Abstract
Many natural systems have inspired the design of engineering artifacts. Example systems include a broad array of products inspired by pseudo-fractal structures found in nature, including bronchial tubes, watersheds, lightning, and leaf veins. Notably, these systems all resolve a volume-to-point flow problem, i.e., diffusive transportation of heat, energy or fluid into a single flow channel from an initial dispersal throughout a substrate volume (coalescence). Flow diffuses slowly throughout the bulk of the substrate medium towards channels. Once in the channels, flow proceeds rapidly towards the sink. These channels are branched and converge at a single point. Several engineering design problems require a volume-to-point flow solution, e.g., the internal conductive cooling of a microchip. This work introduces a novel geometry-based transport analysis, referred to as path length analysis, for evaluating the performance of systems designed for volume-to-point flow. The analysis is inspired by two principles. First, diffusive flow tends to follow the "path of least resistance." Second, the effort required to diffuse material or energy along a path is proportional to the length of the path. Novel channel configurations may be developed using these two principles. These configurations have a pseudo-fractal type structure and are significantly more efficient than the state-of-the-art for cooling problems such as the conductive cooling of a microchip. An extensive finite element analysis confirms the performance for the example of microchip cooling. The primary results include (1) path length optimization leads to high performing structures with a 'natural' appearance, and (2) path length analysis facilitates a new understanding and design tool for analyzing volume-to-point flow problems.