Constructal optimization of arborescent structures with flow singularities

This article presents analytical resolutions of the problem of optimal channel size distribution for arborescent (ramified, branched, tree-like) networks used as flow distributors or collectors. The distributor network connects a single inlet port to an array of outlet ports distributed over a specified square or rectangular surface (point-to-surface problem), and the reverse for a collector. The optimization problem is formulated as follows: find the distribution of channel radii that minimizes total viscous dissipation (or entropy production, pumping power, pressure drop) under constraints of uniform irrigation and of total volume of channels (or of average residence time) and the assumption that the flow is split equally between branches at each junction. With respect to earlier work, the present approach does not assume Poiseuille flow, but accounts instead for singular pressure losses in T-type junctions. Different situations are investigated, with the pressure loss coefficient being either a constant for the whole structure, or depending on local flow conditions. The case where both Poiseuille and singular pressure losses are present is also addressed. Also two types of networks are taken as illustrations: a dichotomic tree (a branch divides into two sub-branches), and a tetratomic tree (a branch divides into four sub-branches). The analytical results are in the form of scaling relations between the different levels of the arborescence, and in the form of distributions for channel radii, dissipation, pressure drop, porous volumes. A so-called “constructal function” is introduced that depends on the network structure and on the pressure loss relation, and is calculable as the sum of a geometric series. All explicit relations for the above quantities may then be expressed in a compact fashion in terms of the constructal function. A remarkable property of the optimized structures concerns the distribution of dissipation density (i.e. dissipation divided by volume): when the singular pressure loss coefficient is independent of local flow conditions, this quantity is uniform over the whole structure, thus satisfying “equipartition of entropy production”. A weaker form of this property is found when the singular pressure drop coefficient depends on local conditions (parity dependence).