Fluid Phase Equilibria 226 (2004) 109120

Correlations for second and third virial cuan

Key Lab l Engin

ber 200er 2004

Abstract

A modifie coefficprinciple. Co ng modreliable and e presWeber corre

The Webe ce it digases. The n the Orepresent th ich wacorrelations paramacentric fact 2004 Else

Keywords: Virial coefficients; PVT; Nonpolar fluids; Haloalkanes; Polar fluids

1. Introdu

The therculated frodependenction of statuseful exprties of gasecoefficientssent the nothe virial cdirectly tovirial coeffiinteractionficient giveand so on.properties cbined with

CorrespoE-mail a

0378-3812/$doi:10.1016/j

http://www.paper.edu.cnction

modynamic properties of gases may be easily cal-m a knowledge of the virial coefficients and theire on temperature. The density explicit virial equa-e, truncated after the third virial coefficient, is aession for calculating the thermodynamic proper-s for reduced densities less than 0.5. The virialare basic thermodynamic properties that repre-

nideal behavior of real gases. The importance ofoefficients lies in the fact that they are relatedthe interactions between molecules. The secondcient represents the departure from ideality due to

s between pairs of molecules, the third virial coef-s the effects of interactions of molecular triplets,All of the equilibrium gas-phase thermodynamican be calculated from the virial coefficients com-the ideal gas heat capacity. The fourth and higher

nding author. Tel.: +86 10 6279 6318; fax: +86 10 6277 0209.ddress: yyduan@mail.tsinghua.edu.cn (Y.-Y. Duan).

virial coefficients usually contribute little to the densities ofgases and have relatively large uncertainties; therefore, mosteffort has been focused on obtaining the second and thirdvirial coefficients.

Virial coefficients are usually derived from experimen-tal measurements such as (a) PVT measurements, (b) speedof sound measurements, (c) JouleThomson measurements,(d) refractive index and relative permittivity measurementsand (e) vapor pressure and enthalpy of vaporization mea-surements. In recent years, many researches have noted thatmost older PVT data, which historically have been the mainsource of the virial coefficients, were not corrected for physi-cal adsorption effects and, therefore, the results for the secondvirial coefficients are too negative at subcritical temperaturesalong with larger errors in the experimental data for the thirdvirial coefficients.

Many kinds of correlations have been developed to cal-culate the second virial coefficients. However, most sufferfrom adsorption effects; consequently, the correlations mustbe modified using new, high-quality experimental data. Thenew correlation was developed to solve these problems and

see front matter 2004 Elsevier B.V. All rights reserved..fluid.2004.09.023

Long Meng, Yuan-Yuan Doratory of Thermal Science and Power Engineering, Department of Therma

Received 23 June 2004; received in revised form 17 SeptemAvailable online 6 Novemb

d form of the well-known Tsonopoulos correlation for second virialmparisons with the new, high-quality experimental data and existisatisfactory for nonpolar compounds. The results also show that thlations for second virial coefficients of polar fluids.r correlation for the third virial coefficients was also improved sinew correlation gives a satisfactory fit for nonpolar compounds as

e literature data for the third virial coefficients of polar fluids, whfor the second and third virial coefficients need the same additionalor and reduced dipole moment.vier B.V. All rights reserved.oefficients of pure fluids, Lei Lieering, Tsinghua University, Beijing 100084, PR China4; accepted 27 September 2004

ients was developed based on the corresponding-statesels show that the present correlation is more accurate,

ent work is roughly equivalent to the Tsonopoulos and

d not well represent the experimental data of nonpolarrbey and Vera correlation did and can also accuratelys well represented by the Weber correlation. The twoeters, such as the critical temperature, critical pressure,

110 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120

enhance the prediction accuracy and reliability based on thecorresponding-states form, which accounts for nonsphericaleffects and dipole moment effects, while neglecting chemicalassociation

For theuncertaintiis scarce. Ttions usedrections tobest knownlected and evirial coeffias we can.

