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Technical Manual on Water Distribution System

Author(s): Jalgaonkar Bhagyashri Ramesh, Vikas Sharma, Mukesh Kumar Mehla, Yadvendra Pal Singh
Abstract:
1Technical Manual On Water Distribution System
Authors
Jalgaonkar Bhagyashri Ramesh
Ph.D. Research Scholar, Department of Soil and Water Engineering,
College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India
Vikas Sharma
Ph.D. Research Scholar, Department of Soil and Water Engineering,
College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India
Mukesh Kumar Mehla
Ph.D. Research Scholar, Department of Soil and Water Engineering,
College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India
Yadvendra Pal Singh
Ph.D. Research Scholar, Department of Soil and Water Engineering,
College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur, Rajasthan, India
Publication Month and Year: December 2019
Pages: 56
E-BOOK ISBN: 978-81-943354-8-1
Academic Publications
C-11, 169, Sector-3, Rohini, Delhi, India
Website: www.publishbookonline.com
Email: publishbookonline@gmail.com
Phone: +91-9999744933
Preface
Water conservation practices vary depending on the use. Agricultural water conservation practices include soil compaction and leveling; diking to prevent runoff and selection of irrigation rates and schedules to minimize evaporative losses.
During the past fifty years, great progress has been made in the construction and operation of water works. Large aqueducts and reservoirs have been built, the efficiency of pumping engines has been improved, and the purification of water has been greatly advanced.
This manual covers all the areas of water related to agriculture. It includes all inclusive practical aspects of geometrical and hydraulic elements of open channel, flow-measuring devices, energy and momentum coefficients, channel design problems, flow control and distribution structures and also flow profiles.
This is anticipated that, as a laboratory manual for students of irrigation and agricultural engineering, it will be useful for students of engineering at undergraduate, postgraduate and Ph.D. students of agricultural engineering. In addition, the manual will be a valuable reference to professional and agricultural scientists working in the field of soil and water conservation and on farm water management.
It is hope that undergraduate, postgraduate and Ph.D. students of agricultural engineering will find this manual quite useful. In preparation of this manual, authors have received help, suggestions and encouragement from several individuals. The authors are also grateful to all teachers and colleagues of the Department of Soil and Water Engineering for their assistance and having useful discussions in preparation of this manual.
In this manual assignment sheets are being provided for every practical exercise so that the concerned teacher may frame some exercises based on practical for proper understanding by the students. It is expected that critical evaluation of assignment by the teacher will be quite useful in better understanding of the subject by the students.
We hope that readers would provide the scope for future alteration and improvement of this manual by their valuable comments and suggestions.
Contents
S. No.ExercisePage No.
1.Computation and use of Geometrical and Hydraulical Elements of Open Channel01-05
Assignment No. 106-06
2.Use of Flow Measuring Devices, Methods and Their Limitations07-19
Assignment No. 220-20
3.Examination of Velocity Distribution and Calculation of Energy and Momentum Coefficients21-29
Assignment No. 330-30
4.Theories and Methods of Open of Channel Design31-41
Assignment No. 442-42
5.Appraisal of Flow Control and Distribution Structures43-46
Assignment No. 547-47
6.Analysis and Computation of Flow Profiles48-51
Assignment No. 652-52
7.References53-53
List of Figures
Fig. No.Title of FiguresPage No.
1.1Elements of an Open Channel01
2.1Typical Cup-Type Flow Current Meter08
2.2Rectangular Weir09
2.3Cipoletti/Trapezoidal Weir10
2.4V-Notch/Triangular Weir10
2.5Parshall Flume11
2.6Plan and Sectional Views of a Parshall Flume12
2.7A Cut-Throat Flume12
2.8Schematic Sketch Illustrating Flow through Free Flow Orifice15
2.9Measuring Head through Submerge16
3.1Typical Curves of Equal Velocity in Various Channel Sections21
3.2Pressure Distribution in Straight and Curved Channels of Small or Horizontal Slope at the Section under Consideration24
3.3Pressure Distribution in Parallel Flow in Channels of Large Slope24
3.4Pressure Distribution in Curvature Flow in Channel of Large Slope25
3.5Energy in Gradually Varied Open-Channel Flow26
3.6Application of the Momentum Principle28
4.1Kennedy's Silt Theory33
4.2Lacey's Regime Theory36
4.3Channel Section According to Lacey's Regime Theory36
4.4Distribution of Tractive Force in a Trapezoidal Channel Section39
5.1Various Components of Drop Spillway43
5.2Sectional view of a Pipe Drop Spillway44
5.3Structural Details of a Chute Spillway45
6.1Schematic Diagram of Flow Profiles48
List of Table
Table No.TablePage No.
1.1Geometric Elements of Channel Sections03
Practical - 1
Computation and use of Geometrical and Hydraulical Elements of Open Channel
Elements of an Open Channel
Fig 1.1: Elements of an Open Channel
T = Top width of the channel
t = Width of surface when the water is at depth d
D = Depth of channel after free board is added
d = Depth of flow in channel
b = Bottom width of channel
c = Wetted sides of channel
θ = Angle between the sloping side and the horizontal
Definitions
Wetted Perimeter
Wetted perimeter is the sum of the lengths of that part of the channel sides and bottom which are in contact with water. It is computed by following formula:
p = b + c + c
Area of Cross Section
Cross section area "a" refers to the area of the wetted section of the channel.
a=(b+t)d/2
Hydraulic Radius
Hydraulic radius "R" is the ratio between the cross section area of the stream and its wetted perimeter.
R=a/p
Hydraulic Slope
Hydraulic slope "s" of a channel is the ratio of its vertical slope "h" for a length "l" of the channel.
s=h/l
The velocity of flow in channels varies directly with the square root of the hydraulic slope (i.e., V ∝ √s).
Free-Board
Free-board is the vertical distance between the highest water level anticipated in the design and the top of the retaining banks.
Angle of Repose
If a mass of loose dry soil is allowed to fall freely to the ground, the soil heap formed by it will attain a conical shape.
Discharge Capacity of Channels
The discharge capacity of channels is obtained from the following formula:
Q = a × v
In which,
Q = Discharge capacity, cubic meters per second
a = Cross sectional area of the wetted section of the channel, square meter
v = Mean velocity of flow in channel, meters per second
Table 1.1: Geometric Elements of Channel Sections
Numericals
A rectangular channel 2.5 m wide carries water at a depth of 1.3 m. The bed slope of the channel is 0.0036. Calculate the average shear stress on the boundary.
Ans.: Area,a = By = 2.5 × 1.3 = 3.25 square meter
Wetted Perimeter, p = B + 2y = 2.5 + (2 × 1.3) = 5.1 m
Hydraulic Radius, R=a/p =3.25/5.1 = 0.6372 m
Average boundary shear stress,
τo = γ R So
∴ τo = (998 × 9.81) × 0.6372 × 0.0036 = 22.46 Pa
2)A trapezoidal channel has a bed width of 2 m and side slope of 1.5 horizontal: 1 vertical. The channel has a longitudinal slope of 1/4000. If the Manning's coefficient of the channel boundary is 0.018, calculate the mean velocity and discharge in the channel for a depth of 1.4 m.
Ans.: Area, a = (B + my) × y
= (2 + 1.5 × 1.4) × 1.4
∴ a = 5.74 square meter
Wetted Perimeter, p = (B + 2 √(m^2+ 1) × y)
= (2 + 2 √(〖1.5〗^2+ 1) × 1.4)
∴ p = 7.048 m
Hydraulic Radius, R=a/p =5.74/7.048 = 0.814 m
Mean velocity, by Manning's formula:
v = 1/( η) R^(2⁄3) 〖 S〗^(1⁄2)
= 1/0.018 〖0.814〗^(2⁄3) 〖 (1⁄4000)〗^(1⁄2)
∴ v = 0.776 m/s
Discharge, Q = a × v
= 5.74 × 0.766
∴ v = 4.40 m^3/s
Design an earth channel to carry a flow of 56 lit/sec in a loamy sand soil. The slope of the channel bed is 0.1 percent.
