Lin S.D. Water and Wastewater Calculations Manual

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For flow rate in a rectangular channel, area A is

A ⫽ WD (4.43)

where W is the width of the channel.

4.6 Critical depth

Given the discharge (Q) and the cross-sectional area (A) at a particular

section, Eq. (4.42a) may be rewritten for specific energy expressed in

terms of discharge

(4.42b)

If the discharge is held constant, specific energy varies with A and

D (Fig. 4.10). At the critical state the specific energy of the flow is at a

minimum value. The velocity and depth associated with the critical

state are called the critical velocity and critical depth, respectively. At

2gA

1 D

Fundamental and Treatment Plant Hydraulics 285

Energy gradient

Water surface

Channel bed

Datum

/2g

Figure 4.9 Energy of open channel flow.

/2g

Figure 4.10 Channel cross section.

the critical state, the first derivative of the specific energy with respect

to the water depth should be zero:

(4.44)

Since near the free surface

dA ⫽ WdD

then

Where W is the top width of the channel section (Fig. 4.10). For open

channel flow, D

⫽ A/W is known as the hydraulic depth (or mean

depth, D

) for rectangular channels.

Hence for nonrectangular channels

(4.45)

W ⫽ gA

(4.45a)

Then the general expression for flow at the critical depth is

(4.46)

and the critical velocity V

(4.47)

(4.48)

(4.49)

Where H

is the minimum specific energy.

5 D

5 2gA/W 5 2gD

Q 5 A 2gD

Q 5 2gA

/W 5 A 2gA/W

1 1 5 0

W 5

1 1 5 0

2gA

1 Db 5 0

286 Chapter 4

For the special case of a rectangular channel, the critical depth

(4.50)

where q ⫽ discharge per unit width of the channel, m

/s/m.

For rectangular channels, the hydraulic depth is equal to the depth

of the flow. Equation (4.45) can be simplified as

(4.51)

The quantity is dimensionless. It is the so-called Froude number,

F. The Froude number is a very important factor for open channel flow.

It relates three states of flow as follows:

F Velocity Flow state

1 critical flow (V ⫽ speed of surface wave)

< 1 subcritical (V < speed of surface wave)

> 1 supercritical (V > speed of surface wave)

Example: Determine the critical depth of a channel when the water is flow-

ing in a trapezoidal channel with a base width of 1.5 m (5.0 ft) and side slope

of 2 on 1. The flow is 4.0 m

/s (63,400 gpm).

solution: Let h ⫽ the critical depth of the channel

Then base ⫽ 1.5 m

W ⫽ 1.5 m ⫹ 2 ⫻ (1/2 h) m

⫽ (1.5 ⫹ h) m

A ⫽⫻[1.5 m ⫹ (1.5 ⫹ h) m] ⫻ h m

⫽ (1.5 h ⫹ 0.5 h

) m

Q ⫽ 4.0 m

g ⫽ 9.806 m/s

Solve h using Eq. (4.45a)

W ⫽ gA

(4.0)

⫻ (1.5 ⫹ h) ⫽ 9.806 ⫻ (1.5h ⫹ 0.5h

)

(1.5h ⫹ 0.5h

)

/(1.5 ⫹ h) ⫽ 1.63

by trial h ⫽ 0.818 m

⫽ 2.68 ft

V . 2gD

V , 2gD

V 5 2gD

V/!gD

!gD

5 1

5 a

1/3

5 a

1/3

Fundamental and Treatment Plant Hydraulics 287

4.7 Hydraulic jump

When water flows in an open channel, an abrupt decrease in velocity

may occur due to a sudden increase of water depth in the downstream

direction. This is called hydraulic jump and may be a natural phenom-

enon. It occurs when the depth in an open channel increases from a

depth less than the critical depth D

to one greater than D

. Under these

conditions, the water must pass through a hydraulic jump, as shown in

Fig. 4.11.

