9.3385PressureReferenceInstrumentsAppliedforceFistoOil reservoirAdjustableReference/plungerpressure porOilpFigure9.8Deadweighttester.common laboratory standard for the calibration of pressure-measuring devices over thepressure range from 70 to 7 × 107 N/m2 (0.01 to 10,000 psi). A deadweight tester, such asthat shown in Figure 9.8, consists of an internal chamber filled with an oil and a close-fittingpiston and cylinder.Chamber pressure acts on the end of the carefully machined piston.A staticequilibrium exists when the external pressure exerted by the piston on the fluid balances thechamber pressure.This external piston pressure is created by a downward force acting over theequivalent area Ae of the piston. The weight of the piston plus the additional weight ofcalibrated masses are used to provide this external force F.At static equilibrium the pistonfloats and the chamber pressure can be deduced asF(9.7)Cerrorcorrections>P=AeA pressure transducer can be connected to a reference port and calibrated by comparison to thechamber pressure.For most calibrations, the error corrections can be ignored.When error corrections are applied, the instrument uncertainty in the chamber pressure using adeadweight tester can be as low as 0.005% to0.01% of the reading.Anumber of elemental errorscontributeto Equation 9.7,including air buoyancyeffects,variations in local gravity,uncertainty intheknown mass of thepiston and addedmasses,sheareffects,thermal expansionofthepiston area,and elastic deformation of the piston (1).An indicated pressure,Pi,can be correctedfor gravity effects, e,from Equation 9.6a or 9.6b,and for air buoyancy effects, ez, by(9.8)p=p;(1 +ei +e2)where(9.9)e2=-air/masseThe testerfluid lubricates thepiston so that the piston is partially supportedby the shear forcesin the oil in the gap separating the piston and the cylinder. This error varies inversely with the testerfluid viscosity, so high-viscosity fluids are preferred.In a typical tester, the uncertainty due to thiserroris less than 0.01%of thereading.At high pressures,elastic deformation of thepiston affects theactual piston area. For this reason, the effective area is based on the average of the piston andcylinder diameters
E1C09 09/14/2010 15:4:53 Page 385 common laboratory standard for the calibration of pressure-measuring devices over the pressure range from 70 to 7 107 N/m2 (0.01 to 10,000 psi). A deadweight tester, such as that shown in Figure 9.8, consists of an internal chamber filled with an oil and a close-fitting piston and cylinder. Chamber pressure acts on the end of the carefully machined piston. A static equilibrium exists when the external pressure exerted by the piston on the fluid balances the chamber pressure. This external piston pressure is created by a downward force acting over the equivalent area Ae of the piston. The weight of the piston plus the additional weight of calibrated masses are used to provide this external force F. At static equilibrium the piston floats and the chamber pressure can be deduced as p ¼ F Ae þXerror corrections ð9:7Þ A pressure transducer can be connected to a reference port and calibrated by comparison to the chamber pressure. For most calibrations, the error corrections can be ignored. When error corrections are applied, the instrument uncertainty in the chamber pressure using a deadweight tester can be as low as 0.005% to 0.01% of the reading. A number of elemental errors contribute to Equation 9.7, including air buoyancy effects, variations in local gravity, uncertainty in the known mass of the piston and added masses, shear effects, thermal expansion of the piston area, and elastic deformation of the piston (1). An indicated pressure, pi, can be corrected for gravity effects, e1, from Equation 9.6a or 9.6b, and for air buoyancy effects, e2, by p ¼ pið Þ 1 þ e1 þ e2 ð9:8Þ where e2 ¼ gair=gmasses ð9:9Þ The tester fluid lubricates the piston so that the piston is partially supported by the shear forces in the oil in the gap separating the piston and the cylinder. This error varies inversely with the tester fluid viscosity, so high-viscosity fluids are preferred. In a typical tester, the uncertainty due to this error is less than 0.01% of the reading. At high pressures, elastic deformation of the piston affects the actual piston area. For this reason, the effective area is based on the average of the piston and cylinder diameters. Reference pressure port Oil p Oil reservoir Piston Applied force Adjustable plunger Ae F Figure 9.8 Deadweight tester. 9.3 Pressure Reference Instruments 385
