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1 California State University Long Beach Department of CECEM CE 336 Fluid Mechanics Laboratory Bernoulli’s Theorem Demonstration Experiment #4 Instructor: Dr. Loan Miller Group 3 Group Members: Major(s) Andy Nicky Sok Mechanical Engineering Huu Tran Mechanical Engineering Kevin Alvarez Mechanical Engineering Date: September 29th, 2023 Fall 2023

2 Table of Contents 1. Purpose of Study…………………………………………………… 3 2. Introduction………………………………………………………… 3 3. Theory……………………………………………………………… 4 4. Equipment and Experimental Set-up………………………………. 6 5. Discussion…………………………………………………………. 8 6. Conclusion………………………………………………………… 14 7. References…………………………………………………………. 16

3 Purpose of Study Bernoulli's theorem is tested in this experiment using a venturi meter. Through the venturi meter and the measurement of volumetric flow rates, static head and kinetic head are measured, recorded, and calculated. Thus, the total amount of energy that passes via the venturi pipe can be calculated. Introduction To verify Bernoulli’s Theorem, in this experiment a venturi meter is used to demonstrate the discharge of the fluid which then can be calculated to see if the law of conservation of energy is applied to the steady flow of the fluid. To do this, a series of manometers connected by a venturi tube and equipped with a piezometric head were observed. As the experiment is running, the system needs to be regulated to have a consistent and reliable reading. Check for air bubbles that may be trapped in the system and to remove them by bleeding which is located on top of the venturi meters. Adjust the flow rate of the system by the hydraulic bench to 0.105 and measure the height of the heads on each manometer. Repeat the process but increase the flow rate by 0.05 until the flow rate reaches 0.55. The diameter of the ducts can be used to determine the area of the duct, and from that area, the flow rate and velocity can be calculated along with the velocity and velocity head. Static head and dynamic head can be added together at the end to get the total head.

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4 Theory Bernoulli’s equation: P is the pressure (Pa) ? is the density of fluid (kg/m) v is the velocity of water (m/s) z is the distance from datum to a specific point (m) g is gravity constant (m/s^2) Flow rate: Q is the flow rate (m^3 /s) t is the time to collect water (s) V is the volume of water collected (m^3) Flow Velocity: v is the velocity of flow (m/s) A is the cross-sectional area at specific points (m^2 ) Q is the flow rate (m^3 /s)

5 Dynamic head: ℎ ? is dynamic head (m) g is gravity constant (m/s^2) v is the velocity of flow (m/s) Total head: H is total head (m) h is static head (m) g is gravity constant (m/s^2) v is the velocity of flow (m/s)

6 Equipment & Experimental Set-up Figure 1: Hydraulic Bench to adjust the flow rate of the system. Figure 2: Venturi Meter used for its different pressure areas for each piezometer.

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7 Figure 3: Manometer used to measure the height of each tubing. Figure 4: Venturi Meter Description of each Piezometer.

8 Discussion ● Table of Recorded Data and Results Flow Rate Distance into Duct Area of Duct Static Head Static Head Velocity Dynamic Head Total Head Q A h h v H m^3/s (m) m^2 (mm) (m) (m/s) (m) (m) 0.000105 h1 0.000 0.0005309 242 0.242 0.198 0.00199 0.24399 h2 0.020 0.0004227 242 0.242 0.248 0.00314 0.24514 h3 0.032 0.0002659 238 0.238 0.395 0.00795 0.24595 h4 0.046 0.0002011 230 0.230 0.522 0.01390 0.24390 h5 0.061 0.0002214 230 0.230 0.474 0.01146 0.24146 h6 0.076 0.0002679 234 0.234 0.392 0.00783 0.24183 h7 0.091 0.0003192 238 0.238 0.329 0.00551 0.24351 h8 0.106 0.0003746 240 0.240 0.280 0.00400 0.24400 h9 0.121 0.0004348 240 0.240 0.241 0.00297 0.24297 h10 0.136 0.0004992 242 0.242 0.210 0.00226 0.24426 h11 0.156 0.0005309 242 0.242 0.198 0.00199 0.24399 0.000155 h1 0.000 0.0005309 248 0.248 0.292 0.00434 0.25234 h2 0.020 0.0004227 246 0.246 0.367 0.00685 0.25285 h3 0.032 0.0002659 236 0.236 0.583 0.01732 0.25332 h4 0.046 0.0002011 222 0.222 0.771 0.03029 0.25229 h5 0.061 0.0002214 224 0.224 0.700 0.02498 0.24898 h6 0.076 0.0002679 232 0.232 0.579 0.01706 0.24906 h7 0.091 0.0003192 238 0.238 0.486 0.01202 0.25002 h8 0.106 0.0003746 240 0.240 0.414 0.00873 0.24873 h9 0.121 0.0004348 242 0.242 0.356 0.00648 0.24848 h10 0.136 0.0004992 244 0.244 0.311 0.00491 0.24891 h11 0.156 0.0005309 246 0.246 0.292 0.00434 0.25034 0.000205 h1 0.000 0.0005309 257 0.257 0.386 0.00760 0.26460 h2 0.020 0.0004227 254 0.254 0.485 0.01199 0.26599 h3 0.032 0.0002659 237 0.237 0.771 0.03030 0.26730 h4 0.046 0.0002011 210 0.210 1.020 0.05299 0.26299 h5 0.061 0.0002214 214 0.214 0.926 0.04369 0.25769 h6 0.076 0.0002679 230 0.230 0.765 0.02984 0.25984 h7 0.091 0.0003192 238 0.238 0.642 0.02102 0.25902 h8 0.106 0.0003746 243 0.243 0.547 0.01526 0.25826 h9 0.121 0.0004348 246 0.246 0.471 0.01133 0.25733 h10 0.136 0.0004992 249 0.249 0.411 0.00860 0.25760 h11 0.156 0.0005309 251 0.251 0.386 0.00760 0.25860

