Thermodynamic and Geometric Optimization of Systems with Flow Irreversibilities

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This project extends in several new directions the method of optimizing thermodynamically the geometry of tree-shaped flows between points and volumes. The geometric details of the flow structure will be deduced from the maximization of global thermodynamic performance (e.g., overall flow resistance) subject to global constraints. Examples are the tree-shaped networks that collect and distribute water and electric power over a territory. The method will also be extended to economics, where the flow consists of goods, and the global objective is revenue maximization. In the field of bioheat transfer, the method will be applied to the pairs of fluid trees (arteries and veins) that are present in vascularized tissues. Flows that are shaped as trees are found everywhere in natural systems, animate and inanimate (e.g., lungs, rivers, lighting). They are also prevalent in engineering, economics and society. The objective of this project is to deduce the optimal geometric structure of tree-shaped flows from the principle of maximizing the global performance of the system permeated by the flow. For example, in a tree network for the distribution of water and electric power over an urban area the global objective is minimum flow resistance and minimum cost. In economics, in the distribution or collection of goods over a territory, the objective is the maximization of revenue. In vascularized tissues under the skin, where trees of arteries and veins are arranged in pairs and in counterflow, the objective is the minimization of flow resistance and body heat loss. All these flow structures will be generated and studied from the point of view of geometric optimization based on global thermodynamic optimization.