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Electrostatistic Precipitation 163

The factor P in Eq. (18) is a measure of field distortion as a result of particle charge.
It reduces to unity (i.e., no distortion) for D = 1, and it approaches 3 for large values of
D (i.e., for conducting particles). For most dielectric substances, D is less than 10.

2.3.2. Diffusion Charging
Field charging becomes less important as particle size decreases, and, subsequent-
ly, diffusion charging begins to play a more important role. The role of diffusion
charging can be examined by the following analysis based on the kinetic theory of
gases. It is known from kinetic theory that the density of gas in a potential field varies
according to
N = N exp( − U KT) (22)
0
where N is the initial ion density, U is the ion potential energy, K is the Boltzmann con-
0
stant, and T is the absolute temperature. The potential energy of an ion with charge q ,
i
in the vicinity of a uniformly charged spherical particle, is
U = qq i (23)
4π Kr
0
where r is the distance from the center of the particle to the ion. The number of ions that
strike the particle per second is, from kinetic theory,
2
(Nv 4)(4πa 2 ) = π a Nv i
i
where v is the root mean square velocity of the ion. Thus, the time interval associated with
i
the ion–particle collisions is
ts ( ) = 1 (24)
2
πaNv
i
Assuming that every ion that makes contact with the particle is captured, the particle
charging can be described by
dq
=π aNv (25)
2
dt i
The solution of Eq. (25) with the initial condition q = 0 is

4π K akT  aN q v t 
2
q = 0 ln  0 i i + 1  (26)
d
q  4 KkT 
i 0
Example 2
Estimate the number of electronic charges acquired by a spherical conducting particle
(P = 3) under conditions of field charging and diffusion charging. Assume a particle radius
of 1 µm and a charging time of 0.1 s.
Solution
Repeat calculations for a particle radius of 0.1 µm. Use the following conditions: E =
0
2
2
6
3
2
1 ×10 V/m, N = 1×10 14 ions/m , m = 4×10 −4 m /V s, K = 8.85×10 −12 C /N m , q =
0 i 0 i
1.602×10 −19 C (electronic charge), k = 1.38×10 −23 N m/K, T = 313 K, and ν = 500 m/s.
i