Table of Contents Flow Measurement Flow Measurement... 1 I. Objective... 1 II. Apparatus... 1 III. Principles and Background... 1 Pitot-Static Tubes... 2 Orifice Plates and Unrecoverable Losses... 4 Flow Development... 5 Environmental Effects... 5 IV. Procedure... 6 Velocity Traverse and Differential Pressure Measurement... 6 Recoverable and Non-Recoverable Pressure Drop Measurement... 9 V. Required Data Analysis... 10 VI. References... 11 I. Objective The object of this experiment is to study the performance of an orifice plate flow measurement device mounted in a circular duct. In the first part, of the lab experiment, the orifice plate will be used to determine the volumetric flow through the duct. A series of measurements will also be taken using Pitot-static probes. In the second part of the lab experiment, the recoverable and the non-recoverable pressure drop through the duct will be examined. II. Apparatus 1. a 6 5/8 inch inside diameter clear plastic air duct with fan, orifice flanges, and air straightener; 2. Dwyer 1/8 th inch diameter Pitot-static probes mounted in a quill with a 12 inch Starret scale; 3. Several capacitance-based pressure gauges with digital readouts; 4. an orifice plate with a 3.033 inch diameter bore ( = d/d = 3.033/6.625 = 0.458); 5. a twelve inch ruler; 6. a protractor 7. a relative humidity gauge, an aneroid barometer and thermometer to measure ambient conditions. III. Principles and Background Flow measurements must be made in chemical plants, refineries, power plants, and any other place where the quality of the product or performance of the plant depends on having a precise flow rate. Flow measurements also enter into our everyday lives in the metering of water and natural gas into our homes and gasoline into our cars. In this experiment we will measure the flow of air in a duct. An orifice plate, shown in Figures 1a and 1b, will be used to directly measure the volumetric flow rate. We will also measure the flow velocity field using a Pitot-static probe, also shown in Figures 1a and 1b. Velocity readings Flow Measurement Page 1 of 18
will be taken across the pipe at different radii, and the volumetric flow rate will be calculated from integrating these readings over the pipe cross-sectional area. The flow of fluid in a duct is governed by the conservation equations: conservation of mass, conservation of momentum and conservation of energy. Because the flow in our duct is effectively isothermal, we'll neglect the energy equation for now. Conservation of mass for a control volume with steady-state flow says that mass flow in equals mass flow out. [ v A ] IN = [ v A ] OUT where is the fluid density, v is the average velocity and A is the duct cross-sectional area. For the isothermal case with nearly constant density (only very small pressure changes allowed, or will change according to the ideal gas law), the volumetric flow rate Q = va must be constant along the duct. The momentum equation tells us what happens to pressure along the duct. For the case of steadystate, inviscid (no wall friction) flow along a continuous streamline in a constant density medium, the Bernoulli equation conserves momentum. 2 P1 v1 P2 v2 + + g h 1= + + g h 2 2 2 2 = constant where P is the pressure, g is gravity and h is the fluid elevation at arbitrary points 1,2 along the flow streamline. The difference in pressure between the points is called the recoverable pressure difference because we can get the original pressure back by simply restoring the original velocity and elevation. Any viscous losses, like friction, cannot be predicted with the Bernoulli equation - these are unrecoverable, irreversible losses. Pitot-Static Tubes Recoverable pressure differences can be used to measure fluid velocity. The measurement of velocity by a Pitot-static probe is based on the stagnation of the momentum of fluid in the moving stream to a zero-velocity pressure force at the Pitot-static probeinlet, a relationship that can be derived from the Bernoulli equation when v 1 = v and v 2 (at the probe entrance) goes to zero: 2 v pstagnation pstatic pdynamic fluid 2 where P stagnation is the total pressure at the forward facing inlet to the Pitot-static probe where the velocity becomes zero, P static is the static pressure along the sides of the Pitot-static probe where the velocity is unchanged from the upstream duct velocity v. The pressure difference, P, is called the dynamic pressure because it is related to the change in fluid velocity. We can calculate the duct velocity from the dynamic pressure as, v= P 2 air Note that this expression is only accurate if the P-S tube points directly into v 1 such that all of v 1 is stagnated. If the P-S tube is misaligned, the measured velocity will be too low. Flow Measurement Page 2 of 18
