Analytical Solutions of the One Dimensional Convective-Dispersive Solute Equation

7/28/2019 Analytical Solutions of the One Dimensional Convective-Dispersive Solute Equation

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^nKedstatesiAJl)Departmentofgriculture AgricuituraiResearchServiceTechnicalBulletin Number1661

AnalyticalSolutionsoftheOne-DimensionalConvective-DispersiveSolute TransportEquation


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ABSTRACTM.h.anenuchtennd..lves.982.nalyticalSolutionsfhene-Dimensionalonvective-DispersiveoluteTransportEquation..S.DepartmentofAgriculture,TechnicalBulletinNo.1661,51p.Thisompendiumistsvailableathematicalodelsndassociatedomputerrogramsorolutionfhene-dimen-sionalonvective-dispersiveoluteransportquation.hegoverningransportquationsncludeermsccountingorconvection,iffusionndispersion,ndinearquilibriumadsorption.nsomecases,heeffectsofzero-orderproduc-tionandfirst-orderdecayhavealsobeentakenintoaccount.Numerousanalyticalsolutionsofth egeneraltransportequationhavebeenpublished,bothinwell-knownandwidelydistributedjournalsndnessernowneportsronferenceroceed-ings.histudyringsogetherhemostommonfhesesolutionsinonepublication.Somefheistedolutionshavebeenpublishedpreviously.Manyothers,however,werenotavailableandhavebeenderivedtoakeheistfolutionsmoreomplete.ser-orientedFORTilANVcomputerrogramsfeveralnalyticalolutionsandonenumericalsolutionaregiveninanappendix.listofLaplacetransformsusedtoderiveth eanalyticalsolutionsisprovidedalso.Keywords:altovement,oluteransportmodels,nalyticalsolutions,equilibriumadsorption,degradation,con-vective-dispersiveransport,aplaceransforms,boundaryconditions,miscibledisplacement.

Document D e l i v e r y Servis BranchUSDA, N a t i o n a S A g r i c u l t u r a l L i b r a r y6 i h F l o o r , NL B I d g 1 0 3 0 1 S a i ^ m o r e B i v dB e i t s v i l l e , MD 20705-2351


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C O I ^ f T E N T S1 . Introduction2 Thegoverningtransportequation3 Initialandboundaryconditions 4 Listofanalyticalsolutions

A.olutionsfornoproductionordecay B.olutionsforzero-orderproductiononly 7C.olutionsforsimultaneouszero-orderproductionandfirst-orderdecay 6

5.ffectofboundaryconditions 06.otation 67 Literaturecited 8

4

8. AppendixA. TableofLaplaceTransforms 029 AppendixB. Selectedcomputerprograms 08

IssuedJune19 82


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AnalyticalSolutions oftheOne-DimensionalConvective-DispersiveSoluteTransportEquationByM .Th.vanGenuchtenandW. J.Alves^ 1 . i r T R Q D U C T I O NTherateatwhichachemicalconstituentmovesthroughsoilisdeterminedbyseveraltransportmechanisms.hesemechanismsoftenactsimultaneouslyonth echemicalandmayincludesuchprocessesasconvection,diffusionanddispersion,linearequi-libriumdsorption,ndero-orderrirst-orderroductionandecay.ecausefheanyechanismsffectingolutetransport, ompleteetfnalyticalolutionshouldeavailable,notonlyforpredictingactualsolutetransportinth eieldutlsoornalyzingheransportechanismsthemselves,forexample,inconjuctionwithcolumndisplacementexperiments.Thispublicationlistsmathematicalmodelsandseveralcomputerprogramsorolutionfhene-dimensionalonvective-dispersiveoluteransportquation.umerousnalyticalsolutionsofthisequationhavebeenpublishedinrecentyears,bothinwell-knownandwidelydistributedscientificjournalsandinlesserknownreportsandconferenceproceedings.hispublicationbringstogetherth emostcommonofthesesolutionsinonepublication.Severalofth elistedsolutionshavebeenpublishedpreviously.Manyothers,however,arenewandwerederivedtomaketh elistofsolutionsmorecomplete.ser-orientedFORTRANIVcomputerprogramsofseveralanalyticalsolutionsaregiveninanappen-dix.llprogramsweresuccessfullytestedonanI BM370/155computer.urthermore,esultsofeachprogramwerecomparedwithresultsasedonanumericalolutionofhegoverningtransportquation;hisasoneoheckherogrammingaccuracyofeachsolution.ard-deckcopiesofallcomputerprograms,ncludingthoselistedinappendixB,areavailableuponrequest.

^Researchoilcientistndesearchechnician,respectively,.S.alinityaboratory,500lenwoodrive,Riverside,Calif. 92501.


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2 . H E G O V E R N I N G TRANSPORT E Q U A T I O N T heartialifferentialquationescribingne-dimensionalchemicalransportnderransientluidlowconditionsstakenasgi(6D1 -qc)- Oc+ P S)=yêc+y^ps-Y6- Y3 P [1]

wherecisth esolutionconcentration(ML"^), isth eadsorbedconcentrationMM"^),isheolumetricoistureontent(L^L*"^), isheispersionoefficientL^T"^), shevolumetricluxLT"^),isheorousmediumbulkdensity(ML"^), sheistanceL),nd simeT).hecoefficients nd\ireateonstantsorirst-orderdecayinth eliquidndolidphasesfth esoil(T~^).hecoefficients ndyepresentimilarateonstantsorzero-orderproductionmth etwosoilphases(ML~^T~ândT"^,respectively) Theolutionf1]equiresnxpressionelatingheadsorbedoncentrations)ithheolutiononcentration(c).everalypesofmodelsforadsorptionorionexchangeareavailableforthispurpose,uchasequilibriumandnon-equilibriummodels.nthisstudyonlysingle-ionequilibriumtransportisconsidered,andth egeneraladsorptionisothermisdescribedbyalinear(orlinearized)equationofth eform

s=kc 2]where snmpiricalistributiononstantM""^L^).Substitutionof[2]into[1]gives

iêreth eretardationfactorR isgivenbyR = + pk/e, 4 ]

andwithth enewratecoefficientsyandygivenbyy=y^+ygPk/6 5]Y=Y +YgP/e 6 ]


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Whenth eolumetricmoistureontentndth evolumetricfluxremainonstantnimendpacesteady-statelow),hetransportequationreducesto.2 , ,P ,0c oc ^OC_ -j.X

where =q/9)ishenterstitialrore-waterelocity.Equation7],rtsppropriateimplifications,asfoundwidespreadapplicationinsoilscience,hemicalandenviron-mentalengineering,ndwateresources.omefheknownapplicationsncludeheovementfmmoniumornitratensoils(Gardner1965,eddyetal.1976,israandMishra1977), pesticidemovementKayandlrick967,anGenuchtenandWierenga974),hetransportofradioactivewastematerials(Arnettetal.976,uguidandReeves977),hefixationofcertainironandzincchelates(LahavandHochberg975),ndth eprecipitationanddissolutionofgypsum(Kemperetal.1975,Glasetal.1979,eislingetal.1978)rothersalts(Melamedetal.977).ransportequationssimilarto7 ]avealsobeenappliedtosaltwaterintrusionproblemsncoastalaqui-fersShamirandHarleman966,othermalandcontaminantpollutionofriversandlakes(Cleary1971,homann1973,aronandWajc1976 iToro974),ndtoconvectiveheattransferproblemsngeneralLykovandMikhailov1961^arslawandJaeger(;959) 3 . INITIAL ANDBOUNDARY CONDITIONSThisompendiumgivesnalyticalsolutionsf7 ]ubjecttovariousinitialandboundaryconditions.hegeneralinitialconditionis

c(x,0)=f(x)t=0)8 ]where(x)anakenseveralforms: constantvaluewithdistance,nxponentiallyncreasingrecreasingunctionwithx,orasteady-statetypedistributionforproductionordecay.wodifferentboundaryconditionscanbeappliedatx= : first-orconcentration-typeboundaryconditionofth eform

c(0,t)=g(t)x=0)9a]orathird-orflux-typeboundaryconditionofth eform^Theyearinitalic,whenitfollowsth eauthor'sname,referstoLiteratureCited,p.98.


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-D11+ve=Vg(t) (x=0)9b]whereg(t)lsocantakeonseveraldistributions,uchasaconstantvalueintime(continuousfeedsolution),apulse-typedistribution,rnxponentiallyncreasingrecreasingfunctionwithtimeotethat9b]doesleadtoconservationofmassnsideasoilcolumn,hereas9a]ayleadtomassbalancerrorshenppliedoisplacementxperimentsnwhichheracerolutionisnjectedt prescribedrate.Theseerrorscanbecomesignificantforrelativelylargevaluesofth eratio(D/v).Forth elowerboundary,th efollowingconditioncanbeapplied

1^( > , t )-0. 10a]

Thisonditionssumesheresencef semi-infiniteoilcolumnhennalyticalolutionsasednhisoundaryconditionresedoalculateffluenturvesromfinitecolumns,omerrorsayentroduced*nlternativeboundaryondition,nehatssedrequentlyoris -placementtudies,isthatofazeroconcentrationgradientatth elowerndofth ecolumn:If (L,t)0 10b]

whereListh ecolumnlength.hiscondition,whichleadstoacontinuousconcentrationdistributionatx=L,hasbeendiscus-sedextensivelyinth eliteratureWehnerandWilhelm956,Pearson1959^anGenuchtenandWierenga1974^ear1979).nourpinion,olearvidenceexistshat10b]eadsoabetterdescriptionofth ephysicalprocessesatandaroundx=Lthan10a]oreover,oundarycondition[9b]doesleadtoadiscontinuousconcentrationdistributionatth ecolumnentrance(x0)nd,such,eemsoontradictheequirementfhavingtohaveacontinuousdistributionatx=L.Inthisstudy,wepresentanalyticalsolutionsforbothlowerboundaryonditions[10a]nd10b]).ecausefherelativelysmallinfluenceofth eimposedmathematicalboundaryconditions,th eanalyticalsolutionsforasemi-infinitesystemshouldrovidelosepproximationsornalyticalolutionsthatareapplicabletoaphysicallywell-definedfinitesystem,especiallyoraboratoryoilolumnshatreotooshort.


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Boundarycondition10a]annotbeappliedtoEq.7 ]orth eparticularcasehen i=0andy>0helowerboundarycon-ditionforasemi-infinitesystemthatissubjecttozero-orderproductiononly(nofirst-orderdecay)is

Ca x( o o , t )=finite. [10c]Table ummarizeshevariousmathematicalmodelsorwhichanalyticalsolutionsaregiveninth enextsection.hegov-erningequationsandassociatedinitialandboundaryconditionsaregroupedintothreecategories:ategoryA,wherethegov-erningtransportequationasoproductionnddecayerms(y= i =0);ategory,orero-orderroductionnly(y^0;i =0);ndategoryC,orimultaneouszero-orderproductionndirst-orderecayy^0,i^0).opecialcategoryisgivenforthosemodelsinwhichth etransportequa-tionhasnly first-orderecayermy=0;i t0).heanalyticalolutionsorheseasesollowmmediatelyfromthosefategory byimplyuttingy=0inhevariousexpressions.similarreductionfromcategoryCtocategoryB,byassuming i =,ismathematicallynotpossiblebecauseofdivisionsbyzero.

Table1.SummaryofmathematicalmodelsforwhichanalyticalsolutionsaregivenGoverningEquation

%t \ 2 %x

Initialconditionf(x)l

Upperboundarycondition

Case Type2 g(t)3Lowerboundary

condition**AlA2A3A4A5A6 \C 2

Cidododo(0


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Table1.SummaryofmathematicalmodelsforwhichanalyticalsolutionsaregivenContinuedGoverningEquation

a t=D ^ * ' - V ^0x2 ^xUpperboundary

Initialconditioncondition

LowerboundaryCase f(x)l Type2 g(t)3 condition**A7 C,+C,e- 1 Cpulse) doA8A9 ^ ~do~Ci 3 ~do~1 C C . "3 do - \ t

~do~Semi-infinite.AlO -do~ do-All ~do~ 1 ~do~ Finite.Al 2 ~do~ 3 do doGoverningEquation

- ,C .c Ct 2 xxB l NA^ 1 Co Semi-inifite.B 2 o 3 do oB3 o 1 o Finite.B4 o~ 3 do doB5 Ci 1 CQ(pulse) Semi-infinite.B6 dS 3 do doB7 o 1 do Finite.B8 o 3 do doB9 ST-ST^ 1 o Semi-infinite.BIO do 3 do oBU do 1 o Finite.B12 do 3 do. doB13 î 1 C Cê " ' ' ' â bo Semi-infinite.B14 o 3 o~B15 do 1 do Finite.B16 o 3 do o

Seefootnotesatendoftable.


