Page 119 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

4.1 Particle Motion 93

With the first assumption, the drag force on the moving particle can now be
described by

F D ¼ 3pld p u vj ð4:2Þ
j
where u is the velocity of air and v is that of the particle.
In Stokes regime, the resistance experienced by a moving particle in a gas is
described by the above equation. In practice, this equation is not perfect. When
particle Re p ¼ 1:0, the error is 12 %. The error can be reduced to 5 % at a Re p ¼ 0:3.
The same drag force can also be calculated using Eq. (4.3),

1 2

F D ¼ C D qu A ð4:3Þ
2 1
which was derived from Newton’s law. Substituting the cross-section area (or
2
rigorously, the frontal area) A ¼ pd =4 into Eq. (4.3) leads to
p

1 2 1 2
F D ¼ C D q u vð Þ pd p ð4:4Þ
g
2 4
Comparing Eqs. (4.2) and (4.4), we can get the drag coefficient in the Stokes
regime,
24l 24
C D ¼ ¼ ð4:5Þ
q d p u vj Re p
j
g
This is the equation for the dashed straight-line portion at the up-left corner of
Fig. 2.4. Note that when we first introduced Fig. 2.4 and Eq. (4.3), we assumed the
solid phase (the sphere) is static and only air moves around the sphere. It would be
easy to understand the concepts in this section by assuming air is quiescent and only
the aerosol particle is moving. Indeed u vj represents the relative motion of
j
particle with respect the air.
Overall, for spherical particles, the following relationships can be used to esti-
mate drag coefficient.

24 Re p 1
8
Re p
>
<

C D ¼ Re p 1 þ 0:15Re 0:687 1\Re p 1000 ð4:6Þ
24 p
>
0:44 Re p [ 1000
:
4.1.3 Dynamic Shape Factor

In the Stokes’ analysis above, the particles were assumed rigid spheres. But in
reality, most of the particles are nonspherical. Being cubic, cylindrical, crystal, or