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1

I

I

Mathematical

Methods

in

Chemical Engineering

Digitized by tlie Internet Archive in

2011

littp://www.arcliive.org/details/matliematicalmetliOOjens

Mathematica

Methods

in

Chemical Engineering V. G.

JENSON

Lecturer

Department of Chemical Engineering Tfie University, Birmingliam, England

and G. V.

JEFFREYS

Senior Lecturer

Department of Chemical Engineering Faculty of Technology

The University, Manchester, England

1963

ACADEMIC PRESS LONDON

and

NEW YORK

ACADEMIC PRESS

INC.

(LONDON) LTD

BERKELEY SQUARE HOUSE BERKELEY SQUARE

LONDON, W.l

U.S. Edition published by

ACADEMIC PRESS 1 1 1

INC.

FIFTH AVENUE

NEW YORK, NEW YORK

Copyright

©

10003

1963 by Academic Press Inc. (London) Ltd

Third printing 1966

All Rights Reserved

No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the pubhshers

Library of Congress Catalog Number: 63-22092

Printed in U.S.A.

PREFACE The development of new more complex and

is becoming and development

processes in the chemical industry

increasingly expensive.

If the research

of the process can be carried out with confidence, the ultimate design will be exact, and therefore the plant will operate more economically. In all facets of such a project, mathematics, which is the language of the quantitative, plays a vital role. Therefore training in mathematical methods is of the utmost importance to chemical engineers. The present text has been written with these ideas in mind, and we would emphasize that our aim is to present mathematics in a form suitable for the engineer rather than to teach engineering to mathematicians. To the pure mathematician an elegant proof is an end in itself, but to the chemical engineer it is merely a means to an end. Consequently this book only sketches demonstrations of the validity of theorems, to encourage the reader to have more confidence in the use of the technique for the solution of engineering problems. In addition, an attempt has been made to sort out the useful from the trivial and flamboyant of the wide variety of mathematical techniques available. The material presented in this book is based on various undergraduate and post graduate courses given in the Chemical Engineering Department of the University of Birmingham. Many of the worked examples have been selected from research work carried out in the department, and these are supplemented with problems taken from the chemical engineering literature. Some chapters of the book (notably Chapters 4 and 5) are almost entirely mathematical, but wherever possible the text has been illustrated with chemical engineering applications.

more

The book was written when both authors were lecturers in the Chemical Engineering Department of the University of Birmingham. It is hoped that the text will encourage chemical engineers to make greater use of mathematics in the solution of their problems. We wish to express our thanks to Professor J. T. Davies for initiating the venture, and to Professor F. H. Garner, and Professor S. R. M. Ellis for their interest and advice in the preparation of the text. Birmingham July,

1963

V.G.J, and G.V.J.

CONTENTS PREFACE Chapter

I

THE MATHEMATICAL STATEMENT OF THE PROBLEM Introduction Representation OF THE Problem Solvent Extraction in Two Stages Solvent Extraction IN A^ Stages Simple Water Still WITH Preheated Feed Unsteady State Operation Salt Accumulation in a Stirred Tank Radial Heat Transfer through a Cylindrical

1.1

1.2 1.3

1.4

.

1.5

1.6 1.7

.... .... .

1.8

Conductor

1 1

3

4 6 8 11

14 16

1.10

Heating A Closed Kettle Dependent and Independent Variables, Parameters

1.11

Boundary Conditions

18

1.12

Sign Conventions Summary OF the Method OF Formulation

21

1.9

1.13

17

19

Chapter 2

ORDINARY DIFFERENTIAL EQUATIONS Introduction Order and Degree

2.1

2.2 2.3

2.4 2.5

2.6 2.7

Order Differential Equations Second Order Differential Equations Linear Differential Equations Simultaneous Differential Equations Conclusions First

....

23 23 24 33 41

66 72

Chapter 3

SOLUTION BY

SERIES

3.1

Introduction

3.2

Infinite Series

3.3

Power

3.4 3.5

Simple Series Solutions Method of Frobenius

3.6

Bessel's

3.7

Properties of Bessel Functions

Series

Equation

.

.

74 74 79 86 90 106 113

5

CONTENTS

Vlll

Chapter 4

COMPLEX ALGEBRA 4.1

Introduction

4.2

The Complex Number The Arg AND Diagram

4.3

4.6

Principal Values Algebraic Operations on the Argand Diagram Conjugate Numbers

4.7

De Moivre's Theorem

4.4 4.5

4.8

4.9

117 117 118 119 .

123

124 125 126

The mu Roots of Unity Complex Number Series

—

Trigonometrical Exponential Identities The Complex Variable 4.12 Derivatives OF A Complex Variable 4.13 Analytic Functions 4.10

.

.

.

.

.

.128 128

4.11

.130 131

4.14 Singularities

132

Integration of Functions of Complex Variables, and Cauchy's Theorem 4.16 Laurent's Expansion and the Theory of Residues 4.

.120

1

.

137

.142

Chapter 5

FUNCTIONS AND DEFINITE INTEGRALS 5.1

5.2 5.3

5.4 5.5

5.6

Introduction The Error Function The Gamma Function The Beta Function Other Tabulated Functions which are Defined by Integrals Evaluation of Definite Integrals .

.

.

149 149 151

154 157

.159

Chapter 6

THE LAPLACE TRANSFORMATION 6.1

Introduction

163

6.2

The Laplace Transform The Inverse Transformation

163

6.3

6.4 6.5

167

Properties of the Laplace Transformation The Step Functions

.

174 179

6.8

Convolution Further Elementary Methods of Inversion Inversion of the Laplace Transform by Contour

6.9

Integration Application of the Laplace Transform to Automatic

6.6

6.7

.

Control Theory

.170

.

.180 182 188

contents

IX

Chapter 7

VECTOR ANALYSIS

7.7

Introduction Tensors Addition and Subtraction of Vectors Multiplication of Vectors Differentiation of Vectors Hamilton's Operator, Integration of Vectors and Scalars

7.8

Standard

7.1

7.2 7.3

7.4 7.5

7.6

.

.

.

V

Identities

Curvilinear Coordinate Systems 7.10 The Equations of Fluid Flow 7.11 Transport of Heat, Mass, and Momentum 7.9

199

200 203 210 216 218 222 227 228

.

231

236

Chapter 8

PARTIAL DIFFERENTIATION

AND

PARTIAL DIFFERENTIAL EQUATIONS

....

8.1

Introduction

8.2

Interpretation of Partial Derivatives Formulating Partial Differential Equations Boundary Conditions Particular Solutions of Partial Differential Equations Orthogonal Functions Method of Separation of Variables

8.3

8.4 8.5

8.6

8.7

8.9

The Laplace Transform Method Other Transforms

8.10

Conclusions

8.8

.

.

....

.

.

.

.

.

238 239 245 252 259 269 272 290 302 306

Chapter 9 FINITE DIFFERENCES 9.1

Introduction

9.2

The Difference Operator, Other Difference Operators.

9.3

9.4 9.5

9.6 9.7 9.8

A

Interpolation Finite Difference Equations Linear Finite Difference Equations Non-Linear Finite Difference Equations Differential-Difference Equations .

307 307 311

315 321 322 331

338

1

X

CONTENTS

Chapter 10

TREATMENT OF EXPERIMENTAL RESULTS Introduction Graph Paper Theoretical Properties Contour Plots Propagation of Errors

10.1

10.2 10.3

10.4

.

10.5

Curve Fitting Numerical Integration

10.6 10.7

Chapter

349 349 354 355 356 360 369

1

NUMERICAL METHODS 11.1

Introduction

11.2

First

11.3

Order Ordinary Differential Equations Higher Order Differential Equations (Initial Value .

.

Type) 1 1

.4

11.5

11.6 11.7

380 380 385

Higher Order Differential Equations (Boundary Value Type) Algebraic Equations Difference-Differential Equations Partial Differential Equations

....

388 397 406 409

Chapter 12

MATRICES 12.1

Introduction

12.2

The Matrix Matrix Algebra

12.3

12.4 12.5

12.6 12.7 12.8

12.9

Determinants of Square Matrices and Matrix Products The Transpose of a Matrix Adjoint Matrices Reciprocal of a Square Matrix

.... .

The Rank and Degeneracy of a Matrix The Sub-matrix

12.10 Solution OF Linear 12.11

437 438 439

Matrix

Algebraic Equations

Series

12.12 Differentiation

and Integration of Matrices

Lambda-Matrices 12.14 The Characteristic Equation 12.15 Sylvester's Theorem 12.16 Transformation OF Matrices 12.17 Quadratic Form 12.13

.

....

443 443 444 444 446 448 448 449 451 452 454 457 459 461

contents

xi

Application to the Solution of Differential Equations 12.19 Solutions of Systems of Linear Differential Equations 12.20 Conclusions 12.18

.

.

.

.

.

.

463 .

.

465 472

Chapter 13

OPTIMIZATION 13.1

13.2 13.3

13.4 13.5

13.6

Introduction Types of Optimization Analytical Procedures

The Method of Steepest Ascent The Sequential Simplex Method Dynamic Programming

.

.

.

.

.

.

.

.

473 474 475 483 485 486

Chapter 14

COMPUTERS 14.1

Introduction

14.2

Analogue Computers Active Analogue Computers

14.3

14.4 14.5

Passive

Digital Computers Comparison of the Uses of Analogue and Digital

Computers

492 493 496 505 509

PROBLEMS

511

APPENDIX

532

SUBJECT INDEX

.543

Chapter

I

THE MATHEMATICAL STATEMENT OF THE PROBLEM 1.1.

Introduction

consists of performing experiments and This may be done quantitatively by taking accurate measurements of the system variables which are subsequently analysed and correlated, or qualitatively by investigating the general behaviour of the system in terms of one variable influencing another. The first method is always desirable, and if a quantitative investigation is to be attempted it is

Nearly

all

applied

science

interpreting the results.

better to introduce the mathematical principles at the earliest possible stage,

may influence the course of the investigation. This is done by looking for an idealized mathematical model of the system. The second step is the collection of all relevant physical information in the form of conservation laws and rate equations. The conservation laws of chemical engineering are material balances, heat balances, and other energy balances; whilst the rate equations express the relationship between flow rate and driving force in the fields of fluid flow, heat transfer, and diffusion of matter. These are then appUed to the model, and the result should be a mathematical equation which describes the system. The type of equation (algebraic, diff*erential, finite difference, etc.) will depend upon both the system under investigation, and the detail of its model. For a particular system, if the model is simple the equation may be elementary, whereas if the model is more refined the equation will be a more difficult type. The appropriate mathematical techniques are then appHed to this equation and a result is obtained. This mathematical result must then be interpreted through the original model in order to give it a physical signisince they

ficance.

In this chapter, only the simplest problems will be considered and the

The more complicated models be introduced throughout the book in the chapters dealing with the particular mathematical techniques which are needed for the completion of

ideas of simple models will be introduced. will

the solutions. 1.2.

Representation of the Problem

A

simple example of the application of these ideas will be given first. The apparatus shown diagrammatically in Fig. 1.1 is to be used for the continuous extraction of benzoic acid from toluene, using water as the extracting solvent.

The two streams and the mixture

vigorously,

are fed into a tank is

then

pumped

A

where they are stirred B where it is allowed

into tank

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

2

to settle into

are

removed

two

layers.

separately,

The upper toluene layer and the lower water layer and the problem is to find what proportion of the

benzoic acid has passed into the solvent phase. Benzene + benzoic ac

Water

q OO

Benzene + benzoic acid

Water + benzoic acid

Single-stage mixer settler

Fig. 1.1.

The problem is idealized in Fig. 1.2 where the two tanks have been combined into a single stage. The various streams have been labelled and two material balances have already been used, (a) conservation of toluene, and (b) conservation of water. These flow rates have been expressed on a solute R ft/min toluene

—

R ft^nnin toluene

>—

X Ib/ft^benzoic acid

c Ib/ft^ benzoic acid S ft/min

water S ft/min

water

y Ib/ft-^ benzoic acid

Fig. 1.2.

Idealized single-stage solvent extraction

free basis to simplify the analysis. The concentration of benzoic acid in each stream has also been stated and this completes the mathematical model. So far, it has been assumed that all flow rates are steady, and that toluene and water are immiscible. Further assumptions are now made that the feed concentration c remains constant, and that the mixer is so efficient that the two streams leaving the stage are always in equilibrium with one another. This last fact can be expressed mathematically by

y where

m

is

= mx

(1.1)

the distribution coefficient.

The equation

is

now

derived from the model by writing

down

a

mass

balance for benzoic acid.

Input of benzoic acid

Output of benzoic acid

= Re Ib/min = i?x -f 5y Ib/min

Since benzoic acid must leave at the same rate as

Rc = The

pair of equations (1.1)

it

enters,

Kx^Sy

and

(1.2)

contain four

(1.2)

known

quantities

1.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

5

c, m) and two unknown quantities (x, y), and they can be solved unknowns as follows. Re = Rx-\-mSx

(R, S,

the

X

Re

=

V

,

R + mS'

^

= mRc R + mS

Therefore, the proportion of benzoic acid extracted '-l

(1.3) ^

^

is

-^^

=

(1.4)

R + mS = 12R, m =

Re

for

if 5" 1/8, and c = 0-1; then and the proportion of acid extracted is 60%. At this stage, it can be seen that even in this simple problem, two dimensionless groups which are characteristic of the problem, have arisen quite

As

X

=

a numerical example,

0-04,

y

=

0-005,

naturally as a result of the investigation. (x

and

E=

Putting

= RlmS

(1.5)

Sy/Rc, equation (1.4) becomes

£= That

is, the proportion extracted dimensionless group a.

1.3.

l/(a

+ l)

(1.6)

governed solely by the value of the

is

Solvent Extraction in

The above example

Two

Stages

now

be reconsidered, but two stages will be used for the extraction of benzoic acid instead of one stage. Each stage still consists of two tanks, a mixer and a settler, with counter-current flow through the stages. The idealized flow system is shown in Fig. 1.3 where the symbols will

R

R c

Stage S

^1

Stage 2

S

1

S

*o

y-,

Fig.

^2

Idealized two-stage extraction

1.3.

have the same meaning as in the previous example, and the diff'erent concentrations in a particular phase are distinguished by suffixes. In accordance with chemical engineering practice, the suffix denotes the number of the stage from which the stream is leaving. The assumptions which were made above are made again, and equation (1.1) is still vahd for each stage separately, giving

yi

A

= mxi

benzoic acid mass balance

is

now

y2

Output of acid (Ib/min)

(1.7)

taken for each stage.

Stage Input of acid (Ib/min)

= mx2 1

Rc + Sy2 Rx^

+ Sy^

Stage 2

Rx^ Rx2 + Sy2

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

4

The

fact that benzoic acid

gives the

must enter and leave a stage

at the

same

rate,

two equations

Rc + Sy2 = Rx^-hSy^ Rx^ Using equations

(1.7) to

= Rx2 + Sy2

ehminate x^ and y2, the inter-stage concentrations,

Rc + mSx2 = (Ryilm) + Syi

Ryjm = Rx2 + mSx2

and Eliminating

>^i

between equations

R{Rc + mSx2) R^c

.-.

(1.8)

and

(1.8)

(1.9)

(1.9),

= (R + mS) (Rx2 + mSx2) = X2(R^+mRS + m^S^)

""^^

>^l=

R^ + mlLm^S^

mRc(R + mS) n2^^pe, ^2e2 R^ + mRS + m^S

Again, the proportion of benzoic acid extracted

^'-''^

(1-11)

is

_ mS(R + mS) Re ~ R^ + mRS + m^S^

Sy^

Introducing the dimensionless groups equation (1.12) becomes a+ a^

E

1

+a+1

(1.12)

and a again from equation

(1.5),

a^-l a^-1

Using the same numerical example as before, i.e. S = 12R, m = 1/8, and c = 0-1 then X2 = 0-021, ^^^ = 0-0066, and the proportion extracted is 79%. A greater degree of extraction has thus been obtained with two stages than with one stage, everything else being the same. ;

1.4.

Solvent Extraction in

N

Stages

This improved extraction leads to the consideration of more than two stages in the extraction system. The algebraic treatment was quite simple for one stage, only requiring the solution of two equations in two unknowns. The application to two stages involved the solution of four equations in four unknowns, and following the same procedure for A^ stages, it would be necessary to solve 2N equations in 27V unknowns. This is too laborious and would require an individual solution for every integer value of A^, and more advanced mathematical techniques are obviously needed to reduce the work. One method using matrix algebra will be illustrated in Chapter 12, but a second method, using finite differences and anticipating the contents of

Chapter 9

will

be used here.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

1.

The general arrangement is as shown in Fig. 1.4 where the flow rates of two streams are still denoted by R and S, and the benzoic acid concentrations by X and y. The suffix notation is again used to distinguish between the

R

1

yj

^

\

1

2

^2

\y

\

X,

1

\

A

\

t

^n-1

n

Yn-I

1

t

Ym-I 1

t

^N-2

N-1 ^N

\

^N-1

1

N

°n S

Fig.

1.4.

^N

R

Idealized A^-stage extraction

the diff'erent states of each stream, the suffix denoting the stage

which the

This time, a benzoic acid material balance the general stage n of the system.

applied to

stream has just

left.

= Rx„ _ + Sy„ + = Rx„ + Sy„

Input of acid (lb/mm)

^

Output of acid (Ib/min)

is

^

Since entry rate and exit rate must be equal,

Rx„.,

The

+ Sy„^,=Rx„ + Sy,

distribution coefficient equation (1.1) .'.

and equation

(1.14)

yn

=

is

still

(1.14)

valid for

any value of n

'^^^n

becomes

Rx„.i

+ mSx„+i =

Introducing a again from equation

(R + mS)x„

(1.5),

x„_i+x„+i =(oc+l)x„ or in standard form, x„ +

This

is

i-(a +

l)x,

+ x„_i=0

(1.15)

a second order linear finite diff*erence equation and the

of solution will be discussed in Chapter

9.

The solution

is

method

quoted here for

F

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

6

completeness,

viz.

y^

and

this

may

extracted, E,

= mx, =

mc(a^^^-a")/(a^-'^-l)

be verified by substitution into equation is

(1.15).

(1.16)

The proportion

given by

which gives equation

(1.13) for the special case

Table

1.1.

Proportion Extracted

(S =

N

60

in

2.

N Stages

1/8)

2

3

5

10

78-9

87-7

95-2

99-4

1

EiVo)

12/?,

m=

N=

Table 1 1 gives a few values of E for different values of A^ for the parsystem considered. This shows how the proportion extracted increases with the number of stages, and indicates that more than ten stages are likely to be wasteful whereas one stage gives a poor degree of extraction. This type of problem will be continued in Chapter 13, where the most economical number of stages will be determined by financial considerations. .

ticular

1.5.

Simple

Water

Still

with Preheated Feed

Figure 1.5 illustrates a distillation apparatus consisting of a boiler B with a constant level device C, fed with the condenser cooHng water. The steam is condensed in and collected in the receiver D. Some of the latent heat of

A

^—

C

B Fig.

1.5.

Water

still

with heat exchanger

evaporation is returned to the boiler by preheating the feed. Denoting the condenser feed rate by F Ib/h and the temperature by Tq° F, the exit water temperature by r°F, the excess water over-flow rate by Wlb/h and the

1.

distillation

THE MATHEMATICAL STATEMENT OF THE PROBLEM by

rate

G Ib/h,

7

performance of the apparatus can be

the

investigated.

Input of water to the still (Ib/h) Output of water from the still (Ib/h) Output of steam from the still (Ib/h)

F=W + G

:.

= F = W = G (1.18)

H

Btu/h, the latent heat of If heat is supplied to the boiler at a rate evaporation of water is L Btu/lb, and 0°F is taken as the datum temperature, a heat balance can be taken over the boiler.

= H+ {F- W)T = 212G+LG = H + Gr (212 + L)G

Heat input (Btu/h) Heat output (Btu/h) .-.

(1.19)

where equation (1.18) has been used to eliminate W. Equation (1.19) contains two unknown quantities, G and J, and another equation is needed to complete the description. This is obtained from a heat balance over the condenser.

Heat gained by cooHng water (Btu/h) Heat lost by condensing steam (Btu/h)

= F{T— Tq) = G{L + 212 — T)

F(T-To) = G{L + 212-T) In deriving equation (1.20) it has been assumed that the distillate the exit water temperature.

Eliminating

G

between

and

(1.19)

(1.20) is

cooled to

(1.20),

H T=To + HIF

F(T-To) =

From

(1.19)

and

(1.21)

(1.21),

H=

G(2n + L-To-HIF)

FH "

^^ F(212 + L-To)-H

^^'^^^

This equation gives the rate of distillation in terms of the heat input, and the temperature and flow rate of the coohng water. If an attempt is made to interpret equation (1.22) for a constant heat supply rate and constant feed temperature, it is seen that as F is decreased, G increases. In fact when F = i//(212+L — Tq) it appears that G is infinite, which is a physical impossibility. Reference to equation (1.21) resolves the difficulty as follows. As F decreases, T increases; but T cannot exceed 212° F as

it

leaves the condenser,

and To .-.

F

does not satisfy equation (1.19) becomes If

this

this gives the restriction

+ HIF=T ^212 F^H/(212-To)

inequahty,

T

will

H = LG :.

G = HjL

(1.23)

remain constant at 212 and

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

8

The temperature restriction has a further influence in that the amount of steam produced by the boiler exceeds the capacity of the condenser. Denoting the rate of collection of distillate by Z), equation (1.20) becomes

F(212-To)

D = (212-ro)F/L

.*.

The complete

solution

is

D

for

F^

F(2n + L-To)-H

and

this solution is illustrated

when

i.e.

H 1(212- Tq)

FH

that

F=

(1.24)

thus given by equations (1.22), (1.23), and (1.24);

D = (212- To)F/L and

= DL

for

F^HI{2n- To)

by the continuous Hne

This shows

in Fig. 1.6.

///(212 — Fq), the rate of collection of distillate

is

a

maximum

H/L. The above analysis has been made without reference to heat losses, and an attempt to allow for these would lead to a much more compHcated model.

at a value

A

would suggest that heat losses in the feed line to would be detrimental, and more serious at low values of F; but heat losses in the vapour line can have two effects. If F is greater than the optimum value, losses from the vapour line will be detrimental, but if F is smaller than the optimum value, heat losses from the vapour line will actually qualitative investigation

the boiler

increase the yield. line

On

the basis of these considerations, the second, dotted

has been drawn in Fig.

Fig. 1.6.

1.6.

1.6.

Variation of distillation rate with feed rate

Unsteady State Operation

In the examples considered so far, the system has been in a steady state, allowing the material entering the system to be equated to the material leaving the system and this has always given algebraic equations. In unsteady state problems, however, time enters as a variable and some properties of the system become functions of time. In the application of conservation laws it is no longer true that the rate of entry of material will equal the rate of exit, since an allowance must be made for material accumulating within the

1.

system.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

9

The general conservation law now becomes

INPUT -OUTPUT = ACCUMULATION The example

of Section 1.2 will

now

be reconsidered as

it

(1.25) starts

from the

following situation. Assume that the single stage contains V^ ft^ of toluene, V2 ft^ of water, and no benzoic acid. Assume, as before, that the mixer is so efficient that the compositions of the two liquid streams are in equilibrium at all times, and, in addition, that a stream leaving the stage is of the same

composition as that phase in the stage

This

c

ibya^

v,,x

xlb/a^

Sftymin

^2'y

S

R

aymin Ib/ft^

Time dependent

state of the

1.7,

aymin

aymin

Fig. 1.7.

which shows the

all illustrated in Fig.

R

y Ib/ft^

now

is

single-stage extraction

system at a general time

6,

where x and y are

functions of time.

Since the flow rates of water and toluene are constant, V^ and V2 will remain constant. The conditions are always changing, and so the material balance must be applied during a small time increment 36. Any function of 6 can be expanded by Taylor's theorem,t and this allows the state of the system at a time 6 + 36 to be expressed in terms of its state at time 6. In this case,

it is

helpful to

draw up

Table

1.2.

the following table.

Condition of the System Before and After a Time Increment

77?^

Property of the System

Flow rate of toluene phase Flow rate of water phase Volume of toluene phase in stage Volume of water phase in stage Cone, of acid

in entrance toluene

Cone, of acid in

exit toluene

Cone, of acid in

exit

6

6+39

R

R

S

S

Vi

V2 c

c

X

'*> dv

water

y

Amount

of acid in toluene layer

Vix

V.xArV.pe

Amount

of acid in water layer

Viy

V2y+V2pe

Mathematical functions

exist which cannot be expanded by Taylor's theorem as but these do not normally occur in chemical engineering applications, and the statement in the text should not cause any difficulty.

t

shown

in Section 3.3.7,

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

10

During a time

interval SO,

Input of acid

= Re 39

Output of acid

= r(x + \

--^^S9)s0 + s( y Y J

2de

--^^do]d9

2d9

J

= Vi-77,S9+V2^S9

Accumulation

d9 Since input — output

=

d9

accumulation

Rcd9-R(x + \%^^y^-^(y +

Taking the

+

^^^^V =

dx ^^ ^1 ~r^^^

— d9++ d9

dy

v.— Vi'^.SO ^d9

limit as 59 -^ 0,

Rc-R.-Sy==v/^^+V,f^

(1.26)

In stating the output, the arithmetic mean of the concentrations was taken, but the later calculations show that this was not necessary; the same equation (1.26) would have been obtained by using the concentrations at time 9 instead. Only the first two terms were taken in the Taylor series for x and y, and these are all that are necessary since the further terms which contain a factor (S9y will disappear when the hmit 39-^0 is taken. Equation (1.26) must now be solved in conjunction with the equihbrium relationship (1.1). Eliminating ;j;,

Rc-Rx-mSx =

dx — + mFi —

dx Fi

do

do

Rearranging,

dx

dO

Rc-{R + mS)x~ and

V\

+ mV2

integrating,

-\n{Rc-(R + mS)x} '

R^^

9

"F. + mF,"^^

^^'^^^

integration A can be evaluated by using the given initial system that the stage contains no benzoic acid at zero time; i.e.

The constant of state of the

when

= 0, x = -\nRc_ " R + mS~ \nRc-]n{Rc-(R + mS)x} = (R + mS)9l(V,-^mV2) 9

:.

(1.28)

or in exponential form:

Re

L

/

R + mSW

,,

^^,

THE MATHEMATICAL STATEMENT OF THE PROBLEM

I.

11

= 0, satisfies the differenThis equation satisfies the condition x = 0, at equation (1.26) for any positive value of 9, and when 0->oo, x-i'RcKR + mS). This is the steady state solution which has already been obtained in equation (1.3). tial

1.7.

Two

Salt Accumulation in a Stirred Tank

A

tank by the following example. of water. A stream of brine containing 2 Ib/ft^ of salt is fed into the tank at a rate of 3 ft^/min. Liquid flows from the tank at a rate of 2 ft^/min. If the tank is well agitated, what is the salt concentration in the tank when the tank contains 30 ft^ of brine ? further points are illustrated

contains 20

1.7.1.

ft^

Simple Treatment

Firstly, the following

simple model of this system can be set up. Liquid it leaves, so that Hquid accumulates in the tank

enters at a faster rate than

of 1 ft^/min. The increase in tank contents is 30 — 20 = 10 ft^, and hence the operation will last for 10 min. During this 10 min, 30 ft^ of brine enter carrying 60 lb of salt, and 20 ft^ of brine leave. Assuming that the final concentration in the tank is X, and that the concentration of the outlet stream increases linearly with time, the following material balance can be estabhshed. at a rate

Input of salt during 10 min (lb) Output of salt during 10 min (lb) Accumulation of salt in the tank

Using equation

(lb)

= = =

60 20 (X/l) 30Jif

(1.25),

60-10X = 30Z X = l-501b/ft^

(1.30)

.-.

This is a very simple model of the system which yields an answer by elementary algebra. It does, however, contain a further assumption beyond those given in the question; the assumption concerns the time variation of the outlet concentration. This additional assumption need not be made if a more detailed model is taken in a similar manner to that in the previous section. 1.7.2.

More

The

Detailed Treatment

of the system at a general time 6 is shown in Fig. 1.8, where are functions of 6. Again, a table can be constructed showing the state of the system at time 6 and at time 6 + 39.

both

state

V and x

^'mm 3fty, 2lb/ft^

c

Ib/ft^

_^2ftymin X Ib/ft^

Fig. 1.8.

Accumulation of

salt in

surge tank

— MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

12

Table

1.3.

The Condition of the System Before and After a Time Increment

Property of the System

e

d

+

Brine input rate (ft^/min) Input salt concentration (Ib/ft^) Brine output rate (ft^/min)

3

3

2 2

2 2

Output

X

salt

Volume of

concentration

(lb/ft 3)

-

dV

(- de

Vx

Salt content of tank (lb)

»-

"->

V

liquid in tank (ft^)

+

3d

\ /

dx

\

)\

de

J

During a time interval dO, Input of brine Output of brine

=

Accumulation of brine

= iv+—-SO] — V

2>de

259

de

\

J

dV 339-239 = —-39

.-.

d9

A

material balance must also be satisfied by the

Input of salt Output of salt

Accumulation of Since input — output

=

salt

= =

3.239

2x39

= (^ +

-;i^

^^j (x

+ -^39) - Vx

accumulation,

dV

656-2x50

salt, viz.

dV dx

dx

=Vx + x^5e+ V--5e + -7^:77;^-^'^ do dU du d9

•

dV

6-2x =

x

d9

•

dVdx dx 39 +F— + d9 d9 d9

^^

(1.32) ^

^

term in equation (1 .32) vanishes. In deriving concentration in the outlet stream at time 9 has been used in the output term since, as in Section 1 .6, any correction to this contains a factor 39 which will lead to its elimination when the limit 39-^0 is taken. The first important point is now evident; the accumulation consists of two terms. The one term is due to a volume change, the other is due to a con-

Taking the

limit 39 -^ 0, the last

this equation, the salt

;

centration change, and both are essential in the solution.

1.

of

THE MATHEMATICAL STATEMENT OF THE PROBLEM

Equations (1.31) and (1.32) must now be solved for x and Equation (1.31) can be solved immediately;

Fas

13

functions

6.

V = A+e But when 6

=

0,

V=

20

V=

:,

Substituting equations (1.31)

and

20 + 9

(1.33)

(1.33) into (1.32),

6~2x =

x + {20 + e) ^ dO

Rearranging,

and

dx

de

6-3x

20 + 6

integrating,

}\n{6-3x) = \n(20 + e) + B where

B

is

the constant of integration. 9

=

But when x

0,

=

(1.34)

|ln6 = ln20 + B Eliminating B, and combining the logarithms,

= In (1 + 0-05^) = l-0-5x (1 + 0-050)-^ x = 2-2(l + 0-05^)-2

-iln(l-0-5x) /. .-.

(1.35)

concentration in the tank at any time, and equation (1.33) gives the volume of brine in the tank. From (1.33) it is seen that F = 30 when 6 = 10, as in Section 1.7.1, and therefore from (1.35)

Equation (1.35) gives the

salt

.'.

Z=

2-2(2/3)'

X=

1-407 lb/ft'

(1.36)

Comparison of this result with the previous result (1.30), shows that the first simple model has a significant error of 6-6% due to the additional assumption of a hnear variation of x with 9; an assumption which is shown to be incorrect by equation (1.35). The two treatments of this example illustrate the important point that the solution of a problem depends upon the choice of model. A simple model needs many assumptions and yields an approximate answer quickly, whereas a more complicated model needs fewer assumptions and yields a more accurate answer by more advanced mathematical techniques. It is evident that the most accurate result will be obtained by making the minimum number of assumptions consistent with obtaining a tractable mathematical problem. In this way the mathematical techniques which are available to an investigator control the applicability of any theoretical prediction made by him. It is therefore desirable that the widest selection of mathematical techniques should be at his disposal.

mathematical methods in chemical engineering

14 1.8.

Radial Heat Transfer through a Cylindrical Conductor

two problems the properties of the system were functions of There are many problems of a similar nature, where the properties of the system are functions of position instead of time, and the following example illustrates the application of the method to this type of problem. Two concentric cyhndrical metallic shells are separated by a solid material. If the two metal surfaces are maintained at different constant temperatures, what is the steady state temperature distribution within the In the last

time.

separating material ?

This

a steady state problem, but some of the properties of the system

is

depend upon the position at which they are measured. In this case, the temperature (T) and the heat flow rate per unit area (Q) are both functions of the radius (r), and the heat balance must be related to a space interval between r and r + dr as shown in Fig. 1.9. Following the same procedure as before, but considering variations of r instead of 6, Table 1.4 can be compiled

Fig.

Table

1 .9.

1 .4.

Radial heat flow through cylindrical conductor

of the System on Either Side of a Space Increment

77?^ Condition

Property of the System

Temperature

Heat

transfer area/unit length

Radial heat flux density Total radial heat flow

r

r

dr

T + fsr dr

T

2n(r

Inr

+

Sr)

«+f"

Q InrQ

+

2n(r

+

Sr)(Q

+

^

^r)

1.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

Considering the element of thickness

dr,

Heat input to inner surface

= InrQ

Heat output from outer surface

=

Accumulation of heat

=

Since input— output

=

2n{r-\-dr)

I \Q +

dQ

\

~T~ ^fj

accumulation,

2nrQ-2n(r + Sr)(Q +

rQ-rQ- r^ 3r -

:.

15

^sA=0

QSr

-

^-^{drY

=

Cancelling out rQ, dividing by Sr, and taking the limit Sr-*0,

r^ + Q = But

Q

is

related to

(1.37)

T by

2=-^^ where k

is

Equation

the thermal conductivity.

d^T

,

dT

,

d^T ,

becomes,

dT

= +__ "^

(1.39)

dr

dr^

k

(1.37)

^

dr

dr^ or

(1.38)

Referring to Table 1.4, the change in heat flow rate two terms in this example, one due to a change in temperature gradient, and the other due to a change in the area of conduction. This problem can be solved by treating (1.39) as a homogeneous second order differential equation (see Section 2.4.3), or by solving equations (1.37) since

is

constant.

across the element gives rise to

and

(1.38) in succession.

Equation (1.37)

dQ_

gives,

_dlr

"""^

e" .*.

or

where In

A

is

\nQ= -\nr + \nA Q = A/r

chosen for the constant of integration. Putting

this value

of

Q

into equation (1.38),

--

-k

—

r

dr

dr

kdT

r

A

\nr= -{kTIA) + B

(1.40)

MATHEMATICAL METHODS

16

CHEMICAL ENGINEERING

IN

Equation (1.40) involves two constants of integration, can be evaluated from the fixed boundary temperatures.

At

r

at

r

and

T= T=

a, b,

B, and these

V

and

\na-\nb = k{T^-To)IA A = k(T, - To)/(ln a-\nb)

.-.

B = (T,\na-To\nb)l{T,-To)

.-.

Putting these values of

- To) In r

^ =

B into equation (1.40), (In 6 - In a)T+ T^ In a-To\nb and

= {T-To)(\nb-\na) + (T,-To)\na T—Tn In r — Ina In 6 - In a Ti - To 1.9.

^^'^^^

T,}

lna= -{kTolA)-^B \nb= -(kTJA) + B

.-.

(Ti

= =

A and

(1.42)

Heating a Closed Kettle

The problems considered

so far, have only used material or heat balances of the equations. In the following example, the equation is derived from a rate equation. closed kettle of total surface area A ft^ is heated through this surface lb by condensing steam at temperature T^ °F. The kettle is charged with of liquid of specific heat C Btu/lb, at a temperature of To^F. If the process is controlled by a heat transfer coefficient h Btu/h ft^ °F, how does the temperature of the liquid vary with time ? in the derivation

A

M

Consider a time interval 39.

Heat output (Btu)

= hA(T,-T)Se =

Accumulation (Btu)

=

Heat input (Btu)

Since input — output

=

MC -77.39

accumulation,

hA(T,- T)39 =

MC--39

(1.43)

du Rearranging,

dT T,-T and

hAd9

MC

integrating, In

(T,-T) = (hAIMC)9 + B

(1.44)

1.

B

THE MATHEMATICAL STATEMENT OF THE PROBLEM

known

can be evaluated by using the

fact that

= 0,T =

when

17

Tq

-\n{T,-To) = B

.-.

MC

T.-To

In this problem, equation (1.43) could have been written down immediately from the definition of the heat transfer coefficient, thus avoiding the necessity of taking a heat balance. 1.10.

Any

Dependent and Independent Variables, Parameters

is an algebraic equation involving These symbols fall into three classes, (a) independent variables, dependent variables, and (c) parameters.

solution of a differential equation

symbols. (b)

1.10.1.

Independent Variables

These are quantities describing the system which can be varied by choice during a particular experiment independently of one another. Examples are time and coordinate variables. 1.10.2.

Dependent Variables

These are properties of the system which change when the independent variables are altered in value. There is no direct control over a dependent variable during an experiment. The relationship between independent and dependent variables is one of cause and effect; the independent variable measures the cause and the dependent variable measures the effect of a particular action. Examples of dependent variables are temperature, concentration, 1.10.3.

and

efficiency.

Parameters

This

is

by

far the largest group, consisting

mainly of the characteristic

and the physical properties of the materials. The group contains all properties which remain constant during an individual experiment, but since different constant values can be taken by a property during different experiments, the correct term for them is " parameters ". Examples are overall dimensions of the apparatus, flow rates, heat transfer coefficients, thermal conductivity, specific heat, density, and initial or boundary values of the dependent variables. Referring to Section 1.6, the solution to the problem of starting a single stage hquid-liquid extraction system is given by equations (1.1) and (1.29). properties of the apparatus

^c

f

1

and by ehminating

y

= mx

-

exp

I-

(1.1)

R + mS

\1

e]

}

(1.29)

x,

mRc

f

/

R + mS

X]

,

,

:

MATHEMATICAL METHODS

18

The symbols can be

IN

CHEMICAL ENGINEERING

classified as follows

Independent variable

9

Dependent variables

x,

Parameters

m, R,

y S,

c,

V^, V2

The parameters

are all fixed in value during an experiment, but can be varied between experiments for comparison purposes. is the only variable whose value can be altered during an experiment, x and y vary during an experiment and between experiments; their values depend upon the choice of values for both the parameters and the independent variable.

The main use of tiation process.

We

this classification is in the interpretation

of the differen-

usually differentiate a dependent variable with respect

an independent variable, and occasionally with respect to a parameter. differentiating either (1.29) or (1.47) with respect to 0, an expression giving the rate of change of concentration with time can be obtained, but differentiating (1.1) with respect to the dependent variable x, gives dyjdx = w, an almost useless piece of information which does not throw any hght on the behaviour of the apparatus. So far, the problems considered in Sections 1.2, 1.3, 1.4, and 1.5 have been equations between parameters since no independent variables were involved, and hence no dependent variables could be present. Each experiment found a set of related constant values, the only variation could be between experiments when some of the parameters may be altered in value. The problems of Sections 1.6, 1.7, and 1.9 involved time as an independent variable, and in Section 1.8, a radial coordinate was the single independent variable. These problems all gave rise to ordinary differential equations. When more than one independent variable is needed to describe a system, the usual result is a partial differential equation, and this type of problem will be dealt with in Chapter 8. to

By

1.11.

Boundary Conditions

An

ordinary differential equation usually arises in any problem which involves a single independent variable. The general solution of this differential equation will contain arbitrary constants of integration, the number of constants being equal to the order of the differential equation (see Chapter 2). To complete the solution of a particular problem, these arbitrary constants have to be evaluated. In formulating the equation, the conservation law (1.25) is applied to an infinitesimal increment of the independent variable, and this yields a differential equation. There is usually a restriction on the range of values which the independent variable can take and this range describes the scope

of the problem. Special conditions are placed on the dependent variable at these end points of the range of the independent variable. These are naturally called " boundary conditions ", and are used to evaluate the arbitrary constants in the solution of the differential equation.

:

1.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

19

These conditions have already been used in the examples considered In Section 1.8, a second order differential equation

eadier in this chapter.

was derived, and its solution (1.40) contains two arbitrary constants B. The independent variable in this problem is r, and the boundaries The boundary conditions are given by are given by r = a, and r = b. equations (1.41) in the form of restrictions on the dependent variable (T) at the boundaries defined by values of the independent variable (r). In the other examples in Sections 1.6, 1.7, and 1.9, 6 was the independent variable, all of the equations were of first order, and each solution only contained one arbitrary constant. When time is the independent variable, the boundary condition is frequently called the " initial condition ". This (1.39)

A and

is

because

it

of the apparatus.

specifies the starting state

most frequently used boundary conditions in heat transfer are (1) Boundary at a fixed temperature, T = Tq. (2) Constant heat flow rate through the boundary, dT/dx = A, (2a) Boundary thermally insulated, dT/dx = 0. (3) Boundary cools to the surroundings through a film resistance described by a heat transfer coefficient, kdTjdx = h{T— Tq). k is the thermal conductivity, h is the heat transfer coefficient, and Tq is the temperature of

The

three

the surroundings.

Boundary conditions be described

will

in

when they

problems involving partial arise in Chapter 8.

1.12.

Most

differential equations

Sign Conventions

sign difficulties are a result of trying to think too deeply about them,

The temptation to anticipate the very strong but it must be resisted. The formulation of equations can be reduced to a set of rules which can be systematically apphed and although it is difficult to frame such a set of rules to cover the wide variety of problems which can arise, an attempt has been made in the next section. The slavish observance of rules is not to be generally recomthus causing a state of complete confusion.

solution of the problem

is

mended, but it is best in the first instance to do just this because it completely eliminates any possibiHty of anticipating properties of the solution, and ensures that the signs are correct. The resulting equation can frequently be subjected to a physical sign check after it has been estabHshed. All terms must be treated as positive during the formulation, and negative signs will only occur due to two causes. (a) The first cause is the negative sign in the general conservation law (1.25), viz.

Input — Output

=

Accumulation

The second cause

is by the definition of the rate equations governing heat transfer, mass transfer, and fluid flow. This will be illustrated below for the case of heat transfer.

(b)

When an

independent variable

is

defined for a system, an origin and

positive direction have to be included in the definition.

This has not been a

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

20

problem so

serious

direction

is

far,

since

when time

is

the independent variable,

the natural one of successive events

and

its

origin

is

its

always

taken as the instant when the initial conditions are estabhshed. The radial coordinate (r) the other independent variable which has occurred is measured outwards from the axis of the body of revolution, the axis being the origin. In other cases it is always necessary to specify the origin of the independent

and also its positive direction. Turning to heat flow, the quantity of heat (Q) conducted past a point through unit cross-sectional area in unit time is counted as positive if it flows in the same direction as the positive direction of the independent variable (coordinate), and negative if it flows in the negative direction of the coordinate. According to Fourier's Law, the rate of conduction of heat is proportional to the temperature gradient, and the direction of the flow is down the temperature gradient as shown in Fig. 1.10 where the two possible variable

f + ve/

-^-ve

^

T

\^^

1 1

O + ve

O-ve

\

—^x Sign convention in heat transfer

Fig. 1.10.

cases are shown.

On

the

left

and as a

hand

side of the diagram, the temperature

heat flows from right to

left and is diagram where the temperature gradient is negative and the heat flow positive. In both cases, the equation can be written in the consistent form

gradient

is

negative.

positive

The

reverse

is

result, the

true in the right

hand

side of the

(1.48)

dx where k

This equation has already been used (1.38), but was of no importance. For mass transfer, material diff^uses down a concentration gradient, and for fluids, the flow is down a pressure gradient. By analogy with the Fourier Law, these rate equations also contain a negative sign in the definition. In conclusion, if the rules which have been illustrated in this chapter are followed, and the solution is not anticipated, the signs will take care of themselves. This can be illustrated by repeating the example of Section 1.7 with a diff'erent boundary condition. Instead of the tank containing 20 ft^ of water The only initially, let it contain 20 ft^ of brine of concentration 5 Ib/ft^. is

the conductivity.

in that application the negative sign

physical difference that this makes,

is

that the brine concentration will

now

1.

THE MATHEMATICAL STATEMENT OF THE PROBLEM

21

decrease with time instead of increase. Mathematically, only equation (1.34) need be altered. The list of properties in Table 1.3 will be unaltered, and the change in salt concentration with time is still adequately described by

dxjdO without

artificially

changing

its

sign.

In this

new problem, dxjdO

is

a negative quantity since x decreases with time and it would be a mistake further to say that the output salt concentration dXO + dO is x-(dxld9)S6. which is the theorem Taylor's from seen mistake can be why this is a reason

now

A

governed by the sign of 39 which is positive. modified problem, and the solution is equation (1.34) is reached, and this now becomes:

origin of this term; the sign

Table

is

1.3 is therefore still true for this

identical until

=5 -|ln(-9) = ln20 + 5

=

when .-.

0,

x

(1.49

Here, the logarithm of a negative number is involved due to integrating — 3a')~^ in an unconventional manner. This need not give rise to any difficulty if In (-1) is treated as an imaginary quantity by analogy with (6

J(— 1), and B

is

ehminated as before by subtraction.

:.

\n(^^) = -3 In (1 + 0-05^) .-.

This

how

is

x

= 2 + 3/(1 + 0-050)^

and it illustrates and how the diff'erent appropriate particular solutions from the general

the correct solution to this modified problem,

signs can be allowed to take care of themselves,

boundary conditions

select

solution of the diff'erential equation.

1.13. (1)

about (2)

Summary of the Method of Formulation

Pick a mathematical model of the process by making assumptions ideal behaviour. Define dependent variables to measure the properties being

its

investigated. (3) Define one or more independent variables in terms of which the dependent variables can be expressed. (4) Define the parameters of the system which particularize the general relationship between the dependent and independent variables. (5) Establish the state of the system at a typical point defined in terms of the independent variables. (6) Take an infinitesimal increment in each of the independent variables, and establish the state of the system at neighbouring points in terms of the state at the typical point by using Taylor's theorem. (7) Apply the relevant conservation laws and ment of the system defined by (6).

rate equations to the ele-

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

22

Cancel out terms where possible, including the incremental intervals. the Hmit as the size of the increments tends to zero. These rules include the possibility that there may be more than one independent variable, resulting in the formulation of a partial differential equation. The treatment of such problems will be given in Chapter 8. Although only the first two terms of Taylor's theorem are usually needed in practice, the infinite series of terms is effectively included, since the later terms involve incremental quantities raised to high powers and these disappear when the incremental quantities tend to zero. The solution given by the analysis is related to the model of the system being investigated, and not to the system itself. The assumptions made in order to derive an equation for the model may or may not be vahd, and if the result does not agree with the experiment, then the model is probably wrong and not the mathematics. As shown in Section 1.7, a more complicated model gives a more accurate result, but frequently, if the model is too detailed, the equation derived defies rigorous mathematical analysis. This difficulty can be overcome by using a numerical method to find an approximate solution to the difficult equation as shown in Chapter 11, or by simplifying the model with a further assumption. It must be realized, however, that as the number of assumptions increases, the likeUhood of the result agreeing with practice decreases. Tables 1.2, 1.3, and 1.4 have been compiled in this chapter to facilitate the description of the method of formulating equations. These tables should be compiled mentally and throughout the rest of the book they will be omitted. If any reader finds difficulty in formulating the equations in the rest of the book, it is suggested that similar tables should be constructed. (8)

(9)

Take

Chapter 2

ORDINARY DIFFERENTIAL EQUATIONS Introduction

2.1.

An

equation relating a dependent variable to one or more independent variables by means of its differential coefficients with respect to the independent variables is called a differential equation. If there is only one independent variable the equation is said to be an "ordinary differential equation". If there are two or more independent variables and the equation contains differential coefficients with respect to each of these, the equation is said to be a "partial differential equation". Thus

is

an ordinary

differential

equation whereas ;

"^

"*"

sx^

2r + 6sx

III

IV

2.

