed that a steady-state film model is applicable and that the film thickness is unchanged. Diffusivity of CO2 in N2 D 16 ð 1000026 m2 /s. Diffusivity of CO2 in H2 D 35 ð 1000026 m2 /s.
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MASS TRANSFER
Solution For a binary system, Maxwell’s Law gives: 0002
dCA D FCA CB uA 0002 uB
dy
For an ideal gas mixture: 0002RT
dCA D FCA CB dy
0002
(equation 10.77)
N0A N0 0002 B CA CB
0003
For a gas absorption process: N0B D 0 if B is insoluble or:
0002
dCA FCB 0 D N dy RT A
i
from Stefan’s law (equation 10.30): N0A D 0002D or:
0002
CT dCA CB dy
1 CB dCA D N0A dy D CT D N0A
1 CT 0002 CA D CT
ii
iii
Comparing equations (i) and (ii): F 1 D RT DCT or:
FD
RT RT or D D DCT FCT
Applying Maxwell’s law to a multicomponent system gives: 0002
dCA D FAB CA CB uA 0002 uB C FAC CA CC uA 0002 uC C Ð Ð Ð dy
For B, C . . . insoluble N0B , N0C . . . D 0 Writing: then: ∴
FAB D
RT RT , and FAC D DAB CT DAC CT
RT dCA RT D N0A CB C N0 CC C Ð Ð Ð . dy DAC CT DAC CT A 0002 0003 N0A CB dCA CC D 0002 C C ÐÐÐ dy CT DAB DAC
0002RT
From equation (iii), using an effective diffusivity D0 for a multicomponent system gives: 0002
1 CT 0002 CA dCA D N0A 0 dy D CT
iv
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Comparing equations (iii) and (iv), then: 0011 0012 1 CB 1 CC D C C ... D0 CT 0002 CA DAB DAC D where xB0 , xC0 Ð Ð Ð D
xB0 x0 C C C ... DAB DAC CB CC > ... CT 0002 CA CT 0002 CA
D mole fraction of B, C . . . in mixture of B, C For absorption of CO2 from mixture with N2 , the concentration driving force is: CA D CT 0.3 0002 0 D 0.3CT where CT is the total molar concentration The mass transfer rate for an area A D AN0A D CA where: or:
CBm is the log mean of 1 ð CT and 0.7CT CBm D
CT 0002 0.7CT CT D 0.841CT and: D 1.189 CT CBm ln 0.7CT
Mass transfer rate, AN0A D 0.3CT or:
D CT Ð ÐA L CBm
16 ð 1000026 CT A 1.189 A D 5.71 ð 1000026 D 0.10 kmol/s L L
CT A D 0.0175 ð 106 kmol/m2 . L
For absorption of CO2 from mixed stream, stream composition must be calculated. 100 moles stream 1 ! 30 moles CO2 70 moles N2 100 moles stream 2 ! 20 moles CO2 50 moles N2 30 moles H2 200 moles mixture ! 50 moles CO2 120 moles N2 30 moles H2 100 moles mixture ! 25 moles CO2 60 moles N2 15 moles H2 xN0 2 D
60 15 D 0.8 xH0 2 D D 0.2 75 75
Diffusivity of CO2 in mixture is given by: 1 0.8 0.2 D C 0 00026 D 16 ð 10 35 ð 1000026 1000026 1 1 C D 0.0557 s/m2 D 0 D 20 175 D0 D 17.9 ð 1000026 m2 /s
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Thus:
CA D CT 0.25 0002 0 D 0.25CT CT 1 0002 0.75 CT D 0.869CT D 1.150 1 C ln 0.75 Bm
CBm D and the mass transfer rate: ANA0 D 0.25CT
17.9 ð 1000026 ACT 1.150A D 5.15 ð 1000026 kmol/s D 0.090 kmol/s L L
PROBLEM 10.45 What is the penetration theory for mass transfer across a phase boundary? Give details of the underlying assumptions. From the penetration theory, the mass transfer rate per unit area NA is given in terms of the concentration difference CA between the interface and the bulk fluid, the molecular diffusivity D and the age t of the surface element by: D CA kmol/m2 s (in SI units) NA D 0010t What is the mean rate of transfer if all elements of the surface are exposed for the same time te before being remixed with the bulk? Danckwerts assumed a random surface renewal process in which the probability of surface renewal is independent of its age. If s is the fraction of the total surface renewed per unit time, obtain the age distribution function for the surface and show that the mean mass transfer rate NA over the whole surface is: p NA D DsCA kmol/m2 s, in SI units
In a particular application, it is found that the older surface is renewed more rapidly than the recently formed surface, and that after a time s00021 , the surface renewal rate doubles, that is it increases from s to 2s. Obtain the new age distribution function.
Solution Assuming the age spread of the surface ranges for t D 0, to t D 1, consider the mass transfer per unit area in each age group is t to t C dt and so on. Then the mass transfer to surface in age group t to t C dt is: D CA t00021/2 dt D 0010 Thus the total mass transfer per unit area is:
D CA 0010
0001
te
t 0
00021/2
dt D
0004 0005te D t1/2 D CA 1 CA te1/2 D2 0010 0010 2 0
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The average mass transfer rate is: 0015
1 D D 1/2 CA te 2 D2 CA te 0010 0010te In the steady state if ft is the age distraction function of the surface, then: surface in age group t to t C dt D ft dt and:
surface in age group t 0002 dt to t D ft 0002 dt dt
Surface of age t 0002 dt to t which is destroyed is that not entering the next age group ft 0002 dt dt 0002 ft dt D s fft 0002 dt dtg dt 0
As dt ! 0, then: 0002f t dt dt D sft dt dt est f0 t C est sft D 0 est ft D K
Integrating gives:
ft D Ke0002st 0006 0002st 00071 1 e K 0002st As the total surface D 1, K e dt D K D 0002s 0 s 0 then:
and:
i
0001
∴KDs
ft D se0002st
ii
The mass transfer rate into the fraction of surface in the age group t 0002 t dt is: D CA se0002st dt 0010t The mass transfer rate into the surface over the age span t D 0 to t D 1 is: 0001 1 D D 00021/2 0002st CA s CA t e dt D D 0010 0010 0 p s Putting D st D ˇ2 , then: t00021/2 D ˇ and:
s dt D 2ˇ dˇ. p 0001 1p 0001 1 s 0002ˇ2 2ˇ dˇ 0010 0010 2 2 0002ˇ2 e Dp D e dˇ D p Ð ID ˇ s s 0 s 2 s 0
Thus the mass transfer rate per unit area for the surface as a whole is: D 0010 p CA s D DsCA 0010 s With the new age distribution function: 1 surface renewal rate/area D s 0< />< ,=' s=' 1=' /><=' t=' 1,=' surface=' renewal=' rate=' area=' d=' 2s='>='>
ft D Ke0002st ft D K0 e00022st
from equation (ii)
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00071/s e0002st K D 1 0002 e00021
0002s s 0 0 0006 00022st 00071 0001 1 1 e K0 00022 Fraction of surface of age to 1 D K0 e e00022st dt D K0 D s 00022s 1/s 2s 1/s
Fraction of surface of age 0 to
1 DK s
0001
0006
1/s
e0002st dt D K
K K0 1 0002 e00021 C e00022 s 2s At t D 1/s , both age distribution functions must apply, and: The total surface is unity or:
Ke00021 D K0 e00022 or: K0 D K Ð e K Ke 00022 1 0002 e00021 C e s 2s 000e K 000e K D 1 0002 e00021 C 12 e00021 D 1 0002 12 e00021 s s
Thus, for the total surface: 1 D
K D s1 0002 12 e00021 00021
Thus: Thus 0 < t <
1 s
1 < /><1>1>
K0 D se1 0002 12 e00021 00021
000e00021 0002st ft D s 1 0002 12 e00021 e
000e00021 00022st
000e00021 100022st e D s 1 0002 12 e00021 e ft D se 1 0002 12 e00021
PROBLEM 10.46 Derive the partial differential equation for unsteady-state unidirectional diffusion accompanied by an nth-order chemical reaction (rate constant k ): ∂ 2 CA ∂CA D D 2 0002 k CnA ∂t ∂y where CA is the molar concentration of reactant at position y at time t. Explain why, when applying the equation to reaction in a porous catalyst particle, it is necessary to replace the molecular diffusivity D by an effective diffusivity De . Solve the above equation for a first-order reaction under steady-state conditions, and obtain an expression for the mass transfer rate per unit area at the surface of a catalyst particle which is in the form of a thin platelet of thickness 2L. Explain what is meant by the effectiveness factor ; for a catalyst particle and show that it is equal p to 1/f tanh f for the platelet referred to previously where f is the Thiele modulus L k /De . For the case where there is a mass transfer resistance in the fluid external to the particle (mass transfer coefficient hD ), express the mass transfer rate in terms of the bulk concentration CAo rather than the concentration CAS at the surface of the particle. For a bed of catalyst particles in the form of flat platelets it is found that the mass transfer rate is increased by a factor of 1.2 if the velocity of the external fluid is doubled.
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The mass transfer coefficient hD is proportional to the velocity raised to the power of 0.6. What is the value of hD at the original velocity? k D 1.6 ð 1000023 s00021 ,
De D 1000028 m2 /s
catalyst pellet thickness 2L D 10 mm.
Solution (i) The partial differential equation for unsteady-state diffusion accompanied by chemical reaction is derived in Volume 1 as equation 10.170 (ii) The molecular diffusivity D must be replaced by an effective diffusivity De because of the complex internal structure of the catalyst particle which consists of a multiplicity of interconnected pores, and the molecules must take a tortuous path. The effective distance the molecules must travel is consequently increases. Furthermore, because the pores are very small, their dimensions may be less than the mean free path of the molecules and Knudsen diffusion effects may arise (iii) Equation 10.170 is solved in Volume 1 to give equation 10.199 for a catalyst particle in the form of a flat platelet (iv) The effectiveness factor is the ratio of the actual rate of reaction to that which would be achieved in the absence of a mass-transfer resistance. For a platelet, it is evaluated in terms of the Thiele modulus as equation 10.202 (v) For the case, where there is an external mass transfer resistance, the reaction rate is expressed in terms of the bulk concentration as equation 10.222: Rv D
k CAo 1/; C k L/hD
For k D 1.6 ð 1000023 s00021 , De D 1000028 m2 /s, L D 5 ð 1000023 m: 0015 0015 k 1.6 ð 1000023 0016DL D 5 ð 1000023 D2 De 1000023 ;D ∴
0.96 1 1 D 0.48 tanh f D tanh 2 D f 2 2 1 1 1 D D D 0.260 ð 106 . 00023 00023 ;k L 0.48 ð 1.6 ð 10 ð 5 ð 10 3.84 ð 1000026
If the original value of mass transfer coefficient is hD , the new value at twice original velocity D hD 2 0.6 D 1.516 hD . Given that the overall rate is increased by a factor of 1.2: & 1 1 D 1.2 1 1 6 6 0.260 ð 10 C 0.260 ð 10 C 1.516hD hD 0002 0003 1 6 0.260 ð 10 C hD 0002 0003 D 1.2 1 0.260 ð 106 C 0.66 hD
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MASS TRANSFER
0.260 ð 106 C
1 1 D 0.312 ð 106 C 0.792 hD hD 1 D 0.25 ð 106 and: hD D 4.0 ð 1000026 m/s . hD
PROBLEM 10.47 Explain the basic concepts underlying the two-film theory for mass transfer across a phase boundary and obtain an expression for film thickness. Water evaporates from an open bowl at 349 K at the rate of 4.11 ð 103 kg/m2 s. What is the effective gas-film thickness? The water is replaced by ethanol at 343 K. What will be its rate of evaporation in kg/m2 s if the film thickness is unchanged? At the surface of the ethanol, what proportion of the total mass transfer will then be attributable to bulk flow? Data. Vapour pressure of water at 349 K D 34 mm Hg Vapour pressure of ethanol at 343 K D 544 mm Hg Neglect the partial pressure of vapour in the surrounding atmosphere Diffusivity of water vapour in air D 26 ð 1000026 m2 /s Diffusivity of ethanol in air D 12 ð 1000026 m2 /s Density of mercury D 13,600 kg/m3 Universal gas constant R D 8314 J/kmol K
Solution For evaporation for a free surface, Stefan’s law is applicable or: N0A D 0002D N0A D
Integration gives:
D
CT dCB CT dCT D CD CB dy CB dy
(from equation 10.30)
D CB CT ln 2 y2 0002 y1 CB 1 CT D CA1 0002 CA2
y2 0002 y1 CBm
For water: Vapour pressure, PA1 D 301 mm Hg at 349 K At 349 K:
CA1 D
0.301 ð 13,600 ð 9.81
PA1 D RT 8314 ð 349
D 0.0138 kmol/m3
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CT D
P 0.760 ð 13,600 ð 9.81
D RT 8314 ð 349
D 0.0350 kmol/m3
For air:
CB1 D CT 0002 CA1 D 0.0350 0002 0.0138
D 0.0212 kmol/m3 CB2 D CT D 0.0350 kmol/m3 .
and:
N0A D
The evaporation rate of water is: D
26 ð 1000026 ð 0.0350 ln y2 0002 y1
0002
0.0350 0.0212
0003
0.456 ð 1000026 kmol/m2 s y2 0002 y1
But: N0A D 4.11 ð 1000023 kg/m2 s D 0.228 ð 1000023 kmol/m2 s. y1 2 ð 1000023 m D 2 mm giving a film thickness of: y2 0002 D For ethanol: Vapour pressure D 541 mm Hg at 343 K. At 343 K:
CA1 D
0.541 ð 13,600 ð 9.81
PA1 D RT 8314 ð 343
D 0.0253 kmol/m3 0.760 ð 13,600 ð 9.81
P D CT D RT 8314 ð 343
D 0.0356 kmol/m3 CB1 D CT 0002 CA1 D 0.0103 kmol/m3 and:
CB2 D CT D 0.0356 kmol/m3 0002 0003 0002 0003 12 ð 1000026 0.0356 0 The evaporation rate of ethanol, NA D ð 0.0356 ð ln 0.002 0.0103 D 0.265 ð 1000023 kmol/m2 s D 12.2 ð 1000023 kg/m2 s
The total flux at any location D diffusional flux C bulk flow bulk flow diffusional flux CB D10002 D10002 total flux total flux CT At the ethanol surface, the proportion of flux due to bulk flow is: 10002
0.0103 D 0.71 0.0356
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SECTION 11
The Boundary Layer PROBLEM 11.1 Calculate the thickness of the boundary layer at a distance of 75 mm from the leading edge of a plane surface over which water is flowing at a rate of 3 m/s. Assume that the flow in the boundary layer is streamline and that the velocity u of the fluid at a distance y from the surface can be represented by the relation u D a C by C cy 2 C dy 3 , where the coefficients a, b, c, and d are independent of y. The viscosity of water is 1 mN s/m2 .
Solution At a distance y from the surface: u D a C by C cy 2 C dy 3 . When y D 0, u D 0, and hence a D 0. The shear stress within the fluid: R0 D 0003 ∂u/∂y yD0 and since ∂u/∂y is constant for small values of y, ∂2 u/∂y 2 yD0 D 0. At the edge of the boundary layer, y D υ and u D us , the main stream velocity. ∂u/∂y D 0 and u D by C cy 2 C dy 3 ∴
∂u/∂y D b C 2cy C 3dy 2 and ∂2 u/∂y 2 D 2c C 6dy When y D 0, ∂2 u/∂y 2 D 0, and hence c D 0. When y D υ, u D bυ C dυ3 D us
and:
∂u/∂y D b C 3dυ2 D 0
∴
b D 00033dυ2
∴
d D 0003us /2υ3 and b D 3us /2υ The velocity profile is given by, u D 3us y/2υ 0003 us /2 y/υ 3
or:
u/us D 1.5 y/υ 0003 0.5 y/υ 3
(equation 11.12)
The integral in the momentum equation 11.9 is now evaluated, and substituting from equations 11.14 and 11.15 into equation 11.9:
υ/x D 4.64 Re00030.5 x Rex D 0.075 ð 3 ð 1000/1 ð 1000033 D 225,000 υ/x D 4.64 ð 225,00000035 D 0.00978 and:
υ D 0.00978 ð 0.075 D 0.000734 m or 0.734 mm 285
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PROBLEM 11.2 Water flows at a velocity of 1 m/s over a plane surface 0.6 m wide and 1 m long. Calculate the total drag force acting on the surface if the transition from streamline to turbulent flow in the boundary layer occurs when the Reynolds group Rex D 105 .
Solution See Volume 1, Example 11.1
PROBLEM 11.3 Calculate the thickness of the boundary layer at a distance of 150 mm from the leading edge of a surface over which oil, of viscosity 50 mN s/m2 and density 990 kg/m3 , flows with a velocity of 0.3 m/s. What is the displacement thickness of the boundary layer?
Solution See Volume 1, Example 11.2
PROBLEM 11.4 Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm diameter at 0.003 m3 /s. What is the velocity of the benzene at the edge of the laminar sub-layer? Assume fully developed flow exists within the pipe.
Solution See Volume 1, Example 11.3
PROBLEM 11.5 Air is flowing at a velocity of 5 m/s over a plane surface. Derive an expression for the thickness of the laminar sub-layer and calculate its value at a distance of 1 m from the leading edge of the surface. Assume that within the boundary layer outside the laminar sub-layer, the velocity of flow is proportional to the one-seventh power of the distance from the surface and that the shear stress R at the surface is given by:
R/0013us2 D 0.03 us 0013x/ 00030.2 where 0013 is the density of the fluid (1.3 kg/m3 for air), is the viscosity of the fluid (17 ð 1000036 N s/m2 for air), us is the stream velocity (m/s), and x is the distance from the leading edge (m).
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THE BOUNDARY LAYER
287
Solution The shear stress in the fluid at the surface: R D 0003 ux /y From the equation given: ∴
R D 0.030013us2 /us 0013x 0.2 ux D 0.030013us2 y/ /us 0013x 0.2
If the velocity at the edge of the laminar sub-layer is ub , ux D ub when y D υb . ∴
ub D 0.030013us2 υb / /us υ0013 0.2
υb /υ D 33.3 ub /us /us υ0013 0.8
and:
From equation 11.24, the velocity distribution is given by:
υb /υ 1/7 D ub /us
ub /us 7 D 33.3 ub /us /us υ0013 0.8
and hence:
ub /us D 1.65 /us υ0013 0.115
or:
D 1.65 Re00030.115 υ Substituting 0.376x 0.8 /us 0013 0.2 for υ from equation 11.29:
ub /us D 1.65[0.376us 0013x 0.8 0.2 / us0.2 00130.2 ]00030.115 D 1.65/0.3760.115 us0.8 x 0.8 00130.8 / 0.8 00030.115 D 1.85 Re00030.09 x Now: From equation 11.31: ∴
υb /υ D ub /us 7 D 74.2 Rex0.63
υ/x D 0.376 Re00030.2 x
υb /x D 74.2 ð 0.376 / Rex0.63 Rex0.2 D 27.9 Re00030.83 x
In this case:
Rex D 1 ð 5 ð 1.3/17 ð 1000036 D 3.82 ð 105 υb D 1.0 ð 27.9 3.82 ð 105 00030.83 D 6.50 ð 1000034 m or 0.65 mm
PROBLEM 11.6 Obtain the momentum equation for an element of the boundary layer. If the velocity profile in the laminar region can be represented approximately by a sine function, calculate the boundary layer thickness in terms of distance from the leading edge of the surface.
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Solution The derivation of the momentum equation for an element of the boundary layer is presented in detail in Section 11.2 and the final expression is: 0001
l
us 0003 ux ux dy
0003Ro D 0013∂/∂x 0
A sine function may be developed as follows. When y D 0, ux D 0 and when y D υ, ux D us . ux D us sin ay
Thus:
and when y D υ, sin ay D 0016/2 or aυ D 0016/2 and a D 0016/2υ. ∴
ux D us sin 0016y/2d
and over the range 0 < y < υ,
ux /us D sin[ 0016/2 y/υ ] The integral in the momentum equation may now be evaluated for the laminar boundary layer considering the ranges 0 < y < υ and υ < y < l separately. 0001 υ 0001 l ∴
us 0003 ux ux dy D us2 f1 0003 sin[ 0016/2 y/υ ]gfsin[ 0016/2 y/υ ]g dy 0
0
0001
l
us 0003 us us dy
C υ
0001
D us2
υ
[sin 0016y/2υ 0003 sin2 0016y/2υ ] dy 0
D us2 [0003[cos 0016y/2υ ]/ 0016/2υ 0003 y/2 C sin 0016y/υ / 20016/υ ]υ0 D us2 υ[ 2/0016 0003 1/2 ] R0 D 0003 ∂ux /∂y yD0 D 0003 us 0016/2υ and substituting in the momentum equation: 0002 0003 00040005 0006 2 1 0003 ∂x D us 0016/2υ 0013∂ us2 υ 0016 2 ∴
υ dυ D 00162 dx/0013us 4 0003 0016
∴
υ2 /2 D [00162 / 4 0003 0016 ] x/0013us
and: ∴
υ D 4.80 x/0013us 0.5
υ/x D 4.80 /x0013us 0.5 D 4.80 Re00030.5 x
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THE BOUNDARY LAYER
PROBLEM 11.7 Explain the concepts of “momentum thickness” and “displacement thickness” for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this ‘thermal thickness’ in terms of the thicknesses of the velocity and temperature boundary layers. Similar forms of cubic equations may be used to express velocity and temperature variations with distance from the surface. For a Prandtl number, Pr, less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 00031/3 . Work out the ‘thermal thickness’ in terms of the thickness of the velocity boundary layer for a value of Pr D 0.7.
Solution Consideration is given to the streamline portion of the boundary layer in Section 11.3 where, assuming: ux D uo C ay C by 2 C cy 3 (equation 11.10) it is shown that the equation for the velocity profile is:
ux /us D 1.5 y/υ 0003 0.5 y/υ 3
(equation 11.12)
The equivalent equation for the thermal boundary layer will be:
001a/001as D 1.5 y/υt 0003 0.5 y/υt 3 where υt is the thickness of the thermal boundary layer. The heat flow is given by: 0001 l QD Cp 0013ux T dy 0
0001
l
[1.5 y/υt 0003 0.5 y/υt 3 ][1.5 y/υ 0003 0.5 y/υ 3 ] dy
D us Ts Cp 0013 0
This is made up of two components: the heat flow through the thermal boundary layer: 0001 υt /υ D us Ts Cp 0013 f 2.25y 2 /υ Ð υt 0003 0.75y 4 / υ3 υt 0003 0.75y 4 / υ Ð υ3t 0
6
C 0.25y /υ2 υ5t g d y/υ and the heat flow through the velocity boundary layer between y D υt and y D υ: 0001 1 D us Ts Cp 0013υ [1.5y/υ 0003 0.5 y/υ 3 ] d y/υ
υt /υ
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Thus, putting
D υt /υ , the heat flow becomes: 00020001 x Q D us Ts Cp 0013υ
2.25 y/υ 2 / 0003 0.75 y/υ 4 1/ C 1/ 0001
3
C 0.25 y/υ 6 / 3 d y/υ
0
l
0005 1.5 y/υ 0003 0.5 y/υ d y/υ 3
C x
D us Ts Cp 0013υ[ 0.75 3 / 0003 0.15 5 1/ C 1/ 3 C 0.036 7 / 3 C 0.75 1 0003
2
0003 0.125 1 0003
D us Ts Cp 0013υ 0.625 0003 0.15
2
4
]
C 0.0107 4
The heat flow from υ to υŁt in the absence of boundary layers D υ 0003 υŁt us Ts 0013Cp . ∴
υ 0003 υŁt D υ 0.625 0003 0.15
and:
υŁt /υ D 0.375 C 0.15
2
2
C 0.0107 4
0003 0.0107
4
When < 1, then D Pr 00030.33 and neglecting the 4 term, an approximate value is:
υŁt /υ D 0.375 C 0.15Pr 00030.67 When Pr D 0.7, Pr 0.67 D 0.788 and:
υŁt /υ D 0.375 C 0.15/0.788 D 0.185
(Since this is much less than 1, neglecting the
4
term is justified.)
PROBLEM 11.8 Explain why it is necessary to use concepts, such as the displacement thickness and the momentum thickness, for a boundary layer in order to obtain a boundary layer thickness which is largely independent of the approximation used for the velocity profile in the neighbourhood of the surface. It is found that the velocity u at a distance y from the surface can be expressed as a simple power function (u / y n ) for the turbulent boundary layer at a plane surface. What is the value of n if the ratio of the momentum thickness to the displacement thickness is 1.78?
Solution The first part of this problem is discussed in Section 11.1. If the displacement and the momentum thicknesses are υŁ and υm respectively, then: 0001 υ 0001 υ the momentum flux D uy dy0013uy D us dy0013us 0
0001
and:
the mass flux D
0001
υ
υm υ
uy dy0013 D 0
us dy0013 υŁ
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THE BOUNDARY LAYER
0001 ∴
υ
uy2 dy D us2 υ 0003 υm
0
0001
υ
uy dy D us υ 0003 υŁ
and: 0
0001
1
uy /us 2 d y/υ D 1 0003 υm 0003 υ
0
0001
1
uy /us d y/υ D 1 0003 υŁ /υ
and: 0
0002 000500060002 0005 0001 1 0001 1
υm /υŁ D 1 0003
uy /us 2 d y/υ 10003
uy /us d y/υ
∴
0
0
If uy /us D y/υ n , then: 0001 1 0001 1
uy /us d y/υ D
y/υ n d y/υ D 1/ n C 1 0
0001
0001
1
and:
uy /us d y/υ D 0
∴
0 1
y/υ 2n d y/υ D 1/ 2n C 1
0
υm /υŁ D [1 0003 1/ 2n C 1 ]/[1 0003 1/ n C 1 ] D 2 n C 1 / 2n C 1 When υm /υŁ D 1.78: 1.78 D 2n C 2/ 2n C 1 D 3.56n C 1.78 D 2n C 2
and:
n D 0.22/1.56 D 0.141 or approximately 1/7 .
PROBLEM 11.9 Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface.
Solution Using the nomenclature of Fig. 11.5, the argument presented in Section 11.2 results in the expression known as the momentum equation, given in equation 11.9, which may be expressed as: 0001 l ux us 0003 ux dy D 0003R0
i 0013∂/∂x 0
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
If the velocity within the boundary layer may be represented by a sine function: 0001
the integral: 0
ux /us D sin[ 0016/2 y/υ ] 0001 1 l ux us 0003 ux dy D us2 υ
ux /us 1 0003 ux /us d y/υ
ii
0
and, substituting from equation (ii): 0001 l 0001 1 ux us 0003 ux dy D us2 υ [sin 0016/2 y/υ 0003 sin2 0016/2 y/υ ] d y/υ 0
0
0007
D us2 υ
1 0003 2/0016 cos[ 0016/2 y/υ ] 0 0003 0.5
0001
1 D us2 υf 2/0016 0003 0.5 C 1/0016 sin 0016y/υ 0 g
1
1 0003 cos 0016y/υ d y/υ
0
D us2 υ 2/0016 0003 0.5 D 0.1366us2 υ From equation (ii): ∴
ux D us sin[ 0016/2 y/υ ] dux /dy D us 0016/2υ cos[ 0016/2 y/υ ]
and when y D 0:
dux /dy yD0 D 0016/2 us /υ But:
R0 D 0003 dux /dy yD0 D 0003 0016/2 us /υ
Therefore, substituting in equation (i): 0013∂/∂x 0.1366us2 υ D 0016/2 us /υ υ dυ D /0013us 0016/0.2732 dx υ2 /x D x/0013us 0016/0.2732 υ2 /x 2 D /0013us x 0016/0.1366 and:
υ/x D 4.80 Re00030.5 x
PROBLEM 11.10 Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface can be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of the surface, when the velocity profile is expressed as a sine function.
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THE BOUNDARY LAYER
Solution The derivation of the momentum equation is given in Section 11.2 to give: 0001 l ∂ 0013 ux us 0003 ux dy D 0003R0 D dux /dy yD0 (equation 11.9) (i) ∂x 0 If the velocity profile is a sine function, then:
ux /us D sin[ 0016/2 y/υ ] Differentiating:
1/us ∂ux /∂y D 0016/2υ cos[ 0016/2 y/υ ] When y D υ, ux D us and ∂ux /∂y D 0 Differentiating again:
1/us ∂2 ux /∂y 2 D 000300162 /4υ2 sin[ 0016/2 y/υ ] When y D 0, ux D 0 and ∂2 ux /∂y 2 D 0 Substituting, noting that ux D us when y > υ: 0001 l 0001 υ ux ux 0003 us dy D us2 sin y/υ 0016/2 [1 0003 sin y/υ 0016/2 ] dy 0
0001
D us2 υ D us2 υ
0
1
[sin y/υ 0016/2 0003 0.5 1 0003 cos y0016/υ ] d y/υ 0
000b 1 1 1 0003 2/0016 cos y/υ 0016/2 0 0003 0.5 y/υ 0 C 0.5 1/0016 sin y0016/υ 0
D us2 υ
2/0016 0003 0.5 C 0 D 0.1366us2 υ
(as in Problem 11.8)
Substituting in equation (i): 0013∂ 0.1366us2 υ /∂x D 0016/2 us /υ ∴
υdυ D
0016/2 /0.1366 /0013us dx
If υ D 0 when x D 0: υ2 /x D 11.5 x/0013us2 and:
υ/x 2 D 11.5 /0013us x and υ/x D 3.39 Re00030.5 x
The displacement thickness, υŁ is given by: 0001 υ 1 us υ 0003 υŁ D us sin[ y/υ 0016/2 ] dy D us 2υ/0016 0003 cos[ y/υ 0016/2 ] 0 0
∴
and: ∴
υ 0003 υŁ D 2υ/0016 D 0.637υ υŁ D 0.363υ
υŁ /x D 0.363 ð 3.39 Re00030.5 D 1.23 Re00030.5 x x
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PROBLEM 11.11 Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface.
Solution The total mass flowrate through plane 1–2 is:
l 0
0013ux dy.
The total momentum flux through plane 1–2 is:
l 0
0013ux2 dy
000f ∂ 000e l 0013u dy dx x ∂x 0 ∂ 000e l 2 000f 0013u dy dx. Change in momentum flux from 1–2 to 3–4 is: ∂x 0 x
Change in mass flowrate from 1–2 to 3–4 is:
Change in momentum flux is attributable, in the absence of pressure gradient, to: (a) Momentum of fluid entering through 2–4 Since all this fluid has velocity us , the momentum flux is: 0007 00020001 l 0005
∂ 0013ux dy dx us ∂x 0 (b) Force due to shear stress at surface D R0 dx. Thus a momentum balance gives: 00070001 l
00070001 l
∂ ∂ 0013ux2 dy dx D 0013ux dy dx.us C R0 dx. ∂x ∂x 0 0 00070001 l
∂ ∴ 0013 ux us 0003 ux dy D 0003R0 ∂x 0 Representing velocity within boundary layer by a sine function, then: 0016 y ux D sin us 2υ Thus:
0001
l 0
0003 0004 ux y ux 10003 d u u υ s s 0 0001 1 0016 y y 0016 y D us2 υ sin Ð 0003 sin2 d 2 υ 2υ υ 0 0001
ux us 0003 ux dy D us2 υ
1
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THE BOUNDARY LAYER
00100002
D
2 0016y 0003 cos 0016 2υ
us2 υ 0010
0005l
1 0003 2 0
0002
0001
1
1 0003 cos 0
0016y y d υ υ
0011
0005l 0011
2 1 1 y 0003 C sin 0016 0016 2 0016 υ 0 0007
2 1 D us2 υ D 0.1366us2 υ. 0003 0016 2 0016 0016 y dux D us Ð cos dy 2υ 2υ 0004 0003 0016 u dux s D Ð dy yD0 2 υ 0003 0004 dux 0016 us . R0 D 0003 D0003 dy yD0 2 υ D us2 υ
Differentiating:
and: Thus: ∴ ∴ ∴
0013
∂ 0016 us
0.1366us2 υ D ∂x 2 υ 0003 0004 0016 υ dυ D dx. 0013us 0.2732 0003 0004 υ2 x 0016 D 2 0013us 0.2732 0003 0004 2 υ 0016 D x2 0013us x 0.1366 υ D 4.796 Re00031/2 x x
and:
PROBLEM 11.12 Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of the surface, when the velocity profile is expressed as a sine function.
Solution The momentum flux across 1–2 is: The change from 1–2 to 3–4 is: 0013
L 0
0013ux2 dy
∂ 000e L 2 000f u dy dx ∂x 0 x
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
The mass flux across 1–2 is:
L 0
0013ux dy.
The change from 1–2 to 3–4 is: 0013
000f ∂ 000e L u dy dx. x ∂x 0
The rate of momentum entering through 2–4 is: 00070001 L
∂ 0013 ux us dy dx, ∂x 0 Assuming us 6D f x , then: ∂P D 0. ∂x A momentum balance gives: 00070001 L
00070001 L
∂ ∂ 0013 ux2 dy dx D 0013 ux us dy dx C R0 dx ∂x ∂x 0 0 00070001 L
0003 0004 ∂ux ∂ ux us 0003 ux dy D 0003R0 D for a Newtonian fluid or: 0013 ∂x ∂y yD0 0 The sine function is: ux D K sin ky, so
∂2 ux ∂ux D Kk cos ky and Kk 2 sin ky ∂y ∂y 2
is satisfied for all finite values of K and k. The boundary conditions are: y D 0 ux D 0 yD0
∂2 ux D0 ∂y 2
y D υ ux D us yDυ Thus:
K D us
∂ux D0 ∂y
0016 0016 and k D . 2 2υ ∂ux 0016us 0016y 0016y ux D us sin , D cos 2υ ∂y 2υ 2υ kD
Hence: 00070001 L
D0
and: y
Thus:
00070001 υ ∂ 0016y 0016 y 0016us sin 1 0003 sin dy D ∂x 2υ 2υ 2υ 0 00070001 1 0012
0013 ∂ 0016y 0016 y y 0016 υ 0003 sin2 sin d D ∂x 2 υ 2υ υ 2υ0013us 0
0013us2
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THE BOUNDARY LAYER
0002 0005 1 1 y 0016 0016y y 0003 C cos 0016 sin d D 2υ 2 2 υ υ 2υ0013us 0 0002 00051 ∂ 2 0016 0016y 1y 1 y D υ cos 0003 C sin 0016 ∂x 0016 2υ 2υ 20016 υ 0 2υ0013us 0002 0005 ∂ 2 0016 1 D υ 0003 ∂x 0016 2 2υ0013us 0003 0004 2 4 υ 0016x 00031 D assuming υ D 0 at x D 0 0016 2 0013us 0003 00040003 0004 υ2 20016 D x2 0013us x
4/0016 0003 1 υ p D 23.1 Re00031/2 D 4.80 Re00031/2 x x x 0002 0003 0004 0005 0001 υ 0016 y 2υ 2υ 0016y υ Ł
υ 0003 υ us D us sin dy D us 0003 cos D us 2 υ 0016 2 υ 0016 0 0 0003 0004 2 υŁ D υ 1 0003 0016
∂ υ ∂x
∴
and :
00070001
297
1
υŁ D 0.363. υ
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SECTION 12
Momentum, Heat and Mass Transfer PROBLEM 12.1 If the temperature rise per metre length along a pipe carrying air at 12.2 m/s is 66 deg K, what will be the corresponding pressure drop for a pipe temperature of 420 K and an air temperature of 310 K? The density of air at 310 K is 1.14 kg/m3 .
Solution For a pipe of diameter d, the mass flow D 0003u00050006d2 0007/4 and the rate of heat transfer, q D 0003u00050006d2 /40007Cp T where T is the temperature rise. Also q D hA0003Tw 0002 Tm 0007 D h0006dl0003Tw 0002 Tm 0007 W where Tw and Tm are the mean wall and fluid temperatures. Thus, 0003h/Cp 0005u0007 D dT/4l0003Tw 0002 Tm 0007 From equation 12.102: R/0005u2 D dT/40003Tw 0002 Tm 0007l Substituting in equation 3.18: 0002P D 4dT0003l/d00070005u2 /40003Tw 0002 Tm 0007 D 0003T0005u2 l0007/0003Tw 0002 Tm 0007 D 000366 ð 1.14 ð 12.22 ð 1.00007/0003420 0002 3100007 D 101.8 0003N/m2 0007/m
PROBLEM 12.2 It is required to warm a quantity of air from 289 K to 313 K by passing it through a number of parallel metal tubes of inner diameter 50 mm maintained at 373 K. The pressure drop must not exceed 250 N/m2 . How long should the individual tubes be? The density of air at 301 K is 1.19 kg/m3 and the coefficients of heat transfer by convection from the tube to air are 45, 62, and 77 W/m2 K for velocities of 20, 24, and 30 m/s at 301 K respectively.
Solution From equations 12.102 and 3.18: 0002P D 40003h/Cp 0005u00070003l/d00070005u2 D 4hlu/Cp d ∴
250 D 40003hlu/Cp ð 0.0500007 or h D 3.125Cp /lu W/m2 K
(i)
298
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The heat transferred to the air D u00030006d2 /400070005Cp 0003T2 0002 T1 0007 D u00030006 ð 0.0502 /400071.19Cp 0003313 0002 2890007 D 0.056Cp u W This is equal to: h0006dl0003Tw 0002 Tm 0007 D h0006 ð 0.050l0003373 0002 3010007 D 11.3hl W ∴
11.31hl D 0.056Cp u or h D 0.0050Cp u/l From equation (i):
(ii)
0003Cp /l0007 D 0.32hu
and substituting in equation (ii): h D 00030.005 ð 0.32hu2 0007 and u D 25 m/s For this velocity, interpolation of the given data gives a value of h D 64 W/m2 K. ∴
0003Cp /l0007 D 00030.32 ð 64 ð 250007 D 512 J/kg K m For air:
and hence:
Cp D 1000 J/kg K l D 00031000/5120007 D 1.95 m
PROBLEM 12.3 Air at 330 K, flowing at 10 m/s, enters a pipe of inner diameter 25 mm, maintained at 415 K. The drop of static pressure along the pipe is 80 N/m2 per metre length. Using the Reynolds analogy between heat transfer and friction, estimate the temperature of the air 0.6 m along the pipe.
Solution See Volume 1, Example 12.2.
PROBLEM 12.4 Air flows at 12 m/s through a pipe of inside diameter 25 mm. The rate of heat transfer by convection between the pipe and the air is 60 W/m2 K. Neglecting the effects of temperature variation, estimate the pressure drop per metre length of pipe.
Solution From equations 3.18 and 12.96, 0002P D 40003h/Cp 0005u00070003l/d00070005u2 Taking Cp D 1000 J/kg K and l D 1 m, then: 0002P D 4000360/10000005 ð 12000700031.0/0.02500070005 ð 122 D 115.2 N/m2 per metre
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PROBLEM 12.5 Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter and the pressure drop over a 4 m length is found to be 1.5 kN/m2 . Using the Reynolds analogy, by how much would the air temperature be expected to fall over the first metre of pipe length if the wall temperature there is kept constant at 295 K? Viscosity of air D 0.018 mN s/m2 . Specific heat capacity of air D 1.05 kJ/kg K.
Solution (Essentially, this is the same as Problem 9.40 though, here, an alternative solution is presented.) From equations 3.18 and 12.102: 0002P D 40003h/Cp 0005u00070003l/d00070005u2 . For a length of 4 m: 1500 D 40003h/Cp 0005u000700034.0/0.05000070005u2 ∴
0003hu/Cp 0007 D 4.69 kg/ms2
(i)
The rate of heat transfer D h0006dl0003Tm 0002 Tw 0007, which for a length of 1 m is: 0003h0006 ð 0.050 ð 1.0000700030.50003320 C T2 0007 0002 2950007 D 0.157h00030.5T2 0002 1350007 The heat lost by the air D u00030006d2 /400070005Cp 0003T1 0002 T2 0007 D u00030006 ð 0.0502 /400070005Cp 0003320 0002 T2 0007 D 0.00196u0005Cp 0003320 0002 T2 0007 ∴
80.10003h/Cp 0005u0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007
Substituting from equation (i): 80.100034.69/0005u2 0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007
(ii)
From equation 12.139: 0003h/Cp 0005u0007 D 0.0320003du0005/0015000700020.25 D 0.03200030015/du000500070.25 At 320 K and 101.3 kN/m2 , 0005 D 000328.9/22.400070003273/3200007 D 1.10 kg/m3 ∴ ∴
0003h/Cp 0005u0007 D 0003hu/Cp 0007/00030005u2 0007 D 4.69/00031.10u2 0007 4.69/00031.10u2 0007 D 0.032[0.018 ð 1000023 /00030.050 ð 1.10u0007]0.25 and u D 51.4 m/s Substituting, in equation (ii): 000380.1 ð 4.690007/00031.10 ð 51.42 0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007 and T2 D 316 K
The temperature drop over the first metre is therefore 4 deg K which agrees with the solution to Problem 9.40. (It may be noted that, in those problems, an arithmetic mean temperature difference is used rather than a logarithmic value for ease of solution. This is probably justified in view of the small temperature changes involved and also the approximate nature of the Reynolds analogy.)
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PROBLEM 12.6 Obtain an expression for the simple Reynolds analogy between heat transfer and friction. Indicate the assumptions which are made in the derivation and the conditions under which you would expect the relation to be applicable. The Reynolds number of a gas flowing at 2.5 kg/m2 s through a smooth pipe is 20,000. If the specific heat of the gas at constant pressure is 1.67 kJ/kg K, what will the heat transfer coefficient be?
Solution The derivation of the simple Reynolds analogy and its application is presented in detail in Section 12.8. For a Reynolds number of 2.0 ð 104 , from Fig. 3.7, 0003R/0005u2 0007 D 0.0032 for a smooth pipe. From equation 12.102: 0003h/Cp 0005u0007 D 0.0032 0005u D 2.5 kg/m2 s and hence:
h D 00030.0032 ð 1670 ð 2.50007 D 13.4 W/m2 K
PROBLEM 12.7 Explain Prandtl’s concept of a ‘mixing length’. What parallels may be drawn between the mixing length and the mean free path of the molecules in a gas? The ratio of the mixing length to the distance from the pipe wall has a constant value of 0.4 for the turbulent flow of a fluid in a pipe. What is the value of the pipe friction factor if the ratio of the mean velocity to the axial velocity is 0.8?
Solution Transfer by molecular diffusion is discussed in Section 12.2 and the concept of the mixing length in Section 12.3.2. By analogy with kinetic theory, the eddy kinematic viscosity, E, is given by: E / 0017E uE (equation 12.18) where 0017E is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. As shown in equation 12.21: uE / 0017E jdux / dyj Combining this with the previous equation: E / 0017E 00030017E j dux / dyj0007 Putting the proportionality constant equal to unity, E D 00172E j dux / dyj
(equation 12.23)
In the absence of momentum transfer by molecular movement, the shear stress is given by: Ry D 0002Ed00030005ux 0007/dy D 0002000500172E jdux /dyjdux /dy.
(equation 12.20)
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Since near a surface dux /dy is positive and assuming Ry is approximately constant at a value at the pipe wall, that is Ry D Ro D 0002R, then: 0001
or:
R D 000500172E 0003dux /dy00072 (equation 12.26)
0003R/00050007 D 0017E 0003dux /dy0007
p
Here, 0003R/00050007, the shearing stress or friction velocity is usually denoted by uŁ . Since from equation 12.35, 0017E D 0.4y, then: uŁ D 0.4y dux /dy Rearranging:
dux /uŁ D dy/0.4y
and integrating:
0003ux /uŁ 0007 D 2.5 ln y C const.
(i)
At y D r: 0003umax /uŁ 0007 D 2.5 ln r C const. const. D 0003umax /uŁ 0007 0002 2.5 ln r
or: Substituting in equation (i):
0003umax /uŁ 0007 0002 2.5 ln r D 0003ux /uŁ 0007 0002 2.5 ln y 0003umax 0002 ux 0007/uŁ D 2.5 ln0003r/y0007 0002 r 0003200060003r 0002 y0007 dyux 0007/0006r 2 uD
and: The mean velocity:
(ii)
0
0002
and dividing by r:
uD2
1
00031 0002 y/r0007 d0003y/r0007ux 0
Substituting for ux from equation (ii): 0002 1 uD2 00031 0002 y/r0007 d0003y/r00070003umax C 2.5uŁ ln0003y/r00070007 0
D2
0003 0004
[umax C 2.5 ln0003y/r0007][0003y/r0007 0002 0.50003y/r00072 ] 0002
0002u 0007
1
00051 0
0006 2.50003r/y0007[0003y/r0007 0002 0.50003y/r0007 ]d0003y/r0007 2
Ł 0
0004 00051 D 2 umax 00030.50007 0002 2.5uŁ 0003y/r0007 0002 0.250003y/r00072 0 D umax 0002 3.75uŁ 0001 When 0003u/umax 0007 D 0.8, u D 0003u/0.80007 0002 3.75u 0003R/0005u2 0007 0001 ∴ 0003R/0005u2 0007 D 0.0667 and 0003R/0005u2 0007 D 0.00444
PROBLEM 12.8 The velocity profile in the neighbourhood of a surface for a Newtonian fluid may be expressed in terms of a dimensionless velocity uC and a dimensionless distance y C from
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the surface. Obtain the relation between uC and y C in the laminar sub-layer. Outside the laminar sub-layer, the relation is: uC D 2.5 ln y C C 5.5 At what value of y C does the transition from the laminar sub-layer to the turbulent zone occur? In the “Universal Velocity Profile”, the laminar sub-layer extends to values of y C D 5 and the turbulent zone starts at y C D 30 and the range 5 < y C < 30, the buffer layer, is covered by a second linear relation between uC and ln y C . What is the maximum difference between the values of uC , in the range 5 < y C < 30, using the two methods of representation of the velocity profile? Definitions: uC D 0003ux /uŁ 0007, y C D 0003yuŁ 00050007/0015 and uŁ2 D R/0005 where ux is the velocity at a distance y from the surface, R is the wall shear stress and 0005 and 0015 are the density and viscosity of the fluid respectively.
Solution As discussed in Section 12.4.2, if the velocity gradient dux / dy approaches a constant value near the surface, 0003d2 ux /dy 2 0007 approaches zero and R D 0015ux /y. ∴
uŁ2 D 0015ux /00030005y0007
and, as given in equation 12.39: 0003ux /uŁ 0007 D yuŁ 0005/0015 D y C uC D y C
Hence: C
C
C
(equation 12.40) C
Since u D 2.5 ln y C 5.5 and y D 2.5 ln y C 5.5, then solving by trial and error, the transition from the laminar sub-layer to the turbulent zone occurs when: y C D 11.6 and uC D 2.5 ln 11.6 C 5.5 D 11.62 For the buffer layer, uC D a ln y C C a0 C
C
C
(equation 12.41) C
When y D 5, u D 5 and when y D 30, u D 2.5 ln 30 C 5.5 D 14 ∴
5 D a ln 5 C a0
∴
a0 D 5 0002 a ln 5 D 5 0002 1.609a
and:
14 D a ln 30 C a0 D a ln 30 C 5 0002 1.609a
∴
9 D 3.401a 0002 1.609a and a D 5.02
and:
a0 D 5 0002 00031.609 ð 5.020007 D 00023.08
The difference between the two values of uC is a maximum when y C D 11.6. From the two-layer theory: uC D 11.6 From the buffer-layer theory: uC D 5.02 ln 11.6 0002 3.08 D 9.2 and hence, the maximum difference is 000311.6 0002 9.20007 D 2.4
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PROBLEM 12.9 Calculate the rise in temperature of water flowing at 4 m/s through a smooth 25 mm diameter pipe 6 m long. The water enters at 300 K and the temperature of the wall of the tube can be taken as approximately constant at 330 K. Use: (a) (b) (c) (d)
The simple Reynolds analogy, The Taylor-Prandtl modification, The buffer layer equation, Nu D 0.023 Re0.8 Pr 0.33 .
Comment on the differences in the results so obtained.
Solution See Volume 1, Example 12.3.
PROBLEM 12.10 Calculate the rise in temperature of a stream of air, entering at 290 K and flowing at 4 m/s through the tube maintained at 350 K; other conditions remaining the same as detailed in Problem 12.9.
Solution See Volume 1, Example 12.4.
PROBLEM 12.11 Air flows through a smooth circular duct of internal diameter 0.25 m at an average velocity of 15 m/s. Calculate the fluid velocity at points 50 mm and 5 mm from the wall. What will be the thickness of the laminar sub-layer if this extends to uC D y C D 5? The density of air may be taken as 1.12 kg/m3 and the viscosity of air as 0.02 mN s/m2 .
Solution See Volume 1, Example 12.1.
PROBLEM 12.12 Obtain the Taylor–Prandtl modification of the Reynolds analogy for momentum and heat transfer, and give the corresponding relation for mass transfer (no bulk flow). An air stream at approximately atmospheric temperature and pressure, and containing a low concentration of carbon disulphide vapour, is flowing at 38 m/s through a series of 50 mm diameter tubes. The inside of the tubes is covered with a thin film of liquid and
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both heat and mass transfer are taking place between the gas stream and the liquid film. The film heat transfer coefficient is found to be 100 W/m2 K. Using a pipe friction chart and assuming the tubes to behave as smooth surfaces, calculate: (a) the film mass transfer coefficient, and (b) the gas velocity at the interface between the laminar sub-layer and the turbulent zone of the gas. Specific heat of air D 1.0 kJ/kg K. Viscosity of air D 0.02 mN s/m2 . Diffusivity of carbon disulphide vapour in air D 1.1 ð 1000025 m2 /s. Thermal conductivity of air D 0.024 W/m K.
Solution The Taylor–Prandtl modification of the Reynolds analogy for heat transfer and mass transfer is discussed in Section 12.8.3 and the relevant equations are: For heat transfer: or:
(equation 12.119)
0003h/Cp 0005us 0007 D #/00031 C ˛0003Pr 0002 100070007
For mass transfer: or:
0003R/0005u2 0007 D 0003h/Cp 0005us 000700031 C ˛0003Pr 0002 100070007 D # 0003R/0005u2 0007 D 0003hD /us 000700031 C ˛0003Sc 0002 100070007 D #
(i) (equation 12.120)
0003hD /us 0007 D #/00031 C ˛0003Sc 0002 100070007
(ii)
Taking the molecular mass of air as 29 kg/kmol and atmospheric temperature as 293 K, the density, 0005 D 000329/22.400070003273/2930007 D 1.206 kg/m3 . ∴
Re D du0005/0015 D 000350 ð 1000023 ð 38 ð 1.2060007/00030.02 ð 1000023 0007 D 114,570
and from Fig. 3.7: # D 0003R/0005u2 0007 D 0.0021. From Table 1.3, the Prandtl number, Pr D Cp 0015/k D 00031.0 ð 103 ð 0.02 ð 1000023 0007/0.024 D 0.833 From Table 1.3, the Schmidt number, Sc D 0015/0005D D 00030.02 ð 1000023 0007/00031.206 ð 1.1 ð 1000025 0007 D 1.508 Substituting in equation (i): 0003100/00031.0 ð 103 ð 1.206 ð 3800070007 D 0.0021/00031 C ˛00030.833 0002 100070007 ∴
0.00218 D 0.0021/00031 0002 0.167˛0007 and ˛ D 0.22
Substituting in equation (ii): 0003hD /380007 D 0.0021/00031 C 0.2200031.508 0002 100070007 D 0.00189 and hD D 0.072 m/s The gas velocity at the interface of the laminar sub-layer and the turbulent zone, ub may also be estimated from: 0003ub /u0007 D 2.32 Re00020.125 or:
(equation 12.60)
ub D 000338 ð 2.3200070003114,570000700020.125 D 20.6 m/s
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PROBLEM 12.13 Obtain the Taylor–Prandtl modification of the Reynolds’ analogy between momentum and heat transfer and give the corresponding analogy for mass transfer. For a particular system a mass transfer coefficient of 8.71 ð 1000026 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub-layer terminates, to the stream velocity. Molecular diffusivity D 1.5 ð 1000029 m2 /s. Viscosity D 1 mN s/m2 . Density D 1000 kg/m3 . Thermal conductivity D 0.48 W/m K. Specific heat capacity D 4.0 kJ/kg K.
Solution The Taylor–Prandtl modification to heat and mass transfer is discussed in Section 12.8.3 resulting in the modified Lewis relation: hD D 0003h/Cp 00050007[1 C ˛0003Pr 0002 10007]/[1 C ˛0003Sc 0002 10007] 3
(equation 12.121)
00023
In this case:
Pr D Cp 0015/k D 00034 ð 10 ð 1 ð 10 0007/0.48 D 8.33
and:
Sc D 0015/0005D D 00031 ð 1000023 0007/00031000 ð 1.5 ð 1000029 0007 D 667
Substituting: 00038.71 ð 1000026 0007 D [2730/00034000 ð 10000007][1 C ˛00038.33 0002 10007]/[1 C ˛0003667 0002 10007] 0.01276 D 00031 C 7.33˛0007/00031 C 666˛0007 and ˛ D 0.844
PROBLEM 12.14 Heat and mass transfer are taking place simultaneously to a surface under conditions where the Reynolds analogy between momentum, heat and mass transfer may be applied. The mass transfer is of a single component at a high concentration in a binary mixture, the other component of which undergoes no net transfer. Using the Reynolds analogy, obtain a relation between the coefficients for heat transfer and for mass transfer.
Solution The solution to this problem is presented in Sections 12.8.1 and 12.8.2 and the relation between the coefficients for heat transfer and mass transfer is: h D 0003CBw /CT 0007Cp 0005hD
(equation 12.112)
PROBLEM 12.15 Derive the Taylor–Prandtl modification of the Reynolds analogy between momentum and heat transfer. In a shell and tube type condenser, water flows through the tubes which are 10 m long and 40 mm diameter. The pressure drop across the tubes is 5.6 kN/m2 and the effects of entry and exit losses may be neglected. The tube walls are smooth and flow may be taken
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as fully developed. The ratio of the velocity at the edge of the laminar sub-layer to the mean velocity of flow may be taken as 2 Re00020.125 , where Re is the Reynolds number in the pipeline. If the tube walls are at an approximately constant temperature of 393 K and the inlet temperature of the water is 293 K, estimate the outlet temperature. Physical properties of water: density D 1000 kg/m3 , viscosity D 1 mNs/m2 , thermal conductivity D 0.6 W/m K, specific heat capacity D 4.2 kJ/kg K.
Solution The Taylor–Prandtl modification of the Reynolds analogy to heat transfer, discussed in Section 12.8.3, leads to: St D 0003h/Cp 0005u0007 D 0003R/0005u2 0007/[1 C ˛0003Pr 0002 10007]
(equation 12.117) (i)
From equation 3.23: 0003R/0005u2 0007 Re2 D 0002Pf d3 0005/00034l00152 0007 D 00035600 ð 000340/100000073 ð 10000007/00034 ð 10 ð 00031 ð 1000023 00072 0007 D 8,960,000 From Fig. 3.8: Re D 62,000 and R/0005u2 D 8,960,000/000362,00000072 D 0.0023 The ratio of the velocity at the edge of the laminar sub-layer to the mean velocity of flow is: ˛ D 2 Re00020.125 D 2/00036200000070.125 D 0.5035 The Prandtl group, Pr D Cp 0015/k D 00034200 ð 1 ð 1000023 0007/0.6 D 7.0 and from equation (i), the Stanton group, St D 0003R/0005u2 0007/[1 C ˛0003Pr 0002 10007] D 0.0023/[1 C 0.503500037.0 0002 10007] D 0.000572 ∴
h D 0.000572Cp 0005u D 0.000572Cp Re 0015/d D 00030.000572 ð 4200 ð 62000 ð 1 ð 1000023 0007/000340/10000007 D 3724 W/m2 K.
The area for heat transfer per unit length of pipe D 0003400006/10000007 D 0.040006 D 0.126 m2 /m and making a heat balance over unit length of pipe dl: ∴ ∴
hAdl0003Tw 0002 T0007 D 0005uCp Ac dT D 0003Re 0015/d0007Cp Ac dT 3724 ð 0.126 dl0003393 0002 T0007 D 000362000 ð 1 ð 1000023 /000340/100000070007 ð 4200 ð 00030006/40007000340/100000072 dT
∴
0.0572 dl D dT/0003393 0002 T0007
Integrating:
0002
0002
10
T0
dl D
0.0572 0
dT/0003393 0002 T0007 293
00030.0572 ð 100007 D ln[0003393 0002 2930007/0003393 0002 T0 0007] ∴
1.772 D 100/0003393 0002 T0 0007 and T0 D 336.6 K
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PROBLEM 12.16 Explain the importance of the universal velocity profile and derive the relation between the dimensionless derivative of velocity uC , and the dimensionless derivative of distance from the surface y C , using the concept of Prandtl’s mixing length 0017E . It may be assumed that the fully turbulent portion of the boundary layer starts at y C D 30, that the ratio of the mixing length 0017E to distance y from the surface, 0017E /y D 0.4, and that for a smooth surface uC D 14 at y C D 30. If the laminar sub-layer extends from y C D 0 to y C D 5, obtain the equation for the relation between uC and y C in the buffer zone, and show that the ratio of the eddy viscosity to the molecular viscosity increases linearly from 0 to 5 through this buffer zone.
Solution The importance of the universal velocity profile is discussed in Section 12.4. From equation 12.18, for isotropic turbulence, the eddy kinematic viscosity, E ˛ 0017E uE where 0017E is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. The momentum transfer rate per unit area in a direction perpendicular to the surface at position y is then: Ry D 0002E d00030005ux 0007/ dy and for constant density,
(equation 12.20)
0002Ry /0005 D E dux / dy
or: where
Ry D 0002E0005 dux / dy
0001
0003Ry /00050007, the friction velocity, may be denoted by uŁ and then uŁ2 D Edux /dy
Assuming E D 0017E uE , that is a proportionality constant of unity, and uE D 0017E j dux / dyj, then: uŁ2 D 00172E 0003dux /dy0007j0003dux /dy0007j and hence near the surface where 0003dux /dy0007 is positive: uŁ D 0017E 0003dux /dy0007 Assuming 0017E D Ky: uŁ dy/y D Kdux Integrating:
ux /uŁ D 00031/K0007 ln y C B where B is a constant
or:
ux /uŁ D 00031/K0007 ln0003yuŁ 0005/00150007 C B0
(equation 12.28)
(equation 12.29)
Since 0003uŁ 0005/00150007 is constant, B0 is also constant and, writing the dimensionless velocity term, 0003ux /uŁ 0007 as uC and the dimensionless derivative of y0003yuŁ 0005/00150007 as y C , then: uC D 00031/K0007 ln y C C B0 Given that K D 0.4, then:
(equation 12.30)
uC D 2.5 ln y C C B0
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Given that for a smooth surface, uC D 14 at y C D 30, then: B0 D 14 0002 2.5 ln 30 D 5.5 uC D 2.5 ln y C C 5.5
and:
(equation 12.37)
For molecular transfer in the laminar sub-layer near the wall, from Section 12.4.2: Ry D 00020015ux /y 00030002Ry /00050007 D uŁ2 D 0015ux /0003y00050007
or: ∴
0003ux /uŁ 0007 D y0005uŁ /0015 and uC D y C
(equation 12.40)
If the buffer zone stretches from y C D 5 to y C D 30 at which uC is 5 and 14 respectively, then in equation 12.41: uC D a ln y C C a0 or:
5 D a ln 5 C a0 and 14 D a ln 30 C a0 uC D 5.0 ln y C 0002 3.05
and:
(equation 12.42)
From equation 12.46, the velocity gradient, duC /dy C D 5/y C From equation 12.61: Ry D 000200030015 C E00050007dux /dy 0001 and substituting uŁ D 0003Ry /00050007: 0003dux /dy0007 D uŁ2 /0003E C 0015/00050007 ∴
and:
C
C
0003du /dy 0007 D 00030015/0005000700031/[E C 0015/0005]0007 D 5/y E/00030015/00050007 D 0003y C /50007 0002 1
(equation 12.62) C
(equation 12.63)
Hence as y C goes from 5 to 30, the ratio of the eddy kinematic viscosity to the kinematic viscosity goes from 0 to 5.
PROBLEM 12.17 Derive the Taylor–Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. If the relationship between the Nusselt number Nu, Reynolds number Re and Prandtl number Pr is: Nu D 0.023 Re0.8 Pr 0.33 calculate the ratio of the velocity at the edge of the laminar sub-layer to the velocity at the pipe axis for water 0003Pr D 100007 flowing at a Reynolds number of 10,000 in a smooth pipe. Use the pipe friction chart.
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Solution The derivation of the Taylor–Prandtl modification of the Reynolds analogy between heat and momentum transfer is presented in Section 12.8.3 and the result is summarised as: St D h/0003Cp 0005us 0007 D 0003R/0005us2 0007/[1 C ˛0003Pr 0002 10007] (equation 12.117) or:
0003R/0005us2 0007 D [h/0003Cp 0005us 0007][1 0002 ˛00031 0002 Pr0007]
For turbulent pipe flow, us is approximately equal to 0003umean /0.820007 and: 0.820003R/0005u2 0007 D [h/0003Cp 0005u0007][1 0002 ˛00031 0002 Pr0007] When Re D 10,000, then from Fig. 3.1, 0003R/0005u2 0007 D 0.0038, and for Pr D 10: 00030.82 ð 0.00380007 D St[1 0002 ˛00031 0002 100007] or: But: and:
St00031 C 9˛0007 D 0.0031
(i)
Nu D 0.023 Re0.8 Pr 0.33 St D Nu/0003Re ÐPr0007 D 0.023 Re00020.2 Pr 00020.67 D 0.023000310,000000700020.2 000310000700020.67 D 0.000777
Hence, substituting in equation (i): 0.00077700031 C 9˛0007 D 0.0031 and ˛ D 0003ub /us 0007 D 0.33
PROBLEM 12.18 Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl’s concept of a “Mixing Length” (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length 0017E is equal to 0.4 times the distance y from the surface and that the dimensionless velocity uC is equal to 5.5 when the dimensionless distance y C is unity. Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer υb is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by Prandtl’s one seventh power law. Blasius Equation: 0003 0006 us υ0005 00020.25 R D 0.0228 0005us2 0015 where 0005, 0015 are the density and viscosity of the fluid, us is the stream velocity, and υ is the total boundary layer thickness.
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Solution The Universal Velocity Profile is discussed in detail in Section 12.4, and in the region where eddy transport dominates (y C > 30) and making all the stated assumptions: uC D 2.5 ln y C C 5.5
(equation 12.37)
If, in the laminar sub-layer (from equation 12.40), uC D y C then: y C D 2.5 ln y C C 5.5 and, solving by trial and error: y C D 11.6 D usŁ υb 0005/0015 p p Since uŁ D 0003R/00050007, then: υb D 11.60015/00030005uŁ 0007 D 11.60015/ 0003R00050007 2
(from equation 12.44) (i)
00020.25
But, from the Blasuis equation: R/0005u D 0.02280003us υ0005/00150007
p p and substituting for R in equation (i), υb D 000311.60015/ 0005000700031/us 0005000700031/0.0228000700020.5 0003us υ0005/001500070.125 and:
0003υb /υ0007 D 76.80003us υ0005/0015000700020.875
Using Prandtl’s one seventh power law, 0003ub /us 0007 D 0003υb /υ00071/7 D 0003υb /υ00070.143 But: ∴
R D 0015ub /υb D 0.02280005us 2 0003us υ0005/0015000700020.25 00030015/υb 0007us 0003υb /υ00070.143 D 0.02280005us 2 0003us υ0005/0015000700020.25
and:
0003υb /υ0007 D 82.380003us υ0005/0015000700020.875
The ratio of the values obtained using the two approaches to the problem is: 000382.38/76.80007 D 1.073
PROBLEM 12.19 Obtain the Taylor–Prandtl modification of the Reynolds analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be, given the following physical properties? Diffusivity D D 5 ð 1000029 m2 /s. Thermal conductivity, k D 0.6 W/m K. Specific heat capacity Cp D 4 kJ/kg K. Density, 0005 D 1000 kg/m3 . Viscosity, 0015 D 1 mNs/m2 . Assume that the ratio of the velocity at the edge of the laminar sub-layer to the stream velocity is (a) 0.2, (b) 0.6. Comment on the difference in the two results.
Solution The discussion of the Reynolds analogy is presented in Section 12.8 and consideration of mass transfer in Sections 12.8.2 and 12.8.3 leads to the modified Lewis relation which
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may be written as: hD [1 0002 ˛00031 0002 Sc0007] D 0003h/Cp 00050007[1 0002 ˛00031 0002 Pr0007] (equation 12.121) (i) The Schmidt group, Sc D 0015/0005D D 00031 ð 1000023 0007/00031000 ð 5 ð 1000029 0007 D 200 The Prandtl group, Pr D Cp 0015/k D 00034 ð 103 ð 1 ð 1000023 0007/0.6 D 6.67 and:
0003h/Cp 00050007 D 4000/00034 ð 103 ð 10000007 D 0.001
Thus, in equation (i): hD [1 0002 ˛00031 0002 2000007] D 0.001[1 0002 ˛00031 0002 6.670007] and:
hD D 0.00100031 C 5.67˛0007/00031 C 199˛0007 m/s
(ii)
When ˛ D 0.2, then from equation (ii), hD D 0.00100031 C 1.1340007/00031 C 39.80007 D 5.2 ð 1000025 m/s When ˛ D 0.6, then from equation (ii), hD D 0.00100031 C 3.4020007/00031 C 119.40007 D 3.6 ð 1000025 m/s It is worth noting that even with a very large variation in ˛ (threefold in fact) the change in the mass transfer coefficient is less than 50%.
PROBLEM 12.20 By using the simple Reynolds analogy, obtain the relation between the heat transfer coefficient and the mass transfer coefficient for the gas phase for the absorption of a soluble component from a mixture of gases. If the heat transfer coefficient is 100 W/m2 K, what will the mass transfer coefficient be for a gas of specific heat capacity 1.5 kJ/kg K and density 1.5 kg/m3 ? The concentration of the gas is sufficiently low for bulk flow effects to be negligible.
Solution From Section 12.8.1, the heat transfer coefficient is given by: 0003R/0005u2 0007 D h/0003Cp 0005us 0007
(equation 12.102)
and the mass transfer coefficient by: 0003R/0005u2 0007 D hD /us
(equation 12.103)
Hence:
(equation 12.105)
In this case:
hD D h/0003Cp 00050007 hD D 100/00031.5 ð 103 ð 1.50007 D 0.044 m/s
PROBLEM 12.21 The velocity profile in the neighbourhood of a surface for a Newtonian fluid may be expressed in terms of a dimensionless velocity uC and a dimensionless distance y C from
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the surface. Obtain the relation between uC and y C in the laminar sub-layer. Outside the laminar sub-layer, the relation takes the form: uC D 2.5 ln y C C 5.5 At what value of y C does the transition from the laminar sub-layer to the turbulent zone occur? In the Universal Velocity Profile, the laminar sub-layer extends to values of y C D 5, the turbulent zone starts at y C D 30 and the range 5 < y C < 30, the buffer layer, is covered by a second linear relation between uC and ln y C . What is the maximum difference between the values of uC , in the range 5 < y C < 30, using the two methods of representation of the velocity profile? Definitions: uC D
ux uŁ
yC D
yuŁ 0005 0015
uŁ2 D R/0005 where ux is velocity at distance y from surface R is wall shear stress 0005, 0015 are the density and viscosity of the fluid respectively.
Solution Laminar sub-layer If the velocity gradient approaches constant value in the laminar sub-layer, then 2 d ux /dy 2 ! 0 and: ux y 0015ux D 0005y
RD0015 uŁ2 and:
ux uŁ y0005 D uŁ 0015
or: uC D y C , by definition
The point of interaction of uC D y C and:
uC D 2.5 ln y C C 5.5
is given by:
y C D 2.5 ln y C C 5.5
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Evaluating: y C : yC
RHS
10 15 20 25 8 7 12 11 11.5 11.6
11.26 12.27 13.00 13.5 10.7 10.4 11.71 11.5 11.6 11.62
and: y C D 11.6 The equation for buffer zone may be written as: uC D A ln y C C B When y C D 5, uC D 5 When y C D 30, uC D 2.5 ln 30 C 5.5 Thus:
5 D A ln 5 C B 5.5 C 2.5 ln 30 D A ln 30 C B
Substracting:
0.5 C 2.5 ln 30 D A ln 6
Thus: A D 00030.5 C 2.5 ln 300007/ln 6 D 5.02 and: B D 5 0002 A ln 5 D 00023.08 The difference in the two values of uC is a maximum when y C D 11.6. From the two-layer theory: uC D 11.6 From the buffer-layer theory: uC D 5.02 ln C11.6 0002 3.08 D 9.2 The maximum difference in the two values of uC is then: 2.4
PROBLEM 12.22 In the universal velocity profile a “dimensionless” velocity uC is plotted against ln y C , where y C is a “dimensionless” distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity × kinematic viscosity), the ratio of the mixing length 00030017E 0007 to the distance 0003y0007 from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for duC /dy C in terms of y C . In the buffer zone the ratio of duC /dy C to y C is twice the value calculated above. Obtain an expression for the eddy kinematic viscosity E in terms of the kinematic viscosity 00030015/00050007 and y C . On the assumption that the eddy thermal diffusivity EH and the eddy kinematic
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viscosity E are equal, calculate the value of the temperature gradient in a liquid flowing over the surface at y C D 15 (which lies within the buffer layer) for a surface heat flux of 1000 W/m2 . The liquid has a Prandtl number of 7 and a thermal conductivity of 0.62 W/m K.
Solution a) In the region where eddy effects considerably exceed molecular contributions: E × 0015/0005 Outside the laminar sub-layer and the buffer-layer but still close to the surface: R D E00050003dux /dy0007 where R is shear stress at surface Writing E as 0017E uE , and approximating, uE D 0017E jdux /dyj gives: 0003 00060003 0006 R dux dux D 00172E 0005 dy dy where the modulus sign is dropped as dux / dy is positive near a surface. p Putting uŁ D R/0005, the shearing stress velocity: 0003 0006 dux 2 2 uŁ D 00172E dy uŁ D 0017E
dux dy
Using the Prandtl approximation 0017E D 0.4 y gives: dux uŁ D 0.4 y dy ∴
dux uŁ D 2.5 dy y
Writing y C D yuŁ 0005/0015 and uC D ux /uŁ , then: uŁ and: b) In the buffer zone: It is given that:
duC uŁ D 2.5 dy C yC duC 2.5 D C dy C y
0003 0006 2.5 5 duC D2 D C C C dy y y
The shear stress is given by: R D 00030015 C E00050007 ∴
2
uŁ D
R D 0005
0003
0015 CE 0005
0006
dux dy dux dy
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and: Hence:
0006 C 0015 du uŁ CE u D 0005 dy C 0015/0005uŁ 0003 0006 C 0003 0006 0015 du 0015 5 0015 D CE CE D C 0005 0005 dy 0005 yC Ł2
0003
When y C D 15, then: 0015/0005 D 00030015/0005 C E0007/3 and:
E D 20015/0005
iii) For heat transfer in buffer zone: q D 00020003k C Cp 0005EH 0007
dT dy
dT dy When y C D 15, then putting E D 20015/0005 gives:
Writing EH D E gives: q D 00020003k C Cp 0005E0007
q D 00020003k C 2Cp 00150007 Thus:
Putting:
dT dT D 0002k00031 C 2Pr0007 dy dy
dT 0002q D dy k00031 C 2Pr0007 k D 0.62 W/m K,
Pr D 7 and 0002q D 1000 W
then: dT/dy D 108 deg K/m or 0.108 deg K/mm
PROBLEM 12.23 Derive an expression relating the pressure drop for the turbulent flow of a fluid in a pipe to the heat transfer coefficient at the walls on the basis of the simple Reynolds analogy. Indicate the assumptions which are made and the conditions under which it would be expected to apply closely. Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter and the pressure drop over a 4 m length is found to be 150 mm water gauge. By how much would the air temperature be expected to fall over the first metre if the wall temperature there is 290 K? Viscosity of air D 0.018 mN s/m2 . Specific heat capacity 0003Cp 0007 D 1.05 kJ/kg K. Molecular volume D 22.4 m3 /kmol at 1 bar and 273 K.
Solution If a mass of fluid, m, situated at a distance from a surface, is moving parallel to the surface with a velocity of us , and it then moves to the surface, where the velocity is zero, it will give up its momentum mus in time t. If the temperature difference between the mass of fluid and the surface is 0s , then the heat transferred to the surface is 0003mCp 0s 0007 and over a surface of area, A: 0003mCp 0s 0007/t D 0002qA where q is the heat transferred from the surface per unit area per unit time.
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If the shear stress at the surface is R0 , the shearing force over area A is the rate of change of momentum or: 0003mus 0007/t D R0 A and Cp 0s /us D q/R0 Writing R0 D 0002R, the shear stress acting on the walls and h as the heat transfer coefficient between the fluid and the surface, then: 0002q/0s D h D 0002R0 Cp /us D RCp /us
or
h/Cp 0005us D R/0005u2
From equation 3.19, the pressure change due to friction is given by: 0002P D 40003R/0005u2 00070003l/d000700030005u2 0007 and substituting from equation 12.102: 0002P D 40003h/Cp 0005u00070003l/d000700030005u2 0007 D 40003hu/Cp 00070003l/d0007 The Reynolds analogy assumes no mixing with adjacent fluid and that turbulence persists right up to the surface. Further it is assumed that thermal and kinematic equilibria are reached when an element of fluid comes into contact with a solid surface. No allowance is made for variations in physical properties of the fluid with temperature. A further discussion of the Reynolds analogy for heat transfer is presented in Chapter 12. Density of air at 320 K D 000329/22.400070003273 ð 3200007 D 1.105 kg/m3 . The pressure drop: 0002P D 150 mm water D 00039.8 ð 1500007 D 1470 N/m2 l D 4.0 m,
d D 0.050 m
In equation 3.23: 0002Pd3 0005/00034l00152 0007 D 00031470 ð 0.0503 ð 1.1050007/[4 ð 4.000030.018 ð 1000023 00072 ] D 3.192 ð 107 From Fig. 3.8, for a smooth pipe: Re D 1.25 ð 105 and from Fig. 3.7: R/0005u2 D 0.0021 The heat transfer coefficient: h D 0003R/0005u2 0007Cp 0005u D 00030.0021 ð 1.05 ð 103 00070005u D 2.2050005u W/m2 K Mass flowrate of air, G D 0005u00030006/400070.0502 D 0.001960005u kg/s Area for heat transfer, A D 00030006 ð 0.050 ð 1.00007 D 0.157 m2 . GCp 0003T1 0002 T2 0007 D hA0003Tm 0002 Tw 0007 where T1 and T2 are the inlet and outlet temperatures and Tm the mean value taken as arithmetic over the small length of 1 m. ∴ 00030.001960005u ð 1.050 ð 103 00070003320 0002 T2 0007 D 00032.2050005u ð 0.157000700030.50003320 ð T2 0007 0002 2900007
and:
T2 D 316 K
The drop in temperature over the first metre is therefore 4 deg K.
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SECTION 13
Humidification and Water Cooling PROBLEM 13.1 In a process in which benzene is used as a solvent, it is evaporated into dry nitrogen. The resulting mixture at a temperature of 297 K and a pressure of 101.3 kN/m2 has a relative humidity of 60%. It is required to recover 80% of the benzene present by cooling to 283 K and compressing to a suitable pressure. What must this pressure be? Vapour pressures of benzene: at 297 K D 12.2 kN/m2 : at 283 K D 6.0 kN/m2 .
Solution See Volume 1, Example 13.1
PROBLEM 13.2 0.6 m3 /s of gas is to be dried from a dew point of 294 K to a dew point of 277.5 K. How much water must be removed and what will be the volume of the gas after drying? Vapour pressure of water at 294 K D 2.5 kN/m2 . Vapour pressure of water at 277.5 K D 0.85 kN/m2 .
Solution When the gas is cooled to 294 K, it will be saturated and Pw0 D 2.5 kN/m2 . From Section 13.2: mass of vapour D Pw0 Mw /RT D 00072.5 ð 18/00078.314 ð 294 D 0.0184 kg/m3 gas. When water has been removed, the gas will be saturated at 277.5 K, and Pw D 0.85 kN/m2 . At this stage, mass of vapour D 00070.85 ð 18/00078.314 ð 277.5 D 0.0066 kg/m3 gas Hence, water to be removed D 00070.0184 0003 0.0066 D 0.0118 kg/m3 gas or:
00070.0118 ð 0.6 D 0.00708 kg/s
Assuming the gas flow, 0.6 m3 /s, is referred to 273 K and 101.3 kN/m2 , 0.00708 kg/s of water is equivalent to 00070.00708/18 D 3.933 ð 1000034 kmol/s. 318
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1 kmol of vapour occupies 22.4 m3 at STP, and:
volume of water removed D 00073.933 ð 1000034 ð 22.4 D 0.00881 m3 /s
Assuming no volume change on mixing, the gas flow after drying D 00070.60 0003 0.00881 D 0.591 m3 /s at STP .
PROBLEM 13.3 Wet material, containing 70% moisture on a wet basis, is to be dried at the rate of 0.15 kg/s in a counter-current dryer to give a product containing 5% moisture (both on a wet basis). The drying medium consists of air heated to 373 K and containing water vapour with a partial pressure of 1.0 kN/m2 . The air leaves the dryer at 313 K and 70% saturated. Calculate how much air will be required to remove the moisture. The vapour pressure of water at 313 K may be taken as 7.4 kN/m2 .
Solution The feed is 0.15 kg/s wet material containing 0.70 kg water/kg feed. Thus water in feed D 00070.15 ð 0.70 D 0.105 kg/s and dry solids D 00070.15 0003 0.105 D 0.045 kg/s. The product contains 0.05 kg water/kg product. Thus, if w kg/s is the amount of water in the product, then: w/0007w C 0.045 D 0.05 or w D 0.00237 kg/s and:
water to be removed D 00070.105 0003 0.00237 D 0.1026 kg/s.
The inlet air is at 373 K and the partial pressure of the water vapour is 1 kN/m2 . Assuming a total pressure of 101.3 kN/m2 , the humidity is: H1 D [Pw /0007P 0003 Pw ]0007Mw /MA
(equation 13.1)
D [1.0/0007101.3 0003 1.0]000718/29 D 0.0062 kg/kg dry air The outlet air is at 313 K and is 70% saturated. Thus, as in Example 13.1, Volume 1: Pw D Pw0 ð RH/100 D 00077.4 ð 70/100 D 5.18 kN/m2 and:
H2 D [5.18/0007101.3 0003 5.18]000718/29 D 0.0335 kg/kg dry air
The increase in humidity is 00070.0335 0003 0.0062 D 0.0273 kg/kg dry air and this must correspond to the water removed, 0.1026 kg/s. Thus if G kg/s is the mass flowrate of dry air, then: 0.0273G D 0.1026 and G D 3.76 kg/s dry air In the inlet air, this is associated with 0.0062 kg water vapour, or: 00070.0062 ð 3.76 D 0.0233 kg/s
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Hence, the mass of moist air required at the inlet conditions D 00073.76 C 0.0233 D 3.783 kg/s
PROBLEM 13.4 30,000 m3 of cool gas (measured at 289 K and 101.3 kN/m2 saturated with water vapour) is compressed to 340 kN/m2 pressure, cooled to 289 K and the condensed water is drained off. Subsequently the pressure is reduced to 170 kN/m3 and the gas is distributed at this pressure and 289 K. What is the percentage humidity after this treatment? The vapour pressure of water at 289 K is 1.8 kN/m2 .
Solution At 289 K and 101.3 kN/m2 , the gas is saturated and Pw0 D 1.8 kN/m2 . Thus from equation 13.2, H0 D [1.8/0007101.3 0003 1.8]000718/MA D 00070.3256/MA kg/kg dry gas, where MA is the molecular mass of the gas. At 289 K and 340 kN/m2 , the gas is in contact with condensed water and therefore still saturated. Thus Pw0 D 1.8 kN/m2 and: H0 D [1.8/0007340 0003 1.8]000718/MA D 00070.0958/MA kg/kg dry gas
At 289 K and 170 kN/m2 , the humidity is the same, and in equation 13.2: 00070.0958/MA D [Pw /0007170 0003 Pw ]000718/MA Pw D 0.90 kN/m2
or: The percentage humidity is then:
D [0007P 0003 Pw0 /0007P 0003 Pw ]0007100Pw /Pw0
(equation 13.3)
D [0007170 0003 1.8/0007170 0003 0.90]0007100 ð 0.90/1.8 D 49.73%
PROBLEM 13.5 A rotary countercurrent dryer is fed with ammonium nitrate containing 5% moisture at the rate of 1.5 kg/s, and discharges the nitrate with 0.2% moisture. The air enters at 405 K and leaves at 355 K; the humidity of the entering air being 0.007 kg moisture/kg dry air. The nitrate enters at 294 K and leaves at 339 K. Neglecting radiation losses, calculate the mass of dry air passing through the dryer and the humidity of the air leaving the dryer. Latent heat of water at 294 K D 2450 kJ/kg. Specific heat capacity of ammonium nitrate D 1.88 kJ/kg K. Specific heat capacity of dry air D 0.99 kJ/kg K. Specific heat capacity of water vapour D 2.01 kJ/kg K.
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Solution The feed rate of wet nitrate is 1.5 kg/s containing 5.0% moisture or 00071.5 ð 5/100 D 0.075 kg/s water. ∴
flow of dry solids D 00071.5 0003 0.075 D 1.425 kg/s If the product contains w kg/s water, then: w/0007w C 1.425 D 00070.2/100
and:
or
w D 0.00286 kg/s
the water evaporated D 00070.075 0003 0.00286 D 0.07215 kg/s
The problem now consists of an enthalpy balance around the unit, and for this purpose a datum temperature of 294 K will be chosen. It will be assumed that the flow of dry air into the unit is m kg/s. Considering the inlet streams:
(i) Nitrate: this enters at the datum of 294 K and hence the enthalpy D 0. (ii) Air: G kg/s of dry air is associated with 0.007 kg moisture/kg dry air. ∴
enthalpy D [0007G ð 0.99 C 00070.007G ð 2.01]0007405 0003 294 D 111.5G kW
and the total heat into the system D 111.5G kW. Considering the outlet streams:
(i) Nitrate: 1.425 kg/s dry nitrate contains 0.00286 kg/s water and leaves the unit at 339 K. ∴
enthalpy D [00071.425 ð 1.88 C 00070.00286 ð 4.18]0007339 0003 294 D 120.7 kW
(ii) Air: the air leaving contains 0.007 G kg/s water from the inlet air plus the water evaporated. It will be assumed that evaporation takes place at 294 K. Thus: enthalpy of dry air D G ð 0.990007355 0003 294 D 60.4G kW enthalpy of water from inlet air D 0.007G ð 2.010007355 0003 294 D 0.86G kW enthalpy in the evaporated water D 0.07215[2450 C 2.010007355 0003 294] D 185.6 kW and the total heat out of the system, neglecting losses D 0007306.3 C 61.3G kW. Making a balance:
111.5G D 0007306.3 C 61.3G
or
G D 6.10 kg/s dry air
Thus, including the moisture in the inlet air, moist air fed to the dryer is: 6.1000071 C 0.007 D 6.15 kg/s
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Water entering with the air D 00076.10 ð 0.007 D 0.0427 kg/s. Water evaporated D 0.07215 kg/s. Water leaving with the air D 00070.0427 C 0.07215 D 0.1149 kg/s Humidity of outlet air D 00070.1149/6.10 D 0.0188 kg/kg dry air.
PROBLEM 13.6 Material is fed to a dryer at the rate of 0.3 kg/s and the moisture removed is 35% of the wet charge. The stock enters and leaves the dryer at 324 K. The air temperature falls from 341 K to 310 K, its humidity rising from 0.01 to 0.02 kg/kg. Calculate the heat loss to the surroundings. Latent heat of water at 324 K D 2430 kJ/kg. Specific heat capacity of dry air D 0.99 kJ/kg K. Specific heat capacity of water vapour D 2.01 kJ/kg K.
Solution The wet feed is 0.3 kg/s and the water removed is 35%, or: 00070.3 ð 35/100 D 0.105 kg/s If the flowrate of dry air is G kg/s, the increase in humidity D 00070.02 0003 0.01 D 0.01 kg/kg or:
0.01G D 0.105
and G D 10.5 kg/s
This completes the mass balance, and the next step is to make an enthalpy balance along the lines of Problem 13.5. As the stock enters and leaves at 324 K, no heat is transferred from the air and the heat lost by the air must represent the heat used for evaporation plus the heat losses, say L kW. Thus heat lost by the inlet air and associated moisture is: [000710.5 ð 0.99 C 00070.01 ð 10.5 ð 2.01]0007341 0003 310 D 328.8 kW Heat leaving in the evaporated water D 0.105[2430 C 2.010007310 0003 324] D 252.2 kW. Making a balance: 328.8 D 0007252.2 C L or
L D 76.6 kW
PROBLEM 13.7 A rotary dryer is fed with sand at the rate of 1 kg/s. The feed is 50% wet and the sand is discharged with 3% moisture. The entering air is at 380 K and has an absolute humidity of 0.007 kg/kg. The wet sand enters at 294 K and leaves at 309 K and the air leaves at 310 K. Calculate the mass flowrate of air passing through the dryer and the humidity of the air leaving the dryer. Allow for a radiation loss of 25 kJ/kg dry air. Latent heat of water at 294 K D 2450 kJ/kg. Specific heat capacity of sand D 0.88 kJ/kg K. Specific heat capacity of dry air D 0.99 kJ/kg k. Specific heat capacity of vapour D 2.01 kg K.
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Solution The feed rate of wet sand is 1 kg/s and it contains 50% moisture or 00071.0 ð 50/100 D 0.50 kg/s water. ∴
flow of dry sand D 00071.0 0003 0.5 D 0.50 kg/s If the dried sand contains w kg/s water, then: w/0007w C 0.50 D 00073.0/100 or w D 0.0155 kg/s
and:
the water evaporated D 00070.50 0003 0.0155 D 0.4845 kg/s.
Assuming a flowrate of G kg/s dry air, then a heat balance may be made based on a datum temperature of 294 K. Inlet streams:
(i) Sand: this enters at 294 K and hence the enthalpy D 0. (ii) Air: G kg/s of dry air is associated with 0.007 kg/kg moisture. ∴
enthalpy D [0007G ð 0.99 C 00070.007G ð 2.01]0007380 0003 294 D 86.4G kW
and:
the total heat into the system D 86.4G kW.
Outlet streams:
(i) Sand: 0.50 kg/s dry sand contains 0.0155 kg/s water and leaves the unit at 309 K. ∴
enthalpy D [00070.5 ð 0.88 C 00070.0155 ð 4.18]0007309 0003 294 D 7.6 kW
(ii) Air: the air leaving contains 0.07 G kg/s water from the inlet air plus the water evaporated. It will be assumed that evaporation takes place at 294 K. Thus: enthalpy of dry air D G ð 0.990007310 0003 294 D 15.8m kW enthalpy of water from inlet air D 0.007G ð 2.010007310 0003 294 D 0.23G kW enthalpy in the evaporated water D 0.4845[2430 C 2.010007310 0003 294] D 1192.9 kW, a total of 000716.03G C 1192.9 kW (iii) Radiation losses D 25 kJ/kg dry air or 25G kW and the total heat out D 000741.03G C 1200.5 kW. Mass balance:
86.4G D 000741.03G C 1200.5 or G D 26.5 kg/s Thus the flow of dry air through the dryer D 26.5 kg/s and the flow of inlet air D 000726.5 ð 1.007 D 26.7 kg/s As in Problem 13.5, water leaving with the air is: 000726.5 ð 0.007 C 0.4845 D 0.67 kg/s and humidity of the outlet air D 00070.67/26.5 D 0.025 kg/kg.
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PROBLEM 13.8 Water is to be cooled in a packed tower from 330 to 295 K by means of air flowing countercurrently. The liquid flows at the rate of 275 cm3 /m2 s and the air at 0.7 m3 /m2 s. The entering air has a temperature of 295 K and a humidity of 20%. Calculate the required height of tower and the condition of the air leaving at the top. The whole of the resistance to heat and mass transfer can be considered as being within the gas phase and the product of the mass transfer coefficient and the transfer surface per unit volume of column 0007hD a may be taken as 0.2 s00031 .
Solution Assuming, the latent heat of water at 273 K D 2495 kJ/kg specific heat capacity of dry air D 1.003 kJ/kg K specific heat capacity of water vapour D 2.006 kJ/kg K then the enthalpy of the inlet air stream is: HG1 D 1.0030007295 0003 273 C H 00072495 C 2.0060007295 0003 273 From Fig. 13.4, when 0013 D 295 K, at 20% humidity, H D 0.003 kg/kg, and: HG1 D 00071.003 ð 22 C 0.00300072495 C 00072.006 ð 22 D 29.68 kJ/kg In the inlet air, the humidity is 0.003 kg/kg dry air or 00070.003/18/00071/29 D 0.005 kmol/kmol dry air. Hence the flow of dry air D 00071 0003 0.0050.70 D 0.697 m3 /m2 s. Density of air at 295 K D 000729/22.40007273/295 D 1.198 kg/m3 . and hence the mass flow of dry air D 00070.697 ð 1.198 D 0.835 kg/m2 s and the mass flow of water D 275 ð 1000036 m3 /m2 s or 0007275 ð 1000036 ð 1000 D 0.275 kg/m2 s. The slope of the operating line, given by equation 13.37 is: LCL /G D 00070.275 ð 4.18/0.835 D 1.38 The coordinates of the bottom of the operating line are: 0013L1 D 295 K and HG1 D 29.7 kJ/kg Hence, on an enthalpy–temperature diagram (Fig. 13a), the operating line of slope 1.38 is drawn through the point (29.7, 295). The top point of the operating line is given by 0013L2 D 330 K, and from Fig. 13a, HG2 D 78.5 kJ/kg. From Figs 13.4 and 13.5 the curve representing the enthalpy of saturated air as a function of temperature is obtained and drawn in. This plot may also be obtained by calculation using equation 13.60. The integral: 0001 dHG /0007Hf 0003 HG
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Curve for saturated air (Hf vs qf)
Enthalpy (HG kJ/kg)
300
250
200
150
HGvs qG
100
(qL 2'HG 2 )
50 (qL1'HG1) 295
300
Operating line (HGvsqL) 305
310
315
320
325
330
Temperature (q K)
Figure 13a.
is now evaluated between the limits HG1 D 29.68 kJ/kg and HG2 D 78.5 kJ/kg, as follows: HG
0013
Hf
0007Hf 0003 HG
1/0007Hf 0003 HG
29.7 40 50 60 70 78.5
295 302 309 316 323 330
65 98 137 190 265 408
35.3 58 87 130 195 329.5
0.0283 0.0173 0.0115 0.0077 0.0051 0.0030
From a plot of 1/0007Hf 0003 HG and HG the area under the curve is 0.573. Thus: 0001 HG2 height of packing, z D [dHG /0007Hf 0003 HG ]G/hD a0017 (equation 13.53) HG1
D 00070.573 ð 0.835/00070.2 ð 1.198 D 1.997, say 2.0 m In Fig. 13a, a plot of HG and 0013G is obtained using the construction given in Section 13.6.3. and shown in Fig. 13.16. From this plot, the value of 0013G2 corresponding
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to HG2 D 78.5 kJ/kg is 300 K. From Fig. 13.5 the exit air therefore has a humidity of 0.02 kg/kg which from Fig. 13.4 corresponds to a percentage humidity of 90%.
