TRIBHUVAN UNIVERSITY INSTITUTE OF ENGINEERING PULCHOWK CAMPUS
LABORATORY REPORT ON BERNOULLI EQUATION
DATE: AUGUST 26th, 2009
SUBMITTED BY:
SUBMITTED TO:
Brian Campbell Davis
Dr. Rajendra Shrestha
065/MSR/502
IOE, ME Department
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Introduction The converging-diverging nozzle test apparatus (Figure 1.1) is used to show the validity of the Bernoulli Equation. It is also used to show the validity of the continuity equation where the fluid flow is relatively incompressible. The data taken will also show the presence of fluid energy losses, often attributed to friction and the turbulence and eddy currents associated with a separation of flow from the conduit walls.
Figure 1.01 – Overall test setup for Bernoulli Equation validation experiment
Figure 1.02 – Test section positioned as “Converging” test section
Figure 1.03 – Test section positioned as “Diverging” test section
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Laboratory Procedure This is the procedure used for the experiment in this report. The manual for the test apparatus (Appendix A), may be consulted for further information regarding the equipment and the procedure. 1. Plug the test apparatus into the mains and close the circuit breakers. 2. Place wood block under the apparatus stand, raising up the right side of the platform 3. Measure the height of the platform at both ends, and the distance between measurement points. 4. Measure the distance between centerline of the manometers in the array. 5. Position the test section in the converging or diverging direction and record. 6. Open the air bleed valve at the top of the manometers. 7. Turn on the pump and adjust the globe valve to achieve a flow rate that results in all manometers being in a readable range. 8. Close the air bleed valve at the top of the manometers. 9. Repeat steps 6 to 8 until all manometers are in a readable range. 10. Drop the stop plug into the basin. 11. Record the time for a specific amount of water to accumulate in the basin. 12. Open the stop plug in the basin and let the water drain out. 13. Repeat steps 10 to 12 and additional 3 times, resulting in 4 sets of flow rate data. 14. Move the dynamic pressure probe in line with the first static pressure port. 15. Record the manometer data for the dynamic pressure probe and static pressure port. 16. Move the dynamic pressure probe in line with the next static pressure port. 17. Record the manometer data for the dynamic pressure probe and static pressure port. 18. Repeat steps 16 to 17 until data has been taken inline with each static pressure port. 19. Turn off the pump. 20. Repeat steps 5 to 19 using different flow rates for both the converging and diverging configuration of the test section. (Minimum two tests for each configuration) configuratio n) 21. Open the circuit breakers and remove the plug from the mains.
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Experimental Data During the testing performed, some testing runs did not produce valid data. This is why the numbering of testing runs is not continuous. Test #
1
Section Orientation
Flow Rate Volume (L) 3 3 3 3
Test #
Data Time (s) 35.44 37.51 36.99 38.54
Test #
Data Time (s) 34.93 35.96 36.73 36.21
Static + Dynamic Manometer (mm H20)
a b c d e f
89 74 57 39 11 25
69 71 71 66 64 26
Order in Flow
Port
Static Manometer (mm H20)
Static + Dynamic Manometer (mm H20)
1 2 3 4 5 6
a b c d e f
125 105 87 70 40 55
105 105 105 100 99 65
Order in Flow
Port
Static Manometer (mm H20)
Static + Dynamic Manometer (mm H20)
1 2 3 4 5 6
a b c d e f
93 84 75 68 55 56
72 72 72 72 70 55
1 2 3 4 5 6
Converging
4
Section Orientation
Flow Rate Volume (L) 3 3 3 3
Port
Static Manometer (mm H20)
Order in Flow
3
Section Orientation
Flow Rate Volume (L) 3 3 3 3
Converging
Data Time (s) 60.20 61.50 61.76 61.76
Converging
4
Test #
5
Section Orientation
Flow Rate Volume (L) 3 3 3 3
Test #
Data Time (s) 28.22 29.26 27.97 28.22
Port
Static Manometer (mm H20)
Static + Dynamic Manometer (mm H20)
f e d c b a
160 43 60 55 92 125
158 158 155 135 100 108
Order in Flow
Port
Static Manometer (mm H20)
Static + Dynamic Manometer (mm H20)
1 2 3 4 5 6
f e d c b a
70 35 40 50 45 65
66 68 65 66 36 45
Order in Flow 1 2 3 4 5 6
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Section Orientation
Flow Rate Volume (L) 3 3 3 3
Diverging
Data Time (s) 43.95 46.01 44.98 45.23
Diverging
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Theoretical Calculations The first and most crucial calculation is determining the incline of the apparatus. This incline not only effects the elevation of fluid flow, but it also tilts the array of manometers, making a correction calculation necessary to achieve accurate readings for all pressure head data taken. The three measurements taken were the height of the platform at both ends (H i1 and Hi2) and the distance between the two height measurements (L i). Using the following followin g equations, the incline angle ( β) was determined in degrees and then converted to radians for ease of use.
