A Generalized Correlation for the Prediction of Phase Behaviour in Supercritical Systems

M. A . TREBBLE and P. M. SIGMUND

Department of Chemical and Petroleum Engineering, The University of Calgary, 2500 University Dr. N. W., Calgary, Alberta, Canada. T2N IN4

A generalized two parameter model is presented which uses molecular properties of pure fluids to predict interaction parameters in the attractive term of the Peng-Robinson equation of state. The predictive method is based on a consider- ation of London (dispersive) forces and includes attractive forces between polar and non-polar molecules that result from induction. Polar-polar effects and quantum forces are omitted. A large body of experimental vapor-liquid equilibrium data measured at near and supercritical conditions (that excludes quantum components and polar-polar systems) was used to calibrate the generalized model. Overall bubble point pressure deviations calculated using the proposed gener- alization were 5.66% for 3240 data points which compares to average deviations of 3.27% obtained by using regressed binary interaction parameters. Average vapor mole fraction deviations were just under 0.0 1 using both the generalized and the regressed interaction parameters. The sensitivity of predicted phase envelopes to dispersive and inductive term in the generalized correlation is shown graphically for several systems. Comparisons are also made to another recent interaction parameter generalization presented by Nishiumi et al. (1988).

On prksente un modele gCnCralise a deux parametres qui utilise les propriCtCs molCculaires de fluides purs pour pre- dire les parametres dinteraction dans le terme dattraction de ICquation dCtat de Peng-Robinson. La mkthode prCdic- tive est basee sur une considkration des forces (dispersives) de London et inclut les forces attractives entre les molecules polaires et non polaires qui rksultent de Iinduction. Les effets polaires-polaires et les forces quantiques sont omis. On a utilise un vaste ensemble de donnCes expkrimentales dCquilibre liquide-vapeur mesurees pres des conditions super- critiques et aux conditions supercritiques (excluant les composantes quantiques et les systemes polaires-polaires) pour calibrer le modkle gknCralisC. Les Ccarts globaux de pression du point de bullL calcules a Iaide de la generalisation proposee, sont de 5,66% pour 3243 points de donnees, ce qui se compare aux Ccarts moyens de 3,27% obtenus par regression de paramktres dinteraction binaires. Les ecarts moyens de la fraction molaire de la vapeur sont juste en-dessous de 0,Ol a la fois avec les parametres dinteraction ginkralises et les parametres obtenus par regression. La sensibilit6 des enveloppes de la phase dispersCe aux termes de dispersion et dinduction dans la correlation gknkralisee est decrite sous forme graphique pour plusieurs systemes. Des comparaisons sont tgalement Ctablies avec une genCralisation des parametres dinteraction presentee rCcemment par Nishiumi et al. (1 988).

Keywords: supercritical systems, phase behaviour, binary interaction parameters, dispersive forces, induction.

he need for binary interaction parameters in equation T of state modelling is well documented (Soave, 1979; Katz and Firoozabadi, 1978). However, perhaps because of the desirability of individual calibration of binary pairs (that unquestionably gives better fits of the data), generalized methods of estimation are still in the early stages of develop- ment. Generalized approaches to the estimation of interac- tion coefficients have been proposed for the Redlich-Kwong equation of state by Chueh and Prausnitz as early as 1967. Their correlation was stated to apply to n-alkane mixtures and was expressed in terms of a ratio of critical volumes. Later, Katz and Firoozabadi (1978) developed a generalized correlation for the prediction of methane-n-alkane interac- tion coefficients applicable to the Peng-Robinson equation of state. The correlation variable was the density of the heavy component at 15.5C. Kato et al. (1981) presented a corre- lation specifically for carbon dioxide-n-alkane binaries as a function of temperature and the n-alkane acentric factor. A new procedure for predicting binary interaction parameters for the Peng-Robinson equation of state (EOS) has recently been presented by Nishiumi et al. (1988). Their procedure utilizes critical volume ratios and acentric factor differences. The model contains five adjustable constants which are used in a variety of combinations to fit specific pairs of compound families,

