Application of a generalized van der Waals equation of state to several nonpolar mixtures

A generalized van der Waals equation of state, applied recently (Nguyen Van Nhu and Kohler, 1995) to the calculation of excess properties and phase equilibria for the mixture methane + ethane, is now extended to several nonpolar binary mixtures.Improved mixing rules for the van der Waals attractive term and for the correction term are proposed. With these mixing rules, the equation gives good agreement for vapour-liquid and liquid-liquid equilibria over a large temperature range for 29 binary mixtures. The agreement of mixture volumes and cross second virial coefficients is also satisfactory.     

A new model and extension of Wong–Sandler mixing rule for prediction of (vapour + liquid) equilibrium of polymer solutions using EOS/GE

The cubic equation of state (CEOS) is a powerful method for calculation of (vapour + liquid) equilibrium (VLE) in polymer solutions. Using CEOS for both the vapour and liquid phases allows one to calculate the non-ideality of polymer solutions based on a single EOS approach. However, the traditional mixing rules are not appropriate to extend the CEOS to non-ideal mixtures such as polymer solutions. Several authors have applied the EOS/G^E approach to predict (vapour + liquid) equilibria in polymer solutions, however, incorporating an appropriate excess Gibbs free energy for the new mixing rule is a major step. In this research, the NRTL-NRF model was extended in terms of volume fraction of polymer and solvent (instead of mole fraction), then equilibrium calculations were carried out using PRSV EOS and Wong–Sandler mixing rules. Using the adjustable parameters as a function of solution temperature, the NRTL-NRF model can be used as a predictive model. In comparison with NRTL model, the results of the new NRTL-NRF model show better accuracy.     

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.     

Application of GC-PPC-SAFT EoS to amine mixtures with a predictive approach

The GC-PPC-SAFT equation of state (EoS) is a combination of a group contribution method [S. Tamouza et al., Fluid Phase Equilib. 222-223 (2004) 67-76; S. Tamouza et al., Fluid Phase Equilib. 228-229 (2005) 409-419] and the PC-SAFT EoS [J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 40 (2001) 1244-1260] which was adapted to the polar molecules [D. Nguyen-Huynh et al., Fluid Phase Equilib. 264 (2008) 62-75]. It is here applied to the vapour pressure and liquid molar volume of primary, secondary and tertiary amines and their mixtures with n-alkanes, primary and secondary alcohols, using previously published group parameters. The mixing enthalpy is also evaluated for the binary systems. Binary interaction parameters k_ij are computed using a group-contribution pseudo-ionization energy, as proposed by Nguyen-Huynh [D. Nguyen-Huynh et al., Ind. Eng. Chem. Res. 47 (2008) 8847-8858]. A unique corrective parameter for the cross-association energy between amines and alcohols is used.The agreement with experimental data in correlation and prediction were found rather encouraging. The mean absolute average deviation (AAD) on bubble pressure is about 3.5% for pure amines. The mean AAD on the vapour-liquid equilibria (VLE) are respectively 2.2% and 5.5% for the amine mixtures with n-alkanes and alcohols. The AADs on saturated liquid volume are about 0.7% for the pure compounds and 0.9% for the mixtures. Prediction results are qualitatively and quantitatively accurate and they are comparable to those obtained with GC-PPC-SAFT on previously investigated systems.     

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.     

Thermodynamic calculation of supercritical-fluid equilibria: New mixing rules for equations of state

Simple cubic equations of state with conventional mixing rules have played an important role in the calculation of phase equilibria and other thermodynamic properties of non-polar fluid mixtures. In the application of supercritical fluids to separation processes, volumetric as well as phase equilibrium properties are very important for rational process design.

Heyen (1980) proposed a cubic equation of state which shows better accuracy in the calculation of volumetric properties, compared to the Peng-Robinson equation of state. In order to apply his equation to polar mixtures, Heyen recently proposed a density-independent mixing rule, but this does not obey the universally-observed quadratic mixing rule of the second virial coefficient in the low-density limit.

This paper proposes a new density-dependent mixing rule for the Heyen equation of state. The Heyen equation of state with our new mixing rule appears to calculate the phase equilibria and the volumetric properties of CO₂-containing non-polar as well as polar mixtures with good accuracy.     

