Calculation of excess enthalpies with the CCOR equation of state

Fluid Phase Equilibria

Calculation

100 ( 1994) 139- 15 1

of excess enthalpies with the CCOR equation of state H .-M . Lin *qa M.-J.

’ Laboratory

Lee b

of Renewable

b Department

Resources Engineering, 1295 Potter Center. Purdue University. West Lafayette, IN 47907-1295. USA of Chemical Engineering, National Taiwan Institute of Technology, Taipei, 106, Taiwan Received 15 November 1993; accepted in final form 15 March 1994

-. Abstract The cubic chain-of-rotators (CCOR) equation of state was applied to calculate excess enthalpies for a variety of binary mixtures. The systems of interest include non-polar and polar compounds with emphasis on asymmetric mixtures under diverse conditions in both gaseous and liquid states. The calculations were, in general, satisfactory to within acceptable accuracy for a majority of mixtures, although significant deviations from experimental data were found for some systems. Possible reasons for the errors were discussed. Keywords:

Theory;

Equation

of state; Cubic; CCOR;

Enthalpy;

Polar; Non-polar

1. Introduction Excess enthalpy is an important thermodynamic property in the design and development of chemical, petroleum, natural gas and many other processes. Various methods are available for its calculation, and some appear to be fairly accurate within certain limitations. However, none have proved satisfactory over a diversity of conditions. Of these methods, equations of state have been the most widely used. While the equations are successful in phase equilibrium calculations, they are not as reliable for excess enthalpies, particularly for polar mixtures in the liquid region. Current calculations of liquid enthalpies are still based heavily upon the approaches from different versions of solution theories, although several studies (Adachi and to extend the Sugie, 1988; Casielles et al., 1989; Chen et al., 1991) have recently attempted

* Corresponding

author.

037%3812/94/$07.00 0 1994 - Elsevier SSDIO378-3812(94)02508-X

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140

H.-M.

Lin. M.-J.

Lee 1 Fluid Phase Equilibria

100 (1994) 139S1.51

equations of state for these mixtures. The purpose of the present work is to examine the feasibility of a simple cubic equation of state to represent this excess property for the entire range of conditions that are experimentally available. The equation used in this work, the CCOR, differs conceptually from the van der Waals equation and its modifications (e.g. the Redlich-Kwong equation) in that the CCOR equation was developed to accord with the results from molecular theories. The calculated results from the CCOR equation are compared with experimental excess enthalpies for a variety of binary mixtures in both gas and liquid regions. Particular attention is given to the asymmetric polar/non-polar, polar/polar and aqueous mixtures for which no method has yet been found accurate. As evidenced by the comparison, the CCOR equation appears to represent well the majority of mixtures. However, substantial discrepancies between calculated and experimental data were obtained for some systems. Possible sources of errors are discussed.

2. The CCOR equation of state The CCOR equation (Lin et al., 1983; Kim et al., 1986) is given by Eq. (A3) of the Appendix. Unlike the van der Waals equation and versions of it such as the Redlich-Kwong and the Soave equations, the CCOR equation is constrained to an approximation in accordance with the results of statistical methanics. The first term of the right-hand side of Eq. (A3) is a simplification of the Carnahan-Starling equation for a hard-sphere fluid, while the second term accounts for the contribution to pressure due to the rotational motion of a polyatomic molecule. The last two terms simulate the attractive force contributions. This equation has successfully been applied to calculate phase equilibria for a number of mixtures over a wide range of conditions (Guo et al., 1985b; Lin et al., 1985; Kim et al., 1986). The equation was later modified by Lee et al. (1986) to improve the calculations in the critical region for polar substances and their mixtures. The CCOR equation is extended in this work to calculate the excess enthalpies (He”) for non-polar/non-polar, non-polar/polar and polar/polar mixtures of both gas and liquid phases. The formulation of the H”” in terms of the equation is detailed in the Appendix.

