25-1

Ki = equilibriumratio, yi xi L = ratioofmolesofliquidtomolesoftotalmixture N = molefractioninthetotalmixtureorfeed w = acentricfactor P = absolutepressure,kPa(abs) Pk = convergencepressure,kPa(abs),psia

SECTION 25

Phase Equilibria

FIG. 25-1Nomenclature

P* = vaporpressure,kPa(abs) R = universalgasconstant,(kPa(abs)m3)/(kmoleK) T = temperature,KorC V = ratioofmolesofvaportomolesoftotalmixture xi = molefractionofcomponentiintheliquidphase yi = molefractionofcomponentiinthevaporphaseSubscripts i = component

VAPOR-LIQUID EQUILIBRIATheequilibriumratio(Ki)ofacomponentiinamulticompo-

nentmixtureofliquidandvaporphasesisdefinedastheratioof themole fractionof that component in thevaporphase tothatintheliquidphase.

Ki= yi xi Eq 25-1

Foranidealsystem(suchasidealgasandidealsolution),thisequilibrium ratio becomes the ratio of the vapor pressure ofcomponentitothetotalpressureofthesystem.

Ki= Pi*

P Eq 25-2Thissectionpresentsanoutlineproceduretocalculatethe

liquidandvaporcompositionsofatwo-phasemixtureinequi-librium using the concept of a pseudobinary system and theconvergencepressureequilibriumcharts.DiscussionofCO2sep-aration,alternatemethodstoobtainKvalues,andequationsofstatefollow.

K-DATA CHARTSThesechartsshowthevapor-liquidequilibriumratio,Ki,for

useinexampleandapproximateflashcalculations.Thechartsdonotgiveaccurateanswers,particularlyinthecaseofnitro-gen.Theyareincludedonlyforillustrativepurposesandtosup-portexampleflashcalculationsandquickestimationofK-val-uesinotherhandcalculations.Thesechartsshouldnotbeusedindesigncalculations.

PreviouseditionsofthisdatabookpresentedextensivesetsofK-databasedupontheGPAConvergencePressure,Pk,meth-od.AcomponentsK-dataisastrongfunctionoftemperatureandpressureandaweakerfunctionofcomposition.Theconver-gence pressuremethod recognizes composition effects in pre-dicting K-data. The convergence pressure technique can beusedinhandcalculations,anditisstillavailableascomputercorrelationsforK-dataprediction.

Availabilityofcomputers,coupledwiththemorerefinedK-valuecorrelationsinmodernprocesssimulators,hasmadethepreviousGPAconvergencepressurechartsoutdated.CompletesetsofthesechartsareavailablefromGPAasaTechnicalPub-lication,TP-22.

Data for N2-CH4 andN2-C2H6 show that the K-values inthesesystemhavestrongcompositionaldependence.Thecom-ponentvolatilitysequenceisN2-CH4-C2H6andtheK-valuesarefunctions of the amount ofmethane in the liquid phase. Forexample,at123Cand2070kPa(abs),theK-valuesdepend-inguponcompositionvaryfrom:

N2 CH4 C2H6 10.2 0.824* 0.0118 3.05 0.635 0.035*

where*indicatesthelimitinginfinitedilutionK-value.Seeref-erence5forthedataonthisternary.Thechartsretainedinthiseditionrepresentroughly12%ofthechartsincludedinpreviouseditions.Thesechartsareacompro-misesetforgasprocessingasfollows:

a. hydrocarbons3000psiaPk[20700kPa(abs)] b. nitrogen2000psiaPk[13800kPa(abs)] c. hydrogensulfide3000psiaPk[20700kPa(abs)]Thepressures ina. through c. above refer to convergence

pressure,Pk,ofthechartsfromtheTenthEditionofthisdatabook.Theyshouldnotbeusedfordesignworkorrelatedactivi-ties.Again,theyareinthiseditionforillustrationandapproxi-mationpurposesonly;however,theycanbeveryusefulinsucharole.Thecriticallocuschartusedintheconvergencepressuremethodhasalsobeenretained(Fig.25-8).

TheGPA/GPSAsponsorsinvestigationsinhydrocarbonsys-temsofinteresttogasprocessors.Detailedresultsaregiveninthe annual proceedings and in various research reports andtechnicalpublications,whicharelistedinSection1.

25-2

Example 25 -1BinarySystemCalculationTo illustrate theuse of binary systemK-value charts, as-

sumeamixtureof60kmolsofmethaneand40kmolsofethaneat87Cand345kPa(abs).Fromthechartonpage25-10,theK-valuesformethaneandethaneare10and0.35respectively.

Solution StepsFromthedefinitionofK-value,Eq25-1:

KC1 = yC1 xC1

= 10

KC2 = yC2 xC2

= 0.35

Rewritingforthisbinarymixture:1yC11xC1

=0.35

Solvingtheaboveequationssimultaneously:xC1 =0.0674yC1 =0.674

Alsobysolvinginthesameway:xC2 =0.9326yC2 =0.326To find the amount of vapor in themixture, let v denote

kmolsofvapor.Summingthekmolsofmethaneineachphasegives:

kmolsC1+C2=100kmolskmolsC1 + kmolsC1=60kmolsinvapor inliquid(yC1v)+(xC1[100v])=60kmols(0.674v)+(0.0674[100v])=60kmolsv=87.8kmolsThemixtureconsistsof87.8kmolsofvaporand12.2kmols

ofliquid.

FLASH CALCULATION PROBLEMTheproblembelowillustratesthecalculationofmulticom-

ponentvapor-liquidequilibriumusingtheflashequationsandtheK-charts indetail.Thevariablesaredefined inFig.25-1.NotethattheK-valueisimpliedtobeatthermodynamicequi-librium.

Asituationofreproduciblesteadystateconditionsinapieceofequipmentdoesnotnecessarilyimplythatclassicalthermo-dynamic equilibrium exists. If the steady composition differsfromthatforequilibrium,thereasoncanbetheresultoftime-limitedmasstransferanddiffusionrates.Thiswarningismadebecause it isnotatallunusual for flowrates throughequip-menttobesohighthatequilibriumisnotattainedorevenclose-lyapproached.Insuchcases,equilibriumflashcalculationsasdescribedherefailtopredictconditionsinthesystemaccurate-ly,andtheK-valuesaresuspectedforthisfailurewheninfacttheyarenotatfault.

