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variable turbulent Prandtl number for heat transfer to supercritical fluids

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InternationalJournalofHeatandMassTransfer92(2016)792–806ContentslistsavailableatScienceDirectInternationalJournalofHeatandMassTransferjournalhomepage:www.elsevier.com/locate/ijhmtAnewformulationofvariableturbulentPrandtlnumberforheattransfertosupercriticalﬂuids

YoonY.Bae

KoreaAtomicEnergyResearchInstitute,9-111Daedeokdaero,Yuseong,Daejeon,34057,RepublicofKoreaarticleinfoabstract

Whenaﬂuidatsupercriticalpressureapproachesthepseudo-criticaltemperatureitexperiencesastrongvariationinphysicalpropertiesputtingapplicabilityofvariousturbulentﬂowmodelingsinquestion.Earliernumericalcalculationsshowed,withoutexception,unrealisticover-predictions,assoonastheﬂuidtemperatureapproachedthepseudo-criticaltemperature.Theover-predictionsmighthavebeenresultedeitherfromaninapplicabilityofwidelyusedturbulencemodelsorfromanunrealistictreatmentoftheturbulentPrandtlnumber(Prt)asaconstant.Recentresearch,bothnumericalandexperimental,indicatesthatPrtisverylikelyafunctionofﬂuid–thermalvariablesaswellasphysicalproperties,whenthegradientsofphysicalpropertiesofaﬂuidaresigniﬁcant.ThispaperdescribestheprocedureforanewformulationofPrtwhichvarieswithphysicalpropertiesandﬂuid–thermalvariables.TheapplicationofthevariablePrtwassurprisinglysuccessfulinreproducingtheﬂuidtemperatureinsupercriticalﬂuidsﬂowinginsmall-diameterverticaltubesrangingfrom4.57to20mm.Ó2015ElsevierLtd.Allrightsreserved.Articlehistory:Received6February2015Receivedinrevisedform15September2015Accepted15September2015Keywords:TurbulentPrandtlnumberReynoldsanalogyMixedconvectionSupercriticalpressureStrongpropertyvariation1.IntroductionAnaccurateestimationoftheheattransferrateortemperatureofthecoolantchannelisessentialforthedevelopmentofasuper-criticalpressurewatercooledreactor(SCWR)[1].Methodsforpre-dictingtheheattransferratetoorfromsupercriticalﬂuidsﬂowinginaverynarrowpassagearenotsatisfactoryandhaveyettobeestablished.Thetwokindsofﬂuid,waterandcarbondioxide(CO2),aremediumsofinterestandlotofworksfortheinvestiga-tionarebeingconductedforapplicationsinareassuchasSCWR,Braytoncycleandcompactprintedcircuitheatexchangers.Anum-berofcorrelationsforthepredictionoftheheattransferrateinﬂu-idsatsupercriticalpressureshavebeenproposedbyvariousresearchers,butmostofthemareapplicableﬂuidsinaforcedcon-vectionregime,asshowninthereviewpapersbyChengandSchu-lenberg[2]andPioroandDuffey[3].Thecorrelationsavailableinliteraturepredicttheheattransferratewithareasonableaccuracyinaforcedconvectionregime;however,inamixedconvectionregime,allofthosecorrelationsfailorpartiallysucceedtoproduceaccuratepredictions,andthevariationissolargethattheirappli-cationtothedesignneedstobeverycautious.SincemostoftheearlierworkshavebeensummarizedbyPioroandDuffey[3],severalselectedrecentworksareintroducedhere.Efforts,bothexperimentalandanalytical,havebeenmadetofor-mulateareliablecorrelationforamixedconvectionheattransferbyresearcherssuchasWattsandChou[4],JacksonandHall[5],Jacksonetal.[6],BaeandKim[7],Baeetal.[8],Bae[9]andJackson[10].Zhuetal.[11]investigatedtheheattransfercharacteristicsofsteam–waterﬂowingupwardintubesatsub-andsuper-criticalpressuresintherangeof13–30MPa.Yangetal.[12]performedanexperimentonheattransfertosupercriticalwaterﬂowinginverticalannularchannels,andevaluatedfourcorrelationsagainstthedata.Lietal.[13]reportedrecentexperimentalresultsfromthesupercriticalwaterheattransfertestfacilitySWAMUPatShanghaiJiatongUniversity.Zhaoetal.[14]reportedexperimentalresultsfromthesameresearchgroupwithdifferentconditionsonlytorevealthattheexistingheattransfercorrelationsdidnotcorrectlyreproducetheheattransferrate.Inadditiontotheexperimentalefforts,alargenumberofnumericalworkshavebeenperformedtosimulatetheﬂowandthermalﬁeldinaﬂuidatsupercriticalpressures,andindoingso,theapplicabilityofvariousturbulencemodelswasexamined.Forbothforcedandmixedconvectionregimes,experimentalandnumericalinvestigationsofthethermalandﬂowﬁeldatsupercrit-icalpressurewasperformedbyLichtetal.[15].Theyconﬁrmedthatforthesimplecaseofdeterioration,numericalsimulationsusingthecommercialCFDcodeFluentofferedaqualitativeinsightintochangesinﬂuidtemperatureandturbulentvelocitiesrespon-siblefortheaxialevolutionofthewalltemperature.Choetal.[16]examinedthreeturbulencemodels,RNGk–e,SSTk–xandonetypeE-mailaddress:yybae@kaeri.re.krhttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.09.0390017-9310/Ó2015ElsevierLtd.Allrightsreserved.Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806793NomenclatureA+ClCe1,Ce2cpDGGkhkpPkPrPrtPrt,oqrRReTu,vu+xyy+effectiveviscoussublayerthicknessconstantintheturbulentviscosityconstantsintransportequationforespeciﬁcheattubediametermassﬂux

productionofturbulenceduetobuoyancyenthalpy

turbulentkineticenergypressure

productionofturbulenceduetoshearPrandtlnumber

turbulentPrandtlnumber(variable)

Prtbeforeadjustmentwithadditionalfunctionsheatﬂux

radialcoordinatetuberadius

Reynoldsnumbertemperature

velocityinxandrdirectionnon-dimensionalu,u=usaxialcoordinate

distancefromthewall

non-dimensionaldistancefromwall,yus=m

yþTBLyþr¼0:8Ryþupeak󰀁a~a

y+attheturbulentboundarylayeredge(300)

y+atr=0.8Ry+at@u=@y¼0

Reynoldsaveragequantity(a:dummy)Favreaveragequantity(a:dummy)

Greeksymbols

turbulentthermaldiffusivity

bvolumetricexpansioncoefﬁcientedissipationrateofturbulentkineticenergyjvonKarmanconstantl,ltmolecularandturbulentviscositym,mtmolecularandturbulentkinematicviscosityqdensityrk,remodelconstantsforturbulentdiffusionofk,ertstandardturbulentPrandtlnumber(=0.9)stshearstress

