Geoscience Reference
In-Depth Information
Three approximate expressions for F.x/are considered below.
1. The Lushnikov-Kulmala (LK) approximation (Lushnikov and Kulmala
2004a
)
(see Eq.
3.69
):
r
1
C
4
1
!
:
1
2
x
2
F
LK
.x/
D
(3.86)
The ideas on the derivation of this equation are given below
2. The Fuchs-Sutugin (FS) approximation (Fuchs and Sutugin
1971
): in deriving
this equation these authors divided the space into two parts: the free molecule
zone and the diffusion zone. They then used the principle of constancy of the total
flux. The radius of the limiting sphere (the spherical surface dividing the space
into free molecular and diffusion zones) is found from the numerical solution of
the BGK (Bhatnagar et al.
1954
) kinetic equation obtained by Sahni (
1966
). In
addition, they replaced ˛.a;R/ by ˛
fm
. Their final result is widely known:
x.x
C
1:13/
4.x
C
3/
:
F
FS
.x/
D
(3.87)
3. Dahneke's (D) approximation (Dahnecke
1983
):
x
2
4.x
C
2/
:
F
D
.x/
D
(3.88)
The last two approximations are discussed by Seinfield and Pandis (
2006
).
Figure
3.3
compares these three approximation. It is seen that the difference is
minor.
3.3.6.2
Concentration Jump
Let us write down the concentration profile in the free molecule regime (Eq.
3.47
):
n
fm
.r/
D
n
1
.n
1
n
C
/b
.r/:
(3.89)
The concentration profiles are presented in Fig.
3.2
. Figure
3.4
shows the
dependence of the reduced concentration jump
a
= on
a
. Figure
3.5
displays
the size dependence of the trapping efficiency. Here we compare our results with
the numerical results of Loylaka et al. (
1989
) and two semiempirical formulas. The
difference between these results does not exceed 10%.
The synthetic concentration profile obtained by sewing the free-molecular and
diffusion profiles (Fig.
3.6
) is compared to the results of calculations obtained by
Williams and Loylaka (
1991
) and by Loylaka et al. (
1989
), where the Boltzmann
equation was solved numerically. Both these curves reproduce identical fluxes but
display a different behavior at small particle sizes.
