Modeling of association effects in mixtures of carboxylic acids with associating and non-associating components

O. Ferreira, T. Fornari, E. A. Brignole and S. B. Bottini

Planta Piloto de Ingeniería Química (UNS-CONICET)
CC 717, 8000 Bahía Blanca, Argentina.
sbottini@plapiqui.edu.ar

Abstract ¾ The group contribution with association equation of state GCA-EOS has been applied to calculate thermodynamic properties of pure compounds and mixtures of carboxylic acids with paraffins, alcohols, water and gases, at low and high pressures. Two associating groups, OH and COOH, were defined. Self- and cross-association in these mixtures were quantified through two parallel COOH/COOH and OH/OH associations. The validity of this approach is supported by an excellent representation of pure compound properties (vapor pressures and compressibility factors) and phase equilibria in mixtures of (associating + inert) and (associating + associating) components at low and high pressures.

Keywords ¾ Association. Group Contribution. Carboxylic Acids.

I. INTRODUCTION

Association and solvation effects, when present, play a major role in the properties of pure compounds and mixtures. They are particularly important in carboxylic acids, where association is present even at low vapor densities.
In the present work the group contribution with association equation of state GCA-EOS (Gros et al., 1996) is applied to calculate phase equilibrium properties in mixtures of carboxylic acids, alcohols, water and gases, at low and high pressures.
The association term in the GCA-EOS model is based on Wertheim’s theory for fluids with highly directed attractive forces as applied in the SAFT equation (Chapman et al, 1990), and follows a group contribution approach. Gros et al. (1996) have used a single hydroxyl (OH) associating group to represent association effects in alcohols, water and their mixtures. In this work the application of the GCA-EOS is extended to mixtures containing carboxylic acids, by defining a new COOH associating group. To allow a straightforward extension of the model to multicomponent mixtures containing OH and COOH associating groups, a new approach is proposed to solve the cross association problem.

II. SELF- AND CROSS-ASSOCIATION IN MIXTURES WITH CARBOXYLIC ACIDS

Carboxylic acids present a high degree of non-ideality even at low pressures, which can be ascribed to the formation of oligomers in both, liquid and vapor phases. Generally only the formation of dimers is considered, and it is possible to find in the literature (Gmehling et al, 1982a) the values of the vapor phase dimerization constants for a number of carboxylic acids.
In order to model association by the GCA-EOS model, it is necessary to determine the number of associating groups, the number of active sites in each group and the values of the corresponding association strengths. The association strength is a function of two parameters: the volume (k ) and the energy (e /k) of association. In this work a new associating group (COOH) was defined as having one associating site that self-associates by double hydrogen bonding (Fig. 1).

Fig.1. Schematic representation of self and cross association in carboxylic acids.

