Heat Transfer along a Rod
Heat Transfer along a Rod
This Demonstration shows the solution to the heat equation for a one-dimensional rod. The rod is initially submerged in a bath at 100 degrees and is perfectly insulated except at the ends, which are then held at 0 degrees. This is a a Sturm–Liouville boundary value problem for the one-dimensional heat equation
∂u(x,t)
∂t
2
∂
∂
2
x
with boundary conditions , , and , where is time, is distance along the rod, is the length of the rod, and .
u(0,t)=0
u(L,t)=0
u(x,0)=f(x)
t
x
L
f(x)=100
The solution is of the form
u(x,t)=sin
∞
∑
i=1
A
n
nπ
L
-t
2
k
nπ
L
e
where is the conductivity parameter (a product of the density, thermal conductivity, and specific heat of the rod) and
k
A=f(x)sinxdx
2
L
L
∫
0
nπ
L
If you increase the number of terms , the solution improves as long as the time is small. As (the final state), the entire rod approaches a temperature of 0 degrees. You can see the effect of the thermal properties by varying the conductivity parameter .
n
t∞
k