Conduction through a spherical shell.

[kristen32123]

How do I solve for the temperature at a certain time for conduction through a spherical shell? I have a shell 5 cm thick with a radius of 0.5m. I am starting with Q=KA(T2-T1)/L and I am completely lost as to where to go with it.

Thanks!

 

[Mentz114]

Imagine the sphere has lines of latitude and longitude drawn on it. The (differential element of ) area of a rectangle defined by these lines is ( working in polar coords $r, \theta, \phi$)

$$dA = rd\theta*rsin\theta d\phi$$

so, using your formula

$$dQ = rd\theta*rsin\theta d\phi*K(T_1-T_2)/0.05$$

Now integrate out $d\theta, d\phi$ between 0 and 2*pi, and that gives the formula for the whole shell.

Of course you don't actually need to integrate - just think about it and the answer is obvious.

This solution is not exact because we're ignoring the difference in area btween the outer and inner surfaces ...

To solve exactly one needs a differential dL of thickness. Not a trivial problem.