Instrumentation and control

3Flow Measurement

Principles of Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Basic Flow Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Inferential Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Velocity Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Magnetic Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Vortex Shedding Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Turbine Meters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Ultrasonic Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Mass Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Volumetric Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Positive Displacement Meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Physical Properties of Fluids & Gases . . . . . . . . . . . . . . . . . . . . . . . . 61

English & SI Units of Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . 61

Fundamental Constants & Conversion Factors. . . . . . . . . . . . . . . . . 62

Flow Conversion Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Gas Compressibility Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Critical Values for Some Gases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Head Losses in Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Specific Heats of Fluids and Gases . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Volume Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Reynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Flowmeter Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

new chap 3 flow.qxd 3/2/2006 11:10 AM Page 55

Compensation of Linear Volumetric Meter Signals . . . . . . . . . . . . . 77

Compensation of Rotameter Signals . . . . . . . . . . . . . . . . . . . . . . . . . 78

Compensation of Differential Pressure Meters . . . . . . . . . . . . . . . . . 79

Differential Pressure Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Head Type Flowmeter Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Open Channel Flow Measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Magnetic Flowmeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Ultrasonic Flowmeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

ANSI/ISA Standard Flow Equations for Sizing Control Valves . . . 101

An ‘Old Timer’s’ Tips for Approximate Plant Calculations . . . . . . . 116

56 ISA Handbook of Measurement Equations and Tables


Principles of Flow

Basic Flow Equation

whereVfr = volumetric flow rateA = cross-sectional area of flowv– = average flow velocity

This equation applies in all cases. Ifflow is in a pipe, the cross-sectionalarea can be found in piping hand-books.

Flow is laminar or turbulent,depending on the flow rate and vis-cosity. This can be predicted by cal-culating the Reynolds number,which is the ratio of inertial forcesto viscous forces:

Re = 123.9 pVD/u

where: Re = Reynolds numberp = density in lbs./ft.3

V = average velocity in ft/sec.D = pipe diameter in inchesu = viscosity in centipoises

Reynolds numbers below 2,000indicate laminar flow; above 4,000,turbulent flow. However, somevelocity meters require valuesabove 20,000 to be absolutely cer-tain the flow is truly turbulent and agood average velocity profile isestablished that can be measuredfrom a single point on the flow pro-file. Most liquid flows are turbulent,while highly viscous flows like

polymers or very low flow rates arelaminar.

Typical flow measurements candetermine: average velocity, velocityat one point, volume of materialflowing, and/or the mass of mate-rial. Velocity measurements, in par-ticular, require the flow streamvelocity to be relatively consistentacross the diameter of the pipe. Lessthan fully turbulent flow createslower velocities near the pipe wall.

Fittings, valves—anything otherthan straight, open pipe upstreamof the sensor—will cause velocityvariations across the diameter ofthe pipe. To achieve uniform flow,different types of flowmetersrequire straight pipe runs upstreamand downstream of the measure-ment. These run requirements areexpressed as a certain number ofstraight, open pipe diameters. Forexample, for a 6-inch pipe, 20 diam-eters would be 10 feet. There are noconsistent recommendations evenfor a particular flowmeter type; it isbest to follow the manufacturer’srecommendations. Recommenda-tions vary from 1 to 20, or evenmore, upstream diameters and asmaller number of downstreamdiameters.

Flow measurements can begrouped into four categories:

1. Inferential methods

2. Velocity methods

3. Mass methods

4. Volumetric methods

V A vfr =

Chapter 3/Flow Measurement 57


Inferential Methods

Placing an obstruction in the flowpath causes the velocity to increaseand the pressure to drop. The dif-ference between this pressure andthe pressure in the pipe can beused to measure the flow rate ofmost liquids, gases, and vapors,including steam. In turbulent flow,the differential pressure is propor-tional to the square of flow rate.

An orifice plate is the most com-mon type of obstruction, and, infact, differential pressure across anorifice is used more than any othertype of flow measurement. Theinstalled base of orifice meters isprobably as great as all other flowmeters combined. The orifice plateis a metal disc with typically around hole in it, placed betweenflanges in the pipe. Differentialpressure can be measured at thepipe flanges directly upstream anddownstream of the orifice or fur-ther upstream and downstream.The calculation formulas of differ-ential pressure for a given orificesize and given location of the pres-sure taps are well developed, so nofield calibration based on actualflow is needed (although the dP cellmay have to be calibrated).

Orifice flow measurements are rel-atively cheap to purchase but haverelatively high installation costs.They have high operating costsbecause they create a fairly largeunrecoverable pressure loss. Also,they have low turndown, in partdue to the squared relationship.

Orifices are suitable for high tem-perature and pressure, and are bestfor clean liquids, gases, and lowvelocity steam flows. They requirelong straight runs upstream anddownstream. They are subject to anumber of errors, such as flowvelocity variations across the pipeand wear or buildup on the orificeplate. Because of these errorsources, they are not generally veryaccurate even when highly accu-rate differential pressure transmit-ters are used.

Other types of obstructions includeventuris and flow tubes which haveless unrecoverable flow loss. Apitot tube is a device that can beinserted in large pipes or ducts tomeasure a differential pressure.

Inferential Mass Flow Measurement

Density of an Ideal Gas

whereDi = density of an ideal gasP = pressureM = molecular weightR = the universal gas constantT = temperature

DPMRTi =



Density of an Imperfect Gas

wherePim = density of an imperfect gasP = pressureM = molecular weightR = the universal gas constantT = temperatureZ = compressibility

Velocity Methods

Magnetic Flowmeters

Magnetic flowmeters depend onthe principle that motion between aconductor (the flowing fluid) and amagnetic field develops a voltagein the conductor proportional to thevelocity of the fluid.

Coils outside the pipe generate apulsed DC magnetic field. Materialto be measured flows through themeter tube, which is lined with anon-conductive material such asTeflon, polyurethane, or rubber.Measuring electrodes protrudethrough the liner and contact thefluid and sense the generated voltage.

The flowing fluid must be conduc-tive, but there are very few otherrestrictions; most aqueous fluidsare suitable. There are fewerReynolds number limitations. Theinstrument is the full diameter ofthe pipe, so there is no pressure

loss. A wide range of sizes areavailable—from very small (1/8inch, for example) up to 10 feet indiameter. The flowing material canbe liquids, slurries and suspendedsolids, and there are minimumstraight run requirements.

Vortex Shedding Flowmeters

Vortex shedding flowmeters meas-ure the frequency of vortices shedfrom a blunt obstruction, called a“bluff body,” placed in the pipe. Asthe flow divides to go around thebluff body, vortices are created oneach side of the divided stream.The rate of vortex creation is pro-portional to the stream velocity.Since each vortex represents anarea of low pressure, the presence-then-absence of low pressures iscounted and the count is propor-tional to the velocity.

Vortex flowmeters provide goodmeasurement accuracy with liq-uids, gases, or steam and are toler-ant of fouling. They have highaccuracy at low flow rates; themeasurement is independent ofmaterial characteristics. They requirelong runs of straight pipe. Eventhough the accuracy of vortexmeters is often stated as a percentof flow rate rather than of full scalewhich does indicate higher accura-cies, below a certain flow rate theycannot measure at all. At some lowflow rate the Reynolds number willbe low enough so no vortices willbe shed.

PPMRTZim=



Turbine Meters

Turbine meters use a multi-bladedrotor supported by bearings in thepipe. The flowing fluid drives therotor at a speed proportional to thefluid velocity. Movement of therotor blades is sensed by a mag-netic pickup outside the pipe. Thenumber of blade tips passing thepickup is counted to get rotorspeed.

These meters have high accuracyfor a defined viscosity. They aresuitable for very high and low tem-peratures and high pressures.However, they are sensitive to vis-cosity changes, and the rotor iseasily damaged by going too fast aspeed. Because of the relativelyhigh failure rate of their movingparts, they are not used as much asin the past.

Ultrasonic Flowmeters

Ultrasonic flowmeters send soundwaves through the flowing stream.They can measure either theDoppler shift as ultrasonic wavesare bounced off particles in theflow stream, or the time differentialof ultrasonic waves with the flowstream compared to against theflow stream. Either method gives asignal which is proportional to flowvelocity. The Doppler methodworks with liquids with suspendedsolids, and the Transit time methodworks with liquids and gases. Inboth methods, the signal is propor-tional to flow velocity.

Ultrasonic meters are non-invasivebut are relatively low accuracy.Because clamp-on ultrasonicmeters are easy to install, they canbe used temporarily to verifyanother flowmeter permanentlyinstalled in the pipe. Since thesame meter can do a variety ofsizes, they are particularly costeffective in large sizes.

Mass Methods

Mass flowmeters measure actualmass flow. While it is possible tocalculate mass flow from a velocityor inferential measurement andother variables like temperature forknown fluids, only one meter typecommonly measures liquid massdirectly, the Coriolis meter. Thismeter used to be applied only forwhen highly accurate, mass flowwas required. Now with lowerprices, a wider range of configura-tions and easier installation, it isbeing applied more routinely.

The heart of a Coriolis meter is atube(s) that is vibrated at resonantfrequency by magnetic drive coils.When fluid flows into the tube dur-ing the tube’s upward movement,the fluid is forced to take on the ver-tical momentum of the vibratingtube. Therefore, as the tube movesupwards in the first half of thevibration cycle, the fluid enteringthe tube resists the motion of thetube and exerts a downward force.Fluid in the discharge end of themeter has momentum in the oppo-site direction, and the difference inforces causes the tube to twist. This



tube twist is sensed as a phase dif-ference by sensors located on eachend of the tube arrangement, andtwist is directly proportional tomass flow rate.

In addition to having high accuracyand a true mass flow measure-ment, Coriolis meters have noupstream and downstream straightrun requirements, are independentof fluid properties, are low mainte-nance, and have a turndown ratioof as much as one hundred. Whilethe meters originally were onlyavailable in a double U-shape, theyare now available in a variety ofconfigurations and sizes.

Volumetric Methods

Positive Displacement Meters

This type of meter separates theflow stream into known volumesby vanes, gears, pistons ordiaphragms, then counts the seg-mented volumes. They have good-to-excellent accuracy, can measureviscous liquids, and have nostraight run requirements. How-ever, they do have a non-recover-able pressure loss, and their mov-ing parts subject to wear.

