ce 2250 lab 3 report -...

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CE 2250

Fluid Mechanics Lab

Dr. Heather D. Smith

Lab 3

Fluid Forces

Group 4

Austin McLean

Doug Meek

Nik Melancon

Joseph Mingo

Date Tested: February 6, 2013 Date Submitted: February 20, 2013

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Introduction

By determining the force produced when the water jet strikes the different vanes the

similarity to how much force a support structure can withstand can be seen. Also, the amount of

force pushed on to a turbine or pump can be calculated. The control volume method will be used

by monitoring the amount of fluids in and out of the system to find the expected force. A

physical application could be a jet turbine where the fluid is air. A specific type of turbine is the

Pelton wheel that has a water jet pointed tangentially to the wheel caught by vanes that generate

a torque causing the wheel to spin.

Apparatus

For the tests to be accomplished, both a Cussons P6233 Impact of Jets Apparatus and

Cussons P6100 Hydraulics Bench were used. An open/close valve allows water to be fed into the

water supply connection in the Impact of Jets Apparatus. The water passes through a removable

nozzle inside the apparatus and exits through drain holes in the base of the apparatus and into the

weir channel below. To balance the forces of the water jets, circular brass weights are applied to

a weight carrier which sits on the weight platform. A flag is attached beside the weight platform

for recording the height of the platform. At the end of the weir channel there is a large basin that

uses an open/close valve to collect water from the apparatus. This system allows for the

measuring of that water with a manometer located on the side of the basin. Two different target

vanes are used: flat and semi-spherical, along with two different size nozzles: 5 mm and 8 mm.

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Figure 1: Impact of Jets Apparatus diagram

Figure 2: AutoCAD drawing of the Impact of Jets Apparatus

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Theory

The focus of this lab is the summation of the forces applied by moving fluids to solid

obstacles. To perform this lab, the Conservation of Linear Momentum is assumed with a control

volume analysis. In order to use this principle, a few assumptions need to be made. They are: the

jet is a free jet open to the atmosphere, the target is smooth and therefore has no shear effect, the

fluid velocity out of the jet is constant, and the fluid is incompressible. Making these

assumptions, the Conservation of Linear Momentum can be written as:

∑ �� = �� − �� (1)

where � is the force applied to the target, �� is the mass flow rate coming out of the control

volume, �� is the velocity coming out of the control volume, �� is the mass flow rate coming

in to the control volume, and � is the velocity coming in to the control volume.

Since mass flow rate, �� , is equal to �� and since all of the other forces cancel except for

the force acting through the vertical nozzle, equation 1 can be rewritten as:

� = �� = �� − �� (2)

where �� is the force acting on the target solely in the y direction, � is the density, and � is the

volumetric flow rate.

Since the out flowing velocity, ��, can be related to the in flowing velocity, �, equation 2 can

be rewritten as:

� = �� 1 − cos �� (3)

where� is the angle between the extension of the velocity vector to the out flowing direction

(anglebetween � and ��).

Flat Vane

For the flat vane, the direction of �� is only in the x direction. Therefore � is equal to

90° from the vertical. Substituting 90° into equation 3 yields:

� = �� 1 − cos 90°� (4)

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� = �� 1 − 0� (5)

� = �� (6)

Semi-spherical Vane

For the semi-spherical vane, the direction of �� has both x and y components. In this

case, � is equal to 135° from the vertical. Substituting 135° into equation 3 yields:

� = �� 1 − cos 135°� (7)

� = �� '1 + √** + (8)

� = 1.7071�� (9)

This says that because the flow is redirected by the curvature of the vane, the force

impacted by a semi-spherical vane is 1.7071 times higher than the force impacted by a flat vane.

Height of Target

As the fluid moves out of the nozzle and closer to the target, it loses some of its kinetic

energy to potential energy. To calculate the velocity coming out of the nozzle, ��., Bernoulli’s

equation is used to relate ��. and �:

�* = ��.

* − 201 (10)

where 0 is the gravitational acceleration and 1 is the distance between the nozzle and the vane.

Procedure

In order to calculate the force applied to the target, the distance between the nozzle and

the target must be known. To find this distance, two heights must be measured. First, the weight

carrier is positioned on the weight platform. The height of the uncompressed platform is

measured and recorded on the datasheet. Then, an equilibrium position is obtained by putting

520 g onto the weight carrier. The height of the compressed platform is measured and recorded.

