LIST OF SYMBOLS
Symbol and meaning а
а а
as А
Аа
Ар А,
Aj1 Ар
Асс (Асс ),.f Ap1'te
А
А
AL Ь
Ь Ь ЬС Ьр
Constant in Наll- Yarborough equation of state Constant in specific-heat equation Constant in gas viscosity equation Acceleration along а stream tube Area Annular area between float and tapered wall of а variable-area flowmeter Deadweight-tester piston area Throat area of а critical nozzle Effective area of float in а variable-area flowmeter Pipe area Ассшасу. combined precision and bias errors Reference-condition accuracy
U.S. units
SI unitst
Btu/(lbm ·mol·oR)
J* I(kg'mol' К)
ftl S2
m/s 2
ft ft 2
m2 m2
in 2 ft2 f1'
mm 2 m2 m2
f1'
m2
%
%
%
%
f1'
m2
2
Plate area in viscosity derivation equation Constant in Redlich-Kwong equation of state Constant in Ostwald power-law equation Constant in liquid viscosity equation Constant in Наll- Yarborough equation of state Constant in equation for specific heat at constant pressure Constant in general form of discharge-coefficient equation Slope constant in liquid-bulkmodulus equation Frequency coefficient for pulsating flow
tExcept for dimensionless or defined SI unit symbols. as in in the (ех! with а superscript asterisk. as in F ,'; .
Тк •
symbols that apply
(о
SI units are shown
FLUID
PROPERТIES
Data оп physical properties is often required for calcu!ations of base flow rates and pipe 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 and temperature, and whether сопесtiопs need to Ье considered depends оп the design objective. In тапу cases, properties are assumed constant at design conditions, and сопесtiопs 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 better for !iquids and gases than for mixtures. In тапу app!ications, particu!ar!y for high inert 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 pure substances and rnixtures. For illustrative purposes shaded areas оп graphs in this chapter 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 more соттоп to use temperature and pressure measurements to calcu!ate density. The reciprocal 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. The iпtепеlаtiопshiрs of pressure, temperature, and specific уо!ите are a!so important because of the law of сопеsропdiпg states. From these relationships, the fluid state сап 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 words уарог and gas are often used interchangeably because they are thermodynamically identical. Historically, the term уарог has Ьееп used to designate а substance, such as 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 under the 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 flow measurement and to discuss typical temperature- and pressure-measuring deviees. Тhis information will Ье used in subsequent ehapters in the development of the engineering 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 ОЬ m ), 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 1 lb f will accelerate 1 lЬm at 32.17405 ftls 2 , а dimensional conversion constant сап Ье derived as
F =
1 gc
-та
(3.1)
т
1
ft
lb f = lЬ .ft/(lbf 'S 2 ) lЬ т ~ The constant gc has the same value as standard gravity go, defined at sea lеуеl and 450 latitude, but it has the dimensions of lb m ·ft/{1bf ·s 2 ). It is, therefore, а dimensional 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 and weight 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 alter the 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 particular influence quantity depend оп the sensitivity of а particular ftowmeter to that quantity and whether or not а calculation correction сап Ье made. For newtonian ftuids, velocity profiles сап usually Ье brought into acceptable limits Ьу the instaHation of sufficient straight pipe or, for shorter lengths, with ftow conditioners. However, other infiuence quantities тау require the installation of pulsating dampers or the use 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 following sections.
VELOC/TY PROFILE Velocity profile is probably the most important (and least understood) infiuence quantity. ТЬе effects of swirl, nonnewtonian ftuids, and nonaxisymmetric profiles оп а rneter's performance are not only difficult to analyze, but they cannot easily ье duplicated in а laboratory.
Newtonian Fluids ТЬе rheological behavior of а ftuid determines whether it is classified as newtonian or 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 to the applied stress. ТЬе constant of proportionality is the absolute viscosity defined in СЬар. 2. АН gases, most liquids, and fine mixtuгes of spherical particles in liquids and 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. Special laboratory 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 specifying а flowmeter than [ог апу other process-monitoring application. It is estimated (Нау ward, 1975) that at least 100 flowmeter types are commercial1 у available, and new
types are being continually introduced. Meters аге chosen оп the basis of cost, line size, the fluid being metered, its state (gas, уарor, ог liquid), meter range, and desired accuracy. Fumess (1993) reviews the British Standard 7405 (1991) and summarizes as follows: With so тапу different types of flowmeters available from so тапу sources of supply, flowmeter selection is becoming increasingly difficult.... ТЬе new BS 7405 classifies closed conduit flowmeters into 1О major groups and this grouping was used in (Ье basic layout of the standard. More (Ьап 45 variables were identified as the most important factors in selection.
