Representation of the vapor-liquid equilibrium of the ternary system carbon dioxide - propane - methanol and its binaries with a cubic equation of state : a new mixing rule

Erroneous liquid phase splitting is often predicted when correlating the vapor-liquid equilibrium of alcohol-alkane systems with a cubic equation of state. It is shown that even the local-composition mixing rules are not sufficiently flexible for an accurate representation of the propane-methanol system, and a new empirical three-parameter mixing rule is introduced. The ternary system carbon-dioxide - propane - methanol is well-predicted from the binary data.     

A generating equation for mixing rules and two new mixing rules for interatomic potential energy parameters

A generating equation for the mixing rules of interatomic potential energy parameters is proposed. It is demonstrated that this equation can, indeed, reproduce many popular mixing rules. A weighting matrix is used with the generating equation. This weighting matrix approach is superior to the present status of mixing rule development. A systematic framework is given for devising new mixing rules and/or comparing them. Two new mixing rules, which are more accurate than the available rules in the literature, are proposed. These rules are capable of reproducing the collision diameter and well-depth parameters for the binary values of noble gases to within their experimental uncertainties.     

Equations of state for strongly nonideal fluid mixtures: Application of local compositions toward density-dependent mixing rules

A local-composition, two-fluid model has been developed for equation-of-state calculations of fluid-phase equilibria for asymmetric mixtures; it is applicable to any equation of state of the van der Waals form. A modification of the quasichemical theory of Guggenheim is applied to mixtures at all fluid densities. Desirable boundary conditions are met at low densities, at high densities, and at high temperatures.In effect, the local-composition model uses density-dependent mixing rules. It contains no new adjustable binary parameters and can be extended to multicomponent mixtures without ternary (or higher) parameters. It appears that, when compared to conventional one-fluid models, significant improvement may be obtained in predictions for vapor-liquid equilibria of typical asymmetric mixtures.     

Density Prediction for Aqueous Multicomponent Solutions Obeying Isopiestic Relation and Isopycnotic Mixing Rule

The density predictive method based on isopiestic relation has been revised. The accuracy of density prediction based on the isopiestic mixing rule has been compared with that based on the isopycnotic mixing rule on ten aqueous ternary electrolyte systems, one quaternary electrolyte system, and one aqueous ternary nonelectrolyte system at 25°C. Of the ten ternary electrolyte systems, five systems show better predictive accuracy by the isopycnotic method, two by isopiestic relation, and three systems show comparable results. For the aqueous quaternary electrolyte and ternary nonelectrolyte systems, the isopycnotic method gives better predictive accuracy. The overall average error of density prediction based on isopiestic mixing rule is 0.00031, while that based on isopycnotic mixing rule is 0.00017.     

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.     

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。     

An equation of state for hydrogen fluoride

Using a similar approach as Lencka and Anderko [AIChE J. 39 (1993) 533], we developed an equation of state for hydrogen fluoride (HF), which can correlate the vapor pressure, the saturated liquid and vapor densities of it from the triple point to critical point with good accuracy. We used an equilibrium model to account for hydrogen bonding that assumes the formation of dimer, hexamer, and octamer species as suggested by Schotte [Ind. Eng. Chem. Process Des. Dev. 19 (1980) 432]. The physical and chemical parameters are obtained directly from the regression of pure component properties by applying the critical constraints to the equation of state for hydrogen fluoride. This equation of state together with the Wong–Sandler mixing rule as well as the van der Waals one-fluid mixing rule are used to correlate the phase equilibria of binary hydrogen fluoride mixtures with HCl, HCFC-124, HFC-134a, HFC-152a, HCFC-22, and HFC-32. For these systems, new equation of state with the Wong–Sandler mixing rule gives good results.     

