characteristics of sharp weirs and the hydraulic jump

2

TABLE OF CONTENTS EXPERIMENT ON V NOTCHAbstractIntroductionMaterials and apparatusProcedureObservationsTabulated dataDiscussionsGraph workEXPERIMENT ON BROAD CRESTED WEIRAbstractIntroductionMaterials and apparatusProcedureObservationsTabulated dataDiscussionsGraph workExplanation on graph workEXPERIMENT ON HYDRAULIC JUMPAbstractIntroductionMaterials and apparatusProcedureObservationsTabulated dataDiscussionsGraph work

ConclusionsRecommendations

Bibliography

3

A series of experiments were carried out in order to analyze the properties of sharp crested weirs. The findings of these experiments, tabulated data, result discussions, graph work and conclusions have been included to help illuminate the relevance of certain phenomena. These experiments have been discussed as below.

THE V-NOTCH EXPERIMENT

ABSTRACT

The results of an experiment carried out to investigate the relationship between the discharge and head above the notch are presented with much focus on the rates of actual and theoretic flow. The experiment was carried out to allow the observation of the properties of the v-notch in relation to discharge, and the coefficient of discharge. The experiment was conducted under tight conditions on a control volume at the point of discharge of water at the notch. The properties of the flowing water; i.e. the temperature and density were put into consideration properties of the V-notch were also determined and recorded. These properties included width of channel, height of crest, half angle of notch, and the crest level. The K value was also gotten from these properties. The data collected in this experiments were used in the calculation of; the K value of the v-notch, actual discharge, theoretical discharge, coefficient of discharge for each stage and the coefficient for each stage. To help relate certain values collected, certain graphs were plotted to indicate the relationships. These graphs included;

1. A graph of head (H) on the abscissa and the actual discharge on ordinate on log-log graph paper and the following relationship indicated. The mean value of the coefficients of each stage and the mean value of the coefficients of discharge were used after putting aside doubtful data.

2. A graph of theoretical discharge on abscissa and the actual discharge on ordinate was also drawn on a section paper.

3. The coefficient of discharge on abscissa was plotted against H/Z on ordinate on a section paper.

Herein discussed are the procedures, results, and theories behind the experiments. Also indicated are the recommendations that have been put across.

INTRODUCTION

4

A weir is an overflow structure extending across a stream or channel and normal to the direction of flow. They are usually categorized based on their shape as either sharp crested or broad crested. In this experiment, focus is laid on a v-notch which is a sharp crested type of weir. It is used as a simple flow measuring tool due to its mechanics of operation. The V-notches used in measurement of discharge are designed and calibrated using standards that have been laid to ensure minimum errors. The flow predicted from this weir is inaccurate when the lower side of the stream wets the downstream chamber-face. Standards may indicate the minimum as a fixed distance from the base of the V also known as the crotch.

The theoretical discharge is related to the variables of this experiment as follows;

Qt=815

√2g . H52 . tanθ=K 'H

52

Where;

g = gravitational acceleration

H = head above notch

θ= half-angle of notch

K ' = 815

√2g. tanθ

After the theoretical and actual discharges were determined, the coefficient of discharge could be obtained according to the following relationship;

Cd=Qa

Qt

Where;

QaRepresents the actual discharge obtained by the discharge measurement device/ the gravimetric method

The head may be converted to the actual discharge;

Qa=CdQt815Cd√2g . H

52=Cd K

'H52

Replacing 815Cd√2 g by K,

Qa = K . H52

5

Since from the experiment Qa and H are measured, K is obtained from the following equation;

K =Q a

H52.

When logarithmic scale paper is applied to Qa = K. H52, K is determined based on an

H-Q graph. Applying the logarithmic operation to this equation, the following is obtained;

Log Qa = log K + 52

log H.

From this relationship, we can conclude that when the experimental data are joined

by a straight line with a gradient of52

, the actual discharge corresponding to H = 1m

gives the value of K

According to British Standard 3680 part 4A, the discharge equation for a v-notch with the angle between 20°∧100° is as indicated below;

Qa=815

CB√2g .H B

52 . tan θ

Where; CB= coefficient of discharge varying with the value of Z/B and H/Z

HB = head which enables for the effective viscosity and surface tension.

Using Kh for the effects,

HB = H + Kh

The value of Kh is a constant value of 0.00085m for a corresponding range of H/Z and Z/B and is neglected in this experiment as it is very small.

PROCEDURE

The width of the approach channel and the height of the crest were measured using the steel tape measure. The temperature of the water was then measured. The crest level of the V-notch was then measured using a hook gauge after the channel was filled up to the crest level with water The operation of the steady water supply system was started and a small discharge set with the gate valve after which the

6

water level was measured with the hook gauge after the flow became steady. The discharge was measured using the bucket, stop watch and the weighing balance. The discharge was increased a little for several times ensuring that the discharge is not too high to hinder water collection by the bucket. Procedures four and five were repeated and all data tabulated as will be indicated herein.

