Fundamentals of Valve Sizing for Liquid

7/28/2019 Fundamentals of Valve Sizing for Liquid

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technical

monograph30

Fundamentals of Valve

Sizing for Liquids

Marc Riveland

Senior Research Engineer


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Fundamentals of Valve

Sizing for Liquids

Notations

A = cross sectional flow areaCv = flow coefficientd = diameter of valve inletD = diameter of adjacent pipingFd = valve style modifierFF= critical pressure ratioFL = pressure recovery coefficientFp = piping correction factorFR = Reynolds number factorg = gravitational accelerationgc = gravitational constantG = liquid specific gravityHI = available head lossKB = Bernoulli coefficientKI = available head loss coefficientKm = pressure recovery coefficientN1 = units factorN2 = units factorN4 = units factorP = pressurePc = absolute thermodynamic critical pressureq = heat transferred out of fluidRev= Reynolds numberrc = critical pressure ratioQ = flow rateU = internal energy of fluidV = velocityw = shaft work done by (or on) fluidZ = elevationp = fluid densityn = kinematic viscosity

General Subscripts:

1 = upstream2 = downstreamv = vaporvc = vena contracta

IntroductionValves are selected and sized to perform a specificfunction within a process system. Failure to performthat given function, whether it is controlling a processvariable or simple on/off service, results in higherprocess costs. The sizing function thus becomes acritical step to successful process operation.

This paper focuses on correctly sizing valves for liquidservice.

Liquid Sizing Equation Background

This section presents the technical substance of theliquid sizing equations. The value of this lies in not onlya better understanding of the sizing equations, but alsoin knowledge of their intrinsic limitations andrelationship to other flow equations and conditions.

The flow equations used for sizing have their roots inthe fundamental equations which describe thebehavior of fluid motion. The two principle equationsare the energy equation and the continuity equation.

The energy equation is equivalent to a mathematicalstatement of the first law of thermodynamics. Itaccounts for the energy transfer and content of the

fluid. For an incompressible fluid (e.g. a liquid) insteady flow, this equation can be written as:

V22gc

) P ) gZ* w) q)U+ constant(1)

where all terms are defined in the nomenclaturesection. The three terms in parenthesis are allmechanical (or available) energy terms and carry aspecial significance. These quantities are all capableof directly doing work. Under certain conditions morethoroughly described later, this quantity may alsoremain constant:

V2

2gc) P ) gZ+ constant

(2)

This equation can be derived from purely kinematicmethods (as opposed to thermodynamic methods) andis known as Bernoullis equation.

The other fundamental equation which plays a vitalrole in the sizing equation is the continuity equation.This is the mathematical statement of conservation ofthe fluid mass. For steady flow conditions(onedimensional) this equation is written as:

VA+ constant(3)

Using these fundamental equations, we can examinethe flow through a simple fixed restriction such as thatshown in figure 1. We will assume the following for thepresent:

1. The fluid is incompressible (a liquid)2. The flow is steady


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Figure 1. Flow through a Simple Fixed Restriction

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3. The flow is onedimensional4. The flow can be treated as inviscid (explained later)5. No change of fluid phase occurs

As seen in figure 1, the flow stream must contract topass through the reduced flow area. The point alongthe flow stream of minimum cross sectional flow area

is the vena contracta. The flow processes upstream ofthis point and downstream of this point differsubstantially, so it is convenient to consider themseparately.

The process from a point several pipe diametersupstream of the restriction to the vena contracta isvery nearly ideal for practical intents and purposes(thermodynamically isentropic). Under this constraint,Bernoullis equation applies and we see that nomechanical energy is lostit merely changes fromone form to the other. Furthermore, changes inelevation are negligible since the flow streamcenterline changes very little, if at all. Thus, energy

contained in the fluid simply changes from pressure tokinetic. This is quantified when considering thecontinuity equation. As the flowstream passes throughthe restriction, the velocity must increase inverselyproportional to the change in area. For example, fromequation 3:

Vvc +(constant)

Avc

(4)

Using upstream conditions as a reference, thisbecomes:

Vvc + V1A1Avc(5)

Thus, as the fluid passes through the restriction, thevelocity increases. Applying equation 2 and neglectingelevation changes (again using upstream conditionsas a reference):

V12

2gc) P1 +

Vvc2

2gc) Pvc

(6)

Inserting equation 5 and rearranging, results in:

Pvc + P1 *V12

2gc A

1Avc

2

* 1(6a)

Thus, at the point of minimum cross sectional area, wesee that fluid velocity is at a maximum (from equation5) and fluid pressure is at a minimum (from equation6).

