Wall Boiling Models.docx
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WallBoilingModels
17.5.16. WallBoilingModels
17.5.16.1. Overview
Theterm“subcooledboiling”isusedtodescribethephysicalsituationwherethewalltemperatureishighenoughtocauseboilingtooccuratthewalleventhoughthebulkvolumeaveragedliquidtemperatureislessthanthesaturationvalue.Insuchcases,theenergyistransferreddirectlyfromthewalltotheliquid.Partofthisenergywillcausethetemperatureoftheliquidtoincreaseandpartwillgeneratevapor.Interphaseheattransferwillalsocausetheaverageliquidtemperaturetoincrease,however,thesaturatedvaporwillcondense.Additionally,someoftheenergymaybetransferreddirectlyfromthewalltothevapor.ThesebasicmechanismsarethefoundationsofthesocalledRensselaerPolytechnicInstitute（RPI）models.
InANSYSFLUENT,thewallboilingmodelsaredevelopedinthecontextoftheEulerianmultiphasemodel.Themultiphaseflowsaregovernedbytheconservationequationsforphasecontinuity（Equation 17–119）,momentum（Equation 17–120）,andenergy（Equation 17–126）.ThewallboilingphenomenonismodeledbytheRPInucleateboilingmodelofKurualandPodowski[190]andanextendedformulationforthedepartednucleateboilingregime（DNB）byLavievilleetal[200].
Thewallboilingmodelsarecompatiblewiththreedifferentwallboundaries:
isothermalwall,specifiedheatflux,andspecifiedheattransfercoefficient（coupledwallboundary）.
Specificsubmodelshavebeenconsideredtoaccountfortheinterfacialtransfersofmomentum,mass,andheat,aswellasturbulencemodelsinboilingflows,asdescribedbelow.
Tolearnhowtosetuptheboilingmodel,pleaserefertoIncludingtheBoilingModel.
17.5.16.2. RPIModel
AccordingtothebasicRPImodel,thetotalheatfluxfromthewalltotheliquidispartitionedintothreecomponents,namelytheconvectiveheatflux,thequenchingheatflux,andtheevaporativeheatflux:
（17–274）
Theheatedwallsurfaceissubdividedintoarea
,whichiscoveredbynucleatingbubblesandaportion
,whichiscoveredbythefluid.
∙Theconvectiveheatflux
isexpressedas
（17–275）
where
isthesinglephaseheattransfercoefficient,and
and
arethewallandliquidtemperatures,respectively.
∙Thequenchingheatflux
modelsthecyclicaveragedtransientenergytransferrelatedtoliquidfillingthewallvicinityafterbubbledetachment,andisexpressedas
（17–276）
Where
istheconductivity,
istheperiodictime,and
isthediffusivity.
∙Theevaporativeflux
isgivenby
（17–277）
Where
isthevolumeofthebubblebasedonthebubbledeparturediameter,
istheactivenucleatesitedensity,
isthevapordensity,and
isthelatentheatofevaporation,and
isthebubbledeparturefrequency.Theseequationsneedclosureforthefollowingparameters:
∙AreaofInfluence
Itsdefinitionisbasedonthedeparturediameterandthenucleatesitedensity:
（17–278）
Notethatinordertoavoidnumericalinstabilitiesduetounboundempiricalcorrelationsforthenucleatesitedensity,theareaofinfluencehastoberestricted.Theareaofinfluenceislimitedasfollows:
（17–279）
Thevalueoftheempiricalconstant
isusuallysetto4,howeverithasbeenfoundthatthisvalueisnotuniversalandmayvarybetween1.8and5.ThefollowingrelationforthisconstanthasalsobeenimplementedbasedonDelValleandKenning'sfindings[77]:
（17–280）
and
isthesubcooledJacobnumberdefinedas
（17–281）
∙FrequencyofBubbleDeparture
ImplementationoftheRPImodelnormallyusesthefrequencyofbubbledepartureastheonebasedoninertiacontrolledgrowth（notreallyapplicabletosubcooledboiling）[64]
（17–282）
∙NucleateSiteDensity
Thenucleatesitedensityisusuallyrepresentedbyacorrelationbasedonthewallsuperheat.Thegeneralexpressionisoftheform
（17–283）
HeretheempiricalparametersfromLemmertandChawla[205]areused,where
and
.Otherformulationsarealsoavailable,suchasKocamustafaogullariandIshii[184]where
（17–284）
Here
Where
isthebubbledeparturediameterandthedensityfunctionisdefinedas
（17–285）
∙BubbleDepartureDiameter
Thedefaultbubbledeparturediameter（mm）fortheRPImodelisbasedonempiricalcorrelations[190]andiscalculatedas
（17–286）
whileKocamustafaogullariandIshii[184]use
（17–287）
with
beingthecontactangleindegrees.
