Systems Lab 1[1]

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Lab 1: Impact of Jet

Rob Eakins1, Jon Nastav2, and Matt Harrison3

Georgia Institute of Technology, Savannah, GA, 31407

The goal of this lab experiment was to observe and verify the principle of conservation oflinear momentum. This is done by showing that the time rate of change in linear momentumis equal to the net flow rate of fluid through the control surface summed with the time rateof change in linear momentum of the control volume contents. Different masses placed onthe apparatus are raised to the same, measured, level of height by the water jet with the flowrate being controlled by opening and closing the flow rate valve. Four target plates, each ofdifferent geometry, were used to demonstrate and compare the effect of varying deflectionangles. The experimental results showed a higher value of force was needed to lift the targetplates back to their equilibrium resting point.

Nomenclature

=v1 = Fluid velocity

A = Cross sectional area of the jetm = Mass flow rate of fluid = Density of the fluid

g = Acceleration due to gravityh = Height of the deflector

Ry = Theoretical force of jet streamFa = Experimental force of jet streamv1 = Corrected jet stream velocityVV = Volumetric flow rate of the fluidQ = Volume of fluid collectedt = Time

K = trailing-edge (TE) nondimensional angular deflection rate

I. Introduction

HE objective of this experiment is to verify the conservation of linear momentum by calculating the amount offorce applied to four different target plates by a water jet. Each of the target plates used will have a different

geometry and will be tested for three different masses on the weight pan. By controlling the flow rate valve theforce from the jet will elevate the target plate back into its original position. This flow rate is then measured andused to calculate the forces of the water jet acting on the target plates. Using the acquired values for the forces, afree body diagram and balance of forces will verify the principle of conservation of linear momentum. Theexperimental results showed small variation from the theoretical results; causes of these discrepancies can beattributed to multiple sources of error.

T

II.Apparatus and Procedure

A. Apparatus:An Armfield F1-10 Hydraulic Bench was used as the base for the jet impact apparatus. The hydraulic bench

provided a volumetric storage tank for the experiment and supplied water for the jet stream. An Armfield F1-16Impact of Jet apparatus was used to impinge the jet impact on various target plates. As can be seen in Figure A, the

jet apparatus was enclosed in an acrylic enclosure to contain the deflected fluid and prevent splashing. With theseapparatuses, a set of weights was used on the weight pan of the jet impact apparatus. To take measurements ofheight a 0.01 resolution digital caliper was used and to take measurements of time a digital stopwatch was used. Adigital scale was also used to measure the actual mass of the weights used in the experiment.

1 Undergraduate Student, 210 Technology Circle, Savannah, GA 31407

Georgia Institute of Technology

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Figure A. Images of Armfield F1-16 Impact of Jet apparatus

B. Procedure.

The first task of the lab was to wash the target plates for use in the jet impact apparatus. Once the target plateswere washed the jet impact apparatus was placed on the top channel of the Armfield F1-10 Hydraulic Bench and theflexible tube from the base was connected to the quick release valve at the bottom of the apparatus. The top plate ofthe jet impact tank was removed by unscrewing the three knurled nuts. Once the top was removed one of the fourtarget plates was screwed onto the vertical shaft in the transparent tank and the top was replaced. The apparatus wasthen leveled by adjusting the feet of the apparatus. The height of the level gauge was adjusted and aligned with thedatum line found on the weight pan. The weight pan was gently oscillated to ensure accuracy. Each individualweight was weighed using a digital scale and then the weights were placed on the weight pan. The flow valve wasthen closed and the service pump was started. The flow valve was opened gradually until the force of the jet streamhad lifted the weight pan high enough for the datum line and level gauge to be realigned. Once equilibrium wasattained the weight pan was oscillated gently to ensure accuracy. A stopwatch was used to time the length of thetime that the flow valve was open. This value was used to divide the volume of water collected in the volumetric

storage tank during the experiment to derive the flow rate used to raise the weight pan. This list of procedures wasthen repeated for three various amounts of weights on the weight pan and for each of the four different geometries oftarget plates.

III.Theory

The volumetric flow rate, V, can be found by dividing the volume of water collected, Q, by the amount of timethe water was allowed to run, t. Each run was limited to a 60 second run time to enforce uniformity and helpminimize error or difference in calculations. Once the volumetric flow rate is calculated it can then be divided bythe cross-sectional area of the nozzle,A, to compute the jet velocity, v1. Equation (1) and (2) show the procedure forfinding the volumetric flow rate and the jet velocity.

V=QtV= Qt(1)

v1=VA1v1= VA1(2)

After the jet velocity is calculated Bernoullis Equation, shown in Equation (3) is used to calculate the corrected jetvelocity, v1.