Clearly then, meter selection is difficult and requires а knowledge of the process as 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, selection bases, 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, are selected 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 are аН differential producers. When same other ftowmeter is selected, it is usual1y because of ап obstructianless feature, wider range, or а tendency against freezing ог condensate buildup in lead lines ог because the fluid is abrasive, dirty, or made up of more than опе component (slurry). It is probably true that аll new ftowmeters must, а! least initially, compete in applications where the thin concentric orifice has proved less than satisfactory. Although orifice ftowmeters continue to account [ог 80 + регсеп! of installed process plant meters, the past 8 to 1О years have seen а gradual shift in meter
INTRODUCтION ТО ТНЕ DIFFERENTIAL PRODUCER
Тhe differential-producing fl.owmeters are the most widely used in industrial process-measurement and control applications. ТЬе square-edged concentric orifice is selected for 80 percent of аН liquid, gas, and vapor (steam) applications. This chapter 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 to measure 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 primary elements began in the seventeenth century. А! (Ье start of the seventeenth century, Castelli and Топicеlli laid the foundation for the theory of differential producers with the concepts that the rate of flow is equal to the velocity times the pipe area and that the discharge through ап orifice varies 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 work was developed into the first commercial flowmeter in 1887 Ьу Clemens Herschel. Herschel's laboratory work defined the dimensions of the Herschel venturi and laid the foundation for future lаЬоrаюrу investigations to determine the relationships between geometry and differential pressure for the other differential producers. In 1913, Е. О. Hickstein (1915) presented early data оп orifice flowmeters with pressure 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 as those for D-and-DI2 and уепа contracta taps. In 1916, Е. G. Bailey delivered а paper оп the measurement of steam with orifice flowmeters, and in 1912 experimental work Ьу Thomas R. Weymouth of the United Natural 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 or reference conditions which prevailed when the flow-coefficient information was obtained. ТЬе condition of the pipe, mating of pipe sections, pressure-tap design, straight lengths of pipe preceding and following the primary element, and lead lines that transmit the differential pressure to the secondary measuring element аН affect measurement 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 not Ьееп successful.
PIPING Reference Piping
ISO Standard 51 67t (1991) gives requirements following items:
[ог
reference piping conceming the
1. 2. 3. 4.
Visual condition of the outside of the pipe as to both straightness and circularity Visual condition of the intemal surface of the pipe Reference-condition relative roughness for the intemal surface (see ТаЫе 5.6) Location of measurement planes and пиmЬег of measurements for the determination of the average pipe diameter D 5. Circularity of а specified length of pipe preceding the primary element tSubsequently ISO 5167 was developed into ANSI/ ASME MFC-3M.
DIFFERENTIAL PRODUCERS: ENGINEERING EQUATIONS
ТЬе
sizing and fiow-rate equations for аН differential producers are identical. ТЬеу are developed from theoretical assumptions, modified Ьу correction factors based оп empirical evidence, and further altered based оп geometric considerations relative to fixed-geometry devices. This chapter develops the engineering equations and presents 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 derived Ьу applying Newton's seeond law to the fluid element shown in Fig. 9.1a. ТЬе sum
of the three forees in the direetion of flow is equated to the mass of the element times its aeeeleration. ТЬе external forees aeting оп the fluid element in the direction of flow are: net driving foree produeed Ьу the static pressure acting over the element's upstream 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 1.
ТЬе
These forees are expressed in differential form, using the English engineering system of units, as дР!
