Overview
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.
Principles
A control volume (CV) is defined by an imaginary closed surface, dubbed the control surface (CS), defined arbitrarily to study the balance of mass in a system. Figure 1A shows an example of a control volume containing a region of flow going through a flow passage. The details of flow in the passage are irrelevant since we are merely interested in obtaining measures of the mass inflow, outflow, and its rate of change inside the flow passage. All these effects can be summarized in the equation for conservation of mass in integral form [1, 2]:
(1)
The first term on the right-hand side of equation (1) represents the rate of change of mass inside the control volume, while the second term represents the net flux of mass through the control surface. The vector difference 
is the relative velocity between the CV and the flow, and the vector 
is the unit outward normal to the area differential. The dot product between the relative velocity and represents the velocity component that crosses the CS, and henceforth contributes to the exchange of mass. The sign of this dot product is negative where the mass flux is directed into the CV and positive where it is directed away from the CV.
Figure 1. Schematic of basic configuration. (A) Smooth intake for a centrifugal fan. The control volume is defined as the inside profile of the passage. The solid walls are excluded from the control volume but their boundary conditions are kept in the control surface (i.e. no penetration and no slip). Port 1 is defined as the entrance side of the passage, while port 2 is defined as the cross-sectional plane that coincides with the tip of the Pitot tube. Flow goes from left to right. (B) Pitot-static system and schematic of data acquisition system. Please click here to view a larger version of this figure.
For the present demonstration, we have the configurations shown in Figure 1A, where a fixed CV follows the contour of as smooth contraction at the intake of a centrifugal fan. The flow through this CV is steady, thus the rate of change of mass inside the control volume is zero. Hence, the first term on the right-hand side of equation (1) vanishes. Also, the CV is attached to the contraction, which is fixed in space and has no velocity, making 
. Therefore, the net flux of mass through this CV is zero and equation (1) simplifies to:
(2)
Considering the configuration in Figure 1A, mass flows into the CV through port 1 and leaves the CV through port 2. Consequently, the surface integral on the right-hand side of equation (2) can be divided into two independent integrals, one for each port. The sign of the dot product is negative in port 1 because the flow goes toward the CV, and positive in port 2 because the flow goes away from the CV. Without assuming that the velocity is homogeneously distributed in either port, let us make 
and 
the respective velocity profiles taking into account that both are what remain after taking the dot product. That is, the magnitude of the velocity component parallel to the area vector, . Finally, pressure changes along the contraction are not significant enough to induce observable changes in density. Henceforth, we can consider the density as constant. Under these circumstances, equation (2) would simplify to:
(3)
Note that, since mass is conserved, the mass flux, 
, is the same through both ports. Given the structure of these relations, each integral in equation (3) expresses the volumetric flow rate,
, through its corresponding port, and this fact helps to define the average velocity,
, for a given port:
(4)
Under inviscid conditions, the velocity at port 2 could be expressed in terms of the conditions outside of the intake using Bernoulli's equation along the central streamline (see Figure 1A for reference):
(5)
Here, the effect of height vanishes on the central streamline because it is horizontal, and is negligible in the other streamlines because the fluid is air, which has a very small specific weight. Also, the initial point on the central streamline is sufficiently far away from the inlet that its velocity is zero. Given that equation (5) is for the idealized inviscid case, this value of velocity will be the same all across port 2. In reality, boundary layer growth affects the velocity profile and makes it non-homogeneous. To account for this effect, the ideal estimation is compared with experimental measurements via the "Discharge Coefficient". This coefficient is defined as the ratio between the measured average velocity and the inviscid velocity for a given cross section of the flow:
(6)
The discharge coefficient, 
, depends on the geometry and the Reynolds number. Once determined, could be used in conjunction with equations (4) and (5) to determine the flow rate across port 2 based on its cross-sectional area and an easy-to-measure pressure differential:
(7)
When putting equations (4), (5), and (6) together, and considering that port 2 is circular, we obtain the following relationship for :
(8)
It is clear from equation (8) that knowledge of the velocity profile is necessary to obtain the discharge coefficient. To this end, we will use velocimetry by Pitot - static probes. As shown in Figure 1B, the Pitot tube brings the flow to a stop sensing the total pressure,
, which is the addition of the static and dynamic pressures at a given point. On the other hand, the static probe at the wall senses the static pressure alone. From Bernoulli's equation applied at a given radial position, the total pressure is just Bernoulli's constant. At port 2, this principle can be expressed by the following relation at an arbitrary radial position:
(9)
Here, we are neglecting the effect of vertical position because our flow passage is horizontal. In summary, one obtains the following relationship for the magnitude of the velocity at a given position 'r' within the pipe:
(10)
The pressure difference 
is directly measured by the pressure transducer depicted in Figure 1B, and the velocity profile is obtained by traversing the Pitot tube along the radial coordinate of the pipe. Note that these velocity measurements are carried out at discrete positions, hence, these data points should be used to solve the integral in equation (8) numerically using either the trapezoidal or the Simpson's rule [1]. Once the value of this integral is obtained, it should be plugged into equation (8) together with the measured value of , the density, and the radius of the duct, to obtain the value of for that particular flow condition. Upon repeating this experiment for different flow conditions, we will obtain a scatter plot that could be used to determine a relationship between and . This relationship can then be substituted in equation (7) to fully determine the flow rate,, as a function of solely .