The process from the vena contracta point to a pointseveral diameters downstream is not ideal andequation 2 no longer applies. By arguments similar toabove, it can be reasoned (from the continuityequation) that as the original cross sectional area is

restored, the original velocity is also restored. Becauseof the nonidealities of this process, however, the totalmechanical energy is not restored. A portion of it isconverted into heat which is either absorbed by thefluid itself or dissipated to the environment. Let usconsider equation 1 applied from several diametersupstream of the restriction to several diametersdownstream of the restriction:

U1 )V1

2

2gc)

P1 )

gZ1gc

+ q + U2 )V2

2

2gc)

P2 )

gZ2gc

+ w

(7)

No work is done across the restriction, so the workterm drops out. The elevation changes are negligible,so the respective terms cancel each other. We cancombine the thermal terms into a single term, HI:

V12

2gc) P1 +

V22

2gc) P2 )HI

(8)

The velocity was restored to its original value so thatequation 8 reduces to:

P1 + P2 )HI(9)

Thus, the pressure decreases across the restriction

and the thermal terms (internal energy and heat lost tothe surroundings) increase.

Losses of this type are generally proportional to thesquare of the velocity (references 1 and 2), so it isconvenient to represent them by the followingequation:

HI+ KIV2

2(10)


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Figure 2. ISA Flow Test Piping Configuration

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In this equation the constant of proportionality, KI, iscalled the available head loss coefficient, and isdetermined by experiment.

From equations 9 and 10 it can be seen that thevelocity (at location 2) is proportional to the squareroot of the pressure drop. Volume flow rate can bedetermined knowing the velocity and correspondingarea at any given point so that:

Q+ V2A2 +2(P1 * P2)

KIA2

(11)

Now, letting:

+ Gw

and, defining:

Cv + A22

wKI

(12)

where G is the liquid specific gravity, equation 11 maybe rewritten as:

Q + CvP1 * P2

G

(13)

Equation 13 constitutes the basic sizing equation usedby the control valve industry, and provides a measureof flow in gallons per minute (GPM) when pressure inpounds per square inch is used. Sometimes it may bedesirable to work with other units of flow orindependent flow variables (pressure, density, etc.).The equation fundamentals are the same for suchcases and only constants or form are different.Reference 3 provides an excellent summary of thevariant forms of the liquid flow equation.

Determination of Flow Coefficients

Rather than experimentally measure KI and calculateCv, it is more straightforward to measure Cv directly.

In order to assure uniformity and accuracy, the

procedures for both measuring flow parameters anduse in sizing are addressed by industrial standards.The currently accepted standards are sponsored bythe Instrument Society of America (ISA) as given inreference 4.

Measurement of Cvand related flow parameters iscovered extensively in reference 4 and is reviewedonly briefly here.

The basic test system configuration is shown in figure2. Specifications, accuracies and tolerances are givenfor all hardware installation and data measurementssuch that coefficients can be calculated to an accuracy

of approximately5%. Fresh water at approximately68oF is circulated through the test valve at specified

pressure differentials and inlet pressures. Flow rate,fluid temperature, inlet and differential pressure, valvetravel and barometric pressure are all measured andrecorded. This yields sufficient information to calculatethe following sizing parameters (the next sectionexplains the meaning and use of these factors):

Flow coefficient (Cv)Pressure recovery coefficient (FL)Piping correction factor (Fp)Reynolds number factor (FR)

In general, each of these parameters depends on the

valve style and size, so multiple tests must beperformed accordingly. These values are thenpublished by the valve manufacturer for use in sizing.

Basic Sizing Procedure

The procedure by which valves are sized for normal,incompressible flow is straightforward. Again, to insureuniformity and consistency, a standard exists which


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delineates the equations and correction factors to beemployed for a given application (reference 5).

The simplest case of liquid flow application involvesthe basic equation developed earlier. Rearrangingequation 13 so that all of the fluid and process relatedvariables are on the right side of the equation, we

arrive at an expression for the valve Cv required forthe particular application:

Cv +Q

P1*P2G

(14)

It is important to realize that valve size is only oneaspect of selecting a valve for a given application.Other considerations include valve style and trimcharacteristic. Discussion of these features fallsoutside the scope of this monograph. Other sources,such as references 6 and 7 make a thorough

presentation.

Once a valve has been selected and Cv is known, theflow rate for a given pressure drop, or the pressuredrop for a given flow rate, can be predicted bysubstituting the appropriate quantities into equation 13.

Many applications fall outside the bounds of the basicliquid flow applications just considered. Rather thandevelop special flow equations for all of the possibledeviations, it is possible (and preferred) to account fordifferent behavior with the use of simple correctionfactors. These factors, when incorporated, change theform of equation 13 to the following (reference 5):

Q + (N1FpFR)CvP1 * P2

G

(15)

All of the additional factors in this equation areexplained in the following sections.

Choked Flow

Equation 13 would imply that, for a given valve, flowcould be continually increased to infinity by simplyincreasing the pressure differential across the valve. Inreality, the relationship given by this equation holds foronly a limited range. As the pressure differential isincreased, a point is reached where the realized massflow increase is less than expected. This phenomenoncontinues until no additional mass flow increaseoccurs in spite of increasing the pressure differential(figure 3). This condition of limited maximum massflow is known as choked flow. To understand moreabout what is occurring and how to correct for it when

Figure 3. Typical Flow Curve Showing Relationship BetweenFlow Rate Q and Imposed Pressure DifferentialDP

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sizing valves, it is necessary to return to some of thefluid flow basics discussed earlier.