New mixing rule for predicting multi-component gas adsorption

The present work describes a predictive model for ascertaining the multi-component gas adsorption equilibria. The model utilizes special form of covolume-dependent (CVD) mixing which is combined with the generalized form of 2-D EOS. Four well known 2-D EOSs; van der Waals, Soave-Redlich-Kwong, Peng-Robinson, Eyring along with the modified CVD mixing rule were used to predict the total adsorption of several binary and ternary systems. Based on the concept of the CVD mixing rule, it was inspired that CVD mixing rule could be a binding bridge between the molecular size and the molecular interaction. To show this, the ratio of the classical mixing rule %AAD to the CVD mixing rule %AAD were plotted versus the difference of the collision or the Leonard-Jones diameters of the gas molecules in the mixtures. It shows that there is a criterion between the CVD and the classical mixing rules in terms of molecular size difference. It seems that, Δσ _LJ≈0.60 Å is the criterion. The CVD mixing rule is approximately predominant in the region of Δσ _LJ≥0.60 Å, whilst, region of Δσ _LJ≤0.60 Å is nearly governed by the classical mixing rule. All predictions by the new mixing rule and the classical mixing rule were compared with the experimental data from the case studies. The new form of the mixing rule is in good agreement with the experimental data even for the non-ideal systems; hence provides a powerful framework to predict multi-component gas adsorption.     

Viscosity Mixing Rules for Binary Systems Containing One Ionic Liquid

In this work the applicability of four of the most commonly used viscosity mixing rules to [ionic liquid (IL)+molecular solvent (MS)] systems is assessed. More than one hundred (IL+MS) binary mixtures were selected from the literature to test the viscosity mixing rules proposed by 1) Hind (Hi), 2) Grunberg and Nissan (G–N), 3) Herric (He) and 4) Katti and Chaudhri (K–C). The analyses were performed by estimating the average (absolute or relative) deviations, AADs and ARDs, between the available experimental data and the predicted ideal mixture viscosity values obtained by means of each rule. The interaction terms corresponding to the adjustable parameters inherent to each rule were also calculated and their trends discussed.     

Application of a maximum likelihood method using implicit constraints to determine equation of state parameters from binary phase behavior data

The maximum likelihood method has been modified to allow the use of implicit constraints in the determination of model parameters from experimental data and the associated experimental uncertainties. The use of the implicit constraints greatly facilitates phase behavior modeling since models typically are complex functions of pressure, temperature, volume, and composition. The new form of the maximum likelihood method is illustrated using recently obtained data for the binary systems m-cresol + quinoline, tetralin + quinoline, and m-cresol + tetralin. Parameters were obtained for six different mixing rules used in conduction with the Soave-Redlich-Kwong equation of state. The mixing rules ranged from a simple interaction parameter to more complex rules involving temperature and volume dependencies.     

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.     

Phase equilibrium calculations for unprocessed well streams containing hydrate inhibitors

It is shown that the phase distribution of methanol and water between a hydrocarbon gas phase, a hydrocarbon liquid phase and an aqueous phase can be represented using the Soave-Redlich-Kwong equation with a non-conventional mixing rule for the a-parameter suggested by Huron and Vidal. Model parameters are estimated from data for binaries of the type methanolhydrocarbon and waterhydrocarbon. New experimental data are presented for two reservoir fluids and for one model system. The paper further presents a phase equilibrium algorithm for calculating the phase boundaries and the equilibrium compositions at the phase boundary for a system consisting of a gas, a liquid and a mixed aqueous phase.     

An Empirical Approach for Estimating the Density of Multicomponent Aqueous Solutions Obeying the Linear Isopiestic Relation

An empirical approach is presented for the density of aqueous multicomponentsolutions conforming to the linear isopiestic relation. This approach can be usedto estimate the densities of multicomponent systems from data on the constituentbinary subsystems at the same water activity. Predicted and measured densitiesfor 22 mixtures have been compared, using the simple Young's rule, theisopycnotic mixing rule of Teng and Lenzi, and the present method. The present methodand Young's rule give the most accurate predictions for strong electrolyte mixtureswithout common ions and for the mixtures with strong ion complexes, respectively.There is no universal best method for the strong electrolyte mixtures with commonions. An extensive comparison has also been given between apparent molarvolume predictions by Young's rule and by the new method. The two rules arerelatively better for the strong electrolyte mixtures without common ions andmixtures containing the transition metal chlorides, respectively. However, neitheris universally better for mixtures of strong common-ion electrolytes.     