3. Results and discussion Table 1 illustrates the calculated results in comparison with experimental data for some non-polar or slightly polar systems, with an emphasis on asymmetric mixtures of significant dissimilarity in molecular size and/or shape. Not included in the Table are numerous mixtures that consist of small molecules (such as CH4 + N? and CH4 + Ar) or show no peculiar differences in characteristics of the constituent molecules. These mixtures were well represented by the equation. The results of Table 1 reveal a clear tendency that large discrepancies occur when the degree of molecular dissimilarity in the mixtures become more significant. The errors are also relatively apparent for mixtures containing long-chain molecules such as n-hexadecane. Asymmetric

H.-M. Table 1 Non-polar

or slightly

Lin, M.-J. Lee 1 Fluid Phase Equilibria

100 (1994) 139-151

141

polar mixtures a

Mixture

kW

(1) +(2)

AAD

Bias

W)

(W

10.7 5.2 1.3 4.9 8.8 3.3 91.9 6.7 2.3 5.6 9.3 5.0 2.1 0.7 3.3 12.2 16.2 28.7 29.8 40.7 38.0 27.0 42.6 52.1

-1.5 -0.9 -0.5 1.9 0.9 0.8 -41.4 0.6 -1.7 -2.2 -0.7 1.0 -0.4 -0.2 -0.4 -2.6 0.2 -4.8 -5.1 -11.3 -8.0 6.5 6.0 6.1

Data pts.

Source b

77 89 26 38 38 44 32 28 8 21 13 12 11 5 21 23 22 17 21 25 24 12 13 11

1 2 3 2 2 2 2 4 5 6 7 7 7 5 4 4 4 4 4 4 4 7 7 7

ccl, c + cs2 = Ccl, ’ + n -hexane Ccl, ’ + benzene Ccl, ’ + n-heptane Ccl, ’ + n-octane Ccl, ’ + isooctane Ccl, ’ + n -hexadecane n-pentane + benzene n-hexane + I-heptene Cyclohexane + benzene Cyclohexane + I-MeNap Cyclohexane + 2-MeNap Cyclohexane + quinoline Benzene + 1-heptene Benzene + n-octane Benzene + n-undecane Benzene + n-dodecane Benzene + n-tetradecane Benzene + n-pentadecane Benzene + n -hexadecane Benzene + n-heptadecane Benzene + 1-MeNap d Benzene + 2-MeNap e Benzene + quinoline

d ’

283-323 293-313 298 2933313 293-313 293-313 2933313 298 298 298 298 309-318 298 298 298 298 298 298 298 298 298 298 309-318 298

-0.0032 0.0097 0.0063 0.0026 - 0.0080 - 0.0039 -0.0987 0.0492 0.0013 0.0428 0.0571 0.0230 0.0511 0.0156 0.0176 0.0175 - 0.0297 -0.0878 -0.0635 - 0.0772 -0.0961 -0.0190 -0.0214 -0.0172

APressure range: l-300 bar for the Ccl, + CS, system. All other mixtures are at atmospheric pressure. b 1, Siddiqi and Lucas (1983); 2, Harsted and Thomsen (1974); 3, Larkin and McGlashan (1961); 4, Diaz Pena and Menduina (1974); 5, Letcher and Baxter (1987); 6, Nicolaides and Eckert (1978); 7, Azner et al. (1985). ‘Using optimal A, and A? as reported by Guo et al. (1985a). d I-Methylnaphthalene. e 2-Methylnaphthalene.

in the equation for the prediction mixtures are sensitive to mixing rules, which are incorporated of equation constants of the mixture from the properties of constitutent components. The rules used in the CCOR equation are listed in the Appendix as Eq. (A4). These rules were developed to mimic the interaction behavior between two simple or near-simple molecules, but are oversimplified for molecules with a more complex nature of interactions in the mixtures. Their reliability, even for simple systems, has been the subject of much discussion over the years. More realistic rules to account for the effects of molecular dissimilarities are needed for improving accuracy. Note also that the calculations were made with only one adjustable interaction constant, k,,. Kim et al. (1986) suggested two constants, k,, and kcv, to give better representation of phase equilibria for all mixtures. Significant improvement was indeed obtained for most mixtures with two adjustable interaction constants, as will be discussed later.

142

H.-M.

Lin, M.-J.

