Phase Equilibrium Modelling for Mixtures with ... - ACS Publications

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Ind. Eng. Chem. Res. 2008, 47, 5660–5668

Phase Equilibrium Modelling for Mixtures with Acetic Acid Using an Association Equation of State Nu´ria Muro-Sun˜e´, Georgios M. Kontogeorgis,* Nicolas von Solms, and Michael L. Michelsen Centre for Phase Equilibrium and Separation Processes (IVC-SEP), Department of Chemical Engineering, Technical UniVersity of Denmark, Lyngby, Denmark

Acetic acid is a very important compound in the chemical industry with applications both as solvent and intermediate in the production of, e.g., polyesters. The design of these processes requires knowledge of the phase equilibria of mixtures containing acetic acid and a wide variety of compounds over extended temperature and pressure ranges. From the scientific point of view, modeling of such equilibria is challenging because of the complex association and solvation phenomena present. In this work, a previously developed association equation of state (cubic-plus-association, CPA) is applied to a wide variety of mixtures containing acetic acid, including gas solubilities, cross-associating systems (with water and alcohols), and polar chemicals like acetone and esters. Vapor-liquid and liquid-liquid equilibria are considered for both binary and ternary mixtures. With the exception of a somewhat inferior performance for the water-acetic acid VLE, which does not seem to affect substantially the performance for the multicomponent systems studied, CPA performs satisfactorily in most cases, using a single interaction parameter over extensive temperature ranges. For accurate description of water-acetic acid, use of the Huron-Vidal mixing rule for the energy parameter of CPA can yield a satisfactory correlation at the cost of more interaction parameters. 1. Introduction Acetic acid is used as a solvent (in aqueous solution) in the production of intermediates of polyesters, vinyl acetate monomer, and many other applications in the chemical industry. Mixtures of acetic acid with esters, ethers, alcohols, and gases are of interest, and an accurate modeling of the phase equilibria of these systems is important in process optimization. 1.1. Previous Modeling Attempts. The association of acetic acid has been studied with a variety of methods, including spectroscopic and quantum chemistry/ab initio approaches,1–3 chemical models specifically accounting for the dimerization of organic acids,4,5 lattice-fluid models,6,7 and various association models which belong to the SAFT family.8–11 Most of these investigations are limited either to pure acetic acid, acetic acid-alkanes, or the acetic acid-water system. Very few of these studies include investigations for mixtures with acetic acid and various chemicals including gases as well as multicomponent mixtures. Moreover, the performance of most models for water-acetic acid is not satisfactory for industrial applications. 1.2. Objective of the Current Work. The main objective of this work is to use the cubic-plus-association (CPA) equation of state for modeling phase equilibria of a wide variety of mixtures containing acetic acid and over a wide range of temperatures and pressures. Because of the variety of the applications and compounds involved in the industrial processes mentioned above, satisfactory modeling should include the following cases: • description of phase equilibria of acetic acid with various nonpolar (hydrocarbons, gases), polar (acetates, ethers), and associating (alcohols, water) compounds; • predictions of various types of phase equilibria (VLE, SLE, and LLE) over extensive temperature and pressure ranges; • use of a minimum number of adjustable interaction parameters in order to ensure predictive capabilities of the * Corresponding author. E-mail: [email protected].

model, especially prediction of ternary and multicomponent phase equilibria based solely on parameters obtained from binary data; and • accurate description of the water/acetic acid VLE. CPA first appeared in the literature in 1996,12 and previous applications have been recently reviewed.13,14 Thus, a short description of the model is presented in Appendix A, and for more information the reader is referred to refs 12-14. Organic acids pose an interesting challenge for thermodynamic models because of their strong associating behavior, with both self- and cross-association. Organic acids have been previously modeled with CPA using the 1A association scheme,15 and this assumption is also adopted here (for the water-acetic acid system, we will also test the 2B scheme for the acid). When referring to association schemes, we employ the terminology of Huang and Radosz for the SAFT equation of state,16 and the association schemes used in this work are

Figure 1. P-x-y diagram for hydrogen (1)-acetic acid (2) at 25, 50, and 75 °C. A constant binary interaction parameter of kij ) -0.3 was employed at all temperatures. Experimental data are from ref 18.