To obtain an estimate of the volumetric flow in the duct from a series of pitot-static tube velocity measurements, one must integrate the velocity over the duct area. Q= vavg A= v A da There are a number of different methods for approximating the above integral. The simplest method is to divide the duct cross-section into a number of equal area sectors, and measure the "average" velocity at the center of each sectors. We can then estimate the velocity by calculating the sum: Numsectors Numsectors v Q = * Numsectors i 1 i vi Ai Apipe* Apipe vavg i 1 The above method only works if the positions of the velocity measurements are carefully chosen. Figure 4 shows how to split the pipe into 6, 12 or 24 equal area sectors. The specific radial positions are given in the figure. The dynamic pressure, P, can be measured using capacitance-based differential pressure (DP) cells or manometers. A manometer relates the pressure difference to the difference in height of two columns of liquid supported by the respective pressures. The equations of hydrostatics tell us that if a manometer is connected to a Pitot-static tube the dynamic pressure will be given by P = g h, where is the manometer fluid density and h is the difference in height of the fluid columns. The DP cells convert pressure force acting over the surface area of a plate to a movement of the plate to a varying electrical capacitance, which may be displayed or digitally acquired. The gages are calibrated in "inches-of-water", an antiquated but common pressure unit which corresponds to the pressure exerted by a one-inch vertical displacement of water at standard conditions. It is easy to imagine how experimenters, using water-filled manometers, chose this as a unit of pressure measurement. We can convert units of "inches-of-water" to Pascals by the following conversion: kpa = 4.019 "Inches-of-water" The dimensions of the Pitot-static probe can be important in assuring that the probe gives an accurate measure of the velocity. The diameter of the Dwyer Pitot-static probe is 1/8th of an inch. To minimize the blockage effects of the Pitot-static probe on the measured flow, the manufacturer recommends that this tube be used in ducts with an inside diameter of three inches or more. This ensures that the blockage of the probe does not significantly change the duct velocity at the probe static ports, causing an error in static pressure measurement. The length of the axial tip of the Pitot-static probe is also critical. In this tube the side ports used to sense the static pressure in the flowing air are five probe diameters from the end of the tube. This requires a smooth end design to prevent disturbances from this leading edge from altering the static pressure to be measured by the side ports. In most Pitot-static probe designs, a minimum distance of eight diameters is recommended to remove this effect. The bend in the tube is also a minimum distance from these side ports so that minimal interference will occur. In this tube, the bend is eight diameters behind the side ports. A shorter distance could produce a higher static pressure reading than is present in the stream. Flow Measurement Page 3 of 18
Textbook descriptions of Pitot-static probes usually describe their use in a laminar flow. What happens when Pitot-static probes are used in time-varying turbulent flows? The pressure difference associated with the fluctuation velocity must move a mass in the pressure sensor to measure the pressure change associated with a given velocity change. The measurement devices are thus second-order mechanical systems with their own natural frequency and damping ratio. If the frequency of the velocity fluctuation is much faster than the natural frequency of the measuring system, then it will display the average value of the fluctuating signal. This will only hold true for moderately turbulent flows (less than 10% turbulence intensity) because the velocity vector must remain approximately parallel to the Pitot-static probe. Duct flows typically have low enough turbulence intensities that the effect of turbulence can be neglected, but disturbed regions of flow near sharp edges or area changes can prevent good readings. Orifice Plates and Unrecoverable Losses Unlike the pitot tube, which uses local recoverable pressure to find velocity at points in the duct, many processes apply obstruction flow meters to measure volumetric flow rate for the entire duct. Obstruction flow meters effectively block part of the duct area, causing an increase in velocity and therefore a change in recoverable pressure according to the Bernoulli equation. Volumetric flow is evaluated by measuring the pressure difference between the upstream and downstream sides of the obstruction, which is an orifice in our experiment. If we try to use the Bernoulli equation here, however, we will be disappointed. The flow through an orifice is not inviscid and the pressure difference is only partially