Environmental Engineering Reference
In-Depth Information
has a maximum at some distance from the particle, whereas the particle electric
potential
(
R
) drops monotonically with distance from the particle. The connec-
tion between these quantities is determined by the Poisson equation (6.52).
Let us consider this connection from another standpoint. According to the Gauss
theorem, the particle electric field strength is
'
Z
(
R
)
e
R
2
E
D
,
and the electric potential at distance
R
from the particle is
Z
1
Z
(
r
)
r
2
'
(
R
)
D
dr
.
R
This gives the following connection between the reduced particle electric potential
and the number density difference:
1
R
Δ
Z
1
1
r
D
1
r
N
N
0
r
2
dr
.
X
D
(6.55)
R
In reality, this equation is identical to the Poisson equation (6.52). In particular, at
small values of
X
where
N
0
X
the solution of this equation is given by (6.53),
as well as by the Poisson equation (6.52). In addition, multiplication of (6.55) by
R
and its double differentiation leads to the Poisson equation (6.52), that is, (6.52)
and (6.55) are equivalent.
We thus have that the particle charge in a dense ionized gas and the screening of
this charge near the particle are determined by the Fuks approach, which is valid at
high plasma density (6.5). As the number density of electrons and ions increases
and criterion (6.5) is violated, the general concept of the Fuks approach for electron
and ion fluxes is conserved and is expressed by (6.50). These formulas are valid at
distances from the charged particle where screening of the particle charge is weak,
and hence the range of these distances is shortened as the number densities of
electrons and ions increase. Nevertheless, (6.50) and (6.51) may be used for eval-
uation of both the particle charge and its screening by surrounding electrons and
ions if the above formulas are used in the right-hand side of (6.55). In this case the
particle charge and screening parameters are determined by competition between
two dimension parameters, the particle radius
r
0
and the Debye-Hückel radius
r
D
,
and the Fuks limit corresponding to the criterion
Δ
N
D
r
D
r
0
.
(6.56)
The inverse ratio is valid in the positive column of gas discharge of not low pres-
sure if the effective particle radius is taken as the radius of curvature for walls.
The conditions were given above to generalize the Fuks approach, and they may be
used in the course of numerical solutions of the above equations. Unfortunately,
in contemporary evaluations (e.g., [98-102]) all these conditions are not fulfilled.