Lab 3 - Flow Over Weirs

Lab 3 - Flow Over Weirs Allison Encarnacion Miguel Felix Adam Ahmadpour Donato Gonzalez California State University Long Beach CE 336 - Section 10 Loan Miller 9/22/2023

Purpose The purpose of this experiment was to examine how weirs can control the flow rate across a channel. Introduction A weir is a partial obstruction on a channel bottom over which fluid must flow. In this experiment a V-notch and a rectangular weir were used as their different geometric designs would lead to different flow rates. Our goal was to determine how each geometric design would affect the flow rate through the channel. The experiment started off with a rectangular weir. The hydraulic bench would automatically calculate the flow rate as the inflow rate was changed. When a steady flow rate was maintained, the height of the water and the flow rate were recorded. To avoid errors, the flow had to be steady before getting the data. The flow rate and height of the water level was taken for ten flow rates. The same process was repeated for the V-notch weir. Theory ? ? 𝛾 + 𝑧 ? + 𝑉 ? 2 2𝑔 = ? ? 𝛾 + 𝑧 ? + 𝑉 ? 2 2𝑔 𝐻 + ? 𝑤 + 𝑉 1 2 2𝑔 = 0 + (𝐻 + ? 𝑤 − ℎ) + ? 2 2 2𝑔 𝑢 2 = √2𝑔 (ℎ + 𝑉 1 2 2𝑔 ) ? 1 = ∫𝑢 2 𝑑𝐴 = ∫ 𝑢 2 ℎ=𝐻 ℎ=0 ℓ𝑑ℎ = ∫ 𝑢 2 ℎ=𝐻 ℎ=0 𝑏𝑑ℎ ? 1 = √2𝑔 𝑏 ∫ (ℎ + 𝑉 1 2 2𝑔 ) 1 2 𝐻 0 𝑑ℎ ? 1 = 2 3 √2𝑔 𝑏 [(𝐻 + 𝑉 1 2 2𝑔 ) 3 2 − (0 + 𝑉 1 2 2𝑔 ) 3 2 ]

• locking and adjustment nuts • scale/ruler • weir carrier • instrument carrier • delivery nozzle • point gauge • V-notch weir wall • rectangular weir wall • flow-rate adjustment knob Discussion

Rectangle ℎ 𝑜 = 14.4 mm Flow Rate (L/s) h (mm) 1 0.097 24.5 2 0.179 32.1 3 0.257 38.2 4 0.336 43.2 5 0.415 48.3 6 0.495 53.1 7 0.575 57.4 8 0.655 61.4 9 0.735 65.5 10 0.815 69.1 Triangle ℎ 𝑜 = 15.7 mm Flow Rate (L/s) h (mm) 1 0.09 31.9 2 0.17 37.8 3 0.25 42.2 4 0.33 45.8 5 0.41 48.8 6 0.49 51.5 7 0.571 54.52 8 0.651 57.8 9 0.731 59.7 10 0.811 61.6 ? 𝑤 = 7.78 cm Table 3: Data Table for Rectangular Weir Weir Plate Height Datum H eight Water L evel Actual Flow Rate Height Above N otch Theoretical Flow rate Experimental Discharge Coefficient Theoretical Discharge Coefficient P w h o h Q H Q t C dr C dr ' m m m 𝑚 3 𝑠 ⁄ m 𝑚 3 𝑠 ⁄ unitless unitless 0.0778 0.0147 0.0245 0.000097 0.0098 0.0000859 1.13 0.62 0.0778 0.0147 0.0321 0.000179 0.0174 0.0002033 0.88 0.63 0.0778 0.0147 0.0382 0.000257 0.0235 0.0003191 0.81 0.63 0.0778 0.0147 0.0432 0.000336 0.0285 0.0004262 0.79 0.64 0.0778 0.0147 0.0483 0.000415 0.0336 0.0005456 0.76 0.64 0.0778 0.0147 0.0531 0.000495 0.0384 0.0006666 0.74 0.65 0.0778 0.0147 0.0574 0.000575 0.0427 0.0007817 0.74 0.65 0.0778 0.0147 0.0614 0.000655 0.0467 0.0008940 0.73 0.66 0.0778 0.0147 0.0655 0.000735 0.0508 0.0010143 0.72 0.66 0.0778 0.0147 0.0691 0.000815 0.0544 0.0011240 0.73 0.66 Table 4: Data Table for V-notch Weir