recoverable. Downstream of the orifice flow separation occurs, creating recirculating eddies that affect the downstream pressure. We need a different equation to account for these unrecoverable losses. A general equation for unrecoverable pressure drop is P unrecoverable = f L D v ( 2 2 )+ k v ( 2 2 r ) where f is an empirical term called a friction factor that accounts for wall friction losses over a duct of length L and diameter D, and k is a term called a form loss coefficient that accounts for losses caused by a change in duct configuration like the orifice plate. The velocity v r is calculated at the smallest area where the form loss occurs, the orifice diameter in this case. Both k and f depend on a characteristic called the Reynolds number, Re = v D where is the fluid viscosity. Reynolds number is an important scaling parameter for fluid flows. It is often used to predict, whether flow is laminar, with Re less than about 5000, or turbulent when Re is greater than about 5000. The friction factor can be evaluated using a table like that shown in Figure 5 given Reynolds number. The form loss must be empirically estimated for specific objects. Because orifice plates are used often for flow measurement, engineers have, over time, developed very detailed instructions, called standards, on how to make plates that give repeatable results. If these instructions are followed, as they are for the orifice in this experiment, the volumetric flow is given by the orifice flow equation, Flow Measurement Page 4 of 18
Q= K o A o 2 P air where Q is the volumetric flow rate of air, A o is the orifice cross-sectional area and K o is the orifice flow coefficient. Note that this is nearly the inverse of the unrecoverable pressure drop equation given before, and K o is related to but not the same as k. The orifice flow coefficient is a function of the ratio of the orifice diameter to the duct diameter, = d/d, and the Reynolds number for flow in the duct. A graph of values for K o for different Reynolds numbers is shown in Figure 2a. This figure is for square-edged orifices with flange taps that are spaced one inch in front of and one inch behind the orifice plate. The Reynolds number, Re d1, is based on the duct diameter. Unlike the earlier unrecoverable pressure drop equation, this equation accounts for both recoverable and unrecoverable effects. Alternatively, the following equation can be used for determining volumetric flow rate and follows a more generalized form. Discharge coefficient can be determined from Figure 2b. Q C A d o 2 p / 1 air 4 Flow Development Whenever the velocity profile in a duct is perturbed, it will eventually recover to a steady profile as it traverses the duct. This is called flow development. When the duct flow goes through the orifice, it forms a high velocity jet downstream. The pressure in this jet is lower than the upstream pressure, because of unrecoverable viscous effects (recirculating eddies) and recoverable effects (increased velocity). The duct diameter is the same, both upstream and downstream of the orifice, so we expect that the jet downstream of the orifice will eventually expand. After some distance, the velocity profile in the duct will look just like the upstream profile. At this point, the recoverable component of pressure drop will have recovered, because velocity is restored. As the jet is expanding, the flow is called a developing flow. In this region the profile is changing along the duct and there is a radial velocity component. It can be difficult to take measurements in developing flows. Once the velocity profile has stabilized, and no longer changes with distance along the duct, the flow is fully developed. Environmental Effects The accuracies of both the Pitot-static probe velocity measurement and the orifice flow measurement are directly related to the accuracy with which the density of the fluid in the duct is known. Since air can be treated as an ideal gas at atmospheric pressures, its density is directly proportional to its pressure and inversely proportional to its temperature as defined by the ideal gas law. The ideal gas law states that: p RT or p v RT where p is the gas pressure; v is the gas specific volume, which is the reciprocal of the gas density ; R is the gas constant for air; and T is the gas temperature. Both the pressure and temperature are required in absolute scales. Thus, the density of dry air is: Flow Measurement Page 5 of 18
