Development of a Universal Group Contribution Equation of State. 2

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Ind. Eng. Chem. Res. 2002, 41, 3489-3498

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Development of a Universal Group Contribution Equation of State. 2. Prediction of Vapor-Liquid Equilibria for Asymmetric Systems Jens Ahlers and Ju 1 rgen Gmehling* Lehrstuhl fu¨ r Technische Chemie (FB9), Carl von Ossietzky Universita¨ t Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany

The PSRK method is a well-established group contribution equation of state (GCEOS) for the prediction of vapor-liquid equilibria (VLE) and other properties. A generalized GCEOS has been developed to overcome the still remaining weaknesses of the PSRK model, which is of great interest for industrial chemistry. A generalized volume-translated Peng-Robinson (PR) equation of state (EOS) has been combined with the group contribution method UNIFAC by means of a simple gE mixing rule, derived from the PSRK one. In this new GCEOS, the FloryHuggins term of the PSRK mixing-rule and the combinatorial part of the UNIFAC model are skipped simultaneously, which leads to a much better description for asymmetric systems. The predictions are further improved by using a quadratic mixing rule for the parameter b with an exponent of 3/4 for the binary parameter bij in the combination rule. Temperature-dependent group interaction parameters are fitted to VLE over a wide range of temperature and pressure. The results of the new GCEOS were investigated for asymmetric systems of alkanes with different gases (CH4, C2H6, CO2, CO, H2, and H2S). The results are compared with PSRK and PSRK(Li). The obtained results show that the new GCEOS leads to a much better description of asymmetric systems. Moreover, the new GCEOS provides reliable predictions of excess enthalpies (hE) and excess volumes (vE) as shown for the system ethane + carbon dioxide. Introduction The PSRK model,1 combining the Soave-RedlichKwong (SRK) equation of state (EOS) with the group contribution method UNIFAC, allows reliable predictions of vapor-liquid equilibria (VLE) and gas solubilities.2 The range of applicability was extended to systems with strong electrolytes by introduction of the group contribution method LIFAC.3 Because of its large range of applicability and the reliability of the predicted results, the PSRK model has become an important tool for the development and design of separation processes and other engineering applications. However, the PSRK model still shows some weaknesses, which are discussed in the literature: (i) Deviations between experimental and predicted saturated liquid densities are similar to those of the basic EOS (SRK). (ii) The Mathias-Copeman R function shows a foolish course at higher reduced temperatures. (iii) The prediction of asymmetric systems often delivers unsatisfying results. (iv) Poor results are obtained for the prediction of excess enthalpies (hE) and activity coefficients at infinite dilution (γ∞). To overcome these weaknesses, a new group contribution EOS has been developed consisting of the following modules: (a) a volume-translated Peng-Robinson (PR) EOS, which delivers a better reproduction of liquid volumes, (b) the Twu R function, which shows a reasonable temperature extrapolation, (c) improved mixing rules for the parameters a and b of the EOS allowing * Corresponding author. Tel.: +49-441-798-3831. Fax: +49441-798-3330. E-mail: [email protected]. Website: http://www.uni-oldenburg.de/tchemie.

the prediction of asymmetric systems, and (d) temperature-dependent interaction parameters (simultaneously fitted to VLE, hE and γ∞). The PSRK model should be replaced completely by the group contribution EOS VTPR. However, before the required group interaction parameters have to be fitted for a large number of main group combinations. New Group Contribution of State In part 1,4 the description of liquid densities of 44 pure compounds of different families (alkanes, aromatics, ketones, alcohols, and refrigerants) has been improved significantly by changing the EOS from SRK to a generalized volume-translated PR (VTPR):

P)

a(T) RT (1) v + c - b (v + c)(v + c + b) + b(v + c - b)

The translation parameter c was determined by the difference of the experimental and calculated densities at a reduced temperature Tr ) 0.7:

c ) vexp - vcalc

(Tr ) 0.7)

(2)

The values are listed in Table 1. If no liquid densities are available, c can be obtained from critical data:4

c ) 0.252

RTc (1.5448zc - 0.4024) Pc

(3)

The deviations between experimental and predicted liquid densities have been reduced from 13.3% (SRK) to 4.1% (VTPR) over the temperature range Tr ) 0.31.0 (Figure 1).