Ans.: Q = 56 lit/sec = 56/1000 = 0.056 cum/sec
Permissible side slopes in loamy sand soil = 1.5:1
Tan θ = 1/1.5 = 0.666
θ = 35'45'; θ/2 = 16°51'
∴ Tan θ/2 = 0.3029
Assuming the depth to be 25 cm,
b = 2d tan θ/2 = 2 × 25 × 0.3029
∴ b = 15 cm
Top width, T = 1.5 × 25 + 15 + 1.5 × 25 = 90 cm
Area, a = (90+15)/2 × 25 = 1312 sq cm = 0.1312 sq m
The section has insufficient capacity,
Assuming, b = 25 cm, a = 0.156 sq m, p = 1.15 m & R = 0.1358 m
v = 1/(0.0225 ) 〖0.1358〗^(2⁄3) 〖 0.001〗^(1⁄2)
∴ v = 0.372 m^3/s
The velocity is within permissible limits,
Q = a × v = 0.156 × 0.372 = 0.058 cum/sec
∴ Q = 58 lit/sec
The section is suitable.
Thus, the design dimensions are: d = 25 cm, Freeboard = 5 cm (assumed on basis of d), b = 25 cm and side slope = 1.5:1.
What size of a circular drainage pipe is needed to carry 1.10 m^3/s of discharge when flowing half full? The pipe is laid at a slope of 0.0004 and the Manning's η for the pipe can be taken as 0.018.
Ans.: Let, D = Diameter of the pipe,
y = Depth of flow = D/2
Area,a = 1/(2 ) × (πD^2)/(4 ) = (πD^2)/(8 )
Wetted Perimeter,p = πD/(2 )
Hydraulic Radius,R=a/p = (((πD^2)/(8 )))⁄((πD/(2 )) ) = D/4
By Manning's formula:
Discharge,Q = a × 1/( η) R^(2⁄3) 〖 S〗^(1⁄2)
1.10 = ((πD^2)/(8 )) × 1/0.018 × (D/(4 ))^(2⁄3) 〖(0.0004)〗^(1⁄2)
∴ D^(8⁄3) = 6.353
∴ D = 2 m
Assignment No. 1
For a flow in a rectangular channel of width 5 m and depth of flow of 2.3 m the Darcy-Weisbach friction factor is estimated to be 0.02. Estimate the values of Chezy's C and Manning's η.
Water flows at a uniform depth of 2 m in a trapezoidal channel having a bottom width 6 m, side slopes 2 horizontal to 1 vertical. If it has to carry a discharge of 65 m3/s, compute the bottom slope required to be provided.
Practical - 2
Use of Flow Measuring Devices, Methods and Their Limitations
Several devices are used for measuring irrigation water on the farm. They are grouped into four categories:
Volumetric measurements
Velocity-Area methods
Measuring structures (i.e. Orifices, Weirs and Flumes)
Tracer methods (i.e. Dilution and Radioisotope method)
Volumetric Measurements
A simple method of measuring small irrigation streams is to collect the flow in a container of known volume for a measured period of time. The time required to fill the container is reckoned with a stopwatch or the second’s hand of a wristwatch. The rate of flow is measured by the formula:
Discharge rate,lit/sec=(Volume of container,lit)/(Time required to fill,sec)
Velocity-Area Method
The rate of flow passing a point in a pipe or open channel is determined by multiplying the cross sectional area of the flow section at right angles to the direction of flow by the average velocity of water.
Discharge = Area × Velocity
Q = a × v
In which,
Q = Discharge rate, cubic meters per second,
a = Area of cross section of channel or pipe, square meter,
v = Velocity of flow, meters per second.
Float Method
The float method of making a rough estimate of the flow in a channel consists of noting the rate of movement of a floating body. A long necked bottle partly filled with water or a block of wood may be used as the float. A straight section of the channel about 30 m along with fairly uniform cross section is selected. Several measurements of depth and width are made within the trial section to arrive at the average cross sectional area. A string is stretched across each end of the section at right angles to the direction of flow. The flow is placed in the channel, a short distance upstream from the trial section. The time the float needs to pass from the upper to the lower section is recorded. To obtain the rate of flow, this average velocity (measured velocity × coefficient) is multiplied by the average cross sectional area of the stream.
Current Meter Method
Fig 2.1: Typical Cup-Type Flow Current Meter
The velocity of water in a stream or river may be measured directly with a current meter and the discharge estimated by multiplying the mean velocity of water by the area of cross section of the stream. The current meter is a small instrument containing a revolution wheel or vane that is turned by the movement of water. It may be suspended by a cable for measurement in deep stream or attached to a rod in shallow stream. The number of revolution of the wheel in a given time interval is obtained and the corresponding velocity is reckoned from a calibration table or graph of the instrument.
Measuring Structures
In farm irrigation practice, the most commonly used devices for measuring water are weirs, parshall flumes, orifices and meter gates.
Weirs
Weirs are used to measure the flow in an irrigation channel, or the discharge of a well or canal outlet at the source. The basic formula for calculating discharge through a weir is:
Q = CLHm
In which,
Q = Discharge capacity, cubic meters per second
C = A coefficient dependent on the nature of the crest and approach conditions
L = Length of crest
H = Head of crest
m = An exponent, depending upon the weir opening
The value of C and m are determined by carefully conducted calibration tests. Weirs are of following types:
Rectangular Weir
Fig 2.2: Rectangular Weir
The rectangular weir takes its name from the shape of the notch. They are used to measure comparatively large discharges. It has a horizontal crest and vertical sides. They may be either contracted rectangular weirs. They have sharp crest and are beveled on the downstream side only. The sides are not beveled. The discharge through rectangular weirs may be computed by the Franci's formula stated below:
Suspended Rectangular Weir
Q = 0.0184 LH3/2
In which,
Q = Discharge capacity, cubic meters per second
L = Length of crest, cm
H = Head over the weir, cm
Contracted Rectangular Weir (With end Contractions at Both Sides)
Q = 0.0184 (L - 0.2H) H3/2
Cipoletti/Trapezoidal Weir
Fig 2.3: Cipoletti/Trapezoidal Weir
The Cipoletti weir is a contracted trapezoidal weir, in which each side of the notch has a slope of 1 horizontal to 4 vertical. It is named after its inventor Cesare Cipoletti, an Italian engineer. The weir has sharp crest and sharp sides, beveled from the downstream side only. It is commonly used to measure medium discharges. The discharge through a Cipoletti weir is computed by the following formula:
Q = 0.0186 LH3/2
In which,
Q = Discharge capacity, cubic meters per second,
L = Length of crest, cm,
H = Head over the crest, cm.
V-Notch/Triangular Weir
Fig 2.4: V-Notch/Triangular Weir
The 90o V-notch weir is commonly used to measure small and medium size stream. The advantages of the V-notch weir are its ability to measure small flows accurately. It has both its sides sharp, beveled from the downstream side only.
The discharge can be computed by:
Q = 0.0138 H5/2
In which,
Q = Discharge capacity, cubic meters per second
H = Head, cm
For heads lower than 5 cm, the weir should preferably be calibrated to obtain the discharge.
Flumes
Parshall Flume
Fig 2.5: Parshall Flume
Fig 2.6: Plan and Sectional Views of a Parshall Flume
The Parshall flume is an open channel type-measuring device that operates with a small drop in head, adopting the venturi principle. The loss of head for free flow limit is only about 25% of that for weir. It is a self-cleaning device. Sand or silt in the flowing water does not affect its operation or accuracy. A Parshall flume consists of three principal sections:
A converging or contracting section at the upstream end leading to
A constricted section or throat
A diverging or expanding section downstream
Parshall flumes allow reasonably accurate measurement even when partially submerged. The velocity of the approaching stream has very little influence on its operation. Discharge through the flume can occur under either free flow or submerged flow condition.
Cut-Throat Flume
Fig 2.7: A Cut-Throat Flume
The flume has a flat bottom, vertical walls and a zero length through section. Since it has no throat section, it was given the name 'Cut-Throat' by the developers. The major advantage of a cut-throat flume is economy, due to simplicity in condition. Any flume length between 45 cm to 3 m can be used, while throat widths between 2.5 cm to 1.8 m have been investigated. The cut-throat flume can operate as either a free flow or a submerged flow structure. Under free flow conditions, critical depth occurs near minimum width (w), which is called the flume throat or the flume neck. The relationship between flow rate Q and upstream depth of flow ha. In a cut-throat flume under free flow conditions is given by the following experimental relationship:
Q = C1 han L
In which,
Q =Flow rate
C1 =Free flow coefficient, which is the value of Q when ha is 1.0 foot, which is the slope of the free flow rating curve when plotted on logarithmic
n =Exponent, whose value depends only on the flume length (L)
The value of n is a constant for all cut-throat flumes of the same length, regardless of the throat width w.
