Environmental Engineering Reference
In-Depth Information
density of drops. The recombination coefficient of two oppositely charged drops in
a gas is given by the Langevin formula (6.10) [64]:
k
rec
Dj
Z
C
Z
j
e
(
K
C
C
K
) ,
(6.109)
where
Z
C
and
Z
are the charges of positive and negative drops and
K
C
and
K
are their mobilities (
K
C
D
K
dr
). On the basis of
the
charge distribu
tion
functionfordrops(6.15),wehavefortheaveragepositive
K
D
j
Z
C
j
and negative
j
Z
j
drop charges
r
r
0
T
π
j
Z
C
jDj
Z
C
jD
.
e
2
Using the expression for the mobility of drops in a gas for case (6.5),
e
2
k
dr
D
,
r
0
4
π
η
10
4
g/(cms) is the air viscosity at room temperature, we obtain
for the average rate constant for recombination for a test charged drop
where
η
D
1.85
T
k
rec
D
.
(6.110)
2
π
2
η
As is seen, the rate constant for recombination for a test charged drop is indepen-
dent of the drop size (we assume all the drops have identical radius
r
0
)andis
10
11
cm
3
/s .
k
rec
D
1.1
We now find an optimal drop size by equalizing the typical time
τ
D
2/
N
dr
k
dr
rec
and the time
2kmisatypical
cloud size. Taking the average water content in the atmosphere (7 g/kg of air), we
have
τ
D
L
/
w
for falling of drops in a cloud, where
L
fal
10
5
s
1
a
r
0
3
1
2
N
dr
k
rec
D
rec
D
1.2
,
τ
10
8
s
1
r
0
a
3
1
τ
w
L
D
fal
D
6
.
From this we obtain for the size of charged water drops when they reach the lower
edge of a cloud
r
0
m.
We note that in considering electric phenomena in the atmosphere, we simpli-
fied the problem and extracted the principal mechanism of Earth charging where
separation of charges in the atmosphere results from falling of charged water
drops. In reality, this process is more complex. In particular, convective motion
of atmospheric air leads to mixing of atmospheric layers [177], and a charged layer
may be transported to other altitudes up to 100 km. This can cause breakdown and
lightning at high altitudes.
3
μ
