Acknowledgements I wish to express my deepest gratitude to my fellow group members, namely; Busawon Heetendr, Lubrun Veeresh and Baichoo Ikhlaas, who played a vital role in allowing me to better understand this experiment and to carry it out correctly. Furthermore, I would like to acknowledge, with much appreciation, the crucial roles of the Laboratory Technicians of the Fluid Mechanics Laboratory and my module lecturer, Mr Deejaysing Jogee, without whom the conducted experiment would not have been performed as successfully as it has been.
Table of Contents Abstract ................................................................................................................................................... 1 1.0 Introduction ....................................................................................................................................... 2 2.0 Aim and objectives ........................................................................................................................... 2 2.1 Aim ............................................................................................................................................... 2 2.2 Objectives ..................................................................................................................................... 2 3.0 Literature Review.............................................................................................................................. 3 3.1 Components of a rectangular weir ................................................................................................ 3 3.2 Derivation of equation for the theoretical discharge over a rectangular weir ............................... 3 3.3 Assumptions made in derivation of equation ................................................................................ 5 3.4 Coefficient of discharge ................................................................................................................ 5 4.0 Methodology ..................................................................................................................................... 6 4.1 Description of apparatus ............................................................................................................... 6 4.2 Experimental Procedures .............................................................................................................. 6 5.0 Experimental Precautions ................................................................................................................. 6 6.0 Data Collection and Analysis............................................................................................................ 7 6.1 Data Collection ............................................................................................................................. 7 6.1.1 Dimensions of rectangular weir ............................................................................................. 7 6.1.2 Volume of water collected at first pump speed ...................................................................... 7 6.1.3 Volume of water collected at second pump speed ................................................................. 7 6.1.4 Volume of water collected at third pump speed..................................................................... 8 6.2 Data Analysis ................................................................................................................................ 8 6.2.1 Average flow rate for first pump speed.................................................................................. 8 6.2.2 Average flow rate for second pump speed ............................................................................. 9 6.2.3 Average flow rates for third pump speed ............................................................................... 9 6.2.4 Determination of discharge coefficient ................................................................................ 10 7.0 Observation and Discussion ............................................................................................................ 11 8.0 Limitations of methodology ............................................................................................................ 11 9.0 Recommended improvements ......................................................................................................... 12
10.0 Health and Safety .......................................................................................................................... 12 11.0 Recommendations (application of results in engineering practice) .............................................. 12 12.0 Dissemination of knowledge ......................................................................................................... 13 13.0 Contribution of Team Members and Resource ............................................................................. 14 13.1 Contribution of team members during conduction of experiment ............................................ 14 13.2 Contribution of team members for report writing ..................................................................... 14 13.3 Summary ................................................................................................................................... 14 13.4 Resource persons ...................................................................................................................... 15 14.0 Conclusion .................................................................................................................................... 15 References ............................................................................................................................................. 16
List of figures Figure 1: Components of the rectangular weir........................................................................................ 3 Figure 2: Orifice...................................................................................................................................... 3 Figure 3: Section at sill. .......................................................................................................................... 4 Figure 4 - Graph of average discharge against h3/2. .............................................................................. 10
List of tables Table 1: Dimensions of weir ................................................................................................................... 7 Table 2 - Data collection for first pump speed........................................................................................ 7 Table 3 - Data collection for second pump speed. .................................................................................. 7 Table 4 - Data collection for third pump speed. ..................................................................................... 8 Table 5 - Flow rates for first pump speed. .............................................................................................. 8 Table 6 - Average flow rate for first pump speed. .................................................................................. 8 Table 7 - Flow rates for second pump speed. ......................................................................................... 9 Table 8 - Average flow rate for second pump speed. ............................................................................. 9 Table 9 - Flow rates for third pump speed. ............................................................................................. 9 Table 10 - Average flow rate for third flow speed. ............................................................................... 10 Table 11 - Data required for plotting of graph. ..................................................................................... 10 Table 12 - Task distribution during the conduction of the experiment. ................................................ 14
Abstract This report discusses about the flow characteristics over a rectangular weir. The discharge coefficient of the fluid flow (water in this case) has been defined through this experiment. In order to achieve the required objective, volumes of water, corresponding to a particular pump speed, have been collected over a certain time period and the respective flow rates have been calculated. Graphs have been plotted from the data obtained for analysing the flow characteristics. It has been confirmed experimentally that the relationship between the discharge and height of water above datum is linear. Based on this graph, the discharge coefficient has also been determined to be 0.68, which exceeds the usual value. An attempt to explain the unusual high value for the discharge coefficient has been made, taking into account the equipment used. Key words: rectangular weir, volume of water, flow rates, pump speeds, discharge coefficient.
