Conservation of Mass and Energy in Fluid Mechanics: Venturi Experiment, Papers of Thermodynamics

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ChE 2412

Conservation of Mass and Energy

Submitted by: Meghna Mandaliya

Due date: January 31 , 2024

Instructors: Prof. M. Kwok & Dr. D. Roach

Letter of Transmittal Meghna Mandaliya Student, Dept. Of Engineering PO Box 5050 Saint John, NB E2L 4L February 2 , 2024 Prof. M. Kwok & Dr. D. Roach, PEng Dept. Of Engineering PO Box 5050 Saint John, NB E2L 4L Dear Prof. Kwok and Dr. Roach, I am pleased to submit the enclosed engineering lab report titled "Conservation of Mass and Energy" as part of the Fluid Mechanics Lab course (ChE 2412) as partial fulfillment of my grade in this course. The purpose of this report is to study the two fundamental concepts of Fluid Mechanics: mass and volume flowrates, and the conservation of energy. These concepts are applied to the flow of water through a converging-diverging nozzle, more commonly known as Venturi. Sincerely, Meghna Mandaliya

Table of Contents

• Letter of Transmittal
• Abstract
• Table of Contents..........................................................................................................
• Introduction
  • Purpose
  • Objectives
  • Background Theory
• Apparatus and Procedure
  • Apparatus
  • Procedure
• Results.......................................................................................................................
• Discussion
• Conclusion
• References
• Appendix A
• Appendix B
• Appendix C

Introduction

Purpose

This experiment introduces the study of Fluid Mechanics and two of its fundamental concepts: mass and volume flow rates, and energy conservation. These will be applied to the flow of water through a converging-diverging nozzle, more commonly known as a Venturi.

Objectives

The objectives of this exercise are to:

1. Study the relationship between flow rate, flow area, and fluid speed using the 1D assumption and apply it to the flow of water through Venturi.
2. Experimentally verify the Bernoulli principle (conservation of energy).
3. Perform flow rate measurements with a hydraulic bench
4. Perform pressure measurements with piezometers.

Background Theory

A standard technique for measuring pressure involves the use of liquid columns in vertical or inclined tubes. Pressure measuring devices based on this technique are called manometers. Three common types of manometers include the piezometer tube, the U-tube manometer, and the inclined-tube. In this experiment we used a manometer with 11 piezometer tubes. Although the piezometer tube is a very simple and accurate pressure measuring device, it has several disadvantages. It is only suitable if the pressure in the container is greater than atmospheric pressure otherwise air would be sucked into the system, and the pressure to be measured must be relatively small so the required height of the column is reasonable. Also, the fluid in the container in which the pressure is to be measured must be a liquid rather than a gas (Gerhart et al., 2021). The relationship between flow rate, flow area, and fluid speed using the 1D assumption can be written as: 𝑄 = 𝑉𝐴 ............................................................ eq. 1 Where, Q = Volumetric Flow rate of fluid

𝜌𝑉𝑖^2 2

• 𝜌𝑔ℎ𝑖 = 𝐶.......................................... eq. 5 Where, p = pressure ρ = density of the fluid V = fluid velocity g = gravitational acceleration h = height of the position on the streamline measured relative to a reference datum C = Bernoulli constant For a fluid, in addition to kinetic and potential energies, energy may also be stored as pressure. The pressure is similar to a spring under compression; the energy is stored in it and can be released to produce some other form of energy. If we produce a change in one of the forms of energy, the others must change for the total energy of the particle to be unchanged. To produce such redistributions of the energy in pipe flows we often change the flow area. Since the energy at all points on the streamline must be the same, we can write: 𝑝 1 + 𝜌𝑉 12 + 𝜌𝑔ℎ 1 = 𝐶 = 𝑝 2 + 𝜌𝑉 22 + 𝜌𝑔ℎ 2 ................................... eq. 6 (D. Roach, 2022).

