Correlacion para segundo y tercer coeficiente virial de sustancias puras

Fluid Phase Equilibria 226 (2004) 109–120

Correlations for second and third virial coefficients of pure fluids

Long Meng, Yuan-Yuan Duan∗, Lei Li

Key Laboratory of Thermal Science and Power Engineering, Department of Thermal Engineering, Tsinghua University, Beijing 100084, PR China

Received 23 June 2004; received in revised form 17 September 2004; accepted 27 September 2004Available online 6 November 2004

Abstract

A modified form of the well-known Tsonopoulos correlation for second virial coefficients was developed based on the corresponding-statesprinciple. Comparisons with the new, high-quality experimental data and existing models show that the present correlation is more accurate,reliable and satisfactory for nonpolar compounds. The results also show that the present work is roughly equivalent to the Tsonopoulos andWeber correlations for second virial coefficients of polar fluids.

The Weber correlation for the third virial coefficients was also improved since it did not well represent the experimental data of nonpolargases. The new correlation gives a satisfactory fit for nonpolar compounds as the Orbey and Vera correlation did and can also accuratelyr . The twoc al pressure,a©

K

1

cdtutcstdvifiapb

ofmostthird

en-edents,nts

mea-ed thatainysi-condureshird

cal-ffer

mustThes and

0d

epresent the literature data for the third virial coefficients of polar fluids, which was well represented by the Weber correlationorrelations for the second and third virial coefficients need the same additional parameters, such as the critical temperature, criticcentric factor and reduced dipole moment.2004 Elsevier B.V. All rights reserved.

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

. Introduction

The thermodynamic properties of gases may be easily cal-ulated from a knowledge of the virial coefficients and theirependence on temperature. The density explicit virial equa-

ion of state, truncated after the third virial coefficient, is aseful expression for calculating the thermodynamic proper-

ies of gases for reduced densities less than 0.5. The virialoefficients are basic thermodynamic properties that repre-ent the nonideal behavior of real gases. The importance ofhe virial coefficients lies in the fact that they are relatedirectly to the interactions between molecules. The secondirial coefficient represents the departure from ideality due tonteractions between pairs of molecules, the third virial coef-cient gives the effects of interactions of molecular triplets,nd so on. All of the equilibrium gas-phase thermodynamicroperties can be calculated from the virial coefficients com-ined with the ideal gas heat capacity. The fourth and higher

∗ Corresponding author. Tel.: +86 10 6279 6318; fax: +86 10 6277 0209.E-mail address:[email protected] (Y.-Y. Duan).

virial coefficients usually contribute little to the densitiesgases and have relatively large uncertainties; therefore,effort has been focused on obtaining the second andvirial coefficients.

Virial coefficients are usually derived from experimtal measurements such as (a)PVTmeasurements, (b) speof sound measurements, (c) Joule–Thomson measurem(d) refractive index and relative permittivity measuremeand (e) vapor pressure and enthalpy of vaporizationsurements. In recent years, many researches have notmost olderPVTdata, which historically have been the msource of the virial coefficients, were not corrected for phcal adsorption effects and, therefore, the results for the sevirial coefficients are too negative at subcritical temperatalong with larger errors in the experimental data for the tvirial coefficients.

Many kinds of correlations have been developed toculate the second virial coefficients. However, most sufrom adsorption effects; consequently, the correlationsbe modified using new, high-quality experimental data.new correlation was developed to solve these problem

378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2004.09.023

110 L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109–120

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

For the third virial coefficients, due to the relatively largeuncertainties in the experimental measurements, reliable datais scarce. Theoretical calculations with potential energy func-tions used are tedious and, for accurate results, require cor-rections to the assumption of pairwise additivity which are atbest known only approximately. In this work, we have col-lected and examined the limited amount of experimental thirdvirial coefficients now available and correlated them as bestas we can.