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Table1.SummaryofmathematicalmodelsforwhichanalyticalsolutionsaregivenContinuedGoverningEquation

^t XC

UpperboundaryInitialCondition

conditionI LowerboundaryCase f(x)l Type 2 g(t)3 condition**

C l NA^ 1 Co Semi-infinite.C2 do~ 3 -dl- doC3 do 1 do Finite.C4 do 3 do doC5 Ci 1 C-(pulse)5 Semi-infinite.C6 dS 3 do doC7 do~ 1 do Finite.C8 do 3 do doC9 ST-ST7 1 do Semi-infinite.CIO do~ 3 do doCll ~do~ 1 do Finite.C12C13 doî 31 __doCCê " ^dodoSemi-infinite.C14 do 3 do-C15 do 1 do Finite.C16 ~do~ 3 do- do

^f(x)inequation[ 8 ] .2'!'forafirst-typeboundarycondition(equation[9a]);' 3 'forathird-typeboundarycondition(equation[9b]).3g(t)inEq. 9 a ]or[9b].Êquation[ 1 0 a ]or[10c]forasemi-infinitesystem;equation 1 0 b ]forafinitesystem.Îndicatesapulse-typeapplication:

g(t) 10( 0


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4 . L I S T O F ANALYTICAL S O L U T I O N S Thisectionresentsnalyticalolutionsf7],ithrwithoutth etworateterms,subjecttoth einitialandboundaryconditionssummarizedintable .everalofth elistedsolu-tionsaveeenpublishedpreviously.thers,owever,erenotavailableandhavebeenderivedtomaketh elistascom-pleteaspossible.aplacetransformtechniquesweregenerallyusedoderivehosenewsolutionshatreapplicabletoasemi-infiniteystemboundaryonditions10a]r10c]).Appendix istssefulaplaceransforms,anyfhemunpublished.Inspectionofth evariousanalyticalsolutionsshowsthatallsolutionsforafinitesystem,thatis,thosebasedonboundarycondition10b],renheormfnfiniteeries.heseseriessolutionsconvergeslowlyforrelativelylargevaluesofth edimensionlessgroup

vL/D [111where sfteneferredosheolumnecletumber.UsingLaplacetransformtechniquesinasimilarwayasshownbyBrenner(2962,approximatesolutionswerederivedthatprovideaccurateanswersforth elargerP-values.hesuggestedrangeofapplicationofth eapproximatesolutionsis^>'-'so r

vL>100

(P>5 f4 0T/R)

(P>100)

[12a]

[12b]whicheverconditionismetfirst.hedimensionlessvariableTin12a],alledth enumberofporevolumeswhenusedincon-junctionwithcolumndisplacementstudies,isgivenby

T=vt/L. [13]

Conditions12a]nd12b]ereobtainedempiricallybycom-paringnumerousresultsbasedonseriesandapproximatesolu-tions.henth econditionsaresatisfied,naccuracyofatleastourignificantlacesillebtainedithheapproximatesolutions.hencondition12a]r12b]snotsatisfied,werecommendthatth eseriessolutionsbeused.nthatcase,onlyabout4to10termsofth eseriesareneededtoassureasimilaraccuracyoffoursignificantdigits.


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SolutionsforNoorDecay2A l .overning R | = D- - v - ^Equation ^^x ^ ^ *

Initial and BoundaryConditionsc ( x , 0 )- C^

(C 0


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A 2 . Governing R= D r- - vEquation ^ " ^x^ ^ ^ Initial and BoundaryConditionsc ( x , 0 )=C ^

(vC 0< tt ^ where

2 ^ /2*/ ^x 1 T fR x t"|^ /V. f (Rx t) ,A(x,t)-erfcIT +( )xp[^g^]2 j1 + )exp(vx/D) erfc[Rx vtl2(DRt)^J

10


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2A3. GoverningR|=D-VEquation ^^x^^InitialandBoundaryConditionsc(x,0)=C^

(C 0 -^15=0

1 1


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ApproximateSolution

A(x,t) y: PRX vt"| 1 / / T x N * FRX vt* ] T -erfc+Texp(vx/D)rfc22 (DRt)^2j 22 (DRt) /2 j[R(2L-x)vtl. 2(DRt/''2 J+1 -2+1(21 Aj ^,p(^L/D) erfc2 '/2,V tvVL/OT!_ Vtv-^ ^ ^Pl-D-4Dt


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A4. GoverningEquation t g^2 X

InitialandBoundaryConditionsc(x,0)=C ,

( - D I f + v c ) 1 vCx= 0O 0


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ApproximateSolution(Brenner1962)

A(x, 2 '/2L2(DRt) ' 2JV 2

22I.-X .3|i+^C 2L - x+:|)'j exp(,I . /D) .rfc fc isl+vt]L 2(DRt) '2 J

14


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2A5. GoverningR||=D-V||Equationx

Initial andBoundaryConditionsC 0


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A6. GoverningEquation^ 8c ^cc3t2x3xInitialndoundary Conditionsc(x,0) =

C X,(-D|+VC) vC

x=0 V O

0 t.

II(.,t)Analytical Solution [seelsoost (1952, p. 50) andindstromandoersma (1971)]c(x,t) =

C^ C -2 )A(x,t) C - ) B(x,t) 0 t.S^ r 2 ^ A(x,t) C ^ - ^ ) B(x,t) - ^ B(x,t-t ) tt^

whereA(x,t) jerfc R(x-x )-vt2(DRtr2 nDRD R / . _ ^vt.x+x +) ]

1(x+x ) 2 "i* " D " ^DR^ exp(vx/D) erfc R(x+x.) vt '2(DRt)'2B(x,t) 4 , [BX tl .V2 ^ /2) exp[. T M > . 1 4DRt

-y1 + + )exp(vx/D) erfc[Rx+vtl2(DRty2j16


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A7. Governing R =D TZEquation ^ x " ^ ^^

InitialandBoundaryConditionsc(x,0) =C j ^+C 2e"^^

C 0


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2A 8 . Governing l lj D-M-vl Equation ^ ^x^ ^ " ^Initial and Boundary Conditions

c(x,0)= C ^ + C ^ e " " ' '!vC 0< tt

( . . t ) - 0AnalyticalSolutionc(x,t)=(^1o 1^(x,t) 2(x,t)t ^ j ^1* ^^o 1^(x,t) 2(x,t) -(x,t-t^)>

where2 ^*/: I " VL I . ,V tvRX --Vt) ,A(x.t) = ^HriTT/J " ^" ^ ^ ^ " " P ^D R E1 ^ FRX t" I . .v__t.

L2(DRt) '2j2 -j1 + )exp(vx/D) erfc

B(x.t) xp(%+ -ax)|l-+4(1+ )exp(^+ax) erfc2D1 ^.j^fx v+2aD)t1^2(DRt/^2 J [RX v+2aD)t1L 2(DRt)' '2 J)y-Qexp(vx/D) erfc|_2(DRt)'2J

18


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A9. Governing R|r*D- -v| Equation ^x^ ^ Initial and Boundary Conditionsc(x,0)= C ^

c ( 0 , t )=C+ C , " ^ * ^' a b

H < " . ^ ) - AnalyticalSolution[ s e e Marino(1974a) f o rt w ospecialc a s e s ]c ( x , t ) = C ^ +( C ^ - C ^ )A(x,t)- H C ^ B(x,t)whereA(x,t)= - ^ erfc T + T Txp(vx/D)erfc r22(DRt)^2j 22(DRt)/2jB(x,t) -" ^ * ^

+jexpl-ii erfcL2(DRt) ' 2jand

,, 4XDR/2y- (1"^ V

19


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Al, GoverningEquation t g^2 x

InitialandBoundaryConditions

x= 0= v(Ca e^^ )a b

c(x,0)=C ,( - D I I + v c )I I ( < , t ) = 0 AnalyticalSolutionc(x,t)=C ,+( CC , )A(x,t)+ C ,B(x,t)1 a 1where 2 2A(x.t)= 4erfcF^^^I+( 2 L 2 ( D R t ) ^ 2 j ^R. (Rx -vt)4DRt

Y ( 1+ - )exp(vx/D)erfc. / '. . \ ~^t 1 V r(v-y)xi Rx-ytKx,t)=e {TTZZTT expL^,^^ erfcl-

2XDR

L 2 ( D R t ) ' 2 j

L 2 ( D R t ) ^ 2 jxp(vx/D)erfcandy= V( 1--)V

V 2

20


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2A l 1 . Governing R 7 = D : " " v -Equationx

Initial and Boundary Conditionsc(x,0)= C ^ c(0,t)= CC ." ' ^ * ^aIf< - - ^ > AnalyticalSolutionc ( x , t )= C ,+( CC . )A(x,t)+ C .B(x,t)1 L 1where 2 ^ ^ D tA(x.t)= -E(^,x)exp[2 " - --j-\m= l RB(x,t)= e " " ^ ^[ B ^ ( x , t )- B2(x,t)]

2_ , . .X L R , v x v B( x , t )= + } , ~i n = l f2 , v L _ \L R ,

o ^ ^ D t^y r2 /VLv rvx ,^ Vt m-(^.x) ^ + (^) exp[^+ x t - ^ - 2-]B2(x,t)- 5 ;m= l2 ,vLv _XLR,^ P m ^ 4D^ Dand xE(,x) 2 ^ 8in(-f-)m Lr / ^ 2 v L . v L ,

2 1


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The eigenvalues^ arethe positiverootsofot( + =O m m uThetermB j ^ C x )convergesmuchslowerthantheotherterms nt h eseriessolution.h i s e r m ,however,canb eexpressedinanalternativeform t h a ti s much easiert o evaluate:, ^ ^ P ^ 2 D ^ V - + v ^^ 2DB( x )=U+C-^)exp(-yL/D)]wherey= V( 1)

V

A p p r o x i m a t e S o l u t i o nA ( x . t )= j e r f c f ' ^ ^ ^ l +j e x p ( v x / D ) e r f c l"^ " ^ ^ 1[ 2 ( D R t ) ' 2 J ^2 ( D R t ) ' 2 j

. , v(2L-x) v ^ t , / / T ^ ^ rR(2L-x)+ v t 12' ' 2,V tvVL R /O T - l _Vt. 1- ^ ^Pt^-4DF^2L-X+-^)

B C x . t )= e " ' ' ^ B 3 ( x , t ) / B ^ ( x )w h e r e

2 2


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c .,p,(:aO| Jzt,.cp^irL^] "2 ^^P^^^^ ^ ' ^^^ l -

B^(x) = ( )exp(-yL/D)and

V

rR(2L-x) +vt" |L 2(DRt)/2 J

2 3


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A12. GoverningEquation

2^ de -3c 9c3xInitialandBoundaryConditions

c(x,0)=C ,

( - D I f + v c )x=0 a b3cdx(L.t)=0AnalyticalSolutionc(x,t) - C g ^ )(x,t) ^(x,t)where

A(x.t) 1 - E( .x)exp[ -|5B(x,t) -"^*" [Bj(x) -^Cx.t)]Bj(x) 1+ I m-1

2 /- . X L vx.^ ^ m 4D^

E(e,'^> I - f) 3 exp(g+Xt vit4DRB2(x,t)and

2 L2R

m-1 r 2 VL)^IR,E (B ,x)m

2vL rvL . /m-p-m ^ m"^^D "(>J2rn2VLv vL r 2/VLv2 4


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Theeigenvalues areth epositiverootsof

" ^ m vLr. Ot() -_-+_=0Th etermB j ^ ( x )convergesmuchslowerthantheothertermsintheseriessolution.histerm,however,canbeexpressedinanalternativeform thatismucheasiertoevaluate:B,(x) exp 2 D ^ V - t v ^^ " " P ^ 2 Dy+v_y-v)wherey=V 1)

VApproximateSolution

r / 2A/ ^N ^ -vt Vt. (Rx-Vt)A(K,t). jerfcj +C )xpl- 4 P R,-5'^^-*TI*45^-*^>'I-

B(x,t)=e ^B3(x,t)/B^(x)where

. jr.\ frR(2L-x)+vtlp(vL/D)erfc TL 2 ( D R t ) ' 2 J

B3(x,t)=( )exp[-^^^lerfcL 2 ( D R t ) ' 2 j2 5


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L 2 ( D R t ) ' ^ 2 J -255"P< *">i=P^^-^lL2(DEt ) ' 2 j

L 2(DRt)^2 J

(y+v)2(DRt//2 J - exp[-to02,iZkj erfcp(2L-x)-ytlL 2(DRt)/2 Jy.v)2--DB^(x) = 1 --telL.exp(-yL/D)(y+v) 6


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SolutionsforZero- Production Only

Bl. Governing ^2^ ^^Equation Dj ^d^ " * " > "^^ (Steady-state) dxBoundaryConditionsc(0)=C^4 ^ ( )=finitedxAnalyticalSolutionc(x)=c^o V

2 7


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B2. Governing 2Equation D - ^ - 1 - v~ + y =0(Steady-state) dxBoundaryConditions( - D 4^ + v c ) I = v C ^ dx Inx= 0( o o )=finitedx

AnalyticalSolutionc(x)=C^^OV

28


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B3. Governing ^2 ^ ^^Equation Dj-v+ y=0(Steady-state) dxBoundaryConditions

c(0)=Coi| ( L ). 0AnalyticalSolution

X ^2m vL /Vx> Jlavnic ( x ) - c ^ + "

wheretheeigenvalues ^ j ^ arethepositiverootsof

Theseriessolutionconvergesooslowlytobeofmuchusenumerically.nalternativeandmoreattractivesolutionisgivenbyc(x)-C^+IJ+ If|exp(-:^)-exp[- ]}

2 9


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, de de 34. Governing D - -v - ^- y =0Equation dx(Steady-state) BoundaryConditions( - D 4^ + v c ) = v C^^ x= 0f f ^ ^ > = AnalyticalSolution

c ( - D - )(V m t ^ m ^(-r^ + 2 D ^ ^ " ^ - ^ ^ ^ ^ ^ ^ 2 0 ^ =1 2^ v L . ^ vL, 2 v L .I^m-^^2D^ '"DI m-'^2D^Wheretheeigenvalues ^ ^ arethepositiverootsof

Theeriesolutionconvergesooslowlytobeofmuchusenumerically.nalternativeandmoreattractivesolutionisgivenbyY^ ^Y^l ^..r^(^~L)- ^ ^ ^=^o ^ ^ ^~ " " P '~

30


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B5. Governing R| =D -v||+yEquationxInitialandBoundaryConditionsc(x,) =

1 0 to c(0,t) =(o > oIl,t) =initeAnalytical Solution (Carslawndaeger 1959, p. 388)c(x,t)

where

C ^ C ^ ^ ) A(x,t) +(x,t)1i

A(x,t) =7 7erfc j +T Txp(vx/D) erfc T22(DRt) /2j 22(DRt) /2jT > / ^ A YL (Rx-vt) ^ FRX tl

(Rx+t), . --exp(vx/D) erfc2 v

[Rx+vt"|2(DRt//2j

31


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B 6 . Governing R|f"D- -v||+YEquationx

Initial and Boundary Conditionsc(x,0)= C ^

/vC 0


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2B7. Governing R | =D - 2 - 1 -,21+Equation ^x InitialandBoundaryConditionsc(x,0)=C^

