fluid mechanics rajeevan sir ii-handouts

Fluid Mechanics 4/4/2012

Dr. B. Rajeevan 1

2K6CE 404 FLUID MECHANICS ‐ II

Dr. B. Rajeevan Assistant Professor

Department of Civil Engineering Government College of Engineering Kannur

Mob: +91 9495 333 088 E‐mail: [email protected] Contact Hours: 4 pm – 5 pm

Introduction

• Requirement – Fluid Mechanics – I

• Sessional – 50 marks

– 2 assignments = 2 x 10 = 20

– 2 tests = 2 x 15 = 30

Total = 50 marks

Register for the course at www.gcek.ac.in/moodle

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Reference Books


Introduction

• Transportation of Liquids

– Closed Conduits – Top Closed

• Pipes and Tunnels

– Open Channels – Top Open

• Streams , Rivers and Canals

Flow of Liquids in open channels or closed conduits with free surface is known as free surface flow or open

channel flow.


Definitions

Open Channel Flow = Free Surface Flow


Definitions

Pressurised Flow = Closed Conduit Flow = Pipe Flow



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Definitions

Combined Free Surface and Pressurised Flow


Definitions

Hydraulic Grade Line – HGL

Energy Grade Line ‐ EGL


Classification of Flows


Steady & Unsteady Flows

Flow velocity versus time ‐‐‐ ?????


Uniform & Non‐uniform Flows

Flow velocity at any instant of time does not vary within the length of channel

Non‐uniform flow = Varied flow


Varied Flow

Gradually Varied Flow

Rapidly Varied Flow

Flow Depth with distance ‐????



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Laminar & Turbulent Flows

Liquid particles move in definite smooth paths ‐ Viscous force dominates Liquid particles move in irregular paths ‐ Inertial force dominates

Reynolds's Number


Laminar & Turbulent Flows

Pipe Flow L = Pipe Diameter Open Channel Flow L = Hydraulic radius or Hydraulic Depth Hydraulic depth = Flow area/Top surface width Hydraulic radius = Flow area/Wetted perimeter Re= 600 – Laminar to Turbulent in Open Channel Flow

Laminar Free Surface Flow is rare


Subcritical, Supercritical, and Critical Flows

Fr= 1 – Critical Flow

Fr< 1 – Subcritical Flow

Fr> 1 – Supercritical Flow


Channels ‐ Terminology

Channels Natural

Artificial

Canal

Flume

Chute

Tunnel

Culvert

Long channel with Long channel with mild slope excavated

in ground

Channel above ground

Channel with steep bottom slope

Channel excavated through hills

Short channel running partially full



The depth of flow, y, at a section is the vertical distance of the lowest point of the channel section from the free surface. The depth of flow section, d, is the depth of flow normal to the direction of flow. The stage, Z, is the elevation or vertical distance of free surface above a specified datum



Table 1: Properties of Typical Channel Sections



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Velocity Distribution


Velocity variation with depth


Kinetic Energy Correction Factor V = Instantaneous velocity Vm = Mean velocity


Momentum Correction Factor V = Instantaneous velocity Vm = Mean velocity


Example ‐ 1

Considering unit width of channel,


Example – 1 – cont’d....



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Example – 1 – cont’d....

END 4 April 2012 25 Dr. B. Rajeevan

Homework

1.

2.

Figure.


Homework

3.

4.


Assignment 1


UNIFORM FLOW

• Flow depth does not change with length

• Normal Depth ‐ ????

• Component of weight of water cause acceleration

• Shear stress at boundaries cause deceleration

• Imbalance between these forces causes non‐uniformity in flow


Uniform and Non‐uniform flows



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Flow Resistance Equations

• Chezy’s equation

• Manning’s formula


Chezy’s equation

Assumptions

1) Steady;

2) the slope of the channel bottom is small;

3) Prismatic.


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Resolving all forces in the direction of flow, we get,

Chezy’s equation cont’d...

DEFINITION SKETCH



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CHEZY FORMULA, 1769 4 April 2012 37 Dr. B. Rajeevan


Dimension of Chezy’s coefficient, C is L0.5T‐1

Divide by ‘g’ to make C dimensionless


Darcy‐Weisbach equation

Pipe Flow

Surface

Smooth

Transition

Rough



Pipe Flow

Moody chart – variation of


Darcy‐Weisbach equation – Open channels

Open Channel = Conduit cut into two



Manning’s Formula

Manning’s Formula


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Sticky Note

what is M1 and Ms


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UNIFORM FLOW

• Flow depth does not change with length

• Normal Depth ‐ ????

• Component of weight of water cause cancelation

• Shear stress at boundaries cause deceleration

• Imbalance between these forces causes non‐uniformity in flow


Uniform and Non‐uniform flows


Flow Resistance Equations

• Chezy’s equation

• Manning’s formula


Chezy’s equation

Assumptions

1) Steady;

2) the slope of the channel bottom is small;

3) Prismatic.