In 1957, Pitzer and Curl[1] proposed a very successful cor-relation for the second virial coefficients of nonpolar gases,which was the basis for several later correlations. O’Connelland Prausnitz[2], Tsonopoulos[3,4], Tarakad and Danner[5], Orbey[6], Weber[7], and Hayden and O’Connell[8]presented various modified Pitzer–Curl correlations whichrefitted the coefficients of the Pitzer–Curl correlation, addedpolar and hydrogen bonding terms, applied new parameters,and so on. However, most inevitably suffered from adsorptioneffects. A large amount of second virial coefficient data havebecome available since 1980, some of which were correctedfor physical adsorption effects, as mentioned in Dymond eta se-l , po-l ases.

l omt y.H oft rausn hichd s andG iresa i’smc r note -t ses.V hes thev sseac abil-i olarht t thee

2

pou-l ol-

lowing widely used expression:

Br = BPc

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

where

f (0) = 0.1445− 0.330

Tr− 0.1385

T 2r

− 0.0121

T 3r

− 0.000607

T 8r

(2a)

f (1) = 0.0637+ 0.331

T 2r

− 0.423

T 3r

− 0.008

T 8r

(2b)

f (2) = a

T 6r

(3)

In Eq. (1), Tr (=T/Tc) is the reduced temperature,Pcand Tc are the critical pressure and critical temperature,R= 8.314472 J mol−1 K−1 is the universal gas constant, andω is the acentric factor.f(0) was obtained by fitting data forsmall spherical molecules (ω = 0), such as argon. Thenf(1)

was obtained from data for larger, nonspherical, nonpolarmolecules (ω �= 0), such as butane and octane.f(2) was ob-tained from data for nonhydrogen bonding polar molecules.Initially, these functions gave good agreement with data forBof nonpolar gases, especially at reduced temperatures below0.75. But more recent, more accurate values fromPVT andspeed of sound measurements have indicated that the result-i( unityf etera entµ

µ

w e(i om-p unds( e.HT sureda ipolem gT mi Fort in thert

a

a

T

a

f

l. [9]. This work presents a new correlation using theected data of Dymond’s compilation for nonpolar gasesar haloalkanes and other nonhydrogen bonding polar g

For the third virial coefficients, Rowlinson[10] calcu-ated the third virial coefficients of polar molecules frhe Stockmayer potential[11], assuming pairwise additivitirschfelder et al.[12] showed that it is a strong function

he reduced dipole moment for polar gases. Chueh and Pitz [13] proposed a corresponding-states correlation woes not allow calculations in the absence of data. Santirande[14] proposed a modified correlation which requdditional dipole polarizability of a molecule and Bondolecular volume[15]. The correlation of Pope et al.[16]

an only be used for compounds with an acentric factoxceeding 0.1. In 1983, Orbey and Vera[17] provided a paricularly simple and effective correlation for nonpolar gaan Nhu et al.[18] gave a correlation which was linked to tecond virial coefficients with additional knowledge ofirial coefficients of hard convex body molecule. The Bond Reich[19] correlation and the Besher and Lielmezs[20]orrelation were not generalized and had limited applicty. Weber [7] presented a successful correlation for paloalkanes adapting the model of Van Nhu et al.[18] using

he critical volume, but the result does not well represenxperimental data for nonpolar fluids.

. Second virial coefficient

For the gaseous second virial coefficient data, Tsonoos [3,4] modified the Pitzer–Curl equation to give the f

-

ng B’s were too negative at subcritical temperatures[9]. Eq.3) is very important at reduced temperatures less thanor representing the behavior of polar fluids. The paramis proposed to be a function of the reduced dipole momr, defined as follows:

r = µ2Pc

1.01325T 2c

(4)

here µ is the dipole moment in Deby1D = 3.33564× 10−30 C m), Tc is in kelvins and Pc isn pascals. The second virial coefficients of polar counds are more negative than those of nonpolar compofor the sameTr and ω). Thus, a should be negativowever,a usually becomes slightly positive forµr < 100.his spurious behavior arises from the fact that the meacentric factor is also affected by the presence of the doment, causing an overcorrection when calculatinB.

o solve this problem, Weber[7] deleted the last tern Eq. (2b), which is assumed to cause the problem.he halogenated methanes and ethanes of interestefrigeration industry, Tsonopoulos[4] and Weber[7] gavehe following expressions:

Tsonopoulos= −2.188× 10−11µ4r − 7.831× 10−21µ8

r (5)

Weber= −9 × 10−7µ2r (6)

sonopoulos[3] also gave

Tsonopoulos= −2.140× 10−4µr − 4.308× 10−21µ8r (7)

or other nonhydrogen bonding polar compounds.

