Page 78 - Computational Modeling in Biomedical Engineering and Medical Physics

Shape and structure morphing of systems with internal flows 65

energy current q i 5 _m i c p ΔT t;i c p is the specific heat of blood and ΔT i is the tempera-
ture difference between the two currents at level i. Such a counterflow provides a lon-
gitudinal temperature gradient ΔT i /L i . Its companion convection heat current
between the vessels and through the tissue in between is
2

_ m i c p ΔT i
q i 5 ; ð2:28Þ
h i p i L i
where h i and p i are the total heat transfer coefficient between the currents and, respec-
tively, the contact perimeter between the two currents. An order of magnitude analy-
sis of Eq. (2.28) yields
q i L i k
ΔT i B ;
h i p i c 2 ð2:29Þ
p
where k B k f (thermal conductivities of blood and tissue, respectively) and h i B k/D i .
The resistance of the fluid inside the channel is of the order B D i /k f and the resistance
of the solid tissue in between the two tubes scales as B t i /k, where t i is of the same
order of magnitude with D i .
The two fluid currents make a single convective tree in counterflow, with zero net
mass flux, which spans from the internal, metabolic temperature of the animal (at i 5 0)
to the skin temperature. Using the flux conservation laws, for mass flow rate, N i _m i 5 _m 0
(constant), and heat rate, N q 5 q (constant), where N is number of branches, the
0
i
i
i
overall temperature difference ΔT (constant) associated to the warm blood vasculariza-
tion is then (Bejan, 2001a,b)
n n
X q 0 k X
ΔT 5 ΔT i B N i L i ;
2 2
_ m c ð2:30Þ
i50 0 p i50
i
which, using L i11 /L i 5 f, L i 5 L 0 f i , L n 5 L 0 f n , and N i 5 2 , yields an estimation of the
total heat rate
2 kL n f 2n 2f n11 2 1
q 0 B q o ðÞ :
2 ð2:31Þ
_ m 0 c ΔT 2f 2 1ð Þ
p
For the reason that both q 0 and _m 0 are proportional with the metabolic rate it fol-
lows that their ratio, q 0 = _m o , does not depend on the size of the body, n. Moreover,
from the standing point of the constructal method, the length of the elemental volume
L n is assumed constant. It was shown (Bejan, 2001a,b) that the order of magnitude of
the hosting volume is
n n11
2 1 2 f
3
VBL ;
n ð2:32Þ
f 1 2 f