Chapter 2, Problem 2.63P

A spherical particle of radius $r_1$ experiences uniform thermal generation at a rate of $\stackrel{.}{q}.$ The particle is encapsulated by a spherical shell of outside radius $r_2$ that is cooled by ambient air. The thermal conductivities of the particle and shell are $k_1$ and $k_2,$ respectively, where $k_1=2k_2.$

1. By applying the conservation of energy principle to spherical control volume A, which is placed at an arbitrary location within the sphere, determine a relationship between the temperature gradient $dT/dr$ and the local radius r, for $0\le r\le r_1.$
2. By applying the conservation of energy principle to spherical control volume B, which is placed at an arbitrary location within the spherical shell, determine a relationship between the temperature gradient $dT/dr$ and the local radius r, for $r_1\le r\le r_2.$
3. On $T-r$ coordinates, sketch the temperature distribution over the range $0\le r\le r_2.$