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164 6 Separation of Particles from a Gas

r 2 and r 1 , respectively. Assume there is no leakage and the gas is incompressible, the
flow rate Q is constant from the inlet through the annular chamber and the outlet.

6.4.1 Cyclone Fractional Efficiency

Assume the airflow within the cyclone is laminar. There is a critical radius, r c ,
where a particle to be located at this position eventually reaches the inner surface of
the outer body of the cyclone. Then, similar to the analysis for gravity settling
chamber and electrostatic precipitator, the fractional particle separation efficiency is

r 2 r c r 2 r c
g d p ¼ ¼ ð6:40Þ
r 2 r 1 W
At steady state, the average tangential velocity of the airflow is

Q Q
u g ¼ ¼ ð6:41Þ
WH ð r 2 r 1 ÞH
With the laminar flow assumption, the airflow does not mix along the radial
direction. The radial component of velocity of the particles can be derived from
Newton’s Second Law,

dv r
m p ¼ F C F D ¼ 0 ð6:42Þ
dt
where v r is the radial speed of the particle. Assume a spherical particle with a
density q and a diameter of d p , the centrifugal force, F C , and the drag force F D
p
exerted on the particle at any position r are, respectively
3 2
v 2 q pd v
p
h p h
F C ¼ m p ¼ ð6:43Þ
r 6 r
F D ¼ 8 pld p v r ð6:44Þ
In this equation, we ignored the Cunningham coefficient because cyclones are
used primarily for separating particles with large sizes. In the Stokes region,
F C ¼ F D , and it leads to

2 2
q d v
p p h
v r ¼ ð6:45Þ
18lr
It is commonly assumed that the particle follows air stream along tangential
direction, that is, v ¼ u . However, the exact description of air tangential speed u h
h
h
depends on the researcher.