FluidPhmeEquilibria,70 (1991)293-308 ElsevierSciencePublishersB.V.,Amsterdam 293 Prediction of Viscositiesof Kydrocarbon Mixtures Kim Aasberg-Petersen,Kim Knudsen, and Aage Fr edensf und EngineeringResearchCenterWC-SEP, Institut for Xemiteknik, the Technical University of Denmark, DK 2800 Lyngby, Denmark. Keywords: viscosity, hydrocarbons, high pressure, prediction ABSTRACT A new modelfor predictionof the viscositiesof hydrocarbons includingoilandgasmixturesispresented.Themodelis basedontheprincipleofcorrespondingstateswithmethane anddecaneasreferencecomponents.The viscosityof a given componentormixtureisdeterminedfromthereducedvis- cositiesofthereferencecomponentsusingthemolecular weightas an interpolation parameter. Themodelhasbeenusedfor predictionofviscositiesof bothpurecomponentsand mixturesoverlarge pressureranges andforreduced temperaturesabove0.476. Theresultsarein goodagreement with the experimental data. The new modelcom- paresfavorably with earlier published methods, which use only onereference component. Finally,the modelhasbeentestedon datafor6 oil mix- turesfromtheNorthSea.Themeandeviationbasedon34 experimental points was 6.4 %. INTRODUCTION Accurateknowledgeofbothgasandliquidviscositiesover widerangesoftemperatureandpressurearerequiredfor designoftransporteguipmentinthechemicalandpetrochemi- calindustry.Mostmodelswhichhavebeenpublishedinthe literatureareonly suitedforeithertheliquidorthevapor phase.One exceptionis themodeldevelopedbyPedersenetal. (19871, whichisbasedontheprincipleofcorresponding stateswithmethaneasthereferencecomponent.Thismodel usuallyyieldsaccuratepredictionsofreservoirfluidvis- cosities, but it may overestimate the viscosities of pure com- ponentsand of well defined hydrocarbon mixtures. It is the purpose of this paper to describe a new modelfor predictingtheviscositiesofhydrocarbonmixturesbothat ambient conditions and at high temperatures and pressures. The 0378-3812/91/$03.500 1991ElsevierSciencePublishersB.V.Al1rightsreserved 294 objectivewas to develop a methodaccurate both for reservoir fluids and for hydrocarbon mixtures with well defined composi- tion. THE MODEL Many viscositymodels basedon the principle of corresponding stateswithasingle referencecomponent havebeendeveloped (e.g.Pedersenet al.,1987).Reliablepredictionsformix- tureswithspeciessimilartothereferencecomponentare usuallyobtained using this method. However,for systems with componentsconsiderably differentin size andshapesubstan- tialdeviationsmayoccur.Itisin thisworksuggestedto introduceasecondreferencecomponentandcalculatethe reducedviscosity(q,) of a given component(denoted x) using the following expression: lwrx=lnnrl + Mwx-Mw1 ln(nr2/nrl) (1) Mw2 - R% MW is the molecular weight and subscripts 1 and 2 refer to the referencecomponents.Thefunctionalformofeq. (1) was originallysuggestedbyTejaandRice(1981)usingthe acentricfactorinstead ofMW.Thisisnotpossibleinthe presentwork,sinceodecreaseswithincreasingmolecular weightfor heavy oil fractions. The reduced properties are determined from: Er = E/ECE=T,P,n (2) Subscriptsrand c indicatereducedand criticalproperties, respectively.Thefollowingexpressionisusedtoevaluate the critical viscosity(Pedersen et al., 1989): T)C = c.Mwl/2p2/3T-l/6 CC Cis a constant. From eqs.