# Heat equation applied to a rod

1. Feb 11, 2009

### dirk_mec1

1. The problem statement, all variables and given/known data

http://img444.imageshack.us/img444/7641/20240456gw8.png [Broken]

2. Relevant equations
http://img14.imageshack.us/img14/5879/63445047rj2.png [Broken]

Note that the rightside of the rod is insulated.

3. The attempt at a solution
I get this model:

$$\frac{ \partial{u} }{ \partial{t} } = \kappa \frac{ \partial{ ^2 u} }{ \partial{x^2} } +s$$

$$u(0,t)=u_0$$
$$\frac{ \partial{u}} { \partial{x} } = 0$$

In steady state this gives: $$u(x) = \frac{- s}{ \kappa} \frac{1}{2}x^2 + \frac{s}{ \kappa } L x + u_0$$

But if I calcute than the asked u' at x=0:

I get:

$$\frac{du}{dx} = \frac{s}{ \kappa} L$$

Is this correct?

Last edited by a moderator: May 4, 2017
2. Feb 12, 2009

### dirk_mec1

What I don't understand is what do they mean by "total heat supply"? I presume they mean s (=source). But I get a different answer out of my equation.

3. Feb 12, 2009

### Mapes

$$\frac{ \partial{u}} { \partial{x} } = 0$$

means nothing on its own; we need to specify a location:

$$\left(\frac{ \partial{u}} { \partial{x} }\right)_{x=L} = 0$$

For the heat supply question: we need to distinguish the total heat S from the heat per length $s=S/L$ that goes into the differential equation. By applying Fourier's conduction law, your answer indicates a total heat flow of $sL=S$, which is correct. The units will always confirm whether S or s is being used appropriately.

4. Feb 12, 2009

### dirk_mec1

You're right but I couldn't get this in latex. Note that the notation you are using isn't the right one either there should be a large bar at the right hand side something like this: $$|_{x=L}$$

Of ocurse, how could I overlooked that!