Determine the radial temperature distribution T(r) through a spherical shell with inner radius R1 and outer radius R2 with surface temperatures T1 and T2, respectively. Assume T1 > T2, and then sketch the relation between temperature and radius from R1 to R2.

The relation is such  that the product of  radial temperature gradient and the square of radius is constant or the radial temperature gradient is inversely proportional to the square of radius.

Explanation:

For this consider following assumptions

By using the the Fourier Law in radial direction(spherical coordinates) is given as

Here A_r is the area of the surface, which for the case of sphere is 4πr^2. Thus the equation becomes

Now in the steady state condition

This is true due to no internal heat generation and no state change. i.e.

As the energy balance is valid, this indicates

As the value of r is changing and it is same at two points, this indicate that the function is independent of r such that

This implies that the function is constant i.e.

As -4πk are already constant, this can be true iff

This relation indicates that the product of  radial temperature gradient and the square of radius is constant or the radial temperature gradient is inversely proportional to the square of radius.

The distribution is indicated in the attached diagram along with the initial sketch.

3.2x10^7