L. Meng et al. / Fluid Phase Equilibria 226 (2004) 109–120 111

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

f (0) = 0.13356− 0.30252

Tr− 0.15668

T 2r

− 0.00724

T 3r

− 0.00022

T 8r

(8)

f (1) = 0.17404− 0.15581

Tr+ 0.38183

T 2r

− 0.44044

T 3r

− 0.00541

T 8r

(9)

Eqs.(8) and (9)were determined by fitting experimental datawhich mainly came from the compilation of Dymond et al.[9]. The first term,f (0), was determined by fitting theB datafor Ar, Kr and Xe, which have nearly zero acentric factors.f (1) was determined by fitting the data for theC1–C8 nor-mal alkanes, oxygen, nitrogen, carbon dioxide and benzene.Since the acentric factor of argon is not strictly equal to zero,but is−0.0022, the final determination off (0) was made byiterative repetition. To reduce the adsorption effects,f (0) wasdetermined by heavily weighting thePVT data of Gilgen etal. [21] and the speed of sound data of Estrada-Alexandersa ee-m sultsw ondew ur-t gree-m tiono i-a aa icala eter-m ingt byc o thelt nsf

g-a wertt thatt ulosc res.

ane,b unds,p peri-m red n byD a-t weres cer-

Fig. 1. Br = f (0)(Tr): (- - -) Tsonopoulos[3]; (· · ·) Pitzer–Curl [1]; (—)present work.

Fig. 2. Deviations of measured second virial coefficient data for ar-gon (ω =−0.0022) from the present correlation: (�) Gilgen et al.[21];(©) Estrada-Alexanders and Trusler[22]; (- - -) Tsonopoulos[3]; (· · ·)Pitzer–Curl[1]; (-·-·-) Weber[7].

tainties, which led to their unreliable values.Fig. 2shows thatthe present results are in excellent agreement with the exper-imental data for argon over the entire range of temperatures,while the other correlations are obviously too negative atsubcritical temperatures. For example, the values calculated

Fig. 3. Deviations of measured second virial coefficients for methane fromthe present correlation: (�) Roe[25]; (�) Haendel et al.[26]; (�) Trusler[27]; (©) Michels et al.[28]; (�) Trappeniers et al.[29]; (�) Hou et al.[30]; (♦) Douslin et al.[31]; (�) Holleran[32]; (- - -) Tsonopoulos[3]; (· · ·)Pitzer–Curl[1]; (-·-·-) Weber[7].

nd Trusler[22] for argon, since they are in excellent agrent with each other. Comparison of our estimated reith the recommended values for argon given by Dymt al.[9] shows that our correlation can representB of argonithin the experimental uncertainties for 75 to 1000 K. F

hermore, our calculated results also are in excellent aent with those derived from the potential energy funcf Aziz and Slaman[23]. Hence,f (0) provides a more relble basis for establishingf (1) and f (2). The data of Guptnd Eubank[24], which have been corrected for the physdsorption effects for butane, were weighted heavily to dine f (1). f (0) andf (1) decrease so steeply with decreas

emperature forTr < 0.75 thatB could not be representedubic polynomials over the entire temperature range, sast terms in Eqs.(8) and (9)were added. In addition, aT−1

rerm was also added tof (1). The three different correlatioor f (0) are compared inFig. 1.

In Fig. 1, the values ofB of our correlation are less netive than those of the Tsonopoulos correlation but lo

han those of the Pitzer–Curl correlation forTr < 0.75, wherehe adsorption effects are significant. The results showhe present correlation effectively improves the Tsonopoorrelation with its too negative values at low temperatu

The second virial coefficients for argon, methane, ethutane and benzene, representative nonpolar comporedicted by the present correlation are compared to exental data inFigs. 2–6. First, the selection of the literatuata was made taking account of the recommendatioymond et al.[9]. Not all the data available in the liter

ure were shown in the figures, since some older datauffered from physical adsorption effects or had large un


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]; (· · ·) Pitzer–Curl[1]; (-·-·-) Weber[7].