UsingtherelationshipsKi=

yi xi Eq 25-3L+V=1.0 Eq 25-4Bywriting amaterial balance for each component in the

liquid,vapor,andtotalmixture,onemayderivetheflashequa-tioninvariousforms.Acommononeis,

xi = Ni

L+VKi=1.0

Eq 25-5

Otherusefulversionsmaybewrittenas

L= Ni

1+(V/L)KiEq 25-6

yi =KiNi

L+VKi Eq 25-7Atthephaseboundaryconditionsofbubblepoint(L=1.00)anddewpoint(V=1.00),theseequationsreduceto

KiNi=1.0(bubblepoint) Eq 25-8and

Ni/Ki=1.0(dewpoint) Eq 25-9Theseareoftenhelpfulforpreliminarycalculationswhere

thephaseconditionofasystematagivenpressureandtem-peratureisindoubt.IfKiNiandNi/Kiarebothgreaterthan1.0,thesystemisinthetwophaseregion.IfKiNiislessthan1.0,thesystemisallliquid.IfNi/Kiislessthan1.0,thesystemisallvapor.Example 25-2Atypicalhighpressureseparatorgasisusedforfeedtoanaturalgasliquefactionplant,andapreliminarystepintheprocessinvolvescoolingto30Cat4140kPa(abs)to liquefyheavierhydrocarbonspriortocoolingtolowertem-peratureswherethesecomponentswouldfreezeoutassolids.

Solution StepsThe feedgas composition is shown inFig.25-3.The flash

equation25-5issolvedforthreeestimatedvaluesofLasshownincolumns3,4,and5.ByplottingestimatedLversuscalcu-latedxi,thecorrectvalueofLwherexi=1.00isL=0.030,whosesolutionisshownincolumns6and7.Thegascomposi-tionisthencalculatedusingyi=Kixiincolumn8.Thiscorrectvalueisusedforpurposesofillustration.Itisnotacompletelyconvergedsolution,forxi=1.00049andyi=0.99998,columns7and8ofFig.25-3.Thiserrormaybetoolargeforsomeapplica-tions. Example 25-3DewPointCalculation

Agasstreamat40Cand5500kPa(abs)isbeingcooledinaheatexchanger.Findthetemperatureatwhichthegasstartstocondense.

Solution StepsTheapproachtofindthedewpointofthegasstreamissim-

ilartothepreviousexample.Theequationfordewpointcondi-tion(Ni/Ki=1.0) issolvedfortwoestimateddewpointtem-peratures as shown in Fig.25-4. By interpolation, thetemperatureatwhichNi/Ki=1.0isestimatedat41.4C.

Notethattheheaviestcomponentisquiteimportantindewpointcalculations.Formorecomplexmixtures,thecharacter-

25-3

izationoftheheavyfractionasapseudocomponentsuchashex-aneoroctanewillhaveasignificanteffectondewpointcalcula-tions.

Carbon DioxideEarlydataonCO2systemsusedtoprepareK-data(Pk=)

chartsforthe1966Editionwerenotconsistent.Later,experi-ence showed that at low concentrations of CO2, the rule ofthumb

KCO2=KC1KC2 Eq 25-10couldbeusedwithaplusorminus10%accuracy.DevelopmentsintheuseofCO2forreservoirdrivehaveledtoextensiveinves-tigationsinCO2processing.SeetheGPAresearchreports(list-edinSection1)andtheProceedingsofGPAconventions.Thereversevolatilityathighconcentrationofpropaneand/orbu-tanehasbeenusedeffectivelyinextractivedistillationtoeffectCO2 separation frommethane and ethane.23 In general, CO2liesbetweenmethaneandethaneinrelativevolatility.

Separation of CO2 and MethaneTherelativevolatilityofCO2andmethaneattypicaloperat-

ingpressuresisquitehigh,usuallyabout5to1.Fromthisstand-point,thisseparationshouldbequiteeasy.However,atprocess-ingconditions,theCO2willformasolidphaseifthedistillationiscarriedtothepointofproducinghighpuritymethane.

Fig.25-5 depicts the phase diagram for themethane-CO2binary system.21 The pure component lines formethane andCO2vapor-liquidequilibriumformtheleftandrightboundariesof the phase envelope. Each curve terminates at its criticalpoint;methaneat83C,4604kPa(abs)andCO2at31C,7382kPa (abs).Theunshadedarea is thevapor-liquidregion.Theshadedarea represents the vapor-CO2 solid regionwhich ex-tendstoapressureof4860kPa(abs).

Because the solid region extends to a pressure above themethanecriticalpressure,itisnotpossibletofractionatepuremethanefromaCO2-methanesystemwithoutenteringthesolidformationregion.Itispossibletoperformalimitedseparation

FIG. 25-2Sources of K-Value Charts

Component

ChartsavailablefromsourcesasindicatedConvergencepressures,kPa(abs)[psia]

BinaryData 5500 6900 10300 13800 20700 34500 69000

[800] [1000] [1500] [2000] [3000] [5000] [10,000]Nitrogen * Methane * Ethylene Ethane * Propylene Propane * iso-Butane n-Butane * iso-Pentane n-Pentane * Hexane * Heptane * Octane Nonanex Decane Hydrogensulfide Carbondioxide

**UseKCO2 = KC1KC2

* BinarydatafromPrice&Kobayashi;Wichterle&Kobayashi;Stryjek,Chappelear,&Kobayashi;andChen&Kobayashi Drawnfor1972Editionbasedonavailabledata Reusedfrom1966Edition Reusedfrom1957Edition PreparedforSecondRevisions1972Editionorrevised** LimitedtoCO2concentrationof10molepercentoffeedorless

Note:Thechartsshowninboldoutlinearepublishedinthiseditionofthedatabook.ThechartsshownintheshadedareaarepublishedinaseparateGPATechnicalPublication(TP-22)aswellasthe10thEdition.

25-4

ofCO2andmethaneifthedesiredmethanecancontainsignifi-cantquantitiesofCO2.

Atanoperatingpressureabove4860kPa(abs),themethanepurityislimitedbytheCO2-methanecriticallocus(Fig.25-6).For example, operating at 4930kPa (abs), it is theoreticallypossibletoavoidsolidCO2formation(Fig.25-7and16-36).Thelimitonmethanepurityisfixedbytheapproachtothemixturecritical. In this case, the critical binary contains 6% CO2. Apracticaloperatinglimitmightbe10-15%CO2.

Oneapproachtosolvethemethane-CO2distillationproblemis to use extractive distillation (See Section 16,HydrocarbonRecovery).Theconceptistoaddaheavierhydrocarbonstreamto the condenser in a fractionation column. About 10GPA re-search reportspresentdataonvariousCO2 systems thatarepertinenttothedesignofsuchaprocess.

