Bernoulli\'s theorem experimental guide PDF

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lab report on bernoulli's theorem...

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Abstract The objective of this experiment is to investigate the validity of the Bernoulli’s equation when applied to the steady flow of water in a Venturi tube. According to Bernoulli’s equation, it states that the total energy of a fluid at steady state in the Venturi tube is conserved. A hydraulic bench fitted with a Venturi tube is used. The experiment was carried out in three different flow rates. Calibration graphs of total head calculated and total head readings of each flow rate against positions in the Venturi tube were plotted in order to make a comparison. Both total heads were not constant throughout the experiment as expected. These results showed that the Bernoulli equation was not valid in practice as the total energy of the steady-flowing fluid in a Venturi tube was not conserved. Energy loss cannot be prevented in any flow rates. However, the Bernoulli’s equation is most ideal when the flowrate is low as less energy is lost due to friction or separation flow of flowing fluid. Introduction and Theory Bernoulli’s theorem relates the pressure, velocity and elevation in a moving fluid. The Bernoulli’s principle states that an increase in the velocity of an inviscid and incompressible fluid occurs simultaneously with a decrease in pressure (E. Alexander, 2017). Inversely, the pressure will increase with a decrease in the velocity of fluid. However, it is only valid under the assumption that the steady-flowing fluid is adiabatic, incompressible, frictionless and non-viscous (Khan, 2018). The principle can be explained by Bernoulli’s equation which was derived from the law of conservation of total energy of a fluid. It is written as shown in Equation 1 below. 𝑢2 𝑝 + + ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 𝜌

𝑬𝒒(𝟏)

u is the velocity of fluid flow, p is the static pressure of fluid, 𝜌 is the density of fluid, h is the potential head. The total fluid head is the sum of velocity head, static pressure head and potential head. These values are obtained by dividing equation 1 by g which is the gravitational acceleration. 𝑝 𝑢2 + + ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2𝑔 𝜌𝑔 𝑢2 2𝑔

is the velocity head and

𝑝 𝜌𝑔

𝑬𝒒(𝟐)

is the static pressure head. From equation 2, it shows that the total

fluid head will remain constant under the conditions as mentioned earlier. The h value can be assumed to be zero if the Venturi tube is on a levelled surface which means there is no height difference. Each term represents energy per unit weight of fluid. (Coulson and Richardson’s, 1999) The diffuser efficiency can be calculated as shown in equation 3 below. It shows the ratio of difference of fluid pressure between the divergence section and convergence section in the Venture tube (University of Bath, 2018). 𝜂=

𝛥 𝑝 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡 𝑝1 − 𝑝5 = 𝛥 𝑝 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡 𝑝5 − 𝑝6

𝑬𝒒(𝟑)

The objective of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a Venturi tube by using a hydraulic bench. The hypothesis of this experiment is that the total energy of a fluid at a steady state in a Venturi tube is conserved. The hydraulic bench is designed to measure the amount of discharge (MXR, 2018). The Venturi tube is an accurately machined clear duct of varying circular cross sections that water flows through. It is made of a convergent and divergent section with different lengths and angles. The entry cone and an exit cone have an angle of 21° and 14°, respectively as shown in Figure 1. It is used to measure flow rate (Kang, Shih-Chung, 2013). This tube is also connected to manometers with pipes. The static pressure distribution is measured by the provided wall pressure tapings 1

along both sections of the Venturi tube. The static pressure head is then shown on manometer 1 to 6.

Figure 1: Venturi Tube connected to manometers with pipes highlighting the convergent and divergent sections 2. Method

Figure 2: Venturi tube with associated dimension The pump was switched on before setting up the rig by fully opening the bench valve and outflow valve. The rig was flushed until the system was filled up with water. The top valve was then closed. The hose provided was inserted into the air bleed screw valve after it was opened. The top valve was opened while the outflow valve was then closed to eliminate air bubbles by flushing through the manometers. The top valve was closed again. After removing the hose, the air bleed screw value was closed and the pump was switched off. The top valve and air bleed screw were opened slowly to decrease water levels in manometers down to 30mm before the pump was switched on again. Three different flow rates which included maximum, minimum and medium were carried out in the experiment. The maximum flow rate had the largest gap between the highest and lowest water level in the manometer whereas the minimum flow rate smallest gap between the highest and lowest water level in the manometer. The flow rate was manipulated by using the outflow and bench valves until the water levels in manometers are different. When the water levels were stabilized, six readings (h1 to h6) of static head were recorded. The volumetric flow rate was determined by recording the time taken to fill the tank from zero to six liters which was shown on the sight glass of level gauge. The digital stopwatch was started when the ball valve was closed in the tank. The stopwatch was stopped exactly when water level reached 6 liters. This measurement was repeated twice to obtain an average value. The total pressure head distribution was measured by traversing the total pressure probe along the length of test section which had marked intervals. Two readings at the pressure tapping on the head and end of pipe were recorded. One reading was recorded in the converging section while four readings were recorded in the diverging section. In the narrowest section, one reading was recorded as well.

