17. Diffusion_equation_in_2D_and_3D

# 17. Diffusion_equation_in_2D_and_3D - 4.5 The Diffusion...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

Ask a homework question - tutors are online