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vv FLOW MEASUREMENT AND ENERGY LOSS IN PIPES Lab 1 Brookelynn Vizzerra Jonathan Zacarias Carolina Santos Keith McCabe University of Idaho CE322-Hydraulics Table of Contents Table of Contents Introduction: .................................................................................................................................................... 2 Theory: ........................................................................................................................................................... 3 Approach: ....................................................................................................................................................... 6 Results: .......................................................................................................................................................... 7 Discussions: .................................................................................................................................................. 10 Reference: .................................................................................................................................................... 11 Appendix 1: ................................................................................................................................................... 12 Appendix 2: ................................................................................................................................................... 13 1|Page Introduction: This lab was designed to better understand pressure and energy differences for incompressible fluids flowing through pipes by using the Scott Fluid Circuit System, represented in Figure 1. In doing this we had to accomplish (1) the calibration of the Venturi Flow Meter, (2) determine the friction factor for a pipe at two Reynolds numbers using the Darcy-Weisbach diagram, and (3) determine the head loss at three different valve openings and pipe configurations. The main findings that helped us accomplish the lab were several head loss measurements using the independent differential manometers and the given max flow equation, several weight and time measurements to reach the desired one-hundred pounds, and a temperature measurement. To organize this report, we will be using a variety of tools such as excel and Mathcad to produce our graphs and results. We will first explain the theory behind this lab and continue onto the approach we used followed by the results and discussion. Figure 1: Scott Fluid Circuit System 2|Page Theory: (1) Calibration of the Venturi Flow Meter A Venturi Flow Meter is used to compare the head loss between a regular diameter and a reduced diameter within a pipe. By reducing the area, the velocity will increase to satisfy the principle of continuity and the pressure will decrease to satisfy the conservation of mechanical energy. For this lab we used the manometers, measuring the differential head, and the correction factor to find the flow rate. The flow-rate equation can be derived from using the Bernoulli equation between the two different diameter points: π12 π1 π22 π2 + + π§1 = + + π§2 + βπ 2π πΎ 2π πΎ Eq. (1) Where p = Pressure V = Velocity z = Height g = Gravitational constant γ = Specific gravity of water hl = Head loss Because of the short distance and no elevation change, the head loss is now considered to be negligible and the z1 and z2 can cancel out. In this equation the pressure head (p/γ) plus the elevation head (z) for each point can be considered respectively as h1 and h2, giving: π12 π22 + β1 = + β2 2π 2π Eq. (2) By noting that A1 and A2 are area, we can now apply the continuity equation and rewrite the previously reduced Bernoulli equation as: π22 π΄22 π22 ( 2) = + (β2 − β1 ) 2π π΄1 2π Eq. (3) In solving V2 we find that: π2 = √2π(ββ) π΄2 √(1 − 22 ) π΄1 Eq. (4) By multiplying A2, we can reach the flow-rate as our final goal: 3|Page π= π΄2 √2π(ββ) √1 − Eq. (5) π΄22 π΄12 The final discharge function given within the lab was: Q = KA√2π(ββ) Eq. (6) Where K is the discharge coefficient: πΎ= πΆπ √1 − πππ πΆπ = π΄22 π΄12 π΄ππ‘π’ππ πππ πβππππ π‘βπππππ‘ππππ πππ πβππππ Eq. (7) With these equations we can accurately generate the calibration curve for the Venturi Flow Meter. The K must be experimentally found and the theoretical values must be corrected for head loss incurred between the two points measured. (2) Determination of the Darcy-Weisbach Friction Factor When a flow is fully developed, steady state, and incompressible within a constant diameter pipe we can use the Darcy-Weisbach equation. This equation uses the friction factors to describe the head loss in the pipe. To derive, we first start with the momentum equation applied to a control volume. This momentum equation states that the sum of the external forces acting on the object is equal to the rate of momentum change in the control volume plus the rate