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Chapter3Steady-StateConductionMultipledimensions
IsothermsandtemperaturegradientThetemperaturegradientvaluegradTisavectordirectedtothesidewithgreatestvaluesoftemperatures.Theisothermalsurface-thisisarealorimaginarygeometricsurface,wherethetemperatureisthesameineachpointatagiventime.Thecrossingofisothermalsurfacesbyplanesgivesthelineswhichcalledisotherms.
2DHeatConductionEquationSteadyStateOneDimensionNoHeatSourceConstantkLaplace’sEquationUseSeparationofVariablesmethodtosolveit.
2DHeatConductionConsidertheboundaryconditionas:SeparationConstant
2DHeatConductionTheformofsolutiondependonthesignofλ2.λ2=0λ20λ20
2DHeatConductionSubstituteInEigenvalues
2DHeatConduction
2DHeatConductionConsidertheboundaryconditionas:SeparationConstant
2DHeatConduction
ConstructionofaFourierExpansionMultiplebynπx/Wandintegratefromx=0tox=W
ConstructionofaFourierExpansion
GeneralizationUsingthePrincipleofSuperpositionSuperpositionPrinciple
TheConductionShapeFactorManymulti-dimensionalheatconductionproblemsinvolveheatflowbetweentwosurfaces,eachofuniformtemperature,withanyothersurfacespresentbeingadiabatic.DefineShapeFactorS:AplaneslabAConcentricCylinderAConcentricSpherefor
TheConductionShapeFactor
TheConductionShapeFactor
TheConductionShapeFactor
TheConductionShapeFactor(Example)Asmalllaboratoryfurnaceisintheformofacubeandisinsulatedwitha10cmlayeroffiberglassinsulation,withaninsideedge30cmlong.Iftheonlysignificantresistancetoheatflowacrossthefurnacewallisthisinsulation,determinethepowerrequiredforsteadyoperationatatemperatureof600Kwhentheoutercasingtemperatureis350K.Thethermalconductivityofthefiberglassinsulationatthemeantemperatureof475Kisapproximately0.11W/mK.Thereare6sides,12
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