Chapter 9Pressure and Velocity Measurements9.1INTRODUCTIONThis chapter introduces methods to measure the pressure and the velocity within fluids.Instrumentsand procedures for establishingknown values of pressurefor calibration purposes,as well as varioustypes oftransducersforpressuremeasurement,arediscussed.Pressureis measuredin static systemsand in moving fluid systems.We alsodiscuss well-established methods measuring thelocal and full-field velocity within a moving fluid. Finally, we present practical considerations, including commonerror sources, for pressure and velocity measurements. Although there are various practical teststandards for pressure, many of which are applied to a specific application or measuring device, theAmericanSocietyofMechanicalEngineers'PerformanceTestCode(ASMEPTC)19.2providesanoverview of basic pressure concepts and measuring instruments thathasbecomethe acceptedstandard (1)Upon completion of this chapter, the reader will be ableto.explain absolute and gauge pressure concepts and describe the working standards thatmeasure pressure directly,.explainthephysicalprinciplesunderlyingmechanicalpressuremeasurementsandthevarioustypes of transducers usedtomeasurepressure,explain pressure concepts related to static systems or with moving fluids.·analyze thedynamicbehavior of pressure system responsedue totransmission lineeffects,and:explain the physical principles underlying various velocity measurement methods and theirpractical use.9.2PRESSURECONCEPTSPressurerepresents a contactforce per unit area.It acts inwardly,and normally to a surface.To betterunderstand the origin and nature of pressure,consider the measurementof pressure at the wall of avessel containingan ideal gas.Asa gas moleculewith someamount ofmomentumcollides with thissolid boundary,itrebounds off in a different direction.From Newton's second law, weknow that thischange in linear momentumof themoleculeproduces an equal butopposite (normal, inward)forceon the boundary. It is the net effect of these collisions averaged over brief instants in time that yieldsthe pressure sensed at the boundary surface.Because there are so many molecules per unit volume375
E1C09 09/14/2010 15:4:52 Page 375 Chapter 9 Pressure and Velocity Measurements 9.1 INTRODUCTION This chapter introduces methods to measure the pressure and the velocity within fluids. Instruments and procedures for establishing known values of pressure for calibration purposes, as well as various types of transducers for pressure measurement, are discussed. Pressure is measured in static systems and in moving fluid systems. We also discuss well-established methods measuring the local and full- field velocity within a moving fluid. Finally, we present practical considerations, including common error sources, for pressure and velocity measurements. Although there are various practical test standards for pressure, many of which are applied to a specific application or measuring device, the American Society of Mechanical Engineers’ Performance Test Code (ASME PTC) 19.2 provides an overview of basic pressure concepts and measuring instruments that has become the accepted standard (1). Upon completion of this chapter, the reader will be able to explain absolute and gauge pressure concepts and describe the working standards that measure pressure directly, explain the physical principles underlying mechanical pressure measurements and the various types of transducers used to measure pressure, explain pressure concepts related to static systems or with moving fluids, analyze the dynamic behavior of pressure system response due to transmission line effects, and explain the physical principles underlying various velocity measurement methods and their practical use. 9.2 PRESSURE CONCEPTS Pressure represents a contact force per unit area. It acts inwardly, and normally to a surface. To better understand the origin and nature of pressure, consider the measurement of pressure at the wall of a vessel containing an ideal gas. As a gas molecule with some amount of momentum collides with this solid boundary, it rebounds off in a different direction. From Newton’s second law, we know that this change in linear momentum of the molecule produces an equal but opposite (normal, inward) force on the boundary. It is the net effect of these collisions averaged over brief instants in time that yields the pressure sensed at the boundary surface. Because there are so many molecules per unit volume 375
