THE GENERALIZED MAXIMUM LIKELIHOOD METHOD APPLIED TO HIGH PRESSURE PHASE EQUILIBRIUM

THE GENERALIZED MAXIMUM LIKELIHOOD METHOD APPLIED TO HIGH PRESSURE PHASE EQUILIBRIUM¹

Lúcio CARDOZO-FILHO², Luiz STRAGEVITCH³, Fred WOLFF²; M. Angela A.MEIRELES⁴, ^*

SUMMARY

The generalized maximum likelihood method was used to determine binary interaction parameters between carbon dioxide and components of orange essential oil. Vapor-liquid equilibrium was modeled with Peng-Robinson and Soave-Redlich-Kwong equations, using a methodology proposed in 1979 by Asselineau, Bogdanic and Vidal. Experimental vapor-liquid equilibrium data on binary mixtures formed with carbon dioxide and compounds usually found in orange essential oil were used to test the model. These systems were chosen to demonstrate that the maximum likelihood method produces binary interaction parameters for cubic equations of state capable of satisfactorily describing phase equilibrium, even for a binary such as ethanol/CO₂. Results corroborate that the Peng-Robinson, as well as the Soave-Redlich-Kwong, equation can be used to describe phase equilibrium for the following systems: components of essential oil of orange/CO₂.

Keywords: Phase equilibria; supercritical extraction; Peng-Robinson; Soave-Redlich-Kwong.

RESUMO

APLICAÇÃO DO MÉTODO DA MÁXIMA VEROSSIMILHANÇA GENERALIZADO AO EQUILÍBRIO DE FASES A ALTAS PRESSÕES. Foi empregado o método da máxima verossimilhança generalizado para determinação de parâmetros de interação binária entre os componentes do óleo essencial de laranja e dióxido de carbono.

Foram usados dados experimentais de equilíbrio líquido-vapor de misturas binárias de dióxido de carbono e componentes do óleo essencial de laranja. O equilíbrio líquido-vapor foi modelado com as equações de Peng-Robinson e de Soave-Redlich-Kwong usando a metodologia proposta em 1979 por Asselineau, Bogdanic e Vidal. A escolha destes sistemas teve como objetivo demonstrar que o método da máxima verosimilhança produz parâmetros de interação binária, para equações cúbicas de estado capazes de descrever satisfatoriamente até mesmo o equilíbrio para o binário etanol/CO₂. Os resultados comprovam que tanto a equação de Peng-Robinson quanto a de Soave-Redlich-Kwong podem ser empregadas para descrever o equilíbrio de fases para o sistemas: componentes do óleo essencial de laranja/CO₂.

Palavras-chave: Eqilíbrio de fases; extração supercrítica; Pen-Robinsosn; soave-Redlich-Kwong.

1 ¾ INTRODUCTION

In modeling supercritical extraction of essential oils with carbon dioxide, oil solubility in solvent is an important parameter used for the selection of process operational conditions such as temperature, pressure, carbon dioxide flow rate and raw material/solvent relationship. Solubility calculation using cubic equations of state (EOS) is a typical high-pressure phase equilibrium problem.

Satisfactory representation of complex mixtures which form essential oils using EOS demands knowledge of binary interaction parameters. On the other hand, determination of the best binary interaction parameters is based on a nonlinear fitting of measured experimental equilibrium data to models that potentially represent phase equilibria. Several modifications have been proposed for adapting the Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) equations to describe specific systems. In general, these modifications introduce alterations in the mixing rules and, for systems such as alcohols/CO₂, in the attractive parameter dependence of the acentric factor. These modifications are not general and ones that can be appropriate for one system may prove inadequate for others. Essential oils are formed by a mixture of chemically different components, for example, alcohols, aldehydes, terpenes, etc. Therefore, any one of previous the modifications in EOS may be satisfactory for all the binary system mixtures. Thus, it is important to focus on which methodology fits the model parameters because improving by the optimization method, one can obtain parameters that better describe systems of interest to the food, pharmaceutical and fine chemistry industries.