10.1021/ie071205k CCC: $40.75  2008 American Chemical Society Published on Web 06/25/2008

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5661 Table 1. Association Schemes Used in This Work (from Ref 16); The Expressions for the Unbonded Site Fractions XA are Given for the 1A, 2B, and 4C Schemes XA approximations

type

∆ approximations

1A 2B

∆AA + 0 ∆AA ) ∆BB ) 0 ∆AB + 0 ∆AA ) ∆AB ) ∆BB ) ∆CC ) ∆CD ) ∆DD ) 0 ∆AC ) ∆AD ) ∆BC ) ∆BD + 0

4C

XA

XA ) XB

- 1 + √(1+4F∆)⁄2F∆ - 1 + √(1+4F∆)⁄2F∆

X A ) XB ) XC ) XD

- 1 + √(1+8F∆)⁄4F∆

Table 2. CPA Pure-Compound Parameters for the Associating Compounds Used in This Work; Water is Modeled as a “4C” Molecule, Acetic Acid is Modeled as “1A”, and Propanol, Acetone, and Isobutyl Acetate are Modeled as “2B” (See Table 1) compound

Tc (K)

b (L/mol)

a0 (bar L2 mol-2)

c1

εAB (bar L mol-1)

water propanol isobutyl acetate acetic acid acetone

647.29 536.78 561.8 591.95 508.20

0.014515 0.064110 0.11172 0.046818 0.0592

1.2277 11.9102 17.52187 9.11957 7.8643

0.67359 0.91709 0.86195 0.4644 0.99510

166.55 210.00 184.10 403.23 111.73

β

AB

× 103

69.2 8.10 49.41 4.52 289

Table 3. CPA Pure-Compound Parameters for the Inert Compounds Used in This Work; All Optimizations are Made in the Tr Range 0.5-0.9 family

ethers esters

a

compound

a0 (bar L2mol-2)

b (L mol-1)

c1

∆P (%)

∆F (%)

acetic anhydride benzene p-xylene DnPEa DiPEa EPEa methyl formate methyl acetate ethyl acetate isobutyl acetate n-butyl acetate

23.31410 17.876 29.31663 26.38364 24.33523 21.76524 10.52239 14.59349 18.88000 28.07736 29.23732

0.08728 0.07499 0.1098 0.11578 0.11578 0.10034 0.05006 0.06625 0.08338 0.11631 0.11788

1.03323 0.7576 0.86256 0.92963 0.94902 0.84978 0.79622 0.87185 0.94265 1.03889 1.02499

0.55 0.90 0.12 0.52 0.88 0.89 0.51 0.57 0.88 0.93 0.81

1.79 1.00 0.41 0.8 1.24 0.45 0.55 0.45 0.52 0.62 0.96

DnPE ) di-n-propylether, DiPE ) diisopropylether, and EPE ) ethyl propyl ether.

presented in Table 1. The 1A scheme was chosen because it provided the best results for mixtures of acetic acid with hydrocarbons15 including azeotropic behavior and gave the best performance for the pure acid.15 Previous publications have only considered pure acetic acid and mixtures with hydrocarbons,15 ternary water-acetic acid-hexane LLE,17 while recently a few results for crossassociating systems with water and alcohols have been presented.14 In accordance with previous investigations,13,14 alcohols are modeled using the two-site (2B) scheme and water is modeled using the four-site (4C) scheme. Tables 2 and 3 present the CPA parameters for all associating and inert compounds considered in this work. The parameters are optimized based on vapor pressures and liquid-density data generated from the Design Institute for Physical Property Data (DIPPR) correlations. For all gases considered in this work (hydrogen, methane, carbon dioxide, carbon monoxide, and nitrogen), critical properties and acentric factors are used. The optimized parameters for n-alkanes are available in ref 13. The next section presents results for gas solubilities in acetic acid mixtures and acetic acid-hydrocarbon mixtures, followed by selected results for mixtures of acetic acid with polar and associating compounds (esters, ethers, acetone, and alcohols). Then a detailed study of the key acetic acid-water system is presented, including the modeling difficulties and the development of a modification of the CPA equation of state needed for the correlation of this mixture. Finally, predictions are shown for three multicomponent systems (VLE and LLE), followed by our conclusions. 2. Results and Discussion 2.1. Mixtures with Gases and Hydrocarbons. Mixtures of acetic acid with inert gases and with hydrocarbons have been

