250 Class Note for M E 345 at PSU
250 Class Note for M E 345 at PSU
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Date Created: 02/06/15
Linear Velocity Measurement Author John M Cimbala Penn State University Latest revision 10 June 2009 Introduction Velocity is a vector that consists of a magnitude speed and a direction Linear velocity is de ned as the rate of change of the position vector with time at an instant in time Technically velocity is always a vector while speed is always a scalar However most people erroneously do not distinguish properly between velocity and speed In this learning module we discuss various ways to measure the velocity of solid objects and owing uids We are concerned here only with linear velocity not angular velocity Two words are used interchangeably to describe the measurement of velocity anemometry and velocimetry Velocity of solid objects Linear velocity transducer LVT Doppler radar velocity measurement There are several instruments used to measure the linear velocity of a solid object and we highlight several of them here A linear velocity transducer L VT is an inductive device that is similar in principle to the linear variable displacement transducer LVDT discussed previously It utilizes the link between electricity and magnetism as found by H A Lorentz namely if a magnetic eld moves near an electrical wire current ows through the wire Whereas an LVDT measures displacement an LVT measures speed An LVT consists of a rod called the core a permanent magnet and two electrical coils as sketched to the right The core slides inside a hollow cylindrical tube called a bobbin as sketched A DC voltage is generated when the core moves Since the two coils are wrapped with opposite polarity and since the magnet also has two poles north and south the south pole induces a voltage primarily in coil 1 and the north pole primarily in core 2 It turns out that the voltage is proportional to the speed of the core and is relatively independent of position within some limited range near the center typically about i15 to either side of center Although the range is limited LVTs are used in some types of machinery like milling machines In Spring semester 2008 two former M E 345 students James Weyand and Collin Julius used a velocity coil to measure the muzzle speed of an air cannon see picture to the right Core Bobbin DC output voltage V0 You are familiar with the Doppler shift for sound waves Namely a noise such as a car horn moving towards you or away from you has an apparently higher or lower pitch or frequency since the wavelength of the sound that reaches your ears is compressed or stretched due to the relative motion Doppler radar works by the same principle but with radio waves instead of sound waves When radio waves strike a moving object the frequency of the re ected radio waves is altered in a similar manner as the sound waves A radarDoppler velocimeter is sketched to the right Here is how it works 0 Radio waves of wavelength h are transmitted incident waves towards a moving object o The object moves with velocity Vat angle 6 relative to the radar unit as sketched o The radio waves re ect off the moving object and are sensed by a radio wave detector receiver that is also mounted on the radar unit Moving obj ect g o The detector measures the frequency of the re ected radar beam and the unit compares the frequency of the transmitted and re ected beams 0 Some trigonome reveals that the Doppler frequency shift AfD de ned as the change in frequency of I p i W the radar beam is 7 Doppler radar units are used by police to measure the speed of automobiles and they are also used in professional sports to measure the speed of baseballs etc Laser light can be used in place of radio waves Then the device is called a laser Doppler veloeimeter Example Given Don is driving 67 mph in a 55 mph zone A police officer nails him with a radar gun that uses a frequency of 10000 MHZ Angle 6 is 10 at the moment of the reading To do Calculate the Doppler frequency shift Solution 0 For any kind of electromagnetic wave wavelength and frequency f are related by the speed of light which is 29979 x 108 ms in a vacuum we approximate c as 300 x 108 ms in air 0 Thus c f and the above equation for the Doppler frequency shift is 2 67 391 hr 10 ZVcosef m 003 10 000Mh216093m 1hr A fD 3 3x108 ms 1mile 3600 s 0 Or since there are 106 HZ in one MHZ AfD 1970 HZ to three signi cant digits Discussion For small angles cos6 is nearly 1 but as the car gets closer to the officer the effect of the angle becomes more and more important and is difficult to take into account That is why police officers like to use radar guns on a straight line nearly facing oncoming traf c 00019664 MHZ VelocitV measurement using displacement and acceleration sensors Velocity is often not measured directly but rather by calculation from measurement of