Lee 1 Fluid Phase Equihhria

100 (1994) 139-151

Other factors can also contribute to the errors in the calculations, in addition to the imperfection of the equation itself (and its synergetic mixing rules): (a) owing to the nature of measurements, excess enthalpies are difficult to determine with accuracy experimentally, and in many instances the experimental errors are systematic; (b) the calculations require reliable values of critical temperature, critical pressure and acentric factor for each of the constituent compounds. For some substances (such as quinoline and I-methylnaphthalene), these properties are not known experimentally, or in some cases, the reported data in the literature are inconsistent. Empirical methods were thus applied for their estimation, and these could cause significant uncertainty. The calculations are particularly susceptible to acentric factor, which is a key parameter in the evaluation of all equation constants of the CCOR. The acentric factor has been used extensively as a measure of acentricity to characterize non-simple fluids, but its applicability to complex substances (particularly long-chain hydrocarbons) that deviate substantially from normal fluid behavior is much in doubt. Further, reliable data for acentric factor are not available for many substances with high boiling points. Table 2 shows the results for some polar substances in mixtures with non-polar or slightly polar compounds. Several methanol- or ethanol-containing mixtures are not reported, for the reason that these mixtures were well represented. An exception is the mixture of ethanol + CCL+ for which the CCOR fails. As shown in the Table, the deviations for this particular system are excessively large and lopsided. Nevertheless, the calculations are generally accurate for this class of mixtures. For polar substances, two equation coefficients, A, and A>, of the temperature-dependent (equation constant) “CX” were modified to account for the polar nature of interactions, as discussed in the Appendix. Guo et al. (1985a) determined a set of these values for each substance from its vapor pressure data for 45 compounds. These optimal values were used in the present calculations. The reliability of A, and Az values, as determined from experimental vapor pressures, depends on the quality of the pressure data used. The values can be biased if the vapor pressure data are not highly accurate over a sufficient range of temperatures. The enthalpy calculations become erroneous as a result of using inadequate A, and A2 values. For all other substances that do not have available values, the generalized correlations of Guo et al. ( 1985a) were applied for the estimation. The calculations for some of the polar/polar mixtures are compared with experimental data in Table 3. The agreement is excellent except for the systems of methanol + chlorobenzene and ethanol + chlorobenzene. The deviations in both systems are substantially biased. The reason for the failure of the CCOR equation to represent these two mixtures is not clear. A possible explanation is that the values of A, and A1 used for chlorobenzene are unreliable. However, the calculations for acetonitrile + chlorobenzene are accurate to within 4% average absolute deviation (AAD). Tables 4 and 5 report the results for aqueous mixtures at atmospheric and high pressures, respectively. These mixtures are of interest in chemical processes, petroleum and natural gas productions, and also in environmental systems. The CCOR equation is satisfactory for all the aqueous mixtures of Table 4 with an adjustable parameter over the temperature range of 363-423 K. Good results were also obtained for high-pressure systems, as shown in Table 5. Only the mixtures with long-chain alkanes (particularly n-octane) are not well represented. Again, one adjustable constant, k,,,, was used in all the calculations for the entire ranges of

H.-M. Table 2 Polar + non-polar

Lin, M.-J.

Lee 1 Fluid Phase Equilibria

or slightly polar mixtures

Mixture

k “‘1

298-323 298 308 308 308 308 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 298 308-323 348 3133333 313-333 313-333

143

a

(1) +(2) CH,Cl, = + Ccl, c CHCl, ’ + benzene CHCl, ’ + toluene CHCl, ’ + m-xylene CHCl, ’ + o-xylene CHCl, ’ + p-xylene CClF,-CC&F + Ccl, c CCIF2-CC&F + CH,-CCI, Ethanol ’ + Ccl, ’ 1-Propanol c + m-xylene 1-Propanol ’ + dipropylamine I-Butanol ’ + m-xylene I-Butanol ’ + dipropylamine I-Pentanol ’ + n-heptane 1-Pentanol ’ + dipropylamine 1-Hexanol ’ + o -xylene 1-Hexanol ’ + dipropylamine 1-Heptanol + n-hexane I-Heptanol + n-heptane I-Heptanol + n-octane 1-Heptanol + n -nonane 1-Heptanol + n-decane 1-Heptanol + dipropylamine I-Octanol ’ + o-xylene I-Octanol ’ + dipropylamine Aniline ’ + cyclohexane Furfural + cyclohexane Ammonia ’ + argon Ammonia ’ + nitrogen Ammonia ’ + methane