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Procedure
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).
). Hence, the reading of this pressure transducer will be directly (
). Enter this value in Table 1.
). Hence, the reading of this transducer will be directly (
). Enter this value in Table 1.
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.
, as the abscissas (Figure 3).
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.
Conservation of mass is a well-known physical principle that is used together with the control volume approach for engineering of many mechanical systems. Control volume analysis of mass conservation is particularly useful to estimate flow rates for large-scale hydraulic structures, such as dams, water treatment plants, or water distribution systems. This method is usually applied as an initial step to give the engineer an idea of the dominant mass exchange in a system. An outlet valve, for example, constructed through the face of the dam is routinely used to control the flow of water. Since mass is conserved, the net mass flux through a control surface, and the rate of change of mass inside the enclosed volume must be in balance. This video will illustrate how to apply the control volume method for mass conservation to calibrate a smooth contraction as a flow meter.
General principles of the control volume method for conservation of linear momentum were discussed in our previous video. Here, we illustrate this approach for conservation of mass. Consider the flow passage in the schematic, consisting of a centrifugal fan with the smooth contraction at the intake. How does the control volume analysis for conservation of mass apply to our system? First, let's take an imaginary closed surface, called Control Surface, to define a control volume containing a region of flow. Next, let's write the general equation for conservation of mass. The first term of the equation represents the rate of change of mass inside the control volume. This term is zero in our case, because the flow through our control volume is steady. Since the control volume is attached to the contraction, the second term of the equation simplifies. This is the net flux of mass through the Control Surface. For our system, the mass flows into the control volume through port one, and leaves the volume through port two. Assuming a constant fluid density along the contraction, and solving the dot product between the fluid velocity and the unit area vector, the equation simplifies further. Since mass is conserved, the mass flux is the same through both ports. Next, knowing the mass flux and the volumetric flow rate through a given port, the average velocity for the port can be obtained. For inviscid fluids, the velocity at port two is constant across the section of the port. This velocity can be calculated using the Bernoulli's equation along the central streamline. If you need to review the Bernoulli's equation, you can watch out previous video. The fluid pressure at port one is the atmospheric pressure. We also assume that the fluid velocity at port one is zero, since it is close enough to the external, quiescent environment. Then the velocity at port two for an inviscid fluid is given by this formula. The velocity profile is nonhomogeneous. In reality, due to boundary layer growth, when a fluid flows near a solid wall, the fluid in contact with the boundary assumes the velocity of the wall. As the distance from the wall increases, the flow velocity gradually recovers until reaching the velocity of the free stream. This region of velocity change near a wall is called the Boundary Layer, and takes place because of the action of viscosity. To account for this effect, the ideal estimation is compared with experimental measurements using the discharge coefficient. For a circular port, like the one used in our example, this coefficient can be calculated if we know the radial velocity profile across the flow passage. The velocity profile can be measured using a Pitot-static probe. If you need to review the working principle of a Pitot-static probe, you can watch our previous video. A Pitot tube brings the flow to a stop, sensing the total pressure, which at any point inside the fluid has two components: A static component, and a dynamic component. The static probe at the wall senses only the static pressure. Applying Bernoulli's equation at port two, the velocity at a given position, r, within the pipe can be determined. The velocity profile is obtained by traversing the Pitot tube along the radial coordinate of the pipe, and by measuring with the pressure transducer, the pressure difference. Finally, the flow rate across port two can be determined using the discharge coefficient together with the passage's cross-sectional area and the pressure difference measured with a second transducer. Now that you understand how to use the control volume method for mass conservation to analyze a flow system, let's apply this method to calibrate a flow passage and to determine its discharge coefficient.
