Lab Report: Application of Bernoulli Equation

Introduction

The aim of this laboratory report is to demonstrate an understanding of Bernoulli's equation by examining the flow of water through a convergent and divergent pipe. Bernoulli's equation relates the pressure, speed, and height of any two points in a pipe during a steady flow of fluid. It is based on the conservation of energy, where the sum of kinetic energy, pressure energy, and potential energy remains constant between two points in the flow. The Bernoulli equation can be expressed as:

(1) p1ρg + v1^2/2 + z1 = p2ρg + v2^2/2 + z2 (m)

Where:

• p: Pressure at the chosen point
• ρ: Density of the fluid at all points
• g: Acceleration due to gravity
• v: Fluid flow speed at a point on a streamline
• z: Elevation of the point above a reference plane

For horizontal flow, potential energy remains constant, simplifying Bernoulli's equation to:

(2) p1ρg + v1^2/2 = p2ρg + v2^2/2 (m)

The assumptions for applying Bernoulli's theorem include continuous flow, incompressible flow, and minimal viscous friction.

The flow in the tube is governed by the continuity equation:

(3) A1v1 = A2v2 (m^3/s)

Where A is the cross-sectional area and v is the flow velocity.

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This equation was used to calculate volumetric flow rate and velocity of the fluid. This report uses measurements from experiment runs 2 and 3 to establish static pressure head and total head values.

Apparatus and Procedure

The following equipment was used in this experiment:

• Hydraulic bench
• Convergent and divergent pipe with static pressure tappings
• Flow measurement by timed volume collection
• Stopwatch
• Spirit Level
• Manometer
• Pitot tube

The experiment was conducted as follows:

1. Ensure that the test segment has a 14° tapered section that converges in the flow direction.
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2. Position the outflow tube above the volumetric tank to allow timed processing.
3. Attach the inlet of the test rig to the flow source, open the bench valve and flow control valve, and fill the pump with water.
4. Calculate the highest permitted volume flow rate based on the measurement of maximum height (h1) and minimum height (h5).
5. Adjust the manometer level using the air bleeding screw and the hand pump.
6. Pump air through the air valve using the hand pump and then close the screw to maintain pressure.
7. Take measurements, including flow rate and manometer readings, while ensuring minimal timing errors.
8. To calculate the volume flow rate at the optimal flow rate, close the ball valve and time how long it takes to collect a known amount of fluid in the tank.
9. Repeat the experiment at least twice to ensure consistency and calculate the average volume flow rate.
10. Record total pressure at various points in the test section using a total pressure probe.
11. Repeat these measurements for different volume flow rates.

Experimental Results

Experiment Run 2

Experiment Run 3

Discussions

The experiment in run 2 showed similarities between theoretical and experimental values, where the measured and calculated values of pressure head are plotted. Although the values do not overlap, the trend is similar. The minor differences may be attributed to simplifications in the equation, not accounting for friction losses in real fluids, or systematic errors in the experiment, such as faulty equipment or parallax error in reading values.

In run 3, there are limited similarities between the theoretical and measured values. The differences may arise from variations in the volume of water collected. To improve accuracy, future experiments should control the volume of water collected.

Conclusion

The experiments successfully demonstrated the application of Bernoulli's equation in fluid dynamics. While the theoretical model may not always reflect real fluids due to friction losses, it provides valuable insights into the behavior of fluids in real-world scenarios.

Bibliography

1. Energyeducation (2018). Bernoulli's equation - Energy Education. Available at: https://energyeducation.ca/encyclopedia/Bernoulli's_equation
2. Khan Academy (n.d.). What is Bernoulli's equation? (article). Available at: https://www.khanacademy.org/science/physics/fluids-fluid-dynamics/a/what-is-bernoullis-equation
3. Lumenlearning.com. (2019). Bernoulli's Equation | Physics. Available at: https://courses.lumenlearning.com/physics/chapter/12-2-bernoullis-equation/
4. Talay, T.A. (1975). Introduction to the aerodynamics of flight. Available at: https://practicalaero.com/wp-content/uploads/2010/04/NASA-SP-367.pdf
5. TecQuipment (2019). Daniel Bernoulli: Bernoulli's Principle and Equation. Available at: https://www.tecquipment.com/knowledge/2019/daniel-bernoulli-bernoullis-principle-and-equation
6. Wiki2 (2016). Bernoulli's principle - Wikipedia Republished WIKI 2. Available at: https://wiki2.org/en/Bernoulli's_principle