C 0


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E( ,x) = 6 X 2^sin(-^)

l^'F^'-ilT heeigenvalues ^ areth epositiverootsof

m


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, . vR(2L-x)-DR R ,-,T j.ts vL.[t '02L-X) ] exp() erfc2v

DRv(x-L), -= exp[^^tr-]erfc2 v [R(2L-x)vt]2(DRt/'^2 JtR(2L-x) -t]2(DRtr2J35


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B 8 . G o v e r n i n g R||"D--V||+YEquationsxInitialandBoundaryConditionsc(x,0)=C^ ft 0

2^ ^Dt./ \ ,T/ \VX vtlA(x.t) =1-1E( ,x) exp[-^-j --J-]m"lB(x,t) =j(x) -2(x,t),, , % E(,>x)4-expCg)B^(x ) = I

T2^ ^Dt1 7 / -n \LVX V m '^im^'^^D^^PflD-DR-77-1B^Cx) = I 2-36


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and2vL r ^ ,'^m..vL,^m^v,E(^,x). _JL- L___LD_t m - ' ^ 2 D > - " i l lt m - ' W

Theeigenvalues ^ ^ arethepositiverootsof

ThetermBj(x),whichalsoappearsinthesteady-statesolution(caseB4),onvergesmuchslowerthantheothertermsintheseriessolution.histerm,however,canbeexpressedinanalternativeformthatismucheasiertoevaluate:V

ApproximateSolutionRx -vt)^,4DRt

1j ( 1+- + )exp(vx/D)erfcL2(DRt)^2j2 / 2- L( \ n..^- tvi rVL ^_ tv "^^SR-> ^ D^2L-x+)]expl - 45^(2L-x+)-- [2L-X +- +-^(ZL-x+) exp(vL/D)erfcrR(2L-x)+vtlL 2 ( D R t ) ^ 2J

3 7


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(_/2^ . 2DR,(Rx -t)^,(X;;R> (Rx + Vt +-) exp[-,nD.^4TIDR' ^^^ ^ " " r ^ ^ ^ I ' lDRT+ < =R (Rx+t) ,,,^,+ I7-9 T ;^exp(vx/D) erfc 2 ., 2DR 2vDRV(x-L),-2 gxpt D ^ ^ ^ ^ * ^ 2 v

[R x+vt"! 2(DRtr2jrR(2L-x) t" |L 2 (DRtr2 J2

+ 2 H -2 ^-^-g-)(2L-x-)2v'^D3+- (2L-x ) ] exp(vL/D) erfcl D - ""2(DRt)*'rR(2L-x) +vtlL 2(DRt)^2 J6D

^HB} ^^P t-D-4DF^2L^ l-

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B9. Governing R|.D -v||+yEquationx

InitialandBoundaryConditions

c(x,0)=C^+C 0


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2BIO. Governing R-|r=D-v|j+Y

EquationxInitialandBoundaryConditionse ( x . O )=c .-fXZl

V!vCti 2 o i 0 0 c

O


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2Uli. Governing R|r=D--v|+YEquationxInitialConditionc(x,0)=A(x)Notehathenitialonditionisfheameormashesteady-stateolutionforheameoundaryonditionscaseB3).

BoundaryConditions!C 0 -

AnalyticalSolution/ A(x)+(C ^-C^)B(x,t)


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ApproximateolutionA(x,t) =4Ierfcfe-llj;!]+1exp(vx/D) erfcr- Lj; lL2(DRt)/2j 22(DRt) /2j

+12 v(2L:20Ajexp(vL/D) erfcF^^^ - ),/ ]L 2 (DRt) ' 2 J V2

42


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.12. Governing R|=D -v||+yEquationx

InitialConditionc(x,0)=A(x)

Notehathenitialonditionsfheameormashesteady-stateolutionorheameoundaryonditionscaseB4).BoundaryConditions

C


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^m ^m vL 4DApproximateSolutionB(x,t)=i 2 2 01 ^^^ffoc-vtl Vtv (R x-vt)

[ Rx+ V tl2 ( D R t ) ^ 2 j 1x - ^ t 2( 1+ "^ * " -^^exp(vx/D)erfc

2/ 2 ^ ^IDT 1 1 - ^ 4D^2L-x+ )]exp[ - ;^(2L-x+)

Vr o T j . 3 v t v , - ^ v t v / / T N N ^ rR(2L-x)+ v t l- - r[2L-X +-^ +-T;^2L-X+-5-) exp(vL/D)erfc - i ^n 2 R 4 D R2 ( D R t ) / 2 J4 4


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2B13. Governing R |^ =D- -v | + yEquationxInitialandBoundaryConditionsc(x,0)=C^c(0,t)=C^+ C^ " ' ' ' ^

II ( , t )= f i n i t eAnalyticalSolutionc(x,t)=C^+(C^-C^)A(x,t)+ C^B(x,t)+E(x,t)whereA(x,t)-^

B(x.t)=e-^^Uexp[l:iJg^]erfcL 2 ( D R t ) ^ 2 j

r ? r -^ Y L (Rx-vt) - FRX -tlE(x,t) = -< t ^ rerfc T ^ ^2(DRt) / 2j(Rx+vt) . / ^ . x ^ FRX t- T Texp(vx/D) erfc r^ ^2(DRt)^2

andy (1~)

45


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2B14. Governing R |r =D- -v| + vr .t ,2 XEquationxInitialandBoundaryConditionsc(x,)=C^

(_D+VC) =(C +, e"'^'') XQbIl(,t) =initeAnalyticalolutionc(x,t) =. +C -,)A(x,t) , B(x,t) +E(x,t)11where 2 ^ ^ 2A(x) =^erfcf *^!+^-^) exp[- < ^ > 1 AU.t; 22(DRt)/2j RDRt J [Rx_+_vt"l2(DRt/^ 2 "i * "^^ "^^ exp(vx/D) erfc

22j-j exp(vx/D) erfcL2(DRt) '2j

46


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E(x,t) XLj. 1/1,DR. - IRx tntvHRx t) erfc2v' L2(DRt)'2j-0"


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2B15.overning R|f=I - -' |f+Y

EquationxInitialandBoundaryConditionsc(x,0)=C^c(0,t)=C^+ C^e"^^

Analytical Solutionc(x,t) =, + (C ^ -,)(x,t) + . B(x,t)F(x,t)1where

2^ DtrV x V tA(x.t) = 1-1 E(^.x) exp[^-io--Vlm= lB(x,t) =" - ^ * " [B (x) -^Cx.t)]

2 , ^ A L vxv 0 E(_,,x) r- exp(-;r-)B l(x) = 1 + I ^ D -'-^20'o2 ^Dt^/ \ r 2 /VL\ vx , , V t1 - E(^.x) [^ ) ] expl2+t- --2-1m= l1 vL. AL^R,F(x,t) = (x) - Cx.t)

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o .() J -exp(^)F j C x )=2D_- 2^ ^Dt! ? / o yL rVx Vt mi(Pm'^) D P2D-4DR-X" F2(x,t)- ^ -and

X2 ^ sin(-5^)E(m'^>^-^ Trr2 VLv LiTheeigenvalues ^ j ^ arethe positiver o o t so f Thet e r m sj ^ ( x )andF j ^ C x )convergemuchslowerthantheothertermsinheseriessolution. Both e r m s ,however, a n eexpressednlternative forms t h a t a r euchasier t oevaluate: ^ ^ P ^ 2 D ^ S r f v ^^ ^ P ^ 2 DB( x ) U+(^)exp(-yL/D)lwhere

V 2V '

andy V( 1- )

FjW.J:+i||exp(-ii)-.xp[: il

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ApproximateSolutionA(x,t)= ^ [RX t" | 1 / ,^ . ^ FRX vt"|=rerfc +Texp(vx/D) erfc j22 ( D R t ) / 2 j 22 ( D R t ) / 2 jrR(2L-x) vt" |L 2 ( D R t ) ^ 2 J ^2(DRt)

2 '/ 2,V tvVL,^_Vtx 1 B(x,t) ="''^3(x,t)/B^(x)

(y-v)_,


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- 4^>'' ^- - ^-P -4IF'I^ ^,.K2 (..-..Z|) expCl) erfc[M2^ ]

2v[R(2L-x) -vtlI

. 2(DRt//2 J)

51


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B16. Governing R|^-D- -v| +YEquationx

InitialandBoundaryConditionsc(x,0) = ,( - D If*v c ) x0 v(C +a e"^^)aH< ' ' > Analyticalolutionc(x,t) = C ^ ^ )(x,t) ^(x,t)F(x,t)where

A(x,t) - I E(a^,x) exp[m= l vx2D 2 ^ e^DtV m 4DR L2R B(x.t) -"^ ^ [Bj(x) - B2(x,t)]Bj(x) ^

( ^m'^) i^^-^^i) 1 exp[||+Xt 4DRB^Cx.t) L^Rm= l rg2 VLRi^ ^m 4D''F(x,t) = (x) -F^Cx.t)

52


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F , ( x , t )=. 2^ D tr . / - / , \' ^ rVx t m2m=l^2 vL.and

E(. X )- ^ ^ ^ ^ 2 D L2vL - ' ^ ^ m vL m

T h e eigenvalues^aret h e positive rootso f

Theterms ^ C x )andj ( x )convergemuchslowerthant h eothert e r m s nt h eseriesolution. BothB j C x )andF ^ C x ) ,however,canb eexpressedi nlternativeforms h a ta r emucheasiert oevaluate:B( x )= ^i -i p


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ApproximateSolutionlu

.2*


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B (x) = 1 - ^y~''\ exp(-yL/D)andF(x,t) = < tyi(Rx t) erfcix iVRx t2(DRt)^2. ]2DRs-x;;?^) Rx+vt+- )xpi- (Rx -t)'4DRt

oj. r * ^ D R . (Rx+vt) 1,_. . FRX vtl["R(2L-X) -t" ! Rv(x-L)i ^ -jxp[ r'-]erfc2v^2(DRt)DR r(2L-x) . V ,t... , 3vt.+2 ^ ^D* "9


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C . SolutionsforSimultaneousZero-order ProductionandFirst-order Decay

dc doCl Governing D" Ai C + y OEquation d x(Steady-state) Boundar yConditionsc(0)=Co

(. )- 0AnalyticalSolution

, ( , ) . X . ( C ^ - J t , ...li iwhere

u =V( 1+-)

5 6


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C2. Governing ^2^ ^^Equation Dj"^"di c +y0(Steady-state) dx

BoundaryConditions( - D |+ vc)

( . , . 0vCx0 ^

AnalyticalSolution(GershonandNir1969/ ^/n Y\ 2v r(v-u)xi\i\i u+v 2Dwhere

V

5 7


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C3, Governing 2Equation Dj" AC +y^Ô(Steady-state) dxBoundaryConditionsc(0)=C o|(u.AnalyticalSolutionc ( x )= +( C ^ - ^)A(x)where

A(x)=1-1 22^ " ^ ^ fo2 /VLv vL,Q2 .vLx uL,'Pm""%2D^tPm^^2Dândwheretheeigenvaluesârethepositiverootsof

Theaboveseriessolutionconvergestooslowlytobeofmuchusenumerically.hefollowingequivalentexpressionforA(x)ismucheasiertoevaluater (v-u)Xi /U^Vv r(vHl)x uL,

A(x)= [ 1+(~ ) exp(-uL/D)]where

V( 1* )'''

5 8


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C4, Governing , 2Equation Dj""^Tc +y'^O(Steady-state) dx

BoundaryConditions( - D is * v c ) =vCx=0dcdx( L )-0Analytical Solutionc(x) = +C - )(x)\i o ^ i whereA(x) =

1 - Iin=l.2vL. /liXx r r s.vL j/ mvvxv^T-^ % s(-r^ 2D ^ " ^ ^ r ^ ^ PW

22r r . 2 VLv vL,2 vLvr 2 vLv Li P m - ^ ^ 2 D ^ - ^ - D ^t P m " ^ ^ 2 D ^ ^ ^ m ^ ^ 2 D ^ " ^ Vandwheretheeigenvalues ^ j ^ arethepositiverootsof

Theaboveseriessolutionconvergestooslowlytobeofmuchusenumerically.hefollowingequivalentexpressionforA(x)ismucheasierto use(seealsoGershonandNir1969)

A(x)

where

r(v-u)xi /U-Vv(v-Hi)x-2uLiexp[^p-l Hr )xp[^jUHv (u-v)TnMU = V (1 +MV 2

59


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rye OC 0 CCC 5 . U)verning R =^" ^Equationx

InitialandBoundaryConditionsc(x,0)=C^

C 0


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2C6. Governing R l ^ =D " "^ S|"A < ^ +Y

EquationxInitialandBoundaryConditionsc(x,0)=C^ ivC 0


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V(v+u)x, ^ TRXutl4. V ^ /VX U tvFRX vtlI5 -P


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2C7. Governing |f "^ "" ^Ix " ^

EquationxInitialandBoundaryConditionsc(x,0)=C^

C 0to

Analytical Solution (SelimndMansell 1976)c(x,t) = + (C. - )A(x,t) + (C - ) B(x,t)t < toX+ (C. -^ )A(x,t) +C - ) B(x,t) - B(x,t-t ) t > tiipLpwhere 2^ ^Dtw XT/ \VX Lit vtA(x.t) = I E( ,x) expir-- -MTIT ni=l iB(x,t) = (x) - iyi.t)

Bl(x)=1-1 .2 E( ,x) exp( )22m " 2 0 ^ ^""' l!+( ) + 1 ,2,

2(x,t) = );

6 3


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and X 2^in(-5^)E(,x) =r/n2 /VL. , vL,

T heigenvalues j j jarehepositiveoots ofvL m ^^m^ * 2D=

T hetermBj(x),whichalsoappearsinth esteady-statesolution(case3),onvergesmuchslowerthanth eothertermsinth esolution.histerm,however,canbeexpressedinanalterna-tiveformthatismucheasiertoevaluate:r(v-u)Xi , /U-Vv r(v+u)x uLi

B( x ) 1 1 +(^)exp(-uL/D)]Approximate SolutionA(x.t) -xp(-jit/R)l -jerfcF ^ ^1(2(DRt)/2j