OBSERVATIONS AND RESULTS

This was the viewed cross section of the v-notch weir. The height H measured using the hook gauge and recorded was taken as the height from the crest to the liquid surface.

crest

It was observed that when the flow became steady, an aligned stream of water flowed out of the notch with a constantly placed Nappe. However, it was also observed that some head was lost when water flowed along the wall of the notch lowering the level of expected efficiency.

Water trickling down the wall of the device

The head loss due to that amount of water was taken to be minimal and was therefore not used in the computation of the results due to the complexity of acquiring the exact quantities of loss involved.

The weir was observed as below in a side view. The nappe increased with increasing range from the device and with increase in discharge. It also increased the ventilation. This aspect can be applied in aeration of water during water treatment process in treatment systems.

7

Turbulence occurring at the bottom of the collection

The results that were obtained and recorded are as below;

Fundamental Data

Properties of waterTemperature 20°C

Density (ρ) 1000 kg/m3

Mass of bucket 0.65 kg

Properties of V-notch

Width of channel (B) 0.6 m

Height of crest (Z) 0.117 m

Half angle of notch (θ) 45°

K’ ¿ 2.362Crest level (gauge) 0.224 m

Stage

Actual Discharge Point gauge

HZ

Theoretical

discharge

×10−3m3/ s

Cd K

Total

mass

kg

Mass of

water

kg

Volume

×10−3m3Time

sec

Discharge

×10−3m3/ s

Mean discharg

e (Q❑)

×10−3m3/ s

ReadingM

Head

(H)m

1

7.1 6.45 6.45 5.28 1.222

1.632 0.1540.07

00.59

83.062 0.533 1.2597.2 6.55 6.55 3.71 1.765

12.8 12.15 12.15 6.36 1.910

2

11.5 10.85 10.85 5.26 2.063

2.140 0.1480.07

60.65

03.761 0.569 1.34410.6 9.95 9.95 4.72 2.108

9.8 9.15 9.15 4.07 2.248

3

9.9 9.25 9.25 3.82 2.421

2.430 0.1440.08

00.68

44.276 0.568 1.34210.2 9.55 9.55 4.02 2.376

10.4 9.75 9.75 3.91 2.494

4

10.2 9.55 9.55 3.58 2.668

2.694 0.1410.08

30.70

94.688 0.575 1.3579.4 8.75 8.75 3.28 2.668

10.7 10.05 10.05 3.66 2.746

5

9.4 8.75 8.75 2.99 2.926

3.014 0.1380.08

60.73

55.123 0.588 1.3909.4 8.75 8.75 2.89 3.028

10.5 9.85 9.85 3.19 3.088

6

11.2 10.55 10.55 2.87 3.676

3.651 0.1330.09

10.77

85.900 0.619 1.46212.0 11.35 11.35 3.13 3.626

- - - - -

Mean valueCdm

=0.575

Km=1.358

Operation Data

8

DISCUSSION AND CONCLUSIONS FROM RESULTS

From the results obtained and tabulated above, certain aspects have been derived. The graphs herein plotted have been used to explain concepts that the experiment was expected to bring out.

LOG-LOG GRAPH OF H AGAINST Qa

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

-1.18

-1.16

-1.14

-1.12

-1.1

-1.08

-1.06

-1.04

-1.02

-1

-0.98

f(x) = 0.332035183127441 x − 1.22591582378626

Graph of log H against log Qa

log Qa

log

H

This graph should produce a straight line. However due to errors in the experiment, this is not the case. However if the trend line is drawn, it should produce a straight line whose gradient should be arithmetically equal to Cd*K

Where; Cd = coefficient of discharge of the v-notch

K = coefficient of the v-notch

In this case the gradient resulting from the line of best fit is 0.332 indicating that Cd*K = 0.332.

The intercept of the Log H axis should indicate the log of the crest level of the v-notch.

For the relationship between Q and H to be linear, the intercept, where H=0, needs to be taken into consideration.

9

GRAPH OF Qt AGAINST Qa

0 2 4 6 8 10 120

2

4

6

8

10

12

f(x) = NaN x + NaNgraph of Qt against Qa

actual discharge, Qa

theo

retic

al d

ischa

rge,

Qt

The graph of Qt against Qa drawn above produces a gradual curve due to erroneous data collected. It was, however, expected to produce a straight line whose gradient was to give the coefficient of discharge of the v-notch.

In this case, if a trend line is drawn as a line of best fit, its gradient is obtained as 0.533. therefore the Cd obtained from this graph is 0.533.

