化工熱力學(xué)授課教案

Chapter3VolumetricPropertiesofPureFluidsContents：Introduction3.1PVTbehaviorofpuresubstances3.2VirialEquationsofstate3.3theidealgas3.4ApplicationoftheVirialEquations3.5Cubicequationofstate3.6GeneralizedCorrelationsforGasesIntroductionWhytostudythischapter?VeryimportantforindustryPressureVolumetemperatureTheycanbeusedtocalculatevolumetricpropertiesVeryeasytobemeasuredVeryimportantforindustryPressureVolumetemperatureTheycanbeusedtocalculatevolumetricpropertiesVeryeasytobemeasuredWehopeHowcanweobtaintheseinformations?PVTTworoutes:PVTmeasuringdirectlycalculatingwithotherparameterswhichcanbemeasureddirectly3.1PVTbehaviorofpuresubstancesBymeasuringthevaporpressure,volumeandtemperatureofapuresubstanceswhendifferentphasesachieveequilibrium,wecandrawthecurveofP-V-TV=constantV=constantPTdiagramPVdiagramT=constantTVdiagramP=constantPTLiquidSolidCriticalpointSublimationcurvePTLiquidSolidCriticalpointSublimationcurveFusioncurveVaporizationcurveFluidC12Forexample:PTdiagramofwaterwhenV=constantTwopoints:1)C—thecriticalpoint PC—thecriticalpressureTC—thecriticaltemperatureCpT2PCCpT2temperatureatwhichapurechemicalspeciescanexistinvapor-liquidequilibrium2)2—thetriplepointwherethethreephasescoexistinequilibriumAccordingtothephaserule,F=1-3+2=0DegreesoffreedominvariantpointDegreesoffreedomAccordingtotheSublimationcurvephaseruleAccordingtotheSublimationcurvephaseruleF=1-2+2=1FusioncurveVaporizationcurveSublimationcurveFourregions:SolidregionAccordingtothephaserule,F=1-1+2=2VaporregionAccordingtothephaserule,F=1-1+2=2liquidregionFluidregion3.1.2PV(1)showingonlyliquid,gasregionswhenV=constantVaporVaporLiquidVPSaturatedliquidliquid/vaporSaturatedvaporTT1VaporVCTTCPCliquid/vaporLiquidPCriticalpointSaturatedvaporcurveSaturatedliquidcurveVT>TC>T1>T2>T3T2T3(2)showingsolid,liquid,gasregionswhenV=constantPPCVCliquid/vaporVaporSolid/liquidsolid/vaporSaturatedvaporcurvePVLiquidCriticalpointsolidSaturatedliquidcurveSolidcurveSublimationcurveThetriplephasecurvePVSS/LPVSS/LLCVV/SV/LFivecurves：saturatedliquidcurvesaturatedvaporcurvesolidcurvesublimationcurveTriplephasecurveSevenregions：solidregionliquidregionvaporregionsolid/liquidregionliquid/vaporregionsolid/vaporregionsup-criticalfluidregionvaporSaturatedvaporcurveSaturatedliquidcurveT℃vaporSaturatedvaporcurveSaturatedliquidcurveT℃0.003155vm3/kgP=22.09MPaP=15MPaP=1MPaThinking(3-1)Onepoint：thecriticalpointTwocurves：saturatedliquidcurvesaturatedvaporcurveFourregions：liquidregionvaporregionliquid/vaporregionsup-criticalfluidregionThinking(3-2)：DiscribeprocessbelowinpTorpVdiagram.Liquefyingsuperheatedvapor(p1,T1)