Constructal tree network for fluid flow between a finite-size volume and one source or sink

Abstract

The ‘constructal theory’ of formation of structure in nature is extended to fluid-flow systems. The fluid flow path between one point and a finite-size volume (an infinite number of points) is optimized by minimizing the overall flow resistance when the flow rate and the duct volume are fixed. The solution is constructed as a sequence of optimization and organization steps. The sequence has a definite time direction: it begins with the smallest building block (elemental system, with flow by volumetric diffusion), and proceeds toward larger building blocks (assemblies, with flow collected in ducts). Optimized at each level are the shape of the assembly, the number of constituents (ie, smaller assemblies), and the distribution of the duct volume. It is shown that the ducts of the optimized assemblies form a tree-like structure, in which every architectural detail is deterministic. It is also shown that the structure cannot be determined when the time direction is reversed, from large elements toward smaller elements. The general importance of the constructal law (access-optimization principle) in physics, biology and economics is discussed.