CEE 320L Lab_03_Group 9

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CEE 320 Fluid Mechanics Fundamentals Lab Calibration of a Venturi Flow Meter Almard Pasion Jason Nguyen Eulamai Tingco Kailah Vergara University of Hawaii at Manoa CEE 320L Fluid Mechanics November 03, 2023

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) EXECUTIVE SUMMARY The aim of this project was to evaluate the Venturi flow meter, which enabled an accuracy assessment and the effectiveness of the Venturi apparatus. Through the utilization of mathematical equations, the team validated whether the anticipated calculated values for the flow rate closely corresponded to the readings obtained from the instrumentation with roughly an 8% difference between values. These findings contributed to a deeper comprehension of the connection between pressure differentials and flow discharges. By processing the raw data obtained from the Venturi flow meter and associated equipment, the team conducted several calculations to determine the flow discharge, pressure differential, Reynolds number, and calibration constant. The group concluded elevations in pressure resulted in heightened flow discharge. Additionally, the Reynolds number had a negligible impact on the calibration constant. Page i

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) TABLE OF CONTENTS EXECUTIVE SUMMARY………………... ……………………………………………………………...i 1 INTRODUCTION .................................................................................................................. 1 1.1 B ACKGROUND .................................................................................................................... 1 1.2 R EASON FOR E XPERIMENT ................................................................................................ 1 1.3 T HEORY ............................................................................................................................. 2 1.4 O BJECTIVE ......................................................................................................................... 2 2 APPROACH ........................................................................................................................... 2 2.1 T EST S ETUP AND I NSTRUMENTATION ................................................................................ 2 2.2 T EST S PECIMENS ............................................................................................................... 6 2.3 T EST P ROCEDURE .............................................................................................................. 6 3 RESULTS ................................................................................................................................ 7 3.1 T ABLES .............................................................................................................................. 7 4 ANALYSIS/ERROR ............................................................................................................. 11 4.1 T ABLES ............................................................................................................................ 11 4.2 T HE VERIFICATION OF Q AND ∆ P THROUGH COMPUTATION ..................................... 11 4.3 T HE ADVANTAGES OF L OG - LOG PLOTTING ...................................................................... 11 4.4 P ROPAGATED E RROR ........................................................................................................ 12 4.5 E RRORS ( RANDOM , SYSTEMATIC , ETC .) ........................................................................... 12 5 CONCLUSIONS/RECOMMENDATIONS ....................................................................... 12 6 REFERENCES ..................................................................................................................... 13 7 APPENDIX ........................................................................................................................... 14 TABLE OF FIGURES Figure 2-1: Venturi Pump ................................................................................................................ 3 Figure 2-2: Adjust flow valves ........................................................................................................ 4 Figure 2-3: Pressure gauge ............................................................................................................. 4 Figure 2-4: Weighing tank with flowing water ................................................................................ 5 Figure 2-5: Water pipe system ......................................................................................................... 5 Figure 2-6: Pressure gauge in operation during the tests ................................................................. 6 Figure 3-1: Linear relationship between the Reynold’s number and the calibration constant ........ 9 Figure 3-2: Log-log graph – the relationship between pressure difference and the volumetric flow rate ................................................................................................................................................. 10 TABLE OF TABLES Table 2-1: Material list ..................................................................................................................... 6 Page ii

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) Table 3-1 - Experimental Values for the water drainage time over 50 lb weight intervals and the corresponding change in pressure for 7 trials .................................................................................. 7 Table 3-2 - Calculated flow rate, calibration factor, initial velocity, and Reynolds number form experiment values (data passed on Table 4.1) ................................................................................. 8 Table 3-3 - Comparison of calculated flow rate values ................................................................... 9 Table 3-4 - Propagated error for each measurement ...................................................................... 10 Table 7-1 – Raw experimental data ............................................................................................... 17 Page iii

