A simple and unified algorithm to solve fluid phase equilibria using either the gamma–phi or the phi–phi approach for binary and ternary mixtures

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Computers and Chemical Engineering

j our na l h o me p ag e: w w w . e l s e v i e r . c o m / l o c a t e / c o m p c h e m e n g

A simple and unified algorithm to solve fluid phase equilibria using either the gamma–phi or the phi–phi approach for binary and ternary mixtures

Romain Privat

^a,∗

, Jean-Noël Jaubert

^a

, Yannick Privat

^b

aÉcoleNationaleSupérieuredesIndustriesChimiques,UniversitédeLorraine,LaboratoireRéactionsetGéniedesProcédés–UPR3349,1rueGrandville,BP20451,NancyCedex9, France

bENSCachan,AntennedeBretagne– AvenueRobertSchumann,35170Bruz,France

a r t i c l e i n f o

Articlehistory:

Received21September2012 Receivedinrevisedform 15November2012 Accepted16November2012 Available online 5 December 2012

Keywords:

Phase-equilibriumcalculation Algorithm

Binarymixtures Ternarymixtures Phi–phi Gamma–phi

a b s t r a c t

Anewalgorithmisproposedforcalculatingphaseequilibriainbinarysystemsatafixedtemperature andpressure.Thisalgorithmisthenextendedtoternarysystems(inwhichcase,themolefractionofone constituentinagivenphasemustbefixedinordertosatisfytheGibbs’phaserule).Thealgorithmhas theadvantageofbeingverysimpletoimplementandinsensitivetotheprocedureusedtoinitializethe unknowns.Mostsignificantly,thealgorithmallowsthesamesolutionproceduretobeusedregardless ofthethermodynamicapproachconsidered(–ϕorϕ–ϕ),thetypeofphaseequilibrium(VLE,LLE,etc.) andtheexistenceofsingularities(azeotropy,criticalityandsoon).

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Severalindustrially importantchemical applicationssuchas single- and multi-stage processes (including flash, distillation, absorption,extraction,etc.)bringtwofluidphasesintocontact(e.g.

twoliquidphasesoraliquidphaseandavaporphase).Simulation andoptimizationof theseprocesses requirethattheproperties oftheequilibrium statesbecalculated,mainly,thetemperature T,thepressurePandthecompositionsofthecoexistingphases.

Robustandefficientalgorithmsforphase-equilibriumcalculations arethereforeneeded.TheGibbs’phaserulestatesthatcalculating two-phaseequilibriuminabinarysystemrequirestwointensive andindependentprocessvariablestobespecifiedwhicharetypi- callychosenfromT,P,x_i(themolefractionofcomponentiinthe firstphase)andx_i(themolefractionofcomponentiinthesecond phase).Thephase-equilibriumproblemthenconsistsofsolvinga setoftwoequilibriumequationswithrespecttotwounknowns:

₁(T,P,x₁)=₁(T,P,x₁)

₂(T,P,x₁)=₂(T,P,x₁)

(1)

∗Correspondingauthor.Tel.:+33383175128;fax:+33383175152.

E-mailaddress:romain.privat@univ-lorraine.fr(R.Privat).

whereidenotesthechemicalpotentialofcomponentiinagiven phase.Thesuperscripts anddenotethefirstandsecondequi- librium phases,respectively. Theprocedurefor solvingthetwo equilibriumequationsisstronglyrelatedto(i)thechoiceofthe twospecifiedvariables andto(ii)thenatureoftheequilibrium (liquid–liquid or liquid–vapor) involved. There are six possible combinations of two variables among thefour aforementioned ones(T, P,x_i and x_i).Four ofthem are oftenacknowledged to be of practical interest since theycan be easilygeneralized to p-componentsystemswithp>2.Thisiswhymanypapersandtext- booksproposesolutionproceduresforonlythefollowingfourcases (Michelsen,1985;Sandler,2006;Smith&VanNess,1975):

•Calculation of x_i (or x_i) and P, given x_i (or x_i) and T. For vapor–liquidequilibrium(VLE),thesekindsofcalculationsare calledbubble-pointpressureanddew-pointpressurecalculations.

•Calculationofx_i (orx_i)andT,givenx_i (orx_i)andP.ForVLE, thesekindsofcalculationsarecalledbubble-pointtemperature anddew-pointtemperaturecalculations.

Thetwo remainingcombinationswhich arescarcely– ifever– consideredare:

•CalculationofTandP,givenx_iandx_i.

•Calculationofx_iandx_i,givenTandP.

0098-1354/$–seefrontmatter© 2012 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.compchemeng.2012.11.006

Thefirstcalculationwhichrequiresthatthemolefractionsinthe twophasesbespecified,isclearlyofnopracticalinterest.Indeed, inthiscase,temperatureandpressurewhicharecertainlytheeas- iestprocessvariablestoworkwith,arenotspecifiedbutbecome accessibleaftersolvingthephase-equilibriumproblem.

Thesecondcombination(wherethevariablesTandParespec- ified)appearstobeparticularlyconvenient.Togiveanexample, calculationofthecomplete isothermalorisobaricbinaryphase diagramsisstraightforwardusinganalgorithm withT andPas specifiedvariables,asopposedtoanalgorithmusingx_iandx_i as specifiedvariables.