10.1021/ie020047o CCC: $22.00 © 2002 American Chemical Society Published on Web 06/18/2002

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Table 1. Twu r Parameters L, M, and N of Noll8 and Translation Parameters c for the PR EOSa component

L

M

methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane carbon dioxide carbon monoxide

0.945 43 0.212 25 0.960 66 0.176 56 0.752 33 0.965 06 0.939 55 0.833 98 1.489 36 0.655 06 0.899 25 0.916 87

1.245 25 0.872 04 0.998 76 0.852 13 0.897 51 0.942 50 0.909 56 0.871 85 1.142 17 0.825 30 0.986 79 0.930 52

N

c [m3/mol]

0.424 15 4.33 × 10-6 1.701 00 4.69 × 10-6 0.609 46 5.11 × 10-6 2.272 41 5.00 × 10-6 0.934 40 3.48 × 10-6 0.816 37 1.30 × 10-6 0.901 35 -0.59 × 10-6 1.079 83 -3.25 × 10-6 0.627 32 -7.84 × 10-6 1.465 50 -12.27 × 10-6 0.768 05 1.60 × 10-6 0.454 85 3.90 × 10-6

a For the alkanes n-undecane to n-eicosane, the R parameters listed by Twu et al.7 are used.

Figure 2. Deviations between experimental and predicted liquid densities for the system ethanol (1) + 2-methylpentane (2) as a function of composition at 298.15 K. Experimental data:19 (9) PSRK, (b) VTPR, (2) VTPR (parameter cethanol ) -1.91 × 10-6 m3/mol, obtained by eq 2 for Tr ) 0.7). Table 2. Generalized Twu-Bluck-Cunningham-Coon r Parameters

Figure 1. Deviations between experimental and predicted liquid densities for 44 pure compounds (alkanes, aromatics, alcohols, refrigerents, and ketones) with the Peng-Robinson (PR), SoaveRedlich-Kwong (SRK), and generalized volume-translated PR (VTPR) EOSs.

It should be pointed out that the VTPR EOS is not able to reproduce densities precisely enough near the critical point, because the slope dv/dT increases extremely in this region. Therefore, it cannot be recommended to apply the VTPR EOS for the calculation of liquid densities of pure compounds and mixtures in the temperature range Tr ) 0.8-1.0. However, Ahlers and Gmehling4 have developed a PR EOS with a temperature-dependent volume translation called T-VTPR, which allows the description of saturated liquid densities up to the critical point, but when this approach is used, problems may occur for the description of the PvT behavior at high pressures.4,5 A recent investigation has shown that, within the temperature range of Tr ) 0.3-0.8, the deviation only amounts to 2.1% (VTPR) in comparison to 11.3% (SRK); if the c parameters are adjusted to experimental densities (eq 2), the deviation is even less than 2%. For the description of mixtures, a linear mixing rule for the parameter c is suggested. As a consequence of the improved description of densities for pure compounds, of course, much better results are also obtained for mixtures. The deviations between experimental and predicted liquid densities for the system ethanol + 2-methylpentane at 298.15 K, for instance, amount to 10.4% for PSRK and 1.4% for the VTPR EOS. Figure 2 shows the deviations as a function of composition. If the parameter c for the compound ethanol is calculated by eq 2 using the experimental density at Tr ) 0.7 (Fexp ) 727.8 kg/m3; cethanol ) -1.91 × 10-6 m3/mol), the deviations only amount to 0.4%.

