# Lecture 6 Steady-state conduction student - 1 MECE 4364...

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MECE 4364 Heat Transfer Prof. Dong Liu Department of Mechanical Engineering University of Houston 1 Lecture 6 – Sep 9, 2010 Lecture 6 Conduction 2

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Lecture 6 Heat Equation: Different Coordinate Systems p TT T T ckkk q txx y y zz ∂∂ ρ ∂∂ ∂ ∂ ∂ ∂ ∂ ⎛⎞ = +++ ⎜⎟ ⎝⎠ ± 2 11 p T T ck r k k q trx r r z z ∂∂∂ φ = ++ + ± 2 22 111 sin sin p T T r k k q r θ ∂φ ∂φ θ ∂θ ∂θ = + ± Cartesian Cylindrical Spherical 3 ( ) kT ⋅∇ Vector form of heat equation for constant k Lecture 6 1D Heat Generation: Heat Equations Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates 0 dd T kq dx dx += ± 1 0 T kr q rdr dr ± 2 2 1 0 T kr q rd r d r ± tube wall solid cylinder (solid rod) spherical shell solid sphere 4 ( ) 0 kT q ∇⋅ ∇ + = ±
Lecture 6 1D Heat Generation: Plane Wall Consider one-dimensional, steady-state conduction in a plane wall of constant k , uniform generation : 2 2 0 dT kq dx + = ± General Solution: () 2 12 2 q Tx x Cx C k ⎛⎞ = −+ + ⎜⎟ ⎝⎠ ± Notes: • Where is the maximum temperature? • Heat flux is not uniform Î function of position ( x ) • Temperature distribution is not bounded by boundaries Î internal temperature can be greater/less than surface temperatures Specific Solution: symmetric, Dirichlet boundary conditions T s,1 = T s,2 = T s 22 2 1 2 s qL x T kL = ± 5 0 0 x dT dx = = ( ) 0 kT q ⋅∇ += ± Lecture 6 1D Heat Generation: Plane Wall

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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 6 Steady-state conduction student - 1 MECE 4364...

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