Connection between zero-pressure mixing rules and infinite-pressure mixing rules

Infinite-pressure mixing rules and zero-pressure mixing rules are related by a very simple equation. The zero-pressure models and the infinite-pressure models are interchangeable through this connection. This paper will show how to convert the Huron–Vidal infinite-pressure mixing rule to a zero-pressure mixing rule, or vice versa. The MHV1 zero-pressure mixing rule will be reformulated to satisfy second virial coefficient constraint. The Twu–Coon and Wong–Sandler infinite-pressure mixing rules will be modified to improve their accuracy for reproducing the incorporated G^E model. A simplification of the Twu–Coon–Bluck [TCB(r)] zero-pressure mixing rule will be presented. The MHV1 mixing rule will be shown to be a special case of TCB(r).     

Applicability of cubic equation of state mixing rules on correlation of excess molar volume of non-electrolyte binary mixtures – Part II

The excess molar volume (V?^E) data of the 24 binary highly non-ideal mixtures containing dicyclic ethers (593 data points) were correlated by the Peng–Robinson–Stryjek–Vera (PRSV) cubic equation of state (CEOS) coupled with two different classes of mixing rules: (i) the composition dependent van der Waals (vdW) mixing rule and (ii) the excess free energy mixing rules (CEOS/G?^E) based on the approach of the Gupta–Rasmunssen–Fredenslund (GRF), as well as the Twu–Coon–Bluck–Tilton (TCBT) mixing rule; both rules with the NRTL equation as the G?^E model. The results obtained by these models show that the type of applied mixing rules, including the number and position of interaction parameters are of great importance for a satisfactory correlation of V?^E data. The GRF mixing rules gave mostly satisfactory results for V?^E correlation of the non-ideal binary systems available at one isotherm of 298.15?K, while for the correlation in temperature range from 288.15 to 308.15?K the TCBT model can be recommended.     

A local composition extension of the van der Waals mixing rule for PR cubic equation of state

New mixing rules (VWLC-I and II) capable of connecting van der Waals (VDW) to CEOS/A^E mixing rule models were developed. These models are able to incorporate the same multi-component mixture parameters obtained for the van der Waals and CEOS/A^E models simultaneously. The VWLC mixing rules directly incorporate local compositions into the cubic equations of state (CEOS). The energy parameters required for the local compositions are calculated from the CEOS parameters. The Peng–Robinson (PR) CEOS was used for this study. Binary interactions parameters were obtained by adjusting the vapor pressure of the binary mixture for several low and high-pressure systems. The predictive capabilities of the VWLC mixing rules were tested by vapor–liquid equilibria calculations for low and high-pressure multicomponent systems. The results were compared with the predictions of the VDW mixing rule and a Huron–Vidal (HV) kind of CEOS/A^E-NRTL mixing rule. The VWLC mixing rules are consistent models giving good results in a broad range of pressures and temperatures in binary and multicomponent mixtures. They compare favorably with the CEOS/A^E-NRTL mixing rule for low-pressure systems. In high-pressure ternary systems VWLC-I and II give good predictions, much better, in fact, than the CEOS/A^E-NRTL mixing rule.     

Thermodynamic modeling of saturated liquid compositions and densities for asymmetric binary systems composed of carbon dioxide,alkanes and alkanols

The present study mainly focuses on the phase behavior modeling of asymmetric binary mixtures. Capability of different mixing rules and volume shift in the prediction of solubility and saturated liquid density has been investigated. Different binary systems of (alkane + alkanol), (alkane + alkane), (carbon dioxide + alkanol), and (carbon dioxide + alkane) are considered. The composition and the density of saturated liquid phase at equilibrium condition are the properties of interest. Considering composition and saturated liquid density of different binary systems, three main objectives are investigated. First, three different mixing rules (one-parameter, two parameters and Wong–Sandler) coupled with Peng–Robinson equation of state were used to predict the equilibrium properties. The Wong–Sandler mixing rule was utilized with the non-random two-liquid (NRTL) model. Binary interaction coefficients and NRTL model parameters were optimized using the Levenberg–Marquardt algorithm. Second, to improve the density prediction, the volume translation technique was applied. Finally, Two different approaches were considered to tune the equation of state; regression of experimental equilibrium compositions and densities separately and spontaneously. The modeling results show that there is no superior mixing rule which can predict the equilibrium properties for different systems. Two-parameter and Wong–Sandler mixing rule show promoting results compared to one-parameter mixing rule. Wong–Sandler mixing rule in spite of its improvement in the prediction of saturated liquid compositions is unable to predict the liquid densities with sufficient accuracy.     

