Experiment 1: Bernoulli’s theorem demonstration Abstract This experiment is carried out to investigate the validity of Bernoulli’s theorem when applied to the steady flow of water in tapered duct and to measure the flow rates and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The Bernoulli’s theorem (Bernoulli’s theorem, 2011) relates the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar. In order to demonstrate the Bernoulli’s theorem, the F1-15 Bernoulli’s Apparatus Test Equipment is used in this experiment. The pressure difference taken is from h1-h5 where the values of h1-h5 are determined to be 50m, 100m and 150m respectively. The time to collect 0.003m3 water in the tank was measured. Lastly the flow rate, velocity, dynamic head, and total head were calculated using the data of the results and from the data given for both convergent and divergent flow. Based on the results taken, it has been analysed that the velocity of the fluid is increase when it flowing from a wider to narrower tube regardless the type of flow and pressure different. The velocity is increased as the pressure different is increased for both types of flow. Once the velocity of fluid is increased thus the dynamic head is also increased. Then, the total head value for convergent flow is the highest at h1 and the lowest at h5 while the opposite situation is happened in divergent flow.
Introduction
Figure 2.1: Apparatus used to investigate the validity of Bernoulli equation
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Figure 2.2: Apparatus used to investigate the validity of Bernoulli equation Figure 2.1 and figure 2.2 show the apparatus used to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapered duct. The apparatus consists of a clear acrylic duct of varying circular cross section, known as a Venturi. The duct has a series of wall tappings that allows measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. These tappings are connected to a manometer bank incorporating a manifold with air bleed valve. Pressurisation of the manometers is facilitated by a hand pump. The apparatus is mounted on an Armfield hydraulics bench. This base board has feet which may be adjusted to level the apparatus and flow control valves are used to control the flow rate through the Venturi. Figure 2.3 and Table 2.1 respectively show the positions (in mm) of the pressure tapings and the dimensions of the cross-sections. The duct has an upstream taper of 14˚ and a downstream taper of 21˚.
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Figure 2.3: Dimension of the Venturi and positions of the pressure tappings Tapping Position A B C D E F
Manometer Height h1 h2 h3 h4 h5 h5
Diameter of cross-section (mm) 25.0 13.9 11.8 10.7 10.0 25.0
Table 2.1: The dimension of cross section A level glass is provided as part of the base. The inlet pipe terminates in a female coupling which may e connected directly to the bench supply. A flexible hose attached to the outlet pipe which should be directed to the volumetric measuring tank on the hydraulics bench. A flow control valve is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve, and the bench supply control valve. The test section is arranged so that the characteristics of flow through both a converging and diverging section can be studied. Water is fed through a hose connector and is controlled by a flow regulator valve at the outlet of the test section. The Venturi can be demonstrated as a means of flow measurement and the discharge coefficient can be determined.
Objectives 1. To investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapered duct. 2. To measure flow rates and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. 3
Theory
Figure 4.1: Pipe of varying cross section (Giles, R.V,1994) Bernoulli's law indicates that, if an inviscid fluid is flowing along a pipe of varying cross section (refer to figure 3.1), then the pressure is lower at constrictions where the velocity is higher, and higher where the pipe opens out and the fluid stagnates. The wellknown Bernoulli equation is derived under the following assumptions: 1. fluid is incompressible ( density ρ is constant ); 2. flow is steady: 3. 4. along a streamline; Then, it is expressed with the following equation:
(equation 4.1) Where (in SI units): p = fluid static pressure at the cross section in N/m2. ρ = density of the flowing fluid in kg/m3 g = acceleration due to gravity in m/s2 (its value is 9.81 m/s2 = 9810 mm/s2) v = mean velocity of fluid flow at the cross section in m/s z = elevation head of the center of the cross section with respect to a datum z=0 h* = total (stagnation) head in m 4
The terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv ), and elevation head (z), respectively. The sum of these terms is known as the total head (h*). According to the Bernoulli’s theorem of fluid flow through a pipe, the total head h* at any cross section is constant (based on the assumptions given above). In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones. Other form of Bernoulli’s equation If the tube is horizontal the difference in height can be disregarded, z1 = z2. Hence,
(equation 4.2) Total pressure head Water at constant head from a tank is allowed to flow through a horizontal pipe line of varying cross section. The pressure heads h1,h2,etc are measured form a probe with an end hole facing into the flow such that it brings the flow to rest locally at the probe end. Thus, h˚ = h +
(equation 4.3)
h1˚ = h2˚ Volume flow rate
(Equation 4.4)
Velocity measurement The velocity of the flow is measured by measuring the volume of the flow, V, over a time period, t. Thus gives the rate of volume flow: Qv = V/t m3/s, which in turn gives the velocity of the flow through a defined area, A i.e. v = Qv / A (equation 4.5)
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Continuity equation For an incompressible fluid, conservation of mass requires that volume is also conserved. A1V1 = A2V2 (equation 4.6)
Apparatus and equipments In order to complete the demonstration of Bernoulli’s theorem, several apparatus are needed. They are as follows: 1. The F1-10 Hydraulic Bench which allows flow by timed volume collection to be measured.
