Constructal Three-Dimensional Trees for Conduction Between a Volume and One Point

This paper extends to three-dimensional heat conduction the geometric “constructal” method of minimizing the overall thermal resistance between a finite-size volume and a small heat sink. The volume contains (i) low-conductivity material that generates heat at every point, and (ii) a small amount of high-conductivity material that must be distributed optimally in space. The given volume is covered in a sequence of building blocks (volume sizes) that starts with the smallest volume element, and continues toward larger assemblies. It is shown that the overall shape of each building block can be optimized for minimal volume-to-point resistance. The relative thicknesses of the high-conductivity paths can also be optimized. These optima are developed analytically and numerically for the smallest elemental volume and the first assembly. The high-conductivity paths form a tree network that is completely deterministic.