Equation Of State Mixing Rules Research Articles

Measurement and correlation of density and viscosity of n-alcohol–water mixtures at temperatures up to 618 K and at pressures up to 40 MPa

Densities and viscosities of methanol-water, ethanol-water and n-propanol-water mixtures were measured over the entire range of compositions at temperatures from 523.2 to 618.2 K and at pressures up to 40 MPa. The excess molar volume calculated from the measured density changed negative to positive with increasing alcohol composition at some conditions studied, and this behavior was observed at lower temperature with increasing alkyl chain length. Maxima in composition dependence of viscosity, which is generally observed in ambient conditions, were not present at 618.2 K. Viscosities could be correlated with the Eyring's theory to within 6.4% at temperatures up to 476.2 K. The Peng-Robinson equation of state and Redlich-Kister mixing rule were used to obtain the excess Gibbs energy included in the Eyring's theory, and two binary interaction parameters in the mixing rules were determined by fitting of viscosity data. The densities were predicted using the volume translated Peng-Robinson equation of state to within 3.2% with parameters being determined by viscosity data at temperatures up to 476.2 K. A switching of fitting parameters from binary interaction parameters of the equation of state mixing rule to a proportionality constant, σ, of Eyring's theory was introduced, and viscosities and densities of the normal alcohol-water mixtures could be correlated and predicted to within 4.6% and 8.9%, respectively at temperatures above 476.2 K.

Correlation of vapor-liquid equilibria for binary mixtures with free energy-based equation of state mixing rules: Carbon dioxide with alcohols, hydrocarbons, and several other compounds

The correlation of vapor-liquid equilibrium data for high-pressure carbon dioxide systems is of interest in a number of industrial applications, including supercritical extraction. Here, we consider the correlation of data for 12 binary systems of carbon dioxide separately with alcohols, with hydrocarbons, and with acetone, benzene, and water. The Wong-Sandler (W-S) and modified Huron — Vidal first order (MHV1) free energy-based equation of state mixing rules (the W-S and MHV1 models) were used in the calculations. Both combined equation of state+free energy models generally resulted in good correlations of the experimental data over wide ranges of temperature and pressure with temperature — independent parameters. However, for the carbon dioxide+water system, the W-S model produced an 11% average absolute deviation in pressure, while no parameter that resulted in an AAD in pressure of less than 20% could be found for the MHV1 model.

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.

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.

Equation of state mixing rule based on activity coefficient model: explicit solution for `finite pressure approach'

An explicit solution has been obtained for the mixing rule of parameter a of the cubic equation of state (EOS) using `finite pressure approach' to connect EOS with an activity coefficient model (ACM). The result is expressed in terms of Huron–Vidal solution and a correction, which is dependent on the repulsive term of EOS. Applications to multicomponent mixtures containing volatile components have been considered.

A zero-pressure cubic equation of state mixing rule for predicting high pressure phase equilibria using infinite dilution activity coefficients at low temperature

The infinite dilution activity coefficients ( γ i ∞) of a solute in a solvent are important data in process separation calculations. These values reflect the degree of non-ideal solution behavior of the solute in the solvent. This paper investigates the use of infinite dilution activity coefficients in cubic equation of state mixing rules for the prediction of phase behavior at high pressures. A mixing rule recently developed by Twu and Coon has been extended from infinite pressure to zero pressure. The methodology for extending the infinite-pressure Twu–Coon mixing rule was developed so that the zero-pressure Twu–Coon mixing rule reproduces the excess Gibbs free energy, as well as liquid activity coefficients of any activity models, with extremely high accuracy without requiring any additional binary interaction parameters. We compare the performance of this new mixing rule with the MHV1 and Wong–Sandler mixing rules for its ability to use γ i ∞ in the prediction of high pressure phase behavior for strongly non-ideal systems.