10

GRAPH OF H/Z AGAINST Cd

0.52 0.54 0.56 0.58 0.6 0.62 0.640

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Graph of H/Z against Cd

Cd

H/Z

11

0.52 0.54 0.56 0.58 0.6 0.62 0.640

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Graph of H/Z against Cd with line of best fit

Cd

H/Z

These graphs herein drawn have been used to indicate just how possible it is to relate calculated data and experimented data. The differences attained are however minimal since it is highly expected that during the experiment, a lot of data is lost randomly or due to observer carelessness. It is also recommended that the equipments be repaired from leakages so as to reduce errors in data collected.

It would also be expected that more time should be issued for extensive study on these principles before experimenting and reporting on the same.

12

THE BROAD-CRESTED WEIR EXPERIMENT

ABSTRACT

Discharge characteristics of a broad crested weir defined by a laboratory test are described. Broad crested weirs may be classified as short, normal or long depending on the water-surface profile over the weir. The discharge equation is obtained by dimensional analysis, and the coefficient of discharge is related to dimensionless ratios that describe the geometry of the channel and the relative influence of the forces that determine the flow pattern.

There is no involvement of new experimental work. The broad crested weir used comprises of a square edge on one side and a rounded edge on the other used for the measurement of low flows. The weirs then broaden to a wide rectangular section at higher flow depths. By positioning the flow appropriately, the burble of flow that results from the rounding of the edge can be investigated. The inflow and outflow processes on the broad crested weir are also investigated. They are normally used as flow of water measuring device in irrigation canals. The purpose of this paper is to present data that will be of use in the design of hydraulic structures for flow control and measurement. A series of steps were followed in this test to indicate the effects of width and step height of broad crested weirs in a rectangular cross-sectional channel on the values of the coefficient of discharge and the approaching coefficients of velocity. The experiment was done severally for a range of different discharges. The sill-referenced heads at the approach channel and at the tail channel were measured in each experiment. The results obtained indicate a discontinuity that occurs in head discharge ratings because of the sudden change of shape of the section width which results into a break in slope when the flow enters the other section.

INTRODUCTION

A weir is a simple device used for measuring discharge and control of flow in open channels. The techniques that are used in making discharge measuring equipment at gauging stations are very important. Portable instruments like weirs, flumes, floats, and volumetric tanks are commonly used for this purpose. Discharges are measured from very small channels like ditches to very large river channels. Broad crested weirs operate under the theory that critical flow conditions are crested above the weir. As such, the depth of water above the weir is equivalent to the critical depth. Critical condition is created when the relationship between the inertial and gravitational forces of

13

flow is equal to 1.0. This occurs when the velocity of flow equals the velocity of the wave (celerity), C =√ gy. This relationship is referred to as Froude Number (Fr). When flow is critical, Fr = 1.0

Fr= V

√gy

Where, y = depth of water. At critical point of flow, it becomes yc

V= average flow velocity

G= gravitational force exerted on flowing liquid

Fr= Resulting Froude number

A lot of research has been carried out on weirs to determine their properties in relation to flow of water. This study was also extended to include the relationship between the discharges obtained from calculations and those obtained from experimental data. Jan et al researched on the same and came up with equations that described flow in simple broad crested weirs and in compound broad crested weirs too. His equations indicated that the values obtained from calculations and the measured ones are less than 3% for flows over broad crested weirs under the experimental conditions. According to Sarker and Rhodes (2004) works, rectangular broad crested weir experiments were performed over laboratory scales and the results obtained compared with numerical calculation results obtained using commercial software. From these, it was found out that for a given flow rate, it was excellently possible to predict the upstream flow depth and the flow profile over the broad crested weir that was rapidly varied was reproduced quite well. It was however, not easy to predict the downstream depth after the energy losses since the determination of this energy is widely based on certain assumptions. The top of the broad crested weir which is opposed to the direction of flow corresponds to a channel inlet whereas the bottom corresponds to the overflow.

Traditionally, the weir discharge is determined from a single depth measurement on the crest. This elementary method, however, is not satisfactory when an accurate discharge determination is required. The flow depth does not correspond to that obtained everywhere on the weir crest. The location of the control, or critical depth, section, is not constant, but varies with the discharge, weir geometry and crest roughness.

14

The flow rate over the broad-crested weir can be represented by the following equation;

In which case;

Q = actual volumetric flow rate (m3/s ¿

Cd= coefficient of discharge

G = gravity

B = width of channel

H = total energy head of the flow upstream measured relative to the weir-crested elevation

y0 = depth of water upstream

yc = critical depth of flow in the channel

h1 = energy head upstream relative to the top of the broad-crested weir

H=h1 + v02

2g

In actual application of the broad-crested weir concept, it is more convenient to use h1 in the equation instead of H. the equation of discharge is then effected by a coefficient of velocity as indicated below.

15

According to French (1985), a proper operation of the broad crested weir is achieved when flow conditions are restricted to an operation range of 0.08<h1<0.33 where L is the length of the weir.