47

ORDINARY DIFFERENTIAL EQUATIONS

Substitution of these expressions for the differential coefficients into / gives (2r + 6sx)

- 4{q + 2rx + 3sx^) + 4(p + qx + rx^ + sx^) =

Equating coefficients of equal powers of

V

4x + 8x^

x,

2r-4q + 4p = 6s-8r + 4^ = 4

4r— 12s =

=

8

q

=

4s

=

r

.'.

yp

.-.

The complementary function

6,

y (iii)

(/)(x) /5

=

y^ + y^

r

or

s

and

10,

integral

is

of

e'"''

=

{A +

=

The

first

where

1

VI is

VII

Bx)e''''

J and

r

be multiples of

will

VIII

The form of

are constants.

form because

e''"'.

Therefore,

all

if

the

the differen-

the particular

to be

=

Hence

ar^/"".

(2.55)

oce^^

be

differential coefficient will

be

=

(A + Bx)e^'' + 7 + lOx + 6x^ + 2x^

y,

coefficient will

p

is

the form Te^^\

assumed

3s

2

for this case of equal roots

particular integral will also be of exponential tial coefficients

= =

= l + 10x + 6x^ + 2x^ y,

and the complete solution

or

are''^,

and the second

differential

substitution into the original differential

equation (2.36) gives (Pr^

or

Evaluation of a from the be determined. (iv)

(f)(x) is

of the form

+ Qr + Ryue''' = Te''' T a = WZTTT^:^^ Pr'^ + Qr + R

known

G

sin

(2.56)

(2-57)

constants enables the particular integral to

nx +

H cos nx; where G and H are constants, The form of the same type be multiples of sin nx or cos nx.

either of which could be zero, whilst nisa. non-zero constant.

particular integral will also be a trigonometrical function of the

because all the differential coefficients will Therefore assume the particular integral to be j^p

The

= L sin nx +

first differential coefficient

-— dx and the second

M cos nx

of equation (2.58) will be

= nL cos nx — nM sm nx

differential coefficient

d^ d^y dx

= —n(L sin nx + ,

M cos nx)

(2.58)

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

48

Substitution in the original differential equation gives

L=

{R-n^P)G + nQH (2.59)

M = {R-n^P)H-nQG {R-n'^Pf + n^Q''

(2.60)

M

Equations (2.59) and (2.60) enable the constants L and to be evaluated and thus the particular integral can be determined. The above four forms of the particular integral that would be obtained from each type of expression of 4){x) are summarized in Table 2.1.

Table

2.1.

Normal Forms of Particular

Right-Hand Side of Equation ^{x)

Particular

Coefficient

Integral

Values

Differential

A

A

constant

Integrals

-§

constant

K

C

To

A

A

polynomial a^-^a^x^- ... +G,j.T"

polynomial

be

in

An

exponential

coefficients.

T

exponential

J^rx

the

equation

differential

and equating

An

by

determined

substituting

"^

Pr'-^Qr-rR

(R-n'P)G+nQH Trigonometrical

Trigonometrical

G

sin

nx+H cos nx

P, Q, and

R

L

sin

nx^-M cos nx

(R-n^Py+n'Q^

(R-n'P)H-nQG (R-n'Py+n'Q'

refer to the coefficients in the differential equation (2.36).

Modified procedure when a term in the particular integral duplicates a complementary function. In each of the above forms of the particular integral, it is possible for one or more terms to be identical with one or more terms in the complementary function. When this occurs the general procedure is to multiply the assumed form of the particular integral by the independent variable, which will destroy the similarity of the terms. However, it is possible that this multiphcation by the independent variable will make other terms in the particular integral and complementary function identical. If this should happen the assumed form of the particular integral should be multiplied by the square of the independent variable, and if similarities still exist the assumed form should be multiplied by the third (v)

term

in the

.

2.

ORDINARY DIFFERENTIAL EQUATIONS

49

power of independent variable and so on until no identical terms remain between the complementary function and the particular integral. Identical terms in both parts of the solution of the differential equation

can

arise in

trial

form

many

For instance, when

ways.

(pix) is

a constant, the

first

for the particular integral will be a constant according to Table 2. 1

If in addition,

one of the roots of the auxiliary equation

the complementary function will also be a constant.

is

zero one term in

Hence

the modified

form of the particular integral should be a constant multiplied by the independent variable, in which case the similarity will have been removed. The following example will make the procedure clear.

trial

Example The

5.

+ 6^ = 18 3^ dx^ dx

Solve

auxiliary equation

I

is

3m^-6m =

II

and the roots are

m= The complementary function

=

y,

m=

and

2

is

Ae"''''

+ Be"'''' = A + Be^""

Since one term in the complementary function integral

is

III

a constant, the particular

cannot be a constant even though the right-hand side of equation

I

contains a constant.

Therefore

let

the

form of the particular y

integral be

= Cx

IV

— =-C dx

Then

^=

and

dx"^

Substitution in equation

I

gives

V

3x + A + Be^''

VI

C=

or

and the particular Integral

is

=

18

(3xO) + 6(C) 3

3x.

Therefore the complete solution

y

is

=

The above complication can arise also when a root of the auxihary equation is equal to the coefficient of the independent variable in the exponential term on the right-hand side of the equation. In this case too, an example will make the procedure clear. Therefore consider

Example

6.

Solve

3—^ + dx"^

10-^ - 8v dx

=

7e"^*

I

MATHEMATICAL METHODS IN CHEMICAL ENGINEERINQ

50

The

auxiliary equation

is

= (3m-2)(m + 4) = m = 2/3 or —4

3m^ + 10m-

8

.*.

II

and the complementary function is ^,^_^g2x/3_^^g-4x

The second term

jjj

of the same form as that given in Table 2.1 for the particular integral of equation I. Therefore, take the modified form in equation III

is

=

y

Cxe"^^

IV

for the particular integral.

ax

-^ = (16x-8)Ce-^"

and

ax

Substitution in equation solution of equation

y

Another way

I

and solving for

C gives C = - J,

and the complete

I is

=

y^ + y^

=

Ae^""^

+ Be-'^'' -^xe-"^""

V

which the above comphcation can appear, is when the roots of the auxiliary equation are both equal to the value m, the coefficient in the exponent in a term of the form Ce'"'^. Thus for an auxiliary equation with equal roots the complementary function will be {A-\-Bx)e'^^ where the first term is similar to the expression Ce""^. Hence the proposed form of the particular integral would be Ex^e"^^, since multiplying by x alone would give a form similar to the second term of the complementary function. E would be evaluated in a manner similar to that illustrated in the above in

examples. Finally, when the roots of the auxiliary equation are complex and the right-hand side of the diiferential equation contains the terms

{A the modified trial

sin

nx

form of the particular x(a sin nx

and the evaluation of a and 2.5.5.

j5 is

Particular Integrals by the

The symbol

+ B cos nx)e^^

(2.61)

integral will be

+ P cos nx)e^''

(2.62)

as above.

Method of Inverse Operators

(dy/dx) representing the differential coefficient signifies that ;^ has been carried out with respect to the

the operation of diflferentiation of

independent variable x. In fact if there is no doubt whatever of the independent variable, it is acceptable to write simply "/)>'" for the differential coefficient implying that the differentiation process has been carried out. Hence it is convenient to refer to the symbol *'i)" as the "differential operator", and prefixing it to a variable means that a differentiation has been

2.

ORDINARY DIFFERENTIAL EQUATIONS

51

D

carried out with respect to an obvious independent variable. The letter alone has no significance and must be placed in front of the dependent variable to signify the differential coefficient.

Dy = dyjdx

I.e.

D{Dy)

Similarly

D{D^y)

and

D"y

so that It

= D^y = d^yjdx^ = D^y = d^yjdx^ = d'^yjdx''

should be noted that

{Dyf = {dyldxf without ambiguity.

The above symbohc representation in terms of the differential operator can be extended to expressions involving different order differential coefficients.

may be

Thus the expression,

written

D^y + Wy + 2y = (D^ + 3D + l)y and the

latter

may

(2.64)

also be factorized,

{p + l){D + l)y

(2.65)

D

can be treated as an it appears that the differential operator ordinary algebraic quantity with certain Umitations. The three basic laws of algebra are the following. (a) The Distributive Law. This states that,

Hence

A(B + C) = AB + AC which also appHes to the

differential operator

D(w +

t;

+ w)

D,

(2.66) viz.

= Du + Dv + Dw

(2.67)

In fact the distributive law was appHed to the operator in equation (2.64). (b) The Commutative Law. This states that,

AB = BA which does not in general apply to the

(2.68)

D. It is conventional to consider that the differential operator differentiates every term which follows it, and not the terms preceding it. Transferring a term past the operator thus alters the value of the expression. For example

Dxy but

it is

xDy

true that the expression (2.65) can be given the equivalent forms,

{D + \){D-\-2)y

The operator (c)

7^

differential operator

will thus

commute with

The Associative Law. This

=

{D-\-2){D + \)y

itself

but not with variables.

states that,

{AB)C = A{BC)

(2.69)

MATHEMATICAL METHODS

52

which does not

CHEMICAL ENGINEERING

IN

in general apply to the differential operator

how

ing two equations illustrate

far the

is

D. The follow-

obeyed.

= (DD)y = {Dx)y + xDy

D{Dy) D{xy)

but

law

(2.70)

In the first term on the right-hand side of equation (2.70) the closing of the bracket cuts y off from the operator thus destroying the simple equality. The basic laws of algebra thus apply to the pure operators, but the relative order of operators and variables must be maintained. Also, the influence of the operator must not be hindered by brackets as in equation (2.70) unless this is specifically desired. Consequently these properties can be utilized particularly in the solution of linear differential equations thus simplifying the solving procedure. Some of these properties will be illustrated in order to familiarize the reader with the techniques. Application of the Differential Operator to Exponentials. Let the differential coefficient of e^"" be required. This expression could be written

=

De^^

Also for the second

(2.71)

differential coefficient,

=

D^eP""

and

pe'""

p^e^^

(2.72)

for the nth differential coefficient, J^n^px

In general,

if

/(i))

pn^px

^

some polynomial of

is

(2.73)

the differential operator,

it

is

possible to write

/(D)6^-=/(p)e^I.e.

the coefficient of

iff(D)

is

x

replaces

each of these new terms

and the

D in the function.

expanded and then each term

differentiated

equation (2.64) y

=

will

This

is

4)

quite obvious, for

in the expansion multiplied

by

e^"",

be similar to the left-hand side of equation (2.73)

form similar

e^"",

(D^

to the right-hand side. Specifically, then the expression can be written

+ 3D + 2)e^^ =

The above concept can be extended exponentials.

(2.

For example,

if

(p' to

+ 3p + 2)e^^

if in

(2.75)

more complex functions involving

the differential coefficient of ye^""

is

required,

then D(ye''')

=

e'^'Dy

+ yDe^"" =

Similarly the second differential coefficient

D^yeP"")

Similarly

it

^^^(D

+ p)y

(2.76)

would be

= DleP\D + p)y'] = eP\D{D + p)y + {D + p)y .Die^"") = e'^iD + p)(D + p)y = e''(D + p^y

(2.77)

follows by comparison of equations (2.76) and (2.77) that DXye'"")

=

e'^'iD

+ pfy

(2.78)

2.

or in even

ORDINARY DIFFERENTIAL EQUATIONS

more general terms, iff(D)

is

53

a polynomial in D,

f{D)(yen==e'J(D + p)y The above property of most important as will be seen

the differential

(2.79)

operator involving exponentials

is

later.

Finally, the operation performed on the exponential function in equations (2.71) to (2.79) can be extended to trigonometrical functions by using

and trigonometrical functions Let the nth differential coefficient of sin px be

the complex relationships between exponential to be given in Chapter 4.

required.

Now

where "

D" (sin px)

Im

= d" Im e'^^ = ImZ)V^^ = Im(ip)"e'^"

(2.80)

(2.81)

" represents the imaginary part of the function

The evaluation of equation Put n = 2m to make Jt even.

(2.81)

and because

which follows it. depends upon whether n is odd or even.

e'^""

:.

D^'"(smpx)

= cos x + sin x = (-p^)"'smpx i

(2.82)

Similarly, for n odd, Z)^'" +

= (-p^Tpcospx

^(sinpx)

The corresponding formulae way are

for differentiating cos px,

(2.83)

which can be

derived in the same

D^"" (cos px)

D^'^^'icospx)

= (- p^y" cos px = -{-p^Tpsinpx

(2.84) (2.85)

The principles involved in equations (2.80) to (2.85) can be extended to operations on more comphcated trigonometrical functions by considering real or imaginary parts as shown. This is left to the initiative of the reader in attempting the examples at the end of the book. The Inverse Operator. The operator

D

signifies differentiation,

i.e.

Z)[J/(x)Jx] =/(x)

(2.86)

so that

jf(x)dx

=

(2.87)

D-^f(x)

which suggests that the reciprocal of the operator

D placed before a function

implies integration of that function with respect to an obvious independent

Thus i)~Ms the "inverse operator" and is an integrating operator. Because of its relationship to the differential operator D it would be expected that it can be treated as an algebraic quantity in exactly the same manner as D. This will now be considered. Equations (2.74) and (2.79) can be extended to an infinite series of positive powers of D, so that if the function of D can be expanded in ascending powers of D, the equations still apply. variable.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

54

Example

7.

Solve II

Transferring the operator to the other side of the equation,

^

-'

-

-m c-

4(1 :.

III

D-4

y^-illHiD)HiDr + iiD)' + -']e'^

IV

Performing the operations on e^^ gives

using the binomial expansion.

V

3^=-ie'^[i+K(i)'+(i)' + ..-] The

series in the brackets is

a geometrical progression whose

sum

to infinity

is 2.

is

VI

3^=-ie'"

.-.

the particular integral of equation

I.

Because the inverse operator in equation III can be expanded in a series of positive powers of Z), it satisfies the condition stated above. Thus, the general property given in equation (2.74) could have been appHed much earlier to equation III. Viz.

^

'

2-4

D-4

as before.

In the above example, equation (2.74) has been used in the form, e^^

KD)

—

=

e^^

(2.88) ^

f(p)

However, if/(/?) = 0, e^^/f(p) is infinite and reference has more general equation (2.79) in order to resolve this (p) = 0, (D—p) must be a factor of f(D). Hence f(D)

where n

is

to be

made

makes

(/>(/?)

to the

When

difficulty.

= (D-pn(D)

the lowest integer which

^

(2.89) finite

and not

zero.

Then

^px

^px

(D-py^(D)

f(D)

1

eP""

(D-pyct^ip)

by applying equation

(2.74).

AppHcation of equation

(2.79) with

y

=

1

gives 1

_„

e^" 1

f(D)

(t>{p)

D"

(2.90)

2.

D

is

ORDINARY DIFFERENTIAL EQUATIONS power

raised to a negative

in equation (2.90),

and

55

this implies integra-

tion.

J—eP^ =

•

It is

--—

(2.91)

unnecessary to include the arbitrary constants arising in the integra-

tions since these are included in the

Example

8.

complementary function.

Solve

^- 8^ + dx

16>'

=

6xe

dx^

In terms of the differential operator, equation

{D^-W + i6)y and the particular

integral

The complementary function

6xe^^

II

4.

=

x^g^*

=

(yH-Bx)e^^

III

is

j,

y

becomes

is

6

and the complete solution

I

= (Z)-4)V =

IV

is

= yc+yp = (^+bx+x>'^^

v

Application of the Inverse Operator to Trigonometrical Functions. Trigonometrical functions can be written as the real or imaginary parts of e'^"" in which case they can be treated in that form by the inverse operator. The analytical procedure to follow would be very similar to that given above for real exponential functions combined with the principles presented in equations (2.80) to (2.85). Therefore no further discussion need be given to this particular topic.

Application of the Inverse Operator to Polynomial Functions. The inverse operator can be made to operate on the polynomial function (/)(x) by expanding the function of the inverse operator in terms of by the usual binomial series for negative exponents. The method is illustrated by the following example.

D

Example

9.

Solve

d^y .2

dx^

"

dy J-

dx

^y

=

4x''

+ 3x^

Introducing the differential operator, equation

I

becomes

{D''-D-6)y = (D-3)(D + 2)>; =

4x^

+ 3x2

II

3—

MATHEMATICAL METHODS

56

and the

particular integral

is

(4x-*

(D-3)(D + 2)

+ 3x^)

Expanding each term of the

D

D^ "^27 9

1

1

yp=

-7

3"^

+

— + 243

35 ^2

36

216

5

4x^ + 3x2

D^

12x^ "^

6

D

4+T

(4x^

65

-

(4x^

+ 3x2)

+ 3x2)

IV

V

7776

1296 7(24x + 6)

13x24

216

1296

36

III

+ "^32 16

+ 6x

(4x^+3x2)

by the binomial theorem gives

D^

D

—D +

(2 + Z)).

(3-D)

partial fractions

+

144x^

CHEMICAL ENGINEERING

1

1 >';

IN

-0

VI

+ 36x2 + 132x-10 216

The complementary function

is

Ae^^' +

y,

and the complete solution y 2.5.6.

=

yc

+ yp =

Be

VII

is

Ae^'' + Be~^''

Particular Integrals by the

(72x3

+ 18x2 + 66x-5)/108

Method of

VIII

Variation of Parameters

This method of determining the particular integral of a differential equation is a very elegant procedure, but it depends on the complete complementary function being known and therefore its appHcabihty is somewhat limited. Furthermore when apphed to differential equations of order higher than three the solution of the simultaneous equations of the parameters is time-consuming, laborious and above all hazardous to the unfamihar and inexperienced.

Consider once again the general differential equation of second order Let P be unity and Q and R be functions of the independent variable. The equation can therefore be written (2.36).

dy

d'y

-^^-\-Q{x)~+R{x)y = and

let

(j>{x)

(2.92)

the complementary function of this equation be y,

= Au + Bv

(2.93)

where A and B are the arbitrary constants and u and v are functions of the independent variable x. Now assume that the particular integral can be expressed,

yp=f,{x)u+f,{x)v

(2.94)

2.

ORDINARY DIFFERENTIAL EQUATIONS

57

where the functions of the independent variable /^(x) and/2(x) are called the parameters. There will be two parameters for a second order, three for a third order and so on. Then •

-p=nix)u+Mx)u'+n(x)v+f2(x)v'

(2.95)

dx

where/',

and

u'

v'

signify the first differential coefficient of/, u

and

v with

Since in equation (2.94), two variables have been introduced to represent one variable, it is legitimate to assume another arbitrary relationship between them. It is convenient at this point to assume that, respect to x.

=

f[{x)u+n(x)v in order to reduce the complexity of the

equation (2.94).

(2.96)

second

differential coefficient

of

Hence,

^V._

u'Ux) + u /;(x) + v"Mx) + v'n(x)

(2.97)

dx-

Substitution of equations (2.94), (2.95)

and

(2.97) into (2.92)

and rearranging

gives

A(x)lu" + Q{x)u' + R{x)uli +f^(x)[v'' + Q{x)v' + R(x)v]

+ + Q(x)u' + R(x)u] and

But

[u"

and

V are solutions of the

[v"

u'fi(x)

+ + v'n(x) =

+ Q(x)v' + R(x)v]

reduced equation when

cP(x)

(2.98)

are both zero because u (/)(x) is

taken to be zero.

Therefore, u'fi{x)

and solving equations

(2.96)

and

+ vyi(x) =

(2.99)

(l)(x)

(2.99) simultaneously gives

dfiix)

dx

v(t)(x)

(2.100)

— uv'

vu'

djp^-u^

and

dx

vu

—uv

The parameters fi{x) and fjix) can be obtained from integration. The final complete solution will be y

=

ufi(x)

+ vf2(x) + Aii + Bv

which includes the complementary function, equation

Example

10.

these equations by

(2.102) (2.93).

Solve

2x^

d^_—^

dx^

dy X -^ dx

+

y

=

x^e

""

I

Equation I is Hnear, and the solution therefore consists of a particular added to the complementary function. The left-hand side is homogeneous, and the complementary function can be found using the substitu-

integral

tion

X

=

e^

given in Section 2.4.3.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

58

The complementary function

the solution of

is

fl_.d_y + y = 2-^-3-^ i.e.

= Ae^ + Be^^ = Ax + Bx^

y^ y^

:.

II

dt

dt^

III

Applying the method of variation of parameters,

=

yp

where u and

let

ux + vx^

IV

v are functions of x.

dy

_,

^

dx

du

,dv

dx

dx

V

du ,dv X— + x^— = dx

^

Put

VI

dx

d^y

du

dx^

dx

_^,,

,

_,dv

,

VII Substituting V,

VI and VII

dx

into equation

xu

\vx^-\-x^'^--

Ix'^

dx

I

gives

— hvx^

-\-

ux -\- vx^

=

x^e~*

dx

which can be simpHfied

to,

2;C^ + dx

x4" = Xe-'

VIII

dx

Subtracting equation VI from equation VIII gives

du

x—- = xe dx M

:.

= -e-*

IX

Substituting this expression into equation VI,

xdv

xe

''

+ X'-- = dx

v=

.-.

-jx^e-^'dx

X

integral in equation X can be expressed in terms of the "error function" by the methods to be given in Chapter 5. The solution is

The

V

Putting equations

IX and XI

= x*e~*— ivTT ed\/x into equation

IV

gives the particular integral,

yp= —xe~''-\-xe~^ — \\lnxQd\lx = — TT^ erfV X

W

The complete

solution

XI

XII

is

y

= Ax + Bx''- i Vttx

erfVx

XIII

2.

ORDINARY DIFFERENTIAL EQUATIONS

59

In the above paragraphs, methods of finding the particular integral and complementary function have been presented for a second order linear These methods are quite general and are equally differential equation. the

The second order applicable to any order linear differential equation. equation was chosen simply for convenience in order to bring out the salient The same treatment would principles in the solution of these equations. have to be given to a third, fourth or higher order equation. 2.5.7.

Illustrative

Problems

Linear differential equations arise very frequently in the solution of chemical engineering problems. Their appearance is not surprising when one considers that chemical engineers are chiefly concerned with the prediction of rates of transfer of heat, momentum and material. Consequently the method of solving such problems will now be illustrated by three examples taken from the various important fields of the subject.

Problem

1.

Simultaneous Diffusion and Chemical Reaction

in

a Tubular

Reactor

A

tubular chemical reactor of length

is

L and

-0 ft^ in

cross section

is

order chemical reaction in which a material converted to a product B. The chemical reaction can be represented,

A

employed to carry out a

1

first

A-^B and the

specific reaction rate constant is

k h"^.

is u ft^/h, the of A is assumed to be constant at D ft^/h, determine the concentration of ^ as a function of length along the reactor. It may be assumed that there is no volume change during the reaction, and that steady state conditions are established.

feed concentration of

A

is Cq,

and the

If the feed rate

diffusivity

Solution

Take a coordinate x to specify the distance of any point from the inlet of the reactor section, let c denote the variable concentration of A in the entry section (x0), as

shown

in Fig. 2.4.

The concentration

Reaction section

L

vary in the

i

J

I

Entry section

will

|_

6x Fig. 2.4.

Tubular reactor analysis

entry section due to diffusion, but will not vary in the section following the reactor, t t Wehner,

J.

F.

and Wilhelm, R. H. Chem. Eng.

Set.

6 ,89 (1956).

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

60

A

material balance can be taken over the element of length dx at a x from the inlet.

distance

X Bulk flow of

A

x-\-5x

uy+u

;

dy -D^ dx

Diff'usionof^

The accumulation

in this case

is

dy -y dx

dx

-D'^+^[ -D'^ dx dx\

\

Sx

I

zero, but the input

must exceed the output

to supply the reaction taking place within the element.

Rate of removal of

A by

= ky dx

reaction

of unit cross-sectional area. Alternatively, this term may be considered as either an output or an accumulation and the general conservation equation (1.25) applied directly. Thus, since the reactor

uy-D-ldx

is

[uy \

-D-^ + '-(-D^dx\ =kySx

+ u-^dx] dx

/

dx

I

dx\

dxj

I

J

SimpHfying, dividing by dx, and rearranging, d'^y

dy

dx^

dx

D-j^^-u-{:-ky =

^

II

Similarly, the material balance in the entry section gives

dc _ d^c = D— ^-M— dx dx

III

which can also be obtained from II by removing the reaction term. Equations II and III are both second order Hnear differential equations with zero right-hand side, thus the complementary function is the complete solution in both cases.

The

auxiliary equation of equation II

.'.

where

is

Dm^-um-k = m = u{l±a)l2D a

= Vl + 4/cD/w^

IV

V VI

Therefore, the solutions of equations II and III are

y

and

= ^expg(l + a)]+Bexpg(l-a)]

c^a + pexpiuxjD)

VII VIII

2.

ORDINARY DIFFERENTIAL EQUATIONS

which contain four arbitrary constants, A, B, a and

p.

61

The four boundary

conditions are at

X

at

X

= — 00, = 0,

at

X

=

c c

IX

Co

dc

y dy

dx

dx

X XI

0,

dy at

= =

X

=

XII

dx

The

first

condition specifies the state of the feed stream, and the second The third condition, taken with equation

ensures continuity of composition.

X,

necessary to conserve material at the boundary assuming that the equal in both sections. The final condition forbids diffusion

is

diffusivities are

out of the reactor and is necessary as a conservation law for the section following the reactor. A full argument supporting the last condition is given in the reference quoted at the start of the problem. Equations IX to XII give respectively,

= Co +p = A+B 2p = A(i + a) + B(l-a) a

oi

+ a) exp

—

[uL

XIII

XIV

XV

1

(1

+ fl) +B(l-fl)exp

[i"-">]

=

XIV and XV gives = A{l-a) + B{l + a) 2co XVI and XVII for A and B gives

XVI

Eliminating a and ^ from XIII,

Solving

.

2co(a-l)

I

XVII

uLa\ XVIII

K B where

+ l)

2co(a

/uLa\

XIX

K

K = (a + 1)^ exp (uLallD) -(a- 1)^ exp - uLajlD)

XX

(

Putting these values of

y

r'^p©

(a

A and B

into equation VII gives the final result,

+ l)exp'»-w(r-e-««)

terms of the other variables and using the fact

,-,-V-...(^) — r

and then

Substituting the above value of >'o into equation XIII

XXI

XXII

v

into equation

gives

mz XXIII Equation XXIII gives the weight fraction of glycerine

in the extract

a function of column height h. Allowing for the solubility of water in tallow, taking

and using the data given

L= .-.

r

=

4-544,

flow rates,

in the problem, gives

G=

8540,

p

mean

phase as

=

Solving equation XXIII for

0-198,

Ka

with

3760, q

=

H=

>;o

=

0-188

0-00348 Ka, 72-0

ft,

v

=

l

+

?^ Ka

gives the value of the

mass

transfer coefficient as

Ka = 14-2 lb glycerine per hour per ft^ of Ka = 14-2, values of y, y'^ and z can

XXIV

With the value be determined as functions of column height with the aid of equations XXIII, VI, V, IV. The results of the arithmetical calculations are shown in graphical form in Fig. 2.6. Figure 2.6. shows that the chemical reaction is virtually complete in the bottom 30 ft of the column, or 40% of the column.

t

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

66

73-5c

67-5

cerine at interfoce

e

O'l

in

extract

0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9

10

Weight fraction Fig. 2.6.

Concentration distributions in hydrolysing column

The above analysis of a continuous hydrolysing column was abstracted from "The Analysis of a Continuous Fat Hydrolysing Plant", by Jeffreys, Jenson and Miles.

2.6.

Simultaneous Differential Equations

These are groups of differential equations containing more than one dependent variable but only one independent variable, and in these equations, all the derivatives of the different dependent variables are with respect to the one independent variable. This distinguishes an equation which is a member of a group of simultaneous differential equations, from a partial differential equation which will contain more than one independent variable

and consequently

will also contain partial derivatives.

All the derivatives in

the simultaneous differential equations will be total derivatives.

The

basis of the solution of simultaneous differential equations

is

alge-

braic ehmination of the variables until only one differential equation relating

two of the variables remains. The final differential equation is solved in the ordinary way by one of the conventional methods. If the equation is Hnear with constant coefficients, one of the methods already presented in this chapter will be suitable, otherwise a method to be given later must be employed. t Jeffreys, Jenson

and Miles, Trans.

Inst.

Chem. Eng. 39, 389

(1961).

ORDINARY DIFFERENTIAL EQUATIONS

2.

67

The variables to be eliminated depend on the set of equations, and the ultimate information required from the mathematical analysis. The variable may be the independent variable or one of the dependent variables. These possibihties will 2.6.1.

now

be considered.

Elimination of the Independent Variable

Consider the pair of simultaneous differential equations,

and

dx ^=/i(^,>')

(2.103)

^^•^^(^'^'^

^^-^^^

is the independent which x and y are the dependent variables and two functions oi x and y. Equations of this kind can be solved by ehminating the independent variable by dividing one equation by the other, say equation (2.104) by equation (2.103) to give

in

variable. fi{x,y) and/2(x,>^) are

(2.105)

T=^tF\ which

is

an ordinary

differential

equation in x and y, and would be solved

as such.

Example 1 solved in paragraph 2.3.3 was really one of the above type. EUmination of the time gave a homogeneous equation which was solved by standard solution methods for homogeneous equations. The above method of ehminating the independent variable is restricted to first order first degree equations in which the independent variable appears only in the derivative. Consequently its scope is very Hmited. 2.6.2.

Elimination of

One

or

More Dependent

Variables

This approach has considerably wider application, however it becomes very involved with equations of high order and therefore it would be better to make use of matrices if there are many equations. The matrix methods of solving large sets of simultaneous equations are presented in Chapter 12, whereas the following paragraphs will be restricted to the simpler situations. Thus the independent variable can be eHminated by, (a) Taking advantage of the algebraic properties of the differential operator. (b) Systematic ehmination.

By Use of the

It has been pointed out that the of algebra and therefore can be used in the ehmination process. The method is best illustrated by an example.

(a)

operator

Differential Operator.

D obeys the fundamental laws

Example

1.

Solve

+ D-6)y + {D^ + 6D + 9)z = (D2 + 3D-10)>' + (D2-3D + 2)z = {D^

and

I

II

MATHEMATICAL METHODS

68

Equations

I

and

II

IN

CHEMICAL ENGINEERING

can be written

= (D-2)(D-h5)y + iD-2){D-l)z=0 by (D + 5) and IV by iD + 3) to give (D + 3)(D + 5)(D-2)>' + (D + 5)(D + 3)^z = (D + 3)(D + 5)(D-2)>' + (D + 3)(D-2)(D-l)2 = (D-b3)(D-2)y + {D + 3yz

and Multiply

III

and

III

IV

V VI

Elimination of y and simplifying the remainder gives

(llD + 13)(D + 3)z

whose solution where

A and B

=

VII

is

are arbitrary constants.

Substitution of z from equation VIII into equation

(D2 + D-6)y

The terms involving Be add up The particular integral of y is

to zero.

F

1

D' + D-6

=

IX

£e-^^^/'^

(n) -(-n-) + ^J^

^"^

yp

gives

r/13\2

E=

where

=

I

~

'

5

13X/11

D-2

D + 3j

XI

Ge-'''^''

G= -^E^^A

where

The complementary function

is

y,

and the complete solution y

=

X

=

He^''+Je-^''

XII

is

He^''+Je-^''+^Ae-^^''^^^

(b) Systematic Elimination.

This term

is

proposed for the eUmination

amount of algebra involved

to a minimum. method consists of setting up a table indicating the number of times and in what form each variable appears in the simultaneous equations. Then the variable that appears in the simplest manner is first eliminated by making the appropriate substitution. Following this the second least frequent variable is removed from the remaining equations by the appropriate substitution, and the substitution process is continued until only one

process which reduces the Essentially the

equation remains. Generally terms which appear as derivatives are left to the final steps of the substitution process. The following example will demonstrate the technique.

Example 2.t 10000 Ib/h of sulphuric acid (specific heat 0-36) is to be cooled in a two-stage countercurrent cooler of the following type. Hot acid t Suggested

Birmingham.

by W.

M. Crooks, Dept.

of Chemical

Engineering,

University

of

:

2.

ORDINARY DIFFERENTIAL EQUATIONS

69

at 174°C is fed to a tank where it is well stirred in contact with cooUng coils. The continuous discharge from this tank at 88°C flows to a second stirred tank and leaves at 45°C. CooHng water at 20°C flows into the coil of the second tank and thence to the coil of the first tank. The water is at 80°C as To what temperatures would the it leaves the coil of the hot acid tank. if due to trouble in the supply, the cooling water suddenly stopped for one hour? On restoration of the water supply, water is put on the system at the rate Calculate the acid discharge temperature after one hour. of 10000 Ib/h. The capacity of each tank is 10000 lb of acid and the overall coefficient of heat transfer in the hot tank is 200 C.H.U./h ft^ °C and in the colder tank 130 C.H.U./h ft^ °C. These coefficients may be assumed constant.

contents of each tank rise

Solution

The steady and the steady

coohng system are shown in Fig. 2.7, data calculated from the illustrated conditions are

state conditions of the state

Water rate before failure of supply Intermediate water temperature between the two tanks Heat transfer area of coil in hot tank Heat transfer area of coil in cold tank

80

o

40^*

20° ^

[

.

7740

Ib/h.

40°C. 69-3 ft^.f 95-8

ft^.

_ /

/

1

88°

i Fig. 2.7.

= = = =

g

45°

Sulphuric acid cooling system

Water Fails

Water flow

is

zero.

For the purpose of the analysis

acid flow rate be

M Ib/h

acid specific heat be

C

feed acid temperature be Tq °C acid temperature ex tank 1 be T^

= = =

let.

10000 0-36

174

°C acid discharge temperature be T2 °C capacity of each tank be time be 6 hours.

V lb

10000

t In practice, these figures would be rounded off to 70 and 96 but to reduce errors of calculation the extra figure has been retained (see Chapter 10).

f

^

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

70

Then

the heat balances are as follows.

Output Accumulation

Input

Tank

MCTq - MCT^ =

1

FC—

I

dO

AT'

MCT^ - MCT2 = VC—^

Tank 2

II

dO These are simultaneous first order = solved in succession. Because becomes

M

differential equations,

V in

but they can be

the numerical example, equation

I

which has the solution,

To-Ti = and when 6

=

T^

0,

=

Ke-'=n4-Ti

88°C the steady

Therefore when 9

=

state temperature.

K=

.-.

IV

S6

1*0,

V

Ti= 174-86e-^'°= 142-4°C Similarly for tank 2,

from equation

II,

174-86^-^-72 =

d_T2

-^

VI

dO This

is

factor

a

first

order linear equation which can be solved using the integrating

e^.

and solution of equation VI

Substitution of the integrating factor

T2

=

which gives for

gives

= 174-(860 + 129>-^

VII

1-0,

T2

=

VIII

94-9°C

Water Supply Restored

Water flow derivation

be

^3

rate

let

is

now

10000 Ib/h but for the purpose of the following Ib/h and let water supply temperature

the water rate be

W

°C.

water temperature ex tank 2 be t2°C. water temperature ex tank 1 be /i°C.

Then

the heat balances are as follows.

Accumulation

Input

Output

1

Wt2 + MCTo

- (Wt^+MCT^) =

Tank 2

Wt^ + MCT^

-

rl'T

Tank

(Wt2 + MCT2)

KC— du dT.

= VC--^ du

IX

X

ORDINARY DIFFERENTIAL EQUATIONS

2.

The heat

transfer rate equations for the

71

two tanks are

{T,-t,)-{J,-h) 1 A r ^('-'^)^ = n ^'^' [ln(r.-r,)-ln(r,-J vv(,

and

^'


oo, the series

is

termed "conver-

gent".

S„-*±co as «->oo, the series is termed "divergent". In other cases, the series is termed "oscillatory". When considering oscillatory series, the value of S„ riiay oscillate between either finite or infinite limits and a further classification can be based on this (b) If (c)

distinction.

In the application of series to differential equations, u„ will be a function of X, consequently S„ will also be a function of x. Because x may take positive or negative values, or even complex values it is desirable to allow u„ to be complex. The properties of complex numbers will be investigated in Chapter 4, but for those readers who are not familiar with the elementary properties of complex numbers, the term "absolute value" refers to the numerical magnitude of the number and is signified by vertical fines. |i/„|

The

series (3.1)

is

=

u„.

said to be "absolutely convergent" if the series .|"l|

is

absolute value of

+

|"2|

+

|"3|

+

+

-..

(3.2)

lWn|.

convergent.

Example

1.

w;,

.*.

This series

is

S„

=

z"

= z + z^ + z^ +

S„-

ratio z.

II

1-z

|z|oo as

«^oo,

therefore the series diverges if z

positive, or oscillates for all other values

(c) If

I

^_^ 1-z

and

+ z"

common

a geometrical progression with

..

,.,

z

Hence the

=

S„

1,

=

of

is

real

z.

n which tends to infinity as n increases indefinitely.

series is divergent.

(d) If |z| = 1 and z 7^ 1, the series oscillates within finite limits. This series thus illustrates all four types of behaviour. 3.2.1. Properties

of Infinite Series

The following

are three useful properties of infinite series, a series contains only positive real numbers or zero, then it must be either convergent or divergent; it cannot oscillate. (ii) If a series is convergent, then w„-^0 as «- 00. This is a property of a (i)

If

:

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

76

convergent series it is not a proof of convergence. A divergent series for v^hich u„-^0 is illustrated in example 2, part (b) below. (iii) If a series is absolutely convergent, then it is also convergent. Statement (iii) can be proved as follows by referring to (i). If u„ is real, put u„ = a„-b„ where a„ = i(\u„\ + u„) and b„ = i (\u„\-u„). Thus a„, the series of positive terms is separated from — b„, the set of negative terms so that neither a„ nor b„ is negative. ;

/.

fli

+ fl2 + «3 +

...

+ fl„ ^

|wi|

+ |w2| +

..-

+ |Wn| ^ S

Since the series a„ only contains positive or zero terms, and the sum cannot exceed S, then statement (i) above proves that ^a„ is convergent. Similarly,

^b„

is

convergent.

between +S and —S. If u„ is complex, a by separating real and imaginary parts. Thus, absolute convergence implies more than mere convergence.

must tend to some

finite limit

similar proof can be constructed

3.2.2.

Comparison Test

Only convergent series will be of importance in the solution of chemical engineering problems, but the following tests will be stated in full as they determine convergence or divergence. It is important to appreciate in these tests that if

shown that a particular series is not convergent, mean that it is divergent; it may oscillate.

it is

not necessarily

The comparison (i)

test is the simplest

If |w„|

shown below, and applying

the above

algebraic inequaUty, gives

Xj_J.4.i4.i4.i4._l_4.J^

L._I_4.JL-

= (i+iV)+(i+iV)+(i+TV)+(i+TV)+* >i + i4--2-4-^4-J- = 1 (l)

.-.

The the

+ (i + i + i) + (i+...+lV) + (TV+-. + 4V) + .- > 1 + 1 + 1 + 1 +

rth bracket contains 3'"^ terms centred first

terms is greater than to exceed any chosen

i{3''— 1)

can be made

on

3^

"'".

By taking

r.

finite value,

Therefore, the

...

sum of sum

sufficient terms, the

and hence the

series is

divergent. (c)

p

1

for

n>

1

by comparison with part

(b) using the test

of

(ii).

Ratio Test

This also consists of two parts

^

(i)If for all (ii)

n>N,

then the series

is

k>l

absolutely convergent.

N, then the series is divergent, means that when I + l/n has been subtracted from the ratio of terms, and the result has been multipHed by n^ if the function

where \f(n) This

\

test

successive

;

f(n) remains finite as n-^oo, then the series is divergent. These last three tests fall into a natural order of increasing resolving

power, each one testing the behaviour of the ratio of successive terms in greater detail. As mentioned in Section 3.1, a series is of little practical value unless it is rapidly convergent; consequently, the more powerful tests are rarely necessary. 3.2.6.

The Integral Test

If /(x)

x>N, if

is

where

and only

if

a positive integrable function of x which decreases for all A^ is some positive integer, then the series ^/(«) converges f(x) dx exists.

;

SOLUTION BY SERIES

3.

79

Each of the above tests is sufficient to prove convergence or divergence, but the large number of series for which they are inconclusive indicates that all of the tests need not necessarily be satisfied by any particular series. The following example illustrates an inconclusive case which is nevertheless convergent.

Example 4

The

successive values of w„/w„+i are 4

3 2'

3'

where

8

9

16

9'

4>

2 7'

27 8

>

*••

after the first term, alternate values are greater

and

less

than unity.

Therefore, the ratio test "proves" the series to be alternately convergent and divergent.

The the

value 3.2.7.

is

series

sum of

of odd terms

the even terms

is

a geometric progression with sum to infinity 2 Therefore, the series I is convergent to the

is 3.

5.

Alternating Series

an alternating

series if

Wl~"2 + W3~"4+--w„>0 for all n. If, in

successive terms decreases for

all

(3-3)

addition, the magnitude of

values of n and w„->0 as «->oo, then the

series is convergent.

Proof S'2„

=

(Wl-W2) + (W3-W4)+- + (W2m-l-«2m)

is positive and the next term W2m + added, 52^ and S2ni+i must both be greater than zero.

Since each term in the brackets

^2m+l

i

J^iust

be

= Ui-{U2-U^)-(u^-U5)-...-(U2m-U2,n+l)

Again, each term in the brackets is positive, and removing the last term to obtain S2m reduces the value further. Therefore, 5*2^ and Sj^ + i are both less than Wj.

^

.*.

Both S2ni and S2m+i increase as they are separately convergent.

5„

^

for all n.

Wi

m increases

but never exceed u^, and hence Because t/„^0 as «-»oo, S2m and S2m+i must both converge to the same limit S. Therefore the alternating series

(3.3) is convergent. 3.3.

Consider the power

Power

Series

series 00

ao If

\aJa„+i\^R

+ aiZ + a2Z^+

...

=

(3-4)

Y.^n^"

as «-^oo, then the ratio test can be applied as follows.

|w„+i

a„ + iz"-'^

a„+i

\z\

\z\

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

80

as «-> 00. Therefore, if

|z|

< R,

\uju„ +

unity and will be convergent by part

i

\

(i)

will tend to some limit greater than of the ratio test. In detail, if |z| =

R-e where

some small

e is

positive

number

R

less

than R,

R

R=

1

+

>

1

+

R-£ (s/R)

= l+(e/i?) in Section 3.2.3, and substituting into equation (3.5) proves that the series (3.4) is convergent. If |z|>i?, a similar proof shows that the series (3.4) is divergent. Therefore, ^^^z" is convergent for |z|R. R is termed the "radius of convergence" of the power series and as shown above, R is the hmit of the ratio of successive coefficients. If \aJa„+i\-^co as «->oo, then the power series will always converge. If \aJa„ + j^\-^0 as «->oo, then the power series will only converge if z = 0; i.e. if all terms are zero.

Taking k

3.3.1.

Binomial Series

The

series

obtained by expanding

(1

+z)^

is

H.,, + £(£zi),.^K£zll(£zlV^ 1.2.3

1.2

where real.

(

j

signifies the

binomial

coefficient, z

be complex, but p

=

p(p-l)(p-2)(...)(p-n + 1.2.3.

a„

=

«n+l Therefore, the binomial series

n

+1 1

p-n is

l)

...n

as M -> 00

convergent for

|z|


^ can be determined for any value of x no matter how small, but not for zero. The series expansion of sin x, equation (3.10) can be used .

3.

SOLUTION BY SERIES

83

in this example, giving

-{'44-> x^

X"

In the limit as x-^0, it can be is valid for all x no matter how small. seen that y-^ 1 and this is the obvious value to take for y when x = 0. This process of resolving indeterminate fractions can be repeated k times

which

provided that k'n

+

i

=

aiyn + a2yl + a^yl +

Substitution of the series II into equation

I

and

...

II

equating the coefficients of

3.

like

SOLUTION BY SERIES

85

powers of j„_i, gives

Ca^-Eal-D A — a^ — ai a->,

^

«4

=

=

^

A-a,-al = Ca2 + (a2-E)(2a^a2 + ^al) + 3ala2a^ A-a^-aj

Ca„ + (a2-E)Y,aiaj-\-a2Y.aiaja,, + a^Y^i^j^kai +

-"

+ ^nnai"~^a2 III

where J^a^ay signifies the coefficient of y„"^^ in the expression of y„ + ^^ using equation II;^^^^^^^;^ signifies the coefficient of >^„""^^ in j^„ + i^; etc. In a numerical example which was attempted by the authors, the first few terms decreased quite rapidly and the method appeared to be successful. It was necessary to prove that the series was convergent by using the result of Section 3.3 that if \a„/a„+i\-^R as n-^co, then the series would be convergent for \y„.i\

This result

1.

is

physically

wrong

as

may be

seen by inspecting equation II. For small values of ;^„_i, >^„ will be greater than y„_^ and the column must therefore be generating the transferable gas rather than absorbing

it.

is useless for one particular case a>l, and part (a) throws serious doubt upon the convergence of the series, even though the first few terms decrease quite rapidly. The method should not therefore be used on an equation of the type I.

Part (b) shows that the method

£'°

=

1,

and k

small,

3.4.

Simple Series Solutions

In Section 2.5.1, the Hnear differential equation with constant coefficients

was solved and shown to have solutions for the complementary function of the types >;

y

and

By

y

= = =

referring to Sections 3.3.2

in ascending

y

power

=

A{\

series

.lie'"^^

+ yl2e'"^"

(2.45)

{A + Bx)e'"'' {A cos a.x

and

of x.

-\-

B sin ax)e^''

3.3.4, all

Thus

(2.49)

of these solutions can be expanded

for the

first

case,

+ m^x-^{m\x^ + ,..) + B{\ + m2X + \mlx^ + ...)

(3.22)

3.

SOLUTION BY SERIES

87

This series form of y can be accepted as a solution of the equation provided that the differential equation is satisfied by it and the series is convergent. It can be shown that the first condition is satisfied by differentiating the series term by term and substituting the result into the differential equation when all terms will cancel out. The series can be checked for convergence by any of the tests given in Section 3.3. Because the linear differential equation with constant coefficients always

vaHd series solution, it is natural to expect at least some differenequations with variable coefficients to possess series solutions. Since the majority of series cannot be summed, it is to be expected that some solutions must be left in series form. The general second order linear differential equation can be expressed in possesses a

tial

the

form

dv

d^y

-^ + PW^_ +

QW>; =

(3.23)

lfP(x) and Q{x) can be expanded in a convergent ascending power series in then the method of solution is as follows.

X,

P(x)

eW Put

y .*.

—ax

= Po + PiX + P2^2 + = Qo + QiX-\-Q2X^ + = ao + aiX + a2X^ +

(3.25)

...

(3.26)

...

=

ai

+ 2a2X + 3a2X^ +

d^y .*.

(3.24)

..-

—"2 =

2(22

ux

...

+ 6(33 X+ 12^4 X+...

and

substitute these series into equation (3.23).

(2fl2

+ 6«3X+12fl4X^ + ...) + (Po+PiX + P2x2 + ...)X x(«i+2a2^ + 3a3X^ + ...) + (2o + 2i^ + 22^^ + ...)x x{ao + aiX + a2X^ + ...) =

(3.27)

equation (3.26) is to be the solution of equation (3.23), then all terms in equation (3.27) must cancel out. Thus, coefficients of Hke powers of x can be equated. If

+ Poa, + Qoao = + 2Pofl2 + i'ifli + Qo«i + Qi«o = 12a4 + 3Pofl3 + 2Pia2 + P2ai + eo«2 + 6i«i + 62«o = .-.

and and

2a2

6fl3

(3.28) (3.29) (3.30)

etc.

Equation (3.28) can be rearranged to determine a2 in terms of a^ and Oq thus,

ci2=-iPoa,-iQoao Substituting equation (3.31) into equation (3.29) of 1.

When

;

a,+ i{r + c +

.*.

The

first

solution

is

given by c

l)= -a,

=

2,

fl,+ i(r

and by successive

(unless c

=

0, r

=

V

1)

thus

+ 3)= -a,

substitution,

_(-ir^

"'*'-

(r+3)!