--dSdА
as
far the net pressure foree,
(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 to the installation requirements given in СЬар. 8 and the details presented in this chapter. This ehapter is eoncerned with differential produeers that аге usually sized to produee а seleeted differential at а design flow rate. In СЬар. 11, design information is presented for fixed-geometry deviees, for whieh the differential (ог, for а target flowmeter, the foree оп the target) must Ье detennined for the design flow rate. " Тhe graphs presented for diseharge eoefficients and gas expansion faetors were developed 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 concentric square-edged orifice. ТЬе pressure first increases, beginning at approximately 0.5 D upstream, and then decreases to а minimum at the уепа contracta. From this point, the pressure recovers to the initial upstream pressure (less pressure losses due 10 friction and energy 10sses). ТЬе specifie (ар 10cation discharge-coefficient equation 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 of pressure-gradient data. Pressure-tap spacing requirements for flange, D and D12, and 2-tD and 8D taps ме given in Fig. 10.2. Individual and annular-slot corner-tap design requirements are 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 orifiee flanges. 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 between grooved flanges (Fig. 1О.4Ь). ТЬе outside diameter of the paddle type varies with the pipe sehedule size to assure eoneentricity when instal1ed between the flange OOlts.
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 limited dimensional seleetivity~ therefore, the differential pressure ог target faree, rather than 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-rate equations 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 target faree, and whether ап expansion factor is required. Table 11.1 presents the necessary modifications to these equations. ТЬе neeessary graphs and equations and examples 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, pipe diameter, 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
(9.103)
where d 2 = f3 2D 2 has Ьееп substituted. With eonstant fluid properties, design URV differential, 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 тау involve the use of а dedicated microprocessor that receives several measurement signals and calculates the flow rate. Compensation for pressure and/or temperature variations оп chart indications mау mеап using pneumatic ог electronic analog computers. 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 presents some 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. Differential pressure is always measured. ТЬе unmeasured-variables term includes а unit соп version factor and аН factors assumed to Ье constant; the measured variables are quantities that must Ье measured for the desired ассигасу (see СЬар. 4). ТЬе ип measured variables аге combined into а meter-coefficient factor F мс 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 and temperature) or are derived from other measurements (such as the Reynolds-number correction, which is derived from the flow rate, and the gas expansion factor, which is derived from differential- and absolute-pressure measurements). Depending оп process variations, the designer detennines which variables must Ье measured and which 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 pipe and 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 velocity increases. Since the mass flow per unit area (mass Оих) is а function of both density and velocity, а critical area exists at which the mass Оих is at а maximum. In this area, the velocity is sonic, and further decreasing the downstream pressure will not increase the mass flow. This is referred to as choked or critical flow. For liquids, if the 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 will not 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 the aero8pace industry has developed а critical nozzle with а downstream diffu8er (уеп turi) recovery section that gives minimum overalI pressure 1088 to maintain critical flow. 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 pressure Рп decreases, the throat velocity Vt increases until а critical pressure ratiot is reached at which the throat velocity is sonic. Further decreases in the downstream pressure will not increase the mass flow rate. ТЬе flow is referred to as subsonic down to the critical pressure ratio, and critical below this ratio. lп critical flow the throat velocity is always sonic, but the velocity increases in the diffuser section, where а normal shock front occurs. Depending оп the downstream pressure, four flow 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 ftow rate divided Ьу the fluid density аге linear flowmeters. Either the operating principle yields 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 the pipeline 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 for flowing and base volumes and for mass flow are the 5аmе. With turbine and vortex flowmeter5, 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 in shape, are liпеаг оуег 20: 1 ог 3О: 1 flow-rate ranges, and, Ьепсе а теап meter coefficient (К factor) is given. Positive-displacement flowmeters аге usual]y саli brated in the desired volumetric units and, through suitabIe internal gearing, directly display the total уоlите. For turbine and vortex flowmeters, the integrated count i5 electronical]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 frequency for 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 factor anу volume units) is defined as К
fHZ
- -
F,v -
qv
pulses unit volume
- --=------
(14.1 )
METER INFLUENCE QUANTITIES
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 from nhese conditions. These are геfепеd to as influence quantities and mау Ье related 11:0 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 vortex Iflowmeters are considered proprietary designs апд, in general, the реrfолnanсе Ье I1:ween differing designs will not ье (Ье same. ТЬе infonnation presented is availabIe 6п 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 with Ia 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 the reported influence quantities for the Coriolis mass flowmeters. DijJerential producers (огфсе, nozzle, аnd venturi). TabIes 15.2 and 15.3 and Figs. 15.8 to 15.23 give the reported influence quantities for the orifice, flow nozzle, 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 the reported in:fluence quantities for turbine flowmeters. Ultrasonic .flowтeter. ТаЫе 15.7 and Figs. 15.45 to 15.52 present some of the reported influence quantities for turbine flowmeters. Vortex.flowтeter. ТаЫе 15.8 and Figs. 15.52 to 15.64 give some ofthe reported influence quantities for turbine flowmeters.