2. The F1-15 Bernoulli’s Apparatus Test Equipment.
3. A stopwatch measurement.
for
timing
the
flow
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Procedures Part 1: Equipment set up 1. The Bernoulli’s equation apparatus is set up on the hydraulic bench so that the base is horizontal. 2. The test-section is ensured to have the 14-tappered section converging in the direction of the flow. 3. The rig outflow tube is positioned above the volumetric tank. 4. The rig inlet is connected to the bench flow supply, the bench valve and apparatus flow control valve are closed and pump is started. 5. Gradually, the bench valve is opened to fill the test rig with water. 6. In order to bleed air pressure tapping point and manometers both the bench valve and the rig flow control valve are closed. The air bleed screw is opened and the cap from the adjacent air valve is removed. 7. A length of small-bore tubing from the air valve is connected to the volumetric tank. 8. The bench valve is opened and allowed to flow through the manometer to purge all air from them. 9. Then, the air bleed screw is tightened; the bench valve and test rig flow control valve are partly opened. 10. Next, the air bleed is opened slightly to allow air to enter the top of the manometers. The screw is re-tightened when the manometer reach a convenient height. Part 2: Taking a set of results 1. The h1-h5 is set to be 50ml using air bleed screw. 2. After the specific volume of h1-h5 is reached the ball valve is closed and the time taken to accumulate 3L of fluid in tank is measured. 3. Steps 1 and 2 are repeated different level of h1-h5 which are 100ml and 150ml. 4. Then the test section is reversed to get the diverging flow. 5. The test section is removed by unscrewing the two couplings and being reversed. 6. The couplings are tightened. 7. Steps 1 until 3 are repeated for diverging section.
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Results Convergent flow 1) h1-h5 = 50mm Volume collected = 0.003 m3 Time to collect = 46.5 sec Flow rate = 6.452 × 10-5 m3/sec
Area of Duct (m2)
Static Head (m)
Velocity (m/s)
Dynamic Head (m)
Total Head (m)
h1
Distance into duct (m) 0.00
490.9 × 10-6
0.160
0.131
8.75 ×
0.1608
h2
0.0603
151.7 × 10-6
0.145
0.425
9.21 ×
0.1542
h3
0.0687
109.4 × 10-6
0.135
0.589
1.77 ×
0.1527
h4
0.0732
89.9 × 10-6
0.120
0.717
2.62 ×
0.1462
h5
0.0811
78.5 × 10-6
0.110
0.821
3.44 ×
0.1444
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2) h1-h5 = 100mm Volume collected = 0.003 m3 Time to collect = 30.55 sec Flow rate = 9.8 × 10-5 m3/sec Distance into duct (m)
Area of Duct (m2)
Static Head (m)
Velocity (m/s)
h1
0.00
490.9 × 10-6
0.190
0.2
2.04 ×
0.19204
h2
0.0603
151.7 × 10-6
0.165
0.6
1.83 ×
0.1833
h3
0.0687
109.4 × 10-6
0.140
0.89
4.04 ×
0.1804
h4
0.0732
89.9 × 10-6
0.115
1.09
6.06 ×
0.1756
h5
0.0811
78.5 × 10-6
0.080
1.25
7.96 ×
0.1596
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Dynamic Head (m)
Total Head (m)
3) h1-h5 = 150mm Volume collected = 0.003 m3 Time to collect = 25.65 sec Flow rate = 1.1696 × 10-4 m3/sec
Area of Duct (m2)
Static Head (m)
Velocity (m/s)
Dynamic Head (m)
Total Head (m)
h1
Distance into duct (m) 0.00
490.9 × 10-6
0.210
0.2383
2.89 ×
0.2129
h2
0.0603
151.7 × 10-6
0.175
0.771
3.03 ×
0.2053
h3
0.0687
109.4 × 10-6
0.140
1.069
5.82 ×
0.1982
h4
0.0732
89.9 × 10-6
0.105
1.301
8.63 ×
