Fluid Mechanics Lab

EN 1029 Laboratory Laboratory FM Declaration: In submitting this report, I hereby declare that, except where I have made clear and full reference to the work of others, this submission, and all the material (e. g. text, pictures, diagrams) contained in it, is my own work, has not previously been submitted for assessment, and I have not knowingly allowed it to be copied by another student. In the case of group projects, the contribution of group members has been appropriately quantified. I understand that deceiving, or attempting to deceive, examiners by passing off the work of another as my own is plagiarism.I also understand that plagiarising another's work, or knowingly allowing another student to plagiarise from my work, is against University Regulations and that doing so will result in loss of marks and disciplinary proceedings. I understand and agree that the University’s plagiarism software ‘Turnitin’ may be used to check the originality of the submitted cou rsework. Contents 1. Introduction 2. Theory 2. 1 Impact of a Water Jet 2. 2 Flow Through a Venturi Meter 3. Experimental procedures and results 3. 1 Experimental procedure – Impact of a Water Jet 3. Experimental procedure – Flow through a Venturi Meter 3. 3 Results– Impact of a Water Jet 3. 4 Results– Flow through a Venturi meter 4. Discussion 4. 1 Impact of a water jet 4. 2 Venturi meter 5. Conclusion 6. References Appendices Abstract Rate of flow was measured in two different experiments, Impact of a water jet and flow through a Venturi meter. The main objective was to calculate the change in momentum and energy loss in flow which was put under pressure. The experiment showed that results obtained can significantly defer from the theory if energy losses are not neglected. 1.Introduction Water is the most commonly used resource of renewable energy. In 21st century, hydropower is used in more than 150 countries around the world. It is also the most effici ent method of producing energy with 90% efficiency output. Impact of a Water Jet is used to show how mechanical work can be created from water flow. When a fluid is put under pressure, the pressure gives it high velocity in a jet. Jet strikes the vanes of the turbine wheel. This wheel then rotates under the impulse created by the water jet hitting the vanes. Venturi meter is used to measure discharge along a pipe.In this experiment, when pressure is dropped, there is an increase in velocity. Pressure magnitude is dependent on rate of flow, so by measuring the pressure drop, discharge can be calculated. Main objective of both experiments is to calculate rate of flow under pressure. 2. Theory 2. 1 Impact of a Water Jet From impulse-momentum change equation it can be assumed that force is generated due to the change in momentum of the water. In other words, force equals the difference between the initial and final momentum flow. Arrangement of jet impact apparatus used is given in Figu re 1 Figure 1Jet impinging on a vane is shown in Figure 2. Control volume V is used, bounded by a control surface S. The entering velocity is u1 (m/s) and it’s in the x –direction. The vane deflects water jet and the leaving velocity is u2 inclined at an angle ? 2 to the x – direction. Pressure over the surface of the jet, apart from the part where it flows over the surface of the vane is atmospheric. The change in direction is due to force generated by pressure and shear stress at the vane’s surface. The mass flow rate is . Mass flow rate is the mass of substance which passes through a given surface per unit time [kg/s].Experiment was done for flat and hemispherical vane. Figure 2 Force on the het in the direction x is FJ (N), then momentum equation in the s- direction is FJ =(u2 cos ? 2 – u1) (1) From Newton's Action- Reaction law, force F on the vane is equal and opposite to Fj F =(u1 – u2 cos ? 2 ) (2) For flat plate ? 2 = 90Â ° so cos ? 