Geoscience Reference
In-Depth Information
Here .a/ is a function depending on the nature of the chemical process and
independent of
J
. An example of such function is given below. If the chemical
process inside the particle is nonlinear, then the function .a/ depends on
J
and
J
is then a solution to the transcendent equation Eq.
3.32
.
The central problem is thus to find ˛.a/. Equations
3.31
and
3.32
allow also for
the consideration of normal condensation and evaporation. In this case A molecules
are the same as the molecules of the host particle and ˛.a/ is referred to as the
condensational efficiency.
3.3.1.2
Flux Matching for Condensation
Next, we extend the flux-matching theory to the case of condensation of neutral
molecules onto the particle surface with n
C
¤
0 and S
p
1. To this end we
generalize Eq.
3.31
as follows:
J
D
˛.a;R/.n
R
n
C
/;
(3.33)
where n
R
is the vapor concentration at a distance
R
from the particle center.
Indeed, the total flux
J
is independent of
R
, and we have the right to consider
the condensation from any finite distance. It is important to emphasize that n
R
is (still)
an arbitrary value
introduced as a boundary condition at the distance
R
(also arbitrary) to a kinetic equation that is necessary to solve for defining the
generalized condensational efficiency ˛.a;R/.Thevalueof˛.a;R/ does not
depend on n
R
n
C
because of the linearity of the problem.
Assuming that we know the exact vapor concentration profile n
exact
.r/ corre-
sponding to the given flux
J
from infinity, we can express
J
in terms of n
exact
as
follows:
J
D
˛.a;R/.n
exact
.R/
n
C
/:
(3.34)
If we choose
R
as sufficiently large, then the diffusion approximation reproduces
the exact vapor concentration profile:
J
4DR
C
n
1
;
n
exact
.R/
D
n
c
.R/
D
(3.35)
with n
c
.r/ being the steady-state vapor concentration profile corresponding to a
given total molecular flux
J
.
Combining Eqs.
3.33
,
3.34
,and
3.35
gives
J
D
˛.a;R/
n
1
n
C
:
J
4DR
(3.36)
