The Bernoullis Experiment - Lab Report Example

Lab report on Bernoulli’s experiment Summary The major aim for conducting the experiment was to establish a steady flow and measure the flow rate. In order to meet this objective, the aspects measured include average time taken to collect the water, the volumetric flow rate per se, and the pressure difference representing the static head and velocity. Intuitively, the process involved the use of combination of weight bench (hydraulic bench), to take the two readings, venture meter and manometer. Eleven points were selected along the venture meter whereby piezometric head values for each was measured and explored in the “procedure” section. In which case, the Bernoulli’s theorem majorly relies on the use of classical venture as can be seen from the experiment. The experiment demonstrates succinctly that the venture is a means for establishing flow measurement and, from which, the discharge coefficient can be derived. From the results, it is true to say that the pressure difference increases as the level of water in the manometer tubes increases. Chapter 1: Introduction Bernoulli’s principle named after its founder, Daniel Bernoulli, asserts that “as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. In which case, the use of Bernoulli’s equation guides the major concepts in the aforementioned definition. The equation makes use of the relation between pressure, velocity and elevation (height). This is different to the continuity equation, which only applies a relation between speed of the fluid and the cross sectional area measured for a pipe in which the fluid moves. It is worth noting that there are certain restrictive conditions to which the principle can hold. This include, the fluid used must be frictionless at a constant density; in which case, steady flow is a must. Since frictional effects are mostly significant when very close to the boundary layers, the use of Bernoulli’s approximation is feasible in outside the boundary layers. In which case, the flow outside is guided by the combination of two forces, pressure and gravity. The concept conveyed can be explored on the basis of law of conservation of energy. As a result, movement of fluid from wider section of a pipe to a constriction, triggers a greater distance movement of corresponding volume at the constriction thereby leading to an increase in speed. The aspects explained therein, constituting work done in both wider and narrower section can be summarized into the product of the pressure and volume. The greater speed evident leads to higher kinetic energy of the given volume in the narrower pipe. Going by the law of energy conservation, the observed increase of kinetic energy is thereby balanced by a decrease in the product of pressure-volume as shown in the Bernoulli’s equation below (Durgaiah 2002, 50). Kinetic energy + potential energy + flow energy = constant Theory The center line, while considering the assumptions that the fluid used must be frictionless at a constant density; in which case, steady flow is a must: From the above Bernoulli’s tequation, the total head (h) is obtained from summing up the pressure head (h), velocity head (hv) , and elevation head (z), respectively as depicted on the left hand side of the equation. Consequently, the theorem supports the assumption that for a fluid flowing through a pipe, the total head at any point is constant. Objectives To measure the piezometric head values for all eleven points To confirm the feasibility of Bernoulli equation when applied to a steady flow. CHAPTER 2: APPARATUS AND EXPERIMENTAL PROCEDURE 2.1 Apparatus Venturi meter Stop watch Weight bench Weight of 6 kg 2.2 Procedures The Bernoulli’s equation apparatus was set up on the weight bench. 11-tapered sections converging in the direction of flow were selected. The mass, at the weight bench was set at 0.006 kg constant for all the points. The outflow tube was then positioned right above the volumetric tank while the rig inlet was connected to the bench flow supply. The bench valve was closed and the pump was then started. The bench valve was opened gradually while filling water in the test rig. The bench valve was opened and permitted to flow through the manometer. The air bleed was opened to allow entry of air to the top of the manometer. After reaching a convenient height, the screw was tightened again. CHAPTER 3: CALCULATIONS AND RESULTS Part of the results as obtained from excel worksheet: Figure 1: Table Depicting Data Obtained Position A B C D E F G H I J K Diameter (mm) 26 23.2 18.4 16 16.8 18.47 20.16 21.84 23.53 25.24 26 Area mm2 530.9 422.7 265.9 201.1 221.1 268 318.8 375 435 500.8 530.9 Area m2 0.0005309 0.0004227 0.000266 0.0002011 0.0002211 0.000268 0.0003188 0.000375 0.000435 0.0005008 0.0005309 Test   Mass (kg) Time (s) Q Distance from A (mm) 0 20 32 46 61 76 91 106 121 136 156 1 (i) 0.006 10.47 0   0.268 0.254 0.155 0.005 0.024 0.111 0.158 0.187 0.205 0.219 0.226 (ii) 7.09   1.287 1.617 2.570 3.398 3.091 2.550 2.144 1.822 1.571 1.365 1.287 Avg time   8.78   0.268 0.219 0.016 -0.236 -0.134 0.021 0.118 0.183 0.227 0.258 0.268 2 (i) 0.006 10.46 0   0.243 0.234 0.168 0.068 0.079 0.137 0.169 0.188 0.200 0.210 0.248 (ii) 9.12   1.154 1.450 2.305 3.048 2.772 2.287 1.922 1.634 1.409 1.224 1.154 Avg time   9.79   0.243 0.204 0.040 -0.162 -0.081 0.044 0.123 0.175 0.210 0.235 0.243 3 (i) 0.006 16.93 0   0.224 0.219 0.178 0.117 0.122 0.157 0.177 0.189 0.197 0.202 0.205 (ii) 19.03   0.629 0.789 1.255 1.659 1.509 1.245 1.047 0.890 0.767 0.666 0.629 Avg time   17.98   0.224 0.212 0.164 0.104 0.128 0.165 0.188 0.204 0.214 0.222 0.224 Figure 2: Picture Form Depicting The Whole Table Sample Calculations Test 1 Q = MASS/AVG.time = 0.006/8.78 = 0.00006834 ( = Q/area = 0.00006834/0.0005309 = 1.287 Hn theoretical = hn experiment – = 0.268 – In order to obtain the graph for Bernouli’s experiment, the following table was made for the the three tests: FIGURE 3: Table For Graph for BERNOULLIS EXPERIMENT     A B C D E F G H I J K Test 1 h experimental (mm) 268 254 155 5 24 111 158 187 205 219 226 Test 2 h experimental (mm) 243 234 