CO2-Ethane SeparationTheseparationofCO2andethanebydistillationislimited

FIG. 25-3Flash Calculation at 4140 kPa and 30C

Component

Column1 2 3 4 5 6 7 8

Feed Gas Composition

30C 4140 kPa

Trial values of L Final L = 0.030L = 0.020 L = 0.060 L = 0.040L

L + VkiLiquid Vapor

Ni KiNi L + V Ki

Ni L + V KiNi L + V Ki

xi = Ni L + V Ki yi

C1 0.9010 3.7 0.24712 0.25466 0.25084 3.61900 0.24896 0.92117CO2** 0.0106 1.23 0.00865 0.00872 0.00868 1.22310 0.00867 0.01066C2 0.0499 0.41 0.11830 0.11203 0.11508 0.42770 0.11667 0.04783C3 0.0187 0.082 0.18633 0.13642 0.15751 0.10954 0.17071 0.01400iC4 0.0065 0.034 0.12191 0.07068 0.08948 0.06298 0.10321 0.00351nC4 0.0045 0.023 0.10578 0.05513 0.07249 0.05231 0.08603 0.00198iC5 0.0017 0.0085 0.06001 0.02500 0.03530 0.03825 0.04445 0.00038nC5 0.0019 0.0058 0.07398 0.02903 0.04170 0.03563 0.05333 0.00031C6 0.0029 0.0014 0.13569 0.04730 0.07014 0.03136 0.09248 0.00013C7+* 0.0023 0.00028 0.11334 0.03817 0.05712 0.03027 0.07598 0.00002TOTALS 1.0000 1.17121 0.77714 0.89834 1.00049 0.99998C7 0.00042C8 0.00014*AverageofnC7+nC8properties

**KC1KC2

FIG. 25-4Dew Point Calculation at 5500 kPa (abs)

Component

Column1 2 3 4 5

Feed Estimated T = 45C Estimated T = 40C

Ni KiNi

KiKi

Ni Ki

CH4 0.854 2.73 0.313 2.75 0.311CO2 0.051 0.866 0.059 0.910 0.056C2H6 0.063 0.275 0.229 0.300 0.210C3H8 0.032 0.070 0.457 0.080 0.400 1.000 1.058 0.977

KCO2calculatedasKC1KC2

Linearinterpolation:Tdew=40[40(45)] 1.0000.977 =41.4C 1.0580.977 AlternativeiterateuntilNi/Ki=1.0

25-5

bytheazeotropeformationbetweenthesecomponents.Anazeo-tropic composition of approximately 67%CO2, 33% ethane isformedatvirtuallyanypressure.24

Fig.25-7 shows the CO2-ethane system at two differentpressures.Thebinaryisaminimumboilingazeotropeatbothpressureswithacompositionofabouttwo-thirdsCO2andone-thirdethane.Thus,anattempttoseparateCO2andethanetonearlypurecomponentsbydistillationcannotbeachievedbytraditional methods, and extractive distillation is required.26(SeeSection16,HydrocarbonRecovery)

Separation of CO2 and H2SThedistillativeseparationofCO2andH2Scanbeperformed

withtraditionalmethods.TherelativevolatilityofCO2toH2Sisquitesmall.WhileanazeotropebetweenH2SandCO2doesnotexist,vapor-liquidequilibriumbehaviorforthisbinaryap-proachesazeotropiccharacterathighCO2concentrations25(SeeSection16,HydrocarbonRecovery).

FIG. 25-5Phase Diagram CH4-Ch2 Binary21

FIG. 25-6Isothermal Dew Point and Frost Point Data for Methane-Carbon Dioxide32

25-6

PHASE EQUILIBRIA METHODSNumerousprocedureshave beendevised to predict phase

equilibria(K-values)andthecorrespondingphysicalpropertiesoftheassociatedphases.Theseinclude:

Equationsofstate(EOS) Activitycoefficientmodels Electrolyticmodels Combinationsofequationsofstatewithliquidtheoryor

withtabulardata Correspondingstatescorrelations.Anumberofmethodscanbeusedforthepurposeofphase

equilibriaandthermodynamicpropertyprediction.Inmoderntimes,calculationsarenottypicallyexecutedbyhand,butin-steadaresolvedbytheuseofthermodynamicsimulationsoft-ware(commercialorproprietary).Thissectiondescribesseveralofthemorepopularprocedurescurrentlyavailable.Itdoesnotpurporttobeall-inclusiveorcomparative.

Equations of State (EOS)Equationsofstatehaveappealforpredictingthermodynam-

icpropertiesbecausetheyprovideinternallyconsistentvaluesforallpropertiesinconvenientanalyticalform.Thesectionbe-lowdiscussesthebasiccapabilitiesofEOS,historicaldevelop-ment,andrecentadvances.

EOS CapabilitiesThefollowingsummarizesthebasiccapabilitiesanddescribestheapplicabilityforsomeofthemorecommonlyusedEOSmethods.

Althoughoriginallydevelopedtodescribesimplegases,EOShaveprovenreliableforpropertypredictionofmosthydrocarbon-basedfluids.

ThesimplecubicEOSaregenerallylimitedtopre-dictionofthermodynamicpropertiesandphaseequilib-

riaforidealorslightlynon-idealsystems;theyarenotsuitableforrepresentationofhighlynon-idealsystems(e.g.,methanol/watersystems).

Theytypicallyareappliedonlytohydrocarbonmix-tureswithrelativelylowconcentrationsofnon-polarorslightlypolarfluids.

Recent advancements havemade cubicEOS suit-ableforhandlinghighconcentrationsofCO2,H2S,andN2.

Applicableforpredictionofphaseequilibriaforpurecom-ponents (VLE) andmixtures (VLE andVLLE) and forpredictionofallthermodynamicpropertiesforvaporandliquidphases.

Originally developed for handling of pure compo-nents, but inclusionanduse of variousmixing rules,whichincorporatebinaryinteractionparameters,haveallowedtheextensionofusetobinaryandmulticompo-nentmixtures.

Usefuloverwiderangesof temperatureandpres-sure, including subcritical, supercritical, and retro-graderegions.

Requireminimalpurecomponentdata.Experimentalbi-narydata canbeused to tune binary interactionpa-rameters,usuallybyregressionofexperimentaldata.

Major EOS types include cubic, virial, correspondingstates, and multi-parameter. Descriptions of the morecommonlyusedcubicandvirialtypesareincludedbelow:

Cubic EOS (e.g., van der Waal, Redlich-Kwong,Soave-Redlich-Kwong,PengRobinson)

Explicitinpressure(P)withrespecttotempera-ture(T)andvolume(V).Theyhaveseparatetermsto correct ideal gas predictions for attraction andrepulsionforcesbetweenthemolecules(correctingtherealvaporpressureandvolumepredictions,re-spectively). When considering the pressure andtemperaturefixed,theEOScanbealgebraicallyre-arrangedtogivearelationshipforVthatisacubic(3rdorder)polynomial.

TheseEOSwillincludeotherparameters,spe-cifictoeachchemicalspeciesthataregenerallyde-terminedfromthecriticalproperties,PcandTc,forthe chemical species. Additional temperature-de-pendentfunctionscanbeaddedtomoreaccuratelymatch pure component behavior (i.e., a tempera-turedependentfunctioncorrelatedtotheaccentricfactor(w)isnormallyusedtobettermatchapurecomponentsvaporpressureversustemperaturebe-havior).

