Page 440 - Water and wastewater engineering

GRANULAR FILTRATION 11-13

TABLE 11-2
Formulas used to compute the clean-water headloss through a granular porous medium
Equation Definition of terms
Carmen-Kozeny (Carmen, 1937) d grain size diameter, m
f friction factor
2
1 L a 2
h f g acceleration due to gravity, 9.81 m/s
L
3 dg h L headloss, m

k filtration constant, 5 based on sieve
2
h = 11 L a f p openings, 6 based on size of separation
L
3 g d
g k headloss coefficient due to viscous forces,
dimensionless
1 k i headloss coefficient due to inertial forces,
.
f 150 175
R dimensionless
L depth of filter bed or layer, m
d a R Reynolds number
R
p fraction of particles (based on mass)

within adjacent sieve sizes
Fair-Hatch (Fair and Hatch, 1933)
S shape factor (varies between 6.0 for
1 ( ) 2 L a spherical particles and 8.5 for crushed
2
hL kvS
3 d 2 g materials)
a superficial (approach) filtration
2
⎛
(1 ) 2 L a 6 ⎞ p velocity, m/s
hL kv ⎜ ⎟ ∑

3 g ⎝ ⎠ dg 2 porosity
viscosity, Pa · s
2
Ergun (1952a) kinematic viscosity, m /s
2 density of water, kg/m 3
(1 L a
h k v ) particle shape factor (1.0 for spheres, 0.82
L
3 2
gd for rounded sand, 0.75 for average sand,

0.73 for crushed coal and angular sand)
2
Lv a
1 −
k i
3
gd

dimensional analysis. The Carmen-Kozeny, Fair-Hatch, and Rose equations are appropriate for
sand filters when the Reynolds number does not exceed 6. For larger filter media, where higher
velocities are used, the flow may be in the transitional range where these equations are not adequate.
The Ergun equation is recommended for these cases (Cleasby and Logsdon, 1999).
The headloss through a clean stratified-sand filter with uniform porosity described by Rose
(1945) serves as an example for illustration:
2
.
1 067( ) ( D) (C )(f) (11-9)
v a
D
h ∑
L
()( g)( ) 4 d g

where h L frictional headloss through the filter, m
v a approach velocity (also known as face velocity, filtration rate, or loading rate ),

3
2
m/s (or m /s · m of surface area)

D depth of filter sand, m
C D drag coefficient