Equation (1.00)
Hi1 = 3.8 cm Hi2 = 9.0 cm Li = 36.5 cm
Figure 1.04 – Angle of incline diagram
The full expression for the Energy Grade Line, or Total Head, is shown below in the familiar form of the Bernoulli Equation. For each location in the closed conduit, the summation of the three terms will equal the total head.
Equation (1.01)
Neglecting friction losses, the total head will not change in the conduit, this is the basic conclusion of the Bernoulli Equation. Equation (1.02)
The first term, or Static Pressure Head at port x, can be read directly on the array of manometers for any port location: a, b, c, d, e, f.
Equation (1.03)
The first two terms, or Static and Dynamic Pressure Head, can be read on the manometer furthest to the right. This reading will be taken from whatever location the dynamic
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pressure probe is located at. For this experiment, the probe is moved inline with one of the static pressure ports and the reading is taken.
Equation (1.04)
In order to determine the experimental dynamic pressure head, Equation 1.03 is subtracted from Equation 1.04.
Equation (1.04)
The above Equation 1.04 is also the governing equation for determining the dynamic pressure head using a pitot pitot tube, also known as a pitot probe. For determining the theoretical dynamic pressure head, we use the continuity theorem and the assumption of incompressible flow to arrive at the following equations for dynamic pressure head. Equation (1.05)
Equation (1.06)
Due to the incline of the conduit, the elevation head (z) cannot be neglected. In this case, the difference in elevation is the important figure and the first port is taken as the datum where z = 0. The remaining five static pressure port locations are all at a higher elevation than the first port, resulting in the negative sign in the equation. The difference in elevation can be determined by using the appropriate trigonometric relation and the angle of incline ( β), where l x is the distance between the port and the datum. Equation (1.07) For the converging nozzle, the first port is “a” and the remaining ports listed in order are “b, c, d, e, f”. For the diverging nozzle, the first port is “f” and the remaining ports listed in order are “e, d, c, b, a”. Combining the previous equations, we now have a Total Head that can be calculated from the experimental dynamic pressure head and the theoretical dynamic pressure head. Equation (1.08)
Equation (1.09) 7
It is important to note that the difference here in “Experimental” and “Theoretical” only refers to the derivation of the dynamic pressure head. In the results section, the experimental dynamic pressure head and the theoretical dynamic pressure head will also be compared to test the validity of the assumption of incompressibility. incompressibil ity. Also in the results section we will be able to qualitatively and quantitatively show the losses in the total head, due to friction and other minor losses. The modified Bernoulli Equation for these losses between points 1 and 2 is shown below.
Equation (1.10)
Equation (1.11)
In addition to using trigonometry to determine the change in elevation head, it must also be used to properly adjust the manometer readings (Figure 1.05). This is because the manometer array is attached to the same platform as the conduit, and it has been tilted at the same angle ( β). The horizontal position of the manometers for ports “a, b, c, d, e, f”, and the dynamic probe are not switched when the nozzle is changed from converging to diverging. The manometer readings are relative to each other, due to this, the x-y datum, will be the zero of the first manometer (port “a”).
Figure 1.05 – Trigonometric adjustments necessary to determine actual manometer height
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For port “a”, a single adjustment is all that is necessary. Equation (1.12) Equation (1.13) For the other ports, the zero of the manometer is not at the datum, so a modified version of Equation 1.12 must be used to before the same Equation 1.13 is used to determine the actual height of the manometer fluid. For a generic generic port “x”, the following equations apply where l m is the distance between centerlines of the manometer “x” and the manometer “a”. Equation (1.14) Equation (1.15)
Due to equal spacing, l m can be expressed in the following equation, where “x -1” is the manometer spaces between the manometer “x” and the datum manometer of port a.
Equation (1.16) Furthermore, the static + dynamic pressure head manometer is always the furthest from the datum manometer of port “a”, giving the following equation.
Equation (1.17) Using this system of equations, the manometer readings taken during the experiment can be converted into the actual fluid heights that they represent.
