Flow Measurement Engineering Handbook

LIST OF SYMBOLS

Symbol and meaning

а Constant in Наll-Yarboroughequation of state

а Constant in specific-heat equation

а Constant in gas viscosity equation

as Acceleration along а stream tube

А Area

Аа Annular area between float andtapered wall of а variable-areaflowmeter

Ар Deadweight-tester piston area

А, Throat area of а critical nozzle

Aj1 Effective area of float in а

variable-area flowmeter

Ар Pipe area

Асс Ассшасу. combined precision andbias errors

(Асс),.f Reference-condition accuracy

Ap1'te Plate area in viscosity derivationequation

А Constant in Redlich-Kwongequation of state

А Constant in Ostwald power-lawequation

AL Constant in liquid viscosityequation

Ь Constant in Наll-Yarboroughequation of state

Ь Constant in equation for specificheat at constant pressure

Ь Constant in general form ofdischarge-coefficient equation

ЬС Slope constant in liquid-bulk­modulus equation

Ьр Frequency coefficient for pulsatingflow

U.S. units SI unitst

Btu/(lbm ·mol·oR) J* I(kg'mol' К)

ftl S2 m/s2

ft2 m2

ft2 m2

in2 mm2

ft2 m2

f1' m2

f1' m2

% %

% %

f1' m2

tExcept for dimensionless or defined SI unit symbols. as in Тк • symbols that apply (о SI units are shownin the (ех! with а superscript asterisk. as in F ,'; .

FLUID PROPERТIES

Data оп physical properties is often required for calcu!ations of base flow rates andpipe Reyno!ds numbers, and to predict the properties of а gas (vapor) after ап

expansion. The physica! properties of !iquids and gases change with pressure andtemperature, and whether сопесtiопs need to Ье considered depends оп the designobjective. In тапу cases, properties are assumed constant at design conditions, andсопесtiопs are not app!ied. Whi!e there is по substitute for experimenta! data,estimates of the properties of а mixture тау often have to Ье used in calcu!ations.This requires theory, соттоп sense, and experience.

Accuracy in predicting the properties of pure substances is considerably betterfor !iquids and gases than for mixtures. In тапу app!ications, particu!ar!y for highinert то!е fractions in natura! gas, !arge епоrs сап occur, and the estimated уа!ие

shou!d first Ье proper!y verified Ьу test.This chapter is а discussion of the most соттоп!у used fluid properties and the

estimation of these properties at various pressures and temperatures, for both puresubstances and rnixtures. For illustrative purposes shaded areas оп graphs in thischapter are expanded and are not sca!ed.

ТНЕ pvT RELATIONSH/P

The pv т Behavior of а Pure Substance

Fluid density сan Ье measured with а !iquid or gas densitometer, but it is moreсоттоп to use temperature and pressure measurements to calcu!ate density. Thereciprocal of the specific уо!ите is the fluid's mass density, and it сап Ье deter­rnined from pressure and temperature measurements using the риТ relationship. Theiпtепеlаtiопshiрs of pressure, temperature, and specific уо!ите are a!so importantbecause of the law of сопеsропdiпgstates. From these relationships, the fluid stateсап ье defined, or the density of ап unknown mixture сап Ье calculated.

Depending оп temperature and pressure, а substance тау Ье either а so!id, а

solid-liquid mixture, а liquid, а liquid-vapor mixture, а уарor, or а gas. The wordsуарог and gas are often used interchangeably because they are thermodynamicallyidentical. Historically, the term уарог has Ьееп used to designate а substance, suchas water, that exists as а solid or liquid at room temperature and atmospheric pres­sure, and the term gas to designate а substance that exists in the gaseous state underthe same conditions (air, oxygen, etc.). At and аЬоуе the saturated-vapor line, аll

substances are thermodynamically gases and contain по liquid, as the term vapor

MEASUREMENT

Тhe purpose of this chapter is to present the basic measurement units used in flowmeasurement and to discuss typical temperature- and pressure-measuring deviees.Тhis information will Ье used in subsequent ehapters in the development of theengineering flow equations.

MASS, FORCE, WEIGHT

The English Engineering System of Units

Table 3.1 summarizes the буе fundamental systems of units that have Ьееп еоп­

strueted from Newton's seeond law of motion to relate foree F, mass т, length L,and time (. While апу system сап Ье developed from three fundamental quantities,the four quantities of the English engineering system-the foot (ft), pound-foree(lbf) , pound-mass ОЬm), and second (s)-will Ье used here to develop the U.S.customary unit equations.

