Lab 1: Measuring Air Velocity 5 February 2010
J. Cool
Lab 1: Measuring Air Velocity – Accuracy and Precision
by
Joe Cool
5 February 2010
Department of Mechanical Engineering
University of Wisconsin-Madison
1513 University Avenue
Madison, WI 53706-1572
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Lab 1: Measuring Air Velocity 5 February 2010
J. Cool
Executive Summary
In this experiment, two measurement procedures were used to assess the velocity of air expelled by a hairdryer. The first method utilized a U-tube monometer and the second used an electronic diaphragm gage (EDG). The precision and accuracy of the measurement systems were evaluated to determine the advantages and disadvantages of each system.
Much of this lab involved identifying the difference in accuracy and precision between the various measurements. Each measurement involved a level of uncertainty that had to be quantified. The respective uncertainties of each measurement were propagated through all of the calculations to give a final uncertainty value for the velocity of the air exiting the hairdryer. The velocity of the air stream generated by the hairdryer determined from each of the devices was:
U-Tube Manometer: 10.4 ± 2.8 (m/s)
Electronic Diaphragm Gage: 9.15 ± 1.16 (m/s)
The average velocity measured by the manometer was 13.6% higher than the velocity measured by the EDG. In addition, the uncertainty interval of the manometer was 2.7 times larger than the uncertainty interval of the EDG. The confidence limits of the manometer exhibit a wider spread than that of the EDG. This indicated the EDG produced more precise measurements than the manometer.
The simple principle of operation for a U-Tube Manometer is more accurate because it does not rely on indirect measurements, because pressure is measured directly. The Electronic Diaphragm Gauge, on the other hand, relies on the indirect measurement of voltage which is converted into pressure by a calibration equation. Less direct measurements have a higher potential for inaccuracy than measurements that are more direct.
Table of Contents
1.0 Introduction / 4
1.1 Background / 4
1.2 Objectives / 6
1.3 Overview / 6
2.0 Procedure and Apparatus / 7
2.1 Experimental setup – Hardware / 7
2.1.1 Experimental setup – U-Tube Manometer / 7
2.1.2 Experimental setup – Electronic Diaphragm Gage / 8
2.2 Experimental setup – Software / 9
2.3 Experimental setup – Procedure / 9
2.3.1 Procedure – U-Tube Manometer / 10
2.3.2 Procedure – Electronic Diaphragm Gage / 10
2.4 Analysis / 10
2.4.1 Analysis – U-Tube Manometer / 11
2.4.2 Analysis – Electronic Diaphragm Gage / 12
3.0 Results / 14
3.1 Results – U-Tube Manometer / 14
3.2 Results – Electronic Diaphragm Gage / 15
4.0 Discussion / 18
5.0 Conclusions and Recommendations / 19
5.1 Summary of Results / 19
5.2 Recommendations / 19
References / 20
Nomenclature
Xi / Measurement / ------
Sx / Standard Deviation / ------
N / number of data points / ------
x / Average / ------
P / Pressure / Pa
h / Height / M
V / Velocity / m/s
g / acceleration due to gravity / m/s2
T / Temperature / ˚C
Q / Resolution / mV/bit
v / Voltage / V
x’ / true value / ------
tν,P / variable values / ------
Greek Symbols
Sym. / Definition / UnitsΔ / change in / ------
ρ / Density / Kg/m3
ν / degrees of freedom / ------
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Lab 1: Measuring Air Velocity 5 February 2010
J. Cool
1.0 Introduction
The motivation of this experiment was to record and evaluate measurements of air velocity from a blow dryer stream. The two devices used in this lab to experimentally determine air velocity were a U-tube Manometer and an Electronic Diaphragm Gauge. The U-Tube Manometer and Electronic Diaphragm Gauge were used to calculate velocity by means of their output measurements of pressure and voltage respectively. In order to determine an experimental value of the hairdryer velocity stream, the concepts of accuracy and precision of the measurements were evaluated.
1.1 Background
The measurement of pressure to determine a fluid’s velocity can be important in many engineering applications. For instance, many airplanes use Pitot-Static Tubes. These tubes use the same fundamental concepts (explained below) used in this lab to determine the speed of the aircraft relative to the air around it.
