Page 122 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

96 4 Properties of Aerosol Particles

4.2.1 Particle Acceleration

Consider a particle with constant mass m that is released in quiescent air with an initial
velocity of zero. Newton’s second law of motion must hold at any instant t [ 0.

~ ð4:14Þ
X d~ vðtÞ
F ¼ m
dt
where ~ vðtÞ is the particle velocity in the static air at time t, and the mass of the
particle is considered as a constant when there is no evaporation or growth. In this
case only two forces, a constant force of gravity and a drag force, act on the falling
particles. The drag force depends on the particle velocity at any instant, ignoring
additional acceleration of the surrounding air.
At any instant, the drag force is given by Stokes’ law. Taking the positive
direction downward, the above vector equation can be described using magnitudes

X 3pld p vðtÞ dvðtÞ
F ¼ mg F D ) mg ¼ m ð4:15Þ
C c dt
Note that in this equation we ignored the bouyant force. This is valid for typical
condtions, when the aerosol particle density is much great than that of the air.
Rearranging the above equation and integration with the initial condition of v =0
at t = 0 leads to

Z t vtðÞ
Z
3pl d p dvðtÞ
dt ¼ h i ð4:16Þ
mC c vðtÞ mgC c
0 0 3pld p
3
1
Integrating both sides and replacing m with pqd leads to
6 p
2
0 1
q p d gC c
t 18l vðtÞ
¼ ln @ A ð4:17Þ

q p d 2 C c q p d 2 gC c
p
18l 18l
If we define a constant
2
q d C c
p p
s ¼ ð4:18Þ
18l
then we get the settling velocity of the particle, vðtÞ, at any time, t.
t
h i
vðtÞ¼ gs 1 exp ð4:19Þ
s