Convection, conduction and heating problem

[Bob Smith]

So this is my current problem:

I have block of solid material that separates two bodies of gas. The temperature of all three is known (and presumably all different). The solid material is also being heated, equally throughout its volume.

I want to know the instantaneous heat transfer between these three bodies (as the rate of heating and all three temperatures are in permanent flux).

Currently I am just using convection to simulate the heat transfer between the three bodies, but this leaves me with a uniform temperature in the solid block, when I'd expect the block to be warmer on whichever side has the warmer body of gas. This solution 'works' but I would like to make it more accurate.

How can I calculate the temperature gradient in the block? I've been reading up on conduction but the problems all seem to assume a constant scenario and the temperature of the solid is dictated purely by the temperatures of the two bodies of gas, which doesn't match my scenario.

Would I then use the two surface temperatures for convection to the two bodies of gas, rather than the average temperature (which would match my current uniform temperature? at least to begin with)?

Suggestions appreciated.

 

[Mapes]

How about writing a differential energy balance in the solid material, i.e.,

k\frac{\partial^2 T}{\partial x^2}+J=\rho c\frac{\partial T}{\partial t}

where J is the volumetric heat generation and x is the horizontal distance? The boundary conditions would be convection on either side. This equation describes the spatial and temporal temperature variations in the solid material, given some assumptions such as uniformity in the vertical direction and temperature-independent material properties.