17.5.16.3. Non-equilibriumSubcooledBoiling
WhenusingthebasicRPImodel,thetemperatureofthevaporisnotcalculated,insteaditisfixedatthesaturationtemperature.InordertomodeldifferentboilingregimeslikeDNBandcriticalheatflux,itisnecessarytoincludethevaportemperatureinthesolutionprocess.Thewallheatpartitionisnowmodifiedasfollows:
（17–288）
Here
isthediffusiveheatfluxofthevaporbubblephase,
isthevaporheattransfercoefficientbasedonturbulentwallfunctions.Thefunction
dependsonthelocalliquidvolumefractionwithsimilarlimitingvaluesastheliquidvolumefraction.Lavievilleetal[200]proposedthefollowingexpression:
（17–289）
Here,thecriticalvalueforthevaporfractionis
17.5.16.4. InterfacialMomentumTransfer
Theinterfacialmomentumtransfermayincludefourparts:
drag,lift,virtualmassandturbulentdriftforces（alldescribedinConservationEquations,InterphaseExchangeCoefficients,andTurbulenceModels.Inthewallboilingmodels,thevirtualmassforceismodeledusingthestandardcorrelationimplementedintheEulerianmultiphasemodelwithinANSYSFLUENT,whilespecificsub-modelshavebeenimplementedfordrag,lift,andturbulentdriftforces.Also,user-definedoptionsareavailableforbothdragandliftforces.
17.5.16.4.1. InterfacialArea
Theinterfacialareaisanimportantparameterforthedragandtheheattransferprocess.Fordispersedboiling,theinterfacialarea,basedonthediameterofthebubble,wouldbeenough.However,asbubblecoalescencetakesplace,thisneedstobemodified.Thefollowingoptionsareincluded:
（17–290）
（17–291）
（17–292）
17.5.16.4.2. InterfacialDragForce
TheinterfacialdragforceiscalculatedusingthestandardmodeldescribedinInterphaseExchangeCoefficients（anddefinedinthecontextoftheinterfacialareainEquation 17–290）isofthegeneralform
（17–293）
Wherethedragcoefficient
isdeterminedbychoosingtheminimumoftheviscousregime
andthedistortedregime
,definedasfollows:
（17–294）
Thebubblediameter
canbeaconstantvalue,aUDF,oracorrelationfunctionoflocalsubcooling
[190]:
（17–295）
17.5.16.4.3. InterfacialLiftForce
ThecoefficientfortheinterfacialliftforceiscalculatedusingthecorrelationproposedbyMoragaetal.[262]:
（17–296）
Where
.Theliftcoefficientcombinestheopposingactionoftwoliftforces:
theclassicalaerodynamicsliftforceresultingfrominteractionbetweenbubbleandliquidshear,andthelateralforceresultingfrominteractionbetweenbubbleandvortexesshedbybubblewakes.Here
isthebubbleReynoldsnumber,and
isthebubbleshearReynoldsnumber.
TheformulationproposedbyTomiyamaetal.[441]isalsoavailablewiththeliftcoefficientexpressedas
,where
（17–297）
and
（17–298）
where
isthebubbleReynoldsnumberand
（17–299）
istheEötvosnumber,with
asthegravitationalaccelerationand
thesurfacetensionnumber.
17.5.16.4.4. TurbulenceDriftForce
IntheANSYSFLUENTEulerianmultiphasemodel,thegeneralcorrelationforturbulencedriftforce（turbulentdispersion）isbasedonSimonin[351].DuetonumericalinstabilitiesthisforceisnowincludedintheRhie&Chowinterpolation[323]forthevolumefluxcalculations.Simonin’s[351]approachcanbealsousedfortheboilingmodel.However,forcompletenessoftheRPImodel,thedefaultfortheturbulentdriftforceisgivenby
（17–300）
Wheretheturbulentdispersioncoefficient
is,bydefault,setto1.0
17.5.16.5. InterfacialHeatTransfer
17.5.16.5.1. VaportoLiquidHeatTransfer
Asthebubblesdepartfromthewallandmovetowardsthesubcooledregion,thereisheattransferfromthebubbletotheliquid,thatisdefinedas
（17–301）
Where
istheinterfacialareadefinedbyEquation 17–291and
istheheattransfercoefficientbasedontheRanz-Marshallcorrelation[315]
（17–302）
17.5.16.5.2. SuperheatedLiquidtoVaporHeatTransfer
Theinterfacetovaporheattransferiscalculatedusingtheconstanttimescalereturntosaturationmethod[200].Itisassumedthatthevaporretainsthesaturationtemperaturebyrapidevaporation/condensation.Theformulationisasfollows:
（17–303）
Where
isthetimescalesettoadefaultvalueof0.05and
istheisobaricheatcapacity.
17.5.16.6. MassTransfer
17.5.16.6.1. MassTransferFromtheWalltoVapor
Theevaporationmassflowisappliedatthecellnearthewallanditisderivedfromtheevaporationheatflux,Equation 17–303
（17–304）
17.5.16.6.2. InterfacialMassTransfer
Theinterfacialmasstransferdependsdirectlyontheinterfacialheattransfer.Assumingthatalltheheattransferredtotheinterfaceisusedinmasstransfer（i.e.evaporationorcondensation）,theinterfacialmasstransferratecanbewrittenas:
（17–305）
17.5.16.7. TurbulenceModels
Turbulencequantitiesdependonthemodelsselectedfortheproblem,i.e.Mixturemodel,DispersedModelorModelperphase.Fortheconventionalmixturek-epsilonmodels,twoadditionalte
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