Submerged Flow Analysis
When the flow conditions are such that the downstream flow depth (hb), is raised to the extent that the flow depths at every point through the structure become greater than the critical depth, resulting in a change in upstream depth, the flume is operating under submerged flow condition and requires that the two flow depths be measured, on upstream (ha) and the other downstream (hb) from the flume neck. The submergence (S), is defined as the ratio often expressed as a percentage of the downstream depth to the upstream depth.
S = hb/ha
Broad-Crested Rectangular Weirs
The following are the basic components of weirs and flumes used in earth channels: entrance to approach channel, a converging transition section, throat, a diverging transition section and stilling basin. A suitable riprap section (usually of stone paving) is provided in the downstream channel section to prevent the erosion of channel bed and sides.
Truncated Flumes
Shortening of the full length structure is possible by deleting the diverging section and the tail water section if the head loss over the section exceeds 0.4 times the head causing the flow, as measured from the crest of the throat section. Such a structure is called a truncated flume. Discharge can be calculated by following formula:
Q = bq
In which,
Q = Discharge over the weir, cubic meters per second,
b = Width of the weir, m
q = Discharge per meter width of the weir, square m/ sec.
Triangular Throated Flumes
Triangular throated flumes are adapted to measure a wide range of discharge, including low flows like return flows to drainage system and operational spillage from irrigation systems. It also permits wide variations in the rate of flow. The ratio between the maximum and minimum flow (Qmax/Qmin) is high in triangular throated flumes. The drop in water surface, required to obtain a constant discharge of water passing over the flume structure (modular structure), irrespective of the water levels in the upstream and downstream of the channel, is as low as 10% of the design head of flow over the structure.
Portable Flumes for Earthen Channels
Portable long-throated flumes are designed to measure comparatively low flows such as those of irrigation furrows and on-farm surface water distribution systems when a permanent point for measurement is not required. They are made of sheet metal with suitable bracing.
Orifices
Orifices in open channels are usually circular or rectangular openings in a vertical bulkhead through which water flows. The edges of the openings are sharp and often constructed of metal. The cross section area of the orifice is small in relation to stream cross section. These conditions allow complete contraction of the stream flow and the velocity of approach becomes negligible.
Free Flow Orifices
Fig 2.8: Schematic Sketch Illustrating Flow through Free Flow Orifice
Free flow orifice plates can be used to measure comparatively small streams like the flow into border strips, furrows or check basins. They consist of sheet iron, steel or aluminum plates that contain accurately machined circular openings or orifices usually ranging in diameter from 2.5cm to 7.5cm. The edge of the orifice need not be sharpened in case of thin walled orifice plates. A plastic scale may be fixed directly on the upstream face of the orifice plate with its zero reading coinciding with the centre of the orifice. The discharge through an orifice is calculated by the formula:
Q = 0.16 × 10-3 a √2gH
In which,
Q = Discharge through orifice, lit/sec
A = Area of cross section of the orifice, square centimeter
g = Acceleration due to gravity, cm/sec2
H =Depth of water over the centre of the orifice (on the upstream side) in case of free flow orifice, or the difference in elevation between the water surface at the upstream and downstream faces of the orifice plate in case of submerged orifice, cm
Submerged Orifices
Submerged orifices may be of two types:
Those having orifices of fixed dimensions
Those in which the height of opening may be varied
A standard submerged orifice has fixed dimensions. The opening is sharp edged and usually rectangular, with the width being 2 to 6 times the height. The adjustable submerged orifice is one in which the height of opening and head may be varied to suit the requirements.
Fig 2.9: Measuring Head through Submerge
Meter Gate
A meter gate is basically a modified submerged orifice so arranged that the orifice is adjustable in area. It is manufactured commercially and are used to control the water flowing from one channel to another. They may serve to measure the rate of flow, if the head and area of opening can be determined and the gate has been calibrated. Meter gates are commonly used by canal irrigation agencies. Normally the gate and the opening into the gate are fully open.
Tracer Method
The tracer methods of water measurement are independent of stream cross section and are suitable for field measurements without installing fixed structures. In this methods, a substance (tracer) is concentrated from is introduced into the flowing water and allowed to thoroughly mix with it. The concentration of the tracer is measured at a downstream section. Since only the quality of water necessary to accomplish the dilution is involved. There is no need to measure velocity, depth, head, cross sectional area or any other hydraulic factors usually considered in discharge measurement.
Dilution Method
In the dilution method of flow measurement, a relatively large quantity of chemical or dye, called a tracer is dissolved in a small quantity of water and placed in a bottle so that the tracer solution can be discharged at a known rate into the water flowing in a channel or pipe. The concentration given here (Weight of tracer / Weight of water) of tracer in the bottle. The rate of injection is q1. To account for any tracer which might already be in the upstream flow, the original tracer concentration may be noted as C0. The discharge at the downstream is the discharge Q at the upstream plus q1 which is the quantity of tracer added. The concentration of tracer at the downstream station may be designed as C2. The above relationship may be expressed mathematically as follows:
QC0 + q1C1 = (Q + q1) C2 Or QC0 + q1C1 = QC2+ q1C2
Rearranged the terms,
QC0 - QC = q1C2 - q1C1
Simplifying,
Q (C0 - C2) = q1 (C2 - C1) Or Q = (〖 C〗_2 -〖 C〗_1)/(〖 C〗_0 - 〖 C〗_2 )
Salt concentration, as such, is difficult to measure directly.
Radioisotope Method
Radioisotopes may be used in place of chemical ore dye tracer and the degree of dilution determined by counting the gamma ray emission from the diluted isotope solution (the downstream flow) using Geiger counters or scintillation counters. In the 'pulse' or 'total' count method, a known amount of radioisotope is introduced into the flow in a relatively short time. At the measuring station downstream, where the isotope is thoroughly mixed, the concentration of the radioisotope tracer is determine from the gamma ray emission detected and counted by the counter.
Q = FA/N
In which,
Q = Volume of water flowing in the channel per unit time
F = Counts per unit of radioactivity per unit volume of water per unit of time
A = Total units of radioactivity to be introduced for each discharge measurement
N = Total counts
Numericals
Using Franci's formula, compute the discharge of a rectangular weir 45 cm long with a head of 12 cm, under the following conditions:
With no end contraction
With one end contraction
With two end contraction
Ans.:
With no end contraction
Q = 0.0184 LH3/2
= 0.0184 × 45 × 123/2
∴ Q = 34.4 lit/sec
With one end contraction
Q = 0.0184 (L - 0.1H) H3/2
= 0.0184 (45 - 0.1 × 12) 123/2
∴ Q = 33.5 lit/sec
With two end contraction
Q = 0.0184 (L - 0.2H) H3/2
= 0.0184 (45 - 0.2 × 12) 123/2
∴ Q = 32.6 lit/sec
The discharge rate of a shallow tube well was estimated by collecting the pumped water in a drum and noting the average time required to fill the drum. The measured time intervals, required to fill the drum to its 200 liters capacity mark were 27 sec, 28 sec, 27.5 sec. Estimate the discharge rate of the well.
Ans.:
Average value of filling time = (27+28+27.5)/3 = 27.5 sec
Discharge rate = 200/27.5 = 7.25 lit/sec
= (7.25 × 60 × 60)/1000
∴ Discharge rate = 26.1 cumec/hr
A rectangular notch has a discharge of 21.5 cubic m/min, when the head of water is half the length of the notch. Assume Cd = 0.6.
Ans.:
Q = 21.5 cubic m/min = 21.5/60 = 0.358 cubic m/sec
H = b/2 = 0.5b and Cd = 0.6
The discharge over the rectangular notch (Q)
0.358 = 2/3 Cd b √2g (H)3/2
= 2/3 × 0.6 × b √(2 ×9.81) (0.5 b)^(3/2)
0.358 = 0.626 b5/2
b5/2 = 0.358/0.626 = 0.572 or
b = 0.8 m
During an experiment in a laboratory, 280 lit of water flowing over a right-angled notch were collected in one min. If the head of water over the still is 100 mm, calculate the coefficient of discharge of the notch.
Ans.:
Q = 280 lit/min = 0.28 cumec/min = 0.0047 cumec/sec
H = 100 mm = 0.1 m
The discharge over the triangular notch (Q) =
0.0047 = 8/15 Cd √2g tan θ/2 H5/2
= 8/15 Cd √(2 ×981) tan 45° × (0.1)5/2
0.0047 = 0.0075 Cd
Cd = 0.0047/0.0075
∴ Cd = 0.627
Assignment No. 2
A rectangular weir of 4.5 m long has a 300 mm head of weir. Determine the discharge over the weir, if coefficient of discharge is 0.6.