at which the momentum flows out of the control volume: ∑∫πΉ = π ∫ πππ∀ + ∫ πππππ΄ ππ‘ Eq. (8) Because the velocity and density are constant over time, we should note here that the momentum equation should be equal to zero: ∑πΉ = 0 Eq. (9) In studying the momentum equation, there are three forces that have to be considered: (1) Pressure, (2) Shear, and (3) Weight. The pressure force (Fp) in this circuit system is the difference of pressure between taps twenty-four and thirty-two (see figure 1) with given pipe diameter (D): π πΉπ = (π1 − π2 ) π· 2 4 Eq. (10) Shear force (Fs), the effect of the wall on the fluid, varies with respect to a change in length. Shear force is acting in the opposite direction of the flow so the value will be negative: πΉπ = −πππ·βπΏ Eq. (11) 4|Page Weight force (Fw), the weight of the fluid in the pipe, changes with respect to length and depth. As it acts in the direction of flow, we will also have a negative value: π βπ§ πΉπ = −πΎ ( π·2 ) βπ ( ) 4 βπ Eq. (12) Combining the above forces into the momentum equation we produce: (π1 + πΎπ§1 ) − (π2 + πΎπ§2 ) = 4βππ π· Eq. (13) In order to introduce the head loss, we now need the energy equation: ∝ π12 π1 π22 π2 + + π§1 + βπ =∝ + + π§2 + βπ‘ + βπ 2 πΎ 2 πΎ Eq. (14) Here, we know α = 1 because the velocity profiles are uniformly distributed. Also, h p and ht are equal to zero because we are not dealing with any pumps or turbines within the system. We now have the Bernoulli Equation with head loss, called the energy equation: π12 π1 π22 π2 + + π§1 = + + π§2 + βπ 2 πΎ 2 πΎ Eq. (1) From this, V1 and V2 are equal and therefore drop out of the equation. With some slight rearranging we now have: (π1 + πΎπ§1 ) − (π2 + πΎπ§2 ) = βπ πΎ Eq. (16) In comparing equations (13) and (16) the left side is the same so we may conclude that: βπ = 4ππΏ πΎπ· Eq. (17) Taking the specific weight (γ) and the local shear stress (τ) as an empirical value, we can yield the final Darcy-Weisbach equation: πΏ π2 2π·π βπ = π π· 2π where π = βπ πΏπ 2 Eq. (18) Eq. (19) The Darcy-Weisbach equation along with the Moody diagram can be used to determine the friction factor. In doing this we will also need the Reynolds number determined from the flowrate (Q), pipe diameter (D), and the kinematic viscosity (ν): π π = ππ· 4π = π ππ·π Eq. (20) (3) Head Loss at Different Valve Openings: Different valve openings causes head loss to change with respect to velocity. The head difference between the two sides of the gate valve is measured by the second manometer while the first manometer is used by the Venturi Flow Meter to obtain the flow rate. In order to 5|Page Eq. (21) obtain the head loss measurements, the two separate manometers must be connected. To calculate K, it is evaluated by the following equation: 2 Kπ οh = 2ππ΄2 By arranging the equation into a more suitable manner, we are able to calculate K: 2 K= 2gπ΄ ββ π Eq. (22) 2 The data points that are recorded should cover a wide range of flow rate. With these known theories, we expect our data to verify the theories. The differences between our theoretical and actual values can be affected due to the aging of the pipes used. Approach: For the first experiment, our goal was to calibrate the Venturi Flow Meter. We began by obtaining a stopwatch to record time, a thermometer in order to obtain viscosity, and a large bucket attached to a scale to measure the weight of our discharge. We then had to vent the air pockets that were trapped in the circuit system by using the finger screws at the top of each independent differential manometer. The screws were then re-tightened to trap the air as manometer fluid. The head was sustained at a constant height maintaining a constant level of water inside the reservoir that would pump water through the system. We first obtained our max flow rate by fully closing valve fifty-two and fully opening valve fortyfive. We noted the time it took to fill the bucket near capacity. For this particular experiment we used a capacity of 100 pounds. At the exact time this weight was reached, we obtained the head loss (βh) given from the manometers. Because we were required ten pairs of readings we chose to reduce the max flow rate by 10% each time and take new numbers. From this equation: Q max ο½ KA 2 gοh Eq. (6) Our βh will change. We then had to adjust valve forty-five for the new βh which in turn changed the flow rate. We re-filled the bucket to capacity noting the new time. These steps were repeated until the ten measurements were taken and the graph is represented in the result section below. For the second experiment, we focused solely on the volumetric flow-rate through pipe three at two different flow rates. Here, both the fill valve and valve forty-five were fully closed so there was no discharge and the flow was recirculated through the system. Valve fifty-two was used to control the flow rates and again the lines were freed of air bubbles. The volumetric flow rates were determined