Heat Conduction in a Thin Rod Click here for an example.

In a Nut Shell: Heat conduction in a thin rod is governed by the following

partial differential equation:

∂u/∂t = k ∂²u/∂x² ---------------------- (1)

where u = u(x,t) = the temperature distribution in the rod

x = the position along the rod

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

and

k is the thermal diffusivity of the rod (material property)

The desired outcome is to predict the temperature distribution, u(x,t), in the rod

subject to the boundary (end) conditions given an initial temperature distribution

in the rod, u(x,0) = f(t).

Since the partial differential equation is second order in its derivative with respect to

x, you will need two boundary conditions. The p.d.e is first order in its derivative

with respect to t, so you need one initial condition. See the figure below.

The common boundary conditions at the ends of the rod are as follows:

a. Specified temperature at an end x = 0 u(0,t) = To

b. Specified temperature at end x = L u(L,t) = T₁

c. Insulated condition at x = 0 ∂u(0,t)/ ∂x = 0

d. Insulated condition at x = L ∂u(L,t)/ ∂x = 0

or any combination of these boundary conditions. i.e. If the temperature is specified

at x = L and the rod is insulated at x = 0, then the appropriate boundary conditions

are ∂u(0,t)/ ∂x = 0 and u(L,t) = 0. Other linear combinations might be like.

C₁ u(0,t) + C₂ u_x (0,t) = F(t) and C₃ u(L,t) + C₄ u_x (L,t) = G(t)

where C₁ , C₂ , C₃ , and C₄ are known constants and where u_x = ∂u/∂x .

Click here to continue with discussion of heat condition in a thin rod.

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