Applicability of cubic equation of state mixing rules on correlation of excess molar volume of Non-Electrolyte binary mixtures

The excess molar volume V ^E data of the binary liquid systems were correlated by the Peng–Robinson–Stryjek–Vera equation of state coupled with two different types of mixing rules: composition dependent van der Waals mixing rule (vdW) and the mixing rule based on the Gupta–Rasmussen–Fredenslund method (GRF), with the NRTL equation as G ^E model. The results obtained by these models show that type of applied mixing rule, a number and position of interaction parameters are of great importance for a satisfactory correlation of V ^E data. The GRF mixing rules coupled with the NRTL model gave mostly satisfactory results for V ^E correlation of the nonideal binary systems of diverse complexity.     

Comparison of different mixing rules for prediction of density and residual internal energy of binary and ternary Lennard–Jones mixtures

Mixing rules are necessary when equations of state for pure fluids are used to calculate various thermodynamic properties of fluid mixtures. The well-known van der Waals one-fluid (vdW1) mixing rules are proved to be good ones and widely used in different equations of state. But vdW1 mixing rules are valid only when molecular size differences of components in a mixture are not very large. The vdW1 type density-dependent mixing rule proposed by Chen et al. [1] is superior for the prediction of pressure and vapor–liquid equilibria when components in the mixture have very different sizes. The extension of the mixing rule to chain-like molecules and heterosegment molecules was also made with good results. In this paper, the comparison of different mixing rules are carried out further for the prediction of the density and the residual internal energy for binary and ternary Lennard–Jones (LJ) mixtures with different molecular sizes and different molecular interaction energy parameters. The results show that the significant improvement for the prediction of densities is achieved with the new mixing rule [1], and that the modification of the mixing rule for the interaction energy parameter is also necessary for better prediction of the residual internal energy.     

超额自由能型汽液相平衡混合规则

较全面地介绍了近几年来发展的各种典型的超额自由能型汽液相平衡混合规则。该类混合规则吸取了状态方程法和活度系数法在相平衡预测方面的优点,并将对于极性体系预测能力非常强的活度系数模型直接应用于状态方程法的相平衡预测中,实现了向高温区的良好外推和对超临界和亚临界组分的连续准确描述。依次发展的HV型、WS型和TC型三个大类的超额自由能型混合规则中,TC型混合规则的预测精确度要略高于HV型、WS型混合规则的预测精确度,而HV型、WS型混合规则的预测精确度大致相当。从发展的角度看,这些超额自由能型混合规则还要接受三元以上体系的汽液相平衡和液液相平衡预测的考验。另外,如何将超额自由能型混合规则扩展到多参数方程来提高相平衡预测精度,也是超额自由能型混合规则的一个值得关注的发展方向。     

Mixing rules for van-der-Waals type equation of state based on thermodynamic perturbation theory

A concept based on the thermodynamic perturbation theory for a `simple fluid' has been applied to the attractive term of a van-der-Waals type equation of state (EOS) to derive a simple mixing rule for the a parameter. The new mixing rule is a small correction to the original one-fluid approximation to account for the influence of particles of j-type on the correlation function of ii-type in a mixture consisting of particles of i and j types. The importance of the correction has been shown by comparison of the calculated results for binary mixtures of Lennard–Jones fluids with the data obtained by numerical method (Monte-Carlo simulation). The new mixing rules can be considered as a flexible generalization of the conventional mixing rules and can be reduced to the original v-d-W mixing rules by defaulting the extra binary parameters to zero. In this way the binary parameters already available in the literature for many systems can be used without any additional regression work. Extension of the new mixing rules to a multicomponent system do not suffer from `Michelsen–Kistenmacher syndrome' and provide the correct limit for the composition dependence of second virial coefficients. Their applicability has been illustrated by various examples of vapor–liquid and liquid–liquid equilibria using a modified Patel–Teja EOS. The new mixing rules can be applied to any EOS of van-der-Waals type, i.e., EOS containing two terms which reflect the contributions of repulsive and attractive intermolecular forces.     