pair air Rair Tair The pressure of the air in the laboratory will be measured with a barometer, while temperature is measured with a thermometer. We are also at a latitude of 41 degrees, so the acceleration due to gravity in Akron is approximately 9.79 m/s 2. A further complication is the slight effect of humidity on the density of air. Water vapor is less dense than air, so humid air is less dense than dry air as represented by the ideal gas equation. We can account for this by finding the mass ratio,, of water vapor mass to dry air mass in the air and then correcting for the difference in the gas constant R, which is 0.4615 kj / kg K for water vapor compared to 0.2870 kj / kg K for dry air as given by humid air 1 + = dry air [ ] = dry air Rvapor 1 + ( ) Rair 1 + [ ] 1 + 1.608 One can determine the mass ratio,, from the psychrometric chart (Figure 3) as a function of the temperature of the air and the relative humidity,, which is the ratio of the vapor pressure of the partially saturated humid air to the vapor pressure of fully saturated air at the given temperature. IV. Procedure The experiment will be conducted in two parts. In the first part, flow rate measurements will be made using both the orifice and Pitot-static probe traverses and the results will be compared. Here we will see the development of the flow downstream of the orifice. In the second part, the static pressure port of the Pitot-static probe will be used to study the recoverable and unrecoverable components of static pressure drop across the orifice plate and along the duct. Differential pressure readings are to be taken across the orifice taps and the pitot-static tube ports with the DP cells. Using these pressure readings and the dimensions of the duct and orifice, calculation of flow through the duct will be possible. The room temperature, barometric pressure and relative humidity will be measured so that accurate estimates of the density of the air in the duct may be made for the velocity calculations. Velocity Traverse and Differential Pressure Measurement 1. Build a VI to measure two channels of data. a. Set up the DAQ Assist to read two voltage channels, AI0 and AI1. Set voltage range to -10 10 V and continuously collect 1000 samples at 1000 Hz. b. Split the signal into the two channels using the Split Signals command. Express Sig Manip Split Signals. Drag down on the icon to show both outputs. c. Add a statistics command to both channels. Mathematics Prob & Stat S.D. & Variance. d. Add numeric indicators to show the mean and standard deviation of the signals. Flow Measurement Page 6 of 18
2. a. Make sure that the fan and the duct sections are assembled together without gaps or leaks. The orifice plate should be installed with the sharp edge facing UPSTREAM and the chamfered edge facing DOWNSTREAM. Record the orifice parameters from the tag on the plate. The flow straightener should be installed at the fan end of the duct. Be sure that all access ports other than the one to be used at the moment are sealed. Make sure that the duct is fully opened by removing the plate at the end of the duct. b. Measure the inside diameter of the duct -average several angles. c. Connect the high-pressure hose of a 5" DP cell to the flange tap at the UPSTREAM side of the orifice and the low-pressure hose to the flange tap on the DOWNSTREAM side of the orifice. Verify that the DP cell is also connected to the data acquisition board. Zero the DP cell readout with the TARE control. 3. Record the temperature, barometric pressure and relative humidity from the weather station in the laboratory. These data will be used to determine the air densities for the orifice flow calculation and the Pitot-static probe velocity calculations. 4 a. Calibrate the data acquisition system to be certain the DP cell and the data acquisition system agree. With no flow in the duct, take 10000 samples at 1000 samples/sec. Record the mean - it should be close to zero. If not, record the bias under noflow conditions. b. Next, turn the fan on and read the pressure difference on the DP cell display. It will oscillate in value. Note the time it takes to cycle and try to determine an average reading by eyeballing. Pinching the hoses to the DP cell may help stabilize the reading. Now sample the signal and record the mean value. Make sure that the total sample time is long enough to average out any cyclical fluctuations in the pressure. The data acquisition may give a reading different than the DP cell. If so, divide the average DP cell reading by the DAQ system mean measurement Flow Measurement Page 7 of 18
