sizing of throttling devices for two phase flow

Sizing of Throttling Device forGas/Liquid Two-Phase FlowPart 2: Control Valves, Orices,and NozzlesRalf Dienera and Jurgen Schmidtba BASF AG, Inorganic Chemicals Europe, Ludwigshafen, Germanyb BASF AG, Safety Engineering, Ludwigshafen, Germany; [email protected] (for correspondence)

Published online 19 January 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/prs.10035

The calculation of the mass ow rate through throt-tling devices is difcult when handling two-phase ow,especially when boiling liquids ow into these ttings.Control valves and orices are typically oversized inindustry and the control range of those valves oftendoes not t the control requirements. In this paper theHNE-DS method is proposed for the sizing of controlvalves, orices, and nozzles in two-phase ow. It ex-tends the -method, originally developed by Leung, byadding a boiling delay coefcient to include the degreeof thermodynamic nonequilibrium at the start of thenucleation of small vapor mass fractions upstream ofthe tting. The additional introduction of a slip correc-tion factor, to take account of hydrodynamic nonequi-librium (slip), also makes it possible to calculate reli-ably the ow rate through control valves and orices inboth ashing and nonashing ow.

In Part 2 the HNE-DS method for short nozzles,orices, and control valves is considered. Part 1 de-scribes the sizing of safety valves using the samemethod. Additionally, the derivation of the HNE-DSmodel is given there in detail. The predictive accuracyof the HNE-DS model has been checked with referenceto more than 1300 sets of experimental data. 2005American Institute of Chemical Engineers Process SafProg 24: 2937, 2005

INTRODUCTIONThe sizing of control valves, orices, and nozzles for

the ow of gases, noncondensing vapor and nonvapor-izing liquids is described in the standards ISO 5167 and

IEC 60534 [1, 2]. The current standards are sufcientlyaccurate for the sizing of these devices for single-phaseow, although they do not contain reliable recommen-dations for two-phase mixtures composed of vapor andliquid. At present there is no appropriate standard ei-ther nationally or internationally.

In the chemical and petrochemical industries andalso in power plants and offshore facilities, however,such a standard is frequently needed. In these plantsliquids are often pumped from tanks or pipeline net-works into parts of the plants having relatively lowpressures. Under this circumstance the feed rate iscontrolled by means of a control valve (see Figure 1). Asafety valve must as a general rule be installed toprotect the plant against overpressure, for instance inthe event that the outlets have become blocked or havebeen inadvertently closed. The size of this safety deviceis based on the maximum feed rate through the controlvalvethe control valve is part of the safety concept.This complicates any replacement of the control valve.Therefore, the ow rate is in practice usually limited byan additional orice, which is tted downstream fromthe control valve. The control valve is then no longerrelevant to safety and it can be replaced by any othervalve without having an effect on the safety concept.Overall, the design engineer is confronted with the taskof estimating the ow rate through the control valve ororice as accurately as possible to determine the size ofthe safety valve on the low-pressure equipment (Figure1). The sizing task is divided into two steps:

1. Sizing a relief valve for two-phase ow (Part 1, seeLiterature Cited [19]) 2005 American Institute of Chemical Engineers

Process Safety Progress (Vol.24, No.1) March 2005 29

2. Sizing a control valve or orice for two-phase ow(Part 2)

Both steps are based on the same method: the HNE-DS(homogeneous nonequilibrium method developed bythe authors Diener and Schmidt). This is an extended-method, originally developed by Leung. In this arti-cle the HNE-DS method is described for sizing controlvalves, orices, and nozzles.

LITERATURE REVIEWA few methods are described in the literature for

two-phase ow with sufcient accuracy for the deter-mination of the mass ow rate through throttling de-vices. These computational methods, however, arevery resource intensive. The calculation requires pre-cise knowledge of the internal geometry of the tting,such as the contour of a throttle cone in a control valve,not usually given by the manufacturer. In addition,physical properties data such as densities, viscosities,enthalpies, and entropies of the vapor and liquidphases as a function of pressure and temperature overa broad parametric range are needed. In many casesthese are not available or they must be determined bymeans of costly measurement procedures.

