MSE 223 Lab Report 3 Templatev2

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MSE 223 – Introduction to Fluid Mechanics - Laboratory Report - Energy Equation in a Venturi Tube Group number: G9 Experiment date: March 8 th , 2016 Report due date: March 15 th , 2016 Saeed Shokoya 301260740 signature_________________________________ Thomas Krammer 301252999 signature_________________________________ Michael Graves 301220150 signature_________________________________ Harrison Handley 301267789 signature_________________________________ Paul Kosteckyj 301247173 signature_________________________________ Jared Northway 301251045 signature_________________________________

1. Introduction The objective of this laboratory is to illustrate Bernoulli’s Law as depicted through a Venturi tube, as well as to determine the flow rate factor given by the variable (K). In order to test these principles we uses a Venturi tube with six wall pressure taps as well as a Pitot tube that is longitudinally mounted. As the cross sectional areas vary along the Venturi tube the pressures will also vary. Bernouilli’s principle is defined as: “a principle in hydrodynamics: the pressure in a stream of fluid is reduced as the speed of the flow is increased.” [1] Bernoulli's Law relates total energy, static pressure and velocity of a flow to show that the total energy of a steady, frictionless and incompressible flow is constant. In this laboratory we must use Bernoulli’s Law and Bernoulli’s Equation to illustrate the pressure differences in the Venturi Tube. In addition, we must utilize Bernoulli’s Equation to determine the flow rate of the fluid. Bernoulli’s Law can be applied to both gaseous fluids as well liquid fluids; however, in this laboratory we tested the principles on liquids. “The Venturi effect is the phenomenon that occurs when a fluid that is flowing through a pipe is forced through a narrow section, resulting in a pressure decrease and a velocity increase.” [2] The Venturi tube allows us to experiment and determine the changes in pressures and velocities of the fluid we pass through the differing cross sectional areas thus allowing us to apply Bernoulli’s equations to determine our unknowns. The Venturi effect works by having a pressure increase over a decreased surface area thus creating a vacuum in the water. As the pressure in the fluid decreases the kinetic energy in the fluid increases. 1

2. Methodology The objective of this experiment is to determine the flow rate factor ( K ) from the volumetric flow rate and the drop in pressure inside the venturi tube. The purpose of this is to familiarize ourselves with Bernoulli’s principle - the relationship between the flow velocity of a fluid and its pressure. The apparatus in Fig. 1 was used to obtain measurements, of which the venturi tube, pressure gauges and axially movable pitot tube were of the most importance. Figure 1 . Components of experimental apparatus 2

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The procedure of the experiment consisted of multiple measurements including one over time. The overall process is as follows. Water was being continually supplied to the first inlet valve on the left of Fig. 1 (from the hydraulics bench pump), which we controlled by opening & closing the inlet valves. The same was done for the outlet valve. As we waited for the initial water flow to become increasingly laminar, we made sure there were no bubbles present in the tube to fault our data. Once the flow was laminar, the pitot tube was closed completely (into the venturi tube) and the water levels in the 6 pressure gauges were regulated using the inlet and outlet valves. Levels in the gauges were measured and recorded as h piz . After that, the pitot tube within the venturi nozzle was sequentially moved perpendicular to the line of action of each pressure gauge, and a measurement was taken at each point from the large single water manometer gauge ( h tot ¿ . Finally, using a stopwatch, we measured the time it took to raise the water level in the tank by 10 liters. With this time, the volumetric flow rate was calculated ( Q ∈ Liters seconds ). Figure 2 . Cross-sectional areas of the venturi tube The experiment presents an applied understanding of the Bernoulli equation (1), or the principle of conservation. It is used to calculate the pressure difference in the tube: P ρ + V 2 2 + gz = Constant (1) where g is the gravitational acceleration, z is the elevation, V is velocity and P is the static pressure. This equation is valid theoretically when the flow is steady, frictionless and incompressible, and no work occurs. The static pressures of the fluid are constant as well. An increase in velocity leads to a reduction in pressure in a flowing fluid, and vice versa. The Bernoulli equation can also be related to the h tot and h piz values that were measured from the single large manometer gauge on the right of Fig. 1 and the smaller pressure gauges respectively. The relationship is described by the formulae: 3

