Thermodynamic modeling of solubilities of various solid compounds in supercritical carbon dioxide: Effects of equations of state and mixing rules

No one can ever deny the significance of calculations of solubilities of industrial solid compounds in supercritical CO₂ in separation processes. In this work, the Peng-Robinson (PR) and the Esmaeilzadeh-Roshanfekr (ER) equations of state (EoS) along with several mixing rules including the Wong-Sandler (WS), the covolume dependent (CVD) and the van der Waals one (VDW1) and two (VDW2) fluid mixing rules are applied to evaluate the solubilities of 52 mostly used solid compounds in supercritical carbon dioxide. Besides, the Van-Laar excess Gibbs energy (G^ex) model is applied in phase behavior calculations by the WS mixing rule. The optimal values of the proposed thermodynamic model parameters are evaluated using the DE (differential evolution) optimization strategy. The absolute average deviations of the model results from 1776 experimental data points and the optimal values of the adjustable parameters of the model are reported to investigate the capabilities of combinations of each equation of state with different mixing rules in calculations of the solubilities. The results indicate that the combination of the ER EoS with the WS mixing rule leads to more accurate results (AAD = 9.0%) compared with other ones.     

A novel equation of state (EOS) for prediction of solute solubility in supercritical carbon dioxide: Experimental determination and correlation

Solubility data of organophosphorous metal extractants in supercritical fluids (SCF) are crucial for designing metal extraction processes. We have developed a new equation of state (EOS) based on virial equation including an untypical parameter as BP/RT, reduced temperature and pressure for prediction of solute solubility in supercritical carbon dioxide (SC CO₂). Solubility experimental data (solubility of tributylphosphate in SC CO₂) were correlated with the two cubic equations of state (EOS) models, namely the Peng–Robinson EOS (PR‐EOS) and the Soave–Redlich–Kwong EOS (SRK‐EOS), together with two adjustable parameter van der Waals mixing and combining rules and our proposed EOS. The AARD of our EOS is significantly lower than that obtained from the other EOS models. The proposed EOS presented more accurate correlation for solubility data in SC CO₂. It can be employed to speed up the process of SCF applications in industry.     

Modeling of solid–liquid equilibria for polyethylene and polypropylene solutions with equations of state

We modeled solid–liquid equilibria (SLEs) in polyethylene and polypropylene solutions with a Soave–Redlich–Kwong (SRK) cubic equation of state (EOS) and a perturbed‐chain statistical associating fluid theory (PC‐SAFT) EOS. Two types of mixing rules were used with SRK EOS: The Wong–Sandler mixing rule and the linear combination of the Vidal and Michelsen mixing rules (LCVM), both of which incorporated the Bogdanic and Vidal activity coefficient model. The performance of these models was evaluated with atmospheric‐pressure and high‐pressure experimental SLE data obtained from literature. The basic SLE equation was solved for the equilibrium melting temperature instead of for the composition. The binary interaction parameters of SRK and PC‐SAFT EOS were estimated to best describe the experimental equilibrium behavior of 20 different polymer–solvent systems at atmospheric pressure and 31 other polymer–solvent systems at high pressure. A comparison with experimental data showed that SRK–LCVM agreed very well with the atmospheric SLE data and that PC‐SAFT EOS was more efficient in high‐pressure conditions. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Estimation of solubility of solids in supercritical carbon dioxide

Peng-Robinson equation of state(PR EOS)was chosen for modeling the thermodynamic be-havior of supercritical(SC)-CO_2/Solid systems.The necessary critical constants and acentric factorof the solute were obtained by the Sigmund and Trebble(1992)method based on the molecular weightand boiling temperature,and the vapor pressure of the solute was calculated by its meltingtemperature and heat of fusion.This approach compared very favorably with the conventional corres-ponding state theory,but without using critical constants and vapor pressure of solutes.Four mixingrules were tested for the calculation of solid solubility in SC-CO_2.van der Waals(vdW)mixing rulewith one parameter was considered to be most suitable for the estimation of solubility.A simplecorrelation was developed for the SC-CO_2/solid binary interaction coefficient k_(ij) with the meltingtemperature of pure solutes.The solubilities of solids in SC-CO_2 were estimated for eleven binarysystems at various temperatures,the total absolute average     