Apart from these complex methods there are alsosizing methods, which are easier to use and in line withactual practice in the chemical and petrochemical in-dustries, but these exhibit certain limits in application.Most of the control valves are sized by means of the IEC60534-2-1 standard [2]. Any limitation in mass owcapacity attributed to cavitation or ashing of a dis-tinctly subcooled liquid is taken into account by em-pirically developed correction factors. If two-phasemixtures or boiling liquids are to be considered in theinlet of the valve, the standard is not applicable. As aconsequence, in practice the ow coefcients for sin-gle-phase gas and single-phase liquid are added withthe mass ow rate of each phase as a weighting factor.This so-called addition model is a frozen ow consid-eration where no momentum, heat, or mass transfer is

taken into consideration. In general, this method maylead to highly underestimated or distinctly overesti-mated mass ow rates. Sheldon and Shuder [3] haverecommended a variable correction factor to obtainless uncertain results. Nevertheless, in the case of boil-ing liquids or mixtures of ashing liquids and vaporsthe results are still unsatisfactory. Diener [4] developeda more sophisticated physical model including hydro-dynamic and thermodynamic nonequilibrium ow. It-eration procedures and a numerical integration of theequations are necessary for calculating the mass owrate. Additionally, pressure- and temperature-depen-dent property data are needed (density, enthalpy, andentropy), which in practice are often not available.

The most common method for calculating the massow rate in the case of a gas/liquid two-phase ow fortechnical purposes is the -method, originally devel-oped by Leung [5, 6]. It is based on the homogeneousow model in which gas and liquid ow at the samevelocity and are uniformly distributed over the owcross section. Both phases are in hydrodynamic equi-librium and the phase boundary is (theoretically) ofinnite size. Under real conditions, however, theseassumptions are true only for the limiting case in sprayor wet vapor ow having just a few drops of liquid inthe vapor. Nevertheless, the model is used for thelower vapor content range and in ow without a phasetransition, such as a mixture of air and water. It stillprovides acceptable results for the practitioner even inthe median range of vapor contents. The reason for thisis the acceleration in the valve that mixes phases thor-oughly and as a result they are largely homogeneouslydistributed.

UNCERTAINTIES OF THE -METHODIn two-phase ow the real mass ow rate can be

determined only by experiments, for example mea-surements made at the Technical University of Ham-burg-Harburg, in the department of Prof. Dr. L. Friedel,and by SAMSON AG (Frankfurt, Germany). There, themass ow rate was measured when mixtures of steamand boiling water were passed through control valves.The test valves used had nominal diameters of 25, 50,and 80 mm and had different types of valve cones(V-port cone, parabolic cone, and perforated cone).The test setups and the measuring methods and tech-nology employed are described by Diener [4]. In Figure2 the mass ow rates calculated by the -method ofLeung (homogeneous equilibrium model) are plottedagainst the measured mass ow rates. If these data setswere in agreement with each other all points would lieon the diagonal. The deviations, however, are far re-moved from the diagonal line and are distributed asym-metrically about the 100% error limit. The maximumdeviation between measured and calculated values is500%; that is, the true ow rate through the controlvalve is about six times larger than the calculated rate.A downstream safety valve, sized using these calcu-lated ow rates through the control valve, would be fartoo small and the calculated relief cross section wouldlikewise be about six times smaller than the crosssection needed. The method is, therefore, unsuitable

Figure 1. Typical layout of production vessel fed by acontrol valve with a safety valve on top to avoid aninadmissible vessel overpressure. The size of thesafety valve is determined by the maximum feedthrough the control valve.

30 March 2005 Process Safety Progress (Vol.24, No.1)

for sizing calculations for inlet ow conditions involv-ing boiling liquids with only low vapor contents.

RECOMMENDATION FOR AN EXTENDED METHOD (HNE-DS)

Mass Flux through an Ideal NozzleTo take the boiling delay into account the HNE-DS

method was developed. It extends the -method by anequation for the boiling delay coefcient. All equationsneeded for the application of the HNE-DS method aregiven in Tables 1 and 2. The basic idea for this modelwas outlined in 1998 [7].

In general, the mass ow rate through nozzles, con-trol valves, and orices may be calculated by

MCV/orif corrACV/orifmid with corr CV/orifS (1)

where ACV/orif is the valve seat/orice area of the tting,mid is the mass ux through an adiabatic, ideal formednozzle in case of frictionless ow, CV/orif is the owcoefcient, is the slip correction factor, and S is asafety factor. The ow coefcient for nozzles equals theso-called velocity coefcient with values of 0.95 to 1.

