Biomedical Engineering Reference
In-Depth Information
transfer and perfusion between the fluid and solid, must be modeled. For the
interfacial heat transfer, Newton's cooling law may be adopted as
1
V
∂x
j
n
j
dA
=
a
f
h
f
f
k
f
∂T
s
T
−
T
(1.23)
A
int
where
a
f
and
h
f
are the specific surface area and interfacial heat transfer
coecient, respectively. For the bundle of vascular tubes of radius
R
,wehave
a
f
=2
ε/R
and
h
f
=
Nu
(
k
f
/
2
R
), such that
a
f
h
f
=
Nu
εk
f
/R
2
, where
Nu
is the Nusselt number based on the local diameter of the vascular tube. If
the local porosity
ε
and specific surface area
a
f
are provided for the complex
tissue-vascular structure, we may estimate the interfacial heat transfer coef-
ficient using
h
f
=
Nu
(
k
f
a
f
/
4
ε
). Roetzel and Xuan (1998) set
Nu
= 4.93 for
both arterial and venous blood vessels. We may appeal to a numerical exper-
iment proposed by Nakayama et al. (2002) for complex porous structures.
As for modeling the blood perfusion term, we may refer back to Figure 1.3
and note that the transcapillary fluid exchange takes place between the blood
and the surrounding tissue. However, the fluid lost from the vascular space will
be compensated by the flow of extravascular fluids and lymph from the tissue
to vascular space. It is quite reasonable to assume that extravascular fluids
and all lymph in the tissue space have the same temperature as the tissue
itself. Thus, we assume that the transcapillary fluid exchange takes place at
the rate of
ω
(m
3
/sm
3
) and model the blood perfusion term as
ρ
f
c
p
f
u
j
Tn
j
dA
=
ρ
f
c
p
f
ω
s
1
V
f
T
−
T
(1.24)
A
int
Note that the perfusion rate
ω
, unlike that of Pennes, varies locally, and
we assume that its local value is provided everywhere. Pennes found that his
model fits the experimental data for
ω
=2
10
−
4
(m
3
/sm
3
). The
perfusion rate varies spatially. In general, it is not an easy task to do
in vivo
measurements for living tissues. Charny (1992) in his review describes how to
measure the perfusion rate in terms of the effective thermal conductivity.
Furthermore, the surface integral terms
10
−
4
to 5
×
×
k
V
A
int
Tn
j
dA
present the tortuosity heat fluxes, which are usually small, as con-
vection dominates over conduction (see e.g., Nakayama et al. 2001). There-
fore, their effects may well be absorbed in effective thermal conductivities,
as done by Xuan and Roetzel (1997). Having modeled the terms associated
with dispersion, interfacial heat transfer, blood perfusion, and tortuosity, the
individual macroscopic energy equations may finally be written for the blood
and tissue phases as follows.
For the blood phase:
V
A
int
k
f
Tn
j
dA
and
−
εk
f
∂
f
f
f
∂
T
∂
∂x
j
∂
∂x
j
T
∂
T
f
=
ερ
f
c
p
f
+
ρ
f
c
p
f
u
j
T
+
εk
dis
jk
∂t
∂x
j
∂x
k
a
f
h
f
s
− ρ
f
c
p
f
ω
s
f
f
−
T
−
T
T
−
T
(1.25)
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