Math Help

Heat Conduction in a Circular or Semi-Circular Plate

In a Nut Shell: The governing equation for heat conduction in a circular or
semi-circular plate is:

∂u/∂t = k [∂²u/∂r²+ (1/r) ∂u/∂r + (1/r²) ∂²u/∂θ²] ----------------------- (1)

where u = u(r,θ,t) = the temperature in the plate at any time t

r,θ = the location in the plate

t = the time at which the temperature at x is u(r,θ,t)

and

k is the thermal diffusivity of the material

When applied to a plate, the desired outcome is to predict the temperature

distribution, u(r,θ,t), in the plate as a function of time.

For steady-state heat conduction, ∂u/∂t = 0. So the steady-state temperature

distribution in the plate is governed by Laplace’s equation:

∂²u/∂r²+ (1/r) ∂u/∂r + (1/r²) ∂²u/∂θ²= 0 ------------------------ (2)

Use the method of separation of variables to solve (2) subject to the boundary conditions.

Consider a thin semi-circular plate of radius a shown below. The objective is to find

the steady state temperature distribution given boundary conditions on the boundary of

the plate.

Return to Differential Equations Notes