Biomedical Engineering Reference
In-Depth Information
Comparison of the foregoing equation against our expression (1.48) for
the tissue reveals that the perfusion term
ρ
f
c
p
f
ω
a
(
v
) is missing.
Obviously, they did not retain the term describing the transcapillary fluid
a
T
−
T
exchange via arterial-venous anastomoses, namely,
A
int
ρ
f
c
p
f
u
j
Tn
j
dA/V
=
ρ
f
c
p
f
ω
f
.
If they did, they would have obtained our expression (48), which
may be rearranged in their form as
T
(1
+
a
a
h
a
+
ρ
f
c
p
f
ω
a
(
s
s
ε
)
ρ
s
c
s
∂
T
∂
∂x
j
ε
)
k
s
∂
T
a
s
)
−
=
−
T
−
T
(1
∂t
∂x
j
+
a
v
h
v
−
ρ
f
c
p
f
ω
a
(
v
s
)+(1
T
−
T
−
ε
)
S
m
(1.55)
In their model, the convection-perfusion parameters, namely, (
a
f
h
f
±
ρ
f
c
p
f
ω
, are replaced by the interfacial convective heat transfer coecients,
a
f
h
f
. This difference should not be overlooked since the perfusion heat sources
could be quite significant for the bioheat transfer in the extremities, as Chato
(1980) demonstrated using his model.
1.5.6 Weinbaum-Jiji Model and Bejan Model
Weinbaum and Jiji (1979) considered bioheat transfer between a paired coun-
tercurrent terminal artery and vein. They took account of the vascular struc-
ture in which vessel number density, velocity, and diameter vary significantly
from the deep tissue layer toward the skin layer. Later, Weinbaum and Jiji
(1985) proposed a simplified model in which an effective thermal conductivity
tensor is introduced as a function of the local blood velocity. They claimed
that the perfusion heat source vanishes within the capillary bed and derived
a single equation to describe the steady-state tissue temperature variations,
which, when the vessels are in parallel to the temperature gradient, reduces to
(1
d
+(1
ρ
f
c
p
f
a
R
2
s
d
dx
u
ε
)
k
s
+
πε
a
2
σ
T
−
−
ε
)
S
m
= 0
(1.56)
(1
−
ε
)
k
s
dx
where
σ
is a geometrical factor of the vessel structure, whereas
R
is the local
radius of the vessel. It is seen that the longitudinal effective thermal con-
ductivity due to countercurrent flow is proportional to the square of blood
mass flow rate. It is also interesting to note that the concept of the longitudi-
nal effective thermal conductivity in countercurrent heat transfer was already
explicit in Bejan (1979) in which he presented a novel method for thermal
insulation system optimization. Bejan (1979) seems to be the first to point
out the relationship associated with the square of the mass flow rate and the
longitudinal effective thermal conductivity by convection. His expression is a
simple one:
(
m
f
c
pf
)
2
UP
s
d
T
Q
=
−
(1.57)
dx
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