Environmental Engineering Reference
In-Depth Information
Solving the set of two equations (6.114) under the assumption that
N
e
N
,
we have for the number densities of charged particles
k
23
k
17
ν
k
23
k
17
ν
ion
ion
ion
)
2
[O
3
]
2
N
e
D
,
N
D
[O
3
] .
(6.115)
(
ν
ν
ν
ν
at
at
ion
The subscripts in the rate constants correspond to the process numbers in Ta-
ble 6.6. Expressions (6.115) show that a reverse connection between processes re-
stores electrons. As a result, there are stable currents in atmospheric air subjected
to an electric field. At
ion
, an instability develops that leads to breakdown
in pure air. This will occur at
E
ν
D
ν
at
25 kV/cm, which is the breakdown strength of
dry air in the case of uniform currents. However, the sequence of processes spec-
ified above leads to breakdown at smaller electric fields. An increase of the ozone
number density leads to an increase of the electron number density, which, in turn,
causes an increase of the ozone number density. Then at certain sets of parameter
values an explosive instability can develop, meaning that breakdown has occurred.
We now seek the threshold for this instability.
For this purpose, we analyze the third equation in (6.114). Substituting it for the
electron number density in (6.115), we have
D
d
[O
3
]
dt
D
N
0
τ
[O
3
]
τ
C
k
ef
[O
3
]
2
,
(6.116)
where
N
0
is the ozone number density in the absence of the electric field.
The effective rate constant determined by
D
M
τ
2
k
23
ν
ν
ion
26
k
ef
D
k
17
(
ν
ν
ion
)
2
at
is the combination of the rate constants for the corresponding processes. Un-
der typical atmospheric conditions we have
N
0
10
12
cm
3
. The effective
rate constant
k
ef
depends strongly on the electric field strength. It has the val-
ues 8
10
13
cm
3
/s at
E
/
N
10
11
cm
3
/s at
E
/
N
D
30 Td, 2
D
50 Td, and
10
10
cm
3
/s at
E
/
N
6
80 Td.
In the stationary case, (6.116) has two solutions, of which one is stable. The in-
stability corresponds to imaginary values for the solutions. Hence, the instability
that leads to breakdown corresponds to the condition
D
)
1
.
k
ef
(4
N
0
τ
(6.117)
10
6
s)
because ozone is produced at high altitudes (40-80 km) and its loss results from
chemical reactions (ozone cycles) or from transport to the Earth's surface. The pa-
rameter
The real lifetime of ozone molecules in the atmosphere is rather long (
τ
in condition (6.117) refers to the time during which an ozone molecule
is located in a region with a large electric field strength. If we take the size of such a
region as the cloud size
L
τ
1 km, and the velocity of transport from this region to
be the wind velocity
v
1m/s,thenwehave
τ
10
3
s, and condition (6.117) gives