Recall that as a liquid passes through a reducedcrosssectional area, velocity increases to a maximumand pressure decreases to a minimum. As the flowexits, velocity is restored to its original value while thepressure is only partially restored, thus creating apressure differential across the device. As thispressure differential is increased, the velocity throughthe restriction increases (thus increasing flow) and thevena contracta pressure decreases. If a sufficientlylarge pressure differential is imposed on the device,the minimum pressure may decrease to or below thevapor pressure of the liquid under these conditions.When this occurs, the liquid becomes

thermodynamically unstable and partially vaporizes.The fluid now consists of a mixture of liquid and vaporwhich is no longer incompressible.

While the exact mechanisms of liquid choking are notfully confirmed, there are parallels between this andcritical flow in gas applications. In gas flows the flowbecomes critical (choked) when the fluid velocity isequal to the acoustic wave speed at that point in thefluid. Pure incompressible fluids have very high wavespeeds so, practically speaking, they do not choke.Liquid/gas or liquid/vapor mixtures, however, typicallyhave very low acoustic wave speeds (actually lowerthan that for a pure gas or vapor) so that it is possiblefor the mixture velocity to equal the sonic velocity andchoke the flow.

Another way of viewing this phenomenon is toconsider the density of the mixture at the venacontracta. As the pressure decreases, the density ofthe vapor phase, and hence the mixture, decreases.Eventually this decrease in density of the fluid offsetsany increase in the velocity of the mixture, to the pointwhere no additional mass flow is realized.


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Figure 4. Generalized rcCurve

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It is necessary to account for the occurrence of chokedflow during the sizing process to insure againstundersizing a valve. In other words, we need to knowthe maximum flow rate a valve can handle under agiven set of conditions. To this end, a procedure wasdeveloped (reference 8) which combines the controlvalve pressure recovery characteristics with thethermodynamic properties of the fluid to predict themaximum usable pressure differential, that is, thepressure differential at which the flow just chokes.

Figure 4. Generalized rcCurve

A pressure recovery coefficient can be defined as:

Km +P1 * P2

P1 * Pvc(16)

Under choked flow conditions it is established inreference 8 that:

Pvc + rcPv(17)

The vapor pressure, Pv, is determined at inlettemperature since the temperature of the liquid doesnot change appreciably between the inlet and the venacontracta. The term rc is known as the critical pressureratio and is another thermodynamic property of the

fluid. While it is actually a function of each fluid and theprevailing conditions, it has been established that datafor a variety of fluids can be generalized according tofigure 4 (references 5 and 8) or the following equation(reference 6) without significantly compromisingoverall accuracy:

rc + FF+ 0.96* 0.28PvcPc

(19)

The value of Km is determined individually by test foreach valve style and accounts for the pressurerecovery characteristics of the valve.

By rearranging equation 16, the pressure differential atwhich the flow chokes can be determined is called theallowable pressure differential:

(P1 * P2)allowable+ Km(P1 * rcPv)(20)

When this allowable pressure differential is used inequation 13, the choked flow rate for the given valvewill result. If this flow rate is less than the requiredservice flow rate, the valve is undersized. It is thennecessary to select a larger valve and repeat thecalculations using the new values for Cvand Km.

The equations supplied in the sizing standard(reference 5) are in essence the same as thosepresented in this paper, except the nomenclature hasbeen changed. In this case:

Qmax+ N1FLCvP1 * FFPv

G

(21)

where:

FL = Km

FF = rc

N1 = units factor

CavitationClosely associated with the phenomenon of chokedflow is the occurrence of cavitation. Simply stated,cavitation is the formation and collapse of cavities inthe flowing liquid. It is of special concern in sizingcontrol valves because left unchecked it can produceunwanted noise, vibration, and material damage.

As discussed earlier, vapor can form in the vicinity ofthe vena contracta when the local pressure dropsbelow the vapor pressure of the liquid. If the outletpressure seen by the mixture as it exits the controlvalve is greater than the vapor pressure, the vaporphase will be thermodynamically unstable and willrevert to a liquid. The entire liquidvaporliquid phasechange process is known as cavitation, although it isthe vaportoliquid phase change which is the primarysource of the damage. During this phase change, amechanical attack occurs on the material surface inthe form of high velocity microjets and shock waves.Given sufficient intensity, proximity, and time, thisattack can remove material to the point where thevalve no longer retains its functional or structuralintegrity. Figure 5 shows an example of such damage.


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Figure 5. Typical Cavitation Damage

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Figure 6. Comparison of High and Low Recovery Valves

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Because cavitation and the damage due to it arecomplex processes, accurate prediction of key eventssuch as damage, noise, and vibration level is difficult.Thus sizing valves for cavitation conditions requiresspecial considerations.

The concept of pressure recovery plays a key role incharacterizing a valves suitability for cavitationservice. A valve which recovers a significantpercentage of the pressure differential from inlet to thevena contracta is appropriately termed a high recoveryvalve. Conversely, if only a small percent is recoveredits classified as a low recovery valve. These two arecontrasted in figure 6. If identical pressure differentialsare imposed on a high recovery valve and a lowrecovery valve, all other things being equal, the highrecovery valve will have a relatively low vena contractapressure. Thus, under the same conditions, the highrecovery valve will more likely cavitate. On the otherhand, if flow through each is such that the inlet and

Figure 7. Pressure Profiles for Flashing and Cavitating Flows

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vena contracta pressures are equal the low recoveryvalve will have the lower collapse potential (P2Pvc),so that cavitation intensity will generally be less.