and then input this ratio as the gain for the DAQ system. Sample again to see if the DP cell and DAQ measurements coincide. If not, keep trying. c. Take a long sample - long enough to average long-term fluctuations of the DP cell that you have observed. Record the mean value and standard deviation of the orifice pressure drop and then turn the fan off. 5. Insert the Pitot-static probe quill in the duct at a vertical location near the end of the duct, far from the orifice plate or other obstruction. Now attach the two hoses from a 5-inch DP cell to the Pitot-static probe. Be sure to connect the highpressure hose to the total pressure tap of the Pitot-static probe. This is the center tube of the device and is the tap that rises axially from the quill. Connect the low-pressure hose to the static pressure tap of the Pitot-static probe. This is the tap that comes out from the side of the tube. It is connected to the outer tube of the Pitot-static probe. Check that this DP cell is also connected to the data acquisition system. NOTE: The DP cell is designed to record only positive pressure differences (that s why the ports are labeled high and low). A negative pressure difference on the DP cell will produce a negative reading but it is not accurate and therefore the hoses have to be switched in order to measure a positive pressure difference. However, the recorded pressure difference may be recorded with a negative sign in order to account for the switching of the hoses. 6. Align the probe tip to point directly upstream toward the fan, which should be into the flow. Maintain this alignment of the Pitot-static probe while taking all velocity measurements. Bring the Pitot-static probe to the bottom of its stroke to make it touch the inside of the duct wall. Record the location of a convenient marker on the top of the Pitot-static probe (such as the bottom of the hose) that has a reading greater than 6.6" on the displacement scale. This is your reference bottom position in inches. 7 a. Calculate the scale readings for the 12 vertical positions indicated in Figure 4. Remember that the P-S tube has a diameter of 1/8", so your initial velocity measurement will be 1/16" away from the wall. Double-check your positions. You must take readings at appropriate positions, or data analysis will be difficult. b. Turn the fan on and sample the P-S tube DP cell output at each of the 12 positions across the diameter of the duct. Be sure to sample long enough. Record the mean value and standard deviation at each position. 8. Repeat steps 5 and 6 at a location just downstream of the orifice plate. If you are in the developing flow region, you may get a reading that is negative. If so, rotate the P-S tube to face downstream and note in your notebook that the velocity calculated at that point will be negative (toward the fan) rather than positive when integrating to find volumetric flow. 9. In order to evaluate potential error in the measurement caused by aligning the pitot tube off-axis, the range of angles must be determined for the P-S tube. Using the protractor, measure the alignment of the P-S tube to determine the maximum offangle at which measurements were taken. When performing the data analysis, Flow Measurement Page 8 of 18
use Figure 6 to determine the uncertainty in the pressure measurement due to this alignment error. Recoverable and Non-Recoverable Pressure Drop Measurement 1. a. Use a simple static probe to measure the sum of the recoverable and unrecoverable static pressure drop along the tube. First, measure the positions of each pressure port along the length of the duct relative to the fan outlet. b. Next, connect the high-pressure hose of the 5-inch DP cell to the static pressure port of the Pitot-static probe. Leave the low-pressure port of the gauge open to the atmosphere. Zero the DAQ system again by adjusting the bias (if necessary). c. Turn the fan ON. Starting at the farthest upstream location (closest to the fan), insert the static probe to the centerline of the duct, and align it with the flow. The position is not critical but the alignment is. Sample the static pressure at this location and record the mean value and standard deviation. d. Move the static probe to the next downstream port and repeat the measurement. Continue until you have readings for the entire length of the duct. Note: the indicated pressure can become negative downstream of the orifice plate. The DP cell isn't designed to read negative pressure, so switch the hoses. Remember that a positive measurement now means a negative pressure, so be sure you record it that way. Don't forget to switch the hoses back. 2 a. Measure the total duct non-recoverable pressure drop as a function of flow rate by taking the difference between the static pressure readings of a far upstream (near the fan) and a far downstream static probe. Position two Pitot-static probes in the center of the pipe; one far upstream and one far downstream of the orifice plate. Choose locations that are in regions of fully developed flow, away from any obstructions. Connect the high-pressure hose of the 5-inch DP cell to the static pressure port of the UPSTREAM Pitot-static probe. Connect the lowpressure hose to the static pressure port of the DOWNSTREAM Pitot-static probe. The DP cell will now indicate the pressure difference between the probes. b. Make sure to record two channels (the pressure drop across the orifice and the pressure difference between the static pressure upstream and downstream). Turn the fan ON and sample the output of the DP cell connected