whenporT=constantTT1T2P1pTLiquidVaporSolidCP1pVTCCT2T1Thinking(3-3)DiscribeprocessbelowinpTorpVdiagramHeatingsuperheatedvaporvapor(p1,T1)whenP=constantTT2P1pTLiquidVaporSolidCT1P1pVTCCT2T1Thinking(3-4)DiscribeprocessbelowinpTorpVdiagramHeatingsuperheatedvaporvapor(p1,T1)whenV=constantPP1T2pTLiquidVaporSolidCT1P1pVTCCT2T13.1.4P-V-TdiagramofpuresubstanceSolidSolidregionSolid/liquidLiquidregionLiquid/vaporVaporregionCriticalpointTriplephasecurveSolid/vaporThinking(3-5)TheprojectionofPVTdiagramPPPVdiagramPTdiagramBCALABPTVCSVVLPTABCSGL-GS-GS-LS/LV/LV/SPVTdiagram3.1.5CriticalBehaviorConstant-volumecurvesinthesingle-phaseregionsVVLPTABCSGL-GS-GS-LpTLiquidVaporSolidCP1Thechangethatoccurwhenapuresubstanceisheatedinasealedtubeofconstantvolume3）mixtureofliquidandvaporJK2）3）mixtureofliquidandvaporJK2）SubheatedvaporG1）SubcooledliquidEpTCVpTCVT1TCCT2pFEEF(J,K)HGNQKJNQGHFortheregionswhereasinglephaseexistsIsothermalcompressibilityIsothermalcompressibilityVolumeexpansivityForliquidThinking(3-6)featuresofκandβforliquid?3.2VirialEquationsofstate3.2.1IntroductiontoEOSForapuresubstanceFmax=?Fmax=C-P+2=1-1+2=2thatis:Forapuresubstance,fixingtwoofitspropertiesfixesalltheothers,andthusdeterminesitsthermonadymicstateWecanwrite:TheequationofstateTheequationofstate(EOS)ValueofEOSEOS的價(jià)值：1）精確地代表相當(dāng)廣泛范圍內(nèi)的PVT數(shù)據(jù)，大大減少實(shí)驗(yàn)測(cè)定工作量。2）可直接計(jì)算不做實(shí)驗(yàn)測(cè)定的其它熱力學(xué)性質(zhì)3）進(jìn)行相平衡的計(jì)算但其通用性差，要在廣泛的氣體密度范圍內(nèi)即用于極性物質(zhì)，又用于非極性物質(zhì)，又要精度，很難做到。ThenecessaryconditionsofEOS1）Accordingwithcriticalcondition2）P→0（V→∞）,accordingwiththelawofidealgasT1pT1pTCCT2VThePVproductforagasorvapormaybeexpressedasafunctionofPbyapowerseriesThePVproductforagasorvapormaybeexpressedasafunctionofPbyapowerseriesLetWherea,B′,C′,etc.,areconstantsforagiventemperatureandagivenchemicalspecies3.2.3TwoformsoftheVirialEquationCompressibilityfactorororB,B′issecondvirialcoefficientsC,C′isthirdvirialcoefficients,etc.ForagivengasthevirialcoefficientsarefunctionsoftemperatureonlyTheVirialequationistheonlyonehavingafirmbasisintheoryFortheexpansionin1/VthetermB/V——interactionsbetweenparisofmoleculesthetermC/V2——three-bodyinteractions,etc.Sincetwo-bodyinteractionsaremanytimesmorecommonthanthree-bodyinteractions;three-bodyinteractionsaremanytimesmorenumerousthanfour-bodyinteractions,etc.,thecontributionstoZofthesuccessivelyhigher-orderedtermsdecreaserapidly.3.4ApplicationoftheVirialEquationstheVirialEquationsTheVirialEquationsareinfiniteseries.TheVirialEquationsareinfiniteseries.Forengineeringpurposes,theirconvergenceshouldbeveryrapid.Thinking(3-6)Atlowpressure,theisothermsarenearlystraightlines,andapproximatetotangentatP→Atlowpressure,theisothermsarenearlystraightlines,andapproximatetotangentatP→0ThatistangentcanreplacetheisothermsatlowpressureThetangentatP→0isForexample:Compressibility-factorgraphformethaneApplicationrange1)ThevirialequationcanbeusedtorepresenttheP-V-Tbehaviorofvapor,butnotsuitabletoliquid2)WhenP<1.5Mpa，twoequationsasfollowedcanbeusedforengineeringpurpose3)When1.5Mpa<P<5Mpa，thevirialequationtruncatedtothreetermsoftenprovidesexcellentresults4)Athighpressure,lowtemperatureandhighrequestofprecision，thevirialequationtruncatedtomoretermscanprovidesexcellentresults應(yīng)用范圍與條件維里方程是一個(gè)理論狀態(tài)方程，其計(jì)算范圍應(yīng)該是很寬闊的，但由于維里系數(shù)的缺乏，使維里方程的普遍性和通用性受到了限制。在使用維里方程時(shí)應(yīng)注意：A.用于氣相PVT性質(zhì)的計(jì)算，對(duì)液相不適用；B.P<1.5Mpa時(shí)，用兩截項(xiàng)維里方程計(jì)算,可滿足工程要求；C.1.5Mpa<P<5Mpa時(shí)，用三截項(xiàng)維里方程計(jì)算；D.高壓，精確度要求高時(shí)，可根據(jù)情況，多取幾項(xiàng)。目前采用維里方程計(jì)算氣體PVT性質(zhì)時(shí)，一般最多采取三項(xiàng)。這是由于多于三項(xiàng)的維里方程中的常數(shù)奇缺，所以多于三項(xiàng)的維里方程一般不大采用。Example3.7Solution3.7(1)ForanidealgasZ=1Z=1(2)SolvingvirialequationtrunctedtwotermsforVgivesZ=0.9014Z=0.9014(3)SolvingvirialequationtrunctedthreetermsforVTofacilitateiteration,writeequationaboveas:V0=3934cm3mol-1V1=3539cm3mol-1V2=3495cm3mol-1……V=3488cm3mol-1[when(Vi+1-Vi)<ε]Z=0.88663.5CubicequationofstateCubicequationofstatesarethepolynomialequationsthatarecubicinmolarvolumeIncludemainly:（1）VanDerWaalsequation（2）R-K（Redlich-Kwong）equation（3）S-R-K(Soave-Ridlich-Kwang)equation（4）P-R（Peng-Robinson）equation（5）P-T（Patel-Teja）equation3.5.1ThevanderWaalsEquationofStateThefirstpracticalcubicequationofstatewasproposedbyJ.D.vanderWaalsin1873Wherea,bareallconstanta/V2:壓力校正項(xiàng)b：體積校正項(xiàng)Whena=b=03.5.2AVDWEquationofStateGenericCubicEquationofStateVDWEquationofStateGenericCubicEquationofStateGenericCubicEquationofStatePeng-Robinson(PR)equationPeng-Robinson(PR)equationRedlich-Kwong(RK)equationSoave-Redlich-Kwong(SRK)equationPatel-Teja(PT)equation3.5.3DeterminationofEquation-stateParametersTheconstantsinanequationofstateforaparticularsubstancemaybeevaluatedbytworoutesAfittoavailablePVTdataThecriticalconstantsForcubicequationofstate,suitableestimatesareusuallyfoundfromvaluesforthecriticalconstantsWherecrWherecrdenotesthecriticalpoint1)VanderWaalsequationMethodoneMethodoneAtthecriticalpointMethodtwoAtthecriticalpointTerm-by-termcomparison2)GenericCubicequationαα(Tr)isadimensionlessfunctionΨandΩarepurenumbers,independentofsubstanceanddeterminedforaparticularequationofstatefromthevaluesassignedtoσandε3）Redlich-Kwong(RK)equationRK方程能較成功地用于氣相P-V-T的計(jì)算，但液相的效果較差，也不能預(yù)測(cè)純流體的蒸汽壓(即汽液平衡)。