In Table 1, the calculation procedure for the massow rate through an ideal nozzle mid is summarized:the required variables of state and physical propertydata are specied for the (maximum permissible) stag-nation condition pin, Tin in front of the tting, such asin an upstream vessel or pipe network. In the case of avery large pressure drop in front of a control valve ororice it may be necessary to determine an imaginarystagnation condition at the inletthat is, an isentropicash calculation from the static inlet conditions to actitious total condition is made. The compressibilityfactor (Eq. 4) and the critical pressure ratio crit (Eq.5) are then determined for a homogeneous ow inthermodynamic equilibrium (N 1). Based on this rstestimate the boiling delay factor N is calculated usingEq. 6b with sufcient accuracy so that the compress-ibility factor may be corrected taking account of theboiling delay. By analogy with the method of Henryand Fauske [8] a value for the exponent a 3/5 is

recommended for throttling devices where the accel-erational pressure drop dominates the frictional pres-sure drop, that is, in typical short nozzles, orices, andcontrol valves. If the area ratio of the nozzle or oricetends to 1 (small ow contraction), the boiling delayexponent decreases. Therefore, the exponent a be-comes 2/5 for safety valves and for long nozzles anddiffusors a value of a 0 is recommended (also see theAppendix in the companion article, Part 1).

In principle, the HNE-DS method can also be ap-plied to venturies. In the absence of detailed experi-mental data with low-quality two-phase ow, the ex-ponent a cannot precisely be specied. From atheoretical perspective, the value will be expected tobe 2/5 or less.

With the aid of the compressibility factor , includ-ing the boiling delay coefcient N, the critical pressureratio crit can now be determined more accurately. Bycomparing this with the actual pressure ratio in opera-tion 0 (Eq. 2), the ow condition in the narrowest owcross section (critical or subcritical pressure ratio) canbe determined (Eq. 7). The corresponding pressureratio crit (critical ow) or 0 (subcritical ow) is thenused to calculate the expansion coefcient (Eq. 8).With this coefcient the mass ux in the narrowest owcross section of a (adiabatic) throttling device in africtionless ow mid is obtained (Eq. 9).

The equation for the compressibility factor N con-tains no additional physical properties and iterationsare unnecessary. In contrast with the distinctly morecomplex nonequilibrium model of Henry and Fauske[8] the mass ux is dened as a continuous function ofthe vapor mass ow quality and there is no need forderivatives of property data functions (cf. Figure 3). Inthis gure the mass ow densities calculated by bothmethods for the ow of steam and water through anideal nozzle are plotted as a function of the stagnationvapor mass ow qualities at inlet pressures of 10, 1, and0.1 MPa (100, 10, and 1 bar). The calculated results arealmost identical. Equally high accuracy is also obtainedusing the refrigerant R12 (cf. Figure 4) whose physicalproperties are very different from those of water. Theenthalpy of vaporization for R12 is smaller by morethan a factor of 10 and its heat capacity is lower by afactor of 4. Nevertheless, the results of both computa-tional methods are in good agreement, at least in theregion of low vapor mass ow qualities up to about10%. Although the new HNE-DS method is consider-ably simpler to apply, the predictive accuracy is similarto that from the more complicated method of Henryand Fauske.

Flow Rate through Control Valves and OricesTo determine the mass ow rate through a control

valve or orice, the ow correction factor corr has tobe specied. It represents the ratio of the true massow rate through the throttling device in comparison toa frictionless ow through an adiabatic ideal nozzle.

Table 2 is a compilation of the equations for deter-mining the mass ow rate through throttling deviceswith high acceleration of the uid, such as controlvalves and orices, on the basis of the (ideal) mass uxcalculated using the relationships in Table 1. The dis-

Figure 2. Accuracy of reproduction of control valvemass ow rates by the -method of Leung for vapor/liquid ow with low vapor content.


charge coefcient for control valves CV is determinedby the value of Kvs (Eq. 11a). The determination of thedischarge coefcient for orices, that is, the so-calledcontraction coefcientthe area ratio of the vena con-tracta and the inlet pipeis based on the dischargecoefcients for the ow of pure vapor and pure liquid.Idelchik [9] has recommended the following pressureloss coefcient for sharp-edged orices in fully devel-oped turbulent ow (Re 104):

pipe dpipedVC 2

12 1orif,l2

12 (10)This loss coefcient can be recalculated into a dis-charge coefcient for pure liquid ow. The compress-ibility dependency of the discharge coefcient is recal-culated from experiments of Perry [10] performed withpure gas ow. A trigonometric function with the pres-sure ratio between outlet pressure and inlet stagnationpressure describes the wide range of experimental val-

ues quite well (Eq. 13). Results of Eq. 13 are almost asaccurate as calculated values based on the more com-plex model of Benedict [11] for contraction coefcients.The discharge coefcients have been validated withexperimental values of BASF for inlet pressures up to300 bar.

