Water and Wastewater Engineering

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11-16 WATER AND WASTEWATER ENGINEERING

The last column is summed and the headloss calculated using Equation 11-9 :



1 067 0 0025 0 5

082 981

.. .

()()

()(

m/sm

m/s

224

5 2

045

75 025

1 0119 10 75 0

)( )

()

()(



 225 076



).

Comments:

1 . This headloss is large for clean bed filtration. It should be less than 0.6 m for the specified

loading rate. Comparison of the effective size and uniformity coefficient for this sand (from

E xample 11-1 ) with the typical values in Table 11-1 reveals that the uniformity coefficient

is too high. As noted later in this chapter (s

ee p. 11-22), the loading rate is high for a stan-

dard sand filter. Either the loading rate should be lowered, the fraction of fines should be

reduced, or some combination of less fines and a lower loading rate should be employed.

2. Note that the equation used to calculate C

changed when the Reynolds number dropped

below 0.5.

3. As may be noted from the number of significant figures presented, this calculation was

performed on a spreadsheet.

Terminal Headloss. As the filter clogs, the headloss will increase so that the results calculated

using the clean bed eq

uations are the minimum expected headlosses. There is no method to pre-

dict the increase in headloss as the filter becomes plugged with accumulated solids without full-

scale or pilot plant data. The phenominological model ( Equations 11-7 and 11-8 ) provides a way

to obtain estimates of the tim

e to reach terminal headloss using pilot plant data. More typically,

terminal headloss pressure is selected based on experience and the hydraulic profile of the entire

treatment plant.

Backwashing Hydraulics

The expansion of the filter bed during backwash is calculated to provide a starting point in

determining the placement of the backwash troughs above the filter bed. Fair and Geyer (1954)

developed the following relationship to predict the depth of the expanded bed:





()()

()





(11-11)

where D

 depth of the expanded bed, m

  porosity of the bed

D  depth the unexpanded bed, m

f  mass fraction of filter media with expanded porosity



 porosity of expanded bed

The conditions during backwash are turbulent. A representative model equation for estimating 

is that given by Richardson and Zaki (1954):





⎛

⎝

⎜

⎞

⎠

⎟

0 2247

(11-12)

GRANULAR FILTRATION 11-17

where v

is the backwash velocity and the Reynolds number is defined as

R 

()()vd



(11-13)

The ex panded porosity is calculated for each fractional s ize of the media and summed for the

entire bed. An approximation technique uses the 60th percentile diameter ( d

60%

in m) to calculate

the Reynolds number,

R 

()( )vd

s%60



(11-14)

and then calculates the expanded porosity in one step for the entire bed depth (Cleasby, 1972). A

more sophisticated model developed by Dharmarajah and Cleasby (1986) is also available.

The determination of D

i s not straightforward. From Equation 11-12 , it is obvious that

the expanded bed porosity is a function of the settling velocity. The particle settling velocity is

determined by Equation 10-8 in Chapter 10. To solve Equation 10-8, the drag coefficient ( C

)

must be calculated. The drag coefficient is a function of the Reynolds number, which, in turn, is

a function of the settling velocity. Thus, the settling velocity is needed to find the settling veloc-

ity! To resolve this dilemma, the settling velocity must be estimated. Knowing the sand grain

diameter and specific gravity, Figure 11-7 can be used to obtain a first es timate for the settling

velocity to use in calculating the Reynolds number. The calculated value of the Reynolds number

is compared to the estimate, and the procedure is iterated until the estimate and the c

alculated

value of R are the same.

E xample 11-3 illustrates the calculation procedure to estimate D

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0005 0.001 0.0015 0.002 0.0025

S.G. = 4

S.G. = 2.6

S.G. = 1.6

Particle diameter, m

Esrimated setting velocity, m/s

0.003 0.0035 0.004

FIGURE 11-7

Particle settling velocity estimation chart. S. G.  specific gravity.

11-18 WATER AND WASTEWATER ENGINEERING

Example 11-3. Determine the depth of the expanded sand filter bed being designed for Ottawa

Island ( Example 11-2 ).

Solution. To begin, select a backwash rate. To retain the finest sand grains used in building the

filter, the backwash rate must not wash out particles with a diameter of 0.000126 m (0.0126 cm).

Using Figure 11-7 , find that for a 0.0126 cm partic

le with a specific gravity of 2.65, the terminal

settling velocity is approximately 1 cm/s (864 m/d).

The computations are shown below.

