Constructal-theory network of conducting paths for cooling a heat generating volume

Abstract

This paper develops a solution to the fundamental problem of how to collect and ‘channel’ to one point the heat generated volumetrically in a low conductivity volume of given size. The amount of high conductivity material that is available for building channels (high conductivity paths) through the volume is fixed. The total heat generation rate is also fixed. The solution is obtained as a sequence of optimization and organization steps. The sequence has a definite time direction, it begins with the smallest building block (elemental system) and proceeds toward larger building blocks (assemblies). Optimized in each assembly are the shape of the assembly and the width of the newest high conductivity path. It is shown that the paths form a tree-like network, in which every single geometric detail is determined theoretically. Furthermore, the tree network cannot be determined theoretically when the time direction is reversed, from large elements toward smaller elements. It is also shown that the present theory has far reaching implications in physics, biology and mathematics.