The Behavior of the Laminar Fluid Motion along a Converging-Diverging Tube - Assignment Example

Laboratory Report: Fluid Mechanics— Verification of Bernoulli’s Equation The behavior of the laminar fluid motion along a converging-diverging tube of known cross-sectional area, at different flow rates, is investigated, and the mass as well as energy conservation laws are verified in this experiment. Another objective of performing it is to measure the loss of energy due to viscous resistance to the fluid the motion in case of higher flow rates. The results obtained from the experiment not only verify the basic laws and characteristics of a fluid in motion, but also determines the losses incurred in energy due to resistive forces. Introduction When a fluid undergoes motion, there are certain laws it must satisfy. Two of the basic laws pertain to the conservation laws for mass and energy. While the conservation of mass leads to Continuity Equation, conservation of energy gives us Bernoulli’s Principle. (1) Continuity Equation: the amount of fluid volume per unit time must remain constant as the fluid cannot be compressed (no density variation) and the mass flowing in to a certain area per unit time must always be same to that going out from that area in exactly the same time. Thus, leading to the important equation Q = A1 ? u1 = A2 ? u2 .....(1) where A1 and u1 are the cross-sectional area and fluid velocity respectively at the location ‘1’; and A2 and u2 are those at ‘2’ along the Venturi Tube shown below. (2) Bernoulli’s Equation: similarly, when we take into account the conservation of total energy of the water in the Tube, which happens to be one of the most fundamental conservation principles in Physics, we obtain the Bernoulli’s equation. It says that the Total Energy, consisting of the Kinetic, Potential, and Pressure components, is always constant-- P1/?g + u12/2g + Z1 = P2/?g + u22/2g + Z2 + HL .....(2) where ‘HL’ denotes the Head Lost because of fluid resistance, ‘Z1’ & ‘Z2’ describing Potential Energy (or, Head), which is constant here as the axis of the Tube lies parallel to the ground and hence may be assumed to be zero, and P1 & P2 are pressures of the fluid at ‘1’ and ‘2’ respectively. The 1st term on either side of the Eqn. (2) above, known as “Static Head” or “Pressure Head”, denoting the energy due to the fluid pressure; the 2nd term defined as “Velocity Head”, denoting the kinetic energy; and the 3rd term named as “Potential Head”, denoting potential energy, combine together to give the total energy or “Head” of the fluid in motion. Apparatus Used (1) Venturi Tube or Venturimeter (2) Stopwatch (3) Graduated Beaker 1 2 3 4 5 6 7 8 9 10 11 Venturimeter or Venturi Tube Methodology Step 1: Known volume of water (10 litres in the first case) is allowed to flow in through the inlet, located at position marked as ‘1’ Step 2: when the water flows through the Venturi Tube and goes out through the valve at ‘11’, it is collected in the graduated beaker and the volume is noted down along with the total time taken to travel through the Tube. Subsequently, Q, the Volumetric Flow Rate is measured using the Eqn. (1) Step 3: simultaneously, the heights of the water columns in each of the 11 capillary tubes connected to the Venturimeter at 11 different locations are also observed carefully and put the relevant column in the Observation Table 1 below Step 4: the Velocity Head and Pressure Head as given by Eqn. (2) are also calculated for different fluid velocities at different locations, and the values are tabulated in Columns 6 & 8 respectively Step 5: finally, the total energy or Head of the fluid is measured just by adding the Velocity Head and the Pressure Head since the Potential Head remains unchanged Step 6: all the 5 steps mentioned above are repeated for 20 litres and 25 litres and the measurements are recorded