High pressure isothermal vapor-liquid equilibria for the binary system of carbon dioxide (CO<Subscript>2</Subscript>)+1,1,1-trifluoroethane (R-143a)

Isothermal vapor-liquid equilibrium data for the binary mixture of carbon dioxide (CO₂)+1,1,1-trifluoroethane (HFC-143a) were measured within the temperature range of 273.15–333.15 K. The data in the two-phase region were measured by using a circulation-type equilibrium apparatus in which both vapor and liquid phases are continuously recirculated. The experimental data were correlated with the Peng-Robinson equation of state (PR-EOS) using the Wong-Sandler mixing rules combined with the NRTL excess Gibbs free energy model. The values calculated by the PR EOS with the W-S mixing rules show good agreement with our experimental data.     

PREDICTION OF MOLAR VOLUMES, VAPOR PRESSURES AND SUPERCRITICAL SOLUBILITIES OF ALKANES BY EQUATIONS OF STATE

A generalized cubic equation of state which can represent all the cubic equations is introduced and thermodynamic property relations for it are presented. Five cubic equations of states with respective mixing rules are used to predict molar volumes and vapor pressures of pure alkanes (from methane till n-tritriacontane) and solubilities of solid wax components (high molecular weight alkanes) in supercritical solvents. They are the RK (Redlich-Kwong), MMM (Mohsennia-Modarress-Mansoori), RM (Riazi-Mansoori), PR (Peng-Robinson), and SRK (Soave-Redlich-Kwon) equations of state. The experimental data necessary to compare the equations of state are taken from the literature. It is demonstrated that the SRK equation of state is more accurate for predicting vapor pressures of alkanes. The RM equation of state is shown to be more accurate for predicting molar volumes of saturated and sub-cooled liquid alkanes as well as molar volumes of alkanes in their supercritical condition. For the solubility of wax components in supercritical solvents it is shown that the MMM equation of state gives the least AAD% for the 270 data points of 10 binary systems studied consisting of a high molecular weight alkane and supercritical ethane and carbon dioxide.     

超临界二氧化碳与二苯醚相平衡研究

为了解决煤焦油及其轻质化产物中二苯醚的萃取精馏分离问题以及为后续的高效分离过程提供工程数据,本文自行设计并搭建了一套流动法可视化高温高压相平衡测定装置,对二苯醚-二氧化碳体系的相平衡数据进行了测定,测定温度为313.15K、333.15K和353.15K,测定压力为8～18MPa,并使用Peng-Robinson (PR)和Soave-Redlich-Kwong（SRK）状态方程结合Quadratic、Adachi-Sugie、Mathias-Klotz-Prausnitz混合规则对的超临界二氧化碳与二苯醚相平衡数据进行了关联计算,关联结果表明PR和SRK状态方程结合AS混合规则可以获得较为准确的关联结果,而传统二次型混合规则得到的计算结果误差相对较大。此外,关联计算还表明,选择合适的混合规则后,不同状态方程的关联结果相差不大。这表明,对于低挥发性的液体与二氧化碳这类非对称混合物体系,相平衡模拟计算的关键在于选取合适的混合规则。     

Unified equation of state based on the lattice fluid theory for phase equilibria of complex mixtures part I. Molecular thermodynamic framework

Consistent calculation of fugacities of fluid mixtures remains as one of the most important subjects in contemporary molecular thermodynamics. In practice, equations of state (EOSs) and g^E-models have been used. However, most EOSs are erroneous for condensed phases at high densities and g^E-models are inapplicable for pressuresensitive systems. Recently to remedy the shortcomings in both approaches, there has been a surge of new g^E-EOS mixing rules. By equating any set of EOS and g^E-models, the limitations in both approaches could be resolved significantly. However, the self-consistency in the underlying concept of those mixing rules remains controversial. During the last several years, the present authors proposed a new lattice-fluid EOS and its simplification relevant to phase equilibrium calculations. Without employing any g^E-EOS mixing rule and with only two parameters for a pure component and one adjustable interaction energy parameter for a binary mixture, results obtained to date demonstrated that the EOSs are quantitatively applicable to a great variety of phase equilibrium properties of mixtures, especially, for complex and/or macromolecular systems. In the present article we summarize the EOSs and extended the applications to liquid-liquid Equilibria. In part I, we discussed briefly the molecular thermodynamic aspects of general derivation of the EOS and a brief discussion of applying the EOSs to pure fluids while the illustrative application to various real mixture systems is discussed in part II.     

