Formulating the BV problem

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March 22nd 2011, 12:25 PM
dwsmith

Formulating the BV problem

In a rod length L with insulated lateral surface and thermal constants $c, \ \rho, \ \kappa, \ K$ heat is generated at a rate uniformly proportional to the temperature, that is, at a rate $ru(x,t)$ per unit volume per unit time, where r is a constant and $u(x,t)$ is the temperature function of the rod. The ends of the rod are maintained at temperature 0 and initially the rod has uniform temperature 1.

Formulate an initial-BV problem for determining $u(x,t)$

$\text{D.E.}: \ u_t=ru_{xx}$
$\displaystyle\text{B.C.}=\begin{cases}u(0,t)=0\\u( L,t)=0\end{cases}$
$\text{I.C.}: \ u(x,0)=1$

I am not sure what to do with the r so I just put it in front of u_{xx}. Is that correct? Also, I need a K, c, and rho somewhere but not sure where and why.
March 22nd 2011, 01:49 PM
Aryth

r does belong there. It is known as Thermal Diffusivity.

It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

$r = \frac{K}{\rho c_p}$
March 22nd 2011, 01:51 PM
dwsmith

Quote:

Originally Posted by Aryth

r does belong there. It is known as Thermal Diffusivity.

It relates thermal conductivity and volumetric heat capacity together. It specifically shows how conductivity changes in comparison to the thermal bulk. As to why it belongs there, I don't know. Maybe seeing the equation will help you figure it out.

$r = \frac{K}{\rho c_p}$

But by making that substitution, I will lose r. In the final solution, there is an r, rho, K, and c.
March 24th 2011, 12:41 PM
dwsmith

The DE is

$\displaystyle\text{D.E.}: \ u_t=ku_{xx}+\frac{r}{c\rho}u$