A Simple Algorithm for Vapor

ASimpleAlgorithmforVapor?Liquid?LiquidEquilibriumComputation

P.SwapnaReddyandK.YamunaRani*

ProcessDynamicsandControlGroup,ChemicalEngineeringSciences,IndianInstituteofChemicalTechnology,Hyderabad500607,India

*SupportingInformation

PhaseequilibriacalculationsplayanimportantroleinmanyanintervalNewtonmethodwithgeneralizedbisection,whichdesignandsimulationcasestudies,especiallyindistillationandlocatesallrealrootswithinspeci?edintervalsoftheunknowns.reactivedistillationprocesses.ThisisparticularlytrueforGLOPEQ,whichisacomputationaltoolforthephaseandsystemsthatexhibittwoliquidphases,sincemethodsbasedchemicalequilibriumproblem,hasbeenintroducedbyonlocalsolvers(e.g.,Newtontype)mayconvergetotrivialMcDonaldandFloudas.6Manymethodsbasedonglobalsolutions.Generally,phaseequilibriacalculationsconsistofoptimization(YushanandZhihong,7Tessieretal.8)havebeenvapor?liquidequilibria(VLE)andvapor?liquid?liquiddevelopedtodetectanyunstablesolutionexploitingsu?cientequilibria(VLLE),basedonthehomogeneousandheteroge-stabilityconditions.Methodsforcalculatingchemicalequili-neousbehaviorofliquidmixtures.TheVLEdataoftherelatedbriuminmixturescontainingmorethanoneliquidphasehaveaqueoussystemsunderambientconditionsareplentifulinthebeendevelopedbyFurzer.9OkasinskiandDoherty10developedliterature,whiletheVLLEdataareratherlimited.Multiphaseamethodthatenablestheexistenceandlocationofbothcoexistencemayoccurinthemixturesofwaterandesters,homogeneousand

whicharetheproductsofesteri?cationofacidswithalcohols.11heterogeneousreactiveazeotropes.MullerandMarquardtdevelopedane?cienthybridapproachthatSeveralalgorithmsforthedeterminationofphasestabilitycombinesaglobalphasestabilityanalysisbasedonintervalhavebeenthesubjectofresearchoverthepastthreedecades.methodswithalocalmethodfordynamicsimulationproblems.Mostofthealgorithmsarebasedonlocalsolutionmethods.Nichitaetal.12appliedthetunnelingmethodforsolvingBuzzietal.1solvedthephasestabilityproblembyaseriesof13phasestabilityproblems,whereasSrinivasandRangaiahhaveusedlocal?ashcalculations.Michelsen2determinedthephasestateevolutionarymethodsto?ndtheoptimalsolutiontophasebyalocalanalysisofthetangentplanemethod.Generally,theequilibriaproblems.Recently,Kangasetal.14developedthelocalsolutionmethodswiththeirlowcomputationaldemandmodi?edboundedhomotopiestosolvephasestabilityanalysisarewell-suitedforsteady-stateordynamicsimulationsfortheproblemsinliquid?liquidequilibrium(LLE)casesforphase-heterogeneous?ashcalculation,buttheyprovetobeinfeasiblesplittingcalculations.BausaandMarquardt15proposedainsomecases,becauseoftheimproperlyguessedinitialvalues.homotopycontinuationmethodfordeterminationofVLLEToovercomethisproblem,globalmethodsbasedonglobalphasecalculationsthatincorporatesaprioriknowledgeaboutoptimizationhavebeendeveloped.Abranch-and-boundalgorithm,whichisslightlyslowerbutcanprovidecompletereliabilityforsolvingphasestabilityanalysis,hasbeenReceived:September27,2011

Revised:June26,2012developedbyMcDonaldandFloudas.3WhitsonandAccepted:July12,2012

Michelsen4showedtheinterpretationsofnegativephasePublished:July12,2012

©http://wendang.chazidian.com/10.1021/ie2022064|Ind.Eng.Chem.Res.2012,51,10719?10730