Determining velocity via pressure measurement is common in aircraft and marine applications, but for other applications it may be advantageous to measure velocity using other methods. One device for measuring fluid velocity, commonly used by weather stations, is the Cup Anemometer [1]. This device consists of evenly spaced hemispherical cups cantilevered out from a vertical central axis. Fluid travelling past the cups creates a moment about the central axis and the device begins to spin. A measurement of revolutions per time is then used to compute the fluid’s average velocity over that time. A much simpler device for measuring fluid velocity is a sphere hung from a string. If the density and drag coefficient of the sphere are known, then one can measure the angle that the string deviates from the vertical direction, and determine the fluid velocity using simple fluid mechanics concepts [2].
Figure 1: Accuracy vs. Precision [3]
The difference between accuracy and precision was also carefully considered in this laboratory. Accuracy represents how close a data point is the the true value not relative to any other data points. Precision is the extent to which a data point can be reproduced by the same process independent of it’s accuracy. See figure 1 for an illustration distinguishing accuracy and precision. The image on the left shows accuracy because the average of the data points lies close to the center of the bull’s eye. The image on the right shows precision because the data points can easily be reproduced.
1.2 Theory
Pressure-based velocity measurements originate from fluid mechanics, and more specifically the Bernoulli equation. The Bernoulli equation states that for inviscid flow, along a streamline, with no externally applied work, and an incompressible fluid (or compressible fluid at low Mach number) mechanical energy is conserved. The Bernoulli equation between two states is commonly written:
(1)
In this experiment gravity was neglected. After removing the gravity terms and rearranging, Equation 1 becomes:
(2)
Equation 2 directly relates a change in pressure between two states (ΔP) to a change in the square of the velocity at each respective state.
Figure 2: U-Tube Manometer
In this experiment, the pressure difference between states was found using a U-tube manometer (figure 2, above). U-tube manometers utilized the fact that fluid pressure varies linearly with depth, and that all fluids at the same depth must share the same pressure. Consider the
manometer shown. Using fluid statics principles it is known that:
P2 = P0 + ρwgh (3)
The air stream leaving the hairdryer is at atmospheric pressure (P1 = P0) and travels at some velocity V1. When the stream reaches the air/water interface the velocity of air is reduced to zero (V2 = 0) and pressure increases to P2. Substituting Equation 3 into Equation 2, along with the given information, then rearranging yields:
(4)
This is the equation used to calculate the velocity of the stream leaving the hairdryer as a function of the manometer height reading.
1.3 Objectives
The objectives of this experiment were to:
· Measure pressure and voltage from a U-tube Manometer and Electronic Diaphragm Gauge, respectively, and relate these outputs to pressures and velocities
· Analyze the pressure and velocity measurements by evaluating accuracy and precision of each measurement device
· Quantify both systematic and random uncertainties of the measured pressure
· Determine the random and systematic uncertainties of the calculated velocities through propagation
1.4 Overview
In the following pages, experimental procedure and a description of the apparatus is given, followed by an analysis of the collected data and interpretation of the results. Finally, the results are discussed and then conclusions and recommendations for the experiment are given.
2.0 Procedure and Apparatus
2.1 Experimental Setup – Hardware
Table 1: Electronic Instrumentation Used in the Experiment
Instrumentation Used / Attributes of InstrumentationNational Instruments CompactDAQ / USB 8 slot chassis
National Instruments Model 9215 / 4-Channel, 100 kS/s, 16-bit, ±10 V Simultaneous Sampling Analog Input Module
Electronic diaphragm gage / Attributes outlined above
Tenma 72-7660 / DC power supply
Tenma digital multimeter / 72-410A
Vidal Sassoon 1875 W hairdryer / nozzle attached, high fan, low heat
2.1.1 Experimental Setup – U-Tube Manometer
As seen in figure 3 (below), a cold air stream was produced by a hairdryer on high outset setting and blown into a U-Tube Manometer made from a plastic hose. The hairdryer used in the lab
Figure 3: U-Tube Manometer set-up used to measure Pressure
was an 1875 W Vidal Sassoon hairdryer with a nozzle attachment, a variable fan setting, and variable heat setting. When the hairdryer air stream was blown into the U-Tube Manometer, a height difference between the right and left sides of the device occurred. The height difference was measured using a ruler with millimeter graduations. The measured height difference is the differential pressure measured in inches of water. Using this experimentally measured pressure differential, the initial velocity of the air stream from the hairdryer was calculated using Bernoulli’s Equation.
2.1.1 Experimental Setup – Electronic Diaphragm Gage
The Electronic Diaphragm Gage, in figure 4 (below), was used to measure velocity in a manner similar to the manometer.
