Methane adsorption characteristics on coal surface above critical temperature

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ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

ContentslistsavailableatScienceDirect

ColloidsandSurfacesA:Physicochemicaland

EngineeringAspects

journalhomepage:/locate/colsurf

a

MethaneadsorptioncharacteristicsoncoalsurfaceabovecriticaltemperaturethroughDubinin–AstakhovmodelandLangmuirmodel

ShixiongHaoa,b,WeiChua, ,QianJianga,XiaopengYua,b

ab

DepartmentofChemicalEngineering,SichuanUniversity,Chengdu610065,China

DepartmentofChemicalEngineering,SichuanUniversityofScience&Engineering,Zigong643000,China

highlights

graphical

abstract

Methodsforestimatingpseudo-saturationvaporpressurewereinvestigated.

Aprocedureofdeterminingtheparameter,k,inps=pc(T/Tc)kwasproposed.

Thein uenceofthecharacteristiccurveformonthepredictionwasinvestigated.

Anewformofthecharacteristiccurvewasproposed.

article

info

abstract

Articlehistory:

Received16October2013Receivedinrevisedform11December2013

Accepted24December2013Available online 3 January 2014

Keywords:Coal

Methane

D–Aequation

Pseudo-saturationvaporpressureCharacteristiccurve

Toacquiretherequiredinformationoftheadsorptionsystemundersupercriticalconditions,fourdifferentrankcoalswereselectedasstudiedsamples.ThetexturesofthecoalswerecharacterizedthroughN2adsorptionat77K.Theirsurfacemorphologieswereanalyzedbyscanningelectronmicroscopy(SEM).Thehighpressureadsorptiondataofmethaneonthecoalswereobtainedatsupercriticaltemperatures.ThedatawereanalyzedusingtheDubinin–Astakhov(D–A)andLangmuirmodels.Themethodsforestimatingpseudo-saturationvaporpressurewereinvestigated.Andthein uenceofthecharacteristiccurveformontheD–Aequationpredictionwasinvestigated.TheresultsshowthattheconstantsderivedbymatchingtheexperimentaldatatotheLangmuirmodelmightlackphysicalsigni cance,thoughtheLangmuirmodelwasofthecorrectqualitativeformtorepresenttheisothermsofmethaneoncoals.Themethodforestimatingpseudo-saturationvaporpressuresproposedbySchwarzfailedtorendertheexperimentaldatatofallontoonecharacteristiccurve.Amodi edprocedureofdeterminingthevalueofparameterkinSchwarz’sequation,i.e.ps=pc(T/Tc)kwasproposed.Itwasfoundthatthemodi edapproachgavethemostsuitabletemperature-independentcharacteristiccurveswithdeterminationcoef cientR2>0.9943.Theformofthecharacteristiccurvecouldin uencetheD–ingcubicpolynomialascharacteristiccurveformwouldresultinanabnormalpredictioni.e.theadsorptionamountwouldincreasewiththepressuredroppingwhenthepressureislessthanapproximately0.9MPaforXingq-1–5#andXujd-1#,0.8MPaforQingh-2–3#,and0.5MPaforLeiy-1#.AnewformofthecharacteristiccurvededucedfromD–Aequationwasproposed.ThepredictionuncertaintyofD–Aequationbyusingthisnewcharacteristiccurveformislessthan2.43%.

© 2013 Elsevier B.V. All rights reserved.

1.Introduction

Correspondingauthor.Tel.:+862885403836;fax:+862885461108.E-mailaddress:chuwei1965@(W.Chu).

Adsorptionphenomenaareincreasinglyutilizedtoperformdesiredbulkseparationorpuri cationpurposes[1–5].Coalisporousmaterial.Coalbedmethane(CBM)isunconventionalnatu-ralgasassociatedwithcoal.ThekeycomponentofCBMisCH4(CH4

0927-7757/$–seefrontmatter© 2013 Elsevier B.V. All rights reserved./10.1016/j.colsurfa.2013.12.047

S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

105

contentbeing88–98%[6]).CBMisretainedincoalbedsmainlyinadsorbedstate[7,8].CBMisnotonlythemajorrootofcoalminedisasterandatmosphericpollutionsource,butalsoavaluablenon-renewableenergy[9].TheinterestbothinrecoveryofCH4fromcoalseamandinoutbursthazardsrelatedtocoalmininghasledtoextensivestudyofgassorptionincoal[7,9–12].

Inrecentyears,increaseinatmosphericCO2concentrationhasraisedconcernaboutclimatechangeandledtoworldwideeffortsonreductionofCO2emissions[7,13–18].AlthoughseveraloptionsforCO2sequestrationarebeingconsidered,apotentiallyattractiveapproachisthestorageofCO2indeep,unmineablecoalbeds[7,19].SuchcoalbedsfrequentlycontainlargeamountsofrecoverableCH4,andtherecoveryofthisgascanbeenhancedbyinjectingCO2intothecoalbeds[7,20].TheinjectionofCO2canservedualpurposestosequesterlargeamountsofCO2andtosimultaneouslyprovideanincreasedsupplyofCH4.Consequently,someofthesequestrationcostscanberecoveredinthevalueoftheCH4produced[7,19].

