Rectangular Weir and Hydraulic Jumps 1 Florida Institute of Technology CVE 3033 Lab 9 Lab #9: Rectangular Weir and Hydraulic Jumps Objectives 1. 2. 3. 4. Calculate flow over a rectangular weir Simulate a hydraulic jump in the water channel Take measurements of flow depths at the specified locations Calculate the critical depths, Froude numbers, head losses, energy losses and flowrate in the jump Introduction The hydraulic jump is a natural phenomenon that occurs when shallow and rapid flow, called supercritical, is forced to change to a deep and slower flow, called subcritical, by an obstruction to the flow. This abrupt change in flow condition is accompanied by considerable turbulence and loss of energy. This phenomenon is governed by the Froude number which balances the inertial forces in a fluid against the gravitational forces. The Froude number is defined as follows: 𝑣 𝐹= √𝑔𝑦 Where F = Froude’s number v = velocity of flow y = depth of flow g = gravitational acceleration Therefore, Supercritical (shallow and rapid) flow: F > 1 Critical Flow: F = 1 Subcritical Flow: F < 1 Table 1: Froude numbers and their associated flow conditions are defined below < 1.0 No jump since flow is already sub-critical 1.0 to 1.7 An undular jump, with about 5% energy dissipation 1.7 to 2.5 A weak jump with 5% to 15% energy dissipation 2.5 to 4.5 Unstable, oscillating jump, with 15% to 45% energy dissipation 4.5 to 9.0 Stable, steady jump with 45% to 70% energy dissipation > 9.0 Rough, strong jump with 70% to 85% energy dissipation Diagrammatic representations of these jumps may be seen in Figure 1. Fall 2021 Rectangular Weir and Hydraulic Jumps 2 Figure 1: Classification of Hydraulic Jumps The hydraulic jump may be illustrated with the aid of a specific energy diagram (SED) as shown in Figure 2. The flow enters the jump at supercritical velocity, 𝑣1 , and depth 𝑦1 , that has a specific energy of 𝐸 = 𝑦1 + 𝑣1 2 . The kinetic energy term, 2𝑔 𝑣1 2 2𝑔 , is predominant. As the depth of flow increases through the jump, the specific energy decreases. Flow leaves the jump area at subcritical velocity with the potential energy predominant. Figure 2: Specific Energy Diagram (SED) Fall 2021 Rectangular Weir and Hydraulic Jumps 3 Nomenclature Table 2: List of Symbols and Description Parameter Nom. Units yc ft y1 ft y2 ft Depth taken at subcritical section of flow; look for where the flow has calmed after the jump Weir/channel Width bw ft Width of weir is taken to be the same as the width of the channel Weir Height hw ft Height of weir (req'd if upstream height is recorded as the route to get critical depth) Cross-sectional Area A1 (ft2) Cross-sectional Area at depth y1, supercritical region. A2 (ft2) Cross-sectional Area at depth y2, subcritical region. Gravitational Constant g (ft/s2) Acceleration due to gravity = 32.2 (ft/s2) Flowrate Qc (ft3/s) Flowrate calculated using depth yc Flowrate Q1 (ft3/s) Flowrate calculated using depth y1 Velocity v1 (ft/s) Velocity; recall Continuity Eqn.: Q = vA, manipulate so that the velocity can be had to calculate Froude's Number F1 or Fr Dless Froude Numbers give an indication of the type of and other characteristics associated with the jump. hL (ft) Conjugate Depth y2/y1 Exp. (ft) Conjugate depth after the hydraulic jump; experimental value. Conjugate Depth y2/y1 Theo. (ft) Conjugate depth after the hydraulic jump; theoretical value calculated by applying the momentum