DEFINITION SKETCH



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CHEZY FORMULA, 1769 4 April 2012 7 Dr. B. Rajeevan


Dimension of Chezy’s coefficient, C is L0.5T‐1

Divide by to make C dimensionless



Pipe Flow

Surface

Smooth

Transition

Rough



Pipe Flow



Darcy‐Weisbach equation – Open channels

Open Channel = Conduit cut into two



Manning’s Equation

Manning’s Formula



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Most Economical Channel Section

• Max Discharge for a given

– Flow area, A;

– Resistance coefficient, n

– Bottom slope, S

• For a given area, Q is max when V is max

• V is max when R is max(for a given S and n)

• R is max when P is min


Most Economical Rectangular Channel Section


y

B

Rectangular channel section is most economical when depth of flow is equal to half the bottom width of hydraulic radius is equal to half the depth of flow.

Most Economical Trapezoidal Channel Section


Half the top width = sloping side length

Most Economical Trapezoidal Channel Section


Hydraulic Radius, R= Half the flow depth

Most Economical Triangular Channel Section


Homework 2.1

Most Economical Circular Channel Section


Condition for Maximum Discharge


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Condition for Maximum Mean Velocity of Flow

Homework 2.2

Computation of Uniform Flow


K = Conveyance of the channel section When Manning’s formula is used, Also,

Section Factor

Normal Depth, yn

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The depth of flow at which uniform flow is maintained in a channel

Worked out Examples

EXAMPLE 1

An irrigation channel of trapezoidal section, having side slopes 3

H: 2 V, is to carry a flow of 10 cumecs on a longitudinal slope of 1

in 5000. The channel is to be lined for which the value of friction

coefficient in Mannings’ formula is n = 0.012. Find the dimensions

of the most economical section of the channel.



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Example 1 ‐ Solution




For most economical channel Section,

Also,

Using Manning’s formula,

Worked out Examples

EXAMPLE 2

Water flows at a uniform depth of 2 m in a trapezoidal channel having a bottom width 6 m, side slopes 2 H: 1 V. If it has to carry a discharge of 65 m3/s, compute the bottom slope required to be provided. Take Manning’s n = 0.025.




Specific Energy


Total Energy per Unit weight

Specific Energy (E) of flow at any section is defined as the energy per unit weight of water measured with respect to the channel bottom as the datum.

Specific Energy



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Specific Energy Curve


Definitions

• Critical Depth,

• Critical Velocity,

• Alternate Depths,

• Subcritical flow or tranquil flow

• Supercritical flow or rapid flow


Critical Depth




For a given specific energy the discharge in a given channel section is maximum when the flow is in the critical state.

Specific Force


Specific Force



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Minimum Specific Force


Critical Flow Computations

• For Critical Flow,


For , Zc is a function of depth of flow. Implies, for prismatic channels, there is only one depth of flow, yc, which makes the flow critical. Since, yc is same at all sections of channel, critical flow in prismatic channels is uniform flow.

Conclusions – Critical Flow

• E is minimum for a given Q

• Q is max for a given E

• F is min for a given Q

• Q is max for a given F

• Velocity head = D/2

• Fr = 1


Critical Flow in Rectangular Channels

• Bottom Width, B = Top Width, T

• Let q = discharge per unit width

– Q = q × B

• For critical flow,

•


Critical Flow in Rectangular Channels


Discharge Diagram



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• Critical Flows in

– Triangular Channel Section

– Parabolic Channel Section

– Trapezoidal Channel Section

• Application of Specific Energy and Discharge

Diagrams to Channel Transitions


Exercises

Example 1

An earth canal in good condition is 17 m wide at bottom and has side slope 2 H: 1V. One side slope extends to a height of 7.8 m above the bottom level and the other side extends to an elevation of 1.8 m, then extends flat to a distance of 150 m and rises vertically. If the slope of the canal is 0.7 m per 1610 m, estimate the discharge when the depth of water is 2.5 m. Assume Chezy’s C = 35.


Solution


2

1

17 m 150 m 2

1

2.5 m 0.7 m

Exercises

• Example 2

For a constant specific energy of 1.8 Nm/N, calculate the maximum discharge that may occur in a rectangular channel 5 m wide.




Exercises

• Example 3

A trapezoidal channel has a bottom width of 6 m and side slopes of 2 H: 1 V. If the depth of flow is 1.2 m at a discharge of 10 m3/s, compute the specific energy and the critical depth.



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For critical Flow,

Alternative Method Plot depth versus section factor




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Gradually Varied Flow(GVF)

• Examples of GVF

– Flow upstream of river/dam

– Flow downstream of a sluice gate

– Flow in channels with break in slopes


A steady non‐uniform flow in a prismatic channel with gradual changes in its water surface elevation

GVF‐Examples


GVF‐Examples

Assumptions

• The pressure distribution at any section is hydrostatic

– A gradual change in surface curvature give rise to negligible normal accelerations.

• The resistance to flow at any depth is given by corresponding uniform flow equation with slope replaced with energy slope.


Assumptions‐cont’d...