--jexp(vx/D) erfc2 1 r^ . V(2L-X) . V t,Tr.\" ^ "DR exp(vL/D) erfcL2(DRt)'2j rR(2L-x) +vtlL 2(DRt)^2 J

2 '2 \, .vv^LT , vt. ,1B(x,t) -2(x,t)/B^(x)where

6 4


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1(v-u)x,.B3(x,t) = xp[ 2 ] erfcL2(DRt)'2J 4expli!g2,rfcL2(DRt) ' 2J(uzi) expl -^g-'"" ^^,- rR(2L-x) -tl(u+v) * -D_ 2(DRt)'2 J ^(u+v)2(u-v) ^D2(DRt)'2 J v^vL lit. . rR(2L-x) +vtl

B (x) - 1< ) exp(-uL/D)andu ( 1 + )2'

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C8. GoverningEquation

ac ^3^c a c' ^ a t=^^^2- 3x-^ ' ^ - ^ ^InitialandBoundaryConditionsc(x,0)=C ,

vC(-D| H -VC) x=00

0twhere

v xA(x,t)= E(^,x)exp[2pm= l R 4DR2

B(x,t)=B^(x)-B^ix.t)

Bj(x)=1 U L E(^,x)exp(g)D 2" > = !fg2++ ^ ' LB2(x,t) = I B(3.x)l3f.(^)]exp[||-^-|^m m L2 Ri n = l 2 2^ ^ m ^ 4D^ D

6 6


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ando / ^ X - X2Xilr/eo8(-^)+|^sin(-^)] E(3^,x)=- _D__

T h eeigenvaluesret h epositiverootso f 2 D, m vL cot( - - ^ r +- =0m m vL 4D

ThetermB , ( x ) ,whichalso appearsi nt h esteady-statesolution( c a s e 4 ) ,convergesmuchslowerthant h eothertermsint h eseriessolution.his e r m ,however,canb eexpressed nanalternativeform t h a ti s much easiert oevaluate:r(v-u)x, , /U-Vv r(v+u)x-ZuL,, ^^P[-2D-^^ - ^ ^ P t 2 DB ^ ( x )=rU+V (U-V) . T/^v1

Approximate SolutionA(x,t) =exp(-pt/R) 1crfcf w 122(DRt/'^2j2 ^ /2,v t.(Rx -vt) ,-DR> ^^P tDRTJ

2 +|-1 + + )exp(vx/D) erfcL2(DRt ) ' 2 j2 '/2/^V tv r, . V , _Vt.,VL/OTVt. ,-lDR-> fl D^2L-x-^)] exp(^- (2L-x_) ]

+^2L-X+|i+ 2L -X+ ) ] exp(vL/D) erfc[R(2L-x) +vt"jl2(DRt/ /2 Jl67


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B(x,t) 2(x,t)/B (x,t)wherer . / N(v-u)xi - [EX tl


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C 9 . Governing R | ^ D " ^ |j "1 ^ ^ + yEquationxInitial Conditionc(x,0) = (x)

= *=l-whereu V ( 1 V

( v - - u ) x ,2D ^

Note h a t h einitialcondition s f h e a m eform s h esteady-statesolutionfor h esameboundaryconditions c a s eC l ) .BoundaryConditions

(C


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2CIO. Governing R |f=D- -v - f ^ -y c+yEquationx

InitialConditionc(x,0) A(x)

Y r . Y\ 2v r(v-u)xi

whereV9VV

Notehatheinitialcondition softhesameformasthesteady-stateolutionforthesameboundaryconditionscaseC2).BoundaryConditions

/vC^(-DH+VC) =


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c i l . Governing R||-=D -v||-yc+ YEquationxInitialConditionc(x,0)= A(x)

where

r ( v - u ) x , U - v .(v+u)x- 2 u L ,- 1+( C) P^~2D-J ^ ^^^ ^ P ^D"^ ' ^l+(^)exp(-uL/D)]

V 2u= V( 1+A!|)V

Note h a t h einitialcondition s f h e a m eform s h esteady-statesolution o r h e a m eboundaryconditions c a s eC 3 ) .BoundaryConditions

c(0,t)=C< t to\0> toAnalytical SolutionA(x) C^ ^ ) B(x,t)t^

c(x,t).A(x) C - ) B(x,t) - B(x,t-t )to 10

where A(x)i sexactlythe initialcondition,and whereB(x,t)=B ^ ( x )- B^Cx.t)with

7 1


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B^(x)=1-1 ~

9 2^ ^Dt(^.x)I^+ (^) exp[ - -_-B^Cx.t)= ^and

2psIn(-J!-)

Theeigenvalues ^ ^ ^ arethepositiverootsof

m^^^m^'l-OThetermB j ^ ( x ) ,whichalsoappearsinthesteady-statesolution(caseC3),onvergesmuchslowerthantheothertermsinthesolution.histerm,however,canbeexpressedinanalterna-tiveformthati smucheasiertoevaluate:

r(v-u)x, U-Vv r(v-Ki)x Li^ ^^Pt-2^^ ^ ^^P^-2DJB( x ) [ 1+( )exp(-uL/D)]ApproximateSolutionB(x,t)=B2(x,t)/B^(x)whereBjCx.t)-Iexp(ig^lerfcL2(DRt)'2j

7 2


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. 1 r(v+u)x, ^FRX h Utl

v ^ r V L u t , ,r R (2L- x ) + v t l-=rexp-fT --^ ]erf c-ij^ " ^ " ^ ^ L 2 ( D R t ) / 2 JB ^ ( x )=1 +(^)exp(-uL/D)

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2C12.overning |f= - -v|f-J i c + yEquationx

InitialConditionc(x,0)= A(x)

f(v-u)x, /U-v. (v+u)x-2uLi= X+( cX_)^^P^~lD~ " ^^ ^^P^ 2D

^U+VU-V)/ T N \ 1where

VNotethatheinitialcondition softhesameformashesteady-stateolutionforthesameboundaryconditionscaseC4).BoundaryConditions

!vC


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.2Bj(x) = 1 -

T-/0r,,2 . /VL ,VXitt1 B^Cx . t ) = ^

=12 .vL . L lL 1 im+%> - ^ D and

Theeigenvalues ^ ^ areth epositiverootsof

E(,x)=

' ^ m vLot()-+ T T :=0^m m vL 4 DT hetermB^Cx),hichalsoappearsnth esteady-statesolu-tion(caseC4),onvergesmuchslowerthanth eothertermsinth eseriessolution.histerm,however,canbeexpressedinanalternativeformthatismucheasiertoevaluate:

r(v-u)x, /U-Vv (v- K i )x-2uLi, ^^pt-2r-i ^^p tDB,(x)=rU+V (U-V)/TAM

ApproximateSolutionB(x,t)=B2(x,t)/B^(x,t)whereT i ( ^ \r(v-u)x, ^TRX- u t " |

V r ( v - K i ) X i ^FR X+ utl

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2 , VVXit,^lD ^ P^ -D R ^ " '

V rV(2L-x) V t , ^ V 1VLLtvL2(DRt)'2j [R(2L-x) +vt]2(DRt)^2V , t . fVL utOT j.vt.- i D ^xDR> ^^P - - R-DF^2L^) ]

. v(u-v) r(v+u)x uL , j .+ ^ ^"xp[-^-erfc(u+v)^"2(DRt)'v(u+v)(v-u)x+uL , ^ ^ '^PlD^^^^ (u-v) rR(2L-x) -ut" |L 2(DRt)^2 J[R(2L-x)ut" 2(DRt)^2 Jan d .2B (x) = 1- ii= exp(-uL/D)(u+v)

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C13.overning R1 =D -v|^-HC + YT - i j ot z oxEquationx

InitialandBoundaryConditionsc(x,0)=C^

c(0,t)=C C ," ' ' ^ aI f ( - . t ) - 0AnalyticalolutionseelearyndUngs1974 )ndMarino(1574b)forsomespecialcases]c(x,t)= +(C,-^)A(x,t)+(C ^-^)B(x,t)+ C ,E(x,t)where

A(x,t) = expf^it/R)l -jerfc] "g 1 (2(DRt)^2j- r -exp(vx/D) erfc T > 22 (DRt) /2 j)

B(x,t) = ;exp[^ / ] erfc rL2(DRt) '2 je.pl- -'=c 1(v4-u)xT2 " ^^"20^ L2(DRt) ' 2j

r . / - ^ ^^t 1(v-w)XiFRX-Wtl[Rx+wtl). 2 ( D R t ) ' / 2 j ]. 1(v-h^)xT ^ " 2 ^ ^ ^ ' 2D^ an dith

77


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u v (1+- )V V 2 w [1+ i\i-R)]V V 2

78


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C14. Governing ^ff "^ "^fs "^ ^ " * " ^EquationxInitialandBoundaryConditionsc(x,0)=C , ( - D II+ vc)


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^ 2L2(DRt) '2j

u ^\ _"^t)/' V (v-w)xi PRXwtl[Rx+wtl2(DRt//2j) .V V(V+W)X,+( )exp[-2-] erfc

and

u ( 1 +- )V 4D/2 w [1 +-=^(ti R)]V

80


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C15. Governing l |f=D -v|j-iiC+ YEquationxInitialandBoundaryConditionsc(x,0)=C^

c(,t)=C^+ C ^ ^eT^^

I f ( ^ ' ^ > = 'AnalyticalSolutionc(x,t) = + (C- )A(x,t) +C - )B(x,t) +, F(x,t)[X R) ^ i|i|i+ (C ^^)A(x,t) + (C - ) B(x,t) . e"'' (^ i R)

|ib p .where 2 ^DtwT.fr.VXit vtlA(x,t) = I E( .x) expl - _-^ -IIl~l j KB(x,t) = B^(x) -B2(x,t)

2 - E(.x) exp(||)Bj(x)=1-1" " ' Ie^'- 1

2x t v t ^ m ^ * ^ E( .x) [^+(f ) 1 exp[|f- -|j -- ]B2(x,t) = I

^ P m D^81


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F(x,t)= e " ^ * "[Fj(x)-F2(x,t)]

F( x ) = 1-12

~ ^ ( ^ m ' ' ^ ^ ^^P^lD^m=l2 ,vL. ( t i-XR)L^.^ ^ m ^ % +

22 D tV''> ' " -^and 2D D

6X23^8in(^)E(3,x) " ^^^^


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4D/2 w [1 + (|i-R)]V

Approximate Solution[Rx tlA(x,t) =xp(-(it/R)


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rU-Vx^ 4 ^ " " ^ " ^i+7^ exp(-uL/D)andF(x,t) -"'^''F3(x,t)/F^(x)whereF3(x.t) = iexp[i3 ] erfcP^^-: i-St'3vx,ty = Yxp i20"

, 1(V+W)Xi" 2 P l 2D ^ L2(DRt)'2j[ " R X wtl+_(wil). ,pr(v+w)x- 2 w L , ._,J(2L-X) -wt1*2(w+v) PlD"^^ ^L 2(DRt)'2 J- . (w+v) (v-w)x+w L , ^ rR(2L-x)wtl

2 -^/-vL MX ^^ t rR(2L-x)vtl___exp(--\+X t)rfc[T JF (x) - ( )exp(-wL/D) 84


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2 C 1 6 . Governing R - 5 =^- 5 - ycy ^ .t , . oxEquationxInitialandBoundaryConditions

c(x,0)=0(-D| +VC)| v(C C . ^^)3x ^ a bx=0t( ^ "=AnalyticalSolutionc(x,t)=^+C-I)(x,t) CI) B(x,t) +, F(x,t)u *R)ppp1+C-, -I) A(x,t) C -I) B(x,t) ,e"'^*' (y R) yD )jywhere 2 ^Dtw \T./0 NVx yt V mA(x.t) = I E(3 ,x) exp[ -__]m= l RB(x,t) = B,(x) -B,(x,t)

E(*x) exp( )Bi(x) =1-1 - 5_D_ m=l ro Z .vL. yL ,o22 ^Dt1 7 /o \ ro2 . .vLv vx yt V m ^(^'^> f^m^% ^ ^P t 2D-R- 4DR- ^B^Cx. t ) = I

85


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F(x,t)= e ' - ^ ' ^ L F j x )- F^x.t)]

F( x )=1-12

'% " ^ W" ^ D2 ^ D t

r i i KF( x , t )=I " = ! 2 ^VL/ ( y- XR)L

and X X _[ _c o s ( - ^ - )+ C - ^ r ^ )sin(-=) ] , - D m^ " ^ m L ' 2 D ' LE(, x )=-

T h eeigenvaluesret h epositiverootso f m^D ^

ThetermsB,(x)and,(x)convergemuchslowerthantheothertermsint h eseriesolution. BothB ^ i x )andF^Cx),however,canb eexpressed nlternativeformsthataremucheasiert oevaluate( c a s eC 4 ) :B , ( x )

exp[


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V. 4D ^ / 2w-vll+-|(p-XR)]

V^proximateSolution

A(x,t) exp(-pt/R)1 -Y i .j.f TRX tl2' ^""UCDR^J\_ (t^R x-vt)^ ' 'TTDR'' ^"^^DRt J L2(DRt)'2J 2 '/2 T-DF(2L-X) ] - ^2L-x+ + 2L-x+:|)']exp(vL/D) erfoF^ iL-^L 2(DRt) vtw B(x,t) =2(x,t)/B^(x,t)where

[ R X utl2(DRt?2j

[Rx vt"|2(DRt?y.(v+u)x,, VVXit.D

.2.+JL. r (2L-x)v_t_L t ,_^.rR(2L-x)vt1 L 2 (DRt) '2 J

87


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-- ;;^> ^^P^D R 4Dt^yD irDR,v(u-v) ,, .iZHli-JHi]erfc Mi=^ f|+i ''P^ 2 D2 (DRt) '2 J(u+v)2 ^ 2 D2 (DRt) '2 J(u-v)B (X ) - -exp(-uL/D)4u+v)^ andF(x,t) -"^ ^ (x.t)/F^(x)whereF 3( X . C ,. C ^ ) X P . ^ ^ 1 -gS

RX vtv^^vx M t t) erfc-/L2 (DRt )^2J ^zFm""P^"^"" v ^v(2 L -x). t .Y I

i-RCZL-X) 4 -t" l 2 ) 1 D(y-XR) TTDR 88


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vwzvle,pt(v4^)x-w L j ^ ^ ^ rR(2L-x) -wtl(wfv)2D2(DRt)/2 J(w-v)2D2(DRt)'2 J ( ^2 F ,(x) - 1 --i^=^exp(-wL/D)^wfv)^

89


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5 . E F F E C T O F B O U N D A R Y C O N D I T I O N S Inhisectioneil lresenteveralalculatedolutedistributionss unctionfistancendime.pecialattentionwillbegiventoth eeffectsofth eappliedupperandloweroundaryonditions.heesultsreeneralizedymakingus eofth efollowingdimensionlessvariables

P=vL/D T=vt/L z=x/L14 ]wherePisth ecolumnPecletnumber, isth enumberofdis-placedporevolumes,nd isth ereduceddistance.omaketh esolutionsforasemi-infinitesystemapplicabletoafiniteprofileofengthLforexample, laboratoryoilolumn),th ereduceddistancecannotexceedone(0


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1 . 00.8-

9. 6e r u. 4ozoo 0.2"

lili>4"-\V=0.25 A2 /\"\ A4\-

Al, A3

- P=5 1

A2, A41 ^

0 . 2 0.4 0 . 6 0 . 8REDUCEDDISTANCE,Z

1 . 0

REDUCEDDISTANCE,ZFigure.alculatedconcentrationdistributionsforR=landP-valuesf nd0,espectively.heurveswerebtainedithhenalyticalolutionsfcases1,A2,A3,andA4 .