Figure1.FlowchartfortheproposedVLLEalgorithm.

thephasediagram.Themethodisbasedonapreprocessingtoobtain

step,inwhichallheterogeneousregionsaredetermined.Allthe18phasecompositionsandfractions.HanandRangiahextendedthisapproachbyintroducingapseudo-aforementionedmethodsarebasedontwo-stepcalculations,temperaturetodevelopamethodapplicabletobothVLEandnamely,determinationofphasestabilityregionsthroughLLE,whichistestedsuccessfullyonseveralVLEandLLEpreprocessingstepandsubsequentcomputationofphaseexamples.Furthermore,HanandRangiah19proposedanequilibria.equation-solvingmethodformultiphaseequilibriumcompu-Therehavebeensomeapproachesreportedintheliteraturetationsrequiringthesolutionofasingleminimizationthatdonotrequiresequentialdeterminationofphasestabilityproblemviatheintroductionofsomephasecharacteristicandphaseequilibriacomputations,butdocomputethevariables.solutionsimultaneously.BullardandBiegler16proposedaInthepresentstudy,analternativeapproachexploitingstrategy17forVLEbasedontheirconstrainedsolutionlocallowcomputationalloadfeaturesisproposedforVLLEapproachbyintroducingapseudo-pressureparametertocomputation,subsequentlyevaluatingthestabilityof

formulateaminimizationproblemwithlinearobjective4thephasebytheinterpretationofWhitsonandMichelsen.Di?erentfunctionandnonlinearconstraints,whichisessentiallyasystemsincludingthewater?toluene?anilinesystem,thesolutiontothenonlinearconstraintequations.Theconstraintethyleneglycol?laurylalcohol?nitromethanesystem,andequationsincludethenecessaryphaseequilibriumequations,(nonreactive)mixturesencounteredinesteri?cationreactionsdependingonthenumberofphases,andmassbalancearestudied,whichareofgreatimportanceinmanychemicalconstraints.Withoutaprioriphaseidenti?cation,theprocessindustries,andaretypicallyproducedfromtheminimizationproblemissolvedbyaversionofsuccessivereactionsofalcoholsandacidunderacidicconditions.Thelinearprogrammingmethodwithprotectivearti?cialvariables,(nonreactive)mixturesencounteredinesteri?cation

reactions

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Table1.ModelsforVLEComputation

phasesatagivenpressure,whichisrequiredforVLE

AntoineEquation

computation.Theapproachisbasedonminimizationofthedeviationbetweenthecalculatedbubble-pointtemperaturesofln(Psat)=A+

B

PsattheassumedtwoliquidphasesandthedeviationbetweentheT+C

[Pa],T[K]

twosetsofcalculatedvapor-phasecompositionsattheUNIQUACEquationcomputedtwobubble-pointtemperatures,throughVLElnγi=lnγCi+lnγR

icalculationsforbubble-pointtemperaturesofthetwoliquidphasesatthegivenpressure.ThisisachievedonthebasisoflnγCi=ln

ψiz

x+

2qln?iψsteepestdescentmethodwithadaptiveweightingfactorswithi

iψ+li?i∑xjlj

ixi

j

respecttocompositionsandtemperature.Theinitialguess?nc

isbasedonasetsatisfyingmaterialbalanceconstraint,andlnγR

i=qi?nc

?1?ln∑?jτji??∑

?jτij

?

whilecheckingforconvergence,materialbalanceconstraintisj=1

j=1

∑?

k=1?kτkj??re-evaluated.Incaseofviolationofmaterialbalanceconstraint,lz

i=theiterativecomputationisrepeatedconsideringthelatest2

(ri?qi)?(ri?1)phasefraction.Thus,theprocedureiscontinueduntilz=10,

zis the coordination number

convergence,withrespecttoequivalenceofboththevaporphases,aswellaswithrespecttomaterialbalance,isachieved.τ??uji?uii??

ji=exp?????RT???

