This work focuses on an analytical approach to understand optimal material utilization in metal matrix heat exchangers. An objective of this work is to develop a bridge between a fully defined and discrete structure to that of a functionally graded porous media. A porous media heat exchanger is a structure which uses porous material, such as a metal foam, to achieve large convective surface areas in a small volume while also using the media as a conductive path from the heat source or sink. Therefore, a functionally graded porous media heat exchanger has a porosity that is specified as a function of position. Constructal theory is used here to develop increasingly complex convective fin structures, optimized at each level of complexity, which have a resulting characteristic of 2-D functional grading. The approach described here is developed from first principles by using Fourier’s law to develop analytical solutions and seeks to yield an optimized heat exchanger configuration that maximizes total heat transfer subject to a fixed amount conductive material, total volume, and flow condition.

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