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) 1 Introduction 1.1 Background In the realm of engineering, you'll be introduced to a variety of flow meter instruments, and one of these instruments utilized in our experiment was the Venturi flow meter. The Venturi flow meter gauges the flow rate of a fluid by narrowing the cross-sectional area in the flow path, thereby generating a pressure differential. At this point of pressure variance, the fluid is directed through a pressure recovery exit section, where it can recover as much as 80% of the pressure differential created at the narrowest point. This phenomenon adheres to the Bernoulli Equation. To assess and validate the performance of this specific flow meter, a standard procedure known as calibration is employed. Calibration is the method of configuring an instrument to provide results for samples within an acceptable range. The Venturi effect refers to the phenomenon of fluid pressure drop that occurs when a fluid passes through a narrow segment of a pipe. As the velocity of the fluid increases within the constriction, there is a corresponding decrease in pressure. The underlying concept corresponds to the continuity equation in the field of fluid mechanics. In the case of an incompressible fluid, it is important that the mass flow rate remains constant at all locations within a pipe, even when alterations in cross-sectional area occur. 1.2 Reason for Experiment Accuracy is of utmost importance in all flow measurement applications. Flow rate measurements are significant in critical applications such as municipal water delivery and chemical processing, as any inaccuracies might have severe consequences. The utilization of a well calibrated venturi tube effectively reduces measurement errors and uncertainty. Technicians Page 1

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) conduct calibration testing on the venturi meter at different flow rates to determine the calibration constant C prior to its installation. 1.3 Theory In this laboratory experiment, we investigated the Venturi Flow Meter's capability to measure flow while minimizing pressure loss. The unique design and structure of the Venturi facilitated the precise measurement of a small pressure differential, which was then compared to the flow discharge. To determine the pressure differential, we measured pressure both upstream and at the throat of the Venturi flow meter. 1.4 Objective The primary goals of this laboratory experiment encompassed enhancing the precision in the measurement of flow values obtained from the Venturi flow meter. Furthermore, the experiment sought to reaffirm the fundamental connection between flow rates and pressure differentials, thereby contributing to a more comprehensive understanding of the Venturi flow meter's performance characteristics. 2 Approach 2.1 Test Setup and Instrumentation To begin with, the experiment necessitated the use of a Venturi pump to induce a pressure differential within the water flow. Figure 2-1 provides an illustration of the actual Venturi pump employed during the experiment. In addition, valves were affixed to the pipeline to regulate the water flow and maintain a consistent flow rate, with Figure 2-2 depicting the specific valve responsible for adjusting the water flow rate. This valve was positioned downstream of the Venturi pump. To monitor the pressure differential, a pressure gauge was affixed to the Venturi Page 2

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) pump, enabling the measurement of the pressure disparity between the pipe and the Venturi throat. Figure 2-3 presents a visual depiction of the pressure gauge utilized in the experiment, complete with multiple tubes for calculating an average pressure difference. The quantification of water flow involved the use of a weight scale, which measured the quantity of water passing through the pipe. Figure 2-4 showcases the weight scale employed in this capacity. Lastly, a network of water pipes interconnected all the instruments, facilitating the seamless flow of water through the entire experimental setup. Figure 2-5 illustrates the tubing that served as the conduit for the experiment, ensuring the transportation of water to and from the respective instruments. Figure 2.6 shows the pressure gage in operation during the testing phase of the experiment. For an overview of the experimental setup and the instruments used, please refer to Figures 2-1, 2-2, 2-3, 2-4, 2-5, and 2-6. Figure 2-1: Venturi Pump Page 3

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) Figure 2-2: Adjust flow valves. Figure 2-3: Pressure gauge Page 4

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) Figure 2-4: Weighing tank with flowing water. Figure 2-5: Water pipe system Page 5

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) Figure 2-6: Pressure gauge in operation during the tests. 2.2 Test Specimens Table 2-1: Material list Material List Venturi meter pump Flow valve Pressure gage Weight scale/Weighing tank Stopwatch 2.3 Test Procedure The procedure commenced by activating both the pump and the downstream valve. During this phase, water circulated from the sump through the piping system, traversing the Venturi, and then returning to the sump by way of the open valve leading to the weighing tank. The valve near the pump was fine-tuned to establish the initial ten discrete flow cycles through the system. Students were divided into two groups, with one stationed near the weighing tank and the other positioned near the pressure gauge. The dump valve for the weighing tank was sealed to collect Page 6

