Ind. Eng. Chem. Res. 2000, 39, 2623-2626
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Use of a Predictive Cubic Equation of State To Model New Equilibrium Data of Binary Systems Involving Fatty Acid Esters and Supercritical Carbon Dioxide Jean-Noe1 l Jaubert* Laboratoire de Thermodynamique des Se´ parations, Institut National Polytechnique de Lorraine, Ecole Nationale Supe´ rieure des Industries Chimiques, 1, Rue Grandville, 54000 Nancy, France
Lucie Coniglio De´ partement de Chimie-Physique des Re´ actions, Institut National Polytechnique de Lorraine, Ecole Nationale Supe´ rieure des Industries Chimiques, 1, Rue Grandville, 54000 Nancy, France
Christelle Crampon Laboratoire d’Etudes et d’Applications de Proce´ de´ s Se´ paratifs, Universite´ d’Aix-Marseille, Case 512, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France
Very recently, Jaubert et al. proposed a purely predictive model for the phase equilibria computation of mixtures involving fatty acid esters (FAE) and supercritical carbon dioxide (SCCO2). In this research note, the capability of the model to predict the isothermal phase diagrams of three binary systems recently measured by Crampon et al. is checked. The results obtained are very encouraging because the proposed model is able to predict the experimental bubblepoint and dew-point pressures with an average deviation of 7%. Introduction In 1999, Jaubert et al.1 developed a purely predictive model to compute phase equilibria of mixtures involving fatty acid esters (FAE) and supercritical carbon dioxide (SC-CO2). This model was able to correlate the 317 bubble points found in the literature of binary systems containing SC-CO2 and a FAE with an overall average deviation of 12.8%. Moreover, an average deviation of ∆P% ) 9.8% was observed with 24 experimental bubble-point pressures for ternary systems. In their paper, Jaubert et al.1 made dew-point measurements of SC-CO2/fish oil systems. The fish oil studied contained 30 different fatty acid ethyl esters (FAEE) and their model was able to predict these dew-point pressures with an average deviation of 5.4%. Such a model is based on a cubic equation of state in which the critical parameters and the acentric factor are calculated through specific correlations. The binary interaction parameters (kij) are temperature dependent and are estimated through a group contribution method (GCM). This GCM was initially developed by Abdoul et al.2 to describe the behavior of hydrocarbon systems. It has been extended to ester compounds by Jaubert et al.1 On one hand, in their paper, Jaubert et al.1 included all the available binary systems in their database to extend the GCM to ester compounds. This means they did not keep experimental data to check the capability of their model to predict the phase behavior of binary systems not included in their database. In other words, their model was only used to correlate but never used to predict phase diagrams of binary systems. * To whom correspondence should be addressed. Fax: (+33) 383.35.08.11.Phone: (+33)383.17.50.81.E-mail: jaubert@ensic. u-nancy.fr.
On the other hand, very recently, Crampon et al.3 measured more than 300 bubble- (or dew-) point pressures of four binary systems. These data were unfortunately published 1 month after Jaubert et al.1 published their own paper. The four investigated binary systems are CO2 + methyl oleate (m18-1ω9), CO2 + ethyl stearate (e18-0), CO2 + ethyl myristate (e14-0), and CO2 + ethyl palmitate (e16-0). For the two first systems, older data were available at the same temperatures in the literature. Moreover, the data, on one hand, for the system CO2 + methyl oleate measured by Crampon et al.3 are in very good agreement with the previously published data. This means that the deviations obtained for this system are similar to the ones given in our previous paper.1 It is thus not necessary to predict these data. On the other hand, the data by Crampon et al.3 concerning the system CO2 + ethyl stearate are quite different (especially in the vicinity of the critical point) from those previously published. For this reason, it looks interesting to predict these data in this research note. By the end of the paper, 226 new bubble- (or dew-) point pressures are predicted for three different binary systems. The aim of this research note is thus to show how the thermodynamic model developed by Jaubert et al.1 is able to predict these new data. Thermodynamic Model The phase equilibria calculations of binary systems involving CO2 and a FAE were performed by using the Peng-Robinson4 equation of state (PR-EOS) with classical mixing rules. To obtain a purely predictive model, the binary interaction parameters were estimated through the group contribution concept.
