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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:
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.
This equation was used to calculate volumetric flow rate and velocity of the fluid.
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This report uses measurements from experiment runs 2 and 3 to establish static pressure head and total head values.
The following equipment was used in this experiment:
The experiment was conducted as follows:
| Volume of Water Collected (ml) | Time Taken to Collect Water (s) | Volumetric Flow Rate (μm^3/s) | Location | Diameter (mm) | Area of Cross-Section (mm^2) | Velocity v (m/s) | Manometer Identifier | Static Head h (mm) | h v^2/(2g) (mm) | Total Head (mm) Measured |
|---|---|---|---|---|---|---|---|---|---|---|
| 500 | 5.15 | 97.0874 | A | 25 | 0.4909 | 0.1978 | 1 | 181 | 182.9938 | 184 |
| 500 | 5.41 | 92.421 | B | 13.9 | 0.1517 | 0.6397 | 2 | 164 | 184.8635 | 181 |
| 500 | 5.37 | 93.11 | C | 11.8 | 0.1094 | 0.8451 | 3 | 146 | 182.4031 | 180 |
| 500 | 5.28 | 94.697 | D | 10.7 | 0.0899 | 1.0355 | 4 | 129 | 183.6484 | 180 |
| 500 | 5.30 | 94.34 | E | 10 | 0.0785 | 1.2057 | 5 | 106 | 180.0958 | 174 |
| Volume of Water Collected (ml) | Time Taken to Collect Water (s) | Volumetric Flow Rate (μm^3/s) | Location | Diameter (mm) | Area of Cross-Section (mm^2) | Velocity v (m/s) | Manometer Identifier | Static Head h (mm) | h v^2/(2g) (mm) | Total Head (mm) Measured |
|---|---|---|---|---|---|---|---|---|---|---|
| 390 | 3.15 | 123.8 | A | 25 | 0.4909 | 0.2522 | 1 | 260 | 263.2424 | 267 |
| 365 | 3.15 | 115.9 | D | 10.7 | 0.0899 | 1.2886 | 4 | 118 | 202.6351 | 267 |
| 365 | 2.96 | 123.3 | E | 10 | 0.0785 | 1.5700 | 5 | 57 | 182.6387 | 260 |
| 485 | 4.22 | 114.9 | F | 25 | 0.4909 | 0.2341 | 6 | 132 | 134.7940 | 150 |
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.
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.
Lab Report: Application of Bernoulli Equation. (2024, Jan 02). Retrieved from https://studymoose.com/document/lab-report-application-of-bernoulli-equation
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