T r > 1b

T r e 1a

generalized R parameter

R(0)

R(1)

R(0)

R(1)

Lgeneralized Mgeneralized Ngeneralized

1.511 442 2.788 270 0.161 590

0.567 879 0.774 270 2.575 067

0.401 219 4.963 070 -0.200 000

0.024 955 1.248 089 -8.000 000

a New generalized Twu R parameters on the basis of the parameters fitted by Noll.8 b Original generalized Twu R parameters.7

Critical data are used to calculate the parameters a and b:

R2Tc2 aii(T) ) 0.45724 R(T) Pc bi ) 0.0778

RTc Pc

(4)

(5)

Twu et al.6 have published an exponential R function for a reliable reproduction of the vapor pressure of the pure component:

R(T) ) TrN(M-1) exp[L(1 - TrNM)]

(6)

The values of the parameters N, M, and L, which have been determined by adjustment at the vapor pressure curve of the considered pure component, are given in Table 1. If no parameters are available or T > Tc, the generalized R function for the PR EOS, developed by Twu et al.,7 can be used. When the R values, calculated by eq 6, for different compounds are represented as a function of the acentric factor ω, for each temperature a curve with a y-axis intercept R(0) and a slope R(1) R(0) is obtained:

R(T) ) R(0) + ω(R(1) - R(0))

(7)

In the following step, the quantities R(0) and R(1), which only depend on the temperature, are represented as a function of the reduced temperature Tr. The resulting curves for R(0) and R(1) can be regressed with the help of eq 6. The resulting generalized parameters Ngeneralized,

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Mgeneralized, and Lgeneralized are listed in Table 2. While Twu et al.7 only used alkane compounds from methane up to n-eicosane for the development of the generalized R function, Noll8 fitted Twu parameters L, M, and N for more than 65 compounds of different families including alkanes, aromatics, ketones, alcohols, ethers, refrigerants, and some gases. These parameters were used for calculating new generalized parameters Ngeneralized, Mgeneralized, and Lgeneralized for Tr < 1 (Table 2). For the supercritical region (Tr > 1), the Twu parameters7 were adopted (Table 2). These were obtained by Twu by fitting R values to gas solubility data of methane and hydrogen in hydrocarbon liquids to reproduce Henry’s constants. At very high temperatures, the value of the chosen R function should approach zero (no interaction between the molecules). Because the Mathias-Copeman function,9 which is used in the PSRK model, does not do justice to this thermodynamic sensible reflection, the Twu function is used in our new GCEOS. Derivation of the gE Mixing Rule Kontogeorgis and Vlamos10 have given a convincing explanation why PSRK fails by predictions of asymmetric systems: the difference between the combinatorial part of the SRK EOS and the combinatorial part of the group contribution method UNIFAC increases with a higher degree of asymmetry of the considered system. Li et al.11 have introduced an empirical correction into the PSRK model, which allows a reliable prediction of asymmetric systems up to pentatetracontane (C45). They defined so-called effective van der Waals volumes R/k and surface areas Q/k for the subgroups CH3, CH2, CH, and C by multiplying the values of the original R and Q values by an empirical correction factor f(nc) 1/2

f(nc) ) 1.0 - 0.36983nc

3/4

+ 1.0287nc 5/4

1.0199nc + 0.41645nc

-

whereby nc corresponds to the number of C atoms in the molecule. This correction delivers very good results for asymmetric systems. Based on the PSRK mixing rule and the Li correction, a simple mixing rule for the parameter a was derived, which allows reliable predictions for asymmetric systems. The PSRK mixing rule can be derived theoretically from the expression of the Helmholtz energy:12

a bRT

)

aii

∑i xib RT +

+

RT

i

b

∑i xi ln b

i

A

(9)

The excess Gibbs energy gE0 of the UNIFAC model13 consists of a combinatorial and a residual part:

gE0 ) gEcomb + gEres

(10)

The combinatorial part is calculated by the following expression:

gEcomb/RT )

whereby Vi is the volume to mole fraction ratio and Fi the surface area to mole fraction ratio:

Vi )

ri

∑j

; Fi )

xjrj

qi

∑j

∑i xi ln Vi + 5∑i xiqi ln Fi

(11)

(12)

xjqj

The Flory-Huggins term in eq 9 can be rearranged:

∑i

xi ln

b

)-

bi

∑i

xi ln Vi′; Vi′ )