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) water, and at this juncture, the first group documented the initial and final weights, noting the time interval using a stopwatch. Simultaneously, the second group recorded the pressure differential readings, which served as the basis for the gauge reading. 3 Results 3.1 Tables Table 3-2 - Experimental Values for the water drainage time over 50 lb weight intervals and the corresponding change in pressure for 7 trials. Trials ΔW ( lb ) ΔP ( psi ) ΔP ( psf ) Δt , ( s ) 1 50 4.3 619.2 6.24 2 50 3.8 574.2 7.8 3 50 3.4 489.6 8.01 4 50 3 432 7.42 5 50 2.5 360 7.67 6 50 1.9 273.6 8.81 7 50 0.6 86.4 15.64 Table 3-3 - Calculated flow rate, calibration factor, initial velocity, and Reynolds number form experiment values (data passed on Table 4.1) Trials Q ( ft 3 / s ¿ C V 1 ( ft s ) Re 1 0.13 0.9 18.5 1988.3 2 0.103 0.78 14.8 1590.6 Page 7

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) 3 0.1 0.79 14.4 1548.9 4 0.109 0.91 15.6 1672.1 5 0.104 0.962 15.04 1617.6 6 0.09 0.961 13.1 1408.3 7 0.05 0.963 7.3 793.3 Figure 3-7: Linear relationship between the Reynolds’ number and the calibration constant. Table 3-4 - Comparison of calculated flow rate values. Trials Q 2 =CK( ∆ P 1 / 2 ¿ ( ft 3 / s ¿ Q 1 = ∆ P ¿ α β ¿ Percent Difference (%) 1 0.13 0.12 8.0 2 0.103 0.114 10.1 3 0.1 0.11 9.5 4 0.107 0.103 9.79 Page 8

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) 5 0.104 0.096 8.0 6 0.09 0.086 4.54 7 0.05 0.053 5.83 % Difference Mean 7.97 Figure 3-8: Log-log graph – the relationship between pressure difference and the volumetric flow rate. Table 3-5 - Propagated error for each measurement ( ∆t )/ t (s) ( ∆W )/ W (lb) ∆ ( ∆ P )/ ∆ P (psi) ∆C / C C ∆C 0.00160256 0.01 0.01162791 0.01741652 0.9 0.015675 0.00128205 0.01 0.01315789 0.017861 0.78 0.013932 0.00124844 0.01 0.01470588 0.01860138 0.79 0.014695 0.00134771 0.01 0.01666667 0.01968105 0.91 0.01791 0.00130378 0.01 0.02 0.02130378 0.962 0.020494 0.00113507 0.01 0.02631579 0.02429297 0.961 0.023346 0.00063939 0.01 0.08333333 0.05230606 0.963 0.050371 Page 9

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) ∆ ( ∆W ) = 0.5 lb , ∆ ( ∆t ) = 0.01 s , ∆ ( ∆P ) = 0.05 psi 4 Analysis/Error 4.1 Tables 3.1 – 3.4 The raw experimental data gathered throughout the seven trials is displayed in Table 3.1: The constant weight of water in the weighing tank, denoted as ΔW, is 50 pounds. The measured pressure difference, expressed in psi, between the venturi input and throat is called ΔP. As the flow rate is lowered, these falls. Table 3.2 consists of the volumetric flow rate, or Q, is determined using ΔW and Δt. The calibration constant C is determined by utilizing the values of Q and ΔP. Based on Q, V1 is the starting velocity in the input pipe. The Reynolds’ number, or Re, indicates whether a flow is turbulent or laminar. The measured Q values and those derived from the ΔP vs. Q relationship are contrasted in Table 3.3. The calibration is confirmed by the near match. The propagating measurement uncertainty based on errors in ΔW, Δt, and ΔP is displayed in Table 3.4. Reasonably minor mistakes are shown by ΔC/C values. 4.2 The verification of Q and ∆ P through computation. Within an 8% discrepancy, the flow rates Q2 computed using the calibration constant equation nearly correspond to the flow rates Q1 found using the ΔP vs. Q connection. This confirms the basic idea between pressure drop and flow rate by showing that the measured ΔP values agree well with the measured flow rates Q. Calculation approximations or tiny measurement errors may be the cause of slight discrepancies. The calibration of the venturi meter is validated overall by the agreement. 4.3 The advantages of Log-log plotting. Over a broad range of values, the power law relationship between ΔP and Q is linearized by log-log scaling. This facilitates the generation of the formula linking the variables by making Page 10

