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02_chap_wang.qxd 05/05/2004 12:39 pm Page 61
Fabric Filtration 61

K = ε 3 cS 2 (3)

where ε is porosity or fraction void volume (dimensionless), c is a flow constant, K is
the Kozeny coefficients, and S is the specific surface area per unit volume of porous
−1
media [ft −1 (m )]. Values of the Kozeny constant can be estimated using the free-sur-
face model (2). Assuming a random orientation averaging two cross-flow fibers and one
parallel fiber and assuming that a cloth medium behaves like a bed of randomly oriented
cylinders, the constant for flow parallel to the cylinder is obtained by

  1 2 
ε
c = 2ε 3 (  1 − ) 2 ln − 3 + ( 4 1 − ) − (1ε − )  (4)
ε


  1 − ε 
and when flow is at right angles to the cylinder,
   1  1 − (1 − ) 

2

ε
c = 2ε 3 (  1 − )  −  (5)
ε ln
ε
   1 −  ε 1 + (1 − ) 
2


As the system is operated, cake deposits on the fabric, producing an additional flow
resistance proportional to the properties of the granular cake layer. The resistance to
fluid flow owing to cake build-up usually amounts to a significant portion of the total
flow resistance. This resistance increases with time as the cake thickness increases. This
additional resistance (∆P ) is typically of the same order of magnitude as the residual
2
resistance (∆P ) and can be expressed as
1
∆P = K v Lt (6)
2
2
2
where ∆P is the change in pressure drop over time interval t [in. H O (cm H O)], K
2 2 2 2
is the cake-fabric filter resistance coefficient,
 in. of water   cm of water 
 1b dust ft )(  or  kg dust m )( 
2
2
 (  m ft min)    (  m min)  
3
3
v is fluid velocity [ft/min (m/min)], L is inlet solids concentration [lb/ft (kg/m )], and
t is time (min). An expression for the cake–fabric filter resistance coefficient using the
Kozeny–Carman procedure has been derived for determining flow through granular
media (2):
k  µ

2
K = ( 3.2 × 10 )   f S    1 −  ε (7)
−
3
2
 g  ρ p   ε 3 
where k is the Kozeny–Carman coefficient, which equals approx 5 for a wide variety of
fibrous and granular materials up to a porosity equal to about 0.8, ε is the porosity or
fraction void volume in cake layer (dimensionless), µ is fluid viscosity [lb /(s ft)], ρ
f m P
3
is the true density of solid material (lb /ft ), and the S is the specific surface area/unit
m
−1
volume of solids in the cake layer (ft ). This equation shows that as the particles being
filtered become smaller in diameter, the porosity of the cake decreases and consequently,
K increases. The net result of the larger cake–fabric filter resistance coefficient (K ) is
2 2
that the pressure drop increases as porosity decreases.