displacement or acceleration Displacement sensors 0 In a previous learning module we discussed methods to measure displacement distance traveled using a displacement sensor 0 By fundamental de nition velocity is the time derivative of displacement 39 0 So theoretically we could calculate velocity by taking the time derivative of displacement measurements from a displacement sensor potentiometer LVDT laser displacement meter etc 0 However there is an inherent problem with this technique 7 namely the process of dijj erentiation of a signal ampli es the noise in the system decreases the signaltonoise ratio 0 Thus velocity measurement by differentiation of displacement data is generally not a wise choice unless the displacement sensor has an extremely high signaltonoise ratio 0 However displacement can be used to very accurately measure average velocity 0 For example a runner or race car driver can easily calculate his or her average speed by timing with a stopwatch how long it takes to travel some known distance see photo to the right 0 In such cases we are not interested in measuring velocity as a function of time but rather only the average velocity over a particular period of time 0 Optical sensors such as photodeteetors are often used to indicate the time at which a moving object crosses the optical path of the sensor 0 With two such sensors located at a known distance apart the average speed of the object is easily calculated as V o This technique is often used for measuring the velocity of very fastmoving objects like bullets I Acceleration sensors 0 In some instruments an accelerometer sensor is available 7 it measures acceleration as a function of time o By fundamental definition velocity is the time integral of acceleration th dt where V0 is the velocity at time to and we integrate from time to to some later time t 0 Unlike differentiation the process of integration decreases the noise in the system increases the signal tonoise ratio 0 Thus velocity measurement by integration of acceleration data is generally a wise choice Velocity ofa owing uid I The velocity of a owing uid is more difficult to measure than that of a solid object for several reasons 0 The uid is often not visible air or not transparent e g oil milk 0 The uid is usually not moving as a solid body but rather individual uid particles move relative to each other 7 velocity is a function of spatial location within the uid ow 0 Velocity probes placed in the ow often disturb the very ow we are trying to measure 0 The ow is often unsteady I Nevertheless engineers have invented many devices that accurately measure uid ow velocity Some of these are discussed here I As you may recall from your study of uid mechanics there are two ways to describe uid motion 0 In the Lagrangian method we follow the movement of individual uid particles 7 we describe the location position velocity and acceleration of a particular uid particle as a function of time o In the Enlerian method we define a control volume and describe the velocity field and acceleration field as functions of space and time within that control volume Lagrangian velocity measurements I Lagrangian velocity measurements consist of following or tracking a uid particle that is marked or identi ed in some way dye in liquids smoke in air tiny soap bubbles in air tiny particles in water 7 it is assumed that the tiny particles move with the uid this is called seeding the ow Some call this timeof ight velocity measurement The veloc39 of the uid particle is calculated by differentiation of position I A simple example is throwing a oating ball into a river and timing how long it takes to move a certain distance thus inferring the velocity of the water surface I An interesting application of the Lagrangian velocity measurement technique is the monitoring of glacier velocity as in the photo to the right Markers are placed on the glacier and then their movement is tracked in time either by surveying equipment or nowadays by global positioning systems Particle image velocimetry I A hightech velocity measurement device for uid ows is called Lighbguidc particle image velocimetry PI V Here is a brief description of delivefy f laser sheet how PIV works 7 Stereoscopic o The uid is seeded w1th tiny particles that are so small that Maiquot a calm sump they move with the uid 10w o A doublepulse laser illuminates a region of ow under study X r l a and a digital camera sometimes two separate cameras quot I39 p 7 39 records two images 7 timed with the two ashes pulses of laser light Illuminated particles appear as bright spots on the 39 photographs because of the ashes of laser light 19mm 0 The displacement of illuminated particles is then determined by analyzing interrogating the two digital photographs with m 39 jmory Flew Of View sophisticated image processing software Distance As between the two bright spots is