Modified Redlich-Kwong and Peng-Robinson Equations

Supercritical fluids can be used as the promising solvents in many applications such as separation, purification and particle sizing of pharmaceuticals, cosmetics, food supplements, natural products, etc. In the supercritical processes, the solubility of the solutes in supercritical fluid is a key parameter for designing optimized operating conditions. The experimental solubilities of the solid compounds and their mixtures in supercritical fluids are limited due to the difficulties of experimental measurements and also time-consuming and costly nature of these measurements. Therefore, it is desirable to develop the predictive and reliable methods for estimating the solubility of solid compounds in supercritical fluids. One way to achieve this aim is using Equation of the State (EoSs). The cubic EoSs are flexible and reliable according to their accuracy. Although considerable progresses in the development of equations of state were reported in the literature, the application of the EoSs is still limited because of their complexity. Additionally, the semiempirical models do not have theoretical basis, but they are widely used in industrial and engineering applications due to their simplicity1–4.. In recent years, some of researchers have worked on the prediction of solid solubility in supercritical fluids by using cubic Equations of the State (EOSs). Khamda et al.1 investigated the cefixime trihydrate and oxymetholone solubilities in supercritical carbon dioxide. They also used semi-empirical correlation and the Peng-Robinson Equation of State (PR EOS) for modeling of these solubilities. Park et al.2 investigated the equilibrium solubilities of two biocides, climbazole, and triclocarban in supercritical carbon dioxide. Subsequently, they applied PR EOS and quasi-chemical nonrandom lattice fluid model for these systems. Chen et al.5 reported the experimental solubilities of cinnamic acid, phenoxyacetic acid and 4-methoxyphenylacetic acid in supercritical carbon dioxide. In order to model these solubilities, they also used Soave-Redlich-Kwong (SRK) and PengRobinson (PR) equations of state. De Zordi et al.6 studied the solubility behavior of pharmaceutical compounds Abstract


Introduction
Supercritical fluids can be used as the promising solvents in many applications such as separation, purification and particle sizing of pharmaceuticals, cosmetics, food supplements, natural products, etc. In the supercritical processes, the solubility of the solutes in supercritical fluid is a key parameter for designing optimized operating conditions. The experimental solubilities of the solid compounds and their mixtures in supercritical fluids are limited due to the difficulties of experimental measurements and also time-consuming and costly nature of these measurements. Therefore, it is desirable to develop the predictive and reliable methods for estimating the solubility of solid compounds in supercritical fluids. One way to achieve this aim is using Equation of the State (EoSs). The cubic EoSs are flexible and reliable according to their accuracy. Although considerable progresses in the development of equations of state were reported in the literature, the application of the EoSs is still limited because of their complexity. Additionally, the semi-empirical models do not have theoretical basis, but they are widely used in industrial and engineering applications due to their simplicity [1][2][3][4] ..
In recent years, some of researchers have worked on the prediction of solid solubility in supercritical fluids by using cubic Equations of the State (EOSs). Khamda et al. 1 investigated the cefixime trihydrate and oxymetholone solubilities in supercritical carbon dioxide. They also used semi-empirical correlation and the Peng-Robinson Equation of State (PR EOS) for modeling of these solubilities. Park et al. 2 investigated the equilibrium solubilities of two biocides, climbazole, and triclocarban in supercritical carbon dioxide. Subsequently, they applied PR EOS and quasi-chemical nonrandom lattice fluid model for these systems. Chen et al. 5 reported the experimental solubilities of cinnamic acid, phenoxyacetic acid and 4-methoxyphenylacetic acid in supercritical carbon dioxide. In order to model these solubilities, they also used Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) equations of state. De Zordi et al. 6 studied the solubility behavior of pharmaceutical compounds Modified Redlich-Kwong and Peng-Robinson Equations of State for Solubility Calculation of Solid Compounds in Supercritical Carbon dioxide containing antioxidants, antibiotics, steroids and antiinflammatory in supercritical fluids. They used a model based on activity coefficients and they determined the parameters of the model as a function of the pharmaceutical compound properties. Housaindokht et al. 3 applied various modified Peng-Robinson equations of state to model the solubility of solid compounds in supercritical carbon dioxide. They also determined interaction parameters for these systems. Cheng et al. 7 investigated the solubility of ergosterol in supercritical carbon dioxide. They used the Peng-Robinson Equation of State (EOS) in combination with the one-parameter and two-parameter van der Waals mixing rules to fit the experimental solubility data. Spiliotis et al. 8 studied the prediction of the liquid and solid aromatic hydrocarbons solubility in supercritical CO 2 with the Linear Combination of the Vidal and Michelsen (LCVM) and Modified Huron-Vidal Two (MHV2) models. Yazdizadeh et al. [4] applied the Peng-Robinson (PR) and the Esmaeilzadeh-Roshanfekr (ER) (EoSs) in combination with Wong-Sandler (WS), the Covolume Dependent (CVD) and the van der Waals one (vdW1) and two (vdW2) fluid mixing rules and the Van-Laar excess Gibbs energy (G ex ) model to model the solubilities of solid compounds in supercritical carbon dioxide.
In this work, the modified RK (bRK) and modified PR (bPR) equations of state in combination with the van der Waals zero (vdW0) mixing rule were proposed for calculation of solid solubilities in supercritical carbon dioxide. The Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) (EoSs) in combination with the van der Waals one (vdW1) and Wong-Sandler (WS) mixing rules were also applied for modeling the solubilities of solid compounds in supercritical carbon dioxide. To identify the advantages of the new proposed models in predicting solubilities of solid compounds in supercritical carbon dioxide, the results of proposed models were compared with the results of the conventional SRK and PR Equations of State (EoSs).