considered in this work. Results for binary systems of acetic acid with each of the gases hydrogen, methane, carbon dioxide, carbon monoxide, and nitrogen are shown in Figures 1–5. The experimental data in Figures 1–4 are from Jo´nasson et al.,18 while those of Figure 5 are from Efremova et al.19 While the results for carbon dioxide and carbon monoxide are excellent (pure predictions, kij ) 0), a large but temperatureindependent interaction parameter was required for hydrogen and nitrogen (equal to -0.3 and 0.15, respectively). The results for methane are not as satisfactory as for the other systems. The model does not predict the correct temperature dependency of the gas solubility.

Figure 2. P-x-y diagram for methane (1)-acetic acid (2) at 25, 50, and 75 °C. The correct temperature trend on the solubility of the gas in the acid is predicted by the model only when temperature-dependent parameters are used. A temperature-dependent binary interaction parameter kij is thus required. Experimental data are from ref 18.

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Figure 3. P-x-y diagram for carbon dioxide (1)- acetic acid (2) at 25, 50, and 75 °C. The lines are pure predictions. Experimental data are from ref 18.

Figure 4. P-x-y diagram for carbon monoxide (1)- acetic acid (2) at 25, 50, and 75 °C. The lines are pure predictions. Experimental data are from ref 18. Table 4. Binary Interaction Parameters for LLE of Hydrocarbons-Acetic Acid and Percentage Deviation between Experimental and Calculated Concentrations; Experimental Data from Ref 44 system acetic acetic acetic acetic acetic acetic

acid-octane acid-nonane acid-decane acid-undecane acid-dodecane acid-cyclohexane

P (kPa)

kij

∆x, acetic (%)

101.32 101.32 101.32 101.32 101.32 101.32

0.046 0.043 0.0408 0.0390 0.0379 0.077

6.52 7.96 4.11 4.02 9.14 1.20

In particular, the acetic acid/nitrogen system shows some peculiarities. CPA correlates well the experimental data at the two lowest temperatures (323.2 and 373.2 K), while the results are less satisfactory at the higher temperatures. With respect to this, it is worth mentioning that the experimental data at the higher temperatures present an unexpected behavior, because the solubility of nitrogen decreases with increasing temperature except for at the temperature of 423.2 K. Further investigation may be required here with respect to the validity of the experimental data. Mixtures of acetic acid with hydrocarbons have been also investigated. Because acetic acid/alkane VLE has been previously extensively studied,15 emphasis is given here on acetic

