Example Of Report On Bernoulli’s Experiment
Experiment conducted on [overtype date of experiment here]
Lab Group Number: [Overtype your lab group number here]
List of Tables 2
List of Figures 2
List of Appendices 2
Chapter 1 Introduction 4
Notes on writing the report 4
Notes on using the template 4
Chapter 2 Apparatus and experimental procedure 6
Experimental Procedure 6
Chapter 3 Calculations and results 8
Chapter 4 Analysis and Discussion 9
Chapter 5 Conclusion 11
List of Tables
List of Figures
Figure 1.1 This is figure title style 5
List of Appendices
Appendix A Title of Appendix A
The aim of the experiment was to use Bernoulli’s Theorem to compare how the flow of a real-world fluid compares to the theoretically predicted flow. The fluid used in the experiment was water. The Bernoulli’s Theoreom experiment was carried out by a venture meter, manometer and a hydraulic bench. The experimental measurements included the static, dynamic and total head using the manometer. The velocity was also measured. Three experiments using the same methodology were carried. The results were graphed on three separate graphs with the Head on the y-axis and the Distance from A along the x-axis. The results were compared between the three graphs. On each graph the experimental head was graphed with the theoretical head so the experimental results can be compared with the theoretical calculations. The experiment proved that water is not an ideal fluid. The of water in the real world is impacted by the amount of friction, temperature in the lab and other physical factors in the environment.
Bernoulli’s Princple describes the behaviour of a fluid dynamic system. Bernoulli described the behaviour “as the the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases” (Bernoulli 1738 cited in Gratten-Guinness 2000). Bernoulli developed an equation for the ideal flow of a an inviscid fluid. An invisced fluid has no viscosity or negligible viscosity. The Ideal flow of a fluid is not slowed by any physical factors found in the real physical world. The purpose of the equation is to represent the energy partitioning for an ideal flow of an inviscid fluid that cannot be compressed and is flows at steady state (in a steady flow).
Bernoulli’s equation for an ideal flow was derived by using geometry principles ( (See fig. 1) In the modern day the equation is a way to represent Hydraulic Grade Line and Engery Line in a system. The Energy Line referes to the amount of total head to the fluid. The various amounts of flow energies are assumed to be constant while the flow moves along a pipe from one cross-section to the next. To be more exact the sum of the energy remains constant while the liguid moves. The equation follows the law of conservation of energy. Pipes change size from large diameters to narrow diameters, the velocity of the flow is higher in the narrow parts of pipe than in the larger areas in order to move equal distances.
The kinetic energy in the part of the pipeline that has a narrow cirucmference because the velocity of the flow is faster, the volume is the same between the narrow and wide parts to the pipeline so the pressure is the factor that must decrease. The conservation law of energy teaches that if the kinetic energy is increasing, than a balance must be kept with the volume and pressure.
kinetic energy + potential energy + flow energy = constant, so Bernoulli’s Constant, H equals the total Head. (See fig. 1)
Figure 1 Bernoulli’s constant
Figure 2 Bernoulli Equation for Ideal Flow
The variables in the Bernoulli Equation (See fig. 2) are listed below.
z = elevation head of the center of the cross section with respect to a point z=0; or the vertical distance from the point called datum
p = fluid static pressure at the cross section (N/m2)
g= gravitational acceleration (m/s2)(value is 9.81 m/s2 = 9810 mm/s2)
v = mean fluid velocity at the cross section (m/s)
h* = total head (m); pressure head+velocity head +elevation head
γ = specific weight of water
L = Length
The objectives of the lab were:
Apparatus and experimental procedure
Calculations and results
Figures 3, 4 and 5 below represent the difference between the theoretical values and the experimental values for water flow in three Laboratory Trials of the Bernoulli’s Effect.. In each figure the difference between the real and the ideal figure can be recognized because the blue data points represent the experimental data and the red data points represent the ideal (theoretical) data points.
Figure 3 Bernoulli’s Principle Experiment 1
Experiment One showed that the ideal and theoritcal flow behaviours were very similar at a distance from A of approximately zero to 50 m. At 50 m from a the both the experimental and theoretical curves are at their lowest point. as the distance from a increase the two lines diverge and continue to diverge until the end of the experiment. At the end of the experiment the difference between the the experimental and theoretical data increased to approximately 0.041 m.
Benoulli’s Principle was demonstrated in the experiment because as the pressure increased the velocity decreased. Q was equal to 4.918 X 10-4 m3/s the velocity increased when the tube the water was flowing through became narrower. At point A (zero distance from A) the theortical and experimental were both equal to 0.270 m with a measured velocity at 0.926 m/s. The v increased until 46 mm from A and then decreased to 0.982m/s at 136 mm from A.
