How to distribute a finite amount of insulation on a wall with nonuniform temperature

This paper shows that it is possible to distribute a finite amount of insulation in an optimal way that minimizes the overall heat transfer rate from a nonisothermal wall to the ambient. The optimal insulation thickness for a plane wall varies as the square root of the local wall-ambient temperature difference. Corresponding variational-calculus results are developed for cylindrical walls covered with insulation. The heat loss reduction associated with using the optimal thickness is greater when the wall is plane, as opposed to cylindrical, and when the wall temperature variation in the x direction has a greater second derivative, d2 T/dx 2. It is shown finally that the best insulation for a single-phase stream suspended in an environment of different temperature is the insulation with uniform thickness.