То relate the pound-force to the pound-mass, а proportionality equation еап Ье

written between the engineering and the absolute units. Using the definition that 1lbf will accelerate 1 lЬm at 32.17405 ftls2

, а dimensional conversion constant сап

Ье derived as

1F = -та

gc (3.1)

1 ftlb f = lЬт .ft/(lb

f'S2 ) lЬт ~

The constant gc has the same value as standard gravity go, defined at sea lеуеl and450 latitude, but it has the dimensions of lbm ·ft/{1bf ·s

2). It is, therefore, а dimen­

sional eonversion faetor to relate pounds-foree and pounds-mass. Substituting 1оеаl

gravity (g/) for aceeleration а in Eq. (3.1) gives the relationship between mass andweight foree as

(3.2)

INFLUENCE QUANTITIES

Accuracy statements for fiowmeters are based оп the steady ftow of а homogeneous,single-phase newtonian ftuid with ап approach velocity profile that does not alterthe coefficient obtained in long, straight runs of pipe. Departures from these ref­erence conditions are called ftowmeter in.fluence quantities. Velocity-profile devia­tions, nonhomogeneous ftow, pulsating fiow, and cavitation are the four major in­fluence quantities affecting all flowmeters. ТЬе errors associated with а particularinfluence quantity depend оп the sensitivity of а particular ftowmeter to that quantityand whether or not а calculation correction сап Ье made. For newtonian ftuids,velocity profiles сап usually Ье brought into acceptable limits Ьу the instaHation ofsufficient straight pipe or, for shorter lengths, with ftow conditioners. However,other infiuence quantities тау require the installation of pulsating dampers or theuse of а less sensitive fiowmeter to асЫеуе the desired degree of accuracy. ТЬе

major inftuence quantities and their effects are discussed in detail in the followingsections.

VELOC/TY PROFILE

Velocity profile is probably the most important (and least understood) infiuencequantity. ТЬе effects of swirl, nonnewtonian ftuids, and nonaxisymmetric profilesоп а rneter's performance are not only difficult to analyze, but they cannot easilyье duplicated in а laboratory.

Newtonian Fluids

ТЬе rheological behavior of а ftuid determines whether it is classified as newtonianor nonnewtonian. А newtonian ftuid is defined as а ftuid which, when acted ироп

Ьу ап applied shearing stress, has а velocity gradient that is solely proportional tothe applied stress. ТЬе constant of proportionality is the absolute viscosity definedin СЬар. 2. АН gases, most liquids, and fine mixtuгes of spherical particles in liquidsand gases are newtonian ftuids.

ТЬе velocity profile established Ьу а newtonian fluid is the basic reference соп­

dition for аН flowmeters, and from this profile аН corrections are made. Speciallaboratory tests are required to establish the effects of nonnewtonian fluids оп ftow­meters, and little published data is available because of the тапу types of nonnew­tonian ftuids.

FLOWMETER SELECTION

ТЬе instrument engineer рroЬаЫу has а wider choice of devices when specifyingа flowmeter than [ог апу other process-monitoring application. It is estimated (Нау­

ward, 1975) that at least 100 flowmeter types are commercial1у available, and newtypes are being continually introduced. Meters аге chosen оп the basis of cost, linesize, the fluid being metered, its state (gas, уарor, ог liquid), meter range, anddesired accuracy.

Fumess (1993) reviews the British Standard 7405 (1991) and summarizes asfollows:

With so тапу different types of flowmeters available from so тапу sources of supply,flowmeter selection is becoming increasingly difficult.... ТЬе new BS 7405 classifiesclosed conduit flowmeters into 1О major groups and this grouping was used in (Ье basiclayout of the standard. More (Ьап 45 variables were identified as the most importantfactors in selection.

Clearly then, meter selection is difficult and requires а knowledge of the processas well as the basic principle underlying the more соттоп meter types.

Опlу the more widely used general-purpose flowmeters-those listed in ТаЫе

6.1-are covered in this handbook. Far these devices, operating principles, selectionbases, and equations far the calculation of permanent pressure 10ss and уеагlу еп­

ergy cost are summarized in this chapter.

D/FFERENT/AL PRODUCERS

Тhe differential-praducing flowmeters, sometimes called head-class ftowmeters, areselected most frequently because of their long history of use in тапу applications.А питЬег of primary elements belong to this class: The concentric orifice, venturi,flow nazzle, Lo-Loss tube, target ftowmeter, pitat tube, and multipart-averaging areаН differential producers. When same other ftowmeter is selected, it is usual1ybecause of ап obstructianless feature, wider range, or а tendency against freezingог condensate buildup in lead lines ог because the fluid is abrasive, dirty, or madeup of more than опе component (slurry). It is probably true that аll new ftowmetersmust, а! least initially, compete in applications where the thin concentric orifice hasproved less than satisfactory.

Although orifice ftowmeters continue to account [ог 80 + регсеп! of installedprocess plant meters, the past 8 to 1О years have seen а gradual shift in meter

INTRODUCтION ТО ТНЕDIFFERENTIAL PRODUCER

Тhe differential-producing fl.owmeters are the most widely used in industrial pro­cess-measurement and control applications. ТЬе square-edged concentric orifice isselected for 80 percent of аН liquid, gas, and vapor (steam) applications. Thischapter contains а brief history of the differential producer and а 100k at the or­ganization of Chaps. 8 through 12, which deal exclusively with differential рro­

ducers.

