Biomedical Engineering Reference
In-Depth Information
from (1.61b). Using equations (1.59) and (1.60), we may replace both inter-
facial and perfusion heat source terms in the energy equation (1.48) for the
tissue by the blood convection terms as
(1
+
a
a
h
a
(
s
d
dx
ε
)
k
s
d
T
a
s
)+
a
v
h
v
(
v
s
)
−
T
−
T
T
−
T
dx
a
v
)+(1
+
ρ
f
c
p
f
ω
a
(
T
−
T
−
ε
)
S
m
(1
s
d
dx
ε
)
k
s
d
T
d
dx
(
ε
a
a
a
+
ε
v
v
v
)
=
−
−
ρ
f
c
p
f
u
T
u
T
dx
+(1
−
ε
)
S
m
= 0
(1.64)
a
=
v
and use equation
As we note the continuity relationship
ε
a
u
−
ε
v
u
(1.63) for the last expression in (1.64), we finally have
(1
d
+(1
ε
)
k
s
+2
ρ
f
c
p
f
ε
a
a
2
s
u
d
dx
T
−
−
ε
)
S
m
= 0
(1.65)
a
f
h
f
dx
which we find almost identical to Bejan's equation (1.58), as we note the
overall heat transfer coecient corresponds to
1
h
a
+
=
h
f
2
U
=
(1.66)
h
v
It is most interesting to find that the foregoing relationship for the lon-
gitudinal effective thermal conductivity holds for all cases, with or without
perfusion bleed-off sources, as long as the local values are used to evaluate the
effective thermal conductivity by convection.
1.6 Effect of Spatial Distribution of Perfusion Bleed-Off
Rate on Total Countercurrent Heat Transfer
As an example for illustration, we shall consider Chato's one-dimensional
problem of countercurrent heat transfer as schematically shown in Figure 1.5.
Chato (1980) assumed the constancy of the perfusion bleed-off rate
ω
a
, namely,
a linear decrease in the arterial flow rate, and that all of the bleed-off fluid
that leaves the artery reenters the vein at the same location. We shall relax his
assumption, allowing the spatial variation of
ω
a
so as to investigate its effect
on the total countercurrent heat transfer. Let us assume that the perfusion
bleed-off rate
ω
a
follows:
ω
a
=(1+
n
)
ω
a
x
L
n
(1.67)
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