# 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