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and Rate-Limited Isotherms With First-Order Loss. 2. An Approximated Simulation After a Pulsed Input. KEITH R. LASSEY. Institute of Nuclear Sciences, Lov•'er ...
WATER RESOURCES RESEARCH, VOL. 24, NO. 3, PAGES 351-355, MARCH 1988

UnidimensionalSoluteTransportIncorporatingEquilibrium and Rate-Limited

Isotherms With First-Order

Loss

2. An ApproximatedSimulationAfter a PulsedInput KEITH R. LASSEY Institute of Nuclear Sciences,Lov•'erHurt, New Zealand

I developandillustratea simpleapproximation to thesimulated concentration historyat a positionin a unidimensional flowpathdownstream of an idealized(Diracdeltafunction)pulseinput.The simulation modelincorporates combinedFreundlichand rate-limitedisotherms as previously formulated.The approximation circumvents the numericalintegrationwhichthe simulationwouldotherwise require;it is particularlyreliableeitherfor largeP•cletnumberor for minorkineticinfluence uponexchange or for both,andit is superior to invokingthe"localequilibrium assumption" asan approximant. INTRODUCTION

dispersioncan be ignored (infinite P•clet number) or (2) sorption is everywheresufficientlyrapid instead of rate-limited as •paper 1")presented analyticalsolutionsto the unidimension- to be locally equilibrated (the "Freundlich limit," FL), the at dispersion-advection (DA) equation, with sorption con- cumbersomeintegral becomesanalytically integrable, thereby trolledby a rate-limited LA isotherm (LA after Lapidus and significantlyeasingthe computerizedsearchfor an optimum

Thepreceding paper[Lassey,thisissue](henceforthcitedas

Aroundson) in tandem with a linear Freundlich isotherm. The latterisothermadds no further complicationto the mathemat-

parameter vector.

The objectiveis to developand exemplifyan approximation to the aforementionedintegral which is accuratein either the •s, beingmanifestedonly in a retardation factor R. The solutions werepresentedfor a pulsedinput flux into a semi-infinite nondispersiveor the Freundlich limits but which remains usefulwhen one or both of these situations is only approxiflowpath,and,by superposition, for a more generalinput. mately realized. This is accomplishedafter a scrutiny of the In thispaperI investigatesimulationsof the dispersantconintegrand concerned. centration downstreamof a pulsedinput, or, equivalently,the This objective is similar to that of Jennings and Kirkner simulated distribution of transit times along a flow path. I referto such simulations at a particular position as a con- [1984] and that of Bahr and Rubin [1987], hereinafterreferred centrationhistory after Bahr and Rubin [1987], a plot of to as B&R. B&R examined the DA equation with LA isotherm (and with alternative isotherms)writing in the form of •vhich wastermed a C diagram by Danck•,erts[ 1953]. /he terminology and notation of paper 1 is adopted the FL plus correctionterm (the "kinetically influencedterm," throughout, and equations from that paper are succinctly or KIT}. B&R term this approach the SKIT method ("separacitedas,for example,"(1-19)" (equation (19) of paper 1). The tion of the kinetically influencedterm"), and in their parlance dimensionless positionand time are denotedX and T (equa- the FL is the "local equilibrium assumption"(LEA). In the tion{1-4}},and P denotesthe longitudinalPficletnumbercor- notation of'paper 1 the SKIT equationis an elimination of Q responding to downstreamposition X -- 1. It is appropriate to from (1-1), (1-2): employrescaleddimensionlessrate constantsto complement therescaling (1-5) of positionand time from •, z in paper 1. •nX2 •,cx:ordingly define

. 72 (-;

•1' = Pt•'I

•c2' -- P•-'2

A* = PA

(1)

• C=R}:t.+A*C+KIT (3)

KIT = (x•* + •:2')- tRvI•

+ A*

Therescaled pulsestrength Fo* = Fo/P

a A*1? •)C

(2)

has a useful interpretation as a concentration scale;RFo* is Loss terms were absentfrom the B&R analysis,and •:•*, •2' thetracerconcentration whichwouldresultfrom distributing are just their Damk6hler numbersDal, Da2 whosesum (the thetracer uniformly throughout themobilefluid. B&R "prefixdenominator") determines the importanceof the /he simulated concentration history(I-32), (1-33)involves KIT. B&R investigatethe parameterdomains in which the anintegral (ofa modifiedBessel function)whoseevaluation is KIT can be ignoredso that the LEA approximatesthe origrequired at a succession of times;eachevaluationrequires inal equations,at leastfor a stepfunctioninput. multiple integrand evaluations to effectnumerical integration. INTEGRAND BEHAVIOR

Fitting a measured breakthrough curvethenrequires iterative

recomputation of the simulatedhistory in order to optimize

The simulated concentration history downstream of a

•a ß unknown parametervector.However,in the eventthat (1) pulsedinput flux of strengthFo* takesthe form (see(1-32), il-33))

Copyright 1988 bytheAmerican Geophysical Union.