(1) and(2) the followingequa- tions may be derived for determination of the viscosity: cx~l(T1+)2(T2,P2P~clK II= . nc1 nl(T1,P1) *nc21 K=(MWx -ml)/w2-ml)(5) 71 andnz areevaluatedatconditionscorrespondingtothe reduced temperature and pressure of component x: Ti = T'Tci/Tcxi=1,2 (6) pi = P. Pci/Pcxi=3.,2 295 (7) Themodeloutlinedintheaboveisextendedtomixtures usingthefollowing mixingrulesfor T, andP, (MO andGub- bins,1976): % 5 ~i~jNN[ (T=i/Pci)13+ (Tcj/P,j) j3 13IT,wiT,j1 12 Tc== 7 (81 I P ZiZj[(Tci/Pci)1'3+(T,j/Pcj) 1313 ij NN 8.4~~i~ji(Tci/Pci)"~4(Tcj/pcj)1'313[TciTcj 112 (9) NN Z I: ZiZj [ (Tc i /Pc i )13+Tc j /Pc j )l J 313 ij 1 Pedersen et al.(1987) noted that large molecules presentin a mixturemakearelativelygreatercontributiontothevis- cositythandothesmallerones.Thefollowingexpressions are therefore used for the mixture molecular weight: . Mw mix = MWn + 0.00867358 NN ?+I = z ziM+I: ziMwi ii N (+.56079_m;.56079) (10) (111 Mwn =BziMwi(MJi has the unit g/mole) (12) i Theconstantsin eq.(10) are determinedbyregressionusing experimental viscosity data for binary mixturesand oils. Methaneandn-decaneareusedasreferencecomponentsin this work.Decane waschosen,sinceit is thelargestalkane for whichasufficientamountofexperimentalviscositydata is available. Methaneis a natural choice due to its presence in large molefractions in mostreservoirfluid mixtures.The reference component correlations are given in the appendix. RESULTS Allcalculationsreportedin thispaperwereperformedst reducedtemperatures above 0.476 correspondingto the reduced freezing point of methane. Mean deviations between observed 296 and predictedviscositiesfor 19 pure components are givenin tables1 and 2. The results for toluene and n-C,* are Compared to the modelof Pedersenet al.(1987) in figs. 1 and 2. The bestagreementwiththeexperimentaldataisobtainedusing the new model. Experimentalandcalculatedviscositiesarecomparedin table3forthebinarymixturesincludedintheparameter estimationforthemixturemolecularweight.Thecalculated viscositiesfortheCO&-C,,andn-CB/toluene mixturesare showninfigs.3and4togetherwiththeresultsusingthe modelofPedersenetal.Thelowestdeviationsareagain obtained using the approach with two reference components, Intable4predictedviscositiesusing boththenewmodel and the methodof Pedersen et al. are compared to the measured valuesfor 6 oil mixtures.The experimental data(Pedersen et al.,1984) were measuredat pressuresabove the bubblepoints oftheoils.Itisseenthatthe2approachesdifferonly slightlyin accuracy. Inordertotestthepredictivecapabilitiesofthenew model,calculationswereperformedfor mixturesnotincluded in theparameterestimation.Figs.5and6 showtheresults using both modelsfor the n-C&benzenesystem and for an equal weight% mixtureof n-c,,, n-qz,n-C,,, and n-c,,. Thelowest deviationsare obtained using the approach with two reference components.Infig.7itis demonstratedthatthenewmodel mayalsobeusedforpredictingtheviscositiesofnatural gases over large temperature and pressure ranges. 1.20 1.00 ~0.80 ;o.,o 8 g0.40 0.20 0.00 1 ____________^--C------ _____---- 1_, ____--a;--po 0 n nn 0 100 200 300 400 5 PRESSURE(BAR) 0 Fig. 1. Experimentaland predicted viscositiesfor toluene. Exp. data: Stephan and Lucas(1978). 