PROBLEM 13.9 Water is to be cooled in a small packed column from 330 to 285 K by means of air flowing countercurrently. The rate of flow of liquid is 1400 cm3 /m2 s and the flowrate of the air, which enters at 295 K with a humidity of 60% is 3.0 m3 /m2 s. Calculate the required height of tower if the whole of the resistance to heat and mass transfer can be considered as being in the gas phase and the product of the mass transfer coefficient and the transfer surface per unit volume of column is 2 s00031 . What is the condition of the air which leaves at the top?
Solution As in Problem 13.8, assuming the relevant latent and specific heat capacities: HG1 D 1.0030007295 0003 273 C H 00072495 C 2.0060007295 0003 273 From Fig. 13.4, at 0013 D 295 and 60% humidity, H D 0.010 kg/kg and hence: HG1 D 00071.003 ð 22 C 0.01000072495 C 44.13 D 47.46 kJ/kg In the inlet air, water vapour D 0.010 kg/kg dry air or 00070.010/18/00071/29 D 0.016 kmol/kmol dry air. Thus the flow of dry air D 00071 0003 0.0163.0 D 2.952 m3 /m2 s. Density of air at 295 K D 000729/22.40007273/293 D 1.198 kg/m3 . and mass flow of dry air D 00071.198 ð 2.952 D 3.537 kg/m2 s. Liquid flow D 1.4 ð 1000033 m3 /m2 s and mass flow of liquid D 00071.4 ð 1000033 ð 1000 D 1.4 kg/m2 s. The slope of the operating line is thus: LCL /G D 00071.40 ð 4.18/3.537 D 1.66 and the coordinates of the bottom of the line are: 0013L1 D 285 K,
HG1 D 47.46 kJ/kg
From these data, the operating line may be drawn in as shown in Fig. 13b and the top point of the operating line is: 0013L2 D 330 K,
HG2 D 122 kJ/kg
Again as in Problem 13.8, the relation between enthalpy and temperature at the interface Hf vs. 0013f is drawn in Fig. 13b. It is seen that the operating line cuts the saturation curve, which is clearly an impossible situation and, indeed, it is not possible to cool the water to 285 K under these conditions. As discussed in Section 13.6.1, with mechanical draught towers, it is possible, at the best, to cool the water to within, say, 1 deg K of the wet
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327
350
Enthalpy (HG kJ/kg)
300 250 200 150 (qL2'HG2)
100 50 (qL1'HG1)
0 280 290 300 310 320 330 Temperature (q K )
Figure 13b.
bulb temperature. From Fig. 13.4, at 295 K and 60% humidity, the wet-bulb temperature of the inlet air is 290 K and at the best water might be cooled to 291 K. In the present case, therefore, 291 K will be chosen as the water outlet temperature. Thus an operating line of slope: LCL /G D 1.66 and bottom coordinates: 0013L1 D 291 K and HG1 D 47.5 kJ/kg is drawn as shown in Fig. 13c. At the top of the operating line: 0013L2 D 330 K and HG2 D 112.5 kJ/kg As an alternative to the method used in Problem 13.8, the approximate method of Carey and Williamson (equation 13.54) is adopted. At the bottom of the column: HG1 D 47.5 kJ/kg,
Hf1 D 52.0 kJ/kg
∴ H1 D 4.5 kJ/kg
Hf2 D 382 kJ/kg
∴ H2 D 269.5 kJ/kg
At the top of the column: HG2 D 112.5 kJ/kg,
At the mean water temperature of 0.50007330 C 291 D 310.5 K: HGm D 82.0 kJ/kg, ∴
Hfm D 152.5 kJ/kg
∴ Hm D 70.5 kJ/kg
Hm /H1 D 15.70 and Hm /H2 D 0.262
and from Fig. 13.17: f D 0.35 (extending the scales)
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350
Enthaolpy (HG kJ/kg)
300 250 200 150 100 50
0 280 290 300 310 320 330 Temperature (q k)
Figure 13c.
Thus: 0001
HG2
[dHG /0007Hf 0003 HG ]G/hD a0017
height of packing, z D
(equation 13.53)
HG1
D 00070.35 ð 3.537/00072.0 ð 1.198 D 0.52 m Due to the close proximity of the operating line to the line of saturation, the gas will be saturated on leaving the column and will therefore be at 100% humidity. From Fig. 13c the exit gas will be at 306 K.
PROBLEM 13.10 Air containing 0.005 kg water vapour/kg dry air is heated to 325 K in a dryer and passed to the lower shelves. It leaves these shelves at 60% humidity and is reheated to 325 K and passed over another set of shelves, again leaving with 60% humidity. This is again reheated for the third and fourth sets of shelves after which the air leaves the dryer. On the assumption that the material in each shelf has reached the wet bulb temperature and that heat losses from the dryer can be neglected, determine: (a) the temperature of the material on each tray, (b) the rate of water removal if 5 m3 /s of moist air leaves the dryer,
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(c) the temperature to which the inlet air would have to be raised to carry out the drying in a single stage.
Solution See Volume 1, Example 13.4
PROBLEM 13.11 0.08 m3 /s of air at 305 K and 60% humidity is to be cooled to 275 K. Calculate, using a psychrometric chart, the amount of heat to be removed for each 10 deg K interval of the cooling process. What total mass of moisture will be deposited? What is the humid heat of the air at the beginning and end of the process?
Solution At 305 K and 60% humidity, from Fig. 13.4, the wet-bulb temperature is 299 K and H D 0.018 kg/kg. Thus, as the air is cooled, the per cent humidity will increase until saturation occurs at 299 K and the problem is then one of cooling saturated vapour from 299 K to 275 K. Considering the cooling in 10 deg K increments, the following data are obtained from Fig. 13.4: 0013 (K)
0013w (K)
% Humidity
H
Humid heat (kJ/kg K)
Latent heat (kJ/kg)
305 299 295 285 275
299 299 295 285 275
60 100 100 100 100
0.018 0.018 0.017 0.009 0.0045
1.032 1.032 1.026 1.014 1.001
2422 2435 2445 2468 2491
At 305 K:
the specific volume of dry air D 0.861 m3 /kg the saturated volume D 0.908 m3 /kg
and hence the specific volume at 60% humidity D [0.861 C 00070.908 0003 0.86160/100] D 0.889 m3 /kg Thus:
mass flow of moist air D 00070.08/0.889 D 0.090 kg/s
Thus the flowrate of dry air D 0.090/00071 C 0.018 D 0.0884 kg/s. From Fig. 13.4, specific heat of dry air (at H D 0 D 0.995 kJ/kg K. ∴
enthalpy of moist air D 00070.0884 ð 0.9950007299 0003 273 C 00070.018 ð 0.0884 ð[4.180007299 0003 273 C 2435] C 0.090 ð 1.0320007305 0003 299 D 6.89 kW
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At 295 K: Enthalpy of moist air D 00070.0884 ð 0.9950007295 0003 273 C 00070.017 ð 0.0884 ð [4.180007295 0003 273 C 2445] D 5.75 kW At 285 K: Enthalpy of moist air D 00070.0884 ð 0.9950007285 0003 273 C 00070.009 ð 0.0884 ð [4.180007285 0003 273 C 2468] D 3.06 kW At 275 K: Enthalpy of moist air D 00070.0884 ð 0.9950007275 0003 273 C 00070.0045 ð 0.0884 ð [4.180007275 0003 273 C 2491] D 1.17 kW and hence in cooling from 305 to 295 K, heat to be removed D 00076.89 0003 5.75 D 1.14 kW in cooling from 295 to 285 K, heat to be removed D 00075.75 0003 3.06 D 2.69 kW in cooling from 285 to 275 K, heat to be removed D 00073.06 0003 1.17 D 1.89 kW The mass of water condensed D 0.088400070.018 0003 0.0045 D 0.0012 kg/s. The humid heats at the beginning and end of the process are: 1.082 and 1.001 kJ/kg K respectively.
PROBLEM 13.12 A hydrogen stream at 300 K and atmospheric pressure has a dew point of 275 K. It is to be further humidified by adding to it (through a nozzle) saturated steam at 240 kN/m2 at the rate of 1 kg steam: 30 kg of hydrogen feed. What will be the temperature and humidity of the resultant stream?
Solution At 275 K, the vapour pressure of water D 0.72 kN/m2 (from Tables) and the hydrogen is saturated. The mass of water vapour: Pw0 Mw /RT D 00070.72 ð 18/00078.314 ð 275 D 0.00567kg/m3 and the mass of hydrogen: 0007P 0003 Pw0 MA /RT D 0007101.3 0003 0.722/00078.314 ð 275 D 0.0880 kg/m3 Therefore the humidity at saturation, H0 D 00070.00567/0.0880 D 0.0644 kg/kg dry hydrogen and at 300 K, the humidity will be the same, H1 D 0.0644 kg/kg. At 240 kN/m2 pressure, steam is saturated at 400 K at which temperature the latent heat is 2185 kJ/kg. The enthalpy of the steam is therefore: H2 D 4.180007400 0003 273 C 2185 D 2715.9 kJ/kg Taking the mean specific heat capacity of hydrogen as 14.6 kJ/kg K, the enthalpy in 30 kg moist hydrogen or 30/00071 C 0.0644 D 28.18 kg dry hydrogen is: 000728.18 ð 14.60007300 0003 273 D 11,110 kJ
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331
The latent heat of water at 275 K is 2490 kJ/kg and, taking the specific heat of water vapour as 2.01 kJ/kg K, the enthalpy of the water vapour is: 000728.18 ð 0.064400074.180007275 0003 273 C 2490 C 2.010007300 0003 275 D 4625 kJ Hence the total enthalpy:
H1 D 15,730 kJ
In mixing the two streams, 28.18 kg dry hydrogen plus 000730 0003 28.18 D 1.82 kg water is mixed with 1 kg steam and hence the final humidity: H D 00071 C 1.82/28.18 D 0.100 kg/kg
In the final mixture, 0.1 kg water vapour is associated with 1 kg dry hydrogen or 00070.1/18 D 0.0056 kmol water is associated with 00071/2 D 0.5 kmol hydrogen, a total of 0.5056 kmol. ∴
partial pressure of water vapour D 00070.0056/0.5056101.3 D 1.11 kN/m2
Water has a vapour pressure of 1.11 kN/m2 at 281 K at which the latent heat is 2477 kJ/kg. Thus if T K is the temperature of the mixture, then: 00072716 C 15730 D 000728.18 ð 14.60007T 0003 273 C 2.82[4.180007281 0003 273 C 2447 C 2.010007T 0003 281] and T D 300.5 K It may be noted that this relatively low increase in temperature occurs because the latent heat in the steam is not recovered, as would be the case in, say, a shell and tube unit.
PROBLEM 13.13 In a countercurrent packed column, n-butanol flows down at the rate of 0.25 kg/m2 s and is cooled from 330 to 295 K. Air at 290 K, initially free of n-butanol vapour, is passed up the column at the rate of 0.7 m3 /m2 s. Calculate the required height of tower and the condition of the exit air. Data: Mass transfer coefficient per unit volume, hD a D 0.1 s00031 . Psychrometric ratio, 0007h/hD 0017A s D 2.34. Heat transfer coefficients, hL D 3hG . Latent heat of vaporisation of n-butanol, 001b D 590 kJ/kg. Specific heat capacity of liquid n-butanol, CL D 2.5 kJ/kg K. Humid heat of gas: s D 1.05 kJ/kg K. Temperature (K)
Vapour pressure of n-butanol 0007kN/m2
295 300 305 310 315 320 325
0.59 0.86 1.27 1.75 2.48 3.32 4.49
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Temperature (K)
Vapour pressure of n-butanol 0007kN/m2
330 335 340 345 350
5.99 7.89 10.36 14.97 17.50
Solution See Volume 1, Example 13.10
PROBLEM 13.14 Estimate the height and base diameter of a natural draught hyperbolic cooling tower which will handle 5000 kg/s water entering at 300 K and leaving at 294 K. The dry-bulb air temperature is 287 K and the ambient wet-bulb temperature is 284 K.
Solution See Volume 1, Example 13.8
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CHEMICAL ENGINEERING Solutions to the Problems in Chemical Engineering Volume 1
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Coulson & Richardson’s Chemical Engineering Chemical Engineering, Volume 1, Sixth edition Fluid Flow, Heat Transfer and Mass Transfer J. M. Coulson and J. F. Richardson with J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 2, Fourth edition Particle Technology and Separation Processes J. M. Coulson and J. F. Richardson with J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 3, Third edition Chemical & Biochemical Reactors & Process Control Edited by J. F. Richardson and D. G. Peacock Solutions to the Problems in Volume 1, First edition J. R. Backhurst and J. H. Harker with J. F. Richardson Chemical Engineering, Volume 5, Second edition Solutions to the Problems in Volumes 2 and 3 J. R. Backhurst and J. H. Harker Chemical Engineering, Volume 6, Third edition Chemical Engineering Design R. K. Sinnott
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Coulson & Richardson’s
CHEMICAL ENGINEERING J. M. COULSON and J. F. RICHARDSON
Solutions to the Problems in Chemical Engineering Volume 1 By
J. R. BACKHURST and J. H. HARKER University of Newcastle upon Tyne
With
J. F. RICHARDSON University of Wales Swansea
OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI
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Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd
First published 2001 J. F. Richardson, J. R. Backhurst and J. H. Harker 2001
All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 4950 X Typeset by Laser Words, Madras, India
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Contents Preface 1. Units and dimensions
iv 1
2. Flow of fluids — energy and momentum relationships
16
3. Flow in pipes and channels
19
4. Flow of compressible fluids
60
5. Flow of multiphase mixtures
74
6. Flow and pressure measurement
77
7. Liquid mixing
103
8. Pumping of fluids
109
9. Heat transfer
125
10. Mass transfer
217
11. The boundary layer
285
12. Momentum, heat and mass transfer
298
13. Humidification and water cooling
318
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Preface Each of the volumes of the Chemical Engineering Series includes numerical examples to illustrate the application of the theory presented in the text. In addition, at the end of each volume, there is a selection of problems which the reader is invited to solve in order to consolidate his (or her) understanding of the principles and to gain a better appreciation of the order of magnitude of the quantities involved. Many readers who do not have ready access to assistance have expressed the desire for solutions manuals to be available. This book, which is a successor to the old Volume 4, is an attempt to satisfy this demand as far as the problems in Volume 1 are concerned. It should be appreciated that most engineering problems do not have unique solutions, and they can also often be solved using a variety of different approaches. If therefore the reader arrives at a different answer from that in the book, it does not necessarily mean that it is wrong. This edition of the solutions manual relates to the sixth edition of Volume 1 and incorporates many new problems. There may therefore be some mismatch with earlier editions and, as the volumes are being continually revised, they can easily get out-of-step with each other. None of the authors claims to be infallible, and it is inevitable that errors will occur from time to time. These will become apparent to readers who use the book. We have been very grateful in the past to those who have pointed out mistakes which have then been corrected in later editions. It is hoped that the present generation of readers will prove to be equally helpful! J. F. R.
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SECTION 1
Units and Dimensions PROBLEM 1.1 98% sulphuric acid of viscosity 0.025 N s/m2 and density 1840 kg/m3 is pumped at 685 cm3 /s through a 25 mm line. Calculate the value of the Reynolds number.
Solution Cross-sectional area of line D 00030004/400050.0252 D 0.00049 m2 . Mean velocity of acid, u D 0003685 ð 1000036 0005/0.00049 D 1.398 m/s. ∴ Reynolds number, Re D du / D 00030.025 ð 1.398 ð 18400005/0.025 D 2572
PROBLEM 1.2 Compare the costs of electricity at 1 p per kWh and gas at 15 p per therm.
Solution Each cost is calculated in p/MJ. 1 kWh D 1 kW ð 1 h D 00031000 J/s000500033600 s0005 D 3,600,000 J or 3.6 MJ 1 therm D 105.5 MJ ∴ cost of electricity D 1 p/3.6 MJ or 00031/3.60005 D 0.28 p/MJ cost of gas D 15 p/105.5 MJ or 000315/105.50005 D 0.14 p/MJ
PROBLEM 1.3 A boiler plant raises 5.2 kg/s of steam at 1825 kN/m2 pressure, using coal of calorific value 27.2 MJ/kg. If the boiler efficiency is 75%, how much coal is consumed per day? If the steam is used to generate electricity, what is the power generation in kilowatts assuming a 20% conversion efficiency of the turbines and generators? 1
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Solution From the steam tables, in Appendix A2, Volume 1, total enthalpy of steam at 1825 kN/m2 D 2798 kJ/kg. ∴
enthalpy of steam D 00035.2 ð 27980005 D 14,550 kW
Neglecting the enthalpy of the feed water, this must be derived from the coal. With an efficiency of 75%, the heat provided by the coal D 000314,550 ð 1000005/75 D 19,400 kW. For a calorific value of 27,200 kJ/kg, rate of coal consumption D 000319,400/27,2000005 D 0.713 kg/s or:
00030.713 ð 3600 ð 240005/1000 D 61.6 Mg/day
20% of the enthalpy in the steam is converted to power or: 000314,550 ð 200005/100 D 2910 kW or 2.91 MW say 3 MW
PROBLEM 1.4 The power required by an agitator in a tank is a function of the following four variables: (a) (b) (c) (d)
diameter of impeller, number of rotations of the impeller per unit time, viscosity of liquid, density of liquid.
From a dimensional analysis, obtain a relation between the power and the four variables. The power consumption is found, experimentally, to be proportional to the square of the speed of rotation. By what factor would the power be expected to increase if the impeller diameter were doubled?
Solution If the power P D f0003DN 0005, then a typical form of the function is P D kDa Nb c d , where k is a constant. The dimensions of each parameter in terms of M, L, and T are: power, P D ML2 /T3 , density, D M/L3 , diameter, D D L, viscosity, D M/LT, and speed of rotation, N D T00031 Equating dimensions: M: 1 DcCd L : 2 D a 0003 3c 0003 d T : 00033 D 0003b 0003 d Solving in terms of d : a D 00035 0003 2d0005, b D 00033 0003 d0005, c D 00031 0003 d0005 0001 5 3 0002 D N d ∴ PDk D2d Nd d or: that is:
P/D5 N3 D k0003D2 N /000b00050003d NP D k Rem
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Thus the power number is a function of the Reynolds number to the power m. In fact NP is also a function of the Froude number, DN2 /g. The previous equation may be written as: P/D5 N3 D k0003D2 N /000b0005m P / N2
Experimentally: From the equation,
P / Nm N3 , that is m C 3 D 2 and m D 00031
Thus for the same fluid, that is the same viscosity and density: 0003P2 /P1 00050003D15 N31 /D25 N32 0005 D 0003D12 N1 /D22 N2 000500031 or: 0003P2 /P1 0005 D 0003N22 D23 0005/0003N21 D13 0005 In this case, N1 D N2 and D2 D 2D1 . ∴
0003P2 /P1 0005 D 8D13 /D13 D 8
A similar solution may be obtained using the Recurring Set method as follows: P D f0003D, N, , 0005, f0003P, D, N, , 0005 D 0 Using M, L and T as fundamentals, there are five variables and three fundamentals and therefore by Buckingham’s 0004 theorem, there will be two dimensionless groups. Choosing D, N and as the recurring set, dimensionally: 0003 0004 D0006L L0006D 00031 N0006T T 0006 N00031 Thus: 00033
0006 ML M 0006 L3 D D3 First group, 00041 , is P0003ML2 T00033 000500031 0006 P0003 D3 D2 N3 000500031 0006
P
D5 N3
Second group, 00042 , is 0003ML00031 T00031 000500031 0006 0003 D3 D00031 N000500031 0006 0001
Thus:
f
P ,
D5 N3 D2 N
0002
D2 N
D0
Although there is little to be gained by using this method for simple problems, there is considerable advantage when a large number of groups is involved.
PROBLEM 1.5 It is found experimentally that the terminal settling velocity u0 of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle 0003 weight of displaced fluid), W; fluid density, , and fluid viscosity, . Obtain a relationship for u0 using dimensional analysis. Stokes established, from theoretical considerations, that for small particles which settle at very low velocities, the settling velocity is independent of the density of the fluid
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except in so far as this affects the buoyancy. Show that the settling velocity must then be inversely proportional to the viscosity of the fluid.
Solution u0 D kda Wb c d , then working in dimensions of M, L and T:
If:
0003L/T0005 D k0003La 0003ML/T2 0005b 0003M/L3 0005c 0003M/LT0005d 0005 Equating dimensions: M: 0 DbCcCd L : 1 D a C b 0003 3c 0003 d T : 00031 D 00032b 0003 d Solving in terms of b: a D 00031, c D 0003b 0003 10005, and d D 00031 0003 2b0005 ∴
u0 D k00031/d00050003Wb 00050003 b / 00050003000b/000b2b 0005 where k is a constant,
or:
u0 D k0003000b/d 00050003W /000b2 0005b
Rearranging: 0003du0 /000b0005 D k0003W /000b2 0005b where (W /000b2 ) is a function of a form of the Reynolds number. For u0 to be independent of , b must equal unity and u0 D kW/d Thus, for constant diameter and hence buoyant weight, the settling velocity is inversely proportional to the fluid viscosity.
PROBLEM 1.6 A drop of liquid spreads over a horizontal surface. What are the factors which will influence: (a) the rate at which the liquid spreads, and (b) the final shape of the drop? Obtain dimensionless groups involving the physical variables in the two cases.
Solution (a) The rate at which a drop spreads, say R m/s, will be influenced by: viscosity of the liquid, ; volume of the drop, V expressed in terms of d, the drop diameter; density of the liquid, ; acceleration due to gravity, g and possibly, surface tension of the liquid,
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0019. In this event: R D f0003000b, d, , g, 00190005. The dimensions of each variable are: R D L/T, D M/LT, d D L, D M/L3 , g D L/T2 , and 0019 D M/T2 . There are 6 variables and 3 fundamentals and hence 00036 0003 30005 D 3 dimensionless groups. Taking as the recurring set, d, and g, then: d 0006 L,
0006 M/L3 g 0006 L/T2
LDd ∴ M D L3 D d3 ∴ T2 D L/g D d/g and T D d0.5 /g0.5
Thus, dimensionless group 1: RT/L D Rd0.5 /dg0.5 D R/0003dg00050.5 dimensionless group 2: LT/M D d0003d0.5 0005/0003g0.5 d3 0005 D /0003g0.5 d1.5 0005
∴
or:
dimensionless group 3: 0019T2 /M D 0019d/0003g d3 0005 D 0019/0003g d2 0005 0001 0002 0019 0.5 R/0003dg0005 D f 0.5 1.5 , g d g d2 0001 0002 2 R2 0019 Df , dg g 2 d3 g d2
(b) The final shape of the drop as indicated by its diameter, d, may be obtained by using the argument in (a) and putting R D 0. An alternative approach is to assume the final shape of the drop, that is the final diameter attained when the force due to surface tension is equal to that attributable to gravitational force. The variables involved here will be: volume of the drop, V; density of the liquid, ; acceleration due to gravity, g, and the surface tension of the liquid, 0019. In this case: d D f0003V, , g, 00190005. The dimensions of each variable are: d D L, V D L3 , D M/L3 , g D L/T2 , 0019 D M/T2 . There are 5 variables and 3 fundamentals and hence 00035 0003 30005 D 2 dimensionless groups. Taking, as before, d,
and g as the recurring set, then: d 0006 L,
0006 M/L3 g 0006 L/T2
LDd ∴ M D L3 D d3 ∴ T2 D L/g D d/g and T D d0.5 /g0.5
Dimensionless group 1: V/L3 D V/d3 Dimensionless group 2: 0019T2 /M D 0019d/0003g d3 0005 D 0019/0003g d2 0005 0001 0002 0019 and hence: 0003d3 /V0005 D f g d2
PROBLEM 1.7 Liquid is flowing at a volumetric flowrate of Q per unit width down a vertical surface. Obtain from dimensional analysis the form of the relationship between flowrate and film thickness. If the flow is streamline, show that the volumetric flowrate is directly proportional to the density of the liquid.
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Solution The flowrate, Q, will be a function of the fluid density, , and viscosity, , the film thickness, d, and the acceleration due to gravity, g, or:
Q D f0003 , g, , d0005, or: Q D K a gb c dd where K is a constant.
The dimensions of each variable are: Q D L2 /T, D M/L3 , g D L/T2 , D M/LT and d D L. Equating dimensions: M: 0 DaCc L : 2 D 00033a C b 0003 c C d T : 00031 D 00032b 0003 c from which, c D 1 0003 2b, a D 0003c D 2b 0003 1, and d D 2 C 3a 0003 b C c D 2 C 6b 0003 3 0003 b C 1 0003 2b D 3b ∴
or:
Q D K0003 2b00031 gb 100032b d3b 0005 Q
D K0003 2 gd3 /000b2 0005b and Q / 100032b .
For streamline flow, Q / 00031 and:
00031 D 1 0003 2b and b D 1
∴
Q / D K0003 2 gd3 /000b2 0005, Q D K0003 gd3 /000b0005
and:
Q is directly proportional to the density,
PROBLEM 1.8 Obtain, by dimensional analysis, a functional relationship for the heat transfer coefficient for forced convection at the inner wall of an annulus through which a cooling liquid is flowing.
Solution Taking the heat transfer coefficient, h, as a function of the fluid velocity, density, viscosity, specific heat and thermal conductivity, u, , , Cp and k, respectively, and of the inside and outside diameters of the annulus, di and d0 respectively, then: h D f0003u, di , d0 , , , Cp , k0005 The dimensions of each variable are: h D H/L2 Tq, u D L/T, di D L, d0 D L, D M/L3 , D M/LT, Cp D H/Mq, k D H/LTq. There are 8 variables and 5 fundamental dimensions and hence there will be 00038 0003 50005 D 3 groups. H and q always appear however as
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the group H/q and in effect the fundamental dimensions are 4 (M, L, T and H/q) and there will be 00038 0003 40005 D 4 groups. For the recurring set, the variables di , , k and will be chosen. Thus: di
k
0006 L, L 0006 M/L3 M 0006 M/LT, T 0006 0003H/q0005/LT, 0003H/q0005
D di D L3 D d3i D M/L D d3i /di D d2i / D kLT D kdi d2i / D k d3i /000b
Dimensionless group 1: hL2 T/0003H/q0005 D hd2i d2i /000b0003k d3i /000b0005 D hdi /k Dimensionless group 2: uT/L D u d2i /000bdi D di u / Dimensionless group 3: d0 /L D d0 /di Dimensionless group 4: Cp M/0003H/q0005 D Cp d3i /k0003 d3i /000b0005 D Cp /k ∴
hdi /k D f0003di u /000b, Cp /k, d0 /di 0005 which is a form of equation 9.94.
PROBLEM 1.9 Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection. It is found by experiment that, when the flow is turbulent, increasing the flowrate by a factor of 2 always results in a 50% increase in the coefficient. How would a 50% increase in density of the fluid be expected to affect the coefficient, all other variables remaining constant?
Solution For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as: hd/k D f0003du /000b, Cp /k0005 or:
hd/k D K0003du /000b0005n 0003Cp /k0005m
∴
h2 /h1 D 0003u2 /u1 0005n .
Increasing the flowrate by a factor of 2 results in a 50% increase in the coefficient, or: 1.5 D 2.0n and n D 0003ln 1.5/ ln 2.00005 D 0.585. Also:
h2 /h1 D 0003 2 / 1 00050.585
When 0003 2 / 1 0005 D 1.50, h2 /h1 D 00031.5000050.585 D 1.27 and the coefficient is increased by 27%
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PROBLEM 1.10 A stream of droplets of liquid is formed rapidly at an orifice submerged in a second, immiscible liquid. What physical properties would be expected to influence the mean size of droplet formed? Using dimensional analysis obtain a functional relation between the variables.
Solution The mean droplet size, dp , will be influenced by: diameter of the orifice, d; velocity of the liquid, u; interfacial tension, 0019; viscosity of the dispersed phase, ; density of the dispersed phase, d ; density of the continuous phase, c , and acceleration due to gravity, g. It would also be acceptable to use the term 0003 d 0003 c 0005g to take account of gravitational forces and there may be some justification in also taking into account the viscosity of the continuous phase. On this basis:
dp D f0003d, u, 0019, , d , c , g0005
The dimensions of each variable are: dp D L, d D L, u D L/T, 0019 D M/T2 , D M/LT,
d D M/L3 , c D M/L3 , and g D L/T2 . There are 7 variables and hence with 3 fundamental dimensions, there will be 00037 0003 30005 D 4 dimensionless groups. The variables d, u and 0019 will be chosen as the recurring set and hence: d 0006 L, LDd u 0006 L/T, T D L/u D d/u 0019 0006 M/T2 , M D 0019T2 D 0019d2 /u2 Thus, dimensionless group 1: LT/M D d0003d/u0005/00030019d2 /u2 0005 D u/0019 dimensionless group 2: d L3 /M D d d3 /00030019d2 /u2 0005 D d du2 /0019 dimensionless group 3: c L3 /M D c d3 /00030019d2 /u2 0005 D c du2 /0019 dimensionless group 4: gT2 /L D g0003d2 /u2 0005/d D gd/u2 and the function becomes: dp D f0003000bu/0019, d du2 /0019, c du2 /0019, gd/u2 0005
PROBLEM 1.11 Liquid flows under steady-state conditions along an open channel of fixed inclination to the horizontal. On what factors will the depth of liquid in the channel depend? Obtain a relationship between the variables using dimensional analysis.
Solution The depth of liquid, d, will probably depend on: density and viscosity of the liquid,
and ; acceleration due to gravity, g; volumetric flowrate per unit width of channel, Q,
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and the angle of inclination, ', or:
d D f0003 , , g, Q, '0005
Excluding ' at this stage, there are 5 variables and with 3 fundamental dimensions there will be 00035 0003 30005 D 2 dimensionless groups. The dimensions of each variable are: d D L,
D M/L3 , D M/LT, g D L/T2 , Q D L2 /T, and, choosing Q, and g as the recurring set, then: Q D L2 /T g D L/T2
D M/L3
T D L2 /Q L D gT2 D gL4 /Q2 , L3 D Q2 /g, L D Q2/3 /g1/3 and T D Q4/3 /Qg2/3 D Q1/3 /g2/3 M D L3 D 0003Q2 /g0005 D Q2 /g
Thus, dimensionless group 1: d/L D dg1/3 /Q2/3 or d3 g/Q2 dimensionless group 2: LT/M D 0003Q2/3 /g1/3 00050003Q1/3 /g2/3 0005/Q2 g D /Q
and the function becomes: d3 g/Q2 D f0003000b/Q , '0005
PROBLEM 1.12 Liquid flows down an inclined surface as a film. On what variables will the thickness of the liquid film depend? Obtain the relevant dimensionless groups. It may be assumed that the surface is sufficiently wide for edge effects to be negligible.
Solution This is essentially the same as Problem 1.11, though here the approach used is that of equating indices. d D K0003 a , b , gc , Qd , ' e 0005
If, as before:
then, excluding ' at this stage, the dimensions of each variable are: d D L, D M/L3 , D M/LT, g D L/T2 , Q D L2 /T. Equating dimensions: M: 0DaCb
0003i0005
L : 1 D 00033a 0003 b C c C 2d
0003ii0005
T : 0 D 0003b 0003 2c 0003 d
0003iii0005
Solving in terms of b and c then: from (i)
a D 0003b
from (iii)
d D 0003b 0003 2c
and in (ii)
1 D 3b 0003 b C c 0003 2b 0003 4c or: c D 00031/3 ∴
d D 2/3 0003 b
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d D K0003 0003b Ð b Ð g00031/3 Ð Q2/30003b 0005
Thus:
dg1/3 /Q2/3 D K0003000b/ Q0005b and:
d3 g/Q2 D K0003000b/ Q0005b 0003'0005e as before.
PROBLEM 1.13 A glass particle settles under the action of gravity in a liquid. Upon what variables would the terminal velocity of the particle be expected to depend? Obtain a relevant dimensionless grouping of the variables. The falling velocity is found to be proportional to the square of the particle diameter when other variables are kept constant. What will be the effect of doubling the viscosity of the liquid? What does this suggest regarding the nature of the flow?
Solution See Volume 1, Example 1.3
PROBLEM 1.14 Heat is transferred from condensing steam to a vertical surface and the resistance to heat transfer is attributable to the thermal resistance of the condensate layer on the surface. What variables are expected to affect the film thickness at a point? Obtain the relevant dimensionless groups. For streamline flow it is found that the film thickness is proportional to the one third power of the volumetric flowrate per unit width. Show that the heat transfer coefficient is expected to be inversely proportional to the one third power of viscosity.
Solution For a film of liquid flowing down a vertical surface, the variables influencing the film thickness υ, include: viscosity of the liquid (water), ; density of the liquid, ; the flow per unit width of surface, Q, and the acceleration due to gravity, g. Thus: υ D f0003000b, , Q, g0005. The dimensions of each variable are: υ D L, D M/LT, D M/L3 , Q D L2 /T, and g D L/T2 . Thus, with 5 variables and 3 fundamental dimensions, 00035 0003 30005 D 2 dimensionless groups are expected. Taking , and g as the recurring set, then: 0006 M/LT, M D LT ∴ L3 D LT, T D L2 /000b
0006 M/L3 , M D L3 2 g 0006 L/T D 2 L/ 2 L4 D 2 / 2 L3 ∴ L3 D 2 / 2 g and L D 2/3 /0003 2/3 g1/3 0005 ∴
T D 0003000b2 / 2 g00052/3 / D 1/3 /0003 1/3 g2/3 0005
and:
M D 0003000b2 / 2 g00051/3 0003000b1/3 /0003 1/3 g2/3 00050005 D 2 /0003 g0005
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Thus, dimensionless group 1: QT/L2 D Q0003000b1/3 /0003 1/3 g2/3 00050005/0003000b4/3 /0003 4/3 g2/3 00050005 D Q / dimensionless group 2: υL D υ000b2/3 /0003 2/3 g1/3 0005 or, cubing D υ3 2 g/000b2 0003υ3 2 g/000b2 0005 D f0003Q /000b0005
and: This may be written as:
0003υ3 2 g/000b2 0005 D K0003Q /000b0005n
For streamline flow, υ / Q1/3 or n D 1 and hence: 0003υ3 2 g/000b2 0005 D KQ /000b, υ3 D KQ000b/0003 g0005 and υ D 0003KQ000b/ g00051/3 As the resistance to heat transfer is attributable to the thermal resistance of the condensate layer which in turn is a function of the film thickness, then: h / k/υ where k is the thermal conductivity of the film and since υ / 1/3 , h / k/000b1/3 , that is the coefficient is inversely proportional to the one third power of the liquid viscosity.
PROBLEM 1.15 A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables would you expect the falling velocity of the particle to depend? Obtain the relevant dimensionless groups. For particles of a given density settling in a vessel of large diameter, the settling velocity is found to be inversely proportional to the viscosity of the liquid. How would this depend on particle size?
Solution This problem is very similar to Problem 1.13, although, in this case, the liquid through which the particle settles is contained in a narrow vessel. This introduces another variable, D, the vessel diameter and hence the settling velocity of the particle is given by: u D f0003d, , , D, s , g0005. The dimensions of each variable are: u D L/T, d D L, D M/L3 , D M/LT, D D L, s D M/L3 , and g D L/T2 . With 7 variables and 3 fundamental dimensions, there will be 00037 0003 30005 D 4 dimensionless groups. Taking d, and as the recurring set, then: d 0006 L, LDd 3
0006 M/L , M D L3 D d3 0006 M/LT, T D M/L D d3 /d D d2 / Thus: dimensionless group 1: uT/L D u d2 /0003000bd0005 D du / dimensionless group 2: D/L D D/d dimensionless group 3: s L3 /M D s d3 /0003 d3 0005 D s /
and dimensionless group 4: gT2 /L D g 2 d4 /0003000b2 d0005 D g 2 d3 /000b2 Thus:
0003du /000b0005 D f00030003D/d00050003 s / 00050003g 2 d3 /000b2 00050005
In particular, 0003du /000b0005 D K0003g 2 d3 /000b2 0005n where K is a constant.
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For particles settling in a vessel of large diameter, u / 1/000b. But 0003u/000b0005 / 00031/000b2 0005n and, when n D 1, n / 1/000b. In this case: 0003du /000b0005 D K0003g 2 d3 /000b2 0005 du / d3 and u / d2
or:
Thus the settling velocity is proportional to the square of the particle size.
PROBLEM 1.16 A liquid is in steady state flow in an open trough of rectangular cross-section inclined at an angle ' to the horizontal. On what variables would you expect the mass flow per unit time to depend? Obtain the dimensionless groups which are applicable to this problem.
Solution This problem is similar to Problems 1.11 and 1.12 although, here, the width of the trough and the depth of liquid are to be taken into account. In this case, the mass flow of liquid per unit time, G will depend on: fluid density, ; fluid viscosity, ; depth of liquid, h; width of the trough, a; acceleration due to gravity, g and the angle to the horizontal, '. Thus: G D f0003 , , h, a, g, '0005. The dimensions of each variable are: G D M/T, D M/L3 , D M/LT, h D L, a D L, g D L/T2 and neglecting ' at this stage, with 6 variables with dimensions and 3 fundamental dimensions, there will be 00036 0003 30005 D 3 dimensionless groups. Taking h, and as the recurring set then: h 0006 L, LDh 3
0006 M/L , M D L3 D h3 0006 M/LT, T D M/L D h3 /0003h000b0005 D h2 / Thus: dimensionless group 1: GT/M D G h2 /0003 h3 0005 D G/000bh dimensionless group 2: a/L D a/h dimensionless group 3: gT2 /L D g 2 h4 /0003000b2 h0005 D g 2 h3 /000b2 and:
0003G/000bh0005 D f00030003a/h00050003g 2 h3 /000b2 00050005
PROBLEM 1.17 The resistance force on a spherical particle settling in a fluid is given by Stokes’ Law. Obtain an expression for the terminal falling velocity of the particle. It is convenient to express experimental results in the form of a dimensionless group which may be plotted against a Reynolds group with respect to the particle. Suggest a suitable form for this dimensionless group. Force on particle from Stokes’ Law D 30004000bdu; where is the fluid viscosity, d is the particle diameter and u is the velocity of the particle relative to the fluid.
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What will be the terminal falling velocity of a particle of diameter 10 µm and of density 1600 kg/m3 settling in a liquid of density 1000 kg/m3 and of viscosity 0.001 Ns/m2 ? If Stokes’ Law applies for particle Reynolds numbers up to 0.2, what is the diameter of the largest particle whose behaviour is governed by Stokes’ Law for this solid and liquid?
Solution The accelerating force due to gravity D 0003mass of particle 0003 mass of liquid displaced0005g. For a particle of radius r, volume D 40004r 3 /3, or, in terms of diameter, d, volume D 400040003d3 /23 0005/3 D 0004d3 /6. Mass of particle D 0004d3 s /6, where s is the density of the solid. Mass of liquid displaced D 0004d3 /6, where is the density of the liquid, and accelerating force due to gravity D 00030004d3 s /6 0003 0004d3 /60005g D 00030004d3 /600050003 s 0003 0005g. At steady state, that is when the terminal velocity is attained, the accelerating force due to gravity must equal the drag force on the particle F, or: 00030004d3 /600050003 s 0003 0005g D 30004000bdu0 where u0 is the terminal velocity of the particle. u0 D 0003d2 g/18000b00050003 s 0003 0005
Thus:
(i)
It is assumed that the resistance per unit projected area of the particle, R0 , is a function of particle diameter, d; liquid density, ; liquid viscosity, , and particle velocity, u or R0 D f0003d, , , u0005. The dimensions of each variable are R0 D M/LT2 , d D L, D M/L3 , D M/LT and u D L/T. With 5 variables and 3 fundamental dimensions, there will be 00035 0003 30005 D 2 dimensionless groups. Taking d, and u as the recurring set, then: d 0006 L, LDd 3
0006 M/L , M D L3 D d3 u 0006 L/T, T D L/u D d/u Thus: dimensionless group 1: R0 LT2 /M D R0 d0003d2 /u2 0005/0003 d3 0005 D R0 / u2 dimensionless group 2: LT/M D d0003d/u0005/0003 d3 0005 D /0003du 0005 R0 / u2 D f0003000b/du 0005
and:
R0 / u2 D K0003du /000b0005n D K Ren
or:
(ii)
In this way the experimental data should be plotted as the group (R/ u2 ) against Re . For this particular example, d D 10 µm D 000310 ð 1000036 0005 D 1000035 m; s D 1600 kg/m3 ;
D 1000 kg/m3 and D 0.001 Ns/m2 . Thus, in equation (i):
u0 D 000300031000035 00052 ð 9.81/000318 ð 0.0010005000500031600 0003 10000005 D 3.27 ð 1000035 m/s or 0.033 mm/s
When Re D 0.2, du / D 0.2 or when the terminal velocity is reached: du0 D 0.2000b/ D 00030.2 ð 0.0010005/1000 D 2 ð 1000037 or:
u0 D 00032 ð 1000037 0005/d
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u0 D 0003d2 g/18000b00050003 s 0003 0005
In equation (i):
00032 ð 1000037 0005/d D 0003d2 ð 9.81/000318 ð 0.0010005000500031600 0003 10000005 ∴
d3 D 6.12 ð 10000313 d D 8.5 ð 1000035 m or 85 µm
and:
PROBLEM 1.18 A sphere, initially at a constant temperature, is immersed in a liquid whose temperature is maintained constant. The time t taken for the temperature of the centre of the sphere to reach a given temperature 'c is a function of the following variables: Diameter of sphere, d Thermal conductivity of sphere, k Density of sphere,
Specific heat capacity of sphere, Cp Temperature of fluid in which it is immersed, 's . Obtain relevant dimensionless groups for this problem.
Solution In this case, t D f0003d, k, , Cp , 'c , 's 0005. The dimensions of each variable are: t D T, d D L, k D ML/Tq, Cp D L2 /T2 q, 'c D q, 's D q. There are 7 variables and hence with 4 fundamental dimensions, there will be 00037 0003 40005 D 3 dimensionless groups. Taking d, , Cp and 'c as the recurring set, then: d
'c Cp
0006 L, L 3 0006 M/L , M 0006 q, q 0006 L2 /T2 q Cp
D d, D L3 D d3 D 'c D d2 /T2 'c and T2 D d2 /Cp 'c or: T D d/Cp0.5 'c0.5
Thus: dimensionless group 1: t/T D tCp0.5 'c0.5 /d dimensionless group 2: kTq/ML D k0003d/Cp0.5 'c0.5 0005'c /0003 d3 0005d D k'c0.5 /Cp0.5 d3 dimensionless group 3: 's /q D 's /'c
PROBLEM 1.19 Upon what variables would the rate of filtration of a suspension of fine solid particles be expected to depend? Consider the flow through unit area of filter medium and express the variables in the form of dimensionless groups. It is found that the filtration rate is doubled if the pressure difference is doubled. What would be the effect of raising the temperature of filtration from 293 to 313 K?
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The viscosity of the liquid is given by: D 0 00031 0003 0.0150003T 0003 27300050005 where is the viscosity at a temperature T K and 0 is the viscosity at 273 K.
Solution The volume flow of filtrate per unit area, u m3 /m2 s, will depend on the fluid density, ; fluid viscosity, ; particle size, d; pressure difference across the bed, P, and the voidage of the cake, e or: v D f0003 , , d, P, e0005. The dimensions of each of these variables are u D L/T, D M/L3 , D M/LT, d D L, P D M/LT2 and e D dimensionless. There are 6 variables and 3 fundamental dimensions and hence 00036 0003 30005 D 3 dimensionless groups. Taking, d, and as the recurring set, then: d 0006 L, LDd
0006 M/L3 , M D L3 D d3 0006 M/LT, T D M/L D d3 /0003d000b0005 D d2 / Thus: dimensionless group 1: uT/L D u d2 /0003000bd0005 D du / dimensionless group 2: PLT2 /M D Pd0003 d2 /000b00052 / d3 D P d2 /000b2 and the function is:
P d2 /000b2 D f0003du /000b0005
This may be written as:
P d2 /000b2 D K0003du /000b0005n
Since the filtration rate is doubled when the pressure difference is doubled, then: u / P and n D 1, P d2 /000b2 D Kdu / and:
u D 00031/K0005Pd/000b, or u / 1/000b
0003000b293 /000b313 0005 D 0 00031 0003 0.0150003293 0003 27300050005/000b0 00031 0003 0.0150003313 0003 27300050005 D 00030.7/0.40005 D 1.75 ∴
0003v313 /v293 0005 D 0003000b293 /000b313 0005 D 1.75
and the filtration rate will increase by 75%.
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SECTION 2
Flow of Fluids — Energy and Momentum Relationships PROBLEM 2.1 Calculate the ideal available energy produced by the discharge to atmosphere through a nozzle of air stored in a cylinder of capacity 0.1 m3 at a pressure of 5 MN/m2 . The initial temperature of the air is 290 K and the ratio of the specific heats is 1.4.
Solution From equation 2.1: dU D υq 0002 υW. For an adiabatic process: υq D 0 and dU D 0002υW, and for an isentropic process: dU D Cv dT D 0002υW from equation 2.25. As 0007 D Cp /Cv and Cp D Cv C R (from equation 2.27), Cv D R/ 0007 0002 1 ∴
W D 0002Cv T D 0002RT/ 0007 0002 1 D RT1 0002 RT2 / 0007 0002 1000b
and: 0007
RT1 D P1 v1 and RT2 D P2 v2 and hence: W D P1 v1 0002 P2 v2 / 0007 0002 1 0007
P1 v1 D P2 v2 and substituting for v2 gives: W D [ P1 v1 / 0007 0002 1000b] 1 0002 P2 /P1 000700021000b/0007 U D 0002W D [ P1 v1 / 0007 0002 1000b][ P2 /P1 000700021000b/0007 0002 1]
and: In this problem:
P1 D 5 MN/m2 , P2 D 0.1013 MN/m2 , T1 D 290 K, and 0007 D 1.4. The specific volume, v1 D 22.4/29 290/273 0.1013/5 D 0.0166 m3 /kg. ∴
0002W D [ 5 ð 106 ð 0.0166000b/0.4][ 0.1013/5000b0.4/1.4 0002 1] D 00020.139 ð 106 J/kg
Mass of gas D 0.1/0.0166 D 6.02 kg ∴
U D 0002 0.139 ð 106 ð 6.20 D 00020.84 ð 106 J or 0002 840 kJ
PROBLEM 2.2 Obtain expressions for the variation of: (a) internal energy with change of volume, (b) internal energy with change of pressure, and (c) enthalpy with change of pressure, all at constant temperature, for a gas whose equation of state is given by van der Waals’ law. 16
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FLOW OF FLUIDS — ENERGY AND MOMENTUM RELATIONSHIPS
17
Solution See Volume 1, Example 2.2.
PROBLEM 2.3 Calculate the energy stored in 1000 cm3 of gas at 80 MN/m2 at 290 K using STP as the datum.
Solution The key to this solution lies in the fact that the operation involved is an irreversible expansion. Taking Cv as constant between T1 and T2 , U D 0002W D nCv T2 0002 T1 where n is the kmol of gas and T2 and T1 are the final and initial temperatures, then for a constant pressure process, the work done, assuming the ideal gas laws apply, is given by: W D P2 V2 0002 V1 D P2 [ nRT2 /P2 0002 nRT1 /P1 ] 0001 0002 RT2 RT1 Equating these expressions for W gives: 0002Cv T2 0002 T1 D P2 0002 P2 P1 In this example: P1 D 80000 kN/m2 , P2 D 101.3 kN/m2 , V1 D 1 ð 1000023 m3 , R D 8.314 kJ/kmol K, and T1 D 290 K Hence: 0002Cv T2 0002 290 D 101.3R[ T2 /101.3 0002 290/80,000000b] By definition, 0007 D Cp /Cv and Cp D 0012v C R (from equation 2.27) or: Cv D R/ 0007 0002 1 Substituting:
T2 D 174.15 K.
PV D nRT and n D 80000 ð 1000023 / 8.314 ð 290 D 0.033 kmol ∴ U D 0002W D Cv n T2 0002 T1 D 1.5 ð 8.314 ð 0.033 174.15 0002 290 D 000247.7 kJ
PROBLEM 2.4 Compressed gas is distributed from a works in cylinders which are filled to a pressure P by connecting them to a large reservoir of gas which remains at a steady pressure P and temperature T. If the small cylinders are initially at a temperature T and pressure P0 , what is the final temperature of the gas in the cylinders if heat losses can be neglected and if the compression can be regarded as reversible? Assume that the ideal gas laws are applicable.
Solution From equation 2.1, dU D υq 0002 υW. For an adiabatic operation, q D 0 and υq D 0 and υW D Pdv or dU D 0002Pdv. The change in internal energy for any process involving an
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
ideal gas is given by equation 2.25: Cv dT D 0002Pdv D dU. P D RT/Mv and hence: dT/T D 0002R/MCv dv/v By definition: ∴
0007 D Cp /Cv and Cp D Cv C R/M (from equation 2.27) R/MCv D 0007 0002 1 and dT/T D 0002 0007 0002 1 dv/v000b
Integrating between conditions 1 and 2 gives: ln T2 /T1 D 0002 0007 0002 1 ln v2 /v1 or T2 /T1 D v2 /v1 000700021 P1 v1 /T1 D P2 v2 /T2 and hence v1 /v2 D P2 /P1 T1 /T2 and:
T2 /T1 D P2 /P1 000700021000b/0007
Using the symbols given, the final temperature, T2 D T P/P0 000700021000b/0007
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SECTION 3
Flow in Pipes and Channels PROBLEM 3.1 Calculate the hydraulic mean diameter of the annular space between a 40 mm and a 50 mm tube.
Solution The hydraulic mean diameter, dm , is defined as four times the cross-sectional area divided by the wetted perimeter. Equation 3.69 gives the value dm for an annulus of outer radius r and inner radius ri as: dm D 400050006r 2 0002 ri2 0007/200050006r C ri 0007 D 20006r 0002 ri 0007 D 0006d 0002 di 0007 If r D 25 mm and ri D 20 mm, then: dm D 2000625 0002 200007 D 10 mm
PROBLEM 3.2 3
0.015 m /s of acetic acid is pumped through a 75 mm diameter horizontal pipe 70 m long. What is the pressure drop in the pipe? Viscosity of acid D 2.5 mNs/m2 , density of acid D 1060 kg/m3 , and roughness of pipe surface D 6 ð 1000025 m.
Solution Cross-sectional area of pipe D 00060005/4000700060.07500072 D 0.0044 m2 . Velocity of acid in the pipe, u D 00060.015/0.00440007 D 3.4 m/s. Reynolds number D ud/ D 00061060 ð 3.4 ð 0.070007/00062.5 ð 1000023 0007 D 1.08 ð 105 Pipe roughness e D 6 ð 1000025 m and e/d D 00066 ð 1000025 0007/0.075 D 0.0008 The pressure drop is calculated from equation 3.18 as: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 From Fig. 3.7, when Re D 1.08 ð 105 and e/d D 0.0008, R/000bu2 D 0.0025. Substituting: 0002Pf D 00064 ð 0.00250007000670/0.075000700061060 ð 3.42 0007 D 114,367 N/m2 or: 114.4 kN/m2 19
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.3 A cylindrical tank, 5 m in diameter, discharges through a mild steel pipe 90 m long and 230 mm diameter connected to the base of the tank. Find the time taken for the water level in the tank to drop from 3 m to 1 m above the bottom. The viscosity of water is 1 mNs/m2 .
Solution If at any time the depth of water in the tank is h and levels 1 and 2 are the liquid levels in the tank and the pipe outlet respectively, then the energy balance equation states that: 0006u2 /20007 C gz C v0006P2 0002 P1 0007 C F D 0 In this example, P1 D P2 D atmospheric pressure and v0006P2 0002 P1 0007 D 0. Also u1 /u2 D 00060.23/500072 D 0.0021 so that u1 may be neglected. The energy balance equation then becomes: u2 /2 0002 hg C 40006R/000bu2 00070006l/d0007u2 D 0 The last term is obtained from equation 3.19 and z D 0002h. Substituting the known data: u2 /2 0002 9.81h C 40006R/000bu2 0007000690/0.230007u2 D 0 or: from which:
u2 0002 19.62h C 31300006R/000bu2 0007u2 D 0 p 0001 u D 4.43 h/ [1 C 31300006R/000bu2 0007]
In falling from a height h to h 0002 dh, the quantity of water discharged D 00060005/4000752 00060002dh0007 D 19.63dh m3 . p 0001 Volumetric flow rate D 00060005/4000700060.2300072 u D 0.0415u D 0.184 h/ [1 C 31300006R/000bu2 0007], and the time taken for the level to fall from h to h 0002 dh is: 0002 0002 000219.63 dh p [1 C 31300006R/000bu2 0007] D 0002106.7h00020.5 [1 C 31300006R/000bu2 0007] dh 0.184 h ∴ the time taken for the level to fall from 3 m to 1 m is: 0003 1 0002 t D 0002106.7 [1 C 31300006R/000bu2 0007] h00020.5 dh 3 2
R/000bu depends upon the Reynolds number which will fall as the level in the tank falls and upon the roughness of the pipe e which is not specified in this example. The pressure drop along the pipe D h000bg D 4Rl/d N/m2 and R D h000bgd/4l. From equation 3.23: 0006R/000bu2 0007 Re2 D Rd2 / 2 D h000b2 gd3 /4l 2 D 0006h ð 10002 ð 9.81 ð 0.233 0007/00064 ð 90 ð 1000026 0007 D 3.315 ð 108 h Thus as h varies from 3 m to 1 m, 0006R/000bu2 00070006Re00072 varies from (9.95 ð 108 ) to (3.315 ð 108 .)
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FLOW IN PIPES AND CHANNELS
21
If R/000bu2 is taken as 0.002, Re will vary from (7.05 ð 105 ) to (4.07 ð 105 ). From Fig. 3.7 this corresponds to a range of e/d of between 0.004 and 0.005 or a roughness of between 0.92 and 1.15 mm, which is too high for a commercial pipe. If e is taken as 0.05 mm, e/d D 0.0002, and, for Reynolds numbers near 106 , R/000bu2 D 0.00175. Substituting R/000bu2 D 0.00175 and integrating gives a time of 398 s for the level to fall from 3 m to 1 m. If R/000bu2 D 0.00175, Re varies from (7.5 ð 105 ) to (4.35 ð 105 ), and from Fig. 3.7, e/d D 0.00015, which is near enough to the assumed value. Thus the time for the level to fall is approximately 400 s.
PROBLEM 3.4 Two storage tanks A and B containing a petroleum product discharge through pipes each 0.3 m in diameter and 1.5 km long to a junction at D. From D the product is carried by a 0.5 m diameter pipe to a third storage tank C, 0.8 km away. The surface of the liquid in A is initially 10 m above that in C and the liquid level in B is 7 m higher than that in A. Calculate the initial rate of discharge of the liquid if the pipes are of mild steel. The density of the petroleum product is 870 kg/m3 and the viscosity is 0.7 mNs/m2 .
Solution See Volume 1, Example 3.4
PROBLEM 3.5 Find the drop in pressure due to friction in a glazed porcelain pipe 300 m long and 150 mm diameter when water is flowing at the rate of 0.05 m3 /s.
Solution For a glazed porcelain pipe, e D 0.0015 mm, e/d D 00060.0015/1500007 D 0.00001. Cross-sectional area of pipe D 00060005/4000700060.1500072 D 0.0176 m2 . Velocity of water in pipe, u D 00060.05/0.01760007 D 2.83 m/s. Reynolds number D ud/ D 00061000 ð 2.83 ð 0.150007/1000023 D 4.25 ð 105 From Fig. 3.7, R/000bu2 D 0.0017. The pressure drop is given by equation 3.18: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 or:
4 ð 0.00170006300/0.15000700061000 ð 2.832 0007 D 108,900 N/m2 or 1 MN/m2
PROBLEM 3.6 Two tanks, the bottoms of which are at the same level, are connected with one another by a horizontal pipe 75 mm diameter and 300 m long. The pipe is bell-mouthed at each
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
end so that losses on entry and exit are negligible. One tank is 7 m diameter and contains water to a depth of 7 m. The other tank is 5 m diameter and contains water to a depth of 3 m. If the tanks are connected to each other by means of the pipe, how long will it take before the water level in the larger tank has fallen to 6 m? Assume the pipe to be of aged mild steel.
Solution The system is shown in Fig. 3a. If at any time t the depth of water in the larger tank is h and the depth in the smaller tank is H, a relationship between h and H may be found. Area of larger tank D 00060005/4000772 D 38.48 m2 , area of smaller tank D 00060005/4000752 D 19.63 m2 .
dh 7m h
H 75 mm bore
7m
300 m
x
5m
5m
Figure 3a.
When the level in the large tank falls to h, the volume discharged D 00067 0002 h0007 ð 38.48 m3 . The level in the small tank will rise by a height x, given by: x D 38.4800067 0002 h0007/19.63 D 000613.72 0002 1.95h0007 H D 0006x C 30007 D 000616.72 0002 1.95h0007 The energy balance equation is: u2 /2 C gz C v0006P1 0002 P2 0007 D F u2 /2 may be neglected, and P1 D P2 D atmospheric pressure, so that: gz D F D gz C 40006R/000bu2 00070006l/d0007u2 , z D 0006h 0002 H0007 D 00062.95h 0002 16.720007 or: and:
00062.95h 0002 16.720007g D 40006R/000bu2 00070006l/d0007u2 0001 u D [00062.95h 0002 16.720007g/40006R/000bu2 00070006l/d0007]
As the level falls from h to h 0002 dh in time dt, the volume discharged D 38.480006dh0007 m3 . Hence:
time, dt D
00060005/4000700060.07500072 0001
or:
dt D
0001
000238.48 dh [00062.95h 0002 16.720007g/40006R/000bu2 00070006l/d0007]
00022780 dh [40006R/000bu2 00070006l/d0007] p 00062.95h 0002 16.720007
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FLOW IN PIPES AND CHANNELS
If R/000bu2 is taken as 0.002, then: 0003 0003 dt D 000215,740
6
p 7
dh 00062.95h 0002 16.720007
23
and t D 10590 s
Average volumetric flowrate D 38.4800067 0002 60007/10590 D 0.00364 m3 /s Cross-sectional area of pipe D 0.00442 m2 . Average velocity in the pipe D 00060.00364/0.004420007 D 0.82 m/s. Reynolds number D 00061000 ð 0.82 ð 0.750007/1000023 D 6.2 ð 104 . From Fig. 3.7, if e D 0.05 mm, e/d D 0.00067 and R/000bu2 D 0.0025, which is near enough to the assumed value of 0.002 for a first estimate. Thus the time for the level to fall is approximately 10590 s (2.94 h).
PROBLEM 3.7 Two immiscible fluids A and B, of viscosities A and B , flow under streamline conditions between two horizontal parallel planes of width b, situated a distance 2a apart (where a is much less than b), as two distinct parallel layers one above the other, each of depth a. Show that the volumetric rate of flow of A is: 0004 00050004 0005 0002Pa3 b 7 A C B 12 A l A C B where, 0002P is the pressure drop over a length l in the direction of flow. l Fluid B
RA = Shear stress at centre-plane on A CL RB = Shear stress at centre-plane on B
s Fluid A
Figure .
Solution Considering a force balance on the fluid lying within a distance s from the centre plane, then: 0004 0005 dus For A: 0002P0006sb0007 D bl A C RA l ds A where RA is the shear stress at the centre plane, RA 0002P sds 0002 ds or: 0002d s D A l A Integrating:
00060002us 0007A D
RA 0002P s2 0002 s C k1 A l 2 A
RB 0002P s2 0002 s C k2 A l 2 B where RB is the shear stress at the centre plane on B. Similarly for B:
00060002us 0007B D
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Noting that:
RA D 0002RB
At s D a:
0006us 0007A D 0 0006us 0007B D 0
On the centre plane: At s D 0 0006us 0007A D 0006us 0007B AD0002
RA 00060002P0007 a2 C a A l 2 A
BD0002
RB 00060002P0007 a2 C a B l 2 B
Thus:
usA D
0002P 2 RA 0006a 0002 s0007 fa 0002 s2 g 0002 2 A l A
and:
usB D
0002P 2 RA fa 0002 s2 g C 0006a 0002 s0007 0006since RA D 0002RB 0007 2 B l B
Centre line velocity, and: Equating: ∴
0006uA 0007CL D
RA a 0002Pa2 C 2 A l A
RA a 0002Pa2 C 2 B l B 0007 0006 0007 2 0006 1 1 0002Pa 1 1 0002 D aRA C 2l A B A B 0006 0007 0002Pa B 0002 A RA D 2l B C A 0006uB 0007CL D
Substituting: B 0002 A 0002P 2 0002Pa 0006a 0002 s2 0007 0002 0006a 0002 s0007 2 A l 2 A l B C A 0006 0007 0002P A 0002 B 2 2 D 0006a 0002 s 0007 C a0006a 0002 s0007 2 A l A C B
0006us 0007A D
Total flowrate of A D QA is given by: 00060003 a 0004 0005 0007 0003 a A 0002 B 0002Pb 2 2 2 QA D bus ds D 0006a 0002 s 0007 C 0006a 0002 as0007 ds 2 A l A C B 0 0 0004 0005 0004 0005 a A 0002 B 0002Pb s3 s2 D a2 s 0002 C a2 s 0002 a 2 A l 3 A C B 2 0 3 3 A 0002 B a 0002Pb 2a C D 2 A l 3 A C B 2 A 0002 B 0002Pba3 2 C D 2 A l 3 20006 A C B 0007
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25
0002Pba3 4 A C 4 B C 3 A 0002 3 B 2 A l 60006 A C B 0007 3 0002Pba 7 A C B D 12 A l A C B
D
PROBLEM 3.8 Glycerol is pumped from storage tanks to rail cars through a single 50 mm diameter main 10 m long, which must be used for all grades of glycerol. After the line has been used for commercial material, how much pure glycerol must be pumped before the issuing liquid contains not more than 1% of the commercial material? The flow in the pipeline is streamline and the two grades of glycerol have identical densities and viscosities.
Solution Making a force balance over an element distance r from the axis of a pipe whose radius is a, then: 0002P0005r 2 D 0002 0006du/dr000720005rl where u is the velocity at distance r and l is the length of the pipe. Hence: and:
du D 000200060002P/2 l0007r dr u D 000200060002P/4 l0007r 2 C constant
When r D a, u D 0, the constant D 00060002P/4 l0007a2 and hence u D 00060002P/4 l00070006a2 0002 r 2 0007. At a distance r from the axis, the time taken for the fluid to flow through a length l is given by 4 l2 /0002P0006a2 0002 r 2 0007. The volumetric rate of flow from r D 0 to r D r is: 0003 r D 00060002P/4 l00070006a2 0002 r 2 000720005rdr 0
D 00060002P0005/8 l000700062a2 r 2 0002 r 4 0007 Volumetric flowrate over the whole pipe D 0002P0005a4 /8 l and the mean velocity D 0002Pa2 /8 l. The required condition at the pipe outlet is: 00060002P0005/8 l000700062a2 r 2 0002 r 4 0007 D 0.99 0002P0005a4 /8 l from which r D 0.95a. The time for fluid at this radius to flow through length l D 4 l2 /0002Pa2 00061 0002 0.952 0007 D 000641 l2 0007/00060002Pa2 ) Hence, the volume to be pumped D 000641 l2 /00060002P0007a2 00070006000500060002P0007a4 /8 l0007 D 410005a2 /8 D 41000500060.2500072 /8 D 0.10 m3
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.9 A viscous fluid flows through a pipe with slightly porous walls so that there is a leakage of kP, where P is the local pressure measured above the discharge pressure and k is a constant. After a length l, the liquid is discharged into a tank. If the internal diameter of the pipe is d and the volumetric rate of flow at the inlet is Qo , show that the pressure drop in the pipe is given by 0002P D 0006Qo /0005kd0007a tanh al, a D 0006128k /d3 00070.5
where:
Assume a fully developed flow with 0006R/000bu2 0007 D 8 Re00021 .
Solution Across a small element of the pipe, the change in liquid flow is: 0002dQ D kP0005ddl and the change in velocity is: du D 00024kPdl/d R/000bu2 D 8 /ud000b
Also:
and: R D 8 u/d
0006i0007
Making a force balance over the element: 0002dP00060005/40007d2 D R0005ddl D 8 u0005dl 0002dP D 32 u dl/d2
and:
0006ii0007
From equations (i) and (ii): 0002dP/du D 00028 u/kPd and:
PdP D 8 u du/kd
Over the whole pipe: 0006Po2 /2 0002 P2 /20007 D 00068 /kd0007[0006uo2 /20007 0002 0006u2 /20007] u2 D uo2 C 0006P2 0002 Po2 00070006kd/8 0007
and:
Assuming zero outlet pressure as a datum, then substituting for u in equation (ii):
0002 0003 0 0002dp [uo2 C 0006P2 0002 Po2 00070006kd/8 0007] D 32 l/d2 Po
Thus:
0001
00068 /kd0007 sinh
00021
0002
Po
00068 uo2 /kd0007
0002
Po2
D 32 l/d2
0002
and:
Po
0002
[8 uo2 /kd0007 0002 Po2 ] D sinh
0006128 k/d3 0007l
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Writing
0001
27
0006128 k/d3 0007 D a:
00068 uo2 /kd0007 D [a2 0006d3 /128 k0007][8 Qo2 ð 160007/0006kd00052 d4 0007] D Qo2 /0006k 2 d2 00052 a2 0007 and:
Po2 D [0006Qo2 /0006k 2 d2 00052 a2 0007 sinh2 al]/00061 C sinh2 al0007 0002P D Po D 0006Qo /kd00050007a tanh al
or:
PROBLEM 3.10 A petroleum product of viscosity 0.5 m Ns/m2 and density 700 kg/m3 is pumped through a pipe of 0.15 m diameter to storage tanks situated 100 m away. The pressure drop along the pipe is 70 kN/m2 . The pipeline has to be repaired and it is necessary to pump the liquid by an alternative route consisting of 70 m of 200 mm pipe followed by 50 m of 100 mm pipe. If the existing pump is capable of developing a pressure of 300 kN/m2 , will it be suitable for use during the period required for the repairs? Take the roughness of the pipe surface as 0.05 mm.
Solution This problem may be solved by using equation 3.23 and Fig. 3.8 to find the volumetric flowrate and then calculating the pressure drop through the alternative pipe system. From equation 3.23: 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 000670,000 ð 0.153 ð 7000007/00064 ð 100 ð 0.52 ð 1000026 0007 D 1.65 ð 109 From Fig. 3.8, Re D 8.8 ð 105 D 0006700 ð 10.15u0007/00060.5 ð 1000023 0007 and the velocity u D 4.19 m/s. Cross-sectional area D 00060005/400070.152 D 0.0177 m2 . Volumetric flowrate D 00064.19 ð 0.01770007 D 0.074 m3 /s. The velocity in the 0.2 m diameter pipe D 0.074/00060005/400070.22 D 2.36 m/s. The velocity in the 0.1 m diameter pipe D 9.44 m/s. Reynolds number in the 0.2 m pipe D 0006700 ð 2.36 ð 0.2/0.5 ð 1000023 0007 D 6.6 ð 105 . Reynolds number in the 0.1 m pipe D 0006700 ð 9.44 ð 0.1/0.5 ð 1000023 0007 D 1.32 ð 106 . The values of e/d for the 0.2 m and the 0.1 m pipes are 0.00025 and 0.0005 respectively. From Fig. 3.7, R/000bu2 D 0.0018 and 0.002 respectively, and from equation 3.18: 0002Pf D [4 ð 0.0018000670/0.200070006700 ð 2.362 0007] C [4 ð 0.002000650/0.100070006700 ð 9.442 0007] D 255,600 N/m2 D 255.6 kN/m2 In addition, there will be a small pressure drop at the junction of the two pipes although this has been neglected in this solution. Thus the existing pump is satisfactory for this duty.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 3.11 Explain the phenomenon of hydraulic jump which occurs during the flow of a liquid in an open channel. A liquid discharges from a tank into an open channel under a gate so that the liquid is initially travelling at a velocity of 1.5 m/s and a depth of 75 mm. Calculate, from first principles, the corresponding velocity and depth after the jump.
Solution See Volume 1, Example 3.9.
PROBLEM 3.12 What is a non-Newtonian fluid? Describe the principal types of behaviour exhibited by these fluids. The viscosity of a non-Newtonian fluid changes with the rate of shear according to the approximate relationship: 0004 0005 dux 00020.5 a D k 0002 dr where a is the viscosity, and du/dr is the velocity gradient normal to the direction of motion. Show that the volumetric rate of streamline flow through a horizontal tube of radius a is: 0004 0005 0005 5 0002P 2 a 5 2kl where 0002P is the pressure drop over a length l of the tube.
Solution For a power-law fluid, the apparent viscosity is given by equation 3.123. Noting that the velocity gradient dux /dr is negative then: 0005 0004 0004 0005 dux n00021 dux n00021 D k 0002 a D k dy dr Thus, in this problem: n 0002 1 D 00020.5 and: n 0002 1 D 00020.5, giving n D 0.5 The fluid is therefore shear-thinning. Equation 3.136 gives the mean velocity in a pipe: 0004
uD
0002P 4kl
00051 0004 n
0005 nC1 n d n 6n C 2
(equation 3.136)
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29
Putting n D 0.5, the volumetric flowrate Q is given by: 0004 0005 0004 0005 0002P 2 1 0005 2 QD d d3 4kl 10 4 Putting d D 2a, then: QD
0005
5
0004
a
5
0002P 2kl
00052
PROBLEM 3.13 Calculate the pressure drop when 3 kg/s of sulphuric acid flows through 60 m of 25 mm pipe ( D 1840 kg/m3 , D 0.025 N s/m2 ).
Solution Reynolds number D ud/ D 4G/0005 d D 4.30/00060005 ð 0.025 ð 0.0250007 D 6110 If e is taken as 0.05 mm, then: e/d D 00060.05/250007 D 0.002. From Fig. 3.7, R/000bu2 D 0.0046. Acid velocity in pipe D 3.0/[1840 ð 00060005/4000700060.02500072 ] D 3.32 m/s. From equation 3.18, the pressure drop due to friction is given by: 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 4 ð 0.0046000660/0.02500071840 ð 3.322 D 895,620 N/m2 or 900 kN/m2
PROBLEM 3.14 The relation between cost per unit length C of a pipeline installation and its diameter d is given by: C D a C bd where a and b are independent of pipe size. Annual charges are a fraction ˇ of the capital cost. Obtain an expression for the optimum pipe diameter on a minimum cost basis for a fluid of density and viscosity flowing at a mass rate of G. Assume that the fluid is in turbulent flow and that the Blasius equation is applicable, that is the friction factor is proportional to the Reynolds number to the power of minus one quarter. Indicate clearly how the optimum diameter depends on flowrate and fluid properties.
Solution The total annual cost of a pipeline consists of a capital charge plus the running costs. The chief element of the running cost is the power required to overcome the head loss which is given by: hf D 80006R/000bu2 00070006l/d00070006u2 /2g0007 (equation 3.20)
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
If R/000bu2 D 0.04/ Re0.25 , the head loss per unit length l is: hf /l D 800060.04/ Re0.25 00070006l/d00070006u2 /2g0007 D 0.0160006u2 /d00070006 /000bud00070.25 D 0.016u1.75 0.25 /0006000b0.25 d1.25 0007 The velocity u D G/000bA D G/000b00060005/40007d2 D 1.27G/000bd2 ∴
hf /l D 0.01600061.27G/000bd2 00071.75 0.25 /0006000b0.25 d1.25 0007 D 0.024G1.75 0.25 /0006000b2 d4.75 0007
The power required for pumping if the pump efficiency is * is: P D Gg00060.024G1.75 0.25 /000b2 d4.75 0007/* If * D 0.5, P D 0.47G2.75 0.25 /0006000b2 d4.75 0007 W If c D power cost/W, the cost of pumping is given by: 0.47cG2.75 0.25 /000b2 d4.75 The total annual cost is then D 0006ˇa C ˇbd0007 C 0006,G2.75 0.25 0.25 /000b2 d4.75 0007 where , D 0.47c Differentiating the total cost with respect to the diameter gives: dC/dd D ˇb 0002 4.75,G2.75 0.25 /000b2 d5.75 For minimum cost, dC/dd D 0, d5.75 D 4.75,G2.75 0.25 /000b2 ˇb and d D KG0.48 0.43 /000b0.35 where:
K D 00064.75,ˇb00070.174
PROBLEM 3.15 A heat exchanger is to consist of a number of tubes each 25 mm diameter and 5 m long arranged in parallel. The exchanger is to be used as a cooler with a rating of 4 MW and the temperature rise in the water feed to the tubes is to be 20 deg K. If the pressure drop over the tubes is not to exceed 2 kN/m2 , calculate the minimum number of tubes that are required. Assume that the tube walls are smooth and that entrance and exit effects can be neglected. Viscosity of water D 1 mNs/m2 .
Solution Heat load D 0006mass flow ð specific heat ð temperature rise0007, or: 4000 D 0006m ð 4.18 ð 200007 and:
m D 47.8 kg/s
Pressure drop D 2 kN/m2 D 2000/00061000 ð 9.810007 D 0.204 m of water. From equation 3.23, 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 00062000 ð 0.253 ð 10000007/00064 ð 5 ð 1000026 0007 D 1.56 ð 106 If the tubes are smooth, then from Fig. 3.8: Re D 2.1 ð 104 . ∴ water velocity D 00062.1 ð 104 ð 1000023 0007/00061000 ð 0.0250007 D 0.84 m/s.
Cross-sectional area of each tube D 00060005/400070.252 D 0.00049 m2 .
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31
Mass flow rate per tube D 00060.84 ð 0.000490007 D 0.000412 m3 /s D 0.412 kg/s Hence the number of tubes required D 000647.8/0.4120007 D 116 tubes
PROBLEM 3.16 Sulphuric acid is pumped at 3 kg/s through a 60 m length of smooth 25 mm pipe. Calculate the drop in pressure. If the pressure drop falls by one half, what will be the new flowrate? Density of acid D 1840 kg/m3 . Viscosity of acid D 25 mN s/m2 .
Solution Cross-sectional area of pipe D 00060005/400070.0252 D 0.00049 m2 . Volumetric flowrate of acid D 00063.0/18400007 D 0.00163 m3 /s. Velocity of acid in the pipe D 00060.00163/0.000490007 D 3.32 m/s. Reynolds number, ud/ D 00061840 ð 3.32 ð 0.025/25 ð 1000023 0007 D 6120 From Fig. 3.7 for a smooth pipe and Re D 6120, R/000bu2 D 0.0043. The pressure drop is calculated from equation 3.18: 0002Pf D 40006R/000bu2 00070006l/d00070006000bu2 0007 D 4 ð 0.0043000660/0.025000700061840 ð 3.322 0007 D 837,200 N/m2 or 840 kN/m2 If the pressure drop falls to 418,600 N/m2 , equation 3.23 and Fig. 3.8 may be used to calculate the new flow. From equation 3.23 : 0006R/000bu2 0007 Re2 D 0002Pf d3 /4l 2 D 0006418,600 ð 0.0253 ð 18400007/00064 ð 60 ð 252 ð 1000026 0007 D 8.02 ð 104 From Fig. 3.8: Re D 3800 and the new velocity is: u0 D 00063800 ð 25 ð 1000023 0007/00061840 ð 0.0250007 D 2.06 m/s and the mass flowrate D 00062.06 ð 0.00049 ð 18400007 D 1.86 kg/s
PROBLEM 3.17 A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is flat? The shear stress acting within the fluid, Ry , varies with velocity gradient dux /dy according to the relation: Ry 0002 Rc D 0002k0006dux /dy0007 where Rc and k are constants for the material.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution The shearing characteristics of non-Newtonian fluids are shown in Fig. 3.24 of Volume 1. This type of fluid remains rigid when the shear stress is less than the yield stress RY and flows like a Newtonian fluid when the shear stress exceeds RY . Examples of Bingham plastics are many fine suspensions and pastes including sewage sludge and toothpaste. The velocity profile in laminar flow is shown in Fig. 3c. Pipe wall Rw Ry
r
Rc
y
ro
R=o
Plug flow region
Velocity distribution
Figure 3c.
A force balance over the pipe assuming no slip at the walls gives: 0002P0005r 2 D Rw 20005rL, and 0006i0007 0002P/L D 2Rw /r where Rw D shear stress at the wall. A force balance over the annular core where y > r0 gives: 0002P0005y 2 D 20005yLRy Hence:
Ry D yRw /r and y D rRy /Rw
(ii)
when:
Ry D RY and r0 D rRY /Rw
(iii)
∴ from equation (ii):
Integrating:
Ry 0002 RY D 0002k0006dux /dy0007 0004 0005 Ry 0002 RY 1 yRw dux D D 0002 RY 0002 dy k k r
(iv)
0002kux D 0006y 2 Rw /2r0007 0002 RY y C C
When y D r, ux D 0, C D 00060002rRw /20007 C RY r. 0004 0005 r y2 0002 ∴ kux D Rw 0002 RY 0006r 0002 y0007 2 2r Substituting for y from equation (iii) gives: 0004 0005 0004 0005 r RY r RY r 0002 0002 RY r 0002 r and ku0 D 0006Rw 0002 RY 00072 ku0 D Rw 2 Rw 2 Rw 2Rw
(v)
(vi)
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The total volumetric flowrate Q is obtained by integrating the equation for the velocity profile to give: 0003 r Qtotal D 0005y 2 00060002dux /dy0007 dy 0
From equation (iv):
Qtotal
1 D k
0003
0004
r
0005y 0
2
yRw 0002 RY r
0005
0005r 3 dy D k
0004
Rw RY 0002 4 3
0005
m3 /s
Over the central core, the volumetric flowrate Qcore is: Qcore D 0005r02 u0 D 00050006rRY /Rw 00072 u0 (from 0006300070007 From equation (vi): Qcore D 00050006rRY /Rw 00072 0006r/2kRw 00070006Rw 0002 RY 00072 D 00060005r 3 RY2 /2kRw3 00070006Rw 0002 RY 00072 If half the total flow is to be within the central core, then: Qcore D Qtotal /2
and:
00060005r 3 RY2 /2kRw3 00070006Rw 0002 RY 00072 D 00060005r 3 /2k00070006Rw /4 0002 RY /30007 0004 0005 Rw RY 2 2 3 0002 RY 0006Rw 0002 RY 0007 D Rw 4 3
PROBLEM 3.18 2
Oil of viscosity 10 mN s/m and density 950 kg/m3 is pumped 8 km from an oil refinery to a distribution depot through a 75 mm diameter pipeline and is then despatched to customers at a rate of 500 tonne/day. Allowance must be made for periods of maintenance which may interrupt the supply from the refinery for up to 72 hours. If the maximum permissible pressure drop over the pipeline is 3450 kN/m2 , what is the shortest time in which the storage tanks can be completely recharged after a 72 hour shutdown? The roughness of the pipe surface is 0.05 mm.
Solution From equation 3.23:
R 0002Pf d3 2 Re D u2 4l 2
0002Pf D 3450 kN/m2 D 3.45 ð 106 N/m2 , d D 0.075 m, D 950 kg/m3 , l D 8000 m and D 10 m Ns/m2 0.01 Ns/m2 . R ∴ Re2 D 00063.45 ð 106 ð 0.0753 ð 9500007/00064 ð 8000 ð 0.012 0007 D 4.32 ð 105 u2 e/d D 00060.05/750007 D 0.0007 From Fig. 3.8 with 0006R/000bu2 0007 Re2 D 00064.32 ð 105 0007, e/d D 0.0007. Re D 1.1 ð 104 D du/
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∴
u D Re /d D 00061.1 ð 104 ð 0.01/0.0750007 D 1.47 ð 103 kg/m2 s
∴ mass flowrate D 00061.47 ð 103 ð 1000023 ð 3600 ð 24 ð 00060005/400070.0752 0007 D 561 tonne/day
Depletion of storage in 72 h D 0006561 ð 72/240007 D 1683 tonne Maximum net gain in capacity in the system D 0006561 0002 5000007 D 61 tonne/day and the time to recharge the tanks D 00061683/610007 D 27.6 days
PROBLEM 3.19 Water is pumped at 1.4 m3 /s from a tank at a treatment plant to a tank at a local works through two parallel pipes, 0.3 m and 0.6 m diameter respectively. What is the velocity in each pipe and, if a single pipe is used, what diameter will be needed if this flow of water is to be transported, the pressure drop being the same? Assume turbulent flow with the friction factor inversely proportional to the one quarter power of the Reynolds number.
Solution The pressure drop through a pipe is given by equation 3.18: 0004 0005 R l 2 0002P D 4 u u2 d In this case, R/000bu2 D K Re00021/4 where K is a constant. 0004 0005 ud 00021/4 l 2 u Hence: 0002P D 4K d DK
u1.75 l000b0.75 D K0 u1.75 /d1.25 d1.25 0.25
For pipe 1 in which the velocity is u1 , 0002P D K0 u11.75 /0.31.25 and the diameter is 0.3 m. Similarly for pipe 2, 0002P D K0 u11.75 /0.61.25 Hence 0006u2 /u1 00071.75 D 00060.6/0.300071.25 D 2.38 and u2 /u1 D 1.64 The total volumetric flowrate D 1.4 m3 /s D 0005/40006d21 u1 C d22 u2 0007 and substituting for d1 and d2 and u2 D 1.64u, u1 D 2.62 m/s and u2 D 4.30 m/s If a single pipe of diameter d3 is used for the same duty at the same pressure drop and the velocity is u3 , then: 00060005/40007d23 u3 D 1.4 and d23 u3 D 1.78 and: and: Since u1 D 2.62 m/s, then:
000610007
0002P D K1 u31.75 /d31.25 0006u3 /u1 00071.75 D 0006d3 /0.300071.25 0.185u31.75 D 4.5d31.25
000620007
From equations (1) and (2), the required diameter, d3 D 0.63 m
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35
PROBLEM 3.20 Oil of viscosity 10 mNs/m2 and specific gravity 0.90, flows through 60 m of 100 mm diameter pipe and the pressure drop is 13.8 kN/m2 . What will be the pressure drop for a second oil of viscosity 30 mNs/m2 and specific gravity 0.95 flowing at the same rate through the pipe? Assume the pipe wall to be smooth.
Solution For the first oil, with a velocity in the pipe of u m/s then: Re D u ð 00060.90 ð 10000007 ð 0006100/10000007/000610 ð 1000023 0007 D 9000u R 0002Pd3 2 Re D u2 4l 2 D 000613.8 ð 10000007 ð 0.103 ð 900/00064 ð 60 ð 0.012 0007 D 5.2 ð 105 From Fig. 3.8, when 0006R/000bu2 0007 Re2 D 5.2 ð 105 for a smooth pipe, Re D 12000. Hence, velocity u D 000612,000/90000007 D 1.33 m/s. For the second oil, the same velocity is used although the density and viscosity are now 950 kg/m3 and 0.03 Ns/m2 . Hence:
Re D 00061.33 ð 0.10 ð 950/0.030007 D 4220
For a smooth pipe, Fig. 3.7 gives a friction factor, R/000bu2 D 0.0048 for this value of Re. From Equation 3.18: 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 4 ð 0.0048 ð 000660/0.100007 ð 950 ð 1.332 D 1.94 ð 104 N/m2 19.4 kN/m2
PROBLEM 3.21 Crude oil is pumped from a terminal to a refinery through a 0.3 m diameter pipeline. As a result of frictional heating, the temperature of the oil is 20 deg K higher at the refinery end than at the terminal end of the pipe and the viscosity has fallen to one half its original value. What is the ratio of the pressure gradient in the pipeline at the refinery end to that at the terminal end? Viscosity of oil at terminal D 90 mNs/m2 . Density of oil (approximately constant) D 960 kg/m3 . Flowrate of oil D 20,000 tonne/day. Outline a method for calculating the temperature of the oil as a function of distance from the inlet for a given value of the heat transfer coefficient between the pipeline and the surroundings.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution Oil flowrate D 000620,000 ð 10000007000624 ð 36000007 D 231.5 kg/s or 0006231/9600007 D 0.241 m3 /s Cross section area of pipe D 00060005/40007 ð 0.32 D 0.0707 m2 Oil velocity in pipe D 00060.241/0.07070007 D 3.40 m/s Reynolds number at terminal D 00063.40 ð 0.3 ð 960/0.090007 D 10,880 Reynolds number at the refinery is twice this value or 21,760. 0002P D 40006R/000bu2 00070006l/d00070006000bu2 0007
From equation 3.18:
(equation 3.18)
2
00020006P/l0007refinery 0006R/000bu 0007refinery D 00020006P/l0007terminal 0006R/000bu2 0007terminal
and: which, from Fig. 3.7:
D 00060.0030/0.003750007 D 0.80
In a length of pipe dl: 0002dP D 40006R/000bu2 00070006dl/d0007000bu2 N/m2 Energy dissipated D 0002dPQ D 00060005/40007d2 u40006R/000bu2 00070006dl/d00070006000bu2 0007 W where u is the velocity in the pipe. The heat loss to the surroundings at a distance l from the inlet is h0006T 0002 TS 00070005dl W where TS is the temperature of the surroundings and T is the temperature of the fluid. Heat gained by the fluid D 00060005/40007d2 u000bCp dT W where Cp (J/kg K) is the specific heat capacity of the fluid. Thus an energy balance over the length of pipe dl gives: 0006R/000bu2 0007d000bu3 dl D h0006T 0002 Ts 00070005d dl C 00060005/40007d2 u000bCp dT 0006R/000bu2 0007 varies with temperature as illustrated in the first part of this problem, and hence this equation may be written as: A dl D B dl C C dT CdT D dl or: A0002B (where A and B are both functions of temperature and C is a constant). Integrating between l1 and l2 , T1 and T2 gives: 0003 l2 0003 T2 CdT dl D A0002B l1 T1 If T1 , Ts , h and T are known (20 deg K in this problem), the integral may then be evaluated.
PROBLEM 3.22 Oil with a viscosity of 10 mNs/m2 and density 900 kg/m3 is flowing through a 500 mm diameter pipe 10 km long. The pressure difference between the two ends of the pipe is
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106 N/m2 . What will the pressure drop be at the same flowrate if it is necessary to replace the pipe by one only 300 mm in diameter? Assume the pipe surface to be smooth.
Solution D 0.01 Ns/m2 , D 900 kg/m3 , d D 0.50 m, l D 10000 m and 0002P D 1 ð 106 N/m2 . 0006R/000bu2 0007 Re2 D 00060002Pd3 0007/00064l 2 0007
(equation 3.23)
D 00061.106 ð 0.503 ð 900/00064 ð 10000 ð 0.012 00070007 D 2.81 ð 107 From Fig. 3.8, Re D ud000b/ D 1.2 ð 105 u D Re /000bd D 00061.2 ð 105 ð 0.01/900 ð 0.500007 D 2.67 m/s If the diameter of the new pipe is 300 mm, the velocity is then: D 2.67 ð 00060.5/0.300072 D 7.42 m/s Reynolds number D 00067.42 ð 0.30 ð 900/0.010007 D 2.0 ð 105 From Fig. 3.7, R/000bu2 D 0.0018 and from equation 3.18: 0002P D 00064 ð 0.0018 ð 000610000/0.30007 ð 900 ð 7.422 0007 D 1.19 ð 107 N/m2
PROBLEM 3.23 Oil of density 950 kg/m3 and viscosity 1000022 Ns/m2 is to be pumped 10 km through a pipeline and the pressure drop must not exceed 2 ð 105 N/m2 . What is the minimum diameter of pipe which will be suitable, if a flowrate of 50 tonne/h is to be maintained? Assume the pipe wall to be smooth. Use either a pipe friction chart or the Blasius equation 0006R/000bu2 D 0.0396 Re00021/4 0007.
Solution From equation 3.6 a force balance on the fluid in the pipe gives: R D 0002P0006d/4l0007 or:
D 2 ð 105 0006d/4 ð 104 0007 D 5d
Velocity in the pipe D G/000bA D 000650 ð 1000/36000007/0006950 ð 00060005/40007d2 0007 D 0.186/d2 Hence:
R/000bu2 D 5d/0006950 ð 00060.186/d2 00072 0007 D 15.21d5 Re D ud000b/ D 00060.186 ð d2 0007 ð d ð 950/00061 ð 1000022 0007 D 1.77 ð 103 /d
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The Blasius equation is: R/000bu2 D 0.0396 Re00020.25 and hence:
(equation 3.11)
15.21d5 D 0.03960006d/1.77 ð 103 00070.25
and:
d D 0.193 m
In order to use the friction chart, Fig. 3.7, it is necessary to assume a value of R/000bu2 , calculate d as above, check the resultant value of Re and calculate R/000bu2 and compare its value with the assumed value. If R/000bu2 is assumed to be D 0.0030, then 15.21d5 D 0.0030 and d D 0.182 m ∴
Re D 00061.77 ð 103 0007/0.182 D 9750
From Fig. 3.7, R/000bu2 D 0.0037 which does not agree with the original assumption. If R/000bu2 is taken as 0.0024, d is calculated D 0.175 m, Re is 1.0 ð 105 and R/000bu2 D 0.0022. This is near enough giving the minimum pipe diameter D 0.175 m.
PROBLEM 3.24 On the assumption that the velocity profile in a fluid in turbulent flow is given by the Prandtl one-seventh power law, calculate the radius at which the flow between it and the centre is equal to that between it and the wall, for a pipe 100 mm in diameter.
Solution See Volume 1, Example 3.5.
PROBLEM 3.25 A pipeline 0.5 m diameter and 1200 m long is used for transporting an oil of density 950 kg/m3 and of viscosity 0.01 Ns/m2 at 0.4 m3 /s. If the roughness of the pipe surface is 0.5 mm, what is the pressure drop? With the same pressure drop, what will be the flowrate of a second oil of density 980 kg/m3 and of viscosity 0.02 Ns/m2 ?
Solution D 0.01 Ns/m2 , d D 0.5 m and A D 00060005/400070.52 D 0.196 m2 , l D 1200 m, D 950 kg/m3 and: u D 00060.4/0.1960007 D 2.04 m/s. Reynolds number D ud/ D 0006950 ð 2.04 ð 0.50007/0.01 D 9.7 ð 104 e/d D 00060.5/5000007 D 0.001 and from Fig. 3.7, R/000bu2 D 0.0027
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From equation 3.8, 0002P D 40006R/000bu2 00070006l/d0007000bu2 D 00064 ð 0.0027 ð 1200 ð 950 ð 2.042 /0.50007 D 1.03 ð 105 N/m2 0006R/000bu2 0007 Re2 D 0002Pd3 /4l 2 5
(equation 3.23) 3
2
D 00061.03 ð 10 ð 0.5 ð 980/4 ð 1200 ð 0.02 0007 D 6.6 ð 106 From Fig. 3.8, Re D 4.2 ð 104 D 0006980u ð 0.5/0.020007 and:
u D 1.71 m/s
∴
volumetric flowrate D 00061.71 ð 0.1960007 D 0.34 m3 /s
PROBLEM 3.26 Water (density 1000 kg/m3 , viscosity 1 mNs/m2 ) is pumped through a 50 mm diameter pipeline at 4 kg/s and the pressure drop is 1 MN/m2 . What will be the pressure drop for a solution of glycerol in water (density 1050 kg/m3 , viscosity 10 mNs/m2 ) when pumped at the same rate? Assume the pipe to be smooth.
