Impact of a Jet of Water

Experiment #4 Impact of a Jet of Water Stephen Mirdo Performed on September 30, 2010 Report due October 7, 2010

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Table of Contents

Object ………………………………………..………………………….………….…. p. 1

Theory ………………………………………………………… ………………………………………………………………..……….....…pp. ……..……….....…pp. 1 -3

Procedure …………………………………………………………………….………...p. 4

Results ………………………………………………………..……...……………pp. 5 - 6

Discussion and Conclusion …………………………………………………….………p. 7

Appendix ……………………………………………………..…….…....………pp. 8 - 10







Object The object of this experiment was to determine the force exerted by a jet of water on a stationary vane and compare the experimental results to the theoretical results.

Theory A moving stream of fluid carries momentu momentum. m. Momentum is defined as the product of the mass of a body and its velocity. When a moving stream of fluid, or jet, is deflected by a surface, a change in linear linear momentum occurs. This change in linear momentum results in the water jet exerting a force on the surface it is impacting.

ΣF = ∫∫CS V ρ Vn dA

(Equation 1)

Equation 1 is the general equation for linear momentum for a deformable control volume. The components of this equation are V, the velocity of the stream of fluid, ρ, the density of the fluid, and Vn, the velocit velocity y of the fluid normal to the impact impact surface. These components are integrated with respect to the impact surface area and yield the force the jet exerts. To calculate the theoretical force a jet exerts on a surface, first a control volume must be defined. For this experiment, a Pelton bucket will will be used for the vane. A diagram of the force exerted on the Pelton bucket by the jet of fluid is seen in Figure 1 below.







To calculate the force the jet of fluid exerts on the Pelton bucket, a force sum calculation is used. A control volume is defined such that the fluid exits at 90 degrees, normal to the defined control volume. This definition of the control volume allows for no fluid to escape in the y direction, thereby simplifying the force calculat calculation. ion. It is assumed that to the left is positive on the x axis. The velocity of the fluid escaping the Pelton bucket is calculated as follows: o

o

V = Vcos(19 ) = V(cos161 )

(Equation 2)

The force balance equation for this system, with to the left on the x axis being positive is as follows:

ΣFx = Fwater – ρAV(Vout – Vin) = 0

(Equation 3)

Simplifying the equation by factoring the velocity term and substituting Equation 3 into the force equation for the velocity of the escaping jet of fluid acting in the x direction is as follows:

ΣFx = Fwater – ρAV2(1 – cos(161 o)) = 0 (Equation 4) The force the Pelton bucket experiences due to the change in momentum of the jet of water is equivalent to the force of the water. Therefore, the equation for the theoretical force of the jet of fluid exerted on the Pelton bucket is as follows: F = ρAV2(1 – cos(161 o))

(Equation 5)

To calculate the area term, A of Equation 5, calculate the cross sectional area of the exit exit of the nozzle. The V term, velocity, is the fluid jet’s velocity at the exit of the nozzle. To calculate the exit exit velocity, divide the volumetric flow rate of the nozzle, Q, by the cross sectional area A. Q = VA

V=Q/A

Equation 6) ( Equation

A method of determining the actual force exerted by a stream of fluid is to evaluate the moment the force exerts on a beam. As seen in Figure 1 below, a nozzle emits a jet of fluid at a constant constant volumetric volumetr ic flow flow rate that impacts the vane. The vane is connected to beam that will rotate about a pivot as seen seen to the left on the beam. beam. The impacting jet of fluid, having its direction of momentum altered, will exert a force on the vane at a fixed distance from the pivot, do, and cause the beam to rotate counterclockwise. To assess the force exerted by the jet, F, a mass jockey is positioned on the beam







pivot on the beam. beam. The tally, or leveling device, of the apparatus will register in the neutral position when the mass jockey is positioned at the appropriate distance from the pivot to counteract the counterclockwise moment exerted by the jet of fluid.

Figure 2: Jet impact testing apparatus diagram (Source: University of Delaware, Department of Mechanical Engineering)

Any subsequent increase in the volumetric flow rate, Q, will cause a proportional increase to the impact force force the jet exerts on the vane. The increased force equates to an increase in the moment moment exerted by the jet jet around the beam’s pivot. To counteract the increased force, and therefore increased moment, the mass jockey must be shifted further to the right on the beam to a position dwi to counteract the increase. To account for for the force at each flow rate, multiply the weight of the mass jockey by the difference in each displacement with reference to the initial position and divide the product by the distance from the fluid jet to the pivot. F = [ Fw (dwi – dw1) ] / do

(Equation 7)







Procedure Equipment: TQ H1D Volumetric Bench (SN: A7771/2) Pelton Bucket Plate Stopwatch