Following the procedure adopted by Gros et al. (1996) for the hydroxyl (OH) group, the COOH association parameters were determined by reproducing the fraction of non-bonded molecules predicted by the SAFT equation for linear acids from propanoic to decanoic at saturated liquid conditions (Wolbach and Sandler, 1997). The values obtained for the energy and volume of association were e/k^COOH = 6500K and k ^COOH = 0.015 cm³/gmol, respectively. As expected, the COOH energy of association is much larger than that of the OH group (e/k^OH = 2700K), which is in accordance with the higher degree of association of carboxylic acids.
The extension of association theories to mixtures containing more than one associating compound requires the estimation of cross-association parameters and in some cases the numerical solution of a set of non-linear equations. Kraska (1998) presents a revision of the various rigorous and approximate solutions for cross-association models. Usually the problem is solved by setting some combination rule to evaluate the cross-association strength from the corresponding self-associations (in general geometric and/or arithmetic rules are applied) (Suresh et al., 1992, Voutsas et al., 1999).
The large difference between the energies of association of the COOH and OH groups precludes the use of this approach to evaluate cross-association effects in mixtures of carboxylic acids with water and/or alcohols. At room temperature, for example, the COOH self-association strength is ten thousand times that of the OH group. A mean value between such dissimilar quantities would be physically meaningless, and the cross-association strength calculated this way will tend to over-predict the degree of cross-association in a solution where the acid molecules are predominantly self-associated.
In this work a different empirical approach is proposed to evaluate cross-association in these mixtures. From the point of view of association, each carboxylic acid molecule is considered to have two associating groups: one COOH group which self-associates through a double hydrogen bonding, and a certain fraction of OH group that has a remnant association capacity and can self-associate with another OH group (see Fig. 1).
In this way, the cross association problem is replaced by the self-association between the OH group from the alcohol or water, and the fraction of OH group present in the carboxylic acid molecule. Following this methodology, the association of carboxylic acids and mixtures of carboxylic acids with water and/or alcohols is represented by two self-associations in parallel: COOH/COOH and OH/OH. The value of each self-association strength is calculated from the respective COOH and OH association parameters (e/k^COOH = 6500K, k ^COOH = 0.015 cm³/gmol, e/k^OH = 2700K and k ^OH = 0.8621 cm³/gmol). The fraction of associating OH group provided by the carboxylic acid molecules was empirically determined by fitting vapor-liquid equilibrium (VLE) data on binary mixtures of acids with water and alcohols. A fraction of OH group equal to 0.25 gave the best correlation of these data.
The methodology proposed here has the additional advantage of solving the association problem by two explicit mathematical expressions for the fraction of non-bonded COOH and OH associating groups. This makes straight forward the application of the GCA-EOS model to multicomponent mixtures of carboxylic acids, water, alcohols and inert components.

III. ATTRACTIVE-ENERGY PARAMETERS

There are three contributions to the residual Helmholtz energy in the GCA-EOS model: free volume, attractive and associative contributions. The group-contribution attractive term has five pure-group parameters (T^*, q, g*, g’ and g") and four binary interaction parameters (the symmetrical k_ij* and k_ij’ and the asymmetrical non-randomness parameters a _ij and a _ji.).
The COOH functional group was not available in the GCA-EOS parameter table. In this work the attractive-energy parameters for this group and its interactions with the paraffinic (CH₃ and CH₂), alcohol (CHOH, CH₂OH, CH₃OH), water (H₂O), triglyceride (TG) and CO₂ functional groups were determined. Tables 1 and 2 show, respectively, the corresponding pure-group and binary interaction parameters. The experimental information used to fit these parameters include: vapor pressures (P^vap) of pure carboxylic acids; binary low-pressure vapor-liquid equilibria (LPVLE) for mixtures of carboxylic acids with alkanes, alcohols and water; high-pressure vapor-liquid equilibria (HPVLE) for binary mixtures of carboxylic acids with carbon dioxide; and infinite dilution activity coefficients (g ¥ ) of alkanes in mixtures of triacetin with palmitic acid.

Table 1. Pure-group parameters

Table 2. Binary interaction parameters

IV. RESULTS AND DISCUSSION

The correlation of carboxylic acid vapor pressures gave an average relative error of 3% within a reduced temperature range between 0.55 and 0.90 (see Table 3). Table 4, on the other hand, compares experimental vapor compressibility factors (Z) at saturation (Miyamoto et al., 1999) with GCA-EOS predictions. The low values of Z reflect the strong association of carboxylic acids in the vapor phase, even at low pressures. It is interesting to notice that the GCA-EOS is able to follow the slight increase of Z with temperature showed by the experimental data.

Table 3. Pure-component vapor pressures.

*Error = ; NP: number of data points.

Table 4. Vapor phase compressibility factors

Figures 2 to 8 represent vapor-liquid equilibria (VLE) results for some binary mixtures of carboxylic acids with alkanes, carbon dioxide, alcohols and water. Excellent correlation (Fig. 2) and prediction (Fig. 3) of low- and high-pressure VLE were achieved for mixtures of alkanes with low and high molecular weight carboxylic acids. The same is true for mixtures with CO₂, as it can be inferred from the results shown in Fig. 4 (prediction) and Fig. 5 (correlation). Figs. 6 and 7 present some results from the correlation of vapor-liquid equilibria of cross associating mixtures.
Finally, Fig. 8 shows GCA-EOS predictions of high-pressure VLE for the ternary system CO2 + oleic acid + triolein. Even though the CO₂ solvent power predicted by the model is lower than the data measured by Bharath et al. (1992), the selectivity is correct; i.e. the equation shows the higher affinity of CO₂ for the fatty acid.

