Page 72 - Computational Modeling in Biomedical Engineering and Medical Physics

Shape and structure morphing of systems with internal flows 59

longest possible inspire and expire intervals. On the other hand, the main respira-
tion function demands, t 1 and t 2 ought to be finite, as required to ease the O 2
transfer from the inhaled fresh air and the vascularized tissue of the pulmonary pas-
sages, through the surfaces that secede the air passages from the pulmonary vascu-
larized tissue. Oxygen transfer occurs through diffusion, on both sides of the
parting interface, due to the small sizes of both the terminal branching of the air-
ways (alveoli) and the capillary blood vessels. In the region irrigated with blood,
1=2 ,where δ
ð
more restrictive to mass transfer, it is of the order jBDΔC=δ, δB Dt 1 Þ
is the mass penetration depth.
For example, humans’ respiration is at B60 bpm, that is, t 1 B1 s. Oxygen diffusivities
2
in the region with air and the region with blood are 10 25 and 10 29 m /s, respectively
(Hydei et al., 1996). After 1 s, in the alveoli the penetration depth is δ B 3mm, two
orders of magnitude larger than the alveoli scale, B50 μm (Grotberg, 1994).In the
blood irrigated region δB30 μm, that is, of the same order of magnitude as the capillar-
ies (Schmidt-Nielsen, 1972). The amount of Oxygen transferred is m 5 jAt 1
DΔC
mB At 1 ;
1=2 ð2:17Þ
ð Dt 1 Þ
|fflfflfflffl{zfflfflfflffl}
j

where A is the area of the overall contact mass transfer surface of the airways, and the
per cycle averaged mass transfer rate
1=2
t 1
1=2
_ m 5 m= t 1 1 t 2 ÞBAD ΔC ; ð2:18Þ
ð
t 1 1 t 2
is due to the lungs and their muscles, which pose a constraint on the needed inspire
and expire time intervals
1=2
t _ m
1 B 5 K constantÞ:
t 1 1 t 2 AD 1=2 ΔC ð ð2:19Þ
Returning to the average power requirement, _ W, eliminating t 2 and solving
@ _ W=@t 5 0 for minimum power consumption for respiration, yields the optimal inspi-
ration time (Bejan and Errera, 1997).
" ! n #

1=2 1 n
1 2 Kt 21 1 1 B :
1;opt 1=2 ð2:20Þ
Kt 1;opt 2n 1 1
A periodic flow is then required by the minimization of the respiration mechanical
power. The particular flow regime, n, has no significant influence on the order of
22
magnitude of t 1,opt B K . For instance, when the animal is small enough such