DefineAandBRKEOScanbewrittenasthefunctionofcompressibilityfactor(壓縮因子Z的三次方表達(dá)式)Tofacilitateiteration,writeRKEOSaboveas:4）Soave-Redlich-Kwong(SRK)equation與RK方程相比，SRK方程大大提高了表達(dá)純物質(zhì)汽液平衡的能力，使之能用于混合物的汽液平衡計(jì)算，故在工業(yè)上獲得了廣泛的應(yīng)用。DefineAandBSRKEOScanbewrittenasthefunctionofcompressibilityfactor（壓縮因子Z的三次方表達(dá)式）Tofacilitateiteration,writeSRKEOSaboveas:5）Peng-Robinson(PR)equationPR方程預(yù)測(cè)液體摩爾體積的準(zhǔn)確度較SRK有明顯的改善。DefineAandBPREOScanbewrittenasthefunctionofcompressibilityfactor（壓縮因子Z的三次方表達(dá)式）Tofacilitateiteration,writePREOSaboveas:3.5.4EquationofStateofVariousConstants（多常數(shù)狀態(tài)方程）equationofstateofvariousconstantsequationofstateofvariousconstantsMartin-HouequationVirialequationBenedict-Webb-RubinequationMartin-Houequation該方程是1955年Martin教授和我國(guó)學(xué)者候虞鈞提出的。為了提高該方程在高密度區(qū)的精確度，Martin于1959年對(duì)該方程進(jìn)一步改進(jìn)，1981年候虞鈞教授等又將該方程的適用范圍擴(kuò)展到液相區(qū)，改進(jìn)后的方程稱為MH-81型方程whereThegenericequationofMartin-Houwhere方程參數(shù)：皆為方程的常數(shù)，可從純物質(zhì)臨界參數(shù)及飽和蒸氣壓曲線上的一點(diǎn)數(shù)據(jù)求得。其中，MH-55方程中，常數(shù)，MH-81型方程中，常數(shù)。方程使用情況：MH－81型狀態(tài)方程能同時(shí)用于汽、液兩相，方程準(zhǔn)確度高，適用范圍廣，能用于包括非極性至強(qiáng)極性的物質(zhì)（如NH3、H2O），對(duì)量子氣體H2、He等也可應(yīng)用，在合成氨等工程設(shè)計(jì)中得到廣泛使用。Benedict-Webb-RubinEquationWhereA0,B0,C0,a,b,c,αandγareallconstantforagivenfluid第一個(gè)能在高密度區(qū)表示流體P-V-T和計(jì)算汽液平衡的多常數(shù)方程方程參數(shù)：為密度等8個(gè)常數(shù)，由純物質(zhì)的p-V-T數(shù)據(jù)和蒸氣壓數(shù)據(jù)確定。目前已具有參數(shù)的物質(zhì)有三四十個(gè)，其中絕大多數(shù)是烴類。方程使用情況：在計(jì)算和關(guān)聯(lián)輕烴及其混合物的液體和氣體熱力學(xué)性質(zhì)時(shí)極有價(jià)值。在烴類熱力學(xué)性質(zhì)計(jì)算中，比臨界密度大1.8～2.0倍的高壓條件下，BWR方程計(jì)算的平均誤差為0.3％左右，但該方程不能用于含水體系。以提高BWR方程在低溫區(qū)域的計(jì)算精度為目的，Starling等人提出了11個(gè)常數(shù)的Starling式（或稱BWRS式）。該方程的應(yīng)用范圍，對(duì)比溫度可以低到0.3，對(duì)輕烴氣體，CO2、H2S和N2的廣度性質(zhì)計(jì)算，精度較高。例:試用RK、SRK和PR方程分別計(jì)算異丁烷在300K，3.704MPa時(shí)摩爾體積。其實(shí)驗(yàn)值為V=6.081m3/kmol。解從附錄二查得異丁烷的臨界參數(shù)為Tc＝126.2KPc＝3.648MPaω＝0.176(1)RK方程(2)SRK方程3.5.6TheoremofCorrespondingStateDefinitionofreducedtemperatureandreducedPressure1）Thetwo-parametertheoremofcorrespondingstatesAllfluids,whencomparedatthesamereducedtemperatureandreducedpressure,haveapproximatelythesamecompressibilityfactor,andalldeviatefromideal-gasbehaviortoaboutthesamedegree.Thinking(3-7)TransfervdWequationtotwo-parameterequationAttention!Forthesimplefluids,suchasargon,kryptonandxenon,thetwo-parametertheoremofcorrespondingstatesarenearlyexact.Formorecomplexfluids,deviationareobserved.Formorecomplexfluids,introductionofathirdcorresponding-stateparametercanimprovtheresults.themostpopularsuchparameteristheacentricfactorvdW方程的