to the static probes as well as the DP cell connected to the orifice meter. Make sure the static probes are properly aligned with the flow. Record the mean values and standard deviations for each DP cell by exporting the data to a file. From the data we can compare the duct non-recoverable pressure drop (the difference between the static probes) with the total duct flowrate measured by the orifice meter. c. Obstruct the duct outlet, using the gate valve, to reduce the duct flowrate and then repeat the two readings from step 2b. Repeat the measurements for five flowrates with orifice DPs of approximately 2.2, 1.8, 1.5, 1.0, 0.7 and 0.3 inches of water. You don't have to match these values, just use similar spacing between them, and record the values. Note that these values appear unevenly spaced because pressure drop across the orifice plate, your reference Flow Measurement Page 9 of 18
measurement of duct flowrate, is related to the square of the flowrate rather than linearly related. 3. Repeat the room temperature, barometric pressure and humidity measurements for use in the error analysis. Clean up and leave the equipment in an orderly state. V. Required Data Analysis 1 a. Find the volumetric flow in the duct using the mean orifice pressure drop measurement. You must iterate on the orifice coefficient K 0 (from Figure 2) and Re for the duct (which is based on the duct diameter and velocity, not the orifice parameters). Evaluate the volumetric flow rate, average velocity and Reynolds number in the duct. Is the flow laminar or turbulent? b. Evaluate the precision and bias uncertainty in the measured value based on the standard deviation (precision) and manufacturer's uncertainty (bias) on the pressure measurement, variations in the room conditions and the accuracy of the orifice coefficient lookup. 2 a. Make two plots of the duct velocity as a function of duct diameter (using zero as the center of the duct), calculated from the dynamic pressure measured with the pitot-static tube during the two vertical traverses. Assume V=0 at the pipe wall. Do the velocity profiles look symmetric about the center? Do the measured profiles appear to be laminar or turbulent in nature? Can you determine if the flow is fully developed at any of the traverses? b. Estimate the uncertainty in the P-S tube velocity readings based on a the standard deviations and manufacturer's bias in dynamic pressure measurement, reasonable errors in tube position and angle (Figure 6), and variations in the room conditions. 3. Calculate the volumetric flow in the duct by integrating the velocities found from each of the two pitot traverses over the duct area. You will get two values of Q. Compare the integration of each traverse with the volumetric flow found from the orifice. 4. Plot the duct mean static pressure as a function of distance from the fan outlet. Identify in detail the pressure features that relate to the recoverable orifice pressure drop, the unrecoverable orifice pressure drop and the friction pressure drop. 5. Use Figure 5 and the duct Reynolds number to calculate the friction pressure drop expected per unit length of the duct. How does this compare to the measured change in duct mean static pressure observed downstream of the orifice. 6. Plot the duct non-recoverable static pressure drop, measured from the difference in the static probe readings, against the duct volumetric flow rate obtained from the orifice pressure drop measurements in Steps 2b,c. You'll need to calculate the orifice flow from the orifice mean pressure drop at each of the flowrates. Check Re for each flowrate to be sure the orifice coefficient is correct. Show that this plot follows a line of the form (VA ORF ) 2 = C( P N-R ) and then evaluate C. Flow Measurement Page 10 of 18
VI. References 1. Theory and Design for Mechanical Measurements. R.S. Figliola and D.E. Beasley, Wiley, (1991). 2. Fluid Mechanics. F.M. White, McGraw Hill, (1979). 3. Fundamentals of Engineering Thermodynamics. M. J. Moran and H. N. Shapiro, Wiley, (1988). Figure 1a. Schematic of the installation of a Pitot-static probe and a metered orifice plate. Flow Measurement Page 11 of 18
Figure 1b. Detail of the velocities, pressures, and flow patterns through a generalized Bernoulli obstruction metered orifice (White, 1979). Figure 2a. Graph showing the variation of Flow coefficient with Reynold's number. Flow Measurement Page 12 of 18
Figure 2b. Graph showing the variation of discharge coefficient with Reynold's number (White, 1979). Flow Measurement Page 13 of 18
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Figure 4. The 24 equal area sections of the experimental circular duct. Figure 6. The effect of Pitot-Static tube yaw angle of measurements of stagnation and static pressure (White, 1979). Flow Measurement Page 15 of 18
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