in Tables 2 & 3 Step 7: bar diagrams are plotted for Evh, the Velocity Head, against the Tube Locations. Similarly, another graph is plotted for Esh, the Static Head, Vs the Tube locations. Lastly, Etotal, the total energy of the fluid is also plotted for all the three cases, as 100% stack bar, in order to see the percentage sharing of the total energy. Results (i) Volume of water at Inflow = 10 litres Time taken to collect the whole amount of water= 48.9 seconds Hence, we have the Flow Rate = 10/48.9 = 0.2045 litres/sec = 0.2045 X 10-3 m3/sec i.e. Volumetric Flow Rate, Q = 204.5 X 10-6 m3/s Table 1: For inflow = 10 litres S. No. Tube Location x Diameter d (mm) Area A (10-6 m2) Velocity u (= Q/A) (m/s) Velocity Head Evh(= u2/2g) (m) Tube Height h (mm) Static Head Esh (= P/?g) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.4036 0.0083 245 0.245 0.2533 2. 2 22.8 408.3 0.5009 0.0128 242 0.242 0.2548 3. 3 18.5 268.8 0.7608 0.0295 230 0.230 0.2595 4. 4 15.8 196.1 1.0428 0.0555 205 0.205 0.2605 5. 5 16.4 211.2 0.9683 0.0478 209 0.209 0.2568 6. 6 17.8 248.8 0.8219 0.0345 220 0.220 0.2545 7. 7 19.2 289.5 0.7064 0.0255 226 0.226 0.2515 8. 8 20.6 333.3 0.6136 0.0192 232 0.232 0.2512 9. 9 21.8 373.3 0.5478 0.0153 236 0.236 0.2513 10. 10 23.1 419.1 0.488 0.0122 240 0.240 0.2522 11. 11 25.4 506.7 0.4036 0.0083 243 0.243 0.2513 Figure 1.1: Velocity Head for Inflow = 10 litres Figure 1.2: Static Head for Inflow = 10 litres Figure 1.3: Velocity Head (in green) and Static Head (in orange) combined for Inflow = 10 litres (ii) Volume of water at Inflow = 20 litres Time taken to collect the whole amount of water= 59.7 seconds Hence, we have the Flow Rate = 20/59.7 = 0.335 litres/sec = 0.335 X 10-3 m3/sec i.e. Volumetric Flow Rate, Q = 334 X 10-6 m3/s Table 2: For inflow = 20 litres S. No. Position of the tube, x Diameter d (mm) Area A (10-6 m2) Velocity u (= Q/A) (m/s) Velocity Head Evh(= u2/2g) (m) Tube Height h (mm) Static Head Esh (= P/?g) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.6592 0.0222 240 0.240 0.2622 2. 2 22.8 408.3 0.8180 0.0341 232 0.232 0.2661 3. 3 18.5 268.8 1.2426 0.0788 195 0.195 0.2738 4. 4 15.8 196.1 1.7032 0.1480 125 0.125 0.2730 5. 5 16.4 211.2 1.5814 0.1276 135 0.135 0.2626 6. 6 17.8 248.8 1.3424 0.0919 170 0.170 0.2619 7. 7 19.2 289.5 1.1537 0.0679 187 0.187 0.2549 8. 8 20.6 333.3 1.0021 0.0512 200 0.200 0.2512 9. 9 21.8 373.3 0.8947 0.0408 210 0.210 0.2508 10. 10 23.1 419.1 0.7969 0.0324 215 0.215 0.2474 11. 11 25.4 506.7 0.6592 0.0222 225 0.225 0.2472 It may be observed that the Total Energy at the inlet (location ‘1’) was 0.2622 and that at the outlet (location ‘11’) was 0.2472. Hence, the Head Lost due to fluid resistance may be calculated as (HL)20lit = 0.2622 – 0.2472 = 0.015m. .....(S1) Figure 2.1: Velocity Head for Inflow = 20 litres Figure 2.2: Static Head for Inflow = 20 litres Figure 2.3: Velocity Head (in green) and Static Head (in orange) combined for Inflow = 20 litres (iii) Volume of water at the Inflow = 25 litres Time taken to collect the whole of water = 56 seconds Hence, we have the Flow Rate = 25/56 = 0.4464 litres/sec = 0.4464 X 10-3 m3/sec i.e. Volumetric Flow Rate, Q = 446.4 X 10-6 m3/s Table 3: For inflow = 25 litres S. No. Position of the tube, x Diameter d (mm) Area A (10-6 m2) Velocity u (= Q/A) (m/s) Velocity Head Evh(= u2/2g) (m) Tube Height h (mm) Static Head Esh (= P/?g) (m) Etotal = Evh + Esh (m) 1. 1 25.4 506.7 0.8810 0.0396 215 0.215 0.2546 2. 2 22.8 408.3 1.0933 0.0610 210 0.210 0.2710 3. 3 18.5 268.8 1.6607 0.1407 139 0.139 0.2797 4. 4 15.8 196.1 2.2764 0.2644 10 0.010 0.3644 5. 5 16.4 211.2 2.1136 0.2279 35 0.035 0.5779 6. 6 17.8 248.8 1.7942 0.1642 90 0.090 1.0642 7. 7 19.2 289.5 1.5420 0.1213 125 0.125 0.2463 8. 8 20.6 333.3 1.3393 0.0915 150 0.150 0.2415 9. 9 21.8 373.3 1.1958 0.0730 165 0.165 0.2380 10. 10 23.1 419.1 1.0651 0.0579 175 0.175 0.2329 11. 