12v12g=12v1'1g+h12v12g= 12v1'2g+ h(3)


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In Equation (3) g represents the acceleration due to gravity and h is the height of the deflector. By rearrangingBernoullis Equation we can derive Equation (4), which is the simplified equation for corrected jet velocity.

v'1=v12-2ghv1'= v12- 2gh(4)

In order to calculate the theoretical reaction force,Ry, the mass flow rate, m, must be calculated. Equation (5) showshow the mass flow rate can be calculated, where is the density of the fluid.

m=Av'1m= A1v1'(5)

Having the mass flow rate calculated then allows for the theoretical reaction force, Ry, can now be calculated byusing Equation (6).

Ry=mv'1(1+cos)Ry= mv1'(1+ cos)(6)

In Equation (6), is the deflector angle that is defined to be parallel to the jet. In comparison to the theoreticalreaction force, the experimental force applied to the target plate can be calculated using Newtons Second Law,which is given in Equation (7).

Fa=mgFa=mg(7)

In this case, m, is equal to the mass applied to the system and, g, again, is the acceleration due to gravity. Havingcalculated both the theoretical and experimental values they can then be compared.

IV.Results

Figure A displays a plot of the theoretical and actual applied forces of the impinging jet stream on the targetplate plotted against the average jet velocity for the 180180 deflector. As can be seen from these plots the

theoretical and actual applied forces take on similar slopes and relative accuracy with exception of the 180deflector. Reasons for error during the experimentation are discussed in the error analysis and discussion sections.


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The discrepancy of the theoretical force minus the actual applied force was then plotted and can be seen in FigureB. Each of the deflectorsdiscrepancies are placed on the same

plot to illustrate the differences in eachof the measurements. Thesediscrepancies will also be discussed in

the error analysis portion of the report.Figure C shows a surface plot

for the discrepancies shown in FigureB. The absolute value of thetheoretical force minus the actualapplied force was divided by thetheoretical force value and plottedagainst both v and .

V.Discussion

The force of linear momentum has been shown to be true in the experimentperformed. The force required to hold the four different deflector plates up withequivalent down force was different depending on the angle of the deflector. Thelower the degree angle on the deflector plate the greater a force, the flow rate of the jetstream, is required to balance the force of the weight applied. As seen in Figure D,the brass weights on top are used to provide a force down in which the jet streamapplies an equivalent force to return the weight pan back to the level gauge mark. Theforce of gravity calculated from the mass of the weights and knowing the geometry ofthe deflector and the flow rate of the jet stream, the ability to find the balanced force is

possible.


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Figure A. Plots for theoretical and applied forces versus average velocity for all four deflectors.

Figure C. Surface plot of the absolute value of the theoretical force minus the applied force divided by the theoretical

force versus both the jet velocity and deflection angle.

Figure B. Theoretical force minus actual force plotted versus theaverage jet velocity.


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Several factors could have caused error during the experiment to make the theoretical and experimental valuesdissimilar. To begin, the volume of the water that was being displaced was measured in liters. A liter is a largemeasurement for the experiment performed and was a critical value in measuring the flow rate. During datacollection, each mass for the specific deflector was repeated only once. The lab manual instructed to repeat eachmass twice for a total of three readings. The more data collected would lead to a more accurate value. Each readingwas done over a sixty second interval using a manually operated stopwatch without recording the exact time takenfrom beginning to end, although each time the stopwatch was within +/- 1.5 seconds of sixty seconds.

The cross-sectional area,A1, will increase as the updated jet velocity, v,decreases as it leaves the outlet of the jet. The theoretical model accurately

predicts the shape of the measured force on the impact surface except in oneparticular case, the 180 degree deflector, as seen in Figure A. ComparingFigure E with the theoretical slope in Table 2, the experimental slope is similarto the theoretical slope except for in the case of the 180 degree deflector.

Figure E. Force applied versus velocity squared with experimental slope

The 90 deflector is seen as a flat plate that the water deflects off of. Aschematic is shown in Figure F that shows the velocity and areas of the water

jet stream as it leaves the jet nozzle and strikes the deflector.

Figure F. Schematic of cross sectional area A1 and A2 for the 90 degree deflector

The velocity profile of a real jet would be non-uniform. For a non-uniform velocityprofile the jet will not strike the whole plate uniformly because it is not fully developedflow. This would cause slight error in the theoretical calculations. Also, for thedifferent deflectors it will strike at different points not reacting the same way. Forinstance, the lower the deflector angle the less likely the full force of the jet will strikethe deflector. Figure G is a sketch of what a real jet profile coming from the nozzlemay look like. The center of the profile would strike the deflector prior to the edges ofthe stream. From the above conclusion, the theoretical value should be assumedto be higher because it assumes that the velocities of the flow are all uniformand contacting the deflector at the same moment in time.

