Fluid Mechanics Bernoulli Experiment Lab Report PDF

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It is a Bernoulli experiment lab report done at the University Malaya. The experiment is to investigate the validity of Bernoulli’s Theorem as applied to the water flowing in a tapered circular duct by using Bernoulli’s Theorem Demonstration Apparatus....

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1. ABSTRACT The experiment is to investigate the validity of Bernoulli’s Theorem as applied to the water flowing in a tapered circular duct by using the Bernoulli’s Theorem Demonstration Apparatus. In this experiment, water will be allowed to enter and pass through along the venturi tube connected to 8 manometer tubes. The time taken for 5 litres of water flows into the system is obtained and recorded for the calculation of the flow rate of the water. After that, the first manometer tube level is adjusted until it reaches 280mm, and the pressure heads of tube 2 to tube 6 are then recorded by taking the readings of the manometer level. Then, the hypodermic probe is pulled perpendicularly to each manometer tube to obtain probe manometer level which is the experimental total head. The experiment is repeated for 4 times by reducing the first manometer level to 260mm, 240mm, and 220mm to obtain four different sets of readings. Then, the experimental total head is determined and compared with the theoretical total head. The average percentage error obtained from the four sets of data ranges from 4.375% to 9.01%. Then, these results show that the experimental total head has slightly deviated from the theoretical total head. This is because Bernoulli’s Theorem assumed the water to be non-viscous, incompressible, and steady flowing. However, in reality, the water flow may differ from the assumptions, and energy may loss in the system due to friction and heat. Moreover, some experimental errors will also cause the deviation of the experimental results.

2. TITLE U4 Bernoulli’s Theorem

3. INTRODUCTION According to Wikipedia, Bernoulli’s Theorem states that the static pressure of a fluid decreases when the velocity of flowing fluid increases. Bernoulli's Theorem applies the law of conservation of energy, which means, in other words, the sum of mechanical energy, potential energy and kinetic energy remains constant along a streamline. The study of Bernoulli's principle is important in daily life applications such as airplane and insecticide spray. The wings of the airplane are streamlined so that the airplane can be lifted easily. On top of that, an insecticide spray uses Bernoulli’s principle to spread out the insecticide easily over a wider area. In this experiment, Bernoulli’s Theorem demonstration apparatus and Hydraulics Bench are used for the investigation of the validity of Bernoulli’s theorem. The water flow in the experiment is based on three assumptions: the water is non-viscid fluid, incompressible fluid, and steady flow. The purpose of this experiment is to investigate the validity of Bernoulli’s Theorem by using Bernoulli’s Theorem Demonstration Apparatus (F1-15). As mentioned earlier, few assumptions have to been made in Bernoulli’s Theorem. However, in reality, there is no such ideal fluid. In fact, fluid in real life will have its viscosity and compressibility. The flow of fluid will not always steady because there will be fluctuation in the velocity of the fluid in real life when it encounters an obstacle or anything. However, Bernoulli’s equation is still applicable to some real situations when the conditions are approximately fulfilled. In this experiment, the flow rate of water, the velocity of water flow, and the total head are measured using the Bernoulli’s Theorem Demonstration apparatus. In general, the total head is a measure of the sum of velocity head, pressure head, and elevation head. The experimental total head is obtained from the probe manometer level whereas the theoretical total head is obtained by using Bernoulli’s equation and calculation. The experimental results are compared with the theoretical calculation to determine the validity of Bernoulli’s Theorem for convergent flow and divergent flow in this experiment. 1

4. OBJECTIVE To investigate the validity of Bernoulli’s Theorem as applied to the flow of water in a tapered circular duct.

5. THEORETICAL BACKGROUND In this experiment, there are a few assumptions made in deriving Bernoulli’s equations: 1. 2. 3. 4.

The fluid is non-viscous. The fluid flow is frictionless. The fluid flow is steady along a streamline. The fluid density remains unchanged is incompressible.

In other words, these assumptions are made according to the law of conservation of energy and mass. The conservation of energy’s equation is as below:

∑ E ¿=∑ E out E¿ = input energy Eout =output energy In this experiment, there are three forces which generates the energy in the system based on the assumptions made before. These forces are pressure force, kinetic force by fluid and gravitational force. The work done by pressure force is:

∆ W =∆ PV Then, the work done by kinetic force of fluid is:

1 ∆ W = m ∆ v2 2 The work done by gravitational energy is:

∆ W =mg ∆ z Where

W =¿ work done P = the surrounding pressure V = volume of fluid m = mass of fluid v = velocity of fluid flow g = gravitational acceleration z = the vertical elevation head Based on the law of conservation of energy, the total energy in the system remains the same throughout the experiment. Hence, the Bernoulli’s equation can be derived as below:

