Mass Conservation and Flow Rate Measurements

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Mechanical Engineering
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JoVE Science Education Mechanical Engineering
Mass Conservation and Flow Rate Measurements

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13:35 min
April 30, 2023

Genel Bakış

Source: Ricardo Mejia-Alvarez and Hussam Hikmat Jabbar, Department of Mechanical Engineering, Michigan State University, East Lansing, MI

The purpose of this experiment is to demonstrate the calibration of a flow passage as a flowmeter using a control volume (CV) formulation [1, 2]. The CV analysis focuses on the macroscopic effect of flow on engineering systems, rather than the detailed description that could be achieved with a detailed differential analysis. These two techniques should be considered complementary approaches, as the CV analysis will give the engineer an initial basis on which route to pursue when designing a flow system. Broadly speaking, a CV analysis will give the engineer an idea of the dominant mass exchange in a system, and should ideally be the initial step to take before pursuing any detailed design or analysis via differential formulation.

The main principle behind the CV formulation for mass conservation is to replace the details of a flow system by a simplified volume enclosed in what is known as the control surface (CS). This concept is imaginary and can be defined freely to cleverly simplify the analysis. For instance, the CS should 'cut' inlet and outlet ports in a direction perpendicular to the dominant velocity. Then, the analysis would consist of finding the balance between the net mass flux through the CS and the rate of change of mass inside the CV. This technique will be demonstrated with the calibration of a smooth contraction as a flowmeter.

İlkeler

Prosedür

1. Setting the facility Make sure that there is no flow in the facility. Verify that the data acquisition system follows the schematic in Figure 1B. Connect the positive port of pressure transducer #1 (see Figure 1B for reference) to the traversing Pitot tube (). Connect the negative port of this same pressure transducer to the static probe of the intake passage (). Hence, the reading of this pressure transducer will be directly (). Record this transducer's conversion factor from Volts to Pascals (). Enter this value in Table 1. Connect the positive port of pressure transducer #2 (see Figure 1B for reference) to the static probe of the intake passage () using a tee. Leave the negative port of pressure transducer #2 open to the atmosphere (). Hence, the reading of this transducer will be directly (). Record this transducer's conversion factor from Volts to Pascals (). Enter this value in Table 1. Set the data acquisition system to sample at a rate of 100 Hz for a total of 500 samples (i.e. 5s of data). Make sure that channel 1 in the data acquisition system corresponds to pressure transducer #1 (). Enter the conversion factor in the data acquisition system to make sure that the pressure measurement () is directly converted to Pascal. Set the Pitot probe at the end of its travel, where it touches the pipe's wall. Since the probe is 2 mm in diameter, the first velocity point is at a radial coordinate 1 mm away from the wall. That is, at a radial position of mm (here, mm). Figure 2. Experimental setting. (A): Flow passage under study. (B): manual traversing system for the Pitot tube. Please click here to view a larger version of this figure. Table 1. Basic parameters for experimental study. Par ameter Value Parameter Value Flow passage radius (Ro) 82.25 mm Transducer #1 calibration constant (m_p1) 136.015944 Pa/V Transducer #2 calibration constant (m_p2) 141.241584 N/V Local atmospheric pressure 100,474.15 Pa Local temperature 297.15 K P_atm-P_2 311.01 Pa 2. Measurements Turn the flow facility on. Record the reading of pressure transducer #2 in Volts from the digital multimeter. Enter this value in Table 1 as and convert the reading from Volts to Pascals using the factor . Use the data acquisition system to record the reading of (). Enter the value of in Table 2. Use the traversing knob to change the radial position of the Pitot tube according to the value suggested in Table 2. Repeat steps 2.4 and 2.6 until Table 2 is fully populated. Change the flow rate by varying the system's discharge. Repeat steps 2.4 to 2.8 for at least ten different flow rates. Turn the flow facility off. Figure 5. Experimental setting. Perforated plates to restrict flow at the discharge of the flow system. Please click here to view a larger version of this figure. Table 2. Representative results. Velocity measurements. r (mm) PT – P2 (Pa) u (r) (m/s r (mm) PT – P2 (Pa) u(r) (m/s) 2.25 300.35 22.34 12.25 302.84 22.43 22.25 305.82 22.54 32.25 302.34 22.41 42.25 294.88 22.13 52.25 295.37 22.15 62.25 292.88 22.06 68.25 293.63 22.09 72.25 294.13 22.10 75.25 299.60 22.31 77.25 293.13 22.07 79.25 284.67 21.75 80.25 256.31 20.63 81.25 198.33 18.15 3. Data Analysis. Determine the velocity profile using the pressure difference values, PT – P2, from Table 2. Enter the results in Table 2. Plot both the pressure and velocity values from Table 2 using the radius,, as the abscissas (Figure 3). Calculate the integral in equation (8) based on the velocity and radius values from Table 2. Calculate the discharge coefficient for each flow rate using equation (8). Plot the discharge coefficient using as the abscissas. Fit a function to the discharge coefficient, a power law is a good choice. Figure 3. Representative results. (A): Example of measurement of static pressure along the radial coordinate of the flow passage. (B): Velocity distribution determined from the measurements of static pressure. Please click here to view a larger version of this figure.

