Biomedical Engineering Reference
In-Depth Information
equation (1.11) for the solid matrix into the corresponding volume averaged
equations as follows:
For the blood phase:
f
∂
T
∂
∂x
j
f
ερ
f
c
p
f
+
ρ
f
c
p
f
u
j
T
∂t
εk
f
∂
ερ
f
c
p
f
u
j
T
f
f
∂
∂x
j
T
+
k
f
V
=
Tn
j
dA
−
∂x
j
A
int
1
V
k
f
∂T
1
V
+
∂x
j
n
j
dA
−
ρ
f
c
p
f
u
j
Tn
j
dA
(1.19)
A
int
A
int
For the solid matrix phase:
(1
Tn
j
dA
s
s
ε
)
ρ
s
c
s
∂
T
∂
∂x
j
ε
)
k
s
∂
T
k
s
V
−
−
(1
=
−
∂t
∂x
j
A
int
k
f
∂T
1
V
1
V
−
∂x
j
n
j
dA
+
ρ
f
c
p
f
u
j
Tn
j
dA
+(1
−
ε
)
S
m
(1.20)
A
int
A
int
s
is the intrinsic average of the solid matrix temperature. Note that
the dispersion heat flux
ρ
f
c
pf
u
j
T
=
ερ
f
c
pf
u
j
T
f
appears in the volume
averaged energy equation (1.19) for the blood phase, which may well be mod-
eled under the gradient diffusion hypothesis:
where
T
f
ερ
f
c
pf
u
j
T
f
=
εk
dis
kj
∂
T
−
(1.21)
∂x
k
A number of expressions have been proposed for the thermal dispersion
thermal conductivity
k
dis
kj
. Nakayama et al. (2006) obtained a transport equa-
tion for the dispersion heat flux vector, which naturally reduces to the forego-
ing gradient diffusion form. For a bundle of vessels of radius
R
, they obtained
the following expression for the predominant axial component of
k
dis
kj
:
ρ
f
c
pf
2
f
R
f
R
1
48
u
ρ
f
c
pf
u
k
dis
xx
=
k
f
<
1
(capillary blood vessels)
k
f
k
f
(1.22a)
k
dis
xx
=2
.
55
ρ
f
c
pf
7
/
8
f
R
f
R
u
ρ
f
c
pf
u
Pr
1
/
8
k
f
>
1
k
f
k
f
(large arteries and veins)
(1.22b)
To close the foregoing macroscopic energy equations (1.19) and (1.20),
the terms associated with the surface integral, describing the interfacial heat
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