386Chapter9PressureandVelocityMeasurementsExample9.4A deadweight tester indicates 100.00 Ib/in.2 (i.e., 100.00 psi), at 70°F in Clemson, SC (Φ = 34°z=841ft).Manufacturer specificationsfortheeffectivepistonarea were stated at72°F so thatthermal expansion effects remain negligible.Take Yair=0.076Ib/ft andYmass =496 Ib/ft.Correctthe indicated reading for known errors.KNOWNpi=100.00psi;z=841ft;Φ=34°ASSUMPTION Systematic error corrections foraltitude and latitude applyFINDPSOLUTION The corrected pressure is found usingEquation 9.8.From Equation 9.9,thecorrection for buoyancy effects ise2=air/masses =-0.076/496=-0.000154The correction for gravity effects is from Equation 9.6a:e1 = -(2.637× 10-3 cos2Φ+ 9.6×10-8z + 5×10-5)-(0.0010+8×105+5×10-5)=-0.001119FromEquation9.8,thecorrectedpressurebecomesp = 100.00 × (1 - 0.000154 - 0.001119) Ib/in.2= 99.87 b/in.2COMMENT This amounts to correcting an indicated signal for known systematic errors. Herethat correction is~0.13%.9.4PRESSURETRANSDUCERSA pressure transducer is a device that converts a measured pressure into a mechanical or electricalsignal. The transducer is actually a hybrid sensor-transducer. The primary sensor is usually an elasticelement thatdeforms ordeflects underthemeasuredpressure relativeto a referencepressure.Severalcommonelasticelements used, as shown in Figure 9.9,includethe Bourdon tube,bellows, capsule, anddiaphragm. A secondary transducer element converts the elastic element deflection into a readilymeasurable signal such as an electrical voltage or mechanical rotation of a pointer. There are manymethods availabletoperform this transducerfunction, and we examine a few common ones.General categories for pressure transducers are absolute, gauge, vacuum, and differential.These categories reflect the application and reference pressure used. Absolute transducers have asealed reference cavity held at a pressure of absolute zero, enabling absolute pressure measure-ments.Gaugetransducershavethereferencecavityopentoatmosphericpressureandareintendedto measure above or below atmospheric pressure or both.Differential transducers measure thedifferencebetweentwo appliedpressures.Vacuumtransducers area specialform of absolutetransducer for low-pressure measurements.Pressure transducers are subject to some or all of the following elemental errors: resolution,zero shift error, linearity error, sensitivity error, hysteresis, noise, and drift due to environmentaltemperature changes.Electrical transducers are also subjectto loading errorbetween the transduceroutput and its indicating device(see Chapter 6).Loading errors increase the transducer nonlinearity
E1C09 09/14/2010 15:4:53 Page 386 Example 9.4 A deadweight tester indicates 100.00 lb/in.2 (i.e., 100.00 psi), at 70F in Clemson, SC (f ¼ 34, z ¼ 841 ft). Manufacturer specifications for the effective piston area were stated at 72F so that thermal expansion effects remain negligible. Take gair ¼ 0.076 lb/ft3 and gmass ¼ 496 lb/ft3 . Correct the indicated reading for known errors. KNOWN pi ¼ 100:00 psi; z ¼ 841 ft; f ¼ 34 ASSUMPTION Systematic error corrections for altitude and latitude apply FIND p SOLUTION The corrected pressure is found using Equation 9.8. From Equation 9.9, the correction for buoyancy effects is e2 ¼ gair=gmasses ¼ 0:076=496 ¼ 0:000154 The correction for gravity effects is from Equation 9.6a: e1 ¼ ð2:637 103 cos2f þ 9:6 108 z þ 5 105Þ ¼ ð0:0010 þ 8 105 þ 5 105Þ¼0:001119 From Equation 9.8, the corrected pressure becomes p ¼ 100:00 ð1 0:000154 0:001119Þ lb=in: 2 ¼ 99:87 lb=in: 2 COMMENT This amounts to correcting an indicated signal for known systematic errors. Here that correction is 0.13%. 9.4 PRESSURE TRANSDUCERS A pressure transducer is a device that converts a measured pressure into a mechanical or electrical signal. The transducer is actually a hybrid sensor-transducer. The primary sensor is usually an elastic element that deforms or deflects under the measured pressure relative to a reference pressure. Several common elastic elements used, as shown in Figure 9.9, include the Bourdontube, bellows, capsule, and diaphragm. A secondary transducer element converts the elastic element deflection into a readily measurable signal such as an electrical voltage or mechanical rotation of a pointer. There are many methods available to perform this transducer function, and we examine a few common ones. General categories for pressure transducers are absolute, gauge, vacuum, and differential. These categories reflect the application and reference pressure used. Absolute transducers have a sealed reference cavity held at a pressure of absolute