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

Chapter 10Flow Measurements10.1 INTRODUCTIONThe rate at which a fluid moves through a conduit is measured in terms of the quantity known as theflow rate.This chapter discusses some of the most common and accepted methods for measuringflow rate.Flow rate can be expressed in terms of a flow volume per unit time,known as thevolume flow rate,or as a mass flow per unit time,known as the mass flow rate.Flow ratedevices.called flow meters, are used to quantify,totalize, or monitor flowing processes.Type, accuracy,size,pressure drop,pressure losses, capital and operating costs, and compatibility with the fluid areimportant engineering design considerations for choosing a flow metering device. All methods havebothdesirable and undesirablefeaturesthatnecessitatecompromise inthe selection of thebestmethod forthe particular application,and many suchconsiderations are discussed in this chapter.Inherent uncertainties in fluid properties, such as density,viscosity, or specific heat, can affect theaccuracy ofaflowmeasurement.However,some techniques,such asthose incorporated into coriolismass flowmeters,do notrequireknowledgeofexactfluid properties,allowing for highly accuratemassflowmeasurementsindemandingengineeringapplications.Thechapterobjectiveistopresentboth an overview of basic flow metering techniques for proper meter selection, as well as thosedesign considerations important in the integration of a flow rate device with the process system itwill meter.Upon completion of this chapter, the reader will be able to.relate velocity distributiontoflowratewithin conduits·use engineering test standards to select and specify size, specify installation considerations,andusecommonobstructionmeters,.understand the differences between volume flow rate and mass flow rate and the meansrequired to measure each,.describe the physical principles employed in various commercially available types of flowmeters andthe engineeringterms commonto their use,and·clearly describe primary and secondary calibration methods for flow meters.10.2HISTORICALBACKGROUNDThe importance to engineered systems give flow measurement methods their rich history.Theearliestavailableaccounts of flowmeteringwere recordedbyHero of Alexandria (ca.150B.c.)whoproposed a scheme to regulate water flow using a siphon pipe attached to a constant head reservoir.423

E1C10 09/14/2010 13:4:37 Page 423 Chapter 10 Flow Measurements 10.1 INTRODUCTION The rate at which a fluid moves through a conduit is measured in terms of the quantity known as the flow rate. This chapter discusses some of the most common and accepted methods for measuring flow rate. Flow rate can be expressed in terms of a flow volume per unit time, known as the volume flow rate, or as a mass flow per unit time, known as the mass flow rate. Flow rate devices, called flow meters, are used to quantify, totalize, or monitor flowing processes. Type, accuracy, size, pressure drop, pressure losses, capital and operating costs, and compatibility with the fluid are important engineering design considerations for choosing a flow metering device. All methods have both desirable and undesirable features that necessitate compromise in the selection of the best method for the particular application, and many such considerations are discussed in this chapter. Inherent uncertainties in fluid properties, such as density, viscosity, or specific heat, can affect the accuracy of a flow measurement. However, some techniques, such as those incorporated into coriolis mass flow meters, do not require knowledge of exact fluid properties, allowing for highly accurate mass flow measurements in demanding engineering applications. The chapter objective is to present both an overview of basic flow metering techniques for proper meter selection, as well as those design considerations important in the integration of a flow rate device with the process system it will meter. Upon completion of this chapter, the reader will be able to
relate velocity distribution to flow rate within conduits,
use engineering test standards to select and specify size, specify installation considerations, and use common obstruction meters,
understand the differences between volume flow rate and mass flow rate and the means required to measure each,
describe the physical principles employed in various commercially available types of flow meters and the engineering terms common to their use, and
clearly describe primary and secondary calibration methods for flow meters. 10.2 HISTORICAL BACKGROUND The importance to engineered systems give flow measurement methods their rich history. The earliest available accounts of flow metering were recorded by Hero of Alexandria (ca. 150 B.C.) who proposed a scheme to regulate water flow using a siphon pipe attached to a constant head reservoir. 423

424Chapter10FlowMeasurementsThe earlyRomans developedelaborate water systems to supplypublicbaths and privatehomes.Infact, SextusFrontinius(A.D.40-103),commissioner of water worksforRome,authored atreatise ondesign methods for urban water distribution systems. Evidence suggests that Roman designersunderstood correlation between volumeflow rate and pipe flow area. Weirs were used to regulatebulk flow through aqueducts, and the cross-sectional area of terra-cotta pipe was used to regulatefresh running water supplies to individual buildings.Following a number of experiments conducted using olive oil and water, Leonardo da Vinci(1452-1519) first formally proposed the modern continuity principle: that duct area and fuidvelocity were related to flow rate.However,most of his writings were lost until centuries later,andBenedetto Castelli (ca. 1577-1644),a student of Galileo, has been credited in some texts withdeveloping the same steady,incompressible continuity concepts in his day.Isaac Newton (1642-1727),DanielBernoulli (1700-1782),andLeonhardEuler(1707-1783)built themathematical andphysical bases on which modern flow meters would later be developed. By the nineteenth century,the concepts of continuity,energy,and momentum were sufficiently understood for practicalexploitation.Relations between flow rate and pressure losses were developed that would permit thetabulation of the hydraulic coefficients necessary for the quantitative engineering design of manymodernflowmeters.10.3FLOWRATECONCEPTSThe flow rate through a conduit, be it a pipeline, duct, or channel, depends on fluid density,averagefluid velocity, and conduit cross-sectional area. Consider fluid flow through a circular pipe of radiusri and having a velocity profile at some axial pipe cross section given by u(r, ).The mass flow ratedepends on the average mass flux flowing through a cross-sectional area; that is, it is the averageproduct of fluid density times fluid velocity and area, as given bym=pu(r,e)dA=pUA(10.1)The pipe area is simply A = r. We see that to directly make a mass flow rate measurement, thedevice must be sensitive to the area-averaged mass flux per unit volume, pU, or to the fluid masspassing through it per unit time. Mass flow rate has the dimensions of mass per unit of time (e.g.,units of kg/s, Ibm/s, etc.).The volume flow rate depends only on the area-averaged velocity over a cross section of flow asgiven byudA=UA(10.2)Q:So to directly measure volume flow rate requires a device that is sensitive either to the averagevelocity,U,or to thefluid volume passing through it per unit time.Volumeflow rate has dimensionsof volume perunit time (e.g., units of m'/s, ft"/s, etc.).The difference between Equations 10.1 and 10.2 is quite significant, in that either requires avery different approachto its measurement: one sensitive to the productof density and velocity ortomass rate,and the other sensitive only to the average velocity or to volume rate.In the simplest case,where density is a constant,the mass flowrate can be inferred bymultiplying themeasured volumeflow rateby the density.But in the metering of manyfluids,this assumption may not be goodenough

