Chapter 3, Problem 3.103P

Radioactive wastes are packed in a thin-walled spherical container. The wastes generate thermal energy nonuniformly recording to the relation $\stackrel{.}{q}={\stackrel{.}{q}}_o\left[1-{\left(r/r_o\right)}^2\right]$ where is the local rate of energy generation per unit volume, $\stackrel{.}{q}$ is a constant, and $r_o$ is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at $T_{\infty }$ and provides a uniform convection coefficient h.

Determine the temperature distribution, $T\left(r\right),$ in the container. Express your result in terms of ${\stackrel{.}{q}}_o,r_o,T_{\infty },h,$ and the thermal conductivity k of the radioactive wastes.