Bankston 1 Flowmeter Discharge Coefficient Estimation Elizabeth Bankston Team 1 Abstract An Edibon FME18 Flow Meter demonstration system was used to obtain experimental values for this experiment. The data obtained was used to calculate energy loss, volumetric flow rate, the square root of pressure drop, and estimate the discharge coefficient for a venturi flowmeter, orifice flowmeter and a rotameter. The energy loss for a rotameter remained constant at around 0.0023 kj/kg, no matter what the flow rate was. The venturi had a similar theme at around 0.0001 kj/kg. The orifice meter lost more energy as the flow rate increased, with the maximum energy lost being 0.001 kj/kg. For a rotameter, flow rate had no effect on the square root of pressure drop; while increasing flow rate increased the square root of pressure drop for both the venturi meter and the orifice meter. The estimated discharge coefficient for the venturi meter and orifice meter were, respectively, 0.656 and 0.680. For the venturi meter, the estimated discharge coefficient was off by 0.324, which indicated that the system s venturi meter was operating at a lower efficiency than it should have. The large error could also be due to technical errors or human errors. For the orifice meter, the estimated discharge coefficient was off by 0.07, which indicated that the system s orifice meter was operating more efficiently than was expected. Introduction In any piping system, a fluid will travel through an array of pipe fittings, valves, and elbows. When a fluid flows through such components, a pressure drop occurs due to the frictional effects caused by changing the pathway of the fluid. Obstruction flowmeters are devices used to measure fluid flow rate, by obstructing the flow and using a differential pressure transducer to record the pressure before and after the obstruction (Cengel & Cimbala, 2014). Orifice, venturi and nozzle flowmeters are all common types of obstruction flowmeters. An orifice meter is one of the most common flow meters used (Wilhelm et al., 2004) and one of the cheapest obstruction flowmeters (Cengel & Cimbala, 2014). The orifice meter is a plate with a hole, significantly smaller than the inside pipe diameter, which is then placed inside a pipe (Wilhelm et al., 2004). One of the negatives to using an orifice meter is that it causes a sudden change in flow area, which leads to significant head loss or permanent pressure loss (Cengel & Cimbala, 2014). Figure 1 shows the typical orifice flow pattern where P1-P2 indicates significant head loss and P1-P3 indicates the permanent head loss.
Bankston 2 Figure 1: Pressure drop pattern caused by an orifice meter (Cengel & Cimbala, 2010). A rotameter consists of a vertical tapered conical tube with a float inside that is free to move. The float rises and falls as the fluid flows through the tapered tube, so that the float weight, drag force, and buoyancy force balance each other and the net force acting on the float is zero (Cengel & Cimbala, 2014). The flow rate is then determined by comparing the height of the float to the scale on the side of the tube. Figure 2 shows how a rotameter balances out the forces so that the net force equals zero and the scale to determine the flow rate. It is also of note that rotameters have significantly more energy recovery than orifices, due to the more gradual obstruction in a rotameter than in an orifice (Ramirez et al.). Although rotameters have more energy recovery than orifices do, rotameters typically have an accuracy of + 5 percent; which means that rotameters are not appropriate for applications that require precision measurements (Cengel & Cimbala, 2014). Figure 2: Typical operating system of a Rotameter ( Flow: The Basics of Rotameters, 2002).
Bankston 3 A venturi meter is the most accurate obstruction meter but also the most expensive as well (Cengel & Cimbala, 2014). The venturi meter uses gradual contraction and expansion to prevent flow separation and swirling (Cengel & Cimbala, 2014). A venturi meter is also able to recover a high percent of pressure loss (Ramirez et al.), thus making it more efficient than an orifice or a rotameter. Objective The objectives of this experiment were 1) to quantify energy losses due to flow through venturi, orifice and rotameter flowmeters and 2) to estimate the discharge coefficients for venturi and orifice flowmeters. Methods For this experiment, an Edibon FME18 Flow Meter demonstration system was used to collect data regarding fluid flow rate. The system consists of a venturi meter, an orifice meter, a rotameter, eight manometers connected to the system at critical points, as well as a dump valve system and a stop watch. Manometers 1, 2, and 3 connect, respectively, to the inlet, neck, and the outlet of the venturi meter. Manometers 4 and 5 connect to the inlet and outlet of the rotameter. Manometers 6, 7, and 8 connect, respectively, to the inlet, just after the orifice obstruction, and further down-stream of the orifice meter obstruction. The dump valve system and stop watch were used to calculate the flow rate. To turn on the fluid flow and to change the flow rate, two black knobs were turned. It was important to watch the manometer heights when turning the knobs; this was to prevent the manometer heights from rising higher or falling lower than the manometer would allow, which would cause the system to break. From the FME18 Flow Meter demonstration system, manometer heights, rotameter flow rates, and outgoing volumes of water with their corresponding times were recorded. The experiment was completed six times; thus six sets of heights, flow rates, and outgoing volumes with their times were recorded. Using the information above allows pressure at each critical point, flow rate, energy losses, and the square root of pressure drop to be calculated for.
