Lin S.D. Water and Wastewater Calculations Manual

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Sedimentation is a solid—liquid separation by gravitational settling.

There are four types of sedimentation: discrete particle settling (type 1),

ﬂocculant settling (type 2), hindered settling (type 3), and compression

settling (type 4). Sedimentation theories for the four types are discussed

in Chapter 6 and elsewhere (Gregory and Zabel, 1990).

The terminal settling velocity of a single discrete particle is

derived from the forces (gravitational force, buoyant force, and drag

force) that act on the particle. The classical discrete particle set-

tling theories have been based on spherical particles. The equation

is expressed as

(5.60)

where u ⫽ settling velocity of particles, m/s or ft/s

g ⫽ gravitational acceleration, m/s

or ft/s

⫽ density of particles, kg/m

or lb/ft

r ⫽ density of water, kg/m

or lb/ft

d ⫽ diameter of particles, m or ft

⫽ coefﬁcient of drag

The values of drag coefﬁcient depend on the density of water (␳), rel-

ative velocity ( µ), particle diameter (d), and viscosity of water (µ), which

gives the Reynolds number R as:

(5.61)

The value of C

decreases as the Reynolds number increases. For R

less than 2 (or 1), C

is related to R by the linear expression as follows:

(5.62)

At low values of R, substituting Eq. (5.61) and (5.62) into Eq. (5.60)

gives

(5.63)

This expression is known as the Stokes’ equation for laminar ﬂow

conditions.

u 5

gsr

2 rdd

18 m

R 5

rud

u 5 a

4gsr

2 rdd

1/2

Public Water Supply 385

In the region of higher Reynolds numbers (2 < R < 500⫺1000), C

becomes (Fair et al., 1968):

(5.64)

In the region of turbulent ﬂow (500⫺1000 < R < 200,000,) the C

remains approximately constant at 0.44. The velocity of settling parti-

cles results in Newton’s equation (ASCE and AWWA 1990):

(5.65)

When the Reynolds number is greater than 200,000, the drag force

decreases substantially and C

becomes 0.10. No settling occurs at this

condition.

Example: Estimate the terminal settling velocity in water at a temperature

of 15°C of spherical silicon particles with speciﬁc gravity 2.40 and average

diameter of (a) 0.05 mm and (b) 1.0 mm.

solution:

Step 1. Using the Stokes’ equation (Eq. (5.63)) for (a)

From Table 4.1a, at T ⫽ 15°C

␳ ⫽ 999 kg/m

, and m ⫽ 0.00113 N ⋅ s/m

d ⫽ 0.05 mm ⫽ 5 ⫻ 10

⫺5

(a)

Step 2. Check with the Reynolds number (Eq. (5.61))

(a) The Stokes’ law applies, since R < 2.

5 0.075

R 5

rud

999 3 0.00169 3 5 3 10

0.00113

5 0.00169 m/s

9.81 m/s

s2400 2 999d kg/m

s5 3 10

18 3 0.00113 N

s/m

u 5

gsr

2 rdd

18 m

u 5 1.74 c

2 rdgd

1/2

1 0.34

386 Chapter 5

Step 3. Using the Stokes’ law for (b), d ⫽ 1 mm ⫽ 0.001 m

Step 4. Check the Reynold number

Assume the irregularities of the particles ⫽ 0.85

Since R > 2, the Stokes’ law does not apply. Use Eq. (5.60) to calculate u.

Step 5. Using Eqs. (5.64) and (5.60)

Step 6. Recheck R

Step 7. Repeat Step 5 with new R

u 5 0.155 sm/sd

4 3 9.81 3 1401 3 0.001

3 3 0.76 3 999

5 0.76

141

!141

1 0.34

5 141

R 5

frud

0.85 3 999 3 0.188 3 0.001

0.00113

u 5 0.188 sm/sd

4 3 9.81 3 s2400 2 999d 3 0.001

3 3 0.52 3 999

4gsr

2 rdd

5 0.52

1 0.34 5

508

!508

1 0.34

5 508

R 5

frud

0.85 3 999 3 0.676 3 0.001

0.00113

5 0.676 sm/sd

u 5

9.81s2400 2 999ds0.001d

18 3 0.00113

Public Water Supply 387

Step 8. Recheck R

Step 9. Repeat Step 7

(b) The estimated velocity is around 0.15 m/s, based on Steps 7 and 9.

