Prediction of methane and carbon dioxide solubility in water(3)

S.Hashemietal./FluidPhaseEquilibria246(2006)131–136133

Platteeuw[21]:

µHw=µMT

w+RT

νmln 1

θmj

(3)

m

j

whereµMTwreferstothechemicalpotentialofwaterinthehypo-thetical(empty)hydratelattice,νmisthenumberofcavitiesof

typemperwatermoleculeinthelatticeandθmjisthefractionofcavitiesoftypemoccupiedbygascomponentj.Thefractionaloccupancyisde nedbythefollowingLangmuirexpression:θmj=

C1+mjfjkmkk

(4)

whereCmjistheLangmuirconstantandfjisthefugacityofcomponentjinthephasewithwhichthehydratephaseisinequilibrium.Therefore,inthecaseofhydrate–liquidwaterequi-librium,fugacityofthehydrateforminggascomponentinEq.(4)isequaltothatintheliquidwaterphase.Inthiswork,Lang-muirconstantsneededtoevaluatefractionaloccupancywereobtainedusingthecorrelationproposedbyParrishandPrausnitz[7].TheequilibriumrelationforwaterfromEq.(2)becomes:

µLw=µH

w

(5)

Thedifferenceinchemicalpotentialofwaterintheempty

hydratelatticeandthatinthepureliquidstateatthesystemtemperatureandpressureis:

µMTw µLw0= µMTw

L0

(6)

TherighthandsideofEq.(6)iscommonlyrepresentedby

Holderetal.[6]:

µMTw L0 µMT L0(T0) p νMTw L0

=w

0+pdp

T hMTw

L0

TdT(7)

0Eqs.(6)and(7)canbecombinedtogivethechemicalpotential

ofwaterintheemptyhydratephase:

µMTw µMT L0(T) pMT L0

=w

0 νwdp 0+p0

T hMTw L0µLTdT+w0(8)0Thechemicalpotentialofwaterinthehydratephasecanbe

obtainedfromEqs.(3)and(8).

µHw µMT L0(T) p νMT L0 T h L0

=w00+wMTpdp wdT0T0

+µL

w0

+

νmln 1 θmj (9)m

j

where

hMTT

w

L0

=

hMTw

L0

(T0)+ CMTPw

L0

TdT(10)

hMT L0(T0)and µMT L0(T)weretakenfromtheworkofHolderwetal.Theseparametersw

0havealsobeenre-optimizedinthisworkusing(H–Lw–V)experimentaldata.Therestofparam-etersweretakenfromtheworkofHolderetal.Thechemicalpotentialofwaterintheliquidsolutionphaseis:µsolw=µLw0

+RTlnaw

(11)

whereawistheactivityofwater.Atequilibrium,thechemical

potentialofwaterinthehydratephasehastobeequaltothatintheliquidsolutionphase.Thus,atequilibrium,Eqs.(9)and(11)give

µMTw L0(T0) p νMT L0 T hMT L0

0+wpdw

dT lnaw0p T0

=

ν mln 1 θmj

(12)m

j

Isofugacitycriteriondoesnotnecessarilyleadtotheminimum

oftheGibbsenergy.AllcomponentsexistinginthesystemhavetobedistributedamongallpossiblephasessuchthattheGibbsenergyisatminimum.TheGibbsenergyofthesystemasamixturepropertyisobtainedbythefollowingequation:

G= Xjiµji(13)

i,j

whereXij

isthemolarcompositionofcomponentiinphasej.In

derivingEq.(13),thetotalnumberofmolesissetequaltoone.3.Resultsanddiscussion

Modelpredictionsofmethanesolubilityinliquidwaterfortemperaturesbetween274and285Kandforpressuresof35,50and65bararepresentedinFig.2usingtheoriginalparame-ters.AsshowninFig.2,hydrate–liquidwaterphasesarestableatthelowertemperatures.Themaximumamountofgasthatcanbedissolvedinaliquidatacertaintemperatureandpres-sureisthesolubility.In(H–Lw)equilibrium,hydrate

crystals

parisonofcalculatedmethanesolubilityinliquidwaterbythe

presentmodelusingtheoriginalparameterswithexperimentaldataofServioandEnglezos[18].