In 1957,relation forwhich wasand Prausn[5], Orbeypresented vrefitted thepolar and hand so on. Heffects. A lbecome avfor physicaal. [9]. Thilected datalar haloalk

For thelated the tthe StockmHirschfeldethe reducednitz [13] prdoes not alGrande [14additionalmolecularcan only beexceedingticularly simVan Nhu etsecond virivirial coeffiand Reich [correlationity. Weberhaloalkanethe criticalexperiment

2. Second

For thelos [3,4] m

lowing widely used expression:

Br = BPcRTc

= f (0)(Tr)+ f (1)(Tr)+ f (2)(Tr) (1)

e

= 0.14

= 0.06

= aT 6r

q. (1)Tc are.3144the acl spheobtainculesd fromlly, thnpolaBut mof so

s wer

verypreseroposefined

1.013

e 3.335scals.ds arethe sever, a

spuriotric facent, colve t. (2balogeeratiollowi

opoulos

er = opoul

opoulos

ther no

http://www.paper.edu.cns and quantum effects.third virial coefficients, due to the relatively largees in the experimental measurements, reliable dataheoretical calculations with potential energy func-are tedious and, for accurate results, require cor-the assumption of pairwise additivity which are at

only approximately. In this work, we have col-xamined the limited amount of experimental thirdcients now available and correlated them as best

Pitzer and Curl [1] proposed a very successful cor-the second virial coefficients of nonpolar gases,

the basis for several later correlations. OConnellitz [2], Tsonopoulos [3,4], Tarakad and Danner[6], Weber [7], and Hayden and OConnell [8]arious modified PitzerCurl correlations whichcoefficients of the PitzerCurl correlation, addedydrogen bonding terms, applied new parameters,owever, most inevitably suffered from adsorption

arge amount of second virial coefficient data haveailable since 1980, some of which were correctedl adsorption effects, as mentioned in Dymond ets work presents a new correlation using the se-of Dymonds compilation for nonpolar gases, po-anes and other nonhydrogen bonding polar gases.third virial coefficients, Rowlinson [10] calcu-

hird virial coefficients of polar molecules fromayer potential [11], assuming pairwise additivity.r et al. [12] showed that it is a strong function ofdipole moment for polar gases. Chueh and Praus-oposed a corresponding-states correlation which

low calculations in the absence of data. Santis and] proposed a modified correlation which requiresdipole polarizability of a molecule and Bondisvolume [15]. The correlation of Pope et al. [16]

used for compounds with an acentric factor not0.1. In 1983, Orbey and Vera [17] provided a par-

ple and effective correlation for nonpolar gases.al. [18] gave a correlation which was linked to theal coefficients with additional knowledge of thecients of hard convex body molecule. The Bosse19] correlation and the Besher and Lielmezs [20]were not generalized and had limited applicabil-[7] presented a successful correlation for polar

s adapting the model of Van Nhu et al. [18] usingvolume, but the result does not well represent theal data for nonpolar fluids.

virial coefcient

gaseous second virial coefficient data, Tsonopou-odified the PitzerCurl equation to give the fol-

wher

f (0)

f (1)

f (2)

In EandR= 8 issmalwas

moletaineInitiaof no0.75.speeding B(3) isfor rea is pr, d

r =

wher(1D =in papoun(forHowThisacen

mom

To sin Eqthe hrefrigthe fo

aTson

aWeb

Tson

aTson

for o45 0.330Tr

0.1385T 2r

0.0121T 3r

0.000607T 8r (2a)

37+ 0.331T 2r

0.423T 3r

0.008T 8r

(2b)

(3)

, Tr (=T/Tc) is the reduced temperature, Pcthe critical pressure and critical temperature,

72 J mol1 K1 is the universal gas constant, andentric factor. f(0) was obtained by fitting data forrical molecules ( = 0), such as argon. Then f(1)ed from data for larger, nonspherical, nonpolar( = 0), such as butane and octane. f(2) was ob-

data for nonhydrogen bonding polar molecules.ese functions gave good agreement with data for Br gases, especially at reduced temperatures below

ore recent, more accurate values from PVT andund measurements have indicated that the result-e too negative at subcritical temperatures [9]. Eq.important at reduced temperatures less than unitynting the behavior of polar fluids. The parametered to be a function of the reduced dipole momentas follows:

2Pc25T 2c

(4)

is the dipole moment in Debye64 1030 C m), Tc is in kelvins and Pc isThe second virial coefficients of polar com-

more negative than those of nonpolar compoundsame Tr and ). Thus, a should be negative.

usually becomes slightly positive for r < 100.us behavior arises from the fact that the measuredtor is also affected by the presence of the dipoleausing an overcorrection when calculating B.his problem, Weber [7] deleted the last term), which is assumed to cause the problem. Fornated methanes and ethanes of interest in then industry, Tsonopoulos [4] and Weber [7] gaveng expressions:

= 2.188 10114r 7.831 10218r (5)9 1072r (6)

os [3] also gave= 2.140 104r 4.308 10218r (7)nhydrogen bonding polar compounds.