Figure 4 Bernoulli’s Principle Experiment 1
Experiment Two showed that the ideal and theoritcal flow behaviours were dissimilar for the duration of the experiment. The shape of the line is similar but the experminetal shows a longer head when the theoretical is at its lowest point. The two lines intersect at approximately 0.15 m head. After that point the two lines diverge increasingly. The theoretical showed a longer head than the experimental after the two lines intersect. At the end of the experiment the difference between the experimental and theoretical data increased to approximately 0.032 m.
Benoulli’s Principle was demonstrated in the experiment because as the pressure increased the velocity decreased. Q was equal to 3.95 x 10-4 m3/s the velocity increased when the tube the water was flowing through became narrower. At point A (zero distance from A) the theortical and experimental were both equal to 0.250 m with a measured velocity at 0.744 m/s. Then v increased to its highest speed at 1.964 ms when the distance was about 50mm. After that point the velocity decreased until the end of the experiment when it measured 0.788 m/s.
Figure 5 Bernoulli’s Principle Experiment 1
Experiment Three showed that the ideal and theoritcal flow behaviours were similar until the distance from A was approximately 100 m., and then the two lines diverge. The shape of the lines once again are similar but the experminetal shows a longer head than the theoretical after the two lines intersect at about 46 m from A. At the point the head measured 0.126 m for the experimental and about 0.118 m for the theoretical. After that point the two lines diverge increasingly. The theoretical showed a longer head than the experimental after the two lines intersect. At the end of the experiment the difference between the experimental and theoretical data increased to approximately 0.019 m.
Benoulli’s Principle was demonstrated in the experiment because as the pressure increased the velocity decreased. Q was equal to 3.224 x 10-4 m3/s the velocity increased when the tube the water was flowing through became narrower. At point A (zero distance from A) the velocity measured 0.607 m/s. Then v increased to its highest speed of 1.604 ms when the distance was about 46 mm from A. After that point the velocity decreased until the end of the experiment when it measured 0.644 m/s.
Analysis and Discussion
In Experiments 1 through 3 Bernoulli’s Principle based on the conservation of energy was observed because the velocity of the water became faster as the pipe narrowed and then decreased when the circumference of the pipe became larger. Experiment 2 which was carried out at the medium pressure compared to experiments 1 and 3, was the experiment when the theoretical and the experimental showed the most difference.
Experiments 1 and 2 showed a similar range of head length. In Experiment 1 the range was from 0.270 m at the beginning to 0 at the lowest point and then increased to 0.223 m. the line graph for Experiment 2 showed a similar range. Experiment 3 used the lowest amount of pressure and the range of the head was also the lowest. The experimental head started at 0.230 m, reached a low point of 0.126 and then increased until the end of the experiment to only 0.209.
Experiment 2 shows the least good fit of the experimental and theoretical data. Experiment 2 used 3.95 x 10-4 m3/s of pressure, whereas Experiment 1 used the highest and Experiment 3 the least amount of pressure. Experiment 1 and 3 both demonstrated a good fit to the theoretical data. The lines were the same until after reaching a low point at abut 46 to 50 m, the lines representing experimental theoretical diverged.
The source of error that most impacted the experiments was error by the human factor. Error can enter the experiment due to inexperience in setting up the equipment and also due to carrying out the expermint. The readings may have been taken incorrectly due to problems of reading the correct level of static head.
Conclusion and Recommendations
The experiment was very satisfactory and showed the Venturi Effect as expected. The fit of the experimental to theoretical data was better than expected. Physical factors effect real measurements of flow such as friction when the water is flowing. If the water is warm the molecules expand and the velocity is expected to change.
Bernoulli’s Principle is very important in the study of hydrology. Derivations of the equation are used for practical purposes in engineering, chemisty and physics. One environmental application is to determine groundwater flow. Another is to calculate the water flow from reservoirs for use in city water systems. The designers of pipelines when water is used to produce electricity need to apply Bernoulli’s Principle.
Batchelor, G.K. (2001). An Introduction to Fluid Mechanics. NYC: Cambridge University Press. available from books.google
Craft, T.J. (n.d.) Inviscid Flows. Fluid Mechanics, Teaching Material [online] available from http://cfd.mace.manchester.ac.uk/tmcfd
Grattan-Guinness, I. (2000). Daniel Bernoulli. and the varieties of mechanices in the 18th century. NAW, [online] 5(1) available from http://www.nieuwarchief.nl/serie5/pdf/naw5-2000-01-3-242.pdf [21 Feb. 2005]
Mathieu, J. and Scott., J. (2000). An Introduction to Turbulent Flow. NYC: Cambridge University Press.
White, F.M. (1998). Fluid Mechanics. 7th ed.NYC: McGraw-Hill Higher Educaton
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