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Shape and structure morphing of systems with internal flows 63

the interaction between its two biological machines exists, quantified by Eq. (2.27),
where N is specific to each stationary state, throughout the underlying circadian cycle. 7
From the TFS perspective, this CPS activity may be characterized as a thermodynamic
system by computing its power, entropy, and efficiency.
In the constructal philosophy, the structure and shape of each of the two parts
(heart and lungs) is the result of purpose and constraints, and in the constraints
class, each of them constraints the other. Thus together with the other body’sparts
they work optimally (synchronized), as a whole. Both rhythmicity and synchroni-
zationgodowntothe elementalcell, the excitable (here, cardiac) tissues
(Chapter 4: Electrical Activity of the Heart). The malfunction of one of them trig-
gers the disordered function of the other one, as an attempt of the whole to adapt
to the new situation (Zemlin et al., 2009).

2.5 The effect of body size
2
The respiration frequency decreases with the animal’s size as _m=A , Eq. (2.26).To
predetermine its relation with the body size two empiric or theoretical laws that relate
the metabolic rate (or _m) and the area of the contact surface for mass transfer, A, with
the body mass, M, respectively, are needed.
As said, allometric laws (Chapter 1: Physical, Mathematical, and Numerical
Modeling) are power laws, largely accepted, between geometric and functional,
flow parameters of the living creatures (Kleiber, 1932; Metabolic Rate and
Kleiber's Law, 2020; Murray 1926a,b; Peters, 1983; Schmidt-Nielsen, 1984).
Their predetermination on a purely theoretical basis was a true endeavor. One of
the most difficult to confirm is the proportionality that exists between the meta-
bolic rate and the body mass, or volume V,raisedto3/4 power.
Because the heat leak to the environment through convection is proportional to
the body’s surface, the metabolic rate has to be proportional to the square of the char-
1/3
acteristic length, V , squared that is with the body mass, or volume raised to power
2/3. However, this heat transfer theoretical result is infirmed through observations on
the mammals and birds, which show off an exponent much closer to 3/4 than 2/3 for
bodies of small size reptiles, amphibians, and fish. The 3/4 law does not interrelate the
cold blood vertebrates although these small size bodies are equally dominated by arbo-
rescent flow structures. From another standpoint, other theoretical studies are based on

7
Circadiam comes from latin: circa (around), and diem (day). The circadian rhythm is a natural, internal
process that regulates the sleep wake cycle and repeats roughly every 24 hours (Brain Basics:
Understanding Sleeping, 2020). It can refer to any biological process that displays an endogenous,
entrainable oscillation of about 24 hours.