Biomedical Engineering Reference
In-Depth Information
Cutaneous layer
Venous vessel
Arterial vessel
Deep tissue layer
FIGURE 1.4
Schematic view of countercurrent heat exchange near the skin surface.
All dependent variables in the microscopic governing equations for the
arterial blood, venous blood, and tissue phases are decomposed in this man-
ner, and then these governing equations are integrated over the local control
volume. After some manipulations following Nakayama and Kuwahara (2008)
and Nakayama et al. (2008), we obtain the volume averaged set of the gov-
erning equations, which can be written assigning the subscripts
a
,
v
, and
s
to
arterial blood vessels (arteries and arterioles), venous blood vessels (veins and
venules), and tissue, as follows:
For the arterial blood phase:
a
∂ε
a
u
j
+
ω
a
= 0
(1.37)
∂x
j
a
1
ρ
∂
p
ν
K
aij
ε
a
a
ω
a
u
i
int
= 0
−
−
u
j
−
(1.38)
∂x
i
a
∂
T
∂
∂x
j
ε
a
a
a
ε
a
ρ
f
c
p
f
+
ρ
f
c
p
f
u
j
T
∂t
ε
a
k
a
∂
T
a
∂x
j
a
∂
∂x
j
∂
T
a
s
)
a
ρ
f
c
p
f
ω
a
=
+
ε
a
k
disa
jk
−
a
a
h
a
(
T
−
T
−
T
∂x
k
(1.39)
For the venous blood phase:
v
∂ε
v
u
j
+
ω
v
= 0
(1.40)
∂x
j
v
1
ρ
∂
p
ν
K
vij
ε
v
v
ω
v
u
i
int
= 0
−
−
u
j
−
(1.41)
∂x
i
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