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

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

E1C10 09/14/2010 13:4:37 Page 426 measurement position. There are several options in selecting measuring positions for differentshaped 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 velocitymeasuring 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

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