Comparison of Void Fraction Correlations

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<p>ComparisonofvoidfractioncorrelationsfordierentowpatternsinhorizontalandupwardinclinedpipesMelkamuA.Woldesemayat,AfshinJ.Ghajar*SchoolofMechanicalandAerospaceEngineering,OklahomaStateUniversity,Stillwater,OK74078,USAReceived1June2006;receivedinrevisedform13September2006AbstractA comparisonof theperformanceof 68void fractioncorrelationsbased on unbiaseddata set (2845data points)cov-eringwiderangeofparametersthanpreviousassessmentswasmade.Acomprehensiveliteraturesearchwasundertakenfor the available void fraction correlations and experimental void fraction data. After systematically rening the data, theperformance of the correlations in correctly predicting the diverse data sets was evaluated. Comparisons between the cor-relationsweremadeandappropriaterecommendationsdrawn.Theanalysisshowedthatmostofthecorrelationsdevel-oped are very restricted in terms of handling a wide variety of data sets. Based on the observations made, an improved voidfraction correlation which could acceptably handle all data sets regardless of ow patterns and inclination angles was sug-gested. Itwasshownthatthiscorrelationhasthebestpredictivecapabilitythanall thecorrelationsconsideredinthisstudy.2006ElsevierLtd.Allrightsreserved.Keywords: Voidfraction;Two-phase;Comparison;Correlations;Horizontalow;Upwardinclinedow1.IntroductionTwo-phaseow, theoretically, isthesimultaneousowoftwoofanyofthethreediscretephases(solid,liquidorgas)ofanysubstanceorcombinationofsubstances.Practicalapplicationsofagasliquidow,ofa single substance or two dierent components, are commonly encountered in the petroleum, nuclear and pro-cess industries. The two phases may be of dierent components and/or there could be a phase change due toevaporationandcondensationofasingleuid.Intheindustrialapplicationswheretwo-phaseowexists,thetaskofsizingtheequipmentforgathering,pumping,transportingandstoringsuchatwo-phasemixturerequirestheformidabletaskofpredictingthephasedistributioninthesystemfromgivenoperatingconditions. Oncethis phasedistributionis known,theproblemmaybesimpliedinsuchawaythatitcouldbeapproachedandtackledinasimilarfashionanalogoustosingle-phaseow.0301-9322/$-seefrontmatter 2006ElsevierLtd.Allrightsreserved.doi:10.1016/j.ijmultiphaseow.2006.09.004*Correspondingauthor.Tel.:+14057445900;fax:+14057447873.E-mailaddress:ghajar@ceat.okstate.edu(A.J.Ghajar).InternationalJournalofMultiphaseFlow33(2007)347370www.elsevier.com/locate/ijmulowOneofthecriticalunknown parametersinvolvedinpredicting thepressurelossand heat transfer(seeforexample Kim and Ghajar, 2006) in any gasliquid system is the void fraction (e) or liquid holdup (1 e) whichis the volume of space occupied by the gas or liquid, respectively. The seemingly benign issue of determiningthe phase distribution from input conditions in a given pipe is complicated as a result of the slippage betweenthe gas and the liquid phases. Owing to the complexity and lack of