atSubscriptseeffective(molecular+turbulent)oinletwwall

oflow-Reynoldsnumbermodel,againsttheexperimentaldataobtainedforatubeandannuluswithanequivalenthydraulicdiameterof4.4mm,andreportedthattheperformanceofthethreemodelswaspartiallysuccessful.Heetal.[17]thoroughlyinvestigatedlow-Reynoldsnumberturbulencemodelsandcon-cludedthatboththelowReynoldsnumberk–emodelsandtheV2Fmodelswereabletocapturethegeneraltrendsoftheinterest-ingwalltemperaturebehaviorobservedwithanupwardﬂowinsomeexperimentswithaﬂuidatapressurejustabovethecriticalvalue,whilethedetailedvariationofthewalltemperaturepre-dictedusingeachmodelwasratherdifferentfromthatintheexperiments.Theyalsofoundthattheeffectontheheattransferwasalmostentirelyduetotheshearproductioneffectcausedbythedistortionofthemeanﬂowasaresultofthestronginﬂuenceofthebuoyancy.Zhangetal.[18]successfullyreproducedusingamodiﬁedversionofalow-ReynoldsturbulencemodelthedatafromaDNScalculationandanexperimentbyemployinganalge-braicﬂuxmodelincalculatingtheturbulenceproductionbasedonthebuoyancy.However,itsapplicationtootherconditionsisstilltobeproven,andthecalculationdomainwastoosmalltogen-eralizetheresults.Zhangetal.[19]comparedtheexperimentalheattransferdatainsupercriticalﬂuidsinacirculartubewiththecalculationresultsobtainedbyemployingsixdifferentturbu-lencemodelsandfoundthattheReynoldsstressmodel(RSM)gavethebestagreementwiththeexperimentaldata,especiallywiththedeterioratedones.TheresultofRSMwasnotmuchdifferentfromthatofRNGk–e,anditsapplicabilityshouldbeconsideredinpar-allelwiththefactthatitrequiressolvingadditionalequations.Jar-ominandAnglart[20]numericallyperformedasensitivityanalysisoftheheatedwalltemperatureandvelocitydistributionintheCFDsimulationoftheupwardﬂowofsupercriticalwater.Theyclaimedthatk–xturbulencemodelsuccessfullysimulatedtheinitialtem-peraturepeakneartheinletandonsetofdeterioration,butwithouttherecoveryofheattransferfromdeteriorationatthepipeexit.Insummary,thenumericalworksperformedthusfaronlytoprovethatallcurrentturbulencemodelsareapplicabletothecaseswithlimitingconditions.Inthiscontext,abreak-throughisneededtosimulatetheﬂuidthermalbehaviorinaﬂuidwithseverepropertyvariation,andfocushasbeengiventoPrtratherthantryingtoimprovetheturbulencemodeling.Withoutexceptionsallnumericalworksperformedthusfarstruggledinsimulatingﬂowandthermalﬁeldswithstrongprop-ertyvariation,thatis,strongbuoyancyoracceleration.Mostoftheturbulencemodelingsweredevelopedbasedonanincom-pressibleandconstant-propertyﬂow.Afterwards,manyeffortshavebeenmadetoextendthemodelsdevelopedforconstantpropertyﬂowstovariablepropertyﬂows,however,theyfocusedonhighspeedﬂoworcompressibleﬂows.Sincethepropertiesofﬂuidsatsupercriticalpressurechangewithtemperaturethanwithpressure,adirectapplicationofthetheorydevelopedforhighspeedorcompressibleﬂowtothecasesofsupercriticalﬂuidsshouldbeverycautious.Asexpected,theapplicationofthevari-ableproperty(mainlydensity)versionofturbulencemodelingwasnotusedinsimulatingahighlybuoyantﬂowofﬂuidsatsuper-criticalﬂow,especiallyinreproducingﬂuidtemperatures.Prtisaproductofapureperspectiveofsimilarityinappearancebetweenthemomentumandenergyequations,andanassumptionthattheirbehaviorwouldbethesameor,atleast,verysimilartoeachother.Accordingly,itwastreatedasunityoranexperimentally-obtainedvalueslightlysmallerthanunityof0.8–0.9.Formostofﬂuidﬂowswithbarelyvaryingﬂuidproperty,ithassuccessfullyworkedandproducedreasonableresults,butneverinthecasesofstrongpropertyvariation.Inthisregard,theauthordecidedtorevisittheReynoldsanalogy,whichconnectsthemomentumandenergyequationsinthenameofthePrt,andtriedtoﬁndanypos-sibilityofextendingittobeappliedtothecaseofstrongpropertyvariation.Inallnumericalworksintroducedabove,includingearlierworksnotmentionedhere,Prtwastreatedasaconstantorfunc-tionofPrandtlnumber.AconstantPrt,whichwasnotsuccessfulinpredictingtheheattransferinadeteriorationregime,doesnotseemtoproperlyrepresentthephysicsinthesupercriticalﬂuidinaverticaltubeunderaheatﬂuxhighenoughtocausedeteriora-tion.ThePrtishighlyunlikelytobeaconstantwhentheﬂuidprop-ertiesexperiencesubstantialvariations.QuarmbyandQuirk[21]measuredtheeddydiffusivityinairﬂowingthroughaplaintube794Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806underhighheatﬂux,andsuggestedanequationforPrt.Daietal.[22]performedexperimentsofbuoyantturbulentplumewithcar-bondioxideandhexaﬂuoride,inwhichthedensityratiobetweenplumeandatmospherewas1.51and5.06,andfoundthatPrtdecreasedfrom0.8atthecenterlinetoavaluesmallerthan0.1neartheplumeedge.Itprobablytellsusthatthedensityratioplaysanimportantroleintheturbulentheattransfer.Kangetal.[23]experimentallyshowedthatthevalueofPrtvariedbetween0.4and1.4.KangandIaccarino[24]andnumericallybyusingLESrevealedthatPrtshowedawidevariationrangingfrom0.1to20.TheeffectofPrthasbeenstudiedbyMohseniandBazargan[25],showingthatavalueofPrtsmallerthanunityresultsinabetteragreementwiththeexperimentaldata,foradeterioratedheattransfer.Liuetal.[26]proposedanewformulationofPrt,whichdependsonthetemperaturegradient,forthecomputationoftheﬁlmcoolingeffectiveness.Theircalculationsclearlyshowedthatthetemperature-dependentPrtproducedbetterresultsthanusingaconstantPrt.ConsideringtheaboveevidencesofPrtbeingfarfromunityinthecaseofsteeppropertygradient,thereissufﬁcientreasontobelievethatPrtmightbeafunctionofﬂowandthermalvariables,aswellasthephysicalproperties.Inthispaper,anattemptwasmadetodevelopanewformula-tionforPrt,whichvariesdependingontheﬂowandthermalvari-ablesaswellasthephysicalpropertiessuchasdensity,speciﬁcheatandtemperature.AfterthecompletionofthedevelopmentofPrtitsapplicabilitywasexaminedagainsttheexperimentaldataforwater,carbondioxideandR22.2.Numericalmethod2.1.GoverningequationsandturbulencemodelInthepresentstudy,averticallyupwardﬂowingﬂuidinauni-formlyheatedtubewasconsidered.Theﬂowwasassumedtobesteadyand2-Daxi-symmetric.Theverticalupwarddirectionwasalignedinthepositivexdirectionandtheradialcoordinatewasr.Inthecircumferentialdirection,thevelocitycomponentsaswellasthegradientofallﬂowandthermalpropertieswereassumedtobezero.Thatis,thereisnoswirl.Thegoverningequationsemployedinthepresentstudywerecontinuityequation,ensembleaveragedNavier–Stokesequation,energyequation,two-equationturbulencemodel.Thegoverningequationsforavelocityﬁeldinacylindricalcoordinatewithatwo-equationk–eturbulencemodelareasfollows[17]:1󰀁@r@xðrq󰀁~uÞþ@@rðrq󰀁~vÞ󰀃¼0ð1Þ1󰀁@󰀁u~2Þþ@ðrq󰀁u~v~Þ󰀃¼À@pþq󰀁g2@󰀁󰀄@~u󰀅󰀃r@xðrq@r@xxþrl󰀁eþ1@󰀁󰀄r@x@x~@󰀅󰀃r@rrl󰀁@ue@rþv~@xð2Þ1󰀁@r@xðrq󰀁~uv~Þþ@@rðrq󰀁v~2Þ󰀃¼À@p@rþq󰀁g1@󰀁rþr@xrl󰀁󰀄e@~v@~u󰀅󰀃@xþ@þ2@󰀁󰀄r@rrl󰀁~󰀅󰀃r󰀁ee@v@rÀ2lv~r2ð3Þ1󰀁@r@xðrq󰀁u~~hÞþ@@rðrq󰀁v~~hÞ󰀃(\"¼þ1@󰀄l󰀁l󰀁t󰀅@~h#rrþ\"@xPrPrt@xþ@󰀄l󰀁l󰀁t󰀅@~h#)@rrPrþPrt@rð4Þ1󰀁@󰀆󰀇@󰀆~r@xr󰀁q~u~kþ@rr󰀁q~v~h󰀇󰀃¼@\"󰀄@xl󰀁þl󰀁t󰀅r@~k#k@xþ1@\"󰀄r@rrl󰀁þl󰀁t󰀅r@~k#󰀁󰀁k@rþqPkþqGkÀq󰀁~eð5Þ1󰀁@r@xðr󰀁qu~~eÞþ@@rðrq󰀁v~~eÞ󰀃¼@󰀁󰀄@xl󰀁þl󰀁t󰀅@~re󰀃1@󰀁󰀄e@xþr@rrl󰀁þl󰀁t󰀅@~re󰀃e@rþq󰀁Ce~~1fee~e21kðPkþGkÞÀq󰀁Ce2fe2~kð6ÞThetildeandbarabovethevariablesindicatetheensembleandFavreaverage,respectively(f0¼FÀF,f00¼FÀ~F,F~¼qF=q).Foraverticalupwardﬂowinatube,gx¼Àgandgr¼0.Theeddyvis-cositywasmodeledbythemodeladoptedearlierinRef.[27].TheReynoldsstresstensorÀqu0iu0j(consideredpracticallynotmuchdifferentfromÀ󰀁qug00iu00j)wasmodeledthroughaBoussi-nesqueapproximation.Àqu0i¼l󰀄~jt@u@u~i󰀅u0j@xiþ@xÀ23dijq~kð7ÞjTheproductionofturbulenceenergybyinteractionwiththemeanﬂow(\"andturbulencePkwasdeﬁnedas󰀄P~󰀅󰀄󰀅2󰀄~󰀅2#󰀄󰀅)k¼mt2@u2@xþ@~v@rþv@~urþ@rþ@v~2@xð8ÞThegravitationalproductionGk¼q0u0igi=qorGk¼Àbgiu0it0wasmodeledusingthegeneralizedgradientdiffusionhypothesis(GGDH)[28]asshownbelow.Àu0it0¼c~kh~@Teu0iu0j@xjð9ÞInthiswork,thelow-Reynoldsnumberturbulencemodelpro-posedbyMyongandKasagi[29](willbereferredtohereinafterasMK)wasused.Inthelow-Reynoldsturbulencemodel,theeddyviscosityisdeﬁnedasfollows:l󰀁t¼q󰀁C~lfk2l~eð10ÞThecoefﬁcientsappearingintheeddyviscosityanddissipationequation aredeﬁned!asfollows:f󰀁l¼1þ3:451Àexp󰀄þÀy󰀅󰀃Re1=2tAþð11Þfe1¼1ð12Þ(\"f1À2󰀄Re󰀅2#)t󰀁󰀄þ󰀅󰀃2e2¼9expÀ61ÀexpÀy5ð13ÞIntheMyong–Kasagimodel[29],A+,themodiﬁedeffectivesub-layerthickness,inEq.(11)is70.TheconstantsfortheMKturbu-lencemodelaresummarizedinTable1.Althoughthereisnouniquedeﬁnitionofthenormalizedwalldistancey+,mostofthecomputationalworksperformedsofar,regardlessofcompressibleorincompressiblecases,employedtheTable1

