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Introduction : Bernoulli's law states that if a non-viscous fluid is flowing along a pipe of varying cross section, then the pressure is lower at constrictions where the velocity is higher, and the pressure is higher where the pipe opens out and the fluid stagnate. Many people find this situation paradoxical when they first encounter it (higher velocity, lower pressure). This is expressed with the following equation: p v2 z h * Constant g 2g (1.8) Where, p = Fluid static pressure at the cross section ρ = Density of the flowing fluid g = Acceleration due to gravity v = Mean velocity of fluid flow at the cross section z = Elevation head of the center at the cross section with respect to a datum h* =Total (stagnation) head The venturi meter consists of a venturi tube and differential pressure gauge. The venturi tube has a converging portion, a throat and a diverging portion as shown in the figure below. The function of the converging portion is to increase the velocity of the fluid and lower its static pressure. A pressure difference between inlet and throat is thus developed, which pressure difference is correlated with the rate of discharge. The diverging cone serves to change the area of the stream back to the entrance area and convert velocity head into pressure head. In metering practice, this non-ideality is accounted by insertion of an experimentally determined discharge coefficient, Cd that is termed as the coefficient of discharge. With Z 1 = Z2 in this apparatus, the discharge coefficient is determined as follow: Cd Qa Qi Discharge coefficient, Cd usually lies in the range between 0.9 and 0.99. Objectives : 1. To determine the discharge coefficient of the venturi meter (1.20) 2. To measure flow rate with venturi meter 3. To demonstrate Bernoulli’s Theorem Procedure : Part 1: Discharge Coefficient Determination 1. The General Start-up Procedures was performed. 2. The hypodermic tube from the test section was withdrawed. 3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted. 4. After the level stabilizes, the water flow rate using volumetric method was measured and the manometers reading was record. 5. Step 4 with at least three decreasing flow rates was repeated by regulating the venturi discharge valve. 6. The actual flow rate, Qa from the volumetric flow measurement method was obtained. 7. The ideal flow rate, Qi from the head difference between h1 and h3 using Equation 1.18 was calculated. The chart of “Qa vs Qi” was plotted and finally obtain the discharge coefficient was obtained, Cd which is the slope (c) Part 2: Flow Rate Measurement with Venturi Meter 1. The General Start-up Procedures were performed. 2. The hypodermic tube from the test section was withdrawed. 3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted. 4. After the level stabilizes, the water flow rate using volumetric method was measured and the manometers reading was record. 5. Step 4 with at least three decreasing flow rates was repeated by regulating the venturi discharge valve. 6. The venturi meter flow rate of each data were calculated by applying the discharge coefficient obtained. 7. The volumetric flow rate with venturi meter flow rate were compared. 8. Data Analysis: Throat Diameter, D3 (mm) 16.0 Inlet Diameter, D3 (mm) 26.0 Throat Area, At (m2) 2.011 x 10-4 Inlet Area, Ai (m2) 5.309 x 10-4 g (m/s2) 9.81 ρ (kg/m3) 1000 Part 3: Bernoulli’s Theorem Demonstration 1. The General Start-up Procedures were performed. 2. All manometer tubings are properly connected to the corresponding pressure taps and are air-bubble free were checked. 3. Discharge valve to the maximum measurable flow rate of the venturi was adjusted. 