E1C10 09/14/2010 13:4:37 Page 424 The early Romans developed elaborate water systems to supply public baths and private homes. In fact, Sextus Frontinius (A.D. 40–103), commissioner of water works for Rome, authored a treatise on design methods for urban water distribution systems. Evidence suggests that Roman designers understood correlation between volume flow rate and pipe flow area. Weirs were used to regulate bulk flow through aqueducts, and the cross-sectional area of terra-cotta pipe was used to regulate fresh running water supplies to individual buildings. Following a number of experiments conducted using olive oil and water, Leonardo da Vinci (1452–1519) first formally proposed the modern continuity principle: that duct area and fluid velocity were related to flow rate. However, most of his writings were lost until centuries later, and Benedetto Castelli (ca. 1577–1644), a student of Galileo, has been credited in some texts with developing the same steady, incompressible continuity concepts in his day. Isaac Newton (1642– 1727), Daniel Bernoulli (1700–1782), and Leonhard Euler (1707–1783) built the mathematical and physical bases on which modern flow meters would later be developed. By the nineteenth century, the concepts of continuity, energy, and momentum were sufficiently understood for practical exploitation. Relations between flow rate and pressure losses were developed that would permit the tabulation of the hydraulic coefficients necessary for the quantitative engineering design of many modern flow meters. 10.3 FLOW RATE CONCEPTS The flow rate through a conduit, be it a pipeline, duct, or channel, depends on fluid density, average fluid velocity, and conduit cross-sectional area. Consider fluid flow through a circular pipe of radius r1 and having a velocity profile at some axial pipe cross section given by uðr; uÞ. The mass flow rate depends on the average mass flux flowing through a cross-sectional area; that is, it is the average product of fluid density times fluid velocity and area, as given by m_ ¼ ðð A ruðr; uÞdA ¼ rUA ð10:1Þ The pipe area is simply A ¼ pr2 1. We see that to directly make a mass flow rate measurement, the device must be sensitive to the area-averaged mass flux per unit volume, rU, or to the fluid mass passing through it per unit time. Mass flow rate has the dimensions of mass per unit of time (e.g., units of kg/s, lbm/s, etc.). The volume flow rate depends only on the area-averaged velocity over a cross section of flow as given by Q ¼ ðð A udA ¼ UA ð10:2Þ So to directly measure volume flow rate requires a device that is sensitive either to the average velocity, U, or to the fluid volume passing through it per unit time. Volume flow rate has dimensions of volume per unit time (e.g., units of m3 /s, ft3 /s, etc.). The difference between Equations 10.1 and 10.2 is quite significant, in that either requires a very different approach to its measurement: one sensitive to the product of density and velocity or to mass rate, and the other sensitive only to the average velocity or to volume rate. In the simplest case, where density is a constant, the mass flow rate can be inferred by multiplying the measured volume flow rate by the density. But in the metering of many fluids, this assumption may not be good enough 424 Chapter 10 Flow Measurements

42510.4VolumeFlowRateThroughVelocityDeterminationto achievenecessaryaccuracy.This can bebecausethedensity changes or itmaynotbe well known,such as in thetransportofpolymers or petrochemicals,orbecause small errors in densityaccumulateinto large errors, such as in the transport of millions of cubic meters of product per day.Theflow character can affectthe accuracy of a flow meter.Theflow through a pipe or duct canbecharacterized asbeinglaminar,turbulent,oratransitionbetweenthetwo.Flowcharacterisdetermined through the nondimensional parameter known as the Reynolds number, defined byUdi40Red,(10.3)TdivVwhere v is the fluid kinematic viscosity and d, is the diameter for circular pipes. In pipes, the flow islaminarwhenRed,<2000 and turbulent at higherReynolds number.TheReynolds number is anecessaryparameterinestimatingflowrate whenusing severalofthetypes offlowmetersdiscussed.Inestimating the Reynolds number in noncircular conduits, the hydraulic diameter, 4rh, is used in placeof diameter dj, where rh is the wetted conduit area divided by its wetted perimeter.10.4VOLUMEFLOWRATETHROUGHVELOCITYDETERMINATIONVolume flow rate can be determined with directknowledge of the velocity profile as indicated byEquation 10.2. This requires measuring the velocity at multiple points along a cross section of aconduit to estimate the velocity profile.Forhighest accuracy, several traverses should be made atdiffering circumferential locations to account for flownonsymmetry.Methods fordetermining thevelocity at a point include any of those previously discussed in Chapter 9. Because it is a tediousmethod,this procedure is most often used for the one-time verification or calibration of system flowrates.For example, the procedure is often used in ventilation system setup and problem diagnosis,where the installation of an in-line flow meter is not necessary because operation does not requirecontinuous monitoring.When using this technique in circular pipes, a number of discrete measuring positions n arechosenalongeachof mflowcross sectionsspacedat360/mdegreesapart,suchasshowninFigure 10.1. A velocity probe is traversed along each flow cross section, with readings taken at each360°3j=Figure10.1Lcationof nmeasurements alongm radial lines ina pipe

