The Group Contribution Concept: A Useful Tool To Correlate Binary

In this paper, a purely predictive model for the phase equilibria computation ... These authors combined the MHV1-modified UNIFAC mixing rule with the...
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Ind. Eng. Chem. Res. 1999, 38, 5011-5018

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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 Jean-Noe1 l Jaubert* and Lucie Coniglio Institut National Polytechnique de Lorraine, Ecole Nationale Supe´ rieure des Industries Chimiques, 1 Rue Grandville, 54000 Nancy, France

In this paper, a purely predictive model for the phase equilibria computation of mixtures involving fatty acids (FA) and supercritical carbon dioxide (SC-CO2) is proposed. In a first step, the phase equilibria modeling of FA/CO2 binary systems were performed using the Peng-Robinson equation of state. To obtain a purely predictive model, the group contribution method developed by Abdoul et al. which allows the prediction of the binary interaction parameters (kij) was extended to FA compounds. In a second step, the proposed model was used to predict the phase equilibria data of ternary and quaternary systems found in the literature. In a last step, bubble-point and dewpoint pressures of systems involving CO2 and multicomponent mixtures of FA were measured in our laboratory, and the previous model was used in order to check its capability in predicting these data. Introduction Supercritical fluid extraction (SCE) of fish oil related compounds using carbon dioxide as the solvent has become a subject of intense research for applications in both food and biochemical industries. Because of the scarceness of experimental data (supercritical fluidliquid equilibria and pure solute physical properties), a purely predictive thermodynamic model plays an important role in the study of SCE processes. It is obvious that as SCE processes operate around phase boundaries the model also needs to be reliable and accurate. The recent literature review made by Jaubert et al.1 has shown that most of the high-pressure vapor-liquid equilibria (VLE) calculations involving CO2 and fish oil compounds are performed by means of cubic equation of states2,3 (EOS). Even if, as shown by Guo et al.,4 precise results may be obtained by fitting the binary interaction parameters kij at each temperature, such an approach is absolutely not predictive and thus out of interest for our study. In return, Coniglio et al.5 have worked on the prediction of systems involving SC-CO2 and fish oil related compounds. These authors combined the MHV1-modified UNIFAC mixing rule with the SRK EOS. Moreover, a quadratic expression with a constant lij value for the mixture b parameter was used. Their model was thus fully predictive, and precise results were obtained for both binary and ternary systems. By the end, in 1999, Jaubert et al.1 extended the Abdoul-Rauzy-Pe´neloux group contribution equation of state (ARP GC EOS) to mixtures containing CO2 and fatty acid esters (FAE) by defining the ester chemical function as a new group. Such a model combines at constant packing fraction the Peng-Robinson (PR) EOS and a Van Laar type Gibbs energy model. The best * To whom correspondence should be addressed. Fax: (+33) 383.35.08.11.Phone: (+33)383.17.50.81.E-mail: [email protected].

advantage of this approach comes from the group contribution method (GCM) initially developed by Abdoul et al.6 in 1991 to predictsas a function of temperaturesthe binary interaction parameter Eij of the Van Laar model. Seventeen groups were initially defined by Abdoul et al.,6 and new groups were added by Fransson et al.7 and by Berro et al.8 The use of this GCM renders fully predictive the thermodynamic model. In this paper, it was decided to check the applicability of the constant packing fraction approach to correlate the phase equilibria of binary systems involving CO2 and fatty acids (FA). This is a great challenge because it is well-known that cubic EOS are not able to properly correlate the properties of polar and associating mixtures. Because the acid chemical function (OdCsOH) was not defined by Abdoul et al.,6 our first task was thus to extend this GCM by adding the acid chemical function (COOH) as a new group. In a second step, the fully predictive model thus obtained was used to predict the experimental data of ternary and quaternary systems found in the literature. In a last step, dew-point and bubble-point pressures of mixtures involving CO2 and eight fatty acids were measured in our laboratory, and our model was used to predict these data. Comparison with the previous predictive method developed by Coniglio et al.5 and based on the MHV1-modified UNIFAC mixing rule is also given. Experimental Section Apparatus and Procedure. High-pressure bubble and dew points of mixtures including CO2 and eight different FA were measured using the static type apparatus shown in Figure 1. This equipment consists of a high-pressure view cell with variable volume built of stainless steel, a magnetic stirrer, a camera (Hitachi model KP-C551), and a TV monitor. The cell, designed by Top Industries S.A., is equipped with a sapphire