Physical Properties of Fluids &

Gases

When measuring flow, physicalproperties of fluids and gases aresignificant when designing sys-tems and measuring performance.Properties of fundamental impor-tance include:

1. Temperature

2. Pressure

3. Liquid State

4. Gaseous State

5. Density

6. Viscosity

7. Specific Gravity

Depending on the type flowmeterused, and application, the follow-ing properties may also be impor-tant:

1. Vapor Pressure

2. Boiling Point

3. Electrical Conductivity

4. Sonic Conductivity

5. Velocity

6. Specific Heat

English & SI units of

Measurement

Many manufacturers publish theirdata in both the English system(which uses inches, pounds,degrees Fahrenheit, and relatedunits) and Système Internationaled’Unités (SI), an improved metricsystem (which uses centimeters,meters, grams, degrees Celsius,and related units). Degrees Celsiusis also called degrees Centigrade, aFrench word. Celsius and Centi-grade are completely interchange-able terms.



The equation to convert degreesFahrenheit to degrees Celsius is:

The equation to convert degreesCelsius to degrees Fahrenheit is:

wheretc = temperature in degrees Celsius

Volume Flow Rate

wherea = volume flow rated = distanceV = measured volumet = time in seconds

Mass Flow Rate

wherea = mass flow rated = distanceM = measured masst = time in seconds

Fundamental Constants and

Conversion Factors

1 psi = 6.895 kPa

1 kPa = 0.1450 psi

1 bar = 100 kPa

1 bar = 14.50 psi

1 MPa = 145.0 psi

1 psi = 27.73 inches of water at °F or C

1 psi = 2.310 feet of water at °F or C

1 kPa = 7.5 mm of water at °F or C

1 kPa = 4.019 inches of water at °F or C

1 lb/ft3 = 16.026 kg/m3

1 lb/ft3 = 0.016026 kg/liter

1 kg/l = 0.0624 lb/ft3

1 lb/ft-sec = 0.000672 centipoise

Fluid Pressure

Absolute pressure is the actualpressure of the fluid with respect toa perfect vacuum, regardless of theatmospheric pressure on the out-side of the container.

Gauge pressure is the fluid pres-sure with respect to the atmos-pheric pressure outside its con-tainer.

adMdt

=

adVdt

=

t tc= +1 8 32.

tt

c =− 321 8.

English Metric Flow Units

Quantity English Metric

Volume ft3/min m3/sec

Mass lb/min kg/sec

Pressure psig kPa, bar

Temp. °F °C, K

Density lb/ft3 kg/m3



Differential pressure is the dif-ference between two pressures.Note that gauge pressure is actu-ally a differential pressure betweenfluid pressure and atmosphericpressure.

Fluid Density

Density is defined as the mass ofthe fluid per unit volume (ρ = m/V).In the English system, density istypically expressed in pounds percubic foot, where the pounds rep-resent mass rather than force. Inthe metric system, density is typi-cally expressed in kilograms percubic meter or kilograms per liter.Equivalence formulas are:

1 lb/ft3 = 16.026 kg/m3

1 lb/ft3 = 0.016026 kg/l

1 kg/l = 0.0624 lb/ft3

Temperature changes have a sig-nificant effect on liquid densities.The effect of pressure is normallyso small it can be ignored. In gen-eral, liquids expand as temperatureincreases, and thus the densitydecreases.

Gases can greatly vary in densitywith both pressure and tempera-ture changes, as well as differencesin molecular weight. The Ideal GasLaw incorporates both Charles’Law, which states that the densityof a gas at constant temperature isdirectly proportional to its absolutepressure, and Boyle’s Law, whichstates the density of a gas of con-stant pressure is inversely propor-

tional to its absolute temperature.The Ideal Gas Law is:

PV = nRT

whereP = absolute pressureV = volumen = mass/molecular weight R = Universal Gas ConstantT = absolute temperature



Flow Conversion Table

To Convert from To Multiply by:

cm3 ft3 0.00003531467

cm3 in3 0.06102374

cm3 m3 0.0000001

cm3 mm3 1000

cm3 gallon 0.0002641721

cm3 quart (liquid) 0.001056688

cm3/sec ft3/min 0.00211888

cm3/sec liter/hr 3.6

ft3 cm3 28,316.847

ft3 in3 1728

ft3 m3 0.028316847

ft3 gallon 7.480519

ft3 liter 28.316847

ft3/hr cm3/sec 7.865791

ft3/hr liter/min 0.4719474

ft3min cm3/sec 471.9474

ft3/min gallon/sec 0.1246753

ft3/sec m3/hr 101.9406

ft3/sec gallon/min 448.8312

ft3/sec liter/min 1699.011

in3 cm3 16.387064

in3 ft3 0.0005787037

in3 m3 0.000016387064

in3 gallon 0.004329004

in3 liter 0.016387064

cm3/min cm3/sec 0.2731177

m3 cm3 100,000

m3 ft3 35.31467



Flow Conversion Table (cont.)


m3 in3 61,023.74

m3 gallon 264.1721

m3 liter 1000

m3/kg ft3/lb 16.01846

mm3 cm3 0.001

mm3 in3 0.00006102374

°F °C 0.5555556

°F K 0.5555556

Dram (fluid) cm3 3.696691

Dram (fluid) in3 0.2255859

Dram (fluid) milliliter 3.696691

Dram (fluid) oz (fluid) 0.125

ft/hr m/sec 0.00008466667

ft/min km/hr 0.018288

ft/min m/sec 0.00508

ft/sec km/hr 1.09728

ft/sec m/min 18.288

ft/sec m/sec 0.3048

ft/poundal Joule 0.0421401

ft/poundal kg/m 0.00429740

ft/poundal liter/atm 0.000415891

gallon cm3 3785.412

gallon ft3 0.13368056

gallon in3 231

gallon Dram (fluid) 1024

gallon liter 3.785412

gallon oz 128

gallon/min ft3/hr 8.020834





gallon/min ft3/sec 0.002228009

gallon/min m3/hr 0.2271247

gallon/min liter/sec 0.06309020

gram Dram 0.56438339

gram grain 15.432358

gram kgm 0.001

gram milligram 1000

gram oz (liquid) 0.035273962

gram lb 0.002046226

gram/cm3 kgm/m3 1000

gram/cm3 kgm/liter 1

gram/cm3 lb/ft3 62.42796

gram/cm3 lb/in3 0.03612729

gram/cm3 lb/gallon 8.345404

gram/liter gram/cm3 0.001

gram/liter kgm/m 1

gram/liter lb/ft3 0.0624280

gram/liter lb/gallon 0.0083454

gram/force Dyne 980.665

gram/force Newton 0.00980665

Joule ft-lb force 0.737562

Joule kg-force-meter 0.101972

Joule Newton-meter 1

Kelvin °F 1.8

Kelvin °C 1

Kelvin °Rankin 1.8

kg oz (fluid) 35.273962

kg lb 2.2046226





kg/m3 gram/liter 1

kg/m3 lb/ft3 0.06242796

kg/m3 lb/in3 0.00003612729

kg/force Dyne 0.0000980665

kg/force Newton 9.80665

kg/force lb/force 2.20462

kg/force Poundal 70.9316

kPa lb/ft2 20.8854

kPa lb/in2 0.1450377

liter cm3 1000

liter ft3 0.03531467

liter in3 61.02374

liter m3 0.001

liter Dram 270.5122

liter gallon 0.26417205

liter oz (fluid) 33.81402

liter quart (fluid) 1.056688

liter/min ft3//hr 2.118880

liter/min ft3/sec 0.0005885778

liter/min gallon/hr 15.85032

liter/min gallon/sec 0.004402868

liter/sec ft3//hr 127.1328

liter/sec ft3/min 2.118880

liter/sec gallon/hr 951.0194

liter/bar Joule 100

MPa bar 10

MPa Newton/mm3 1

meter ft 3.2808399





meter in 39.37007874

millibar Pa 100

milligram Dram 0.0005643834

milligram oz (fluid) 0.00003527396

milligram lb 0.00000220462

milligram/liter lb/ft3 0.00006242796

milligram/force Dyne 0.980665

milligram/force Newton 0.00000980665

milligram/force/cm Dyne/cm 0.980665

milligram/force/cm Newton/m 0.000980665

milligram/force/in Dyne/cm 0.386089

milligram/force/in Newton/m 0.000386089

mm in 0.03937008

Newton Dyne 0.00001

Newton kg/force 0.1019716

Newton Poundal 7.23301

Newton lb/force 0.224809

Newton/meter ft/lb force 0.737562

Newton/meter Joule 1

Newton/meter kg/meter force 0.1019716

oz (fluid) Dram 8

oz (fluid) gallon 0.0078125

oz (fluid) lb 0.0625

oz (fluid) cm3 29.57353

oz (fluid) in3 1.8046875

oz (fluid) milliliter 29.57353

oz (fluid) quart 0.03125

Pascal Newton/m2 1



Flow Conversion Table (cont.)To Convert from To Multiply by:

Pascal Newton/mm2 0.000001

Pascal Poundal/ft2 0.671969

Pascal lb/ft2 0.0208854

Pascal lb/in2 force 0.000145038

pint cm3 473.1765

pint in3 28.875

pint liter 0.4731765

pint oz (fluid) 16

lb Dram 256

lb gram 7000

lb kg 0.45359237

lb ton (U.S.) 0.0005

lb/ft3 kg/m3 16.01846

lb/ft3 lb/in3 0.0005787037

lb/in3 gram/cm3 27.679905

lb/in3 lb/ft3 1,728

lb/ft kg/m 1.488164

lb/ft/hr Pascal/sec 0.0004133789

lb/ft/sec Pascal/sec 1.488164

lb/gallon gram/cm3 0.1198264

lb/gallon gram/liter 119.8264

lb/gallon kg/m3 119.8264

lb/gallon lb/ft3 7.480519

Poundal gram/force 14.0981

Poundal Newton 0.1382550

Poundal lb/force 0.031081

psi lb/in2 force 1

quart (fluid) liter 0.94635295

quart (fluid) in3 57.75

quart (fluid) cm3 946.35295

ton (U.S.) kg 907.18474

ton (U.S.) ton (metric) 0.90718474



Gas Compressibility Factors

The True Gas (or “Real Gas”) Law(Non-Ideal Gas Law)

whereP = absolute pressureV = volumeZ = normalized compressibilityn = mass/molecular weightR = universal gas constantT = absolute temperature

