Heat transfer through a cylindrical shell

[Ian Baughman]

Homework Statement

An infinitely long cylindrical shell has an inner radius a and outer radius b. If the inside is maintained at a temperature T_a and the outside at a temperature T_b, determine the rate of heat flow per unit length between inner and outer surfaces assuming the shell has a thermal conductivity k.

Homework Equations

[/B]
H = -KA((T_H-T_C)/L)

The Attempt at a Solution

[/B]
1) I said let T_H = T_a and T_C = T_b
2) I let L = b-a so my new expression is:

H = -KA((T_a-T_b)/(b-a))​

3) My issue here is I can not figure out what to use for the cross sectional area A. In the example I was using as reference the heat was flowing through the pipe not the outer shell of it so the cross sectional area was easy to calculate.
4) My idea was to use 2πr and multiply it by the length of the shell but since it is an infinitely long cylindrical shell that wouldn't make sense to do.

 

[Orodruin]

Your formula is only applicable to heat transfer through a flat surface. To find the stationary state in a different geometry you have to solve the Laplace equation for that geometry.

Also note that you are asked for the transfer per unit length of the cylinder.

 

[Ian Baughman]

Your formula is only applicable to heat transfer through a flat surface. To find the stationary state in a different geometry you have to solve the Laplace equation for that geometry.

Also note that you are asked for the transfer per unit length of the cylinder.

Do you mean like starting with the definition of heat transfer, H = (dQ/dT) or H = -KA(dT/dx), and solving from there using Laplace?

 

[Orodruin]

Do you mean like starting with the definition of heat transfer, H = (dQ/dT) or H = -KA(dT/dx), and solving from there using Laplace?

That differential equation is only valid for heat transfer in one dimension (flat geometry). You cannot apply it here unless your shell is very thin. In general you need to solve the heat equation in cylinder coordinates - which for a stationary situation is equivalent to laplace equation.