The discharge coefcient for two-phase ow (Eq.15) is specied similar to the procedure specied inSchmidt and Westphal [12, 13], that is, weighting thedischarge coefcients for vapor and liquid ow by themean void fraction in the narrowest ow cross section(Eq. 14). It is not recommended to weight it by the voidfraction at inlet stagnation condition because that maylead to a signicant error. For example, at the expan-sion of boiling water from a tank at a pressure of 10 barinto the atmosphere, the steam content at inlet stagna-tion condition (tank) is zero, whereas the steam con-tent at the narrowest ow cross section is greater than90 Vol %.

In addition to boiling delay the difference in velocity

Table 1. Determination of mass ux for frictionless ow through an adiabatic throttling device (such as nozzle,orice, control valve, safety valve).

State variables and property data pin, Tin, pout, hv,in, cpl,in, vg,in, vl,in, xin

Pressure ratios 0poutpin

p0pin

critpcritpin

pVCpin

(2)

Homogeneous specic volumeof mixture

vin xinvg,in (1 xin)vl,in (3)

Compressibility factor(equilibrium condition, N 1) N1

xinvg,invin

cpl,inTinpin

vinvg,in vl,inhv,in

2(4)

Critical pressure ratio(equilibrium condition, N 1)

crit 0.55 0.217 ln N1 0.046 (ln N1)2 0.004

(ln N1)3

(5)

N1 2 crit2 N1

2 2 N1 1 crit2 2 N1

2 critN1 2 2 N1

2 1 crit 0Compressibility factor

(nonequilibrium condition,N 1)

xinvg,in

vin

cpl,inTinpinvin

vg,in vl,inhv,in 2

N (6a)

N xin cpl,inTinpin vg,in vl,inhv,in2 ln 1crita

(6b)

a 3/5 orices, control valves, short nozzlesa 2/5 safety valves (see Part 2), control valve (high lift)a 0 long nozzles, orice with large area ratios

Critical pressure ratio 2 crit 0.55 0.217 ln 0.046 (ln )

2 0.004 (ln )3 (7)

2 crit2 (2 2)(1 crit)

2 22ln(crit) 22(1 crit) 0

Expansion coefcient critical 0 crit f crit

subcritical 0 crit f 0

ln1 11

1 1 1(8)

Mass ux for isentropicfrictionless ow mid 2pinvin (9)


between the gaseous and liquid phases (slip)the so-called hydrodynamic nonequilibriumshould also betaken into consideration (see the derivation in Appen-dix A). For this purpose Simpson et al. [14] species thetwo-phase multiplier, which is based on the effectivespecic volume by Lottes [15]. The multiplier has beenvalidated with a large volume of measured data forow through orices and valves.

Although, in comparison with thermodynamic non-equilibrium, the effect of hydrodynamic nonequilib-rium is relatively moderate, it can nevertheless give rise

to deviations of about 3050% in mass ow rates. Thiswould lead to a mass ow rate larger than that calcu-lated without taking the nonequilibrium into consider-ation and is thus not conservative. As an example, themeasured discharge capacities through control valvespresented in Figure 2 were recalculated using theHNE-DS model with and without taking the slip cor-rection coefcient into account (see Figures 5 and 6).Even in the case of no slip correction, the deviationsbetween measured and calculated values are distinctlysmaller than those calculated with the original -meth-

Table 2. Determination of discharge capacity through control valves and orices.