Normalized pressure and temperature values (Z)

whereTr = reduced temperaturePr = reduced pressureT = absolute temperatureP = absolute pressureTc = critical temperaturePc = critical pressure

TTT

PPP

rc

rc

=

=PV ZnRT=


Values of the Universal Gas Constant (R)

Mass Pressure Volume Temperature R Value

lb psia ft3 °Rankine 10.73

lb psfa ft3 °Rankine 1554

kg kPa (abs) m3 Kelvin 8.314

kg kPa (abs) liter Kelvin 8.314

kg kg/cm3 liter Kelvin 84.78

kg bars liter Kelvin 83.14


Critical Values for Some Gases

Gas Mol. Wt. Tc-°F Pc-psia Tc-°C Pc-kPa

Acetic Acid 60 1071 840 595 5792

Acetylene 26 556 911 309 6280

Ammonia 17 730 1640 405 11,310

Argon 40 272 705 151 4860

Benzene 78 1011 702 562 4840

Butane 58 765 551 425 3800

Carbon Dioxide 44 548 1072 304 7390

Carbon Monoxide 28 239 507 133 3500

Carbon Tetrachloride 154 1001 661 556 4560

Chlorine 71 751 1118 417 7709

Cyclohexane 84 997 594 554 4100

Decane 142 1115 312 619 2150

Ethane 30 550 708 305 4880

Ethanol 46 929 927 516 6390

Ethyl Chloride 64.5 829 764 460 5270

Ethyl Either 74 839 522 466 3600

Ethylene 28 509 748 283 5160

Helium* 4 (24) (151) (13.3) (1050)

Heptane 100 972 377 540 2600

Hexane 86 914 436 508 3010

Hydrogen* 2 (74) (306) (41) (2110)

Hydrogen Chloride 36.5 584 1200 324 8270

Hydrogen Cyanide 27 822 735 457 5070

Methane 16 343 673 191 4640

Methanol 32 924 1450 513 10,000

Methyl Chloride 50.5 749 967 416 6670

Neon* 20 (95) (498) (52) (3430)



*Pseudo-critical values shown.

Critical Values for Some Gases (cont’d.)

Gas Mol. Wt. Tc-°F Pc-psia Tc-°C Pc-kPa

Nitric Oxide 30 323 955 179 6590

Nitrogen 28 227 492 126 3390

Nonane 128 1072 336 596 2320

Octane 114 1025 362 569 2500

Oxygen 32 278 730 154 5030

Pentane 72 847 486 470 3350

Propane 44 666 617 370 4250

Propanol 76 914 779 508 5370

Propylene 42 658 662 365 4562

Sulfur Dioxide 64 775 1142 430 7870

Sulfur Trioxide 80 885 1228 491 8470

Toluene 92 1069 612 594 4220

Water 18 1165 3206 647 22,100



Head Losses in Pipes

Head loss consists of two primarycomponents: friction losses, causedby the walls of a pipe, and minorlosses. A fluid’s viscosity and flowturbulence both contribute to friction loss. The Darcy-Weisbachformula can be used to calculatefriction losses in circular pipes:

wheref = friction factorL = pipe lengthV = average velocityd = internal diameterg = gravity

Friction factor can be determinedby knowing the relative roughnessof the pipe, solving for the Reynoldsnumber, and using the MoodyChart found in most fluid mechan-ics books. To determine the Rey-nolds number, use the followingequation:

wherev = viscosity

Minor losses are caused by achange in flow pattern, caused bybends in a pipe, a sudden changein a pipe diameter, valves, etc.Tables in many fluid mechanicsbooks provide minor head loss val-

ues for different types of bends,valves, elbows, tees etc. Minorchanges (hm) are small when com-pared to friction losses in largepipelines. They can be calculatedusing this equation:

wherehm = minor changeK = minor head loss coefficient

Influence of Viscosity onFlowmeter Performance

Reynolds number for flow in apipe.

whereD = pipe diameterv– = average flow velocityKv = Kinematic viscosityMf = mass flowva = absolute viscosity

Re = =DvK

MDvv

f

a

4π

hmKV

g=

2

2

Re =Vdv

hf =f L Vd 2g

2



Specific Heats of Fluids andGases

Specific heat is the amount of energyrequired to increase the temperatureof one unit of mass of a material byone degree. Common units are calo-ries/gram – °C, joules/gram – °C, andBTU/pound – °F.

Specific heat is important whencomputing heat flow from a massflow measurement and differentialtemperature. The equation is:

where:Q = heat flow rateW = mass flow rateCp = specific heat∆T = temperature difference (for

example, inlet and outlet of aheater)

Liquids have only one form of spe-cific heat (Cp). Gases have twoforms: Cp, measured at constantpressure, and Cv, measured at con-stant volume. The ratio of Cp/Cvis important when designing differ-ential pressure flowmeters for gasflow.

Differential pressure meters use anequation based on velocity change.Velocities are inversely propor-tional to the inlet cross-sectionalarea and the restriction throat area:

whereVfr = volumetric flow rateA1 and A2 = cross-sectional areasof inlet and throatv1 and v2 = velocities at inlet andthroat

The preceding equation is true forliquids. Gases, however, will expanddue to lower pressure at the throat.As a result, a correction factor, Y, isincluded in gas flow equations.Called the Gas Expansion Factor, itdepends on line pressure, differen-tial pressure, meter geometry andthe isentropic exponent for the par-ticular gas at operating conditions.

V A v A vfr = =1 1 2 2

Q C Tp= ∆W



Volume Flow Rate

whereVfr = volumetric flow rateA = area of tubev– = average velocity of fluid

Reynolds Numbers

Reynolds Number

whereRe = Reynolds number p = fluid densityv– = average velocity of fluidD = a dimensionµ = absolute fluid viscosity

Pipe Reynolds Number

whereReD = Pipe Reynolds NumberVgpm = volume flow rate, gallons per minuteG = liquid specific gravity µcp= fluid viscosity, centipoise Din = inside pipe diameter, inches

ReDgpm

cP in

V G

D=

3160

µ

Re =pvD

µ

V Avfr =



Flowmeter Accuracy

Percent of Actual Flow Rate

% of RateFlow Uncertainty x 100

Instantaneous Flow Rate= ±


Q D1

V1D2

V2D3

V3

Z2 = Z1

Head Due

to Elevation

Z1

Datam

Flow

ww

P1

P2

Pressure

Head

v12

Velocity Head

Total Head

2g

v22

1 2

v

P

PvHead


Percent of Full Scale Flow

Percent of Maximum Differential Pressure (dP)

Compensation of Linear Volumetric Meter Signals

Volumetric Flow

whereQ = the volumetric flow rateK = the factor which scales the signal to flow rateFt = the thermal expansion of the meter due to temperature

Mass Flow

whereW = mass flowρ = fluid densityQ = volumetric flow rateK = the factor which scales the signal to flow rateFt = the thermal expansion of the meter due to temperature

Gas Expansion Factor (Y)

whereY = gas expansion factor∆P = the differential pressureP = absolute pressure

Y = −∆

1 (constant)P

P

W = =ρ ρx Q

signalKFt

( )( )

Q =(signal)

KFt

% of Maximum dPdP Uncertainty x 100

Maximum dP= ±

% of Full ScaleFlow Uncertainty x 100

Full Scale Flow Rat= ±

ee



Compensation of RotameterSignals

Liquids

whereW = mass flowK = a flow coefficientρ = fluid densityρ

f= float density

Gases

whereW = mass flowK = a flow coefficientρ = fluid density

W = K(signal) ρ

W = −

K ff

(signal) ( )ρ ρ ρρ


Flowmeter Range

Average Coefficient Selected by

Manufacturer for Meter Total Range

Mete

r C

oeff

icie

nt

Flow Range Over Which

Meter Will be Used

Process Maximum

Flow Rate

Flowmeter

Minimum

Flowmeter

Maximum

Recommended Average

Coefficient for Actual

Flow Range

± 5

% R

ate

± 1

% R

ate

Process Minimum


Compensation for Differential Pressure Meters

whereFa = expansion factor for meters calibrated at 60°Fα = coefficient of expansion of the flow restriction materialTb =base temperature

*Values shown are for water; may be higher for other liquids.

*Values shown are for water; may be higher for other liquids.

F Ta b= +1 2α( )

Differential Pressure Meters

Change Liquid Gas

Density up 1% -0.5 -0.5

Temp. up 10°C at -100°C * +3.0

Temp. up 10°C at 20°C +0.1* +1.7

Temp. up 10°C at 200°C +0.6* +1.0

Press. up 1 psig at -10 psig 0.0 -10.0

Press. up 1 psig at 0.0 psig 0.0 -3.5



Meter Expansion, T up 100°C -0.2 -0.2

Meter Factor changes up 1% +1.0 +1.0

Volumetric Meters

Change Liquid Gas

Density up 1% -1.0 -1.0

Temp. up to 10°C at -100°C * +6.0

Temp. up 10°C at 20°C +0.2* +3.4

Temp. up 10°C at 200°C +.06* +2.0








Effect of Fluid Properties on Flowmeter Accuracy


Differential Pressure Flowmeters

Differential pressure (DP) flowme-ters—also known as “head-typemeters”—are widely applied whenaccurate fluid flow measurementsin pipes are required at reasonablecosts. DP devices have a flowrestriction in the line that causes adifferential pressure, or “head,”between the two measurementlocations. Of all the head-typemeters, the orifice flowmeter is themost widely applied device.

Head Type Flowmeter Elements

Head type flowmeters are based onthe energy exchange which occurswhen the flow area changesbetween the velocity (kinetic)energy and the pressure energyfound in the flowing fluid. The

“Bernoulli Equation” states that thetotal energy in a flowing fluid isconserved after accounting for themechanical work done by the fluid(such as with a turbine) or on thefluid (by a pump) along with anyheat lost or gained from the sys-tem. This means that any of thethree energy forms normally con-sidered in this context; potential(elevation), kinetic, and pressurecan be converted into any of theother forms. The increase in flowvelocity is converted into adecrease in the pressure. This pres-sure difference is called ‘head’ andis used to infer the flow rate. Whenthe flow area returns to the originalsize then most of the pressure isconverted back into velocity exceptfor the losses due to turbulence(see Figure 3-1). The figure is anattempt to show the relationship

Rotameters

Change Liquid Gas

Density up 1% -.04 -.05

Temp. up 10°C at -100°C * +3.0

Temp. up 10°C at 20°C +0.2* +1.7

Temp. up 10°C at 200°C +0.6* +1.0








* Values shown are for water; may be higher for other liquids.


between velocity changes and theresulting pressure pattern. Note thepressure change is proportional tothe velocity change squared. Thismeans that at lower flow rates thepressure difference is less sensitiveto flow changes. Any analysis of errors must consider the effectsof this.

