Biomedical Engineering Reference
In-Depth Information
microcirculation. Internal forced convection is concerned with fluids that are bound on all
surfaces (e.g., flow through blood vessels, the respiratory tract, or heat exchangers) and is
driven by some pressure gradient (e.g., the heart, a pump, or a turbine). For these types of
situations, the velocity profile, which has been described in earlier chapters, and the tem-
perature profile are of primary interest. From these two profiles, we can then obtain heat-
transfer rates and other properties of interest.
We will begin by developing the temperature profile for a fluid in cylindrical coordi-
nates. The temperature within the fluid must satisfy the energy conservation laws, which
states that the actual energy transported through a fluid must be equal to the energy trans-
ported through a fluid that has a uniform temperature profile. This is analogous to analyz-
ing fluids as if they are invisicid and making statements about viscous fluids from this
analysis. Mathematically, the Conservation of Energy equation states that
ð
E
fluid
5
mc
p
T
m
5
ρc
p
TðrÞuðrÞdA
ð
7
:
28
Þ
A
m
is the fluids mass flow rate,
c
p
is the fluids specific heat at constant pressure (nor-
mally assumed to be a constant),
T
m
is the mean fluid temperature,
where
ρ
is the fluid density
(also, normally assumed to be a constant), and
T
(
r
) and
u
(
r
) are the temperature and veloc-
ity profile in radial coordinates, respectively. In this analysis, we assume that the velocity
and temperature profiles are only functions of the radial direction.
Equation 7.28
can be
rearranged to calculate the mean temperature of the fluid, if needed.
Solving
Equation 7.28
for a steady two-dimensional (Cartesian coordinates) flow of a
fluid with constant fluid properties with negligible stresses, the solution to the energy bal-
ance equation is
2
T
2
T
@y
2
ρc
p
u
@T
@x
1
v
@
T
5
k
@
@x
2
1
@
ð
7
:
29
Þ
@y
or in radial coordinates
u
@
T
ρrc
p
dr
k
r
@
T
@r
ð
7
:
30
Þ
@z
5
Equation 7.29 or 7.30
can be used to solve for the temperature profile within a fluid as the
Navier-Stokes equations were used to determine the velocity profile of the fluid. Note that
some information about velocity is needed to solve for the temperature because this is a
coupled convection problem.
Using this same type of energy balance analysis, we can obtain general heat transfer
equations when the blood vessel can be assumed to have either a constant surface temper-
ature or a constant surface heat flux. In most biological applications, neither the surface
temperature nor the surface heat flux will be constant, but we may be able to use these
simplifications to approximate certain conditions and obtain meaningful results. In the
absence of any work by the fluid, the Conservation of Energy can be solved to obtain the
heat transfer rate,
Q
, to or from the fluid. This is represented as
Q
5
mc
p
ðT
o
2
T
i
Þ
ð
7
:
31
Þ
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