100 (1994) 139-151

0.0370 - 0.0278 - 0.0529 - 0.0722 -0.0732 -0.0758 0.0184 0.0209 -0.0501 - 0.0248 -0.0014 0.0017 -0.1354 0.0293 -0.1161 0.0093 -0.1151 0.0055 0.0101 0.0087 0.0045 - 0.0002 - 0.0826 0.0291 -0.0822 0.0778 0.0754 0.1148 0.1941 0.1764

AAD (%)

Data pts.

Source b

(“/u)

9.3 4.0 13.6 11.3 9.8 10.1 1.8 3.7 86.1 30.5 7.7 27.6 3.4 42.9 6.9 25.6 6.6 21.2 25.5 33.6 25.5 22.6 15.7 24.0 14.7 10.4 7.9 13.6 9.9 6.4

-0.3 2.2 -0.9 -0.9 0.2 -0.4 0.1 -1.3 -84.7 -11.0 0.5 -4.7 1.6 -27.9 0.9 -8.7 0.4 - 10.4 -9.3 - 5.4 -7.9 - 12.6 -3.8 -4.6 -1.6 -3.6 6.8 -0.6 0.7 -0.7

40 13 10 10 9 10 17 15 21 21 15 21 23 25 24 21 24 17 21 13 19 21 17 15 19 24 12 75 30 52

1 2 2 2 2 2 3 3 4 5 6 5 6 7 6 8 6 6 6 6 6 6 6 8 6 8 9 10 11 11

Bias

a Pressure range: l-300 bar for the CH,Cl, + Ccl, system; 8- 142 bar for mixtures containing ammonia. All others are at atmospheric pressure. b 1, Siddiqi and Lucas (1983); 2, Rastogi et al. (1971); 3. Dohnal and Patterson (1984); 4, Nagata and Tamura (1984): 5, Rodriguez-Nunez et al. (1985); 6, Sarmiento et al. (1985); 7, Hamam et al. (1984); 8, Rodriguez-Nunez et al. (1984); 9, Nicolaides and Eckert (1978); 10, Naumowicz and Woycicki (1984); 11, Naumowicz and Woycicki (1986). ‘Using optimal A, and A2 as reported by Guo et al. (1985a).

temperature and pressure that are experimentally available. Some of the data are within the region of the critical state, which presents an additional degree of difficulty in calculating with accuracy. Kim et al. (1986) discussed the need for two adjustable interaction constants in the CCOR equation to describe the phase equilibrium behavior of asymmetric mixtures. Table 6 demonstrates the calculated results with two constants, k,, and kcv, for some mixtures of highly

144

H.-M.

Table 3 Polar + polar mixtures

Lin, M.-J.

at atmospheric

Lee /Fluid

Phase Equilibria

100 (1994) 139-151

pressure

Mixture

k ai1

(1) +(2)

fK)

CClF,-CC&F + CH,-CF, CClF,-CC&F + CHCl,-CH,Cl b CClF,-CCl,F + CHCIZ-CHCl, Methanol b + chlorobenzene b Ethanol b + chlorobenzene b Ethanol b + acetone b 1-Propanol b + methyl butanoate 1-Butanol b + methyl butanoate 1-Butanol b + propyl acetate b I-Pentanol b + methyl butanoate I-Pentanol b + propyl acetate b I-Hexanol b + propyl acetate b I-Heptanol b + propyl acetate b 1-Octanol b + propyl acetate b Acetonitrile b + chlorobenzene b

298 298 298 298 298 298-323 298 298 298 298 298 298 298 298 298

0.0268 0.0564 0.0212 - 0.0838 - 0.0475 0.0309 0.0277 0.0239 0.0299 0.0376 0.0414 0.0397 0.0660 0.0612 0.0292

AAD

Bias (%)

Data pts.

