Constructal trees of circular fins for conductive and convective heat transfer

This paper extends to three dimensions and to convective heat transfer the constructal method of minimizing the thermal resistance between a volume and one point. In the first part of the paper, the heat flow mechanism is conduction, and the heat generating volume is occupied by low conductivity material (k 0) and high conductivity inserts (k p). At the elemental-volume level the inserts are shaped as constant-thickness disks. In the first assembly the disks are mounted on a common stem of k p material. The internal and external geometric aspect ratios of the elemental volume and the first assembly are optimized numerically subject to volume constraints. In the second part of the paper the interstitial spaces once occupied by k 0 material are bathed by forced convection. The k p inserts function as ‘fin material’, and the first assembly becomes a fin bush with cylindrical symmetry. The geometry of the first assembly is optimized subject to total volume and solid volume constraints. The optimal number of circular fins, the optimal external shape of the assembly, the optimal ratio of the central stem diameter divided by the circular fin thickness, and the maximized global conductance of the assembly are reported as functions of the external flow (pressure drop number) and volume fraction of fin material. It is also shown that the optimization sequence can be shortened (albeit with approximate results) by adopting at the first-assembly level the internal geometric aspect ratios that were optimized independently at the elemental-volume level.