Chapter 2 Heat Conduction Equation2-21 We consider a thin element of thickness Δxin a large plane wall (see Fig. 2-13 in the text). The density of the wall is ρ, the specific heat is C, and the area of the wall normal to the direction of heat transfer is A. In the absence of any heat generation, an energy balanceon this thin element of thickness Δxduring a small time interval Δt can be expressed as &&QQEtxxx−=+ΔΔΔelementwhere ΔΔΔΔEEEmCTTCAxTttttt telement=−=ΔT−=−+++()ρSubstituting, CAxTTtx−++ΔΔΔΔDividing by AΔxgives −−=−1AxCtxTaking the limit as and yields Δx→0Δt→0tTCxTkAxA∂=⎟⎠⎞⎜⎝⎛1since, from the definition of the derivative and Fourier’s law of heat conduction,
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