A weir of 8 m long is to be built across a rectangular channel to discharge a flow of 9 m3/s. If the maximum depth of water on the upstream side of the weir is to be 2 m, what should be the height of the weir? Adopt Cd = 0.62.
A 30 m long weir is divided into 10 equal bays by vertical posts each of 0.6 m width. Using Franci's formula, calculate the discharge over the weir under an effective head of 1m.
Practical - 3
Examination of Velocity Distribution and Calculation of Energy and Momentum Coefficients
Velocity Distribution in Channel Section
Owing to the presence of a free surface and to the friction along the channel wall, the velocities in a channel are not uniformly distributed in the channel section. The measured maximum velocity in ordinary channels usually appears to occur below depth; the closer to the banks, the deeper is the maximum. In a broad, rapid and shallow stream or in a very smooth channel, the maximum velocity may often because the curvature of the vertical-velocity-distribution curve to increase. A surface wind has very little effect on velocity distribution.
Fig 3.1: Typical Curves of Equal Velocity in Various Channel Sections
Wide Open Channel
Under the condition, the sides of the channel have practically no influence on the velocity distribution in the central region and the flow in the central region can therefore be regarded as two-dimensional in hydraulic analyses. The central region exists in rectangular channels only when the width is greater than 5 to 10 times the depth of flow, depending on the condition of surface roughness. Thus, wide-open channels can safely the defined as a rectangular channel whose width is greater than 10 times the depth of flow.
Measurement of Velocity
The channel cross sections is divided into vertical strips by a number of successive verticals and mean velocities in verticals are determined by measuring the velocity at 0.6 of the depth in each vertical or where more reliable results are required by taking the average of the velocities of 0.2 and 0.8 of the depth. When the stream is covered with ice, the mean velocity is no longer close to 0.6 of the water depth still gives reliable results. The average of the mean velocities in any two adjacent verticals multiplied by the area between the verticals gives the discharge.
Velocity-Distribution Coefficients
Because of non-uniform distribution of velocities over a channel section, the velocity head of an open channel flow is generally greater than the value computed according to the expression v2/2g, where v is the mean velocity. When the energy principle is used in computation, the true velocity head may be expressed as α v2/2g, where α is known as the energy coefficient or Coriolis coefficient, in honor of G. From the principle of mechanics, the momentum of the fluid passing through a channel section per unit time is expressed by βωQv/g, where β is known as the momentum coefficient or Boussinesq coefficient, after J. Boussinesq who first proposed it, ω is the unit weight of water; Q is the discharge and v is the mean velocity.
Determination of Velocity-Distribution Coefficients
Let, ∆A be an elementary area in the whole water area A and ω the unit weight of water; then the weight of water passing ∆A per unit time with a velocity v is ωv∆A. The kinetic energy of water passing ∆A per unit time is ωv3∆A/2g. This is equivalent to the product of the weight ωv∆A and the velocity head v2/2g. The total kinetic energy for the whole area as A, the mean velocity as v and the corrected velocity head for the whole area as αv2/2g, the total kinetic energy is ∑v2A/2g. Equating this quantity with ∑ωv3∆A/2g and reducing we get,
α = (∫v^3 dA)/(v^3 A) ≈ (∑v^3 ∆A)/(v^3 A)
The momentum of water passing ∆A per unit time is the product of the mass ωv∆A/g and the velocity v or ωv2∆A/g. The total momentum is ∑ωv2∆A/g. Equating this quantity with the corrected momentum for the whole area or βωAv2/g and reducing we get,
β = (∫v^2 dA)/(v^2 A) ≈ (∑v^2 ∆A)/(v^2 A)
The energy and momentum coefficients can be computed by the following formulas:
α = 1 + 3 ᗴ2 – 2 ᗴ3
β = 1 + ᗴ2
Where,
ᗴ = VM/ v-1
VM = The maximum velocity
v = The mean velocity
Pressure Distribution in a Channel Section
The pressure at any point of the cross section of the flow in a channel of small slope can be measured by the height of the water column in a piezometer tube installed at the point. The pressure at any point on the section is directly proportional to the point below the free surface and equal to the hydrostatic pressure corresponding to this depth. In other words, the distribution of pressure over the cross section of the channel is the same as the distribution of hydrostatic pressure; that is the distribution is linear and can be represented by a straight line AB. The application of the hydrostatic law to the pressure if the flow filaments have no acceleration components in the plane of cross section. This type of flow is theoretically known as parallel flow that is such that the streamlines have neither substantial curvature nor divergence. Pressure coefficient is expressed by
α' = 1/Qy ∫_0^A▒hvdA = 1 + 1/Qy ∫_0^A▒cvdA
Where, Q is the total discharge and y is the depth of flow. It can easily be seen that α' is greater than 1.0 for concave flow, less than 1.0 for convex flow and equal to 1.0 for parallel flow.
Fig 3.2: Pressure distribution in straight and curved channels of small or horizontal slope at the section under consideration. h = Piezometric head; hs = Hydrostatic head; and c = Pressure head correction for curvature (a) Parallel flow; (b) Convex flow; (c) Concave flow
Effect of Slope on Pressure Distribution
With reference to a straight sloping channel of unit width and slope angle θ, the weight of the shaded water element of length dL is equal to ωycosθdL. The pressure due to this weight is ωycos2θdL. The unit pressure is equal to ωycos2θ and the head is
h = ycos2θ or h = dcosθ
Where,
d = ycosθ, the depth measured perpendicularly from the water surface.
Fig 3.3: Pressure Distribution in Parallel Flow in Channels of Large Slope
Fig 3.4: Pressure Distribution in Curvature Flow in Channel of Large Slope
Energy in Open Channel Flow
It is known as elementary hydraulics that the total energy in foot-pounds per pound of water in any streamline passing through a channel section may be expressed as the total head in feet of water, which is equal to the sum of the elevation above a datum, the pressure head, and the velocity head. A streamline of flow in a channel of large slope may be written as:
H = ZA + dAcosθ + (V_A^2)/2g
Where,
ZA = Elevation of point A above the datum plane
dA = Depth of point A below the water surface measured along the channel section
θ = Slope angle of the channel bottom
(V_A^2)/2g = Velocity head of the flow in the streamline passing through A.
The total energy at the channel section is,
H = Z + dcosθ + α V^2/2g
For channels of small slope, θ ≈ 0. Thus, the total energy at the channel section is
H = Z + d + α V^2/2g
According to the principle of conservation of energy, the total energy head at the upstream section 1 should be equal to the total energy head at the downstream section 2 plus the loss of energy hf between the two sections; or
Z1 + d1cosθ + α1 (V_1^2)/2g = Z2 + d2cosθ + α2 (V_2^2)/2g + hf
The equation applies to parallel or gradually varied flow for a channel of small slope, it becomes
Z1 + y1 + α1 (V_1^2)/2g = Z2 + y2 + α2 (V_2^2)/2g + hf
These two equation two equation is known as the energy equation when α1 = α2 = 1 and hf = 0. Equation becomes:
Z1 + y1 + (V_1^2)/2g = Z2 + y2 + (V_2^2)/2g = Constant
This is the well-known Bernoulli's equation.
Fig 3.5: Energy in Gradually Varied Open-Channel Flow
Momentum in Open Channel Flow
The momentum of the flow passing a channel section per unit time is expressed by βωQv/g, where β is the momentum coefficient, ω is the unit weight of water in lb/ft3, Q is the discharge in cfs and v is the mean velocity in fps. According to Newton's second low of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the resultant of all the external forces that are acting on the body.
Qω/g (β_2 V_2- β_1 V_1 )= P_1-P_2+ wsinθ- F_f
Where,
Q, ω and V = as previously defined, with subscripts referring to sections 1 and 2, P_1 and P_2 = Resultant of pressures acting on the two sections,
w = Weight of water enclosed between the sections,
F_f = Total external force of friction and resistance acting along the surface of contact between the water and the channel.
The above equation is known as the momentum equation force coefficient and expressed by,
β' = 1/(AZ ̅ ) ∫_0^A▒hdA = 1 + 1/(AZ ̅ ) ∫_0^A▒cdA
Where,
Z ̅ = Depth of the centroid of the water area A below the free surface
h = Pressure head on the elementary area dA
c = Pressure head correction
It can easily be seen that β' is greater than 1.0 for concave flow, less than 1.0 for convex flow and equal to 1.0 for parallel flow.
