Biomedical Engineering Reference
In-Depth Information
Here
δ
c
is the diffusion boundary layer thickness given as
√
π Dt
δ
c
=
(6.50)
The corresponding diffusion force is defined as
3
πμd u
D
C
c
F
d
=
(6.51)
The total number of particles that are deposited in an interval
dt
is given as
C
0
D
dN
=
Jdt
=
πt
dt
(6.52)
The total number of particles that are deposited per unit area in the time interval 0
to
t
may be obtained by integrating Eq. (6.52). Thus,
C
0
4
Dt
π
N
=
(6.53)
The analysis presented here may be used to estimate the particle deposition in a tube.
For a constant velocity gas flow in a tube of length
L
and radius
R
, the residence time
is
t
L/u
, where
u
is the gas velocity. Assuming that the wall deposition process
is similar to that of a uniform concentration near a wall, and using (6.53) it follows
that
=
DL
uR
2
C
out
C
in
=
4
√
π
1
−
(6.54)
Equation (6.54) provides an approximate estimate of the change in the concentration
of particles along a tube. A detailed analysis of duct flow leads to
C
out
C
in
=
DL
uR
2
2
.
56
φ
2
/
3
0
.
177
φ
4
/
3
,
−
+
+
=
1
1
.
2
φ
φ
(6.55)
6.6
Turbulent Particle Dispersion
Particles that are transported under a turbulent flow are subjected to the character-
istics of turbulence from the fluid. The additional coupling and linking of the fluid
turbulence to the particle equations increases the complexity of the CFPD modelling.
In this chapter we will pay particular attention to the Lagrangian particle trajectory
analysis because it allows a more intuitive understanding of turbulent transport and
dispersion of particles to be introduced. However we begin by describing the general
Eulerian two-fluid model approach.
The turbulent equations of motion for a single phase were derived in Chap. 5.
It was shown that representing turbulence for a single fluid itself was difficult and
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