• The bottom slope of the channel is very small

• Prismatic

• = 1

• n is independent of depth of flow

The slope of the channel bottom may be assumed small if it is less than 5 percent. In such a case, sin tan , in which = angle of the channel bottom with horizontal, and the flow depths measured vertically or normal to the bottom are approximately the same.


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Differential Equation


DATUM


Wide Rectangular Channel


Using Chezy’s Equation


For rising water surface


For falling water surface

HOMEWORK

Classification of Bottom Slopes


For a given channel with a known Q = Discharge, n = Manning coefficient, and S0 =Channel bed slope,

yc = critical water depth and yn = Uniform flow depth can be computed. There are three possible relations between yn and yc as

1) yn > yc , 2) yn < yc , 3) yn = yc .

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Sticky Note

Dynamic equation for GVF


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For horizontal and adverse slope channels, uniform flow depth

yn does not exist.




• Critical

• Mild

• Steep

• Horizontal

• Adverse


Zones


Zones


Zones



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Zones


Water Surface Profiles


– M‐curve

– S‐curve

– C‐curve

– H‐curve

– A‐curve





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Backwater and Drawdown Curves

• Depth of flow increases in the direction of flow

(dy/dx is +ve) – curve (Zone 1 & 3)

• Depth of flow decreases in the direction of flow

(dy/dx is ‐ve) – curve (Zone 2)


Equation of GVF



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Characteristics of Surface Profiles


Characteristics of Surface Profiles


Self study

• Surface Profiles in Critical sloped channels

• Surface Profiles in Horizontal sloped channels

• Surface Profiles in Adverse sloped channels


Example 1

• Flow depth in a section of the non‐uniform flow reach of the channel is 2.9 m. Determine the type of flow profile in the channel. Take yc = 2.63 m and yn = 3.17 m.



• Given, y = 2.9 m ; yc = 2.63 m and yn = 3.17 m.

• Since yn > yc, slope is mild.

• Also, yc < y < yn, profile is in Zone 2.

• Hence it is M2 curve.


Example 2

• A rectangular channel with a bottom width of 4 m and a bottom slope of 0.0008 has a discharge of 1.5 m3/s. In a gradually‐varied flow in this channel, the depth at a certain location is found to be 0.3 m. Assuming n = 0.016, determine the type of GVF profile.



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Solution


Step 1: Determine normal depth, yn

Step2: Determine critical depth, yc

Step 3: Compare given y with normal depth and identify the slope.

Step 4: Compare normal depth and critical depth with given depth

and determine the type of curve.

Solution


Solving by trial and error,

Critical Depth,

Type of Profile

Practical Examples


Practical Examples


Practical Examples


Practical Examples



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Why ?

• All major hydraulic engineering activities

• Determination of the effect of a hydraulic

structure on the channel

• Inundation of land

• Estimation of the flood zone


Methods

• Step Method

• Graphical Integration Method

• Direct Integration Method


Step Method


Step Method


Steps



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Graphical Integration Method


Direct Integration Method – Bresse’s Method

• Wide rectangular channels

• Chezy’s equation is used



Example 1 A rectangular channel 7.5 m wide has a uniform depth of flow

of 2 m and has a bed slope of 1 in 3000. If due to weir constructed at the downstream end of the channel, water surface at a section is raised by 0.75 m, determine the water surface slope with respect to horizontal at this section. Assume Manning’s n =0.02.


7.5 m

2 m

Solution



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Example 2 A rectangular channel 10 m wide carries a discharge of 30

cumecs. It is laid at a slope of 0.0001. If at a section in this channel the depth is 1.6 m, how far (upstream or downstream) from the section will the depth be 2.0 m? Take Manning’s n = 0.015.


Solution



Self study

• Direct Integration Method

Backhmeteff method

Chow method



END OF MODULE –II (GVF)



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• Stream lines in Uniform flow and GVF are parallel – acceleration negligible – pressure distribution hydrostatic


SHALLOW WATER THEORY

In Rapidly varied flow, the sectional area of flow changes abruptly within a short distance. Turbulent eddying loss is more important than boundary friction in this case. Hydraulic jump is a typical example of rapidly varied flow.


Rapidly Varied Flow (RVF)

• Streamlines have sharp curvatures – nonparallel‐Non hydrostatic pressure distribution

• Flow profile discontinuous due to rapid change of flow depth

• Analyzed using Boussinesq and Fawer assumptions


Assumptions

• In the Boussinesq assumption, the vertical

flow velocity is assumed to vary linearly from

zero at the channel bottom to the maximum

at the free surface.

• In the Fawer assumption, this variation is

assumed to be exponential.


Assumptions

• Before and after jump formation flow is uniform

and pressure distribution is hydrostatic

• The length of jump is small – loss due to friction

neglected

• Component of weight of water along flow

direction is neglected



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Characteristics of RVF

• Streamlines are not parallel

• Variation in the cross‐sectional shape and size, due to change in the flow direction


Fig. 4.1 Definition sketch for abrupt drop


General Equation of Hydraulic Jump


General Equation of Hydraulic Jump



Conjugate Depths

Hydraulic jump in rectangular channels


Relation between conjugate depths


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Example 1

• A horizontal rectangular channel 4 m wide carries a discharge of 16 cumecs. Determine whether a hydraulic jump may occur at an initial depth of 0.5 m. If a jump occurs, determine the sequent depth to this initial depth. Also, determine the energy loss in the jump.