P_1=1/2 ωby_1^2 and P_2=1/2 ωby_2^2
Assume, F_f = ωh_f^' by ̅
Where,
h_f^' = Friction head and y ̅ is the average depth or (y1 + y2)/2.
The discharge through the reach may be taken as the product of the average velocity and the average area or
Q = 1/2 (V_1- V_2 )by ̅
In addition, it is evident that the weight of the body of water is
W = ωby ̅L and sinθ = (Z_1- Z_2)/L
Substituting all the above expressions for the corresponding items in equation and simplifying,
Z1 + y1 + β1 (V_1^2)/2g = Z2 + y2 + β2 (V_2^2)/2g + hf'
The item hf measures the internal energy dissipated in the whole mass of the water in the reach, whereas the item hf' in the momentum equation measures the losses due to external forces exerted on the water by the walls of the channel.
Fig 3.6: Application of the Momentum Principle
Numericals
Velocity distribution in a rectangular channel of breadth B and depth yo was approximated as v = k√y, where k is a constant. Compute the average velocity v, α and β of the channel.
Ans.: We have the equation of average velocity in a rectangular channel, i.e.
v = 1/y_0 ∫_0^(y_0)▒vdy = 1/y_0 ∫_0^(y_0)▒〖ky^(1/2) dy〗
v = k/y_0 2/3 y^(3/2) |■(y_0@0)┤ = 2/3 ky_0^(1/2)
Now, α = 1/y_0 ∫_0^(y_0)▒〖( (ky^(1/2) )^3)/v^3 dy〗 = 1/y_0 ∫_0^(y_0)▒〖( (ky^(1/2) )^3)/(2/3 ky^(1/2) )^3 dy〗
Simplifying, α = 1.35
β = 1/y_0 ∫_0^(y_0)▒〖( ky^(1/2))/(2/3 ky_0^(1/2) ) dy〗
Simplifying, β = 1.25
In the measurement of discharge in a river, it was found that the depth increases at a rate of 0.5 m per hour. If the discharge at that section is 15 m3/sec and the surface width of river is 15m, estimate the discharge at a section 1.2 km upstream.
Ans.: This is an example of unsteady open channel flow in 1 D. Continuity equation is:
∂Q/∂Z+ T ∂y/∂t=0 or (Q_2-〖 Q〗_1)/∂x=- T ∂y/∂t
Q_2-〖 Q〗_1=(- T ∂y/∂t) ∂x
Q_1=Q_2+ T (∂y/∂t) ∂x
Q_1=15+15 (0.5/(60 ×60)) ×(1.2 ×1000)
∴Q_1=(15+2.5)
∴Q_1=17.5 m3/Sec
Assignment No. 3
The velocity distribution in a rectangular channel of 2 m depth is given by v = (y/y_0 )^(1/2). Find α and β.
Practical - 4
Theories and Methods of Open of Channel Design
Channel is one of the most important components of hydraulic structures in the field of water resources engineering used to convey water for irrigation, power production, water supply, storm disposal and some other important purpose.
Types of Channel to be Designs
There are two methods of channel to be designed are:
Non-Erodible or Rigid Boundary Channels
Erodible or Mobile Boundary Channels
Roughness coefficient plays an important role in the design of channel shape to carry the design discharge. The roughness or resistance coefficient is dependent on many factors like surface roughness, flow conditions, vegetation cover, channel irregularity, channel alignment, silting, scouring, obstruction to flow and depth flow.
Design of Non-Erodible or Rigid Boundary Channels
The design this type of channel, Manning's and Chezy's equation are used. Based on these two uniform flow formulae, its relationship with discharge, materials forming the bed and sides of the channel, amount of silt or sediment transported by the flowing water without deposition. The following two methods are used to design this type of channel:
Method of Best Hydraulic Section or Economic Section
The continuity equation is: Q = a × v
Using Manning's equation for average velocity v
Q = A/( η) 〖(A/( P))^(2⁄3) S〗_b^(1⁄2)
P = (〖 S〗_b^(3⁄4))/( 〖η Q〗_ ^(3⁄2) ) A5/2
P = K A5/2
The wetted perimeter P is minimum when area A is minimum, so cost of excavation and the lining used to prevent seepage and erosion will be minimum. Thus, the design of channel considering P to be minimum is called method of economic section.
Method of Permissible Velocity
The permissible velocity for clear water varies from 0.45 m/s for fine sand to 1.52 m/s for cobbles and shingles and water transporting colloidal silts varies from 0.752 m/s to 1.68 m/s. Therefore, it is difficult to select the exact value.
Steps Required for Design by Permissible Velocity
Assume a permissible velocity within the limit of 0.61 m/s to 0.91 m/s. If smaller silt is to be transported by water in the canal with lining, it is better to assume higher side of permissible velocity. This will also increase the discharge.
Fix the bed slope based on topography of the field.
Select Manning's η based on roughness of the bed from Chow's table, if the Nikuradse's equivalent sand roughness size ks are known from table or curve for type of surface of the channel.
Side slope z H: 1 V for trapezoidal (or rectangular) may be assumed. 1 H: 1 V is better.
Q, z, Sb and v are known, if y is the design depth and B is the bottom width are area of a trapezoidal channel is A = (B + zy) y (z = 0 for rectangular channel, B = 0 for triangular channel).
By continuity, A =Q/(V ) i.e. (B + zy) y = Q/(V ).
Using Manning's equation, v = 1/( η) 〖(A/( P))^(2⁄3) S〗_b^(1⁄2)
i.e. v = 1/( η) 〖(((B + zy) y)/( B + zy √(1+ z^2 ) ))^(2⁄3) S〗_b^(1⁄2)
Solve for two unknowns B and y given in above equation.
Provide Free Board (FB) at least 5% of depth for low Q and 30% of depth for higher Q or select in between these two.
Design of Erodible or Mobile Boundary Channels
The flow in erodible channel is influenced by many physical factors. Therefore, precise design of such channel is difficult due to complexity of physical factors and filed conditions. The stability of erodible channel, which governs the design is dependent on properties of material forming the channel body than the hydraulic principles of flow in the channel.
Design by Regime Approach
A channel in which neither silting nor scouring takes place is called regime channel or stable channel. This stable channel is said to be in a state of regime if the flow is such that silting and scouring need no special attention. The basis of designing such an ideal channel is that whatever silt has entered the channel at its canal head, it is always kept in suspension and not allowed to settle anywhere along its course.
Kennedy's Silt Theory
Fig 4.1: Kennedy's Silt Theory
The theory says that, the silt carried by flowing water in a channel is kept in suspension by the eddy current rising to the surface.
Assumptions Regarding Kennedy’s Silt Theory
The eddy current is generated because of friction between flowing water and the roughness of the canal bed
The quality of the suspended silt is proportional to bed width
The theory is applicable to those channels, which are flowing through the bed consisting of sandy silt or same grade of silt
Critical Velocity Based on Kennedy’s Silt Theory
Critical velocity is the mean velocity, which will just make the channel free from silting and scouring. The velocity is based on the depth of the water in the channel. The general form of critical velocity is as follow:
Vo = CDn
Where,
Vo = Critical velocity,
D = Full supply depth,
C & n = Constants which found to be 0.546 and 0.64 respectively.
Thus, Equation rewritten as follows:
Moreover, this equation further improved upon realization that silt grade influences critical velocity. Therefore, a factor termed as critical velocity ratio introduced and the equation became as follows:
Where,
M = Critical velocity ratio which equal to actual velocity (V) divided by critical velocity (Vo).
Limitations of Kennedy’s Silt Theory
Trial and error method used for the canal design using Kennedy’s Silt Theory
There is no equation for bed slope assessment, so the equation developed by Kutter used to compute bed slope
The ratio of channel width (B) to its depth (D) has no significance in Kennedy’s Silt Theory
There is not perfect definition for salt grade and salt charge
Complex phenomenon of silt transportation is not fully accounted and only critical velocity ratio (m) concept is considered sufficient
Lindley's Regime Theory
Lindley (1919) defined the regime concept as the dimensions, width, depth and gradient of a channel to carry a given supply (of water) loaded with a given silt that were all fixed by nature.