Overall, the experiments theoretical values and experimental data seem to bewell aligned in the previous mentioned ways. The 180 deflector has mucherror due to some unknown reason. The results confirm the principle of

conservation of linear momentum.

VI.Error Analysis

The jet velocities overall error can be calculated using equation?, With thegiven uncertainties of Q= +/- 0.5 L, for the flow rate, t= +/- 0.5 s, for timeand A= +/- 0.01 m2, for the cross sectional area.

v1= v1Q1A1t+ v1A1Qt+ v1tQA1


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Figure D. Armfield apparatus with brasweights on the weight pan

Figure G. Sketch of non-uniform profile assuming zero velocity at edge


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The real jet velocities overall error can be calculated using equation? with the given uncertainties of g=0.01 forgravity and h= +/- 0.01 for the height.

v1'= v1'v1gh+ v1'gv1h+ v1'hv1gUsing equation ? the overall theoretical reaction force can be found with the uncertainties

Appendix

Table 1. Raw data and calculated values from the experiment and table provided in lab manual

DeflectorAngle, , ()

Water Collected,Q, (m^3)

CollectionTime, t, (s)

Height ofDeflector, h, (m)

Mass Applied,M, (kg)

Jet Velocity,v1, (m/s)

Corrected JetVelocity, v1', (m/s)

180 0.01 60 0.02828 0.09952 3.315727981 3.311460712

180 0.014 60 0.02166 0.19914 4.642019174 4.63968555

180 0.017 60 0.01595 0.29876 5.636737568 5.635322565

180 0.011 60 0.02828 0.09952 3.647300779 3.643421877

180 0.014 60 0.02166 0.19914 4.642019174 4.63968555

180 0.018 60 0.01595 0.29876 5.968310366 5.966973992120 0.011 60 0.02799 0.09952 3.647300779 3.643461675

120 0.0155 60 0.02166 0.19914 5.139378371 5.13727068

120 0.018 60 0.0322 0.29876 5.968310366 5.965612175

120 0.01 60 0.02799 0.09952 3.315727981 3.311504499

120 0.015 60 0.02166 0.19914 4.973591972 4.971413994

120 0.018 60 0.0322 0.29876 5.968310366 5.965612175

90 0.015 60 0.02844 0.09952 4.973591972 4.970732049

90 0.017 60 0.02508 0.14916 5.636737568 5.634512438

90 0.019 60 0.02177 0.19914 6.299883164 6.298155117

90 0.012 60 0.02844 0.09952 3.978873577 3.975298095

90 0.016 60 0.02508 0.14916 5.30516477 5.302800509

90 0.019 60 0.02177 0.19914 6.299883164 6.298155117

30 0.018 60 0.02844 0.09952 5.968310366 5.965927306

30 0.023 60 0.02508 0.14916 7.626174356 7.624529842

30 0.026 60 0.02177 0.19914 8.620892751 8.619630028

30 0.017 60 0.02844 0.09952 5.636737568 5.634214267

30 0.025 60 0.02508 0.14916 8.289319953 8.287807025

30 0.026 60 0.02177 0.19914 8.620892751 8.619630028


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Table 2. Raw data and calculated values from the experiment and table provided in lab manual

TheoreticalSlope

TheoreticalForce, Ry, N

Applied Force,Fa, N

Errorin v2

Errorin v1'

Overall Errorin Ry

Overall Errorin Fa

Rv-Fa

0.020183598 1.102399645 0.9762912 0.126108445

0.020183598 0.434485904 1.9535634 -1.519077496

0.020183598 0.640967715 2.9308356 -2.289867885

0.020183598 0.26792764 0.9762912 -0.70836356

0.020183598 0.434485904 1.9535634 -1.519077496

0.020183598 0.718632551 2.9308356 -2.212203049

0.091190682 1.210539245 0.9762912 0.234248045

0.091190682 2.40666344 1.9535634 0.45310004

0.091190682 3.245342188 2.9308356 0.314506588

0.091190682 1.000002674 0.9762912 0.023711474

0.091190682 2.253773787 1.9535634 0.300210387

0.091190682 3.245342188 2.9308356 0.314506588

0.027742846 0.685475151 0.9762912 -0.290816049

0.027742846 0.880772395 1.4632596 -0.582487205

0.027742846 1.100468754 1.9535634 -0.853094646

0.027742846 0.438420055 0.9762912 -0.537871145

0.027742846 0.780120318 1.4632596 -0.683139282

0.027742846 1.100468754 1.9535634 -0.853094646

0.058019006 2.065029207 0.9762912 1.088738007

0.058019006 3.372845293 1.4632596 1.909585693

0.058019006 4.310697374 1.9535634 2.357133974

0.058019006 1.841776817 0.9762912 0.8654856170.058019006 3.985194706 1.4632596 2.521935106

0.058019006 4.310697374 1.9535634 2.357133974

An appendix, if needed, should appear before the acknowledgements.