0.1913
h5
0.0811
78.5 × 10-6
0.060
1.490
0.1132
0.1732
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Divergent flow 1) h1-h5 = 50m Volume collected = 0.003 m3 Time to collect = 32.15 sec Flow rate = 9.33 × 10-5 m3/sec
Area of Duct (m2)
Static Head (m)
Velocity (m/s)
Dynamic Head (m)
Total Head (m)
h1
Distance into duct (m) 0.00
490.9 × 10-6
0.145
0.190
1.84 ×
0.1468
h2
0.0603
151.7 × 10-6
0.120
0.615
1.93 ×
0.1393
h3
0.0687
109.4 × 10-6
0.100
0.853
3.71 ×
0.1371
h4
0.0732
89.9 × 10-6
0.095
1.038
5.49 ×
0.1499
h5
0.0811
78.5 × 10-6
0.095
1.189
7.21 ×
0.1671
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2) h1-h5 = 100m Volume collected = 0.003 m3 Time to collect = 23.35 sec Flow rate = 1.285 × 10-4 m3/sec Area of Duct (m2)
Static Head (m)
Velocity (m/s)
Dynamic Head (m)
Total Head (m)
h1
Distance into duct (m) 0.00
490.9 × 10-6
0.165
0.262
3.50 ×
0.1685
h2
0.0603
151.7 × 10-6
0.120
0.847
3.66 ×
0.1566
h3
0.0687
109.4 × 10-6
0.070
1.175
7.04 ×
0.1404
h4
0.0732
89.9 × 10-6
0.065
1.429
0.1041
0.1691
h5
0.0811
78.5 × 10-6
0.065
1.637
0.1366
0.2016
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3) h1-h5 = 150m Volume collected = 0.003 m3 Time to collect = 17.95 sec Flow rate = 1.671 × 10-4 m3/sec
Area of Duct (m2)
Static Head (m)
Velocity (m/s)
Dynamic Head (m)
Total Head (m)
h1
Distance into duct (m) 0.00
490.9 × 10-6
0.180
0.340
5.89 ×
0.1859
h2
0.0603
151.7 × 10-6
0.125
1.102
6.19 ×
0.1869
h3
0.0687
109.4 × 10-6
0.050
1.527
0.1188
0.1688
h4
0.0732
89.9 × 10-6
0.035
1.859
0.1761
0.2111
h5
0.0811
78.5 × 10-6
0.035
2.129
0.2310
0.2660
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Sample Calculations Convergent flow Pressure different (h1-h5)
=
Flow rates
=
50 m
=
=
Velocity
6.452 × 10-5 m3/sec
=
=
=
Dynamic head
0.131 m/s
=
=
=
Total head
m
=
Static Head + Dynamic Head
=
0.160 m +
=
0.16087 m
14
m
Discussions This experiment is carried out to investigate the validity of Bernoulli’s equation when applied to the steady flow of water in a tapered duct. Apart from that, this experiment is also to measure the flow rates and both static and total pressure heads in a rigid convergent and divergent tube of known geometry for a range of steady flow rates. Based on the calculation made after the experiment, the velocity of the flowing fluid is increased as the fluid flows from a wider to narrower pipe regardless the pressure difference and type of flow of each result taken. These can be seen in all results tables.
From the analysis of the results, can be concluded that for both type of flow, be it convergent or divergent, the velocity increases as the pressure difference increases. For instance, the velocities at pressure head h1 at pressure difference of 50 millimetres, 100 millimetres and 150 millimetres for convergent flow are 0.131m/s, 0.20 m/s and 0.2383 m/s respectively, which are increasing. The same goes to divergent flow, whereby the velocities are increasing when the pressure difference between h1 and h5 is increased at which the velocities at pressure head h1 at pressure difference of 50 millimetres, 100 millimetres and 150 millimetres are 0.190m/s, 0.262 m/s and 0.340 m/s respectively. Note that for divergent flow, the water flows from pressure head h5 to h1, which is from narrow tube to wider tube. As the velocity increase the dynamic head is calculated to be increased too.