MulticomponentmixturesaretreatedwiththesameEOSparametersthataredeterminedforthepurecomponentspresentinthemixture.Theequa-tionsusedtoblendthepurecomponentvaluesarereferred toas mixing rules,which often includebinaryinteractionparameterstoaccountfornon-ideal interactions between pairs of unlike mole-cules.

The EOSs are generally not tuned to purecomponent liquid density data, so they give poorrepresentationsofliquidmolarvolume/liquidden-

FIG. 25-7Vapor-Liquid Equilibria CO2-C2H621

25-7

sity(Therearetechniquestoimprovethis,suchasintroducingavolumetranslationterm;seeRecentEOSAdvancementsectionbelow).

Notgenerallyaccurateforpolarcompoundsorlongchainhydrocarbons (Thereare techniques toimprovethis,suchasintroductionofahigherordertemperaturedependencyontheattractionparam-eterorasymmetricmixingrules;see RecentEOSAdvancementsectionbelow).VirialEOS(e.g.,BWR,BWRS,Lee-Kessler-Plocker)

Explicitinpressure(P)withrespecttotempera-ture(T)andvolume(V).Expandedintermsofvol-umeraisedtopowersmuchhigherthan3rdorder.

Because of the larger number of parametersthat can be tuned to pure component data, theseEOS canbemoreaccurate than cubicEOSwhencalculatingliquiddensities.

For pure components, these EOS can give amore accurate representation than cubicEOS forallthermodynamicdata.However,thedetermina-tionoftheseparametersismorecomplex,requiresmoreexperimentaldata,andmayrequireacompli-catedprocedureforfittingthatexperimentaldata.

Whenappliedtomulticomponentmixtures,thelargenumberofpurecomponentparametersmustbe blended, perhaps each with their own mixingrules.Inaccuraciesassociatedwiththeapplicationofthesemixingrulesmaymakethemixtureprop-ertiesnomoreaccurate thanwhatonewouldgetfromasimplercubicEOS.

Notusuallyapplicableforpolarsystems.Computerized corresponding states methods

may be based on virial EOS for reference fluids.Thecorrespondingstatesmethodsthenprovidetheframeworktoblendthepropertiesofthereferencefluidstogivevaluesforamulticomponentmixture.OtherapplicableEOStypes

EOSforAssociatingSystemsEOS that include terms for the physical

forces(attraction/repulsion)andanassociatingterm that takes into accounthydrogen bond-ing;seeKontogeorgisandFolas34forreviewsofassociatingEOS.

TheSAFT(StatisticalAssociateFluidThe-ory) family of EOS are based on Wertheimsperturbation theory and can be applied to awiderangeoffluidsincludinglongchaincom-ponentsandhydrogenbonding(e.g.,hydrocar-bon-alcohol-watersystems).

CPA (Cubic-Plus-Association) EOS com-binesacubicequationofstate(e.g.,SRK)fordescribingphysicalinteractionswithanasso-ciationtermsimilartoSAFT.HighlyAccuratePureComponentEOS

Typically apply only to utility systemswithin a facility, not to the main processingandseparationtrains.

ExamplesincludeNBSSteamTables,Spanand Wagner EOS (CO2), Wagner and PrussEOS(Water),andheattransferfluidmodels.

Historical Development of EOS for Phase EquilibriaTwopopularstateequationsforK-valuepredictionsaretheBenedict-Webb-Rubin(BWR)equationandtheRedlich-Kwongequation.

TheoriginalBWRequation17useseightparametersforeachcomponentinamixtureplusatabulartemperaturedependenceforoneoftheparameterstoimprovethefitofvapor-pressuredata. This original equation is reasonably accurate for lightparaffinmixturesatreducedtemperaturesof0.6andabove.8Theequationhasdifficultywithlowtemperatures,non-hydro-carbons,non-paraffins,andheavyparaffins.

ImprovementstotheBWRincludeadditionaltermsfortem-perature dependence, parameters for additional compounds,andgeneralizedformsoftheparameters.

Starling20hasincludedexplicitparametertemperaturede-pendence inamodifiedBWRequationthat iscapableofpre-dictinglightparaffinK-valuesatcryogenictemperatures.

TheRedlich-Kwongequationhastheadvantageofasimpleanalyticalformwhichpermitsdirectsolutionfordensityatspec-ifiedpressureandtemperature.Theequationusestwoparame-tersforeachmixturecomponent,whichinprinciplepermitspa-rametervaluestobedeterminedfromcriticalproperties.

However, as with the BWR equation, the Redlich-KwongequationhasbeenmadeusefulforK-valuepredictionsbyem-piricalvariationoftheparameterswithtemperatureandwithacentricfactor11,18,19andbymodificationoftheparameter-com-bination rules.15,19 Considering the simplicity of theRedlich-Kwongequationform,thevariousmodifiedversionspredictK-valuesremarkablywell.

Interaction parameters for non-hydrocarbons with hydro-carboncomponentsarenecessaryintheRedlich-Kwongequa-tion to predict theK-values accuratelywhenhigh concentra-tions of non-hydrocarbon components are present. They areespeciallyimportantinCO2fractionationprocesses,andincon-ventionalfractionationplantstopredictsulfurcompounddis-tribution.

TheChao-Seadercorrelation7usestheRedlich-Kwongequa-tionforthevaporphase,theregularsolutionmodelforliquid-mix-turenon-ideality,andapure-liquidpropertycorrelationforeffectsof component identity, pressure, and temperature in the liquidphase.Thecorrelationhasbeenappliedtoabroadspectrumofcompositionsattemperaturesfrom50Fto300Fandpressuresto2000psia.Theoriginal(P,T)limitationshavebeenreviewed.12

PrausnitzandChuehhavedeveloped16aprocedureforhigh-pressure systemsemployingamodifiedRedlich-Kwongequa-tion for the vapor phase and for liquid-phase compressibilitytogetherwithamodifiedWohl-equationmodelforliquidphaseactivity coefficients.Complete computer program listings aregivenintheirbook.Parametersaregivenformostnaturalgascomponents.Adleretal.alsousetheRedlich-KwongequationforthevaporandtheWohlequationformfortheliquidphase.6

Thecorrespondingstatesprinciple10isusedinalltheproce-duresdiscussedabove.Theprincipleassumesthatthebehaviorofallsubstancesfollowsthesameequationformsandequationparametersarecorrelatedversusreducedpropertiesandacen-tricfactor.Analternatecorrespondingstatesapproachistore-ferthebehaviorofallsubstancestothepropertiesofareference

25-8

substance, these properties being given by tabular data or ahighlyaccuratestateequationdevelopedspecificallyfortheref-erencesubstance.Thedeviationsofothersubstancesfromthesimplecritical-parameter-ratiocorrespondencetothereferencesubstancearethencorrelated.Mixturerulesandcombinationrules,asusual,extendtheproceduretomixturecalculations.Leland and co-workers have developed9 this approach exten-sivelyforhydrocarbonmixtures.