Bankston 4 The manometer heights were converted to meters before being used in equation (1), which calculates pressure at the corresponding manometer location. After calculating the pressure, energy loss was calculated for each flow meter in the system using equation (2); which was derived from the energy equation when Wturbine and Wpump equal zero, V1 and V2 are equal, and z1 and z2 are equal (equation (3)). P = ρ g h (Pa) (1) E Loss = P ρ (kj kg ) (2) P 1 + α V 2 1 + g z ρ 2 1 + W pump = P 2 + α V 2 2 + W ρ 2 turbine + E loss (3) Where: P = Pressure of the fluid at a certain point (Pa), ρ = density of water (kg/m 3 ), g = gravitational acceleration (m/s 2 ), h = manometer height (m) ELoss = energy loss (kj/kg), P = pressure drop and in this case permanent pressure loss (kpa), α = kinetic energy correction factor (unit less), V = fluid velocity (m/s), z = elevation (m), Wturbine = power required to run a turbine, Wpump = power required to run a pump. It is important to take note of two things: 1) pressure should be converted to kilopascals (kpa), so that conversion factor (i) can be used and result in kilojoules per kilogram (kj/kg), and 2) that the pressure drop ( P) is different for a venturi meter, an orifice meter and a rotameter.
Bankston 5 Pressure drop for a venturi meter, an orifice meter, and a rotameter, when calculating energy loss, can be found in table 1. 1 kj = 1 kpa m3 (i) kg Table 1: Pressure drops used for calculating energy loss. Venturi Orifice Rotameter P P1 P3 P6 P8 P4 P5 Energy losses for all three flowmeters was then plotted against the calculated flow rate. Flow rate was calculated using equation (3) and the square root of pressure drop was calculated using equation (4). Q = V t 1m3 1000 L ( m3 s ) (3) Q meter = P (4) Where: Q = volumetric flow rate (m 3 /s), V = volume (L), t = time (s), pressure drop. Qmeter = volumetric flow rate at the flow meter desired, also called square root of It is also important to note that the pressure drop for calculating the square root of pressure drop is different than the pressure drop used to calculate energy loss for the venturi and orifice meters (table 2).
Bankston 6 Table 2: Properties of Venturi and Orifice Flowmeters. P A1 (m 2 ) A2 (m 2 ) Venturi P1 P2 8.04 x 10-4 3.14 x 10-4 Orifice P6 P7 9.62 x 10-4 2.83 x 10-4 The calculated volumetric flow rate was then plotted against the square root of the pressure drop for each meter. After graphing the calculated flow rate versus the square root of pressure drop, equation (5) and table 2 were used to calculate the actual discharge coefficient for a venturi and an orifice meter. Slope = C d (A 2 ) 2 ρ(1 [ A 2 A1 ]2 ) (5) Where: Slope = the slope of the plotted line for either the venturi or orifice, Cd = flow discharge coefficient (unit less), ρ = density of fluid (kg/m 3 ), A2 = area for desired flowmeter taken from table 2, A1 = area for same desired flowmeter taken from table 2. Results & Discussion When Energy loss for the rotameter was plotted against the calculated volumetric flow rate, it was observed that energy loss remained nearly constant no matter what the flow rate was (figure 3). When energy loss for a venturi meter was plotted against the calculated flow rate, it was observed that there was an increase in energy loss, more than there was for the rotameter, but it still showed a near horizontal linear trend overall (figure 3). The rotameter and venturi
Energy losses (E) [kj/kg] Bankston 7 meter showed similar energy losses; however, the orifice meter had the most energy loss as flow rate increased (figure 3). 0.0025 0.002 0.0015 Venturi Rotameter 0.001 Orifice y = 2.3135x R² = 0.8875 0.0005 0 y = 0.2925x R² = 0.8659 0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045 Volumetic flow rate (Q) [m^3/s] Figure 3: Energy losses (kj/kg) versus flow rates for a Venturi, Rotameter, and Orifice flow meter. When calculating for energy loss, the pressure drops used were different than the pressure drops used to calculate the flow rate at a specific flow meter. Calculating energy loss required the permanent pressure loss; while calculating the flow rate at a specific flow meter required the pressure drop across the flowmeter desired (figure 1). This is because just after the fluid moves through the flowmeter, there is a large pressure drop, but as the fluid continues to travel through the pipe some pressure is recovered. The difference between the inlet pressure and the pressure downstream of the outlet is the permanent pressure loss; so when calculating energy loss, the pressure recovered can be ignored, but the immediate pressure drop after an obstruction determines the flow rate at that flowmeter. When the calculated flow rate was plotted against the square root of pressure drop for the rotameter, it was observed that the data created a vertical line (figure 4). This was suggests that the pressure drop remains constant no matter what the flow rate is. Trend lines were created for the venturi and orifice data to determine the slope of the line, which is used to calculate the estimated discharge coefficient (figure 4). Equation (5) and table 2 were used to estimate the
