Chapter 2, Problem 2.15P

Suppose that a pipe carrying a hot fluid with an external temperature of $T_i$ and outer radius $r_i$ is to be insulated with an insulation material of thermal conductivity k and outer radius $r_o$. Show that if the convection heat transfer coefficient on the outside of the insulation is $\bar{h}$ and the environmental temperature is $T_{\infty }$, the addition of insulation actually increases the rate of heat loss if $r_o<k/\bar{h}$, and the maximum heat loss occurs when $r_o=k/\bar{h}$. This radius, $r_{cr}$ is often called the critical radius.