Shapefactorsareusedtoaccountfordeparturefromsimplecorrespondingstatesrelationships,withtheusualreferencesub-stancebeingmethane.TheshapefactorsaredevelopedfromPVTandfugacitydataforpurecomponents.Theprocedurehasbeentestedoverareducedtemperaturerangeof0.4to3.3andforpres-sures to 4000psia. Sixty-two components have been correlatedincludingolefinic,naphthenic,andaromatichydrocarbons.

TheSoaveRedlich-Kwong(SRK)13isamodifiedversionoftheRedlich-Kwongequation.Oneoftheparametersintheorig-inalRedlich-Kwongequation,a,ismodifiedtoamoretempera-turedependentterm.Itisexpressedasafunctionoftheacen-tric factor. The SRK correlation has improved accuracy inpredicting the saturation conditions of both pure substancesandmixtures.Itcanalsopredictphasebehaviorinthecriticalregion, although at times the calculations become unstablearoundthecriticalpoint.Lessaccuracyhasbeenobtainedwhenapplyingthecorrelationtohydrogen-containingmixtures.

Peng and Robinson14 similarly developed a two-constantequation of state in 1976. In this correlation, the attractivepressure term of the semi-empirical van derWaals equationhasbeenmodified.Itpredictsthevaporpressuresofpuresub-stancesandequilibriumratiosofmixtures.

Inapplyinganyoftheabovecorrelations,theoriginalcriti-cal/physicalpropertiesusedinthederivationshouldbeinsertedintotheappropriateequations.Itiscommonforonetoobtainslightlydifferentsolutionsfromdifferentcomputerprograms,evenforthesamecorrelation.Thiscanbeattributedtodiffer-ent pure component and binary parameters, iteration tech-niques, convergence criteria, and initial estimation values,amongother itemsasdescribed in theRecentEOSAdvance-mentssub-sectionbelow.Determinationandselectionofapar-ticular equation of stateand interactionparametersmustbedonecarefully,consideringthesystemcomponents,theoperat-ingconditions,etc.

Recent EOS Advancements Whilesomeofthefunda-mental,basicequationofstateformsareincludedattheendofthissection,therehavebeenmanyadvancementsinthepredic-tion of phase equilibria and thermodynamic properties sincethelastupdateofthissection(pre-1990).Asaresult,andduetotheextensiveuseofcommercialsimulationtools,resultswhichdiffersomewhatwilllikelybeobtainedforthesameEquationofState,dependingonthesoftwarechosen,andevenoptionsse-lectedwithinthesoftware.InadditiontothoseitemslistedintheHistoricaldevelopmentofEOSsectionabove,thisislarge-lyduetotheadvancementsmadeinapplicationofEOSmeth-ods.Thefollowingisabriefsummaryofsomebasicreasonsforthesedifferencesfromonesoftwarepackagetoanother,alongwith a general description of advanced applications of EOSmethods.

ImprovedmixingrulesAnumberofdifferentmixingrulescanbeappliedto

anEOS,somemuchmorecomplexthanothers.Ingen-eral,morecomplexmixingrulesallowfortherangeofapplicability of anEOS extended further beyond the

available experimental data; however, more experi-mentaldataisrequiredtoallowforaproperfitofthemixingequation.

ApplicationofmorecomplexmixingrulescanmakeEOSmethodsadequateforpolar/non-idealsystems.

Specifically,ActivityCoefficientmethods havebeenuseddirectlyinsomemixingrulestomoreac-curately predict binary interactions of mixtureswithpolarandnon-polarcomponentsathighpres-sure,despite theActivityCoefficientmethodonlybeing fit to available low pressure experimentaldata(i.e.,Wong-Sandler).

EnhancedbinaryinteractionparametersGroupcontributionmethodshavebeendevelopedto

estimatebinaryinteractions(e.g.PredictiveSRK)andgreatly improve predictions especially for mixtureswithpolarandnon-polarcomponents.

Interaction parameters are typically fitted to ex-perimentaldataforeachspecificEOSandmixingrulecombination.Inturn,morequalityexperimentaldatainthepressure,temperature,andcompositionalregionof a particular application of interest allows for en-hanced binary interaction parameters and improvedEOSpredictions.However,fittinginteractionparame-terstodifferentsetsofdatawillresultininconsistentpredictionsfromonetooltoanother.

Binary interactionparameters are often tempera-turedependent,andmaybe fitbydiffering tempera-ture dependency forms, for which proper choice canimpactEOSperformance.Theabilityforausertospec-ifybinaryinteractionparametersisincludedinmanyof commerciallyavailable simulationproducts.A toolthatallowsfornon-constantspecification(e.g.,includestemperature dependence) will result in improved re-sults.

AdditionalequationtermsAdditionofextendedoradvancedalphafunctions

(intermolecularattraction)toimprovefittingofvaporpressure,whichcanimprovetheabilityoftheEOStohandle polar/non-ideal systems. Addition of volumetranslation parameters allow for better prediction ofliquiddensitiesfortheEOS.

LiquidphasepropertyhandlingModificationofthehandlingofthetermdescribing

real volume ofmolecules/intermolecular repulsion al-low for better prediction of liquid densities for theEOS.

Solidshandling(e.g.,ice,hydrate,solidCO2,solidhydro-carbon)

While EOS are used to represent the fugacity ofcomponents in a fluid phase (vapor and liquid), theycanbecombinedwithmodelsrepresentingthefugacityinthesolidphasetomodelVSEandLSE.Seethesec-tiontitledEquationsofStateandtheSolidPhasebe-lowformoredetaileddiscussion.

There are a number of multi-parameter equations (i.e.,GERG35),thatcurrentlyexistandareabletomodelsystemstowithinexperimentalerror.However,duetothecomplexityandcomputingpowerrequiredforthese,theyarenotoftenusedin

25-9

facilitydesignsimulations.Oneobvioususeofanequationofthisnatureistogeneratepseudo-experimentaldatafromwhichnewbinaryinteractionparameterscanberegressedforaspe-cific system (P, T, and composition), and in turn used in an EOS toimproveitsreliability.

