Chapter 3, Problem 3.96P

A cylindrical shell of inner and outer radii, $r_i$ and $r_o,$ respectively, is filled with a heat-generating material that provides a uniform volumetric generation rate $\left(W/m^3\right)$ of $\stackrel{.}{q}.$ The inner surface is insulated, while the outer surface of the shell is exposed to a fluid at $T_{\infty }$ and a convection coefficient h.

1. Obtain an expression for the steady-state temperature distribution $T\left(r\right)$ in the shell. expressing your result in terms of $r_i,r_o,\stackrel{.}{q},\text{}h,T_{\infty },$ and the thermal conductivity k of the shell material.
2. Determine an expression for the heat rate, $q\text{'}\left(r_o\right),$ at the outer radius of the shell in terms of $\stackrel{.}{q}$and shell dimensions.