Fig. 5. Deviations of measured second virial coefficients for butane fromthe present correlation: (�) Gupta and Eubank[24]; (- - -) Tsonopoulos[3];(· · ·) Pitzer–Curl[1]; (-·-·-) Weber[7].

from the Tsonopoulos correlation are too negative below areduced temperature of 0.85, and this difference increases to19 cm3 mol−1 at 81 K, where the uncertainty is expected tobe no more than 5 cm3 mol−1. For methane (ω = 0.01142), anearly spherical molecule, the present work also gives very

Fig. 6. Deviations of measured second virial coefficients for benzene fromthe present correlation: (�) Sherwood and Prausnitz[37]; (©) Waelbroeck[38]; (�) Bich et al.[39]; (♦) Bich et al.[40]; (�) Connolly and Kandalic[41]; (- - -) Tsonopoulos[3]; (· · ·) Pitzer–Curl[1]; (-·-·-) Weber[7].

satisfactory results, while the other correlations have the sameproblems as with argon: too negative forTr < 1.6 and too posi-tive forTr < 1.6. The Tsonopoulos correlation gives a positivedeviation of 1.25 cm3 mol−1 at 673 K, where the experimen-tal uncertainty is no more than 0.1 cm3 mol−1. The data inFig. 3 also show that the present correlation forf (0) accu-rately represents theB data for methane.

The comparison inFig. 4for ethane shows that the presentwork appears to give the best results over the entire range,except that it is slightly negative forTr < 0.8, where the exper-imental uncertainties are also large and only one set of data isavailable. However, the Tsonopoulos correlation is too neg-ative over the whole range, and the deviation increases con-siderably asTr decreases from 0.9 to 0.7. The Pitzer–Curlcorrelation is also too negative at subcritical temperatures,as is the Weber correlation at supercritical temperatures. Thedata for butane inFig. 5 also show that our correlation is inexcellent agreement with the experimental data of Gupta andEubank[24] which was successfully corrected for 90–95%of the adsorption errors, while the errors of the Tsonopoulosand Weber correlations increase at reduced temperatures lessthan 0.75. A similar situation is found for benzene inFig. 6with the present correlation giving the best results, while theother correlations have large deviations at low temperatures.

In addition, the four correlations for other kinds of nonpo-l eda

e re-l u-lup n-s edf -p

B

W ts rfl theo s.S datah ngly,t opinga

a

I trile,a ta forp wereu f

ar fluids, such as Kr, CO2, O2, N2, etc., were also comparctually, just not shown with figures in this paper.

For polar fluids, the data analysis gave almost the samationship betweenf (2) andTr as found earlier by Tsonopoos, so the form of Eq.(3) was used forf (2). NewB data weresed in the regression to obtain the optimum value fora ofolar haloalkanes, listed inTable 1, to establish the relatiohip betweena andµr. The second virial coefficient obtainrom the Stockmayer potential[11] for polar molecules is exanded as

(T ) = b{ϑ1(T ) + ϑ2(T )µ2r + ϑ3(T )µ4

r + ϑ4(T )µ6r + · · ·}

(10)

eber[7] indicated that the first two terms of Eq.(10)are noufficient to give the optimum value fora of strongly polauids, therefore,µ4

r andµ6r terms were added to correlate

ptimum value fora of both weakly and strongly polar fluidince the 1990s, a large amount of new experimentalave been reported for R22, R32, and R134a. Accordi

hese three substances were heavily weighted in develnew fit for haloalkanes:

= −1.1524× 10−6µ2r + 7.2238× 10−11µ4

r

− 1.8701× 10−15µ6r (11)

n addition, since we took ketones, aldehyde, acetonind ethers as nonhydrogen bonding compounds, the daropanone, acetonitrile, acetaldehyde, diethyl ether, etc.sed to generate a different dependence ofa as a function o


Table 1Optimum values and RMSD for some polar fluids

Substance µr Optimuma RMSD (cm3 mol−1)