acid/alkane LLE and acetic acid/aromatic hydrocarbons VLE. Table 4 presents all LLE correlation results, while Figures 6 and 7 show some typical results. The following comments summarize our observations: (1) CPA describes satisfactorily acetic acid/alkane LLE away from the critical consolute area. As expected, the results are less satisfactory as the critical point is approached. (2) Equally satisfactory LLE results are obtained whether the critical properties or the fitted parameters are used in CPA for alkanes (see Appendix A), although the model’s performance using the latter values is slightly better. (3) There is, to our knowledge, no experimental indication of immiscibility between acetic acid and aromatic hydrocarbons; the only LLE data found were those for formic acid with benzene. This observation, in combination with the fact that acetic acid is immiscible with n-alkanes at lower temperatures, would probably indicate that solvation between acetic acid and aromatics is of importance. However, as Figure 7 indicates, inclusion of solvation makes little difference. This is further verified by the small positive interaction parameters obtained. This observation is in agreement with the CPA behavior for alcohols with aromatic hydrocarbons, where again the solvation was of minor importance in the VLE calculations (though somewhat more pronounced when infinite dilution activity coefficients were considered). 2.2. Polar Chemicals and Alcohols. Phase equilibria of mixtures containing acetic acid with a wide variety of chemicals have been considered: esters, ethers, acetone, and self-associating polar compounds like alcohols. These systems are characterized by a variety of molecular interactions, including strong dispersion, polar, hydrogen bonding, and cross-association effects. Although esters and ethers are polar compounds, they are primarily considered inert in this work. i.e.. non self-associating. In accordance with previous investigations,20,21 acetone is modeled as a 2B self-associating fluid. Selected results are shown in Figures 8–10 for three esters (methyl acetate, ethyl acetate, and isobutyl acetate), Figure 11 for acetone, and Figures 12–14 for one ether, one alcohol, and acetic anhydride. Overall, the performance of CPA is satisfactory for these systems. Specifically, a small temperature-independent binary interaction parameter was required to correlate the data for the systems with methyl- and ethyl acetate. However, with isobutyl acetate, use of just a single kij was inadequate. Good results were obtained if it was assumed that isobutyl acetate was selfassociating. This was surprisingly not needed for butyl acetate, where satisfactory description of butyl acetate/acetic acid VLE is obtained even when the ester is considered inert. Figure 11 for acetic acid-acetone shows that excellent predictions (kij ) 0) are obtained when acetone is considered self-associating. CPA successfully captures the negative deviations from Raoult’s law. The Elliott combining rule30 is used. Similarly good results are obtained for the acetic acid/alcohol and ethers considered, as shown for a few systems in Figures 12 and 13. Other mixtures, e.g., diisopropyl ether/acetic acid perform similarly. The small negative interaction parameters may indicate a slight underestimation of the cross-association. In the acetic acid/alcohol mixtures, the Elliott combining rule (Appendix A) is used for describing the cross-association. 2.3. Water-Acetic Acid and Ternary Phase Equilibria. The water-acetic acid VLE has been studied systematically because of its industrial importance. The separation process of acetic acid/water mixtures, especially the recovery of acetic acid from water, is an important industrial process, usually done by

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Figure 5. P-x-y diagram for the system N2 (1)-acetic acid (2). Experimental data at 323.2 K (0), 373.2 K(2), 423.2 K (•), and 473.2 K (O), from ref 19. CPA calculations with kij ) 0.1515 at 323.2 K (black line), 373.2 K (blue line), 423.2 K (purple line), and 473.2 K (red line).

Figure 6. LLE for acetic acid-n-dodecane. The full line indicates CPA calculations using alkane parameters fitted to vapor pressures and liquid densities. The dashed line indicates CPA calculations using critical properties for the alkane (kij ) 0 is the optimum value in this case).

Figure 7. VLE for the system acetic acid-benzene at 101.33 kPa. Experimental data are from refs 35 (∆) and 36 (2). CPA calculations without solvation, kij ) 0.0434 (s), and with solvation, kij ) 0.05 and βAiBj ) 0.0019 (---).

distillation or by liquid-liquid extraction and for which phase equilibria (VLE as well as LLE) data are required. The process design is sensitive to the VLE data, particularly at the higher pressures and temperatures. This is because, at the water-rich end, the relative volatilities approach unity. A short discussion of the various interactions and associating schemes in water-acid-hydrocarbons is of interest. Results for ternary mixtures with acetic acid, water, and aromatic hydrocarbons are shown later. The difference between the various association schemes is not just in the different numbers of association sites (four for water, one for acid). The water sites are specified so that only sites of different types can bond. The single acid site is assumed to bind to all other site typessanother acid site and both types of water sites. There is more than one association scheme that can be used for acids, but the general consensus is that the 1A scheme is most appropriatesespecially

Figure 8. T-x-y diagram for acetic acid (1)-methyl acetate (2) at P ) 1 atm. Symbols are for experimental data, ())37 and (∆),22 and the lines are for the CPA calculations with kij ) 0 (---) and kij ) -0.0698 (s).

