Thermodynamic Modeling of the Water+ Acetic Acid+ CO2 System

Jan 5, 2007 - Modeling of the water + acetic acid + CO2 system with the cubic plus association (CPA) equation of state. (EoS) is presented involving ...
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Ind. Eng. Chem. Res. 2007, 46, 932-938

Thermodynamic Modeling of the Water + Acetic Acid + CO2 System: The Importance of the Number of Association Sites of Water and of the Nonassociation Contribution for the CPA and SAFT-Type Models Christophoros A. Perakis,* Epaminondas C. Voutsas, Kostis G. Magoulas, and Dimitrios P. Tassios Laboratory of Thermodynamics and Transport Phenomena, School of Chemical Engineering, National Technical UniVersity of Athens, 9, Heroon Polytechniou Street, Zografou Campus, 15780 Athens, Greece

Modeling of the water + acetic acid + CO2 system with the cubic plus association (CPA) equation of state (EoS) is presented involving correlation of the binary data and prediction of the ternary one, yielding very satisfactory results. The number of necessary association sites of water in CPA is investigated with an emphasis on the very difficult to model water + acetic acid binary. The results are compared with those of two SAFTtype models, the original and the simplified one (SSAFT), which use the same association term as CPA. The nonassociation contribution (physical part) in the CPA and the two SAFT models is examined, based on the results using the same association sites for water. It is concluded that the optimum number of sites for water depends on the other compounds present and, furthermore, that the nonassociation contribution of SAFT or SSAFTsin spite of its complexitysdoes not offer any advantage over the simple cubic EoS. 1. Introduction Water is the molecule on which life on Earth mostly depends, and therefore, it is encountered in almost every industrial application. Its importance lies primarily in the ability of its molecules to strongly associate with forces of an electrostatic nature (Coulombic), i.e., to form hydrogen bonds. It is due to these hydrogen bonds that water is liquid and not gas at room temperature and that it has such a high boiling point relative to other molecules of similar molecular weight. In order to describe satisfactorily the phase behavior of water or other hydrogen-bonding compounds, one has to take into account, apart from the repulsive or attractive (dispersion or polar) physical forces between molecules (physical part), the hydrogen bond. In the beginning of the previous decade, Chapman et al.1,2 and Huang and Radosz,3 by extending Wertheim’s theory,4,5 developed the statistical associating fluid theory (SAFT) equation of state (EoS). The SAFT EoS combines two terms in the total residual Helmholtz energy of a system. The first term includes the contribution of the physical forces and is obtained by accounting for hard-sphere repulsive interactions, dispersive interactions, and chain connectivity; the second term accounts for the contribution of association (hydrogen bonds). Several modifications of the physical part of this EoS have been proposed, such as the hard-sphere SAFT (SAFTHS) by Jackson et al.,6,7 the simplified one (SSAFT) by Fu and Sandler,8 the Lennard-Jones one (SAFT-LJ) by Mu¨ller and Gubbins,9 the variable-range one (SAFT-VR) by Gil-Villegas et al.,10 and the perturbed chain one (PC-SAFT) by Gross and Sadowski.11 More details can be found in two recent reviews by Mu¨ller and Gubbins12 and Economou.13 A different approach was used in this laboratory for such systems by combining a cubic EoS for the physical interactions with the association part of the SAFT EoS for the hydrogen bonding between molecules, leading to the cubic plus association (CPA) EoS.14 [The software for VLE calculations with the CPA EoS is available upon request from the authors at the labora* To whom correspondence should be addressed. Tel.: +30 210 772 3230. Fax: +30 210 772 3155. E-mail: [email protected].