Solution Cross-sectional area of pipe D 000600060005/40007 ð 0.052 0007 D 0.00196 m2 Water velocity, u D 4/00061000 ð 0.001960007 D 2.04 m/s. Reynolds number, Re D 00062.04 ð 1000 ð 0.05/1 ð 1000023 0007 D 102,000 From Fig. 3.7, R/000bu2 D 0.0022 From equation 3.18, 0002P D 00064 ð 0.0022 ð 0006l/0.050007 ð 1000 ð 2.042 0007 D 732l For glycerol/water flowing at the same velocity: Re D 00062.4 ð 1050 ð 0.05/1 ð 1000022 0007 D 10,700 From Fig. 3.7, and: ∴
and
R/000bu2 D 0.0037 P D 00064 ð 0.0037 ð 0006l/0.050007 ð 1050 ð 2.042 0007 D 1293l.
0002Pglycerol / 0002 Pwater D 00061293l/732/l0007 D 1.77 0002Pglycerol D 00061.77 ð 1 ð 106 0007 D 1.77 ð 106 N/m2
PROBLEM 3.27 A liquid is pumped in streamline flow through a pipe of diameter d. At what distance from the centre of the pipe will the fluid be flowing at the average velocity?
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Solution A force balance on an element of fluid of radius r gives: du 0002P0005r 2 D 000220005rl dr Pr Pr 2 du D0002 or 0002 u D C constant dr 2 l 4 l 0004 0005 Pa2 When r D d/2 D a, u D 0 and the constant D 0002 0002 4 l 0002P 2 ∴ 0006a 0002 r 2 0007 uD 4 l
or:
0002
The flow, dQ, through an annulus, radius r and thickness dr is given by: dQ D 20005r dr ur 0002P 2 2 0006a 0002 r 0007 D 20005r dr 4 l 0004 0005 4 0002P Q D 0005a 8 l
and: The average velocity is:
uav D Q/0005a2 D
0002Pa2 8 l
The radius at which u D uav is: 0002Pa2 0002P 2 D 0006a 0002 r 2 0007 8 l 4 l from which: r 2 D a2 /2 D d2 /8 and r D 0.35d
PROBLEM 3.28 Cooling water supplied to a heat exchanger flows through 25 mm diameter tubes each 5 m long arranged in parallel. If the pressure drop over the heat exchanger is not to exceed 8000 N/m2 , how many tubes must be included for a total flowrate of water of 110 tonne/h? Density of water D 1000 kg/m3 . Viscosity of water D 1 mNs/m2 . Assume pipes to be smooth-walled. If ten per cent of the tubes became blocked, what would the new pressure drop be?
Solution R 0002Pd3 ð Re2 D 2 u 4l 2
(equation 3.23)
D 00068000 ð 00060.02500073 ð 10000007/00064 ð 5 ð 00061 ð 1000023 00072 0007 D 6.25 ð 106
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From Fig. 3.8: Re D ud/ D 5 ð 104 ∴ u D 00065 ð 104 ð 1 ð 1000023 0007/00061000 ð 0.0250007 D 2.0 m/s
Flowrate per tube D 00062.0 ð 00060005/40007 ð 0.0252 0007 D 0.982 ð 104 m3 /s Total flowrate D 0006110 ð 10000007/00061000 ð 36000007 D 0.3056 m3 /s ∴ Number of tubes required D 0.3056/00060.982 ð 1000024 0007 D 31.1 or 32 tubes
If 10 per cent of the tubes are blocked, velocity of fluid D 00062.0/0.90007 D 2.22 m/s Re D 00065 ð 104 0007/0.9 D 5.5 ð 104 and, from Fig. 3.7, R/000bu2 D 0.00245. From equation 3.18, pressure drop is: 0002P D 00064 ð 0.00245 ð 00065/0.0250007 ð 1000 ð 2.222 0007 D 9650 N/m2 , an increase of 20.6%
PROBLEM 3.29 The effective viscosity of a non-Newtonian fluid may be expressed by the relationship: 0004 0005 dux 00 a D k 0002 dr where k 00 is constant. Show that the volumetric flowrate of this fluid in a horizontal pipe of radius a under isothermal laminar flow conditions with a pressure drop 0002P/l per unit length is: 0004 0005 20005 7/2 0002P 1/2 QD a 7 2k 00 l
Solution In Section 3.4.1 of Volume 1 it is shown that for any fluid, the shear stress, Rr , at a distance r from the centre of the pipe may be found from a force balance for an element of fluid of length l across which the pressure drop is 0002P by: 0004 0005 r 0002P 0002P0005r 2 D 20005rl00060002Rr 0007 or 0002 Rr D (equation 3.7) 2 l The viscosity is related to the velocity of the fluid, ux , and the shear stress, Rr , by: (from equation 3.4)
Rr D a 00060002dux /dr0007 If, for the non-Newtonian fluid, a D k 00 00060002dux /dr0007 then:
Rr D k 00 00060002dux /dr000700060002dux /dr0007 D k 00 00060002dux /dr00072
Combining the two equations for Rr : k 00 00060002dux /dr00072 D
r 2
0004
0002P l
0005
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0005 0002P 1/2 1/2 r dr 2k 00 l 0004 0005 2 0002P 1/2 3/2 ∴ 0002ux D r C constant. 3 2k 00 l 000e 0004 000f 0005 2 0002P 1/2 3/2 a When r D a (at the wall), ux D 0 and the constant D 0002 3 2k 00 l 0004 0005 0002P 1/2 2 3/2 0006a 0002 r 3/2 0007r and: ux D 2k 00 l 3
or:
0002
dux D dr
0004
The volumetric flowrate Q is: 0004 0005 0003 Q 0003 a 0003 0002P 1/2 2 a 3/2 dQ D 20005rux dr D 20005 0006ra 0002 r 5/2 0007 dr 2k 00 l 3 0 0 0 0004 0005 0004 0005 0004 0005 0002P 1/2 2 3 7/12 20005 0002P 1/2 7/2 a D a D 20005 2k 00 l 3 14 7 2k 00 l
PROBLEM 3.30 Determine the yield stress of a Bingham fluid of density 2000 kg/m3 which will just flow out of an open-ended vertical tube of diameter 300 mm under the influence of its own weight.
Solution The shear stress at the pipe wall, R0 , in a pipe of diameter d, is found by a force balance as given Volume 1, Section 3.4.1: 00060002R0 00070005 dl D 00060002P000700060005/40007d2 or:
0002R0 D 00060002P00070006d/4l0007
(equation 3.6)
If the fluid just flows from the vertical tube, then: 0002P/l D g and:
0002R0 D g0006d/40007 D 00062000 ð 981 ð 0.30007/4 D 1472 N/m2
PROBLEM 3.31 A fluid of density 1200 kg/m3 flows down an inclined plane at 15° to the horizontal. If the viscous behaviour is described by the relationship: 0004 0005 dux n Ryx D 0002k dy
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where k D 4.0 Ns0.4 /m2 and n D 0.4, calculate the volumetric flowrate per unit width if the fluid film is 10 mm thick.
Solution Flow with a free surface is discussed in Section 3.6 and the particular case of laminar flow down an inclined surface in Section 3.6.1. For a flow of liquid of depth υ, width w and density down a surface inclined at an angle 4 to the horizontal, a force balance in the x direction (parallel to the surface) may be written. The weight of fluid flowing down the plane at a distance y from the free surface is balanced by the shear stress at the plane. For unit width and unit height: 0002Ryx D g sin 4y Ryx D 0002k0006dx /dy 0007n and k0006dux /dy0007n D g sin 4y Substituting k D 4.0 Ns0.4 /m2 , n D 0.4, D 1200 kg/m3 and 4 D 15° : 4.00006dux /dy00070.4 D 00061200 ð 9.81 ð sin 15° 0007y or: 0006dux /dy D 762y0007 and dux /dy D 1.60 ð 107 y 2.5 ux D 4.57 ð 106 y 3.5 C constant When the film thickness y D υ D 0.01 m, ux D 0. Hence 0 D 0.457 C c and c D 00020.457. ∴
ux D 4.57 ð 106 y 3.5 0002 0.457 The volumetric flowrate down the surface is then: 0003 Q 0003 w0003 0 dQ D ux dw dy 0
or, for unit width: Q/W D
00100
0
0.01 00064.57
0.01
ð 106 y 3.5 0002 0.4570007 dy D 0.00357 0006m3 /s0007/m
PROBLEM 3.32 A fluid with a finite yield stress is sheared between two concentric cylinders, 50 mm long. The inner cylinder is 30 mm diameter and the gap is 20 mm. The outer cylinder is held stationary while a torque is applied to the inner. The moment required just to produce motion is 0.01 Nm. Calculate the torque needed to ensure all the fluid is flowing under shear if the plastic viscosity is 0.1 Ns/m2 .
Solution Concentric-cylinder viscometers are in widespread use. Figure 3d represents a partial section through such an instrument in which liquid is contained and sheared between the stationary inner and rotating outer cylinders. Either may be driven, but the flow regime
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Pointer Retaining spring
ro ri
h Rotating outer cylinder
Stationary inner cylinder Figure 3d.
Partial section of concentric-cylinder viscometer
which is established with the outer rotating and the inner stationary is less disturbed by centrifugal forces. The couple transmitted through the fluid to the suspended stationary inner cylinder is resisted by a calibrated spring, the deflection of which allows calculation of the torque, T, and hence the inner wall shearing stress Ri is given by: T D 0002Ri ri 20005ri h This torque T originates from the outer cylinder which is driven at a uniform speed. On the inner surface of the outer cylinder the shear stress is Ro and:
∴
and:
T D 0002Ro ro 20005ro h 0002T Ro D 20005ro2 h Ri D
0002T 20005ri2 h
For any intermediate radius r, the local shear stress is: 0004 20005 0004 20005 ro r 0002T Rr D D Ro 2 D Ri i2 20005r 2 h r r In this example, ri D 0.015 m, r2 D 0.035 m, h D 0.05 m and T D 0.01 Nm which just produces motion at the surface of the inner cylinder. Using these equations: Ri D T/000620005ri2 h0007 D [0.01/000620005 ð 0.0152 ð 0.050007] D 141.5 N/m2
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As motion just initiates under the action of this torque, this shear stress must equal the yield stress and: RY D 141.5 N/m2 If all the fluid is to be in motion, the shear stress at the surface of the outer cylinder must be at least this value and the shear stress at the inner cylinder will be higher, and will be given by: Ri D Ro 0006ro /ri 00072 D [141.500060.035/0.01500072 ] D 770 N/m2 The required torque is then: T D Ri ð 20005ri2 h D 0006770 ð 20005 ð 0.0152 ð 0.050007 D 0.054 Nm
PROBLEM 3.33 Experiments with a capillary viscometer of length 100 mm and diameter 2 mm gave the following results: Applied pressure (N/m2 ) 1 ð 103 2 ð 103 5 ð 103 1 ð 104 2 ð 104 5 ð 104 1 ð 105
Volumetric flowrate (m3 /s) 1 ð 1000027 2.8 ð 1000027 1.1 ð 1000026 3 ð 1000026 9 ð 1000026 3.5 ð 1000025 1 ð 1000024
Suggest a suitable model to describe the fluid properties.
Solution Inspection of the data shows that the pressure difference increases less rapidly than the flowrate. Taking the first and the last entries in the table, it is seen that when the flowrate increases from 1 ð 1000027 to 1 ð 1000024 m3 /s, that is by a factor of 1000, the pressure difference increases from 1 ð 103 to 1 ð 105 N/m2 that is by a factor of only 100. In this way, the fluid appears to be shear-thinning and the simplest model, the power-law model, will be tried. From equation 3.136: Q D 00060005/40007d2 u D 00060002P/4kl00071/n [n/00066n C 20007]00060005/40007d00063nC10007/n Using the last set of data: 1.0 ð 1000024 D [00061 ð 105 0007/00064k ð 0.10007]1/n 00060005/800070006n/00063n C 10007000700062 ð 1000023 000700063nC10007/n or:
Q D K00060002P00071/n
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A plot of Q against 0002P on logarithmic axes, shown in Figure 3e, gives a slope, 00061/n0007 D 1.5 which is constant over the entire range of the experimental data. This confirms the validity of the power-law model and, for this system: n D 0.67 10−4
Slope, 1/n = 1.5
Q (m3/s)
10−5
10−6
10−7 103
104
105
106
−∆P (N/m ) 2
Figure 3e.
The value of the consistency coefficient k may be obtained by substituting n D 0.67 and the experimental data for any one set of data and, if desired, the constancy of this value may be confirmed by repeating this procedure for each set of the data. For the last set of data: Q D 00060005/40007d2 u D 00060002P/4kl00071/n [n/00066n C 20007]00060005/40007d00063nC10007/n 5
1.5
(from equation 3.136)
4.5
Thus: 1 ð 1000024 D [00061 ð 10 0007/00064 ð 0.1k0007] 00061/9000700060005/4000700060020007 and:
k D 0.183 Nsn m2
In S.I. units, the power-law equation is therefore: R D 0.1830006dux /dy00070.67 or:
7 D 0.183,P 0.67
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PROBLEM 3.34 Data obtained with a cone and plate viscometer, cone half-angle 89° cone radius 50 mm, were: cone speed (Hz) measured torque (Nm) 4.6 ð 1000021 7 ð 1000021 1.0 3.4 6.4 3.0 ð 10
0.1 0.5 1 5 10 50
Suggest a suitable model to describe the fluid properties.
ω r0 r θ
r0 r
dr
Figure 3f. Cone and plate viscometer
Solution A cone and plate viscometer, such as the Ferranti–Shirley or the Weissenberg instruments, shears a fluid sample in the small angle (usually 4° or less) between a flat surface and a rotating cone whose apex just touches the surface. Figure 3f illustrates one such arrangement. This geometry has the advantage that the shear rate is everywhere uniform and equal to ω/ sin 4, since the local cone velocity is ωr and the separation between the
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solid surfaces at that radius is r sin 4 where ω is the angular velocity of rotation. The shear stress Rr acting on a small element or area dr wide will produce a couple 000220005r dr Rr r about the axis of rotation. With a uniform velocity gradient at all points in contact with the cone surface, the surface stress R will also be uniform, so that the suffix can be omitted and the total couple about the axis is: 0003 r0 2 0002C D 20005r 2 R dr D 0005r03 R 3 0 The shear stress within the fluid can therefore be evaluated from this equation. In this problem, 4 D 1° D 0.0175 rad and r0 D 0.05 m When the cone speed is 0.1 Hz, ω D 20005 ð 0.1 D 0.628 Hence the shear rate, ω/ sin 4 D 00060.628/0.01750007 D 36 s00021 3c The shear stress is given by: R D 20005703 When c D 4.6 ð 1000022 Nm, R D 00063 ð 4.6 ð 1000022 /000620005 ð 0.053 00070007 D 176 N/m2 The remaining data may be treated in the same way to give: Cone speed (Hz)
Shear rate (s00021 )
Torque (Nm)
Shear stress (N/m2 )
0.1 0.5 1 5 10 50
36 180 360 1800 3600 18000
0.46 0.70 1.0 3.4 6.4 30.0
1760 2670 3820 13000 24500 114600
These data may be plotted on linear axes as shown in Fig. 3.24 or on logarithmic axes as in Fig. 3.26 given here as Figs 3g and 3h, respectively. It will be seen from Fig. 3g that linear axes produce an excellent straight line with an intercept of 1500 N/m2 and this indicates a Bingham plastic type of material whose characteristics are described by equation 3.122 dux jRy j 0002 Ry D p (equation 3.122) dy From Fig. 3g, the slope is p D 6.4 Ns/m2 and the graph confirms Bingham plastic behaviour.
PROBLEM 3.35 Tomato pur´ee of density 1300 kg/m3 is pumped through a 50 mm diameter factory pipeline at a flowrate of 0.00028 m3 /s. It is suggested that in order to double production: (a) a similar line with pump should be put in parallel to the existing one, or
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25000
Shear stress (N/m2)
20000
15000
10000
Slope = 6.4 Ns/m2
5000 Intercept = 1500 N/m2 0
500 1000 1500 2000
3000
Shear rate (s−1)
Figure 3g.
Shear stress (N /m2)
100,000
10,000
1000 10
100
1000
10,000
100,000
Shear rate (s−1)
Figure 3h.
(b) a large pump should force the material through the present line, or (c) a large pump should supply the liquid through a line of twice the cross-sectional area. Given that the flow properties of the pur´ee can be described by the Casson equation: 0004
1/2
00060002Ry 0007
1/2
D 00060002RY 0007
dux C 0002 c dy
00051/2
where RY is a yield stress, here 20 N/m2 , c is a characteristic Casson plastic viscosity, 5 Ns/m2 , and dux /dy is the velocity gradient, evaluate the relative pressure drops of the three suggestions, assuming laminar flow throughout.
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Solution The Casson equation is a particular form of equation 3.122 which applies to a number of foodstuffs as well as tomato puree.
or: and:
0004 0005 dux 1/2 00060002Ry 00071/2 D 00060002RY 00071/2 C 0002 c dy 0004 00051/2 dux 0002 c D 00060002Ry 00071/2 0002 00060002RY 00071/2 dy
dux D 00060002RY 0007 C 00060002Ry 0007 0002 20006Ry RY 00071/2 dy 0012 1 0011 dux D 0002 00060002Ry 0007 C 00060002RY 0007 0002 00062Ry RY 00071/2 dy c
0002 c
The Rabinowitch–Mooney equation gives the total volumetric flowrate Q through the pipe as: 0003 Rw Q 1 D0002 0006Ry 00072 f0006Ry 0007 dRy (equation 3.149) 0005a3 c Rw3 0 where a is the pipe radius and Rw is the stress at the wall. Substituting for f0006Ry 0007: 1 Q D 3 0005a c Rw3
0003 0
Rw
1/2
0006Ry3 C Ry2 RY C 2Ry5/2 RY 0007 dRy
000e 000fR w Ry4 Ry3 RY 1 4 7/2 1/2 D C C Ry RY c Rw3 4 3 7 0 RY 4 1 Rw 1/2 C C Rw1/2 RY D c 4 3 7
In this problem, RY D 20 N/m2 , c D 5 Ns/m2 , Q D 2.8 ð 1000024 m3 /s, a D 0.025 m and substituting these values, Rw D 030.24 N/m2 . From equation 3.138, Rw D 0006D/4000700060002P/l0007 D 30.24 N/m2 and:
00060002P/l0007 D 2420 0006N/m2 0007/m
For case (a), the pressure drop will remain unchanged. For case (b), the flowrate D 2Q and substituting 2Q for Q enables Rw to be recalculated as 98.0 N/m2 and (0002P/l) to be determined as 7860 0006N/m2 0007/m. p For case (c), the flowrate D 2Q and the pipe diameter D a 2. Again recalculation of Rw gives a value of 14.52 N/m2 and 00060002P/l0007 D 821 0006N/m2 0007/m.
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PROBLEM 3.36 The rheological properties of a particular suspension can be approximated reasonably well by either a “power law” or a “Bingham plastic” model over the shear rate range of 10 to 50 s00021 . If the consistency k is 10 Nsn /m2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham plastic model? What will be the pressure drop when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power law model? Calculate the centre line velocity for this pressure drop for the Bingham plastic model and comment on the result.
Solution See Volume 1, Example 3.10.
PROBLEM 3.37 Show how, by suitable selection of the index n, the power-law may be used to describe the behaviour of both shear-thinning and shear-thickening non-Newtonian fluids over a limited range of shear rates. What are the main objections to the use of the power law? Give some examples of different types of shear-thinning fluids. A power-law fluid is flowing under laminar conditions through a pipe of circular crosssection. At what radial position is the fluid velocity equal to the mean velocity in the pipe? Where does this occur for a fluid with an n-value of 0.2?
Solution Steady state shear-dependent behaviour is discussed in Volume 1, Section 3.7.1. du (equation 3.4) dy 0004 0005n du (equation 3.119) For a non-Newtonian power law fluid, R D k dy 0004 0005n00021 0004 0005 du du du D a Dk dy dy dy For a Newtonian fluid,
RD
0004
where the apparent viscosity a D k
du dy
0005n00021
When n < 1, shear-thinning behaviour is represented n > 1, shear-thickening behaviour is represented n D 1, the behaviour is Newtonian.
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For shear-thinning fluids, a ! 1 at zero shear stress and a ! 0 at infinite shear stress. Paint often exhibits shear thinning behaviour as its apparent viscosity is very high while in the can and when just applied to a wall but its apparent viscosity is very low as the brush applies it to the surface when it flows readily to give an even film. Toothpaste remains in its tube and on the brush when not subjected to shear but when sheared, as it is when the tube is squeezed, it flows readily through the nozzle to the brush. For a fluid flowing in a pipe of radius a, length l with a central core of radius r, a force balance gives: dur n 2 0002P0005r D k 0002 20005rl dr 0002Pr dur n Dk 0002 2l dr 0002P 1/n 1/n dur or: 0002 r D dr 2kl nC1 0002P 1/n n r n CC Integrating: 0002ur D 2kl nC1 1/n nC1 0002P n a n When r D a, ur D 0 and C D 0002 2kl nC1 1/n nC1 nC1 0002P n ∴ ur D a n 0002r n 2kl nC1 The mean velocity is u given by the volumetric flow/area 0003 Q 0003 a 1 1 dQ D 2 20005r dr ux or: uD 2 0005a 0 0005a 0 0004 0005 0003 a nC1 2nC1 0002P 1/n n 1 n r0002r n 20005 ∴ uD a dr 0005a2 2kl nC1 0 0004 0005 nC1 0002P 1/n n ∴ uD a n 2kl 3n C 1 When the mean velocity D average velocity, then: 0004 0005 0004 0005 nC1 nC1 nC1 0002P 1/n n 0002P 1/n n a n D a n 0002r n 2kl 3n C 1 2kl nC1
or:
When n D 0.2 then:
r nC1
nC1 n D a 3n C 1 0004 0005 n nC1 r 2n D a 3n C 1
10002
r D a
0004
0.4 1.6
0005 0.2 1.2
D 0.794
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PROBLEM 3.38 A liquid whose rheology can be represented by the power-law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow is 1 m/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n? Water, of viscosity 1 mNs/m2 flowing through the pipe at the same mean velocity gives rise to a pressure drop of 104 N/m2 compared with 105 N/m2 for the non-Newtonian fluid. What is the consistency (“k” value) of the non-Newtonian fluid?
Solution In problem 3.37, the mean velocity, u, is shown to be: 0004
uD
0002P 2kl
00051/n
a
nC1 n
n 3n C 1
and the velocity at any distance y from the pipe axis is: 0004
ur D
0002P 2kl
00051/n
nC1 nC1 n a n 0002r n nC1
The maximum velocity, umax , will occur when y D 0 and: 0004 0005 nC1 0002P 1/n n a n umax D 2kl nC1 ∴
As shown previously:
umax 1.2 3n C 1 D D and n D 0.111 u 1.0 nC1 0004 0005 0002P 1/n nC1 n uD a n 2kl 3n C 1
When n D 0.111 for the non-Newtonian fluid, 0002P D 105 N/m2 and u D 1 m/s 0004 5 00059 10 ∴ 1D a10 ð 0.083 2kl When n D 1 for water, 0002P D 104 N/m2 and u D 1 m/s and k D . 0004 40005 10 ∴ 1D a2 ð 0.25 2 l 0004 4 00059 10 a18 ð 3.81 ð 1000026 or: 1D 2 l 0004 4 00059 0004 5 00059 10 10 ∴ a10 ð 0.083 D a18 ð 3.81 ð 1000026 2kl 2 l
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From which, when a D 0.0025 m and D 1 ð 1000023 Ns/m2 , k D 6.24 Nsn /m2
PROBLEM 3.39 Two liquids of equal densities, the one Newtonian and the other a non-Newtonian “powerlaw” fluid, flow at equal volumetric rates down two wide vertical surfaces of the same widths. The non-Newtonian fluid has a power-law index of 0.5 and has the same apparent viscosity, in SI units, as the Newtonian fluid when its shear rate is 0.01 s00021 . Show that, for equal surface velocities of the two fluids, the film thickness for the Newtonian fluid is 1.125 times that of the non-Newtonian fluid.
Solution For a power-law fluid:
0004
RDk 0004
Dk
du dy du dy
0005n
(equation 3.121) 0005n00021 0004
dux dy
0005
D a 0006dux /dy0007
where a is the apparent velocity D k0006dux /dy0007n00021 For a Newtonian fluid: 0004 0005 dux RD dy
(equation 3.122)
(equation 3.3)
When n D 0.5 and 0006dux /dy0007 D 0.01, D a and: a D D k0006dux /dy0007n00021 D k00060.01000700020.5 D 10 k D and k D 0.1 . The equation of state of the power-law fluid is therefore: R D 0.1 0006dux /dy00070.5 For a fluid flowing down a vertical surface, length l and width w and film thickness S, at a distance y from the solid surface, a force balance gives: 0006S 0002 y0007wl000bg D Rwl D k0006dux /dy0007n wl g 1/n dux D 0006S 0002 y00071/n or: dy k 0004 0005 g 1/n nC1 n and: ux D 0006S 0002 y0007 n 0002 C const. k nC1 g 1/n nC1 n When y D 0, ux D 0 and the constant D S n K nC1 g 1/n n nC1 nC1 S n 0002 0006S 0002 y0007 n and: ux D k nC1
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At the free surface where y D S: us D
g 1/n 0004
k
n nC1
0005
S
nC1 n
The volumetric flowrate Q is given by: 0005 0003 s g 1/n 0004 n 0005 0004 nC1 nC1 QD w dy S n 0002s0002y n k nC1 0 g 1/n 0004 n 0005 2nC1 Dw S n k 2n C 1
(i)
(ii)
For the non-Newtonian fluid, k D 0.1 , n D 0.5 and equation (ii), when expressed in S.I. units, becomes: 0004 0005 0004 00052 g 2 gs2 4 Dw ð 0.25 s D 25 w (iii) 0.1 For the Newtonian fluid, n D 1 and k D and substituting in equation (ii): 0004 0005 g 3 ð 0.33SN QDw
(iv)
where SN is the thickness of the Newtonian film. For equal flowrates, from equations (iii) and (iv): 0004
25w
000bgs2
or:
00052
0004
0005 g 3 SN 0004 0005 g 3 SN D 75 S4
D 0.33w
(v)
For equal surface velocities, the term 0006000bg/K0007 in equation (i) can be substituted from equation (v) and: 0004
0005 g 2 For the non-Newtonian fluid: us D ð 0.33S3 0.1 0004 3 00052 SN D 100 ð 0.33S3 75S4 6 D 0.00592SN /S5 0004
0005 g 2 ð 0.5SN 0004 3 0005 SN 2 5 D ð 0.5SN D 0.0067SN /S4 75S4
For the Newtonian fluid: us D
and:
SN /S D 00060.00667/0.005920007 D 1.126
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PROBLEM 3.40 A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 ð 104 N/m2 . When the flowrate is doubled, the pressure drop increases by a factor of 1.5. A pitot tube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Any equations used must be derived from the basic relation between shear stress R and shear rate: R D k0006,0007 P n
radial distance from centre of pipe (s mm)
velocity (m/s)
0 10 20 30
0.80 0.77 0.62 0.27
Solution For a power-law fluid: At the initial flowrate:
4 ð 104 D kun
With a flow of:
6 ð 104 D k00062u0007n 1.5 D 2n
Dividing: and hence:
n D 0.585
For the power-law fluid:
dux n R D k dy
A force balance on a fluid core of radius s in pipe of radius r gives:
or:
Integrating:
Rs 20005sl D 0002P0005s2 0004 0005 dux n Ps Rs D k 0002 D0002 ds 2l 0004 0005 P 1/n 1/n dux D 0002 s 0002 ds 2kl 0004 0005 0004 0005 nC1 n P 1/n 0002ux D 0002 s n C constant 2ks nC1
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When s D r, ux D 0 (the no-slip condition): 0004
00051 0004 0005 nC1 0002P n n r n and hence the constant D 0002 2kl nC1 0004 00051/n 0004 0005 nC1 nC1 n P Substituting: ux D 0002 r n 0002s n 2kl nC1 0004 0005 0004 0005 nC1 0002P 1/n n On the centre line: uCL D r n 2kl nC1 s ux n and hence: D10002 uCL r s 2.71 at when n D 0.585 : D10002 r
nC1
The following data are obtained for a pipe radius of r D 35 mm: experimental
radius s (mm)
s 2.71 ux D10002 uCL r
ux (m/s)
ux /uCL
0 10 20 30
1 0.966 0.781 0.341
0.80 0.77 0.62 0.27
1 0.96 0.77 0.34
Thus, the calculated and experimental values of ux /uCL agree within reasonable limits of experimental accuracy.
PROBLEM 3.41 A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2 ) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient 0002P/l and volumetric flowrate Q is: 4 1 000500060002P0007r 4 1 0002 X C X4 QD 8l p 3 3 where l is the pipe radius, p is the plastic viscosity, and X is the ratio of the yield stress to the shear stress at the pipe wall. Calculate the flowrate for this pipeline when the pressure drop is 600 kN/m2 . It may be assumed that the flow is laminar.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution For a Bingham-plastic material, the shear stress Rs at radius s is given by: 0004 0005 dux Rs 0002 RY D C p 0002 0006Rs 0002 RY 0007 ds dux D 0 0006Rs 0003 RY 0007 ds The central unsheared core has radius rc D r0006RY /R0007 (where r D pipe radius and R D wall shear stress) since the shear stress is proportional to the radius s. In the annular region:
1 dux 1 s D 0002 0006R 0002 RY 0007 D 0002P 0002 RY 0006from a force balance0007 ds p p 2l 0004 0005 0003 1 s2 0002P 0002 RY s C constant 0002ux D 0002 dux D p 4l For the no-slip condition: ux D 0, when s D r 0004 0005 r2 1 0002P 0002 RY s C constant Thus: 0D p 4l 0006 0007 1 0002P 2 2 and: us D 0006r 0002 s 0007 0002 RY 0006r 0002 s0007 p 4l Substituting:
0002P D
2R l/r
us D
1 p
0006
0007 R 2 0006r 0002 s2 0007 0002 RY 0006r 0002 s0007 2r
(i)
The volumetric flowrate through elemental annulus, dQA D us 20005sds 0006 0007 0003 r 1 R 2 Thus: QA D 0006r 0002 s2 0007 0002 RY 0006r 0002 s0007 20005sds 2r rc p 0004 2 2 0005 0004 0005 r 1 r s s4 s3 RY rs2 20005 0002 0002 R 0002 D p 2r 2 4 R 2 3 rc Writing
RY RY D X and rc D r , then : R R 0006 0004 4 0005 0004 3 0005 0004 0005 1 r r r4 r3 X4 r 4 20005 1 X2 r 4 0002 0002 0002 R 0002X 0002 QA D p 2r 2 4 2 3 2r 2 4 0004 3 2 00050007 r X r 3 X3 0002 CX 2 3 0006 0007 20005R 3 1 1 1 1 1 1 D r 0002 X 0002 X2 C X4 C X3 0002 X4 p 8 6 4 8 2 3
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FLOW IN PIPES AND CHANNELS
D
0005Rr 3 4 p
0006
5 1 0002 4/3X 0002 2X2 C 4X3 0002 X4 3
59
0007
(ii)
In the core region Substituting: s D rc D 0006RY /R0007r D Xr in equation (i) for the core velocity uc gives: 0006 0007 R 2 1 0006r 0002 X2 r 2 0007 0002 RY 0006r 0002 Xr uc D p 2r 0006 0007 Rr 1 Rr 2 00061 0002 X 0007 0002 X00061 0002 X0007 0002 D f200061 0002 X2 0007 0002 4X C 4X2 0007g p 2 4 p D
0014 Rr 0013 2 0002 4X C 2X2 4 p
The flowrate through the core is: uc 0005rc2 D uc 0005X2 r 2 D Qc Thus:
Qc D D
Rr 0005X2 r 2 f2 0002 4X C 2X2 g 4 p Rr 3 0005 f2X2 0002 4X3 C 2X4 g 4 p
The total flowrate is: 0006QA C Qc 0007 D Q and:
QD
0014 0005Rr 3 0013 1 0002 43 X C 13 X4 4 p
Putting
RD
0002Pr then : 2l QD
When: Then:
000500060002P0007r 4 f1 0002 43 X C 13 X4 g 8l p
0002P D 6 ð 105 N/m2 , l D 200 m d D 40 mm and r D 0.02 m. R D 0002P
r 0.02 D6ð ð 105 D 30 N/m2 2l 400
p D 0.150 Ns/m2 RY D 14.35 N/m2 and: Thus:
RY 14.35 D D 0.478 R 30 0006 0007 1 4 00060005000700066 ð 105 000700060.0200074 3 1 0002 ð 0.478 C 00060.4780007 QD 8 ð 200 ð 0.150 3 3 XD
D 0.000503 m3 /s
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SECTION 4
Flow of Compressible Fluids PROBLEM 4.1 A gas, having a molecular weight of 13 kg/kmol and a kinematic viscosity of 0.25 cm2 /s, flows through a pipe 0.25 m internal diameter and 5 km long at the rate of 0.4 m3 /s and is delivered at atmospheric pressure. Calculate the pressure required to maintain this rate of flow under isothermal conditions. The volume occupied by 1 kmol at 273 K and 101.3 kN/m2 is 22.4 m3 . What would be the effect on the required pressure if the gas were to be delivered at a height of 150 m (i) above, and (ii) below its point of entry into the pipe?
Solution From equation 4.57 and, as a first approximation, omitting the kinetic energy term: 0002P2 0001 P1 0004/vm C 40002R/u2 00040002l/d00040002G/A00042 D 0 At atmospheric pressure and 289 K, the density D 000213/22.400040002273/2890004 D 0.542 kg/m3 Mass flowrate of gas, G D 00020.4 ð 0.5420004 D 0.217 kg/s. Cross-sectional area, A D 0002000e/4000400020.2500042 D 0.0491 m2 . Gas velocity, u D 00020.4/0.04910004 D 8.146 m/s ∴
G/A D 00020.217/0.04910004 D 4.413 kg/m2 s
Reynolds number,
Re D du/0010 D 00020.25 ð 8.146/0.25 ð 1000014 0004 D 8.146 ð 104
From Fig. 3.7, for e/d D 0.002, R/u2 D 0.0031 v2 D 00021/0.5420004 D 1.845 m3 /kg v1 D 000222.4/1300040002298/27300040002101.3/P1 0004 D 190.5/P1 m3 /kg
and:
vm D 00020.923P1 C 95.250004/P1 m3 /kg
Substituting in equation 4.57: P1 0002P1 0001 101.30004103 /00020.923P1 C 95.250004 D 400020.0031000400025000/0.25000400024.72600042 and:
P1 D 111.1 kN/m2 60
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61
The kinetic energy term D 0002G/A00042 ln0002P1 /P2 0004 D 00024.41300042 ln0002111.1/101.30004 D 1.81 kg2 /m4 s2 This is negligible in comparison with the other terms which equal 5539 kg2 /m4 s2 so that the initial approximation is justified. If the pipe is not horizontal, the term g dz in equation 4.49 must be included in the calculation. If equation 4.49 is divided by v2 , this term on integration becomes gz/v2m . ∴
vm D 00020.923 ð 111.1 C 95.250004/111.1 D 1.781 m3 /kg vair D 000224.0/290004 D 0.827 m3 /kg
As the gas is less dense than air, vm is replaced by 0002vair 0001 vm 0004 D 00010.954 m3 /kg. ∴
gz/v2m D 00029.81 ð 150/0.9542 0004 D 1616 N/m2 or 0.16 kN/m2
(i) If the delivery point is 150 m above the entry level, then since gas is less dense, P1 D 0002111.1 0001 0.160004 D 110.94 kN/m2 (ii) If the delivery point is 150 m below the entry level then, P1 D 0002111.1 C 0.160004 D 111.26 kN/m2
PROBLEM 4.2 Nitrogen at 12 MN/m2 pressure is fed through a 25 mm diameter mild steel pipe to a synthetic ammonia plant at the rate of 1.25 kg/s. What will be the pressure drop over a 30 m length of pipe for isothermal flow of the gas at 298 K? Absolute roughness of the pipe surface D 0.005 mm. Kilogram molecular volume D 22.4 m3 . Viscosity of nitrogen D 0.02 mN s/m2 .
Solution Molecular weight of nitrogen D 28 kg/kmol. Assuming a mean pressure in the pipe of 10 MN/m2 , the specific volume, vm at 10 MN/m2 and 298 K is: vm D 000222.4/2800040002101.3/10 ð 103 00040002298/2730004 D 0.00885 m3 /kmol
Reynolds number, ud/0010 D d0002G/A0004/00100004. A D 0002000e/4000400020.02500042 D 4.91 ð 1000013 m2 . ∴ 0002G/A0004 D 00021.25/4.91 ð 1000013 0004 D 2540 kg/m2 s
and:
Re D 00020.025 ð 2540/0.02 ð 1000013 0004 D 3.18 ð 106
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From Fig. 3.7, for Re D 3.18 ð 106 and e/d D 00020.005/250004 D 0.0002, R/u2 D 0.0017 In equation 4.57 and neglecting the first term: 0002P2 0001 P1 0004/vm C 40002R/u2 00040002l/d00040002G/A00042 D 0 or:
P1 0001 P2 D 4vm 0002R/u2 00040002l/d00040002G/A00042 D 4 ð 0.0088500020.00170004000230/0.02500040002254000042 D 466,000 N/m2 or 0.466 MN/m2
This is small in comparison with P1 D 12 MN/m2 , and the average pressure of 10 MN/m2 is seen to be too low. A mean pressure of 11.75 kN/m2 is therefore selected and the calculation repeated to give a pressure drop of 0.39 MN/m2 . The mean pressure is then 000212 C 11.610004/2 D 11.8 MN/m2 which is close enough to the assumed value. It remains to check if the assumption that the kinetic energy term is negligible is justified. Kinetic energy term D 0002G/A00042 ln0002P1 /P2 0004 D 0002254000042 ln000212/11.610004 D 2.13 ð 105 kg2/m4s2 The term 0002P1 0001 P2 0004/vm , where vm is the specific volume at the mean pressure of 11.75 MN/m2 D 00020.39 ð 106 0004/0.00753 D 5.18 ð 107 kg2 /m4 s. Hence the omission of the kinetic energy term is justified and the pressure drop D 0.39 MN/m2
PROBLEM 4.3 Hydrogen is pumped from a reservoir at 2 MN/m2 pressure through a clean horizontal mild steel pipe 50 mm diameter and 500 m long. The downstream pressure is also 2 MN/m2 and the pressure of this gas is raised to 2.6 MN/m2 by a pump at the upstream end of the pipe. The conditions of flow are isothermal and the temperature of the gas is 293 K. What is the flowrate and what is the effective rate of working of the pump? Viscosity of hydrogen D 0.009 mN s/m2 at 293 K.
Solution Neglecting the kinetic energy term in equation 4.55, then: 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 where P1 D 2.6 MN/m2 and P2 D 2.0 MN/m2 . Thus:
v1 D 000222.4/200040002293/273000400020.1013/2.60004 D 0.468 m3 /kg
If Re D 107 and e/d D 0.001, from Fig. 3.7, R/u2 D 0.0023.
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63
Substituting: 00022.02 0001 2.62 00041012 /00022 ð 2.6 ð 106 ð 0.4680004 C 400020.002300040002500/0.0500040002G/A00042 D 0 from which G/A D 111 kg/m2 s. ∴
Re D d0002G/A0004/0010 D 00020.05 ð 111/00020.009 ð 1000013 0004 D 6.2 ð 105
Thus the chosen value of Re was too high. If Re is taken as 6.0 ð 105 and the problem reworked, G/A D 108 kg/m2 s and Re D 6.03 ð 105 which is in good agreement. A D 0002000e/4000400020.0500042 D 0.00197 m2 and:
G D 108 ð 0.00197 D 0.213 kg/s
The power requirement is given by equation 8.71 as 00021/00150004GP1 v1 ln0002P1 /P2 0004 If a 60% efficiency is assumed, then the power requirement is: D 00021/0.60004 ð 0.213 ð 2.6 ð 106 ð 0.468 ln00022.6/20004 D 00021.13 ð 105 0004 W or 113 kW
PROBLEM 4.4 In a synthetic ammonia plant the hydrogen is fed through a 50 mm steel pipe to the converters. The pressure drop over the 30 m length of pipe is 500 kN/m2 , the pressure at the downstream end being 7.5 MN/m2 . What power is required in order to overcome friction losses in the pipe? Assume isothermal expansion of the gas at 298 K. What error is introduced by assuming the gas to be an incompressible fluid of density equal to that at the mean pressure in the pipe? 0010 D 0.02 mNs/m2 .
Solution If the downstream pressure D 7.5 MN/m2 and the pressure drop due to friction D 500 kN/m2 , the upstream pressure D 8.0 MN/m2 and the mean pressure D 7.75 MN/m2 . The mean specific volume is: vm D 000222.4/200040002298/273000400020.1013/7.750004 D 0.16 m3 /kg and:
v1 D 000222.4/200040002298/273000400020.1013/8.00004 D 0.15 m3 /kg
It is necessary to assume a value of R/u2 , calculate G/A and the Reynolds number and check that the value of e/d is reasonable. If the gas is assumed to be an incompressible fluid of density equal to the mean pressure in the pipe and R/u2 D 0.003, the pressure drop due to friction D 500 kN/m2 is: ∴
and
0002500 ð 103 /0.160004 D 400020.0030004000230/0.0500040002G/A00042 G/A D 658 kg/m2 s. Re D d0002G/A0004/0010 D 00020.05 ð 658/0.02 ð 1000013 0004 D 1.65 ð 106
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
From Fig. 3.7 this corresponds to a value of e/d of approximately 0.002, which is reasonable for a steel pipe. For compressible flow: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
Substituting: 0002G/A00042 ln00028.0/7.50004 C 00027.52 0001 8.02 00041012 /00022 ð 8.0 ð 106 ð 0.150004 C 400020.0030004000230/0.0500040002G/A00042 D 0 from which: G/A D 667 kg/m2 s and G D 667 ð 0002000e/4000400020.0500042 D 1.31 kg/s Little error is made by the simplifying assumption in this particular case. The power requirement is given by equation 8.71: D 00021/00150004GP1 v1 ln0002P1 /P2 0004 If the compressor efficiency D 60%, power requirement D 00021/0.60004 ð 1.31 ð 8.0 ð 106 ð 0.15 ln00028/7.50004 D 00021.69 ð 105 0004 W or 169 kW
PROBLEM 4.5 A vacuum distillation plant operating at 7 kN/m2 pressure at the top has a boil-up rate of 0.125 kg/s of xylene. Calculate the pressure drop along a 150 mm bore vapour pipe used to connect the column to the condenser. The pipe length may be taken as equivalent to 6 m, e/d D 0.002 and 0010 D 0.01 mN s/m2 .
Solution From vapour pressure data, the vapour temperature D 338 K and the molecular weight of xylene D 106 kg/kmol. In equation 4.55: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 Cross-sectional area of pipe, A D 0002000e/4000400020.1500042 D 1.76 ð 1000012 m2 G/A D 00020.125/1.76 ð 1000012 0004 D 7.07 kg/m2 s The Reynolds number, is ud/0010 D d0002G/A0004/0010 D 00020.15 ð 7.07/00020.01 ð 1000013 0004 D 1.06 ð 105 From Fig. 3.7, with e/d D 0.002 and Re D 1.06 ð 105 , 0002R/u2 0004 D 0.003. Specific volume, v1 D 000222.4/10600040002338/27300040002101.3/7.00004 D 3.79 m3 /kg.
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FLOW OF COMPRESSIBLE FLUIDS
65
Substituting in equation 4.55: 00027.000042 ln00027/P2 0004 C 0002P22 0001 72 0004 ð 106 /2 ð 7 ð 103 ð 3.79 C 4 ð 0.00300026/0.15000400027.0700042 D 0 where P2 is the pressure at the condenser 0002kN/m2 0004. Solving by trial and error: P2 D 6.91 kN/m2 ∴
0002P1 0001 P2 0004 D 00027.0 0001 6.910004 D 0.09 kN/m2 or 90 N/m2
PROBLEM 4.6 Nitrogen at 12 MN/m2 pressure is fed through a 25 mm diameter mild steel pipe to a synthetic ammonia plant at the rate of 0.4 kg/s. What will be the drop in pressure over a 30 m length of pipe assuming isothermal expansion of the gas at 300 K? What is the average quantity of heat per unit area of pipe surface that must pass through the walls in order to maintain isothermal conditions? What would be the pressure drop in the pipe if it were perfectly lagged? 0010 D 0.02 mNs/m2 .
Solution At high pressure, the kinetic energy term in equation 4.55 may be neglected to give: 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 Specific volume at entry of pipe, v1 D 000222.4/2800040002300/273000400020.1013/120004 D 0.00742 m3 /kg Cross-sectional area of pipe, A D 0002000e/4000400020.02500042 D 0.00049 m2 ∴ G/A D 00020.4/0.000490004 D 816 kg/m2 s.
Reynolds number, d0002G/A0004/0010 D 0.025 ð 816/00020.02 ð 1000013 0004 D 1.02 ð 106 If e/d D 0.002 and Re D 1.02 ð 106 , R/u2 D 0.0028 from Fig. 3.7. Substituting: 0002122 0001 P22 00041012 /00022 ð 12 ð 106 ð 0.007420004 D 400020.00280004000230/0.0250004000281600042 and: P2 D 11.93 MN/m2 and: pressure drop D 000212.0 0001 11.930004 D 0.07 MN/m2 0005 70 kN/m2 The heat required to maintain isothermal flow is given in Section 4.5.2 as Gu2 /2. The velocity at the high pressure end of the pipe D volumetric flow/area D 0002G/A0004v1 D 0002816 ð 0.00720004 D 6.06 m/s and the velocity in the plant is taken as zero. Thus:
Gu2 /2 D 0.4 ð 00026.0600042 /2 D 7.34 W
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Outside area of pipe D 000230 ð 000e ð 0.0250004 D 2.36 m2 . Heat required D 00027.34/2.360004 D 3.12 W/m2 This low value of the heat required stems from the fact that the change in kinetic energy is small and conditions are almost adiabatic. If the pipe were perfectly lagged, the flow would be adiabatic and the pressure drop would then be calculated from equations 4.77 and 4.72. The specific volume at the low pressure end v2 to be calculated from: 0001 0002 0003 00040001 0002 00032 0004 0002 0003 0016 0001 1 P1 A 2 0016 C1 v1 v2 2 80002R/u 00040002l/d0004 D C ln 10001 0001 20016 v1 G v2 0016 v1 (equation 4.77) For nitrogen, 0016 D 1.4 and hence: 0001 0002 0003 00040001 0002 0003 0004 1.4 0001 1 12 ð 106 1 2 0.00742 2 800020.00280004000230/0.0250004 D C 10001 2 ð 1.4 0.00742 816 v2 1.4 C 1 0005 v2 0006 ln 0001 1.4 0.00742 Solving by trial and error, v2 D 0.00746 m3 /kg. Thus: 0002 0003 0002 0003 0002 0003 0002 0003 1 G 2 2 0016 0016 1 G 2 2 v1 C P1 v1 D v2 C P2 v2 2 A 0016 00011 2 A 0016 00011
(equation 4.72)
Substitution gives: 000281600042 00020.0074200042 /2 C [1.4/00021.4 0001 10004]12 ð 106 ð 0.00742 D 000281600042 00020.0074600042 /2 C [1.4/00021.4 0001 10004]P2 ð 106 ð 0.00746 and: P1 D 11.94 MN/m2 The pressure drop for adiabatic flow D 000212.0 0001 11.940004 D 0.06 MN/m2 or 60 kN/m2
PROBLEM 4.7 Air, at a pressure of 10 MN/m2 and a temperature of 290 K, flows from a reservoir through a mild steel pipe of 10 mm diameter and 30 m long into a second reservoir at a pressure P2 . Plot the mass rate of flow of the air as a function of the pressure P2 . Neglect any effects attributable to differences in level and assume an adiabatic expansion of the air. 0010 D 0.018 mN s/m2 , 0016 D 1.36.
Solution G/A is required as a function of P2 . v2 cannot be found directly since the downstream temperature T2 is unknown and varies as a function of the flowrate. For adiabatic flow,
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FLOW OF COMPRESSIBLE FLUIDS
v2 may be calculated from equation 4.77 using specified values of G/A and substituted in equation 4.72 to obtain the value of P2 . In this way the required data may be calculated. 0001 0002 0003 00040001 0002 00032 0004 0002 0003 0016 0001 1 P1 A 2 0016 C1 v1 v2 2 C ln 10001 0001 80002R/u 00040002l/d0004 D 20016 v1 G v2 0016 v1
(equation 4.77) 0.50002G/A00042 v21 C [0016/00020016 0001 10004]P1 v1 D 0.50002G/A00042 v22 C [0016/00020016 0001 10004]P2 v2 (equation 4.72) or:
0.50002G/A00042 0002v21 0001 v22 0004 C [0016/00020016 0001 10004]P1 v1 D P2 [0016/00020016 0001 10004]v2
When P2 D P1 D 10 MN/m2 , G/A D 0. If G/A is 2000 kg/m2 s, then: Re D 00020.01 ð 2000/0.018 ð 1000013 0004 D 1.11 ð 106 When e/d D 0.0002, R/u2 D 0.0028 from Fig. 3.7 and: v1 D 000222.4/2900040002290/273000400020.1013/100004 D 0.0083 m3 /kg
Substituting in equation 4.77: 0001
0002 00032 0004 0.36 1 10 ð 106 C 800020.00280004000230/0.010004 D 2 ð 1.36 0.0083 2000 0001 0004 0002 0003 0.0083 2 2.36 0005 v2 0006 ð 10001 0001 ln v2 1.36 0.0083
and: v2 D 0.00942 m3 /kg. Substituting for v2 in equation 4.72 gives: P2 D [0.50002200000042 00020.00832 0001 0.009422 0004 C 00021.36/0.36000410 ð 106ð0.0083 ]/00021.36/0.360004 ð 0.00942 and: P2 D 8.75 MN/m2 . In a similar way the following table may be produced. G/A0002kg/m2 s0004
v2 0002m3 /kg0004
P2 0002MN/m2 0004
0 2000 3000 3500 4000 4238
0.0083 0.00942 0.012 0.0165 0.025 0.039
10.0 8.75 6.76 5.01 3.37 2.04
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5000
G /A (kg / m2s)
4000
3000
2000
1000
0 10
9
8
7
4 3 6 5 Pressure P2 (MN/m2)
2
1
0
Figure 4a.
These data are plotted in Fig. 4a. It is shown in Section 4.5.4, Volume 1, that the maximum velocity which can p occur in a pipe under adiabatic flow conditions is the sonic velocity which is equal to p 0016P2 v2 . From the above table 0016P2 v2 at maximum flow is: 0007 1.36 ð 2.04 ð 106 ð 0.039 D 329 m/s The temperature at this condition is given by P2 v2 D RT/M, and: T2 D 000229 ð 0.039 ð 2.04 ð 106 /83140004 D 227 K The velocity of sound in air at 227 K D 334 m/s, which serves as a check on the calculated data.
PROBLEM 4.8 Over a 30 m length of 150 mm vacuum line carrying air at 293 K, the pressure falls from 1 kN/m2 to 0.1 kN/m2 . If the relative roughness e/d is 0.002, what is approximate flowrate?
Solution The specific volume of air at 293 K and 1 kN/m2 is: v1 D 000222.4/2900040002293/27300040002101.3/1.00004 D 83.98 m3 /kg
It is necessary to assume a Reynolds number to determine R/u2 and then calculate a value of G/A which should correspond to the original assumed value. Assume a Reynolds number of 1 ð 105 .
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When e/d D 0.002 and Re D 105 , R/u2 D 0.003 from Fig. 3.7. 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
Substituting: 0002G/A00042 ln00021.0/0.10004 C 00020.12 0001 12 0004 ð 106 /00022 ð 1 ð 103 ð 83.980004 C 400020.0030004000230/0.1500040002G/A00042 D 0 and: 0002G/A0004 D 1.37 kg/m2 s. The viscosity of air is 0.018 mN s/m2 . ∴ Re D 00020.15 ð 1.370004/00020.018 ð 1000013 0004 D 1.14 ð 104
Thus the chosen value of Re is too high. When Re D 1 ð 104 , R/u2 D 0.0041 and G/A D 1.26 kg/m2 s. Re now equals 1.04 ð 104 which agrees well with the assumed value. G D 1.26 ð 0002000e/40004 ð 00020.1500042 D 0.022 kg/s
Thus:
PROBLEM 4.9 A vacuum system is required to handle 10 g/s of vapour (molecular weight 56 kg/kmol) so as to maintain a pressure of 1.5 kN/m2 in a vessel situated 30 m from the vacuum pump. If the pump is able to maintain a pressure of 0.15 kN/m2 at its suction point, what diameter of pipe is required? The temperature is 290 K, and isothermal conditions may be assumed in the pipe, whose surface can be taken as smooth. The ideal gas law is followed. Gas viscosity D 0.01 mN s/m2 .
Solution Use is made of equation 4.55 to solve this problem. It is necessary to assume a value of the pipe diameter d in order to calculate values of G/A, the Reynolds number and R/u2 . If d D 0.10 m, A D 0002000e/4000400020.1000042 D 0.00785 m2 ∴
G/A D 000210 ð 1000013 /0.007850004 D 1.274 kg/m2 s
and
Re D d0002G/A0004/0010 D 0.10 ð 1.274/00020.01 ð 1000013 0004 D 1.274 ð 104
For a smooth pipe, R/u2 D 0.0035, from Fig. 3.7. Specific volume at inlet, v1 D 000222.4/5600040002290/27300040002101.3/1.50004 D 28.7 m3 /kg 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0 (equation 4.55) Substituting gives: 00021.27400042 ln00021.5/0.150004 C 00020.152 0001 1.52 0004 ð 106 /00022 ð 1.5 ð 103 ð 28.70004 C 00020.00350004000230/0.10000400021.27400042 D 000116.3 and the chosen value of d is too large.
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A further assumed value of d D 0.05 m gives a value of the right hand side of equation 4.55 of 25.9 and the procedure is repeated until this value is zero. This occurs when d D 0.08 m or 80 mm.
PROBLEM 4.10 In a vacuum system, air is flowing isothermally at 290 K through a 150 mm diameter pipeline 30 m long. If the relative roughness of the pipewall e/d is 0.002 and the downstream pressure is 130 N/m2 , what will the upstream pressure be if the flowrate of air is 0.025 kg/s? Assume that the ideal gas law applies and that the viscosity of air is constant at 0.018 mN s/m2 . What error would be introduced if the change in kinetic energy of the gas as a result of expansion were neglected?
Solution As the upstream and mean specific volumes v1 and vm are required in equations 4.55 and 4.56 respectively, use is made of equation 4.57: 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/00022RT/M0004 C 40002R/u2 00040002l/d00040002G/A00042 D 0 R D 8.314 kJ/kmol K and hence: 2RT/M D 00022 ð 8.314 ð 103 ð 2900004/29 D 1.66 ð 105 J/kg The second term has units of 0002N/m2 00042 /0002J/kg0004 D kg2 /s2 m4 which is consistent with the other terms. A D 0002000e/4000400020.1500042 D 0.0176 m2 ∴
G/A D 00020.025/0.01760004 D 1.414
and
Re D d0002G/A0004/0010 D 00020.15 ð 1.4140004/00020.018 ð 1000013 0004 D 1.18 ð 104
For smooth pipes and Re D 1.18 ð 104 , R/u2 D 0.0040 from Fig. 3.7. Substituting in equation 4.57 gives: 00021.41400042 ln0002P1 /1300004 C 00021302 0001 P12 0004/1.66 ð 105 C 4 ð 0.0040000230/0.15000400021.41400042 D 0 Solving by trial and error, the upstream pressure, P1 D 1.36 kN/m2 If the kinetic energy term is neglected, equation 4.57 becomes: 0002P22 0001 P12 0004/00022RT/M0004 C 40002R/u2 00040002l/d00040002G/A00042 D 0 and P1 D 1.04 kN/m2 Thus a considerable error would be introduced by this simplifying assumption.
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FLOW OF COMPRESSIBLE FLUIDS
PROBLEM 4.11 Air is flowing at the rate of 30 kg/m2 s through a smooth pipe of 50 mm diameter and 300 m long. If the upstream pressure is 800 kN/m2 , what will the downstream pressure be if the flow is isothermal at 273 K? Take the viscosity of air as 0.015 mN s/m2 and assume that volume occupies 22.4 m3 . What is the significance of the change in kinetic energy of the fluid?
Solution 0002G/A00042 ln0002P1 /P2 0004 C 0002P22 0001 P12 0004/2P1 v1 C 40002R/u2 00040002l/d00040002G/A00042 D 0
(equation 4.55)
The specific volume at the upstream condition is: v1 D 000222.4/2900040002273/27300040002101.3/8000004 D 0.098 m3 /kg
G/A D 30 kg/m2 s ∴
Re D 00020.05 ð 300004/00020.015 ð 1000013 0004 D 1.0 ð 105
For a smooth pipe, R/u2 D 0.0032 from Fig. 3.7. Substituting gives: 00023000042 ln0002800/P2 0004 C 0002P22 0001 8002 0004 ð 106 /00022 ð 800 ð 103 ð 0.0980004 C 400020.003200040002300/0.05000400023000042 D 0 and the downstream pressure, P2 D 793 kN/m2 The kinetic energy term D 0002G/A00042 ln0002800/7930004 D 7.91 kg2 /m4 s2 This is insignificant in comparison with 69,120 kg2 /m4 s2 which is the value of the other terms in equation 4.55.
PROBLEM 4.12 If temperature does not change with height, estimate the boiling point of water at a height of 3000 m above sea-level. The barometer reading at sea-level is 98.4 kN/m2 and the temperature is 288.7 K. The vapour pressure of water at 288.7 K is 1.77 kN/m2 . The effective molecular weight of air is 29 kg/kmol.
Solution The air pressure at 3000 m is P2 and the pressure at sea level, P1 D 98.4 kN/m2 . v dP C g dz D 0 v D v1 0002P/P1 0004
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
dP C g dz D 0 P1 v1 P P1 v1 ln0002P2 /P1 0004 C g0002z2 0001 z1 0004 D 0
and: and:
v1 D 000222.4/2900040002288.7/27300040002101.3/98.40004 D 0.841 m3 /kg. ∴
98,400 ð 0.841 ln0002P2 /98.40004 C 9.8100023000 0001 00004 D 0 P2 D 68.95 kN/m2
and:
The relationship between vapour pressure and temperature may be expressed as: log P D a C bT When,
T D 288.7, P D 1.77 kN/m2
and when,
T D 373, P D 101.3 kN/m2
∴
log P D 00015.773 C 0.0209 T When P2 D 68.95, T D 364 K.
PROBLEM 4.13 A 150 mm gas main is used for transferring gas (molecular weight 13 kg/kmol and kinematic viscosity 0.25 cm2 /s) at 295 K from a plant to a storage station 100 m away, at a rate of 1 m3 /s. Calculate the pressure drop if the pipe can be considered to be smooth. If the maximum permissible pressure drop is 10 kN/m2 , is it possible to increase the flowrate by 25%?
Solution If the flow of 1 m3 /s is at STP, the specific volume of the gas is: 000222.4/130004 D 1.723 m3 /kg. The mass flowrate, G D 00021.0/1.7230004 D 0.58 kg/s. Cross-sectional area, A D 0002000e/4000400020.1500042 D 0.0176 m2 ∴
G/A D 32.82 kg/m2 s 0010/ D 0.25 cm2 /s D 0.25 ð 1000014 m2 /s
and ∴
0010 D 00020.25 ð 1000014 000400021/1.7230004 D 1.45 ð 1000015 N s/m2 Re D 00020.15 ð 32.82/1.45 ð 1000015 0004 D 3.4 ð 105
For smooth pipes, R/u2 D 0.0017, from Fig. 3.7. The pressure drop due to friction is: 40002R/u2 00040002l/d00040002G/A00042 D 400020.001700040002100/0.150004000232.8200042 D 4883 kg2 /m4 s2 and: 0001P D 00024883/1.7230004 D 2834 N/m2 or 2.83 kN/m2 .
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73
If the flow is increased by 25%, G D 00021.25 ð 0.580004 D 0.725 kg/s G/A D 41.19 kg/m2 s and:
Re D 00020.15 ð 41.90004/00021.45 ð 105 0004 D 4.3 ð 105
and, from Fig. 3.7, R/u2 D 0.00165 The pressure drop D 400020.0016500040002100/0.150004000241.1900042 1.723 D 4.33 kN/m2 (which is less than 10 kN/m2 ) It is therefore possible to increase the flowrate by 25%.
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SECTION 5
Flow of Multiphase Mixtures PROBLEM 5.1 It is required to transport sand of particle size 1.25 mm and density 2600 kg/m3 at the rate of 1 kg/s through a horizontal pipe 200 m long. Estimate the air flowrate required, the pipe diameter, and the pressure drop in the pipe-line.
Solution For conventional pneumatic transport in pipelines, a solids-gas mass ratio of about 5 is employed. Mass flow of air D 00011/50003 D 0.20 kg/s and, taking the density of air as 1.0 kg/m3 , volumetric flowrate of air D 00011.0 ð 0.200003 D 0.20 m3 /s In order to avoid excessive pressure drops, an air velocity of 30 m/s seems reasonable. Ignoring the volume occupied by the sand (which is about 0.2% of that occupied by the 2 air), the cross-sectional area of pipe p required D 00010.20/300003 D 0.0067 m , equivalent to a pipe diameter of 00014 ð 0.0067/00050003 D 0.092 m or 92 mm. Thus a pipe diameter of 101.6 mm (100 mm) would be specified. From Table 5.3 for sand of particle size 1.25 mm and density 2600 kg/m3 , the freefalling velocity is: u0 D 4.7 m/s p In equation 5.37, 0001uG 0004 us 0003 D 4.7/[0.468 C 7.25 00014.7/26000003] D 6.05 m/s The cross-sectional area of a 101.6 mm i.d. pipe D 00010005 ð 0.10162 /40003 D 0.0081 m2 . ∴
and:
air velocity, uG D 00010.20/0.00810003 D 24.7 m/s us D 000124.7 0004 6.050003 D 18.65 m/s
Taking the viscosity and density of air as 1.7 ð 1000045 N s/m2 and 1.0 kg/m3 respectively, the Reynolds number for the air flow alone is: Re D 00010.102 ð 24.7 ð 1.00003/00011.7 ð 1000045 0003 D 148,000 and from Fig. 3.7, the friction factor D 0.002.
74
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FLOW OF MULTIPHASE MIXTURES
0004Pair D 4 0001l/d00030011u2
(equation 3.18)
D 00014 ð 0.0020001200/0.1020003 ð 1.0 ð 24.72 0003 D 9570 N/m2 or 9.57 kN/m2 assuming isothermal conditions and incompressible flow. 00010004Px / 0004 Pair 00030001us2 /F0003 D 00012805/u0 0003 ∴
(equation 5.38)
0004Px D 0001280500010004Pair 0003F0003/0001u0 us2 0003 D 00012805 ð 9.57 ð 1.00003/00014.7 ð 18.652 0003 D 16.4 kN/m2
PROBLEM 5.2 Sand of a mean diameter 0.2 mm is to be conveyed in water flowing at 0.5 kg/s in a 25 mm ID horizontal pipe 100 m long. What is the maximum amount of sand which may be transported in this way if the head developed by the pump is limited to 300 kN/m2 ? Assume fully suspended heterogeneous flow.
Solution See Volume 1, Example 5.2.
PROBLEM 5.3 Explain the various mechanisms by which particles may be maintained in suspension during hydraulic transport in a horizontal pipeline and indicate when each is likely to be important. A highly concentrated suspension of flocculated kaolin in water behaves as a pseudohomogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald–de Waele power-law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced, even though the flowrate of suspension is kept constant. Explain how this is possible in “slug” flow, and estimate the possible reduction in pressure gradient for equal volumetric, flowrates of suspension and air.
Solution If u is the superficial velocity of slurry, then: For slurry alone: The pressure drop in a pipe of length l is: Kun l. If the air: slurry volumetric ratio is R, there is no slip between the slurry and the air and the system consists of alternate slugs of air and slurry, then: The linear velocity of slurry is 0001R C 10003u
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
1 RC1 Assuming that the pressure drop is the sum of the pressure drops along the slugs, then: 0001 0002 l n the new pressure drop is: Kf0001R C 10003ug D Kun l0001R C 10003n00041 RC1 Fraction of pipe occupied by slurry slugs is
Kun l0001R C 10003n00041 pressure gradient with air D D 0001R C 10003n00041 pressure gradient without air Kun l
Then:
rD
For
n D 0.15 and: R D 1 r D 200040.85 D 0.55
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SECTION 6
Flow and Pressure Measurement PROBLEM 6.1 Sulphuric acid of density 1300 kg/m3 is flowing through a pipe of 50 mm internal diameter. A thin-lipped orifice, 10 mm diameter, is fitted in the pipe and the differential pressure shown by a mercury manometer is 10 cm. Assuming that the leads to the manometer are filled with the acid, calculate (a) the mass of acid flowing per second, and (b) the approximate loss of pressure caused by the orifice. The coefficient of discharge of the orifice may be taken as 0.61, the density of mercury as 13,550 kg/m3 , and the density of water as 1000 kg/m3 .
Solution See Volume 1, Example 6.2.
PROBLEM 6.2 The rate of discharge of water from a tank is measured by means of a notch, for which the flowrate is directly proportional to the height of liquid above the bottom of the notch. Calculate and plot the profile of the notch if the flowrate is 0.1 m3 /s when the liquid level is 150 mm above the bottom of the notch.
Solution The velocity of fluid discharged as a height h above the bottom of the notch is: 0001 u D 00052gh0007 The velocity therefore varies from zero at the bottom of the notch to a maximum value at the free surface. For a horizontal element of fluid of width 2w and depth dh at a height h above the bottom of the notch, the discharge rate of fluid is given by: 0001 dQ D 00052gh00072wdh If the discharge rate is linearly related to the height of the liquid over the notch, H, w will be a function of h and it may be supposed that: w D khn where k is a constant. 77
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Substituting for w in the equation for dQ and integrating to give the discharge rate over the notch Q then: 0002 H 0001 Q D 2 00052g0007 k hn h0.5 dh 0
0002 0001 D 2 00052g0007 k
H
hnC0.5 dh
0
0001 D 2 00052g0007 k[1/0005n C 1.50007]H0005nC1.50007
Since it is required that Q / H: n C 1.5 D 1 and:
n D 00040.5 0001 Q D 2 00052g0007 kH
Thus:
Since Q D 0.1 m3 /s when H D 0.15 m: 0001 k D 00050.1/0.150007[1/00052 00052g0007] D 0.0753 m1.5
Thus, with w and h in m:
w D 0.0753h00040.5
and, with w and h in mm:
w D 2374h00040.5
and using this equation, the profile is plotted as shown in Figure 6a.
500
400
h (mm)
300
200
100
300
200
100
0
100
200
300
Distance from centre line, w (mm)
Figure 6a.
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79
PROBLEM 6.3 Water flows at between 3 l and 4 l/s through a 50 mm pipe and is metered by means of an orifice. Suggest a suitable size of orifice if the pressure difference is to be measured with a simple water manometer. What is the approximate pressure difference recorded at the maximum flowrate?
Solution Equations 6.19 and 6.21 relate the pressurepdrop to the mass flowrate. If equation 6.21 is used as a first approximation, G D CD A0 0011 00052gh0007. For the maximum flow of 4 l/s, G D 4 kg/s. The largest practicable height of a water manometer will be taken as 1 m and equation 6.21 is then used to calculate the orifice area A0 . If the coefficient of discharge CD is taken as 0.6, then: 0001 4.0 D 0.6A0 ð 1000 00052 ð 9.81 ð 1.00007, A0 D 0.0015 m2 and d0 D 0.0438 m The diameter, d0 , is comparable with the pipe diameter and hence the area correction term must be included and: [1 0004 0005A0 /A1 00072 ] D [1 0004 000543.82 /502 00072 ] D 0.641. Therefore the value of A0 must be recalculated as: 0003 4.0 D 0.6A0 ð 1000 00052 ð 9.8 ð 1.00007/[1 0004 0005A0 /A1 00072 ] from which A0 D 0.00195 m2 and d D 0.039 m or 39 mm 0003
0003
[1 0004 0005A0 /A1 00072 ] D
[1 0004 0005392 /502 00072 ] D 0.793
Substituting in equation 6.19: 0001 4.0 D 00050.6 ð 0.001950007 ð 1000 00052 ð 0.00100050004P0007/0.7930007
and:
0004P D 12320 N/m2 or 12.3 kN/m2
PROBLEM 6.4 The rate of flow of water in a 150 mm diameter pipe is measured by means of a venturi meter with a 50 mm diameter throat. When the drop in head over the converging section is 100 mm of water, the flowrate is 2.7 kg/s. What is the coefficient for the converging cone of the meter at that flowrate and what is the head lost due to friction? If the total loss of head over the meter is 15 mm water, what is the coefficient for the diverging cone?
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution The equation relating the mass flowrate G and the head loss across a venturi meter is given by: 0004 CD A0 2v0005P1 0004 P2 0007 GD (equation 6.19) v 1 0004 0005A0 /A1 00072 G D CD 0011 0003
A 1 A2
0005A21 0004 A21 0007 0001 G D CD 0011C0 00052ghv 0007
0001
00052v0005P1 0004 P2 00070007
(equation 6.32) (equation 6.33)
where C0 is a constant for the meter and hv is the loss in head over the converging cone expressed as height of fluid. A1 D 00050016/4000700050.1500072 D 0.0176 m2 A2 D 00050016/4000700050.0500072 D 0.00196 m2 0001 C0 D 00050.0176 ð 0.00196/ 00050.01762 0004 0.001962 00070007 D 0.00197 m2 ∴
hv D 0.1 m p 2.7 D 0005CD ð 1000 ð 0.001970007 00052 ð 9.81 ð 0.100007 and CD D 0.978
In equation 6.33, if there were no losses, the coefficient of discharge of the meter would be unity, and for a flowrate G the loss in head would be 0005hv 0004 hf 0007 where hf is the head loss due to friction. 0001 Thus: G D 0011C0 [2g0005hv 0004 hf 0007] Dividing this equation by equation 6.33 and squaring gives: 1 0004 0005hf /hv 0007 D C2D and hf D hv 00051 0004 C2D 0007 ∴
hf D 10000051 0004 0.9782 0007 D 4.35 mm
0 If the head recovered over the diverging cone 0001 is hv and the coefficient of discharge for 0 0 0 0 the converging cone is CD , then G D CD 0011C 00052ghv 0007 If the whole of the excess kinetic energy is recovered as pressure energy, the coefficient 0 C0D will equal unity and 0003 G will be obtained with a recovery of head equal to hv plus some quantity hf0 , G D 0011C0 [2g0005hv0 C hf0 0007 Equating these two equations and squaring gives: 0
0
CD2 D 1 C 0005hf0 /hv0 0007 and hf0 D hv0 0005CD2 0004 10007 Thus the coefficient of the diverging cone is greater than unity and the total loss of head D hf C hf0 . Head loss over diverging cone D 000515.0 0004 4.350007 D 10.65 mm
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81
The coefficient of the diverging cone C0D is given by: 0001 G D C0D 0011C0 00052ghv0 0007 and:
hv0 D 0005100 0004 150007 D 85 mm p 2.7 D 0005C0D ð 1000 ð 0.001970007 00052 ð 9.81 ð 0.0850007 or C0D D 1.06
PROBLEM 6.5 A venturi meter with a 50 mm throat is used to measure a flow of slightly salt water in a pipe of inside diameter 100 mm. The meter is checked by adding 20 cm3 /s of normal sodium chloride solution above the meter and analysing a sample of water downstream from the meter. Before addition of the salt, 1000 cm3 of water requires 10 cm3 of 0.1 M silver nitrate solution in a titration. 1000 cm3 of the downstream sample required 23.5 cm3 of 0.1 M silver nitrate. If a mercury-under-water manometer connected to the meter gives a reading of 221 mm, what is the discharge coefficient of the meter? Assume that the density of the liquid is not appreciably affected by the salt.
Solution If the flow of the solution is x m3 /s, then a mass balance in terms of sodium chloride gives: 0005x ð 0.05850007 C 000520 ð 1000046 ð 58.50007 D 0.1375000520 ð 1000046 C x0007 x D 0.0148 m3 /s
and:
or, assuming the density of the solution is 1000 kg/m3 , the mass flowrate is: 00050.0148 ð 10000007 D 14.8 kg/s For the venturi meter, the area of the throat is given by: A1 D 00050016/40007000550/100000072 D 0.00196 m2 and the area of the pipe is: A2 D 00050016/400070005100/100000072 D 0.00785 m2 From equations 6.32 and 6.33: 0003 C0 D A1 A2 / 0005A21 0004 A22 0007 D 0.00204 m2 h D 221 mm Hgunder-water D 0.221000513500 0004 10000007/1000 D 2.78 m water 0001 and hence: 14.8 D 0005CD ð 1000 ð 0.002040007 00052 ð 9.81 ð 2.780007 and:
CD D 0.982
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 6.6 A gas cylinder containing 30 m3 of air at 6 MN/m2 pressure discharges to the atmosphere through a valve which may be taken as equivalent to a sharp edged orifice of 6 mm diameter (coefficient of discharge D 0.6). Plot the rate of discharge against the pressure in the cylinder. How long will it take for the pressure in the cylinder to fall to (a) 1 MN/m2 , and (b) 150 kN/m2 ? Assume an adiabatic expansion of the gas through the valve and that the contents of the cylinder remain at 273 K.
Solution Area of orifice D 00050016/4000700050.00600072 D 2.828 ð 1000045 m2 . The critical pressure ratio wc is: (equation 4.43) wc D [2/0005k C 10007]k/0005k000410007 Taking k D 001a D 1.4 for air, wc D 0.527. Thus sonic velocity will occur until the cylinder pressure falls to a pressure of: P2 D 0005101.3/0.5270007 D 192.2 kN/m2 . For pressures in excess of 192.2 kN/m2 , the rate of discharge is given by: 0003 G D CD A0 0005kP1 /v1 000700052/0005k C 1000700070005kC10007/0005k000410007 (equation 6.29) p If k D 1.4, G D 1.162 ð 1000045 0005P1 /v1 0007 If Pa and va are atmospheric pressure and the specific volume at atmospheric pressure respectively, Pa va D P1 v1 and v1 D Pa va /P1 Pa D 101,300 N/m2 and va D 000522.4/290007 D 0.773 m3 /kg ∴
and:
v1 D 0005101,300 ð 0.773/P1 0007 D 000578,246/P1 0007 0003 G D 1.162 ð 1000045 0005P12 /78,2460007 D 4.15 ð 1000048 P1 kg/s
If P1 is expressed in MN/m2 , then: G D 0.0415 P1 kg/s. For pressures lower than 192.2 kN/m2 : G2 D 0005A0 CD /v2 00072 2P1 v1 0005k/k 0004 10007[1 0004 0005P2 /P1 00070005k000410007/k ]
(equation 6.26)
3
v2 D va D 0.773 m /kg
P2 D Pa D 101,300 N/m2 v1 D Pa va /P1
Substituting gives:
2
G D 2.64 ð 1000044 [1 0004 0005Pa /P1 00070.286 ]
Thus a table of G as a function of pressure may be produced as follows:
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FLOW AND PRESSURE MEASUREMENT
P < 192.2 kN/m2
P > 192.2 kN/m2
P (MN/m2 )
G (kg/s)
P (MN/m2 )
G (kg/s)
0.1013 0.110 0.125 0.150 0.175
0 0.0024 0.0039 0.0053 0.0062
0.2 0.5 1.0 2.0 6.0
0.0083 0.0208 0.0416 0.0830 0.249
These data are plotted in Fig. 6b, from which discharge rate is seen to be linear until the cylinder pressure falls to 0.125 MN/m2 . 0.25
G (kg /s)
0.20
0.15 0.010
0.10
G (kg /s)
Low pressure range 0.005
0.05 0
0.1 0.2
0.3 0.4 0.5
P1 (MN/m2) 0
1
2
3
4
5
6
P1 (MN/m2)
Figure 6b.
If m is the mass of air in the cylinder at any pressure P1 over the linear part of the curve, G D dm/dt D 0.0415P1 . ∴
dt D dm/0.0415P1 m D 000529/22.400070005P1 /0.10130007 ð 30 D 383.4P1 kg
∴
and
dt D 383.4dm/0.0415m D 9240d m/m t D 9240 ln0005m1 /m2 0007
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
At 6 MN/m2 and 1 MN/m2 , the masses of air in the cylinder are 2308 and 383.4 kg respectively. ∴
The time for the pressure to fall to 1 MN/m2 D 9240 ln00052308/3834.40007 D 16,600 s 00054.61 h0007
As 0.15 MN/m2 is still within the linear region, the time for the pressure to fall to this value is 34,100 s .00059.47 h0007
PROBLEM 6.7 2
Air, at 1500 kN/m and 370 K, flows through an orifice of 30 mm2 to atmospheric pressure. If the coefficient of discharge is 0.65, the critical pressure ratio 0.527, and the ratio of the specific heats is 1.4, calculate the mass flowrate.
Solution If the critical pressure ratio wc is 0.527 (from Problem 6.6), sonic velocity will occur until the pressure falls to 0005101.3/0.5270007 D 192.2 kN/m2 . For pressures above this value, the mass flowrate is given by: 0001 G D CD A0 0005kP1 /v1 0007[2/0005k C 10007]0005kC10007/0005k000410007 (equation 6.29) 0001 p If k D 1.4, G D CD A0 00051.4P1 /v1 000700052/2.400072.4/0.4 D CD A0 00050.468P1 /v1 0007 P1 D 1,500,000 N/m2 v1 D 000522.4/2900070005370/27300070005101.3/15000007 D 0.0707 m3 /kg p Substituting gives: G D 00050.65 ð 30 ð 1000046 0007 00050.486 ð 1,500,000/0.07070007 D 0.061 kg/s
and:
PROBLEM 6.8 Water flows through an orifice of 25 mm diameter situated in a 75 mm pipe at the rate of 300 cm3 /s. What will be the difference in level on a water manometer connected across the meter? Viscosity of water is 1 mN s/m2 .
Solution See Volume 1, Example 6.1.
PROBLEM 6.9 Water flowing at 1.5 l/s in a 50 mm diameter pipe is metered by means of a simple orifice of diameter 25 mm. If the coefficient of discharge of the meter is 0.62, what will
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be the reading on a mercury-under-water manometer connected to the meter? What is the Reynolds number for the flow in the pipe? Density of water D 1000 kg/m3 . Viscosity of water D 1 mN s/m2 .
Solution Mass flowrate, G D 00051500 ð 1000046 ð 10000007 D 1.5 kg/s. Area of orifice, A0 D 00050016/4000700050.02500072 D 0.00049 m2 . Area of pipe, A1 D 00050016/4000700050.05000072 D 0.00196 m2 . Reynolds number D 0011ud/
D d0005G/A1 0007/ D 0.0500051.5/0.001960007/00051 ð 1000043 0007 D 3.83 ð 104
0001 The orifice meter equations are 6.19 and 6.21; the latter being used when [1 0004 0005A0 /A1 00072 ] approaches unity. 0001 0001 [1 0004 0005A0 /A1 00072 ] D [1 0004 0005252 /502 00072 ] D 0.968 Thus: p Using equation 6.21, G D CD A0 0011 00052gh0007 gives: 0001 1.5 D 0.62 ð 0.00049 ð 1000 00052 ð 9.81h0007, and h D 1.24 m of water
Using equation 6.19 in terms of h gives: 0001 1.5 D 00050.62 ð 0.00049 ð 1000/0.9680007 00052gh0007 and h D 1.16 m of water
This latter value of h should be used. The height of a mercury-under-water manometer would then be 1.16/0005000513.55 0004 1.000007/1.000007 D 0.092 m or 92 mm Hg.
PROBLEM 6.10 What size of orifice would give a pressure difference of 0.3 m water gauge for the flow of a petroleum product of density 900 kg/m3 at 0.05 m3 /s in a 150 mm diameter pipe?
Solution As in previous problems, equations 6.19 and 6.21 may be used to calculate the flow through an orifice. In this problem the size of the orifice is to be found so that the simpler equation will be used in the first instance. 0001 G D CD A0 0011 00052gh0007 (equation 6.21) G D 00050.05 ð 9000007 D 45.0 kg/s 0011 D 900 kg/m3
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h D 0.3 m of water or 00050.3/0.90007 D 0.333 m of petroleum product CD D 0.62 (assumed) p 45.0 D 00050.62 ð A0 ð 9000007 00052 ð 9.81 ð 0.3330007
∴
A0 D 0.3155 m2 and d0 D 0.2 m.
Thus:
This orifice diameter is larger than the pipe size so that it was clearly wrong to use the simpler equation. 0003 Thus: G D CD A0 0011 [2gh/00051 0004 0005A0 /A1 00072 0007] (equation 6.19) A1 D 00050016/4000700050.1500072 D 0.0177 m2 0003 45.0 D 00050.62 ð A0 ð 9000007 [2 ð 9.81 ð 0.33/00051 0004 0005A0 /0.017700072 0007]
∴
A0 D 0.154 m2 and d0 D 0.14 m
Thus:
PROBLEM 6.11 The flow of water through a 50 mm pipe is measured by means of an orifice meter with a 40 mm aperture. The pressure drop recorded is 150 mm on a mercury-underwater manometer and the coefficient of discharge of the meter is 0.6. What is the Reynolds number in the pipe and what would the pressure drop over a 30 m length of the pipe be expected to be? Friction factor, ! D R/0011u2 D 0.0025. Density of mercury D 13,600 kg/m3 . Viscosity of water D 1 mN s/m2 . What type of pump would be used, how would it be driven and what material of construction would be suitable?
Solution Area of pipe, A1 D 00050016/4000700050.0500072 D 0.00197 m2 . Area of orifice, A0 D 00050016/4000700050.0400072 D 0.00126 m2 . h D 150 mmHg under water D 0.15 ð 000513600 0004 10000007/1000 1.88 m of water. 1 0004 0005A0 /A00072 D 0.591, and hence: 0003 G D CD A0 0011 [2gh/00051 0004 0005A0 /A00072 0007] (equation 6.19) 0001 D 00050.6 ð 0.00126 ð 10000007 2 ð 9.81 ð 1.88/0.591 D 5.97 kg/s Reynolds number, 0011ud/ D d0005G/A1 0007/ D 0.0500056.22/0.001970007/00051 ð 1000043 0007 D 1.52 ð 105 The pressure drop is given by: 0004P/v D 40005R/0011u2 00070005l/d00070005G/A00072 D 400050.00250007000530/0.05000700055.97/0.0019700072 D 5.74 ð 107 kg2 /m4 s2
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0004P D 5.74 ð 107 ð 00051/10000007 D 00055.74 ð 104 0007 N/m2 or 57.4 kN/m2 Power required D head loss (m) ð G ð g D 00055.74 ð 104 /1000 ð 9.81000700055.97 ð 9.810007 D 343 W For a pump efficiency of 60%, the actual power requirement D 0005343/0.60007 D 571 W. Water velocity D 5.97/00050.00197 ð 10000007 D 3.03 m/s. For this low-power requirement at a low head and comparatively low flowrate, a centrifugal pump, electrically driven and made of stainless steel, would be suitable.
PROBLEM 6.12 A rotameter has a tube 0.3 m long which has an internal diameter of 25 mm at the top and 20 mm at the bottom. The diameter of the float is 20 mm, its effective density is 4800 kg/m3 , and its volume 6.6 cm3 . If the coefficient of discharge is 0.72, at what height will the float be when metering water at 100 cm3 /s?
Solution See Volume 1, Example 6.4.
PROBLEM 6.13 Explain why there is a critical pressure ratio across a nozzle at which, for a given upstream pressure, the flowrate is a maximum. Obtain an expression for the maximum flow for a given upstream pressure for isentropic flow through a horizontal nozzle. Show that for air (ratio of specific heats, 001a D 1.4) the critical pressure ratio is 0.53 and calculate the maximum flow through an orifice of area 30 mm2 and coefficient of discharge 0.65 when the upstream pressure is 1.5 MN/m2 and the upstream temperature 293 K. Kilogram molecular volume D 22.4 m3 .
Solution The reasons for critical pressure ratios are discussed in Section 4.4.1. The maximum rate of discharge is given by: 0003 (equation 6.29) Gmax D CD A0 0005kP1 /v1 000700052/0005k C 1000700070005kC10007/0005k000410007 For an isentropic process, k D 001a D 1.4 for air. The critical pressure ratio,
wc D 00052/k C 10007k/0005k000410007
(equation 4.430007
Substituting for k D 001a D 1.4, wc D 00052/2.400071.4/0.4 D 0.523
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The maximum rate of discharge is given by equation 6.29. P1 D 1.5 ð 106 N/m2 A0 D 30 ð 1000046 m2 k D 1.4 and CD D 0.65 At P1 D 1.5 MN/m2 and T1 D 293 K, the specific volume v1 is: v1 D 000522.4/2900070005293/273000700050.1013/1.50007 D 0.056 m3 /kg 0001 Substituting, Gmax D 0.65 ð 30 ð 1000046 00051.4 ð 1.5 ð 106 /0.056000700052/2.400072.4/0.4 D 0.069 kg/s
PROBLEM 6.14 A gas cylinder containing air discharges to atmosphere through a valve whose characteristics may be considered similar to those of a sharp-edged orifice. If the pressure in the cylinder is initially 350 kN/m2 , by how much will the pressure have fallen when the flowrate has decreased to one-quarter of its initial value? The flow through the valve may be taken as isentropic and the expansion in the cylinder as isothermal. The ratio of the specific heats at constant pressure and constant volume is 1.4.
Solution From equation 4.43: the critical pressure ratio, wc D [2/0005k C 10007]k/0005k000410007 D 00052/2.400071.4/0.4 D 0.528 If the cylinder is discharging to atmospheric pressure, sonic velocity will occur until the cylinder pressure has fallen to 0005101.3/0.5280007 D 192 kN/m2 The maximum discharge when the cylinder pressure exceeds 192 kN/m2 is given by: 0004 0005 00060005kC10007/0005k000410007 kP1 2 Gmax D CD A0 (equation 6.29) v1 0005k C 10007 If Pa and va are the pressure and specific volume at atmospheric pressure, then:
and:
Gmax
1/v1 D P1 /Pa va 0004 0005 00060005kC10007/0005k000410007 kP12 2 D C D A0 Pa va k C 1 0003 D CD A0 P1 [0005k/Pa va 000700052/k C 10007]0005kC10007/0005k000410007
If G350 and G192 are the rates of discharge at 350 and 192 kN/m2 respectively, then: G350 /G192 D 0005350/1920007 D 1.82 or:
G192 D 0.55G350
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For pressures below 192 kN/m2 : 0007 0005 0006
0005 00060005k000410007/k k P2 C D A0 2P1 v1 10004 GD v2 k00041 P1
(equation 6.26)
Substituting for 1/v1 D P1 /Pa va and v2 D va gives: 0007 0005 0006
0005 00060005k000410007/k k P2 C D A0 GD 2Pa va 10004 va k00041 P1 and:
G2 D 0005CD A0 /va 00072 2Pa va [k/0005k 0004 10007][1 0004 0005P2 /P1 00070005k000410007/k ] D 0005CD A0 /va 00072 2Pa va ð 3.5[1 0004 0005P2 /P1 00070.286 ]
When P1 D 192 kN/m2 , G192 D 0.55G350 , P2 , atmospheric pressure, 101.3 kN/m2 and: 00050.55G350 00072 D 0005CD A0 /va 00072 2Pa va ð 3.5[1 0004 0005101.3/19200070.286 ] When the final pressure P1 is reached, the flowrate is 0.25G350 . ∴
00050.25G350 00072 D 0005CD A0 /va 00072 2Pa va ð 3.500051 0004 0005101.3/P1 00070.286 0007
Dividing these two equations gives: 0005 0006 0.55 2 1 0004 0005101.3/19200070.286 D 0.25 1 0004 0005101.3/P1 00070.286 and:
P1 D 102.3 kN/m2
PROBLEM 6.15 Water discharges from the bottom outlet of an open tank 1.5 m by 1 m in cross-section. The outlet is equivalent to an orifice 40 mm diameter with a coefficient of discharge of 0.6. The water level in the tank is regulated by a float valve on the feed supply which shuts off completely when the height of water above the bottom of the tank is 1 m and which gives a flowrate which is directly proportional to the distance of the water surface below this maximum level. When the depth of water in the tank is 0.5 m the inflow and outflow are directly balanced. As a result of a short interruption in the supply, the water level in the tank falls to 0.25 m above the bottom but is then restored again. How long will it take the level to rise to 0.45 m above the bottom?
Solution The mass flowrate G is related to the head h for the flow through an orifice when the area of the orifice is small in comparison with the area of the pipe by: 0001 G D CD A0 0011 00052gh0007 (equation 6.21)
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If h is the distance of the water level below the maximum depth of 1 m, then the head above the orifice is equal to 00051 0004 h0007 and: 0001 G D CD A0 0011 [2g00051 0004 h0007] When the tank contains 0.5 m of water, the flowrate is given by: 0001 G D 00050.6 ð 00050016/4000700050.0400072 ð 1000 00052 ð 9.81 ð 0.500070007 D 2.36 kg/s The input to the tank is stated to be proportional to h, and when the tank is half full the inflow is equal to the outflow, or:
2.36 D 0005K ð 0.50007 and K D 4.72 kg/ms
p p Thus the inflow D 4.72h kg/s and the outflow D CD A0 0011 2g 00051 0004 h0007 kg/s. p p p The net rate of filling D 4.72h 0004 CD A0 0011 2g 00051 0004 h0007 D 4.72h 0004 3.34 00051 0004 h0007 Time to fill the tank D (mass of water/rate of filling) D 1 ð 1.5 ð 00050.45 0004 0.250007 ð 1000/rate D 300/rate The time to fill from 0.25 to 0.45 m above the bottom of the tank is then: 0002 0.75 300dh p time D 0.55 4.72h 0004 3.34 00051 0004 h0007 This integral is most easily solved graphically as shown in Fig. 6c, where the area under the curve D 0.233 s/m and the time D 0005300 ð 0.2330007 D 70 s.
3.0
1/(4.72−3.34√ 1−h)
2.5
2.0
1.5
Area under curve = 0.233 s/m
1.0
0.5
0 0.55
0.55
0.65 h (m)
0.70
0.70
Figure 6c.
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PROBLEM 6.16 The flowrate of air at 298 K in a 0.3 m diameter duct is measured with a pitot tube which is used to traverse the cross-section. Readings of the differential pressure recorded on a water manometer are taken with the pitot tube at ten different positions in the crosssection. These positions are so chosen as to be the mid-points of ten concentric annuli each of the same cross-sectional area. The readings are: Position Manometer reading (mm water) Position Manometer reading (mm water)
1 18.5 6 14.7
2 18.0 7 13.7
3 17.5 8 12.7
4 16.8 9 11.4
5 15.7 10 10.2
The flow is also metered using a 150 mm orifice plate across which the pressure differential is 50 mm on a mercury-under-water manometer. What is the coefficient of discharge of the orifice meter?