Apparatus and Procedure The piezometer tubes in the manometer of this experiment enable us to make sure that the pressure at every cross-sectional area remains constant. The control and supply valves help us to manually adjust pressure levels in each piezometer tube so that we have good reference levels. The hydraulics bench helps us to collect the same mass of water for every trial for uniform results.

Apparatus

This experiment involved the use of a hydraulics bench, Venturi, stopwatch and total of 30 kg weight. Figure 1 Flow diagram of the apparatus, showing the path of water followed during the experiment

Figure 4 A complete apparatus set up.

Procedure

1. The piezometer tubes in the manometer were already leveled at the start of the experiment.
2. We checked the supply and control valve to make sure that they were closed.
3. We turned on the pump and opened the supply valve with caution since we did not want the water level in piezometer tubes to go above 240 mm.
4. Then, we gradually opened the control valve until there was sufficient flow and bubbles from the piezometers had cleared out.
5. We increased the flow using the control valve until the level in piezometer D was nearly zero.
6. We used supply valve and control valve to maintain piezometer A at 240 mm and piezometer D at 0 mm without having any air bubbles in any tubes.
7. We measured the time to collect 30 kg of water using the hydraulics bench and a stopwatch and noted our reading in our field notes.
8. Once water is collected, we noted down the piezometer levels.
9. We adjusted the control valve until the level of piezometer D was approximately 90 mm and then adjusted the supply valve until the level of piezometer A was approximately 240 mm. If the level of piezometer A rose above 240 mm, we reduced it by adjusting the supply valve and then readjusting the control valve.

10. Repeated steps 7 and 8.
11. For the final trial, we adjusted the control valve until the level of piezometer D was approximately 150 mm and then adjusted the supply valve until the level of piezometer A was approximately 240 mm. If the level of piezometer A rose above 240 mm, we reduced it by adjusting the supply valve and then readjusting the control valve.
12. Repeated steps 7 and 8.
13. Closed the supply valve and turned off the pump.
14. Closed the control valve (D. Roach, 2022). Results Field notes taken during the laboratory can be found in Appendix A. Sample calculations can be found in Appendix B, and Appendix C shows the excel sheet. The raw data can be found in Appendix A. Mass flow rates are calculated by using the following relationship: 𝑚𝑑𝑜𝑡 = 𝑚𝑎𝑠𝑠 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 𝑡𝑖𝑚𝑒 ........................................... eq. 6 The volumetric flow rate can be calculated by manipulating Equation 2, 𝑄 = 𝑚𝑑𝑜𝑡 𝜌 The estimated average velocity at a point in the Venturi, Vi, can be calculated from Equation 1, 𝑉𝑖 = 𝑄 𝐴𝑖 where Ai is the cross-sectional area at the point of interest. The pressure at each of the pressure taps is calculated from: 𝑝𝑖 = 𝜌𝑔ℎ𝑖........................................................... eq. 7

0.974 1991 341 C 1.288 1501 831 D 1.168 1540 792 E 0.966 1825 508 F 0.811 1942 390 G 0.691 2031 302 H 0.596 2080 253 J 0.518 2109 223 K 0.488 2129 204 L Figure 5 Plot for Velocity of Fluid in each trial vs cross-sectional area of the venturi at a point Figure 5 represents a graph of estimated average velocity as a function of cross-sectional area for all three flow rates. These velocities and cross-sectional areas are calculated using the data noted in field notes by an excel spreadsheet.

0.000000 0.000100 0.000200 0.000300 0.000400 0.000500 0. Velocity (m/s) Cross-sectional Area (m^2)

Velocity vs Cross-sectional Area

Trial 1 Trial 2 Trial 3

Figure 6 Plot for Velocity of fluid in each trial vs locations on the Venturi Figure 6 represents the estimated average velocity at each of the locations specified in Table 1 of lab 1 handout posted on D2L page of CHE 2412 for all three flow rates. Figure 7 Plot for Pressure vs Locations on the Venturi 0

1

2

-75 -60 -45 -30 -15 0 15 30 45 60 75 90 105 120 Velocity (m/s) Locations on the Venturi (mm)

Velocity vs Locations on the Venturi

Trial 1 Trial 2 Trial 3

Figure 9 Plot for Energy lost vs Locations on the Venturi Figure 9 represents a plot of the lost energy, ΔC, at each of the locations specified in Table 1 of lab 1 handout posted on D2L page of CHE 2412 for all three flow rates.