10.1021/ie990544d CCC: $18.00 © 1999 American Chemical Society Published on Web 12/06/1999

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Figure 1. Apparatus used for dew- and bubble-point measurement: 1, equilibrium cell; 2, window in sapphire; 3, CO2 tank; 4, removable screwed cap allowing one to feed in the fatty acids; 5, thermostated water outlet; 6, thermostated water inlet; 7, TV connected to the cine camera aimed at observing inside the cell; 8, stirring magnet; 9, graduated piston allowing the measurement of the cell volume; 10, manual system allowing the pressure to vary by changing the cell volume.

window to enable a direct visualization of the medium. The temperature inside the cell is regulated thanks to a thermostated bath. The highest working temperature is equal to 150 °C. When the manual pump is moved, the pressure inside the cell may vary from 1 to 700 bar and is measured with an accuracy of 1 bar thanks to a pressure gauge. The total cell volume is recorded with an accuracy of 0.1 mL on the graduated piston connected to the cell. Such a volume may vary between 1.7 and 12 cm3. At the beginning of each experiment, the stainless steel screw is removed; that is, the top of the cell is opened. A given amount of each of the eight FA selected in this study is thus introduced into the cell. Because at ambient conditions the FA are solid, an analytical balance was used to determine precisely the amount of each FA introduced in the cell. The cell containing the mixture of FA being opened, the CO2 tank is slightly opened to send CO2 inside the cell and to overrun air outside. A few seconds later, the cell which now contains the acids and a small amount of CO2 is closed thanks to the steel screw. The CO2 tank is also closed and the cell is thermostated at low temperature (around 10 °C). Once the temperature and pressure are uniform inside the cell, the CO2 bottle is once again opened and a given amount of CO2 spontaneously enters the cell because of the pressure difference between the cell and the CO2 tank. After the CO2 bottle is closed, the cell is left until the pressure and the temperature stabilize. The initial temperature and pressure are recorded. The initial total cell volume is simply read on the graduated piston connected to the cell. At low temperature, the acids remain solid and do not mix with the CO2. It is thus easy to determine the amount of CO2 introduced in the cell from the knowledge of the density of each acid (given by Akzo Nobel) and from the knowledge of the CO2 density (taken in IUPAC tables9). Once the global cell composition (amount of CO2 and amount of each FA) is known, the cell is thermostated at different temperatures. For a given temperature, the piston is manually moved very slowly to reduce the total volume and to increase pressure until the cell becomes monophasic. In fact, once the pressure is stabilized, it is slowly decreased until a second phase appears. The medium is

Table 1. Mass Composition of the Nine-Component Systems Investigated in This Work experiment no. 1

2

mass content/g CO2 a8-0 a10-0 a12-0 a14-0 a16-0 a18-0 a18-1 a20-0 CO2 a8-0 a10-0 a12-0 a14-0 a16-0 a18-0 a18-1 a20-0

8.9408 0.0393 0.0622 0.0650 0.0692 0.0668 0.0666 0.0002 0.0007 2.7088 0.7045 0.5835 0.6491 0.6248 0.6318 0.6422 0.0019 0.0065

thus successively compressed and expanded to define the most narrow range of pressure for the phase transition. The stabilization has to be reached after each modification of pressure, but because of the small cell volume and the efficiency of the stirring, the equilibration does not require more than 3 h. The pressure gauge thus indicates the saturation pressure of the system, i.e., the pressure at which a given amount of matter thermostated at a given temperature becomes monophasic. The experimental accuracy on such a measurement was found to be approximately 1 bar because the cine camera connected to the cell allows a very precise location of the phase transition. Experimental Results. In this study, two experiments were performed. For each of them, the global composition of the cell is given in Table 1. In the first experiment, the CO2 mole fraction is equal to 0.9914, and at each temperature, through the camera images, it was possible to observe the disappearance of the bottom phase (the one having the highest density) meaning that dew-point pressures were measured. In return, in the second experiment for which the CO2 mole fraction is equal to 0.764, the opposite phenomenon

