Biomedical Engineering Reference
In-Depth Information
Fig. 5.2 a
Pipe.
b
trachea
with two cross-sectional
slices are shown.
Cross
section
1 is taken just after the
larynx and has a
cross-sectional area of
1.5 cm
2
.
Cross section
2is
taken in the main trachea
region and has a
cross-sectional area of
3.0 cm
2
a
b
.
m
=
u
A
A
1
= 1.5cm
2
1
1
1
cross-section 1
A
2
= 3.0cm
2
cross-section 2
.
m
=
u
A
2
2
2
For an incompressible flow such as inhaled air, density can be treated as a constant;
therefore, we can simplify the above equation by dividing out
ρ
and
x
·
y
·
1
which gives
∂u
∂x
+
∂v
∂y
0
=
(5.4)
Equation (5.4) is the statement of conservation of mass, which is also known as
the continuity equation for a steady two-dimensional flow of a fluid with constant
density. If we repeat the derivation in the
z
-direction, we then get the relation under
three-dimensions
∂u
∂x
+
∂v
∂y
+
∂w
∂z
0
=
(5.5)
While the derivation shown is for steady flow, Eqs. (5.4) and (5.5) are valid for steady
as well as unsteady flows for incompressible fluids.
Physical Interpretation
Before the availability of imaging diagnostic tools (e.g.
MRI, CT scans), early studies of the trachea airway were based on a uniform pipe
model. Additionally, in most Fluid Mechanics textbooks, the principal of mass con-
servation is often explained by a fluid flowing in a uniform pipe. Therefore, let us
consider flow in the trachea using both a uniform pipe geometry and a realistic CT-
scan based model shown below in Fig.
5.2
. The mass entering both geometries can be
denoted by the mass flow rate
m
1
, which is equivalent to the product of the density,
inlet velocity and cross-sectional area, i.e.
ρ u
1
A
1
.
This mass must be equal to the
mass flow rate leaving the pipe, which is denoted by
˙
m
2
given as
ρ u
2
A
2
.Ifwe
consider the uniform pipe, then both the inlet and outlet cross-sections of the pipe
are equal, i.e.
A
1
=
A
2
. Based on the mass conservation, the outlet velocity
u
2
must
˙
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