understanding of the basic underlying phys-icsoftheproblem, themajorityoftheanalysesweremoreinclinedtowardsempiricalcorrelations.Forthesakeof simplicityboththetheoretical modelsaswell astheempirical correlationsthatarecollectedfromtheliteratureanddiscussedherewouldbesimplyreferredtoasvoidfractioncorrelations.The design engineer is faced with the dicult task of choosing the right correlation among the plethoraof correlations available. The fact that there are plenty of correlations available would not be a concern had itnot been for the fact that most of the correlations have some form of restrictions attached to them. The idea ofcombining dierent correlations which claim to do well for the specic ow characteristics for which they aredeveloped in trying to take care of all the design need would entail its own diculty. For instance, one of themost common restrictions to the correlations, ow pattern dependency, is sometimes a purely subjective judg-ment of theinvestigatorespeciallyforthosepointsonornearowpatternboundaries. Moreover, oneisbound to run into some discontinuity when switching from one correlation to the other for dierent practicaloperating conditions. The other pitfall is that the majority of the void fraction data and also the correlationsare primarily developed for mostly horizontal orientation though some vertical ows and inclined cases havebeeninvestigated.Thepurposeofthisworkistondavoidfractioncorrelationthatcouldacceptablypredictmostoftheexperimental datacollectedforallinclinationanglesaswellasthedierentowpatternswithoutresortingbacktoverycomplexexpressionsthat requireiterativeschemeswithmultiplesolutions. Acomprehensivesearch is made in the open literature to collect allthe available void fractioncorrelations as wellas the mea-suredvoidfractiondatafromvarioussources. Comparisonsofpredictiveperformanceandpromiseofthecapabilityofthecollectedvoidfractioncorrelationswasmadeusingthecollecteddatafromtheliterature.Based on the analysis, the best performing correlations were selected and recommendations were drawn ontheirstrengthsandweaknesses.Adetailedstudyoftheperformanceofseveralofthetopperformingcorre-lations together with the data points which they could not handle well was made. Introducing correction fac-tors whichtake intoaccount the physical nature of the datasets, animprovedversiontoone of thesecorrelations was suggested that could handle data points which failed to be captured by the originalcorrelation.2.CorrelationsSixtyeightcorrelationshavebeenconsideredinthisanalysisforwhichwehavecompleteinformationtocalculate (predict) void fraction based on input parameters in our database, see Table 1. In presenting the cor-relations, dierent categorizing criteria were sought into which the developed correlations might convenientlyfall. Rather than going after physical parameters which most of the time are too narrow, we have followed theworkof Vijayanet al.