Solution Cross-sectional area of duct D 00050016/4000700050.300072 D 0.0707 m2 . Area of each concentric annulus D 0.00707 m2 . If the diameters of the annuli are designated d1 , d2 etc., then: 0.00707 D 00050016/4000700050.32 0004 d21 0007 0.00707 D 00050016/400070005d2 0004 d22 0007 0.00707 D 00050016/400070005d22 0004 d23 0007 and so on, and the mid-points of each annulus may be calculated across the duct. For a pitot tube, the velocity may be calculated from the head h as u D For position 1, h D 18.5 mm of water.
p
00052gh0007
The density of the air D 000529/22.400070005273/2980007 D 1.186 kg/m3 .
and:
h D 000518.5 ð 1000043 ð 1000/1.1860007 D 15.6 m of air 0001 u D 00052 ð 9.81 ð 15.60007 D 17.49 m/s
In the same way, the velocity distribution across the tube may be found as shown in the following table. Mass flowrate, G D 00051.107 ð 1.1860007 D 1.313 kg/s For the orifice, [1 0004 0005A0 /A1 00072 ] D [1 0004 00050.15/0.300072 ] D 0.938 h D 50 mm Hg-under-water D 00050.05 ð 000513.55 0004 10007 ð 1000/1.1860007 D 529 m of air p and: 1.313 D CD 00050016/4000700050.1500072 ð 1.186 00052 ð 9.81 ð 529/0.9380007 and CD D 0.61
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Position
1 2 3 4 5 6 7 8 9 10
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Distance from axis of duct (mm)
Manometer reading Water (mm)
Air (m)
24 57 75 89 101 111 121 130 139 147
18.5 18.0 17.5 16.8 15.7 14.7 13.7 12.7 11.4 10.2
15.6 15.17 14.75 14.16 13.23 12.39 11.55 10.71 9.61 8.60
Air velocity (u m/s)
Velocity ð area of annulus 0005m3 /s0007
17.5 17.3 17.0 16.7 16.1 15.6 15.1 14.5 13.7 13.0
0.124 0.122 0.120 0.118 0.114 0.110 0.107 0.103 0.097 0.092 Total D 1.107
The velocity profile across the duct is plotted in Fig. 6d. Centre line of duct
Duct wall
20
Velocity (m/s)
15
10
5
0
20
40
60
80 100 120 140
Distance from duct axis (mm)
Figure 6d.
PROBLEM 6.17 Explain the principle of operation of the pitot tube and indicate how it can be used in order to measure the total flowrate of fluid in a duct. If a pitot tube is inserted in a circular
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cross-section pipe in which a fluid is in streamline flow, calculate at what point in the cross-section it should be situated so as to give a direct reading representative of the mean velocity of flow of the fluid.
Solution The principle of operation of a pitot tube is discussed in Section 6.3.1. It should be emphasised that the pitot tube measures the point velocity of a flowing fluid and not the average velocity so that in order to find the average velocity, a traverse across the duct is necessary. Treatment of typical results is illustrated in Problem 6.16. The point velocity is p given by u D 00052gh0007 where h is the difference of head expressed in terms of the flowing fluid. For streamline flow, the velocity distribution is discussed in Section 3.3.4 and: us /uCL D 1 0004 0005s2 /r 2 0007
(equation 3.32)
where us and uCL are the point velocities at a distance s from the wall and at the axis respectively and r is the radius of the pipe. The average velocity is: uav D umax /2
(equation 3.36)
When us D uav D umax /2, us /umax D 0005umax /20007/umax D 1 0004 0005s2 /r 2 0007 and:
0.5 D s2 /r 2 from which s D 0.707 r
PROBLEM 6.18 The flowrate of a fluid in a pipe is measured using a pitot tube, which gives a pressure differential equivalent to 40 mm of water when situated at the centre line of the pipe and 22.5 mm of water when midway between the axis and the wall. Show that these readings are consistent with streamline flow in the pipe.
Solution For streamline flow in a pipe, a force balance gives: 0004P0016r 2 D 0004 and:
0004u D
du 20016rl dr
0004P r 2 0004P r and 0004 u D C constant. 2 l 2 l 2
When r D a (at the wall), u D 0, the constant D 0004Pa2 /4 l and:
uD0004
P 2 0005a 0004 r 2 0007 4 l
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The maximum velocity, umax D
0004Pa2 4 l u
and:
umax
D10004
r 2
a
When r D a/2, u/umax D 0.75. The pitot tube is discussed in Section 6.3.1 and: p uDk h
(from equation 6.10)
At the centre-line, u D umax and h D 40 mm. p ∴ umax D K 40 D 6.32 K At a point midway between the axis and the wall, u D u1/2 and h D 22.5 mm. p ∴ u1/2 D K 22.5 D 4.74 K u1/2 /umax D 00054.74 K/632 K0007 D 0.75 and hence the flow is streamline.
PROBLEM 6.19 Derive a relationship between the pressure difference recorded by a pitot tube and the velocity of flow of an incompressible fluid. A pitot tube is to be situated in a large circular duct in which fluid is in turbulent flow so that it gives a direct reading of the mean velocity in the duct. At what radius in the duct should it be located, if the radius of the duct is r? The point velocity in the duct can be assumed to be proportional to the one-seventh power of the distance from the wall.
Solution An energy balance for an incompressible fluid in turbulent flow is given by: u2 /2 C gz C vP C F D 0
(equation 2.55)
Ignoring functional losses and assuming the pitot tube to be horizontal, 0005u22 0004 u12 0007/2 D 0004v0005P2 0004 P1 0007 If the fluid is brought to rest of plane 2, then:
and:
0004u12 /2 D 0004v0005P2 0004 P1 0007 0001 p u1 D 2v0005P2 0004 P1 0007 D 2gh
(equation 6.10)
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If the duct radius is r, the velocity uy at a distance y from the wall (and s from the centreline) is given by the one-seventh power law as: y 1/7 uy D us (equation 3.59) r where us is the velocity at the centreline. The flow, dQ, through an annulus of thickness dy1 distance y from the axis is: y 1/7 dQ D 20016sdyus r Multiplying and dividing through by r 2 gives: s y 1/7 y d dQ D 20016r 2 us r r r
y 1/7 y y or, since s D 0005r 0004 y0007: D 20016r 2 us 1 0004 d r r r 0002 1 000e 1/7 8/7 000f y y y The total flow is: Q D 20016r 2 us 0004 d r r r 0 000e 000f 7 y 8/7 7 y 15/7 1 D 20016r 2 us 0004 D 0.8170016r 2 us 8 r 15 r 0 The average velocity, uav D Q/0016r 2 D 0.817us uy D uav , 0.817us D us 0005y/r00071/7
Thus: ∴
0005y/r0007 D 0.243 and s/r D 0.757
PROBLEM 6.20 A gas of molecular weight 44 kg/kmol, temperature 373 K and pressure 202.6 kN/m2 is flowing in a duct. A pitot tube is located at the centre of the duct and is connected to a differential manometer containing water. If the differential reading is 38.1 mm water, what is the velocity at the centre of the duct? The volume occupied by 1 kmol at 273 K and 101.3 kN/m2 is 22.4 m3 .
Solution As shown in section 6.2.5, for a pitot tube: u12 /2 C P1 v D u22 /2 C P2 v u2 D 0, and hence, u1 D
p 20005P2 0004 P1 0007v
Difference in head D 38.1 mm of water ∴ P2 0004 P1 D 0005000538.1/10000007 ð 1000 ð 9.810007 D 373.8 N/m2
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The specific volume, v D 000522.4/4400070005373/27300070005101.3/202.60007 D 0.348 m3 /kg p ∴ u1 D 2 ð 373.8 ð 0.348 D 16.1 m/s
PROBLEM 6.21 Glycerol, of density 1260 kg/m3 and viscosity 50 mNs/m2 , is flowing through a 50 mm pipe and the flowrate is measured using an orifice meter with a 38 mm orifice. The pressure differential is 150 mm as indicated on a manometer filled with a liquid of the same density as the glycerol. There is reason to suppose that the orifice meter may have become partially blocked and that the meter is giving an erroneous reading. A check is therefore made by inserting a pitot tube at the centre of the pipe. It gives a reading of 100 mm on a water manometer. What does this suggest?
Solution From the reading taken from the pitot tube, the velocity in the pipe, and hence the mass flowrate, can be calculated. From the orifice meter, the mass flowrate can also be calculated and compared with the accurate value. 0001 For the pitot tube, u D 2gh (equation 6.10) where u D umax at the pipe axis, and the head loss h is in m of the liquid flowing. Now: ∴
h D 0005100/10000007 ð 00051000/12600007 D 0.0794 m of glycerol p umax D 2 ð 9.81 ð 0.0794 D 1.25 m/s
Reynolds number D 00051260 ð 1.25 ð 0.05/0.050007 D 1575 ∴
uav D 0.5umax D 0.63 m/s
(equation 3.36)
Mass flowrate D 00050.63 ð 1260 ð 00050016/40007 ð 0.052 0007 D 1.56 kg/s For the orifice meter: 0004 A0 2v0005P1 0004 P2 0007 (equation 6.190007 mass flowrate, G D CD v 1 0004 0005A0 /A1 00072 0004 2gh D C D A0 0011 1 0004 0005d0 /d00074 0004 2 ð 9.81 ð 0005150/10000007 2 D 0005CD ð 00050016/40007 ð 0.038 ð 12600007 D 2.99CD 1 0004 00050.038/0.0500074 ∴ CD D 00051.56/2.990007 D 0.53 which confirms that the meter is faulty.
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PROBLEM 6.22 The flowrate of air in a 305 mm diameter duct is measured with a pitot tube which is used to traverse the cross-section. Readings of the differential pressure recorded on a water manometer are taken with the pitot tube at ten different positions in the cross-section. These positions are so chosen as to be the mid-points of ten concentric annuli each of the same cross-sectional area. The readings are as follows: Position Manometer reading (mm water) Position Manometer reading (mm water)
1 18.5 6 14.7
2 18.0 7 13.7
3 17.5 8 12.7
4 16.8 9 11.4
5 15.8 10 10.2
The flow is also metered using a 50 mm orifice plate across which the pressure differential is 150 mm on a mercury-under-water manometer. What is the coefficient of discharge of the orifice meter?
Solution For a pitot tube, the velocity at any point in the duct is: 0001 u D 2gh
(equation 6.10)
where h is the manometer reading in m of the fluid which flows in the duct. ∴
h D 0005reading in mm water/10000007 ð 00050011w /0011air 0007 m
The total volumetric air flowrate is given by: Q D 0005area of duct ð average velocity0007 00100001 2gh D area ð 00051/100007 00100001 0001 manometer reading D 00050016/40007 ð 0.3052 ð 0.1 ð 00050011w /0011a 0007/1000 ð p p p p 0001 0001 D 0.00102 00050011w /0011a 00070005 18.5 C 18.0 C 17.5 C Ð Ð Ð 10.20007 D 0.0394 00050011w /0011a 0007 For the orifice meter, the volumetric flowrate is given by: 0001 A0 2gh Q D CD 0001 1 0004 0005A0 /A1 00072 A0 D 00050016/40007 ð 0.152 D 0.0177 m2 , A1 D 00050016/40007 ð 0.3052 D 0.0731 m2 , h D 50 mm Hg under water
∴
∴
D 00050.05000513.6 0004 1.00007/1.00007 D 0.63 m of water D 0.6300050011w /0011a 0007 m of air. 0001 0.0177 Q D CD ð 0001 2 ð 9.81 ð 0.6300050011w /0011a 0007 1 0004 00050.0177/0.073100072 0001 D 0.066 00050011w /0011a 0007 ð CD p p 0.0394 00050011w /0011a 0007 D 0.066 00050011w /0011a 0007 ð CD and CD D 0.60
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PROBLEM 6.23 The flow of liquid in a 25 mm diameter pipe is metered with an orifice meter in which the orifice has a diameter of 19 mm. The aperture becomes partially blocked with dirt from the liquid. What fraction of the area can become blocked before the error in flowrate at a given pressure differential exceeds 15 per cent? Assume that the coefficient of discharge of the meter remains constant when calculated on the basis of the actual free area of the orifice.
Solution If two sections in the pipe are chosen, 1 being upstream and 2 at the orifice, then from an energy balance: u12 /2 C P1 v D u22 /2 C P2 v
(from equation 2.55)
and G, the mass flowrate D u2 A2 /v D u1 A1 /v ∴
or:
0005u22 /200070005A2 /A1 00072 C P1 v D u22 /2 C P2 v u22 D
20005P1 0004 P2 0007v 1 0004 0005A2 /A1 00072
The volumetric flowrate, Q D CD A2 u2 ∴
Q2 D C2D A22 ð
20005P1 0004 P2 0007v 1 0004 0005A2 /A1 00072
D 2C2D 0005P1 0004 P2 0007vA21 A22 /0005A21 0004 A22 0007 or:
Q D K0003
A1 A2
000510007
A21 0004 A22
If the area of the orifice is reduced by partial blocking, the new orifice area D rA2 where f is the fraction available for flow. The new flowrate D 0.85 Q when the error is 15 per cent and: KA1 fA2 0.85Q D 0003 A21 0004 f2 A22
000520007
A1 D 00050016/40007 ð 252 D 491 mm2 A2 D 00050016/40007 ð 192 D 284 mm2 ∴ Dividing equation (2) by equation (1) and substituting gives: 0001 f 00054912 0004 2842 0007 0.85 D 0001 00054912 0004 r 2 2842 0007
from which f D 0.89 or 11 per cent of the area is blocked.
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99
PROBLEM 6.24 Water is flowing through a 100 mm diameter pipe and its flowrate is metered by means of a 50 mm diameter orifice across which the pressure drop is 13.8 kN/m2 . A second stream, flowing through a 75 mm diameter pipe, is also metered using a 50 mm diameter orifice across which the pressure differential is 150 mm measured on a mercury-underwater manometer. The two streams join and flow through a 150 mm diameter pipe. What would you expect the reading to be on a mercury-under-water manometer connected across a 75 mm diameter orifice plate inserted in this pipe? The coefficients of discharge for all the orifice meters are equal. Density of mercury D 13600 kg/m3 .
Solution As in Problem 6.23: u22
p 20005P1 0004 P2 0007v 20005P1 0004 P2 0007v D and Q D CD A2 2 1 0004 0005A1 /A2 0007 1 0004 0005A2 /A1 00072
For pipe 1, A2 D 00050016/40007 ð 0.052 D 0.00196 m2 , A1 D 00050016/40007 ð 0.102 D 0.00785 m2 , 0005P1 0004 P2 0007 D 13,800 N/m2
∴
For pipe 2,
v D 00051/10000007 D 0.001 m3 /kg 0004 2 ð 13,800 ð 0.001 Q1 D CD ð 0.00196 D 0.011CD 1 0004 00050.00196/0.0078900072 p p 20005P1 0004 P2 0007v D 2gh (equation 6.10)
A2 D 0.00196 m2 , A1 D 00050016/40007 ð 0.0752 D 0.0044 m2 Head loss, h D 150 mm Hg-under-water or 0005150/10000007 ð [000513,600 0004 10000007/1000] D 1.89 m water. 0004 2 ð 9.81 ð 1.89 ∴ Q2 D CD ð 0.00196 D 0.0133CD 1 0004 00050.00196/0.004400072 Total flow in pipe 3, Q3 D 0005Q1 C Q2 0007 D 00050.011CD C 0.0133CD 0007 D 0.0243CD For pipe 3, A2 D 00050016/40007 ð 0.0752 D 0.0044 m2 , A1 D 00050016/40007 ð 0.152 D 0.0176 m2 0004 p 2 ð 9.81 ð h and: Q3 D CD ð 0.0044 D 0.020CD h 2 00051 0004 00050.0044/0.01760007 0007 p ∴ 0.0243CD D 0.020CD h and: or
h D 1.476 m of water 00051.476 ð 10000007/0005000513600 0004 10000007/10000007 D 117 mm of Hg-under-water.
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PROBLEM 6.25 Water is flowing through a 150 mm diameter pipe and its flowrate is measured by means of a 50 mm diameter orifice, across which the pressure differential is 2.27 ð 104 N/m2 . The coefficient of discharge of the orifice meter is independently checked by means of a pitot tube which, when situated at the axis of the pipe, gave a reading of 100 mm on a mercury-under-water manometer. On the assumption that the flow in the pipe is turbulent and that the velocity distribution over the cross-section is given by the Prandtl one-seventh power law, calculate the coefficient of discharge of the orifice meter.
Solution For the pitot tube: uD
0001
2gh
(equation 6.10)
where h is the manometer reading in m of the same fluid which flows in the pipe. ∴
h D 0005100/10000007 ð 0005000513.6 0004 1.00007/1.00007 D 1.26 m of water p The velocity at the pipe axis, u D 00052 ð 9.81 ð 1.260007 D 4.97 m/s For turbulent flow, the Prandtl one-seventh power law can be used to give: uav D 0.82 ð uaxis ∴
(equation 3.60)
uav D 0.82 ð 4.97 D 4.08 m/s
For the orifice meter, the average velocity is: 0004 20005P1 0004 P2 0007v uD 1 0004 0005A2 /A1 00072 A2 D 00050016/40007 ð 0.052 D 0.00196 m2 , A1 D 00050016/40007 ð 0.152 D 0.0177 m2 , v D 0.001 m3 /kg ∴
uav D 6.78 m/s
The coefficient of discharge D 0005uav from pitot0007/0005uav from orifice meter0007. D 00054.08/6.780007 D 0.60
PROBLEM 6.26 Air at 323 K and 152 kN/m2 flows through a duct of circular cross-section, diameter 0.5 m. In order to measure the flowrate of air, the velocity profile across a diameter of the duct is measured using a pitot-static tube connected to a water manometer inclined at an angle of cos00041 0.1 to the vertical. The following results are obtained:
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Distance from duct centre-line (m)
Manometer Reading hm (mm)
0 0.05 0.10 0.15 0.175 0.20 0.225
104 100 96 86 79 68 50
101
Calculate the mass flowrate of air through the duct, the average velocity, the ratio of the average to the maximum velocity and the Reynolds number. Comment on these results. Discuss the application of this method of measuring gas flowrates, with particular emphasis on the best distribution of experimental points across the duct and on the accuracy of the results. Take the viscosity of air as 1.9 ð 1000042 mN s/m2 and the molecular weight of air as 29 kg/kmol.
Solution If hm is the manometer reading, the vertical manometer height will be 0.1hm (mm of water). For a pitot tube, the velocity at any point is: 0001 u D 2gh (equation 6.10) where h is the manometer reading in terms of the fluid flowing in the duct. Thus:
h D 00050.1hm /10000007 ð 00050011w /0011air 0007 0011air D 000529/22.400070005152/101.300070005273/3230007 D 1.64 kg/m3
∴
and:
h D 00050.1hm /1000000700051000/1.640007 D 0.061hm 0001 0001 u D 2 ð 9.81 ð 0.061hm D 1.09 hm (m/s)
If the duct is divided into a series of elements with the measured radius at the centre-line of the element, the velocity of the element can be found from the previous equation and the volumetric flowrate calculated. By adopting this procedure across the whole section, the required values may be determined. For example, at 0.05 m, where hm D 10 mm, Inner radius of element D 0.025 m Outer radius of element D 0.075 m Area of element ∴
D 001600050.0752 0004 0.0252 0007 D 0.0157 m2 0001 p u D 00051.09 hm 0007 D 1.09 100 D 10.9 m/s
Volumetric flowrate in the element D 000510.9 ð 0.01570007 D 0.171 m3 /s
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The following table is constructed in the same way. Distance Outer Inner from duct radius of radius of centre line element element (m) (m) (m) 0 0.05 0.10 0.15 0.175 0.20 0.225 0.25
0.025 0.075 0.125 0.1625 0.1875 0.2125 0.2375 0.25
Area of element 0005m00072
0 0.025 0.075 0.125 0.1625 0.1875 0.2125 0.2375
0.00196 0.0157 0.0314 0.0339 0.0275 0.0314 0.0353 0.0192 0011
hm Velocity (mm) u (m/s) 104 100 96 86 79 68 50 0
D 0.1964 m2
Volumetric flowrate Q m3 /s
11.1 10.9 10.7 10.1 9.7 8.9 7.7 0
0.0218 0.171 0.336 0.342 0.293 0.279 0.272 0 0011
D 1.715 m3 /s
Average velocity D 00051.715/0.19640007 D 8.73 m/s Mass flowrate D 00051.715 ð 1.640007 D 2.81 kg/s uav /umax D 00058.73/11.10007 D 0.79 Re D 00058.73 ð 1.64 ð 0.050007/00051.9 ð 1000045 0007 D 3.77 ð 104 The velocity distribution in turbulent flow is discussed in Section 3.3.6 where the Prandtl one-seventh power law is used to give: uav D 0.82umax
(equation 3.63)
This is close to that measured in this duct though strictly it only appears at very high values of Re. Reference to Fig. 3.14 shows that, at Re D 3.8 ð 104 , the velocity ratio is about 0.80 which shows remarkably good agreement.
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SECTION 7
Liquid Mixing PROBLEM 7.1 A reaction is to be carried out in an agitated vessel. Pilot plant experiments were performed under fully turbulent conditions in a tank 0.6 m in diameter, fitted with baffles and provided with a flat-bladed turbine. It was found that satisfactory mixing was obtained at a rotor speed of 4 Hz, when the power consumption was 0.15 kW and the Reynolds number was 160,000. What should be the rotor speed in order to retain the same mixing performance if the linear scale of the equipment is increased 6 times? What will be the power consumption and the Reynolds number?
Solution See Volume 1, Example 7.3.
PROBLEM 7.2 A three-bladed propeller is used to mix a fluid in the laminar region. The stirrer is 0.3 m in diameter and is rotated at 1.5 Hz. Due to corrosion, the propeller has to be replaced by a flat two-bladed paddle, 0.75 m in diameter. If the same motor is used, at what speed should the paddle rotate?
Solution For mixing in the laminar region, the power requirement is: P D k 0 N2 D3
(equation 7.17)
where k 0 D 1964 for a propeller and 1748 for a flat paddle. Thus, for a propeller 0.3 m in diameter rotating at 1.5 Hz: P D 00041964 ð 1.52 ð 0.32 0006 D 119.3 W and for a paddle, 0.75 m in diameter using the same motor: 119.3 D 00041748N2 ð 0.753 0006 and N D 0.403 Hz (24 rpm) 103
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PROBLEM 7.3 Compare the capital and operating costs of a three-bladed propeller with those of a constant speed six-bladed turbine, both constructed from mild steel. The impeller diameters are 0.3 and 0.45 m respectively and both stirrers are driven by a 1 kW motor. What is the recommended speed of rotation in each case? Assume operation for 8000 h/year, power costs of £0.01/kWh and interest and depreciation at 15%/year.
Solution The capital cost of an impeller, C D FM CB Pn where FM is a factor for the material of construction which for mild steel D 1.0, CB is a base cost 0004£0006, P is the power (kW) and n is an index (0004). For the propeller: CB D 960£000419900006 and n D 0.34 ∴
C D 00041.0 ð 960 ð 10.34 0006 D £960
Interest and depreciation D 0004960 ð 15/1000006 D £144/year Operating costs D 00041 ð 0.01 ð 80000006 D £80/year
0001
a total of £224/year
For the turbine: CB D 3160£000419900006 and n D 0.10 ∴
C D 00041.0 ð 3160 ð 10.10 0006 D £3160
Interest and depreciation D 00043160 ð 15/1000006 D £474/year Operating costs D 00041 ð 0.01 ð 80000006 D £80/year
0001
a total of £554/year.
In equation 7.13, k 0 D 165 for a propeller and 3245 for a turbine. For the propeller: P D 165N3 D5 or:
1000 D 165N3 0.35 and N D 13.5 Hz (810 rpm)
For the turbine: P D 3245N2 D5 or:
1000 D 00043245N3 0.455 0006 and N D 2.54 Hz (152 rpm)
PROBLEM 7.4 In a leaching operation, the rate at which solute goes into solution is given by: dM/dt D k0004cs 0004 c0006 kg/s where M kg is the amount of solute dissolving in t s, k 0004m3 /s0006 is a constant and cs and c are the saturation and bulk concentrations of the solute respectively in kg/m3 . In a pilot test on a vessel 1 m3 in volume, 75% saturation was attained in 10 s. If 300 kg of a solid
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LIQUID MIXING
containing 28% by mass of a water soluble solid is agitated with 100 m3 of water, how long will it take for all the solute to dissolve assuming conditions are the same as in the pilot unit? Water is saturated with the solute at a concentration of 2.5 kg/m3 .
Solution The mass of solute M, dissolving in t s is: dM/dt D k0004cs 0004 c0006 kg/s
(i)
For a batch of solution, V m3 in volume: dM D Vdc and substituting for dM in (i): Integrating:
V dc/dt D k0004cs 0004 c0006/V (ii)
ln00040004cs 0004 c0 0006/0004cs 0004 c00060006 D kt/V
where c0 is the concentration of the solute when t D 0. For pure water, c0 D 0 kg/m3 when t D 0 and hence equation (ii) becomes: c D cs 00041 0004 e0004kt/V 0006 kg/m3
(iii)
For the pilot test, the batch volume, V D 1 m3 , and cs D 2.5 kg/m3 at saturation. When t D 10 s, 75% saturation is achieved or: c D 00042.5 ð 75/1000006 D 1.875 kg/m3 Therefore, in equation (iii): 1.875 D 2.500041 0004 e000410k/1 0006 and k D 0.138 For the full-scale unit, the batch volume, V D 100 m3 . Mass of solute present D 0004300 ð 28/1000006 D 84 kg and c D 000484/1000006 D 0.84 kg/m3 . Therefore, in equation (iii): 0.84 D 2.500041 0004 e00040.138t/100 0006 and t D 297 s
PROBLEM 7.5 For producing an oil-water emulsion, two portable three-bladed propeller mixers are available; a 0.5 m diameter impeller rotating at 1 Hz and a 0.35 m impeller rotating at 2 Hz. Assuming turbulent conditions prevail, which unit will have the lower power consumption?
Solution Under turbulent conditions, the power requirements for mixing are given by: P D kN3 D5
(equation 7.13)
In this case: P1 D 0004k13 ð 0.55 0006 D 0.03125k and P2 D 0004k23 ð 0.355 0006 D 0.0420k ∴
P1 /P2 D 00040.03125k/0.0420k0006 D 0.743
Thus the 0.5 m diameter impeller will have the lower power consumption; some 75% of that of the 0.35 m diameter impeller.
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PROBLEM 7.6 A reaction is to be carried out in an agitated vessel. Pilot-plant experiments were performed under fully turbulent conditions in a tank 0.6 m in diameter, fitted with baffles and provided with a flat-bladed turbine. It was found that satisfactory mixing was obtained at a rotor speed of 4 Hz, when the power consumption was 0.15 kW and the Reynolds number 160,000. What should be the rotor speed in order to retain the same mixing performance if the linear scale of the equipment is increased 6 times? What will be the power consumption and the Reynolds number?
Solution See Volume 1, Example 7.3.
PROBLEM 7.7 Tests on a small scale tank 0.3 m diameter (Rushton impeller, diameter 0.1 m) have shown that a blending process between two miscible liquids (aqueous solutions, properties approximately the same as water, i.e. viscosity 1 mN s/m2 , density 1000 kg/m3 ) is satisfactorily completed after 1 minute using an impeller speed of 250 rev/min. It is decided to scale up the process to a tank of 2.5 m diameter using the criterion of constant tip-speed. (a) What speed should be chosen for the larger impeller? (b) What power will be required? (c) What will be the blend time in the large tank?
Solution a) In the small scale tank, the 0.1 m diameter impeller is rotated at 250 rev/min or: 0004250/600006 D 4.17 Hz. The tip speed is then: 0013DN D 00040013 ð 0.1 ð 4.170006 D 1.31 m/s If this is the same in the large scale tank, where D D 00042.5/30006 D 0.83 m, then: 1.31 D 00040013 ð 0.83 ð N0006 from which the speed of rotation to the larger impeller, N D 0.346 Hz or 20.8 rev/min b) In the large scale tank: and 0015 D 1 ð 1000043 Ns/m2 . Thus,
N D 0.346 Hz,
D D 0.83 m,
0014 D 1000 kg/m3
Re D D2 N0014/0015 D 00040.832 ð 0.346 ð 10000006/00041 ð 1000043 0006 D 238,360.
From Fig. 7.6, for a propeller mixer, the Power number, Np D 0.6. Thus:
0.6 D P/0014N3 D5
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and:
107
P D 0.60014N3 D5 D 00040.6 ð 1000 ð 0.3463 ð 0.835 0006 D 9.8 W
c) In the smaller tank: Re D D2 N0014/0015 D 00040.12 ð 4.17 ð 10000006/00041 ð 1000043 0006 D 41700 In Equation 7.22, the dimensionless mixing time is: 0018m D Ntm D kRe or for tm D 60s: 00044.17 ð 100006 D k ð 41700 and:
k D 0.0060
Thus in the larger tank: Ntm D 0.0060 Re or: and:
0.346 tm D 00040.0060 ð 238,3600006 tm D 4140 s or 1.15 min
PROBLEM 7.8 An agitated tank with a standard Rushton impeller is required to disperse gas in a solution of properties similar to those of water. The tank will be 3 m diameter (1 m diameter impeller). A power level of 0.8 kW/m3 is chosen. Assuming fully turbulent conditions and that the presence of the gas does not significantly affect the relation between the Power and Reynolds numbers: (a) What power will be required by the impeller? (b) At what speed should the impeller be driven? (c) If a small pilot scale tank 0.3 m diameter is to be constructed to test the process, at what speed should the impeller be driven?
Solution (a) Assuming that the depth of liquid D tank diameter, then: volume of liquid D 00040013D2 /40006 H D 00040013 ð 32 ð 30006/4 D 21.2 m3 With a power input of 0.8 kW/m3 , the power required be the impeller is: P D 00040.8 ð 21.20006 D 17.0 kW (b) For fully turbulent conditions and 0015 D 1 mN s/m2 , and the power number, from Fig. 7.6 is approximately 0.7. On this basis: P/0014N3 D5 D 0.7 or: from which:
000417.0 ð 103 0006/00041000N3 ð 15 0006 D 0.7 N D 2.90 Hz or 173 rev/ min
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(c) For the large tank, from Equation 7.13: P D kN3 D5 or:
000417.0 ð 103 0006 D k ð 2.903 ð 15
from which:
k D 697
Thus, for the smaller tank, assuming power/unit volume is constant: volume of fluid D 00040013/400060.32 ð 0.3 D 0.021 m3 and:
power supplied, P D 00040.021 ð 0.8 ð 103 0006 D 17 W
Thus, for the smaller tank: 17 D 697 N3 ð 0.15 and:
N D 13.5 Hz or 807 rev/ min .
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SECTION 8
Pumping of Fluids PROBLEM 8.1 A three-stage compressor is required to compress air from 140 kN/m2 and 283 K to 4000 kN/m2 . Calculate the ideal intermediate pressures, the work required per kilogram of gas, and the isothermal efficiency of the process. It may be assumed that the compression is adiabatic and interstage cooling is provided to cool the air to the initial temperature. Show qualitatively, by means of temperature–entropy diagrams, the effect of unequal work distribution and imperfect intercooling, on the performance of the compressor.
Solution It is shown in Section 8.3.4 that the work done is a minimum when the intermediate pressures Pi1 and Pi2 are related to the initial and final pressures P1 and P2 by: Pi1 /P1 D Pi2 /Pi1 D P2 /Pi2
(equation 8.45)
P1 D 140 kN/m2 and P2 D 4000 kN/m2 . ∴
P2 /P1 D 28.57
∴
Pi2 /Pi1 D P2 /Pi2 D
p 3
28.57 D 3.057,
2
Pi1 D 428 kN/m , and:
Pi2 D 1308 kN/m2
The specific volume of the air at the inlet is: v1 D 000622.4/2900070006283/27300070006101.3/1400007 D 0.579 m3 /kg
Hence, for 1 kg of air, the minimum work of compression in a compressor of n stages is: 0004 0001 0002 00030001 00020006 000310007/n
P2 W D nP1 v1 00031 (equation 8.46)
00031 P1 Thus: W D 00063 ð 140,000 ð 0.579000700061.4/0.40007[000628.5700070.4/3ð1.4 0003 1] D 319,170 J/kg The isothermal work of compression is: Wiso D P1 V1 ln0006P2 /P1 0007
(equation 8.36)
D 0006140,000 ð 0.579 ln 28.570007 D 271,740 J/kg The isothermal efficiency D 0006100 ð 271,7400007/319,170 D 85.1% 109
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Compression cycles are shown in Figs 8a and 8b. The former indicates the effect of various values of n in PVn D constant and it is seen that the work done is the area under the temperature–entropy curve. Figure 8b illustrates the three-stage compressor of this problem. The final temperature T2 , found from T2 /T1 D 0006P2 /P1 00070006 000310007/ , is 390 K. The dotted lines illustrate the effect of imperfect interstage cooling.
Adiabatic temperature
Rise Temperature, T
P2
n>γ P1
T2
n=γ n<γ n=1
T1
Line of constant pressure Work done where PV n = constant n<γ 0
Entropy, S
Figure 8a.
4000 kN/m2
1308 428
Indicates imperfect cooling Temperature (K)
140 390
Delivery for imperfect cooling
283 Inlet Delivery Work done stage 3
0
Work done stage 2
Work done stage 1
Entropy (kJ/kg K)
Figure 8b.
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PROBLEM 8.2 A twin-cylinder, single-acting compressor, working at 5 Hz, delivers air at 515 kN/m2 at the rate of 0.2 m3 /s. If the diameter of the cylinder is 20 cm, the cylinder clearance ratio 5%, and the temperature of the inlet air 283 K, calculate the length of stroke of the piston and the delivery temperature.
Solution For adiabatic conditions, PV D constant P2 /P1 D 0006T2 /T1 0007 /0006 000310007 or T2 D T1 0006P2 /P1 00070006 000310007/
and:
Thus the delivery temperature D 2830006515/101.300070.4/1.4 D 500 K The volume handled per cylinder D 00060.2/20007 D 0.1 m3 /s. Volume per stroke D 00060.1/50007 D 0.02 m3 /s at 515 kN/m2 . Volume at the inlet condition D 00060.02 ð 2830007/500 D 0.0126 m3 /s. From equation 8.42, 0.0126 D Vs [1 C c 0003 c0006P2 /P1 00071/ ] where c is the clearance and Vs the swept volume. Thus: ∴
0.0126 D Vs [1 ð 0.05 0003 0.050006515/101.300071/1.4 ] and Vs D 0.0142 m3 0006000f/4000700060.200072 ð stroke D 0.0142 and the stroke D 0.45 m
PROBLEM 8.3 A single-stage double-acting compressor running at 3 Hz is used to compress air from 110 kN/m2 and 282 K to 1150 kN/m2 . If the internal diameter of the cylinder is 20 cm, the length of stroke 25 cm, and the piston clearance 5%, calculate: (a) the maximum capacity of the machine, referred to air at the initial temperature and pressure, and (b) the theoretical power requirements under isentropic conditions.
Solution The volume per stroke D 00062 ð 0006000f/4000700060.200072 ð 0.250007 D 0.0157 m3 The compression ratio D 00061150/1100007 D 10.45. The swept volume Vs is given by: 0.0157 D Vs [1 C 0.05 0003 0.05000610.4500071/1.4 ] and Vs D 0.0217 m3
(equation 8.42)
The work of compression/cycle is: W D P1 0006V1 0003 V4 00070006 / 0003 10007[0006P2 /P1 00070006 000310007/ 0003 1]
(equation 8.41)
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and substituting for 0006V1 0003 V4 0007, gives: W D P1 Vs [1 C c 0003 c0006P2 /P1 00071/ ][ / 0003 10007][0006P2 /P1 00070006 000310007/ 0003 1] 0.286
D 0006110,000 ð 0.0157000700061.4/0.40007[000610.450007
(equation 8.43)
0003 1] D 5781 J
The theoretical power requirement D 00063 ð 57810007 D 17,340 W or 17.3 kW The capacity D 00063 ð 0.01570007 D 0.047 m3 /s
PROBLEM 8.4 Methane is to be compressed from atmospheric pressure to 30 MN/m2 in four stages. Calculate the ideal intermediate pressures and the work required per kilogram of gas. Assume compression to be isentropic and the gas to behave as an ideal gas. Indicate on a temperature–entropy diagram the effect of imperfect intercooling on the work done at each stage.
Solution The ideal intermediate pressures are obtained when the compression ratios in each stage are equal. If the initial, intermediate, and final pressures from this compressor are P1 , P2 , P3 , P4 , and P5 , then: P2 /P1 D P3 /P2 D P4 /P3 D P5 /P4 D P5 /P1 as in problem 8.1. P5 /P1 D 000630,000/101.30007 D 296.2 and: Hence:
0006P5 /P1 00070.25 D 4.148 P2 D 4.148P1 D 00064.148 ð 101.30007 D 420 kN/m2 P3 D 4.148P2 D 1.74 MN/m2 P4 D 4.148P3 D 7.23 MN/m2
The work required per kilogram of gas is: 00030001 0002 0004
P5 0006 000310007/n
00031 W D nP1 V1
00031 P1
(equation 8.46)
For methane, the molecular mass D 16 kg/kmol and the specific volume at STP D 000622.4/160007 D 1.40 m3 /kg. If D 1.4, the work per kilogram is: W D 00064 ð 101,300 ð 1.40000700061.4/0.40007[0006296.200070.4/00064ð1.40007 0003 1] D 710,940 J/kg or 711 kJ/kg The effect of imperfect cooling is shown in Figs 8a and 8b.
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PROBLEM 8.5 An air-lift raises 0.01 m3 /s of water from a well 100 m deep through a 100 mm diameter pipe. The level of the water is 40 m below the surface. The air consumed is 0.1 m3 /s of free air compressed to 800 kN/m2 . Calculate the efficiency of the pump and the mean velocity of the mixture in the pipe.
Solution See Volume 1, Example 8.6
PROBLEM 8.6 In a single-stage compressor: Suction pressure D 101.3 kN/m2 . Suction temperature D 283 K. Final pressure D 380 kN/m2 . If each new charge is heated 18 deg K by contact with the clearance gases, calculate the maximum temperature attained in the cylinder.
Solution The compression ratio D 0006380/101.30007 D 3.75. On the first stroke, the air enters at 283 K and is compressed adiabatically to 380 kN/m2 . Thus:
T2 /T1 D 0006P2 /P1 00070006 000310007/
D 3.750.286 D 1.459
Hence, the exit temperature is: T2 D 00061.459 ð 2830007 D 413 K The clearance volume gases which remain in the cylinder are able to raise the temperature of the next cylinder full of air by 18 deg K leaving the cylinder and its contents at 0006283 C 180007 D 301 K. After compression, the exit temperature is: T D 0006301 ð 3.750.286 0007 D 439.2 K On each subsequent stroke, the inlet temperature is always 301 K and hence the maximum temperature attained is 439.2 K.
PROBLEM 8.7 A single-acting reciprocating pump has a cylinder diameter of 115 mm and a stroke of 230 mm. The suction line is 6 m long and 50 mm diameter and the level of the water in the suction tank is 3 m below the cylinder of the pump. What is the maximum speed at which the pump can run without an air vessel if separation is not to occur in the suction line? The piston undergoes approximately simple harmonic motion. Atmospheric pressure is equivalent to a head of 10.4 m of water and separation occurs at pressure corresponding to a head of 1.22 m of water.
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Solution The tendency for separation to occur will be greatest at the inlet to the cylinder and at the beginning of the suction stroke. If the maximum speed of the pump is N Hz, the angular velocity of the driving mechanism is 2000fN radian/s. The acceleration of the piston D 00060.5 ð 0.2300070006000fN00072 cos 00062000fN0007 m/s2 . The maximum acceleration, when t D 0, is 4.54 N2 m/s2 . Maximum acceleration of the liquid in the suction pipe is: 00060.115/0.0500072 00064.54N2 0007 D 24.02 N2 m/s Accelerating force on the liquid D 000624.02N2 0006000f/2000700060.0500072 ð 6 ð 10000007. Pressure drop in suction line due to acceleration D 000624.02N2 ð 6 ð 10000007 D 1.44 ð 105 N2 N/m2 D 00061.44 ð 105 N2 /1000 ð 9.810007 D 14.69 N2 m of water Pressure head at the cylinder when separation is about to occur: 1.22 D 000610.4 0003 3.0 0003 14.69 N2 0007 m of water and : N D 0.65 Hz
PROBLEM 8.8 An air-lift pump is used for raising 0.8 l/s of a liquid of density 1200 kg/m3 to a height of 20 m. Air is available at 450 kN/m2 . If the efficiency of the pump is 30%, calculate the power requirement, assuming isentropic compression of the air 0006 D 1.40007.
Solution Volume flow of liquid D 800 cm3 /s or 800 ð 1000036 m3 /s Mass of flowrate of liquid D 0006800 ð 1000036 ð 1200 D 0.96 kg/s Work done per second D 00060.96 ð 20 ð 9.810007 D 188.4 W Actual work of expansion of air D 0006188.4/0.30007 D 627.8 W. The mass of air required per unit time is: W D Pa va m ln0006P/Pa 0007 D Pa Va ln0006P/Pa 0007
(equation 8.49)
where Va is the volume of air at STP, Thus:
627.8 D 101,300Va ln0006450/101.30007 and Va D 0.0042 m3
The work done in the isentropic compression of this air is: P1 V1 [ /0006 0003 10007][0006P2 /P1 00070006 000310007/ 0003 1] D 0006101,300 ð 0.0042000700061.4/0.40007[0006450/101.30007
(equation 8.37) 0.286
0003 1] D 792 J
Power required D 792 J/s D 792 W or 0.79 kW.
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PROBLEM 8.9 A single-acting air compressor supplies 0.1 m3 /s of air (at STP) compressed to 380 kN/m2 from 101.3 kN/m2 pressure. If the suction temperature is 288.5 K, the stroke is 250 mm, and the speed is 4 Hz, find the cylinder diameter. Assume the cylinder clearance is 4% and compression and re-expansion are isentropic 0006 D 1.40007. What is the theoretical power required for the compression?
Solution See Volume 1, Example 8.3.
PROBLEM 8.10 Air at 290 K is compressed from 101.3 to 2000 kN/m2 pressure in a two-stage compressor operating with a mechanical efficiency of 85%. The relation between pressure and volume during the compression stroke and expansion of the clearance gas is PV1.25 D constant. The compression ratio in each of the two cylinders is the same and the interstage cooler may be taken as perfectly efficient. If the clearances in the two cylinders are 4% and 5% respectively, calculate: (a) (b) (c) (d)
the the the the
work of compression per unit mass of gas compressed; isothermal efficiency; isentropic efficiency 0006 D 1.40007; ratio of the swept volumes in the two cylinders.
Solution See Volume 1, Example 8.4.
PROBLEM 8.11 Explain briefly the significance of the “specific speed” of a centrifugal or axial-flow pump. A pump is designed to be driven at 10 Hz and to operate at a maximum efficiency when delivering 0.4 m3 /s of water against a head of 20 m. Calculate the specific speed. What type of pump does this value suggest? A pump built for these operating conditions has a measured overall efficiency of 70%. The same pump is now required to deliver water at 30 m head. At what speed should the pump be driven if it is to operate at maximum efficiency? What will be the new rate of delivery and the power required?
Solution Specific speed is discussed in Section 8.2.3 of Volume 1, where it is shown to be Ns D NQ1/2 /0006gh00073/4 . This expression is dimensionless providing that the pump speed, throughput, and head are expressed in consistent units.
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In this problem, N D 10 Hz, Q D 0.4 m3 /s, and h D 20 m. Thus:
Ns D 000610 ð 00060.400070.5 /00069.81 ð 2000070.75 0007 D 0.121
Reference should be made to specialist texts on pumps where classifications of pump types as a function of specific speed are presented. A centrifugal pump is suggested here. Q / N and Q1 /Q2 D N1 /N2 and: Thus: and:
2
(equation 8.15) 2
h / N and h1 /h2 D 0006N1 /N2 0007
(equation 8.16)
000620/300007 D 000610/N2 00072 from which N2 D 12.24 Hz 0.4/Q2 D 000610/12.240007 from which Q2 D 0.49 m3 /s
Power required D 00061/n00070006mass flow ð head ð g0007 D 00061/0.7000700060.49 ð 1000 ð 30 ð 9.810007 D 206 W
PROBLEM 8.12 A centrifugal pump is to be used to extract water from a condenser in which the vacuum is 640 mm of mercury. At the rated discharge, the net positive suction head must be at least 3 m above the cavitation vapour pressure of 710 mm mercury vacuum. If losses in the suction pipe account for a head of 1.5 m, what must be the least height of the liquid level in the condenser above the pump inlet?
Solution The system is illustrated in Fig. 8c. From an energy balance, the head at the suction point of the pump is: hi D 0006P0 /0017h0007 C x 0003 0006ui2 /2g0007 0003 hf
Condenser pressure = P0 kN/m2 Liquid level
xm
Pump Velocity = ui
Figure 8c.
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The losses in the suction pipe D 1.5 m, and 0006ui 2 /2g0007 D hf D 1.5 The net positive suction head (NPSH) is discussed in Section 8.2.3 where it is shown that: NPSH D hi 0003 0006Pv /0017g0007 where Pv is the vapour pressure of the liquid being pumped. The minimum height x is then obtained from: 3 D 0006P0 /0017g0007 C x 0003 1.5 0003 0006Pv /0017g0007 P0 D 0006760 0003 6400007 D 120 mm Hg D 16,000 N/m2 Pv D 0006760 0003 7100007 D 50 mm Hg D 6670 N/m2 0017 D 1000 kg/m3 , g D 9.81 m/s2 ∴
x D 3 C 1.5 0003 000616,000 C 66700007/00061000 ð 9.810007 D 3.55 m
PROBLEM 8.13 What is meant by the Net Positive Suction Head (NPSH) required by a pump? Explain why it exists and how it can be made as low as possible. What happens if the necessary NPSH is not provided? A centrifugal pump is to be used to circulate liquid of density 800 kg/m3 and viscosity 0.5 mN s/m2 from the reboiler of a distillation column through a vaporiser at the rate of 400 cm3 /s, and to introduce the superheated liquid above the vapour space in the reboiler which contains liquid to a depth of 0.7 m. Suggest a suitable layout if a smooth-bore 25 mm pipe is to be used. The pressure of the vapour in the reboiler is 1 kN/m2 and the NPSH required by the pump is 2 m of liquid.
Reboiler P = l kN / m2 0.7 m
h0
Vaporiser
Figure 8d.
Solution See Volume 1, Example 8.2
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PROBLEM 8.14 1250 cm3 /s of water is to be pumped through a steel pipe, 25 mm diameter and 30 m long, to a tank 12 m higher than its reservoir. Calculate the approximate power required. What type of pump would you install for the purpose and what power motor (in kW) would you provide? Viscosity of water D 1.30 mN s/m2 . Density of water D 1000 kg/m3 .
Solution For a 25 mm bore pipe, cross-sectional area D 0006000f/4000700060.02500072 D 0.00049 m2 . Viscosity, u D 00061250 ð 1000036 0007/0.00049 D 2.54 m/s. Re D 0017ud/001e D 00061000 ð 2.54 ð 0.0250007/00061.3 ð 1000033 0007 D 48,900
and
From Table 3.1 the roughness e of a steel pipe will be taken as 0.045 mm. Hence e/d D 00060.046/250007 D 0.0018. When e/d D 0.0018 and Re D 4.89 ð 104 , from Fig. 3.7 R/0017u2 D 0.0032 The pressure drop is then calculated from the energy balance equation and equation 3.19. For turbulent flow of an incompressible fluid: u2 /2 C gz C v0006P2 0003 P1 0007 C 40006R/0017u2 00070006l/d0007u2 D 0 The pressure drop is: 0006P1 0003 P2 0007 D 0017[u2 /2 C gz C 40006R/0017u2 00070006l/d0007u2 ] D 0017f[0.5 C 40006R/0017u2 00070006l/d0007]u2 C gzg since the velocity in the tank is equal to zero. Substituting: 0006P1 0003 P2 0007 D 1000f[0.5 C 400060.0032000630/0.0250007]2.542 C 00069.81 ð 120007g D 219,500 N/m2 or 219.5 kN/m2 Power D G[0006u2 /20007 C gz C F] D 0006kg/s00070006m2 /s2 0007 D 0006m2 /s00070006N/m2 0007 00033
(equation 8.60)
5
D 00061.25 ð 10 000700062.195 ð 10 0007 D 275 W If a pump efficiency of 60% is assumed, the pump motor should be rated at 0006275/0.60007 D 458 W. A single stage centrifugal pump would be suitable for this duty.
PROBLEM 8.15 Calculate the pressure drop in, and the power required to operate, a condenser consisting of 400 tubes, 4.5 m long and 10 mm internal diameter. The coefficient of contraction at the entrance of the tubes is 0.6, and 0.04 m3 /s of water is to be pumped through the condenser.
Solution Flow of water through each tube D 00060.04/4000007 D 0.0001 m3 /s. Cross-sectional area of each tube D 0006000f/4000700060.0100072 D 0.0000785 m2 .
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Water velocity D 00060.0001/0.00007850007 D 1.273 m/s. Entry pressure drop is: 0001 00022 0017u2 1 Pf D 0003 00031 2 Cc
(from equation 3.78)
D 00061000 ð 1.2732 /20007[00061/0.60007 0003 1]2 D 360 N/m2 Re D 0017ud/001e D 00061000 ð 1.273 ð 0.010007/00061.0 ð 1000033 0007 D 1.273 ð 104 If e is taken as 0.046 mm from Table 3.1, then e/d D 0.0046 and from Fig. 3.7: R/0017u2 D 0.0043. The pressure drop due to friction is: Pf D 40006R/0017u2 00070006l/d000700060017u2 0007
(equation 3.18)
D 00064 ð 0.004300064.5/0.01000700061000 ð 1.2732 00070007 D 12,540 N/m2 Total pressure drop across one tube D 000612,540 C 3600007 D 12,900 N/m2 or 12.9 kN/m2 0001 0002 0001 0002 pressure volumetric If tubes are connected in parallel, power required D ð drop flowrate D 000612,900 ð 0.040007 D 516 W
PROBLEM 8.16 75% sulphuric acid, of density 1650 kg/m3 and viscosity 8.6 mN s/m2 , is to be pumped for 0.8 km along a 50 mm internal diameter pipe at the rate of 3.0 kg/s, and then raised vertically 15 m by the pump. If the pump is electrically driven and has an efficiency of 50%, what power will be required? What type of pump would you use and of what material would you construct the pump and pipe?
Solution Cross-sectional area of pipe D 0006000f/4000700060.0500072 D 0.00196 m2 . Velocity, u D 3.0/00061650 ð 0.001960007 D 0.93 m/s. Re D 0017ud/001e D 00061650 ð 0.93 ð 0.050007/8.6 ð 1000033 D 8900 If e is taken as 0.046 mm from Table 3.1, e/d D 0.00092. From Fig. 3.7, R/0017u2 D 0.0040. Head loss due to friction is: hf D 0003Pf /0017g D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
2
D 00064 ð 0.00400070006800/0.05000700060.93 /9.810007 D 22.6 m Total head D 000622.6 C 150007 D 37.6 m.
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Power D 0006mass flowrate ð head ð g0007
(equation 8.61)
D 00063.0 ð 37.6 ð 9.810007 D 1105 W If the pump is 50% efficient, power required D 00061105/0.50007 D 2210 W or 2.2 kW. For this duty a PTFE lined pump and lead piping would be suitable.
PROBLEM 8.17 60% sulphuric acid is to be pumped at the rate of 4000 cm3 /s through a lead pipe 25 mm diameter and raised to a height of 25 m. The pipe is 30 m long and includes two right-angled bends. Calculate the theoretical power required. The density of the acid is 1531 kg/m3 and its kinematic viscosity is 4.25 ð 1000035 m2 /s. The density of water may be taken as 1000 kg/m3 .
Solution Cross-sectional area of pipe D 0006000f/4000700060.02500072 D 0.00049 m2 Velocity, u D 00064000 ð 1000036 /0.000490007 D 8.15 m/s. Re D 0017ud/001e D ud/0006001e/00170007 D 00068.15 ð 0.0250007/00064.25 ð 1000035 0007 D 4794 If e is taken as 0.05 mm from Table 3.1, e/d D 0.002 and from Fig. 3.7, R/0017u2 D 0.0047. Head loss due to friction is given by: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.00470007000630/0.025000700068.152 /9.810007 D 152.8 m and z D 25.0 m From Table 3.2, 0.8 velocity heads 0006u2 /2g0007 are lost through each 90° bend so that the loss through two bends is 1.6 velocity heads or 00061.6 ð 8.152 0007/00062 ð 9.810007 D 5.4 m. Total head loss D 0006152.8 C 25 C 5.40007 D 183.2 m. Mass flowrate D 00064000 ð 1000036 ð 1.531 ð 10000007 D 6.12 kg/s. From equation 8.61 the theoretical power requirement D 00066.12 ð 183.2 ð 9.810007 D 11,000 W or 11.0 kW.
PROBLEM 8.18 1.3 kg/s of 98% sulphuric acid is to be pumped through a 25 mm diameter pipe, 30 m long, to a tank 12 m higher than its reservoir. Calculate the power required and indicate the type of pump and material of construction of the line that you would choose. Viscosity of acid D 0.025 N s/m2 . Density D 1840 kg/m3 .
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Solution Cross-sectional area of pipe D 0006000f/4000700060.002500072 D 0.00049 m2 . Volumetric flowrate D 1.3/00061.84 ð 10000007 D 0.00071 m3 /s. Velocity in the pipe, u D 00060.00071/0.000490007 D 1.45 m/s Re D 0017ud/001e D 000600061.84 ð 10000007 ð 1.45 ð 0.0250007/0.025 D 2670 This value of the Reynolds number lies within the critical zone. If the flow were laminar, the value of R/0017u2 from Fig. 3.7 would be 0.003. If the flow were turbulent, the value of R/0017u2 would be considerably higher, and this higher value should be used in subsequent calculation to provide a margin of safety. If the roughness is taken as 0.05 mm, e/d D 00060.05/250007 D 0.002 and, from Fig 3.7, R/0017u2 D 0.0057. The head loss due to friction, hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.0057000630/0.025000700061.452 /9.810007 D 5.87 m z D 12 m so that the total head D 17.87 m. The theoretical power requirement, from equation 8.61, is: power D 000617.87 ð 1.3 ð 9.810007 D 227 W If the pump is 50% efficient, actual power D 0006227/0.50007 D 454 W A PTFE lined centrifugal pump and lead or high silicon iron pipe would be suitable for this duty.
PROBLEM 8.19 A petroleum fraction is pumped 2 km from a distillation plant to storage tanks through a mild steel pipeline, 150 mm in diameter, at the rate of 0.04 m3 /s. What is the pressure drop along the pipe and the power supplied to the pumping unit if it has an efficiency of 50%? The pump impeller is eroded and the pressure at its delivery falls to one half. By how much is the flowrate reduced? Density of the liquid D 705 kg/m3 . Viscosity of the liquid D 0.5 mN s/m2 . Roughness of pipe surface D 0.004 mm.
Solution Cross-sectional area of pipe D 0006000f/400070.152 D 0.0177 m2 . Velocity in the pipe D 00060.04/0.01770007 D 2.26 m/s. Reynolds number D 00060.705 ð 1000 ð 2.26 ð 0.150007/00060.5 ð 1000033 0007 D 4.78 ð 105 e D 0.004 mm, e/d D 00060.004/1500007 D 0.000027 and from Fig. 3.7, R/0017u2 D 0.00165 The pressure drop is: 0003Pf D 40006R/0017u2 00070006l/d000700060017u2 0007
(equation 3.18)
0003Pf D 00064 ð 0.00165000700062000/0.1500070006705 ð 2.262 0007 D 316,900 N/m2 or 320 kN/m2
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If the pump efficiency is 50%, power D 0006head ð mass flowrate ð g0007/0.5 D pressure drop (N/m2 0007 ð volumetric flowrate (m3 s0007/0.5 D 0006316,900 ð 0.040007/0.5 D 25,350 W or 25.4 W If, due to impeller erosion, the delivery pressure is halved, the new flowrate may be found from: 0006R/0017u2 0007 Re2 D 0003Pf d3 0017/4l001e2 (equation 3.23) The new pressure drop D 0006316,900/20007 D 158,450 N/m2 and: 0006R/0017u2 0007 Re2 D 0006158,450 ð 0.153 ð 7050007/00064 ð 2000 ð 0.52 ð 1000036 0007 D 1.885 ð 108 From Fig. 3.8, when 0006R/0017u2 0007 Re2 D 1.9 ð 108 and e/d D 0.000027, Re D 3.0 ð 105 and:
00063.0 ð 105 0007 D 0006705 ð 0.15 ð u0007/00060.5 ð 1000033 0007 and: u D 1.418 m/s
The volumetric flowrate is now: 00061.418 ð 0.01770007 D 0.025 m3 /s
PROBLEM 8.20 Calculate the power required to pump oil of density 850 kg/m3 and viscosity 3 mN s/m2 at 4000 cm3 /s through a 50 mm pipeline 100 m long, the outlet of which is 15 m higher than the inlet. The efficiency of the pump is 50%. What effect does the nature of the surface of the pipe have on the resistance?
Solution Cross-sectional area of pipe D 0006000f/400070.052 D 0.00196 m2 Velocity of oil in the pipe D 00064000 ð 1000036 0007/0.00196 D 2.04 m/s. Re D 0017ud/001e D 00060.85 ð 1000 ð 2.04 ð 0.050007/00063 ð 1000033 0007 D 2.89 ð 104 If the pipe roughness e is taken to be 0.05 mm, e/d D 0.001, and from Fig. 3.7, R/0017u2 D 0.0031. Head loss due to friction is: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.003100070006100/0.05000700062.042 /9.810007 D 10.5 m The total head D 000610.5 C 150007 D 25.5 m The mass flowrate D 00064000 ð 1000036 ð 8500007 D 3.4 kg/s Power required D 000625.5 ð 3.4 ð 9.81/0.50007 D 1700 W or 1.7 kW The roughness of the pipe affects the ratio e/d. The rougher the pipe surface, the higher will be e/d and there will be an increase in R/0017u2 . This will increase the head loss due to friction and will ultimately increase the power required.
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PROBLEM 8.21 600 litres/s of water at 320 K is pumped in a 40 mm i.d. pipe through a length of 150 m in a horizontal direction and up through a vertical height of 10 m. In the pipe there is a control valve which may be taken as equivalent to 200 pipe diameters and other pipe fittings equivalent to 60 pipe diameters. Also in the line there is a heat exchanger across which there is a loss in head of 1.5 m of water. If the main pipe has a roughness of 0.0002 m, what power must be delivered to the pump if the unit is 60% efficient?
Solution Mass flowrate of water D 0006600 ð 1000036 ð 10000007 D 0.6 kg/s. Cross-sectional area of pipe D 0006000f/400070.042 D 0.00126 m2 . Velocity of water in the pipe D 0006600 ð 1000036 /0.001260007 D 0.476 m/s. Re D 0017ud/001e D 00061000 ð 0.476 ð 0.040007/00061 ð 1000033 0007 D 1.9 ð 104 . If e D 0.0002 m, e/d D 0.005, and from Fig. 3.7, R/0017u2 D 0.0042. The valve and fittings are equivalent to 260 pipe diameters which is equal to 0006260 ð 0.040007 D 10.4 m of pipe. The equivalent length of pipe is therefore 0006150 C 10.40007 D 160.4 m. The head loss due to friction is: hf D 40006R/0017u2 00070006l/d00070006u2 /g0007
(equation 3.20)
D 00064 ð 0.004200070006160.4/0.04000700060.4762 /9.810007 D 1.56 m ∴
total head D 00061.56 C 1.5 C 100007 D 13.06 m.
and:
power required D 000613.06 ð 0.6 ð 9.810007/0.6 D 128 W
PROBLEM 8.22 A pump developing a pressure of 800 kN/m2 is used to pump water through a 150 mm pipe 300 m long to a reservoir 60 m higher. With the valves fully open, the flowrate obtained is 0.05 m3 /s. As a result of corrosion and scaling the effective absolute roughness of the pipe surface increases by a factor of 10. By what percentage is the flowrate reduced? Viscosity of water D 1 mN s/m2 .
Solution 800 kN/m2 is equivalent to a head of 80,000/00061000 ð 9.810007 D 81.55 m of water. If the pump is required to raise the water through a height of 60 m, then neglecting kinetic energy losses, the head loss due to friction in the pipe D 000681.55 0003 600007 D 21.55 m. The flowrate under these conditions is 0.05 m3 /s.
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The cross-sectional area of the pipe D 0006000f/400070.152 D 0.0177 m2 . Velocity of the water D 00060.05/0.01770007 D 2.82 m/s. Head loss due to friction is: hf D 80006R/0017u2 00070006l/d00070006u2 /2g0007 Thus:
21.55 D 80006R/0017u2 00070006300/0.15000700062.822 /00062 ð 9.8100070007
and:
R/0017u2 D 0.0033
(equation 3.20)
Re D 0017ud/001e D 00061000 ð 2.82 ð 0.150007/1000033 D 4.23 ð 105 From Fig. 3.7, e/d is 0.003. If, as a result of scaling and fouling, the roughness increases by a factor of 10, the new value of e/d D 0.03. Fig. 3.7 can no longer be used since the new velocity, and hence the Reynolds number, is unknown. Use is made of equation 3.23 and Fig. 3.8 to find the new velocity. The maximum head loss due to friction is still equal to 21.55 m as the pump head is unchanged. Thus:
21.55 m D 000621.55 ð 1000 ð 9.810007 D 211,410 N/m2 0006R/0017u2 0007 Re2 D 0003Pf d3 0017/4l001e2
(equation 3.23)
D 0006211,410 ð 0.153 ð 1000/4 ð 300 ð 1000036 0007 D 6.0 ð 108 From Fig. 3.8, Re D 2.95 ð 105 when e/d D 0.03. Hence the new velocity D 00062.95 ð 105 ð 1000033 0007/00061000 ð 0.150007 D 1.97 m/s Reduction in flow D 000610000062.82 0003 1.970007/2.820007 D 30.1 per cent.
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SECTION 9
Heat Transfer PROBLEM 9.1 Calculate the time taken for the distant face of a brick wall, of thermal diffusivity, DH D 0.0042 cm2 /s and thickness l D 0.45 m, initially at 290 K, to rise to 470 K if the near face is suddenly raised to a temperature of 0006 0 D 870 K and maintained at that temperature. Assume that all the heat flow is perpendicular to the faces of the wall and that the distant face is perfectly insulated.
Solution The temperature at any distance x from the near face at time t is given by: 0006D
ND1 0001
p p
00041000bN 0006 0 ferfc[ 2lN C x000b/ 2 DH t000b] C erfc[2 N C 1000bl 0004 x/ 2 DH t000b]g
ND0
(equation 9.37) and the temperature at the distant face is: 0006D
ND1 0001
p
00041000bN 0006 0 f2 erfc[ 2N C 1000bl]/ 2 DH t000bg
ND0
Choosing the temperature scale such that the initial temperature is everywhere zero, 0006/20006 0 D 470 0004 290000b/2 870 0004 290 D 0.155 p DH D 0.0042 cm2 /s or 4.2 ð 1000047 m2 /s, DH D 6.481 ð 104 Thus: 0.155 D
ND1 0001
and l D 0.45 m
00041 erfc 347 2N C 1000b/t0.5
ND0
D erfc 347t00040.5 0004 erfc 1042t00040.5 C erfc 1736t00040.5 Considering the first term only, 347t00040.5 D 1.0 and t D 1.204 ð 105 s The second and higher terms are negligible compared with the first term at this value of t and hence: t D 0.120 Ms (33.5 h)
125
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PROBLEM 9.2 Calculate the time for the distant face to reach 470 K under the same conditions as Problem 9.1, except that the distant face is not perfectly lagged but a very large thickness of material of the same thermal properties as the brickwork is stacked against it.
Solution This problem involves the conduction of heat in an infinite medium where it is required to determine the time at which a point 0.45 m from the heated face reaches 470 K. The boundary conditions are therefore: 0006 D 0,
t D 0;
0006 D 00060 , t > 0
0006 D 870 0004 290 D 580 deg K, 0006 D 0,
x D 1,
for all values of x x D 0, t > 0
t>0
0006 D 0,
x D 0, t D 0 0002 2 0003 ∂ 0006 ∂2 0006 ∂2 0006 ∂0006 D DH C C ∂t ∂x 2 ∂y 2 ∂z2 D DH
∂2 0006 ∂x 2
The Laplace transform of:
(for unidirectional heat transfer) 0004 1 0006e0004pt dt 0006 D 0006N D
(equation 9.29) (i)
0
d2 0006N p N 0006N tD0 00060004 D 2 dx DH DH p p Integrating equation (ii): 0006N D B1 ex p/DH C B2 e0004x p/DH C 0006tD0 /p p p p p d0006N and: D B1 p/DH ex p/DH 0004 B2 p/DH e0004x p/DH dx 0004 1 N 0006 0 e0004pt dt D 0006 0 /p In this case, 0006 t>0 D
and hence:
0005
and:
xD0
∂0006 ∂t
0006
0
0004
1
D t>0 xD0
0
0002
∂0006 ∂t
(ii) (iii) (iv)
0003
e0004pt dt D 0
Substituting the boundary conditions in equations (iii) and (iv): 0006N t>0 D 0006 0t>0 /p D B1 C B2 C 0006tD0 /p or xD0
and: ∴
0005
B1 C B2 D 0006 0t>0 /p
xD0
∂0006N ∂t
0006
xD0
p p D 0 D B1 p/DH e1 0004 B2 p/DH e00041 t>0 xD0
p B1 p/DH D 0 and B1 D 0,
B2 D 0006 0t>0 /p xD0
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p
p p D 0006 0 p00041 e0004k p where k D x/ DH p p The Laplace transform of p00041 e0004k p D erfc k/2 t (from Volume 1, Appendix). 0007 x 0 and: 0006 D 0006 t>0 erfc p 2 DH t xD0
0006N D B2 e0004x
From (iii),
p/DH
(v)
When x D 0.45 m, 0006 D 470 0004 290 D 180 deg K, and hence in (v), with DH D 4.2 ð 1000047 m2 /s, p
180/580 D erfcf[0.45/ 6.481 ð 1000044 ][1/ 2 t000b]g D 0.31 p ∴
0.45/6.481 ð 1000044 /2 t D 0.73 t D 2.26 ð 105 s or
and:
0.226 Ms 62.8 h000b
As an alternative method of solution, Schmidt’s method is used with the construction shown in Fig. 9a. In this case x D 0.1 m and it is seen that at x D 0.45 m, the temperature is 470 K after a time 20 t. In equation 9.43: t D 0.1000b2 / 2 ð 4.2 ð 1000047 D 1.191 ð 104 s and hence the required time, t D 20 ð 1.191 ð 104 D 2.38 ð 105 s D 0.238 Ms 66.1 h The difference here is due to inaccuracies resulting from the coarse increments of x. 900
870
800 9 7
Temperature (K)
16
700 19 17
14 12 10
5
8
3
6
15
600
13 11 20 18 16
500
4
1
9 7
14
470
19
2
12
17 10 20 18
400
16 19
14
17
12
5
15 13
8
11 6
3
9
15 13
300
290
1.5
10
1.4
1.3
1.2
1.1
9
10
11
1.0
0.9
0.8
0.7
11 9 7
0.6
10
7
4
8 5 6
0.5
0.4
0.3
0.2
0.1
0
Distance from hot face (m)
Figure 9a.
PROBLEM 9.3 Benzene vapour, at atmospheric pressure, condenses on a plane surface 2 m long and 1 m wide maintained at 300 K and inclined at an angle of 45° to the horizontal. Plot the thickness of the condensate film and the point heat transfer coefficient against distance from the top of the surface.
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Solution At 101.3 kN/m2 , benzene condenses at Ts D 353 K. With a wall temperature of Tw D 300 K, the film properties at a mean temperature of 327 K are: 0018 D 4.3 ð 1000044 N s/m2 , 0019 D 860 kg/m3 , k D 0.151 W/m K and 001a D 423 kJ/kg D 4.23 ð 105 J/kg Thus: s D f[40018k Ts 0004 Tw x]/ g sin 001c001a00192 g0.25 00044
D f[4 ð 4.3 ð 10
(equation 9.168)
ð 0.151 353 0004 300000bx]/ 9.81 sin 45° ð 4.23 ð 105 ð 8602 g0.25
D 2.82 ð 1000044 x 0.25 m Similarly: h D f 00192 g sin 001c001ak 3 /[40018 Ts 0004 Tw x]g0.25
(equation 9.169)
D f 8602 ð 9.81 sin 45° ð 4.23 ð 105 ð 0.1513 /[4 ð 4.3 ð 1000044 353 0004 300000bx]g0.25 D 535x 00040.25 W/m2 K Values of x between 0 and 2.0 m in increments of 0.20 m are now substituted in these equations with the following results, which are plotted in Fig. 9b.
Thickness of film (s mm)
750 700
0.25
650
0.20
600
0.15
550
0.10
500
0.05
Heat transfer coefficient (h W/m2 K)
800
450 0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Distance from top of surface (x m) Figure 9b.
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x (m)
x 0.25
x 00040.25
s (m)
h
W/m2 K000b
0 0.1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0 0.562 0.669 0.795 0.880 0.946 1.000 1.047 1.088 1.125 1.158 1.189
1 1.778 1.495 1.258 1.136 1.057 1.000 0.956 0.919 0.889 0.863 0.841
0 1.58 ð 1000044 1.89 ð 1000044 2.24 ð 1000044 2.48 ð 1000044 2.67 ð 1000044 2.82 ð 1000044 2.95 ð 1000044 3.07 ð 1000044 3.17 ð 1000044 3.27 ð 1000044 3.35 ð 1000044
1 951 800 673 608 566 535 512 492 476 462 450
PROBLEM 9.4 It is desired to warm 0.9 kg/s of air from 283 to 366 K by passing it through the pipes of a bank consisting of 20 rows with 20 pipes in each row. The arrangement is in-line with centre to centre spacing, in both directions, equal to twice the pipe diameter. Flue gas, entering at 700 K and leaving at 366 K, with a free flow mass velocity of 10 kg/m2 s, is passed across the outside of the pipes. Neglecting gas radiation, how long should the pipes be? For simplicity, outer and inner pipe diameters may be taken as 12 mm. Values of k and 0018, which may be used for both air and flue gases, are given below. The specific heat capacity of air and flue gases is 1.0 kJ/kg K. Temperature (K)
Thermal conductivity k(W/m K)
Viscosity 0018(mN s/m2 )
250 500 800
0.022 0.044 0.055
0.0165 0.0276 0.0367
Solution Heat load, Q D 0.9 ð 1.0 366 0004 283 D 74.7 kW Temperature driving force, 00061 D 700 0004 366 D 334 deg K, 00062 D 366 0004 283 D 83 deg K and in equation 9.9, 0006m D 334 0004 83000b/ ln 334/83 D 180 deg K
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Film coefficients
Inside: hi di /k D 0.023 dG0 /0018000b0.8 Cp 0018/k000b0.4 00042
di D 12 mm or 1.2 ð 10
(equation 9.61)
m.
The mean air temperature D 0.5 366 C 283 D 325 K and k D 0.029 W/m K. Cross-sectional area of one tube D $/4 1.2 ð 1000042 2 D 1.131 ð 1000044 m2 Area for flow D 20 ð 20000b1.131 ð 1000044 D 4.52 ð 1000042 m2 . Thus, mass velocity G D 0.9/ 4.52 ð 1000042 D 19.9 kg/m2 s. At 325 K, 0018 D 0.0198 mN s/m2 or 1.98 ð 1000045 N s/m2 Cp D 1.0 ð 103 J/kg K Thus: hi ð 1.2 ð 1000042 / 2.9 ð 1000042 D 0.023 1.2 ð 1000042 ð 19.9/1.98 ð 1000045 0.8 ð 1.0 ð 103 ð 1.98 ð 1000045 /0.029000b0.4 0.4138hi D 0.023 1.206 ð 104 0.8 0.683000b0.4
and hi D 87.85 W/m2 K
Outside: 0.6
Cp 0018/k000b0.3 ho do /k D 0.33Ch do G0 /0018000bmax
do D 12.0 mm
or
(equation 9.90)
1.2 ð 1000042 m
G0 D 10 kg/m2 s for free flow G0max D YG0 / Y 0004 do where Y, the distance between tube centres D 2do D 2.4 ð 1000042 m. ∴
G0max D 2.4 ð 1000042 ð 10.0000b/ 2.4 ð 1000042 0004 1.2 ð 1000042 D 20 kg/m2 s At a mean flue gas temperature of 0.5 700 C 366 D 533 K,
0018 D 0.0286 mN s/m2 or 2.86 ð 1000045 N s/m2 , k D 0.045 W/m K and Cp D 1.0 ð 103 J/kg K ∴
Remax D 1.2 ð 1000042 ð 20.0000b/ 2.86 ð 1000045 D 8.39 ð 103
From Table 9.3, when Remax D 8.39 ð 103 , X D 2do , and Y D 2do , Ch D 0.95. Thus: ho ð 1.2 ð 1000042 / 4.5 ð 1000042 D 0.33 ð 0.95 8.39 ð 103 0.6 ð 1.0 ð 103 ð 2.86 ð 1000045 /0.045000b0.3 or:
0.267ho D 0.314 8.39 ð 103 0.6 0.836000b0.3
and ho D 232 W/m2 K
Overall: Ignoring wall and scale resistances, then: 1/U D 1/ho C 1/hi D 0.0114 C 0.0043 D 0.0157 and:
U D 63.7 W/m2 K
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Area required
In equation 9.1, A D Q/U0006m D 74.7 ð 103 / 63.7 ð 180 D 6.52 m2 . Area/unit length of tube D $/4 12 ð 1000042 D 9.43 ð 1000043 m2 /m and hence: total length of tubing required D 6.52/ 9.43 ð 1000043 D 6.92 ð 102 m. The length of each tube is therefore D 6.92 ð 102 / 20 ð 20 D 1.73 m
PROBLEM 9.5 A cooling coil, consisting of a single length of tubing through which water is circulated, is provided in a reaction vessel, the contents of which are kept uniformly at 360 K by means of a stirrer. The inlet and outlet temperatures of the cooling water are 280 K and 320 K respectively. What would be the outlet water temperature if the length of the cooling coil were increased by 5 times? Assume the overall heat transfer coefficient to be constant over the length of the tube and independent of the water temperature.
Solution (equation 9.1)
Q D UATm
where Tm is the logarithmic mean temperature difference. For the initial conditions: Q1 D m1 ð 4.18 320 0004 280 D U1 A1 [ 360 0004 280 0004 360 0004 320000b]/ [ln 360 0004 280000b/ 360 0004 320000b] or: and:
167.2m1 D U1 A1 80 0004 40000b/ ln 80/40 D 57.7U1 A1
m1 /U1 A1 D 0.345
In the second case, m2 D m1 , U2 D U1 , and A2 D 5A1 . ∴
Q2 D m1 ð 4.18 T 0004 280 D 5U1 A1 [ 360 0004 280 0004 360 0004 T000b]/ ln 360 0004 280000b/ 360 0004 T000b
or:
4.18 m1 /U1 A1 T 0004 280000b/5 D 80 0004 360 C T000b/[ln[80/360 0004 T000b]
Substituting for m1 /U1 A1 , 0.289 T 0004 280 D T 0004 280000b/[ln 80/ 360 0004 T000b] or:
ln[80/ 360 0004 T000b] D 3.467 and
T D 357.5 K
PROBLEM 9.6 In an oil cooler, 216 kg/h of hot oil enters a thin metal pipe of diameter 25 mm. An equal mass of cooling water flows through the annular space between the pipe and a
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
larger concentric pipe; the oil and water moving in opposite directions. The oil enters at 420 K and is to be cooled to 320 K. If the water enters at 290 K, what length of pipe will be required? Take coefficients of 1.6 kW/m2 K on the oil side and 3.6 kW/m2 K on the water side and 2.0 kJ/kg K for the specific heat of the oil.
Solution Heat load
Mass flow of oil D 6.0 ð 1000042 kg/s. and hence, Q D 6.0 ð 1000042 ð 2.0 420 0004 320 D 12 kW Thus the water outlet temperature is given by: 12 D 6.0 ð 1000042 ð 4.18 T 0004 290 or T D 338 K Logarithmic mean temperature driving force
In equation 9.9: 00061 D 420 0004 338 D 82 deg K, and:
00062 D 320 0004 290 D 30 deg K
0006m D 82 0004 30000b/ ln 82/30 D 51.7 deg K
Overal coefficient
The pipe wall is thin and hence its thermal resistance may be neglected. Thus in equation 9.8: 1/U D 1/ho C 1/hi D 1/1.6 C 1/3.6000b
and U D 1.108 kW/m2 K
Area
In equation 9.1, A D Q/U0006m D 12/ 1.108 ð 51.7 D 0.210 m2 Tube diameter D 25 ð 1000043 m (assuming a mean value) area/unit length D $ ð 25 ð 1000043 ð 1.0 D 7.85 ð 1000042 m2 /m and the tube length required D 0.210/ 7.85 ð 1000042 D 2.67 m
PROBLEM 9.7 The walls of a furnace are built of a 150 mm thickness of a refractory of thermal conductivity 1.5 W/m K. The surface temperatures of the inner and outer faces of the refractory are 1400 K and 540 K respectively. If a layer of insulating material 25 mm thick of
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HEAT TRANSFER
thermal conductivity 0.3 W/m K is added, what temperatures will its surfaces attain assuming the inner surface of the furnace to remain at 1400 K? The coefficient of heat transfer from the outer surface of the insulation to the surroundings, which are at 290 K, may be taken as 4.2, 5.0, 6.1, and 7.1 W/m2 K for surface temperatures of 370, 420, 470, and 520 K respectively. What will be the reduction in heat loss?
Solution Heat flow through the refractory,
Q D kA T1 0004 T2 /x
(equation 9.12)
Thus for unit area,
Q D 1.5 ð 1.0 1400 0004 T2 / 150 ð 1000043 D 14,000 0004 10T2 W/m2
(i)
where T2 is the temperature at the refractory–insulation interface. Similarly, the heat flow through the insulation is: Q D 0.3 ð 1.0 T2 0004 T3 / 25 ð 1000043 D 12T2 0004 12T3 W/m2
(ii)
The flow of heat from the insulation surface at T3 K to the surroundings at 290 K, is: Q D hA T3 0004 290 or T3 0004 290 hW/m2
(iii)
where h is the coefficient of heat transfer from the outer surface. The solution is now made by trial and error. A value of T3 is selected and h obtained by interpolation of the given data. This is substituted in equation (iii) to give Q. T2 is then obtained from equation (ii) and a second value of Q is then obtained from equation (i). The correct value of T3 is then given when these two values of Q coincide. The working is as follows and the results are plotted in Fig. 9c. Q = 14000 − 10 T2
Q (W/m 2)
10000 8000 6000 4000
4050 W/m 2
Q = h (T3 − 290)
2000
662 K 0
300
350
400
450 500 T3 (K)
550
600
650
700
750
Figure 9c.
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T3 (K)
h (W/m2 K)
Q D h T3 0004 290 (W/m2 )
T2 D T3 C Q/12 (K)
Q D 14,000 0004 10T2 (W/m2 )
300 350 400 450 500 550 600 650 700 750
3.2 3.9 4.7 5.6 6.5 7.8 9.1 10.4 11.5 12.7
32 234 517 896 1355 2028 2821 3744 4715 5842
302.7 369.5 443.1 524.7 612.9 719.0 835.1 962.0 1092.9 1236.8
10,973 10,305 9569 8753 7871 6810 5649 4380 3071 1632
A balance is obtained when T3 D 662 K, at which Q D 4050 W/m2 . In equation (i):
4050 D 14,000 0004 10T2 or T2 D 995 K
Thus the temperatures at the inner and outer surfaces of the insulation are 995 K and 662 K respectively With no insulation, Q D 1.5 ð 1.0 1400 0004 540000b/ 150 ð 1000043 D 8600 W/m2 and hence the reduction in heat loss is 8600 0004 4050 D 4550 W/m2 or:
4540 ð 100000b/8600 D 52.9%
PROBLEM 9.8 A pipe of outer diameter 50 mm, maintained at 1100 K, is covered with 50 mm of insulation of thermal conductivity 0.17 W/m K. Would it be feasible to use a magnesia insulation, which will not stand temperatures above 615 K and has a thermal conductivity of 0.09 W/m K, for an additional layer thick enough to reduce the outer surface temperature to 370 K in surroundings at 280 K? Take the surface coefficient of heat transfer by radiation and convection as 10 W/m2 K.
Solution For convection to the surroundings
Q D hA3 T3 0004 T4 W/m where A3 is area for heat transfer per unit length of pipe, m2 /m000b. The radius of the pipe, r1 D 50/2 D 25 mm or 0.025 m. The radius of the insulation, r2 D 25 C 50 D 75 mm or 0.075 m.
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The radius of the magnesia, r3 D 75 C x D 0.075 C 0.001x m where x mm is the thickness of the magnesia. Hence the area at the surface of the magnesia, A3 D 2$ 0.075 C 0.001x m2 /m and Q D 10[2$ 0.075 C 0.001x000b] 370 C 280 D 424.1 C 5.66x W/m
(i)
For conduction through the insulation
Q D k 2$rm l T1 0004 T2 / r2 0004 r1
(equation 9.22)
where rm D r2 0004 r1 / ln r2 /r1 . ∴
Q D 0.17[2$ ð 1.0 r2 0004 r1 ] 1100 0004 T2 /[ r2 0004 r1 ln 0.075/0.025000b] D 0.972 1100 0004 T2 W/m
(ii)
For conduction through the magnesia
In equation 9.22: Q D 0.09[2$ ð 1.0 r3 0004 r2 ] T2 0004 370000b/[ r3 0004 r2 ln 0.075 C 0.001x000b/0.075] D 0.566 T2 0004 370000b/ ln 1 C 0.013x000b
(iii)
For a value of x, Q is found from (i) and hence T2 from (ii). These values are substituted in (iii) to give a second value of Q, with the following results: x (mm)
Q D 424.1 C 5.66x (W/m)
T2 D 1100 0004 1.028Q (K)
Q D 0.566 T2 0004 370000b/ ln 1 C 0.013x (W/m)
5.0 7.5 10.0 12.5 15.0 17.5 20.0
452.4 466.6 480.7 494.9 509.0 523.2 537.3
635 620 606 591 577 562 548
2380 1523 1092 832 657 531 435
From a plot of the two values of Q, a balance is attained when x D 17.5 mm. With this thickness, T2 D 560 K which is below the maximum permitted and hence the use of the magnesia would be feasible.
PROBLEM 9.9 In order to heat 0.5 kg/s of a heavy oil from 311 K to 327 K, it is passed through tubes of inside diameter 19 mm and length 1.5 m forming a bank, on the outside of which steam is condensing at 373 K. How many tubes will be needed?
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In calculating Nu, Pr, and Re, the thermal conductivity of the oil may be taken as 0.14 W/m K and the specific heat as 2.1 kJ/kg K, irrespective of temperature. The viscosity is to be taken at the mean oil temperature. Viscosity of the oil at 319 and 373 K is 154 and 19.2 mN s/m2 respectively.
Solution Heat load
Q D 0.5 ð 2.1 327 0004 311 D 16.8 kW Logarithmic mean driving force
00061 D 373 0004 311 D 62 deg K, 00062 D 373 0004 327 D 46 deg K ∴ in equation 9.9,
0006m D 62 0004 46000b/ ln 62/46 D 53.6 deg K
A preliminary estimate of the overall heat transfer coefficient may now be obtained from Table 9.18. For condensing steam, ho D 10,000 W/m2 K and for oil, hi D 250 W/m2 K (say). Thus 1/U D 1/ho C 1/hi D 0.0041, U D 244 W/m2 K and from equation 9.1, the preliminary area: A D 16.8 ð 103 / 244 ð 53.6 D 1.29 m2 The area/unit length of tube is $ ð 19.0 ð 1000043 ð 1.0 D 5.97 ð 1000042 m2 /m and:
total length of tubing D 1.29/ 5.97 ð 1000042 D 21.5 m
Thus:
number of tubes D 21.5/1.5 D 14.3, say 14 tubes
Film coefficients
The inside coefficient is controlling and hence this must be checked to ascertain if the preliminary estimate is valid. The Reynolds number, Re D di G0 /0018 D 19.0 ð 1000043 G0 /0018 At a mean oil temperature of 0.5 327 C 311 D 319 K, 0018 D 154 ð 1000043 N s/m2 . Area for flow per tube D $/4 19.0 ð 1000043 2 D 2.835 ð 1000044 m2 . ∴
total area for flow D 14 ð 2.835 ð 1000044 D 3.969 ð 1000043 m2
and hence: Thus:
G0 D 0.5/ 3.969 ð 1000043 D 1.260 ð 102 kg/m2 s Re D 19.0 ð 1000043 ð 1.260 ð 102 / 154 ð 1000043 D 15.5
That is, the flow is streamline and hence:
hi di /k 0018s /0018000b0.14 D 2.01 GCp /kl000b0.33
(equation 9.85)
At a mean wall temperature of 0.5 373 C 319 D 346 K, 0018s D 87.0 ð 1000043 N s/m2 .
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137
The mass flow, G D 0.5 kg/s. ∴
hi ð 19.0 ð 1000043 /0.14 87.0 ð 1000043 /154 ð 1000043 0.14 D 2.01
0.5 ð 2.1 ð 103 / 0.14 ð 1.5 0.33
or:
0.13hi ð 0.923 D 2.01 ð 16.6 and hi D 266 W/m2 K
This is sufficiently close to the assumed value and hence 14 tubes would be specified.
PROBLEM 9.10 A metal pipe of 12 mm outer diameter is maintained at 420 K. Calculate the rate of heat loss per metre run in surroundings uniformly at 290 K, (a) when the pipe is covered with 12 mm thickness of a material of thermal conductivity 0.35 W/mK and surface emissivity 0.95, and (b) when the thickness of the covering material is reduced to 6 mm, but the outer surface is treated so as to reduce its emissivity to 0.10. The coefficients of radiation from a perfectly black surface in surroundings at 290 K are 6.25, 8.18, and 10.68 W/m2 K at 310 K, 370 K, and 420 K respectively. The coefficients of convection may be taken as 1.22 0006/d000b0.25 W/m2 K, where 0006(K) is the temperature difference between the surface and the surrounding air and d(m) is the outer diameter.
Solution Case (a)
Assuming that the heat loss is q W/m and the surface temperature is T K, for conduction through the insulation, from equation 9.12, q D kAm 420 0004 T000b/x The mean diameter is 18 mm or 0.018 m, and hence: Am D $ ð 0.018 D 0.0566 m2 /m x D 0.012 m ∴
q D 0.35 ð 0.0566 420 0004 T000b/0.012 D 693.3 0004 1.67T000bW/m
(i)
For convection and radiation from the surface, from equation 9.119: q D hr C hc A2 T 0004 290 W/m where hr is the film coefficient equivalent to the radiation and hc the coefficient due to convection given by: hc D 1.22[ T 0004 290000b/d]0.25 where d D 36 mm or 0.036 m ∴
hc D 2.80 T 0004 290000b0.25 W/m2 K
If hb is the coefficient equivalent to radiation from a black body, hr D 0.95hb W/m2 K The outer diameter is 0.036 m and hence: A2 D $ ð 0.036 ð 1.0 D 0.1131 m2 /m ∴
q D [0.95hb C 2.80 T 0004 290000b0.25 ]0.1131 T 0004 290 D 0.1074hb T 0004 290 C 0.317 T 0004 290000b1.25 W/m
(ii)
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Values of T are now assumed and together with values of hb from the given data substituted into (i) and (ii) until equal values of q are obtained as follows: T q D 693.3 0004 1.67T hb 0.1074hb T 0004 290 0.317 T 0004 290000b1.25 q 2 (K) (W/m)
W/m K (W/m) (W/m) (W/m) 300 320 340 360 380 400
193.3 160.0 126.7 93.3 60.0 26.7
6.0 6.5 7.1 7.8 8.55 9.55
6.5 20.9 38.1 58.7 82.7 112.8
5.7 22.2 42.1 64.2 87.9 113.0
12.2 43.1 80.2 122.9 170.6 225.8
A balance is obtained when T D 350 K and q D 106 W/m.
Case (b)
For conduction through the insulation, x D 0.006 m and the mean diameter is 15 mm or 0.015 m. ∴
Am D $ ð 0.015 ð 1.0 D 0.0471 m2 /m
∴
q D 0.35 ð 0.0471 420 0004 T000b/0.006 D 1154 0004 2.75T W/m
(i)
The outer diameter is now 0.024 m and A2 D $ ð 0.024 ð 1.0 D 0.0754 m2 /m The coefficient due to convection is: hc D 1.22[ T 0004 290000b/0.024]0.25 D 3.10 T 0004 290000b0.25 W/m2 K The emissivity is 0.10 and hence hr D 0.10hb W/m2 K ∴
q D [0.10hb C 3.10 T 0004 290000b0.25 ]0.0754 T 0004 290 D 0.00754hb T 0004 290 C 0.234 T 0004 290000b1.25 W/m
(ii)
Making the calculation as before: T q D 1154 0004 2.75T hb2 0.0075hb T 0004 290 0.234 T 0004 290000b1.25 q 2 (W/m) (W/m) (W/m) (K) (W/m)
W/m K 300 320 340 360 380 400
329.0 274.0 219.0 164.0 109.0 54.0
6.0 6.5 7.1 7.8 8.55 9.55
0.5 1.5 2.7 4.2 5.8 7.9
4.2 16.4 31.1 47.4 64.9 83.4
4.7 17.9 33.8 51.6 70.7 91.3
A balance is obtained when T D 390 K and q D 81 W/m.
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PROBLEM 9.11 A condenser consists of 30 rows of parallel pipes of outer diameter 230 mm and thickness 1.3 mm with 40 pipes, each 2 m long in each row. Water, at an inlet temperature of 283 K, flows through the pipes at 1 m/s and steam at 372 K condenses on the outside of the pipes. There is a layer of scale 0.25 mm thick of thermal conductivity 2.1 W/m K on the inside of the pipes. Taking the coefficients of heat transfer on the water side as 4.0 and on the steam side as 8.5 kW/m2 K, calculate the water outlet temperature and the total mass flow of steam condensed. The latent heat of steam at 372 K is 2250 kJ/kg. The density of water is 1000 kg/m3 .
Solution Overall coefficient
1 1 1 xw xr D C C C U hi ho kw kr
(equation 9.201)
where xr and kr are the thickness and thermal conductivity of the scale respectively. Considering these in turn, hi D 4000 W/m2 K. The inside diameter, di D 230 0004 2 ð 1.3 D 227.4 mm or 0.2274m. Therefore basing the coefficient on the outside diameter: hio D 4000 ð 0.2274/0.230 D 3955W/m3 K For conduction through the wall, xw D 1.3 mm, and from Table 9.1, kw D 45 W/m K for steel and kw /xw D 45/0.0013 D 34615 W/m2 K The mean wall diameter D 0.230 C 0.2274000b/2 D 0.2287 m and hence the coefficient equivalent to the wall resistance based on the tube o.d. is: 34615 ð 0.2287/0.230 D 34419 W/m2 /K For conduction through the scale, xr D 0.25 ð 1000043 m, kr D 2.1 W/m K and hence: kr /xr D 2.1/0.25 ð 1000043 D 8400 W/m2 K The mean scale diameter D 227.4 0004 0.25 D 227.15 mm or 0.2272 m and hence the coefficient equivalent to the scale resistance based on the tube o.d. is:
8400 ð 0.2272/0.230 D 8298 W/m2 K ∴
and:
1/U D 1/3955 C 1/8500 C 1/34419 C 1/8298 D 5.201 ð 1000044 U D 1923 W/m2 K
Temperature driving force
If water leaves the unit at T K: 00061 D 372 0004 283 D 89 deg K, 00062 D 372 0004 T000b
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and in equation 9.9: 0006m D [89 0004 372 0004 T000b]/ ln[89/ 372 0004 T000b] D T 0004 283000b/ ln[89/ 372 0004 T000b] Area
For 230 mm o.d. tubes, outside area per unit length D $ ð 0.230 ð 1.0 D 0.723 m2 /m. Total length of tubes D 30 ð 40 ð 2 D 2400 m and hence heat transfer area, A D
2400 ð 0.723 D 1735.2 m2 .