E1C10 09/14/2010 13:4:37 Page 425 to achieve necessary accuracy. This can be because the density changes or it may not be well known, such as in the transport of polymers or petrochemicals, or because small errors in density accumulate into large errors, such as in the transport of millions of cubic meters of product per day. The flow character can affect the accuracy of a flow meter. The flow through a pipe or duct can be characterized as being laminar, turbulent, or a transition between the two. Flow character is determined through the nondimensional parameter known as the Reynolds number, defined by Red1 ¼ U d1 v ¼ 4Q pd1v ð10:3Þ where v is the fluid kinematic viscosity and d1 is the diameter for circular pipes. In pipes, the flow is laminar when Red1 < 2000 and turbulent at higher Reynolds number. The Reynolds number is a necessary parameter in estimating flow rate when using several ofthetypes of flow meters discussed. In estimating the Reynolds number in noncircular conduits, the hydraulic diameter, 4rH, is used in place of diameter d1, where rH is the wetted conduit area divided by its wetted perimeter. 10.4 VOLUME FLOW RATE THROUGH VELOCITY DETERMINATION Volume flow rate can be determined with direct knowledge of the velocity profile as indicated by Equation 10.2. This requires measuring the velocity at multiple points along a cross section of a conduit to estimate the velocity profile. For highest accuracy, several traverses should be made at differing circumferential locations to account for flow nonsymmetry. Methods for determining the velocity at a point include any of those previously discussed in Chapter 9. Because it is a tedious method, this procedure is most often used for the one-time verification or calibration of system flow rates. For example, the procedure is often used in ventilation system setup and problem diagnosis, where the installation of an in-line flow meter is not necessary because operation does not require continuous monitoring. When using this technique in circular pipes, a number of discrete measuring positions n are chosen along each of m flow cross sections spaced at 360/m degrees apart, such as shown in Figure 10.1. A velocity probe is traversed along each flow cross section, with readings taken at each j = 2 j = m j = 1 . . . i = 1 2 r 3 n 360° m Figure 10.1 Lcation of n measurements along m radial lines in a pipe. 10.4 Volume Flow Rate Through Velocity Determination 425

426Chapter10FlowMeasurementsmeasurement position. There are several options in selecting measuring positions for different-shaped ducts, and such details are specified in available engineering test standards (1,2, 4, 15).Thesimplest method is to divide theflow area into smaller equal areas, making measurements at thecentroidofeachsmall areaandassigningthemeasuredvelocitytothatarea.Regardlessoftheoptionselected,theaverageflowrateisestimatedalongeachcrosssectiontraversedusingEquation10.2and thepooled meanoftheflow rates forthe mcross sections calculatedto yieldthe bestestimateofthe duct flow rate. Uncertainty is assessed both by repeating the measurements and by analyzing forspatial variation effects. Example 10.1 illustrates this method for estimating volume flow rate.Example10.1A steady flow of air at 20°C passes through a 25.4-cm inside diameter (i.d.) circular pipe. A velocity-measuring probe is traversed along three cross-sectional lines = 1, 2, 3) of the pipe, andmeasurements are made at four radial positions (i = 1, 2, 3, 4) along each traverse line, such thatm=3andn=4.Thelocationsforeachmeasurementareselectedatthecentroidsofequallyspacedareal increments as indicated below (1, 3).Determine the volume flow rate in the pipe.Ug (m/s)rlriRadial Location, iLine 1 (j= 1)Line 2 (j = 2)Line 3(j = 3)18.780.35368.718.6226.316.200.61246.2633.693.743.790.790640.93541.241.201.28KNOWNUu(r/ri)fori=1,2,3,4; j=1,2,3di=25.4cm(A=Td/4=0.051m2)ASsUMPTIONsConstant and steadypipeflowduringall measurements;IncompressibleflowFINDVolumeflowrate,QSOLUTION Theflowrate is found by integrating thevelocityprofile across theduct alongeachline and subsequently averaging the three values. For discrete velocity data, Equation 10.2 iswritten along each line, m = 1, 2, 3, as4Q, = 2Urdr~2UgrAr1=1where Ar istheradial distance separating eachposition of measurement.This can be furthersimplified since the velocities are located at positions that make up the centroids of equal areas:AU