(2000) toclassifythe correlations intofour categories as shown in Table 1. These are:Slipratiocorrelations: Ageneral expressionforthistypeofcorrelationwasputforwardbyButterworth(1975) as a function of the ratios between wetness fraction (1 x) and x the quality or dryness fraction,where x is dened as the ratio of gas ow rate to the total ow rate; the ratios of densities of the gas and liquidphase (qG and qL); and the ratios of the viscosities of the liquid and gas phase (lL and lG). Most of the cor-relations in this category have a xed constant term in the expression for the void fraction. However, the con-stant term in the expressions of Smith (1969), ASM, and Premoli et al. (1970), APRM, are much more complexas shown in Table 1. For example the constant term APRM in Premoli et al.s correlation is a complex functionofthetotalmassux(G),surfacetension(r),andaccelerationofgravity(g).KeHcorrelations:Thesecorrelationsareaconstantorsomefunctionalmultipleoftheno-sliporhomoge-neousvoidfraction,eH.Banko(1960)namedtheconstant, K,aowparameterwhichwasshowntobeafunctionof thepressure(P). Hughmark(1962) improvedthis parameter introducingtheFroudenumber(Fr), basedonthetwo-phase mixture velocity(UM). Theliquidinput content (k) andReynolds number(Re) were alsoincorporatedintothe owparameter expression. Kowalczewski (IsbinandBiddle, 1979)348 M.A.Woldesemayat,A.J.Ghajar/InternationalJournalofMultiphaseFlow33(2007)347370Table1VoidfractioncorrelationsconsideredforthisstudyAuthor/source VoidfractioncorrelationSlipratiocorrelationsHomogeneous e 1 1xx_ _qGqL_ _lLlG_ _ _ _1LockhartandMartinelli(1949) e 1 0:281xx_ _0:64 qGqL_ _0:36lLlG_ _0:07_ _1Fauske(1961) e 1 1xx_ _qGqL_ _0:5_ _1Fujie(1964) e 1 68947:57=Pe_1_ _1xx_ _qGqL_ _ _ _1Thom(1964) e 1 1xx_ _qGqL_ _0:89lLlG_ _0:18_ _1Zivi(1964) e 1 1xx_ _qGqL_ _0:67_ _1TurnerandWallis(1965) e 1 1xx_ _0:72 qGqL_ _0:4lLlG_ _0:08_ _1Baroczy(1966) e 1 1xx_ _0:74 qGqL_ _0:65lLlG_ _0:13_ _1Smith(1969) e 1 ASM1xx_ _qGqL_ _ _ _1where ASM 0:4 0:6qLqG 0:41xx_ __ _=1 0:41xx_ _ _ _Premolietal.(1970) e 1 APRM1xx_ _qGqL_ _ _ _1where APRM 1 F1y1yF2 yF2_ _, F1 1:578Re0:19LqLqG_ _0:22F2 0:0273WeLRe0:51LqLqG_ _0:08; y 1xx_ _qGqL_ _ _ _1; WeL G2DrqL; ReL GDlLChisholm(1973) e 1 1 x 1 qLqG_ __1xx_ _qGqL_ __ _1Madsen(1975) e 1 1xx_ _b qGqL_ _0:5_ _1where b 1 logqLqG_ _log1xx_ _ _ _1SpeddingandChen(1984) e 1 2:221xx_ _0:65 qGqL_ _0:65_ _1Chen(1986) e 1 0:181xx_ _0:6 qGqL_ _0:33lLlG_ _0:07_ _1HamersmaandHart(1987) e 1 0:261xx_ _0:67 qGqL_ _0:33_ _1PetalazandAziz(1997) e 1 0:735l2LU2SG=r2_ _0:0741xx_ _0:2 qGqL_ _0:126_ _1Zhaoetal.(2000) e 1 e0:125 1xx_ _0:875 qGqL_ _0:875lLlG_ _0:875_ _1KeHcorrelationsArmand(1946) e = 0.833eHChisholm(1983),Armand(1946) e 1eH1eH0:5 eHBanko(1960) e = [0.71 + (1.45 102)P]eHwherePinMPaHughmark(1962) e Re1=6Fr1=8k1=4_ _eHwhere Re GD1e lLelG; Fr U2MgD, k USLUSLUSGNishinoandYamazaki(1963) e 1 1xxqGqL_ _0:5e0:5Kowalczewski(1964)ae eH 0:71 eH0:5Fr0:0451 PPc_ _Guzhovetal.