The marker flag is then moved until it is exactly level with the platform. Now the experiment is

ready to be performed. The first weight on the data sheet is added to the weight carrier. The

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pump is started and water flow is established by opening the bench regulating valve until it is

fully open. The platform should rise above equilibrium. The bench regulating valve is slowly

closed in order to bring the platform back to equilibrium. This means that the weight applied to

the top is balanced by the force being applied to the target. Once equilibrium is established, the

velocity being applied to the target is estimated. To do this, the velocity coming out of the nozzle

must be measured. This is done by measuring the flow rate in two ways. The first way is to use

the weir attached to the channel. The height of the water leaving the channel is measured and

recorded (in liters/minute) on the datasheet. The second way is to record a fixed volume over a

certain amount of time. This is done by filling the tank and recording the time it takes to reach a

known volume. One person stands by with a stopwatch and another person operates the drainage

valve. When the person with the timer is ready, the valve operator closes the valve that drains the

tank by turning it 90° clockwise and the timer is started. The time is measured until either the

volume reaches 15 L or until 1 minute has elapsed. Record the time and the volume reached on

the datasheet. Reopen the valve to empty the tank. The next weight on the data sheet is then

added to the weight carrier and the experiment is repeated. When all weights on the datasheet

have been tested, the bench regulating valve is closed and the pump is turned off. The nozzle

type is switched and the experiment is repeated for each new weight. Then, the nozzle diameter

size is switched and the experiment is repeated again for both the flat and the semi-spherical

nozzle types.

Results

The following tables (Tables 1-4) show all of the calculated values for each vane type;

the nozzle diameter and vane type organizes them. The tables include calculations for Flow Rate

from Weir (m^3/s), Flow Rate from Volume (m^3/s), % Difference in Flow Rate, Average Flow

Rate (m^3/s), Nozzle Velocity (m/s), Inflow Velocity (m/s), Impulse Momentum (N),

Theoretical Force (N), Experimental Force (N), and Experimental Error (%). Fexp/Fth at the

bottom of the table is the Experimental Force / Theoretical Force. When the % Difference in

Flow rate is less than 125% then the Average Flow Rate is used. Otherwise, the Average Flow

Rate was taken as the Volume Flow Rate.

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Table 1: 8mm Diameter Nozzle with Flat Vane

Table 2: 8 mm Diameter Nozzle with Semi-Spherical Vane

Trial 1 2 3 4 5 6

Weight, W (g) 1040 950 850 750 650 550

Volume, V (L) 15 15 15 15 15 8

Time, t (s) 28.4 32.5 37.6 47.7 57.0 60.0

Flow Rate from Weir

(L/min) 31.0 28.0 24.0 19.0 15.0 8.0

Flow Rate from Weir

(m^3/s) 0.00052 0.00047 0.00040 0.00032 0.00025 0.00013

Flow Rate from Volume

(m^3/s) 0.00053 0.00046 0.00040 0.00031 0.00026 0.00013

Difference in Flow Rate

(%) 97.82 101.11 100.27 100.70 95.00 100.00

Average Flow Rate

(m^3/s) 0.00052 0.00046 0.00040 0.00032 0.00026 0.00013

Nozzle Velocity (m/s) 10.39 9.23 7.95 6.28 5.10 2.65

Inflow Velocity (m/s) 10.37 9.21 7.92 6.24 5.06 2.56

Impulse Momentum (N) 5.42 4.27 3.16 1.97 1.30 0.34

Theoretical Force (N) 5.42 4.27 3.16 1.97 1.30 0.34

Experimental Force (N) 5.10 4.22 3.24 2.26 1.28 0.29

Experimental Error (%) 5.83 1.27 -2.37 -14.61 1.70 13.73

Fexp/Fth 0.94166 0.98726 1.02372 1.14606 0.98303 0.86273

8mm Diameter Nozzle with Flat Vane

Trial 1 2 3 4 5 6

Weight, W (g) 1320 1230 1060 890 720 550

Volume, V (L) 15 15 15 15 15 5

Time, t (s) 29.9 32.6 39.2 47.0 60.0 60.0

Flow Rate from Weir

(L/min) 31.0 29.0 22.5 18.0 14.0 7.0

Flow Rate from Weir

(m^3/s) 0.00052 0.00048 0.00038 0.00030 0.00023 0.00012


(m^3/s) 0.00050 0.00046 0.00038 0.00032 0.00025 0.00008


(%) 102.99 105.04 98.00 94.00 93.33 140.00

Average Flow Rate

(m^3/s) 0.00051 0.00047 0.00038 0.00031 0.00024 0.00010






Experimental Error (%) -52.53 -57.77 -86.35 -91.62 -70.68 -58.05

Fexp/Fth 1.52526 1.57771 1.86352 1.91618 1.70684 1.58049

8 mm Diameter Nozzle with Semi-Spherical Vane

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Table 3: 5 mm Diameter Nozzle with Flat Vane