Experiment: 1) Close the drain in the catch basin of the jet impact apparatus. Open the flow flow control valve, turn the pump on and set the flow flow rate to a desired value. Using the sight glass on the side of the device, measure the time taken for the volume in the catch basin to rise rise from zero to an arbitrary volume. Divide this volume by the time taken to achieve iit. t. This value is a flow rate, Q. 2) Obtain a theoretical velocity by dividing the flow rate Q by the cross-sectional area of the nozzle. nozzle. This obtained value is is the theoretical velocity of the fluid fluid exiting the nozzle. 3) Use the theoretical velocity to calculate a theoretical moment exerted on the beam by the jet. 4) Turn off the pump of the apparatus. Ensure that the flow flow control valve has been been closed and open the drain of the catch basin so as to let the fluid in the system return to the pump. 5) Move the mass jockey to the zero position and calibrate the lever arm of the testing apparatus by adjusting the spring via the thumb screw. 6) Turn the pump back back on and open the flow control valve. valve. Adjust the flow rate until a deflection of the lever arm is noticed. Leave the pump pump running and move move the mass jockey to the right until the moment created by the water jet has been balanced. Record this distance. Do not not adjust adjust the spring. 7) Increase the flow rate of the water jet by an increment of two liters per minute. Again, slide the jockey mass to the right until the moment has been balanced. Record this value. 8) Repeat Step 7 for a desired number of data points.









Results Table 1: Known values used to determine theoretical and actual values of force exerted by a jet of water 0.600 kg Jockey Mass 0.010 m Nozzle Diameter 3 1000 kg/m Density of Water 0.150 m Distance from Pivot to Vane Table 2: Theoretical values of the force exerted by a jet of water obtained by using Equation 2 and Equation 3. Volumetric Volumetric Velocity Flow Flow Force (N) (m/s) 3 (LPM) (m /s) 10 1.67E-04 2.12 0.6054 12 2.00E-04 2.55 0.8718 14 2.33E-04 2.97 1.1866 16 2.67E-04 3.40 1.5499 18 3.00E-04 3.82 1.9615 20 3.33E-04 4.24 2.4217 22 3.67E-04 4.67 2.9302 24 4.00E-04 5.09 3.4872 26 4.33E-04 5.52 4.0926 28 4.67E-04 5.94 4.7465 30 5.00E-04 6.37 5.4487 Table 3: Experimental values of the force exerted by a jet of water obtained by using Equation 2 and Equation 4. Volumetric Volumetric dw dwi-dw1 Velocity Flow Flow Force (N) (mm) (m) (m/s)







6.0000

5.0000

4.0000 ) N ( e 3.0000 c r o F

2.0000

1.0000

0.0000 10

15

20

25

30

Flow Rate (LPM) Actua l

T he he or or et et ic ica l

Figure 3: Theoretical and actual force exerted by a jet of water on a Pelton bucket

35







Discussion and Conclusion The Pelton bucket is a component of the Pelton wheel that is found in impulse turbines. The raised ridge between between the two hollows of the device essentially splits a jet of o water into two and deflects each nearly 180 . This deflection prevents the jet jet from spraying out of and away from the device. By controlling the velocity velocity vector of the fluid jet, the Pelton bucket is able to extract more energy from the moving fluid by changing its linear momentum. The fluid that leaves the Pelton bucket has a small fraction of the velocity it came in with, concluding that it has a very efficient design. It was that the theoretical and experimental forces have a significant percentage of error between them, as seen in Table 4 below. below. Most of this error is due to the theoretical calculations neglecting the force of gravity gravity on the jet of water. When the jet leaves leaves the nozzle, it will have the calculated velocity obtained by using Equation 3. However, after the fluid has obtained any height above the nozzle, the force of gravity acts on it and decreases the velocity. The reduction in velocity velocity can be determined by using Bernoulli’s equation, where V is the respective velocity, g is gravitational acceleration and Δz is the distance between the nozzle and the vane. 2

V

vane

2

=V

nozzle

– 2gΔz

Table 4: Percent error of experimental and theoretical force of a jet of water Experimental Force

Theoretical Force

% Error

0.4709

0.6054

22.22%

0.2747

0.8718

68.49%

0.5494

1.1866

53.70%

0.9025

1.5499

41.77%

1.2949

1.9615

33.98%

1.8050

2.4217

25.46%













Appendix Data Usage Sample Calculation of Velocity at 10 LPM -2

3

2

V = Q/A = [(10 LPM * 1.5e m /L * min/60 s)] / [(π/4) [(π/4) * (0.010m) ] = 2.12 m/s

Sample Calculation of Experimental Force at 12 LPM 2

F = 0.600 kg * 9.81 m/s * (0.007 m) / 0.15 m = 0.2747 N

Sample Calculation of Theoretical Force at 20 LPM 3

2

2

o

F = 1000 kg/m * (π/4) * (0.010m) * (4.24 m/s) * (1-cos(161 ) = 2.4217 N









Bibliography rd

Introduction to Fluid Mechanics, 3 Edition William S. Janna (1993) Fundamentals of Material Science and Engineering: An Integrated Approach W.D. Callister, Jr and D.G. Rethwish (2008) A Manual for the Mechanics of Fluid Laboratory William S. Janna (2008)

Impact of a Jet of Water

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