10.1 Overﬂow rate

For sizing the sedimentation basin, the traditional criteria used are

based on overﬂow rate, detention time, weir loading rate, and horizon-

tal velocity. The theoretical detention time is computed from the volume

of the basin divided by average daily ﬂow (plug ﬂow theory):

(5.66)

where t ⫽ detention time, h

24 ⫽ 24 h/d

V ⫽ volume of basin, m

or million gallon (Mgal)

Q ⫽ average daily ﬂow, m

/d or Mgal/d (MGD)

The overﬂow rate is a standard design parameter which can be deter-

mined from discrete particle settling analysis. The overﬂow rate or sur-

face loading rate is calculated by dividing the average daily ﬂow by the

total area of the sedimentation basin as follows:

(5.67)

where u ⫽ overﬂow rate, m

/(m

⋅ d) or gpd/ft

Q ⫽ average daily ﬂow, m

/d or gpd

A ⫽ total surface area of basin, m

or ft

l and w ⫽ length and width of basin, respectively m or ft

u 5

t 5

24V

u 5 0.149 sm/sd

4 3 9.81 3 1401 3 0.001

3 3 0.83 3 999

5 0.83

116

!116

1 0.34

5 116

R 5

0.85 3 999 3 0.155 3 0.001

0.00113

388 Chapter 5

For alum coagulation, u is usually in the range of 40 to 60 m

/(m

⋅ d)

(or m/d) (980 to 1470 gpd/ft

) for turbidity and color removal. For lime

softening, the overflow rate ranges 50 to 110 m/d (1230 to 2700

gpm/ft

). The overﬂow rate in wastewater treatment is lower, rang-

ing from 10 to 60 m/d (245 to 1470 gpm/ft

). All particles having a set-

tling velocity greater than the overﬂow rate will settle and be removed.

It should be noted that rapid particle density changes due to tem-

perature, solid concentration, or salinity can induce density current

which can cause severe short-circuiting in horizontal tanks (Hudson,

1972).

Example: A water treatment plant has four clariﬁers treating 4.0 MGD

(0.175 m

/s) of water. Each clariﬁer is 16 ft (4.88 m) wide, 80 ft (24.4 m) long,

and 15 ft (4.57 m) deep. Determine: (a) the detention time, (b) overﬂow rate,

2.5 times the basin width.

solution:

Step 1. Compute detention time t for each clariﬁer, using Eq. (5.66)

(a)

Step 2. Compute overﬂow rate u, using Eq. (5.67)

(b)

Step 3. Compute horizontal velocity v

(c)

5 0.387 ft/min 5 11.8 cm/min

v 5

92.83 ft

/min

16 ft 3 15 ft

5 781 gpd/ft

5 31.83 m

u 5

1,000,000 gpd

16 ft 3 80 ft

5 3.447 h

t 5

16 ft 3 80 ft 3 15 ft

5570 ft

5 92.83 ft

/min

5 5570 ft

Q 5

4 MGD

1,000,000 gal

1 ft

7.48 gal

1 day

24 h

Public Water Supply 389

Step 4. Compute weir loading rate u

(d)

10.2 Inclined settlers

Inclined (tube and plate) settlers are sedimentation units that have

been used for more than two decades. A large number of smaller diam-

eter (20 to 50 mm) tubes are nested together to act as a single unit and

inclined with various angles (7⬚ to 60⬚). The typical separation distance

between inclined plates for unhindered settling is 2 in (5 cm) with

inclines of 3 to 6 ft (1 to 2 m) height. The solids or ﬂocs settle by gravi-

tational force. It is not necessary to use tubes and can take various

forms or plates also. The materials are lightweight, generally PVC or

ABC plastic (1 m ⫻ 3 m in size).

Tube settlers have proved as effective units. However, there is a ten-

dency of clogging.

