Optimization of tree-shaped fluid networks with size limitations

In this paper, we show how to minimize the pumping power requirement in a fluid tree-shaped network under different size constraints (volume, surface, length). The Lagrange multiplier method is applied to obtain a problem formulation in which the pipe diameters do not appear explicitly. It is found that such a formulation exists for both volume and surface constrained networks. In Y-shaped junctions, optimal angles of branching and diameter ratios are determined. A different approach aiming at minimizing the network global cost (summation of size and pumping costs) is presented. It is showed that the geometrical features of the network are the same when one minimizes the global cost rather than minimizing pumping power under constraint. An optimal allocation of cost between pumping and size limitation was found. Finally, we extend the global cost minimization approach to the design of a porous architecture. This article provides fundamental tools for the designers of fluid tree-shaped networks.