ABSTRACT

A thermodynamic consistency test developed by the authors is applied to high pressure phase equilibrium data of ternary mixtures containing a compressed gas and two solid solutes. The experimental data have been modeled using the Peng-Robinson equation of state with classical van der Waals mixing rules. These type of mixtures are ussualy highly asymmetric and present inherent difficulties of analysis, in particular because of the dispersion of the data and the very low concentration of the solutes in the compressed gas phase. Despite this, the model employed allows correlating the solubility of the solids in the high pressure gas with deviations much lower than other models. The complex mixture carbon dioxide+capsaicin+b-carotene is used as a study case. It is shown that the thermodynamic test can be applied with confidence and consistency or inconsistency of the data can be accurately determined. <>INTRODUCTION

Solubility data of solids in high pressure gases reported in the literature usually present large discrepancies, which have been attributed to different reasons by different researchers. Even though we cannot be certain on the true reasons for the large discrepancies, we need to decide whether a set of solubility data is acceptable enough as to be used in process design, analysis and simulation. Therefore, methods to test inherent inaccuracies of the experimental data have become necessary. The so-called “thermodynamic consistency tests”, based on the Gibbs-Duhem equation, are the usual ways of checking experimental phase equilibrium data. Methods applicable to low pressure data have been widely published [1], but studies on thermodynamic consistency tests for high pressure phase equilibrium data are scarce and the only attempt for ternary mixtures we know is that recently presented by the authors [2]. The cases of interests in this study are mixtures containing two solid solutes dissolved in a high pressure solvent. With some appropriate assumptions the mole fraction of the solute "i" in the high pressure solvent, or solubility, at the temperature T, can be reduced to [3, 4]:

 

(1)

In eqn. (1), j_i is the fugacity coefficient of the solid component in the high pressure gas, P_i^sub is the sublimation pressure of the pure solid, V_i^s is the molar volume of the solid, T is the absolute temperature, and R is the ideal gas constant.

 

For a multicomponent homogeneous gas mixture at constant temperature, the Gibbs-Duhem equation can be written as [5]:

 

(2)

being y_i  the mole fraction of component "i" in the gas phase, P is the system pressure, Z is the compressibility factor of the gas mixture, and j_i is the fugacity coefficient of component “i”. If more than one solute is present in the high pressure gas mixture, one equation for each solute component must be written. For a given solute “j” the equation is:

 

(3)

The Peng-Robinson equation with the van der Waals mixing rule [6] has been used in this study to correlate the experimental ternary high pressure solubility data and to determine the fugacity coefficients j_I needed in eqn. (3). The model is designated here as PR/vdW. <>Consistency Criteria

To define the criteria for thermodynamic consistency it is first required that the model be able to correlate the data within acceptable deviations. The model is accepted and the consistency test is applied if deviations between calculated and experimental solubility values for each and every point are within -20% to 20%. After the model is found appropriate, it is required that the Gibbs-Duhem equation is fulfilled [2, 7]. In eqn. (3 the left hand side is designated by A_Pj and the expression inside the summation term on the right hand side by A_jj. That is:

 

	(4)
	(5)

Thus, if a set of data is considered to be consistent A_Pj should be equal to A_jj for j=2 and j=3 within acceptable defined deviations. To set the margins of errors, an individual percent area deviation [%DA_j] _i  between experimental and calculated values is defined as:

 

(6)

The consistency method requires that the difference between the integral on the left hand side of eqn. (3) and the sum of the integrals on the right hand side is between defined margins (between -20% and +20%), to declare the data as being thermodynamically consistent. In a previous work on binary mixtures the authors have clearly explained how these limits of acceptance for the solubility and the areas, were determined [7]. The data are considered to be thermodynamically consistent (TC) if the deviations in correlating the solute solubility and all the individual deviations in the areas are within the defined limits. The data are considered to be thermodynamically inconsistent (TI) if the deviations in correlating the solute solubility are within the established limits but the individual deviations in the areas are outside the established limits, for more than 25% of the data points in the data set. If only some few data points (less than 25% of the original points), give deviations outside the accepted range the set of data is declared to be not fully consistent (NFC).