The objective of this work was to study the application of the generalized maximum likelihood method [9] for determining the binary interaction parameters between carbon dioxide and essential oil components. Vapor-liquid equilibrium was modeled with PR and SRK using the Asselineau et al [1] formulation. The Asselineau et al [1] methodology has the advantage of representing all basic phase equilibrium problems simultaneously. Quadratic mixing rules with two interaction parameters were used. The choice generalized maximum likelihood method was chosen due to possibility the of considering experimental uncertainties and to its versatility in fitting parameters while simultaneously using several sets of data. Thus, isothermal, as well as isobar, data can be used for mixtures with any number of components.

2 ¾ METHODOLOGY

In the generalized maximum likelihood method, it is necessary to supply physical model equation parameters that should be optimized. In the present work, the equations are given in Table 1. Fugacity coefficients (j_i) were determined with the PR and SRK equations with the following mixing rules:

(1)

(2)

where e k_ij and k"_ij are the binary interaction parameters.

¹

(a

,b

) pair can be represented by (T, N

^V

), (P, N

^V

) and (T,P) depending on phase equilibria problem.

There are a few problems to be considered when dealing with phase equilibria at high pressures. A crucial one is the initial parameter estimate. The Nelder-Mead and, particularly, the maximum likelihood methods are very sensitive to parameter initial estimates. For a high pressure region, if good initial estimates are not available the optimization process can oscillate and fails to converge. Since Raoult’s law is a good initial estimate at low pressure, a computational strategy can be to start at a low pressure and slowly increase to high pressure regions. This procedure can avoid trivial solutions in the vicinity of the critical point.

To avoid convergence problems, the following procedure was adopted: for each choice of (k_ij and k"_ij), a complete equilibrium curve was calculated for a group of incrementally small, evenly spaced equilibrium points. Computation started at low pressure using Raoult’s law as the initial estimate and continued up to the maximum experimental pressure. Next, calculated points closer to experimental data were used for objective function computation. This procedure considerably increased the computational efforts, as compared to the conventional procedure, but numerical stability was increased since initial estimate dependence for binary interaction parameters diminished. This procedure was repeated until parameters were optimized. The process of optimization used the following objective function:

(3)

where k represents the array of experimental data, j represents the array of experimental points for a given experimental data point, and i represents the array of components array the mixture.

3 ¾ RESULTS

The methodology described was applied to some binary systems formed by orange essential oil components and carbon dioxide. Table 2 shows available experimental data for binary systems of interest. Pure component properties are supplied in Table 3.

TABLE 2.
Systems selected to evaluate k_ij e k"_ij parameters

Systems	Number of points	T (K)	P (bar)	Type of data	Reference
CO2/Ethanol	10	313	60 -81	P-T-x-y	YOON et al [11]
CO2/Ethanol	23	313-333	5 -105	P-T-x-y	SUZUKI et al [10]
CO2/Linalool	13	313-333	40 -110	P-T-x-y	IWAI et al [4]
CO2/a -Pinene	34	313-328	78 -95	P-T-x-y	PAVLÍCEK et al [6]
CO2/a -Pinene	23	313	46 -75	P-T-y	RICHTER; SOVOVÁ [8]
CO2/Limonene	20	308-323	30 -95	P-T-x-y	GIACOMO et al [3]
CO2/Limonene	15	313-333	39 -103	P-T-y	IWAI et al [5]

TABLE 3.
Properties of pure components.

Component	PC (bar)	TC (K)	Teb2 (K)	w
Ethanol	61.48a	513.92a	351.65a	0.6452a
Linalool	25.82b	635.99d	472.15c	0.7617e
a -Pinene	28.90 b	630.87 d	429.35	0.3242e
Limonene	27.56 b	661.11 d	451.15c	0.3170e
CO2	73.82a	304.19a	194.7c	0.2276a

^a

Data from DIPPR [2];

^b

P

_c

calculated using JOBACK [7];

^c

T

_eb

from CRC [12];

^d

T

_c

calculated using JOBACK [7] and T

_eb

from CRC [12];

^e

Lee -Kesler [7] method used to calculate acentric factor.

Calculated binary interaction parameters are shown in Table 4. All available experimental data points were used (Table 2). Table 5 contains global standard deviation of calculated pressure and the vapor phase composition for the experimental data used.

TABLE 4.
Calculated interaction parameters.