Control Valve Orice

Data from Table 1 mid vl,in, vin, , mid, crit, 0Geometric data Kvs dorif (orice diameter), dpipe

Dischargecoefcient

ref 1000 kg/m3,

orif,l1

1 0.7071 d orif2d pipe2 (11b)ACV Kvs ref2pref 11a c 58 38 orif,l (12)

orif,gc orif,l

2

c orif,l2

cosout (13)

0 crit f crit 0 crit f 0 (14)

Vg

Vg Vl 1

vl,in

vin1 1 1Aorif

4d orif

2 orif,g 1 orif,l (15)

Slip correction(two-phasemultiplier, [14])

vinve,in vinvl,in 1 xinvg,invl,in1/6

11 xinvg,invl,in5/6

1

1/2 (16)Mass ow rate Mout,S1 (CV/orif ACV/orif) mid (17)

S safety factor (recommended values 11.3)

Mass ow rate tobe discharged

Mout MCV/orif,S1S MCV/orif,S11.3 (18)

Figure 3. Comparison of the Henry/Fauske model with the HNE-DS method for steam/water ow.


od: the mean logarithmic deviation is just 32%, whereasthe variance of the logarithmic deviation is now only38% (denition of statistic numbers; see Appendix C).This is much better by comparison with the values of126 and 138%, respectively, obtained when the orig-inal -method is used. However, deviations of up to100% are still possible. When the slip correctionfactor is applied, the accuracy of reproduction isagain signicantly lowered (see Figure 6). The devi-ations are distributed almost symmetrically about thediagonal. The mean logarithmic deviation is 5%(that is, the mass ow rate is slightly high). Thus, onaverage, the calculation approach yields slightly con-servative results. At 17% the variance of the logarith-mic deviation is relatively low. With a few exceptionsthe deviations are less than 30%. To allow for a

conservative estimate, a safety margin of 30% (factorof S 1.3) is proposed for the determination of themaximum mass ow rate. In Figure 7 steam/waterorice data from Friedrich [16] are presented. Again,the accuracy of reproduction of the new HNE-DSmethod is excellent.

In contrast to the data of ashing steam/water ow,Figure 8 shows the comparison of calculated and mea-sured data for the ow of nonashing air and water.The mean logarithmic deviation is only 21% for all ofthe 723 data points. Figure 8 underlines the quality ofthe slip correction factor because the boiling delayfactor does not come into play.

Figure 4. Comparison of the Henry/Fauske model, the HNE-DS, and the original method for R12vapor/liquid ow.

Figure 5. Accuracy of reproduction of control valvemass ow rates by means of the HNE-DS methodwithout slip correction in vapor/liquid ow havinglow vapor content.

Figure 6. Accuracy of reproduction of control valvemass ow rates by means of the HNE-DS methodwith slip correction for steam/water ow having lowvapor content.


SUMMARYThe -method, originally developed by Leung, is

extended by a boiling delay coefcient to take accountof the delayed boiling of a liquid (thermodynamic non-equilibrium) in a depressurizing ow process. The ex-tension led to the new HNE-DS method, which is just aseasy to use as that developed by Leung, and requiresphysical properties only at the stagnation condition.Resource-intensive equations of state and derivationsof physical property functions are not needed; nor, asa rule, are iterations necessary. Only in the case of lowcompressibility factors ( 2) is it advisable to deter-mine the critical pressure ratio by means of the implicitequation.

The advantage of the HNE-DS model is that it can beapplied equally to several throttling devices, such ascontrol valves, orices, nozzles, and safety valves (Part1). The overall reproducibility of the HNE-DS modelhas been checked with reference to more than 1300

sets of steam/water experimental data and more than700 measurements with air/water. Additionally, theHNE-DS model was proven for the ow through throt-tling devices with the refrigerant R12, CO2/CO2 vapor,and N2/N2 vapor. In any case, the HNE-DS modelprovides excellent results, even at very low mass owqualities at the inlet of the throttling device.

NOMENCLATUREorif orice discharge coefcient for two-

phase owCV control valve discharge coefcient

for two-phase owCV/orif discharge coefcient for a valve or an

orice for two-phase ow (controlvalve V CV; orice V orice)

ACV/orif seat area of the control valve (ACV/orif ACV) or orice (ACV/orif Aorif)

a boiling delay exponent (see Table 1)cpl,in specic heat capacity of the liquid at

stagnation statedorif orice diameterdpipe pipe diameter

void fraction in the narrowest owcross section of the throttling device

slip correction factorcorr ow correction factor

pressure ratio (ratio of real pressurein the narrowest ow cross sectionand the inlet stagnation pressure)

0 back pressure ratio at the outlet ofthe control valve or orice (ratio ofback pressure and the inlet stagna-tion pressure)

crit critical pressure ratio (ratio of criticalpressure in the narrowest ow crosssection and the inlet stagnation pres-sure)

hv,in latent heat of vaporization at stagna-tion state

KVS liquid discharge factor for fullyopened control valve

Figure 7. Comparison of mass ow rate through orices according to HNE-DS and measured by Friedrich [16].