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120 111

Eq. (1) is also used in this work. For nonpolar fluids, Eqs.(2a) and (2b) are redefined here as

f (0) = 0.1

0

f (1) = 0.1

0

Eqs. (8) anwhich main[9]. The firfor Ar, Krf (1) was dmal alkaneSince the abut is 0.0iterative repdeterminedal. [21] andand Truslerment withwith the reet al. [9] shwithin thethermore, oment withof Aziz anable basisand Eubankadsorptionmine f (1).temperaturcubic polynlast terms iterm was afor f (0) are

In Fig. 1ative thanthan thosethe adsorptthe presentcorrelation

The secbutane andpredicted bmental datadata was mDymond eture were ssuffered fro

. Br = f (0)(Tr): (- - -) Tsonopoulos [3]; ( ) PitzerCurl [1]; ()t work.

. Deviations of measured second virial coefficient data for ar-=0.0022) from the present correlation: () Gilgen et al. [21];

strada-Alexanders and Trusler [22]; (- - -) Tsonopoulos [3]; ( )Curl [1]; (---) Weber [7].

es, which led to their unreliable values. Fig. 2 shows thatresent results are in excellent agreement with the exper-tal data for argon over the entire range of temperatures,

the other correlations are obviously too negative atitical temperatures. For example, the values calculated

. Deviations of measured second virial coefficients for methane fromesent correlation: () Roe [25]; () Haendel et al. [26]; () Trusler) Michels et al. [28]; () Trappeniers et al. [29]; () Hou et al.) Douslin et al. [31]; () Holleran [32]; (- - -) Tsonopoulos [3]; ( )Curl [1]; (---) Weber [7].

http://www.paper.edu.cn3356 0.30252Tr

0.15668T 2r

.00724T 3r

0.00022T 8r

(8)

7404 0.15581Tr

+ 0.38183T 2r

.44044T 3r

0.00541T 8r

(9)

d (9) were determined by fitting experimental dataly came from the compilation of Dymond et al.

st term, f (0), was determined by fitting the B dataand Xe, which have nearly zero acentric factors.etermined by fitting the data for the C1C8 nor-s, oxygen, nitrogen, carbon dioxide and benzene.centric factor of argon is not strictly equal to zero,022, the final determination of f (0) was made byetition. To reduce the adsorption effects, f (0) wasby heavily weighting the PVT data of Gilgen etthe speed of sound data of Estrada-Alexanders[22] for argon, since they are in excellent agree-

each other. Comparison of our estimated resultscommended values for argon given by Dymondows that our correlation can represent B of argonexperimental uncertainties for 75 to 1000 K. Fur-ur calculated results also are in excellent agree-

those derived from the potential energy functiond Slaman [23]. Hence, f (0) provides a more reli-for establishing f (1) and f (2). The data of Gupta

[24], which have been corrected for the physicaleffects for butane, were weighted heavily to deter-f (0) and f (1) decrease so steeply with decreasinge for Tr < 0.75 that B could not be represented byomials over the entire temperature range, so the

n Eqs. (8) and (9) were added. In addition, a T1rlso added to f (1). The three different correlationscompared in Fig. 1., the values of B of our correlation are less neg-

those of the Tsonopoulos correlation but lowerof the PitzerCurl correlation for Tr < 0.75, whereion effects are significant. The results show thatcorrelation effectively improves the Tsonopouloswith its too negative values at low temperatures.

ond virial coefficients for argon, methane, ethane,benzene, representative nonpolar compounds,

y the present correlation are compared to experi-in Figs. 26. First, the selection of the literatureade taking account of the recommendation by

t al. [9]. Not all the data available in the litera-hown in the figures, since some older data werem physical adsorption effects or had large uncer-

Fig. 1presen

Fig. 2gon (() EPitzer

taintithe pimenwhilesubcr

Fig. 3the pr[27]; ([30]; (Pitzer

112 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120

Fig. 4. Deviations of measured second virial coefficients for ethane from thepresent correlation: () Estrada-Alexanders and Trusler [33]; () Douslinand Harrison [34]; () Pompe and Spurling [35]; () Jaeschke [36]; (- - -)Tsonopoulos [3]; ( ) PitzerCurl [1]; (---) Weber [7].