ThedesignofoptimalrecoveryofCH4andCO2sequestrationincoalbedsreliesgreatlyontheavailabilityofhigh-pressure,supercriticaladsorptiondataforgassorptionaswellasreliableadsorptionmodelsthatarecapableofaccuratepredictionsofadsorptionphenomena[20].Tosimulatereservoirconditions,lab-oratorysorptionexperimentsaregenerallyconductedatelevatedtemperature,usuallybetween293and323K[7,11].However,thecoalbedtemperatureincreaseslinearlywithminingdepthincreas-ing[21].

MethanesorptionisothermsincoalarecommonlyIUPACtypeI[7].TheLangmuirmodeliswidelyusedintheCBMindustrybecauseofitssimplicityandprovidingareasonable ttomostexperimentaldata[7].However,theassumptionofanenergeticallyhomoge-neoussurfaceasproposedbyLangmuirtheoryisnottrueforcoal[11].Furthermore,theLangmuirmodelcannotpredictmethaneadsorptionamountatdifferenttemperaturesfromoneadsorptionisotherm.Thismeansthattoobtainmethanecapacityatdifferenttemperaturestherewouldbealotofexperimentstodo.

TheDubinin–Astakhov(D–A)modelhasbeencommonlyappliedtothedescriptionoftypeIisotherms[22–24].TheD–Aequationwasmainlydevelopedfortheadsorptionofvaporsbelowthecriticalpoint[22].However,experimentsofgasadsorptiononporoussolidshaveshownthatthereisnoabruptchangeintheadsorptionduringthetransitionfromsub-criticaltosuper-criticalconditions[22].ThissuggeststhattheD–Aequationcanbeempir-icallyappliedtosuper-criticalgasesaswell[22].ThenicefeatureofusingD–Aequationfordescriptionofgasadsorptionisthatasinglecharacteristiccurvewouldbeobtainedifthecharacteris-ticenergyisindependentoftemperature[22].Ifthecharacteristiccurveforanadsorptionsystemisknown,thenadsorptionuptakeoftheadsorbateatdifferenttemperaturescouldbepredicted.

TheproblemsofapplyingtheD–Aequationtosupercriti-cal uidsare[11,22,25]:(1)theestimationofthesaturationvaporpressure,ps.Abovethecriticaltemperature,Tc,thecon-ceptofsaturationvaporpressuredoesnotexist,hencetheuseofpseudo-saturationvaporpressureswasproposed;(2)theden-sityoftheadsorbedphaseatagiventemperature.Inordertoobtainthevolumeofadsorbedphase,itisnecessarytohaveavalueofadsorbedphasedensity.Theadsorbedphasedensityisnotdirectlymeasurableandasaconsequence,itsvalueisapproxi-mated;and(3)theformofthetemperature-invariantcharacteristiccurvetobeutilized.Thein uenceofthemethodsforestimatingpseudo-saturationvaporpressuresoncharacteristiccurvesofacti-vatedcarbon-methaneadsorptionsystemshasbeenreported[26].However,toourknowledge,nostudyhasbeenreportedforcoal-methanesystems,andnostudyhasbeenreportedonthein uenceofthecharacteristiccurveformontheD–Aequationpredictiontoo.Toavoidtheseproblemsabove,Kimetal.[27]usedgasdensityinsteadofgaspressure(andadsorbedphasedensity

Table1

Proximateandultimateanalysisofthecoals.

Sample

Proximateanalysiswt.%

Ultimateanalysiswt.%

Ash

VM

FC

Moisture

C

H

O

N

Xingq-1–5#8.4029.8371.170.8077.426.5814.601.40Qingh-2–3#1.6621.4778.530.6884.666.477.081.79Xujd-1#11.7111.6688.340.6788.703.564.353.39Leiy-1#

18.45

5.55

94.45

0.15

90.98

1.26

5.53

2.23

Ashwascalculatedonadrybasis;Volatilematter(VM)and xedcarbon(FC)werecalculatedonadried,ash-freebasis.

ratherthanpseudo-saturationvaporpressure)tomodifytheclas-sicDubinin–Radushkevich(D–R)equation,andthenappliedthemodi edD–Requationtothegasesadsorptiononcoalsundersupercriticalconditions.However,themodi edD–RequationisnotconvenienttouseinCBMindustryforthegasdensityisnotasintuitiveaspressure.

