Optimal distribution of viscous dissipation in a multi-scale branched fluid distributor

This paper examines some theoretical aspects of the optimal design of multi-scale fluid distributors or collectors, built on a binary or quaternary branching pattern of pores. The design aims to distribute uniformly a fluid flow over a specified square surface (uniform irrigation) while simultaneously minimizing the residence time, the residence-time distribution, the pressure drop and the viscous dissipation, leading to an optimization problem of the pore-size distribution, for both length and diameter. For the binary branching, the uniform distribution of outlet points requires a particular, non-monotonous scaling law for pore lengths, and this distinguishes the structure from fractal branching patterns that have been studied previously. The quaternary branching allows a fractal-type structure (constant scale ratios for both pore length and radius). An important general result is established: in the optimal pore-size distribution, the density of viscous dissipation power (W⋅m−3) is uniformly distributed over the volume at all scales.