Inclined settling systems can be designed as cocurrent, countercur-

rent, and cross-ﬂow. Comprehensive theoretical analyses of various ﬂow

geometries have been discussed by Yao (1976). The ﬂow velocity of the

settler module (v) and the surface loading rate for the inclined settler

(u) (Fig. 5.6) are calculated as (James M. Montgomery Consulting

Engineering, 1985):

(5.68)

(5.69)

where v ⫽ velocity of the water in settlers, m/s or ft/s

Q ⫽ ﬂow rate, m

/s or MGD

A ⫽ surface area of basin, m

or ft

␪ ⫽ inclined angle of the settlers

u ⫽ settling velocity, m/s or ft/s

w ⫽ width of settler, m or ft

H ⫽ vertical height, m or ft.

Example: Two ﬂocculators treat 1.0 m

/s (22.8 MGD) and remove ﬂocs large

than 0.02 mm. The settling velocity of the 0.02 mm ﬂocs is measured in the

laboratory as 0.22 mm/s (0.67 in/min) at 15°C. Tube settlers of 50.8 mm (2

in) square honeycombs are inclined at a 50° angle, and its vertical height is

u 5

AsH cos u 1 w cos

v 5

A sin u

5 25,000 gpd/ft 5 310 m

2.5w

1,000,000 gpd

2.5 3 16 ft

390 Chapter 5

1.22 m (4 ft). Determine the basin area required for the settler module and

the size of each ﬂocculator at 15°C.

solution:

Step 1. Determine the area needed for the settler modules

Q ⫽ (1 m

/s)/2 ⫽ 0.5 m

/s ⫽ 30 m

/min

w ⫽ 50.8 mm ⫽ 0.0508 m

H ⫽ 1.22 m

␪ ⫽ 50°

Using Eq. (5.69)

Step 2. Determine A

In practice, the actual conditions in the settlers are not as good as under con-

trolled laboratory ideal conditions.

A safety factor of 0.6 may be applied to determine the designed settling

velocity. Thus

suse 240 m

A 5 238.6 m

u 5 0.6 3 0.00022 m/s 5

0.0315

0.5s0.0508d

As1.22 3 0.643 1 0.0508 3 0.643

u 5

AsH cos u 1 w cos

Public Water Supply 391

Figure 5.6 Schematic diagram of settling section.

Step 3. Find surface loading rate Q/A

Q/A ⫽ (0.5 ⫻ 24 ⫻ 60 ⫻ 60 m

/d)/240 m

⫽ 180 m

/(m

⋅ d)

⫽ 3.07 gpm/ft

(Note: 1m

/(m

⋅ d) ⫽ 0.017 gpm/ft

Step 4. Compute ﬂow velocity in the settlers, using Eq. (5.68)

v ⫽ Q/A sin␪ ⫽ 180/0.766

⫽ 235 (m/d)

⫽ 0.163 m/min

⫽0.0027 m/s

Step 5. Determine size of the basin

Two identical settling basins are designed. Generally, the water depth of the

basin is 4 m (13.1 ft). The width of the basin is chosen as 8.0 m (26.2 ft). The

calculated length of the basin covered by the settler is

l ⫽ 240 m

/8 m ⫽ 30 m ⫽ 98 ft

In practice, one-fourth of the basin length is left as a reserved volume for

future expansion. The total length of the basin should be

Step 6. Check horizontal velocity

Q/A ⫽ (30 m

/min)/(4 m ⫻ 8 m)

⫽ 0.938 m/min ⫽ 3 ft/min

Step 7. Check Reynolds number (R) in the settler module

11 Filtration

The conventional ﬁltration process is probably the most important

single unit operation of all the water treatment processes. It is an oper-

ation process to separate suspended matter from water by ﬂowing it

5 30 , 2000, thus it is in a lamella flow

R 5

s0.0027 m/sds0.0127 md

0.000001131 m

Hydraulic radius R 5

0.0508

4 3 0.0508

5 0.0127 m

30 m 3

5 40 ms131 ftd

392 Chapter 5

through porous ﬁlter medium or media. The ﬁlter media may be silica

sand, anthracite coal, diatomaceous earth, garnet, ilmenite, or ﬁnely

woven fabric.