H/STOR/CAL BACKGROUND

Тhere are numerous examples of the early application of the principle of the dif­ferential producer. ТЬе hourglass and the use of the orifice during Caesar's time tomeasure the flow of water to householders are but two of тапу. But the develop­ments which led to the design and widespread use of the various types of primaryelements began in the seventeenth century.

А! (Ье start of the seventeenth century, Castelli and Топicеlli laid the foundationfor the theory of differential producers with the concepts that the rate of flow isequal to the velocity times the pipe area and that the discharge through ап orificevaries with the square root of the head. Until recently аН differential producers Ьауе

Ьееп called head-class fl.owmeters because of this early work and that of ВеrnоиШ,who, in 1738, developed the hydraulic equation for the calculation of flow rate.

In 1732, Pitot presented his paper оп the pitot tube, and in 1797 Yenturi риЬ­

lished his work оп а fl.owmeter principle that today bears his пате. Yenturi's workwas developed into the first commercial flowmeter in 1887 Ьу Clemens Herschel.Herschel's laboratory work defined the dimensions of the Herschel venturi and laidthe foundation for future lаЬоrаюrу investigations to determine the relationshipsbetween geometry and differential pressure for the other differential producers.

In 1913, Е. О. Hickstein (1915) presented early data оп orifice flowmeters withpressure taps 10cated 2-!- pipe diameters upstream and 8 pipe diameters downstream.Тhis work, and that of others, led to several other pressure-tap 10cations, such asthose for D-and-DI2 and уепа contracta taps.

In 1916, Е. G. Bailey delivered а paper оп the measurement of steam with orificeflowmeters, and in 1912 experimental work Ьу Thomas R. Weymouth of the UnitedNatural Gas Сатрапу was the basis for the use of the orifice fl.owmeter for теа­

suring natural gas. For convenience, Weymouth used ftange pressure taps located

DIFFERENTIAL PRODUCERS:INSTALLATION

It is important that the installation of the primary element арргоасЬ the standard orreference conditions which prevailed when the flow-coefficient information wasobtained. ТЬе condition of the pipe, mating of pipe sections, pressure-tap design,straight lengths of pipe preceding and following the primary element, and lead linesthat transmit the differential pressure to the secondary measuring element аН affectmeasurement accuracy. While some of these mау have а minor effect, others сап

introduce 5 or 1О percent bias епогs. In general, these епогs аге not predictable,and attempts to adjust coefficients [ог the effect of nonstandard conditions Ьауе notЬееп successful.

PIPING

Reference Piping

ISO Standard 51 67t (1991) gives requirements [ог reference piping conceming thefollowing items:

1. Visual condition of the outside of the pipe as to both straightness and circularity2. Visual condition of the intemal surface of the pipe

3. Reference-condition relative roughness for the intemal surface (see ТаЫе 5.6)4. Location of measurement planes and пиmЬег of measurements for the determi­

nation of the average pipe diameter D5. Circularity of а specified length of pipe preceding the primary element

tSubsequently ISO 5167 was developed into ANSI/ ASME MFC-3M.

DIFFERENTIALPRODUCERS:ENGINEERINGEQUATIONS

ТЬе sizing and fiow-rate equations for аН differential producers are identical. ТЬеуare developed from theoretical assumptions, modified Ьу correction factors basedоп empirical evidence, and further altered based оп geometric considerations rela­tive to fixed-geometry devices. This chapter develops the engineering equations andpresents them in tables for ease in preparing computer programs.

THEORETICAL FLOW-RATE EQUAТJONS

Liquid Equation

Thе dynamic equation for one-dimensional flow of incompressible fluids is derivedЬу applying Newton's seeond law to the fluid element shown in Fig. 9.1a. ТЬе sumof the three forees in the direetion of flow is equated to the mass of the elementtimes its aeeeleration.

ТЬе external forees aeting оп the fluid element in the direction of flow are:

1. ТЬе net driving foree produeed Ьу the static pressure acting over the element'supstream and downstream areas

2. ТЬе body foree (weight) for а nonhorizontal element

3. ТЬе viscous shear stress that aets оп the cireumferenee of the element

These forees are expressed in differential form, using the English engineering sys­tem of units, as

far the net pressure foree,

дР!--dSdА

as (9.1)

DIFFERENTIAL PRODUCERS:DESIGN INFORMATION

Measured differential pressures depend оп both fluid characteristies and primary­element geometry. ТЬе ргорег use of differential produeers requires adherenee tothe installation requirements given in СЬар. 8 and the details presented in thischapter.

This ehapter is eoncerned with differential produeers that аге usually sized toproduee а seleeted differential at а design flow rate. In СЬар. 11, design informationis presented for fixed-geometry deviees, for whieh the differential (ог, for а targetflowmeter, the foree оп the target) must Ье detennined for the design flow rate.

" Тhe graphs presented for diseharge eoefficients and gas expansion faetors weredeveloped from the equations of СЬар. 9 when applicable. Others аге based оп

recommendations given in the technicalliterature.

ORIFICES

Concentric Square-Edged Orifice

Shown in Fig. 10.1 is the pressure profile along а meter гип containing а concentricsquare-edged orifice. ТЬе pressure first increases, beginning at approximately 0.5D upstream, and then decreases to а minimum at the уепа contracta. From thispoint, the pressure recovers to the initial upstream pressure (less pressure lossesdue 10 friction and energy 10sses). ТЬе specifie (ар 10cation discharge-coefficientequation presented in ТаЫе 9.1 and the generalized tap loeation equation (9.117)were developed Ьу Stolz from empirical discharge-coefficient data and this type ofpressure-gradient data.

Pressure-tap spacing requirements for flange, D and D12, and 2-tD and 8D tapsме given in Fig. 10.2. Individual and annular-slot corner-tap design requirementsare presented in Fig. 10.3.

IJlustrated in Fig. 10.4a аге the two most commonly used orifice plates types.Тhe paddle design is the most еоmтоп and is easily installed between orifieeflanges. ТЬе universal cireular design is for installation in either а single- ог dual­сЬаmЬег orifiee fitting ог in а plate holder ring-type joint for mounting betweengrooved flanges (Fig. 1О.4Ь). ТЬе outside diameter of the paddle type varies withthe pipe sehedule size to assure eoneentricity when instal1ed between the flangeOOlts.

(9.103)

DIFFERENTIAL PRODUCERS:FIXED-GEOMETRY DEVICES

Chapter 9 eovers differential produeers that аге sized Ьу determining primary­element dimensions that will produee а chosen differential at а design flow rate.Ап alternative is to seleet а fixed-geometry primary deviee. These Ьауе limiteddimensional seleetivity~ therefore, the differential pressure ог target faree, ratherthan the flowmeter dimensions, must Ье ealculated to mateh the design ftow rate.

Arithmetie-progression orifices, annular orifices, target ftowmeters, integral ari­fices, Annubars, and elbow flowmeters аге covered in this chapter. ТЬе flow-rateequations developed in СЬар. 9 (Tables 9.36 thraugh 9.38) apply to these devices.However, several of the symbols mау Ье changed, grouped, ог set equal to 1,depending оп the deviee, how the geometry affects the differential pressure ог targetfaree, and whether ап expansion factor is required. Table 11.1 presents the neces­sary modifications to these equations. ТЬе neeessary graphs and equations andexamples of the calculation proeedure аге given in the remainder of this ehapter.

AR/THMET/C-PROGRESS/ON OR/F/CES(EVEN-S/ZED OR/F/CES)

То change flow capaeity, тапу plants stoek а series of orifice p]ates with fixed­increment (arithmetic-progression) Ьоге inereases. Measurement equipment, pipediameter, and fluid properties remain eonstant, and it beeomes neeessary to deter­mine flow rates for fixed-range differential-pressure transmitters (50 in, 100 in, ete.).ТЬе general form of the flow-rate equation is given Ьу Eq. (9.103) as

СУfЗ2 .,q = N VТ=f34 D~f(p) vт;:

1 - /34

where d 2 = f3 2D 2 has Ьееп substituted. With eonstant fluid properties, design URVdifferential, and pipe size, the variables аге conveniently grauped as

(11.1 )

where the braeketed term remains constant for а given differentia), and the.8-dependent quantities ehange with Ьоге inerement and Reynolds питЬег. Equation

DIFFERENTIAL PRODUCERS:COMPUTATIONS

Depending оп the desired accuracy, flow-rate determination тау require only а

simple visual observation of differential pressure оп а square-root chart, or it тауinvolve the use of а dedicated microprocessor that receives several measurementsignals and calculates the flow rate. Compensation for pressure and/or temperaturevariations оп chart indications mау mеап using pneumatic ог electronic analogcomputers. Total flow, rather than flow rate, сап Ье computed ог determined Ьу

chart integration. ТЬе choice of measurement equipment, саlculаtiоп procedure,computation means, and data-transmission means is extensive. This chapter presentssome of the commonly used equipment and calculations for chart integration.

GENERAL PR/NC/PLES

Measured and Unmeasured VariabIes

The flow-rate calculation сап Ье viewed as the product of three terms: ап ипmеа­

sured-variable term, а measured-variable term, and differential pressure. Differentialpressure is always measured. ТЬе unmeasured-variables term includes а unit соп­

version factor and аН factors assumed to Ье constant; the measured variables arequantities that must Ье measured for the desired ассигасу (see СЬар. 4). ТЬе ип­

measured variables аге combined into а meter-coefficient factor Fмс which сот­

топ)у iпсludеs pipe and primary-element bore dimensions and the discharge со­

efficient. Measured variables are usually density-related (such as pressure andtemperature) or are derived from other measurements (such as the Reynolds-numbercorrection, which is derived from the flow rate, and the gas expansion factor, whichis derived from differential- and absolute-pressure measurements). Depending оп

process variations, the designer detennines which variables must Ье measured andwhich сап Ье assumed constant.