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 43.58 0.00245 12.5 7.3 7.1R236ea 25.90 −0.00078 21.6 25.7 28.1Propanone 149.03 −0.03410 38.1 35.32-Butanone 111.29 −0.02313 65.5 74.12-Pentanone 84.42 −0.01803 163.6 142.83-Pentanone 92.84 −0.01308 65.9 76.9Dimethyl ether 55.98 −0.01752 64.2 54.2Diethyl ether 21.80 −0.00449 63.8 44.7Diisopropyl ether 14.26 −0.00277 68.7 71.0Ethanol 191.44 −0.04482 38.0 36.7Acetonitrile 249.48 −0.12116 175.5 178.5

µr for other nonassociated polar compounds:

a = −3.0309× 10−6µ2r + 9.503× 10−11µ4

r

− 1.2469× 10−15µ6r (12)

Here we roughly ignored the associating effects of nonhy-drogen bonding compounds. Both equations are plotted inFig. 7. The values ofa for haloalkanes are mostly positivefor µr < 100 as already mentioned above. Although we have

Fig. 7. Dependence of the polar parametera on the reduced dipole momentµr: (—) Eq.(11); (- - -) Eq.(12); (©) optimum values for haloalkanes; (�)o

developed the most reliable correlations forf (0) and f (1),approaching the true values, Eq.(3) is perhaps not the bestpolynomial form sincef (2) is also expected to be effected bythe dipole’s direction and location within the molecule whichare important at lowµr values. The predicted second virialcoefficients of R22, R32, R134a and R123 are compared withexperimental data inFigs. 8–11. The results show how the

Fig. 8. Deviations of measured second virial coefficients for R22 from thepresent correlation: (�) Zander[42]; (©) Lisal et al.[43]; (�) Schramm andWeber[44]; (�) Natour et al.[45]; (♦) Schramm et al.[46]; (�) Demiriz etal. [47]; (�) Haendel et al.[48]; (�) Haworth and Sutton[49]; (�) Esper eta
ptimum values for other nonhydrogen bonding polar fluids. l. [50]; (- - -) Tsonopoulos[3]; (-·-·-) Weber[7].


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]; (—) Weber’s 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 data even at low temperatures for the polar moleculesin Figs. 8–11. The Tsonopoulos correlation has negative er-rors while the Weber correlation has positive errors at lowtemperatures, which are similar to the results for nonpolarfluids. These errors are probably caused by the inaccuracy ofthe nonpolar terms in their correlations.

The root mean square deviations (RMSD) for 28 kindsof polar molecules, calculated using the data in Dymond etal. [9], are listed inTable 1. These results show that in gen-eral, all three correlations are roughly equivalent for polarhaloalkanes with most errors in line with the estimated exper-imental uncertainties. The deviations are significantly greaterthan the estimated uncertainties for only R11 and R141b, andboth this work and Tsonopoulos correlation have large errorsin calculating the data for other nonhydrogen bonding polarmolecules. So, these results suggest that although the currentcorrelation uses modified nonpolar terms, this work does notgive significantly improved predictions for the second virialcoefficients of polar fluids due to the weakness of Eq.(3).

3. Third virial coefficient

ffi-c se ofi t ande thedo thee ad-v s,a tionw t theP ngt

C

w fµ

c am-e ard.D onw er thets iona ot

C

wS ndsa for

Since the experimental values of the third virial coeients are often very much in error and scarce becaunherent difficulties in avoiding systematic measuremenvaluation errors, general correlations which directly fitata are not appropriate. The Van Nhu et al.[18] model isften considered to be better than most for correlatingxperimental data for the third virial coefficients with theantage that the uncertainty in the individual coefficientBndC, is offset to a large extent by their close associahen the truncated virial equation is used to represenVTdata. Weber[7] successfully simplified the model usi

he critical volume as a parameter:

= Ch + (B − Bh)2δcϑ(Tr) (13)

hereBh = b, Ch = 0.625b2, b = 0.36vc, δc is a function or andϑ(Tr) is a strong functions ofTr, andvc is the criti-al volume. However, the additional critical volume parter makes the use of the truncated virial equation awkwifferent pieces of information are required dependinghether the equation is truncated after the second or aft

hird virial coefficient. To solve this problem,Bh andCh areubstituted into Eq.(13) and then both sides of the equatre multiplied by (Pc/RTc)2. Eq.(13)is then transformed int

he corresponding-states form:

r = 0.081Z2c + (Br − 0.36Zc)