Fig. 5. Deviathe present co( ) PitzerC

from the Treduced tem19 cm3 mobe no morenearly sphe

Fig. 6. Deviathe present co[38]; () Bic[41]; (- - -) Ts

satisfactory results, while the other correlations have the sameproblems as with argon: too negative forTr < 1.6 and too posi-tive for Tr < 1.6. The Tsonopoulos correlation gives a positivedeviation otal uncertaFig. 3 alsorately repre

The comwork appeexcept thatimental uncavailable. Hative over tsiderably acorrelationas is the Wedata for buexcellent aEubank [24of the adso

eber0.75.the prcorre

additi

http://www.paper.edu.cnand Wthanwithother

Intions of measured second virial coefficients for butane fromrrelation: () Gupta and Eubank [24]; (- - -) Tsonopoulos [3];url [1]; (---) Weber [7].

sonopoulos correlation are too negative below aperature of 0.85, and this difference increases to

l1 at 81 K, where the uncertainty is expected tothan 5 cm3 mol1. For methane ( = 0.01142), arical molecule, the present work also gives very

tions of measured second virial coefficients for benzene fromrrelation: () Sherwood and Prausnitz [37]; () Waelbroeckh et al. [39]; () Bich et al. [40]; () Connolly and Kandaliconopoulos [3]; ( ) PitzerCurl [1]; (---) Weber [7].

lar fluids, sactually, ju

For polalationship blos, so the fused in thepolar haloaship betwefrom the Stpanded as

B(T ) = b{

Weber [7] isufficient tfluids, thereoptimum vSince the 1have beenthese threea new fit fo

a = 1.15 1.87

In additionand etherspropanone,used to genf 1.25 cm3 mol1 at 673 K, where the experimen-inty is no more than 0.1 cm3 mol1. The data inshow that the present correlation for f (0) accu-sents the B data for methane.parison in Fig. 4 for ethane shows that the present

ars to give the best results over the entire range,it is slightly negative for Tr < 0.8, where the exper-ertainties are also large and only one set of data isowever, the Tsonopoulos correlation is too neg-

he whole range, and the deviation increases con-s Tr decreases from 0.9 to 0.7. The PitzerCurlis also too negative at subcritical temperatures,ber correlation at supercritical temperatures. The

tane in Fig. 5 also show that our correlation is ingreement with the experimental data of Gupta and] which was successfully corrected for 9095%

rption errors, while the errors of the Tsonopouloscorrelations increase at reduced temperatures lessA similar situation is found for benzene in Fig. 6esent correlation giving the best results, while thelations have large deviations at low temperatures.on, the four correlations for other kinds of nonpo-uch as Kr, CO2, O2, N2, etc., were also comparedst not shown with figures in this paper.r fluids, the data analysis gave almost the same re-etween f (2) and Tr as found earlier by Tsonopou-orm of Eq. (3) was used for f (2). New B data wereregression to obtain the optimum value for a of

lkanes, listed in Table 1, to establish the relation-en a andr. The second virial coefficient obtainedockmayer potential [11] for polar molecules is ex-

1(T )+ 2(T )2r + 3(T )4r + 4(T )6r + }(10)

ndicated that the first two terms of Eq. (10) are noto give the optimum value for a of strongly polarfore,4r and6r terms were added to correlate the

alue for a of both weakly and strongly polar fluids.990s, a large amount of new experimental data

reported for R22, R32, and R134a. Accordingly,substances were heavily weighted in developingr haloalkanes:

24 1062r + 7.2238 10114r01 10156r (11)

, since we took ketones, aldehyde, acetonitrile,as nonhydrogen bonding compounds, the data foracetonitrile, acetaldehyde, diethyl ether, etc. wereerate a different dependence of a as a function of

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120 113

Table 1Optimum values and RMSD for some polar fluids

Substance r Optimum a RMSD (cm3 mol1)Present work Tsonopoulos Weber

R11 3.97 0.00614 33.4 35.3 28.2R12 7.16 0.00171 10.2 16.9 8.9R13 10.92 0.00856 4.7 6.0 4.2R22 76.76 0.00469 8.3 11.9 14.2R23 144.77 0.01469 4.9 3.7 3.6R32 180.95 0.02586 7.8 5.4 7.9R40 136.80 0.01053 10.7 10.2 13.0R41 198.08 0.05129 5.7 6.3 8.9R114 7.93 0.00264 23.3 35.4 18.3R115 6.68 0.01404 11.0 13.5 12.2R141b 77.50 0.00132 54.8 37.0 7.7R142b 109.29 0.00452 22.6 15.3 21.4R123 31.84 0.00091 6.5 22.8 76.2R124 49.36 0.00069 10.8 5.7 21.3R125 75.82 0.00069 7.7 4.5 4.5R134a 121.17 0.00740 3.0 6.2 3.7R143a 169.91 0.01703 6.4 14.9 11.4R152a 152.76 0.01661 22.0 9.9 7.3R227ea 12.5 7.3 7.1R236ea 21.6 25.7 28.1Propanone 38.12-Butanone 65.52-Pentanone 163.63-Pentanone 65.9Dimethyl ethe 64.2Diethyl ether 63.8Diisopropyl e 68.7Ethanol 38.0Acetonitrile 175.5

r for othe

a = 3.03 1.24

Here we rodrogen bonFig. 7. Thefor r < 10

Fig. 7. Depenr: () Eq. (optimum valu

http://www.paper.edu.cn43.58 0.0024525.90 0.00078

149.03 0.03410111.29 0.02313

84.42 0.0180392.84 0.01308

r 55.98 0.0175221.80 0.00449

ther 14.26 0.00277191.44 0.04482249.48 0.12116r nonassociated polar compounds:

09 1062r + 9.503 10114r69 10156r (12)ughly ignored the associating effects of nonhy-ding compounds. Both equations are plotted invalues of a for haloalkanes are mostly positive

0 as already mentioned above. Although we have

dence of the polar parameter a on the reduced dipole moment11); (- - -) Eq. (12); () optimum values for haloalkanes; ()es for other nonhydrogen bonding polar fluids.

developedapproachinpolynomiathe dipolesare importacoefficientsexperiment

Fig. 8. Deviapresent correlWeber [44]; (al. [47]; () Hal. [50]; (- - -)35.374.1

142.876.954.244.771.036.7

178.5the most reliable correlations for f (0) and f (1),g the true values, Eq. (3) is perhaps not the bestl form since f (2) is also expected to be effected bydirection and location within the molecule whichnt at low r values. The predicted second virialof R22, R32, R134a and R123 are compared with

al data in Figs. 811. The results show how the

tions of measured second virial coefficients for R22 from theation: () Zander [42]; () Lisal et al. [43]; () Schramm and) Natour et al. [45]; () Schramm et al. [46]; () Demiriz etaendel et al. [48]; () Haworth and Sutton [49]; () Esper etTsonopoulos [3]; (---) Weber [7].

114 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120

Fig. 9. Deviations of measured second virial coefficients for 32 from thepresent correlation: () Qian et al. [51]; () Sato et al. [52]; () Defibaugh etal. [53]; () Sun et al. [54]; () Weber and Goodwin [55]; (- - -) Tsonopoulos[3]; (---) Weber [7].

Fig. 10. Deviations of measured second virial coefficients for R134a fromthe present correlation: () Schramm et al. [46]; () Tillner-Roth and Baehr[56]; () Qian et al. [57]; () Bignell and Dunlop [58]; () Goodwin andMoldover [59]; () Beckermann and Kohler [60]; () Weber [61]; (- - -)Tsonopoulos [3]; (---) Weber [7].

Fig. 11. Deviations of measured second virial coefficients for R123 fromthe present correlation: () Schramm and Weber [44]; () Goodwin andMoldover [62]; () Webers equation [63]; (- - -) Tsonopoulos [3]; (---)Weber [7].

present correlation is obviously better than the previous cor-relations.

The present predicated results agree well with the experi-mental datain Figs. 8rors whiletemperaturfluids. Thethe nonpol

The rooof polar mal. [9], areeral, all thhaloalkaneimental uncthan the estboth this win calculatimolecules.correlationgive significoefficients

3. Third v

Since thcients areinherent dievaluationdata are nooften consexperimentvantage thaand C, is owhen the tPVT data. Wthe critical

C = Ch +where Bh =r and (Tcal volumeeter makesDifferent pwhether thethird virialsubstitutedare multiplthe corresp

Cr = 0.081where Br =Since the care in the

http://www.paper.edu.cneven at low temperatures for the polar molecules11. The Tsonopoulos correlation has negative er-the Weber correlation has positive errors at lowes, which are similar to the results for nonpolarse errors are probably caused by the inaccuracy ofar terms in their correlations.t mean square deviations (RMSD) for 28 kindsolecules, calculated using the data in Dymond etlisted in Table 1. These results show that in gen-ree correlations are roughly equivalent for polars with most errors in line with the estimated exper-ertainties. The deviations are significantly greaterimated uncertainties for only R11 and R141b, andork and Tsonopoulos correlation have large errorsng the data for other nonhydrogen bonding polarSo, these results suggest that although the currentuses modified nonpolar terms, this work does notcantly improved predictions for the second virialof polar fluids due to the weakness of Eq. (3).

irial coefcient

e experimental values of the third virial coeffi-often very much in error and scarce because offficulties in avoiding systematic measurement anderrors, general correlations which directly fit thet appropriate. The Van Nhu et al. [18] model is

idered to be better than most for correlating theal data for the third virial coefficients with the ad-t the uncertainty in the individual coefficients, Bffset to a large extent by their close associationruncated virial equation is used to represent the

eber [7] successfully simplified the model usingvolume as a parameter:

(B Bh)2c(Tr) (13)b, Ch = 0.625b2, b = 0.36vc, c is a function ofr) is a strong functions of Tr, and vc is the criti-. However, the additional critical volume param-the use of the truncated virial equation awkward.ieces of information are required depending onequation is truncated after the second or after the

coefficient. To solve this problem, Bh and Ch areinto Eq. (13) and then both sides of the equation

ied by (Pc/RTc)2. Eq. (13) is then transformed intoonding-states form:

Z2c + (Br 0.36Zc)2c(Tr) (14)B(Pc/RTc), Cr =C(Pc/RTc)2 and Zc = Pcvc/RTc.ritical compressibility factors of most compoundsrange of 0.230.29, particularly 0.250.27 for

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120 115

haloalkanes, we simply let Zc = 0.26 in Eq. (14). Thus, thecritical volume parameter is not needed, and the character-istic information required for calculating C is the same asthose requilation repregases and h

Cr = c0+(Bwhere c0 =

f0(Tr) =(

1

f1(Tr) =(

The first terargon, carbhave zero rfitting the dR152a, R32the reducedment withthe Webershould bemoment, hthe reducedVan Nhu etthe resultstal data ofthe locatioatures withpresent corf1(Tr) needWeber corrcoefficientsas classical

Figs. 12and Vera coC for nonpSince thesmall polarsubstancesbe seen forTr > 1.5.

The resuR32, R134the Weberin Figs. 16lent agreemrepresentsmental uncis not for pof the prestages of the

2. Third virial coefficients of argon: () Holborn and Otto [64]; ()ls et al. [65]; () Whalley et al. [66]; () Kalfoglou and Miller [67];ilgen et al. [21]; () Tanner and Masson [68]; () Onnes and Crom-[9]; () Crain and Sonntag [69]; () Michels et al. [70]; () Estrada-nders and Trusler [71]; () Eq. (15); (- - -) Orbey and Vera [17]; (---)[7].

3. Third virial coefficients of nitrogen: () Roe [25]; () Zhang et al.) Michels et al. [73]; () Holborn and Otto [64]; () Michels et al.) Canfield et al. [75]; () Onnes and Van Urk [9]; () Hoover et al.) Duschek et al. [77]; () Otto et al. [78]; () Eq. (15); (- - -) Orbeyera [17]; (---) Weber [7].

. Third virial coefficients of carbon dioxide: () Patel et al. [79]; ()ovich and Masalov [80]; () Glowka [81]; () Pfefferle et al. [82];olste et al. [83]; () Butcher and Dadson [84]; () Duschek et al.) Katayama et al. [86]; () Michels and Michels [87]; () Holste et]; () Eq. (15); (- - -) Orbey and Vera [17]; (---) Weber [7].

http://www.paper.edu.cnred for calculating B. Therefore, the new corre-senting the third virial coefficients for nonpolaraloalkanes is

r c1)2[f0(Tr)+ 4r f1(Tr)] (15)5.476 103, c1 = 0.0936;

094.051 3334.145T 0.1r

+ 3389.848T 0.2r

1149.580T 0.3r

)

(16)

2.0243 0.85902Tr

) 1010 (17)

m, f0(Tr), was determined by fitting the C data foron dioxide, methane, nitrogen and benzene, whicheduced dipole moments. f1(Tr) was determined byata for some haloalkanes, such as R134a, R143a,and R23. Noted that f0(Tr) is equal to 0.174 whentemperature is equal to unity, which is in agree-

the calculated value at the critical temperature incorrelation. Weber suggested that the polar termcorrelated using the cube of the reduced dipoleowever, we got better results with the quartic of

dipole moment which is similar to the case ofal. [18]. Although f1(Tr) was somewhat arbitrary,

show that it accurately represents the experimen-haloalkanes. Furthermore, as Weber also found,n of the maximum in C shifts to lower temper-

increasing reduced dipole moment. In fact therelation describes this phenomenon so well thatnot be a function of r as was necessary in the

elation. The calculated results for the third virialfor argon, carbon dioxide, nitrogen and methane,nonpolar gases, are shown in Figs. 1215.