297 TABLE1 Meandeviationsin predicted pure component viscosities.N is the number of experimental points and T and P are the tempera- ture and pressure ranges. ComponentN T(N) P (Bar)Dev(%IRef =2 C3 i-C 5 n-C 5 n-C7 n-C 9 "-=ll n-=12 "'%4 "-%5 "-'16 ""'18 cyclo-c6 157308-4401-700 196175-480l-350 84280-520l-600 70320-520l-500 70300-520l-500 55300-470l-500 52310-500l-500 56313-500l-500 9333-373l-400 10353-408l-400 6353-373l-400 7373-408l-400 49290-500l-500 7.3a 11.1a 11.4a 11.6a 12.8a 5.0a 7.0a 7.8 a,b 13.8b 19.6 b,c 29.7b 38.6 b,c 46.9a a: Stephan and Lucas(1978); b: Ducoulombier et al.(1986); c: Hogenboom et al.(1967). TABLE2 Meandeviationsinpredictedpurecomponentviscositiesfor aromatic components. N,T,P and references: see table 1. ComponentN T(W P (Bar)Dev(%)Ref Benzene51290-500l-40012.9a Toluene53295-500l-40011.0a n-ethyl- benzene45300-500l-40010.2a n-butyl benzene9333-373l-40014.0b n-hexyl benzene9333-373l-40024.8b n-octyl benzene6353-373l-40034.9b 298 2.50 2.00 c z.1.50 5 8 1.00 2 0.50 0.00 I 1I 02004000 PRESSURE(BAR) IO Fig. 2. Experimentaland predicted viscositiesfor n-Cu. Exp. data: Stephan and Lucas(1978). TABLE3 Comparisonbetweenexperimentaland correlatedbinarymixture viscositiesby the new model. MixtureN T(E) P(Bar)Dev(%)ref YC289170-30017-3484.4d c1'c3122311-411l-5518.3e Cl/n-Cl0101292-43198-41710.7f n-C7/n-Cl06293-313l-4008.0b n-Clo/n-Cl624313-353l-4007.2b Toluene/n-C820303-3681.014.4g CG2/n-Cl057311-40367-3465.1h Total4197.4- b: see table 1; d: Diller(1984); e: Giddings et al.(1966): f: Knapstad(1986); g: Ling and Van Winkle(1958); h: Cullicl and Mathis(1984); 0. 10- oooooEXP15%CO1 UaaoPEXP50Rco1 _THISWORK _______PEDERSEN 0.001I8 01002003004 PRESSURE(BAR) 299 Fig.3. Experimentaland predicted viscositiesat T=403X far CC&b-C,, mixtures. Exp. data: Cullick and Mathis(1984). 0.60 oooooEXP _THISWORV ________PEDERSEN 0.00 3105 T3 (t?) 4)O Fig. 4. Experimerital .and predicted viscositiesat atmospheric pressurefor a mixture of 41.8 % toluene and 58.2'% n-Ca. Exp. data: Ling and Van Winkle(1958). 00008$Xp THISWBWK -PEDERSEN________ T(K) 0 Fig.5.Experimental andprediabed $fq&j;y$g@&$!esat saturation point pressure for an equime~ar z&&eFe(?fa=& a?$ benzene. Exp. data; Medani and Hasan (1978). 2.50 2.00 lz Z.1.50 t= iii E:1.00 9 0.50 0.00 Fig. 6. Experimental and predicted viscosities for an cquai weight % mixture of n-C,,, n-C,*,n-C,,, and n-c,,.Exp.data: Ducoulombier et al. (19$(j), 301 TABLE 4 Comparisonbetweenexperimental and predicted viscositiesfor oil mixtures.Exp. data: Pedersen et al. (1984). Deviations(%) OilN T(K) P (Bar)n-range(cP) 1RCThis work 16387.6188-3320.38-0.463.64.8 26366.5275-3900.40-0.473.14.2 35366.0270-3970.32-0.375.412.2 45347.8234-3420.42-0.514.210.8 55342.0263-3451.12-1.239.03.8 67371.0204-4020.30-0.365.34.5 Total 345.16.4 0.02 - oooooEXPT=3tlK onoaoEXP t T=444K _THISWORK 0.00i,,III 01002003004005006 PRESSURE(BAR) 10 Fig.7.Experimentalandpredictedviscositiesofanatural gas. Exp, data: Lee et al.