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3. The Motion of Particles
Drag force
Spherical particle, Re < 1
FD  3  d V
Drag coefficient
d particle diameter
V flow velocity
FD
24
CD  1

2
Re
2  aV A
A projected area
 2
d
4
Case 1: With slip
3  d V
FD 
Cc
Cc
is Cunningham correction factor
For d > 0.1 mm
For d > 0.01 mm
 2.52 
Cc  1  

d


  
 0.55  
Cc  1     2.514  0.8exp  

d 
 d 

Case 2: High Re, Re > 1
Case 3: Nonspherical particle
3  dV V 
FD 
Cc

dV
Shape/type
spherical
fiber (L/d = 4)
quartz dust
fused alumina
talcum (platelet)
is shape factor
is equivalent volume diameter

1
1.32 (axis perpendicular to flow)
1.07 (axis parallel to flow)
1.36
1.04-1.49
2.04
Motion under gravity
3  dV V 
FD 
Cc
Equation of motion
dVy
dt

Vy

g 0
Particle relaxation time or time constant
2

d
m

 p
3 d 18
Terminal settling velocity
 pd 2 g
VTS  g 
18
Mechanical mobility
2

d
VTS
p
B

mg 18m
Terminal settling velocity with slip, shape factor
2
 Cc   p d g
VTS   
   18
Motion under electrical forces
q  ne
FE  E q  ne E
q particle charge
n number of charge
e electron charge = 1.6x10-19 C
E electric field
In equilibrium
FE  FD
3  d V
ne E 
Cc
Terminal electrical velocity
Electrical mobility
ne E Cc
VTE 
3  d
VTE neCc
Z

E 3  d
Relation between VTE and E for two particle sizes
Motion under thermal gradients
Thermophoretic force -> Temperature gradient
Thermophoretic velocity
VT  k1T
Motion under no external force
Equation of motion
Velocity
dV
m
 3  d V
dt
V  V0 exp  t  
Traveling distance
t
t
0
0
x  t    V dt   V0 exp  t   dt
 V0 e  t 
Stopping distance, t >>
S  V0
 B mV0
 p d 2V0

18
Similarity in particle motion
1. Reynolds number (Re) must be equal
2. Stokes number (Stk) must be equal
Stk 
stopping distance
characteristic length
S

D
 pd U

18 D
2
With slip
 pCc d 2U
Stk 
18 D
Particle motion for several values of Stokes number
3. When gravity is important, gravitational parameter (G) must be equal
VTS
G
U
To determine if inertia or gravity is more important, use Froude number (Fr)
Stk V 2
Fr =

G
gD
Aerodynamic diameter
Aerodynamic diameter (da) is the diameter of a spherical particle of density
0 = 1 g/cm3 which has the same terminal settling velocity in air as the particle
of interest.
12
 p 
da  d p  
 0 
Stokes diameter (ds) is the diameter of a spherical particle that has the same
density and terminal settling velocity in air as the particle of interest.
12
 b 
da  ds  
 0 
 b is the bulk density
Comparison of equivalent volume diameter, Stokes diameter, and aerodynamic diameter.
Inertial impaction
Stokes number
2

d
U
p p U Cc
Stk 

Dj 2
9 D j
D j is the jet diameter
Collection efficiency characteristics of an impactor
Collection efficiency characteristics of an impactor: Ideal -v- real
Diffusion (Brownian motion)
Random motion of an aerosol particle in still air
Fick’s first law
dn
J   DB
dx
J
DB
n
x
is the particle flux (# particles per unit area per unit time)
is the diffusion coefficient
is the number of particles
is the direction of motion
Stokes-Einstein derivation
k T Cc
DB 
 kT B
3  d
RMS and average velocity
12
crms
 3k T 


 m 
12
 18k T 

  d 3 
 p 
12
 8k T 
c 


m


12
 48k T 
 2
   d 3 
p


Diffusion-related properties of standard-density spheres at 293 K
dp
(micron)
DB
(m2/sec)
B
(m/N sec)
c
(m/sec)
0.00037*
2.0x10-5
-
460
0.01
5.4x10-8
1.3x1013
4.4
0.1
6.9x10-10
1.7x1011
0.14
1.0
2.7x10-11
6.8x109
0.0044
10.0
2.4x10-12
6.0x108
0.00014
* diameter of air molecule
Deposition by diffusion
Aerosol particle collides and sticks to the surface
Fick’s second law
dn
d 2n
 DB 2
dt
dx
Boundary and initial conditions
n  0, t   0 , t  0
n  x,0   n0 , x  0
Solution
 x 
n  x, t   erf 
 2 D t 
B 

Concentration profile for a stagnant aerosol of 0.05-mm particles near a wall
General form of the concentration profile near a wall
Cumulative number of particle deposited per unit area during time t
12
 DBt 
N  t   2n0 




Deposition velocity: velocity that particles move to a surface and is analogous to
the terminal settling velocity due to gravity.
Vdep
J

n0
Cumulative deposition of particles on a horizontal surface during 100 sec.
Cumulative deposition
dp
(micron)
Diffusion
2
(#/m )
Settling
2
(#/m )
Ratio
Diffusion/settling
0.001
0.01
0.1
1.0
10
100
2.6x104
2.6x103
3.0x102
59
17
5.5
0.68
6.9
88
3500
3.1x105
2.5x107
3.8x104
380
3.4
1.7x10-2
5.5x10-5
2.2x10-7
Diffusion of aerosol particles on the tube wall
Penetration for circular tube
23
, m  0.009
nout 1  5.50 m  3.77 m
P

nin 0.819exp  11.5m   0.0975exp  70.1m  , m  0.009
Deposition parameter
4 DB L DB L
m

2
 dt U
Q
L
dt
U
Q
is the length of the tube
is the diameter of the tube
is the average velocity
is the flow rate
Penetration for rectangular tube
1  2.96 m 2 3  0.4 m
, m  0.005
P
0.910exp  7.54 m   0.0531exp  85.7 m  , m  0.005
Peclet number: another dimensionless parameter used in diffusion motion
UD
Pe 
DB
D
is the characteristic length
Penetration of aerosol particles in a tube.
Fractional loss to the walls by diffusion for an aerosol flowing through
a 1-m-long tube
dp
Flow rate (L/min)
(micron)
0.1
1.0
10
0.001
0.01
0.1
1.0
1.000
0.428
0.029
0.003
0.978
0.108
0.006
0.0008
0.422
0.025
0.001
0.0002