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166 6 Separation of Particles from a Gas

Substituting r c back into Eq. (6.40) leads to the description of the fractional
efficiency as
i 1=2
2
h
2 q p d p Qh 2
r 2 r
ð
2 9lH ln r 2 =r 1 Þ
g d p ¼ ð6:53Þ
r 2 r 1
It seems like we have now a mathematical description but it requires a certain
calibration because h 2 is unknown. In addition, it is unlikely that there is a laminar
flow in an industrial scale cyclone. Alternatively, we can use the empirical model
developed by Lapple [19] that follows.

6.4.1.2 Lapple Model

A semi-empirical model was developed by Lapple [19]. The average radial speed is
described in terms of the migration time. It can be considered as the average
terminal speed in the centrifugal field. On average, before the particle reaches the
inner surface of the cyclone outer wall, corresponding to r ¼ r 2 , the average ter-
minal speed is

r 2 r c
v r ¼ ð6:54Þ
t
where t is the corresponding residence time. Combination of Eqs. (6.45) and (6.54)
leads to

2 2
q d v
p p h
r 2 r c ¼ t ð6:55Þ
18lr
Substituting Eq. (6.55) into (6.40) leads to the fractional efficiency of the par-
ticles with diameter d p
2 2
q d v
r 2 r c p p h
g d p ¼ ¼ t ð6:56Þ
W 18lrW
The residence time depends on the engineering design of the cyclone. Many
models of commercial cyclones have been developed since the end of the nine-
teenth century. Typical cyclones are classified into four basic categories, as depicted
in Fig. 6.7, based on airflow direction:
(1) reverse flow with a tangential inlet (involute),
(2) reverse flow with a guide vane inlet (vane-axial),
(3) uniflow with a tangential inlet, and
(4) uniflow with guide vanes.