Before starting the experiment, familiarize yourself with the layout of the lab and the equipment inside the facility. First, make sure that there is no flow inside the facility by checking the main switch. Then check that the jet's lid is covered. Now start setting up the data acquisition system by following the diagram described in the 'Principles' section. Connect the positive port of the first pressure transducer to the traversing Pitot tube. Connect the negative port of the transducer to the static probe of the intake passage. Hence, the reading of this pressure transducer will give you, directly, the pressure difference PT - P2. Record the transducer's conversion from volts to pascals. Next, connect the positive port of the second pressure transducer to the using Pitot tube using a T connection. Leave the negative port of the transducer open to the atmosphere. The reading of this pressure transducer will give you the pressure difference. Record the transducer's conversion factor from volts to pascals. Measure the flow passage radius with a ruler. Also collect the data for the atmospheric pressure and the temperature at your location from the National Weather Service website. Record these values in a parameters table together with the values for the conversion factors of the two pressure transducers. Now, set the data acquisition system to sample at a rate of 100 Hertz, for a total of 500 samples in order to obtain five seconds of data. Make sure the channel zero in the data acquisition system corresponds to the first pressure transducer. Then enter the conversion factor in the system to have the pressure readings directly in pascal. Now, enter the conversion factor for the second pressure transducer, and ensure that this pressure transducer corresponds to channel one in the data acquisition system. Set the Pitot probe at the end of its travel, where it touches the pipe's wall. Since the probe is two millimeters in diameter, the first measurement will be performed at a radial coordinate one millimeter away from the wall.
After the data acquisition system is set up, turn on the flow facility. Now you are ready to start data acquisition. Record the reading of the second pressure transducer using a digital multimeter. Convert this value from volts to pressure units using the conversion factor, and record it in the parameters table. For the current position of the Pitot tube, use the data acquisition system to record the pressure difference given by the first transducer. Record this value in the results table. Change the radial position of the Pitot tube using the traversing knob. Measure the pressure difference at this position inside the flow passage with the data acquisition system. Repeat this step for difference radial positions across the flow passage, and record the readings in the results table. Next, change the flow rate inside the passage by varying the system's discharge. To this end, plates with perforations of different diameters are placed at the discharge of the system to restrict the flow at different levels. Measure the pressure difference for different radial positions inside the flow passage, and repeat this step for at least ten different values of the flow rate. At the end of your experiment, remember to turn the flow facility off.
At each position, r, of the Pitot tube across the diameter of the flow passage, you have a measurement of the difference between total pressure and the static pressure. For each data point, calculate the flow velocity and enter its value in the results table. Repeat for all data points in the table, and then plot the velocity profile across the pipe. Now, calculate the discharge coefficient. To do so, first you need to plot the product between velocity and radius in function of radius. Since the velocity measurements are carried out at discrete positions, the integral in the formula for the discharge coefficient must be solved numerically using, for example, the trapezoidal rule. Next, calculate the discharge coefficient using the value of the integral together with the values recorded in the parameter table for the fluid density, the passage radius, and the measured difference between the atmospheric pressure and the static pressure at port two. Repeat these calculations for each set of data corresponding to every experimental value of the flow rate inside the passage. Now, take a look at your results.
Make a scatter plot of the discharge coefficients for different flow rates versus the values of the square root of one minus the pressure ratio. Fit a power-law function to the scatter plot, and determine a general relationship between the discharge coefficient, and the ratio between the static pressure at the flow passage and the local atmospheric pressure. Next, substitute this relationship in the equation for the flow rate. Here the density can be expressed for convenience in terms of atmospheric pressure and absolute temperature using the ideal gas law. This expression of the flow rate was, thus, developed to keep its validity under changes in local atmospheric conditions, passage size, and unit system. In summary, to calibrate a passage as a flow meter, it is necessary to establish a relationship between the flow rate and an easy-to-measure variable such as pressure difference.
The control volume method for mass conservation has a wide range of applications across the field of mechanical engineering. A Venturi tube is a device used in confined flows to determine flow rate based on pressure changes between two different sections of the passage. The method presented in this video can be used to correct for boundary layer effects inside the Venturi tube, and determine the device's discharge coefficient. The control volume analysis for mass conservation can be used to assess flow rate for large-scale hydraulic systems by comparing the depth of flow before and after the flow restrictions.
