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 Squareedged 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