Water Flow in Open Channels

8/12/2019 Water Flow in Open Channels

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CHAPTER 6:

Water Flow in Open Channels

University of Palestine

Engineering Hydraulics

2 nd semester 2010-2011

1



Introduction.

Type of Open Channels.

Types of Flow in Open Channels.

Flow Formulas in Open Channels.

Most Economical Section of Channels.

Energy Principles in Open Channel Flow.

Non-uniform Flow in Open Channels.

Hydraulic Jump.

Content

2



Open channel hydraulics, a subject of great importance to

civil engineers, deals with flows having a free surface inchannels constructed for water supply, irrigation, drainage,and hydroelectric power generation; in sewers, culverts,and tunnels flowing partially full; and in natural streamsand rivers.

An open channel is a duct in which the liquid flows with afree surface.

This is in contrast with pipe flow in which the liquidcompletely fills the pipe and flow under pressure.

The flow in a pipe takes place due to difference of pressure(pressure gradient), whereas in open channel it is due tothe slope of the channel bed (i.e.; due to gravity).

Introduction

3



It may be noted that the flow in a closed conduit is notnecessarily a pipe flow. It must be classified as openchannel flow if the liquid has a free surface.

for a pipe flow:

The hydraulic gradient line (HGL) is the sum of theelevation and the pressure head (connecting the watersurfaces in piezometers).

The energy gradient line (EGL) is the sum of the HGLand velocity head.

The amount of energy loss when the liquid flows fromsection 1 to section 2 is indicated by hL.

Introduction

4



Pipe system

Introduction

5



For open channel flow :

The hydraulic gradient line (HGL) corresponds to the water surface line (WSL); the free water surface issubjected to only atmospheric pressure which iscommonly referred to as the zero pressure reference inhydraulic engineering practice.

The energy gradient line (EGL) is the sum of the HGLand velocity head.

The amount of energy loss when the liquid flows fromsection 1 to section 2 is indicated by hL. For uniformflow in an open channel, this drop in the EGL is equalto the drop in the channel bed.

Introduction

6


http://slidepdf.com/reader/full/water-flow-in-open-channels 7/83Open Channel

Introduction

7



Based on their existence, an open channel can be naturalor artificial :

Natural channels such as streams, rivers, valleys , etc.These are generally irregular in shape, alignment androughness of the surface.

Artificial channels are built for some specific purpose,such as irrigation, water supply, wastewater, waterpower development, and rain collection channels.These are regular in shape and alignment with uniform

roughness of the boundary surface.

Type of Open Channels

8




9




10



Based on their shape, an open channel can be prismatic ornon-prismatic:

Prismatic channels: a channel is said to be prismatic when the cross section is uniform and the bed slop isconstant.

)Non-prismatic channels: when either the cross sectionor the slope (or both) change, the channel is referred toas non-prismatic. It is obvious that only artificialchannel can be prismatic.

The most common shapes of prismatic channels arerectangular, parabolic, triangular, trapezoidal and circular.


11




The most common shapes of prismatic channels arerectangular, parabolic, triangular, trapezoidal and circular.

12



Types of Flow in Open Channels

The flow in an open channel can be classified into thefollowing types :

A).Uniform and non-uniform flow:

If for a given length of the channel, the velocity of flow,depth of flow, slope of the channel and cross-sectionremain constant, the flow is said to be uniform.

Otherwise it is said to be non-uniform.

Non-uniform flow is also called varied flow which can befurther classified as: Gradually varied flow (GVF) where the depth of the flow changes

gradually along the length of the channel.

Rapidly varied flow (RVF) where the depth of flow changessuddenly over a small length of the channel. For example, when

water flows over an overflow dam, there is a sudden rise (depth) of water at the toe of the dam, and a hydraulic jump forms.

13




14



Uniform Flow Types of Flow in Open Channels

15




B). Steady and unsteady flow: :

The flow is steady when, at a particular section, the depthof the liquid and other parameters (such as velocity, area ofcross section, discharge) do not change with time. In anunsteady flow, the depth of flow and other parameterschange with time.