376Chapter9Pressure and VelocityMeasurementsSystempressureGauge systempressureLocalreferencepressureStandardAbsoluteatmosphericsystempressurepressure101.325kPaabs.Absolute14.696 psiareference760 mm Hg abs.29.92 in Hg abs.pressurePerfectvacuumFigure 9.1 Relative pressure scales.involved (e.g., in a gas there are roughly 1ol molecules per mm). pressure is usually considered tobecontinuous.Factorsthataffectthefrequencyorthenumberofthecollisions,suchastemperatureand fluid density,affect the pressure.Infact, this reasoning is the basis of the kinetic theory fromwhichtheidealgasequationofstateisderivedApressure scale must berelated to molecular activity,since a lack of any molecular activitymustformthelimitof absolutezeropressure.Apurevacuum,whichcontainsno molecules.provides the limit for a primary standard for absolute zero pressure. As shown in Figure 9.1, theabsolute pressure scale is quantified relative to this absolute zero pressure.The pressure understandard atmospheric conditions is defined as 1.01320x 10°Pa absolute (where1Pa=1N/m)(2).This is equivalentto101.32kPa absolute1atm absolute14.696 Ib/in.?absolute (written as psia)1.013 bar absolute(where 1bar = 100kPa)The term"absolute"might be abbreviated as"a"or“"abs."Also indicated in Figure 9.1 is a gauge pressure scale. The gauge pressure scale is measuredrelativeto some absolutereference pressure,which is defined in a manner convenient to themeasurement.The relation between an absolutepressure,Pabs,and its correspondinggaugepressure,Pgauge, is given by(9.1)Pgauge = Pabs - Powhere po is a reference pressure.A commonly used reference pressure is the local absoluteatmosphericpressure.Absolutepressureis a positive number.Gaugepressure can be positiveornegative depending on the value of measured pressure relative to the reference pressure. Adifferential pressure, suchasp-P2,isarelativemeasureof pressure
E1C09 09/14/2010 15:4:52 Page 376 involved (e.g., in a gas there are roughly 1016 molecules per mm3 ), pressure is usually considered to be continuous. Factors that affect the frequency or the number of the collisions, such as temperature and fluid density, affect the pressure. In fact, this reasoning is the basis of the kinetic theory from which the ideal gas equation of state is derived. A pressure scale must be related to molecular activity, since a lack of any molecular activity must form the limit of absolute zero pressure. A pure vacuum, which contains no molecules, provides the limit for a primary standard for absolute zero pressure. As shown in Figure 9.1, the absolute pressure scale is quantified relative to this absolute zero pressure. The pressure under standard atmospheric conditions is defined as 1.01320 105 Pa absolute (where 1 Pa ¼ 1 N/m2 ) (2). This is equivalent to 101.32 kPa absolute 1 atm absolute 14.696 lb/in.2 absolute (written as psia) 1.013 bar absolute (where 1 bar ¼ 100 kPa) The term ‘‘absolute’’ might be abbreviated as ‘‘a’’ or ‘‘abs.’’ Also indicated in Figure 9.1 is a gauge pressure scale. The gauge pressure scale is measured relative to some absolute reference pressure, which is defined in a manner convenient to the measurement. The relation between an absolute pressure, pabs, and its corresponding gauge pressure, pgauge, is given by pgauge ¼ pabs p0 ð9:1Þ where p0 is a reference pressure. A commonly used reference pressure is the local absolute atmospheric pressure. Absolute pressure is a positive number. Gauge pressure can be positive or negative depending on the value of measured pressure relative to the reference pressure. A differential pressure, such as p1 p2, is a relative measure of pressure. Gauge system pressure 101.325 kPa abs. 14.696 psia 760 mm Hg abs. 29.92 in Hg abs. Local reference pressure System pressure Perfect vacuum Absolute system pressure Absolute reference pressure Standard atmospheric pressure Figure 9.1 Relative pressure scales. 376 Chapter 9 Pressure and Velocity Measurements