Heat load
The cross-sectional area for flow/tube D $/4 0.230000b2 D 0.0416 m2 /tube. Assuming a single-pass arrangement, there are 1200 tubes per pass and hence area for flow D 1200 ð 0.0416 D 49.86 m2 . For a velocity of 1.0 m/s, the volumetric flow D 0.1 ð 49.86 m3 /s and the mass flow D 1000 ð 4.986 D 4986 kg/s. Thus the heat load, Q D 4986 ð 4.18 T 0004 283 D 20,840 T 0004 283 kW or 2.084 ð 107 T 0004 283 W. Substituting for Q, U, A, and 0006m in equation 9.1:
2.084 ð 107 T 0004 283 D 1923 ð 1735.2 T 0004 283000b/ ln[89/ 372 0004 T000b] or:
ln[89/ 372 0004 T000b] D 0.1601 and T D 296 K
The total heat load is, therefore, Q D 20,840 296 0004 283 D 2.71 ð 105 kW and the mass of steam condensed D 2.71 ð 105 /2250 D 120.4 kg/s.
PROBLEM 9.12 In an oil cooler, water flows at the rate of 360 kg/h per tube through metal tubes of outer diameter 19 mm and thickness 1.3 mm, along the outside of which oil flows in the opposite direction at the rate of 6.675 kg/s per tube. If the tubes are 2 m long and the inlettemperatures of the oil and water are 370 K and 280 K respectively, what will be the outlet oil temperature? The coefficient of heat transfer on the oil side is 1.7 kw/m2 K and on the water side 2.5 kW/m2 K and the specific heat of the oil is 1.9 kJ/kg K.
Solution In the absence of information as to the geometry of the unit, the solution will be worked on the basis of one tube — a valid approach as the number of tubes effectively appears on both sides of equation 9.1. If Tw and To are the outlet temperature of the water and the oil respectively, then:
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141
Heat load
Q D
360/3600 ð 4.18 Tw 0004 280 D 0.418 Tw 0004 280 kW for water and:
Q D
75/1000 ð 1.9 370 0004 To D 0.143 370 0004 To kW for the oil.
From these two equations, Tw D 406.5 0004 0.342To K Area
For 19.0 mm o.d. tubes, surface area D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and for one tube, surface area D 2.0 ð 0.0597 D 0.1194 m2 Temperature driving force
00061 D 370 0004 Tw ,
00062 D To 0004 280000b
and in equation 9.9: 0006m D [ 370 0004 Tw 0004 To 0004 280000b]/[ln 370 0004 Tw / To 0004 280000b] D 650 0004 Tw 0004 To / ln 370 0004 Tw / To 0004 280 Substituting for Tw : 0006m D 243.5 0004 0.658To / ln 0.342To 0004 36.5000b/ To 0004 280 K Overall coefficient
hi D 2.5 kW/m2 K di D 19.0 0004 2 ð 1.3 D 16.4 mm Therefore the inside coefficient, based on the outside diameter is: hio D 2.5 ð 16.4/19.0 D 2.16 kW/m2 K Neglecting the scale and wall resistances then: 1/U D 1/2.16 C 1/1.7 D 1.052 m2 K/kW and:
U D 0.951 kW/m2 K
Substituting in equation 9.1 gives: 0.143 370 0004 To D 0.951 ð 0.1194 243.5 0004 0.658To / ln 0.342To 0004 36.5000b/ To 0004 280 ∴
ln 0.342To 0004 36.5000b/ To 0004 280 D 0.523 and To D 324 K
PROBLEM 9.13 Waste gases flowing across the outside of a bank of pipes are being used to warm air which flows through the pipes. The bank consists of 12 rows of pipes with 20 pipes, each
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0.7 m long, per row. They are arranged in-line, with centre-to-centre spacing equal, in both directions, to one-and-a-half times the pipe diameter. Both inner and outer diameter may be taken as 12 mm. Air with a mass velocity of 8 kg/m2 s enters the pipes at 290 K. The initial gas temperature is 480 K and the total mass flow of the gases crossing the pipes is the same as the total mass flow of the air through them. Neglecting gas radiation, estimate the outlet temperature of the air. The physical constants for the waste gases, assumed the same as for air, are: Temperature (K)
Thermal conductivity (W/m K)
Viscosity (mN s/m2 )
250 310 370 420 480
0.022 0.027 0.030 0.033 0.037
0.0165 0.0189 0.0214 0.0239 0.0260
Specific heat D 1.00 kJ/kg K.
Solution Heat load
The cross area for flow per pipe D $/4 0.012000b2 D 0.000113 m2 and therefore for 12 ð 20 D 240 pipes, the total flow area D 240 ð 0.000113 D 0.027 m2 . Thus:
flow of air D 8.0 ð 0.271 D 0.217 kg/s
which is also equal to the flow of waste gas. If the outlet temperatures of the air and waste gas are Ta and Tw K respectively, then: Q D 0.217 ð 1.0 Ta 0004 290 kW or 217 Ta 0004 290 W and: from which:
Q D 0.217 ð 1.0 480 0004 Tw kW Tw D 770 0004 Ta K
Area
Surface area/unit length of pipe D $ ð 0.012 ð 1.0 D 0.0377 m2 /m. Total length of pipe D 240 ð 0.7 D 168 m and hence the heat transfer area, A D 168 ð 0.0377 D 6.34 m2 . Temperature driving force
00061 D 480 0004 Ta 00062 D Tw 0004 1290000b
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or, substituting for Tw :
00062 D 480 0004 Ta D 00061
Thus in equation 9.9:
0006m D 480 0004 Ta
Overall coefficient
The solution is now one of trial and error in that mean temperatures of both streams must be assumed in order to evaluate the physical properties. Inside the tubes: a mean temperature of 320 K, will be assumed at which, k D 0.028 W/m K, 0018 D 0.0193 ð 1000043 N s/m2 , and Cp D 1.0 ð 103 J/kg K Therefore: hi di /k D 0.023 di G/0018000b0.8 Cp 0018/k000b0.4
(equation 9.61)
hi ð 0.012/0.028 D 0.023 0.012 ð 8.0/0.0193 ð 1000043 0.8 ð 1 ð 103 ð 0.0193 ð 1000043 /0.028000b0.4 ∴
hi D 0.0537 4.974 ð 103 0.8 0.689000b0.4 D 41.94 W/m2 K
Outside the tubes: The cross-sectional area of the tube bundle D 0.7 ð 20 1.5 ð 0.012 D 0.252 m2 and hence the free flow mass velocity, G0 D 0.217/0.252 D 0.861 kg/m2 s. From Fig. 9.27: Y D 1.5 ð 0.012 D 0.018 m and therefore:
G0max D 0.861 ð 0.018000b/ 0.081 0004 0.012 D 2.583 kg/m2 s
At an assumed mean temperature of 450 K, 0018 D 0.0250 ð 1000043 N s/m2 and k D 0.035 W/m K. ∴
Remax D 0.012 ð 2.583000b/ 0.0250 ð 1000043 D 1.24 ð 103 From Table 9.3: for X D 1.5do and Y D 1.5do , Ch D 0.95. In equation 9.90:
ho 0.012/0.035 D 0.33 ð 0.95 1.24 ð 103 0.6 1.0 ð 103 ð 0.0250 ð 1000043 /0.035000b0.3 ∴
ho D 0.914 ð 71.8 ð 0.7140.3 D 59.3 W/m2 K Hence, ignoring wall and scale resistances: 1/U D 1/41.91 C 1/59.3 D 4.07 ð 1000042 U D 24.57 W/m2 K
and: Thus, in equation 9.1:
217 Ta 0004 290 D 24.57 ð 6.34 480 0004 Ta from which:
Ta D 369.4 K
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With this value, the mean air and waste gas temperatures are 330 K and 440 K respectively. These are within 10 deg K of the assumed values in each case. Such a difference would have a negligible effect on the film properties and recalculation is unnecessary.
PROBLEM 9.14 Oil is to be heated from 300 K to 344 K by passing it at 1 m/s through the pipes of a shelland-tube heat exchanger. Steam at 377 K condenses on the outside of the pipes, which have outer and inner diameters of 48 and 41 mm respectively, though due to fouling, the inside diameter has been reduced to 38 mm, and the resistance to heat transfer of the pipe wall and dirt together, based on this diameter, is 0.0009 m2 K/W. It is known from previous measurements under similar conditions that the oil side coefficients of heat transfer for a velocity of 1 m/s, based on a diameter of 38 mm, vary with the temperature of the oil as follows: Oil temperature (K) Oil side coefficient of heat transfer (W/m2 K)
300 74
311 80
322 97
333 136
344 244
The specific heat and density of the oil may be assumed constant at 1.9 kJ/kg K and 900 kg/m3 respectively and any resistance to heat transfer on the steam side may be neglected. Find the length of tube bundle required?
Solution In the absence of further data, this problem will be worked on the basis of one tube. Heat load
Cross-sectional area at the inside diameter of the scale D $/4 0.038000b2 D 0.00113 m2 . ∴
volumetric flow D 0.00113 ð 1.0 D 0.00113 m3 /s
and: ∴
mass flow D 0.00113 ð 900 D 1.021 kg/s heat load, Q D 1.021 ð 1.9 344 0004 300 D 85.33 kW
Temperature driving force
00061 D 377 0004 300 D 77 deg K, 00062 D 377 0004 344 D 33 deg K and, in equation 9.9:
0006m D 77 0004 33000b/ ln 77/33 D 52 deg K
Overall coefficient
Inside: The mean oil temperature D 0.5 344 C 300 D 322 K at which hi based on di D 0.038 m D 97 W/m2 K.
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Basing this value on the outside diameter of the pipe: hio D 97 ð 0.038/0.048 D 76.8 W/m2 K Outside: From Table 9.17, ho for condensing steam will be taken as 10,000 W/m2 K. Wall and scale: The scale resistance based on d D 0.038 m is 0.0009 m2 K/W or: k/x D 1/0.0009 D 1111.1 W/m2 K. Basing this on the tube o.d., k/x D 1111.1 ð 0.038/0.048 D 879.6 W/m2 K. 1/U D 1/hio C 1/ho C x/k
(equation 9.201)
D 0.0130 C 0.0001 C 0.00114 or:
U D 70.2 W/m2 K
Area
A D Q/U0006m D 85.33 ð 103 / 70.2 ð 52 D 23.4 m2 Area per unit length of pipe D $ ð 0.048 ð 1.0 D 0.151 m2 /m and length of tube bundle D 23.4/0.151 D 154.9 m A very large tube length is required because of the very low inside film coefficient and several passes or indeed a multistage unit would be specified. A better approach would be to increase the tube side velocity by decreasing the number of tubes in each pass, though any pressure drop limitations would have to be taken into account. The use of a smaller tube diameter might also be considered.
PROBLEM 9.15 It is proposed to construct a heat exchanger to condense 7.5 kg/s of n-hexane at a pressure of 150 kN/m2 , involving a heat load of 4.5 MW. The hexane is to reach the condenser from the top of a fractionating column at its condensing temperature of 356 K. From experience it is anticipated that the overall heat transfer coefficient will be 450 W/m2 K. Cooling water is available at 289 K. Outline the proposals that you would make for the type and size of the exchanger, and explain the details of the mechanical construction that you consider require special attention.
Solution A shell-and-tube unit is suitable with hexane on the shell side. For a heat load of 4.5 MW D 4.5 ð 103 kW, the outlet water temperature is: 4.5 ð 103 D m ð 4.18 T 0004 289000b
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In order to avoid severe scaling in such a case, the maximum allowable water temperature is 320 K and hence 310 K will be chosen as a suitable value for T. Thus:
4.5 ð 103 D 4.18m 310 0004 289 and m D 51.3 kg/s
The next stage is to estimate the heat transfer area required. (equation 9.1)
Q D UA0006m where the heat load Q D 4.5 ð 106 W U D 450 W/m2 K 00061 D 356 0004 289 D 67 deg K,
00062 D 356 0004 310 D 46 deg K
and from equation 9.9, 0006m D 67 0004 46000b/ ln 67/46 D 55.8 deg K No correction factor is necessary as the shell side fluid temperature is constant. ∴
A D 4.5 ð 106 / 450 ð 55.8 D 179.2 m2 A reasonable tube size must now be selected, say 25.4 mm, 14 BWG.
The outside surface area is therefore $ ð 0.0254 ð 1.0 D 0.0798 m2 /m and hence the total length of tubing required D 179.2/0.0798 D 2246 m. A standard tube length is now selected, say 4.87 m and hence the total number of tubes required D 2246/4.87 D 460. It now remains to decide the number of tubes per pass, and this is obtained from a consideration of the water velocity. For shell and tube units, u D 1.0 0004 1.5 m/s and a value of 1.25 m/s will be selected. The water flow, 51.3 kg/s D 51.3/1000 D 0.0513 m3 /s. The tube i.d. is 21.2 mm and hence the cross-sectional area for flow/tube D
$/4 0.0212000b2 D 0.000353 m2 . Area required to give a velocity of 1.25 m/s D 0.0513/1.25 D 0.0410 m2 and hence number of tubes/pass D 0.0410/0.000353 D 116 and number of passes D 460/116 ³ 4. As the shell side fluid is clean, triangular pitch might be suitable and 460 ð 25 mm o.d. tubes on 32 mm triangular pitch with 4 tube side passes can be accommodated in a 0.838 m i.d. shell and still allow room for impingement plates. The proposed unit will therefore consist of: 460, 25.4 mm o.d. tubes ð 14 BWG, 4.87 m long arranged in 4 tube side passes on 32 mm triangular pitch in a 0.838 m i.d. shell. The general mechanical details of the unit are described in Section 9.9.1 of Volume 1 and points of detail are: (i) impingement baffles should be fitted under each inlet nozzle; (ii) segmental baffles are not usually fitted to a condenser of this type; (iii) the unit should be installed on saddles at say 5° to the horizontal to facilitate drainage of the condensate.
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PROBLEM 9.16 A heat exchanger is to be mounted at the top of a fractionating column about 15 m high to condense 4 kg/s of n-pentane at 205 kN/m2 , corresponding to a condensing temperature of 333 K. Give an outline of the calculations you would make to obtain an approximate idea of the size and construction of the exchanger required. For purposes of standardisation, 19 mm outside diameter tubes of 1.65 mm wall thickness will be used and these may be 2.5, 3.6, or 5 m in length. The film coefficient for condensing pentane on the outside of a horizontal tube bundle may be taken as 1.1 kW/m2 K. The condensation is effected by pumping water through the tubes, the initial water temperature being 288 K. The latent heat of condensation of pentane is 335 kJ/kg. For these 19 mm tubes, a water velocity of 1 m/s corresponds to a flowrate of 0.2 kg/s of water.
Solution The calculations follow the sequence of earlier problems in that heat load, temperature driving force, and overall coefficient are obtained and hence the area evaluated. It then remains to consider the geometry of the unit bearing in mind the need to maintain a reasonable cooling water velocity. As in the previous example, the n-pentane will be passed through the shell and cooling water through the tubes. Heat load
Q D 4.0 ð 335 D 1340 kW assuming there is no sub-cooling of the condensate. As in Problem 9.15, the outlet temperature of the cooling water will be taken as 310 K, and for a flow of G kg/s: 1340 D G ð 4.18 310 0004 288 or G D 14.57 kg/s Temperature driving force
00061 D 333 0004 288 D 45 deg K, and:
00062 D 333 0004 310 D 23 deg K
0006m D 45 0004 23000b/ ln 45/23 D 32.8 deg K
Overall coefficient
Inside: For forced convection to water in tubes: hi D 4280 0.00488T 0004 1000bu0.8 /di0.2 W/m2 K
(equation 9.221)
where T, the mean water temperature D 0.5 310 C 288 D 299 K; u, the water velocity will be taken as 1 m/s — a realistic optimum value, bearing in mind the need to limit the
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pressure drop, and di D 19.0 0004 2 ð 1.65 D 15.7 mm or 0.0157 m. ∴
hi D 4280 0.00488 ð 299 0004 1000b1.00.8 /0.01570.2 D 4280 ð 0.459 ð 1.0000b/0.436 D 4506 W/m2 K
or, based on the outer diameter, hio D 4.506 ð 0.0157000b/0.019 D 3.72 kW/m2 K Wall: For steel, k D 45 W/m K and x D 0.00163 m and hence: x/k D 0.00163/45 D 0.0000362 m2 K/W or 0.0362 m2 K/kW Outside: ho D 1.1 kW/m2 K Ignoring scale resistance: 1/U D 1/ho C x/k C 1/hio D 0.9091 C 0.0362 C 0.2686 and:
U D 0.823 kW/m2 K
Area
Q D UA0006m and hence:
A D 1340/ 0.823 ð 32.8 D 49.6 m2
Outer area of 0.019 m diameter tube D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and hence total length of tubing required D 49.6/0.0597 D 830.8 m. Thus with 2.5, 3.6, and 5.0 m tubes, the number of tubes will be 332, 231 or 166. The total cooling water flow D 14.57 kg/s and for u D 1 m/s, the flow through 1 tube is 0.20 kg/s ∴
the number of tubes/pass D 14.57/0.20 D 73
Clearly 3 passes are usually to be avoided, and hence 2 or 4 are suitable, that is 146 or 292 tubes, 5.0 or 2.5 m long. The former is closer to a standard shell size and 166 ð 19 mm tubes on 25.4 mm square pitch with two tube side passes can be fitted within a 438 mm i.d. shell. In this event, the water velocity would be slightly less than 1 m/s in fact 1 ð 146/166 D 0.88 m/s, though this would not affect the overall coefficient to any significant extent. The proposed unit is therefore 166 ð 19 mm o.d. tubes on 25.4 mm square pitch 5.0 m long with a 438 mm i.d. shell. In making such calculations it is good practice to add an overload factor to the heat load, say 10%, to allow for errors in predicting film coefficients, although this is often taken into account in allowing for extra tubes within the shell. In this particular example, the fact that the unit is to be installed 15 m above ground level is of significance in limiting the pressure drop and it may be that in an actual situation space limitations would immediately specify the tube length.
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PROBLEM 9.17 An organic liquid is boiling at 340 K on the inside of a metal surface of thermal conductivity 42 W/m K and thickness 3 mm. The outside of the surface is heated by condensing steam. Assuming that the heat transfer coefficient from steam to the outer metal surface is constant at 11 kW/m2 K, irrespective of the steam temperature, find the value of the steam temperature would give a maximum rate of evaporation. The coefficients of heat transfer from the inner metal surface to the boiling liquid which depend upon the temperature difference are: Temperature difference between metal Heat transfer coefficient metal surface surface and boiling liquid (deg K) to boiling liquid kW/m2 K 22.2 27.8 33.3 36.1 38.9 41.7 44.4 50.0
4.43 5.91 7.38 7.30 6.81 6.36 5.73 4.54
Solution For a steam temperature Ts K, the heat conducted through the film of condensing steam, Q D hc A Ts 0004 T1 , or: Q D 11 ð 1.0 Ts 0004 T1 D 11.0 Ts 0004 T1 kW/m2
(i)
where T1 is the temperature at the outer surface of the metal. For conduction through the metal, Q D kA T1 0004 T2 /x D 42 ð 1000043 ð 1.0 T1 0004 T2 /0.003 D 14.0 T1 0004 T2 kW/m2
(ii)
where T2 is the temperature at the inner surface of the metal. For conduction through the boiling film: Q D hb T2 0004 340 D hb T2 0004 340 kW/m2
(iii)
where hb kW/m2 K is the film coefficient to the boiling liquid. Thus for an assumed value of T2 the temperature difference T2 0004 340 is obtained and hb from the table of data. Q is then obtained from (iii), T1 from (ii), and hence Ts from (i) as follows:
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T2 (K)
T2 0004 340 (K)
hb
kW/m2 K000b
Q
kW/m2
T1 (K)
Ts (K)
362.2 367.8 373.3 376.1 378.9 381.7 384.4 390.0
22.2 27.8 33.3 36.1 38.9 41.7 44.4 50.0
4.43 5.91 7.38 7.30 6.81 6.36 5.73 4.54
98.4 164.3 245.8 263.5 264.9 265.2 254.4 227.0
369.2 379.5 390.9 394.9 397.8 400.7 402.6 406.2
378.1 394.4 413.3 418.9 421.9 424.8 425.7 426.8
It is fairly obvious that the rate of evaporation will be highest when the heat flux is a maximum. On inspection this occurs when Ts D 425 K.
PROBLEM 9.18 It is desired to warm an oil of specific heat 2.0 kJ/kg K from 300 to 325 K by passing it through a tubular heat exchanger containing metal tubes of inner diameter 10 mm. Along the outside of the tubes flows water, inlet temperature 372 K, and outlet temperature 361 K. The overall heat transfer coefficient from water to oil, based on the inside area of the tubes, may be assumed constant at 230 W/m2 K, and 0.075 kg/s of oil is to be passed through each tube. The oil is to make two passes through the heater and the water makes one pass along the outside of the tubes. Calculate the length of the tubes required.
Solution Heat load
If the total number of tubes is n, there are n/2 tubes in one pass on the oil side, that is the oil passes through 2 tubes in traversing the exchanger. The mass flow of oil is therefore D 0.075 ð n/2 D 0.0375n kg/s and the heat load: Q D 0.0375n ð 2.0 325 0004 300 D 1.875n kW Temperature driving force
00061 D 361 0004 300 D 61 deg K, 00062 D 372 0004 325 D 47 deg K and, in equation 9.9: 0006m D 61 0004 47000b/ ln 61/47 D 53.7 deg K In equation 9.213: X D 00062 0004 00061 / T1 0004 00061 and Y D T1 0004 T2 / 00062 0004 00061 where T1 and T2 are the inlet and outlet temperatures on the shell side and 00061 and 00062 are the inlet and outlet temperatures on the tube side.
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∴
X D 325 0004 300000b/ 372 0004 300 D 0.347
and:
Y D 372 0004 361000b/ 325 0004 300 D 0.44
151
For one shell side pass, two tube side passes, Fig. 9.71 applies and F D 0.98. Area
In equation 9.212, A D Q/UF0006m D 1.875n/ 0.230 ð 0.98 ð 53.7 D 0.155n m2 . The area per unit length based on 10 mm i.d. D $ ð 0.010 ð 1.0 D 0.0314 m2 /m and total length of tubing D 0.155n/0.0314 D 4.94n m. Thus the length of tubes required D 4.94n/n D 4.94 m.
PROBLEM 9.19 A condenser consists of a number of metal pipes of outer diameter 25 mm and thickness 2.5 mm. Water, flowing at 0.6 m/s, enters the pipes at 290 K, and it should be discharged at a temperature not exceeding 310 K. If 1.25 kg/s of a hydrocarbon vapour is to be condensed at 345 K on the outside of the pipes, how long should each pipe be and how many pipes would be needed? Take the coefficient of heat transfer on the water side as 2.5, and on the vapour side as 0.8 kW/m2 K and assume that the overall coefficient of heat transfer from vapour to water, based upon these figures, is reduced 20% by the effects of the pipe walls, dirt and scale. The latent heat of the hydrocarbon vapour at 345 K is 315 kJ/kg.
Solution Heat load
For condensing the organic at 345 K, Q D 1.25 ð 315 D 393.8 kW If the water outlet temperature is limited to 310 K, then the mass flow of water is given by: 393.8 D G ð 4.18 310 0004 290 or G D 4.71 kg/s Temperature driving force
00061 D 345 0004 290 D 55 deg K, 00062 D 345 0004 310 D 35 deg K Therefore in equation 9.9, 0006m D 55 0004 35000b/ ln 55/35 D 44.3 deg K. No correction factor is necessary with isothermal conditions in the shell. Overall coefficient
Inside: hi D 2.5 kW/m2 K. The outside diameter D 0.025 m and di D 25 0004 2 ð 2.5000b/103 D 0.020 m. Basing the inside coefficient on the outer diameter: hio D 2.5 ð 0.020/0.025 D 2.0 kW/m3 K
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Outside: ho D 0.8 kW/m2 K and hence the clean overall coefficient is given by: 1/Uc D 1/hio C 1/ho D 1.75 m2 K/kW or Uc D 0.572 kW/m2 K Thus allowing for scale and the wall: UD D 0.572 100 0004 20000b/100 D 0.457 kW/m2 K Area
In equation 9.1:
A D Q/U0006m D 393.8/ 0.457 ð 44.3 D 19.45 m2
Outside area D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and hence total length of piping D 19.45/0.0785 D 247.6 m. 4.71 kg/s water 4.71/1000 D 0.00471 m3 /s and hence cross-sectional area/pass to give a velocity of 0.6 m/s D 0.00471/0.6 D 0.00785 m2 Cross-sectional area of one tube D $/ 0.020000b2 D 0.000314 m2 . Therefore number of tubes/pass D 0.00785/0.000314 D 25. Thus: with 1 tube pass, total tubes D 25 and tube length D 247.6/25 D 9.90 m with 2 tube passes, total tubes D 50 and tube length D 247.6/50 D 4.95 m with 4 tube passes, total tubes D 100 and tube length D 247.6/100 D 2.48 m A tube length of 2.48 m is perhaps the most practical proposition.
PROBLEM 9.20 An organic vapour is being condensed at 350 K on the outside of a bundle of pipes through which water flows at 0.6 m/s; its inlet temperature being 290 K. The outer and inner diameters of the pipes are 19 mm and 15 mm respectively, although a layer of scale, 0.25 mm thick and of thermal conductivity 2.0 W/m K, has formed on the inside of the pipes. If the coefficients of heat transfer on the vapour and water sides are 1.7 and 3.2 kW/m2 K respectively and it is required to condense 0.025 kg/s of vapour on each of the pipes, how long should these be, and what will be the outlet temperature of water? The latent heat of condensation is 330 kJ/kg. Neglect any resistance to heat transfer in the pipe walls.
Solution For a total of n pipes, mass flow of vapour condensed D 25n ð 1000043 kg/s and hence load, Q D 0.025n ð 330 D 8.25n kW.
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For a water outlet temperature of T K and a mass flow of G kg/s: 8.25n D G ð 4.18 T 0004 290 kW or
G D 1.974n/ T 0004 290 kg/s
(i)
00061 D 350 0004 290000b, 00062 D 350 0004 T and hence in equation 9.9: 0006m D [ 350 0004 290 0004 350 0004 T000b]/ ln[ 350 0004 290000b/ 350 0004 T000b] D T 0004 290000b/ ln[60/ 350 0004 T000b] deg K. Considering the film coefficients: hi D 3.2 kW/m2 K, ho D 1.7 kW/m2 K and hence: hio D 3.2 ð 0.015000b/0.019 D 2.526 kW/m2 K. The scale resistance is:
x/k D 0.25 ð 1000043 /2.0 D 0.000125 m2 K/W or 0.125 m2 K/kW Therefore the overall coefficient, neglecting the wall resistance is given by: 1/U D 1/hio C x/k C 1/ho D 0.5882 C 0.125 C 0.396 D 1.109 m2 K/kW or U D 0.902 kW/m2 K Therefore in equation 9.1: A D Q/U0006m D 8.25n/f0.902 T 0004 290000b/ ln[60/ 350 0004 T000b]gm2 D
4.18G T 0004 290 ln[60/ 350 0004 T000b] D 4.634G ln[60/ 350 0004 T000b]m2 0.902 T 0004 290000b
(ii)
The cross-sectional area for flow D $/4 0.015000b2 D 0.000177 m2 /tube. G kg/s G/1000 D 0.001G m3 /s and area/pass to give a velocity of 0.6 m/s D 0.001G/0.6 D 0.00167G m2 . ∴
number of tubes/pass D 0.00167G/0.000177 D 9.42G
(iii)
Area per unit length of tube D $ ð 0.019 ð 1.0 D 0.0597 m2 /m. ∴ total length of tubes D 4.634G ln[60/350 0004 T000b]/0.0597 D 77.6G ln[60/350 0004 T000b]m
length of each tube D 77.6G ln[60/350 0004 T000b]/n m and, substituting from (i), tube length D 77.6 ð 1.974n ln[60/ 350 0004 T000b]/ n T 0004 290000b] D 153.3 ln[60/ 350 0004 T000b]/ T 0004 290 m
(iv)
The procedure is now to select a number of tube passes N and hence m in terms of n from (iii). T is then obtained from (i) and hence the tube length from (iv). The following results are obtained:
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No. of tube passes N 1 2 4 6
Total tubes n
Outlet water temperature T (K)
Tube length (m)
308.6 327.2 364.4 401.6
3.05 3.99 — --
9.42G 18.84G 37.68G 56.52G
Arrangements with 4 and 6 tube side passes require water outlet temperatures in excess of the condensing temperature and are clearly not possible. With 2 tube side passes, T D 327.2 K at which severe scaling would result and hence the proposed unit would consist of one tube side pass and a tube length of 3.05 m. The outlet water temperature would be 308.6 K.
PROBLEM 9.21 A heat exchanger is required to cool continuously 20 kg/s of water from 360 K to 335 K by means of 25 kg/s of cold water, inlet temperature 300 K. Assuming that the water velocities are such as to give an overall coefficient of heat transfer of 2 kW/m2 K, assumed constant, calculate the total area of surface required (a) in a counterflow heat exchanger, i.e. one in which the hot and cold fluids flow in opposite directions, and (b) in a multipass heat exchanger, with the cold water making two passes through the tubes, and the hot water making one pass along the outside of the tubes. In case (b) assume that the hot-water flows in the same direction as the inlet cold water, and that its temperature over any cross-section is uniform.
Solution The heat load, Q D 20 ð 4.18 360 0004 335 D 2090 kW and the outlet cold water temperature is given by: 2090 D 25 ð 4.18 T2 0004 300 or T2 D 320 K Case (a)
00061 D 360 0004 320 D 40 deg K,
00062 D 335 0004 300 D 35 deg K
and in equation 9.9: 0006m D 40 0004 35000b/ ln 40/35 D 37.4 deg K As the flow is true counter-flow, no correction factor is necessary and F D 1.0. Therefore in equation 9.150: A D Q/UF0006m D 2090/ 2.0 ð 1.0 ð 37.4 D 27.94 m2
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Case (b)
Again, 0006m D 37.4 K. In equation 9.212: X D 00062 0004 00061 / T1 0004 00061 D 320 0004 300000b/ 360 0004 300 D 0.33 Y D T1 0004 T2 / 00062 0004 00061 D 360 0004 335000b/ 320 0004 300 D 1.25 Hence, from Fig. 9.71, F D 0.94 and in equation 9.212: A D 2090/ 2.0 ð 0.94 ð 374 D 29.73 m2
PROBLEM 9.22 Find the heat loss per unit area of surface through a brick wall 0.5 m thick when the inner surface is at 400 K and the outside at 310 K. The thermal conductivity of the brick may be taken as 0.7 W/m K.
Solution Q D kA T1 0004 T2 /x
(equation 9.12)
D 0.7 ð 1.0 400 0004 310000b/0.5 D 126 W/m2
PROBLEM 9.23 A furnace is constructed with 225 mm of firebrick, 120 mm of insulating brick, and 225 mm of building brick. The inside temperature is 1200 K and the outside temperature 330 K. If the thermal conductivities are 1.4, 0.2, and 0.7 W/m K, find the heat loss per unit area and the temperature at the junction of the firebrick and insulating brick.
Solution If T1 K and T2 K are the temperatures at the firebrick/insulating brick and the insulating brick/building brick junctions respectively, then in equation 9.12, for conduction through the firebrick: Q D 1.4 ð 1.0 1200 0004 T1 /0.255 D 6.22 1200 0004 T1 W/m2
(i)
For conduction through the insulating brick: Q D 0.2 ð 1.0 T1 0004 T2 /0.120 D 1.67 T1 0004 T2 W/m2
(ii)
and for conduction through the building brick: Q D 0.7 ð 1.0 T2 0004 330000b/0.225 D 3.11 T2 0004 330 W/m2
(iii)
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The thermal resistances of each material, (x/kA), are: firebrick D 1/6.22 D 0.161; insulating brick D 1/1.67 D 0.60; building brick D 1/3.11 D 0.322 K/Wm2 ; and in equation 9.18:
1200 0004 330 D 0.161 C 0.60 C 0.322000bQ or:
Q D 803.3 W/m2
T firebrick/T D x/kA000bfirebrick / x/kA ∴
1200 0004 T1 / 1200 0004 330 D 0.161/ 0.161 C 0.60 C 0.322 D 0.161/1.083 and:
T1 D 1071 K
Similarly for the insulating brick:
1071 0004 T2 / 1200 0004 330 D 0.60/1.083 and:
T2 D 589 K
PROBLEM 9.24 Calculate the total heat loss by radiation and convection from an unlagged horizontal steam pipe of 50 mm outside diameter at 415 K to air at 290 K.
Solution Outside area per unit length of pipe D $ ð 0.050 ð 1.0 D 0.157 m2 /m. Convection
For natural convection from a horizontal pipe to air, the simplified form of equation 9.102 may be used: hc D 1.18 T/do 0.25 In this case: T D 415 0004 290 D 125 deg K and do D 0.050 m. ∴
hc D 1.18 125/0.050000b0.25 D 8.34 W/m2 K Thus, heat loss by convection: qc D hc A T1 0004 T2 D 8.34 ð 0.157 415 0004 290 D 163.7 W/m
Radiation
An extension of equation 9.118 may be used. Taking the emissivity of the pipe as 0.9: qr D 5.67 ð 1000048 ð 0.9 4154 0004 2904 ð 0.157 D 181.0 W/m and the total loss is 344.7 W/m length of pipe.
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PROBLEM 9.25 Toluene is continuously nitrated to mononitrotoluene in a cast-iron vessel 1 m in diameter fitted with a propeller agitator of 0.3 m diameter driven at 2 Hz. The temperature is maintained at 310 K by circulating cooling water at 0.5 kg/s through a stainless steel coil of 25 mm outside diameter and 22 mm inside diameter wound in the form of a helix of 0.81 m diameter. The conditions are such that the reacting material may be considered to have the same physical properties as 75% sulphuric acid. If the mean water temperature is 290 K, what is the overall heat transfer coefficient?
Solution The overall coefficient Uo based on the outside area of the coil is given by equation 9.201: 1/Uo D 1/ho C xw /kw do /dw C 1/hi do /di C Ro C Ri do /di where dw is the mean pipe diameter. Inside
The coefficient on the water side is given by equations 9.202 and 9.203: hi D k/d 1 0004 3.5d/dc 0.023 di u0019/0018000b0.8 Cp 0018/k000b0.4 u0019 D 0.5/[ $/4 ð 0.0222 ] D 1315 kg/m2 s
where:
di D 0.022 m, dc D 0.80 m and for water at 290 K: k D 0.59 W/m K, 0018 D 0.00108 Ns/m2 , and Cp D 4180 J/kg K. ∴
hi D 0.59/0.022 1 C 3.5 ð 0.22/0.80 ð 0.023 0.022 ð 1315/0.00108000b0.8 ð 4180 ð 0.00108/0.59000b0.4 D 0.680 26,780000b0.8 7.65000b0.4 D 5490 W/m2 K
Outside
In equation 9.204:
ho dv /k 0018s /0018000b0.14 D 0.87 Cp 0018/k000b0.33 L 2 N0019/0018000b0.62 For 75% sulphuric acid: k D 0.40 W/m K, 0018s D 0.0086 N s/m2 at 300 K, 0018 D 0.0065 N s/m2 at 310 K, Cp D 1880 J/kg K and 0019 D 1666 kg/m3 ∴ ho ð 1.0/0.40 0.0086/0.0065000b0.14 D 0.87 1880 ð 0.0065/0.40000b0.33
ð 0.32 ð 2.0 ð 1665/0.0065000b0.62 ∴
and:
2.5ho 1.323000b0.14 D 0.87 30.55000b0.33 46,108000b0.62 ho D 0.348 ð 3.09 ð 779000b/1.04 D 805.5 W/m2 K
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Overall
Taking kw D 15.9 W/m K and Ro and Ri as 0.0004 and 0.0002 m2 K/W respectively, then in equation 9.201: 1/Uo D 1/805.5 C 0.0015/15.9 0.025/0.0235 C 1/5490 0.025/0.022 C 0.0004 C 0.0002 0.025/0.022 D 0.00124 C 0.00010 C 0.00021 C 0.00040 C 0.00023 D 0.00218 m2 K/W Uo D 458.7 W/m2 K
and:
PROBLEM 9.26 7.5 kg/s of pure iso-butane is to be condensed at a temperature of 331.7 K in a horizontal tubular exchanger using a water inlet temperature of 301 K. It is proposed to use 19 mm outside diameter tubes of 1.6 mm wall arranged on a 25 mm triangular pitch. Under these conditions the resistance of the scale may be taken as 0.0005 m2 K/W. Determine the number and arrangement of the tubes in the shell.
Solution The latent heat of vaporisation of isobutane is 286 kJ/kg and hence the heat load: Q D 7.5 ð 286 D 2145 kW The cooling water outlet should not exceed 320 K and a value of 315 K will be used. The mass flow of water is then: 2145/[4.18 315 0004 301000b] D 36.7 kg/s In order to obtain an approximate size of the unit, a value of 500 W/m2 K will be assumed for the overall coefficient based on the outside area of the tubes. 00061 D 331.7 0004 301 D 30.7 deg K,
00062 D 331.7 0004 315 D 16.7 deg K
and from equation 9.9: 0006m D 30.7 0004 16.7000b/ ln 30.7/16.7 D 23.0 deg K. Thus, the approximate area D 2145 ð 103 / 500 ð 23.0 D 186.5 m2 . The outside area of 0.019 m diameter tubes D $ ð 0.019 ð 1.0 D 0.0597 m2 /m and hence the total length of tubing D 186.5/0.0597 D 3125 m. Adopting a standard tube length of 4.88 m, number of tubes D 3125/4.88 D 640. With the large flow of water involved, a four tube-side pass unit is proposed, and for this arrangement 678 tubes can be accommodated on 25 mm triangular pitch in a 0.78 m i.d. shell. Using this layout, the film coefficients are now estimated and the assumed value of U is checked. Inside
Water flow through each tube D 36.7 678/4 D 0.217 kg/s.
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The tube i.d. D 19.0 0004 2 ð 1.67 D 15.7 mm the cross-sectional area for flow D $/4 0.0157000b2 D 0.000194 m2 and hence the water velocity: u D 0.217/ 1000 ð 0.000194 D 1.12 m/s. From equation 9.221 : hi D 4280[ 0.00488 ð 308 0004 1]1.120.8 /0.01570.2 D 4280 ð 0.503 ð 1.095000b/0.436 D 5407 W/m2 K or, based on outside diameter: hio D 5407 ð 0.0157000b/0.019 D 4468 W/m2 K or 4.47 kW/m2 K Outside
The temperature drop across the condensate film, Tf is given by: (thermal resistance of water film C scale)/ total thermal resistance D 0006m 0004 Tf /0006m or:
1/4.47 C 0.5000b/ 1/0.500 D 23.0 0004 Tf /23.0 Tf D 14.7 deg K
and:
The condensate film is thus at 331.7 0004 14.7 D 317 K. The outside film coefficient is given by: ho D 0.72[ k 3 00192 g001a000b/ jdo 0018Tf ]0.25
(equation 9.177) p At 317 K, k D 0.13 W/m K, 0019 D 508 kg/m3 , 0018 D 0.000136 N s/m2 and j D 678 D 26.0. ∴
ho D 0.72[ 0.133 ð 5082 ð 9.81 ð 286 ð 103 /
26 ð 19.0 ð 1000043 ð 0.000136 ð 14.5000b]0.25 D 814 W/m2 K or 0.814 kW/m2 K
Overall
1/U D 1/4.47 C 1/0.814 C 0.50 D 1.952 U D 0.512 kW/m2 K or 512 W/m2 K
and:
which is sufficiently near the assumed value. For the proposed unit, the heat load: Q D 0.512 ð 678 ð 4.88 ð 0.0597 ð 23.0 D 2328 kW or an overload of:
2328 0004 2145000b100/2145 D 8.5%
PROBLEM 9.27 37.5 kg/s of crude oil is to be heated from 295 to 330 K by heat transferred from the bottom product from a distillation column. The bottom product, flowing at 29.6 kg/s is to
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be cooled from 420 to 380 K. There is available a tubular exchanger with an inside shell diameter of 0.60 m, having one pass on the shell side and two passes on the tube side. It has 324 tubes, 19 mm outside diameter with 2.1 mm wall and 3.65 m long, arranged on a 25 mm square pitch and supported by baffles with a 25% cut, spaced at 230 mm intervals. Would this exchanger be suitable?
Solution Mean temperature of bottom product D 0.5 420 C 380 D 400 K. Mean temperature of crude oil D 0.5 330 C 295 D 313 K. For the crude oil at 313 K: Cp D 1986 J/kg K, 0018 D 0.0029 N s/m2 , k D 0.136 W/m K and 0019 D 824 kg/m3 . For the bottom product at 400 K: Cp D 2200 J/kg K. Heat loads: tube side: Q D 37.5 ð 1.986 330 0004 295 D 2607 kW. shell side: Q D 29.6 ð 2.20 420 0004 380 D 2605 kW.
Outside coefficient
Temperature of wall D 0.5 400 C 313 D 356.5 K and film temperature, Tf D 0.5 400 C 356.5 D 378 K. At 378 K, 0019 D 867 kg/m3 , 0018 D 0.0052 N s/m2 , and k D 0.119 W/m K Cross-sectional area for flow D shell i.d. ð clearance ð baffle spacing000b/pitch D 0.60 ð 0.0064 ð 0.23000b/0.025 D 0.353 m2 (assuming a clearance of 0.0064 m). ∴
G0max D 29.6/0.0353 D 838.5 kg/m2 s Remax D 0.019 ð 838.5000b/0.0052 D 306.4
and:
Therefore in equation 9.90, taking Ch D 1:
ho ð 0.019/0.119 D 0.33 ð 1.0 3064000b0.6 2200 ð 0.0052/0.119000b0.3 or:
ho D 2.07 ð 125 ð 3.94 D 1018 W/m2 K or 1.02 kW/m2 K
Inside coefficient
Number tubes per pass D 324/2 D 162. Inside diameter D [19.0 0004 2 ð 2.1000b]/1000 D 0.0148 m
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and cross-sectional area for flow D $/4 0.0148000b2 D 0.000172 m2 per tube or:
0.000172 ð 162 D 0.0279 m2 per pass. ∴
G0 D 37.5/0.0279 D 1346 kg/m2 s In equation 9.61:
hi ð 0.0148/0.136 D 0.023 0.0148 ð 1346/0.0029000b0.8 1986 ð 0.0029/0.136000b0.4 hi D 0.211 6869000b0.8 42.4000b0.4 D 1110 W/m2 K
or, based on the outside area, hio D 1110 ð 0.0148000b/0.019 D 865 W/m2 K hio D 0.865 kW/m2 K.
or: Overall coefficient
Neglecting the wall and scale resistance, the clean overall coefficient is: 1/Uc D 1/1.02 C 1/0.865 D 2.136 m2 K/kW The area available is A D 324 ð 3.65 ð $ ð 0.019 D 70.7 m2 and hence the minimum value of the design coefficient is: 1/UD D A0006m /Q 00061 D 420 0004 330 D 90 deg K, and: ∴
00062 D 380 0004 295 D 85 deg K
0006m D 90 0004 85000b/ ln 90/85 D 87.5 deg K 1/UD D 70.7 ð 87.5000b/2607 D 2.37 m2 K/kW
The maximum allowable scale resistance is then: R D 1/UD 0004 1/Uc D 2.37 0004 2.136 D 0.234 m2 K/kW This value is very low as seen from Table 9.16, and the exchanger would not give the required temperatures without frequent cleaning.
PROBLEM 9.28 A 150 mm internal diameter steam pipe, carrying steam at 444 K, is lagged with 50 mm of 85% magnesia. What will be the heat loss to the air at 294 K?
Solution In this case: di D 0.150 m, do D 0.168 m and dw D 0.5 0.150 C 0.168 D 0.159 m. ds D 0.168 ð 2 ð 0.050 D 0.268 m and dm (the logarithmic mean of do and ds ) D 0.215 m.
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The coefficient for condensing steam including any scale will be taken as 8500 W/m2 K, kw as 45 W/m K, and kl as 0.073 W/m K. The surface temperature of the lagging will be assumed to be 314 K and hr C hc to be 10 W/m2 K. The thermal resistances are therefore:
1/hi $d D 1/ 8500 ð $ ð 0.150 D 0.00025 mK/W
xw /kw $dw D 0.009/ 45$ ð 0.159 D 0.00040 mK/W
xl /kl $dm D 0.050/ 0.073$ ð 0.215 D 1.0130 mK/W
1/ hr ð hc ds D 1/ 10 ð 0.268 D 0.1190 mK/W Neglecting the first two terms, the total thermal resistance D 1.132 mK/W. From equation 9.261, heat lost per unit length D 444 0004 294000b/1.132 D 132.5 W/m. The surface temperature of the lagging is given by: T lagging000b/T D 1.013/1.132 D 0.895 and:
T lagging D 0.895 444 0004 294 D 134 deg K
Therefore the surface temperature D 444 0004 134 D 310 K which approximates to the assumed value. Assuming an emissivity of 0.9: hr D 5.67 ð 1000048 ð 0.9 3104 0004 2944 / 310 0004 294 D 3.81 W/m2 K. For natural convection: hc D 1.37 T/ds 0.25 D 1.37[ 310 0004 294000b/0.268]0.25 D 3.81 W/m2 K. ∴
hr C hc D 9.45 W/m2 K which again agrees with the assumed value. In practice forced convection currents are usually present and the heat loss would probably be higher than this value. For an unlagged pipe and T D 150 K, hr C hc would be about 20 W/m2 K and the heat loss, Q/l D hr C hc $d0 T D 20$ ð 0.168 ð 150 D 1584 W/m. Thus the heat loss has been reduced by about 90% by the addition of 50 mm of lagging.
PROBLEM 9.29 A refractory material with an emissivity of 0.40 at 1500 K and 0.43 at 1420 K is at a temperature of 1420 K and is exposed to black furnace walls at a temperature of 1500 K. What is the rate of gain of heat by radiation per unit area?
Solution In the absence of further data, the system will be considered as two parallel plates.
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The radiating source is the furnace walls at T1 D 1500 K and for a black surface, e1 D 1.0. The heat sink is the refractory at T2 D 1420 K, at which e2 D 0.43. Putting A1 D A2 in equation 9.150: q D e1 e2 9 T41 0004 T42 / e1 C e2 0004 e1 e2 D 1.0 ð 0.43 ð 5.67 ð 1000048 15004 0004 14204 / 1.0 C 0.43 0004 0.43 ð 1.0 D 2.44 ð 1000048 9.97 ð 1011 /1.0 D 2.43 ð 104 W/m2 or 24.3 kW/m2
PROBLEM 9.30 The total emissivity of clean chromium as a function of surface temperature, T K, is given approximately by: e D 0.38 1 0004 263/T000b. Obtain an expression for the absorptivity of solar radiation as a function of surface temperature, and calculate the values of the absorptivity and emissivity at 300, 400 and 1000 K. Assume that the sun behaves as a black body at 5500 K.
Solution It may be assumed that the absorptivity of the chromium at temperature T1 is the emissivity of the chromium at the geometric mean of T1 and the assumed temperature of the sun, T2 where T2 D 5500 K. Since:
e D 0.38 1 0004 263/T
i 0.5
then, taking the geometric mean temperature as 5500T1 : a D 0.38f1 0004 [263/ 5500T1 0.5 ]g
ii000b
For the given values of T1 , values of e and a are now calculated from (i) and (ii) respectively to give the following data: T1 300 400 1000
T1 T2 0.5 1285 1483 2345
e 0.047 0.130 0.280
a 0.300 0.312 0.337
PROBLEM 9.31 Repeat Problem 9.30 for the case of aluminium, assuming the emissivity to be 1.25 times that for chromium.
Solution In this case:
and:
e D 1.25 ð 0.38 1 0004 263/T1 D 0.475[ 1 0004 263/T1 ]
i000b
a D 0.475 0004 1.66T00040.5 1
ii000b
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The following data are obtained by substituting values for T1 in equations (i) and (ii): T1 300 400 1000
T1 T2 0.5 1285 1483 2345
e 0.059 0.163 0.350
a 0.378 0.391 0.422
PROBLEM 9.32 Calculate the heat transferred by solar radiation on the flat concrete roof of a building, 8 m by 9 m, if the surface temperature of the roof is 330 K. What would be the effect of covering the roof with a highly reflecting surface, such as polished aluminium, separated from the concrete by an efficient layer of insulation? The emissivity of concrete at 330 K is 0.89, whilst the total absorptivity of solar radiation (sun temperature D 5500 K) at this temperature is 0.60. Use the data for aluminium from Problem 9.31 which should be solved first.
Solution The emission from a body with an emissivity, e, at a temperature T is given by: I D e9T4 Thus, for the concrete: I D 0.89 ð 5.67 ð 1000048 ð 3304 D 598.5 W/m2 Taking T D 330 K as the equilibrium temperature, the energy emitted by the concrete must equal the energy absorbed and, since the absorptivity of concrete, a D 0.60, the solar flux is then: Is D 598.5/0.6 D 997.4 W/m2 which approximates to the generally accepted figure of about 1 kW/m2 . With a covering of polished aluminium, then using the data given in Problem 9.31 and an equilibrium surface temperature of T K, the absorptivity is: a D 0.475 0004 1.66T0.5 and, with an area of 8 ð 9 D 72 m2 , the energy absorbed is:
1.0 ð 103 ð 72 0.475 0004 1.66T0.5 D 3.42 ð 104 0004 1.20 ð 105 T0.5 W
(i)
The emissivity is given by: e D 0.475 1 0004 263/T D 0.475 0004 125T00041 and the energy emitted is:
72 ð 5.67 ð 1000048 T4 0.475 0004 125T00041 D 1.94 ð 1000046 T4 0004 5.10 ð 1000044 T3 W
(ii)
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Equating (i) and (ii): 1.94 ð 1000046 T4 0004 5.10 ð 1000044 T3 C 1.20 ð 105 T0.5 D 3.42 ð 104 or:
5.67 ð 10000411 T4 0004 1.49 ð 1000048 T3 C 3.51T0.5 D 1
Solving by trial and error, the equilibrium temperature of the aluminium is: T D 438 K . Substituting T D 438 K in (i), the energy absorbed and emitted is then 2847 W which represents an increase of some 375 per cent compared with the value for the concrete alone.
PROBLEM 9.33 A rectangular iron ingot 15 cm ð 15 cm ð 30 cm is supported at the centre of a reheating furnace. The furnace has walls of silica-brick at 1400 K, and the initial temperature of the ingot is 290 K. How long will it take to heat the ingot to 600 K? It may be assumed that the furnace is large compared with the ingot, and that the ingot is always at uniform temperature throughout its volume. Convection effects are negligible. The total emissivity of the oxidised iron surface is 0.78 and both emissivity and absorptivity may be assumed independent of the surface temperature. (Density of iron D 7.2 Mg/m3 . Specific heat capacity of iron D 0.50 kJ/kg K.)
Solution As there are no temperature gradients within the ingot, the rate of heating is dependent on the rate of radiative heat transfer to the surface. In addition, since the dimensions of the ingot are much smaller than those of the surrounding surfaces, the ingot may be treated as a black body. Volume of ingot D 15 ð 15 ð 30 D 6750 cm3 or 0.00675 m3 . Mass of ingot D 7.2 ð 103 ð 0.00675 D 48.6 kg. For an ingot temperature of T K, the increase in enthalpy D d mCp T000b/dt or mCp dT/dt where t is the time and Cp the specific heat of the ingot. The heat received by radiation D A9a T4f 0004 T4 where the area, A D 4 ð 30 ð 15 C
2 ð 15 ð 15 D 2250 cm2 or 0.225 m2 . The absorptivity a will be taken as the emissivity D 0.78 and the furnace temperature, Tf D 1400 K. Thus: or:
mCp dT/dt D A9a T4f 0004 T4 0004 t 0004 dT mCp 600 dt D 4 4 aA9 0 290 Tf 0004 T
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0002
∴
tD
00030002 0003
48.6 ð 0.50 ð 103 1
0.78 ð 0.225 ð 5.67 ð 1000048
4 ð 14003 0002 0003600 T
1400 C T C 2 tan00041 ð ln D 200 s
1400 0004 T 1400 290
PROBLEM 9.34 A wall is made of brick, of thermal conductivity 1.0 W/m K, 230 mm thick, lined on the inner face with plaster of thermal conductivity 0.4 W/m K and of thickness 10 mm. If a temperature difference of 30 K is maintained between the two outer faces, what is the heat flow per unit area of wall?
Solution For an area of 1 m2 , thermal resistance of the brick:
x1 /k1 A D 0.230/ 1.0 ð 1.0 D 0.230 K/W
thermal resistance of the plaster:
x2 /k2 A D 0.010/ 0.4 ð 1.0 D 0.0025 K/W
and in equation 9.18: 30 D 230 C 0.0025000bQ or Q D 129 W
PROBLEM 9.35 A 50 mm diameter pipe of circular cross-section and with walls 3 mm thick is covered with two concentric layers of lagging, the inner layer having a thickness of 25 mm and a thermal conductivity of 0.08 W/m K, and the outer layer a thickness of 40 mm and a thermal conductivity of 0.04 W/m K. What is the rate of heat loss per metre length of pipe if the temperature inside the pipe is 550 K and the outside surface temperature is 330 K?
Solution From equation 9.22, the thermal resistance of each component is: r2 0004 r1 /k 2$rm l Thus for the wall: r2D 0.050/2 C 0.003 D 0.028 m r1 D 0.050/2 D 0.025 m and:
rm D 0.028 0004 0.025000b/ ln 0.028/0.025 D 0.0265 m.
Taking k D 45 W/m K and l D 1.0 m the thermal resistance is: D 0.028 0004 0.025000b/ 45 ð 2$ ð 0.0265 ð 1.0 D 0.00040 K/W.
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For the inner lagging: r2 D 0.028 C 0.025 D 0.053 m r1 D 0.028 m and:
rm D 0.053 0004 0.028000b/ ln 0.053/0.028 D 0.0392 m.
Therefore the thermal resistance D 0.053 0004 0.028000b/ 0.08 ð 2$ ð 0.0392 ð 1.0 D 1.2688 K/W For the outer lagging: r2 D 0.053 C 0.040 D 0.093 m r1 D 0.053 m and:
rm D 0.093 0004 0.053000b/ ln 0.093/0.053 D 0.0711 m
Therefore the thermal resistance D 0.093 0004 0.053000b/ 0.04 ð 2$ ð 0.0711 ð 1.0 D 2.2385 K/W From equation 9.19: Q D 550 0004 330000b/ 0.0004 C 1.2688 C 2.2385 D 62.7 W/m
PROBLEM 9.36 The temperature of oil leaving a co-current flow cooler is to be reduced from 370 to 350 K by lengthening the cooler. The oil and water flowrates, the inlet temperatures and the other dimensions of the cooler will remain constant. The water enters at 285 K and oil at 420 K. The water leaves the original cooler at 310 K. If the original length is 1 m, what must be the new length?
Solution For the original cooler, for the oil:
Q D Go Cpo 420 0004 370000b
and for the water:
Q D Gw Cpw 310 0004 285000b
∴
Go Cp /Gw Cp D 25/50 D 0.5
where Go and Gw are the mass flows and Cpo and Cpw the specific heat capacities of the oil and water respectively. 00061 D 420 0004 285 D 135 deg K, 00062 D 370 0004 310 D 60 deg K for co-current flow, and from equation 9.9: 0006m D 135 0004 60000b/ ln 135/60 D 92.5 deg K If a is the area per unit length of tube multiplied by the number of tubes, then: A D 1.0 ð a m2 and in equation 9.1: Go Cp 420 0004 370 D Ua 92.5 or Go Cp /Ua D 1.85 For the new cooler, for the oil: Q D Go Cpo 420 0004 350 and for the water,
Q D Gw Cpw T 0004 285000b
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where T is the water outlet temperature. Thus:
T 0004 285 D Go Cp /Gw Cp 420 0004 350 D 0.5 ð 70000b
and:
T D 320 K
∴ 00061 D 420 0004 285 D 135 deg K, 00062 D 350 0004 320 D 30 deg K, again for co-current flow, and from equation 9.9: 0006m D 135 0004 30000b/ ln 135/30 D 69.8 deg K
In equation 9.1: Go so 420 0004 350 D Ual69.8 ∴
l D Go Cp /Ua ð 1.003 D 1.85 ð 1.003 D 1.86 m
PROBLEM 9.37 In a countercurrent-flow heat exchanger, 1.25 kg/s of benzene (specific heat 1.9 kJ/kg K and density 880 kg/m3 ) is to be cooled from 350 K to 300 K with water which is available at 290 K. In the heat exchanger, tubes of 25 mm external and 22 mm internal diameter are employed and the water passes through the tubes. If the film coefficients for the water and benzene are 0.85 and 1.70 kW/m2 K respectively and the scale resistance can be neglected, what total length of tube will be required if the minimum quantity of water is to be used and its temperature is not to be allowed to rise above 320 K?
Solution Heat load:
For the benzene: Q D 1.25 ð 1.9 350 0004 300 D 118.75 kW. In order to use the minimum amount, water must leave the unit at the maximum temperature, 320 K. Thus for G kg/s water: 118.75 D G ð 4.18 320 0004 290 or G D 0.947 kg/s Temperature driving force
00061 D 350 0004 320 D 30 deg K, 00062 D 300 0004 290 D 10 deg K and in equation 9.9: 0006m D 30 0004 10000b/ ln 30/10 D 18.2 deg K. In the absence of further data, it will be assumed that the correction factor is unity. Overall coefficient
Inside: hi D 0.85 kW/m2 K or based on the tube o.d., hio D 0.85 ð 22/25 D 0.748 kW/m2 K.
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Outside: ho D 1.70 kW/m2 K. Wall: Taking ksteel D 45 W/m K, x/k D 0.003/45 D 0.00007 m2 K/W or 0.07 m2 K/kW. Thus neglecting any scale resistance: 1/U D 1/0.748 0004 1/1.70 C 0.07 D 1.995 m2 K/kW U D 0.501 kW/m2 K
and: Area
In equation 9.1: A D Q/U0006m D 118.75/ 0.0501 ð 18.2 D 13.02 m2 . Surface area of a 0.025 m o.d. tube D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and hence total length of tubing required D 1302/0.0785 D 165.8 m
PROBLEM 9.38 Calculate the rate of loss of heat from a 6 m long horizontal steam pipe of 50 mm internal diameter and 60 mm external diameter when carrying steam at 800 kN/m2 . The temperature of the surroundings is 290 K. What would be the cost of steam saved by coating the pipe with a 50 mm thickness of 85% magnesia lagging of thermal conductivity 0.07 W/m K, if steam costs £0.5/100 kg? The emissivity of both the surface of the bare pipe and the lagging may be taken as 0.85, and the coefficient h for the heat loss by natural convection is given by: h D 1.65 T000b0.25 W/m2 K where T is the temperature difference in deg K. The Stefan-Boltzmann constant is 5.67 ð 1000048 W/m2 K4 .
Solution For the bare pipe
Steam is saturated at 800 kN/m2 and 443 K. Neglecting the inside resistance and that of the wall, it may be assumed that the surface temperature of the pipe is 443 K. For radiation from the pipe, the surface area D $ ð 0.060 ð 6.0 D 1.131 m2 and in equation 9.119: qr D 5.67 ð 1000048 ð 0.85 ð 1.131 4434 0004 2904 D 1714 W.
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For convection from the pipe, the heat loss: qc D hc A Ts 0004 T D 1.65 443 0004 290000b0.25 ð 1.131 443 0004 290 D 1.866 443 0004 290000b1.25 D 1004 W and the total loss D 2718 W or 2.71 kW For the insulated pipe
The heat conducted through the lagging ql must equal the heat lost from the surface
qr C qc . Mean diameter of the lagging D [ 0.060 C 2 ð 0.050 C 0.060]/2 D 0.110 m at which the area D $ ð 0.110 ð 6.0 D 2.07 m2 and in equation 9.12: ql D 0.07 ð 2.07 443 0004 Ts /0.050 D 1280 0004 2.90Ts W where Ts is the surface temperature. The outside area D $ 0.060 C 2 ð 0.050 ð 6.0 D 3.016 m2 and from equation 9.119 : qr D 5.67 ð 1000048 ð 0.85 ð 3.016 T4s 0004 2904 D 1.456 ð 1000047 T4s D 1030 W and:
qc D 1.65 Ts 0004 290000b0.25 ð 3.016 Ts 0004 290 D 4.976 Ts 0004 290000b1.25 W
Making a heat balance:
1280 0004 2.90Ts D 1.456 ð 1000047 T4s 0004 1030 C 4.976 Ts 0004 290000b1.25 or:
4.976 Ts 0004 290000b1.25 C 1.456 ð 1000047 T4s C 2.90Ts D 2310
Solving by trial and error: Ts D 305 K and hence the heat lost D 1280 0004 2.90 ð 305 D 396 W. The heat saved by lagging the pipe D 2712 0004 396 D 2317 W or 2.317 kW. At 800 kN/m2 , the latent heat of steam is 2050 kJ/kg and the reduction in the amount of steam condensed D 2.317/2050 D 0.00113 kg/s or:
0.00113 ð 3600 ð 24 ð 365 D 35,643 kg/year
∴
annual saving D 35,643 ð 0.5000b/100 D £178/year
It may be noted that arithmetic mean radius should only be used with thin walled tubes, which is not the case here. If a logarithmic mean radius is used in applying equation 9.8, Ts D 305.7 K and the difference is, in this case, negligible.
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PROBLEM 9.39 A stirred reactor contains a batch of 700 kg reactants of specific heat 3.8 kJ/kg K initially at 290 K, which is heated by dry saturated steam at 170 kN/m2 fed to a helical coil. During the heating period the steam supply rate is constant at 0.1 kg/s and condensate leaves at the temperature of the steam. If heat losses are neglected, calculate the true temperature of the reactants when a thermometer immersed in the material reads 360 K. The bulb of the thermometer is approximately cylindrical and is 100 mm long by 10 mm in diameter with a water equivalent of 15 g, and the overall heat transfer coefficient to the thermometer is 300 W/m2 K. What temperature would a thermometer with a similar bulb of half the length and half the heat capacity indicate under these conditions?
Solution The latent heat of dry saturated steam at 170 kN/m2 and 388 K D 2216 kJ/kg. Therefore heat added to the reactor D 2216 ð 0.1 D 221.6 kJ/s D 221.6 kW which is equal to the increase in enthalpy, dH/dt. The enthalpy of the contents, neglecting the heat capacity of the reactor and losses D mCp dT/dt D 700 ð 3.8 dT/dt or 2660 dT/dt kW ∴
2660 dT/dt D 221.6
and the rate of temperature rise, dT/dt D 0.083 deg K/s. At time t, the temperature of the reactants is: T D 290 C 0.083t K
(i)
The increase in enthalpy of the thermometer is equal to the rate of heat transfer from the fluid, or: (ii)
mCp t dTt /dt D UAt T 0004 Tt where the subscript t refers to the thermometer. ∴
15/1000 ð 4.18 dTt /dt D 0.300 $ ð 0.010 ð 0.100 T 0004 Tt
and:
dTt /dt D 0.0150 T 0004 Tt deg K/s
At time t s, the temperature of the thermometer is therefore: Tt D 290 C [0.0150 T 0004 Tt ]t K
(iii)
When Tt D 360 K, then substituting from equation (i): 360 D 290 C f0.0150[290 C 0.083t 0004 360]gt
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or:
0.00125t2 0004 1.05t 0004 70 D 0 and t D 902 s
Therefore in (i):
T D 290 C 0.083 ð 902 D 364.9 K
With half the length, that is 0.050 m, and half the heat capacity, that is 7.5 g water, then in equation (ii):
7.5/1000 ð 4.18 dTt /dt D 0.300 $ ð 0.010 ð 0.050 T 0004 Tt
dTt /dt D 0.0150 T 0004 Tt
or:
The same result as before and hence the new thermometer would also read 360 K.
PROBLEM 9.40 How long will it take to heat 0.18 m3 of liquid of density 900 kg/m3 and specific heat 2.1 kJ/kg K from 293 to 377 K in a tank fitted with a coil of area 1 m2 ? The coil is fed with steam at 383 K and the overall heat transfer coefficient can be taken as constant at 0.5 kW/m2 K. The vessel has an external surface of 2.5 m2 , and the coefficient for heat transfer to the surroundings at 293 K is 5 W/m2 K. The batch system of heating is to be replaced by a continuous countercurrent heat exchanger in which the heating medium is a liquid entering at 388 K and leaving at 333 K. If the heat transfer coefficient is 250 W/m2 K, what heat exchange area is required? Heat losses may be neglected.
Solution Mass of liquid in the tank D 0.18 ð 900 D 162 kg ∴
mCp D 162 ð 2100 D 340,200 J/deg K
Using the argument given in Problem 9.77: 340,200 dT/dt D 500 ð 1 383 0004 T 0004 5 ð 2.5 T 0004 293 or: or:
D 191,500 0004 500T 0004 12.5T C 3663 D 195,163 0004 512.5T 664 dT/dt D 380.8 0004 T
The time taken to heat the liquid from 293 to 377 K is: 0004 377 dT/ 380.8 0004 T t D 664 293
D 664 ln[ 380.8 0004 293000b/ 380.8 0004 377000b] D 2085 s 0.58 h For the heat exchanger:
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T1 D 388 0004 377 D 11 deg K, T2 D 333 0004 293 D 40 deg K and from equation 9.9: Tm D 40 0004 11000b/ ln 40/11 D 22.5 deg K. Mass flow D 162/2085 D 0.0777 kg/s Heat load: Q D 0.0777 ð 2.1 377 0004 293 D 13.71 kW In equation 9.1: U D 250/1000 D 0.250 kW/m2 K, The area required: A D 13.71/ 0.250 ð 22.5 D 2.44 m2 .
PROBLEM 9.41 The radiation received by the earth’s surface on a clear day with the sun overhead is 1 kW/m2 and an additional 0.3 kW/m2 is absorbed by the earth’s atmosphere. Calculate approximately the temperature of the sun, assuming its radius to be 700,000 km and the distance between the sun and the earth to be 150,000,000 km. The sun may be assumed to behave as a black body.
Solution The total radiation received D 1.3 kW/m2 of the earth’s surface. The equivalent surface area of the sun is obtained by comparing the area of a sphere at the radius of the sun, 7 ð 105 km and the area of a sphere of radius (radius of sun C distance between sun and earth) or: A1 /A2 D 4$ 7 ð 105 2 /4$ 150 ð 106 C 7 ð 105 2 D 2.16 ð 1000045 . Therefore radiation at the sun’s surface D 1.3ð 103 /2.16ð 1000045 D 6.03 ð 107 W/m2 . For a black body, the intensity of radiation is given by equation 9.112: 6.03 ð 107 D 5.67 ð 1000048 T4
and
T D 5710 K
PROBLEM 9.42 A thermometer is immersed in a liquid which is heated at the rate of 0.05 K/s. If the thermometer and the liquid are both initially at 290 K, what rate of passage of liquid over the bulb of the thermometer is required if the error in the thermometer reading after 600 s is to be no more than 1 deg K? Take the water equivalent of the thermometer as 30 g, the heat transfer coefficient to the bulb to be given by U D 735 u0.8 W/m2 K. The area of the bulb is 0.01 m2 where u is the velocity in m/s.
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Solution If T and T0 are the liquid and thermometer temperatures respectively after time t s, then: dT/dt D 0.05 K/s and hence T D 290 C 0.05t When t D 600 s, T 0004 T0 D 1. ∴
T D 290 C 600 ð 0.05 D 320 K and T0 D 319 K Balancing: Gt C0019t dT0 /dt D UA T 0004 T0
∴ ∴ ∴
30/1000000b4.18 dT0 /dt D UA 290 C 0.05t C T0 dT0 /dt C 7.98UAT0 D 2312UA 1 C 0.000173t
0004 e7.98UAt e7.98UAt 0004 0.000173 dt e7.98UAt T0 D 2312UA 1 C 0.000173t 7.98UA 7.98UA D 290 1 C 0.000173t000be7.98UAt 0004 0.050
e7.98UAt Ck 7.98UA
When t D 0, T0 D 290 K and k D 0.00627/UA. ∴
T0 D 290 1 C 0.000173t 0004 0.00627/UA 1 0004 e00047.98UAt
When t D 600 s, T0 D 319 K. ∴ ∴
and:
319 D 320 C 0.00627/UA 1 0004 e00044789UA 00044789UA D ln 1 0004 159.5UA UA D 00040.000209 ln 1 0004 159.5UA000b
Solving by trial and error: UA D 0.00627 kW/K. A D 0.01 m2 and hence: U D 0.627 kW/m2 K or 627 W/m2 K ∴
627 D 735u0.8
and
u D 0.82 m/s
PROBLEM 9.43 In a shell-and-tube type of heat exchanger with horizontal tubes 25 mm external diameter and 22 mm internal diameter, benzene is condensed on the outside of the tubes by means of water flowing through the tubes at the rate of 0.03 m3 /s. If the water enters at 290 K and leaves at 300 K and the heat transfer coefficient on the water side is 850 W/m2 K, what total length of tubing will be required?
Solution Mass flow of water D 0.03 ð 1000 D 30 kg/s
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and hence the heat load D 30 ð 4.18 300 0004 290 D 1254 kW At atmospheric pressure, benzene condenses at 353 K and hence: 00061 D 353 0004 290 D 63 deg K,
00062 D 353 0004 300 D 53 deg K
and from equation 9.9: 0006m D 63 0004 53000b/ ln 63/53 D 57.9 deg K No correction factor is required, because of isothermal conductions on the shell side. For condensing benzene, ho will be taken as 1750 W/m2 K. From Table 9.18: hi D 850 W/m2 K or, based on the outside diameter, hio D 850 ð 22/25 D 748 W/m2 K. Neglecting scale and wall resistances: 1/U D 1/1750 C 1/748 D 0.00191 m2 K/W and:
U D 524 W/m2 K or 0.524 kW/m2 K
Therefore, from equation 9.1: A D 1254/ 0.524 ð 57.9 D 41.3 m2 . Outside area of 0.025 m tubing D $ ð 0.025 ð 1.0 D 0.0785 m2 /m and total length of tubing required D 41.3/0.0785 D 526 m.
PROBLEM 9.44 In a contact sulphuric acid plant, the gases leaving the first convertor are to be cooled from 845 to 675 K by means of the air required for the combustion of the sulphur. The air enters the heat exchanger at 495 K. If the flow of each of the streams is 2 m3 /s at NTP, suggest a suitable design for a shell-and-tube type of heat exchanger employing tubes of 25 mm internal diameter. (a) Assume parallel co-current flow of the gas streams. (b) Assume parallel countercurrent flow. (c) Assume that the heat exchanger is fitted with baffles giving cross-flow outside the tubes.
Solution Heat load
At a mean temperature of 288 K, the density of air D 29/22.4 273/288 D 1.227 kg/m3 , where 29 kg/kmol is taken as the mean molecular mass of air. ∴
mass flow of air D 2.0 ð 1.227 D 2.455 kg/s.
If, as a first approximation, the thermal capacities of the two streams can be assumed equal for equal flowrates, then the outlet air temperature D 495 C 845 0004 675 D 665 K
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and for a mean specific heat of 1.0 kJ/kg K, the heat load is Q D 2.455 ð 1.0 665 0004 495 D 417.4 kW For gas to gas heat transfer, an overall coefficient of 1/ 1/60 C 1/60 D 30 W/m2 K will be taken using the data in Table 9.17. (a) Co-current flow
00061 D 845 0004 495 D 350 deg K,
00062 D 675 0004 665 D 10 deg K
and in equation 9.9: 0006m D 350 0004 10000b/ ln 350/10 D 95.6 deg K. Therefore in equation 9.1: A D 417.4 ð 103 / 30 ð 95.6 D 145.5 m2 . For 25 mm i.d. tubes an o.d. of 32 mm will be assumed for which the outside area D
$ ð 0.032 ð 1.0 D 0.1005 m2 /m and total length of tubing D 145.5/0.1005 D 1447 m. At a mean air temperature of 580 K: 0019 D 29/22.4 273/580 D 0.609 kg/m3 . ∴ volume flow of air D 2.445/0.609 D 4.03 m3 /s. For a reasonable gas velocity of say 15 m/s: area for flow D 4.03/15 D 0.268 m2 . Cross-sectional area of one tube D $/4000b0.0252 D 0.00050 m2 . ∴ number of tubes/pass D 0.268/0.00050 D 545, each of length D 1447/545 D 2.65 m.
In practice, the standard length of 2.44 m would be adopted with 1447/2.44 D 594 tubes in a single pass. (b) Countercurrent flow
In this case, 00061 D 845 0004 665 D 180 deg K, 00062 D 675 0004 495 D 180 deg K, and 0006m D 180 deg K In equation 9.1: A D 417.4 ð 103 / 30 ð 180 D 77.3 m2 and total length of tubing D 77.3/0.1005 D 769 m. With a velocity of 15 m/s, each tube would be 769/545 D 1.41 m long. A better arrangement would be the use of 769/2.44 D 315 tubes, 2.44 m long, though this would give a higher velocity and hence an increased air side pressure drop. With such an arrangement, 315 ð 32 mm o.d. tubes could be accommodated in a 838 mm i.d. shell on 40 mm triangular pitch.
(c) Cross flow
As in (b), 0006m D 180 deg K. From equation 9.213: X D t2 0004 t1 / T1 0004 t1 D 665 0004 495000b/ 845 0004 495 D 0.486 and:
Y D T1 0004 T2 / t2 0004 t1 D 845 0004 675000b/ 665 0004 495 D 1.0
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Thus, assuming one shell pass, two tube-side passes, from Figure 9.71: F D 0.82 and 0006m F D 0.82 ð 180 D 147.6 K Thus, in equation 9.212: A D 417.4 ð 103 / 30 ð 147.6 D 94.3 m2 and: total length of tubing D 94.3/0.1005 D 938 m. Using standard tubes 2.44 m long, number of tubes D 938/2.44 D 384 or 384/2 D 192 tubes/pass. The cross-sectional area for flow would then be 192 ð 0.00050 D 9.61 ð 1000042 m2 and the air velocity D 4.03/9.61 ð 1000042 D 41.9 m/s. This is not excessive providing the minimum acceptable pressure drop is not exceeded. The nearest standard size is 390 ð 32 mm o.d. tubes, 2.44 m in a 940 mm i.d. shell arranged on 40 mm triangular pitch in two passes.
PROBLEM 9.45 A large block of material of thermal diffusivity DH D 0.0042 cm2 /s is initially at a uniform temperature of 290 K and one face is raised suddenly to 875 K and maintained at that temperature. Calculate the time taken for the material at a depth of 0.45 m to reach a temperature of 475 K on the assumption of unidirectional heat transfer and that the material can be considered to be infinite in extent in the direction of transfer.
Solution This problem is identical to Problem 9.2 except for slight variations in temperature, and reference may be made to that solution.
PROBLEM 9.46 A 50% glycerol–water mixture is flowing at a Reynolds number of 1500 through a 25 mm diameter pipe. Plot the mean value of the heat transfer coefficient as a function of pipe length, assuming that: Nu D 1.62 Re Pr d/l000b0.33 . Indicate the conditions under which this is consistent with the predicted value Nu D 4.1 for fully developed flow.
Solution For 50% glycerol–water at, say, 290 K: 0018 D 0.007 N s/m2 , k D 0.415 W/m K and Cp D 3135 J/kg K. ∴ ∴
h ð 0.025000b/0.415 D 1.62[ 1500 ð 3135 ð 0.007/0.415 0.025/l000b]0.33 h D 26.89 1983/l000b0.33 D 330/l0.33 W/m2 K
h is plotted as a function of l over the range l D 0–10 m in Fig. 9d. When Nu D 4.1: h D 4.1k/d D 4.1 ð 0.415000b/0.025 D 68.1 W/m2 K. Taking this as a point value, l D 330/68.1000b3 D 113.8 m
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Mean heat transfer coefficient (h W/m2 K)
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
330
250
200
150 0
1
2
3
4 5 6 7 Pipe length (l m)
8
9
10
Figure 9d.
which would imply that the flow is fully developed at this point. For further discussion on this point reference should be made to the turbulent flow of gases in Section 9.4.3.
PROBLEM 9.47 A liquid is boiled at a temperature of 360 K using steam fed at 380 K to a coil heater. Initially the heat transfer surfaces are clean and an evaporation rate of 0.08 kg/s is obtained from each square metre of heating surface. After a period, a layer of scale of resistance 0.0003 m2 K/W, is deposited by the boiling liquid on the heat transfer surface. On the assumption that the coefficient on the steam side remains unaltered and that the coefficient for the boiling liquid is proportional to its temperature difference raised to the power of 2.5, calculate the new rate of boiling.
Solution When the surface is clean, taking the wall and the inside resistances as negligible, the surface temperature will be 380 K. Thus:
Q D h0 A Ts 0004 T000b
where Q D GL, G kg/s is the rate of evaporation of fluid of latent heat L J/kg, A D 1 m2 , and Ts and T are the surface and fluid temperature respectively. ∴
0.08L D ho ð 1.0 380 0004 360 or ho D 0.004L ho / Ts 0004 T000b2.5
or:
ho D k 0 380 0004 360000b2.5 D 1.79 ð 103 k 0
∴
k 0 D 0.004L/ 1.79 ð 103 D 2.236 ð 1000046 L
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When the scale has formed, the total resistance is: 0.0003 C 1/[2.236 ð 1000046 L Ts 0004 360000b2.5 ] D 0.0003 C 4.472 ð 105 L 00041 Ts 0004 360000b00042.5 For conduction through the scale: GL D 380 0004 Ts /0.0003 D 3.33 ð 103 380 0004 Ts
(i)
For transfer through the outside film: GL D t 0004 360000b/[4.472 ð 105 L 00041 Ts 0004 360000b00042.5 ] 0004 2.236 ð 1000046 L Ts 0004 360000b3.5
(ii)
and for overall transfer: GL D 380 0004 360000b/[0.0003 C 4.472 ð 105 L 00041 Ts 0004 360000b00042.5 ]
(iii)
Inspection of these equations shows that the rate of evaporation G is a function not only of the surface temperature Ts but also of the latent heat of the fluid L. Using equations (i) and (ii) and selecting values of T in the range 360 to 380 K, the following results are obtained: Surface temperature Ts (K)
Mass rate of evaporation G (kg/s)
Latent heat of vaporisation L (kJ/kg)
362 364 366 368 370 372 374 376 378 380
0.000025 0.00029 0.0012 0.0033 0.0071 0.013 0.023 0.036 0.055 0.080
2,400,000 186,000 39,600 12,200 4710 1990 869 364 121 0
At a boiling point of 360 K it is likely that the liquid is organic with a latent heat of, say, 900 kJ/kg. This would indicate a surface temperature of 374 K and an evaporation rate of 0.023 kg/s. A precise result requires more specific data on the latent heat.
PROBLEM 9.48 A batch of reactants of specific heat 3.8 kJ/kg K and of mass 1000 kg is heated by means of a submerged steam coil of area 1 m2 fed with steam at 390 K. If the overall heat transfer coefficient is 600 W/m2 K, calculate the time taken to heat the material from 290 to 360 K if heat losses to the surroundings are neglected. If the external area of the vessel is 10 m2 and the heat transfer coefficient to the surroundings at 290 K is 8.5 W/m2 K, what will be the time taken to heat the reactants over the same temperature range and what is the maximum temperature to which the reactants can be raised? What methods would you suggest for improving the rate of heat transfer?
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Solution Use is made of equation 9.209: ln[ Ts 0004 T1 ]/ Ts 0004 T2 D UAt/GCp ∴
ln[ 390 0004 290000b/ 390 0004 360000b] D 600 ð 1.0t/ 1000 ð 3.8 ð 103 ln 3.33 D 0.000158t and t D 7620 s 2.12 h000b
or:
The heat lost from the vessel: QL D hAv T 0004 Ta , where Ta is the ambient temperature. ∴
QL D 8.5 ð 10.0 T 0004 290 D 85.0T 0004 24650 W
Heat from the steam D heat to the reactants C heat losses ∴
UA Ts 0004 T D GCp dT/dt C 85.0T 0004 24650 600 ð 1.0 390 0004 T D 1000 ð 3.8 ð 103 dT/dt C 85.0T 0004 24650 0004 t 0004 T2 dt D 5548 dT/ 3777.6 0004 T 0
∴
T1
t D 5548 ln[ 377.6 0004 T1 / 337.6T2 ] D 5548 ln[ 377.6 0004 290000b/ 377.6 0004 360000b] D 8904 s 2.47 h000b
The maximum temperature of the reactants is attained when the heat transferred from the steam is equal to the heat losses, or: UA Ts 0004 T D hAv T 0004 Ta Thus:
600 C 1.0 390 0004 T D 8.5 ð 10.0 T 0004 290 and T D 378 K
The heating-up time could be reduced by improving the rate of heat transfer to the fluid, by agitation of the fluid for example, and by reducing heat losses from the vessel by insulation. In the case of a large vessel there is a limit to the degree of agitation and circulation of the fluid through an external heat exchanger is an attractive alternative.
PROBLEM 9.49 What do you understand by the terms “black body” and “grey body” when applied to radiant heat transfer? Two large parallel plates with grey surfaces are situated 75 mm apart; one has an emissivity of 0.8 and is at a temperature of 350 K and the other has an emissivity of 0.4 and is at a temperature of 300 K. Calculate the net rate of heat exchange by radiation per square metre taking the Stefan–Boltzmann constant as 5.67 ð 1000048 W/m2 K4 . Any formula (other than Stefan’s law) which you use must be proved.
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Solution The terms “black body” and “grey body” are discussed in Sections 9.5.2 and 9.5.3. For two large parallel plates with grey surfaces, the heat transfer by radiation between them is given by putting A1 D A2 in equation 150 to give: q D [e1 e2 9/ e1 C e2 0004 e1 e2 ] T41 0004 T42 W/m2 In this case: q D [ 0.8 ð 0.4 ð 5.67 ð 1000048 / 0.8 C 0.4 0004 0.8 ð 0.4000b] 3504 0004 3004 D 0.367 ð 5.67 ð 1000048 ð 6.906 ð 109 D 143.7 W/m2
PROBLEM 9.50 A longitudinal fin on the outside of a circular pipe is 75 mm deep and 3 mm thick. If the pipe surface is at 400 K, calculate the heat dissipated per metre length from the fin to the atmosphere at 290 K if the coefficient of heat transfer from its surface may be assumed constant at 5 W/m2 K. The thermal conductivity of the material of the fin is 50 W/m K and the heat loss from the extreme edge of the fin may be neglected. It should be assumed that the temperature is uniformly 400 K at the base of the fin.
Solution The heat lost from the fin is given by equation 9.254: Qf D hbkA000b00061 tan hmL where h is the coefficient of heat transfer to the surroundings D 5 W/m2 K, b is the fin perimeter D 2 ð 0.075 C 0.003 D 0.153 m, k is the thermal conductivity of the fin D 50 W/mK, A is the cross-sectional area of the fin D 0.003 ð 1.0 D 0.003 m2 , 00061 is pthe temperature p difference at the root D T1 0004 TG D 400 0004 290 D 100 deg K, m D hb/kA D
5 ð 0.153000b/ 50 ð 0.003 D 2.258 and L is the length of the fin D 0.075 m. ∴ Qf D 5 ð 0.153 ð 50 ð 0.003000b[110 tanh 2.258 0004 0.075000b] D 0.339 ð 100 tanh 0.169 D 6.23 W/m
PROBLEM 9.51 Liquid oxygen is distributed by road in large spherical insulated vessels, 2 m internal diameter, well lagged on the outside. What thickness of magnesia lagging, of thermal conductivity 0.07 W/m K, must be used so that not more than 1% of the liquid oxygen evaporates during a journey of 10 ks (2.78 h) if the vessel is initially 80% full? Latent heat of vaporisation of oxygen D 215 kJ/kg. Boiling point of oxygen D 90 K. Density of liquid oxygen D 1140 kg/m3 . Atmospheric temperature D 288 K. Heat transfer coefficient from the outside surface of the lagging surface to atmosphere D 4.5 W/m2 K.
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Solution For conduction through the lagging: Q D 4$k T1 0004 T2 / 1/r1 0004 1/r2
(equation 9.25)
where T1 will be taken as the temperature of boiling oxygen D 90 K and the tank radius, r1 D 1.0 m. In this way, the resistance to heat transfer in the inside film and the walls is neglected. r2 is the outer radius of the lagging. ∴
Q D 4$ ð 0.07 90 0004 T2 / 1/1.0 0004 1/r2 W
(i)
For heat transfer from the outside of the lagging to the surroundings, Q D hA T2 0004 Ta where h D 4.5 W/m2 K, A D 4$r22 and Ta , ambient temperature D 288 K. ∴
Q D 4.5 ð 4$r22 T2 0004 288 D 18$r22 T2 0004 288 W
(ii)
The volume of the tank D 4$r13 /3 D 4$ ð 1.03 /3 D 4.189 m3 . ∴
volume of oxygen D 4.189 ð 80/100 D 3.351 m3 and mass of oxygen D 3.351 ð 1140 D 3820 kg
∴
mass of oxygen which evaporates D 3820 ð 1/100 D 38.2 kg 38.2/ 10 ð 103 D 0.00382 kg/s
or: ∴ heat flow into vessel: ∴ In (ii)
Q D 215 ð 103 ð 0.00382 D 821 W 821 D 18$r22 T2 0004 288 and T2 D 288 0004 14.52/r22
Substituting in (i): 821 D 4$ ð 0.07[90 0004 288 C 14.52/r22 ]/ 1 0004 1/r2 or:
r22 0004 1.27r2 C 0.0198 D 0 and r2 D 1.25 m
Thus the thickness of lagging D r2 0004 r1 D 0.25 m.
PROBLEM 9.52 Benzene is to be condensed at the rate of 1.25 kg/s in a vertical shell and tube type of heat exchanger fitted with tubes of 25 mm outside diameter and 2.5 m long. The vapour condenses on the outside of the tubes and the cooling water enters at 295 K and passes through the tubes at 1.05 m/s. Calculate the number of tubes required if the heat exchanger is arranged for a single pass of the cooling water. The tube wall thickness is 1.6 mm.
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Solution Preliminary calculation
At 101.3 kN/m2 , benzene condenses at 353 K at which the latent heat D 394 kJ/kg. ∴
heat load: Q D 1.25 ð 394 D 492 kW
The maximum water outlet temperature to minimise scaling is 320 K and a value of 300 K will be selected. Thus the water flow is given by: 492 D G ð 4.18 300 0004 295 or: ∴
G D 23.5 kg/s [or 23.5/1000 D 0.0235 m3 /s] area required for a velocity of 1.05 m/s D 0.0235/1.05 D 0.0224 m2
The cross-sectional area of a tube of 25 0004 2 ð 1.6 D 21.8 mm i.d. is:
$/4 ð 0.02182 D 0.000373 m2 and hence number of tubes required D 0.0224/0.000373 D 60 tubes. The outside area D $ ð 0.025 ð 2.5 ð 60 D 11.78 m2 00061 D 353 0004 295 D 58 deg K,
00062 D 353 0004 300 D 53 deg K
and in equation 9.9: 0006m D 58 0004 53000b/ ln 58/53 D 55.5 deg K ∴
U D 492/ 55.5 ð 11.78 D 0.753 kW/m2 K
This is quite reasonable as it falls in the middle of the range for condensing organics as shown in Table 9.17. It remains to check whether the required overall coefficient will be attained with this geometry. Overall coefficient
Inside: The simplified equation for water in tubes may be used: hi D 4280 0.00488T 0004 1000bu0.8 /di0.2 W/m2 K.
(equation 9.221)
where T D 0.5 300 C 295 D 297.5 K u D 105 m/s and di D 0.0218 m ∴
hi D 4280 0.00488 ð 297.5 0004 1000b1.050.8/0.2180.2 D 4322 W/m2 K or 4.32 kW/m2 K Based on the outside diameter: hio D 4.32 ð 0.218/0.025 D 3.77 kW/m2 K
Wall: For steel, k D 45 W/m K, x D 0.0016 m and hence: x/k D 0.0016/45 D 0.000036 m2 K/W
or
0.036 m2 K/kW
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Outside: For condensation on vertical tubes: ho 00182 /k 3 00192 g000b0.33 D 1.47 4M/0018000b00040.33
(equation 9.174)
The wall temperature is approximately 0.5 353 C 297.5 D 325 K, and the benzene film temperature will be taken as 0.5 353 C 325 D 339 K. At 339 K: k D 0.15 W/m K, 0019 D 880 kg/m3 , and 0018 D 0.35 ð 1000043 Ns/m2 . With 60 tubes, the mass flow of benzene per tube, G0 D 1.25/60 D 0.0208 kg/s. For vertical tubes, M D G0 /$do D 0.0208/ $ ð 0.025 D 0.265 kg/ms ∴ ho [ 0.35 ð 1000043 2 /0.152 ð 8802 ð 9.8]0.33 D 1.47[4 ð 0.0208/ 0.35 ð 1000043 ]00040.33 ∴
1.699 ð 1000044 ho D 1.47 ð 1.62 ð 1000041 ho D 1399 W/m2 K or 1.40 kW/m2 K
and: Overall: Neglecting scale resistances:
1/U D 1/hio C x/k C 1/ho D 0.265 C 0.036 C 0.714 D 1.015 m2 K/kW and:
U D 0.985 kW/m2 K
This is in excess of the value required and would allow for a reasonable scale resistance. If this were negligible, the water throughput could be reduced. On the basis of these calculations, 60 tubes are required.
PROBLEM 9.53 One end of a metal bar 25 mm in diameter and 0.3 m long is maintained at 375 K and heat is dissipated from the whole length of the bar to surroundings at 295 K. If the coefficient of heat transfer from the surface is 10 W/m2 K, what is the rate of loss of heat? Take the thermal conductivity of the metal as 85 W/m K.