DISCUSSIONS AND PROOFS
А.1 NEWТON'S МЕТНОО
FOR ТНЕ
APPROX/MATE SOLUT/ON OF NUMER/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 technique that requires fewer estimates than other methods. In тапу cases the initial solution is sufficiently accurate, and а single calculation сап Ье used. ТЬе calculations are readily programmable оп hand calculators, dedicated microprocessors, or central computers. As ап example of the use of Newton's method, consider а 2-in (50-тm) orifice flowmeter operating at а Reynolds питЬег of 10,000, [ог which the flow equation reduces to
q
= 4019 0.8884 . + 075 q"
(A.l)
In this equation, the first constant (4.019) is the calculated flow rate at ап infinite Reynolds 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 Ье successively substituted until the relationship is satisfied. Instead, Eq. (A.l) сап Ье rearranged into а function equation as F = 4019 . ТЬеп, to solve асе substituted ТЬе values
+ 0.8884_ 075 q q"
(А.2)
Eq. (А.2) [ог the flow rate q, successive estimates of the flow rate into Eq. (А.2) until F is calculated to Ье zero. of F for several flow rates аге given in ТаЫе А.1, beginning with the infinite flow rate. These pairs of values are shown plotted in Fig. А.l. ТЬе zero crossing provides the zero root of Eq. (А.2), which is the desired flow rate. Its value сап Ье read as 4.316; when substituted into Eq. (А.l) ог (А.2), this value satisfies the equality. The number of iterations (ог estimates or guesses) is reduced if the equation of the 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
То
Multiply
Ьу
ACCELERATION теи, ре' second2 (m/S2) теи, ре, вecond 2 (mJS2) теи, ре' second2 (т/а 2 )
ftJs 2 free fall, standard (g)
in/S2
3.048 9.806 2.540
ООО*Е
- 01
650*Е
+ 00
ООО·Е
- 02
ANGLE radian (rad) radian (rad) radian (rad)
degree (angle) JDinute (angle) вecond (angle)
1.745 329 Е - 02
2.908 882 4.848 137
Е
Е
- 04 - 06
AREA mete,z (т2 )
9.290
3О4*Е
(т 2 ) (ш2 ) (т 2 )
6.451
6ОО*Е
ft2 in2
mete,z
шi 2 (international) Шi2 (U.S. ашvеу)
mete,z
mete,z
BENDING
meter (N . ш) meter (N· т) пеwtoп теи, (N· т) пеwtoп meter (N . ш) пеwtoп meter (N . т) пеwtoп
BENDING
Е Е
MOMENТ OR
i.OOO ООО*Е - 07
9.806 650·Е + 00 7.061 552 Е - 03 1.129 848 Е - 01 1.355 818 Е + 00
TORQUE PER UNIТ LENGTH
пеwtoп шеter ре'
lbr·ftlin lbr·in/in
+ 06 + 06
MOMENТ OR ТORQUE пеwtoп
dyne'cm kf,.m oz,·in lb,·in lbr·ft
2.589 988 2.589 998
- 02 - 04
пеwtoп
meter (N . т/т) 5.337 866 Е meter ре, meter (N· m/m) 4.448 222 Е
+ 01 + 00
ENERGY (lNCLUDES WORK)
British tbermal unit (Internationa1 ТаЫе) joule (J) British tbermal unit (тean) jou1e (J) British thermal unit (thermochemical) joule (J) tFac:torв with
an
aвteriak are епd.
1.055 056
Е
1.055 87 Е 1.054 350 Е
+ 03 + 03 + 03
GENERALIZED FLUID PROPERТIES Absolute pressure Pf (psia)
-200
о
200
400
600
800
1000
1200
2.6
2.4
() ~17
,,~I--I.-
2.2
.,,1~(Q9~
а. 2.0
.......
$-~(,
~
~ 18
~'l.rv.J
..
1.4
u
1.2
~
~
.... ..... [ / ';;i~ ~r;. . . r~ ~ ....... ~~
~
10
11:
а::
....~~~
v.. . ~~..........wX.... r/.~
~~
.~ v.
0.8
06
v v'/
0.4
. .~
...