4. After the level stabilizes, the water flow rate using volumetric method was measured. 5. The hypodermic tube (total head measuring) connected to manometer #G was slided gently, so that its end reaches the cross section of the Venturi tube at #A. Wait for some time and the readings from manometer #G and #A were noted down. The reading shown by manometer #G is the sum of the static head and velocity heads, i.e. the total (or stagnation) head (h*), because the hypodermic tube is held against the flow of fluid forcing it to a stop (zero velocity). The reading in manometer #A measures just the pressure head (hi) because it is connected to the Venturi tube pressure tap, which does not obstruct the flow, thus measuring the flow static pressure. 6. Step 5 for other cross sections (#B, #C, #D, #E and #F) was repeated. 7. Step 3 to 6 with three other decreasing flow rates were repeated by regulating the venturi discharge valve. 8. The velocity, ViB was calculated using the Bernoulli’s equation where; Vi 2 g (h8 hi ) 9. The velocity, ViC was calculated using the continuity equation where Vi_Con = Qav / Ai 10. The difference between two calculated velocities were determined. Result : Part 1: Discharge Coefficient Determination Qav hA hB hC hD hE hF h A - hC Qi (LPM) (mm) (mm) (mm) (mm) (mm) (mm) (m) (LPM) 12.15 131 126 90 113 118 123 0.041 11.69 11.93 138 133 99 120 124 130 0.039 11.40 11.72 152 147 115 136 139 145 0.037 11.11 11.11 173 168 136 157 160 166 0.037 11.11 10.22 209 201 178 195 198 203 0.031 10.17 Part 2: Flow Rate Measurement with Venturi Meter Qav hA hB hC hD hE hF (LPM) (mm) (mm) (mm) (mm) (mm) (mm) 12.15 131 126 90 113 118 123 11.93 138 133 99 120 124 130 11.72 152 147 115 136 139 145 11.11 173 168 136 157 160 166 10.22 209 201 178 195 198 203 Qav(LPM) hA - hC (m) Qi(LPM) Calculated Flow Rate (LPM) Error(%) 12.15 0.0041 11.69 12.04 0.91 11.93 0.0039 11.40 11.75 1.51 11.72 0.0037 11.11 11.45 2.30 11.11 0.0037 11.11 11.45 3.06 10.22 0.0031 10.17 10.48 2.54 Part 3: Bernoulli’s Theorem Demonstration Cross Section Using Continuity Using Bernoulli equation equation Difference h*=hG hi ViB = Ai = ViC = ViB-ViC (mm) (mm) √[2*g*(h* - hi )] π Di2 / 4 Qav / Ai (m/s) (m/s) (m2) (m/s) # A 150 131 0.61 0.00053 0.38 0.23 B 159 126 0.80 0.00037 0.55 0.25 C 160 90 1.17 0.00020 1.01 0.16 D 160 113 0.96 0.00031 0.65 0.31 E 159 118 0.90 0.00038 0.53 0.37 F 155 123 0.79 0.00053 0.38 0.41 Calculation : Part 1: Discharge Coefficient Determination Qi (LPM) A 2 Qi A2V2 A2 1 2 A1 1 / 2 p1 p 2 Z1 Z 2 2 g 1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.041) 0.5}=11.69 1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.039) 0.5}=11.40 1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.037) 0.5}=11.11 1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.037) 0.5}=11.11 1/ 2 1000*60*{0.0082 * π[1-(0.008/0.013) 4]-0.5(2*9.81*0.031) 0.5}=10.17 According to the graph, Cd=1.0303 Part 2: Flow Rate Measurement with Venturi Meter Calculated Flow Rate (LPM) Error(%) 1.0303*11.69=12.04 | [(12.04-12.15)/12.15] |*100% =0.91 1.0303*11.40=11.75 | [(11.75-11.93)/11.93] |*100% =1.51 1.0303*11.11=11.45 | [(11.45-11.72)/11.72] |*100% =2.30 1.0303*11.11=11.45 | [(11.45-11.11)/11.11] |*100% =3.06 1.0303*10.17=10.48 | [(10.48-10.22)/10.22] |*100% =2.54 Part 3: Bernoulli’s Theorem Demonstration ViB = √[2*g*(h* π - hi )] Ai = ViC = ViB-ViC Di2 Qav / Ai (m/s) /4 (m2) (m/s) (m/s) √[2*9.81*(150-131 )]=0.61 π / 4=0.00053 0.00020/0.00053=0.38 0.61-0.38=0.23 √[2*9.81*(159-126 )]=0.80 π 21.62/4=0.00037 0.00020/0.00037=0.54 0.80-0.54=0.26 √[2*9.81*(160-90 )]=1.17 262 π 162 / 4=0.00020 0.00020/0.00020=1.00 1.17-1.00=0.17 √[2*9.81*(160-113 )]=0.96 π 202 / 4=0.00031 0.00020/0.00031=0.65 0.96-0.65=0.31 √[2*9.81*(159-118 )]=0.90 