Multiple Length Scales for Maximal Heat Transfer Density in Forced and Natural Convection

The present paper is a review of a new conceptual design for the maximization of heat transfer density (i.e., heat transfer rate per unit of volume) in channels installed in a fixed volume. The volume is filled optimally with parallel equidistant heated blades. The optimal spacing between two blades is such that the thermal boundary layers merge at the end of the channel. The blades can be cooled either by laminar natural or forced convection. Unheated volumes of fluid near the tips of the boundary layers are eliminated through the insertion of new generations of smaller blades. Based on the same principle, new generation of even smaller blades are added stepwise to the multi-scale structure. The optimal length of each family of new blades is determined based on the assumption that the flow downstream the smallest plates is not disturbed. This allows us to predict the exact height where the thermal boundary layer of two distinct generations will merge. As the number length of scales increase, the volume-averaged heat transfer density increases. The results show that the improvement associated with optimal insertion of the first and second generations of scales is significant. Diminishing returns are also observed as the complexity increases, meaning that the contribution of each smaller scale is less important than the contribution of the preceding scale. The theoretical results (i.e., the optimal spacings, optimal lengths, maximal heat transfer density and the cutoffs) are validated numerically.