The most common head type flowelement is the orifice plate (see Fig-ure 3-2). Most commonly this is around flat plate with a round holebored in the center.

There are several reasons for this:1. The physics of the orifice

plate are well known andthere is a large researchdatabase.

2. The geometry of a sharpedge round orifice in around plate in a round pipeis easily to replicate andmeasure.

3. International and nationalstandards exist.

4. Many purchase and custodycontracts specify orificemeters.

5. It is inexpensive to makesignificant changes in themeter calibration by replac-ing the orifice plate withone of a different bore.

The orifice meter can be very accu-rate, but only if the design, installa-tion, and maintenance are donevery well and closely adhere to the


Figure 3-2: Orifice Plate

Figure 3-1: Velocity and Pressure Profiles across Orifice Plate


standards. For custody transfer(sale) of fluids this is justified. Manyother orifice meter applications areused for less demanding applica-tions and are installed with theunderstanding that uncertaintiesare increased by compromised butless expensive installation. The keyto decisions around this is “thevalue of the measurement.”

The orifice equation, (simplified):

whereQ is flow rated is orifice boreC is the orifice coefficienth is head across orificeρ flowing fluid density

C for the orifice plate is defined inan equation as a complex functionof Beta and Reynolds number. Anaverage value of 0.61 can be usedfor preliminary designs andapproximations. This approxima-tion is valid only for Beta ratio (ratioof bore to pipe inside diameter) inthe range of 0.2 to 0.5 and forReynolds numbers between 10,000and 100,000. For larger bore diam-eters, larger Beta ratio (β), it is nec-essary to compensate for the veloc-ity of approach, and the equationused is:

The effect of Beta is less than 5%for Beta less than 0.55.

The orifice plate is installedbetween “orifice flanges” withpressure taps (see Figure 3-3). Ori-fice installations differ dependingon the application and size. Somespecial orifice fittings allow the ori-fice plate to be removed andreplaced without stopping flow.Note also that even when reportingflow in terms of volume the differ-ential pressure signal is a functionof the fluid density and that uncer-tainty increase as a function of thedensity uncertainty.

Other “tappings” are used. “Cor-ner taps” measure the pressures atthe faces of the orifice plate.“Radius” or “D, D/2” taps sense thepressures at one pipe diameterupstream and one half pipe diame-ters downstream. The orifice coeffi-cient is different for each type oftapping. For calculation details, seestandards. For mechanical details,see manufacturers’ catalogs.

Q d C h=−

• • •1

1 42

βρ

Q d C h= • • •2 ρ


Figure 3-3: Orifice Plate

Installation


The venturi tube (Figure 3-4) isanother head meter elementshown in the standards. Becausethe inlet and the outlet provide asmooth change in flow path it hasthe characteristic of a smaller totalpressure loss for a given flow. It isalso thought to be less sensitive towear and to upstream flow distur-bances. Most venturis are made tothe geometries shown in the stan-dards. A number of standarddesigns are made and each has aspecific flow coefficient. It is morecomplex to fabricate than a simpleorifice run and thus tends to bemore expensive. The orifice equa-tions are used with coefficients onthe order of 0.9 to 0.98.

The flow nozzle (Figure 3-5) isanother head type flow element. Itis available in a number of con-structions. Permanent flow pres-sure losses are less than for the ori-fice plate and greater than for aventuri. Most often the designs

shown in the standards are used.The orifice equations apply, withthe appropriate coefficient. Severalstandard designs are available.

The Pitot tube (Figure 3-6) convertsall the velocity energy at one pointinto pressure head. Since the flowis measured at only one point anyvariations in the flow pattern


Figure 3-4: Venturi Tube

Figure 3-5: Flow Nozzle

Installation


across the pipe are not discovered.The Averaging Pitot Tube has mul-tiple sensing points and averagesthe pressure. The head developedis less than the orifice plate. Somecommercial designs have highercoefficients.

The Elbow Meter (Figure 3-7)measures the difference in pres-sure on the inside radius of anelbow compared to the outside.The differential generated is rela-tively small unless the velocity andthe fluid density are both relativelyhigh.

A number of other designs for headflow elements are available com-mercially. See the catalogs. Theseall are based on the same physics.Some are more tolerant to solids inthe flowing stream. At least onedesign has a body in the streamwhich moves as the flow changes.

Bernoulli’s Equation at Each Flow

Cross-section

whereP = static pressure (force per unit

area)ρ = fluid densityv– = average fluid velocityg = acceleration due to gravityz = elevation head of the fluid

from a reference datum

Incompressible Fluids

The relationship between velocityand fluid flows for incompressiblefluid in a close conduit is:

wheresubscripts refer to sections 1 and 2

Flow Rate for Compressible

Fluids

Particularly for gases, versus liquids,a change in temperature and pressure results in a change in volume, so flow rate units areexpressed in actual volume or stan-

Q A v A v= × = ×1 1 2 2

P vg

zρ

+ + =2

2constant


Figure 3-6: Pitot Tube

Figure 3-7: Elbow Meter


dard volumetric flow rates. In theU.S., cubic foot is the most com-monly used unit for gas volume. InISO 5024 for natural gas or petro-leum gas, standard pressure andtemperature are 14.696 psia and59°F (15°C). For ANSI/API 2530 thebase pressure and temperature are14.73 psia and 60°F (15.5°C). Basepressures and temperatures canvary by industry, country, andmutually agreed contractual terms.

When gas densities at the flowingcondition and base condition areknown, flow rates in actual andbase conditions are:

where(Qscf)b = flow rate in standardcubic feet per second at theselected base conditionQacfs = volumetric flow rate inactual cubic feet per secondρf = density of fluid at the flow-ing conditionρb = density of fluid at the basecondition

Flow Rate Through a Hole

of a Tank

whereQ = flow rateA = cross-section area of the holeC = flow coefficient (typical 0.60) g = acceleration due to gravityh = height of liquid

Open Channel Flow Measurement

Triangular or V-Notch Weir

whereQ = flow rate

H = head on the weir

K = a constant

for cfs, K = 2.50 tan

for mgd, K = 1.62 tan

for gpm, K = 1120 tan

whereα = angle of triangular opening

cfs = ft3 per secondmgd = million gallons per daygpm = gallon per minute

Triangular (V-Notch) Sharp Crest

Weir

2Hmax

Minimum

2Hmax

Minimum

Hmax

α2

α2

α2

Q KH= 2 5.

Q A C gh= 2

Q A C gh= 2



Maximum Recommended Flow Rates for Triangular Weirs

V-Notch Angle Maximum Head, ft. cfs mgd gpm

22.5° 2.0 2.81 1.82 1260

30.0° 2.0 3.82 2.47 1710

45.0° 2.0 5.85 3.78 2630

60.0° 2.0 8.16 5.28 3660

90.0° 2.0 14.10 9.14 6330

120.0° 2.0 24.50 15.80 11,000

Minimum Recommended Flow Rates for Triangular Weirs

V-Notch Angle Minimum Head, ft. cfs mgd gpm

22.5° 0.2 0.009 0.006 4.04

30.0° 0.2 0.012 0.008 5.39

45.0° 0.2 0.019 0.012 8.53

60.0° 0.2 0.26 0.017 11.70

90.0° 0.2 0.045 0.029 20.20

120.0° 0.2 0.077 0.050 34.80



Rectangular Weir with End

Contractions

whereQ = flow rateH = head on weirL = crest length of weirK = a constantfor cfs, Q = 3.33(L - 0.2H)H1.5

for mgd, Q = 2.15(L - 0.2H)H1.5

for gpm, Q = 1500(L - 0.2H)H1.5

Rectangular Weir W/O End

Contractions

wherefor cfs, Q = 3.33LH1.5

for mgd, Q = 2.15LH1.5

for gpm, Q = 1500LH1.5Rectangular Sharp-Crested Weir

Q = KLH1.5

Q = K(L - 0.2H )1.5


2Hmax

Minimum

2Hmax

Minimum

Hmax

2Hmax

Minimum

Hmax

LCrest Length

LCrest Length


Maximum Recommended Flow Rates for Rectangular Weirs with End Contractions

Crest Length, ft Maximum Head, ft cfs mgd gpm1.0 0.50 1.06 0.685 476

1.5 0.75 2.92 1.890 1310

2.0 1.00 5.99 3.870 2690

2.5 1.25 10.50 6.770 4710

3.0 1.50 16.50 10.70 7410

4.0 2.00 33.90 21.90 15,200

5.0 2.50 59.20 38.30 26,600

6.0 3.00 93.40 60.40 41,900

8.0 4.00 192.00 124.00 86,200

10.0 5.00 335.00 217.00 150,000

Minimum Recommended Flow Rates for Rectangular Weirs with End Contractions

Crest Length, ft Minimum Head, ft cfs mgd gpm1.0 0.2 0.286 0.185 128

1.5 0.2 0.435 0.281 195

2.0 0.2 0.584 0.377 262

2.5 0.2 0.733 0.474 329

3.0 0.2 0.882 0.570 396

4.0 0.2 1.180 0.762 530

5.0 0.2 1.480 0.955 664

6.0 0.2 1.770 1.150 794

8.0 0.2 2.370 1.530 1060

10.0 0.2 2.970 1.920 1330



Maximum Recommended Flow Rates for Rectangular Weirs without End Contractions

Crest Length, ft Maximum Head, ft cfs mgd gpm

1.0 0.50 1.18 0.761 530

1.5 0.75 3.24 2.10 1450

2.0 1.00 5.66 4.30 2990

2.5 1.25 11.60 7.52 5210

3.0 1.50 18.40 11.90 8560

4.0 2.00 37.70 24.30 16,900

5.0 2.50 65.80 42.50 29,500

6.0 3.00 140.00 67.10 46,700

8.0 4.00 213.00 138.00 95,600

Minimum Recommended Flow Rates for Rectangular Weirs Without End Contractions

Crest Length, ft Minimum Head, ft cfs mgd gpm

1.0 0.2 0.298 0.192 134

1.5 0.2 0.447 0.289 201

2.0 0.2 0.596 0.385 267

2.5 0.2 0.745 0.481 334

3.0 0.2 0.894 0.577 401

4.0 0.2 1.190 0.770 534

5.0 0.2 1.490 0.962 669

6.0 0.2 1.790 1.160 803

8.0 0.2 2.380 1.540 1070



Maximum Recommended Flow Rates for Cipolletti WeirsCrest Length, ft Minimum Head, ft cfs mgd gpm

