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Rapid Filtration 361

design. The spreadsheet may be used for design of a generic Analysis
under-drain system. Any combination of design variables may The first task is to determine the air flow, that is, Q(air
be selected to assess the effect on orifice flows and the wash), required by the system. The second task is to cal-
culate the pressure required at the outlet of the compres-
pressure surface for the manifold system as a whole.
sor. This is done as depicted in Figure 12.30. First the
The objective of manifold design is to have uniform orifice
pressure in the bubbles as they emerge from any given
flowoverthewholemanifoldsystem.Themaximum orificeflow
orifice is the static pressure of the water, that is,
occurs at the first lateral and the first orifice while the minimum
p(bubble) ¼ r w gh(water) plus atmospheric pressure. Then
orifice flow is at the last lateral and the last orifice. The goal of the pressure losses due to pipe friction must be calculated,
manifold design is to minimize the difference between these two for example, by Equation D.32, along with any other
extremes in orifice flow but with the constraints that the sizes of losses such as bends. This gives, by the Bernoulli relation,
the header, laterals, orifices and the number of laterals, and the pressure at the compressor exit. Knowing the absolute
orifices should be within practical guidelines. As a corollary to pressure at the compressor exit, the power can be calcu-
objectiveofhavingnear-uniformorificeflowsisthatthepressure lated by the equation for an adiabatic compression, that is,
surface should have a minimum difference between the first Equation D.75.
and last orifice. Solution
1. Determine the air flow, Q(air wash),
12.4.4.1.1 Hydraulic Grade Line for Backwash
System Q(air-wash) ¼ Loading(air-wash) A(filter)
Figure 12.30, a three-dimensional perspective drawing for back- ¼ 1:52 m = min (STP) (4:267 m 6:096 m)
3
wash, applies also to the air wash and illustrates the idea of 3 3
¼ 39:52 m = min (1396 standard ft = min )
relative pressure losses. For air, with pressure dimensions
applied to the Bernoulli relation, we may think of a ‘‘pneumatic’’
2. Determine the pressure, p 2 . which is the friction loss
grade line (PGL, which has pressure as the energy dimension, and other losses in the pipe system plus the water
3
that is, FL=L ) instead of an hydraulic grade line (HGL, which pressure at the depth of submergence, that is,
has length as the energy dimension, that is, FL=F).
The Darcy-Weisbach equation is applicable; for gases p 2 ¼ Dp(friction) þ rwgh(water)
2
the form is D p ¼ f (L=D) r(gas) (v =2). Equation D.23,
0.5
Q(orifice) ¼ A(orifice)C D [2gDh] , is for orifice flow. Table where
CDD.2 may be adapted as a spreadsheet solution to calculate
the power of an adiabatic compression required for the com- v 2
L
Dp(friction) ¼ f r(air) (D:43)
pressor (default compressor and motor efficiencies for 0.70 D 2g
for each, respectively, are incorporated in the spreadsheet).
The orifice coefficient of 0.61 is also a default value, as is, f, 3. Calculate the theoretical compressor power for an
the pipe friction coefficient. The pressure for an air wash adiabatic compression by Equation D.72, that is,
system is determined by the water depth at the orifice, the
Dp(orifice) for the required air flow per orifice, and the friction P ¼ Q(air) p 1 (k=(k 1){(p 2 =p 1 ) [(k 1)=k] 1} (D:59)
and minor losses through the laterals, manifold, and other
4. Assume the compressor efficiency is say 0.7 and the
pipes, which gives the pressure, p 2 for the compressor; p 1
pressure is the ambient air pressure. The air flow, Q(air for motor efficiency is 0.7 to calculate actual power
one filter) combined with the ratio, p 2 =p 1 , and inlet tempera- required by the compressor and the power required
by an electric motor.
ture, permits calculation of the compressor power.
Comments
Example 12.6 Compressor Flow and Pressure The value of ‘‘P’’ can be calculated most expediently by
for Air Wash means of the spreadsheet, Table CDD.5.

Given 12.4.4.2 Types of Backwash Systems
Suppose a filter bed is 4.27 6.10 m (14 20 ft) plan area The three kinds of backwash systems are (Monk, 1987) as
and that the air wash capacity in terms of surface loading follows: (1) direct pumping, (2) pump and reservoir, and (3)
2
3
2
3
should be 1.524 m =m =min (STP) (5 ft =ft =min). Let tem-
perature be 258C and let atmospheric pressure be sea self backwashing. In direct pumping, filtered water is pumped
level, 101,325 kPa. Let the depth of water in the filter from the clear-well and pressurizes the under-drain system. In
box be say 3.0 m (9.84 ft), that is, from the orifices to the the ‘‘pump-and-reservoir’’ type, filtered water is pumped from
crests of the backwash water troughs. Let the temperature the clear-well to separate reservoir, usually elevated. A self
be say 208C and assume an altitude of 1585 m (5200 ft). backwashing system utilizes the head from a tailwater over-
Required flow with an associated intermediate reservoir before the
Compressor flow capacity and discharge pressure and clear-well. This intermediate reservoir has an overflow weir
power. to the clear-well that has high enough crest elevation to