Prediction of gas–solid equilibrium using equations of state

The correlation of experimental data of solubility of solids in supercritical solvents using equations of state (EOS) is a well known and relatively simple approach. Normally, cubic EOS with quadratic mixing rules (or modifications of them) may be satisfactorily used for fitting purposes and much research is being done in order to improve the predictive capabilities of present models. Usually, models are assessed in terms of goodness of fit of solubility data; however, in spite of the pressure and temperature conditions of supercritical extraction processes, the critical behavior predicted by models used for correlating data is rarely analyzed. In this work, the analysis of critical lines in the presence of solid–fluid equilibria is proposed as an additional test of models because it gives the correct topology of predicted phase diagrams. It is shown that multiphase equilibria is normally predicted in ranges of interest for supercritical extraction. Multiphase equilibria is not obvious when a stability analysis is ignored, inducing possible modeling pitfalls. In addition, multiphase behavior affects the prediction of stable solid–fluid equilibria. Examples and phase diagrams are discussed in detail using a modified van der Waals EOS and quadratic mixing rules.

Equation of State Mixing Rules That Incorporate Only the Residual Part of the UNIQUAC Model

New mixing rules that incorporate the residual part of the UNIQUAC activity coefficient model and the correct quadratic concentration dependence of the second virial coefficient are presented. Following our recent approach, only the attractive part of the excess Helmholtz energy derived from the equation of state (EOS) is related to the excess Gibbs energy. It is recognized that, in the extension of this concept to the UNIQUAC model, the interpretation of the physical contributions of the combinatorial and residual parts must be considered. We conclude that the attractive forces are related only to the residual part of the UNIQUAC model. This allows the formulation of a class of mixing rules that are independent of the repulsive term in the EOS and are applicable to both cubic and noncubic EOS. The proposed mixing rules are applied to different low-pressure systems, exhibiting vapor−liquid and liquid−liquid equilibria, and to two near critical and supercritical systems.

CEOS/AE mixing rules constrained by vdW mixing rule and second virial coefficient

AbstractA new class of mixing rules, with van der Waals one‐fluid mixing rule and second virial coefficient constraints, was developed for cubic equations of state (EOS) for application to various asymmetric, highly nonideal chemical systems. This approach evolved from previous research in the EOS field. The resulting mixing rule is practical in application, yet more flexible than Huron–Vidal or Wong–Sandler mixing rules and depends only on composition and temperature. It also avoids the problems associated with the Wong–Sandler mixing rules. A cubic EOS with the new mixing rules was applied successfully to complex mixtures. The new rules reduce to van der Waals mixing rules when the parameters in the nonrandom excess Helmholtz free energy are set to zero. Since many times the classic mixing rules worked very well for nonpolar systems, it is extremely desirable that the composition‐dependent mixing rules reduce to the classic mixing rules. Preserving this capability also ensures that the binary interaction parameters for the classic quadratic mixing rules in the DECHEMA Chemistry Data Series for mixtures of low boiling substances and other existing databases for systems involving hydrocarbons and gases can be used directly in the new EOS mixing rules, in addition to liquid‐activity coefficient model parameters currently reported in the Series.

Matching equation of state mixing rules to activity coefficient model expressions

Procedures for deriving equation of state mixing rules based on matching the zero pressure excess Gibbs energy to that of a previously parameterized activity coefficient model are discussed. I have investigated the conditions required for accurate reproduction of the original model and provide a simple graphical tool for evaluating the error of the equation of state mixing rule. An extension of the MHV-2 mixing rule to incorporate mixtures containing components with very low volatility is suggested.

A comparison of various cubic equation of state mixing rules for the simultaneous description of excess enthalpies and vapor-liquid equilibria

Recent new mixing and combining rules for cubic equations of state (EOS) have extended the range of such equations to the accurate description of the vapor-liquid equilibria (VLE) of highly nonideal mixtures. However, the simultaneous correlation and/or prediction of vapor-liquid equilibrium (VLE) and liquid mixture excess enthalpies ( H ex) by either activity coefficient models or equations of state has been a difficult problem in applied thermodynamics. In this communication, we re-examine this problem using a modified version of the Peng-Robinson equation of state and the two-parameter van der Waals one-fluid, Wong-Sandler and modified Huron-Vidal mixing rules. For comparison, the direct use of activity coefficient models is also considered. In each case a temperature dependence of the model parameters is introduced in an attempt to represent simultaneously VLE and H ex behavior. Four highly nonideal binary mixtures (2-propanol + water, methanol + benzene, benzene + cyclohexane, and acetone + water) are considered. The results indicate that while all the models can accurately correlate VLE and H ex data separately, attempting to predict the values of one property with parameters obtained from the other does not give satisfactory results with any model. Also, we find that the simultaneous correlation of both VLE and H ex with the EOS models at one temperature is possible, but extrapolations to other temperatures with parameters obtained in this way did not result in accurate predictions of either VLE or H ex. The main problem appears to be that the excess free energy (activity coefficient) models used are not capable of representing both VLE and H ex over a range of temperatures, and so equations of state that incorporate these free energy models have the same shortcoming.