2δcϑ(Tr) (14)

hereBr = B(Pc/RTc), Cr = C(Pc/RTc)2 andZc = Pcvc/RTc.ince the critical compressibility factors of most compoure in the range of 0.23–0.29, particularly 0.25–0.27


haloalkanes, we simply letZc = 0.26 in Eq.(14). Thus, thecritical volume parameter is not needed, and the character-istic information required for calculatingC is the same asthose required for calculatingB. Therefore, the new corre-lation representing the third virial coefficients for nonpolargases and haloalkanes is

Cr = c0+(Br − c1)2[f0(Tr) + µ4r f1(Tr)] (15)

wherec0 = 5.476× 10−3, c1 = 0.0936;

f0(Tr) =(1094.051− 3334.145

T 0.1r

+ 3389.848

T 0.2r

− 1149.580

T 0.3r

)

(16)

f1(Tr) =(

2.0243− 0.85902

Tr

)× 10−10 (17)

The first term,f0(Tr), was determined by fitting theC data forargon, carbon dioxide, methane, nitrogen and benzene, whichhave zero reduced dipole moments.f1(Tr) was determined byfitting the data for some haloalkanes, such as R134a, R143a,R152a, R32 and R23. Noted thatf0(Tr) is equal to 0.174 whenthe reduced temperature is equal to unity, which is in agree-ment with the calculated value at the critical temperature inthe Weber correlation. Weber suggested that the polar terms ipolem ic oft e ofV y,t imen-t und,t -a t thep l thatf heW irialc ane,a

ya n ofC ies.S esents polars canb forT

23,R witht tionil telyr peri-m ationi hoseo n-t irial

Fig. 12. Third virial coefficients of argon: (♦) Holborn and Otto[64]; (�)Michels et al.[65]; (�) Whalley et al.[66]; (�) Kalfoglou and Miller[67];(�) Gilgen et al.[21]; (©) Tanner and Masson[68]; (�) Onnes and Crom-melin [9]; (�) Crain and Sonntag[69]; (�) Michels et al.[70]; (�) Estrada-Alexanders and Trusler[71]; (—) Eq.(15); (- - -) Orbey and Vera[17]; (-·-·-)Weber[7].

Fig. 13. Third virial coefficients of nitrogen: (♦) Roe[25]; (�) Zhang et al.[72]; (�) Michels et al.[73]; (�) Holborn and Otto[64]; (�) Michels et al.[74]; (©) Canfield et al.[75]; (�) Onnes and Van Urk[9]; (�) Hoover et al.[76]; (�) Duschek et al.[77]; (�) Otto et al.[78]; (—) Eq.(15); (- - -) Orbeyand Vera[17]; (-·-·-) Weber[7].

Fig. 14. Third virial coefficients of carbon dioxide: (♦) Patel et al.[79]; (�)Vukalovich and Masalov[80]; (�) Glowka [81]; (�) Pfefferle et al.[82];(�) Holste et al.[83]; (©) Butcher and Dadson[84]; (�) Duschek et al.[85]; (�) Katayama et al.[86]; (�) Michels and Michels[87]; (�) Holste etal. [88]; (—) Eq.(15); (- - -) Orbey and Vera[17]; (-·-·-) Weber[7].

hould be correlated using the cube of the reduced doment, however, we got better results with the quart

he reduced dipole moment which is similar to the casan Nhu et al.[18]. Althoughf1(Tr) was somewhat arbitrar

he results show that it accurately represents the experal data of haloalkanes. Furthermore, as Weber also fohe location of the maximum inC shifts to lower tempertures with increasing reduced dipole moment. In facresent correlation describes this phenomenon so wel

1(Tr) need not be a function ofµr as was necessary in teber correlation. The calculated results for the third v

oefficients for argon, carbon dioxide, nitrogen and meths classical nonpolar gases, are shown inFigs. 12–15.