Figure 9. P-x-y diagram for acetic acid (1)-ethyl acetate (2) at three temperatures from T ) 323.15 to 373.15 K. A binary interaction parameter of kij ) -0.0404 was used at all three temperatures. Symbols are for experimental data23 at 323.2 K ()), 343.2 K (∆), and 373.2 K (0), and the lines are the CPA calculations with kij ) -0.0404 at 323.2 K (black line), 343.2 K (red line), and 373.2 K (green line).

when considering the pure acid and acid/alkanes.15 Finally, we model the solvation with benzene by assuming that benzene has a single association site that does not self-associatesso it is not like a 1A site. This introduces an additional binary interaction parameter, since the solvation strength is adjusted to fit the LLE behavior in the benzene-water binary system. No solvation has been assumed between acetic acid and benzene in this work, in accordance to the discussion in the previous section. Figure 15 shows VLE calculations for the system acetic acid-water at 462 K using CPA and the 1A scheme for the acid. Calculations are compared to the experimental data from Freeman and Wilson.26 Only qualitatively correct behavior is obtained using a single binary interaction parameter. Similar results are obtained at other temperatures. In all cases, large

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Figure 10. P-x-y diagram for acetic acid (1)-isobutyl acetate (2) at T ) 390.15 K. Symbols ()) are experimental data24 at 390.15 K. The lines are the CPA calculations with kij ) 0, for nonassociating acetate (black line) and for acetate with a 2B association scheme (red line).

Figure 11. P-x-y diagram for the system acetic acid (1)-acetone (2) at T ) 303.15 and 323.15 K. Acetone is modeled as a 2B self-associating molecule, with parameters from ref 20. The curves are predictions. Experimental data are from ref 25.

• Correction factors have been introduced to the association strength. • An additional site has been introduced on acetic acid that could cross-associate with water but not self-associate. • Different interactions between the acetic acid and the positive/negative water sites have been implemented. None of the above modifications led to satisfactory improvement. Moreover, the overall performance of CPA seems similar (a bit better) to that of various SAFT-type approaches as reported in the literature.3,8,27 Preliminary results with other SAFT-type approaches28–31 have not led to improved representation of the water-acetic acid VLE. 3. CPA-Huron-Vidal Modification As neither improvements in the association contribution nor the functional form of the physical term seem to improve the results, a different approach has been followed by modifying the mixing rules of the energy parameter. In contrast to the SAFT family, CPA provides us with the possibility for a fairly straightforward extension of the physical part. Thus, a Huron-Vidal mixing rule can be used together with a modified nonrandom two-liquid (NRTL) expression instead of the van der Waals one-fluid mixing rule, and the equations involved are presented in Appendix A. This approach introduces additional parameters in the model. Three parameters are needed, while the NRTL nonrandomness parameter is fixed to 0.3. Two of the three adjustable parameters are in the energy term of NRTL, while an additional binary interaction parameter acting on the acid-water cross-association energy is introduced. Some results are shown in Figures 16 and 17. The improvement over the conventional CPA is significant. The difficulty in describing the VLE can be ascribed to large, competing terms of opposite sign. For the data set at atmospheric pressure, we found, somewhat surprisingly, that the fraction of nonbonded sites for acetic acid is 4 times lower at infinite dilution in water than in pure acetic acid (0.007 vs 0.028). This roughly corresponds to a difference by a factor of 4 in the fugacity coefficient and stresses the sensitivity to the mechanisms and expressions used for the cross-association. In addition, the NRTL contribution corresponds to an infinite dilution activity coefficient for acetic acid below 0.1. These negative effects on the activity of acetic acid in the water-rich end are then countered by a large, positive contribution from the physical term in the equation of state. 4. Ternary VLE and LLE

Figure 12. VLE for the system di-n-propyl ether-acetic acid at 101.32 kPa. Experimental data ()) are from ref 38. CPA calculations with kij ) 0 (s) and kij ) -0.0283 (---).

negative interaction parameters are required. Assuming that the acid is a 2B associating molecule can improve the correlation slightly. However, this approach was not explored further since, for the sake of consistency, generality, and especially purecomponent considerations, it was decided to model acids as 1A throughout this work. Various attempts have been undertaken in order to improve these results with CPA: • Different pure acetic acid parameters have been regressed based on vapor pressures and liquid-density data.