0.2
\.4111"'/
LJ
О
1/
v/
/
1/
v~
./
-~~~/
~~~/ ~V,
/
~......
..
.
I )
1/
V
1/
V
V~
/
1/
1/
1/
~
1/
1/
~' ....
г---~
3.2
~~vv ~ 3.0 V ~ V
~
",
...
~
,,~
.~
L.o
"
~
V, ......
,.....
~ .~~
,
..... ,~
2.6 c.~ И L/ 2.4 ~~~ vv ...
...~
U
I-
~ VVv~~~
V ....
VVV~V
1.6
V[/~[/~~~ [.... Vv~~
....
1 / ·..... 1-:00
'/[/1.' :~"'"~ ~Г/ V~ ~';::rJ:III ~~
~
::J
+-
е
1.4
ILI
1.2
+-
1.0
U ILI U
0.8
l/r~ ~.~~
~~
2.2
....... ~V 2.0 t--!" 1,.;"[..;[,.;' 10" 1.8 ........
~~r~
V VI/I . . . . . . 10' ~~V Vi;'" '/~ v--"'~.~ Vl .... v~~
r':""
~ ~ 2.8
г7 ,..'~;/ .... ~/"'7 7f / ~()y~...., ...... Q ~
с..." .. ~fl;
1/ 1/
/ ... /
~ ~
l~~/
........
"
1/"'"
v'~
~ :I~ ~~Г7' 3.4
r/v . . . . . ~~ . .""
1/
u
-g
~ 1717....
11'
5.
ILI
VI.-L.7'~~ [/VI.7,
~
~
L7"Г r ~
Vv о; '"
v
~O
~ 1.6
~e
а..
Е
ILI
~ ~
ILI а:
0.6
,..
~.,
0,4
0.2 -200
о
200
400
Figure weight.
О.l
в(ю
600
Temperature
Т,
1000
1200
(OF)
Estimated reduced pressure and temperature fгош molecular
(From GPSA. 1979; used
и'irh
permission.)
LIQUID DENSIТV AND SPECIFIC GRAVIТV
0.95
® Buft1!f"fot ® Coconutoil © C01tonseed oil
I(H
'""'"
.:: ~~
''''''
r--~
~
~ 1-1--~
Vl.l
~!V -~~
@Polmoil
I':[) t-~~ r-.t: r-. ....... ..... ~ "-- t-.r--.t--., ~
-
Izд
~~ ~~
j
~
® Peanutoil ® Soybeon oil © VeqetobIe oi 1 ® Wheot-germoil
---""
F=::: ==a.iro... ~
'-
~.,.. ::-~ "~ .... h
..!?J
h
?
I
0.85
50
75
100
125 150 Temperature (OF)
Figure Е.l Specific gravities of fats and oils. with perтiss;on.)
1.06
(Froт
175
~
200
I"'--~
225
Fischer & Porter Catalog /О-А-54; used
® Вlиe size solution ® Rosin size solutlOn (4.8 - 5.0 % soli ds) ® Bluing solution ® Sveen glue r-I--I© Nylon size solution ® wateг-disрегsibIе оит, PLF 50 ~}@ Rosin size solutions ... r-::-: ® Rosin size solution (4'r.solidsl
-1"--
1.04
(е)
16-
f'" 1.02 ~
~ 1.00
~~I--
~
(G)
1(01
..,;al""';; "'-~
'u
--1--
-r-;-
g 0.96 50
r-,......
IlEJ
V"BJ
u> 0.98
u...
r-r- I-t-1'-. ti1~ r-~ ..... h ~
Т
I
75
100
1
~tt: I""-~ 100-
Ir.:jy
l' I
125 150 Temperature (OF)
-
f\....r-. "- t-~
'~ 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
1000
1 1 ] J
,
8
" -
3
1\
~'I," 100 ;:-
8
~
6 -..с ~
j
'>, u
i
5 4
'"
--_.... . . Jlm-
3 2
i
'" .....
с
~
10
'\.
!\
r'I~ ~~
1 .....
i"'l:" ~~
'1
..,;"'1-00....
,K
;,:""00
......
.....
............
;::: ~~
P_nutoil
€>
",
,
~
~~
{.9
"
'"r-... ~~'" lJt
-!--
-!--
FiIh-оil sofuЫe concentrate
i".,
~ ~
\
(р
V
«1>
Q) Wh..t-germ oil
""""
'~~
~~ ВIIJII
-1-
ф Veget.ыeoil
"
"-
~
- ....
