Page 77 - Computational Modeling in Biomedical Engineering and Medical Physics

64 Computational Modeling in Biomedical Engineering and Medical Physics

the optimization of the pumping mechanical power minimization in tree flows in
physiology and river morphology (Thompson, 1942; West et al., 1997).
However, the minimum pumping power and the minimal corporal heat loss are parts
of the same optimization principle: the constructal principle or how to be best fit, and it
brings in the metabolic rates of all vertebrates, with warm blood (3/4) and with cold blood
(2/3). Here theheattransferisnot considered individually,exclusively, butitiscombined
with the pumping power minimization in fluid constructal trees (Bejan, 2000a,b).
The minimization of the fully developed Hagen Poiseuille flow resistance of a fluid
tree network (each mother tube, of length L i and diameter D i , bifurcates into two identi-
cal daughter tubes, of lengths L i11 and diameters D i11 ) when limiting the total volume of
the tubes, unveils the optimal ratio of the successive diameters D i11 /D i 5 2 21/3 .This
finding, the old Murray law, is particularly robust because it is independent of the lengths
(L i , L i11 ) and of the relative positions of the three (mother and daughters) tubes. It is not
even essential to recognize the constructal method as foundation of this structure. The
heat transfer analysis may be conducted by accepting, heuristically, this tree network
(Bejan, 2000a,b) or the one in (Westetal.,1997)—it is as an assumption only.
The blood vessels trees and the superposition of the venous and arterial trees is so inti-
mate that each tube of one tree is in counterflow with a tube of the other (Fig. 2.12,left),
and the reality of counterflow of blood and other fluids (Fig. 2.12,right)are widely recog-
nized in physiology (Schmidt-Nielsen, 1984; Vogel, 1988). In the analysis of the heat cur-
rent between the root and the canopy in a warm-blooded animal of concern is the heat
lost through the subcutaneous level, and heat is transferred laterally, between the arterial
(warmer) and venous (colder) currents (Section 2.3.4).
In the volume delimited by the adiabatic boundary the enthalpy of the warmer
fluid is larger than that of the colder fluid, the counterflow conveys longitudinally the

Figure 2.12 Trees of counterflow convective trees.