Figures 18–21 present predictions for one ternary VLE system and two ternary LLE mixtures. In all of these calculations, the original CPA approach has been used for water-acetic acid, i.e., using the van der Waals one-fluid mixing rules and not the Huron-Vidal approach. First, the ternary system acetic acid-water-carbon dioxide at 60 bar and 333.1 K is shown in Figure 18, with vapor in Figure 18a and liquid in Figure 18b. The carbon dioxide gas/ liquid ratio at 25 °C is shown in Figure 19. The ternary data are from the group of Maurer.32 The lines are predictions based on correlation of the binary systems; thus, for water-acetic acid, kij ) -0.223, and for carbon dioxide-acetic acid, kij ) 0. In the binary system carbon dioxide-water, there is no kij, but carbon dioxide solvates in water as described previously.14 Excellent agreement is obtained with the experimental data.

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5665

Figure 13. (left) VLE for the system acetic acid-propanol at 93.99 kPa (experimental data are from ref 39 ())). CPA calculations at 93.99 kPa with CR-1 and with kij ) -0.0269 (s) and kij ) -0.0078 (---). (right) VLE at 101.32 kPa (experimental data are from refs 40 (∆) and 41 (2). CPA calculations at 101.32 kPa with CR-1 and kij ) -0.0269 (s) and kij ) -0.0444 (---).

Figure 14. Binary VLE for acetic acid with acetic anhydride. Experimental data from ref 42 (333.15 and 353.15 K) and ref 43 (365.15 K).

Figure 16. VLE in the system acetic acid (1)-water (2) at 462 K. Experimental data are from ref 26. The calculations are performed with CPA using Huron-Vidal mixing rules and the following parameter values: A12 ) -8.5, A21 ) -981.6, and D12 ) 0.601. The aij value is fixed to 0.3. The 1A scheme is used for acetic acid.

Figure 15. VLE in the system acetic acid (1)-water (2) at 372.8 K. Experimental data are from ref 26. The calculations are performed with CPA using Elliott’s combining rule (ECR) and k12 ) -0.223. The 1A scheme is used for acetic acid.

Finally, two water-aromatic hydrocarbon-acetic acid ternary LLE systems were studied (water-benzene-acetic acid and water-xylene-acetic acid). The ternary data are digitized from Suresh and Beckman.33 The kij’s for the binary systems acetic acid-water and acetic acid-benzene were taken from VLE data. The water-benzene and water-xylene binary systems were modeled including solvation, as discussed recently.34 Despite the rather poor performance of the binary water-acetic acid system, the ternary systems studied were modeled quite well, as can be seen in Figures 20 and 21. 5. Conclusions An association equation of state, the cubic-plus-association (CPA), has been applied to mixtures containing acetic acid and a large variety of chemicals ranging from gases and inert

Figure 17. VLE in the system acetic acid (1)-water (2) at 1 bar. Experimental data are from ref 45. The calculations are performed with CPA using Huron-Vidal mixing rules and the following parameter values: A12 ) -8.5, A21 ) -981.6, and D12 ) 0.601. The aij value is fixed to 0.3. The 1A scheme is used for acetic acid.

hydrocarbons, polar nonassociating compounds like ethers and esters, up to hydrogen bonding substances like alcohols and water. Very satisfactory vapor-liquid and liquid-liquid equilibria are achieved in most cases using a single binary-specific interaction parameter. One notable exception is the water-acetic acid VLE, where satisfactory results can only be achieved by adding a local-composition excess Gibbs energy model in the mixing rule for the energy parameter and at the cost of more

5666 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

Figure 20. Ternary LLE in the system acetic acid (1)-benzene (2)-water (3) at atmospheric pressure. The dotted lines are the experimental tie-lines. The solid pink lines are model predictions. Experimental data are from ref 33. The interaction parameters are as follows: kij ) -0.223 for water-acetic acid, kij ) 0.0355 and βAiBj ) 0.079 for water-benzene, and kij ) 0.035 for acetic acid-benzene (value estimated at 292.15 K, no solvation assumed).

