Lin S.D. Water and Wastewater Calculations Manual

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solution:

Step 1. Determine ratio A

: A

Step 2. Calculate C

Step 3. Calculate H

␥ ⫽ 13.55 at 20⬚C

H ⫽ ␥y ⫽ 13.55 ⫻ 24.6 cm ⫽ 333 cm

⫽ 3.33 m

Step 4. Estimate discharge Q

Assuming C

⫽ 0.98

5 0.091sm

/sd

5 0.98 3 0.162 3 0.0707 22 3 9.81 3 3.33

Q 5 C

22gH

5 1/ 2sA

2 1 5 1/ 2s6.25d

2 1

5 0.162

5 6.25

s12 cmd

5 113 cm

5 0.0113 m

s30 cmd

5 707 cm

5 0.0707 m

Fundamental and Treatment Plant Hydraulics 295

Figure 4.12 ASME nozzle coefficients.

This value needs to be verified by checking the corresponding Reynolds

number R of the nozzle.

Step 5. Calculate R

⫽ Q/A

⫽ 0.091 m

/s/0.0113 m

⫽ 8.053 m/s

The kinematic viscosity at 20⬚C is (Table 4.1a)

␯⫽1.007 ⫻ 10

⫺6

then

Using this R value, with d

⫽ 0.4, the chart in Fig. 4.12 gives

⫽ 0.977

Step 6. Correct Q value

5.4 Discharge measurements

Oriﬁces.

The discharge of an orifice is generally expressed as

(4.62)

where Q ⫽ flow rate

⫽ coefficient of discharge

A ⫽ area of orifice

g ⫽ gravity acceleration

H ⫽ water height above center of orifice

All units are as Eq. (4.58a)

Example: The water levels upstream and downstream are 4.0 and 1.2 m,

respectively. The rectangular orifice has a sharp-edged opening 1 m high and

1.2 m wide. Calculate the discharge, assuming C

⫽ 0.60.

Q 5 C

A 22gH

5 0.0907 m

Q 5

0.977298

0.98

3 0.091 m

5 9.6 3 10

R 5

8.053 m/s 3 0.12 m

1.007 3 10

296 Chapter 4

solution:

Step 1. Determine area A

A ⫽ 1 m ⫻ 1.2 m ⫽ 1.2 m

Step 2. Determine discharge Q

Weirs. The general discharge equation for a rectangular, horizontal

weir is

(4.63a)

where Q ⫽ flow rate, ft

/s or m

L ⫽ weir length, ft or m

H ⫽ head on weir, ft or m

⫽ the water surface above the weir crest

⫽ coefficient of discharge

In the US customary units, let y be the weir height in feet.

(US customary units) (4.64a)

(SI units) (4.64b)

The theoretical equation for a rectangular streamlined weir is (ASCE

& WEF, 1992)

(4.63b)

Q 5

5 1.78 1 0.24

5 3.32 1 0.40

Q 5 C

3/2

5 5.34 m

5 0.60 3 1.2 m

22 3 9.81 m/s

3 s4 2 1.2d m

Q 5 C

A 22gsh

2 h

Fundamental and Treatment Plant Hydraulics 297

where g is the gravitational acceleration rate. This formula reduces to

Q ⫽ 3.09 LH

3/2

(for U.S. customary units) (4.63c)

or Q ⫽ 1.705 LH

3/2

(for SI units) (4.63d)

Example 1: A rectangular flow control tank has an outflow rectangular weir

1.5 m (5 ft) in length. The inflow to the box is 0.283 m

/s (10 ft

/s). The crest

of the weir is located 1.2 m above the bottom of the tank. Find the depth of

water in the tank.

solution:

Step 1. Determine head on weir H

Using Eq. (4.63d)

Q ⫽ 1.705 LH

3/2

0.283 ⫽ 1.705(1.5)H

3/2

⫽ 0.111

H ⫽ 0.23 (m)

Step 2. Calculate water depth D

D ⫽ 1.2 m ⫹ H ⫽ 1.2 m ⫹ 0.23 m

⫽ 1.43 m

Example 2: In a rectangular channel 4 m (13 ft) high and 12 m (40 ft) wide,

a sharp-edged rectangular weir 1.2 m (4 ft) high without end contraction will

be installed. The flow of the channel is 0.34 m

/s (12 ft

/s). Determine the

length of the weir to keep the head on the weir 0.15 m (0.5 ft).

solution:

Step 1. Compute velocity of approach at channel, V

Head due to velocity of approach

Step 2. Use weir formula (Eq. (4.63d))

