132S.Hashemietal./FluidPhaseEquilibria246(2006)

131–136

Fig.1.Phasediagramofmethane–watersystem[1,18].

drivingforcearoundthegasbubbleisassumedtobethedif-ferencebetweenthegashydrateformersolubilityatthegasbubble–liquidinterfaceanditsconcentrationinthebulkliq-uidwater.Hence,vapor–liquidwater(Lw–V),vapor–liquidwater–hydrate(H–Lw–V)andhydrate–liquidwater(H–Lw)equilibriumarekeyconceptsinthemodelingofthesesystems.Moreover,inordertousegashydratesforcarbondioxideseques-tration,(H–Lw)solubilitypredictionisrequiredtoassessthestabilityofcarbondioxidehydrateslocatedatthebottomoftheocean.(Lw–V)and(H–Lw–V)equilibriumhavebeenthesubjectofyearsofstudy[6–17,1].However,thereareonlyfewpapersdiscussingthesolubilityofgasinwaterinthehydrate–liquidwatertwo-phasezone.Thusageneralmodelforpredictingthesolubilitywherehydrateandliquidwaterareinequilibriumissoughtinthisstudy.

Duetosimplicityandlackofanapplicablemodel,Henry’sLawhasbeenemployedrecentlytopredictthesolubilityinhydrate–liquidwaterequilibrium[18–19]whereitwasassumedthatgassolubilityat(H–Lw)equilibrium,i.e.at(Pexp,Texp),isequaltothatat(Peq,Texp)inP–Tequilibriumdiagram,seeFig.1[1,18].PeqandTexparelocatedonthethree-phaseline.Hence,TexpisalsoequaltoTeqandthesolubilitypredictedwiththisassumptionwouldbethatatthe(H–Lw–V)equilibriuminsteadof(H–Lw).

OnewayofdeterminingsolubilityistheuseofaGibbsenergy–solubilitydiagram[20].Thisapproachseeksthesol-ubilityatwhichtheGibbsenergycurvehasthelowestvalue.MinimumofGibbsenergydemonstrateswhichphasesaresta-bleatacertaintemperatureandpressure.Althoughthetheorybehindthegraphicalapproachisidenticaltothatofthemostcommonlyusedfugacityorchemicalpotential,itcannotbelookeduponasapracticalmodelforsolubilityprediction.

TherehavealsobeensomeattemptstodevelopamodeltopredictthesolubilitybasedonthevanderWaalsandPlatteeuw[21]model.AmongthisgroupistheworkofHanda[22]whichemploystheactivitycoef cienttorepresentthechemicalpoten-tialofwaterintheliquidphase.Handaderivedthechemicalpotentialofwaterinliquidandhydratephasesintermsofthepressureatequilibrium.Thisledtoanequationillustratingthedependencyofgassolubilityintheliquidphasetopressure.Kimetal.[23]andYangetal.[24,25]alsoemployedthevanderWaalsandPlatteeuwmodelalongwiththenon-randomlat-tice uidhydrogenbondingequationofstate,whichrequires

severalphysicalparameters,binaryinteractionparametersandhydrogenbondingenergyandentropytobe tted.Inthesestud-ies,hydrate–liquidwaterequilibriumdiscussionswereprimarilydirectedtowardsthein uenceofpressureonmethaneandcar-bondioxidesolubility.Thesolubilitydidnotsigni cantlyvaryoverthepressurerangeinvestigatedbecausethecompressibilityofbothliquidwaterandhydrateissmall.ZatsepinaandBuffet[26]appliedthevanderWaalsandPlatteeuwmodelcoupledwiththeParrishandPrausnitz[7]modelandtheTrebble–Bishnoiequationofstate[27]toshowthattemperaturevariationsaremoresigni cantthanpressurevariationsforestablishingtheequilibriumconditionsinmarinesediment.Theaccuracyoftheirmodelwasnottestedwithexperimentaldata.BallardandSloan[28]extendedthevanderWaalsandPlatteeuwmodelwhichnowallowsforthedistortionofhydrateduetothepresenceofaguestmolecule.Sincethestandardhydratevolumediffersfromthevolumeoftheequilibriumhydrate,thereshouldbeanenergychangethatisproportionaltothedifferenceinvolume.Theyaccountforthisdistortionviaanactivitycoef cient.How-ever,assumingaconstanthydratevolumeforpressureslowerthan200bardoesnotleadtoanysigni canterror[29].

Inthiswork,thevanderWaalsandPlatteeuw[21]andHolder[6]modelswereemployedalongwiththeTrebble–Bishnoiequa-tionofstate[27],whichisafour-parametercubicequationofstate.TheTrebble–Bishnoiequationofstatewaschosenduetoitssimplicitycomparedtothenon-randomlattice uidhydrogenbondingequationofstate,anditsrelativesuccessinpredicting(Lw–V)equilibriumofamethane–watersystemand(H–Lw)equilibriumofacarbondioxide–watersystemoverothercubicequationsofstatesuchasPeng–Robinson[30]andSoave–Redlich–Kwong[31].ThePeng–RobinsonandSoave–Redlich–Kwongequationsofstatewerefoundtofailincharacterizingphaseequilibriaofamethane–watersystem[32].Thegashydrateformersolubilitypredictionresultshavebeenreportedusingtheoriginal[32]aswellasthere-optimizedmix-ingrulebinaryinteractionparametersobtainedinthiswork.

2.Theory

Forthree-phasehydrate–liquidwater–vaporequilibrium,thebasicequationsfortheequilibriumconditionare:

µLi=µV

i(i=1,N)(1)µLi=µHi(i=1,N)

(2)

whereNisthetotalnumberofcomponents.Forthree-phaseequilibriumcalculations,Eqs.(1)and(2)aresolvedsimultane-ously.Vapor–liquidwaterequilibriumandhydrate–liquidwaterequilibriumarede nedbysolvingEqs.(1)and(2),respectively.Thechemicalpotentialofacomponentinthevapororliquidphasemaybecalculatedusingasuitableequationofstate,Trebble–Bishnoiinthisstudy.ThechemicalpotentialofwaterinthehydratephaseisgivenbyvanderWaalsand