Atthispoint, itisimportanttohighlightthatcalculatingthe compositionsoftwophasesinequilibriumatafixedTandPdiffers notablyfromtheclassical(P,T)-flashcalculation.Indeed,calculat- ingx_iandx_iatgivenTandPonlyinvolvesphasevariables(T,Pand themolefractionsinthetwophases)butdoesnotinvolveanyover- allvariable(theproportionofthephases,overallmolefractions, etc.).Incontrast,aflashalgorithminvolvesbothphasevariables andoverallvariables.Morespecifically:

•ina(P,T)-flashalgorithm,thevariablesTandParespecifiedalong withtheoverallcomposition(bycomparison,thecalculationof x_iandx_i atagivenTandPonlyrequiresTandPtobespecified),

•ina(P,T)-flashalgorithm,thetwophase-equilibriumequations arecoupledwithmaterialbalances(whereasthecalculationof x_iandx_i atagivenTandPonlyinvolvesthephase-equilibrium equations),

•ina(P,T)-flashalgorithm,theunknownsarex_iandx_iaswellas, themolarproportionofagivenphase(whereasforthecalculation ofx_iandx_i atagivenTandP,x_iandx_i arethesoleunknowns).

•AtagivenTandP,dependingontheoverallcomposition,a(P, T)-flashalgorithmcanindicatewhetherthesystemismadeupof asinglephase(molarproportionsofthephases<0or>1)ortwo phases(molarproportions∈[0;1]).Thecalculationofx_iandx_i is thenonlyperformedinthelattercase.

Insummary,a(P,T)-flashalgorithmandthecalculationofx_iand x_i atagivenTandPdonotinvolvethesameinputvariables,the sameunknownsandthesamesetofequationstosolve.Inaddition, contrarytoa(P,T)-flashalgorithm,thecalculationofx_iandx_i at agivenTandPallowsuncouplingthephase-equilibriumproblem andtheresolutionofthemass-balanceequationswhichmayhave twointerests:

–the dimension of the system of equations to solve becomes smallerwhichmayfacilitateitsresolution;

–thephase-equilibriumproblemtakestheadvantageousformof apseudo-linearsystem,aswillbeprovedinthenextsections.

Inthispaper,anewalgorithmisproposedtocalculatephase equilibria in binary systems at a specified T and P. As will be shown,thealgorithmcanbeusedeitherwiththe–ϕortheϕ–ϕ approachandallowsVLEorLLE(liquid–liquidequilibria)tobecal- culatedaswell.Thisprocedureisconsideredtobeuniversal,asit unifiesthevariousapproachesandkindsoffluid-phaseequilib- riumcalculations.Thegreatsimplicityofthealgorithmfacilitates easyimplementation.Thealgorithmcanbeusedtocalculatecom- plexphasediagramsexhibitingazeotropyorcriticality,andisthus intendedforaudiencesinacademicorindustrialenvironmentswho usephase-equilibriumthermodynamicsasatoolandwishtohave attheirdisposalauniquealgorithmtoperformeitherVLEorLLE calculations.

Therefore,thealgorithmcanbeusedtomodel,e.g.adistilla- tioncolumn,astrippingprocess,aliquid–liquidextractionorany separationunitinvolvingfluid–fluidphaseequilibria.

Thefinalsectionofthepapershowshowthisalgorithmcanbe simplyextendedtoternarysystemsbyspecifyingathirdvariable.

2. Generalformalismofthetwo-phaseequilibrium probleminbinarysystemsatspecifiedtemperatureand pressure

2.1. The–ϕapproach

In the present case, a liquid-activity-coefficient model (also knownasmolarexcessGibbsenergymodel)isusedtoevaluate thethermodynamicpropertiesoftheliquidphase.Anequationof state(EoS)isusedforthegaseousphase.

•Forabinarysysteminliquid–liquidequilibriumatafixedtem- peratureTandpressureP,Eq.(1)maybewrittenasfollows:

x₁·1(x₁)=x₁·1(x₁)

(1−x₁)·₂(x₁)=(1−x₁)·₂(x₁)

(2)

wherex₁andx₁arethemolefractionsofcomponent1ineach ofthetwoliquidphasesinequilibriumandidenotestheactiv- itycoefficientofcomponentiinaliquidphase._iisclassically derivedfromanexcessGibbsenergymodel,suchthat:RTln_i= (∂G^E/∂ni)_T,P,nj

/

=iwhereG^EisthetotalmolarGibbsenergyofthe liquidphase and n_i, theamountofcomponent iin theliquid phase.Thesekindsofmodelarepressure-independentsothat, theliquid-activity coefficients dependonly onthemolefrac- tionsandthetemperature.Forthesakeofclarity,onlythetwo unknownsx₁andx₁inthephase-equilibriumproblematfixedT andPappearwithinparentheses,inEq.(2).

