Constructal theory: from thermodynamic and geometric optimization to predicting shape in nature

This is a review of recent theoretical developments toward predicting macroscopic organization (the occurrence of shape and structure) in natural flow systems, animate and inanimate. The starting point is the question of how to optimize the access between one point and a finite volume (i.e., an infinite number of points). If the volume is an electronic device that generates heat uniformly, then access optimization means minimum thermal resistance between the volume and a point-size heat sink. Similarly, if the volume must be bathed at every point by a flow (e.g., air flow in the lung, or blood flow in a capillary bed), optimal access means minimum flow resistance between the volume and a source or sink. The main discovery is purely geometric: any finite-size portion of this composite can have its shape optimized such that its overall resistance to flow is minimal. Consequently, the optimal-access solution for the total volume is obtained by optimizing volume shape at every length scale, in a sequence that begins with the smallest building block (elemental system), and proceeds toward larger building blocks (assemblies, constructs). The solution is constructed, hence the “constructal” name of the associated theory. The paths form a tree network in which every single geometric detail is determined theoretically. The tree network cannot be determined theoretically when the time direction is reversed, from large elements toward smaller elements. The constructal principle is further illustrated for fluid flow between a volume and one point, for minimum-time travel between an area and one point, and for minimum-cost economics structures.