PR SRK
Kijx102 Kijx102 Kijx102 Kijx102
CO2/Ethanol	9.048	-1.414	8.543	-1.412
CO2/Linalool	4.281	-3.156	4.363	-3.249
CO2/a-Pinene	9.482	-2.820	10.28	-2.805
CO2/ Linomene	10.15	1.960	9.921	-1.415

TABLE 5.
Global deviation in relation to experimental data used in parameter calculations

PR SRK
System	Number of points	P(bar)x101	y1x103	P (bar)x101	y1x103
CO2(1)/ Ethanol(2)	31	5.0	3.9	5.7	3.6
CO2(1)/ Linalool(2)	13	0.8	2.2	0.9	2.4
CO2(1)/a-Pinene(2)	47	4.0	1.0	3.3	0.5
CO2(1)/ Limonene(2)	26	3.0	0.5	1.1	0.5

FIGURE 1.
Comparison of equilibrium isotherms calculated with PR equation (k_i,j = 9.048x10^-2 and k_i,j = -1.414x10^-2) and experimental data for CO₂/ethanol. SRK results overlap those of PR

In Figure 1 the calculated curves and the corresponding experimental points are presented for two equilibrium isotherms (313.4 and 333.4 K) for carbon dioxide/ethanol. Analyzing the results in Figure 1 and Table 5, it can be observed that both equations satisfactorily described phase equilibrium for CO₂/ethanol. Results for this binary system should be emphasized, in view of the recognized difficulty in representing it by cubic state equations. This fact corroborates the superiority of the maximum likelihood method, when compared to other optimization methods that do not take experimental uncertainties into account.

FIGURE 2.
Comparison of equilibrium isotherms calculated with SRK equation ( k_i,j = 4.363x10^-2 and k_i,j = -3.249x10^-2) and experimental data for CO₂/linalool. PR results overlap those of SRK.

Figure 2 shows experimental points and predicted equilibrium isotherms (313.2, 323.2 and 333.2 K) for CO₂/linalool. In Figure 3 calculated experimental data points are presented for two equilibrium isotherms (313.2 and 323.2 K) for CO₂/limonene. Figure 4 presents results for three equilibrium isotherms (313.15, 323.15 and 323.15 K) for CO₂/a-pinene. Results in Figures 1 to 4 and Table 5 demonstrate that the PR, as well as the SRK, equation satisfactorily describes phase equilibrium for the binary systems studied.

FIGURE 3.
Comparison of equilibrium isotherms calculated with PR equation (k_i,j = 10.15x10^-2 and k_i,j = -1.960x10^-2) and experimental data for CO₂/linomene. SRK results overlap those of PR

FIGURE 4.
Comparison of equilibrium isotherms calculated with SRK equation (k_i,j = 10.28x10^-2 and k_i,j = -2.805x10^-2) and experimental data for CO₂/a-pinene. PR results overlap those of SRK.

4 ¾ CONCLUSIONS

Deviations among experimental data and calculated data with binary interaction parameters determined in this work are smaller than those published in the original references. This fact corroborates the superiority of the maximum likelihood method when compared to other optimization methods that do not take experimental uncertainties into account.

The proposed methodology for calculation of binary interaction parameters usually avoids the convergence difficulties mentioned for problems of phase equilibria at high pressures.

Analyzing the results in Table 5, it is verified that the PR and SRK equations satisfactorily represent the binary systems analyzed. Results for the CO₂/ethanol system should be emphasized, in view of the recognized difficulty in representing it by cubic state equations.

5 ¾ REFERENCES

[1] ASSELINEAU, L.; BOGDANIC, G.; VIDAL, J. A Versatile Algorithm For Calculating Vapor-Liquid Equilibria. Fluid Phase Equilibria. vol. 3, pp. 273-290, 1979.

[2] DANNER, R. P.; DAUBERT, T. E. Data Compilation Tables of Properties of Pure Compounds. DIPPR. The Pennsylvania State University, 1984.

[3] GIACOMO, G.; BRANDANI, V.; RE, G. D.; MUCCIANTE, V. Solubility of Essential Oil Components in Compressed Supercritical Carbon Dioxide. Fluid Phase Equilibria. vol. 52, pp. 405-411, 1989.

[4] IWAI, Y.; HOSOTANI, N.; MOROTOMI, T.; KOGA, Y.; ARAI, Y. High-Pressure Vapor-Liquid Equilibria for Carbon Dioxide + Linalool. J. Chem. Eng. Data. vol. 39, pp. 900-902, 1994.

