CE 331 L Mauriot Lab Report 4

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CE 331 – INTRODUCTION TO FLUIDS MECHANICS AND HYDRAULICS LABORATORY Section Number: L-M02 Experiment Number: 4 Experiment: Pressure Transducers Submitted by: Louis Mauriot Submitted to: Zada Tawalbeh Date Experiment Performed: 02/23/23 Date Experiment Report Submitted: 03/02/23 Name of People Who Participated: Louis Mauriot Jarret Lee Dante Salazar John Paul Slape

Table of Contents Introduction ................................................................................................................................................. 2 Objectives .................................................................................................................................................... 3 Experimental Set-Up .................................................................................................................................... 3 Diagram 1: Schematic of experimental set up and wiring for pressure transducer and submerged water logger ........................................................................................................................................ 5 Results ......................................................................................................................................................... 5 Discussion and Conclusion ........................................................................................................................... 8 Appendix ................................................................................................................................................... 11 References ................................................................................................................................................. 12 List of Figures Figure 1: Response of Transducer Figure 2: Relationship between Head and Voltage Output Figure 3: Raw Sample Calculations for Increasing Head in Hypothetical Tank Figure 4: Raw Sample Calculations for Decreasing Head in Hypothetical Tank List of Tables Table 1: Coefficients Table Table 2: Increasing Pressure Head Raw Data Table 3: Decreasing Pressure Head Raw Data List of Graphs Graph 1: Increasing Water Depth vs. Voltage Graph 2: Decreasing Water Depth vs. Voltage Graph 3: Water Depth vs. Pressure Gage Graph 4: Water Depth vs. PS9805 Water Level 1

Introduction A pressure transducer is an instrument that estimates pressure head with the assistance of a voltmeter and batteries. This process is possible since the current flowing through, powered by the batteries, passes through a resistor which influences the amount of resistance that travels through. The latter is influenced by the amount of pressure appliance. Since we are studying fluid mechanics and hydraulics, the pressure we will be measuring will be of water. To make this happen, the resistor is therefore placed in direct contact with the water located inside a water column. The resistor is placed at the very bottom of the water column. As more water is added, the more pressure the resistor will be submitted to. The amount of pressure directly affects the amount of resistance. Since we have a constant amount of current flowing through our set up, our conclusion was that resistance was proportional to voltage. With the experimental apparatus we had available, there was no way to directly measure the resistance. However, by utilizing the voltmeter, we could obtain the direct relationship between the pressure head and the electrical signal and therefore obtain the actual measurement of any unknown head. Objectives Mastering the use of a pressure transducer and utilizing it to determine the two sets of coefficients a and b for the linear relationship between pressure head and transducer to follow up with the determination of the head of a column of water of unknown depth and finally mastering the accuracy of a PS9805 Water level to estimate pressure head were the objectives of this experiment. Experimental Set-Up 2

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For this lab we were equipped with the following experimental apparatus: - A standard differential pressure transducer - One 12-V battery - A set of connecting wires - 1 Voltmeter - An 8 ft. water column with water level control tubes on the side - A PS9805 Water level transducer and Campbell Scientific® logging software - A connecting data cable - A laptop computer We started by pouring water into the water column, one foot heigh at a time. For each foot of water poured, the head data collected by the PS9805 Water level and Campbell Scientific® logging software was recorded. This process was repeated until we reached 5 feet high. Another group measured, using an alternate water column measure the voltmeter and pressure readings at each foot of water poured using a voltmeter and a pressure gage. All data was recorded in the tables which can be found in the Appendix of this lab report. 3

Diagram 1: Schematic of experimental set up and wiring for pressure transducer and submerged water logger Results Figure 1: Response of Transducer V output = V 1 + R ( P ) Where: V output = the voltageoutput by thetransducer ,volts V 1 = aconstant calledthe null voltage,volts R ( P ) = the responseof transducer ¿ pressure,unitless 4

Figure 2: Relationship between Head and Voltage Output V output = a + b∙ H ↔H = V output − a b Where: V output = the voltageoutput by thetransducer ,volts H = the pressure head ∨ head ,ft a ∧ b = linear calibrationcoefficients,unitless Graph 1: Increasing Water Depth vs. Voltage Graph 2: Decreasing Water Depth vs. Voltage 5

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Graph 3: Water Depth vs. Pressure Gage Graph 4: Water Depth vs. PS9805 Water Level 6

Table 1: Coefficients Table Regression Coefficient Correlation Coefficient (R 2 ) Increasing Water Depth vs. Voltage 0.656 0.9997 Decreasing Water Depth vs. Voltage 0.856 0.9352 Water Depth vs. Pressure Gage 0.4073 0.9981 Water Depth vs. PS9805 Water Level Head 1.0326; 1.039 0.9979; 0.998 Discussion and Conclusion Pressure and depth are important to distinguish and understand in the domain of fluid hydraulics since we discovered that as the depth increases, the pressure increases as well as we collected data in this laboratory experiment. Pressure is a fundamental factor of the mathematical form of Bernoulli’s principle as we can see in the following equation: P 1 + 1 2 ρV 1 2 + ρg h 1 = P 2 + 1 2 ρV 2 2 + ρgh 2 Where: P 1,2 = Pressure (Pa) This principle can be applied in cases where conducts such as pipes change size causing a change in pressure (Khan Academy, 2018). A second application of pressure is in the automobile industry. The brake pressure is measured in PSI and depends on the hydraulic fluid that is brake fluid. It serves as a direct transfer of force into pressure (O’Reilly Auto Parts, 2023). 7