BASIC EQUATION OF STATE FORMSRefertooriginalpapersformixingrulesformulticomponent

mixtures.

van der Waals30

Z3(1+B)Z2+AZAB=0 A = aP R2T2 B = bP RT a = 27R

2T2c 64Pc b = RTc 8Pc

Redlich-Kwong28

Z3Z2+(ABB2)ZAB=0 A = aP R2T2.5

B = bP RT a = 0.42747 R

2T2c.5 Pc b = 0.0867 RTc Pc

Soave Redlich-Kwong (SRK)13

Z3Z2+(ABB2)ZAB=0 A = aP R2T2

B = bP RT a=aca ac =

0.42747 R2T2c Pc

a12=1+m(1Tr12) m=0.48+1.574w0.176w2 b =0.08664 RTc Pc

Peng Robinson31

Z3(1B)Z2+(A3B22B)Z(ABB2B3)=0 A = aP R2T2

B = bP RT

a = 0.45724 R2T2c a Pc

a12=1+m(1Tr12) m=0.37464+1.54226w0.26992w2 b = 0.0778 RTc Pc

Benedict-Webb-Rubin-Starling (BWRS)20, 29

P = RT+ BoRTAoCo Do Eo 1 V T2 + T3T4 V2

+ bRTa d 1 +aa+ d 1 T V3 T V6

+ c 1 1+ v2 V3 T2 V2

Note:w,theacentricfactorisdefinedinSection23.

Other Phase Equilibria MethodsActivity Coefficient Models Anothercommonmethod

used for the purpose of phase equilibria and thermodynamicproperty prediction is the use of Activity Coefficientmodels.ThefollowingisabriefsummaryofthebasiccapabilitiesanddescribestheapplicabilityforsomeofthemorecommonlyusedActivityCoefficientmethods.

Activitycoefficientmodelsarethebestmethodforrepre-sentationofhighlynon-idealand/orpolarsystems(i.e.,aqueous systems, amines,NH3, caustic,CO2,H2S) andarethereforetypicallyusedinthechemicalsindustry.

Whilethesemodelsaregenerallyonlyapplicabletopre-diction of phase equilibria for binary andmulticompo-nentmixtures (VLE and LLE), they are not for phaseequilibriaofpurecomponents.However,theydorequirehighqualitypurecomponentpropertypredictions (e.g.,vaporpressure).DependingonthespecificActivityCoef-ficientmethod,itmaynotalwaysallowforLLEpredic-tionbecausetuningthemodelstoVLEspecificdataorLLEspecificdatamayresultindrasticallydifferentpa-rameters.Forthisreason,VLLEpredictionsmustalsobeusedwithcaution.

Applicable for prediction of thermodynamic propertiesfortheliquidphaseonly.Vaporpropertiesareunreliableandmust be calculated using anothermethod; histori-callythishasbeendoneusingidealgasassumptionsforthevaporphase,butcommonlyincludesmoreadvancedEOSmethods,asdescribedintheMixedModelssectionbelow.

Limitedtosystemswithinthepressureandtemperaturerangesoftheexperimentaldataitiscorrelatedagainst.

Thesemodelsareonlysuitableforlowtomoderatepressuresystems,typicalinthechemicalsindustry,be-causeactivitycoefficientsdependontemperature,butareindependentofpressure,whilemutualsolubilitiesareinfactdependantonpressureinhighpressureLLEsystems.

At typicaloperatingpressures, theuseofavaporpressure is not appropriate for light gases above thecriticalpointandinsteadtheselightgasesaretreatedasHenryscomponents,whereHenryslawcoefficientsarederivedfromexperimentalgassolubilitydata.

25-10

Extrapolationoutsidetheexperimentaldatarangeisnotrecommended.

Requiresaseparatemodel todetermine liquiddensity.ThePoyntingcorrectionisusedtoaccountforpressuredependenceofacomponentsliquidphaseactivity.

Someexamples of commonActivityCoefficientmodels in-clude:ChienNull,NRTL,Margules,UNIFAC,UNIQUAC,vanLaar,andWilson.

Electrolytic Models Electrolyticmodelsareasub-setofActivityCoefficientmodels.ThegeneralpurposeofElectrolyticmodelsistohandlesystemswheredissociationofcomponentsisimportant.Ingeneral,thesecomponentsdonotdirectlypar-ticipateinVLE(i.e.,ionsorsolidsthatdonotdissolveorvapor-ize),buttheyofteninfluenceactivitycoefficientsofotherspe-ciesbyreactionorinteractionandinturn,indirectlyparticipateinimpactingthephaseequilibriaofthesystem.Thesemodelsgenerallyrequirehighqualityexperimentdataforthespecificapplicationtheyarebeingappliedto(e.g.,aminegastreatingsystems).

Mixed Models Mixedmodelsarecommonlyusedinor-dertocombinethestrengthsofthevariousmethodsdescribedabove.ThesemodelscommonlyuseanEOSmethodforphaseequilibriaandpredictionofvaporphasethermodynamicprop-erties,butuseanActivityCoefficientorElectrolytemethodfordetermination of thermodynamicproperties and/or density ofthe liquid phase(s). However, because of the use ofmultiplemethods,thesemodelsdonotalwaysproduceconsistentpredic-tions,especiallyatornearthecriticalpoint.

Practical Application of Phase Equilibria Methods

General Considerations Whilephaseequilibriaoftyp-icalhydrocarbonsystemsismodeledverywellwithcurrentlyavailable commercial computer simulation tools, there aremanyareaswherecarefulattentiontomethodselectionisneed-ed.Somesystems,duetothethermodynamiccomplexity,aredifficulttomodelusingthebasicmethodsdescribedabove,evenwithmoreand/orbetterquality experimentaldata.Someex-amplesofthesecomplexsystemsinthegasprocessingindustryareshownbelow.Theequilibriaandthermodynamicpropertyresults obtained using a specific method for these systemsshouldbecarefullyevaluatedandcomparedtocommercialex-periencewhenusedinaprocessdesign:

Complex Systems to Model Operating conditions that cause divergence from ideal

fluidbehaviorSelfassociating/dimerizing(i.e.,aqueoussolutions)Dissociating(i.e.,aminegastreating,sourwater)Sterically hindered dissociating compounds (i.e.,

certainamines)Nearcriticalandsupercriticalfluids(i.e.,supercrit-

ical natural gas compression, supercritical H2S/CO2compression,hydrocarbonsystemsnearcritical)

Combinations of non-polar and polar compounds(i.e.,hydrocarbonsystemswithsourorproducedwater,dehydration or dew point depression with glycols ormethanol)

Hydrogen bonding (i.e.,methanol-water-hydrocar-bonsystems)

NaturalGasDewPointCalculations Otherconsiderationsthatimpactphaseequilibria

Reactive(i.e.,aminegastreating,caustictreating)Mass transfer limited (i.e., tertiary and hindered

amines)Absorption(i.e.,CO2andH2Sadsorptioninphysical

orchemicalsolvents)Overall,itisimportanttounderstandthecapabilitiesand

limitationofeachmethodofrepresentingphaseequilibriaandprediction of thermodynamic properties, and each methodsspecificapplicabilitytothegasprocessingindustryforproperchoiceanduse.However, it shouldbenoted that tuningofamethodtoqualityexperimentaldataintheregionofoperationorinterestisperhapsmoreimportantthanthemethodchoiceitself,specifically,thechoiceofmixingrulesandqualityofbi-nary interaction parameters, which can typically be readilymodifiedincommercialsoftware.