Source a

(“/) 1.9 2.8 5.0 48.0 63.1 9.7 6.9 4.5 7.4 1.0 4.0 4.1 1.1 3.4 3.9

-0.6 -1.1 -0.4 -29.6 -29.2 2.2 -1.8 -2.4 -2.8 0.7 -1.5 -1.6 0.4 -2.1 -1.8

15 15 14 18 17 25 15 17 21 17 23 21 17 19 16

1 1 1 2 2 3 4 4 4 4 4 4 4 4 2

a 1, Dohnal and Patterson (1984); 2, Nagata and Tamura (1986); 3, Nicolaides al. (1985). b Using optimal A, and A, as reported by Guo et al. (1985a). Table 4 Aqueous

mixtures

Component

Hydrogen Nitrogen co CGZ Methane Ethane Ethylene Propane Propylene n-Butane n-Pentane n-Hexane Cyclohexane Benzene n-Heptane n-Octane

at atmospheric

(2)

pressure

et

a

k 01,

373-423 373-423 364-403 363-393 373-423 363-393 363-393 363-393 363-393 363-393 363-423 363-423 363-393 363-393 363-423 363-423

and Eckert (1978); 4, Fernandez

0.8026 0.6639 0.7849 0.6650 0.6667 0.7752 0.6976 0.7542 0.6650 0.6541 0.6003 0.5177 0.4859 0.4250 0.4474 0.4403

AAD

Bias (%)

Data pts.

Source b

(%) 7.4 7.4 9.5 6.5 6.8 6.2 6.6 4.0 5.4 4.7 4.8 5.6 3.9 4.1 5.3 2.8

-1.1 -2.8 -2.5 -0.4 -1.2 -1.6 -1.7 0.8 -1.3 - 1.8 1.5 -3.3 -1.4 0.8 0.3 0.4

59 57 39 40 69 40 40 40 42 40 45 41 48 40 71 45

1 2 3 3 1 4 4 5 5 5 6 6 7 7 6 6

a Using optimal A, and A, as reported by Guo et al. (1985a) for water. b 1, Smith et al. (1983); 2, Richards et al. (1981); 3, Smith and Wormald (1984); 4, Lancaster and Wormald 5, Lancaster and Wormald t 1986); 6, Smith et al. (1984); 7, Wormald and Lancaster (1985).

(1985);

H.-M. Lin, M.-J. Lee 1 Fluid Phase Equilibria 100 (1994) 139-151 Table 5 Aqueous Component

mixtures

at high pressures

a

k at/

(2) pbar)

Hydrogen Nitrogen co CO, Methane Ethane Ethylene Propane n-Butane n -Pentane n -Hexane n -Heptane n-Octane

448-698 448-698 473-698 448-698 448-698 448-698 448-698 448-698 448-698 448-698 448-698 448-698 498-648

145

3.6-112 3.5-126 7.6-122 4.0-134 3.5-126 5.5-122 4.6-66 5.33137 6.3-130 5.5-118 3.8-126 4.1-126 3.4-146

0.1577 0.1521 0.1597 0.2431 0.2607 0.4382 0.3389 0.4907 0.4816 0.4445 0.3282 0.3726 0.4988

AAD

Bias (%)

Data pts.

Source b

(“/) 4.9 6.6 4.1 6.5 7.2 6.1 8.5 7.1 9.7 9.7 8.1 13.8 19.2

-1.7 -4.1 -1.6 0.2 -2.3 -0.2 -1.9 0.5 -3.1 -1.4 1.5 -2.3 -5.9

57 74 22 137 77 86 32 102 67 67 69 73 183

1 2 3 3 4 5 5 6 6 7 I 7 7

a Using optimal A, and A, as reported by Guo et al. (1985a) for water. b 1, Wormald and Coiling (1985); 2, Wormald and Colling (1983); 3, Wormald et al. (1986); 4, Wormald and Colling (1984); 5, Lancaster and Wormald (1987); 6, Wormald and Lancaster (1986); 7, Wormald et al. (1983).

dissimilar molecules. The improvement over the use of one adjustable constant. is evident. Only three systems, methanol + chlorobenzene, ethanol + chlorobenzene and ethanol + CCL, were little affected. Their deviations are all excessively biased, regardless of whether one or two adjustable constants are used. The exact cause of these systematic deviations from experimental data is not known, although some possible sources of error were discussed in previous sections.