ConstantsintheMyong–Kasagiturbulencemodel.

ClCe1Ce2rtrkreCh0.091.401.800.91.41.30.3Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806795deﬁnitionyþ¼usy=mw,whereus¼ðsw=qwÞ1=2usingwallproper-ties.Instead,inthiswork,analtereddeﬁnitionbasedonthelocalpropertiessuchasyþ¼ðsw=qÞ1=2y=m,whichiscalledthesemi-localscaling,wasused.Theonlyreasonforthechoicewasthatthecomputationalresultsobtainedinthisworkwiththelatterdef-initionagreedwiththeexperimentalresultsbetterthanthosewiththeformerdeﬁnition.Whenonlyanarrowregionnearwallisheatedtohightemperature,theuseofus¼ðsw=qwÞ1=2willresultsinanunrealisticallyhighvalueofy+,makingflapproachunitymorerapidlyasy+increases.Whenthephysicalpropertiesconsid-erablyvarythedeﬁnitionofusandy+greatlyinﬂuencesthenumericalresultsthroughthedampingfunctionandcoefﬁcientsinvolvingy+asanindependentvariable.Anexactassessmentoftheeffectofthedeﬁnitionofusandy+isnotavailableatthemomentandleftforafurtherresearch.Huangetal.[30]alsoindi-catedthatthesemi-localscalingwasthebestintheinterpretationoftheirDNScalculationresultsofcompressibleturbulentchannelﬂows.Foysietal.[31]reportedthatauseofthesemi-localscalingimprovedthesimilarityinturbulence,althoughglobalsimilaritywasnotachieved.2.2.TurbulentPrandtlnumberReynolds[32]reviewedmorethanthirtywaysforpredictingPrtandtheSchmidtnumber(Sct).Kays[33]examinedthethen-availableexperimentaldataonPrtforthetwo-dimensionalTBL.Prthasbeentreatedasaconstantaround0.9orunityinmostofearliernumericalworks.However,thereareothercaseswherePrtisfarfromunity.Prtbecomesapproximately0.7foranaxi-symmetriccaseofaheatedjet,whileplanarjetdataindicateavalueof0.5;inathermallydevelopingwallboundedTBL,Prtisaroundunityonlyinthecoreboundarylayer,anditdecreasestoavaluelessthan0.5[34].InaTBL,Prtdecreasesfromaround1.5inthesub-layerto0.7attheouteredgeoftheboundarylayer[35].AccordingtotheexperimentdataprovidedbyDaietal.[22],thevalueofPrtreachesassmallas0.05.TheaboveevidencestronglyimpliesthatPrtcanhardlybeunityoraconstantveryclosetoitatleastinthecaseofheatingorcoolingofﬂuidexperi-encingsubstantialpropertyvariation.PrtispurelyaproductoftheReynoldsanalogy,whichclaimsthatthemechanismofturbulentheattransferwouldbeverysimilartothatoftheturbulentmomentumtransfer.AnassessmentofPrtforthecase(C)publishedin[36]wasmadeandtheresultisshowninFig.1.Evidently,Prthasvaluesmuchlessthanunityinsidetheturbulentboundarylayer,especiallyinthebufferandviscoussublayer.Thepeaksareduetothestronggradi-entofvelocityandtemperature,whichisinevitableirregularityoriginatedfromtheapplicationofgradientdiffusionhypothesisintheturbulentmomentumandenergydiffusion.ItshouldbenotedthatthevaluesofPrtcontinuouslydecreasetowardsthewallFig.1.VariationofPrtwithy+atthreeaxiallocations.G=166.62kg/m2s,q=30.87kW/m2,P=8.0MPa,d=2mm.After[36].Fig.2.Conﬁgurationofthetubeusedinthiswork.Table2