Vc = 0.57 y0.57 and Vc = 0.27 B0.377
The equation will be:
0.57 y0.57 = 0.27 B0.377
y=(0.27/0.57)^(1/0.57) 〖(B^0.377 ) 〗^(1/0.57)
y = 0.27 B^0.66
Ranga Raju and Misri's Simplified Design
Kennedy's method is tedious. They replaced Ganguillet-Kutters by Manning's equation:
v = 1/η 〖R^(2⁄3) S〗_b^(1⁄2)
and Kennedy's equation, V = 0.55 mr y0.64
This two equation provide for determination of unknowns y, B, Sb if Q, mr, η and z are known. Side slope of trapezoidal channel in this method is usually taken as 1/2 H : 1 V. Then
A = By + 0.5 y2
P = B + 2y √(1+ 〖0.5〗^2 ) = B + 2y √1.25 = B + 2.236 y
R=A/P = (By + 0.5 y2)/(B + 2.236 y) = (y (B_N + 0.5) )/(B_N + 2.236 )
Assuming non-dimension breadth, BN = B/y
A = y2 (B/y + 0.5) = y2 (BN + 0.5)
v =Q/(A ) = Q/(〖y 〗^2 (B_N + 0.5) )
Manning's equation becomes, S_b^(1⁄2) = V_n/( R^(2⁄3) ) = Q_n/( 〖y 〗^2 (B_N + 0.5) R^(2⁄3) )
Substituting R in equation get, Sb = (Q^2 n^2 〖〖(B〗_N + 2.236)〗^(4/3))/( 〖y 〗^(16/3) (B_N + 0.5)^(10⁄3) )
Q/(〖y 〗^2 (B_N + 0.5) )=0.55 〖y 〗^0.64 m_r
y = ((1.818 Q)/(m_r (B_N + 0.5) ))^0.378
Eliminating y from equation we get (Q^0.02 S_b)/(〖m_r〗^2 η^2 ) is a function of non-dimensional breadth (B/y).
Steps Required in Design by Ranga Raju and Misri's
For a given discharge, η and mr, a suitable value of Sb is chosen based on topography
Compute (Q^0.02 S_b)/(〖m_r〗^2 η^2 ). The value of mr and η are taken based on experience. Commonly adopted value of mr is 0.9 to 1.1 and η from 0.02 to 0.025
Read the corresponding value of BN = B/y from the abscissa. Check where obtained B/y is close to the value. If not very close, adjust Sb
Calculate y from equation
Find B from the equation B = BN y
This method is much simpler than the method based on Garret's diagram and results are more accurate in the sense that the method is free from interpolation errors.
Lacey's Regime Theory
Fig 4.2: Lacey's Regime Theory
Fig 4.3: Channel Section According to Lacey's Regime Theory
In this theory, Lacey described in detail concept of regime conditions and rugosity coefficient. It may be seen that for a channel to achieve regime condition following three conditions have to be fulfilled:
Channel should flow uniformly in “incoherent unlimited alluvium” of same character as that transported by the water
Silt grade and silt charge should be constant
Discharge should be constant
Lacey's Regime Equation
On the basis of arguments, Lacey plotted a graph between regime velocity (V) and hydraulic mean radius (R) and gave the relationship:
V= K.R1/2
Where,
K = Constant.
It can be seen here that power of R is a fixed number and needs no alteration to suit different conditions. Lacey recognized the importance of silt grade in the problem and introduced a concept of function ‘f’ known as silt factor. He adjusted the values of such that it also came under square root sign. Thus, it gave scalar conception. Above equation is thus modified as
V = K×√(f×R)
The three fundamental equations are: V = 0.639 ×√(f×R)
Where,
V = Regime velocity in m/sec.
Af2 = 141.2 V5
V = 10.8 R2/3 S1/3
Where,
S = Slope of water surface.
These four equations are the Lacey's basic regime flow equation.
Steps Required in Design by Lacey’s Theory
Values of discharge Q, sand size of d in mm, side slope z H: 1 (if not given assume 1/2 H: 1 V or 1 H: 1 V) or angle of repose of soil to reduce earth pressure are given or known:
From the known sediment size d in mm, find silt factor by equation F = 1.76 √d
Find the velocity Vc from known Q and F from equation Vc = (QF2/140)1/6
Find the area from continuity equation, A =Q/V_c
Now, A = (B + zy) y when channel is trapezoidal
P=2y √(1+ z^2+ B) = 4.75 √Q
Solve for two unknowns B and y given in above equation
Find bed slope Sb by equation S_b=F^(5⁄3)/( 3450 Q^(1⁄6) )
Limitations of Lacey’s Theory
The Lacey’s work is based on field observations and empirically derived equations and therefore it cannot be said to be theory in strict sense
Regime equations in their derived from cannot be applied universally as they hold good mostly for the regions whose data was taken for study
Like Kennedy’s theory, even though perfect definition of silt grade and silt charge is not given most of the equations are based on the silt factor ‘f’
In practice regime condition stated by Lacey is very rarely achieved and that too after a long period
The field observations have shown limited acceptance of the concept of semi-elliptical section of a regime channel
Complex phenomenon of sediment concentration and transport has not been scientifically considered
The Tractive Force Method
Assume a control volume of area A, wetted perimeter P and distance (channel length) Δx, with channel head drop ΔH: Assume a channel slope So then the equation becomes:
So = ΔH/Δx = tan θ
The total weight of the water in the control volume (gravitational force) is: W = γ A Δx
The tractive force is: Ft = W sinθ ≅ W tan θ = γ A Δx So
The resisting force is: Fr = τo P Δx
Equating tractive and resisting force leads to: τo = γ (A/P) So = γ R So
For hydraulically wide channels (R ≅ y): τo = γ y So
The distribution of the tractive stress varies along the wetted perimeter. At the channel bottom, the maximum tractive force is the full value: γySo. At the channel sides, the maximum tractive force is only: 0.78 γySo.
Fig 4.4: Distribution of Tractive Force in a Trapezoidal Channel Section
Tractive Force Ratio
Two forces are acting on a soil particle of submerged weight Ws resting on the side of a channel, with tractive stress τs and angle of inclination of the slope φ: The tractive force: aτs (a = effective area of the particle). The gravity force component: Ws sinφ. The resultant acting force is:
Fa = (Ws2sin2φ + a2τs2)1/2
When this acting force is large enough, the particle will move. At equilibrium, the resistance force is equal to the acting force. The resistance force is equal to the normal force Ws cosφ times the coefficient of friction tanθ, where θ is the angle of friction of the material: The resisting force is:
Fr = Ws cosφ tanθ
Therefore, Ws cosφ tanθ = (Ws2sin2φ + a2τs2)1/2
Squaring both sides: Ws2 cos2φ tan2θ = Ws2sin2φ + a2τs2
a2τs2 = Ws2 cos2φ tan2θ - Ws2sin2φ
τs2 = (Ws2/a2) cos2φ tan2θ [1 - (tan2φ/ tan2θ)]
τs = (Ws/a) cosφ tanθ [1 - (tan2φ/ tan2θ)]1/2
This is the shear stress on the sides of a channel with side slope angle φ. The force balance on a level surface, with φ = 0, is (cosφ = 1; sinφ = 0): Ws tanθ = a τL
Therefore, the shear stress that causes impending motion on a level surface is: τL = (Ws/a) tanθ
The tractive force ratio is defined as: K = τs/τL
K = cosφ [1 - (tan2φ/ tan2θ)]1/2
This ratio K is only a function of φ and θ. Simplifying:
K = {cos2φ [1 - (tan2φ/ tan2θ)]}1/2
K = [cos2φ - (cos2φ tan2φ/ tan2θ)]1/2
K = [cos2φ - (sin2φ/ tan2θ)]1/2
K = [cos2φ - (sin2φ cos2θ /sin2θ)]1/2
K = {(1 - sin2φ) - [sin2φ (1 - sin2θ) /sin2θ]}1/2
K = {(1 - sin2φ) - [(1 - sin2θ) (sin2φ /sin2θ)]}1/2
K = {(1 - sin2φ) - [(sin2φ /sin2θ) - sin2φ]}1/2
K = [1 - (sin2φ /sin2θ)]1/2
Values of the angle of repose of non-cohesive material (sand). The diameter is that of a particle for which 25% (by weight) of the material is larger.
Numericals
Assume an earth channel on a grade of 0.10%, depth of water 40 cm, bottom width 60 cm and side slope 1.5 to 1. Calculate the velocity of flow and carrying capacity of the channel.
Ans.: Side slope = 1.5 to 1 for flow depth of 40 cm.