Ce(PX,PT)/Fo* = P exp(-tct*T)fl(PX, PT) + W(X, T)

l•aper number 7W4843. (•3-1397/88/007W-4843505.00

(5• 35!

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scales of relative error.

The secondterm is the contribution of multiple adsorption-

in the limit P-,, cc (see(1-39)),therebysimplifyingthenondis-

desorption'

persivelimit of W(X, T).

In the FL (namely,•.'•*, •.-:*--• oc,with finite * •,*• siderablesimplificationalso occurs(see(1-40)):the leading term in (!) vanishes, and the integrand(2) againapproaches Dirac 6 function,thistime peakedabouty = y•, where

W(X, T)=P•'2* f•fl(PX, Py) exp (-az - a2)azl•(a•a 2) dy (6)

y,, = T/REL

with a•, a2 dependentuponthe integratingvariabley: ax = •'•*y

a2 = K2*(T -- y)

(7)

Here I• is the modified Besselfunction (1-31), and the choice of concentration measure (resident or flux) determines the form of the dispersivespread(DS) functionfl of (1-35). In writing down (5)-(7) I have dropped the overall multiplier exp (--A'T), effectivelysetting A* = 0.

As the P•clet number increases,so Pfl(PX, Py) becomes increasinglypeaked{abouty = X), with peak width vanishing

RFL= (}C•*+

2 , 2

The existence of thispeakis evidentfromtheBessel funct},c• ':• asymptote

l•(Z)z_. • -+«rr-•/2Z-3/,,exp(2Z•'2} for thenthe integrand in (6) is dominated by thestrong½ peakedGaussianfactor

exp { - [(•'• ,y)•/2whose width vanishes in the FL.

(Ke,(T- y)11/212•

L•.•'•.

L!NIDIM•,NSR•N^L S(•LUlt:,TRANSP(•R r, 2

353

Fig. 2. As for Figure 1, but for P = 100,; = 0.1, 1.0.

My approximation is motivatedby integrandbehaviorin

with

•hose nondispersive and FL extremes;where one or both of d•ese extremesis approached,the integrand exhibitsa strong

peakaboutan intermediatevalue of y. For sucha strongly pe•ed integrandit becomesan increasinglygood approxi•.mti.on to extract from the integral the slowly varying terms, evaluating them at the location of the peak •the "Laplace a:pproximation" [Erdklyi, 1970]). I adopt this strategyand nu•,•r'.•a!lyexamineits adequacy.

•2 •. •j f •- (.V}, • • -1 • -.•lp.F-3 +

; = •'• ,.,•2 ••2

(14} (15)

•, = [;•T -

In the nondis•rsive or Freundlich limits, the •ak width a vanishes'when either P or • is merely large, • measur• the narrownessof this Gaussianlike •ak. My approximation is the leadingterm of an asymptoticexpansionof the integral for

thelargeparametera-2.

T•E APPROXIMATION TO W(X, T)

For brevity, denote the integrand {6• as

Withoutlossof generality,selectX = 1 (i.e., x = L). Theasymptote{9} dominatesthe Besselfunction behavior evenfor modest arguments, Z >• 10. Thus, except for the

:•ma!test ,•aluesof •,'•*, •:2' and of elapsedtime T, behaviorof rotegrand {6}is dominatedby exp [-f(y)], in which

•{•'} 1} 2.+ [[&.,.y], . =P(Y-• 43' 2 [•c2.{T_ y)],,2

;T2ff:- 3}-1

{1l)

W=

•3'} exp I-f

{y}] dy

Performing a changeof integrating variable y•

u2 =.flY)-.!'(;)

{.16}

u, where

sign{u)= sign(y- .f)

and extracting all nonexponential thctors out of the integral tevaluatingthem at u = 0}, producesthe approximation

Thefirstterm or "P6cletcomponent"of {11) either emerges

ilrom fl•4P,Py} or is rapidlyapproachedby fla(P, Py) at large Pec•tnumbers and/orsmally. The secondtermor "isotherm

G(y} dy =

W m1•.;2} t 2oGle}{1 + err [{•f}t,.2]}

{17}

where

component" of ( 11} is from ( 10}and peaksat y = 3'•.

