Page 79 - Computational Modeling in Biomedical Engineering and Medical Physics

66 Computational Modeling in Biomedical Engineering and Medical Physics

which paves the path to a theoretical (constructal) relation between the metabolic rate
and the total volume. In the limit, Eq. (2.31),(2f ) n11 .. 1and f n11 ,, 1 q 0 B 2 n
n
and V B (2/f ) ,which yields
log q 0 3
5 ;
logV 4 ð2:33Þ
3/4
which means that q 0 B V , or (because q 0 = _m o does not depend on the body size)
3=4
_ mBM ; ð2:34Þ
which is, in fact, the allometric law.
This two fold geometric optimization with the two restrictions and the pairing of
tubes in constructs larger than the preset elemental volume is the essence of the con-
structal method. The constructal theory combines two ideas—(1) the optimized tree
for minimum pumping power and subject to spatial constraints, and (2) the convection
heat transfer or, better, the insulation characteristic to the two counterflow fluid
trees—and reinstates geometry in the place it deserves, in physiology, river morphol-
ogy, and any other domain where macroscopic structure and form define the flow sys-
tem outside equilibrium.
The convection thermal resistance of the counterflow trees, R1, resides within the
animal, Fig. 2.12 bottom. This resistance acts in parallel with the internal resistance, R2,
which models the conductive heat loss through the solid tissue. Outside the animal, the
heat current passes through the body surface that is exposed to the environment.
For the cold blood vertebrates, the temperature drop on R1 is minimal and, when
changes in the environment temperature intervene, the leading resistance is R3. In
2/3
consequence the heat loss rate and the metabolic rate follow closely V . For warm
blood animals, the body side of the skin poses a significant thermal resistance. In larger
mammals R1 , R2 and the heat current is transferred by the convection tree, and
the metabolic rate matches the predominant tendency V 3/4 .
The lung is also a convection currents tree, the result of two overlapping air trees:
the inspiration and the expiration flows. The convection tree is made of heat currents
and constitutes a corridor for heat flow of the same type as the tree considered above.
During the inspiration the cold air warms up progressively along the passages through
which it travels. During the expiration, the warm air cools down progressively along
the same passages. The tissues of passages walls act as a regenerative heat exchanger in
engineering terminology. Supplementary to the convection tree for thermal insulation,
and relying on the same inlet outlet mechanism, the lung acts as a tree path for mini-
mizing the water loss (Nield and Bejan, 1999).
The thickness of the tissue penetrated by the mass diffusion through the respiration
or the heartbeat is proportional to t 1/2 . The volume or mass of the tissue penetrated