3/2

- 0.05536nc (nc < 45) (8)

gE0

Figure 3. Change of the quotients ralkane/rethane and balkane/bethane in dependence of the degree of asymmetry of the system: (O) parameter b (PR EOS); (2) relative van der Waals volume parameter r with Li correction; (0) relative van der Waals volume parameter r (original values).

bi

∑j xjbj

(13)

With bi, the covolume of the EOS, and ri, the relative van der Waals volume of component i, the PSRK model contains two parameters with a similar meaning but different values. Figure 3 shows the ratio of the volume parameters of alkanes to ethane as a function of the number of C atoms in the alkane. Additionally, the values which are obtained with the Li correction are given. With increasing difference in the molecule size (from decane upward), the ratio b/bethane rises exponentially, while the ratio r/rethane increases linearly. However, a good agreement between the ratios of b/bethane and r*/r*ethane of the Li correction is observed. Therefore, the parameter ri in eq 12 is replaced by bi. From this, it follows that the Flory-Huggins term in eq 13 and the first summation in eq 11 can be skipped. Furthermore, the second summation in eq 11 can be cancelled because of the negligible contribution of this part. Then the PSRK mixing rule simplifies to the following expression:

a b

)

∑i

xi

aii bi

+

gEres A

(14)

The partial molar derivative a j i, which is required for calculating the fugacity coefficients, can be derived from eq 14:

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a ji )

Ind. Eng. Chem. Res., Vol. 41, No. 14, 2002

( ) ∂nTa ∂ni

(

)b

T,P,nj*i

RT ln γi

+

A

(

bi

gEres A

+

)

aii

+

)

aii xi (b h i - b) (15) bi

∑i

As a consequence, the relative van der Waals volume parameter is not required anymore. The parameter A only depends on the chosen EOS and reference state. Using the PR EOS, A depends on the packing fraction 1/u:

A)

u + (1 - x2) 1 ln 2x2 u + (1 + x2)

(16)

In our approach a constant packing fraction 1/u is assumed (excess volume vE ) 0): L

vL vi ) u ) ui ) b bi

(17)

The parameter u was obtained by determining the ratio vL/b for more than 75 compounds, as described by Fischer and Gmehling12 for the SRK EOS. The liquid volume vL at 1 atm was chosen as the reference state. For the PR EOS, the calculations delivered a value of 1.224 98 for the parameter u; therefore, the value of the parameter A amounts to -0.530 87. Modification of the Mixing Rule for b Chen et al.14 have shown that the b mixing rule has a sensitive influence on the description of asymmetric systems. Instead of a linear mixing rule, which is used in the PSRK model, Chen proposes a quadratic mixing rule:

b)

∑i ∑j xixjbij

(18)

whereby the binary parameter bij is calculated by means of the following combination rule:

bij3/4 )

bi3/4 + bj3/4 2

(19)

The exponent 3/4 was adopted from the group contribution method modified UNIFAC (Dortmund),15 where the same exponent is used for the relative van der Waals volume parameter ri in the combinatorial part. Theoretical investigations of Deiters16 led to a similar exponent of 0.8. The exponent 3/4 significantly improves the prediction of asymmetric systems, as can be seen from Figure 4. Temperature-Dependent Parameters The VLE of gas-alkane systems cover a large temperature and pressure range. Therefore, temperaturedependent group interaction parameters are necessary for a reliable description. For this reason, the ψ function, used in the model modified UNIFAC (Dortmund), is adopted:

Figure 4. Influence of the b mixing rule on the description of asymmetric systems. Experimental and predicted VLE for ethane (1) + n-decane (2) at T ) 444.26 K (b, O),20 ethane (1) + n-hexatriacontane (2) at T ) 423.20 K (2),21 ethane (1) + n-eicosane (2) at T ) 340.00 K (9),22 (s) this work using eqs 18 and 19, and (- ‚‚‚ -) this work using a linear b mixing rule.