1|Page
1.0 Introduction Weirs are structures on which overflow of water occurs. Weirs can be defined as an obstruction in an open channel over which flow takes place. The weirs are constructed to measure the volumetric rate of flow of water in rivers or streams. A notch is an opening on the side of a tank such that the free surface is found below the upper edge of the opening. Weirs and notches can be of two type namely, rectangular and V-notch. A weir or notch may be considered as a large orifice and thus the head over the sill at the outflow can be related to the discharge by using energy principles.
2.0 Aim and objectives 2.1 Aim The aim of this experiment is to determine the value of Cd and verify the equation of flow over a rectangular weir given by: 𝑄=
3 2 𝐶𝑑 𝑏 √2𝑔 (ℎ)2 3
Where cd is the coefficient of discharge. b is the width of weir h is the height of water above crest g is the acceleration due to gravity 2.2 Objectives The objectives of the experiment have been set as follows:
Obtain different values for the flow rate in a laboratory hydraulic bench by changing the speed and hence the volume of water flowing and measuring the corresponding heights, h of water above the crest.
Plot a graph of Q /cm3s-1 against h3/2 /cm to determine the value of Cd.
2|Page
3.0 Literature Review 3.1 Components of a rectangular weir The difference between a weir and a notch is that the notch is a small size while the weir is of a bigger one used on large scale. Notches are usually made in a plate while weirs are made of masonry or concrete. The different components of the rectangular weir are shown in Figure 1. Free water surface
Nappe
Sill / Crest
Figure 1: Components of the rectangular weir
As described in the figure above the nappe is a sheet of water flowing through a notch. The sill or crest is the top of the weir over which water flows. 3.2 Derivation of equation for the theoretical discharge over a rectangular weir A rectangular in used in an open channel flow. The head water flowing above the weir is measured and it is correlated to the flow rate of water. Therefore the equation for the rectangular weir gives the flow rate of water as a function of the head over the weir. Consider an orifice with a jet shown in Figure 2. 1 h d
2
Figure 2: Orifice
Applying Bernoulli’s equation between point 1 and 2: P1/𝜌g + v12/ 2g + h = P2/𝜌g + v22/ 2g + h Since the pressure at 1 and 2 are atmospheric pressure and v1 = 0 0+0+h = 0+ v22/ 2g
3|Page
∴ h = v22/ 2g The theoretical velocity (Torricelli’s Theorem) is given by: v2 = √2𝑔ℎ
Consider the section at the sill shown in Figure 3
h dh dh
b Figure 3: Section at sill.
Let, H = Height of water above sill of the notch b = Length of weir Consider a horizontal strip of water of width L and thickness 𝛿h at a depth of h below the free surface as shown above. Area = 𝑏 × 𝛿ℎ The theoretical velocity of water through the strip as given by Torricelli’s theorem: Velocity = √2𝑔ℎ Using the equation of continuity, Q = AV Discharge through strip, dQ = b × 𝑑ℎ × √2𝑔ℎ
…. (1)
The total discharge, over the whole notch is the found by integration of equation (1) from h=0 at the free surface to h = H at the bottom of the notch. Therefore, 𝐻
Q = ∫0 𝑏 × √2𝑔ℎ 𝑑ℎ 𝐻
= √2𝑔 × 𝐶𝑑 × 𝑏 ∫0 √ℎ 𝑑ℎ 1 +1
ℎ2
=√2𝑔 × 𝐶𝑑 × 𝑏 [ 1
+1
]
2
4|Page
Q = 2/3 × 𝒃 × √𝟐𝒈 (𝑯)𝟑/𝟐 (Theoretical discharge) The actual discharge is considered to be less than the theoretical discharge and hence, it is changed by the introduction of a coefficient of discharge, Cd. Therefore, Actual discharge = Cd × Theoretical discharge Q = 2/3 × 𝑪𝒅 × 𝒃 × √𝟐𝒈 (𝑯)𝟑/𝟐 3.3 Assumptions made in derivation of equation
Water discharges over the weir from the surface of a large reservoir, so it can be assumed that the velocity of approach is negligible and the pressure is atmospheric. There is no energy loss. Flow is one dimensional. The fluid is incompressible.
