Fluid Mechanics: The Continuity Equation - Lab Report Example

Laboratory Report: Fluid Mechanics The rate of flow of water will be determined by analyzing its flow through pre determined cross section. The designed tube to determine the rate of flow of water is a converging and a diverging tube. The rate of flow alters are different sections of the tube. The diameter will help to determine the flow of water. Introduction The major goal that is to be determined through the experiment is to analyse the flow manner of water through an enclosed pipe. For the evaluation of the flow, the Bernoulli’s equation and Continuity equation will be utilised. The Continuity Equation depicts the mathematical form of the “law of conservation of mass”. It is a fact that an enclosed body like a pipe make the flow of water or any other fluid to have no flow velocity components in regular direction, as, the enclosed body makes the flow to be also the tangential direction (Sahu and Sahu, 2009). Utilizing the law of conservation of mass, the above phenomenon can be mathematically interpreted as: Q = A1 × v1 = A2 × v2 eq1 On the other hand, the Bernoulli’s equation well depicts the change in the flow of fluid at two different points (Sahu and Sahu, 2009). Regarding the above situation, the Bernoulli’s equation can be written as: P1/ρg + v12/2g + Z1 = P2/ρg + v22/2g + Z2 + HL eq2 The theory can be applied with the availability of the Venturi-meter as the apparatus. Venturi meter is used to determine the rate of flow of any fluid if the motion of the fluid remains uniform. The Venturi-meter is such designed that a tube with smaller diameter lies between two tubes of wider diameter. All three are connected. At each of the three sections, narrow tubes are fused that lie perpendicular to the main apparatus as given in Figure 1. The three vertical fused tubes have the responsibility to determine the pressure of fluid at three sections. Practically, 11 monometers are utilized in the experimentation of the flow rate of the fluid. The apparatus is such designed to maintain the flow of water properly. On the other hand, stop watch is also utilized to measure the interval of time, as rate is measured with reference to time. Figure 1: Venturimeter Results S No. Position of the tube, x Diameter d (mm) Area A (10-6 m2) Velocity v (= Q/A) (m/s) Velocity Head Evh(= v2/2g) (m) Tube Height h (mm) Static Head Esh (= P/ρg) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.26 0.0035 245 0.245 0.2485 2. 2 22.8 408.3 0.32 0.0052 245 0.245 0.2502 3. 3 18.5 268.8 0.49 0.0123 240 0.24 0.2523 4. 4 15.8 196.1 0.68 0.0236 230 0.23 0.2536 5. 5 16.4 211.2 0.63 0.0203 230 0.23 0.2503 6. 6 17.8 248.8 0.53 0.0143 235 0.235 0.2493 7. 7 19.2 289.5 0.46 0.0108 237 0.237 0.2478 8. 8 20.6 333.3 0.40 0.0082 242 0.242 0.2502 9. 9 21.8 373.3 0.36 0.0066 244 0.244 0.2506 10. 10 23.1 419.1 0.32 0.0052 245 0.245 0.2502 11. 11 25.4 506.7 0.26 0.0035 245 0.245 0.2485 Table 1: 10 L inflow 10 L inflow Time 75.55 sec Flow rate = = 10/75.55 = 0.1324 litres/sec = 0.1324 X 10-3 m3/sec Volumetric Flow Rate, Q = 132.4 X 10-6 m3/s Figure 1.1: 10 litres Velocity Head for Inflow 10 litres Static Head for Inflow Velocity Head (in blue) Static Head (in red) in Total Energy for 10 litres Inflow 20 litres Inflow Time 59.08 sec flow rate = 20/59.08 = 0.3385 litres/sec = 0.3385 X 10-3 m3/sec Volumetric Flow Rate, Q = 338.5 X 10-6 m3/s S. No. Position of the tube, x Diameter d (mm) Area A (10-6 m2) Velocity v (= Q/A) (m/s) Velocity Head Evh(= v2/2g) (m) Tube Height h (mm) Static Head Esh (= P/ρg) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.67 0.0229 243 0.243 0.2659 2. 