11 25.4 506.7 0.8810 0.0396 187 0.187 0.2266 It is observed in this case as well the Total Energy of the fluid is different at the entry and the exit points, or the inlet and the outlet of the Venturi Tube. As it may be found from the Table 3 above that Etotal, the Total Energy at the location ‘1’ was 0.2546 whereas that at ‘11’ was 0.2266, the Head Lost due to viscous resistance is measured to be (HL)25lit = 0.2546 – 0.2266 = 0.028m. .....(S2) Figure 3.1: Velocity Head for Inflow = 25 litres Figure 3.2: Static Head for Inflow = 25 litres Figure 3.3: Velocity Head (in green) and Static Head (in orange) combined for Inflow = 25 litres Discussion The Head Lost measured in case of inflow volumes of 20 litres, and 25 litres, is an important parameter as far as dynamic characteristics of the fluid is concerned. It may be considered as a good indicator of viscous dissipation. It may be observed from the Tables 1 & 2 as well as Figs. 1.1 & 2.1 that when the Venturimeter is filled with 20 litres of water, as against 10 litres in the first case, the fluid Velocity Head increases from 0.055 to 0.15. This increase in the Kinetic Energy of the fluid causes stronger viscous resistance by surrounding fluid layers, and hence a Head Loss of 0.015m (result from S1). As the volume is increased further to 25 litres, the fluid velocity increases too and so is the Head Lost due to the resistance, which is measured to be 0.028m (result from S2). Another interesting behavior that may be noted from the Figs 1.3, 2.3, and 3.3 is that the percentage shares of the Velocity Head and the Static Head in the region of the smaller cross-sectional area (location ‘4’) are poles apart in case of higher Flow Rate (when inflow volume is either 20 litres or 25 litres). This effect is more prominent in the third case where the Pressure Head at the middle reduces drastically (Fig. 3.3). Definitely, it signifies the onset of turbulence in the fluid flow as its velocity starts accelerating and simultaneously viscous dissipation getting stronger and stronger. Besides, it is observed from all the Tables and corresponding graphs that both the Continuity Equation as given by the Eqn. (1), and the Bernoulli’s Equation as given by the Eqn. (2), are satisfied by the water used inside the Venturimeter in our case. The fluid velocity and hence Velocity Head increase as the cross-sectional area of the Venturimeter decreases from the position ‘1’ to ‘4’, and then it again increases as the fluid moves from the location ‘4’ to ‘11’ (Fig. 1.1, 2.1, 3.1)— confirming that the velocity and cross-sectional areas are inversely proportional, and reaffirming the validity of the Continuity Equation. Similarly, it is also evident from the Figs. 1.1 & 1.2, 2.1 & 2.2, and 3.1 & 3.2 that the fluid pressure inside the Venturimeter varies in such a way that it increases at those locations where the velocity decreases, to compensate for the loss of Kinetic Energy. Again, it starts decreasing the pressure (and Pressure Head) when the Kinetic Energy is increased—confirming Bernoulli’s Equation. Conclusion Apart from verifying two of basic laws of fluid dynamics, our objective of carrying out this experiment was to find out if there was any energy loss as a result of fluid resistance, caused due to its viscosity. This energy loss is measured to be of appreciable amount (in fact, of the same order to the kinetic energy), when the fluid starts moving at higher velocity. Hence it may be concluded that higher the velocity of the fluid, greater will be the energy loss due to resistance due to other fluid layers. Though the fluid behaves as an ideal laminar fluid, satisfying both the Continuity Equation and the Bernoulli’s Equation, in all the three cases discussed above, it may be noted here that as soon as velocity increases, turbulence starts setting in (when Q = 446.4 X 10-6 m3/s). Read More