Low pressure vapor–liquid equilibrium in ethanol + congener mixtures using the Wong–Sandler mixing rule

Phase equilibrium in binary ethanol mixtures found in alcoholic beverage production has been analyzed using a cubic equation of state (EoS) and suitable mixing and combining rules. The main objective of the study is the accurate modeling of the congener concentration in the vapor phase (substances different from ethanol), considered to be an important enological parameter in the alcohol industry. The Peng–Robinson (PR) equation of state has been used and the Wong–Sandler (WS) mixing rules, that include a model for the excess Gibbs free energy, have been incorporated into the equation of state constants. In the Wong–Sandler mixing rules the van Laar (VL) model for the excess Gibbs energy has been used. This combination of equations of state, mixing rules and combining rules are commonly applied to high pressure phase equilibrium and have not yet been treated in a systematic way to complex low pressure ethanol mixtures as done in this work. Nine binary ethanol + congener mixtures have been considered for analysis. Comparison with available literature data is done and the accuracy of the calculations is discussed, concluding that the model used is accurate enough for engineering applications.     

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.     

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.     

Application of the Perturbed-Chain SAFT equation of state to polar systems

The Perturbed-Chain SAFT (PC-SAFT) equation of state is applied to pure polar substances as well as to vapor–liquid and liquid–liquid equilibria of binary mixtures containing polar low-molecular substances and polar co-polymers. For these components, the polar version of the PC-SAFT model requires four pure-component parameters as well as the functional-group dipole moment. For each binary system, only one temperature-independent binary interaction k_ij is needed. Simple mixing and combining rules are adopted for mixtures with more than one polar component without using an additional binary interaction parameter. The ability of the model to accurately describe azeotropic and non-azeotropic vapor–liquid equilibria at low and at high pressures, as well as liquid–liquid equilibria is demonstrated for various systems containing polar components. Solvent systems like acetone–alkane mixtures and co-polymer systems like poly(ethylene-co-vinyl acetate)/solvent are discussed. The results for the low-molecular systems also show the predictive capabilities of the extended PC-SAFT model.     

Bubble pressures and saturated liquid molar volumes of trichlorofluoromethane—chlorodifluoromethane mixtures. Representation of refrigerant-mixtures vapor—liquid equilibrium data by a modified form of the Peng—Robinson equation of state

Experimental vapor—liquid equilibrium data and saturated liquid molar volumes of chlorodifluoromethane—trichlorofluoromethane binary mixtures have been obtained at four temperatures (298.15, 323.15, 348.15 and 373.15 K) using apparatus described previously.The experimental vapor—liquid equilibria are represented well by a modified form of the Peng—Robinson equation of state with one interaction parameter, but the mean deviation between the calculated and experimental densities is 5%.Vapor—liquid data for binary refrigerant mixtures from the literature are treated using the modified form of the Peng—Robinson equation of state with one adjusted interaction parameter in the mixing rule for a. The representation is fair and is not improved by introducing an additional parameter in the mixing rule for b.     

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.     

Application of a volume-translated Peng-Robinson equation of state on vapor-liquid equilibrium calculations

A volume-translated Peng-Robinson (VTPR) equation of state (EOS) is developed in this study. Besides the two parameters in the original Peng-Robinson equation of state, a volume correction term is employed in the VTPR EOS. In this equation, the temperature dependence of the EOS energy parameter was regressed by an improved expression which yields better correlation of pure-fluid vapor pressures. The volume correction parameter is also correlated as a function of the reduced temperature. The VTPR EOS includes two optimally fitted parameters for each pure fluid. These parameters are reported for over 100 nonpolar and polar components. The VTPR EOS shows satisfactory results in calculating the vapor pressures and both the saturated vapor and liquid molar volumes. In comparison with other commonly used cubic EOS, the VTPR EOS presents better results, especially for the saturated liquid molar volumes of polar systems. VLE calculations on fluid mixtures were also studied in this work. Traditional van der Waals one-fluid mixing rules and other mixing models using excess free energy equations were employed in the new EOS. The VTPR EOS is comparable to other EOS in VLE calculations with various mixing rules, but yields better predictions on the molar volumes of liquid mixtures.     