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5013 Table 2. Experimental Dew-Point and Bubble-Point Pressures Measured as a Function of Temperaturea experiment no.

type of measurement

1

dew-point pressures

2

bubble-point pressures

Table 3. Estimated Pure-Component Physical Properties

temp/ °C

measured saturation pressure/bar

40 50 60 70 80 90 100 110 120 40 50 60 70 80 90 100 110 120

173 ( 1 198 ( 1 212 ( 1 229 ( 1 243 ( 1 254 ( 1 267 ( 1 273 ( 1 278 ( 1 120 ( 1 139 ( 1 156 ( 1 174 ( 1 192 ( 1 207 ( 1 220 ( 1 234 ( 1 246 ( 1

a For each experiment the constant global composition of the cell is given in Table 1.

took place (disappearance of the vapor phase) meaning that bubble-point pressures were measured. For each experiment, nine temperatures ranging from 40 to 120 °C were selected. The corresponding saturation pressures (dew or bubble points) are summarized in Table 2. When these experimental saturation pressures are plotted versus temperature, it is possible to build for each global composition (two in this study) a phase envelope in the pressure-temperature (P/T) plane. Thermodynamic Model As stated in the Introduction, the phase equilibria calculations of mixtures involving CO2 and FA were performed by combining at constant packing fraction the PR EOS and the Van Laar gE model. All of the details of this thermodynamic model are stated in this section. Pure-Component Equation of State. The cubic EOS used for all calculations is the PR3 EOS:

P)

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

with

b ) 0.07780RTc/Pc

and

To precisely calculate the vapor pressures of pure components, the following Soave2 type a(T) function, developed by Rauzy,10 was used:

[

( ()

R2Tc2 T 1+m 1Pc Tc

Tc/K

Pc/bar

acentric factor

a6-0 a8-0 a10-0 a12-0 a14-0 a16-0 a18-0 a18-1 a18-1ω9 a18-2ω6 a20-0

663.00 696.73 719.94 740.11 758.40 775.33 791.33 790.77 790.77 790.21 806.67

31.998 25.615 21.583 18.647 16.414 14.659 13.242 13.526 13.526 13.821 12.076

0.6938 0.7186 0.8091 0.8991 0.9844 1.0648 1.1397 1.1304 1.1304 1.1209 1.2091

a First letter: (a) for acid. First number: the carbon atom number of the molecule. Second number: the number of double bonds in the fatty acid chain. Last number (after ω): the position of the terminal double bond relative to the nonpolar end of the fatty acid chain.

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). As pointed out by many authors, the FA decompose before reaching their normal boiling temperature and the three properties required by the EOS (Tc, Pc, and ω) have to be estimated. The approach developed in 1995 by Coniglio et al.5 and also used by Jaubert et al.1 was followed here. The estimated physical properties of all of the FA investigated in this work are summarized in Table 3 with the exception of the most volatile FA (the caproic acid a6-0), for which the values given in Table 3 are in fact experimental values. It is worth noticing that the use of group contribution methods implies that some isomers which only differ by the location of the double bonds have exactly the same properties. Mixing Rules. In this section, it is briefly recalled how the mixing rules of the PR EOS may be defined by introducing at constant packing fraction an excess Gibbs energy function. From classical thermodynamics, it is well-known that the fugacity coefficient φi of component i in a homogeneous phase is given by

ln φi )

R ) 8.314 41 J‚mol-1‚K-1

a(T) ) 0.457236

compound designationa

0.445075

)]

2

with

m ) 6.812553 × [x1.127539 + 0.517252ω - 0.003737ω2 - 1] To perform phase equilibria calculations with such an EOS, it is necessary to estimate for each pure component its critical temperature and pressure and its acentric factor.

