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Approximate Operational Calculusin Chemical- EngineeringSTUART W. CHURCHILL

University of Michigan, Ann Arbor, Michigan

Approximations are suggested to extend the usefulness of operational calculus in solvingboundary-value problems of interest to the chemical engineer. General approximationsare outlined and specific ones illustrated. The use of computing machines with operationalcalculus is R SOonsidered.

Operational calculus has been founduseful in many characteristic problems inchemical engineering. The applicationsconsidered by Marshall and Pigford 6) ndthose noted in recent annual reviews (8 t o1 1 ) include diffusion, conduction,convection,distillation, kinetics, and process control.Unfortunately, present operational tech-niques are not always adequate for the veryproblems for which, in principle, operationalcalculus holds the most promise. Theselimitations are aggravated by the notoriousdifficulty encountered by the mathematicianin establishing proofs in operational calculusfor procedures that he suspects to be valid.Approximation can be used t o surmountsome of the limitations in operationaltheory. In addition to permitting solutionof complex and otherwise intractableproblems, approximation may indicateuseful simplifications that are not apparentfrom an exact solution. The use of approxi-mation has been rather slighted in theliterature of operational calculus, presum-ably because of the taint imposed by lackof rigor. Although numerous specificexamples having a physical or mathematicalbasis have been reported, only McLachlan(7) and Doetsch 3) have attempted togeneralize the use of approximation, andthen only for the limiting conditions of longand short times in transient problems.Since the purpose of this paper is toacquaint the chemical engineer with thepossibilities and advantages of a mathe-matical technique, rather than t o developnew mathematics, all proofs and mostdetails will be omitted. Approximationswill generally be justified on physicalgrounds. The engineer is seldom disturbedby his inability to establish rigorously thevalidity or uniqueness of a solution if itsatisfies physical tests.Consideration will be limited to theLaplace transformation, which is the mostextensively used form of operational cal-culus. The use of approximation with othertransforms is also advantageous, andequivalent proceduresare readily developed.THE LAPLACE TRANSFORMATION

If a function F ( t ) is multiplied by e-8:and integrated with respect to t fromzero to infinity, a new function, f ( s ) , isobtained. This operation is called theLaplace transformation of F ( t ) , and f(s) iscalled the Laplace transform; i.e.,3 ( F ( t ) ]= I r n e - F ( t ) d t = f(s) (1)

0

Th e inverse operation is designated

Operational calculus was developedempirically by Heaviside as a set of rulesof procedure. Subsequently a n extensivetheory was developed, and the mathe-matical foundations, limitations, andformal applications are well treated inmodern texts 2, 7 , 14 . Although trans-forms have been derived and tabulatedfor many functions and mathematicaloperations (4 ) ,problems of interest moreofkn than not yield functions not to befound in such tables.

Letting the transform variable, s, bea complex variable permits expression ofthe inverse transformation in terms of th e

Linear ordinary differential equationswith constant coefficients are thus trans-formed into algebraic equations, solutionof which followed by inversion thenyields a solution to the differential equa-tion. Other operational properties permitsimplification and solution of a few morecomplex types of differential equations.The transformation of simple partialdifferential equations yields ordinarydifferential equations, which may beeasier to solve. The solution of theordinary differential equation involvinga transformed function may possibly besolvable by a second transformation.Multiplicat ion of th e transforms of twofunctions corresponds to a particularintegration

complex inversion integral. P 1 < \

where

y = a sufficiently large, fixed value of x.This proper ty permits solution of certainintegral equations as well as additionaldifferential equations.

s = x + i yThe theorv of residues and line integrals The operationcan be used to evaluate the foregoingintegral. In many cases the problem isreduced to one of finding the singularitiesin f s ) . where

S ( F ( t - m)}= e - m s f ( s ) (8)The complex inversion integral can bewritten as a real integral,

where u = real part of f s), and v =imaginary par t of f(s), but in this formthe integration is generally too difficultto be performed analytically.The m ost important operational prop-erty of the Laplace transformation arisesfrom the derivative

