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Steady state heat equation in concentric spherical shells

  1. Jul 19, 2011 #1
    1. The problem statement, all variables and given/known data
    2. Relevant equations
    3. 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" [Broken]) 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?
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jul 20, 2011 #2


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    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.
  4. Jul 20, 2011 #3
    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.
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