Biomedical Engineering Reference
In-Depth Information
Fig. 5.9
Temperature variation with time for a steady flow rate in the nasal cavity
these derivatives, consider a
steady flow
of air, initially at 20
◦
C, through the nasal
cavity. The mucous walls of the nasal cavity are maintained at a surface temperature
of 37
◦
C throughout the geometry (Fig.
5.9
).
If we took temperature measurements at Point C over time, then the temperature
profile would be represented by
T(t)
1
which is constant because the flow that passes
Point C will have the same temperature. This means that the local acceleration
derivative in the energy equation is zero:
∂T
∂t
=
0
Now if we follow a fluid particle along its trajectory as it travels from A through C and
then B, then there is some variation in the temperature profile which is represented by
T(t)
2
. The temperature from point A through to B shows the air temperature heating
up because of the surrounding walls. This spatial variation is represented by the
convective derivative, which describes the temperature variation from one point to
another:
u
∂T
v
∂T
∂x
+
∂y
=
0
If we consider the local acceleration derivative during an unsteady inhalation cycle,
then
∂T
∂t
0 and will vary with time—specifically the air temperature will rise more
rapidly when the inhalation is slowest. If we follow a fluid particle for the convective
derivative, the air temperature will fluctuate not only with the inhalation cycle but
also with the temperature variation in the spatial coordinates
x
,
y
, and
z
. The total
derivative of energy is, therefore, the instantaneous time rate of change at a given
point following a moving fluid element. This is made up of the local derivative, which
is the time rate of change at the given point, and the convective derivative, which
is the time rate of change due to the spatially changing fluid property as the fluid
element moves to that given point within the flow field.
The remaining term in Eq. (5.21) represents the heat flow due to conduction (the
diffusion
), where the thermophysical property
k
is the thermal conductivity of the
fluid. To examine the physical interaction of the diffusion terms with the convective
=
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