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Lecture 12 Transient conduction

# Lecture 12 Transient conduction - 1 MECE 4364 Heat Transfer...

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1 MECE 4364 Heat Transfer MECE 4364 Heat Transfer Prof. Dong Liu Prof. Dong Liu Department of Mechanical Engineering Department of Mechanical Engineering University of Houston University of Houston 1 Lecture 12 – Oct 5, 2010 Transient Conduction: Transient Conduction: Spatial Effects and the Role of Spatial Effects and the Role of Analytical Solutions Analytical Solutions

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2 Plane Wall 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 T T x t α = (5.26) ( ) ,0 i T x T = (5.27) 0 0 x T x = = (5.28) ( ) , x L T k h T L t T x = = (5.29) Existence of seven independent variables: ( ) , , , , , , i T T x t T T k h α = (5.30) How may the functional dependence be simplified? The Semi-Infinite Solid A solid that is initially of uniform temperature T i and is assumed to extend to infinity from a surface at which thermal conditions are altered. ( ) , i T x t T → ∞ = Second boundary condition
3 Semi Semi- Infinite Solid The Semi-Infinite Solid A solid that is initially of uniform temperature T i and is assumed to extend to infinity from a surface at which thermal conditions are altered. Special Cases : Case 1: Change in Surface Temperature (T s ) ( ) ( ) 0, ,0 s i T t T T x T = = ( ) , x erf 2 t s i s T x t T T T α = (5.57) ( ) s i s k T T q t πα ′′ = (5.58) Semi Semi- Infinite Solid (cont.) Infinite Solid (cont.) ( ) ( ) 1 2 2 2 / , exp 4 erfc 2 o i o q t x T x t T k t q x x k t α π α α ′′ = ′′ (5.59) Case 2: Uniform Heat Flux ( ) s o q q ′′ ′′ = ( ) 0 0, x T k h T T t x = = ( ) 2 2 , 2 2 i i T x t T x erfc T T t hx h t x h t exp erfc k k k t α α α α = + + ⎦ ⎢ (5.60) Case 3: Convection Heat Transfer ( ) , h T

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