Volumeteric flow rate (Q) [m^3/s] Bankston 8 discharge coefficient for both the venturi meter and the orifice meter, which yielded 0.656 and 0.602 as the respective discharge coefficients (table 3). 0.00045 0.0004 0.00035 0.0003 0.00025 0.0002 0.00015 0.0001 0.00005 0 y = 1E-05x R² = 0.972 y = 9E-06x R² = 0.998 Venturi Orifice Rotameter 0 10 20 30 40 50 60 Square root of pressure drop [Pa] Figure 4: Volumetric flow rate versus the square root of the pressure drop across either the Venturi meter, the Orifice meter or the Rotameter. Table 3: Data comparison of Theoretical and estimated Discharge Coefficients for Venturi and Orifice Flowmeters. Venturi Orifice Discharge Coefficient (Cd) Discharge Coefficient (Cd) Theoretical Cd 0.959 Theoretical Cd 0.602 Estimated Cd 0.656 Estimated Cd 0.680 In table 3 the theoretical discharge coefficient is shown for both the venturi and orifice meters, which are calculated by using either equations (6), (7), and (8) or equations (9), (7), and (8). C d = 0.5959 + 0.0312 β 2.1 0.184 β 8 + 91.71 β2.5 Re 0.75 (unit less) (6)
Bankston 9 Re = ρ V avg L μ (unit less) (7) β = A 2 A 1 (unit less) (8) C d = 0.9975 6.53 β0.5 Re 0.5 (unit less) (9) Where: Cd = Theoretical discharge coefficient (unit less), β = Momentum-flux correction factor (unit less), Re = Reynolds number (unit less), Vavg = average velocity (m/s), L = characteristic length (m), μ = kinematic viscosity (kg/m*s). It was observed that the theoretical discharge coefficient for the venturi meter was higher than the estimated discharge coefficient; while the theoretical discharge coefficient for the orifice was less than the estimated discharge coefficient (table 3). The theoretical discharge coefficient for the venturi was with in 0.021 of the accepted theoretical discharge coefficient value, 0.98 (Cengel & Cimbala, 2014). The theoretical discharge coefficient for the orifice was within 0.008 of the accepted theoretical discharge coefficient value, 0.61 (Cengel & Cimbala, 2014). This means that the venturi meter had more energy losses than it was supposed to, since the estimated discharge coefficient was off by 0.324 from the accepted theoretical discharge coefficient. This could be because of technical errors within the system or human errors when setting up the experiment. Since the orifice estimated discharge coefficient was off by 0.07 from the accepted theoretical discharge coefficient, it means that the energy loss was about as much as was expected.
Bankston 10 Conclusion Using the data collected from a FME18 Flow Meter demonstration system, it was possible to calculate the energy losses and discharge coefficients for a venturi meter and an orifice meter. As flow rate increases, energy loss tends to increase as well, for the orifice and venturi meters; whereas the rotameter tends to have a stable level of energy loss, no matter the flow rate. The estimated discharge coefficient for a venturi meter was much lower than the theoretical discharge coefficient; while the estimated discharge coefficient for an orifice meter was higher than the theoretical discharge coefficient. The lower than expected estimated discharge coefficient indicates that the venturi meter was operating at a lower efficiency than it should have, while the orifice was operating at a higher efficiency than it should have. References Cengel, Y. A., & J. M. Cimbala. Fluid Mechanics: Fundamentals and Applications. McGraw- Hill Companies Inc., 2010. Web. 1 November, 2015. Cengel, Y. A., & J. M. Cimbala. Fluid Mechanics: Fundamentals and Applications. New York: the McGraw-Hill Companies Inc., 2014. Book. 1 November, 2015. Flow: The Basics of Rotameters sensors ONLINE. 2002. Web. 1 November, 2015. http://www.sensorsmag.com/sensors/flow/the-basics-rotameters-1068 Ramirez, B. C., G. D. N. Maia, A. R. Green, D. W. Shike, L. F. Rodríguez, &R. S. Gates. Technical note: DESIGN AND VALIDATION OF A PRECISION ORIFICE METER FOR VENTILATION RATE CONTROL IN OPEN-CIRCUIT RESPIRATION CHAMBERS. TRANSACTIONS OF THE ASABE. N.P. N.D. 57(6): 1865-1872. Web. 1 November 2015. Wilhelm, L. R., D. A. Suter, & G. H. Brusewitz. Fluid flow. Chapter 4 in Food & Process Engineering Technology. N.P. 2004: 65-110. St. Joseph Michigan: ASAE (Rev. Aug. 2005). Web. 1 November 2015.