The balance between the hydraulic forces P

and P

, represented by

the two triangles and the momentum flux through sections 1 and 2, per

unit width of the channel, can be expressed as

⫺ P

⫽ ␳q (V

⫺ V

) (4.52)

where q is the flow rate per unit width of the channel. Substituting the

following quantities in the above equation

we get

since

1 D

dsD

2 D

d 5

2 D

d 5 rq a

; V

; P

288 Chapter 4

/2g

∆E

Figure 4.11 Hydraulic jump.

then

(4.53)

This equation may be rearranged into a more convenient from as

follows:

(4.54)

where F

is the upstream Froude number and is expressed by

(4.55)

If the hydraulic jump is to occur, F

must be greater than one; i.e. the

upstream flow must be supercritical. The energy dissipated (⌬E) in a

hydraulic jump in a rectangular channel may be calculated by the equation

(4.56)

Example 1: A rectangular channel, 4 m wide and with 0.5 m water depth,

is discharging 12 m

/s flow before entering a hydraulic jump. Determine the

critical depth and the downstream water depth.

solution:

Step 1. Calculate discharge per unit width q

Step 2. Calculate critical depth D

from Eq. (4.50)

Step 3. Determine upstream velocity V

Step 4. Determine the upstream Froude number

!gD

6.0

!9.81 3 0.5

5 2.71

3 m

0.5 m

5 6.0 m/s

5 0.972 m

5 a

1/3

5 a

3 m

/s 3 3 m

9.81 m/s

1/3

q 5

12 m

4 m

5 3sm

s or m

/sd

⌬E 5 E

2 E

2 D

!gD

5 a

1 2F

1/2

5 D

1 D

Fundamental and Treatment Plant Hydraulics 289

Step 5. Compute downstream water depth D

from Eq. (4.54)

Example 2: A rectangular channel 9 ft (3 m) wide carries 355 ft

/s (10.0 m

/s)

of water with a water depth of 2 ft (0.6 m). Is a hydraulic jump possible? If so,

what will be the depth of water after the jump and what will be the horsepower

loss through the jump?

solution:

Step 1. Compute the average velocity in the channel

Step 2. Compute F

from Eq. (4.55)

Since F

> 1, the flow is supercritical and a hydraulic jump is possible.

Step 3. Compute the depth D

after the hydraulic jump

Step 4. Compute velocity V

after jump

Step 5. Compute total energy loss ⌬E (or ⌬h)

5 D

5 2 1

19.72

2 3 32.2

5 8.04 sftd

355 ft

9 ft 3 6.03 ft

5 6.54 ft/s

5 6.03 ft

5 2 cA0.25 1 2 3 2.46

1/2

2 0.5d

5 D

cA0.25 1 2F

1/2

2 0.5d

!gD

19.72 ft/s

!32.2 ft/s

3 2 ft

5 2.46

355 ft

9 ft 3 2 ft

5 19.72 ft/s

5 D

1 2F

1/2

5 0.5 ca

1 2 3 2.71

1/2

5 1.68smd

290 Chapter 4

Step 6. Compute horsepower (hp) loss

5 Flow Measurements

Flow can be measured by velocity methods and direct discharge meth-

ods. The measurement flow velocity can be carried out by a current

meter, Pitot tube, U-tube, dye study, or salt velocity. Discharge is the

product of measured mean velocity and cross-sectional area. Direct

discharge methods include volumetric gravimeter, Venturi meter, pipe

orifice meter, standardized nozzle meter, weirs, orifices, gates,

Parshall flumes, etc. Detail flow measurements in orifices, gates,

tubes, weirs, pipes, and in open channals are discussed by Brater

et al. (1996).

5.1 Velocity measurement in open channel

The mean velocity of a stream or a channel can be measured with a cur-

rent meter. A variety of current meters is commercially available. An

example of discharge calculation with known mean velocity in sub-cross

section is presented in Chapter 1.

5.2 Velocity measurement in pipe ﬂow

Pitot tube. A Pitot tube is bent to measure velocity due to the pressure

difference between the two sides of the tube in a flow system. The flow

velocity can be determined from

(4.57)

where V ⫽ velocity, m/s or ft/s

g ⫽ gravitational acceleration, 9.81 m/s

or 32.2 ft/s

⌬h ⫽ height of the fluid column in the manometer or a different

height of immersible liquid such as mercury,

m or ft

V 5 22g⌬h

5 54.4 hp

Loss 5

⌬EgQ

550

1.35 ft 3 62.4 lb/ft

3 355 ft

550 ft

lb/hp

⌬E 5 E

2 E

5 8.04 ft 2 6.69 ft 5 1.35 ft

5 D

5 6.03 1

6.54

2 3 32.2

5 6.69 sftd

Fundamental and Treatment Plant Hydraulics 291

Example: The height difference of the Pitot tube is 5.1 cm (2 in). The specific

weight of the indicator fluid (mercury) is 13.55. What is the flow velocity of

the water?

solution 1: For SI units

Step 1. Determine the water column equivalent to ⌬h

⌬h ⫽ 5.1 cm ⫻ 13.55 ⫽ 69.1 cm ⫽ 0.691 m of water

Step 2. Determine velocity

solution 2: For British units

Step 1.