Morespecificinformationrelativetophaseequilibriameth-odscanbefoundinGoodwinetal36andKontogeorgisandFo-las.34

Dew Point CalculationAthermodynamicdewpointisdefinedas thatpointwhere liquid firstappears fromthegasphase. AnEOSmodelactuallycalculatesthispointwithex-actlyzeroliquiddropout.Inreality,fornaturalgasthispointcannotdirectlybemeasured fromexperimentalmethods,butcanbeestimatedfromPVTdatatakenverynearthedewpointenvelope by extrapolating liquid dropout data to 0.0 volumepercentliquid.

In pipeline operations, one can consider a practical dewpoint that represents a small volume of liquid condensationwhichdoesnotimpactpipelineperformance,usuallyrepresent-ingatraceofliquidonthepipewall.ThispracticaldewpointiswhatisactuallymeasuredbytheBureauofMineschilledmir-rordevice.

InrecentworkfortheGPAbyBullin,et.al.,(RR-213),thepracticaldewpointwasdefinedas0.00027m3liquidper1000m3gas.Asanaturalgasgetsleaner,thedifferencebetweentheEOSpredicteddewpointof0.0volumepercentliquidandthepracticaldewpointof0.00027m3 liquidper1000m3gascanincrease to amuch as 5.6C.The practical dewpoint can berepresentedbythetemperature inaBureauofMineschilledmirrordevicewheredropletsbeginto form.Thus,whenEOSmodelsareused,boththethermodynamicdewpointof0.0vol-umepercentliquidandthepracticaldewpointof0.00027m3liquidper1000m3gasshouldbeconsideredwhenadjustingtheEOSparameterstomatchtheexperimentalvalues,andalsotobetterdeterminetheconditionsthatimpactplant/pipelineper-formance.Forleangaseswherethereismorethana2.2to5Cdifferenceinthetwovalues,itmaybenecessarytoonlyfittothepracticaldewpointvaluetoevaluateplant/pipelineperfor-mance.

ThisalsopointstothefactthatwhenfittinganEOSmodeltoanexperimentallyobtaineddewpointvalue,itshouldnotbeassumedthatthereporteddewpoint(experimental)representsthethermodynamicdewpointof0.0volumepercentliquid,un-less of course it has been confirmed to be extrapolated frommultipleexperimentalpointswithinthetwophaseenvelope.

25-11

It cannotbe overemphasized that If theEOS is fit to thewrongdewpointconditionforsomenaturalgascompositions(thermodynamicversuspractical),itcansignificantlyaffecttheamountofliquidsthatarepredictedtocondenseinplantopera-tions or in pipelines, affecting overall designs.Also for thesegaseswithasignificantdifferenceinthedewpointvalue,thevolumepercentliquiddatawilldisplayanupwardcurvatureasthethermodynamicdewpointisneared.ThismayresultinarequirementforamoreexpandedcompositiontobeusedintheEOSmodel,perhapsevenaboveC12+.

Amine Treating Aminetreatingisoneofthemostcom-monunitoperationsinthegasprocessingindustry.Thereareavarietyofaminesolutiontypes(i.e.primary,secondarytertiary,sterically hindered, mixtures, with additives), and solutionstrengths,allofwhichpossesscharacteristicsthatcausediver-gencefromidealfluidbehavior(selfassociating/dimerizing,dis-sociatingandpotentiallystericallyhindered,andcombinationsofpolarandnon-polar compounds). Inaddition theyarealsocomplicatedsystemstomodelbeingreactive,sometimesmasstransferlimited,absorption/desorptionsystems.Thiscombina-tion of system attributes presents challenges for predictingequilibrium and thermodynamic properties accurately. Allthermodynamic phase equilibriummodels depend on experi-mentalthermodynamicequilibriadata.Manytimestechnologysuppliers,andor/simulationsupport companies,willuse realplantdatatoverifyamodelinareaofinterest.Someofthemostcommonmodelsusedtodescribethephaseequilibriaofaminesystemsaredescribedbelow.37Choiceoftheappropriatemodelshouldbeconsideredcarefully.

Kent-Eisenburg(1976)38ModelBasedondefiningchemi-calreactionequilibriaintheliquidphase.Assumesallac-tivityandfugacitycoefficientsareunity(i.e.,idealsolutionsandidealgases)andforcesafitbetweenexperimentalandpredictedvaluesbytreatingtwoofthereactionequilibriumconstantsasvariables(e.g.,theaminedissociationreactionand the carbamate formation reation). This approach istypicallyonlyapplicabletosystemswithsingleacidgases(CO2orH2S,butnotboth)andunlessamodifiedapproachis used, to primary and secondary amines. The modelshouldnotbeextrapolatedasitsthermodynamicbasisisweakanddoesnotsupportextrapolation.

DeshmukhandMather(1981)39MethodBasedonDe-bye-Huckel theory and is thermodynamically rigorous.SimilartotheKent-EisenburgModelinuseofchemicalreactions,butactivity coefficientsareestimatedon thebasisofion-ion,ion-molecule,andmolecule-moleculein-teraction parameters. Uses either the Suave-Redlich-KwongorPeng-RobinsonEOS for thevaporphase.AnactivitycoefficientdescriptionofHenryslawisusedtodescribethephysicalsolubilityofacidgasesintheaminesolvents.Binaryinteractionparametersmustbeadjust-edinordertofitfinalVLEmodeltoexperimentaldata.

LiandMather(1994)40MethodBasedonPitzersGibbsexcessenergyequations41andaccountsforthevariousionicandmolecularspeciesintheliquidphasewhenacidgasisdissolvedintoamixedaminesolution.Thisisamoremod-ernrenditionofanactivitycoefficientapproach.

Austgenetal.(1991)42ElectrolyticNRTLbasedmodelThis is also anactivity coefficientmodel in the vein ofthosediscussedintheElecrolyticandMixedmodelsec-tionsabove.Itisoneofthemostthermodynamicallyrig-orousmodels,butrequiresuseofbinaryandternaryin-teraction parameters in regressing the model toexperimentaldata.

TheDeshmukhandMather,LiandMather,andAustgenmethodsalluseactivitycoefficientmodelsfortheliquidphase,andEOSmodelsforthevaporphase.Modelparametersmustbe found for each by regressing the models to experimentaldata.Theprimarylimitationisnotthemodelsthemselves,butthehighlyvariable,andsometimesunknownqualityoftheex-perimental data. Because the experimental data is often thelimitationandnotthemodel,thismayallowanyoftheactivitycoefficientmodelstobejustasreliableastheothersforagivenapplication.