In the case of the experiment that was operated, a v-notch of known dimensions and coefficient was used thus making the calculations easier. The actual discharge was measured by the v-notch whereby;

Qa=K vH v

52

K v=815

Cdv √2 g tanθ

Where, H v=¿head above V-notch

Cdv=¿Coefficient of discharge of V-notch

θ=¿Half-angle of v-notch

K v=¿Coefficient of V-notch

The values of Cdv and K v that are used in this experiment were obtained from the v-notch experiment.

Normally, the total head relative to the crest level of the broad crested weir at section one is given by;

E=H 1−Z+V 12

2g=H 1−Z+ 1

2g(Qa

B H 1

)2

Where H 1 = depth at section 1

V 1 = velocity of flow at section 1

Z = height of weir

B = width of weir

The specific energy at the weir is equal to the total head, E. there exists a relationship between the specific energy and the depth at the control section (critical depth, H c as indicated in the equation below:

16

E=32H c

If the critical depth is measured, it becomes easy to calculate the specific energy and the discharge of the weir. Determining the exact position of the critical flow is, however, the difficult task. In order to determine the coefficient of discharge in this experiment, it becomes necessary to adopt the upstream depth and the approaching velocity in order to use the same in the following calculationQa/B H 1.

After the actual discharge and the depth hsve been determined, it now becomes possible to calculate the Froude number so as to help determine the states of flow tht are involved. The following formulae are used;

Velocity of flow, V = Qa

BH

Celerity, u = √ gH

Froude number is given as a relationship between the velocity of flow and the celerity.

Fr=Vu

=Qa

BH √gH

The values obtained from this relationship are used in classifying flow according to the following criterion;

Fr > 1.0 supercritical flow

Fr = 1.0 critical flow

Fr < 1.0 supercritical flow

The control section therefore is located at the point where the critical flow occurs, i.e. where Fr = 1.0

Critical flow is normally assumed to occur at the weir crest.

MATERIALS USED

1. A steady water supply system2. A round-nose broad-crested weir with rubber packings3. An adjustable slope rectangular open channel with point gauges4. A v-notch with a hook gauge

17

5. A steel tape measure6. A thermometer

PROCEDURE

After the dimensions of the broad crested weir and the distances from section 2A to 2F were measured, the slope of the channel was set as zero. The temperature of the water was then measured after which the crest level of the broad crested weir and the channel bed level were measured using the point gauges. The crest level of the v-notch was measured using the hook-notch, then water was filled up to the crest level. The operation of the steady water supply system was started and a small discharge set. The head above the v-notch was measured after the flow became steady.

The depth of flow in the upstream where the weir does not exert influence on the water surface was measured. The change of state of flow by the broad crested weir was observed and the control flow section identified by letting a drop of water fall on the flow surface. The discharge was then increased a little and the procedure skipped back to the stage of measuring the head on the v-notch until the last one. At one flow rate, the depth at sections 2A to 2F were measured, recorded and used to calculate the discharge over the weir.

Observations Made

for this experiment, photographs and diagrams were collected and included in this report to support the theoretical and analytical discussions of the tabulated data. The layout and change in states of flow were also indicated as shown by the images herein included. The flow was observed to fluctuate over the weir as shown below.

18

The flow depth was observed to reduce suddenly fall as the water flew over the weir indicating a change in the state of flow. These changes observed can be attributed to the analysis of the graph work that is also included in this report.

Results Obtained

***FUNDAMENTAL DATA***

Properties of water Temperature 19 °CDensity (ρ) 1000 kg/m3

Dimensions of broad crested weir

Width (B) 0.3mLength(L) 0.3mHeight (z) 0.15m1-0.006L/B 0.994m

Crest level (point gauge)

0.696m

Property of channel Bed level (point gauge) 0.546mProperties of v-notch Half angle of V-notch 45°

Coefficient of discharge (Cdv)

0.567

Coefficient (KV) 1.358Crest level(hook gauge) 0.216m

*****OPERATION DATA*****

19

V-notch

Stage

ReadingM

Head

(HV)

Discharge(Qa)×10-

3m3/s

Readingm

Depth

(H1)m

H1-

ZM

L/(

H1-Z)