(1966). DISCUSSION Since only the critical temperature, the critical pressure, and the molecular weight are needed for determinationof pure componentviscosities,thenew modelis entirelypredictive. Itisthereforeworthnotingthatpredictionsusingthenew methodaresuperiortothoseobtainedusingthemodelof Pedersen et al. (1987), which uses three parameters correlated 302 topurecomponent data.Thisdemonstratesthat thenew model withtworeferencecomponentsisanimprovementcomparedto using the properties of methane alone. Theagreementbetween experimentaland predictedpurecom- ponentviscositiesis goodin mostcases. Thelargest devia- tions occur for the alkylbenzenes and alkanes with the longest chainlength.Thisindicatesthatthemodelshouldbeused withcautionif significant extrapolation beyond n-C,, is per- formed. From table 1 it is observed that the predicted results deviatesubstantially from experimentfor cyclohexane. This is acommonobservationformodels,whichusealkanesas referencecomponents(see e.g. Pedersen et al., 1984). The agreement between experimental and predicted results for thebinarymixtures givenin table3is in generalsatisfac- tory.Thelargest deviationoccursforthehighlyasymetric mixtureofmethaneanddecane.Verylowdeviationsare obtainedfor the CO,/n-C,, mixturealthoughthe concentration ofCO2 isas high as50 %(see fig.3). Thisindicatesthat themodelisabletopredicttheinfluenceofCO2 uponthe viscosityof reservoir fluids. However,further investigations areneededto verifythis,especiallysincen-C,, isoneof the reference components used in the model. Theverysatisfactory resultsobtainedfor the mixturesof toluene/n-Cs and benzene/n-C6 are truepredictions,since the differencebetweenthemolaraverageandweightaverage molecularweightsisquitesmall(seeeq.10).Theresults show thatthe new modelis able to handle mixtureswithboth paraffinsandaromaticcomponentswithbetteraccuracythan themodelofPedersenetal.(1987).Notealsothecon- siderableimprovementobtainedforthemixtureofn-C,,, n- C,z, n-c,,, and n-C,,. CalculatedviscositiesusingthemodelofPedersenetal. areslightlymoreaccurate thanthoseobtained usingthenew methodfor the oil mixtures. However, since the uncertaintyof theexperimentaldeterminationoftheplusfractionsofthe oilsisnotlessthan10%(Pedersenetal.,1989),the observed difference could well be purely coincidental. Further investigationson alargernumberofoil mixturesareneces- sarytodeterminewhichmodelisthesuperiorforreservoir fluid viscosity predictions. CONCLUSIONS In thispapera new modelfor predictionsof the viscosities of hydrocarbonmixtures has been presented. The modelis based ontheprincipleofcorrespondingstateswithmethaneand decaneasreference components.Themolecularweightis used astheinterpolation parameter.Besidesthecorrelationsfor the referencefluid viscosities the new model contains no par- ameters adjusted to pure component experimental data. Anumberof predictionsof purecomponentand mixturevis- cositieshavebeenperformedoverlargepressurerangesat reducedtemperaturesinexcessof0.476.Formostofthe systemsstudied.theagreementwithexperimentis verygood. However,the model is not recommendedfor mixtures withlarge concentrationsof naphtenic components. 303 The new method has been compared to an earlier method ofthe sametype withonly one referencecomponent. For oil mixtures the two models perform equally Well. For almost all other mix- tures the predictions have been considerably improved. ACKNOWLEDGEMENT Oneoftheauthors(KAaP) isgratefultotheNorwegianoil company Norsk Hydro for financial support for this project. REFERENCES Adachi,Y.,Lu,B.C.