In this work we are interested in developing an extrapola- tive model with ties to intermolecular forces. Since equa- tions of state are a convenient means of representing observable P-V-T behaviour, it is of interest to examine the relationship between the parameters used in these equations

and those used in describing intermolecular forces. The study is limited to the practical problem of considering these forces of attraction in terms of deviations from classical EOS mixing rules which presume pairwise additivity of molecular inter- actions. We begin with the Peng-Robinson equation of state and the usual mixing prescriptions:

a . . . . . . . . . (1) RT p = - -

(V - b) V ( V + b) + b(v - b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a = ccx; xj a;j (2)

h = EX; b; ( 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ajj = ( a ; a j p 5 (1 - 6,) . . . . . . . . . . . . . . . . . . . (4)

As summarized by Kwak and Mansoori (1989, the work of Leland and others (1968a, b, 1969) gives a statistical mechanical argumentation for the relationship between a and b in the van der Waals EOS, and to both the depth of the minimum in an intermolecular potential (c), and the separation distance (u) at which the potential is zero, as follows:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a (x (T3 (5)

bcx ( T ~ (6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

For the Redlich-Kwong (1949) EOS, Kwak and Mansoori show the variation of a with E and (T as:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a 0: ~ . ~ 0 ~ (7)

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, DECEMBER, 1990 1033

In order to relate the intermolecular potential functions to the equation of state parameters, we refer to the work of Hudson and McCoubrey (1 959) who derived an expression that related the well depth (E) to a sixth power potential of attraction via the London theory of dispersive forces given by :

Introducing a geometric mean mixing rule on E and an arith- metic mean mixing rule on 1.7, they suggested the following relationships:

(10) 'Ti] = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u, + 'TL

2

Now if we take aij 0: E :; a;, a' 0: Vci , and u t cc VCij , then Equations (4), (7), and (9) can be combined to give:

where:

The relationship between the interaction parameter 6, and the critical volumes of the molecules is therefore seen to be analogous to that described by Chueh and Prausnitz (1967) for representing interactions in a binary mixture:

where I9 represents an adjustable parameter to account for variability in the exponent of the attractive potential.

To include forces of induction between non-polar and polar molecules we include additional potentials as described by Prausnitz et al. (1986):

2 -3 4)"' aJ (aIpJ + 4 = - I' 2 (I l + 4 ) r; r;

. . . . . . . . . . . . . . . . . . . . . (14)

where the first term is the London potential and the last two terms describe attractive potentials resulting from induction. Assuming superposition of the potential terms and a propor- tionality between polarizability (a) and critical volume ( Vc), a semi-empirical expression for the interaction parameter is given by:

- 3(a,QJ2 + Q~Q, ' ) 2 r t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (15)

Figure 1 - Polarizibility versus critical volume (cc denotes cm').

Support for the proportionality between the polarizability and the critical volume is shown by Figure 1 which is a plot of polarizability versus critical volume for some normal hydrocarbons up to dodecane.

The resulting generalization is therefore a two constant (0, 7) description of a binary interaction parameter limited to pairs of non-polar - non-polar and non-polar - polar molecules. Potential functions describing forces between polar molecules are also summarized by Prausnitz et al. (1986) but were not included in this work. It is likely that a single binary interaction parameter will prove insufficient to match phase behaviour in polar-polar systems in any case.

Determination of model parameters

In order to calibrate values of I9 and 7 in Equation (15), it was necessary to obtain and screen a large vapor-liquid equilibrium data base. The Gas Processor's Association data base developed by Daubert (1986) was selected for use in this application and some minor screening was performed as described by Trebble (1990). Most of the data contained in the data base are at conditions which correspond to a super- critical regime for the lighter constituent. Regressed values for binary interaction parameters for three cubic EOS, including the Peng-Robinson equation, were also given by Trebble (1990). Excluding quantum components and binaries in which both molecules had dipole or quadrupole moments, reduced the number of binary systems to 72 and the number of pure component constituents to 28. The 28 pure compo- nents are listed with their associated physical properties in Table 1. Critical properties and dipole moments were obtained from Reid et al. (1987) while quadrupole moments were obtained from Prausnitz et al. (1986).