Thus, it is apparent that the lower pressure recovery

devices are more suited for cavitation service.

The possibility of cavitation occurring in any liquid flowapplication should be investigated by checking for thefollowing two conditions:

1. The service pressure differential is approximatelyequal to the allowable pressure differential, and

2. The outlet pressure is greater than the vaporpressure of the fluid.

If both of these conditions are met the possibility existsthat cavitation will occur. Because of the potentiallydamaging nature of cavitation, sizing a valve in this

region is not recommended. Special purpose trims andproducts to control cavitation should be considered.Because of the great diversity in the design of thisequipment it is not possible to offer general guidelinesfor sizing them. Refer to specific product literature formore information.

Flashing

Flashing shares some common features with chokedflow and cavitation in that the process begins withvaporization of the liquid in the vicinity of the venacontracta. However, in flashing applications, thepressure downstream of this point never recovers to avalue which exceeds the vapor pressure of the fluid sothat the fluid remains in the vapor phase. Schematicpressure profiles for flashing and cavitating flow arecontrasted in figure 7.

Flashing is of concern not only because of its ability tolimit flow through the valve, but also because of thehighly erosive nature of the liquidvapor mixture.Typical flashing damage is smooth and polished in


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Figure 8. Typical Flashing Damage

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Figure 9. Viscous Flow Correction Factors

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appearance (figure 8) in stark contrast to the rough,cinderlike appearance of cavitation (figure 5).

If P2 < Pv, or there are other service conditions toindicate flashing, the standard sizing procedure shouldbe augmented with a check for choked flow.

Furthermore, suitability of the particular valve style forflashing service should be established with the valvemanufacturer.

Viscous Flow

One of the assumptions implicit in the sizingprocedures presented to this point is that of fullydeveloped, turbulent flow. Turbulent flow and laminarflow are flow regimes which characterize the behaviorof flow. In laminar flow all fluid particles move parallelto one another in an orderly fashion and with no mixingof the fluid. Conversely, turbulent flow is highly randomin terms of local velocity direction and magnitude.While there is certainly net flow in a particulardirection, instantaneous velocity components in alldirections are superimposed on this net flow.Significant fluid mixing occurs in turbulent flow. As istrue of many physical phenomena, there is no distinctline of demarcation between these two regimes, so athird regime of transition flow is sometimesrecognized.

The physical quantities which govern this flow regimeare the viscous and inertial forces, the ratio of which isknown as the Reynolds number. When the viscousforces dominate (a Reynolds numbers below 2000) theflow is laminar, or viscous. If the inertial forcesdominate (a Reynolds number above 3000) the flow isturbulent, or inviscid.

Consideration of these flow regimes is importantbecause the macroscopic behavior of the flowchanges when the flow regime changes. The primarybehavior characteristic of concern in sizing is thenature of the available energy losses. In earlierdiscussion it was asserted that, under the assumptionof inviscid flow, the available energy losses wereproportional to the square of the velocity.

In the laminar flow regime, these same losses arelinearly proportional to the velocity; in the transitionalregime, these losses tend to vary. Thus, for equivalentflow rates, the pressure differential through a conduit

or across a restriction will be different for each flowregime.

To compensate for this effect (the change inresistance to flow) in sizing valves a correction factorwas developed (reference 9). The required Cv can bedetermined from the following equation:

Cvreqd

+ FRCvrated(22)

The factor FR is a function of the Reynolds numberand can be determined from a simple nomographprocedure (reference 10), or by calculating theReynolds number for a control valve from the followingequation and determining FR from figure 9 (reference9).

Rev +N4FdQ

nFL12Cv12 1N2

(FL)2Cvd22 ) 1

14

(23)

To predict flow rate or resulting pressure differential,the required flow coefficient is used in place of therated flow coefficient in the appropriate equation.

When a valve is installed in a field piping configurationwhich is different than the specified test section, it isnecessary to account for the effect of the alteredpiping on flow through the valve. (Recall that thestandard test section consists of a prescribed length ofstraight pipe up and downstream of the valve.) Fieldinstallation may require elbows, reducers, and tees,which will induce additional losses immediatelyadjacent to the valve. To correct for this situation, twofactors are introduced: Fp and Flp. The former is usedto correct the flow equation when used in theincompressible range, while the latter is used in the


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choked flow range. The expressions for these factorsare:

Fp + SKN2 Cvd22

) 1*12

(24)

FIp + FLFL2KIN2 Cvd22

) 1*12

(25)

The term K in equation 24 is the sum of all losscoefficients of all devices attached to the valve and theinlet and outlet Bernoulli coefficients. Bernoullicoefficients are coefficients to the velocity head term inthe energy and Bernoulli equations, which account forchanges in the kinetic energy as a result of acrosssectional flow area change. They are calculatedfrom the following equations.

