# Flux of a uniform rod

• February 6th 2011, 01:18 PM
dwsmith
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?

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

How do I obtain the flux at x = L?
• February 7th 2011, 02:34 PM
dwsmith
Quote:

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?

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

How do I obtain the flux at x = L?

$\displaystyle\text{D.E.}: \ \frac{d^2u}{dx^2}=0\Rightarrow u(x)=xk+g$

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

$u(0)=g=C$

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

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

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

How can I obtain the solution from all this?
• February 7th 2011, 03:06 PM
zzzoak
From BC we get

$g=C$

$k_1=-C/L$

and

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

Flux is

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

$
C=FL/ \kappa .
$