9 1


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leastfoursignificantplacesbysolutionsforasemi-infinitesystemaslongaszisrestrictedto0


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Figure .ffectofPonth econcentrationatx=LandforT=.hecurveswereobtainedwithth eanalyt-icalsolutionsofcasesAl,A2,A3,andA4 .

0.48.24.0APORE VOLUME,

08

H402-

r 1 1 ' ^ i *'' 1 " /y- B '// -1 1 _ /// Al.A4 - II - - I -- I -

-L1-.J- i-

0.42604PORE VOLUME,

08

2 0.6-zui4 iO02

-TP60

JW _04 08 \z Te 2 ^ 0 ZPORE VOLUME.

Figure3. EffectofPoncalculatedeffluentcurvesforcasesAl,A2,A3,andA4.9 3


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ParlangeandStarr{1975.heerrorsntroducedbyapprox-imatingth esolutionofA4byth emuchsimplersolutionofAlareaboutth esameasth edifferencesbetweenth ecurvesAlandA4nfigure.econd,heurvesorAlnfigure3arelocatedexactlybetweenthoseforA2andA3.nequationformthiscanbeexpressedas^A3=2c^i-CA2=L)16 ]

wherehesubscriptsAl,2,ndA3refertoth eappropriateanalyticalsolutions.hislastproperty,whichisextremelyaccurateoraluesf hatreotoomall,ollowsdirectlyromhepproximateolutionofase3.imilarrelationspplyorllpproximateolutionsor initesystemandafirst-typeboundaryconditionatx=0(thatis,alsofornonzerovaluesof\,i ,andy) .orexample,forcaseC7onehas^C 7'2cc5-c^^.x=L)17 ]

Theabovediscussionofheboundaryeffectsisrestrictedtocaseswhereth eproductionanddecaytermsarezero.imilareffectsfheoundaryonditionsanlsoeemonstratedwheneithery- orbotharenonzero.nlyafewcommentsfortheseaseswillbegivenhere.heeffectsofth eboundaryconditionsaregenerallymorepronouncedforth especialcaseofzero-orderproductiononly(y^0,L=0) .hisisshowninfigure4whereth esteady-statesolutionsofcasesBltoB4areplottedfortwovaluesofhecolumnPecletnumber.esultsaregivenforC andavalueofoneforth edimensionlessratetermYyL/v. 18 ]

T heifferencesetweenheourolutionsreonsiderable,especiallywhenPequals5.otethatth esolutionforcaseBlisindependentofP.Theffectsfheoundaryonditionsreenerallyesssignificanthen,ndditionoero-orderroduction,hechemicalisalsosubjecttofirst-orderdecay.igure5showsth esteady-statesolutionsofasesloC4,ortwovaluesofy,andor aluefneorheimensionlessecayconstant

I l=^ l L / v . 19 ]T heurvesorhewoaluesfyare,nhisarticularexample,sjnmnetricwithrespecttoth elinec= . Notethat.

94


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02 0 . 4 0 . 6 0 . 8 1 . 0REDUCEDDISTANCE,Z

Figure4. ffectofPonsteady-stateconcentrationdistributionsforcasesBl,. B 2 ,B3,andB4,

1 . 6

2-i . (

06

04 -02-

- T IT T I_ C 2 ^;; ^ ^ \- \ ^ _ yyy /2' y -< yc\ -xT -NN\. X-o - I^\ ^v -- C2 \ \ : > ; 4 - . ; - ^^**^


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ataPecletnumberof20,hefiniteandsemi-infini tesolu-tionsareessentiallythe sameovertheregion0


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uPore volume( T = vt/L).

Yore-water velocity,w w =[ v ^ + 4 D ( ^ - \ R ] ' ^ 2XistanceX jonstanti n severalinitialconditions( t a b l e1 ) .y ( v \mr .zeduced distance( z= x/L).aecayconstanti n severalinitialconditions( t a b l e1 ) .-th eigenvalueYeneralzero-orderratecoefficient for produc-t i o n .Yero-ordersolid phaseratecoefficientforproduction.Yero-orderliquidphaseratecoefficientforproduction.Yimensionlesszero-orderratecoefficient(Y= Y ^ / v ) .9olumetricmoisturecontent.\ecayconstantin severalboundaryconditions( t a b l e1 ) .\ xeneralfirst-orderratecoefficientf o rdecay.| iirst-ordersolid phaseratecoefficientf o rdecay.| iirst-orderliquidphaseratecoefficientf o rdecay. Iimensionlessfirst-orderratecoefficient( = ^ i L / v ) .pulk density

9 7


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7 . I T E R A T U R E C I T E DAbramowitz,.,ndtegun,.A.970.andbookofmathe-maticalfunctions. DoverPublications,NewYork..Arnett,R.,Deju,R.A.,Nelson,.W.,andothers. 1976.Conceptualandmathematicalmodelingfth eHanfordgroundwaterflowregime.eportNo.ARH-ST-140,tlanticRichfieldHanfordCo.,Richland,Wash.Baron,.,ndWajc,..976.hermalpollutionofth eScheldtstuary.n;..ansteenkisteeditor).ystemsimulationnwateresources,orth-Hollandublishingo.,Amsterdam,p.193-213.Bastian,W.C,andLapidus,L.956.ongitudinaldiffusioninionexchangendhromatographiecolumns.initeolumn.JournalofPhysicalChemistry 60:816-817.Bear,J.972.ynamicsoffluidsinporousmedia.mericanElsevierPublishingCo.,NewYork.979.nalysisofflowagainstdispersioninporousmedia-Comments.JournalofHydrology0:381-385.Brenner,H.962.hediffusionmodeloflongitudinalmixinginbedsffiniteength.umericalvalues.hemicalEngi-neeringScience 17:229-243.Carslaw,H.S.andJaeger,J.D.959.onductionofheatinsolids. Secondedition. OxforduniversityPress,London.Cleary,R.W.971.nalogsimulationofthermalpollutioninrivers. In:SimulationCouncilProceedings l(2):41-45. j j Adrian,.D.973.nalyticalsolutionofth econvective-dispersivequationorationdsorptionnsoils.oilScienceocietyofAmericaProceedings7:197-199.3^Ungs,M.J.974.nalyticallongitudinaldis-persionmodelinginsaturatedporousmedia.ummaryreprintofpaperpresentedatth eFallAnnualMeetingofth eAmericanGeo-physicalUnion,SanFrancisco.DiToro,D.M.974.erticalinteractionsinphytoplanktonAnsymptoticigenvaluenalysis.roceedingsfhe7th Conference,reatakesesearch,nternationalssociationGreatLakesResearch,p.17-27.

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Duguid,..,ndeeves,.977.comparisonofmasstransportsingveragendransientainfalloundaryon-ditions.[ nW..ray,..inder,nd..rebbia(editors).initeelementsinwaterresources,entechPress,London,p.2.25-2.35.Gardner,W.R.965.ovementofnitrogeninsoil.r nW.V.BartholomewandF.E.Clark(editors).Soilnitrogen.gronomy10:550-572.mericanSocietyofAgronomy,Madison,Wis.Gershon,..,andNir,A.969.ffectofboundarycondi-tionsofmodelsontracerdistributioninflowthroughporousmediums. WaterResourcesResearch5:830-840.Glas,..,lute,.,ndcWhorter,..979.is-solutionandtransportofgypsuminsoils:.heory.oilScienceSocietyofAmericaJournal43:265-268.Jost,W.952.iffusioninsolids,liquids,gases.cademicPress,NewYork.Kay,.D.,andElrick,D.E.967.dsorptionandmovementoflindaneinsoils. SoilScience104:314-322.Keisling,T.G.,Rao,P.S.C.,andJessupR.E.978.er-tinentriteriafordescribingth edissolutionofgypsumbedsinlowingater.oilcienceocietyfAmericaJournal42:234-236.Kemper,W.D.,Olsen,J .,andDemooy,C.J.975.issolutionrateofgypsuminflowinggroundwater.oilScienceSocietyofAmericaProceedings39:458-463.Lahav,.,ndHochberg,.975.ineticsofixationofironandincappliedaseEDTA,eHDDHA,ndZnEDTAinth esoil. SoilScienceSocietyofAmericaProceedings39:55-58.Lapidus,L.,andAmundson,N.R.952.athematicsofadsorp-tioninbeds.VI.heeffectsoflongitudinaldiffusioninionexchangendhromatographieolumns.ournalfhysicalChemistry56:984-988.Lindstrom,..,aque,.,reed,..,ndoersma,.1967.heorynheovementfomeerbicidesnsoils:Lineardiffusionandconvectionofchemicalsinsoils.ournalofEnvironmentalScienceandTechnology1:561-565.^ j ^ [Boersma,L.971.theoryonth emasstransportofreviouslyistributedhemicalsn ateraturatedsorbingporousmedium. SoilScience111:192-199.

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j^ ber he t tinger,.975.noteonaLaplacetransformpairassociatedwithmasstransportinporousmediaandheattransportproblems.IAM,JournalofAppliedMathe-matics29:288-292.Lykov,A.v.,andMikhailov,Y.A.961.heoryofenergyandmasstransfer. Prentice-Hall,EnglewoodCliffs,N.J.Marino,..974a.ongitudinalispersionnaturatedporousmedia.ournalofth eHydraulicsDivision,Proceedingsofth eAmericanSocietyofCivilEngineers100:151-157.1974b.istributionofcontaminantsinporousmediaflow.aterResourcesResearch10:1013-1018.Mason,M.ndWeaver,W.924.hesettlingofsmallparti-clesinafluid. PhysiologicalReviews23:412-426.Melamed,D.,Hanks,R.J .,andWillardson,L.S.977.odelofaltlowinsoilwithasource-sinkterm.oilScienceSocietyofAmericaJournal41:29-33.Misra,,andMishra,B.K.977.iscibledisplacementofnitratendhloridenderieldonditions.oilcienceSocietyofAmericaJournal41:496-499.Ogata,A.,andBanks,R.B.961.solutionofth ediffer-entialquationfongitudinalispersionnporousedia.U.S.GeologicalSurveyProfessionalPaper11-A,A1-A9.Parlange,J.Y.,andStarr,J.L.975.ineardispersioninfiniteolumns.oilcienceocietyofAmericaProceedings39:817-819.andStarr,J.L.978.ispersioninsoilcolumns:Effectofboundaryconditionsandirreversiblereactions.oilScienceSocietyofAmericaJournal42:15-18.Pearson,J.R.A.959.noteonth eDanckwerts'boundaryconditionorontinouslowreactors.hemicalngineeringScience10:281-284.Reddy,K..,Patrick,Jr.,W.H.,andPhillips,R.E.976.Ammoniumdiffusionasafactorinnitrogenlossromfloodedsoils. SoilScienceSocietyofAmericaJournal40:528-533.Selim,H.M.,andMansell,R.S.976.nalyticalsolutionofth eequationfortransportofreactivesolute.aterResourcesResearch12:528-532.

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Shamir,..,ndHarleman,...966.umericalandanalyticalsolutionsofdispersionproblemsinhomogeneousandlayeredquifers.eporto.9,ydrodynamicsaboratory,Mass.Inst.Tech.Cambridge,Mass.Thomann,..973.ffectfongitudinaldispersionondynamicwaterqualityresponseoftreamsndrivers.aterResourcesResearch9:355-366.vanGenuchten,M.Th.977.nth eaccuracyandefficiencyofseveralnumericalschemesforsolvingth econvective-dispersiveequation.n_W..ray,..inder,nd..rebbia(editors),initeelementsnwateresources,entechPress,London,p.1.71-1.90..980.eterminingtransportparametersfromsolutedisplacementxperiments.esearcheporto.18,.S .SalinityLaboratory,Riverside,Ca..981.nalyticalsolutionsforchemicaltransportwithsimultaneousadsorption,ero-orderproductionandfirst-orderdecay.ournalofHydrology49:213-233.andWierenga,.. 1974. Simulationofone-dim-ensionalsolutetransferinporousmedia. NewMexicoAgri-culturalExperimentStationBulletinNo.628,LasCruces.Qj^ray,W.G.978.nalysisofsomedispersioncorrectedumericalchemesorolutionfheransportequation.nternationalournalfumericalethodsnEngineering12:387-404.Wehner,.F.,ndWilhelm,R.H.956.oundaryconditionsofflowreactor. ChemicalEngineeringScience6:89-93.

