《工程测试与信号处理》课程教学资源（文献资料）Pressure and Velocity

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

380Chapter9PressureandVelocityMeasurementsClosed endMercury vapor atvapor pressureMeniscusScaleLiquidmercuryTTTTTTTTTTGlasstubeGlass cylinderZeroingreservoirpointerAdjustingscrewFigure9.4Fortin barometer.change the vapor pressure,for temperature and altitude effects on the weight ofmercury,and fordeviations from standard gravity (9.80665 m/s2 or 32.17405 f/s).Correction curves are providedbyinstrumentmanufacturersBarometers areusedaslocal standardsforthemeasurementof absoluteatmosphericpressure.Under standard conditions for pressure temperature and gravity, the mercury rises 760 mm (29.92in.)above the reservoir surface.The U.S.National Weather Service alwaysreports a barometricpressure that has been corrected to sea-level elevation.ManometerA manometeris an instrumentusedto measure differential pressure based onthe relationshipbetween pressure and the hydrostatic equivalent head of fluid. Several design variations are

E1C09 09/14/2010 15:4:52 Page 380 change the vapor pressure, for temperature and altitude effects on the weight of mercury, and for deviations from standard gravity (9.80665 m/s2 or 32.17405 ft/s2 ). Correction curves are provided by instrument manufacturers. Barometers are used as local standards for the measurement of absolute atmospheric pressure. Under standard conditions for pressure temperature and gravity, the mercury rises 760 mm (29.92 in.) above the reservoir surface. The U.S. National Weather Service always reports a barometric pressure that has been corrected to sea-level elevation. Manometer A manometer is an instrument used to measure differential pressure based on the relationship between pressure and the hydrostatic equivalent head of fluid. Several design variations are Glass cylinder reservoir Zeroing pointer Adjusting screw Mercury vapor at vapor pressure Closed end Meniscus Liquid mercury Glass tube Scale Figure 9.4 Fortin barometer. 380 Chapter 9 Pressure and Velocity Measurements

9.3381PressureReferenceInstruments+日上Figure 9.5 U-tube manometer.available,allowingmeasurementsrangingfrom theorder of0.001mmofmanometerfluid to severalmeters.The U-tube manometer in Figure 9.5 consists of a transparent tube filled with an indicatingliquid of specific weight Ym.This forms two free surfaces of the manometer liquid. The difference inpressures P, and p2 applied across the two free surfaces brings about a deflection, H, in the level ofthe manometer liquid.For a measured fluid of specific weight ,the hydrostatic equation can beapplied tothemanometerof Figure9.5asPi =P2+x+mH-(H +x)which yields the relation between the manometer deflection and applied differential pressure,(9.5)Pi-P2=(Ym-)HFrom Equation 9.5,the static sensitivity of the U-tube manometer is given byK =1/(m-).To maximize manometer sensitivity,we want to choose manometer liquidsthat minimize the value of (Ym-).From a practical standpoint the manometer fluid must notbe soluble with the working fluid. The manometer fluid should be selected to provide adeflection that is measurable yet not so great that it becomes awkward to observe.A variation in the U-tubemanometer is the micromanometer shown in Figure 9.6.Thesespecial-purpose instruments are used to measure very small differential pressures,down to0.005mmH,O (0.0002 in.H,O).In the micromanometer,themanometer reservoirismovedup or down until the level of the manometer fluid within the reservoir is at the same level as a setmark within a magnifying sight glass. At that point the manometer meniscus is at the set mark, andthis serves as a reference position.Changes in pressure bring about fluid displacement so that thereservoir mustbe moved upor down tobring the meniscus back to the setmark.The amount of thisrepositioning is equal to the change in the equivalent pressure head. The position of the reservoir iscontrolled by a micrometer or other calibrated displacement measuring device so that relativechanges in pressurecan bemeasured withhigh resolution