•Forliquid–vaporequilibriumatafixedtemperatureTandpres- sureP,Eq.(1)isequivalentto:

P·y1·C1(y1)=P^sat₁ ·x1·1(x1)

P·(1−y₁)·C₂(y₁)=P^sat₂ ·(1−x₁)·₂(x₁) (3) wherex1andy1 arethemolefractionsofcomponent1inthe liquidandvaporphases,respectively;P^sat_i isthevaporpressure ofthepurecomponenti;thecoefficientC_iisdefinedas:

C_i(y₁)=ϕˆ^gas_i (y₁)

ϕ^sat_i ·F_P,i (4)

whereϕˆ^gas_i isthefugacitycoefficientofcomponentiinthegas phase,ϕ_i^satisthefugacitycoefficientofpureiatitsvaporpressure attemperatureTandF_P,iistheclassicalPoyntingcorrectionfactor forcomponentiatTandP(oftensetto1atlow tomoderate pressures).Eq.(3)canbealternativelywrittenasfollows:

x₁·₁(x₁)=y₁·[P·C₁(y₁)/P₁^sat]

(1−x1)·2(x1)=(1−y1)·[P·C2(y1)/P₂^sat]

(5)

Asbefore, only the two unknowns (x₁ and y₁)in the phase- equilibriumproblematafixedTandPappearwithinparentheses inEq.(5).

2.2. Theϕ–ϕapproach

This approach requires the use of a pressure-explicit EoS to model both phases in equilibrium. The method offers the undeniable advantage of a unique formalism for

liquid–liquid,liquid–vapor andmoregenerallyfluid–fluidphase equilibria.Eq.(1)isthenwrittenasfollows:

x₁·ϕˆ₁=x₁·ϕˆ₁

(1−x₁)·ϕˆ₂=(1−x₁)·ϕˆ₂ (6) where x₁ and x₁ are themolefractions ofcomponent 1 inthe two fluid phasesin equilibrium;ϕˆ_i andϕˆ_i denotethefugacity coefficientsofcomponentiineachfluidphase.

For a binarysystem,thepressure-explicitEoSexpressesthe pressureofaphaseasafunctionofthetemperatureTofthephase, themolefractionz₁ofcomponent1andthemolarvolumevofthe phase.Henceforth,thepressure-explicitEoSwillbedenotedbythe function:P_EoS(T,v,z₁).Expressionsforthefugacitycoefficientsare derivedfromtheEoSandarethusnaturalfunctionsofthevariables T,vandz1:ϕˆi(T,v,z1).

Consequently,thefugacitycoefficientsϕˆ_iandϕˆ_i ofcomponent iinthetwophasesinequilibriumareaccessibleatafixedtemper- atureT0andpressureP0,providedthatthemolarvolumes(vand v)andthemolefractionsofcomponent1(x₁andx₁)areknown:

ϕˆ_i=ϕˆ_i(T₀,v,x₁)andϕˆ_i =ϕˆ_i(T₀,v,x₁).Themolarvolumeofafluid phasecan becalculated bysolvingtheEoS,giventhetempera- ture,thepressureandthecompositionofthephase.Therefore,at fixedT₀ andP₀,theunknownsv andvareobtainedbysolving thetwoequations:P₀=P_EoS(T₀,v,x₁)andP₀=P_EoS(T₀,v,x₁)(by assumingknownvaluesofmolefractionsx₁ andx₁).

Finally,thephase-equilibriumproblematafixedtemperature T₀andpressureP₀usingaϕ–ϕapproach,isreducedtosolvingaset offourequationswithrespecttofourunknowns(v,x₁,v,x₁):

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

x₁ ·ϕˆ1(v,x₁)=x₁·ϕˆ1(v,x₁)

(1−x₁)·ϕˆ2(v,x₁)=(1−x₁)·ϕˆ2(v,x₁) P0=PEoS(v,x₁)

P0=PEoS(v,x₁)

(7)

Onceagain,onlytheunknownsintheproblemappearwithin parentheses.

2.3. Writingofthephase-equilibriumproblemusingaunique formalism

Eqs.(2),(5)and(6)canbewritteninthefollowingunifiedform:

x₁·F₁ =x₁·F₁

(1−x₁)·F₂ =(1−x₁)·F₂ (8) ExpressionsforthefunctionsF₁,F₁,F₂andF₂aregiveninTable1 andareselectedbasedonthechosenapproach andthekindof phase-equilibriumcalculation(VLE,LLE,etc.)beingconsidered.

Table1

ExpressionsoffunctionsF1andF2appearinginEq.(8).

Approach: –ϕapproach ϕ–ϕapproach

Typeofphase- equilibrium calculation:

VLE LLE Fluid–fluid

equilibrium

F₁ 1 ₁ ϕˆ₁

F₁ P·C₁/P^sat_{1 1} ϕˆ₁

F₂ 2 ₂ ϕˆ₂

F₂ P·C2/P^sat_{2 2} ϕˆ₂

3. Derivationofauniquealgorithmforphase-equilibrium calculationinbinarysystemsatafixedtemperatureand pressure

3.1. Generalsolutionprocedure

Eq.(8)canbeequivalentlywrittenas:

x₁ ·F₁ −x₁·F₁=0

−x₁·F₂ +x₁·F₂=F₂−F₂ (9) Sucha systemis highlynonlinear andthecoefficients F₁,F₁,F₂ andF₂dependonx₁andx₁.Usingmatrixnotations,Eq.(9)canbe alternativelywrittenas:

F₁ −F₁

−F₂ F₂

A(X)

x₁ x₁

X

=

0 F₂−F₂

B(X)

(10)

orequivalently,providingmatrixAcanbeinverted:

X=[A(X)]⁻¹B(X) (11)

with:

A⁻¹= 1 det(A)

F₂ F₁ F₂ F₁

(12)

Consequently,thevaluesofthemolefractionsx₁andx₁atiter- ation(k+1)canbededucedfromthevaluesofthequantitiesF₁,F₁, F₂ andF₂atiterationkasfollows:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

x₁^(k+1)=

F₁ (F₂−F₂) F₁ F₂−F₁ F₂

(k)

x₁^(k+1)=

F₁ (F₂−F₂) F₁ F₂−F₁ F₂

(k)

=x₁^(k+1)

F₁ F₁

(k) (13)

Thisiterativeschemerepresentsthemainresultofthispaper andisactuallyanapplicationoftheclassicaldirect-substitution methodtothetwo-phaseequilibriumproblem.Algorithmsbased onthefollowingprocedurecanbeusedtocalculatephaseequilib- ria:

1Provideinitialestimatesfor theunknownsx₁ andx₁.Setthe iterationcountertok=0.

2CalculatethecoefficientsF₁,F₂,F₁andF₂forthecurrentiteration k.

3Defineaconvergencecriterionıasfollows:

ı=[|x₁F₁−x₁F₁|+|(1−x₁)F₂−(1−x₁)F₂|]^(k) (14) Ifıislowenough,thenconvergenceisreachedandtheprocedure isterminated.

4Updatetheunknownsx₁andx₁usingEq.(13);updatetheitera- tioncounterto(k=k+1)andreturntostep2.

Thesubsequentsectionsillustratetheefficiencyoftheiterative schemeanddiscussitsrangeofapplicability.

3.2. Preliminaryremarksregardingtheconvergenceofthe procedureforthegeneralsolution

Nextsectionsareintendedtoconvincethereaderthatthepro- posedformulationofthetwo-phaseequilibriumproblemcoupled withthesolutionprocedurecandealwithcomplexscenarioseffi- cientlyandhasawiderangeofapplicability.However,itisclearly

notpossibletoguarantyconvergenceinallcases.Itisindeedwell knownthatfixed-pointmethods –includingdirect-substitution methods–canpotentiallyfailduetotheexistenceofdiverging sequencesthatdependonthenatureoftheequationtosolveand theinitialguessestothesolution.Inthisregard,Luciaetal.have pointedoutthatevenchaoticandperiodicbehaviorscanbeexhib- itedbyfixed-pointmethods(Lucia,Guo,Richey&Derebail,1990), thusemphasizingthecomplexityofaconvergencestudy.There- fore,itseemsverydifficult,ifnotimpossible,todetermineapriori forwhichcases,themethodwillconvergeordiverge.InAppendices AandB,somemathematicalresultsontheconvergenceoffixed- pointmethodsareappliedtotwo-phaseequilibriumproblems;a generalmethodologyforstudyingfailedcasesisalsoprovided.

3.3. Numericalexample1:calculationofliquid–liquidphase equilibriumusinganactivity-coefficientmodel

Inordertoillustratethesimplicityoftheiterativeschemegiven byEq.(13),letuscalculatethecompositionsx₁andx₁oftwoliq- uidphasesinequilibriumatT=300Kforafictitiousbinarysystem (1)+(2),using the–ϕapproach. Thenon-idealityof theliquid phaseisaccountedforbyaVan-Laartypeactivity-coefficientmodel (Smith&VanNess,1975):

g^E(x₁)= A12A21x1x2

A12x1+A21x2

, with

⎧ ⎪

⎨

⎪ ⎩

x2=1−x1

A₁₂=9000Jmol⁻¹ A₂₁=7000Jmol⁻¹

(15)

TheactivitycoefficientsofthetwospeciesarederivedfromtheVan Laarg^Emodel:

i=exp

Aij

RT

Ajixj

A_ijx_i+A_jix_j

2

,with

i=/jand(i,j)∈{1;2}²

R=8.314472Jmol⁻¹K⁻¹ (16) ThealgorithmproposedfortheLLEcalculationatafixedtempera- tureusingthe–ϕapproachisasfollows:

1Provideinitialestimatesfortheunknowns,forinstance:x₁⁽⁰⁾= 0.01andx₁⁽⁰⁾=0.99.Settheiterationcounterto:k=0.

2Calculate

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

₁^(k)=₁(x₁^(k)) ₁^(k)=₁(x₁^(k)) ₂^(k)=2(x₁^(k)) ₂^(k)=2(x₁^(k))

fromEq.(16).

3Defineı=[|x₁₁−x₁₁|+|(1−x₁)₂−(1−x₁)₂|]^(k).

Ifı<(whereistheprecisionafforded,e.g.=10⁻⁶),thena solutionisreachedandtheprocedureisterminated.

4Updateunknownsx₁ andx₁usingEq.(13)andTable1(calcula- tionofaLLEusinga–ϕapproach).Theiterativeschemecanbe writtenas:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

x₁^(k+1)=

₁·(₂−₂) ₁·₂−₁·₂

(k)

x₁^(k+1)=

₁·(₂−₂) ₁·₂−₁·₂

(k)

=x₁^(k+1)

_{1 1}

(k) (17)

5Setk=k+1.Returntostep2.