Vapor–liquid equilibria predictions of hydrogen–hydrocarbon mixtures with the Huron–Vidal mixing rule

An excess Gibbs-equation of state (G^E-EoS) framework based on the Huron–Vidal mixing rule, has been applied to study vapor–liquid equilibria (VLE) of hydrogen–hydrocarbon mixtures. The mixing rule couples the Peng–Robinson–Stryjek–Vera (PRSV) EoS with a local composition solution model. The solution model is based on one-fluid theory treatment and assigns a single energy parameter to each binary pair. This energy parameter relates to the preference of the molecules for like to unlike interactions. The allocation of a system's number of interactions to the individual species in a binary mixture, incorporates the use of size parameters which gain significance only in the liquid phase. In a two parameter form, the framework has been used for the simultaneous data reduction of a large number of binary and several ternary hydrogen–hydrocarbon mixtures. These systems were taken over an extended range of pressures and temperatures. Results from the data reduction are reported in both tabular and graphical forms. Correlations for the model parameters have been identified with the acentric factor of the hydrocarbon in hydrogen–hydrocarbon binary mixtures. In a fully predictive mode, the model has shown to describe well VLE of binary hydrogen–linear alkane systems. Comparisons of these results with calculations from the Peng–Robinson (PR) EoS and the classical mixing rule (vdW) are included.     

Effect of mixing rule boundary conditions on high pressure (liquid + liquid) equilibrium prediction

We examine the prediction of high pressure (liquid + liquid) equilibrium (LLE) from the Peng–Robinson equation with three excess Gibbs free energy (G^ex)-based mixing rules (MR): the first order modified Huron–Vidal (MHV1), the Wong–Sandler (WS), and a hybrid of these two (referred to as G^exB₂). These mixing rules differ by the boundary conditions used for determination of the temperature and composition dependence of parameters a and b in the PR EOS. The condition of matching the excess Gibbs free energy from the EOS at zero pressure to that from the G^ex model, used in MHV1 and G^exB₂ MR, leads to a similar miscibility gap from PR EOS and the G^ex model used. On the other hand, the condition of matching excess Helmholtz energy from the EOS at infinite pressure to that from the G^ex model, used in the WS MR, shows remarkable deviations. The condition of quadratic composition dependence in the second virial coefficient (B₂), used in WS and G^exB₂ MR, allows for both positive and negative values in the molar excess volume. Depending on the mixture, either the increase or decrease of the miscibility gap with pressure can be observed when the WS or the G^exB₂ MR is used. The condition of linear combination of molecular sizes of each component used in the MHV1 MR, however, often leads to small, positive molar excess volumes. As a consequence, the predicted LLE from using the MHV1 MR are insensitive to pressure. Therefore, we find that the G^exB₂ mixing rule provides the best predictive power for the LLE over a wide range of temperature and pressure.     

用二元交互作用函数预测高压下多组分体系的气液平衡

用和体系的状态有关且满足不变性条件的二元交互作用函数,结合F函数修改的立方状态--方程FRKS方程,预测高压下多组分体系的气液平衡.选择15个三元体系及其组分二元系来检验方法的可行性,这些体系覆盖了从简单的接近理想溶液行为的体系到高度非理想体系.计算结果表明,该方法不仅能相当精确地关联各种类型二元系的气液平衡,而且能在仅用组分二元系参数的条件下较准确地预测所考察的所有三元体系的气液平衡     

Measurements of vapour-liquid equilibria,densities, viscosities and dielectric properties of cyclohexanone + triethylamine mixtures with respect to keto-enol tautomerism

Densities, ?, kinematic viscosities ν, refractive indices n_D referred to the sodium D-line, static relative permittivities ε, specific conductances κ and vapour-liquid equilibrium data for cyclohexanone + triethylamine mixtures have been determined at two temperatures. Excess properties of mixing calculated from these data indicate that in such mixtures considerable amounts of enol-amine associates are formed. The permittivity data indicate that the formation of associates is an exothermic process, which also has a marked influence on the thermodynamic excess properties of the mixtures. The enol-amine associates may undergo electrolytic dissociation. A common evaluation of static relative permittivities and specific conductances shows that for the mixture the conductance data are affected considerably by the formation of undissociated ion pairs in the sense of Bjerrum theory, especially at concentrations of the mixture where it is less polar. The conductance data are also influenced considerably by the fact that the ion pairs are solvated if the molar ratio of triethylamine exceeds 0.5.     

