Page 441 - Water and wastewater engineering

11-14 WATER AND WASTEWATER ENGINEERING

f mass fraction of sand particles of diameter, d g
d g geometric mean diameter of sand grains, m

shape factor

2
g acceleration due to gravity, m/s

porosity
The drag coefficient is defined in Equations 10-10 and 10-11 . The Reynolds number is used
to calculate the drag coefficient. The sand grain diameter is multiplied by the shape factor to
account for nonspherical sand grains. The summation term is included to account for stratifica-
tion of the sand grain sizes in the filter. The size distribution of the sand particles is found from a
sieve analysis. The mean size of the material retained between successive sieve sizes is assumed
to correspond the geometric mean size of successive sieves. It is calculated as

.
d d )
d g ( 12 0 5 (11-10)
where d g geometric mean diameter of grain size distribution between sieves, mm
d 1 , d 2 diameter of upper and lower sieve openings, mm

A cursory examination of the Rose equation reveals the following important relationships:
3
• The headloss is directly proportional to the square of the filtration or loading rate (m of
2
water/d · m of filter surface area or m/d or m/h), so small increases in loading rate are
amplified.
• Headloss is inversely proportional to the diameter of the sand grains.
• The porosity, which is assumed constant through the bed, plays a very strong inverse role
in the headloss.

From a design point of view, given the design flow rate ( Q ), the filtration rate ( v a ) may be
adjusted by adjusting the surface area of the filter box. The sand grain size distribution is speci-
fied by the effective size and uniformity coefficient. The headloss can be reduced by limiting the
amount of fines in the distribution of sizes.
The porosity of the sand plays a significant role in controlling the headloss, and it is not in
the control of the designer. Although other media, such as anthracite coal, provides an alternative
means of adjusting the porosity, the range of porosities is not great. A further confounding fac-
tor is that all the headloss equations assume a uniform porosity through the depth of the bed—an
unlikely occurrence once the bed is stratified.
A subtle but important variable that is not evident from a cursory examination of the equa-
tion is that of the water temperature. The viscosity is used in calculating the Reynolds number
and it, in turn, is a function of the water temperature.
Example 11-2 illustrates the computation of the clean bed headloss for a stratified sand filter.

Example 11-2. Estimate the clean filter headloss in Ottawa Island’s proposed new sand filter
using the sand described in Example 11-1 , and determine if it is reasonable. Use the following
2
3
assumptions: loading rate is 216 m /d · m , specific gravity of sand is 2.65, the shape factor is 0.82,
the bed porosity is 0.45, the water temperature is 10 C, and the depth of sand is 0.5 m.