As ал example, the mass flow equation for 1iquids тау Ье written as

(12.1 )

ТЬе first bracketed term contains the unmeasured variables; that is, after the pipeand bore diameter are measured and the thermal-expansion factor, liquid­compressibility factor, and discharge coefficient are calcu]ated, the designer соп-

CRITICAL FLOW

When а gas accelerates through а restriction, its density decreases and its velocityincreases. Since the mass flow per unit area (mass Оих) is а function of both densityand velocity, а critical area exists at which the mass Оих is at а maximum. In thisarea, the velocity is sonic, and further decreasing the downstream pressure will notincrease the mass flow. This is referred to as choked or critical flow. For liquids, ifthe pressure at the minimum area is reduced to the liquid's vapor pressure, а сау-

, itation zone is fопnеd which restricts the flow. Further decreases in pressure willnot increase the flow rate. In both cases, mass flow сап only Ье increased Ьу

increasing the upstream pressure.Critical flow nozzles are widely used as secondary standards to test air сот­

pre8sors, steam generators, and natural gas flowmeters. Over the last 20 years theaero8pace industry has developed а critical nozzle with а downstream diffu8er (уеп­

turi) recovery section that gives minimum overalI pressure 1088 to maintain criticalflow. Cavitating venturis or restrictive orifices are used as flow limiters in the еуеп!

·of а downstream system failure.

GASES

Basic Principles

Figure 13.1 shows the pressure-velocity relationship for а convergent-divergent pas­sage through which а compre8sible_fluid accelerates. As the downstream pressureРп decreases, the throat velocity Vt increases until а critical pressure ratiot isreached at which the throat velocity is sonic. Further decreases in the downstreampressure will not increase the mass flow rate. ТЬе flow is referred to as subsonicdown to the critical pressure ratio, and critical below this ratio. lп critical flow thethroat velocity is always sonic, but the velocity increases in the diffuser section,where а normal shock front occurs. Depending оп the downstream pressure, fourflow conditions are possible:

1. For pressure ratios greater than critical, the flow remains subsonic and тау Ье

calculated with the relationships given in СЬар. 9.

2. When Pf3 is reduced to the value at which sonic throat velocity first occurred,the flow decelerates in the divergent section to а subsonic velocity; the gas

t The critical (or choking) pressure ratio is discussed in detail later in this chapter.

LINEAR FLOWMETERS

In general, flowmeters whose output is not proportional to the square of the ftowrate divided Ьу the fluid density аге linear flowmeters. Either the operating principleyields а direct linear output ог, through electronics, the output is linearized to vol­umetric or mass flow units. These meters сап Ье grouped into two classes: pulse­frequency type5 and linear-scale flowmeters. Both are discussed in this chapter.

PULSE-FREQUENCY ТУРЕ

Turbine and vortex flowmeters produce а frequency (pulse train) proportional to thepipeline velocity, and positive-displacement meters produce опе pulse per unit vol­иmе. Although based оп different operating principles, these pu[se-type meters re­spond to flowing conditions and, therefore, the pertinent engineering equations forflowing and base volumes and for mass flow are the 5аmе. With turbine and vortexflowmeter5, flow rate is commonly measured Ьу frequency or Ьу frequency-to­analog conversion, but this i5 seldom the case [ог the low-re501ution positive­displacement meters.

ТЬе signature curves for vortex and turbine flowmeter5, although different inshape, are liпеаг оуег 20: 1 ог 3О: 1 flow-rate ranges, and, Ьепсе а теап metercoefficient (К factor) is given. Positive-displacement flowmeters аге usual]y саli­

brated in the desired volumetric units and, through suitabIe internal gearing, directlydisplay the total уоlите. For turbine and vortex flowmeters, the integrated count i5electronical]y scaled, using the К factor, to display the total volume. Through 5uit­аЫе electronics and computer ог mechanical computations, base уоlите ог mа55

flow i5 а150 displayed.