Solution


Example 2

• In a rectangular channel there occurs a jump corresponding to = 2.5. Determine the critical depth and head loss in terms of the initial depth,



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Solution


Types of Hydraulic Jump

• Undular jump

• Weak jump

• Oscillating jump

• Steady jump

• Strong jump



Applications of Hydraulic Jump

• Dissipation of excess energy

• Raised water level

• Increases the weight on apron

• Increases the discharge through sluices

• Mixing of chemicals


SURGES • Moving wave which makes abrupt changes in depth of flow.

• Moving Hydraulic Jump

• Sudden opening and closing of gates

• Positive or negative

– Increase or decrease in depth of flow


A surge is a moving wave front which results in an abrupt change of the depth of flow. It is a rapidly varied unsteady flow condition.


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Positive Surge


Definition Sketch for Surge Movement


Consider the movement of a positive surge wave in x‐direction in an open channel having an irregular cross section as shown in Figure above. Here, as the surge moves with an absolute velocity, Vw, flow depth becomes equal to y2 behind the surge. Undistributed flow depth ahead of the surge is y1. The corresponding flow velocities behind and ahead of the slope front are V2 and V1 respectively. The surge has been created due to a sudden change of flow rate from Q1 to Q2. In this context, the problem definition for surge computation is: given Q1,y1,Q2 and channel slope parameters, determine the surge wave velocity, Vw and the surge height, y2‐y1. Equations for computing the above are based on the basic principles of conservation of mass and momentum.

Assumptions

Following assumptions are made in the derivation. Channel is horizontal and frictionless; Pressure distribution is hydrostatic at locations away from the front; Velocity is uniform within the cross section, at location away from the front; Change in the flow depth at the front occurs over a very short distance; wave shape, height, and wave velocity do not change as the wave propagates in the channel; water surfaces behind and ahead of the wave front are parallel to the bed


Derivation of Equations

We first choose a control volume encompassing the wave front. This control volume can be made stationary by superimposing a constant velocity, Vw (equal to the absolute velocity of surge wave) in the negative x‐direction. Thus the unsteady flow of previous Figure may be transformed to steady flow in the Figure that follows, and the principles of conservation of mass and momentum can be applied to a steady flow situation.


Surge movement viewed as steady flow


Applying continuity equation to the control volume of above Figure, we get


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in which, ρ = density of water; A2 = flow area behind the wave and A1 = flow area ahead of the wave.

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (1)

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (2)

Equation (2) can also be written as

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (3)

Another way of writing the continuity equation is

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (4)

Since ρ is a constant, Eq. (1) may be written as


Applying momentum equation to the control volume

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (5)

The channel is prismatic, horizontal and frictionless. Therefore, the only force acting on the control volume is pressure force. Pressure force acts in the positive x ‐ direction at the inlet section and in the negative x ‐ direction at the outlet section. Equation (5) can be written as

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (6)

= depth to the centroid of inlet section of the C.V.

=depth of the centroid of outlet section.


Substitution of Eq. (2) in Eq (6) leads to

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (7)

Substitution of Eq. (3) in Eq. (7) and subsequent simplification leads to

‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (8)

Here, wave is propagating in the downstream direction. Therefore, Vw should be greater than V1.


‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (9)

‐‐‐‐‐‐‐‐‐‐‐‐‐(10)

Now, substitution of Eq. (4) in Eq. (7) and subsequent simplification leads to

‐‐‐‐‐‐‐‐‐‐‐‐ (11)


Equations (10) and (11) can be used to determine the surge wave velocity and the surge height, if we know the values of undisturbed flow depth, y1, flow rate before the surge, Q1, and the flow rate after the surge, Q2. Equations (10) and (11) are non‐linear equations. They can be solved by an appropriate numerical technique. For rectangular channels, Eqs. (10) and (11) simplify to the following.

‐‐‐‐‐‐‐‐‐‐‐‐ (12)

‐‐‐‐‐‐‐‐‐‐‐‐ (13)



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Self Study

• Positive Surge – Case b

• Negative Surge

• Location of Hydraulic Jump


Energy Dissipators

• Stilling basins

• Flip Buckets

• Roller Buckets


Stilling Basin


The hydraulic jump is used for energy dissipation in a stilling basin

Head less than 50 m

Chute blocks

Baffle blocks

End sills

Stilling Basin


• The chute blocks serrate the flow entering the basin and lift up part of the jet. This produces more eddies increasing energy dissipation, the jump length is decreased, and the tendency of the jump to sweep out of the basin is reduced.

• The baffle blocks stabilize the jump and dissipate energy due to impact.

• The sill stabilizes the jump and inhibits the tendency of the jump to sweep out.