9 Flow Rate Distance into Duct Area of Duct Static Head Static Head Velocit y Dynamic Head Total Head Q A h h v H m^3/s (m) m^2 (mm) (m) (m/s) (m) (m) 0.000255 h1 0.000 0.0005309 267 0.267 0.480 0.01176 0.27876 h2 0.020 0.0004227 263 0.263 0.603 0.01855 0.28155 h3 0.032 0.0002659 237 0.237 0.959 0.04688 0.28388 h4 0.046 0.0002011 196 0.196 1.268 0.08198 0.27798 h5 0.061 0.0002214 203 0.203 1.152 0.06761 0.27061 h6 0.076 0.0002679 226 0.226 0.952 0.04617 0.27217 h7 0.091 0.0003192 239 0.239 0.799 0.03253 0.27153 h8 0.106 0.0003746 247 0.247 0.681 0.02362 0.27062 h9 0.121 0.0004348 251 0.251 0.586 0.01753 0.26853 h10 0.136 0.0004992 254 0.254 0.511 0.01330 0.26730 h11 0.156 0.0005309 257 0.257 0.480 0.01176 0.26876 0.000301 h1 0.000 0.0005309 280 0.280 0.567 0.01638 0.29638 h2 0.020 0.0004227 275 0.275 0.712 0.02584 0.30084 h3 0.032 0.0002659 238 0.238 1.132 0.06531 0.30331 h4 0.046 0.0002011 180 0.180 1.497 0.11423 0.29423 h5 0.061 0.0002214 190 0.190 1.359 0.09420 0.28420 h6 0.076 0.0002679 224 0.224 1.123 0.06433 0.28833 h7 0.091 0.0003192 241 0.241 0.943 0.04532 0.28632 h8 0.106 0.0003746 252 0.252 0.803 0.03290 0.28490 h9 0.121 0.0004348 259 0.259 0.692 0.02442 0.28342 h10 0.136 0.0004992 262 0.262 0.603 0.01853 0.28053 h11 0.156 0.0005309 266 0.266 0.567 0.01638 0.28238 0.000355 h1 0.000 0.0005309 294 0.294 0.669 0.02279 0.31679 h2 0.020 0.0004227 288 0.288 0.840 0.03594 0.32394