3.4 Coefficient of discharge The theoretical discharge needs to be modified due to the following: 1. Loss of had due to friction 2. Free surface velocity is not zero 3. The actual flow area over the crest is less than the theoretical flow area due to crest and surface contraction. 4. The streamlines curve near the weir producing centrifugal effects. These deviations from the ideal flow is corrected by the introduction of the coefficient of discharge, Cd. The Cd values for rectangular weirs varies with the head and height of weir from the bottom of the channel. Experimental values obtained by J.B Francis in 1850 showed that the Cd may be taken as 0.622 in nearly all flow problems.
5|Page
4.0 Methodology 4.1 Description of apparatus The apparatus used consists of a channel with a rectangular notch at the end. Water is allowed to over the rectangular notch and it is then collected in a trough and volume collected can be measured. Furthermore, a stopwatch was required. 4.2 Experimental Procedures 1. The channel was allowed to fill with water until the free surface was at the same level as the crest of the weir by switching on the pump found at the base of the apparatus. 2. The hook gauge was adjusted such that the pointer just touched the surface of the water and the hook gauge reading was taken. This corresponded to the zero mark on the gauge. 3. The pump was set at the first speed such that the water began to flow over the weir. 4. The hook gauge was readjusted and new height, h was obtained for the new level of water. 5. The volume of water collected for a specified time was determined using a measuring cylinder and a stopwatch. 6. The procedure was repeated until 8 readings were obtained. 7. The pump was set at a new speed and procedures 4 to 6 were repeated.
5.0 Experimental Precautions
While collecting water in the trough care should be taken to minimize the loss of water.
The trough should be cleaned dry before each experiment otherwise any residual fluid would change the value for the actual volume collected.
A larger trough should be used as the speed of the pump is increased so as to avoid over flowing of water in the trough.
Splashing of water should be avoided while collecting water in trough to prevent loss of volume.
6|Page
6.0 Data Collection and Analysis 6.1 Data Collection 6.1.1 Dimensions of rectangular weir The dimensions of the cross section of the rectangular weir were measured using a ruler and shown in the table below. Table 1: Dimensions of weir
Side of weir Width Height of crest from the bottom of channel
Measurement taken /mm 60 49
6.1.2 Volume of water collected at first pump speed Height, h = 15 mm Table 2 - Data collection for first pump speed.
Experiment No. 1 2 3 4 5 6 7 8 9 10
Volume / cm³ 1455 950 835 960 885 860 980 1042 935 1025
Time / s 6.45 4.39 3.57 4.26 4.01 4.00 4.50 4.71 4.26 4.81
6.1.3 Volume of water collected at second pump speed Height, h = 22 mm Table 3 - Data collection for second pump speed.
Experiment No. 1 2 3 4 5 6 7 8 9 10
Volume / cm³ 2630 2420 1510 1600 1350 1475 1092 1150 1210 1300
Time / s 6.61 6.42 4.00 3.98 3.65 3.89 3.20 3.02 3.13 3.43
7|Page
6.1.4 Volume of water collected at third pump speed Height, h = 31 mm Table 4 - Data collection for third pump speed.
Experiment No. 1 2 3 4 5 6 7 8
Volume / cm³ 4190 3235 2960 3380 3650 2410 3730 3590
Time / s 6.24 4.83 4.01 5.01 5.42 3.55 5.06 5.34
6.2 Data Analysis 6.2.1 Average flow rate for first pump speed Table 5 - Flow rates for first pump speed.
Volume / cm³ 1455 950 835 960 885 860 980 1042 935 1025
Time / s 6.45 4.39 3.57 4.26 4.01 4.00 4.50 4.71 4.26 4.81
Flow rate, Q / cm³s¯¹ 225.58 216.40 233.89 225.35 220.70 215.00 217.78 221.23 219.48 213.10
For the first pump speed, the flow rates that were deemed to be most consistent are as follows. Table 6 - Average flow rate for first pump speed.
