Environmental Engineering Reference
In-Depth Information
Substituting these expressions into the above kinetic equation for the charge dis-
tribution function of clusters, we reduce the kinetic equation to the form
D
C
C
D
exp(
Zx
)
Zx
exp(
Zx
)
D
C
(
Z
1)
x
f
(
Z
)
D
1
f
(
Z
1)
1
exp[(
Z
1)
x
]
D
(
Z
C
1)
x
C
1
f
(
Z
C
1) .
(6.15)
exp[(
Z
C
1)
x
]
This kinetic equation may be used for determination of the charge distribution
function for clusters. We consider first the case when along with criterion (6.5), the
criterion
x
1 holds true, which is inverse with respect to criterion (6.13). In this
case clusters are mostly neutral, and a small number of clusters have charge
Z
D
˙
1, whereas the probability for a cluster having charge
j
Z
j
2isexponentially
small. Indeed, from the kinetic equation (6.15) it follows that
D
C
(
e
2
x
f
(2)
f
(1)
D
1)
D
C
2
D
e
x
1)
D
,
2(
D
C
C
D
e
2
x
)(
e
x
where
f
(
Z
)with
Z
2 is exponentially small. The same conclusion relates to
Z
2. This allows us to restrict our attention to neutral and singly charged ions.
We have
f
(0)
1, and from (6.14) we have
D
C
(
e
x
f
(1)
f
(0)
1)
D
C
D
x
f
(1)
D
D
e
x
)
x
D
,
(
D
C
C
D
(
e
x
f
(
1)
f
(0)
1)
D
D
C
x
f
(
1)
D
D
C
e
x
)
x
D
.
(
D
C
From this we find for the average charge of particles
D
2
D
2
D
C
D
x
C
Z
D
f
1
f
1
D
.
In the case
D
C
1.
We now consider the case when criterion (6.5) holds true along with criteri-
on (6.13). It is convenient to introduce a new variable
z
D
D
Δ
D
D
C
,thisformulagives
Z
D
2
Δ
D
/(
Dx
)
D
xZ
,andbecause
Z
is a whole number and
x
1, this variable is taken to be continuous. Then we
represent the kinetic equation (6.15) in the form
D
F
(
z
)
e
z
]
f
(
z
)[
D
C
F
(
z
)
C
C
f
(
z
x
)
D
C
F
(
z
x
)
x
)
e
z
C
x
C
f
(
z
C
x
)
D
F
(
z
C
D
0,
where
z
1
.
Expanding this equation over the small parameter
x
F
(
z
)
D
e
z
1 and accounting for the
first two expansion terms, we reduce the kinetic equation to the form
D
D
e
z
f
(
z
)
F
(
z
)
dz
2
D
D
C
e
z
f
(
z
)
F
(
z
)
d
2
d
dz
x
2
D
0 .
(6.16)