eqn. to the hydraulic jump. E1 (ft) Energy entering jump; Specific Energy calculated at depth, y1 E2 (ft) Energy entering jump; Specific Energy calculated at depth, y2 Ec (ft) Energy entering jump; Specific Energy calculated at depth, yc %Eloss (%) Difference/loss in energy as a result of the jump; energy dissipated from jump:= %𝐸𝑙𝑜𝑠𝑠 = ℎ𝐿 ⁄𝐸1 × 100 Qalt (ft3/s) Critical Depth Sequent Depth Froude Number Headloss Specific Energy Energy Loss Flowrate Fall 2021 Description Depth taken from top of weir (approximately middle) to surface of water Depth taken at supercritical section of flow; usually lowest elevation of the free surface wrt channel bottom Note Bernoulli's Equation Flowrate calculated using equation 11. Rectangular Weir and Hydraulic Jumps 4 Hydraulic jumps dissipate a large amount of energy in open channel flows. This makes them very useful in canal, dam and spillway designs. However, assistance is sometimes needed to make the hydraulic jumps occur at the desired locations near spillways. This can be done by increasing the surface roughness, adding a baffle wall, or sloping the basin floor. As shown above, the best design range for the Froude number is between 4.5 and 9.0. In this range, a well-balanced steady jump will occur with a large amount of energy dissipation. A jump with a Froude number between 2.5 and 4.5 is the worst design case since a jump in this range will create large waves that could cause structural damage. Theory 1. When F =1, then 𝐹 = 𝑣 can be rewritten as √𝑔𝑦 𝑣2 𝑦 = 2𝑔 Eqn. 1 2. Assuming that the specific energy of the flow system is constant, by Bernoulli’s equation: 𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 (𝑆𝑢𝑝𝑒𝑟𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙) = 𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 (𝑆𝑢𝑏𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙) 3. Therefore, 𝑃1 𝛾 𝑣2 + 2𝑔1 + 𝑧1 = 𝑝2 𝛾 𝑣2 + 2𝑔2 + 𝑧2 + ℎ𝐿 Eqn. 2 Where P = pressure, 𝛾 = unit weight of the fluid, z = elevation from a datum. 4. From Figure 2 above, 𝑦1 = 𝑃1 𝛾 + 𝑧1 𝑦2 = and 𝑝2 𝛾 + 𝑧2 5. And substituting the above into Equation 2, we can define a new quantity, which is the total head with respect to a local datum: 𝑣2 𝑣2 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 𝑦1 + 2𝑔1 = 𝑦2 + 2𝑔2 + ℎ𝐿 Eqn. 3 6. Taking the left side of the Eqn. 3 above, 𝑣2 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 𝐸𝑡 = 𝑦1 + 2𝑔1 Eqn. 4 7. From conservation of mass, the continuity eqn. may be expressed as: 𝑄1 𝑄1 2 2 𝑣1 = 𝑡ℎ𝑒𝑛, 𝑣1 = 2 𝐴1 𝐴1 8. And letting A1 = 𝑏𝑦1 for a rectangular open channel where b = bw. Substituting all of this into Eqn. 4 yields 𝑄2 𝐸𝑡 = 𝑦1 + 2𝑔𝑦 1 𝑄2 9. Letting K = 2𝑔𝑏 𝑤 2 2𝑏 2 𝑤 Eqn. 4a , we get 𝐾 𝐸𝑡 = 𝑦1 + 𝑦 1 2 Eqn. 5 10. If Q, g and bw are constants, then K is a constant, and if we plot Eqn. 5 on a y vs. Et graph, we will get the curve in Figure 3. Fall 2021 Rectangular Weir and Hydraulic Jumps 5 Figure 3: y vs. Et graph Determining head loss: 11. From Equation 3: 𝑣2 𝑣2 𝑦1 + 2𝑔1 = 𝑦2 + 2𝑔2 + ℎ𝐿 12. Therefore, headloss ℎ𝐿 = (𝑦1 − 𝑦2 ) + 𝑄 𝑄 1 2 (𝑣1 2 −𝑣2 2 ) Eqn. 6 2𝑔 13. Now, 𝑣1 = 𝐴 𝑎𝑛𝑑 𝑣2 = 𝐴 , and substituting these values into Eqn. 6 (𝑄 2 ) ℎ𝐿 = (𝑦1 − 𝑦2 ) + [ 2𝑔 1 1 1 2 × [(𝐴 2 ) − (𝐴 2 )]] Eqn. 7 14. Predicting 𝑦2 given 𝑦1 : Applying the momentum equation to the hydraulic jump: 𝑦2 = 15. Taking 𝐹1 = 𝑣1 √𝑔𝑦1 = √𝑔𝑏 𝑄2 𝑤 2 𝑦1 3 𝑦1 2 8𝑄 2 × [√1 + 𝑔𝑏 𝑤 2 𝑦1 3 − 1] Eqn. 8 , Eqn. 8 can be rewritten as: 𝑦2 = 𝑦1 2 × [√1 + 