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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)
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
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:
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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.
- Spring '10