Gas solubilities in ionic liquids using a generic van der Waals equation of state

A family of modified van der Waals equations of state (vdW EOS) is extremely useful for many industrial applications. For example, the generic Redlich-Kwong (RK) EOS or its modification by Soave (SRK EOS) and Peng-Robinson (PR EOS) are still of popular use in industry to the present day. These two most popular (“cubic”) EOSs are based on modifications [1/(V² + bV), or 1/(V² + 2bV − b²)] of the volume dependence on the attractive part of the original van der Waals EOS [1/V²] and also modifications of the temperature dependence of the attractive “a(T)” parameter of the original EOS (constant a). It is extremely rare in actual EOS applications to use the volume dependence of the original van der Waals EOS. In the present phase equilibrium calculations, we employ such a generic vdW EOS, P = RT/(V − b) − a(T)/V², with our well-tested mixing rule for multi-component mixtures. Using the same form of the “a(T)” parameter and the mixing rule, it has been found that all generic RK, PR, and vdW EOSs can present the phase behaviors (temperature-pressure-composition diagrams) equally well. It is shown that experimental gas solubility data (CO₂, CF₃-CFH₂, SO₂, and NH₃) in room-temperature ionic liquids are well correlated with the present EOS model, and also that the phase behaviors such as LLE (liquid-liquid separations) are satisfactorily predicted.     

Extending the range of COSMO‐SAC to high temperatures and high pressures

The range of the predictive Gibbs energy of solvation model, COSMO‐SAC, is extended to large ranges of density, pressure, and temperature for very nonideal mixtures by combining it with an equation of state (EOS) using the Wong‐Sandler mixing rule. The accuracy of isothermal vapor‐liquid equilibria (VLE) calculations based on using the predictive COSMO‐SAC model and separately the correlative NRTL model is compared, each combined with three different forms of the Peng‐Robinson equation of state. All the models considered require the value of the EOS mixing rule binary parameter k_ij. The NRTL model also requires three other parameters obtained from correlation low pressure VLE data. The PRSV + COSMO‐SAC model is showed, with its one adjustable parameter obtained from low temperature data leads good predictions at much higher temperatures and pressures. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1806–1813, 2018     

Experimental determination and activity coefficient based models for mixture solubilities of nitrophenol isomers in supercritical carbon dioxide

The experimental solubilities of the mixture of nitrophenol (m- and p-) isomers were determined at 308, 318 and 328 K over a pressure range of 10–17.55 MPa. Compared to the binary solubilities, the ternary solubilities of m-nitrophenol increased at 308, 318 and 328 K. The ternary solubilities of p-nitrophenol increased at 308 K, while the ternary solubilities decreased at lower pressures and increased at higher pressure at 318 and 328 K. The solubilities of the solid mixtures in supercritical carbon dioxide (SCCO₂) were correlated with solution models by incorporating the non-idealities using activity coefficient based models. The Wilson and NRTL activity coefficient models were applied to determine the nature of the interactions between the molecules. The equation developed by using the NRTL model has three parameters and correlates mixture solubilities of solid solutes in terms of temperature and cosolute composition. The equation derived from the Wilson model contains five parameters and correlates solubilities in terms of temperature, density and cosolute composition. These two new equations developed in this work were used to correlate the solubilities of 25 binary solid mixtures including the current data. The average AARDs of the model equations derived using the NRTL and Wilson models for the solid mixtures were found to be 7% and 4%, respectively.     