Optimum values of I9 and 9 were determined by minimizing bubble point pressure deviations (00 and deviations in vapor phase composition (Or) resulting from fixed T, x bubble point calculations. More specifically the objective function ( E l ) was defined by the following equations:

Bubble Point at Fixed T, x ; i = n

El = [of" + DY;] . . . . . . . . . . . . . . . . . . (16) i = l

1034 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, DECEMBER. 1990

TABLE 1 Pure Component Properties

Coniponent

nitrogen hydrogen sulfide carbon dioxide methane ethane propane 1-butene n-butane isobutane n-pentane 2-methyl butane neo-pen tane cyclohexane n-hrxane toluene methylcyclohexane n-heptane m-xylene p-x ylene ethy lbenzene ethy lcyclohexane n-octane n-propy lbenzene meutylene propy lcyclohexane n-nonane n-decane hexadecane

N2 H2S c 0 2 c1 c 2 c 3 C4ENE NC4 IC4 NC5 IC5 NEOC5 CYC6 C6 TOL ClCYC6 c 7 MXYL PXYL C2BEN C2CYC6 C8 C3BEN MESI C3CYC6 c 9 c10 C16

126.2 373.2 304.1 190.4 305.4 369.8 419.6 425.2 408.2 469.7 460.4 433.8 553.5 507.5 591.8 572.2 540.3 617.1 616.2 617.2 609.0 568.8 638.2 637.3 639.0 594.6 617.7 722.0

3.39 8.94 7.38 4.60 4.88 4.25 4.02 3.80 3.65 3.37 3.39 3.20 4.07 3.01 4.10 3.47 2.74 3.54 3.51 3.60 3.00 2.49 3.20 3.13 2.80 2.29 2.12 1.41

89.8 0.0372 98.6 0.0971 93.9 0.2390 99.2 0.0109 148.3 0.0979 203.0 0.1518 240.0 0.1905 255 .O 0.1994 263.0 0.1847 304.0 0.2522 306.0 0.2286 303 .O 0.1956 308.0 0.2089 370.0 0.2990 316.0 0.2641 368.0 0.2359 432.0 0.3494 376.0 0.3261 379.0 0.3222 374.0 0.3027 450.0 0.2401 492.0 0.3977 440.0 0.3454 429.0 0.3999 450.0 0.2615 548.0 0.4488 603.0 0.4902 960.0 0.7338

-0.474

- 1.360 0.285

-0.206

0.095

0.032

0.032

0.095

0.126

0.095 0.032 0.126

0.032

0 = 0.20

0 regressed 6..

I I

F vci vcj / vcij

Figure 2 - Generalization for non-polar pairs.

where:

DP = 100.0. I P, - Pculc/ l P , . . . . . . . . . . . . . (17) DY = 100.0 * I yrxp - yculc 1 . . . . . . . . . . . . . . . . . . (18)

An optimum 19 value of 0.2 is shown in Figure 2 along with independently regressed binary interaction parameters for all of the non-polar pairs contained in the data base. The value of 7 determined from the minimization was 9OpJ- mol- where di ole and quadrupole values are expressed as 10iRpJiPcm32 and 1026pJ12 cm5I2 respectively.

Discussion of results

Table 2 presents a summary of the regression analysis for all of the binary systems evaluated. Values of pressure and composition deviations were evaluated with both zero (ZERO) interaction parameters and with the regressed (REG) interaction parameters presented by Trebble (1990). Table 3 includes corresponding results comparing the generaliza- tion of Nishiumi, Arai and Takeuchi (NAI) with Equation (15), hereafter referred to as the TS generalization. Tables 2 and 3 have been filed with the Depository for Unpublished Data, CISTI, National Research Council of Canada, Ottawa, Ontario, K1A OS2, and can be obtained from the depository or from the authors upon request. A summary of these tables shows average errors in bubble point pressures (DP) and vapor mole fractions (DY) for the entire data base as follows:

ZERO 6, DP = 10.70 DY = 1.57 PTS = 3199 REG 6ij DP = 3.27 DY = 0.89 PTS = 3270 NAT 6ij DP = 4.43 DY = 0.96 PTS = 3269 TS 6 , DP = 5.66 DY = 0.99 PTS = 3240

It is evident from the overall deviations that the NATgener- alization yields lower errors than the TS generalization, but that both represent most systems very well. The NATgener- alization requires some 30 constants, however, in order to fit nine families of compounds, some of which include only one member.