h tot = P ρg + V 2 2 g + z ( 2 ) h piz = P ρg + z ( 3 ) where their difference from a common reference point inside the venturi nozzle results in the velocity at the respective cross sections in Fig. 2 ( V measured ). Inside the venturi tube in Fig. 2 lays a duct of varying cross-sectional area which is where the respective elevations ( z ) come into play. asured = √ 2 g ( h tot − h piz ) ( 4 ) We can achieve the calculation of the flow rate factor ( K ) by understanding how the venturi apparatus works. The venturi tube provides the ideal conditions for a flow rate measurement, in which there is a pressure loss ∆ P between the largest cross sectional area of the tube and the smallest. Since we have the means to measure this pressure difference from the difference between the largest pressure value minus the smallest out of the 6 gauges in h piz , and the flow rate has been determined using the stopwatch, we can easily calculate the K factor, as shown in the calculations section. The units of K are L s √ ¯ ¿¿ where Q is in L / s and the pressure difference in √ ¯ ¿ . ¿ K = Q √ ∆ P ( 5 ) Calculations Table 1 - Measured and Calculated Data 1 2 3 4 5 6 Cross Sectional Area (mm 2 ) 338.60 233.50 84.60 170.20 255.20 338.60 Piezometric Head (mm) 267.00 243.00 39.00 170.00 193.00 200.00 Total Head (mm) 275.00 272.50 269.00 242.00 235.00 227.00 Measured Velocity (m/s) 0.396 0.761 2.124 1.189 0.908 0.728 Calculated Velocity (m/s) 0.458 0.664 1.832 0.911 0.607 0.458 Measured Velocity Head (mm) 8.00 29.50 230.00 72.00 42.00 27.00 Calculated Velocity Head (mm) 10.68 22.46 171.12 42.28 18.81 10.68 4

Time to pump 10L (s) 64.51 Flow Rate, Q (L/s) 0.155 Flow Rate, Q (m 3 /s) 1.55x10 -4 The Cross Sectional Area for each portion of the venturi tube was given in the lab manual. Once as the water flow was properly regulated the measurements of both the piezometric and total head were taken. It should be noted that the values of the piezometric and total head varied by approximately ±20mm. This is most likely a result of the water pump. The measured velocity is calculated using equation (3) as follows: V measured , 1 = √ 2 g ( h tot , 1 − h piz , 1 )= √ 2 × 9.81 × ( 0.275 − 0.267 ) = 0.40 [ m / s ] ( 6 ) The calculated velocity is based on a series of calculations relating the timed average volumetric flowrate for 10L. Q = Volume Time = 10 L 64.51 s = 0.155 [ L / s ] = 1.55 × 10 − 4 [ m 3 / s ] ( 7 ) V calculated , 1 = Q A 1 = 1.55 × 10 − 4 0.3386 = 0.458 [ m / s ] ( 8 ) The measured velocity head is based on the difference between the total head and the piezometric head. h meas. V , 1 = h tot , 1 − h piz, 1 = 0.275 − 0.267 = 8.00 [ mm ] (9) The calculated velocity head is based on a part of the equation for total head of a flow and the velocities found using the timed average volumetric flowrate. h calc.V , 1 = V 1 2 2 × 9.81 = 0.458 2 2 × 9.81 = 10.68 [ mm ] ( 10 ) The flow rate factor (K) of the venturi tube is calculated as follows: Since point 1 and 6 have the same cross sectional area and are at opposite ends of the venturi tube, h piz = P ρg ( 11 ) ∆ P 1 → 6 = ρg h piz , 1 − ρg h piz , 6 = 1000 × 9.81 × ( 0.267 − 0.200 ) = 657.27 [ N m 2 ] ( 12 ) 657.27 [ N m 2 ] = 6.57 × 10 − 3 ¿ (13) 5

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K = Q √ ¯ ¿ = 0.155 √ 0.00657 = 1.912 ¿¿ (14) 3. Results and Discussion 1) Why does the energy grade line slope downward in the direction of flow? What are the assumptions that lead to a horizontal total energy line? The energy grade line will slope downward in the direction of the flow due to the effects of friction. Over time the friction will oppose the flow causing the total energy to drop more and more. Assuming that the system is ideal which would allow for no friction would be the cause of a horizontal flow line. 2) Is it possible for the pressure to increase in a uniform pipe flow? What happens in the case of a contracted pipe? 6 Figure 3 - Fluid Velocity throughout the Venturi Tube Figure 4- Total, Piezometric, and Velocity Heads through the Venturi Tube. 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Fluid Velocity Distribution through Venturi Tube Measured Velocity (m/s) Calculated Veloc- ity(m/s) Points of Specified Cross Sections Velocity (m/s) 0 1 2 3 4 5 6 7 0 50 100 150 200 250 300 Total, Piezometric, and Velocity heads Total Head (mm) Piesometric Head (mm) Measured Velocity Head (mm) Calculated Velocity Head (mm) Points of Specified Cross Sections Head (mm)