^

/x^

.3

x^

x'

The second

solution, v^hen

yi

= 2ao(e"*-l + x) c = 0, becomes

= ao-aoX-ha2X^2

=

\

^41--)

'^° (21 -3"!

^2^^

-— +

ao(i-x) + 2a2(e-''-l

VI

^2^"^

3.4

+ x)

VII

All coefficients after the first two can be found in terms of ^2 and there are thus two arbitrary constants in the second solution making three in all. Inspection shows that the series associated with ^2 in equation VII is identical with the first solution as given

One

by equation VI.

thus been obtained twice, and in Case Illb

when a

coefficient

determinate, the series obtained by using the smaller root

complete solution.

The complete

solution of equation

I

is

solution has

becomes

in-

in fact the

can therefore be

written

y

— Iqi, and B =

= A{.-x) + Be-''

where

A =

3.5.5.

Summary of the Method of Frobenius

Qq

VIII

2a2 in equation VII.

The examples given above have been specially selected so that the series solutions could all be summed. The solutions can be checked quite easily by differentiation and substitution, thus demonstrating that the series method gives correct results. It must be emphasized that the methods of Chapter 2 should always be tried before resorting to the method of Frobenius.

Too much time should not be devoted

to trying to

sum

the

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

100

series obtained because on many occasions the solution of problems of engineering importance will not be expressible in closed analytical form. The method has only been described for expressing the solution as a series of powers of x. Solutions in powers of {x — Xq) can also be obtained by moving the origin along the x axis to Xq and then proceeding as before. The method will be appHcable if the convergence conditions are satisfied at

the

new origin. The differential equation

—^ + xt{x)—+ ^^W:r: dx

^

z^ dx^

^My = ^

(3.33)

can be solved by putting

y

=

(ao^O)

I.a„x"^^

(3.36)

F(x) and G(x) can be expanded in a convergent series of non-negative powers of x for all \x\

4 ft

U tn

T-^ '

^

^

I

t.;t

^"

I

—

Jl

Gas preheater

Fig. 3.2.

Output

Input

-kA

By conduction

A

is

dT

dx

dx

d /

is

dT\ dx

upC^[

T+

—Sx

-

the cross-sectional area of the pipe, and

steady state the accumulation

^

dx

ndh{T^ -T)Sx

Wall heat transfer

where

dT

upC^T

By mass flow

4in dia

d

its

diameter.

Since in

zero,

T'T-i

-kA-— + upC„T + 7idh(T^-T)3x ^ dx

— kA— dx dx

dT\

,

/

,

dT

\

d^T dT ndh(T^-T)= -kA—-^ + upC -— ^

dx

dx"-

Rearranging,

upC.dT

d^T

ndh^^

^^

^

II

dx'

Putting

t

= T^-T, d^t

dx^

upC^

dt

kA dx

-

4/2

— kd

r

=

III

Inserting the numerical values,

—4 -688o4^-3000x-^r = dx"^

dx

IV

3.

Put X

=

SOLUTION BY SERIES

z^ to rationalize the coefficient of

an attempt

is

made first

c

V

dz

dz"^

the results for the

t,

z—,-il + 13160z^)^-12000zh =

:.

If

105

to solve this equation

few terms

=

by the method of Frobenius,

are,

or 2 from the indicial equation

Taking

c

=

a^=0 a^

=

2 ^3

3440^0

«4

= =

800^0 7-9

x lO^ao

coefficients are increasing at an alarming rate, yet a test of equation V shows that the series must be convergent. This is a case mentioned earher where a convergent series is of no practical value since more than 100 terms would have to be calculated to determine even the first solution. The trouble arises because two of the coefficients in equation IV are much larger than the coefficient of d^t/dx^. The second derivative arises from the gas conduction term and if this is neglected, equation IV becomes

The

+ 0'436x-^t = ^ dx

VI

which can be solved by separating the variables. Viz.

= aexp(-0-872x^) = 600 — 70 = 530. at x = 0, is = 530exp(-0-872x^) T = 600-530exp(-0-872x")

VII

r

The boundary condition .-.

?

VIII

f

.-.

The exit gas temperature is therefore 507°F. The approximation of neglecting the second

differential term in equation IV can be checked by comparing the neglected term with one of the other terms. Differentiating VIII twice gives

-^= -23Mx-^exp(-0-872x^) dx

-4 = 23M(ix-3/2 + 0-436x-^)exp(-0-872x^) dx

d^t Lc.r.r.dt

'6880

dx^l

which

ix-3/2-0-436x-' ^

dx

6880x"" 1

1

13760X

15780;c"

small except when x is small. 10"^, the error ratio = 7^%.

is

If X = The error has thus been made over a justifies the

above

result.

negligible part of the pipe

and

this

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING,

106

3.6.

The

differential

equation

x^4^ + known

Equation

Bessel's

X

^+

(x^

- k^)y =

(3.48)

where A: is a positive or zero This equation arises so frequently in practical problems that it has been studied in great detail and this section will be devoted to giving an outUne of the derivation of the various solutions by using the method of Frobenius. Because the equation occurs so frequently, the series solutions have been standardized and tabulatedf thus eliminating the necessity of working through the series solution for each individual problem. Many other second order differential equations can be reduced to Bessel's equation (3.48) by a suitable substitution, and they too can then be solved by using the standard tables. Applying the method of Frobenius by putting is

as Bessel's equation of order k,

constant.

y

and

its

The

indicial equation

differential

= ^a„x"^^

(3.36)

forms (3.37) and (3.38) into equation

(3.48).

is

aQc{c —

l)

c

=

+ aQC — k^aQ =

which has the two roots

The

difference

between the roots

/c

is

or

-k

(3.49)

thus 2k, and the type of solution will

depend upon the nature of k. Equating coefficients of x^'^^, fli(c .'.

Because c /:

=

!,

+ l)cH-ai(c + l)-k^fli =0 a^(c + l + k)(c+l-k)=0

= ±k, and k^O,

c= — ^, c+1— =

then c+l+/:>0, but for the special case Therefore,

k

=

i

relation

is

obtained by equating coefficients of x''^^

unless

The recurrence

0.

/:

andc=-i,

ai=0

aXr + c)(r + c-l) + aXr + c)-k^a, + a,_2 =0 aXr + cy-a,k'=-a,_2 .-.

••

(3.50)

^r — 2

^'~ (r

(3.51)

(3-52) /"3

c-jA

+ c + k){r + c-k)

t For example, Jahnke and Emde, 'Tables of Functions with Formulae and Curves," Dover Publications, New York. Or Watson, "Theory of Bessel Functions," Cambridge

University Press (1922).

SOLUTION BY SERIES

3.

The

first

relation

=

solution for c

k, will

be normal in

Combining equations

(3.51)

and

=

ai r

all

and the recurrence

cases

becomes r(r

and

107

can be replaced by 2

w

(3.53) gives

=

^3

m

by

(w- 1),

^5

=

...

=0

(3.55)

in the recurrence relation (3.54)

^"" Replacing

+ 2/c)

(w — 2),

2m(2m + 2/c) etc.,

and making successive substitutions as

before,

^2m

(-ir^o

^

22'"m!(m + fc)(m + /c-l)(...)(/c+l)

m!r(m + + l) /c

where r(A:+l) is the Gamma function or generalized factorial whose properties will be discussed in Chapter 5.

The

first

solution of Bessel's equation

is

thus

„rom!r(m + /c+l) The

Bessel function of the

Defining a

new

first

kind of order k

arbitrary constant, the

first

is

defined as

solution of equation (3.48)

is

thus

y,=AUx) The form of

the second solution, for c

= —

(3.57) A:,

depends on whether or not

2k is an integer, and the method of solution diverges into four channels corresponding with the cases of the method of Frobenius. 3.6.1.

Case I (2k

is

not an integer or zero)

In this case, the second solution is obtained from the in equation (3.56) by —k, giving the complete solution

y 3.6.2.

Case II (k

=

=

first

by replacing k

AJ,(x) + BJ.,(x)

(3.58)

0)

Returning to the recurrence relation (3.53) and putting «r

= -«r-2 (r + c)'j

A:

=

0,

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

108

= from equation (3.51), r can be replaced by expression for the coefficient of the general term is Since a^

^""

2m

again and the

(-ir^o

^

(2m + c)2(2m + c-2)2(...)(c + 2)'

Defining u{x,c) as given in Section 3.5.2,

"^""'"^

^

^^"^^^

Jo (2m + c)^(2m + c-2)^(...)(c + 2)^ and putting

Differentiating equation (3.59) with respect to c

c

=

gives the

second solution, l^\2m

"-'^

(-irao(ix)

,n«-,-l-l-...-i

(m !)

=

2

m

3

=-H'"'-i/"sr(-^-^)] The expression in brackets is the Neumann form of the second solution, but an alternative form named after Weber is more frequently used. The Weber form is obtained by adding (y — In 2) Jo{x) to the Neumann form and multiplying the result by (2/n). Thus y2

=

BYo(x)

where

l^oW

=

2ri...i..x . [ln(ix) + 7] Jo(x) .

....

..-,

7r„,=

71

is

the

and

y

Weber form of is

which

=

+- +

y

lim (1

m-coV

2

is,

V

^

Case Ilia {k

is

is

-+...

+

m

=

c

= —k,

1

,

+:;+•••

+.2.

m

2

Inm

)

=0-5772...

/

=

is

thus (3.60)

an integer) first

recurrence relation in which ^2* appears

=

-«2fc-2

ci2]ik-^c){2>k-^c)=

-ajk-i

the left-hand side

is

zero but the right-hand side

is

not zero

an even coefficient. The method for Case Ilia given must therefore be followed.

because a2k-2 Section 3.5.4

\

AJo(x) + BYo(x)

a2ji2k-^cf-a2kk^

When

( 1

defined by

3

Ik is an even integer and the from equation (3.52),

or

.

/ (mi;

i

solution of equation (3.48) with k

y

3.6.3.

(-l)'"(ix)''"/,

the Bessel function of the second kind of order zero,

Euler's constant

The complete

2 -- ^ E

is

in

— SOLUTION BY SERIES

3.

The eventual

109

solution using

IS

y,

=

= -Bl\nix + y]Ux)--

BY,{x)

^

^—TT^^tt^)"""'

— y — — [2 1+am + k

^^

nj^o

m\(m + k)\

+

...

+

ml

—+ +-+ 1

...

m + kj

where y is again Euler's constant, and the Weber form has been found by adding agiy — In 2) J^ix) to the Neumann form and multiplying by (2/7r). Ykix) is the Weber form of the Bessel function of the second kind of order k. The complete solution of equation (3.48) is thus

= AUx) + BY,{x)

y

when k 3.6.4.

is

an

Case Illb {2k

The

first

(3.61)

integer. is

an odd integer)

recurrence relation in which a 2k appears

M/c + c)(3/c + c)=-a2/c-2 When

is,

from equation

(3.52),

(3.62)

= —k,

the left-hand side is zero, but the right-hand side is an odd and in all cases considered so far, the odd coefficients have been zero. This must now be investigated further. If A: = |, equation (3.50) shows that a^, the critical coefficient, is indeterminate and the solution follows Case Illb giving two normal power series. If A: = 3/2, the critical recurrence relation (3.62) becomes c

coefficient

«3(T + c)(f+c)=

and

-a,

=

Continuing this argument, ajk will 0, aQ is indeterminate. always be indeterminate if 2k is odd, and the solution again consists of two since a^

normal

series.

y

The general

k

is

(3.63)

=

AJk(x)

+ BJ.k(x)

(3.58,63)

not an integer or zero, or

y

k

AJk(x) + BJ_ki^)

solution of Bessel's equation (3.48) can therefore be written

y if

=

=

AJk(x) + BYk(x)

(3.60,61)

an integer or zero. Examples of the use of the above Bessel functions occur quite frequently in chemical engineering when the problem involves partial differen-

if

is

tial equations, but their use in ordinary differential equations is rare in chemical engineering but more frequent in other branches of engineering. Rather than present an irrelevant example at this point, the reader is referred

to

Example

3 in Section 8.7.5

form one stage

in the solution

and Section

8.7.3,

where the above solutions

of certain partial differential equations.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

110

Modified BesseVs Equation

3.6.5.

now been solved for all real values of x, and sum of two linearly independent functions of x

BesseFs equation (3.48) has the solution given as the

each multiplied by an arbitrary constant. Two kinds of Bessel function have been defined both of which are needed to express the solution if k is an integer or zero, but two versions of only the first kind if k is not an integer or zero.

A

further type of differential equation can be solved with the aid of

Bessel functions

;

it is

+ ^^.-i^''^ ^^)y = ^^-J^ dx'^ dx

(3.64)

Equation (3.64) can be obtained from Bessel's equation (3.48) by replacing / = ^(—1). The solution of equation (3.64) can thus be

X with ix where written

y if

k

is

=

AJk{ix) + BJ_,,{ix)

(3.65)

not an integer or zero, or

y

=

AJ„(ix) + BY^(ix)

(3.66)

k

is an integer or zero. In Section 2.5.2 it was found that the solution of the second order linear differential equation with constant coefficients could be expressed more conveniently in terms of real trigonometric functions than complex exponential functions when the auxiliary equation had complex roots. Here, instead of using the Bessel functions of a complex variable, it is similarly convenient to define modified Bessel functions of a real variable so that the above solutions can be expressed in real form. The necessary definitions are

if

/,(x)

=

e-*""A(ix)

(3.67)

\2-^J _ y ~ „%m\r(m + k+l)

K,{x)

=

e*'*^ ""'[A(ix) + iT,(.x)]

(3.68) *-,^

(-l)'^i[ln(ix) + v]/,(x) + i

2=

(-l)'"(/c-m-l)! ^

'

m ,

^

^J^o

The

(-ir(ixy'"^^[^

y if

k

is

k

is

is

'-(ixy

1,1

ml

1

1

m-\-k\

thus

= Ah(x) + BI.,(x)

(3.69)

not an integer or zero, or

y if

1

[2

m\im-\-k)\

solution of equation (3.64)

\, fn\

=

AI,(x) + BK,(x)

(3.70)

an integer or zero.

Ik(x) is called the

and KfXx)

is

modified Bessel function of the

first

kind of order

the modified Bessel function of the second kind of order k.

k,

SOLUTION BY SERIES

3.

111

Example, Heat Loss Through Pipe Flanges. Two thin wall metal pipes of 1 in and joined by flanges | in thick and 4 in diameter, are carrying steam at 250°F. If the conductivity of the flange metal k = 11^ Btu/h ft^ °F ft~^ and the exposed surfaces of the flanges lose heat to the surroundings at Ti = 60°F according to a heat transfer coefficient h = 1 Btu/h ft^ °F, find the rate of heat loss from the pipe, and the proportion which leaves the rim of the flange. external diameter

Solution. circular face is

It is

only necessary to consider one flange with one exposed

and an exposed rim.

more convenient

Because

all

dimensions are in inches,

it

to use inches instead of feet for the length dimension,

therefore take a radial co-ordinate r measured in inches

from the

axis of the

pipe and consider the heat balance over an element of width dr, as

shown

in Fig. 3.3.

Pipe flange

Fig. 3.3.

^ = -2n\r ^ ^

Input ^

The of

k

dT

nkr

dT

n

dr

12

dr

—=

^ , , Output

nkrdT dTl =-__ + _^___J,, + __(T-r,)

Accumulation

=

factors 12,

d

nkr

\

,

Inrhdr ^

and 144 are necessary because k and h are defined

in

terms

feet.

Simplification of the heat balance gives

dT

d'-T

Putting y equation.

= T-T^, and x

-=

dx^

Comparison with equation

h

r^(h/6k) gives the standard form of the

+

(3.64)

X -^

dx

-

X

V=

shows that equation

II

II is

a modified Bessel

MATHEMATICAL METHODS

112

According to equation

equation of zero order.

y

The boundary conditions

r

at

=

.

=

(3.70) the solution

is

thus

AIo(x) + BKo{x)

III

i,T = 250

IV

are,

at

and

CHEMICAL ENGINEERING

IN

=

"

2,

k dT h = J^"^ J^^CT-TJ ^

V

where equation V states that the heat conducted through the metal to the rim must equal the heat lost from the rim surface to the surrounding air. Changing the variables, and introducing the numerical values, equations IV and V become X = 00195, at VI y = 190

and

x

at

dy

=

0-078,

-f-

=

-0-0195y

VII

ax

and using the properties end of this chapter,

Differentiating equation III, (3.99),

and

(3.100), Hsted at the

-^ ax Substituting equations VIII

= AI,{x)-BK,(x)

and

A/i(0-078)-BXi(0-078)

and substituting equation

(3.90),

III into

=

III into

(3.91),

VIII

equation VII gives

-0-0195[/4/o(0-078) + BKo(0-078)]

IX

equation VI gives

yl/o(0-0195) + BKo(0-0195) =

190

X

Finding the values of the various Bessel functions from tables! and solving gives equations IX and

X

A=

B=

186-5,

0-8636

Putting these values into equation III and reverting to the original variables gives the temperature distribution

T=

60+186-5/o(0-039r) + 0-8636Ko(0-039r)

The heat conducted from

the pipe by the flange

is

XI

given by

nk (dT\

= = and the heat

lost

M27[BXi(0-0195)-^/i(00195)_, 47-81 Btu/h

through the rim :2

6 t "British Association (1937).

is

given by

\drj, = 2

Mathematical Tables," Vol. VI.

Cambridge University Press

3.

= =

4-507

SOLUTION BY SERIES

113

[5Ki (0-078) -^/i(0-078)]

16-62 Btu/h

35% of this through the rim. The above problem is similar to the problem of a flat circular cooling fin, which has been solved by Gardner! who presents solutions in the form of Bessel functions for the efficiencies of a variety of shapes of extended Therefore, the pipe loses 47-8 Btu/h through each flange, and is

lost

cooHng

surfaces.

Properties of Bessel Functions

3.7.

known properties and relationdo Bessel functions, although these are not so widely known. Only a few of the more useful properties will be listed here, and the reader is referred to any of the specialized books on Bessel functions given at the end of the chapter, for a more comprehensive Hst. The formulae will be given in sections on limiting behaviour, differential properties, and integral proJust as trigonometric functions have well

ships, so

perties. 3.7.1.

Behaviour Near the Origin

As x-^O, Jk{x)

and

same Hmit as its leading term. and are therefore zero when x is zero for k>0.

the Bessel function tends to the

IJ^x) start at x^

Therefore,

when

=

/c

Jo(0)

0,

^.(0)

when

/c>

The ambiguous

0,

= =

/o(0) /,(0)

= =

1

(3.71)

(3.72)

J-,(0)=±/_,(0)=±oo

(3.73)

tions (3.56) or (3.67)

resolved in any particular case by reference to equawhere the sign is determined by the sign of r(A:+l)

given in Section

In

sign

5.3.

is

all

physical cases,

if

x =

is

a point in the system,

no physical property can be infinite and the part of the solution containing J-ilx) or I-u.{x) must be eHminated anyway. The above physical argument also appHes to Y^{x) and K^{x) which are both infinite at the origin for all values of k. If k is not zero, the infinity is due to a negative power of x; if k is zero, the infinity is due to the term In x. Therefore, for all values of k, -y,(0)

=

X,(0)

=

If the origin is a point in the calculation only physically permissible solutions. 3.7.2.

(X)

(3.74)

field,

then /^(x) and 4(jc) are the

Asymptotic Behaviour for Large X

If x>/:, then the term k'^ in Bessel's equation (3.48) can be neglected comparison with x^ in the last term. Equation (3.48) becomes ,

t

d^d

Gardner, K. A.

dy

T

^

Trans. A.S.M.E. 67, 621 (1945).

in

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

114

A

substitution suggested by the

first

=

x\v

z

ax d'z •*

^ Substituting (3.76)

dx^

'

and

-r2

two terms

,d'y

= x'-—^ + dx^

=

^

.

'

_,dy

ijZ. X " dx

_

1V--3/2

-A + ^-r-ly

(3.77)

(3.77) into equation (3.75) gives

d^z dx^

is

(3.76)

ax

x--^ + Again, IjAx

is

(ix

^+x)z =

small compared with x'lf

d^ + ^

x^\. z

=

(3.78)

dx'

This

is

the equation of simple harmonic

z

motion which has the solution

= ^cosx + 5sinx

Substituting back into equation (3.76), >^

= ^x~^cosx + 5x~^sinx

(3.79)

With the above simple treatment, it has been shown that for x>l and x'^k, the Bessel functions behave as damped oscillations of period 271. The evaluation of A and B to simulate the behaviour of the defined Bessel functions is an involved matching procedure, the result of which is

A similar procedure may be adopted with equation (3.64) for the modified Bessel functions.

Thus, 4(x) as

x

is

The

result

is

I,{x)--^e^

(3.82)

e-^ k.m-Jy, 2x

(3.83)

the only Bessel function of the four which diverges to infinity

increases; the other three decay to zero.

:

3.

SOLUTION BY SERIES

115

Differential Properties

3.7.3.

By

differentiating the series (3.56)

x

—

Jk{(xx)

The corresponding formulae X

^

term by term

it

can be shown that

=

kJk{ccx)-(xxJk + i(ccxy

=

(XX Jk -

1

(ax) — kJ,,(ccx)

Yk(ax)

d

—

.

for the other Bessel functions are

=

kYk(ccx)

-ccxYk+ i(ax)

axy,_i(ax)-/c7^(ax)

X

(3 84)

Ikiocx)

=

kI^{ccx)-^ccxIk+i(ocx) kl^iccx) + ccxlk+ i(ocx) ax/fc_

d

X -— K^(ax) dx

(3.85)

i(ax)- /c/fc(ax)

(3.86)

— OLxKk axKj^ + i(ax)

=

kKjJioix) kKk((xx)

=

-axKk.i{oix)-kK,,(ax)

(3.87)

Integral Properties

3.7.4.

J

J

J

axV^_ i(ax) dx =

x^J^iax)

(3.88)

oix%_ i(ax) dx =

x'Y,(ax)

(3.89)

ax'^/fc.

i(ax)

dx

=

xH^iax)

Jax%_i(ax) Jx = -x%(ax)

(3.90) (3.91)

The following integrals are needed for demonstrating the orthogonahty of the Bessel functions (see Section 8.6). Since only J^{x) and 4(x) are finite at the origin and therefore the only functions of practical importance at the origin, the orthogonality integrals need only be quoted for these two functions. A,

Uctx)J,{px)xdx

=

-^^ [i+X2 +

(xi+X2) + 1(^1

is

(4.2)

as follows.

l>2

+ ^2)

(4.3)

Subtraction gives

Z1-Z2 =

+ i>i-X2-iy2 = (xi-X2) + /(yi->^2)

M.M.C.E.

Xi

117

(4.4) 5

MATHEMATICAL METHODS

18

IN

CHEMICAL ENGINEERING

Multiplication gives

(4.5)

Division gives Zi

^

Z2

+ t>i ^ xi + iyi X2-iy2 X2 + I>2 ^2 + ^>2'^2-i>2 XiX2 + yiy2 yxX2-y2^ + ') xl + yl ^2 + >^2 xx

/

(4.6)

The first three operations are quite straightforward but in the division operation the complex number was eHminated from the denominator by multiplying both numerator and denominator by a specially chosen complex number. This number which is obtained by reversing the sign of the imaginary part of the denominator, is called the "complex conjugate" of the denominator, and Section 4.6 is devoted to the further properties of conjugate numbers.

The two complex numbers

z^ and Z2 will be equal if their difference is and by inspection of the subtraction operation (4.4) it can be seen that only if Xj— ^2 = and yi—y2 = 0. That is, z^ and Z2 are only Zi— Z2 = equal if their real and imaginary parts are separately equal.

zero,

In each of the algebraic operations presented in equations (4.3) to (4.6) was identical to that which would be followed the same operation was being performed on a real number. The resulting

the mathematical procedure if

number was

in general

complex containing both a

real

and an imaginary

part; although either part could be zero.

4.3

The Argand Diagram

Early in the nineteenth century Argand suggested that a complex number could be represented by a line in a plane in much the same way as a vector The value of the complex number could be is represented (c.f. Chapter 7). expressed in terms of two axes of reference, and he suggested that one axis be called the real axis and the other axis arranged perpendicular to the first

5

Q

T 1

1

y

4

\

-

1

,

,1

I

-5 -4 -3

-2

1

-1

Fig. 4.1.

N 5/1

!

q1

2




The Argand diagram

S

COMPLEX ALGEBRA

4.

119

be called the imaginary axis. Then a complex number z = x + iy could be represented by a line in the plane having projections x on the real axis and y on the imaginary axis. Such a line is illustrated in Fig. 4.1. Figure 4.1 is called the Argand diagram and the plane in which the line is drawn is called the Argand plane of z. The Hne OP represents the complex number 4 + 3/ whose real part has a value of 4 and imaginary part 3. On the other hand, the Hne OQ represents the complex number 4/— 3 with real part —3 and imaginary part 4. The lengths of OP and OQ are equal but the complex numbers they represent are unequal because the real parts and the imaginary parts of each are different. This emphasizes the fact that any equation involving complex numbers is equivalent to two real equations obtained by separately equating the real and imaginary parts. 4.3.1

Modulus and Argument

For each of the complex numbers represented by the 5. That is,

lines

OP

and OQ,

the lengths of the Hnes are

x/42

+ 3^ = V(-3)2 + 42 =

5

(4.7)

The value of the length of the line representing the complex number is called the "modulus" or "absolute value" of the number and is usually written in the alternative forms,

=

r

The inchnation of the positive real axis

and

is

|z|

= mod z = V^^ + y^

line representing the

(4.8)

complex number to the "phase" of z,

called the "amplitude", "argument", or

is

usually written,

= amp z =

6

arg z

Solving equations (4.8) and (4.9) for

=

x

From

the above

it

rcos6

=

tan" ^ (y/x)

(4.9)

x and y gives 2indy = rsin6

(4.10)

can be seen that a complex number can be expressed x+iy, or with the aid of equations

in terms of cartesian co-ordinates as

(4.10) in polar co-ordinates as

z

where

cis

is

=

rcosO

+

ir sin

=

+

r(cos

i

sin 9)

=

r cis 9

(4.11)

a convenient notation which should be read "cos 9 + i sin 0".

4.4.

Principal Values

When

a complex number is expressed in polar co-ordinates, the principal value of 9 is always implied unless otherwise stated. That is in Fig. 4.1, 9p

9q If the angle is

equal to (0-645

= =

tan- \3I4) tan"

H- 4/3)

= =

37°

143°

= =

0-645 radians 2-495 radians

its principal value amp (4 + 3/) would be and amp (4/- 3) would be equal to (2-495 +

not restricted to

+ 2«7c)

radians,

5—2

MATHEMATICAL METHODS

120

CHEMICAL ENGINEERING

IN

Inn) radians, where n could be zero or an integer. When the complex number is represented on the Argand diagram the principal value is the smaller of the two angles between the positive real axis and the hne in question. The sign of the angle depends upon the sense of rotation from the

and this implies that the principal value lies in shown in Fig. 4.2. It is sometimes more convenient

positive real axis,

the range

—

to choose

to

71

+

71

as

Fig. 4.2.

Principal value of the argument

a different range of values for 9 but the range is chosen here for consistency with the definitions to be used in the theory of complex variables at the end of this chapter.

4.5.

Algebraic Operations on the Argand Diagram

In Section 4.2 the addition, subtraction, multiplication, and division of

two complex numbers were carried out. If the Argand diagram describes all the properties of complex numbers it should be possible to carry out the above algebraic operations on the diagram. Thus consider Fig. 4.3a in which the complex numbers represented by the lines OP and OQ in Fig. 4.1 are redrawn. If z^ is the sum of Zj represented by OP and Z2 represented by

OQ

then,

or

Z3

=

Zi

+ 22 =

(xi

X3

=

xi

+ X2

and

+ x2) + i(yi + y2) >'3

=

(4-12)

j;i+>^2

Equations (4.12) give the co-ordinates of Z3 on the Argand diagram as shown in Fig. 4.3a by the line OR. Using the same numerical values as before,

X3

= 4-3 = .*.

It is easily

shown

Z3

that the point

the parallelogram

OPQR

1

=

and y^ X3

= 3+4 =

+ i>3 = l +

R can be located

as in Fig. 4.3a.

'7i'

7 (4.13)

geometrically by completing

COMPLEX ALGEBRA

4.

121

In the same way, the subtraction of two complex numbers can be expressed in the form of the addition of Zj to minus Z2 where

_Z2 = -(4i-3) = 3-4i

(4.14)

By completing the parallelogram OPQT in Fig. 4.3b to locate the diagonal OT, the difference between the complex numbers z^ and Z2 (i.e. Z4 = Zj — Z2)

--6 Addition and subtraction on the Argand diagram

Fig. 4.3.

is

obtained.

The co-ordinates of

Z4 can be verified algebraically as before,

thus,

Z4

From

=

=

Zi4-(-Z2)

(4

+ 3) + i(3-4) =

7-i.

the elementary geometrical property of triangles that the

the absolute lengths of any side, there follows the

two

sides of a triangle

important |^i

sum of

must exceed the third

result,

+ ^2h|^i| + |^2|

(4.15)

Argand diagram, and division of two terms of polar coordinates. Thus

In order to illustrate multipHcation and division on the it

is

first

necessary to

show how the

complex numbers are expressed Z5

or

Z5

where

=

=

ZiZ2

=

Tj r2 [(cos ^1

= =

ri r2

rir2(cos0i

[cos (01

r5(cos05

+

cos 62

in

multiplication

+ sin 6 1) (cos 6 2 + sin 6 2) — sin 01 sin ^2) + ^(sin Oi cos 62 +

+ 62) +

i

i

i

sin (O^

(4.16)

cos 0^ sin ^2)]

+ ^2)]

isin^s)

=

r.5-/1,2 r, r-

and

65

=

9^

+ 62

(4.17)

Therefore, to multiply two complex numbers, it is necessary to multiply the moduli and add the arguments. Hence the multiplication of two complex

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

122

numbers can be expressed on an Argand diagram by multiplication of the two real numbers r^ and r2 to give the absolute length r^, and drawing the line OU of length r^ at an angle (^1 + ^2) with the positive real axis as illustrated in Fig. 4.4.

°\

f\ a\ ^

^1

u"

Fig. 4.4.

The

P

^2

"^

Multiplication and division

division of z^

by

Zj

on the Argand diagram

can be expressed in terms of polar coordinates

as follows. Zi

ri(cos0i

4-

isin^i)

Z2

r2(cos02

+

isin02)

ri(cosOi

+

i

~

sin ^1) (cos 02

r2(cos^02

=

— [cos (01 - 62) +

i

+

—

^'sin02)

sin^02)

sin {6^

- ^2)]

(4.18)

where the denominator was again made real by multiplying numerator and denominator by the complex conjugate of the denominator. Therefore, to divide two complex numbers it is necessary to divide the moduli and subtract the arguments. Hence the division of two complex numbers can be expressed on the Argand diagram by dividing the real number r^ by the real number ^2, to give an absolute length of r^, then drawing the line OV at an angle {0^ — 9 2) with the positive real axis. This is

illustrated in Fig. 4.4.

In the multipHcation and division operations 0^ and Q2 were the principal

However {O1+O2) ^^^ (^1 — ^2) need not be the principal values of the arguments of z^ and Zg. Thus consider the complex numbers Zj = (3/— 4) and Z2 = (/—I) with arguments 01 = 143°, and 02 = 135°. By equation (4.5) Z1Z2 = 1-7/, and 05 = 0i + 02 = 278°. The principal value of the argument lying between —180° and + 180° is —82° and this value should be used in subsequent calculations values of the arguments of the numbers.

unless there

is

some

over-riding consideration (e.g. see Section 4.14.3).

COMPLEX ALGEBRA

4.

It

has been shown that

123

possible to represent simple addition, sub-

it is

and division of two complex numbers, on the Argand diagram. Therefore it is obvious that by a series of geometric operations it is possible to describe any process involving complex numbers on an Argand diagram provided that they consist of a combination of the above four operations. traction,

multiplication

Conjugate Numbers

4.6.

Two complex numbers

such as x+iy and x—iy of which the real and imaginary parts are of equal magnitude, but in which the imaginary parts are of opposite sign are said to be conjugate numbers. On the Argand diagram they can be considered to be mirror images of each other in the real axis. Conjugate numbers have already been used in the definition of the division process. Usually, the conjugate of a complex number z is written as z, and throughout this text conjugates of complex numbers will be written accordingly. If a

complex number possesses a conjugate, the imaginary part cannot be the sum and the product of a complex number with its con-

However

zero.

jugate are always real.

Thus,

+ iy) + {x- iy) = 2x (4.19) and (x + iy) (x - iy) = x^- (iy)^ = x^ + y^ (4.20) /^ — = 1. The division of a complex i.e. the final result is a real number since (x

number by

its conjugate will not produce a real number. These relations between sums and products can be extended to more than one complex number and its conjugate. Thus if w and z be two complex numbers whose conjugates are w and z respectively, let

w= then

vv

also let

z

then

z .*.

.'.

But

= = =

ri(cos e ri(cos 6

r2(cos

(f)

r2(cos

(j)

= ri(cos 9 + = (r^ cos 9 +

+ + —

i

sin 9)

(4.21)

i

sin 9)

(4.22)

i

sin

0)

(4.23)

i

sin

(j>)

(4.24)

+ r2(cos + sin 0) r2 cos 0) + i{r^ sin 9 + r2 sin w + z = (r cos 9 r2 cos — i{r^ sin 9 + r2 sin — sin w-{-z = r i(cos 9 — sin 9) + r2(cos = (ri cos + r2 cos - i(ri sin 9 + r2 sin w-\-z

1

i

-{•

sin 9)

(j)

i

(/>)

i

(j)

i

(/>)

(4.26)

(j))

(4.27)

equations (4.26) and (4.27)

w+z = w+z That

(4.25)

(j))

(/>)

From

(j))

the conjugate of the

sum of two complex numbers

(4.28)

is equal to the conjugates of the complex numbers. In fact this principle can be extended to any number of complex numbers. is,

sum of the

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

124

In a similar manner product of wz. That is

it

can be shown that

jP

p Also

if

number q

the complex

is

q q

number p

= wz = wz

= =

is

the

(4.29) (4.30)

related to

then

the complex

if

>v

and

z

by

v^lz

(4.31)

vv/z

(4.32)

to the reader to verify equations (4.29) to (4.32).

It is left

De Moivre's Theorem

4.7.

This important theorem can be enunciated as follows: For all rational is one of the values of (cos 0-\-i sin 0)". The proof of De Moivre's theorem depends on the value of n. That is, on whether « is a positive or negative integer, or a real fraction. Irrational numbers such as n are not included, but this is an academic restriction. The product of two complex numbers z^Z2 was shown to be values of n, cos nd-\-i sin nQ

ri

r2[cos(0i

and there

in Section 4.5,

+ ^2) +

i'

sin (^i

+ ^2)]

was stated that the algebraic process could be

it

extended to evaluating multiple products. Hence,

Z1Z2Z3

= =

+ 02) + ^sin(0i + 02)](cos03 + isin^3) (4.33) — sin {0^ + O^) sin ^3 + Ti r2 r3[cos (01 + O2) cos ^3 + {cos 03 sin (9^ + ^2) + sin ^3 cos {6^ + ^2)}] = ri r^ r3[cos (0^ + 02 + ^3) + sin {0^ + 02 + ^3)] (4.34) rir2r3[cos(0i

i

i

The above algebraic operations can be extended when it can be shown that ZiZ2...z„ If zi

=

=

rir2...r„[cos(0i

Z2-...

=

+ 02+

...

+0„)

+

to n

ism(0i

complex numbers

+ 02+

...0„)]

(4.35)

z„,

then equation (4.35) becomes z"

or

=

(cos

where « If «

is

is

+

r"(cos

+

i

i

sin 0)"

a positive integer. a negative integer, say (cos

+

=

" But using

De

=

—m,

sin 0)"

i

=

sin 0)"

r"(cos

n6

+

cos nO

i

+ sin

i

sin «0)

nO

(4.36)

(4.37

then sin 0)"'"

(cos

+

i

(cos0

+

isin0)"'

^"^'^^^

Moivre's theorem for a positive integer as given by equation

(4.37),

(cos

+

'

=

1 /

sin 0)"*

cos

m0 +

i

sin

m0

(4.39

:

COMPLEX ALGEBRA

4.

By a

125

by equation

direct application of the division rule as given

cos

mO +

i

cos

mO

sin

(-me) + and

Collecting together equations (4.38), (4.39),

sin

i

(4.18),

- mO)

(

(4.40)

(4.40), gives the required

result,

(cos 9

+

sin 6)"

i

=

+

cos nO

n9

sin

i

(4.41)

De

Moivre's theorem for a negative integer. a fraction, say p/q, where p and q are both integers with q positive but p either positive or negative, then by De Moivre's theorem

which proves

Finally, if «

for

an

is

integer, '

Pn

COS -9 \ q

(

+ .

P ^' isin-9 •

•

]

Q

=

cos p9

=

(cos9

Taking the ^th root of each side of equation

cos- 9

+

i

+

p9

sin

i

/

sin- 9

=

+

i

9y

sin

(4.42)

(4.42) gives

(cos 9

+

sin

i

9)^^^^

(4.43)

Equation (4.43) proves De Moivre's theorem for fractional values of n; hence, the theorem is proved for all rational values of n.

The

4.8.

mn

Roots of Unity

The «th root of a complex number z is z^^" which can be expressed in terms of polar coordinates, and it follows from De Moivre's theorem that r^^"(cos9

+

i

9 + 2s7i

sin 9y^"

cos

.

.

h

sm

I

9 + 2snl

n

=

^^ ^^^

(4.44)

n

]

where the real number 5 («- 1). 0, 1, 2, .... Equation (4.44) expresses the nth root of any number z real or complex in its most general form, where the argument is not restricted to its principal value. Therefore, if z = 1 -0, r = 1 -0, and 9 = 0, the nth. root of unity is ,

1^/"

=

cos

—+

i

—

sm

n or

more

specifically,

(4.45)

n

by substituting values of

s into equation (4.45) the nth

roots of unity are .

cos

+ sm z

^

=

,

271

^

1 -0,

cos

n

,

h

i

sm

2{n-\)n

cos -^

—

n

2n

—n .

,

cos

.

An — + n

.An

i

2{n-\)n

+ sm -^^

—

i

n

sm

—n

,

...

^^ ^^^ (4.46)

Inspection of equation (4.46) shows that all the n roots are different, and are situated at the n corners of a regular «-sided polygon centred at the origin.

1

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

126

De

Moivre's theorem as expressed by equation (4.44) can be used to find number and this is illustrated in the following example. the nth root of any complex

Example. Find the square root of 4 + Sv — 3. Using the equations (4.8) and 4,9) to express z in terms of polar coordinates gives

r^

=

+ (3^/5y = 16 + 9x5 = 61 r = 7-81 and r^ = 2-195

4^

.-.

e

= .*.

Using equation

=

tan-^(3V5/4) cos iO

(4.44), the

=

tan"

1-6771

^

and

0-8695,

59° 12'

III

=

0-4939

IV

two square roots of 4 + 3/\/5

are

±2-795(0-8695 + 0-49390

or

+(2-43

V

+ 1-380

Complex Number

4.9.

If the

II

=

iO

sin

I

Series

complex number p^ can be expressed

as

Pm = ^m + iv„

(4.45)

where both u^ and v^ are real, and if the complex number sum of n complex numbers p^, then

Zn=

t,

m=l

The summations

Pm=

t,

+

u^

i

m=l

z„ is

t, v„

Y

Mm

=

and

^n>

(4.46)

m=l

^ u^ and ^ v„ are both sums of real numbers m= 1

equal to the

Y

^m

m=

=

yn

and

if

(4-47)

and as n tends to infinity both x„ and y„ tend to finite limits denoted by x and y respectively, then the two series in equation (4.47) are convergent. Their sum must therefore be convergent, or 00

00

z=\imz„= Y n-»oo m=

^m 1

+

i

Y. v^

m=

= x + iy

(4.48)

1

Therefore, for a complex series to be convergent, both the real part and the imaginary part of the series must each be convergent. The convergence of complex series is usually discussed by referring to the associated series consisting of the absolute values of the individual terms.

That

is,

the convergence of the series given by equation (4.46) ^„

is

= Pi+P2+... +Pn

(4.46)

discussed by referring to the series S„

=

|Pi|

+ |p2|+...+|p„|

(4.49)

COMPLEX ALGEBRA

4.

5

If S„ tends to a finite limit

as «-^oo, then the series (4.46)

But equation

absolutely convergent.

(4.

IPi-rPi] .*.

^

is

said to be

5) gives

1

\Pi\-^\P2\

+ P2 + P3I ^

IP1

127

+ |p2l + |P3|

|Pl|

etc.

N^|Pl| + b2|+...|Pn|

.-.

and

it

follows that

if

the series (4.46)

is

(4.50)

absolutely convergent, then

it

must

also be convergent.

The convergence tests discussed in Chapter 3 are applicable to series of complex numbers and therefore need not be repeated here, but an invaluThe able result can be obtained from the exponential series as follows. series for a complex exponent z was given in Chapter 3 as e^

= l+z +

+ ^ n

+

^'+... 2

which is convergent for (3.7) becomes

,...,

+

values of

all

(3.7)

...

In particular,

z.

„.L.i + L +

if

2

=

z>,

equation

(4.51)

...

Each of the series in equation (4.52) can be recognized by comparing with equations (3.10) and (3.11) in Chapter 3. Thus or

more

e'^

=

e^

+ «>

cos y

+

i

sin

(4.53)

y

generally e^

=

=

e^(cos

+

y

i

sin y)

(4.54)

Equation (4.53) can also be proved in the following neat fashion. Consider f(y)

and

=

e" '^(cos y

+

i

sin y)

(4.55)

differentiate with respect to y.

.*.

-— dy

=

e~'y(— siny

+

icosy)

= =

e~'^(— sin>'

+

icosy

—

—

ie~'y(cosy icos>'

+

= ^ (constant) (4.55) gives /(>^) = 1. e~'^(cosy + isiny) = A =

but when

^^

=

0,

equation /.

Rearranging gives equation

f(y)

(4.53).

cos

>'

+

i

sin

>'

=

e'^

isinj;)

siny)

Integrating, •*.

+

1

— mathematical methods in chemical engineering

128

4.10. It

Trigonometrical

Identities

has just been shown that e'^

and

—Exponential

similarly

it

= cosy +

isiny

(4.53)

isiny

(4.56)

can be shown that e~'y

= cosy —

Addition of equations (4.53) and (4.56) gives

+ e-'y = 2cosy g-'y e'^ + cos>^ = e^y

or

(4.57)

Subtraction of equation (4.56) from equation (4.53) gives giy—^-iy = 2isin>'

or

sin>'

=

——

(4.58)

The identities developed above between real trigonometrical functions and imaginary exponential functions can be extended to include hyperbolic functions. Thus, using equation (4.57) to find the cosine of a pure imaginary quantity,

cos IX

=

——

=

-

which defines the hyperbohc cosine function, cos ix

=

—+ — =

e^^

-

*

e

cosh X

(4.59)

Similarly, -;2,

sin ix

=

=

=

i

i

sinh x

(4.60)

2

2i

which defines the hyperbolic sine function. The above relationships can be extended to define a hyperbolic tangent function as, tan ix

=

sin ix

=

cos ix .*.

tan ix

=

i

i

sinh x

—=

;

i

tanh x

cosh X

tanh X

(4.61)

from the correspondby using equations (4.59), (4.60), and (4.61).

All properties of hyperbohc functions can be obtained

ing trigonometrical identities

The Complex Variable In the paragraphs above, the complex number z has been expressed in terms of the fixed real quantities x and y by the relation z = x + iy. If however the quantities x and y are variables, then so is z. In fact the quantity z 4.11.

4.

COMPLEX ALGEBRA

129

since z is related to the variables x and x and y can be expressed by a curve in a The plane is called the "z plane".

becomes a "complex variable", and y\ the way in which z varies with plane having coordinates x and y. If the

variable

w

complex variable z

related functionally to another

is

complex

so that

\^=f{z)

(4.62)

and the complex variable w can be expressed by the

r^al quantities u and v u + iv, then when z varies in the z plane, w will follow a curve corresponding to equation (4.62) in the "w plane" whose coordinates are u and V. Since z = a: + z> and w = u + iVyU and i; must be real functions of both X and y. This will become clear in the following example.

so that

vv

=

Example. If z

= x + iy,

=

and w

w + /i;, where

w= express u and y in terms of Substitute x, y, u, .*.

and u

x and

v for

+ iv —

z2

I

y.

w and (x

z in equation

+ iyy =

I.

x^ + 2ixy — y^

II

Equating real and imaginary parts gives u

=

x^ — y^

and

t;

=

III

2x>'

which determine u and v as functions of x and y. Because the complex variable w has a functional relationship to z expressed by equation (4.62), derivatives of w with respect to z will exist provided that /(z) is continuous in the region where the derivatives are required. Thus a complex variable vv = /(z) is said to be continuous at a point p if /(/?) has a finite value at z = p, and if the limit of /(z) as z approaches /? is /(/?). That is, lim/(z)=/(p)

(4.63)

If/(z) = z"^ and the point/? is the origin, the value of/(z) at the origin depends upon the way in which the origin is approached. For instance, if z approaches zero along the positive real axis,

lim z"^

whereas

if

=

00

(4.64)

z approaches zero along the negative real axis,

lim

z-^= -00

(4.65)

The Hmit of z~^

as z approaches zero does not therefore exist, and the function z~^ has a "singularity" at z = 0. On the other hand, if /(z) = z'^, the function is continuous everywhere in the z plane but becomes infinite as z becomes infinite. Furthermore, z"^ has only one value at any given value of z and therefore it is said to be a "single valued" function.

more than one value valued" function.

However,

if

w is

for each value of

z,

related to z

and w

is

by w

=

z^, vv

can have

then said to be a

"many

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

130

can arise therefore in a number of ways when treating complex many of these can be resolved by confining the function to regions in the complex plane in which the functional relationships apply. This fact will be better understood later in the chapter. Difficulties

variables, but

4.12.

Derivatives of a Complex Variable

Consider the complex number

w=f(z) and

to be a continuous function,

Then

the partial derivative of

w

(4.62)

w =

let

u+

iv

with respect to

and z = x + iy sls x can be obtained

before. in

two

ways, thus

dw rdx

du

=

dx

(4.66)^

dx

dw^dldz^df

and

^^^^^

dx since dz/dx

,dv

+ i^

-z-

=

1

.

dz dx

dz

Equating these two expressions,

w

Similarly, differentiating

df

du

dz

dx

.

cv

dx

with respect to y gives

dw

du

dv

dy

dy

dy

,, ,^,

dw df dz df — = _— = j— dz dy dz dy .

and

(4.70)

since dz/dy

,df

_du

,dv

dz

dy

dy

"

(Ai\\

Eliminating dfjdz from equations (4.68) and (4.71) gives

Since w, v, x, and y are must be equated.

,du

dv

,df

du

,dv

dx

dx

dz

dy

dy

all real,

the real

= ^ dx dy ??!

• ' '

If equations (4.73) are satisfied,

single valued function

engineering problems. certain restrictions

and imaginary parts of equation

and

—=-— dy

(4.72)

(4 73)

dx

the derivative dw/dz becomes a unique in the mathematical solution of

which can be used

Hence the

derivative of a

which derivatives of

real variables

complex variable has do not have and these

COMPLEX ALGEBRA

4.

131

They are known

are described by equations (4.73).

Riemann" conditions and they must be plex number to have any meaning.

"Cauchy-

as the

satisfied for the derivative

of a com-

Analytic Functions

4.13.

function w = f(z) of the complex variable z = x + iy is called an analytic or regular function within a region R, if at all points Zq in the region

A

it

satisfies (i)

the following conditions.

valued in the region R. has a unique finite value. has a unique finite derivative at Zq which

It is single

(ii)

It

(iii)

It

Riemann

satisfies

the Cauchy-

conditions.