8𝐹1 2 − 1] Eqn. 9 16. For rectangular channels: 3 𝑞2 𝑦𝑐 = √2𝑔 where q = Q/bw 17. Simplifying this equation, we get: 𝑞2 𝑦𝑐 = =≫ 𝑞 = √𝑦𝑐 3 𝑔 𝑔 18. Now substituting for q – Q/bw yields: 𝑄 = 𝑏𝑤 √𝑦𝑐 3 𝑔 3 Fall 2021 Eqn. 10 Rectangular Weir and Hydraulic Jumps 6 19. Determining Q given 𝑦2 and 𝑦1 for a rectangular channel with no slope: From Eqn. 9, solve for Fr 𝑦2 1 = [√1 + 8𝐹1 2 − 1] 𝑦1 2 20. Use Fr to yield velocity, v using 𝑣 𝐹𝑟 = √𝑔𝑦1 21. Now, Solve for Q: 𝑄 = 𝑣𝑏𝑤 𝑦1 Eqn. 11 Apparatus • Flow Demonstration Channel as shown in Figure 2 • Rectangular weir • Water • Screw driver • Stopwatch. Ruler The flow demonstration channel unit consists of a transparent flume, a headtank with adjustable undershot gate, a movable tailgate, a reservoir, a circulating pump, a flow meter and flow control valves. A motorized jacking system adjust the slope of the channel bed. The complete assembly is mounted on a castered frame for easy portability. All of the wetted parts of the equipment are made of non-corrosive materials. The only utility required for operation is electric service. Figure 4: Flow Demonstration Channel Water is allowed to flow over the weir and later hindered by a trap door. Rulers will be used to measure the depths for the supercritical and subcritical flows. The trapdoor at the end of the flow tank acts as a barrier to the flow and thereby results in the hydraulic jump. In this experiment, when the trapdoor is lifted, the flow on the trapdoor is faster than the horizontal flow. The horizontal flow at the down-end of the tank is suddenly slowed considerably as compared to the up-end of the tank. The hydraulic jump will then be simulated as shown: Fall 2021 Rectangular Weir and Hydraulic Jumps 7 Figure 5: Simulated Hydraulic Jump: as more flow is slowed than sped-up, the hydraulic jump region would travel upstream very quickly (like the crest of a wave). Procedure 1. Adjust the slope indicator to the zero mark. 2. Ensure that all the dials are off. 3. Plug in the apparatus. 4. Level the apparatus. 5. Ensure that the tailgate horizontal by adjusting knob near the control panel. 6. Measure and record the dimensions of the rectangular weir on data sheet provided. 7. Place the weir in the channel and attach with screws. 8. Open the small and large orifice knobs slightly. 9. Turn on the pump. 10. Close the small orifice knob. 11. Slowly open the large orifice knob to increase flow in the channel. 12. Set the flow of the water to a certain level. Next, lift the trapdoor up to a sufficient angle to cause a hydraulic jump. 13. Measure the different levels 𝑦1 and 𝑦2 of the hydraulic jump as it rolls back through the length of the flow tank. Measure the critical depth 𝑦𝑐 . Record these values. Use your Data Sheet and the Nomenclature Table as a guide. 14. Repeat steps 12 and 13 for 4 more trials of varying flows (such that 𝑦𝑐 is observed). Fall 2021 Rectangular Weir and Hydraulic Jumps 8 Florida Institute of Technology CVE 3033 Lab 9 Lab #9: Rectangular Weir and Hydraulic Jumps DATA SHEET Name: _____________________________ Date Performed: ______________________ Lab Partners: ___________________________________________________________________ Table 3: Lab Data Width of weir, bw Acceleration due to Gravity, g (ft) (ft/s2) yc Trial y1 y2 (in) (ft) (in) (ft) (in) (ft) A1 (ft2) A2 (ft2) Q1 (ft3/s) v1 (ft/s) F1 hL (ft) E1 (ft) E2 (ft) Ec (ft) %Eloss % Qalt (ft3/s) 1 2 3 4 5 6 7 Trial 1 2 3 4 5 6 