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Heat current through a spherical shell

  1. Aug 7, 2005 #1
    A spherical shell has inner radii a and outer radii b. The temperatures at the inner and outer surfaces are T2 and T1. The thermal conductivity of the shell material is k. I have to derive an equation for the total heat current through the shell.

    The equation for heat current through a rod is H=k*A*DeltaT/L where L is the length of the rod.

    For this sperical shell the area which is perpendicular to the flow of the heat is changing with the radius. So I have to integrate the area. Am I right here?
    I have tried to integrate the area by doing:
    A=integrate(4*pi*r^2) from a to b
    But I end up with something far from the right answer, don't want to get into that.
    Could anyone please give me a hint to this problem?

    The right answer is H=4*pi*k*a*b*DeltaT/(b-a)
  2. jcsd
  3. Aug 7, 2005 #2


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    Staff: Mentor

    Along the lines of the example H=k*A*DeltaT/L

    the current is the area integral of the heat flux.

    In general, heat flux is given by -k*[itex]\nabla T[/itex], and [itex]\nabla T[/itex] can be approximated by [itex]\frac{\Delta T}{\Delta x}[/itex], where x is the generalized length dimension.

    In the case of the sphere, one would apply [itex]\frac{\Delta T}{\Delta r}[/itex].

    Also the area midway between the inner and outer surfaces may be given the the 4[itex]\pi[/itex]*R2, where R = [itex]\sqrt{r_i r_o}[/itex], i.e. the geometric mean radius.

    Properly integrating the problem should also give the same answer, but the limits of integration are not (a, b), since one is considering a spherical surface.
  4. Aug 8, 2005 #3
    Thank you. :smile:

    Just one question. How do you derive the mean of the radius?
    Last edited: Aug 8, 2005
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