15 show that Eq. (15) agrees well with the Orbeyrrelation in giving a satisfactory representation ofolar fluids within the experimental uncertainties.equation of Weber was established to represent

substances, it does not describe these nonpolarwell. For example, visible positive deviations canTr = 1.01.5 with obvious negative deviations for

lts obtained with the present correlation for R23,a, R143a, R152a, R125 are shown together withcorrelation and the Orbey and Vera correlation21. Figs. 1621 show that Eq. (15) is in excel-ent with the Weber correlation which accurately

the data for polar haloalkanes within the experi-ertainties. Since the Orbey and Vera correlationolar fluids, their results are not as good as thoseent work. Thus, Eq. (15) incorporates the advan-other two correlations to represent the third virial

Fig. 1Miche() GmelinAlexaWeber

Fig. 1[72]; ([74]; ([76]; (and V

Fig. 14Vukal() H[85]; (al. [88

116 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120

Fig. 15. Third virial coefficients of methane: () Kleinrahm et al. [89]; ()Michels and Nederbragt [90]; () Trusler et al. [91]; () Dymond et al. [9];() Haendel et al. [26]; () Hoover et al. [92]; () Schamp et al. [93]; ()Douslin et al. [31]; () Roe [25]; () Pope et al. [16]; ( ) Trappeniers etal. [29]; ( ) Holleran [32]; () Eq. (15); (- - -) Orbey and Vera [17]; (---)Weber [7].

Fig. 16. ThirdRasskazov et() Eq. (15);

coefficientsHowever, tgood. The rof r is on

Fig. 17. Thirdand Los [98];[100]; () De(---) Weber

Fig. 18. Third virial coefficients of R134a: () Qian et al. [57]; ()Yokozeki et al. [97]; () Goodwin and Moldover [59]; () Eq. (15); (- --) Orbey and Vera [17]; (---) Weber [7].

http://www.paper.edu.cnvirial coefficients of R23: () Timoshenko et al. [94]; ()al. [95]; () Lange and Stein [96]; () Yokozeki et al. [97];(- - -) Orbey and Vera [17]; (---) Weber [7].

for both nonpolar fluids and polar haloalkanes.he predictions for weakly polar fluids are not asesults for R124 as an example, for which the valuely 49.36, are shown in Fig. 22. The result shows

virial coefficients of R32: () Sato et al. [52]; () Kuznetsov() Zhang et al. [99]; () Yokozeki et al. [97]; () Fu et al.fibaugh et al. [53]; () Eq. (15); (- - -) Orbey and Vera [17];

[7].

Fig. 19. ThirWeber and D() Eq. (15);

that none othe experimsurements orepresentatnation of C

Fig. 20. ThirTamatsu et al[17]; (---) Wd virial coefficients of R143a: () Yokozeki et al. [97]; ()efibaugh [101]; () Nakamura et al. [102]; () Gillis [103];(- - -) Orbey and Vera [17]; (---) Weber [7].

f the three correlations can accurately representental data derived from the speed of sound mea-f Gillis [103]. The errors are probably due to poor

ion of B for weakly polar fluids since the determi-was related to B as expressed by Eq. (15). It is

d virial coefficients of R152a: () Yokozeki et al. [97]; (). [104]; () Gillis [103]; () Eq. (15); (- - -) Orbey and Veraeber [7].

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120 117

Fig. 21. Third virial coefficients of R125: () Yokozeki et al. [97]; () Yeet al. [105]; () Gillis [103]; () Boyes and Weber [106]; () Duarte-Garzaet al. [107]; () Eq. (15); (- - -) Orbey and Vera [17]; (---) Weber [7].

interestingment with tonly one ot

Eq. (15)for nonpolreduced temto a large nua similar prBh(T) andterm of theNhu emphaof Tr

The corpolar and a(Br obtaineingly, althoavailable, Etal data abogive satisfaassociationprediction.

Fig. 22. ThirdWeber [108];

Fig. 23. Third virial coefficients of ammonia: () Adam and Schramm[109]; () Glowka [81]; () Eq. (15); (- - -) Orbey and Vera [17]; (---)Weber [7].

iscussion

e imphey aal gasureme

of aboons, inally aated afit to

onpolases te acc

lationgas R, theting ths for te, progasesost n

http://www.paper.edu.cnto note that the predicted value is in good agree-he only data point of Boyes and Weber [108], withher set of data available.has a positive minimum at Tr equal to about 5

ar molecules, which is certainly wrong for highperatures. However, this error does not contributemerical error inChere. The Weber correlation hasoblem, which we attribute to the assumption thatCh(T) are constants, and neglecting an important

Van Nhu model for high temperatures. As Vansized, his model is probably wrong for the limit

.

relation was also compared to data for stronglyssociating substances. The results for ammoniad using Eq. (10)) are shown in Fig. 23. Surpris-ugh no experimental data for low temperatures areq. (15) is in good agreement with the experimen-ve 300 K. However, all three correlations cannotctory predictions for water, which indicates that aterm should be added to Eq. (15) to improve the

4. D

Ththat tof remeas

racyrelatinorm

truncgoodfor ndecre

Thcorre

polargasesresen

mateethanpolarFor mvirial coefficients of R124: () Gillis [103]; () Boyes and() Eq. (15); (- - -) Orbey and Vera [17]; (---) Weber [7].