measured and the speed is determined by where At is the known time interval between laser pulses The direction of the particle movement is also determined by image processing and therefore the velocity of the illuminated particle is calculated There are two basic types of PIV Standard PI V two rapid laser ashes with a very small At followed by a long pause Cinema or cinemagraphie PI V one laser ash per camera frame with no pause but longer At 0 There are both 2D and 3D PIV systems 2D systems use one camera and measure ow velocity in a plane illuminated by a laser light sheet 3D systems use two cameras stereoscopic photography to measure the velocity in the plane of the laser light sheet and also in the direction normal to the plane of the light sheet as sketched Although PIV is fundamentally Lagrangian the final result is a velocity field which is an Eulerian result Eulerian velocity measurements Eulerian velocity measurements involve a probe or sensor of some kind sitting in a uid ow With Eulerian techniques instead of tracking individual marked uid particles we measure the velocity of the uid that happens to be owing past the sensor at the time of measurement If the sensor is at a xed location in the uid ow it measures the velocity at that point as a function of time If the sensor is traversed moved around in the ow eld and the ow eld is steady we can map the steady or timeaveraged velocity eld as a function of spatial location There are various Eulerian velocity measurement devices some of which are described here Laser Doppler velocimetry An optical technique involving a laser is laser Doppler velocimetry LD V also called laser veloeimetry L V or laser Doppler anemometry LDA Like PIV LDV measures the velocity of small seed particles as they move through the uid However whereas PIV is a Lagrangian technique following the motion of individual particles LDV is an Eulerian technique since the velocity is measured at a fixed point in the ow A basic single velocity component dualbeam LDV system is sketched and described below l huludelcclur Receiving zr lens I 4 Beam spliucr Sending lens Laser l a 7 g lii Mcusurcmenl V Mirror L 4 volume Bragg cell 0 The laser beam is split by a halfsilvered mirror called a beam splitter into two parallel laser beams of equal intensity 0 Both beams pass through a converging lens that focuses the beams at a point in the ow actually a small volume called the F39mg 1 lines measurement volume or the focal volume 0 When the two beams cross the waves interfere with each other A I creating a bright and dark fringe pattern as sketched to the right 0 Laser light is scattered by small seed particles that pass through Law Measurcmcm the measurement volume The scattered light intensity is bright mums volume then dark then bright etc as the particle moves through the fringe pattern 0 The scattered laser light is collected by a receiving lens and photodetector o The photodetector converts uctuations in light intensity into a uctuating voltage signal 0 Finally a signal processor determines the frequency f of the voltage signal and thus the frequency f of the scattered light 7 l W Fringe lines 39Zsi pjZ is the wavelength of the laser light and or is the angle between the two beams as sketched above 0 The speed of the seed particle turns out to be linearly proportional to the frequency of the uctuating 11 ht mtens1t and 1s 1ven b V s w g y g y f 2 s1 nol 2 In practice velocity is measured in more than one direction simultaneously 0 2D LDV systems employ two separate laser beams of two different wavelengths or colors and the respective fringe patterns are in two usually mutually orthogonal directions 0 3D LDV systems employ three separate laser beams three colors and are used to measure the three dimensional velocity at the location of the measurement volume Laser Doppler velocimetry is somewhat of a misnomer since we do not actually calculate a Doppler shift Finally a Bragg cell is used on one of the beams to shift its frequency slightly causing the fringe pattern to move With this device it is possible to distinguish between positive and negative velocities through the focal volume 0 Mathematically the spacing s between fringe lines is obtained from trigonometry s where Thermal hot wire and hot lm anemometrv A thermal anemometer works under the principle that the rate of convective heat transfer from a hot object to the surrounding uid increases as the speed of the uid owing around the object increases A simple example is blowing at your spoon full of soup to cool it down 7 the rate of convection heat transfer and thus the rate of cooling of the soup is increased by blowing cooler air over it Thermal anemometers consist of an electrically heated sensor as sketched to the right along with appropriate electronics Thermal anemometers have extremely small sensors and thus they can be used to measure the