Figure 8. Accuracy of reproduction of control valvemass ow rates by means of the HNE-DS methodwith slip correction for air/water ow having lowvapor content.


expansion factor or outow functionmid mass ux through an adiabatic fric-

tionless nozzle (mcrit Mid /ACV/orif)MV/orif mass ow rate through a control

valve or orice, which has to be dis-charged from the pressurized system

N boiling delay factorpin stagnation or inlet stagnation pres-

sure (see Figure 1)pout back pressure at the outlet of the

control valve or orice (the pressurethat exists at the outlet of a throttlingdevice)

pcrit critical pressure at choking condi-tions

pVC pressure in the narrowest ow crosssection (uid-dynamic pressure oc-curring in the narrowest ow crosssection of the throttling device)

pref reference pressure difference (0.1MPa)

ref reference density (1000 kg/m3)

Sabs variance of the absolute deviationsSln variance of the logarithmic devia-

tionsS safety factor (recommended value

11.3)Tin inlet stagnation temperature (see Fig-

ure 1)vg,in specic gas volume at inlet stagna-

tion statevin mixture-specic volume at inlet stag-

nation statevl,in specic liquid volume at inlet stagna-

tion state compressibility factor

xin inlet stagnation mass ow quality,that is, the ratio of the gas mass owrate to the total mass ow rate of atwo-phase mixture at stagnation state

X ln mean logarithmic deviationXi,abs absolute deviation between experi-

mental and calculated valueXi,ln logarithmic deviation between ex-

perimental and calculated valueYi,calc calculated value, e.g., mass ow rateYi,exp experimental value, e.g., mass ow

rate

APPENDIX A: DEFINITION OF THE SLIP FACTORThe ideal mass ow rate of a frictionless homoge-

neous ow through an adiabatic nozzle is dened bythe HNE-DS model as

mid 2pinvin (A1)whereby is the expansion coefcient of the uid.Considering the slip between gas and liquid phasethe velocity ratio of the averaged gas and liquid veloc-itieswould lead to an increased mass ow rate be-cause of the increase in density of the ow:

mid,slipmid

vinve,inf mid,slip mid with vinve,in (A2)The slip-corrected specic volume may be developedby a momentum balance (so-called momentum-spe-cic volume) including the slip factor K according to[14]

ve,in xinvg,in K 1 xinvl,inxin 1 xinK K vg,invl,in

5/6

(A3)

Rearranging Eq. A3 leads to a momentum specic vol-ume model based on the specic volume of bothphases at inlet stagnation condition.

ve,in vl,in1 xinvg,invl,in1/6

1 1 xinvg,invl,in

5/6

1

(A4)

APPENDIX B: APPLICATION LIMITS OF THE MODELThe application range of the HNE-DS method is

exactly the same as that for the original -method [5, 6].Special emphasis should be given to the followingassumptions:

Validity of the ClausiusClapeyron equation. It isproven for single-component vapor/liquid sys-tems, but also usable for multicomponent vapor/liquid systems, if the difference of the boilingpoint from each component is less the 100 C.

Vapor phase behaves as an ideal gas. This holds,if the stagnation pressure is less than or equal tohalf of the thermodynamic critical pressure of thecomponent (pred p/pc 0.5) and the temper-ature is less or equal to 0.9 times the criticaltemperature (Tred T/Tc 0.9). Otherwise, a realgas coefcient has to be introduced into themethod.

In general, the HNE-DS method is applicable to everythrottling device in industrial processes. The designengineer needs to assume the contraction rate withinthe throttling device and the relaxation time for heattransfer between both phases. In short throttling de-vices, with large depressurization, an exponent a of 3/5is recommended as a rst estimate, whereas in less-pronounced nonequilibrium ows a lower value forthe exponent is recommended.

APPENDIX C: DEFINITION OF STATISTIC NUMBERSThe average predictive accuracy of the models is

based on the values obtained for the variance of thelogarithmic deviations between the experimental andcalculated values (Table C1). Moreover, the mean log-arithmic deviation characterizing the average under- oroverprediction of the experimental values is depictedfor the sake of completeness. The advantages of using


these parameters are already discussed by Govan [17],Friedel [18], and Diener [4] and showed in the past toallow for a balanced description of the merits of eachcorrelation.