10.1021/ie0000360 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/16/2000
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Table 1. Estimated Pure Components Physical Properties compound
Tc/K
Pc/bar
acentric factor
ethyl myristate (e14-0) ethyl palmitate (e16-0) ethyl stearate (e18-0)
743.408 765.177 781.540
14.5194 13.1288 11.9813
0.8259 0.9019 0.9774
Pure Components Equation of State. The PengRobinson4 equation of state (EOS) may be written as
P)
a(T) RT v - b v(v + b) + b(v - b) RTc with b ) 0.077 80 Pc and R ) 8.31441 J‚mol-1‚K-1
To calculate precisely the vapor pressures of the pure components, the following Soave5-type function developed by Rauzy6 was used, 2
[
( ()
2
R Tc T 1+m 1Pc Tc with m ) 6.812 553
a(T) ) 0.457 236
0.445075
)]
2
[x1.127 539 + 0.517 252ω - 0.003 737ω2 - 1] Prior to performing phase equilibria calculations with such an equation of state, it is necessary to estimate the critical temperature and pressure and the acentric factor for each pure component. Estimation of the Pure Component Properties. In the case of CO2, the physical properties are known from experimental measurements (Tc ) 304.21 K, Pc ) 73.795 bar, and ω ) 0.225). But as pointed out by many authors, the fatty acid esters decompose before reaching their normal boiling temperature and therefore the three properties required by the EOS (Tc, Pc, and ω) have to be estimated. The approach developed in 1995 by Coniglio et al.,7 and used by Jaubert et al.,1,8,9 was followed. The estimated physical properties of all the esters investigated in this work are summarized in Table 1. Mixing Rules. For a mixture m, containing p compounds of mole fraction xi, classical mixing rules were used: p
bm )
xibi ∑ i)1 p
am(T) )
p
∑ ∑[xai(T)aj(T)xixj(1 - kij(T))] i)1 j)1
The kij are binary parameters that characterize the interactions of unlike molecules. In this study, kij values were calculated using the GCM initially developed by Abdoul et al.2 for hydrocarbons and extended to fish oil components by Jaubert et al.1,8 The following expressions have to be used:
Eij(T) - (δi - δj)2 xai(T) with δi ) kij(T) ) 2δiδj bi Parameter Eij(T) is estimated through the GCM by
Table 2. A0kl and B0kl Parameters (in J‚cm-3) for the Five Groups Appearing in Mixtures of CO2 and FAE (The 10 Values in Bold Were Adjusted by Jaubert et al.1 and the 10 Other Ones Were Determined by Abdoul et al.2) A0kl
CO2 (1)
CH3 (2)
CH2 (3)
HC) (4)
CO2 (1) CH3 (2]) CH2 (3) HC) (4) COO (5)
0 150.37 140.08 32.60 -305.4
0 26.76 25.83 328.05
0 19.85 355.16
0 -1750.04
B0kl
CO2 (1)
CH3 (2)
CH2 (3)
HC) (4)
CO2 (1) CH3 (2) CH2 (3) HC) (4) COO (5)
0 176.05 270.77 -3.43 -603.94
0 46.11 39.46 1297.09
0 47.38 1347.19
0 -2608.25
Table 3. Deviations of the Equilibrium Pressure (Bubble- or Dew-Point) When the Model Developed by Jaubert et al.1 Is Used To Predict the Vapor-Liquid Equilibria Measurements Performed by Crampon et al.3 a number of data temp./K points
system
∆P%
CO2 + e14-0
313.15 323.15 333.15
26 27 27
6.98 6.46 5.78 average: 6.40
CO2 + e16-0
313.15 323.15 333.15
25 26 26
5.43 6.89 7.54 average: 6.64
CO2 + e18-0
313.15 323.15 333.15
23 23 23
8.56 8.72 10.09 average: 9.12 global average deviation: 7.31
N (100.0/N)∑i)1 |(Pexp
∆P% ) - Pcal)/Pexp| where Pexp and Pcal are respectively the experimental and the calculated bubble-point or dew-point pressures. N is the number of experimental data points. a
Eij(T) ) -
1 ng
ng
∑ ∑(Rik - Rjk)(Ril - Rjl)Akl(T) 2k)1l)1
where ng is the number of different groups present in a solution of molecules i and j. Rik is the fraction of molecule i occupied by group k. Akl(T), where k and l are two different groups, is a temperature-dependent function:
Akl(T) ) A0kl
( ) 298.15 T
rkl
with rkl )
( ) { B0kl A0kl
- 1 and
A0kl ) A0lk B0kl ) B0lk
}
A0kl and B0kl are constant parameters. To apply this GCM to mixtures containing FAEE and CO2, the five following groups must be defined: CO2 (1), CH3 (2), CH2 (3), HC ) (4), and OdC-O (5). The corresponding A0kl and B0kl parameters are given in Table 2. Experimental Database. In 1999, Crampon et al.3 measured 226 bubble-point and dew-point pressures for three binary systems (CO2 + e14-0; CO2 + e16-0; CO2 + e18-0). Each time three different temperatures were considered. All these data were predicted in this study.
Ind. Eng. Chem. Res., Vol. 39, No. 7, 2000 2625
Figure 1. Isothermal phase diagrams predicted with the thermodynamic model developed by Jaubert et al.1: (s) predicted phase envelope and (+) experimental bubble-point or dew-point pressure. (a) System CO2 + e14-0 at T/K ) 313.15; (b) system CO2 + e14-0 at T/K ) 323.15; (c) system CO2 + e14-0 at T/K ) 333.15; (d) system CO2 + e16-0 at T/K ) 313.15; (e) system CO2 + e16-0 at T/K ) 323.15; (f) system CO2 + e16-0 at T/K ) 333.15; (g) system CO2 + e18-0 at T/K ) 313.15; (h) system CO2 + e18-0 at T/K ) 323.15; (i) system CO2 + e18-0 at T/K ) 333.15.