zero, enabling absolute pressure measurements. Gauge transducers have the reference cavity open to atmospheric pressure and are intended to measure above or below atmospheric pressure or both. Differential transducers measure the difference between two applied pressures. Vacuum transducers are a special form of absolute transducer for low-pressure measurements. Pressure transducers are subject to some or all of the following elemental errors: resolution, zero shift error, linearity error, sensitivity error, hysteresis, noise, and drift due to environmental temperature changes. Electrical transducers are also subject to loading error between the transducer output and its indicating device (see Chapter 6). Loading errors increase the transducer nonlinearity 386 Chapter 9 Pressure and Velocity Measurements
3879.4PressureTransducersMotionP2psMotionMotion[P1C-shaped Bourdon tubeBellowsDiaphragmQ2MotionMotionCorrugated diaphragmCapsuleFigure 9.9 Elastic elements used as pressure sensors.over its operating range.When this is a consideration, a voltage follower (see Chapter 6)can beinserted at the output of thetransducerto isolatetransducer load.Bourdon TubeTheBourdontubeis a curvedmetal tubehaving an elliptical cross section thatmechanicallydeforms under pressure. In practice, one end of the tube is held fixed and the input pressure appliedintermally.Apressuredifferencebetween theoutsideof thetube and theinsideof thetubebringsabouttubedeformationandadeflectionof thetubefreeend.Thisactionof thetubeunderpressurecan be likened to the action of a deflated balloon that is subsequently inflated.Themagnitudeof thedeflection of the tube end is proportional to the magnitude of the pressure difference. Severalvariations exist, such as the C shape (Fig.9.9), the spiral, and the twisted tube. The exterior of thetube is usually open to atmosphere (hence, the origin of the term “"gauge"pressure referring topressure referenced to atmospheric pressure), but in some variations the tube may be placed within asealedhousing and the tube exteriorexposed to some otherreferencepressure,allowing for absoluteand fordifferential designs.TheBourdontubemechanical dial gaugeisa commonlyusedpressuretransducer.A typicaldesign is shown in Figure 9.10, in which the secondary element is a mechanical linkage that convertsthe tube displacement intoa rotation ofa pointer.Designs existfor loworhigh pressures,includingvacuum pressures, and selections span a wide choice in range.The best Bourdon tube gauges haveinstrument uncertainties as lowas 0.1%of thefull-scaledeflectionofthegauge,withvalues of0.5%to 2% more common.But the attractiveness of this device is that it is simple,portable,and robust,lastingforyears ofuse
E1C09 09/14/2010 15:4:53 Page 387 over its operating range. When this is a consideration, a voltage follower (see Chapter 6) can be inserted at the output of the transducer to isolate transducer load. Bourdon Tube The Bourdon tube is a curved metal tube having an elliptical cross section that mechanically deforms under pressure. In practice, one end of the tube is held fixed and the input pressure applied internally. A pressure difference between the outside of the tube and the inside of the tube brings about tube deformation and a deflection of the tube free end. This action of the tube under pressure can be likened to the action of a deflated balloon that is subsequently inflated. The magnitude of the deflection of the tube end is proportional to the magnitude of the pressure difference. Several variations exist, such as the C shape (Fig. 9.9), the spiral, and the twisted tube. The exterior of the tube is usually open to atmosphere (hence, the origin of the term ‘‘gauge’’ pressure referring to pressure referenced to atmospheric pressure), but in some variations the tube may be placed within a sealed housing and the tube exterior exposed to some other reference pressure, allowing for absolute and for differential designs. The Bourdon tube mechanical dial gauge is a commonly used pressure transducer. A typical design is shown in Figure 9.10, in which the secondary element is a mechanical linkage that converts the tube displacement into a rotation of a pointer. Designs exist for low or high pressures, including vacuum pressures, and selections span a wide choice in range. The best Bourdon tube gauges have instrument uncertainties as low as 0.1% of the full-scale deflection of the gauge, with values of 0.5% to 2% more common. But the attractiveness of this device is that it is simple, portable, and robust, lasting for years of use. p2 p2 p1 p2 p1 p1 p2 Motion Motion Motion C-shaped Bourdon tube swolleB mgarhpaiD Motion Corrugated diaphragm Capsule p1 p1 p2 Motion Figure 9.9 Elastic elements used as pressure sensors. 9.4 Pressure Transducers 387