Figure 18. Ternary VLE in the system carbon dioxide-water-acetic acid. Vapor-phase compositions are shown in (a), and liquid-phase compositions are shown in (b). The lines are predictions based on correlations of the individual binary systems. Experimental data are from ref 32. The interaction parameters are as follows: kij ) -0.223 for water-acetic acid, kij ) 0 for carbon dioxide-acetic acid, and kij ) 0 and βΑιBj ) 0.2 for water-carbon dioxide.

Figure 21. Experimental data (points) and CPA calculations (s) for the ternary system: p-xylene-acetic acid-water at 298 K. Experimental data are from ref 33. The interaction parameters are as follows: kij ) -0.223 for water-acetic acid, kijj ) -0.0133 and βΑιBj ) 0.0667 for water-pxylene, and kij ) 0.0171 for acetic acid-p-xylene.

represent several ternary systems with acetic acid, both VLE and LLE, using only a single interaction parameter per binary system. Acknowledgment The authors wish to acknowledge the financial support from the companies supporting the CHIGP (Chemicals in Gas Processing) consortium (BP, Statoil, TOTAL, and Mærsk Oil and Gas). Appendix A: CPA Equation of State The CPA equation of state (EoS), proposed by Kontogeorgis et al.,12–14 can be expressed for mixtures in terms of pressure P, as follows: P)

Figure 19. Ternary VLE in the system carbon dioxide-water-acetic acid. Carbon dioxide gas/liquid ratio as a function of carbon dioxide liquid mole fraction. Interaction parameters are the same as in the caption of Figure 18.

adjustable binary parameters, especially if broad temperature ranges should be covered. However, CPA can adequately

R(T) 1 RT ∂ ln g RT 1+F Vm - b Vm(Vm + b) 2 Vm ∂F

(

)∑ x ∑ × i

i

Ai

(1 - XAi)

(1)

The key element of the association term is XA, which represents the mole fraction of the molecule i not bonded at site A, while xi is the mole fraction of component i. XA is related to the association strength ∆AiBj between two sites belonging to two

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5667

b)

∑xb

i i

i

For extending the CPA EoS to mixtures of two associating compounds, e.g., alcohols or glycols with water, combining rules for the association energy (εAiBj) and the association volume (βAiBj) are required. Over the years, different combining rules have been suggested; however, in this work, only the CR-1 and the Elliott combining rules (ECR), described below, are used. Only these two combining rules have been found successful in previous applications. The expressions of the cross-association energy and crossassociation volume parameters with CR-1 are as follows: Figure A1. Various possible associations that can occur in the ternary system water-benzene-acetic acid. Only one of each type is shown for clarity. Benzene is modeled as weakly associating in water (solvating).

different molecules, e.g., site A on molecule i and site B on molecule j, determined from XAi )

1

1+F

∑x∑x j

j

Bj

AiBj Bj∆

(2)

Ai

where the association strength ∆ Bj in CPA is expressed as

[ ( ) ]

∆AiBj ) g(F) exp

εAB - 1 bijβAiBj RT

(3)

with the radial distribution function 1 g(F) ) 1 - 1.9n

1 and n ) bF 4

while bi + bj 2 Finally, the energy parameter of the EoS is given by a Soavetype temperature dependency, while b is temperature independent: bij )

a(T) ) a0(1 + c1(1 - √Tr))2 Tr ) T/Tc where Tc is the critical temperature. In the expression for the association strength ∆AiBj, the parameters εAiBj and βAiBj are called the association energy and the association volume, respectively. These two parameters are only used for associating components, and the three additional parameters of the SRK term (a0, b, c1) are the five purecompound parameters of the model. They are obtained by fitting vapor-pressure and liquid-density data. For inert (not selfassociating) components, e.g., hydrocarbons, only the three parameters of the SRK term are required, which can either be obtained from vapor pressures and liquid densities or be calculated in the conventional manner (critical data, acentric factor). When the CPA EoS is used for mixtures, the conventional mixing rules are employed in the physical term (SRK) for the energy and covolume parameters. The geometric mean rule is used for the energy parameter aij. The interaction parameter kij is, in the applications for self-associating mixtures, e.g., alcohol, water, glycol, or acid with n-alkanes, the only binary adjustable parameter of CPA: a)