TheiterationsfortheLLE calculationatT=300Karedetailedin Table 2. The proposed algorithm can be used to constructthe entireLLEphasediagram,whichexhibitsanuppercriticalsolu- tiontemperature(UCST).Todoso,thealgorithmhastoberunat intermediatetemperaturesbetweenT_min=300K(forinstance)and Tmax=UCST.Thesameinitialestimatesforx₁andx₁canbeused foralltheLLEcalculations(followingthefirststepoftheproposed LLEcalculationalgorithm).Alternatively,inordertoincreasethe speedofconvergence,theinitialestimatesforagivenLLEcalcu- lationattemperatureTcanbemadeatatemperatureclosetothe temperatureunderconsideration.

For the binary system (1)+(2), the coordinates of the upper liquid–liquid critical point are analytically found to be:

UCST482.95K and x_1,crit0.4073. The complete LLE diagram, fromlowtemperaturestotheUCSTandcalculatedusingthepro- posedalgorithm,isshowninFig.1(a).

Speedofconvergenceoftheproposedalgorithm:

Theproposedalgorithmisazeroth-ordermethodsincederiva- tivesofthemodel donot updatethemolefractions withinthe iterative process (see Eq. (13)). A majordrawback of this kind ofmethodistheslowconvergencethatisobserved.Thisfeature isillustratedinFig.1(b)showingthatfortemperaturesfarfrom theUCST,severaldozensofiterationsareneededforconvergence.

WhenapproachingtheUCST,convergenceiseventuallyreached afteraverylargenumberofiterations.Atthecriticaltemperature, thisnumberbecomesnearlyinfinite.

Inaddition,despiteofverypoorinitialestimates,theproposed algorithmisobservedtoberemarkablyrobustandfacilitatesaccu- ratedeterminationofthecompositionsofthetwoliquidphasesin equilibrium,evenattemperaturesclosetotheUCST.Furthermore, thecalculationprocedurerequires quiteareasonable computa- tiontime: theFortran 90 program used tocalculate theentire LLE diagramin Fig.1(a)runsin less than1/10s, using anIntel Xeon E5345^© CPU. Note however that stronglyinadequate ini- tialestimatesmayleadinsomerarecases,totheso-calledtrivial solution,producingfluidphasesofidenticalcompositions.When constructingtheentirephasediagram,suchsituationscanbeeas- ilyavoidedbyperformingaseriesofLLEcalculationsatincreasing Table2

IterationdetailsoftheLLEcalculationperformedinSection3.3.

Iteration x₁ x_{1 1 1 2 2} logı

0.01000 0.99000

1 0.02789 0.93862 28.67378 1.00849 1.00356 12.69554 −0.46

2 0.03248 0.92352 27.54304 1.01330 1.00482 11.90531 −0.99

3 0.03379 0.91845 27.23057 1.01517 1.00521 11.65221 −1.48

4 0.03417 0.91668 27.13949 1.01585 1.00533 11.56517 −1.95

5 0.03429 0.91605 27.11251 1.01610 1.00536 11.53453 −2.43

6 0.03432 0.91583 27.10447 1.01619 1.00537 11.52365 −2.89

7 0.03433 0.91575 27.10207 1.01622 1.00538 11.51977 −3.36

8 0.03434 0.91572 27.10135 1.01623 1.00538 11.51839 −3.82

9 0.03434 0.91571 27.10114 1.01623 1.00538 11.51790 −4.28

10 0.03434 0.91571 27.10107 1.01624 1.00538 11.51772 −4.74

11 0.03434 0.91571 27.10105 1.01624 1.00538 11.51766 −5.19

12 0.03434 0.91571 27.10105 1.01624 1.00538 11.51764 −5.65

13 0.03434 0.91571 27.10105 1.01624 1.00538 11.51763 −6.10

temperatureswheretheinitialestimatesforaLLEcalculationata certaintemperaturearegivenbythecompositionsx₁andx₁from theLLEcalculationatthetemperatureimmediatelybelowthepre- scribedtemperature.Theseefficientinitialestimates reducethe numberofiterationsrequiredfortheconvergence(seeFig.1(b)) andmoreimportant,generallypreventthetrivialsolutionfrom beingreached,eveninthevicinityofthecriticaltemperature.

3.4. Numericalexample2:calculationofliquid–vaporphase equilibriumusinganactivity-coefficientmodelfortheliquid phaseandtheidealgasequationofstateforthevaporphase

As an illustration of this case, we consider the binary sys- tem acetone (1)+chloroform (2) which is known to exhibit a negativeazeotrope.TheNRTL(Non-RandomTwo-Liquid)activity- coefficientmodel(Renon&Prausnitz,1968)isusedtorepresent thenon-idealityoftheliquidphase:

g^E(x₁) RT =x1x2

21exp(−˛₂₁)

x₁+x₂exp(−˛₂₁)+ ₁₂exp(−˛₁₂) x₂+x₁exp(−˛₁₂)

,

with

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

x₂=1−x₁ 12=b12/T 21=b21/T b₁₂=209.38K b21=−431.47K

˛=0.1831

(18)