1.0 0.50 1.19 0.789 534

1.5 0.75 3.28 2.120 1470

2.0 1.00 6.73 4.350 3020

2.5 1.25 11.80 7.600 5300

3.0 1.50 18.60 12.000 8350

4.0 2.00 38.10 24.600 17,100

5.0 2.50 66.50 43.000 29,800

6.0 3.00 105.00 67.800 47,100

8.0 4.00 214.00 139.000 96,000

10.0 5.00 375.00 243.000 168,000

Minimum Recommended Flow Rates for Cipolletti Weirs

Crest Length, ft Minimum Head, ft cfs mgd gpm

1.0 0.2 0.301 0.195 135

1.5 0.2 0.452 0.292 203

2.0 0.2 0.602 0.389 270

2.5 0.2 0.753 0.487 338

3.0 0.2 0.903 0.584 405

4.0 0.2 1.200 0.778 539

5.0 0.2 1.510 0.973 678

6.0 0.2 1.810 1.170 812

8.0 0.2 2.410 1.560 1080

10.0 0.2 3.010 1.950 1350



Trapezoidal or Cipolletti Weir

wherefor cfs, Q = 3.37LH1.5

for mgd, Q = 2.18LH1.5

for gpm, Q = 1510LH1.5

Flumes

whereQ = flow rateH = head pressure, point Ha

n = constant power, dependenton throat width and units

K = a constant, dependent onthroat width

Triangular (V-Notch) Sharp Crest

WeirQ = KH1.5

Q = KLH1.5

Discharge Equations for Parshall Flumes(W = Throat Width in Feet)

Width Cubic Feet/Second

MillionGallon/Day Gallon/Minute

1 in Q = 0.338H1.55 Q = 0.3218H1.55 Q = 152H1.55

2 in Q = 0.676H1.55 Q = 0.437H1.55 Q = 303H1.55

3 in Q = 0.992H1.547 Q = 0.641H1.547 Q = 445H1.547

6 in Q = 2.06H1.58 Q = 1.33H1.547 Q = 925H1.58

9 in Q = 3.07H1.53 Q = 1.98H1.53 Q = 138H1.53

10 to 50 feet Q =(3.69W+2.5)H1.65

Q = (2.39W+ 1.61)H1.6

Q =(1660W+1120)H1.6


2Hmax

Minimum

2Hmax

Minimum

Hmax


Maximum Recommended Flow Rates for H Flumes

H Flume Size, ft Minimum Head, ft cfs mgd gpm

.50 0.50 0.375 0.224 156

.75 0.75 0.957 0.619 430

1.00 1.00 1.970 1.270 884

1.50 1.50 5.420 3.500 2430

2.00 2.00 11.100 7.170 4980

2.50 2.50 19.300 12.500 8660

3.00 3.00 30.700 19.800 13,800

4.50 4.50 84.500 54.600 37,900

Minimum Recommended Flow Rates for H Flumes

H Flume Size, ft Minimum Head, ft cfs mgd gpm

.50 0.2 0.0004 0.0003 0.180

.75 0.2 0.0006 0.0004 0.269

1.00 0.2 0.0007 0.0005 0.314

1.50 0.2 0.0011 0.0007 0.494

2.00 0.2 0.0014 0.0009 0.628

2.50 0.2 0.0018 0.0012 0.808

3.00 0.2 0.0021 0.0014 0.942

4.50 0.2 0.0031 0.0020 1.390


Ha

Hb

AR

DP WH

CFlow

2/3 A

General Flume Configuration


Maximum Recommended Flow Rates for Trapezoidal Flumes

Flume Type MinimumHead, ft cfs mgd gpm

Large 60° V 0.45 0.198 0.128 88.8

2 in., 45° WSC 0.77 1.820 1.180 817.0

12 in., 45°SRCRC 1.29 7.080 4.580 3180.0

Minimum Recommended Flow Rates for Trapezoidal Flumes

Flume Type MinimumHead, ft cfs mgd gpm

Large 60° V 0.14 0.010 0.006 4.37

2 in., 45° WSC 0.10 0.023 0.015 10.30

12 in., 45°SRCRC 0.20 0.160 0.103 71.80



Target Flowmeters

Mass Flow Rate in Terms of

Target Force

whereF = target forceρ = fluid densityD, dτ = pipe and target diameters,respectivelyK = constant that includes targetblockage

Bτ = dD

τ


M FD d

d

PF

=−

=

(constant) x x

(constant) x x

π

π

ττ

τ

2

2

2

ρ

DDB

B

KD F

1 2−

=

τ

τ

τρ

Pipe Target

d

D

Force

Idealized Flow Streamlines Past a Circular Disc


Rate of Heat Loss Flowmeter

whereqt = rate of heat loss per unit time∆T = mean temperature elevationof wired = diameter of wirek = thermal conductivity of fluidstreamCv = specific heat of fluid stream atconstant volumeρ = density of fluid streamv = average velocity of fluidstream

Temperature Rise Flowmeter

whereW = mass flowH = heat(power) input∆T = temperature changeCp = specific heat at constant temperature

W =∆

HT Cp*

qt T k kC dvv= ∆ +[ ( ) ]/2 1 2ρπ


T∆

H

H

T1

T1

T2

T2

ACDCAC

Thermocouple

++

Thomas Flowmeter

Laub Flowmeter

Rate of Heat Loss Flowmeter

Section 2

Section 1

Float

Tube

Flow In

Flow Out

Fundamental Operation of a

Variable Area Flowmeter


Note: 1/16 to 1/2 in. glass tube meters with ANSI class 150 flanged connec-tions would be limited to a rating of 270 psig (1826 kPa) at 100°F by theANSI code rating.

Warning: Do not use glass in hazardous applications. Derate gaspressure ratings due to damage and deterioration in use. Even avery small scratch on the end of a glass tube increases the chanceof breakage due to stress and leads to failures.

Typical Pressure Ratings for Glass Tube MetersSize Inches psig kPa

1/16-1/4 250-500 1724-3448

1/2 300 2069

3/4 200 1379

1 180 1241

1 1/2 130 896

2 100 690

3 70 483

Typical Range of Tube Flow Rates

Size Inches Water Air

1/8 0.5-200 cc/min 50-7500 scc/min

1/4 100-2000 cc/min 4000-34000 scc/min

3/8 0.13-0.55 gpm 0.75-2.4 scfm

1/2 0.25-4.0 gpm 1-20 scfm

3/4 1.9-5.0 gpm 8-20 scfm

1 4.0-20 gpm 20-45 scfm

1 1/2 9.0-50 gpm 38-112 scfm



Magnetic Flowmeters

Because they have no protrusionsinto the flow stream, magneticflowmeters offer the advantage ofnot obstructing flow – unless theirsize is less than that of the pipelineitself. Improvements in ease-of-use/installation and reduced costshave made miniature DC magneticflowmeters more popular.

Principle of Operation:

Faraday’s Law of ElectromagneticInduction is the underlying princi-ple of many electrical devices andalso applied to electrical powergeneration. It states that the magni-tude of the voltage induced in a conductive medium movingthrough a magnetic field, and at aright angle to the field, is directlyproportional to the product of themagnetic flux density (B), thevelocity of the medium (v–), andpath length (L) between the probes.

Magnetic flowmeters apply Fara-day’s law, as follows: when a con-ductive liquid passes through ahomogenous field, a voltage isgenerated along a path betweentwo electrodes positioned withinthe magnetic field on oppositesides of the pipe. The path length isthe distance between the two elec-trodes. If the magnetic field (B) isconstant and the distance (D)between the electrodes is fixed, theinduced voltage is directly propor-tional to the velocity (v–) of the liquid.

For a more detailed explanation ofmagnetic flowmeters, see ISA’sbook, Industrial Flow Measure-ment, 3rd Edition, edited by DavidW. Spitzer.

E = constant x B x D x v

E = constant x B x L x v



*Conductivity too low for magnetic flowmeter

Electrical Conductivity of Aqueous Solutions, inMicrosiemens/cm

Chemical Name Formula Temp., °C Conductivity inMicrosiemens/cm

Acetic Acid CH3CO2H 18 1.08 x 103

4.00 x 10-2*

Ammonia NH3 15 8.67 x 102

1.93 x 102

Calcium Chloride CaCl2 18 6.43 x 104

1.37 x 105

Hydrochloric Acid HCl 15 3.95 x 105

6.62 x 105

Hydrofluoric Acid HF 18 1.98 x 104

3.41 x 105

Nitric Acid HNO3 18 3.12 x 105

4.90 x 105

Phosphoric Acid H3PO4 15 5.66 x 104

9.79 x 104

Sodium Carbonate Na2CO3 18 4.51 x 104

8.36 x 104

Sodium Hydroxide NaOH 18 4.65 x 104

8.20 x 104

Sulfuric Acid H2SO4 18 2.09 x 105

1.07 x 105



*Conductivity too low for magnetic flowmeter**Low conductivity application

*Conductivity too low for magnetic flowmeter

Conductivities of Miscellaneous LiquidsName Temp., °C Conductivity in

Microsiemens/cm

Black Liquor 93 5000

Fuel Oil – <10-7*

Water, New York City 25 72

Electrical Conductivity of Pure Liquids

Chemical Name Temp., °C Conductivity in Microsiemens/cm

Carbon Tetrachloride 18 4.0 x 10-2*

Ethyl Alcohol 25 0.0013*

Furfural 25 1.5**

Glycol 25 0.3**

Methyl Alcohol 18 0.44**



Ultrasonic Flowmeters

Primarily a flowmeter technologyfor liquid and some gas applica-tions, ultrasonic flowmeters useacoustic waves, or vibrations, tomeasure flow traveling through apipeline. Some designs permitmeasurements external to the pipe,while other designs require thesensor to be in contact with theflowstream. Doppler and Transit-Time are two of the more popularultrasonic flowmeter types.