Questions 1. What is the advantage of using a V-notch (i.e., triangular) weir over the rectangular weir? The advantage of using the V-notch weir over the rectangular weir is that it can provide more accurate outcomes. The experimental flow rate for the V-notch was closer to the theoretical flow rate than the rectangular weir was. 2. What is the coefficient of discharge C d ? What is the typical range of values for C d ? The coefficient of discharge is the ratio between the actual flow and the theoretical flow rate. It shows us how effectively the flow device can convert potential energy into kinetic energy while considering the friction losses. The typical range of values for Cd of the triangular weir are 0.58 to 0.62. 3. List the assumptions used to derive the theoretical discharge formula for the weirs. When we applied the principles of continuity and Bernoulli’s equations to a specific streamline within a fluid flow system, we relied on several underlying assumptions to facilitate our analysis: 1. Uniform Velocity Upstream: Firstly, we assumed that the velocity distribution of the fluid upstream of the weir plate exhibits uniformity. In other words, the flow velocity is consistent across this section. Furthermore, we considered the upstream velocity to be negligible, effectively approaching zero. This assumption simplifies the initial conditions for our analysis. 2. Parallel Streamlines at the Nappe: Moving to the region where the flowing water forms a sheet, known as the nappe, we assumed that the fluid streamlines within this section remain parallel to one another. This assumption allows us to make an important inference: that the pressure at the nappe can be approximated as atmospheric pressure. We treat the flowing water at the nappe as if it were a jet with a pressure head of zero. 3. Non-Uniform Velocity Profile at the Nappe: Within the nappe itself, we acknowledge that the velocity profile of the fluid is not uniform. Instead, the velocity distribution varies along this section.

4. Elevation Datum at the Crest: To establish a consistent reference point, we adopted the elevation at the crest of the weir as our datum. This means that all elevations in our analysis are referenced relative to this specific point. 5. Negligible Energy Losses: Finally, in our analysis along this streamline, we made the simplifying assumption that there are no significant energy losses occurring. This assumption is particularly useful when applying Bernoulli’s equation to describe the fluid’s behavior at any two points along this path. Discuss Results: 1. For rectangular weir, plot experimental C d vs H (scatter plot). Draw trend line and discuss the relationship. Compare the experimental results of discharge coefficient C d to that of the theoretical discharge coefficient C d . Figure 1: Rectangular Weir: C d vs. H As for our results there is no need for a correction factor, except for the 1.13 value. As we understand that our experimental Cd values should align with our theoretical values, the value 1.13, does not align with the theoretical values. Typically, the value should be less than or equal to 1. Having a value greater than 1 would indicate that something in the experiment setup or the data analysis process such as measurements or calculations, may need a correction.

Figure 3: Rectangular Weir: Q vs H As the flow rate increases so will the distance above the notch. As the distance above the notch expands, the gap between the actual and the theoretical flow rate also increases. The separation as shown in figure 3 seems to be increasing linearly because the height of the notch may be relatively constant. Also, the actual flow rate seems to fall short of the theoretical values due to the presence of frictional losses. 4. Plot Q vs H for V-notch weir (scatter plot). Draw the trend line and discuss the relationship. Figure 4: V-Notch Weir: Q vs H

The relationship between the Flow rate Q and the height of the notch shows that as you increase the height of the notch the flow tends to increase as well. It is ideal that the actual flow rate would fall short from the theoretical flow rate, this is due to friction factors that are present in the system. As the height of the notch increases, the gap between the actual flow and the theoretical flow will also increase. The gap seems to lower between the actual and the theoretical when the flow rates are lower. 5. What are the limitations of the theory? A couple of limitations of the theory would include steady flow and no energy losses along the flow path. The assumption of steady flow suggests that the flow rate remains constant over time but in our scenario, we can see that the flow may not always be steady. Therefore, measurements will drastically change resulting in inaccurate data. For energy loss, we must assume that the energy in the fluid will remain constant with having no friction losses or turbulence. 6. Why would you expect wider variations of C d values at lower flow rate? Having a reduction in the flow or the velocity would allow the water or fluid to advance creating more friction losses. Slight changes to the water flow would result in larger changes in the C d values. Wider variations only occur at lower flow rates. Conclusion This experiment demonstrates the characteristics of fluid flow over weirs of different surface areas. The advantage of using a V-notch weir over a rectangular weir is that a V-notch has a lower notch area, so the depth of the V-notch is higher which leads to a more accurate measurement for low flow rates. The coefficient of discharge (C d ) is the ratio of actual discharge versus the ideal discharge. The typical range for the coefficient of discharge is from 0.95 to 0.99 and will always typically be less than one. The assumptions used to derive the theoretical discharge formula are that the fluid is ideal (no viscosity), with steady incompressible flow.