Page 200 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

6.5 Filtration 175

later on, the equations for diffusion and interception were further simplified by Lee
and Liu [20]. Our discussion is focused on the staggered array models as follows.
In the staggered array model, air approaches the cylinders with a uniform speed
of U 0 . The diameter of the uniform cylinders is d f . In the cylindrical coordinates
defined by r; hÞ with origin at the center of the cylinder, the corresponding
ð
dimensionless stream line function is

sinh A a
r
3
W ¼ þ Br þ 2r lnr ðÞ
2Y r 2
1

1 a 1 Kn f þ aKn f
2
A ¼ ð6:68Þ
1 þ Kn f

ð 1 aÞ Kn f 1
B ¼
1 þ Kn f
where a is called the solidity of the filter, it is the ratio of solid volume to the entire

filter bulk volume; r ¼ 2r d f is a dimensionless distance from the center of the
fiber; the hydrodynamic factor Y is described as
lna 3 a Kn f 2a 1Þ 2 a 2
ð
Y ¼ þ þ ð6:69Þ
2 41 þ Kn f 1 þ Kn f 41 þ Kn f 4
In this equation, the Knudsen number is defined as the ratio of the mean free path
of the gas molecules to the radius of the filter fiber. Similar to Eq. (4.8), it is
expressed as

Kn f ¼ 2k d f ð6:70Þ

It quantifies the slip effect in particle filtration; Kn f is significant for d f <2 µm.
The mean free path of air (λ) under standard conditions is about 0.066 µm. In this
case, 2λ/d f < 0.066 % when d f >2 µm.
For cases with a 1 and Kn f 1, the streamline function becomes

sinh 1

W ¼ r þ 2r lnr ð6:71Þ
2Y r
For a typical filtration process under normal conditions, Kn f 1, and the
Kuwabara hydrodynamic factor in Eq. (6.69) can be simplified by letting Kn f ! 0:

lna 3 a 2
Y ¼ þ a ð6:72Þ
2 4 4