Vapor–liquid equilibrium measurements and correlations for an azeotropic system of ethane + hexafluoroethane

The research on the isothermal vapor–liquid phase behavior for the ethane (R170) + hexafluoroethane (R116) system is presented in this paper. The vapor–liquid equilibrium (VLE) data were measured at four temperatures 189.31, 192.63, 247.63 and 252.80 K with an apparatus based on recirculation method. The experimental results were correlated with the Peng–Robinson equation of state using two types of mixing rules, the Panagiotopoulos–Reid mixing rule and the Huron–Vidal mixing rule involving the NRTL model. The calculated data using the regressed parameters were compared with the previous measured results, and good agreements can be observed.     

Calculation of critical lines of hydrocarbon/water systems by extrapolating mixing rules fitted to subcritical equilibrium data

The phase behavior of water/hydrocarbon mixtures in a wide range of pressures is important for various applications ranging from reservoir engineering to environmental engineering. In this work, mutual solubility and critical loci of hydrocarbon/water systems are calculated using the Peng–Robinson–Stryjek–Vera cubic equation of state with four mixing rules: (1) van der Waals mixing rules with one binary interaction parameter (vdW-1), (2) van der Waals mixing rules with asymmetric composition dependent binary interaction parameter (vdW-A), (3) Wong–Sandler mixing rules (WS) and (4) second-order modified Huron–Vidal mixing rules (MHV2). It was found that the parameters obtained from correlating liquid–liquid equilibria using different mixing rules would lead to prediction of completely different forms of critical behavior. Unusual branches of critical loci were found with WS and MHV2 mixing rules. Therefore, equation of states models must be used with extreme caution when applied for predicting phase behavior over wide ranges of temperatures and pressures.     

Thermophysical properties of aqueous solutions of the 1-ethyl-3-methylimidazolium tricyanomethanide ionic liquid

Thermophysical behavior of the binary system [water + 1-ethyl-3-methylimidazolium tricyanomethanide ionic liquid (IL)] was thoroughly characterized through systematic measurements of (vapor + liquid) equilibria (water activity a_w), mixing enthalpy, density, viscosity, and refractive index. The measurements were performed in the entire composition range and/or specifically in the highly dilute IL region, at T = 298.15 K or as a function of temperature in the range from (288.15 to 318.15) K. Effective experimental methods minimizing IL sample consumption, using flow arrangements, instrument couplings and high degree of automation were preferably employed. In particular, the a_w determination based on the chilled-mirror dew point technique and implemented by an AquaLab 4TE instrument was identified as a generally superior approach to study VLE of (water + IL) systems. Excess thermodynamic properties (Gibbs free energy, enthalpy, heat capacity, and volume) and property deviations from the linear mixing rule (viscosity, refractive index) were evaluated, Padé approximants being used to correlate adequately their complex composition dependences. The extensive a_w data were processed by a two-step procedure fitting first the temperature dependence at each isopleth and subsequently the composition dependence at each isotherm. Good estimates could be thus obtained for derivative thermal properties (enthalpy, heat capacity). Alternatively, the water activity and excess enthalpy data were correlated simultaneously by a NRTL-type model, providing their compact, thermodynamically consistent and adequate representation. Despite small absolute values of excess Gibbs free energy (G^E), the system is revealed to be highly nonideal, the small G^E resulting from close compensation of its large enthalpy and entropy contributions. Large endothermic effects and an enhanced increase of entropy upon mixing found for this system indicate relative weakness of interactions between unlike molecules and a massive structure breakage in the solution. Positive values of excess volume and negative values of viscosity and refractive index deviations found in the major part of the composition range corroborate this general energetic and structural pattern, although the situation appears to be more complicated in the highly dilute IL region, where these properties congruently exhibit a sign inversion.     

