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160 Chung-Shin J. Yuan and Thomas T. Shen

where the cylinder is assumed to have zero potential and r is assumed negligible with
0
respect to r . Using Eq. (5) to eliminate σ in Eq. (10) yields
1 i
σ ln r r )
1
V = V + p r + i ( 1 0 (11)
2
c
0
1
4 K 8π Km V
0 0 1 0
The principal effect of additional space charge as a result of particles is that, for a
fixed voltage between the discharge and collecting electrodes, there is a reduction in cur-
rent. Particle space charge may be calculated using Eq. (21). In recapitulating the various
expressions for field strength, note that the simplest expression, Eq. (3), neglects any
form of space charge; this is tantamount to negligible corona current. The next expression,
Eq. (4), considers the presence of ionic space charge. The final expression, Eq. (9),
recognizes both ionic and particle space charge.
The above results hold true for the coaxial wire–cylinder geometry only. In trying to
obtain similar results for the duct or wire–plate geometry, the solution of Poisson’s equa-
tion poses some mathematical difficulties. One way around this is to find approximate
solutions for low corona current cases. It can be shown that (14,15)

4π Km
0(
i = b ln( d' r ) VV − ) (12)
0
1
V
1
c
0
2
0
where b is the wire-to-plate spacing and d′ is a parameter given by
d′= 4 π for b c ≤ 1.0
b
where c is the wire-to-wire spacing. The wire–plate geometry where b/c ≤ 1.0 covers
most of the practical duct-precipitation cases. The corona-starting voltage V in Eq. (12)
c
is given by
′
V = r E ( d r ) (13)
ln
c
0
c
0
The average plate-current density is given by
i = i 2 c (14)
1
a
The case of uniformly distributed particle space charge can be expressed by
σ b 2 ln dr ′ ( )
V = V + p i + 0 (15)
c
2 K 1 4 pK m V
0 0 i 0
Up to this point no mention has been made of calculating the corona-starting field
intensity E or the sparkover field. It is very difficult to calculate reliable values of these
c
parameters by using atomic data. Practical values are best established by observing var-
ious similar installations. The reader is referred to Table 1 in Section 3.1 for some typical
electrical parameter of practical electrostatic installations.
An empirical evaluation of E for round wires and outer electrodes of arbitrary shape
c
can be obtained from (15)
r ′)
/
E =′  A′ + B′ ( δ 12  (16)
δ
0
c