Sieve no. Dia., m

Estimated

, m/s

Estimated

Reynolds

number C

, m/s

Fraction

retained

Reynolds

number Exponent

Expanded

porosity f/(1

)

8–12 0.002 0.3376.43 0.56 0.277891 0.053 425.23 0.412 0.255 0.071

12–16 0.00142 0.2 178.18 0.70 0.209215 0.171 227.300.387 0.309 0.247

16–20 0.001 0.15 94.11 0.90 0.15441 0.146 118.14 0.362 0.371 0.232

20–30 0.000714 0.1 44.80 1.32 0.107828 0.204 58.91 0.338 0.448 0.369

30–40 0.000505 0.07 22.18 2.06 0.072715 0.176 28.10 0.314 0.5370.380

40–50 0.000357 0.05 11.20 3.38 0.047723 0.119 13.04 0.290 0.635 0.326

50–70 0.000252 0.03 4.74 6.78 0.028313 0.059 5.46 0.266 0.758 0.244

70–100 0.000178 0.02 2.23 1

3.09 0.017121 0.0312.33 0.245 0.877 0.252

100–140 0.000126 0.015 1.19 23.34 0.01079 0.007 1.04 0.226 0.983 0.412

Sum  0.966 Sum  2.53

The estimated settling velocities in the third column were found from Figure 11-7 . The Reynolds

number was then computed with this estimated velocity. For the first row:

R 

()()( ) ( )( )( ) d v



082 0002 030

130

.. .

m m/s

7710

376 435







m/s

Note that the shape factor, sand particle diameter, and viscosity are all the same as in Example 11-2 .

The drag coefficient ( C

) is calculated in the same fashion as Example 11-2 . The settling velocity is

calculated using Equation 10-8 assuming the density of water is 1,000 kg/m

. For the first row:





2 3

4981 2650 1 000

m/s kg/m kg/m()( )(., ,

0 002

3 0 55838 1 000

)( )

()( )( )

kg/m

⎡

⎣

⎢

⎤

⎦

⎥

0 2778839. m/s

The density of the sand grain is the product of the specific gravity (from Example 11-2 ) and the

density of water:



()( )2 65 1 000 2 650

., ,kg/m kg/m

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

R 







()( )0 002 0 2778839

1 307 10

425

.. s

. s

mm/

m /

..22

0.1

 1.83

and

0.2247 R

0.1

 0.41

and





041

001

0 277884

0 255

m/s

⎛

⎝

⎞

⎠

The first row of the last column is then

0053

10255

0 0711











()

where 0.053 is the mass frac tion of sand having a geometric mean diameter of 0.002 m, that is,

between sieve numbers 8 and 12.

U sing Equation 11-11 with a porosity of 0.45 and an undisturbed bed depth of 0.5 m from

E xample 11-2 , the depth of the expanded be

d is then

 

()()()

1045 0 5 2 53 0070. . . .696 or .mm

Comments:

1 . The expansion ratio is D

/ 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 wou ld be D

 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

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:

F a Dg

see

 mg ()()()()()1





(11-15)

where F

 gravitational force of the expanded filter bed, N

m  mass of bed, kg/m

g  acceleration due to gravity, 9.81 m/s



 density of media, kg/m

  density of water, kg/m



 porosity of expanded bed

a  cross-sectional area of filter bed, m

 depth of expanded bed, m

11-20 WATER AND WASTEWATER ENGINEERING

Converting to pressure by dividing by the area of the bed and converting units of pressure (N/m

)

to units of head (m):

see





()()()



 



(11-16)

11-6 GRANULAR FILTRATION PRACTICE

Filter Type

The selection of the type of filter is strongly influenced by the characteristics of the raw water

and treatment processes that precede and follow the filter. Deep bed anthracite monomedia filters

accumulate headloss at a low rate, and they have a high capacity for accumulating solids. This

leads to long filter runs. However, when the water is not conditione

d properly because of high

variability in the raw water quality ( Figure 11-8 ), improper chemical dosage, or lack of opera-

tor vigilance, the performance degrades dramatically. Dual-media filters provide a more robust

0.6

Headloss, m

1.2

1.8

2.4

6555 3025300 200 85 80

20 40 60

Time, hr

Turbidity units

80 100 120 140

FIGURE 11-8

T urbidity values at filtration rates of 117 and 293 m

/d  m

. In the upper graph, the dotted curve is for 293 m

/ d  m

; the solid,

117 m

/d  m

. In the lower graph, the numbers at the top are the raw-water turbidity values at the time shown during the run; the

dashed curve is for filter influent; the dotted for filter effluent at 117 m

/d  m

; and the solid for filter effluent at 293 m

/ d  m

( Source: G. G. Robeck, K. A. Dostal, and R. L. Woodward, “Studies of Modification in Water Filtration,” Jour. AWWA,

56(2), February 1964: 198.)