Velocity of flow(v1)m/s

Velocity

head(V1

2/2g)m

×10-4

Specific

energy(E)m

×10-2

Theoretical discharge(Qt)×10-

3m3/s

Cd Cdt

1 0.132 0.084

2.778 0.729 0.183

0.033

9.091

0.0309

0.4858

3.30485

3.064 0.9067

0.954

2 0.100 0.116

6.223 0.738 0.187

0.037

8.108

0.069

2.438 3.7243

3.666 0.958

3 0.128 0.088

3.12 0.733 0.187

0.037

8.108

0.034

0.6127

3.7061

3.639 0.857

0.958

4 0.127 0.089

3.21 0.735 0.189

0.039

7.692

0.0357

0.6486

3.9065

3.938 0.815

0.960

5 0.124 0.092

3.49 0.737 0.191

0.041

7.317

0.0388

0.7667

4.1077

4.246 0.822

0.961

6 0.120 0.096

3.878 0.739 0.193

0.043

6.977

0.0431

0.9467

4.3094

4.563 0.8498

0.963

7 0.115 0.101

4.403 0.743 0.197

0.047

6.383

0.0489

1.22 4.7122

5.2168

0.8440

0.966

8 0.112 0.104

4.737 0.746 0.2 0.05

6.00

0.0526

1.413 5.0141

5.726 0.8272

0.961

9 0.110 0.106

4.968 0.746 0.2 0.05

6.00

0.0552

1.554 5.0155

5.729 0.8672

0.961

10 0.100 0.116

6.224 0.755 0.209

0.059

5.085

0.0692

2.439 5.9244

7.354 0.846

0.971

20

TABLE 6.2 DATA SHEET

Selected stage 10

Actual discharge Qa m3/s 6.224*10-3

Crest level of weir (m) 0.696

Width of the weir (B)0.3

OPERATION DATA

Section Distance from

section 2A

Water level(point gauge)

m

Depth(H)M

Velocity of

flow(v)m/s

Propagation

Velocity(u)m/s

Froude number(Fr

)

2A 0.00 0.748 0.052 0.3990 0.7141 0.55872B 0.05 0.74 0.044 0.4715 0.6569 0.71782C 0.10 0.734 0.038 0.546 0.6104 0.89452D 0.15 0.731 0.035 0.5928 0.5858 1.01192E 0.20 0.729 0.033 0.6287 0.5689 1.10512F 0.25 0.721 0.025 0.8297 0.4951 1.6758

21

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Y-Values

Log Qa

Log

E

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.850

0.10.20.30.40.50.60.70.80.9

f(x) = 0.421577586172224 x + 0.374022596901658R² = 0.457366930352443

graph of log specific energy against log of Qa with trend line

Y-ValuesLinear (Y-Values)

log Qa

Log

E

22

0.002 0.003 0.004 0.005 0.006 0.007 0.0080.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

f(x) = 6.28900320061414 x + 0.167940203488735R² = 0.990911683523423

depth H1 against actual discharge Qa

Depth H1Linear (Depth H1)


H1 (m

)

0.002 0.003 0.004 0.005 0.006 0.007 0.0080.17

0.1750.18

0.1850.19

0.1950.2

0.2050.21

0.215

f(x) = 6.28900320061414 x + 0.167940203488735

depth H1 against actual discharge Qa with trend-line



H1 (m

)

23

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

f(x) = 0.246143688409021 x − 0.0156790639865338R² = 0.354250738011931

Graph of H1 against Cd


Cd

H1

0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90.17

0.175

0.18

0.185

0.19

0.195

0.2

0.205

0.21

0.215

f(x) = 0.246143688409021 x − 0.0156790639865338

Graph of H1 against Cd with trendline


Cd

H1

THE HYDRAULIC JUMP EXPERIMENT

DETERMINATION OF THE PROPERTIES AND APPLICATIONS OF THE HYDRAULIC JUMP

ABSTRACT

The results of an experimental investigation of the phenomenon of the hydraulic jump are presented, with focus on the dependence on the flow rates and the level of opening on the sluice gate. The experiment was carried out so as to open up some information about the hydraulic jump. It was centered on observing its physical appearance and understanding the relationship between the depth and the specific energy under a condition of constant discharge. During the formation of the hydraulic jump, there is an unknown amount of energy that is lost. To determine the possible value of this energy, there is need to apply the Momentum equation since the energy equation cannot be applied at this point. A control volume is used by enclosing the jump as will be indicated herein and also discussed using the continuity equation. Data is collected for this purpose. The data collected in this experiment include the fundamental and operation data. The temperature and density properties of the water, the dimensions of the channel and the properties of the v-notch are measured out as the fundamental data. At varying volumetric rates of flow, the operational data is collected. This data include the head, discharge and water surface levels at different stages on the V-notch. The water surface levels at two different levels on the rectangular channel are also measured out at the three different stages using the same discharge.

While the values of the fluid depth after and before the jump and Froude number inside the jump strongly depend on variations in flow rate and levels of openings on the sluice gate, this dependence does not seem to be strong for the Froude number outside the jump. The collection of these data will be highly instrumental in the determination of energies in the system using the knowledge of momentum and the momentum equation and conservation of mass. Herein illustrated and discussed is the whole process that led to the determination of the required energies and their possible applications in the field of engineering.