• 0 200 400 600 800 1000 1200 1400 1600 -80 -60 -40 -20 0 20 40 60 80 100 120 Energy Lost (J) Locations on the Venturi (mm)

Energy Lost vs Locations on the Venturi

Trial 1 Trial 2 Trial 3

Discussion The results obtained from this experiment explain a lot about the Law of Conservation of mass and energy under 1D assumption. In figure 5 , all the three trials for different mass flow rates show a similar pattern. When the cross-sectional area of the venturi increases, the velocity of the fluid decreases. This relationship is a consequence of the conservation of mass, as the total mass flow rate of the fluid remains constant throughout the Venturi tube. In simpler terms, as the fluid moves from the narrower to the wider section, the increase in the cross-sectional area allows the fluid to spread out, causing a reduction in velocity. Assumptions used to obtain these results are: Incompressibility: The fluid is assumed to be incompressible, meaning its density remains constant throughout the flow. Steady Flow: The experiment assumes steady-state conditions, meaning that the flow parameters (such as velocity and pressure) do not vary with time. Negligible Viscosity: Viscous effects within the fluid are assumed to be negligible. While viscosity is a fundamental property of fluids, in this case, its impact on the overall flow is considered small, allowing for a simplified analysis. No Heat Transfer: The experiment focuses solely on mass flow rates and velocities, neglecting any heat transfer considerations since the pressure is small throughout the experiment. The plot of the estimated average velocity as a function of cross-sectional area looks like a hyperbolic curve. In Figure 6 , we see that the velocity steadily increases before 0 on x-axis and then starts to decrease. As discussed before, increase in velocity indicates decrease in cross-sectional area and vice versa. We also observe that at the first and last location on the Venturi, the velocity of the fluid remains the same. By looking at the respective cross-sectional area at each location on the Venturi given in Table 1 of lab 1 handout posted on D2L page of CHE 2412, our observations confirm the law of conservation of mass. Bernoulli's equation relates the fluid velocity, gravitational potential energy, and pressure at different points along a streamline. In the Venturi tube, as the cross-sectional area decreases, the fluid velocity increases according to the continuity equation. According to Bernoulli's equation, an increase in fluid velocity is associated with a decrease in pressure. Therefore figure 8 shows a decrease in pressure in the narrower section and an increase in pressure in the wider section of the Venturi tube. The relationship between figure 6 and figure 7 is interconnected through the principles of fluid dynamics, with changes in velocity inversely affecting pressure according to

Conclusion Through the application of the 1D assumption, the experiment successfully investigated the relationship between flow rate, flow area, and fluid speed. The study provided a foundation for understanding how decrease in cross-sectional area leads to increase in pressure and velocity of fluid. The experiment aimed to validate the Bernoulli principle that asserts the conservation of energy in fluid flow. Although we did not get ideal results related to loss of energy across the Venturi tube, this experiment helped us understand the factors that lead to discrepancies in data in the real world. The experiment's practical aspect involved using a hydraulics bench for accurate and controlled flow rate measurements. This not only demonstrated the application of theoretical concepts in a real-world setting but also highlighted the significance of precision and instrumentation in fluid mechanics experiments. The utilization of piezometers for pressure measurements contributed crucial data on pressure variations within the Venturi tube. In conclusion, the experiment successfully achieved its objectives by combining theoretical knowledge with hands-on practical applications and enhanced the understanding of fluid dynamics, particularly in the context of Venturi flow, and demonstrated the importance of experimental verification in validating fundamental principles.

References Gerhart, A. L., Hochstein, J. I., & Gerhart, P. M. (2021). Munson, Young and Okiishi's fundamentals of Fluid Mechanics. Wiley. Roach, D. (2022). Handout Lab #1 - Conservation of Mass and Energy.