S ( F ' ( t ) ]= s - F(+O) (5)Differentiation of F ( t ) hus corresponds to

F ( t ) = 0 for t < 0permits solution of certain finite-differ-ence equations and finite-difference-differential equations such as thoseencountered in equilibrium-stage opera-tions.From these examples, several limita-tions on the usefulness of the operationa lmethod are apparent or implied:1. The functions encountered in theequations and boundary conditions mustbe transformable.2. The transformed equations andboundary conditions must be solvable.3. It must be possible to invert thetransformed solution.

multiplication of the transform- f(s) bys and subtraction of F(+O). Equation (5)in turn leads to an expressioll for thetransform of the nth derivative:The restrictions On thetransformation are not very serious andare satisfied by most functions encoun-tered in engineering problems. On theS ( F ( t ) = s f(s) - -'F(+O)

- Sn-*F (+O) - - F -'(+O)other hand, t h e class of equations whichare simplified by transformation is ratherlimited. W ith such equat ions the relative

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advantage of the operational method overother classical methods depends largely onthe na ture of the boundary conditions.In particular, if the values of the de-pendent variable to be transformed andits derivatives a t t = O are known, theoperational method is a t least worthlooking into. Thc simplification gainedby transformation is oftcn negated bydifficulties in inverting the transformedsolution.

integral, Equation 4), an be evaluatedgraphically, numerically, or by quadra-ture, as suggested in Problem 111.6 . Approximate values may be usedfor the singularities when residue theoryis used to carry out the inversion. Thisapproximation is not used in any of theillustrative problems,

The corresponding exact solutions are

e - k ~c, = k,C [-----, k , +1k , - k , (21)ILLUSTRATIVE PROBLEMS

Four problems were chosen to illustratethe use of approximation with theLaplace transformation. The equationsin the problems are streamlined by theimplicit inclusion of physical factors andconstants in generalized variables.Although the greatest utili ty of approx-imation is in complex problems thatcannot be solved rigorously, relativelysimple problems were deemed best forillustration. In several cases the natureand validity of the approximations areapparent by direct comparison with arigorous solution. The techniques usedin these simple problems are of coursedirectly applicable in more complex ones.

APPROXIMATE INVERSIONThe inversion process may be simplifiedor expedited by the following methods,one or more of which are applicable inmost problems.1. The transformed function may beexpanded in series. The entire series maythen be inverted or appropriate termsmay be discarded before inversion. Fre-quently several different expansions arefeasible, yielding solutions or indicatingapproximations of different utility. Theuse of series expansion is illustrated in

Problems I, 11, and 111, which follow.2. A complicated function may berepresented empirically by some arbi-trary function which is more readilyinverted, as indicated in Problem 111.This method of approximation can seldombe justified mathematically but is worthtrying if the solution can be tested.(Similarly, boundary conditions andgenerating functions may be approxi-mated by other functions more susceptibleto transformation, as illustrated inProblem IV.)3. The inversion may be carried outonly for particular or limiting values of aparameter, as indicated in Problems Iand 111, or of a nontransformed inde-pendent variable, as indicated in Problem11. This procedure may be expressedmathematically as

andc, = c, - 1cn-,kn-, . k,C,

e - k , t.[l kn-l-kl) k~,-~-kl). ( h - k , )+ . . I 23)In the following approximations s

will be treated as a real variable.Short-time Approximation

For very short times 1st will be verylarge with respect to all the rate con-stants. Then

Problem IJeffreys 5) has indicated the use ofapproximation i n the solution of t he se tof simultaneous rate equat ions describinga radioactive decay series. He also dis-cusses the physical interpretat ion of t heapproximate results in some detail. Thegeneral problem is represented by theequations

C,c1 -+(25)

d c i-- = -klCl with C, = 6, at t = 0d t(12)