instantaneous velocity at a point in the ow without appreciably disturbing the ow The time response of the sensor is fast also due to its small size There are two primary types of thermal anemometer probes o If the sensing element is a wire typically a few microns in diameter and about a millimeter long it is called a hatwire probe and the measurement system is called a hatwire anemometer o If the sensing element is a thin metallic lm typically less than 01 micron thick it is called a hat lm probe and the measurement system is called a hat lm anemometer The lm is mounted on a ceramic support that is much larger typically about 50 microns in diameter Hot lm sensors have poorer spatial resolution and frequency response because of their larger size but are much more rugged compared to hotwire sensors Because of this hot lms are commonly used in water and other liquids while hot wires are more commonly used in air and other gas ows The most common thermal anemometer system is the constanttemperature anemometer CTA As its name implies the sensor is heated by electric current to a constant temperature typically about 200 C The sensor wants to cool down as uid velocity increases but electronic controls a Wheatstone bridge maintain the temperature by varying the electric current that passes through the sensor The higher the velocity the higher the rate of heat transfer required to keep the sensor at its fixed temperature and therefore the higher the current passing through the sensor Electric current I Flow chOCity VV Wire support Sensor 2 thin wire uppmximutclyl mm long with a diameter of 5 p111 The electrical power supplied to the sensor by electrical resistance heating is Wel m Istmsor E2 YRSSHW where Rsensor is the electrical resistance of the sensor hot wire or hot lm and E is the voltage across the sensor Assumin that all of this supplied electrical power heats the wire the power balance results in King is law E a b V where constants a b and n are calibrated for a given sensor Finally the voltage is measured and is calibrated as a function of velocity across the probe Thermal anemometers can be used to measure one two or three velocity components simultaneously by using probes with a one b two or c three sensors respectively as sketched below u b 139 Pitot and Pitotistatic probes A less sophisticated Eulerian velocity measurement technique involves probes that infer velocity by measuring pressure APitutpmbe is just a tube with a pressure tap at the stagnation point that measures stagnation pressure APitntxtaticpmbe has both a stagnation pressure tap and several circumferential static pressure taps and it measures both 9 Some authors call Pitotstatic probes PitutDarcypmbes since i Darcy was actually the rst to combine the two pressure measurements into one single probe Fluid velocity is inferred from the Pitot probe measurement of stagnation pressure by assuming that the static We consider here incompressible ow and we assume that the probe is aligned into the ow as sketched below for both the a Pitot probe and the b Pitotstatic probe Piiul probe stagnation and static pressures A closeup photograph of a Pitotstatic probe is shown to the right pressure is known or measured with a separate pressure tap or probe while the Pitotstatic probe measures both pressures PItohstutlc probe V V gt gt Slagnulion Srugnulion Suuic pressure PI assure pressure To slalic pressure meter To smgnarion pmssurc mcrcr 1 To smgnmion pressure mcrcr II b A typical application is sketched to the right 7 measurement of velocity in awind tunnel At sufficiently high velocities the Bernoulli equation is a reasonable approximation viscous effects are neglected and thus I Pitobslalic probe 2 where pis the uid density Flow Salic pressurc P1 Stagnation pressure PI of the static pressure taps in the probe andz is vertical elevation Since location 1 is a stagnation point the velocity there V1 is zero The probe is assumed to be small compared to the length scale associated with changes in ow velocity 7 we assume that velocity Vz just outside the boundary layer above the static pressure holes in the probe is equal to velocity Vupstream of the probe which we are trying to measure Since locations 1 and 2 are in close proximity we neglect changes in elevation In fact if the probe is horizontal as in the sketch to the right the average elevation of all the static pressure holes at location 2 is identical to that at the stagnation point Thus with V1 0 V2 V andzl zz the Bernoulli equation Wind tunnel wall Flexible lnbing 1 1 Dil39l39clcutial pressure lrzlnsduccr or manomcrcr m Incasulc Pr 7 P3 reduces to the Pitutfnrmula The two pressures are measured by some kind ofpressure sensor as discussed in a previous learning module In the above sketch the difference between stagnation pressure P1 and static pressure P2 is measured by a differential pressure transducer 