Biomedical Engineering Reference
In-Depth Information
v
∂
T
∂
∂x
j
ε
v
v
v
ε
v
ρ
f
c
p
f
+
ρ
f
c
p
f
u
j
T
∂t
ε
v
k
v
∂
v
v
∂
∂x
j
T
∂
T
v
s
)
v
ρ
f
c
p
f
ω
v
=
+
ε
v
k
disv
jk
−
a
v
h
v
(
T
−
T
−
T
∂x
j
∂x
k
(1.42)
For the solid tissue phase:
s
ε
)
ρ
s
c
s
∂
T
(1
−
∂t
(1
+
a
a
h
a
(
s
∂
∂x
j
ε
)
k
s
∂
T
(1.43)
a
s
)+
ρ
f
c
p
f
ω
a
a
=
−
T
−
T
T
∂x
j
v
s
)+
ρ
f
c
p
f
ω
v
v
+(1
+
a
v
h
v
(
T
−
T
T
−
ε
)
S
m
where
ε
a
and
ε
v
are the volume fractions of the arterial blood and that of the
venous blood, respectively, such that
ε
=
ε
a
+
ε
v
. The terms associated with
the surface integral are modeled as
+
ν
f
∂u
i
n
j
dA
=
1
V
f
p
ρ
f
∂x
j
+
∂u
j
ν
f
K
ij
ε
f
−
−
u
j
(1.44)
∂x
i
A
int
which is simply Darcy's law and
ρ
f
u
j
n
j
dA/V
=
ρ
f
ω
(1.45)
A
int
is the mass flow rate per unit volume through the interface
A
int
, modeled in
terms of the perfusion bleed-off rate
ω
(1/sec). The perfusion bleed-off rate
ω
describes the volume rate of the fluid per unit volume, bleeding off to the solid
matrix through the interfacial vascular wall. Thus, the momentum bleed-off
rate is modeled as
ρ
f
u
i
u
j
n
j
dA/V
=
ρ
f
ω
u
i
int
(1.46a)
A
int
where
u
i
int
is the velocity vector averaged over the interface. Likewise, the
enthalpy bleed-off rate is modeled as
f
ρ
f
c
p
f
u
j
Tn
j
dA/V
=
ρ
f
c
p
f
ω
T
(1.46b)
A
int
For the interfacial heat transfer, Newton's cooling law is adopted as
∂x
j
n
j
dA
=
a
f
h
f
f
1
V
k
f
∂T
s
T
−
T
(1.47)
A
int
where
a
f
and
h
f
are the specific surface area and interfacial heat transfer coef-
ficient, respectively. Furthermore,
k
dis
jk
is the thermal dispersion conductivity
tensor, as introduced in Nakayama et al. (2006).
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