(a) Without velocity of approach

Q ⫽ 1.705 LH

3/2

0.34 ⫽ 1.705 LH

3/2

h 5

snegligibled

V 5

0.34 m

12 3 4

5 0.007 m/s snegligibled

298 Chapter 4

0.34 ⫽1.705 L (0.15)

1.5

L ⫽ 3.43(m)

⫽ 11.2 ft

(b) Including velocity of approach

For freely discharging rectangular weirs (sharp-crested weirs), the

Francis equation is most commonly used to determine the flow rate. The

Francis equation is

Q ⫽ 3.33 LH

3/2

(4.63e)

where Q ⫽ flow rate, ft

L ⫽ weir length, ft

H ⫽ head on weir, ft

⫽ the water surface above the weir crest

For constracted rectangular weirs

Q ⫽ 3.33 (L ⫺ 0.1nH )H

3/2

(US customary units) (4.63f)

Q ⫽ 1.705 (L ⫺ 0.1 nH ) H

3/2

(SI units) (4.63g)

where n ⫽ number of end constractions.

Example 3: Conditions are the same as in Example 2, except that the weir

has two-end constractions. Compute the width of the weir.

solution:

Step 1. For constracted weir, without velocity of approach

Q ⫽ 1.705 (L ⫺ 0.2H)H

3/2

0.34 ⫽ 1.705 (L ⫺ 0.2 ⫻ 0.15) ⫻(0.15)

1.5

L ⫽ 3.40 (m)

⫽ 11.1 ft

Step 2. Discharge including velocity of approach

Since velocity head is negligible

L ⫽ 3.4 m

Q 5 1.075 sL 2 0.2Hd

[

sH 1 hd

3/2

2 h

3/2

]

L 5 3.4 m

Q 5 1.705 L

[

s0.15 1 hd

1.5

2 h

1.5

]

Fundamental and Treatment Plant Hydraulics 299

Example 4: Estimate the discharge of a weir 4 ft long where the head on the

weir is 3 in.

solution:

By the Francis equation

Q ⫽ 3.33 LH

3/2

⫽ 3.33 ⫻ 4 ⫻ (3/12)

3/2

⫽ 1.66 (ft

/s)

For the triangular weir and V-notch weir, the flow is expressed as

(4.65)

where ␪ ⫽ weir angle

⫽ discharge coefficient, calibrated in place

⫽ 1.34 for SI units

⫽ 2.50 for US customary units

Different values have been proposed

The triangular weir is commonly used for measuring small flow rates.

Several different notch angles are available. However, the 90⬚ V-notch

weir is the one most commonly used. The discharge for 90⬚ V-notch with

free flow is

Q ⫽ 2.5 H

2.5

(4.66)

where ␪ ⫽ discharge, ft

H ⫽ head on the weir, ft

Detailed discussion of flow equations for SI units for various types of

weir is presented by Brater et al. (1996).

Example 5: A rectangular control tank has an outflow rectangular weir

8 ft (2.4 m) in length. The crest of the weir is 5 ft (1.5 m) above the tank bottom.

The inflow from a pipe to the tank is 10 ft

/s (0.283 m

/s). Estimate the water

depth in the tank, using the Francis equation.

solution:

Q ⫽ 3.33 LH

3/2

Rewrite equation for H

Example 6: A rectangular channel 6 ft (1.8 m) wide has a sharp-crested weir

across its whole width. The weir height is 4 ft (1.2 m) and the head is 1 ft

(0.3 m). Determine the discharge.

H 5 a

3.33L

2/3

5 a

3.33 3 8

2/8

5 0.52 sftd

Q 5 C

a tan

5/2

300 Chapter 4

solution:

Step 1. Determine C

using Eq. (4.64)

Step 2. Compute discharge Q

Q ⫽ C

3/2

⫽ 3.32 ⫻ 6 ⫻ (1)

3/2

⫽ 19.92 (ft

/s)

Example 7: A circular sedimentation basin has a diameter of 53 ft (16m).

The inflow from the center of the basin is 10 MGD (0.438 m

/s). A circular

effluent weir with 90⬚ V-notches located at 0.5 ft (0.15 m) intervals is

installed 1.5 ft (0.45 m) inside the basin wall. Determine the water depth

on each notch and the elevation of the bottom of the V-notch if the water

surface of the basin is at 560.00 ft above mean sea level (MSL).

solution:

Step 1. Determine the number of V-notches

Diameter of the weir

d ⫽ 53 ft ⫺ 2 ⫻ 1.5 ft ⫽ 50 ft

Weir length

l ⫽ ␲d ⫽ 3.14 ⫻ 50 ft ⫽ 157 ft

No. of V-notches

Step 2. Compute the discharge per notch

5 0.0493 sft

/sd notch

5 31,847

gal/d

notch

1 ft

7.48 gal

1 day

86,400 s

Q 5

10,000,000 gal/d

314 notches

5 31,847 sgal/dd/notch

n 5 157 ft 3

2 notches

5 314 notches

5 3.22 1 0.40

5 3.22 1 0.40 3

5 3.32

Fundamental and Treatment Plant Hydraulics 301

Step 3. Compute the head of each notch from Eq. (4.66)

Q ⫽ 2.5 H

2.5

H ⫽ (Q/2.5)

1/2.5

⫽ (Q/2.5)

0.4

⫽ (0.0493/2.5)

0.4

⫽ 0.21 (ft)

Step 4. Determine the elevation of the bottom of the V-notch

Elevation ⫽ 560.00 ft ⫺ 0.21 ft

⫽ 559.79 ft MSL

Parshall ﬂume.

The Parshall flume was developed by R. L. Parshall in

1920 for the British measurement system. It is widely used for meas-

uring the flow rate of an open channel. It consists of a converging sec-

tion, a throat section, and a diverging section. The bottom of the throat

section is inclined downward and the bottom of the diverging section is

inclined upward. The dimensions of the Parshall flume are specified in

the British system, not the metric system. The geometry creates a crit-

ical depth to occur near the beginning of the throat section and also gen-

erates a back water curve that allows the depth H

to be measured from

observation well a. There is a second measuring point H

located at the

downstream end of the throat section. The ratio of H

to H

is defined

as the submergence of the flume.

Discharge is a function of H

. The discharge equation of a Parshall

flume is determined by its throat width, which ranges from 3 in to 50 ft.

The relationship between the empirical discharge and the gage reading

for various sizes of flume is given below:

Throat width W, ft Discharge equation, ft

/s Flow capacity, ft

3 in (0.25 ft) Q ⫽ 0.992 0.03–1.9

6 in (0.5 ft) Q ⫽ 2.06 0.05–3.9

9 in (0.75 ft) Q ⫽ 3.07 0.09–8.9

1–8 ft Q ⫽ 4 W up to 140

10–50 ft Q ⫽ (3.6875 W ⫹ 2.5)

SOURCE: R. L. Parshall (1926)

When the ratio of H

to H

exceeds the following values:

0.5 for flumes of 1, 2, and 3 in width,

0.6 for flumes of 6 and 9 in width,

0.7 for flumes of 1 to 8 ft width,

0.8 for flumes of 8 to 50 ft width.

The flume is said to be submerged. When a flume is submerged, the

actual discharge is less than that determined by the above equations.

The diagram presented in Figs. 4.13 and 4.14 can be used to determine

1.6

1.522W

0.26

1.53

1.58

1.547

302 Chapter 4

discharges for submerged Parshall flumes of 6 and 9 in width, respec-

tively. Figure 4.15 shows the discharge correction for flumes with 1 to 8

ft throat width and with various percentages of submergence. The cor-

rection for the 1-ft flume can be applicable to larger flumes by multiplying

by a correction factor of 1.0, 1.4, 1.8, 2.4, 3.1, 4.3, and 5.4 for flume sizes

of 1, 1.5, 2, 3, 4, 6, and 8 ft width, respectively. Figure 4.16 is used for

determining the correction to be subtracted from the free-flow value for

a 10-ft Parshall flume. For larger sizes, the correction equals the value

from the diagram multiplied by the size factor M. Figures. 4.13–4.16

are improved curves from earlier developments (Parshall 1932, 1950).

Fundamental and Treatment Plant Hydraulics 303

Figure 4.13 Diagram for determining rate of submerged flow through a 6-in Parshall

flume (US Department of the Interior, 1997 ).

Figure 4.14 Diagram for determining rate of submerged flow through a 9-in Parshall

flume (US Department of the Interior, 1997).

Example 1: Calculate the discharge through a 9-in Parshall flume for the

gage reading H

of 1.1 ft when (a) the flume has free flow, and (b) the flume

is operating at 72% of submergence.

solution:

Step 1. Calculate Q for free flow for (a)

Q ⫽ 3.07 ⫽ 3.07 (1.1)

1.53

⫽ 3.55 (ft

/s)

1.53

304 Chapter 4

Figure 4.15 Diagram for determining correction to be subtracted from the free discharge

to obtain rate of submerged flow through 1- to 8-ft Parshall flumes (US Department of the

Interior, 1997).

Figure 4.16 Diagram for determining correction to be subtracted from discharge flow to

obtain rate of submerged flow through 10-to 50- ft Parshall flumes (US Department of the

Interior, 1997 ).