Only functions which satisfy these conditions can be utilized in pure and applied mathematics, and therefore the whole of the following treatment with analytic functions. However, when a mathematical analysis of an engineering problem involving complex functions is being undertaken, care should be exercised to confirm that the function satisfies the criteria stated above. The following examples illustrate the application of these will deal

rules.

Example 1. If w = Riemann conditions and

But

w =

z^

w =

u

= +

u

,.

J^ ox

show

z^,

that the function satisfies the

state the region

(x /v,

=

+ iyY =

wherein the function

— 3xy^

real

and

= 3x^-3/

du

-

dy

By inspection of equations III and IV Riemann conditions (4.73) are satisfied. of

z,

w

is finite;

hence the function

w =

parts, II

= 3x^-3/ ? dy

III

^— ox

-^

and imaginary 3x^y — y^

v

dv

— = — 6xy

and

Cauchy-

analytic.

+ 3ix^y — 3xy^ — iy^

x^

and equating

x^

is

=

=

IV

6xy ^

can be seen that the CauchyFurthermore, for all finite values z^ is analytic in any region of finite it

size.

Example

2.

If

iv

= z"^ show

Riemann conditions and

that the function satisfies the Cauchy-

state the region

w=

-

=

z

wherein the function

—i— = -^^^^ x-\-iy

x^

X x^

+ y^

and

V

—V ^ x^ + y^ ^

analytic.

I

+ y^

=

is

'

II

MATHEMATICAL METHODS

132

From

equations

CHEMICAL ENGINEERING

IN

II,

du

x^ v^ — y^-x"-

dv

III

-

dx~ dy~{x' + yy

Equations

III

du

dv

dy

dx

—Ixy {x^

+ y^f

and IV show that the Cauchy-Riemann conditions are

satis-

but consider the behaviour at the origin. dujdx is obtained by keeping ;; constant, and to reach the origin, y must be constant at zero; hence equation II becomes

fied at a general point

1

=-

w

X

-=-

:.

dx

which tends to negative

infinity as

V

X^

x tends

to zero through either positive or

negative values. II

Similarly, dvldy at the origin must be evaluated simphfies to

when x =

0.

Equation

_ -1 y

which tends to positive negative values.

=

'-^

•

••

dy

y tends

infinity as

1 /

VI

to zero through either positive or

Therefore, from equations

V

and VI, dujdx and dvjdy

diverge to opposite extremes at the origin, and one half of the Cauchy-

Riemann conditions the condition

is

not

satisfied.

du Xdy

= -

The Cauchy-Riemann conditions becomes

It

can be shown that the other half of

is satisfied, i.e.

infinite at the origin,

dv^ =

VII

dx are not fully satisfied at the origin,

and the function w

= z"Ms

everywhere in the z plane with the exception of the one point z

4.14.

w

therefore analytic

=

0.

Singularities

has been shown in the examples above that the function w = z^ is = oo whilst the function w = z~^ is analytic everywhere except at z = 0. In fact no function except a constant is analytic throughout the complex plane, and every function of a complex variable has one or more points in the z plane where it ceases to be analytic. These points are called "singularities", and they may be classified as follows. It

analytic everywhere except at z

COMPLEX ALGEBRA

4.

(a) Poles

133

or unessential singularities.

(b) Essential singularities.

Branch points.

(c)

The

4.14.1.

(c) arises

Poles or Unessential Singularities

A pole becomes tion

and (b) are characteristic of single valued functions with multivalued functions. Each will be discussed below.

singularities (a)

whereas

a point in the complex plane at which the value of a function Thus w = z"^ is infinite at z = 0, and therefore the funcz"^ is said to have a pole at the origin. However, in the neighis

infinite.

w =

bourhood of the

origin, this function

is finite

and consequently analytic but

w = z"-^ is said to be "first order" whilst the function w = z~^ possesses a second order pole. The order of a pole is determined in the following manner. If w = /(z) becomes infinite at the point z = a, then define at the pole

it

ceases to be analytic.

^(z)

=

in

(z-a)"/(z)

(4.74)

it is possible to find a finite value of n which makes then the pole of /(z) has been "removed" in forming The order of the pole is defined as the minimum integer value of n for

where n

is

an

integer.

g{z) analytic at z g{z).

The pole

which

g(z)

is

=

If

a,

analytic at z

=

«.

The function

w=

-—^——

(4.75)

where p and q are both positive and finite, is a function containing multiple That is, it contains a pole of order/? at the origin, and a pole of order ^ at z = ^. If either p ox q'ls not an integer, the order of the pole is the next higher integer than p ox q respectively. poles.

4. 1 4.2.

Essential Singularities

Certain functions of complex variables have an infinite number of terms which all approach infinity as the complex variable approaches a specific

These could be thought of as poles of infinite order, but as the removed by multiplying the function by a finite factor, they cannot be poles. This type of singularity is called an essential singularity and is portrayed by functions which can be expanded in a descending power series of the variable. The classical example is e^'^ which has an value.

singularity cannot be

essential singularity at z

=

0.

/(^)

Similarly the series

= Tw^-«)"'' =

(4.76)

n

has an essential singularity at z = a. Essential singularities can be distinguished from poles by the fact that they cannot be removed by multiplying by a factor of finite value. Thus consider the function

w=

e^l'

(AJl)

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

134

Using equation

expand the exponential function,

(3.7) to

w=

1

1

+-+ z

Since

w

is

1

is

some ,\

+—

—

1

...

-

+

(4.78)

...

^

n\z"

2!z^

remove the

infinite at the origin, try to

by z^ where p

The

—-^ +

singularity

^

by multiplying

finite integer.

z^w

= zP+z^-^ +

—J- +

...

—

j-

+

(4.79)

...

equation (4.79) continues indefinitely and eventually n will be Equation (4.79) thus consists of a finite number of positive powers of z, followed by an infinite number of negative powers of z. Since all terms of equation (4.79) are positive, then z^w->oo as z-^0. It is therefore impossible to find a finite value of p which will remove the singularity in e^^' at the origin. The singularity is thus "essential". Finally, it can be shown that the behaviour of a function in the neighbourhood of a pole and in the neighbourhood of an essential singularity is quite different. At a pole, the modulus of the function increases symmetrically about the pole to give an infinite pinnacle at the pole whereas at an essential singularity, the function behaves very peculiarly. For example, Mclachlanf has shown that the function e^^' does not have the characteristics of a pole as z goes to zero. Thus for a fixed value of the argument of z in the range < ^ < ^tt the modulus traces out a spiral of ever increasing = the function oscillates. This behaviour radius in the w plane; whilst at is most unlike that of a function near a pole. series in

greater than p.

;

Branchpoints

4.14.3.

The

singularities described above arise from the non-analytic behaviour of single valued functions. However multivalued functions such as w = z^^" (where az is a positive integer) frequently arise in the solution of engineering problems. Therefore this type of function must be accommodated in the theory of complex variables. Thus consider the function

w= and

let

z vary in such a

way

that

it

z^

(4.80)

follows a circular path whose centre

the origin. Converting to polar coordinates using equations (4.1 z

where

r is

=

1)

and

(4.81)

re''

a constant and 9 a variable.

is

(4.53),

Substituting equation (4.81) into

equation (4.80) gives

w=

y/re^''

(4.82)

Hence, when the variable z makes one traverse of the circumference of the circle in the z plane, 6 passes through 2n radians, but w with a constant

modulus of V^

will

only pass through n radians by equation (4.82).

"Complex Variable Theory and Transform Calculus," t Mclachlan, N. W. Cambridge University Press (1955).

If z p.

16.

4.

of the circle in the z plane with 6 changing from 2k complete the circle as shown dotted in the w plane Consequently for any value of z represented by a point on the

makes a second to

47r,

circuit

the function

of Fig.

4.5.

135

COMPLEX ALGEBRA

w

will

W

plane

Vhn

Tri \

\

/

\

Fig. 4.5.

Branch point of

vv

=

r*

y

/

on the 2 and w complex planes

circumference of the circle in the z plane, there will be two corresponding values of w represented by points in the w plane. One value will be on the solid line semicircle where OrfxJy

which can be interpreted as a volume as shown in Fig.

Fig. 5.2.

=

5.2.

If the variables

Evaluation of erf oo

X and y are changed to polar coordinates x

(5.4)

rcos6,

r

y

and 9 using the equations

=

rsin6

(5.5)

then the element of area (dx dy) has to be replaced by (r dr dO). Also, the range of integration has to be altered from the square OABC, to the quadrant which results in an error denoted by e. Thus, equation (5.4) can

OAC

5.

FUNCTIONS AND DEFINITE INTEGRALS

151

be written Rnll

P = \] The volume represented by

maximum

height of exp

and as R-*co,

(

e-'-'rdrdO

+

s

has a base area which

e

(5.6)

is less

— R^). Thus e\.

Repeated applications of equation (5.14) for integer values of n gives r(n)

=

(n-l)(^i-2)(...)(2)(l)r(l)

= {n-l)\

(5.15)

by using equation (5.13). The gamma function is thus a generalized factorial, and for positive integer values of n, the gamma function can be replaced by a factorial as given by equation (5.15). For non-integer values of «, the defining integral (5.12) has been evaluated numerically for l0

and

R^oo

x~^ dx -—= Ini 1

(5.29)

+^

have been taken.

Using equation

(4.58)

to introduce the trigonometrical function gives the result,

rx-^^_^ 1

J

(5.30)

+x

sin ^71

Putting this value for the integral into equation (5.24) gives

B{\-q,q)

=

^^^

(5.31)

sm^Ti

0, the area will remain constant at unity,

and

this limit defines the unit

lit-b)

=

impulse function as

lim[S,(0-S,^,(0]/c

(6.23)

c-*0

Taking the Laplace transform of both ^[I(t

b)']

=

lim

sides,

using equation (6.22) gives (6.24)

\-

cs

By

L'Hopital's rule, Section 3.3.8, this hmit becomes

^U(t-b)']

=

e-''

(6.25)

Frequently, the unit impulse function is called the "Dirac delta function'* and is written S(t — b). It expresses the rate of change of the unit step function. 6.5.3.

The Staircase Function

is shown in Fig. 6.4 where it can be seen that the function is formed by the successive addition of unit step functions at 0, b, 2b, 3b, .... etc.

This

MATHEMATICAL METHODS

176

CHEMICAL ENGINEERING

IN

S(b.t)

4

32 1

2b

1b

Fig. 6.4.

The

3b

staircase function

Hence the Laplace transform of the staircase function together the transforms of each step function. Thus 2bs

bs

5

The be

infinite series in

summed

5

(l

will

be

+

...

5

— nbs

+ e-^' + e-^^' +

equation (6.26)

is

+ ...)ls

(6.26)

a geometrical progression which can

to give the result

^[5(^0] = Use

obtained by adding

nbs

+ =

is

made of

(6.27)

^^^::^

these functions later in the text but the use of the unit

step function will be illustrated

by the following example.

Example. A control valve of the type shown in Fig. 6.5 is actuated by ranging from 3 to 15 Ib/in^ gauge operating on a 16-in diameter diaphragm whose effective area is equivalent to 100 square inches. The air pressure

Air

^^

Spring

Diaphragm

Becking

Valve stem

plote

Fig. 6.5.

Control valve

dead weight of the moving parts of the valve,' allowing for the is estimated to be 300 lb weight, the stiffness of the spring is 600 lb/in and the damping constant is estimated to be 17-0 lb

effective

friction in the gland, etc.,

If the total

sec/in.

valve

if

lift

of the valve

is

2-0

in,

predict the response of the

the controlling air pressure suddenly changes

12 Ib/in^ gauge.

from 6

Ib/in^

to

6.

THE LAPLACE TRANSFORMATION

177

Solution

A

force balance

Input force

= AP A

on the valve

is

So(t)

Output force

,2

= m -rj.

movable parts

(a) to

overcome

(b) to

overcome resistance of spring

=

kx.

to

overcome damping resistance

~

^~T'

(c)

inertia of

dx

Therefore at equilibrium

dx

d^x

m -^ +

c

—+

di^^'dt

Let a

=

c/2mco where

Then equation

I

co

=

/ex

= APASo(t)

s/k/m

becomes

— ^ + 2aco — + co^x = — dt The Laplace transform of equation (s^

+

sx(0)

ms

In order to ease the calculations, air pressure

let

x(0)

=

+ 2acox(0) + x'(0)

the origin of

= =

of 6 Ib/in^ corresponds to x .-.

11

II is

APA

+ loiws + a)^)x =

So(0

m

dt

x'(0)

111

x be changed so that an

0,

and therefore

APA ms(s^ + 2acos + co^) The

inverse transform of equation

APA

r

where

tan(/)

X can now be evaluated problem. Thus

APAlk =

m=

"

=

a/Vl —

as a function of

/

partial fractions as 1

/

a^

VI

by using the data given in the

1

300 lb dead weight

V "

IV can be obtained by

e~*"^

IV

V

m

2mco

"

=

300

32x12

=

0-781 lb sec^/in

0-'

2 X 0-781 X 27-72

"

°'^^^

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

178

a^

=

taiKp

=

0-920 0-393

=

0-427

0-920 .*.

(j)

=

0-404 radians

V gives Table 6.2 and hence Fig. 6.6. measured from the top of the valve, Fig. 6.6 has been drawn to show valve opening X. Substituting these values into equation

Because x

is

Table

6.2.

Evaluation of Equation

^

UJt

0-92

1

1

2 3

4 5

6 7 8

9 10

0-9196 0-8698

0870

0-7337 0-4953 0-3343 0-2257 0-1523 0-1028 0-0694 0-0469 0-0316 0-0214

X

V

01346 -0-7075 -0-9910 -0-4939

1-237

0-3929 0-9697 0-7819

0-960 0-933 0-963

-0-0220 -0-5883

1-013

0-6-

(Ot

Fig. 6.6.

0-000 0-362 0-933

Control valve response

1-224 1-075

1007

yx

1-500 1-138

0-567 0-263 0-276 0-425 0-540 0-567 0-537 0-493 0-487

6.

THE LAPLACE TRANSFORMATION

179

Therefore, the response of the valve is lOjll'll = 0-36 sec. That is, in 0-36 sec the position of the valve alters by one inch and has stabilized at its

new

position. 6.6.

It

Convolution

frequently happens in solving problems by the Laplace transformation

that the final transform of the equation is a product of two easily identifiable transforms, but these are difficult to resolve in the form of a summation. When this arises, the inverse transformation can be accomplished by the

method of "convolution". Space does not permit the derivation of the convolution integral, and its use will be explained. Any reader

therefore only the mechanics of interested in the derivation

Thus

if

factors

advised to consult

is

the transform of the equation g{s)

and

h{s)

whose inverse

Thomson!

or Churchill^.

composed of the transforms are recognizable from

is

and

J{s)

this is

Laplace transform tables, the inverse transform of J{s)

is

obtained as

follows.

Let the inverse transform of g{s) be be h{t).

g{t),

and the inverse transform of

h{s)

Then

since

r/ x

/(s)

=

-/ n r/ ^ g{s) h(s)

(6.28)

t

fit)

t

= ^-\g{s)Ks)\ =jgix)Kt-T)dT

=jh(z)git-x)dr

(6.29)

Equation (6.29) gives the convolution integral of /(r) and imphes its symmetry. The convolution integral is frequently given the shorthand form f(t)

The use of the convolution

= g(t)h*(t)

(6.30)

an inverse transform

integral to find

will

be

illustrated in the next example.

—+ ly

Example. Find the inverse transform of 7^

775.

(s^

Transform number 16

in the tables

m= then

g(t)

=

shows that

nis)

=

^^

h(t)

=

cost

if

Therefore, by the convolution integral (6.29), t

f(t)

=

cos T COS (t —x)dx J

I

r

=i t X

Thomson, W.

T. Churchill, R. V.

J

[cos

t

+

COS (2t - 0] dx

"Laplace Transformation." Longmans Green & Co. "Operational Mathematics." McGraw Hill Book Co.

II

MATHEMATICAL METHODS

180

/(O

IN

CHEMICAL ENGINEERING

= ilrcost + i sin (2t - 0]o = i(rcosr + isin/ + isinO = i(rcosr + sinO

III

IV

The answer given by equation IV checks with the answer given number 23.

directly

in the tables

6.7.

Further Elementary Methods of Inversion

Three further methods for finding the inverse Laplace transform will be given in this section. These methods will often yield answers more quickly than either partial fractions or convolution, and they can also be used when previous methods fail. 6.7.1.

Using the Properties of the Transformation

was shown in Section 6.4.2 that dividing the transform of a function and /(/) by s was equivalent to integrating the function between the Hmits /. Thus the power of s in the denominator of f{s) can be increased in an integral manner by successive applications of equation (6.21) as illustrated in the next example. When this process is combined with the shifting theorem (Section 6.2.2), a powerful method of inversion results. It

Example 1. ¥'md f(t) if J{s) = \ls\s+a). Transform number 8 shows that if Jy{s)

=

\l{s-\-a),

then f^{t)

=

e~°\

Application of equation (6.21) gives

s(s-\-a)

s t

:.

Mt) =

je-'"dt

1-eII

a

A further application

of equation (6.21) to this result gives

^n s

^

'^'^'s^s + a)

t

a

J 1 t

a

+

L

1

(at

Therefore,

if j(i)

=

— a Jo

+ e-'^'-l)

Jls^{s + a), then

m=

(e-''

+ at-l)la

III

6.

Expansion

6.7.2.

in

THE LAPLACE TRANSFORMATION

181

a Descending Power Series

be noted in the tables that as |5|^cx), then/(5)->0 for every transalthough an apphcation of L'Hopital's rule is often necessary to show this. The first seven transforms in the table also give an inversion for any negative power of 5; and hence, if/(5) can be expanded in a series containing only negative powers of s, these first few transforms will determine a series form for/(0. An ascending power series will always contain positive powers of the variable after a certain stage, so that to restrict the series to negative powers, Inspection of transform a descending power series of s must be used. numbers 1 to 7 shows that the termwise inversion will lead to an ascending power series in t. Referring back to Section 3.3, if this series has a finite radius of convergence, then the result will only be vaHd for small values of t. Hence, the initial behaviour of a system can be determined by inverting the The method of obtaining first few terms of such a power series expansion. the series will be illustrated by the following examples. It will

form

pair,

Example

2.

Determine /(O

The logarithmic expansion ing powers of

=

\ff{s) (3.8)

In [1

appHed

+ (ajsY].

to f{s) gives a series of descend-

s. Viz.

Transforming each term separately using number

{atl_(atl

?[ The

series in

3,

(at)^ (atl

gives

1

III

2

4

!

6

!

!

equation III can be recognized (equation (3.11)) giving the

result

= -(cosat-l)

f(t)

which confirms transform number 105 in the

Example

3,

Determine /(O

=

if /(^)

IV

tables.

l/s^(s+a^).

In order to obtain a descending power series, s must be considered to be must be rearranged in the following manner before the binomial theorem can be appUed. Thus large so that the term (s + a^)

1

a^

/,

o*

a^

a^ S^i M.M.C.E.

'

s'i

S*i

\

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

182

Applying transform number 6 to each term of equation

=

2t^

2~t'^

2't'^

x^Ti

1.3V7r

1.3.5V7r

2^

2

2^t'^ 1.

3.5.7^7:

(20-

_(201 1.3.5

1.3

II gives

III

1.3.5.7

Equation III is likely to be a more useful inversion formula for the given transform than that shown in the tables as number 41. Other Series Expansions

6.7.3.

more convenient on occasions

to expand f{s) in descending powers apply the shifting theorem; and many other expansions are also possible in terms of a wide variety of functions. The only restriction on any series expansion is that it should be convergent for sufficiently large values of s, and the following example illustrates such an expansion in terms of exponential functions. It is

of

(s

+ a) and

Example

4.

In a note published by T.

Woodt,

the following transform

requires inversion /(^)

= s + alpe-

Following the method given in the pubhcation, equation

'

I

can be rearranged

as 7(s)

1

= s

Provided /?e"'^^
/.> 00,

Gauss'

/(«)-* J/(/+l) which

test.

Similarly,

it

= (0)(3)a, ^3 = = a-j = ag = =

(3)(2)a, i.e. .*.

a^

...

Hence, when / = 1, g = a^m is a solution of equation (8.122). This can easily be verified by substitution, and the solution is finite when m = I. The solution

is

thus physically reasonable.

can readily be seen that if / is an odd integer, there will be a value of p making the right-hand side of equation (8.129) zero and thus giving a polynomial of finite degree. Similarly, if / is any even integer, there will be a value of/? making the right-hand side of equation (8.128) zero with similar effect. These solutions are called the "Legendre polynomials" and are given the It

.

8.

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

symbol Pi{m). The solution of equation (8.122)

is

thus

g=^APim) + BQim) where QJim) denotes

the infinite series.

solution exists mathematically, but

grounds and

it is

279

(8.132)

remember that this invariably unreasonable on physical It is as

well to

be rejected at this stage. Combining equations (8.119) and (8.132) according to equation (8.114) it

will

gives the particular solution

T=

(>l,r'

+ B,r-^-i)P,(cos0)

(8.133)

Again, any number of these solutions can be superposed for integer values of / giving 7

= 1

{Ay + Bir-'-'^)Picose) f =

(8.134)

Equation (8.122) can be written in the form d_

dm

VI

0)

+ 17;

VII VIII

A2b^ + B2b-^=iT^ Aib' + Bib-^-^

=

IX

X

(for/>2)

Although there are an infinite number of equations and unknowns in the above set, they can be solved in pairs. Thus equations V and VII can be solved for Aq and Bq, and equation VI with / = 1 and equation VIII can be solved for A^ and B^. Rather than complete the algebra at this stage, it is more efficient to find the required answer in terms of Ai and Bi, then solve sufficient

The a

is

of equations

rate at

V

which the

to

X

for the constants required.

solid conducts heat

away from

the sphere of radius

given by jt

Q=^ -kilna^'f^)

sin 9

d9

XI

8.

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

and substituting into

Differentiating equation (8.134) with respect to r

equation

XI

gives i^

Q=

281

00

llAia'-'-(l + l)Bia-'-^}Pi(cose)smede

£

-27r/cfl' I

+

= -2nka^

[lAia'-'-(l + l)Bia-'-^]

f

1

Pi(m)dm

f

XII

But the integral occurring in equation XII is the orthogonality integral if Pq(w) = 1 is inserted into it. Therefore, all terms after the first are zero in the series in equation XII.

Q=

.-.

47ik

XIII

Bo

and hence only Bq need be determined from the Thus

Ao + Boa~^ =

of equations

set

V

to X.

Ti

Ao + Bob-'=:T2+iT4 4 _

Bo(a-'-b-')=T,-T,-\T^

Bo=~(T,-T^-\T^) —

:.

b

XIV

a

Substituting into equation XIII gives

e = ^(r,-r,-iT,) b—a Therefore,

if

Q

is

plotted as ordinate

XV

and {Ti-T2-^ T^)

as abscissa, for a

variety of heat supply rates, the gradient will be 4nkab/(b — a)

from which

the thermal conductivity (k) can be evaluated. 8.7.3.

Equations Involving Three Independent Variables

The steady state flow of heat in a cyHnder equation in cylindrical polar coordinates (7.125)

d^T 2

do)^

+

J_^ + + 0)^ Z2i^ 0) do d9^

is

]^dT

djj

T:i7:

dz

^=

governed by Laplace's

o

(8-138)

where there are three independent variables a>, 6, and z. This equation has to be solved in stages and solutions are first sought in the form T=f(co,e).g(z) Substituting this

form into equation .25

(8.139)

(8.138) gives

+ -5r,s + -i^9+/9" =

o

Dividing by fg and separating the terms dependent upon z

ir^ + i^ + ±^]=Z^=_,2

(8140)

MATHEMATICAL METHODS

282

CHEMICAL ENGINEERING

IN

where -v^ is the chosen separation constant. Each side of equation (8.140) must still be constant because a function of co and 9 is equated to a function of z only. The second half of equation (8.140) gives

^2-^9 =

(8.141)

=

(8.142)

which has the solution g

The other

A,e'' + B,e-'"

half of equation (8.140) gives the partial differential equation

.2^

J

co^T-^

^-^

+ co-^+ ,

,..

,

^ + co^vy=0

^ ^

,

_2,.2

(8.143)

which has only two independent variables. Applying the rr.-thod of separation of variables again, by looking for solutions of the type

=

f(co,e) .-.

Dividing by

FG

F(co).G(e)

(o^F"G + coFG + FG"-\-coh^FG

and separating the

+

(8.144)

=

(8.145)

variables gives

(o^v^

=

—^ = k^

(8.146)

Solving the second part as before gives

G^Ak cos kd + Bk sin kO The

first

(8.

147)

part of equation (8.146) gives (o^F" + coF'

+ ((o^v^-k^)F =

Changing the variable by substituting cov

.'.

=

x

x^^ + x^ + (x^-k'')F = dx

(8.148)

(8.149)

„{x)ydx

Eigen-functions not Orthogonal

Sometimes the variables

will separate in the partial differential

equation

to give an equation of Sturm-Liouville type, but the

boundary conditions are eigen-functions are not orthogonal. If an attempt is

such that the resulting made to use the orthogonality relationships, the general equation (8.154)

is

MATHEMATICAL METHODS

284

CHEMICAL ENGINEERING

IN

obtained, but the terms in the infinite sum are not zero. Instead of each equation (8.154) being solvable for A„, each equation contains all of the ^^s and thus an infinite set of equations, each containing an infinite number of unknowns is obtained. Such a system of equations is normally insoluble and an approximate solution has to be found by assuming that A-j, A^, .... etc. are all zero and solving the first six equations for the first six coefficients. The accuracy can be improved by taking an increasing number of equations and coefficients, but the effort is rarely worth while and six equations

is

the normal practical limit. There are however

ing an exact solution

Example

2.

and these are

A solid rectangular

and maintained

two ways of

find-

slab at a uniform temperature Tq has

The temperature of one exposed

four edges thermally insulated. raised to

still

illustrated below.

face

its is

temperature of the other exposed the temperature distribution varies

at Tj, whilst the

Find how is held constant at Tq. with time. Because the edges are insulated, heat will only flow in one direction and equation (8.47) will describe the process. The solution follows the method given in Section 8.7.1 identically until the particular solutions (8.100) and (8.101) arise. The boundary conditions can be written: face

= 0, = L,

X

at

and

x

at

These do not

satisfy the

r= T=

To

I

T^

II

conditons of orthogonality given in Section

Substituting equations

I

and

8.6.

II into the particular solution (8.100) gives

To

= Ao

III

T,=Ao + BoL Hence, the particular solution

IV

is

V

T=^To + (T,-To)xlL and

this should be recognized as the steady state solution. Equation V is now used to define a new variable Z thus,

Z=

VI

r-To-(Ti-ro)x/L

Substituting into the differential equation (8.47) gives the

dx^

dt

which is the same as equation (8.47) with conditions I and II become

= 0, x = L,

x

at

and

at

The equation and boundary conditions

new equation

T replaced

by Z.

The boundary

Z= Z= in

terms of

VIII

IX

Z

are

now

identical with the corresponding equations in Section 8.7.1.

solution

is

given by equation (8.111) with

T replaced

by Z.

completely Hence, the

In the present

I

-

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

8.

problem, the

final

boundary condition that the temperature

is

285

Tq at

all

points initially becomes

Z = -(Ti

X with

Comparing equation

when

Tq)xIL

/

X

=

equation (8.107) shows that

/oW=

XI

-{T,-To)xlL

L [ J

nnx nnx T^-Tol r X cos — /oW sin —L— dx = nn L .

.

•

C

^

cos J

I

=

1^ nnx — dx — L JQ

Zii::Z2r(_i)"L]

XII

nn by integrating by parts and simplifying.

Substituting equation XII into the

solution (8.111) gives

Z=

y „=

—{-mT,-To)sm'^e-"'^'^'^'^'

XIII

L

nn

Returning to the original variables by using equation VI,

^^:^ = Ti-Tq

-+

L

—

-(-l)"sin E= onn L

e-"^'^^-/^^

XIV

„

The above method, which should be adopted when

the boundary condiand II will not give an orthogonal set of eigen-functions, involves finding any particular solution (such as V) which satisfies the conditions and defining a new variable (Z) so that the new solution consists of an orthogonal set of eigen-functions. In this example, the first eigen-function is given by equation V and provided it is removed from the others an orthogonal set remains and the method can be pursued as if it were normal. tions I

;

Example 3. In the study of flow distribution in a column packed with woven wire gauze,t it has been observed that the hquid tends to aggregate at the walls. If the column is a cyhnder of radius b ft and the feed to the column distributed within a central core of radius a ft with velocity Uq ft/sec, determine the fractional amount of liquid on the walls as a function of distance from the inlet in terms of the parameters of the system. is

The problem is illustrated in Fig. 8.8. where the vertical velocity C/ ft/sec considered to be a function of the radial coordinate r and the vertical coordinate z measured from the inlet. It is assumed that a normal diffusion equation is

dU V=-D— dr appHes to the velocity distribution. fluid velocity ties.

D

will

t Porter,

and

D ft is

F ft/sec

I

is

the horizontal

component of

the coefficient governing the equalization of veloci-

be a property of the packing and of the particular fluid used.

K. E. and Jones, M. C.

Trans. Instn.

Chem.

Engrs., 41, 240 (1963).

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

286

Taking a material balance over the element shown Input ^

Output

in Fig. 8.8 gives

dU

= InrU dr — D

-—-Inrbz dr

= 2nrUdr +

dU —d (InrU Sr)dz '-D—-2nrSz oz

Accumulation

dr

[D

dU — InrSz] \

J

= d / dU \ - -(D-—2nrdz]dr ^Q

{InrU dr)dz

or\dr dU ^d / dU\ 'Tz=''drV^)

dz

•*•

"

dz

The boundary conditions z

=

~

II

J

III

[dr^ '^r'di^J

Flow

Fig. 8.8.

At

d /

dr\dr

in a

packed column

are as follows. 0,


a,

a,

U = Uq

^

IV and

at

r

=

0,

U

U=

must be

finite

V

boundary condition at r = b must allow fluid to accumulate on and a reasonable assumption is that there is a capacity 2nbk ft^/ unit length at r = b. It is further assumed that fluid which has diff*used to the wall is equally distributed in this capacity, so that the same flow rate exists in this fictitious layer on the wall as in the packing in contact with the

The

final

the walls

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

8.

287

This condition can be stated symbolically thus,

wall.

r

at

=

dU

dU

dr

oz

-D-—2nb = 2nbk--

b,

dU dU = k-^D— oz or

i.e.

VI

The problem is thus to solve equation III, satisfy the initial condition IV, and satisfy the boundary conditions V and VI. Trying the method of separation of variables by putting

VII

U=f{r)g{z) .-.

= D(f"g+f'glr)

fg'

Dividing by Dfg gives

^J.1±PL=_,2 Dg where -a^

is

VIII

f The

the chosen separation constant.

first

part of the equation

gives

^ + a'Dg =

dz

which has the solution

=

g

is real and has been chosen The second part of equation VIII gives

indicating that a

.2*^ J

^^J

+ X

IX

A'e-''"^'

r-f

correctly.

„2«2 + a^rY=0

X

,

Equation X can be transformed to the standard form of Bessel's equation (3.48) by putting x = ar. The solution is thus

= a

#

0,

ifa

=

0.

if

AJo(ar) + Byo(ar)

XI

or

/=^o + ^olnr Since Yq{0)

Bq

= B=

0.

= — oo

The

and InO

= — oo,

XII

boundary condition

V asserts

that

solutions of equation III are thus

if

oi

if

a

= 0, # 0,

U = Ao "'^'^^ 1/ = AJoio^r) e

XIII

XIV

Boundary condition VI is satisfied by solution XIII, and putting equation into equation VI gives

XIV

D^a7i(a6)e-"'^^ by using equations Simplifying equation

(3.88)

and

=

-kA^oi''DJo((xb)e-

''"'''

XV

(3.97) to differentiate the Bessel function.

XV gives Ji(a„ b) + ka„Jo(oi„ b)

=

XVI

MATHEMATICAL METHODS

288

IN

CHEMICAL ENGINEERING

which determines the eigen-values a„. There are an infinite number of solutions of equation XVI, the first few of which have been tabulated.! The most general solution of equation III is therefore obtained by adding all

Thus

permissible particular solutions together.

U = Ao+

XVII

Y. A„Jo{oc„r)e-'^'^'

where the eigen-functions are zero order Bessel functions of the first kind. It can be shown by rearranging equation X that it is of Sturm-Liouville type with a weighting function r. Boundary condition VI is not one of the types Hsted in Section 8.6 and hence the eigen-functions will not be orthogonal. Nevertheless, the usual method can be followed by putting z = and multiplying equation XVII by r Jo(oc„r) and integrating. This gives b

b

\

Ul^orJo(^r.r)dr

=

{

''

00

AorJo(cc^r)dr

+ "

b

b

X "

^n ^

f

Jo(^„

r)Jo(of,, r)r

It is

simpler to evaluate the terms of equation XVIII separately.

first

term becomes b

\

^r

XVIII

b

Thus

the

a

^\z = ofJo{^mr)dr=

(

U qvJ ^i^x^r) dr

b

= by using equation (3.88) with k

=

Similarly, the next

\.

XIX

U,aJ,{oi^a)loi,„

term gives

b

\

= AqB

AorJo(o

is

I

dt

the hydrauhc diffusivity.

The initial pressure in the formation is removed at a constant rate q, then

where k

rj

/^

II

^r

the thickness of the formation,

and

/i is

the

coefficient of viscosity. t Crank,

J.

Carslaw, H.

S.

Press (1947).

"The Mathematics of Diffusion." Oxford University Press (1956). and Jaeger, J. C. " Conduction of Heat in Solids." Oxford University

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

296

Transforming equation

I

by the same procedure as given in Example

1

dp _ po p= 2+7^--p=-r dr

ni

rV^

IV

d^p

s

1

4-

dr^

rj

t]

Putting

=

X x^

which

is

—

- x^p= x— dx

—^ + dx^

V

s

the modified form of Bessel's equation (3.64) having the solution

p==AIo(x) + BKo(x) + pols

VI

Equation (3.82) shows that as x->oo, Iq(x)-^ co and hence

^ =

p=^BKo(x) + pols Transforming boundary condition

s

Using equation (3.71) that x->0 gives

VII

II gives

= lin,2_^l^

i

r-o

/o(0)

=

and

1,

VIII

dr

A^

and taking the

limit of equation (3.68)

as

IX

Ko(x)^-\nx Therefore, near the base of the well, equation VII becomes

p^ -B\nx =

X

-Blnir^Js/rj)

Substituting into equation VIII, therefore

q -

.

=

InrkhB

,.

lim

InkhB

=

5^^ifi^_« 2nkhs

^^ XI

XII

s

Hence the transformed solution VII becomes p

= —a Ko(r^/slri)

Po 1

Since integration of a function with respect to

transform by

s,

as

shown by equation

,

Xlll

s

s

(6.21),

t is

equivalent to dividing

its

transform number 117 in the

tables enables equation XIII to be inverted, thus t

P--JI-^e-^'"''dt + p, Putting

and hence

t

=

dt

=

XIV

XV

r^l4rjz

r

4r]z^

8.

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

297

20

XVI

dz + pi

where Zq

=

r^jArjt.

(5.44), the solution

Comparing the integral in equation XVI with equation can be written in terms of a tabulated function. Thus P

Example

= 4nkh

EK-r^l4r]t) + Po

XVII

Gas absorption accompanied by a first order reaction in a wall column is to be used for the absorption of a gas (A). The gas is consumed in the liquid phase by a pseudo first order reaction Develop expressions giving the point in terms of the gas concentration. absorption rate and effective penetration depth as a function of distance from the hquid inlet. Figure 8.9 shows a section through the liquid film where z is the distance from the liquid inlet, and x is a coordinate measured inwards from the

falling film.

5.

A wetted

Velocity distribution in a falling film

Fig. 8.9.

surface of the film which it is

is

assumed to be of uniform thickness L. Although

a poor assumption which depends

velocity distribution v{x) will be

upon

inlet conditions, the

assumed to

parabolic

exist right at the inlet

and

remain unchanged throughout the length of the column. The gas is assumed pure so that there will be no gas phase resistance and the surface concentration of A in the liquid will be constant at Cq, For any element of the hquid (3x Sz) to be in equilibrium, the amount of A carried in the z direction by bulk flow, plus the amount diffusing in the x direction, must be sufficient to supply the chemical reaction proceeding within the element.

Expressing this fact symbolically.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

298

dc

Input rate

=

vcSx — Dzr^^ ox

Output

=

vcdx

rate

Consumption where

c

is

rate

=

+

dc d ~-(vcSx)5z — D^-dz ox dz

+

d /

t-

dx\

dc

\

dx

J

—D—dz]Sx

kc dx Sz

the concentration of

A

the specific reaction rate constant.

D is its diffusivity, and k is Completing the balance and cancelling

in the liquid,

appropriate terms gives

dz

Transforming equation

I

dx^

by the method previously explained gives

— svc + D——^ =

kc

dx^

D-7^-(/c + st?)c =

II

dx

But V = V(l—x^/L^) where V is the interfacial velocity, and equation II would have to be integrated using the method of Frobenius to obtain an accurate solution. However, the penetration of A into the liquid will be small for small values of z and as a first approximation it will be assumed that x^L so that v can be replaced by Fin equation 11. Thus

d^

D-j-:2-(k + sV)c

=

III

dx'

which is a linear second order differential equation with constant coefficients having the solution c

As x A'

=

=

X'exp(xy^) + Bexp(-xy^)

increases, the concentration of 0.

At

:c

=

0,

A

must decrease

the boundary condition gives c

=

to zero,

Cq

Cq/s

c,

Using transform 82 and the shifting theorem (Section

6.2.2),

IV and hence

8.

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

299

Applying the property given in equation (6.21) to equation VI and inverting equation

V

gives

l~V~

fx

x^V

(

Putting

a^

= x^F/4D

and

p^

=

kz\

^^

^

klV

VIII

to simplify the algebra, gives z

c

It is diflScult to

=

a

J

/

o'^

Co

evaluate this integral, but

and hence

r^

=

its

form can be changed by putting

— + p^z-2aP z

dr=(^-^^-^)dz

and into equation

IX

to give

z

00

z

c

/.

=

Coe-^'^eifcf-^

-

py/zj

-

CqP

j

-p=exp( -

- - P^z\dz

XI

of P were reversed in equation IX, the integral would not be any way. Hence the sign of P may be reversed in equation XI

If the sign

altered in

without altering the

result.

Thus z

c

=

Coe^-'erfc^-^

V^sT^ +

Adding equations XI and XII

Co;8

J

-^exp(--^ - jJ^z) dz

together, dividing the result

XII

by 2 and returning

300

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

to the original

symbols through equation VIII yields the

+ The point absorption

iexp(xy^)erfcg7£^Vy)

rate per unit area

R= Transforming equation

final result

XIV

"I

is

given by

XIV x=

gives

R= -D — dx

and substituting

for c

XIII

from equation

_

Dco

V

XV x=

gives

k\

IV (

XVI Inverting equation XVI by using transform 38 (Section 6.2.2), therefore

R= :.

Cos/oVe'

kz/V

and the

shifting

theorem

'k

V Vt.^JP'"'^^^'^"'] R = Co^/kD [erfV/cz/F + V^/^exp (- fcz/F)]

XVII

In order to determine the depth of penetration of the solute, some must be set for its concentration to be negligible. This hmit

arbitrary limit

Fig. 8.10.

chosen as c/cq

=

Approximation for

erfc

y

and equation XIII gives an implicit expression for x as a function of film length z. llie relationship is simple enough to solve by trial and error except for small values of z but an approximate solution can be found for this region as follows.

is

002,

the penetration depth

Figure 8.10 shows {2l\ln)e~^^ plotted against y so that erfc>' is the If y is large it is reasonable to suppose that erfc y is some

shaded area.

8.

PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS

301

constant multiple of the area of the triangle formed as shown by the tangent to the curve. Putting

h=-^e-'" dh

7'-

dy~

XVIII

-y'

=

2yh

but from the triangle,

dh

_ -h

dy a

:.

a

=

XIX

iy

The approximation can be written crfc y

where

K is

XX

= iKha

some proportionality constant greater than unity. and a from equations XVIII and XIX gives

Substituting for h

erfc3;

Evaluating

K=

1-71.

K by putting y Hence

=

= —^e"^' on both

1-5

XXI

sides of equation

erfcy=^(0-484/3;)e-^'

XXI

gives

XXII

Ify = 2-5, the error in equation XXII is less than J%, and if >^ = 1, erfc y = 0-178 whereas (0-484/;;)e->'' = 0-157. Hence equation XXII should only be used for >^>1. Introducing dimensionless coordinates into equation XIII by putting

X = xyjklb c

and Z^

=

kzjV

XXIII

ie-erfc(|-z)+i.^erfcg + z)

Co

Using equation XXII gives c

Co

0-242

— X

2Z

^

0-242

+

z

—X

2Z

0-242Z

is

vahd

+z

(-.?-)

X^

/

expi-7^-Z'j

(|i-) which

exp

^

4Z

for

^>1+Z

XXIV

:

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

302

The penetration depth 0-002

Solving by

Z^

for

1

trial

Z

is

thus given by

(^ - ^^) = ^'^^^^ ^^P ( ~ ^ ~ ^0

and error using equation

XXV

for

Z
^o

or

+ Azo

more

=

(9.1)

=

and

Zi

-Zq =

is,

If

.

.

;

Az„_i (9.3)

(9.3) gives

=

(ji-yo) + (zi-Zo)

(yi

+ Zi)-{yo + Zo) =

AC^'o

+ Zo)

(9.4)

generally A(>'„

That

.

(p{xo+h)-(l)(xo)

A

A: is

+ z„) =

+ Az„

A>^„

(9.5)

obeys the "distributive law of algebra". a finite constant A{ky„)

= ky„^,-ky„ =

kAy„

(9.6)

A can be interchanged without altering the result. That obeys the "commutative law of algebra" with regard to constants. Finally, A obeys the "index law of algebra". That is so that k and

AP+^

=

A^A^

=

A«A^

is,

A

(9.7)

This will be further explained below. 9.2.2.

Differences of Second

and Higher Orders

In Section 9.2 a sequence of was constructed thus

first

differences of the function

(yi-yo)'Ay2-yi)i---'Ayn-yn-i)

The

=

Ayo',Ayii"-iAyn-i

y = f(x) (9.8)

differencing process described by equation (9.8) can be repeated to give

Ay,-Ayo = A'yo But from the relationship of Ay to y

AVo =

A>'i-A3;o

=

it

(9.9)

can be seen that

(}^2->^i)-(>'i->^o)

=

Second difference relationships of the type given

>'2-2>'i+>'o

(9.10)

in equation (9.10) are

309

FINITE DIFFERENCES

9.

derivatives in differential equations by finite method of solution. This will be numerical differences prior to using a

employed to replace second

illustrated later in Section 11.4.1 et seq.

The procedure developed

in the

above paragraph to obtain an expression y can be extended to differences of

for the second difference of a variable

higher order.

For example the A'>'o

and the fourth

difference

=

third difference of yQ

is

>^3-3y2 + 3>^i-yo

is

AVo =

>'4-4j3

+ 6>^2-4yi + >^o

In general terms for any variable y„ in a sequence, the

^""yn^ ym^n-

(9.11)

{Vjym^n-l+

(9.12)

wth

difference

is

(9.

[^

13)

In equation (9.13) the coefficients of the expansion on the right of the equation are the same as those obtainable from a binomial expansion of

{\-xT. 9.2.3.

Difference Tables

The most convenient way of expressing the first, second and higher order differences of a function is by means of difference tables. Such a table is illustrated below for a function y = f{x). Table ariable

>'o

xi

yi

xi

yi

1

.

A

Function

xq

9.

The Difference Table A2

A3

A4

^...Ayo

t^yi

A4;;0 i^yi

XI

>'3

^ X4

—"

"

.

A>'3

yA

In the above table, the second column gives the value of the function at specified values of the variable x. The third column having the heading A is constructed from the values of y in the second column by subtracting y^ from 3^1, y^ from y2, and so on to give the column of first differences. The fourth column of second differences is obtained from the first differences in the third column by subtracting the entry above from the one below. The difference table is completed by repeating the procedure outlined above. This will be illustrated by the following example. M.M.C.E.

11

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

310

Example. Construct a difference table of >^ = f{x) from the following data and state what conclusions can be drawn from the result.

X

1-0

y

7-567

M 12-159

1-2

he difference table of these data

X

y

10

7-567

11

12-159

1-3

17-399

23-339

1-4

1-5

30-031

37-527

1-6

45-879

is

Ai

A2

A3

A4

4-592 0-648

0-052

5-240 1-2

17-399

1-3

23-339

1-4

30-031

1-5

37-527

1-6

45-879

0-700

0052

5-940 0-752

0-052

6-692

0-804 0-052

7-496 0-856 8-352

Inspection of this difference table shows that all the 3rd differences have the same value which from the elementary properties of a polynomial suggests that the data represents a function of the third degree.

Difference tables are very useful for assessing scientific data whether it be interpolation, extrapolation or fitting an equation to a curve of experimetal data. These appUcations will be illustrated in Section 9.4 after more of the background has been established. However, certain properties of difference tables will be stated at this point. These are (i) If the value of a function is known at equally spaced values of the independent variable of the function, or if the difference between successive values of the function is known, all the columns of higher differences can be constructed. (ii) If one complete column of differences of any order, and one item of each column on the left of the complete column is known, all the columns of the lower differences and the values of the function itself can be determined within the range of the independent variable. (iii) If the members of the column of nih. differences have the same constant value, the data from which the difference table was constructed is derived from a polynomial of degree n. These rules can be tested by considering the example given above. Thus A^jK = 0-052 and the last member of the A^ column is 0-856. Therefore the last but one member will be (0-856-0-052) = 0-804. In a similar manner each item in the A^ column can be inserted and thereafter the A^ column can be completed if one of its members is given.

9.

9.2.4.

311

FINITE DIFFERENCES

Finite Difference of a Product or Quotient

On

occasion it is necessary to find the difference of a product of two Thus, if two functions of an independent variable «, which can have integer values only, are written as/(«) and («), then the first difference of the product of these two functions is functions.

=f(n + l)(t>(n + l)-f(n)ct>(n) = /(« + !) l(t>(n + 1) - (Pin)-] + ct>{n) [/(« + 1) -/(«)] = f{n + l)A0(n) + {n)^f(n)

A[/(n)(/>(n)]

(9.14)

or grouping the terms differently, A[/(n)(/)(n)]

=

(i>{n

+ \)^f{n)+f{n)^4>{n)

(9.15)

which demonstrates the similarities between finite difference calculus

and

infinitesimal calculus in respect of the differential coeflficient of a product of

two functions. If the first difference of the quotient of /(«)

and

(«) is

desired, a similar

procedure gives,

Kn + l) f(n) L^WJ cPin + 1) (n] l)j)(n)-(/)(n + l)/(n) ^ /(n + (/)(« + l)0(n) ^ (») U{n + 1) -/(n)] -/(n) 4)(n + \)(i>(n) \fin)}

\_ct>{n

^

+ 1) - 4>{n)\

(l>(nW(n)-f(n)A^(n) (/>(«

+ l)0(n)

^

•

^

which again demonstrates the similarity between finite difference calculus and infinitesimal calculus. However, it should be observed that the denominator of equation (9.16) does not contain a squared term and in this respect differs shghtly from the differential coefficient of a quotient.

9.3.

The

Other Difference Operators

above sections are, to be precise, called "forward differences" and are the ones most generally used for the solution of engineering problems. Consequently they are simply referred to as the first, second or «th differences and the prefix "forward" is omitted when it is clearly understood which type of difference is meant. However, in certain interpolation formulae the terms "backward difference" and "central difference" are employed and therefore their meanings must be understood. In actual fact there is no basic difference between a forward difference and a backward difference since both express the difference between the higher and next lower value of the function, and the significance between these terms is differences presented in the

11-2

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

312

best understood by reference to Table 9.1.

