Is the Heat Equation Model for an Insulated Rod Correct?

[dirk_mec1]

Homework Statement

http://img444.imageshack.us/img444/7641/20240456gw8.png

Homework Equations

http://img14.imageshack.us/img14/5879/63445047rj2.png

Note that the rightside of the rod is insulated.

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?

 

[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.

 

[Mapes]

Your answer looks fine. Note, though, that the equation

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

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.

 

[dirk_mec1]

Your answer looks fine. Note, though, that the equation

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

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

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

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: $$|_{x=L}$$

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.

Of ocurse, how could I overlooked that!