Table 4 compares the results of the Katz and Firoozabadi correlation for interaction parameters between methane and n-alkanes to the results of the NAT and TS generalizations. The graphical correlation of Katz and Firoozabadi was linearized with the following expression:

THE (ANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, DECEMBER, 1990 1035

TABLE 4 Comparisons of Correlations for Methane-Alkane Binaries

Katz NAT

Alkane 1, DP DY DP DY

c2 0.0000 1.13 0.27 1.80 0.38 C3 0.0059 3.64 0.46 2.25 0.42 NC4 0.0159 3.98 0.84 6.36 0.84 IC4 0.0131 3.35 1.64 1.67 1.03 NC5 0.0220 4.12 0.64 4.08 0.61 IC5 0.0211 6.64 3.27 6.99 3.14 NEOC5 0.0175 2.85 3.42 2.87 2.89 NC6 0.0262 8.62 0.94 6.63 1.03 NC7 0.0294 2.58 0.75 3.70 0.69 NC8 0.0318 4.53 0.53 2.01 0.48 NClO 0.0354 3.64 0.20 2.54 0.20

TS

DP DY

1.30 0.32 3.04 0.45 3.84 0.84 3.03 1.54 4.26 0.65 6.64 3.27 2.66 3.35 8.07 0.95 2.81 0.72 2.21 0.50 2.89 0.19

Average 4.20 0.74 3.72 1.06 3.70 1.16

TABLE 5 Comparisons of Correlations for C0,-Alkane Binaries

Kato NAT TS

Alkane DP DY DP DY DP DY

CI c3 NC4 NC5 NC7 NClO ClCYC6 C2CYC6 NC16

Average

1.95 0.66 3.77 0.70 6.23 1.13 1.22 0.69 3.93 1.25 5.43 1.54 2.93 0.96 5.79 1.65 8.16 2.04 2.74 1.32 4.09 1.64 6.09 1.82 5.57 0.81 3.58 0.74 2.85 0.71 2.63 0.47 4.39 0.60 4.10 0.60

16.98 1.76 4.29 1.33 3.76 1.29 16.89 1.75 5.54 0.98 6.38 0.90 13.07 1.31 10.87 1.51 14.30 0.37

5.83 0.98 5.14 1.08 6.94 1.16

6,J = 0.13p(kg/dm3 at 15.5"C) - 0.06 . . . . . . . . (19) Table 5 gives a similar comparison to the generalization of Kato et al. (1981) for systems of carbon dioxide with hydrocarbons.

Figures 3 through 8 show calculated and experimental phase equilibrium isotherms in a pressure-concentration format. The figures indicate the sensitivity of the calculated phase envelopes to interaction coefficients over a range of conditions that represent the effects of variations in molecular size, molecular type and temperature in the equation of state. Table 6 gives the contribution of each of the terms in Equa- tion (15) (disperse and inductive) for the binary pairs used in the example systems.

The sensitivity of calculated phase behaviour to interac- tion parameters as a function of dispersive effects of molecular size difference is illustrated for two n-alkane com- binations. In this work the effect of molecular size differ- ence is expressed as a critical volume ratio. The results are shown in Figures 3 and 4 by way of comparison calcula- tions for the species pairs, methane-n-decane and ethane-n- decane. The negligible quadruple and dipole effects between these pairs facilitates a comparison of how the critical volume ratio of a binary pair changes both computed and observed phase behavior. As shown in Figure 3, the simple quadratic mixing rule (zero ai j ) expressed in Equation (4) results in underestimation of saturation pressures for the methane- decane system, while hij (0.052) predicted from Equation ( I 5) yields excellent agreement with experiment. Examination

48.0, I 1 I I I

6i, = 0.052 - 40.0.- 6 =o.ooo .................... il

52.0- -

24.0- -

16.0- -

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Mole Fraction Methane

Figure 3 - P-z Diagram for methane and decane at 37.7"C. 8000.0 I I I 1

Figure 4 - P-z Diagram for ethane and decane at 71.0"C.