E1C10 09/14/2010 13:4:37 Page 426 measurement position. There are several options in selecting measuring positions for different￾shaped ducts, and such details are specified in available engineering test standards (1, 2, 4, 15). The simplest method is to divide the flow area into smaller equal areas, making measurements at the centroid of each small area and assigning the measured velocity to that area. Regardless of the option selected, the average flow rate is estimated along each cross section traversed using Equation 10.2 and the pooled mean of the flow rates for the m cross sections calculated to yield the best estimate of the duct flow rate. Uncertainty is assessed both by repeating the measurements and by analyzing for spatial variation effects. Example 10.1 illustrates this method for estimating volume flow rate. Example 10.1 A steady flow of air at 20C passes through a 25.4-cm inside diameter (i.d.) circular pipe. A velocity￾measuring probe is traversed along three cross-sectional lines ( j ¼ 1, 2, 3) of the pipe, and measurements are made at four radial positions (i ¼ 1, 2, 3, 4) along each traverse line, such that m ¼ 3 and n ¼ 4. The locations for each measurement are selected at the centroids of equally spaced areal increments as indicated below (1, 3). Determine the volume flow rate in the pipe. Uij (m/s) Radial Location, i r/r1 Line 1 ( j ¼ 1) Line 2 ( j ¼ 2) Line 3 ( j ¼ 3) 1 0.3536 8.71 8.62 8.78 2 0.6124 6.26 6.31 6.20 3 0.7906 3.69 3.74 3.79 4 0.9354 1.24 1.20 1.28 KNOWN Uijð Þ r=r1 for i ¼ 1; 2; 3; 4; j ¼ 1; 2; 3 d1 ¼ 25:4 cm A ¼ pd2 1=4 ¼ 0:051 m2 ASSUMPTIONS Constant and steady pipe flow during all measurements; Incompressible flow FIND Volume flow rate, Q SOLUTION The flow rate is found by integrating the velocity profile across the duct along each line and subsequently averaging the three values. For discrete velocity data, Equation 10.2 is written along each line, m ¼ 1, 2, 3, as Qj ¼ 2p Z r1 0 Urdr  2p X 4 i¼1 UijrDr where Dr is the radial distance separating each position of measurement. This can be further simplified since the velocities are located at positions that make up the centroids of equal areas: Qj ¼ A 4 X 4 i¼1 Uij 426 Chapter 10 Flow Measurements

PressureDifferentialMeters42710.5Then, the mean flow rate along each line of traverse isQl=0.252m2/sQ2=0.252m2/sQ3=0.254m/sThe average pipe flow rate is the pooled mean of the individual flow rates30=(Q)9,=0.253m2/sExample10.2Referring to Example 10.1,determine a value of the systematic standard uncertainty in mean flowrate due to the measured spatial variation.KNowNDataofExample10.1overthree(m=3)traversesections.SOLUTION The systematic standard uncertaintyin mean flowrate due to error introduced bythe measured spatial variations is estimated by(e, - (0) /20.0012S(0)b-0.0007m2/swithV=2V3V3Vm10.5PRESSUREDIFFERENTIALMETERSThe operating principle of a pressure differential flow rate meter is based on the relationshipbetweenvolumeflowrateandthepressuredropAp=P,-P2,betweentwolocationsalongtheflowpath,(10.4)Qα (pPi-p2)"wheren=1forlaminarflowoccurringbetween thepressuremeasurement locations andn=/2forfullyturbulentflow.Understeadyflowconditions,anintentionalreductioninflowareabetweenlocations1and2causes a measurablelocal pressuredrop across this flowpath.This reduced flowarea leadsto a concurrent local increase invelocity due toflow continuity (conservation ofmass)principles.The pressure drop is in part due to the so-called Bernoulli effect, the inverse relationshipbetween local velocityand pressure,but is alsoduetoflowenergylosses.Pressuredifferential flowrate meters that use area reduction methods are commonly called obstruction meters.ObstructionMetersThreecommonobstructionmeters aretheorificeplate,theventuri,and theflownozzle.Flowareaprofiles ofeach are shown in Figure 10.2.These meters are usually inserted in-line with a pipe, suchas between pipe flanges.This class of meters as a whole operates using similar physical reasoning torelate volume flow rate to pressure drop.Referring to Figure 10.3, consider an energy balancebetweentwocontrol surfaces foran incompressiblefluidflowthrough thearbitrarycontrol volume

E1C10 09/14/2010 13:4:37 Page 427 Then, the mean flow rate along each line of traverse is Q1 ¼ 0:252 m3 /s Q2 ¼ 0:252 m3 /s Q3 ¼ 0:254 m3 /s The average pipe flow rate Q is the pooled mean of the individual flow rates Q ¼ hQi ¼ 1 3 X 3 j¼1 Qj ¼ 0:253 m3 /s Example 10.2 Referring to Example 10.1, determine a value of the systematic standard uncertainty in mean flow rate due to the measured spatial variation. KNOWN Data of Example 10.1 over three (m ¼ 3) traverse sections. SOLUTION The systematic standard uncertainty in mean flow rate due to error introduced by the measured spatial variations is estimated by bQ ¼ shQi ffiffiffiffi mp ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 3 j¼1 Qj  hQi 2 =2 s ffiffiffi 3 p ¼ 0:0012 ffiffiffi 3 p ¼ 0:0007 m3 /s with n ¼ 2 10.5 PRESSURE DIFFERENTIAL METERS The operating principle of a pressure differential flow rate meter is based on the relationship between volume flow rate and the pressure drop Dp ¼ p1  p2, between two locations along the flow path, Q / p1  p2 ð Þn ð10:4Þ where n ¼ 1 for laminar flow occurring between the pressure measurement locations and n ¼ ½ for fully turbulent flow. Under steady flow conditions, an intentional reduction in flow area between locations 1 and 2 causes a measurable local pressure drop across this flow path. This reduced flow area leads to a concurrent local increase in velocity due to flow continuity (conservation of mass) principles. The pressure drop is in part due to the so-called Bernoulli effect, the inverse relationship between local velocity and pressure, but is also due to flow energy losses. Pressure differential flow rate meters that use area reduction methods are commonly called obstruction meters. Obstruction Meters Three common obstruction meters are the orifice plate, the venturi, and the flow nozzle. Flow area profiles of each are shown in Figure 10.2. These meters are usually inserted in-line with a pipe, such as between pipe flanges. This class of meters as a whole operates using similar physical reasoning to relate volume flow rate to pressure drop. Referring to Figure 10.3, consider an energy balance between two control surfaces for an incompressible fluid flow through the arbitrary control volume 10.5 Pressure Differential Meters 427