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) it simple to identify the exponents α and β. Higher flow rates would result in the values being squeezed on a normal scale. For improved viewing, log-log spreads them out. Log - log scale also provides information about possible measurement mistakes or departures from the best possible theoretical behavior. 4.4 Propagated Error Based on errors in W, t, and ΔP, the partial derivates were deduced to estimate the relative error ΔC/C. The partial derivatives can be found in the appendix of this lab report. 4.5 Errors (random, systematic, etc.) Random errors expected from uncertainty in reading analog scales and stopwatches. Approximations in calculations and curve fitting could also contribute to errors. Systematic errors could occur in pressure or flow measurements if instruments are mis-calibrated. 5 Conclusions/Recommendations Through the “Calibration of a Venturi Flow Meter” experiment, we explored the Venturi flow meter’s performance by the recorded pressure, time, weight, calibration factor, initial velocity, and Reynolds number. The experiment reaffirmed the essential connection between the flow rates and pressure differentials, to understand the Venturi flow meter’s capabilities and significance in various applications. Examining the calculated results, the following can be concluded:  Based on table 3-3, the mean percentage difference is around 8%.  Based on table 3-4, the average calibration factor is 0.90. Page 11

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3)  The relationship between the volumetric flow rate (Q) and Pressure differential (ΔP) was determined by the exponential function derived. The relationship was shown in Figure 4-2.  The calculated flow rates Q2 closely align with the measured flow rates Q1 with an 8% discrepancy, which confirmed the relationship between pressure drop and flow rate.  The advantage of using a log-log scale when overseeing the range of the values collected is that it compressed it into a more manageable linear range. Compared to utilizing a power function, it allowed for the development of a formula which would accurately calculate the flow rates based on the data collected such as pressure. Recommendations that can be used to improve the experiment if done again:  More than one person is to record the time the trails. With the average of the recorded time for each trial, the time data would be more accurate.  To ensure an accurate time measure when the tank is filled to a particular weight, the water in the tank should be removed for each trial for little errors to occur. 6 References Gerhart. (n.d.). Video library . Gerhart, Hochstein: Munson, Young and Okiishi’s Fundamentals of Fluid Mechanics, 8th Edition - Instructor Companion Site. https://bcs.wiley.com/hebcs/Books? action=resource&bcsId=9930&itemId=1119080703&resourceId=39618 Hyoungsu, Park (2023). Lab3_F23. Retrieved from Page 12

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) https://laulima.hawaii.edu/access/content/group/MAN.XLSCEE320hp.202410/Lab/Lab3/la b3_F23.pdf 7 Appendix A-1 Equations: Bernoulli Equation: P 1 + 1 2 ρv 1 2 + z 1 = P 2 + 1 2 ρv 2 2 + z 2 Alternative version: P 1 + 1 2 ρv 1 2 + ρgh 1 = P + 1 2 ρv 2 2 + ρg h 2  P: pressure  ρ : density  v : Fluid Velocity  g : Force of gravity  h : height Volumetric Discharge: Q = ∆W γ ∆t  Q: Volumetric Discharge  γ : Specific weight of water (9.807 kN / m 3 )  ∆W : Weight flow rate  ∆t : Avg measured time. Pertinent Overall Equation: Q = CK √ ∆P , where K and C can be found using: K = A 2 √ 2 g γ [ 1 − ( D 1 D 2 ) ] C = Q K √ ∆ P Page 13

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3)  Q: Volumetric Discharge  C: Calibration constant  K: Constant for a given venturi installation  ∆ P : Pressure difference * Q = β ( Δ p ) α ß Relationship between Q and Δ p . Initial Velocity: D 1 ¿ 2 π 4 ¿ V 1 = Q A 1 = Q ¿  V 1 : Initial Velocity  Q: Volumetric Discharge  A 1 : Area of section 1 Flow Discharge Difference: ∆Q = | Q 2 − Q 1 | Q 2 + Q 1 2 × 100%  ∆Q : Flow discharge difference  Q 2 : Experimental values of flow rate or volumetric discharge  Q 1 : Graphed values of flow rate or volumetric discharge Reynolds Number : R e = VD ν  R e : Reynolds number  V : Fluid Velocity Page 14

