Chapter4Unsteady-StateConduction4-1INTRODUCTIONApplicationofseparation-ofvariablesmethodinthedeterminationoftemperaturedistributioninaninfiniteplatesubjectedtosuddencoolingofsurfaces.AssumingresultsinFromboundarycondition[b],C1=0Fromboundarycondition[c]orFinalseriesformofthesolutionis4-2LUMPED-HEAT-CAPACITYSYSTEMTimeconstant(時間常數)Energybalance:Whenthetimeequalstotimeconstant,ApplicabilityofLumped-CapacityAnalysisCharacteristicdimension:4-3TRANSIENTHEATTRANSFERINASEMI-INFINITESOLIDTheinitialtemperatureofthesemi-infinitesolidisTi,thesurfaceissuddenlyloweredtoT0.SeekanexpressionfortheTdistributioninthesolidasafunctionoftime.TheproblemissolvedbyLaplace-transformtechnique.Gausserrorfunction:ConstantsurfacetemperatureAtsurfacetheheatflowisHeatflowatanyxposition:ConstantHeatFluxonSemi-InfiniteSolidEnergyPulseatSurface4-4CONVECTIONBOUNDARYCONDITIONSForasemi-infinitesolidwithaconvectionboundaryconditionThesolutionis:HeislerChartsTheBiotandFourierNumbersApplicabilityoftheHeislerChartsInLumpedHeatCapacityanalysis,characteristicdimensioncanbedefinedasThetimeconstantbecomes4-5MULTIDIMENSIONALSYSTEMSGoverningeq.Initialandboundaryconditions:Definition:Governingeq.Initialandboundaryconditions:Initialandboundaryconditions:Forplate1withthickness2L1Initialandboundaryconditions:Forplate2withthickness2L2TobeshownthatDimensionlesstemperaturedistributioncanbeexpressedasaproductofthesolutionsforthetwoplateproblemsInasimilarmanner,Conclusion:HeatTransferinMultidimensionalSystems4-6TRANSIENTNUMERICALMETHODForone-dimensionalproblem:BoundaryconditionsifConvergencecondition:ForwardandBackwardDifferencesForwarddifferenceandexplicitformulationBackwarddifferenceand

implicitformulation向前差分：將時間步長末時節點的溫度用時間步長起點時周圍節點的溫度表示的差分方法。向后差分：空間微分用當前時刻溫度表示的差分方法。4-7THERMALRESISTANCEANDCAPACITYFORMULATIONForwarddifference:Stabilityrequirement:Considerationonround-offerrorHeatsour