Determine the steady-state temperature distribution between two concentric spheres with...

Determine the steady-state temperature distribution between two concentric spheres with radii 1 and 4, respectively, if the temperature of the outer sphere is maintained at 80° and the inner sphere at 0? (see Exercise 1.5.12).

Reference Exercise 1.5.12:

Assume that the temperature is spherically symmetric, u = u(r, t), where r is the distance from a fixed point (r₂ = x₂ + y₂ + z₂). Consider the heat flow (without sources) between any two concentric spheres of radii a and b.

(a) Show that the total heat energy is

(b) Show that the flow of heat energy per unit time out of the spherical shell at

A similar result holds at r = a.

(c) Use parts (a) and (b) to derive the spherically symmetric heat equation