Hierarchical transport networks optimizing dynamic response of permeable energy-storage materials

Channel widths and spacing in latticelike hierarchical transport networks are optimized to achieve maximum extraction of gas or electrical charge from nanoporous energy-storage materials during charge and discharge cycles of specified duration. To address a range of physics, the effective transport diffusivity is taken to vary as a power, m , of channel width. Optimal channel widths and spacing in all levels of the hierarchy are found to increase in a power-law manner with normalized system size, facilitating the derivation of closed-form approximations for the optimal dimensions. Characteristic response times and ratios of channel width to spacing are both shown to vary by the factor 2/m between successive levels of any optimal hierarchy. This leads to fractal-like self-similar geometry, but only for m=2 . For this case of quadratic dependence of diffusivity on channel width, the introduction of transport channels permits increases in system size on the order of 10;{4} , 10;{8} , and 10;{10} , without any reduction in extraction efficiency, for hierarchies having 1, 2 and, 8 levels, respectively. However, we also find that for a given system size there is an optimum number of hierarchical levels that maximizes extraction efficiency.