Page 67 - Computational Modeling in Biomedical Engineering and Medical Physics

54 Computational Modeling in Biomedical Engineering and Medical Physics

These theoretical considerations concur with the allometric laws qualitatively and
quantitatively. The allometric law accepted for the respiratory contact surface in mam-
mals is (Weibel, 1972; Schmidt-Nielsen, 1984)

0:98
A M
D3:31 ; ð2:8Þ
1m 2 1kg
2
where A and M are presented in m and in kg, respectively. In the ambit of the mam-
3
mals corporal masses (10 23 .. .10 kg) it may be approximated by
A M
B3 :
m 2 kg ð2:9Þ
^
^
3
3
The theoretical relation A i BM i suggests A i BM i /(ρλ D 0 ), and for ρB10 kg/m ,
2
and D 0 B10 μm, for the smallest of the length scale (alveoli) yields
2
A i B 10 M i m :
λ kg ð2:10Þ
Because λ B 3, this theoretical formula is, essentially, the same empiric correlation
Eq. (2.9).
Fig. 2.7 (Morega and Proa, 2004) was obtained using Figs. 2.5 and 2.6. The overall
^
b
contact surface is nearly proportional with φ V i where the exponent b B 0.9, inde-
i
pendent of λ.
The constructal theory may thus predict the allometric relations between the size
of the body, the metabolic rate, the breathing frequencies, and heart beatings.
Counterflow convection trees

Living systems, tissues and animals are vascularized by pairs of trees of wider vessels (arter-
ies, veins) interlaced in counterflow, which are also heat convection paths (Keller and
Seiler, 1971; Wulff, 1974; Chato, 1980; Chen and Holmes, 1980),where theheat fluxis
relative to the temperature gradient of the ensemble. For instance, the inhalation/exhala-
tion function of the lung relies on two countercurrent streams, which facilitate the heat
drain to lower temperatures serving in the mean time as a thermal insulation structure—
the better, the better the thermal contact between the two counterflows is. The most
important convective configuration of vessels is that in which the two vessels are perfectly
interlaced such that in each pair of parallel tubes, embedded in an interstitial space, the
flows are in countercurrent, for example, Fig. 2.8,left(Morega and Proca, 2004). Fig. 2.8,
right shows the same interlaced constructal concept applied to concurrent, redundant elec-
tric power distribution networks (Morega et al., 2008).
The ratios between the diameters of the succeeding tubes and their lengths mini-
mize the flow resistance under the constraints of the total volume of the tubes and the