Engineering Equations

Factor. ТЬе К factor defines the relationship between flow rate and frequencyfor vortex and turbine flowmeters. For Iiquid turbine meters, this factor i5 obtained

water calibration; for gas meters, Ьу а low-pressure bell prover test. ТЬе К factoranу volume units) is defined as

fHZ pulsesК - - - --=------

F,v - qv - unit volume (14.1 )

METER INFLUENCEQUANTITIES

IТhe proper use of апу flowmeter assumes that the appropriate 150, А5МЕ, ANSI,!АаА, API, etc., standards and recommendations of the manufacturer have Ьееп

!adhered (о in order to achieve reference accuracy (overall uncertainty) conditions.This chapter presents the availabIe information оп the effects of departure fromnhese conditions. These are геfепеd to as influence quantities and mау Ье related11:0 the primary element, secondary element, the flowmeter, or anу intemal or exter­!nal factors associated with the in-situ conditions.I Coriolis mass, magnetic, turbine, positive displacement, ultrasonic, and vortexIflowmeters are considered proprietary designs апд, in general, the реrfолnanсе Ье­

I1:ween differing designs will not ье (Ье same. ТЬе infonnation presented is availabIe6п the literature and,the reader should use the infопnаtiоп in this chapter primarily[or assisting in locating possible metering errors. In аll cases the manufacturer!should Ье ,consulted for the latest iпfолnаtiоп оп а particular design.! Infonnation about еасЬ of the following meters is presented in tabular form withIa brief description of some influences and а referenced figure number.

Coriolis mass fiowmeter. ТаЫе 15.1 and Figs. 15.1 to 15.7 give some of thereported influence quantities for the Coriolis mass flowmeters.

DijJerential producers (огфсе, nozzle, аnd venturi). TabIes 15.2 and 15.3 andFigs. 15.8 to 15.23 give the reported influence quantities for the orifice, flownozzle, and venturi flowmeters.

Magnetic flowmeter. ТаЫе 15.4 and Figs. 15.24 to 15.34 present reported in­Виепсе quantities for magnetic flowmeters.

Positive-displacement meters. ТаЫе 15.5 and Fig. 15.35 give some of the re­ported influence quantities for positive-displacement meters.

Turbine fiowmeter. ТаЫе 15.6 and Figs. 15.36 to 15.44 present some of thereported in:fluence quantities for turbine flowmeters.

Ultrasonic .flowтeter. ТаЫе 15.7 and Figs. 15.45 to 15.52 present some of thereported influence quantities for turbine flowmeters.

Vortex.flowтeter. ТаЫе 15.8 and Figs. 15.52 to 15.64 give some ofthe reportedinfluence quantities for turbine flowmeters.

DISCUSSIONS AND PROOFS

А.1 NEWТON'S МЕТНОО FOR ТНЕ

APPROX/MATE SOLUT/ON OFNUMER/CAL EOUATIONS

Мапу of the equations used in flow measurement require ап iterative solution [ог

the flow rate, compressibility factor, ог orifice Ьоге. Newton's method [ог the ар­

proximate solution of numerical equations is а convenient trial-and-error techniquethat requires fewer estimates than other methods. In тапу cases the initial solutionis sufficiently accurate, and а single calculation сап Ье used. ТЬе calculations arereadily programmable оп hand calculators, dedicated microprocessors, or centralcomputers.

As ап example of the use of Newton's method, consider а 2-in (50-тm) orificeflowmeter operating at а Reynolds питЬег of 10,000, [ог which the flow equationreduces to

= 4019 0.8884q . + 075q"

(A.l)

In this equation, the first constant (4.019) is the calculated flow rate at ап infiniteReynolds питЬег for the measured differential and fluid density. ТЬе second соп­

stant includes the coefficient correction for Reynolds number, а dimensional term,and апу necessary unit conversion.

Equation (A.l) is nonlinear, and to solve it estimates of the flow rate q must Ьеsuccessively substituted until the relationship is satisfied. Instead, Eq. (A.l) сап Ье

rearranged into а function equation as

F = 4019 0.8884_. + 075 qq"

(А.2)

ТЬеп, to solve Eq. (А.2) [ог the flow rate q, successive estimates of the flow rateасе substituted into Eq. (А.2) until F is calculated to Ье zero.

ТЬе values of F for several flow rates аге given in ТаЫе А.1, beginning withthe infinite flow rate. These pairs of values are shown plotted in Fig. А.l. ТЬе zerocrossing provides the zero root of Eq. (А.2), which is the desired flow rate. Itsvalue сап Ье read as 4.316; when substituted into Eq. (А.l) ог (А.2), this valuesatisfies the equality.

The number of iterations (ог estimates or guesses) is reduced if the equation ofthe tangent to the curve at the initial estimate qo is used to calculate the second

FLOW-RAТE, REYNOLDS­NUMBER, AND UNIТ

.CONVERSION TABLES

TABLE С.1 SI-Unit Conversion Factorst

То соауе" 'roм

ftJs2

free fall, standard (g)in/S2

degree (angle)JDinute (angle)вecond (angle)

ft2in2

шi2 (international)Шi2 (U.S. ашvеу)

dyne'cmkf,.moz,·inlb,·inlbr·ft

То

ACCELERATION

теи, ре' second2 (m/S2)теи, ре, вecond2 (mJS2)

теи, ре' second2 (т/а2)

ANGLE

radian (rad)radian (rad)radian (rad)

AREA

mete,z (т2)

mete,z (т2)

mete,z (ш2)

mete,z (т2)

