# Heat conduction in a beam with variable x-section

1. Feb 25, 2012

### schliere

Say we have a laterally insulated beam and some boundary conditions at either end, be it convective or fixed-temperature, but the cross-sectional area is variable. If the cross-section were constant I'd just say it were a 1-D problem, but I'd imagine that having the cross-sectional area be a function of the lateral distance would change this. Conceptually, how would you model it using the heat equation and boundary conditions?

Or does anyone know of literature to explain this?

2. Feb 27, 2012

### OldEngr63

Assume that the rate of change of cross section is "slow" and re-derive the 1-D heat transfer equation.

3. Feb 29, 2012

### schliere

Thanks for the guidance! I was able to derive this:

$\rho \text{Cp} \frac{\partial T}{\partial t}=\frac{1}{A(x)}\frac{\partial }{\partial x}\left(k A(x)\frac{\partial T}{\partial x}\right)+\dot{q}(x)$

where $A(x)$ is the cross-sectional area of the beam as a function of the lateral distance, $\dot{q}$ is the internal heat generation as a function of the lateral distance, $\rho$ is the density of the material, $\text{Cp}$ is the specific heat of the material, and $k$ is the thermal conductivity.

4. Feb 29, 2012

### OldEngr63

Looks like you are headed in the correct direction.

5. Feb 29, 2012

### Mech_Engineer

You might also consider looking at the limit of an infinite number of stacked rectangular blocks of increasing cross-sectional area, each with length L/n. It could be you can find a simple solution based on the average cross-sectional area across the beam.

Since conduction's resistance in the 1-D case is approximated as (L/(K*A)), reducing the length of intermediate slices will linearly reduce the resistance, and decreasing the area will linearly increase the resistance.

6. Mar 1, 2012

### schliere

I tried finding a solution for an average cross-sectional area but there was a rather large error, especially when using a largely variable cross-sectional area, like one proportional to x^4 (which was in the problem that made me wonder about it).

Also, that formula for resistance is based on a constant cross-section, and doesn't hold up at all when you have a variable area. I think what you're talking about would wind up being much more complicated than using the heat equation I found, not to mention the fact that I'm talking about not necessarily talking about steady state or no-heat-generation situations.

7. Mar 1, 2012

### Mech_Engineer

Well then you're off and running! Good luck.