7 a Utube manometer or other pressure sensor can be used instead Some advantages of the Pitotstatic probe are 0 It is simple to use 0 It is inexpensive compared to some of the other more sophisticated velocity measurement techniques 0 It is highly reliable since it has no moving parts Some disadvantages of this velocity measurement technique include o The probe must be aligned reasonably straight into the ow or else signi cant errors may result The probe disturbs the ow unlike the LDV Pitotstatic probes are not very responsive to uctuating velocity elds In other words they have a poor frequency response compared to LDV or hot wire anemometry Nevertheless Pitotstatic probes have been used successfully for decades to measure ow velocity in both gases and liquids For example the next time you y on an airplane look for a Pitot or Pitotstatic probe under the wing or on the fuselage Ice buildup on the Pitot probes is believed to have contributed to the crash ofAir France Flight 447 on June 12009 7 an Airbus A330200 In many cases the static pressure is atmospheric or nearly atmospheric and thus only the stagnation pressure measurement is needed to determine the velocity This can be accomplished with a Pitot tube An example is shown to the right 7 a Pitot gauge designed to measure re hydrant velocity along with a photo of it being used Rotating velocimeters In many practical applications it is not necessary to use small sensors to measure ow velocity For example weather stations measure the velocity of the wind and the velocimeters can be rather large compared to Pitotstatic probes or the measurement volume of an LDV system Rotating velocimeters are devices that infer velocity by measuring the rotation rate of a rotating turbine and there are seveml varieties The most common rotating velocimeter used by weather stations is the cup anemometer as shown to the right The one shown here has three cups but fourcup anemometers are also common Cup anemometers work on the principle that the open concave side of the cup experiences a higher dmg force than does the closed convex part Thus when the wind blows the sha rotates in a preferred direction To infer the wind speed some type of mechanism converts sha rotation speed into a voltage a current or a series of pulses that is calibrated to display the wind speed in the desired units mihr ms kmhr knots etc Some handheld cup anemometers are also commercially available as shown to the right There are some obvious disadvantages of cup anemometers 0 They are quite large compared to the other point measurement devices discussed previously 7 cup anemometers measure the average wind speed over the volume occupied by the cups 0 Most cup anemometers are designed only for horizontal ows although this is usually not a problem with wind speed measurements 0 They do not measure the direction of the wind Threecup anemometers are therefore usually coupled with aweather vane device that measures the wind direction independently of it speed Another very popular type of rotating velocimeter is the turbine anemometer also called avane anemometer This device employs a rotating turbine to infer velocity Amrbme anemumeterwurks un the same pnnclple as the eup memumeter Except that the turban luuks hke a small dueted wrnduw fan mnnmg haekwards as seen m the pretures heluwnght The mere aeeurate turban memumeters usually have a separate turban pumun and electmmcs puruun as shuwn tn the nearnght hut the all m unequot unts are mere eunyement tn use see preture tn the far nght Atemperature sensur Lhanmstur Dr thermueuuple rs typreauy rneluded wrth amrbme anemumeter 5 that huth an temperature and ar speed are measured sxmultaneuusly Sueh deyrees are eaned lhzrmamzm mzlas Turban memumeters are smaner and mure eunyenent tn use than eup memumeters and they ran he turned m my drreetaun tn face the ar law 7 even yerueany 1fnecesszry Turban memumeters are u m used by HVAC heatrng yenulauun and ar eundruunrny engrneers tn measure the ar uw m duets and nut ufau supp1y drffusers ete The maur drsadyantage rs thatthe measured velumty rs rntegrated uyerthe face area quhe turhrne wheh rs durte large eumpared tn the measurement srze ufa Fxtutrstan pruhe hut wrre Dr LDV measurement vulume Elecumnzgneu vein ems A deyree eaned an electrarmgnz r tensesmar ean he used tn measure aurd velumty m candudmg mds The pnn ple uf uperatrun rs that when amameue eld rs apphed Lhruugh an elecmcally eundueung aurd mutrun quhe aurd rnduees ayedtage aeruss Wm eleetrudes plaeed m the law The uutputyedtagers greatest furhdghly eundueung mds sueh ashqmd meta1s Seawaterhas any hdgh eunduetanee and eleetrumameue yeluermeters are u en used m uneanugzphy expenments where they are ealedezeasumgmz will mm The strength quhe uutputyedtage rnereaseswrth speed mdrfpruperly eahhrated canmfer aurd velunty
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