V. CONCLUSIONS

In this work, association effects in carboxylic acids were represented by a group contribution approach. Cross-association effects in mixtures of carboxylic acids with alcohols and water, were taken into account by solving two parallel self-association problems. With this approach and using a single set of parameters, good representation was obtained for pure component properties and phase equilibria in mixtures of carboxylic acids with inert compounds, alcohols and water at low and high pressures. The group contribution nature of the GCA-EOS allows it to be applied to mixtures for which experimental information is scarce or not available. In this respect, the GCA-EOS is a useful tool to explore process conditions for the extraction and fractionation of fatty acids and other fatty oil derivatives with supercritical fluids.

Fig. 2. VLE of heptane (1) + propanoic acid (2):
5 Exp. (Gmehling et al., 1982a, pg. 215); ¾ GCA-EOS.

Fig. 3. VLE of ethane (1) + oleic acid(2): 5 Exp.(Peter et al., 1991); ¾ GCA-EOS.

Fig. 4. VLE of CO₂ (1)+ butanoic acid (2): · 5 ´ Exp
(Byun et al., 2000), ¾ GCA-EOS.

Fig. 5. VLE of CO₂ (1)+ palmitic acid(2):
5 Exp. (Bharath et al., 1993); ¾ GCA-EOS.

Fig. 6. VLE of 1-propanol (1)+ propanoic acid (2): 5 Exp. (Gmehling et al., 1982, pg.486); ¾ GCA-EOS.

Fig. 7. VLE of H₂O (1)+ propanoic acid(2): 5 Exp. (Gmheling and Onken, 1977a, pgs.222 and 224); ¾ GCA-EOS.

Fig. 8. VLE of the system CO2 + oleic acid + triolein:
O · Experimental (Bharath et al, 1992); ¾ GCA- EOS.

APPENDIX

There are three contributions to the residual Helmholtz function (A^res) in the GCA-EoS model: free volume (A^fv), attractive (A^att) and associative (A^assoc).

A^res = A^fv + A^att + A^assoc (A1)

The free volume contribution is represented by the extended Carnahan-Starling equation for mixtures of hard spheres developed by Mansoori and Leland (1972):

(A / RT)fv =	3(l1l2 / l3) (Y - 1) + (l32 / l23)	(A2)
	(-Y + Y2 - ln Y) + n lnY

with:

and

n_i being the number of moles of component i, NC the number of components in the mixture, n the total number of moles, V the total volume and d_i the hard-sphere diameter per mol of species i.

The following generalized expression gives the temperature dependence of the hard-sphere diameter:

d_i = 1.065655 d_ci{1 - 0.12 exp[-2T_ci / (3T)]} (A3)

where d_ci and T_ci are, respectively, the critical hard-sphere diameter and critical temperature of component i. The value of d_ci can be determined from the critical properties:

d_ci = (0.08943 RT_ci / P_ci)^1/3 (A4)

or it can be calculated by fitting data on the vapor pressure of species i (Skjold-Jørgensen, 1984). For high molecular weight compounds (for which T_c and P_c are unknown and vapor pressure data is unavailable) the d_ci value can be determined by fitting experimental data on infinite dilution activity coefficients of alkanes in compound i (Bottini et al., 1999).

