# Model of heat flow in a sphere

1. Dec 4, 2005

### capslock

I have derived expression for the heat flow along a bar with cross-sectional area A, given by 'Q = -KA dT/dx' where K is thermal conductivity constant and T and x refer to temperature and distance measured from the high temperature end of the bar.

I understand this. My problem is when I try to apply it to a sphere:

Say we have a spherical heat source of radius a at the centre of a solid sphere of radius b > a. Take the sphere as having thermal conductivity constant K. The source emits heat equally in all directions at a rate of Q per second. The outside of the outer sphere is help at constant temperature T_0.

How would I determine the temperature at the surface of the heat emitting sphere using the original differential equation?

Best Regards, James.

2. Dec 5, 2005

### mezarashi

Stick to first principles, which is the equation you have started with:
Assume equilibrium conditions.

$$\dot{Q} = KA\frac{dT}{dr}$$

where dT/dr is the temperature gradient at any distance r. The area of conduction at any distance r is $$A = 4\pi r^2$$.

Separate your variables and integrate accordingly. In case you didn't know, this same procedure can be used to derive your more familiar equation:

$$\dot{Q} = KA\frac{T_2-T_1}{x_2-x_1}$$

3. Dec 5, 2005

### Tide

Don't forget that heat flows from hotter to colder! :)