168 68 79 137 169 188 200 210 248 Test 3 h experimental (mm) 224 219 178 117 122 157 177 189 197 202 205 From the table, a graph of hn against position from A was made to help in depicting and exploring more on the Bernouli’s experiment. The three graphs with different lines represent the three tests carried out: Figure 4: Graph of hn against position from A The above graph depicts the pattern existing for the relationship between hn and position in the venture meter tube. It shows that the pattern assumed for all the three tests is similar as it starts at a higher point at position A, then drops to lowest point at D then again rise to relatively higher point at K, for all. Further, the graph also indicates that test 1 has both the highest point and lowest point while test 2 has relatively higher and lower point and 3 is the last in this category. This confirms the uniformity of all the tests. CHAPTER 4: ANALYSIS AND DISCUSSION The major objective of this lab is to confirm the validity of the Bernoulli equation when setting up a steady flow. Further, the experiment was set to measure the flow rates and the piezometric heads (both static and total pressure heads) contained in a convergent and divergent tube for a range of steady flow rates. Intuitively, the experiment makes use of bernoulli’s principle which makes it nature based on the relation between velocity and pressure for a frictionless fluid. In order to meet this objective, the aspects measured include average time taken to collect the water, the volumetric flow rate per se, and the pressure difference representing the static head and velocity. Intuitively, the process involved the use of combination of weight bench (hydraulic bench), to take the two readings, venture meter and manometer. Eleven points were selected along the venturi meter whereby piezometric head values for each was measured and explored in the “procedure” section. From this, the venturi meter allowed for deriving a relation between the flow rates, therein, and pressure measurement under the guidance of Bernoulli’s equation. The result obtained from this experiment confirms that an increase in pressure difference triggers a corresponding increase in flow rates of water thereby leading to increase in velocities. This was observed for both the convergent and divergent flow. As shown, whenever there was pressure difference increase the result was a rise in the water level contained in the manometer tube. Intuitively, there is a difference in velocity for fluid flowing through wider and lower constrictions. The velocity of the fluid increases whenever it flows from a wider constriction to a narrower constriction. The table in the results section confirms this, as depicted in the observed increase in the velocity of water moving in the tapered duct as the area of the duct increases. It is also worth noting that this was observed regardless of difference in pressure and the flow type. Analysis of the results concludes that a decrease in water flow rate leads to a decrease in the velocity of water. As can be shown in the results the velocity for the cross sections increased as the diameter decreased; point A had the largest diameter (26 mm) but with slowest flow rate and hence the velocity thereby was also low while point D (16 mm) had the tightest diameter but with highest flow rate and highest velocity. In which case, the fast flow rate was evident at points of smaller tubes while the slowest was in the bigger tubes. For A, the difference was 0.268 while for D the velocity difference was 3.398 and this was confirmed for all the three tests. Consequently, this confirms that the velocity of the fluid flowing in a tube is affected by the diameter since a bigger tube triggers a larger difference in velocity while a smaller tube results to the difference in velocity, VIB and VIC is smaller. Even though the experiment was completed successfully, the presence of zero error must have occurred while taking the measurement for each test. In which case, while taking the readings the level of static head might have not been allocated appropriately. Consequently, this might have triggered slight effect on the calculations and this can be evident at position K where the graph defied the perceived uniform pattern for the three tests. Consequently, the experiment confirms the validity of Bernoulli’s principle when applied to steady flow of fluid contained in tapered duct. In which case, there is increase in absolute velocity values along similar channel. Any slight difference witnessed from the actual value can be contributed to insignificant errors witnessed positioning of the eye while taking the readings. There might have been slight diversion in positioning the eye parallel to scale. In order to solve this error for future experiment, the water level must be allowed to settle and be stable. CHAPTER 5: CONCLUSION AND RECOMMENDATION 5.1 Conclusion The lab successfully confirms that the readings obtained for each tube increase as a result of increase observed in the pressure difference. Consequently, for convergent and divergent flow, the Bernoulli’s equation is valid as both cases obey the provision of the equation. For both case, as the time taken for collected water increases so does the pressure difference increases. Intuitively, for the same channel, the total head pressure increases with increase in velocity for both convergent and divergent flow. 5.2 Recommendation The trap bubbles should all be removed before starting the experiment The experiment should be repeated several times to arrive at an average value which guarantees more accuracy Control the valve with more attention to ensure that constant values are obtained for the pressure difference The water meniscus must be allocated with much attention by ensuring that the eye is positioned parallel to the scale while taking the manometer readings. This ensures avoidance of parallax error. Avoid, by any means, leakage of water from the instrument used in the experiment. Bibliography Durgaiah, D. R. (2002). Fluid mechanics and machinery. New Delhi: New Age International. Read More