(

)

anm + bnmT + cnmT 2 ψnm ) exp T

(20)

The objective function F of the fitting routine is formulated in the same way as described by Holderbaum and Gmehling.1 With the Dortmund Data Bank (DDB), a comprehensive database was available for fitting the required group interaction parameters, which are listed in Table 3. The required Q-values of the subgroups are listed in Table 4. Results and Discussion In the following figures, experimental and calculated VLE, predicted with the new GCEOS VTPR, are shown. These predictions are compared with the results obtained using the original PRSK and the Li version (PSRK(Li)). Alkane + Alkane Systems. Alkane + alkane systems are very suitable to prove if the newly developed GCEOS shows the desired potential because no interaction parameters are used (gEres ) 0), which can overcome shortcomings in the basic structure of the model. Besides the models PRSK and PSRK(Li), additionally the LCVM mixing rule,17 which was developed for the reliable prediction of asymmetric systems, is considered. In this paper the LCVM mixing rule is combined with the PR EOS and the Twu R function. Figure 5 shows results for the system ethane + n-decane. Concerning the phase equilibrium at 511 K, the prediction at the critical point of the mixture (x1 ≈ 0.7) and the vapor phase fraction with the new EOS and the LCVM model is better than the prediction with PSRK. PSRK(Li) only slightly corrects the PSRK prediction. Considerable larger improvements are observed for higher asymmetric systems, e.g., for the system ethane + n-octacosane (Figure 6). In this case, the PSRK prediction fails. The best predictions of the initial slopes are obtained by the GCEOS developed in this work and the LCVM model, which both provide significantly better results than the predictions of the PSRK(Li) model.

Ind. Eng. Chem. Res., Vol. 41, No. 14, 2002 3493 Table 3. Group Interaction Parameters for the Newly Developed GCEOS n CH2 CH2 CH2 CH2 CH2

m CO2 CH4 H2S H2 CO

anm/K 941.82 177.41 893.01 280.31 191.53

cnm/K-1

bnm

amn/K 10-2

-3.6180 -1.0450 -3.1342 -0.21676 0.28932

0.34290 × -0.15683 × 10-4 0.1302 × 10-2 -0.11678 × 10-2

-6.4843 -82.636 742.31 480.34 4.4228

cmn/K-1

bmn -0.2200 × 0.24879 -5.7074 -3.1762 -0.44705

10-1

0.980 00 × 10-3 0.227 76 × 10-2 0.126 50 × 10-1 0.600 77 × 10-2

Table 4. Q Values of the Subgroups for the Newly Developed GCEOSa no. of no. of subgroups subgroup Q value subgroups subgroup Q value 1 2 3 4 112 a

CH3 CH2 CH C CO

0.8480 0.5400 0.2280 0.0000 0.8280

113 114 117 118

H2 H2S CO2 CH4

0.4160 1.2020 0.9820 1.1240

The subgroups are numbered following the PSRK matrix.

Figure 7. Experimental and predicted VLE for methane (1) + n-hexadecane (2). Experimental data: T ) 623.15 K (b, O),25,26 T ) 573.15 K (2, 4),26 T ) 423.15 K (9, 0),26 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 5. Experimental and predicted VLE for ethane (1) + n-decane (2). Experimental data: T ) 510.93 K (2, 4),20 T ) 344.26 K (b, O),23 (- -) PSRK, (‚‚‚) PSRK(Li), (- ‚ -) LCVM, and (s) this work.

Figure 8. Experimental and predicted VLE for methane (1) + n-dotriacontane (2). Experimental data: T ) 623.15 K (b),27 T ) 523.15 K (2),27 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 6. Experimental and predicted VLE for ethane (1) + n-octacosane (2). Experimental data: T ) 573.15 K (2),24 T ) 348.20 K (b),21 (- -) PSRK, (‚‚‚) PSRK(Li), (- ‚ -) LCVM, and (s) this work.