Principle of Operation –

Doppler Flowmeter:

In 1842, the Austrian physicistChristian Johann Doppler first pre-dicted frequencies of receivedsound waves depend on themotion of the source, relative to thereceiver. For example, anapproaching fire engine’s sirensounds higher pitched than afterthe siren passes by. That’s becausean approaching fire engine’s veloc-ity packs the sound waves moreclosely together, while the soundwaves move further apart as thefire engine speeds away.

To measure flow in a pipe, onetransducer typically transmits anultrasonic beam of approximately0.5 MHz into the flow stream of aliquid containing sonically reflec-tive materials such as solid parti-cles or bubbles. These movingmaterials alter the frequency of thebeam received at a second trans-ducer. The frequency can be usedto develop an analog or digital sig-nal proportional to flow rate.

Basic equations for a Dopplerflowmeter are:

Snell’s law:

Therefore:

where:VT = Sonic velocity of transmittermaterialθT = Angle of transmitter sonicbeamK = Calibration factorVF = Flow velocity∆F = Doppler frequency changeVS = Sonic velocity of fluidfT = Trasmitted frequencyθ = Angle of fT entry in liquid

Principle of Operation –

Transit-Time Flowmeter:

Also called “time of flight” and“time of travel,” transit-timeflowmeters measure the differencein travel time between pulses trans-mitted along and against the fluidflow. Pulses are typically beamed ata 45° angle in the pipe, with oneclamp-on transducer locatedupstream of the other. Each trans-ducer alternately transmits andreceives bursts of ultrasonicenergy. The difference in the transittimes in the upstream (TU) versus

Vf

fX

VSin

K fFT

T

T=

∆= ∆

θ

SinV

SinV

T

T S

θ θ=

∆ =f fT2 sin VV

F

Sθ



the downstream (TD) directions canbe used to calculate the flowthrough the pipe.

where:TU = Upstream transit timeTD = Downstream transit timeVF = Liquid flow velocityCO = Velocity of sound in fluid.

Flow Equations for SizingControl ValvesANSI/ISA–75.01.01–2002 (IEC 60534-2-1 Mod)

Scope

ANSI/ISA-75.01.01-2002 includesequations for predicting the flowcoefficient of compressible andincompressible fluids through con-trol valves. The equations forincompressible flow are based onstandard hydrodynamic equationsfor Newtonian incompressible flu-ids. They are not intended for usewhen non-Newtonian fluids, fluidmixtures, slurries, or liquid-solidconveyance systems are encoun-tered.

At very low ratios of pressure dif-ferential to absolute inlet pressure(∆P/P1), compressible fluids

behave similarly to incompressiblefluids. Under such conditions, thesizing equations for compressibleflow can be traced to the standardhydrodynamic equations for New-tonian incompressible fluids. How-ever, increasing values of ∆P/P1result in compressibility effects thatrequire that the basic equations bemodified by appropriate correctionfactors. The equations for com-pressible fluids are for use with gasor vapor and are not intended foruse with multiphase streams suchas gas-liquid, vapor-liquid or gas-solid mixtures.

For compressible fluid applica-tions, this part of ANSI/ISA-75.01.01-2002 is valid for all valves.However, manufacturers of somevalves with xT ≥ 0.84have reportedminor inaccuracies. Caution mustalso be exercised when applyingthe equations for compressible flu-ids to gaseous mixtures of com-pounds, particularly near phaseboundaries.

The accuracy of results computedwith the equations in this standardwill be governed by the accuracy ofthe constituent coefficients and theprocess data supplied. Methods ofevaluating the coefficients used inthe equations presented here aregiven in ANSI/ISA-75.02-1996. Thestated accuracy associated with thecoefficients in that document is ± 5% when Cv/d2 <0.047 N18. Rea-sonable accuracy can only bemaintained for control valves ifCv/d2 <0.047 N18.

TL

C V

TL

C V

Vk T T

T T

UO F

DO F

FU D

U D

=−

=+

=⋅ −

⋅

cos

cos

( )

θ

θ



SymbolsSymbol DescriptionC Flow coefficient (Kv, Cv)

CI Assumed flow coefficient for iterative purposes

d Nominal valve size

D Internal diameter of the piping

D1 Internal diameter of upstream piping

D2 Internal diameter of downstream piping

Do Orifice diameter

Fd Valve style modifier

FF Liquid critical pressure ration factor

FL Liquid pressure recovery factor of a control valve withoutattached fittings

FLP Combined liquid pressure recovery factor and piping geometryfactor of a control valve with attached fittings

FP Piping geometry factor

FR Reynolds number factor

Fγ Specific heat ratio factor

Gg Gas specific gravity (ratio of density of flowing gas to density ofair with both at standard conditions, which is considered in thispractice to be equal to the ratio of the molecular weight of gasto molecular weight of air

M Molecular mass of flowing fluid

N Numerical constants

P1 Inlet absolute static pressure measured at point A

P2 Outlet absolute static pressure measured at point B

PC Absolute thermodynamic critical pressure

Pr Reduced pressure (P1,P2)

Pv Absolute vapor pressure of the liquid at inlet temperature

∆P Differential pressure between upstream and downstream pres-sure taps (P1 - P2)

Q Volumetric flow rate (see note 5)

Rev Valve Reynolds number

T1 Inlet absolute temperature

Tc Absolute thermodynamic critical temperature

Tr Reduced temperature (T1/Tc)



ts Absolute reference temperature for standard cubic meter

W Mass flow rate

x Ration of pressure differential to inlet absolute pressure (∆P/ P1)

xT Pressure differential ratio factor of a control valve withoutattached fittings at choked flow

xTP Pressure differential ratio factor of a control valve with attachedfittings at choked flow

Y Expansion Factor

Z Compressibility factor

ν Kinematic viscosity

ρ1 Density of fluid at P1 and T1ρ1ρ0 Relative density (ρ1ρ0 = 1.0 for water at 15°C)

γ Specific heat ratio

ζ Velocity head loss coefficient at a reducer, expander or other fit-ting attached to a control valve or valve trim

ζ1 Upstream velocity head loss coefficient of fitting

ζ2 Downstream velocity head loss coefficient of fitting

ζB1 Inlet Bernoulli coefficient

ζB2 Outlet Bernoulli coefficient

Note 1 To determine the units for the numerical constants, dimen-sional analysis may be performed on the appropriate equationsusing the units given in Table 1

Note 2 1 bar = 102 kPa = 105 Pa

Note 3 1 centistoke = 10-6 m2/s

Note 4 These values are travel-related and should be stated by themanufacturer.

Note 5 Volumetric flow rates in cubic meters per hour, identified by thesymbol Q, refer to standard conditions. The standard cubicmeter is taken at 1013.25 mbar and either 273 K or 288 K (see Table 1).



Normative references

The following normative docu-ments contain provisions which,through reference in this text, con-stitute provisions of this part ofANSI/ISA-75.01.01-2002. All norma-tive documents are subject to revi-sion, and parties to agreementsbased on this part of ANSI/ISA-75.01.01-2002 are encouraged toinvestigate the possibility of apply-ing the most recent editions of thenormative documents indicatedbelow. Members of IEC and ISOmaintain registers of currently validInternational Standards.

IEC 60534-1:1987, Industrial-process control valves – Part 1:Control valve terminology andgeneral considerations

IEC 60534-2-3:1997, Industrial-process control valves – Part 2:Flow capacity – Section 3: Testprocedures

ANSI/ISA-75.02-1996, Control ValveCapacity Test Procedures

ANSI/ISA-75.05.01-2001, Control ValveTerminology

Definitions

For the purpose of ANSI/ISA-75.01.01-2002, definitions given inIEC 60534-2-1 apply with the addi-tion of the following:

3.1 valve style modifier Fdthe ratio of the hydraulic diameterof a single flow passage to thediameter of a circular orifice, thearea of which is equivalent to thesum of areas of all identical flowpassages at a given travel. Itshould be stated by the manufac-turer as a function of travel.

Installation

In many industrial applications,reducers or other fittings areattached to the control valves. Theeffect of these types of fittings onthe nominal flow coefficient of thecontrol valve can be significant.

In sizing control valves, using therelationships presented herein,the flow coefficients calculatedare assumed to include all headlosses between points A and B, asshown below.


Control valve with or without fittings

Flow

BA

Pressuretap

/2/1Pressuretap

/1 = two nominal pipe diameters

/2 = six nominal pipe diameters

Reference pipe section for sizing


Sizing equations for incompressible fluids

The equations listed below identifythe relationships between flowrates, flow coefficients, relatedinstallation factors, and pertinentservice conditions for controlvalves handling incompressible flu-ids. Flow coefficients may be calcu-lated using the appropriate equa-tion selected from the ones givenbelow.

Turbulent flow

The equations for the flow rate of aNewtonian liquid through a controlvalve when operating under non-choked flow conditions are derivedfrom the basic formula as given inIEC 60534-2-1.

Non-choked turbulent flow

Non-choked turbulent flow withoutattached fittings:[Applicable if ]

The flow coefficient shall be deter-mined by

Eq. 1

NOTE 1: The numerical constantN1 depends on the units used inthe general sizing equation and thetype of flow coefficient: Kv or Cv.

Non-choked turbulent flow withattached fittings[Applicable if

]

The flow coefficient shall be deter-mined as follows:

Eq. 2

Choked turbulent flow

The maximum rate at which flowwill pass through a control valve atchoked flow conditions shall be cal-culated from the following equa-tions:

Choked turbulent flow withoutattached fittings[Applicable if

]

The flow coefficient shall be deter-mined as follows:

Eq. 3

Choked turbulent flow withattached fittings[Applicable if

]

The following equation shall beused to calculate the flow coeffi-cient:

Eq. 4

Non-turbulent (laminar and

transitional) flow

The equations for the flow rate of aNewtonian liquid through a controlvalve when operating under non-turbulent flow conditions are

CQ

N F P F PLP F v=

−1 1

1 0ρ ρ/

∆ ≥ −P F F P F PLP P F v( / ) ( )21

CQ

N F P F PL F v=

−1 1

1 0ρ ρ/

∆ ≥ −P F F P F PLP P F v( / ) ( )21

CQ

N F PP=

∆1

1 2ρ ρ/

∆ < − P F F P F PLP P F v( / ) ( )21

C NP

=∆1

1 0ρ ρ/


∆ < −( )P F P F PL F v2

1


derived from the basic formula asgiven in IEC 60534-2-1. This equa-tion is applicable if Rev < 10,000.