E1C09 09/14/2010 15:4:53 Page 381 available, allowing measurements ranging from the order of 0.001 mm of manometer fluid to several meters. The U-tube manometer in Figure 9.5 consists of a transparent tube filled with an indicating liquid of specific weight gm. This forms two free surfaces of the manometer liquid. The difference in pressures p1 and p2 applied across the two free surfaces brings about a deflection, H, in the level of the manometer liquid. For a measured fluid of specific weight g, the hydrostatic equation can be applied to the manometer of Figure 9.5 as p1 ¼ p2 þ gx þ gmH  gðH þ xÞ which yields the relation between the manometer deflection and applied differential pressure, p1  p2 ¼ ð Þ gm  g H ð9:5Þ From Equation 9.5, the static sensitivity of the U-tube manometer is given by K ¼ 1=ðgm  gÞ. To maximize manometer sensitivity, we want to choose manometer liquids that minimize the value of (gm  g). From a practical standpoint the manometer fluid must not be soluble with the working fluid. The manometer fluid should be selected to provide a deflection that is measurable yet not so great that it becomes awkward to observe. A variation in the U-tube manometer is the micromanometer shown in Figure 9.6. These special-purpose instruments are used to measure very small differential pressures, down to 0.005 mm H2O (0.0002 in. H2O). In the micromanometer, the manometer reservoir is moved up or down until the level of the manometer fluid within the reservoir is at the same level as a set mark within a magnifying sight glass. At that point the manometer meniscus is at the set mark, and this serves as a reference position. Changes in pressure bring about fluid displacement so that the reservoir must be moved up or down to bring the meniscus back to the set mark. The amount of this repositioning is equal to the change in the equivalent pressure head. The position of the reservoir is controlled by a micrometer or other calibrated displacement measuring device so that relative changes in pressure can be measured with high resolution. H x p1 p2 m Figure 9.5 U-tube manometer. 9.3 Pressure Reference Instruments 381

382Chapter9PressureandVelocityMeasurementsPiInclined tubeMeniscMicrometeradjustingscrewSetmark王ReservoirFlexibletubeFigure 9.6 Micromanometer.The inclined tube manometer is also used to measure small changes in pressure.It is essentiallya U-tube manometer with one leg inclined at an angle , typically from 10 to 30 degrees relative tothe horizontal.As indicated in Figure 9.7,a change in pressure equivalent to a deflection of height Hin a U-tube manometer would bring about a change in position ofthemeniscus in the inclined leg ofL =H/sin e. This provides an increased sensitivity over the conventional U-tube by the factor of1/sin 0.A number of elemental errors affect the instrument uncertainty of all types of manometers.These include scale and alignment errors, zero error, temperature error,gravity error, and capillaryand meniscus errors.The specific weight of the manometer fluid varies with temperaturebutcan becorrected. For example, the common manometer fluid of mercury has a temperature dependenceapproximatedby133.084848.707[1b/ft3]Yig= 1 +0.000 [N/m] 1+ 0.000101(T 32)with T in C or °F, respectively.A gravity correction for elevation z and latitude corrects forgravity erroreffects using the dimensionless correction,(9.6a)ej = -(2.637 ×10-3 cos2Φ + 9.6 × 108 z + 5 × 10-5)us=-(2.637×10-3cos2Φ+2.9×10-8z+5×10-5(9.6b)where Φ is in degrees and z is in feet for Equation 9.6a and meters in Equation 9.6b. Tube-to-liquidcapillary forces lead to the development of a meniscus.Although the actual effect varies with purityof themanometer liquid, these effects can be minimized by using manometer tubebores of greaterthan about 6 mm (0.25 in.).In general, the instrument uncertainty in measuring pressure can be aslowas0.02%to0.2%ofthereading

E1C09 09/14/2010 15:4:53 Page 382 The inclined tube manometer is also used to measure small changes in pressure. It is essentially a U-tube manometer with one leg inclined at an angle u, typically from 10 to 30 degrees relative to the horizontal. As indicated in Figure 9.7, a change in pressure equivalent to a deflection of height H in a U-tube manometer would bring about a change in position of the meniscus in the inclined leg of L ¼ H/sin u. This provides an increased sensitivity over the conventional U-tube by the factor of 1/sin u. A number of elemental errors affect the instrument uncertainty of all types of manometers. These include scale and alignment errors, zero error, temperature error, gravity error, and capillary and meniscus errors. The specific weight of the manometer fluid varies with temperature but can be corrected. For example, the common manometer fluid of mercury has a temperature dependence approximated by gHg ¼ 133:084 1 þ 0:00006T N=m3   ¼ 848:707 1 þ 0:000101ð Þ T  32 lb=ft3   with T in C or F, respectively. A gravity correction for elevation z and latitude f corrects for gravity error effects using the dimensionless correction, e1 ¼ ð2:637 103 cos2f þ 9:6 108 z þ 5 105 ÞUS ð9:6aÞ ¼ ð2:637 103 cos2f þ 2:9 108 z þ 5 105 Þmetric ð9:6bÞ where f is in degrees and z is in feet for Equation 9.6a and meters in Equation 9.6b. Tube-to-liquid capillary forces lead to the development of a meniscus. Although the actual effect varies with purity of the manometer liquid, these effects can be minimized by using manometer tube bores of greater than about 6 mm (0.25 in.). In general, the instrument uncertainty in measuring pressure can be as low as 0.02% to 0.2% of the reading. Inclined tube Meniscus Set mark Micrometer adjusting Reservoir screw Flexible tube H p1 p2 Figure 9.6 Micromanometer. 382 Chapter 9 Pressure and Velocity Measurements