S

_ _ -dCz - -k,C, + klC,d t (13)_-C3 - -k ,C, + k2C2 (14)d t

etc., and finally

4. Approximate inversions for largeand small values of t can sometimes beobtained by letting s assume small andlarge values, respectively; i.e.,F ( t )E S-'(f(s)) (10)t -m 8 0and

for the end product, which does notdecompose.Transformation yields the correspond-ing equations Moderate-time Approximation

Expanding the transforms in series bydivision yields a useful form for moder-ately short times corresponding to Is1greater than any k . ThusOne or both of these approximations areillustrated in Problems I, 11, and 111.Equation (10) is not valid if any of thesingularities in f(s)occur in th e half of thecomplex plane where the real value of sis positive. Thus formal use of thisapproximation may lead to error; how-ever, it is usually easier to detect anerroneous solution than to find thesingularities.5 . The real form of the inversion

a.nd

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.[ - k l + IC2 + k J + .. ] 34)s(35)

and6, = c, 1 - k , t + - -2

- Icltr + ][ 3

k 2 k lCotc, = ___2

k,_,k,-Z klCotn--ln - l) (39), =

1- ( k , + k + . k n - d t + .nThe first terms of t he series solutions arethe very short-time solutions. The seriesthemselves are obviously those obtainedfrom series expansion of the exactsolutions. Thus in this problem a short-time approximation of any desiredaccuracy can be obtained by letting/ s [ o and dropping higher order terms.Long- t ime Appr oximatio n

For long times s will be small withrespect to some of the rate constants .If it is assumed th at all the rate constantsexcept kz are much larger than Is1C,k ,c1 and C, -0 (40, 41)

and C, + C 0 C k a t (42, 43)

k Ck3and C , e-k2r (44, 45)

into Equation (48) and simplifying givesk2COcn ---f s(s + k2) - G 1 - 7 )

u ( r ,s) [l snd Cn -+ CO[l- - k t ] 46, 47) S4;Equation (41) does not follow directlyfrom Equation (40) but is apparent fromEquation (12) or (20).These approximatesolutions can, of course, be obtained fromthe exact solution. Thus the formalprocedure used to obtain the approxima-tions is substantiated in this case eventhough it is not in general rigorous.Problem II

The representation of a transformedsolution by an asymptotic series toexpedite inversion or to yield a moresatisfactory solution upon inversion hasbeen used widely. The following examplefrom Carslaw and Jaeger ( I ) also illus-tra tes inversion at a single value of theuntransformed variable.The transformed solution for thetemperature in a long cylindrical rod,initially at zero temperature but main-tained a t unit temperature a t the surfaceafter time zero, can be written in gener-alized terms as

The details of the derivation of Equation(48) are given in reference 1. The exactsolution is

(49)

Although all the terms of Equation (51)could be inverted, only the first termneed be considered for short times if ris not too small:

Short- time Appro ximation for CenterPutting r = 0 in Equation (48) andthen introducing the positive exponen-tials of the asymptotic series only in thedenominator gives

] (53)l -a - -For short times, only the first term needsto be inverted and

1/stT(0, g efi 1/4 ;;) (54)Prob lem

Sleicher and Churchill (19) used anumber of techniques to invert th e follow-ing expression for the transient tempera-ture of a sphere in a dispersion of spheresexposed to a radiant flux

(55)

The details of the derivation of Equationwhere a, are the roots of Jo a,)= 0.Short- time Appro ximatio n

Equation (49) is inconvenient fort < 0.02 because the series convergesvery slowly. For short times and large sonly the positive exponentials of theasymptotic series

are significant. Introducing these terms

( 5 5 ) and the physical significance-of thevariables are given in the foregoingreference.Short-time Approximation

For large s and L > 1(56)1s ( l + i+%(s) --

Equation (56) is also obtained by lettingL -+ 00 which corresponds to reductionof the problem to one of a single spherein an infinite medium.Integral Inversion

The inverse transform of Equation (56)can be written in terms of t he real integral

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The integral can be evaluated graphicallyfor a series of values of t and 4. Accurateevaluation a t small values of t / d isdifficult owing to a sharp peak in theintegrand at g = 2 /1 - (I/%$). Thisdifficulty can be minimized by rearrang-ing Equation (57) as

or by taking a mean valuc of the expo-nential at the peak; i.c.,T Ei 1 - exp{ - (1 - k)} 59)As an example of the difficulty that canarise from an improper approximation,it will be noted that neglecting l/+ withrespect to 2 in the denominator of theintegrand produces a zero in the denomi-nator at y = 1 and thus makes theintegrand infinite.

