Physica A 384 (2007) 719–724 Parabolic scaling of tree-shaped constructal network Diogo Queiros-Conde a,Ã , Jocelyn Bonjour b , Wishsanuruk Wechsatol c , Adrian Bejan c a E ´ cole Nationale Supe´rieure de Techniques Avance´es, 32 Bd Victor, 75015 Paris, France b CETHIL-UMR 5008, CNRS INSA-Lyon Univ. Lyon 1, 9 rue de la Physique, F-69621 Villeurbanne, France c Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, USA Received 21 July 2006; received in revised form 3 April 2007 Available online 18 May 2007 Abstract 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. r 2007 Elsevier B.V. All rights reserved. Keywords: Scale-dependent fractals; Constructal theory; Networks; Entropic skins geometry 1. Introduction Networks and multi-scale approaches represent important tools in physics and life sciences. Mandelbrot’s pioneering proposal of fractal geometry [1] has been widely applied, but in spite of its incontestable importance in science, two main limitations have been identified: (i) the number of decades (between 0.5 and 2 decades) where fractality is evidenced is small [2]; (ii) over this limited scale range, the fractal dimension appears to be scale-dependent (which is paradoxical) in a significant number of studies [3–7]. In order to overcome these limitations, we propose here a new geometrical background combining constructal theory [8], which is a method to generate flow architectures such as tree-shaped networks, and entropic skins geometry [7] (ESG), which describes deviations from scale invariance by a diffusion equation through scale space. Constructal theory [8] is a theoretical view of how the configuration of flow systems would be generated everywhere, in nature and engineering. In this theory, the generation of flow configuration is reasoned on the basis of a physics principle of maximization of flow access in flow systems that are free to morph. For example, ARTICLE IN PRESS www.elsevier.com/locate/physa 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.05.037 Ã Corresponding author. E-mail address: diogo.queiros-conde@ensta.fr (D. Queiros-Conde).