C). Laminar and turbulent flow:

The flow in open channel can be either laminar orturbulent. In practice, however, the laminar flow occurs

very rarely. The engineer is concerned mainly with

turbulent flow. In the case of open channel Reynold’s number is defined as:

16




Recall that Reynold’s number is the measure of relative effects

of the inertia forces to viscous forces. 17




18




D). Sub-critical, critical, and supercritical flow:

The criterion used in this classification is what is known byFroude number, Fr, which is the measure of the relativeeffects of inertia forces to gravity force:

19




D). Sub-critical, critical, and supercritical flow:

The criterion used in this classification is what is known byFroude number, Fr, which is the measure of the relativeeffects of inertia forces to gravity force:

20



Flow Formulas in Open Channels

In the case of steady-uniform flow in an open channel, thefollowing main features must be satisfied:

The water depth, water area, discharge, and the velocitydistribution at all sections throughout the entire channellength must remain constant, i.e.; Q , A , y , V remainconstant through the channel length.

The slope of the energy gradient line (S), the water surfaceslope (S ws), and the channel bed slope (S0) are equal.

S = S ws = S0

D Water Surface

T.E.L

channel bed

21

l l h l



Flow Formulas in Open Channels The depth of flow, y , is defined as the vertical distance

between the lowest point of the channel bed and the free

surface. The depth of flow section, D , is defined as the depth of

liquid at the section, measured normal to the direction offlow.

D

Water Surface

T.E.L

channel bed

Unless mentioned otherwise, the depth of flow and the depth of f low section will be assumed equal. For uniform flow the depth attains a constant valueknown as the normal depth, yn 22

l l h l



S RC V h

1.The Chezy Formula(1769)

C is the Chezy coefficient (Chezy’s resistance factor), m 1/2/s, a dimensionalfactor which characterizes the resistance to flow .

Pwetted

Awetted Radiushydraulic

slopebed S

Rh


Many empirical formulas are used to describe the flow in openchannels

The Chezy formula is probably the first formula derived foruniform flow. It may be expressed in the following form

23

Fl F l i O Ch l



2. The Manning Formula: (1895)

2/13/21S R

nV h

tCoefficienM

PwettedAwetted Radiushydraulic

anning n

slopebed S

Rh


where n = Manning’s coefficient for the channel roughness,m-1/3/s.

Substituting manning Eq. into Chezy Eq, we obtain the Manning’s formula for uniform flow:

24

Fl F l i O Ch l




25

Fl F l i O Ch l



3. The Strickler Formula:


where kstr = Strickler coefficient, m1/3/s ComparingManning formula and Strickler formulas, we can see that

26

Fl F l i O Ch l



Example 1

1 . 5

m

3.0m

2

1

open channel of width = 3m as shown, bed slope = 1:5000,d=1.5m find the flow rate using Manning equation, n=0.025.

sVAQ

V

P

A R

P

A

S Rn

V

h

h

/m84.49538.0

m/s538.05000

1927.0025.0

1

927.0708.9

9

9.70835.132

m95.1935.0

1

3

3

2

22

2

3

2


27

Fl F l i O Ch l



Example 2

open channel as shown, bed slope = 69:1584, find the flow rateusing Chezy equation, C=35.


28

Fl F l i O Ch l



sVAQ

V

P

A

R

P

A

S RC V

h

h

/m84.11352.1627.0

m/s7.01584

69.0917.035

917.018.177

52.162

m177.1804.552.28.166.38.115072.0

m52.16215072.06.32

52.272.08.1652.2

2

04.552.2

3

2222

2


Example 2 cont.

29

Fl F l i O Ch l




Example 3: Group workThe cross section of an open channel is a trapezoid with a ottom

width of 4 m and side slopes 1:2, calculate the discharge if thedepth of water is 1.5 m and bed slope = 1/1600. Take Chezyconstant C = 50.

30

M t E i l S ti f Ch l



During the design stages of an open channel, the

channel cross-section, roughness and bottom slope are given.

The objective is to determine the flow velocity , depth

and flow rate, given any one of them. The design ofchannels involves selecting the channel shape andbed slope to convey a given flow rate with a givenflow depth. For a given discharge, slope and

roughness, the designer aims to minimize thecross-sectional area A in order to reduceconstruction costs

Most Economical Section of Channels

31




A section of a channel is said to be most economical

when the cost of construction of the channel isminimum.