4. Conclusions The CCOR equation was applied to calculate excess enthalpies for a variety of mixtures in both gas and liquid states. The emphasis was on mixtures that have not been successfully represented by existing correlation methods. The CCOR equation was found satisfactory for the majority of systems to within the acceptable accuracy in engineering applications. These systems include polar/non-polar and polar/polar mixtures for which most of the simple equations of state fail. However, the equation did not work as well for highly asymmetric mixtures. The reason is attributed, at least in part, to the mixing rules, which are oversimplified to describe adequately the complex behavior of interactions between molecules of significant dissimilarity. More reliable rules are needed for improving accuracy. Large deviations were also obtained for the mixtures that contain long-chain molecules. The problem, in addition to the mixing rules, could probably be traced back to the association of the CCOR equation with the acentric factor, which is essential to the evaluation of all equation constants. While the acentric factor is a useful parameter for near-normal fluids, it probably fails to account for the non-central nature of intermolecular forces for long-chain molecules.

H.-M.

146 Table 6 Calculations

with two adjustable

Lin, M.-J,

Lee / Fluid Phase Equilibria

interaction

Mixture (1) +f2)

k 01,

CHCI, d + toluene Ccl, a + n-hexadecane Benzene + quinoline Benzene + I-MeNap Benzene + 2-MeNap Benzene + IZ-undecane Benzene + n -dodecane Benzene + n-tetradecane Benzene + n-pentadecane Benzene + n-hexadecane Benzene + n-heptadecane Methanol d + chlorobenzene d Ethanol a + Ccl, a Ethanol d + chlorobenzene a Propanol d + m-xylene I-Butanol a + m-xylene I-Pentanol a + n-heptane 1-Hexanol d + o -xylene I-Heptanol + n-hexane I-Heptanol + n-heptane I-Heptanol + n-octane I-Heptanol + n-nonane I-Octanol d + 0 -xylene

0.0705 -0.0191 - 0.0346 -0.0328 - 0.0536 - 0.0283 - 0.0698 -0.1707 -0.1749 -0.1849 -0.2236 - 0.0244 -0.0293 -0.0191 0.1317 0.2129 0.0618 0.1221 0.0703 0.0571 0.0522 0.1845 0.1014

constants

100 (1994) 139-151

for some highly asymmetric

mixtures

kc, (“/u)

AAD (%)

Bias

0.1343 - 0.0803 -0.0185 -0.0148 - 0.0326 -0.0120 -0.0403 - 0.0792 -0.1051 - 0.0984 -0.1138 0.0612 0.0210 0.0302 0.1730 0.2413 0.0969 0.1314 0.0669 0.0541 0.0513 0.2043 0.0832

7.2 25.7 12.1 16.5 9.9 10.4 9.8 13.2 8.0 9.2 10.9 47.0 76.3 59.1 18.4 8.3 25.5 7.6 13.2 12.5 17.8 15.8 6.5

-3.0 -5.1 3.6 10.0 - 1.5 -0.4 3.4 2.5 2.3 3.1 4.5 -25.0 -73.9 - 29.0 -12.1 -4.8 - 18.5 -5.7 -9.9 -11.2 -16.1 -4.9 -4.8

__

d Using

optimal

A, and A2 as reported

by Guo et al. (1985a).

Nevertheless, the calculations with the CCOR equation, in general, appear to be satisfactory, considering the simplicity of the cubic equation, the complex nature of molecular interactions in mixtures and the uncertainties of experimental data.