Figs. 12–15show that Eq.(15)agrees well with the Orbend Vera correlation in giving a satisfactory representatiofor nonpolar fluids within the experimental uncertaint

ince the equation of Weber was established to reprmall polar substances, it does not describe these nonubstances well. For example, visible positive deviationse seen forTr = 1.0–1.5 with obvious negative deviationsr > 1.5.

The results obtained with the present correlation for R32, R134a, R143a, R152a, R125 are shown together

he Weber correlation and the Orbey and Vera correlan Figs. 16–21. Figs. 16–21show that Eq.(15) is in excel-ent agreement with the Weber correlation which accuraepresents the data for polar haloalkanes within the exental uncertainties. Since the Orbey and Vera correl

s not for polar fluids, their results are not as good as tf the present work. Thus, Eq.(15) incorporates the adva

ages of the other two correlations to represent the third v


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. Third virial coefficients of R23: (©) Timoshenko et al.[94]; (�)Rasskazov et al.[95]; (�) Lange and Stein[96]; (♦) Yokozeki et al.[97];(—) Eq.(15); (- - -) Orbey and Vera[17]; (-·-·-) Weber[7].

coefficients for both nonpolar fluids and polar haloalkanes.However, the predictions for weakly polar fluids are not asgood. The results for R124 as an example, for which the valueof µr is only 49.36, are shown inFig. 22. The result shows

Fig. 17. Third virial coefficients of R32: (♦) Sato et al.[52]; (�) Kuznetsovand Los[98]; (©) Zhang et al.[99]; (�) Yokozeki et al.[97]; (�) Fu et al.[100]; (�) Defibaugh et al.[53]; (—) Eq. (15); (- - -) Orbey and Vera[17];(-·-·-) Weber[7].

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].

Fig. 19. Third virial coefficients of R143a: (♦) Yokozeki et al.[97]; (©)Weber and Defibaugh[101]; (�) Nakamura et al.[102]; (�) Gillis [103];(—) Eq.(15); (- - -) Orbey and Vera[17]; (-·-·-) Weber[7].

that none of the three correlations can accurately representthe experimental data derived from the speed of sound mea-surements of Gillis[103]. The errors are probably due to poorrepresentation ofB for weakly polar fluids since the determi-nation ofC was related toB as expressed by Eq.(15). It is

Fig. 20. Third virial coefficients of R152a: (�) Yokozeki et al.[97]; (©)Tamatsu et al.[104]; (�) Gillis [103]; (—) Eq. (15); (- - -) Orbey and Vera[17]; (-·-·-) Weber[7].


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].

interesting to note that the predicted value is in good agree-ment with the only data point of Boyes and Weber[108], withonly one other set of data available.

Eq. (15) has a positive minimum atTr equal to about 5for nonpolar molecules, which is certainly wrong for highreduced temperatures. However, this error does not contributeto a large numerical error inChere. The Weber correlation hasa similar problem, which we attribute to the assumption thatBh(T) andCh(T) are constants, and neglecting an importantterm of the Van Nhu model for high temperatures. As VanNhu emphasized, his model is probably wrong for the limitof Tr → ∞.

The correlation was also compared to data for stronglypolar and associating substances. The results for ammonia(Br obtained using Eq.(10)) are shown inFig. 23. Surpris-ingly, although no experimental data for low temperatures areavailable, Eq.(15) is in good agreement with the experimen-tal data above 300 K. However, all three correlations cannotgive satisfactory predictions for water, which indicates that aassociation term should be added to Eq.(15) to improve theprediction.

FW

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

4. Discussion

The importance of the virial coefficients lies in the factthat they are very useful for representing thePVT behaviorof real gases at low densities. Although good-qualityPVTmeasurements in the gas phase have an experimental accu-racy of about 0.1% in density, corresponding-states type cor-relations, including the present one and Weber’s work, do notnormally achieve this accuracy. The virial equation of state,truncated after the third virial coefficient, can provide a verygood fit to precisePVT data for densities up to about 0.5ρcfor nonpolar gases. For polar gases, this maximum densitydecreases to about 0.25ρc or even lower.