Flowinletconditionsforthecasesstudiesinthispaper.CasesW1-aLietal.[13]W1-bW1-cW1-dW2,Lietal.[13]W3,Shitsman[42]W4,Shitsman[43]W5Vikhrevetal.[44]C1[45]C2[45]C3,unpublishedKAERIdata1C4,unpublishedKAERIdata1F1,Morietal.[46]F2,Morietal.[46]F3,Morietal.[46]1FluidWaterPressure(MPa)25.0Inlettemp(K)574Massﬂux(kg/m2s)7.87.87.87.84734303804934004006005004004001000Heatﬂux(kW/m2)542.567775991847132036057030509060102590D(mm)7.6Reo65,663CO225.024.523.326.58.128.128.127.75574598398.4334.1282282283278.4365.6291.3312.47.681620.44.574.574.574.44.44.44.439,32441,73226,71421,58619,30319,30329,47821,99625,694929729,026R225.55.55.5Thesedatahavebeenproducedwiththosepublishedin[45]underthesameresearchprogramexecutedbytheauthor.ThedataweredepositedintheOECD/NEAdatabankafterathoroughexaminationbytheparticipantsinthecoordinatedresearchprogram(CRP)supportedbyIAEA.TheresultsoftheCRPwasdocumentedintheIAEATECDOCseries,IAEA-TECDOC-1746in2014.796Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806Fig.3.Griddependencyforthecaseofcarbondioxideatsupercriticalpressureﬂowingupwardinacirculartube[45].contrarytothewell-knownexperimentalevidences,were,withoutexceptionPrtincreasedtowardthewall.Firstofall,letusexaminethestepstakentoderivetheReynoldsanalogyfortheturbulentheattransfer(thediscussionofKaysetal.[37]waslargelyfollowed.).Imagineanelementofﬂuidofmassdmthatmovesintheydirectionatdistance‘(whichisa‘‘mixinglength”).Letusassumethattheeffectivevelocityofﬂuidintheypﬃﬃﬃﬃﬃﬃﬃ(=RÀr)directionisCv02.Accordingtothemomentumtheorem,theshearforceisequaltotherateofmomentumtransfer.Then,theeffectiveshearstressispﬃﬃﬃﬃﬃﬃﬃ~󰀉dq~󰀁󰀉jdq󰀁j@uust󰀉þ‘¼Cv021þ󰀉~󰀉󰀁󰀉du󰀁@yqqqSimilarly,theeffectiveheatﬂuxis󰀄󰀉󰀉󰀅ð17Þ 󰀉~󰀉󰀉󰀉󰀉!~pﬃﬃﬃﬃﬃﬃﬃ~󰀉~󰀉󰀉󰀉󰀉󰀉_00󰀁qTTTdqdcjdcjjdqjdqdcp󰀉t󰀉þ󰀉p󰀉þpþ󰀉󰀉‘@T¼Cv021þ󰀉þ󰀉󰀉󰀉󰀉󰀉~~~󰀉@y󰀁cp󰀁dTcpdTcpqqqcpqdTð18ÞPleasenotethattheincrementalvaluesinEqs.(17)and(18),weretakenasabsolutevalues.Thissomewhatadhocarrangementcanbejustiﬁedifwerecallthattheturbulentmixinghasnodirec-tionalpreference.Positiveandnegativegradientofrelevantvari-ableswillequallyplayamixingordiffusingrole.Thetermsexpressedasaratiobetweenincrementalvariationoftheﬂuidpropertyorﬂowvariablecanbeeasilyreplacedwithderivativeswithrespecttotheradialcoordinate.However,theincremental~needtobemodeled.~,dq󰀁,dcpanddTvariationssuchasduNow,webegintoderiveanexpressionforthemixinglength,whichholdsfromthewalltocenterlineofatubeorpipe.Anempir-icalequationformt=mfortheentireregionoutsidetheviscoussub-layerinatube[37]is~Þ¼Cv02ðq~þu~dq~dq󰀁þdu󰀁Þ󰀁u󰀁dust¼¼Cv02dðqSimilarly,theeffectiveheatﬂuxisFApﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃð14Þ_00qt¼C~Þ󰀁cpTv02dðqpﬃﬃﬃﬃﬃﬃﬃ󰀆󰀇~~~~~~02󰀁󰀁󰀁󰀁󰀁󰀁¼CvqcpdTþqTdcpþcpTdqþqdcpdTþcpdqdTþTdqdcpð15ÞpﬃﬃﬃﬃﬃﬃﬃIfthefunctionalrelationsofincrementalvariationofthevari-~withrespecttoeachotheracrossdistance~,dq󰀁,dcpanddTables,du‘areknown,theexactvalueofPrtcanbecalculated.Inaconven-tionalapproach,thepropertyvariationwasgenerallyneglectedundertheassumptionofconstantpropertiesornegligiblevariationofthem.However,foraﬂuidatsupercriticalpressure,theextra~dcpþcpT~dq~dq~dq󰀁þdu󰀁þ󰀁andq󰀁TtermsinEqs.(14)and(15),u~þcpdq~þT~dq󰀁dT󰀁dcp,arenolongernegligible,andcanpossi-󰀁dcpdTqblybegreaterthantheremainingterms.Thesecond-ordertermsshouldnotbeneglected,ifanextremelyaccuratepredictionisrequired,sincetheyamounttogreaterthan1%atsomelocations.Whenthemixingdistanceissufﬁcientlysmall,theincrements~,dq~,dT󰀁anddcpcansafelybeexpressedasdu󰀆r󰀇2mtjyþ󰀆r󰀇1þ2¼1þRRm6󰀁󰀃ð19ÞAty¼0,mt=m¼jyþc=3.ThisvalueandcorrespondingmixinglengthwillprevailinmostoftheTBLexceptviscoussublayer.Inaviscousandlog-lawregion,themixinglengthissimplydeter-minedbythewell-knownfunctionproposedbyVan-Driest,‘¼jy½1ÀexpðÀyþ=AþÞ󰀃,andem=misdeﬁnedbysimplemixinglengththeoryas~󰀉mt‘2󰀉@u󰀉¼󰀉󰀉mm@y󰀉󰀉󰀉ð20Þ~¼‘du~@u;@y~¼‘dT~@T;@y󰀁¼‘dq󰀁@q;@yanddcp¼‘@cp@yð16ÞFromEqs.(19)and(20),‘max,whichisvalidinmostpartoftheTBL,isInthiswork,theradialgradientsaremuchlargerthantheaxialones,onlyradialderivativesareretained.WhenweinsertEq.(16)intoEqs.(14)and(15),weobtainafterlittlerearrangement,‘max¼\"󰀄󰀅󰀉󰀉À1󰀉@u~󰀉mjyþc󰀉󰀉󰀉󰀉3@y#1=2ð21ÞFinally,themixinglengthinapipeisdeterminedasfollows.Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806797‘¼minð‘VanDriest;‘maxÞð22ÞBydeﬁnitionofmomentumandthermaldiffusivity,st~q_00~q¼mt@u@y;q󰀁t@Tcp¼at@yð23ÞWithouttheextratermsinthebracketsoftheright-handsideEqs.(17)and(18),Prmpﬃﬃﬃﬃﬃﬃofﬃt¼mt=atwouldbeunitysincet¼at¼Cv02.However,withtheextratermsPrtwouldnotsimplybecomeunity,butendsupwithaquitedifferentvaluefromunity.Theincrementalvariationsofthevariablesd~u,dq󰀁,dcpandd~Tcanbeexpressedas‘multipliedbycorrespondingdifferentialswithrespecttoy.TheinsertionofthoseexpressionsintoEqs.(17)and(18),andfromthedeﬁnitionoftheturbulentPrandtlnumber,resultsinthebasicvariableturbulentPrandtlnumberPrt;oexpressedas¼1þ~u󰀉󰀆󰀇.󰀆󰀇󰀉󰀉󰀉Prq󰀁󰀉󰀉@󰀁@q@~u󰀉󰀉þ‘󰀁󰀉󰀉@󰀁@qy󰀉󰀉t;o1þ~T󰀉q󰀁󰀉󰀉󰀆@@cp@q󰀁󰀇.󰀆y@~T󰀇󰀉@y󰀉󰀉þ~T󰀉󰀉󰀆@cp󰀇.󰀆y@~T󰀇󰀉@y@y󰀉󰀉‘󰀉q󰀉󰀉@cp󰀉@y󰀉󰀉þ‘󰀉q󰀁󰀉󰀉@󰀁q󰀉󰀉‘~T󰀉󰀉󰀁󰀆p󰀉@yþcp@y󰀉þq󰀁c󰀉@cq@y@~T󰀇À1󰀉@y󰀉p󰀉@y󰀉󰀉ð24ÞWater, Upward, P = 26 MPa, G = 7.9 kg/m2s, q = 542.5 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=3.5 m, MK 750700Tpc, 661.53 KK,T650600 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb Calculation, Taxis5500.00.51.01.52.02.53.03.54.0x, m(a) 10000 y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.53.03.54.0x, m(b) Fig.4.CaseW1-a,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=26MPa,G=7.9kg/m2s,q=542.5kW/m2.DatafromLietal.[13].Inapracticalsense,thetermsincludingthemixinglengthinEq.(24)canbeneglected.ThevalueofPrt,oapproachesunityorevenalargenumberwhenthevalueofy+issmallerthan10.Intheviscosity-dominatedregion,turbulentheatﬂuxwillalsobesuppressedasmuchastheturbulentmomentumtransfer.Therefore,itwasassumedthatthePrtdiesdowntowardthewall,followingthesamemanneroftheturbulentmomentumtransfer.Theaboveargumentallowsustointroducethefollowingfunction,whichisexactlythesameasthedampingfunctionoriginallyproposedbyVanDriest,astheﬁrstfactortobeaddedtothevariablePrt¼1Àexp󰀄,o.Àþ󰀅fy1Aþð25ÞConsideringthatallgradientsinﬂowvariablesandﬂuidproper-tieswillbereducedtozeroaroundthetubecenterline,Prtisexpectedtoapproachastandardvalue,say,rt,inthisdirectionasitdoestowardthewall.Therefore,thefollowingfunctionisintroduced󰀁totake󰀄thisassumption󰀅󰀃intoaccount.f51þtanhBÀyþ2¼0:10ð26ÞWater, Upward, P = 26 MPa, G = 7.9 kg/m2s, q = 677 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.5 m, MK 750700KT,pc, 661.53 KT650600 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb Calculation, Taxis5500.00.51.01.52.02.53.0x, m(a) 10000 y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.53.0x, m(b) Fig.5.CaseW1-b,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=26MPa,G=7.9kg/m2s,q=677kW/m2.DatafromLietal.