Page 190 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

6.4 Cyclone 165

6.4.1.1 Crawford Model
Crawford [6] derived a formula by applying fluid dynamics to the air phase, which
leads to

Q 1
v h ¼ u h ¼ ð6:46Þ
H ln r 2 =r 1 Þ r
ð
Substituting Eq. (6.46) into (6.45) leads to

q d 2 Q 2 1
p p
v r ¼ 3 ð6:47Þ
18l H ln r 2 =r 1 Þ r
ð
Over an infinitesimal period of time, dt, the particle moves outward along radial
direction a distance of dr ¼ v r dt and an arc length along the tangential direction,
rdh ¼ v h dt. For the same dt

dr rdh
dt ¼ ¼ ð6:48Þ
v r v h
It leads to

dr v r
¼ ð6:49Þ
rdh v h
Substituting Eqs. (6.46) and (6.45) into (6.49) leads to

rdr q d 2 Q
p p
¼ ð6:50Þ
dh 18l H ln r 2 =r 1 Þ
ð
where the right-hand side of this equation is constant for fixed particle size and
cyclone configuration. For a particle entering the cyclone at r ¼ r c when h ¼ 0, its
radial position in the cyclone is defined by

" 2 #
q d Q
2 2 p p
r r ¼ h ð6:51Þ
c
9l Hln r 2 =r 1 Þ
ð
When the particle reaches the collecting wall, r ¼ r 2 , and the corresponding
angle h 2 is determined by
" 2 #
q d Q
2 2 p p
r r ¼ h 2 ð6:52Þ
2 c
9l H ln r 2 =r 1 Þ
ð