# Heat Equation

• Jan 27th 2011, 05:01 PM
dwsmith
Heat Equation
A rod of length L, cross section A, whose lateral surface is insulated, is made of a material of thermal constants c, p, k. Heat is produced electrically at a rate of $\beta$ per unit volume. The ends are kept at temperature T and initially the rod is at temperature zero. Formulate the initial-boundary value problem for the temperature in the rod.

My book is of no help on how to do this.
• Jan 28th 2011, 07:36 AM
Ackbeet
See if your school's library has Carslaw and Jaeger's Conduction of Heat in Solids. That should help quite a bit.
• Jan 28th 2011, 02:21 PM
dwsmith
Quote:

Originally Posted by dwsmith
A rod of length L, cross section A, whose lateral surface is insulated, is made of a material of thermal constants c, p, k. Heat is produced electrically at a rate of $\beta$ per unit volume. The ends are kept at temperature T and initially the rod is at temperature zero. Formulate the initial-boundary value problem for the temperature in the rod.

My book is of no help on how to do this.

Here is what I got:

$\displaystyle \text{D.E.} \ \ u_{t}=ku_{xx}, \ \ \ 0\leq x\leq L, \ \ \ t>0$

$\displaystyle \text{B.C.} \ \ u(0,t)=T \ \ \text{and} \ \ u(L,t)=T \ \ \ t>0$

$\displaystyle \text{I.C.} \ \ u(x,0)=0, \ \ \ 0\leq x\leq L$
• Jan 28th 2011, 02:22 PM
dwsmith
Quote:

Originally Posted by Ackbeet
See if your school's library has Carslaw and Jaeger's Conduction of Heat in Solids. That should help quite a bit.

I will have to look for that later. Is it long or short and sweet?
• Jan 28th 2011, 02:31 PM
Ackbeet
I would definitely agree with your initial and boundary conditions. As for the DE itself, I'm not sure that the thermal constant k of the rod is necessarily the same as the heat transfer coefficient, k, in the Heat Equation. In addition, I'm not seeing where the heat generation comes into your equations. It strikes me that your DE might not be homogeneous: don't there have to be source terms in there somewhere? I suppose the question to be asked is this: is the heat being generated all along the rod (as seems likely)? Or is it just being generated at the ends?

Here's the Carslaw and Jeager book on Amazon. It's about 520 pages.
• Jan 28th 2011, 02:33 PM
dwsmith
Quote:

Originally Posted by Ackbeet
I would definitely agree with your initial and boundary conditions. As for the DE itself, I'm not sure that the thermal constant k of the rod is necessarily the same as the heat transfer coefficient, k, in the Heat Equation. In addition, I'm not seeing where the heat generation comes into your equations. It strikes me that your DE might not be homogeneous: don't there have to be source terms in there somewhere? I suppose the question to be asked is this: is the heat being generated all along the rod (as seems likely)? Or is it just being generated at the ends?

I wish could answer that question but I don't know. My book tends to throw the user to the wolves.
• Jan 29th 2011, 03:30 AM
Ackbeet
The Wiki on the Heat Equation and Thermal Diffusivity show you how to construct the constant you need multiplying the second-order spatial derivative.

Based on Model Problem XX.6 on this webpage, I would hazard a guess at something like

$u_{t}=\alpha\,u_{xx}+\dfrac{\beta}{AL}.$