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Results & Conclusion Test 1: Converging Nozzle 110.00
100.00
90.00
80.00 ) m m (
d a e H
Exp Total Theo Total
70.00
Contant EGL Static + Elev
60.00
50.00
40.00
30.00 0
1
2
3
4
5
6
7
Port Number
Figure 1.06 – Test 1: Head vs. Port Number for converging section and flow rate = 4.85 L/min
Test 3: Converging Nozzle 150.00
140.00 130.00
120.00
110.00
) m m (
Experimental Theoretical
d 100.00 a e H
Contant EGL Static + Elev
90.00
80.00 70.00 60.00
50.00 0
1
2
3
4
5
6
7
Port Number
Figure 1.07 – Test 3: Head vs. Port Number for converging section and flow rate = 5.01 L/min
10
Test 4: Converging Nozzle 110.00
105.00
100.00
) m m (
d a e H
95.00
Experimental Theoretical Contant EGL Static + Elev
90.00
85.00
80.00
75.00 0
1
2
3
4
5
6
7
Port Number
Figure 1.08 – Test 4: Head vs. Port Number for converging section and flow rate = 2.94 L/min
Test 5: Diverging Nozzle 210.00
190.00
170.00
150.00 ) m m (
Experimental Theoretical
d 130.00 a e H
Contant EGL Static + Elev
110.00
90.00
70.00
50.00 0
1
2
3
4
5
6
7
Port Number
Figure 1.09 – Test 5: Head vs. Port Number for diverging section and flow rate = 6.33 L/min
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Test 6: Diverging Nozzle 110.00
100.00
90.00
) m m (
d a e H
80.00
Experimental Theoretical Contant EGL Static + Elev
70.00
60.00
50.00
40.00 0
1
2
3
4
5
6
7
Port Number
Figure 1.10 – Test 6: Head vs. Port Number for diverging section and flow rate = 4.00 L/min
Test 1: Converging Nozzle 110.00
100.00
90.00
80.00 ) m m (
d a e H
Exp Total Theo Total
70.00
Contant EGL Static + Elev
60.00
50.00
40.00
30.00 0
20
40
60
80
100
120
140
160
Port Location (mm)
Figure 1.11 – Test 1: Head vs. Port Location for converging section and flow rate = 4.85 L/min
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Test 3: Converging Nozzle 150.00
140.00
130.00
120.00 ) m110.00 m (
Experimental Theoretical
d a e H100.00
Contant EGL Static + Elev
90.00
80.00
70.00
60.00 0
20
40
60
80
100
120
140
160
Port Location (mm)
Figure 1.12 – Test 3: Head vs. Port Location for converging section and flow rate = 5.01 L/min
Test 4: Converging Nozzle 110.00
105.00
100.00
) m m (
d a e H
95.00
Experimental Theoretical Contant EGL Static + Elev
90.00
85.00
80.00
75.00 0
20
40
60
80
100
120
140
160
Port Location (mm)
Figure 1.13 – Test 4: Head vs. Port Location for converging section and flow rate = 2.94 L/min
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Test 5: Diverging Nozzle 200.00
180.00
160.00
) 140.00 m m (
Experimental Theoretical
d a e H120.00
Contant EGL Static + Elev
100.00
80.00
60.00 0
20
40
60
80
100
120
140
160
Port Location (mm)
Figure 1.14 – Test 5: Head vs. Port Location for diverging section and flow rate = 6.33 L/min
Test 6: Diverging Nozzle 110.00
100.00
90.00 ) m m (
d a e H
Experimental Theoretical
80.00
Contant EGL Static + Elev
70.00
60.00
50.00 0
20
40
60
80
100
120
140
160
Port Location (mm)
Figure 1.15 – Test 6: Head vs. Port Location for diverging section and flow rate = 4.00 L/min
Due to the flexible nature of the dynamic probe, in some place it was pushed to the edge of the conduit, giving falsely low data. For both converging and diverging configurations, configurations , test point 1 is inaccurate due to this issue. For diverging configuration
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only, test point 5 is inaccurate due to this issue. For analyzing the fluid flow, the other data points will be considered. All three of the converging nozzle tests show good correlation between the Experimental Total Head and the Theoretical Total Head. The overall generally flat slope of the data points in the converging section shows a low frictional head loss there. While there are not multiple data points in the diverging section for this configuration, it still can be seen that most of the head loss is in the diverging section. Specifically, Specificall y, most of the losses are between ports “e” and “f” (ordered 5 and 6), which shows that the larger loss is experienced in the expansion portion of the conduit. The two diverging nozzle tests do not show as good a correlation between the Experimental Total Head and the Theoretical Total Head. The Experimental Total Head, which is taken from the moving pressure probe, is similar to data taken in the converging nozzle tests. It is the data from the static pressure ports, ports, which result in the Theoretical Total Head, which are too low. It is concluded that during the expansion, there is separation of flow causing a reduction in flow cross sectional area and an increase in velocity and dynamic head. This increase in dynamic head explains why the theoretical dynamic head is less, resulting in a lower Theoretical Total Head than the Experimental Total Head. In the converging configuration, while the expansion is steeper, the diverging configuration shows that even when the expansion is more gradual than the contraction, the expansion causes most of the losses in the conduit. This is shown directly in the diverging configuration charts that plot the total head vs. x-distance in mm.
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Appendix A: Lab Apparatus Manual
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