While for alkane + alkane systems no parameters are used, for the following systems, new group interaction parameters had to be fitted. But, all systems which belong to one main group combination (e.g., carbon

dioxide + alkane) are predicted with the same group interaction parameters, which are listed in Table 3. Methane + Alkane Systems. The system CH4 + n-hexadecane in Figure 7 is characterized by the fact that the behavior at the critical point as well as the vapor fraction is better described by the new GCEOS in comparison to PSRK(Li). The original PSRK model predicts pressures which are much to high. A good agreement between experimental and predicted values of the new GCEOS and the PSRK(Li) model is also obtained for the system CH4 + n-dotriacontane (Figure 8), while large deviations are obtained for PSRK even at low mole fractions. Carbon Monoxide + Alkane Systems. Figures 9 and 10 illustrate that moderate (e.g., CO + n-dodecane) as well as strong (CO + n-hexatriacontane) asymmetric

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Figure 9. Experimental and predicted VLE for carbon monoxide (1) + n-dodecane (2). Experimental data: T ) 344.30 K (2),28 T ) 410.90 K (b),28 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 12. Experimental and predicted VLE for hydrogen (1) + n-tetradecane (2). Experimental data: T ) 328.15 K (2),31 T ) 473.15 K (b),31 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 10. Experimental and predicted VLE for carbon monoxide (1) + n-hexatriacontane (2). Experimental data: T ) 473.05 K (2),29 T ) 572.95 K (b),29 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 13. Experimental and predicted VLE for hydrogen (1) + n-hexatriacontane (2). Experimental data: T ) 473.05 K (2),29 T ) 573.15 K (b),29 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 11. Experimental and predicted VLE for hydrogen (1) + ethane (2). Experimental data: T ) 135.15 K (9, 0),30 T ) 149.50 K (b, O),30 T ) 169.40 K ([, ]),30 T ) 189.55 K (2, 4),30 T ) 212.15 K ([, ])30, (- -) PSRK, and (s) this work.

systems are well predicted with the new GCEOS. PSRK(Li) shows worse results and the original PSRK unsatisfying results.

Figure 14. Experimental and predicted VLE for carbon dioxide (1) + n-pentadecane (2). Experimental data: T ) 316.00 K (2, 4),32 T ) 292.00 K (b, O),32 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Hydrogen + Alkane Systems. Fitting new group interaction parameters should improve the description of asymmetric as well as nonasymmetric systems. The new GCEOS delivers a reliable prediction for the system

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Figure 15. Experimental and predicted VLE for carbon dioxide (1) + n-hexatriacontane (2). Experimental data: T ) 573.25 K (2),33 T ) 473.35 K (9),33 T ) 423.20 K (b),34 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 17. Experimental and predicted VLE for hydrogen sulfide (1) + n-hexadecane (2). Experimental data: T ) 523.15 K (2),36 T ) 373.15 K (b),36,37 T ) 323.15 K (9, 0),36,37 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

Figure 16. Experimental and predicted VLE for ethane (1) + hydrogen sulfide (2). Experimental data: T ) 283.15 K (2, 4),35 T ) 255.32 K (b, O),35 T ) 199.93 K (9, 0),35 (- -) PSRK, and (s) this work.

Figure 18. Experimental and predicted VLE for hydrogen sulfide (1) + n-eicosane (2). Experimental data: T ) 423.25 K (2),37 T ) 323.00 K (b),37 (- -) PSRK, (‚‚‚) PSRK(Li), and (s) this work.

H2 + ethane (Figure 11) even up to 600 MPa. Significant improvements in comparison to PSRK and PSRK(Li) are also achieved for H2 + n-tetradecane (Figure 12). Larger deviations between experimental and predicted VLE are only observed for strong asymmetric systems, such as that shown in Figure 13 for the system H2 + nhexatriacontane. Carbon Dioxide + Alkane Systems. Also the VLE behavior of CO2 + n-pentadecane (Figure 14) is much better predicted by the new GCEOS. Both PSRK and PSRK(Li) fail to predict the phase equilibrium at 316 K. The phase equilibria with n-hexatriacontane (Figure 15) are slightly better described with PSRK(Li) than with the new GCEOS, but for both models, the improvements are obvious when compared to PSRK. Hydrogen Sulfide + Alkane Systems. For fitting new group interaction parameters, the already available original PSRK parameters can be used as initial values. Astonishing results were observed for H2S + alkane systems for the new GCEOS using the original PSRK parameters because with the new GCEOS generally better results are obtained than with PSRK or PSRK(Li), for which the parameters were fitted. Improvements are achieved for the system H2S + ethane (Figure

