Parabolic scaling of tree-shaped constructal network

We investigate the multi-scale structure of a tree network obtained by constructal theory and we propose a new geometrical framework to quantify deviations from scale invariance observed in many fields of physics and life sciences. We compare a constructally deduced fluid distribution network and one based on an assumed fractal algorithm. We show that: (i) the fractal network offers lower performance than the constructal object, and (ii) the constructal object exhibits a parabolic scaling explained in the context of the entropic skins geometry based on a scale diffusion equation in the scale space. Constructal optimization is equivalent to an equipartition of scale entropy production over scale space in the context of entropic skins theory. The association of constructal theory with entropic skins theory promises a deterministic theory to explain and build optimal arborescent structures.