Page 135 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

4.4 Particle Coagulation 109

4p a þ bÞ D A þ D B Þ
ð
ð
Ka; bÞ ¼ ð4:40Þ
ð
aþb 4 D A þD B Þ
ð
þ
aþbþd AB ð aþbÞc AB
where a and b are the radii of the particles of concerns, c AB is the average particle
thermal velocity, d AB is the jump distance, and D i is the diffusivity of the particle of
size i ð¼A or BÞ. If we use particle diameters, d pA and d pB , instead of the radii,
Eq. (4.40) becomes

2p d pA þ d pB D A þ D B Þ
ð

Kd pA ; d pB ¼ ð4:41Þ
ð
d pA þd pB 8 D A þD B Þ
þ
ð d pA þd pBÞc AB
d pA þd pB þ2d AB
The particle diffusion coefficient is calculated using
kTC c
D i ¼ ð4:42Þ
3pld pi
and the mean thermal velocity are calculated using
2 2 1=2

c AB ¼ c þ c B ð4:43Þ
A
where the mean thermal velocity of the particle can be calculated using the equation
for gas molecules (Eq. 2.7) by replacing the mass of a molecule with the mass of the
particle.

r ffiffiffiffiffiffiffiffi
8kT
c i ¼ ð4:44Þ
pm i
Similar to Eq. (4.43), the jump distance, d AB , is determined as

2 2 1=2
d AB ¼ d þ d ð4:45Þ
A B
where the individual jump distance is calculated with
1:5
3

2 2
d pi þ l i d þ l
pi i
d i ¼ d pi ð4:46Þ
3d pi l i
8D i
l i ¼ ð4:47Þ
pc i
It is obviously a very tedious work to accurately predict the particle size change
by polydisperse coagulation. Readers are referred to the literature for more com-
plicated models for polydisperse coagulation.