In early times, a slow sand ﬁlter was used. It is still proved to be

efficient. It is very effective for removing flocs containing microor-

ganisms such as algae, bacteria, virus, Giardia, and Cryptosporidium.

Rapid ﬁltration has been very popular for several decades. Filtration

usually follows the coagulation–ﬂocculation–sedimentation processes.

However, for some water treatment, direct ﬁltration is used due to the

high quality of raw water. Dual-media ﬁlters (sand and anthracite,

activated carbon, or granite) give more beneﬁts than single-media ﬁl-

ters and became more popular; even triple-media ﬁlters have been

used. In Russia, up-ﬂow ﬁlters are used. All ﬁlters need to clean out

the medium by backwash after a certain period (most are based on

head loss) of ﬁltration.

The ﬁlters are also classiﬁed by allowing loading rate. Loading rate

is the ﬂow rate of water applied to the unit area of the ﬁlter. It is the

same value as the ﬂow velocity approaching the ﬁlter surface and can

be determined by

v ⫽ Q/A (5.67a)

where v ⫽ loading rate, m

/(m

⋅ d) or gpm/ft

Q ⫽ ﬂow rate, m

/d or ft

/d or gpm

A ⫽ surface area of ﬁlter, m

or ft

On the basis of loading rate, the ﬁlters are classiﬁed as slow sand ﬁl-

ters, rapid sand ﬁlters, and high-rate sand ﬁlters. With each type of ﬁlter

medium or media, there are typical design criteria for the range of load-

ing rate, effective size, uniform coefﬁcient, minimum depth require-

ments, and backwash rate. The typical loading rate for rapid sand ﬁlters

is 120 m

/(m

⋅ d) [83 L/(m

⋅ min) or 2 gpm/ft

]. For high-rate ﬁlters, the

loading rate may be four to ﬁve times this rate.

Example: A city is to install rapid sand ﬁlters downstream of the clariﬁers.

The design loading rate is selected to be 160 m

/(m

⋅ d) (2.7 gpm/ft

). The

design capacity of the waterworks is 0.35 m

/s (8 MGD). The maximum sur-

face per ﬁlter is limited to 50 m

. Design the number and size of ﬁlters and cal-

culate the normal ﬁltration rate.

solution:

Step 1. Determine the total surface area required

A 5

0.35 m

/s s86,400 s/dd

160 m

5 189 m

Public Water Supply 393

Step 2. Determine the number (n) of ﬁlters

Select four ﬁlters.

The surface area (a) for each ﬁlter is

a ⫽ 189 m

/4 ⫽ 47.25 m

We can use 7 m ⫻ 7 m or 6 m ⫻ 8 m, or 5.9 m ⫻ 8 m (exact)

Step 3. If a 7 m ⫻ 7 m ﬁlter is installed, the normal ﬁltration rate is

11.1 Filter medium size

Before a filter medium is selected, a grain size distribution analysis

should be performed. The sieve size and percentage passing by weight

relationships are plotted on logarithmic-probability paper. A straight

line can be drawn. Determine the geometric mean size (m

) and geo-

metric standard deviation size (␴

). The most common parameters

used in the United States to characterize the filter medium are effec-

tive size (ES) and uniformity coefficient (UC) of medium size distri-

bution. The ES is that the grain size for which 10% of the grain (d

)

are smaller by weight. The UC is the ratio of the 60-percentile (d

)

to the 10-percentile. They can be written as (Fair et al., 1968; Cleasby,

1990):

(5.70)

(5.71)

The 90-percentile, d

, is the size for which 90% of the grains are smaller

by weight. It is interrelated to d

as (Cleasby 1990)

⫽ d

(10

1.67 log UC

) (5.72)

The d

size is used for computing the required ﬁlter backwash rate for

a ﬁlter medium.

Example: A sieve analysis curve of a typical ﬁlter sand gives d

⫽ 0.54 mm

and d

⫽ 0.74 mm. What are its uniformity coefﬁcient and d

UC 5 d

5 s

1.535

ES 5 d

5 m

1.282

v 5

0.35 m

/s 3 86,400 s/d

4 3 7 m 3 7 m

5 154.3 m

/sm

n 5

189 m

50 m

5 3.78

394 Chapter 5