&/'=./'{TI -J}•} {18} It ca.n easilybe shownthatf {y}possesses a singlestationary •nt• a minimumat y = 3•whichliesbetween1 and y• aswe!! This result does not depend upon the validity of expansion aswahinthe integrationrange [0, T]. That minimumis 1o- 113} throughout [0, T]. With the err function replaced by c'•tedv,'here

unity, t17• is the leading term of the well-known Lapla• asymptotic expansion[e.g., Erd•lyi, 1970]. The erf function f'(.•} = 0 (12) assuresunit•rmity t•r f--• T i i.e., •'2' >>•* -• '• }. •h•chcanbe solvednumericall)'for )?(e.g.,see,appendix). The desiredapproximationis thus

When thespreadof transittimesisdominated by kineticinfiuP;} exp[-,fl.•}]d•i•{•2th2} e•.ce•, theP6cletcomponent determines ), whichisnearunity; W(1,T} = {x/2}• •aP•'2*(•SP, conversely, when transit times are spreadpredominantlyby ' I1 + err [{•,f}• 2]• {19}

:hydrodynamic dispersion (asat theFL},theisotherm compo;=m• thedominantinfluenceon ), whichis near y•.

in which

A MacLaurin expansion of./'(3'}aboutits minimumpro-

f(Y) ='f{);:) +• o- +O[ly-.•}3] (13}

(20)

i•(Z) = exp( - 2'Z'•')[•(Z)

354

LASSEY'UNIDIMEN$iONALSOLUTETRANSPORT,2

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Fig. 3. As for Figure 1, but for •c• /•c2 = 0.1, 10 and ( = 1, 100.

in (6) is evaluatedvia an adaptiveSimpson's rule.Thepractiasymptotic expressions for fl and it, so that the products(20) cal evaluation of(19) is addressedin the appendix. naturally appear in practical numerical algorithms. Note that The approximationis generallyreliableto withinabout The exponential functions in (20) exactly cancel those in the

the product 0.to(P,P.½)is nonsingular at .f • 0. It can be verified that the approximation (19) converges uniformly to the nondispersiveand Freundlich limits for .• 1 and .½--•y•, respectively. A NUMERICAL

SURVEY

or better wherever 0'/3•< 0.1. The somewhat erratic accuracy for T < 1 in some cases arises from the role of the leading

termin (5),whichis dominantat earlytimes;theduration dominance increases withP but decreases with •cx*.It isquite impractical to citeboundsfor a reliableapproximation inthe three-dimensional parameter spaceof •cx*,•c2',P; fortunately, 0'/.f seemsa usefulindicatorof reliabilityand is in fact

Figures1-3 displaysimulatedconcentrationhistoriesfor Cr and for various combinations of P, •c•*, •c:*. Errors incurred by-productcalculationof the approximationitself. by employingthe approximation(19) are also displayed,toIn general, the approximation is dependable for P > 100 getherwith valuesof j5and 0.(as j7/T and 0'/35). It is instructive and/or for• > 100(atleastfor•:x*and•c2'within a factor 10). to record the analytical result a

A comparison of theutilityof myapproximation withthat

P

c3'-• (37/T) = 2T2(c/37) 2

of B&R

(21)

whichguaranteesthat 37/Tis a decreasingfunctionof T. All calculationsutilize rational approximationsboth to the modified Besselfunction [Blair and Edwards, 1974] and to the complementaryerror function [Hart et al., 1968]. The integral

is instructive.

B&Rconsider a stepinputofstrength Coandasa criterion foraccuracy of theirLEAselect thetimesnapshot atwhich the concentration historypassesthrough0.•Co. The accura•

withwhichtheconcentration at thistimeisreproduced by

LœAapparently depends upon•:•*+ •:* andP.Accuracy

LASSEY:UNIDIMENSIONAL SOLUTETRANSPORT, 2

a5thin 2% at P = 1 and •c•*+ •c2'> 65,but thislowerlimit Then increases toward•- 240asP increases to 100.ThustheLEA is }ess reliable asP increasesrathe opposite ofmyapproximation whose reliabilityhascomparable boundsfor P--- 1 and im-

355

p[¥r2(y - 2 _ 1)+ sz3]

•>{S) =

•T2+ Pz3

{s•m-* g, •,•"•-, •) via proves asP increases. Thereason forthedivergence atlarger Iterationproceedsto convergence

?canbeattributedto thefollowing. The LEA (beingmy FL)

rsequivalent to myapproximation in which37isreplaced by land, insignificantly, the errorfunction replaced by unity). At eachiteration a numericallystabley is computedfrom s as However, my approximationexplicitlypermits.9 to move

away from3k toward1 as the P•cletcomponent off(y) assumes moreimportance.Thus my approximation explicitly

w = T(v2 + 4)-•,2

improves upontheLEAin cases wheretheP6clet component

z = •(T/w +/') = 2/(T/w- v)

isasignificant or majorinfluence on•.

y= w/z CONCLUSIONS

The expression chosenfor z dependsupon the signof v.