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These steps were repeated for all three flow rates. The test section was now reversed by the demonstrator. The steps to set up the rig and to obtain static head, total head and volumetric flow rate were repeated with three different flow rates again. After the results were recorded, the pump was switched off and system was then drained. 3. Results and Calculations The time taken for the collection of 6 liters fluid was recorded in Table 1. The calculation of volumetric flow rate is shown in equation 4 below. 𝑉𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝐹𝑙𝑜𝑤 𝑅𝑎𝑡𝑒 =

6 𝑇𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑓𝑜𝑟 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 6 𝑙𝑖𝑡𝑒𝑟𝑠 𝑓𝑙𝑢𝑖𝑑

Eq (4)

Table 1: Time taken to collect 6 liters of fluid with digital stopwatch Test Section Forward

Time taken for collection of 6 liters fluid (s) Type of Flow Rate 1 2 Average Maximum 42.96 43.66 43.31 Minimum 53.04 51.41 52.23 Medium 53.04 50.81 51.93 Maximum 41.88 40.50 41.19 Minimum 102.65 103.10 102.88 Medium 48.72 47.22 47.97

Run No. 1 2 3 4 5 6

Reversed

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =

𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝐶𝑟𝑜𝑠𝑠 − 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎

𝑬𝒒(𝟓)

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 2 2𝑔

𝑬𝒒 (𝟔)

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝐻𝑒𝑎𝑑 =

𝑇𝑜𝑡𝑎𝑙 𝐻𝑒𝑎𝑑 = 𝑆𝑡𝑎𝑡𝑖𝑐 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝐻𝑒𝑎𝑑

Volumetric Flow Rate (m3 s-1) 0.000139 0.000115 0.000116 0.000146 0.000058 0.000125

Volumetric Flow Rate (liter s-1) 0.139 0.115 0.116 0.146 0.058 0.125

𝑬𝒒 (𝟕)

The velocity and velocity head were calculated using Equation 5 and Equation 6. The total head was then calculated using Equation 7. The calculated values were recorded in Table 2 to Table 6 which are shown in Appendix A. The total head reading was recorded in the same tables as well.

Total Head Calculated (m)

Figure 3 and Figure 4 were plotted according to Table 2, Table 3 and Table 4. 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

Position (-) Maximum Flow Rate

Minimum Flow Rate

Medium Flow Rate

Figure 3: Graph of total head calculated against position in forward test section (run1-3) 3

Total Head Reading (m)

0.300 0.250 0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

Position (-) Maximum Flow Rate

Minimum Flow Rate

Medium Flow Rate

Figure 4: Graph of total head reading against position in the forward test section (run 1-3)

Total Head Calculated (m)

Figure 5 and Figure 6 were then plotted according to Table 5, Table 6 and Table 7. 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

Position (-) Maximum Flow Rate

Minimum Flow Rate

Medium Flow Rate

Total Head Reading (m)

Figure 5: Graph of total head calculated against position in reversed test section (run4-6) 0.300 0.250 0.200 0.150 0.100 0.050 0.000 0

1

2

3

4

5

6

7

Position (-) Maximum Flow Rate

Minimum Flow Rate

Medium Flow Rate

Figure 6: Graph of total head reading against position in the reversed test section (run4-6) Table 2 shows the diffuser efficiency which was calculated using Equation 8. η is the diffuser efficiency. Δpdivergent is the change in pressure in the diverging section. Δpconvergent is the change in pressure in the converging section.