List of symbols AAD a, b, c, d A,rAz AX, A4 G 3G CR H k alJ k GJ

average absolute deviation parameters in the CCOR equation of state constants in LX(T) for T, I 1 constants in a(T) for T,. > 1 constants in y(T) parameter in the CCOR equation of state molar enthalpy (J mol-‘) binary interaction constant for a, binary interaction constant for c,

H.-M. Lin, M.-J. Lee 1 Fluid Phase Equilibria 100 (1994) 139-151

P R T V

147

pressure (bar) gas constant (J mol-’ K-l) temperature (K) molar volume (cm3 mol-‘) mole fraction compressibility factor

>

Greek letters temperature-dependent temperature-dependent general variable acentric factor variables as defined

c(

Y tI

variable variable

in equation in equation

in the CCOR

equation

parameter parameter

a c

of state

Subscripts critical property i-component i-j pair molecules j-component mixture reduced property residual property

C

i ij j m r rs

Superscript ex

excess property

Appendix: Excess enthalpy (II”“) with the CCOR equation By definition,

wx = H - i

x,H, = H,, -

i=l

i

xiHrs.L

where H,, represents state by

molar residual

H,,=RT*

-+RT(Z-1)

The CCOR

equation

(Al)

1=1

enthalpy.

The residual

enthalpy

is related to an equation

of state (Lin et al., 1983; Kim et al., 1986) is given

of

(W

H.-M. Lin, M.-J. Lee 1 Fluid Phase Equilibria 100 (1994) 139-151

148

p = RT[l

+ 0.77(b/V)] V - 0.42b

+ CR

1 V(V+c) _

a

bd

(A3)

V( V + c)( V - 0.42b)

This equation contains five equation constants. Of these constants, a and c are temperaturedependent while b, CR and d relate to the acentric factor of a substance (Kim et al., 1986). Two coefficients, C, and C,, are associated with the temperature-dependent equation for c, and four (A, and A2 at reduced temperature T, I 1; A3 and A4 at T, > 1) for a. All these coefficients were generalized with the acentric factor by Kim et al. (1986) for non-polar and slightly polar substances. For polar fluids, Guo et al. (1985a) determined a set of values of A, and A2 for each substance from its vapor pressure data, while other constants were kept unchanged. These optimal values of Al and A2 were also correlated by Guo et al. (1985a) in terms of acentric factor and dipole moment. In this work, the coefficients Al and A2 were calculated from the correlations only if the values were not available. For a mixture, the equation constants are evaluated from the following mixing rules: 8, = i i l=lj=l

(A4)

X,XJO,

where f3 refers to a, b, c, d or CR for a mixture. for ai, and d,, a, = ( ’ d,

=

bJ)taraJ

8, was represented

( A4b)

and the arithmetic

mean rule for b,, cv and Cz, (A4c)

b, = (b, + bJ)/2

c;

=

mean rule

( A44

)o.5

(4d,)05

‘Z, =(l

by the geometric

-k,,)(c, (CR

Introducing

+

(A‘W

+cJ)/2

c3/2 the CCOR

-&‘“(V-:42b,)]

( A4e) equation

into Eq. (A2), we obtain

(A5)

H.-M.

Lin, M.-J.

Lee 1 Fluid Phase Equilibria

100 (1994) 139-151

149

where Ql = [0.42&(0.42& Qz = - (0.42b,

+ 2c,) + c;] -’

(A5a)

+ 2cm)Ql

(A5b)

The partial derivatives appearing in Eqs. (A5c) and (A5d) are respectively given by the following equations:

where

act.

- = UP,‘<-0.5A,,~;25T-‘.2s aT

+ 4A,,T/Tz,),

for T,,I 1

acl

-A = CX~[(A~~/T,,) - 0.5A,,/(TTci)o.5], for T,,> 1 aT and ac, aYI aT-Cc' (> FT = -c,,Y,C~, C2,T'c21 -"/(T,,)c'2~ with

The parameters k,, in Eq. (A5c) and k,, in Eq. (A5d) are adjustable interaction constants, were determined for each binary mixture by the least-squares regression of experimental enthalpies in deviation from calculated values. The constants were reported in the Tables. 6 was calculated with both k,, and k,,,, while all other Tables used only k,,(as k,, was be zero).

which excess Table set to

References Adachi, Y. and Sugie, H. 1988 A new method to predict thermodynamic properties of mixtures by means of a cubic equation of state. J. Chem. Eng. Jpn., 21: 57-63. Aznar, E., Ruiz, B. and Gutierrez Losa, C., 1985. Excess molar enthalpies at 298.15 K of (cyclohexane + c( methylquinoline) and of (benzene + c( methylquinoline). J. Chem. Thermodyn., 17: 1121- 1126.

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