Mob Fraction Carbon Dioxide

Figure 5 - P-z Diagram for CO, and decane at 104.4"C.

1036 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, DECEMBER, 1990

TABLE 6 Interaction Parameter Contributions for Example Systems

~

v c , vc, V' ', Q, Dispersive Inductive ' I Binary cm3/mol cm'/mol cm3/mol p ~ " , . cm", term term

c1-CIO 99.2 603 .O 279.6 0.0521 o.Ooo0 0.0521 c2-c 10 148.3 603.0 324.3 -0.206 0.0319 O.OOO3 0.0322 co2-c 10 93.9 603.0 274.2 -1.360 0.0552 0.0474 0.1026 CO2-Cl6 93.9 960.0 374.0 -1.360 0.0842 0.0330 0.1172

Dipsenive Term = 1.0 - Inductive Term = 90 p J - ' . mol-'

h,, = Dispersive Term + Inductive Term 10 0 I 1 I 1 I

6,, = 0.103 -

0.0 1 I I 1 I 0.0 0.2 0.4 0.6 0.8 1.0

Mole Fraction Carbon Dioxide Figure 6 - P-z Diagram for CO, and decane at 238.0"C.

of Figure 4, calculated for the system ethane-n-decane, shows that as the critical volume ratio more close!y approaches unity, the dispersive term in the correlation overcorrects the 6, (0.032). For some binary systems, including methane- ethane, ethane-propane and propane-decane, slightly nega- tive 6; regressed from the individual n-alkane binary studies have been reported as the critical volume ratios approach unity. Since the proposed generalization considers only attractive forces, the predicted values of 6, are always posi- tive and will necessarily overpredict the interaction parameter in the above systems.

A sample system illustrating the additional effects of molecular moments and temperature on the sensitivity of cal- culated phase behavior to generalized interaction parameters is shown for carbon dioxide in combination with n-decane in Figures 5 through 7. Carbon dioxide was chosen for the illustration not only because of its significance as an extrac- tion agent, but also to show the effect of a large quadrupole on computed and observed phase behavior. As shown in the figures it is necessary to consider both the dispersive (0.055) and inductive (0.048) contributions in the generalized inter- action correlation in order to adequately match the observed phase behaviour data (6i, = 0.103). It is interesting to note that the temperature independent correlation appears to apply over a wide range of phase behavior observation. Other than an apparent overestimation of decane solubility in the vapor phase (that increases with temperature), no other significant systematic deviation in computed behaviour with tempera- ture is noticeable.

18.C

15.c

12.c

n

I B e 0.c v

VI

B 6.a

3.0

0.0

6.. =

6.. 5

6 = li

0.055 .__.__._.____.......

I I I I I

VE 0.78 0.80 0.82 0.84 0.86 Mole Froction Carbon Dioxide

Figure 7 - P-y Diagram for CO, and decane at 238.0"C. 12.0 I I I I 1 1

blj = 0.m - 10.0.- 6. . = 0.004 ....................

6. = 0.000 -__------- i 11

8.0.-

n

2 z " 6.0-1 v

i! L f

4.0- -

0.0 0.2 0.4 0.6 0.8 1.0 M& Froction Carbon Dioxide

Figure 8 - P-z Diagram for CO, and hexadecane at 35.0"C.

Figure 8 illustrates a comparison of experimental data and predictions for a binary system of carbon dioxide and hex- adecane. The overestimation of the binary interaction may be attributed to the initial assumption, used in developing Equation (9), that the ionization potential ratio is close to unity. Using ionization totentiah from Prausnitz et al. (1986), the ratio of 2 ( Z i 4 ) 5 / ( Z i + 4 ) is as follows for pairs

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68, DECEMBER, 1990 1037

of hydrocarbons with carbon dioxide: C02 - methane 0.9997 C02 - propane 0.9949 C02 - heptdne 0.9906 C02 - mesitylene 0.9708

As the ionization potential of the heavy molecule continues to decrease, the effect may become much larger and the con- tribution to (,] from the dispersive term would tend to zero. The correlation also gives larger deviations for the systems nonane-hydrogen sulphide, ethylcyclohexane-hydrogen sul- phide, and propylcyclohexane-hydrogen sulphide. Again this may be a function of large ionization potential differences since the 6, values for these systems are also overpredicted. The carbon dioxide-hexadecane system may therefore represent an upper limit for the safe application of this gener- alization. The introduction of a second binary interaction parameter, to account for correction in the b term, is an obvious next step and is currently being evaluated for extending the correlation to larger molecules.

