ترع University of South Florida LEARNING OBJECTIVES . CGN 3021 L E3: Discharge Over Weirs Study relationships between head and discharge of a weir. Compare discharge from a rectangular and a triangular (V-notch), sharp-edged weir. Identify characteristics of flow over a rectangular and a triangular (V-notch), sharp- edged weir. Determine the coefficient of discharge for a rectangular and a triangular (V-notch), sharp-edged weir. THEORY OF THE EXPERIMENT The weir is a device long used for the measurement and control of water flow in open channels. Small river, stream or pond discharges are often determined by measuring water heights upstream of a calibrated weir. For a rectangular weir of width b, the energy considerations can be applied to derive the theoretical discharge equation, where H is the height of the water above the weir edge, g is acceleration due to gravity and Q is the volumetric flow rate. Rectangular Weir (theoretical): Q = ²√2gb H¹.5 Equation 1 For a V-notch weir of angle 6 (entire angle of weir) the same procedure can be used to develop the discharge equation V-notch Weir (theoretical): Q=√2g tan(H².5 Equation 2 (A slightly different version of the V-notch weir equation can be found in some references, where the entire angle of the weir is defined as 20; in such derivations 8/2 is replaced by 8.) However, the theoretical derivations do not account for the actual details of the shape and extension of the "nappe" (sheet of water) as it spills over the weir opening, so a correction factor is introduced. This correction factor is called the coefficient of discharge (Ca), so the working equations are: Rectangular Weir (working): Q=C₁3√√2gb H¹.5 Equation 3 V-notch Weir (working): Q = C₁ √2g tan(H².5 Equation 4 Each weir has its own Ca value, which may have some dependence on the height of water above the weir edge (H), but this may be determined by experiment. In this lab you will vary the volumetric flow rate and measure the height, i.e., collect Q and H values. You will use these experimentally determined values to calculate the coefficient of discharge while also examining how well the discharge equations describe the relationship between Q and H. Notice that in each weir equation the height above the weir edge (H) is raised to a power (let this exponent be n). A consequence of this is that even a small change in n can make a dramatic difference in the predicted flow over the weir. The data collected in this experiment can be used to directly examine how closely the theoretical exponents (n = 1.5 or 3/2 for the rectangular Spring 22 ترع University of South Florida CGN 3021 L weir, n = 2.5 or 5/2 for the V-notch weir) model the actual flow over the weirs. To see how this is done, generalize Equation 3 by replacing the theoretical exponent 1.5 with n, then take the logarithm of both sides to get: From Equation 3: Q = C₁ √2gb Hm Take logarithm: logQ = log (C₁3√2gb H™) Separate variable H: log (Q) = log (C₁² √2gb) + log(H¹) Rectangular Weir log form: log (Q) = n log(H) + log (C₁² √2gb) Equation 5 This shows that if the theoretical relationship is correct, a plot of log (Q) vs. log(H) will take the form of a straight line, with slope = n and y-intercept = (C√2gb). Thus determining the slope and y-intercept values from the log (Q) vs. log(H) plot of the experimental data can be used to obtain the exponent n and the coefficient of discharge Cd. The same approach can be used for the V-notch weir. Specifically: 1. then re-arrange: V-notch Weir log form: log(Q) = n log(H) + log (C₁-√2g tan()) Equation 6 15 EXPERIMENTAL PROCEDURE Observe the weir plates. The "crest" of a weir is the base of the weir where water begins to overflow the weir. a. Measure and record the width of the rectangular notch and the angle of the V-notch, which is best found by measuring the depth and width of the V. b. Tracing the weirs may allow for more accurate measurements. 2. Establish the datum of the hook gage (i.e. the hook gage reading corresponding to the level of the crest of the notch). a. See Figure 1 for the basic weir apparatus located in the channel of the Hydraulics Bench. Place the Vernier hook and point gage about halfway between the stilling baffle and the weir, positioning the numbered scales to face the bench controls. b. Turn the pump switch to ON and admit water from the bench supply to the channel by turning the bench control valve to the left. Fill the channel until the level is approximately at the bottom of the notch (L.e., the crest of the weir), then close the bench control valve by turning it all the way to the right and switch off the pump (this prevents draining). c. Using a small beaker, carefully bail water out from behind the stilling baffle until the water surface in the channel lies just at the crest of the weir. View the weir at eye level from upstream in the channel to determine where the water surface lies in relation to the crest of the weir. Spring 22 ترع University of South Florida Siling baffle d. When the correct level has been obtained, wait for the water to settle, and then adjust the hook gage so that the datum is zero. Do this by using the coarse adjustment nut and fine adjustment screw to bring the point of the hook just to the surface, where a small uplifting will be visible. The uplifting is best