16), although for the pure component H2S the generalized R parameters from Table 2 were used. The generalized parameters correctly reproduce the vapor pressure of the pure component H2S. For the system H2S + n-hexadecane (Figure 17), the slopes of the pressure with composition are predicted very well with the new GCEOS. Also, for the system H2S + n-eicosane (Figure 18), satisfying results are observed with PSRK(Li) and the new GCEOS in contrast to PSRK. System of CO2 + Ethane. As can be seen in Figure 19, not only the VLE behavior of this system but also the azeotropic data are described well with the developed GCEOS in the temperature range covered. Moreover, the model is able to predict the critical line of CO2 + ethane. It should be mentioned that the same parameters are used for all other CO2 + alkane systems (e.g., Figures 14 and 15). Besides the prediction of VLE, the fitted group interaction parameters can be used to calculate excess enthalpies and excess volumes to prove the temperature and pressure dependence of the excess Gibbs energy:

( ) ∂gE/T ∂1/T

P,x

) hE

(21)

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Figure 19. Experimental and predicted VLE for carbon dioxide (1) + ethane (2). Experimental data (b, O) at T ) 298.15 K,38 T ) 291.15 K,38 T ) 263.15 K,39 T ) 220.00 K,40 azeotropic data (0),41,42 critical data (2),19 (s) this work (VLE), (‚‚‚) this work (azeotropic points), and (- -) this work (critical line).

Figure 20. Experimental and predicted excess enthalpies for carbon dioxide (1) + ethane (2). Experimental data: T ) 300.10 K, P ) 7.5 MPa (2);43 T ) 293.15 K, P ) 7.58 MPa ([);44 T ) 272.10 K, P ) 4.5 MPa (9);43 T ) 272.10 K, P ) 11.0 MPa, (b);43 (s) this work.

( ) ∂gE ∂P

T,x

) vE

(22)

Experimental data for these thermodynamic properties are scarce. However, for the system CO2 + ethane, a few experimental data are available. The heats of mixing depend on the temperature as well as the pressure. Both dependencies are correctly reproduced by the new GCEOS (Figure 20): On the one hand, with increasing temperatures from 272.1 up to 300.1 K, higher excess enthalpies are observed. On the other hand, increasing pressures (from 4.5 to 11.0 MPa) lead at constant temperatures (272.1 K) to lower excess enthalpies. Both dependencies are predicted by the new GCEOS VTPR. Furthermore, a good agreement between experimental and predicted excess volume effects is observed (Figure 21). The increase of the volume change ∆v at 291.6 K and 58.5 bar, which nearly corresponds to the azeotropic pressure, at a mole fraction of about 0.5 (Figure 22) is caused by a phase transition18 because the system becomes supercritical. When the pressure is raised to 62 bar, the mixture is subcritical again and, as a consequence, ∆v becomes much smaller. Because

Figure 21. Experimental and predicted excess volumes for carbon dioxide (1) + ethane (2). Experimental data: T ) 300.15 K, P ) 7.5 MPa ([);43 T ) 270.15 K, P ) 4.5 MPa (b);43 T ) 270.15 K, P ) 11.0 MPa (2);43 (s) this work.

