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13 ConceptsofThermalAnalysis

13.1 IntroductionTherearethreedifferenttypesofheattransfer:conduction,convection,andradiation.Atemperaturedifferencemustexistforheattransfertooccur.Heatisalwaystransferredinthedirectionofdecreasingtemperature.Temperatureisascalar,butheatfluxisavectorquantity.

Conductiontakesplacewithintheboundariesofabodybythediffusionofitsinternalenergy.Thetemperaturewithinthebody,T,isgiveninunitsofdegreesCelsius[C],Fahrenheit[F],Kelvin[K],orRankin[R].Itsvariationinspacedefinesthetemperaturegradientvector,T,withunitsof[K/m]say.Theheatfluxvector,q,isdefinebyFouriersConductionLaw,asthethermalconductivity,k,timesthenegativeofthetemperaturegradient,q=kT.Thermalconductivityhastheunitsof[W/mK]whiletheheatfluxhasunitsof[W/m2].Theconductivity,k,isusuallyonlyknowntotwoorthreesignificantfigures.Forsolidsitrangesfromabout417W/mKforsilverdownto0.76W/mKforglass.

Aperfectinsulatormaterial(k 0)willnotconductheat;thereforetheheatfluxvectormustbeparalleltotheinsulatorsurface.Aplaneofsymmetry(wherethegeometry,kvalues,andheatsourcesaremirrorimages)actsasaperfectinsulator.Infiniteelementanalysis,allsurfacesdefaulttoperfectinsulatorsunlessyougiveaspecifiedtemperature,aknownheatinflux,aconvectioncondition,oraradiationcondition.

Convectionoccursinafluidbymixing.Herewewillconsideronlyfreeconvectionfromthesurfaceofabodytothesurroundingfluid.Forcedconvection,whichrequiresacoupledmasstransfer,willnotbeconsidered.Themagnitudeoftheheatfluxnormaltoasolidsurfacebyfreeconvectionisqn=hAh(ThTf)wherehistheconvectioncoefficient,Ahisthesurfaceareacontactingthefluid,Thistheconvectingsurfacetemperature,andTfisthesurroundingfluidtemperature,respectively.Theunitsofhare[W/m

2K].Itsvaluevarieswidelyandisusuallyknownonlyfromonetofoursignificantfigures.Typicalvaluesforconvectiontoairandwaterare525and5001000W/m2K,respectively.

Radiationheattransferoccursbyelectromagneticradiationbetweenthesurfacesofabodyandthesurroundingmedium.Itisahighlynonlinearfunctionoftheabsolutetemperaturesofthebodyandmedium.Themagnitudeoftheheatfluxnormaltoasolidsurfacebyradiationisqr=Ar(Tr

4Tm4).HereTristhe

absolutetemperatureofthebodysurface,Tmistheabsolutetemperatureofthesurroundingmedium,Aristhebodysurfaceareasubjectedtoradiation,=5.67x108W/m2K4istheStefanBoltzmannconstant,andisasurfacefactor(=1foraperfectblackbody).

Transient,orunsteady,heattransferintimealsorequiresthematerialpropertiesofspecificheatatconstantpressure,cpin[kJ/kgK],andthemassdensity,in[kg/m

3].Thespecificheatistypicallyknownto2or3significantfigures,whilethemassdensityisprobablythemostaccuratelyknownmaterialpropertywith4to5significantfigures.

13.1.1 OnedimensionalthermalstructuralanalogyTheonedimensionalgoverningdifferentialequationfortransientheattransferthroughanareaA,ofconductivitykx,density,specificheatcpwithavolumetricofheatgeneration,Q,forthetemperatureTattimetis(kxT/x)/x+Q(x)=cpT/t,for0xLandtimet0.Itrequiresinitialconditionstodescribethebeginningstate,andboundaryconditionsforlatertimes.Forasteadystatecondition(T/t=0)thetypicalboundaryconditionsofoneofthefollowing:

1. Tprescribedat0andL,or2. Tprescribedatoneendandaheatsourceattheother,or

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3. Tprescribedatoneendandaconvectionconditionattheother,or4. Aconvectionconditionatoneendandaheatsourceattheother,or5. Aconvectionconditionatbothends.

Thesethermalconditions,in1D,arerelatedtothedisplacementsandstressinanaxialbarassummarizedinTable131.

Table131Termsofthe1DthermalstructuralanalogyThermalAnalysisItem,[units],symbol StructuralAnalysisItem,[units],symbolUnknown:Temperature[K],T Unknown:Displacements[m],uGradient:TemperatureGradient[K/m],T Gradient:Strains[m/m],Flux:Heatflux[W/m2],q Flux:Stresses[N/m2],Source:HeatSourceforpoint,line,surface,volume[W],[W/m],[W/m2],[W/m3],Q

Source:Forceforpoint,line,surface,volume[N],[N/m],[N/m2],[N/m3],Q

Restraint:Prescribedtemperature[K],T Restraint:Prescribeddisplacement[m],u

Reaction:Heatflowresultant[W],Q Reaction:Forcecomponent[N],Q

MaterialProperty:Thermalconductivity[W/mK],k MaterialProperty:Elasticmodulus[N/m2],E

MaterialLaw:Fourierslaw MaterialLaw:Hookeslaw

13.1.2 ThreedimensionalformulationInthe3DcasethedifferentialequationbecomestheanisotropicPoissonEquation(seeChapter16).Thatis,theabovediffusionterm(secondderivativesinspace)isexpandedtoincludederivativeswithrespecttoyandz,timestheircorrespondingthermalconductivityvalues.