tory’s web address: http://ttpl.chemeng.ntua.gr/.] The computation time is thus greatly reduced, while the results are similar or even better than those obtained with SAFT, as has been shown in several publications.15-18 An important decision that must be made when using the SAFT or CPA EoSsor other association modelssis the number of hydrogen-bonding (association) sites present in the molecule. For example, theory says that water has four sites where it can form hydrogen bonds: two at the unbonded electron pair of oxygen and two at the hydrogen atoms. However, some experimental results19 suggest that only three sites of water form hydrogen bonds due to steric hindrance of the two association sites on the oxygen atom, which may not allow the formation of two hydrogen bonds. Since the matter is still ambiguous, various researchers have adopted either the three-site or the foursite or even the two-site approach to model the phase equilibria of associating mixtures containing water, using SAFT-type models.3,8,20-31 The number of water sites has been also investigated by this laboratory using the CPA EoS,16-18,32,33 indicating that the appropriate number of sites for water in CPA is four. The second factor affecting the performance of an EoS in mixtures containing associating compounds is the term that describes the nonassociation contribution (physical term). For example, Yakoumis et al.16 and Voutsas et al.,17 by comparing the CPA with the original SAFT of Huang and Radosz in the correlation of water-containing binary mixtures of alkanes and alkenes using a four-site water, obtained better results with the former. They concluded that the SAFT EoS, despite the complexity of its physical term as compared to a simple cubic EoS, does not offer any advantage over the latter, at least for the systems they examined. The importance of these two factors, the number of association sites and the physical term, is further demonstrated here in modeling the water + acetic acid + CO2 system with the CPA EoS. We consider a three- and a four-site model for water and evaluate its performance in describing the phase equilibrium behavior of the water/CO2 and water/acetic acid systems and in the prediction of the ternary system. Finally, the results for the water/acetic acid binary system are compared with those of

10.1021/ie0609416 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/05/2007

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Fu and Sandler,8 who used the original and simplified SAFT models with a three-site water, to evaluate the contribution of the physical term in the two models. 2. The Model The compressibility factor of the CPA EoS is given by

ZCPA ) ZPR + Zassoc

(1)

where the physical part of the compressibility factor, ZPR, is that of the Peng-Robinson (PR) EoS:

ZPR )

V/(V - b) - aV RT(V(V + b) + b(V - b))

(2)

and the association part, Zassoc, is given by the following expression:

Zassoc ) -

(

1 2

1+F

)∑∑

∂(ln g) ∂F

i

xi[d(Ai)](1 - XAi) (3)

Ai

where F is the molar density of the mixture, xi is the analytical mole fraction of the component i in the mixture, and d(Ai) is the number of association sites of type A on molecule i. Also, XAi is the mole fraction of the molecule i not bonded at site A, which is calculated by the following equation:

XAi ) (1 + F

xjXB ∆A B )-1 ∑i ∑ B j

i j

(4)

j

The association strength ∆AiBj in CPA is expressed as

( ( ) )

∆AiBj ) g exp g)

AiBj - 1 bijβAiBj RT

1 - (y/2) , (1 - y)3

y)

(5)

bF 4

where AiBj and βAiBj are the association energy and volume between site A of molecule i and site B of molecule j, respectively. CPA EoS requires three parameters for pure nonassociating compounds, namely, Tc′, Pc′, and ω′, while for associating compounds two more hydrogen-bonding parameters, ABand βAB, are also required. In both cases the CPA parameters are determined by fitting pure-compound vapor pressure and liquid volume data. For the extension of the CPA EoS to mixtures, the classical van der Waals mixing rules are applied to a and b of the physical part with the following combining rules:

aij ) xaiaj(1 - kij),

bij ) (bi + bj)/2

(6)

Furthermore, combining rules for the cross-association energy and volume parameters are needed between different associating molecules. In this work the geometric mean will be used for both parameters:

AiBj ) xAiBiAjBj(1 - lij),

βAiBj ) xβAiBiβAjBj

(7)

Finally, for mixtures containing one associating compound and a compound that cross-associates but does not self-associate (i.e., solvates), the cross-association strength in the mixture is given by multiplying the association strength of the associating compound by a factor (solvation factor):

∆AiBj ) ∆Aisij

(8)