Inthepresentwork,methaneadsorptiondataonfourdiffer-entrankcoalswereobtainedatsupercriticaltemperatures.TheD–Aequationisappliedtomethane-coalisothermdataatele-vatedtemperature.Thepurposesofthecurrentstudyare:(1)todeterminewhichempiricalmethodpresentedintheliteraturetoevaluatethepseudo-saturationvaporpressureismostsuitableforanalyzingtheadsorptiondataabovetheTcofmethane.Thecri-terionusedinouranalysisisthatthedatashouldbereducedtotemperature-independentplots;and(2)todeterminetheformofthetemperature-invariantcharacteristiccurve.Thecriterionispre-dictionaccuracy.

2.Experimental

2.1.Materials

Inthiswork,Xingqingcoal(bituminouscoal,QinghaiChina,labeledasXingq-1–5#),Qinghuacoal(bituminouscoal,QinghaiChina,labeledasQingh-2–3#),Xujiadongcoal(bituminouscoal,HunanChina,labeledasXujd-1#),andLeiyangcoal(anthracite,HunanChina,labeledasLeiy-1#)wereused.Theproximateanal-ysisandultimateanalysisforthecoalsaregiveninTable1.Theoxygencontentwasdeterminedbydifference[28].Thesulphurinthefourcoalswasnotdetected.Thecoalsampleswerecrushedtoparticleswithsizeof60–80mesh.

2.2.Methods

2.2.1.2.2.1.Characterization

TexturalcharacterizationofthecoalsampleswasconductedusingaNOVA1000esurfaceareaandporesizeanalyzer(Quanta-chromeCompany,USA)withN2(at77K)asadsorbate.Priortoanalysis,samplesweredegassedat383Kfor24h[29].Thespe-ci csurfaceareawascalculatedbythemultipointBETmethod[30,31];themicroporevolumewasdeterminedbytheD-Requation[31,32];thetotalporevolumeVtwasevaluatedfromthenitrogenadsorptionatp/p0=0.98–0.99[33];andtheporesizedistributionwascalculatedbythebythenonlocaldensityfunctionaltheory(NLDFT)method[31].

Surfacemorphologywasinvestigatedbyscanningelectronmicroscopy(SEM)(JEOL/EOJSM-5900,Japan).

2.2.2.Adsorptionofmethane

Theadsorptionisothermswereobtainedthroughanequilib-riumvolumetricmethod.Thedetailedexperimentalmethodwaspresentedelsewhere[29].Toavoidthein uenceofwaterinthecoalsonthecalibrationofthereferencecellvolume,thesamplesweredriedinanovenovernightat383Kbeforemethaneadsorp-tion[29,32].

106S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

Theprecisionofthehigh-precisionpressuretransducerusedintheexperimentsis0.05%ofthefullscalevalue(maximumpressure10MPa).ThetemperaturesensorsarePt100 RTDswithanesti-mateduncertaintyof0.1K.ThelinesexposedtoairwerewrappedwithheatingtapecontrolledbyaPIDcontroller.Thepuritiesofthemethaneandheliumusedintheexperimentsare99.99and99.999%,respectively.

2.2.3.Isothermmodels

Alltheconventionalisothermequationswereoriginallyderivedforabsoluteadsorptionamount(nabsolute,mmol/g),thereforeitshouldbenabsolutetobeappliedforevaluatingthethermody-namicpropertyofadsorbedphase[34].However,thenabsolutethatrevealingthetruemasscon nedintheadsorbedphasecannotbedirectlymeasured.Theso-calledexcessadsorptionamount(nexcess,mmol/g)thatisapproachedfromtheexperimentalequilibriumdataisusedforthecalculationofnabsolute[27,29,31]:

nadsolute

=

nexcess

1 (1)

gasadsorbed

where gasand adsorbedarethedensityofthefreegasphaseandtheadsorbedphase,respectively(ing/cm3).

Amajorproblemindeterminingabsolutesorptionisthat adsorbedcannotbedetermineddirectlyandcertainassumptionshavetobemade.Sahaetal.[35]proposedanewmethodtoaccountfortheadsorbedphasevolume.Inthiswork,weusetheSaha’smethodtocalculatethedensityoftheadsorbedphase( adsorbed).TheSaha’smethodisasfollows[35],

adsorbed= bexp[ (T Tb)]

(2)

where bisthedensityoftheliquidmethaneatthenormalboil-ingpoint(0.4224g/cm3);Tbisnormalboilingpointofmethane(111.67K);and isthethermalexpansioncoef cientoftheadsorbedphasethatcanbeexpressedasfollows

[31,35,36].

=

1

vadsorbed

adsorbed

1(3)

where adsorbedistheadsorbedphasespeci cvolume.

ngmuirmodel.TheformoftheLangmuirequationusedhereinisexpressedas[22]:

nabsolutebp

(4)

0=1+bp

wheren0isthelimitingadsorptioncapacity(inmmol/g);pistheequilibriumpressure(absolutepressureinMPa);bistheLangmuirconstant(inMPa 1).n0andbaretheadsorptionparametersthatareoptimizedfromtheleast-squarescriteriausingtheexperimen-taldata.