Sonuçlar

For a given restriction of the flow at the fan's discharge, Figure 3A shows the measurements of dynamic pressure () at different radial locations inside the pipe after traversing with the Pitot tube. These values were used to determine the local velocity at those radial locations, and the results are shown in Figure 3B. After using the trapezoidal rule on these data to solve…

Applications and Summary

We demonstrated the application of control volume analysis of conservation of mass to calibrate a flow passage as a flow meter. To this end, we demonstrated the use of a Pitot-static system to determine the flow rate across the flow passage using integration over the velocity profile. Then, the concept of discharge coefficient was incorporated to account for the effect of boundary layer growth near the walls of the flow passage. Based on a set of velocity measurements for different flow rates, we developed a regression t…

Referanslar

  1. White, F. M. Fluid Mechanics, 7th ed., McGraw-Hill, 2009.
  2. Munson, B.R., D.F. Young, T.H. Okiishi. Fundamentals of Fluid Mechanics. 5th ed., Wiley, 2006.
  3. Chapra, S.C. and R.P. Canale. Numerical methods for engineers. Vol. 2. New York: McGraw-Hill, 1998.
  4. Buckingham, E. Note on contraction coefficients of jets of gas. Journal of Research, 6:765-775, 1931.
  5. Lienhard V, J.H. and J.H. Lienhard IV. Velocity coefficients for free jets from sharp-edged orifices. ASME Journal of Fluids Engineering, 106:13-17, 1984.

DEŞİFRE METNİ

1. Setting the facility Make sure that there is no flow in the facility. Verify that the data acquisition system follows the schematic in Figure 1B. Connect the positive port of pressure transducer #1 (see Figure 1B for reference) to the traversing Pitot tube (). Connect the negative port of this same pressure transducer to the static probe of the intake passage (). Hence, the reading of this pressure transducer will be directly (). Record this transducer's conversion factor from Volts to Pascals (). Enter this value in Table 1. Connect the positive port of pressure transducer #2 (see Figure 1B for reference) to the static probe of the intake passage () using a tee. Leave the negative port of pressure transducer #2 open to the atmosphere (). Hence, the reading of this transducer will be directly (). Record this transducer's conversion factor from Volts to Pascals (). Enter this value in Table 1. Set the data acquisition system to sample at a rate of 100 Hz for a total of 500 samples (i.e. 5s of data). Make sure that channel 1 in the data acquisition system corresponds to pressure transducer #1 (). Enter the conversion factor in the data acquisition system to make sure that the pressure measurement () is directly converted to Pascal. Set the Pitot probe at the end of its travel, where it touches the pipe's wall. Since the probe is 2 mm in diameter, the first velocity point is at a radial coordinate 1 mm away from the wall. That is, at a radial position of mm (here, mm). Figure 2. Experimental setting. (A): Flow passage under study. (B): manual traversing system for the Pitot tube. Please click here to view a larger version of this figure. Table 1. Basic parameters for experimental study. Par ameter Value Parameter Value Flow passage radius (Ro) 82.25 mm Transducer #1 calibration constant (m_p1) 136.015944 Pa/V Transducer #2 calibration constant (m_p2) 141.241584 N/V Local atmospheric pressure 100,474.15 Pa Local temperature 297.15 K P_atm-P_2 311.01 Pa 2. Measurements Turn the flow facility on. Record the reading of pressure transducer #2 in Volts from the digital multimeter. Enter this value in Table 1 as and convert the reading from Volts to Pascals using the factor . Use the data acquisition system to record the reading of (). Enter the value of in Table 2. Use the traversing knob to change the radial position of the Pitot tube according to the value suggested in Table 2. Repeat steps 2.4 and 2.6 until Table 2 is fully populated. Change the flow rate by varying the system's discharge. Repeat steps 2.4 to 2.8 for at least ten different flow rates. Turn the flow facility off. Figure 5. Experimental setting. Perforated plates to restrict flow at the discharge of the flow system. Please click here to view a larger version of this figure. Table 2. Representative results. Velocity measurements. r (mm) PT – P2 (Pa) u (r) (m/s r (mm) PT – P2 (Pa) u(r) (m/s) 2.25 300.35 22.34 12.25 302.84 22.43 22.25 305.82 22.54 32.25 302.34 22.41 42.25 294.88 22.13 52.25 295.37 22.15 62.25 292.88 22.06 68.25 293.63 22.09 72.25 294.13 22.10 75.25 299.60 22.31 77.25 293.13 22.07 79.25 284.67 21.75 80.25 256.31 20.63 81.25 198.33 18.15 3. Data Analysis. Determine the velocity profile using the pressure difference values, PT – P2, from Table 2. Enter the results in Table 2. Plot both the pressure and velocity values from Table 2 using the radius,, as the abscissas (Figure 3). Calculate the integral in equation (8) based on the velocity and radius values from Table 2. Calculate the discharge coefficient for each flow rate using equation (8). Plot the discharge coefficient using as the abscissas. Fit a function to the discharge coefficient, a power law is a good choice. Figure 3. Representative results. (A): Example of measurement of static pressure along the radial coordinate of the flow passage. (B): Velocity distribution determined from the measurements of static pressure. Please click here to view a larger version of this figure.