428Chapter10FlowMeasurementsFlowFlowFigure 10.2 Flow area pro-ForwardForwardfilesofcommonobstructionViewViewmeters.(a)Square-edged(a)(b)orificeplatemeter.(b)American Society ofMechanical EngineersFlow(ASME) long radius nozzle(c) ASME Herschel venturi山山meter.(c)EddyrecirculationregionsControlvolumeVenacontracta①?!streamlinesPiP2Figure10.3Control volumeconceptas appliedbetweentwostreamlinesforflowthroughanobstructionmeter.shown.Undertheassumptionsofincompressible,steadyand one-dimensionalflowwithnoexternalenergytransfer, theenergyequation is2+5-8++hu(10.5)2g2gwherehzi-,denotes the head losses occurringbetween control surfaces 1and 2.For incompressible flows,conservation of mass between cross-sectional areas 1 and 2 givesU, = U,42(10.6)AThen, substituting Equation 10.6 into Equation 10.5 and rearranging yields the incompressiblevolumeflow rate,A22(P1-P2)+2ghL1-2(10.7)Q, = U,A2 =pV1- (A2/A,)2

E1C10 09/14/2010 13:4:37 Page 428 shown. Under the assumptions of incompressible, steady and one-dimensional flow with no external energy transfer, the energy equation is p1 g þ U2 1 2g ¼ p2 g þ U2 2 2g þ hL12 ð10:5Þ where hL12 denotes the head losses occurring between control surfaces 1 and 2. For incompressible flows, conservation of mass between cross-sectional areas 1 and 2 gives U1 ¼ U2 A2 A1 ð10:6Þ Then, substituting Equation 10.6 into Equation 10.5 and rearranging yields the incompressible volume flow rate, QI ¼ U2A2 ¼ A2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ð Þ A2=A1 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p1  p2 ð Þ r s þ 2ghL12 ð10:7Þ Flow Flow Flow Forward View Forward View (a) (c) (b) Figure 10.2 Flow area pro- files of common obstruction meters. (a) Square-edged orifice plate meter. (b) American Society of Mechanical Engineers (ASME) long radius nozzle. (c) ASME Herschel venturi meter. Eddy recirculation regions Vena contracta streamlines Control volume p1 p2 1 d1 d2 d0 2 Figure 10.3 Control volume concept as applied between two streamlines for flow through an obstruction meter. 428 Chapter 10 Flow Measurements

10.5PressureDifferential Meters429wherethe subscriptI emphasizesthatEquation10.7givesanincompressibleflowrate.Laterwedrop the subscript.When the flow area changes abruptly,the effective flow area immediately downstream of thearea reduction is not necessarily the same as the pipe flow area.This was originally investigated byJean Borda(1733-1799)and illustrated inFigure10.3.Whena fluid cannotexactlyfollowa suddenarea expansion dueto its own inertia, a central core flow called the vena contracta forms that isbounded by regions of slower moving recirculating eddies.Thepressure sensed withpipe wall tapscorresponds to the higher moving velocity within the vena contracta with its unknown flow area, A2.Toaccountfor thisunknown,we introducea contraction coefficientCewhere C=A2/Ao,withAobased on themeter throat diameter,intoEquation 10.7.This givesC.Ao2(P1- p2) + 2ghL-2(10.8)QrPV1 - (CeAo/A,)2Furthermore thefrictional head losses can be incorporated into a friction coefficient, C,such thatEquation10.8becomes2(pi - p2)CfCeAo(10.9)QrpV1-(C(A0/A1)2For convenience, the coefficients are factored out of Equation 10.9 and replaced by a singlecoefficientknown as the discharge coefficient, C.Keeping in mind that the ideal flowrate wouldhave no losses and no vena contracta, the discharge coefficient represents the ratio of the actual flowrate through a meter to the ideal flow rate possible for the pressure drop measured, that is,C=Ql/Qi.Reworking Equation 10.9 leads to the incompressibleoperating equation2Ap-24p(10.10)KoAQI= CEAowhere E,known as the velocity of approach factor, is defined by11(10.11)F/1-(Ao/A,)2V1-β4with the beta ratio defined asβ=dold,and where Ko =CE is called the flow coefficient.The discharge coefficient and the flow coefficient are tabulated quantities found in teststandards (1, 3, 4). Each is a function of the flow Reynolds number and theβ ratio for eachparticular obstruction flow meter design, C =f(Red, β) and Ko = f(Red,β).CompressibilityEffectsIn compressible gas flows,compressibility effects in obstruction meters can be accountedforbyintroducing the compressible adiabatic expansion factor, Y.Here Y is defined as the ratio of theactual compressiblevolumeflow rate,Q,divided by the assumedincompressibleflow rateQrCombiningwithEquation10.10yieldsQ=YQ,=CEAoYV2Ap/Pi(10.12)