[13].798Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806Theconstant10isanarbitraryvaluegiventomakesurethatthefunctionf2variessmoothlyaroundyþ¼B.ThevalueofBshouldbecarefullydeterminedtomakesurethatbeyondthepointofyþ¼B,theﬂowisinthewakeregion.PrtderivedhereisvalidonlyintheTBLandnotinthewakeregion,wherethemixinglengththeoryisnolongervalid.InaTBL,thelog-lawandwakeregionareseparatefromeachotherataroundyþ¼B,andthemomentumdiffusivityasymptoticallyreachesaconstantvalueasitentersthewakeregion,ascanbefoundinthetextbookdealingwithTBL[34,37].Thereisnoreasonnottobelievethatthemechanismofturbulentheatexchangediffersfromthatofthemomentumexchange.TheexperimentaldataprovidedbyQuarmbyandQuirk[21]andDaietal.[22]showthatitmaybenaturaltoassumethatPrtasymptot-icallyconvergesto0.9asyapproachesR,saythetubecenterlineoraxis.InthecaseofTBLonaﬂatplate,theboundaryconditionforeattheTBLouteredgeisgivenasafreestreamvalue.ThiscannotbethecaseforaTBLinapipe,sinceeiscalculatedasapartofthesolutionratherthanbeingdeﬁnedorgiven.Ascanbeseeninref-erences[38,39],mt=masymptoticallyapproachesaconstantvalue,wheretheoreticallyeverypropertyorvariableshouldhavenearlyzerogradients.ItisstronglyindicatedthatPrthasaconstantvalueWater, Upward, P = 26 MPa, G = 7.9 kg/m2s, q = 759 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.5 m, MK 800 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb750 Calculation, TaxisK700,TTpc, 661.53 K6506005500.00.51.01.52.02.53.0x, m(a) 10000 y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.53.0x, m(b) Fig.6.CaseW1-c,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=26MPa,G=7.9kg/m2s,q=759kW/m2.DatafromLietal.[13].beingclosetoaconventionalvalueof0.9;thisisconsistentwiththefactthatPrtasymptoticallyapproachesaconstantinTBLonaﬂatplatewithzeropressuregradient[40].Finally,withtheincorporationofthefunctionsintroducedabove,thevariablePrttakesthefollowingform.Prt¼rtÀf1f2ðrtÀPrt;oÞð27ÞEq.(27)wastestedinthepresentnumericalsimulationofvar-iousexperimentaldatawithwater,carbondioxideandR22,espe-ciallythosedataplaguedwithheattransferdeteriorationaccompanyingasuddenincreaseinwalltemperature.2.3.NumericalprocedureThepresentnumericalstudywasperformedusinganin-housecode,aversionmodiﬁedfromtheoneprovidedbyFerzigerandPeric[41].Basically,theSIMPLEalgorithmwithasinglepressurecorrectionstepwasapplied.Allvariableswereassignedtothecol-locatedgrids.Diffusiveﬂuxeswerediscretizedbybusingthepowerlawscheme;while,forconvectiveﬂuxes,linearinterpola-tionwasallowedtobeblendedwithanupwindapproximation.TheresultingmatrixofthevariableswasiterativelysolvedbyWater, Upward, P = 26 MPa, G = 7.9 kg/m2s, q = 918.1 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9, L0=0.5 m, LQ=2.5 m, MK Experiment, Tw Experiment, Tb Calculation, T+900w (TBL_edge: y=300) Calculation, Tb Calculation, Taxis Calculation, Tw (TBL_edge: r=0.8R)800K,T700Tpc, 661.53 K6000.00.51.01.52.02.53.0x, m(a) 10000 y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.53.0x, m(b) Fig.7.CaseW1-d,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=26MPa,G=7.9kg/m2s,q=918kW/m2.DatafromLietal.[13].Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806799Stone’sSIP(stronglyimplicitprocedure)method.Thecomputa-tionalobjectwasaverticalupwardﬂowofwaterorcarbondioxideorR22inuniformlyheatedcirculartubeswithseveraldifferentdiameters.TheconﬁgurationofthetubeisshowninFig.2.Toobtainathermallyanddynamicallyfullydevelopedturbu-lentvelocitydistributionattheinletoftheheatedsection,aveloc-itydistributionproportionaltoa1/7powerofreducedradiuswasassumedatthetubeentrance;andasufﬁcientlylongﬂowdevelop-ingregionwithalengthof0.5mwasprovidedinfrontoftheheatedsection.Thelengthoftheheatedsectionwascontrolledtoinsurethattheﬂuidtemperatureattheendissufﬁcientlyhigherthanthecriticaltemperature.Thecalculationdomainwasdiscretizedintorectangulargrids.Thegridwasreﬁnedintothewallintheradialdirection.Aftertry-ingseveralvaluesofyþvalueofy+P(theattheﬁrstnodefromthewall),yþP<0:5wasfoundtobeoptimumwitharesultofreason-ablyaccurateconvergedsolutions.Whentheheatﬂuxisveryhigh,thesituationgetsmorecomplex.Asthewalltemperatureapproachesthepseudo-criticaltemperature,anoscillationappears.Thisisconsideredpurelynumericalnoiseandhasnothingtodowithphysics.AﬁnermeshassmallasyþP¼0:002reducedtheamplitudeoftheoscillation,butdidnotremoveitcompletely,Water, Upward, P = 25 MPa, G = 473 kg/m2s, q = 471 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT= 9, L0=0.5 m, LQ=3.0 m1000 Experiment, Tw Experiment, Tb Calculation, Tw , modified TBL outer edge Calculation, T+900w, TBL outer edge: yr=0.8R Calculation, Tb Calculation, Taxis800K,T700Tpc, 661.53 K6005000.00.51.01.52.02.53.03.5x, m(a) 10000 y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.53.03.54.0x, m(b) Fig.8.CaseW2,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=25MPa,G=473kg/m2s,q=471kW/m2.DatafromLietal.[13].resultinginaconvergenceproblem.Theoscillationwassuppressedbyintentionallylimitingtheunrealisticsuddentemperaturejumpintheimmediateneighborcell,suchthat,½TÀ0:5ðTfrontþTrearÞ󰀃<5K.Theadjustmentlikethisispurelyarbitrarywithoutanyphys-icalbackground,andshouldbedeterminedcasebycase.However,thisdidnotaffecttheoverallcalculations,whileconsiderablyalle-viatingtheoscillation.Thenumberofradialnodesgreaterthan50didnotconspicu-ouslyimprovethepredictions,whilerequiringsubstantiallyalongertimeforconvergence.Theaxialgriddependencywasinves-tigatedbycomparing50,100and150nodesforthecaseofcarbondioxidewithmassﬂuxof400kW/m2andheatﬂuxof50kW/m2(caseC2inTable2),andtheresultsareshowninFig.3.Theresultswith100and150axialnodesdidnotshowanynoticeableimprovementascomparedwiththeresultwith50nodes.SincethepurposeofthisworkwasnottoobtainaccuratenumericalresultsbuttoexaminetheapplicabilityofnewlydevelopedPrt,agrid,100(A)Â50(R),whichallowsreasonableaccuracywhilerequiringminimumcomputationalworks,waschosen.GriddependencycheckedforthecaseofwaterwasnearlysimilartothecaseofcarbondioxideandR22.Generallyspeaking,theWater, Upward, P = 24.5 MPa, G = 380 kg/m2s, q = 360 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9, L0=0.5 m, LQ=1.5 m, MK 700 Experiment, Tw Experiment, Tb Calculation, Tpc, 663.4 Kw Calculation, TTb Calculation, Taxis600K,T5004000.00.51.01.52.0x, m(a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R)1000 Sublayer thicknessy+100100.00.51.01.52.0x, m(b) Fig.9.CaseW3,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=24.5MPa,G=380kg/m2s,q=360kW/m2.DatafromShitsman[43].800Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806calculationsstronglydependontheradialnumberofnodesandweaklyontheaxialone.TheﬂowconditionsexaminedinthisworkaresummarizedinTable2.Allcasesselectedforthenumericalexperimentinthispaperareforheattransferdeterioration,sinceithasbeenprovedthattheexperimentaldatawithoutheattransferdeteriorationcanbesimulatedquiteaccuratelywiththewell-knownturbulencemodelsemployingaconstantPrt[16,19].ItshouldbenotedthattheReynoldsnumbersattheinletforallcasesareconsideredintherangeoffullyturbulentﬂow.Thefollowingconvergencecriterionwassetasthesumofthenetﬂuxofeachﬂowvariable(exceptpressure)normalizedbythesumoflocalﬂuxbeinglessthan10À6.However,forsomecases,theconvergencecriterionisrelievedtoalargervalueof10À3sinceitcouldnotbesatisﬁedowingtotheoscillation.Nevertheless,attheoutlet,thevelocityandenthalpywerecontinuallyrenewedsothatthemassandenergyconservationsweresatisﬁedwithin1%error.TheﬂuidpropertieswerecalculatedfromatableconstructedfromtheNISTstandardreferencedatabase[47]withasuitableintervalofenthalpysothatthetemperatureatthevicinityoftheWater, Upward, P = 23.3 MPa, G = 430 kg/m2s, q = 320 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9, L0=0.5 m, LQ=2.0 m, MK 800 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb Calculation, TaxisK,T700Tpc, 651.7 K6000.00.51.01.52.02.5x, m (a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100.00.51.01.52.02.5x, m (b) Fig.10.CaseW4,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=23.3MPa,G=430kg/m2s,q=320kW/m2.DatafromShitsman[43].pseudo-criticaltemperature(Tpc)isresolvedasﬁnerthan0.1K.Theﬂuidpropertieswerelinearlyinterpolatedfromthetable.3.Resultsanddiscussions3.1.WaterInthissection,theresultsofanumericalsimulationontheexperimentsperformedusingwaterasaworkingﬂuidarepre-sented.Thedataweredigitizedfrompublishedpapers.Fig.4showsthetemperaturedistributionandthevalueofy+alongthetubewallandaxisundertheconditionsofp=26.5MPa,G=7.9kg/m2s,q=542.5kW/m2s.Inthetitleofplate(a)theacronymGGDHimpliesthegeneralizedgradientdif-fusionhypothesis,andJPRT=9thecurrentlyusedformulaforPrt.InallgraphssimilartoFig.4,thesolidlinesfromthetoprep-resentthewall,bulkandcenterlinetemperature,respectively.ConsideringthatthevalueofA+isabout26forafullydevelopedTBLonaﬂatplatwithoutapressuregradientand70inalow-Reynoldsnumberturbulencemodel,thevalueofA+isgivenasWater, Upward, P = 26.5 MPa, G = 493 kg/m2s, q = 570 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9, L0=0.5 m, LQ=6.0 m, MK Experiment, Tw Experiment, Tb1000 Calculation, Tw, modified viscous sublayer Calculation, T+w, A=70 900 Calculation, Tb Calculation, Taxis800KT,700pc, 663.4 KT6005004003000123456x, m (a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100100123456x, m (b) Fig.11.CaseW5,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=26.5MPa,G=493kg/m2s,q=570kW/m2.DatafromVikhrevetal.[44].Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–80630880125620u+Prt415210 x, m 0.3 (unheated) 0.6u+ = 5.6 log(y+)+4.9 0.9(u+ = ln(y+)/0.41+5.0) 1.2 1.5 1.8 2.1 2.4u+ = y+ 2.7 3.01.00.80.60.40.20.0u, m/s5000.0000.0040.111010010000.1110y+1001000r, my+(a)(b)(c)Fig.12.Distributionofu,u+andPrtalongthetube.p=26MPa,G=7.9kg/m2s,q=918.1kW/m2.DatafromLietal.[13].26and70intheregionsoftheunheatedandheatedpartsofthetestsectionforallcalculations.Theagreementbetweenthecalcu-lationandmeasurementisextremelygood.TheagreementwasrelativelypoorintheregionnearthepointoftheheatingfrontCO2, Upward, P = 8.12 MPa, G = 400 kg/ms, g = 30 kW/m32022end.Thisisprobablyduetothefactthatthethermalboundarybeginsdevelopingfromthepointofheatinputinthecalculation,whileitisalreadydevelopedtosomeextentduetothethermalconductioninthedirectionofthetubeinlet.Asthethermalbound-Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.0 m, MK400CO2, Upward, P = 8.12 MPa, G = 400 kg/ms, g = 50 kW/m22Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.0 m, MK310Tpc, 308.4 K380360 Experiment, Tw Calculation, Tw Calculation, Taxis Experiment, Tb Calculation, TbT,K300T,K290340320300Tpc, 661.53 K Experiment, Tw Calculation, Tw Calculation, Taxis0.00.51.01.52.0 Experiment, Tb Calculation, Tb2802.53.0x, m2800.00.51.01.52.02.5(a) y+ at umax1000 y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness y+ at umaxx, m(a) y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000y+100y+100100.00.51.01.52.02.53.0100.00.51.01.52.02.5x, mx, m(b)Fig.13.CaseC1,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=8.12MPa,G=400kg/m2s,q=30kW/m2.DatafromBaeetal.[45].(b)Fig.14.CaseC2,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=8.12MPa,G=400kg/m2s,q=50kW/m2.DatafromBaeetal.[45].802Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806arylayerdevelops,thewalltemperatureincreases.TheouteredgeoftheTBLwasdeﬁnedasthelineofy+correspondingtor=0.8R,hereinafterreferredtoasyþr¼0:8R[38].Beyondthisline,theregionbecomesawakeregionwherethetheoryoftheTBLnolongerholds,andaccordingly,Prtdevelopedherecannotbeapplied.ThisdeﬁnitionoftheTBLouteredgeworksverywellwhentheﬂowremainsinastateofnormalormildheattransferdeterioration.Whenheattransferdeteriorationbecomessevere,yþr¼0:8RnolongerrepresentstheTBLouteredge,andtheedgemovesintotheTBL.ItcanclearlybeseeninFig.5(b)whencomparedwithFig.4(b).Anopensquarerepresentsthelocationof@u=@y¼0,hereinafterreferredtoasyþupeak.Astheheatﬂuxincreasesfrom542.5to677kW/m2s,asshowninFig.5,theheattransferdeteriorationisseenapparently;andyþuþpeakgetsclosertoyr¼0:8R.Theresultwithastrongerheatﬂuxofq=759kW/m2swiththesamemassﬂuxisshowninFig.6.Asexpected,theheattransferdeteriorationismoreapparent.Itshouldbenotethatyþr¼0:8RstillplaysaroleoftheTBLouteredge.However,whentheheatﬂuxincreasestoq=918kW/m2s(seeFig.7),yþupeakcloselyapproachesyþr¼0:8R.Inthiscasethewakeregionpenetrateswellbeyondyþr¼0:8R;andthislinenolongerrepresentstheTBLouteredge.OnceCO222, Upward, P = 8.12 MPa, G = 600 kg/ms, g = 90 kW/mGrid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.0 m, MK Experiment, Tw Experiment, Tb400 Calculation, Tw Calculation, Tb Calculation, TaxisK,T350300Tpc, 306.4 K0.00.51.01.52.02.5x, m(a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R)1000 Sublayer thicknessy+100100.00.51.01.52.02.5x, m(b) Fig.15.CaseC3,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=8.12MPa,G=600kg/m2s,q=90kW/m2.DatafromKAERI’sunpublisheddata.avelocitypeakpenetratestheTBL,thedownstreamwillexperi-encethedisturbancesresultingfromthepenetration.Foraremedyofthisunacceptablesituation,yþTBL¼300wascho-senasthelineoftheTBLouteredgeoncetheTBLwaspenetratedbyyþuThecasewithyþpeak.TBL¼300apparentlyagreeswiththeexperimentaldatabetterthanthecasewiththelineofr=0.8R.ThelinesofyþTBL¼300andyþr¼0:8Rwillmergesufﬁcientlyfardown-streamwheretheﬂowﬁeldreturnstotheonecorrespondingtoatypicalconstant-propertyturbulentﬂowinacirculartube.IftheboundarybetweenTBLandwakeregionispreciselydeﬁned,amoreaccuratepredictionofthewalltemperaturecanbeachieved.However,atthemoment,noexperimentalinformationisavailableforanaccurateestimationoftheboundarybetweentheTBLandwakeregion.IncasesW1b–d,thereproductionofthewalltemperaturerecoveryafteraheattransferdeteriorationwasinsurprisinglygoodagreementwiththeexperimentaldata.Thusfar,wehavediscussedthecasesofnormalormildheattransferdeterioration.Wenowexaminethecaseswithrelativelysevereheattransferdeterioration.Fig.8showstheresultsforthecaseofp=25MPa,G=473kg/m2sandq=471kW/m2.SevereheattransferdeteriorationisCO22, Upward, P = 7.75 MPa, G = 500 kg/ms, g = 60 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9 , L0=0.5 m, LQ=2.0 m, MK400 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb Calculation, Taxis350K,T300Tpc, 661.53 K0.00.51.01.52.02.5x, m(a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R)1000 Sublayer thicknessy+100100.00.51.01.52.02.5x, m(b) Fig.16.CaseC4,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=7.75MPa,G=500kg/m2s,q=60kW/m2.DatafromKAERI’sunpublisheddata.Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806803apparent.Ascanbeseeninthelowerplate(b),thelineyþupeakpen-etrateddeepintotheTBL.ThispenetrationwillobviouslyalterthevelocityproﬁleintheTBLtoresultinthemodiﬁcationofcorre-spondingthermalbehavior.ThewalltemperaturedistributionbasedonyþTBL¼yþr¼0:8Rdidnotagreewellwiththeexperimentaldata.ThisdiscrepancymightstemfromthefactthatyþTBL¼yþr¼0:8RcorrespondstothelocationwellbeyondtheTBLedgeanddeepinthewakeregion.TheeffectoftheTBLlocationwillbediscussedfurtherinSections3and4.InFig.9,caseW3ofp=24.5MPa,G=380kg/m2sandq=360kW/m2isshown.ThewalltemperaturesremainedunderTpc.Thecalculatedresultqualitativelyagreeswiththeexperimen-taldata.Itshouldbenotedthatthecalculationdidnotpickuptheﬁrstpeakoftheexperiment.Itispresumedthatthepeakisweaklyrelatedtothestrongpropertyvariation,sinceitappearedwellbelowTpc.InFig.10,caseW4ofp=23.3MPa,G=430kg/m2s,q=320kW/m2isshown.Theagreementisqualitativelygood,butitwasnotabletocapturetheextremelyhighvalue.Thecaseofthemostsevereheattransferdeterioration,caseW5ofp=26.5MPa,G=493kg/m2s,q=570kW/m2,isshowninFig.11.Again,asincaseW3showninFig.9,theﬁrstpeakwasnotcap-tured,whilethesecondpeakwascapturedqualitativelywell,althoughthelocationofthepeakdidnotcoincide.Anextrasimu-lationwasconductedforcaseW5withadomainincludingonlytheR22, Upward, P = 5.5 MPa, G = 400 kg/m2s, q = 10 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT= 9, JTBL=1 , L0=0.5 m, LQ=3.0 m385 Experiment, Tw Experiment, Tb380 Calculation, Tw Calculation, Tb Calculation, TaxisK375Tpc 374.5 K,T3703650.00.51.01.52.02.53.03.5x, m(a) umax did not appear y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000+y100100.00.51.01.52.02.53.03.5x, m(b) Fig.17.CaseF1,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=5.5MPa,G=400kg/m2s,q=10kW/m2.DatafromMorietal.[46].ﬁrstpeak,reducingtheaxialheatedlengthofthedomainfrom6mto1m.Itiscommensuratetoagridreﬁnementsincewemain-tainedthenumberofaxialgrids.Thisgridreﬁnementdidnotcap-turetheﬁrsteither,anditmayindicatethatthepeakappearinginthiscasedidnotoriginatefromaseverepropertyvariation.ItshouldbenotedthattheﬁrstpeakappearedincasesW3andW5didnotappeareitherinotherwaterexperimentsorintheCO2experiments.Thetypicaldistributionsofu,u+andPrtforcaseW1-dareshowninFig.12.Ascanbeseeninplate(a),thevelocityuattheexitstillshowsanovershoot.Thelow-densityregionwithanover-shootinitiallyappearedintheregionveryclosetotheviscoussub-layerexpandsintheradialdirectionastheﬂuidmovesdownstream.Thevelocitydistributionwilleventuallyresumeatypicaloneforaturbulentboundarylayerintheregionsufﬁcientlyfardownstream.Inplate(b),thedistributionofu+asafunctionofy+isshown.Asﬂuidenterstheheatedsection,theu+proﬁlesud-denlystartstodeviatefromthestandardlog-lawproﬁle.Astheﬂuidmovesfurtherdownstreamtotheexit,theu+proﬁlereturnstothestandardone,whileinthiscase,theu+proﬁleattheexitstillshowedtheinﬂuenceofthevelocityovershoot.Thedecreaseinthevalueofu+nearthecenterlinecorrespondstothevelocityover-shootandincreaseoffrictionvelocityduetodensitydecrease.ThedistributionofPrtisshowninplate(c).ThePrtshowedaveryR22, Upward, P = 5.5 MPa, G = 400 kg/m2s, q = 25 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT= 9, JTBL=1 , L0=0.5 m, LQ=4.0 m450 Experiment, Tw Experiment, Tb Calculation, Tw Calculation, Tb Calculation, Taxis400Tpc 374.5KK,T35030001234x, m(a) y+ at umax y+ at tube axis1000 y+ at TBL edge (r=0.8R) Sublayer thickness+y1001001234x, m(b) Fig.18.CaseF2,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=5.5MPa,G=400kg/m2s,q=25kW/m2.DatafromMorietal.[46].804Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806smallvalueinthelog-lawregion,anditgraduallyreturnedtoastandardvalueof0.9towardthewallandcenterline.TheinnerandouterboundariesofthelowPrtregionstronglyinﬂuencedthenumericalsimulation,andafurtherstudyisrequiredforamorerobustsimulation,althoughcurrentlyusedboundaries,A+=70andyþr¼0:8R,servedverywellforthepurposeofthispaper.3.2.CarbondioxideInthissection,theresultsofanumericalsimulationfortheexperimentsperformedusingcarbondioxideasaworkingﬂuidarepresented.Thecalculationresultsforcarbondioxidearelittledifferentfromthoseforwater.Acaseofnearlynormalheattransferwiththeconditionsofp=8.12MPa,G=400kg/m2s,q=30kW/m2isshowninFig.13.ThecalculationcloselyfollowedtheexperimentaldatawithaslightoverestimationintheregionwhereTbulkiscloseTpc.Ascanbeseeninthelowerplate(b),avelocitypeakappeared.However,thevaluesofyþupeakweretoolargerthanthatforr=0.8Rtoexertanyinﬂuenceonthethermalbehavior.Fig.14showsacasewithamoresevereheattransferdeterioration.Astheheatﬂuxincreasedto50kW/m2,withthesamemassﬂux,aseverewallR22, Upward, P = 5.5 MPa, G = 1000 kg/m2s, q = 90 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT= 9, JTBL=1 , L0=0.5 m, LQ=2.5 m Experiment, T Experiment, T440wb Calculation, Tw Calculation, Tb420 Calculation, Taxis400KTpc, 374.5 K,T3803603403203000.00.51.01.52.02.53.0x, m(a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness1000+y100100.00.51.01.52.02.53.0x, m(b)Fig.19.CaseF3,(a)temperaturedistribution,(b)distributionofy+alongthetubewallandaxis.p=5.5MPa,G=1000kg/m2s,q=90kW/m2.DatafromMorietal.[46].temperatureriseappeared.Intheregionofthefrontendheating,yþuasmallvalueofyþpeakreachedwellbelow70.Suchupeakwillinﬂu-encetheﬂuid–thermalbehaviorintheTBLanywaybymodifyingit,asdiscussedearlierforcaseW2.yþTBL¼yþr¼0:8Rworkedverywell,andnomodiﬁcationwasneeded,differentfromcaseW2.Toseetheeffectofdifferentmassandheatﬂux,caseC3ofG=600kg/m2s,q=90kW/m2wassimulated,andtheresultsareshowninFig.15.Thecalculatedwalltemperaturesagreeverywellwiththeexperimentaldata.AscanbeseenincasesC2andC3,theaccu-racyofreproducingtherecoveriesfromtheheattransferdeterio-rationwassurprising.CaseC4,whoseresultsareshowninFig.16,waschosentoinvestigatetheeffectofpressure.Thepres-sureincaseC4is7.75MPa,whichisclosertotheTpcthanthatofcasesC1–C3.Theresultsagreedwellwiththeexperimentaldata,asincasesC1–C3.3.3.Freon(HCFC22)Inthissection,theresultsofanumericalsimulationfortheexperimentsperformedusingR22,atypeofFreon,astheworkingﬂuidsarepresented.Thedataweredigitizedfromthepaperpre-Water, Upward, P = 25 MPa, G = 473 kg/m2s, q = 471 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT= 9, L0=0.5 m, LQ=3.0 m950 Experiment, Tw Experiment, Tb Calculation, T, TBL and A+w modified 900 Calculation, Tw, Original Calculation, T850b Calculation, Taxis800K,T750700650Tpc, 661.53 K6005500.00.51.01.52.02.53.03.54.0x, m(a) y+ at umax y+ at tube axis y+ at adjusted TBL edge y+ at 0.8R Sublayer thickness1000y+100100.00.51.01.52.02.53.03.54.0x, m(b) Fig.20.CaseW2.EffectofthemodiﬁcationofsublayerthicknessandTBLouteredge.p=25MPa,G=473kg/m2s,q=471kW/m2.DatafromLietal.