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10 h3 0.032 0.0002659 238 0.238 1.335 0.09085 0.32885 h4 0.046 0.0002011 160 0.160 1.766 0.15889 0.31889 h5 0.061 0.0002214 176 0.176 1.603 0.13103 0.30703 h6 0.076 0.0002679 220 0.220 1.325 0.08948 0.30948 h7 0.091 0.0003192 242 0.242 1.112 0.06304 0.30504 h8 0.106 0.0003746 258 0.258 0.948 0.04577 0.30377 h9 0.121 0.0004348 266 0.266 0.816 0.03397 0.29997 h10 0.136 0.0004992 272 0.272 0.711 0.02578 0.29778 h11 0.156 0.0005309 278 0.278 0.669 0.02279 0.30079 Flow Rate Distance into Duct Area of Duct Static Head Static Head Velocity Dynamic Head Total Head Q A h h v H m^3/s (m) m^2 (mm) (m) (m/s) (m) (m) 0.000405 h1 0.000 0.0005309 310 0.310 0.763 0.02966 0.33966 h2 0.020 0.0004227 300 0.300 0.958 0.04678 0.34678 h3 0.032 0.0002659 234 0.234 1.523 0.11824 0.35224 h4 0.046 0.0002011 136 0.136 2.014 0.20680 0.34280 h5 0.061 0.0002214 160 0.160 1.829 0.17054 0.33054 h6 0.076 0.0002679 214 0.214 1.512 0.11646 0.33046 h7 0.091 0.0003192 244 0.244 1.269 0.08205 0.32605 h8 0.106 0.0003746 263 0.263 1.081 0.05957 0.32257 h9 0.121 0.0004348 275 0.275 0.931 0.04421 0.31921 h10 0.136 0.0004992 282 0.282 0.811 0.03355 0.31555 h11 0.156 0.0005309 289 0.289 0.763 0.02966 0.31866 0.000451 h1 0.000 0.0005309 330 0.330 0.849 0.03678 0.36678 h2 0.020 0.0004227 317 0.317 1.067 0.05801 0.37501

11 h3 0.032 0.0002659 237 0.237 1.696 0.14663 0.38363 h4 0.046 0.0002011 110 0.110 2.243 0.25645 0.36645 h5 0.061 0.0002214 141 0.141 2.037 0.21148 0.35248 h6 0.076 0.0002679 207 0.207 1.683 0.14441 0.35141 h7 0.091 0.0003192 244 0.244 1.413 0.10174 0.34574 h8 0.106 0.0003746 270 0.270 1.204 0.07387 0.34387 h9 0.121 0.0004348 283 0.283 1.037 0.05483 0.33783 h10 0.136 0.0004992 293 0.293 0.904 0.04161 0.33461 h11 0.156 0.0005309 301 0.301 0.849 0.03678 0.33778 0.000500 h1 0.000 0.0005309 350 0.350 0.942 0.04520 0.39520 h2 0.020 0.0004227 333 0.333 1.183 0.07130 0.40430 h3 0.032 0.0002659 234 0.234 1.880 0.18022 0.41422 h4 0.046 0.0002011 80 0.080 2.487 0.31520 0.39520 h5 0.061 0.0002214 123 0.123 2.258 0.25992 0.38292 h6 0.076 0.0002679 200 0.200 1.866 0.17750 0.37750 h7 0.091 0.0003192 247 0.247 1.566 0.12505 0.37205 h8 0.106 0.0003746 277 0.277 1.335 0.09079 0.36779 h9 0.121 0.0004348 294 0.294 1.150 0.06739 0.36139 h10 0.136 0.0004992 306 0.306 1.002 0.05114 0.35714 h11 0.156 0.0005309 315 0.315 0.942 0.04520 0.36020 Flow Rate Distance into Duct Area of Duct Static Head Static Head Velocity Dynamic Head Total Head Q A h h v H m^3/s (m) m^2 (mm) (m) (m/s) (m) (m) 0.000550 h1 0.000 0.0005309 374 0.374 1.036 0.05470 0.42870 h2 0.020 0.0004227 355 0.355 1.301 0.08628 0.44128 h3 0.032 0.0002659 231 0.231 2.068 0.21807 0.44907

12 h4 0.046 0.0002011 48 0.048 2.736 0.38140 0.42940 h5 0.061 0.0002214 100 0.100 2.484 0.31451 0.41451 h6 0.076 0.0002679 192 0.192 2.053 0.21477 0.40677 h7 0.091 0.0003192 249 0.249 1.723 0.15131 0.40031 h8 0.106 0.0003746 285 0.285 1.468 0.10986 0.39486 h9 0.121 0.0004348 306 0.306 1.265 0.08154 0.38754 h10 0.136 0.0004992 320 0.320 1.102 0.06188 0.38188 h11 0.156 0.0005309 332 0.332 1.036 0.05470 0.38670 Table 1 : Recorded Data and Results ● Sample Calculations Using the first flow rate for the sample calculation: Figure 5. The venturi meter section with Area, Distance from Datum and corresponding diameters of the section (all measurements are in mm) All millimeter is converted to meter. ⮚ Average velocity through the pipe section, at each location of the piezometric tube:

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13 v = Q A = 0.000105 0.00053093 = ¿ 0.198 ( m s ) ⮚ Dynamic Head (Velocity Head) at each location of the piezometric tube: v 2 2 g = v 2 2 ∗ 9.81 = 0.198 2 2 ∗ 9.81 = ¿ 0.00199 ( m ) ⮚ Total energy (H) at each location of the piezometric tube: H = P γ + z + v 2 2 g = static head + 0 + dynamichead H = h + v 2 2 g = 0.242 + 0.00199 = 0.24399 ( m ) See Table 1 for results of calculations. 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 0.03 0.08 0.13 0.18 0.23 0.28 0.33 0.38 HGL vs Distance x Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Distance x (m) Static Head (m) Figure 6. HGL vs. distance along the duct for the recorded flows.

14 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.2 0.25 0.3 0.35 0.4 0.45 0.5 EGL vs Distance x Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Distance x (m) Total Head (m) Figure 7. EGL vs. distance along the duct for the recorded flows. Conclusion

15 The purpose of this laboratory experiment is to empirically verify Bernoulli's theorem using a Venturi meter, demonstrating the principle of conservation of energy in fluid flow and its practical applications in fluid dynamics and engineering. Questions 1. What is a venturi meter? What is advantage of using a venturi meter over a weir (exp #2) or orifice (exp #3) to measure discharge? A Venturi meter is a precision flow measurement device with a gradually narrowing tube that allows for accurate flow rate determination by measuring pressure differences. Using a Venturi meter offers advantages over weirs and orifices for measuring discharge because it provides higher accuracy across a broader range of flow rates, reduces head loss, is less sensitive to changes in fluid properties, and is less prone to cavitation. 2. What are the assumptions of Bernoulli’s equation? Bernoulli's equation assumes the fluid is incompressible, with a constant density, and that the flow is steady, devoid of unsteadiness over time. Additionally, it presumes an inviscid and irrotational flow, meaning no internal friction or rotation occurs within the fluid. The equation also relies on the conservation of energy principle, asserting that the total mechanical energy of a fluid particle remains constant along its path, given no external forces apart from gravity. Discussion of Results

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16 1. How HGL and EGL changed along the length of the duct? At the outset, at the Venturi meter's entrance, the HGL stood at a higher level, indicating the fluid's pressure head. As the fluid gained speed while passing through the narrow throat of the Venturi meter, the HGL declined due to the intensified fluid velocity, consistent with Bernoulli's principle. Upon exiting the throat and entering the diverging section, the HGL experienced a partial rebound but stayed below its initial level. This alteration in HGL showcased the transformation of pressure energy into kinetic energy as the fluid accelerated, followed by a partial restoration of pressure energy. The Energy Grade Line (EGL) exhibits an initial increase followed by a decrease along the length of the duct. This phenomenon reflects energy transformations within the flowing fluid. Initially, as the fluid accelerates in the constricted throat, pressure energy is converted into kinetic energy, causing the EGL to rise. However, in the diverging section, only a partial conversion of kinetic energy back into pressure energy occurs, resulting in a net energy loss and causing the EGL to decrease. This pattern demonstrates the principles of energy conservation. 2. Discuss on the validity of Bernoulli's equation for Convergent flow and Divergent flow. Bernoulli's equation demonstrated its accuracy for convergent flow within the Venturi meter's throat. The equation's assumptions aligned well with the conditions in this section, where fluid velocity increased, leading to a decrease in pressure. Precise pressure measurements affirmed the equation's validity, confirming Bernoulli's theorem..

17 In the divergent flow section, Bernoulli's equation continued to be relevant but needed specific attention. It assumed adiabatic conditions and took into account the fluid's deceleration as it expanded. Although the pressure decrease was not as dramatic as in the throat, the experiment still showcased a measurable reduction in pressure energy along the diverging section, consistent with the principles of Bernoulli's theorem. References Sultana, R. (2017). Lab # 4: Bernoulli’s Theorem Demonstration. In CE 336 Fluid Mechanics Lab Manual (pp. 21–30). essay, CSULB Fluid Mechanics Lab Coordinator. Armfield, 2012, “Bernoulli’s Theorem Demonstration”, Instruction Manual. Munson, B. R., T. H. Okiishi, W. W. Huebsch, A. P. Rothmayer, 2012, “Fundamentals of Fluid Mechanics”, 7th edition, John Wiley, Chapter 8. CE 336 Fluid Mechanics Student Manual, 1993, CSULB.