Prediction of vapor-liquid equilibria at high pressures using activity coefficients at infinite dilution

A modified procedure of using the Huron-Vidal mixing rule based on available activity coefficients at infinite dilution at low pressures is shown to be useful for making vapor-liquid equilibrium predictions for non ideal mixtures at high temperatures and pressures. We compare the performances of this mixing rule (called HVID model), coupled with the SRK equation of state and a reduced UNIQUAC model, with those of two other equation of state mixing rules, one by Dahl and Michelsen (MHV2) and another by Wong and Sandler (WS). Six binary systems are used in this study, and we find that the errors in the predicted pressure with the new mixing rule are, on the whole, less than those obtained when using the MHV2 model. Compared with the WS mixing rule, the HVID model performs slightly better.

Some Properties of Equation of State Mixing Rules Derived from Excess Gibbs Energy Expressions

Several recent proposals for using excess free energy models in constructing mixing rules for cubic equations of state have been analyzed. The reproduction of the base model behavior at the correlating temperature and the nature of the extrapolation to higher temperatures have been examined. The Wong−Sandler, LCVM, and MHV-1 mixing rules in an equation of state result in excess Gibbs free energies with contributions in addition to those from the base activity coefficient model. As a result, considerable degradation of calculated equilibrium behavior can occur. The excess enthalpies produced by these mixing rules are generally more positive than would be found from the temperature derivative of the G excess models, with only the MHV-2 mixing rule giving a reasonably quantitative reproduction. The success of the Wong−Sandler mixing rule in extrapolating to higher temperatures appears to derive in part from this positive addition to the excess enthalpy.

On the combination of equation of state and excess free energy models

In this communication we review the bases for a collection of equation of state mixing rules which have been developed that combine activity coefficient (or excess free energy of mixing) models with equations of state. We show that combining equations of state and activity coefficient models at infinite pressure produces mixing rules that are free of ad hoc assumptions, and are easy to understand and implement. In contrast, when the link is made at zero or low pressure, problems arise because the liquid volume appears in the mixing rule. In principle, this requires solving the equation of state for the liquid volumes of each of the pure components and for the mixture. To avoid such calculations, and to deal with the problem that a liquid phase solution to the equation of state may not exist for one or more of the pure components at the temperature of interest, several ad hoc procedures have been proposed. It is these procedures that differentiate between the various zero (low) pressure mixing rules. Each of the mixing rules reviewed here is successful for at least some range of temperatures, though some of their shortcomings are discussed. Finally, two new variants of this class of mixing rules are proposed, of which one is tested and found to be simple, and at least as satisfactory as presently used zero-pressure mixing rules.

An enthalpy-based cubic equation of state mixing rule for cross-prediction of excess thermodynamic properties of hydrocarbon and halogenated refrigerant mixtures

Mixing rules based on excess enthalpy of a binary fluid mixture were derived for Soave's equation. Specifically, the mixture SRK equation was cast in a form with complete separation of equation of state constants from thermodynamic variables P, V, and T yielding an equation of state with two volume parameters and one energy-volume parameter. An expression for the binary excess enthalpy was obtained, and the infinite-pressure limit was determined to provide a composition and temperature explicit expression for the SRK parameter a M; a conventional linear mixing rule was used for b M. This set of mixing rules was used with published excess enthalpy data to predict VLE for eleven binary systems comprised of simple aliphatic and halogenated hydrocarbons. A systematic analysis of the results by type of system shows the new mixing rule to be marginally superior to the original Soave polynomial mixing rules with k 12 = 0 or to an athermal form of the new mixing rule for a M ( H E = 0) for simple hydrocarbon systems. However, a marked improvement was observed for nonideal mixtures containing halogenated compounds for which conventional mixing rules with k 12 = 0 for the liquid phase generally fail. Moreover, when only a few excess enthalpy data for the single liquid phase (or supercritical fluid region) which could be fit to a simple polynomial expansion were available, excellent predictions were usually observed. Finally, the new mixing rules produce an approximately quadratic composition dependence of the vapor mixture second virial coefficient at low pressure as is required by statistical mechanics.