Pressure Concepts3779.2Free surfaceat ho,PoPo= p(ho)p(h) = Po + yhfhp(h)Surface of area, Aat depth, hFluid specific weight, Figure9.2 Hydrostatic-equivalent pressurehead and pressure.Pressurecanalsobedescribedintermsofthepressureexerted ona surfacethatis submergedina column of fluid at depth h, as depicted in Figure 9.2.From hydrostatics, the pressure at any depthwithin a fluid of specific weight y can be written as(9.2)Pabs(h) = p(ho) + yh = po + yhwhere Po is the pressure at an arbitrary datum line at ho, and h is measured relative to ho. The fluidspecific weight is given by =pg where p is the density. When Equation 9.2 is rearranged, theequivalent head of fluid at depth h becomes(9.3)h = [Pabs(h) - p(h)]/= (Pabs - Po) /The equivalent pressure head at one standard atmosphere (po = O absolute) is760mmHgabs=760torrabs=1atmabs=10,350.8mmH,0abs=29.92inHgabs=407.513inH2absThe standard is based on mercury (Hg) with a density of 0.0135951 kg/cm’ at 0°C and water at0.000998207kg/cm2at20C(2).Example 9.1Determine the absolute and gauge pressures and the equivalent pressure head at a depth of 10 mbelowthefree surfaceof a pool of water at 20°C.KNOWNh=10m;whereho=0isthefreesurfaceT=20°℃PH20 = 998.207kg/m3Specificgravityofmercury,Shx=13.57ASSUMPTI0Np(ho)=1.0132×10’PaabsFIND Pabs Pgauge, and h
E1C09 09/14/2010 15:4:52 Page 377 Pressure can also be described in terms of the pressure exerted on a surface that is submerged in a column of fluid at depth h, as depicted in Figure 9.2. From hydrostatics, the pressure at any depth within a fluid of specific weight g can be written as pabsðhÞ ¼ pðh0Þ þ gh ¼ p0 þ gh ð9:2Þ where p0 is the pressure at an arbitrary datum line at h0, and h is measured relative to h0. The fluid specific weight is given by g ¼ rg where r is the density. When Equation 9.2 is rearranged, the equivalent head of fluid at depth h becomes h ¼ pabsðhÞ pðhÞ=g ¼ pabs p0 ½ ð ð Þ =g 9:3Þ The equivalent pressure head at one standard atmosphere (p0 ¼ 0 absolute) is 760 mm Hg abs ¼ 760 torr abs ¼ 1 atm abs ¼ 10; 350:8 mm H2O abs ¼ 29:92 in Hg abs ¼ 407:513 in H2O abs The standard is based on mercury (Hg) with a density of 0.0135951 kg/cm3 at 0C and water at 0.000998207 kg/cm3 at 20C (2). Example 9.1 Determine the absolute and gauge pressures and the equivalent pressure head at a depth of 10 m below the free surface of a pool of water at 20C. KNOWN h ¼ 10 m; where h0 ¼ 0 is the free surface T ¼ 20C rH2O ¼ 998:207 kg/m3 Specific gravity of mercury, SHg ¼ 13.57 ASSUMPTION p(h0) ¼ 1.0132 105 Pa abs FIND pabs, pgauge, and h h h p(h) Surface of area, A at depth, h Fluid specific weight, + p(h) = p0 + h Free surface at h0 p , p0 0 = p(h0) Figure 9.2 Hydrostatic-equivalent pressure head and pressure. 9.2 Pressure Concepts 377
378Chapter9PressureandVelocityMeasurementsSOLUTIONTheabsolutepressureisfounddirectlyfromEquation9.2.Usingthepressureatthe free surface as the reference pressure andthe datum line for ho,the absolute pressure must bePabs(h) = 1.0132 × 105 N/m2 +(997.4kg/m)(9.8m/s)(10 m)1 kg-m/N-s=1.9906 ×105N/m2absThis is equivalent to 199.06 kPa abs or 1.96 atm abs or 28.80 Ib/in.2 abs or 1.99 bar abs.The pressure can be described as a gauge pressure by referencing it to atmospheric pressure.FromEquation9.1,p(h) = Pabs - Po = h=9.7745×10*N/m2which is also equivalent to 97.7 kPa or 0.96 atm or 14.1 Ib/in.2 or 0.98 bar.Wecan express this pressureas an equivalent column of liquid,(1.9906×105)- (1.0132×105)N/m2h= Pabs - Po(998.2kg/m3)(9.8m/s2)(1N-s2/kg-m)pg.