KBinlet

+ 1* (dD)4

(26a)

KBoutlet

+ (dD)4 * 1

(26b)

Thus, if reducers of identical size are used at the inletand outlet, these terms cancel out.

The term KI in equation 25 includes the losscoefficients and Bernoulli coefficient on the inlet sideonly.

In the absence of test data or knowledge of losscoefficients, loss coefficients may be estimated frominformation contained in other resources such asreference 3.

The factors Fp and FIp would appear in flow equations(15) and (21) respectively as follows:

For incompressible flow:

Q+

FpCv

P1 * P2G (27)

For choked flow:

Qmax+ FICvP1 * FFPv

G

(28)

Summary

It has been shown that a fundamental relationshipexists between key variables (P1, P2, Pv, G, Cv, Q) forflow through a device such as a control valve.Knowledge of any four of these allows the fifth to becalculated or predicted. Furthermore, adjustments tothis basic relationship are necessary to account forspecial considerations such as installed pipingconfiguration, cavitation, flashing, choked flow, andviscous flow behavior. Adherence to these guidelineswill insure correct sizing and optimum performance.

References

1. Streeter, Victor L., and E. Benjamin Wylie, Fluid

Mechanics, 7th Ed., McGrawHill Book Company,New York, 1979.

2. Olson, Reuben M., Essentials of Engineering FluidMechanics, 3rd Ed., Intext Educational Publishers,New York, 1973.

3. Flow of Fluids Through Valves, Fittings, and Pipe,Crane Company, New York, 1978.

4. Instrument Society of America, Control ValveCapacity Test Procedure, ANSI/ISAS75.02, 1981,Research Triangle Park, North Carolina.

5. Instrument Society of America, Control ValveSizing Equations, ANSI/ISAS75.01, 1977,

Pittsburgh, Pennsylvania.

6. Schafbuch, Paul M., Fundamentals of FlowCharacterization(Technical Monograph 29), FisherControls International, Inc., Marshalltown, Iowa, 1985.

7. Control Valve Handbook, Fisher Controls Company(Fisher Controls International, Inc.), Marshalltown,Iowa, 1977.

8. Stiles, G.F., Development of a Valve SizingRelationship for Flashing and Cavitation Flow,Proceedings of the First Annual Final ControlElements Symposium, Wilmington, Del., May 1416,1970.

9. Stiles, G.F. Liquid Viscosity Effects on ControlValve Sizing, 19th Annual Symposium onInstrumentation for the Process Industries, Texas A &M, 1964.


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Fisher Controls International, Inc.205 South Center StreetMarshalltown, Iowa 50158 USAPhone: (641) 7543011Fax: (641) 7542830Email: [email protected]: www.fisher.com

D350408X012 / Printed in U.S.A. / 1985

The contents of this publication are presented forinformational purposes only, and while every effort has

been made to ensure their accuracy, they are not to be

construed as warranties or guarantees, express or implied,

regarding the products or services described herein or

their use or applicability. We reserve the right to modify or

improve the designs or specifications of such products at

any time without notice.

EFisher Controls International, Inc. 1974, 1985;

All Rights Reserved

Fisher and FisherRosemount are marks owned by

Fisher Controls International, Inc. or

FisherRosemount Systems, Inc.

All other marks are the property of their respective owners.


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(FISHER] Fisher Controls

technicallllonograph31

Fundamentals of ValveSizing for GasesFloyd D. Jury


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a f Valvefa r Gases

valve sizing can be both expensive and inA valve that's too small will not pass the re-and the process will be starved. A valve that is

will not only be more expensive, but it can also leadinstability and other problems.

e days of selecting a valve based upon the size of theare gone forever. Selecting the correct valve size forapplication requires a knowledge of process con

that the valve will actually see in service. Thefor using this information to size the valve is bas-

a combination of theory and experimentation..,..

floefforts in the development of valve sizing theorythe problem of sizing valves for liquid flow.was one of the early experimenters wh o

the science of fluid flow theory to liquid flow.experimental modifications to this theory have

a useful liquid flow equation.Q gp m Cv-VLlP/G (1)Q gpm Liquid flow in gpmCv Valve sizing coefficientLlP Valve pressure drop

G Liquid specific gravity

equation rapidly became widely accepted fo r sizingon liquid service and most manufacturers of valvespublishing Cv data in their catalogs.

s inevitable that the valves, which had worked so wellwould sooner or later be used to control the flow

gases, such as air.

just as inevitable that the good results obthe Cv equation would strongly tempt its use to

the flow of gas.

Modi f ied C v Equat ionIn order to use the liquid flow equation for air it wasnecessary to make two modifications. The first step was tointroduce a conversion factor to change flow units fromgallons-per-minute to cubic-feet-per-hour. The second stepwas to relate liquid specific gravity in terms of pressure,which would be more meaningful for gas flow. The resultwas the Cv equation revised for the flow of air at 60F.

(2)14

G,eneralizing this equation to handle any gas at anytemperature requires only a simple modification factor basedupon Charles' Law for gases

The term 52 0 represents the product of the specific gravityand temperature of air at standard conditions. The specificgravity is one or unity. In absolute units, the standardtemperature is 52 0 0 R which corresponds to 60F. The Gan d T represent th e specific gravity and absolutetemperature of any gas.The apparent simplicity of Equation (3) can obscure theserious problems that develop from indiscriminately usingthis simple conversion without being aware of its ratherstrict limitations that result from compressibility effects andcritical flow.