As head increases, pressure increases as well. The relationship between these two latter can be defined by the utilized fluid’s specific gravity. The pressure/head relationship can be qualified as being dependent on which fluid is chosen. The relationship between pressure and head often arises when working with pumps. (What Is the Difference between Pressure and Head?, 2016). Based on observations from collected data in this lab experiment, null voltage represents a null pressure and head. Based on the resulting graphs obtained for the voltage vs. water depth from a visual point of view, the relationship between these chosen variables does seem to be linear. This is confirmed by the chosen linear trendline that seems to fit the graph accurately and follows all data points. The V= a+b·H equation looks like a good representation of the corresponding linear relationship observed on each data set. Firstly, we can refer back to the basic core definition of a linear function. The equation is always under the following form: y = ax + b For each graph, the resulting linear equation can be applied to this definition as follows: y = 0.656x + 1.3833 We can observe here that our “a” is 0.656 and our “b” is 1.3833. Let’s now examine the Voutput equation: V= a+b·H We can rearrange it and obtain it under this form: V=b·H + a 8

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This form of the equation lines up our initial linear function definition where we have b as “a”, H as our “x” variable, and a as “b”. Therefore, we can conclude that the V= a+b·H is a good representation of the relationship between the voltage and head for each data set. In every theoretical problem, for each foot of water, this following equation can be used to obtain the actual pressure (in PSI) (Communications, n.d.): Pressure = 14.7 psi + 0.445 x psi Where: 14.7 psi is the reference pressure at sea level. 0.445 is the coefficient per each extra foot above sea level, this is the amount of extra pressure applied. x is the amount of feet above sea level Knowing this, we can compare both slopes from each equation. We obtained in our lab experiment a coefficient of 0.4073 whereas the actual theoretical equation is 0.445. This latter was our expected value. Comparing both, the actual numerical numbers being only slightly different by a few digits therefore most likely having a very small percentage error, this experimental obtained value could be considered acceptable. The slope and intercept of this line equation for the graph of the head measured by the PS9805 level transducer that the sensor itself is thoroughly accurate but not perfect. We can utilize the linear regression coefficients obtained in a multiple linear regression model to try and obtain more accurate readings from this sensor. The assumptions that affected the accuracy of our estimate of the unknown height were the data we had already collected in our tables, between the voltmeter reading and the assumption it 9

corresponded to the PS9805 head reading for the corresponding reference theoretical increasing/decreasing head. The linearity of the coefficients used for the unknown water depth aren’t completely linear. We can refer to the correlation coefficients from each graph. Ideally, if the slope was perfectly linear, they would be equal to 1. Both coefficients are very close to 1 but not quite equal therefore contributing to our source of errors. In conclusion, this lab allowed us to master the core purpose of a pressure transducer and allowed us to understand the different relationships that exist between voltage, pressure head and pressure. Naturally, there were some sources of error accumulated along the way that started with reading the water levels directly on the water column. The PS9805 pressure transducer ended up being fairly accurate but we discovered through data analysis that it has its limits. Appendix Figure 3: Raw Sample Calculations for Increasing Head in Hypothetical Tank V output = a + b∙ H ↔H = V output − a b = 4 − 1.3833 0.656 = 3.99 volts Figure 4: Raw Sample Calculations for Decreasing Head in Hypothetical Tank V output = a + b∙ H ↔H = V output − a b = 4 − 0.65 0.856 = 3.91 volts Table 2: Increasing Pressure Head Raw Data Increasing Pressure Head (ft) Gage Pressure (PSI) Voltmeter Reading (Volts) PS9805 Head (ft) 10

0 0 1.4 0 1 0.3 2 1.28 2 0.8 2.72 2.28 3 1.2 3.35 3.25 4 1.6 4.01 4.2388 5 2.1 4.66 5.259 Table 3: Decreasing Pressure Head Raw Data Decreasing Pressure Head (ft) Gage Pressure (PSI) Voltmeter Reading (Volts) PS9805 Head (ft) 5 2.1 4.66 5.269 4 1.6 4.01 4.28 3 1.2 3.35 3.29 2 0.8 2.72 2.27 1 0.3 2 1.28 0 0 0 0 References What is the difference between pressure and head? (2016, May 31). GemmeCotti. https://www.gemmecotti.com/pump-pressure-vs-head/ Communications, G. E. O. of M. and. (n.d.). Calculating Underwater Pressure . Van.physics.illinois.edu. Retrieved February 28, 2023, from https://van.physics.illinois.edu/ask/listing/2233 Khan Academy. (2018). What is Bernoulli’s equation? Khan Academy. https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-bernoullis- equation O’Reilly Auto Parts . (2023). Oreillyauto.com. https://www.oreillyauto.com/shop/b/oil--chemicals--- fluids/maintenance-chemicals/brakes/brake-fluid/371c20e7cfcc New Mexico State University. Retrieved from class website: CE 331 Lab 4 Pressure Transducers 11

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