2 = 0. Therefore F =u1 (3) For the hemispherical cup, it’s assumed that ? 2 = 180Â ° so cos ? 2 = -1,so F =( u1 + u2 )(4) The effect of change of elevation on jet speed and the loss of speed due to friction over the surface of the vane is neglected.Therefore u1 = u2. So, F=2u1(5) If resistance forces are neglected, this is the maximum possible value of force on the hemispherical cup. Rate at which momentum enters the control volume, or rate of flow of momentum in the jet, is detonated by symbol J. J =u1(6) For the flat plate rate of flow of momentum in the jet is equal to the force on the vane. This is shown in equation (3). F=J(7) For the hemispherical cup, maximum possible value of the force is from equation (5) F=2J (8) If the velocity of the jet is uniform over it’s cross section it can be concluded that =? 1A (9) 2. 2 Flow Through a Venturi Meter Piezometer tubes were bored into a wall and links were made from a each of these to perpendicular manometer tubes, w hich were placed in front of a millimetre scale. Venturi meter is shown in Figure 3 Figure 3 It’s assumed that the fluid used is frictionless and incompressible, fluid flow is steady, and energy equation was derived along a stream line. Bernoulli’s theorem states that u122g+ h1 = u222g+ h2 = un22g+ hn (10) From continuity equation Q=U1A1=U2A2=UnAn(11) here Q is discharge rate( m3/s), and A is cross-sectional area of the pipe(m2) substituting for u1 gives u222ga2a12+ h1 = u222g+h2 (12) Solving equation (3) for u2 gives u2 =2g(h1-h2)1-a2a12 (13) From equation (4) Q=a22g(h1-h2)1-a2a12 (14) In previous equation it was assumed there was no energy loss in the flow and the velocity was constant. In reality, there is some energy loss and velocity is not uniform. Equation (5) is therefore corrected to Q=Ca22g(h1-h2)1-a2a12(15) Where C is the coefficient of the meter.Its value usually lies in within range 0. 92 to 0. 99. Ideal pressure distribution is given in equation (7) hn-h1 u222g=a2a12-a2an2 (16) 3. Experimental procedures and results 3. 1 Experimental procedure – Impact of a Water Jet The apparatus shown in Figure 1 was levelled and lever was balanced, with jockey weight at zero setting. Weight of the jockey was measured. Diameter of the nozzle, height of the vane above the nozzle and the distance from the pivot of the lever to the centre of the vane were recorder. Water was then released through the supply valve and flow rate increased to maximum.The force on the vane displaces the lever, which is then restored to its balanced position by sliding the jockey weight along the lever. The mass flow rate can be established by gathering of water over a timed interval. Additional readings are then taken at number of reducing flow rates. The most efficient way of reducing flow is to place jockey weight precisely at desired position, and then adjust the flow control valve to bring the lever to the balanced position. Range of settings of the jockey posi tion may be separated efficiently into uniform steps. 3. Experimental procedure – Flow through a Venturi Meter The objective of this experiment is to establish the coefficient of the meter C. Bench vale and control vale should be open so water can flow to clear air pockets from the supply system. The control valve is then progressively closed, so the meter is exposed to a steadily increasing pressure. This will cause water to pass up the tubes. When water levels have risen to a suitable height, the bench valve is slowly closed, so that, as both valves are lastly shut of, the meter is left holding static water under adequate pressure.Amounts were then recorded for values of (h1 –h2) and discharge value Q is recorded. The rate of flow is measured by gathering of water in weighing tank, whilst values of h1 and h2 were read from the scale. Similar readings may be taken at a sequence of reducing values of h1 –h2. About 6 readings, proportionately spread in the range of 250 mm to zero. By reading off from all the tubes at any of the settings used, the pressure distribution along the length of the Venturi meter may be logged. 