CO2-C2H6共沸物分离的立方型状态方程选取

汽液平衡热力学模型的准确选取对CO₂-C₂H₆共沸物分离流程的设计和操作分析至关重要。在汽液平衡实验数据的基础上,依据逸度平衡原则,评估vdW、RK、SRK和PR立方型状态方程结合vdW、Margles和CVD混合规则预测CO₂纯物质、CO₂-C₂H₆共沸物和n-C₅H₁₂-CO₂-C₂H₆三元体系汽液平衡的可靠性,采用平均绝对误差的方法进行状态方程的选取。结果表明：SRK状态方程计算CO₂纯物质汽液平衡性质的精度最高;PR状态方程结合Margles混合规则可以准确计算CO₂-C₂H₆共沸体系汽液平衡特性;对于n-C₅H₁₂-CO₂-C₂H₆三元体系,SRK状态方程结合Margles混合规则计算精度明显优于vdW、RK和PR状态方程。通过试差迭代法优化CO₂-C₂H₆共沸体系和n-C₅H₁₂-CO₂-C₂H₆三元体系的二元交互作用参数,状态方程的计算精度得到明显提高。     

Comparison between the artificial neural network,SAFT and PRSV approach in obtaining the solubility of solid aromatic compounds in supercritical carbon dioxide

The aim of this study is to model the solubilities of solid aromatic compounds in supercritical carbon dioxide (SCCO₂) using feed-forward artificial neural network (ANN). Temperature, pressure, critical properties and acentric factor of each solute have been used as independent variables of ANN model. The parameters of multi-layer perceptron (MLP) network have been adjusted by back propagation learning algorithm using experimental data which have been collected from various literatures. In order to find the optimal topology of the MLP, different networks were trained and examined and the network with minimum absolute average relative deviation percent (AARD%), mean square error (MSE) and suitable regression coefficient (R²) has been selected as an optimal configuration. By this procedure a single hidden layer network composed of nineteen hidden neurons has been found as an optimal topology. Sensitivity error analyses confirmed that the optimal ANN can predict experimental data with an excellent agreement (AARD% = 4.99, MSE = 7.08 × 10⁻⁷ and R² = 0.99699). Capability of the proposed ANN model has compared with those published results which have obtained by SAFT combined with eight different mixing rules (one, two and three parameters mixing rules) and PRSV equation of state (EOS). The best presented overall AARD% for SAFT approach with one, two and three parameters mixing rules are 16.15, 12.32% and 7.65%, respectively while PRSV EOS showed AARD% of 21.10%. The results emphasize that the proposed ANN model can predict the solubilities of solid aromatic compounds in SCCO₂ more accurate than SAFT and PRSV EOS.     

一种新的WS型混合法则

立方形状态方程被广泛应用于相平衡计算中．绝大多数状态方程是结合简单的van der Waals混合法则来进行VLE数据关联的．然而对于应用这些方程关联高度非理想混合物的复杂的相行为,仅用van der waals单维流体混合法则是不够的．甚至在van der Waals混合法则引入另外一个参数都不能够精确地关联含有极性物质的混合物．     

Phase behavior of mixture of supercritical CO2 + ionic liquid: Thermodynamic consistency test of experimental data

Various models have been applied composed of the Peng‐Robinson equation of state (PR‐EoS) and the Soave‐Redlich‐Kwong equation of state (SRK‐EoS) associated with three mixing rules including the following: Wong‐Sandler (WS), van der Waals one (vdW1), and van der Waals two (vdW2) for phase behavior modeling of mixtures of supercritical CO₂ + different ionic liquids in vapor–liquid equilibrium (VLE) region. It has been found that the PR EoS implying the WS mixing rule can be used as a reliable thermodynamic model to perform a thermodynamic consistency test on the experimental data of phase behaviors of the supercritical CO₂ + ionic liquid systems (19 commonly‐used ionic liquids have been studied). The results show that 40% of the experimental data seem to be thermodynamically consistent, 55.5% seem to be thermodynamically inconsistent, and 4.5% seem to be not fully consistent. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3892–3913, 2013     

汽液相平衡计算混合法则中协体积项b的改进

混合物汽液相平衡通常需要引入合适的混合法则才能更好地进行关联。针对目前混合法则中协体积项b的改进提出了一种新的修正协体积项b的方法。该方法以Mie势能理论、London色散力理论为基础,同时引入Leach等分子形状系数进行修正,这样协体积项b中的交互作用参数l_ij形式上是纯组分分子形状系数θ和φ及临界参数的函数,无需实验数据拟合得到,从而真实反映了二元混合物的实际混合情况。将修正的协体积项b运用于MHV1、LCVM及HV混合法则并结合PR NRTL模型对不同种类的16个体系汽液相平衡进行了计算,并与采用b ∑x_ib_i的MHV1、LCVM及HV混合法则结合PR NRTL模型计算的结果进行比较,结果表明运用修正的协体积项b的混合法则+PR+NRTL模型所得的计算结果精度优于采用b ∑x_ib_i的MHV1、LCVM及HV混合法则 PR NRTL模型的计算结果。     