1 2 ∆ W = m ∆ v + mg ∆ z+P ∆ V 2 1 1 2 2 m v 1 +mg z 1+P1 V = m v 2 +mg z 2 +P2 V 2 2

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Dividing all the terms by the volume, V and substituting density of fluid , ρ=

m , V

1 1 2 2 ρ v + ρg z 1 +P1= ρ v 2 + ρg z2 +P2 2 1 2 Then, all the terms are divided by density and gravitational acceleration to obtain the Bernoulli’s equation in head form for the experiment, 2 P2 P 1 v2 2 v1 + z 1+ = +z 2+ ρg ρg 2 g 2g

where

P ρg v2 2g

= the static pressure head = the velocity head In this experiment, the duct remains horizontal and no elevation happen, thus

z 1= z 2 . Therefore, the Bernoulli’s equation in this experiment can be written as: 2 2 v 1 P1 v 2 P2 + = + 2 g ρg 2 g ρg Then, the total head can be measured experimentally by placing the hypodermic probe perpendicularly to each of the manometer tubes. On the other hand, the total head can be calculated theoretically by the equation as below:

Total Head , H =

2 P v + 2 g ρg

The rate of fluid flow is measured and calculated by collecting the volume of entered fluid, V over the time taken, t: Rate of fluid flow, Q=

V t

By using the equation of continuity, the velocity of fluid flow, v can be derived by dividing the rate of fluid flow over by the cross-sectional area of the manometer tube, A:

v=

Q A

6. APPARATUS i) ii) iii)

Hydraulics Bench Bernoulli’s Theorem demonstration apparatus Stopwatch

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7. PROCEDURES

Figure 1: The Experimental Setup (Source: https://uta.pressbooks.pub/appliedfluidmechanics/chapter/experiment-2/)

1. The experiment is set up as Figure 1. 2. The apparatus is levelled on the Hydraulic Bench by adjusting the feet. 3. A small amount of wetting agent is injected into the test section, and the test section is ensured to have 14 degrees of angle converging in the direction of flow. 4. The total head probe is withdrawn before the release of the couplings to reverse the test section. 5. The ball valve is opened to let the water out from the system so that there is no water in the system. The exit valve is opened. 6. The pump of the Hydraulic Bench is turned on. 7. All the manometer tubes are ensured to be levelled. Once the manometer tubes are levelled, the left valve is opened to let water enter the system. 8. The air-bleed valve is opened to discharge the air from the system and all connecting pipes 9. The pressure in the system is controlled by using the exit valve and the inlet feed to obtain the largest convenient difference between the highest and lowest manometer levels. 10. The water outlet in the system is turned off by closing the ball valve. 11. The stopwatch is started when the flow meter is at 1 litre and stopped when it is at 6 litres. The reading of the stopwatch is taken for the calculation of the flow rate of water. 12. The readings of all the manometer tubes are taken and recorded.

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13. The total head is measured by pulling the hypodermic probe perpendicularly to each tube from test section h1 to h6 . The readings of the total heads of all manometer tubes are taken and recorded. 14. The steps are repeated four times by changing the flow rates of water to obtain different readings. 15. Once the experiment is done, the ball valve is opened to allow the water to exit the system and the water inlet is turned off. The exit valve is then closed and the pump of the Hydraulic Bench is turned off.

8. RESULTS AND DISCUSSION Set 1 First manometer level = 280mm Volume of water= 5 L Time = 50.69 s Table 1: Set 1 (First manometer level 280 mm)

Test sectio n

No .

h1 h2 h3 h4 h5 h6

1 2 3 4 5 6

Diameter, D(

×10

−2

m) 2.50 1.39 1.19 1.07 1.00 2.50

Flow area, A (x −4

10 m ) 4.909 1.520 1.094 0.899 0.785 4.909

Flow rate, Q(

Fluid velocity,

v

m 3 /s

(m/s)

0.2009 0.6489 9.86 0.9016 4 1.0972 ×10 1.2566 0.2009

Pressure Head,

Velocity Head,

P ρg

v 2g

(m)

(m) 0.002 0.021 0.041 0.061 0.080 0.002

0.280 0.258 0.240 0.218 0.183 0.198

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Calculated Total Head, H T (m)

Measured Total Head,

HE (m)

