Biomedical Engineering Reference
In-Depth Information
ε
a
/K
a
=
ε
v
/K
v
. Equation (1.70), however, indicates that the blood pressure
difference increases for either small perfusion bleed-off rate
E
or large expo-
nent
n
, which may result from aging.
Following Chato (1980), we note that the axial conduction terms in the
blood energy equations are negligibly small as compared to the convection
and perfusion terms. Then, the energy equations (1.50) and (1.51) along with
the foregoing velocity distributions reduce to
d
a
d
(
x/L
)
=
T
N
a
v
)
−
E
L
1+
n
(
T
−
T
(1.72)
1
−
N
+(1+
n
)
E
x
L
n
v
d
(
x/L
)
=
d
T
v
a
)
(
T
−
T
(1.73)
E
x
L
1+
n
1
−
where
a
f
h
f
L
ρ
f
c
p
f
u
0
N
=
(1.74)
is the number of heat transfer units. The boundary conditions are given by
x/L
=0:
a
=
a
0
T
T
(1.75)
v
L
(1.76)
A series of numerical integrations were carried out for various sets of three
important dimensionless parameters, namely, the dimensionless perfusion rate
E
, the number of heat transfer units
N,
and the exponent
n
. Thus, the tem-
perature profiles along the vessel axes are obtained for the case of
n
= 0 and
presented in Figures 1.7(a) and 1.7(b) for a physiological range of
E
and
N
val-
ues. The results appear to be in perfect agreement with the exact expressions
reported by Chato (1980). The difference between the present curves for
n
=0
and those based on Chato's solution is indiscernible in the figure. Naturally,
a better blood circulation (i.e., larger
E
) results in warming the venous blood
eciently. The figures show that its eciency as a heat exchanging system
increases with
N
.
The temperature profiles along the vessel axes for the case of
n
= 5 are
presented in Figures 1.8(a) and 1.8(b). It is interesting to note that the arterial
blood temperature for the case of nonzero
n
always stays higher than that
for the case of
E
= 0 (i.e., without perfusion) even at the end of the vessel.
Following Chato (1980), we shall evaluate the total heat transfer from the
artery to vein in terms of
q
a−v
=
ρc
p
f
u
0
(
v
=
x/L
=1:
T
T
v
0
−
v
L
)
T
T
(1.77)
or its dimensionless form, namely,
q
a−v
v
0
−
v
L
L
)
=
T
T
(1.78)
a
0
−
v
a
0
−
v
L
ρc
p
f
u
0
(
T
T
T
T
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