Page 186 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

6.3 Electrostatic Precipitation 161

At the surface of the particle, where r ¼ d p 2; Eq. (6.32) leads to,
2
2K E ne

N i d p 2 ¼ N i0 exp ð6:33Þ
d p kT
The number flux of ions can be determined by assuming that all the ions are
captured when they strike the particle.

At r = d p 2, the number of ions loss from the air to the surface of the particle is
then described as

dn N i0 c i pd 2 p 2K E ne 2
¼ exp ð6:34Þ
dt 4 d p kT
where n is the number of ions moved from air to the surface of the particle,
.
c i = mean thermal speed of the ions and (pd 2 4) stands for the surface area of the
p
particle. Integration of the above equation leads the number of ions charged on
the particles at time t
2
d p kT d p K E c i pe N i0
nðtÞ¼ 2 ln 1 þ t ð6:35Þ
2e K E 2kT
where c i is the mean thermal speed of ions. Under normal condition, the mean
thermal speed of ion c i is about 239 m/s [41]. k is Boltzmann constant
2
2
9
(1.38 × 10 −23 J/K), K E =9 10 Nm /C , and N i0 is ion concentration.
6.3.2.2 Field Charging

With the presence of electric field, ions are forced to move along the direction of the
electric field. Leading to a high rate of collision between the ions and the particles.
This is referred to as field charging mechanism. The number of ions charged to a
particle by field charging depends on the properties of the particle, its size, and the
intensity of the electric field, E:
Ed 2 t
p 3e r
nðtÞ¼ ð6:36Þ
4eK E 2 þ e r t þ s
6
where E= intensity of the electric field with a typical value of 10 V/m, e r ¼ rel-
ative permittivity or dielectric constant of the particle with respect to vacuum, and
e r ¼ e=e 0 ; e 0 ¼ 8:854 10 12 C/Vm is the permittivity of a vacuum. The per-
mittivity of typical particles can be found in handbooks. τ is the charging constant
and it varies with the field condition.