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CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3)  D : Diameter  ν : Fluid Kinematic Viscosity Error Propagation – from W, t, and ∆ P to C: ∆C = ∂f ∂W ∆W + ∂ f ∂t ∆t + ∂f ∂ ( ∆ P ) ∆ ( ∆ P )  W: measured weight  t: measured time  ∆ P : measured pressure differential Partial Derivatives for Error Propagation: Given: C = f ( W ,t ,∆ P ) = W Kγt √∆ P , ∆C = ∂f ∂W ∆W + ∂ f ∂t ∆t + ∂f ∂ ( ∆ P ) ∆ ( ∆ P ) ∂ f ∂W ∆W = ( W Kγt √ ∆ P ) d dW = W ( 1 Kγt √ ∆ P ) d dW = ( 1 Kγt √ ∆ P ) ( ∂ f ∂t ∆W C ) = ( 1 Kγt √ ∆ P ) W Kγt √ ∆ P = ∆W W ∂f ∂t ∆ t = ( W Kγt √ ∆ P ) d dt = ( W Kγ √ ∆ P ) t − 1 d dt = − W Kγ √ ∆ P ( t − 2 ) ( ∂ f ∂t ∆t C ) = − W Kγ √ ∆P ( t − 2 ) W Kγt √ ∆ P = − − W Kγt 2 √ ∆P W Kγt √ ∆ P = − ∆t t = | − ∆t t | = ∆t t Page 15

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) ∂ f ∂ ( ∆P ) ∆ ( ∆ P )= ( W Kγt √ ∆ P ) d d ∆P = − W 2 Kγt √ ∆ P 3 ( ∂ f ∂t ∆ ( ∆ P ) C ) = − W 2 Kγt √ ∆ P 3 W Kγt √ ∆ P = − 1 ∆ P ( ∆ P ) 2 ∆ P = | − 1 ∆ P ( ∆ P ) 2 ∆P | = 1 ∆ ( ∆ P ) 2 ∆ P ∨ 1 2 ∆ ( ∆ P ) ∆P Therefore, with the partial derivatives and the calibration constant, the relative error formula is: ∆C C = ¿ ∆W W + ∆t t + 1 ∆ ( ∆ P ) 2 ∆ P . A-2 Sample Calculations: Table 3.1 *Using sample 1. psi to psf: 0.6 psi ( 144 ¿ 2 ft 2 )= 86.4 psf Table 3.2 *Using sample 1. Q = ∆W γ ∆t = 50 lb 62.41 lb ft 3 ( 6.24 s ) = 0.128389 C = 0.13 f t 3 / s 0.006 √ 86.4 psf = 0.96 3 V 1 = 0.13 f t 3 / s π 4 ( 2 12 ft ) 2 = 7.3 ft / s Re = 18.5 ft / s ( 2 12 ft ) 0.00155 = 793.3 Table 3.3 Q 2 = 0.963 ⋅ 0.0057 √ 86.4 = 0.05 f t 3 / s *Using sample 7. Q 1 = 0.0082 ⋅ 86.4 0.4179 = 0.053 f t 3 / s *Using sample 7. Page 16

CEE 320L Fluid Mechanics Fall 2023 Laboratory (No.3) % difference: | Q 2 − Q 1 | Q 2 + Q 1 2 × 100% = | 0.05 − 0.053 | 0.05 + 0.053 2 × 100% = 5.8252% *Using sample 7. x = ∑ x n = 8 + 10.1 + 9.5 + 9.79 + 8 + 4.54 + 5.83 7 = 7.96642 Table 3.4 *Using sample 1. ( ∆t ) t ( s ) = 0.01 6.24 = 0.00160256 ( ∆W ) W ( lb ) = 0.5 50 = 0.01 ∆ ( ∆ P ) ∆ P ( psi ) = 0.05 4.3 = 0.01162791 ∆C C = ( ∆W ) W + ( ∆t ) t + 1 2 ∆ ( ∆ P ) ∆ P = 0.01 + 0.00160256 + 1 2 ( 0.01162791 ) = ¿ 0.01741652 ∆C = ∆C C ( C ) = 0.01741652 ( 0.9 ) = 0.015675 A-2 Raw Data: Table 7-6 – Raw experimental data Trials Weight (lb) Time (s) Gage Pressure (psi) 1 50 6.24 4.3 2 50 7.8 3.8 3 50 8.01 3.4 4 50 7.42 3 5 50 7.67 2.5 6 50 8.81 1.9 7 50 15.64 0.6 Page 17

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