Non-turbulent flow withoutattached fittings

The flow coefficient shall be calcu-lated as follows:

Eq. 5

Non-turbulent flow with attachedfittings

For non-turbulent flow, the effect ofclose-coupled reducers or otherflow disturbing fittings is unknown.While there is no information onthe laminar or transitional flowbehavior of control valves installedbetween pipe reducers, the user ofsuch valves is advised to utilize theappropriate equations for line-sizedvalves in the calculation of the FRfactor. This should result in conser-vative flow coefficients since addi-tional turbulence created by reduc-ers and expanders will furtherdelay the onset of laminar flow.Therefore, it will tend to increasethe respective FR factor for a givenvalve Reynolds number.

Sizing equations for compressible

fluids

The equations listed below identifythe relationships between flowrates, flow coefficients, relatedinstallation factors, and pertinentservice conditions for controlvalves handling compressible flu-

ids. Flow rates for compressible flu-ids may be encountered in eithermass or volume units and thusequations are necessary to handleboth situations. Flow coefficientsmay be calculated using the appro-priate equations selected from thefollowing.

The flow rate of a compressiblefluid varies as a function of the ratioof the pressure differential to theabsolute inlet pressure (∆P/P1), des-ignated by the symbol x. At valuesof x near zero, the equations in thissection can be traced to the basicBernoulli equation for Newtonianincompressible fluids. However,increasing values of x result inexpansion and compressibilityeffects that require the use ofappropriate factors.

Turbulent flow:

Non-choked turbulent flow

Non-choked turbulent flow withoutattached fittings[Applicable if x<F7xT]

The flow coefficient shall be calcu-lated using one of the followingequations:

Eq. 6

Eq. 7 CN PY

T ZxM

=W

8 1

1

CN Y xP

=W

6 1 1ρ

CQ

N F PR=

∆1

1 0ρ ρ/



Eq. 8a

Eq. 8b

Non-choked turbulent flow withattached fittings[Applicable if x < Fγ xTP]

The flow coefficient shall be deter-mined from one of the followingequations:

Eq. 9

Eq. 10

Eq. 11a

Eq. 11b

Choked turbulent flow

The maximum rate at which flowwill pass through a control valve atchoked flow conditions shall be cal-culated as follows:

Choked turbulent flow withoutattached fittings[Applicable if x ≥ Fγ xT]

The flow coefficient shall be calcu-lated from one of the followingequations:

Eq. 12

Eq. 13

Eq. 14a

Eq. 14b

Choked turbulent flow withattached fittings[Applicable if x ≥ Fγ xTP]

The flow coefficient shall be deter-mined using one of the followingequations:

Eq. 15

Eq. 16

Eq. 17a

Eq. 17b CQN F P

G T Z

F xP

g

TP=

0 667 7 1

1

. γ

CQN F P

MT ZF xP TP

=0 667 9 1

1

. γ

CN F P

T ZF x MP TP

=W

0 667 8 1

1

. γ

CN F F x PP TP

=W

0 667 6 1 1. γ ρ

CQ

N P

G T Z

F xg

T=

0 667 7 1

1

. γ

CQ

N PMT ZF xT

=0 667 9 1

1

. γ

CN P

T ZF x M

=W

0 667 8 1

1

1. γ

CN F x PT

=W

0 667 6 1 1. γ ρ

CQ

N F PY

G T Z

xP

g=7 1

1

CQ

N F PYMT Z

xP=

9 1

1

CN F PY

T ZxMP

=W

8 1

1

CN F Y xPP

=W

6 1 1ρ

CQ

N PY

G T Z

xg=

7 1

1

CQ

N F PYMT Z

xP=

9 1

1



Non-turbulent (laminar and tran-

sitional) flow

The equations for the flow rate of aNewtonian fluid through a controlvalve when operating under non-turbulent flow conditions arederived from the basic formula asgiven in IEC 60534-2-1. These equa-tions are applicable if Rev < 10,000(see Equation 28). In this subclause,density correction of the gas isgiven by (P1 + P2)/2 due to non-isentropic expansion.

Non-turbulent flow withoutattached fittings

The flow coefficient shall be calcu-lated from one of the followingequations:

Eq. 18

Eq. 19

Non-turbulent flow with attachedfittings

For non-turbulent flow, the effect ofclose-coupled reducers or otherflow-disturbing fittings is unknown.While there is no information onthe laminar or transitional flowbehavior of control valves installedbetween pipe reducers, the user ofsuch valves is advised to utilize theappropriate equations for line-sizedvalves in the calculation of the FR

factor. This should result in conser-vative flow coefficients since addi-tional turbulence created by reduc-ers and expanders will furtherdelay the onset of laminar flow.Therefore, it will tend to increasethe respective FR factor for a givenvalve Reynolds number.

Determination of Correction

Factors

Piping geometry factor (FP)

The piping geometry factor (FP) isnecessary to account for fittingsattached upstream and/or down-stream to a control valve body. TheFP factor is the ratio of the flow ratethrough a control valve installedwith attached fittings to the flowrate that would result if the controlvalve was installed withoutattached fittings and tested underidentical conditions which will notproduce choked flow in eitherinstallation. To meet the accuracyof the FP factor of ±5%, the FP fac-tor shall be determined by test inaccordance with ANSI/ISA-75.02-1996.

When estimated values are permis-sible, the following equation shallbe used:

Eq. 20

In this equation, the factor Σ ξ is thealgebraic sum of all of the effectivevelocity head loss coefficients of all

F

NC

d

P =

+

1

12

12

2Σξ

CQ

N FMT

P P PR=

∆ +22

1

1 2( )

CN F

TP P P MR

=∆ +

W

27

1

1 2( )



fittings attached to the controlvalve. The velocity head loss coeffi-cient of the control valve itself isnot included.

Eq. 21

In cases where the piping diame-ters approaching and leaving thecontrol valve are different, the ξBcoefficients are calculated as fol-lows:

Eq. 22

If the inlet and outlet fittings areshort-length, commercially avail-able, concentric reducers, the ξ1and ξ2 coefficients may be approx-imated as follows:

Eq. 23 Inlet reducer:

Eq. 24 Outlet reducer (expander):

Eq. 25 Inlet and outlet reducers ofequal size:

The FP values calculated with theabove ξ factors generally lead to theselection of valve capacities slightlylarger than required. This calcula-

tion requires iteration. Proceed bycalculating the flow coefficient C fornon-choked turbulent flow.

NOTE: Choked flow equations andequations involving FP are notapplicable.

Next, establish C1 as follows:

Eq. 26 C1 = 1.3C

Using C1 from Equation 26, deter-mine FP from Equation 20. If bothends of the valve are the same size,FP may instead be determinedfrom Figure 2. Then, determine if

Eq. 27

If the condition of Equation 27 issatisfied, then use the C1 estab-lished from Equation 26. If the con-dition of Equation 27 is not met,then repeat the above procedureby again increasing C1 by 30%.This may require several iterationsuntil the condition required inEquation 27 is met.

Reynolds Number Factor (FR)

The Reynolds number factor FR isrequired when non-turbulent flowconditions are established througha control valve because of a lowpressure differential, a high viscos-ity, a very small flow coefficient, ora combination thereof.

The FR factor is determined by divid-ing the flow rate when non-turbulentflow conditions exist by the flow ratemeasured in the same installationunder turbulent conditions.

CF

CP

≤ 1

ξ ξ1 2

2 2

1 5 1+ = −

.dD

ξ12

2 2

1 0 1= −

.dD

ξ11

2 2

0 5 1= −

.dD

ξBdD

= −

14

∑ = + + −ξ ξ ξ ξ ξ1 2 1 2B B



Liquid Pressure Recovery Factors

(FL) or (FLP)

Liquid pressure recovery factorwithout attached fittings (FL)

FL is the liquid pressure recoveryfactor of the valve without attachedfittings. This factor accounts for theinfluence of the valve internalgeometry on the valve capacity atchoked flow. It is defined as theratio of the actual maximum flowrate under choked flow conditionsto a theoretical, non-choked flowrate which would be calculated ifthe pressure differential used wasthe difference between the valveinlet pressure and the apparentvena contracta pressure at chokedflow conditions. The factor FL maybe determined from tests in accor-dance with ANSI/ISA-75.02-1996.

Combined liquid pressure recoveryfactor and piping geometry factorwith attached fittings (FLP)

FLP is the combined liquid pressurerecovery factor and piping geome-try factor for a control valve withattached fittings. It is obtained inthe same manner as FL .

To meet a deviation of ±5% for FLP,FLP shall be determined by testing.When estimated values are permis-sible, the following equation shallbe used:

Here Σξ1 is the velocity head losscoefficient, ξ1 + ξB1, of the fittingattached upstream of the valve asmeasured between the upstreampressure tap and the control valvebody inlet.

Liquid critical pressure ratio factor(FF)

FF is the liquid critical pressureratio factor. This factor is the ratioof the apparent vena contractapressure at choked flow conditionsto the vapor pressure of the liquidat inlet temperature. At vapor pres-sures near zero, this factor is 0.96.

Values of FF may be approximatedfrom the following equation:

Expansion Factor Y

The expansion factor Y accounts forthe change in density as the fluidpasses from the valve inlet to thevena contracta (the location justdownstream of the orifice wherethe jet stream area is a minimum). Italso accounts for the change in thevena contracta area as the pressuredifferential is varied. Theoretically,Y is affected by all of the following:

a) ratio of port area to bodyinlet area;

b) shape of the flow path;

c) pressure differential ratio x ;

FF

FN

C

d

LPL

L

=

+ ∑

12

21 2

2

( )ξ

FF

FN

C

d

LPL

L

=

+ ∑

12

21 2

2

( )ξ



d) Reynolds number; and

e) specific heat ratio γ.

The influence of items a), b), c), ande) is accounted for by the pressuredifferential ratio factor xT, whichmay be established by air test.

The Reynolds number is the ratioof inertial to viscous forces at thecontrol valve orifice. In the case ofcompressible flow, its value is gen-erally beyond the range of influ-ence, except where the flow rate orthe CV is very low or a combinationof both exist.

The pressure differential ratio xT isinfluenced by the specific heat ratioof the fluid.

Y may be calculated using the fol-lowing equation:

The value of x for calculation pur-poses shall not exceed FγXT. If x >FγXT, then the flow becomeschoked and Y = 0.667.

Pressure Differential Ratio Factor

(xT) or (xTP).