Standardized Stilling Basins

• St. Anthony Falls stilling basin;

• Stilling basins developed by the U.S. Bureau of

Reclamation (each suitable for a certain range

of head)

• A basin recommended by the U.S. Army Corps

of Engineers


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Stilling Basin


U.S. Army Corps of Engineers stilling basin

Stilling Basin


Stilling Basin


Flip Buckets

• The flip bucket energy disspator is suitable for sites where the tail water depth is low (which would require a large amount of excavation if a hydraulic jump dissipator were used) and the rock in the downstream area is good and resistant to erosion.

• The flip bucket, also called ski‐jump dissipator, throws the jet at a sufficient distance away from the spillway where a large scour hole may be produced. Initially, the jet impact causes the channel bottom to scour and erode. The scour hole is then enlarged by a ball‐mill motion of the eroded rock pieces in the scour hole. A plunge pool may be excavated prior to the first spill for controlled erosion and to keep the plunge pool in a desired location.

• A small amount of the energy of the jet is dissipated by the internal turbulence and the shearing action of the surrounding air as it travels in the air. However, most of the energy of the jet is dissipated in the plunge pool.


Flip Bucket


Flip Bucket


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Roller Bucket

• A roller bucket may be used for energy dissipation if the downstream depth is significantly greater than that required for the formation of a hydraulic jump.

• In this dissipator, the dissipation is caused mainly by two rollers: counterclockwise roller near the water surface above the bucket and a roller on the channel bottom downstream of the bucket.

• The movement of these rollers along with the intermixing of the incoming flows results in the dissipation of energy.


Roller Bucket



Plunge Pool A plunge pool is an energy dissipating device located at the outlet of a spillway. Energy is dissipated as the discharge flows into the plunge pool. Plunge pools are commonly lined with rock riprap or other material to prevent excessive erosion of the pool area. Discharge from the plunge pool should be at the natural streambed elevation. Typical problems may include movement of the riprap, loss of fines from the bedding material and scour beyond the riprap and lining.

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Turbines

• Machines to convert hydro‐power to mechanical energy

• Mechanical energy generated by turbines is used to run electric generators to develop electric power ‐ Hydroelectric power

• Cheaper – compared to oil and coal

• Environment friendly


Elements of Hydroelectric Power Plants

• Head race

• Penstocks

• Tail race

• Forebay


Hydroelectric Power Plants



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General Layout with Reaction Turbine


General Layout with Impulse Turbine


Advantages of Hydroelectric Power

• Hydroelectricity is a renewable energy resource. • Hydroelectricity is one of the most efficient energy sources because most

of the kinetic energy of the water is converted to electrical energy. • No greenhouse gases or other dangerous gases are produced so there is

no damage of this kind to the environment. • No fuel is needed, therefore the price of hydroelectricity will not change if

the price of fuel increases. • Hydroelectric plants are generally less expensive to run than other

generating plants. • Electricity can be generated almost straight away compared to coal‐fired

power stations which take several hours to start. • Electricity can be stored for later use by using excess production to pump

water to a higher altitude facility until it is released again to generate electricity.

• Hydroelectric plants only need a turbine and generator where as coal‐fired stations need a furnace, boiler, condenser, cooling towers etc.


Disadvantages of Hydroelectric Power

• The construction of hydroelectric plants is expensive.

• Hydroelectric plants are site specific. In other words you can't build them just anywhere.

• Hydroelectric plants can have a detrimental effect on the river flow and water supply. The construction of hydroelectric plants usually means that areas of land will be flooded. This means that habitats for animals and plants are lost. People living in the area may also lose their land.


Head

• Head

– Gross Head (H1) – Difference between head and tail races

– Net Head (H) – Head at entrance to turbine

• = H1 Losses(hf)


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Losses of Energy


Efficiency

• Hydraulic efficiency

• Mechanical efficiency

• Volumetric efficiency

• Overall efficiency


Hydraulic Efficiency


Mechanical Efficiency


Volumetric Efficiency


Overall Efficiency


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Sticky Note

Power delivered to runner ----------------------------- power supplied at inlet


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TURBINES

Impulse

Pelton Wheel

Reaction

Francis

Kaplan

Classification of Turbines


TURBINES

Tangential flow

Radial flow Axial flow Mixed flow


Pelton wheel Kaplan turbine


TURBINES

High head (> 250 m)

Medium head (60 – 250 m)

Low head (< 60 m)


Pelton wheel Francis Turbine Kaplan Turbine


TURBINES

Specific speed (8.5 ‐ 30)

Medium head (50 ‐ 340)

Low head (255 ‐ 860)


Pelton wheel Francis Turbine Kaplan Turbine


Runner

Vane/Bucket/Blade

Impulse Turbine

• A nozzle at the end of penstock transforms water under a high head into a powerful jet. The momentum of this jet is destroyed by striking the runner, which absorbs the resulting force. If the velocity of the water leaving the runner is nearly zero, all of the kinetic energy of the jet will be transformed into mechanical energy, so the efficiency is high.



