# 3040 notes.pdf - ME 3040 Heat Transfer Dr Sandip Dutta...

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ME 3040- Heat TransferDr. Sandip DuttaTransient Conduction Heat Transfer
Transient ConductionTransient ConductionA heat transfer process for which the temperature varies with time, as wellas location within a solid.It is initiated whenever a system experiences a change in operating conditions.It can be induced by changes in:surface convection conditions ( ), ,h TSolution TechniquesTheLumped Capacitance MethodExact SolutionsThe Finite-Difference Methodsurface radiation conditions ( ),surrh ,Ta surface temperature or heat flux, and/orinternal energy generation.
Lumped Capacitance MethodThe Lumped Capacitance MethodBased on the assumption of a spatially uniform temperature distributionthroughout the transient process.Why is the assumption never fully realized in practice?General Lumped Capacitance Analysis:Consider a general case, which includes convection,radiation and/or an appliedheat flux at specified surfacesas well as internal energy generation,,,,,,s cs rs hAAAHence, . ,Tr tT t
Lumped Capacitance Method (cont.)First Law:inoutstgdEdTcEEEdtdtAssumingenergy outflow due to convection and radiation andinflow due to an applied heat flux ,sq,,,surgss hs crs rdTcq AhATTh ATTEdt• May hand hrbe assumed to be constant throughout the transient process?How must such an equation be solved?(5.15)
Special Case (Negligible Radiation)Special Cases(Exact Solutions, )  0iTTNegligible Radiation,/:TTb a,,//gs css hahAc bq AEcThe non-homogeneous differential equation is transformed into a homogeneous equation of the form:dadt Integrating from t = 0 to anytand rearranging,/exp1expiiTTb aatatTTTT(5.25)To what does the foregoing equation reduce as steady state is approached?How else may the steady-state solution be obtained?
Special Case (Convection)Negligible Radiation and Source Terms,0,0 :grshhEq,s cdTchATTdt (5.2),0is ctcdhAdt ,exps ciihATTtTTcexpttThe thermal time constantis defined as,1ts cchA(5.7)ThermalResistance, RtLumped ThermalCapacitance, CtThe change in thermal energy storagedue to the transient process isout0tstEQEdt  ,0ts chAdt 1expittc (5.8)Note: ; = TTddT(5.6)
Special Case (Radiation)Negligible Convection and Source Terms,0,0 :grshh EqAssuming radiation exchange with large surroundings and a= e,44,surs rdTcATTdte ,44sur0is rTTtAcdTTTdtesursur3,sursursur1n1n4is riTTTTctATTTTTe