Results and Discussion For each one of the experiments performed by Crampon et al.,3 a bubble-point pressure or a dew-point pressure algorithm was used to compute the equilibrium pressure. This means that the global composition of the cell and the temperature were the input data whereas the pressure was the calculated output. The deviations observed are summarized in Table 3. To better visualize the deviations obtained with the predictive model, isothermal phase diagrams were calculated for each system at each temperature. These nine phase envelopes are shown in Figure 1. Such a figure clearly shows that whatever the system, the CO2 mole fraction, and the temperature, the proposed model is always able to predict with quite high accuracy the equilibrium pressure. For these three systems, the global average deviation of 7.31% is much lower than the deviation previously obtained by Jaubert et al.,1 which was equal to 12.8%. The reason is that the systems considered by Jaubert et al.1 came from different authors who often did not agree each other.
It is also important to notice that in their paper Crampon et al.3 correlated their data with the same version of the PR-EOS as the one used in this research note. They however used a quadratic expression for the covolume. By fitting, for each system and each temperature, two binary interaction parameters (kij and lij), they were able to correlate the data used in this study with an overall deviation of 5.3%. Such a deviation is only 2% lower than the deviation obtained in this work. This is another indicator of the quality of the proposed model. Conclusion In their previous paper, Jaubert et al.1 have extended the group contribution method initially developed by Abdoul et al.2 to ester compounds. They have thus developed a purely predictive method to estimate phase equilibria of multicomponent mixtures involving FAE and CO2. With such a model, phase equilibria of binary systems were correlated with an average accuracy of 12.8%.
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In this note, to test the quality of the proposed model, 226 equilibrium pressures measured by Crampon et al.3 for three binary systems were predicted with an average deviation of 7.3%. This confirms that the proposed model is particularly suitable to predict the phase behavior of systems containing a FAE and CO2. The authors are convinced that such a good result could be obtained because the kij parameters are temperature dependent. Nomenclature a(T) ) temperature-dependent function of the equation of state A0kl, B0kl ) constant parameters allowing the calculation of the binary interaction parameters b ) covolume Eij ) binary interaction energy kij ) binary interaction parameter of the equation of state m ) shape parameter P ) pressure Pc ) critical pressure R ) ideal gas constant T ) temperature Tc ) critical temperature v ) volume Greek Letters ω ) acentric factor Rik ) fraction occupied by group k in the molecule i
Literature Cited (1) Jaubert, J.-N.; Coniglio, L.; Denet, F. From the Correlation of Binary Systems Involving Supercritical CO2 and Fatty Acid
Esters to the Prediction of (CO2-Fish Oils) Phase Behavior. Ind. Eng. Chem. Res. 1999, 38 (8), 3162-3171. (2) Abdoul, W.; Rauzy, E.; Pe´neloux, A. Group-Contribution Equation of State for Correlating and Predicting Thermodynamic Properties of Weakly Polar and Non-associating Mixtures. Binary and Multicomponent Systems. Fluid Phase Equilibr. 1991, 68, 47102. (3) Crampon, C.; Charbit, G.; Neau, E. Phase Equilibrium of Fish Oil Ethyl Esters with Supercritical Carbon Dioxide. J. Supercrit. Fluids 1999, 16, 11-20. (4) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. (5) Soave, G. Equilibrium Constants From a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197-1203. (6) Rauzy, E. Les Me´thodes Simples de Calcul des Equilibres Liquide Vapeur sous Pression. Ph.D. Dissertation, University of Aix-Marseille III, Marseille, France, 1982. (7) Coniglio, L.; Knudsen, K.; Gani, R. Model Prediction of Supercritical Fluid-Liquid Equilibria for Carbon Dioxide and Fish Oil Related Compounds. Ind. Eng. Chem. Res. 1995, 34, 24732483. (8) Jaubert, J. N.; Coniglio, L. The Group Contribution Concept: A Useful Tool To Correlate Binary Systems and To Predict the Phase Behavior of Multicomponent Systems Involving Supercritical CO2 and Fatty Acids. Ind. Eng. Chem. Res. 1999, 38 (12), 5011-5018. (9) Jaubert, J. N.; Coniglio, L. Model Prediction of VaporLiquid Equilibria of Mixtures of Crude Fish Oil Fatty Acid Ethyl Esters and Supercritical Carbon Dioxide. Entropie 2000, 224/225, 69-74.
Received for review January 3, 2000 Revised manuscript received April 6, 2000 Accepted May 16, 2000 IE0000360
