Prior analyses and experiments have demonstrated that varying or scaling the number of fluid channels in each layer of a stacked multi-layer heat sink yields distinct advantages over traditional single-layer designs which use channels with high aspect ratios. Specifically, a design which implements scaling in order to vary the porosity (or equivalently, the number of channels) from one layer to the next allows a given thermal performance to be realized at a lower pressure drop than the corresponding non-scaled design. While previous studies use porous media theory for their analytical foundation just as the current studies do, they also focused on heat exchangers with machined channels, and have consequently been limited to discrete variation of the porosity (or number of channels) in each layer. Extending this approach by allowing for continuous porosity variation provides a generalized and powerful design method for scaled multi-layer heat exchangers by mathematically modeling them using two-equation volume averaged quantities. This approach also yields insight into the fundamental design parameters which control the performance of this class of heat exchangers, and suggests that their performance-governing parameters are similar to those which govern fin performance.
This paper focuses primarily on the thermal behavior of these liquid cooled heat sinks which employ various scaling functions to define the spatial variation of porosity. Comparisons are made with available closed-form solutions as well as results available from prior studies of discretely-scaled designs. The results indicate that scaling laws which have not been investigated previously can likely yield further performance improvements relative to prior designs.