Environmental Engineering Reference
In-Depth Information
∂
∂
V
t
1
∂
∂
p
z
μ
ρ
(
)
z
2
(5.6)
+
V
⋅∇
V
= −
+
f
+ ∇
V
z
z
z
ρ
where
∂
∂
1
∂
∂
∂
∂
(5.7)
V
⋅∇ =
V
+
r
V
+
V
r
θ
z
r
θ
z
and the Laplacian operator in cylindrical coordinates is deined as
2
2
2
1
∂
∂
⎛
⎜
∂
∂
⎞
⎟
+
1
∂
∂
+
∂
∂
2
∇ =
r
(5.8)
2
2
r
r
r
r
θ
z
If the low is steady, then the time derivatives in Equations 5.4 through 5.6 can be ignored.
5.2.2 b
oundary
c
onditions
The aforementioned system of four nonlinear partial differential equations can only be solved ana-
lytically when a number of assumptions are made for very simple low geometries. Therefore, in
most cases, numerical methods are required to determine a solution for the velocity and pressure
ields. Numerical solution of the equations of motion requires knowledge about the velocity or
pressure at some or all boundaries of the low geometry. The nature of the low characteristics and
geometry determines which boundary conditions need deining.
As low has been studied in a vast variety of geometric lung models, it is dificult to recommend
boundary conditions that are appropriate in every circumstance. In almost all cases, however, it
is assumed that the no-slip condition (
V
at the wall = 0) exists along the walls of any airway. The
deinition of any inlet and outlet boundary conditions is not as straightforward. For instance, the low
velocity at the inlet may be designated steady (time-invariant) or unsteady and may be uniform over
the diameter of the inlet or vary with radial direction. In addition, in different cases, the velocity, the
velocity gradient, the pressure, or the pressure gradient may be deined. In airway low modeling,
the velocity proile at the exit of an airway is often assumed parabolic, and the conducting airway is
assumed suficiently long enough for the low to be fully developed.
2
In some simpliied cases, only the inlet conditions may need to be speciied. For example, if one
assumes that the pressure in the luid varies only in the axial direction, then the mathematical char-
acter of the equations changes and only the inlet velocity proile is needed to obtain a solution.
3
In
this case the outlet velocity proile is determined as part of the model solution.
5.2.3 i
dealized
v
elocity
P
roFiles
The velocity proiles present in real lung airways are determined by a great many factors
including ventilatory conditions, lung morphology, and the airway generation(s) being considered.
Experimental studies
4-6
have shown that these proiles may be skewed and, in general, geometrically
complex. However, by assuming idealized velocity proiles in individual airway generations,
fundamental equations of luid motion can be reduced and the development of simpliied
expressions for deposition via different mechanisms can be obtained. Idealized velocity proile
types include laminar fully developed (parabolic), laminar undeveloped (plug), and turbulent. The
type of velocity proile selected determines the appropriate expressions for particle deposition (e.g.,
via sedimentation, diffusion, or inertial impaction) in a given airway. Examples of such expressions
were presented in detail in Chapter 3.
Martonen
7
presented recommendations for velocity proiles in different generations of lung air-
ways, based on previously published experimental measurements.
8
At the entrance to the trachea, it
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