GRANULAR FILTRATION 11-21

design when the raw water is improperly conditioned. Rapid sand filters have been in use for

over a century. They have proven reliable but are limited in that the finest sand is on the top. As

a result, the smallest pore spaces are also on the top. Therefore, most of the particles will clog in

the top layer of the filter and only a small portion of the filter depth will be used. Because of the

clogging of the

small pore spaces, headloss is quickly accumulated and the length of the filter run

is shorter than with the coarse media.

Softened grou ndwater has the most consistent water quality over long periods of time. In

addition, the precipitate floc is very tough. Each of the three different media filters perform well.

D ual- or multimedia filters are favored when the raw water source is a large lake such as

Lake Michigan. They are recommended for rivers with seasonal flooding, that res

ults in large,

rapid changes in turbidity over short periods of time.

Monomedia filters are preferred when the water s ource characteristics change only slowly

with time and when chemical conditioning can keep pace with the changes.

Number

For smaller plants ( 8,000 m

/ d), the minim um number of filters is two. For plants with a de-

sign capacity greater than 8,000 m

/ d, the minimum number of filters is four. A rule-of-thumb

estimate for larger plants may be made using Kawamura’s suggestion (2000):

N  0 0195

0 5

()Q

(11-17)

where N  total number of filters

Q  maximum design flow rate, m

/ d

Example 11-4. Estimate the number of filters for Ottawa Island’s new water treatment plant

( Examples 11-1 , 11-2 , 11-3 ) if the maximum day design flow rate is 18,400 m

/ d.

Solution. Using Equation 11-17 , the number of filters is

N 0 0195 18 400 2 65

0 5

., .

()

However, the design guidance for plants with a design capacity greater than 8,000 m

/ d is a mini-

mum of four filters.

Comment. Four filters provide more flexibility in operation.

Filtration Rate

When a plant has a small number of filters, the filtration rate in the remaining filters increases

dramatically when one filter is taken off-line for backwashing or maintenance. A sudden increase

in the filtration rate in those filters in service may result in particle detachment and an increase

in turbidity in the effluent. This condition must be analyzed when the filtration rate is

selected. If

the design filtration rate is to be maintained, the nominal filtration with all units in service will be

proportionally lower. GLUMRB (2003) specifies that the filters shall be capable of meeting the

plant design capacity at the approved design filtration rate with one filter removed from service.

The capital cost of the filter is directly related to the filtration rate because higher filtration

rates result in a

smaller area for the filter bed. With proper coagulation, inc lusion of polymer

11-22 WATER AND WASTEWATER ENGINEERING

addition, and good settling, dual-media filters can readily achieve satisfactory results at filtra-

tion rates up to 25 m

/h · m

of surface area (25 m/h or 600 m

/ d · m

). However, filter effluent

quality tends to degrade at filtration rates above 12.5 m/h (3 00 m

/ d · m

) with weak alum floc

without polymer (MWH, 2005).

Generally, conservative design filtration rates are 7.5 m/h for rapid sand filters, 15 m/h for

dual-media filters, and 25 m/h for deep, coarse monomedium filters that have polymer added as a

filter aid. Approximate clean bed headlosses for

common filter beds and filtration rates are given

in Table 11-3 .

Dimensions. The area of a filter bed may be estimated as



(11-18)

where A  area of a bed, m

Q  maximum day flow rate, m

/ d

N  number of beds

q  filtration rate, m

/ d · m

Although some extremely large plants may employ filter areas up to 200 m

, a common upper

bound for large plants is 100 m

. The general range in area is 25 to 100 m

with an average of

about 50 m

(Kawamura, 2000; MWH, 2005).

Filters are generally composed of two cells per filter box to form a bed. Generally, the gul-

let bisects the box to form the two cells. The width of a filter cell should be less than 6 m so that

“off-the-shelf” wash water troughs may be used. The suggested length-to-width ratio of a cell is

in the range of 2:1 to 4:1 (K

awamura, 2000).

The filter box depth is on the order of 4 to 8 m to provide space for the underdrain system,

media, and headloss. Due to construction costs, filter designs rarely provide more than 2 to 3 m of

available head through the filter bed. Experience indicates that effluent turbidity begins to inc

rease

when the net headloss is over 1.8 m for well-conditioned floc and dual-m edia filters, and may

be less than 0.8 m for poorly conditioned floc (Cleasby and Logsdon, 1999; Kawamura, 2000;

MWH, 2005).

TABLE 11-3

Approximate clean bed headlosses for common filter beds

Type of filter bed Filtration rate, m/h Headloss, m

Standard rapid sand5 0.3

Standard rapid sand 7.5 0.45

Standard dual media 10 0.3

Standard dual media 12.5 0.45

Standard dual media 20 0.6

Standard dual media 25 0.75

(Source: Adapted from Kawamura, 2000.)