3. 3 Results– Impact of a Water Jet Two sets of readings were taken, one for the flat plate other for the hemispherical plate.Table 1 contains readings for the flat plate and Table 2 results for the hemispherical plate. These tables can be found in Appendix 2. Mass flow is calculated by dividing the Quantity (kg) by Time (s) taken to collect water. Quantity should be converted to m3 where 1 kg water will be 1/1000 m3. e. g. If quantity is 30 kg, time taken is 52. 69 s, mass flow is 0. 569 103 x Q. Using the equation (9), u1 can be calculated. From uo2 = u12 – 2gs , uo can be deduced. For flat plate J can be calculated using equation (6). F is calculated from F X 150 = W x yData from Table 1 and 2 are plotted on a graph to give a comparison between forces and rate of momentum flow of the impact. Graph is present ed in Figure 4. Additional information are given in Apendex 2 Figure 4 (Series 1-flat plate, Series 2- hemispherical plate) 3. 4 Results– Flow through a Venturi meter Two sets of data were compared. Values shown in Table 4 are measurements of h1 and h2 at different discharges. In this part of the experiment C is assumed to be constant over a range of measurement. Closer inspection of Table 4 shows C is not constant as Q varies.Piezometer measurements are recorded in Table 5 and compared with ideal pressure distribution given In Table 3. Figure 5 Graph shown in Figure 5 gives variation of (h1 -h2)1/2 With Q. Equation of the graph line is y= 0. 581 x h1-h2=0. 581 x Qx 103 Q =5. 81 x 10-4h1-h2 (16) Substitute (16) in equation (15) to get a value of C. C= 0. 604 Figure 6 shows both ideal and set of results obtained in the experiment. Series 1 shows ideal pressure distribution, and series2 shows obtained results. Figure 6 4. Discussion 4. 1 Impact of a water jet Theory compares we ll with the experiment considering that the two lines have different gradients.In theory, gradients of lines are significantly steeper, and this might be because an error in the experiment occurred. Likely errors that might have occurred are measurements of mass of jockey weight; distance L from centre of vane to pivot of lever or diameter of water jet emerging from nozzle. If Mass of jockey was wrongly logged by 0. 001kg, Force on the vane would have 2% error. The graph that was obtained shows force on the hemisphere us less than twice the flat plate. This can be concluded from the line gradient. This implication is supported by the theory.In theory, no friction losses or any other kind of energy losses were included in equations. In the actual experiment, there were some energy losses like friction over the surface of the vane and effect of change of elevation on jet speed. It was assumed that velocity of the jet was uniform over its cross section, which would imply ideal flow. It ’s likely that this was not the case, and momentum gained by the change in velocity. 4. 2 Venturi meter Value of C determined in table A is higher than it theoretically should be. This is probably due errors that occurred in the experiment, like parallax rror. Air in pipes could have also caused an error in the experiment. Value of C obtained from Figure 5 gives a more realistic value of 0. 604. The difference between the ideal pressure results and values recorded in the experiment is acceptable considering the coefficient of the meter C that is not included in ideal pressure distribution. Flow of 1x 10-3 m3/s is expected to lie on a negative hn-h1u222g value. 