Methane adsorption characteristics on coal surface above critical temperature
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ColloidsandSurfacesA:Physicochem.Eng.Aspects444 (2014) 104–113
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ColloidsandSurfacesA:Physicochemicaland
EngineeringAspects
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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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