17. Diffusion_equation_in_2D_and_3D

17. Diffusion_equation_in_2D_and_3D - 4.5 The Diffusion...

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Unformatted text preview: 4.5 The Diffusion Equation in 2-D and 3-D For the thin bar case, there is one spatial coordinate (x). The derivation of Section 4.4 led to: 1-D diffusion eqn. for bar temperature, T(x,t) Two Dimensions Consider a rectangular thin flat plate with insulated top and bottom. • • • ∆ ∆ element y thickness, h Use Cartesian coordinates Volume element, ∆ ∆∆ Insulation means that heat flow, occurs only in the (x,y) plane. i.e.: ̂ ̂ (2-D) side view x top view Derivation in 2-D leads to: qy 2-D diffusion eqn. for plate temperature, T(x,y,t) qx For a thin circular flat plate, we need to use polar coordinates to describe the plate shape. Derivation leads to: Element has sides ∆r and r∆θ y θ thickness, h x , ̂ 1 1 for T(r,θ,t) ̂ r The general statement for 1-D, 2-D, and 3-D cases in any coordinate system is: the Laplacian operator T Extensions to include sources/sinks etc. carry through from Section 4.4. e.g. with a source: T S ρc ...
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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