Environmental Engineering Reference
In-Depth Information
where
C
is the normalization constant and
f
n
n
max
(
t
).
We have the following relations for the number density of clusters
N
cl
and the
total number density of bound atoms
N
b
:
D
0if
n
>
Z
f
n
dn
,
N
b
Z
nf
n
dn
.
N
cl
D
D
Using (6.72) for the size distribution function for clusters, we get for these param-
eters
3
3
Cn
1/3
4
Cn
4/3
N
cl
D
max
,
N
b
D
max
.
On the basis of the average cluster size
n
D
N
b
/
N
cl
,weobtain
n
max
4
n
D
.
(6.73)
Thus, within the framework of the model we have at the end of the process from
the second balance equation of set (6.69)
n
max
4
n
max
4
KN
m
N
a
N
b
D
N
m
D
N
cl
D
τ
.
From this and (6.71) we find for the maximum cluster size
n
max
at the end of this
process and the duration
τ
of the cluster growth process [112, 114]
3.2
G
1/4
N
m
k
0
1.2
G
3/4
n
max
D
,
τ
D
.
(6.74)
These formulas describe cluster parameters at the end of the cluster growth pro-
cess. Because
G
1, a typical cluster size is large,
n
1, and the reduced nucle-
ation time
τ
is large compared with a typical time of (
k
0
N
m
)
1
for attachment of
one atom.
6.3.3
Cluster Growth in Coagulation and Coalescence
Figure 6.18 shows the mechanisms of clusters growth [111]. Figure 6.18a corre-
sponds to the above case of conversion of a gas or vapor consisting of free atoms
into a gas of clusters when the vapor pressure exceeds the saturated vapor pressure.
Figure 6.18b corresponds to the coagulation mechanism due to joining of clusters
that proceeds according to the scheme
M
n
m
C
M
m
!
M
n
,
(6.75)
and is an important process for the Earth's atmosphere [115]. Note that although a
buffer gas does not partake in the coagulation process, it plays a stabilization role.
As is seen, the coagulation process proceeds at a small number density
N
of free