You've just watched Jove's Introduction to Control Volume Analysis for Mass Conservation. You should now understand how to apply this method to measure the flow rate across a flow passage and determine the discharge coefficient of the system. Thanks for watching.
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Results
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 equation (4) for the average velocity, we obtained a value of
m/s. On the other hand, the value of 
from Table 1 was used to determine the ideal velocity from equation (5): 
m/s. Hence, the discharge coefficient for this flow condition is: 
. This value is shown in Figure 4 as a red triangle.
After repeating this experiment twenty-nine more times, we obtained the scatter plot shown in Figure 4. This data can be well represented by a power law of
:
(11)
The reason for this choice of argument is to make sure that the leading constant stays dimensionless, and hence this correlation would still be valid regardless of the system of units used for the pressure. This function can be substituted in equation (7) to obtain the flow rate as a function of
:
(12)
Here, all the constants of equations (7) and (11) were lumped into a single dimensionless constant:
. Consequently, equation (12) is valid for any system of units as long as the variables are consistently assigned the corresponding units. For convenience, the density from equation (7) was expressed in terms of the atmospheric pressure and absolute temperature using the ideal gas law. Equation (12) is valid for different atmospheric conditions as it accounts for changes in local pressure and temperature (T and Patm). Also, as long as geometric similarity is conserved, this equation would be valid for passages of different sizes as accounted by the radius R.
Figure 4. Representative results.
: Discharge coefficients determined at different flow rates. 
: Discharge coefficient determined with the velocity measurements demonstrated herein. - : Power law fitted to the experimental data.
Figure 6. Representative results. Plot of the product between velocity and radius.
Table 3. Representative results. Discharge coefficient.
| √(1-P2/Patm ) | Cd |
| 0.019 | 0.735 |
| 0.020 | 0.761 |
| 0.025 | 0.795 |
| 0.026 | 0.808 |
| 0.029 | 0.826 |
| 0.032 | 0.835 |
| 0.039 | 0.855 |
| 0.041 | 0.862 |
| 0.042 | 0.873 |
| 0.044 | 0.880 |
| 0.047 | 0.891 |
| 0.049 | 0.899 |
| 0.049 | 0.917 |
| 0.050 | 0.924 |
| 0.050 | 0.903 |
| 0.051 | 0.909 |
| 0.052 | 0.927 |
| 0.053 | 0.917 |
| 0.054 | 0.926 |
| 0.054 | 0.935 |
| 0.055 | 0.924 |
| 0.056 | 0.940 |
| 0.060 | 0.953 |
| 0.063 | 0.967 |
| 0.064 | 0.972 |
| 0.065 | 0.975 |
| 0.066 | 0.977 |
| 0.067 | 0.983 |
| 0.069 | 0.985 |
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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 that expresses the discharge coefficient as a function of the ratio between the static pressure at the flow passage and the local atmospheric pressure,. Finally, this regression was incorporated into an equation for the flow rate through the passage as a function of. This equation was developed to keeps its validity under changes in local atmospheric conditions, passage size, and unit system.
Control volume analysis for mass conservation offers many alternatives to calibrate flow passages as flowmeters. For example, perforated plates, nozzles, and venturi tubes are used in confined flows to determine flow rate based on pressure changes between two different sections of the passage. And similar to our example, these devices need to be characterized with a discharge coefficient that corrects for boundary layer effects.
In flow through open channels, control volume analysis for mass conservation can also be used to assess flow rate by comparing the depth of flow before and after flow restrictions such as spillways, partially open gates, or cross-section reductions. The main significance of these applications is that hydraulic structures for water distribution, control, and treatment are of very large scales that would preclude the use of other flow devices.
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References
- White, F. M. Fluid Mechanics, 7th ed., McGraw-Hill, 2009.
- Munson, B.R., D.F. Young, T.H. Okiishi. Fundamentals of Fluid Mechanics. 5th ed., Wiley, 2006.
- Chapra, S.C. and R.P. Canale. Numerical methods for engineers. Vol. 2. New York: McGraw-Hill, 1998.
- Buckingham, E. Note on contraction coefficients of jets of gas. Journal of Research, 6:765-775, 1931.
- 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.