In that table

it

will

be seen that

column refers to yg and the various differences are obtained from successive members of the series starting from yQ. These are the forward differences and they are shown by the sohd arrow. On the other hand, the last entry in each column is the first, second and third forward difference of each successively lower member of the series as shown by the dotted arrow. These entries in the table can be relabelled as backward the

first

entry in each

differences by using the following definition.

^yn

The second backward

difference

V^y, It is

in

now clear that

Table

A

all

= yn-yn-i = is

^yn-i

(9.17)

written similarly thus,

= yn-2y„.,+y„.2 =

^^yn-i

(9.18)

of the backward differences shown by the dotted arrow

9.1 refer to y4..

third type of difference

known

as the "central difference"

is

symbol-

ized by prefixing d to the function.

Thus 'n-i

Central differences would be represented by a horizontal line in Table 9.1. To sum up, forward differences should be used when data from the initial parts of a table are to be assessed and backward differences used when

data from the end of a table are to be considered. This is particularly true and extrapolation purposes. Finally, for interpolation near the centre of a collection of data central differences should be employed. There are two further important difference operators which will be described in the following sub-sections. for interpolation

The operator

9.3.1.

E

The operation of changing the value of a function to correspond with a change of one increment in the independent variable is denoted by E. That

is

In a similar

EyQ

=

y^

E(Ey,.2)

=

E'y„.2

(9.20)

manner y„

=

Ey„_,

=

=

.••

= £>o

(9.21)

Equation (9.21) shows that E" operates on yQ n times to increase the value of the function corresponding to an increase of /7 increments in the independent variable.

The exponent n Thus

in

E" can take any positive or negative integer

E-'KxQ)=KxQ-h) where the exponent (-1)

signifies that the

function

lower value of the independent variable. The operator E has significance only when

value.

(9.22) is

reduced to the next

it is placed before a function or variable, and in this respect it is similar to other operators. Also, E can be treated hke an algebraic symbol in that it obeys the distributive, commu-

9.

tative,

313

FINITE DIFFERENCES

The reader

and index laws of algebra.

is

asked to verify these for

himself.

Since E and A are both finite difference operators that obey the same laws of algebra, it would be expected that they must be inter-related. The relationship between them can be shown as follows.

yn^,=Ey, yn^,-yn

=

(9.23) (9.24)

^yn

Substituting equation (9.23) into (9.24) gives

{E-\)y„

=

^y„

(9.25)

which can be written as an identity between operators thus,

£ = 1+A

(9.26)

This simple relationship is most important because it often allows algebraic expressions in terms of the one operator to be simplified in terms of the other by using ordinary algebraic manipulations. Thus, if x is the independent variable that is capable of taking the values Xq {xq + h) {xq + 2h) ;

'^'•'*'''°

/(xo

By

+ ft) =

;

;

(9.27)

£/(xo)

Taylor's theorem (see Section 3.3.7)

EKx,) where

D

= /(xo + h)= /(xo) + hDfixo) +

is

^' D'fixo)

the differential operator d/dx.

+

^ D'f(xo) +

.

.

.

(9.28)

From equation (9.28) it can be seen

that 2

/i,n\3

/(xo)

The terms

in the square bracket

=

£/(xo)

=

(l

+ A)/(xo)

(9.29)

form the expansion of the exponential

function e^^, and therefore equation (9.29), provides a relationship between the differential operator and the finite difference operators. That is

E=\ + ^ = The

(9.30)

e''^

three symbols in equation (9.30) are all operators and they can be treated In the mathematical relations developed to show

like algebraic quantities.

the relationship between them, the analysis has been confined to the first order of the operator. This was done for convenience only, and for any value of .„ ^„ ,, ,^^ ^« ^..

w

E"'

=

{l

+ ^Y =

e'"^^

(9.31)

In addition, any polynomial of these symbols having constant coefficients represents an operation on a function. Thus

£7(^o)

= (l+A)7(xo) = /(xo) + A/(xo) + A^/(^o) + (7) (2)

=

[i

^'Kxo) + (3)

+ (7)a + (-)a^ + (-)a3+...]/(xo)

...

(9.32)

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

314

Equation (9.32) is known as "Newton's forward difference formula". The polynomial within the square bracket is a polynomial operator and m is This polynomial operator only has any number, integer or otherwise. significance when it operates on a function such as/(x) in equation (9.32). This result finds extensive use in interpolation (Section 9.4). In a similar manner it is possible to produce an expansion of £"" in terms of the operator V. Thus by equation (9.26)

=G-^)'= and if the last expression on /(xo) the result is £7(^o)

in

(l-Afi-')-"

equation (9.33)

is

(9.33)

expanded and made to operate

= (l-A£-')"7(xo)

+ ...{^^''^~^'^K'f{xo-rh) + ... Using equations £-/(>.)

(9.17, 18) to introduce

backward

(9.34)

differences,

("^') v/w+^^^ ("*;"')s'-/(>.)+ -/(«.)+ (7) v/(»o) +

.(7)v.(-')v=.(-^)v ^+ Equation (9.35)

is

known

as

/(^o)

...

(9.35)

"Newton's backward difference formula" and Again m can be any number,

equation (9.32) is used for interpolation. integer or otherwise. like

The Difference Quotient

9.3.2.

The difference quotient was introduced by Norlund (1920) in order to show the similarity between finite difference calculus and infinitesimal calculus. The difference quotient is defined by the equation

^ /(xo + ft)-/(xo) On

occasion

it is

(9 3^^

h

h

referred to as "Norlund's Operator".

2

Similarly

A would

A/(x

)

=A

be the second difference quotient and

p+ o

^)-/fa) l

is

written

_ fixo + 2h)-2f(xo + h)+f(xo)

^^^^^

These concepts can be extended to the mth difference quotient, thus

f

/(^o)

=

~

\f(xo

+ mh) + (7)/(^o + '^^^h) +

+

(2

)/(-^o

+ '^^^^h) +... +/(xo)]

(9.38)

:

9.

315

FINITE DIFFERENCES

Equation (9.38) shows the relationship between the difference operator and which may be summarized as

difference quotient

AJ(xo)

=

h^'Afixo)

(9.39)

h

Thereafter the relationship of the difference quotient with the other finite difference operators follows from the above sections which may be summarized thus A"'

= (E-IT =

(e""^

- IT =

h'"A

(9.40)

h

Finally, it can be shown that the difference quotient obeys the distributive law for addition, the index law for positive integers and the commutative law for constants. It is left for the reader to verify this statement.

9.4.

Interpolation

It frequently happens that the engineer is confronted with a collection of data relating two variables, such as the pressure inside a vessel corresponding The pressure is thus some unknown to the temperature of its contents. function of the temperature, and he is required to predict the pressure at some temperature in between or outside the limits of the data he possesses. If the pressure is required within the limits of the data, the mathematical procedure is called "interpolation"; whereas if the point lies outside the range of the data, the mathematical process is called "extrapolation". Interpolation or extrapolation depends on the assumption that the functional relationship between the two variables is continuous over the range of the independent variable being considered. Therefore in order to be able to interpolate or extrapolate, the engineer must obtain this functional relationship or some other functional relationship that approximates to it. For this purpose the trends in the data must be considered from the point of view of the experimental results and the mathematical analysis. For example, the chemical engineer would be famihar with the general shape of a vapour pressure-temperature curve and its position on a graph from the experimental data. The mathematician is able to assign an approximate equation to this curve and thereafter interpolation is straightforward. Usually the experimental data is approximated by a polynomial and the degree of the polynomial can often be estimated by preparing a difference table from the collected data. The difference column that gives an approximately constant value, gives the degree of the polynomial which can be fitted to the data by selection of the constants.

9.4.1.

Newton's Formulae

When

the data points are available at equal intervals of the independent Newton's difference formulae can be employed and the procedure best illustrated by the following examples.

variable, is

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

316

Example

Estimate the value o^ Jq{z) at z Jq{z) given in the following table. 1.

Jo(z)

The following

difference table

z

n

is

4-0

3-8

-0-4026

-0-3918

-0-3643

-0-3202

3-479 from the values of

3-6

3-4

3-2

z

=

-0-3971

constructed from the data supplied.

^Mz)

.ov^y

A3/o(r)

^'-Mz)

1

2 3

4

From the above table it can be seen that Jo{z) can be approximated by a polynomial of second degree within the range of data supplied. (a) Using forward differences

and

Then by Newton's forward

+ mh) =

=

ZQ

Zq

=

3-2

m=

.-.

Jo(zo

z

+ mh and h

=

0-2

1-395

difference equation (9.32)

Joizo)

+

[^j

AJo(zo)

+ (^jA^Joizoj

Substituting from the difference table gives Jq(z)

=

-0-3202 + (l-395x -00441)

+

1-395 X 0-395

x 0-0166

= -0-3771 (b)

For

Using backward

differences.

this interpolation, Zq will

be selected at 3-8 because AVo(z)

will give

a value derived from Jq(3-4). Consequently the interpolation formula will include differences obtained from the neighbourhood of /o(3*479) as shown

by the

Thus Then

lines in the difference table.

Zo

=

3-8

3-479

=

3-8

let

.-.

Inserting these figures into

m=

and

+m

h

=

0-2

x 0-2

-1-605

Newton's backward difference formula

(9.35)

9.

+ m/i) =

o(zo

=

317

FINITE DIFFERENCES

+ (7) V^(^o) + ("^J^) VVo(zo;

Jo(^o)

1-605

-0-4026

+

x-O-OlOS] 1

+ =

1-605 X -0-605 X 0-0167' [-

-0-3772

/o(3 '479) predicted by the forward diflference formula agrees very well with the result obtained using the backward difference formula. The result estimated from Chistovaf for /o(3-479) is -0-3772 which agrees

The value of

with both

results.

Either of the difference formulae can be used for interpolation, but for

extrapolation the forward difference formula must be used.

Example

2.

Using the data given in the table below, estimate the vapour ammonia vapour at 167°F. Latent heat of ammonia

pressure and density of

=

544 Btu/lb. °

Temperature

Pressure Ib/in^

From

70

80

90

128-8

153-0

180-6

100 211-9

110 247-0

the above data the following difference table

^p

P

I

70

128-8

80

1530

is

120 286-4

130 330-3

140 379-1

prepared bJ-p

ts}p

lA'l 3-4

27-6

90

180-6

100

211-9

110

247-0

120

286-4

130

330-3

140

379-1

0-3

3-7

31-3

01 3-8

351

0-5 4-3

39-4

0-2 4-5

43-9

0-4 4.9

48-8

The fourth

differences t^^p have not

been calculated because A^;? are very contribution to the estimated pressure. Inspection of the difference table shows that the last entry in the A^/? column (i.e. 0-4) is derived from p = 247-0 Ib/in^ corresponding to 110°F. Theresmall and will

make

very

little

fore 110°F will be taken as the base temperature.

p =f(t) =f(to + mh)

Then t Chistova

E.

A. " Tables of Bessel Functions." Pergamon Press, London (1959).

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

318

where

=

t

167°,

=

167

.-.

=

^o

110 +

10m

Newton's forward difference formula P

10° 5-7

(9.32) gives

(T)a.o.0AVo + 0A'Po

+

=

Po

=

^.^r. 247-0

— 5-7

+

.^,

X 39-4

—-— x ^^ — x ,, + 5-7x4-7x3-7 1x2x3

5-7x4-7

+

247-0 + 224-58

0-4

4-5

1x2

1

= =

= orm =

and h

110°

+ 60-28 + 6-61

538-5 Ib/in^

The value extrapolated from Perry f

538.0 Ib/in^.

is

In order to find the density of the

ammonia vapour

it is

convenient to

use the Clausius Clapeyron equation J

dp

SH

^

dT

nv,-vd

where Tis the temperature in degrees Rankine. Assuming that the volume of the liquid is negUgible compared with the volume of the vapour gives p ^

=

—= 1

V,

T dp SH dT

III

dp/dT can be evaluated from the difference table by taking the logarithm of equation (9.30). Thus /i£)

Interpreting

D

as

=

ln(l

+ A) = A-iA2+iA^-...

d/dT and substituting equation IV into

P

IV III gives

V

= ;J^(Ap-iA^P+iA'p-...)

The values of the differences in the bracket of equation V must be determined at the required temperature of 167°F. This can be done by extrapolating each column of differences in the table using the same method as the extrapolation for/? at 167°F. at

t

=

167°F,

p =

Substituting into equation

The

538-5, A/?

V

results are

=

70-4,

=

10 X 544 .-.

This value t Perry,

is

J.

more H.

reliable

(Ed.).

6-8,

A^ =

6-8

0-4\

l.r. ___(70.4-+ -)

p

.

=

0-0099

Ib/ft^

than that obtained from the ideal gas laws.

" Chemical Engineers Handbook," 3rd edition.

(1950). X

Bransom,

S.

0-4

gives

627 '

P

A^ =

H. " Applied Thermodynamics." Van Nostrand (1960).

McGraw

Hill

9.

9.4.2.

319

FINITE DIFFERENCES

BesseVs Interpolation Formula

This formula

is

for use with central differences along a horizontal line in

can be seen in Table 9.1 that values are only available are filled by taking an arithmetical average of the two values adjacent to the one required in the same column and denoting the value by fxd, jxd^, .... etc., depending upon the order of the difference. It is customary to choose a fractional value of m which is less than unity and use the formula

the difference table.

It

in alternate columns.

The gaps

EJ(xo)=f(xo) + mdf(xo + ih)

(m + l)m(m-l)(m-2)

f

^^^^

[fiS'f(xo

m-i

,,^ + "^-^S'Kxo + ih)^ + ih) ,

,

4!

+ 9.4.3.

(9.41)

...etc.

Lagrange's Interpolation Formula

Quite often, the data are available at unequal intervals of the independent variable and this precludes the use of a normal difference table.f Lagrange's method fits a polynomial to the data but otherwise has no connection with finite difference calculus: it is included in this chapter merely for completeness within this section concerning interpolation. The fourth degree polynomial which passes through the four points (xq, yo), (^1, yi), (X2, y2)^ (x^, y^) can be written

(x-Xi)(x-X2)(x-X3) ^(^o-^i) (^0-^2) (^0-^3)

(x-Xo)(x-X2)(x-X^) _^

\xi-Xo)(Xi-X2)(Xi-X^) (x-Xo)(x-Xi)(x-X3)

^

^(X2-Xo)(X2-Xi)(X2-X3)

(^3-^0) (^3-^1) (^3 -^2)

and

this

form

is

known

as "Lagrange's interpolation formula".

more than four points

The

exten-

obvious the numerator and denominator of each fraction is extended to include the further values of x„ and the complete equation is extended by adding further terms to include all values of >^„. It is to be noted that the term corresponding to the appropriate y„ is sion to

t It is

is

is

fairly

:

possible to calculate in terms of " divided differences " where the first difference ratio of By to bx, but the method used herein is considered to be more

found from the

satisfactory.

t

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

320

omitted from each numerator and its counterpart from the denominator. The use of many more than four points is not recommended due to the

computational

effort.

Example 3. Find the thermal conductivity of propane at 100 atm. and 210°F from the following data given by Leng and Comings. rCF)

P(atm)

^(Btu/h ft2.°F

96-7

0-0490

131-5

00518

88-9

0-0440 0-0466

154

189

131-8

140-9

0-0402 0-0435

95-3

0-0353

123-0

00376

96-7

222

284

ft-i)

From the graph given in the paper, other results show that in this region the conductivity varies approximately linearly with pressure so that a linear interpolation is adequate in this direction.

A set of values for conductivity and temperature at a pressure of 100 atm. can be found from the above values as follows. At 154°F and 100 atm., k

100-0 -96-7

=

0-0490

=

0-0493

Hence the following

+

(0-0518-0-0490)

table can be compiled.

T

154

189

222

284

k

0493

0-0447

0-0404

0-0357

Substituting these values into equation (9.42) gives the thermal conductivity of propane at 100 atm. and 210°F. Thus

k

=

(210-189)(210-222)(210-284) 0-0493

(154-189)(154-222)(154-284)

+

(210-154)(210-222)(210-284) 0-0447 (189 -154) (189 -222) (189 -284)

t Leng, D. E. and Comings, E.

W.

I.E.C. 49

(ii),

2042, 1957.

9.

/c

(210-154)(210-189)(210-284)

+

0-0404

+

0-0357

(222 - 154) (222 - 189) (222 - 284)

(210-154)(210-189)(210-222) (284 -154) (284 -189) (284 -222)

= 0-0419 Btu/hft^°F ft"

difference equation

finite

^

Finite Difference Equations

9.5.

A

321

FINITE DIFFERENCES

is

a relationship between an independent

and sucThus an equation of the type

variable that can take discrete values only, a dependent variable, cessive differences of the dependent variable.

+ «l>^l + «2>'2 + ---+«n>'n = W equation if y = f{x) and x can take the

'30>'0

(9-43)

values Xq, Xq + H, a finite difference XQ + nh only. In equation (9.43), Oq, a^, aj, .... a„ can be constants or functions of the independent variable, (pix) could be a constant or a function of X. Equation (9.43) can therefore be written is

....

+ aJ{xo + nh) = 0(x) (a„E" + a„.,E"-' + ... + ao)f(xo) = (P(x) /(xq + rh) = EJ(xq). But £ = 1 + A,

«o/(^'o)

or

because .-.

+ «i/(^o + /i) +

.

(p„A" + ;7„_iA"-^

.

.

+ ...+;.o)/K) =

0W

where the coefficients p^ are obtained from a^. Equations different forms of the same finite difference equation.

(9.44)

(9.45)

(9.43) to (9.45) are

Finite difference equations are similar to differential equations in that they

can be classified by "order" and "degree". Thus the order of a finite difference equation is determined by the order of the highest difference. For example, the equation (9.46) yn^i-ciyn = ^ is

of order

1

because

it

involves only the

first

difference,

(£-a)j„ = [A-(a-l)]>'„

On

the other

=

(9.47)

hand the equation yn-ciy,-,

is

i.e.

+ by,.2 =

of order 2 because the highest difference [A2

is

^

of order

(9.48) 2,

thus

+ (2-«)A + (l-a + 6)]y„_2 =

(9.49)

should be noted from equations (9.47) and (9.49) that it is the difference between the highest and lowest members of the dependent variable that

It

determines the order of the equation, and not the highest indicated by the suffix to the dependent variable.

member

as

MATHEMATICAL METHODS

322

CHEMICAL ENGINEERING

IN

The degree of a finite difiference equation is the highest degree of the dependent variable, or any of its differences in the equation. Equations containing y„, Ay„ and A"y„ are of the first degree and are said to be Hnear, whereas equations containing products of the type >'„>'„_ i or their differences are of degree 2. Linear difference equations are by far the most common, and they will be treated initially. Linear Finite Difference Equations

9.6.

Equation

(9.43)

coefficients Qq, a^,

is ....

a linear a„

may

finite difference equation of order n and the be functions of the independent variable or

constants.

The solution of this type of equation is similar to a Hnear differential equation in that the complete solution is made up of a complementary solution and a particular solution. Proof of this can be obtained in a manner very similar to that

shown

and However, each part of the complete solution

in Section 2.5.1 for differential equations,

therefore will not be repeated. will 9.6.

be discussed.

L The Complementary

Solution

For ease of manipulation, consider the second order

linear finite difference

equation with constant coefficients.

yn^2-Ay„^, + By„ = In the same

way

(t>(n)

as for differential equations, the

the solution of equation (9.50)

when

(j)(n) is

zero,

is

i.e.

yn^2-Ay„^,+By„ = But equation

(9.50)

complementary solution

(9.51)

(9.51) can be written in the form

(E^-AE + B)y„ =

(9.52)

Since the polynomial within the brackets can be treated as an algebraic quantity,

it

can be factorized. .-.

{E-p,)(E-p,)y„=^0

(9.53)

The two factors could be in either order since the operator E will commute with constants such as pj and P2', hence there are two independent solutions of equation (9.51) satisfying

or It

(0

{E-p,)y„

(ii)

(JE:-P2)>'„

= =

can be seen by substitution that the solution of

and the solution of

Hence

(i) is

yn

=

A'pl

(9.54)

>„

=

B'pl

(9.55)

(ii) is

the general solution of equation (9.51) y„

=

A'p\

is

+ B'pl

given by (9.56)

9.

323

FINITE DIFFERENCES

where A' and B' are two arbitrary constants that would be evaluated by the boundary conditions of the problem. It will be noticed that equation (9.56) is the complementary solution of the original second order equation and that it contains two arbitrary constants. In a similar manner it can be shown that an nth order equation will provide a complementary solution containing n arbitrary constants; which again shows the similarity to the complementary function of a Unear differential

equation.

Equation (9.52)

is

known

difference equation (9.50). plex.

as the "characteristic equation" of the finite

roots

Its

may be

different or equal, real or

com-

pi and p2 are real, the final complementary solution. If the roots are complex,

The case of

different roots

is

given above, and

if

equation (9.56) is they will occur in conjugate pairs thus, p^

= a+

ij?

=

=

p2

re'"^;

(x-ip

=

re'"'^

(9.57)

In this case, the solution (9.56) can be written in the form y^

=

r\A' cos

n

+

by using De Moivre's theorem (Section solution

when

B' sin n0) 4.7).

This

(9.58) is

the complementary

the roots of the characteristic equation are complex

and

is

similar to the solution of the corresponding differential equation. Finally,

when

the roots of the characteristic equation are equal, the

complementary solution takes the form y„

=

{A'

+ B'n)p"

(9.59)

again Hke the comparable form of the differential equation (Section 2.5.2b). 9.6.2.

The Particular Solution

This can invariably be found either by the method of undetermined coor the method of inverse operators both of which are very similar to the corresponding method for Hnear differential equations. Since these methods were described in Section 2.5, they will only be repeated briefly here with reference to the second order finite difference equation efficients

;

{E^-AE + B)y„ =

A particular

solution of this equation

(9.50)

{n)

is

by the method of inverse operators. The operator can be factorized as the above sub-section, and separated into partial fractions. Thus

E'-AE + B ~ {E-p,){E-P2) ~ {E-pd ~ where a

=

l/ipi-Pz)-

Each

partial fraction

(E-p^)

on the right-hand

in

^^'^^^

side of

MATHEMATICAL METHODS

324

IN

CHEMICAL ENGINEERING

equation (9.61) can be expanded as follows -1 1 -f

£-Pi

i-p,/

(1-Pi) a

+ --—,-...

(i-Pi)

1-Pi

a

and

E-P2

(1-P2)L

(9.62)

(1-Pi)

A^ A 1-^— + 7^—V2--I I-P2 (I-P2)

(9-63)

of the operators are most convenient for finding the a polynomial in «, because only a finite number of terms will be needed in the expansions. When (j){n) = ka", an alternative procedure is advantageous. Thus,

The forms

(9.62, 63)

particular solution if

(/)(a?) is

and

Ea"

=

E^'a"

=

a"^^

=

a.a"

a""^"

=

a'".a"

In general,

f(E)a"=f(a)a"

(9.64)

provided /(£) can be expanded as a polynomial in E. Equation (9.64) is the key to a particular solution when (/)(«) = ka". The usual difficulties can arise when/(fl) is infinite, but these can be overcome by techniques similar to those given in Section 2.5.5. It should be noted, however, that the equation analogous to equation (2.79) takes the form f(E)(a"y„)

=

a"f(aE)y„

(9.65)

where a and E appear as a product instead of a sum on the right-hand side. Equation (9.65) is of Httle practical value since it does not lead to any simplification in the form of the operator. Hence, rewriting equation (9.50) with (/>(«) = ka",

(E^-AE + B)y„ = and using equation

a'^

(9.66)

(9.64), the particular solution is

yn

provided

ka"

— Aa + B ^

0.

=

2 .„+ Bo a^-Aa

(9-67)

This exceptional case can be treated in a similar

manner to the corresponding differential equation by using equation (9.65) and introducing the difference operator A using equation (9.26). In concluding this sub-section on the determination of particular solutions of finite difference equations,

should be pointed out that a particular when 4^+l_ A

—Y~

A^'^'''

the equiUbrium constant.

That

—

1

is

= K^X^

ym

where y„ and x^ are the mole fractions of the solute

in the gas

and

liquid

phases respectively. Y,

Gi

•-o/o

1

1

V m-1 (Y™

X„_',

I'Cu-l

^m

m

+

1

A^ N

r

1 ^N^1 Yn +

1

.

Ln X^

Fig. 9.1

Considering Fig.

9.1

and taking a material balance over

plate

m gives

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

326

But

yrrr

G^=Y„Gs +

and x„ L„

i

= X^ Lq

= -^"l {^^ ''-m{hrya:„

or

K.

(

Substituting equation III into equation I to eliminate

y _

II

Ill

X^ and X^.^

gives

^m+l+(^ff,-i/-Km-iG^-i)y^_^ l

+ (LJiC„GJ

But (L„_i/Ar„_iG;„_i) = ^;„-i, the absorption factor for plate m—\\ and ^„, the absorption factor for plate m. Therefore equation IV can be written as

(LJKJj^ =

Taking a mean value of the absorption factor as suggested by Edmistert or Horton and Franklin| equation V becomes

y„+i-(i+^)y^+^y^_i = o

vi

A

is the mean value of the absorption factor between the bottom and top of the column. Putting equation VI into operational form and factorizing gives

where

[_E^-{l

+ A)E + A-\Y^ = {E-\){E-A)Y^ =

VII

which has the solution

=

y^

Ci

+ C2.4'"

VIII

where C^ and C2 are arbitrary constants.

Now since

Yq

A

is

IX

assumed constant; and equation VIII gives

+ C2

X

+ C2^^"'^ y^+i-yo = C2(A^^^-l)

XI

Yo

y^+j

Similarly, .-.

and

= KXo

= =

C, Ci

Y^+^-Y^

also

=^

C2(A^''^-A)

XII XIII

Dividing equation XIII by equation XII gives

y Y which

is

the well

known Kremser

AN+i_A AN+l_i

equation.

W. C. I.E.C. 35, 837 (1943). G. and Franklin, W. B. I.E.C. 32, 1384 (1940).

t Edmister, t Horton,

_y -Y

XIV

9.

327

FINITE DIFFERENCES

Simultaneous Linear Difference Equations

9.6.3.

The

analysis presented in the sections above has been confined to one dependent variable. However, when two or more dependent variables are present in a system of difference equations it is usually possible to ehminate all of these variables except one by the normal rules of algebra. Thus consider the following examples.

Example

2. In a proposed chemical process 10 000 Ib/h of a pure liquid be fed continuously to the first of a battery of two equal sized stirred tank reactors operating in series. If both vessels are to be maintained at the same constant temperature so that the chemical reaction

A is to

kz

ki

A->B-C of the vessels that will give the

will take place, estimate the size

yield of the product B.

and k2

=

0-05

min"^

The

I

specific reaction rate constant k^

at the temperature of the reactors,

=

and the

maximum 0-1

min~^

fluid density

constant at 60 Ib/ft^. Let the concentration of leaving any vessel n be C^ „, the concentration of B be Cfi „, and of C be Cc^n- Then a material balance over stage n is

A

gives for:

component A, where 6

is

the nominal holding time in the reactor;

component

B,

^B,n-l~^B,n—k2CB^rfi — k^C^nd Solution of equation II is obtained in the usual way. Thus k2e = ^. Then /. x^ ^ (l + a)C^.„-C^.„-i =

+ a)-l]Q,„ = C^,n = K,p'[ constant and pi =

III let

where K^ is an arbitrary Cyin from equation VI into equation [(l

The complementary

V

The

is

VI 1/(1+ a).

and rearranging

Substituting for gives

+ i5)£-l]Q„_i=aXip"i

solution of equation

Cs,n

where K2

III

the arbitrary constant

=

VII

VU is

K2P"2

and P2

a and

IV

[£(l

or

k^O

=

=

VIII

1/(1+ fi).

particular solution of equation VII can be written

_

oiKipl

"(l + «Pi-l

=

-T^-

IX

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

328

by using equation

(9.64).

The complete CB,n

Since the

initial

feed to the reactors

K2 =

=

X

pure A, at «

is

=

0,

C^ = C^

X gives XIII

-7r^(p"l-p"2)

must be a maximum for « = 2 and maximum by varying n at fixed 9,

to find the

than by varying 6 at fixed

Hence, by differentiating equation XIII,

n.

dCB„

ccC.Q

dn

jS-a

C^

„

Pi

Inpi

Pi

lnp2

Pi 1

+a

1

+P

p2

XV

1

+ 0-1^

1

+ 0-05^

1

1

and

,llO:05gyjn(l+0.05^)

.hen

XIV

1

1

Since

=

(Pl\np,-pl\np2)

.

^nd

XII

The condition required is some value of 6. It is easier

that

o

XI

^C^,o

Substituting these values into equation

CB,n

thus

= C^

Kj^

and

is

K2p"2+^p\

=

.'.

solution of equation VII

\1 + 0'W)

In (1

^^^

+ 0-10)

Following the methods of Section 11.5 to solve

this

by successive approxima-

tion gives the result

= •

*

.'.

XVII

0-456

Reactor volume

60x60y

Volumetric feed rate

10,000

Optimum volume of each

=

reactor

^ ,^,

0-456 is

l-27ft^.

Example 3. 8540 Ib/h of an animal fat are to be hydrolyzed and extracted column using 3760 Ib/h of water.f If the column is to operate

in a spray

under counter current flow conditions, the percentage hydrolyzable glycerine is 8-53% by weight, and the glycerine in the fatty acid leaving the tower is 0-24% by weight, calculate the number of theoretical stages in the column. in the fat

t Jefferys, G. V., Jenson, V.

G. and Miles,

F. R.

Trans. Instn.

Chem. Eng.

39, 389 (1961).

9.

FINITE DIFFERENCES

329

A mass balance will give a glycerine concentration in the Sweetwater of 18-8%, and the total weight of fat phase held up in the column is 12 200 lb. The distribution ratio of glycerine between water and fat is 10-32, and the reaction rate constant is 10-2 h~^. Solution.

plate

column

In order to determine N, the number of theoretical stages, the illustrated in Fig. 9.2 is considered and the following symbols

are used.

L G

Ib/h fat phase rising through the column.

H

lb fat phase held up per stage. weight fraction of glycerine in raflfinate. weight fraction of glycerine in extract. weight fraction of unreacted fat in rafRnate. lb fat required to produce 1 -0 lb of glycerine. pseudo first order reaction rate constant expressed in terms of fat concentration (h~^).

X y z

w k

Ib/h water phase descending through the column.

Cj.

^ Ar

n-1

n

^

+

1

N

F

1 FK5. 9.2

A

glycerine balance over stage n gives

Gj„_i

and an equivalent column gives

+ Lx„ + + i

kH — w

z„

=

Gy^ + Lx^

glycerine balance between plate n

-Z;v +

i

+ Gy„_i =

G};^

+ Lfx„ +

I

and the base of the

II

1

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

330

The equilibrium

relation

is

yn

where

m

is

= ^^n

Substitution of {zjw) from equation of X by the use of equation III gives

mG^

mG/L =

Equation

V

a,

kH/L =

a linear

is

into equation

I

mG

kH fm

,

Letting

III

the equilibrium ratio.

\

subsidiary equation

[E^-(a + P+l)E + oc(l + p)']y„ = and the complementary solution y„

The

particular solution

=

^

gives

The

equation.

finite difference

and elimination

mG

and rearranging IV

p,

II,

is

VI

is

A(x"

+ B(l + py

VII

is

(E-cc){E-l-P)

C=

where

- ^z^+i

j?^a>;^

j

Because

C

is

independent of n and

£ = + A, 1

£"

can be replaced by unity.

Hence equation VIII becomes

-C

a-1 The complete

solution of equation

V is

then

a—

with the boundary conditions that at n = 0, ^ = 0; and at n = A'+l, X = 0. Substituting these boundary conditions into equation gives

X

\

and

The

a-1

;[«''+ '-(l

+ ^Sf+'J

B = - /?0VZ!!1£^±>) [..r'^'"\..1 a-1 J[a''*'-(,l+pf*'\ V final solution after simplifying is

_

mzjv^i

r

^"-K^TTDr

^

(a-l)a^-)?(l+^f ] (l-a+^)(l + ^rJ

9.

Substituting the data supplied into equation

a

—L

10-32x3760

_kH _

10-2x12200

where 12 200/

=

gives ^

=

^, 4'54

_ "

14-6

8540

^~~L~ A^^

XI

mG =

=

331

FINITE DIFFERENCES

8540

H, the hold-up per

xN

N

stage.

Then from equation XI /10-32xO-0853\

^.oo

^

4-54^

3-54 X 4-54^ - (14-6/Ar) (1

+

+ 14-61 Nf

XII

(^'--)(-W gives

A^

9.7.

=

2-8 (by trial

and

error)

Non-Linear Finite Difference Equations

These types of equation occasionally arise in the solution of engineering problems and they are difficult and sometimes impossible to solve. However, first order non-Unear equations can be solved graphically, and some second order equations can be solved by special substitutions. These methods will be discussed separately below. 9.7.1.

Graphical Solution

Consider the

first

order non-linear difference equation

y„^,-yi + Ay„+B = where

A

and

B are

constants,

ing to their subscripts.

and rearrange

it

(9.68)

to separate the terms accord-

Thus

y„^,==y'„-Ay„-B

(9.69)

By

selecting arbitrarily a suitable set of values for y„, a corresponding set of values of >^„+i can be calculated from equation (9.69). Hence a table of

values of ^'^ and y„+i can be compiled and the results plotted on rectangular coordinates in the form y„vsy„+i. The curve AB in Fig. 9.3 illustrates equation (9.69). In order to find the solution, the diagonal jv„ = y„+i shown by the line PQ is constructed. Starting at the boundary condition yQ at

A

(of coordinates

>'o,

yi) and drawing the ordinate, the point

the diagonal where yo = >'i. From C, a horizontal line at the point D. The coordinates of meet the curve

AB

is

D

C

is

located

on

drawn back to are (y^, y2) by

equation (9.66) and hence y2 is evaluated. By continuing this stepwise procedure to A^ steps, the value of y^+i can be obtained. Thus the value of >^ corresponding to any value of n can be obtained. This method will work for all first order equations provided that they can be separated into a simple form such as equation (9.69). It is usually easier to choose values for the variable on the more complicated side of the

I

MATHEMATICAL METHODS

332

IN

CHEMICAL ENGINEERING

h +

Graphical solution of a

Fig. 9.3.

l

finite difference

equation

equation and solve for the other variable. Thus it is simpler to choose a set of values for y„ and solve equation (9.69) for y„+i, than assume values for y„ + and solve the quadratic for y„. i^

Example 1. 1000 Ib/h of ethyl alcohol is to be esterified by reacting with 850 Ib/h of acetic acid in a battery of continuous stirred tank reactors, each of 30 ft^ capacity and maintained at 100°C. If the equilibrium relation is such that 75-2% of the acid will be esterified, estimate the number of reactors required for 60 conversion. At 100°C the specific reaction rate for esterification is 4-76x10"'^ 1/g mole min and that for the hydrolysis of the ester is 1 -63 x 10""* 1/g mole min. The density of the reaction mixture can be assumed constant at 54 Ib/ft^. Solution. In a battery of A^ stirred tank reactors, consider a material balance on reactor m. That is,

%

is the concentration of reactant A leaving vessel /w, r is the rate of reaction, q is the volumetric flow rate, and V is the volume of one reactor. For a second order reaction

where C^„,

A + B^C + D the rate of chemical reaction r

and

if

is

'A,

^A,m

— {^l^A^B^^l^C^Wnfi

IV

the nominal holding time in the vessel. concentration of reactant B exceeds that of reactant A by an c initially, this difference will be maintained throughout the system. is

If the

amount

III

in the effluent

the vessel.

where

is

= ki^CACB-k2CcCD

assumed to be perfect, the concentration of the comfrom the vessel will be the same as the contents of Therefore equation I becomes

the agitation

ponents

II

9.

That

is,

in

m

any reactor

HNITE DIFFERENCES

333

the concentration of reactant

B

is

(C^^„ + c).

By

the stoichiometry of the reaction the concentration of each product Substituting these concentrations of components B, C, and equation IV and rearranging gives

D

is

into

V is a first order non-linear finite difference equation which will be solved graphically for the total number of reactors required to convert 60% of the acetic acid. Thus assuming the feed is uniformly mixed, the Equation

initial

concentration of acetic acid

850x54 ^-'^

^^'

=

C^^o c

.-.

V

^^ ^ lb

= "185^ =

Similarly,

is

=

_ 24-8x454 ^ ^e moles p^ ^^^^^^ ^ 6-63—

10*18 g moles/1.

10-18-6-63

3-55 g moles/1.

_^

30-0x54x60

^

'=

.

=^^-^""'

1850

Substituting these values into equation

C^,.-i

=

V gives

= Q.. + [4-76xlO-^QJC^., + 3-55)-l-63x xlO->63-C^.J^]52-6 = 0-0164Ci,^+ 1-202 C^,^-0-376 VI

Arbitrary values of C^„ given in the table were selected and substituted into equation VI to give the corresponding values of C^ „,_i.

^A,m-t

•^A,m

^A,m-i

is

6-6

8-27

60

7-43

5-0 4-0

30

6 05 4-70 3-38

2-0

2 09

plotted against

C^ ^

in Fig. 9.4;

and

starting with the feed

of 6.63 g moles/1, the number of reactors The required conversion is 60%,

sition

.*.

acid in final effluent

.".

number of

=

0-40x6-63

is

=

2-65 g moles/1;

reactors to produce this effluent

Tillert extended the graphical procedure presented t Tiller, F.

M. Chem.

Eng. Prog. 44, 299 (1948).

compo-

''stepped off" as shown.

is 7.

above to second order

334

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

equations.

Thus consider the second order non-linear

finite

difference

equation

yn^2-^Pyn^i where

R

and

Q

and

are functions of >'„

1

1

+ Qyn = R P can be a

1

1

1

1

/

8 Acetic acid

in

feed

A/

V

/h

4

E


'„+i.

^

1

2

-

/

/

2

1

/

/

1 1

'

Fig. 9.4.

2

3

4

1

1

5

6

7

Graphical reactor analysis

The graphical procedure is as follows. On rectangular coordinates plot {R—Qy^v^y^ as represented by the curve AB in Fig. 9.5. Similarly plot [/? — (2+^)>^„ + i] vs>'„+i on the same scales and let this curve be represented by AC. Finally, plot the diagonal y^ vs >^„, represented by OD. Two

R-(P+0)y

R-Oy

Fig. 9.5.

Graphical solution of a nonlinear second order difference equation

I

I

9.

boundary conditions

will

be

335

FINITE DIFFERENCES

known

for a second order equation; yo will be

one, and y^ can be evaluated by using an overall material or energy balance for the problem. Mark off yo and yi on the abscissa and draw ordinates

from each to the curve AC. The vertical difference between the curves at yi corresponds to -Py^ and if this is added to (R—Qyo) at L to give the point M, the vertical distance from the abscissa to corresponds to y„+2 ^Y equation (9.70). Drawing a horizontal line to the diagonal at T and a vertical line to the axis locates y2 in its correct position. It cuts the curves AB and AC to give —Py2 which on adding to the AB curve at y^ gives the value of y^ which is located in its correct position relative to y2 by drawing a horizontal line from V to the diagonal. This establishes the stepwise procedure and can be continued to locate y„. In Fig. 9.5 the stepwise procedure is indicated by the arrows. The graphical procedure given above for second order non-linear difference equations is of limited application. Thus it cannot be applied to the solution of equations in which product terms appear from more than one level of >'. For instance, in Example 3, Section 3.3.9, equation I could not be solved by this graphical method because on rearrangement to the form

M

y„^,-{A-Cy„.,+Ey„)y„ + {B-Dy„.,)y„.,=0

(9.71)

P = A-Cy„.,+Ey„

(9.72)

the function

be noticed that P contains terms involving y„ and y„-i. The term Cynyn-i cannot be transferred to the other term without upsetting the condition to be satisfied by Q in equation (9.70). Hence equation (9.71) cannot be solved by the graphical method proposed by Tiller; the numerical method to be presented in Section 11.4.1 must be used. It will

9.7.2.

Analytical Solution

At the present time only a hmited number of non-linear finite difference equations can be solved analytically and these are solvable because they can be transformed into linear equations. Thus consider the following example. Example

2.

Solve the equation

Converting equation

I

to logarithmic form,

log>'„+2

+

logy„

=

II

21og);„+i

Letting "n

equation

II

=

logy„

becomes the second order «n +

linear equation

2-2Mn+l+Wn =

Solution of equation III by the methods already explained Wn

or

logy„

= =

+ C2W Ci + C2n

Ci

IH is

IV

V

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

336

Now

Ci=logX andC2 = logB y^ = AB"

let

then

VI

The solution of the above equation depended on the linearization of equation I and unless such a transformation can be found it is not possible to solve the equation.

A

non-linear finite difference equation of second order that frequently is the Riccati finite difference equation.

appears in engineering problems

yn+iyn + Ay„^,+By, + C

where A, B and C are constants. by the following method. Let

and

It is

yn

=

+d

u„

(9.74)

and rearrange thus

+ S){u„ + S) + A(u„^,+S) + B(u„ + d) + C = + (B + S)u„ + lS'' + (A + B)S-\-C']=0

u„+,u„ + (A + d)u„ + , If S is

(9.73)

converted to a Hnear difference equation

substitute equation (9.74) into (9.73)

(u„^,

=

(9.75)

chosen so that S^-\-(A

and equation

(9.75)

is

+ B)S-\-C =

(9.76)

divided by (u„+^u„), then

(B +

S)— + (A + d)- +1 =

Putting

x„

=

lK =

ll(y„-3)

(9.77)

(9.78)

then equation (9.77) becomes

x„^,+Px, + Q = which

is

a

first

order linear

finite difference

P = ^±^

B+S

(9.79)

equation with

J-ande= ^ B+S

Equation (9.79) can be solved by the methods already given, to yield the final solution

y„-3 where the constant problem. This

is

K

is

A+S {- B + 3 L

.A

1

^

A + B + 23

(9.80)

from the boundary conditions of the example below.

to be evaluated

illustrated in the

Example 3. A benzene-toluene feed containing 60 mole per cent of benzene is fed continuously to a distillation column. If there are 9 plates between the reboiler and the feed plate, and the top product contains 98 mole per cent benzene whilst the liquid leaving the base of the column contains 2 mole per cent benzene, estimate the overall plate efficiency of the column. The feed enters the column at its boihng point and the relative volatility of benzene to toluene can be considered to be constant at 2-3. The reflux ratio is

3-0.

9.

Consider 100 lb moles feed and

and

let

D=

W D+W

= =

iOO

.'.

337

FINITE DIFFERENCES

moles

=

distillate;

moles residue. I

+ 0-02(100-D) and W = 39-6 lb moles.

60

II

0-98Z)

from which D = 60-4 lb moles mass balance between the bottom of the column and some plate n

A

stripping section

Lx, + , where

L G

is is

= Gy„-^Wx^

III

the molar reflux down the column and the molar vapour rate up the column.

Since the relative volatility

X and y on any

plate

is

constant, the equilibrium relationship between

is

Substituting equation

IV

"^"

=

y^

IV

into equation III gives

aGx„ _ Wx„ "^^"l+(a-l)x/

-"--" +

,-n-[

L = F + RD =

Now and

in the

is

L(a-l)

(100 + 3 X 60-4)

J

=

^"-L(a-1) =

V ^

VI

281-2 lb moles/h

letting

^=

^~ = C=

^=-^ =

a—

0-769

1-3

1

aG + (a-l)Wx^ _

(2-3

x 4 x 60-4) + (l-3 x 39-6 x 0-02)

~

L(a-l)

281-2 X 1-3

1-523

Wx^ = 39-6x0-02 ^^^^^ ^^ = 0-0022 281-2x1-3 L(a-l)

equation VI becomes the Riccati equation*

x„^,x„ + Ax„^,-Bx„-C

The

solution of equation VII

in the

above

text.

Thus

is

=

obtained by following the procedure set out

let

VIII

X„-d where 3

is

obtained from the equation S^

or

VII

+ (A-B)d-C = (5^-0-754(5 =

0-0022

.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

338

The

roots of this equation are 5

The complete

=

or -0-003

0-757

solution of equation VII

is

A+d Selecting the root S

when « =

0, x„

=

= —0-003 and

XI

A-B + 26

1>

inserting the

boundary condition that

0-02 gives

K = 42-2 The value of « corresponding

to x„

=

0-60

is

given by

0-7661" 1-66

=

+

42-2

1-32

XII

1-526 as «

=

7-0 ideal stages.

Subtracting the reboiler from this

overall plate efficiency as 6-0/9

9.8.

=

number

gives the

67%.

Differential-Difference Equations

It has been shown in the above sections that chemical engineering equipment consisting of a number of stages can be analysed by means of finite difference equations when the operating conditions are steady. However,

when

of equipment is subjected to a step change in operating started up, or is being shut down, the compositions of the streams passing through these stages change with time. This results in the presence of differential terms in addition to the finite difference terms in what is known as a "differential-difference" equation. This type of equation will be treated in this section, but it should be pointed out that on many occasions the final equation describing the process is too complex for analytical solution, and it is necessary to make a "stage-to-stage" analysis with the aid of a computer. The solution of such problems will be treated in Chapter 1 1 The analytical solution of differential-difference equations is accomplished by converting this type of equation into a finite difference form by means of the Laplace transformation. The transformed equation is solved and inverted to show how the conditions of the process streams passing through the equipment vary with time during the transient period. The technique to apply, where possible, is illustrated by the following examples. this type

conditions, or

is

Example 1. A system consists of A'^ stirred tanks each of volume v ft^ arranged in cascade. If each tank initially contains pure water and a salt stream of concentration ^o Ib/ft^ is fed to the first tank at a rate R ft^/h, calculate what the output concentration from the last tank should be as a function of time if the stirring is 100% efficient. Use the result to compare the transient behaviour of all systems of total volume Nv

= V (constant).

Solution.

339

FINITE DIFFERENCES

9.

With the normal convention regarding

subscripts, a salt balance

over the «th stage gives

Rx„.,-Rx„ =

v^

I

Taking the Laplace transformation of equation I by using equation (6.11) to remove the derivative, and remembering that the system initially contains

no

salt,

Rx„.i-Rx„ =

.-.

Solving this Unear

finite difference

vsx„

II

equation by the method of Section 9.6

gives III

\R + vs/ Because the feed composition

is

constant,

Xq

=

IV

XqIs

s=^f_^y

•

V

'"-7(«f5)" From

transform number 10 in the table at the end of the book,

[(„_!)! Hence, using equation

(s

J

+ af

(6.21), t

Integrating the right-hand side of equation VII by parts

[s{s

+ ar\

(N-i)\[

=

Inverting equation

r

1

'

_^N-l^-at

iN-2^-at

fl(N-l)!

a\N-2y.

„'

times gives

^1

}'

a

-t^-^e'"^ — — + a(N-2)\J a{N-l)\

VI by using equation

*''-^o[l

^

'

a

(N— 1)

t^'-^e-^'dt

^-at '"

+ a""

VIII,

-

'

(jv-1)!

j

-. a^

VIII

,

MATHEMATICAL METHODS IN CHEMICAL ENGI^fEERING

340

^S

~

^0 ""^0

Rtiv

^

Rt

,

,N-1

{Rt/vy

{RtIvY

IX

which is the solution to the first part of the problem. For the second part, if K = Nv, equation IX becomes

1 H

In particular,

if

A'^

=

NRt {NRtVf ^^^ + + V 2!

if

A'^

=

+

{NRtVf-^ —^ {N-\)\

X

1

'n

and

...