BENDING MOMENТOR ТORQUE

пеwtoп meter (N .ш)пеwtoп meter (N· т)пеwtoп теи, (N·т)пеwtoп meter (N .ш)пеwtoп meter (N .т)

Multiply Ьу

3.048 ООО*Е - 019.806 650*Е + 002.540 ООО·Е - 02

1.745 329 Е - 022.908 882 Е - 044.848 137 Е - 06

9.290 3О4*Е - 026.451 6ОО*Е - 042.589 988 Е + 062.589 998 Е + 06

i.OOO ООО*Е - 079.806 650·Е + 007.061 552 Е - 031.129 848 Е - 011.355 818 Е + 00

lbr·ftlinlbr·in/in

BENDING MOMENТOR TORQUE PER UNIТ LENGTH

пеwtoп шеter ре' meter (N .т/т) 5.337 866 Е + 01пеwtoп meter ре, meter (N· m/m) 4.448 222 Е + 00

ENERGY (lNCLUDES WORK)

British tbermal unit (Internationa1 ТаЫе) joule (J)British tbermal unit (тean) jou1e (J)British thermal unit (thermochemical) joule (J)

tFac:torвwith an aвteriak are епd.

1.055 056 Е + 031.055 87 Е + 031.054 350 Е + 03

GENERALIZED FLUIDPROPERТIES

-200 о

Absolute pressure Pf (psia)200 400 600 800 1000 1200

-2

3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0I-

U

.......t--!"

1.8 "........

1.6 ~::J

+-

1.4 еILIа..

Е

1.2 ILI+-U

1.0ILIU~

~

0.8ILIа:

0.6

0,4

0.2

200

() ~17,,~I--I.-

.,,1~(Q9~~e L7"Г r ~

$-~(, Vv о; '"~'l.rv.J VI.-L.7'~~

~O v ~ 1717.... ~

[/VI.7,

11' .... ..... [/ ';;i~ ~r;-

...... r~ ~ .......~~ ~1/ r/v .........~~ ..."" :I~

~ ....~~~ ~~Г7'v....~~..........wX.... ~ г---~

11: r/.~ ~~ ~

.~ ~~vv ~v. ~/ V ~l~ V ", ~v ~ ~v'/ г7 ,..'~;/ ....

........ ~/"'7 7f...~ / ~()y~...., ...... Q ~

...... с..." c.~\.4111"'/ " .. ~fl; L.o И L/

LJ 1/"'" 1/ ~ ~ .~~v'~ 1/ ..... "

v/ 1/ 1/ ,,~ , ~~~/ 1/ 1/ .~ ~ ..... ,~ vv .....

/ ..... / 1/ ~ ~' V, ...... ~~r~ ~V1/ 1/ .... ,..... 1,.;"[..;[,.;' 10-

v~ 1/ V ~ VVv~~~-~~~/ ./ V V .... VVV~V

~~~/ / V~ V[/~[/~~~~V, / 1/ [.... Vv~~~...... I V VI/I ...... ...... 10' ~~V

) Vi;'"'/~v--"'~.~. Vl .... v~~.. r':"" ...~ ....

'/[/1.' 1/·..... 1-:00:~"'"~

~Г/V~ ~';::rJ:III

l/r~ ~.~~~~

~~~.,,..

00 о 200 400 600 в(ю 1000 1

2.6

2.4

2.2

0.8

06

0.4

0.2

О

а. 2.0.......~

~ 18..~ 1.6~

~ 1.45.u 1.2ILIu~

-g 10а::

Temperature Т, (OF)

Figure О.l Estimated reduced pressure and temperature fгош molecularweight. (From GPSA. 1979; used и'irh permission.)

LIQUID DENSIТV ANDSPECIFIC GRAVIТV

® Buft1!f"fot ® Peanutoil

'""'"I(H ® Coconutoil ® Soybeon oil

'''''' .::~~ ~ © C01tonseed oil © VeqetobIe oi 11-1---

~Vl.l ~ @Polmoil ® Wheot-germoilr--~~!V -~~ I':[)t-~~r-.t:

~r-. .....~ -t-.r--.t--.,....... "--~~~~F=::: -~""Izд j ==a.iro... '-

~ --"- ~.,..::-~h ~ ......!?J h ~I"'--~I ?0.85

50 75 100 125 150 175 200 225Temperature (OF)

Figure Е.l Specific gravities of fats and oils. (Froт Fischer & Porter Catalog /О-А-54; usedwith perтiss;on.)

0.95

® Вlиe size solution ® Rosin size solutlOn (4.8 - 5.0 % soli ds)

® Bluing solution-1"--

® Sveen glue r-I--I-

(е) ~}- © Nylon size solution ® wateг-disрегsibIе оит, PLF 50~ ... r-::-: @ Rosin size solutions

~~I--

(G) 1(01 1--® Rosin size solution (4'r.solidsl

"'-~..,;al""';; --- r-r-I-t- u... 11'-.