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𝜂=

𝛥𝑝𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡 𝛥𝑝𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡

𝑬𝒒(𝟖)

Table 2: Diffuser Efficiency for run 1 to run 6 Run

Δ pconvergent 1 2 3 4 5 6

Δ pdivergent 0.2 0.12 0.16 0.215 0.08 0.145

0.1 0.08 0.075 0.035 0.03 0.055

Diffuser Efficiency, η 0.5 0.666667 0.46875 0.162791 0.375 0.37931

4. Discussion From Figure 2, the total head readings remained constant only until the total pressure probe reached the position 5 while the total head readings in Figure 4 decreased as the total pressure probe moved along the length of test section. According to principle of Bernoulli’s equation as shown in Equation 2, the total head should always stay constant. Due to the assumption that the steady-flowing fluid is adiabatic, incompressible, frictionless and non-viscous in Bernoulli’s Theorem, the energy loss through friction of flowing water against the walls of the Venturi tube was not taken into account. Thus, the results obtained were affected. The initial constant total head readings in the converging section from Figure 4 proved that the Bernoulli’s Theorem is valid only if the fluid have straight parallel stream lines and negligible turbulence. However, the total head decreased in the diverging section. Based on the continuity equation of continuity shown in Equation 8, it states that for an incompressible fluid flowing in a tube of varying cross-section, the mass flow rate must be the same throughout the tube (Ariena, 2014) . Thus, the velocity of fluid increases when it is at the narrowest section. In the diverging section, the velocity decreased and caused turbulence. 𝜌1 𝐴1 𝑉1 = ρ2 𝐴2 𝑉2

𝑬𝒒(𝟖)

‘ρ’ is the density, ‘A’ is the cross-sectional area and ‘V’ is the velocity. Figure 1 and Figure 3 showed that the total head calculated at each maximum flow rate decreased the most from position 1 to position 6. This is because the Bernoulli’s Theorem is least valid when the flow rate is maximum. The fluid with high velocity does not have straight parallel stream lines. The fluid with high velocity will cause turbulence to occur. Turbulence causes more resistance in flow. Thus, greater energy losses by fluid in high velocity. 𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 =

𝑇𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 1 – 𝑡𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 6 𝑇𝑜𝑡𝑎𝑙 ℎ𝑒𝑎𝑑 𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 𝑖𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 1

𝑥100%

Eq (9)

The decrease in percentage of total head calculated from the head of the pipe to the end of the pipe was calculated by using Equation 9. From Table 3 to Table 5 in Appendix A, the total head calculated decreased by an average of 31.6% in run 1 to run 3 where the diverging section is angled at 14o. From Table 6 to Table 8 in Appendix A the total head calculated decreased by an average of 52.2% in run 4 to run 6 where the diverging section is angled at 21o. According to Table 2, the diffuser efficiency in run 1 to run 3 where the divergent section has an angle of 14o also has a higher value than run 4 to run 6 where the divergent section has an angle of 21o. This showed that a steeper angle of the diverging section will have a greater decrease of total head calculated from the head of the pipe to the end of the pipe which means more energy is lost when the diverging section is 21o due to separation flow of flowing fluid (Stokes, 2018). The flow rate of fluid also affects the total energy loss in Venturi tube. When the flow rate increases, the increased velocity of fluid will cause more turbulence effect. Therefore, more energy is lost due to the turbulence effect. 5

There are many possible sources of error that can lead to inaccurate data collection. Parallax error may occur when reading manometers. Some of the readings were rounded up or down to the nearest 5 since the smallest interval was 5mm. The eye level must be perpendicular to the reading when recording the data to reduce parallax error, by reading at the right meniscus. It was difficult to decide the water level in the manometers as the water level fluctuates. Thus, the static pressure head obtained might not be accurate. To improve this, multiple readings should have been recorded for each manometer to obtain average value which is closer to the true measurement. Due to unfamiliar lab equipment such as hydraulic bench and the Venturi tube, it was time consuming to handle it. Every task should be distributed evenly among group members to ensure the experiment completed on time.

5. Conclusion By applying Bernoulli equation to the steady flow of water in a Venturi tube, the results showed that the equation is not valid in this experiment. The hypothesis made was inaccurate as the total energy of a fluid at a steady state in a Venturi tube was not conserved. It is impossible to satisfy all the conditions in Bernoulli’s equation as there are significant energy losses due to friction which must be taken into account. Thus, there are energy losses at all three flow rates. However, the fluid with a low flow rate which is in the converging section where the cross-sectional area is large and velocity is low shows the most ideal as it has the least variation as shown in Figure 3 to Figure 7. The fluid only has straight parallel stream lines and less energy loss due to friction when the velocity is low. The energy loss was most significant when the flow rate is at maximum. The results of this experiment can be improved if precautions are taken during the experiment to minimize both the experimental error and propagation error. The Bernoulli’s theorem is invalid when the angle of diverging section is steep and when the flow rate of fluid is high as it will cause more energy to be lost.