LITERATURE CITED1. ISO 5167, Measurement of uid ow by means of

pressure differential devices inserted in circularcross-section conduits running full, Beuth Verlag,Berlin, 2003.

2. DIN EN 60534-2-1, Stellventile fur die Prozessrege-lungTeil 2-1: Durchusskapazitat; Bemessungs-gleichungen fur Fluide unter Einbaubedingungen(IEC 60534-2-1:1998); Ausgabe 200003.

3. Sheldon, C.W. and Schuder, C.B., Sizing controlvalves for liquidgas mixtures, Instruments andControl Systems, 38 (1965).

4. Diener, R., Berechnung und Messung der Massen-durchsatzcharakteristik von Stellventilen beiZweiphasenstromung (Calculation and measure-ment of the mass ow rate characteristics of controlvalves in two-phase ow), Fortschr.-Ber. Series 7,No. 388, 2000.

5. Leung, J.C., A generalized correlation for one-com-ponent homogeneous equilibrium ashing chokedow, AIChE Journal, 32 (1986), 17431746.

6. Leung, J.C., Similarity between ashing and non-ash-ing two-phase ows, AIChE Journal 36 (1990), 797800.

7. Diener, R. and Schmidt, J., Extended -methodapplicable for low inlet mass ow qualities, 13thMtg. ISO/TC185/WG1, Ludwigshafen, Germany,June 15, 1998.

8. Henry, R.E. and Fauske, H.K., The two-phase crit-ical ow of one-component mixtures in nozzles,orices, and short tubes, Journal of Heat Transfer,93 (1971), 179187.

9. Idelchik, I.E., Handbook of hydraulic resistance,3rd ed., CRC Press, Boca Raton, FL, 1994.

10. Perry, J.A., Critical ow through sharp-edged ori-ces, Transactions of ASME, 71 (1949), 757764.

11. Benedict, R.P., Fundamentals of pipe ow, Wiley,New York, 1980.

12. Schmidt, J. and Westphal, F., PraxisbezogenesVorgehen bei der Auslegung von Sicherheitsven-tilen und deren Abblaseleitungen fur die Durch-stromung mit Dampf/Flussigkeits-GemischenTeil 1 (Practical procedure for the sizing of safetyvalves and their relief lines for the ow of vapor/liquid mixturesPart 1), Chemie Ingenieur Tech-nik, 69 (1997), No. 6.

13. Schmidt, J. and Westphal, F., PraxisbezogenesVorgehen bei der Auslegung von Sicherheitsven-tilen und deren Abblaseleitungen fur die Durch-stromung mit Dampf/Flussigkeits-GemischenTeil 2 (Practical procedure for the sizing of safetyvalves and their relief lines for the ow of vapor/liquid mixturesPart 2), Chemie Ingenieur Tech-nik, 69 (1997), No. 8.

14. Simpson, H.C., Rooney, D.H., and Grattan, E., Twophase ow through gate valves and orice plates,Int Conf on the Physical Modelling of Multi-PhaseFlow, Coventry, UK, April 1920, 1983.

15. Lottes, P., Expansion losses in two-phase ow,Nuclear Science and Engineering 9 (1961), 2631.

16. Friedrich, H., Durchu durch einstuge Dusen beiverschiedenen thermodynamischen Zustanden, En-ergie 10 (1960), 411419.

17. Govan, A.H., A note on statistical methods for com-paring measured and calculated values, HTFSRS767-1 (1988), 315323.

18. Friedel, L., Kriterien fur die Beurteilung der Vorher-sagegenauigkeit von halbempirischen Berech-nungsmodellen (Criteria for the evaluation of thepredictive accuracy of halfempirical models), Che-mie Ingenieur Technik, 53 (1981), No. 1.

19. Diener, R., and Schmidt J. Sizing of throttlingdevice for gas/liquid two-phase ow Part 1:safety valves, Process Safety Progress, 23 (2004)335344.

Table C1. Denition of statistical numbers used to characterize the average predictive accuracy of the models.

Statistical Number Denition

Variance of logarithmic deviations Sln exp i1n X i,ln2n f 1 1 Xi,ln ln Yi,expYi,calcVariance of absolute deviations Sabs i1n X i,abs2n f 1 Xi,abs Yi,exp Yi,calcMean logarithmic deviation X

ln exp1n

i1

n

Xi,ln 1 Xi,ln lnYi,expYi,calc


sizing of throttling devices for two phase flow

Documents