Fig. 24. Denswork: () 105() 105.15 Kortance of the virial coefficients lies in the factre very useful for representing the PVT behaviores at low densities. Although good-quality PVTnts in the gas phase have an experimental accu-ut 0.1% in density, corresponding-states type cor-cluding the present one and Webers work, do not

chieve this accuracy. The virial equation of state,fter the third virial coefficient, can provide a veryprecise PVT data for densities up to about 0.5car gases. For polar gases, this maximum densityo about 0.25c or even lower.uracies of the present correlation and the Weberfor the nonpolar gases argon and nitrogen and134a are shown in Figs. 2426. For these three

present correlation is better than Webers in rep-e gas-phase densities over the whole range. Esti-

he density error for the nonpolar gases methane,pane, butane, carbon dioxide and oxygen and theR143a, R125a, R32 and R22 are shown in Table 2.onpolar fluids and polar haloalkanes, the presentity deviations of argon from EOS of Tegeler et al. [110]; present.15 K, () 181.15 K, () 303.15 K, ( ) 453.15 K; Weber [7]:

, () 181.15 K, () 303.15 K, ( ) 453.15 K.

118 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120

Table 2Estimated density uncertainties of nonpolar and polar fluids

Temperature range Density uncertainty of nonpolar fluids (%) Density uncertainty of polar fluids (%)

Tr = 0.60.9Tr = 0.91.2Tr >1.2

Fig. 25. Denswork: () 78() 128.15 K

Fig. 26. DenBaehr [112];( ) 373.15 K,338.15 K, ( )

correlationpredictingis often sligature and w

5. Conclu

A modifithird virialthe second

he noavaila.56nt for

http://www.paper.edu.cnPr < 0.6 0.6 < Pr < 0.7 Pr > 0.7

0.2 0.2 0.3 2.00.1 0.2 1.0

and tbestTr = 0preseity deviations of nitrogen from EOS of Span et al. [111]; present.15 K, () 128.15 K, () 253.15 K; Weber [7]: () 78.15 K,, () 253.15 K.

sity deviations of R134a from EOS of Tillner-Roth andpresent work: () 228.15 K, () 298.15 K, () 338.15 K,() 449.15 K; Weber [7]: () 228.15 K, () 298.15 K, ()373.15 K; ( ) 449.15 K.

is roughly equivalent to the Weber correlation inthe gas-phase nonideality. The Weber predictionhtly better than this work near the critical temper-orse at supercritical temperatures.

sions

ed correlation was developed for the second andcoefficients of nonpolar and polar fluids. For

virial coefficients, the simple spherical term f (0)

The represthat of thehavior (poswork is neeing f (0) ansubstances

For thedependencreasonable

Cr = f (0)(where 2 inonpolar flwhich wasboth have twork which

Additioels for mixis feasible aper binary

List of syma paB, C sef (0), f (1),

tioP prR unT te

Greek lette di

1Dr re

ac

Subscriptsc cr

cal caexp exr rePr < 0.3 0.3 < Pr < 1.0

0.2 0.5 20.5 2

npolar term f (1) very successfully represent theble data within the experimental imprecision foror higher. The results give confidence that thems for f (0) and f (1) are closer to the final form.entation of the polar term, f (2), is not as good asnonpolar terms as evidenced by the spurious be-itive value of a) for weakly polar fluids. Furtherded to improve the model for f (2) without affect-d f (1), and extend the correlation to associated.third virial coefficients, Eq. (13) gives an implicite on through the second virial coefficient, so theform of the empirical correlation should be

Tr)+ 2f (1)(Tr)+ f (2)(Tr, r) (18)s used instead of . Eq. (18) accurately representsuids; however it does not work well for polar fluids,the same as the correlation for B. Perhaps B and Che same unknown mechanism that requires further

should start with the second virial coefficient.nal work is also needed to provide improved mod-tures. The extension of the correlation to mixturesnd would involve only one interaction coefficient

system.

bolsrameter of polar contribution to B, f (2)cond and third virial coefficientsf (2) dimensionless functions of Tr in B correla-nsessureiversal gas constant

mperature

rspole moment, in Debye

= 3.335641030 C mduced dipole moment

entric factor

itical propertylculated resultperimental resultduced property (not including r)

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109120 119

Acknowledgements

This work was supported by the National Natural ScienceFoundationEducation

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Correlations for second and third virial coefficients of pure fluidsIntroductionSecond virial coefficientThird virial coefficientDiscussionConclusionsAcknowledgementsReferences