(1967) e 0:81eH1 exp2:2Frp(continuedonnextpage)M.A.Woldesemayat,A.J.Ghajar/InternationalJournalofMultiphaseFlow33(2007)347370 349Table1(continued)Author/source VoidfractioncorrelationLoscherandReinhardt(1973)be eH 0:71 eH0:5Fr0:0451 PPc_ _GreskovichandCooper(1975) e 1 0:671sinh0:263Fr0:5_ _ _ _1eHMoussaliae 1 30:4USG=USL116011:6USG=USL13:2USG=USLKutucugluae eH 1 eH0:5Fr0:21 PPc_ _2Czopetal.(1994) e = 0.285 + 1.097eHArmandMassinace = (0.833 + 0.167x)eHDriftuxcorrelationsFilimonovetal.(1957) e = USG/(UM + UGM)where UGM 0:65 0:0385PDH0:063_ _0:25ForP &lt; 12.7UGM 0:33 0:00133PDH0:063_ _0:25ForP P12.7Dimentievetal.de 1:07j0:8gD0:25HqGqLqG_ _0:23For jgqGqLqG_ _0:56 3:7ore 1:9j0:34gD0:25HqGqLqG_ _0:09For jgqGqLqG_ _0:5&gt; 3:7Wilsonetal.(1961) e 0:56157F0:62086 qGqLqG_ _0:0917LD_ _0:11033ForF &lt; 2e 0:68728F0:41541 qGqLqG_ _0:10737LD_ _0:11033ForF P2Nicklinetal.(1962) e USG=1:2UM 0:35 gDpHughmark(1965) e = USG/1.2UMGregoryandScott(1969) e = USG/1.19UMRouhaniandAxelsson(1970) e x=qGC0xqG1xqL_ _ UGMG_ _1where UGM 1:18qLp_ _grqL qG 0:25RouhaniI C0 = 1 + 0.2(1 x)RouhaniII C0 1 0:21 xgD0:25 qLG_ _0:5Bonnecazeetal.(1971) e USG_1:2UM 0:35 gDp1 qGqL_ _ _ _MattarandGregory(1974) e = USG/(1.3UM + 0.7)KokalandStanislav(1989) e USG_1:2UM 0:345gDqLqGqL_ _ _Dixee USGUSG1 USLUSG_ _qGqL_ _0:1____2:9gr qLqG q2L_ _0:25______1Toshibaee = USG/(1.08UM + 0.45)Sunetal.(1980) e USG0:82 0:18PPc_ _1UM 1:41grqLqG</p> <p>q2L_ _0:25_ _1Jowittee USG_1 0:796 exp 0:061qLqG__ _ _ _UM 0:034qLqG_ 1_ _ _ _Bestionee USG_UM 0:188gDqLqG</p> <p>qG_ _0:5_ _Inoueetal.(1993) e = USG/[(0.00676P + 1.026)UM + (0.0051m+0.0691)(0.0942P2 1.99P + 12.6)]MaierandCoddington(1997) e = USG/[(CMC1P + CMC2)UM + {(v1P2+ v2P + v3)G + (v4P2+ v5P + v6)}]whereCMC1 = 0.00257,CMC2 = 1.0062,v1 = 6.73 107,v2 = 8.81 105,v3 0:00105; v4 0:00563; v5 0:123; v6 0:8GeneralcorrelationsSterman(1956) e 0:2U4SGq0:6G qLqG0:4gr_ _0:2d1D_ _0:25Ifd1D&gt; 1 )d1D 1where d1 260q0:2GqLqG0:7rg_ _0:5Flanigan(1958) e 1 3:063U1:006SG</p> <p>1ChisholmandLaird(1958) e 1 0:8= 1 21Xtt1X2tt_ _ _ _1:75350 M.A.Woldesemayat,A.J.Ghajar/InternationalJournalofMultiphaseFlow33(2007)347370andLoscherandReinhardt(Friedel,1977)introducedapressureratiocorrectionbasedonthermodynamiccriticalpressure(Pc)forsinglecomponentows.Table1(continued)Author/source VoidfractioncorrelationHoogendoorn(1959) e1e 0:6USG1 e1eUSLUSG_ _ _ _0:85Wallis(1969) e 1 X0:8tt</p> <p>0:38NealandBanko(1965) e 1:25USGUM_ _1:88U2SLgD_ _0:2Beggs(1972)ehe0 1 Csin1:8h 13sin31:8h_ Mukherjee(1979) e 1 exp C1 C2sinh C3sin2h C4lLqLr30:25_ _USGqLgr_ _0:25_ _C5USLqLgr_ _0:25_ _C6Gardner(1980)1e1e0:5 1:7USGq0:5LqLqGgr0:25 P0:16y_ _2=3Gardner(1980)2e1e0:5 11:2USGq0:5LqLqGgr0:25 P0:3y_ _2=3Tandonetal.(1985) For50 &lt; ReL &lt; 1125e 1 1:928Re0:315L0:151Xtt2:85X0:476tt_ _ _ _10:9293Re0:63L0:151Xtt2:85X0:476tt_ _ _ _2ForReL &gt; 1125e 1 38Re0:088L0:151Xtt2:85X0:476tt_ _ _ _1 0:0361Re0:176L0:151Xtt2:85X0:476tt_ _ _ _2El-Boheretal.(1988) e 1 0:27e0:69HFr0:177 lLlG_ _0:378ReWeL_ _0:067_ _1where Fr U2SLgD; Re qGUSLDlL; We qLU2SLDrMinamiandBrill(1987) e exp lnZ19:218:7115_ 4:3374_ _where Z1 1:84U0:575SLUSGD0:0277q0:5804Lg0:3696r0:1804_ _0:25P101325_ _0:05l0:1LKawajietal.