Thread: Flux of a uniform rod

1. Flux of a uniform rod

The end x = 0 of a uniform rod is maintained at a constant temperature and the end x = L is maintained at temperature zero. After the steady state has been reached, it is found that the flux out of the end of the rod at x = L has the value F. What is the temperature at x = 0?

$\displaystyle \displaystyle\text{B.C.}=\begin{cases} u(0,y)=C\\ u(L,y)=0\end{cases}$

How do I obtain the flux at x = L?

2. Originally Posted by dwsmith
The end x = 0 of a uniform rod is maintained at a constant temperature and the end x = L is maintained at temperature zero. After the steady state has been reached, it is found that the flux out of the end of the rod at x = L has the value F. What is the temperature at x = 0?

$\displaystyle \displaystyle\text{B.C.}=\begin{cases} u(0)=C\\ u(L)=0\end{cases}$

How do I obtain the flux at x = L?
$\displaystyle \displaystyle\text{D.E.}: \ \frac{d^2u}{dx^2}=0\Rightarrow u(x)=xk+g$

$\displaystyle u(L)=Lk+g=0\Rightarrow g=-Lk$

$\displaystyle u(0)=g=C$

Also, "...empirical physical law which states that the heat flux at any point is proportional to the temperature gradient at that point."

$\displaystyle \displaystyle F(x,t)=-\kappa\frac{\partial u}{\partial x}(x,t), \ \ \kappa>0$

The solution to my question is $\displaystyle \displaystyle\frac{FL}{\kappa}$

How can I obtain the solution from all this?

3. From BC we get

$\displaystyle g=C$

$\displaystyle k_1=-C/L$

and

$\displaystyle u(x)=-(C/L)x+C.$

Flux is

$\displaystyle F=-\kappa \frac{du}{dx}|_{x=L}=\kappa \; C/L$

$\displaystyle C=FL/ \kappa .$