(

)

∂Ares/RT ∂ni

- ln z

(1)

T,V,nj*i

where z is the compressibility factor of the phase and Ares is the difference between the Helmholtz energy of the real phase and the Helmholtz energy of an ideal gas mixture having the same temperature T, the same volume V, and the same composition as the real mixture. By defining the packing fraction of a phase as η ) b/v, where b is the covolume and v the molar volume, one obtains

Ares(T,η,n) )n RT

∫0η

z(T,η,n) - 1 dη η

(2)

To compute the fugacity coefficients through eq 1, it is thus necessary to be able to estimate the compressibility factor z(T,η,n) of a phase. According to the reference at constant packing fraction, Pe´neloux et al.11 have shown that z(T,η,n) could be expressed by

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z(T,η,x) )

xizi(T,η) + zE(T,η,x) ∑ i)1

zE(T,η,x) )

(

)

E η ∂A (T,η,x) RT ∂η

T,x

(3)

(5)

According to the previous work of Abdoul et al.,6 the following function was selected for F(η):

F(η) )

1 + η(1 + x2) 1 ln 2x2 1 + η(1 - x2)

(6)

A Van Laar like gE model was also selected: p

p

∑∑xixjbibjEij(T)

gE(T,x) )

1 i)1j)1 2

p

xibi ∑ i)1

with

{

Eij(T) ) Eji(T) (7) Eii(T) ) 0.0

In eq 7, the covolume bi of each pure component is calculated through the PR EOS whereas the binary interaction parameter Eij(T), which is usually fitted on experimental data, was computed by the GCM developed by Abdoul et al.6 As explained in the Introduction, the group COOH (acid chemical function) is, however, not available, and our first task was thus to add such a group. Abdoul’s Group Contribution Method. To predict the binary interaction parameters Eij(T) of the Van Laar gE model (eq 7) in terms of group contributions, Abdoul et al.6 used the following equation initially proposed by Redlich et al.:12

Eij(T) ) -

1

ng ng

∑ ∑(Rik - Rjk)(Ril - Rjl)Akl(T)

2k)1l)1

( ) 298.15 T

rkl

with

(4)

zi is the compressibility factor of pure component i calculated by a given equation of state (not necessarily the same one for each pure component). In this study, zi was estimated by the PR EOS whatever the pure component considered (CO2 or FA). Moreover, in eq 4, the molar excess Helmholtz energy AE that is the difference between the molar Helmholtz energy of the real solution and the molar Helmholtz energy of the corresponding ideal solution may be expressed as explained by Pe´neloux et al.11 as the product of two functions. The first one is a classical excess Gibbs energy function (Van Laar, NRTL, UNIQUAC, ...) depending only on temperature and composition, whereas the second only depends on the packing fraction. This means

AE(T,η,x) ) gE(T,x) F(η)

Akl(T) ) A0kl

(8)

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:

( ) {

rkl ) and

B0kl A0kl

-1

A0kl ) A0lk (9) B0kl ) B0lk

A0kl and B0kl are constant parameters determined by Abdoul et al.6 for 17 different groups. To apply this GCM to mixtures containing FA and CO2, the five following groups must be defined: CO2 [1], CH3 [2], CH2 [3], HCd [4], and COOH [5]. The first four groups were previously defined by Abdoul et al.6 Correlation of Binary Systems Involving CO2 and FA Extension of Abdoul’s Group Contribution Method to Fatty Acid Compounds. To predict the phase behavior of mixtures involving FA and CO2, it is necessary to define as a new group the acid chemical function (HOsCdO) and to determine the interactions between this new group (number 5) and the four other groups (CO2, CH3, CH2, and HCd). The eight following 0 0 parameters required thus to be estimated: A1-5 , A2-5 , 0 0 0 0 0 0 A3-5, A4-5, B1-5, B2-5, B3-5, and B4-5. Moreover, Abdoul et al.6 did not consider experimental VLE data of molecules with several double bonds in the regression of the interaction parameter between groups 1 (CO2) and 4 (HCd). In their previous paper, Jaubert et al.1 decided, however, to re-estimate the two following 0 0 and B1-4 . These two recent values parameters: A1-4 were used in this paper. In this study, the eight previously described parameters were fitted in order to reproduce experimental phase equilibria data of binary systems involving mixtures of CO2 and FA. For these systems, 145 experimental data points were found in the literature.13-18 Most of these data include the knowledge of the temperature, pressure, and liquid (x) and gas (y) phase compositions. A summary of these data points is given in Table 4. The parameter estimation was performed using a P/T flash algorithm. The objective function Np