5. Conclusion From both experiments it can be concluded that the flow was not ideal and there were significant energy losses that differ obtained results from theoretical results.In the impact of a water jet experiment it was proven that force on a flat plate is less than the force on the hemispherical plate. Therefore change in momentum flow was smaller. In the Venturi meter experiment it was shown that ideal pressure distribution differs from obtained results because energy losses effect the results. The errors in both experiments can affect the results significantly an lead to wrong assumptions. References Fluid Mechanics, Third Edition? JF Douglas, JM Gasiorek, JA Swafield? Longman Mechanics of Fluids? BS Massey, Van Nostrant Reinhold? Chapman & HallAppendix 1-Nomenclature Symbol| Quantity| SI units| F| Force| N| J| Rate of flow of momentum| N| u| velocity| m/s| | Mass flow rate| Kg/s| D| Diameter| m| h| height| m| A| Cross-section area| m2| ?| Angle of elevation| degrees| ?| density| Kg/m3| Appendix 2-Raw data Impact of a water jet Diameter of nozzleD= 10. 0 mm Cross sectional area of nozzle A =? D24=7. 85 x 10-5 m2 Height of vane above nozzle tips= 35 mm = 0. 035 m Distance from centre of vane to pivot of leverL= 150 mm Mass of jockey weightM= 0. 600 kg Weight of jockey weightW =Mg = 0. 600 x9. 81 =5. 89 NQuantity (kg)| T(s)| y(mm)| 103 x Q(m3/s)| U1(m/s)| U0(m/s)| J(N)| F(N)| 30| 52. 69| 65| 0. 569| 7. 25| 7. 20| 4. 13| 2. 55| 30| 57. 81| 55| 0. 519| 6. 61| 6. 56| 3. 43| 2. 16| 30| 61. 28| 45| 0. 490| 6. 24| 6. 18| 3. 06| 1. 77| 15| 22. 76| 35| 0. 659| 8. 40| 8. 36| 5. 54| 1. 37| 15| 28. 12| 25| 0. 533| 6. 80| 6. 75| 3. 62| 0. 98| 15| 37. 09| 15| 0. 404| 5. 15| 5. 08| 2. 08| 0. 59| 15| 75. 09| 5| 0. 200| 2. 54| 2. 40| 0. 51| 0. 196| Table 1 Quantity(kg)| T(s)| y(mm)| 103 x Q(m3/s)| U1(m/s)| U0(m/s)| J(N)| F(N)| 30| 52. 87| 120| 0. 567| 7. 23| 7. 18| 8. 24| 4. 71| 30| 56. 8| 105| 0. 527| 6. 72| 6. 67| 7. 08| 4. 12| 30| 60. 78| 90| 0. 494| 6. 29| 6. 24| 6. 21| 3. 53| 15| 21. 75| 75| 0. 690| 8. 79| 875| 6. 07| 2. 94| 15| 24. 60| 60| 0. 610| 7. 77| 7. 73| 9. 48| 2. 35| 15| 28. 32| 45| 0. 530| 6. 75| 6. 70| 7. 16| 1. 77| 15| 37. 32| 30| 0. 402| 5. 12| 5. 05| 4. 12| 1. 18| Table 2 Venturi Meter Piezometer Tube No. N| Diameter of cross-section(mm)| Areaa(m2)| | | | A(1)BCD(2 )EFGHJKL| 26. 0023. 2018. 4016. 0016. 8018. 4720. 1621. 8423. 5325. 2426. 00| 0. 0005310. 0004230. 0002660. 0002010. 0002220. 0002680. 0003190. 0003750. 0004350. 00050. 000531| 0. 150. 6900. 8701. 0000. 9520. 8660. 7940. 7330. 6800. 6340. 615| 0. 1430. 2260. 5721. 0000. 8230. 5630. 3970. 2880. 2140. 1610. 143| 0. 000-0. 083-0. 428-0. 857-0. 679-0. 420-0. 253-0. 145-0. 070-0. 0180. 000| Table 3 Quantity (kg)| T(s)| h1(mm)| h2(mm)| 103 x Q(m3/s)| (h1- h2)(mm)| (h1 -h2)1/2(mm)1/2| C| 12| 17. 67| 346| 20| 0. 679| 0. 326| 0. 571| 1. 236| 12| 17. 53| 346| 20| 0. 685| 0. 326| 0. 571| 1. 248| 12| 17. 60| 346| 20| 0. 682| 0. 326| 0. 571| 1. 242| 12| 20. 69| 330| 84| 0. 580| 0. 246| 0. 496| 1. 216| 12| 18. 40| 330| 84| 0. 652| 0. 246| 0. 496| 1. 367| 12| 19. 5| 330| 85| 0. 616| 0. 246| 0. 496| 1. 212| 12| 21. 36| 324| 114| 0. 562| 0. 210| 0. 458| 1. 275| 12| 20. 90| 324| 114| 0. 574| 0. 210| 0. 458| 1. 303| 12| 21. 13| 324| 114| 0. 568| 0. 210| 0. 458| 1. 289| 12| 20. 00| 336| 58| 0. 600| 0. 278| 0. 527| 1. 183| 12| 18. 31| 336| 58| 0. 655| 0. 278| 0. 527| 1. 292| 12| 19. 16| 336| 58| 0. 628| 0. 278| 0. 527| 1. 239| 6| 12. 23| 310| 176| 0. 491| 0. 134| 0. 366| 1. 395| 6| 12. 32| 310| 176| 0. 487| 0. 134| 0. 366| 1. 342| 6| 12. 28| 310| 176| 0. 489| 0. 134| 0. 366| 1. 389| 6| 17. 11| 298| 224| 0. 351| 0. 074| 0. 272| 1. 342| 6| 18. 5| 298| 224| 0. 317| 0. 074| 0. 272| 1. 212| 6| 18. 03| 298| 224| 0. 334| 0. 074| 0. 272| 1. 277| 6| 0| 296| 296| 0| 0| 0| 0| 6| 0| 296| 296| 0| 0| 0| 0| 6| 0| 296| 296| 0| 0| 0| 0| Table 4 Piezometer Tube No. | Q=0. 682 x 10-3u222g – 0. 587 m| | hn(mm)| hn – h1(m)| hn-h1u222g| A(1)| 346| 0. 000| 0| B| 328| -0. 018| -0. 0307| C| 202| -0. 144| -0. 245| D(2)| 20| -0. 326| -0. 555| E| 52| -0. 294| -0. 501| F| 142| -0. 204| -0. 348| G| 190| -0. 156| -0. 266| H| 224| -0. 122| -0. 208| J| 246| -0. 100| -0. 170| K| 264| -0. 082| -0. 140| L| 268| -0. 078| -0. 133| Table 5 Appendix C