0.282 0.279 0.281 0.279 0.263 0.200

0.275 0.270 0.265 0.260 0.255 0.210

Table 2: Percentage error of Set 1

No. 1 2 3 4 5 6

Test section

h1 h2 h3 h4 h5 h6

Experimental Total Head (m) 0.275 0.270 0.265 0.260 0.255 0.210

Average percentage error of Set 1 =

Theoretical Total Head (m) 0.282 0.279 0.281 0.279 0.263 0.200

Percentage error (%) 2.48 3.23 5.69 6.81 3.04 5.00

2.48 + 3.23 + 5.69 + 6.81 + 3.04 + 5.00 % 6 = 4.375 %

Set 2

First manometer level = 260mm Volume of water= 5 L Time = 50.96 5

Table 3: Set 2 (First manometer level 260 mm)

Test sectio n

No .

h1 h2 h3 h4 h5 h6

1 2 3 4 5 6

Diameter, D(

×10

−2

m) 2.50 1.39 1.19 1.07 1.00 2.50

Flow area, A (x −4

10 m ) 4.909 1.520 1.094 0.899 0.785 4.909

Flow rate, Q(

Fluid velocity,

v

m 3 /s

(m/s)

0.1999 0.6455 9.81 0.8969 2 1.0914 ×10 1.2499 0.1999

Pressure Head,

Velocity Head,

P ρg

v2 2g

(m) 0.260 0.240 0.225 0.205 0.190 0.183

(m) 0.002 0.022 0.041 0.061 0.081 0.002

Calculated Total Head, H T (m)

Measured Total Head,

HE (m) 0.260 0.263 0.265 0.263 0.260 0.225

0.262 0.261 0.266 0.266 0.270 0.185

Table 4: Percentage error of Set 2

No.

Test section

h1 h2 h3 h4 h5 h6

1 2 3 4 5 6

Experimental Total Head (m) 0.260 0.263 0.265 0.263 0.260 0.225

Theoretical Total Head (m) 0.262 0.261 0.266 0.266 0.270 0.185

Percentage error (%) 0.76 0.77 0.38 1.13 3.70 21.62

0.76 + 0.77 + 0.38 + 1.13+ 3.70 + 21.62 % 6 = 4.727 %

Average percentage error of Set 2 =

Set 3

First manometer level = 240mm Volume of water= 5 L Time = 47.72 s Table 5: Set 3 (First manometer level 240 mm)

No . 1 2 3 4 5 6

Test sectio n

h1 h2 h3 h4 h5 h6

Diameter, D(

×10−2 m) 2.50 1.39 1.19 1.07 1.00 2.50

Flow area, A (x −4

10 m ) 4.909 1.520 1.094 0.899 0.785 4.909

Flow rate, Q( 3

m /s

Fluid velocity,

v (m/s)

0.2135 0.6895 1.04 8 0.9580 ×10 1.1657 1.3350 0.2135

Pressure Head,

Velocity Head,

P ρg

v2 2g

Calculated Total Head, H T (m)

(m) 0.240 0.210 0.190 0.160 0.115 0.135

(m) 0.002 0.024 0.047 0.069 0.091 0.002

0.242 0.234 0.237 0.229 0.206 0.137

Measured Total Head,

HE (m) 0.235 0.235 0.235 0.233 0.233 0.185

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Table 6: Percentage error of Set 3

No.

Test section

h1 h2 h3 h4 h5 h6

1 2 3 4 5 6

Experimental Total Head (m) 0.235 0.235 0.235 0.233 0.233 0.185

Average percentage error of Set 3 =

Theoretical Total Head (m) 0.242 0.234 0.237 0.229 0.206 0.137

Percentage error (%) 2.89 0.43 0.84 1.75 13.11 35.04

2.89 + 0.43 + 0.84 + 1.75 +13.11 +35.04 % 6 = 9.01 %

Set 4 First manometer level = 220mm Volume of water= 5 L Time = 56.53 s Table 7: Set 4 (First manometer level 220 mm)

Test sectio n

No .

h1 h2 h3 h4 h5 h6

1 2 3 4 5 6

Diameter, D(

×10

−2

m) 2.50 1.39 1.19 1.07 1.00 2.50

Flow area, A (x

10−4 m ) 4.909 1.520 1.094 0.899 0.785 4.909

Flow rate, Q(

m 3 /s

Fluid velocity,

v (m/s)

0.1802 0.5819 8.84 0.8085 5 ×10 0.9839 1.1268 0.1802

Pressure Head,

Velocity Head,

P ρg

v2 2g

(m) 0.220 0.200 0.184 0.163 0.128 0.145

(m) 0.002 0.017 0.033 0.049 0.065 0.002

Calculated Total Head, H T (m)