1 0 1


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APPENDIX A.TABLE OFL A P L A C E T R A N S F O R M S

^0(s)=L"^^F(t)dtThefollowingabbreviationsareusedInthetable:A = - e x p ( - 1^)B = erfcC^^)

2C=exp(at-ax)erf 0 ( 2 7 1 " " "a/t)2D=exp(at+ax)erfCC-^TT +a/t)

f(s) F(t)

^-x/s " ^2t-x/se/s A-x/ses B -x/ss/s 2t--x/s2x2t)B - xtA2s

1 0 2


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APPENDIXA.TableofLaplaceTransformsContinuedf(s)(t)^4t)''^î"erfc(^) (n=0,l,2,...)l+n/2^' """"'2/ts/ -x/s/sea2s-a A +4 ( C - D )e"^"^28-a (C+ D)-x/se/ va/s(s-a)

(s-a')

e( C-D )

/se" " ^ * ^ Â+7^(1-ax+2a^t)C2

^^"^ 7^( 1+ax+2a^t)D4 a-x/s^ " ^(2at-x)C+-^(2at+ x)D-x/s ~ -Â--ij ( 1+ax-2a^t)C

/s(s-a2) ^ " ^ ^+-Xr ( 1-ax-2a^t)D4 a - ^

/ -x/s/seXa+/s2t-x/sea+/s ( -a)A+ aA -aD103


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APPENDIX A.Tableo fLaplaceTransformsContinuedf(s)(t)

-x/se/s(a+/s)-x/ses(a+/s)( B-D)^x^s

a1122-x/s 7(14-ax+at+-=rax)Be Jas(a+/s) --jD--|( 2 + ax)Aa a

8-x/sr n-1 I

ye~ 1' ^f C+ 4 - ( 3 +2ax+4at )D-atA(8-a^)(a+/8) ^ * -x/s,A+TC-7-^(l+2ax+ 4at )D(8-a^)(a+/8)* **x/s^- C+- ( - 1+2ax+4a^t)D- - | A/s(s-a)(a+/s) 4a 4a-x/s ,3 ,3ea 4a^+- 3-2ax-4a^t)D2s(s-a)(a+/s) a a

0 4


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APPENDIXA.Tableo f Laplace TransformsContinuedf ( s )( t )( s - a ( a + / s )

(a+/s)^

-^- ( 1 +ax + 2 a ^ t )Ae " ^ " ^ ^ 1. ,2l}, 6 a ^ +^4at- 2ax - 1 )C

1 6 a ^ ^ [ 4 a ^ t -1 + 2a^(x+ 2 a t ) ^ lD

' ^ ^ 1+ 2 a ^ t )A - a(2+ax + 2 a ' ^ t )De"^^"2(a+/s)^ ( 1 + ax + 2 at )D -2 a t A- x / se/s(a+/s)^ 2 t A -( x +2at)D

( - 1 + ax+ 2 a ^ t )D + -| B - A8(a+/s)-x/s AtA _ _1( 2 + , , )Bea ' ^ ^ - ^ ^ > '4( - 2 + a x + 2 a 2 t )Da

-^ ( 3+ 2ax+ a ^ t + - | a^x^)B-x/ses^(a+/s)^ +-^ ( - 3 + a x + 2 a ^ t )D - -j ( 6+ a x ) tAa1 0 5


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106

APPENDIXA.Tableo f Laplace TransformsContinuedf ( s )( t )

, -x/s I - ( 3+ ax+ 2 a ^ t )A + - ^ ^ C /sea(s-a2)(a+/s)2 _ _i^ ^^ g^^+^ ^ ^ 2 ^ + 2a2(x+ 2 a t ) 2 ]D^- - r - ( 1 +ax+ 2 a ^ t )A/ Q2 2 a- x / s 8ae2 2(s-a) ( a + / s ) + [ _ i+ 2ax + 8 a ^ t + 2a^(x+ 2 a t ) ^ lD8 a ^

-x/s - ( - 1 +ax+ 2 a ^ t )A + - C 2 a 2 a 3/s(s-a) ( a + / s )- i r -[ 1- 2ax + 2 a ' ^ ( x+ 2 a t ) ' ^ ]D8 a - ^

1 6 a ^ ^ [ 1+a(4a^t- l ) ( x + 2 a t )4 a^(x + 2 a t ) ^ ]D22/, , x 2^(s-a ( a + / s ) 1 6 a ^ ^ ( 1+ax - 2 a ^ t )C 1 2 a 3^ [ - 3 + 4a^t +a^(x + 2at)^]A

- ! ^ ^ - ^[ 1 + 2ax + S a ' ^ t +|- ( x + 2 a t ) ^ ]D( a + / s ) - ^ 2- at(4+ ax + 2 a t )Ae " ^ ' ^ ^( 2 +ax+ 2a^t)A(a+/s)^ -[ x+ 3 a t+ " I ( x + 2 a t ) ^ ]D


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APPENDIXA.TableofLaplaceTransformsContinuedf(8) F(t)

-x/se I t+ I "(x+ 2at)^]D -t(x+2at)A/s(a+/8)^

-x/se -[ - 3 + 3ax+ 14a\ + 2a^(x + 2at)^]A12a^(s-a^)(aVs)^ +- [ C +(2ax-1 )D ]

1 6 a - ^"ifci ^ ^ ' ^ " ^ 2at)(x+6at)+2a(x + 2at)^]D

/ -x/s/se(a+/s)^

2t[ 2+ 2ax + a\ +|-( x +2at)^]A-[ x +at+a(x+ 2at)(x+ 3at)

2 + (x+ 2 a t ) - ^ lD-x/se [ t+j (x+ 2at)(x+ 4at)+ f(x+ 2at)^]D

-1[3x+1 0at +a(x+2at)^]A(a+/s)^

-x/s y[ 4 t+(x+ 2at)^]A-[t(x+ 2at)+ - i (x+2at)^]D0

/s(a+/s)^

-x/s 1 - n2(a+b)^ 2(a-b)(s-a^)(b+/s) +^^exp(b^t+ bx)erfc( +b/t)a-b

107


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APPENDIX B .ELECTED C O M P U T E R P R O G R A M S Thisappendixcontainsaseriesoftableslistinguser-orientedcomputerrogramsfeveralkeyanalyticalolutionsfheone-dimensionalconvective-dispersivetransportequation.ach programsugmentedithamplenputatandssociatedlistingsfhecomputerprintout.hesampleprogramson-sideredarethoseforcasesAl(togetherwithA2),A3,B14,an dC8.numericalcomputersolution(Nl)isalsoprovided.hissolutionmaybeusedforthosecaseswherenoanalyticalsolu-tionisavailable.Table (page111)liststh emostsignificantvariablesinth ecomputerprograms.henamesofsimilarvariablesindifferentprogramshavebeenkeptheamewheneverpossible.ablelistsheamplenputatausedorth efiveomputerpro-grams.listingofth efunctionEXF,whichiscommontoal lprogramsxceptl,siveneparatelynable.hisfunctionwilleiscussedelow.istingsfherogramsthemselves,ogetherithheomputerutput,reivenntables5and6forcaseAl,tables and8forcaseA3,tables9and10forcaseB14,tables11and12forcaseC8,andtables13and14forcaseNl(thenumericalsolution).ThefunctionEXF(A,B),whichappearsinal lprogramsexceptNl,islistedintable4.hisfunctiondefinesth eproductofth eexponentialfunction(exp)andth ecomplementaryerrorfunction(erfc)asfollows

EXF(A,B)-exp(A)erfc(B) Bl]where

erfc(B)=- /"exp(-T^)dx. B2]T woifferentpproximationsresedorXF(A,B).or0


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aj.2548296 3 2=-.28449673 3-1.421414 3 4=-1.453152851.061405 [B5]

andorB>3seealsoequation7.1,14]fAbramowitzandStegun1970):EXF(A,B)j-exp(A-B^)/(B+0.5/(B+l./(B+B6]

1.5/(B +2./(B +2.5/(B +1.)))))).FornegativevaluesofB,th efollowingadditionalrelationisused:

EXF(A,B)=2exp(A)-EXF(A,-B), B7]ThefunctionEXF(A,B)annotbeusedforverysmallorverylargevaluesofitsargumentsA,B,hefunctionreturnszeroforth efollowingtwoconditions:

|A| 170 A-B^l>170B8]B0TheomputerrogramsorhenalyticalolutionsrellwrittenindoubleprecisionFORT R A NIV;heyproduceanswers

thatavenccuracyfteastourignificantigits.Initially,omeroblemserencounteredithnccurateevaluationofth eapproximatesolutionsforth efinitesystems,especiallythosethatareapplicabletoflux-typesoilsurfaceboundaryonditionscases4 ,8).heseapproximatesolu-tionsequirehedditionndubstractionferyargenumbers,eadingoargeoundoffrrorsndnverallaccuracyofatmostthreesignificantplaceswhenP>100.hefollowingrocedure,irstsuggestedbyBrenner1962)^asusedtoderivealternativeandmoreeasilyevaluatedformsofth eapproximatesolutions.Asnexample,onsiderth eapproximatesolution ^ ^ofaseA4 . Thissolutioncanbewritteninth eform

c^4-c^2 ^G(x,t) B9] where^shenalyticalolutionfase2ndhereG(x,t)isgivenby

2/ 2

109


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exp(vL/D)erfc^ "' ' ^^i/^1BIO]L 2(DRt)^2JThepproximateolutionssednl yorelativelyargevaluesfhergumentnherfc-functionfBIO].suitableasymptoticexpansionforerfcistherefore(equation7.1.23ofAbramowitzandStegun1970):erfc(B)- ^^[l H ("D"1 >3... .(2m^l) ] )SubstitutingBll]ntoBIO]ndcombiningappropriatetermsallowseveralfheeadermsnheeriesoecancelled. Additionalsimplificationleadstoth enewform

2 / 2o/ . \ /^vtv .vL R ,^- vtv

-^1' 2 D t r o T (m-l)vti:-l)'^^I1.3....(2m-l)]( > [2L-X-^]1 ^ ^ S S : i B 1 2 ] "='2L-X+^)

Thisseriesexpansionconvergesrapidly;atmostfivetermsofth eseriesareneededtogenerateanswersthathaveanaccuracyof4significantdigits.nimportantadvantageofB12]sthathexpressionowanevaluatedasilyningleprecisionrithmeticithoutffectingheour-placeaccuracy.owever,th edoubleprecisionformatofth ecomputerprogramsaseenetainedorheresent.herevernecessary,symptoticexpansionssimilartoB12]orcaseA4wereerivedlsoorhetherasesnvolving initesystem;theyhavebeenincludedinth ecomputersolutions.T henumericalolutionNl,istedintable3,sbasedonalinearfiniteelementapproximationofth espatialderivativesinth eransportequationandathird-orderfinitedifferenceapproximationofth etimederivative.hetheoreticalbasisofthisarticularschemesdiscussedlsewhere(vanGenuchten1977anGenuchtenandGray978)ndwillnotbereviewedhere.heprogramassumeshathenodalspacingDELX)andth etimeincrement(DELT)remainconstant.

110


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Table 2.List of th eost significant variables inhecomputerprogramsVariableA P R X

BETA

CC(I)COCA,C B

CICONC

DefinitionVariableondicatefheolutionorsemi-infiniteystemanesedopprox-imateth esolutionforafinitesystem:PRX=x/L-0.9+8/P. (A3,C8).Dummyvariableforth eI-theigenvalue,G(I).(A3,C8).Dummyvariableforconcentration, .Nodalvaluesofconcentration(Nl).Constantinputconcentration,CConstants(C^,C^)inseveralboundaryconditions(seetable1). (B14,Nl).Constantinitialconcentration,C^* Concentration,c.

CONS(V,D,R,...) Subroutinetocalculateth econcentrationforafiniteprofile(A3,C8).DDBND

DELTDELXDONE

DT DXDZERO

Dispersioncoefficient.Constant(a)inseveralboundaryconditions(seetable1). (B14,C8).Timeincrementinnumericalsolution(Nl).Nodaldistanceinnumericalsolution(Nl).First-orderratecoefficientfordecay,y.(C8,Nl).Incrementintimeforcomputerprintout.Incrementindistanceforcomputerprintout.Zero-orderateoefficientforproduction,y. (B14,Nl).

I l l


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Table.List of theostcomputerprogramsContinued significantariablesnheVariable DefinitionElGENl(P) Subroutinetoalculateheirst0eigen-values(.)forheeriesolutionffiniterofileith irst-typeoundarycondition(A3).E1GEN3(P) Subroutinetoalculatehefirst0igen-values( . )forheeriesolutionffiniterofileith hird-typeoundarycondition(C8).EXF(A,B)G(I)KINIT

Functiontocalculateexp(A)erfc(B).Vectorontainingheirst0igen-val-ues( 3)fortheseriessolutions(A3,C8).Inputodefortheinitialconditioninthenumericalsolution.fKINIT=-1,thecon-stantinitialconcentration(CI)isread n :ifINIT=,henitialoncentrationisspecifiedintheprogramitself; fKINIT=1 ,thendividualodalaluesfheoncen-tration,C(I),arereadinseparately(Nl).

KSURF Inputcodefortheupperboundaryconditionintheumericalolution.fSURF ,first-typeboundaryconditioni sspecified;ifKSURF= ,athird-typeboundaryconditionisspecified(Nl).N Numberoftermsintheseriessolution; fNequalszerointheprintout,heapproximatesolutionwasused(A3,C8).NC Numberfxamplesonsiderednachro-gram.NE Number felementsinthenumericalsolution(Nl).NN Numberofnodesinthenumericalsolution:NN =NE+1(NI).NSTEPS Numberoftimestepsinthenumericalsolu-tion(Nl).

