Material Distribution Optimization in a Metal Matrix Heat Sink Using a Constructal-Design Inspired Unit T-Cell

Abstract

Constructal principles are used to investigate the optimization of material utilization in a metal matrix heat sink that maximizes the total heat transfer rate through the base of heat sink. This approach utilizes a two-dimensional geometry to examine spatial heat flow and optimal material distribution in a metal matrix in the plane perpendicular to the coolant flow direction. The matrix is composed of multiple layers of conductive tees built up from the smallest constituent, the unit T-cell. The unit cell consists of a conductive tee-shaped geometry with the two rectangular void regions each making up half of a cooling channel. The horizontal boundaries must match the temperature and heat flux at the boundaries of the neighboring unit cells as this is a conjugate conduction/convection problem. The geometry of the unit cell is characterized by aspect ratios of channel width to height, overall cell width to height, and channel height to cell height. The matrix structure is assembled by stacking unit cells into multiple layers where the number of cells in each layer is an integer multiple of the cells contained in the lower layer. The solution is obtained for optimal T-cell geometric parameters under a set of predetermined constraints including overall volume, solid fill fraction, and number of layers. When a large number of stacked unit cells are considered, the results describe the ideal spatial distribution of porosity and pore sizes for two dimensional functionally graded metal-matrix heat sink. These results will lead to a better understanding of the role played by the porosity distribution in a metal-matrix heat sink and may be applied to the analysis, optimization, and design of more effective heat sinks.