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Plan view of a Pelton turbine installation (courtesy Voith Siemens Hydro Power Generation).


Pelton Wheel



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Work Done and Efficiencies




Self study



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Reaction Turbine

• Only a part of the energy of water available at the turbine entrance is converted to KE and a substantial part remains as pressure energy.

• Change from pressure to KE energy takes gradually while the runner moves. For this change to take place, the runner must be encased to contain the water pressure (or suction), or they must be fully submerged in the water flow.


Reaction Turbine cont’d …

• Reaction turbines are acted on by water, which changes pressure as it moves through the turbine and gives up its energy.

• Newton's third law describes the transfer of energy for reaction turbines.

• Most water turbines in use are reaction turbines and are used in low (< 30m) and medium (30 300m)head applications.



Francis Turbine

• Francis turbines are radial flow reaction turbines, with fixed runner blades and adjustable guide vanes, used for medium heads.

• In the high speed Francis the admission is always radial but the outlet is axial.

• Francis turbines can be set in an open flume or attached to a penstock.

•


Francis Turbine cont’d …

• For small heads and power open flumes are commonly employed.

• Steel spiral casings are used for higher heads, designing the casing so that the tangential velocity of the water is constant along the consecutive sections around the circumference.

• Small runners are usually made in aluminum bronze castings. Large runners are fabricated from curved stainless steel plates, welded to a cast steel hub.



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A Francis turbine runner, rated at nearly one million hp (750 MW), being installed at the Grand Coulee Dam, United States.

Parts of Francis Turbine


Part Name Purpose

Scroll Casing/Spiral Casing

Provide an even distribution of water around runner‐leads to constant velocity of water‐ c/s area gradually decreased. Made of cast steel/plate steel/concrete/concrete and steel

Speed Ring/Stay Ring

Upper and lower rings held together by stay vanes Directs water from the scroll case to guide vanes Resists the load imposed upon it by internal water pressure and weight of turbine & generator to foundation Made of cast iron/cast steel/fabricated steel

Stay vanes No of stay vanes = half the no of guide vanes

Guide vanes Fixed on the periphery of runner Regulates the quantity of water supplied to the runner Airfoil shaped Made of cast steel/stainless steel/plate steel

Parts of Francis Turbine


Part Name Purpose

Runner Series of curved vanes(16 to 24 in number) evenly arranged Water enters the runner radially and leaves axially – creates a force to rotate the runner Made of cast iron/ cast steel/mild steel/stainless steel

Shaft Made of forged steel Used to transfer the torque created by runner to generator

Draft Tube Water from runner to tail race via draft tube Made of cast steel/Plate steel/Concrete Airtight Lower end submerged below the tail water level Permits negative/suction head to be developed so that the turbine can be placed above the tail water level Converts kinetic energy to pressure energy

Draft Tube


Draft Tube

• In reaction turbines, to reduce the kinetic energy still remaining in the water leaving the runner a draft tube or diffuser stands between the turbine and the tail race.

• A well‐designed draft tube allows, within certain limits, the turbine to be installed above the tailwater elevation without losing any head.

• As the kinetic energy is proportional to the square of the velocity one of the draft tube objectives is to reduce the outlet velocity.

•


Draft Tube

• An efficient draft tube would have a conical section but the angle cannot be too large, otherwise flow separation will occur. The optimum angle is 7° but to reduce the draft tube length, and therefore its cost, sometimes angles are increased up to 15°.

• Draft tubes are particularly important in high‐speed turbines, where water leaves the runner at very high speeds. In horizontal axis machines the spiral casing must be well anchored in the foundation to prevent vibration that would reduce the range of discharges accepted by the turbine.



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What is the purpose of draft tube in

hydraulic turbines?

Draft tube has following purpose :‐ 1. It makes possible the installation of the turbine above the tail race level without the loss of head. 2. the velocity of water at the runner outlet is very high. By employing a draft tube of increasing cross sectional area, the discharge takes place at a much lower velocity and thus, a part of the kinetic energy that was going as a waste is recovered as a gain in the pressure head, and this increases the efficiency of the turbine. 3.The draft tube prevents the splashing of water coming out of the runner and guides the water to the tail race.


Work Done and Efficiencies


Kaplan Turbine


Kaplan Turbine


Kaplan Turbine


Kaplan Turbine

• Kaplan and propeller turbines are axial‐flow reaction turbines, generally used for low heads.

• Large Kaplan turbines have adjustable runner blades and may or may not have adjustable guide‐ vanes.

• If both blades and guide‐vanes are adjustable it is described as "double‐regulated".

• If the guide‐vanes are fixed it is "single‐regulated".

• Unregulated propeller turbines are used when both flow and head remain practically constant, and are most common in micro‐hydro applications.



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Dr. B. Rajeevan 1





Pumps • A pump is a machine which converts mechanical energy to

fluid energy, the fluid being incompressible. This action is opposite to that in hydraulic turbines.

• A pump is a device used to move fluids, such as gases, liquids or slurries.