Series InversionFormally carryingindicated in Equation

and inversion gives

out the division(56) yields

quite unsound mathematically, bu t whichleads to a useful and verifiable result. Duecaution should be exercised in using thisformal procedure because a very smallerror in representing f(s) may lead t o amuch larger error in F ( t ) .Long-time Approximation

For long times and L >> 34, as s+ 0and

3tT + - L3The form of Equat ion (64) and hence ofEquation (63) can also be rationalizedphysically.General Approximation

It is apparent that the expressionu = - 1- + 4-)

s 1 + + s L syields asymptotic solutions for both longand short times provided that > 400and L3 >> 34. Direct numerical compari-son shows that Equation (65) differsfrom Equation (55) by no more than S ,even in the intermediate range of realvalues of s where both the long- and short-time terms are appreciable. Inversion ofEquation (65) yields

Again an approximation which is unjusti-fiable mathematically appears to give aphysically acceptable solution.

- 1 Problem IVTransient conduction in a semiinfiniteslab with radiation and convection fromthe surface and generation of energy dueto a chemical reaction at the surface canbe used to illustrate the approximation ofboundary conditions. The boundary-value problem can be written. I (61)

Unfortunately, Equation (61) convergesslowly except for very small t .Parametric Approximation

Practical values of the parameter 4were found to exceed 400. For+ > 400,4 s never more tha n 2.5% of (1 +4s) for all real values of s fromzero to infinity. If s neglected in(1+6 +s), the inversion is readilyperformed and gives

(62)= 1 - ff/@Equation (62), which can also be ob-tained by physical reasoning, is foundto differ from Equation (57) by less than3% for all t at + = 400. This is anexample of a formal procedure which is

h[T,- TI + c ( T a 4- T4)+ A e - B / TdTdxk - = 0 at x = 0 (68)

and2 = 0 at, t = 0 (69)

Approximate Representation of Radiat ionreplaced by a linear expressionThe awkward radiation term can be

h,[T, - 7 1 = n [ T a 4- T4 ] (70)where

h, = d + TI[T,' + T 2 ] (71)

For any limited range of T the variationof h, will be slight and a mean valuc canbe selected which leads to only a slighterror.Approximate Representation of Generation

Similarly, over a moderate range oftemperature the exponential expressioncan be replaced by a linear expressionile-"?' a + b7 (72)with the coefficients a and b chosenempirically.

Solut ionAfter introduction of the approximateboundary conditions the problem can berewritten in t he following simplified form:

(73)T = 0 at = 0 (74)

T , - T + z = O at Z = O (75)where

dT

Z = ( h + h , - b ) ~ / k (76)i= ( h + h , - b) dtlkpc (77)

and

The rewritten problem can be trans-formed and the transformed problemsolved t o give7Toe- :z

1)1c = (79)Equation (79) is then inverted to give

MACHINE INVERSIONA general card program has beendeveloped ( I S ) for the inversion of the

Laplace transform of functions whichcan be expressed as a rational algebraicfunction a ( s ) / b ( s ) with a numerator ofany order up t o fifteen and a denominatorof any order up to sixteen. A furtherrestriction is th at the poles of the functionbe of the following types: first-orderpoles, real or complex; second-order poles,real only; and first-, second-, or third-order poles a t the origin. The processdescribed requires that the poles beknown in advance. However, the rootsof such equations can also be found toany desired degree of accuracy by routinemachine computation, and so the entireoperation can be programmed.

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