7 Trial 1 2 3 4 5 Fall 2021 y2/y1 y2/y2 Exp. (ft) Theo. (ft) Rectangular Weir and Hydraulic Jumps 9 Calculations/Sketch: Fall 2021 Rectangular Weir and Hydraulic Jumps 1 Lab Report Format and Guidelines Section Description Points Points Available Awarded □ Executive Summary Introduction Procedure/ Equipment Analysis/ Results Fall 2021 Provide a summary (250 words minimum) of this lab, which includes only the key points of your procedure, analyses and conclusions. □ What were the major steps in your data collection? □ Include any problems encountered during your lab that compromised the data collection. □ What were the results once you analyzed your data? □ You must include specific data and results as part of this executive summary. □ What are your major conclusions? □ Spend some extra time to organize and edit the information you wish to leave with your freshman reader. The executive summary should be written last. Use MS Word’s “Word Count” tool and include your word count at the end of your Executive Summary. □ What were the goals of the lab? □ Give some introductory material on open flow in channels and the purpose of hydraulic jumps to familiarize your reader with the topic. □ Include a picture and description of the equipment used to perform this experiment. Label if necessary. □ Briefly provide a list or paragraph of the steps performed during the lab. Prepare an Excel table for each set of your results. Sample calculations for each equation should be handwritten (or typed) and included in the Appendix. Don’t forget your units! Also, please label all tables, figures and/or charts! Calculations and Analysis of Data (20 pts): □ Calculate the flowrate, Qc, at depth 𝑦𝑐 , using Eqn.___. □ Calculate velocity, v1 using Eqn. ___. □ Calculate and tabulate Froude’s Number (F1) using 𝑦1 values □ Calculate and tabulate the headlosses of the various trials using Eqn. ___ □ Calculate and tabulate the y2/y1 ratios for each trial. □ Using the values of Froude’s Number (F1) calculated in Step 1, obtain the theoretical y2/y1 ratios for each trial using Eqn. ___. □ Calculate and tabulate specific energy (E1) using 𝑦1 values, etc.; Eqn. ___. □ Calculate and tabulate percent energy loss using equation: = %𝐸𝑙𝑜𝑠𝑠 = ℎ𝐿 ⁄𝐸1 × 100 □ Calculate the flowrate, Qalt, using Eqn.___. Should be the same as Q1 in step 1. 10 5 0 50 Rectangular Weir and Hydraulic Jumps 2 Discussion Conclusion Graphs of Data (15 pts) □ Specific Energy Diagram: Plot 𝑦1 , 𝑦2 and 𝐸𝑡 values for each trial on a y vs. 𝑬𝒕 graph using arithmetic scale and compare the graph with the theoretical graph. Show the critical depth. □ Plot hL vs. Q for each trial. Add an appropriate trendline and show the equation. □ Attach the original data sheet from your lab. Be sure that all work is shown including formulas and units of measurements. □ Discuss the classification of flow in an open channel: uniform flow, non-uniform flow; classify the hydraulic jump. Include if possible a sketch of the flow observed in the lab and label accordingly. □ Discuss the importance of hydraulic jumps citing a realworld application. □ Do the values of Froude’s Number make sense with what was seen in the lab? Did the Froude’s Numbers follow the same trend as shown in Figure 1? Identify each trial with a type of jump. □ What is the importance of an SED? Do the Et values make sense? □ Is the appearance of the specific energy diagram as expected? Explain. □ Explain