=10mH,09.3PRESSUREREFERENCEINSTRUMENTSThe units of pressure can be defined through the standards of the fundamental dimensions of mass,length, and time.In practice,pressure transducers are calibrated by comparison against certainreference instruments, which also serve as pressure measuring instruments.This section discussesseveral basic pressure reference instruments that can serve either as working standards or aslaboratoryinstruments.McLeod GaugeThe McLeod gauge, originally devised by Herbert McLeod in 1874 (3), is a pressure-measuringinstrumentand laboratoryreferencestandardusedtoestablishgaspressures inthesubatmosphericrange of 1 mm Hg abs down to0.1 mm Hg abs.Apressure that is below atmospheric pressure is alsocalled a vacuum pressure. One variation of this instrument is sketched in Figure 9.3a, in which thegauge is connected directlyto the low-pressure source.The glass tubing is arranged so that a sampleof thegas at an unknown low pressure can be trapped by inverting the gauge from the sensingposition, depicted as Figure 9.3a, to that of the measuring position, depicted as Figure 9.3b. In thisway,the gas trapped within the capillary is isothermally compressed by a rising column of mercury.Boyle'slawisthenusedtorelatethetwopressuresoneither sideofthemercurytothedistanceoftravel of the mercury within the capillary. Mercury is the preferred working fluid because of its highdensityandverylowvaporpressureAttheequilibrium andmeasuringposition,thecapillarypressure,p2,isrelatedtotheunknowngas pressure to be determined, pi, by P2 = p,(V/V2) where V, is the gas volume of the gauge inFigure 9.3a (a constant for a gauge at any pressure), and V2 is the capillary volume in Figure 9.3b.But V2=Ay,whereAis theknowncross-sectional areaof thecapillaryand yis thevertical lengthof
E1C09 09/14/2010 15:4:52 Page 378 SOLUTION The absolute pressure is found directly from Equation 9.2. Using the pressure at the free surface as the reference pressure and the datum line for h0, the absolute pressure must be pabsðhÞ ¼ 1:0132 105 N=m2 þ 997:4 kg=m3 ð Þ 9:8 m=s2 ð Þð Þ 10 m 1 kg-m=N-s2 ¼ 1:9906 105 N=m2 abs This is equivalent to 199.06 kPa abs or 1.96 atm abs or 28.80 lb/in.2 abs or 1.99 bar abs. The pressure can be described as a gauge pressure by referencing it to atmospheric pressure. From Equation 9.1, pðhÞ ¼ pabs p0 ¼ gh ¼ 9:7745 104 N=m2 which is also equivalent to 97.7 kPa or 0.96 atm or 14.1 lb/in.2 or 0.98 bar. We can express this pressure as an equivalent column of liquid, h ¼ pabs p0 rg ¼ 1:9906 105 1:0132 105 N=m2 998:2 kg=m3 ð Þ 9:8 m=s2 ð Þ 1 N-s2 ð Þ =kg-m ¼ 10 m H2O 9.3 PRESSURE REFERENCE INSTRUMENTS The units of pressure can be defined through the standards of the fundamental dimensions of mass, length, and time. In practice, pressure transducers are calibrated by comparison against certain reference instruments, which also serve as pressure measuring instruments. This section discusses several basic pressure reference instruments that can serve either as working standards or as laboratory instruments. McLeod Gauge The McLeod gauge, originally devised by Herbert McLeod in 1874 (3), is a pressure-measuring instrument and laboratory reference standard used to establish gas pressures in the subatmospheric range of 1 mm Hg abs down to 0.1 mm Hg abs. A pressure that is below atmospheric pressure is also called a vacuum pressure. One variation of this instrument is sketched in Figure 9.3a, in which the gauge is connected directly to the low-pressure source. The glass tubing is arranged so that a sample of the gas at an unknown low pressure can be trapped by inverting the gauge from the sensing position, depicted as Figure 9.3a, to that of the measuring position, depicted as Figure 9.3b. In this way, the gas trapped within the capillary is isothermally compressed by a rising column of mercury. Boyle’s law is then used to relate the two pressures on either side of the mercury to the distance of travel of the mercury within the capillary. Mercury is the preferred working fluid because of its high density and very low vapor pressure. At the equilibrium and measuring position, the capillary pressure, p2, is related to the unknown gas pressure to be determined, p1, by p2 ¼ p1ð81=82Þ where 81 is the gas volume of the gauge in Figure 9.3a (a constant for a gauge at any pressure), and 82 is the capillary volume in Figure 9.3b. But 82 ¼ Ay, where A is the known cross-sectional area of the capillary and y is the vertical length of 378 Chapter 9 Pressure and Velocity Measurements
