Heat current through a spherical shell

[Swatch]

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?
Thanks

The right answer is H=4*pi*k*a*b*DeltaT/(b-a)

 

[Astronuc]

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*$\nabla T$, and $\nabla T$ can be approximated by $\frac{\Delta T}{\Delta x}$, where x is the generalized length dimension.

In the case of the sphere, one would apply $\frac{\Delta T}{\Delta r}$.

Also the area midway between the inner and outer surfaces may be given the the 4$\pi$*R², where R = $\sqrt{r_i r_o}$, 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.

 

[Swatch]

Thank you.

Just one question. How do you derive the mean of the radius?