But the cost of construction of a channel depends on

excavation and the lining. To keep the cost down or

minimum, the wetted perimeter, for a given discharge,

should be minimum.

This condition is utilized for determining the

dimensions of economical sections of different forms of

channels.


32




Most economical section is also called the best

section or most efficient section as the discharge,passing through a most economical section of channelfor a given cross sectional area A, slope of the bed S0and a resistance coefficient, is maximum.


Hence the discharge Q will be maximum when the wetted perimeter P is minimum.

33




The most ‘efficient’ cross-sectional shape is determined

for uniform flow conditions. Considering a givendischarge Q, the velocity V is maximum for theminimum cross-section A. According to the Manningequation the hydraulic diameter is then maximum.

It can be shown that:1. the wetted perimeter is also minimum,2. the semi-circle section (semi-circle having its

centre in the surface) is the best hydraulic section


34




Most Economical Rectangular Channel

Because the hydraulic radius is equal to the water crosssection area divided by the wetted parameter, Channelsection with the least wetted parameter is the besthydraulic section

Rectangular section


35




DBA BD2P

D

A 2DP

0dD

dP

222 2 02 D

D B

D

A

D

A

dD

dP

D

B 2

2

B D

Most Economical Rectangular ChannelMost Economical Section of Channels

36




D ) Dn(B A

212 n D B P

Dn

D

A B

212 n D )nD

D

A( P

0dD

dP

012 2

2 nn

D

A

dD

dP n

D

An12

2

2

D Dn Bn

D DnD)(Bn 2 12 2

2

2

Dn2Bn1D 2

or

Most Economical Section of ChannelsMost Economical Trapezoidal Channel

37




DOF

The best side slope for Trapezoidal section

3

1 k 60

0dk

dP

k


Other criteria for economic Trapezoidal section

38




Circular section

2sin8 4

22 d d A

d r P 2

d D 95.0154

Maximum Flow using Manning

Maximum Flow using Chezy

d D 94.0151

d D 81.075.128

Maximum Velocity using Manning or Chezy


Most Economical Circular Channel

39





40




Example 4Circular open channel as shown d=1.68m, bed slope = 1:5000, find theMax. f low rate & the Max. velocity using Chezy equation, C=70.

sVAQ

V

m P

A R

d P

d d A

S RC V

h

h

/m496.117.269.0

m/s69.05000

1485.070

485.05.4

17.2

m4.568.1180

154

m17.21542sin8

68.1

180154

4

68.12sin

84

3

22222

154Max. flow rate


41




m/s748.05000157.070

57.03775.3

93.1

m 378.368.1180

75.128

m93.175.1282sin

8

68.1

180

75.128

4

68.12sin

84

22222

V

m P

A R

d P

d d A

S RC V

h

h

75.128Max. Velocity


Example 4 cont.

42




Trapezoidal open channel as shown Q=10m3/s, velocity =1.5m/s,

for most economic section. find wetted parameter, and the bedslope n=0.014.

m D

D D D A

DkD B A

mV

Q A

B D

D B D

kD Bk D

78.1

667.6)2

36055.0(

667.65.1

10

6055.0

2

2322

31

2

21

2

2

2

Example 5


43




m P

k D D P

k D B P

49.72

3178.12)78.1(6055.0

126055.0

12

2

2

2

To calculate bed Slope

6.1941:1

5.189.0014.0

1

89.0

49.7

667.6

m49.7

m667.6

1

3

2

2

3

2

S

S V

P

A R

P

A

S Rn

V

h

h

Example 5

cont.


44




Use the proper numerical method to calculate uniform water

depth flowing in a Trapezoidal open channel with B = 10 m, asshown Q=10m3/s if the bed slope 0.0016, n=0.014. k = 3/2. to aprecision 0.01 m, and with iterations not more than 15.Note: you may find out two roots to the equation.

1

ManningFrom

2

2

2

1

2/1

3/2

22

2/13/2

S

P

A

n A

Q

P

A R

DnD B P

nD B B D A

S RnV

h

h

Example 6:


45




sm A

QV

m A

m D

/326.07.30

10

7.30)28.2(2

328.210

28.2

22

Example 6 cont.