Pressure differential ratio factorwithout fittings (xT)

xT is the pressure differential ratiofactor of a control valve installedwithout reducers or other fittings. Ifthe inlet pressure P1 is held con-stant and the outlet pressure P2 isprogressively lowered, the massflow rate through a valve will

increase to a maximum limit, a con-dition referred to as choked flow.Further reductions in P2 will produceno further increase in flow rate.

This limit is reached when the pres-sure differential x reaches a valueof FγXT. The limiting value of x isdefined as the critical differentialpressure ratio. The value of x usedin any of the sizing equations andin the relationship for Y shall beheld to this limit even though theactual pressure differential ratio isgreater. Thus, the numerical valueof Y may range from 0.667, when x= FγXT, to 1.0 for very low differen-tial pressures.

The values of xT may be estab-lished by air test. The test proce-dure for this determination is cov-ered in ANSI/ISA-75.02-1996.

Pressure differential ratio factorwith attached fittings (xTP)

If a control valve is installed withattached fittings, the value of xTwill be affected. To meet a devia-tion of ±5% for xTP , the valve andattached fittings shall be tested as aunit. When estimated values arepermissible, the following equationshall be used:

NOTE: Values for N5 are given inTable 1 at end of Flow chapter,page 113.

x

xF

x

NC

d

TP

T

P

T i=

+

15

12

2ξ

Yx

F xT T= −1

3



In the above relationship, xT is thepressure differential ratio factor fora control valve installed withoutreducers or other fittings. ξ1 is thesum of the inlet velocity head losscoefficients (ξ1 + ξB1) of the reduceror other fitting attached to the inletface of the valve.

If the inlet fitting is a short-length,commercially available reducer, thevalue of ξ may be estimated usingEquation 23.

Specific Heat Ratio Factor Fγγ

The factor xT is based on air nearatmospheric pressure as the flow-ing fluid with a specific heat ratio of1.40. If the specific heat ratio for theflowing fluid is not 1.40, the factorFγ is used to adjust xT . Use the fol-lowing equation to calculate thespecific heat ratio factor:

Compressibility Factor Z

Several of the sizing equations donot contain a term for the actualdensity of the fluid at upstreamconditions. Instead, the density isinferred from the inlet pressure andtemperature based on the laws ofideal gases. Under some condi-tions, real gas behavior can deviatemarkedly from the ideal. In thesecases, the compressibility factor Zshall be introduced to compensatefor the discrepancy. Z is a functionof both the reduced pressure andreduced temperature (see appro-priate reference books to deter-

mine Z ). Reduced pressure Pr isdefined as the ratio of the actualinlet absolute pressure to theabsolute thermodynamic criticalpressure for the fluid in question.The reduced temperature Tr isdefined similarly. That is

TTTr

C= 1

PPPr

C= 1

F γγ

=1 40.



Table 1 - Numerical constants NFlow coefficient C Formulae unit

Constant Kv Cv W Q P,∆∆P ρρ T d, D νν

N1 1 x 10-1 8.65 x 10-2 - m3/h kPa kg/m3 - - -1 8.65 x 10-1 - m3/h bar kg/m3 - - -

1 - gpm psia lbm/ft3 - - -

N2 1.60 x 10-3 2.14 x 10-3 - - - - - mm -

8.90 x 102 - - - - - in -

N4 7.07 x 10-2 7.60 x 10-2 - m3/h - - - - m2/s1.73 x 104 - gpm - - - - cS2.153 x 103 - scfh - - - - cS

N5 1.80 x 10-3 2.41 x 10-3 - - - - - mm -1.00 x 103 - - - - - in -

N6 3.16 2.73 kg/h - kPa kg/m3 - - -3.16 x 101 2.73 x 101 kg/h - bar kg/m3 - - -

6.33 101 lbm/h - psia lbm/ft3 - - -N7 4.82 4.17 - m3/h kPa - - - -

(t = 15.6°C) 4.82 x 102 4.17 x 102 - m3/h bar - - - -1.36 x 103 - scfh psia - - - -

N8 1.10 9.48 x 10-1 kg/h - kPa - K - -1.10 x 102 9.48 x 101 kg/h - bar - K - -

1.93 101 lbm/h - psia - R - -

N9 2.46 x 101 2.12 x 101 - m3/h kPa - K - -(t = 0°C) 2.46 x 103 2.12 x 103 - m3/h bar - K - -

6.94 x 103 - scfh psia - R - -

N9 2.60 x 101 2.25x 101 - m3/h kPa - K - -(ts = 15°C) 2.60 x 103 2.25 x 103 - m3/h bar - K - -

7.32 x 103 - scfh psia - R - -

N18 8.65 x 10-1 1.00 - - - - - mm -6.45 x 102 - - - - - in -

N19 2.5 2.3 - - - - - mm -9.06 x 10-2 - - - - - in -

N22 1.73 x 101 1.50 x 101 - m3/h kPa - K - -(ts = 0°C) 1.73 x 103 1.50 x 103 - m3/h bar - K - -

4.92 x 103 - scfh psia - R - -

N22 1.84 x 101 1.59 x 101 - m3/h kPa - K - -(ts = 15°C) 1.84 x 103 1.59 x 103 - m3/h bar - K - -

5.20 x 103 - scfh psia - R - -

N27 7.75 x 10-1 6.70 x 10-1 kg/h - kPa - K - -(ts = 0°C) 7.75 x 10-1 6.70 x 10-1 kg/h - bar - K - -

1.37 x 101 lbm/h - psia - R - -

N32 1.40 x 102 1.27 x 102 - - - - - mm -1.70 x 101 - - - - - in -


NOTE: Use of the numerical constants provided in this table together with the practical metric andU.S. units specified in the table will yield flow coefficients in the units in which they are defined.


Table 2 - Physical Contants1)

Gas or vapor Symbol M γγ Fγγ Pc2) Tc

3)

Acetylene C2H2 26.04 1.30 0.929 6,140 309

Air - 28.97 1.40 1.000 3,771 133

Ammonia NH3 17.03 1.32 0.943 11,400 406

Argon A 39.948 1.67 1.191 4,870 151

Benzene C6H6 78.11 1.12 0.800 4,924 562

Isobutane C4H9 58.12 1.10 0.784 3,638 408

n-Butane C4H10 58.12 1.11 0.793 3,800 425

Isobutylene C4H8 56.11 1.11 0.790 4,000 418

Carbon dioxide CO2 44.01 1.30 0.929 7,387 304

Carbon monoxide CO 28.01 1.40 1.000 3,496 133

Chlorine Cl2 70.906 1.31 0.934 7,980 417

Ethane C2H6 30.07 1.22 0.871 4,884 305

Ethylene C2H4 28.05 1.22 0.871 5,040 283

Fluorine F2 18.998 1.36 0.970 5,215 144

Freon 11 (trichloromonofluormethane) CCl3F 137.37 1.14 0.811 4,409 471

Freon 12 (dichlorodifluoromethane) CCl2F2 120.91 1.13 0.807 4,114 385

Freon 13 (chlorotrifluoromethane) CClF 104.46 1.14 0.814 3,869 302

Freon 22 (chlorodifluoromethane) CHClF2 80.47 1.18 0.846 4,977 369

Helium He 4.003 1.66 1.186 ,229 5.25

n-Heptane C7H16 100.20 1.05 0.750 2,736 540

Hydrogen H2 2.016 1.41 1.007 1,297 33.25

Hydrogen chloride HCl 36.46 1.41 1.007 8,319 325

Hydrogen fluoride HF 20.01 0.97 0.691 6,485 461

Methane CH4 16.04 1.32 0.943 4,600 191

Methyl chloride CH3Cl 50.49 1.24 0.889 6,677 417

Natural gas 4) - 17.74 1.27 0.907 4,634 203

Neon Ne 20.179 1.64 1.171 2,726 44.45

Nitric oxide NO 63.01 1.40 1.000 6,485 180

Nitrogen N2 28.013 1.40 1.000 3,394 126

Octane C8H18 114.23 1.66 1.186 2,513 569

Oxygen O2 32.00 1.40 1.000 5,040 155

Pentane C5H12 72.15 1.06 0.757 3,374 470

Propane C3H8 44.10 1.15 0.821 4,256 370

Propylene C3H6 42.08 1.14 0.814 4,600 365



1) Constants are for fluids (except for steam) at ambient temperature and atmosphericpressure.

2) Pressure units are kPa (absolute).3) Temperature units are in K.4) Representative values; exact characteristics require knowledge of exact constituents.

Saturated steam - 18.016 1.25 -1.32 4)

0.893 -0.943 4)

22,119 647

Sulphur dioxide SO2 64.06 1.26 0.900 7,822 430

Superheated steam - 18.016 1.315 0.939 22,119 647




An ‘Old Timer’s’ Tips for Approximate Plant Calculations

Cullen Langford, a self-described “Old Timer”, who provided considerablehelp with the Head Type Flowmeter Elements, portion of this chapter, hasgenerously agreed to share with ISA Handbook readers the “approximateplant calculations, for preliminary design and checking” shown below. Dr.Langford advises that readers use them with care, however, because “theseare valid only for normal situations.”

Units, definitions

Valve not corrected for fittings, choking, etc.

Orifice, approximate discharge coefficient Cd,=0.61 for 0.2<β<0.55 and10,000<Rd 100.000

Pitot tube, Cd =0.65, approx, coefficients vary

Wedge, Cd = 0.46 for h/D = 0.5

Rd

Fluid Velocity

v gpm D= •0 48 2. /

Rd d cP= • •( )6 32. /W

W = • • • •360 2D C hd ρ.

W = • • • •360 2D C hd ρ

W

W

W

= • • • •

= • • •

= • • •

360

360

360

2

2

2

d C h

d C h

h C d

d

d

d

ρ

ρ

/( )

/( )

ρ

C P C Pv v= • • = • • •W W/( . ) .63 2 63 2∆ ∆ρ ρ

W, pph P, psi h inwc

ρ, Lb/cuft R=19.316 g=grav, 32.16 ft/s2

m, mol wt D or d, inches v, ft/s


Hydraulic Horse Power, the power to pump, or the power lost to turbulence.

Hydraulic Head: in feet of fluid head

Absolute temperatureGas Density, T in K, p in psia

ρ = •mP R T/

K C R F0 0 0 0273 16 459 69= + = +. .

H v g= 2 2/

HP psid= • •( )W / 946 ρ



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