Fob )

exp xi,CO ∑ [( i)1

) ( 2

2

cal exp cal - xi,CO + yi,CO - yi,CO 2 2 2

)] 2

(10)

in which Np is the number of experimental data points xCO2 and yCO2 are respectively the mole fraction of CO2 in the liquid and in the gas phase was minimized using the minimization technique of Nelder and Mead.19 The values of the eight fitted parameters are summarized in Table 5. The calculated isothermal binary phase diagrams are shown in Figure 2 for the six binary systems at selected temperatures. For possible comparison with the previous work by Coniglio et al.,5 the deviations were calculated using a bubble-point algorithm and are summarized in Table 6. The obtained results (see Table 6 and Figure 2) show that the thermodynamic model developed in this work gives accurate results and that the deviations on the bubble-point pressures are 3.5% smaller than those obtained in 1995 by Coniglio et al.5 The gas-phase composition is also better calculated with the proposed

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5015 Table 4. Experimental Binary Data Base type of data and corresponding no. of data experimental points for the whole temp binary system temp/K CO2 + a6-0 CO2 + a12-0

CO2 + a16-0

CO2 + a18-1ω9

CO2 + a18-2ω6 CO2 + a20-0

313.15 353.15 333.15 353.15 373.20 423.20 473.20 353.15 373.15 373.20 423.20 473.20 313.15 327.25 333.15 350.05 353.15 374.25 400.95 313.15 333.15 373.20 423.20 473.20

ref

T, P, x, y T, P, x T, P, x, y T, P, x

8 2 22 9

13

T, P, x, y T, P, x

9 13

13, 14

T, P, x, y T, P, x

43 12

15-18

T, P, x, y

12

15

T, P, x, y T, P, x

4 11

14

13, 14

Table 5. A0kl and B1kl Parameters (in J‚cm-3) for the Five Groups Appearing in Mixtures of CO2 and FA.a The eight values in bold were regressed in this study, the two underlined values were determined by Jaubert et al.,1 and the last 10 were given by Abdoul et al.6 A0kl

CO2 [1]

CH3 [2]

CH2 [3]

HCd [4]

CO2 [1] CH3 [2] CH2 [3] HCd [4] COOH [5]

0 150.37 140.08 32.60 -448.83

0 26.76 25.83 -111.18

0 19.85 -25.39

0 -891.84

B0kl

CO2 [1]

CH3 [2]

CH2 [3]

HCd [4]

CO2 [1] CH3 [2] CH2 [3] HCd [4] COOH [5]

0 176.05 270.77 -3.43 85.44

0 46.11 39.46 -14.07

0 47.38 -22.65

0 3411.21

model. From an industrial point of view, deviations of 10% may be regarded as acceptable. This means our goal has been achieved. To check the reliability and the predictive capability of our model, it was decided to predict the behavior of ternary, quaternary, and multicomponent systems. Indeed, Abdoul et al.6 developed their GCM to predict with high accuracy the phase behavior of very complex mixtures such as petroleum fluids (see, for example, Jaubert et al.20). Moreover, working on mixtures of FAE, Jaubert and co-workers1,21 showed that such a model could lead to small deviations even when the number of components becomes as high as 30. It is thus possible to expect a good prediction of the phase behavior of systems containing many different FA. Phase Equilibria Prediction of Multicomponent Systems In this section, the proposed model is used to predict first the phase behavior of a ternary and a quaternary system and second the behavior of multicomponent systems with a larger number of components.