Measured Total Head,

HE (m) 0.216 0.215 0.218 0.215 0.213 0.180

0.222 0.217 0.217 0.212 0.193 0.147

Table 8: Percentage error of Set 4

No. 1 2 3 4 5 6

Test section

h1 h2 h3 h4 h5 h6

Experimental Total Head (m) 0.216 0.215 0.218 0.215 0.213 0.180

Average percentage error of Set 4 =

Theoretical Total Head (m) 0.222 0.217 0.217 0.212 0.193 0.147

Percentage error (%) 2.70 0.92 0.46 1.42 10.36 22.45

2.70 + 0.92+ 0.46 + 1.42 + 10.36 + 22.45 % 6 = 6.385 %

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DISCUSSION Based on all the tables above, the velocity head increases when the pressure head decreases for the convergent flow. The results obtained obey Bernoulli’s theorem, and thus Bernoulli’s Theorem is valid for the convergent flow. For the divergent flow, the results obtained have a relatively high percentage error. The divergent flow in set 2 shows the pressure head decreases with velocity head decreases. Theoretically, the pressure head should be increase when the velocity head decreases in divergent flow. Hence, Bernoulli’s Theorem is not valid for divergent flow in this experiment. Besides, the results show that the velocity of the water flow increases when the flow area decreases. This is a consequence of the continuity equation as the continuity equation states that the flow rates at any two points must be the same, as long as there is no fluid added or discharged. Hence, the smaller the diameter of the manometer tube, the greater the velocity head. Based on the results obtained, the average percentage error obtained is between the range 4.375% to 9.01%, which means the experimental total head has slightly deviated from the theoretical total head. In fact, the measured total head should be the same as the calculated total head based on Bernoulli’s Theorem. Hence, these deviations are mainly due to the water flow does not exactly follow all the assumptions made in Bernoulli’s Theorem. Firstly, the fluid is assumed to be non-viscous, which means the viscosity of the fluid is equal to zero. However, in reality, all real fluid except superfluid has their respective viscosities. Likewise, the water has its own viscosity as other fluids. Therefore, water flow in this experiment will experience shearing force and friction due to its viscosity, thus causes the deviation of the results. Secondly, Bernoulli’s Theorem assumes there is no loss of energy due to friction in the experiment. In fact, during the water flows in the Bernoulli’s Theorem Apparatus, energy is lost due to friction and shearing forces between the wall of the tubes and the water. Hence, the energy in the experiment is not conserved. Furthermore, when water flows in the tapered circular duct, it will cause the loss of energy as energy can be transferred through the wall of the circular duct to another wall. As a result, the probe manometer levels obtained are different from the theoretical calculated total head. Thirdly, the flow of water is assumed to be steady. However, in the real case, the velocity of water flows may be varied with time due to the existence of turbulent flow in this experiment. When the water flow is converged and diverged in the experiment, it will cause turbulence. Hence, the speed and direction of water flow vary. Therefore, there is a change of mechanical energy of the fluid in the system and causes the total head probe to be different from the theoretical total head. Apart from that, the water in the experiment is assumed to be incompressible. The water is indeed essentially incompressible in reality because the density of water is almost constant throughout the experiment. Hence, the change of density of the fluid can be ignored for an incompressible fluid. Therefore, the water can be assumed as an incompressible fluid in this experiment. After all, some experimental errors may occur and affect the experimental results. Parallax error may happen when taking readings of the scales of the manometer level of each tube. This happens when the eyes are not placed perpendicularly to the scale readings, and thus causes inaccurate readings. This error can be solved by making sure the eyes are placed perpendicularly to the scale readings when taking the readings of the manometer level. Besides, the presence of air bubbles inside the manometer tubes will cause an inaccurate 8

reading of the manometer level. Therefore, the air bubbles should be discharged completely first in this experiment by using the air-bleed valve.

9. CONCLUSION In conclusion, the velocity of the fluid flow increases when the pressure of the fluid decreases. From the results obtained, Bernoulli’s Theorem is valid for the convergent flow as the velocity of the fluid increases when the pressure of the fluid decreases. However, the results show that Bernoulli’s Theorem is not valid for the divergent flow.

10. REFERENCES Bernoulli's principle. (2020, 20 December). In Wikipedia. Retrieved 26 December 2020 from https://en.wikipedia.org/wiki/Bernoulli's_principle Boundless Physics. (n.d.). Lumen. Retrieved 26 December 2020 https://courses.lumenlearning.com/boundless-physics/chapter/fluids-in-motion/

from

Cengel, Y. A., & Cimbala, J. M. (2006). Fluid mechanics: Fundamentals and applications. Boston: McGraw-Hill. Experiment #2: Bernoulli's Theorem Demonstration. (2019, 26 December). Applied Fluid Mechanics Lab Manual. Retrieved 26 December 2020 from https://uta.pressbooks.pub/appliedfluidmechanics/chapter/experiment-2/ Incompressible flow. (2020, 03 December). In Wikipedia. Retrieved 26 December 2020 from https://en.wikipedia.org/wiki/Incompressible_flow

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