• A pump displaces a volume by physical or mechanical action.

• One common misconception about pumps is that they create pressure. Pumps alone do not create pressure; they only displace fluid, causing a flow. Adding resistance to flow causes pressure.

• Pumps fall into two major groups: positive displacement pumps and rotodynamic pumps. Their names describe the method for moving a fluid.


Classification of Pumps


Positive displacement pumps

• The principle of action, in all positive displacement pumps, is purely static. These pumps are also called as ‘static pumps’.

• The pumps, operated under this principle, are reciprocating, screw, ram,plunger, gear, lobe, perialistic, diaphram, radial piston, axial piston etc.


Rotodynamic pumps

• In rotodynamic pumps, however, the energy is transferred by rotary motion and by dynamic action.

• The rotating blade system imparts a force on the fluid, which is in contact with the blade system at all points, thereby making the fluid to move i.e., transferring mechanical energy of the blade system to kinetic energy of the fluid.


RECIPROCATING PUMPS PUMPS



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Working Principle of Reciprocating Pump


Components of reciprocating pumps

• Components of reciprocating pumps:‐ a) Piston or plunger: – a piston or plunger that reciprocates in a closely fitted cylinder. b) Crank and Connecting rod: – crank and connecting rod mechanism operated by a power source. Power source gives rotary motion to crank. With the help of connecting rod we translate reciprocating motion to piston in the cylinder. c) Suction pipe: – one end of suction pipe remains dip in the liquid and other end attached to the inlet of the cylinder. d) Delivery pipe: – one end of delivery pipe attached with delivery part and other end at discharge point. e) Suction and Delivery valves: – suction and delivery valves are provided at the suction end and delivery end respectively. These valves are non‐return valves.


WORKING OF RECIPROCATING PUMP

• Operation of reciprocating motion is done by the power source (i.e. electric motor or i.c engine, etc).

• Power source gives rotary motion to crank;

• with the help of connecting rod we translate reciprocating motion to piston in the cylinder (i.e. intermediate link between connecting rod and piston).

• When crank moves from inner dead centre to outer dead centre vacuum will create in the cylinder.

• When piston moves outer dead centre to inner dead centre and piston force the water at outlet or delivery value.


EXPRESSION FOR DISCHARGE OF THE RECIPROCATING PUMP


Where: – Q: – discharge in m3/sec A: – cross‐section of piston or cylinder in m2 L: – length of stroke in meter N: – speed of crank in r.p.m

CENTRIFUGAL PUMPS PUMPS


Introduction

• Centrifugal pumps are the most widely used of all the turbo machine (or rotodynamic) pumps.

• This type of pumps uses the centrifugal force created by an impeller which spins at high speed inside the pump casing.



Dr. B. Rajeevan 3

Components

• Stationery

• Rotary


Stationery Components

a) Casing: – It is an air tight passage surrounding the impeller. It is designed in such a way that the kinetic energy of the water discharged at the outlet of the impeller is converted into pressure energy before the water leaves the casing and enters the delivery pipe. Types of casing:‐

Volute casing: – It is spiral type of casing in which area of flow increase gradually. The increase in area of flow decreases the velocity of flow and increases the pressure of water. Vortex casing: – if a circular chamber is introduced between casing and the impeller, the casing is known as vortex casing. Casing with guide blades: – the impeller is surrounded by a series of guide blades mounted on a ring know as diffuser.

b) Suction pipe: – a pipe whose one ends is connected to the inlet of the pump and other end dip into water in a sump. c) Delivery pipe: – a pipe whose one end is connected to the outlet of the pump and other end is involved in delivering the water at a required height.


Rotary Components

Impeller: – It is the main rotating part that provides the centrifugal acceleration to the fluid. Classification of impeller:

a) Based on direction of flow: ∙ Axial‐flow: – the fluid maintains significant axial‐flow direction components from the inlet to outlet of the rotor. ∙ Radial‐flow: – the flow across the blades involves a substantial radial‐flow component at the rotor inlet, outlet and both. ∙ Mixed‐flow: – there may be significant axial and radial flow velocity components for the flow through the rotor row.

b) Based on suction type: ∙ Single suction: – liquid inlet on one side. ∙ Double suction: – liquid inlet to the impeller symmetrically from both sides.

c) Based on mechanical construction: ∙ Closed: – shrouds or sidewall is enclosing the vanes. ∙ Open: – no shrouds or wall to enclose the vanes. ∙ Semi‐open or vortex type.


Working Principle of Centrifugal Pump


WORKING

• Water is drawn into the pump from the source of supply through a short length of pipe (suction pipe). Impeller rotates; it spins the liquid sitting in the cavities between the vanes outwards and provides centrifugal acceleration with the kinetic energy.

• This kinetic energy of a liquid coming out an impeller is harnessed by creating a resistance to flow. The first resistance is created by the pump volute (casing) that catches the liquid and shows it down.

• In the discharge nozzle, the liquid further decelerates and its velocity is converted to pressure according to BERNOULLI’S PRINCIPAL.