INTRODUCTION

A classical jump (hydraulic jump in a rectangular section) is a phenomenon that involves the water surface rising, surface rollers forming, intense mixing occurring, air being entrained and energy being dissipated at the transition whenever flow changes from supercritical to subcritical.

This experiment was purposed to observe the hydraulic jump phenomenon and to compare measured flow depths with theoretical results based on the application of the momentum principle and continuity. In the laboratory flume, the flow is regulated from the upstream end by a sluice gate to develop a shallow and rapid supercritical flow. Gradually, a hydraulic jump is formed at the transition point

25

where the downstream subcritical and the upstream supercritical flows coincide. It occurs analogously to the shock wave phenomenon in aerodynamics where a supersonic flow meets a subsonic flow and a shock front develops at the transition between the two flow regimes. When observed on a control volume, its properties and capabilities can be observed and analyzed. An observation of the same may be covered by the following illustration.

The channel is taken to be having a uniform width, b.

Using the above illustrated diagram, the continuity equation may be expressed as;

Q = b v1h1=bv2h2-----------------------------------------------------------------------------(i)

Whereby Q represents the discharge, v represents the velocities at the different sections. ’h’ values indicate the heights at section just before and exactly after the jump. The momentum equation taking into consideration the hydrostatic forces and the momentum fluxes ignoring the frictional forces at the bottom and side surface of the channel is;

12ρgbh1

2=12ρgbh2

2=ρQ (v1−v2)------------------------------------------------------------(ii)

The flowing water is homogeneous thus the density is taken to be constant together with the gravitational force acting on it. Taking a momentum function to be equal to;

M= v2h2 g

+h2

2----------------------------------------------------------------------------------(iii)

26

Then using equation (i), it is correct to suggest that equation (ii) implies that

M 1=M 2-----------------------------------------------------------------------------------------(iv)

The relationship between the water depths before and after the jump may be

expressed from equation (i) and (ii) as; h1h2

=12(√1+8 Fr22−1) or;

h2h1

=12(√1+8 Fr12−1)

Where Fr1=v1

√gh1 and Fr2=

v2√gh2

;

For a hydraulic jump, the upstream flow is supercritical and Fr1>1. On the other hand, the Froude number Fr2 of the downstream subcritical flow needs to satisfy Fr2=𝑉2 𝑔ℎ2<1

We can further express the principle of conservation of mass in this open channel section as

h1+v12

2g=h2+

v22

2 g+hl

And show that the “head loss” hlfor hydraulic jump is calculated as;

hl=(h2h1)

3

4 h1h2

Therefore the energy loss at a hydraulic jump becomes a simple function of the relative depths of flow in question. This leads to an increased ease of hydraulic jump function application. This does not, however, eliminate the need to access the significance of the result as far as precision is concerned. Severe hydraulic jumps may lead to very large energy losses and the dissipation of this energy may be used relevantly in different fields of engineering.

MATERIALS AND METHODS

APPARATUS

For the purpose of this experiment, the following apparatus were needed and therefore provided for the said purpose. The availability and effectiveness of these apparatus did much in contributing into the final observations made since great losses would lead to gross errors that would be difficult to adjust. The equipments used include:

1. An adjustable-slope rectangular open channel with point gauges.

27

2. A steady water supply system.3. A v-notch with a hook gauge.4. A sluice gate with rubber packings.5. An adjustable-height suppressed weir.6. A steel tape measure.7. A thermometer.

METHOD

For collection of values that are close to an accurate, it was expected that the procedures were followed keenly with a high degree of attention so as to ensure accuracy of collected data. Failure to follow the right procedure or a tendency to skip certain steps may lead to gross errors that are not desirable in this case. Below are the steps that were followed during the experiment.

PROCEDURE

After the equipment was set-up as illustrated by the following diagram, the steps were followed as follows.

The sluice gate was put on the open channel with rubber packings in the space between the gate and the channel bed after which the channel bed was set to be horizontal so that the slope will be zero. The width of the channel was then measured with the steel tape measure and the channel bed levels at sections one and two were also measured using the point gauges. The temperature of the water was measured after the crest level of the v-notch that was pouring water into the channel was measured.

28

It was then that the operation of the steady water supply system was started with the opening on the sluice gate left as 0.009m. The head above the v-notch was measured and the water surface levels at sections one and two also measured using the point gauges. The change in flow was observed. The opening height of the sluice gate was increased by 0,002m two more times and other hydraulic jumps created by adjusting the height of the suppressed weir. The sixth Procedure was repeated twice simultaneously after procedure seven to observe values for states B and C. The discharge was increased for stages two and three and corresponding values for states A, B and C recorded as the experiment proceeded.

