Conservation of EnergybyDan SchwarzSchool of EngineeringGrand Valley State UniversityEGR 365 – Fluid MechanicsSection 01Instructor: Dr. S. FleischmannJune 5, 2007OutlineI. Purpose Statementa. Bernoulli’s equation was used to describe the conservation of energy for gravity driven fluid flow.b. The accuracy of Bernoulli’s equation was determined through experimentation.II. Backgrounda. The experimental system shown in Figure 1 produces a flow of water that is driven by gravity.Figure 1: The experimental system exemplifies gravity driven fluid flow. b. Bernoulli’s equation can be used to describe the fluid flow of this system and kinematics can be used to describe the path of the fluid stream. The horizontal range of the fluid stream is described by Equation 1. See Appendix A for details.  02 yyhx (1)c. Experimental Methodi. The tank was filled with water.ii. A long strip of tape was placed on the side of the tank with depth marks every 2 inches. (The pressure head at each mark was determined by adding the distance between bottom of the tank and the nozzle to the depthof tank at the mark)iii. A ruler was placed directly beneath the nozzle to measure the range of the water stream. A plumb bob was used to align the ruler.d1 = 10.5625ind2 = 0.25inhxy = 8.75inz1z2iv. The range of the water stream was measured with the ruler each time the water height in the tank was level with one of the 2 inch markings on the tape.v. Range and pressure head measurements were recorded at 6 different marks.III. Results / Discussiona. The measurements taken from the experimental procedure are compared with the predicted ranges in Table 1. The actual range of the water stream was lower than the predicted range because of energy losses sustained in the nozzle. The large discrepancies indicate that the energy loss in the nozzle was very large.Table 1: Predicted and experimental water stream ranges are given for the system.Hieght, h (in)Actual Range,x (in)PredictedRange, x (in)% Discrepancy52.4375 11.75 42.8473%50.4375 11.25 42.0273%48.4375 11.00 41.1773%46.4375 10.75 40.3273%42.4375 10.00 38.5474%38.4375 9.50 36.6874%b. The range of the water stream was plotted as a function of the pressure head in Figure 1. The figure indicates that the actual and predicted data trends are approximately parallel. The large vertical shift between the data trends is due to the energy loss from the nozzle which resulted in low experimental ranges. Figure 2: Range of the water stream as a function of pressure head.IV. Conclusionsa. The main source of experimental error was caused by the oscillating spray of the water stream. However, the spray of the water stream was not significant enough to account for the large discrepancies as shown by the error bars in Figure 2.b. There was no propagated error in this experiment since the range and pressure head measurements were taken directly c. The large discrepancy was caused by energy losses in the nozzle. These losses reduced the kinetic energy of the exiting water which reduced the actual range significantly. V. Appendicesa. Appendix A – Question 1i. Find the range equation when initial velocity is negligible.1. Begin with Bernoulli’s equation.222212112121gzvpgzvp2. If the diameter of the tank is large relative to the diameter of the nozzle, then the velocity of the water in the tank is negligible compared to the velocity of the water exiting the tank. The static pressure of the water is the same at any point in the system. Cancelinappropriate terms.222121gzvgz 3. Rearrange the equation to solve for the initial velocity of the water stream exiting the tank, 2v. ghzzgv 222124. The path of the water stream can be described in the x and y directions using kinematic equations.020221ygtyxtvx5. Rearrange the vertical kinematic equation to solve for time, t. gyyt02 6. Substitute the initial velocity of the stream (step 3) into the kinematic equation for horizontal range (step 4). Also substitute thetime equation (step 5) into the horizontal range equation. For simplicity, set the origin of the coordinate system at the origin of the water stream.  002022 yyhgyyghx ii. Find the range equation when initial velocity is not negligible.1. Begin with the conservation of mass equation.2211AvAv2. Substitute the relationship 4/2dA into the conservation of mass equation and cancel any redundant terms.222211dvdv 3. Rearrange the equation to solve for the velocity of water in the tank, 1v.2221221vddvv 4. Substitute this equation in the Bernoulli equation and cancel inappropriate terms.22214222121gzvgzv 5. Rearrange the equation to solve for the initial velocity of the water stream exiting the tank, 2v. 4212ghv6. The path of the water stream can be described in the x and y directions using kinematic equations.020221ygtyxtvx7. Rearrange the vertical kinematic equation to solve for time, t. gyyt02 8. Substitute the initial velocity of the stream (step 5) into the kinematic equation for horizontal range (step 6). Also substitute thetime equation (step 7) into the horizontal range equation. For simplicity, set the origin of the coordinate system at the origin of the water stream.     4004120212yyhgyyghxb. Appendix B – Question 2i. Determine if  is small enough to consider the velocity of water in the tank negligible.1. Find  for the system.02367.05625.1025.012inindd2. Find  41 for the system. 19999.002367.011443. is small enough to consider the velocity of water in the tank negligible.c. Appendix C – Question 3i. Identify the variables in Equation 1.1. The dependant variable is the range of the water stream, x.2. The independent variable is the pressure head of the tank, h.3. There is no propagated error in this experiment because the range can be measured directly.ii. Develop a plan for experimentation.1. Both x and h will be measured directly using a yard stick2. Since the path of the fluid stream is described by a quadratic, at least 3 points of x and h measurements must be taken to fit the data.3. Since the stream scatters, 5 or 6 points will be measured to improve the accuracy of the fit line.d. Appendix D – Question 4i. The kinetic energy of the water stream will be