using the Venturi Flow Meter and the corresponding manometer from the first experiment. A second flow manometer was attached to determine the total head loss within the system, which was largely due to the friction in the pipes. This head loss was then used to find the friction factor value of the pipe using the Darcy-Weisbach equation. This 6|Page experiment was completed at the top of the range tested in task one and then again at half as large to calculate a better friction factor value. The measurements were taken with the manometer and the difference between water levels within the manometer. For the last experiment, the goal was to determine the head loss characteristics of the three different gate valve openings. Again, the water was recirculating, valve forty-five and the fill valve were fully closed, and valve fifty-two was used to control the flow rates. At the three different positions; fully open, two turns open, and one turn open, the flow rates were changed according to our βh numbers gathered in task one. The first two positions required five readings whereas the last one only required three because the manometer was too short when the flow was high. Head loss was measured from the manometer and the friction factor, determined in the last experiment, was used to calculate the K value of the valve. As with any experiment, error was involved to some extent. The scale used was accurate to the nearest one pound so we could attribute half pound errors in measurement. Another error noticed was the manometer readings. When taking the values the water was not stabilized. The last possible error was the accuracy of the timer. This would account for a fairly small impact and should not affect the laboratory results. Results: (1) Venturi Flow Meter Calibration In the lab, we calibrated the Venturi Flow Meter based on the data that we calculated by using the difference in the height of the water column and the equation: Q = KA√2π(ββ) Eq. (6) By recording the weight of the water in the bucket over an amount of elapsed time, we were able to record the flow rate of the system. This in turn provided us with a theoretical flow rate and an actual flow rate. Figure 2 shows the curve of the flow rate values that we recorded for the max value of flow rate, and decreased by ten percent with each new reading. Figure 2: Flow Rate vs Differential Head 7|Page Figure 3: Flow Rate vs. K-value as a function of Reynolds Number By decreasing the flow rate, we were able to obtain a more accurate flow rate. The uncertainty in accuracy of the values was due to human error. The uncertainty in elevation readings and flow rates are evaluated by the following equation: ππ ππ 2 ππ‘ 2 √ = ( ) +( ) π π€ π‘ Eq. (23) By arranging the equation into a more suitable arrangement: 2 2 πΏπ πΏπ √ ππ ≤ ( ππ) + ( ππ) πΏπ πΏπ‘ Eq. (24) Since there is one variable used, the variance in the manometer levels are due to turbulences in the system. (2) Darcy-Weisbach Friction Factor Determination To obtain the Reynolds number, friction factors, and the error of propagation for the second portion of the lab, where we calculated the max flow rate and half of the max flow rate by 8|Page looking at the elevation change and head loss in manometers, the three following equations the below were used: Friction factor: π= π·2πβπ π·2πβπ π΄2 = πΏπ 2 πΏπ 2 Eq. (25) Reynolds number: π π = 4π ππ·π Eq. (20) Error of Propagation: ππ ππ ππ < | ππ| + | ππ| ππ ππ Eq. (26) The kinematic viscosity number that we obtained was directly inferred from the temperature of 56 degrees Fahrenheit. Once the kinematic viscosity value was found, we were able to calculate Reynolds number. By interpolation, we were able to find the value of π to be 1.309*10-5 ft2/s. The error of propagation for the friction factor of the two different flow rates was evaluated by the following equation: ππ ππ ππ ππ ≤ | πβπ | + | ππΏ| + | ππ| πβπ ππΏ ππ Eq. (27) This equation, which is based on the flow rate, length of the pipe, and the difference in head loss values, gives the error as a result. The error of propagation for the Reynolds number for the two different flow rates were evaluated by the equation: ππ π = | ππ π ππ| ππ Eq. (28) This equation, based on the flow rate, gives the error as a result. After these values for the two separate flow rates were calculated, we were then directed to plot the results on a standard Moody Diagram which can be seen in Figure 3. Figure 4: Moody Diagram In using the Moody Diagram, the uncertainty bounds should be taken into account and change the values accordingly. From referencing the book, we were able to use a relative roughness for the Drawn Copper to be 0.0015. Table 1 details the values that we were able to calculate. (3) Gate Valve Head Loss Characteristics The final portion of the Lab was to obtain flow rate against head loss at valve settings of fully open, two turns open, and one turn open. To gain a valid representative sample, we