Table 4: 5 mm Diameter Nozzle with Semi-Spherical Vane

Trial 1 2 3 4 5 6

Weight, W (g) 820 780 720 660 600 550

Volume, V (L) 15 15 15 11 9 6

Time, t (s) 51 58 69 60 61 64

Flow Rate from Weir

(L/min) 15.0 14.0 12.5 10.0 8.0 6.0

Flow Rate from Weir

(m^3/s) 0.00025 0.00023 0.00021 0.00017 0.00013 0.00010


(m^3/s) 0.00029 0.00026 0.00022 0.00018 0.00015 0.00009


(%) 85.00 90.22 95.83 90.91 90.37 106.67

Average Flow Rate

(m^3/s) 0.00027 0.00025 0.00021 0.00018 0.00014 0.00010






Experimental Error (%) 21.83 17.10 14.80 11.67 21.49 37.80

Fexp/Fth 0.78172 0.82902 0.85200 0.88327 0.78509 0.62204


Trial 1 2 3 4 5 6

Weight, W (g) 1010 910 820 730 640 550

Volume, V (L) 15 15 15 10 8 5

Time, t (s) 56 59 70 61 61 62

Flow Rate from Weir

(L/min) 15 14 12 10 8 5

Flow Rate from Weir

(m^3/s) 0.00025 0.00023 0.00020 0.00017 0.00013 0.00008


(m^3/s) 0.00027 0.00025 0.00021 0.00016 0.00013 0.00008


(%) 93.33 91.78 93.33 101.67 101.67 103.33

Average Flow Rate

(m^3/s) 0.00026 0.00024 0.00021 0.00017 0.00013 0.00008







Fexp/Fth 0.82583 0.74162 0.79064 0.87020 0.77849 0.51079

5mm Diameter Nozzle with Semi-Spherical Vane

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Figure 3 shows the experimental force value vs. the theoretical force calculated for each

trial. The graph shows that the 8mm spherical series has the greatest values of all four vanes and

the 5mm flat and semi-spherical have nearly the same values. From this one can conclude that

the 8mm diameter vane produces more force than the 5mm diameter vane, and the semi-

spherical produces more force than the flat vane.

Figure 3: Experimental Force vs. Theoretical Force

Figure 4 shows the Volumetric Flow Rate vs. The Experimental / Theoretical Ratio for

each trial. From the chart it can be seen that both 5mm diameter vanes show a smaller volumetric

flow rate. Both of the 8mm diameter vanes show a fairly level slope for the Force Ratio.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0.00 2.00 4.00 6.00 8.00

Th

eo

reti

cal

Fo

rce

(N

)

Experimental Force (N)

Experimenal Force vs Theoretical Force

8mm Flat

8mm Semi-Spherical

5mm Flat

5mm Semi-Spherical

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Figure 4: Flow Rate vs. Force Ratio

Figure 5 shows the calculated impact momentum vs. the experimental force measured in

the lab. The slope of the trend line for each series can be used for calculating the experimental

force from the impact momentum on the vanes.

0

0.5

1

1.5

2

2.5

0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

Ex

pe

rim

en

tal

Fo

rce

/Th

eo

reti

cal

Fo

rce

Average Flow Rate (m^3/s)

Flow Rate vs Force Ratio

8mm Flat

8mm Semi-Sphere

5mm Flat

5mm Semi-Sphere

y = 0.9851x

R² = 0.9891

y = 1.6232x

R² = 0.9739

y = 0.8152x

R² = 0.9914

y = 1.3574x

R² = 0.9882

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ex

pe

rim

en

tal

Fo

rce

(N

)

Impact Momentum (N)

Impact Momentum vs Experimental

Force

8mm Flat

8mm Semi-Spherical

5mm Flat

5mm Semi Spherical

Linear (8mm Flat)

Linear (8mm Semi-

Spherical)

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Figure 5: Impact Momentum vs. Experimental Force

The table below (Table 5) shows the nozzle size, vane type, theoretical slope,

experimental slope, and percent error.

Table 5: Theoretical and Experimental Slope

Conclusion

From the tests run, it was found that the larger the diameter of the nozzle from which the

water comes out of, the larger the force it will exert on its target as seen in figure 3 with the

comparison of nozzles. Also, it was found that the force was greater when the target was semi-

spherical as opposed to a flat surface; this confirms equation 9 which states that the force is

1.7071 times the force from a flat vane. It can also be seen from figure 4 that as the flow rate

increases for each vane type and nozzle size, the experimental force approaches the theoretical

force calculated. One can also conclude that the 5 mm nozzle is almost identical for both types of

vanes, which can be assumed that the smaller the nozzle size, smaller force exerted the more

likely it is for the theoretical to match the experimental force. Also, the flat vane is the most

consistent between theoretical and experimental, using an 8 mm nozzle. It can be seen in figure 5

that the semi-spherical vanes transfer greater force than the flat vanes, and that it is not

dependent on nozzle size. Finally, by looking at table 5 it can be assumed that some human error

on calculations, imprecise measurements, estimating the water level in the weir channel, and

time keeping would be the reason for the percent error being high. However, the table shows that

the larger the nozzle size the smaller the percent error for the forces.