Optimization of automatically generated rules for predicting the presence and absence of substructures from MS and MS/MS data

A pattern-recognition/artificial-intelligence program, referred to as MAPS (Method for Analyzing Patterns in Spectra), was recently developed to identify the relationships that exist between substructures and the characteristic features they produce in the spectra from mass spectrometry (MS) and successive mass spectrometry (MS/MS). MAPS has been extended to utilize these relationships to formulate exclusion rules as well as inclusion rules, so that the absence of recognized substructures can be predicted as well as their presence. The potential usefulness of each MS and MS/MS spectral feature in such rule formulation is characterized by correlation and uniqueness factors. The correlation factor expresses the degree of correlation between a feature and a specific substructure; the uniqueness factor expresses the uniqueness of a feature with respect to that substructure. Features with high correlation factors are most use for predicting the absence of substructures, whereas features with high uniqueness factors are most useful for predicting their presence. Feature intensity-data have been found to improve the inclusion-rule performance and degrade the exclusion-rule performance. Criteria for optimizing the predictive abilities of both rule types are discussed.     

Mixing rules for hardsphere chain mixtures and their extension to heteronuclear hardsphere polyatomic fluids and mixtures

Mixing rules are very important for the calculation of fluid properties using different equations of state. In order to find the theoretical lead of the mixing rule for the size parameter, a mixing rule [1] for hardsphere mixtures has been proposed on the basis of Carnahan-Starling equation and Boublik-Mansoori equation. As its extension, mixing rules for hardsphere chain mixtures are proposed in this work. A mixing rule for the segment number (or chain length) is derived on the limitation of the equality of segment diameters, from the first order thermodynamic perturbation theories (TPT1) for pure chain fluids and for chain mixtures. Meanwhile, the mixing rule for the segment diameter is the same as the mixing rule for hardsphere mixtures on the limitation of monomer mixtures. The two mixing rules are checked together over wide ranges of conditions for hardsphere chain mixtures and compared with the first order thermodynamic perturbation theory (TPT1) and also with simulation data available in literature. An another interesting usage of new mixing rules is to describe the heteronuclear hardsphere polyatomic pure fluids, which consist of hardspheres with different segment diameters as in methane and ethane in which carbon and hydrogen atoms are looked as bonded spheres, and heteronuclear hardsphere polyatomic mixtures. The comparison with simulation data shows the validity of the mixing rules.     

Acoustic,viscometric and optical properties of binary mixtures of tetrahydrofuran with 1-propanol and 2-propanol

The values estimated from various mixing rules for the ultrasonic velocity, viscosity and refractive index have been compared with the respective values measured earlier at 293, 303, and 313?K over the entire mole fraction range of two binary mixtures of tetrahydrofuran (THF) with 1-propanol (1-p) and 2-propanol (2-p). There is an excellent agreement between the experimental values of ultrasonic velocity and of refractive index with the respective values obtained from the mixing rules. The mixing rules for viscosity provide values agreeing broadly with those obtained from experimental measurements. The relative merits and interrelations of these mixing rules are discussed.     

The experiments and correlations of the solubility of ethylene in toluene solvent