2.2.3.2.Dubinin–Astakhov(D–A)model.TheD–Amodelforadsorp-tionofmethaneontononhomogeneouscarbonaceoussolidsisexpressedas[22,31]:

W t

=(5)

0exp A

W=

nabsolute×10 3×MCH4

(6)

adsorbed

A=RTln

ps

(7)

where3Wisthevolumeofadsorbedphaseatequilibrium(incm/g);W0isthelimitingvolumethatthegascanoccupyintheadsorbent

Table2

Texturalcharacteristicsofthecoals.

Sample

SBET(m2/g)

Vmic(mm3/g)

Vt(mm3/g)Xingq-1–5#2.5810.44644.403Qingh-2–3#2.4550.37233.177Xujd-1#2.6350.51433.850Leiy-1#

7.486

3.1190

13.770

(incm3/g);Aistheadsorptionpotential(inJ/mol);Eisthechar-acteristicenergyoftheadsorptionsystem(inJ/mol);MCH4isthemolarmassofmethane(ing/mol);Risthegasconstant;andtisthestructuralheterogeneityparameter.

InordertocalculatetheadsorptionpotentialA,thevalueofthevaporpressurepsatagiventemperatureshouldbeknown.Satu-rationvaporpressuresareusedforadsorbateswhentemperaturesarebelowTc[22].However,aboveTc,thereisnoconceptofthevaporpressure;hencetheuseofpseudo-saturationvaporpressureswasproposed.Inthisstudy,weusethreeapproachesthatexistedintheliteraturestoevaluatethepseudo-saturationvaporpressureaboveTc[11,22,26]:

(1)UseAntoineequationforthesaturationvaporpressureandextrapolateittotemperaturesabovethecriticaltemperature[22,26]:

lnps=CB

(8)

whereB,C,andDareAntoineparameters.

(2)UsetheDubinin’smethod[11,22]:

p s=pc

T 2

(9)

c

wherethesubscriptcreferstothecriticalpoint(formethane,pcis4.5992MPa,andTcis190.56K).

(3)UsetheAmankwahandSchwarzmethod[22,26]:

p s=pc

T k

(10)

c

wherekisaparameterspeci ctotheadsorbate–adsorbentsystem.3.Resultsanddiscussion

3.1.Texturalcharacterization

Table2summarizesthetexturalparameterscalculatedfromthe

nitrogenisotherms.TheporesizedistributionsarepresentedinFig.S2.FromFig.S2,itcanbeobservedthattheXingq-1–5#,Qingh-2–3#,andXujd-1#havesimilarporesizedistribution.However,theporesizedistributionoftheLeiy-1#issigni cantlydifferentfromtheothers.

Fig.S1showstheSEMimagesofthetypicalcoalsamples.ItwasobservedthatthesurfacemorphologyofLeiy-1#showsalittledifferentfromtheXujd-1#.TheparticlesofLeiy-1#aresmallerandarrangemoretightlythanthoseoftheXujd-1#.ThedifferenceinthesurfacemorphologyisinagreementwiththeSBETandVmicdataofcoals.Iftheparticlesaresmallerandarrangemoretightly,theSBETorVmicofcoalwillbehigher.

ngmuircorrelations

Fig.1showsthemethaneuptakedatameasuredat303,313,

and323Kforthesamples.ThesolidlinesinFig.1areLangmuir tcurves.Theregressedadsorptionparameters(n0andb)arelistedinTable3.

AccordingtotheassumptionsoftheLangmuirmodel,thesat-urationcapacitynoissupposedtorepresenta xednumberof

S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

107

ngmuircurve tstomethaneisothermdataforsamples:(a)Xingq-1–5#;(b)Qingh-2–3#;(c)Xujd-1#;and(d)Leiy-1#.Solidlinesarecurve ts.

surfaceadsorptionsite.Itshouldthereforebeatemperature-independentconstant[37].However,asshowninTable3,thesaturationcapacitynoisafunctionoftemperature.Similarobser-vationsweremadebyZhouandZhou[38]andClarksonetal.[11].Ruthven[37]heldthattheconstantsderivedbymatch-ingtheexperimentaldatatotheLangmuirmodelmightlackphysicalsigni cance,thoughtheLangmuirmodelwasofthecorrectqualitativeformtorepresentatypeIisotherm.Sat-urationcapacitynobeingafunctionoftemperatureinZhouandZhou’s,Clarksonetal.’sandourstudiescon rmRuthven’sview.

3.3.Determinationofcharacteristiccurves

Table3

AdsorptionparametersoftheLangmuirmodelforadsorptionisothermsofmethaneonthecoalsat303,313,and323K.