Ternary System. The SC-fluid-liquid equilibrium data determined by Zou et al.22 on the ternary system CO2 + a18-1ω9 + a18-2ω6 at 40 and 60 °C were predicted with the proposed model. The results obtained are shown in Figure 3. An average deviation of ∆P% ) 25.2% was obtained on the 39 experimental bubble-point pressures, whereas for the same system, Coniglio et al.5 report a deviation of 28.8%. The deviations observed are higher than those obtained for each one of the binary systems CO2 + a18-1ω9 and CO2 + a18-2ω6. The first reason is that the slope of the experimental phase diagram (see Figure 3) is very steep. This means that a small error on the liquid-phase composition may lead to high deviations on the corresponding bubble-point pressure. Moreover, for the binary systems, the data points measured by Zou et al.15 also lead to deviations much higher than those corresponding to other authors (see, for example, the binary system CO2 + a18-2ω6, for which Zou et al.15 are the unique authors to have made measurements). This feature means the experimental data points by Zou et al.15 are not in good agreement with other measurements. It is thus not surprising that the deviations observed on this ternary system are quite high. Quaternary System. The SC-fluid-liquid equilibrium data determined by Brunner17 on the quaternary system CO2 + n-hexane + a18-1ω9 + a18-0 at temperatures close to 102 °C were also predicted. An average deviation of ∆P% ) 8.5% was observed on the six experimental bubble-point pressures. The composition of the corresponding gas phase was predicted with an average deviation of 100|∆y|av ) 0.16. For the same system, Coniglio et al.5 report deviations of ∆P% ) 8.2% and 100|∆y|av ) 0.39. Here, both predictive models are quite equivalent though the model developed in this work is able to better predict the gas-phase composition. The two acids present in this quaternary system have structures very similar (the same carbon atom number) to those contained in the ternary system. Because, in this case, the deviations observed are similar to the ones obtained on the binary systems, it is once more possible to argue that the measurements made by Zou et al.22 on the ternary system are probably not very accurate or even consistent. Multicomponent Systems. The predictive model developed in this work was used to predict the experimental dew-point and bubble-point pressures measured in this study (see Table 2). From our knowledge, the model by Coniglio et al.5 is the unique one published in the literature to be purely predictive too. To compare these two models, predictions of the measured experimental saturation pressures were also made with Coniglio’s model. The results are summarized in Table 7 and can be shown in Figure 4. The P/T phase envelopes shown in Figure 4 are the locus of the bubblepoint and/or dew-point pressures as a function of temperature for a mixture, the global composition of which is known. In such diagrams, the global composition of the system remains constant, whereas the pressure and temperature vary. The system is monophasic outside the phase envelope and polyphasic inside. It becomes clear that the purely predictive model developed in this work is able to predict the phase behavior of complex mixtures with quite high accuracy even when a large amount of CO2 (experiment 1) is introduced in the cell. What is very important to outline is that the deviations observed on multicom-

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Figure 2. Calculated isothermal phase diagrams using parameters given in Tables 3 and 5.+: experimental bubble-point pressures; /: experimental dew-point pressures. (a) System CO2 + a6-0 at T/K ) 353.15. (b) System CO2 + a12-0 at T/K ) 353.15. (c) System CO2 + a16-0 at T/K ) 373.15. (d) System CO2 + a18-1ω9 at T/K ) 333.15. (e) System CO2 + a18-2ω6 at T/K ) 313.15. (f) System CO2 + a20-0 at T/K ) 473.2.

ponent systems (9.2% on average) are very similar to those obtained on binary systems (9.3% on average). This feature means that the purely predictive model developed in this work may be used to predict with high confidence the behavior of fatty acids whatever the temperature, pressure, and amount of CO2 considered. In return, Coniglio’s model may lead to excellent results (less than 3% deviation) but only when the amount of CO2 is not too high. The fact that Coniglio’s

model could lead to high deviations when the CO2 mole fraction approaches unity has previously been pointed out by Jaubert et al.1 Conclusion In this study, binary systems involving SC-CO2 and FA were correlated with the PR EOS using the approach at constant packing fraction initially developed by Prof. Pe´neloux. To do so, the group contribution method

Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 5017 Table 6. Comparison of Bubble-Point Pressures and Gas Composition Calculated by Different Methods system: CO2 + a6-0 a12-0 a16-0 a18-1ω9 a18-2ω6 a20-0 average

Coniglio et al.5 (predictive method) ∆P% 100|∆Y|av

this work (predictive method) ∆P% 100|∆Y|av

28.2 10.2 8.5 11.4 19.7 13.5 12.8

10.3 4.8 9.9 8.7 19.6 10.9 9.3

0.53 0.30 0.02 0.29 0.71 0.03 0.28

0.70 0.18 0.06 0.26 0.40 0.02 0.23

a ∆P% and ∆y are respectively the percent average relative deviations on the bubble-point pressures and the average absolute deviations on the gas-phase compositions.

Figure 4. P/T phase envelopes calculated with the predictive model developed in this work: (a) first global composition (see experiment 1 in Table 2); (b) second global composition (see experiment 2 in Table 2). b: experimental dew-point pressures. O: experimental bubble-point pressures. s: predicted dew-point curve. - - -: predicted bubble-point curve. +: calculated critical point which joins the bubble- and dew-point curves. Table 7. Comparison of Two Thermodynamic Models To Predict the Measured Dew- and Bubble-Point Pressures of Two Complex Mixtures Involving CO2 and FA predicted saturation pressure (bar) and deviation (∆P%) with exp. values experiment no.

Figure 3. Prediction of VLE data for the ternary system CO2 (1) + a18-1ω9 (2) + a18-2ω6 (3): (a) T/K ) 313.15; (b) T/K ) 333.15. 0, ), O: experimental dew-point pressures for x2/(x2 + x3) equal to 0.15, 0.50, and 0.75 respectively. 9, [, b: experimental bubblepoint pressures for x2/(x2 + x3) equal to 0.15, 0.50 and 0.75, respectively. - - -, s, -‚-: predicted curves for x2/(x2 + x3) equal to 0.15, 0.50, and 0.75, respectively. At T/K ) 333.15, these three calculated curves are undistinguishable.

developed by Abdoul et al. for the estimation of the binary interaction parameter Eij of the Van Laar excess Gibbs energy model was extended to FA components and a new group, OdCsOH had to be defined. The fully predictive model thus obtained was used to predict the phase behavior of systems involving eight different FA and SC-CO2. The nice results obtained (less than 10% deviations) show that the mixing rule at constant packing fraction is suitable to predict the phase behavior of complex mixtures.

temp/°C

1

40 50 60 70 80 90 100 110 120

2

40 50 60 70 80 90 100 110 120

Coniglio et al.5

this work

225.3 -30.22 248.2 -25.36 267.6 -26.24 283.9 -23.97 286.6 -17.94 296.2 -16.62 296.5 -11.03 308.7 -13.09 311.8 -12.16 average: 19.6% 109.4 8.87 136.8 1.57 160.4 -2.79 180.5 -3.71 197.5 -2.88 212.0 -2.41 224.2 -1.89 234.3 -0.13 242.7 1.34 average: 2.8%

160.1 7.46 171.7 13.28 186.1 12.21 200.3 12.54 213.1 12.30 224.1 11.77 233.1 12.69 240.1 12.07 244.9 11.90 average: 11.8% 135.3 -12.73 141.6 -1.85 153.5 1.59 166.8 4.16 179.8 6.34 192.2 7.16 203.6 7.44 214.1 8.50 223.6 9.10 average: 6.5%

According to this work where Abdoul’s group contribution method has been extended to FA components together with the recent work by Jaubert et al.1 where the same GCM was extended to esters, a useful tool based on the group contribution concept has been developed to predict the phase behavior of any mixtures involving CO2 and fish oil related compounds.

