Heat conduction in a beam with variable x-section

[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?

 

[OldEngr63]

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

 

[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.

 

[OldEngr63]

Looks like you are headed in the correct direction.

 

[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.

 

[schliere]

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.

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.

 

[Mech_Engineer]

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