CHAPTER FIVE

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CHAPTER FIVE

• PRINCIPLES OF OPEN CHANNEL FLOW

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5.1 FUNDAMENTAL EQUATIONS OF FLOW

• 5.1.1 Continuity Equation
• Inflow Outflow
• Area A1 and A2 and Velocity V1 and
  V2
• Area x Velocity (A . V) Discharge, Q
• ie. A1 V1 A2 V2 Q

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5.1.2 Energy Equation

• Energy is Capacity to do work
• Work done Force x Distance moved
• Forms of Energy
• Kinetic Energy - velocity
• Pressure Energy - pressure
• Potential Energy - Height or elevation

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Kinetic Energy (KE)

• Energy possessed by Moving objects.
• Solid Mechanics KE 1/2 m V2
• But Mass W/g where W is the weight
• In hydraulics KE 1/2 . W/g . V2 W
  V2 /2g
• KE per unit weight
• ( W V2 /2g) / W V2 /2g

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Pressure Energy

• Fluid flow under pressure has ability to do work
  and so possesses energy by virtue of its
  pressure.
• Pressure force P. a where P is pressure.
  w specific volume
• If W is the weight of water flowing, then Volume
  W/w
• Distance moved by flow W/w.a (Recall
  Volume/area distance
• Work done Force x distance P.a x
  W/wa WP/w
• Pressure Energy per unit weight P/w

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Potential Energy

• Energy Related to Position
• Wt. of Fluid W at a height Z
• Then Potential Energy W Z
• Potential Energy per unit wt. Z
•  
• Total Energy available is the sum of the three
• E P/w V2 /2g Z The
  Bernoulli Equation.

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Bernoulli Theorem

• Total Energy of Each Particle of a Body of Fluid
  is the Same Provided That No Energy Enters or
  Leaves the System at Any Point.
• Division of Available Energy Between Pressure,
  Kinetic and Position May Change but Total Energy
  Remains Constant.
• Bernoulli Equation Is Generally Used to Determine
  Pressures and Velocities at Different Positions
  in a System.
• Z1 V12/2g P1 /w Z2 V22 /2g P2 /w

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5.2 UNIFORM FLOW IN OPEN CHANNELS

• 5.2.1 Definitions
• a) Open Channel Duct through which Liquid
  Flows with a Free Surface - River, Canal
•  
• b) Steady and Non- Steady Flow In Steady
  Flows, all the characteristics of flow are
  constant with time. In unsteady flows, there are
  variations with time.

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Uniform and Non-Uniform Flow

• In Uniform Flow, All Characteristics of Flow Are
  Same Along the Whole Length of Flow.
• Ie. Velocity, V1 V2 Flow Areas, A1
  A2
• In Uniform Channel Flow, Water Surface is
  Parallel to Channel Bed. In Non-uniform Flow,
  Characteristics of Flow Vary along the Whole
  Length.

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More Open Channel Terms

• d) Normal Flow Occurs when the Total Energy
  line is parallel to the bed of the Channel.
•  
• f)Uniform Steady Flow All characteristics of
  flow remain constant and do not vary with time.

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Parameters of Open Channels

• a) Wetted Perimeter, P The Length of
  contact between Liquid and sides and base of
  Channel
  
• P b 2 d d normal depth
• b)Hydraulic Mean Depth or Hydraulic Radius (R)
  If cross sectional area is A, then R A/P,
  e.g. for rectangular channel, A b d, P b
  2 d

Area, A
d
b
Wetted Perimeter
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Empirical Flow Equations for Estimating Normal
Flow Velocities

• a) Chezy Formula (1775) Can be derived
  from basic principles. It states that
• Where V is velocity R is hydraulic radius and
  S is slope of the channel. C is Chezy
  coefficient and is a function of hydraulic radius
  and channel roughness.

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Manning Formula (1889)

• Empirical Formula based on analysis of various
  discharge data. The formula is the most widely
  used.
•  
• 'n' is called the Manning's Roughness Coefficient
  found in textbooks. It is a function of
  vegetation growth, channel irregularities,
  obstructions and shape and size of channel.