?

τii=τjj=1Afterattainingconvergence,iftheobtainedphasefractionisfoundtobeinfeasible(<0or>1),itisinferredthattheliquidψrxiii=

∑?qi=

iximixtureishomogeneousandVLEcomputationissu?cienttojrjxj

∑jqjxj

determinethevapor-phasecompositionandtemperatureattheψiisthedimensionlessvolumefractionofcomponentigivenpressure.

?iisthedimensionlessareafractionofcomponenti

Theproposedalgorithmisshownasa?owchartinFigure1,NRTLEquation

andcanbesummarizedforan-componentsystemasfollows:lnγi=lnγCi+lnγR

i(1)Foragivenpressureandanoverallcombinedliquid

nc

composition(xi)forncomponents(i=1,...,n),guessalnγC=1τjiGjixii=

∑j∑liquidcompositionofoneofthephasesi.e.xp1,i(i=1,...,k=1Gkixk

n),andaphasefractionbetweenthetwoliquidphaseslnγR

(?,de?nedastheratioofthequantityofoneliquidi=

∑

nc

xjGij

?∑nck=1xkτkjj=1

∑?τGkj?

ij??k=1Gkjxk??∑k=1Gkjxk??

phasetothetotalquantityofthetwo-phaseliquidmixture).

G(2)Determinetheliquidcompositionofthesecondphase

ij=exp(?αijτij)τij=abijij+T

basedonmaterialbalanceconstraints,basedontheαij=cij

Gii=1

τii

=0

guessedvalueofthe?rstliquidphasecompositionandphasefraction:

ofaceticacidwithdi?erentalcoholsareconsideredwhichcovertheproductionofimportantsolventslikepropylacetate,butylacetate,amylacetate,hexylacetate,andbutylx(?xp1,i)

p2,i=

xi?propionate.These?vequaternarysystemsexhibitacomplex1??(1)

phasebehaviorwiththedi?erentbinaryandternary(3)Findthebubble-pointandvapor-phasecompositionfor

azeotropes.

eachliquidphasecomposition,usingVLEcalculation(assumingthevaporstobeideal),accordingto2.PROPOSEDALGORITHM

UNIQUACorNRTLequationsasTPhaseequilibriumcomputationsofmulticomponentvapor?Thesecalculationsareperformedbyguessingp1,yp1,i,Tthep2,ybubblep2,i.liquidequilibria(VLE)consistofdeterminationofthevaporpoint?rstandusingtheintervalhalvingmethodtocompositionandbubble-pointtemperatureinequilibriumwithupdatethebubblepointand?ndingtheupdatedvaporaliquidmixtureofgivencomposition,atagivenpressure.Oncomposition,andbycontinuingtheiterativeproceduretheotherhand,vapor?liquid?liquidequilibria(VLLE)untilthesumofthevaporcompositionofallthecomputationforatwo-phaseliquidmixturewithagivencomponentsequals1.

combinedliquidmixturecompositionatagivenpressure(4)Comparethetwobubblepointsandequilibriumvapor

consistsofdeterminationofthecompositionoftwoliquidcompositionsusingtheerrormeasureas

phasesandtheirphasefractions,vapor-phasecomposition,n

andthebubble-pointtemperature.Althoughthereareseverale=(0.01)(Tp1?Tp2)2

+

?ypossibleproblemformulationsbasedontheinformation∑(yp1,ip2,i)2

i=1

(2)

available(suchascombinedliquidcompositionandbubble-pointtemperature,vaporcomposition,andpressureorandifthecomputedevalueislessthanapresetthresholddewpointtemperature,etc.),thepresentstudyisrestrictedvalue,gotonextstep5;otherwise,gotostep6.tothecasewithgivenliquidmixturecompositionand(5)Materialbalanceconstraintisdeterminedas

pressure.

