Constructal tree networks for the time-dependent discharge of a finite-size volume to one point

This paper shows that the time needed to discharge a volume to a concentrated sink can be minimized by making appropriate changes in the geometry of the flow path. The time-dependent flow of heat between a volume and one point is chosen for illustration, however, the same geometric optimization method (the constructal principle) holds for other transport processes (fluid flow, mass transfer, conduction of electricity). There are two classes of geometric degrees of freedom in designing the flow path: the external shape of the volume, and the distribution (amount, location, orientation) of high-conductivity inserts that facilitate the volumetric collection of the discharge. The optimization of flow path geometry is executed in a sequence of steps that starts with the smallest volume elements and proceeds toward larger and more complex volume sizes (first constructs, second constructs, etc.). Every geometric feature is the result of minimizing the time of discharge, or the resistance in volume-to-point flow. The innermost details of the structure have only a minor effect on the minimized time of discharge. The high-conductivity inserts come together into a tree-network pattern which is the result of a completely deterministic principle. The interstices are equally important in this optimal design, as they are occupied by the low-conductivity material in which the energy charge was stored initially. The paper concludes with a discussion of the relevance on this deterministic principle—the constructal law—to predicting structure in natural flow, and to understanding why the geometry of nature is not fractal.