388Chapter9PressureandVelocityMeasurementsBourdontubeClosed endSoringoPinionhhi./MechanicallinkageScaleOpento appliedpressureFigure 9.10 Bourdon tube pressureAppliedpressuregauge.BellowsandCapsuleElementsA bellows sensing element is a thin-walled,flexiblemetal tubeformed into deep convolutions andsealed at one end (Fig. 9.9). One end is held fixed and pressure is applied internally. A differencebetween the internal and external pressures causes the bellows to change in length.Thebellows ishoused within a chamber that can be sealed and evacuated for absolute measurements, ventedthrough a reference pressure port for differential measurements, or opened to atmosphere for gaugepressure measurements.A similar design,the capsule sensing element,is also a thin-walled,flexiblemetal tube whose length changes with pressure, but its shape tends to be wider in diameter andshorter in length (Fig. 9.9).Amechanical linkageis usedto convertthetranslational displacementof thebellows orcapsulesensors into a measurable form.A common transducer is the sliding arm potentiometer (voltage-divider,Chapter 6)found in the potentiometric pressure transducer shown in Figure 9.11.Anothertype uses a linear variabledisplacementtransducer (LVDT; see Chapter12)tomeasure the bellowsIP1xP2R.-E.R.Figure9.11Potentiometer pressureE,transducer
E1C09 09/14/2010 15:4:53 Page 388 Bellows and Capsule Elements A bellows sensing element is a thin-walled, flexible metal tube formed into deep convolutions and sealed at one end (Fig. 9.9). One end is held fixed and pressure is applied internally. A difference between the internal and external pressures causes the bellows to change in length. The bellows is housed within a chamber that can be sealed and evacuated for absolute measurements, vented through a reference pressure port for differential measurements, or opened to atmosphere for gauge pressure measurements. A similar design, the capsule sensing element, is also a thin-walled, flexible metal tube whose length changes with pressure, but its shape tends to be wider in diameter and shorter in length (Fig. 9.9). A mechanical linkage is used to convert the translational displacement of the bellows or capsule sensors into a measurable form. A common transducer is the sliding arm potentiometer (voltagedivider, Chapter 6) found in the potentiometric pressure transducer shown in Figure 9.11. Another type uses a linear variable displacement transducer (LVDT; see Chapter 12) to measure the bellows Spring Bourdon tube Scale Applied pressure Open to applied pressure Pinion Mechanical linkage Closed end A –– A A A Sector Figure 9.10 Bourdon tube pressure gauge. RL Ei p1 p2 Rx Eo x L Figure 9.11 Potentiometer pressure transducer. 388 Chapter 9 Pressure and Velocity Measurements
9.4PressureTransducers389orcapsuledisplacement.TheLVDT design has a high sensitivityandiscommonlyfound inpressuretransducers rated forlow pressures and for small pressure ranges,such as zero to severalhundred mm Hg absolute, gauge, or differential.DiaphragmsAn effective primarypressure element is a diaphragm (Figure9.9), which is a thin elastic circularplate supported about its circumference.The action of a diaphragm within a pressure transducer issimilar to the action of a trampoline; a pressure differential on the top andbottom diaphragm facesacts to deform it. The magnitude of the deformation is proportional to the pressure difference. Bothmembraneandcorrugateddesigns areused.Membranesaremadeof metalornonmetallicmaterial,such as plastic or neoprene. The material chosen depends on the pressure range anticipated and thefluid in contact with it. Corrugated diaphragms contain a number of corrugations that serve toincrease diaphragm stiffness and to increase the diaphragm effective surface area.Pressure transducers that use a diaphragm sensorare well suited for either static or dynamicpressure measurements. They have good linearity and resolution over their useful range. Anadvantage of the diaphragm sensor is that the very low mass and relative stiffness of the thindiaphragm give the sensor a very high natural frequency with a small damping ratio.Hence, thesetransducers can have a very wide frequency response and very short 90% rise and settling times.Thenatural frequency (rad/s)of a circulardiaphragm canbe estimated by (4)Emt?Wn=10.21(9.10)12(1-up)pr4where Em is the material bulk modulus (psi or N/m), t the thickness (in. or m), r the radius (in. or m),p the material density (lbm/in."or kg/m), and up the Poisson's ratio for the diaphragm material. Themaximum elastic deflection of a uniformly loaded, circular diaphragm supported about itscircumference occurs at itscenterand can be estimated by3(pi -p2)(1 - up) r4(9.11)Ymax16Emt3provided that the deflection does not exceed one-third the diaphragm thickness. Diaphragms shouldbe selected so as to not exceed this maximum deflection over the anticipated operating rangeVarious secondary elements are availabletotranslatethis displacementof thediaphragm intoameasurable signal.Several methods arediscussedbelow.StrainGaugeElementsA common method for converting diaphragm displacement into a measurable signal is to sense thestraininducedonthediaphragmsurfaceasitisdisplaced.Straingauges,deviceswhosemeasurableresistance is proportional to their sensed strain (see Chapter 11), can be bonded directly onto thediaphragm, integrated within the diaphragm material or onto a deforming element (such as a thinbeam)attached to the diaphragm so as to deform with thediaphragm and to sense strain.Metal straingauges can be used with liquids. Strain gauge resistance is reasonably linear over a wide range ofstrain and can be directly related to the sensed pressure (5).A diaphragm transducer using straingauge detection is depicted in Figure 9.12
E1C09 09/14/2010 15:4:53 Page 389 or capsule displacement. The LVDT design has a high sensitivity and is commonly found in pressure transducers rated for low pressures and for small pressure ranges, such as zero to several hundred mm Hg absolute, gauge, or differential. Diaphragms An effective primary pressure element is a diaphragm (Figure 9.9), which is a thin elastic circular plate supported about its circumference. The action of a diaphragm within a pressure transducer is similar to the action of a trampoline; a pressure differential on the top and bottom diaphragm faces acts to deform it. The magnitude of the deformation is proportional to the pressure difference. Both membrane and corrugated designs are used. Membranes are made of metal or nonmetallic material, such as plastic or neoprene. The material chosen depends on the pressure range anticipated and the fluid in contact with it. Corrugated diaphragms contain a number of corrugations that serve to increase diaphragm stiffness and to increase the diaphragm effective surface area. Pressure transducers that use a diaphragm sensor are well suited for either static or dynamic pressure measurements. They have good linearity and resolution over their useful range. An advantage of the diaphragm sensor is that the very low mass and relative stiffness of the thin diaphragm give the sensor a very high natural frequency with a small damping ratio. Hence, these transducers can have a very wide frequency response and very short 90% rise and settling times. The natural frequency (rad/s) of a circular diaphragm can be estimated by (4) vn ¼ 10:21 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Emt2 12 1 y2 p rr4 vuut ð9:10Þ where Em is the material bulk modulus (psi or N/m2 ), t the thickness (in. or m),rthe radius (in. or m), r the material density (lbm/in.3 or kg/m3 ), and yp the Poisson’s ratio for the diaphragm material. The maximum elastic deflection of a uniformly loaded, circular diaphragm supported about its circumference occurs at its center and can be estimated by ymax ¼ 3 p1 p2 ð Þ 1 y2 p r4 16Emt3 ð9:11Þ provided that the deflection does not exceed one-third the diaphragm thickness. Diaphragms should be selected so as to not exceed this maximum deflection over the anticipated operating range. Various secondary elements are available to translate this displacement of the diaphragm into a measurable signal. Several methods are discussed below. Strain Gauge Elements A common method for converting diaphragm displacement into a measurable signal is to sense the strain induced on the diaphragm surface as it is displaced. Strain gauges, devices whose measurable resistance is proportional to their sensed strain (see Chapter 11), can be bonded directly onto the diaphragm, integrated within the diaphragm material or onto a deforming element (such as a thin beam) attached to the diaphragm so as to deform with the diaphragm and to sense strain. Metal strain gauges can be used with liquids. Strain gauge resistance is reasonably linear over a wide range of strain and can be directly related to the sensed pressure (5). A diaphragm transducer using strain gauge detection is depicted in Figure 9.12. 9.4 Pressure Transducers 389