Length of wetted side, C = √(〖60〗^2+ 〖40〗^2 ) = 72 cm = 0.72 m
Wetted Perimeter, p = 72 + 40 + 72 = 184 cm = 1.84 m
Top Width, T = 60 + 40 + 60 = 160 cm = 1.6 m
Area of Cross Section, a = ((40+160) × 40)/2 = 4000 cm2 = 0.4 m2
Hydraulic Radius, R=a/p =0.4/1.84 = 0.217 m
Hydraulic Slope, S = 0.1/100 = 0.001
Manning's, η = 0.025
Mean Velocity of Flow, v = 1/( η) R^(2⁄3) 〖 S〗^(1⁄2)
= 1/0.025 〖(0.217)〗^(2⁄3) 〖 (0.001)〗^(1⁄2)
∴ v = 0.456 m/s
Carrying Capacity, Q = a × v
= 0.4 × 0.456
= 0.182 m^3/s
∴ Carrying Capacity = 182 lit/sec
Estimate the mean velocity of flow and carrying capacity of a lined canal water course, rectangular in section with a bottom width of 50 cm and depth of flow of 25 cm (inside). Single layer bricks are laid in cement mortar with a cement plaster 8 mm thick. The slope of the channel bed is 2 m/km.
Ans.: Area of cross section of the channel,
a = 50/100 × 25/100 = 0.125 m2
Wetted Perimeter, p = 0.25 + 0.5 + 0.25 = 1 m
Hydraulic Radius, R=a/p =0.125/1 = 0.125 m
Hydraulic Slope, S = 2/1000 = 0.002
Manning's, η = 0.015
Mean Velocity of Flow, v = 1/( η) R^(2⁄3) 〖 S〗^(1⁄2)
= 1/0.015 〖(0.125)〗^(2⁄3) 〖 (0.002)〗^(1⁄2)
∴ v = 0.746 m/s
Carrying Capacity, Q = a × v
= 0.125 × 0.746
= 0.093 m^3/s
∴ Carrying Capacity = 93 lit/sec
Assignment No. 4
Design a non-erodible boundary channel laid on a slope of 0.0016 with discharge 9.1 m3/s. Assume Manning's η = 0.015 with permissible velocity 1.3 m/s
Design a trapezoidal channel laid on a slope of 0.0016 carrying a discharge of 10.125 m3/s. The channel is to be excavated in earth cutting is non-colloidal coarse gravel and pebbles.
Practical - 5
Appraisal of Flow Control and Distribution Structures
Appropriate structures are integral components of an effective and efficient water conveyance and distribution system at the farm level.
Drop Structure
Drop structures are used to discharge water in a channel from a higher level to a lower one. They may be open type drop or pipe drops.
Fig 5.1: Various Components of Drop Spillway
Open Drop Structures
Open drop structures can be made of timber, concrete or brick or stone masonry
Timber is usually not preferred due to its short life
Low cost drop structures can be built by using discarded drums or barrels of the type used to transport coal tar
The components of a drop structure are the inlet, the vertical overfall section and the outlet
The check gate provided at the inlet of the drop structure is used to control the water surface elevation on the upstream stretch of the channel
The minimum width of the inlet of a drop structure is equal to the bottom width of the irrigation channel
Water enters the structure through the inlet which is in the form of a weir or notch in a wall
Vertical walls, known as cut-off walls, extend down into the soil under the inlet in order to prevent water seepage under the structure
Drop structures often set up eddy currents in the irrigation stream and these currents tend to cause erosion of the channel section immediately downstream from the structure
Pipe Drop Structures
Fig 5.2: Sectional View of a Pipe Drop Spillway
Sometimes construction of an open drop structure is not possible without disturbing an existing bund or dam
In such cases water can be safely discharged from a higher level to a lower one by providing a pipe drop
This type of structure allows the discharge of water through a pipeline, leaving the bund or dam undisturbed
The velocity of flow of water in pipe drop spillways using different size pipes may be calculated from the following relationship obtained by applying the Bernoulli's theorem
Available head = Frictional loss in the pipeline + Velocity head + Head loss at the entrance of the pipe + Head loss at the bend.
I.e. H = 〖4ƒlV〗^2/(2gd )+ V^2/(2g )+ K_1 V^2/(2g )+ K_2 V^2/(2g )
In which,
H = Difference in elevation between the water level at the upstream and downstream ends of the structure (m)
V = Velocity of flow in the pipe (m/sec)
ƒ = Coefficient of friction for the pipe (usually assumed to be about 0.01)
l = Length of pipe (m)
g = Acceleration due to gravity (m/sec^2) i.e. 9.81 m/sec^2
d = Diameter of pipe (m)
K_1,K_2 = Constants (For pipe drop) K_1=0.5 & K_2=0.25
The discharge capacity of the pipe drop structure may be determined by the relationship
Q = a × v
In which,
Q = Discharge capacity, cubic meters per second
a = Cross sectional area of the pipe, square meter
v = Velocity of flow, meters per second
Chute Spillways
Fig 5.3: Structural Details of a Chute Spillway
Chute spillways carry the flow down steep slopes through a lined channel rather than by dropping the water in a free overfall
On steep slopes, chutes are more economical than a series of drop structures required taking the flow down the slope
The chute spillway consists of an inlet, channel section and outlet
The structure may be made of concrete or stone or brick lay in cement mortar
A low cost chute spillway can be made of precast concrete channel sections with a stilling basin at the outlet
The discharge capacity of the channel section of chute spillways may be calculate by using the Manning's formula
Chute channels are usually rectangular in cross section
The length of the tilling basin varies from 1.0 to 1.5 m under the normal ranges of flow obtained in the farm irrigation system
The depth of the stilling basin is about 10 cm below the bed level of the downstream channel
When the high velocity water is showed down to a low velocity in a stilling basin, there is a sudden rise in the depth of flow, known as a hydraulic jump
Hence, the height of the walls in the downstream channel should be increased, in order to prevent overfall
Usually it is sufficient to keep the same depths for the upstream and downstream sections of the channel and deepen the stilling basin floor level
Numericals
Given: H = 1 m, d = 10 cm, ƒ = 0.012, l = 3; determine the discharge capacity of the pipe drop spillway.
H = 〖4ƒlV〗^2/(2gd )+ V^2/(2g )+ K_1 V^2/(2g )+ K_2 V^2/(2g )
Ans.: Substituting the values of the variables;
H = 〖4 ×0.012 ×3 ×V〗^2/(2 ×9.81 ×(10/100))+ V^2/(2 ×9.81 )+ 0.5 × V^2/(2 ×9.81 )+ 0.25 × V^2/(2 ×9.81 )
V = √6.26 = 2.5 m/sec
a = (πd^2)/(4 ) = π/(4 ) × 10/(100 ) × 10/(100 ) = 0.0078 sq. m
Q = a × v
= 0.0078 × 2.5
∴ Q = 19.5 lit/sec
Assignment No. 5
Determine the size of concrete pipe needed in a drop inlet spillway for a peak flow of 3 cu. m per second and a total head of 3 m. Determine the slope to be given to the pipe to flow full. Length of pipe = 12 m, entrance loss coefficient Ke = 0.5 and friction loss coefficient Kc = 0.03.
A 50cm corrugated metal pipe of 75m length is used in construction of drop inlet spillway. Given the head is 2m, entrance loss coefficient is 0.5. Estimate the peak discharge under pipe flow.
Practical - 6
Analysis and Computation of Flow Profiles
A brief description of different types of flow profiles is given in Fig. 6.1. Dotted lines show the profiles near the critical depth and channel bottom, as at these point the streamlines are curved. Moreover, such equations of gradually varied flow are not applicable.
Fig 6.1: Schematic Diagram of Flow Profiles
Mild Slope Profiles
A flow, in which the normal depth (yn) is greater than the critical depth (yc) is called streaming flow and the slope of free water surface is called mild slope or M-profile. There are three types of such profiles as discussed below:
M1 Profile
It is the most important among all the profiles and represents the backwater curve. This type of profile usually occurs, when a dam of a weir is constructed across a mild along channel. In this case y > yn > yc.
M2 Profile
It represents a drawdown curve. This type of profile usually occurs, when the tail of a mild channel is submerged into a reservoir of a depth less than the normal depth. It also occurs, when the cross-section of a mild channel is subjected to a sudden enlargement. In this case yn > y > yc.
M3 Profile
It also represents a backwater curve. This type of profile usually occurs, when a channel after flowing below a sluice flows over a mild channel. In this case yn > yc > y.