Solution Use is made of equation 9.254: Qf D
hbkA000b00061 tanh mL
where the coefficient of heat transfer from the surface, h D 10 W/m2 K; the perimeter, b D $ ð 0.025 C 0.0785 m; the cross-sectional area, A D $/4 ð 0.0252 D 0.000491 m2 ; the thermal conductivity of the metal, k D 85 W/m K; the ptemperature difference at the root, 00061 D 375 0004 295 D 80 deg K; the value of m D hb/kA D p [ 10 ð 0.0785000b/ 85 ð 0.000491000b] D 4.337, and the length of the rod, L D 0.3 m. ∴ Qf D 10 ð 0.0785 ð 85 ð 0.000491000b[80 tanh 4.337 ð 0.3000b]
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D
185
0.0328 80 tanh 1.3011000b
D 14.49 e1.301 0004 e00041.301 / e1.301 C e00041.301 D 14.49 3.673 0004 0.272000b/ 3.673 C 0.272 D 12.5 W
PROBLEM 9.54 A shell-and-tube heat exchanger consists of 120 tubes of internal diameter 22 mm and length 2.5 m. It is operated as a single-pass condenser with benzene condensing at a temperature of 350 K on the outside of the tubes and water of inlet temperature 290 K passing through the tubes. Initially there is no scale on the walls, and a rate of condensation of 4 kg/s is obtained with a water velocity of 0.7 m/s through the tubes. After prolonged operation, a scale of resistance 0.0002 m2 K/W is formed on the inner surface of the tubes. To what value must the water velocity be increased in order to maintain the same rate of condensation on the assumption that the transfer coefficient on the water side is proportional to the velocity raised to the 0.8 power, and that the coefficient for the condensing vapour is 2.25 kW/m2 K, based on the inside area? The latent heat of vaporisation of benzene is 400 kJ/kg.
Solution Area for heat transfer, based on the tube i.d. D $ ð 0.022 ð 1.0 D 0.0691 m2 /m or:
120 ð 2.5 ð 0.0691 D 20.74 m2 .
With no scale
Heat load: Q D 4 ð 400 D 1600 W. Cross-sectional area of one tube D $/4000b0.0222 D 0.00038 m2 and hence area for flow per pass D 120 ð 0.00038 D 0.0456 m2 . ∴
and:
volume of flow of water D 0.0456 ð 0.7 D 0.0319 m3 /s mass flow of water D 0.0319 ð 1000 D 31.93 kg/s
The water outlet temperature is given by, 1600 D 31.93 ð 4.18 T 0004 290 or T D 302 K 00061 D 350 0004 290 D 60 deg K, 00062 D 350 0004 302 D 48 deg K and in equation 9.9, 0006m D 60 0004 48000b/ ln 60/48 D 53.8 deg K. In equation 9.1, U D Q/A0006m D 1600/ 20.74 ð 53.8 D 1.435 kW/m2 K Neglecting the wall resistance, 1/U D 1/hi C 1/h0i
1/1.435 D 1/hi C 1/2.25 and hi D 3.958 kW/m2 K hi is proportional to u0.8 or 3.958 D k 0.7000b0.8 and k D 5.265
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With scale
hi D 5.265u0.8 kW/m2 K, scale resistance D 0.20 m2 K/kW 1/U D 1/ 5.265u0.8 C 0.20 C 1/2.25000b
and: ∴
U D u0.8 / 0.190 C 0.644u0.8 kW/m2 K
that is:
Q D 1600 kW as before.
The mass flow of water is: u ð 0.0456 ð 1000 D 45.6u kg/s and the outlet water temperature is given by: 1600 D 45.6u ð 4.18 T 0004 290 or:
T D 290 C 8.391/u K 00061 D 350 0004 290 D 60 deg K, 00062 D 350 0004 290 0004 8.391000b/u D 60 0004 8.391000b/u
and: 0006m D 60 0004 60 0004 8.391000b/u000b/ ln[60/ 60 0004 8.391000b/u] D 8.391/fu ln[60u/ 60u 0004 8.391000b]g In equation 9.1: 1600 D [u0.8 / 0.190 C 0.644u0.8 ] ð 20.74 ð 8.391/fu ln[60u/ 60u 0004 8.391000b]g or:
1/fu0.2 0.190 C 0.644u0.8 ln[60u/ 60u 0004 8.391000b]g D 9.194
The left-hand side of this equation is plotted against u in Fig. 9e and the function equals 9.194 when u D 2.06 m/s.
9.2
9.194
9.0
Function
8.8 8.6 8.4 2.06
8.2 8.0 0.8
1.0 1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
Water velocity (u m/s)
Figure 9e.
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HEAT TRANSFER
PROBLEM 9.55 Derive an expression for the radiant heat transfer rate per unit area between two large parallel planes of emissivities e1 and e2 and at absolute temperatures T1 and T2 respectively. Two such planes are situated 2.5 mm apart in air. One has an emissivity of 0.1 and is at a temperature of 350 K, and the other has an emissivity of 0.05 and is at a temperature of 300 K. Calculate the percentage change in the total heat transfer rate by coating the first surface so as to reduce its emissivity to 0.025. Stefan–Boltzmann constant D 5.67 ð 1000048 W/m2 K4 . Thermal conductivity of air D 0.026 W/m K.
Solution The theoretical derivation is laid out in Section 9.5.5 and the heat transfer by radiation is given by putting A1 D A2 in equation 9.150 to give: qr D [ e1 e2 9000b/ e1 C e2 0004 e1 e2 ] T41 0004 T42 For conduction between the two planes: (equation 9.12)
qc D kA T1 0004 T2 /x D 0.026 ð 1.0 350 0004 300000b/0.0025 D 520 W/m
2
For radiation between the two planes: qr D [ e1 e2 9000b/ e1 C e2 0004 e1 e2 ] T41 0004 T42 D [ 0.1 ð 0.05 ð 5.67 ð 1000048 / 0.1 C 0.05 0004 0.1 ð 0.05000b] 3504 0004 3004 D 13.5 W/m2 Thus neglecting any convection in the very narrow space, the total heat transferred is 533.5 W/m2 . When e1 D 0.025, the heat transfer by radiation is: qr D [ 0.025 ð 0.05 ð 5.67 ð 1000048 / 0.025 C 0.05 0004 0.025 ð 0.05000b] ð 3504 0004 3004 D 6.64 W/m2 and: qr C qc D 526.64 W/m2 Thus, although the heat transferred by radiation is reduced to 100 ð 6.64000b/13.5 D 49.2% of its initial value, the total heat transferred is reduced to 100 ð 526.64000b/533.5 D 98.7% of the initial value.
PROBLEM 9.56 Water flows at 2 m/s through a 2.5 m length of a 25 mm diameter tube. If the tube is at 320 K and the water enters and leaves at 293 and 295 K respectively, what is the value of the heat transfer coefficient? How would the outlet temperature change if the velocity was increased by 50%?
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Solution The cross-sectional area of 0.025 m tubing D $/4000b0.0252 D 0.000491 m2 . Volume flow of water D 2 ð 0.000491 D 0.000982 m3 /s Mass flow of water D 1000 ð 0.000982 D 0.982 kg/s ∴
Heat load, Q D 0.982 ð 4.18 295 0004 293 D 8.21 kW
Surface area of 0.025 m tubing D $ ð 0.025 ð 1.0 D 0.0785 m2 /m A D 0.0785 ð 2.5 D 0.196 m2
and:
00061 D 320 0004 293 D 27 deg K, 00062 D 320 0004 295 D 25 deg K 0006m D 27 0004 25000b/ ln 27/25 D 25.98 say 26 deg K
and:
In equation 9.1: U D 8.21/ 0.196 ð 26 D 1.612 kW/m2 K An estimate may be made of the inside film coefficient from equation 9.221, where T, the mean water temperature, is 294 K. Thus:
hi D 4280 0.00488 ð 294 0004 1000b2.00.8 /0.0250.2 D 4280 ð 0.435 ð 1.741/0.478 D 6777 W/m2 K or 6.78 kW/m2 K
The scale resistance is therefore given by:
1/1.612 D 1/6.78 C R or: R D 0.473 m2 K/kW With a water velocity of 2.0 ð 150/100 D 3.0 m/s, assuming a mean water temperature of 300 K, then: hi D 4280 0.00488 ð 300 0004 1000b3.00.8 /0.0250.2 D 4280 ð 0.464 ð 2.408/0.478 D 10004 or 10.0 kW/m2 K ∴
1/U D 0.473 C 1/10.0 and U D 1.75 kW/m2 K
For an outlet water temperature of T K: 00061 D 27 deg K, and, taking an arithmetic mean:
00062 D 320 0004 T deg K
0006m D 0.5 27 C 320 0004 T D 173.5 0004 0.5T deg K.
The mass flow of water D 0.982 ð 150000b/100 D 1.473 kg/s, and the heat load, ∴
from which:
Q D 1.473 ð 4.18 T 0004 293 D 6.157T 0004 1804 kW
6.157T 0004 1804 D [1.75 ð 0.196 173.5 0004 0.5T000b] T D 294.5 K
The use of 300 K as a mean water temperature has a minimal effect on the result and recalculation is not necessary.
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PROBLEM 9.57 A liquid hydrocarbon is fed at 295 K to a heat exchanger consisting of a 25 mm diameter tube heated on the outside by condensing steam at atmospheric pressure. The flowrate of hydrocarbon is measured by means of a 19 mm orifice fitted to the 25 mm feed pipe. The reading on a differential manometer containing hydrocarbon-over-water is 450 mm and the coefficient of discharge of the meter is 0.6. Calculate the initial rate of rise of temperature (deg K/s) of the hydrocarbon as it enters the heat exchanger. The outside film coefficient D 6.0 W/m2 K. The inside film coefficient h is given by: hd/k D 0.023 ud0019/0018000b0.8 Cp 0018/k000b0.4 where: u D linear velocity of hydrocarbon (m/s). d D tube diameter (m), 0019 D liquid density
800 kg/m3 , 0018 D liquid viscosity 9 ð 1000044 N s/m2 , Cp D specific heat of liquid (1.7 ð 103 J/kgK), and k D thermal conductivity of liquid (0.17 W/mK).
Solution The effective manometer fluid density, is 200 kg/m3 . The pressure difference across the orifice D 450 mm water or: that is:
450 ð 800/200 D 1800 mm hydrocarbon H D 1.80 m
The area of the orifice D $/4000b0.0192 D 2.835 ð 1000044 m2 In equation 6.21: G D 0.6 ð 2.835 ð 1000044 ð 800 2 ð 9.81 ð 1.80 D 1.36 35.3 D 0.808 kg/s The volume flow D 0.808/800 D 0.00101 m3 /s. Cross-sectional area of a 0.025 m diameter pipe D $/4000b0.0252 D 0.000491 m2 and hence the velocity, u D 0.00101/0.000491 D 2.06 m/s. The inside film coefficient is given by:
hi ð 0.025/0.17 D 0.023
2.06 ð 0.025 ð 800000b/9 ð 1000044 0.8 ð
1.7 ð 103 ð 9 ð 1000044 /0.17000b0.4 or:
hi D 0.1564 4.58 ð 104 0.8 9.0000b0.4 D 2016 W/m2 K or 2.02 kW/m2 K
Neglecting scale and wall resistances: 1/U D 1/6.0 C 1/2.02 and U D 1.511 kW/m2 K For steam at atmospheric pressure, the saturation temperature D 373 K and at the inlet the temperature driving force D 373 0004 295 D 78 deg K.
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The heat flux is: 1.511 ð 78 D 117.9 kW/m2 . For a small length of tube, say 0.001 m, the area for heat transfer D $ ð 0.025 ð 0.001 D 7.854 ð 1000045 m2 and the heat transfer rate D 117.9 ð 7.854 ð 1000045 ð 1000 D 9.27 W. In the small length (0.001 m) of tube, mass of material D 0.000491 ð 0.001 ð 800 D 3.93 ð 1000044 kg and hence temperature rise D [9.27/ 3.93 ð 1000044 ð 1.7 ð 103 ] D 13.9 deg K/s
PROBLEM 9.58 Water passes at a velocity of 1.2 m/s through a series of 25 mm diameter tubes 5 m long maintained at 320 K. If the inlet temperature is 290 K, at what temperature would it leave?
Solution Assuming an outlet water temperature of T K, the mean water temperature is therefore: D 0.5 T C 290 D 0.5T C 145 K. The coefficient may be calculated from: h D 4280 0.00488T 0004 1000bu0.8 /d0.2
(equation 9.221)
D 4280[0.00488 0.5T C 145 0004 1]1.20.8 /0.0250.2 D 25.28T 0004 3028.1 W/m2 K Area for heat transfer D $ ð 0.025 ð 5.0 D 0.393 m2 and the heat load, Q D [1.2 $/4000b0.0252 ð 1000 ð 4.18 ð 103 T 0004 290000b] D 2462T 0004 714,045 W Therefore neglecting any scale resistance:
2462T 0004 714,045 D 25.28T 0004 3028.1000b0.393[320 0004 0.5T C 145000b] from which: and:
T2 C 25.98T 0004 101,851 D 0 T D 306.4 K
[An alternative approach is as follows: The heat transferred per unit time in length dL of pipe, D h ð $ ð 0.025dL 320 0004 Tk W where Tk is the water temperature at L m from the inlet. The rate of increase in the heat content of the water is:
$/4 ð 0.0252 ð 1.2 ð 1000 ð 4.18 ð 103 dT D 2462 dT
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The outlet temperature T0 is then given by: 0004 T0 0004 5 dT D 0.0000319h dL 290 320 0004 T 0 ln 320 0004 T0 D ln 30 0004 0.0001595h D 3.401 0004 0.0001595h
or:
At a mean temperature of say 300 K, in equation 9.221: h D 4280 0.00488 ð 300 0004 1000b1.20.8 /0.0250.2 D 4805 W/m2 K Thus: ln 320 0004 T0 D 3.401 0004 0.0001595 ð 4805 T0 D 306.06 K]
and:
PROBLEM 9.59 Heat is transferred from one fluid stream to a second fluid across a heat transfer surface. If the film coefficients for the two fluids are, respectively, 1.0 and 1.5 kW/m2 K, the metal is 6 mm thick (thermal conductivity 20 W/m K) and the scale coefficient is equivalent to 850 W/m2 K, what is the overall heat transfer coefficient?
Solution From equation 9.201: 1/U D 1/ho C xw /kw C R C 1/hi D 1/1000 C 10.006/20 C 1/850 C 1/1500 D 0.001 C 0.00030 C 0.00118 C 0.00067 D 0.00315 m2 K/W ∴
U D 317.5 W/m2 K or 0.318 kW/m2 K
PROBLEM 9.60 A pipe of outer diameter 50 mm carries hot fluid at 1100 K. It is covered with a 50 mm layer of insulation of thermal conductivity 0.17 W/m K. Would it be feasible to use magnesia insulation, which will not stand temperatures above 615 K and has a thermal conductivity of 0.09 W/m K for an additional layer thick enough to reduce the outer surface temperature to 370 K in surroundings at 280 K? Take the surface coefficient of transfer by radiation and convection as 10 W/m2 K.
Solution The solution is presented as Problem 9.8.
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PROBLEM 9.61 A jacketed reaction vessel containing 0.25 m3 of liquid of density 900 kg/m3 and specific heat 3.3 kJ/kg K is heated by means of steam fed to a jacket on the walls. The contents of the tank are agitated by a stirrer rotating at 3 Hz. The heat transfer area is 2.5 m2 and the steam temperature is 380 K. The outside film heat transfer coefficient is 1.7 kW/m2 K and the 10 mm thick wall of the tank has a thermal conductivity of 6.0 W/m K. The inside film coefficient was 1.1 kW/m2 K for a stirrer speed of 1.5 Hz and proportional to the two-thirds power of the speed of rotation. Neglecting heat losses and the heat capacity of the tank, how long will it take to raise the temperature of the liquid from 295 to 375 K?
Solution For a stirrer speed of 1.5 Hz, hi D 1.1 kW/m2 K. ∴
1.1 D k 0 1.50.67 and k 0 D 0.838 Thus at a stirrer speed of 3 Hz, hi D 0.838 ð 3.00.67 D 1.75 kW/m2 K. The overall coefficient is given by: 1/U D 1/1750 C 0.010/6.0 C 1/1700 C 0.00283 (equation 9.201)
and:
U D 353.8 W/m2 K neglecting scale resistances.
The time for heating the liquid is given by: ln[ Ts 0004 T1 / Ts 0004 T2 ] D UAt/mCp
(equation 9.209)
In this case: m D 0.25 ð 900 D 225 kg and Cp D 3300 J/kg K. ∴
ln[ 380 0004 295000b/ 380 0004 375000b] D 353.8 ð 2.5t/ 225 ð 3300 2.833 D 0.00119t and t D 2381 s 40 min000b
PROBLEM 9.62 By dimensional analysis, derive a relationship for the heat transfer coefficient h for natural convection between a surface and a fluid on the assumption that the coefficient is a function of the following variables: k D thermal conductivity of the fluid, Cp D specific heat of the fluid, 0019 D density of the fluid, 0018 D viscosity of the fluid, ˇg D the product of the coefficient of cubical expansion of the fluid and the acceleration due to gravity, l D a characteristic dimension of the surface, and T D the temperature difference between the fluid and the surface. Indicate why each of these quantities would be expected to influence the heat transfer coefficient and explain how the orientation of the surface affects the process. Under what conditions is heat transfer by natural convection important in Chemical Engineering?
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Solution If the heat transfer coefficient h can be expressed as a product of powers of the variables, then: h D k 0 k a Cpb 0019c 0018d ˇg000be lf Tg where k 0 is a constant. The dimensions of each variable in terms of M, L, T, Q, and q are: heat transfer coefficient, h D Q/L2 Tq thermal conductivity, k D Q/LTq specific heat, Cp D Q/Mq viscosity, 0018 D M/LT density, 0019 D M/L3 the product, ˇg D L/T2 q00041 length, lDL temperature difference, T D q Equating indices: M: L: T: Q: q:
0 00042 00041 1 00041
D 0004b C c C d D 0004a 0004 3c 0004 d C e C f D 0004a 0004 d 0004 2e DaCb D 0004a 0004 b 0004 e C g
Solving in terms of b and c: a D 1 0004 b000b, d D b 0004 c000b, e D c/2000b, f D 3c/2 0004 1000b, g D c/2 and hence: h D k0 or:
0002
0003 0002 00030002 0003 0002 0003c 3c/2 k b c 0018b Cp 0018 b l3/2 0019 ˇg000b1/2 T1/2 c/2 l c/2 0 k C 0019
ˇg D k T k b p 0018c l l k 0018 0002 0003b 0002 3 2 0003c/2 Cp 0018 l 0019 ˇgT hl D k0 k k 00182
where Cp 0018/k is the Prandtl number and l3 00192 ˇgT/00182 the Grashof number. A full discussion of the significance of this result and the importance of free of natural convection is presented in Section 9.4.7.
PROBLEM 9.63 A shell-and-tube heat exchanger is used for preheating the feed to an evaporator. The liquid of specific heat 4.0 kJ/kg K and density 1100 kg/m3 passes through the inside of tubes and is heated by steam condensing at 395 K on the outside. The exchanger heats liquid at 295 K to an outlet temperature of 375 K when the flowrate is 1.75 ð 1000044 m3 /s and to 370 K when the flowrate is 3.25 ð 1000044 m3 /s. What is the heat transfer area and the value of the overall heat transfer coefficient when the flow rate is 1.75 ð 1000044 m3 /s?
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Assume that the film heat transfer coefficient for the liquid in the tubes is proportional to the 0.8 power of the velocity, that the transfer coefficient for the condensing steam remains constant at 3.4 kW/m2 K and that the resistance of the tube wall and scale can be neglected.
Solution i) For a flow of 1.75 ð 1000044 m3 /s: Density of the liquid D 1100 kg/m3 Mass flow
D 1.75 ð 1000044 ð 1100 D 0.1925 kg/s.
Heat load
D 0.1925 ð 4.0 373 0004 295 D 61.6 kW 00061 D 395 0004 295 D 100 deg K,
00062 D 395 0004 375 D 20 deg K
and in equation 9.9: 0006m D 100 0004 20000b/ ln 100/20 D 49.7 deg K Thus, in equation 9.1: U1 A D 61.6/49.7 D 1.239 kW/K ii) For a flow of 3.25 ð 1000044 m3 /s: Mass flow D 3.25 ð 1000044 ð 1100 D 0.3575 kg/s Heat load D 0.3575 ð 4.0 370 0004 295 D 107.3 kW 00061 D 395 0004 295 D 100 deg K,
00062 D 395 0004 370 D 25 deg K
and in equation 9.9: 0006m D 100 0004 25000b/ ln 100/25 D 54.1 deg K Thus in equation 9.1: U2 A D 107.3/54.1 D 1.983 kW/K ∴
U2 /U1 D 1.983/1.239 D 1.60
The velocity in the tubes is proportional to the volumetric flowrate, v cm3 /s and hence hi / v0.8 or hi D k 0 v0.8 , where k 0 is a constant. Neglecting scale and wall resistances: 1/U D 1/h0 C 1/hi D 1/3.4 C 1/k 0 v0.8 and: U D 3.4k 0 v0.8 / 3.4 C k 0 v0.8
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∴
195
U1 D 3.4k 0 ð 1750.8 / 3.4 C k 0 ð 1750.8 D 211.8k 0 / 3.4 C 62.3k 0
and: U2 D 3.4k 0 ð 3250.8 / 3.4 C k 0 ð 3250.8 D 347.5k 0 / 3.4 C 102.2k 0 ∴ ∴ ∴
[347.5k 0 / 3.4 C 102.2k 0 ]/[211.18k 0 / 3.4 C 62.3k 0 ] D 1.60 k 0 D 0.00228 U1 D 3.4 ð 0.00228 ð 1750.8 / 3.4 C 0.00228 ð 1750.8 D 0.136 kW/m2 K
and the heat transfer area, A D 1.239/0.136 D 9.09 m2 .
PROBLEM 9.64 0.1 m3 of liquid of specific heat capacity 3 kJ/kg K and density 950 kg/m3 is heated in an agitated tank fitted with a coil, of heat transfer area 1 m2 , supplied with steam at 383 K. How long will it take to heat the liquid from 293 to 368 K, if the tank, of external area 20 m2 is losing heat to the surroundings at 293 K? To what temperature will the system fall in 1800 s if the steam is turned off? Overall heat transfer coefficient in coil D 2000 W/m2 K. Heat transfer coefficient to surroundings D 10 W/m2 K.
Solution If T K is the temperature of the liquid at time t s, then: heat input from the steam D UA Ts 0004 T D 2000 ð 1 383 0004 T or 2000 383 0004 T W Similarly, heat losses to the surroundings D 10 ð 20 T 0004 293 D 200 T 0004 293 W and, net heat input to the liquid D 2000 383 0004 T 0004 200 T 0004 293 D 824,600 0004 2200T W This is equal to:
Q D mCp dT/dt000b
where m D 0.1 ð 950 D 95 kg and Cp D 3000 J/kg K. ∴
95 ð 3000000bdT/dt D 824,600 0004 2200T000b
or:
129.6 dT/dt D 374.8 0004 T000b
Thus the time taken to heat from 293 to 368 K is: 0004 368 dT/ 374.8 0004 T t D 129.6 293
D 129.6 ln
374.8 0004 293000b/ 374 0004 368 D 1559 s 0.43 h The steam is turned off for 1800s and, during this time, a heat balance gives:
95 ð 3000000bdT/dt D 0004 10 ð 20 T 0004 293 ∴
285,000 dT/dt D 58600 0004 200T or 1425 dT/dt D 293 0004 T000b
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The change in temperature is then given by: 0004 T 0004 dT/ 293 0004 T D 1/1425 368
1800
dt 0
ln
293 0004 368000b/ 293 0004 T D 1800/1425 D 1.263 and T D 311.8 K .
PROBLEM 9.65 The contents of a reaction vessel are heated by means of steam at 393 K supplied to a heating coil which is totally immersed in the liquid. When the vessel has a layer of lagging 50 mm thick on its outer surfaces, it takes one hour to heat the liquid from 293 to 373 K. How long will it take if the thickness of lagging is doubled? Outside temperature D 293 K. Thermal conductivity of lagging D 0.05 W/mK. Coefficient for heat loss by radiation and convection from outside surface of vessel D 10 W/m2 K. Outside area of vessel D 8 m2 . Coil area D 0.2 m2 . Overall heat transfer coefficient for steam coil D 300 W/m2 K.
Solution If T K is the temperature of the liquid at time t s and T1 K the temperature at the outside surface of the vessel, then heat flowing through the insulation is equal to the heat lost by convection and radiation to the surroundings or:
kA/x T 0004 T1 D hc A T1 0004 T0 where hc is the coefficient for heat loss, A the outside surface area of the vessel and T0 the ambient temperature. Thus:
0.05 ð 8/0.050 T 0004 T1 D 10 ð 8 T1 0004 293 T1 D 0.0909T C 266.4 K
and: ∴ Heat loss to the surroundings
D 10 ð 8 0.0909T C 266.4 0004 293000b
D 7.272T 0004 2128 W Heat input from the coil D 300 ð 0.2 393 0004 T D 23580 0004 60T W and net heat input D 23580 0004 60T 0004 7.272T 0004 2128 D 25708 0004 67.3T W which is equal to: Q D mCp dT/dt or:
mCp dT/dt D 25708 0004 67.3T
and:
0.0149mCp dT/dt D 382 0004 T
It takes t D 3600 s to heat the contents from 293 to 373 K, or: 0004 373 3600 D 0.0149mCp dT/ 382 0004 T 293
∴
and:
241610 D mCp ln
382 0004 293000b/ 382 0004 373 D 2.291mCp mCp D 105442 J/K
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If the thickness of the lagging is doubled to 0.100 m, then:
0.05 ð 8/0.100 T 0004 T1 D 10 ð 8 T1 0004 293 T1 D 0.0476T C 279.1 K
and: ∴ Heat loss to the surroundings
D 10 ð 8 0.0476T C 279.1 0004 293000b
D 3.808T 0004 1112 W Heat input from the coil D 300 ð 0.2 393 0004 T D 23580 0004 60T W and net heat input D 23580 0004 60T 0004 3.808T 0004 1112 D 24692 0004 63.808T. ∴
mCp dT/dt D 24,692 0004 63.808T 105442 dT/dt D 24,692 0004 63.808T000b
or:
1652.5 dT/dt D 387 0004 T
Thus, the time taken to heat the contents from 293 to 373 K is: 0004 373 dT/ 387 0004 T t D 1652.5 293
D 1625.5 ln[ 387 0004 293000b/ 387 0004 373000b] D 1652.5 ð 1.904 D 3147 s 0.87 h000b
PROBLEM 9.66 A smooth tube in a condenser which is 25 mm internal diameter and 10 m long is carrying cooling water and the pressure drop over the length of the tube is 2 ð 104 N/m2 . If vapour at a temperature of 353 K is condensing on the outside of the tube and the temperature of the cooling water rises from 293 K at inlet to 333 K at outlet, what is the value of the overall heat transfer coefficient based on the inside area of the tube? If the coefficient for the condensing vapour is 15,000 W/m2 K, what is the film coefficient for the water? If the latent heat of vaporisation is 800 kJ/kg, what is the rate of condensation of vapour?
Solution From equation 3.23: R/0019u2 Re2 D 0004Pf d3 0019/ 4l/00182 Taking the viscosity of water as 1 mN s/m2 0.001 Ns/m2 , then: 0004Pf d3 0019/ 4l/00182 D 20,000 0.025000b3 1000/ 4 ð 10 0.001000b2 D 7,812,500 From Fig. 3.8, for a smooth pipe, Re D 57,000 ∴
and:
du0019/0018 D 0.025u1000000b/0.001 D 57,000 u D 2.28 m/s
∴ Volume flow of water D $ 0.0252 /4000b2.28 D 0.00112 m3 /s
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Mass flow of water D 1000 ð 0.00112 D 1.12 kg/s Heat removed by water D 1.12 ð 4.187 333 0004 293 D 187.6 kW Surface area of tube, based on inside diameter D $ ð 0.025 ð 10 D 0.785 m2 Vapour temperature = 353 K
∴ T1 D 353 0004 293 D 60 deg K
T2 D 353 0004 333 D 20 deg K and from equation 9.9, Tm D 60 0004 20000b/ ln 60/20 D 36.4 deg K From equation 9.1, 187.6 D U ð 0.785 ð 36.4 and the overall coefficient based on the inside diameter is: U D 6.57 kW/m2 K. In equation 9.201, neglecting the wall and scale resistances: 1/U D 1/ho C 1/hi
1/6.57 D 1/15.0 C 1/hi and hi D 11.68 kW/m2 K If the latent heat of condensation is 800 kJ/kg, then assuming the vapour enters and the condensate leaves at the boiling point: rate of condensation D 187.6/800 D 0.235 kg/s
PROBLEM 9.67 A chemical reactor, 1 m in diameter and 5 m long, operates at a temperature of 1073 K. It is covered with a 500 mm thickness of lagging of thermal conductivity 0.1 W/m K. The heat loss from the cylindrical surface to the surroundings is 3.5 kW. What is the heat transfer coefficient from the surface of the lagging to the surroundings at a temperature of 293 K? How would the heat loss be altered if the coefficient were halved?
Solution From equation 9.20, the heat flow at any radius r is given by: Q D 0004k 2$ C 1000bdT/dr W ∴
dr/r D 00042$kl/Q000bdT
Integrating between the limits r1 and r2 at which the temperatures are T1 and T2 respectively: 0004 r2 0004 T2
dr/r D 00042$kl/Q dT r1
∴
T1
ln r2 /r1 D 00042$kl/Q T1 0004 T2
In this case: r1 D 1/2 D 0.50 m, r2 D 0.50 C 500/1000 D 1.0 m, l D 5 m, k D 0.1 W/mK, Q D 3500 W and T1 D 1073 K. ∴
and:
ln 1.0/0.50 D 2$ ð 0.1 ð 5/3500 1073 0004 T2 0.693 D 0.00090 1073 0004 T2 and T2 D 301 K
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The heat flow to the surroundings is: Q D ho Ao T2 0004 To ∴
3500 D ho $ ð 2.0 ð 5.0 301 0004 293 and ho D 14.05 W/m2 K
If this value is halved, that is ho2 D 7.02 W/m2 K, then: Q2 D 7.02 2$ ð 2.0 ð 5.0 T2 0004 293 D 220.5 T2 0004 293 T2 D Q2 /220.5 C 293 K.
and: But:
Q2 /Q1 D T1 0004 T2 2 / T1 0004 T2 1
∴
Q2 /3500 D [1073 0004 Q2 /220.5 0004 293]/ 1073 0004 301 0.000286Q2 D 0.00130 780 0004 0.00454Q2 Q2 D 3473 W — a very slight reduction in the heat loss.
and:
In this case, T2 D 3473/220.5 C 293 D 308.7 K.
PROBLEM 9.68 An open cylindrical tank 500 mm diameter and 1 m deep is three-quarters filled with a liquid of density 980 kg/m3 and of specific heat capacity 3 kJ/kg K. If the heat transfer coefficient from the cylindrical walls and the base of the tank is 10 W/m2 K and from the surface is 20 W/m2 K, what area of heating coil, fed with steam at 383 K, is required to heat the contents from 288 K to 368 K in a half hour? The overall heat transfer coefficient for the coil may be taken as 100 W/m2 K. The surroundings are at 288 K. The heat capacity of the tank itself may be neglected.
Solution The rate of heat transfer from the steam to the liquid is: Uc Ac 383 0004 T D 100Ac 383 0004 T W where Ac is the surface area of the coil. The rate of heat transfer from the tank to the surroundings D UT AT T 0004 288 where UT is the effective overall coefficient and AT the surface area of the tank and liquid surface. In this case: UT AT D 10
$ ð 0.5 ð 1 C $/4000b0.52 C 20 $/4000b0.52 D 21.6 W/K. ∴ rate of heat loss D 21.6 T 0004 288 W ∴ net rate of heat input to the tank D 100Ac 383 0004 T 0004 21.6 T 0004 288 W.
This is equal to mCp dT/dt, where the mean specific heat, Cp D 3000 J/kgK. Volume of liquid D 75/100 $/4000b0.52 ð 1 D 0.147 m3
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Mass of liquid: m D 0.147 ð 980 D 144.3 kg and: ∴
mCp D 144.3 ð 3000 D 432,957 J/K. 432,957 dT/dt D 100Ac 383 0004 T 0004 21.6 T 0004 288 D 38,300Ac C 6221 0004 100Ac C 21.6000bT 0004 ∴
368
dT/
38,300Ac C 6221000b/ 100Ac C 21.6 0004 T 288
0004
1800
D
100Ac C 21.6000b/432,957000b
dt 0
∴
lnf[ 38,300Ac C 6221000b/ 100Ac C 21.6 0004 288]/ [ 38,300Ac C 6221000b/ 100Ac C 21.6 0004 368000b]g D 0.00416 100Ac C 21.6 This equation is solved by trial and error to give: Ac D 5.0 m2 .
PROBLEM 9.69 Liquid oxygen is distributed by road in large spherical vessels, 1.82 m in internal diameter. If the vessels were unlagged and the coefficient for heat transfer from the outside of the vessel to the atmosphere were 5 W/m2 K, what proportion of the contents would evaporate during a journey lasting an hour? Initially the vessels are 80% full. What thickness of lagging would be required to reduce the losses to one tenth? Atmospheric temperature D 288 K. Boiling point of oxygen D 90 K. Density of oxygen D 1140 kg/m3 . Latent heat of vaporisation of oxygen D 214 kJ/kg. Thermal conductivity of lagging D 0.07 W/m K.
Solution Volume of the vessel Volume of liquid oxygen Mass of liquid oxygen Surface area of unlagged vessel Heat leakage into the vessel
∴ ∴
D $d3 /6 D $ ð 1.823 /6 D 3.16 m3 D 80/100000b3.16 D 2.53 m3 D 2.53 ð 1140 D 2879 kg D $ ð 1.822 D 10.41 m2 D hc A T1 0004 T2 D 5.0 ð 10.41 288 0004 90 D 10,302 W or 10.3 kW
Evaporation rate of oxygen D 10.3/214 D 0.048 kg/s Evaporation taking place during 1 h D 0.048 ð 3600 D 173.3 kg
which is 100 ð 173.3/2879 D 6.02% of the contents In order to reduce the losses to one tenth, the heat flow into the vessel must be 1.03 kW and this will be achieved by reducing the temperature driving force to:
288 0004 90000b/10 D 19.8 deg K.
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In this case the outside temperature of the lagging will be 288 0004 19.8 D 268.2 K and the temperature drop through the lagging will be 268.2 0004 90 D 178.2 deg K. Thus, the heat flow through the lagging is: 1030 D kA/x000bTlagging D 0.07 ð 10.41/x000b178.2 from which the thickness of the lagging, x D 0.126 m or 126 mm This calculation does not take into account the increase in the surface area at the lagging surface since it was assumed to be that of the tank, 10.41 m2 . In practice, it will be larger than this and, if this is taken into account, the reasoning is as follows: Radius of the tank D 1.82/2 D 0.91 m ∴ For a lagging thickness of x m, the new radius is 0.91 C x000bm and the surface area is: 4$ 0.91 C x000b2 m2 ∴ convective heat gain D 5.0 ð 4$ 0.91 C x000b2 288 0004 T D 1030 W and the outside temperature of the lagging:
T D 288 0004 16.39/ 0.91 C x000b2 K
(i)
The heat flow through the lagging (taking an arithmetic mean area and neglecting the curvature), 1030 D 0.07/x000b4$ 0.91 C x/2000b2 T 0004 90 Substituting for T from (i) into (ii): 1030 D 0.07/x000b4$ 0.91 C x/2000b2 198 0004 16.39000b/ 0.91 C x000b2 Solving by trial and error: x D 0.151 m or 151 mm
PROBLEM 9.70 Water at 293 K is heated by passing it through a 6.1 m coil of 25 mm internal diameter pipe. The thermal conductivity of the pipe wall is 20 W/m K and the wall thickness is 3.2 mm. The coil is heated by condensing steam at 373 K for which the film coefficient is 8 kW/m2 K. When the water velocity in the pipe is 1 m/s, its outlet temperature is 309 K. What will the outlet temperature be if the velocity is increased to 1.3 m/s, if the coefficient of heat transfer to the water in the tube is proportional to the velocity raised to the 0.8 power?
Solution The surface area of the coil D $do l D $
25 C 2 ð 3.2000b/1000000b6.1 D 0.602 m2 i) When the water velocity is 1 m/s: Area for flow D $d2i /4 D $ 25/1000000b2 /4 D 0.00049 m2 Volume flow D 0.00049 ð 1.0 D 0.00049 m3 /s
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Taking the density as 1000 kg/m3 , Mass flow of water D 1000 ð 0.00049 D 0.491 kg/s and taking the mean specific heat as 4.18 kJ/kg K, Heat load D 0.491 ð 4.18 309 0004 293 D 32.83 kW With steam at 373 K,
T1 D 373 0004 293 D 80 deg K, T2 D 373 0004 309 D 64 deg K
and from equation 9.9, Tm D 80 0004 64000b/ ln 80/64 D 71.7 deg K Therefore from equation 9.1, the overall coefficient, U D 32.83/ 0.602 ð 71.7 D 0.761 kW/m2 K or 761 W/m2 K. From equation 9.201, and neglecting any scale resistance: 1/U D 1/hi C 1/h0 C x/k In this case, ho D 8 kW/m2 K D 8000 W/m2 K, k D 20 W/mK, x D 3.2 mm or 0.0032 m and hi D Ku0.8 where u D 1 m/s and K is a constant. ∴
1/761 D 1/K10.8 C 1/8000 C 0.0032/20 0.00131 D 1/K C 0.000125 C 0.00016 and K D 976
ii) When the velocity is 1.3 m/s Volume flow of water D 0.00049 ð 1.3 D 0.000637 m3 /s Mass flow of water
D 1000 ð 0.000637 D 0.637 kg/s
∴ Heat load
D 0.637 ð 4.18 T 0004 293 D 2.663 T 0004 293 kW or 2663 T 0004 293 W
The inside coefficient, hi D 976 ð 1.30.8 D 1204 W/m2 K and the overall coefficient, U is given by: 1/U D 1/1204 C 1/8000 C 0.0032/20 ∴
U D 896.4 W/m2 K T1 D 373 0004 293 D 80 deg K and T2 D 373 0004 T deg K.
Thus, from equation 9.9: Tm D 80 0004 373 C T000b/ ln[80/ 373 0004 T000b] D T 0004 293000b/ ln[80/ 373 0004 T000b] deg K and in equation 9.1: 2663 T 0004 293 D 896.4 ð 0.602 T 0004 293000b/ ln[80/ 373 0004 T000b] ln 80/ 373 0004 T D 0.2026 and: T D 307.7 K .
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PROBLEM 9.71 Liquid is heated in a vessel by means of steam which is supplied to an internal coil in the vessel. When the vessel contains 1000 kg of liquid it takes half an hour to heat the contents from 293 to 368 K if the coil is supplied with steam at 373 K. The process is modified so that liquid at 293 K is continuously fed to the vessel at the rate of 0.28 kg/s. The total contents of the vessel are always being maintained at 1000 kg. What is the equilibrium temperature which the contents of the vessel will reach, if heat losses to the surroundings are neglected and the overall heat transfer coefficient remains constant?
Solution Use is made of equation 9.209: ln
Ts 0004 T1 / Ts 0004 T2 D UA/mCp t In this case: Ts D 373 K, T1 D 293 K, T2 D 368 K, m D 1000 kg and t D 0.5 h or 1800 s ∴
ln
373 0004 293000b/ 373 0004 368 D UA/Cp 1800/1000 D 2.773 UA/Cp D 1.54 kg/s
and:
For continuous heating, assuming UA/Cp is constant and losses are negligible then: Q D UA Ts 0004 T D mCp T 0004 T1 where T is the temperature of the contents. ∴
UA 373 0004 T D 0.28Cp T 0004 293000b
UA/Cp 373 0004 T D 0.28T 0004 82.04
Substituting for UA/Cp :
1.54 ð 373 0004 1.54T D 0.28T 0004 82.04 and:
T D 360.7 K
PROBLEM 9.72 The heat loss through a firebrick furnace wall 0.2 m thick is to be reduced by addition of a layer of insulating brick to the outside. What is the thickness of insulating brick necessary to reduce the heat loss to 400 W/m2 ? The inside furnace wall temperature is 1573 K, the ambient air adjacent to the furnace exterior is at 293 K and the natural convection heat transfer coefficient at the exterior surface is given by ho D 3.0T0.25 W/m2 K, where T is the temperature difference between the surface and the ambient air. Thermal conductivity of firebrick D 1.5 W/m K. Thermal conductivity of insulating brick D 0.4 W/m K.
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Solution The conduction through the firebrick is given by: Q D 400 D 1.5 ð 1.0/0.2 1573 0004 T2
(equation 9.12)
and T2 , the temperature at the firebrick/insulating brick interface, is: T2 D 1579.7 K. For the natural convection to the surroundings: Q D ho A T3 0004 Ta 400 D 3.0T0.25 ð 1.0 T3 0004 293000b
or: but T3 0004 293 D T and:
400 D 3.0T1.25 ∴
T D 133.30.8 D 50.1 deg K
and the temperature at the outer surface of the insulating brick, T3 D 293 C 50.1 D 343.1 K. Thus, applying equation 9.12 to the insulating brick: 400 D 0.4 ð 1.0/x 1519.7 0004 343.1 and the thickness of the brick, x D 1.18 m
PROBLEM 9.73 2.8 kg/s of organic liquid of specific heat capacity 2.5 kJ/kg K is cooled in a heat exchanger from 363 to 313 K using water whose temperature rises from 293 to 318 K flowing countercurrently. After maintenance, the pipework is wrongly connected so that the two streams, flowing at the same rates as previously, are now in co-current flow. On the assumption that overall heat transfer coefficient is unaffected, show that the new outlet temperatures of the organic liquid and the water will be 320.6 K and 314.5 K, respectively.
Solution i) Countercurrent flow Heat load, Q D 2.8 ð 2.5 363 0004 313 D 350 kW ∴ water flow D 350/4.18 318 0004 293 D 3.35 kg/s
T1 D 363 0004 318 D 35 deg K, T2 D 313 0004 293 D 20 deg K and, from equation 9.9, Tm D 45 0004 20000b/ ln 45/20 D 30.83 deg K.
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From equation 9.1: 350 D UA ð 30.83 and:
UA D 11.35 kW/K.
ii) Co-current flow Heat load, Q D 2.8 ð 2.5 363 0004 T D 7.0 363 0004 T kW for the organic and for the water, Q D 3.35 ð 4.18 T0 0004 293 D 14.0 T0 kW where T and T0 are the outlet temperatures of the organic and water respectively. ∴
and:
363 0004 T D 14.0/7.0 T0 0004 293 T0 D 474.5 0004 0.5T K. T1 D 363 0004 293 D 70 deg K, T2 D T 0004 T0
and from equation 9.9, Tm D 70 0004 T 0004 T0 / ln[70/ T 0004 T0 ] deg K. In equation 9.1: 70 360 0004 T D 11.35 70 0004 T C T0 / ln[70/ T 0004 T0 ] ∴
7.0 360 0004 T D 11.35 70 0004 T C 474.5 0004 0.5T000b/ ln[70/ T 0004 474.5 C 0.5T000b]
or: 0.617 360 0004 T D 544.5 0004 1.5T000b/ ln[70/ 1.5T 0004 474.5000b] Solving by trial and error, T D 319.8 K which is very close to the value suggested, 320.6 K. ∴ The outlet temperature of the water is: T0 D 474.5 0004 0.5 ð 319.8 D 314.6 K which agrees almost exactly with the given value. Thus for co-current flow: Q D 7.0 363 0004 319.8 D 302.4 kW T1 D 70 deg K (as before), T2 D 319.8 0004 314.6 D 5.2 deg K and from equation 9.9, Tm D 70 0004 5.2000b/ ln 70/5.2 D 24.92 deg K ∴ in equation 9.1: 302.4 D UA ð 24.92 and:
UA D 12.10 kW/K
which is in relatively close agreement with the counter-current value.
PROBLEM 9.74 An organic liquid is cooled from 353 to 328 K in a single-pass heat exchanger. When the cooling water of initial temperature 288 K flows countercurrently its outlet temperature is 333 K. With the water flowing co-currently, its feed rate has to be increased in order to give the same outlet temperature for the organic liquid, the new outlet temperature of the water is 313 K. When the cooling water is flowing countercurrently, the film heat transfer coefficient for the water is 600 W/m2 K.
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What is the coefficient for the water when the exchanger is operating with cocurrent flow if its value is proportional to the 0.8 power of the water velocity? Calculate the film coefficient from the organic liquid, on the assumptions that it remains unchanged, and that heat transfer resistances other than those attributable to the two liquids may be neglected.
Solution i) For countercurrent flow: T1 D 353 0004 333 D 20 deg K and T2 D 328 0004 288 D 40 deg K. ∴ From equation 9.9: Tm D 40 0004 20000b/ ln 40/20 D 28.85 deg K. ii) For co-current flow:
T1 D 353 0004 288 D 65 deg K and T2 D 328 0004 313 D 15 deg K. ∴
Tm D 65 0004 15000b/ ln 65/15 D 34.1 deg K.
Taking countercurrent flow as state 1 and co-current flow as state 2, then, in equation 9.1: Q D U1 ATm1 D U2 ATm2 or:
U1 /U2 D 28.85/34.1 D 0.846
(i)
The water velocity, u / 1/T, where T is the rise in temperature of the water, or: ∴
u D K/T where K is a constant. u1 /u2 D T2 /T1 D 313 0004 288000b/ 333 0004 288 D 0.556.
But: hi / u0.8 or: hi D k 0 u0.8 Thus: ∴
hi1 /hi2 D u1 /u2 0.8 D 0.556000b0.8 D 0.625 600/hi2 D 0.625 and hi2 D 960 W/m2 K .
From equation 9.201, ignoring scale and wall resistances: 1/U D 1/ho C 1/hi ∴
1/U1 D 1/ho C 1/600 and U1 D 600ho / 600 C ho
and
1/U2 D 1/ho C 1/960 and U2 D 960ho / 960 C ho
∴ ∴
U1 /U2 D 600/960 960 C ho / 600 C ho D 0.846 (from (i)) 0.625 960 C ho D 0.846 600 C ho and ho D 417 W/m2 K .
PROBLEM 9.75 A reaction vessel is heated by steam at 393 K supplied to a coil immersed in the liquid in the tank. It takes 1800 s to heat the contents from 293 K to 373 K when the outside
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temperature is 293 K. When the outside and initial temperatures are only 278 K, it takes 2700 s to heat the contents to 373 K. The area of the steam coil is 2.5 m2 and of the external surface is 40 m2 . If the overall heat transfer coefficient from the coil to the liquid in the vessel is 400 W/m2 K, show that the overall coefficient for transfer from the vessel to the surroundings is about 5 W/m2 K.
Solution Using the argument in Section 9.8.3, the net rate of heating is given by: mCp dT/dt D Uc Ac Ts 0004 T 0004 Uo Ao T 0004 To where Uc and Uo are the overall coefficients from the coil and the outside of the vessel respectively, Ac and Ao are the areas of the coil and the outside of the vessel and Ts , T and To are the temperature of the steam, the contents and the surroundings respectively. Writing Uc Ac D a and Uo Ao D b: mCp dT/dt D a 393 0004 T 0004 b T 0004 Ta D 393a C bTa 0004 a C b000bT
000e373 Integrating: t D mCp / a C b ln 1/[393a C bTa 0004 a C b000bT000b] Ta s D mCp / a C b ln[ a 393 0004 Ta / 20a 0004 373b C bTa ] s When Ta D 293 K:
1800 D mCp / a C b ln[ 100a000b/ 20a 0004 80b000b]
(i)
When Ta D 278 K:
2700 D mCp / a C b ln[ 115a000b/ 20a 0004 95b000b]
(ii)
Dividing (i) by (ii):
0.667 D ln[5a/ a 0004 4b000b]/ ln[23a/ 4a 0004 19b000b]
(iii)
But Uc D 400 W/m2 K, Ac D 2.5 m2 and hence: a D Uc Ac D 1000 W/K or 1 kW/K. Substituting in (iii): 1.5 D ln[23/ 4 0004 19b000b]/ ln[5/ 1 0004 4b000b] Solving by trial and error: b D 0.2 kW/K or 200 W/K. ∴
Uo ð 40 D 200 and Uo D 200/40 D 5 W/m2 K .
PROBLEM 9.76 Steam at 403 K is supplied through a pipe of 25 mm outside diameter. Calculate the heat loss per metre to surroundings at 293 K, on the assumption that there is a negligible drop in temperature through the wall of the pipe. The heat transfer coefficient h from the outside of the pipe of the surroundings is given by: h D 1.22 T/d000b0.25 W/m2 K where d is the outside diameter of the pipe (m) and T is the temperature difference (deg K) between the surface and surroundings.
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The pipe is then lagged with a 50 mm thickness of lagging of thermal conductivity 0.1 W/m K. If the outside heat transfer coefficient is given by the same equation as for the bare pipe, by what factor is the heat loss reduced?
Solution For 1 m length of pipe: surface area D $dl D $ 25/1000 ð 1.0 D 0.0785 m2 With a negligible temperature drop through the wall, the wall is at the steam temperature, 403 K, and T D 403 0004 293 D 110 deg K. Thus, the coefficient of heat transfer from the pipe to the surroundings is: h D 1.22[110/ 25/1000000b]0.25 D 9.94 W/m2 K. and the heat loss: Q D hA Tw 0004 Ts D 9.94 ð 0.0785 403 0004 293 D 85.8 W/m With the lagging: Q D k 2$rm l T1 0004 T2 / r2 0004 r1
(equation 9.22)
In this case: k D 0.1 W/mK, T1 D 403 K and T2 is the temperature at the surface of the lagging. r1 D 25/1000000b/2 D 0.0125 m r2 D 0.0125 C 50/1000 D 0.0625 m and: rm D 0.0625 0004 0.0125000b/ ln 0.0625/0.0125 D 0.0311 m Thus: Q D 0.1 ð 2$ ð 0.0311 ð 1 403 0004 T2 / 0.0625 0004 0.0125 D 0.391 403 0004 T2 (i) But: Q D 1.22[ T2 0004 293000b/ 2 ð 0.0625000b]0.25 $ ð 2 ð 0.0625 ð 1 T2 0004 293 D 0.806 T2 0004 293000b1.25
(ii)
From (i) and (ii): 0.391 403 0004 T2 D 0.806 T2 0004 293000b1.25 Solving by trial and error: T2 D 313.5 K and hence: Q D 0.39/ 403 0004 313.5 D 35.0 W/m, a reduction of: or:
85.8 0004 35.0 D 50.8 W/m
50.8 ð 100/85.8 D 59.2%
PROBLEM 9.77 A vessel contains 1 tonne of liquid of specific heat capacity 4.0 kJ/kg K. It is heated by steam at 393 K which is fed to a coil immersed in the liquid and heat is lost to the
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surroundings at 293 K from the outside of the vessel. How long does it take to heat the liquid from 293 to 353 K and what is the maximum temperature to which the liquid can be heated? When the liquid temperature has reached 353 K, the steam supply is turned off for two hours and the vessel cools. How long will it take to reheat the material to 353 K? Coil: Area 0.5 m2 . Overall heat transfer coefficient to liquid, 600 W/m2 K. Outside of vessel: Area 6 m2 . Heat transfer coefficient to surroundings, 10 W/m2 K.
Solution If T K is the temperature of the liquid at time t s, then the net rate of heat input to the vessel, Uc Ac Ts 0004 T 0004 Us As T 0004 Ta D mCp dT/dt W where the coefficient at the coil, Uc D 600 W/m2 K, the coefficient at the outside of the vessel, Us D 10 W/m2 K, the areas are: coil, Ac D 0.5 m2 , vessel, As D 6.0 m2 , the temperatures are: steam, Ts D 393 K, ambient, Ta D 293 K, the mass of liquid, m D 1000 kg and the specific heat capacity, Cp D 4.0 kJ/kg K or 4000 J/kg K. Thus: 1000 ð 4000000bdT/dt D 600 ð 0.5 393 0004 T 0004 10 ð 6 T 0004 293 and: ∴
11,111 dT/dt D 376.3 0004 T 0004 T2 t D 11,111 dT/ 376.3 0004 T D 11,111 ln[ 376.3 0004 T1 / 376.3 0004 T2 ]
(i) (ii)
T1
When T1 D 293 K and T2 D 353 K then: t D 11,111 ln 83.3/23.3 D 14,155 s (3.93 h) The maximum temperature to which the liquid can be heated is obtained by putting dT/dt D 0 in (i) to give: T D 376.3 K. During the time the steam is turned off (for a period of 7200 s) a heat balance gives: mCp dT/dt D 0004Us As T 0004 Ta or:
1000 ð 4000000bdT/dt D 0004 10 ð 6 T 0004 293000b
∴
0004
T
Integrating:
66,700 dT/dt D 293 0004 T 0004 dT/ 293 0004 T D 0.000015
353
∴
7200
dt 0
ln
293 0004 353000b/ 293 0004 T D 0.000015 ð 7200 D 0.108 and T D 346.9 K.
The time taken to reheat the liquid to 353 K is then given by (ii): 0004 353 dT/ 376.3 0004 T t D 11,111 346.9
D 11,111 ln[ 376.3 0004 346.9000b/ 376.3 0004 353000b] D 2584 s 0.72 h000b
PROBLEM 9.78 A bare thermocouple is used to measure the temperature of a gas flowing through a hot pipe. The heat transfer coefficient between the gas and the thermocouple is proportional
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to the 0.8 power of the gas velocity and the heat transfer by radiation from the walls to the thermocouple is proportional to the temperature difference. When the gas is flowing at 5 m/s the thermocouple reads 323 K. When it is flowing at 10 m/s it reads 313 K, and when it is flowing at 15.0 m/s it reads 309 K. Show that the gas temperature is about 298 K and calculate the approximate wall temperature. What temperature will the thermocouple indicate when the gas velocity is 20 m/s?
Solution If the gas and thermocouple temperatures are Tg and Tk respectively, then the rate of heat transfer from the thermocouple to the gas: Q1 D Ku0.8 Tg 0004 T000b
(i)
where K is a constant and u the gas velocity. Similarly, the rate of heat transfer from the walls to the thermocouple is: Q2 D k 0 Tw 0004 T W
(ii)
where k 0 is a constant and Tw is the wall temperature. At equilibrium:
Q1 D Q2 and u0.8 D k 0 /k Tw 0004 T000b/ T 0004 Tg
When u D 5 m/s,
T D 323 K and in (iii): 0.8
5 ∴
(iii)
D k 0 /k Tw 0004 323000b/ 323 0004 Tg D 3.624
k 0 /k D 3.624 323 0004 Tg / Tw 0004 323000b
(iv)
When u D 10 m/s, T D 313 K and in (iii): 100.8 D k 0 /k Tw 0004 313000b/ 313 0004 Tg D 6.31 Substituting for k 0 /k from (iv): 6.31 D 3.624 323 0004 Tg Tw 0004 313000b/[ Tw 0004 323 313 0004 Tg ]
(v)
When u D 15 m/s, T D 309 K and in (iii): 150.8 D k 0 /k Tw 0004 309000b/ 309 0004 Tg D 8.73 Substituting for k 0 /k from (iv): 8.73 D 3.624 323 0004 Tg Tw 0004 309000b/[ Tw 0004 323 309 0004 Tg ]
(vi)
If Tg D 298 K, then in (v): 1.741 D [ 323 0004 298 Tw 0004 313000b]/[ Tw 0004 323 313 0004 298000b] or:
Tw 0004 313000b/ Tw 0004 323 D 1.045 and Tw D 533 K.
If Tg D 298 K, then in (vi): 2.409 D 323 0004 298 Tw 0004 309000b/[ Tw 0004 323 309 0004 298000b] or:
Tw 0004 309000b/ Tw 0004 323 D 1.060 and Tw D 556 K
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This result agrees fairly well and a mean value of Tw D 545 K is indicated. In equation (iv):
k 0 /k D 3.624 323 0004 298000b/ 545 0004 323 D 0.408
∴ in equation (iii)
u0.8 D 0.408 545 0004 T000b/ T 0004 298000b
When u D 20 m/s: 10.99 D 0.408 545 0004 T000b/ T 0004 298 and T D 306.8 K.
PROBLEM 9.79 A hydrocarbon oil of density 950 kg/m3 and specific heat capacity 2.5 kJ/kg K is cooled in a heat exchanger from 363 to 313 K by water flowing countercurrently. The temperature of the water rises from 293 to 323 K. If the flowrate of the hydrocarbon is 0.56 kg/s, what is the required flowrate of water? After plant modifications, the heat exchanger is incorrectly connected so that the two streams are in co-current flow. What are the new outlet temperatures of hydrocarbon and water, if the overall heat transfer coefficient is unchanged?
Solution Heat lost by the oil D 0.56 ð 2.5 363 0004 313 D 70.0 kW For a flow of water of G kg/s, heat gained by the water is: 70.0 D G ð 4.18 323 0004 293 and G D 0.56 kg/s i) For countercurrent flow: T1 D 363 0004 323 D 40 deg K, T2 D 313 0004 293 D 20 deg K and from equation 9.9: Tm D 40 0004 20000b/ ln 40/20 D 28.85 deg K In equation 9.1: 70.0 D UA ð 28.85 and UA D 2.43 kW/K. ii) For co-current flow: If To and Tw are the outlet temperature of the oil and water respectively, the heat load is: Q D 0.56 ð 2.5 363 0004 To D 0.56 ð 4.18 Tw 0004 293000b
(i)
508.2 0004 1.4To D 2.34Tw 0004 685.9 and:
Tw D 510.3 0004 0.60To
(ii)
T1 D 363 0004 293 D 70 K T1 D To 0004 Tw D To 0004 510.3 C 0.60To D 1.60To 0004 510.3 K
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and in equation 9.9: Tm D 70 0004 1.60To C 510.3000b/ ln[70/ 1.60To 0004 510.3000b] D 580.3 0004 1.60To / ln[70/1.60To 0004 510.3000b] deg K and: 508.2 0004 1.4To D 2.43 580.3 0004 1.60To / ln[70/ 1.60To 0004 510.3000b]
508.2 0004 1.4To D 2.126 508.2 0004 1.40To / ln[70/ 1.60To 0004 510.3000b] ∴
70/ 1.60To 0004 510.3 D e2.126 D 8.38 and To D 324.2 K .
and in equationT(ii): w D 510.3 0004 0.60 ð 324.2 D 315.8 K .
PROBLEM 9.80 A reaction mixture is heated in a vessel fitted with an agitator and a steam coil of area 10 m2 fed with steam at 393 K. The heat capacity of the system is equal to that of 500 kg of water. The overall coefficient of heat transfer from the vessel of area 5 m2 is 10 W/m2 K. It takes 1800 s to heat the contents from ambient temperature of 293 to 333 K. How long will it take to heat the system to 363 K and what is the maximum temperature which can be reached? Specific heat capacity of water D 4200 J/kgK.
Solution Following the argument of Problem 9.77 and taking ambient temperature as the initial temperature of the mixture, 293 K, then: net rate of heatingD 500 ð 4200000bdT/dt D Uc ð 10 393 0004 T 0004 10 ð 5 T 0004 293 W ∴
2,100,000 dT/dt D 3930Uc 0004 10Uc T 0004 50T C 14,650 D 3930Uc C 14,650 0004 10Uc C 50000bT.
∴ ∴
2,100,000/ 10Uc C 50 dT/dt D
3930Uc C 14,650000b/ 10Uc C 50 0004 T. 000f 333 t D [2,100,000/ 10Uc C 50000b] 293 dT/[ 3930Uc C 14,650000b/ 10Uc C 50 0004 T]
In heating from 293 to 333 K, the time taken is 1800 s and: 1800 D [2,100,000/ 10Uc C 50000b] lnf[ 3930Uc C 14,650000b/ 10Uc C 50 0004 293]/ [ 3930Uc C 14,650000b/ 10Uc C 50 0004 333]g Solving by trial and error: Uc D 61.0 W/m2 K. Thus, net rate of heating is: 2,100,000 dT/dt D 61.0 ð 10 393 0004 T 0004 10 ð 5 T 0004 293 D 254,380 0004 660 T W or:
3182 dT/dt D 385.4 0004 T W 0004 363 ∴ time for heating, t D 3182 dT/ 385.4 0004 T000b
(i)
293
D 3182 ln[ 385.4 0004 293000b/ 385.4 0004 363000b] D 3182 ln 92.4/22.4 D 4509 s 1.25 h000b
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The maximum temperature which can be attained is obtained by putting dT/dt D 0 in (i) which gives: Tmax D 385.5 K .
PROBLEM 9.81 A pipe, 50 mm outside diameter, is carrying steam at 413 K and the coefficient of heat transfer from its outer surface to the surroundings at 288 K is 10 W/m2 K. What is the heat loss per unit length? It is desired to add lagging of thermal conductivity 0.03 W/m K as a thick layer to the outside of the pipe in order to cut heat losses by 90%. If the heat transfer from the outside surface of the lagging is 5 W/m2 K, what thickness of lagging is required?
Solution Outside area of pipe: $dl D $ 50/1000 ð 1.0 D 0.157 m2 /m. Assuming the pipe wall is at the temperature of the steam, that is the resistance of the wall is negligible, then the heat loss is: Q D hA Tw 0004 Ta D 10 ð 0.157 413 0004 288 D 196 W/m With the addition of lagging of thickness, x mm, the required heat flow is 19.6 W/m The diameter of the lagging D 50 C 2x000b/1000 m and the surface area of the lagging D [$ 50 C 2x000b/1000] ð 1.0 D 0.157 C 0.00628x m2 /m. The heat transferred to the surroundings, 19.6 D 5 0.157 C 0.00628x T2 0004 288 W and:
T2 D 49.14 C 1.809x000b/ 0.157 C 0.00628x K
(i)
For conduction through the lagging: Q D k 2$rm l T1 0004 T2 / r2 0004 r1 W
(equation 9.22)
where Q D 19.6 W/m, k D 0.03 W/mK, l D 1.0 m and the steam temperature, T1 D 413 K. r1 D 25 mm or 0.025 m, r2 D 25 C x mm or 25 C x000b/1000 D 0.025 C 0.001x m ∴
rm D 0.025 C 0.001x 0004 0.025000b/ ln[ 0.025 C 0.001x000b/0.025] D 0.001x/ ln 1 C 0.04x m
Thus, in equation 9.22: 19.6 D 0.03 2$ 0.001x/ ln 1 C 0.04x ð 1.0 413 0004 T2 / 0.025 C 0.001x 0004 0.025 ∴
103,981 D [x/ ln 1 C 0.04x000b] 413 0004 T2 /0.001x
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Substituting for T2 from (i): 104 D [413 0004 49.14 C 1.809x000b/ 0.157 C 0.00628x000b]/ ln 1 C 0.04x Solving by trial and error, x D 52.5 mm.
PROBLEM 9.82 It takes 1800 s (0.5 h) to heat a tank of liquid from 293 to 323 K using steam supplied to an immersed coil when the steam temperature is 383 K. How long will it take when the steam temperature is raised to 393 K? The overall heat transfer coefficient from the steam coil to the tank is 10 times the coefficient from the tank to surroundings at a temperature of 293 K, and the area of the steam coil is equal to the outside area of the tank.
Solution Using the argument in Problem 9.77, mCp dT/dt D Uc Ac Ts 0004 T 0004 Us As T 0004 Ta In this case, Ts D 383 K, Ta D 293 K, Us D Uc /10 and Ac D As D A (say). ∴
mCp dT/dt D Uc Ac 383 0004 T 0004 Uc Ac /10 T 0004 293000b
∴
Uc Ac /mCp dt D dT/ 412.3 0004 1.1T000b
On integration: 0004
323
Uc Ac /mCp t D 293
000e323 dT/ 412.3 0004 1.1T D 1/1.1 ln 1/ 412.3 0004 1.1T 293
Since it takes 1800 s to heat the liquid from 293 to 333 K, then: 1800 Uc Ac /mCp D 0.909 ln[ 412.3 0004 1.1 ð 293000b]/[412.3 0004 1.1 ð 323000b] D 0.909 ln 90/57 and:
Uc Ac /mCp D 0.00023071 s00041
On increasing the steam temperature to 393 K,
Heat transferred from the steam D Uc Ac 393 0004 T W Heat lost to the surroundings D Uc Ac /10 T 0004 293 W and: ∴
mCp dT/dt D Uc Ac 393 0004 T 0004 0.1Uc Ac T 0004 293 W dT/ 422.3 0004 1.1T D Uc Ac /mCp dt D 0.0002307 dt
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HEAT TRANSFER
On integration: 0004
323
0.0002307t D 293
dT/ 422.3 0004 1.1T D 1/1.1000b[ln 1/ 422.3 0004 1.1T000b]333 293
Thus, on heating from 293 to 323 K: 0.0002307t D 0.909 lnf[422.3 0004 1.1 ð 293000b]/[422.3 0004 1.1 ð 323000b]g D 0.909 ln 100/67 and:
t D 1578 s 0.44h000b
PROBLEM 9.83 A thermometer is situated in a duct in an air stream which is at a constant temperature. The reading varies with the gas flowrate as follows: air velocity (m/s)
thermometer reading (K)
6.1 7.6 12.2
553 543 533
The wall of the duct and the gas stream are at somewhat different temperatures. If the heat transfer coefficient for radiant heat transfer from the wall to the thermometer remains constant, and the heat transfer coefficient between the gas stream and thermometer is proportional to the 0.8 power of the velocity, what is the true temperature of the air stream? Neglect any other forms of heat transfer.
Solution As with Problem 9.78, a heat balance on the thermometer gives: hw Tw 0004 T D hg T 0004 Tg where hw and hg are the coefficients for radiant heat transfer from the wall and for convection to the gas respectively and Tw , T and Tg are the temperatures of the wall, thermometer and gas, respectively, above a datum of 533 K. When u D 12.2 m/s,
hw Tw 0004 0 D hg 0 C Tg
When u D 7.6 m/s, since hg / u0.8 , When u D 6.1 m/s,
(i)
hw Tw 0004 10 D hg 7.6/12.2000b0.8 000410 C Tg 0.8
hw Tw 0004 20 D hg 6.1/12.2 000420 C Tg
(ii) (iii)
Dividing equation (i) by equation (ii): Tw / Tw 0004 10 D 12.2/7.6000b0.8 Tg / Tg 0004 10 D 1.46Tg / Tg 0004 10000b
(iv) 0.8
and dividing equation (i) by equation (iii): Tw / Tw 0004 20 D 12.2/6.1 Tg / Tg 0004 20 D 1.741Tg / Tg 0004 20000b
(v)
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From (v): ∴
CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Tw Tg 0004 20Tw D 1.741Tw Tg 0004 34.82Tg Tw D 34.82Tg / 20 C 0.741Tg K
(vi)
Substituting for Tw from (vi) into (iv): 34.82Tg / 27.4Tg 0004 200 D 1.46Tg / Tg 0004 10 and Tg D 000411.20 K and hence, the temperature of the gas is 533 0004 11.2 D 521.8 K.
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SECTION 10
Mass Transfer PROBLEM 10.1 Ammonia gas is diffusing at a constant rate through a layer of stagnant air 1 mm thick. Conditions are fixed so that the gas contains 50% by volume of ammonia at one boundary of the stagnant layer. The ammonia diffusing to the other boundary is quickly absorbed and the concentration is negligible at that plane. The temperature is 295 K and the pressure atmospheric, and under these conditions the diffusivity of ammonia in air is 0.18 cm2 /s. Calculate the rate of diffusion of ammonia through the layer.
Solution See Volume 1, Example 10.1.
PROBLEM 10.2 A simple rectifying column consists of a tube arranged vertically and supplied at the bottom with a mixture of benzene and toluene as vapour. At the top, a condenser returns some of the product as a reflux which flows in a thin film down the inner wall of the tube. The tube is insulated and heat losses can be neglected. At one point in the column, the vapour contains 70 mol% benzene and the adjacent liquid reflux contains 59 mol% benzene. The temperature at this point is 365 K. Assuming the diffusional resistance to vapour transfer to be equivalent to the diffusional resistance of a stagnant vapour layer 0.2 mm thick, calculate the rate of interchange of benzene and toluene between vapour and liquid. The molar latent heats of the two materials can be taken as equal. The vapour pressure of toluene at 365 K is 54.0 kN/m2 and the diffusivity of the vapours is 0.051 cm2 /s.
Solution In this solution, subscripts 1 and 2 refer to the liquid surface and vapour side of the stagnant layer respectively and subscripts B and T refer to benzene and toluene. If the latent heats are equal and there are no heat losses, there is no net change of phase across the stagnant layer. This is an example of equimolecular counter diffusion and: NA D 0002DPA2 0002 PA1 /RTL
(equation 10.23)
where L D thickness of the stagnant layer D 0.2 mm D 0.0002 m. 217
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
As the vapour pressure of toluene D 54 kN/m2 , the partial pressure of toluene from Raoult’s law D 1 0002 0.59 ð 54 D 22.14 kN/m2 D PT1 and: PT2 D 1 0002 0.70 ð 101.3 D 30.39 kN/m2 For toluene: NT D 00020.051 ð 1000024 30.39 0002 22.14 /8.314 ð 365 ð 0.0002
D 00026.93 ð 1000025 kmol/m2 s For benzene: PB1 D 101.3 0002 22.14 D 79.16 kN/m2 PB2 D 101.3 0002 30.39 D 70.91 kN/m2 Hence, for benzene: NB D 00020.051 ð 1000024 70.91 0002 79.16 /8.314 ð 365 ð 0.0002
D 6.93 ð 1000025 kmol/m2 s Thus the rate of interchange of benzene and toluene is equal but opposite in direction.
PROBLEM 10.3 By what percentage would the rate of absorption be increased or decreased by increasing the total pressure from 100 to 200 kN/m2 in the following cases? (a) The absorption of ammonia from a mixture of ammonia and air containing 10% of ammonia by volume, using pure water as solvent. Assume that all the resistance to mass transfer lies within the gas phase. (b) The same conditions as (a) but the absorbing solution exerts a partial vapour pressure of ammonia of 5 kN/m2 . The diffusivity can be assumed to be inversely proportional to the absolute pressure.
Solution (a) The rates of diffusion for the two pressures are given by: NA D 0002D/RTL P/PBM PA2 0002 PA1
(equation 10.34)
where subscripts 1 and 2 refer to water and air side of the layer respectively and subscripts A and B refer to ammonia and air. Thus:
PA2 D 0.10 ð 100 D 10 kN/m2 and PA1 D 0 kN/m2 PB2 D 100 0002 10 D 90 kN/m2 and PB1 D 100 kN/m2 PBM D 100 0002 90 / ln100/90 D 94.91 kN/m2
∴
Hence:
P/PBM D 100/94.91 D 1.054 NA D 0002D/RTL 1.05410 0002 0 D 000210.54D/RTL
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219
If the pressure is doubled to 200 kN/m2 , the diffusivity is halved to 0.5D (from equation 10.18) and: PA2 D 0.1 ð 200 D 20 kN/m2 and PA1 D 0 kN/m2 PB2 D 200 0002 20 D 180 kN/m2 and PB1 D 200 kN/m2 ∴
PBM D 200 0002 180 / ln200/180 D 189.82 kN/m2 P/PBM D 200/189.82 D 1.054 i.e. unchanged
Hence: NA D 00020.5D/RTL 1.05420 0002 0 D 000210.54D/RTL, that is the rate is unchanged (b) If the absorbing solution now exerts a partial vapour pressure of ammonia of 5 kN/m2 , then at a total pressure of 100 kN/m2 : PA2 D 10 kN/m2 and PA1 D 5 kN/m2 PB2 D 90 kN/m2 and PB1 D 95 kN/m2 PBM D 95 0002 90 / ln95/90 D 92.48 kN/m2 ∴
P/PBM D 100/92.48 D 1.081 NA D 0002D/RTL ð 1.08110 0002 5 D 00025.406D/RTL
At 200 kN/m2 , the diffusivity D 0.5D and: PA2 D 20 kN/m2 and PA1 D 5 kN/m2 PB2 D 180 kN/m2 and PB1 D 195 kN/m2 ∴
PBM D 195 0002 180 / ln195/180 D 187.4 kN/m2 P/PBM D 1.067 NA D 00020.5D/RTL 1.06720 0002 5 D 00028.0D/RTL Thus the rate of diffusion has been increased by 1008 0002 5.406 /5.406 D 48%.
PROBLEM 10.4 In the Danckwerts’ model of mass transfer it is assumed that the fractional rate of surface renewal s is constant and independent of surface age. Under such conditions the expression for the surface age distribution function is se0002st . If the fractional rate of surface renewal were proportional to surface age (say s D bt, where b is a constant), show that the surface age distribution function would then assume the form: 2 2b/0010 1/2 e0002bt /2
Solution From equation 10.117:
f0 t D sft D 0
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In this problem: s D bt and ebt
2 /2
ft D constant D k
∴
ft D ke0002bt
2 /2
The total area of surface considered is unity and: 0001 1 ∴ ft dt D 1 0
0001
1
∴
ke0002bt
2 /2
dt D 1
0
and by substitution as in equation 10.120: k0010/2b 0.5 D 1 k D 2b/0010 0.5 and ft D 2b/0010 1/2 e0002bt
2 /2
PROBLEM 10.5 By consideration of the appropriate element of a sphere show that the general equation for molecular diffusion in a stationary medium and in the absence of a chemical reaction is: 0002 2 0003 ∂CA ∂ 2 CA ∂ CA 1 ∂ 2 CA 1 2 ∂CA cotˇ ∂CA DD C 2 C 2 C 2 2 C ∂t ∂r 2 r ∂ˇ2 r ∂r r ∂ˇ r sin ˇ ∂00162 where CA is the concentration of the diffusing substance, D the molecular diffusivity, t the time, and r, ˇ, 0016 are spherical polar coordinates, ˇ being the latitude angle.
Solution The basic equation for unsteady state mass transfer is: 00040002 0003 0002 2 0003 0002 2 0003 0005 ∂CA ∂ 2 CA ∂ CA ∂ CA DD C C ∂t ∂x 2 yz ∂y 2 zx ∂z2 xy
(equation 10.67) (i)
This equation may be transformed into other systems of orthogonal coordinates, the most useful being the spherical polar system. (Carslaw and Jaeger, Conduction of Heat in Solids, gives details of the transformation.) When the operation is performed: x D r sin ˇ cos 0016 y D r sin ˇ sin 0016 z D r cos ˇ and the equation for CA becomes: 0006 0002 0003 0002 0003 0007 ∂CA D ∂ 1 ∂ ∂CA 1 ∂ 2 CA ∂CA D 2 r2 C sin ˇ C ∂t r ∂r ∂r sin ˇ ∂ˇ ∂ˇ sin2 ˇ ∂00162
ii
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MASS TRANSFER
which may be written as: 0006 2 0002 0003 0007 ∂CA ∂ CA 2 ∂CA 1 1 υ ∂ 2 CA 2 ∂CA C 1 0002 001c
C DD C ∂t ∂r 2 r ∂r r 2 ∂001c ∂001c r 2 1 0002 001c2 ∂00162 where:
001c D cos ˇ.
iii
iv
In this problem ∂CA /∂t is given by: ∂CA DD ∂t
0002
∂ 2 CA ∂ 2 CA 1 ∂ 2 CA 1 2 ∂CA cotˇ ∂CA C 2 C C C 2 2 2 2 2 2 ∂r r ∂ˇ r ∂r r ∂ˇ r sin ˇ ∂0016
0003
v
Comparing equations (iii) and (v) is necessary to prove that: 1 ∂ r 2 ∂001c
0002 0003 ∂ 2 CA ∂ 2 CA 1 1 ∂ 2 CA 1 cot ˇ ∂CA 2 ∂CA 1 0002 001c
C 2 D C C 2 2 2 2 2 2 2 2 ∂001c r 1 0002 001c ∂0016 r ∂ˇ r ∂ˇ r sin ˇ ∂0016
001c D cos ˇ, 1 0002 001c2 D 1 0002 cos2 ˇ D sin2 ˇ ∂ 2 CA 1 ∂ 2 CA 1 D r 2 1 0002 001c2 ∂00162 r 2 sin2 ˇ ∂00162
∴
It now becomes necessary to prove that: 1 ∂ 2 CA cot ˇ ∂CA 1 ∂ C 2 D 2 2 2 r ∂ˇ r ∂ˇ r ∂001c From equation (iv): ∴
0002 0003 2 ∂CA 1 0002 001c
∂001c
001c D cos ˇ ∂001c/∂ˇ D 0002 sin ˇ
and:
vi
∂2 001c/∂ˇ2 D 0002 cos ˇ 0002 0003 0002 0003 ∂ˇ 1 ∂ 1 ∂ 2 ∂CA 2 ∂CA ∂ˇ 1 0002 001c
D 2 1 0002 001c
2 r ∂001c ∂001c r ∂ˇ ∂ˇ ∂001c ∂001c
vii
viii
Substituting from equation (iv) for 001c from equation (vii) for ∂ˇ/∂001c gives: 0002 0003 0002 0003 1 1 1 ∂ ∂CA 1 ∂CA 1 ∂ 2 D 2 1 0002 cos ˇ
0002 sin ˇ D 2 r ∂ˇ ∂ˇ 0002 sin ˇ 0002 sin ˇ r ∂ˇ ∂ˇ 0002 sin ˇ 0006 0007 2 2 1 1 1 ∂ CA ∂ CA ∂CA cot ˇ ∂CA D 2 0002 sin ˇ 0002 cos ˇ
D 2 C C 2 2 2 r ∂ˇ ∂ˇ 0002 sin ˇ r ∂ˇ r ∂ˇ
PROBLEM 10.6 Prove that for equimolecular counter diffusion from a sphere to a surrounding stationary, infinite medium, the Sherwood number based on the diameter of the sphere is equal to 2.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Solution If the particle has a radius r, and is surrounded by a spherical shell of radius s then, Moles per unit time diffusing through the shell, M is given by: 0002 0003 dCA M D 40010s2 0002D ds At steady state, M is constant and: 0001 s2 0001 CA 2 ds M D 400100002D
dCA 2 s1 s CA1 0002 0003 1 1 M 0002 D 40010DCA1 0002 CA2
s1 s2 If CA1 is the concentration at s1 D r and CA2 is the concentration at s2 D 1, then: M/r D 40010D0002CA
The mass transfer coefficient: hd D
M M D 2 A0002CA
40010r 0002CA
hD 40010r 2 0002CA /r D 40010D0002CA
hD D D/r D 2D/d hD d/D D Sh D 2
PROBLEM 10.7 Show that the concentration profile for unsteady-state diffusion into a bounded medium of thickness L, when the concentration at the interface is suddenly raised to a constant value CAi and kept constant at the initial value of CAo at the other boundary is: 0004nD1 0005 CA 0002 CAo 2 1 z exp0002n2 00102 Dt/L 2 sinnz0010/L . D10002 0002 CAi 0002 CAo L 0010 nD1 n Assume the solution to be the sum of the solution for infinite time (steady-state part) and the solution of a second unsteady-state part, which simplifies the boundary conditions for the second part.
Solution The system is shown in Fig. 10a. The boundary conditions are: At time,
tD0 t>0 t>0
CA D CAo CA D CAi CA D CAo
0< />
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MASS TRANSFER
223
CAi
CAo L y 0
Figure 10a.
Replacing CAi by C0i and CA by C0 where: CA D C0 C CAo and CAi D C0i C CAo , then using these new variables: At:
t D 0 C0 D 0 t > 0 C0 D C0i t > 0 C0 D 0
0< />
The problem states that the solution of the one dimensional diffusion equation is: C0 D steady state solution C
1
exp0002n2 00102 Dt/L 2 An sinn0010y/L
0
where the steady state solution D C0i 0002 C0i y/L. (A derivation of the analogous equation for heat transfer may be found in Conduction of Heat in Solids by H. S. Carslaw and J. C. Jaeger, Oxford, 1960.) 0001 2 L An D initial concentration profile–steady state sinn0010y/L dy L 0 0001 2 L [0 C C0i y/L 0002 C0i ] sinn0010y/L dy D L 0 D 00022C0i /n0010 (this proof is given at the end of this problem).
Hence:
C0 D C0i 0002 C0i y/L 0002 0004
∴
1 2C0i 1 exp0002n2 00102 Dt/L 2 sinn0010y/L
0010 nD0 n
0005 1 21 y 2 2 2 exp0002n 0010 Dt/L sinn0010y/L
C D Co C Ci 0002 Co 1 0002 0002 L 0010 nD0 n
An D
2 L
0001 0
L
[C0i y/L 0002 C0i ] sinn0010y/L dy
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
D
2C0i L2
0001
L
y sinn0010y/L dy 0002 0
2C0i L
0001
L
sinn0010y/L dy 0
0001 0001 2C0i L 2C0i L 1 0002 2 0007 0007 D 2 L L 0 0 0006 0007 0001 L 0001 L L n0010y L n0010y Ly 1 D 0002 cos cos dy 0007 C n0010 L 0 L 0 0 n0010
Putting u D y, du D dy dv D sinn0010y/L dy,
and: 0001 ∴ 0
L
vD0002
n0010y L cos n0010 L
0002 0003 0002 2 0003L L Ly n0010y n0010y L 1 D 0002 0007 C sin cos n0010 L 0 n2 0010 2 L 0
L2 L2 L2 cos n0010 C 2 2 sin n0010 D 0002 00021 n n0010 n 0010 n0010 0002 0003L 0001 L n0010y L L L 2 D 0002 cos cos n0010 C 0007 D0002 n0010 L n0010 n0010 0 0 D0002
L L cos n0010 C n0010 n0010 L L D 0002 00021 n C n0010 n0010 D0002
2C0 2C0i 2C0i 2 D 1 0002 0007 An D 2 i 0007 L L L2
0002 0003 0002 0003 L2 2C0i L L n n 0002 00021 0002 0002 00021 C n0010 L n0010 n0010
D 00022C0i /n0010
PROBLEM 10.8 Show that under the conditions specified in Problem 10.7 and assuming the Higbie model of surface renewal, the average mass flux at the interface is given by: 0007
nD1 0006 00102 1 2 2 2 2 2 NA t D CAi 0002 CAo D/L 1 C 2L /0010 Dt
0002 2 exp0002n 0010 Dt/L
6 n nD1 Use the relation
1 1 D 00102 /6. 2 n nD1
Solution The rate of transference across the phase boundary is given by: NA D 0002D∂CA /∂y yD0
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MASS TRANSFER
According to the Higbie model, if the element is exposed for a time te , the average rate of transfer is given by: 0001 1 te NA D 0002D∂C/∂z zD0 dt te 0 From Problem 10.7, the concentration C is: 0004 0005 1 21 y 2 2 2 C D CAo C CAi 0002 CAo 1 0002 0002 exp0002n 0010 Dte /L sin n0010y/L L 0010 nD0 n 0004 0005 1 ∂C 20010 1 2 2 2 D CAi 0002 CAo 0002 0002 exp0002n 0010 Dte /L cos n0010y/L ∂y L 0010 nD0 L 0004 0005 0002 0003 1 ∂C 1 20010 2 2 2 D CAi 0002 CAo 0002 0002 exp0002n 0010 Dte /L
∂y yD0 L 0010 0 L 0005 0001 0004 1 20010 DCAi 0002 CAo te 1 2 2 2 exp0002n 0010 Dte /L dt NA D 0002 0002 0002 te L 0010 0 L 0 0004 0005te 0002 0003 1 DCAi 0002 CAo
20010 te L2 2 2 2 D0002 0002 0002 0002 2 2 exp0002n 0010 Dte /L
te L 0010 0 L n 0010 D 0
DCAi 0002 CAo
te 0004 0005 0003 0002 1 0002 1 2 0003 0010 2 te L 2 L 0002 0002 0002 2 exp0002n2 00102 Dte /L 2 C 0002 2 2 L 0010 0 n 0010D 0010 0 L n 0010 D 00041 0005
1 1 D 2L 2 00021 2 2 2 NA D CAi 0002 CAo 1 C 2 exp0002n 0010 Dte /L C L 0010 Dte 0 n2 n2 0 D0002
1 00021 0
D
1 00021 0
n2
2 2
D
1
2
exp0002n 0010 Dte /L C
1 00021 1
D 0002 exp000200102 Dte /L 2 C 1 0006
n2
exp0002n2 00102 Dte /L 2 C
1 00021 1
n2
1 1 n2 0
1 1 1 1 exp0002n 0010 Dte /L C C 2 2 n2 n n 0 1 2 2
2
exp0002n2 00102 Dte /L 2 C 1 C 00102 /6
0007 0010 1 2 2 2 2 2 0002 2 exp0002n 0010 Dte /L C 1 0002 exp00020010 Dte /L
6 n 2
Considering the terms 1 0002 exp000200102 Dte /L 2 and Dte /L 2 to be very small so that 000200102 Dte /L 2 is small and exp000200102 Dte /L 2 ! 1. Therefore, 1 0002 exp000200102 Dte /L 2 is
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
approximately zero and:
0007
1 0006 1 D 2L 2 00102 NA D CAi 0002 CAo 1 C 2 0002 2 exp0002n2 00102 Dte /L 2
L 0010 Dte nD1 6 n
PROBLEM 10.9 According to the simple penetration theory the instantaneous mass flux: 0002 00030.5 D NA t D CAi 0002 CAo
0010t What is the equivalent expression for the instantaneous heat flux under analogous conditions? Pure sulphur dioxide is absorbed at 295 K and atmospheric pressure into a laminar water jet. The solubility of SO2 , assumed constant over a small temperature range, is 1.54 kmol/m3 under these conditions and the heat of solution is 28 kJ/kmol. Calculate the resulting jet surface temperature if the Lewis number is 90. Neglect heat transfer between the water and the gas.
Solution The heat flux at any time, f D 0002k∂)/∂x where k is the thermal conductivity, ) the temperature, and y the distance in the direction of transfer. The flux satisfies the same differential equation as ), that is: DH ∂2 f/∂y 2 D ∂f/∂t
y > 0, t > 0
where DH D thermal diffusivity D k/+Cp . This last equation is analogous to the mass transfer equation 10.66: ∂C/∂t D D∂2 C/∂y 2
The solution of the heat transfer equation with f D Fo (constant) at y D 0 when t > 0 is:
y f D Fo erfc p 2 DH t
The temperature rise is due to the heat of solution HS . Heat is liberated at the jet surface at a rate H t D NoA HS , or:
H t D CAi 0002 CAo HS D/0010t 0.5
The temperature rise, T, due to the heat flux H t into the surface is: 0001 L 1 H t 0002 ) d) p p TD +Cp 0010DH 0 ) p CAi 0002 CAo Hs D/DH and: TD +Cp
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MASS TRANSFER
The Lewis number D h/Cp +hD D Pr/Sc D DCp +/k. D/DH D DCp +/k D 90 CAi 0002 CAo D 1.54 kmol/m3 HS D 28 kJ/kmol p T D 1.54 ð 28 90 /1000 ð 4.186 D 0.1 deg K
∴
PROBLEM 10.10 In a packed column, operating at approximately atmospheric pressure and 295 K, a 10% ammonia-air mixture is scrubbed with water and the concentration is reduced to 0.1%. If the whole of the resistance to mass transfer may be regarded as lying within a thin laminar film on the gas side of the gas-liquid interface, derive from first principles an expression for the rate of absorption at any position in the column. At some intermediate point where the ammonia concentration in the gas phase has been reduced to 5%, the partial pressure of ammonia in equilibrium with the aqueous solution is 660 N/m2 and the transfer rate is 1000023 kmol/m2 s. What is the thickness of the hypothetical gas film if the diffusivity of ammonia in air is 0.24 cm2 /s?
Solution The equation for the rate of absorption is derived in Section 10.2.2 as: NA D 0002D/RTL PA2 0002 PA1
(equation 10.23)
If subscripts 1 and 2 refer to the water and air side of the stagnant film and subscripts A and B refer to ammonia and air, then: PA1 D 66.0 kN/m2 and PA2 D 0.05 ð 101.3 D 5.065 kN/m2 D D 0.24 ð 1000024 m2 /s, R D 8.314 kJ/kmol K, T D 295 K and NA D 1 ð 1000023 kmol/m2 s ∴
L D 0002D/NA RT PA2 0002 PA1
D 00020.24 ð 1000024 /1000023 ð 8.314 ð 295 66.0 0002 5.065 D 00020.000043 m
The negative sign indicates that the diffusion is taking place in the opposite direction and the thickness of the gas film is 0.043 mm.
PROBLEM 10.11 An open bowl, 0.3 m in diameter, contains water at 350 K evaporating into the atmosphere. If the air currents are sufficiently strong to remove the water vapour as it is formed
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and if the resistance to its mass transfer in air is equivalent to that of a 1 mm layer for conditions of molecular diffusion, what will be the rate of cooling due to evaporation? The water can be considered as well mixed and the water equivalent of the system is equal to 10 kg. The diffusivity of water vapour in air may be taken as 0.20 cm2 /s and the kilogram molecular volume at NTP as 22.4 m3 .
Solution If subscripts 1 and 2 refer to the water and air side of the stagnant layer and subscripts A and B refer to water vapour and air, then the rate of diffusion through a stagnant layer is: NA D 0002D/RTL P/PBM PA2 0002 PA1
(equation 10.34)
where, PA1 is the vapour pressure of water at 350 K D 41.8 kN/m2 . PA2 D 0 (since the air currents remove the vapour as it is formed.) PB1 D 101.3 0002 41.8 D 59.5 kN/m2 and PB2 D 101.3 kN/m2 . ∴
PBM D 101.3 0002 59.5 / ln101.3/59.5 D 78.17 kN/m2 . and: P/PBM D 101.3/78.17 D 1.296.
∴
NA D 00020.2 ð 1000024 /8.314 ð 350 ð 1000023 1.2960 0002 41.8
D 3.72 ð 1000024 kmol/m2 s D 3.72 ð 1000024 ð 18 D 6.70 ð 1000023 kg water/m2 s Area of bowl D 0010/4 0.32 D 0.0707 m2 Therefore the rate of evaporation D 6.70 ð 1000023 ð 0.0707 D 4.74 ð 1000024 kg/s Latent heat of vaporisation D 2318 kJ/kg Specific heat capacity of water D 4.187 kJ/kg K Rate of heat removal D 4.74 ð 1000024 ð 2318 D 1.10 kW If the rate of cooling D d)/dt K/s, then: water equivalent ð specific heat capacity ð d)/dt D 0.0617
or:
10 ð 4.187 ð d)/dt D 1.10 and d)/dt D 0.026 deg K/s
PROBLEM 10.12 Show by substitution that when a gas of solubility CC is absorbed into a stagnant liquid of infinite depth, the concentration at time t and depth y is: y CC erfc p 2 Dt Hence, on the basis of the simple penetration theory, show that the rate of absorption in a packed column will be proportional to the square root of the diffusivity.
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229
Solution The first part of this question is discussed in Section 10.5.2 and the required equation is presented as equation 10.108. In Section 10.5.2 the analysis leads to equation 10.113 which expresses the instantaneous rate of mass transfer when the surface element under consideration has an age t, or: NA t D CAi 0002 CAo D/0010t
The simple penetration theory assumes that each element is exposed for the same time interval te before returning to the bulk solution. The average rate of mass transfer is then: 0001 0001 1 te CAi 0002 CAo te NA t dt D D/0010t 0.5 dt NA D te 0 te 0 D 2CAi 0002 CAo D/0010te p and the rate of absorption is proportional to D.
PROBLEM 10.13 Show that in steady-state diffusion through a film of liquid, accompanied by a firstorder irreversible reaction, the concentration of solute in the film at depth y below the interface is: k L 0002 y
CA Ci D sinh D CAi k L sinh D if CA D 0 at y D L and CA D CAi at y D 0, corresponding to the interface. Hence show that according to the “film theory” of gas-absorption, the rate of absorption per unit area of interface, NA is given by: NA D kL CAi
ˇ tanh ˇ
p where ˇ D Dk /kL , D is the diffusivity of the solute, k the rate constant of the reaction, KL the liquid film mass transfer coefficient for physical absorption, CAi the concentration of solute at the interface, y the distance normal to the interface and yL the liquid film thickness.