(1987) e 1:05q0:5LUSLgDqLqG0:5_ _0:5SpeddingandSpence(1989)e1e 0:45 0:08 exp1000:25 U2SLUSGUSL_ _0:65Hartetal.(1989) e 1 USLUSG1 108qLqGRe0:726SL_ _0:5_ _ _ _1HuqandLoth(1992) e 1 21x212x 14x1xqLqG1_ _ _ _0:5Abdulmajeed(1996) e = 1 0.528(USGUSL)0.216121(EL)theoForturbulentow(EL)theo = exp(0.9304919 + 0.5285852R 9.219634 102R2+ 9.02418 104R4)Forlaminarow(EL)theo = exp(1.1 + 0.6788495R 0.1232191 102R2 1.778653 103R3+1.626819 103R4)whereR = lnXand X USGqGlLUSLqLlG_ _LqLU2SLqGU2SGwithL = 0.2forturbulentowandL = 1forlaminarowGomezetal.(2000) e = 1 exp (0.45h + 2.48 106ReM)Grahametal.(2001) e 1 1Ft 1Xtt_ _0:321where Ft G2x31xq2GgD_ _0:5aIsbinandBiddle(1979).bFriedel(1977).cLeung(2005).dKataokaandIshii(1987).eCoddingtonandMacian(2002).M.A.Woldesemayat,A.J.Ghajar/InternationalJournalofMultiphaseFlow33(2007)347370 351Drift uxcorrelations: Zuber and Findlay (1965) developed an expression for predicting void fraction tak-ing into consideration the non uniformity in the ow captured by a distribution parameter (C0), and the driftvelocity (UGM), dened as the dierence between the gas phase velocity (UG) and the two-phase mixture veloc-ity(UM).Thesupercialgasvelocity(USG)hasanequallyimportantpartintheexpression.Allthecorrela-tions that fall into this category dier in their terms for the distribution coecient and drift velocity or both.Filimonovet al. (1957) consideredahydraulicdiameter(DH), Dimentievet al. (KataokaandIshii, 1987)dened dimensionless hydraulic diameter DH and gas ux (jg) as DH DHrgqLqG_ _0:5and jg USGqLqGgr_ _0:25_, respectively while Inoue et al. (1993) added total mass dependency (m) in their drift velocityterms. Wilsonetal. (1961) denedamodiedFroudenumber, F USGqLqGgr_ _0:25, aLaplacelengthpara-meter, L rgqLqG_ _0:5andthe pipe diameter, D, intheir correlation. Similarly, Maier andCoddington(1997)curvettedtheirdatawitheightconstants,CMC1,CMC2,v1tov6whichwereintroducedintheirdriftuxtypecorrelation.Generalvoidfractioncorrelations:Thesearemostlyempiricalinnaturewiththebasicunderlyingphysicalprinciplesincorporatedintothedierent physical parameterswhendevelopingthem. ChisholmandLaird(1958), Wallis (1969), Tandonet al. (1985) andGrahamet al. (2001) usedthe Lockhart andMartinelli(1949)parameter, Xtt lLlG_ _0:11xx__0:9qGqL_ _0:5,intheircorrelations.El-Boheretal.(1988)denedtheFroudenumber (Fr) and Weber number (We) based on the supercial liquid velocity (USL), while Graham et al. (2001)included a Froude rate parameter (Ft). Beggs (1972) introduced an inclination correction factor (C) to predictvoidfractionatanypipeinclinationangle(h)asafunctionofthehorizontalvoidfraction.Similarly,Muk-herjee(1979)curvettedhisdata withsixconstants (C1 toC6) forhorizontal andupward inclinationswhileMinami and Brill (1987) correlated their data with a parameter, Z1, for predicting void fraction in horizontalpipes. Abdulmajeed(1996)denedaparameter, L, withdierentvaluesforlaminarandturbulentowtomodifytheLockhartandMartinelli (1949)parameter. Then, avariabl...</p>