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Jet Momentum AE: 341 Fluid Mechanics Laboratory

Author Alejandro Cuara Instructor Mohamed Amine Abassi

February 10, 2020

Jet Momentum

Alejandro Cuara

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Contents Introduction......................................................................................................................................2 Theory..............................................................................................................................................2 Results and discussion.....................................................................................................................5 Conclusion.....................................................................................................................................11 Acknowledgements........................................................................................................................11 References......................................................................................................................................11 Appendices....................................................................................................................................12

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Introduction The name of this lab experiment is Jet Momentum. The purpose of this lab experiment is to determine the magnitude of the force of a water jet upon a flat plate and a hemispherical cup and to compare those values with those calculated using the principle of impulse and momentum. This experiment was performed by Alejandro Cuara, a student at San Diego State University, on campus on January 29th, 2020 as a requirement for the Fluid Mechanics Lab course.

Theory For the Jet Momentum experiment, a waterjet is used to release water vertically on a hemispherical cup or a flat plate while contained in a large cylinder. At the base there is a tube which carries the water to a measuring tank located on the bottom of the bench. The cup/plate is supported by a metal bar above the cylinder which holds a jockey weight. The jockey weight is used to counter the force of the waterjet on the cup/plate and return it to its original zero position. The theoretical force of the waterjet is calculated utilizing the principle of Impulse and Jet Momentum using the Mass Flow rate, velocity of the water, and the deflector angle. Equation 1

m = Mass F = Force u = Velocity vector

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Equation 2 To calculate the force due to the jet on the cup/plate we need to find angle

and use the deflection

. After deriving equation 1 and dividing by the change of time we get the equation for

theoretical force.

= Density = Vector velocity

= Change of mass = Change of time

Jet Momentum

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Equation 3

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Equation 4

= Mass Flow Rate

= Change of mass

= Density = mass/volume

= Change of time

= Vector velocity

= Nozzle diameter

A = Nozzle area Equation 5 Assuming

, (absence of elevation change and energy loss) we obtain the following: .

For

For

*note:

for the flat plate and

for the hemispherical cup.

Equation 6 To find the volumetric flow rate we use:

= Volumetric flow rate This equation will be used to find the mass flow rate from equation 3 and will be helpful in finding the velocity vector.

Jet Momentum To find

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, the Conservation of Mechanical Energy (Bernoulli’s equation) is applied

After the appropriate calculations are completed for theoretical force, it must be compared to the experimental force which is

= Gravity valued at 9.795 m/sec^2. = Distance from the zero position of the jockey weight.

Results and discussion Diameter of nozzle = d = 10 mm = 0.01m Cross sectional area of nozzle = A

= 7.85*10^-5 m^2 Height between the nozzle tip and deflector: h = 35mm = 0.035m Distance from zero position = x For the 90-degree deflector: x = 10mm = 0.01m For the 180-degree deflector: x = 20mm = 0.01m When the jockey weight is moved x-distance, the force F is given by the F_exp equation

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For the 90-degree deflector

= 0.40N For the 180-degree deflector

= 0.80N The mass flow rate,

, is found by timing how long it takes for a known amount of water to fill

the reservoir. The equation is as follows, For the 90-degree deflector

For the 180-degree deflector

Next, the volumetric flow rate can be found using the mass flow rate and density. For the 90-degree deflector

Jet Momentum

Alejandro Cuara

For the 180-degree deflector

Rearranging the volumetric flow rate, we can determine the velocity at the nozzle For the 90-degree deflector

For the 180-degree deflector

The velocity at the impact site can be found using

For the 90-degree deflector

For the 180-degree deflector

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Finally, the theoretical force can be found using the equations for each deflector. For the 90-degree deflector

For the 180-degree deflector

The data is as follows:

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Jet Momentum Data graphically represented:

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Based on the figures shown previously it is clear that the force produced on each deflector is proportional to the momentum of the water in the jet as it hits the deflector. Sample calculations will follow in the appendices section as well as the MatLab code and graphs.

Conclusion The magnitude of the force of a waterjet on two different deflectors was determined and compared to calculated values found by using the principle of Impulse and Momentum. The calculated force is associated with the measured force. The theoretical force equations theorized that the hemispherical force is larger than compared to the flat place and they are correct. The calculated R values indicate that values are close but should not be determined through experiments because of common errors. These include wrong readings, delayed reactions, and user error.

Acknowledgements (Example) I thank San Diego State University for providing the necessary facilities, and Professor Abbasi for the insightful instructions.

References 1. Abassi M. (2018) AE 341 Fluid Mechanics Lab Lecture 1 https://blackboard.sdsu.edu/webapps/blackboard/execute/announcement?

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method=search&context=course_entry&course_id=_374272_1&handle=announcements _entry&mode=view

Appendices

MatLab Graphs (above):

Jet Momentum

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Matlab Code: x = [0.742 1.121 1.591 2.202 2.246 2.888 2.827 3.834 4.065 4.572] y = [0.742 1.121 1.591 2.202 2.246 2.888 2.827 3.834 4.065 4.572] x1 = [0.742 1.121 1.591 2.202 2.246 2.888 2.827 3.834 4.065 4.572] y1 = [0.3918 0.7836 1.1754 1.5672 1.959 2.350 2.742 3.134 3.526 3.918] close all figure(1) scatter (x1,y1) p = polyfit(x1,y1,1) f = polyval(p,x1); hold on plot(x,f,'--r') title ('Experimental Force vs mu_0 for Flat Plate') xlabel ('mu_0 (N)') ylabel ('Experimental Force') hold on legend('Experimental values','Best Fit line') figure(2) scatter (x,y) p = polyfit(x,y,1) f = polyval(p,x); hold on plot(x,f,'--r') title ('Theoretical Force vs mu_0 for Flat Plate') xlabel ('mu_0 (N)') ylabel ('Theoretical Force (N)') hold on legend('Theoretical values','Best Fit line') x3 y3 x4 y4

= = = =

[0.5685 1.042 1.866 2.060 2.511 3.260 3.637 4.707 5.239] [0.783 1.567 2.350 3.134 3.918 4.701 5.485 7.052 7.836] [0.5685 1.042 1.866 2.060 2.511 3.260 3.637 6.470 4.707 5.239] [1.137 2.084 3.733 4.120 5.022 6.521 7.275 12.941 9.414 10.479]

figure(3) scatter (x3,y3) p = polyfit(x3,y3,1) f = polyval(p,x3); hold on plot(x3,f,'--r') title ('Experimental vs mu_0 for Spherical Cup') xlabel ('mu_0 (N)') ylabel ('Experimental Force') hold on legend('Experimental','Best Fit line') figure(4) scatter (x4,y4) p = polyfit(x4,y4,1) f = polyval(p,x4);

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hold on plot(x4,f,'--r') title ('Theoretical Force vs mu_0 for Spherical Cup') xlabel ('mu_0 (N)') ylabel ('Theoretical Force (N)') hold on legend('Theoretical values','Best Fit line')

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