Flow rate, Q / cm³s¯¹
Average flow rate, Qmean/ cm3s-1
216.40 215.00 217.78 219.48 217.17
8|Page
6.2.2 Average flow rate for second pump speed Table 7 - Flow rates for second pump speed.
Volume / cm³ 2630 2420 1510 1600 1350 1475 1092 1150 1210 1300
Time / s 6.61 6.42 4.00 3.98 3.65 3.89 3.20 3.02 3.13 3.43
Flow rate, Q / cm³s¯¹ 397.89 376.95 377.50 402.01 369.86 379.18 341.25 380.79 386.58 379.01
For the second pump speed, the flow rates that were deemed to be most consistent are as follows. Table 8 - Average flow rate for second pump speed.
Flow rate, Q / cm³s¯¹
376.95 377.50 379.18 379.01 378.16
Average flow rate, Qmean/ cm3s-1 6.2.3 Average flow rates for third pump speed Table 9 - Flow rates for third pump speed.
Volume / cm³ 4190 3235 2960 3380 3650 2410 3730 3590
Time / s 6.24 4.83 4.01 5.01 5.42 3.55 5.06 5.34
Flow rate, Q / cm³s¯¹ 671.47 669.77 738.15 674.65 673.43 678.87 737.15 672.28
9|Page
For the third pump speed, the flow rates that were deemed to be most consistent are as follows. Table 10 - Average flow rate for third flow speed.
Flow rate, Q / cm³s¯¹
671.47 674.65 673.43 672.28 672.96
Average flow rate, Qmean/ cm3s-1
6.2.4 Determination of discharge coefficient Table 11 - Data required for plotting of graph.
Pump Speed 1 2 3
Height, h / m 0.015 0.022 0.031
Height, h3/2 x 10-3 / m3/2 1.837 3.263 5.458
Q average x 10-6 / m3s-1 217.17 378.16 672.96
Graph of average discharge against h3/2 0.0008 0.0007
y = 0.1266x - 2E-05
Qaverage / m3s-1
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0 0
0.001
0.002
0.003
Height,
h3/2
0.004
/
0.005
0.006
m3/2
Figure 4 - Graph of average discharge against h3/2.
Attempting to calculate Cd while ignoring y-intercept: Gradient of graph = 0.1211 2 3
𝐶𝑑 × × 0.06 × √2 𝑥 √9.81 = 0.1211 𝐶𝑑 = 0.68
10 | P a g e
7.0 Observation and Discussion Due to the limits of the experiment, the graph plotted did not show the true relationship between the flow rate and the height of the head above the weir edge. The theoretical equation showed a linear relationship passing through the origin with the gradient equal to the constants multiplied by the coefficient of discharge. A straight line was obtained using the set of data to determine their relationship, indicating that it was linear but it did not pass on the origin. From previous studies, it has been shown that the discharge coefficient for a rectangular weir is normally taken as 0.622. The discharge coefficient determined in this case exceeds this value and this may be due to a lower amount of energy losses (about 32 % in this case) during discharge over the weir. Surface tension and turbulence of the flow of water in the weir channel are some factors which may have contributed to the energy losses. However, these parameters are difficult to be controlled, especially in laboratory equipments and thus, they shall always be present. Furthermore, there might have been secondary flows during the discharge, which might have contributed to more water being discharged per unit time. Lastly, it may that the measuring devices used to determine the flow rate over the rectangular weir may themselves be faulty to some extent, leading to systematic errors along with the random errors caused by the observer. This may have influenced the value of the discharge coefficient to a little extent.
8.0 Limitations of methodology The experiment did not show accurate enough results due to the following limitations:
A small bucket was used to gather the volume of water flowing from the weir, limiting the time range in calculating the flow rate to very short, thus affecting the accuracy.
Two testers had to coordinate their actions in order to measure both the flow volume and the time taken; this might have led to random errors due to different reaction times.
A small weir was used, increasing frictional losses and turbulence.
The range of values of heights above the head of the weir was too short.
Only 3 flow rates were available from the apparatus.