~ - - CALCULATED

Q : , ~ ', , ' ACTUAL

LIP 0.02

Figure I. Comparison of Equation (3) and anActual Flow Curve


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A plot of this equation shows a straight line relationshipwhere the slope of the curve is a function of the valve sizingcoefficient, Cv ' The greater the Cv of the valve, the steeperthe slope.An actual flow curve would show good agreement with thetheoretical curve at low pressure drops. However, a significant deviation occurs at pressure drop ratios greater thanapproximately 0.02 because the equation wa s based uponthe assumption of incompressible flow. When the pressuredrop ratio exceeds approximately 0.02 the gas can no longer AIbe considered an incompressible fluid. #'

FLOW---

RESTRICT ION /

~ V E N ACONTRACTA

Figure 2. Vena Contracta IllustrationA much more serious l imitation on this equation involves thephenomenon of critical flow. To help understand criticalflow, a control valve, at any flow opening, can berepresented by a simple restriction in the line. As the f lowpasses through the physical restriction, there is a neckingdown, or contraction, of the f low stream. The minimumcross-sectional area of the flow stream occurs just a shortdistance downstream of the physical restriction at a pointcalled the vena contracta. I n order to maintain a steadyflow of fluid through the valve, it is obvious that the velocitymust be greatest at the vena contracta where the crosssectional area is the least.As the LlP across the valve increases, flow also increases,and the velocity at the vena contracta increases. At some

value of LlP, however, the gas reaches sonic velocity at thevena contracta. Since the gas can't normally travel anyfaster than this l imiting velocity, a choked flow condition isreached known as critical flow.When critical flow is reached, Equation (3) becomes absolutely worthless fo r predicting the flow since the flow nolonger increases with pressure drop. So far, all we have is anequation that deviates significantly from the actual flow fo rpressure drop ratios greater than 0.02 and is totally inaccurate once critical flow is reached.Various valve manufacturers modified the Cv equation evenfurther in an attempt to predict the behavior of gases at bothcritical and subcritical flow conditions. This approach had avery strong economic appeal to the manufacturers since itmeant they would still only have to test their valves on waterto obtain a Cv' The modified equation would then take careof predicting the gas flow. As it turned out, three equationswere developed all of which did a fairly decent job of predicting gas flow through standard globe type valves at pressu redrop ratios less than 0.5.

QQQ

1360Cvy/ (P , -P 2 )P/GT1364Cvy (P , -P 2 )P,IGT

1 3 6 0 C v - y l L l P / G T ~ ( P , +P2 )! 2

(4)(5)(6)

For globe type valves, which were in most common use atthe time, critical flow is reached at a pressure drop ratio ofabout 0.5. In the lo w pressure drop region the slope of theflow curve plotted from any of these three equations is thesame as that established by the original Cv equation (Eq. 3).If the pressure drop ratio is equal to 0.5, each of themodified equations will predict a flow which approximatesthe actual critical flow. At this point, all three of the modifiedequations reduce to the form of a constant times Cv and theabsolute inlet pressure. This indicates that once the criticalpressure ratio is reached, the flow through the valve will nolonger be dependent upon the pressure drop across thevalve. The flow will change only as a function of the inletpressure.

3


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while it looked as though the problem was solved. Lowtype valves, such as those shown in Figure 3

reasonably well with these equations, bu t then alongof high r.ecovery valves such as those

in Figure 4.

Recove ry Valves~flow through a hf'Qh recovery valve is quite streamlined

efficient compared to that in a lo w recovery type valve.o valves have equal flow areas and are passing theflow, the high recovery valve will exhibit much less

drop than the lo w recovery valve. High and lo wto the valve's ability to convert velocity at the

contracta back into pressure downstream of the valve.

FLOW-IP T = = ~ ~ ~ I I ~ ~ i ~ : ~ { l i i i i i i ~ P 2

IIIPT - - ~ ~ - - - - - f - - - - - - - - - -I PI _- 2i /. HIGH

\ I / RECOVERYP2I I./. LOW RECOVERYFigure 5. Comparison of Pressure Profiles for High

an d Lo w Recovery Valves

The pressure profiles fo r tw o valves having the same pres-sure drop and flow rate are shown in Figure 5. If critical flowis imminent, it is obvious that the pressure drop ratio fo r thehigh recovery valve will be much less than for the low recovery valve. While it's true that lo w recovery valves, such as theglobe style valves, exhibit critical flow at a pressu re dropratio of 0.5, the more efficient high recovery valves can ex-hibit critical flow at pressure drop ratios as lo w as 0.15.Nov::" let's consider the case of a high recovery valve and a10llY.$recovery valve that both have the same Cv' Since the initial slop of the flow curve is related to Cv' this portion of thecdrve will be the same for both valves.Since the flow predicted by the critical flow equationdepends directly upon Cv the equation will predict the samecritical flow fo r both valves. We have already seen, however,that the high recovery valve will exhibit critical f low atpressure drop ratios as lo w as 0.15. In other words, themodified Cv equations grossly over-predict the critical flowthrough the high recovery valve.This point is important enough to warrant repeating. A highrecovery valve with the same Cv and tested under similarconditions as a lo w recovery valve will have much lesscritical gas flow capacity. Thus, if the modified Cv equations,