Polymer cyclic olefin copolymer (COC) is produced from the reaction of attaching ethyl groups to the norbornene monomer in liquid phase. The first step of process is dissolving ethylene in a liquid phase where toluene is present as a cosolvent. Thus, the solubility of ethylene in liquid toluene is the most important factor affecting the production of COC. In this study, the solubility of ethylene in toluene was measured in the temperature range from 323.15 to 423.15 K and pressure range from 5 to 25 bar. The experiments were conducted by the method of pressure decaying with a newly designed apparatus. The experimental results show that the solubility of ethylene in toluene increases with increasing pressure but decreases with increasing temperature.The experimental solubility data were expressed in the vapor–liquid equilibrium relationship and correlated fairly well by the bubble–pressure calculation with the Peng–Robinson equation of state (PR EOS) incorporated with the van der Waals one-fluid and the Zhong–Masuoka mixing rules with the consideration of binary interaction parameters. The results showed the van der Waals (vdW-1) mixing rule is slightly better than the Z–M mixing rule for pressure correlation but the Z–M mixing rule is slightly better for vapor composition correlation.A semi-empirical solubility equation with four parameters for the present binary system was proposed in this study. This proposed model estimates the solubility easier and as accurate as the PR EOS does for the present system.     

Modification of Peng Robinson EOS for modelling (vapour + liquid) equilibria with electrolyte solutions

A modification of the extended Peng–Robinson equation of state (PR-EOS) is presented to describe the (vapour + liquid) equilibria of systems containing water and salts. The modification employs three additional terms including a Born term, a Margules term and two terms separately used for estimation of the long-range electrostatic interactions (the Debye–Huckel (DH) or the mean spherical approximation (MSA) terms). Effects of two mixing rules, first, the Panagiotopoulos and Reid mixing rule (PR) and, second, the Kwak and Mansoori mixing rule (KM), on the final values of VLE calculations are also investigated. The results show that the KM mixing rule is more appropriate than the PR mixing rule. The proposed equation of state is used to calculate the (vapour + liquid) equilibrium (VLE) of the systems containing (water + sodium sulphate + carbon dioxide) and (water + sodium chloride + carbon dioxide) at high pressure. The comparison of calculated results with the experimental data shows that a combination of KM mixing rule with the DH term results a more accurate VLE values.     

Extension of Huron-Vidal-type Mixing Rule to Three-parameter Harmens-Knapp Cubic Equation of State

In this paper was extended the HV-type mixing rules to Harmens-Knapp cubic equation state (HK CEOS). The new HV-type mixing rule with HK CEOS was tested for Vapor-liquid equilibrium(VLE) of different polar and nonpolar systems. The tested results are in good agreement with existing experimental data within a wide range of temperatures and pressures. In comparison with the VDW mixing rule, the new mixing rule gives much better predictions for the VLE of nonpolar and polar systems.     

Experimental measurement and modeling of the vapor–liquid equilibrium of carbon dioxide + chloroform

Isothermal bubble and dew points, saturated molar volumes, and mixture critical points for binary mixtures of carbon dioxide+chloroform (trichloromethane) (CO₂/CHCl₃) have been measured in the temperature region 303.15–333.15 K and at pressures up to 100 bar. Mixture critical points are reported at 313.15, 323.15, and 333.15 K. The data were modeled with the Peng–Robinson equation of state using both the van der Waals-1 (vdW-1) mixing rule and the Wong–Sandler (WS) mixing rule incorporating the UNIQUAC excess free energy model. The WS mixing rule provided a better representation of the data than did the vdW-1 mixing rule, though with three adjustable parameters instead of one. The extrapolating ability of both of the mixing rules was investigated. Using the parameters regressed at 323.15 K, the WS mixing rule yielded better extrapolations for the composition dependence at 303.15, 313.15, and 333.15 K than the vdW-1 mixing rule.     

A Computationally Efficient Approach to Applying the SAFT Equation for CO2 + H2O Phase Equilibrium Calculations

The statistical associating fluid theory equation of state (EoS) is employed in a time efficient way for the correlation and prediction of vapor–liquid equilibrium of the CO₂ + H₂O binary system for the temperature (10–100 °C) and pressure (1–600 bar) ranges suitable for simulation of CO₂ geologic sequestration. The effective number of segments and energy parameter are correlated with the reduced temperature. Simple mixing rules are applied to obtain binary interaction parameters. Assigning a fixed H₂O composition in the mixing rule makes the phase equilibrium calculations relatively fast compared to other EoS’s. The results obtained by the model used were found to be in satisfactory agreement with the literature data.