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Best Hydraulic Section or Economic Channel
Section

• For a given Q, there are many channel shapes.
  There is the need to find the best proportions of
  B and D which will make discharge a maximum for a
  given area, A.
• Using Chezy's formula
• Flow rate, Q A
• For a rectangular Channel P b 2d
• A b d and therefore b A/d
• i.e. P A/d 2 d

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Best Hydraulic Section Contd.

• For a given Area, A, Q will be maximum when P
  is minimum (from equation 1)
• Differentiate P with respect to d
• dp/dd - A/d2 2
• For minimum P i.e. Pmin , - A/d2 2
  0
• A 2 d2 ,
• Since A b d ie. b d 2 d2 ie. b
  2 d
• i.e. for maximum discharge, b 2 d OR

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For a Trapezoidal Section
Zd
Zd
Area of cross section(A) b d
Z d2 Width , b A/d - Z d
...........................(1) Perimeter b
2 d ( 1 Z 2 )1/2 From (1), Perimeter
A/d - Z d 2 d(1 Z2 )1/2
1
d
Z
b
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For maximum flow, P has to be a minimum i.e
dp/dd - A/d2 - Z 2 (1 Z2 )1/2 For
Pmin, - A/d2 - Z 2 (1 Z2)1/2 0
A/d2 2 (1 Z2 ) - Z
A 2 d2 ( 1 Z2 )1/2 - Z d2 But
Area b d Z d2 ie. bd Z d2 2 d2 (1
Z2 ) - Z d2 For maximum discharge, b 2
d (1 Z2)1/2 - 2 Z d or
  Try Show that for the best hydraulic
section  
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     NON-UNIFORM FLOW IN OPEN CHANNELS

• 5.3.1 Definition By non-uniform flow, we mean
  that the velocity varies at each section of the
  channel.
• Velocities at Sections 1 to 4 vary (Next
  Slide)
• Non-uniform flow can be caused by
  i)Differences in depth of channel and
• ii) Differences in width of channel.
• iii) Differences in the nature of bed
• iv) Differences in slope of channel and
• v) Obstruction in the direction of flow.

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Variations of Flow Velocities
1
3
4
2
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Non-uniform Flow In Open Channels Contd.

• In the non-uniform flow, the Energy Line is not
  parallel to the bed of the channel.
• The study of non-uniform flow is primarily
  concerned with the analysis of Surface profiles
  and Energy Gradients.

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Energy Line Analysis
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Energy Line Analysis Contd

• For the Energy Line, total head is equal to the
  depth above datum plus energy due to velocity
  plus the depth of the channel. Pressure energy
  is not included because we are working with
  atmospheric pressure.
• ie. H Z d V 2 / 2 g

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Specific Energy, Es

• When we neglect Z, the energy obtained is called
  specific energy (Es).
• Specific energy (Es) d V2 /2g
• Non-uniform flow analysis usually involves the
  energy measured from the bed only, the bed
  forming the datum, and this is called specific
  energy.
• In Uniform flow, the specific energy is constant
  and the energy grade line is parallel to the bed.
  
• In non-uniform flow, although the energy grade
  line always slopes downwards in the direction of
  flow, the specific energy may increase or
  decrease according to the particular channel's
  flow conditions.

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Variation of Specific Energy( Es) with depth (d)
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Variation of Specific Energy( Es) with depth (d)
Contd.

• Es d V 2 /2g, since q v d ie.
• v q/d, Es d q 2 /2 g d2
• For a given q, we can plot the variation of Es
  (specific energy) with flow depth, and use the
  graph to solve the cubic equation above. For a
  given value of Es, there are two values of d,
  indicating two different flow regimes.
• Flow at A is slow and deep (sub-critical) while
  flow at B is fast and shallow (super-critical)

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Variation of Spcific Energy, Es with depth, d
Depth, d
Increasing Flow per unit Width, q
A
C.
B
Specific Energy, Es
Minimum Es
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Critical Depth

• Critical Depth we observe from the graph is the
  depth at which the hydraulic specific energy
  possessed by a given quantity of flowing water is
  minimum.
• CRITICAL FLOW occurs at CRITICAL DEPTH and
  CRITICAL VELOCITY.
• At Critical point C, the value of Es is minimum
  for a given flow rate q.