-Y.,andSugie,H.:"AFour-Parameter Equation of State". Fluid Phase Eq., 11, 1983, p.29. Chou, G.and Prausnitz, J.M.:*@ APhenomenologicalCorrection toanEquationofStatefor theCriticalRegion".. AIChEJ., 35, 1989, p-1487. Cullick,A.S. and Mathis, M.L.:**Densities and Viscositiesof Mixturesof Carbon Dioxide and n-Decanefrom 310 to 403 K and 7 to 30 MPa". J. Chem. Eng. Data, 29, 1984, p.393. Diller, D.E.: 'Measurement of the Viscosity of Compressed Gas- eousandLiquidMethane+EthaneMixtures".J.Chem.Eng. Data, 29, 1984, p-215. Ducoulombier, D., Zhou, H., Boned, C., Peyrelasse, J., Saint-Guirons,H.,and Xans,P.:"Pressure(l-1000 bars)and Temperature(20-100 lC) Dependenceof the Viscosityof Liquid Hydrocarbons". J. Phys. Chem., 90, 1986, p.1692. Giddings,J.G.,Xao,J.T.F.,andKobayashi,R.:"Development of a High-PressureCapillary-Tube Viscometerand Its Applica- tiontoMethane,Propane,andTheirMixturesintheGaseous and Liquid Regions". J. Chem Phys., 45, 1966, p.578. Hanley, H.J.M., McCarty, R.D., and Haynes, W.M.:"Equation for theViscosityandThermalConductivityCoefficientsof Methane". Cryogenics, 15, 1975, p.413. Hogenboom,D.L.,Webb,W.,andDixon,J.A.:"Viscosityof SeveralLiquidHydrocarbonsasaFunctionofTemperature, Pressure, and Free Volumef'. J. Chem. Phys., 46, 1967, p-2586. Jensen,B.H.: "Densities, Viscosities, and Phase Equilibriain Enhancedoil Recovery".Ph.D-Thesis,Institutfor Kemiteknik, the Technical University of Denmark, 1987. Knapstad,B.:"Viscasityofn-Decane-MethaneatElevated PressureandTemperature**. Report,Institut.forUorganisk Kjemi, Norges Tekniske Hogskole, Trondheim Unlversity,1986. 304 Lee, A.L., Gonzalez, M.H., and Eakin, B.E.: "The Viscosityof Natural Gases". JPT, 1966, p.977. Ling, T.D. and Van Winkle, M.:"Properties of Binary Mixtures asa Function of Composition".Ind. Eng.Chem.DataSer.,3, 1958, p.88. McCarty,R.D.:"AModifiedBenedict-Webb-RubinEquationof State for Methane Using Recent Experimental Data". Cryogenics, 14, 1974, p.276. Medani,M.S. and Hasan, M.A.:"Composition Dependenceof Vis- cosityofn-Hexane-BenzeneBinaryMixturesatElevated Temperatures". J. Chem. Eng. Data, 23, 1978, p-34. MO,K.C.andGubbins,K.E.:llConformal SolutionTheoryfor ViscosityandThermalConductivityofMixtures".MolPhys., 31, 1976, p-825. Pedersen,K.S.,Fredenslund,Aa.,Christensen,P.L.,and Thomassen,P.: l'Viscosity of Crude Oils". Chem. Eng. Sci., 39, 1984, p.1011. Pedersen,K.S. and Fredenslund, Aa.: An Improved Correspondig States Model for the Prediction of Oil and Gas Viscositiesand Thermal Conductivitiesll. Chem. Eng. Sci., 42, 1987, p.182. Pedersen,K.S.,Thomassen,P.,andFredenslund,Aa.:"The Propertiesof Oils andNaturalGases".GulfPublishingInc., 1989. Stephan,K.andLucas,K.:'lViscosity ofDenseFluids".The PurdueResearch Foundation,Lafayette,Indiana. PlenumPress, New York, 1978. Teja,A.S.andRice,P.:"GeneralizedCorrespondingstates MethodfortheViscositiesofLiquidMixtures".Ind.Eng. Chem. Fundam., 20, 1981, p-77. APPENDIX Thepurecomponent viscosityiscalculatedasafunctionof temperatureand density.AccordingtoHanleyet.al.