A hollow sphere of wall thickness \(1 \mathrm{~mm}\) and inner diameter of \(20 \mathrm{~mm}\) is exposed to an environment where the inner surface of the sphere is held at a temperature of \(T_{i}=40^{\circ} \mathrm{C}\) while the outer surface is exposed to a convective environment where \(h=100 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\) and \(T_{\infty}=40^{\circ} \mathrm{C}\). The temperature profile within the wall is given generically as:

\[T=-\left(\frac{\dot{q}}{6 k}\right) r^{2}+\frac{C_{1}}{r}+C_{2}\]

a. What are the boundary conditions for the problem?

b. Solve the problem using the above solution to get the temperature distribution in the wall.

c. Where is the location of the maximum temperature?

d. What heat generation rate in the wall of the sphere will lead to a maximum temperature of \(150^{\circ} \mathrm{C}\) if \(k=25 \mathrm{~W} / \mathrm{mK}, C_{1}=-25\), and \(C_{2}=80\) ? (Hint: What sign must the heat generation be?)