No in uniform pipe flow it wouldn’t be possible to increase the pressure. Since pressure changes depend on the flow rate the diameter of the pipe or the velocity of the fluid would have to change to change the pressure. In the case of a contracted pipe the pressure would actually decrease. When the fluid flows through the constriction the velocity of the fluid would increase which would cause the pressure to decrease. 3) What is the effect of decreasing the flow rate on the pressure loss, ΔP? How will it affect the form factor, K? Pressure drop is a mainly a function of velocity and pipe length while the flow rate is a function of diameter and velocity. Decreasing the flow rate would therefore lead to a decrease in the pressure drop. As shown in equation 4 the form factor K will decrease. 3) H ow can flow separation in a pipe be avoided? To avoid flow separation within a pipe it is necessary to remove any adverse pressure gradients. These occurs when the pipe has any type of protrusion in the entrance of the pipe. This is in combination with bends in the pipe developing low pressure at the bend which generates an adverse pressure gradient flow and thus flow separation. Finally, geometric differences in the pipe in terms of diameter will generate flow separation. This most notably occurs when the pipe diameter contracts. Furthermore, when the pipe diameter grows as in a diffuser flow separation can occur but depend))s on the cone angle of the diffuser, where a larger angle results in higher amount of flow separation. 4) In what circumstances would cavitation occur in a flow? Cavitation occurs when there is a restriction of flow results in an area of low pressure. If the fluid is at a low enough pressure for the corresponding temperature, then it will turn into its vapor state. These bubbles of vapor will then return back to their liquid state as the pressure increases. This process most notably occurs at control valves and other regions of constriction where the fluid is at a low pressure. 5) Calculate the Reynolds numbers at A 3 (r 3 = 5.2 mm) & A 6 (r 6 = 10.4 mm). Using the Moody chart in White’s book, determine whether it’s laminar or turbulent. Equation for velocity and Reynolds number in a certain cross section: ˙ Q A 3 ( 15 ) ℜ d = ρV avg d μ ( 16 ) Water at 20  C: ρ = 998 [ kg m 3 ] ∧ μ = 0.001 [ kg ms ] Calculations for A 3 assuming water at 20  C: 7

V 3 , avg = ˙ Q A 3 = 0.000155 [ m 3 s ] π ( 5.2 [ mm ] ) 2 = 1.82463 [ m s ] ( 17 ) ℜ d = ρV avg d μ = 998 ∙ 1.82463 ∙ 2 ∙ 0.0052 0.001 = 18398 ( 18 ) Which is larger than 2000 and is thus turbulent flow. Calculations for A 6 assuming water at 20  C: V 3 , avg = ˙ Q A 3 = 0.000155 [ m 3 s ] π ( 10.4 [ mm ] ) 2 = 0.456158 [ m s ] ( 19 ) ℜ d = ρ V avg d μ = 998 ∙ 0.456158 ∙ 2 ∙ 0.0104 0.001 = 9469 ( 20 ) Which is larger than 2000, and is thus turbulent flow once again. 4. Conclusions/Results The objective of this lab was to demonstrate the flow rate factor and Bernoulli’s law in a venturi tube. Using pressure gauges and a pitot tube in order to collect data throughout the venturi tube. Using the collected piezometric head and the total energy head we can calculate the velocity of the fluid through the tube (using equation 4) at various locations. This value can be compared with calculated values obtained using the continuity equation and the volumetric flow rate. Bernoulli’s equation (1) describes the relationship between the total energy, static pressure, velocity of a flow. The equation shows that the total energy of a steady, frictionless and incompressible flow is constant. Substituting the terms of the equation with heads, a measure of energies per unit weight, we can measure the total energy head (2). This total energy head can be measured through the use of a pitot tube as it describes the height which a liquid would rise within the pitot tube. Another head is the piezometric head, which is based upon the elevation and pressure head of the flow (3). This heads can be used to find the velocity of the flow within the cross-section of the venturi tube. The results of the experiment conclusively showed the relationships of the variables within Bernoulli’s equations, however due to various sources of errors there is a considerable level of uncertainty between theoretical and measured values. The experiment required the measurement of 6 pressure gauges at various locations within the venturi tube, due disturbances in the flow fluctuations in the level of the fluid in the gauges make accurate readings more difficult. As a result, our measured values vary from the calculated values with a maximum of 8

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59% error and a minimum of 14% error. Despite these errors our data conclusively showed a relationship which is consistent to those described in theory as seen in the above figures. References [1] School of Mechatronic Systems Engineering, " Energy Equation in a Venturi Tube," Simon Fraser University. [1]"Definition of BERNOULLI'S PRINCIPLE", Merriam-webster.com , 2016. [Online]. Available: http://www.merriam-webster.com/dictionary/Bernoulli 's%20principle. [Accessed: 12- Mar- 2016]. [2]"The Venturi Effect", Tech-faq.com , 2015. [Online]. Available: http://www.tech- faq.com/venturi-effect.html . [Accessed: 12- Mar- 2016]. 9