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b) Some Properties of Critical Flow Es
(specific energy) d q 2 /2gd 2
..............(1) At critical flow, E has a
minimum value - obtain minimum value by
differentiation dEs/dd 1 - q2 /gd 3
0 ie. q 2 /g dc3 1 , d dc -
critical depth. dc q 2
/g (q 2 g d3 )
  This means that critical depth, d is a
function of flow per unit width, q only. From
above q 2 g d c3 but q
Vc dc ie. Vc2 dc2 g dc3 and
V c2 g dc and Vc2 /2g
1/2 dc - kinetic energy ie.
dc Vc2 /g This means that when the
value of velocity head is double the depth of
flow, the depth is critical. The specific energy
equation (1), now becomes E dc 1/2
dc 3/2 dc dc
2/3 Ec   ie. dc q2 /g
Vc2 /g 2/3 Ec    
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Some Properties of Critical Flow Contd.

• It is also interesting to see how discharge q
  varies with depth, d for a given amount of
  specific energy, E

d
dc
q
Max Discharge
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Variation of Es with d Concluded
Es d q 2 /2g d2 ie. q2 2g d 2 (Es
- d) 2g d2 Es - 2g d3 For maximum q,
dq/dd 0 ie. 2 dq/dd 4 Es g d - 6 g
d2 dq/dd (4 Es g d - 6 g d2)/2
0 ie. 2 Es g d - 3 g d 2 0 ie. dc
2 Es/3 This means that maximum flow occurs
when d 2/3 Es. This equation is similar to
the one above for critical flow. Therefore, if
the specific energy available is Ec, then maximum
flow occurs at 2/3 Ec ie. at the critical
depth.   Summarising When discharge is
constant, critical depth is the depth at minimum
specific energy and when the specific energy is
constant, critical depth is the depth at maximum
discharge.
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      Sub-Critical and Super-Critical Flows

• At the increase of slope of a bed of flow, the
  level of flow drops and velocity of flow
  increases.
• Where a condition exists such that the depth of
  flow is below critical depth, the flow is
  referred to as super critical.
• Super-critical velocity refers to the velocity
  above critical velocity.
• Similarly, sub-critical velocity refers to
  velocity below critical velocity.
• These flow regimes can be represented by the two
  limbs of the depth-specific energy curve.

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Sub-Critical and Super-Critical Flows Contd.
d
Sub-critical
Critical
Super-critical
Es
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Froude Number

• This is a Dimensionless Ratio Characterizing Open
  Channel Flow.
• Froude Number, F V/ gd
• Stream velocity/wave velocity
• When F 1, Flow is critical (d dc
  and V Vc)
• F lt 1, Flow is sub-critical (d gt dc and
  V lt Vc)
• F gt 1, Flow is super-critical(dlt dc and V
  gt Vc)

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Hydraulic Jump

• If a super-critical flow suddenly changes to a
  sub-critical flow, a hydraulic jump is said to
  have occurred.
• The change from super-critical to sub-critical
  flow may occur as a result of an obstruction
  placed in the passage of the flow or the slope of
  the bed provided is not adequate to maintain
  super-critical flow eg. water falling from a
  spillway.

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Diagram of Hydraulic Jump
d
dsub
dc
Water Falling From a Spillway
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Depth and Energy Loss of Hydraulic Jump

• As energy is lost in the hydraulic jump, we
  cannot use the Bernoulli equation to analyse.
• Momentum equation is suited to this case - no
  mention of energy. The aim is to find expression
  for d2

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Depth and Energy Loss of Hydraulic Jump Contd.

d2
V2
d1
V1
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Depth and Energy Loss of Hydraulic Jump Concluded
It can be shown that   Also
the loss of energy (m) during a hydraulic jump
can be derived as   The depths are in m
and Power loss (kW) 9.81 x Flow rate ,
m3/s x Energy loss (m)    
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Hydraulic Jump Concluded

• A hydraulic jump is use
• ful when we require
• i) Dissipation of energy e.g. at the foot of a
  spillway
• ii) When mixing of fluids is required e.g. in
  chemical and processing plants.
• iii) Reduction of velocity e.g. at the base of a
  dam where large velocities will result in
  scouring.
• It is, however, undesirable and should not be
  allowed to occur where energy dissipation and
  turbulence are intolerable.