Calculation of Solubility
To determine the solid solubility in supercritical fluid, the thermodynamic equilibrium is used as follows:

PureSolid
Supercritical where PureSolid i f is the fugacity of each pure solute and Supercritical i f is the fugacity of the solute in supercritical fluid. In this study, the following assumptions were considered to obtain the required expression for performing the phase equilibrium calculations: 1. The solubility of supercritical fluid in the solid phase is neglected. 2. The pure solid fugacity is considered to be equal to the fugacity of the solute i in the mixture. 3. The molar volume of the solid phase is constant. 4. The solid phase fugacity coefficient in saturation condition is considered to be unity. Considering these assumptions, Equation (1) can be expressed as follows: where P and T are pressure and temperature, v denotes the molar volume, R is the universal gas constant. Sat stands for saturation. y and j are mole fraction and fugacity coefficient of the solid solute in supercritical phase, respectively. The saturation vapor pressures at different temperatures were given in Table 1.

The Modified Redlich-Kwong Equation of
State (βRK) The RK EOS 14 can be written as follows: The energy parameter (a) and volume parameter (b) are obtained from the critical properties. The critical properties of pure fluids were listed in Table 2.
In this investigation, a new functionality for energy parameter of Redlich-Kwong EoS as a function of temperature and pressure similar to the work of Heidaryan and Jarrahian 15 and similar to our previous work 20 was proposed to evaluate the solubilities of ten solid compounds in supercritical CO 2 .
( ) In which β is a temperature dependant parameter that can be expressed in terms of reduced temperature as follows: In which i refer to CO 2 or solute. Thus, β 11 -β 13 and β 21β 23 are related parameters to solute and CO 2 , respectively. It worth noting that the b function (including three parameters for each compound) used in this work is different with the β function (including six parameters for each compound) used in the work of Heidaryan and Jarrahian 15 . Therefore, not only our β function is a new function but also our application is different and the proposed models were used for solubility calculation.

The Modified Peng-Robinson Equation of
State (βPR) The PR EOS 14 is expressed as: where a shows the energy parameter and b denotes the volume parameter. The PR EOS parameters are defined in terms of critical properties as follows: The other parameters of PR EOS are expressed as follows:  (12) where ω shows the acentric factor.The subscripts c and r are related to the critical and reduced properties, respectively.
In this work, the modified version of PR EoS is also suggested for determining the phase equilibrium.
Similar to Equation (7), a temperature dependant expression in terms of reduced temperature is considered for b function.
In this work, the van der Waals and Wong Sandler mixing rules were used for the solid-supercritical equilibrium calculations. The van der Waals mixing rule is expressed as follows (1 ) where y i denotes the mole fraction of component i in supercritical phase. If the parameter k ij was taken as zero, the mixing rule was denoted as the vdW0 instead of vdW1 mixing rules. The Wong Sandler mixing rule is written as follows, (20) in which, In this study, the van-Laar activity model 4 was applied for calculating the excess Gibbs energy.

