Biomedical Engineering Reference
In-Depth Information
When the effect of particle inertia is negligible, we have a convection-diffusion
transport equation for the chemical component/species in two-dimensions:
D
∂C
∂x
D
∂C
∂y
∂C
∂t
+
∂
(
uC
)
∂x
∂
(
vC
)
∂y
∂
∂x
∂
∂y
+
=
+
(6.42)
The solution of Eq. (6.42) results in the particle concentration field.
In the absence of flow and when the particle source is far from walls, the mean-
square displacement of Brownian particles in one direction is given as
x
2
(
t
)
=
2
Dt
(6.43)
One terminology used commonly in the aerosol community is the
diffusion velocity
,
which is defined as flux to the wall per unit concentration. That is,
J
C
0
u
D
=
(6.44)
Similarly, a diffusion force may be defined as
F
diff
=
3
πμu
d
/C
c
(6.45)
Analysis
For a one-dimensional case, the diffusion equation given by Eq. (6.42) in
the absence of a flow field becomes
∂C
∂t
=
∂
2
C
∂y
2
D
(6.46)
For an initially uniform concentration of aerosols in the neighbourhood of an ab-
sorbing wall, the initial and boundary conditions are
C
(
y
,0)
=
C
o
,
C
(0,
t
)
=
0 and
C
(
C
o
, where
C
o
is the particles number concentration at the initial time and
far from the wall. The solution to Eq. (6.46) then becomes
∞
,
t
)
=
=
C
0
erf
y
C
(
y
,
t
)
√
4
Dt
(6.47)
Where
ξ
2
√
π
e
−
ξ
2
dξ
;
erf
(
ξ
)
=
erf
(0)
=
0;
erf
(
∞
)
=
1
0
The flux to the wall then is given by
C
o
D
πt
y
=
0
=
D
∂c
∂x
J
=−
(6.48)
The corresponding deposition velocity, which is defined as flux per unit concentra-
tion, is then given by
D
πt
=
J
C
0
=
D
δ
c
u
D
=
(6.49)
Search WWH ::
Custom Search