any variation in yc. □ Discuss hL as a function of Q. □ What sources of error did you encounter in the lab and what magnitude of effect did it have on the results? □ Briefly summarize the highlights of the lab and discuss their significance in the real world □ What did you learn from conducting this lab? □ Give at least one recommendation to improve the accuracy of this lab. Total Points Final Lab 9 Grade Fall 2021 30 5 100 Note: Reformat to clearly distinguish equation numbers, nomenclature, their names and units. Lab 9: Rectangular Weir and Hydraulic Jumps - Results Width of weir, bw Height of weir, hw 0.149 N/A (m) (m) 9.81 (m/s2) Acceleration due to Gravity, g yc Trial 1 2 3 4 5 Trial 1 2 3 4 5 1 2 3 4 5 y1 y2 (cm) (m) (cm) (m) (cm) (m) measured converted measured converted measured converted 4.5 0.045 1 0.010 6.8 0.068 4.7 0.047 1.2 0.012 6.1 0.061 4.5 0.045 1.4 0.014 7.2 0.072 4.6 0.046 1.3 0.013 7.25 0.073 4.25 0.043 0.75 0.008 7.2 0.072 A1 A2 2 2 Qc v1 F1 h L-1 3 (m /s) (m ) (m ) (m/s) (m) calculated calculated calculated calculated calculated calculated 0.001 0.010 0.0045 2.990 9.55 0.39 0.002 0.009 0.0048 2.660 7.75 0.30 0.002 0.011 0.0045 2.136 5.76 0.17 0.002 0.011 0.0046 2.377 6.66 0.22 0.001 0.011 0.0041 3.659 13.49 0.61 y 2 /y 1 Trial 11/1/2021 y 2 /y1 E1 E2 Ec %Eloss Qalt (m3/s) Exp. (m) Theo. (m) (m) (m) (m) % calculated calculated calculated calculated calculated calculated calculated 6.800 13.500 0.466 0.078 0.068 83% 0.004 5.083 10.962 0.372 0.075 0.071 80% 0.005 5.143 8.150 0.246 0.081 0.068 67% 0.004 5.577 9.413 0.301 0.082 0.069 73% 0.005 9.600 19.077 0.690 0.079 0.064 88% 0.004 Parameter Nom. Units Critical Depth yc m Description Depth taken from top of weir (approximately middle) to surface of water Depth taken at supercritical section of flow; usually lowest elevation of the free surface wrt channel bottom y1 m y2 m Weir/channel Width bw m Weir Height hw m A1 (m ) Cross-sectional Area at depth y1, supercritical region. A2 (m2) Cross-sectional Area at depth y2, subcritical region. Gravitational Constant g (m/s ) Acceleration due to gravity = 32.2 (ft/s2) Flowrate Qc (m3/s) Flowrate calculated using depth yc Velocity v1 (m/s) Velocity; recall Continuity Eqn.: Q = vA, manipulate so that the velocity can be had to calculate Froude's Number Froude Number F1 or Fr Dless Froude Numbers give an indication of the type of and other characteristics associated with the jump. Headloss hL (m) Conjugate Depth y 2 /y 1 Conjugate Depth y 2 /y 2 Sequent Depth Crosssectional Area E1 Specific Energy Energy Loss Flowrate E2 Depth taken at subcritical section of flow; look for where the flow has calmed amer the jump Width of weir is taken to be the same as the width of the channel Height of weir (req'd if upstream height is recorded as the route to get critical depth) 2 2 Note Bernoulli's Equation Conjugate depth amer the hydraulic jump; experimental Exp. (m) value. Conjugate depth amer the hydraulic jump; theoretical value calculated by applying the momentum eqn. to the hydraulic Theo. (m) jump. Energy entering jump; Specifc Energy calculated at depth, (m) y1 Energy leaving jump; Specifc Energy calculated at depth, (m) y2 Ec (m) Specifc Energy calculated at critical depth, yc %Eloss (%) Difference/loss in energy as a result of the jump; energy dissipated from jump:= (HL/E1) * 100 Qalt (m3/s) Flowrate calculated using equation 11. Graphs of y vs. E Has to be done per trial