The accuracies of the present correlation and the Webercorrelation for the nonpolar gases argon and nitrogen andpolar gas R134a are shown inFigs. 24–26. For these threegases, the present correlation is better than Weber’s in rep-resenting the gas-phase densities over the whole range. Esti-mates for the density error for the nonpolar gases methane,ethane, propane, butane, carbon dioxide and oxygen and thepolar gases R143a, R125a, R32 and R22 are shown inTable 2.For most nonpolar fluids and polar haloalkanes, the present

F tw(

ig. 22. Third virial coefficients of R124: (�) Gillis [103]; (�) Boyes andeber[108]; (—) Eq.(15); (- - -) Orbey and Vera[17]; (-·-·-) Weber[7].

ig. 24. Density deviations of argon from EOS of Tegeler et al.[110]; presenork: (�) 105.15 K, (�) 181.15 K, (�) 303.15 K, ( ) 453.15 K; Weber[7]:�) 105.15 K, (©) 181.15 K, (�) 303.15 K, ( ) 453.15 K.


Table 2Estimated density uncertainties of nonpolar and polar fluids

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

Pr < 0.6 0.6 <Pr < 0.7 Pr > 0.7 Pr < 0.3 0.3 <Pr < 1.0

Tr = 0.6–0.9 0.2 – – 0.2 –Tr = 0.9–1.2 0.2 0.3 2.0 0.5 2Tr >1.2 0.1 0.2 1.0 0.5 2

Fig. 25. Density deviations of nitrogen from EOS of Span et al.[111]; presentwork: (�) 78.15 K, (�) 128.15 K, (�) 253.15 K; Weber[7]: (�) 78.15 K,(©) 128.15 K, (�) 253.15 K.

Fig. 26. Density deviations of R134a from EOS of Tillner-Roth andBaehr [112]; present work: (�) 228.15 K, (�) 298.15 K, (�) 338.15 K,( ) 373.15 K, (�) 449.15 K; Weber[7]: (�) 228.15 K, (©) 298.15 K, (�)338.15 K, ( ) 373.15 K; ( ) 449.15 K.

correlation is roughly equivalent to the Weber correlation inpredicting the gas-phase nonideality. The Weber predictionis often slightly better than this work near the critical temper-ature and worse at supercritical temperatures.

5. Conclusions

A modified correlation was developed for the second andthird virial coefficients of nonpolar and polar fluids. Forthe second virial coefficients, the simple spherical termf (0)

and the nonpolar termf (1) very successfully represent thebest available data within the experimental imprecision forTr = 0.5–6 or higher. The results give confidence that thepresent forms forf (0) andf (1) are closer to the “final” form.The representation of the polar term,f (2), is not as good asthat of the nonpolar terms as evidenced by the spurious be-havior (positive value ofa) for weakly polar fluids. Furtherwork is needed to improve the model forf (2) without affect-ing f (0) and f (1), and extend the correlation to associatedsubstances.

For the third virial coefficients, Eq.(13)gives an implicitdependence onω through the second virial coefficient, so thereasonable form of the empirical correlation should be

Cr = f (0)(Tr) + ω2f (1)(Tr) + f (2)(Tr, µr) (18)

whereω2 is used instead ofω. Eq.(18)accurately representsnonpolar fluids; however it does not work well for polar fluids,which was the same as the correlation forB. PerhapsB andCboth have the same unknown mechanism that requires furtherwork which should start with the second virial coefficient.

Additional work is also needed to provide improved mod-els for mixtures. The extension of the correlation to mixturesis feasible and would involve only one interaction coefficientper binary system.

LaBf

PRT

Gµ

µ

ω

Sccer

ist of symbolsparameter of polar contribution toB, f (2)

, C second and third virial coefficients(0), f (1), f (2) dimensionless functions ofTr in B correla-

tionspressureuniversal gas constanttemperature

reek lettersdipole moment, in Debye1D = 3.33564×10−30 C m

r reduced dipole momentacentric factor

ubscriptscritical property

al calculated resultxp experimental result

reduced property (not includingµr)


Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (No. 50225622) and the Fok Ying TungEducation Foundation (No. 81051).

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Correlacion para segundo y tercer coeficiente virial de sustancias puras