E1C10 09/14/2010 13:4:37 Page 429 where the subscript I emphasizes that Equation 10.7 gives an incompressible flow rate. Later we drop the subscript. When the flow area changes abruptly, the effective flow area immediately downstream of the area reduction is not necessarily the same as the pipe flow area. This was originally investigated by Jean Borda (1733–1799) and illustrated in Figure 10.3. When a fluid cannot exactly follow a sudden area expansion due to its own inertia, a central core flow called the vena contracta forms that is bounded by regions of slower moving recirculating eddies. The pressure sensed with pipe wall taps corresponds to the higher moving velocity within the vena contracta with its unknown flow area, A2. To account for this unknown, we introduce a contraction coefficient Cc, where Cc ¼ A2=A0, with A0 based on the meter throat diameter, into Equation 10.7. This gives QI ¼ CcA0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ð Þ CcA0=A1 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p1  p2 ð Þ r s þ 2ghL12 ð10:8Þ Furthermore the frictional head losses can be incorporated into a friction coefficient, Cf, such that Equation 10.8 becomes QI ¼ CfCcA0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ð Þ CcA0=A1 2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p1  p2 ð Þ r s ð10:9Þ For convenience, the coefficients are factored out of Equation 10.9 and replaced by a single coefficient known as the discharge coefficient, C. Keeping in mind that the ideal flow rate would have no losses and no vena contracta, the discharge coefficient represents the ratio of the actual flow rate through a meter to the ideal flow rate possible for the pressure drop measured, that is, C ¼ QIactual=QIideal . Reworking Equation 10.9 leads to the incompressible operating equation QI ¼ CEA0 ffiffiffiffiffiffiffiffi 2Dp r s ¼ K0A0 ffiffiffiffiffiffiffiffi 2Dp r s ð10:10Þ where E, known as the velocity of approach factor, is defined by E ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ð Þ A0=A1 2 q ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1  b4 p ð10:11Þ with the beta ratio defined as b ¼ d0/d1, and where K0 ¼ CE is called the flow coefficient. The discharge coefficient and the flow coefficient are tabulated quantities found in test standards (1, 3, 4). Each is a function of the flow Reynolds number and the b ratio for each particular obstruction flow meter design, C ¼ f Red1 ð Þ ; b and K0 ¼ f Red1 ð Þ ; b . Compressibility Effects In compressible gas flows, compressibility effects in obstruction meters can be accounted for by introducing the compressible adiabatic expansion factor, Y. Here Y is defined as the ratio of the actual compressible volume flow rate, Q, divided by the assumed incompressible flow rate QI. Combining with Equation 10.10 yields Q ¼ YQI ¼ CEA0Y ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Dp=r1 p ð10:12Þ 10.5 Pressure Differential Meters 429

430Chapter10FlowMeasurementswhereP,is theupstreamffuid density.WhenY=1,theflow is incompressible,and Equation 10.12reduces to Equation 10.10.Equation 10.12 represents the mostgeneral form of the workingequation for volume flowratedetermination whenusingan obstructionmeter.Theexpansion factor,Y,depends on several values: the β ratio, thegas specific heat ratio,k,andthe relative pressure drop across the meter, (p-p2)pi, for a particular meter type, that is,Y=f[β,k, (pr -P2)/p.l. As a general rule, compressibility effects should be considered if(Pl - P2)/p1 ≥ 0.1.StandardsTheflowbehaviors ofthe most common obstruction meters,namely theorificeplate,venturi,andflow nozzle, have been studied to such an extent that these meters are used extensively withoutcalibration.Values for the discharge coefficients,flow coefficients,and expansion factors aretabulated and available in standard U.S.and international flow handbooks along with standardizedconstruction, installation,and operation techniques(1,3,4,16).Equations 10.10 and 10.12 are verysensitivetopressure taplocation.For steamor gas flows,pressuretaps should be oriented on thetopor side of the pipe; for liquids, pressure taps should be oriented on the side. We discuss therecommended standard taplocations with each meter (1, 4).A nonstandard installation or designrequiresan in situcalibration.Orifice MeterAn orifice meter consists ofa circular platehaving a central hole (orifice).The plate is inserted into apipe so as to effect a flow area change. The orifice hole is smaller than the pipe diameter andarranged to be concentric with the pipe's i.d.The common square-edged orifice plate is shown inFigure 10.4.Installation is simplified by housing the orifice plate between two pipe flanges.Withthis installation technique any particular orifice plate is interchangeable with others of different βvalue.The simplicity of the design allows for a range ofβ values to be maintained on hand at modestexpense.For an orifice meter, plate dimensions and use are specified by engineering standards (1, 4).Equation 10.12 is used with values of Ao and β based on the orifice (hole) diameter, do. The platethickness should bebetween 0.005d, and 0.02dj,otherwiseatapermustbe addedtothedownstream side (1,4).The exactplacement of pressure taps is crucial to use standard coefficients.Standard pressure tap locations include (i)flange taps where pressure tap centers are located25.4mm (1 in.)upstreamand 25.4mm(1 in.)downstream of the nearest orifice face,(2)d andd/2 taps located one pipe diameter upstream and one-half diameter downstream of the upstreamorifice face, and (3) vena contracta taps.Nonstandard tap locations always require in situ metercalibration.Values for the flow coefficient, Ko=(Red,β) and for the expansion factor, Y=f[β,k, (p1- p2)/pi] for a square-edged orifice plate are given in Figures 10.5 and 10.6 basedon the use of flange taps.The relative instrument systematic uncertainty in the discharge coefficient(3)is~0.6%of Cfor0.2≤β≤0.6andβ%ofCfor allβ>0.6.Therelativeinstrumentsystematicuncertainty for the expansion factor is about [4(pi - P2)/p.]% of Y. Realistic estimates of theoverall systematic uncertainty in estimatingQusing an orifice meter arebetween1% (highβ)and3%(lowβ)at highReynolds numberswhen using standard tables.Althoughthe orificeplaterepresents a relatively inexpensive flow meter solution with an easily measurable pressure drop