Theactivitycoefficientsofthetwospeciesintheliquidmixture arethengivenby:

i=exp

x_j²

jiexp(−2˛ji)

[x_i+x_jexp(−˛_ji)]² + ijexp(−˛ij) [x_j+x_iexp(−˛_ij)]²

,

with

i=/j

(i,j)∈{1;2}² (19)

Thevaporpressuresofthepurecompoundsaregivenby:

_log[Psat

1 (t)/bar]=4.2184−1197.01/[(t/^◦C)+228.06]

log[P₂^sat(t)/bar]=3.9629−1106.90/[(t/^◦C)+218.55] (20) ThepressureofthesystemissettoP=1atm.FromSection3.1, theproposedalgorithmcanbewrittenasfollows:

1Provideinitialestimates fortheunknowns x1 and y1. Setthe iterationcounterto:k=0.

2CalculateP₁^sat(T)andP₂^sat(T)usingEq.(20).

3Calculate

₁^(k)=1(x^(k)₁ )

₂^(k)=2(x^(k)₁ ) from Eq. (19). Calculate the coefficients C₁^(k) and C₂^(k) using Eq. (4), the pure component liquid molar volume correlations, thepurecomponent vapor pressurescorrelationsandanappropriateEoSforthegasphase.

4Define ı=[|Py₁C1/P^sat₁ −x11|+|P(1−y1)C2/P^sat₂ −(1−x1)2|]^(k). Ifı<(where,istheprecisionafforded,e.g.=10⁻⁶),thena solutionisreachedandtheprocedureisterminated.

5Updatetheunknownsx1andy1:

⎧ ⎪

⎨

⎪ ⎩

x^(k₁⁺¹⁾ =

C₁ (PC₂−P₂^sat₂) P^sat_{1 1}C₂−P^sat_{2 2}C₁

(k)

y^(k₁⁺¹⁾ =x₁^(k+1)×[P₁^sat₁/(PC₁)]^(k)

(21)

6Setk=k+1.Returntostep3.

Fig.1.(a)Composition-temperatureLLEdiagramcalculatedwithVanLaar’sactivity coefficientmodel.(b)Representationofthenumberofiterationsoftheproposed LLEcalculationalgorithmthatarenecessarytoreachconvergence(convergence criterion:ı<10⁻⁶)withrespecttothetemperature.

Atlowpressures,thecoefficientsC_idefinedinEq.(4)aregen- erallyassumedtobeequalto1(thePoyntingfactorcorrectionis assumedtobeequalto1andtheideal-gasEoSisusedtomodelthe gas-phasebehavior).Theiterativeschemethenreducesto:

⎧ ⎪

⎨

⎪ ⎩

x^(k₁⁺¹⁾=

P−P^sat_{2 2} P₁^sat₁−P₂^sat₂

(k)

y^(k₁⁺¹⁾=x^(k₁⁺¹⁾×

P₁^sat₁/P

(k)

(22)

ItappearsthatthetwoequationsinthesystemgivenbyEq.(22) canbesolvedseparatelyandsuccessively(i.e.thetwoequations canbedecoupled).Theupperequationonlyinvolvestheunknown x1whereasthelowerequationdirectlyyieldsy1asafunctionofx1. Asaresult,theinitialestimatefory₁doesnotaffecttheiterative scheme.

Suchasituation,whichisaparticularcaseofthegeneralsolution procedure previously presented, arises because the VLE calcu- lation in binary systems at a specified T and P using the –ϕ approachatlowpressures,reducestotheresolutionofaunique equation(P−P₁^satx₁₁−P₂^sat(1−x₁)₂=0)withrespecttoasin- gleunknown(x₁).Theiterativeschemetakestheformofaclassical 1-dimensionaldirect-substitutionmethod.

Thisalgorithmwasappliedatt=64^◦C(T=337.15K).Atthistem- peratureandatmosphericpressure,theazeotropicsystemacetone (1)+chloroform(2)exhibitstwodistinctVLE.Thesolutionthatthe

Table3

IterationdetailsforthetwoVLEcalculationsperformedinSection3.4usingtwo differentinitialestimatesforunknownx1.

Iteration x1 y1 logı

0.01000

1 0.20795 0.00636 −0.67

2 0.23475 0.16184 −1.46

3 0.23003 0.18760 −2.20

4 0.23115 0.18298 −2.83

5 0.23090 0.18407 −3.47

6 0.23095 0.18382 −4.11

7 0.23094 0.18388 −4.76

8 0.23094 0.18386 −5.40

9 0.23094 0.18387 −6.04

0.99000

1 0.66050 1.28752 −0.14

2 0.60094 0.75855 −0.94

3 0.62039 0.66233 −1.44

4 0.61130 0.69334 −1.77

5 0.61513 0.67879 −2.15

6 0.61343 0.68491 −2.50

7 0.61417 0.68219 −2.86

8 0.61384 0.68337 −3.22

9 0.61398 0.68286 −3.58

10 0.61392 0.68308 −3.94

11 0.61395 0.68298 −4.30

12 0.61394 0.68303 −4.66

13 0.61394 0.68301 −5.02

14 0.61394 0.68301 −5.38

15 0.61394 0.68301 −5.74

16 0.61394 0.68301 −6.10

proposedalgorithm convergesto,dependsonthechoseninitial estimatefortheunknownx₁.Asanillustration,Table3showsthe resultsrespectivelyobtainedusingx⁽⁰⁾₁ =0.01andx⁽⁰⁾₁ =0.99.