  <>Results and DiscussioN

To the best of our knowledge the only set of data for the ternary mixture CO₂+capsaicin+b-carotene published in the literature is that of Skerget and Knez [8], data that is checked for consistency in this study. The range of pressure of the data is from 10 to 30 MPa and the range of solubility is from 2.4x10^-4 to 4.5x10^-4 in mole fraction for capsaicin and from 1x10^-7 to 6x10^-7 in mole fraction for β-carotene. The Table 1 shows the modeling results for the ternary mixture. This Table shows the deviations between experimental and calculated values for the solute solubilities and the interaction parameters k₁₂, k₁₃, and k₂₃ calculated using ternary solubility data and the PR/vdW model. The model parameters were determined using genetic algorithms as described by the authors [2, 7], method that has been successfully applied in this study. In all cases the deviations for the gas solvent is lower than 1%, so numbers are not shown. Table 2 presents the results of the consistency test for the ternary mixture. Table 1: Calculated deviations and optimum interaction parameters for the ternary mixture CO₂(1)+capsaicin(2)+β-carotene(3) using the PR+VdW model.

 

Table 2: Details of the consistency test for the ternary systems CO₂(1)+capsaicin(2)+β-carotene(3)

 

 

The original data set containing ten points at both temperatures (298 and 313K) showed to be not fully consistent (NFC). This is so because the maximum deviations in the individual areas, designated as Max.[%DA₂]_i  and Max.[%DA₃]_i , are outside the acceptable interval of  -20% to +20%. Once the data with high area deviations are removed, the remaining set of 7 data points at 298K and 8 data points at 313K showed to be consistent. The subindexes 2 and 3 indicate the average deviation with respect to solutes 2 and 3, respectively. Although the average percent deviation is not explicitly considered as a determining parameter for the decision of consistency or inconsistency, the results presented in Table 5 show the expected behavior. <>CONCLUSIONS

According to the results, the following conclusions can be drawn: i) the Peng-Robinson equation with the van der Waals mixing rule can be used to acceptable correlate the experimental ternary high pressure solubility data; ii) the numerical method based on genetic algorithms shows to be efficient for searching a global optimum; and iii) the consistency test allows to determine the thermodynamic consistency of the experimental data. ACKNOWLEDGMENTS

 

The authors thank the support of the National Commission for Scientific and Technological Research (CONICYT-Chile), through the research grant FONDECYT 1040285, the Direction of Research of the University of La Serena-Chile for permanent support through several research grants and the Center for Technological Information (CIT, La Serena-Chile), for computer and library support. <>REFERENCES

[1] Raal, J.D., Mühlbauer A.L., Phase Equilibria: Measurement and Computation, Taylor & Francis, Washington, DC (1988)

[2] Valderrama J.O., Robles P.A., Fluid Phase Equil., 242(1), 93-102  (2006)

[3] Walas S.M., Phase Equilibria in Chemical Engineering, Butterworth Publisher-USA  (1985)

[4] Prausnitz J.M., Lichtenthaler R.N., Gomez de Azevedo E. (1999), Molecular Thermodynamics of Fluid Phase Equilibria, 3^rd edition, Prentice Hall. Englewood Cliffs, NJ - USA (1999)

[5] Bertucco, A.;  Barolo, M.; Elvassore, N.,  AIChE J., 43(2), 547-554 (1997)

[6] Peng D.Y., Robinson D.B., Ind. Eng. Chem. Fundam., 15(1), 59-64 (1976)

[7] Valderrama J.O., Zavaleta J., J. Supercrit. Fluids., 39(1), 20-29 (2006)

[8] Skerget M., Knez Z., J. Agric. Food Chem., 45, 2066-2069 (1997)