E1C10 09/14/2010 13:4:37 Page 430 where r1 is the upstream fluid density. When Y ¼ 1, the flow is incompressible, and Equation 10.12 reduces to Equation 10.10. Equation 10.12 represents the most general form of the working equation for volume flow rate determination when using an obstruction meter. The expansion factor, Y, depends on several values: the b ratio, the gas specific heat ratio, k, and the relative pressure drop across the meter, p1  p2 ð Þp1, for a particular meter type, that is, Y ¼ f b; k; p1  p2 ð Þ=p1 ½ . As a general rule, compressibility effects should be considered if p1  p2 ð Þ=p1  0:1. Standards The flow behaviors of the most common obstruction meters, namely the orifice plate, venturi, and flow nozzle, have been studied to such an extent that these meters are used extensively without calibration. Values for the discharge coefficients, flow coefficients, and expansion factors are tabulated and available in standard U.S. and international flow handbooks along with standardized construction, installation, and operation techniques (1, 3, 4, 16). Equations 10.10 and 10.12 are very sensitive to pressure tap location. For steam or gas flows, pressure taps should be oriented on the top or side of the pipe; for liquids, pressure taps should be oriented on the side. We discuss the recommended standard tap locations with each meter (1, 4). A nonstandard installation or design requires an in situ calibration. Orifice Meter An orifice meter consists of a circular plate having a central hole (orifice). The plate is inserted into a pipe so as to effect a flow area change. The orifice hole is smaller than the pipe diameter and arranged to be concentric with the pipe’s i.d. The common square-edged orifice plate is shown in Figure 10.4. Installation is simplified by housing the orifice plate between two pipe flanges. With this installation technique any particular orifice plate is interchangeable with others of different b value. The simplicity of the design allows for a range of b values to be maintained on hand at modest expense. For an orifice meter, plate dimensions and use are specified by engineering standards (1, 4). Equation 10.12 is used with values of A0 and b based on the orifice (hole) diameter, d0. The plate thickness should be between 0.005 d1 and 0.02 d1, otherwise a taper must be added to the downstream side (1, 4). The exact placement of pressure taps is crucial to use standard coefficients. Standard pressure tap locations include (1) flange taps where pressure tap centers are located 25.4 mm (1 in.) upstream and 25.4 mm (1 in.) downstream of the nearest orifice face, (2) d and d/2 taps located one pipe diameter upstream and one-half diameter downstream of the upstream orifice face, and (3) vena contracta taps. Nonstandard tap locations always require in situ meter calibration. Values for the flow coefficient, K0 ¼ Red1 ð Þ ; b and for the expansion factor, Y ¼ f b; k; p1  p2 ð Þ=p1 ½  for a square-edged orifice plate are given in Figures 10.5 and 10.6 based on the use of flange taps. The relative instrument systematic uncertainty in the discharge coefficient (3) is 0.6% of C for 0:2 b 0:6 and b% of C for all b > 0.6. The relative instrument systematic uncertainty for the expansion factor is about 4ðp1  p2Þ=p1 ½ % of Y. Realistic estimates of the overall systematic uncertainty in estimating Q using an orifice meter are between 1% (high b) and 3% (low b) at high Reynolds numbers when using standard tables. Although the orifice plate represents a relatively inexpensive flow meter solution with an easily measurable pressure drop, 430 Chapter 10 Flow Measurements

43110.5PressureDifferentialMeters12dandd/2pressuretapsOrificeplatePipeflanges25.4mm(1 in.)1reun ae0(Ap)inFigure 10.4 Square-edged orifice meter installed in a pipeline with optional 1 d and /2 d, and flange pressuretaps shown. Relative flow pressure drop along pipe axis is shown.it introduces a large permanent pressure loss, (Ap)ioss =pgh, into the flow system.The pressuredrop is illustrated in Figure 10.4 with the pressure loss estimated from Figure 10.7.Rudimentary versions of the orifice plate meter have existed for several centuries. BothTorricelli and Newton used orifice plates to study the relation between pressure head and efflux fromreservoirs, although neither ever got the discharge coefficients quite right (5).Venturi MeterA venturi meter consists of a smooth converging (21 degrees ± 1 degree)conical contractionto a narrow throat followed by a shallow diverging conical section, as shown in Figure 10.8.The engineering standard venturi meter design uses either a 15-degree or 7-degree divergent