A modification of Wong-Sandler mixing rule for the prediction of vapor-liquid equilibria in binary asymmetric systems

Systems consisting of light components and heavy hydrocarbons are highly asymmetric and industrially important. Design and control of facilities for separation and purification of such mixtures require vapor-liquid equilibrium data. Coupling of the cubic equation of state (EOS) with excess Gibbs energy models (EOS/G ^ex models) failed to represent the vapor-liquid equilibria (VLE) of such systems accurately. The main purpose of this work is to present a modification of Wong-Sandler mixing rule with using the composition dependent binary interaction parameter. Vaporliquid equilibria for 30 binary systems are calculated using the SRK equation of state with proposed model and Wong-Sandler mixing rule. Calculated pressures and mole fractions of vapor phase are compared with experimental data. The average absolute percentage deviation indicates that error involved in the application of modified Wong-Sandler model is less than Wong-Sandler model in most cases.     

A practical equation of state for non‐spherical and asymmetric systems for application at high pressures. Part 2: Extension to mixtures

In part I of this series the pure component PHCT‐DNSK equation of state (EOS) was presented. In this paper the EOS is extended to describe mixtures, particularly asymmetric mixtures containing one or more low molecular weight spherical compound together with one or more high molecular weight chain‐like compound. The EOS utilises theoretically correct mixing rules and is generally able to predict the correct trends quantitatively for binary mixtures, and in most cases outperform other EOSs. With the use of a small, temperature independent, interaction parameter the EOS is able to predict the phase behaviour of the investigated systems qualitatively. The EOS is able to predict the phase behaviour of a multi‐component system containing one or more light components and a range of heavy hydrocarbons with improved accuracy compared to other EOSs at reduced computational times. © 2011 Canadian Society for Chemical Engineering     

A comprehensive comparison among four different approaches for predicting the solubility of pharmaceutical solid compounds in supercritical carbon dioxide

Supercritical technologies have been developed in the food, environmental, biochemical and pharmaceutical product processing during the recent decades. Obtaining accurate experimental solubilities of pharmaceutical compounds in supercritical carbon dioxide (SC-CO₂) and their correlations are highly important and essential for the design of industrial operating units. In this study, the solubilities of six pharmaceutical compounds (Anti-HIV, Antiinflammatory and Anti-cancer) in SC-CO₂ were correlated using four different models: cubic equation of state (EoS) model (SRK and modified-Pazuki EoSs), empirical and semi-empirical models (Chrastil, Mendez-Santiago-Teja, Spark et al. and Bian et al. models), regular solution model coupled with the Flory-Huggins equation, and an artificial neural network-based (ANN-based) model. In EoS calculations, twin-parametric van der Waals (vdW2) and Panagiotopoulos-Reid (mrPR) mixing rules were used for estimating the supercritical solution properties, with three different sets employed for obtaining critical and physicochemical properties of the solid compounds. To evaluate the capabilities of various approaches, a comprehensive comparison was carried out among the four models based on several statistical criteria, including AARD, R_adj and F-value. Results of the analysis of variance (ANOVA) indicated that the ANN-based model provided the best results in terms of correlating the experimental solubility of the pharmaceutical compounds in SC-CO₂.     

紫杉醇在二氯甲烷-超临界二氧化碳混合溶剂中的溶解度

Phase behavior of paclitaxel in solvent mixtures of dichloromethane and supercritical carbon dioxide was investigated using a supercritical phase monitor. Cloud point pressures were determined as a function of temperature, pressure and paclitaxel content from 313.1 to 343.1 K and pressures up to 33.52 MPa. The ternary mixtures exhibit a typical lower critical solution temperature behavior. When paclitaxel content increases, the single-phase region shrinks in size. Three cubic equations of state (Redlich-Kworng, Soave-Redlich-Kwong and Peng-Robinson equation of state) coupled with the van der Waals one-fluid mixing rules were selected to correlate the experimental data. The results indicate that SRK EOS coupled with two binary interaction parameters kij and lij can pre-dict paclitaxel solubility for the best fit of experimental data.