Geoscience Reference
In-Depth Information
Another direction of modeling is the use of the moment method, which reduces
the full rather complicated set of equations describing the kinetics of particle
formation-growth to a finite set of ordinary differential equations for some integral
characteristics of the particle-size spectra (Friedlander
1983
,
2000
; Williams and
Loyalka
1991
). Sometimes the moment methods allow for obtaining exact results
even in practically important cases. In particular, Friedlander (
1983
) established
that free-molecular condensational growth can be described by a set of only four
first-order differential equations. Lushnikov and Kulmala (
1998a
) reported on a
set of condensational coefficients affording the exact description of nucleation-
condensation kinetics in terms of a finite number of moments and applied these
results to the analysis of the condensational growth of dimers (Lushnikov and
Kulmala
1998b
). Rigorous asymptotic analysis of the kinetics of nonbarrier nucle-
ation in the presence of a source of condensable vapor was given by Lushnikov
and Kulmala (
2000
) for algebraic dependencies of the condensation efficiencies on
particle size. Then, this approach was extended to the transition regime of particle
growth (Lushnikov and Kulmala
2001
; Dal Maso et al.
2005
).
Here we consider the development of the aerosol state of the atmosphere
assuming the following:
At the initial moment of time, the source of a condensable vapor is switched on.
The productivity of the source (the number of particles produced per unit volume
at a given time) is I.t/
D
I
0
i.t/,whereI
0
is the dimensionality carrier and the
dimensionless function i.t/describes the time (
t
) dependence of the productivity.
Att
D
0 there exists a foreign aerosol with known size distribution, distributed
over particle masses
g
as N.g;0/
D
N
0
n.g;0/where N
0
D
P
N.g;0/is the
particle number concentration and the particle mass
g
is measured in the units of
the masses of condensable molecules.
The qualitative picture of the development of the situation appears as follows
(Fig.
3.1
) (Friedlander
1983
). At the initial period, concentration grows linearly with
time (for simplicity, we consider i
D
1), then it begins to bend because part of the
vapor condenses onto foreign particles. At t
D
t
when the vapor concentration
reaches a sufficiently high level for the spontaneous nucleation process to start,
newly born particles appear and also begin to consume the vapor. The vapor
concentration thus passes a maximum, whose value is determined either by the
nucleation rate or the rate of vapor losses by condensation onto foreign particles.
The nucleation process stops at t
D
t
when the vapor concentration level drops
lower than the critical value c
and results in the formation of a highly dispersed
mode. The number concentration of the latter depends on the characteristics of
foreign particles: their number concentration and some geometric characteristics
(more complex than simply average particle radius or average particle surface).
At the post-nucleation stage, evolution mainly continues owing to the growth of
foreign particles by simultaneous joining of the vapor molecules and the smaller
nanometric particles. The growing particles reach submicrometer sizes.
Although a huge number of papers considered the condensation process in the
transition regime (extensive citations are given by Lushnikov and Kulmala (
2001
),