GRANULAR FILTRATION 11-23

Example 11-5. In continuing the design of Ottawa Island’s rapid sand filter, determine the area

of each individual filter and the plan (horizontal) dimensions of a filter box. Use the filtration rate

of 216 m

/ d · m

from Example 11-2 .

Solution:

a. From Example 11-4 , the initial estimate of number of filters is 4.

b. Using Equation 11-18 , with all beds in service, the area of a bed is

A 

18 400

4 216

3 2

m /d

filtersm/dm()( )



2296 213

or m /filter.

c . Redundancy capacity for the maximum day with one filter ou t of service must be pro-

vided. The choices are:

1 . Increase the number of beds to five and reduce the area.

2. Increase the number of beds to six and reduce the area.

3. Maintain the number of beds at four but make the area larger. This allow

s for a

lower q during average conditions and meets the design loading rate with one bed

out of service on the maximum day.

4. Switch to a dual-media filter that would allow a higher loading rate.

d. Bec ause of construction and operational considerations, filters are built in pairs. Thus,

alternative (1) is eli

minated. Alternative (2) would be acceptable but would have a capi-

tal cost 50% greater than the four-filter system. Without switching to dual media, option

3 offers the most economical alternative.

e. If each filter is increased by 1/3, the filtration rate with one filter out of service would be

q 

18 400

3 1 333 21 3

m /d

filtersm()()()

660

3 2

. m /dm



Therefore, for a trial calculation assume the area of one filter is (1.3333)(21.3 m

)

 28.4 m

f. As a trial, select a total width of two cells  5.2 m. Each cell has a width of 2.6 m. The

length of each cell is then

L 

28 4

226

()( )

This design meets two design criteria: the cell width is < 6 m and the L:W ratio (2.1:1) is

within the recommended range of 2:1 to 4:1.

11-24 WATER AND WASTEWATER ENGINEERING

g. An assumption of a gullet width of 0.6 m yields the plan view sketched below.

Gullet

2.6 m 2.6 m

5.5 m

0.6 m

Comments:

1 . The assumed gullet width will be checked in Example 11-7 .

2. There are several lengths and widths that will meet the design criteria. The Solver tool*

in a spreadsheet can be used to define the upper and lower bounds of the width per cell.

The spreadsheet and dialog box are shown in Figure 11-9 .

3. The vertical dimension

s of the filter box depends on the media, underdrain and headloss

through the filter. The depth of the filter box will be determined in Example 11-8 .

Media

The selection of the filter type implicitly specifies the media type. The grain size distribution

plays a strong role in the trade-off between headloss (larger media minimizes headloss) and

filtration efficiency (smaller media captures particles better). The primary design criteria are the

effective size ( E ) and the

uniformity coefficient ( U ).

A s shown in Figure 11-5 , a low uniformity coefficient results in better utilization of the filter

bed. The effective size plays a significant role in the headloss. The estimate of the clean bed headloss

can be used to evaluate alternate media specifications. For example, becaus

e clean bed headlosses

range from 0.3 to 0.6 m (Castro et. al., 2005), initial headlosses in excess of 0.6 m imply either that

the filtration rate is too high or that the media has too large a proportion of fine grain sizes.

The media in multimedia filters must be matched to ensure that all the media fluidize at the

same backwash rate so that one media is

not washed out or fails to fluidize. The drag and gravi-

tational forces of the smallest denser media grain size can be balanced with the largest lighter

media grain size by equating the settling velocities of these particle sizes using Equation 10-8 ,

and solving for the ratio of the diameters (Kawamura, 2000):

0 667







⎡

⎣

⎢

⎤

⎦

⎥

(11-19)

*Solver is a “tool” in Excel

. Other spreadsheets may have a different name for this program.

GRANULAR FILTRATION 11-25

FIGURE 11-9

Solver solution for length and width of filter box for Example 11-5 .

Example 11-5

Filter box dimensions for Ottawa Island

Filtration rate

Number of filters

Area =

Width per cell =

Width per box =

L =

L:W =

Solver dialog box inputs

Target cell B11

B11

B17

< or =

> or =

< or =

min

Equal to

By changing

Subject to the constraints

max

18,400

216

28.4

2.66

5.33

BCD

/d − m

where d

 diameter of lighter media, m

 diameter of denser media, m

  density of water, kg/m



 density of medium with diameter d

, kg/m



 density of medium with diameter d

, kg/m

GLUMRB (2003) specifies the following effective sizes and uniformity coefficients:

• For sand, E of 0.45–0.55 mm and U 1.65

• For anthracite coal as a monomedium, E of 0.45–0.55 mm and U 1.65.

• For anthracite coal as a cap, E of 0.8–1.2 mm

and U 1.85.