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Nomenclature AE ) excess Helmholtz energy Ares ) residual Helmholtz energy a(T) ) temperature-dependent function of the EOS A0kl, B0kl ) constant parameters allowing the calculation of the binary interaction parameters b ) covolume Eij ) binary interaction parameter of the Van Laar excess model gE ) excess Gibbs energy kij ) binary interaction parameter of the EOS m ) shape parameter P ) pressure Pc ) critical pressure R ) ideal gas constant T ) temperature Tc ) critical temperature v ) volume xi ) mole fraction z ) compressibility factor Greek Letters φi ) fugacity coefficient η ) packing fraction ω ) acentric factor Rik ) fraction occupied by group k in the molecule i

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(8) Berro, C.; Barna, L.; Rauzy, E. A Group-Contribution Equation of State for Predicting Vapor-Liquid Equilibria and Volumetric Properties of Carbon Dioxide-Hydrocarbons systems. Fluid Phase Equilib. 1996, 114, 63-87. (9) Angus, S.; Armstrong, B.; Reuck, K. M. IUPACsInternational Thermodynamics Tables of the Fluid State Carbon Dioxide; Pergamon Press: Oxford, U.K., 1976. (10) Rauzy, E. Les Me´thodes Simples de Calcul des Equilibres Liquide-Vapeur sous Pression. Ph.D. Dissertation. The French University of Aix-Marseille III, 1982. (11) Pe´neloux, A.; Abdoul, W.; Rauzy, E. Excess Functions and Equations of State. Fluid Phase Equilib. 1989, 47, 115-132. (12) Redlich, O.; Derr, E. L.; Pierotti, G. J. Group Interaction. 1. A Model for Interaction in Solutions. J. Am. Chem. Soc. 1959, 81, 2283-2285. (13) Bharath, R.; Yamane, S.; Inomata, H.; Adschiri, T.; Arai, K. Phase Equilibria of Supercritical CO2 - Fatty Oil Component Binary Systems. Fluid Phase Equilib. 1993, 83, 183-192. (14) Yau, J. S.; Chiang, Y. Y.; Shy, D. S.; Tsai, F. N. Solubilities of Carbon Dioxide in Carboxylic Acids Under High Pressures. J. Chem. Eng. Jpn. 1992, 25 (5), 544-548. (15) Zou, M.; Yu, Z. R.; Kashulines, P.; Rizvi, S. S. H.; Zollweg, J. A. Fluid-Liquid-Phase Equilibria of Fatty Acid Methyl Esters in Supercritical Carbon Dioxide. J. Supercrit. Fluids 1990, 3 (2), 23-28. (16) Yu, Z. R.; Rizvi, S. S. H.; Zollweg, J. A. Phase Equilibria of Oleic Acid, Methyl Oleate, and Anhydrous Milk Fat in Supercritical Carbon Dioxide. J. Supercrit. Fluids 1992, 5, 114-122. (17) Brunner, G. Phesengleichgewichte in Anwsenheit Komprimierter Gase und ihre Bedeutung bei der Trennung Schwerfluchtiger Stoffe. Habilitationsschrift, University of ErlangenNurnberg, Erlangen, Germany, 1978. (18) Bharath, R.; Inomata, H.; Adschiri, T.; Arai, K. Phase Equilibrium Study for the Separation and Fractionation of Fatty Oil Components Using Supercritical Carbon Dioxide. Fluid Phase Equilib. 1992, 81, 307-320. (19) Nelder, J. A.; Mead, R. A Simplex Method for Function Minimization. Comput. J. 1965, 7, 308-313. (20) Jaubert, J. N.; Neau, E.; Pe´neloux, A.; Fressigne´, C.; Fuchs, A. Phase Equilibrium Calculations on an Indonesian Crude Oil Using Detailed NMR Analysis or a Predictive Method To Assess the Properties of the Heavy Fractions. Ind. Eng. Chem. Res. 1995, 34, 640-655. (21) 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, accepted. (22) Zou, M.; Yu, Z. R.; Rizvi, S. S. H.; Zollweg, J. A. FluidLiquid Equilibria of Ternary Systems of Fatty Acids and Fatty Acid Esters in Supercritical CO2. J. Supercrit. Fluids 1990, 3, 8590.

Received for review July 22, 1999 Revised manuscript received October 4, 1999 Accepted October 4, 1999 IE990544D