Coconut oil
Мёner.' oil
.....
-!--
е Sovbunoil
.... ~
~
~ ...... rJ5
J
Butterflt
®
1
-
Ф Cottonteed oil
~~
\
I
@ Codlmroil
~~
2
~
©
~)
'1'\..
I
(1) Castor oil
JI..
4
>-
0t
r\.
6 5
сп
OF LIQUIDS
-~ _!--
=::
1-1-
,
~)
===:~
~
...... r--.....
~Ioo..
..... "'1"'000.
8
r-.. .....
6 5
Г"oo~
4
........
3
,
....... ~
2
......
~
Ioo",.,,~
,.Q) .1
............r--,.
1.0
50
75
юо
125 150 Temperaiure TF (Of)
Figure F.l Kinematic viscosJtles of fats and oils. /О-А-54; used with perтission.)
175
200
225
(From Fischer & Porter Cata/og
ISENТROPIC
EXPONENTS
100
8 1\
б
4
о
.&
3
~,
2
~
'......
O·~5
10
~
~
0.80
5
с
,
~ ~
111
111
f
а.
+о
41
..с
..
'!-...:)
!..С)
8
~~
.~tJ'"
~y
а.
CI)
А
, ...
~~
.....
_Г'-;::~
~
VbI'1Vt::~~ ~~
t7-
,.
'1,.-
~t:::
///
... ~Vh ~ ~~ 11, I ):«~ ///1-.0001 ~ ./ 'Z.~~ f
7
f
11
"" "/
/
"" .1' I
.1'
"
"''' //
.~
"
~
Z
./ ,..",0,. ././ 7 7Г.7 '7 ~.~ ~ ~~ ./ / ./1./ 7' ~.c. ~
' // /" / '/1/ /~ ~//. ~/ /г/ 71 ~, / / V / ~.; 10/, ~ tI / / / /~V7': '!>. ' // /~'l/ V/ / / 1//
r/) r7/
/ v ./
'//
r-.C>.~( /г/~~If~ ~/~ 'l: ~% / , .I;1~~~~~c.P ~/ &CJo~~ ~~t:lC~~ ~ ~~ ~ ~/ lIl.1 /~~~ ~V ~~~~V/'LlV~ ~~ ~ V/ t;~v~~~
~~ /':./ ~~ ~~ /~~ ~~~~V~~"~ '"'./ "' -''' .r /1,/
0,1
8 б
V~ V"/
J>..:i
4
2
"/
(;~
f
/ /
v""fl'
б
f
!.1'7
'.&J"
5
I
u
~ 'и
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1.0
3
I
"'"
-.."'-.
/
j'.:
u
о
1I,~~
/
l'
:t=
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......
~)
1
О o~~~ ~~ !/IJ V~I/[1: Й~~~'""";::: '1,.У" 7),~~ /~V/I /J %~ j ) ~ 1-pi'~::
2
о
u
')IJ
r/5~)
о,)
I~')i ...... -.'
,
Jl 171
4 3
~lIJ ~~ 7'-..L
1/
u
.Е
:-..
"""1:
8 6
"\
I \~
rN. ~ ~
0090
;.
,
\
5
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"./"
7/ /~
./'"'
Z
'"'/" "7 /"/" /" ./
./
. / / ./
/
/ /V7 -;/
/"
"
~ ~ V~ 'l" 1/
lI" V 1/
/
"
./
./
/" /
./
./ / /
/./
7~ '/
5 ~~ ~ V/ [// ~ 4
,/".// / / " /"
V/
./
./
./ ' / /iI'.
/./.
",// '//. ~~ /,~V V/
V//
'h ~ V~ V 3 v ~ ~~ ~ v /,! /'1 ~~ ~ 2 ~V ,1.1
~ ~ // / / ~ / / ~VI ~ V ~ ~ ~ ~~ ~~ jO~ /,1
0.01 0.01
v:
2
3
4 5 6
8 0.1
2
3
4 5 6
8 1.0
2
3 4 5 6
8 10
Reduced pressure Pr
Figure 1.1 Specific-heat pressure-correction factof Р уг> Еor simple fluid, тisler, 1974; all rights resenJed, used l1-'ith perтission.)
UJ
=
О.
(Froт Ed-
