CHEMICAL ENGINEERING

Solutions to the Problems in Chemical Engineering Volume 1

Coulson & Richardson’s Chemical Engineering

Chemical Engineering, Volume 1, Sixth editionFluid Flow, Heat Transfer and Mass TransferJ. M. Coulson and J. F. Richardsonwith J. R. Backhurst and J. H. Harker

Chemical Engineering, Volume 2, Fourth editionParticle Technology and Separation ProcessesJ. M. Coulson and J. F. Richardsonwith J. R. Backhurst and J. H. Harker

Chemical Engineering, Volume 3, Third editionChemical & Biochemical Reactors & Process ControlEdited by J. F. Richardson and D. G. Peacock

Solutions to the Problems in Volume 1, First editionJ. R. Backhurst and J. H. Harkerwith J. F. Richardson

Chemical Engineering, Volume 5, Second editionSolutions to the Problems in Volumes 2 and 3J. R. Backhurst and J. H. Harker

Chemical Engineering, Volume 6, Third editionChemical Engineering DesignR. K. Sinnott

Coulson & Richardson’s

CHEMICAL ENGINEERINGJ. 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

Butterworth-HeinemannLinacre House, Jordan Hill, Oxford OX2 8DP225 Wildwood Avenue, Woburn, MA 01801-2041A 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 publicationmay be reproduced in any material form (includingphotocopying or storing in any medium by electronicmeans and whether or not transiently or incidentallyto some other use of this publication) without thewritten permission of the copyright holder exceptin accordance with the provisions of the Copyright,Designs and Patents Act 1988 or under the terms of alicence issued by the Copyright Licensing Agency Ltd,90 Tottenham Court Road, London, England W1P 9HE.Applications for the copyright holder’s written permissionto reproduce any part of this publication should be addressedto 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

Contents

Preface iv

1. Units and dimensions 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

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 incor-

porates 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.

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 ⊲�/4⊳0.0252D 0.00049 m2.

Mean velocity of acid, u D ⊲685 ð 10�6⊳/0.00049 D 1.398 m/s.

∴ Reynolds number, Re D du�/� D ⊲0.025 ð 1.398 ð 1840⊳/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 ⊲1000 J/s⊳⊲3600 s⊳ 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 ⊲1/3.6⊳ D 0.28 p/MJ

cost of gas D 15 p/105.5 MJ or ⊲15/105.5⊳ 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

2 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

Solution

From the steam tables, in Appendix A2, Volume 1, total enthalpy of steam at 1825 kN/m2D

2798 kJ/kg.

∴ enthalpy of steam D ⊲5.2 ð 2798⊳ 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 ⊲14,550 ð 100⊳/75 D 19,400 kW.

For a calorific value of 27,200 kJ/kg, rate of coal consumption D ⊲19,400/27,200⊳

D 0.713 kg/s

or: ⊲0.713 ð 3600 ð 24⊳/1000 D 61.6 Mg/day

20% of the enthalpy in the steam is converted to power or:

⊲14,550 ð 20⊳/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) diameter of impeller,

(b) number of rotations of the impeller per unit time,

(c) viscosity of liquid,

(d) 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 f⊲DN��⊳, then a typical form of the function is P D kDaNb�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 T�1

Equating dimensions:

M : 1 D c C d

L : 2 D a � 3c � d

T : �3 D �b � d

Solving in terms of d : a D ⊲5 � 2d⊳, b D ⊲3 � d⊳, c D ⊲1 � d⊳

∴ P D k

(

D5

D2d

N3

Nd

�

�d�d

)

or: P/D5N3� D k⊲D2N�/�⊳�d

that is: NP D k Rem

UNITS AND DIMENSIONS 3

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/D5N3� D k⊲D2N�/�⊳m

Experimentally: P / N2

From the equation, P / NmN3, that is m C 3 D 2 and m D �1

Thus for the same fluid, that is the same viscosity and density:

⊲P2/P1⊳⊲D51N3

1/D52N3

2⊳ D ⊲D21N1/D2

2N2⊳�1 or: ⊲P2/P1⊳ D ⊲N22D3

2⊳/⊲N21D3

1⊳

In this case, N1 D N2 and D2 D 2D1.

∴ ⊲P2/P1⊳ D 8D31/D3

1 D 8

A similar solution may be obtained using the Recurring Set method as follows:

P D f⊲D, N, �, �⊳, f⊲P, D, N, �, �⊳ D 0

Using M, L and T as fundamentals, there are five variables and three fundamentals

and therefore by Buckingham’s � theorem, there will be two dimensionless groups.

Choosing D, N and � as the recurring set, dimensionally:

D � L

N � T�1

� � ML�3

]

Thus:

[

L � D

T � N�1

M � �L3D �D3

First group, �1, is P⊲ML2T�3⊳�1� P⊲�D3D2N3⊳�1

�P

�D5N3

Second group, �2, is �⊲ML�1T�1⊳�1� �⊲�D3D�1N⊳�1

��

�D2N

Thus: f

(

P

�D5N3,

�

�D2N

)

D 0

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 � 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

4 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

If: u0 D kdaWb�c�d, then working in dimensions of M, L and T:

⊲L/T⊳ D k⊲La⊲ML/T2⊳b⊲M/L3⊳c⊲M/LT⊳d⊳

Equating dimensions:

M : 0 D b C c C d

L : 1 D a C b � 3c � d

T : �1 D �2b � d

Solving in terms of b:

a D �1, c D ⊲b � 1⊳, and d D ⊲1 � 2b⊳

∴ u0 D k⊲1/d⊳⊲Wb⊳⊲�b/�⊳⊲�/�2b⊳ where k is a constant,

or: u0 D k⊲�/d�⊳⊲W�/�2⊳b

Rearranging:

⊲du0�/�⊳ D k⊲W�/�2⊳b

where (W�/�2) 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,