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Table2.-computerVariablePRTTOTITIMETITLE(I)

T MTOL

VWO

XX(I)XIXLXM

Listofth emostsignificantvariablesinth eprogramsContinuedDefinitionColumnPecletnumber:P*vL/D.Retardationfactor.Dummyvariablefortor(t-t^).Durationoftracerpulseaddedtoprofile,t^.Initialtimeforcomputerprintout.Time,t.Vectorcontaininginformationoftitlecard(inputlabel).Finaltimeforcomputerprintout.Convergencecriterionforseriessolution(A3,C8).Averagepore-watervelocity,v.Dimensionlesstime:WO=vt/x. Equalsnum-berofporevolumesifx*L.Distance,x.Nodalcoordinatesinnumericalsolution(Nl).Initialdistanceforcomputerprintout.Columnlength,L.(A3,C8).Maximumdistanceforcomputerprintout.

113


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Table3.--Sampleinputdataforthe5computerprogramslistedinthisbulletinColumn: 12345678901234567890123456789012345678901234567890123456789012345678901234567890Program Card 1 2 3 4 5 6 7 8

A l 12 2 EXAMPLE Al-1 (P"5)3 1.0 4.0 1 . 0 1000.0 . 0 1 . 04 . 0 2.0 20.0 5.0 5.0 25.05 EXAMPLEA l-26 25.0 37.5 3.0 5.0 . 0 1 . 07 lOO.O . 0 100.0 1 . 0 1 . 0 30.0

A 3 12 2 EXAMPLE A3-1 (P=5)3 1.0 4.0 1 . 0 1000.0 . 0 1 . 0 .00014 . 0 2.0 20.0 20.0 5.0 5.0 25.05 EXAMPLEA3-26 25.0 37.5 3.0 5.0 . 0 1 . 0 .00017 lOO.O . 0 100.0 100.0 1 . 0 1 . 0 30.0B1 4 12 1 EXAMPLEB14-13 25.0 37.5 3.0 . 5 . 2 5 . 0 . 0 10.04 . 0 5. 0 100.0 2 . 5 2 . 5 7 . 5C 8 12 2 EXAMPLEC8-1 (P=5)3 1.0 4.0 1 . 0 1000.0 . 5 . 2 5 . 0 1 . 04 . 0 2.0 20.0 20.0 5.0 5.0 25.0 .00015 EXAMPLEC8-26 25.0 37.5 3 . 0 5.0 5.0 . 2 5 . 0 1 . 07 . 0 5.0 100.0 100.0 2 . 5 2 . 5 12.5 .0001N l 12 2 EXAMPLE A3-1 (P=5)3 40 125 -1 . 5 . 2 5.0 . 0 . 0 . 04 1.0 4.0 1 . 0 . 0 1 . 0 . 0 1000.05 EXAMPLEB14-16 40 125 3 -1 2 . 5 . 1 2 . 5 . 5 . 0 . 2 57 25.0 37.5 3 . 0 . 0 . 0 10.0 1000.0


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4.Fortranlisting o ft h efunctionEXF(A,B)= exp(A)erfc(B)EXF

FU N C T I C NEXPA,BJPURPOSE:TOCALCULATEEXPA)ERFCBJ IM PL I C I TREAL*8A-H,0-Z)EXF=0.0IFIDABSA).GT.170.J.AND.B.LE.0.))RETURNIF(B.NE.O.O)GOTOEXF=DEXP(A)RETURN1C=A-B*BIFi(DABS(C).GT.170.).AND.(B.GT.O.))RETURNIF(C.LT.-170.)GOTO4X=DABSB)IFX.GT.3.0)GOTOT=l./(!+.32759I1*X)Y=T*i.2548296-T*(.2844967~T*(1.421414-T*(1.453152-1.061405*T)J)J GOTO32Y=.5641896/(X+-5/(X+l./


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Table.Fortran listingf computer programAl. T he functionXF is listed intableMAIN

CcC *CNE-DIMENSIQNALCCNVECTIVE-DISPERSlVEEQUATION Al * CCEMI-INFINITEPROFILE *CCPRODUCTIONORCECAY CINEARADSORPTICN R ) * CONSTANTINITIALCONCENTRATIONtCIJ * CNPUTCGNCENTRAT ICN =CO(T.LE.TOJ * C0 (T.GT.TO) *CC IMPLICITREAL*8(A-H,C-Z)DIMENSIONTITLE(20)CC READNUMBEROFCURVESTOdECALCULATEDREAD(5,1000)NCDO^K=1,NCREAD(5,I001)TITLEWRITE{6,1002)TITLEC C READANDWRITEINPUTP ARAMETERSREAD(5,1003)V,D,R,TO,CI,C0READ(5.1003i XI,DX,XM,TI,TTM WRITE(6,1004) V,D,R,TC,CI,COC C D=D/RV=V/RIFiDX.E.O.) DX=1.0IF(DT.EQ.O.) DT=1.0IMAX=(XM+DX-XIJ/DXJMAX=(TM+DT-TI)/DTE=0.0DO 4 J=1,JMAXIF(IMAX.GE.J) WRITE(6,1005)TIME=TI+(J-1)*DTDO 4 I=1,IMAXX=XI+(I-1J*DXVVO=0.0IFiX.Et.O.) GO TO 1 VVO=V*R*TIME/XDO 2 M=l,2A1=0.01 1 6


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MAINA2=0.0T=TIME+(1-M)*T0IFT.LE.O.)GOTOCM=(X-V*T)/DSQRT(4.*D*T)CP=iX+V* T)/0SQRT4.*D*T)Q=V*X/DA1=0.5*(EXF(E,CMJ+EXF(C,CP))A2=0.5*EXF(E,CM)+V*DSCRT{.3183099*1/jJ*EXFi-CM*CM,c)-0.5* (l.+w+V*V1*T/0)*EXF(Q,CP)IF(M.EG.2)GOTO3CONC1=CI+(CO-CI)*A1CONC2=CI+(CO-CI)*A22CONTINUE3CONC1=CONC1-CO*A1C0NC2=C0NC2-C0*A2 4WRITE(6,1006)X,TIMEtVVCCCNCl,C0NC2

F0RMAT(I5)FCRMAT(20A4)F0RMAT(lHl,10X,82iI*)/ll ,1H*,80X,i*/li,I*,9X,ONE-DIMENSIONAL1QNVECTIVE-DISPERSIVECUATION,25X,l*/iiX,1H*OX,1H*/11X,1H*,29X,SEMI-INFINITEROFILE ,50X,1H*/IX,I*,9X,NORODUCT IONND3ECAY',48X,1H*/11X,H*,9X,LINEARDSORPT IONR),0X,1H*/1X,I*,94X,CONSTANTNITIALONCENTRAT IONCI),6X,1H*/1X,IH*,9X,INPUT5C0NCENTRATI0N OT .LE.TO ,37X, lH*/liX,lH*,29X,=T.GT.TOJ6,37X,lH*/llX,lH*,80X,lH*/llX,lH*,20A4,lH*/ilX,lH*,80X,lH*/llX,82(l7H*))0PMAT8F10.0)FORMAT(//ilX,INPUTP ARAMETERS/IIX,16(lH=i//IIX, V=,F12.4,15X,ID= ,F12.4/ilX,R=,F12.4,15X,T0=',F11.4/11X,CJ=,Fii.4,15X,2C0=,F11.4}FORMAT(///IIX, DISTANCE,11X , TIMEt7X,PuKEVOLUME,12X,CONCENTR1ATI0N/14X,(X),i3X,TJ,11X,(VVOJ ,X, F IRST-TYPEBC,4X,THIR2D-TYPEBC)F0RMAT(4X,3(5X,F10.4),3X,F12.4,5X,F12.4J STOP END

1 1 7


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Table.Sample output fromomputer programAl4c3c4c4c3ee:(c3e:(e:e4e4c4c4(3ee4(:Oc:tc4c;tc30e;e:9e:O(:ec:OE:ec4c:c3O(*4t4(4[^ **1t4c4e^

* ON-OIMENSIONAL CONVECTIVE-CISPERSIV EQUATIONSEMI-INFINITEPROFILENOPRODUCTIONANDDECAYLINEARADSORPTIONR)CONSTANTINITIALCONCENTRATION(CDIN PUTCONCENTRATION=CO(T.LE.TO)= 0(T.GT.TO)EXAMPLEAl-1 (P=5)

: c 4 e : 9 e : e c3 t c : c 3g c 4 c : e c 30 c : ^ : e e : e c : i e : Q c : ^ : e c i O c : e e 4 c :9 e 3 e c : j c 4 c 4 e 3 g c * : e aS . : e ( 3 S c : 4 e ^ 4 ( 4 : e c X e 3f i c : i c

INPUTPARAMETERS

V = 1.0000R = 1.0000C I = 0.0 D=.0000TO=000.0000CO= 1.0000

DISTANCE(X)0.02.00004.00006.00008.000010.000012.000014.000016.000018.000020.0000

TIME( T )5.00005.00005.00005.00005.00005.00005.00005.00005.00005.00005.0000

POREVOLUME(WO)0.02.50001.250CC.83330.^2500.50000.41670.35710.3125C.2778G.25C0

CONCENTRATIONFIRST-TYPEBC THIRD-TYPE BC1.00000.90360.77310.62090.46480.32240.2040.12150.06350.03240.014

0.76400.63760.50230^37120.25590.16380.09700.05300.02660.01230.0052

DISTANCE( X )0.02.00004.00006.00008.000010.000012.000014.000016.000018.000020.0000

TIME( T J 10.000010.0000IC.OOOO10.000010.000010.000010.000010.000010.0000lO.QOOO10.0000

POREVCLUME(WO)0.05.000C2.50001.66671.25001.0000C.8333C.71430.625CC.5556C.500C

CONCENTRATIONFIRST-TYPEB C THIRD-TYPEBC1.00000.96260.90860.83770.75170.65440.55120.44l0.3500.26410.1909

0.88450.81980.74240.65480.56100.46570.37380.28950.21610.15510.1070

DISTANCE( X )0.02.0000

TIME(Ti15.000015.0000

POREVOLUME(VVC)CO7.50C0

CONCENTRATIONFIRST-TYPEBC THIRD-TYPEBC1.0000.93650.9818.9003

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4.00006.00008.000010.000012.000014.000016.000018.000020.0000

15.000015.000015.000015.000015.000015.000015.0C0015.000015.0000

3 . 75 C C2.500C1.875C1.50C01.25CC1.C714 0.9375C.8333C.750C

0.95490.91810.87070.81290.74560.67070.59070.50870.4278

0.85490.8004 0.73750.66770.59310.51610. 4 3 9 40.36560.2969

(X)0.02.00004.00006.00008.000010.000012.000014.000016.000018.000020.0000

TIME(T )20.000020.000020.000020.000020.00002C.000020.000020.000020.00002C.000020.0000

POREVOLUME(VVC)COIC.OOOO5.0OC03 . 2 3 3 32.50002.C0GC1.6671.^2861.25001.1111l.OOCO

CONCENTRATIONFIRST-TYPEBC THIRD-TYPEBC1.00000.99020.97560.95510.92780.893 30.85110.80140. 7 4 4 90.68270.6162

0.96300.94160.91420.88010.83920.79160.73790.67880.61570.55010. 4 8 3 7

(X)0.02.00004.00006.00008.000010.000012.000014.000016.000018.000020.0000

T IMET )25.000025.000025.000025.0C0025.000025.000025.000025.000025.000025.000025.0000

PO REVOLUME(VVC)CO12.50006.25004.16673.125C2.50CC2.08331.76571.56251.38891.2500

CONCENTRATIONFIRST-TYPtf1.00000.994 4 0.98600.97400.95780.93680.91030.87800.83990.79600.7467

BCHIRD-TYPEBC0.97760.96460.94760.92610.89950.86770.83040.78790.74040.68860.6334119


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: i :^ :^: :tlii: :(,:i i(l7^:t :^:tt:iii:i iti:ti : ilt:fli:ti: i(li:

****

ONE-DIMENSIONAL CONVECTIVE-CISPERSIVE EQUATIONSEMI-INFINITE PROFILENO PRODUCTION AND DECAY LINEAR ADSORPTION R)CONSTANT INITIAL CONCENTRATION (CDINPUT CONCENTRATION = CO (T.LE.TO)= 0 (T.GT.TO)EXAMPLE Al-2

*

*3 0 c ai e

4c :0 c 4c 4c ec 4e :( :A c 9c :^ (^ Q c :^ e :^ 4c c 3 9e :Q e: ( :9 c 9e :C c ec :0 c # 3 4 c 4 c 41 :(c 4t :(( 3 0 c c 4c :(c : c :te : c J C:^

INrUT PARAMETERS V = R = CI =

25.00003.00000.0D = TO = 37.50005.0000CO = 1.0000

DISTANCE T IME PORE VCLUME CONCENTRAT I ON (X) T) (WO) FIRST-TYPE BC THIRD-TYPE100.0000 1.0000 C.2500 0.0000 0.0000100.0000 2.0000 C.5000 0.0000 0.0000100.0000 2.0000 0.7500 0.0000 0.0000100.0000 4,0000 l.COCO 0.0000 0.0000100.0000 5.0000 1.250C 0.0000 0.0000100.0000 6.0000 1.5000 0.0000 0.0000100.0000 7.0000 1.7500 0.0010 0.0008100.0000 8.0000 2.0000 0.0113 0.0088100.0000 9.0000 2.2500 0.0563 0.0465100.0000 10.0000 2.500C 0.1655 0.1439100.0000 11.0000 2.7500 0.337b 0.3059100.0000 12.0000 3.0000 0.5332 0.4987100.0000 13.0000 3.2500 0.6975 0.6700100.0000 14.0000 3.5CCC 0.7b02 0.74 100.0000 15.0000 3.7500 0.7509 0.7592100.0000 16.0000 4.0CC0 0.6228 O.0474100.0000 17.0000 4.25C0 0.4485 0.4795100.0000 18.0000 4.50C0 0.2840 0.3124100.0000 19.0000 4.7500 0.1607 0.1816100.0000 20.0000 5.00C0 0.0825 0.0956100.0000 21.0000 5.25CC 0.0389 0.0462100.0000 22.0000 5.50C0 0.0171 0.0208100.0000 23.0000 5.7500 0.0071 0.0088100.0000 24.0000 6.0000 0.0028 0.0035100.0000 25.0000 6.2500 0.0010 0.0013100.0000 26.0000 6.5000 0.0004 0.0003100.0000 27.0000 6.7500 0.0001 0.0002100.0000 28.0000 7.0000 0.0000 0.0001100.0000 29.0000 7.2500 0.0000 0.0000100.0000 30.0000 7.5000 0.0000 0.0000