Sour Water StrippingH2S,CO2,andammonia,fromsourwater is performed inmany sournatural gas treatingplants.Commonapplicationsincludecondensedwater(H2SandCO2),sulfur plant tail gas quench tower blowdown (H2S ,CO2, andcausticorammonia)andtoamorelimitedextent,strippingofsour produced water (brine, including NaCl and many othersalts,withH2S,andCO2).Muchoftheliteraturedescribesde-signandoperationofrefinerysourwaterstrippers,whichhandle(H2S CO2, ammonia, cyanides, phenols, and organic acids).Guidelinesspecifictogastreatingaremorelimited.

SourwaterservesasagoodexampleofthestrongeffectofpHonionicdissociation,andofthesmaller,thoughsometimessig-nificant,effectofdissolvedsaltconcentrationsongassolubility.

AprincipalionizationequilibriumforH2Sinaqueoussolu-tion,

H2S H++HS Eq 25-11

isgreatlyaffectedbypH.Theun-dissociatedfractionoftheto-talH2Siscloseto1forapHoflessthan5,about0.33atapHof7.0,andlessthan0.01atapHof9.043.SincethevaporpressureofH2Sisproducedbyitsun-dissociatedfraction,pHinastrip-pingtowerhasagreateffectonthestrippingefficiency.AsH2Sgasisstrippedfromsolution,thedissociatedionsre-combinetoprovidemore un-dissociatedH2Swhich revives the partialpressureabovethesolution.ItshouldbenotedthatthepHofthesolutionchangesasCO2,isstrippedfromthewater.Insomeservices,withH2S,CO2,NH3,and lowsalts, thepHmustbecontrolledto(typicallyto7-8)toinsurethatbothcomponentscan be stripped. Stripping is, of course, enhanced by highertemperatures.

Saltconcentrationoftensofthousandsofppm,asiscom-moninproducedwaters,reducesthesolubilityofH2Sandthusenhancesstrippingbutthisisasmallerinfluencethantem-perature,andmuchsmallerthanpH.ThepHofproducedwateristypicallyaboveneutraltostart,evenwiththedissolvedsalts,becauseofthenaturalbuffersinthewater.ThepHisfurtherelevated,astheCO2isstrippedfromthebrine,sometimesre-sultinginfoulingofthemasstransfermediumwithinthestrip-per.Therefore,acidissometimesaddedinfrontofthestrippertolowerthepHandpreventfouling.

A commonmethod to calculate equilibrium of sourwatersystems is theWilson-API-Sourmethod,andvariants,whichuseamodificationofVanKrevelensapproach toaccount fortheionizationofH2S,CO2andNH3intheaqueouswaterphase(SeeAPIPublication95544andGPARR-5245).Thistypeofmod-el isvalidup toabout50psig,with limitedother ionic (salt)speciespresent. Toextendtherangeof thesourequilibriumprediction,mixedmodelsusingacombinationoftheAPI-Sourmodel,andanEOS(e.g.,PengRobinson)havebeendeveloped.Alternateapproacheswhichaccountfortheinfluenceofotherionic salts present in the solution are electrolytic andmixedelectrolytic/EOSmodels.

25-12

Adiscussionondesignofsourwaterstrippersystemsispre-sentedinSection19.StevensandMosher46provideabroadre-viewofvariationsandoptions,andpotentialproblemsareasindesignandoperationofsourwatersystems.Bechok47isanoldbuttraditionalreferenceforsourwaterequilibriaandstripperdesign.

EQUATIONS OF STATE AND THE SOLID PHASE

Aswillbediscussedinsomedetaillater,equationsofstatearelimitedintheirabilitytopredictequilibriainvolvingsolidphases.Thatmakesitdesirabletohaveawayofcheckingsuchpredictions made in simulations. Good experimental data, ifavailable,isthebestcheck.Somewaysofcheckingand/orcor-rectingsimulated resultsarediscussedbelow,afteran intro-ductiontothegeneraltopicofsolidphases.

The PhaseRule, developed byGibbs back in the 1870s,still serves as a trusty background tomulti-phase equilibria.Fornon-reactivecompounds,theequationis:

F=C+2P Eq 25-12whereCisthenumberofcomponentsorspeciesinthemixture,Pisthenumberofphasespresent,andFisthenumberofde-greesoffreedomofthesystem.Insituationswheresolidsmaybeof concern ingasprocessing, thereareusuallyboth liquidandvaporphasesalsopresent.ThreenumericalexamplesforthePhaseRuleareshowninthetablebelow,followedbydis-cussionoftheirconsequenceswhenallthreephasesco-exist.

C, Number of Components

P, number of Phases

Degrees of

FreedomConsequence

1,singlecomponent 3,V,L,andS 0 UniqueTriplePoint2,twocomponents 3,V,L,andS 1 Triplepointlocusline

3,threecomponents 3,V,L,andS 2 Nolocusline;muchmorecomplexequilibria

Single-Component Triple PointsThefirstexample,forasinglecomponent,isbestknown.A

uniquetriplepointcanbefound,forexample,ontheP-HchartforCO2 in thisSection.Thisuniquepointonsuchachart isusefultobetterunderstandthepresenceofvariousmixturesofphasesinthatareaofthechart.However,thenumericalvaluesofthetriplepointpressureandtemperaturearenotveryrele-vanttoprocessoperations.Itishighlyunlikelythatanyther-modynamic path for a fluid (say, in blowdown) would passthroughthetriplepoint(seetwosuchlinesontheexamplesofusesforP-Hcharts,alsointhisSection).Thus,quantifyingaunique triple point is rarely of help in checking a simulatedprediction.

Two-Component Triple Point Locus LinesIn contrast, ina few industrially important cases,knowl-

edgeofthetriplepointlocuslineforatwo-componentsystemcanprovideaveryusefulcheckonsimulatorpredictions.TheexamplebelowisfromarecentlypublishedstudybyWalteretal48thatchecked,andmadecorrectionsto,simulationsofblow-downofahighpressureacidgaslinecontaining(essentially)onlyCO2andH2S.

0

50

100

150

200

250

300

350

400

450

500

-100 -90 -80 -70 -60 -50

Pre

ssu

re, k

Pa

(ab

s)

Tem pera ture , C

Triple Point locus line 44% H 2S , 56% CO 2 Temp erature C orrectio n

Intheabovediagram,thecontinuouslinemarkedwithtri-anglesisthetriplepointlineforsolidsdepositioninthepres-enceofbothliquidandvaporphasesfromSobocinskiandKu-rata25withthedataforthelinegiveninthereferencedocument.Fortherangeofpressuresandtemperaturesshownhere,thesolidphaseisCO2ice.Atmuchlowerpressuresandtempera-tures(