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 TT x t α ∂∂ = (5.26) ( ) ,0 i Tx T = (5.27) 0 0 x T x = = (5.28) () , x L T kh T L t T x = −= (5.29) • Existence of seven independent variables: ( ) ,, , , , , i TTx t TTk h = (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 T →∞ = Second boundary condition
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3 Semi -Infinite Solid 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 TtTT x T =≠ = ( ) , x erf 2t s is Tx t T TT α ⎛⎞ = ⎜⎟ ⎝⎠ (5.57) ( ) s i s kT T q t πα ′′ = (5.58) Semi Semi -Infinite Solid (cont.) Infinite Solid (cont.) () 1 2 2 2/ ,e x p 4 erfc 2 o i o qt x kt qx x k t απ ′′ −= (5.59) Case 2: Uniform Heat Flux ( ) s o qq = 0 0, x T kh T T t x = ⎡⎤ ⎣⎦ ( ) 2 2 , 2 2 i i x erfc t hx h t x h t exp erfc kk k t αα = −+ + ⎢⎥ (5.60) Case 3: Convection Heat Transfer ( ) , hT
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4 Semi -Infinite Solid Infinite Solid • Important features • Case 1: (1) interior temperature approaches T s as t →∞ (2) heat flux decreases as t -1/2 • Case 2: (1) surface temperature increases as t 1/2 (2) heat flux is constant • Case 3: (1) surface and interior temperatures approach T as t increases (2) heat flux decreases Semi -Infinite Solid Infinite Solid • Two semi-infinite solids, initially at uniform temperatures T A,i and T B,i , are placed in contact at their free surfaces • Assume the contact resistance can be neglected, thermal equilibrium requires At the instant of contact, both surfaces must have the same temperature T s • Question • How to determine T s ?
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This note was uploaded on 01/27/2011 for the course MECE 4364 taught by Professor Lipinglui during the Winter '10 term at University of Houston.

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

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