Sample

T(K)

no(mmol/g)

b(MPa 1)

R20.99480.99470.99490.99500.99480.99470.99510.99490.99490.99490.99480.9947

Xingq-1–5#

Qingh-2–3#

Xujd-1#

Leiy-1#

303313323303313323303313323303313323

0.95481.04071.11490.72470.72780.75960.69800.73580.77481.84891.87111.9093

0.16630.12170.09590.18480.16280.13270.14080.11300.09250.42570.35040.2838

3.3.1.ExtrapolationAntoineequationmethod

TheAntoineparametersB,C,andDinEq.(8)wereobtainedfromregressionanalysisofmethanesaturationvaporpressuredatabelowTc.Themethanesaturationvaporpressurepsatagiventem-peratureT(rangefrom100to190K)wereobtainedfromtheNISTwebsite[39].TherelationshipbetweenTandpsformethaneisshowninFig.S3.ThedotlinesinFig.S3istheAntoineequation ttingcurve.Theregressedadsorptionparameters(B,C,andD)are7.7437,1306.5485,and19.4362,respectively.

InordertoutilizetheD–Aequation,itsthreeparameters(W0,E,andt)wereevaluatedbyanon-linearregressionprocedure.Theinputdatafortheregressionwerethevolumeofadsorbedphase,W,andtheadsorptionpotential,A.

Fig.2showsthecharacteristicplotsforthecoals.ThevolumeofadsorbedphaseWwascalculatedbyEq.(6).Theadsorptionpoten-tialAwascalculatedbyEq.(7),inwhichthepseudo-saturationvaporpressurepswascalculatedbyextrapolationofAntoineequa-tion.

Theformofthetemperature-invariantcharacteristiccurvetobeutilizedisoneoftheimportantproblemsassociatedwiththeapplicationofpotentialtheories[11].AmankwahandSchwarz[26],andClarksonetal.[11]didnotgivetheformofthecharacteristiccurveintheirworks.Lietal.[40],Suetal.[41]andFengetal.[42]usedpolynomialtoexpressthecharacteristiccurve.Althoughthismethodissimple,thequadraticorcubicpolynomialparametersarelackofphysicalmeaning.Inthepresentwork,wededucedan

108S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

Fig.2.Characteristiccurvesformethaneon(a)Xingq-1–5#,(b)Qingh-2–3#,(c)Xujd-1#,and(d)Leiy-1#.ThepswascalculatedbyextrapolationofAntoine’sequation.

equationi.e.Eq.(11)fromtheD–Aequationtoexpressthecharac-teristiccurve.

W=W0exp

t

A

(11)

whereW0,E,andtareparameterstobedeterminedbyregressionanalysis.TheinputdatafortheregressionanalysiswereW,andA.Theregressedadsorptionparameters(W0,E,andt)arereportedinTableS1.ThesolidlinesinFig.2representarethecharacteristiccurvescalculatedfromtheEq.(11)usingtheregressedadsorptionparameters(W0,E,andt).AgarwalandSchwarz[43]de nedthemaximumdeviationofadsorbedphasevolumeataparticularvalueofAdividedbythemaximumadsorptionvolume(×100)asdevia-tion.However,therearescarcelytwodiscretedatapointsofwhichtheAvaluesaresame.Consequently,thedeviationofadsorbedvol-umeataparticularvalueofAisnoteasytodetermineaccurately.Inthispaper,weusethedeterminationcoef cient,R2,toindicatethedeviationofthedata.Obviously,theclosertoonethevalueofR2,thelessdiscretethedata.ThemaximumofR2is0.9888forLeiy-1#,whiletheminimumofR2is0.9811forQingh-2–3#andXujd-1#s.

consideredempirical.Dubinin’sempiricalmethodi.e.Eq.(9)waswidelyusedtocalculatethepseudo-saturationvaporpressureingas-solidadsorptionsystem[26,31,36,44,45].

Fig.3showsthecharacteristicplotsforthecoals.ThesolidlinesinFig.3representthecharacteristiccurvescalculatedfromtheEq.(11)usingtheregressedadsorptionparameters(W0,E,andt)thatarelistedinTableS2.Thepseudo-saturationvaporpressurepswascalculatedbyEq.(9).ThemaximumofR2is0.9936forXujd-1#,whiletheminimumofR2is0.9920forXingq-1–5#.ThisindicatesthatcharacteristiccurvesusingtheDubininmethodforpseudo-saturationvaporpressuredisplaysmallerdevi-ationsthantheAntoineequationextrapolationtechniques(seeFig.S3).