n

Inthisstudy,weproposeanovelapproachbyconsideringm=

∑(xci?xi)2

simpleupdatingmechanismfortheliquidcompositionoftwo

i=1

(3)

10721

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wherexci=(1??)xp2,i+?xthepriorknowledgeregardingthecomponentsthatgetis>0.001,thengotop1,stepi.Ifmaterialbalanceconstraint2withthelatestdistributedbetweenphases,anappropriateinitialguessvaluephasefractionvaluecomputedasshouldbechosen,andbasedontheleastdistributed

ncomponent,thephasefractioncanbechosentoobtaina?=1∑??xi?xp2,i?feasibleliquid-phasecompositionforthesecondphase.Inthis

ni=1??xp1,i?x?

p2,i??(4)case,itisnotpossibletoprovidespeci?cguidelinesregardingthechoiceofinitialguessvalues.otherwise,stopthecomputation.(6)Updatetheliquidcompositionofboththephasesusinga3.SYSTEMSSTUDIED

weightingfactor(eandaweighting1,e2)timesthedeviationofthebubblepointsfactortimesthedeviationoftheTheperformanceoftheproposedalgorithmisevaluatedforthecorrespondingequilibriumvaporcompositiondeviationwater?toluene?anilinesystemanddi?erent(nonreactive)ofthetwophases:mixturesencounteredinesteri?cationreactionstodeterminevapor?liquid?liquidequilibria(VLLE)andvapor?liquidxp1,i=xp1,i+e1(yequilibria(VLE)dataandestimatingthevapor-phasep1,i?yp2,i)+e2(Tp1?Tp2)(5)compositionandbubble-pointtemperatureatagivenpressure.TheVLEmodelsemployedinthepresentstudynamely,xNRTLandUNIQUACalongwiththeAntoineequation,arep2,i=xp2,i+e1(yp2,i?yp1,i)+e2(Tp2?Tp1)(6)reportedinTable1.Theinterval-halvingmethodisusedtowheree1ande2aretheweightingfactors.solveNRTLorUNIQUACmodelsinthepresentstudyto

(7)Basedonthemagnitudeofthedeviations,ifnecessary,determinetheVLEforeachphase.

updatetheweightingfactorsbyapresetfactor,i.e.,3.1.IsopropylAcetateSystem.Thereactionbetween

aceticacid(AA)andisopropanol(IPOH)producesisopropyl

If(abs(Tp1?Tp1old))>5°CorIfacetate(IPAc)andwater(W),asshowninthefollowing(abs(Tequation:

p2?Tp2old))>5°CC2H4O2+C3H8O?C5H10O2+H2Oe1=e1/2Thesystemhasfourpurecomponents,20twohomogeneouse2=e2/2(7)minimumboilingbinaryazeotropes(W-IPOHandIPOH-whereTIPAc),oneminimumboilingheterogeneousbinaryazeotrope

p1oldandTp2oldarethepreviousbubblepointsofthetwophases.Thengotostep3andcontinueuntil(W-IPAc),andoneminimumboilingheterogeneousternaryconvergence.azeotrope(W-IPOH-IPAc).Songetal.21theoreticallypredictedandexperimentallycon?rmedthepresenceofaTheaforementionedapproachdependsontwoparameters,homogeneousreactiveazeotropeintheisopropylacetatei.e.,initialguessesofoneoftheliquidphasecompositionsandsystem.Forthissystem,theVLEandVLLEdataareobtainedphasefraction.Inthisapproach,theinitialguessvaluesforoneusingtheNRTLmodel,andthethermodynamicpropertiesoftheliquidphasecompositionsmainlycontainsasingleandtheinteractionparametersfortheNRTLmodelcomponentandsmallfractionsofothercomponents(for(Gadewaretal.22)arelistedinTableS1intheSupportingexample,0.998,0.001,and0.001forathree-componentInformation.Thesystemisassumedtobeisobaricatmixture).Basedontheseguessvalues,thephasefractionisatmosphericpressure.assumedsuchthatitgivesfeasiblesecond-liquid-phase3.2.n-ButylAcetateSystem.Theesteri?cationofaceticcompositions,aspereq1.Becauseofthedominantpresenceacid(AA)withn-butanol(BuOH)producesn-butylacetateofasinglecomponentinthe?rstliquidphase,areasonable(BuAc)andwater(W),asshowninthefollowingequation:choicefortheinitialguessvalueforphasefractionisthecompositionofthatcomponentintheoverallliquidmixture.C2H4O2+C4H10O?C6H12O2+H2OThe?overallliquidThesystemisassumedtobeisobaricatatmosphericpre-?andvalue(1is?positive,?).Ifthemixture