13.2 ThermalanalysisinputpropertiesThethermalmaterialpropertiesavailableinSWSimulationarelistedinTable133.Onlytheconductivitiesaretheoreticallyneededforasteadystatestudy,butSWSimulationalwaysrequeststhemassdensity.Anytransient(timedependent)thermalanalysisinvolvestheproductofthemassdensityandspecificheat,asseenintheaboveequation.

Table132Isotropicthermalproperties

Table133AnisotropicthermalpropertiesinprincipalmaterialdirectionsSymbol Label Item DENS Massdensityc C Specificheat,atconstantpressurekx KX ThermalconductivityinmaterialXdirectionky KY ThermalconductivityinmaterialYdirectionkz KZ ThermalconductivityinmaterialZdirection

13.3 FiniteElementThermalAnalysis

13.3.1 ThermalrodelementFromtheaboveanalogythematrixequationsofasingleelement(fromsections2.3and2.4)is

Symbol Label Item Application DENS Massdensity Transientc C Specificheat,atconstantpressure Transientk KX Thermalconductivity Steadystateandtransient

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1 1 1 1

wherek kxA/Lmaybereferredtoasthethermalstiffnessoftherodoflength,L,area,A,andthermalconductivitykx.Inthiscase,T correspondstoanodaltemperature,andFcorrespondstotheresultantnodalheatpowerfromthevariousheatsources.ThetypicalunitsoftheabovethreematricesareW/C,C,andW.

13.3.2 AlgebraicequationsThefiniteelementmethodcreatesasetofalgebraicequationsbyusinganequivalentgoverningintegralformthat is integratedoverameshthatapproximatesthevolumeandsurfaceofthebodyof interest. Themeshconsistsofelementsconnectedtonodes. Inathermalanalysis,therewillbeonesimultaneousequationforeachnode.Theunknownateachnodeisthetemperature.Today,atypicalthermalmeshinvolves20,000to100,000nodesand thus temperatureequations. Therestraintsarespecified temperatures (oraconvectioncondition since it includes a specified fluid temperature). The reactions are is the resultant heat powernecessary tomaintaina specified temperature.Allother conditionsadd loador source terms. Thedefaultsurfaceconditionisaninsulatedboundary,whichresultsinazerosource(load)term.

The assembledmatrix equations for thermal equilibrium have exactly the same partitioned form as thestructuralsystemsofsection2.5:

wherenowTgrepresentsthegiven(restrained)nodaltemperatures,Fgrepresentstheknownresultantnodalheatpoweratthenode.Thissystemofequationsissolvedjustasdescribedinsection2.5.ThethermalrestraintsitemsforsteadystateanalysisaregiveninTable134.

Mostprogramsofferonlyatemperaturerestraint.SWSimulationalsoofferstheabilitytodefineanonidealmaterialinterface,asillustratedinFigure131.Thisisoftenneededinpracticeandisreferredtoasacontactresistance.Itbasicallydefinesatemperaturejumpacrossaninterfaceforagivenheatfluxthroughtheinterface.Thenecessaryresistanceinput,R,dependsonvariousfactors.TheRvalueisthesameconceptusedisspecifyinghomeinsulation.Table135givestypicalRvalues,whileTable136citesvaluesofitsreciprocal,theconductance.

Thethermalload(source)itemsforsteadystateanalysisaregiveninTable137.Bothconvectionandradiationrequireinputsoftheestimatedsurfaceconditions.Typicalconvectioncoefficientsaregivenin

Table138.Notethatthereisawiderangeinsuchdata.Therefore,youwilloftenfinditnecessarytorunmorethatonestudytodeterminetherangeofanswersthatcanbedevelopedinyourthermalstudy.Havingsuppliedalltherestraints,loads,andmaterialpropertiesyoucanrunathermalanalysisandcontinueontopostprocessinganddocumentingtheresults.

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Table134Restraintsinsteadystatethermalanalysis

RestraintType GeometricEntities RequiredInput

Temperature Vertexes,edges,facesandparts Temperaturevalueandunits

Contactresistance

Twocontactingfaces Totalthermalresistanceorunitthermalresistance.Seediscussion.

Figure131Idealandthermalcontactresistanceinterfaces

Table135Typicalcontactresistancevalues,Rxe4,[m2K/W]ContactPressure Moderate 100kN/m2 10,000kN/m2Aluminum/aluminum/air 0.5 1.55.0 0.20.4Copper/copper/air 0.1 110 0.10.5Magnesium/magnesium/air 1.53.5 0.20.4Stainlesssteel/stainlesssteel/air 3 625 0.74.0

Table136Typicalcontactconductancevalues,C,[W/m2K]

ContactingFaces(pressureunknown) ConductanceAluminum/aluminum/air 220012000Ceramic/ceramic/air 5003000Copper/copper/air 10,00025,000Iron/aluminum/air 45,000Stainlesssteel/stainlesssteel/air 20003700Stainlesssteel/stainlesssteel/vacuum 2001100

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Table137LoadsforsteadystatethermalanalysisLoadType GeometricEntities RequiredInputConvection Faces Filmcoefficientandbulktemperatureinthedesiredunits

HeatFlux Faces Heatflux(heatpower/unitarea)valueinthedesiredunits

HeatPower Vertexes,edges,facesandparts

Totalheatpowervalueandunits(rateofheatgenerationperunitvolumetimesthepartvolume)

Insulated(Adiabatic)

Faces None.Thisisthedefaultconditionforanyfacenotsubjecttooneofthethreeaboveconditions

Radiation Faces Surroundingtemperature,emissivityvaluesandunits,andviewfactorforsurfacetoambientradiation

Table138Typicalheatconvectioncoefficientvalues,h,[W/m2K]FluidMedium h

Air(naturalconvection) 525A