To plot the complete isobaric phase diagram presented in Fig.2(a),itwasthusnecessarytoproceedintwosteps:theportion ofthephasediagramlocatedonthelefthandside(lhs)oftheneg- ativeazeotropewasobtainedbyperformingasuccessionofVLE calculationsatincreasingtemperatures,usingtheinitialestimate x⁽⁰⁾₁ =0.01systematically.Theportionofthephasediagramlocated ontherighthandside(rhs)wasobtainedusingasimilarproce- durewithx⁽⁰⁾₁ =0.99.Alsonotethatanaccuratedeterminationof theazeotroperequiresthatthetemperaturesteptbeefficiently managedwhencalculatingeachpartofthephasediagram.Asan example,thetemperaturecanbeincreasedusingaconstanttem- peraturestep(e.g.t=0.1K),untila trivialsolution(i.e.x₁=y₁)

is reached(meaningthat thecurrent temperatureis abovethe azeotropictemperature).Thetemperatureisthusreturnedtoits previousvalueandthenincreasedbyusingasmallertemperature step(e.g.t=t/10).Theprocedureisrepeateduntilt<t_min (e.g.t_min=10⁻⁴K).

Itisinterestingtonotethattheproposedalgorithmcanalsobe easilyusedtocalculateVLEwhenthemolefractionsofthecom- pounds inthetwo phasesare greaterthan1 orless thanzero, althoughthisisnotrelevantforengineeringcalculations.Thisfea- tureshowsthatthealgorithmdoesnotnecessarilydivergeatthe specifiedtemperatureandpressurewhennophysicallymeaning- fulVLEexists.Fig.2(b)showsthatconvergenceisreachedafter quiteareasonablenumberofiterationsforeachVLEcalculation.

Whenthephasediagramdoesnotexhibitcriticality,afewdozens ofiterationsaregenerallyrequired.Notethatthisnumberreaches aminimumatthepure-componentVLEpoint.

Onthechoiceofinitialestimatesforx₁:

SincetheVLEcalculationreducestosolveoneequationwith respecttox₁whenusingthe–ϕapproachwithC_i=1,aninitial valueisnotneededfortheunknowny₁andaninitialestimatemay beeasilychosenforx1,asexplainedbelow.

Firstofall,notethatitissimpletochoosetheinitialestimate inmostofthecaseswheretheliquid–vaporphasediagramdoes notexhibitazeotropyorathree-phaseline(oranycombinationof both).InsuchcasesandifaVLEexistsatthespecifiedtemperature andpressure,thenanyvalueofx₁∈]0;1[leadstoacorrectsolution (divergenceproblemsarequitescarce).

Aspreviouslymentioned,phasediagramsexhibitingazeotropy aretrickiertodealwithsincetwoVLEmayexistatafixedtem- perature and pressure. The case of the binary systemacetone (1)+chloroform(2)atatmosphericpressureisreconsideredasan illustration.Threedifferentsituationsmayoccur:

•thecalculationconvergestoasolutionsuchthatx₁>y₁whichis locatedinthelefthandside(lhs)ofthephasediagramshownin Fig.2(a)

•thecalculationconvergestoasolutionsuchthatx₁<y₁whichis locatedintherighthandside(rhs)ofthephasediagramshown inFig.2(a)

•orthecalculationdiverges.

Thethreedomainsassociatedwitheachofthesethreesitua- tionsareshowninFig.3.Thedomainofdivergenceislargeatlow temperatures,narrowswithincreasingtemperatureandreduces toasinglepointattheazeotropictemperature.Thisfigurealso

Fig.2. (a)Isobaricphasediagramoftheacetone(1)+chloroform(2)systemplottedatP=1atmusingthe–ϕapproach.(b)Representationofthenumberofiterationsfor theproposedVLEcalculationalgorithmthatarenecessarytoreachconvergence(convergencecriterion:ı<10⁻⁶)withrespecttothetemperature.Thefullanddottedlines arerespectivelyassociatedwiththecalculationofthelefthandside(lhs)andrighthandside(rhs)ofthephaseequilibriumdiagram.

Fig.3.Influenceontheconvergenceoftheproposedalgorithm,ofthechoseninitial estimateforx1.

illustrateswhytheinitialestimatesx⁽⁰⁾₁ =0.01andx₁⁽⁰⁾=0.99can beusedtoconstructthephasediagramatalltemperatureswithout difficulty.Sincethedomainofdivergencegenerallyoccursinthe middleofthecompositionrange,itisadvisabletosystematically chooseinitialx₁estimatesclosetozeroandone.

Moredetailsonthefixedpointmethodandconvergence-related issuescanbefoundinAppendixA.

3.5. Numericalexample3:calculationofafluid–fluidphase equilibriumusinganequationofstate

Inthissection,thePeng–RobinsonEoS(Peng&Robinson,1976) isusedtomodeltheliquid–vaporphasebehaviorofthebinarysys- temcarbondioxide(1)+n-hexane(2)atT=393.15KandP=40bar.