3.3.2.Dubinin’smethod

ExtrapolationAntoineequationresultsinhighpswhichare40.1840MPafor303.15K,45.3881MPafor313.15K,and50.9029MPafor323.15Krespectively.Flood[43]hasproposedthatintheabsenceofanydirectexperimentalinformationontheadsorbedphase,estimationofpsasthevaporpressureshouldbe

3.3.3.Schwarz’smethod

Theinteractionbetweenanadsorbateandaparticularadsor-bentwouldbeuniquetotheadsorptionsystemstudied.AmankwahandSchwarz[26]proposedEq.(10)toestimatepseudo-saturationvaporpressuresinapplicationoftheD–Aequation.Tocalculatetheadsorptionpotential,A,thevalueofkinEq.(10)shouldbeknown.However,thevalueofkisnotknownapriori.AmankwahandSchwarz[26]usedthefollowingmethodtodeterminethevalueofk.WhenEq.(10)isusedinconjunctionwithEq.(5)(i.e.D–Aequation),afourparameterequationisobtained:

k t

pcT/TcRTW

=exp ln

(12)

S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

109

Fig.3.Characteristiccurvesformethaneon(a)Xingq-1–5#,(b)Qingh-2–3#,(c)Xujd-1#,and(d)Leiy-1#.ThepswascalculatedbyDubinin’smethod.

Eq.(12)whenwrittenintermsofamountadsorbed(mmol/g)is[22,31,36]:

nabsolute

k t

pcT/TcRT

=exp ln

(13)

theparametersn0,E,t,andkareobtainedfromtheregressionanal-ysis.Theinputdatafortheregressionanalysisarenabsolute,T,and

p.

Fig.S4.showsSchwarzcurve ts.Theadsorptionparameters(n0,E,t,andk)obtainedfromtheregressionanalysisarelistedinTable4.Itcanbeobservedthatn0,E,t,andkarefunctionsoftemperature.

Table4

AdsorptionparametersoftheD–AmodelusingSchwarz’smethodtoestimatekforcalculationpseudo-saturationvaporpressure.

Sample

T

n0(mmol/g)

E(kJ/mol)

t

k

Xingq-1–5#

Qingh-2–3#

Xujd-1#

Leiy-1#

303313323303313323303313323303313323

0.59670.52410.55890.48810.51760.39510.40440.37060.41161.59701.54581.6213

4.46424.08454.00706.06435.72274.18564.68644.23993.78547.99128.45158.0852

1.29041.17361.321.56611.63601.20331.33981.21641.34852.09332.14542.4872

1.771.231.772.342.921.121.811.422.093.513.294.47

This ndingisinconsistentwithSchwarz’sresults[26].Theparam-eterEinD–Amodelischaracteristicenergywhichisameasureofthestrengthofinteractionbetweenadsorbateandadsorbent[22].Theadsorptionofmethaneoncoalisphysicaladsorption[46].TheinteractionofmethanemoleculestocoalsurfaceismainlyLon-dondispersionforceswhichareindependentoftemperature[46].Consequently,theparameterEshouldnotvarywithtemperature[22].ThevariationsofEwithtemperaturecouldcomefromthereciprocalin uencebetweentheparameterskandtinEq.(13).ThekvaluesshowninTable4canbeinsertedintoEq.(10)todeterminethepseudo-saturationvaporpressure,ps,whichisusedtocalculatetheadsorptionpotentials,A.Andthencharacteristiccurvescouldbeobtained.ThesecharacteristiccurvesareshowninFig.4.ThesolidlinesinFig.4representthecharacteristiccurvescal-culatedbyusingtheregressedadsorptionparameters(W0,E,andt)thatarelistedinTableS3.ThemaximumofR2is0.9455forXujd-1#,whiletheminimumofR2is0.6914forQingh-2–3#.Thisindi-catesthatusingtheSchwarz’smethodforpseudo-saturationvaporpressureresultsindatapointofcharacteristiccurvesbeingmoredispersethantheAntoineequationextrapolationandDubinin’sempiricalmethods.

3.3.4.Modi edSchwarz’smethod

ItwasobservedthatSchwarz’smethoddidnotresultinsin-glecharacteristiccurvefromFig.4.Inthepresentwork,weimprovedtheSchwarz’smethod.Inthemodi edmethod,thepseudo-saturationvaporpressureisstillcalculatedusingtheEq.(10)proposedbyAmankwahandSchwarz.However,kwhichisusedtocalculatepsisdeterminedbythefollowingprocedure:

110S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

Fig.4.Characteristiccurvesfor(a)Xingq-1–5#,(b)Qingh-2–3#,(c)Xujd-1#,and(d)Leiy-1#.ThepswascalculatedbySchwarz’smethod.

(1)Letkbeequalto1.6,2.0,2.4,...,and4.0respectivelyanda

seriesofpseudo-saturationvaporpressure,ps,whichareusedtocalculatetheadsorptionpotential,A,areobtained;

(2)PlottherelationshipbetweenWandA,andthecharacteristic

curvescorrespondingtodifferentkcouldbeobtained;

(3)FitthecharacteristiccurveswithEq.(11),andthedetermina-tioncoef cientR2couldbeobtained;

(4)PlottherelationshipbetweenR2andk,anddoregressionanal-ysisforthecurves.Then,theoptimizedkwhichthemaximumR2correspondstowouldbeobtained.

Table5

RegressionparametersforthecharacteristiccurvesusingEq.(11)asregressionequationandpsbeingcalculatedbythemodi edSchwarz’smethod.