9.3383PressureReferenceInstrumentsP1InclinedtubeH=LsineTFigure 9.7 Inclined tubeLOmanometer.Example 9.2A high-quality U-tube manometer is a remarkably simple instrument to make. It requires only atransparent U-shaped tube, manometer fluid, and a scale to measure deflection. While a U-shapedglass tube of 6 mm or greater internal bore is preferred, a length of 6-mm i.d. (inside diameter) thick-walled clear tubing from the hardware store and bent to a U-shape works finefor many purposes.Water, alcohol, or mineral oil are all readily available nontoxic manometer fluids with tabulatedproperties. A sheet of graph paper or a ruler serves to measure meniscus deflection. There is a limitto the magnitude of pressure differential that can be measured, although use of a step stool extendsthis range.Tack the components to a board and the resulting instrument is accurate for measuringmanometer deflections down to one-half the resolution of the scale in terms of fluid used!The U-tube manometer is a practical tool useful for calibrating other forms of pressuretransducers in the pressure range spanning atmospheric pressure levels. As one example, the ad-hocU-tube described is convenient for calibrating surgical pressure transducers over the physiologicalpressure ranges.Example9.3An inclined manometer with the inclined tube set at 30 degrees is to be used at 20°C to measure anair pressureof nominal magnitudeof 100N/mrelative to ambient.Manometer"unity"oil (S=1)is to be used.The specific weight of the oil is 9770 ± 0.5% N/m(95%)at 20°C, the angle ofinclination can be set to within 1 degree using a bubble level, and the manometer resolution is 1 mmwith a manometer zero error equal to its interpolation error.Estimate the uncertainty in indicateddifferential pressure at thedesign stage.KNOWNp=100N/m~(nominal)ManometerResolution:1mmZeroerror:0.5mm = 30 ± 1° (95% assumed)Ym=9770±0.5%N/m2(95%)

E1C09 09/14/2010 15:4:53 Page 383 Example 9.2 A high-quality U-tube manometer is a remarkably simple instrument to make. It requires only a transparent U-shaped tube, manometer fluid, and a scale to measure deflection. While a U-shaped glass tube of 6 mm or greater internal bore is preferred, a length of 6-mm i.d. (inside diameter) thick￾walled clear tubing from the hardware store and bent to a U-shape works fine for many purposes. Water, alcohol, or mineral oil are all readily available nontoxic manometer fluids with tabulated properties. A sheet of graph paper or a ruler serves to measure meniscus deflection. There is a limit to the magnitude of pressure differential that can be measured, although use of a step stool extends this range. Tack the components to a board and the resulting instrument is accurate for measuring manometer deflections down to one-half the resolution of the scale in terms of fluid used! The U-tube manometer is a practical tool useful for calibrating other forms of pressure transducers in the pressure range spanning atmospheric pressure levels. As one example, the ad-hoc U-tube described is convenient for calibrating surgical pressure transducers over the physiological pressure ranges. Example 9.3 An inclined manometer with the inclined tube set at 30 degrees is to be used at 20C to measure an air pressure of nominal magnitude of 100 N/m2 relative to ambient. Manometer ‘‘unity’’ oil (S ¼ 1) is to be used. The specific weight of the oil is 9770  0.5% N/m2 (95%) at 20C, the angle of inclination can be set to within 1 degree using a bubble level, and the manometer resolution is 1 mm with a manometer zero error equal to its interpolation error. Estimate the uncertainty in indicated differential pressure at the design stage. KNOWN p ¼ 100 N/m2 (nominal) Manometer Resolution: 1 mm Zero error: 0.5 mm u ¼ 30  1 (95% assumed) gm ¼ 9770  0.5% N/m3 (95%) p2 p1 Inclined tube 0 L H = L sin Figure 9.7 Inclined tube manometer. 9.3 Pressure Reference Instruments 383