The binary parameters kij, lij, and sij in eqs 6-8 are adjustable parameters, and they are fitted to binary phase equilibrium data. For a more detailed description of the model readers are referred to a previous publication.18 3. Pure Compounds To describe the vapor pressure and liquid density of a pure associating compound with the CPA EoS, first one must decide on the number and type of the association sites of its molecules. Carbon dioxide is a compound that does not self-associate; however, there is evidence that it can form weak hydrogen bonds with other associating compounds.34 Therefore, CO2 will be treated as a solvating molecule, which means that it only crossassociates with other associating molecules. As far as acetic acid is concerned, the one-site approach (1A, as defined by Huang and Radosz3) is adopted, which gave satisfactory results in the description of its mixtures with hydrocarbons with CPA.35 Finally, the 4C and 3B configurations3 are examined for water, as has been explained in the Introduction. The parameters for water, acetic acid, and CO2 were fitted in order to describe their vapor pressure and liquid density from about the triple point to the critical point. The DIPPR data compilation36 was used for water and CO2; however, for acetic acid this was not the case. Vapor-liquid equilibrium (VLE) for the water/acetic acid mixture exists at very low pressures, and therefore, to describe the system satisfactorily, the model has to reproduce the vapor pressures of the pure components accurately. However, the DIPPR correlation slightly but systematically underpredicts the vapor pressure of acetic acid at temperatures below 400 K. For this reason, literature experimental data were used for the acetic acid vapor pressure below 400 K. Vapor pressure data above 400 K and liquid density data were generated using the DIPPR correlations. The resulting CPA EoS parameters along with the correlation results are presented in Table 1. 4. Binary Mixtures Binary interaction parameters kij, lij, and sij in eqs 6-8 were fitted to binary phase equilibrium data. Bubble point pressure calculations were performed in the case of the water/acetic acid NP mixture, and F ) (1/2)∑i)1 [(Pexpi - Pcalci)/Pexpi]2 was the objective function that was minimized. Bubble point temperature calculations were also performed for this mixture, but only for prediction purposes. Flash calculations were performed for the rest binaries with the following objective function: F ) NP (1/2)∑i)1 (x2expi - x2calci)2 + (y1expi - y1calci)2, where the second component is always carbon dioxide. 4.1. The Acetic Acid/Water Mixture. Initially, water was treated as a four-site molecule, which we have found to be the appropriate one in the presence of another associating compound, ethanol.18 Nevertheless, the results were bad, as shown in Figure 1. It seems that this association scheme leads to strong cross-association (overpredicts the association strength of water in the mixture), since the bubble point (or dew point) pressures are largely underestimated in the water-rich region. Accordingly, an association scheme with lesser association sites would be more appropriate for water in this case. Consequently, the threesite approach was adopted for the water molecule, which produced very good results, as shown in Figure 1, much better than those presented for the SAFT and SSAFT EoS by Fu and Sandler8 with a three-site water and by Wolbach and Sandler,26

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Table 1. CPA EoS Pure Component Parameters component sites data source T range (K) Tc′ (K) Pc′ (bar) ω′ AB (K) βAB ∆Ps a (%) ∆Vl b (%) a

∆Ps (%) )

1 NP

water 3 DIPPR 278-641 503.33 209.36 0.3288 1341.8 0.2034 0.2 0.5 NP

∑ i)1

water 4 DIPPR 278-641 305.40 135.62 0.1609 1811.3 0.1062 1.5 0.7

acetic acid 1 DIPPR, refs 37-41 298-533 480.45 66.57 0.0022 5243.4 0.0022 1.0 0.6

|Pi exps - Pi calcs| Pi exps

100where NP is the number of experimental data points.

b

∆Vl (%) )

CO2 DIPPR 219-298 306.97 75.29 0.1832 1.9 3.2

1 NP

NP

∑ i)1

|Vi expl - Vi calcl| Vi expl

100.

