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

dw
m = F − F d (29)
e
dt
where m and w are particle mass and migration velocity, respectively, and F and F are
e d
the electrostatic and viscous drag forces acting on the particle, respectively. For a col-
lecting field of E , F is given by qE . Also, assuming laminar flow exists, the viscous
p e p
drag force on a spherical particle is given by Stoke’s law, F = 6πµaw, where µ is the
d
gas viscosity. Thus, Eq. (29) becomes
dw
m = qE − 6π µ aw (30)
p
dt
whose integration velocity yields
 qE    6πµ at  
w =    1 − exp −  (31)
p
 6πµ a   m  
The exponential term is quite negligible for t > 0.01 s. Dropping this term is equivalent
to ignoring the acceleration term on the left-hand side of Eq. (31). Consequently, Eq.
(30) reduces to
qE p
w = (32)
6πµ a

which is a result of equating electrostatic and viscous drag forces. For particles charged
by ion bombardment, the maximum charge attained by the particle is given by Eq. (20).
Thus, Eq. (32) becomes

2 K PaE E p
w = (33)
c
0
3µ
where E is the electric strength of charging field. For single-stage precipitators, the
charging field E and the collecting field E are approximately the same.
c p
The migration velocity is therefore seen to be proportional to the discharge and col-
lecting fields and also the particle radius a, but inversely proportional to gas viscosity
µ. In practice, laminar flow is seldom achieved, and Eq. (32) would overestimate migra-
tion velocities for nonlaminar flow. If the particle size approaches the mean free path of
gas molecules (λ= 6.8 × 10 −8 m in atmospheric air at 25°C), then Eq. (33) must be
multiplied by the Cunningham correction factor,
λ
   a  
C = 1 +   1 26. + 0 40. exp −1 10.  (34)

    λ  
a
This means an increase in migration velocity; for example, for a particle of 0.5 µm
radius in atmospheric air at 25°C, the Cunningham correction factor is 1.17, an increase
of 17% in migration velocity.
Assume that a particle at the discharge electrode must move a distance d to be col-
lected at the collecting electrode. Let the particle velocity in the direction of gas flow
be the same as the gas velocity v, whereas the transverse velocity from the discharge
electrode to the collecting electrode be given by the migration velocity w. The particle