Steep Slope Profiles
A flow, in which the critical depth (yc) is greater than the normal depth (yn) is known as rapid flow and the slope of free water surface is known as steep slope or S-profile. There are three types of such profiles as below:
S1 Profile
It presents a backwater curve. This type of profile usually occurs, when a dam or weir is constructed across steep channel. It also occurs when the tail of a steep channel is submerged into a reservoir of a depth more than the normal depth. In this case y > yc > yn.
S2 Profile
It represents a drawdown curve. This type of profile usually occurs, when the steep slope of channel changes from steep to steeper. It also occurs, when the cross section of a steep channel is subjected to a sudden enlargement. In this case yc > y > yn.
S3 Profile
It also represents the backwater curve. This type of profile usually occurs, when a channel after flowing below a sluice flows over a steep channel. It also occurs when the slope of the channel changes from steeper to steep. In this case yc > yn > y.
Critical Slope Profiles
A flow, in which the normal depth (yn) is equal to the critical depth (yc) is called a critical flow and the slope of free water surface is called critical slope or C-profile. There are two types of such profiles are:
C1 Profile
It represents a backwater curve. This type of profile usually occurs on the critical slope portion, when the slope of the channel changes from critical to mild. In this case, y > yc. However, yc > yn.
C2 Profile
Since in a critical depth profile, the normal depth line and critical depth line coincide therefore, no curve is possible between these lines. However, a line coinciding with these two lines can be drawn to represent C2 profile, which will indicate a uniform critical flow. In this case yn = y = yc. Some others do not mention the C2 profile.
C3 Profile
It is also represents a backwater curve. This type of profile usually occurs at the hydraulic jump. It also occurs, when the channel after flowing below a sluice gate flows over a critical slope channel. In this case yc > y. However, yc = yn.
Horizontal Slope Profiles
In a channel with horizontal bed, the normal depth (yn) of flow is not definite and it may be either below or above the critical depth (yn). The slope of free water surface is known as horizontal slope or S-profile. There are two types of such profiles as below:
H1 Profile
It represents a drawdown curve and is similar to M2 profile. In this case yn > y > yc.
H2 Profile
It represents a backwater curve and is similar to M3 profile. In this case yn > yc > y.
Adverse Slope Profiles
In a channel with adverse slope, the bed of channel rises in the direction of flow. Because of this, there is no definite normal depth line and it is assumed to be above the critical depth line. The slope of free water surface is called adverse or A profile. There are two types of such profiles are:
A1 Profile
It represents a drawdown curve. This type of profile usually occurs, when the cross section of an adverse channel is subjected to a sudden enlargement. In this case, yn > y > yc.
A2 Profile
It represents a backwater curve. This type of profile usually occurs, when a channel after flowing below a sluice flows over an adverse slope. In this case yn > yc > y.
Numericals
Water flows in a triangular channel of side slope 1 horizontal: 1 vertical and longitudinal slope of 0.001. Determine whether the channel is mild, steep 2 critical when a discharge of 0.2 m^3/s flows through it. Assume Manning's η = 0.015. For what range of depths will the flow be on a type 1, 2 or 3 curves.
Ans.: For a depth of flow of y in the channel,
A = y2, T = 2y, R = y2/ 2 √2y = (1/2√2) y
Critical depth, yc = Q^2/g= (A_c^3)/T_c = (y_c^6)/〖2y〗_c = (y_c^5)/2
yc = (〖2Q〗^2/g)^(1/5)=(〖2 (0.2)〗^2/9.81)^(1/5) = 0.382 m
Normal depth yo, Q = a × 1/( η) R^(2⁄3) 〖 S_o〗^(1⁄2)
= 1/( η) 〖y_o^2 (y_o/(2√2))〗^(2⁄3) 〖 S_o〗^(1⁄2)
= 0.5/( η) y_o^(8⁄3) 〖 S_o〗^(1⁄2)
y_o^(8⁄3)= (η Q)/(0.5 〖 S_o〗^(1⁄2) ) = (0.015 × 0.2)/(0.5 ×〖 (0.001)〗^(1⁄2) ) = 0.18974
∴ yo = 0.536 m
Since, yo > yc, the channel is a mild slope channel for this discharge. If y is the depth of flow:
For, M1 Curve y > 0.536 m
M2 Curve 0.536 m > y > 0.382 m
M3 Curve y < 0.382 m
Assignment No. 6
Water is taken from a lake y a triangular channel, with side slope of 1V: 2 H. The channel has a bottom slope of 0.01 and a Manning's roughness coefficient of 0.014. The lake level is 2.0 m above the channel entrance and the channel ends with a free fall. Determine:
The discharge in the channel
The water surface profile and the length of it by using the direct integration method
References
Chaudhry MH. Open Channel Flow. Prentice-Hall NJ, 1993.
Chow Ven T. Open Channel Hydraulic, Mc-Graw Hill Book Co. New York, 1959.
Kinori BZ. Manual of Surface Drainage Engineering. Elsevier Publ. Co. Amsterdam, 1970.
Henderson FM. Open Channel Flow. Macmillan Co. New York, 1966.
USBR. Water Measurement Manual. United States Bureau of Reclamation, 1977.
Madan MD. Open Channel Flow. PHI, 2009.
Khurmi RS. A Textbook of Hydraulics, Fluid Mechanics and Hydraulic Machines-SI Units. S. Chand and company Ltd., 2014.
Subramanya K. Theory and Applications of Fluid Mechanics including Hydraulic Machines. Tata Mcgraw Hill Publishing Co Ltd., 2003.
Micheal AM. Irrigation Theory and Practices. Vikas Publication House Pvt. Ltd; Second edition, 2008.
About the Authors
Jalgaonkar Bhagyashri Ramesh is Ph.D. Research Scholar of the Department of Soil and Water Engineering, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur. Jalgaonkar Bhagyashri Ramesh did her B. Tech. (Agricultural Engineering) from College of Agricultural and Engineering, Dr. Balasaheb Sawant Konkan Krishi Vidyapeeth, Dapoli (MS) and M. Tech. (Irrigation and Water Management Engineering) from Department of Soil and Water Engineering, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur. Jalgaonkar Bhagyashri Ramesh was awarded with Jain Irrigation Medal for standing second in order of merit in M. Tech. (Irrigation and Water Management Engineering) in year 2017. She has published 3 papers, 5 abstracts and 3 articles (Marathi language) in various journals and proceedings. She has also Co-author and author of three book chapters related subjective. She is Life Member of Indian Society of Agrometeorologist, Anand and Indian Society of Agricultural Engineers, New Delhi.
Vikas Sharma is Ph.D. Research Scholar of the Department of Soil and Water Engineering, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur. He did his B. Tech. (Agriculture Engineering) from Uttar Pradesh Technical University (UP) and M. Tech. (Irrigation and Drainage Engineering) from College of Technology, G.B Pant University of Agriculture and Technology, Pantnagar. Vikas Sharma was awarded with Bronze Medal for standing third in order of merit in B.Tech (Agricultural Engineering) in year 2014. He has published 11 papers, 10 abstracts and 4 articles in various journals and proceedings. He is a Member of Indian Society of Agrometeorologist, Anand.
Mukesh Kumar Mehla is Ph.D. Research Scholar in the Department of Soil and Water Engineering, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur. He did his B.Tech. (Agricultural Engineering) and M. Tech. Agril. Engg. (Soil and Water Engineering) from College of Agricultural Engineering and Technology, Chaudhary Charan Singh Haryana Agricultural University, Hisar, Haryana, India. He has been Awarded ICAR-JRF/SRF (PGS) fellowship during Ph.D. He has qualified GATE 2017, AIEEA-PG 2017 and AICE-JRF/SRF (PGS) 2019. He has published 3 papers in various journals and 8 abstracts in conference proceedings. He is Life Member of Indian Society of Agricultural Engineers, New Delhi.
Yadvendra Pal Singh is Ph.D. Research Scholar of the Department of Soil and Water Engineering, College of Technology and Engineering, Maharana Pratap University of Agriculture and Technology, Udaipur. Yadvendra Pal Singh obtained his B. Tech (Agricultural Engineering) from Uttar Pradesh Technical University, Lucknow and M. Tech. (Irrigation and Drainage Engineering) from College of Technology and Engineering, G.B. Pant Agriculture University, Pantnagar (Uttarakhand). He has published 12 papers and 8 abstracts in various journals and proceedings. He has also Co-author and author of three book Chapters related subjective. He is Life Member of Indian Society of Agro meteorologist, Anand.
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