3799.3PressureReferenceInstrumentsZero lineP2PressureMeasuringsensingcapillaryporReferencecapillary(a) Sensing position(b) Indicating positionFigure 9.3 McLeod gauge.the capillary occupied by the gas. With y as the specific weight of the mercury, the difference inpressures is related by P2 - P = yy such that the unknown gas pressure is just a function of y:(9.4)PI = Ay2 /(VI - Ay)In practice,a commercial McLeod gauge has the capillary etched and calibrated to indicateeither pressure, Pi, or its equivalent head, p/, directly. The McLeod gauge generally does notrequire correction. The reference stem offsets capillary forces acting in the measuring capillary.Instrument systematicuncertaintyis on theorderof0.5%(95%)at1mmHgabsand increasesto3%(95%) at 0.1 mm Hg abs.BarometerA barometerconsistsof aninverted tubecontaininga fluid andisusedto measureatmosphericpressure.To create the barometer, the tube, which is sealed at only one end, is evacuated to zeroabsolute pressure.The tube is immersed with the open end down within a liquid-filled reservoir asshown in the illustration of the Fortin barometer in Figure 9.4.The reservoir is open to atmosphericpressure, which forces the liquid to rise up the tube.From Equations 9.2and 9.3, the resulting height of the liquid column above the reservoir freesurface is a measure of the absolute atmospheric pressure in the equivalent head (Eq. 9.3).Evangelista Torricelli (1608-1647), a colleague of Galileo, can be credited with developing andinterpretingtheworkingprinciplesof thebarometerin1644.As Figure9.4 shows, the closed end of the tube is at the vapor pressure ofthebarometric liquidat room temperature. So the indicated pressure is the atmospheric pressure minus the liquid vaporpressure.Mercury is the most common liquid used because it has a very low vapor pressure, and so,for practical use, the indicated pressure can be taken as the local absolute barometric pressure.However,forveryaccurateworkthebarometer needs tobecorrectedfortemperatureeffects,which
E1C09 09/14/2010 15:4:52 Page 379 the capillary occupied by the gas. With g as the specific weight of the mercury, the difference in pressures is related by p2 p1 ¼ gy such that the unknown gas pressure is just a function of y: p1 ¼ gAy2 =ð81 AyÞ ð9:4Þ In practice, a commercial McLeod gauge has the capillary etched and calibrated to indicate either pressure, p1, or its equivalent head, p1/g, directly. The McLeod gauge generally does not require correction. The reference stem offsets capillary forces acting in the measuring capillary. Instrument systematic uncertainty is on the order of 0.5% (95%) at 1 mm Hg abs and increases to 3% (95%) at 0.1 mm Hg abs. Barometer A barometer consists of an inverted tube containing a fluid and is used to measure atmospheric pressure. To create the barometer, the tube, which is sealed at only one end, is evacuated to zero absolute pressure. The tube is immersed with the open end down within a liquid-filled reservoir as shown in the illustration of the Fortin barometer in Figure 9.4. The reservoir is open to atmospheric pressure, which forces the liquid to rise up the tube. From Equations 9.2 and 9.3, the resulting height of the liquid column above the reservoir free surface is a measure of the absolute atmospheric pressure in the equivalent head (Eq. 9.3). Evangelista Torricelli (1608–1647), a colleague of Galileo, can be credited with developing and interpreting the working principles of the barometer in 1644. As Figure 9.4 shows, the closed end of the tube is at the vapor pressure of the barometric liquid at room temperature. So the indicated pressure is the atmospheric pressure minus the liquid vapor pressure. Mercury is the most common liquid used because it has a very low vapor pressure, and so, for practical use, the indicated pressure can be taken as the local absolute barometric pressure. However, for very accurate work the barometer needs to be corrected for temperature effects, which p1 (a) Sensing position Pressure sensing port (b) Indicating position Reference capillary Zero line Measuring capillary p1 p2 y Figure 9.3 McLeod gauge. 9.3 Pressure Reference Instruments 379