E1C10 09/14/2010 13:4:37 Page 431 it introduces a large permanent pressure loss, ð Þ Dp loss ¼ rghL, into the flow system. The pressure drop is illustrated in Figure 10.4 with the pressure loss estimated from Figure 10.7. Rudimentary versions of the orifice plate meter have existed for several centuries. Both Torricelli and Newton used orifice plates to study the relation between pressure head and efflux from reservoirs, although neither ever got the discharge coefficients quite right (5). Venturi Meter A venturi meter consists of a smooth converging (21 degrees 1 degree) conical contraction to a narrow throat followed by a shallow diverging conical section, as shown in Figure 10.8. The engineering standard venturi meter design uses either a 15-degree or 7-degree divergent x d and d/2 pressure taps Orifice plate Pipe flanges 25.4 mm (1 in.) 0 Relative pressure differential d d/2 Δp (Δp)loss d1 d0 Figure 10.4 Square-edged orifice meter installed in a pipeline with optional 1 d and ½ d, and flange pressure taps shown. Relative flow pressure drop along pipe axis is shown. 10.5 Pressure Differential Meters 431

432Chapter10FlowMeasurementsSquare-dj ≥ 58.6 mm (2.3 in.)edged0.3≤β≤0.70.80orifice0.76uoiasa0.72β=0.700.680.600.64Figure10.5Flow coeffi-0.50cients fora square-edged8:38orifice meter having flange0.60pressure taps. (Courtesy103104105105of American Society ofRedyMechanical Engineers,New York, NY; compiled(0.5959 + 0.0312p2.1=0.184p8 + 91.71p2.5Red0.75K.=(1 β4)1/2from data in reference 1.)section(1,4).Themeterisinstalledbetweentwoflangesintendedforthispurpose.Pressuretapsarelocated justaheadof theupstreamcontractionandatthethroat.Equation10.12isusedwithvaluesfor both A and β based on the throat diameter, do-The quality of a venturi meter ranges from cast to precision-machined units. The dischargecoefficient varies little for pipe diameters above 7.6 cm (3 in.). In the operating range 2 × 105≤Red,≤2×10°and 0.4≤β≤0.75,a valueof C=0.984with a systematic uncertaintyof 0.7%(95%)for cast units and C=0.995 with a systematic uncertainty of 1% (95%)for machined unitsshould be used (1,3, 4).Valuesforexpansionfactor are shown in Figure 10.6andhave an instrumentsystematic uncertainty of [(4+ 100β°)(P, -p2)/pi]% of Y(3). Although a venturi meter presentsa much higher initial cost over an orificeplate,Figure 10.7demonstrates that themeter shows amuch smaller permanent pressure loss for a given installation.This translates into lower systemoperating costs forthepump orblowerusedtomovetheflow.The modern venturi meter was first proposed by Clemens Herschel (1842-1930).Herschel'sdesign was based on his understanding of the principles developed by several men, most notablythose of Daniel Bernoulli.However, he cited the studies of contraction/expansion angles and theircorresponding resistance losses by Giovanni Venturi (1746-1822)and later those by James Francis(1815-1892) as being instrumental to his design of a practical flow meter.FlowNozzlesA flownozzle consists of a gradual contraction from the pipe's inside diameter down to a narrowthroat. It needs less installation space than a venturi meter and has about 80% of the initial cost.CommonformsaretheISO1932nozzleandtheASMElongradiusnozzle(1,4).Thelongradius

E1C10 09/14/2010 13:4:37 Page 432 section (1, 4). The meter is installed between two flanges intended for this purpose. Pressure taps are located just ahead of the upstream contraction and at the throat. Equation 10.12 is used with values for both A and b based on the throat diameter, d0. The quality of a venturi meter ranges from cast to precision-machined units. The discharge coefficient varies little for pipe diameters above 7.6 cm (3 in.). In the operating range 2  105 Red1 2  106 and 0:4 b 0:75, a value of C ¼ 0:984 with a systematic uncertainty of 0.7% (95%) for cast units and C ¼ 0:995 with a systematic uncertainty of 1% (95%) for machined units should be used (1, 3, 4). Values for expansion factor are shown in Figure 10.6 and have an instrument systematic uncertainty of 4 þ 100b2 p1  p2 ð Þ=p1  % of Y (3). Although a venturi meter presents a much higher initial cost over an orifice plate, Figure 10.7 demonstrates that the meter shows a much smaller permanent pressure loss for a given installation. This translates into lower system operating costs for the pump or blower used to move the flow. The modern venturi meter was first proposed by Clemens Herschel (1842–1930). Herschel’s design was based on his understanding of the principles developed by several men, most notably those of Daniel Bernoulli. However, he cited the studies of contraction/expansion angles and their corresponding resistance losses by Giovanni Venturi (1746–1822) and later those by James Francis (1815–1892) as being instrumental to his design of a practical flow meter. Flow Nozzles A flow nozzle consists of a gradual contraction from the pipe’s inside diameter down to a narrow throat. It needs less installation space than a venturi meter and has about 80% of the initial cost. Common forms are the ISO 1932 nozzle and the ASME long radius nozzle (1, 4). The long radius 103 104 105 106 Red1 K0 = 1 (0.5959 + 0.0312 2.1 – 0.184 8 + 91.71 2.5Red1 –0.75) (1 – 4) 1/2 Flow coefficient K0 = CE 0.60 0.30 0.40 0.50 0.60 = 0.70 0.64 0.68 0.72 0.76 Square￾edged 0.80 orifice d1 ≥ 58.6 mm (2.3 in.) 0.3 ≤ ≤ 0.7 Figure 10.5 Flow coeffi- cients for a square-edged orifice meter having flange pressure taps. (Courtesy of American Society of Mechanical Engineers, New York, NY; compiled from data in reference 1.) 432 Chapter 10 Flow Measurements