Optimal design and pressure drop in fractal arborescent structures

International audienceDichotomic or polytomic arborescences are used in engineering as fluid distributors or collectors, for example for evenly feeding multi‐tubular heat exchangers or catalytic reactors. They also represent idealized models for natural arborescent structures such as the vascular or the bronchi systems of animals, or the root and branch system of vegetals. Both natural and man‐made such structures obey some kind of optimization principle, in view of handling the topological “point‐to‐surface (volume)” problem. The present contribution aims at studying the optimal size distribution (scaling law) of the channels constituting such an arborescence from the point of view of minimizing the pressure drop under geometric constraints (finite total volume of the pore structure, specified surface or volume to be irrigated). Obviously, the laws governing the flow behavior in the channels considered play an essential role in this optimization, and these laws may be very different, depending on the rheological behavior of the fluid, on the turbulent character of the flow, on the geometry of the channels. It is expected that singularities play an important role, because of the many branching and junctions, at short distances from each other, resulting in essentially non‐established flow regimes. The present optimization takes the singular pressure losses in T‐type junctions into consideration and shows how the total pressure drop, the viscous dissipation and the optimal size distribution between scales are affected by the singularities. The present contribution shows that analytical solutions may be found for this problem under certain assumptions, and these solutions furnish a convenient way to characterize the optimal size distribution and the minimal pressure drop. These solutions are compared to some numerical resolutions and the influence of different parameters, essentially the flow‐rate, is investigated