Next, the total head value for convergent flow is calculated to be decreased from h1 to h5 for all pressure difference in the other word, the total head value for h1 is the highest and while total head value for h5 is the lowest. These can be seen at all results for convergent flow. This situation is the opposite for divergent flow. The total head value for h1 is the lowest while the total head value for h5 is the lowest. All the above situation are best describe the Bernoulli’s equation where; p = fluid static pressure at the cross section in N/m2. ρ = density of the flowing fluid in kg/m3 g = acceleration due to gravity in m/s2 (its value is 9.81 m/s2 = 9810 mm/s2) v = mean velocity of fluid flow at the cross section in m/s z = elevation head of the center of the cross section with respect to a datum h* = total (stagnation) head in m 15
However, after the experiment, there are some errors on the results shown in divergent flow. This situation might due to some error or weaknesses when taking the measurement of each data. One of them is the observer must have not read the level of static head properly, where the eyes are not perpendicular to the water level on the manometer. Therefore, there are some minor effects on the calculations due to the errors. Conclusions In conclusion, the velocity of fluid will increase if the fluid is flowing from a wider to narrower tube and the velocity will decrease in the opposite case regardless of the type of flow and the pressure difference. As the velocity for all cases increases the dynamic head values are also seem to be increased. The total head for convergent flow is decrease from h1 to h5 while the total head value for divergent flow is the lowest at h1 and the highest at h5. But there are some errors occurred in the results for divergent flow. There might be due to some errors occurred during the experiment. However, the results can be improved if some precautions are taken during the experiment for example the eyes level must be placed parallel to the scale when manometer readings are taken. Besides that, the valve is also need to be controlled slowly to stabilize the water level in the manometer. Bernoulli’s theorem has several applications in everyday lives. In certain problems in fluid flows when given the velocities at two points of the streamline and pressure at one point, the unknown is the pressure of the fluid at the other point. In such cases (if they satisfy the required condition for Bernoulli's Equation) Bernoulli's Equation can be used to find the unknown pressure. One such example is the flow through a converging nozzle. Flow enters the nozzle at low speed, accelerates and leaves the nozzle at atmospheric pressure. We have to find the pressure at inlet. The inlet pressure can be simply find by applying Bernoulli's Equation between inlet and outlet points and calculate the unknown pressure assuming that the change in elevation in zero.
In this example there is no change in elevation. The converging nozzle causes fluid to accelerate. From the energy balance feature of the equation we can say the increase in velocity results in the drop in the pressure at the outlet of the nozzle. In conclusion, through this experiment all the objectives are successfully achieved. From all the data and results calculated it is proved the validity of Bernoulli’s equation. All the flow rates and both static and total pressure heads in a rigid convergent/divergent tube are managed to be calculated.
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Recommendation 1. Repeat the experiment several times to get an accurate result. 2. The eye of observers should be placed parallel to the scale of manometer to get an accurate reading. 3. Make sure the bubbles in the manometer are completely removed by adjusting the bleed screw. 4. The valve should be control slowly so that the pressure difference can be maintained. References 1. Fluid Mechanics for Chemical Engineers, Prentice Hall. 2. Douglas, J.F.,Gasiorek, J.M. and Swaffield, J.A., Fluid Mechanics, 3rd ed. Longmans Singapore Publisher, 1995. 3. Giles, R.V., Evett, J.B. and Cheng Liu, Schaumm’s Outline Series Theory and Problems of Fluid Mechanics and Hydraulic, McGraw Hill Intl., 1994. 4. H. Lamb, Hydrodynamics, 6th ed. (Cambridge: Cambridge Univ. Press, 1953), pp. 20-25 5. Bernoulli’s theorem. (2011). In Encyclopædia Britannica. Retrieved from http://www.britannica.com/EBchecked/topic/62615/Bernoullis-theorem 6. "Bernoulli's Theorem" from The Wolfram Demonstrations Project retrieved from http://demonstrations.wolfram.com/BernoullisTheorem/
Appendices
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