Page 69 - Computational Modeling in Biomedical Engineering and Medical Physics

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total volume, and it explains why the metabolism rate (the total heat rate) of an animal
has to be proportional to its size, to the total volume, or to the body mass. It is the
resistance to the loss of body heat that relates the metabolism rate and the size of the
body in all animals, with warm blood or cold blood. At the limit of the small dimen-
sions of the bodies of warm-blooded animals, the heat is transferred mainly through
the conductive tissues and the exponent used in the power law (exponential) is, as
observed, of 1/3. In oysters and amphibians, the resistance is dominated by the out-
ward convection of the body surface, and the exponent is the noticed one, of 2/3. In
mammals and birds, the resistance is dominated by counterflows and the exponent is
the one observed, that is, 3/4 (Bejan, 2001a,b).

2.4 Structure in time: rhythm

All flow systems have the unifying principle of the spatial, geometric shape. Nature how-
ever, exhibits not only spatial shape and structure but also temporal structure: rhythm.
The same is true for the engineered systems. Breathing and respiration have many aspects
in common with the engineered processes controlled by the dynamic transfer through
unsteady diffusion. The lung and the vascularized tissue have similar objectives and con-
straints. The fluid (blood) as soon as admitted inside the smallest passages (alveoli, capillar-
ies) initiates the mass transfer within the surrounding tissues. The effectiveness of this
transport mechanisms decreases in time. The maximization of the mass transfer rate implies
the removal of the old fluid batch such that a new batch produces again higher fluxes.
The most known natural flows that exhibit also rhythm are the breathing and the
heartbeats of animals, surprised by the allomoteric laws (Murray, 1926a,b). For instance,
the frequency of the heartbeats are relative to the body mass of the animal raised to
power of B0.25. A mouse breathes faster than an elephant and the human’s heart beats
faster than the heart of the horse.
An inherent question is why nature selected the pulsating flow rather than the sta-
tionary flow to serve these beings. The existence of finely tuned pulsating flows—fre-
quencies, flow resistances, etc.—may be exposed using the constructal principle and
the findings concurs with Murray’s law ascertainments.

Intermittent heat transfer
A glimpse in the optimal structuring in time of a rhythmic process is offered by the
simple loading/discharging of the electric capacitor, C [F], in a PWM (pulse width
modulated) flyback convertor scheme. The capacitor is aimed to transfer sequentially
power from an “upstream” source to a “downstream” load (Veli et al., 2019). During
the loading phase, C is connected in series with a current limiting resistance, R 1 [Ω],
to be powered by a primary voltage source, U 0 [V]—for simplicity U 0 is assumed con-
stant, or the source is of “infinite” power, concept equivalent to the “thermostat” in