were instructed to obtain five data points for each valve setting, displayed in Figure 4. For the first 9|Page two valve settings, we were able to obtain our representative five data point sample however, for the one turn open setting, we were only able to obtain three representative samples due to the sensitivity of the pressure in the system. Our results proved our theory was correct because when it’s fully open we have a lower head loss due to a bigger area. Q vs. K 1.4 1.2 y = -0.0151x2 + 0.2423x 1 0.8 Full y= + 0.0244x 2 turns K 0.6 -0.0003x2 0.4 1 turn 0.2 0 -10 0 10 20 30 40 50 -0.2 -0.4 Q (cfs) Figure 5: Flow rate vs. change in height The results from the 3 different valve settings were obtained and are displayed in Table 2. Discussions: For the first part of the experiment we were to generate the calibration curve for the Venturi Flow Meter. We used the relationship between Q and βh to find values for K through this equation: π = πΎπ΄√2πββ Eq. (6) To solve for ten different values of Q to find values of βh. With Q and βh known we were able to plot them in excel and compare the graph obtained to that of the theoretical equation. We found that our graph looked like a square root function which fit our theoretical equation. Even though our graph looks correct we know that there are factors of uncertainties involved while obtaining these values. One of these uncertainties would be the person with the stop watch recording the time required to get to one-hundred pounds with different valve openings. Another uncertainty would be that at one measurement we over filled the reservoir and water poured out of the tube which might have caused a Q higher than the other tests. 10 | P a g e For the second part of the lab we used the values Qmax and Q50% to solve for the DarcyWeisbach friction factors and Reynolds numbers. By using the temperature, length, and given diameter we were then able to calculate for these values. After our calculations were made our values for ‘f’ ended up being higher than the theoretical value for drawn copper tubing which is 0.0015. This could be due to the fact that the pipes in the Scott Fluid Circuit System are aged and differ from the pipes used for the theoretical friction factor. The purpose of the last part of our lab was to see if our data was relevant to theory when dealing with gate valves. In theory the bigger the area the flow is going through the lower the head loss will be. In our situation we first started with a fully open gate valve and we had a max head loss of 8.25 inches. As we reduce the flow, our head loss reduced with it. When we did one rotation on the gate valve we then had a head loss of 38.25 inches which proved that going from a fully open valve to closing it by one rotation reduces the area thus causing velocity to increase producing a higher head loss. We then did two full rotations on the gate valve thus increasing head loss even more, proving our theory that different valve openings cause head loss to vary with respect to velocity. Reference: Fundamentals of Hydraulic Engineering Engineering Fluid mechanics, 10th edition Lab handout http://parra.sdsu.edu/ 11 | P a g e Appendix 1- Raw Data Sheets: Attached 12 | P a g e Appendix 2-Sample Calculation: Attached Graph Paper Flow Rate Q value (cfs) Q value (in^3/s) Qmax 0.034 58.031 0.5*Qmax 0.017 28.885 Table 1: Values for Q, f, Re, and errors Friction Factor 0.025 0.028 +/- Error of Propagation Friction Factor 0.001043 0.001217 Fully Open 2 Turns Open Q (cfs) Head Loss Q (cfs) Head Loss (in) (in) 0.0335 8.25 0.0168 45 0.0259 5.5 0.0141 28 0.0196 3 0.0133 14.25 0.0129 1.5 0.0072 7 0.0063 0.25 0.0032 1.75 Table 2: Results of Gate Valve Characteristics Reynolds Number 4.995*104 2.486*104 +/- Error of Propagation Reynolds Number 1,481 689.2 1 Turn Open Q (cfs) Head Loss (in) 0.00167 38.25 0.00142 29 0.00114 19.5 Excel Calculations 1st Q Max 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2nd Q 1 0.5 Time for 100 lbs 47.72 52.97 61.66 68.38 81.82 95.87 124.22 162.18 255.65 476.43 Elevation Change Left 36.5 10.5 Elevation Change Q (cfs) Q (in^3/s) 58.031 52.279 44.911 40.498 33.845 28.885 22.293 17.075 10.832 5.812 0.034 0.030 0.026 0.023 0.020 0.017 0.013 0.010 0.006 0.003 46 37.5 29 22.5 16.5 11.5 7 4 1.8 0.5 Elevation Change Right 46 11.5 Q (in^3/s) K (actual) 58.031 0.043 28.885 0.040 13 | P a g e 3rd Q 1 0.8 0.6 0.4 0.2 Full Q 2 turns 1 turn Diameter Area Velocity 1 Velocity 2 Viscosity Friction factor 1 Friction factor 2 Reynolds 1 Reynolds 2 k (copper) 0.5 0.4 0.3 0.2 0.1 0 0 0 0.785 0.483982 119.903 59.68258 1.31E-05 Elevation Change Left 8.25 5.5 3 1.5 0.25 Elevation Change Left 45 28 14.25 7 1.75 38.25 29 19.5 in in^2 in/s in/s ft^2/s Elevation Change Right 46 29 16.5 7 1.8 kvalues 0.090 0.088 0.089 0.084 0.103 Elevation Change Right 11.5 7 4 1.8 0.5 0.5 0.25 0.125 0.019 0.019 0.020 0.019 0.019 0 0 0 0.065 ft 9.992 4.974 ft/s ft/s 0.0253 0.0278 49953.2 24864.57 0.0015 14 | P a g e