Modeling of phase equilibria with CPA using the homomorph approach

For association models, like CPA and SAFT, a classical approach is often used for estimating pure-compound and mixture parameters. According to this approach, the pure-compound parameters are estimated from vapor pressure and liquid density data. Then, the binary interaction parameters, k_ij, are estimated from binary systems; one binary interaction parameter per system. No additional mixing rules are needed for cross-associating systems, but combining rules are required, e.g. the Elliott rule or the so-called CR-1 rule. There is a very large class of mixtures, e.g. water or glycols with aromatic hydrocarbons, chloroform-acetone, esters-water, CO₂-water, etc., which are classified as “solvating” or “induced associating”. The classical approach cannot be used and the cross-association interactions are difficult to be estimated a priori since usually no appropriate experimental data exist, while the aforementioned combining rules cannot capture the physical meaning of such interactions (as at least one of the compounds is non-self-associating). Consequently, very often one or more of the interaction parameters are optimized to experimental mixture data. For example, in the case of the CPA EoS, two interaction parameters are often used for solvating systems; one for the physical part (k_ij) and one for the association part (β^cross). This limits the predictive capabilities and possibilities of generalization of the model. In this work we present an approach to reduce the number of adjustable parameters in CPA for solvating systems. The so-called homomorph approach will be used, according to which the k_ij parameter can be obtained from a corresponding system (homomorph) which has similar physical interactions as the solvating system studied. This leaves only one adjustable parameter for solvating mixtures, the cross-association volume (β^cross). It is shown that the homomorph approach can be used with success for mixtures of water and glycols with aromatic hydrocarbons as well as for mixtures of acid gases (CO₂, H₂S) with alcohols and water. The homomorph approach is less satisfactory for mixtures with fluorocarbons as well as for aqueous mixtures with ethers and esters. In these cases, CPA can correlate liquid-liquid equilibria for solvating systems using two adjustable parameters. The capabilities and limitations of the homomorph approach are discussed.     

An extension of cubic equations of state to vapor-liquid equilibria in polymer-solvent mixtures

Cubic equations of state (EOS) are extended to describe polymer-solvent vapor-liquid equilibria (VLE). The solvents are described the conventional way using critical parameters. To describe the pure polymers, only the weight-average molecular weight is necessary, though number-average molecular weight, polydispersity and melt density can be incorporated if desired. To extend the model to mixtures, a mixing rule that combines EOS with excess energy models is used. In this formulation, the excess Gibbs energy term is considered in two parts: the classical Flory term for the entropic contributions and a residual term that takes care of specific interactions between the solvent and the polymer. For athermal mixtures that exhibit no such interactions, the residual term drops out and the model becomes completely predictive. Otherwise, for residual contributions, depending upon the complexity of specific molecular interactions anticipated in the mixture, either a single parameter Flory expression or a two-parameter NRTL equation can be used. We conclude that the simple cubic EOS approach presented here is easy to use, yet competes successfully with more sophisticated EOS models developed particularly for polymer solutions. Moreover, it offers more flexibility if one or more parameters are to be tuned to the process data.     

A new formulation of the predictive NRTL-PR model in terms of kij mixing rules. Extension of the group contributions for the modeling of hydrocarbons in the presence of associating compounds

A generalized NRTL model was previously proposed for the modeling of non ideal systems and was extended to the prediction of phase equilibria under pressure according to the cubic NRTL-PR EoS. In this work, the model is reformulated with a predictive k_ij temperature and composition dependent mixing rule and new interaction parameters are proposed between permanent gases, ethane and nitrogen with hydrocarbons, ethane with water and ethylene glycol. Results obtained for excess enthalpies, liquid-vapor and liquid-liquid equilibria are compared with those provided by the literature models, such as VTPR, PPR78, CPA and SRKm. A wide variety of mixtures formed by very asymmetric compounds, such as hydrocarbons, water and ethylene glycols are considered and special attention is paid to the evolution of k_ij with respect to mole fractions and temperature.