OBSERVATIONS MADE AND RESULTS COLLECTED

During the experiment, certain observations were made that were relevant to the expected results. As the discharge was set and water allowed to flow, the conditions of flow after the sluice gate kept changing until a constant condition became dominant. This occurred when the flow became steady. The flow became steady when the height of liquid upstream of the sluice gate became a constant and no further changes were observed on the height. The steady flow was first observed at the input section when the flow through the v-notch ceased to change and the height above the crest became a constant.

As the discharge rate was increased gradually and/or the opening on the gate increased, the type of jump and its characteristics also changed similarly due to an change in the relationship between the velocity of flow and the celerity. This change is illustrated in the diagram below.

29

It was observed that the turbulence at the point of the jump was more violent with smaller gate openings. It took approximately seven minutes for the jump to properly form after which data was taken and recorded at each stage as indicated in the tables below.

RESULTS TABULATION

TITLE: EXPERIMENT DATE: 28 TH MARCH 2014

EXPERIMENTER: NO. :

***FUNDAMENTAL DATA***

Properties of water

Temperature 22 ℃

Density ( ρ )

1000 kg/m3

Dimensions of channel

Width ( B )

0.30 m

Channel Bed Level

Section 1

0.4750 m

Section

30

2 0.4750 m

Properties of V-Notch

Half angle of notch ( θ )

45 °

Coefficient of Discharge ( CdV )

0.575

Coefficient ( KV )

1.36

Crest Level 0.2180 m

***OPERATION DATA***

Stage

State

V-Notch Reading of point gauge

Depth

Reading

m

Head( HV )

M

Discharge

( Q )x 10-

3m3/s

Section 1

m

Section 2

m

Section 1

( H1 )m

Section 2

( H2 )M

1A

0.1304 0.0876

3.0890.489 0.526 0.014 0.051

B 0.482 0.528 0.007 0.053C 0.480 0.536 0.005 0.061

2A

0.1395 0.0785

2.3480.482 0.516 0.007 0.041

B 0.481 0.521 0.006 0.046C 0.480 0.530 0.005 0.055

***CALCULATION***

Stage

State

Velocity Velocity Head

Specific Energy

Froude Number

Actual

( H 2

H 1

)

Theoretic

al

( H 2

H 1

)

Actual

head

loss( ∆E )

m

Theoretical Head loss( hj )

m

Section 1( V1 )

m/s

Section 2( V2 )

m/s

Section 1

(V 2

1

2 g)

m

Section 2

(V 2

1

2 g)

x10-3

m

Section 1

( E1 )M

Section 2

( E2 )m

Section 1

( Fr1 )

Section 2

( Fr2 )

1A 0.735 0.202 0.028 2.080 0.042 0.053 1.983 0.286 3.643 2.349 -

0.0110.018

B 1.471 0.194 0.110 1.918 0.117 0.055 5.613 0.269 7.571 7.454 0.062 0.066C 2.059 0.169 0.216 1.456 0.221 0.062 9.297 0.218 12.20

012.65

70.159 0.144

31

2A 1.118 0.191 0.064 1.860 0.071 0.043 4.266 0.301 5.857 5.554 0.028 0.034B 1.304 0.170 0.087 1.473 0.093 0.047 5.375 0.253 7.667 7.118 0.046 0.058C 1.565 0.142 0.125 1.028 0.130 0.056 7.066 0.193 11.00

09.505 0.074 0.114

These results were recorded without any erasure whatsoever. They have been used in the determination of the values that were used in the plotting of the graphs herein included.

0 0.05 0.1 0.15 0.2 0.250

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

f(x) = − 0.0483478404356256 x + 0.0147907264551792R² = 0.845726402844078

specific energy against depth in section 1 stage 1

s.1 depthLinear (s.1 depth)

depth

spec

ific e

nerg

y

32

At

section 1 it is expected that as the depth of water increases, the specific energy of the flow reduces gradually. It is expected that there should be alternate depths at which the specific energies are the same. This is however not the case as the values collected in the first stage were not sufficient enough to produce the expected graph therefore the relationship was not indicated as expected.

0.052 0.054 0.056 0.058 0.06 0.062 0.0640.046

0.048

0.05

0.052

0.054

0.056

0.058

0.06

0.062

f(x) = 1.11940298507463 x − 0.00843283582089552R² = 0.999466950959488

graph of specific energy against depth at sec-tion2 stage 1

s2 depth1Linear (s2 depth1)

depth

spec

ific e

nerg

y

0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.140

0.0010.0020.0030.0040.0050.0060.0070.008

f(x) = − 0.0331833520809899 x + 0.00925196850393701R² = 0.978908886389202

graph of specific energy against depth at sec-tion 1 stage 2

s.1 depth2Linear (s.1 depth2)

depth

spec

ific e

nerg

y

33

0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.0580

0.01

0.02

0.03

0.04

0.05

0.06

f(x) = 1.06390977443609 x − 0.00444360902255638R² = 0.996975053527859

graph of specific energy against depth at section2 stage 2

s2 se22Linear (s2 se22)

depth

spec

ific e

nerg

y

These graphs derived from depths of flow and specific energy at section indicate that the depth increases as time passes linearly with the specific energy indicating that the two variables are linearly related. The variables are directly proportional.