The attractive contribution to the Helmholtz energy accounts for dispersive forces between functional groups, through a density-dependent, local-composition expression based on the NRTL model (Renon and Prausnitz, 1968):

(A5)

with:

	(A6)
	(A7)

(A8)

Dg_ij = g_ij - g_jj (A9)

z being the coordination number (set equal to 10), vⁱ_j the number of groups of type j in molecule i, q_j the number of surface segments assigned to group j, the total number of surface segments, q _k the surface fraction of group k, g_ij the attractive energy between segments of groups i and j, and a _ij the non-randomness parameter.
The attractive energy g_ij is calculated from the energy between like-group segments trough the following combination rule:

g_ij = k_ij (g_iig_jj)^1/2 (A10)

where the binary interaction parameter k_ij is symmetrical (k_ij = k_ji ). Both, the attractive energy between like segments and the binary interaction parameter are temperature dependent:

g_jj = g^*_jj(1 + g'_jj (T / T^*_j - 1) + g''_jj ln(T / T^*_j)) (A11)

k_ij = k^*_ij {1 + k'_ij ln[2T / (T^*_i + T^*_j)]} (A12)

where T^*_i is an arbitrary but fixed reference temperature for group i; g^*_jj, g'_jj and g''_jj are pure-group energy parameters and k^*_ij and k'_ij are binary group interaction parameters.
The association contribution to the Helmholtz function is calculated with a group contribution expression (Gros et al., 1996) based on Whertheim’s theory (1984a,b, 1986a,b) of associating fluids:

(A13)

where NGA represents the number of associating groups, n_i^assoc the number of moles of associating group i, M_i the number of associating sites assigned to group i and X^(k,i) the mole fraction of group i not bonded at site k. The number of moles of associating group i is:

(A14)

where represents the number of times associating group i is present in molecule m and n_m is the total number of moles of species m; the summation includes all NC components in the mixture.
The mole fraction of group i not bonded at site k is determined from the following expression:

(A15)

The value of X^(k,i) depends on the molar density of associating group j (r_j^assoc = n_j^assoc / V) and the association strength D ^(k,i,l,j) between site k of group i and site l of group j:

(A16)

where the association parameters are the energy (e) and volume (k) of association. In mixtures containing alcohols and/or water, association effects can be computed by the same hydrogen-bonding hydroxyl group characterized by an energy e /k = 2700 K and a volume k = 0.8621 cm³/mol (Gros et al., 1997).

