Chapter 3, Problem 3.74P

The outer surface of a hollow sphere of radius $r_2$ is subjected to a uniform heat flux $q_2^n.$ The inner surface at $r_1$ is held at a constant temperature $T_{s,1}.$

1. Develop an expression for the temperature distribution $T\left(r\right)$ in the sphere wall in terms of $q_2^n,T_s,r_1,r_2,$and the thermal conductivity of the wall material k.
2. If the inner and outer tube radii are $r_1=50\text{mm}$ and $r_2=100\text{mm,}$ what heat flux $q_2^n$ is required to maintain the outer surface at $T_{s,2}=50°\text{C,}$ while the inner surface is at $T_{s,1}=20°\text{C?}$ The thermal conductivity of the wall material is $k=10\text{W/m}\cdot \text{K}\text{.}$