Appendix

Nozzle Size Vane Type Slope, Th Slope, Ex % Error

5mm Flat 1.00 0.82 18.48

5mm Semi-Spherical 1.71 1.36 20.49

8mm Flat 1.00 0.99 1.49

8mm Semi-Spherical 1.71 1.62 4.91

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Trial 1 2 3 4 5 6

Weight, W (g) 1040 950 850 750 650 550

Volume, V (L) 15 15 15 15 15 8

Time, t (s) 28.4 32.5 37.6 47.7 57.0 60.0

Flow Rate from Weir

(L/min) 31.0 28.0 24.0 19.0 15.0 8.0

Flow Rate from Weir

(m^3/s) 0.00052 0.00047 0.00040 0.00032 0.00025 0.00013


(m^3/s) 0.00053 0.00046 0.00040 0.00031 0.00026 0.00013


(%) 97.82 101.11 100.27 100.70 95.00 100.00

Average Flow Rate

(m^3/s) 0.00052 0.00046 0.00040 0.00032 0.00026 0.00013






Experimental Error (%) 5.83 1.27 -2.37 -14.61 1.70 13.73

Fexp/Fth 0.94166 0.98726 1.02372 1.14606 0.98303 0.86273


Trial 1 2 3 4 5 6

Weight, W (g) 1320 1230 1060 890 720 550

Volume, V (L) 15 15 15 15 15 5

Time, t (s) 29.9 32.6 39.2 47.0 60.0 60.0

Flow Rate from Weir

(L/min) 31.0 29.0 22.5 18.0 14.0 7.0

Flow Rate from Weir

(m^3/s) 0.00052 0.00048 0.00038 0.00030 0.00023 0.00012


(m^3/s) 0.00050 0.00046 0.00038 0.00032 0.00025 0.00008


(%) 102.99 105.04 98.00 94.00 93.33 140.00

Average Flow Rate

(m^3/s) 0.00051 0.00047 0.00038 0.00031 0.00024 0.00010






Experimental Error (%) -52.53 -57.77 -86.35 -91.62 -70.68 -58.05

Fexp/Fth 1.52526 1.57771 1.86352 1.91618 1.70684 1.58049

8 mm Diameter Nozzle with Semi-Spherical Vane

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Trial 1 2 3 4 5 6

Weight, W (g) 820 780 720 660 600 550

Volume, V (L) 15 15 15 11 9 6

Time, t (s) 51 58 69 60 61 64

Flow Rate from Weir

(L/min) 15.0 14.0 12.5 10.0 8.0 6.0

Flow Rate from Weir

(m^3/s) 0.00025 0.00023 0.00021 0.00017 0.00013 0.00010


(m^3/s) 0.00029 0.00026 0.00022 0.00018 0.00015 0.00009


(%) 85.00 90.22 95.83 90.91 90.37 106.67

Average Flow Rate

(m^3/s) 0.00027 0.00025 0.00021 0.00018 0.00014 0.00010







Fexp/Fth 0.78172 0.82902 0.85200 0.88327 0.78509 0.62204


Trial 1 2 3 4 5 6

Weight, W (g) 1010 910 820 730 640 550

Volume, V (L) 15 15 15 10 8 5

Time, t (s) 56 59 70 61 61 62

Flow Rate from Weir

(L/min) 15 14 12 10 8 5

Flow Rate from Weir

(m^3/s) 0.00025 0.00023 0.00020 0.00017 0.00013 0.00008


(m^3/s) 0.00027 0.00025 0.00021 0.00016 0.00013 0.00008


(%) 93.33 91.78 93.33 101.67 101.67 103.33

Average Flow Rate

(m^3/s) 0.00026 0.00024 0.00021 0.00017 0.00013 0.00008







Fexp/Fth 0.82583 0.74162 0.79064 0.87020 0.77849 0.51079

5mm Diameter Nozzle with Semi-Spherical Vane

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Bibliography

“lab 3 full”. Youtube.com. 31 January 2013. CE2250lsu. 16 February 2013

<http://www.youtube.com/watch?v=Jl4beBnSao0&feature=youtu.be>

Smith, Heather. “Lab 3 – Fluid Forces.” Moodle2.lsu.edu. Louisiana State University, 31

January 2013. Web. 16 February 2013.

<http://moodle2.lsu.edu/pluginfile.php/394673/mod_resource/content/1/lab_3.pdf>

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