~

^o~"^o

^

Rt/V

XI

4,

V

Y

£>

4Rt

^^^/^ \^

32/Rt\

^/RtY

XII

0-5 h"1h°

Fig. 9.6.

It is difficult

Nv

Output from an

to evaluate equation

into equation

VI

A'^

X

tank system of total volume

in the limit as

N^oo,

but putting

V=

gives

0

which can be expanded by the binomial theorem thus,

s

V

N + l/Ksy (iV +l)(iV + 2) /VsV Vs /VsV _ {N+1){N 3!iV2 r'^ 2N [rJ \r) 3! yj^j

XIII

9.

If

N

is

341

FINITE DIFFERENCES

large, the expression in the

square brackets can be approximated to

give Vs

,

1

/VsV

/VsV3

1

= ^^-K./K

XIV

s

The

inversion of equation

Hence

of the book.

as

XIV

is

number

given by

61 in the table at the

end

N^ oo XV

x^-^XoS,(t)

k=

where

V/R

Equations X, XI, XII, and XIV have been plotted in Fig. 9.6 to show the system responds for different values of A''. The important result emerges that a large number of small stirred tanks in series has approximately the same response to a step change as a single tube of the same volume with plug flow. The converse is also true, that flow in a tube with a small amount of back-mixing can be represented by a chain of stirred tanks of the same total volume.

how

Example 2. Acetic anhydride is to be hydrolyzed in a laboratory stirred tank reactor battery consisting of three vessels of equal size. 1800 ml of anhydride solution of concentration 0-21 g moles/1 is charged into each vessel at 40°C, and 600 ml/min of a solution containing 0-137 g moles/1 is continuously fed to the first reactor. Estimate the time required for the reactor system to settle down to steady state operation.

The

specific reaction rate constant for the hydrolysis at

40°C is 0-38 min~ ^. and consider

Solution. Initially let there be a battery of n reactors in series

a material balance over reactor m.

Let q be the volumetric flow rate. V be the volume of each reaction vessel. C be the concentration of anhydride. of the liquid and the temperature of each reaction vessel considered to be constant, then If the density

Now

V/q

=

9 the holding time,

dC„

and r^

C

dt

Since

all

the vessels are to be of the

6

=

is

kC^.

-cCil") ""'

same

^

size

'

and the battery

is

to be operated

isothermally,

——- =Q(constant) M.M.C.E.

III 12

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

342

Thus carrying out the Laplace transformation on equation

II,

by making

use of

^[^]=5C„(s)-C,(0) from Section

6.2.1, gives

5C„(s)-C,(0) + eC,(s)

-

^C,-i(s)

=

IV

But the vessels all start with the same composition in them, hence C„{0) = Equation IV can be put into the standard form for a first order differ-

0-21.

ence equation thus, 0(s

Solving equation

V

+ Q)C^is)-C^.,(s) =

V

0'2ie

by the methods of Section 9.6 gives the general solution 0*21 9

where

A

an arbitrary constant.

is

m=

When

.

.

Co

=

Co(s)

=

0-137

s

—

0-210

0-137 ••

= ^ + e-(^^ePl

which determines A. Substituting into equation VI and rearranging gives 0-210

0-137 '"^^^

"

sO'^is

+ QT

0-137

'^

0{s

r

+ Q)-\[

^...r

1

0-210

_

1

1

0"(5 +

+ ^.

1

Qt\

—

T

+

...

+

1

VIII

by algebraic division. Each term in the square brackets can be inverted directly by using the tables at the end of the book (transform number 10), and the first term can be inverted by using the property of the transform given by equation (6.21). Hence the solution is

C.=

0-137

0-137

r,

{Qtf

(eO""M

nt

Equation IX is a general expression giving the concentration of the from a stirred tank reactor battery containing n vessels of equal size, each starting from the same initial concentration. Expressions of this

effluent

—

;

343

FINITE DIFFERENCES

9.

type have been derived for different kinds of reaction by Mason and Piretj and the reader is advised to consult these authors for a more extensive analysis.

In the special example considered here, the last vessel to reach steady state will be the third,

and since

V

1800

, = - = t;^=3-0^ 600

^

q

^^ =

2=

=

^ + 0-38

0-713

9

OQ .-.

C3

=

C3

=

2-14

0-014-0-014(l+0-713r + 0-254r^)e-°-'^^^

+ 0-21(1 + 0-333r + 0-056r^)e-°''^'^' .-.

0-014 + (0-196 + 0-060r + 0-0082f2)e-°'''^'

X

X

which gives C3, the composition of the effluent from the a function of time, has been plotted in Fig. 9.7. This shows that the laboratory reactor battery has reached steady state conditions within the limits of experimental detection in about 15 minutes. Equation

final reactor as

—

—

I-'

1

I

1

i

1

1

-

0-20^

0-15

O'lO

\

-

-\

-

0-05 -

^^^-o.^.___ 1

1

1

6

4

-

?~

?^

?—

8

10

12

14

16

Time (min)

Fig. 9.7.

Time

to reach steady state

Example

3. 1000 Ib/h of a solution containing 0-01 lb nicotine per lb of being extracted by 1200 Ib/h of kerosine in a counter current extractor containing 8 stages. If the concentration of the feed hquor is suddenly changed to 0-02 lb nicotine per lb of water, how long will it take for the extractor to settle down to steady state operation under the new conditions ?

water

is

The equiUbrium

relation

is

y = 0-86X where

and

The hold up per

Y is

X is lb

of kerosine

nicotine per lb of water.

stage can be taken to be constant at

400 t

lb nicotine per lb

Mason, D. R. and

lb

water and 200 lb kerosine.

Piret. E. L.

r.E.C. 42, 817 (1950).

:

MATHEMATICAL METHODS

344

1

S

Y.

"

Yn

CHEMICAL ENGINEERING

IN

N

Y...

r^




>

Yn+1

>

h

h

h

X„

R

^1

Fig. 9.8.

s

H

H

H

R

\

K-

Stagewise counterflow system

Solution

The system meanings.

is

and the symbols have the following

illustrated in Fig. 9.8

R Flow

rate of water

S

Flow

rate of kerosine phase (1200 Ib/h)

N

Total number of stages

phase (1000 Ib/h)

X„ Nicotine concentration

(8)

in water phase in «th stage

Y„ Nicotine concentration in kerosine phase in nth stage

m

Distribution ratio (0-86)

h

Hold up of kerosine per stage (200 lb) Hold up of water per stage (400 lb)

H

Xi Original feed concentration

Xp New

A

(0-01 lb/lb)

feed concentration (0-02 lb/lb)

nicotine balance over the nth stage gives

RX„., + SY„^,-RX,-SY„ =

dX.

h-^ + H

dY„ ^^

The equilibrium

relationship gives Yn

Eliminating Y„, Y„ +

i

= mX„

between equations

I

II

and

II

and rearranging, dX."

aZ„^i-(a + l)X„ + Z„_i=/?— where

The plant

= mSIR,

a is initially

p

=

(h

+ mH)IR

III

IV

operating at steady state, and the concentration in first. Thus, putting

each stage must be determined for these conditions dXJdt = 0, equation III becomes

aX„^,-{a+V,X„ + X,n-l The method of Section 9-6

gives the general solution

X„ The boundary conditions at

«

=

0,

«

=

iV

=

VI

A^-B(x-''

for this part of the

problem are

^0 = ^0

1

VII at

+ l,

Ys^,^mX^^,^0

J

^

9.

^=

Hence,

345

FINITE DIFFERENCES

^^

Therefore, equation VI gives the

^=

a^

initial state

^"'

of the system where

A and B

are determined by equations VIII.

For the unsteady state part of the problem, Laplace transformation of equation III. Thus

it

necessary to take the

is

ocX„^,-(x + l)X„ + X„.,=P(sX„-A-Bcc-") where equation VI has been used for the

IX Rearranging,

conditions.

initial

or in operator form,

[a£:^-(a+l+Si5)£:+l]X„_i

= -Ap-B^o^-"

X

Equation X is hnear so that its solution can be expressed as the sum of a complementary function and a particular solution. The complementary function is given by

X„., = Cpl-'+Dpr'

P1+P2 =

where

and

P1P2

=

(a

XI

+ l + si5)/a

XII XIII

1/of

The

particular solutions for the separate terms on the right hand side of be found by the inverse operator method described in Section 9.6.2. Thus for the first term

X can

a-a-l-sj5 + l

s

and for the second term,

B

-BfaCombining equations XI, XIV, and equation X; which is

X„ = Cp\ + Dpi

or

XV

A

+

gives the complete solution of

-4-

—

s

Fir/~^

XVI

5

The boundary conditions can at at

n

=

n

= N+l,

also be transformed thus,

Xo = X,,

0,

X;v+i=0,

.'.

.'.

Xo =

X^^,=0

555

Substituting these boundary values into equation

VIII to eliminate

X,ls

A and Xp

XVI and

] >

XVII

J

using equations

B, gives

^ ^ + A+B = ^ -=C C+D + +D

X'o —

XVIII

MATHEMATICAL METHODS

346

and

o

CHEMICAL ENGINEERING

IN

=

cpr^+^pr^+

=

Cp^,^'+Dp''2-''

^^^°'

XIX

C

Solving equations XVIII and XIX for equation XVI gives the solution

and D, and substituting

N+\ Pi

X„ —

n

Pi

into

_ PiN+l Pin

XX

I

p^'-pV'

XX it must be remembered that pj and p2 by equations XII and XIII as functions of s. Therefore, the method of residues must be used to invert the last term in equation XX. In order to invert equation

are determined

To

simplify the nomenclature, put

The numerator on the right-hand side of equation XXI could become infinite if IP2I became infinite. But if \p2\ is large, equation XIII ensures that IpiI is small. Hence it can be shown that as IP2I -> 00, J{s) -> 0; and the only singularities off{s) arise from the zeros of the denominator. These zeros occur at 5 = and when Pi^"^^ = Pi^^^- The second solution shows that IpiI = IP2I, and since Pi and P2 are the roots of a quadratic equation with real coeflScients, they must be equal, or mutual complex conjugates. Therefore, putting p^

=

re-'^,

p^+i_p^+i ^

= Hence f(s) has

singularities

when

P2

=

XXII

re''

J.N+l^^iN + l)i^_^-iN+l)ie-^

2ir^'-^sm(N + i)e

s

=

0,

XXIII

and when

(N+l)e = kn where k

is

an

XXIV

integer.

Putting equations

XXII

into equations XIII r^

2oircose

=

and XII

gives

XXV

1/a

= + l + sp

XXVI

(x

a + l-2Vacos[/c7r/(iV + l)]

^^^^^

(N+ 1), XXVII gives a distinct set of values of 5 for /: = 0, 1, but repeated values for integer values of k exceeding {N+ 1). When k = 0, or (A^+ 1), equations XXII show that pi = P2, and an application of L'Hopital's rule to equation XXI shows that/(5) remains finite as P2 PiTherefore /(5) has (A^'+l) simple poles: Equation

.

.

.

-

one and

at 5 A'^

=

0,

given by equation

XXVII

with k

=

1

,

2,

.

.

.

A^.

9.

347

FINITE DIFFERENCES

Using equation (4.113) to find the residue of f(s)

at 5

=

a"^

=

5

0,

=

Pi

P2

1,

=

0, gives at

XXVIII

^0= ^-N-i_i = ^^.1^1

•

Substituting equations

XXII

into equation r"

^^^>

=7

-^^^

XXI

gives

sin(iV+l-n)e ^'^^^ sin(iV+l)e

sin(iV + l)0

s

Remembering that is a function of s given by equation equation (4.113) to find the residue of e'^fis) gives r" s

sin

~s(iV+l)

_ ~ *

(N + 1 - n)e 2ar sin 6 e^

cos(iV + l)0

XXV, and XXVII

Combining the

+

l)(a +

to simplify

[„yv .N+l-n + l-n

r

XXXI,

-(a + l-2Vacos0)n

^""^

l-2Vacos0)

residues as given

remainder of equation

XXXI

p

2(-l)^a-^"-^>/^sin0sin(iV+l-n)0 (7V

using

sm{N + l-n)e e'' (N + l)cos(N + l)e{deids)

- r" Using equations XXIV,

XXVI, and

^

L

by equation

(6.43)

J

and inverting the

XX,

n

r^N+l-n + l-n r^/V

1

-i

-i

-]

^^^

^

2(-l)^a-("-^>/^sin^sin(iV+l~n)g

+ (Ap AoJL^

0/ z. Jk=l

(iV

+ l)(a + l-2Vacos0)

^^

(iV+l)(a + l-27acos0)

Equation XXXII is the complete solution to the problem where 6 is given by equation XXIV, and s by equation XXVII. Since the final stage will be the last to reach the new steady state conditions, a graph should be plotted showing Xg as a function of time. From equation XXXII,

+ -L^l .(X.-X,)L^z|^i^.. it

=

1

9(a

1

2yja cos 6)

XXXIII

^ MATHEMATICAL METHODS

348

IN

CHEMICAL ENGINEERING

Using the numerical values given in the problem,

= =

e

a

knl9

h

=

a^

20 40 60

1

2 3

10159

1-868

3-087 4-384 5-603 6-596 7-245

80

0-3528

100 120 140 160

-0-3528 -1-0159 -1-5564 -1-9092

7 8

a-^/2^ 0-8956 2a-7/2 sin2^ sin2 d

9(a

0-2257 0-8743

5

0-544

—s

1-9092 1-5564

4 6

+ mH = R

1-3278,

2 Va cos 6

k

20/c°

mSi'R == 1-032

P .-.

=

+ l-2Vacos

01170

01896

0-4132 0-7500 0-9699 0-9699 0-7500 0-4132

0-1729 0-1469

01170

00059

9)

01149 0809 0-0489 0-0230

= 0-195-0-1896 e^'' + 0-1729e^^'-0-1469e^^^ + + 0- 1 149 e''' - 0-0809 e''' + 0-0489 e''' - 0-0230 e''' + 0-0059 e'''

lOOXg

Equation

XXXIV

XXXIV

has been plotted in Fig. 9.9 from the following cal-

culated values.

^8

t

1

2 3

4 6 8

0972

10

0-0975 0-1012 0-1107 0-1232

12

14 16 18

01470 01640

20 24

0-20

1

x%

t

.

1

0-1752 0-1824

01870 01899 0-1917 0-1929 0-1942 -^

.

,^-c^—

^-'^'''^''^

-

0-18

CO

6

i 6

'"^

CO wo

CO

6

6

9^

CO vb wo

CO wo CO

d^ wo

O

wo

VD WO

VO

o 6

p 6

as

00

CO

6

6

o 6

S °

? =

o 6

VO CO

^ -^

6

CO

6

Co OO

wo

3 6

6

6

2 6

CO oo wo

un

^

\b

CO OS WO

wo CO

VO CO

wo

6

6

6

b

2 CO 6

CO ro CO

oo

CO

6

6

6

6

CO wo

VO

oo

6

6

6

g 6

wo

? en

p

6 in

6 s '^ 6 Tf Tf

wo

o

CO

6

CO

6

6

? "^

^

wo

rg

fS

00 CO

r-

8 6

'"^

6

? ^ wo

(N

oo

wo

6

CO fs

cs

VO

en

6

2 could be evaluated due to b^^ and b^

62,

M.M.C.E.

13

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

374

being eliminated together.

It will

now

be shown that in Gauss' method, the

three coefficients Qq, 02, a^ occurring in equation (10.71) can be evaluated from only three ordinates, if three special values of the abscissa are chosen.

Denoting these three values by x^, ^2, X3, and the corresponding ordinates by y^, y2, y-^, each pair of values must satisfy equation (10.70). Hence

= = =

^1

yi yi It is

now assumed

in the

+ ^l^l + ^2^1 + ^3^1+^4^1+^5^1

(10.72)

+ '=:(l+x2)-i

00

1

00000

0-89445 0-70711 0-55475 0-44722 0-37138 0-31623 0-27473 0-24254

the trapezium rule for three points,

=

1-00000 + 2(0-44722) + 0-24254

=

2-1369

the trapezium rule for nine points,

=

i(l'00000 + 0-24254) + i(0-89445 + 0-70711

=

2-0936

+ 0-55475 + + 0-44722 + 0-37138 + 0-31623 + 0-274

3)

Simpson's rule for three points (10.67) gives

_ ~ = (e)

IN

2 3

(1-00000+ 1-78888 + 0-24254)

2-0209

Simpson's rule for nine points gives

=

iVO'i

=

i(l -00000 + 0-24254) + KO-7071 1

=

2-0941

+ 4>'2 + 2>^3 + 4>^4 + 2>^5 + 4j^6 + 2>'7 + 4>'8 + >'9) X 4 + 0-44722 + 0-3 1623) + + 1(0-89445 + 0-55475 + 0-37138 + 0-27473)

In Gauss' three point method (10.78), if the range of integration is and ± 0-7746. Therefore, if 1, then the ordinates are required at the range of integration is to 4, then ordinates are required at x = 2 and X = 2±2 (0-7746) = 3.5492 and 0-4508. (f)

-1

to

^

= =

A[5(0-91 165 + 0-271 18) + 8(0-44722)] 2-1093

10.

(g)

TREATMENT OF EXPERIMENTAL RESULTS

377

In Gauss' four point method, the ordinates will be required at x = and 3-7222. Equation (10.80) then gives the average

0-2778,, 1-3200, 2-6800,

height within the range of integration. /

= =

4[0-1739(0-96351

+ 0-25945) + 0-3261(0-60386 + 0-34959)]

2-0944

The above

results

can be compared in Table

Table

10,2.

Comparison of Integration Formulae

Method Trapezium

Result

rule (3 pts)

Simpson's

» „

(9 pts) (3 pts)

,,

(9 pts)

Gauss 3 point Gauss 4 point

It

Absolute

Relative error

error

-00011

2-0209 2-0941

-00006

(%)

21 005

00422

21369 20936

00738

3-5

21093

00146

0-03 0-7

2 0944

-00003

0-01

can be seen that Gauss' method

that Gauss' four point

10.2.

method

is

is

the superior one for three points, and

better than either of the other nine point

methods. The reason why Simpson's rule is so poor for three points is illustrated in Fig. 10.9, where the true curve and the parabolic fit can be compared-

FiG. 10.9.

Parabolic approximation for Simpson's rule

The cause of the bad fit is the point of inflection be represented by a quadratic equation. 10.7.4.

at

x

=

0-7071 which cannot

Comparison of Methods

In most cases, the four-point method of Gauss gives an accuracy equivaof Simpson's rule with nine points equally spaced across the same range. Gauss' method is thus more economical in the use of data, but lent to the use

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

378

the calculations are a

little

more complicated because of

the

more

difficult

weighting factors. Gauss' method should have many applications to the investigation of the behaviour of full scale production equipment, where it is necessary to compress all readings into as short a time as possible due to plant fluctuareduction in the number of readings required without loss of tions. accuracy is an obvious advantage. Another example is obtaining a representative sample of a steady state process stream. It is much more efficient

A

Gauss intervals, mix them in the Gauss proportions and analyze the composite sample; than to take twice as many samples at equal intervals and analyze each one separately or mix them all to take a set of samples at the

together in equal proportions.

An

interesting point has

been noted by McDermottj concerning the

integration of experimental results. Figure 10.10 illustrates the true relation-

ship between

two variables x and y as the

Fig. 10.10.

straight

Comparison of the trapezium

rule

Hne AB. The two groups

and Simpson's

rule

of three equally spaced experimental points (a), (b) have been chosen so that the total error in each group is zero, and these are the only two possible groups which satisfy this condition (except for their mirror images in the line AB). It is obvious from the group (a) that the parabolic fit seriously under-estimates the area, whilst the two straight fines joining the points give a closer approximation. For the group (b), the difference between the straight fine area and the parabolic area is smafi, but does show a marginal advantage in favour of the trapezium rule. It is thus apparent that the trapezium rule is more satisfactory than Simpson's rule. This conclusion is in direct contrast to the result demonstrated in Table 10.2, where the higher polynomials gave the better results. These two conflicting conclusions can be reconciled as foUows. For evaluating analytical integrals or integrating accurate experimental data, the methods of Gauss and Simpson's rule should be used; but if the data are t McDermott, C. Birmingham (1962).

M.Sc. Thesis, Chemical Engineering Department, University of

10.

TREATMENT OF EXPERIMENTAL RESULTS

379

The question of accurate the data must be to justify the use of Gauss' method or Simpson's rule is a very difficult one, and the answer must depend upon the For linear relationships the trapezium rule type of curve being fitted. inherently inaccurate, the trapezium rule should be used.

how

appears to be the best for all experimental data, and the simpHcity of the However, this statement does not is a further recommendation. detract from the value of Gauss' method or Simpson's rule for integrating accurate experimental or theoretical data; nor does it invahdate a planned experiment to sample a process at the Gauss intervals.

method

1

Chapter

1

NUMERICAL METHODS Introduction

11.1.

There are many problems in mathematics for which no analytical solution is known. There are also others, for which the analytical solution is tedious and the answer may be in the form of an infinite series that can only be interpreted after much computational effort. A numerical method may be the only one which will yield a solution to the first kind of problem, and may be the most efficient method of solving the second kind of problem. Mistakes are difficult to locate in some analytical solutions and the self-checking features of numerical methods based on successive approximation eliminate this difficulty.

Simpson's rule and Gauss' method, which were described in Chapter 10, can be interpreted as numerical solutions of the first order differential equation

'£=m

(11.1)

These solutions are valuable if /(x) is not a simple function. The purpose of this present chapter is to present numerical solutions for more compHcated types of ordinary differential equations, to locate roots of algebraic and transcendental equations, and solve partial differential equations numerically.

Order Ordinary Differential Equations many first order differential equations were solved, but

First

11.2.

In Section 2.3

if

not hnear and the variables will not separate, none of the methods given can be applied. The problem is thus to solve the equation

is

(11-2)

j^=fi^,y) ax

There are two methods available, one due to Picard which the other due to Runge and Kutta which is numerical. 11.2.1.

Picard' s

It is first

at is

is

algebraic,

and

Method

assumed that a boundary condition of the type x

=

y=^b

a,

available to particularize the solution of equation (11.2). 380

(11.3)

As a

first

11.

NUMERICAL METHODS

381

approximation, in the neighbourhood of the starting point, y can be replaced by b on the right-hand side of equation (11.2). Thus

-j--=f(x,b) Equation

(11.4)

can

now be

(11.4)

integrated with respect to

=

/'^

x giving

jf(x,b)dx

which determines y^^ as a function of

(11.5)

now substituted into the right-hand side of equation (11.2), a further solution y^^^ can be obtained. If y^^^ is

x.

Thus jfix,/'')dx

(11.6)

This process can be continued indefinitely, by putting the latest version of y"^ in the right-hand side of equation (11.2) to determine y^"'^^\ Thus

/"+!)

=

J/(x,y"))Jx

(11.7)

After each integration (11.7), equation (11.3) must be used to evaluate the constant of integration.

Example

1.

If

>^

=

when x =

1

dy

find the value of

is satisfied,

Putting

>'

=

1

and the equation

1,

^

x^-y J

dx

X

y when x

=

in the right-hand side

2.

of equation

I

gives

d_y^_:^-\ dx /i)

Integrating,

X

=

Ix

—

\dx

= ix^-lnx + But y^^

=

1

when x =

Ci

II

= i + i^'-lnx

III

1,

Substituting y^^^ into the

.-.

C,

.'.

y^^

=

i

right-hand side of equation

dj^ _

_

dx

J.

2x

_

X

Inx

2

X

X

1

Inx

2

2x

X

I gives

Integrating, ^(2)

^i^^2_^xnx + Kin xf + C2

IV

382

MATHEMATICAL METHODS N CHEMICAL ENGINEERING

But /^^ =

1

]

when X =

I,

•1

A

=

y^^^

:.

i-\-ix'

-iInx +

i(l

nx^

further cycle of calculation gives

/2> Putting

X =

= | + |x2-Jln x +

i(lnx)'--idnxf

VI

2 in each approximation gives

/!)

=

1-807

1-644 1-670 It is

apparent from these figures that a few further cycles of calculation are lies within the range l-644-^ 1-670; nearer

necessary, but that the true answer to the latter figure.

An is

analytical solution

linear.

so that at

available for this

is

problem because equation

I

Solving by the method of Section 2.3.4 gives

X =

y

= ix^+ix-^

y

=

VII

2,

1-667

VIII

can be demonstrated that Picard's method is converging towards the analytical solution in this particular case by expanding equation VII into a logarithmic series. If x = e\ It

x-'

=

e-'

= '-'" 21-

SI-"'"

^(Inx)^

.

= l-lnx + -^^ ^

(Inx)^^

^+

...

Using equation VII, therefore y

Comparing

=

ix' + i-i^rix

+

i(\nx)^-}(\nx)'

+ ...

IX

V, and VI with the corresponding can be seen that the set of approximate solutions y"^ approaches the true solution as n increases. Picard's method has two disadvantages. Firstly, as successive substitutions are made the amount of work is increased and the integrations become more complex; and secondly, integrals frequently arise which cannot be evaluated analytically and these have to be solved by a numerical method. It is reasonable therefore to use a numerical method from the very beginning. coefficients in equations III,

coefficients in equation IX,

11.2.2.

it

The Runge-Kutta Method

The method

to be described is a composite one, using the principles discovered separately by Runge and by Kutta. The analytical details of the modem method differ from the original development and the algebra is

II.

NUMERICAL METHODS

383

rather tedious; however, the general principles can be outlined by reference

Assuming that equation (11.2) is satisfied, and an initial conyo 2it X = Xq) is given, it is desired to find the value of y when XQ + h where h is some given constant.

to Fig. 11.1. dition (y

X

=

=

iQ+h

-I Fig. 11.1.

Third order Runge-Kutta process

The general solution of equation (11.2) will consist of a family of curves, each curve having a particular value of the constant of integration. Three specially chosen curves of this family (a, b, c) are illustrated in Fig. 11.1; the curve b passing through the point (xq, yo). The gradient of the tangent to the curve at (xq, yo) can be calculated from equation (11.2) thus determining the length Ar^ by k,

=

hf(xo.yo)

(11.8)

yo + k^ is obviously a poor approximation to the desired answer unless the curve happens to be a straight line, and a better approximation can be

obtained from the following reasoning. The gradient of the chord to the curve between Xq and Xq + H should be approximately equal to the gradient of the tangent to the curve at Xq + p. Unfortunately, at this stage of the calculation, the position of the curve at Xo + ih is not known; however, an approximate value for y at x = Xq + ^H is given by y = >^o + 2^i* This point (xq + P, ^^0 + 2^i) liss on a different curve of the family as illustrated by c in Fig. 11.1. The gradient of the tangent to curve c at (xq + ^h, j'o + i^i) can be calculated from equation (11.2) and the length k2 constructed by drawing a line parallel to this tangent through (xq, yo). Thus^ k2

=

hf(xo + iKyo

+ ik,)

(11.9)

For the type of curve illustrated, k2 is also an underestimate. The method of obtaining an even closer estimate for the end ordinate is to find some average gradient of three tangents, one at the start of the interval, one in

384

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

the centre,

and one

All three tangents should be taken to the not possible. The tangents so far, are to the correct

at the end.

correct curve b but this

is

curve b, and to the curve c which lies below b. To compensate for the effect of the latter tangent, the final tangent must be taken to a curve lying above b. On the ordinate at ;c = Xq + H, y = ;^o + ^2 is approximately on curve b,

y

=

>'o

+ ^i

is

below

definitely

j should be above

b.

b,

hence

>'o

+ /c2 + (/c2-/ci)

=

Finding the gradient of the tangent to the curve a which

passes through this point from equation (11.2) gives ^3

= V(^o + ^,yo + 2/c2-/ci)

can be shown analytically that the ordinate dX x through (xo, yo) is given by It

=

y

>^o

(11.10)

=

XQ

+h

to the curve

+ i(/ci+4/c2 + /c3)

(11.11)

where k^, k2, k^ are given by the equations: k,

=

hf(xo.yo)

k2

=

hf{xo

+ ih,yo + ik,)

k3

=

hf(xo

+ h,yo+2k2-k,)

(11.8) (11.9)

(11.10)

" This formula (11.11) is known as the " third order Runge-Kutta formula because the term corresponding to the third derivative term in the Taylor series for

To

y expanded about

illustrate the use

given in example

Example

is

2.

(xq, yo) is correct. it will

of this method,

be applied to the problem

1.

If

>^

=

1

when x =

and the equation

\,

dy

x^ — y

dx

X

value of y when x = 2. Using the above symbols, h = \; hence equation

satisfied, find the

/ci=/(l,l)

Equation (11.9)

now

=

(11.8) gives II

gives

fc2=/(li.l)

=

(9/4) -1

3/2

-^ ~ 6 Putting equations

II

and

III

III into (11.10) gives

4-2i

=

i

IV

NUMERICAL METHODS

11.

Substituting equations

II, III

y

and IV into

385

(11.11) gives the

answer

= l + i(0 + 3Kf) = 1J

V

which is identical with the analytical solution given by equation VIII in example 1. If the range of integration is large, it can be subdivided just as for Simpson's rule, with consequent increase of accuracy. This will be illustrated by an example in the next sub-section. The method of Picard, and the original formulae of Runge and Kutta are given by Piaggio,t together with the theoretical justification of each method. 11.3.

Higher Order Differential Equations (Initial Value Type)

It is a simple matter to convert an nth order differential equation to n simultaneous first order equations. It is only necessary to define (« — 1) new

variables

by ^

dy

=

"^

^2

^x ...2

(11.12)

etc.

W._i

=

dx n-1

and remove all derivatives from the original differential equation except which is replaced by (dw„_ildx). The solution of the set of equations (11.12) with the original differential

(d"yldx")

equation is now very similar to the solution of a single first order equation, provided that the boundary conditions are of initial value type (Section 8.4.5). In this case, at some value x = Xq, values for y, w^, W2 w„_i, will be available and an extension of the Runge-Kutta process can be used. The formulae for the second order differential equation .

.

.

,

g..(„,2) are as follows.

Putting

dy dx equation (11.13) becomes

^ = F{x,y,w) t Piaggio,

H. T. H. " Differential Equations

".

G. Bell

(11.15)

&

Sons Ltd., London (1928).

MATHEMATICAL METHODS

386

IN

CHEMICAL ENGINEERING

From the initial values (xq, yQ, Wq) and the interval be defined as before. Thus k,

=

Xi =

(h) in x, k^

and K^ can

/zwo

(11.16)

/iF(xo,yo,Wo)

(11.17)

where k^ is the first approximation to the change in y and K^ is the corresponding change in w. Proceeding in a similar manner to the first order case, k2

=

h{wo + \K,)

(11.18)

K2 = hF(xo + ih,yo + ik„Wo + iK,) k,

=

X3 =

(11.19)

h(wo + 2K2-K,)

(11.20)

hF{xo + h,yo + 2k2-k,,Wo + 2K2-K,)

(11.21)

The values of y and w

at the

end of the interval are given by

y = yo + i(ki+4k2 + k,) w = Wo+iiK,+4K2 + K^)

Example

and the

initial

value of jv Solution.

1.

Given the

diiferential

(11.23)

equation

x =

conditions that at

when x =

(11.22)

y =

0,

I

and dyldx

=

0,

find the

1.

Putting

dy I

*•

= \, Xq = 0, yg used in succession. Thus Since h

fci

=

x^vv — 2>;^

Jx

8

II

Wq

=

0,

equations (11.16) to (11 .21 ) can be

=

^1 = kz

1,

^w

III

l(°3^)=

-0.2500

= 1(0-0-1250)=

K2 _

^

-0-1250

IV

V

r(0-5)^(-0-125)-2(l)^-| 8

= -0-25391 2k,- k. = -0-2500 2K,-K, = -0-25782 fca = -0-25782

VI VII VIII

IX

NUMERICAL METHODS

11.

i^3

(1)2(- 0-25782) -2(0-7500)2

=

1

=

-0-17285

8

[

Equation (11.22)

now

-,

J

.

X

gives the solution

y .-.

387

y

= 1- -1(0 + 0-50000 + 0-25782) = 0-8737

XI

between k^, k2, and k^ in this example There is and the above solution may not be very accurate. To check this, the integration can be performed in two stages by choosing h = ^ and applying the formulae (11.16) to (11.23) twice. The intermediate and final results are a large relative variation

x

at

and

x

at

= =

0-5,

y

1-0,

>;

= 0-9687, = 0-8779

w = -0-1230

XU

The answers given by equations XI and XII agree to better than and the solution can be written y

=

1

%,

0-878

with reasonable certainty that the third significant figure of the correct solution.

is

within two units

Direct Use of Taylor's Theorem

11.3.1.

The following method has been rendered almost obsolete by the increased availability of digital computers, but for manual calculations with a desk machine it can still give accurate answers quite efficiently. Taylor's theorem (Section 3.3.7) states that

f(x,

+ h)

=f(xo) + hfXxo) + ihY(xo)+

...

+

^^/"\xo) +

...

(11.24)

can be differentiated indefinitely. For a second order differential equation of the form (11.13), with the initial values of >; and dy/dx given at X = Xq, all terms in equation (11.24) can be found as follows. The boundary conditions specify /(xq) and f'(xo), and the differential equation (11.13) can be used to calculate /"(-x^o)- By differentiating equation (11.13) with respect to x, an equation can be developed giving /^^^(xq) in terms of Xq, /(xq), /'(^o)» and /'(xq). Hence P"\xo) can be found by differentiating equation (11.13) n-2 times. Equation (11.24) thus gives a series solution to the non-linear differential equation (11.13). The method will be illustrated by repeating example 1.

if fix)

Example

and the of y

2.

initial

when x =

Given the

differential

conditions that at 1.

x

=

equation

0,

>^

=

1

and dyjdx

=

0, find the

value

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

388 Solution

At X

The

=

given that

0, it is

differential

y

=

y'

=

I

l

II

equation can be rewritten in the form 8/'

Using equations

and

I

II,

III

therefore

=

/'= -0-2500

IV

^y^^'^^x^f + lxy'-Ayy'

V

x

at

= x'>;'-2/

0,

Differentiating equation III gives

atx =

.-.

Differentiating equation 8y(^)

=

V

8/5)

= .-.

gives

+ Axy" + 2y' - Ayy" - A{y'f atx = 0, /^> = 0-1250

+ 6x/^^ + 6/-4;;/^>-12>;V' /^>= -0-1875 atx = 0,

;cV'^^

^ xV^> + 8x/''> + 12/3>-4>'/''>-16//^^-12(/)' /^^= -0-1562 atx = 0, .-.

Substituting into equation (11.24) with h

y .-.

VI

VII VIII

manner,

in this

8^6)

=

x^y^^^ .-.

Continuing

/2^

0,

y

=

1

IX

X XI XII

gives

= 1-0000-01250 + 0-0052-0-0016-0-0002 = 0-8784

XIII

The direct use of Taylor's theorem is apparently simpler than the RungeKutta method in this application, because both calculations have been done manually. However, the successive differentiations performed above cannot be done by a digital computer and a more complicated problem would favour the use of a computer programmed for the Runge-Kutta method using

many

sub-intervals of integration.

11.4.

Higher Order Differential Equations (Boundary Value Type)

With second and higher order differential equations, the boundary may be specified at two different values of the independent variable. In such cases, there is insufficient data available at x = Xq, for either of the above methods to be used directly. However, a trial and error method based on either of the above methods is practicable for second conditions

order equations.

Thus, consider the equation

NUMERICAL METHODS

11.

again, with the

boundary conditions:

x

at

and

389

x

at

= Xo, = x^,

y

=

yo

(11.25)

y

=

yN

(11.26)

N integration

has been assumed that

steps are necessary to cover a value is assumed for /(xo), equation (11.13) can be solved as an initial value problem, thus generating a value for y at x = x^^. This value is unhkely to equal yff, therefore a second choice is made for j'(^o)» resulting in a second value for y slI x = x^^. The value of y at x = x^^ can thus be evaluated as a function of y'(xQ), and the correct value of yXxo) can be found by using an interpolation formula from Chapter 10. The value of >^'(^o) obtained in this way can then be checked by a final solution of an initial value problem. If this method is adopted, the early trials are usually made with a small value of A^ and widely spaced values of >''(^o)- As the solution is refined, the value of iVis increased and the assumed values of >^'(-^o) ^re contained within a narrower band. The calculated internal point values then give the shape of the solution curve {y vs x).

where

it

the distance between the boundaries.

If

Example 1. In example 4, Section 2.4.3, the problem of cooHng a graphite was considered, and the problem was stated in mathematical terms as follows. Find the rate of flow of heat into the water cooler at X = 1 ft when the temperature distribution satisfies electrode

where Tq

=

70,

/cq

=

88-9, a

The boundary conditions

=

0-0180,

and p

=

2-40.

are

at

x

=

0,

T=

2700

II

at

x

=

l,

T=

300

III

and the

rate of flow of heat into the cooler

H=

is

given by

dT

-16-4-

ax

IV

Solution

Since dT/dx is required at x dent variable by putting

=

,

it is

convenient to change the indepen-

= l-x

V

(^o-aT)^-a^^y-^(T-To) =

VI

z

•*•

1

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

390

and the boundary conditions become, z

at

and

z

at

= =

0,

r=

300

VII

1,

T=

2700

VIII

From

equations VII and VIII the average temperature gradient is 2400, but the gradient at x = should be smaller than this because less heat reaches the water cooler than leaves the furnace, and the conductivity is higher at the lower temperature. Therefore, as a first approximation, put

r = 2000 7" dz

Putting

atz

=

=

IX

X

>v

equation VI becomes

dw

oiw^

+ P(T-To)

XI

Uq-olT

dz

Taking one interval and using the method of Section values can be calculated.

11.3, the following

= 2000 = 2434

Xi = ^2 =

1673

2A:2-/ci=2868

1K2-K^=

2477

=

11 543

k^ A:2

= 4477

^3 .-.

T=

.-.

w=

1^3

868-9

300 +-^(2000 + 9736+ 4477)

2000 + i(869 + 6692 + 11 543)

=3002

=

5184

XII XIII

The answer given by equation XIII is suspect because K^ is so different from Ki and K2* It appears that r'(0) has been chosen 10% too large by comparing equations VIII and XII. Therefore T\0) is reduced to 1800 for the second trial, and the interval is halved to reduce the discrepancy between A'3

and K2.

The

calculation for the

.-.

/.

For the second

first

interval gives

k^= 900 iCi = 352-5 k2= 988 1^2 = 476-9 k^ = 1201 X3 = 834-1 =1309 T(i)= 300 + ^(900+3952 + 1201) w(i) = 1800 + -1(352-5 + 1907-6+ 834-1) = 2316

XIV

XV

interval, /ci

/C2

k^ .-.

T(l)

.-.

w(l)

= = =

1158

Ki= 761-5

1348

K2 = X3 =

2009

1232

3948

= 1309 + ^(1158 + 5392 + 2009) = 2735 = 2316 + 1(762+4928 + 3948) =3922

XVI XVII

NUMERICAL METHODS

11.

391

XVI shows that the temperature drop across the electrode is 2435° compared with the true value 2400°. Hence, the assumed value too high. Therefore, putting dTjdx = - 1770 r(0) = 1800 is probably at ;c = 1 into equation IV gives the result Equation

now

U%

if

=

29000Btu/h

XVIII

considered to be sufficiently accurate for the purpose of could be improved further by taking more intervals and choosing T'(0) more accurately, but this is not justified by the accuracy of

This answer

the problem.

is

It

the original data. Finite Difference

11.4.1

Methods

solving higher order ordinary differential An equations of boundary value type, is to replace the differential equation with an equivalent finite difference equation. This technique is most useful for second order equations, or higher order equations replaced by simultaneous second order equations. There are many bases for the method, but the derivation from Taylor's theorem is probably the easiest to follow. The closed range of the independent variable a ^ x ^ b is divided into equal intervals and the points are labelled x„, where Xq = a, and ;c^ = b. The dependent variable (y) can be expressed in the neighbourhood of any point (x„) in terms of x and its derivatives at x„ by means of Taylor's theorem (Section 3.3.7), and if h denotes the increment of x between neighbouring points, alternative

method of

yn^i =/(x„+i)

=fM + hfXx„)+ih'r(x„) +

...

yn=f(x„) yn-i

(11.27)

(11.28)

=/fc-i) =f(x„)-hnx„) + ih'fXx„)-...

(11.29)

terms in h^ and higher degree can be neglected relate neighbouring point values to the derivatives at the central point (x„); and they can be solved to determine the derivatives in terms of the point values. Thus If /z is

chosen

sufficiently small,

These three equations

in equations (11.27, 29).

fM = yn /X^n)

=

(11.30)

^"^'^""/""'

^"--'^;;-^^"= rfa) =

(11.31)

%:l

(11.32)

These equations illustrate the use of Norlund's operator, as may be seen by comparing equations (11.32) and (9.37), and they allow any second order differential equation to be replaced by an equivalent second order difference equation.

The

calculation details

and treatment of boundary conditions

easily explained with reference to

a particular example.

is

more

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

392

Example 2. The cylindrical combustion chamber of a rocket motor has coolant ducts of lenticular cross-section at frequent evenly spaced intervals within the cyhndrical wall. The dimensions of a typical duct are illustrated in Fig. 11.2. Radiant heat is received uniformly by the internal surface of

0-3251

Fig. 11.2.

Rocket motor cooling duct

the combustion chamber at a rate Q = 51 cals/sec cm^ and this heat is conducted into the liquid coolant at a bulk temperature T^ = 130°C according to a variable heat transfer coefficient h cals/sec cm^°C. h can be determined from the Dittus-Boelter equation expressed for gases as /z = aw°*^, where u is the local centre Une velocity and a = 0-00626 for the coolant used. Spikinsf has determined the velocity distribution as

"-h(i^)1 the coordinate measured as shown in Fig. 11.2, and Uq, the axial If the metal conductivity k = 0-107 has a value 100 cm/sec. cals/sec cm^°C cm~^, find the temperature distribution within the chamber

where z

is

velocity,

wall.

Solution

The shape of

the duct is rather compHcated, so it was idealized as shown There are Hnes of symmetry at AE, DG, and JK, and a typical section has been straightened out by distorting the junction CKJBF. In the ideahzed situation, radiant heat enters through AB and BC which is chosen to equal BJ, negligible heat passes through the external surface CD, and the coolant flows over EF and F'G. All areas and heat paths are correct in this model, except for the cjurved path from BF to CF which is only approximately correct. Neglecting the temperature drop through the metal wall with respect to the temperature drop within the coolant film, reduces the problem to the determination of the metal temperature {T) as a function of a single in Fig. 11.3.

coordinate

(x).

t Spikins, D. J., Ph.D. Thesis, Chemical Engineering Department, University of Birmingham, 1958.

NUMERICAL METHODS

1.

r

393

C

B

F

G

F

I Idealized metal section

Fig. 11.3.

Taking a heat balance over an element of metal

in the section

ABFE

gives

d^T

h ,_

dx^

ks

_.

^^--(T-T,) +

Q I

ks

or inserting the data

d^T

0-8

0-00626 X 100^-^

0107

dx^

To simpUfy

X 0-1219

[i'llj

the arithmetic,

it is

_

57

=

II

0-107 X 0-1219

convenient to put

X

=

I'llX

III

e=T-Ts

IV

- 30-8 (l-XT'^ + 7048 = dX section BCFT, where no coolant flows

V

and

^

:,

Similarly, in the

(T-Ts) +

but heat

is

still

absorbed in a thicker section,

d^e

dX'

X

^

Finally, in the section 1-1

v^

*r)

IT)

wo

VO VO

wo

r-

(N

S On ^^

00

oo

t^

S o

-

CO CO

^ oo

a

jq

I

o

OO ON (N

VO

S

O ro

O

m OO

S ovooo»naN—'OvooeN

—

oory->>o

6666666666666666666666666666666 s.

s.

s.

s.

S

S

>
i

8

8

S

S

H H H H

8868SSS88S

8

8

H

>i

8

8

8

8

S

H H H H H

0i(x)

(12.86)

= Ux)

or written in condensed matrix form

fWW = [/oW]W = {0W}

(12.87)

MATHEMATICAL METHODS

464

CHEMICAL ENGINEERING

IN

In equations (12.86) and (12.87) /,y(D) is a polynomial of the differential operator having constant coefficients. The matrix of differential operators {(D) is analogous to the >^.-matrix and in the determination of the complementary function the technique depends upon the substitution of ). for D. However, certain nomenclature are required and in addition to the terms for 2-matrices given in Section 12.13 the following are necessary.

{(D)

is

called the i)-matrix.

|f(Z))| is

called the Z)-determinant

The highest degree of order of the system. 12.18.1.

D

and

is

frequently expressed A(D).

any of the elements of f(D)

in

is

said to be the

Solution by Conversion to an Equivalent System

In some systems of differential equations it is possible to transform form that will permit ready solution of the differenThis is accompHshed by pre-multiplying the Z)-matrix by tial equations. another non-singular Z)-matrix of the same order whose jD-determinant is free from the differential operator D. The following example will clarify the D-msLtrix into such a

the method.

Example. Solve the system of equations

D^y^

+ {D^ + 3D-2)y2 = D^y^ — Dyj =

These equations can be put

D^

(D'

-D

[y,

U

\

Pre-multiplying both sides of equation

I

sinx

form thus

in matrix

+ 3Z)-2)1

P^

x

sm X

by

1

-D

-D

D^ + 2

II

gives 1

-D

D^

-D

D^ + 2

D^

(D'

+ 3D

2)'

>i

1

-D

-D

D^ + 2

III

2D'

-D

.y2

.) (2D2 + 3D-2)

'yi'

-(2D^ + 3D')).

.yi.

{2D^ + ZD-2)y2

X

— D sin X IV

-Dx + (D^ + 2)smx

=

x

V

- cosx

2D^yi-2D^y2-3D^y2 = -1 + sinx Solving equation

y2

V

=

smx

VI

by the methods of Chapter 2 gives

Ae~^''-hBe^''-^x-l + ^(4cosx

-

3sinx)

VII

12.

Integrating equation

465

MATRICES

VI twice and then

for yj

from equation substituting

VII gives y^

=

C-\-Ex — ix'

where A, B, C, and

E are

Ae'^'' + 2Be^'' + -^{3cosx-2ismx)

VIII

the four arbitrary constants required by this third

order system of equations. In the above example the matrix 1

-D

-D

^

D^ + 2

but guided by the aim that all but one of the dependent variables were to be eliminated from one of the equations. This enabled that variable to be evaluated and inserted into the other equation which was subsequently solved by conventional methods. When there are a large number of dependent variables it is necessary to convert the D-matrix into triangular form by pre-multiphcation by another D-matrix whose determinant is a constant. The transformed system must be of the type

was obtained by

7ii(o)

trial,

/12(D)

/13(D)

fln(D)

/22(D)

/23(D)

f2n(D)

/33(D)

yi yz (12.88)

it can be seen that the last row is a function of ;^„ only, row above is a function of >^„ and >^„_i only. Hence the equation of the last row will give an expression for y^ which will be inserted into the equation of the row above to give an expression for j„_i and so on until

In equation (12.88) whilst the

the system has been solved.

12.19.

