4 Laminar film condensation in a vertical tube in

LaminarfilmcondensationinaverticaltubeinthepresenceofnoncondensablegasShripadT.Revankar*,DougPollockSchoolofNuclearEngineering,PurdueUniversity,WestLafayette,IN47907-1290,USAReceived1July2002;receivedinrevisedform1August2004;accepted6September2004Availableonline23November2004AbstractAtwo-dimensional,steadystatemodelofconvectivefilm-wisecondensationofavaporandnoncondens-ablegasmixtureflowingdownwardinsideaverticaltubeisdeveloped.Themasstransferatthefilmandgasinterfaceistreatedasdiffusioncontrolledprocess.Thenoncondensableeffectonthecondensationistakenintoaccountthroughboundarylayeranalysisofspeciesconcentrationandenergybalance.Numericalpre-dictionswereobtainedforthecondensationheattransfercoefficientofturbulentvaporflowassociatedwithlaminarcondensate.Thepredictionswerecomparedwiththeexperimentaldataintheliteraturetoassessthemodel.Noncondensablemassfractionandvapor–noncondensablemixturetemperaturewerepresentedintheformofradialandaxialprofiles.
2004ElsevierInc.Allrightsreserved.Keywords:Laminarfilmcondensation;Noncondensablegas;Verticaltubecondenser;Steamcondensation1.IntroductionManyindustrialsystemsuseverticaltubecondensersandindustrialpracticehasindicatedthat,often,muchhighercoefficientsofheattransferareobtainedwhenvaporsarecondensedinside0307-904X/$-seefrontmatter
2004ElsevierInc.Allrightsreserved.doi:10.1016/j.apm.2004.09.010*Correspondingauthor.E-mailaddress:[e-mail](S.T.Revankar).(2005)341–359NomenclatureVariablesx,y,rcoordinatescpspecificheatDbbinarydiffusioncoefficientDratioofdensities–q1/qgggravitationalconstanthheattransfercoefficienthfgenthalpyjgdiffusivemassfluxkthermalconductivitycoefficientlliquidmmassflowrateMmolecularweightNratioofkinematicviscositiesm1/mgPpressureq*generalizedheatfluxReReynoldsnumberRradiusRcgasconstantTtemperatureuaxialvelocityvradialvelocityWnoncondensablegasmassfractionGreeksymbolsathermaldiffusionfactordfilmthicknesseeddydiffusivityqdensityrsurfacetensionmkinematicviscositylviscositysshearstressSubscriptsiinterfacebbulkgnoncondensablegasssaturationvvaporwwall342S.T.Revankar,D.Pollock/AppliedMathematicalModelling29(2005)341–359tubesratherthanoutside[1].However,inpracticaloperationsofthecondensers,smallamountsofnoncondensablegasmayexistinworkingvaporsduetocharacteristicsofthesystemorchem-icalbreakdownofworkingvapors.Itiswellknownthatthepresenceofnoncondensablegasesinavaporcangreatlyreducetheperformanceofcondensers[2–9].Thisisbecauseofthefactthatthepresenceofnoncondensablegaslowersthepartialpressureofthevapor,thusreducingthesatu-rationtemperatureatwhichcondensationoccurs.Thecondensationofvaporsfromavapor–gasmixtureinatubehasbeenstudiedbyvariousauthors[3,10–20].Theanalysisoftheheatandmasstransferduringcondensationofavaporinthepresenceofanoncondensablegashasgenerallyinvolvedeitherboundarylayeranalysisorheatandmasstransferanalogymethods.Inbothofthesemethodsthecondensationisviewedasoccurringintwointeractingboundarylayers,thevapor–airmixtureandthecondensateboundarylayers.Theboundarylayersolutionscurrentlyavailabledealprimarilywiththeflatplateconfigurationandstagnantatmosphericconditions[21,22].TheanalysisbyheatandmasstransferanalogymodelsfollowthegeneralmethodologyofColburnandHougen[3,14–16].Ghiaasiaanetal.[17]presentedatwo-fluidmodelforcondensa-tioninthepresenceofnoncondensablegasinachannel.Thecondensate-gasinterphaseheat,massandmomentumtransfer,aretreatedusingthestagnantfilmmodel.Thismethodologyrequiresinterphasesurfaceareaconcentrationandliquidandgassidetransfercoefficients.Con-densationinthepresenceofnoncondensablegashasrecentlybeenexperimentallystudiedintubeflowconfigurationbyVierow[23],Ogg[24],Siddique[25]andKuhnetal.[26],becauseofitsapplicationinvariousindustrialsystems.Fromtheliteratureitisevidentthatnoanalyticaltreat-mentexiststhatisfreefromasemi-empiricalapproachandwhichisconsistentwiththeprocessofcondensation(includinglaminarandturbulentfilmflow)insideaverticaltubeinthepresenceofnoncondensablegas.Thepresentworkdealswithconvectivefilm-wisecondensationofavaporandgasmixtureflowingdownwardinsideaverticaltube.Aunifiedprocedureisdevelopedtopredictthecondensationheattransfercoefficientofturbulentvaporflowassociatedwithlaminarcondensate.Thepredictionsarefirstcomparedwithpublishedpuresteamdata[27,28]andthenwiththedataforcondensationwithnoncondensable[23,25].2.AnalysisFig.1showsaschematicofthephysicalmodelconsideredforthecondensationprocesswithdefinedcoordinatesystems.Asaturatedvapor–gasmixture,withconstantpropertiescorrespond-ingtoinletpressureandtemperatureconditions,entersthetubeandisturbulent.Thecondensertubesurfaceisbelowthevaporsaturationtemperature.Thecondensateformsalongthetubesur-faceandthusanannularflowregimeisrealizedinsidethecondensertube.Alongthecondensatefilminterface,temperature,momentumandconcentrationboundarylayersdevelopandrespectivegradientsareformed.Thesegradientsareshownlatertoimpedethecondensation.Thefollowingassumptionsweremadeinthedevelopmentofgoverningequations:1.Theflowistwo-dimensionalandsteadystate.2.Thecrosssectionalgeometryisci