ThePeng–RobinsonEoSisgivenby:

PEoS(T,v,zi)= RT

v−bm− am

v(v+bm)+bm(v−bm) (23) whereP_EoSisthepressureofthemixturecalculatedfromtheEoS,v isthemolarvolumeofthefluid;z_iisthemolefractionofcomponent iinthefluidphaseunderconsideration(zi=xiforaliquidphaseand z_i=y_iforagasphase);andamandbmarethetwoEoSparametersfor themixture,whichfollowtheclassicalmixingrulesgivenbelow:

am=

2 i=1

2 j=1

zizj

aiaj(1−kij) and bm=

2 i=1

zibi (24)

Expressionsforthepure-componentquantitiesa_iandb_iarepro- videdinTable4.Thecriticaltemperatures,criticalpressuresand acentricfactorsofthepurecomponentsthatareneededtoevaluate Table4

ParametrizationofthePeng–RobinsonEoS.

Quantity Expression

Criticalcompacity c 1

3

−1+

³

6√

2+8−

³

6√ 2−8

Universalconstant˝a 8(5 c+1)/(49−37 c) Universalconstant˝b c/( c+1)

Shapefunctionmi 0.37464+1.54226ωi−0.26992ω²_i

ac,i ˝aR²T_c,i²/Pc,i

bi ˝_bRTc,i/Pc,i

˛_i(T)

1+m_i

1−

T/T_c,i

2

ai(T) ac,i˛_i(T)

Table5

Physicalpropertiesofpurecomponents.

Tc/K Pc/MPa ω

CO2 304.2 7.383 0.2236

n-Hexane 507.6 3.025 0.3013

N2 126.2 3.400 0.0377

Methane 190.6 4.599 0.0120

Ethane 305.3 4.872 0.0990

Propane 369.8 4.248 0.1520

thecoefficientsa_iandb_i,aregiveninTable5.Thebinaryinterac- tionparametersareestimatedusingthePPR78group-contribution method(Jaubert&Mutelet,2004;Jaubert,Privat&Mutelet,2010):

k₁₂=k₂₁=0.1178andk₁₁=k₂₂=0.Asexplained inSection 2.2,a pressure-explicitEoSformixturesthatcanrepresentbothaliq- uidandavaporphase,asopposedtoavolume-explicitEoS,uses thetemperatureT,themolarvolumevandthecompositionasvari- ables.Consequently,thefluid–fluidequilibriumproblematafixed TandPinvolvessolvingfourequations(seeEq.(7))withrespect tofourunknowns:themolarvolumesand compositionsof the twophasesinequilibrium. InthePeng–Robinson EoS,thefuga- citycoefficientofagivencompoundiinaphasecharacterizedby thetemperatureT,themolarvolumev,themolefractionsz_iofthe compoundsandthepressureP=P_EoS(T,v,z_i),isgivenby:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

lnϕˆ_i= bi

bm

_Pv

RT −1

−ln

_P(v−bm)

R·T

− am

2√ 2RTbm

ı_i− bi

bm

· ln

v+bm(1+√ 2) v+bm(1−√

2)

with : ıi=2

√a_i am

nc

j=1

zj

aj(1−kij)

(25)

Thefollowingalgorithmisproposedforsolvingaliquid–vapor equilibriumataspecifiedTandP,modeledwithanEoS:

1Provideinitial estimatesfor the unknownmolefractions, for instance:[x₁]⁽⁰⁾=0.01and[y₁]⁽⁰⁾=0.99.Settheiterationcounter to:k=0.

2Calculatethemolarvolumesoftheliquidandgasphasesatthe currentiterationk(denotedbyv^(k)_L andv^(k)_G )bysolvingtheEoSat afixedT,Pandcomposition:

v^(k)_L isthesolutionoftheequation: P=PEoS(T,v^(k)_L ,x^(k)₁ )

v^(k)_G isthesolutionoftheequation: P=PEoS(T,v^(k)_G ,y^(k)₁ ) (26)

Notethat theresolution ofanEoSisa recurringthermody- namicissue,addressedmanytimes(Deiters,2002;Privat,Gani

&Jaubert,2010).ForacubicEoS,asimpleCardano-typemethod canbeused.Inaddition,thesolutionoftheEoSmayproduce eitheroneorthreeroots.WhentheEoShasthreeroots,theliq- uidroothasthesmallestmolarvolumewhereasthevaporroot hasthehighestmolarvolume.

3Calculatethefugacitycoefficientsofallcomponentsinallphases:

∀ⁱ∈{1;2}

ϕˆ_i,L^(k)=ϕˆi(T,v^(k)_L ,x^(k)₁ )

ϕˆ_i,G^(k)=ϕˆi(T,v^(k)_G ,y^(k)₁ )

(27)

4Defineı=[|x₁ϕˆ1,L−y₁ϕˆ1,G|+|(1−x₁)ϕˆ2,L−(1−y₁)ϕˆ2,G|]^(k). Ifı<(whereistheaffordedprecision,e.g.=10⁻⁶),thena solutionisreachedandtheprocedureisterminated.