thenconvergedissplitintotwophasesoffractions

thatliquidvaluecompositionofthe?rstphase(xssure.Thenonreactivesystemhasfourpurecomponents,23

)exhibitsfractiononemaximum-boilinghomogeneousbinaryazeotrope(AA-

aheterogeneousnature,i.e.,consistingoftwoliquidiphasesBuOH),oneminimumboilinghomogeneousbinaryazeotropeandtheabove-mentionedapproach(vapor?liquid?liquid(BuOH-BuAc),andtwoheterogeneousminimumboilingequilibria,VLLE)isusedtocalculatevaporcompositionandbinaryazeotropes(W-BuOHandW-BuAc),andoneminimumtemperatureoftwophases.Otherwise,theliquidmixtureisboilingheterogeneousternaryazeotrope(W-BuOH-BuAc).homogeneousinnatureandnormalVLEiscarriedoutusingForthissystem,theVLEandVLLEdataareobtainedusingtheUNIQUACorNRTLmethodgiveninTable1togettheUNIQUACmodel.Theinteractionparametersforthetheequilibriumvapor-phasecompositionandthebubbleUNIQUACmodelandthethermodynamicpropertiesfrompoint.Venimadhavanetal.24arelistedinTableS2intheSupportingInsystemsthatexhibitanarrowerregionofheterogeneity,orInformation.wheretheliquid-phasecompositionofneitherofthephases3.3.AmylAcetateSystem.Amylacetate(AmAc)iscorrespondstoalmost-puresinglecomponents,thechoiceofproducedbytheesteri?cationofaceticacid(AA)withamylinitialguessvaluesforoneoftheliquid-phasecompositionsalcohol(AmOH),asshowninthefollowingequation:shouldbechosendi?erently.Whileselectingtheinitialguessvalues,itisnecessarytoknowwhichofthecomponentsgetC2H4O2+C5H12O?C7H14O2+H2Odistributedintoboththeliquidphases,andwhichcomponentsAmylacetatesystemisassumedtobeisobaricatatmosphericarepredominantlypresentonlyinoneofthephases.Basedonpressure.Thenonreactivesystemhasfourpurecomponents,25

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Figure2.Ternaryplotswithternaryazeotropes(denotedbysquares,■)andbinaryazeotropes(denotedbytriangles,▲).

onehomogeneousmaximum-boilingbinaryazeotrope(AA-AmOH),3.4.n-HexylAcetateSystem.Hexylacetate(HeAc)istwoheterogeneousminimum-boilingbinaryazeotropes(W-producedbythereactionbetweenaceticacid(AA)andAmOHandW-AmAc),andaheterogeneousminimum-boilingn-hexanol(HeOH),asshowninthefollowingequation:ternaryazeotrope(W-AmOH-AmAc).Forthissystem,theVLE

andVLLEdataareobtainedusingtheNRTLmodelandtheC2H4O2+C6H14O?C8H16O2+H2O

interactionparametersfortheNRTLmodel(Chiangetal.26);

thethermodynamicpropertiesarelistedinTableS3intheThenonreactivesystemhasfourpurecomponents,onehomogeneousminimiumboilingazeotrope(HeOH-HeAc),twoSupportingInformation.heterogeneousminimum-boilingbinaryazeotropes

(W-HeOH

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