Cylindrical trees of pin fins

In this paper, we extend the constructal optimization method to cylindrical assemblies of pin fins. The assembly is arranged as a tree with one stem and many radial branches. The optimization consists of maximizing the global conductance subject to fixed total volume and amount of fin material. The length scale of the spacing between adjacent elemental fins is selected based on earlier results regarding the forced convection of compact electronic packages. The optimized features of the tree construct are the external shape (height/diameter) and the internal ratio between the stem diameter and the diameter of the elemental fins. The paper shows how the geometric optimum responds to changes in the remaining parameters of the design: the volume fraction occupied by fin material, the free stream velocity, and the Prandtl number. The optimized geometry is relatively robust. It is shown that optimized trees with tapered fins have slightly larger global conductances, but nearly the same external and internal aspect ratios. This last comparison also shows that the more efficient trees look more natural.