Biomedical Engineering Reference
In-Depth Information
where
T
o
is the mean outlet temperature and
T
i
is the mean inlet temperature. You may recall
from a previous course in heat transfer that the surface heat flux (
q
s
Þ
can be defined as
Þ
where
h
x
is the local convection heat transfer coefficient,
T
s
is the surface temperature, and
T
m
is the mean fluid temperature. Using
Equations 7.31 and 7.32
, we can now analyze our
idealized conditions for constant surface heat flux or constant surface temperature. In the
case where we can assume that
q
s
5
h
x
ðT
s
2
T
m
Þ
ð
7
:
32
q
s
is a constant, then the rate of heat transfer can also be
expressed as
Q
5
q
s
A
s
5
mc
p
ðT
o
2
T
i
Þ
ð
7
:
33
Þ
where
A
s
πDL
for circular blood ves-
sels with a length of
L
. The mean fluid outflow temperature and the mean surface temper-
ature become
is the area available for heat transfer and is equal to
T
o
5
T
i
1
q
s
A
s
mc
p
T
s
5
T
m
1
q
s
h
which both vary linearly with the surface heat flux. By recognizing that
q
s
and
h
are con-
stants with respect to the z-direction (radial coordinates, along the blood vessel length),
we can then obtain
2
πRq
s
mc
p
T
m
5
T
i
1
z
ð
7
:
34
Þ
by differentiating and equating the equations that describe the linear variation of the mean
outlet temperature and the surface temperature. In
Equation 7.34
,
R
is the blood vessel
radius and
z
is the axial distance along the vessel.
Equations 7.33 and 7.34
can be used to
solve for various heat transfer properties under the constant surface heat flux assumption.
If the conditions are such that we can make the assumption of a constant surface tem-
perature, then the heat transfer and temperature conditions will be modified. Here, the
rate of heat transfer can be expressed as
Q
5
hA
s
ΔT
average
ð
7
:
35
Þ
The average temperature difference can be calculated by
T
i
1
T
o
2
where the second term on the right-hand side is the bulk mean fluid temperature, and this
is the temperature at which all fluid properties are analyzed. We can further relate the
heat-transfer rates to the fluid temperatures by
ΔT
average
T
s
2
ln
T
s
2
T
o
T
s
2
T
i
hA
S
mc
P
52
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