Ch5-Transient Conduction

# Ch5-Transient Conduction - Chapter 5 Transient Conduction...

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Chapter 5 Transient Conduction

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Solution to the Heat Equation for a Plane Wall with Symmetrical Convection Conditions If the lumped capacitance approximation can not be made, consideration must be given to spatial, as well as temporal, variations in temperature during the transient process. For a plane wall with symmetrical convection conditions and constant properties, the heat equation and initial/boundary conditions are: 2 2 1 α = T T x t (5.26) ( 29 ,0 = i T x T (5.27) 0 0 = = x T x (5.28) ( 29 , = - = - x L T k h T L t T x (5.29) Existence of seven independent variables: ( 29 , , , , , , α = i T T x t T T k h (5.30) How may the functional dependence be simplified?

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Dimensionless temperature difference: * i i T T T T θ θ θ - = - * x x L Dimensionless coordinate: Dimensionless time: * 2 t t Fo L α The Fourier Number The Biot Number: solid hL Bi k ( 29 * * , , θ = f x Fo Bi Exact Solution: ( 29 ( 29 * 2 * 1 exp cos n n n n C Fo x θ ζ ζ = = - (5.39a) ( 29 4sin tan 2 sin 2 ζ ζ ζ ζ ζ = = + n n n n n n C
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