Solution The basic equation for diffusion through a film of liquid accompanied by a first-order irreversible reaction is: Dd2 CA /dy 2 D k CA p where a2 D k /D.
or
d2 CA /dy 2 D a2 CA
(equation 10.171) (i)
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The general solution of equation (i) is: CA D A cosh ay C B sinh ay
ii
where A and B are constants. The boundary conditions are: At y D L, CA D 0
iii
At y D 0, CA D CAi
iv
Substituting equation (iii) in equation (ii): 0 D A cosh aL C B sinh aL and substituting equation (iv) in equation (ii): CAi D A C 0 and A D CAi and B D 0002CAi cosh aL/ sinh aL ∴
CA D CAi cosh ay 0002 CAi
cosh aL sinh ay sinh aL
CAi cosh ay sinh aL 0002 cosh aL sinh ay
sinh aL CAi sinh aL 0002 y
D sinh aL p sinh aL 0002 y
sinh k /DL 0002 y
p D CAi D CAi sinh aL sinh k /DL
D
0002
Rate of absorption:
NA D 0002D
dCA dy
0003 yD0
Assuming CA to be small so that bulk flow can be neglected, then: 0002 0003 d sinh aL 0002 y
NA D 0002D dy sinh aL DCAi a cosh aL sinh aL D DCAi a/ tanh aL D DCAi aL/L tanh aL D
kL D D/L ˇ D Dk /kL D k /D L D aL ∴
NA D
kL CAi ˇ tanh ˇ
PROBLEM 10.14 The diffusivity of the vapour of a volatile liquid in air can be conveniently determined by Winkelmann’s method, in which liquid is contained in a narrow diameter vertical tube
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maintained at a constant temperature, and an air stream is passed over the top of the tube sufficiently rapidly to ensure the partial pressure of the vapour there remains approximately zero. On the assumption that the vapour is transferred from the surface of the liquid to the air stream by molecular diffusion, calculate the diffusivity of carbon tetrachloride vapour in air at 321 K and atmospheric pressure from the following experimentally obtained data: Time from commencement of experiment (ks)
Liquid level (cm)
0 1.6 11.1 27.4 80.2 117.5 168.6 199.7 289.3 383.1
0.00 0.25 1.29 2.32 4.39 5.47 6.70 7.38 9.03 10.48
The vapour pressure of carbon tetrachloride at 321 K is 37.6 kN/m2 , and the density of the liquid is 1540 kg/m3 . The kilogram molecular volume is 22.4 m3 .
Solution Equations 10.37 and 10.38 state that: NA D 0002D
CA2 0002 CA1 CT y2 0002 y1 CBm
In this problem, the distance through which the gas is diffusing will be taken as h and CA2 D 0. ∴
NA D DCA /h CT /CBm kmol/m2 s
where CA is the concentration at the interface. If the liquid level falls by a distance dh in time dt, the rate of evaporation is: NA D +L /M dh/dt kmol/m2 s Hence:
+L /M dh/dt D DCA /h CT /CBm
If this equation is integrated, noting that when t D 0, h D h0 , then: h2 0002 h02 D 2MD/+L CA CT /CBm t or: t/h 0002 h0 D +L /2MD CBm /CA CT h 0002 h0 C +L CBm /MDCA CT h0
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Thus a plot of t/h 0002 h0 against h 0002 h0 will be a straight line of slope s where: s D +L CBm /2MDCA CT
or
D D +L CBm /2MCA CT s
The following table may be produced: t (ks) 1.6 11.1 27.4 80.2 117.5 168.6 199.7 289.3 383.1 h 0002 h0 (mm) 2.5 12.9 23.2 43.9 54.7 67.0 73.8 90.3 104.8 t/h 0002 h0
s/m ð 1000026
0.64 0.86 1.18 1.83 2.15 3.52 2.71 3.20 3.66
3.5
t / (h −h0) s/m × 10−6
3.0
2.5
2.0 Slope = 3.04 ×10−7 s/m2
1.5
1.0
0.5
0
20
40
60
80
100
120
(h −h0) mm
Figure 10b.
These data are plotted as Fig. 10b and the slope is: s D 3.54 0002 0.5 1000026 /100 ð 1000023 D 3.04 ð 1000027 s/m2 CT D 1/22.4 273/321 D 0.0380 kmol/m3 M D 154 kg/kmol CA D 37.6/101.3 1/22.4 273/321 D 0.0141 kmol/m3 +L D 1540 kg/m3 CB1 D 0.0380 kmol/m3
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233
CB2 D 0.0380 0002 0.0141 D 0.0239 kmol/m3 ∴
CBm D 0.0380 0002 0.0239 / ln0.0380/0.0239 D 0.0304 kmol/m3
Hence:
D D 1540 ð 0.0304 /2 ð 154 ð 0.0141 ð 0.0380 ð 3.04 ð 1000027
D 9.33 ð 1000026 m2 /s
PROBLEM 10.15 Ammonia is absorbed in water from a mixture with air using a column operating at atmospheric pressure and 295 K. The resistance to transfer can be regarded as lying entirely within the gas phase. At a point in the column the partial pressure of the ammonia is 6.6 kN/m2 . The back pressure at the water interface is negligible and the resistance to transfer can be regarded as lying in a stationary gas film 1 mm thick. If the diffusivity of ammonia in air is 0.236 cm2 /s, what is the transfer rate per unit area at that point in the column? If the gas were compressed to 200 kN/m2 pressure, how would the transfer rate be altered?
Solution See Volume 1, Example 10.3.
PROBLEM 10.16 What are the general principles underlying the two-film penetration and film-penetration theories for mass transfer across a phase boundary? Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. According to the penetration theory, the instantaneous rate of mass transfer per unit area NA t at some time t after the commencement of transfer is given by: D NA t D CA 0010t where CA is the concentration driving force and D is the diffusivity. Obtain expressions for the average rates of transfer on the basis of the Higbie and Danckwerts assumptions.
Solution The various theories for the mechanism of mass transfer across a phase boundary are discussed in Section 10.5. The basic equation for unsteady state equimolecular counter-diffusion is: 00040002 0003 0002 2 0003 0002 2 0003 0005 ∂CA ∂ 2 CA ∂ CA ∂ CA C C (equation 10.67) DD ∂t ∂x 2 yz ∂y z xz ∂z2 xy
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Considering the diffusion of solute A away from the interface in the y-direction this equation becomes: ∂CA ∂ 2 CA DD 2 ∂t ∂y The boundary conditions are: tD0 t>0 t>0
0< /><1 yd0='>1>
CA D CAo CA D CAi CA D CAo
where CAo is the concentration in the bulk of the phase and CAi is the equilibrium concentration at the interface. The instantaneous rate of mass transfer per unit area NA at time t is given by: NA t D CA D/0010t Higbie assumed that every element of surface is exposed to the gas for the same length of time ) before being replaced by liquid of the bulk composition. Amount absorbed in time ): 0001 ) 0001 ) QD NA t d) D CA D/0010) d) D 2CA D)/0010 0
0
000e p p The average rate of absorption: Q/) D 2CA D)/0010 /) D 2CA D/0010)
Danckwerts suggested that each element would not be exposed for the same time but that a random distribution of ages would exist. It is shown in Section 10.5.2 that this age distribution may be expressed ft D se0002st . The average rate of absorption is the value of NA t averaged over all elements of the surface having ages between 0 and 1 is then given by: 0001 1 0001 1 p p 0002s) NA D s NA t e d) D CA D/0010 e0002s) / ) d) D CA Ds 0
0
PROBLEM 10.17 A solute diffuses from a liquid surface at which its molar concentration is CAi into a liquid with which it reacts. The mass transfer rate is given by Fick’s law and the reaction is first order with respect to the solute. In a steady-state process, the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration CA of solute at a depth y from the surface in terms of the molecular diffusivity D and the reaction rate constant k . What is the molar flux at the surface?
Solution 2 2 2 As in Problem p 10.13, the basic equation is: d CA /dy D a CA where a D k /D
Then:
CA D A cosh ay C B sinh ay
(i) ii
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235
The first boundary condition is at y D 0, CA D CAi , and CAi D A. CA D CAi cosh ax C B sinh ax
Hence:
iii
The second boundary condition is that when y D L and: NA D 0002DdCA /dy yD0 D 00022DdCA /dy yDL Differentiating equation (iii): dCA /dy D CAi a sinh ay C Ba cosh ay and:
dCA /dy yD0 D Ba
and:
dCA /dy yDL D aB/2 D CAi a sinh aL C Ba cosh aL
so that:
BD
2CAi sinh aL 1 0002 2 cosh aL
iv
Substituting equation (iv) into equation (iii): 2CAi sinh aL sinh ay 1 0002 2 cosh aL D CAi [cosh ay 0002 2cosh ay cosh aL C sinh aL sinh ay ]/1 0002 2 cosh aL
CA D CAi cosh ay C
D CAi [cosh ay 0002 2 cosh ay C L ] The molar flux at the surface D NA D 0002DdCA /dy yD0 . dCA D CAi [a sinh ay 0002 2a sinh aa C L ] dy dCA /dy yD0 D 00022CAi a2 sinh aL NA D 2DCAi a2 sinh aL a D k /D
NA D 2DCAi k /D sinh L k /D p D 2CAi k sinhL k /D
PROBLEM 10.18 4 cm3 of mixture formed by adding 2 cm3 of acetone to 2 cm3 of dibutyl phthalate is contained in a 6 mm diameter vertical glass tube immersed in a thermostat maintained at 315 K. A stream of air at 315 K and atmospheric pressure is passed over the open top of the tube to maintain a zero partial pressure of acetone vapour at that point. The liquid level is initially 11.5 mm below the top of the tube and the acetone vapour is transferred to the air stream by molecular diffusion alone. The dibutyl phthalate can be regarded as completely non-volatile and the partial pressure of acetone vapour may be calculated from Raoult’s law on the assumption that the density of dibutyl phthalate is sufficiently greater than that of acetone for the liquid to be completely mixed.
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Calculate the time taken for the liquid level to fall to 5 cm below the top of the tube, neglecting the effects of bulk flow in the vapour. 1 kmol occupies 22.4 m3 . Molecular weights of acetone, dibutyl phthalate D 58 and 279 kg/kmol respectively. Liquid densities of acetone, dibutyl phthalate D 764 and 1048 kg/m3 respectively. Vapour pressure of acetone at 315 K D 60.5 kN/m2 . Diffusivity of acetone vapour in air at 315 K D 0.123 cm2 /s.
Solution Considering the situation when the liquid has fallen to a depth h cm below the top of the tube, volume of acetone evaporated D 0010/4 0.6 2 h 0002 1.15 D 0.283h 0002 1.15 cm3 . At this time, the amount of dibutyl phthalate is: 2 ð 1.048/278 D 0.00754 mol and the amount of acetone D [2 0002 0.283h 0002 1.15 ]0.764/58 D 0.0306 0002 0.00372h
∴
8.23 0002 h
0.0306 0002 0.00372h D 0.00754 C 0.0306 0002 0.00372h
10.24 0002 h
0002 0003 8.23 0002 h Partial pressure of acetone D 60.5 kN/m2 10.24 0002 h Molar concentration of acetone vapour at the liquid surface 0002 0003 0002 0003 0002 00030002 0003 60.5 273 1 8.23 0002 h ð ð D 101.3 315 22400 10.24 0002 h 0002 0003 8.23 0002 h D 2.31 ð 1000025 mol/cm3 10.24 0002 h Mole fraction of acetone D
dh 0.764 ð D 0.0132dh/dt mol/cm2 s dt 58 0002 0003 8.23 0002 h 00025 D D/h ð molar concentration at surface D 0.123/h 2.31 ð 10 10.24 0002 h 0002 0003 8.23 0002 h 1 dh D ð 2.84 ð 1000026 0.0132 ∴ dt h 10.24 0002 h 0002 0003 10.24 0002 h dt and: h dh D 8.23 0002 h 4650 Rate of evaporation of acetone: NA D
The time for the liquid level to fall from 1.15 cm to 5 cm below the top of the tube is obtained by integrating this equation: 0003 0003 0001 5 0002 0001 t 0001 5 0002 0001 t 10.24 0002 h 1 16.6 1 h dh D dt D h 0002 2.02 0002 dh D dt 8.23 0002 h 4650 0 h 0002 8.23 4650 0 1.15 1.15 and:
t D 79500 s 79.5 ks ³ 22 h
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PROBLEM 10.19 A crystal is suspended in fresh solvent and 5% of the crystal dissolves in 300 s. How long will it take before 10% of the crystal has dissolved? Assume that the solvent can be regarded as infinite in extent, that the mass transfer in the solvent is governed by Fick’s second law of diffusion and may be represented as a unidirectional process, and that changes in the surface area of the crystal may be neglected. Start your derivations using Fick’s second law.
Solution The mass transfer process is governed by Fick’s second law: ∂CA ∂ 2 CA DD 2 ∂t ∂y
(equation 10.66)
and discussed in Section 10.5.2 The boundary conditions for the crystal dissolving are: When t D 0 t>0 t>0
0< /><1 yd1='>1>
CA D 0 CA D 0 CA D CAs the saturation value
These boundary conditions allow the solution of equation 10.66 using Laplace transforms as the most convenient method: 0001 1 ∂CA ∂CA D dt (equation 10.102) e0002pt dt ∂t 0 0001 1 000f 00101 D e0002pt CA 0 C p e0002pt CA dt D 0 C pCNA (equation 10.103) 0
Taking Laplace transforms of both sides of equation 10.66: ∂2 CNA pCNA D D 2 ∂y ∴
and:
∂2 CNA p 0002 CNA D 0 ∂y 2 D p p CNA D Ae p/D y C Be0002 p/D y
(equation 10.105)
When y D 1, CA D 0 ∴ and A D 0 CNA D 0 When y D 0, CA D CAs and CNA D CAs /po , B D CAs /p p C As 0002 p/D y e ∴ CNA D p p Inverting: CA D CAs erfcy/2 Dt
(See Volume 1, Appendix Table 13)
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0002
Mass transfer rate at the surface D 0002D
0003 yD0
0002 00030012 1 2 y 0002y 2 /4Dt p p e d 0010 y/2pDt
2 Dt 0002 0003 2 1 2 D CAs p 0002 p e0002y /4Dt (equation 10.111) 0010 2 Dt 0003 0002 ∂CA CAs ∴ D 0002p dy yD0 0010Dt 0002 0003 ∂CA D D CAs NA t D 0002D ∂t yD0 0010t 0001 t D Dp The mass transfer in time t D dt D 2 t 0010t 0010 0 p and the mass transfer is proportional to t M1 t1 Thus: D M2 t2
∂CA ∂ D CAs ∂y ∂y
and:
0011
∂CA ∂y
0001
M1 D 5%, M2 D 10%, and t1 D 300 s p 0.5 D 300/t2 and t2 D 1200 s
PROBLEM 10.20 In a continuous steady state reactor, a slightly soluble gas is absorbed into a liquid in which it dissolves and reacts, the reaction being second-order with respect to the dissolved gas. Calculate the reaction rate constant on the assumption that the liquid is semi-infinite in extent and that mass transfer resistance in the gas phase is negligible. The diffusivity of the gas in the liquid is 1000028 m2 /s, the gas concentration in the liquid falls to one half of its value in the liquid over a distance of 1 mm, and the rate of absorption at the interface is 4 ð 1000026 kmol/m2 s.
Solution The equation for mass transfer with chemical reaction is: ∂CA ∂ 2 CA D D 2 0002 k CnA ∂t ∂y
(equation 10.170)
For steady state second order reaction where n D 2: D
d2 CA 0002 k C2A D 0 dy 2
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Putting dCA /dy D q:
239
dq dCA dq d2 CA dq D Dq D 2 dy dy dCA dy dCA
Thus:
Dq
dq 0002 k C2A D 0 dCA
q dq D k /D C2A dCA q2 /2 D k /D C3A /3 C const. In an infinite system at y D 1, CA D 0 and dCA /dy D 0 and hence the constant D 0. 0003 0002 dCA 2 2 k 3 2k 3/2 dCA CA and D0002 C ∴ D dy 3D dy 3D A noting the negative sign since dCA /dy is negative for all values of CA . 2k 00021/2 Thus: 00022CA D 0002 y C constant 3D At the free surface, y D 0 and CA D CAi D constant. ∴
and: or:
00021/2
constant D 00022CAi 0013 0014 p 00021/2 00021/2 2 CA 0002 CAi D 2k /3D y 00021/2
CA
00021/2
0002 CAi
D
p k /6D y
When y D y1 , CA D CAi /2, and: 00021/2
CAi /2 00021/2 0002 CAi
k /6D y1 p 00021/2 When y1 D 1000023 , substituting gives: CAi D 2.42 ð 1000023 k /6D
p p 00021/2 ∴ CAi 0002 2.42 ð 1000023 k /6D D k /6D y D
The mass transfer rate at the interface, where y D 0, is: 0002 0003 dCA 2k D 3/2 NA t D 0002D CAi D dy yD0 3 0002 00033 2k D 1 p ð D D 8.47 ð 108 D2 /k 3 2.42 ð 1000023 k /6D
When D D 1 ð 1000028 m2 /s and NA D 4 ð 1000026 kmol/m2 s : k D 8.47 ð 108 ð 1 ð 1000028 2 /4 ð 1000026 D 212 m3 /kmol s
PROBLEM 10.21 Experiments have been carried out on the mass transfer of acetone between air and a laminar water jet. Assuming that desorption produces random surface renewal with a
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constant fractional rate of surface renewal, s, but an upper limit on surface age equal to the life of the jet, 4, show that the surface age frequency distribution function, ft , for this case is given by: ft D s exp0002st/[1 0002 exp0002st ]
for
0< />< />
ft D 0
for
t > 4.
Hence, show that the enhancement, E, for the increase in value of the liquid-phase mass transfer coefficient is: E D [0010s4 1/2 erfs4 1/2 ]/f2[1 0002 exp0002s4 ]g where E is defined as the ratio of the mass transfer coefficient predicted by conditions described above to the mass transfer coefficient obtained from the penetration theory for a jet with an undisturbed surface. Assume that the interfacial concentration of acetone is practically constant.
Solution For the penetration theory: ∂CA ∂ 2 CA DD 2 ∂t ∂y
(equation 10.66)
As shown in Problem 10.19, this equation can be transformed and solved to give: p p N A D Ae p/D y C Be0002 p/D y C The boundary conditions are: When y D 0,
CA D CAi ,
and when y D 1, ∴
B D CAi /p
CA D 0 and A D 0 p
N A D CAi e0002 p/D y C p 0015 NA dC 1 1 0002pp/D y D 0002CAi e dy D p
From Volume 1, Appendix, Table 12, No 84, the inverse: dCA 1 1 0002y 2 /4Dt D 0002CAi e dy D 0010t 0002 0003 dCA D at time t At the surface: NA t D 0002D D CAi dy yD0 0010t The average rate over a time 4 is: 0001 4 1 D dt D p D 2CAi CAi 4 0010 0 00104 t
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241 p 6 0 and NA D 2CAi 0002 CA0 D/00104 for mass transfer without surface D MASS TRANSFER
In general, CA0 renewal. Random surface renewal is discussed in Section 10.5.2 where it is shown that the age distribution function is: D constant e0002st D ke0002st where s is the rate of production of fresh surface per unit total area of surface. If the maximum age of the surface is 4, then: 0001 4 e0002st dt D 1 K 0
k 0002 [e0002st ]40 D 1 s 1 0002 e0002s4 D s/k and K D 0002 ∴ the age distribution function is:
s 1 0002 e0002s4
s 1 0002 e0002s4
0003
e0002st
The mass transfer in time 4 is: 0001 4 0001 4 0002st D D e s s 0002st p dt CAi 0002 CA0
CAi 0002 CA0
e dt D 0002s4 0002s4 0010t 10002e 0010t 10002e t 0 0 p p The integral is conveniently solved by substituting st D ˇ2 and t D ˇ/ s or s dt D 2ˇ dˇ and dt D 2ˇ dˇ/s p 0001 4p 0001 4 p s 0002ˇ 2ˇ dˇ 0010 p 2 erf s4 0002ˇ2 e Dp erf s4 D CAi 0002 CA0 Ds e dˇ D Then: ˇ s s 0 s 1 0002 e0002s4 0 The enhancement factor E is given by: p p erf s4 p p CAi 0002 CA0 Ds 0002s4 0010s4 erf s4 1 0002 e ED D 21 0002 e0002s4
D CAi 0002 CA0
2 00104
PROBLEM 10.22 Solute gas is diffusing into a stationary liquid, virtually free of solvent, and of sufficient depth for it to be regarded as semi-infinite in extent. In what depth of fluid below the surface will 90% of the material which has been transferred across the interface have accumulated in the first minute? Diffusivity of gas in liquid D 1000029 m2 /s.
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Solution As in the previous problem, the basic equation is: ∂ 2 CA ∂CA DD 2 ∂t ∂y
(equation 10.66)
which can be solved using the same boundary conditions to give the rate of mass transfer at depth, y, NA y,t as: D 0002y 2 /4Dt dCA NA y,t D 0002D D CAi e dy 0010t At some other value of y D L, the amount which has been transferred in time t per unit area is: 0001 t D 0002y 2 /4Dt e CAi dt 0010t 0 This integral can be solved by making the substitution:
so that: and:
ˇ2 D y 2 /4Dt p ˇ D y/2 Dt p t D y 2 /4Dˇ2 , t00021/2 D y/2ˇ D dt D 0002y 2 /2D ˇ00023 dˇ
The amount transferred at depth L is then: 0004 p 0005 D y 2 Dt 0002y 2 /4Dt p y p e 0002 0010 erfc p D CAi 0010 D y 2 Dt 0004 0005 Dt 0002y 2 /4Dt y e D CAi 2 0002 y erfc p 0010 2 Dt 0004 0005 Dt 0002y 2 /4Dt y e 2 0002 y erfc p 0010 2 Dt mass transfer at L D and: mass transfer at y D 0 Dt 2 0010 p p y y 2 2 D e0002y /4Dt 0002 p 0010 erfc p D e0002X 0002 X 0010 erfc X 2 Dt 2 Dt p where X D y/2 Dt Under the conditions in this problem, this ratio D 0.1. erfc X D 1 0002 erf X so that erfc X can be calculated from Table 13 in the Appendix of Volume 1. Values of X will be assumed and the right hand side evaluated until a value of X is found such that the right hand side D 0.1.
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MASS TRANSFER
X
e0002X
erf X
erfc X
p X 0010 erfc X
Right hand side
1 0.9 0.97 0.96
0.368 0.445 0.390 0.398
0.843 0.797 0.830 0.825
0.157 0.203 0.170 0.175
0.278 0.324 0.292 0.297
0.0897 0.121 0.098 0.101
2
∴
X D 0.96 D y 2 /4Dt
∴
y D 0.96 ð 4 ð 1000029 ð 60 0.5 D 4.8 ð 1000024 m or 0.48 mm
PROBLEM 10.23 A chamber, of volume 1 m3 , contains air at a temperature of 293 K and a pressure of 101.3 kN/m2 , with a partial pressure of water vapour of 0.8 kN/m2 . A bowl of liquid with a free surface of 0.01 m2 and maintained at a temperature of 303 K is introduced into the chamber. How long will it take for the air to become 90% saturated at 293 K and how much water must be evaporated? The diffusivity of water vapour in air is 2.4 ð 1000025 m2 /s and the mass transfer resistance is equivalent to that of a stagnant gas film of thickness 0.25 mm. Neglect the effects of bulk flow. Saturation vapour pressure of water D 4.3 kN/m2 at 303 K and 2.3 kN/m2 at 293 K.
Solution Moles transferred, n D DA/L CAs 0002 CA
where CA D concentration (kmol/m3 ), CAs is the saturation value of CA at the surface D is the diffusivity and L is the thickness of the stagnant gas film. If the saturated vapour pressure at the interface is 4.3 kN/m2 and if at any time the partial pressure in the air is PA kN/m2 , then the rate of evaporation is given by: 0006 0002 00030007 dn 1 4.3 0.01 ð 2.4 ð 1000025 273 PA D ð 0002 dt 0.25/1000
22.4 303 101.3 101.3 D 3.81 ð 1000027 4.3 0002 PA kmol/s 1 m3 of air at 303 K and 101.3 kN/m2 is equivalent to 1/22.4 273/293 D 0.0416 kmol Initial moisture content D 0.8/101.3 ð 0.0416 D 3.29 ð 1000024 kmol Final moisture content D 0.9 ð 2.3/101.3 ð 0.0416 D 8.50 ð 1000024 kmol ∴ Water evaporated D 8.50 0002 3.29 ð 1000024 D 5.21 ð 1000024 kmol D 5.21 ð 1000024 ð 18 D 9.38 ð 1000023 kg water
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At a pressure P kN/m2 : n D PA /101.3 1/22.4 273/293 D 4.11 ð 1000024 PA kmol/m3 dn/dt D 4.11 ð 1000024 dPA /dt
and: ∴
4.11 ð 1000024 0001
2.3
∴
0.8
dPA D 3.81 ð 1000027 4.3 0002 PA
dt
dPA D 9.27 ð 1000024 dt 4.3 0002 PA
from which t D 604 s 10 min
PROBLEM 10.24 A large deep bath contains molten steel, the surface of which is in contact with air. The oxygen concentration in the bulk of the molten steel is 0.03% by mass and the rate of transfer of oxygen from the air is sufficiently high to maintain the surface layers saturated at a concentration of 0.16% by weight. The surface of the liquid is disrupted by gas bubbles rising to the surface at a frequency of 120 bubbles per m2 of surface per second, each bubble disrupts and mixes about 15 cm2 of the surface layer into the bulk. On the assumption that the oxygen transfer can be represented by a surface renewal model, obtain the appropriate equation for mass transfer by starting with Fick’s second law of diffusion and calculate: (a) The mass transfer coefficient (b) The mean mass flux of oxygen at the surface (c) The corresponding film thickness for a film model, giving the same mass transfer rate. Diffusivity of oxygen in steel D 1.2 ð 1000028 m2 /s. Density of molten steel D 7100 kg/m3 .
Solution If C0 is defined as the concentration above a uniform datum value: ∂ 2 C0 ∂C0 DD 2 ∂t ∂y
(equation 10.100)
The boundary conditions are: when
t D 0, t>0 t>0
0< /><1 yd0='>1>
C0 D 0 C0 D C0i C0 D 0
The equation is most conveniently solved using Laplace transforms. The Laplace transN 0 of C0 is: form C 0001 1 N0 D e0002pt C0 dt (equation 10.101) C 0
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MASS TRANSFER
N0 ∂C D ∂t
0001
∂C0 dt ∂t 0 0001 0002pt 0 1 D [e C ]0 C p
Then:
1
e0002pt
(equation 10.102) 1
N0 e0002pt C0 dt D pC
(equation 10.103)
0
Since the Laplace transform operation is independent of y, ∂ 2 C0 ∂ 2 C0 D 2 ∂y ∂y 2
(equation 10.104)
Taking Laplace transforms of both sides of equation 10.100: N0 D D pC
From which:
N0 ∂2 C ∂y 2
N0 ∂2 C p N0 0002 C D0 2 ∂y D p p N 0 D Ae p/D y C Be0002 p/D y C
(equation 10.105)
N0 D 0 C and N 0 D C0i /p C and p 0 N 0 D Ci e0002 p/D y C p p N0 dC C0 D 0002 p i e0002 p/D y dy pD
When y D 1, When y D 0, ∴
Inverting:
1 C0 ∂C0 2 D 0002 p i ð p e0002y /4Dt ∂y 0010t D
AD0 B D C0i /p
(See Volume 1, Appendix Table 12) 0002
The mass transfer rate at the surface, NA t D 0002D
∂C0 ∂y
0003
D yD0
C0i
D at time t 0010t
The average rate of mass transfer in time t: 0001 D 1 t 0 D 0 dt D 2Ci Ci t 0 0010t 0010t Taking 1 m2 of surface, the area disrupted by the bubbles per second is: 120 ð 15/10000 D 0.18/s ∴ Average surface age duration D 1/0.18 D 5.55 s
C0i D 0.16 0002 0.03 /100 D 0.0013 kg O2 /kg steel D 0.0013/32 ð 7100 D 0.2885 kmol/m3
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Then:
(a) The mass transfer coefficient D 2
D D 21.2 ð 1000028 /0010 ð 5.55 0.5 0010t D 5.25 ð 1000025 m/s
(b) The mean rate of transfer, NA D 2C0i D/0010t 0.5 D 2 ð 0.28851.2 ð 1000028 /0010 ð 5.55 0.5 D 1.51 ð 1000025 kmol/m2 s N 0i (c) The film thickness L is given by: NA D D/L C and:
L D 1.2 ð 1000028 ð 0.2885 /1.51 ð 1000025 D 2.29 ð 1000024 m D 0.23 mm
PROBLEM 10.25 Two large reservoirs of gas are connected by a pipe of length 2L with a full-bore valve at its mid-point. Initially a gas A fills one reservoir and the pipe up to the valve and gas B fills the other reservoir and the remainder of the pipe. The valve is opened rapidly and the gases in the pipe mix by molecular diffusion. Obtain an expression for the concentration of gas A in that half of the pipe in which it is increasing, as a function of distance y from the valve and time t after opening. The whole system is at a constant pressure and the ideal gas law is applicable to both gases. It may be assumed that the rate of mixing in the vessels is high so that the gas concentration at the two ends of the pipe do not change.
Solution The system and nomenclature are shown in Fig. 10c. y = −L
y = +L y =0
CA = CA 0
CA = 0
Figure 10c.
When time t D 0, For gas A: ∂CA ∂ 2 CA DD 2 ∂t ∂y
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MASS TRANSFER
When When When When
t D 0, t D 0, t>0 t>0
0002L < y < 0 0< />
CA CA CA CA
D CA0 D0 D CA0 D0
CB CB CB CB
D0 D CB0 D0 D CB0
For gas B: ∂CB ∂ 2 CB DD 2 ∂t ∂y When When When When
tD0 tD0 t>0 t>0
0002L < y < 0 0< />
and for all values of y: ∂CA ∂CB C D0 ∂y ∂y As in previous problems, these equations may be solved by the use of Laplace transforms. For y > 0: p p N A D Ae p/D y C Be0002 p/D y C and for y < 0:
p
p N A D A0 e C
p/D y
C B 0 e0002
p/D y
C CA0 /p
The boundary conditions may now be used to evaluate the constants thus: p P 00022 p/D L CA0 /p e p AD0002 21 0002 e00022 p/D L
CA0 /p
p 21 0002 e00022 p/D L
p A0 D 0002B0 e2 p/D L p 00022 p/D L C 1
Be p B0 D e2 p/D L C 1
BD
Substituting these values: 0007 nD1 0006 CA0 2nL C y 2n C 1 L 0002 y p erfc p 0002 erfc CA D 2 nD0 2 Dt 2 Dt
This relation can be checked as follows: 0007 1 0006 n C 1 L CA0 nL CA0 (a) When y D 0: CA D erfc p 0002 erfc p D 2 0 2 Dt Dt (b) When y D L: CA D 0
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PROBLEM 10.26 A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Fick’s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, D, the first-order reaction rate constant k , the film thickness L and the concentration CAS of solute in a saturated solution. The reaction is initially carried out at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K? Reaction rate constant at 293 K D 2.5 ð 1000026 s00021 . Energy of activation for reaction (in Arrhenius equation) D 26430 kJ/kmol. Universal gas constant R D 8.314 kJ/kmol K. Molecular diffusivity D D 1000029 m2 /s. Film thickness, L D 10 mm. Solubility of gas at 313 K is 80% of solubility at 293 K.
Solution See Volume 1, Example 10.11
PROBLEM 10.27 Using Maxwell’s law of diffusion obtain an expression for the effective diffusivity for a gas A in a binary mixture of B and C, in terms of the diffusivities of A in the two pure components and the molar concentrations of A, B and C. Carbon dioxide is absorbed in water from a 25 per cent mixture in nitrogen. How will its absorption rate compare with that from a mixture containing 35 per cent carbon dioxide, 40 per cent hydrogen and 25 per cent nitrogen? It may be assumed that the gas-film resistance is controlling, that the partial pressure of carbon dioxide at the gas–liquid interface is negligible and that the two-film theory is applicable, with the gas film thickness the same in the two cases. Diffusivity of CO2 in hydrogen D 3.5 ð 1000025 m2 /s; in nitrogen D 1.6 ð 1000025 m2 /s.
Solution Maxwell’s Law of Diffusion is discussed in Section 10.3.2 where for a two component gaseous mixture: 0002dPA /dy D FAB CA CB uA 0002 uB
(equation 10.77)
For an ideal gas,
PA D CA RT
(equation 10.9a)
and from equation 10.78:
uA D N0A /CA
(equation 10.9b)
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249
when B is not undergoing mass transfer, or uB D 0, then: 0002RTdCA /dy D FAB CB N0A N0A D 0002
RT dCA RT CT dCA D0002 FAB CB dy FAB CT CB dy
By comparison with Stefan’s Law: N0A D 0002DAB
CT dCA CB dy
Then:
DAB D RT/FAB CT
or:
FAB D
(equation 10.29)
RT DAB CT
Applying to A in a mixture of B and C: 0002dPA /dy D FAB CA CB uA 0002 uB C FAC CA CC uA 0002 uC
For the case where uB D uC D 0: dCA RT RT 0002RT D CB N0A C CC N0A dy DAB CT DAC CT 0002dCA /dy CT CB /DAB C CC /DAC
or:
N0A D
From Stefan’s Law:
N0A D 0002D0
CT dCA dCA /dy CT D0002 CT 0002 CA dy CT 0002 CA /D0
i
ii
where D0 is the effective diffusivity of A in the mixture Comparing equations (i) and (ii): 1 CB CC 1 1 D C D0 DAB CT 0002 CA DAC CT 0002 CA For CO2 in N2 : N0A D D
1 CT CCO2 CT yCO2 DDð (equation 10.33) ð CN2 lm L yN2 lm L
D D 1.6 ð 1000025 m2 /s yN2 1 D 1.0, yN2 2 D 0.75, ∴
yN2 lm D [1 0002 0.75 / ln1/0.75 ] D 0.87 yCO2 D 0.25, CT and L are unknown
CT 1.6 ð 1000025 ð 0.25 D 4.6 ð 1000026 CT /L kmol/m2 s 0.87 L For CO2 in a mixture of H2 and N2 , the effective diffusivity, derived in the first part of the problem, is used to give D0 : Hence:
N0A D
1 1 0.4 0.25 1 C D 2.4 ð 1000025 m2 /s D ð ð D0 3.5 ð 1000025 1 0002 0.35 1.6 ð 1000025 1 0002 0.35
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
N0A D D0
CT CCO2 1 CT yCO2 D D0 ð ð ð CN2 CH2 lm L yN2 CH2 lm L
yCO2 D 0.35 yN2 CH2 l D 1.0, yN2 CH2 2 D 0.65 and:
N0A D 2.4 ð 1000025 ð
C ln1/0.65
ð 0.35 D 1.033 ð 1000025 CT /L 1 0002 0.65
L
∴ The ratio of mass transfer rates D 1.033 ð 1000025 /4.6 ð 1000026 D 2.25
PROBLEM 10.28 Given that from the penetration theory for mass transfer across an interface, the instantaneous rate of mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure time in the Higbie model and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be derived from first principles.
Solution Given that the instantaneous mass transfer rate D Kt00021/2 , then for the Higbie model, the average mass transfer rate for an exposure time te is given by: 0001 1 te 00021/2 Kt dt D 2Kte00021/2 te 0 For the Danckwerts model, the random surface renewal analysis, presented in Section 10.5.2, shows that the fraction of the surface with an age between t and t C dt is a function of t D ft dt and that ft D Ke0002st where s is the rate of production of fresh surface per unit total area. For a total surface area of unity: 0006 0002st 00071 0001 1 e Ke0002st dt D 1 D K D K/s 0002s 0 0 ∴
K D s and ft D se0002st dt
The rate of mass transfer for unit area is: 0001 1 0001 00021/2 0002st Kt ð se dt D Ks 0
1
t00021/2 e0002st dt 0
2
Substituting ˇ D st and s dt D 2ˇ dˇ, then: 0001 1p 0001 1 p p p p s 0002ˇ2 2ˇ 2 e dˇ D K s ð 2 Rate D Ks e0002ˇ dˇ D K s ð 2 ð 0010/2 D K 0010s ˇ s 0 0
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MASS TRANSFER
If the rates from each model are equal, then: p 2Kte00021/2 D K 0010s
or
ste D 4/0010
PROBLEM 10.29 Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the mass transfer rate immediately before mixing is only 16 per cent of that calculated from the theory and the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process takes place every second, what thickness of surface film would account for the discrepancy? Diffusivity of ammonia in water D 1.76 ð 1000029 m2 /s.
Solution For the penetration theory:
When t D 0, When t > 0, When t > 0,
∂CA ∂ 2 CA DD 2 (equation 10.66) ∂t ∂y CA D 0 y D 0, CA D CAi D constant yD1 CA D 0
As shown earlier in problems 10.19 and 10.21, this equation may be transformed and solved to give: p p N A D Ae2 p/D y C Be00022 p/D y C (equation 10.105) When y D 0, y D 1,
N A D CAi /p C NA D 0 C
and hence:
A D 0 and B D CAi /p p N A D CAi e0002 p/D y Hence, C p NA p 0002pp/D y ∂C D 0002CAi e and: ∂y D 0002 0003 ∂CA D CAi in time t (as in Problem 10.21). At the surface, NA t D 0002D D ∂y yD0 0010t For the film, the origin is taken at the interface between the film (whose thickness is L) and the mixed fluid. p p N A D Ae p/D y C Be0002 p/D y Again: C (equation 10.105)
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CAx at y D 0 is now a variable y D 1, CA D 0 p NA D C N Ax e0002 p/D y A D 0 and C NA p N 0002pp/D y dC D0002 CAx e dy D 0016 0017 N1 dC pN CAx D0002 dy D
Hence: and: and:
yD0
To maintain mass balance at the film boundary: D CAi 0002 CAx D 0002D L
0002
∂CA ∂y
0003 yD0
Taking Laplace transforms: D L So:
and:
Hence:
Inverting:
0002
CAi N Ax 0002C p
0016
0003
D 0002D
D CAi D L p N Ax C
0002
NA ∂C ∂y
D C L
0017
D
N Ax pD C
yD0
0003
N Ax pD C
D CAi L p D D p C pD L
p p D CAi 1 e0002 p/D y DL p D p C pD L p 1 1/L D 0002CAi p p e0002 p/D y p D p C p L 0016p 0017 Dt CAi y/L Dt/L2 y ∂CA D0002 e e C p erfc ∂y L L 2 Dt 0016p 0017 0002 0003 ∂CA D Dt/L2 Dt NA yD0 D 0002D D CA i e erfc ∂y yD0 L L 0016p 0017 D Dt/L2 Dt CAi e erfc L L D 0.16 D CAi 0010t N Ax dC D0002 dy
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MASS TRANSFER
or: writing
253
Dt Dt/L2 Dt 1 ð 0.16 D 0.0903 e erfc D 2 2 L L 0010 Dt 2 XD then XeX erfc X D 0.0903 2 L
Solving by trial and error: X D 0.101 When t D 1 s, D D 1.76 ð 1000029 m2 /s, and: L D 0.42 mm.
PROBLEM 10.30 A deep pool of ethanol is suddenly exposed to an atmosphere consisting of pure carbon dioxide and unsteady state mass transfer, governed by Fick’s Law, takes place for 100 s. What proportion of the absorbed carbon dioxide will have accumulated in the 1 mm thick layer of ethanol closest to the surface? Diffusivity of carbon dioxide in ethanol D 4 ð 1000029 m2 /s.
Solution See Volume 1, Example 10.6.
PROBLEM 10.31 A soluble gas is absorbed into a liquid with which it undergoes a second-order irreversible reaction. The process reaches a steady-state with the surface concentration of reacting material remaining constant at CAs and the depth of penetration of the reactant being small compared with the depth of liquid which can be regarded as infinite in extent. Derive the basic differential equation for the process and from this derive an expression for the concentration and mass transfer rate (moles per unit area and unit time) as a function of depth below the surface. Assume that mass transfer is by molecular diffusion. If the surface concentration is maintained at 0.04 kmol/m3 , the second-order rate constant k2 is 9.5 ð 103 m3 /kmol s and the liquid phase diffusivity D is 1.8 ð 1000029 m2 /s, calculate: (a) The concentration at a depth of 0.1 mm. (b) The molar rate of transfer at the surface kmol/m2 s . (c) The molar rate of transfer at a depth of 0.1 mm. It may be noted that if: dCA d2 CA dq D q, then: Dq 2 dy dy dCA
Solution Considering element of unit area and depth dy, then for a steady state process: RATE IN 0002 RATE OUT D REACTION RATE
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0011 0002 0003 0012 d dCA dCA dCA 0002 0002D C 0002D 0002D dy D k2 C2A dy
dy dy dy dy
or:
D
d2 C A D k2 C2A dy 2
d2 CA k2 0002 C2A D 0 dy 2 D Putting:
qD
dCA d2 C A dq dCA dq dq , then: D Dq D Ð dy dy 2 dy dCA dy dCA
dq k2 0002 C2A D 0 dCA D k2 q dq D C2A D 2 k2 C3A q D CK Integrating: 2 D 3 When y D 1, q D 0, CA D 0 and: K D 0 0002 0003 1 dCA 2 k2 C3A D Thus: 2 dy D 3 dCA 2 k2 3/2 Dš C dy 3D A Substituting:
q
As NA D 0002DdCA /dy is positive, negative root must apply and: 2 k2 00023/2 CA dCA D 0002 dy 3D 2 k2 00021/2 Integrating: 00022CA D 0002 yCK 3D 00021/2
When y D 0, CA D CAs and: K D 00022CAs Thus: or:
1 2 k2 y 0002 D 2 3D 00040002 0005 0003 CA 00021/2 1 k2 00021/2 y 00021 D CAs CAs 6D 0002
0003 CAs 1/2 1 k2 00021/2 y. 00021 D CAs CA 6D 00021/2 CA
(i) For the conditions given:
1 k2 D 6D
00021/2 CAs
001500110002 0003 0002 00030012 1 9.5 ð 103 6 1.8 ð 1000029
D 0.938 ð 106 (m/kmol)0.5
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MASS TRANSFER
At depth of 0.1 mm D 1000024 m : 0002
and:
CAs CA
00031/2
00021 0002
CAs CA
1 k2 y D 93.8 6D C1/2
D 93.8 CAs
D 18.8
00031/2
D 19.8 CA D
0.04 D 0.00010 kmol/m3 . 19.82
(ii) The molar transfer rate at surface is: 2 k2 3/2 dCA NA 0002 D D0002 C 0002D . dy 3 D As 2k2 D 3/2 D CAs 3 2 D ð 9.5 ð 103 ð 1.8 ð 1000029 0.04 3/2 3 D 3.38 ð 1000023 ð 0.008 D 2.70 ð 1000025 kmol/m2 s. (iii) The molar transfer rate at depth of 0.1 mm is: 2k2 D 3/2 CA NA D 3 D 3.38 ð 1000023 ð 0.00010 3/2 D 3.38 ð 1000029 kmol/m2 s
PROBLEM 10.32 In calculating the mass transfer rate from the penetration theory, two models for the age distribution of the surface elements are commonly used — those due to Higbie and to Danckwerts. Explain the difference between the two models and give examples of situations in which each of them would be appropriate. (a) In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average mass transfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is p [D/0010t ], where D is the diffusivity. (b) If for unit area of surface the surface renewal rate is s, by how much will the mass transfer coefficient be changed if no surface has an age exceeding 2/s? (c) If the probability of surface renewal is linearly related to age, as opposed to being constant, obtain the corresponding form of the age distribution function.
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It may be noted that:
0001
p
1
e
0002x2
0010 2
dx D
0
Solution (a) If the age distribution function be ft , then the surface in the age group t to t C dt is: ft dt. Then, the surface of age t C dt minus, the surface of age t C dt is the surface destroyed in the dt, or: ft 0002 ft C dt D sft dt 0002f0 t C dt dt D sft dt
or:
f0 t C sft D 0
As dt ! 0, 0018
Using the integrating factor e
sdt
i
D est then: est ft D K (const)
and:
ft D Ke0002st 0006 0002st 00071 0001 1 e K the total surface D 1 D K e0002st dt D K D 0002s 0 s 0
ii
ft D se0002st
and hence:
The mass transfer rate into fraction of surface of age t to t C dt (per unit total area of surface) is: D D 0002st CA se dt D CA e0002st t00021/2 dt 0010t 0010 The mass transfer rate per unit area, NA D Putting st D ˇ2 : and:
Thus:
D CA 0010
0001
1
t00021/2 e0002st dt 0
s dt D 2ˇ dˇ 0001 1 1/2 D s 2 NA D CA e0002ˇ 2ˇ dˇ 0010 ˇ 0 0001 1 sD 2 CA e0002ˇ dˇ D2 0010 0 p sD 0010 p CA D DsCA D2 0010 2 p NA D Ds the mass transfer coefficient D CA
iii
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MASS TRANSFER
(b) For an age range of surface from 0 to 2/s, application of equation (ii) gives: 0006 0002st 00072/s 0001 2/s e K 1DK e0002st dt D K D 1 0002 e00022
0002s 0 s 0 s or: KD 1 0002 e00022 s and: ft D e0002st 1 0002 e00022 Thus NA from equation (ii) is multiplied by factor 1/1 0002 e00022 or: 1 p DsCA D 1.16 times the value in equation 3. NA D 1 0002 e00022 The mass transfer rate per unit total area of surface is: 0017 0001 2/s 0016 0001 2/s D D 0002st CA 1.16se dt D 1.16sCA t00021/2 e0002st dt NA D 0010t 0010 0 0 st D ˇ2 p s 00021/2 t D ˇ
Putting: then: and:
s dt D 2ˇ dˇ 0001
p
2
Integrating: 0
p
s 0002ˇ2 2ˇ 2 e dˇ D p ˇ s s
0001
p
2
2
e0002ˇ dˇ 0
p p 2 0010 Dp erf 2 s 2 p 0010 D erf 2 s
p p 0010 CA erf 2 D 1.16 ð 0.954 DsCA s p D 1.107 DsCA p The mass transfer coefficient, hD D 1.107 Ds, an increase of 10.7% NA D
D 1.16s 0010
(c) For probability of surface renewal being linearly related to age, s D kt (where k is a constant) Equation (i) becomes: f0 t C ktft D 0. 0018 2 The integrating factor is: e kt dt D ekt /2 ekt and:
2 /2
ft D K ft D Ke0002kt
2 /2
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0001
1
0002kt2 /2
0001 2 1 0002kt2 /2 p dt D K e d k/2 t k 0
The total surface is: 1 D K e 0 k t D X: Putting p 20001 1 2 2 0010 0010 0002X2 e dX D K DK , 1DK k 0 k 2 2k 2k then: KD 0010 2k 0002kt2 /2 and the age distribution function is: e 0010
PROBLEM 10.33 Explain the basis of the penetration theory for mass transfer across a phase boundary. What are the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed? (Do not present a solution of the differential equation.) Obtain the age distribution function for the surface: (a) On the basis of the Danckwerts’ assumption that the probability of surface renewal is independent of its age. (b) On the assumption that the probability of surface renewal increases linearly with the age of the surface. Using the Danckwerts surface renewal model, estimate: (c) At what age of a surface element is the mass transfer rate equal to the mean value for the whole surface for a surface renewal rate (s) of 0.01 m2 /m2 s? (d) For what proportion of the total mass transfer is surface of an age exceeding 10 seconds responsible?
Solution (a) Danckwerts age distribution function Dividing the total unit surface into elements each of duration dt, then: dt
t − dt
dt
t
dt
t + dt
t + 2dt
If the fraction of surface in age band t to t C dt is ft dt, then: the fraction of surface in age band t 0002 dt to t will be ft 0002 dt dt.
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The surface not going from t 0002 dt/t to t/t C dt D ft 0002 dt dt 0002 ft dt D 0002f0 t 0002 dt dt dt This will be surface destroyed in time dt D destruction rate ð area of surface ð time interval
D s[ft 0002 dt dt]dt 0
Thus: 0002f t 0002 dt dt dt D s[ft 0002 dt dt]dt As dt ! 0 then: f0 t C sft D 0 Using the Danckwerts model: s D const. est f0 t C sest ft D 0 est ft D K ft D Ke0002st There is no upper age limit to surface 0006 0002st 00071 0001 1 e 0002st Thus total surface D 1 D K e dt D K D K/s or: K D s 0002s 0 0 ft D se0002st . (b) s D at where a is a constant. Thus:
eat
2 /2 0
f t C ateat e
2 /2
at2 /2
ft D 0 ft D K0 2/2
ft D K0 e0002at 0001 0002 0003 0001 1 2 1 0002at2/2 a 0002at2/2 0 0 For unit total surface: 1 D K e dt D K e d t 2 0 pa 0 2 0010 D K0 a 2 2a 2a 0002at2/2 0 and ft D e and: K D 0010 0010 (c) regarding surface renewal as random 0001 and:
1
kt00021/2 se0002st dtCA 0001 1 D CA ks t00021/2 e0002st dt.
For unit total surface, mass transfer (mol/area time) D
0
0
where CA is the concentration driving force in moles per unit volume Putting st D x 2 , then: s dt D 2x dx 0001 1 p 1 2 The mass transfer rate D ksCA x 00021 s e0002x 2x dx. s 0 0001 p 1 p p 0010 p 0002x2 D CA 2k s D CA k 0010s e dx D CA 2k s 2 0
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The mass transfer rate at time t is: CA kt00021/2 Thus the age of surface at which rate D average is given by or: p kt00021/2 D k 0010s tD
100 1 D D 31.8 s 0010s 0010
(d) Surface of age less than 10 seconds The mass transfer taking place into surface of age up to 10 s is given by the same expression as for the whole surface but with upper limit of 10 s instead of infinity 0001 k100002t p 2 or: CA 2k s e0002x dx 0
p when t D 10 s: x D st D 0.01 ð 10 D 0.316 The mass transfer into surface up to 10 s age is then: p 0001 0.316 p p p 0010 2 CA 2k s erf 0.316 D CA k 0010s ð 0.345 e0002x dx D CA 2k s 2 0 p
Thus a fraction: 0.345 is contributed by surface of age 0 0002 10 s and: a fraction: 0.655 by surface of age 10 s to infinity .
PROBLEM 10.34 At a particular location in a distillation column, where the temperature is 350 K and the pressure 500 m Hg, the mol fraction of the more volatile component in the vapour is 0.7 at the interface with the liquid and 0.5 in the bulk of the vapour. The molar latent heat of the more volatile component is 1.5 times that of the less volatile. Calculate the mass transfer rates kmol m00022 s00021 of the two components. The resistance to mass transfer in the vapour may be considered to lie in a stagnant film of thickness 0.5 mm at the interface. The diffusivity in the vapour mixture is 2 ð 1000025 m2 s00021 . Calculate the mol fractions and concentration gradients of the two components at the mid-point of the film. Assume that the ideal gas law is applicable and that the Universal Gas Constant R D 8314 J/kmol K.
Solution In this case: T D 350 K, P D 500 mm Hg D
and:
0002
0003 500 ð 101,300 D 0.666 ð 105 N/m2 760
D D 2 ð 1000025 m2 /s 0002 0003 0.666 ð 105 P CT D D D 0.0229 kmol/m3 RT 8314 ð 350
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MASS TRANSFER
If:
A D MVC
B D LVC, then:
9A D 1.59B N0A 9A D 0002N0B 9B N0B D 0002N0A ,
and:
9A D 00021.5N0A 9B
N0A D 0002DCT
0001
dxA N0A C N0B C CA dy CT
D 0002DCT
dxA C N0A 0002 1.5N0A xA dy
5ð1000024
dxA dy 0001 0.5
dy D 0002DCT 0
0.7
N0A ð 5 ð 1000024
(from equations 10.46a and 10.19)
D 0002DCT
N0A 1 C 0.5xA D 0002DCT N0A
dxA C uF CA dy
i
dxA 1 C 0.5xA
0019 001a0.5 D 0002DCT 2 ln1 C 0.5xA
0.7
1 1.25 ð N0A D 00022DCT ln 1.35 5 ð 1000024 D 1.41 ð 1000024 kmol/m2 s N0B D 0002 2.11 ð 1000024 kmol/m2 s
and:
At the mid-point: y D 2.5 ð 1000024 m
0019 001axA N0A ð 2.5 ð 1000024 D 00022 ð 1000025 ð 0.0229 ð 2 ln1 C 0.5xA
0.7
00024
00024
1.41 ð 10 ð 2.5 ð 10 1 C 0.5xA D0002 1.35 2 ð 1000025 ð 0.0229 ð 2 1.35 D 0.0384 ln 1 C 0.5xA ln
and:
xA D 0.598
The concentration gradient is given by equation (i) or: N0 1 C 0.5xA
dxA D A dy 0002DCT When xA is 0.598, then: dxA 1.41 ð 1000024 1 C 0.5 ð 0.598
D dy 00022 ð 1000025 ð 0.0229 D 400 m00021 D 0.4 mm00021
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dxA dCA D CT D 9.16 kmol/m4 . dy dy
and:
PROBLEM 10.35 For the diffusion of carbon dioxide at atmospheric pressure and a temperature of 293 K, at what time will the concentration of solute 1 mm below the surface reach 1 per cent of the value at the surface? At that time, what will the mass transfer rate (kmol m00022 s00021 be: (a) At the free surface? (b) At the depth of 1 mm? The diffusivity of carbon dioxide in water may be taken as 1.5 ð 1000029 m2 s00021 . In the literature, Henry’s law constant K for carbon dioxide at 293 K is given as 1.08 ð 106 where K D P/X, P being the partial pressure of carbon dioxide (mm Hg) and X the corresponding mol fraction in the water.
Solution ∂CA ∂ 2 CA DD 2 ∂t ∂y where CA is concentration of solvent undergoing mass transfer. The boundary conditions are: y D 0 (interface) yD1 tD0
CA D CAs (solution value) CA D 0 CA D 0
t>0 0< />< />
Taking Laplace transforms then: 0003 0001 10002 ∂CA ∂CA 0002pt D e dt ∂t ∂t 0 0001 1 0002pt 1 0002pept CA dt D 0 C pCNA D [CA e ]0 0002 0
∂ CA ∂ CNA D 2 ∂y ∂y 2 2
Thus:
2
pCNA D D
∂2 CNA ∂y 2
∂2 CNA p 0002 CNA D 0 ∂y 2 D p
p CNA D Ae For t > 0; when y D 1 when y D 0
p/Dy
CA D 0, CA D CAs
C Be0002
p/Dy
N A D 0 ∴ A D 0. C 0006 0002pt 00071 0001 1 e CAs 0002pt N . CAs e dt D CAs D CAs D 0002p 0 p 0
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MASS TRANSFER
Thus: and: Inverting:
CAs D B.1 p CAs 0002pp/Dy. CNA D e p CA y D erfc p CAs 2 Dt
See Table in Volume 1, Appendix
Differentiating with respect to y: 0011 0002 0003 0001 1 1 ∂CA 2 y ∂ 0002y 2 /4Dt p p D e d CAs ∂y ∂y 0010 y/2pDt 2 Dt 1 2 1 2 2 D p Ð p 0002e0002y /4Dt D 0002 p e0002y /4Dt 0010 2 Dt 0010Dt 0011 0012 00021 0002y 2 /4Dt The mass transfer rate at t, y, N D 0002D p CAs e 0010Dt 0002 0003 D 0002y 2 /4Dt ∂CA e when: t > 0, then: 0002D D CAs ∂y yDy 0010Dt D At t > 0 and y D 0, then: N0 D CAs 0010t
i
ii
For a concentrated 1% of surface value at y D 1 mm, CA /CAs D 0.01 and: 0011 0012 1000023 p 0.01 D erfc 2 1.5 ð 1000029 t Writing erf x D 1 0002 erfc x, then: 0.99 D erf12.91t00021/2
1.82 D 12.91t00021/2
From tables
t D 50.3 s The mass transfer rate at the interface at t D 50.3 s is given by equation (ii) as: 0015 D 1.5 ð 1000029 D CAs N0 D CAs 0010t 0010 ð 50.3 D 3.08 ð 1000026 CAs kmol/m2 s The mass transfer rate at y D 1 mm. and t D 50.3 s is given by equation (i) as: N D N0 e0002y
2 /4Dt
00026 /4ð1.5ð1000029 ð50.3
D 3.08 ð 1000026 e000210
CAs
D 1.121 ð 1000027 CAs where CAs is in kmol/m3 .
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Henry’s law constant, K D 1.08 ð 106 , where: K D P/X. Also: P D 760 mm Hg X D Mol fraction in liquid X D 760/1.08 ð 106 D 7.037 ð 1000024 kmol CO2 kmol solution (³ per kmol water)
and:
Water molar density D 1000/18 kmol/m3 0002 0003 00024 1000 CAs D 7.037 ð 10 D 0.0391 kmol/m3 18 NA0 D 1.204 ð 1000027 kmol/m2 s.
When y D 0, then: When y D 1 mm, then:
NA D 4.38 ð 1000029 kmol/m2 s.
PROBLEM 10.36 Experiments are carried out at atmospheric pressure on the absorption into water of ammonia from a mixture of hydrogen and nitrogen, both of which may be taken as insoluble in the water. For a constant mole fraction of 0.05 of ammonia, it is found that the absorption rate is 25 per cent higher when the molar ratio of hydrogen to nitrogen is changed from 1 : 1 to 4 : 1. Is this result consistent with the assumption of a steady-state gas-film controlled process and, if not, what suggestions have you to make to account for the discrepancy? Neglect the partial pressure attributable to ammonia in the bulk solution. Diffusivity of ammonia in hydrogen D 52 ð 1000026 m2 /s Diffusivity of ammonia in nitrogen D 23 ð 1000026 m2 /s
Solution Using Maxwell’s Law for mass transfer of A in B then: 0002dPA D FCA CB uA 0002 uB
dy or:
0002RT Ð
(equation 10.77)
dCA D FCA CB N0A /CA 0002 N0B /CB
dy
For an absorption process N0B D 0 and: 0002
FCB dCA D N0A dy RT
i
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MASS TRANSFER
265
1 CB dCA D N0A Ð (equation 10.30) dy D CT RT RT or: F D ii
Thus: DD FCT DCT For the three-component system, equation 1 may be written: N0 dCA D A FAB CB C FAC CC
0002 dy RT RT RT and FAC D . From (2): FAB D DAB CT DAC CT 0002 0003 CB 1 dCA CC 0 Substituting: 0002 D NA C iii
dy DAB DAC CT Stefan’s law for 3-components system may be written as: dCA CT where D0 is the effective diffusivity N0A D 0002D0 Ð CT 0002 CA dy From Stefan’s Law: 0002
1 CT 0002 CA dCA D N0A 0 Ð dy D CT Comparing equations (iii) and (iv): 1 1 CB 1 CC D Ð C Ð D0 DAB CT 0002 CA DAC CT 0002 CA or:
0002
iv
xB0 x0 C C DAB DAC where xB0 and xC0 are mole fractions of B, C in the “stationary” gas. Taking A as NH3 , B as H2 and C as N2 , then: Case 1 xB0 D 0.5 xC0 D 0.5 0002 0003 0002 0003 1 0.5 0.5 D C D 0.03135 ð 106 s/m2 D0 52 ð 1000026 23 ð 1000026 D
D0 D 31.9 ð 1000026 m2 /s
and:
xBM D Mass transfer rate,
N0A D D
Case 2
and:
0.05 1 0002 0.95 xT D D 0.975, and D 1.026 00021 ln[1/0.95 ] ln0.95
xBm xT xT 0 D0 1 CA D CA D L xBm L xBm xT 1 CA Ð 31.9 L xBm
xB0 D 0.8 xC0 D 0.2 0002 0003 0002 0003 1 0.8 0.2 D C D 0.0241 ð 106 s/m2 D0 52 ð 1000026 23 ð 1000026 D0 D 41.5 ð 1000026 m2 /s
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xT /xBm , L and CA are as in case 1. ∴
N0A D
xT xT D0 1 CA D CA Ð 41.5 L xBm L xBm
41.5 N0A 2 D 1.30 or 1.3 times greater in the second case. D N0A 1 31.9 The observed ratio of two rates is only 1.25. This may be explained by: 1. Steady-state film conditions do not exist and there is some periodic partial disruption of the film. Penetration model ! N0A / D0.5 c/f. / D for film model. In this problem observed result would be accounted for by: N0A / D0m . where 1.3 m D 1.25 or m D 0.85. 2. The assumption of a gas-film controlled process may not be valid. If there is a liquid-film resistance, the effect of increasing the gas-film diffusivity will be less than predicted for a gas-film controlled process. 3. The value of the film thickness L is not the same because of different hydrodynamic conditions (second mixture having a lower viscosity). In this case, the film thickness would be expected to be reduced giving rise to the reverse effect so this is not a plausible explanation. 4. Experimental inaccuracies!
PROBLEM 10.37 Using a steady-state film model, obtain an expression for the mass transfer rate across a laminar film of thickness L in the vapour phase for the more volatile component in a binary distillation process: (a) where the molar latent heats of two components are equal. (b) where the molar latent heat of the less volatile component (LVC) is f times that of the more volatile component (MVC). For the case where the ratio of the molar latent heats f is 1.5. what is the ratio of the mass transfer rate in case (b) to that in case (a) when the mole fraction of the MVC falls from 0.75 to 0.65 across the laminar film?
Solution Case (a): With equal molar latent heats, equimolecular counter diffusion takes place and there is no bulk flow. Writing Fick’s Law for the MVC gives: NA D 0002D
dCA dy
(equation 10.4)
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MASS TRANSFER
Integrating for the steady state, then: NA D D
xA 0002 xA2
CA 1 0002 CA 2 D DCT 1 y2 0002 y1 L
i
Case (b): The net heat effect at the interface must be zero and hence: N0A 9A C N0B 9B D 0 N0B D 0002
and:
9A 0 1 N D 0002 N0A 9B A f
In this case: Total flux of A D Diffusional flux C Bulk-flow 0002 0003 dxA dCA 1 C uF CA D 0002DCT C xA N0A 0002 N0A or: N0A D 0002D dy dy f 0006 0002 00030007 dxA 1 Thus: N0A 1 0002 xA 1 0002 D 0002DCT f dy 0001 L 0001 CA 2 1 0002 0003 dxA dy D 0002DCT N0A 1 CA1 0 1 0002 xA 1 0002 f 0002 000300070007xA 0006 0006 2 DCT 1 0003 ln 1 0002 xA 1 0002 ∴ N0A L D 0002 0002 1 f xA1 10002 f 0002 0003 1 1 0002 xA2 1 0002 DCT f 0003 ln 0002 0003 D0002 ii
1 1 10002 L 1 0002 xA 1 1 0002 f f From equations (i) and (ii):
RD
N0A case b D NA case a
1
1 1 xA1 0002 xA2 10002 f
0002 0003 1 1 0002 xA 2 1 0002 f 0002 0003 ln 1 1 0002 xA 1 1 0002 f
Substituting f D 1.5 then: 10002
1 1 D f 3
xA1 D 0.75 RD For B:
1 1 3
Ð
xA2 D 0.65
1 0002 0.65 ð 1 ln 0.75 0002 0.65 1 0002 0.75 ð
1 3 1 3
D 1.303
R D 1.303/1.5 D 0.869.
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PROBLEM 10.38 Based on the assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration CA of a solute A at a depth y below the interface at a time t after the formation of the interface is given by: 0006 0007 CA y D erfc p CAi 2 Dt
where CAi is the interface concentration, assumed constant and D is the molecular diffusivity of the solute in the solvent. The solvent initially contains no dissolved solute. Obtain an expression for the molar rate of transfer of A per unit area at time t and depth y, and at the free surface (y D 0). In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid. Occasionally, two droplets coalesce forming a single drop of twice the volume. What is the ratio of the mass transfer rate (kmol/s) at a coalesced drop to that at a single droplet when each has fallen the same distance, that is to the bottom of the column? The fluid resistance force acting on the droplet should be taken as that given by Stokes’ law, that is 30010001cdu where 001c is the viscosity of the continuous phase, d the drop diameter and u its velocity relative to the continuous phase. It may be noted that: 0001 1 2 2 erfcx D p ex dx. 0010 x
Solution CA 2 y D erfc p D p CAi 0010 2 Dt
0001
1 p y/2 Dt
e
0002y 2 /4Dt
0002
d
y p 2 Dt
0003
Differentiating with respect to y at constant t gives: 0011 0012 0001 1 1 ∂CA 1 2 ∂ 0002y 2 /4Dt p Dp e dt CAi ∂y 0010 ∂y 2 Dt y/2pDt
Dp
1 0010Dt
0002e0002y
2 /4Dt
1 2 0002e0002y /4Dt D CAi 0010Dt D D bt00021/2 At the interface, when y D 0: NA D CAi 0010t
∂CA D CAi 0002D
Thus: NA t D 0002D ∂y
D 0002y 2 /4Dt e 0010t
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MASS TRANSFER
From Stokes’ Law, the terminal falling velocity of the droplet is given by: 0010 30010001cdu0 D d3 +s 0002 + g 6 or:
u0 D
d2 g +s 0002 + D Kd2 18001c
Thus, the time taken for the droplet to travel the depth H of the rising liquid is: H 1 Kd2 2
Since the liquid is rising at a velocity of 12 Kd2 and the relative velocity is Kd2 0002 12 Kd2 D 12 Kd2 , the mass transfer rate (kmol/m2 s) to droplet at end of travel is: K b d 2H The mass transfer rate to the drop is: K K 3 K 1 2 d0010d D 0010b d D 0010b 0002 p d3 b 2H 2H H 2 For coalesced drops, the new diameter is: 21/3 d The terminal falling velocity is: K22/3 d2 Its velocity relative to the liquid is: K22/3 d2 0002 12 Kd2 D Kd2 22/3 0002 12
Thus:
Time of fall of drop D
H Kd2 22/3
0002 12
K 001b 2/3 1 d 2 0002 2 kmol/m2 s H K 001b 2/3 1 d 2 0002 2 001021/3 d 2 kmol/s Mass transfer rate to drop D bd H 001b K D 0010b 22/3 0002 12 22/3 d3 kmol/s H
Mass transfer rate at end of travel D bd
The ratio of the mass transfer rate for the coalesced drop to the mass transfer rate for the single droplet is then: 001b 22/3 0002 12 22/3 p D D 2.34 1/ 2
PROBLEM 10.39 In a drop extractor, a dense organic solvent is introduced in the form of spherical droplets of diameter d and extracts a solute from an aqueous stream which flows upwards at
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a velocity equal to half the terminal falling velocity u0 of the droplets. On increasing the flowrate of the aqueous stream by 50 per cent, whilst maintaining the solvent rate constant, it is found that the average concentration of solute in the outlet stream of organic phase is decreased by 10 per cent. By how much would the effective droplet size have had to change to account for this reduction in concentration? Assume that the penetration theory is applicable with the mass transfer coefficient inversely proportional to the square root of the contact time between the phases and that the continuous phase resistance is small compared with that within the droplets. The drag force F acting on the falling droplets may be calculated from Stokes’ Law, F D 30010001cduo , where 001c is the viscosity of the aqueous phase. Clearly state any assumptions made in your calculation.
Solution For droplets in the Stokes’ law region, the terminal falling velocity is given by: 0010/6 d3 +s 0002 + g D 30010001cdu0 or:
u0 D
d2 g +s 0002 + D kd2 18001c
The mass transfer rate to the droplet is Kte00021/2 moles per unit area per unit time / Kte1/2 moles per unit area in time te / Kte1/2 d2 moles per drop during time of rise The concentration of solute in drop: / Kte1/2 d2 /d3 / Kte1/2 d00021 . Initial case:
u0 D kd2 Rising velocity of liquid D
kd2 2
Velocity of liquid relative to container D kd2 0002
kd2 kd2 D 2 2
H D te . kd2 /2 2H 00021 2H 00022 Thus the concentration in the drop / K d D C1 d / K kd2 k Time of exposure in height H D
Second case:
New drop diameter D d0 Rising velocity of liquid D 1.5 ð
3 kd2 D kd2 2 4
Rising velocity of drop relative to liquid D kd02 Velocity relative to container D kd02 0002 34 kd2 Time of exposure D
kd02
H 0002 34 d2
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271
Concentration of solute in drop D Kte1/2 d0 00021
1/2 H C2 D K d0 00021 kd0 2 0002 34 d2
∴
Given that: C2 /C1 D 0.9, then:
K
1/2 H d000021
kd02 0002 34 d2
D 0.9 2H 00022 d K k
d000022 D 1.62d00024 d02 0002 34 d2
Squaring gives: Writing R D
d0 , then: d
1 D 1.62 R4 0002 34 R2 R4 0002 34 R2 0002 0.6173 D 0 R D 1.11 or 11.1 per cent increase
PROBLEM 10.40 According to the penetration theory for mass transfer across an interface, the ratio of the concentration CA at a depth y and time t to the surface concentration CAs at the liquid is initially free of solute, is given by y CA D erfc p CAs 2 Dt where D is the diffusivity. Obtain a relation for the instantaneous rate of mass transfer at time t both at the surface y D 0 and at a depth y. What proportion of the total solute transferred into the liquid in the first 90 s of exposure will be retained in a 1 mm layer of liquid at the surface, and what proportion will be retained in the next 0.5 mm? The diffusivity is 2 ð 1000029 m2 /s.
Solution For a rectangular particle:
Thus:
Thiele Modulus D 0016 D 9L 0015 k 5 ð 1000024 D D 500 m00021 9D D 2 ð 1000029
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LD
8 2
D 4 mm or 0.004 m
f D 500 ð 0.004 D 2 tanh f tanh 2 0.96 Thus the effectiveness factor, ; D D D D 0.48 f 2 2 For a spherical particle:
(equation 10.202)
Thiele Modulus D f D 9R RD
4 0010r 3 Volume ro D 3 o2 D Surface 40010ro 3
5 ð 1000023 m 3 5 ð 1000023 500 ð D 0.833 3 fD 9
R
RD
Thus the effectiveness factor,
1 1 coth 3f 0002 2 f 3f 1 1 D coth 2.5 0002 0.833 3 ð 0.8332 D 1.217 0002 0.480 D 0.736
;D
(equation 10.215)
PROBLEM 10.41 Obtain an expression for the effective diffusivity of component A in a gaseous mixture of A, B and C in terms of the binary diffusion coefficients DAB for A in B, and DAC for A in C. The gas-phase mass transfer coefficient for the absorption of ammonia into water from a mixture of composition NH3 20%, N2 73%. H2 7% is found experimentally to be 0.030 m/s. What would you expect the transfer coefficient to be for a mixture of composition NH3 5%, N2 60%, H2 35%? All compositions are given on a molar basis. The total pressure and temperature are the same in both cases. The transfer coefficients are based on a steady-state film model and the effective film thickness may be assumed constant. Neglect the solubility of N2 and H2 in water. Diffusivity of NH3 in N2 D 23 ð 1000026 m2 /s. Diffusivity of NH3 in H2 D 52 ð 1000026 m2 /s.
Solution For case 1: 1 The effective diffusivity D is given by: 0 D D 0
0002
73/80 23 ð 1000026
0003
0002
C
7/80 52 ð 1000026
0003
(from equation 10.90)
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MASS TRANSFER
D0 D 24.2 ð 1000026 m2 /s 0003 1 D0 The mass transfer coefficient is: L log mean 1 and 0.8 0016 001700021 0.2 24.2 ð 1000026 1 D D ð 27 ð 1000026 D 0.030 m/s 1 L L ln 0.8 or:
and hence:
0002
L D 0.90 ð 1000023 m
Case 2: 1 The effective diffusivity D is given by: 0 D D 0
0002
60/95 23 ð 1000026
0003
0002
C
35/95 52 ð 1000026
0003
D0 D 28.9 ð 1000026 m2 /s 0002 0003 D0 1 The mean transfer coefficient is: Ð L log mean 1 and 0.95 001700021 0016 0002 0003 28.9 ð 1000026 0.05 D D 0.033 m/s. 1 0.9 ð 1000023 ln 0.95
or:
PROBLEM 10.42 State the assumptions made in the penetration theory for the absorption of a pure gas into a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the far boundary of the liquid. Starting with Fick’s second law of diffusion, obtain an expression for (i) the concentration, and (ii) the mass transfer rate at a time t and a depth y below the surface. After 50 s, at what depth y will the concentration have reached one tenth the value at the surface? What is the mass transfer rate (i) at the surface, and (ii) at the depth y, if the surface concentration has a constant value of 0.1 kmol/m3 ?
Solution CA y D erfc p CAS 2 Dt
(equation 10.108)
Differentiating with respect to y: 0011 0002 0003 0001 1 1 ∂CA 2 y ∂ 0002y 2 /4Dt p p D e d CAS ∂y dy 0010 y/2pDt 2 Dt D 0002p Thus:
NA y,t
1
e0002y
2 /4Dt
0010Dt 0016 0017 0011 0012 D 0002y 2 /4Dt 1 0002y 2 /4Dt D 0002D 0002 p CAS D CAS e e 0010t 0010Dt
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
When t D 50 s and CA /CAS D 0.1, then:
or:
y 0.1 D erfc p 00029 2 10 ð 50 y 0.9 D erf p 2 Dt
From Table 13 in the Appendix of Volume 1, the quantity whose error fraction D 0.9 is: y p D 1.16 2 Dt D CAS At the surface: NA yD0,t D 0010t 0015 1000029 ð 0.1 D 0010 ð 50 D 0.252 ð 1000026 kmol/m2 s. At a depth y:
2
NA D 0.252 ð 1000026 ð e00021.16 D 0.0656 ð 1000026 kmol/m2 s.
PROBLEM 10.43 In a drop extractor, liquid droplets of approximately uniform size and spherical shape are formed at a series of nozzles and rise countercurrently through the continuous phase which is flowing downwards at a velocity equal to one half of the terminal rising velocity of the droplets. The flowrates of both phases are then increased by 25 per cent. Because of the greater shear rate at the nozzles, the mean diameter of the droplets is, however, only 90 per cent of the original value. By what factor will the overall mass transfer rate change? It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may be neglected. From the penetration theory, the concentration CA at a depth y below the surface at time t is given by: 0006 0007 0001 1 CA y 2 2 D erfc p where erfc X D p e0002x dx CAS 0010 X 2 Dt
where CAS is the surface concentration for the drops (assumed constant) and D is the diffusivity in the dispersed (droplet) phase. The droplets may be assumed to rise at their terminal velocities and the drag force F on the droplet may be calculated from Stokes’ Law, F D 30010001c du.
Solution Case 1: For a volumetric flowrate Q1 , the numbers of drops per unit time is: Q1 1 6 0010d3
D
6Q1 0010d31
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MASS TRANSFER
275
The rising velocity is given by a force balance: 30010001cd1 u D 16 0010d31 +1 0002 +2 g or:
u1 D
d21 g +1 0002 +2 D Kd21 relative to continuous phase 18001c
The downward liquid velocity is 12 Kd21 The upward droplet velocity relative to container is: Kd21 0002 12 Kd21 D 12 Kd21 and the time of contact during rise through height H is: tc D
H 1 2 2 Kd1
.
(i)
The mass transfer rate is: 0002D∂CA /∂y . 0011 0012 0011 0002 00030012 0001 1 2 y y ∂ ∂ 1 ∂CA 0002y 2 /4Dt p p erfc p D e d D Thus: CAS ∂y ∂y ∂y 0010 y/2pDt 2 Dt 2 Dt 0001 1 1 ∂ 2 2 Ð p 0002p D e0002y /4Dt dt. ∂y 2 Dt 0010 y 0002 0003 CAS 0002y 2 /4Dt ∂CA CAS ∂CA D 0002p D 0002p or: e ∂y ∂y yD0 0010Dt 0010Dt The mass transfer rate at the surface is: (moles/area ð time). 0002 0003 D CAS 0002D 0002 p CAS D 0010t 0010Dt The mass transfer in time te1 is: 0001 te1 D D 1/2 1/2 00021/2 CAS te1 CAS D Kte1 t dt D 2 0010 0010 0 Substituting from equation (i): p 2H Mass transfer in moles per unit area of drop / p Kd1 The mass transfer per drop is proportional to: p H 00021 2 p H d1 d1 / 2 d1 2 K K The mass transfer per unit time D Mass transfer per drop ð drops/time p H H Q1 6Q1 d1 ð or: proportional to: 2 / 8.48 3 K K 0010d21 0010d1
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Case 2: diameter D 0.9d D d2 , Q2 D 1.25Q1 The number of drops per unit time is: 7.5Q1 10.29Q1 6Q2 D D 3 3 00100.9d1
0010d2 0010d31 Rising velocity D Kd21 D K0.9d1 2 D 0.81Kd21 Downward liquid velocity D
5 4
ð 12 Kd21 D 0.625 Kd21 relative to continuous phase
Rising velocity relative to container D 0.81 Kd21 0002 0.625 Kd21 D 0.185Kd21 Contact time, te2 D
H 0.185Kd21
Mass transfer per drop Mass transfer per unit time
H 1 / 2.325 0.185K d1
H 00021 d K 1 H 00021 H 2 d1 ð 0.9d1 / 1.883 d1 / 2.325 K K H 10.29Q1 d1 ð / 1.883 K 0010d31 H Q1 / 19.37 K 0010d21
Mass transfer in time te2 per unit area /
kte1/2 2
/
Thus, the factor by which mass transfer rate is increased is: 19.37/8.48 D 2.28
PROBLEM 10.44 According to Maxwell’s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble, and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. Carbon dioxide is absorbed in alkaline water from a mixture consisting of 30% CO2 and 70% N2 , and the mass transfer rate is 0.1 kmol/s. The concentration of CO2 in the gas in contact with the water is effectively zero. The gas is then mixed with an equal molar quantity of a second gas stream of molar composition 20% CO2 , 50%, N2 and 30% H2 . What will be the new mass transfer rate, if the surface area, temperature and pressure remain unchanged? It may be assumed that a steady-state film model is applicable and that the film thickness is unchanged. Diffusivity of CO2 in N2 D 16 ð 1000026 m2 /s. Diffusivity of CO2 in H2 D 35 ð 1000026 m2 /s.
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MASS TRANSFER
Solution For a binary system, Maxwell’s Law gives: 0002
dCA D FCA CB uA 0002 uB
dy
For an ideal gas mixture: 0002RT
dCA D FCA CB dy
0002
(equation 10.77)
N0A N0 0002 B CA CB
0003
For a gas absorption process: N0B D 0 if B is insoluble or:
0002
dCA FCB 0 D N dy RT A
i
from Stefan’s law (equation 10.30): N0A D 0002D or:
0002
CT dCA CB dy
1 CB dCA D N0A dy D CT D N0A
1 CT 0002 CA D CT
ii
iii
Comparing equations (i) and (ii): F 1 D RT DCT or:
FD
RT RT or D D DCT FCT
Applying Maxwell’s law to a multicomponent system gives: 0002
dCA D FAB CA CB uA 0002 uB C FAC CA CC uA 0002 uC C Ð Ð Ð dy
For B, C . . . insoluble N0B , N0C . . . D 0 Writing: then: ∴
FAB D
RT RT , and FAC D DAB CT DAC CT
RT dCA RT D N0A CB C N0 CC C Ð Ð Ð . dy DAC CT DAC CT A 0002 0003 N0A CB dCA CC D 0002 C C ÐÐÐ dy CT DAB DAC
0002RT
From equation (iii), using an effective diffusivity D0 for a multicomponent system gives: 0002
1 CT 0002 CA dCA D N0A 0 dy D CT
iv
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Comparing equations (iii) and (iv), then: 0011 0012 1 CB 1 CC D C C ... D0 CT 0002 CA DAB DAC D where xB0 , xC0 Ð Ð Ð D
xB0 x0 C C C ... DAB DAC CB CC > ... CT 0002 CA CT 0002 CA
D mole fraction of B, C . . . in mixture of B, C For absorption of CO2 from mixture with N2 , the concentration driving force is: CA D CT 0.3 0002 0 D 0.3CT where CT is the total molar concentration The mass transfer rate for an area A D AN0A D CA where: or:
CBm is the log mean of 1 ð CT and 0.7CT CBm D
CT 0002 0.7CT CT D 0.841CT and: D 1.189 CT CBm ln 0.7CT
Mass transfer rate, AN0A D 0.3CT or:
D CT Ð ÐA L CBm
16 ð 1000026 CT A 1.189 A D 5.71 ð 1000026 D 0.10 kmol/s L L
CT A D 0.0175 ð 106 kmol/m2 . L
For absorption of CO2 from mixed stream, stream composition must be calculated. 100 moles stream 1 ! 30 moles CO2 70 moles N2 100 moles stream 2 ! 20 moles CO2 50 moles N2 30 moles H2 200 moles mixture ! 50 moles CO2 120 moles N2 30 moles H2 100 moles mixture ! 25 moles CO2 60 moles N2 15 moles H2 xN0 2 D
60 15 D 0.8 xH0 2 D D 0.2 75 75
Diffusivity of CO2 in mixture is given by: 1 0.8 0.2 D C 0 00026 D 16 ð 10 35 ð 1000026 1000026 1 1 C D 0.0557 s/m2 D 0 D 20 175 D0 D 17.9 ð 1000026 m2 /s
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MASS TRANSFER
Thus:
CA D CT 0.25 0002 0 D 0.25CT CT 1 0002 0.75 CT D 0.869CT D 1.150 1 C ln 0.75 Bm
CBm D and the mass transfer rate: ANA0 D 0.25CT
17.9 ð 1000026 ACT 1.150A D 5.15 ð 1000026 kmol/s D 0.090 kmol/s L L
PROBLEM 10.45 What is the penetration theory for mass transfer across a phase boundary? Give details of the underlying assumptions. From the penetration theory, the mass transfer rate per unit area NA is given in terms of the concentration difference CA between the interface and the bulk fluid, the molecular diffusivity D and the age t of the surface element by: D CA kmol/m2 s (in SI units) NA D 0010t What is the mean rate of transfer if all elements of the surface are exposed for the same time te before being remixed with the bulk? Danckwerts assumed a random surface renewal process in which the probability of surface renewal is independent of its age. If s is the fraction of the total surface renewed per unit time, obtain the age distribution function for the surface and show that the mean mass transfer rate NA over the whole surface is: p NA D DsCA kmol/m2 s, in SI units
In a particular application, it is found that the older surface is renewed more rapidly than the recently formed surface, and that after a time s00021 , the surface renewal rate doubles, that is it increases from s to 2s. Obtain the new age distribution function.
Solution Assuming the age spread of the surface ranges for t D 0, to t D 1, consider the mass transfer per unit area in each age group is t to t C dt and so on. Then the mass transfer to surface in age group t to t C dt is: D CA t00021/2 dt D 0010 Thus the total mass transfer per unit area is:
D CA 0010
0001
te
t 0
00021/2
dt D
0004 0005te D t1/2 D CA 1 CA te1/2 D2 0010 0010 2 0
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
The average mass transfer rate is: 0015
1 D D 1/2 CA te 2 D2 CA te 0010 0010te In the steady state if ft is the age distraction function of the surface, then: surface in age group t to t C dt D ft dt and:
surface in age group t 0002 dt to t D ft 0002 dt dt
Surface of age t 0002 dt to t which is destroyed is that not entering the next age group ft 0002 dt dt 0002 ft dt D s fft 0002 dt dtg dt 0
As dt ! 0, then: 0002f t dt dt D sft dt dt est f0 t C est sft D 0 est ft D K
Integrating gives:
ft D Ke0002st 0006 0002st 00071 1 e K 0002st As the total surface D 1, K e dt D K D 0002s 0 s 0 then:
and:
i
0001
∴KDs
ft D se0002st
ii
The mass transfer rate into the fraction of surface in the age group t 0002 t dt is: D CA se0002st dt 0010t The mass transfer rate into the surface over the age span t D 0 to t D 1 is: 0001 1 D D 00021/2 0002st CA s CA t e dt D D 0010 0010 0 p s Putting D st D ˇ2 , then: t00021/2 D ˇ and:
s dt D 2ˇ dˇ. p 0001 1p 0001 1 s 0002ˇ2 2ˇ dˇ 0010 0010 2 2 0002ˇ2 e Dp D e dˇ D p Ð ID ˇ s s 0 s 2 s 0
Thus the mass transfer rate per unit area for the surface as a whole is: D 0010 p CA s D DsCA 0010 s With the new age distribution function: 1 surface renewal rate/area D s 0< />< ,=' s=' 1=' /><=' t=' 1,=' surface=' renewal=' rate=' area=' d=' 2s='>='>
ft D Ke0002st ft D K0 e00022st
from equation (ii)
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MASS TRANSFER
00071/s e0002st K D 1 0002 e00021
0002s s 0 0 0006 00022st 00071 0001 1 1 e K0 00022 Fraction of surface of age to 1 D K0 e e00022st dt D K0 D s 00022s 1/s 2s 1/s
Fraction of surface of age 0 to
1 DK s
0001
0006
1/s
e0002st dt D K
K K0 1 0002 e00021 C e00022 s 2s At t D 1/s , both age distribution functions must apply, and: The total surface is unity or:
Ke00021 D K0 e00022 or: K0 D K Ð e K Ke 00022 1 0002 e00021 C e s 2s 000e K 000e K D 1 0002 e00021 C 12 e00021 D 1 0002 12 e00021 s s
Thus, for the total surface: 1 D
K D s1 0002 12 e00021 00021
Thus: Thus 0 < t <
1 s
1 < /><1>1>
K0 D se1 0002 12 e00021 00021
000e00021 0002st ft D s 1 0002 12 e00021 e
000e00021 00022st
000e00021 100022st e D s 1 0002 12 e00021 e ft D se 1 0002 12 e00021
PROBLEM 10.46 Derive the partial differential equation for unsteady-state unidirectional diffusion accompanied by an nth-order chemical reaction (rate constant k ): ∂ 2 CA ∂CA D D 2 0002 k CnA ∂t ∂y where CA is the molar concentration of reactant at position y at time t. Explain why, when applying the equation to reaction in a porous catalyst particle, it is necessary to replace the molecular diffusivity D by an effective diffusivity De . Solve the above equation for a first-order reaction under steady-state conditions, and obtain an expression for the mass transfer rate per unit area at the surface of a catalyst particle which is in the form of a thin platelet of thickness 2L. Explain what is meant by the effectiveness factor ; for a catalyst particle and show that it is equal p to 1/f tanh f for the platelet referred to previously where f is the Thiele modulus L k /De . For the case where there is a mass transfer resistance in the fluid external to the particle (mass transfer coefficient hD ), express the mass transfer rate in terms of the bulk concentration CAo rather than the concentration CAS at the surface of the particle. For a bed of catalyst particles in the form of flat platelets it is found that the mass transfer rate is increased by a factor of 1.2 if the velocity of the external fluid is doubled.
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
The mass transfer coefficient hD is proportional to the velocity raised to the power of 0.6. What is the value of hD at the original velocity? k D 1.6 ð 1000023 s00021 ,
De D 1000028 m2 /s
catalyst pellet thickness 2L D 10 mm.
Solution (i) The partial differential equation for unsteady-state diffusion accompanied by chemical reaction is derived in Volume 1 as equation 10.170 (ii) The molecular diffusivity D must be replaced by an effective diffusivity De because of the complex internal structure of the catalyst particle which consists of a multiplicity of interconnected pores, and the molecules must take a tortuous path. The effective distance the molecules must travel is consequently increases. Furthermore, because the pores are very small, their dimensions may be less than the mean free path of the molecules and Knudsen diffusion effects may arise (iii) Equation 10.170 is solved in Volume 1 to give equation 10.199 for a catalyst particle in the form of a flat platelet (iv) The effectiveness factor is the ratio of the actual rate of reaction to that which would be achieved in the absence of a mass-transfer resistance. For a platelet, it is evaluated in terms of the Thiele modulus as equation 10.202 (v) For the case, where there is an external mass transfer resistance, the reaction rate is expressed in terms of the bulk concentration as equation 10.222: Rv D
k CAo 1/; C k L/hD
For k D 1.6 ð 1000023 s00021 , De D 1000028 m2 /s, L D 5 ð 1000023 m: 0015 0015 k 1.6 ð 1000023 0016DL D 5 ð 1000023 D2 De 1000023 ;D ∴
0.96 1 1 D 0.48 tanh f D tanh 2 D f 2 2 1 1 1 D D D 0.260 ð 106 . 00023 00023 ;k L 0.48 ð 1.6 ð 10 ð 5 ð 10 3.84 ð 1000026
If the original value of mass transfer coefficient is hD , the new value at twice original velocity D hD 2 0.6 D 1.516 hD . Given that the overall rate is increased by a factor of 1.2: & 1 1 D 1.2 1 1 6 6 0.260 ð 10 C 0.260 ð 10 C 1.516hD hD 0002 0003 1 6 0.260 ð 10 C hD 0002 0003 D 1.2 1 0.260 ð 106 C 0.66 hD
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MASS TRANSFER
0.260 ð 106 C
1 1 D 0.312 ð 106 C 0.792 hD hD 1 D 0.25 ð 106 and: hD D 4.0 ð 1000026 m/s . hD
PROBLEM 10.47 Explain the basic concepts underlying the two-film theory for mass transfer across a phase boundary and obtain an expression for film thickness. Water evaporates from an open bowl at 349 K at the rate of 4.11 ð 103 kg/m2 s. What is the effective gas-film thickness? The water is replaced by ethanol at 343 K. What will be its rate of evaporation in kg/m2 s if the film thickness is unchanged? At the surface of the ethanol, what proportion of the total mass transfer will then be attributable to bulk flow? Data. Vapour pressure of water at 349 K D 34 mm Hg Vapour pressure of ethanol at 343 K D 544 mm Hg Neglect the partial pressure of vapour in the surrounding atmosphere Diffusivity of water vapour in air D 26 ð 1000026 m2 /s Diffusivity of ethanol in air D 12 ð 1000026 m2 /s Density of mercury D 13,600 kg/m3 Universal gas constant R D 8314 J/kmol K
Solution For evaporation for a free surface, Stefan’s law is applicable or: N0A D 0002D N0A D
Integration gives:
D
CT dCB CT dCT D CD CB dy CB dy
(from equation 10.30)
D CB CT ln 2 y2 0002 y1 CB 1 CT D CA1 0002 CA2
y2 0002 y1 CBm
For water: Vapour pressure, PA1 D 301 mm Hg at 349 K At 349 K:
CA1 D
0.301 ð 13,600 ð 9.81
PA1 D RT 8314 ð 349
D 0.0138 kmol/m3
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
CT D
P 0.760 ð 13,600 ð 9.81
D RT 8314 ð 349
D 0.0350 kmol/m3
For air:
CB1 D CT 0002 CA1 D 0.0350 0002 0.0138
D 0.0212 kmol/m3 CB2 D CT D 0.0350 kmol/m3 .
and:
N0A D
The evaporation rate of water is: D
26 ð 1000026 ð 0.0350 ln y2 0002 y1
0002
0.0350 0.0212
0003
0.456 ð 1000026 kmol/m2 s y2 0002 y1
But: N0A D 4.11 ð 1000023 kg/m2 s D 0.228 ð 1000023 kmol/m2 s. y1 2 ð 1000023 m D 2 mm giving a film thickness of: y2 0002 D For ethanol: Vapour pressure D 541 mm Hg at 343 K. At 343 K:
CA1 D
0.541 ð 13,600 ð 9.81
PA1 D RT 8314 ð 343
D 0.0253 kmol/m3 0.760 ð 13,600 ð 9.81
P D CT D RT 8314 ð 343
D 0.0356 kmol/m3 CB1 D CT 0002 CA1 D 0.0103 kmol/m3 and:
CB2 D CT D 0.0356 kmol/m3 0002 0003 0002 0003 12 ð 1000026 0.0356 0 The evaporation rate of ethanol, NA D ð 0.0356 ð ln 0.002 0.0103 D 0.265 ð 1000023 kmol/m2 s D 12.2 ð 1000023 kg/m2 s
The total flux at any location D diffusional flux C bulk flow bulk flow diffusional flux CB D10002 D10002 total flux total flux CT At the ethanol surface, the proportion of flux due to bulk flow is: 10002
0.0103 D 0.71 0.0356
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SECTION 11
The Boundary Layer PROBLEM 11.1 Calculate the thickness of the boundary layer at a distance of 75 mm from the leading edge of a plane surface over which water is flowing at a rate of 3 m/s. Assume that the flow in the boundary layer is streamline and that the velocity u of the fluid at a distance y from the surface can be represented by the relation u D a C by C cy 2 C dy 3 , where the coefficients a, b, c, and d are independent of y. The viscosity of water is 1 mN s/m2 .
