Chapter Four: Single Particle Dynamics
Stokes’s Law and Corrections
Aerodynamic Diameter
Particle Relaxation Time and Stop Distance
High Reynolds Number
A. Stokes’s Law
Def.: Drag coefficient
Drag coefficient = (Drag Force/Projected Area)/(Dynamic Pressure)
F /A
CD  D 2
V / 2
Based on experimental results as shown above:
24
24
for Re < 1, or the Stokes’s law and where Re is the
CD 

Re d pV p
Reynolds number,  and  are the density and viscosity of air, dp is
the particle diameter, and Vp is the particle velocity relative to air.
C D  0.44 for Re > 1000, or the Newton’s law; and in the transition region
CD 
24
( 1  0.15Re0.687 )
Re
For a spherical particle under Stokes’s law, which is generally valid for the aerosol
in the ambient atmosphere, the drag force is:
FD  3d pV p , for Re < 1 or the Stokes’s law
Assumptions for the above relationships:
1. rigid spherical particle
2. Stokes’s law or inertial force is much smaller than viscous force
3. continuum fluid
4. free flow without wall effects
5. the density of air is constant or low Mach number flow
6. steady state flow
Corrections for Stokes law:
1. Slip correction for non-continuum flow:
FD 
3d pV p
Cc
C c  1  2.52
CC  1 

d
and
for dp  0.1 m

d 

2.34

1.05
exp

0.39

 for all particle sizes
d 
  

where Cc is the Cunningham slip correction factor.
2. near wall correction for non-symmetrical flow pattern:
FD 
3d pV p
Cc
k and k  1 
9 dp /2
, where h is the distance
16 h
between the center of particle and the surface
3. dynamic shape factor: correction for non-spherical particle
FD 
3d pV p
Cc
k where  is the shape factor as shown below
B. Aerodynamic Diameter
At terminal settling velocity, the drag force is equal to the gravitational force
3d pV p
Cc
(  p   )d p g
3
k 
6
and V p  Vs , thus
(  p   )d p g C c
Vs 
, for Re < 1.0
18
k
2
 pd p g
C
If (  p   ) and c  1 , Vs 
18
k
2
Effect of Pressure on Terminal Setting Velocity of Standard Density Spheres at 293K
Particle
VTS at the Indicated Pressure(m/s)
Diameter
P=0.1 atm
P=1.0 atm
P=10 atm
(m)
-8
-9
0.001
6.910
6.910
6.910-10
0.01
0.1
1
10
100
6.910-7
7.010-6
8.810-5
0.0035
0.29
7.010-8
8.810-7
3.510-5
0.0031
0.25
8.710-9
3.510-7
3.110-5
0.0029
0.17
Definition: Mobility B is the ratio of the terminal velocity of a particle to the steady
V
1
state force producing that velocity, thus, B 
for large particles.

FD 3d p
C. Particle Relaxation Time and Stop Distance
The particle velocity in a still air when released from rest can be derived as (based
on Newton’s 2nd law):
dV p
mp
 m p g  FD  m p g  3d pV p with V p (t  0)  0
dt
 pd p2 g
 pd p2
t / 
(1  e
) where  p 
Thus, V p 
= particle relaxation time
18
18
p
Thus, particle relaxation time is the characteristic time for particle to transit from
one state to another state.
Relaxation Time for Standard Density Particles at Standard Conditions
Particle Diameter
(m)
Relaxation Time
(s)
0.01
0.1
1.0
10.0
7.010-9
9.010-8
3.510-6
3.110-4
100
3.110-2
Because the particle relaxation time is generally very small, the particle dynamics
are generally assumed to be in equilibrium. That is, the transition period is
generally not taken into account for small particles.
The horizontal distance particle traveled when injected horizontally into a still air
is
mp
d 2xp
dt 2
  FD  3d pV p  3d p
dx p
dt
with x p (t  0)  0 is the particle position, Thus
 pd p2
t / 
t / 
xp 
V po (1  e
)   pV po(1  e
)
18
p
where V po is the initial particle velocity.
p
Therefore, the maximum traveling distance or the stop distance S is S   pV po
Note that the above equation is only valid for Stokes flow. Mercer (1973)
derived the following approximate equation within 3% difference for initial
Reynolds number up to 1500 as
 p  1/ 3
 Re1/0 3  
S
 Re0  6 arctan 

g 
 6 
d
Stopping Distance, Initial Reynolds Number, and Time to Travel 95 Percent of
the Stopping Distance for Standard Density Spheres with an Initial Velocity of
10 meters per Second
Particle
Diameter
(m)
0.01
0.1
1.0
10.0
100
Stopping
Distance ,a
V0=10 m/s
Re0
0.0066
0.066
0.66
6.6
66
(mm)
7.010-5
9.010-4
0.035
2.3 b
127 b
Time to Travel
95% of Stopping
Distance a
(s)
2.010-8
2.710-7
1.110-5
8.510-4b
0.065b
D. High Reynolds Number
The relationship between drag force and velocity is no longer linear at high
Reynolds number. Therefore, the determination of the terminal settling velocity is
more complicated than those in Stokes flow. There are two different methods to
determine the terminal settling velocity: iteration and graphical method.
1. Iteration method
C D  f (Re)
 4 p d p g 

V s  
 3C D  
1/ 2
and CD is function of Reynolds number
Therefore, the iteration method is to start with a guess for Vs, and then compute
the CD as given at the beginning in this chapter. The computed CD is then used
to calculate the Vs’ as the above equation. The iteration is repeated until the
difference between Vs’ and Vs is within acceptable error.
2. Graphical method
CD is function of Reynolds number as shown at the beginning in this chapter
4  p d p g
3
C D Re 
2
3 2
 K  const
Therefore, the interception of the two lines as shown below is the answer.