BC

12 0


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7.Fortran listing o fcomputerprogram A 3 .hefunction EXFi slisted in 4

MAIN

*NE-DIM ENSI CNAlCCNVECTIVEDISPERSIVEQUATION AJ IRST-TYPEBOLNDARYC ONDITI ON *INITEP R O FILE *OP R O DU CTI CNCC E C AY INEARADSORPTI CN( R ) *ONSTANTINITIALCCNCENTRATIN(CIJ *NPUTC ON C ENTRATION =C D(T.LE.TO ) *0T.GT.TO) *

IMPLICITREAL*8(A-H,C - 2}C GMMCNG ( 2 0 )D IMENSI ONTITLE20 ) READNUMBERO FCURVESTOB EGENERATEDR E A 0 5 , 1 0 0 0 )NC

DO4K=1,NCR EAD( 5 , 1 0 0 1 )TITLEWRITE(6, 1 0 0 2 )TITLER EADANDWRITEINPUTPARAMETERSR EAD( 5 , 1 0 0 3 )V,D,RtTO,C ItC 0,TOLREAD5,1003) XI,DX,XM,XL,TI.T,TM WRITE(6,1004) V,D,RfTC,CI,CO,XL,TOLD=D/RV=V/RIF(OX.Q.O.) DX=1.0IF(DT.EQ.O.) DT=1.0XM=DM IN 1 ( XM ,XL )P=V*XL/DIMAX=(XM+DX-XI)/DXJMAX=TM+DT-TI)/DTIFP.LE.100.) CALL EIGENKP)DO 4 J=1,JMAXTIME=TI+(J-1)*DTIF(IMAX.GE.J) WRITE(6,1005)DO 4 I=1,IMAXX=XI+(I-1)*DX

12 1


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c c

MINvva=o.oIFX.EQ.O.)GOTOVVO=V*R*TIME/X1DO2M=l,2C=0.0T=TIME+(1-M)*T0IF(T.LE.C.)GOTO2CALLCCNS(C,V,D,X,T,XL,TOL,NJ IF(M.EQ.2)GOTO3CCNC=CI+iCO-CIi*C2CONTINUE3C C N C = C C N C-CO* C4RITE(6,1006) X,TIME,VVCCCNC,N

1000ORMATdS)1001ORMAT(20A4}1002F0RMAT(1H1,10X,82(IH*)/I IX ,1H*,80X ,1H*/11X,IH*,9X,'ONE-DIMENSIONAL1CCNVECTIVE-DIS PERSIVEECUATIN',25X,1H*/11X,1H*,80X,lH*/llX,lH* t92X,FIRST-TYPEBOUNDARYCCNDITI ON ,42X,1H*/1IX,1H*,9X,'FINITEPROFI3LE,57X, lH*/llX,lH*,8CX,lh*/llX,lH* ,9X, NPRODUCTIONC f tDECAY,494X,1H*/11X,1H*,9X,LINEARACSORPTIONR ,50X,1H*/11X,IH* ,9X,CONST5ANTINI T I ALCONCENTRAT ICN(CI),36X,1H*/11X,1H*,9X,INPJTCCNCNTR 6 A T I 0 N=CO(T.LE.TO) ,37X,1H*/11X,1H*,29X,= T.GT.TOJ,37X,1H*7/llX,lH*,80X,lH*/llX,lH*,20A4,lH*/llX,lH*,80X,iH*/liX,d2(lH*iJ1003FORMAT(8F10.0)1004F0RMAT(//11X,INPUTP ARAMETERS'/IIX,161H= J//IIX,V=,F12.4,15X,ID=,F12.4/11X,R=,F12.4,15X,T0=,F11.4/11X,CI=,F11.4,15X,2C0=,F11.4/11X,XL=,F11.4,15X,T0L=,F10.6i1005FORMAT(///IIX , DISTANCE,IIX, TIME ,7X,POKEVOLUME,6X,CNCENTRAIT ION .SX,NUMBER/14X,(X),13X, (TJ,11X,(VVOi,14X,


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EIGENlSUBROUTINEEIGENKP) PURPOSE:TOCALCULATETHEEIGENVALUES IMPLICITREAL*8CA-H,0-Z)COMMONG(20)BETA=0.1DC41=1,20J=01J=J+1IF(J.GT.15)GOTO3DELTA=-0.2*(-0.5)**J2ET 2 = BETABETA=BETA+DELTAA=BET2*0C0S(BET2)+0.5*P*DSIN(BET2)B=BETA*DC0S(BETA)+0.5*P*0SINCBETA)IF(A*B) 1,3,23G(I)=(BET2*B-BETA*AJ/(E-A)4BETA=BETA+0.2WRITE(6,1000) (G(I),I=l,20i1000 F0RMAT(//11X,CALCULATED EIGENVALUESVliX,22i1H=J/(8X,5F12.6/JJRETURN END

12 3


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C C N SSUBROUTINECONSC,V,D,X,T,XL,TOL,I)CC PURPOSE:TOCALCULATECONCENTRAT IONC C IM PLICITREAL*8(A-H,C-Z)CCMMCNG(20)1=0P=V*XL/DQ=V*X/DAPRX=X/XL-0.9+8./P IFAPRX.LT.O.)GOTOIF(


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8.Sampleoutputfromcomputerprogram A3

ONE-DIMENSIONALCONVECTIVE-CISPERSIVEQUATIONFIRST-TYPEBOUNCAKYCCNDITICNFINITEPROFILENOPRCUCTIONORDECAYLINEARADSORPTION R )CONSTANTINITIALCONCENTRATION(CDINPUTCONCENTRATION=CO(T.LE.TOi= 0(T.6T.T0)EXAMPLEA3-1 (P=5)

PARAMETERS = 1.0000 = 1.0000I = 0.0L = 20.0000

D 4.0000T O = 1000.0000CO = 1.0000TOL = 0.000100EIGENVALUES

2.380644 5.163306.1515641.2149064.310123 20.5414623.6671866.7965649.928469 36.1972729.3333822.4702985.607854 51.8844265.0232768.1624211.301816

TIME PORE VCLUME CONCENTRATION NUMBER (X) (T) (WO) (C) OF TERMS0.0 5.0000 CO 1.0000 02.0000 5.0000 2.500C 0.9036 64.0000 5.0000 1.2500 0.7731 6 6.0000 5.0000 C.8333 0.6209 6 8.0000 5.0000 C.625C 0.4648 610.0000 5.0000 C.50C0 0.3ZZ5 6 12.0000 5.0000 0.4167 0.2064 6 14.0000 5.0000 C.3571 0.1216 6 16.0000 5.0000 C.3125 0.0661 6 18.0000 5.0000 C.2778 0.0348 6 20.0000 5.3000 C.250 0.0240 6

TIME PCBE VOLUME CONCENTRATION NUMBER (X) (T) (WO) (C) OF TERMS 0.0 10.0000 CO 1.0000 02.0000 10.0000 5.COCO 0.9626 6 4.0000 10.0000 2.50CC 0.9086 612 5


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6.0000 10.0000 1.6667 0.8378 6 8.0000 10.0000 1.250C 0.7520 610.0000 10.0000 l.OCOC 0.6553 612.0000 IC.OOOO C.8323 0.5536 6 14.0000 10.0000 C.7143 0.4544 6 16.0000 10.0000 C.6250 0-3666 618.0000 10.0000 0.5556 0.3013 6 20.0000 10.0000 C.5C0 0.2747 6

DISTANCE TIME POPE VCLME CONCENTRATION NUMBER (X) (T) (WO) C) OF TERMS 0.0 15.0000 CO 1-0000 02.0000 15.0000 7.500C 0.*819 64.0000 15.0000 3.7500 0.9553 6 6.0000 15.0000 2.50C0 0.9189 48.0000 15.0000 1.875C 0.6726 410.0000 15.0000 1.5000 0.8170 412.0000 15.0000 1.2500 0.7544 414.0000 15.0000 1.071 0.6889 416.0000 15.0000 C.9375 0.6271 418.0000 15.0000 0.8333 .5788 420.0000 15.0000 C.7500 0.5586 6

DISTANCE TIME PORE VOLUME CONCENTRATION NUMBER (X) (T) (VVC) (CJ OF TERMS 0.0 20.0000 CO 1-0000 02.0000 20.0000 IC.CCCO 0.9905 44.0000 20.0000 5.00CO 0-9764 46.0000 2C.0000 3.3333 0.9569 48.0000 20.0000 2.5000 0.9316 410.0000 20.0000 2.0000 0.9005 412.0000 20.0000 1.6667 0.8648 414.0000 20.0000 1.4286 0.8266 416.0000 20.0000 1.25C0 0.7899 418.0000 20.0000 1.1111 0.7608 420.0000 20.0000 1.0000 0.7485 4

DISTANCE TIME PORE VCLUME CONCENTRATION NUMBER (X) (T) (VVC) (C) OF TERMS 0.0 25.0000 CO 1.0000 02.0000 25.0000 12.50CC 0.9949 44.0000 25.0000 6.2500 0.9872 46.0000 25.0000 .1667 0.9766 48.0000 25i0000 2-1250 0.9626 410.0000 25.0000 2.50CC 0.94i>5 412.0000 25.0000 2.0823 0.9255 414.0000 25.0000 1.7857 0.9041 416.0000 25.0000 1.5625 0.8833 418.0000 25.0000 1.3889 0.8668 420.0000 25.0000 1.25C0 0.8598 4

12 6


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:ti:^:^:^:^:^^:^^^:fliii::îîiiiîtîl;î(lt:i^^î^:^:li:i^^^:(li^ ***

*

ONE-DIMENSIONALCONVECT IVE-CISPERSIVhEgATI NFIRST-TYPEBOUNDARYCCNDITICNF I N I T EP ROFILENOPRODUCTIONO i< DECAy LINEARADSORPTION(R )CONSTANTI N I T I A LCONCENTRATION(CDIN P UTCONCENTRATION=CO(T.LE.TOi= 0(T.GT.TO)EXAMPLEA3-2 *

*#:c*****:cXc************** *** Jit************:ec****^*^

INPUT PARAMETERSV= 25.3C00R= 3.0000CI= 0.0XL= 100.0000

D = 37.5000TO= 5.0000CO= 1.0000TOL = C.000100CALCULATEDIGENVALUES3.050337.102126.15669C2.2151145.27819118.3464121.4199934.49893C7.5830530.67208333.7656706.8634679.965076^3.0701436.17832749.2893142.4028195.5185856.6363811.75o003DISTANCE T IME P O REVOLUME CONCENTRATION NUMBER(XJ (T) LVVCJ C) OFTERMS100.0000 1.0000 C.250O 0.0000 0100.0000 2.0000 C.5000 0.0000 0100.0000 3.0000 C.750C 0.0000 0100.0000 4.0000 l.OOCO 0.0000 0100.0000 5.0CO0 1.2500 0.0000 0100.0000 6.0000 1.500C 0.0000 0100.0000 7.0000 1.7500 0.0013 0100.0000 8.0000 2.COO0 0.0138 0100.0000 9.0000 2.25CC 0.0660 0100.0000 IC.OOOO 2.50C0 0.1872 0100.0000 11.0000 2.750C 0.3697 0100.0000 12.0000 3.0000 0.5677 0100.0000 13.0000 3 .2500 0.7250 0100.0000 14.0000 3.5000 0.7920 0100.0000 15.0000 3.7500 0.7427 0100.0000 16.0000 ^.OOCO 0.5983 0100.0000 17.0000 ^.2500 0.4174 0100.0000 18.0000 4.5000 0.2557 0100.0000 19.0000 4 .7500 0.1399 0

127


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l O O . C O O 0 . 0 0 0 0. 0 0 C C1 0 0 . 0 0 0 01 . 0 0 0 0. 2 5 C 01 0 0 . 0 0 0 02 . 0 0 0 0. 5 C C Cl O O . O O C O3 . 0 0 0 0. 7 5 C Cl O O . C O O OA . 0 0 0 0. C C C Cl O C . O C O O5 . 0 0 0 0. 2 5 0 C1 0 0 . 0 0 0 06 . 0 0 0 0. 5 0 C C1 0 0 . 0 0 0 07 . 0 0 0 0. 7 5 C 01 0 0 . 0 0 0 08 . 0 0 0 0.COCOl O O . t O O O9 . 0 0 0 0. 2 5 0 01 0 0 . 0 0 0 00 . 0 0 0 0. 5 C C 0 0 . 0694 00 . 0 3 1 7 00 . 0 1 3 5 00 . 0 0 5 4 00 . 0 0 2 0 120 . 0 0 0 7 100 . 0 0 0 3 100 . 0 0 0 1 1 00 . 0 0 0 0 100 . 0 0 0 0 1 00 . 0 0 0 0 101 2 8


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.Fortran listing of computer program14. T he functionXF is listed in MINNt-DIMENSIONALCCNVECTIVE-DIS PERSIVEEQUATION 614 *HIRD-TYPEBOUNCARVCONDITION *EHI-INF INITEFRGFILE *ER-CROERPR CDLCTIN(DZERO) * INEARA DSOR P T I C N R J ONSTANTI N I T I ALCCNCENTRAT ION(CD NPUTC ON C EN T R A T I C N=CA+CB*EXPi-OBND*T) *

IM PLICITREAL*8A-H,C-ZJDIMENSIONTITLE(20)READNUMBEROFCURVESTOBECALCULATEDREAD(5,1000)NCDO4K=1,NCREA0(5,1001)TITLEWPITE(6,1002)TITLEREADANDW R I TEINPUTPA R AMETERSREA0(S),1003)V,D,R,DZf


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c c

MAINVVO=V*R*T/X1 P=V*X/DS=DSQRT(4.*D*T)A1=X-V*TA2=X+V*T AM=0.5*EXF(

Analytical Solutions of the One Dimensional Convective-Dispersive Solute Equation

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