Question: Insulation placed on a small-diameter pipe or electrical wire to prevent heat loss may actually increase the heat loss instead. This occurs because as the insulation radius rz increases, the insulation thermal resistance increases but the outside surface area is increased at the same time. The result is that the rate of heat loss will increase with increasing r2 until a maximum heat loss is reached, after which further increase in rz lowers rate of beat loss.

if the inside surface temperature of the surroundings are at TZ, find the value of insulation, T1, is known, and the rz for maximum rate of heat loss. Assume that T1, T2, k, h, and r1 are constant.