viewed from the side with your eye close to the water level. Raise or lower the left-hand movable scale to bring the zero line to the zero line of the larger right-hand scale. The datum is now established at zero at the crest of the weir notch. Delivery nozzle CGN 3021 L Sliding mast Point Figure 1: Diagram of the weir apparatus. Fine adjustment nut Scale Vemier Instrument carrier Weir plate (Vor U) Thumb nuts Weir carrier Spring 22 3. Record a series of measurements of discharge and head for the weir. a. Tum on the pump again and open the bench control valve to the left. The valve is sensitive once it is open; quarter or half-tums will significantly change flow rate. b. To maintain steady state flow over the weir, minimize bodily contact with the tank, which may create waves or sloshing. c. Open the bench control valve until maximum discharge over the weir is attained. For the V-notch weir, this occurs when the surface of the water is at the top of the weir without overtopping. For the rectangular weir, the maximum discharge should be at about 65 mm of head (near the top of the channel walls) because beyond this level the water begins to flow around the edges of the stilling baffle and makes the surface of the water uneven. d. Allow the water to settle and then raise the hook gage until the point just makes an uplifting on the surface. Record the head at the maximum discharge by reading where the ترع University of South Florida CGN 3021 L Spring 22 left-hand zero line is located on the right-hand scale. Next, take a measurement of discharge over the weir by dropping the dump valve ball into the drain and watching the sight gage. Use a stopwatch to time how long it takes to accumulate 5 liters in the volumetric tank (or some other volume of water that takes between 20 and 40 seconds to accumulate). Open the dump valve to prepare for the next reading. e. Take readings of roughly equal decrements in head, about 1-1.5 mm for the V-notch, and about 5 mm for the rectangular weir. Slightly reduce the flow until the point of the hook has emerged from the water at approximately the desired decrement (this will require patience to allow the water to still, and good judgment by sight). Allow flow to stabilize before recording the head, volume collected, and time. Lower the hook and point gage until the point coincides with the new water surface level. With further decrements of head and flow, the waiting time for 10 liters of water to collect becomes impractical, so adjust the procedure for smaller collected volumes of water. Eight different discharges for each weir should be sufficient, but take note of the following caveat: Readings should be discontinued when the level has fallen to a point at which the nappe of water over the weir edge ceases to "spring clear" of the notch plate. That is, do not attempt to apply the weir equations to a flow of water over the weir which is too slow to project outward and form an air gap between the water and the weir plate. This is likely to occur when the head is reduced to about 20 mm for the rectangular notch and to about 30 mm for the V-notch. RESULTS 1. Fill in the data in Tables 1 and 2. 2. Determine angle 8 (entire angle of weir) and the value of tan(8/2) for the V-notch. DATA ANALYSIS AND QUESTIONS FOR LAB REPORTS 1. Plot the experimentally observed discharge rate versus head (Q vs. H) in SI units for each of the Rectangular Weir and the V-notch Weir. On each plot also include curves generated when using the theoretical discharge equation (Equations 1 and 2) for the range of H from zero to the maximum H observed. 2. Consider that the magnitude of head above the bottom of the weir opening (i.e., the crest) is the driving force. For any given head value, is the actual weir flow rate (Q) greater than or less than the theoretical flow rate? Do you therefore expect the correction factor Ca (coefficient of discharge) to be a value larger than 1 or smaller than 1? 2. Plot log (Q) vs. log (H) for the Rectangular Weir and the V-notch Weir data. For this graph, Q and H should be in SI units of (m³/s and m, respectively). Remember that by convention in mathematics a plot of log (Q) vs. log (H) should have log (Q) on the vertical axis and log (H) on the horizontal axis. Use the Trendline option to display the equation of a best-fit straight line for each plot. 3. Determine the values of Ca and n for both weirs from the plots of log (Q) vs. log(H) by obtaining the value of slope and y-intercept of a best-fit straight line and using Equations 5 and 6 as discussed in the Theory of the Experiment section. Compare to the theoretical exponents given (n = 1.5 or 3/2 for the rectangular weir; n = 2.5 or 5/2 for the V-notch weir). ترع University of South Florida DATA SHEET Notch Width: Notch Height: Table 1: V-notch Weir data Water Levels 1 2 3 4 5 6 7 8 Water Levels 1 2 3 4 5 6 7 16.5cm 7.75CM 8 Head Volume Time (s) (mm) (L) 23 33 зем Notch Width: Table 2: Rectangular Weir data Head Volume (mm) CGN 3021 L SSSS 20 31 5 24.94 12.ST 10.53 8.26 14.13 11.62 10.26 35 40 44 10 46 10 48 10 SI 10 8.84 5 5 5 5 11.02 35 43 So 52 10 54 10 58 10 Time и и 21.87 13.86 5 8.31 6.69 12.61 11.99 10.66 90 Notch angle 8: Value of tan(₂): 45 (m³/s) O log Q (Q is in m³/s) log Q (Q is in m³/s) log H (H is in m) log H (H is in m) Spring 22