# Lecture 4 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 4 – Sep 2, 2010 1D Steady-State Conduction 2

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Heat Equation: Different Coordinate Systems p TT T T ckkk q txx yy 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 ± 1-D steady state conduction with no heat generation ± Common geometries: ± Plane wall: Cartesian coordinate ( x ) where the heat flow is only in the direction perpendicular to the wall ± Tube wall: cylindrical coordinates ( r ) where there is only radial heat flow through the wall of a tube ± Spherical shell: spherical coordinates ( r ) where there is only radial heat flow through the shell of a sphere 1-D Steady Conduction 4
Plane Wall Governing Equation: Dirichlet Boundary Conditions: Solution: Heat Flux: Heat Flow: T (0) = T s ,1 ; T ( L ) = T s ,2 d dx k dT dx ⎟ = 0 k d 2 T dx 2 = 0 for uniform k Notes: for one-dimensional, steady-state conduction in a plane wall w/o heat generation, regardless of the B.C. regardless of the B.C. ± the temperature varies linearly with x ± the heat flux is constant ( why? ) ± the heat transfer rate is constant ( why? ) 5 Tube Wall ² Cylindrical and spherical systems often experience temperature gradients in the radial direction only and may be treated as 1-D.

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

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