2 22.8 408.3 0.83 0.0351 237 0.237 0.2721 3. 3 18.5 268.8 1.26 0.081 200 0.2 0.281 4. 4 15.8 196.1 1.73 0.1527 130 0.13 0.2827 5. 5 16.4 211.2 1.6 0.1306 140 0.14 0.2706 6. 6 17.8 248.8 1.36 0.0944 173 0.173 0.2674 7. 7 19.2 289.5 1.17 0.0698 192 0.192 0.2618 8. 8 20.6 333.3 1.02 0.0531 205 0.205 0.2581 9. 9 21.8 373.3 0.91 0.0423 215 0.215 0.2573 10. 10 23.1 419.1 0.81 0.0335 221 0.221 0.2545 11. 11 25.4 506.7 0.67 0.0229 230 0.23 0.2529 Table 2 Total Energy at the inflow valve 0.2659 Outflow valve was 0.2529 Head Lost due to fluid resistance: (HL)20lit = 0.2659 – 0.2529 = 0.013m Velocity Head Inflow = 20 litres Static Head Inflow = 20 litres Velocity Head blue Static Head red in Total Energy for 20l Inflow 25l Inflow Time 53.77 sec Flow rate = 25/53.77 = 0.4649 litres/sec = 0.4649 X 10-3 m3/sec Volumetric Flow Rate, Q = 464.9 X 10-6 m3/s S. No. Position of the tube, x Diameter d (mm) Area A (10-6 m2) Velocity v (= Q/A) (m/s) Velocity Head Evh(= v2/2g) (m) Tube Height h (mm) Static Head Esh (= P/ρg) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.67 0.0229 243 0.243 0.2659 2. 2 22.8 408.3 0.83 0.0351 237 0.237 0.2721 3. 3 18.5 268.8 1.26 0.081 200 0.2 0.281 4. 4 15.8 196.1 1.73 0.1527 130 0.13 0.2827 5. 5 16.4 211.2 1.6 0.1306 140 0.14 0.2706 6. 6 17.8 248.8 1.36 0.0944 173 0.173 0.2674 7. 7 19.2 289.5 1.17 0.0698 192 0.192 0.2618 8. 8 20.6 333.3 1.02 0.0531 205 0.205 0.2581 9. 9 21.8 373.3 0.91 0.0423 215 0.215 0.2573 10. 10 23.1 419.1 0.81 0.0335 221 0.221 0.2545 11. 11 25.4 506.7 0.67 0.0229 230 0.23 0.2529 Table 3 There appears to be discrepancy in the total energy. This is because the inflow valve was 0.2793 and the outflow was 0.2402 Fluid resistance impacted the Head Loss: (HL)25lit = 0.2793 – 0.2402 = 0.0391m. Velocity Head Inflow = 25 L Static Head Inflow = 25 L Velocity Head blue Static Head red Total Energy for Inflow 25 L Conclusion The experiment proves both the Continuity equation and the Bernoulli’s equation. As the different in the flow occurs due to the difference in the diameter of the tubes, the velocity of the water is found to be more in the pipe with lesser diameter that proves the Continuity equation to be correct and valid. On the other hand, the pressure in the three sections are different that makes the venturi-meter to fluctuate, which verifies the Bernoulli’s equation. It is concluded from the experiment that head lost is caused by the resistance in the flow of the fluid. The increase in the velocity of the fluid is due to the fact the venturi-meter tries to maintain the rate of flow. In this way, it can be said that the energy that makes a fluid flow become constant. I is clearly determined that the flow rate of fluid changes from 132.4 X 10-6 m3/s, 10 litre volume to 464.9 X 10-6 m3/s, 25 litre volume. On the other hand, as the rate of flow increase the viscous resistance also increases which, consequently amplifies the Head lost from 0 at the volume of 10L to about 0.013m at the volume of 20L and 0.0391m at the volume of 25L. From the experiment, it can be concluded that the static head and velocity head deform in decreased cross sectional regions of the fluid flowing. That changes the behaviour of the flow to turbulent from laminar. References Handbook of Piping Design By Sahu, Sahu G.K. (2009) Read More