Acknowledgments

The preferred spelling of the word acknowledgment in American English is without the e after the g.Avoid expressions such as One of us (S.B.A.) would like to thank Instead, write F. A. Author thanksSponsor and financial support acknowledgments are also to be listed in the acknowledgments section.

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Periodicals1Vatistas, G. H., Lin, S., and Kwok, C. K., Reverse Flow Radius in Vortex Chambers, AIAA Journal, Vol. 24, No. 11,

1986, pp. 1872, 1873.


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2Dornheim, M. A., Planetary Flight Surge Faces Budget Realities,Aviation Week and Space Technology, Vol. 145, No. 24,9 Dec. 1996, pp. 44-46.

3Terster, W., NASA Considers Switch to Delta 2, Space News, Vol. 8, No. 2, 13-19 Jan. 1997, pp., 1, 18.All of the preceding information is required. The journal issue number (No. 11 in Ref. 1) is preferred, but the month (Nov.)

can be substituted if the issue number is not available. Use the complete date for daily and weekly publications. Transactionsfollow the same style as other journals; if punctuation is necessary, use a colon to separate the transactions title from the journaltitle.

Books4Peyret, R., and Taylor, T. D., Computational Methods in Fluid Flow, 2nd ed., Springer-Verlag, New York, 1983, Chaps. 7,

14.5Oates, G. C. (ed.),Aerothermodynamics of Gas Turbine and Rocket Propulsion, AIAA Education Series, AIAA, New York,

1984, pp. 19, 136.6Volpe, R., Techniques for Collision Prevention, Impact Stability, and Force Control by Space Manipulators,

Teleoperation and Robotics in Space, edited by S. B. Skaar and C. F. Ruoff, Progress in Astronautics and Aeronautics, AIAA,Washington, DC, 1994, pp. 175-212.

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Proceedings7Thompson, C. M., Spacecraft Thermal Control, Design, and Operation, AIAA Guidance, Navigation, and Control

Conference, CP849, Vol. 1, AIAA, Washington, DC, 1989, pp. 103-1158Chi, Y., (ed.),Fluid Mechanics Proceedings , SP-255, NASA, 1993.9Morris, J. D. Convective Heat Transfer in Radially Rotating Ducts,Proceedings of the Annual Heat Transfer Conference,

edited by B. Corbell, Vol. 1, Inst. Of Mechanical Engineering, New York, 1992, pp. 227-234.

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Reports, Theses, and Individual Papers10Chapman, G. T., and Tobak, M., Nonlinear Problems in Flight Dynamics, NASA TM-85940, 1984.11Steger, J. L., Jr., Nietubicz, C. J., and Heavey, J. E., A General Curvilinear Grid Generation Program for Projectile

Configurations, U.S. Army Ballistic Research Lab., Rept. ARBRL-MR03142, Aberdeen Proving Ground, MD, Oct. 1981.12

Tseng, K., Nonlinear Greens Function Method for Transonic Potential Flow, Ph.D. Dissertation, Aeronautics andAstronautics Dept., Boston Univ., Cambridge, MA, 1983.

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13Richard, J. C., and Fralick, G. C., Use of Drag Probe in Supersonic Flow,AIAA Meeting Papers on Disc [CD-ROM], Vol.1, No. 2, AIAA, Reston, VA, 1996.

14Atkins, C. P., and Scantelbury, J. D., The Activity Coefficient of Sodium Chloride in a Simulated Pore SolutionEnvironment, Journal of Corrosion Science and Engineering [online journal], Vol. 1, No. 1, Paper 2, URL:

http://www.cp/umist.ac.uk/JCSE/vol1/vol1.html[cited 13 April 1998].15Vickers, A., 10-110 mm/hr Hypodermic Gravity Design A, Rainfall Simulation Database [online database], URL:

http://www.geog.le.ac.uk/bgrg/lab.htm[cited 15 March 1998].

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16TAPP, Thermochemical and Physical Properties, Software Package, Ver. 1.0, E. S. Microware, Hamilton, OH, 1992.

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Docket No. P-01-1532, filed 11 Feb. 1979.

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