Geoscience Reference
In-Depth Information
We solve this equation with respect to
J
and obtain the expression for ˛.a/
:
˛.a;R/
1
C
˛.a/
D
:
(3.37)
˛.a;R/
4DR
Equation
3.37
is exact if R
l,wherel is the mean free path of condensing
molecules in the carrier gas. To find ˛.a;R/and
R
we must call on the approxima-
tions.
The free-molecule expression approximates ˛.a;R/.
˛.a;R/
˛
fm
.a;R/;
(3.38)
where ˛
fm
.a;R/ is the trapping efficiency in the free molecule zone.
The radius
R
of the limiting sphere is found from the condition: “the diffusion
flux from the diffusion zone is equal to the diffusion flux from the free molecule
zone” The diffusion flux is defined from Fick's law. Hence,
ˇ
ˇ
ˇ
ˇ
rDR
D
ˇ
ˇ
ˇ
ˇ
rDR
dn
fm
.r/
dr
dn
c
.r/
dr
;
(3.39)
where n
fm
.r/ is the vapor concentration profile found in the free-molecule zone
for a<r<Rand n
c
.r/ is the concentration profile in the diffusion zone. The
distance
R
separates the zones of the free-molecule and the continuous regimes.
The total flux of A in the free-molecule zone is equal to the total flux in the
diffusion zone.
3.3.2
Solving the Kinetic Equation
Now we will solve the kinetic equation
@f
s
@r
D
0
s
v
r
(3.40)
For the following it is convenient to introduce the notation,
r
D
L
r
L
2
;
C
D
L
2
L
a
D
1
C
D
L
a
L
2
(3.41)
Here .x/ is the Heaviside step function (.x/
D
1 at x
1 and 0 otherwise).
The factor
cuts off the molecules flying past by the target particle. Then the
Maxwell boundary condition has the form:
1
S
p
f
1
C
:
1
2
S
p
n
C
f
1
.a;E;L/
D
(3.42)