π 222 / 4=0.00038 0.00020/0.00038=0.53 0.90-0.53=0.37 √[2*9.81*(155-123 )]=0.79 π 262 / 4=0.00053 0.00020/0.00053=0.38 0.79-0.38=0.41 Discussion : Part 1 : The Qi was calculated by the equation: A 2 Qi A2V2 A2 1 2 A1 1 / 2 p1 p 2 Z1 Z 2 2 g 1/ 2 According to the Qav from the data, use Qav and Qi to plot the graph and then find the slope which is the discharge coefficient, Cd. In Bernoulli's law, the pressure of pipe would become smaller when the diameter decrease, which is shown good in the all data. According to the theory, discharge coefficient, Cd usually lies in the range between 0.9 and 0.99. However, the calculation of Cd in part 1, Cd=1.0303, which is larger than 0.99. Here are some reasons why the error generated: 1. The error come from the leaking of the machine, in the experiment, not only the tube, but also the pump was leaking. 2.The error of manometers reading and the measure of flow rate Qav., according to the data of the result, there is a data that different Qav have the same hA - hC. which is impossible in the real life. 3.There still had some bubbles in the plastic transfer tube effect the manometers reading. The way to reduce those error are : Check the function of device carefully before the experiment and make sure the air bubbles are escape from the pipe. Testing five times of the time then use the average time to measure the flow rate Qav. Part 2 : Use the Cd which we find in part 1 and the formula : Cd Qa Qi to calculate the calculated flow rate. Then use the equation :error = | (Qa-Qav)/Qa |*100% to find the error. According to the data, some of the calculated flow rate are larger than Qav. , but if the calculated flow rate were calculated by using the ideally discharge coefficient, the Qav would become larger than the calculated flow rate. So the error of the calculated flow rate was come form the error of the Cd calculation. Part 3 : Use the Bernoulli equation : ViB =√[2*g*(h* - hi )] and the data we collect to calculate the flow rate ViB. After that, use the Continuity equation : Ai = π Di2 / 4 and ViC = Qav / Ai to calculate the flow rate ViC. Finally, compare two flow rate.In Bernoulli's law, the velocity of flow rate would increase when the diameter decrease, which is shown well in the data in the part 3. The two flow rate should be same with each theoretically. However, after the calculation, all of the ViB were larger than ViC. In Bernoulli equation part, the error can only come from the manometers reading of hi and h*. In Continuity equation part, the error is come from the calculation of Qav. The error of Qav is due to the reading and the time measure. Use the average time to calculate Qav can decrease the amount of error. Activity the error of the Continuity equation part should be less than the Bernoulli equation part if the Qav is calculated by the average time, the Qav would become almost same with the ideally flow rate. However there still have those error in the manometers reading of hi and h*, which make the Bernoulli equation part more inaccurate. Conclusion : In Bernoulli's law, the pressure is lower at constrictions where the velocity is higher, and the pressure is higher where the pipe opens out and the fluid stagnate. In the pipe, the velocity is increasing when the diameter is decreasing, the velocity become higher when the diameter become lower. According to the result, the Bernoulli equation is proven. Reference : Applied Fluid Mechanics 5th Edition, Robert L. Mott, Prentice-Hall Elementary Fluid Mechanics 7th Edition, Robert L. Street, Gary Z. Watters, John K. Vennard, John Wiley & Sons Inc. Fluid mechanics 4th Edition, Reynold C. Binder Fluid Mechanics with applications, Anthony Esposito, Prentice-Hall International Inc.