Solution At a distance y from the surface: u D a C by C cy 2 C dy 3 . When y D 0, u D 0, and hence a D 0. The shear stress within the fluid: R0 D 0003 ∂u/∂y yD0 and since ∂u/∂y is constant for small values of y, ∂2 u/∂y 2 yD0 D 0. At the edge of the boundary layer, y D υ and u D us , the main stream velocity. ∂u/∂y D 0 and u D by C cy 2 C dy 3 ∴
∂u/∂y D b C 2cy C 3dy 2 and ∂2 u/∂y 2 D 2c C 6dy When y D 0, ∂2 u/∂y 2 D 0, and hence c D 0. When y D υ, u D bυ C dυ3 D us
and:
∂u/∂y D b C 3dυ2 D 0
∴
b D 00033dυ2
∴
d D 0003us /2υ3 and b D 3us /2υ The velocity profile is given by, u D 3us y/2υ 0003 us /2 y/υ 3
or:
u/us D 1.5 y/υ 0003 0.5 y/υ 3
(equation 11.12)
The integral in the momentum equation 11.9 is now evaluated, and substituting from equations 11.14 and 11.15 into equation 11.9:
υ/x D 4.64 Re00030.5 x Rex D 0.075 ð 3 ð 1000/1 ð 1000033 D 225,000 υ/x D 4.64 ð 225,00000035 D 0.00978 and:
υ D 0.00978 ð 0.075 D 0.000734 m or 0.734 mm 285
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PROBLEM 11.2 Water flows at a velocity of 1 m/s over a plane surface 0.6 m wide and 1 m long. Calculate the total drag force acting on the surface if the transition from streamline to turbulent flow in the boundary layer occurs when the Reynolds group Rex D 105 .
Solution See Volume 1, Example 11.1
PROBLEM 11.3 Calculate the thickness of the boundary layer at a distance of 150 mm from the leading edge of a surface over which oil, of viscosity 50 mN s/m2 and density 990 kg/m3 , flows with a velocity of 0.3 m/s. What is the displacement thickness of the boundary layer?
Solution See Volume 1, Example 11.2
PROBLEM 11.4 Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm diameter at 0.003 m3 /s. What is the velocity of the benzene at the edge of the laminar sub-layer? Assume fully developed flow exists within the pipe.
Solution See Volume 1, Example 11.3
PROBLEM 11.5 Air is flowing at a velocity of 5 m/s over a plane surface. Derive an expression for the thickness of the laminar sub-layer and calculate its value at a distance of 1 m from the leading edge of the surface. Assume that within the boundary layer outside the laminar sub-layer, the velocity of flow is proportional to the one-seventh power of the distance from the surface and that the shear stress R at the surface is given by:
R/0013us2 D 0.03 us 0013x/ 00030.2 where 0013 is the density of the fluid (1.3 kg/m3 for air), is the viscosity of the fluid (17 ð 1000036 N s/m2 for air), us is the stream velocity (m/s), and x is the distance from the leading edge (m).
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287
Solution The shear stress in the fluid at the surface: R D 0003 ux /y From the equation given: ∴
R D 0.030013us2 /us 0013x 0.2 ux D 0.030013us2 y/ /us 0013x 0.2
If the velocity at the edge of the laminar sub-layer is ub , ux D ub when y D υb . ∴
ub D 0.030013us2 υb / /us υ0013 0.2
υb /υ D 33.3 ub /us /us υ0013 0.8
and:
From equation 11.24, the velocity distribution is given by:
υb /υ 1/7 D ub /us
ub /us 7 D 33.3 ub /us /us υ0013 0.8
and hence:
ub /us D 1.65 /us υ0013 0.115
or:
D 1.65 Re00030.115 υ Substituting 0.376x 0.8 /us 0013 0.2 for υ from equation 11.29:
ub /us D 1.65[0.376us 0013x 0.8 0.2 / us0.2 00130.2 ]00030.115 D 1.65/0.3760.115 us0.8 x 0.8 00130.8 / 0.8 00030.115 D 1.85 Re00030.09 x Now: From equation 11.31: ∴
υb /υ D ub /us 7 D 74.2 Rex0.63
υ/x D 0.376 Re00030.2 x
υb /x D 74.2 ð 0.376 / Rex0.63 Rex0.2 D 27.9 Re00030.83 x
In this case:
Rex D 1 ð 5 ð 1.3/17 ð 1000036 D 3.82 ð 105 υb D 1.0 ð 27.9 3.82 ð 105 00030.83 D 6.50 ð 1000034 m or 0.65 mm
PROBLEM 11.6 Obtain the momentum equation for an element of the boundary layer. If the velocity profile in the laminar region can be represented approximately by a sine function, calculate the boundary layer thickness in terms of distance from the leading edge of the surface.
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Solution The derivation of the momentum equation for an element of the boundary layer is presented in detail in Section 11.2 and the final expression is: 0001
l
us 0003 ux ux dy
0003Ro D 0013∂/∂x 0
A sine function may be developed as follows. When y D 0, ux D 0 and when y D υ, ux D us . ux D us sin ay
Thus:
and when y D υ, sin ay D 0016/2 or aυ D 0016/2 and a D 0016/2υ. ∴
ux D us sin 0016y/2d
and over the range 0 < y < υ,
ux /us D sin[ 0016/2 y/υ ] The integral in the momentum equation may now be evaluated for the laminar boundary layer considering the ranges 0 < y < υ and υ < y < l separately. 0001 υ 0001 l ∴
us 0003 ux ux dy D us2 f1 0003 sin[ 0016/2 y/υ ]gfsin[ 0016/2 y/υ ]g dy 0
0
0001
l
us 0003 us us dy
C υ
0001
D us2
υ
[sin 0016y/2υ 0003 sin2 0016y/2υ ] dy 0
D us2 [0003[cos 0016y/2υ ]/ 0016/2υ 0003 y/2 C sin 0016y/υ / 20016/υ ]υ0 D us2 υ[ 2/0016 0003 1/2 ] R0 D 0003 ∂ux /∂y yD0 D 0003 us 0016/2υ and substituting in the momentum equation: 0002 0003 00040005 0006 2 1 0003 ∂x D us 0016/2υ 0013∂ us2 υ 0016 2 ∴
υ dυ D 00162 dx/0013us 4 0003 0016
∴
υ2 /2 D [00162 / 4 0003 0016 ] x/0013us
and: ∴
υ D 4.80 x/0013us 0.5
υ/x D 4.80 /x0013us 0.5 D 4.80 Re00030.5 x
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THE BOUNDARY LAYER
PROBLEM 11.7 Explain the concepts of “momentum thickness” and “displacement thickness” for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this ‘thermal thickness’ in terms of the thicknesses of the velocity and temperature boundary layers. Similar forms of cubic equations may be used to express velocity and temperature variations with distance from the surface. For a Prandtl number, Pr, less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 00031/3 . Work out the ‘thermal thickness’ in terms of the thickness of the velocity boundary layer for a value of Pr D 0.7.
Solution Consideration is given to the streamline portion of the boundary layer in Section 11.3 where, assuming: ux D uo C ay C by 2 C cy 3 (equation 11.10) it is shown that the equation for the velocity profile is:
ux /us D 1.5 y/υ 0003 0.5 y/υ 3
(equation 11.12)
The equivalent equation for the thermal boundary layer will be:
001a/001as D 1.5 y/υt 0003 0.5 y/υt 3 where υt is the thickness of the thermal boundary layer. The heat flow is given by: 0001 l QD Cp 0013ux T dy 0
0001
l
[1.5 y/υt 0003 0.5 y/υt 3 ][1.5 y/υ 0003 0.5 y/υ 3 ] dy
D us Ts Cp 0013 0
This is made up of two components: the heat flow through the thermal boundary layer: 0001 υt /υ D us Ts Cp 0013 f 2.25y 2 /υ Ð υt 0003 0.75y 4 / υ3 υt 0003 0.75y 4 / υ Ð υ3t 0
6
C 0.25y /υ2 υ5t g d y/υ and the heat flow through the velocity boundary layer between y D υt and y D υ: 0001 1 D us Ts Cp 0013υ [1.5y/υ 0003 0.5 y/υ 3 ] d y/υ
υt /υ
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
Thus, putting
D υt /υ , the heat flow becomes: 00020001 x Q D us Ts Cp 0013υ
2.25 y/υ 2 / 0003 0.75 y/υ 4 1/ C 1/ 0001
3
C 0.25 y/υ 6 / 3 d y/υ
0
l
0005 1.5 y/υ 0003 0.5 y/υ d y/υ 3
C x
D us Ts Cp 0013υ[ 0.75 3 / 0003 0.15 5 1/ C 1/ 3 C 0.036 7 / 3 C 0.75 1 0003
2
0003 0.125 1 0003
D us Ts Cp 0013υ 0.625 0003 0.15
2
4
]
C 0.0107 4
The heat flow from υ to υŁt in the absence of boundary layers D υ 0003 υŁt us Ts 0013Cp . ∴
υ 0003 υŁt D υ 0.625 0003 0.15
and:
υŁt /υ D 0.375 C 0.15
2
2
C 0.0107 4
0003 0.0107
4
When < 1, then D Pr 00030.33 and neglecting the 4 term, an approximate value is:
υŁt /υ D 0.375 C 0.15Pr 00030.67 When Pr D 0.7, Pr 0.67 D 0.788 and:
υŁt /υ D 0.375 C 0.15/0.788 D 0.185
(Since this is much less than 1, neglecting the
4
term is justified.)
PROBLEM 11.8 Explain why it is necessary to use concepts, such as the displacement thickness and the momentum thickness, for a boundary layer in order to obtain a boundary layer thickness which is largely independent of the approximation used for the velocity profile in the neighbourhood of the surface. It is found that the velocity u at a distance y from the surface can be expressed as a simple power function (u / y n ) for the turbulent boundary layer at a plane surface. What is the value of n if the ratio of the momentum thickness to the displacement thickness is 1.78?
Solution The first part of this problem is discussed in Section 11.1. If the displacement and the momentum thicknesses are υŁ and υm respectively, then: 0001 υ 0001 υ the momentum flux D uy dy0013uy D us dy0013us 0
0001
and:
the mass flux D
0001
υ
υm υ
uy dy0013 D 0
us dy0013 υŁ
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THE BOUNDARY LAYER
0001 ∴
υ
uy2 dy D us2 υ 0003 υm
0
0001
υ
uy dy D us υ 0003 υŁ
and: 0
0001
1
uy /us 2 d y/υ D 1 0003 υm 0003 υ
0
0001
1
uy /us d y/υ D 1 0003 υŁ /υ
and: 0
0002 000500060002 0005 0001 1 0001 1
υm /υŁ D 1 0003
uy /us 2 d y/υ 10003
uy /us d y/υ
∴
0
0
If uy /us D y/υ n , then: 0001 1 0001 1
uy /us d y/υ D
y/υ n d y/υ D 1/ n C 1 0
0001
0001
1
and:
uy /us d y/υ D 0
∴
0 1
y/υ 2n d y/υ D 1/ 2n C 1
0
υm /υŁ D [1 0003 1/ 2n C 1 ]/[1 0003 1/ n C 1 ] D 2 n C 1 / 2n C 1 When υm /υŁ D 1.78: 1.78 D 2n C 2/ 2n C 1 D 3.56n C 1.78 D 2n C 2
and:
n D 0.22/1.56 D 0.141 or approximately 1/7 .
PROBLEM 11.9 Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface.
Solution Using the nomenclature of Fig. 11.5, the argument presented in Section 11.2 results in the expression known as the momentum equation, given in equation 11.9, which may be expressed as: 0001 l ux us 0003 ux dy D 0003R0
i 0013∂/∂x 0
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
If the velocity within the boundary layer may be represented by a sine function: 0001
the integral: 0
ux /us D sin[ 0016/2 y/υ ] 0001 1 l ux us 0003 ux dy D us2 υ
ux /us 1 0003 ux /us d y/υ
ii
0
and, substituting from equation (ii): 0001 l 0001 1 ux us 0003 ux dy D us2 υ [sin 0016/2 y/υ 0003 sin2 0016/2 y/υ ] d y/υ 0
0
0007
D us2 υ
1 0003 2/0016 cos[ 0016/2 y/υ ] 0 0003 0.5
0001
1 D us2 υf 2/0016 0003 0.5 C 1/0016 sin 0016y/υ 0 g
1
1 0003 cos 0016y/υ d y/υ
0
D us2 υ 2/0016 0003 0.5 D 0.1366us2 υ From equation (ii): ∴
ux D us sin[ 0016/2 y/υ ] dux /dy D us 0016/2υ cos[ 0016/2 y/υ ]
and when y D 0:
dux /dy yD0 D 0016/2 us /υ But:
R0 D 0003 dux /dy yD0 D 0003 0016/2 us /υ
Therefore, substituting in equation (i): 0013∂/∂x 0.1366us2 υ D 0016/2 us /υ υ dυ D /0013us 0016/0.2732 dx υ2 /x D x/0013us 0016/0.2732 υ2 /x 2 D /0013us x 0016/0.1366 and:
υ/x D 4.80 Re00030.5 x
PROBLEM 11.10 Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface can be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of the surface, when the velocity profile is expressed as a sine function.
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THE BOUNDARY LAYER
Solution The derivation of the momentum equation is given in Section 11.2 to give: 0001 l ∂ 0013 ux us 0003 ux dy D 0003R0 D dux /dy yD0 (equation 11.9) (i) ∂x 0 If the velocity profile is a sine function, then:
ux /us D sin[ 0016/2 y/υ ] Differentiating:
1/us ∂ux /∂y D 0016/2υ cos[ 0016/2 y/υ ] When y D υ, ux D us and ∂ux /∂y D 0 Differentiating again:
1/us ∂2 ux /∂y 2 D 000300162 /4υ2 sin[ 0016/2 y/υ ] When y D 0, ux D 0 and ∂2 ux /∂y 2 D 0 Substituting, noting that ux D us when y > υ: 0001 l 0001 υ ux ux 0003 us dy D us2 sin y/υ 0016/2 [1 0003 sin y/υ 0016/2 ] dy 0
0001
D us2 υ D us2 υ
0
1
[sin y/υ 0016/2 0003 0.5 1 0003 cos y0016/υ ] d y/υ 0
000b 1 1 1 0003 2/0016 cos y/υ 0016/2 0 0003 0.5 y/υ 0 C 0.5 1/0016 sin y0016/υ 0
D us2 υ
2/0016 0003 0.5 C 0 D 0.1366us2 υ
(as in Problem 11.8)
Substituting in equation (i): 0013∂ 0.1366us2 υ /∂x D 0016/2 us /υ ∴
υdυ D
0016/2 /0.1366 /0013us dx
If υ D 0 when x D 0: υ2 /x D 11.5 x/0013us2 and:
υ/x 2 D 11.5 /0013us x and υ/x D 3.39 Re00030.5 x
The displacement thickness, υŁ is given by: 0001 υ 1 us υ 0003 υŁ D us sin[ y/υ 0016/2 ] dy D us 2υ/0016 0003 cos[ y/υ 0016/2 ] 0 0
∴
and: ∴
υ 0003 υŁ D 2υ/0016 D 0.637υ υŁ D 0.363υ
υŁ /x D 0.363 ð 3.39 Re00030.5 D 1.23 Re00030.5 x x
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PROBLEM 11.11 Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface.
Solution The total mass flowrate through plane 1–2 is:
l 0
0013ux dy.
The total momentum flux through plane 1–2 is:
l 0
0013ux2 dy
000f ∂ 000e l 0013u dy dx x ∂x 0 ∂ 000e l 2 000f 0013u dy dx. Change in momentum flux from 1–2 to 3–4 is: ∂x 0 x
Change in mass flowrate from 1–2 to 3–4 is:
Change in momentum flux is attributable, in the absence of pressure gradient, to: (a) Momentum of fluid entering through 2–4 Since all this fluid has velocity us , the momentum flux is: 0007 00020001 l 0005
∂ 0013ux dy dx us ∂x 0 (b) Force due to shear stress at surface D R0 dx. Thus a momentum balance gives: 00070001 l
00070001 l
∂ ∂ 0013ux2 dy dx D 0013ux dy dx.us C R0 dx. ∂x ∂x 0 0 00070001 l
∂ ∴ 0013 ux us 0003 ux dy D 0003R0 ∂x 0 Representing velocity within boundary layer by a sine function, then: 0016 y ux D sin us 2υ Thus:
0001
l 0
0003 0004 ux y ux 10003 d u u υ s s 0 0001 1 0016 y y 0016 y D us2 υ sin Ð 0003 sin2 d 2 υ 2υ υ 0 0001
ux us 0003 ux dy D us2 υ
1
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THE BOUNDARY LAYER
00100002
D
2 0016y 0003 cos 0016 2υ
us2 υ 0010
0005l
1 0003 2 0
0002
0001
1
1 0003 cos 0
0016y y d υ υ
0011
0005l 0011
2 1 1 y 0003 C sin 0016 0016 2 0016 υ 0 0007
2 1 D us2 υ D 0.1366us2 υ. 0003 0016 2 0016 0016 y dux D us Ð cos dy 2υ 2υ 0004 0003 0016 u dux s D Ð dy yD0 2 υ 0003 0004 dux 0016 us . R0 D 0003 D0003 dy yD0 2 υ D us2 υ
Differentiating:
and: Thus: ∴ ∴ ∴
0013
∂ 0016 us
0.1366us2 υ D ∂x 2 υ 0003 0004 0016 υ dυ D dx. 0013us 0.2732 0003 0004 υ2 x 0016 D 2 0013us 0.2732 0003 0004 2 υ 0016 D x2 0013us x 0.1366 υ D 4.796 Re00031/2 x x
and:
PROBLEM 11.12 Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of the surface, when the velocity profile is expressed as a sine function.
Solution The momentum flux across 1–2 is: The change from 1–2 to 3–4 is: 0013
L 0
0013ux2 dy
∂ 000e L 2 000f u dy dx ∂x 0 x
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
The mass flux across 1–2 is:
L 0
0013ux dy.
The change from 1–2 to 3–4 is: 0013
000f ∂ 000e L u dy dx. x ∂x 0
The rate of momentum entering through 2–4 is: 00070001 L
∂ 0013 ux us dy dx, ∂x 0 Assuming us 6D f x , then: ∂P D 0. ∂x A momentum balance gives: 00070001 L
00070001 L
∂ ∂ 0013 ux2 dy dx D 0013 ux us dy dx C R0 dx ∂x ∂x 0 0 00070001 L
0003 0004 ∂ux ∂ ux us 0003 ux dy D 0003R0 D for a Newtonian fluid or: 0013 ∂x ∂y yD0 0 The sine function is: ux D K sin ky, so
∂2 ux ∂ux D Kk cos ky and Kk 2 sin ky ∂y ∂y 2
is satisfied for all finite values of K and k. The boundary conditions are: y D 0 ux D 0 yD0
∂2 ux D0 ∂y 2
y D υ ux D us yDυ Thus:
K D us
∂ux D0 ∂y
0016 0016 and k D . 2 2υ ∂ux 0016us 0016y 0016y ux D us sin , D cos 2υ ∂y 2υ 2υ kD
Hence: 00070001 L
D0
and: y
Thus:
00070001 υ ∂ 0016y 0016 y 0016us sin 1 0003 sin dy D ∂x 2υ 2υ 2υ 0 00070001 1 0012
0013 ∂ 0016y 0016 y y 0016 υ 0003 sin2 sin d D ∂x 2 υ 2υ υ 2υ0013us 0
0013us2
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THE BOUNDARY LAYER
0002 0005 1 1 y 0016 0016y y 0003 C cos 0016 sin d D 2υ 2 2 υ υ 2υ0013us 0 0002 00051 ∂ 2 0016 0016y 1y 1 y D υ cos 0003 C sin 0016 ∂x 0016 2υ 2υ 20016 υ 0 2υ0013us 0002 0005 ∂ 2 0016 1 D υ 0003 ∂x 0016 2 2υ0013us 0003 0004 2 4 υ 0016x 00031 D assuming υ D 0 at x D 0 0016 2 0013us 0003 00040003 0004 υ2 20016 D x2 0013us x
4/0016 0003 1 υ p D 23.1 Re00031/2 D 4.80 Re00031/2 x x x 0002 0003 0004 0005 0001 υ 0016 y 2υ 2υ 0016y υ Ł
υ 0003 υ us D us sin dy D us 0003 cos D us 2 υ 0016 2 υ 0016 0 0 0003 0004 2 υŁ D υ 1 0003 0016
∂ υ ∂x
∴
and :
00070001
297
1
υŁ D 0.363. υ
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SECTION 12
Momentum, Heat and Mass Transfer PROBLEM 12.1 If the temperature rise per metre length along a pipe carrying air at 12.2 m/s is 66 deg K, what will be the corresponding pressure drop for a pipe temperature of 420 K and an air temperature of 310 K? The density of air at 310 K is 1.14 kg/m3 .
Solution For a pipe of diameter d, the mass flow D 0003u00050006d2 0007/4 and the rate of heat transfer, q D 0003u00050006d2 /40007Cp T where T is the temperature rise. Also q D hA0003Tw 0002 Tm 0007 D h0006dl0003Tw 0002 Tm 0007 W where Tw and Tm are the mean wall and fluid temperatures. Thus, 0003h/Cp 0005u0007 D dT/4l0003Tw 0002 Tm 0007 From equation 12.102: R/0005u2 D dT/40003Tw 0002 Tm 0007l Substituting in equation 3.18: 0002P D 4dT0003l/d00070005u2 /40003Tw 0002 Tm 0007 D 0003T0005u2 l0007/0003Tw 0002 Tm 0007 D 000366 ð 1.14 ð 12.22 ð 1.00007/0003420 0002 3100007 D 101.8 0003N/m2 0007/m
PROBLEM 12.2 It is required to warm a quantity of air from 289 K to 313 K by passing it through a number of parallel metal tubes of inner diameter 50 mm maintained at 373 K. The pressure drop must not exceed 250 N/m2 . How long should the individual tubes be? The density of air at 301 K is 1.19 kg/m3 and the coefficients of heat transfer by convection from the tube to air are 45, 62, and 77 W/m2 K for velocities of 20, 24, and 30 m/s at 301 K respectively.
Solution From equations 12.102 and 3.18: 0002P D 40003h/Cp 0005u00070003l/d00070005u2 D 4hlu/Cp d ∴
250 D 40003hlu/Cp ð 0.0500007 or h D 3.125Cp /lu W/m2 K
(i)
298
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299
The heat transferred to the air D u00030006d2 /400070005Cp 0003T2 0002 T1 0007 D u00030006 ð 0.0502 /400071.19Cp 0003313 0002 2890007 D 0.056Cp u W This is equal to: h0006dl0003Tw 0002 Tm 0007 D h0006 ð 0.050l0003373 0002 3010007 D 11.3hl W ∴
11.31hl D 0.056Cp u or h D 0.0050Cp u/l From equation (i):
(ii)
0003Cp /l0007 D 0.32hu
and substituting in equation (ii): h D 00030.005 ð 0.32hu2 0007 and u D 25 m/s For this velocity, interpolation of the given data gives a value of h D 64 W/m2 K. ∴
0003Cp /l0007 D 00030.32 ð 64 ð 250007 D 512 J/kg K m For air:
and hence:
Cp D 1000 J/kg K l D 00031000/5120007 D 1.95 m
PROBLEM 12.3 Air at 330 K, flowing at 10 m/s, enters a pipe of inner diameter 25 mm, maintained at 415 K. The drop of static pressure along the pipe is 80 N/m2 per metre length. Using the Reynolds analogy between heat transfer and friction, estimate the temperature of the air 0.6 m along the pipe.
Solution See Volume 1, Example 12.2.
PROBLEM 12.4 Air flows at 12 m/s through a pipe of inside diameter 25 mm. The rate of heat transfer by convection between the pipe and the air is 60 W/m2 K. Neglecting the effects of temperature variation, estimate the pressure drop per metre length of pipe.
Solution From equations 3.18 and 12.96, 0002P D 40003h/Cp 0005u00070003l/d00070005u2 Taking Cp D 1000 J/kg K and l D 1 m, then: 0002P D 4000360/10000005 ð 12000700031.0/0.02500070005 ð 122 D 115.2 N/m2 per metre
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CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS
PROBLEM 12.5 Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter and the pressure drop over a 4 m length is found to be 1.5 kN/m2 . Using the Reynolds analogy, by how much would the air temperature be expected to fall over the first metre of pipe length if the wall temperature there is kept constant at 295 K? Viscosity of air D 0.018 mN s/m2 . Specific heat capacity of air D 1.05 kJ/kg K.
Solution (Essentially, this is the same as Problem 9.40 though, here, an alternative solution is presented.) From equations 3.18 and 12.102: 0002P D 40003h/Cp 0005u00070003l/d00070005u2 . For a length of 4 m: 1500 D 40003h/Cp 0005u000700034.0/0.05000070005u2 ∴
0003hu/Cp 0007 D 4.69 kg/ms2
(i)
The rate of heat transfer D h0006dl0003Tm 0002 Tw 0007, which for a length of 1 m is: 0003h0006 ð 0.050 ð 1.0000700030.50003320 C T2 0007 0002 2950007 D 0.157h00030.5T2 0002 1350007 The heat lost by the air D u00030006d2 /400070005Cp 0003T1 0002 T2 0007 D u00030006 ð 0.0502 /400070005Cp 0003320 0002 T2 0007 D 0.00196u0005Cp 0003320 0002 T2 0007 ∴
80.10003h/Cp 0005u0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007
Substituting from equation (i): 80.100034.69/0005u2 0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007
(ii)
From equation 12.139: 0003h/Cp 0005u0007 D 0.0320003du0005/0015000700020.25 D 0.03200030015/du000500070.25 At 320 K and 101.3 kN/m2 , 0005 D 000328.9/22.400070003273/3200007 D 1.10 kg/m3 ∴ ∴
0003h/Cp 0005u0007 D 0003hu/Cp 0007/00030005u2 0007 D 4.69/00031.10u2 0007 4.69/00031.10u2 0007 D 0.032[0.018 ð 1000023 /00030.050 ð 1.10u0007]0.25 and u D 51.4 m/s Substituting, in equation (ii): 000380.1 ð 4.690007/00031.10 ð 51.42 0007 D 0003320 0002 T2 0007/00030.5T2 0002 1350007 and T2 D 316 K
The temperature drop over the first metre is therefore 4 deg K which agrees with the solution to Problem 9.40. (It may be noted that, in those problems, an arithmetic mean temperature difference is used rather than a logarithmic value for ease of solution. This is probably justified in view of the small temperature changes involved and also the approximate nature of the Reynolds analogy.)
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MOMENTUM, HEAT AND MASS TRANSFER
301
PROBLEM 12.6 Obtain an expression for the simple Reynolds analogy between heat transfer and friction. Indicate the assumptions which are made in the derivation and the conditions under which you would expect the relation to be applicable. The Reynolds number of a gas flowing at 2.5 kg/m2 s through a smooth pipe is 20,000. If the specific heat of the gas at constant pressure is 1.67 kJ/kg K, what will the heat transfer coefficient be?
Solution The derivation of the simple Reynolds analogy and its application is presented in detail in Section 12.8. For a Reynolds number of 2.0 ð 104 , from Fig. 3.7, 0003R/0005u2 0007 D 0.0032 for a smooth pipe. From equation 12.102: 0003h/Cp 0005u0007 D 0.0032 0005u D 2.5 kg/m2 s and hence:
h D 00030.0032 ð 1670 ð 2.50007 D 13.4 W/m2 K
PROBLEM 12.7 Explain Prandtl’s concept of a ‘mixing length’. What parallels may be drawn between the mixing length and the mean free path of the molecules in a gas? The ratio of the mixing length to the distance from the pipe wall has a constant value of 0.4 for the turbulent flow of a fluid in a pipe. What is the value of the pipe friction factor if the ratio of the mean velocity to the axial velocity is 0.8?
Solution Transfer by molecular diffusion is discussed in Section 12.2 and the concept of the mixing length in Section 12.3.2. By analogy with kinetic theory, the eddy kinematic viscosity, E, is given by: E / 0017E uE (equation 12.18) where 0017E is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. As shown in equation 12.21: uE / 0017E jdux / dyj Combining this with the previous equation: E / 0017E 00030017E j dux / dyj0007 Putting the proportionality constant equal to unity, E D 00172E j dux / dyj
(equation 12.23)
In the absence of momentum transfer by molecular movement, the shear stress is given by: Ry D 0002Ed00030005ux 0007/dy D 0002000500172E jdux /dyjdux /dy.
(equation 12.20)
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Since near a surface dux /dy is positive and assuming Ry is approximately constant at a value at the pipe wall, that is Ry D Ro D 0002R, then: 0001
or:
R D 000500172E 0003dux /dy00072 (equation 12.26)
0003R/00050007 D 0017E 0003dux /dy0007
p
Here, 0003R/00050007, the shearing stress or friction velocity is usually denoted by uŁ . Since from equation 12.35, 0017E D 0.4y, then: uŁ D 0.4y dux /dy Rearranging:
dux /uŁ D dy/0.4y
and integrating:
0003ux /uŁ 0007 D 2.5 ln y C const.
(i)
At y D r: 0003umax /uŁ 0007 D 2.5 ln r C const. const. D 0003umax /uŁ 0007 0002 2.5 ln r
or: Substituting in equation (i):
0003umax /uŁ 0007 0002 2.5 ln r D 0003ux /uŁ 0007 0002 2.5 ln y 0003umax 0002 ux 0007/uŁ D 2.5 ln0003r/y0007 0002 r 0003200060003r 0002 y0007 dyux 0007/0006r 2 uD
and: The mean velocity:
(ii)
0
0002
and dividing by r:
uD2
1
00031 0002 y/r0007 d0003y/r0007ux 0
Substituting for ux from equation (ii): 0002 1 uD2 00031 0002 y/r0007 d0003y/r00070003umax C 2.5uŁ ln0003y/r00070007 0
D2
0003 0004
[umax C 2.5 ln0003y/r0007][0003y/r0007 0002 0.50003y/r00072 ] 0002
0002u 0007
1
00051 0
0006 2.50003r/y0007[0003y/r0007 0002 0.50003y/r0007 ]d0003y/r0007 2
Ł 0
0004 00051 D 2 umax 00030.50007 0002 2.5uŁ 0003y/r0007 0002 0.250003y/r00072 0 D umax 0002 3.75uŁ 0001 When 0003u/umax 0007 D 0.8, u D 0003u/0.80007 0002 3.75u 0003R/0005u2 0007 0001 ∴ 0003R/0005u2 0007 D 0.0667 and 0003R/0005u2 0007 D 0.00444
PROBLEM 12.8 The velocity profile in the neighbourhood of a surface for a Newtonian fluid may be expressed in terms of a dimensionless velocity uC and a dimensionless distance y C from
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the surface. Obtain the relation between uC and y C in the laminar sub-layer. Outside the laminar sub-layer, the relation is: uC D 2.5 ln y C C 5.5 At what value of y C does the transition from the laminar sub-layer to the turbulent zone occur? In the “Universal Velocity Profile”, the laminar sub-layer extends to values of y C D 5 and the turbulent zone starts at y C D 30 and the range 5 < y C < 30, the buffer layer, is covered by a second linear relation between uC and ln y C . What is the maximum difference between the values of uC , in the range 5 < y C < 30, using the two methods of representation of the velocity profile? Definitions: uC D 0003ux /uŁ 0007, y C D 0003yuŁ 00050007/0015 and uŁ2 D R/0005 where ux is the velocity at a distance y from the surface, R is the wall shear stress and 0005 and 0015 are the density and viscosity of the fluid respectively.
Solution As discussed in Section 12.4.2, if the velocity gradient dux / dy approaches a constant value near the surface, 0003d2 ux /dy 2 0007 approaches zero and R D 0015ux /y. ∴
uŁ2 D 0015ux /00030005y0007
and, as given in equation 12.39: 0003ux /uŁ 0007 D yuŁ 0005/0015 D y C uC D y C
Hence: C
C
C
(equation 12.40) C
Since u D 2.5 ln y C 5.5 and y D 2.5 ln y C 5.5, then solving by trial and error, the transition from the laminar sub-layer to the turbulent zone occurs when: y C D 11.6 and uC D 2.5 ln 11.6 C 5.5 D 11.62 For the buffer layer, uC D a ln y C C a0 C
C
C
(equation 12.41) C
When y D 5, u D 5 and when y D 30, u D 2.5 ln 30 C 5.5 D 14 ∴
5 D a ln 5 C a0
∴
a0 D 5 0002 a ln 5 D 5 0002 1.609a
and:
14 D a ln 30 C a0 D a ln 30 C 5 0002 1.609a
∴
9 D 3.401a 0002 1.609a and a D 5.02
and:
a0 D 5 0002 00031.609 ð 5.020007 D 00023.08
The difference between the two values of uC is a maximum when y C D 11.6. From the two-layer theory: uC D 11.6 From the buffer-layer theory: uC D 5.02 ln 11.6 0002 3.08 D 9.2 and hence, the maximum difference is 000311.6 0002 9.20007 D 2.4
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PROBLEM 12.9 Calculate the rise in temperature of water flowing at 4 m/s through a smooth 25 mm diameter pipe 6 m long. The water enters at 300 K and the temperature of the wall of the tube can be taken as approximately constant at 330 K. Use: (a) (b) (c) (d)
The simple Reynolds analogy, The Taylor-Prandtl modification, The buffer layer equation, Nu D 0.023 Re0.8 Pr 0.33 .
Comment on the differences in the results so obtained.
Solution See Volume 1, Example 12.3.
PROBLEM 12.10 Calculate the rise in temperature of a stream of air, entering at 290 K and flowing at 4 m/s through the tube maintained at 350 K; other conditions remaining the same as detailed in Problem 12.9.
Solution See Volume 1, Example 12.4.
PROBLEM 12.11 Air flows through a smooth circular duct of internal diameter 0.25 m at an average velocity of 15 m/s. Calculate the fluid velocity at points 50 mm and 5 mm from the wall. What will be the thickness of the laminar sub-layer if this extends to uC D y C D 5? The density of air may be taken as 1.12 kg/m3 and the viscosity of air as 0.02 mN s/m2 .
Solution See Volume 1, Example 12.1.
PROBLEM 12.12 Obtain the Taylor–Prandtl modification of the Reynolds analogy for momentum and heat transfer, and give the corresponding relation for mass transfer (no bulk flow). An air stream at approximately atmospheric temperature and pressure, and containing a low concentration of carbon disulphide vapour, is flowing at 38 m/s through a series of 50 mm diameter tubes. The inside of the tubes is covered with a thin film of liquid and
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both heat and mass transfer are taking place between the gas stream and the liquid film. The film heat transfer coefficient is found to be 100 W/m2 K. Using a pipe friction chart and assuming the tubes to behave as smooth surfaces, calculate: (a) the film mass transfer coefficient, and (b) the gas velocity at the interface between the laminar sub-layer and the turbulent zone of the gas. Specific heat of air D 1.0 kJ/kg K. Viscosity of air D 0.02 mN s/m2 . Diffusivity of carbon disulphide vapour in air D 1.1 ð 1000025 m2 /s. Thermal conductivity of air D 0.024 W/m K.
Solution The Taylor–Prandtl modification of the Reynolds analogy for heat transfer and mass transfer is discussed in Section 12.8.3 and the relevant equations are: For heat transfer: or:
(equation 12.119)
0003h/Cp 0005us 0007 D #/00031 C ˛0003Pr 0002 100070007
For mass transfer: or:
0003R/0005u2 0007 D 0003h/Cp 0005us 000700031 C ˛0003Pr 0002 100070007 D # 0003R/0005u2 0007 D 0003hD /us 000700031 C ˛0003Sc 0002 100070007 D #
(i) (equation 12.120)
0003hD /us 0007 D #/00031 C ˛0003Sc 0002 100070007
(ii)
Taking the molecular mass of air as 29 kg/kmol and atmospheric temperature as 293 K, the density, 0005 D 000329/22.400070003273/2930007 D 1.206 kg/m3 . ∴
Re D du0005/0015 D 000350 ð 1000023 ð 38 ð 1.2060007/00030.02 ð 1000023 0007 D 114,570
and from Fig. 3.7: # D 0003R/0005u2 0007 D 0.0021. From Table 1.3, the Prandtl number, Pr D Cp 0015/k D 00031.0 ð 103 ð 0.02 ð 1000023 0007/0.024 D 0.833 From Table 1.3, the Schmidt number, Sc D 0015/0005D D 00030.02 ð 1000023 0007/00031.206 ð 1.1 ð 1000025 0007 D 1.508 Substituting in equation (i): 0003100/00031.0 ð 103 ð 1.206 ð 3800070007 D 0.0021/00031 C ˛00030.833 0002 100070007 ∴
0.00218 D 0.0021/00031 0002 0.167˛0007 and ˛ D 0.22
Substituting in equation (ii): 0003hD /380007 D 0.0021/00031 C 0.2200031.508 0002 100070007 D 0.00189 and hD D 0.072 m/s The gas velocity at the interface of the laminar sub-layer and the turbulent zone, ub may also be estimated from: 0003ub /u0007 D 2.32 Re00020.125 or:
(equation 12.60)
ub D 000338 ð 2.3200070003114,570000700020.125 D 20.6 m/s
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PROBLEM 12.13 Obtain the Taylor–Prandtl modification of the Reynolds’ analogy between momentum and heat transfer and give the corresponding analogy for mass transfer. For a particular system a mass transfer coefficient of 8.71 ð 1000026 m/s and a heat transfer coefficient of 2730 W/m2 K were measured for similar flow conditions. Calculate the ratio of the velocity in the fluid where the laminar sub-layer terminates, to the stream velocity. Molecular diffusivity D 1.5 ð 1000029 m2 /s. Viscosity D 1 mN s/m2 . Density D 1000 kg/m3 . Thermal conductivity D 0.48 W/m K. Specific heat capacity D 4.0 kJ/kg K.
Solution The Taylor–Prandtl modification to heat and mass transfer is discussed in Section 12.8.3 resulting in the modified Lewis relation: hD D 0003h/Cp 00050007[1 C ˛0003Pr 0002 10007]/[1 C ˛0003Sc 0002 10007] 3
(equation 12.121)
00023
In this case:
Pr D Cp 0015/k D 00034 ð 10 ð 1 ð 10 0007/0.48 D 8.33
and:
Sc D 0015/0005D D 00031 ð 1000023 0007/00031000 ð 1.5 ð 1000029 0007 D 667
Substituting: 00038.71 ð 1000026 0007 D [2730/00034000 ð 10000007][1 C ˛00038.33 0002 10007]/[1 C ˛0003667 0002 10007] 0.01276 D 00031 C 7.33˛0007/00031 C 666˛0007 and ˛ D 0.844
PROBLEM 12.14 Heat and mass transfer are taking place simultaneously to a surface under conditions where the Reynolds analogy between momentum, heat and mass transfer may be applied. The mass transfer is of a single component at a high concentration in a binary mixture, the other component of which undergoes no net transfer. Using the Reynolds analogy, obtain a relation between the coefficients for heat transfer and for mass transfer.
Solution The solution to this problem is presented in Sections 12.8.1 and 12.8.2 and the relation between the coefficients for heat transfer and mass transfer is: h D 0003CBw /CT 0007Cp 0005hD
(equation 12.112)
PROBLEM 12.15 Derive the Taylor–Prandtl modification of the Reynolds analogy between momentum and heat transfer. In a shell and tube type condenser, water flows through the tubes which are 10 m long and 40 mm diameter. The pressure drop across the tubes is 5.6 kN/m2 and the effects of entry and exit losses may be neglected. The tube walls are smooth and flow may be taken
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as fully developed. The ratio of the velocity at the edge of the laminar sub-layer to the mean velocity of flow may be taken as 2 Re00020.125 , where Re is the Reynolds number in the pipeline. If the tube walls are at an approximately constant temperature of 393 K and the inlet temperature of the water is 293 K, estimate the outlet temperature. Physical properties of water: density D 1000 kg/m3 , viscosity D 1 mNs/m2 , thermal conductivity D 0.6 W/m K, specific heat capacity D 4.2 kJ/kg K.
Solution The Taylor–Prandtl modification of the Reynolds analogy to heat transfer, discussed in Section 12.8.3, leads to: St D 0003h/Cp 0005u0007 D 0003R/0005u2 0007/[1 C ˛0003Pr 0002 10007]
(equation 12.117) (i)
From equation 3.23: 0003R/0005u2 0007 Re2 D 0002Pf d3 0005/00034l00152 0007 D 00035600 ð 000340/100000073 ð 10000007/00034 ð 10 ð 00031 ð 1000023 00072 0007 D 8,960,000 From Fig. 3.8: Re D 62,000 and R/0005u2 D 8,960,000/000362,00000072 D 0.0023 The ratio of the velocity at the edge of the laminar sub-layer to the mean velocity of flow is: ˛ D 2 Re00020.125 D 2/00036200000070.125 D 0.5035 The Prandtl group, Pr D Cp 0015/k D 00034200 ð 1 ð 1000023 0007/0.6 D 7.0 and from equation (i), the Stanton group, St D 0003R/0005u2 0007/[1 C ˛0003Pr 0002 10007] D 0.0023/[1 C 0.503500037.0 0002 10007] D 0.000572 ∴
h D 0.000572Cp 0005u D 0.000572Cp Re 0015/d D 00030.000572 ð 4200 ð 62000 ð 1 ð 1000023 0007/000340/10000007 D 3724 W/m2 K.
The area for heat transfer per unit length of pipe D 0003400006/10000007 D 0.040006 D 0.126 m2 /m and making a heat balance over unit length of pipe dl: ∴ ∴
hAdl0003Tw 0002 T0007 D 0005uCp Ac dT D 0003Re 0015/d0007Cp Ac dT 3724 ð 0.126 dl0003393 0002 T0007 D 000362000 ð 1 ð 1000023 /000340/100000070007 ð 4200 ð 00030006/40007000340/100000072 dT
∴
0.0572 dl D dT/0003393 0002 T0007
Integrating:
0002
0002
10
T0
dl D
0.0572 0
dT/0003393 0002 T0007 293
00030.0572 ð 100007 D ln[0003393 0002 2930007/0003393 0002 T0 0007] ∴
1.772 D 100/0003393 0002 T0 0007 and T0 D 336.6 K
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PROBLEM 12.16 Explain the importance of the universal velocity profile and derive the relation between the dimensionless derivative of velocity uC , and the dimensionless derivative of distance from the surface y C , using the concept of Prandtl’s mixing length 0017E . It may be assumed that the fully turbulent portion of the boundary layer starts at y C D 30, that the ratio of the mixing length 0017E to distance y from the surface, 0017E /y D 0.4, and that for a smooth surface uC D 14 at y C D 30. If the laminar sub-layer extends from y C D 0 to y C D 5, obtain the equation for the relation between uC and y C in the buffer zone, and show that the ratio of the eddy viscosity to the molecular viscosity increases linearly from 0 to 5 through this buffer zone.
Solution The importance of the universal velocity profile is discussed in Section 12.4. From equation 12.18, for isotropic turbulence, the eddy kinematic viscosity, E ˛ 0017E uE where 0017E is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. The momentum transfer rate per unit area in a direction perpendicular to the surface at position y is then: Ry D 0002E d00030005ux 0007/ dy and for constant density,
(equation 12.20)
0002Ry /0005 D E dux / dy
or: where
Ry D 0002E0005 dux / dy
0001
0003Ry /00050007, the friction velocity, may be denoted by uŁ and then uŁ2 D Edux /dy
Assuming E D 0017E uE , that is a proportionality constant of unity, and uE D 0017E j dux / dyj, then: uŁ2 D 00172E 0003dux /dy0007j0003dux /dy0007j and hence near the surface where 0003dux /dy0007 is positive: uŁ D 0017E 0003dux /dy0007 Assuming 0017E D Ky: uŁ dy/y D Kdux Integrating:
ux /uŁ D 00031/K0007 ln y C B where B is a constant
or:
ux /uŁ D 00031/K0007 ln0003yuŁ 0005/00150007 C B0
(equation 12.28)
(equation 12.29)
Since 0003uŁ 0005/00150007 is constant, B0 is also constant and, writing the dimensionless velocity term, 0003ux /uŁ 0007 as uC and the dimensionless derivative of y0003yuŁ 0005/00150007 as y C , then: uC D 00031/K0007 ln y C C B0 Given that K D 0.4, then:
(equation 12.30)
uC D 2.5 ln y C C B0
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Given that for a smooth surface, uC D 14 at y C D 30, then: B0 D 14 0002 2.5 ln 30 D 5.5 uC D 2.5 ln y C C 5.5
and:
(equation 12.37)
For molecular transfer in the laminar sub-layer near the wall, from Section 12.4.2: Ry D 00020015ux /y 00030002Ry /00050007 D uŁ2 D 0015ux /0003y00050007
or: ∴
0003ux /uŁ 0007 D y0005uŁ /0015 and uC D y C
(equation 12.40)
If the buffer zone stretches from y C D 5 to y C D 30 at which uC is 5 and 14 respectively, then in equation 12.41: uC D a ln y C C a0 or:
5 D a ln 5 C a0 and 14 D a ln 30 C a0 uC D 5.0 ln y C 0002 3.05
and:
(equation 12.42)
From equation 12.46, the velocity gradient, duC /dy C D 5/y C From equation 12.61: Ry D 000200030015 C E00050007dux /dy 0001 and substituting uŁ D 0003Ry /00050007: 0003dux /dy0007 D uŁ2 /0003E C 0015/00050007 ∴
and:
C
C
0003du /dy 0007 D 00030015/0005000700031/[E C 0015/0005]0007 D 5/y E/00030015/00050007 D 0003y C /50007 0002 1
(equation 12.62) C
(equation 12.63)
Hence as y C goes from 5 to 30, the ratio of the eddy kinematic viscosity to the kinematic viscosity goes from 0 to 5.
PROBLEM 12.17 Derive the Taylor–Prandtl modification of the Reynolds analogy between heat and momentum transfer and express it in a form in which it is applicable to pipe flow. If the relationship between the Nusselt number Nu, Reynolds number Re and Prandtl number Pr is: Nu D 0.023 Re0.8 Pr 0.33 calculate the ratio of the velocity at the edge of the laminar sub-layer to the velocity at the pipe axis for water 0003Pr D 100007 flowing at a Reynolds number of 10,000 in a smooth pipe. Use the pipe friction chart.
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Solution The derivation of the Taylor–Prandtl modification of the Reynolds analogy between heat and momentum transfer is presented in Section 12.8.3 and the result is summarised as: St D h/0003Cp 0005us 0007 D 0003R/0005us2 0007/[1 C ˛0003Pr 0002 10007] (equation 12.117) or:
0003R/0005us2 0007 D [h/0003Cp 0005us 0007][1 0002 ˛00031 0002 Pr0007]
For turbulent pipe flow, us is approximately equal to 0003umean /0.820007 and: 0.820003R/0005u2 0007 D [h/0003Cp 0005u0007][1 0002 ˛00031 0002 Pr0007] When Re D 10,000, then from Fig. 3.1, 0003R/0005u2 0007 D 0.0038, and for Pr D 10: 00030.82 ð 0.00380007 D St[1 0002 ˛00031 0002 100007] or: But: and:
St00031 C 9˛0007 D 0.0031
(i)
Nu D 0.023 Re0.8 Pr 0.33 St D Nu/0003Re ÐPr0007 D 0.023 Re00020.2 Pr 00020.67 D 0.023000310,000000700020.2 000310000700020.67 D 0.000777
Hence, substituting in equation (i): 0.00077700031 C 9˛0007 D 0.0031 and ˛ D 0003ub /us 0007 D 0.33
PROBLEM 12.18 Obtain a dimensionless relation for the velocity profile in the neighbourhood of a surface for the turbulent flow of a liquid, using Prandtl’s concept of a “Mixing Length” (Universal Velocity Profile). Neglect the existence of the buffer layer and assume that, outside the laminar sub-layer, eddy transport mechanisms dominate. Assume that in the turbulent fluid the mixing length 0017E is equal to 0.4 times the distance y from the surface and that the dimensionless velocity uC is equal to 5.5 when the dimensionless distance y C is unity. Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer υb is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by Prandtl’s one seventh power law. Blasius Equation: 0003 0006 us υ0005 00020.25 R D 0.0228 0005us2 0015 where 0005, 0015 are the density and viscosity of the fluid, us is the stream velocity, and υ is the total boundary layer thickness.
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Solution The Universal Velocity Profile is discussed in detail in Section 12.4, and in the region where eddy transport dominates (y C > 30) and making all the stated assumptions: uC D 2.5 ln y C C 5.5
(equation 12.37)
If, in the laminar sub-layer (from equation 12.40), uC D y C then: y C D 2.5 ln y C C 5.5 and, solving by trial and error: y C D 11.6 D usŁ υb 0005/0015 p p Since uŁ D 0003R/00050007, then: υb D 11.60015/00030005uŁ 0007 D 11.60015/ 0003R00050007 2
(from equation 12.44) (i)
00020.25
But, from the Blasuis equation: R/0005u D 0.02280003us υ0005/00150007
p p and substituting for R in equation (i), υb D 000311.60015/ 0005000700031/us 0005000700031/0.0228000700020.5 0003us υ0005/001500070.125 and:
0003υb /υ0007 D 76.80003us υ0005/0015000700020.875
Using Prandtl’s one seventh power law, 0003ub /us 0007 D 0003υb /υ00071/7 D 0003υb /υ00070.143 But: ∴
R D 0015ub /υb D 0.02280005us 2 0003us υ0005/0015000700020.25 00030015/υb 0007us 0003υb /υ00070.143 D 0.02280005us 2 0003us υ0005/0015000700020.25
and:
0003υb /υ0007 D 82.380003us υ0005/0015000700020.875
The ratio of the values obtained using the two approaches to the problem is: 000382.38/76.80007 D 1.073
PROBLEM 12.19 Obtain the Taylor–Prandtl modification of the Reynolds analogy between momentum transfer and mass transfer (equimolecular counterdiffusion) for the turbulent flow of a fluid over a surface. Write down the corresponding analogy for heat transfer. State clearly the assumptions which are made. For turbulent flow over a surface, the film heat transfer coefficient for the fluid is found to be 4 kW/m2 K. What would the corresponding value of the mass transfer coefficient be, given the following physical properties? Diffusivity D D 5 ð 1000029 m2 /s. Thermal conductivity, k D 0.6 W/m K. Specific heat capacity Cp D 4 kJ/kg K. Density, 0005 D 1000 kg/m3 . Viscosity, 0015 D 1 mNs/m2 . Assume that the ratio of the velocity at the edge of the laminar sub-layer to the stream velocity is (a) 0.2, (b) 0.6. Comment on the difference in the two results.
Solution The discussion of the Reynolds analogy is presented in Section 12.8 and consideration of mass transfer in Sections 12.8.2 and 12.8.3 leads to the modified Lewis relation which
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may be written as: hD [1 0002 ˛00031 0002 Sc0007] D 0003h/Cp 00050007[1 0002 ˛00031 0002 Pr0007] (equation 12.121) (i) The Schmidt group, Sc D 0015/0005D D 00031 ð 1000023 0007/00031000 ð 5 ð 1000029 0007 D 200 The Prandtl group, Pr D Cp 0015/k D 00034 ð 103 ð 1 ð 1000023 0007/0.6 D 6.67 and:
0003h/Cp 00050007 D 4000/00034 ð 103 ð 10000007 D 0.001
Thus, in equation (i): hD [1 0002 ˛00031 0002 2000007] D 0.001[1 0002 ˛00031 0002 6.670007] and:
hD D 0.00100031 C 5.67˛0007/00031 C 199˛0007 m/s
(ii)
When ˛ D 0.2, then from equation (ii), hD D 0.00100031 C 1.1340007/00031 C 39.80007 D 5.2 ð 1000025 m/s When ˛ D 0.6, then from equation (ii), hD D 0.00100031 C 3.4020007/00031 C 119.40007 D 3.6 ð 1000025 m/s It is worth noting that even with a very large variation in ˛ (threefold in fact) the change in the mass transfer coefficient is less than 50%.
PROBLEM 12.20 By using the simple Reynolds analogy, obtain the relation between the heat transfer coefficient and the mass transfer coefficient for the gas phase for the absorption of a soluble component from a mixture of gases. If the heat transfer coefficient is 100 W/m2 K, what will the mass transfer coefficient be for a gas of specific heat capacity 1.5 kJ/kg K and density 1.5 kg/m3 ? The concentration of the gas is sufficiently low for bulk flow effects to be negligible.
Solution From Section 12.8.1, the heat transfer coefficient is given by: 0003R/0005u2 0007 D h/0003Cp 0005us 0007
(equation 12.102)
and the mass transfer coefficient by: 0003R/0005u2 0007 D hD /us
(equation 12.103)
Hence:
(equation 12.105)
In this case:
hD D h/0003Cp 00050007 hD D 100/00031.5 ð 103 ð 1.50007 D 0.044 m/s
PROBLEM 12.21 The velocity profile in the neighbourhood of a surface for a Newtonian fluid may be expressed in terms of a dimensionless velocity uC and a dimensionless distance y C from
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313
the surface. Obtain the relation between uC and y C in the laminar sub-layer. Outside the laminar sub-layer, the relation takes the form: uC D 2.5 ln y C C 5.5 At what value of y C does the transition from the laminar sub-layer to the turbulent zone occur? In the Universal Velocity Profile, the laminar sub-layer extends to values of y C D 5, the turbulent zone starts at y C D 30 and the range 5 < y C < 30, the buffer layer, is covered by a second linear relation between uC and ln y C . What is the maximum difference between the values of uC , in the range 5 < y C < 30, using the two methods of representation of the velocity profile? Definitions: uC D
ux uŁ
yC D
yuŁ 0005 0015
uŁ2 D R/0005 where ux is velocity at distance y from surface R is wall shear stress 0005, 0015 are the density and viscosity of the fluid respectively.
Solution Laminar sub-layer If the velocity gradient approaches constant value in the laminar sub-layer, then 2 d ux /dy 2 ! 0 and: ux y 0015ux D 0005y
RD0015 uŁ2 and:
ux uŁ y0005 D uŁ 0015
or: uC D y C , by definition
The point of interaction of uC D y C and:
uC D 2.5 ln y C C 5.5
is given by:
y C D 2.5 ln y C C 5.5
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Evaluating: y C : yC
RHS
10 15 20 25 8 7 12 11 11.5 11.6
11.26 12.27 13.00 13.5 10.7 10.4 11.71 11.5 11.6 11.62
and: y C D 11.6 The equation for buffer zone may be written as: uC D A ln y C C B When y C D 5, uC D 5 When y C D 30, uC D 2.5 ln 30 C 5.5 Thus:
5 D A ln 5 C B 5.5 C 2.5 ln 30 D A ln 30 C B
Substracting:
0.5 C 2.5 ln 30 D A ln 6
Thus: A D 00030.5 C 2.5 ln 300007/ln 6 D 5.02 and: B D 5 0002 A ln 5 D 00023.08 The difference in the two values of uC is a maximum when y C D 11.6. From the two-layer theory: uC D 11.6 From the buffer-layer theory: uC D 5.02 ln C11.6 0002 3.08 D 9.2 The maximum difference in the two values of uC is then: 2.4
PROBLEM 12.22 In the universal velocity profile a “dimensionless” velocity uC is plotted against ln y C , where y C is a “dimensionless” distance from the surface. For the region where eddy transport dominates (eddy kinematic viscosity × kinematic viscosity), the ratio of the mixing length 00030017E 0007 to the distance 0003y0007 from the surface may be taken as approximately constant and equal to 0.4. Obtain an expression for duC /dy C in terms of y C . In the buffer zone the ratio of duC /dy C to y C is twice the value calculated above. Obtain an expression for the eddy kinematic viscosity E in terms of the kinematic viscosity 00030015/00050007 and y C . On the assumption that the eddy thermal diffusivity EH and the eddy kinematic
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viscosity E are equal, calculate the value of the temperature gradient in a liquid flowing over the surface at y C D 15 (which lies within the buffer layer) for a surface heat flux of 1000 W/m2 . The liquid has a Prandtl number of 7 and a thermal conductivity of 0.62 W/m K.
Solution a) In the region where eddy effects considerably exceed molecular contributions: E × 0015/0005 Outside the laminar sub-layer and the buffer-layer but still close to the surface: R D E00050003dux /dy0007 where R is shear stress at surface Writing E as 0017E uE , and approximating, uE D 0017E jdux /dyj gives: 0003 00060003 0006 R dux dux D 00172E 0005 dy dy where the modulus sign is dropped as dux / dy is positive near a surface. p Putting uŁ D R/0005, the shearing stress velocity: 0003 0006 dux 2 2 uŁ D 00172E dy uŁ D 0017E
dux dy
Using the Prandtl approximation 0017E D 0.4 y gives: dux uŁ D 0.4 y dy ∴
dux uŁ D 2.5 dy y
Writing y C D yuŁ 0005/0015 and uC D ux /uŁ , then: uŁ and: b) In the buffer zone: It is given that:
duC uŁ D 2.5 dy C yC duC 2.5 D C dy C y
0003 0006 2.5 5 duC D2 D C C C dy y y
The shear stress is given by: R D 00030015 C E00050007 ∴
2
uŁ D
R D 0005
0003
0015 CE 0005
0006
dux dy dux dy
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and: Hence:
0006 C 0015 du uŁ CE u D 0005 dy C 0015/0005uŁ 0003 0006 C 0003 0006 0015 du 0015 5 0015 D CE CE D C 0005 0005 dy 0005 yC Ł2
0003
When y C D 15, then: 0015/0005 D 00030015/0005 C E0007/3 and:
E D 20015/0005
iii) For heat transfer in buffer zone: q D 00020003k C Cp 0005EH 0007
dT dy
dT dy When y C D 15, then putting E D 20015/0005 gives:
Writing EH D E gives: q D 00020003k C Cp 0005E0007
q D 00020003k C 2Cp 00150007 Thus:
Putting:
dT dT D 0002k00031 C 2Pr0007 dy dy
dT 0002q D dy k00031 C 2Pr0007 k D 0.62 W/m K,
Pr D 7 and 0002q D 1000 W
then: dT/dy D 108 deg K/m or 0.108 deg K/mm
PROBLEM 12.23 Derive an expression relating the pressure drop for the turbulent flow of a fluid in a pipe to the heat transfer coefficient at the walls on the basis of the simple Reynolds analogy. Indicate the assumptions which are made and the conditions under which it would be expected to apply closely. Air at 320 K and atmospheric pressure is flowing through a smooth pipe of 50 mm internal diameter and the pressure drop over a 4 m length is found to be 150 mm water gauge. By how much would the air temperature be expected to fall over the first metre if the wall temperature there is 290 K? Viscosity of air D 0.018 mN s/m2 . Specific heat capacity 0003Cp 0007 D 1.05 kJ/kg K. Molecular volume D 22.4 m3 /kmol at 1 bar and 273 K.
Solution If a mass of fluid, m, situated at a distance from a surface, is moving parallel to the surface with a velocity of us , and it then moves to the surface, where the velocity is zero, it will give up its momentum mus in time t. If the temperature difference between the mass of fluid and the surface is 0s , then the heat transferred to the surface is 0003mCp 0s 0007 and over a surface of area, A: 0003mCp 0s 0007/t D 0002qA where q is the heat transferred from the surface per unit area per unit time.
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If the shear stress at the surface is R0 , the shearing force over area A is the rate of change of momentum or: 0003mus 0007/t D R0 A and Cp 0s /us D q/R0 Writing R0 D 0002R, the shear stress acting on the walls and h as the heat transfer coefficient between the fluid and the surface, then: 0002q/0s D h D 0002R0 Cp /us D RCp /us
or
h/Cp 0005us D R/0005u2
From equation 3.19, the pressure change due to friction is given by: 0002P D 40003R/0005u2 00070003l/d000700030005u2 0007 and substituting from equation 12.102: 0002P D 40003h/Cp 0005u00070003l/d000700030005u2 0007 D 40003hu/Cp 00070003l/d0007 The Reynolds analogy assumes no mixing with adjacent fluid and that turbulence persists right up to the surface. Further it is assumed that thermal and kinematic equilibria are reached when an element of fluid comes into contact with a solid surface. No allowance is made for variations in physical properties of the fluid with temperature. A further discussion of the Reynolds analogy for heat transfer is presented in Chapter 12. Density of air at 320 K D 000329/22.400070003273 ð 3200007 D 1.105 kg/m3 . The pressure drop: 0002P D 150 mm water D 00039.8 ð 1500007 D 1470 N/m2 l D 4.0 m,
d D 0.050 m
In equation 3.23: 0002Pd3 0005/00034l00152 0007 D 00031470 ð 0.0503 ð 1.1050007/[4 ð 4.000030.018 ð 1000023 00072 ] D 3.192 ð 107 From Fig. 3.8, for a smooth pipe: Re D 1.25 ð 105 and from Fig. 3.7: R/0005u2 D 0.0021 The heat transfer coefficient: h D 0003R/0005u2 0007Cp 0005u D 00030.0021 ð 1.05 ð 103 00070005u D 2.2050005u W/m2 K Mass flowrate of air, G D 0005u00030006/400070.0502 D 0.001960005u kg/s Area for heat transfer, A D 00030006 ð 0.050 ð 1.00007 D 0.157 m2 . GCp 0003T1 0002 T2 0007 D hA0003Tm 0002 Tw 0007 where T1 and T2 are the inlet and outlet temperatures and Tm the mean value taken as arithmetic over the small length of 1 m. ∴ 00030.001960005u ð 1.050 ð 103 00070003320 0002 T2 0007 D 00032.2050005u ð 0.157000700030.50003320 ð T2 0007 0002 2900007
and:
T2 D 316 K
The drop in temperature over the first metre is therefore 4 deg K.
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SECTION 13
Humidification and Water Cooling PROBLEM 13.1 In a process in which benzene is used as a solvent, it is evaporated into dry nitrogen. The resulting mixture at a temperature of 297 K and a pressure of 101.3 kN/m2 has a relative humidity of 60%. It is required to recover 80% of the benzene present by cooling to 283 K and compressing to a suitable pressure. What must this pressure be? Vapour pressures of benzene: at 297 K D 12.2 kN/m2 : at 283 K D 6.0 kN/m2 .
Solution See Volume 1, Example 13.1
PROBLEM 13.2 0.6 m3 /s of gas is to be dried from a dew point of 294 K to a dew point of 277.5 K. How much water must be removed and what will be the volume of the gas after drying? Vapour pressure of water at 294 K D 2.5 kN/m2 . Vapour pressure of water at 277.5 K D 0.85 kN/m2 .
Solution When the gas is cooled to 294 K, it will be saturated and Pw0 D 2.5 kN/m2 . From Section 13.2: mass of vapour D Pw0 Mw /RT D 00072.5 ð 18/00078.314 ð 294 D 0.0184 kg/m3 gas. When water has been removed, the gas will be saturated at 277.5 K, and Pw D 0.85 kN/m2 . At this stage, mass of vapour D 00070.85 ð 18/00078.314 ð 277.5 D 0.0066 kg/m3 gas Hence, water to be removed D 00070.0184 0003 0.0066 D 0.0118 kg/m3 gas or:
00070.0118 ð 0.6 D 0.00708 kg/s
Assuming the gas flow, 0.6 m3 /s, is referred to 273 K and 101.3 kN/m2 , 0.00708 kg/s of water is equivalent to 00070.00708/18 D 3.933 ð 1000034 kmol/s. 318
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1 kmol of vapour occupies 22.4 m3 at STP, and:
volume of water removed D 00073.933 ð 1000034 ð 22.4 D 0.00881 m3 /s
Assuming no volume change on mixing, the gas flow after drying D 00070.60 0003 0.00881 D 0.591 m3 /s at STP .
PROBLEM 13.3 Wet material, containing 70% moisture on a wet basis, is to be dried at the rate of 0.15 kg/s in a counter-current dryer to give a product containing 5% moisture (both on a wet basis). The drying medium consists of air heated to 373 K and containing water vapour with a partial pressure of 1.0 kN/m2 . The air leaves the dryer at 313 K and 70% saturated. Calculate how much air will be required to remove the moisture. The vapour pressure of water at 313 K may be taken as 7.4 kN/m2 .
Solution The feed is 0.15 kg/s wet material containing 0.70 kg water/kg feed. Thus water in feed D 00070.15 ð 0.70 D 0.105 kg/s and dry solids D 00070.15 0003 0.105 D 0.045 kg/s. The product contains 0.05 kg water/kg product. Thus, if w kg/s is the amount of water in the product, then: w/0007w C 0.045 D 0.05 or w D 0.00237 kg/s and:
water to be removed D 00070.105 0003 0.00237 D 0.1026 kg/s.
The inlet air is at 373 K and the partial pressure of the water vapour is 1 kN/m2 . Assuming a total pressure of 101.3 kN/m2 , the humidity is: H1 D [Pw /0007P 0003 Pw ]0007Mw /MA
(equation 13.1)
D [1.0/0007101.3 0003 1.0]000718/29 D 0.0062 kg/kg dry air The outlet air is at 313 K and is 70% saturated. Thus, as in Example 13.1, Volume 1: Pw D Pw0 ð RH/100 D 00077.4 ð 70/100 D 5.18 kN/m2 and:
H2 D [5.18/0007101.3 0003 5.18]000718/29 D 0.0335 kg/kg dry air
The increase in humidity is 00070.0335 0003 0.0062 D 0.0273 kg/kg dry air and this must correspond to the water removed, 0.1026 kg/s. Thus if G kg/s is the mass flowrate of dry air, then: 0.0273G D 0.1026 and G D 3.76 kg/s dry air In the inlet air, this is associated with 0.0062 kg water vapour, or: 00070.0062 ð 3.76 D 0.0233 kg/s
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Hence, the mass of moist air required at the inlet conditions D 00073.76 C 0.0233 D 3.783 kg/s
PROBLEM 13.4 30,000 m3 of cool gas (measured at 289 K and 101.3 kN/m2 saturated with water vapour) is compressed to 340 kN/m2 pressure, cooled to 289 K and the condensed water is drained off. Subsequently the pressure is reduced to 170 kN/m3 and the gas is distributed at this pressure and 289 K. What is the percentage humidity after this treatment? The vapour pressure of water at 289 K is 1.8 kN/m2 .
Solution At 289 K and 101.3 kN/m2 , the gas is saturated and Pw0 D 1.8 kN/m2 . Thus from equation 13.2, H0 D [1.8/0007101.3 0003 1.8]000718/MA D 00070.3256/MA kg/kg dry gas, where MA is the molecular mass of the gas. At 289 K and 340 kN/m2 , the gas is in contact with condensed water and therefore still saturated. Thus Pw0 D 1.8 kN/m2 and: H0 D [1.8/0007340 0003 1.8]000718/MA D 00070.0958/MA kg/kg dry gas
At 289 K and 170 kN/m2 , the humidity is the same, and in equation 13.2: 00070.0958/MA D [Pw /0007170 0003 Pw ]000718/MA Pw D 0.90 kN/m2
or: The percentage humidity is then:
D [0007P 0003 Pw0 /0007P 0003 Pw ]0007100Pw /Pw0
(equation 13.3)
D [0007170 0003 1.8/0007170 0003 0.90]0007100 ð 0.90/1.8 D 49.73%
PROBLEM 13.5 A rotary countercurrent dryer is fed with ammonium nitrate containing 5% moisture at the rate of 1.5 kg/s, and discharges the nitrate with 0.2% moisture. The air enters at 405 K and leaves at 355 K; the humidity of the entering air being 0.007 kg moisture/kg dry air. The nitrate enters at 294 K and leaves at 339 K. Neglecting radiation losses, calculate the mass of dry air passing through the dryer and the humidity of the air leaving the dryer. Latent heat of water at 294 K D 2450 kJ/kg. Specific heat capacity of ammonium nitrate D 1.88 kJ/kg K. Specific heat capacity of dry air D 0.99 kJ/kg K. Specific heat capacity of water vapour D 2.01 kJ/kg K.
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Solution The feed rate of wet nitrate is 1.5 kg/s containing 5.0% moisture or 00071.5 ð 5/100 D 0.075 kg/s water. ∴
flow of dry solids D 00071.5 0003 0.075 D 1.425 kg/s If the product contains w kg/s water, then: w/0007w C 1.425 D 00070.2/100
and:
or
w D 0.00286 kg/s
the water evaporated D 00070.075 0003 0.00286 D 0.07215 kg/s
The problem now consists of an enthalpy balance around the unit, and for this purpose a datum temperature of 294 K will be chosen. It will be assumed that the flow of dry air into the unit is m kg/s. Considering the inlet streams:
(i) Nitrate: this enters at the datum of 294 K and hence the enthalpy D 0. (ii) Air: G kg/s of dry air is associated with 0.007 kg moisture/kg dry air. ∴
enthalpy D [0007G ð 0.99 C 00070.007G ð 2.01]0007405 0003 294 D 111.5G kW
and the total heat into the system D 111.5G kW. Considering the outlet streams:
(i) Nitrate: 1.425 kg/s dry nitrate contains 0.00286 kg/s water and leaves the unit at 339 K. ∴
enthalpy D [00071.425 ð 1.88 C 00070.00286 ð 4.18]0007339 0003 294 D 120.7 kW
(ii) Air: the air leaving contains 0.007 G kg/s water from the inlet air plus the water evaporated. It will be assumed that evaporation takes place at 294 K. Thus: enthalpy of dry air D G ð 0.990007355 0003 294 D 60.4G kW enthalpy of water from inlet air D 0.007G ð 2.010007355 0003 294 D 0.86G kW enthalpy in the evaporated water D 0.07215[2450 C 2.010007355 0003 294] D 185.6 kW and the total heat out of the system, neglecting losses D 0007306.3 C 61.3G kW. Making a balance:
111.5G D 0007306.3 C 61.3G
or
G D 6.10 kg/s dry air
Thus, including the moisture in the inlet air, moist air fed to the dryer is: 6.1000071 C 0.007 D 6.15 kg/s
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Water entering with the air D 00076.10 ð 0.007 D 0.0427 kg/s. Water evaporated D 0.07215 kg/s. Water leaving with the air D 00070.0427 C 0.07215 D 0.1149 kg/s Humidity of outlet air D 00070.1149/6.10 D 0.0188 kg/kg dry air.
PROBLEM 13.6 Material is fed to a dryer at the rate of 0.3 kg/s and the moisture removed is 35% of the wet charge. The stock enters and leaves the dryer at 324 K. The air temperature falls from 341 K to 310 K, its humidity rising from 0.01 to 0.02 kg/kg. Calculate the heat loss to the surroundings. Latent heat of water at 324 K D 2430 kJ/kg. Specific heat capacity of dry air D 0.99 kJ/kg K. Specific heat capacity of water vapour D 2.01 kJ/kg K.
Solution The wet feed is 0.3 kg/s and the water removed is 35%, or: 00070.3 ð 35/100 D 0.105 kg/s If the flowrate of dry air is G kg/s, the increase in humidity D 00070.02 0003 0.01 D 0.01 kg/kg or:
0.01G D 0.105
and G D 10.5 kg/s
This completes the mass balance, and the next step is to make an enthalpy balance along the lines of Problem 13.5. As the stock enters and leaves at 324 K, no heat is transferred from the air and the heat lost by the air must represent the heat used for evaporation plus the heat losses, say L kW. Thus heat lost by the inlet air and associated moisture is: [000710.5 ð 0.99 C 00070.01 ð 10.5 ð 2.01]0007341 0003 310 D 328.8 kW Heat leaving in the evaporated water D 0.105[2430 C 2.010007310 0003 324] D 252.2 kW. Making a balance: 328.8 D 0007252.2 C L or
L D 76.6 kW
PROBLEM 13.7 A rotary dryer is fed with sand at the rate of 1 kg/s. The feed is 50% wet and the sand is discharged with 3% moisture. The entering air is at 380 K and has an absolute humidity of 0.007 kg/kg. The wet sand enters at 294 K and leaves at 309 K and the air leaves at 310 K. Calculate the mass flowrate of air passing through the dryer and the humidity of the air leaving the dryer. Allow for a radiation loss of 25 kJ/kg dry air. Latent heat of water at 294 K D 2450 kJ/kg. Specific heat capacity of sand D 0.88 kJ/kg K. Specific heat capacity of dry air D 0.99 kJ/kg k. Specific heat capacity of vapour D 2.01 kg K.
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Solution The feed rate of wet sand is 1 kg/s and it contains 50% moisture or 00071.0 ð 50/100 D 0.50 kg/s water. ∴
flow of dry sand D 00071.0 0003 0.5 D 0.50 kg/s If the dried sand contains w kg/s water, then: w/0007w C 0.50 D 00073.0/100 or w D 0.0155 kg/s
and:
the water evaporated D 00070.50 0003 0.0155 D 0.4845 kg/s.
Assuming a flowrate of G kg/s dry air, then a heat balance may be made based on a datum temperature of 294 K. Inlet streams:
(i) Sand: this enters at 294 K and hence the enthalpy D 0. (ii) Air: G kg/s of dry air is associated with 0.007 kg/kg moisture. ∴
enthalpy D [0007G ð 0.99 C 00070.007G ð 2.01]0007380 0003 294 D 86.4G kW
and:
the total heat into the system D 86.4G kW.
Outlet streams:
(i) Sand: 0.50 kg/s dry sand contains 0.0155 kg/s water and leaves the unit at 309 K. ∴
enthalpy D [00070.5 ð 0.88 C 00070.0155 ð 4.18]0007309 0003 294 D 7.6 kW
(ii) Air: the air leaving contains 0.07 G kg/s water from the inlet air plus the water evaporated. It will be assumed that evaporation takes place at 294 K. Thus: enthalpy of dry air D G ð 0.990007310 0003 294 D 15.8m kW enthalpy of water from inlet air D 0.007G ð 2.010007310 0003 294 D 0.23G kW enthalpy in the evaporated water D 0.4845[2430 C 2.010007310 0003 294] D 1192.9 kW, a total of 000716.03G C 1192.9 kW (iii) Radiation losses D 25 kJ/kg dry air or 25G kW and the total heat out D 000741.03G C 1200.5 kW. Mass balance:
86.4G D 000741.03G C 1200.5 or G D 26.5 kg/s Thus the flow of dry air through the dryer D 26.5 kg/s and the flow of inlet air D 000726.5 ð 1.007 D 26.7 kg/s As in Problem 13.5, water leaving with the air is: 000726.5 ð 0.007 C 0.4845 D 0.67 kg/s and humidity of the outlet air D 00070.67/26.5 D 0.025 kg/kg.
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PROBLEM 13.8 Water is to be cooled in a packed tower from 330 to 295 K by means of air flowing countercurrently. The liquid flows at the rate of 275 cm3 /m2 s and the air at 0.7 m3 /m2 s. The entering air has a temperature of 295 K and a humidity of 20%. Calculate the required height of tower and the condition of the air leaving at the top. The whole of the resistance to heat and mass transfer can be considered as being within the gas phase and the product of the mass transfer coefficient and the transfer surface per unit volume of column 0007hD a may be taken as 0.2 s00031 .
Solution Assuming, the latent heat of water at 273 K D 2495 kJ/kg specific heat capacity of dry air D 1.003 kJ/kg K specific heat capacity of water vapour D 2.006 kJ/kg K then the enthalpy of the inlet air stream is: HG1 D 1.0030007295 0003 273 C H 00072495 C 2.0060007295 0003 273 From Fig. 13.4, when 0013 D 295 K, at 20% humidity, H D 0.003 kg/kg, and: HG1 D 00071.003 ð 22 C 0.00300072495 C 00072.006 ð 22 D 29.68 kJ/kg In the inlet air, the humidity is 0.003 kg/kg dry air or 00070.003/18/00071/29 D 0.005 kmol/kmol dry air. Hence the flow of dry air D 00071 0003 0.0050.70 D 0.697 m3 /m2 s. Density of air at 295 K D 000729/22.40007273/295 D 1.198 kg/m3 . and hence the mass flow of dry air D 00070.697 ð 1.198 D 0.835 kg/m2 s and the mass flow of water D 275 ð 1000036 m3 /m2 s or 0007275 ð 1000036 ð 1000 D 0.275 kg/m2 s. The slope of the operating line, given by equation 13.37 is: LCL /G D 00070.275 ð 4.18/0.835 D 1.38 The coordinates of the bottom of the operating line are: 0013L1 D 295 K and HG1 D 29.7 kJ/kg Hence, on an enthalpy–temperature diagram (Fig. 13a), the operating line of slope 1.38 is drawn through the point (29.7, 295). The top point of the operating line is given by 0013L2 D 330 K, and from Fig. 13a, HG2 D 78.5 kJ/kg. From Figs 13.4 and 13.5 the curve representing the enthalpy of saturated air as a function of temperature is obtained and drawn in. This plot may also be obtained by calculation using equation 13.60. The integral: 0001 dHG /0007Hf 0003 HG
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Curve for saturated air (Hf vs qf)
Enthalpy (HG kJ/kg)
300
250
200
150
HGvs qG
100
(qL 2'HG 2 )
50 (qL1'HG1) 295
300
Operating line (HGvsqL) 305
310
315
320
325
330
Temperature (q K)
Figure 13a.
is now evaluated between the limits HG1 D 29.68 kJ/kg and HG2 D 78.5 kJ/kg, as follows: HG
0013
Hf
0007Hf 0003 HG
1/0007Hf 0003 HG
29.7 40 50 60 70 78.5
295 302 309 316 323 330
65 98 137 190 265 408
35.3 58 87 130 195 329.5
0.0283 0.0173 0.0115 0.0077 0.0051 0.0030
From a plot of 1/0007Hf 0003 HG and HG the area under the curve is 0.573. Thus: 0001 HG2 height of packing, z D [dHG /0007Hf 0003 HG ]G/hD a0017 (equation 13.53) HG1
D 00070.573 ð 0.835/00070.2 ð 1.198 D 1.997, say 2.0 m In Fig. 13a, a plot of HG and 0013G is obtained using the construction given in Section 13.6.3. and shown in Fig. 13.16. From this plot, the value of 0013G2 corresponding
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to HG2 D 78.5 kJ/kg is 300 K. From Fig. 13.5 the exit air therefore has a humidity of 0.02 kg/kg which from Fig. 13.4 corresponds to a percentage humidity of 90%.
PROBLEM 13.9 Water is to be cooled in a small packed column from 330 to 285 K by means of air flowing countercurrently. The rate of flow of liquid is 1400 cm3 /m2 s and the flowrate of the air, which enters at 295 K with a humidity of 60% is 3.0 m3 /m2 s. Calculate the required height of tower if the whole of the resistance to heat and mass transfer can be considered as being in the gas phase and the product of the mass transfer coefficient and the transfer surface per unit volume of column is 2 s00031 . What is the condition of the air which leaves at the top?
Solution As in Problem 13.8, assuming the relevant latent and specific heat capacities: HG1 D 1.0030007295 0003 273 C H 00072495 C 2.0060007295 0003 273 From Fig. 13.4, at 0013 D 295 and 60% humidity, H D 0.010 kg/kg and hence: HG1 D 00071.003 ð 22 C 0.01000072495 C 44.13 D 47.46 kJ/kg In the inlet air, water vapour D 0.010 kg/kg dry air or 00070.010/18/00071/29 D 0.016 kmol/kmol dry air. Thus the flow of dry air D 00071 0003 0.0163.0 D 2.952 m3 /m2 s. Density of air at 295 K D 000729/22.40007273/293 D 1.198 kg/m3 . and mass flow of dry air D 00071.198 ð 2.952 D 3.537 kg/m2 s. Liquid flow D 1.4 ð 1000033 m3 /m2 s and mass flow of liquid D 00071.4 ð 1000033 ð 1000 D 1.4 kg/m2 s. The slope of the operating line is thus: LCL /G D 00071.40 ð 4.18/3.537 D 1.66 and the coordinates of the bottom of the line are: 0013L1 D 285 K,
HG1 D 47.46 kJ/kg
From these data, the operating line may be drawn in as shown in Fig. 13b and the top point of the operating line is: 0013L2 D 330 K,
HG2 D 122 kJ/kg
Again as in Problem 13.8, the relation between enthalpy and temperature at the interface Hf vs. 0013f is drawn in Fig. 13b. It is seen that the operating line cuts the saturation curve, which is clearly an impossible situation and, indeed, it is not possible to cool the water to 285 K under these conditions. As discussed in Section 13.6.1, with mechanical draught towers, it is possible, at the best, to cool the water to within, say, 1 deg K of the wet
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327
350
Enthalpy (HG kJ/kg)
300 250 200 150 (qL2'HG2)
100 50 (qL1'HG1)
0 280 290 300 310 320 330 Temperature (q K )
Figure 13b.
bulb temperature. From Fig. 13.4, at 295 K and 60% humidity, the wet-bulb temperature of the inlet air is 290 K and at the best water might be cooled to 291 K. In the present case, therefore, 291 K will be chosen as the water outlet temperature. Thus an operating line of slope: LCL /G D 1.66 and bottom coordinates: 0013L1 D 291 K and HG1 D 47.5 kJ/kg is drawn as shown in Fig. 13c. At the top of the operating line: 0013L2 D 330 K and HG2 D 112.5 kJ/kg As an alternative to the method used in Problem 13.8, the approximate method of Carey and Williamson (equation 13.54) is adopted. At the bottom of the column: HG1 D 47.5 kJ/kg,
Hf1 D 52.0 kJ/kg
∴ H1 D 4.5 kJ/kg
Hf2 D 382 kJ/kg
∴ H2 D 269.5 kJ/kg
At the top of the column: HG2 D 112.5 kJ/kg,
At the mean water temperature of 0.50007330 C 291 D 310.5 K: HGm D 82.0 kJ/kg, ∴
Hfm D 152.5 kJ/kg
∴ Hm D 70.5 kJ/kg
Hm /H1 D 15.70 and Hm /H2 D 0.262
and from Fig. 13.17: f D 0.35 (extending the scales)
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350
Enthaolpy (HG kJ/kg)
300 250 200 150 100 50
0 280 290 300 310 320 330 Temperature (q k)
Figure 13c.
Thus: 0001
HG2
[dHG /0007Hf 0003 HG ]G/hD a0017
height of packing, z D
(equation 13.53)
HG1
D 00070.35 ð 3.537/00072.0 ð 1.198 D 0.52 m Due to the close proximity of the operating line to the line of saturation, the gas will be saturated on leaving the column and will therefore be at 100% humidity. From Fig. 13c the exit gas will be at 306 K.
PROBLEM 13.10 Air containing 0.005 kg water vapour/kg dry air is heated to 325 K in a dryer and passed to the lower shelves. It leaves these shelves at 60% humidity and is reheated to 325 K and passed over another set of shelves, again leaving with 60% humidity. This is again reheated for the third and fourth sets of shelves after which the air leaves the dryer. On the assumption that the material in each shelf has reached the wet bulb temperature and that heat losses from the dryer can be neglected, determine: (a) the temperature of the material on each tray, (b) the rate of water removal if 5 m3 /s of moist air leaves the dryer,
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(c) the temperature to which the inlet air would have to be raised to carry out the drying in a single stage.
Solution See Volume 1, Example 13.4
PROBLEM 13.11 0.08 m3 /s of air at 305 K and 60% humidity is to be cooled to 275 K. Calculate, using a psychrometric chart, the amount of heat to be removed for each 10 deg K interval of the cooling process. What total mass of moisture will be deposited? What is the humid heat of the air at the beginning and end of the process?
Solution At 305 K and 60% humidity, from Fig. 13.4, the wet-bulb temperature is 299 K and H D 0.018 kg/kg. Thus, as the air is cooled, the per cent humidity will increase until saturation occurs at 299 K and the problem is then one of cooling saturated vapour from 299 K to 275 K. Considering the cooling in 10 deg K increments, the following data are obtained from Fig. 13.4: 0013 (K)
0013w (K)
% Humidity
H
Humid heat (kJ/kg K)
Latent heat (kJ/kg)
305 299 295 285 275
299 299 295 285 275
60 100 100 100 100
0.018 0.018 0.017 0.009 0.0045
1.032 1.032 1.026 1.014 1.001
2422 2435 2445 2468 2491
At 305 K:
the specific volume of dry air D 0.861 m3 /kg the saturated volume D 0.908 m3 /kg
and hence the specific volume at 60% humidity D [0.861 C 00070.908 0003 0.86160/100] D 0.889 m3 /kg Thus:
mass flow of moist air D 00070.08/0.889 D 0.090 kg/s
Thus the flowrate of dry air D 0.090/00071 C 0.018 D 0.0884 kg/s. From Fig. 13.4, specific heat of dry air (at H D 0 D 0.995 kJ/kg K. ∴
enthalpy of moist air D 00070.0884 ð 0.9950007299 0003 273 C 00070.018 ð 0.0884 ð[4.180007299 0003 273 C 2435] C 0.090 ð 1.0320007305 0003 299 D 6.89 kW
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At 295 K: Enthalpy of moist air D 00070.0884 ð 0.9950007295 0003 273 C 00070.017 ð 0.0884 ð [4.180007295 0003 273 C 2445] D 5.75 kW At 285 K: Enthalpy of moist air D 00070.0884 ð 0.9950007285 0003 273 C 00070.009 ð 0.0884 ð [4.180007285 0003 273 C 2468] D 3.06 kW At 275 K: Enthalpy of moist air D 00070.0884 ð 0.9950007275 0003 273 C 00070.0045 ð 0.0884 ð [4.180007275 0003 273 C 2491] D 1.17 kW and hence in cooling from 305 to 295 K, heat to be removed D 00076.89 0003 5.75 D 1.14 kW in cooling from 295 to 285 K, heat to be removed D 00075.75 0003 3.06 D 2.69 kW in cooling from 285 to 275 K, heat to be removed D 00073.06 0003 1.17 D 1.89 kW The mass of water condensed D 0.088400070.018 0003 0.0045 D 0.0012 kg/s. The humid heats at the beginning and end of the process are: 1.082 and 1.001 kJ/kg K respectively.
PROBLEM 13.12 A hydrogen stream at 300 K and atmospheric pressure has a dew point of 275 K. It is to be further humidified by adding to it (through a nozzle) saturated steam at 240 kN/m2 at the rate of 1 kg steam: 30 kg of hydrogen feed. What will be the temperature and humidity of the resultant stream?
Solution At 275 K, the vapour pressure of water D 0.72 kN/m2 (from Tables) and the hydrogen is saturated. The mass of water vapour: Pw0 Mw /RT D 00070.72 ð 18/00078.314 ð 275 D 0.00567kg/m3 and the mass of hydrogen: 0007P 0003 Pw0 MA /RT D 0007101.3 0003 0.722/00078.314 ð 275 D 0.0880 kg/m3 Therefore the humidity at saturation, H0 D 00070.00567/0.0880 D 0.0644 kg/kg dry hydrogen and at 300 K, the humidity will be the same, H1 D 0.0644 kg/kg. At 240 kN/m2 pressure, steam is saturated at 400 K at which temperature the latent heat is 2185 kJ/kg. The enthalpy of the steam is therefore: H2 D 4.180007400 0003 273 C 2185 D 2715.9 kJ/kg Taking the mean specific heat capacity of hydrogen as 14.6 kJ/kg K, the enthalpy in 30 kg moist hydrogen or 30/00071 C 0.0644 D 28.18 kg dry hydrogen is: 000728.18 ð 14.60007300 0003 273 D 11,110 kJ
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The latent heat of water at 275 K is 2490 kJ/kg and, taking the specific heat of water vapour as 2.01 kJ/kg K, the enthalpy of the water vapour is: 000728.18 ð 0.064400074.180007275 0003 273 C 2490 C 2.010007300 0003 275 D 4625 kJ Hence the total enthalpy:
H1 D 15,730 kJ
In mixing the two streams, 28.18 kg dry hydrogen plus 000730 0003 28.18 D 1.82 kg water is mixed with 1 kg steam and hence the final humidity: H D 00071 C 1.82/28.18 D 0.100 kg/kg
In the final mixture, 0.1 kg water vapour is associated with 1 kg dry hydrogen or 00070.1/18 D 0.0056 kmol water is associated with 00071/2 D 0.5 kmol hydrogen, a total of 0.5056 kmol. ∴
partial pressure of water vapour D 00070.0056/0.5056101.3 D 1.11 kN/m2
Water has a vapour pressure of 1.11 kN/m2 at 281 K at which the latent heat is 2477 kJ/kg. Thus if T K is the temperature of the mixture, then: 00072716 C 15730 D 000728.18 ð 14.60007T 0003 273 C 2.82[4.180007281 0003 273 C 2447 C 2.010007T 0003 281] and T D 300.5 K It may be noted that this relatively low increase in temperature occurs because the latent heat in the steam is not recovered, as would be the case in, say, a shell and tube unit.
PROBLEM 13.13 In a countercurrent packed column, n-butanol flows down at the rate of 0.25 kg/m2 s and is cooled from 330 to 295 K. Air at 290 K, initially free of n-butanol vapour, is passed up the column at the rate of 0.7 m3 /m2 s. Calculate the required height of tower and the condition of the exit air. Data: Mass transfer coefficient per unit volume, hD a D 0.1 s00031 . Psychrometric ratio, 0007h/hD 0017A s D 2.34. Heat transfer coefficients, hL D 3hG . Latent heat of vaporisation of n-butanol, 001b D 590 kJ/kg. Specific heat capacity of liquid n-butanol, CL D 2.5 kJ/kg K. Humid heat of gas: s D 1.05 kJ/kg K. Temperature (K)
Vapour pressure of n-butanol 0007kN/m2
295 300 305 310 315 320 325
0.59 0.86 1.27 1.75 2.48 3.32 4.49
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Temperature (K)
Vapour pressure of n-butanol 0007kN/m2
330 335 340 345 350
5.99 7.89 10.36 14.97 17.50
Solution See Volume 1, Example 13.10
PROBLEM 13.14 Estimate the height and base diameter of a natural draught hyperbolic cooling tower which will handle 5000 kg/s water entering at 300 K and leaving at 294 K. The dry-bulb air temperature is 287 K and the ambient wet-bulb temperature is 284 K.
Solution See Volume 1, Example 13.8
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