⌬h ⫽ 2/12 ft ⫻ 13.55 ⫽ 2.258 ft

Step 2.

5.3 Discharge measurement of water ﬂow in pipes

Direct collection of volume (or weight) of water discharged from a pipe

divided by time of collection is the simplest and most reliable method.

However, in most cases it cannot be done by this method. A change of

pressure head is related to a change in flow velocity caused by a sudden

change of pipe cross-sectional geometry. Venturi meters, nozzle meters,

and orifice meters use this concept.

Venturi meter. A Venturi meter is a machine-cased section of pipe with

a narrow throat. The device, in a short cylindrical section, consists of

an entrance cone and a diffuser cone which expands to full pipe diam-

eter. Two piezometric openings are installed at the entrance (section 1)

and at the throat (section 2). When the water passes through the

throat, the velocity increases and the pressure decreases. The decrease

of pressure is directly related to the flow. Using the Bernoulli equa-

tion at sections 1 and 2, neglecting friction head loss, it can be seen

that

(4.16)

5 z

1 z

V 5 22g⌬h 5 22 3 32.2 ft/s

3 2.258 ft

5 12.06 ft/s

5 12.06 ft/s 3 0.304 m/ft

5 3.68 m/s

V 5 22g⌬h

5 22 3 9.81 m/s

3 0.691 m

5 3.68 m/s

292 Chapter 4

For continuity flow between sections 1 and 2, Q

⫽ Q

⫽ A

Solving the above two equations, we get

(4.58a)

where Q ⫽ discharge, m

/s or ft

, A

⫽ cross-sectional areas at pipe and throat, respectively,

or ft

g ⫽ gravity acceleration, 9.81 m/s

or 32.2 ft/s

H ⫽ h

⫺ h

⫽ pressure drop in Venturi tube, m or ft

Z ⫽ z

⫺z

⫽ difference of elevation head, m or ft

⫽ coefficient of discharge

For a Venturi meter installed in a horizontal position, Z ⫽ 0

(4.58b)

where

(4.59)

Example: A 6-cm throat Venturi meter is installed in an 18-cm diameter hor-

izontal water pipe. The reading of the differential manometer is 18.6 cm of

mercury column (sp gr ⫽ 13.55). What is the flow rate in the pipe?

solution:

Step 1. Determine A

5 ps18/2d

5 254.4 cm

5 0.2544 m

5 ps6/2d

5 28.3 cm

5 0.0283 m

254.4

28.3

5 8.99

2 A

2 1

Q 5

22gH

2 A

5 C

22gH

5 C

22gsH 1 Z d

2 A

Q 5

22g[sh

2 h

d 1 sz

2 z

2 A

Fundamental and Treatment Plant Hydraulics 293

Step 2. Calculate C

Step 3. Calculate H

Step 4. Calculate Q

Nozzle meter and oriﬁce meter. The nozzle meter and the orifice meter

are based on the same principles as the Venturi meter. However, nozzle

and orifice meters encounter a significant loss of head due to a nozzle

or orifice installed in the pipe. The coefficient (C

) for the nozzle meter

or orifice meter is added to the discharge equation for the Venturi meter,

and needs to be determined by on-site calibration. The discharge for the

nozzle meter and orifice meter is

(4.60)

or, for horizontal installation

(4.61)

The nozzle geometry has been standardized by the American Society of

Mechanical Engineers (ASME) and the International Standards

Association. The nozzle coefficient C

is a function of Reynolds number

(R ⫽ V

/␯) and ratio of diameters (d

). A nomograph (Fig. 4.12)

is available. Values of C

range from 0.96 to 0.995 for R ranging from

5 ⫻ 10

to 5 ⫻ 10

Example: Determine the discharge of a 30-cm diameter water pipe. An

ASME nozzle of 12-cm throat diameter is installed. The attached differential

manometer reads 24.6 cm of mercury column. The water temperature in the

pipe is 20⬚C.

Q 5 C

22gH

Q 5 C

22gsH 1 Z d

5 0.20 sm

/sd

Q 5 C

22gH

5 0.112 3 0.2544 3 22 3 9.81 3 2.52

5 2.52 m

H 5 gy 5 13.55 3

18.6

100

5 0.112

5 1/

2 1 5 1/ 2s8.99d

2 1

294 Chapter 4