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6.5 Filtration 187

h n i
8
1 a d bed þb for d bed [ 2
<
d G d G
a ¼ 6:1 ð6:108Þ
1 12:6 exp for \2
: d bed 3:6d bed d bed
d G d G d G
where d bed = body diameter of packed bed. The coefficients of a, b, and n are
constants and dependent on the shape of the granules. For spherical granules,
a ¼ 1; b ¼ 0:375 and n ¼ 2
By ignoring the ESP effect, the filtration efficiency of a single granule is
described in terms of several dimensionless variables as
2=3 15=8 0:4
3
g ¼ 4A 1=3 D p þ A s N 1=8 d p þ 3:38 10 A s Gr 1:2 d p
sG s vdw
U 0 d G d G d G
ð6:109Þ
where D p = particle diffusion coefficient, U 0 = approaching air speed, d G = granule
diameter, d p = particle diameter, Gr = gravity number, which is defined as the ratio
of gravity settling speed to approaching air speed, N vdw = van der Waals number,
and A s = filter porosity parameter. The first term on the right-hand side characterizes
the diffusion effect, second term for van der walls effect, and the last term for
gravitational effect.
With the definition of gravity settling speed defined in Eq. (6.110)

2
q d gC c
p p
v TS ¼ ð6:110Þ
18l
the gravity numberGr is described as
2
V TS q d gC c
p p
Gr ¼ ¼ ð6:111Þ
U 0 18lU 0
The filter porosity parameter can be described using the solidity, a.

5
2 2a 3
A s ¼ ð6:112Þ
1
5
2 3a 3 þ 3a 3 2a 2
The van der Waals number, N vdw , is described in terms of Hamaker constantH ij

Table 6.2 Hamaker
Particle-media Hamaker constant (J) References
constants
Glass beads–Air 5 10 19 [1]
NaCl particles–Air 0:64 10 19 [2]
Silica–Air 0:65 10 19 [13]