(1975) thisis superior to using T andP asindependent parameters. Insteadofusing the33 parameterMBWR-equationintheform suggestedbyMcCarthy(1974) forthedensitycalculationof methaneand decane a newcorrelationhas been developed.The newcorrelationis computationallymuchfaster,andcontains lessparametersthantheMBWR,whichbecauseoflacking experimentaldataisnecessaryfordecanetoavoidobscure extrapolation. 305 Dsi%j aR GitBidegW&iandf state dS starting point the fol- lowing expreSSiOn is LOUndt obeoptimal f@r evaluation of the indlLdrvolumes ef methane and decane: DC= %,AIS -c-v c,exp (A.1) (A.21 The volumes vArrand vc,ALOare obtained from the cubitiA&S eguatien bf Stat@ .in the form given by Jensen (1987). Sub script c meatis o:fitieaiMlpaf&fW%rCis a Peneloux-type correction. c=RcRTcJije (A-3) dl = (~P/~P)~RT=(A.41 d, is evaluated at saturated liquid conditions for Oi&Ctitical temperatures and atthe critcal isochore for supercritical temperatures as suggester by Chueh and Prausnitz (1989). The fun&ionsd, &t;l88% BE@ betimnfnedwith the aid of the parameters DPlandDP2 W&iiareWEiiUat;rdattheactual temperature and pressure!: DPl = (dp/lr~)~~T, (A.8) DP2 = W2P/Q2)T If DP2 is positive: d2 = DPl d3 * 1 If DP2 is less than zero: (A.61 (A.71 (A.81 (A-9) d3 P-Dp11/2 (A.10) 0 kl t and kr are constants determined byregression. m%.hanethe33-parameterMBWR-equation given For byMcCarthy (1974) w asused as"experimental data". For decane 576 data points in the interval from 294 - 680 K and from 0 - 700 atm. were included in the parameter estimation. In table A.1 the 306 resultingvaluesof ous physical propert Rk,, and k, are given along withvari- &of methane and decane. TABLE A.1 Physicalpropertiesand valuesof R,, k,, andk, for methane and decane. Tf: Freezing point. MethaneDecane T,(K) PC(atm) Mw(g/mole) w pc(g/ml) Tf(W Rc.lOOO kl.1000 k2 190.55617.40 45.3920.18 16.043142.284 0.0080.484 0.16490.2269 90.7243.5 -0.12200.9700 3695.00.1665 0.46690.6376 The viscositycorrelation Thefollowingequationsareusedfor determinationofthe viscosityof the reference components: 0=nk (T) + PI)~(T)+q2(P,T) (A.11) 9 Ik(T) _zGViT(i-4)/3 i=l vl(T) =A+B(C - ln(T/F))2 T)~(T,P)=ii2exp(j,+j4/T) (A.13) (A.14) H2 = -1 + exp(pl(j,+ j3/T312)+ eP12(i, + j6/T+ j,/T2)) (A.15) (A.12) 307 8=(P-P,)/P,(E-16) TheseequationswasoriginallydevelopedbyHanleyetal. (1975) to correlate the methane viscosity. For methane the values of the GV1 - parameters in eq.(4.15) givenby Hanleyetal. were maintained.For decane GV - GV, areequaltozeroandGV, - GV, weretakenfrom theDIPPR- tables(1985). Alltheparametersineqs.(A.l3-A.15)were estimated.In tableA.2dataof the parameterestimationare givenandtableA.3listsallparametersforthepurecom- ponentviscositycorrelation.If the parametersin tableA.3 areusedtheviscosityisobtainedin ALP ifthedensityis givenin g/ems. TABLK A.2 Parameter estimationdata.Nis thenumberofexperimental points. MethaneDecane N T-range(K) P-range(atm.) Dev.(%) 881252 91-523244-477 O-6800-1000 3.13.8 308 TABLEA.3 Valuesofparametersfor thepurecomponentviscositycorrela- tion. MethaneDecane Gvl Gv2 GV3 GV4 GV5 Gv6 GV7 GV8 GV9 A-100 B C F 1 32 33 34 j5 j6 '7 -2090970.2640 2642760.9487 -14728271.0 471640.0 -9491.90.0 1220.00.0 -96.28 0.0 4.274 0.0 -0.08140.0 239460.00248 343.79 81.35 0.4487 5.9583 168.0490.0 -22.768-11.739 30.574 16.092 -14929-18464 1061.5-811.3 -1.47481.9745 290.62898.45 30396 119620