Page 446 - Water and wastewater engineering

GRANULAR FILTRATION 11-19

The expanded bed porosity (next to last column) is calculated from Equations 11-12 and 11-13 .
For the first row d 0.002 and
.
(0 002 m (0 2778839 m/ s
)
)
.
R 425..22
.
1 307 10 6 m 2 s /
0.1
R 1.83
and
0.1
0.2247 R 0.41
and
.
⎛ 001 m/s ⎞ 041
.
e 0 255.
⎝ 0 277884 ⎠
.
The first row of the last column is then
.
f 0053
0 0711
.
.
e ( 1 0 255)
1
where 0.053 is the mass fraction of sand having a geometric mean diameter of 0.002 m, that is,
between sieve numbers 8 and 12.
Using Equation 11-11 with a porosity of 0.45 and an undisturbed bed depth of 0.5 m from
Example 11-2 , the depth of the expanded bed is then

.
.
.
.
D 10 45 )(0 5 m )(2 53 ) 0.696 or 0 70 m
(
e
Comments:
1. The expansion ratio is D e / D 1.40 or a 40% bed expansion.
2. The depth of the expanded bed is the starting point for setting the elevation of the back-
wash troughs. In this case it would be D e D 0.70 m 0.5 m 0.2 m above the
undisturbed bed surface plus a factor of safety.
3. The “one-step” calculation procedure yields a D e of 0.60 m.

The hydraulic headloss that occurs during backwashing is calculated to determine the head
required for the backwash pumps. The loss of head in expanding the filter is calculated as the
gravitational force of the entire expanded bed:

(
)
e
e
F mg ( s )(1 )( a D )( g) (11-15)
g

where F g gravitational force of the expanded filter bed, N
3
m mass of bed, kg/m
2
g acceleration due to gravity, 9.81 m/s

3
s density of media, kg/m
3
density of water, kg/m
e porosity of expanded bed
2

a cross-sectional area of filter bed, m
D e depth of expanded bed, m