對(duì)比形式vdW方程的對(duì)比形式代入對(duì)比參數(shù)無因次化處理ThelogarithmofthereducedvaporpressureofapurefluidisapproximatelylinearinthereciprocalofabsolutetemperatureThisisobservednottobetrue,eachfluidhasitsownS-1.0-1.01/Tr1/Tr=1/0.7=1.43Slope≈-3.2(n-Octane)Slope≈-2.3(ArKrXe)logprsatPitzernotedthatallvapor-pressuredataforthesimplefluids(Ar,Kr,Xe)(SF)lieonthesamelinewhenplottedaslgprsatvs.1/Tr,andthatthelinepassesthroughprsat=-0.1atTr=0.7,TheacentricfactorisdefinedasthisdifferenceevaluatedatTr=0.73）Thethree-parametertheoremofcorrespondingstatesAllfluidshavingthesamevalueofω,whencomparedatthesameTrandPr,haveaboutthesamevalueofZ,andalldeviatefromideal-gasbehaviortoaboutthesamedegree.3.6GeneralizedCorrelationsforGases3.6.1PitzercorrelationforthecompressibilityFactorwhereZ0andZ1arefunctionsofbothTrandPr.Z0—thecompressibility-factorofsimplefluidZ1—thecompressibility-factorofreferencefluidHowcanweobtainZ0,Z1?Z0?Forthesimplefluidsω=000.200.20.40.60.81.01.20.050.10.20.51.02.05.010.0Tr=0.70.91.01.21.54.0Two-phaseregionCompressed-liquid(Tr<1.0)prZ0Z0=F0(Tr,Pr)TheLee/KeslercorrelationTablesandfigureswhichpresentvaluesofZ0andZ1asfunctionsofTrandprZ1?Fornon-simplefluidsAsimplelinearrelationbetweenZandωifvaluesofTr,Praregiven,Z0canbeobtainedAsimplelinearrelationbetweenZandωTheLee/KeslercorrelationTablesandfigureswhichpresentvaluesofZ0andZ1asfunctionsofTrTheLee/KeslercorrelationTablesandfigureswhichpresentvaluesofZ0andZ1asfunctionsofTrandprZ1=F1(Tr,Pr)例2-3計(jì)算一個(gè)125cm3的剛性容器，在50℃和18.745MPa的條件下能貯存甲烷多少克（實(shí)驗(yàn)值是17克解：查出Tc=190.58K,Pc=4.604MPa,ω=0.011Z0?Z1?查表及內(nèi)插:TrPr3.0005.0004.0711.600.84100.86170.85211.700.88090.89840.8860查表及內(nèi)插:TrPr3.0005.0004.0711.600.23810.26310.25151.700.23050.27880.2564使用情況：Pitzer關(guān)系式對(duì)于非極性或弱極性的氣體能夠提供可靠的結(jié)果，誤差，應(yīng)用于極性氣體時(shí)，誤差要增大到5％～10%，而對(duì)于締合氣體和量子氣體，使用時(shí)應(yīng)當(dāng)注意。3.6.2PitzerCorrelationsfortheSecondVirialCoefficientSubstitutionT=Tr·Tc,P=Pr·Pcintothesimplestformofthevirialequation:Thus,Pitzerandcoworkersproposedasecondcorrelation,whichyieldsvaluesforBPc/RTc:ThereducedsecondvirialcoefficentThereducedsecondvirialcoefficentSubstitutionT=Tr·Tc,P=Pr·Pcintothesimplestformofthevirialequation:Thus,Pitzerandcoworkersproposedasecondcorrelation,whichyieldsvaluesforBPc/RTc:Secondvirialcoefficientsarefunctionsoftemperatureonly,andsimilarlyB0andB1arefunctionsofreducedtemperatureonly.Theyarewellrepresentedbythefollowingequations:Attention！01234501234512346780普遍化維利系數(shù)使用區(qū)普遍化壓縮因子使用區(qū)Trpr3.6.3Theconditionunderwhichtheideal-gasequationcanbeusedasareasonableapproximationtorealityTheideal-gasequationisareasonab