It is also interesting to note that a liquid-liquid phase tran- sition in the carbon dioxide-hexadecane system is evident at pressures above 8 MPa. The liquid-liquid split is calcu- lated at almost the same pressure regardless of the interac- tion parameter used even though the phase envelope is extremely sensitive to the magnitude of the interaction parameter.

The TS generalization does appear to underpredict the interaction parameter for most of the binaries containing nitrogen. One possible explanation for this is that the value of the quadrupole for nitrogen is inaccurate. If the value for Q for nitro en were decreased from -0.474 to -0.791 (p5112 ~ r n ~ ~ x l O ~ ~ ) then the overall deviations in DP and DY would drop to 5.37% and 0.99 respectively for 3242 converged bubble point calculations.

For dispersive forces only, a better empirical fit of the data base was obtained by substituting 44.0/(Vcj VCj)O. for 8 in Equation 15 (where V,. is given in cm/mol). The average pressure deviations for this function were 5.5% for 3263 con- verged points which is slightly better than the results for a constant 8 value of 0.20. The other advantage of the 44.0/ (Vcj Vcj). exponent is that it produces a damped exponent for pairs with large size differences which appears to fit the data in these systems better (ie. methane-eicosane regressed 6, is negative) and may in fact correct for the effect of the ionization potential ratio. A constant 8 value will give an ever increasing 6.. as the molecular size differential increases. A constant $was retained at this point since it is more theoretically satisfying and since a b inter- action is envisioned to account more properly for molecular size differences.

Conclusions

A generalized two parameter model for representing binary interactions in the attractive term of the Peng-Robinson equa- tion of state has been developed and evaluated. The model includes both dispersive and inductive force terms and requires molecular information including critical volume, dipole moment, and quadrupole moment.

Results of phase behaviour predictions for 72 binary systems indicate that errors arising from the generalized model are only slightly higher than those exhibited by the component dependent models presented by others (Nishiumi et al., Katz and Firoozabadi, Kato et al.).

This work is seen as a step towards developing a predic- tive model for describing phase behaviour in a wide range of systems including those with large molecular size differ- ences and those with strongly polar constituents, typical of supercritical extraction processes. Although the generalized model has been used with reasonably large molecules (hex- adecane), further development of the model will be neces- sary in order to include much larger molecules, polar-polar intermolecular forces, and hydrogen bonding effects.

Acknowledgements

Financial support received from the Natural Sciences and Engineering Research Council of Canada and from Amoco Canada Petroleum Co. Ltd. is greatly appreciated.

Nomenclature

a b DP DY E l EOS = equation of state I = ionization potential (ev)

kp. = binary interaction parameter in TCv = absolute pressure (Pa) PTS = number of experimental data points Q = quadrupole (pJ - c,~~) r = intermolecular separation distance (cm) R = universal gas constant (Jlmol . K ) REG = regressed value T = temperature (K) v = molar volume (cmirnol) v = total volume (cm3) x = liquid mole fraction y = vapor mole fraction z = overall mole fraction

= equation of state parameter (Pa . ( ~ m ~ / m o l ) ~ ) = equation of state parameter (cm3 / moi) = deviation in bubble point pressure = deviation in vapor mole fraction = objective function defined by Equation (16)

Greek

cy = polarizability (cm3) A,, = binary interaction parameter in a E = intermolecular potential ( p J ) 77 = parameter in Equation (15) ( p J - * mol-) 0 = parameter in Equation (15) p = dipole moment ( p J * . cm3) u = intermolecular separation (cm) d, = attractive potential term w = acentric factor

Subscripts

c = critical calc = calculated exp = experimental value i = component index j = component index

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~

Manuscript received October 12, 1989; revised manuscript received January 26, 1990; accepted for publication February 28, 1990.

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 68. DECEMBER, 1990 1039