Table 2. Binary Interaction Parameters for the CPA EoS Determined by Fitting the Binary Phase Equilibrium Data along with the Deviations Obtained between the Experimental Data and the Correlation Results interaction parameters k12 ) 3 × 10-4(T/K - 298) - 0.2019 l12 ) 2 × 10-4(T/K - 298) - 0.1668

k12 ) 1 × 10-3(T/K - 298) - 0.1087

k12 ) 7.5 × 10-4(T/K - 298) + 0.0722 100s12 ) 1.82 × 10-3(T/K - 298) + 1.84 × 10-3

a

∆f )

1 NP

100∆x2a

i)1

∆Pb (%)

Acetic Acid (1)/Water (2) at 313-390 K 0.89 0.5 1.13 0.8 1.00 0.5 0.68 1.3 1.08 0.80 0.78

∆Ta

T (K)

P (mmHg)

313.15 353.15 363.05 343.2 0.33 0.53 0.20

70 200 760

ref 42 42 43 44 45 45 45

Water (1)/CO2 (2) at 298-348 K 0.09 0.14 0.06 0.29 0.05 0.32 0.07 0.45 0.02 0.22 0.09 0.06

298.15 304.19 323.15 348.15 313.15 333.15

46 46 46 46 47, 48 49

Acetic Acid (1)/CO2 (2) at 313-353 K 0.81 0.04 1.86 0.08 0.57 0.21

313.20 333.20 353.20

49 49 49

NP

∑

100∆y1a

|fiexp - ficalc| where f ) x2, y1, T. b ∆P (%) )

1 NP

NP

∑ i)1

|Piexp - Picalc| 100. Piexp

who used SAFT with a four-site water. Note that, since the available data cover a temperature range from 313 to 390 K, the two interaction parameters were made linearly temperature dependent. Thus the results presented in this figure and those

in Table 2 and Figures 2 and 3 were obtained using these parameters. The results of the last figure indicate that reasonably satisfactory predictions of the relative volatility values, which are very important for separation purposes, are obtained with

Figure 1. VLE for acetic acid/water at 313.15 K with CPA assuming two different association schemes. Experimental data were taken from Lazeeva and Markuzin.42

Figure 2. VLE for acetic acid/water at 760 mmHg with CPA with the 3-1 association scheme. Experimental data were taken from Ito and Yoshida.45

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Figure 5. VLE for acetic acid/CO2 at 353.2 K with CPA.49

Figure 3. Prediction of the relative volatility, R, for the acetic acid/water binary at 760 mmHg.45

Figure 4. VLE for water (1)/CO2 (2) at 323.15 K with

Figure 6. Phase equilibrium prediction of the ternary system water (1) + acetic acid (2) + CO2 (3) at 353.2 K and 101 bar with the CPA EoS.50

CPA.46

model overpredicts the water vapor phase concentration at pressures above 100 bar. Better results can be obtained, however, by treating water as a four-site molecule, as recently demonstrated by the authors in modeling the water + ethanol + CO2 system18 and also shown in Figure 4. 4.3. The Acetic Acid/CO2 Mixture. Two temperaturedependent parameters, namely k12 and s12, were fitted to VLE experimental data of the acetic acid/CO2 mixture from 313 to 353 K. The model describes satisfactorily both the liquid phase and the vapor phase as can be seen from the deviations presented in Table 2 and from Figure 5.

AAE% errors of 6.7, 5.9, and 10.2 at 760, 200, and 70 mmHg, respectively, although the model cannot predict the steep increase of the relative volatility in the very dilute region of water. 4.2. The Water/CO2 Mixture. Since the three-site approach for water yielded satisfactory results for the acetic acid/water mixture, it was adopted also for the water/CO2 mixture. Carbon dioxide was treated as a solvating molecule, and therefore, apart from k12, the solvation factor, s12, was also fitted to binary experimental data. Both parameters were considered to depend linearly on temperature for the temperature range from 298 to 348 K. The correlation resulted in very low values for the solvation factor, and consequently this parameter was set equal to zero, which actually means that CO2 behaves as a nonassociating molecule in this case. The results are presented collectively in Table 2, while for a single isotherm results are also presented graphically in Figure 4. As it is shown, the correlation of the solubility of CO2 in water is very satisfactory; however, the