A small measuring cylinder of capacity of 1L was used for the volumes which reached just below 4L, resulting in the measurements becoming time-consuming.
11 | P a g e
9.0 Recommended improvements To cater for the previously mentioned energy losses, the following improvements are recommended.
Integration of an inbuilt volume measuring system by either calibrating a bucket or the basin itself.
A mechanism could have been built into the system allowing it to discharge over a fixed amount of time.
Use of a larger weir to reduce losses (both water and energy losses).
Addition of more flow rates to increase the range of results.
According to theory, the discharge coefficient varies with the height of water above the datum line. From an engineering point of view, this relationship must be verified through a graph of discharge
coefficient
against
height
of
water.
If a series of rectangular notches with different values of Cd were used, the independent variable would be Cd, and dependent variable for a particular flow speed would be the height of the head. Keeping the flow speed constant, the rectangular notches would have been fitted and the height of the upstream measured.
10.0 Health and Safety
Anti-slippery shoes should be worn to avoid any injury due to any spillage of water on the floor.
The pump should not be operated with wet hands due to risk of electric shocks.
11.0 Recommendations (application of results in engineering practice) Rectangular weirs have a wide range of engineering applications. However, they are mostly used in hydraulic engineering, as for triangular ones. Some of the applications of rectangular weirs and the use the discharge coefficient are listed below. 1. Weirs allow a simple method for hydrologists and engineers to measure the volumetric rate in small to medium-sized streams or in industrial discharge locations. This is of particular use when new water resources are to be investigated and reservoirs or dams are to be built. 2. The discharge coefficient allows engineers to determine how much energy is being lost while water is being discharged from a particular body. This discharge coefficient plays an important role in the determination of the storage capacity of a reservoir or dam. Furthermore, the discharge coefficient allows an adequate design of pumps to be used in piping networks.
12 | P a g e
Note: The same engineering applications have been listed for triangular weirs in the previous report. Weirs, in general, have common engineering applications. The only factor determining the use of a weir of a particular shape may be the amount of water flowing, the amount of energy losses (related to discharge coefficient) and the in-situ conditions of the water body.
12.0 Dissemination of knowledge The measurement of water in large water bodies cannot be done by filling a container and measuring the volume of water stored. This is a highly inaccurate and inadequate practice. Thus, engineers have developped new structures which are able to determine, theoretically, the amount of water or any other fluid passing in a channel or stream per unit time. These structures are called weirs and can exist in many shapes: rectangular, triangular, trapezoidal etc… In simple terms, a weir can be described simply as a barrier, across a water body, used to alter its flowing characteristics. The determination of the volume of water passing over the weir per unit time (volumetric flow rate) is dependent on the following: 1. The geometry of the weir. 2. The depth of water above a reference line on the weir. The geometry of the weir will always be the same. Thus, based on the depth of water above the datum (reference line), engineers are able to calculate the volumetric flow rate through specific equations. As said earlier, the equations provide a means to calculate the theoretical volumetric flow rate. But, in practice, it is almost impossible to obtain such a flow rate. This is mainly because of energy losses which occur while the water passes over the weir. Thus, in order to determine the actual flow rate, investigations are carried out on site. Past studies reveal that there is a discrepancy between the theoretical and actual flow rates. To be able to describe this discrepancy quantitatively (so that on-site investigations are not required time and again), engineers have developped a relationship between the 2 flow rates. This relationship is defined by a factor called the discharge coefficient, CD. The relationship is as follows: Actual flow rate = CD x Theoretical flow rate. The discharge coefficient can be used to determine the amount of energy losses that occur when water passes over the weir. In this experiment, the main aim was to determine this discharge coefficient for a rectangular weir and it was found to be 0.68 and this value exceeds the usual one.
13 | P a g e
Note: Once again, this section is similar as in the previous report due to the fact that both experiments had the same aim.
13.0 Contribution of Team Members and Resource 13.1 Contribution of team members during conduction of experiment Table 12 - Task distribution during the conduction of the experiment.
Activity Setting up apparatus and ensuring necessary precautions are taken. Measurement for time and height of water. Recording raw data. Discharge of excess water from tanks. Measurement of volume of water in measuring cylinders. On-site calculations related to ensure that experiment has been correctly performed.