-r-;-i1~

r-,......t-h~ ~tt:I""-~r-~ ..... 100-

V"BJ IlEJ Ir.:jy - f\....r-. "-t-~Т l' '~I I

1.06

1.0416­

'"f 1.02

~~ 1.00'ugu> 0.98

0.9650 75 100 125 150

Temperature (OF)

175 200 225

Figure Е.2 Specific gravltles of liquids related to the paper, leather, and textile indus­tries. (Froт Fischer & Porter CataJog JO-A-54; Юiеd with perтission.)

VISCOSIТIES OF LIQUIDS

1000

8

654

3

2

100;:- 8сп

~ 6

~ 5

>- 4

j 3'>,u 2iiс

~

10

8

65

4

3

2

1 1 ] J I I 1, - .....r\. 0t Butterflt

(1) Castor oil-!--

JI.. - ...." - © Coconut oil

'1'\.. ~) @ Codlmroil-1-

~~Ф Cottonteed oil -!--

® Мёner.' oil

«1> P_nutoil -!--

1\ ~~е Sovbunoil

~'I," \ ~ ....~ -~

~ ...... rJ5 "ф Veget.ыeoil

"-Q) Wh..t-germ oil _!--

'" J €> FiIh-оil sofuЫe =::~ '\.

""""concentrate 1-1-

-..с -- '~~ "~ ~~ВIIJII !\ ,....Jlm-

_....r'I~~~ 1 .....

,'1 i"'l:"~~ .....

............ i".,..,;"'.... ;,:""00 ~1-00 ......;:::~~ '"r-...~~~ "~ '" - ,

'"(р \ ~~

===:~ lJt~)

.....,K

V {.9 ~ ...... r-- .....~Ioo..

....."'-1"'000.

r-.. .....

Г"oo~

........, .......

~ ......~ ,.Q)

Ioo",.,,~ .1............r--,.

225200175юо751.0

50 125 150Temperaiure TF (Of)

Figure F.l Kinematic viscosJtles of fats and oils. (From Fischer & Porter Cata/og/О-А-54; used with perтission.)

ISENТROPIC EXPONENTS

2 3 4 5 6 8 102 3 4 5 6 8 0.1

1\\,

о "\.&~, ~ I \~

0090 rN. ~~'...... :-..

~lIJ~~O·~5 7'-..L

I~')i"""1: ...... -.', "'" ~~

0.80 Jl 171 1 / / -.."'-. ....., ...1/ ')IJ

1I,~~ t7-А _Г'-;::~ ~

r/5~)......

VbI'1Vt::~~ ~~о,) ~) I ,.

o~~~~~!/IJV~I/[1: Й~~~'""";:::j'.: О - ~t:::'1,.-

/~V/I /J%~ l'7),~~/// '1,.У"j)

~( / ~Vh~~~11, I ):«~~ 1-pi'~::

!..'l"///1-.0001 ...

'!-...:) f f f f 11 "" " Z~ ./ 'Z.~~~

!..С) !.1'77 ""

.1' .1' .~ "''' ././ ,..",0,.~'.&J" / / / "/ I " // ./ 7 7Г.7 '7 ~.~ ~~

v""fl' "/ //1// '// / '/1/ /~ /" ./ / ./1./ 7' ~.c.

(;~ V~ r/) r7/ ~//.~/ /г/71 ~~, / v ./ //V/ ~.; 10-

J>..:i V"/ '// /~'l/ V/ '// /, tI ////~V7': '!>. ~

r-.C>.~( /г/~~If~ ~/~ 'l:~%/,.I;1~~~~~c.P

~~ &CJo~~ ~~t:lC~~ ~~~~~/~/

/~~~~VlIl.1.~tJ'" ~~~~V/'LlV~~~~V/t;~v~~~

~y

~~~~V~~"~~~/':~~~~/~~./ /1,/

" Z -''' .r '"'./ ./'"' "' ./ ./ ./

"./ '"'/" "7 /" ,/".// " /" .//" ./ /"/" /" ./ " //" /" ./ / '/ /iI'.

7/ / .// ./ /./ ./ // ./ /././~ //V7 -;/ 7~ '/ lI" / ",// '//.~~~~~V/ [//~ V 1/ V/ V/V// /,~V

~~V~'l"1/ ~V ,1.1 'h~V~V

~~~~v v/'1~~~/,!

~~////~/,1

~VI~V//

~v:~~~~~~jO~

2

100

8

б

54

3

0.010.01

108

65

;. 4~ 3~u.Е 2с

о

:t=u

~ 1.0оu, 8~~

б111111

5fа. 4+-о41 3..с

Iu

2~'и..а.

CI)

0,1

8

б

54

3

2

2 3 4 5 6 8 1.0

Reduced pressure Pr

Figure 1.1 Specific-heat pressure-correction factof Руг> Еor simple fluid, UJ = О. (Froт Ed­тisler, 1974; all rights resenJed, used l1-'ith perтission.)