Page 442 - Water and wastewater engineering

GRANULAR FILTRATION 11-15

Solution. The computations are shown in the table below.

f
)( )
(C D
Sieve no. % Retained d, m R C D ∑
d
8–12 5.3 0.002 3.1370 9.684551 256.64
12–16 17.1 0.00142 2.2272 13.12587 1,580.7
16–20 14.6 0.001 1.5685 18.03689 2,633.4
20–30 20.4 0.000714 1.1199 24.60549 7,030.1
30–40 17.6 0.000505 0.79208 34.01075 11,853
40–50 11.9 0.000357 0.55995 47.21035 15,737
50–70 5.9 0.000252 0.39526 60.72009 14,216
70–100 3.1 0.000178 0.27919 85.96328 14,971
100–140 .7 0.000126 0.19763 121.4402 6,746.7
Total (C D )(f)/d 75,025

In the first two columns, the grain size distribution from Example 11-1 is rearranged to show
the fraction retained between sieves. The third column is the geometric mean diameter of the
sand grain computed from the upper and lower sieve size. The fourth column is the Reynolds
number computed from Equation 10-9 with the correction for nonspherical sand grains. For the
first row,
()()( ) (0 82. )(0 002. m )(0 0025. m/s )
dv
R
a .
3 137
2
1.307 10 6 m /s
3
The filtration velocity of 0.0025 m/s is the conversion of the filtration loading rate to compatible
units:
3
216 m /dm 2
v 0 0025 m/s.
,
a
86 400 s/d
The kinematic viscosity is determined from Appendix A using the water temperature of
6 2 2
10 C. The factor of 10 is to convert from m /s to m /s.
The drag coefficient is calculated in column 5 using either Equation 10-10 or 10-11 , depend-
ing on the Reynolds number. For the first row,
24 3
C 0 34
.
D
/
R R 12

.
.
76507 1 69380 34 9 6846
.
.
The final column is the product of the fractional mass retained and the drag coefficient divided
by the diameter. For the first row,
(C D )( ) (9 6846. )(0 053. ) 25664. m 1
f
.
d 0 002