46

Variation of flow and velocity with depth in circular pipes



Variation of flow and velocity with depth in circular pipes

47

Energy Principles in Open Channel Flow



g

V y Z EnergyTotal

2

2


D

Water Surface

T.E.L

channel bed

Referring to the figure shown, the total energy of a flowingliquid per unit weight is given by

Where:

Z = height of the bottom of channelabove datum,

y = depth of liquid,V = mean velocity of flow.

If the channel bed is taken as the datum (as shown),then the total energy per unit weight will be.This energy is known as specific energy, Es. Specificenergy of a flowing liquid in a channel is defined asenergy per unit weight of the liquid measured from

the channel bed as datum 48





D

Water Surface

T.E.L

channel bed

The specific energy of a flowing liquid can be re-written inthe form:

49





It is defined as the curve which shows the variation of specific

energy (Es ) with depth of f low y. It can be obtained as follows:

Let us consider a rectangular channel in which a constantdischarge is taking place.

Specific Energy Curve (rectangular channel)

But

g

V y E specific

2

2

2

2

2

A g

Q y E s Or

50






g

V

2

2

H

G

E

The graph between specific energy (x-axis) and depth (yaxis)

may plotted.

51




Referring to the diagram above, the following features can be

observed:1. The depth of flow at point C is referred to as critical depth, yc. It

is defined as that depth of f low of liquid at which the specificenergy is minimum, E min , i.e.; E min @ yc . The flow thatcorresponds to this point is called critical flow (Fr = 1.0).

2. For values of Es greater than E min , there are two correspondingdepths. One depth is greater than the critical depth and the otheris smaller then the critical depth, for example ; Es1 @ y 1 and y 2 These two depths for a given specific energy are called thealternate depths.

3. If the flow depth y > yc , the flow is said to be sub-critical (Fr < 1.0). In this case Es increases as y increases.

4. If the flow depth y < yc , the flow is said to be super-critical (Fr > 1.0). In this case Es increases as y increases.



52




Froude Number (Fr) h

r D g

V F

T

A

WidthSurfaceWater

Area)(WettedFlowof Areah D

T

T

Flow Fr

Sub-critical1 > Fr

Critical1 = Fr

Supercritical1 < Fr

g A

T Q F r 3

22


53




Critical Flow

Subcritical

Super criticalcritical


54




Rectangular Channel

31

2

g

q y

C

h

r D g

V F 1

At critical Flow D g

V

D g

V F h

r For rectangular section

3

1

2

2

g B

Q yc

q=Q/B


a) Critical depth, yc , is defined as that depth of flow of liquid at

which the specific energy is minimum, Emin,

b) Critical velocity, Vc , is the velocity of flow at critical depth.

55




Rectangular Channel


c) Critical, Sub-critical, and Super-critical Flows:Critical f low is defined as the flow at which the specific energyis minimum or the flow that corresponds to critical depth. Referto point C in above figure, E min @ yc .

and therefore for critical flow Fr = 1.0

If the depth flow y > yc , the flow is said to be sub-critical. In this case Esincreases as y increases. For this type of flow, Fr < 1.0 .

If the depth flow y < yc , the flow is said to be super-critical. In this caseEs increases as y decreases. For this type of flow, Fr > 1.0 .

56




Rectangular Channel

gy p p

d) Minimum Specific Energy in terms of critical depth: At (Emin , yc ) ,

57




Other Sections

)2(2

)53(

c

ccc

Dn B

D Dn B E

sin

)2sin2(

16)cos1(

2 d d E c

cc D E

4

5

Trapezoidal section

Circular section

Triangle section

cc D E

2

3 Rectangular section

at critical flow Fr =1 where: 13

2

2 g A

T Q F r

gy p p

58




Example 1

13

2

g A

T Q

1

81.93

14.2

3

124.233.1

23

2

3

2

cc

c

cc

c

D D

D

g DnD B

nD BQ

m31.0c D

Determine the critical depth if the flow is 1.33m3/s. thechannel width is 2.4m

gy p p

59




Example 2

4481.9

253

2

3

1

2

3

1

2

23

1

2

2

B B B g B

Q yc

0.0062

D

0.016

125

2

P

AR

SR n

1V

32

c

32

B D

B

B D

B D B D

cc

c

c

Rectangular channel , Q=25m3/s, bed slope =0.006,determine the channel width with critical flow usingmanning n=0.016

gy p p

60




m B

B

B

B

B B

B B

B B

3

0.006

8

4

0.016

1

4

25

0.006

42

4

0.016

1

4

25

32

35

31

32

32

32

32

Example 2 cont.

gy p p

61

Non-uniform Flow in Open Channels:



p

Non-uniform flow is a flow for which the depth of flow is aried. This

varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF).