∑∑xxa , i j ij

i

j

where aij ) √aiaj(1 - kij)

(4)

εAiBi + εAjBj and βAiBj ) √βAiBiβAjBj 2 The expression of the cross-association strength (∆AiBj) with ECR is εAiBj )

∆AiBj ) √∆AiBi∆AjBj Assuming that the radial distribution function in eq 3 is g(F) ≈ 1 as well as the term exp(εAB/RT) - 1 ≈ exp(εAB/RT), it can be shown that the equivalent expressions for the crossassociation energy and cross-association volume parameters with ECR in eq 3 are as follows: εAiBj )

εAiBi + εAjBj 2

and βAiBj ) √βAiBiβAjBj

√bibj bij

Thus, ECR and Elliott are similar, with the only difference being the second term containing the covolume parameters in the expression for the cross-association volume. CPA can be extended to mixtures with one self-associating compound and one inert compound, where there is possibility for solvation between the two compounds. For example, Folas et al.34 have recently showed that CPA can be successfully extended to systems containing aromatic hydrocarbons, by using a modification of the CR-1 combining rule, allowing, however, the cross association volume βAiBj to be optimized from the experimental data. Thus, the cross-association energy parameter is, for associating aromatic or olefinic mixtures, equal to the value of the associating compound (water, alcohol, or glycol) divided by two: εassociating and βAiBj (fitted) 2 Then, the association strength will be estimated by eq 3, and in this way, the built-in temperature dependency of the crossassociation strength is retained for solvating systems. Calculations have showed that this is important in order to obtain satisfactory results, e.g., for water-aromatic hydrocarbons over large temperature ranges. Figure A1 shows the various crossassociating and solvating schemes between water and either acetic acid or benzene. εAiBj )

CPA-Huron-Vidal This variant of CPA has been developed specifically for water/acetic acid VLE. The most important difference from the CPA equation is that, instead of the mixing rule of eq 4, the classical Huron-Vidal (HV) mixing rule is applied as a mixing rule to the parameter in SRK as follows, a ) b

aii

∑z b i

i

ii

-

gE ln 2

(5)

5668 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

where the NRTL model is used as the activity coefficient model: gE ) RT

∑x

∑xG C

i

i

j

ji ji

j

∑xG j

(6)

ji

j

Gji ) bj exp(-RijCji)

(7)

Aji (8) RT Thus, there are three binary parameters required per binary (Aij, Aji, and Rij ) Rji). There is a fourth binary parameter (Dij) that has nothing to do with the HV mixing rules but rather results from the combining rule for the cross-associating strength (the strength of association between an acid site and a water site): Cji )

∆ij ) Dij√∆ii∆jj

(9)

An importance advantage of the HV mixing rule is that, by setting Rij ) 0 and by choosing Aij and Aji appropriately, the classical one-fluid mixing rule with kij is recovered. This is in line with the philosophy of CPA, which itself reduces to SRK in the absence of association. Abbreviations CPA ) cubic plus association DnPE ) di-n-propylether DiPE ) diisopropylether ECR ) Elliott’s combining rule EPE ) ethyl propyl ether EoS ) equation of state HV ) Huron-Vidal LLE ) liquid-liquid equilibria PC-SAFT ) perturbed-chain statistical associating fluid theory SLE ) solid-liquid equilibria SRK ) Soave-Redlich-Kwong equation of state VLE ) vapor-liquid equilibria

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ReceiVed for reView September 7, 2007 ReVised manuscript receiVed February 28, 2008 Accepted March 12, 2008 IE071205K