Encyclopedia of Environmental Science and Engineering, Volume I and II

TABLE 3

Motion of a single spherical particle

Rep  1 (Stokes) 1  Rep  104 104  Rep (Newton)

drag coefficient, CD 24/Rep

055 48

2

.
.
Rep

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

0.44

drag force,

RCAf Dpf

v



r^2

2

3 pmDpv

055 48

8

2

..

vDpfr D vp

m

pm

⎛

⎝

⎜

⎞

⎠

⎟

0.055prf (vDp)^2

gravitational settling equation of

motion m v

t

ppmgRf

p

f

d

d

 1

r

r

⎛

⎝⎜

⎞

⎠⎟

or,

d

d

v

t

 1 

3

4

r 2

r

r

r

f

p

f

pp

g DCvD

⎛

⎝⎜

⎞

⎠⎟

terminal velocity, vt (dv/dt  0) Dgpp f

2

18

rr

m

() AAA 12 212

11

⎛ +

⎝

⎜⎜

⎞

⎠

⎟⎟



.

A

(^1) fpD
 48. m
r
AgDpf
f
2 ^254 p

.
rr
r
3
12
Dgpp f
f
()
/
rr
r
⎛ 
⎝⎜
⎞
⎠⎟
unsteady motion
time, t
velocity, v
t vv
g vv
t
t
 

t 1 n^0
⎛
⎝⎜
⎞
⎠⎟
t
g CC
p
p Dt t D p
 p

(^2422)
0
t
dRe
Re Re Re
Re
∫
not simple because of Rep  104
at initiation of motion
falling distance, S
St
t
∫ 0 vd
vttgtv v t
g
t 
t
()exp 0 1
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
vt
tt
tg p
t
gp p p
t
t
Re
/ , Re Re / Re
0 d
0
∫

Re
18
initial velocity, : terminal v
2
p
pf
g
pp
t
vD D
vv
r
m
t
r
m
,,: 0 eelocity
Rep 0 , Rept: Rep at v 0 and at vt respectively, CDt: drag coefficient at terminal velocity
20 AEROSOLS
where A p is the projected area of the particle on the flow (
pDp^2 /4), and C D is the drag coefficient of the particle. The
drag coefficient C D depends on the Reynolds number,
ReuDrpfrm/ (13)
where u r is the relative velocity between the particle and air
(  | u  v |, u  velocity of air flow, v  particle velocity),
and m is the viscosity of the fluid.
The motion of a particle having mass m p is expressed by
the equation of motion
m
p t
d
d
v
∑F (14)
where v is the velocity of the particle and F is the force acting
on the particle, such as gravity, drag force, or electrical force.
Table 3 shows the available drag coefficients depending on
C001_002_r03.indd 20C001_002_r03.indd 20 11/18/2005 10:09:10 AM11/18/2005 10:09:10 AM