Such situations occur when control structures are used in the

channel or when any obstruction is found in the channel

Such situations may also occur at the free discharges and when a

sharp change in the channel slope takes place.

The most important elements, in non-uniform flow, that will be

studied in this sectionare:

Classification of channel-bed slopes. Classification of water surface profiles. The dynamic equation of gradually varied flow.

Hydraulic jumps as examples of rapidly varied flow. 62




p

63




pBed Slopes-Classification of Channel

The slope of the channel bed can be classified as:

1) Critical Slope: the bottom slope of the channel is equal to the criticalslope. In this case S0 = Sc or yn = yc .

2) Mild Slope: the bottom slope of the channel is less than the criticalslope. In this case S0 < Sc or yn > yc .

3) Steep Slope: the bottom slope of the channel is greater than thecritical slope. In this case S0 > Sc or yn < yc .4) Horizontal Slope: the bottom slope of the channel is equal to zero(horizontal bed). In this case S0 = 0.0 .

5) Adverse Slope: the bottom slope of the channel rises in the direction

of the flow (slope is opposite to direction of flow). In this case S0 =negative .The first letter of each slope type sometimes is used to indicate theslope of the bed. So the above slopes are abbreviated as C, M, S, H, and

A, respectively. 64




pBed Slopes-Classification of Channel

65




pClassification of Flow Profiles (water surface profiles):

66




pClassification of Flow Profiles (water surface profiles):

67




Classification of Flow Profiles (water surface profiles):

68





69





70

Hydraulic Jump



A hydraulic jump occurs when flow changes from a supercriticalflow (unstable) to a sub-critical flow (stable). There is a sudden

rise in water level at the point where the hydraulic jump occurs.Rollers (eddies) of turbulent water form at this point. Theserollers cause dissipation of energy.

71

Hydraulic Jump



General Expression for Hydraulic Jump:In the analysis of hydraulic jumps, the following assumptions are

made:(1) The length of hydraulic jump is small. Consequently, the loss

of head due to friction is negligible.

(2) The flow is uniform and pressure distribution is due tohydrostatic before and after the jump.

(3) The slope of the bed of the channel is very small, so that the

component of the weight of the fluid in the direction of the flowis neglected.

72

Hydraulic Jump



Hydraulic Jump in Rectangular Channels

But for Rectangular section

73

Hydraulic Jump




74

Hydraulic Jump




75

Hydraulic Jump




76

Hydraulic Jump



77

Hydraulic Jump



78

Hydraulic Jump



Example 1 A 3-m wide rectangular channel carries 15 m 3/s of water at a 0.7 m

depth before entering a jump. Compute the downstrem water depthand the critical depth

72.27.081.9

14.7

/14.77.0

5

366.181.9

5

/s.mm53

15

1

1

1

1

1

3

2

3

gd

V

F

smd

qV

md

q

r

c

md

d

365.2

1)72.2(812

1

7.0

2

22

79

Hydraulic Jump



Example 2

dn = Depth can calculated from manning equation

d1=dn

d2

80

Hydraulic Jump



calsupercritiisflowthe142.108.181.9

63.4

/63.43

15

08.1

004.032

3

01.0

1

3

15

322

3

1

1

11

1

1

1

32

32

gd

V F

smd

V

md d D

D

D

D

D B D P

D BD A

S Rn A

Q

r

n

h

d1=dn

d2

a)

b) md

d

7.1

1)42.1(812

1

08.1

2

22

81

Hydraulic Jump



c)

d1=dn

d2

m g g

E

smd

V

032.0294.27.1

263.408.1

/94.27.13

15

3

15

22

2

2

82



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Water Flow in Open Channels