[5] IWAI, Y.; MOROTOMI, T.; SAKAMOTO, K.; KOGA, Y.; ARAI, Y. High-Pressure Vapor-Liquid Equilibria for Carbon Dioxide + Limonene. J. Chem. Eng. Data. vol. 41, pp. 951-922, 1996.

[6] PAVLÍCEK, J.; RICHTER, M. High Pressure Vapor-Liquid Equilibrium in the Carbon Dioxide - a -Pinene System. Fluid Phase Equilibria. vol. 90, pp. 125-133, 1993.

[7] REID, R.C., PRAUSNITZ, J.M.; POLING, B.E. The Properties of Gases and Liquids, 4th ed. McGraw-Hill Co., New York, USA, 1987.

[8] RICHTER, M.; SOVOVÁ, H. The Solubility of Two Monoterpenes in Supercritical Carbon Dioxide. Fluid Phase Equilibria. vol. 85, pp. 285-300, 1993.

(9) STRAGEVITCH, L.; D'ÁVILA, S. G. A Generalized Maximum Likelihood Method for Estimation of Parameters of Nonlinear Models with Implicit Constraints. Brazilian Journal of Chemical Engineering. vol. 14, no. 1, pp. 41-52, 1997.

[10] SUZUKI, K.; SUE, H.; ITOU, M.; SMITH, R. L.; INOMATA, H.; ARAI, K.; SAITO, S. Isothermal Vapor-Equilibrium Data for Binary Systems at High Pressure: Carbon Dioxide-Methanol, Carbon Dioxide-Ethanol, Carbon Dioxide-1-Propanol, Methanol-Ethanol, Methanol-1-Propanol, Ethane-Ethanol, and Ethane-1-Propanol Systems. J. Chem. Eng. Data. vol. 35, pp. 63-66, 1990.

[11] YOON, J. H.; LEE, H. S.; LEE. H. High-Pressure Vapor-Liquid Equilibria for Carbon Dioxide + Methanol, Carbon Dioxide + Ethanol, and Carbon Dioxide + Methanol + Ethanol. J. Chem. Eng. Data. vol. 38, pp. 53-55, 1993.

[12] WEAST, R. C.; ASTLE, M. J. CRC - Handbook of Data on Organic Compounds, CRC Press, Inc., Florida, 1992, vol. 1 and 2.

6 ¾ NOMENCLATURE

a, b parameters for PR and SRK equations; k_i,j binary interaction parameters for PR and SRK equations; N_C number of components of essential oil; N_L total number of moles in carbon dioxide poor phase; N_V total number of moles in carbon dioxide rich phase; molar fraction of component i in essential oil; P pressure; P_C critical pressure; T temperature; T_eb normal boiling temperature; T_C critical temperature; x_i molar fraction of component i in carbon dioxide poor phase; y_i molar fraction of component i in carbon dioxide rich phase; w acentric factor

6 – Greek letters

a and b specified variables that depend on the equilibrium problem; s experimental uncertainty; fugacity coefficient of component i in carbon dioxide poor phase; fugacity coefficient of component i in carbon dioxide rich phase;

[e-mail]

³

DEQ / UFRS, Rua Luiz Englert s/n, 90040-040 Porto Alegre (Centro), RS, Brazil

⁴

LASEFI-DEA / FEA-UNICAMP, Cx. Postal 6121, 13083-970 Campinas, SP, Brazil,

[e-mail]

^* To whom correspondence should be addressed.

• (9) STRAGEVITCH, L.; D'ÁVILA, S. G. A Generalized Maximum Likelihood Method for Estimation of Parameters of Nonlinear Models with Implicit Constraints. Brazilian Journal of Chemical Engineering. vol. 14, no. 1, pp. 41-52, 1997.

¹ Received for publication in 8/5/97. Accepted for publication in 05/12/97.

²

DEQ / UEM, Av. Colombo, 5790 Bloco D-90 Campus Universitário, 87020-900 Maringá, PR, Brazil,

Publication Dates

• Publication in this collection
  17 Dec 1998
• Date of issue
  Dec 1997

History

• Accepted
  05 Dec 1997
• Received
  08 May 1997