Sample

W0(cm3/g)

E(kJ/mol)

t

R2

Xingq-1–5#Qingh-2–3#Xujd-1#Leiy-1#

0.04860.03480.03110.1090

5.03845.04764.58376.9336

1.44901.47891.37791.9258

0.99430.99420.99460.9948

Theadsorptionforce eldofvariousadsorbentswilluniquelyimpactadsorbate[26].

Fig.S5showstherelationshipbetweenkandR2.Thedatawereregressedusingthequadraticpolynomial.ThesolidlinesinFig.S5representthe ttingcurvesusingtheregressedparameters(a0,a1,anda2)thatarelistedinTableS4.Consequently,theoptimizedvalueofkis2.69forXingq-1–5#,2.24forQingh-2–3#,2.26forXujd-1#,and2.7forLeiy-1#.

Fig.5showsthecharacteristiccurvesofthesamplesforwhichkistakentheoptimizedvalue.ThesolidlinesinFig.5arethe ttingcurves(theblacklinebeingEq.(11) ttingcurveandtheredlinebeingthecubicpolynomial ttingcurverespectively).The ttingparametersbasedontheEq.(11)areshowninTable5.Thecubicpolynomial ttingparametersareshowninTable6.

AsshowninFig.5,Table5and6,themodi edSchwarz’smethodtoevaluatethepseudo-saturationvaporpressureresultsinmuch-improvedsinglecharacteristiccurvesforthesamples.Thevalueofoptimizedkisnotsameforthesamples.Thisisbecausetheadsorbate-adsorbentinteractionisnotsameforthefoursamples.

3.4.Determinationofthecharacteristiccurveformtobeutilized

Theformofcharacteristiccurvetobeutilizedisoneoftheimportantproblemsassociatedwiththeapplicationofpotentialtheories[11].Thecharacteristiccurveisusually ttedbypoly-nomial[40–42,47].Althoughthepolynomial ttingissimple,theparametersobtainedbyregressionanalysishavenophysical

Table6

Cubicpolynomialregressionparametersforthecharacteristiccurves(pswascalcu-latedbythemodi edSchwarz’smethod).

Sample

a0

a1

a2

a3

R2

Xingq-1–5#Qingh-2–3#Xujd-1#Leiy-1#

0.05130.03590.03320.1119

0.0067 0.0038 0.0056 0.0024

2.4305×10 4 4.8588×10 49.7957×10 5 0.00214

4.9921×10 56.1986×10 51.8119×10 51.4417×10 4

0.99430.99470.99460.9958

S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

111

Fig.5.Characteristiccurvesfor(a)Xingq-1–5#,(b)Qingh-2–3#,(c)Xujd-1#,and(d)Leiy-1#.Thepswascalculatedbythemodi edSchwarz’smethod.

meaning.AsshowninFig.5,thedifferentcharacteristiccurveformcouldresultindifferentW0value.TheW0valuesobtainedfromcubicpolynomialregressionarelargerthanthoseobtainedfromEq.(11)regressionforallcoals.Todeterminethecharacteristiccurveformtobeutilized,weusetheregressionparametersinTable5andTable6,topredicttheisothermsat333and343K,respectively.

reportedinTable6intothefollowingequation,i.e.Eq.(16),apre-dictiveisothermequation.i.e.Eq.(17)isobtained.

nabsolute=

W adsorbed

×1000

CH4

=62.5 adsorbeda0+a1A+a2A2+a3A3

(16)

k

RTlnpcT/Tc RTln nabsolute=62.5 adsorbed a0+a1+a2

3

10106

pcT/Tck

2

+a3

RTln

pcT/Tck

3

(17)

109

WhenEq.(11)isusedascharacteristiccurveform,bysubstitut-ingtheregressionparametersvaluesreportedinTable5intoEq.(13),apredictiveisothermequationisobtainedforthesamplesatdifferenttemperaturewhichisasfollows:

k t

pcT/TcRT

lnnabsolute=n0exp

(14)

where

W0 adsorbed

n0=×1000

CH4

ThevalueofkinEqs.(14)and(17)is2.69forXingq-1–5#,2.24forQingh-2–3#,2.26forXujd-1#,and2.70forLeiy-1#,respectively(see3.3.4).Itshouldbenotedthatthedensityofadsorbedphase, adsorbed,isafunctionoftemperaturewhichiscalculatedbyEq.(2).Eqs.(14)and(17)allowforthegenerationofpredictiveisothermswhichcanbecomparedtotheexperimentaldata.Plotsofisothermsshowingacomparisonoftheexperimentalandpre-dictivevaluesareshowninFig.6.ThelinesinFig.6refertothepredictiveisotherms,whilepointsrefertothelaboratorydata.Thepredictionuncertaintyisdeterminedbythemeanrelativedeviation :

(15)

Correspondingly,whencubicpolynomialisusedascharacter-isticcurveform,bysubstitutingtheregressionparametersvalues

N 1 nexp,i ncal,i2 =

i=1

exp,i

(18)

112S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

Fig.6.Experimentalvs.predictiveisothermsofmethaneon:(a)Xingq-1–5#,(b)Qingh-2–3#,(c)Xujd-1#,and(d)Leiy-1#.TheredlinesarepredictiveisothermswhichbasedonEq.(17)andtheblacklinesarethosewhichbasedonEq.(14).

whereNisthenumberofdatapoints.