Solutions of Systems of Linear Differential Equations

Systems of linear differential equations can be solved in a similar manner to single linear differential equations by establishing the complementary function and the particular integral. However, with systems of equations the process will be much more complicated because of the inter-relationship between the dependent variables and the effect of one solution on the particular integrals of the others. Each part of the complete solution will now be considered separately. 12.19.1.

in

The Complementary Function

A system of linear differential equations has been written in matrix form equation (12.87) and the complementary function of this equation will be

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

466

obtained by solving

when {^(x)}

is

The

That

zero.

f(D){y}

is

by solving the equation (12.89)

solution of equation (12.89) will depend

D-matrix, and each latent root

is

= on the

latent roots of the

" said to contribute a " constituent solution

complementary function. Hence the solution of equation (12.89) depends upon whether the latent roots of f(Z)) are distinct or multiple. These to the

be considered separately. Distinct latent roots. Let X^ be any distinct root of by substituting X for D in the D-matrix. Then since

will

(i)

f(D) e^'-* F(A,)

=

e^'^"

f(A,)F(A,)

^{Xr) is a solution of equation (12.89), but as can be written in the form e^*"*

F(A,)

=

=

A(>1)

obtained

=

(12.90)

X^ is

a simple root F(A^)

{MM

(12.91)

so that the constituent solution corresponding to the distinct root X^ arbitrary multiple of (ii)

Then

an

Let

Multiple latent roots. the matrices

X^

be any one of s repeated latent roots.

if

Wo(x,4) =

^'^"F(A,)

WJx,A,) = ;g.e^-F(A.)]=e-(A^,yp(A.)

and it

is

e'^'"^k^.

can be shown by a similar procedure to that given for distinct roots that

f(D){y}=f(D)W,(x,AJ

Wp(x, X^ all satisfy the system of differential {s—\) for obvious reasons. convenient to write the matrix Wp(x, AJ in the form e^'''Zp{x, X^

so that Wo(x, AJ; Wi(x, X^\ equations. In all the above/? .

It is

(12.92)

.

.

=

where

P^^ '-\K) Kp-1)^'f ^-U) ^^-^ + + ^^^ ^^:; U^As) = F^(A,) + ^^^--^ 1! 2!

...

+

x^F(A.)

(12.93)

and there will be s matrices of Z corresponding to the set of repeated roots. Hence since each matrix will have n columns (the order of i{D)) there will be ns constituent solutions, but of these only s solutions will be hnearly independent. Hence for a root X^ from s repeated roots the constituent solution will be {y}

and the

= {^i.W

/c2sW

.

.

.

U^)} e'^'

(12.94)

columns of the type represented by equation (12.94) corresponding may be chosen to be proportional to any s linearly independent columns of the matrices s

to the s equal roots

Zi(x,A,)

= =

FW +

Zp(x, X,)

=

¥'(X,)

Zo(x,^.)

F(A,)

xF(A,)

+

^

^^^''

+

^

'—

—+

x'F(X,)

467

MATRICES

12.

Since p = s—l the elements in the column vector of equation (12.94) will be polynomials in x of degree (s-l). These column vectors are called " modal columns " and the matrix formed from aU the n roots of f(D) is

/C12W k22{x)

/C21W

k2n(x)

K(x)

and

is

known

(12.95)

as the "

modal matrix

Those columns obtained from

".

distinct

roots will have constant elements. Finally if the exponential functions e^"^ are collected into a diagonal matrix M{x) where ?^^*

...

,X2X

Mx M(x) =

(12.96)

fA.fiX

and {C}

is a column of n arbitrary constants, the complementary function of the system is

{y}

=

K(x) Mix) {C}

Example. Solve the following system of

(12.97)

differential equations.

=

(D^ + 4D + 3)y2-{D^ + D + 2)y,

The above equations can be written

(D'-3D + 2)

(D'-l)

_-{D^ + D + 2) The

>l-matrix obtained

rom which The

in matrix

(D2

1

+ 4D + 3)J

from the Z)-matrix

form thus

[yr

is

-(A-1)(A-2)

(A-l)(A + l)j

.-(A2 + A + 2)

(A+1)(A + 3)J

A(A)

=

(A

=

[^2.

- 1)^ (A + 1) (A + 2)

II

latent roots are

^

= -1;

A2=-2;

^4

=

1

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

468

The

respective adjoints are for

>?.i

and X^ '0'

0"

F(/i)

3

[1

=

III .2

6,

_2_

-1

-3'

4

12.

"-r

[1

3]

IV and the adjoints

for

F(A3)

and A4 are

y^.3

"8

"2"

0"

[4

= _4

F'(^)

.1_

2A- 3_

1

-2"

'1

.3

-1.

.1.

[3

-1]

= k4(x) =

k3(x)

{2

1}

{2

l}

and the complementary function 1

-1

2

2(1

4

1

(1

.2

;.= i

and A4 the modal columns are

for the latent roots ^3

2.

'6

VI _2A +

and

0]

-2/'

"22 + 4

=

4

VII

+

l}x

{2

VIII

is

+ x)-

0'

-X

^-2x

+ ^).

rc.1 C2

IX C^

e"

e'_

or the conventional result

>^i=2[C3 + C4(l + x)]e^-C,e-^and

>;2

.Q.

is

= 2Cie-" + 4C2e-'" + [C3 + C4(l + x)]e"

X XI

12.19.2 The Particular Solution

To complete it is

the solution of the system of Hnear differential equations necessary to obtain a particular solution of equation (12.87). Essentially

this solution will

be {y,}

=

F(D) f(D)-'{0(x)}

A(D)

{•/-W}

(12.98)

Hence it is necessary to be able to evaluate the last term in equation (12.98). Therefore consider the case i^(.v) = Re"* where R is a column of constants.

Then

(n) which

is

A((?) i= 0.

r(D)R,«x^,».FW,

\{Dy

the particular solution

when ^(x)

(12.99)

A(0) is

exponential provided that

When

A{9)

=

469

MATRICES

12.

the following procedure

must be carried

out.

Let the

particular solution be

{y^}

where b

is

constant.

=

b[F'(^) + xF(0)] e^"

(12.100)

Then f(D)b[F'(^)

+ xF(^)] e^* =

Re^*

(12.101)

from which he'''{(D

+ e)[¥'(0) + x¥(e)'] = Re'"

(12.102)

Using Taylor's theorem to expand f(D-\-6), and operating with the powers of

D

upon X

gives

be'"[f(0)F'(^) + f' (^)F(0) + xf(^)F(0)]

=

= f \e)F{e) + f{9W(0) = f(^)F(0)

Since

and

=

Re'"

(12.103)

A(0)

A'(^)

equation (12.103) gives the value b

Hence the

particular solution

{yp}

=

=

R/A'(0)

(12.104)

is

^^

[F'(0)

+ xF(0)] e'"

(12.105)

Other forms for (p(x) can be treated in a similar manner to the corresponding forms of (l>(x) in the inverse operator method (Section 2.5.5). Finally, the complete solution is the sum of the complementary function and the particular integral as for Hnear differential equations.

A

Example. battery of A'' stirred tank reactors are arranged to operate isothermally in series. Each reactor has a volume of V ft^ and is equipped with a perfect agitator so that the composition of the reactor effluent is the

same as the tank contents. If initially the tanks contain pure solvent only, and at a time designated ^o» ^ ft^/min of a reactant A of concentration Cq lb moles/ft^ are fed to the first tank, estimate the time required for the

Q

Mh

A

concentration of leaving the tank to be lb moles/ft^. The stoichiometry of the reaction is represented by the equation fci

k2

A:^B^t:C k[

/C2

first order and the feed to the reactor does not contain any product B or C, but is assumed to contain a catalyst that initiates the reaction as soon as the feed enters the first reactor.

All the reactions are

Solution

The concentration of A The concentration of B The concentration of C M.M.C.E.

will will will

be written C^ be written C^ be written Q. 16

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

470

Then a mass balance on any tank n

be

will

V^ = qC^.,-qC„-rV where r is the rate of chemical reaction. For component A this gives II

component B,

for

^^ =

^Q.„-i-(^Q.n+^/c;Q„+F/c2Q.„-F/c,C^.„-F/c,'Cc.„)

at

III

and

for

component C,

qCrn-i-(qCrn+Vk',Cc,„-Vk,CsJ

IV

dt

Dividing equations holding time, gives

C^,

dC^,n dt

dCn -

II, III,

=

dt

and IV by

V

and calhng

{V/q), 9 the

nominal

V

p-(^+^l)Q.n + ^iQ.„

^^ - Q +

k2

+ K^ Cs,„ + hC^,„ + k',Cc,„

VI

and

dCc

-=

dt

~^" ~\e'^ V ^C.n + kzCB,,

VII

Equations V, VI, and VII can be written in matrix form as follows 'A,n

c A,n-l

CB.n

n-1

lQ.hj

C,

(iie+k,)

-k\

-k,

(iie-\-k2+k[) -/C2

n-lj

'

-k'2 (l/^

+ /ci).

"Q,„'

Q,„ Cc.n\ VIII

or

more concisely dC„

C„_i

dt

9

-GC

IX

where C„ and C„_i are column vectors and G is a third order square matrix Equation IX can be applied to the first reactor thus

with constant elements.

^

dt

9

X

The

solution of equation

X is

C,=^" + where K^

is

471

MATRICES

12.

XI

e- z^/

the terms inside

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

478

the square bracket will be greater than zero for all values of a, will

have the same sign as

be a maximum.

If z^x is negative, z will If z^x

is

and d^z/ds^

z^^.

positive, z will be

minimum.

=

In the special case z^^Zyy

z^^, there will be some values for a which

exhibit a point of inflexion whereas all other values of a give a consistent

maximum

or minimum. If z^^Zyy < z^^, the surface has a saddle point. The above principles of maxima and minima have been used by Jenson and Jeffreyst for the evaluation of the optimum number of stages in solvent extraction processes. The analysis for counter current processes is as follows. Consider a feed solution containing Xq lb solute per lb of solute-free

of a countercurrent extraction process at the Let the process contain A'' stages and let an extracting solvent be fed into stage A^ at the rate of B Ib/h on a solutefree basis. The solvent passes countercurrent to the feed liquor through the plant, finally emerging as the final extract from stage 1. The spent feed will be discharged from stage TV with a solute concentration X^. The carrier liquor of the feed will be assumed to be immiscible in the extracting solvent. Assume that the feed rate and the solute content of the feed are constant and let the unit cost of the feed in any system of currency be y per lb of solute. Let the total hourly cost of each stage including operation, fixed charges, and the depreciation of capital be a. Finally let the cost of the solvent, including recovery from the final extract and make-up of spillage losses, be P per lb of solvent. Then on the basis of one hour's operation: liquor to be fed to stage rate of

C

on a

Ib/h

1

solute-free basis.

Cost of feed

is

Cost of solvent Similarly

product

if

X

the value of

is

1

-0 lb

is

yCX^

(13.12)

^B

(13.13)

of extract product, the total value of the

is

IC{X^-X^) and the hourly

profit

from the process

is

(13.14)

given by

P = AC(Xo-X;v)-(aiV + /?B + yCXo) But X^

is

related to

^N = ^0 [

where S

=

—C

-

mB

and

(13.15)

Xq by

m

is

^J^rf ]

"(see Section 9.6.3)

(13.16)

the distribution ratio.

Substitution of equation (13.16) into (13.15) gives

P = XCX in

s^-i °U^+i-i_

which the variables are t Jenson, V. G.

and

-

r

N and

Jeffreys,

G. V.

[yCXo + aN + pB]

B.

The maximum value of P

Brit.

Chem. Eng.

6,

676 (1960).

(13.17)

is

obtained

479

OPTIMIZATION

13.

by the procedure presented above. Thus

dP

dN

]'"S-« = = ACJoS^[ (si/l)2

(13.18)

From which

_ icJ^" g

5^^(5-1)1115 (13.19)

(5'^-'^-l)'

and

dP

^^+1 i\cN-i_/'\r 1^/'c^^_1^c^ /-mS'\ N(S^^'-l)S^-'-(N+l)(S^-l)S" l I

,^^

r

„_^ (13.20)

or

z

—— = ^

r(5^-'^+N-iV5-5)" (13.21)

N+l 1\2 (5'"^^-l)

Am An

In equations (13.19) and (13.21) the cost terms have been separated from the process terms. They are two complex simultaneous algebraic equations that will be difficult to solve analytically, but since A'' is a whole number, an approximate but sufficiently accurate solution can be obtained as follows. The right-hand sides of equations (13.19) and (13.21) can be evaluated for

any pair of values of 5 and

dimensionless cost groups

5 and

.

A^ to give a numerical value for each of the

aC^q

and

— —.

-

Hence by taking a

series

of

AtviXq

N

repeating the calculations, a contour chart can be Such group using 5 as ordinate and as abscissa. contour charts have been prepared by the above authors for values of A^ ranging from to 20 and values of S between and 2; these are shown in Figs. 13.4 and 13.5. In a particular example the basic cost terms would be known enabling the cost groups to be evaluated, and for these values equation (13.19) is satisfied along one curve in Fig. 13.4 and equation (13.21) satisfied along a curve in Fig. 13.5. Both equations can only be satisfied simultaneously at the point of intersection of these two chosen curves. This point is easily located by superimposing the two contour charts as shown in Fig. 13.6 and

pairs of values of

drawn

for each cost

N

the coordinates of the point of intersection give the modified solvent to feed

and number of ideal stages for optimum operation. Figure 13.6 is an accurate version of the graphical solution in which the values of 5 have been restricted to between and 1.0 since for conventional solvent extraction this is considered to be the most applicable range. The procedure applied to the optimization of counter flow extraction using Fig. 13.6 is perfectly general and the figure has universal application for the evaluation of the optimum solvent to feed ratio and number of ideal stages. The technique employed of using contour charts can be extended to other fields of chemical engineering.

ratio

MATHEMATICAL METHODS

480

IN

CHEMICAL ENGINEERING

2-On

5

Fig. 13.4.

5 Fig. 13.5.

iO

15

Contour chart of equation

10

20 (13.19)

20

15

Contour chart of equation

(13.21)

3.

OPTIMIZATION

481

5%

by weight solution of acetaldehyde in If the annual capital and operating costs per stage are estimated to be £1600, the cost of supplying, pumping and regenerating the solvent from the final extract is estimated to be £4-5 per ton, and the value of the acetaldehyde is taken to be £155 per ton; what is the optimum number of stages and quantity of Example.

toluene

is

8000 Ib/h of a

to be treated with water to extract the acetaldehyde.

solvent required for the process. The distribution ratio of weight of acetaldehyde in water to acetaldehyde in toluene may be considered constant at 2-2.

Solution

5%

feed solution gives

CXo = 400 a

=

4s.

12

Xq

=

0-053.

Ib/h.

per day: p

3

4

5

6

=

0-0402s. per lb:

A

=

7

Fig. 13.6.

Optimization chart

l-384s. per lb.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

482

^

=

and

0-0072

XCXo From

-^ =

0-251

hnXo

Fig. 13.6

N=

10 or 11 stages

S=0-9 =

3840 Ib/h of solvent

Problems Involving Four Variables

13.3.3.

The

presented above can be extended to four or minimum values of

analytical procedure

variables.

That

is,

maximum

the

w=f(x,y,z)

(13.22)

can be located by differentiating equation (13.22) and equating to zero.

dw = when

df

^dx +

df -:^dy

ox

oy

Thus

df

+ -fdz =

(13.23)

oz

the conditions

df

df

/ = 0;

/ = 0;

oy

ox

df

/=

(13.24)

oz

are satisfied.

In problems containing four or more variables, the analytical solution becomes increasingly complex and impracticable. Therefore, other techniques of optimization must be applied, and these will be considered in later sections

of

this chapter.

13.3.4. Optimization with

In

a Restrictive Condition

some optimization problems a

relationship

must be

satisfied

by the

In vapour-Hquid equihbria for example, the temperature, pressure, and compositions of liquid and vapour are related by the

independent variables.

One

boiling point condition.

eliminated analytically, but this alternative

method

is

variable, say temperature, can always be

may

not be algebraically convenient, and an

available.

Thus consider equation

(13.22) with the restrictive condition

g{x,y.z)

=

(13.25)

Differentiating equation (13.25) gives ^_^rf,

OX

+

^_^rf,

oy

+ ^?,, =

(13.26)

oz

the optimum point, equations (13.23) and (13.26) must be satisfied for displacements in any direction, and in particular for dx = 0. Hence the equations

At

^dy + ^dz =

(13.27)

oy

oz

^Idy +

= fdz dz

(13.28)

dy

OPTIMIZATION

13.

must be

consistent.

This

is

only possible

= f + ,fi oy

By

relaxing the conditions that

if

and|^ + A^-^=0 oz

oy

it

483

dx

=

(13.29)

oz

and imposing the condition dy

0,

=

0,

follows that

|^ OX

+ A^ =

13.30)

ox

Equations (13.29) and (13.30) can be obtained by optimizing the function

W=f(x,y,z) + Xg(x,y,z) when

the

optimum point

is

(13.31)

given by the solution of the equations

dx

ox

dy

dy

dz

dz

g{x, y,z)

(13.32)

=

which contain the four unknowns (x, y, z, X), X is known as a " Lagrange The above method can be extended to two restrictive conditions by introducing two arbitrary multipliers. multiplier ".

13.4.

The Method of Steepest Ascent

The method of steepest ascent is one of the best known methods of optimization when the number of variables involved is large. The procedure was developed by Box and his co-workers. t Essentially the method consists of defining the criterion to be maximized as the " response " of the process. This may be the profit, the yield, or the output from a process and will depend on a number of factors such as temperature, concentration, pressure, etc. The relationship between the response and these factors can be written in the

form ;/

which

is

=/(xi,X2,X3,...,x„)

called the '"response function

(13.33)

".

The response function can be plotted in terms of two of the factors to give what is known as the " response surface ". Such a surface will be undulating and reaching the optimum will correspond to climbing the highest hill on the surface. Generally speaking the greatest gradient will be in the direction of the maximum, and therefore if the steepest ascent can be t Box,

G. E.

P.

and Wilson, K. B.

/. Statistical

Soc. 1,

B13

(1951).

MATHEMATICAL METHODS

484

IN

CHEMICAL ENGINEERING

XI

Response contour

Fig. 13.7.

located on a response surface, response.

However

it is

it

map

will lead in the direction

of the

optimum

very difficult to represent a surface on paper, con-

sequently it is advantageous to plot the surface in the form of lines of equal response on a graph whose coordinates are two of the variables. These lines are called "response contours " and are shown in Fig. 13.7. The line of steepest ascent starting from any point is obtained by trial by drawing hnes perpendicular to the contours. These lines will show the extent to which the different factors must be varied in order to give a maximum increase in the response. This procedure will only direct the way to the optimum conditions since at the

maximum

is zero, and becomes more

the gradient

the line of steepest ascent

as the

maximum

0-3

0-6

0-4

o E


= =

1/—' V m

^0 - 3

^

>'i

^lx~)? .

/

=

v;;i^

IX

+

-

XVIII

XIX

obtained from the functional equation VIII by considering poUcy with a feed of Then by repeating the procedure given above, it can be

is

the second stage to be using an optimal one-stage

composition x^.

shown

that

V That

is,

nrX

m

for the two-stage policy the ratio of solvent to raffinate rate to each

N

stages, the optimal stage must be the same, or extending this concept to poHcy is obtained by dividing the total quantity of solvent available equally between the A'' stages. The above problem was presented by Aris et a!.'\ and extended to more complex crossflow extractions in which the solvent and the parent liquid in the feed to the process are partially miscible, and the equilibrium is not represented by a straight Hne. Solution of such problems can only be solved by computer and these authors did in fact use a Univac Scientific Computer

Model

1103.

t Aris, R.,

Rudd, D.

F.

and Amundson, N. R. Chem. Eng.

Sci. 12,

88 (1960).

Chapier 14

COMPUTERS 14.1.

Introduction

In engineering mathematics, the need for numerical calculation constantly arises. For instance in the method of Frobenius and separation of variables, terms of an infinite series have to be evaluated for many values of the variables. In addition, the solution of difference equations by graphical methods (Section 9.7.1) is a less accurate but more convenient way than the corres-

ponding numerical method of successive substitution. Much numerical work is involved in inverting and multiplying matrices, but the most arduous Because computation occurs in the numerical methods of Chapter 11. arithmetical operations are basically simple, and desk calculating machines and slide rules are in everyday use, the complete mechanization of arithmetic has been a natural consequence. The development has taken place along two channels leading to analogue computers and digital computers. Whereas the analogue computer is a development of the shde rule principle, the digital computer is a development of the desk calculating machine. The slide rule operates by representing each number by a geometrical length, a particular length on one scale always representing the same series of digits. The basic scales are logarithmic, but most shde rules have double logarithmic scales, and scales with varying lengths representing the logarithmic unit (i.e. squares, cubes). The principle is to replace a number by an equivalent length, and to calculate in terms of these lengths. In the general purpose analogue computer, each number is replaced by an equivalent electrical potential, and the calculation is then performed electrically in terms of these potentials.

The digital computer is a development of the desk calculating machine which is capable of storing its own calculating instructions in the form of a program. Speeds have been increased enormously by using electronic valves and electrical signals instead of revolution counters and mechanical linkages, and improved electronic techniques are constantly increasing the speed of calculation

The

still

further.

between the two computers can be illustrated by The fraction J can in with the desk machine. principle be located exactly on a slide rule; only human error hmits the accuracy. However, the fraction must be approximated by 0-33333333 on an eight-figure desk machine and there is thus a physical Hmit to the accuracy. Although the fraction exists on a slide rule but not on a calculating machine, no-one would suggest that the shde rule was more accurate. This same essential difference

comparing the

slide rule

492

;

14.

COMPUTERS

493

argument applies to a comparison of analogue and digital computers the analogue computer uses a continuous number scale, whereas the digital computer uses a discrete number scale. Analogue and digital computers also use different modes of calculation the analogue calculates in parallel whereas the digital operates in series. In the analogue method, all dependent variables are continuously available and can be displayed graphically on a multibeam oscilloscope, but the digital computer can only perform a set sequence of commands as stored in its program and can only offer discrete values for one variable at a time. Attempts are being made to develop parallel digital computers with multiple arithmetic units, and series analogue computers with memory faciHties. This is leading naturally to the production of hybrid computers. The great advantage of automatic calculation is that once an analogue has been constructed or a program has been written, a whole set of problems with a range of values for many parameters can be solved very quickly, enabling the optimum choice of parameters to be made. Also, particular characteristics of the behaviour of the system can be associated with various controUing parameters thus helping in the instrumentation and choice of ;

control system for a plant.

Passive

14.2.

Analogue Computers

computer to understand, and its mots an inter-connected arrangement of electrical impedances. The differential equation relating the potential at any point in the network to time is chosen to be of the same form as the differential equation it is desired to solve. Provided that the boundary conditions are also analogous, the behaviour of the voltage at a point will be identical to the behaviour of the dependent variable at the analogous point. This

is

by

far the simplest type of

common form

is

V\M R

n+1

—r^AAA R

Typical section of passive analogue circuit

Fig. 14.1.

Consider, for example, the section of an electrical circuit depicted in 14.1. Denoting the electric potential at point n by e„, and using

Fig.

Kirchoff's law

—-e„ o ^n+1o

K or

o e„ -1

.

i^

"-e„ o

R

=C

(14.1)

dt

— = ^^(e„+i — fe_^,-2e. + 2e„

dt

Ao de„

e.^.') en -i)

(14.2)

494

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

which expresses the rate of change of potential at « as a function of the Comparing equations (14.2) and (11.52), it can be seen that e„ will have the same behaviour as r„ if potentials at neighbouring points.

RC =

(Ax)2/a

(14.3)

Equation (14.3) is thus the equation linking the one dimensional heat conduction equation to its electrical analogue (Fig. 14.1), and both sides of the equation have the dimensions of time. It is not essential that equation (14.3) should be satisfied for the use of the analogue, but if

RC =

2(Ax)^/a

(14.4)

the electrical potentials will alter at half the rate of the corresponding temperatures. In moving from equation (14.3) to (14.4), the time scale of the

analogue has been modified.

This technique enables slow processes to be and fast processes to be retarded. Whilst the circuit of Fig. 14.1 is suitable within the range of the problem, special circuits must be constructed to terminate the repeated circuit in conformity with the boundary conditions. The terminal circuits for two boundary conditions are illustrated in Fig. 14.2. The closing of the switch accelerated for investigation purposes,

R/2

R

R

R

hMArrWA •5-

Fig. 14.2.

c±

Analogue

1

C/2.

circuit for

boundary conditions

change in the electric potential at the end of the circuit which corresponds to a sudden change in temperature at one face of a large slab. The reason for the resistor of value R/l is as follows. In the analogue, electrical resistance represents reciprocal thermal conductivity, whereas electrical condensers represent thermal capacity. R and C thus represent the thermal resistance and capacity of a space increment. Considering a symmetrical system in which both faces of the slab are subject to the same step change; circuit (a) would terminate both ends of the system. To maintain the correct total resistance and capacity within the system and maintain symmetry, the first resistor in the chain must have half of the resistance of any other member. Figure 14.2(b) shows the termination of the circuit corresponding to an insulated face. A similar argument to the above explains the condenser of in (a) causes a step

half value.

The analogue the system.

is used by alternately closing the switch and discharging Using a potentiometer circuit, the potential at any point can be

14.

COMPUTERS

495

determined as a function of time, or an oscilloscope can be used to display The benefits from using the analogue are the ease and rapidity and the repeatability of the results. measurements, making of It will be seen that this type of analogue involves the finite difference representation of the prototype system, and hence it intrinsically contains the errors associated with curtailing a Taylor series. the behaviour.

14.2.1.

Electrolytic

Tanks

The flow of electric current in an extended conductor obeys Laplace's equation, hence by studying the potential distribution in such a conductor, any solution of Laplace's equation can be inferred. The most common example of this technique is the electrolytic trough which consists of a shallow flat bottom tank containing an electrolyte with submerged electrodes and insulators to represent the boundaries. By the nature of the system, the tank has a uniformly distributed resistance, but its electrical capacity is negligible. It is thus suitable for boundary value problems associated with Laplace's equation, but not for initial value problems associated with Poisson's equation,

V2r =

aV ct

(14.5)

The analogy is obviously restricted to one or two dimensions, although axi-symmetrical systems can be studied in a tilted tank which is analogous to a typical wedge section from a cylinder. Boundary conditions of type 1 (dependent variable fixed) are represented by electrodes of the same shape as the boundary maintained at the analogous constant potential. Lines of symmetry or other type 2 boundary conditions are represented by insulated boundaries, or electrodes with fixed current supplies. Type 3 boundary conditions have to be imposed by trial and error, matching the potential on the electrode with the current drawn from or supphed by it. The analogue is used by exploring the tank with a fine probe and plotting the equipotentials. The mechanical and electrical systems need careful design. double pantograph is frequently used so that a recording pen will follow an identical path to the probe, but the system must be accurately constructed because both the curvature and spacing of the equipotentials are physically important. Unless the electrolyte has the same depth at all parts of the tank, the equipotentials will be distorted; hence a well supported glass plate is usually used for the bottom of the tank. The electrical system has to be chosen to avoid gas formation on the electrodes, polarization potentials, and yet be at a safe voltage for operation. The probe must be submerged sufficiently to mask the effect of the meniscus, and not draw sufficient current to distort the equipotentials it is trying to locate. It is normal to use about 20 volts at 1000 c/s to ehminate electrolysis and polarization effects without introducing serious phase lags due to the natural capacitance of the system. The probe is often connected to the grid of a triode valve with cathode bias to avoid drawing current from the tank.

A

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

496

tank is being used to simulate a two dimensional flow field, a double probe is needed to diff'erentiate the potential field and obtain a velocity component. It is of vital importance in this connection that neither probe influences the potential distribution in the neighbourhood of the other probe. Carefully designed electronic circuits are also needed to detect the small difference between two larger alternating voltages. For studying solutions of Laplace's equation with awkward shaped boundaries, the electrolytic tank is ideal; the alternative solution using numerical methods (Section 11.7.4) is frequently too large for the rapid store of a digital computer (see Section 14.4.3) and too laborious for manual If the

calculation.

14.3.

Active Analogue Computers

An

analogue computer for general calculations, as opposed to the above wiU be described in this section. The basic unit of such a computer is the " operational amplifier ", which is capable of amphfying D.C. voltages with an amphfication factor of 10^-10^. The size of an analogue computer is usually expressed in terms of the number of operational ampHfiers it contains. Although the high gain amplifier is the basic unit of the computer, its accuracy really depends upon the tolerance of the resistors and condensers in the associated calculating circuits. Three figure accuracy is all that can be expepted even from good quality components, and hence this is the limit to which this type of computer can work. special purpose analogues,

14.3.1. Basic

A mpUfier for A ddition

Figure 14.3 illustrates the arrangement of computing elements around The standard symbol

the operational amplifier during the addition process.

-AAAAr

S.J.

g^o

R \MA/

e3o

v\^A^^

Fig. 14.3.

Amplifier '^

!

Addition

circuit

for the operational amplifier is a triangle, and the diagram illustrates the amplifier with a feed-back resistor {Rp) connecting its output potential {cq) to its input at the " summing junction " (S.J.). Three input resistors of equal

value {R) connect the summing junction to three input potentials (^i, ^2, ^3). Denoting the summing junction potential by e^ and the amplification factor by /i, the following current balance can be obtained at the summing junction.

COMPUTERS

14.

497

es^e,-e^^e^^eo-e_^^^ (14.6)

R

R

R

Rf

because the amplifier draws no current, and the amphfying factor equation

Eliminating

e^

eo= -

liCs

(14.6)

and

between equations

(14.7) (14.7)

.o=-f(^. + ^a + .3)(l +

and rearranging,

~^')

If Rp and R are chosen to have the same value, and ^ equation (14.8) simplifies to

^0=

-(^1

(14.8)

is

sufficiently large,

+ ^2 + ^3)

(14.9)

which shows that the output potential is equal to the sum of the input potentials but of opposite polarity. Provided ^ > 10^, the error caused by assuming that /i is infinite is less than one part in 10"". A typical computer arrangement has a 1 megohm resistor in the feedback circuit, two 1 megohm resistors and two 100 kilohm resistors in the input circuit. Thus the ratio R^jR = 10 in the latter case, resulting in the insertion of a scale factor into the addition. Incidentally, this reduces the accuracy of the addition by a factor of ten and hence ji is usually chosen to

exceed 10^. The standard range of potentials for computing purposes ± 100 volts, hence equation (14.7) shows that

e,< 10"^ the

i.e.

summing junction

is

volts

is

usually

(14.10)

virtually at earth potential during a calculation.

fortunate that equation (14.9) gives addition of the inputs with a sign change, since this faciUtates subtraction. Putting a single input into the It is

R = Rp, results in a sign change only. Hence to subone potential from another, its sign is changed by an ampUfier, and then added in a second amplifier.

addition circuit with tract it is

14.3.2. Basic Amplifier for Integration

To back

obtain an integrating circuit,

it is

only necessary to replace the feedThis is shown in Fig. 14.4

resistor {Rp) in Fig. 14.3 with a condenser.

C

eio

v^ Fig. 14.4.

^^0 SJ.

Integration circuit

where only one input is shown for simpUcity. Assuming that the summing junction is still at earth potential due to the high amphfication factor, a

MATHEMATICAL METHODS IN CHEMICAL ENGI^fEERING

498

similar current balance gives

i--C^

(14.11)

Integrating equation (14.11) gives t

^o

Choosing a

megohm

resistor

= ^Je,dt

(14.12)

and a microfarad condenser

gives

t

eo=-je,dt

(14.13)

in seconds. The physical interpretation of equation a constant potential, the output potential (^q) increases in value by an amount e^ during each second of operation. By connecting other inputs through resistors (R) to the summing junction

where

t

(14.13)

many

is

is

measured

that

if ^i is

variables can be simultaneously

added together and integrated. With

input resistors of different values, the addition can be in different proportions. It should again be noted that equation (14.13) involves a change of polarity indicated by the negative sign in front of the integral.

14.3.3. Multiplication

A

by a Constant

simple earthed potentiometer has the effect of multiplying a potential

by any fractional constant. Thus

in Fig. 14.5,

eo

=

ke,

(14.14)

provided no current is drawn from Cq. The elements in a computer installation are usually chosen so that a much larger current passes down the

(1-k)R

^^

oe

kR

Fig. 14.5.

resistor than

°

Multiplication by a constant

is likely to be drawn from Cq. In addition, facihties are usually provided for setting the potential at Cq at the desired fraction of the reference voltage temporarily applied at e, with Cq under loaded conditions. Thus errors due to loading the potentiometer arm can be ehminated.

14.

4.3.4.

COMPUTERS

499

Multiplication by a Variable

There are four kinds of multiplier in general use; the servo-multiplier, the mark-space multiplier, the quarter-square multiplier,

multipher.

The

first is

and the Hall

effect

electro-mechanical, but the other three are purely

electronic.

The servo-multipHer consists of a small D.C. motor coupled to all of arms of a gang of about six potentiometers. One variable is fed to the end of any potentiometer, as e^ in Fig. 14.5, and the other variable is used to position the armature of the D.C. motor; thus giving an output potential, which is representative of the product, from the wiper arm of the potentiometer. The key to the design of this item of equipment is the positioning of the armature to represent the variable. The standard reference voltage (100 V) is appUed to one of the potentiometer gang and the output from its wiper will be uniquely determined by the position of the D.C. motor armature. This potential is compared electronically with the value of the variable, and a signal is generated in proportion to the discrepancy. The signal is fed to the motor so that its armature rotates in such a direction as to reduce the difference. Hence it can be seen that a strongly coupled feedthe wiper

back of this nature

cause the armature to follow the changing value of the due to the inertia of the armature and coupled wiper arms, but a safe frequency limit is always quoted by the supplier of the instrument. variable.There

is

will

inevitably a time lag

The mark-space

wave of fixed The height {h) is

multiplier operates by generating a square

frequency and wavelength

(A) as illustrated in Fig. 14.6.

V

Fig. 14.6.

Square wave

in

mark-space multiplier

determined by one variable, and the ratio {ajX) is fixed by the other variable. Thus the average height of this curve is a measure of the product of the two variables. Integration of the curve to determine the average height is also performed electronically, and provided A is small, the output will be smooth

and

faithful.

An is

algebraic identity

exploited in the quarter-square multiplier.

The addition and two sub-

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

500 tractions

on the

right

hand

side present Httle difficulty,

and two standard

units for generating the square of a variable are the only new equipment required. It will be shown in Section 14.3.5 that such a function can be

represented as a succession of straight line segments using biased diodes. circuits for the above operations are usually combined into a single unit, so that the details of the above circuitry need not be understood.

The

an electrical potential difference is apphed across one pair of faces of a cubic block of semi-conductor, and a magnetic field is imposed across a second pair of faces, then an electrical potential difference appears across the remaining pair of faces. This is known as the " Hall effect " and the potential difference generated is proportional to both the input potential difference and the magnetic field strength. The device can be used for multiplication by generating a current which is proportional to the one variable and passing it through an electromagnet to produce the magnetic field. If

The other

variable potential

is

applied to the appropriate pair of faces, and

the resultant potential difference will represent the product. All of the above multipliers have their uses, but the servo-multiplier restricted in frequency range,

from

and the Hall

effect multiplier

is

must be shielded

stray magnetic fields.

14.3.5. Function Generation

With the exception of the simple trigonometrical and exponential funcwhich can be obtained by solving Hnear differential equations with

tions

constant coefficients, a device is needed for generating arbitrary functions of a variable. This need arises particularly when experimental data in the form of a graph is to be presented to the computer. Such a device makes use of the rectifying characteristic of a diode. Compared with the input resistors of the operational amplifier, the diode resistance is neghgible to potential differences of one polarity but infinite to the other. Thus if a diode is inserted in any line connecting computing components, only signals of one polarity will be conducted. If the diode is suitably biased using a resistance network, the cut-off point can be varied from zero to either positive or negative potentials.

Basic diode circuit

Consider Fig. 14.7 which shows a diode interupting the passage of an input signal e^ to a summing amplifier. The diode will not conduct until the armature of the potentiometer picks up a positive potential, and hence the output potential ^q

will

be zero until e^ reaches a certain positive potential

:

14.

COMPUTERS

501

determined by both the position of the armature and the fixed negative (—K). The rate of change of Cq with respect to ej is independent of K, therefore the slope of the output versus input line can be chosen by positioning the armature of the potentiometer, and the starting point is then fixed by the value of K. A combination of circuits similar to Fig. 14.7 all feeding the summing junction of the same amplifier from the same source ei results in Cq representing the sum of many straight Hne segments which together will represent a functional relationship between Cq and ej. A unit which utiHzes these ideas in a more efficient circuit is called a " diode function generator ", and to generate a given function it is only necessary to represent the curve by a series of straight line segments and then to set two potentiometers for each segment. potential

14.3.6. Application to Differential It

was shown

Equations

an amplifier can be used to integrate is only one time variable (as

in Section 14.3.2 that

a function with respect to time.

Since there

opposed to three coordinates) it is only possible to integrate with respect to a single variable in an analogue computer. Hence analogue computers are restricted to solving ordinary differential equations. The numerical methods of Section 11.7 can be used to transform many partial differential equations into difference-differential equations and in this form they are amenable to solution by analogue methods; although much computing equipment is required.

The apphcation of analogue computers

to ordinary differential equations

further restricted because the independent variable

is represented by time which is of necessity an open range variable. Hence initial value problems can be solved directly, but boundar>' value problems have to be solved by trial and error using the method presented in Section 11.4. The method will be illustrated by the following example of an initial value problem involving is

linear differential equations.

Example. Solve the

differential equations

+ 2— + — + T^ dx^ dx dx

3>;-7z

=

dz

subject to the initial conditions

x

at

=

z

0,

=

4,

y

=

l,

dy/dx

=

Solution

A signal representing representing

its

>'

or z can always be obtained by integrating a signal In order to solve the given equations, signals

derivative.

representing y, z, dy/dx, dz/dx, and d^y/dx^ must be generated, and the first three can be obtained by integration from the last two. Hence the differential M.M.C.E.

j-7

502

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

equations must be rearranged to generate dzjdx and d^y/dx^ algebraically.

Thus

2= -2i^dx

dx^

dz

=

— -3y + lz

I

dx

_y_52 + 2

II

dx

A simple potentiometer can be used as in

Fig. 14.5 to multiply by a fractional convenient to represent the independent variable (x) by the computer time variable (t) using a scale factor (a) to prevent the coefficients in equations I and II exceeding unity. Thus putting

constant, thus

it is

x

=

at

III

d^y

dy

dz

dr

dt

dt

-4 = -2a~-a~-3a^y + la^z

IV

dz

— = ay — 5az + 2a

V

dt

Choosing the maximum suitable value of a, equations

=

0-2t

_

^ .dy -0-4 -f

X d^y 2

dt^

III,

IV, and

V become VI

^dz

^ -0-2-

dt

-0-12>; + 0-28z

VII

dt

= 0-2};-z + 0-4 ^ dt

VIII

Equation VII can be simpHfied by eliminating dzjdt using equation VIII.

Thus 2 dt""

= -0-4-^ -

0-16y + 0-48z-008

IX

dt

In most analogue systems, there is a Hmit to the electrical potential allowed within the computing circuits and 100 volts will be assumed for this Hmit. The initial values of y and z are 1 and 4 respectively, and assuming a stable system in which all derivatives vanish as ;c oo, >^ = If, z = f. The unit of ^^ and z is therefore chosen as 25 volts. The circuit shown in Fig. 14.8 can thus be constructed. It is selfexplanatory except for the four feeds from the ± 100 V supplies. In order to set the initial conditions, the condenser in the feedback line of the integrating amplifier must be charged to an appropriate potential. The electronic means of doing this usually involves a sign change, and the polarities of the supplies for the initial conditions have been chosen to allow for this. The initial condition part of the circuit is automatically disconnected as the computation starts. Two further connections are made to the supply points to represent the constant terms in equations VIII and IX.

-

COMPUTERS

14.

0-2

—

>

Potentiometer

503

I.e.

Initial

Summer

Fig. 14.8.

condition

Integrator

Active analogue computing circuit

A double beam oscilloscope connected to the outputs of the amplifiers producing ^v and z is the best means of exhibiting the solution, but calculations repeatedly suspended at fixed time intervals and measured by a bridge most accurate solution. Assuming an integrating time constant of one second, as in Section 14.3.2, equation VI shows that 5 seconds of computing time correspond to a unit increase in the value of x. The accuracy of the potentiometer settings can be checked by allowing the calculation to proceed for a long time and comparing the potentials with the known values of y and z at steady state,

circuit give the

14.3.7. Application to

Process Control Systems

In the above sections, the operational amplifier has been shunted with a feedback resistance (Rp) to give an addition operator, or with a condenser (C) to give an integrating operator; but these two circuits can be combined

shown in Fig. 14.9 to represent a further useful operation. usual current balance at the summing junction gives

as

eo

R^R

+C

Taking the

dco (14.15) dt 17-

MATHEMATICAL METHODS

504

IN

CHEMICAL ENGINEERING

Introducing the differential operator (D) and rearranging,

(D + a)eo= -bei where

At

i?

this stage,

it

differential operator

=

l/K^ C

and b

(14.16)

= l/RC

(14.17)

is necessary to establish the equivalence between the (D) and the Laplace transform parameter (s) when the

AAAAr

R

'1o

AAAAr

-o^O S.J.

Fig. 14.9.

initial

Simple transfer function

conditions of the problem are irrelevant.

Putting

/(0)=/'(0)= .... =f"-'\0) = in

equation (6.13) gives

^[DYCO] =

s"f(s)

(14.18)

F(s)f{s)

(14.19)

which can be generalized to give

nFiD)f(t)]

=

showing that the operator F(D) transforms to theorem of Section 6.2.2 is identical to:

f(DXyen = e'y{D + p)y

F{s).

Also, the shifting

(2.79)

Hence, in control theory where it can be assumed running at steady state, the differential operator (D) can be replaced by the Laplace transform parameter (s). Thus equation nan be V>p written writt(=»n (14.16) can with s replaced by D.

that the system

is

initially

(14.20)

and the circuit of Fig. 14.9 represents the transfer function —bl(s + a) with an input signal (ej) and an output signal (eo). The response of the control system characterized by the above transfer function can therefore be studied by altering e^ in any desired manner. Any control system can be simulated by combining circuits of the type shown in Fig. 14.9, and the stability can be studied quickly on the analogue computer. If the various resistors have adjustable values, the best settings for the real control system can be determined by trial and error on the computer without introducing any hazard into the real process.

14.

14.3.8. Application to

COMPUTERS

505

Simulation

It has been shown above that the analogue computer can simulate the behaviour of a control system. This idea can be extended to any process which can be represented in the form of ordinary differential equations. useful application of this nature is to the study of reaction mechanisms and If experimental curves are available to the evaluation of rate constants. exhibit the concentrations of the initial reactants, the intermediate products, and the final products any proposed mechanism can be studied on the computer by displaying the various concentrations on an oscilloscope. By adjusting potentiometer ratios, the rate constants can be varied until the computer gives results of similar appearance to the experimental results. The final settings of the potentiometers will thus determine the reaction rate constants Having determined the rate constants, the in the proposed mechanism. yield of any product can be maximized by trial and error, by feeding different reaction mixtures to the system and allowing the reaction to proceed for different lengths of time. Thus the computer can save time and expense in the way of chemicals and analysis of samples after the initial investigation has been performed experimentally.

A

;

14.4.

Digital Computers

In digital computation, every calculation is reduced to basic fundamentals. All calculations are performed in the " arithmetic unit " which can only

perform the four operations of addition, subtraction, multipHcation, and Even these operations are not fundamental, since the other three operations are performed by repeated subtractions. Nevertheless any one of the above four operations can be performed on a pair of numbers entering the unit. Other operations, such as extracting a square root, must be expressed in terms of the four basic operations, but only two numbers and a result can simultaneously exist in the arithmetic unit. Hence there must be an orderly flow of numbers through the arithmetic unit. The control of this flow is governed by the " program " which refers to each quantity in the calculation by a label called its " address ". The numbers can be considered to be stored in a set of pigeon-holes, each hole having a unique address. The calculation proceeds by extracting numbers from the pigeon-holes, processing division.

them

in the arithmetic unit,

and

re-allocating the results to the pigeon-holes.

One point which must be appreciated

at this stage is that the program itself must also be stored in a section of the pigeon-holes. A piece of stored program information consists of an instruction accompanied by a reference to the address of the next piece of program. The calculation thus proceeds by the logic of the program controlling the passage of numbers through the arithmetic unit, and at the same time maintaining its own relentless progress. The operation of the various parts of the digital computer will be discussed in the

following sub-sections.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

506 14.4.1.

The Binary System

The decimal system of calculation has little to recommend it except The structure of a number in decimal notation is exposed in the

famiHarity.

equation:

723

= 7x10^ + 2x10^ + 3x10°

(14.21)

This notation removes the necessity of having a different symbol for each number by repeating the ten basic symbols in various orders, the position of a digit implying the power of ten by which it is to be multiplied. To store a decimal digit in a computer requires a device having ten distinct There are many devices having two distinct states which is inconvenient. states such as off/on, or charged/discharged, and it is simpler to use these devices for storing numbers. Each digit can only have two values in this system, and the powers of two have to be used instead of the powers of ten. Thus the quantity given in equation (14.21) is written

lOUOlOOU =

2^ + 2^

+ 2^ + 2^ + 2 + 1

(14.22)

binary system. Strictly, 2 and its powers should also be written in binary notation, but there is no ambiguity in equation (14.22) as written. All computers will automatically translate decimal numbers into equivalent binary numbers, so the translation details need not be considered. in the

14.4.2. Floating Point Arithmetic

Equation (14.22) shows that a large number of digits are required to number in normal form. There is a more economical way of establishing an extensive number scale with a restricted number of digits and this will be illustrated in the decimal system. In the normal way, four digits can express any number in the range -^ 9999; but using three digits for a number and one digit for an exponent, the range covered can be increased to 0-1 -)• 10^. Thus express a

723

=

0-723 X 10^

(14.23)

With the convention that the number written Hes between OT and 0-999, and the fourth digit is the exponent, the number in equation (14.23) can be stored as 7233. A number written in this form is known as its *' floating point " form. Both the number and the exponent are written in the binary notation, and a standard number consisting of 40 binary digits is interpreted as follows. The first digit gives the sign of the number, the next 30 digits express the number, the next digit gives the sign of the exponent, and the last 8 digits give the magnitude of the exponent of two. Thus any number in the range

10-^^

motion (7.126) can be expressed

alternative forms

du (a)

and

V^ =

— -maC= ot

-V/ + vV^m

in the

MATHEMATICAL METHODS

522

^-«aC=-Vx-vVaC ot

(b)

49.

CHEMICAL ENGINEERING

IN

= WP + 2"^ + ^

where

X

and

F=-VQ

Two

concentric spherical metaUic shells of radii a and b cm (a < b) are separated by a solid of thermal diffusivity a cm^/sec. The outer surface of the inner shell is maintained at Tq°C and the inner surface of the outer shell at T^^'C, Derive the differential equation governing the un-

steady state temperature distribution in the solid as a function of time and radial coordinate.

Show

^l|^^%^^+ — —

r=

b

a

a

b

r

nnl(b — a).

=

where P

form

that the solution takes the

fi"sinm.-a)}e-^„

=

i

r

Show how B„ can be determined from any

initial

temperature distribution.

50.

A

cyhndrical block of metal at a uniform temperature ro°F has its The temperature of the curved surface is suddenly raised to and maintained at Ti°F. Prove that the tem-

circular faces thermally insulated.

T of any

perature

where

point in the block at any time

T.-T

__2

Ti-To

a

r is the radial

thermal

diffusivity,

-

Jo(a„r)

^t'l

oc„Ji((x„a)

coordinate, a

and

is

/ is

given by:

^.,„,^.

the radius of the cyHnder,

Jo(a„a)

is

of a tracer

is

the

= M.Sc, Birmingham

51. If a liquid

K is

a„ are the positive roots of

1962.

flowing through a fixed bed at a rate v cm/sec when a pulse added, show that the later distribution of the tracer is given

by the solution of

d^C dx^

E

where

is

infinite

the mixing coefficient which operates in the axial direction

Removing x by

(x) only.

dC_dC ~^ dx~ dt

the substitution z

= x — vt, and

assuming an

bed length, solve the above equation.

Show that the maximum concentration of tracer passing a downstream observation station arrives earlier than the residence time by an amount

Ejv'^.

{See

J. J.

Carberry and R. H. Bretton, A.I.Ch.E.J.

4, 367, 1958.)

PROBLEMS 52.

