Page 62 - Computational Modeling in Biomedical Engineering and Medical Physics

Shape and structure morphing of systems with internal flows 49

conduction heat transfer problem (a Poisson problem, Chapter 1: Physical,
Mathematical, and Numerical Modeling). This constructal design embodies and is the
objective, the deterministic result of the physics that governs the system: gradient
driven diffusion, or conduction (heat, mass, electrical current). It is scalable (self-similar)
and robust—higher order ensembles have the same volume-to-point resistance. This
principle may suggest that the core of the tree construction of many animate and inan-
imate systems is one single design principle: the volume-to-point constrained minimi-
zation of the overall flow resistance between one port on the boundary and the finite,
contained volume (territory), which is an infinite number of points—one for all and
all for one, topologically possible in a continuous medium. A design principle so gen-
eral confers predictability to the tree network structure, with its main properties.

Fluid trees
The constructal optimized ensemble, Fig. 2.1 (Morega and Proca, 2004), was
introduced later in physiology, for fluid trees as a three-dimensional convenient,
representative model for the vascular arborescence (Cohn, 1954, 1955).Itwas
known, experimentally, that every mother vessel splits into two smaller, daughter
vessels, as it was known and demonstrated for tubes, based on flow resistance mini-
mization, that the diameter must decrease by a constant factor (2 21/3 )ateach
bifurcation: this result had been derived in Thompson (1942). And the geometric
description of these constructions that is presented without theory in the fractal
geometry, as a heuristic model of the lung bronchial tree (Mandelbrot, 2020) is, in
fact, a two-dimensional rendering of Cohn’s (1954) ramified fluid system (Bejan
and Zane, 2012; Morega, 2013).
To solve the essential problem of the volume-to-point flow, Fig. 2.2 Morega and Bejan
(2005), with minimum resistance (Bejan, 1988, 1997a,b) a two-dimensional representation
of area A was assumed.
The flow through the port on the boundary is connected to each point of the
0 _
territory, and the mass, m ,and volumetric, _mw, flow rates are related through
0 _
000 _
m 5 m A. The volume is a single fluid saturated porous medium with constant
properties. Extending Darcy model (Chapter 1: Physical, Mathematical, and
Numerical Modeling) to a more general case, A is filled here by an inhomoge-
neous porous medium made of a low permeability material (K) and a small vol-
ume fraction of inserts (cracks, open, or filled, etc.) of significantly higher
permeability (K 1 , K 2 ,.. .), of unspecified thicknesses (D 1 , D 2 ,...)and lengths (L 1 ,
L 2 ,...). Thus the elemental cell (A 0 ) is made of the low permeability material
and high permeability strip (K 0 , D 0 ). Each successive higher order ensemble of
volume (A i )isa set ofpreviousorder ensembles ofsize (A i－1 ), which are
tributaries for the collector layer (K i , D i , L i ).