5. Ternary Mixture The linear correlations that were obtained for the interaction parameters of the binary mixtures were used to predict the phase equilibria of the ternary water + acetic acid + CO2 system without any further parameter adjustment. The “3-1 solvating” association scheme was employed, and the experimental data

Table 3. VLE Predictions with the CPA EoS for the Water (1) + Acetic Acid (2) + CO2 (3) Mixture50 T (K)

P range (bar)

NP

100∆y1

100∆x1

100∆y2

100∆x2

313.3 333.1 353.2

51-101 51-162 51-161

11 26 29

0.21 0.09 0.07

1.17 0.88 0.95

0.67 0.51 0.27

1.09 0.70 0.68

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Figure 7. Acetic acid selectivity, S, predictions at 353.2 K with the CPA EoS. Experimental data were taken from Bamberger et al.50

of Bamberger et al.,50 which extend from the low-pressure liquid-vapor region to the high-pressure liquid-liquid region of the water + acetic acid + CO2 mixture, were used to evaluate the model’s performance. Very satisfactory results are obtained as can be seen in Table 3, where the deviations between the model predictions and the experimental data are presented, and, graphically, in Figure 6 where the two-phase region of this system at 353.2 K and 101 bar is shown. Bamberger et al.50 modeled the system using the PR EoS in a modification of Melhem et al.51 coupled with the mixing rule of Panagiotopoulos and Reid and taking also the dimerization of acetic acid in both phases into account by applying the chemical theory. They present their results only graphically, but a visual comparison of them with those obtained here indicates that CPA predictions are superior. To design or simulate a separation process successfully, satisfactory predictions of the acetic acid selectivity, S ) (y2/ x2)/(y1/x1), and distribution coefficient, K2 ) y2/x2, are needed. Reasonably satisfactory predictions of these quantities are obtained as indicated by the typical results presented for 353.2 K in Figures 7 and 8. 6. Discussion and Conclusions The cubic plus association equation of state (CPA EoS) has been applied in the correlation of the binary phase equilibrium data and the prediction of the ternary data for the water + acetic acid + carbon dioxide system at various temperatures and pressures. Very satisfactory results are obtained using a threesite water, a solvating CO2, and a one-site acetic acid. Certain important observations, in addition to the satisfactory prediction of the water + acetic acid + CO2 system, can be made based on the obtained results. First, is the need to change the number of sites for water from four, which has been successfully used with CPA in mixtures with other associating compounds such as alcohols,18,32 to three, in order to obtain satisfactory correlation of the water/acetic acid binary. This may be due to the strong intermolecular interactions between water and acetic acid (strong cross-association) that weakens the selfassociation of the water molecules. Thus, fewer association sites on the water molecule are needed. This observation is also supported by the results of Sum and Sandler,52 who used ab initio calculations to compute molecular interaction energies for water in the presence of ethanol or acetic acid. They report a

Figure 8. Acetic acid distribution coefficient (K ) y/x) values at 353.2 K: CPA predictions vs experimental data.50

