How to Calculate Heat Current in a Spherical Shell?

[TheDemx27]

Homework Statement

A spherical shell has inner and outer radii r_a and r_b, respectively, and the temperatures at the inner and outer surfaces are T_a and T_b. The thermal conductivity of he shell material is k. Derive an equation for the total heat current thought the shell in the steady state. Then calculate the temperature as a function of r, the distance from the center of the shell.

Homework Equations

H=-kA(T_b-T_a)/(r_b-r_a)

The Attempt at a Solution

I know that I'm supposed to use calculus somehow, I write the area A as a function of r, A(r)=4pi*r^2. I don't know what to do from there

 

[TheDemx27]

Oh I think I got it. It is just a separable differential equation, right?

 

[Bystander]

Hint: "Maxwell."

 

[TheDemx27]

That does nothing for me. I was only ever taught the derived forms of maxwell's equations...

Treating it as a separable differential equation H=-kA*(dT/dr) I got
H=k*4pi*(T_b-T_a)/(1/r_b-1/r_a)

 

[Bystander]

https://en.wikipedia.org/wiki/James_Clerk_Maxwell
Theory of Heat

 

[Chestermiller]

That does nothing for me. I was only ever taught the derived forms of maxwell's equations...

Treating it as a separable differential equation H=-kA*(dT/dr) I got
H=k*4pi*(T_b-T_a)/(1/r_b-1/r_a)

This result is right on target, and is the way I would have solved the problem too. Nice job.

 

[TheDemx27]

This result is right on target, and is the way I would have solved the problem too. Nice job.

Thanks for checking me.