Reformulation of Wong‐Sandler mixing rule for cubic equations of state

AbstractThe cubic equation of state mixing rule of Wong and Sandler is reformulated in a way that eliminates one of its parameters and so that it can go smoothly from activity coefficient‐like behavior to the classical van der Waals one‐fluid mixing rule merely by variation of its parameters. The parameters in the reformulated mixing rule can be obtained from correlation of vapor–liquid equilibrium data or from the two infinite dilution activity coefficients for each binary pair in the mixture. When these activity coefficients are estimated from the UNIFAC group contribution model, the mixing rule becomes completely predictive. The correlative and completely predictive forms of this mixing rule are tested here and shown to be accurate for five difficult binary systems and one ternary mixture over large ranges of temperature and pressure.

An equation of state mixing rule for correlating ternary liquid-liquid equilibria

Abstract A new mixing model for cubic equations of states that is theoretically correct in the composition dependence of the second and third virial coefficients is derived. This new mixing rule has been tested in conventional vapor-liquid equilibria calculations. It was found to be as good as activity coefficient models and other equation of state mixing rules for solutions of various kinds of nonideality. It can also be extrapolated over large temperature and pressure ranges. Its salient feature is an extra degree of freedom that accounts for ternary interaction, which enables it to give consistent correlation of vapor-liquid and liquid-liquid equilibria of ternary systems and the corresponding binaries.

Prediction of vapor-liquid equilibria at high pressures using activity coefficient parameters obtained from low-pressure data: a comparison of two equation of state mixing rules

Two recent equation of state mixing rules, one by Dahl and Michelsen (MHV2) and another by Wong and Sandler (W-S), are claimed to be useful for making vapor-liquid equilibrium predictions for nonideal mixtures at high temperatures and pressures using data obtained at low pressures. Here the authors compare the performance of these two mixing rules in predicting the high-pressure phase behavior of nine binary systems using published activity coefficient parameters. They also compare the predictions for two ternary systems using mixing rule parameters obtained from low-pressure binary mixture data. They find that both mixing rules can be used to make reasonable high-pressure vapor-liquid equilibrium predictions from low-pressure data. However, the errors in the predicted pressure with the W-S mixing rule when used with either the Peng-Robinson or Soave-Redlich-Kwong equation of state are, on the average, about half or less those obtained when using the MHV2 mixing rule. Also, the predictions of the MHV2 mixing rule deteriorate significantly as the temperature and pressure range increase, which is not the case with the W-S mixing rule.

Radial distribution functions and their role in modeling of mixtures behavior

Radial distribution (Pair correlation) functions are the primary linkage between macroscopic thermodynamics properties and intermolecular interactions of fluids and fluid mixtures. Numerous theories of mixtures exist based on the definition of the radial distribution functions. Utilization of such theories require the knowledge about mixture radial distribution functions and their relations. Generally the exact radial distribution functions of real fluid mixtures are not available. In this paper a number of approximations for the mixture radial distribution functions are presented. Application of these approximation can result in analytic mixture theories and mixing rules. A uniform statistical mechanical technique is presented through which one can derive equation of state mixing rules based on different RDF approximations. The relative merits of different sets of mixing rules derived by this technique and their applications for modeling of the behavior of mixtures are discussed.

Equation of state mixing rule for nonideal mixtures using available activity coefficient model parameters and that allows extrapolation over large ranges of temperature and pressure

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTEquation of state mixing rule for nonideal mixtures using available activity coefficient model parameters and that allows extrapolation over large ranges of temperature and pressureDavid S. H. Wong, Hasan Orbey, and Stanley I. SandlerCite this: Ind. Eng. Chem. Res. 1992, 31, 8, 2033–2039Publication Date (Print):August 1, 1992Publication History Published online1 May 2002Published inissue 1 August 1992https://doi.org/10.1021/ie00008a027RIGHTS & PERMISSIONSArticle Views1573Altmetric-Citations142LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (781 KB) Get e-Alerts Get e-Alerts