Suppose we have a cylinder of radius R, in an environment with temperature Tenv, and heat is generated in the wire at a rate P per unit volume (for example due to a current - the exact nature is irrelevant). The heat flux from the surface of the wire is A(T(R)-Tenv) for constant A (Newton's law of cooling). Find the temperature T(r).
Thermal diffusion equation,
where κ is the thermal conductivity of the medium, C the heat capacity per unit volume of the medium, and H the rate of production of heat per unit volume in the medium.
In the steady state
The Attempt at a Solution
I'm assuming we want the steady state solution - however, if we had say a cup of tea and we had Newton cooling, the temperature would decay exponentially to Tenv, so the steady state would just be Tenv - is it the fact that we have some heat production P that allows a non trivial steady state?[/B]
Anyway, for some reason I believe I am supposed to convert the heat flux into a heat gained per unit volume, which is done by
If correct, why am I allowed to do this - how do I know it is lost uniformly throughout the cylinder?
Using the laplacian in cylindrical polars (no z or angular dependence)
We need A=0 for T to be finite at r=0, and we need T(R) at r=R so B=T(R)+(HR2/4κ).
However I don't really like leaving T(R) in the answer - how am I supposed to find out what this is?
Thanks for any help with the bold bits :)