Team member(s) Baichoo Ikhlaas, Veeresh Lubrun, Veeramah Avinaash Heetendr Busawon Veeresh Lubrun. Veeramah Avinaash Heetendr Busawon , Baichoo Ikhlaas Busawon. H, Baichoo. I, Veeramah. A, Veeresh. L
13.2 Contribution of team members for report writing This report was written both as a group and individually. The sections whereby all the group members have contributed in writing include the following: 1. Introduction. 2. Aims and Objectives. 3. Literature Review. 4. Methodology. 5. Data Collection. The remaining sections were written individually. 13.3 Summary To complete the experiment and this report within the given time frame, a strategic plan needed to be adopted by all team members to ensure that the work progressed as smoothly as possible and to ensure that everybody are on the same page whenever an activity has been carried out. For this to be possible, the flow pattern had been adopted.
1. Distribution of tasks prior to the conduction of the experiments. 2. Ensuring that all precautions are taken during the experiment.
14 | P a g e
3. Determine the flow rates on-site to see the relative closeness of the calculated values. The 4 closest values were then taken to obtain a mean flow rate for a particular pump speed. 4. Repeating the experiment if the calculations proved the experimental readings to be wrong. 5. Regular meetings to discuss outline of report. 6. Carry out the group work required for this report and ensuring that each member is satisfied with the work. 13.4 Resource persons The experiment conducted needed to be complemented by additional technical information, which were provided to us by the following persons. 1. Mr Seebun – Laboratory Technician at the Fluid Mechanics Laboratory. 2. Mr D.Jogee – Module Lecturer
14.0 Conclusion The discharge coefficient of the triangular weir was found to be 0.68, indicating a loss of energy of nearly 32 %. Past rigorous studies have found that the discharge coefficient for a rectangular weir is normally 0.622 and the determined value exceeds this one, indicating in a reduction of the amount of energy lost. Thus, it can be said that this experiment catered for some of the factors that normally lead to some of the energy losses. If a comparison is made between the discharge coefficients for a triangular weir (previous report) and a rectangular one, it can be deduced that the one for the rectangular weir is lower, thus indicating that there are more energy losses when water is discharged over a triangular weir. It is this comparison that determines which weir shape is to be used for a particular application. In the light of the above, it can be said that the aims and objectives of the experiment have been duly satisfied.
15 | P a g e
References 1. Douglas J.F., Gasiorek J.M., Swaffield J.A., 2003. Fluid Mechanics. 2nd ed. p 112 – 115, Singapore: Longman 2. Kundu. P, 2004, Fluid Mechanics, 3rd ed. p 110 – 114: Elsevier Academic Press, California. 3. University of Mauritius, 2014, Hydraulics Practicals Laboratory Sheets, Faculty of Engineering, Department of Civil Engineering. 4. Jecock, M (2011). River Fisheries and Coastal fish weirs, Heritage Assets [Online] Available from:
https://www.english-heritage.org.uk/publications/iha-river-fisheries-coastal-fish-
weirs/riverfisheriescoastalfishweirs.pdf [Accessed on 11.18.14] 5. Munson B, R, Young D, F, Okiishi, T, H and Huebsch W, W (2009). Fundamentals of Fluid Mechanics, 6th Edition, John Wiley &Sons Inc, USA. 6. Rajput R, K (2013). A textbook of fluid mechanics and hydraulic machines in SI units. S Chand, New Delhi. 7. Bright Hub Engineering, Open Channel Flow Measurement 5: the Rectangular Weir. [Online]. Available at: http://www.brighthubengineering.com/hydraulics-civil-engineering/65880-openchannel-flow-measurement-5-the-rectangular-weir/ [Accessed on 14 December 2014] 8. Douglas J.F., Gasiorek J.M., Swaffield J.A., 2005. Fluid Mechanics. 5th ed., Person Education, England. 9. Durgaiah D. R. 2004, Fluid Mechanics and Machinery. New Age International (P) Limited, New Delhi, India. 10. Sharp Crested Weir 2, Derivation of the discharge equation for a rectangular weir. [Online]. Available at: http://www.jfccivilengineer.com/sharp_crested_weir_2.htm [accessed on: 13 December 2014]
16 | P a g e