SPECIFIC SPEED • speed of an imaginary pump geometrically similar in every respect to the actual pump and capable of delivering unit quantity against a unit head.

• It is denoted by NS


Where: – N: – pump speed in r.p.m Q: – discharge in m3/sec H: – head per stage in meter


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Specific Speed


EFFICIENCIES OF CENTRIFUGAL PUMPS

• Mechanical efficiencies: – It is ratio of the impeller power to the shaft power.

• Hydraulic efficiencies: – It is ratio of the manometric head to the Euler head.

• Volumetric efficiencies:‐ It is ratio of the actual to the theoretical discharge.

• Overall efficiencies: – It is ratio of the water power to the shaft power.


CAVITATION


Definition • Cavitation is the formation and then immediate implosion (inward

bursting) of cavities in a liquid – i.e. small liquid‐free zones ("bubbles") – that are the consequence of forces acting upon the liquid.

• It usually occurs when a liquid is subjected to rapid changes of pressure that cause the formation of cavities where the pressure is relatively low.

• Cavitation is transient unsteady phenomenon characterized by a growth of holes or cavities.

• Cavitation creates problem in operation of all three types of centrifugal pumps viz. radial, mixed and axial flow pumps, whenever high discharge, high rotational speed or low head is encountered.

• Pumps with low specific speed are more susceptible to cavitation as compared to high specific speed pumps.



As the vapor bubbles move along the impeller vanes, the pressure around the bubbles begins to increase until a point is reached where the pressure on the outside of the bubble is greater than the pressure inside the bubble. The bubble collapses. The process is not an explosion but rather an implosion (inward bursting). Hundreds of bubbles collapse at approximately the same point on each impeller vane. Bubbles collapse non‐symmetrically such that the surrounding liquid rushes to fill the void forming a liquid microjet. The micro jet subsequently ruptures the bubble with such force that a hammering action occurs. Bubble collapse pressures greater than 1 GPa have been reported. The highly localized hammering effect can pit the pump impeller. After the bubble collapses, a shock wave emanates outward from the point of collapse. This shock wave is what we actually hear and what we call "cavitation".


Dr. B. Rajeevan 5


Cavitation is a significant cause of wear in some engineering

contexts. When entering high pressure areas, cavitation bubbles

that implode on a metal surface cause cyclic stress. This results in

surface fatigue of the metal causing a type of wear also called

"cavitation".

The most common examples of this kind of wear are pump

impellers and bends when a sudden change in the direction of

liquid occurs.

Cavitation is usually divided into two classes of behaviour: inertial

(or transient) cavitation and non‐inertial cavitation.


Inertial cavitation is the process where a void or bubble in a liquid rapidly collapses, producing a shock wave. Inertial cavitation occurs in nature in the strikes of mantis shrimps and pistol shrimps, as well as in the vascular tissues of plants. In man‐made objects, it can occur in control valves, pumps, propellers and impellers. Non inertial cavitation is the process in which a bubble in a fluid is forced to oscillate in size or shape due to some form of energy input, such as an acoustic field. Such cavitation is often employed in ultrasonic cleaning baths and can also be observed in pumps, propellers, etc.


Since the shock waves formed by cavitation are strong enough to

significantly damage moving parts, cavitation is usually an undesirable

phenomenon.

It is specifically avoided in the design of machines such as turbines or

propellers, and eliminating cavitation is a major field in the study of fluid

dynamics.

Cavitation damage on a valve plate for an axial piston hydraulic pump



Cavitation damage to a Francis turbine.

Types of Cavitation

• Traveling Cavitation: As name suggests, this type of cavitation is not a steady one, it moves from place to place within pump.

• Fixed Cavitation: This type of cavitation is fixed at a place and hardly changes its position.

• Vortex Cavitation: Here vortex i.e. circular flow is generated and thereby occurrence of cavitation.

• Vibratory Cavitation.



Dr. B. Rajeevan 6

Effects of Cavitation

Harmful Effects of Cavitation • Cavitation affects the performance of various hydraulic machines like pumps, turbines etc. This reduces their overall efficiency.

• Noise is generated which is unwanted everywhere but in some cases like submarines noise must not be generated as it may create the problem while hiding.

• Drag force increases in cavitation parts. • Due to braking of bubbles shock waves are produced which generates vibrations. Vibrations are damn dangerous at very high speeds.

• Material damage due to erosion.


Beneficial Effects of Cavitation

• Cavitation can be used for agitation and mixing.

• A cavitation noise boomer can be used as sound source for an echo ranging survey of ocean bottom conditions.

• Jet cavitation can be used very effectively for tunneling through rock.


Methods to Avoid Cavitation Damage

• Primary Design.

• Air Injection.

• Cathodic Protection.

• Hydrogen Evolution.

• Corrosion Inhibitors.


CAVITATION

IN PUMPS


Cavitation in Pumps



Dr. B. Rajeevan 7


CAVITATION

IN TURBINES





Dr. B. Rajeevan 8


For more information see the following link:

http://en.wikipedia.org/wiki/Cavitation

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