Steady state heat equation in concentric spherical shells

[capandbells]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm trying to find the steady state solution to the heat equation for a system of spherical shells (looks like http://correlatingcancer.com/wp-content/uploads/2009/01/nanoshell-thumb.jpg" ) where heat generation Q occurs in the outer shell (so I have two Laplace equations for the inner sphere and the medium outside the system and a Poisson equation for the outer shell). The system is also spherically symmetric, so the equations are just in the radial variable.

I know the solution for the Laplace equation in this case is T(r) = A + B/r and I believe that in the case of the inner region, B must be zero since the solution has to be finite at the origin. This means the solution is a constant. However, if the solution is constant, then the boundary condition on the heat conduction at the inner radius can't hold, since dT/dr = 0. I'm not sure what I'm doing wrong here. Can someone help me out?

 

[Mapes]

There is no steady state solution if there is a non-zero flux inwards. Think about it: the core will get hotter and hotter without limit. Steady state will occur when outwards conduction matches the one half of the generated heat that is conducted inwards, giving a net flux of zero and a core temperature equal to the temperature of the shell where heat is generated.

 

[capandbells]

Thanks. I had been thinking about that, but my brain's natural tendency to avoid hard work made me doubt it. :P I'll get to work on the full heat equation then.