higher intermolecular interaction value for the water-acetic acid pair (strong cross-association) compared to that for the waterethanol pair, and at the same time a lower intermolecular interaction value for the water-water pair (weaker water selfassociation) in the water/acetic acid mixture compared to that in the water/ethanol mixture. Second, mixtures of CO2 with other polar molecules such as water, ethanol, and acetic acid cannot be modeled satisfactorily with either CPA or SAFT EoS by treating CO2 as an inert molecule. Treatment of CO2 as either a solvating or an associating molecule is justified both by this work and by a previous one.18 This is due to the fact that the weak electrostatic interactions coming from the CO2 quadrupoles are not taken into account explicitly by these models. The fact that zero crossinteraction in the water/CO2 pair was obtained here is due to the three-site water that was dictated by the water/acetic acid binary. Of course, it would be more realistic if an explicit polar term was included, as has been suggested by Tumakaka et al.53 and Karakatsani and Economou.54 Third is the substantially improved correlation results for the water-acetic acid system with CPA compared to those obtained with the original SAFT and SSAFT by Fu and Sandler,8 who used also a three-site water model. This suggests that the nonassociation contribution of theses two SAFT models, despite their complexity, do not offer any advantage over a simple cubic EoS. This is in agreement with the findings of Yakoumis et al.16 and Voutsas et al.,17 who reached a similar conclusion by comparing CPA with the original SAFT in the correlation of water-containing binary systems of alkanes and alkenes using a four-site water in both EoS. Acknowledgment This work is financially supported by the project “Heraclitus”. The Project is cofunded by the European Social Fund (75%) and National Resources (25%). List of Symbols a ) energy parameter Ai ) site A on molecule i b ) covolume parameter (dm3 mol-1) Bj ) site B on molecule j d(Ai) ) number of association sites of type A on molecule i g ) radial distribution function

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K ) distribution coefficient kij ) binary interaction parameter lij ) binary interaction for the association energy NP ) number of experimental data points P ) pressure (bar) Ps ) vapor pressure (bar) R ) gas constant (bar dm3 mol-1 K-1) S ) selectivity sij ) solvation factor T ) temperature (K) V ) molar volume (dm3 mol-1) Vl ) saturated liquid volume (dm3 mol-1) xi ) liquid mole fraction of component i XAi ) mole fraction of the molecule i not bonded at site A yi ) vapor mole fraction of component i Z ) compressibility factor Greek Symbols R ) relative volatility β ) association volume parameter ∆ ) association strength (dm3 mol-1)  ) association energy parameter (bar dm3 mol-1) F ) molar density (mol dm-3) ∑A ) summation over all the sites (starting with A) on the molecule ω ) acentric factor Subscripts and Superscripts assoc ) associating or due to association c ) critical calc ) calculated exp ) experimental Literature Cited (1) Chapman, W.; Jackson, G.; Gubbins, K. Phase Equilibria of Associating Fluids: Chain Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1057. (2) Chapman, W.; Gubbins, K.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (3) Huang, S.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (4) Wertheim, M. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1986, 42, 459. (5) Wertheim, M. Fluids with Highly Directional Attractive Forces. IV. Equilibrium polymerization. J. Stat. Phys. 1986, 42, 477. (6) Jackson, G.; Chapman, W.; Gubbins, K. Phase Equilibria of Associating Fluids: Spherical Molecules with Multiple Bonding Sites. Mol. Phys. 1988, 65, 1. (7) Green, D.; Jackson, G. Theory of Phase Equilibria for Model Aqueous Solutions of Chain Molecules: Water + Alkane Mixtures. J. Chem. Soc., Faraday Trans. 1992, 88, 1395. (8) Fu, Y.-H.; Sandler, S. A Simplified SAFT Equation of State for Associating Compounds and Mixtures. Ind. Eng. Chem. Res. 1995, 34, 1897. (9) Mu¨ller, E.; Gubbins, K. An Equation of State for Water from a Simplified Intermolecular Potential. Ind. Eng. Chem. Res. 1995, 34, 3662. (10) Gil-Villegas, A.; Galindo, A.; Whitehead, P.; Mills, S.; Jackson, G.; Burgess, A. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of Variable Range. J. Chem. Phys. 1997, 106, 4168. (11) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (12) Mu¨ller, E.; Gubbins, K. Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches. Ind. Eng. Chem. Res. 2001, 40, 2193. (13) Economou, I. Statistical Associating Fluid Theory: A Successful Model for the Calculation of Thermodynamic and Phase Equilibrium Properties of Complex Fluid Mixtures. Ind. Eng. Chem. Res. 2002, 41, 953.

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ReceiVed for reView July 20, 2006 ReVised manuscript receiVed November 10, 2006 Accepted November 20, 2006 IE0609416