Figure 4: Electronic Diaphragm Gauge Setup
In order to measure pressure using the Electronic Diaphragm Gauge, a data acquisition program was created using LabView 8.6 (see figure 5, below). In the Diaphragm Gauge setup, a BNC connector was connected from the EDG to the NI-9215 DAQ module and another connection was made between the DC power supply and the digital multi-meter. The hairdryer stream was directed into a clear tube in order to produce the pressure difference that the data acquisition system read as a voltage. This voltage was converted to a pressure measurement using a linear calibration provided by the manufacturer.
The Electronic Diaphragm Gauge has several design conditions that need to be considered in the uncertainty calculations. In particular, the span (3.75 V ± 60 mV), the voltage ratiometricity (1.5% * 3.75 V), and the offset (0.25 VDC ± 60 mV) are three systematic uncertainties associated with the gauge specifications [4]. The dominant of these three device specifications is by far the voltage ratiometricity. This can be characterized how far the device deviates from the voltage-pressure linear relationship of 0.25V-4V is equivalent to 0 inH2O - 10 inH2O. Another Electronic Diaphragm Gauge specification of particular interest is the conversion between voltage and pressure.
2.2 Experimental setup – Software
The software packages used for this lab were National Instruments LabVIEW 8.6 and Microsoft Excel 2007 for data acquisition and analysis, respectively. The block diagram used to collect data for the Electronic Diaphragm Gauge is shown in figure 5 (below).
Figure 5: LabVIEW 8.6 Block Diagram used to collect Diaphragm Gauge Voltage
2.3 Experimental setup -- Procedure
A mercury barometer was used to measure the ambient air pressure. A mercury thermometer was used to measure the ambient air temperature and the hairdryer outlet temperature. These measurements were used to calculate the density of air and water.
2.3.1 Procedure – U-Tube Manometer
The first velocity measurement was obtained using the U-tube manometer. One group member held the manometer tubing parallel (±5°) to the hairdryer’s air stream and centered the tubing inside the hairdryer nozzle (±5 mm from center). The hairdryer was then turned on high fan, low heat while another group member read and recorded the height difference of the water in the manometer. This was performed six times, while alternating the dryer operator and manometer reader between each run.
2.3.2 Procedure – Electronic Diaphragm Gage
The second velocity measurement was obtained using the Electronic Diaphragm Gage. The overall procedure was very similar. One group member held the sensor tubing parallel to the hairdryer’s air stream, while another group member recorded the data using LabVIEW. In LabVIEW 1000 samples were collected using a sample rate of 1000 Hz. Another measurement was taken of ambient air using the same method.
2.4 Analysis
After data was collected the accuracy and precision of the measurements was assessed. The following equations were used to generate the statistics needed for data analysis.
The average value of observed data was calculated using the following equations where xi is the ith sample and N is the number of samples:
x=1Ni=1Nxi (5)
The standard deviation of a data set was calculated with this relationship:
Sx=1N-1i=1N(xi-x)2 (6)
The subsequent equation was used to contrive the precision interval of a set of measurements:
±tν,PSx (7)
In equation (5) the correct value of t was selected based on the specified confidence level, P, and the degrees of freedom for the set, ν, which is equal to N-1.
After the average and precision of the collected data is determined the uncertainty of the next data point can be expressed as:
xi=x±tν,PSx (% P) (8)
Some data analysis also required the precision interval for the true mean which can be calculated using the following relationship:
±tν,PSxN (9)
The uncertainty of the true value can then be assessed using the following equation:
x'=x±tν,PSxN (% P) (10)
In this experiment the desired value of air stream velocity was derived from equation (1). In order to determine the uncertainty of the air stream velocity, the combined uncertainty needed to be evaluated. The combined uncertainty can be calculated using the following equation:
ur=i=1J (∂r∂xiuxi)2 (11)
In equation (11) ur is the resulting uncertainty of a function r and uxi is the uncertainty associated with an independent variable in r. J is the number of independent variables.
2.4.1 Analysis – U-Tube Manometer
When experiment was conducted five measurements of Δh were obtained. Equation (9) was used to determine and the uncertainty of the height was found to be Δh=0.0061 ± 0.0032m (P=95%) and Δh=0.0061 ± 0.0053m (P=99%). Please see Appendix A (not included in this example report) for all calculations performed for experiment 1.