Thermodynamics HW Solutions 84

Thermodynamics HW Solutions 84 - Chapter 2 Heat Conduction...

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Chapter 2 Heat Conduction Equation 2-21 We consider a thin element of thickness Δ x in 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 balance on this thin element of thickness Δ x during a small time interval Δ t can be expressed as && QQ E t xx x −= Δ Δ element where Δ Δ ΔΔ EE E m C T T C A x T tt t tt t element =− = Δ T = ++ + () ρ Substituting, C A x TT t x + + Δ Δ Δ Δ Dividing by A Δ x gives = 1 A x C t x Taking the limit as and yields Δ x 0 Δ t 0 t T C x T kA x A = 1 since, from the definition of the derivative and Fourier’s law of heat conduction,
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