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References 1. E. Alexander, D. (2017). Nature's Machines. University of Kansas, Lawrence, KS, United States: Academic Press, pp.51-97. 2. Coulson, J. M. and Richardson, J. F., 1999. Momentum changes in a fluid. In: Coulson’s and Richardson’s Chemical Engineering. Volume 1. Place of publication: Oxford, Elsevier Butterworth-Heinemann, chapter 2.4.2. 3. University of Bath, 2018 CE10185 Chemical Engineering Skills and Practice 1, Student Lab Book 2018-19, pp91-98. 4. MXR., 2018. Bernoulli's Theorem Demonstration Lab Report Uitm | Fluid Dynamics | Pressure Measurement. [online] Available at: https://www.scribd.com/document/188076875/Bernoulli-s-Theorem-Demonstration-LabReport-Uitm [Accessed 29 Nov. 2018] 5. Kang, Shih-Chung, 2013. Development of Virtual Equipment: Case Study of the Venturi Tube Experiment. Journal of Professional Issues in Engineering Education & Practice, 139(4), pp. 281-289.

6. Ariena, A., 2014. Bernoulli’s Principle Demonstration (Lab Report) [Online]. Available from: https://www.scribd.com/doc/246653261/Bernoulli-s-Principle-Demonstration-Lab-Report [Accessed on 18 October 2018]

7. Stokes, N. (2018). [online] Nptel.ac.in. Available at: https://nptel.ac.in/courses/112104118/lecture-15/15-1_mesure_flow.htm [Accessed 3 Dec. 2018].

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Appendix Appendix A: Data measured from Venturi tube for all runs Table 3: Data measured from Venturi tube for run no.1

Position 1.0 1.1 2.0 3.0 4.0 5.0 5.1 6.0

Description Head of Pipe Converging Section Narrowest Section Diverging Section Diverging Section Diverging Section Diverging Section End of Pipe

Crosssectional Diameter Area (m2) (mm)

Velocity Velocity Head (ms-1) (m)

Static Head Readings (m)

Total Head Calculated (m)

Total Head Reading (m)

25.0 0.000491

0.282

0.004

0.250

0.254

0.250

25.0 0.000491

0.282

0.004

0.250

0.254

0.245

10.0 0.000079

1.764

0.159

0.050

0.209

0.240

10.7 0.000090

1.541

0.121

0.060

0.181

0.225

11.8 0.000109

1.267

0.082

0.075

0.157

0.220

13.9 0.000152

0.913

0.042

0.110

0.152

0.195

25.0 0.000491

0.282

0.004

0.150

0.154

0.195

25.0 0.000491

0.282

0.004

0.150

0.154

0.190

Table 4: Data measured from Venturi tube for run no.2

Position 1.0 1.1 2.0 3.0 4.0 5.0 5.1 6.0

Description Head of Pipe Converging Section Narrowest Section Diverging Section Diverging Section Diverging Section Diverging Section End of Pipe

CrossDiameter sectional Area (m2) (mm)

Velocity (ms-1)

Velocity Head (m)

Static Head Reading (m)

Total Head Calculated (m)

Total Head Reading (m)

25.0

0.000491

0.234

0.002793

0.180

0.183

0.250

25.0

0.000491

0.234

0.002793

0.180

0.183

0.250

10.0

0.000079

1.463

0.109098

0.060

0.169

0.245

10.7

0.000090

1.278

0.08323

0.070

0.153

0.235

11.8

0.000109

1.051

0.056271

0.075

0.131

0.230

13.9

0.000152

0.757

0.029225

0.110

0.139

0.200

25.0

0.000491

0.234

0.002793

0.140

0.143

0.205

25.0

0.000491

0.234

0.002793

0.140

0.143

0.200

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Table 5: Data measured from Venturi tube for run no.3

Position 1.0 1.1 2.0 3.0 4.0 5.0 5.1 6.0

Description Head of Pipe Converging Section Narrowest Section Diverging Section Diverging Section Diverging Section Diverging Section End of Pipe

Crosssectional Diameter Area (m2) (mm)

Velocit...