6PP, = LOW RECOVERY--------------------------Q , ~ 6 P = O . 1 5~ P, HIGH RECOVERY--------------------------------

Figure 6. Critical Flow for High and Lo w RecoveryValves with Equal Cv


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intended for low recovery valves, are used to size a highrecovery valve, the critical flow capacity of the valve can beover-estimated by as much as 300 percent.This may sound like a strange circumstance, but it should berealized that for both valves to have the same Cv the highrecovery valve would be much smaller than the low recoveryvalve. The geometry of the valve greatly influences liquidf low; whereas, the critical flow of gas depends essentiallyonly upon the flow area of the valve. Thus, a smaller highrecovery valve will pass less critical gas flow, but its greaterstreamlined flow geometry allows it to pass as much liquidf low as the larger lo w recovery valve.

C g ' A G a s Sizing Coe f f ic ie n tBecause of the problems in using Cv to predict critical flow inboth high and lo w recovery valves, Fisher Controls Company began testing all valves on air as well as water. Fromthese tests, a gas sizing coefficient, Cg , was defined in 1951to relate critical flow to the absolute inlet pressure. Since Cgis experimentally determined for each style and size of valve,it can be used to accurately predict the critical flow for bothhigh and low recovery valves. Equation (7) shows the defining equation for Cg

Cg is determined by testing the valve with 60F air undercritical flow conditions. To make the equation applicable forany gas at anv temperature, the same correction factor canbe used that vCas applied previously to the original Cv equation.

(8)

Fisher no w found itself with two gas sizing equations. Oneequation (Eq. 3) applied only to quite lo w pressure dropratios while the other (Eq. 8) was good only for predictingcritical flow. What about the transition region?In an attempt to pu t the pieces of this puzzle together, theFisher research department conducted thousands of tests onevery different style of valve available including both highand lo w recovery valves as well as some intermediate ones.

When the results of these tests were normalized withrespect to critical flow and then plotted, a very useful factbecame apparent. It was noted that all of the test points inthe sloping portion of the flow curve could be quite closelyapproximated by the first quarter cycle of a standard sinecurve.

Univ e rs a l Ga s Sizing Equat ionBased on this test program, Fisher Controls Companydeveloped, in 1963, a Universal Gas Sizing Equation. Thisequation is universal in the sense that it accurately predictsthe flow for either high or low recovery valves, for any gas

and under any service conditions. This equation incorporatesboth the basic Cv equation and the Cg critical flow equationinto a single, dual-coefficient equation where the ne w factor,C/, is introduced.

Q scfh = ~ 5 2 0 1 G T C g P , S I N ~ 5 9 . 6 4 / C , k v P 7 P J R B d . (9)where:

C, is defined as the ratio of the gas sizing coefficient and theliquid sizing coefficient. It provides an index which tells ussomething about the physical f low geometry of the valve. Inother words, it s numerical value tells us whether the valveis high or lo w recovery or someplace in between. A simple il -lustration will help clarify the relationship between C, andthe valve recovery characteristics.

Example:High Recovery Valve

4680254CgICv4680125418.4

Low Recovery Valve468013 5C/C v4680/13534.7

Assume two valves with identical flow areas. One is a highrecovery valve, and one is low recovery. Since Cg is determined under critical f low conditions it is relatively independent of the recovery characteristics of the valve. The criticalf low is primarily a function of the valve area only. Thus bothvalves will have the same Cg Flow geometry, however, hasa significant influence upon liquid flow. The greater efficiency and better streamlining of the high recovery valve willallow it to pass nearly twice as much liquid flow even withthe same port area. Correspondingly the Cv will be nearlytwice as large as the lo w recovery valve.

This example not only shows ho w C, can vary with valverecovery characterisitics, bu t it also illustrates the typicalrange of C, values. In general, C, values can range fromabout 16 to 37.

Note that Cv' which appears in the denominator, is the factorwhich varies primari ly with the valve's recoverycharacteristics. This example illustrates the general principlethat high recovery valves have low C, values, while lo wrecovery valves have high C, values.

In order to accurately predict the gas f low for any style valve,two sizing coefficients are needed. Cg helps to predict theflow based upon the physical size or flow area, while C, ac-counts for differences in valve recovery characteristics. TheUniversal Gas Sizing Equation (Eq. 9) incorporates both ofthese coefficients. This equation may appear somewhatcomplex upon first encounter, but a look at the tw o ex-tremes of the equation can clear up some of the mystery.