Under the considerations that have been asked for in the laboratory practical manual, the following series of equations have been derived to help explain the phenomena;

Q = A1V1 = A2V2

Bh1V1 = Bh2V2

Therefore, V2=h1V 1h2

(V2)2 = (h1V 1h2

)2 = V 12(h1h2

)2

Change of pressure forces

ΔFp = ϱgA1h12

- ϱgA2h22

= ϱg (Bh1

2

2 - Bh2

2

2)

= ϱgB2

(h12- h2

2)

= ϱgB2

(h1+h2)(h1-h2)

Change in the momentum

34

ΔFm = ϱBh1V 12

-ϱBh2V 22

= ϱB (h1V 12

-h2V 22 ¿

Therefore,

ΔFm = ϱB [(h1V 12

-h2V 12(h1h2

)2]

=ϱBV 12 h1h2

(h1-h2))

From the Newton’s second law of motion;-ΔFm = ΔFp

-ϱBV 12 h1h2

(h1-h2)) = ϱgB2

(h1+h2)(h1-h2)

V 12

=g2

(h1+h2)h2h1

…eqn (a)

Q=AV

Q=B2h1[g2

(h1+h2)h2h1

]

But q=QB

,

Therefore, q2=h12 g2

¿h1+h2)h2h1

=gh1h2(¿¿)…eqn (b)From eqn (1);

0=h1h2+h22-2h1v 1

2

gbasing the above on the quadratic equation ,x=−b+¿¿¿

then a=1 ;b=h1;c=−2h1 v12

gh2=−h1+¿¿¿

2h2=-h1+_√h12+8 v12h1g

2h2=-h1+_h1√1+8 v 12h1g

h2=h1/2[√1+8 v12h1g-1]

butFr=v

√ghthereforeh2h1

=12

¿]…eqn 10.5

h1h2

=12

¿]..eqn 10.6

35

Energy change is given by

ΔE=(v 12−v 22

2g)-(h2-h1)

Substituting for values of V1 and v2

gives

ΔE=(h2−h1)3

4h1h2 …eqn 10.7

CONCLUSION

These experiments carried out under controlled conditions were expected to explain the phenomena of flow through weirs and the characteristics of the resultants. These relationships have been brought out as such though there accuracy may be in a lot of question.

The errors herein obtained may be as a result of negligence and fault of equipment being used. Negligence, on the part of the observer, led to gross errors that were indicated in the broad crested weir experiment. Fault of equipment, e.g. leakages, led to random errors that were evenly distributed in the data collected as brought out in the v-notch and Hydraulic jump experiments.

Recommendations

It is recommended that the equipment in the laboratory should be checked and maintained regularly for the purpose of producing data that are closest to the expected. The researchers should also be given enough time and chance to study wide and experiment repeatedly on these phenomena so as to acquaint themselves properly with the expected procedures in this study.

There should also be sufficient support and supervision by the technicians so as to ensure the procedures are adhered to and to help minimize gross errors.

BIBLIOGRAPHY

1. Discharge relations for Rectangular Broad Crested Weirs Farzin Salmasi, Sanaz Poorescandar. Ali Hosseinzadeh Dalir, Davood Farsadi Zadeh Tabris University2. Equipment for Engineering Education G.U.N.T. Geratebau G.M.B.H, Baisbuttel Germany3. Hydraulic Jumps Sara Connoly, May 4, 2001

36

The college of Wooster.4. Numerical Study of a Turbulent Hydraulic Jump Qun Zhao, Shubhra K. Misra, Ib A. Svendsen and James T. Kirby. Engineering Mechanics conference University of Delaware, Newark, DE5. Discharge Characteristics of Broad- Crested Weirs H. J. Tracy United States Department of the Interior6. Hydrology and Hydraulic Engineering Dept of Civil Engineering, School of Engineering City College of New York7. Open Channel Hydraulics Vent T. E. Chow, 1959, Singapore, McGraw-Hill8. Hydraulics in Civil and Environmental Engineering Andrew Chadwick and John Morfett, 1993 Edmunds Bury Press, Bury ST Edmunds, Suffolk, Great Britain9. Essentials of Engineering Hydraulics Jonas M. K. Dake, 1972 London, Chapman and Hall Ltd10. The Hydraulics Practical Manual Jomo Kenyatta University of Agriculture and Technology Kenya.11. Lecture Notes on Hydraulics Dr. Kazungu Maitaria Department of Civil, Construction and Environmental Engineering Jomo Kenyatta University of Agriculture and Technology