REFERENCES
1. Bermudez, A., O. Ferreira and G. Foco, "Infinite dilution activity coefficients of solvents in fatty oil derivatives", Proc. of Enpromer, Santa Fe, Argentina, 613-618, (2001).         [ Links ]
2. Bharath, R., H. Inomata, T. Adschiri and K. Arai, "Phase equilibrium study for the separation and fractionation of fatty oil components using supercritical carbon dioxide", Fluid Phase Equilibria, 81, 307-320, (1992).         [ Links ]
3. Bharath, R., S. Yamane, H. Inomata, T. Adschiri and K. Arai, "Phase equilibria of supercritical CO2-fatty oil component binary systems", Fluid Phase Equilibria, 83, 183-192, (1993).         [ Links ]
4. Bottini, S.B., T. Fornari and E.A. Brignole, "Phase equilibrium modeling of triglycerides with near critical solvents", Fluid Phase Equilibria, 158-160, 211-218 (1999).         [ Links ]
5. Byun, Hun-soo, Nam-seok Jeon and M.A. Mchugh, "Phase behavior and modeling of supercritical carbon dioxide - acid mixtures", Proc. 5th Int. Symposium on Supercritical Fluids (CD-ROM), Atlanta, Georgia, USA, (2000).         [ Links ]
6. Chapman, W., K. Gubbins, J. Jackson and M. Radosz, "New reference equation of state for associating liquids", Ind. Eng. Chem. Res, 29, 1709-1721, (1990).         [ Links ]
7. Daubert, T.E. and R.P. Danner, Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, Hemisphere Publishing Corporation, (1989).         [ Links ]
8. Gmehling, J. and U. Onken, Vapor Liquid Equilibrium Data Collection. Aqueous-organic systems, Dechema Chemistry Data Series I 1, Frankfurt, Germany, (1977).         [ Links ]
9. Gmehling, J. and U. Onken, Vapor Liquid Equilibrium Data Collection. Organic hydroxyl compounds: alcohols, Dechema Chemistry Data Series I 2a, Frankfurt, Germany, (1977a).         [ Links ]
10. Gmehling, J., U. Onken and W. Arlt, Vapor Liquid Equilibrium Data Collection. Organic hydroxyl compounds: alcohols (suplement 1), Dechema Chemistry Data Series I 2c, Frankfurt, Germany, (1982).         [ Links ]
11. Gmehling, J., U. Onken and P. Grenzheuser, Vapor Liquid Equilibrium Data Collection. Carboxylic acids, anhydrides, esters, Dechema Chemistry Data Series I 5, Frankfurt, Germany, (1982a).         [ Links ]
12. Gmehling, J., U. Onken and W. Arlt, Vapor Liquid Equilibrium Data Collection. Aqueous-organic systems (Suplement 1), Dechema Chemistry Data Series I 1a, Frankfurt, Germany, (1998).         [ Links ]
13. Gros, H., S.B. Bottini and E.A. Brignole, "A group contribution equation of state for associating mixtures", Fluid Phase Equilibria, 116, 537-544 (1996).         [ Links ]
14. Gros, H.P, S.B. Bottini and E.A. Brignole, "High pressure phase equilibrium modeling of mixtures containing associating compounds and gases", Fluid Phase Equilibria, 139, 75-87 (1997).         [ Links ]
15. Kraska, T., "Analytic and fast numerical solutions and approximations for cross association models within statistical association fluid theory", Ind. Eng. Chem. Res., 37, 4889-4892, (1998).         [ Links ]
16. Mansoori, G.A. and T.W. Leland, "Statistical thermodynamics of mixtures", J. Chem. Soc. Faraday Trans. II, 68, 320, (1972).         [ Links ]
17. Miyamoto, S., S. Nakamura, F. Iwai and Y. Arai, "Measurement of vapor phase compressibility factors of monocarboxylic acids using a flow type apparatus", J.Chem.Eng.Data, 44, 48-51, (1999).         [ Links ]
18. Peter, S. and H. Jakob, "The rheological behavior of coexisting phases in systems containing fatty acids and dense gases", The Journal of Supercritical Fluids, 4, 166-172, (1991).         [ Links ]
19. Renon, H and J.M. Prausnitz, "Local compositions in thermodynamic excess functions for liquid mixtures", AIChE J., 14, 135-144 (1968).         [ Links ]
20. Skjold-Jørgensen, S., "Gas solubility calculations II. Application of a new group-contribution equation of state", Fluid Phase Equilibria, 16, 317-351 (1984).         [ Links ]
21. Skjold-Jørgensen, S., "Group contribution equation of state (GC-EOS): A predictive method for phase equilibrium computations over wide ranges of temperature and pressures up to 30 Mpa", Ind. Eng. Chem. Res., 27, 110-118 (1988).         [ Links ]
22. Suresh, S. and J. Elliot, "Multiphase equilibrium analysis via a generalized equation of state for associating mixtures", Ind. Eng. Chem. Res., 31, 2783-2794, (1992).         [ Links ]
23. Voutsas, E.C., I.V. Yakoumis and D.P. Tassios, "Prediction of phase equilibria in water/alcohol/alkane systems", Fluid Phase Equilibria, 158-160, 151-163, (1999).         [ Links ]
24. Wertheim, M.S., "Fluids with highly directional attractive forces. I. Statistical thermodynamics", J. Stat. Phys., 35, 19, (1984a).         [ Links ]
25. Wertheim, M.S., "Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations", J. Stat. Phys., 35, 35, (1984b).         [ Links ]
26. Wertheim, M.S., "Fluids with highly directional attractive forces. III. Multiple attraction sites", J. Stat. Phys., 42, 459, (1986a).         [ Links ]
27. Wertheim, M.S., "Fluids with highly directional attractive forces. IV. Equilibrium polymerization", J. Stat. Phys., 42, 477, (1986a).         [ Links ]
28. Wolbach, J.P. and S.I. Sandler, "Using molecular orbital calculations to describe phase behavior of hydrogen-bonding fluids", Ind. Eng. Chem. Res., 36, 4041-4051, (1997).         [ Links ]