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.     

Vapor-liquid equilibrium predictions of refrigerant mixtures from a cubic equation of state with a G-EoS mixing rule

A modified excess Gibbs energy model which is based on the local composition concept and assigns a single energy parameter per pair of components, is incorporated into the G^E—EoS thermodynamic formalism for vapor-liquid equilibrium (VLE) calculations of simple and complex refrigerant mixtures. One temperature set of data close to 273 K is used to obtain the model's parameters, which are used to extrapolate the VLE at other temperatures and pressures. A one-parameter form of the model based on the Wong-Sandler mixing rule is presented for several simple systems. The physical significance of the model's energy parameter is connected to the preference of the mixture for like to unlike interactions. The model is applied for VLE predictions of the ternary system R14-R23-R13, and the results are compared to calculations using the 3PWS model [H. Orbey. S.I. Sandler, Ind. Eng. Chem. Res. 34 (1995) 2520–2525] and the van der Waals mixing rule. Modelling of a few complex systems with only three data points given at each temperature is shown with a two parameter version of our model on the basis of the Huron-Vidal mixing rule.     

Solubility of methane in octane + ethanol at temperatures from 303.15 K to 333.15 K and pressures up to 12.01 MPa

Solubility of methane in octane + ethanol was measured at temperatures ranging from 303.15 K to 333.15 K and pressures ranging from 2.60 MPa to 12.01 MPa. Experimental data were analyzed using the Soave-Redlich-Kwong equation of state with three types of mixing rules, and the estimated average deviation from the experimental solubility data was less than 3.5 %.     

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.     

Molecular thermodynamics of fluid mixtures containing molecules differing in size and potential energy

Recent computer-simulation work by Shing and Gubbins (1983) for binary mixtures has shown that common semiempirical models (van der Waals n-fluid models) are in error when the molecules of the two components differ appreciably in size; the error is most severe in the dilute region. While perturbation theories are much better, they (like computer simulations) are not as yet useful for engineering work because of prohibitive computer requirements.This work proposes an algebraic expression for the Helmholtz energy of a mixture which gives results in very good agreement with those reported by Shing and Gubbins. This expression, using the local-composition concept, is based on a simplified but realistic picture of a fluid mixture: short-range order and long-range disorder. The proposed expression uses the Mansoori-Carnahan-Starling-Leland equation for the contribution of repulsive forces. For the contribution of attractive forces, it uses a new expression based on not one but several radii for the first-neighbor shell, one radius for each component.With reasonable simplifications, the resulting equation for the Helmholtz energy indicates that the van der Waals “constant” a is a strict quadratic function of mole fraction only at very low densities; at advanced densities, there are small deviations from the quadratic mixing rule. For practical calculations, the computer requirements are nearly the same as those for conventional engineering models.     

Improper matching of solvation energy components in G-based mixing rules

The physical significance of terms in two excess Gibbs free energy (G^ex)-based mixing rules, the modified Huron–Vidal (MHV1) and Wong–Sandler (WS) mixing rule, are examined through the use of solvation free energy. It is found that these mixing rules are in fact matching the charging contributions of solvation in an equation of state (EOS) to the complete solvation free energy in a liquid activity coefficient model (LM). The cavity contributions in the EOS are canceled as a result of the constant liquid molar volume to molecular volume ratio. The underlying idea of G^ex-based mixing rules that the EOS should behave like a LM at some limiting condition breaks down due to such an improper matching of solvation free energy components.