Results and Discussion
In this work, the solubilities of ten solid compounds including Methimazole, Ascorbic acid, Ascorbyl palmitate, Propyl gallate, Aspirin, Fluoranthene, Triclocarban, Hinokitiol, Phenol and Climbazole in supercritical CO 2 were modeled. The experimental solubilities of solids were obtained from the literature 2,[9][10][11][12][13]19 . In order to model the solubilities of these solids in supercritical CO 2 , the Peng Robinson and the SRK equations of state (EOSs) combined with the van der Waals (vdW1) and Wong Sandler (WS) mixing rules were used. To obtain the binary interaction parameters and the parameters of the model for van der Waals (vdW1) and Wong Sandler (WS) mixing rules, the parameters were obtained via regression with the experimental data through the minimization of an objective function. The average absolute relative deviation percent (AARD%), defined by the following expression:  (22) in which N represents the number of experimental points, y i,exp is the experimental solubility data and y i,exp represents the calculated solubility. The average absolute relative deviations percent (AARD%), optimized model and binary interaction parameters were represented in Table 3. Figs. 1-2 compare the calculated solubility results by the sets of PR-vdW1, SRK-vdW1, PR-WS and SRK-WS with the experimental data for Phenol and Triclocarban compounds, respectively. One can see that the performance of WS mixing rule is much better than vdW1 mixing rule. Therefore, the combination of the SRK and the PR EOSs with the WS mixing rule is more suitable for modeling the solubilities of these ten solids in supercritical CO 2 .     Subsequently, to investigate the performance of the proposed EOSs (bRK and bPR EOSs) in combination with the simple mixing rule of vdW0, these proposed equations were applied to model the solubilities of these ten solids in supercritical CO 2 . The results of Average Absolute Relative Deviations Percent (AARD%) and the model parameters were reported in Table 4. Figures  3-4 show the calculated solubility results by the sets of PR-vdW1, SRK-vdW1, βPR-vdW0 and βRK-vdW0 for Phenol and Triclocarban compounds, respectively. As it is shown in Table 4, the accuracy of the proposed models is much better than the combination of SRK and PR EOSs with vdW1 mixing rule, even better than combination of SRK and PR EOSs with WS mixing rule. The calculation results of the models demonstrated that the bRK and bPR EoSs are capable of modeling the solubilities of these ten solid in supercritical CO 2 without using the complicated mixing rule. Therefore, bRK and bPR EOSs in combination with the simple mixing rule (vdW0) are reliable methods for determining the phase equilibrium of (solid + supercritical CO 2 ) systems.
In order to investigate the validity of the proposed EOSs, these proposed models were compared with the models reported in literature. First, the results of the proposed models for seven compounds including Methimazole, Ascorbic acid, Ascorbyl palmitate, Propyl gallate, Aspirin, Fluoranthene and Phenol were compared with the results of Esmaeilzadeh-Roshanfekr (ER) equation of state in combination with vdW1, vdW2, CVD and WS mixing rules [4] . The results of AARD% are presented in Table 5. It can be concluded that the proposed models performed better than the results of Esmaeilzadeh-Roshanfekr (ER) equation of state in combination with vdW1, vdW2, CVD and WS mixing rules. The results of the proposed models (βPR-vdW0 and βRK-vdW0) for three compounds including Propyl gallate, Methimazole and Aspirin were also compared with the results of regular solution model (One-parameter and Two-parameter) and Two commonly used semiempirical equations (Chrastil and Mendes-Santiago and Teja equations) in Table 6. In comparison with the results of these models 16 , the present models (βPR-vdW0 and βRK-vdW0) perform better than regular solution and semi-empirical models. It is found that the present model is reliable for solubility calculations of these ten solids in supercritical carbon dioxide.

Conclusions
In this investigation, the modified RK (bRK) and the modified PR (bPR) equations of state in combination with the vdW0 mixing rule were used to determine the solubilities of ten solid compounds in supercritical CO 2 .
The optimized parameters of the proposed models were determined and reported. Subsequently, the results of these models were compared with the SRK and PR EOSs in combination with VdW1 and WS mixing rules and the other applied models in the literature. It is demonstrated that the relative error (AARD%) between experimental  data and the calculated results by the proposed model is less than 5.1% indicating that the proposed models in this work has higher precision than the models in the literature.