5


16/17

consider the extreme where the valve pressure dropis quite small (L'lP/P, < 0.02). This means that the angle

the sine function will also be quite small in radians. Fromtrigonometry recall that, for small angles, the sine of

e angle can be approximated by the angle itself in radians.this assumption of a small pressure drop ratio, the ungas sizing equation simply reduces to the original Cv(Eq. 3). We already know that this equation fits the

data in the incompressible flow region where thedrop ratio is less than 0.02.

other extreme of the Universal Sizing Gas Equation is inof critical flow. Critical flow is first established at

e point where tl-)e sine function reaches its maximumthe end of the first quarter cycle. At this point the

function is equal to one and the angle is equal to n/2drop ratio at this point is known as the

pressure drop ratio.the critical pressure drop ratio, where the sine function

the Universal Gas Sizing Equation simplyto the critical flow equation (Eq. 8). This Universal

Equation was originally developed to predict theflow fo r any valve style based upon the experimental

determined gas sizing coefficient, Cg summary, the Universal Gas Sizing Equation takes the

Cv equation at one extreme and the critical flow equaat the other extreme and blends the two together with athat fits the experimental data. All of thisone universal equation!

find it more convenient to deal with sinedegrees rather than in radians. This is easily ac

by a simple conversion constant. The new conin the angle becomes 3417 rather than 59.64. Now,

sine angle will be 90 degrees at the critical pressure dropn/2 radians.

As the pressure drop across the valve increases, the sineangle increases from zero up to 90 degrees. If the angle isallowed to increase beyond 90 degrees, the equation wouldpredict a decrease in flow. Since this is not a realistic situation, the angle must be restricted to 90 degrees maximum.Tt'1\ mathematical development of the Universal Gas Sizing&quation shown in Equation (10) is based upon the use ofthe perfect gas laws. The expression -yi520/GT is derivedfrom the equation of state for a perfect gas. While it is truethat no perfect gases, as such, exist in nature, there are amultitude of applications where the perfect gas assu mptionis a useful approximation.

For those special applications where the perfect gasassu mption is not adequate, a more general form of theUniversal Gas Sizing Equation has been developed.Q'b/hr = 1.06-yd,P,CgSINIT34171C,)-viP/PJoeg. (11)where:

Gas, Steam, or Vapor flow (lbs/hr.)Inlet gas density (lbs/ft1)

Equation (11) is known as the density form of the UniversalGas Sizing Equation. It is the most general form and can beused for both perfect and non-perfect gas applications.Steam is the most common application where Equation (11)is used. The steam density can be easily found from published steam tables.

Because steam service applications are so common, aspecial form of the Universal Equation was developed. If thepressure stays below 1000 psig, Equation (12) can be usedwhich simplifies'the calculation.


17/17

Q'b/hr, = [C 5P1(1 + 0 , 0 0 0 6 5 T 5 h U S I N 8 3 4 1 7 / C , ) ~ J D . g ,(12)where:

C5 Steam sizing coefficientT5h Degrees of superheat (0 F)

Equation (11) is more general and can be used in all caseswhere Equation (12) is valid; however. Equation (11) re-quires a knowledge of the steam density (d , ), For steam 'i\below 1000 psig. a constant relationship exists between thegas sizing coefficient (C g ) and the steam sizing coefficient(C 5 ),

(13)

Density changes that occur as the steam becomessuperheated are compensated for by the superheat correction factor that appears in the denominator of Equation (12),Use of Equation (12) eliminates the need for steam tables tolook up the density of superheated steam,

At pressures greater than 1000 psig. the steam begins todeviate significantly from the constant relationship definedin Equation (13) and the superheat correction is no longervalid, At greater pressures. Equation (11) must be used foraccurate results,

Conc lus ionThe Universal Gas Sizing Equation can be used to determinethe flow of gas through any style of valve, Absolute units oftemperature and pressure must be used in this equation,When the critical pressure drop ratio causes the sine angleto be 90 degrees. the equation will predict the value of thecritical flow, For service conditions that would result in anangle of greater than 90 degrees. the equation must bel imited to 90 degrees in order to accurately determine thecritical flow that exists,

The most common use of the Universal Gas Sizing Equationis to determine the proper valve size for a given set of serviceconditions, The first step is to calculate the required Cg byusing the Universal Gas Sizing Equation, The second step isto select a valve from the catalog with a Cg which equals orexceeds the calculated value, Care should be exercised tomake certain that the assumed C, value for the Cg calculation matches the C, for the final valve selection from thecatalog,

It should be apparent by now that accurate valve sizing forgases requires use of the dual coefficients Cg and C" Asingle coefficient is not sufficient to describe both thecapacity and the recovery characteristics of the valve.

This paper has dealt exclusively with the problem of sizingvalves for gas flow. Liquid flow requires a different set ofconsiderations which are discussed in another paper. '

The proper selection and sizing of a control valve for gas ser-vice is a highly technical problem with many factors to beconsidered, Fisher Controls Company provides technical in formation. test data. sizing catalogs. nomographs. sizingsliderules. and even computer programs that remove theguesswork and make valve sizin(1 a simple and accurateprocedure.

Meet the Author ,Floyd Jury, Director of EducationMSME, N.C. State College, 1963;BSME, University of Alabama, 1961.PrevIous associations: Bell TelephoneLaboratOries, Guilford College, ThiokolChemical Cctrp" WOI- TV TelevisionStudios

Documents

Fundamentals of Valve Sizing for Liquid