[13].Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806805sentedbyMorietal.[46].In(b)Fig.17,theresultsforcaseF1ofp=5.5MPa,G=400kg/m2sandq=10kW/m2areshown.Inthiscase,theoverallperformanceofthenumericalsimulationisfairlygoodasexpectedforthenormalheattransfer.Itshouldbenotedthatvelocityovershootdidnotappearinthiscase.Astheheatﬂuxincreasedto25kW/m2,whileotherconditionsremainedthesame,asuddenwalltemperatureriseappearedwhenthebulktempera-tureapproachedTpc,asshownin(b)Fig.18.Theoverallperfor-mancewasreasonablygood;however,exceptfortheregionclosetothetemperaturepeak,thesimulationsigniﬁcantlyunder-estimatedtheexperimentaldata.TheresultsforcaseF3ofp=5.5MPa,G=1000kg/m2sandq=90kW/m2areshownin(b)Fig.19.Inthiscase,thesimulationgenerallyfollowedtheexperi-ment,butanunderestimationwasapparent.InallcasesforﬂuidR22,theapplicabilityofthenewlydevelopedPrtwasalsosuccess-fullyconﬁrmed.3.4.EffectofsublayerthicknessandTBLouteredgeInthissection,theeffectsofthemodiﬁcationoftheviscoussub-layerthickness,A+,andtheTBLouteredge,yþTBL,arediscussed.InWater, Upward, P = 26.5 MPa, G = 493 kg/m2s, q = 570 kW/m2Grid: 100 (L) x 50 (R), GGDH, JPRT=9, L0=0.5 m, LQ=6.0 m, MK Experiment, Tw Experiment, Tb1000 Calculation, Tw, modified viscous sublayer Calculation, T+w, A=70 900 Calculation, Tb Calculation, Taxis800KT,700pc, 663.4 KT6005004003000123456x, m(a) y+ at umax y+ at tube axis y+ at TBL edge (r=0.8R) Sublayer thickness modified Sublayer thickness1000y+100100123456x, m(b) Fig.21.Effectofthemodiﬁcationofsublayerthickness.p=26.5MPa,G=493kg/m2s,q=570kW/m2.DatafromVikhrevetal.[44].Fig.20,theeffectsofthemodiﬁcationofbothA+andyþTBLareshownforcaseW2.Sincethelineofyþupeakpenetratedtheviscoussublayer,itwassuspectedthatbothA+andyþTBLweresigniﬁcantlychangedfromthenormalstate.yþTBLwasnewlydeﬁnedsuchthatitranwiththelineofyþuþpeakwhenyupeak<70andasymptoticallyandslowlyapproachedyþr¼0:8Rasshowninthelowerplate.Thismodiﬁcationistotallyarbitraryandhastobeveriﬁedlaterexperimentally.ThevalueofA+wasalsomodiﬁedtoreﬂecttheseverepenetrationoflineyþupeakintotheviscoussublayersuchthatithadthesamevalueofyþuþpeakwheneveryupeakwassmallerthan70;otherwise,itremainedas70.Thewalltemperaturedistributionsindicatedbysolid(modiﬁed)anddashed(original)linescorrespondtothetwodifferentyþTBLindicatedbythedarkyellow(modiﬁed)andblue(original:yþTBL¼yþr¼0:8R)triangles,respectively.TheresultwiththemodiﬁedyþTBLandA+agreedmuchbetterwiththeexperimentaldata.However,thisadjustmentusinganadhocintroductionofnewdeﬁnitionofyþTBLmightbenotenoughtoexplainthecompli-catedbehaviorintheTBLwithstrongbuoyancy.Thefactorscon-trollingtheﬂuidthermalbehaviorintheTBLincludenotonlythedeﬁnitionoftheTBLouteredgebutalsomanyothersincludingturbulenceproperties.Theresultsofthecomparisonhere,how-ever,convincinglyindicatethatthevaluesofyþ+TBLandAhaveastronginﬂuenceontheﬂuid–thermalbehaviorandeventuallyonthewalltemperature.ItshouldbenotedthatthemodiﬁcationofeitheryþorA+TBLdidnotresultintheexpectedresults.Theinﬂuenceofthevalueofyþ+TBLandAonanothercase,W5,wasalsoexamined.In(b)Fig.21,theeffectofmodiﬁcationofonlyA+isshown.Clearly,themodiﬁcationresultedinabetteragree-mentbetweenthecalculationandexperiment.Theabovetwocaseswerepresentedsolelyforthepurposeofshowingtheinﬂuencesofthemodiﬁcationofyþ+TBLandA,whichwereindeedverystrong.However,itsphysicalbackgroundisstilltobeproventhroughfurthernumericalandexperimentalefforts.ItishopedthatadetailedstudyoftheTBLseverelymodiﬁedbytheirregularvelocitydistributionsuchasM-shapevelocitydistribu-tionwillrevealanycauses,ifany,forthesuddentemperaturejumpanddropinﬂuidsﬂowingunderconditionsofsupercriticalpressureandtemperature.4.ConclusionsThevalueofPrthaslongbeenconsideredtobeunityorclosetoit.Mostofthenumericalworkstopredictthewallandﬂuidtem-peratureﬂowingintubesandchannelsatsupercriticalpressureshavefailedorwerepartiallysuccessfulinalimitedregionofmildpropertyvariation.Inparticular,attemptstosimulatetherecoveryfromthedeterioration,occurringinasupercriticalheattransfer,werefailedwithoutexception.Theauthorsuspectedthatoneofthemajorreasonsforthefailureorinadequacywasaninappropri-atenessofPrt.SeveralexperimentaldataandnumericalresultsalsoindicatedthatPrtcanbeverysmallerorlargerthanunityinaregionofseverepropertyvariation.Inthispaper,anattempthasbeenmadetoformulateanewvariablePrtwithconsiderationoftheeffectofseverephysicalpropertyvariationaswellastheﬂowproperties.Theformulationwasbasicallybasedonthewell-knownmixinglengththeory.ThedevelopedvariablePrtwasappliedtothenumericalestimationoftheavailableexperimentaldataforwater,carbondioxideandR22.Thecalculationresultsagreedsurprisinglywellwiththeexperimentaldata.TherecoveryofheattransferfromdeteriorationthathasneverbeenreproducedwasalsosuccessfullyreproducedinthenumericalcalculationemployingthenewlydevelopedPrt.TheapplicationofnewlydevelopedPrtwouldnotbelimitedtothecaseofsupercriticalpressureﬂuids;andratheritcanbe806Y.Y.Bae/InternationalJournalofHeatandMassTransfer92(2016)792–806appliedforanynumericalsimulationswithﬂuidsexperiencingaseverepropertyvariation.Theclaimsputforwardherearebasedpurelyonthespecula-tionandnumericalexperimentswithoutanyexperimentalevi-dences,whicharestronglyanticipatedinthenearfuturetochallengetheclaimsgivenhere.ConﬂictofinterestNonedeclared.AcknowledgmentsTheauthorwouldliketoacknowledgetheﬁnancialsupportbytheNuclearResearch&DevelopmentProgramoftheNationalResearchFoundationofKorea(NRF)grantfundedbytheKoreangovernment(MSIP).(Grantcode:NRF-2012M2A8A2025682).Dr.J.H.Bae’sreleaseofhisvaluableDNSdatashouldalsobeappreciated.References[1]AtechnologyroadmapforgenerationIVnuclearenergysystems,GIF-002-00,USDOENuclearEnergyAdvisoryCommitteeandtheGenerationIVInternationalForum,December2002.[2]X.Cheng,T.Schulenberg,HeatTransferatSupercriticalPressures—LiteratureReviewandApplicationtoanHPLWR,WissenschaftlicheBerichte(Tech.Report)FZKA6609,ForschungszentrumKarlsruhe,Mai,2001.[3]I.L.Pioro,R.B.Duffey,Experimentalheattransferinsupercriticalwaterﬂowinginsidechannels(survey),Nucl.Eng.Des.235(22)(2005)2407–2430.[4]M.J.Watts,C.T.Chou,Mixedconvectionheattransfertosupercriticalpressurewater,in:Proceedingsofthe7thInternationalHeatTransferConference,München,1982,pp.495–500.[5]J.D.Jackson,W.B.Hall,InﬂuencesofBuoyancyonheattransfertoﬂuidsﬂowinginverticaltubesunderturbulentcondition,in:S.Kakaç,D.B.Spalding(Eds.),TurbulentForcedConvectioninChannelsandBundles,HemispherePublishing,1979,pp.613–0.[6]J.D.Jackson,M.A.Cotton,B.Axcell,Studiedofmixedconvectioninverticaltubes,Int.J.HeatFluidFlow10(1)(19)2–1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