Constructal heat trees at micro and nanoscales

We consider the problem of cooling a two-dimensional heat generating conducting volume with one heat sink, such that the smallest features of the internal structure are so small that the conventional description of conduction breaks down. The effective thermal conductivity exhibits the “size effect,” and is governed by the smallest structural dimension, which is comparable with the mean free path of the energy carriers. According to the constructal method, the development of the internal cooling structure proceeds from small to large, in steps of geometric optimization and assembly. This starts at the elemental level, where there is only one high-conductivity layer for collecting and evacuating the heat. The shape of the smallest volume can be optimized for minimal thermal resistance. Next, a first construct is formed by optimizing the number of assembled elements and the internal geometric features of the assembly. The method is repeated at the second construct level, where several first constructs are grouped so that their global thermal resistance is minimal. The construction reveals an internal multiscale structure shaped as a tree, where the spaces between the smallest branches are ruled by nanoscale heat transfer. It is shown that the transition from regions with nanoscale heat transfer to regions with conventional heat transfer is governed not only by the smallest dimensions, but also by heterogeneity (relative amounts of high- and low-conductivity materials).