523

Considering two dimensionless flow and neglecting the inertia terms in show that the change of variables given by

the Navier-Stokes equation,

_ _d^_dilj

P

dx

dy

dy

dx

= Po-9py + PYf~^yf^y

yields the equations

dx^

^

dy""

p\d^'^ dy^)~~dt Show

that a solution of these equations takes the (l)

i/r

= Ae-^'"-"' cos Kx + My = 5e-'"^+"'sinKx

m = s/K^ + (pnlfi)

if

(This

53.

method has been used to describe flow influidized beds by W. J. Rice and R. H. Wilhelm, A.I.Ch.EJ. 4, 423, 1958.)

The steady laminar flow of a

liquid through a heated cyUndrical pipe has a parabolic velocity distribution if natural convection effects and variation of physical properties with temperature are neglected. If the fluid entering the heated section is at a uniform temperature (tj), and the wall is maintained at a constant temperature (/^), develop Graetz's solution by neglecting the thermal conductivity in the axial direction.

(The solution Trans.

54.

form

Assuming tension

Am.

Inst.

that Stokes' approximation

(y) varies

is

is

reported by T. B. Drew,

Chem. Engrs.

26, 26, 1931.)

vaUd and that the

interfacial

according to y

=

cLCOsO

+P

determine the fall velocity of a spherical droplet of a material of viscosity is the polar angle measured /Zi through a medium of viscosity 112 where ;

from the front of the droplet. Show that there is only motion inside the droplet

a^g^p where a

is

the radius of the drop

>

if

3a

and Ap

is

the density difference.

MATHEMATICAL METHODS

524 55.

CHEMICAL ENGINEERING

IN

The sudden closure of a valve generates a pressure wave within the liquid flowing in the pipe leading to the valve. The passage of this wave causes compression of the liquid and expansion of the pipe. Show that the velocity of the liquid and the pressure are related by the equations dp

p dv

dx

g dt

d'p ,d'p — ^ = c^—^

and where If

c it

the velocity of propagation of the pressure wave. can be assumed that a uniform pipe of length L connects a

is

reservoir at

P(xJ)

x =

=

Po

x

to the valve at

+

4cpvo

^ 2.

{

=

show

L,

.^nx ^sm(2n + — —Tsm(2n + 1)—

-IT

.

^.

net

.

,

7X—

l)

{G, R. Rich, Trans.

56.

that

A.S.M.E. 67, 361, 1945.)

A liquid at a uniform temperature falls as a thin film down a flat vertical surface.

Determine the velocity distribution and the steady

state thick-

ness of the film. If a short section of the wall is maintained at a higher temperature than the Uquid, determine the average heat transfer coefficient in the heated section. The velocity distribution can be simphfied in the thin layer where temperature gradients are important.

57.

Given that the values of sinh x for x

=

are 0-100, 0-150, 0-201, 0-253, 0-305 sinh

58.

X when x

=

0-1, 0-15, 0-2, 0-25, 0-30 and 0-35 and 0-357; estimate the value of

0-328.

The thermal conductivity of methyl chloride

at the temperatures 50°F, 100°F, 150°F, 200°F. 250°F, 300°F, 350°F and 400°F is 0-0057, 0-0069, 0-0081, 0-0092, 0-0104, 0-0116, 0-0127 and 0-0139; estimate the thermal

conductivity at 413°F.

59. Solve the difference equations

(ii)

ynyn+2

(iii)

>'„+4-9y„H-3 + 30>;„+2-44>'n+i+24y„

= =

yn-^i 4''a

60. Solve the difference equation

given that

^o

=

0;

>'i

=

3;

>^2

=

6;

^3

=

36

PROBLEMS 61.

525

of a carrier gas containing Yq lb solute A per lb of carrier gas ideal plates and fed into the base of a plate column containing contacted with Llb/h of solute-free liquid containing Xf^+^lb solute A

G Ib/h

N

is

per lb of solute-free liquor. If the concentration of A in the exit gas and 7^ and X^ respectively, show that the concentration of A in the gas phase leaving any plate n in the column is given by

liquid streams are

where

H

is

Henry's law constant expressed by Y^

=

HXff.

number of plates required to absorb 99% mole ketene moles/h of gas containing 4-36% ketene by volume using 138 lb moles/h of glacial acetic acid. The plate efficiency, of a 3-5 ft diameter tower containing 0-7D weirs and therefore a plate hold-up of 2-09 lb moles Hquid, is 40%. The pseudo Ifirst order reaction rate constant for the chemical absorption process may be taken to be 0-075 sec"-^ and the equihbrium relationship

62. Calculate the

from 135

lb

Y*

= 2X

where Y* is the lb moles ketene/lb mole carrier gas in equilibrium with X lb moles ketene/lb mole of acetic acid in the hquid.

A

(Problem taken from Problem of Chemical Engineering Design, by G. V. Jeffreys; published by Institution of Chemical Engineers, London, 1962.)

is to be produced in a battery of continuous stirred tank operating in series by reacting 2000 Ib/h of butanol with 326 Ib/h of glacial acetic acid containing sufficient sulphuric acid to catalyze the reaction at 100°C. Under these conditions the rate of reaction can be expressed by the equation

63. Butyl acetate

reactors

where k is 0-28 ft^/lb mole.min and C^ is the concentration of acetic acid in lb moles/ft^. If the effective volume of each tank is 10 ft^ and the density of the reaction mixture is assumed to be constant at 48 Ib/ft^, estimate the

number of reaction vessels required in the final discharge

64.

A reactant A

is

if

the concentration of the acetic acid

not to exceed 3

Ib/ft^.

N

is to be converted into a product 5 in a battery of continuous stirred tank reactors of total volume Kft^. If the feed rate is q ft^/min and the concentration of ^ is C^^q show that the production of 5 is a maximum when all the tanks are the same size. The rate of reaction can be taken to be first order and the battery is to operate isothermally.

:

:

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

526 65.

:

8000 Ib/h of a 5

%

by weight solution of acetaldehyde

in toluene

treated with 5000 Ib/h of water in a 6 theoretical stage extraction

is

to be

column

operating under counterflow conditions. If the concentration of the feed is suddenly changed to a 3% by weight solution, how long will it take for the extractor to settle down to steady state operation under the new conditions? conditions

The equiUbrium where

y= X=

relation

is

F=

2-2A^

lb acetaldehyde per lb of water \b acetaldehyde per lb of toluene.

The hold up per

stage can be

assumed to be constant

at

750 lb toluene; 300 lb water.

66.

A block of metal

12 in x 12 in x 2 in has

its

rectangular faces thermally

and its vertical square faces losing heat to the surroundings at 60°F by means of a heat transfer coefficient. It can be assumed that the temperature in the block is uniform and that the properties of the insulated

metal are specific heat c

The following

=

density p

0-10,

=

4201b/ft^

table gives the metal temperature as a function of time

Time(min) Temperature CF)

200

Determine the heat transfer

5

10

15

20

25

30

35

40

186

173

161

151

141

133

125

119

coefficient.

In addition to the experimental readings, the given values are subject to the following errors c,

Assuming

+ 1%;

p,

+

0-1%;

all

dimensions,

that the thermocouples are accurate,

heat transfer coefficient determined

±yV'

how

accurately

is

the

?

Birmingham 1961.

67.

Use the method of averages

to find the best curve of the type:

y which

68.

fits

= Asm(x + B)

the following values.

A and B

are constant.

x

0°

30°

60°

90°

120°

150°

y

0-944

1-242

1-208

0-850

0-264

-0-392

Use the method of

least squares to

Nu =

fit

the best equation of the type

aPr"

to the following data.

J

527

PROBLEMS Pr

Nu

Pr

0-46 0-53 0-63 0-74 4-2 5-6

24-8

100

26-5

17-7

3 5

Nu

Pr

Nu

84-5

32

140 127 189 245 315 380

31-6 70-3

58-5

115 115 150 170 165 193

340 590

95

245

55

28-5

18-6

300

25-3

60-3

41

69 58-4 70-7

370

93 185

480 195

{The information above was collected from various sources by W, L. Friend and A. B. Metzner, A.I.Ch.EJ. 4, 393, 1958.)

69.

leaving a batch distillation column is being continuously reprocessed so that it is difficult to measure directly the amount produced. However, the flow rate can be continuously measured and samples are taken at hourly intervals giving the following results.

The top product

Time

(hours)

More

volatile

1

90 60

ponent (%)

Flow

rate (Ib/min)

(a)

(b)

70.

2

3

4

5

6

7

8

91

91

63

62

89 62

87 60

84 59

80 57

75 57

com92 62

What is the total amount produced during the period of 8 hours ? What proportion of this total is the more volatile component ?

The reaction

rate constant for the

decomposition of a substituted dibasic

acid has been determined at various temperatures as follows.

TCO

500

A:xl04(hr-i)

Use

the

method of

in the

equation

where

T is

701

least squares to

measured

89-4 45-4

7-34

1-08

1010 138

determine the activation energy (E)

in degrees Kelvin.

71. Calculate the first four positive roots

X tan X

of

=

2

to four decimal places.

Show graphically why the method (Section 11.5.3) oscillates about the root Newton's method

yields the

answer

far

of successive approximation

and converges slowly, whereas

more

efficiently.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

528

number of plates required to absorb 95 % mole of a component A from 500 lb moles/h of a gas containing 5 % of ^ by volume using 600 lb moles/h of a 5 % mole solution of the absorbent B. A reacts with B according to a second order irreversible reaction

72. Calculate the

r If the liquid hold

= kC^Cs

up on each tray

is

4 lb moles and gaseous hold up

is

that the difference equation takes the form (9.71). The plate efficiency is 15%, k = 2 x 10^ ft^/lb mole.h, the absorbent solution negligible,

show

has the physical properties of water, and Henry's law constant l-2x 10^ moles/mole.

73.

is

A mathematical model for the washing of a filter cake has been proposed by M. T. Kuo (A.I.Ch.EJ. 6, 566, 1960). It is assumed that the wash Hquor is in plug flow through the pores of the cake, and that a stagnant film of filtrate remains on the pore walls. A mass transfer coefficient governs the passage of solute across the boundary between the two Uquid layers. Show that the process is described by the equations: A2-^ =

hS(c,-C2) del

dci

for




4a'

29 s^

/-

sin af

+ 4a^

cosh ar

—

cos at

sinarsinhaf

2a^ 31

_1

_J.

^{smh at

—

smat)

J. jCcoshflt

—

cos at)

2a 32 s

4

—a 4

2a

33

(1

+ a^/

(s'

+a

r) sin ar

ar cos a

d"

e'

34*

s

—

==6*^(1+2^0

35

(s-a)*

y/nt 1

36

—a —

\l s

\l s

—h

2V t:^^

(e^'-O

1

1

37

/--fle''''erfc(aVO

Vs + a "T^ + ae'^erfCaVO

38

Vs 39 s

+ a'

slnt

yjn

1

40 ^/'s(s-a'')

-e'^'ediay/l) a

1

41

a\ln * L„{t) is the

Laguerre polynomial of degree

n.

J

J

flrV3\

smhat

1

5

30

.

v3sin^-

X [^^^-2

APPENDIX

535

Table of Laplace Transforms {continued)

m

Ks) 2 „1 h''-a

e°''[fc-aerf(aVO]

42

-te'''erfc(6V0

1__

43

e'^^'erfcCaV?)

\/s(Vs + a) 1

44

.«^'gerf(a70-l]

45

Vs(5-fl')(Vs + 6) 46*

+ e^''erfc(feV7)

(1-5)"

nl

= i^2n(V0

(2n)ly/nt n!

47

^2„+l(V^)

-1

48t

4s 49 \/s

+ ay/s-\-b

TW

50 (s

b)t

-,(/c>0)

+ fl)H5 + ^)

.(^'-) 51 (s

te

+ ar(s + b)^

'••"-[/.("-i^')

.(^^')] 52

\ls-\-2a

— \Js -e-''h(.at)

y/s

+ 2a+y/s

7'-'"""'.(^'')

(/c>0)

53

(V5 + a + Vs + 6)2*

• /f„(x)

t /„(x)

is

the Hermite polynomial,

/f„(;c)

=

e'*^(e~'*).

= i~VJix\ where /„ is BesseFs function of the first kind.

MATHEMATICAL METHODS IN CHEMICAL ENGINEERING

536

Table of Laplace Transforms {continued)

m

f(s)

54

(Vs + g + Vs)-''^

(i;>-l)

-e-i"IXiat)

\/s\/s + a 1

55

56

Jo(at)

+ a^-sy

i^s'

{v>-\)

1

57

'

(/c>0) r(/c)

58

59

{^|7^-sf{k>0) {s-y/s^-ay

1

60

2\k

ka"

Uat)

(v>-l)

(k>0) Y{k)

{£)

^'-(^' '

< < when > k when t

t

0)

s

S-fl 101

+ kV)

103* -log(s2

104*

|(e''-0

S—

102* -log(l

-K4)

+ a')(a>0)

4log(s^ + a^)(a>0)

-0)

110

-e^'^'erfcCi^sX/oO)

111

e'''Qifcsfks{k>0)

\

-(i) 7r\/7(r

+ /c) when^O0)

124 a 125

g-s,.) COS s + Ci s sin 5

{t

+ a) 1

7V\

8

SUBJECT INDEX Associative law of Algebra, 51, 52, 206,

213 Autocode, 508 Average point, 362

Absolute convergence, 76 Absolute value of complex numbers, 75, 119 Absorption factor, 325, 326 Acetaldehyde, extraction of, 481-482 Acetic anhydride, hydrolysis of, 341-343 Acrivos and Amundson equation, 471 Addition of matrices, 439-440 Addition of vectors, 203-210 " Address " of numbers, 505 Adsorption coefficients, 365-369 Algebra, complex, 117-148; see also under complex matrix, 439-442 three basic laws, 51, 52, 206, 213

Averages, method in curve fitting, 361364 Axial symmetry, 234-236, 248-250, 276, 354-355

B Benzene, batchwise chlorination 28

—

Benzene toluene, feed column, 336-338

in

of,

26-

distillation

Benzoic acid extraction

from toluene,

1, 2, 3, 5,

9

Algebraic equations

Bernoulli's theorem, 235

by matrices, 448-449 numerical methods of solving, 397406 Alternating series, 79 Ammonia vapour, pressure and density of, 317-318 Amplifier basic for addition, 496-497

Bessel equation, 106-116, 173, 282, 283,

linear, solution

basic for integration,

287, 296, 306 integral properties of,

modified, 110 second kind, 107-109 tables of (Chistova), 317 Bessel's interpolation formula, 319 Beta function, 154-155 argument of, 154 relation with gamma function, 156157

497^98

operational, 496

Amplitude of complex number, 119 Amplitude of elliptic integrals, 157, 158 Analysis of a continuous fat hydrolysing plant, 62-66, 328-331 Analytic functions of complex variable,

Binary system of digital computers, 506 Binomial power series, 80 Binomial theorem, 80, 143, 181, 182, 340 Blockdiagram, 189, 191

131-148 Arbitrary constant, 18, 42, 45, 290, 323 Argand diagram, 118-1 19, 120, 199, 200, 399 algebraic operations on, 120-123

Boundary conditions, analogue

Argument of complex number,

119, 122, 136

gamma function,

152

tion,

1

circuit for, 494,

502

at fixed temperature, 19

beta function, 154

Ascent, steepest

15

1

Bessel functions first kind, 107

—method

constant heat flow rate, 19 initial condition, 19 in Laplace transformation, 1 66 in partial differential equations, 252-

of optimiza-

483-485

259 543

—

—

SUBJECT INDEX

544

Boundary conditions

Columns

cont.

in partial differential equations first

type, function specified,

cont.

252-

253 in Sturm-Liouville equation,

271-

272 integro-differential,

third type,

257

mixed conditions, 256-

257

Schliching, 267

Boundary value problems, 258 Liebmann's method for solution

function, 42-45, 465-

Complementary solution, 322-323 Complex conjugate, 118, 123-124 Complex number, 75, 117-128

Argand diagram

of,

428^29 method of solution

of,

429-

436 solution of higher order diff'erential

equations, 388-397 Branchcut, 137, 155, 185 Branchpoints, 134-137 inversion of Laplace transformation, 185,187-188 Branch principal, 137 184, 186

or

468

absolute values of, 75, 119 operations algebraic on diagram, 120-123 amplitude of, 119, 122, 136

Prandtl, 267

Bromwich path,

algebra, 51, 206,

divergent series, 76

Complementary specified,

thermally insulated, 19 Boundary layer theory

relaxation

Commutative law of

213, 308, 312, 315, 442 Comparison test for convergent

particular solutions suggested by,

265-267 second type, derivative 253-256

addition of, 460 interchange of, 459 multiplication of, 460

Argand

118-119 136 conjugate numbers, 123-124 De Moivre's theorem, 124-125, 323 imaginary part, 117 modulus and argument, 119 nth roots of unity, 125-126 principal values of, 1 19-120

argument

of,

of, 119, 122,

real part, 117 series, 74,

79-82, 126-127

tensors of, 200

trigonometrical, exponential identities,

128

Complex variables, 128-148, 222 analytic functions of, 131-148

Cauchy 's

Cauchy integral formula, 141, 142

theorem, 226

formula,

integral

141,

142 137,

144,

145,

186,

188,

Cauchy-Riemann conditions for derivatives of complex variables, 131, 132, 135, 138

Cayley-Hamilton theorem, 450, 456 Characteristic equation, 454-457 Chloramine synthesis, optimum conditions for, 485 Circle, uniform motion in, 218

Cauchy's theorem, 137, 144, 145, 188, 226 derivatives of, 130-131 evaluation of residues, 144, 145-147 at multiple poles, 147-148 integration of functions of, 137

Laurent's expansion, 142-145

132-147 branch points, 134-137 essential, 133-134

singularities of,

Clausius-Clapeyron equation, 318

poles or unessential, 133, 137-148 theory of residues, 142-145 Components of vectors, 204-205

Code

Computers, 492-510

Clairaut substitutions, 33

autocode of computer, 508 machine, 508 Column matrix, 438-439

active analogue, 496-505,

application tions,

to

509-510 equa-

differential

501-503

SUBJECT INDEX

Computers

cont.

active analogue

cont.

application to process control

systems, 503-504

radius of, 80

basic amplifier for addition, 496-497 basic amplifier for integration, 497-

498 function generation, 500-501 multfplication by a constant, 498 multiplication by a variable, 499-500

analogue, 492-505, 509-510 comparison of analogue with

digital,

509-510 505-510 binary system, 506

digital, 492,

floating point arithmetic,

506-507

input and output devices, 507 storage echelons, 507 sub-routines, 507-508 the program, 508-509 electrolytic tanks, 495-496 passive analogue, 493-496

Conductor transfer radial heat through, 14, 199 temperature distribution in, 199

cylindrical,

flat,

Conformable matrices, 440 Conjugate, complex,

545

Convergence absolute, 76 of complex series, 126

1 1

Convergent series, 75 comparison test for, 76 Gauss' test for, 78 integral test for, 78

Raabe's

test for,

78

ratio test for, 77

Convolution, 179-180 Coordinate systems, curvilinear, 228-231 Coordinates cylindrical polar, 231 cylindrical polar with axial symmetry, 256 spherical polar, 228-231, 248, 250 Cosine integral, 158, 539 Countercurrent extractor, 343 Counter flow extraction, optimization of, 479 Crank-Nicolson method, 422 Cross or vector product, 212-213

Cubic equation, analytical solution 398-399 Curve fitting, 360-369 least square, 364, 369 method of averages, 361-364

Conjugate numbers, 123-124 Conservation law, general, 9, 19, 220, 237, 247, 249 Constant of integration, 34, 35, 151 Constituent solution to complementary function, 466 Continuity equation, 220, 232, 250-252 Contour charts, 479, 480

Degeneracy of a matrix, 446-448 Degeneracy, single, 447 De Moivre's theorem, 124-125, 323 Density of ammonia vapour, 317-318 Dependence, linear of a matrix, 446

Contour

Derivative

integration, 137-148, 155, 156,

158, 162

partial, 218,

inversion in Laplace transform by,

182-188

Contour

plots,

Control systems automatic, 188 closed loop, 190

open loop, 190 stability,

192

stability criteria,

192-198

195-198 Control theory, automatic, 188-198 Control valve, response to varying air pressure, 176-179 stability criteria (Nyquist),

239-242

substantive, 245 total,

355-356

of,

240

Desk calculating machine, 492 Determinantal equation, 453-454 Determinants /)-, 464 and matrix of square matrices products, 443 test, 193,437 Difference(s)

backward, 311, 314, 316 central, 311, 319 first,

307

— SUBJECT INDEX

546 DifFerence(s)

Differential equations, partial, 23, 245-

cont.

forward, 311, 316 fourth, 309 mih, 309 second, 308 tables, 309-310, 316, 317, 318 third, 309 Difference equations characteristic equation, 323

306,

of,

245-252

particular solutions of, 259-267

solution by

method of images, 263-

267 solution by numerical methods, 409-

436

338-348 by numerical methods, 406-^09 finite, 321-338 degree and order of, 321-322 linear finite, 322-331 complementary solution, 322-323 particular solution, 323-326 non-linear finite, 331-338 analytical, 335-338 graphical solution, 331-335 Riccati finite, 336, 337 simultaneous linear, 327-331 Difference formula, Newton's, 314-318 differential,

solution

superposition of solutions, 261-263 Differential equations, second order, linear,

33,41-66, 86-116

auxiliary equation

complex roots equal roots

43-45 43-44

to,

to,

unequal roots to, 43 complementary function, 42-45 illustrative problems of, 59-66 particular integrals, 45-59 inverse operators, 50-56 undetermined coefficients, 46-50 variation of parameters, 56-59 particular solution of systems of, 468-

471 solution by numerical methods, 385-

Difference operator J, 307, 308 £•,312-314

397 solution by series, 86-116

Difference quotient, 314-315 Differential equations, 23-73, 86-116, 163, 169-179, 245-306, 380-397, 409-436, 463-472, 501-503 application of active analogue to, 501503 Bessel's equation, 106-116

complementary function of system 465-468

409-436

formulation

solutions of systems of,

equations,

Differential

non-linear,

dependent

465^71 second

order,

33^1

variable

explicitly,

not

occuring

34

homogeneous, 35^1 problems of, 59-66

illustrative

of,

independent variable not occurring

order and degree, 23 solution by conversion to equivalent system, 464-^65

equations, simultaneous, 66-73, 465-472 Differential operator, 50-56 application to exponentials, 52-53 use in eliminating dependent variables

non-linear, 23

solution by matrices, 463-472

standard methods of solving, 24 with variable coefficients, solution by Laplace transformation, 170-175 Differential equations, first order, 24-33 exact, 24-25 homogeneous, 25-28 linear, 28 use in heat transfer, 30-33 solved by integrating factors, 28-33 solution by numerical methods, 380385

explicitly, 34, 35

Differential

from simultaneous differential equations, 67,68 Bessel's of properties Differential equation, 115 Differential total, 240

Differentiation of matrices, 450-451

Differentiation of vectors, 216-218 Differentiation process, interpretation of, 18,

238-240

Diffusion equation, linear, 295

1

SUBJECT INDEX

Error function, 58, 149-151, 187, 264

Diode, 500

properties of, 151

Diode function generator, 501

Errors, propagation of, 356-360

Dirac delta function, 175 Distillation apparatus, 6 Distillation

column

feed composition, 355 plate efficiency of, 336-338 Distillation

variation

rate,

with feed

rate, 8

Distributive law of algebra, 51, 206, 213,

308,312,315 Dittus-Boelter equation, 392

Divergence, 220 Divergent series, 75

comparison

test for,

Exponential function, 74, 127

76

test for,

optimization, 486-491

Efficiencies,

355

Eigen-functions, 276, 283, 295 not orthagonal, 283-289 Eigen-values, 276, 454 Electrode, graphite temperature profile,

38^1, 389-391 Electrolytic tanks,

through addition, 356-357 through general functional relationship, 359-360 through multiplication and division, 357-358 through subtraction, 357 Ethyl alcohol, esterification of, 332-333 Ethyl benzene, dehydrogenation of, 412-420,422-428 Euler's constant, 108, 109

78 integral test for, 78 Raabe's test for, 78 ratio test for, 77 Dot or scalar product, 211 Droplet, detailed concentration distribution around, 248 Duct, rectangular, velocity ratio of flow through, 430-436 Dummy variable, 149, 151, 156 Dynamic function, 1 89 Dynamic programming method of

Gauss'

547

495^96

Elliptic integral

of first kind, 157 of second kind, 158 Elimination, systematic in simultaneous differential equations, 68-72 Enthalpy, concentration diagram, 209,

210 Error absolute, 356

of measurement, 360 of method, 360 of precision, 360 relative, 357

identities of, 128

transform of, 166 Exponential integral, 159, 539 Exponential power series, 80, 127, 313 Extraction of benzoic acid, 1-6, 9-1 of nicotine, 343-348 Extraction plant, dynamic programming, 486-491 Extrapolation, 315-318

Factor, integrating, 29 Feedback, 191, 496-498 Fictitious points, 417, 428 Finite differences, 307-348 difference quotient, 314-315 equations of, 321-348 of a product or quotient, 311 operator -J, 307-308 -E, 312-314 -Norlund,314,391 second and higher orders of, 308-309 tables of, 309, 310, 316, 317, 318 Fin, temperature distribution of, 100-103 Fixed bed catalytic reactor, 171-175, 412-420, 422-428 Floating point, 506-507

Fluid flow equations

of, 231-236 235-236 ideal round sphere, 265-267 in packed column, 285-289 Stokes approximation, 232-235

ideal,

use of continuity equation for compressible fluid, 250-252

—

1

SUBJECT INDEX

548 Fluid flow

Gauss'

cont.

round sphere, 267-269 with axis of symmetry, 234

viscous,

Geometrical 206-210

Fluid property (temperature), 244 Force field, 224 conservative, 223 Forces, triangle of, 203 Fortran, 508 Fourier integral theorem, 183 law, 20

transforms, 303-305 Frobenius method, 90-109, 277, 298, 492 Functions, orthogonal, 269-289 Functions, potential, 223, 236 Functions and definite integrals, 149-162

function beta, 154-155

convergent or divergent 278 applications of vectors,

test for

series, 78,

Gradient, starting of graph, 354-355 Graphical location of roots of algebraic equations, 399-400 Graphical solution of non-linear finite diff'erence equations, 331-335 Graphite electrode, rate of flow of heat from, 389-391 electrode, temperature profile of, 38-41 thermal conductivity of, 363 Graph paper, 349-354 linear, 350-351 logarithmic, 352 semi-logarithmic, 351 triangular, 352-354 Green's theorem, 227-228

error, 149-151

evaluation of B{\-q,

q),

155

gamma, 151-154 between

relationship

beta

and

H

gamma, 156-157 functions,

tabulated defined by in-

tegrals,

157-159

integral(s)

evaluation of, 159-162 elliptic of 1st kind, 157 of 2nd kind, 158 Laplace transform of, 170-175 sine, cosine and exponential, 158definite,

Gamma function,

107, 151-154, 162 Laplace transforms, 164

relationship with beta function, 156-

157

Gas absorption application of series to, 84-86 in falling film,

297-302

Gas absorption column performance

eff'ect,

500

multiplier, 499

Halogenation of hydrocarbon, 26-28 Hamilton's operator, 200, 218-222, 229

Hankel transform, 305 Heat conduction equation boundary conditions, 19, 252-257, 258 cylindrical polar coordinates, 256

spherical polar coordinates, 276 steady state, 200 steady state linear, 199

159

in

Hall

of,

325-326

transient operations, 407-409

Gas compressor, minimum consumption of energy of, 487^89 Gas heater, tubular, exist gas temperature, 103-105 Gauss' divergence theorem, 226 Gauss' method of numerical integration, 373-377, 380

unsteady state, 236, 237, 244-245 unsteady state linear, 246-248 Heat conduction solutions axial symmetry, 14-16, 276-281 rocket motor coolant duct, 392-397 steady state with axial symmetry, 276281

unsteady state linear, 259-265, 273276, 284-285, 290-295, 410-420, 422-428 Heat content of semi-infinite slab, 262 Heat exchanger, 245 Heat flow, sign convention in, 20

Heat loss from fin, 100-103 in simple water

still,

8

through pipe flanges,

1 1

1

SUBJECT INDEX Heat transfer

coefficient, 19,

549

Interpolation formula

256

319

Heaviside, 163

Bessel's,

Hermite polynomial, 535 Hydrocarbon, halogenation

Lagrange's, 319-321, 457 of,

Newton's, 315-318

26-28

Hydrolysis of acetic anhydride, 341-343 of animal fat in spray column, 62-66,

32B-331 of tallow in spray column, 6266 Hyperbolic and trigonometric inverse functions, 81

Hyperbolic functions, 81, 293 for half integer order of Bessel's identities of, 128 series, 8

I

Images, method

of,

166, 167-170, 179-188 by contour integration, 182-188 convolution integral, 179 in expansions descending power series, 181

further elementary

methods

of,

180-

182 other series expansions, 182 solutions

ot

differential

equations,

169-170

equation, 116

Hyperbolic power

Inverse transformation,

in

solving partial

263-267 Index law of algebra, 308, 313, 315 Indicial equation, 91-100 differential equations,

using partial fractions, 168-169 using properties of transformation, 180 when singularities are poles, 185 when singularity is a branch point, 185 when there are both poles and a branchpoint, 187-188 Isothermal surface, 219 Iteration,

420-428

Bessel's equation, 106

roots different, not integer, 91, 100 roots differing by an integer, 96, 98, 100

Jet laminar,

roots equal, 93, 100

Jury problem, 258

Infinite series,

J

dynamics

of,

38

74-79

Influence equations, 430

Input and output devices in digital computers, 507 Integral test for convergent or divergent series, 78 Integral properties of Bessel's equation,

Kettle, closed, heating of, 16

Kirchoff's Law, 493 Kremser-Brown equation,

325, 326

115 Integrals

area and surface, of vectors, 224

Lagrange multiplier, 483

222 volume, of vectors, 224-225

Lagrange's interpolation formula, 319-

line,

Integration of vectors and scalars, 222-

227 Integration, numerical, 369-379

Integration of matrices, 452 Initial value,

problems,

166,258,302 step-by-step

solution,

method

of

410-428

type equations, solution by numerical

methods, 385-388 Interpolation, 314, 315-321

321,457 Laguerre polynomial, 534 Lambda matrices, 452-454 Laplace equation, 236, 272, 276, 279, 281,428,435,495 Laplace operator, 163-167 Laplace transformation, 142, 157, 163198, 290-302, 338-348 application to automatic control theory, 188-198 control systems, 190-198

—

—

SUBJECT INDEX

550 Laplace transformation convolution, 179-180

Mass

cont.

transfer

cont.

diffusion equation, 273

differentiation of, 170, 173

equation, 233

use of to find inverse transform, 179-180 inverse transforms, 167-170, 180-188, 303 solution of differential equations,

integro-differential

integral,

169-170

boundary

idealization of, 2

using partial fractions, 168-169 method of solving differentialdifference equation, 338-348 method of solving partial differential equation, 290-302 of derivatives, 165-166 of integral of a function, 1 70

representation of,

1

Matrices, 199, 210, 437-472 addition, 439-440

444

adjoint,

application to solution of differential

equations, 463-471 augmented, 449

properties of, 170-174

characteristic equations,

166-167 step functions, 174-179 staircase function, 175-176 unit step, 174-175

column, 438 commutable, 442 conformable, 440 D-, 464 degeneracy of, 446, 447

shifting theorem,

unit impulse, 175

use

of,

con-

257 sign convention in, 20 with axial symmetry, 248-250 Mathematical principles, 1-22 dition,

determinantal equation, 453-454 determinants of square matrices and matrix products, 443 diagonal, 439

176-179

table of, 532-541

transforms, 163-167, 258 Latent roots, 453-459, 462-472 distinct, 453, 458,

454-457

466

differentiation

multiple, 453, 458, 466

Latent column vector, 454 Laurent's expansion, 142-145, 146, 148 Legendre's equation of order " / ", 277 Legendre polynomials, 278-280 L*H6pital's rule, 61, 82, 147, 175, 181,

262,293,416 Liebmann's method for boundary Value

and

integration

of,

450-452 equivalent, 460-461 lambda, 452-454 linear dependence of, 446 matrix algebra, 439-442 matrix series, 449^51 minors of, 446-447 modal, 467

problems, 428-429, 508 Logarithmic power series, 80

multiplication, 440-442

Loop

null,

non-singular, 443

439

closed control system, 190 open control system, 190

orthogonal, 443 polynomial, 450-451

open transfer function, 196

powers

of,

449-450

quadratic form, 461-463

rank

M Magnet,

field strength

by iron

filings,

218 Many- valued functions, 129, 133, 134, 136 Marching problem, 258

Mass

447^48

446 reciprocal of a transposed matrix, 446

row, 438 singular, 443

solution of linear algebraic equations,

transfer

by molecular

of,

reciprocal of a matrix product, 445 reciprocal of a square matrix, 444-

diffusion,

237

448-449

—

SUBJECT INDEX Matrices

cont,

solution of systems of linear differential

equations,

465^71

square, 439

sub-matrix, 448 subtraction, 440 Sylvester's theorem, 457-459 transformation of, 459-461 unit,

439

Matrizant, 452 Medians of triangle, concurrency, 207-

208 Mellin-Fourier theorem,,! 83, 184, 188 Mellin transform, 306 Method of formulation, summary,

Numerical methods, 380-436 algebraic equations, 397-406 analytical solution of cubic equation, 398-399 comparison of methods, 406 graphical location of roots, 399400 improvement of roots by successive approximation, 400-402 Newton's method, 402-406 difference-differential equations, 406409 step-by-step method, 407^09 first

order ordinary differential equations,

higher

21

Minor of a matrix, 446-447 Modal columns, 467 Modulus of complex number,

Modulus of

finite

differential

equations,

boundary value type, 388-397 119, 122,

difference equation,

411

Mole fractions, 325

Moment

of a force, 212 Multiplier hall effect,

380-385

order

385-397

134, 157

499

mark space, 499 quarter square, 499 servo, 499

Murphree

551

plate efficiency, 84

Multi- valued functions, 129, 133, 134, 136

N Navier-Stokes equation, 232-237, 265, 430, 435 Neumann form of Bessel's equation, 108, 109

initial value type, 385-388 use of Taylor's theorem, 387-388

420-428 Liebmann's method for boundary value problems, 428-429 partial differential equations, 409436 Picard's method, 380-382 relaxation methods for boundary value problems, 429-436 Runge-Kutta method, 382-388, 390391, 406-407 step-by-step method, 407-409, 410428 Null matrix, 439 Nyquist criteria of stability in automatic control systems, 195-198 Nyquist diagram, 196 iteration,

Newton formulae, 314-318 method of solving algebraic equations,

402-406 second law of motion, 201, 232 Nicotine, extraction of, 343-348 Norlund's operator, 314, 391 Number(s) absolute value of, 75 " address " of, 505

complex, 75, 117-128 conjugate, 123-124 imaginary, 117

Offset, 192 Oilfield, exploitation of,

295-297

Oliver, settling velocity of suspensions,

403 Operator diff'erential,

50-56, 504 307-308, 311-315

finite difference,

Laplace, 163-167 Optimization, 473-491 analytical procedures, 475-483 involving 2 variables, 475 involving 3 variables, 475-482

—

—

—

1

SUBJECT INDEX

552

Optimization cont. analytical procedures cont. involving 4 variables, 482 with are strictive condition, 482^83 dynamic, 474 dynamic programming, 486-491 method of steepest ascent, 483-485 sequential simplex method, 485-486

474 types, 474 Optimal policy, 487

Partial differential equations

cont.

compounding independent

variables

one variable, 259-261 conclusions, 306 continuity equation, 250-252 formulating, 245-252 Fourier transforms, 303-305 into

Hankel and Mellin transforms, 305306

static,

images, method of, 263-265 initial value type, 258-259, 290-302,

Optimum curtailed, 475

410-428 420-428 Laplace transform, 290-302 step-by-step, 410-420 mass transfer with axial symmetry, 248-250 numerical methods, 409-436 orthogonal functions, 269-272 Sturm-Liouville equation, 270-272 other transforms, 302-306 particular solution suggested by boundary conditions, 265-267 particular solutions, 259-269 separation of variables, 272-289 iteration,

operation, stages for, 478-482

Optimization chart, 481 Origin in graph, 354 Orthogonal functions, 269-272 Orthogonality property, general use of, 283 Orthogonal, eigen-functions not, 283289 Oscilloscope, double beam, 493, 503, 505 Osculating plane of curve, 218 Over-relaxation, 434

eigen-functions

orthogonal,

not

283-289 Parameters, 17-18 of Beta function, 154 use in evaluation of definite integrals, 159 use in Laplace transformation, 163,

168,195,290,504 use in partial derivatives, 239 use in sine integration, 158 variation of, 56-59 Partial

derivatives,

interpretation

of,

239-245 Partial differential equations, 245-306,

409^36 boundary conditions, 252-259 1st type,

2nd

function specified, 252-253

type, derivative specified,

253-

256

equations involving variables,

independent

3

281-283

general use of orthogonality property, 283 steady state heat transfer with axial

symmetry, 276-281 unsteady state linear heat conduction, 246-248, 273-276 superposition of solutions, 261-263 Partial differentiation, 239-245 changing independent variables, 243 between independent relationships variables, 243-245 relationships between partial derivatives, 240-242

—

use in inverting transforms, 168-169

Partial fractions

3rd type, mixed conditions, 256-257

Particular integral, 45-59

integro diff"erential, 257

Particular solution, 323-325, 468-471

boundary value

type, 258-259,

438, 508

Liebmann's method, 428-429 relaxation method, 429-438 solution by computer, 508

428-

Partition of matrix, 448

Picard's

method of solving

differential equations,

1st

380-382

Pipe flanges, heat loss through, 1 Poisson's equation, 43 435, 495 1 ,

1

order

1

SUBJECT INDEX Ratio

Polar coordinates cylindrical, 231 spherical, 228-231

in

order, 133, 155

complex

variables, 133, 141-148

inversion of Laplace transform, 185,

optimum volume

187-188 second order, 133

diff"usion

and chemical

Recurrence relation, 93-109, 277 Relaxation methods for boundary value problems, 429^36, 508

series,

Laplace

transformation, 181 Powers of matrices, 449-450 Prandtl's " boundary layer theory ", 267

Residual, 430 Residual balancing, 434 Residues, 144, 156, 293, 302 evaluation of, 145-147

evaluation at multiple poles, 147-148 theory of, 142-148, 169, 182-188

Response

Pre-multiplication, 442

Pressure distribution in

yield,

reaction in 59-61

Potentiometer, 498-505

by

maximum

412^20,422-428 simultaneous

Potential function, 223, 236

79-86 descending, expansion

for

327-328, 474 Reactors, tubular accuracy of rate of reaction, 359-360 dehydrogenation of ethyl benzene in,

Point conductivity (graphite), 363 Polynomial functions application of inverse operator, 55 Hermite, 538 Laguerre, 534 matrix, 450-451 Ponchon-Savarit method, 209 Post multiplication, 442

Power

convergent or divergent 77 Reactors, tank concentration of effluent from, 469-47 esterification of ethyl alcohol, 332-333 hydrolysis of acetic anhydride, 341343 test for

series,

Poles first

553

oilfield,

295-297

Problems in chemical engineering, 59-66 selected, 511-531 illustrative

Product, inner, 441 Program of computers, 505, 508-509 Propane, thermal conductivity of, 320321 Pythagoras, theorem of, 205

Quadriatic form of matrices, 461-463 semi-axes and principal planes

contours, 484 function, 483 surface, 483

transform 1 89

Reynolds number, 233, 235, 269 Riccati finite diff"erence equation, 336,

338 Right-hand screw rule, 212, 214, 221, 224 Rocket motor cooling duct, temperature distribution of, 392-397 Rodrigue's formula, 279 Routh-Hurwitz criteria of control system stability,

of,

461-463

Routh's

192

rules, 193

Row matrix, 438 Runge-Kutta formula, 384 Runge-Kutta method of solving

Raabe's

convergent or divergent 78 Radial heat transfer through cylindrical conductor, 14 Radius of convergence, 80, 90 Rank of a matrix, 446 Rank of a tensor, 200 test for

1st

order diff"erential equations, 382388, 390-391, 406-407

series,

Saddle point, 476 Salt accumulation in tank, 11-13

SUBJECT INDEX

554 Scalars,201,203 field, 218

Singularities

or dot product, 211, 224 point function, 218, 223, 229, 232 triple product, 215

branch points, 134-137, 185 133-134 of complex variables, 132-137 poles, 133, 141-148, 185, 187-188 Slide rule, 492

variables, 232

Solvent extraction, 1-6, 9-11, 343-348,

integration of, 222-227

essential,

Separation constant, 275, 276, 282 Sequential simplex method of optimiza-

485-486 74-116 alternating, 79

478-482 Sphere(s)

boundary condition

tion,

application in chemical engineering,

83-86 Bessel's equation, 106-116, 173, 282,

287 example, heat loss through pipe flanges. 111

of half-integer order, 116 functions of negative order, 116 modified, 110 properties, 113-116 comparison test, 76 convergent, 75 divergent, 75 Gauss' test, 78 indicial equation, 91-100 functions

infinite,

74-79

properties of, 75-76 integral test, 78

matrix, 449-51

power, 79-86 descending expansion, 181 Raabe's test, 78 ratio test, 77 simple solutions, 86-90 solution by method of Frobenius, 90105 Shifting

theorem,

166-167,

180,

182,

504 Sign conventions, 19-21 in fluid flow,

in heat flow, in

at

porous surface,

257

Series,

20 20

mass transfer, 20

Significant figure, 360

Simpson's Rule for numerical integration, 371-373, 380, 385, 420, 434 Sine integral, 158,539 Single valued function, 129, 130, 133, 134, 136

ideal fluid flow round, 265-267 laminar flow round, 435 thermal conductivity of insulator between two concentric metallic, 279-281 viscous flow round, 267-269 Spherical particles, settling velocity of suspensions of, 403-406 Stagewise counterflow system, 344 Stagewise processes, generalized vector

method for, 208-210 Stars, irregular,

Step-by-step

of

429

method of solution

diff"erence-diff"erential

equations,

407^09 of initial value problems, 410-420 Step functions, 174-179 staircase function, 175-176 unit impulse, 175 unit step, 174-175 use of, 176-179 Stokes approximation, 232-235 stream function, 234, 236 theorem, 137,225-226 Storage echelons, 507 Store backing, 507 inner (core), 507 main (drum), 507 rapid access, 507 Stream function for fluid flow, 236 Stokes', 234 Stress tensor, 201-203, 437 Sturm-Liouville equation, 270-272, 275 279, 283, 288 Sub-routine programs in computer, 507508 Subsidiary equation, 169

SUBJECT INDEX Substantive time derivatives, 245 Successive approximation, improvement of roots of algebraic equations by,

400-402 Sulphuric acid, acid discharge tempera-

68-72

ture in cooling,

Summing junction, Suspensions

of

496, 503 spherical

Sylvester's

particles

403^06

settling velocity of,

Theorem, 450, 457-459, 471

555

Transfer function, 189, 193, 196 in Nyquist diagram, 197 Transformation of matrices, 459-461

Transforms Fourier, 303-305 Hankel, 305 Laplace, 163-168 Laplace, elementary, 165 Laplace, table of, 532-541 Mellin, 306

response, 189

Transpose of a matrix, 443-444 Tallow, continuous hydrolysis of, 62-66, 328-331 Tank system, output from, 338-341 Tartaglia's method of solution of cubic equation, 398 Taylor's theorem,

9, 10, 21, 82, 147,

221,

238, 246, 247, 249, 251, 255, 262, 313, 358, 391, 402, 429, 469, 483,

495 direct use of,

387-388

Temperature surface variation with time in

rectangular slab, 263, 291 total time derivative,

245

variation in all directions, 253

Temperature distribution in semi-iniinite slab, 260, 291 in transverse fin, in wall, 264-265,

100-103 294-295

of rocket motor cooling duct, 392-397 with time in rectangular slab, 284-285, 293 Tensors, 200-203 first rank (vectors), 201 variation

second rank (stress), 201-203 rank of, 200 zero rank (scalars), 201 Test determinants, 193 Test functions, 193

Thermal conductivity, 236 of graphite, 363 of insulator, 279-281 of propane, 320-321

Thermal

diff'usivity,

237, 248, 249

Time increment eff'ect

effect

9

of in salt accumulation, 12 of in unsteady state operation,

matrix, reciprocal of, 446

of product of matrices, 443-444 ofrow vector, 443 Trapezium rule for numerical integration, 370-371 Treatment of experimental results comparison of methods, 377-379 contour plots, 355-356 curve fitting, 360-369 method of averages, 361-364 method of least squares, 364-369 graph paper, 349-354 linear, 350-351 logarithmic, 352 semi-logarithmic, 351 triangular, 352-354 numerical integration, 369-379 Gauss' method, 373-377 Simpson's rule, 371-373 trapezium rule, 370-371 propogation of errors, 356-360 through addition, 356-357 through general functional relationship, 358-360 through multiplication and division, 357-358 through subtraction, 357 sources of error, 360 theoretical properties, 354-355 efficiencies, 355 starting gradient, 354-355 the origin, 354 Triangle, medians concurrent, 207-208 Triangle of forces, 203 Trigonometric functions application of inverse operator, 55 extension of exponential operation, 53,110

—

—

SUBJECT INDEX

556 Trigonometric functions

cont.

identities, 128 in half-integer Bessel's equations, 116 solution by digital computers, 507 Trigonometric and hyperbolic inverse

functions, 81

Trigonometric power Triple product scalar, 215 vector, 215-216

series, 81

u Unit matrix, 439, 442 Unitvectors, 204, 214,240 divergence of, 230 Unity,

Unsteady

vectors

roots of, 125

/2th

state operation, 8-13, 68,

198, 246, 343, 407,

Vector analysis cont. Green's theorem, 226 Hamilton's operator, 218-222 ideal fluid approximation, 235-236 scalar triple product, 215 standard identities, 227 Stokes' approximation, 232-235 Stokes' theorem, 225-226 tensors, 200-203 of first rank (vectors), 201 of second rank, 201-203 of zero rank (scalars), 201 transport of heat, mass and momentum, 236-237 vector triple product, 215

192-

469

addition and subtraction

of,

203-210

and scalars integration of, 222-227 and scalars, line integrals, 222-223 components, 204-205 curl or rotation of, 220-222, 229

Vapour-liquid equilibria, 244 Vapour pressure of ammonia, 317-318 Variable(5)

complex, 128-148 dependent, 17-22, 23, 25, 30, 34, 252, 307, 321, 354 elimination from different equations,

66-73

differentiation of,

216-218

divergence of, 219-220, 229 geometric applications, 206-210 multiplication of, 210-216 partial differentiation of, 218 position 205-206 properties of addition, 206 scalar or dot product, 211

dummy,

149,151, 156 independent, 17-22, 23, 25, 30, 34,

unit vector relationships, 214

vector or cross product, 212, 213

volume

252, 307, 321, 354

integrals,

224-225

boundary value or Jury problem,

Velocity potential, 236

258 changing, 243 closed range, 258

Viscosity, coefficient of, 201, 232 Viscous flow round sphere, 267-269 Vinyl chloride, synthesis of, 171-175 Vorticity distribution, 269, 355 Vorticity of fluid element, 222, 233, 234, 235, 236

compounding

into

one

variable,

259-261 elimination

from

differential

equations, 67

equations involving three, 281-283 initial value or marching problem, 258 open range, 258 method of separation of, 272-289 not truly independent, 243-245 Vector analysis, 199-237 curvilinear coordinate systems, 228231 equations of fluid flow, 231-236 Gauss' divergence theorem, 226

w Water-Still, simple with preheated feed,

6

Weber form of

Bessel's equation, 108,

109

Work, done by

force during displace-

ment, 211

z-plane, 129, 132, 134, 145, 155

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