Theoft-appliedmodelof unidimensional solutetransportas considered in this paper (and in its predecessor) presentsan

3. At convergenceg = •{•, whereupon

g = kVlf -2- 1) integral expression astheconcentration downstream of pulsed f(g) is thenproportionalto g•. The error funcinput. Thepractical evaluation of thisintegralat eachsimula- The exponent tiontimerequiresnumericalintegrationand thereforemulti-

tion argumentin (19} is evaluatedvia the practicalexpression

pieintegrand evaluation. In thispaperI havedeveloped an approximation of wideapplicability andpredictable reliability

af= • 2Tal+•r

whichcircumventsnumerical integration, replacing the inte-

gral witha single integrand evaluation pluserrorfunction. Theapproximationis usefulparticularlyfor large P&let numbers (P >• 100) and/or rapid exchange(large Damk6hler

Acknowledgments.This paper has benefited from critical ap-

praisalsby colleaguesFreemanCook {NZ Soil Bureau,DSIR) and Grahame Weir {Applied Maths Division, DSIR}.

numbers •c•*,•'2')- It shouldalwaysimprovethe "localequilibriumassumption"[Bahr and Rubin, 1987] as an approximant,becausethe former selects an asymptotically correct

REFERENCES

parameter (.•) whereasthe latter adoptsa near-identical for-

Bahr, J. M., and J. Rubin, Direct comparisonof kinetic and local equilibriumformulationsfor solutetransportaffectedby surface

centration history.

Danckwerts, P. V., Continuous flow systems:Distribution of resi-

reactions, Water Resour. Res., 23, 438-452, 1987. mulawitha nonoptimalparameterchoice(37replacedby Blair, J. M., and C. A. Edwards, Stable rational minimax approxiTheapproximation shouldparticularlybenefitcurvefitting mations to the modified Besselfunctions Io(X) and I•(X), Rep. which requiresiterative recomputationof the simulatedconAECL-492& At. Energyof Can. Ltd., Chalk River, Ont., 1974.

dencetimes, Chem.En.q.Sci., 2, 1-13, 1953. APPENDIX: COMPUTING THE APPROXIMATED

W( 1, T)

Theapproximation (19)can be evaluatedfor time T by the following numericalstrategy:

1. Aninitialguesstimate f{o)of f satisfies the appropriate •unds.

2. Equation{12) is solvediterativelyfor y startingwith

½% Thefollowing Newton-Raphson strategy hasprovendependable. Define the iterating function

½(s)= s - 0(s)/½'(s)

½(s)=f'(y)

M•ere sisa convenient iteratingvariabledefinedin termsof y

Erddlyi,A., Uniform asymptoticexpansions of integrals,in Analytic Methodsin MathenmticalPhysics,editedby R. P. Gilbert and R. G. Newton,pp. 149-!68, Gordonand Breach,New York, 1970. Hart, J. F., E. W. Cheney,C. L. Lawson,H. J. Maehly, C. K. Mesztenyi, J. R. Rice, H. G. Thacher,and C. Witzgall, ComputerApproxilnations, SIAM Seriesin AppliedMathematics, editedby R. F. Drenick, H. Hochstadt, and D. Gillette, John Wiley, New York, 1968.

Jennings,A. A., and D. J. Kirkner, Instantaneousequilibriumapproximationanalysis,d. ttydraul. En.q.,110, I700-I717, !984. Lassey,K. R., Unidimensional solutetransportincorporatingequilibrium and rate-limited

isotherms with first-order loss, 1, Model con-

ceptualizations and analyticsolutions,Water Resour.Res.,this issue.

K. R. Lassey,Institute of Nuclear Sciences,Private Bag, Lower Hurt, New Zealand.

$= [•h.'l*)l:2g 4'(K'2*) 1'2][(h.'l*)l/2/g -- (K2*) 1!2] z = [(T -- y)/y]

{ReceivedMarch 2, 1987; revised October 26, 1987:

acceptedNovember4, 1987.}