Figure 22. Experimental and predicted volume effects for carbon dioxide (1) + ethane (2). Experimental data: T ) 291.60 K, P ) 5.85 MPa (2);45 T ) 291.6 K, P ) 6.2 MPa (b);45 (s) this work.

excess enthalpy and volume data are not taken into account in the fitting procedure, the predictions are surprisingly good. Conclusion A new GCEOS (VTPR) has been presented with the aim of replacing the existing PSRK model, which still contains a few weaknesses. A volume-translated PR EOS (VTPR) delivers a much better description of liquid densities of pure components and mixtures in comparison to the SRK EOS. The fitted Twu R-function parameters as well as the generalized form guarantee a good reproduction of the purecomponent vapor pressures. The VTPR EOS is combined with the residual part of the UNIFAC method by a newly derived gE mixing rule. Together with the use of a nonlinear b mixing rule, a distinct improvement of the prediction for asymmetric systems is achieved. The results are comparable or even better than the predictions with the LCVM mixing rule and the Li correction for the PSRK model, respectively. Especially, the behavior at the critical point and the vapor mole fraction can be better described with the new GCEOS, which is, for instance, an important aspect for the development and design of extraction processes with supercritical

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gases. Group interaction parameters for alkanes with CH4, CO, H2, CO2, and H2S are fitted to VLE over a large range of pressure and temperature. These alkane + gas systems play an important part in the petroleum and gas processing industry. The predictions of excess enthalpies and excess volumes for the system CO2 + ethane are surprisingly good. Besides better results for volumetric properties and asymmetric systems, temperature-dependent group interaction parameters (only temperature-independent interaction parameters are used in the UNIFAC model, which is implemented in the PSRK method) for all main group combinations (uniform parameter set) will be fitted simultaneously to VLE, excess enthalpies (hE), and activity coefficients at infinite dilution (γ∞) in the future. The first results for alkanes, aromatics, and ketones, which will be presented in the next paper, are very promising. Moreover, the range of applicability will be extended to other mixture properties such as solid-liquid equilibria (SLE) of eutectic systems. In this context, the Dortmund Data Bank, which today contains approximately 22 400 VLE, 15 600 hE, 36 700 γ∞, and 10 400 SLE data sets, is an important tool for further model development. The next step could be the extension of the GCEOS VTPR to systems with strong electrolytes, which is already realized for the PRSK model.3

u ) inverse packing fraction v ) molar volume [m3/mol] V ) volume to mole fraction ratio VLE ) vapor-liquid equilibrium x ) liquid mole fraction y ) vapor mole fraction z ) compressibility factor

Acknowledgment

Superscripts

The authors thank the “Fonds der Chemischen Industrie” for financial support of J.A. by means of a scholarship.

E ) excess L ) liquid phase ∞ ) infinite * ) Li correction

Greek Letters R ) temperature-dependent function of a(T) γ ) activity coefficient F ) density ω ) acentric factor ψ ) temperature-dependent function in the residual part Subscripts 0 ) at standard condition c ) critical point c ) number of C atoms comb ) combinatorial part gen ) generalized i, j ) components i and j k ) structural group m, n ) main groups m and n r ) reduced res ) residual part calc ) calculated exp ) experimental

Nomenclature a ) cohesive energy parameter of the PR EOS A ) constant of the gE mixing rule anm ) temperature-independent group interaction parameter b ) volumetric parameter of the PR EOS bnm ) linear temperature-dependent group interaction parameter c ) temperature-independent volume correction for the VTPR equation [m3/mol] cnm ) quadratic temperature-dependent group interaction parameter DDB ) Dortmund Data Bank EOS ) equation of state g ) Gibbs energy GCEOS ) group contribution equation of state F ) surface area to mole fraction ratio h ) molar enthalpy [J/mol] L, M, N ) Twu-Bluck-Cunningham-Coon R-function parameters OF ) objective function P ) pressure [MPa] PR ) Peng-Robinson equation of state PSRK ) predictive Soave-Redlich-Kwong r ) relative van der Waals volume parameter of the molecule i R ) general gas constant [J/mol‚K] R ) relative van der Waals volume parameter of the structural group k q ) relative van der Waals surface area parameter of the molecule i Q ) relative van der Waals surface area parameter of the structural group k SRK ) Soave-Redlich-Kwong equation of state T ) absolute temperature [K]

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Received for review January 16, 2002 Revised manuscript received May 2, 2002 Accepted May 8, 2002 IE020047O