Themeanrelativedeviation forthetwodifferentcharacteristiccurveformsisevaluatedandshowninTable7.

AsshowninFig.6andTable7,themeanrelativedeviation forusingEq.(11)ascharacteristiccurveformissmallerthanthatforusingcubicpolynomialascharacteristiccurveform.Inaddition,theuseofcubicpolynomialasthecharacteristiccurveformcouldresultinanabnormalpredictivevaluethattheadsorptionamountincreasewiththepressuredroppingwhenthepressureislessthanapproximately0.9MPaforXingq-1–5#andXujd-1#,0.8MPaforQingh-2–3#,and0.5MPaforLeiy-1#.

ThermodynamicssuggeststhatanadsorptionisothermmustexhibittheHenrylawbehaviorwhenpressureisverylow.Unfortu-nately,theD–AequationsdonothavethecorrectHenrylawwhenthepressureisapproachingzero.TheslopeoftheD–Aadsorption

isothermiszerowhenthepressureapproacheszerowhent>1[22].However,asshowninFig.6,theslopeoftheD–Aadsorptionisothermforusingcubicpolynomialascharacteristiccurveformisin nitywhenthepressureapproacheszero.AlltheseindicatethatusingEq.(11)ascharacteristiccurveformismorereasonablethanusingthepolynomial.

4.Conclusion

Table7

Relativedeviation( )forpredictiveisotherms.

Sample

T(K)

Relativedeviation, (%)

Eq.(11)

Cubicpolynomial

Xingq-1–5#

Qingh-2–3#

Xujd-1#

Leiy-1#

333343333343333343333343

2.091.831.311.272.431.651.792.30

3.215.731.822.126.592.172.324.29

Inthiswork,highpressure(upto8.3MPa)adsorptionisothermsformethanewereobtainedat vetemperaturesforfourdifferentrankcoals.TheLangmuirandD–Aequationshavebeenappliedtomodeltheadsorptionofmethaneoncoalsabovesupercriticaltemperature.

ThetemperaturedependenceoftheLangmuirparameter,n0,indicatesthattheassumptionofanenergeticallyhomogeneoussurfaceofadsorptionisnotstrictlytrueformethaneadsorptiononcoal.

Todeterminewhichmethodtoevaluatethepseudo-saturationvaporpressureismostsuitableforanalyzingtheadsorptiondataabovetheTcofmethane.TheextrapolationofAntoine’sequation,Dubinin’smethodandSchwarz’smethodforestimatingpseudo-saturationvaporpressurewereusedtotestthehypothesisthatthedataforeachadsorptionsystemcouldbecollapsedontosinglecharacteristiccurves.ItwasfoundthatthedeviationofcharacteristiccurvedataforusingDubinin’smethodtoestimatepseudo-saturationvaporwassmallerthanthoseforusingtheothertwomethods.However,Dubinin’smethoddonottakeinto

S.Haoetal./ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113

113

accountuniquenessoftheinteractionbetweenanadsorbateandaparticularadsorbent.Although,Schwarz’smethodaccountsforadsorbate-adsorbateandadsorbate-adsorbentinteractions,theadsorptiondatafailedtobereducedtotemperatureindependentplots.

Amodi edprocedureofdeterminingthevalueofparameter,k,inSchwarz’sequation,i.e.Eq.(10)wasproposed.Whenthisnewprocedurewasappliedtoanalyzethelaboratoryadsorptiondata,theadsorptiondatacanbereducedtosingletemperatureinvariantcharacteristiccurvesforthecoals.

Theformofcharacteristiccurvecouldin uenceD–ingcubicpolynomialascharacteristiccurveformcouldresultinanabnormalprediction.Thatistosaytheadsorptionamountwouldincreasewiththepressuredropping.AnewformofcharacteristiccurvededucedfromD–Aequationwasproposed.ThepredictionuncertaintyofD–Aequationbyusingthischarac-teristiccurveformislessthanthatforusingcubicpolynomialascharacteristiccurveform.

Acknowledgements

ThisworkwassupportedbytheNationalBasicResearchPro-gramofChina(973Program,No.2011CB201202)andtheNationalNaturalScienceFoundationofChina(20776089).

TheauthorsalsothankJingjieLuo,WenjingSun,NingWangandYanyanFengfortheirhelpfuldiscussions.

AppendixA.Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,intheonlineversion,at/10.1016/j.colsurfa.2013.12.047.

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