384Chapter9PressureandVelocityMeasurementsASSUMPTIONSTemperature and capillary effects in themanometer and gravity error in thespecific weights of the fluids are negligible.The degrees of freedom in the stated uncertainties arelarge.FINDudSOLUTION The relation between pressure andmanometerdeflection is given byEquation9.5with H=L sin e:Ap=Pi-P2= L(m=)sin ewhere P2 is the ambient pressure so that Ap is the nominal pressure relative to the ambient.For anominalAp=10oN/m2,thenominal manometerriseLwouldbeApApL=21mmmsin(m-)sinwhere Ym 》 and the value for and its uncertainty are neglected. For the design stage analysis,p =f(m,L, ), so that the uncertainty in pressure, Ap, is estimated byToApTo4p++(ud),(ud)(ud),alaeAt assumed 95% confidence levels,the manometer specific weight uncertainty and angleuncertaintyareestimatedfromtheproblemas(ua)m= (9770N/m2)(0.005)~49N/m3(ua)。=1degree = 0.0175 radThe uncertainty in estimating the pressure from the indicated deflection is due both to themanometer resolution, uo, and the zero point offset error, which we take as its instrument error, ue.Using the uncertainties associated with these errors,(ud)z= /ug +u = V(0.5 mm)2 + (0.5mm)2= 0.7mmEvaluating the derivatives and substituting values gives a design-stage uncertainty interval of ameasuredAp of(ua)ap= (0.26)2 + (3.42) + (3.10) = ±4.6N/m2 (95%)COMMENTAta 30-degree inclination and for this pressure, theuncertainty in pressure isaffected almostequallybytheinstrument inclination andthedeflection uncertainties.Asthemanometer inclination is increased to a more vertical orientation, that is, toward the U-tubemanometer,inclination uncertainty becomesless important andis negligible near 90 degrees.However,foraU-tubemanometer,thedeflectionisreducedtolessthan11mm,a50%reductioninmanometer sensitivity,with an associated design-stage uncertainty of 6.8N/m(95%).DeadweightTestersThe deadweight tester makes direct use of thefundamental definition of pressure as a force perunit area to create and to determine the pressure within a sealed chamber.These devices are a

E1C09 09/14/2010 15:4:53 Page 384 ASSUMPTIONS Temperature and capillary effects in the manometer and gravity error in the specific weights of the fluids are negligible. The degrees of freedom in the stated uncertainties are large. FIND ud SOLUTION The relation between pressure and manometer deflection is given by Equation 9.5 with H ¼ L sin u: Dp ¼ p1  p2 ¼ Lðgm  gÞ sin u where p2 is the ambient pressure so that Dp is the nominal pressure relative to the ambient. For a nominal Dp ¼ 100 N/m2 , the nominal manometer rise L would be L ¼ Dp ð Þ gm  g sin u  Dp gmsin u ¼ 21 mm where gm g and the value for g and its uncertainty are neglected. For the design stage analysis, p ¼ fðgm; L; uÞ, so that the uncertainty in pressure, Dp, is estimated by ð Þ ud p ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qDp qgm ð Þ ud gm  2 þ qDp qL ð Þ ud L  2 þ qDp qu ð Þ ud u  2 s At assumed 95% confidence levels, the manometer specific weight uncertainty and angle uncertainty are estimated from the problem as ð Þ ud gm ¼ 9770 N=m3 ð Þð Þ 0:005 49 N=m3 ð Þ ud u ¼ 1 degree ¼ 0:0175 rad The uncertainty in estimating the pressure from the indicated deflection is due both to the manometer resolution, uo, and the zero point offset error, which we take as its instrument error, uc. Using the uncertainties associated with these errors, ð Þ ud L ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 o þ u2 c q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:5 mmÞ 2 þ ð0:5 mmÞ 2 q ¼ 0:7 mm Evaluating the derivatives and substituting values gives a design-stage uncertainty interval of a measured Dp of ð Þ ud Dp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:26Þ 2 þ ð3:42Þ 2 þ ð3:10Þ 2 q ¼ 4:6N=m2 ð95%Þ COMMENT At a 30-degree inclination and for this pressure, the uncertainty in pressure is affected almost equally by the instrument inclination and the deflection uncertainties. As the manometer inclination is increased to a more vertical orientation, that is, toward the U-tube manometer, inclination uncertainty becomes less important and is negligible near 90 degrees. However, for a U-tube manometer, the deflection is reduced to less than 11 mm, a 50% reduction in manometer sensitivity, with an associated design-stage uncertainty of 6.8 N/m2 (95%). Deadweight Testers The deadweight tester makes direct use of the fundamental definition of pressure as a force per unit area to create and to determine the pressure within a sealed chamber. These devices are a 384 Chapter 9 Pressure and Velocity Measurements