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     

Unified equation of state based on the lattice fluid theory for phase equilibria of complex mixtures part II. Application to complex mixtures

In part I of the present article [Yoo et al., 1995], new rigorous and simplified lattice-fluid equations of state (EOS) were derived and their characteristic features of the molecular thermodynamic foundation were discussed by applying to pure fluids. In this part II, both EOSs were extended to various phase equilibrium properties of mixtures. Comparison of the models with experimental mixture data ranges from density, to equilibria of vaporliquid, vapor-solid and liquid-liquid phases for nonpolar/nonpolar, nonpolar/polar, polar/polar mixtures. Both models were also applied to supercritical fluid phase equilibria and activities of solvents in polymer solutions. With two temperature dependent parameters for pure compounds and one temperature-independent binary interaction energy parameter for a binary mixtures, results obtained to date illustrated that both EOSs are quantitatively applicable to versatile phase equilibria of mixtures over a wide range of temperatures, pressures and compositions.     

Calculation of the solid solubilities in supercritical carbon dioxide using a new G mixing rule

The aim of this study is to develop a new EOS/G^ex-type mixing rule with special attention to calculating the solid solubilities of aromatic hydrocarbons, aliphatic carboxylic acids, aromatic acids, and heavy aliphatic and aromatic alcohols in supercritical carbon dioxide. A volume correction term is applied with a combination of second and third virial coefficients which the equation for the third virial coefficient is quadratic, according to the suggestion by Hall and Iglesias-Silva. In this study, the cubic Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) equations of state have been used to calculate the solid solubilities of 23 solutes in supercritical CO₂, by using six mixing rules, namely, the Wong-Sandler (WS) rule, the Orbey-Sandler (OS) rule, the van der Waals one fluid rule with one (VDW1) and two (VDW2) adjustable parameters, the covolume dependent (CVD) rule and the new mixing rule. In all cases, the NRTL model was chosen as the excess Gibbs free energy model. The coefficients of the NRTL model and the binary interaction parameters of six mixing rules with two EOSs (PR and SRK EOSs) have been determined for 100 data sets of 23 binary systems over a wide range of temperatures and pressures covering more than 970 experimental data points which are reported in the literature. The results show that the PR EOS with the new mixing rule model is more accurate than the PR and SRK EOSs with the other mixing rules for solid solubility calculations in supercritical carbon dioxide.The regressed interaction parameters of the binary system, without any further modification, were then extended to four ternary mixtures, giving satisfactory results of the solid solubilities in supercritical CO₂.     

Taking advantage of available cubic equations of state

All available simple cubic equations of state (EOS) are developed for specific representations. An effort has been made to take advantage of these existing equations for binary vapor-liquid equilibrium (VLE) as well as density calculations by assigning different EOS to different components of the mixture under consideration. A four-parameter cubic equation was used for combining the equations in the calculation. The effect of substance-dependent Ω_ac on vapor-liquid equilibrium calculations was further investigated, using two sets of mixing rules. A criterion for selection of equations for VLE calculation by means of the proposed approach was suggested. The improvement in the prediction of liquid volumes for binary mixtures based on the fitting of pure component liquid volumes was very satisfactory.     

Simultaneous correlation of excess gibbs energy and enthalpy of mixing by the UNIQUAC Equation

Using data for excess Gibbs energy, g^E, and enthalpy of mixing, h^E, temperature dependent parameters of the UNIQUAC equation have been estimated for twenty four systems of binary mixtures. Fifteen of them include data for g^E and h^E at more than one different isotherm. These parameters are later tested in predicting the g^E and h^E data simultaneously and representing the effect of temperature on such data. The UNIQUAC equation with temperature dependent parameters represents larger values of maximum heat of mixing than does the UNIQUAC equation with the parameters independent of temperature.     

Calculation of phase equilibrium for water+carbon dioxide system using nonrandom lattice fluid equation of state

For the geological sequestration of carbon dioxide to prevent global warming, the phase equilibrium data for water and carbon dioxide mixture play an important role in process design and operation. In this work, the nonrandom lattice fluid equation of state with hydrogen bonding (NLF-HB EOS) was applied for the prediction of phase equilibrium of mixtures containing water and carbon dioxide. A new set of pure component parameters for carbon dioxide above critical condition was found and optimum binary interaction parameters were reported to correlate mutual solubility of mixtures. The calculated results were compared with the Peng-Robinson Equation of State with the conventional mixing rule (PR-EOS) and the Wong-Sandler mixing rule (PR-WS-EOS). The calculation results show that NLF-HB EOS can correlate mutual solubility of water+carbon dioxide mixtures with reasonable accuracy within a single theoretical framework.     

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.     

A modified Peng–Robinson equation of state for phase equilibrium calculation of liquefied,synthetic natural gas,and gas condensate mixtures

A modified Peng–Robinson equation of state, MPR2 EOS, is introduced by incorporating a new alpha function and a temperature dependent function for covolume, b. The modified cubic EOS has three input per each substance: critical temperature, critical pressure, and acentric factor. The coefficients of temperature dependence of the alpha and beta functions, relating to the parameters a and b of the new cubic EOS, are obtained by simultaneous fitting of saturated experimental vapour pressure and liquid density data for several pure components. The percent absolute average deviation (AAD%) of 1.38, 4.80 and 2.89 are obtained to correlation of the saturation vapour pressure, liquid density and vapour volume, respectively. Also the ADD% of 2.575 is computed for prediction of saturation enthalpy of vapourisation of the pure compounds. For calculation of phase equilibrium of mixture, the modified PR EOS is used for prediction of liquid density of the LNG mixtures. Also the new EOS is applied for construction of the phase envelop of synthetic natural gas, SNG, mixtures and calculation vapour–liquid equilibria of gas condensates. The results demonstrate that the new MPR2 EOS can be used for calculation of vapour–liquid equilibrium of pure components and mixtures with good accuracy.     

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.     

Modified parameters for the Redlich-Kwong equation of state

Two new parameters are proposed for the Redlich-Kwong equation of state. They are developed for pure components at T>T_c with emphasis placed on the application of the Redlich-Kwong equation to vapor-liquid equilibrium studies. The proposed parameters reflect the departure from the experimental T_c and P_c values, and are related to the conventional parameters. Calculated critical isotherms of pure propane and hydrogen by means of the proposed method and two methods available in the literature are compared and discussed. A set of mixing rules is proposed for correlating vapor-liquid equilibrium data. Critical volumes and acentric factors of pure components are not required in the application of these mixing rules. Applicability of the proposed new parameters and mixing rules is demonstrated by the correlation of vapor-liquid equilibrium data for five binary systems at thirty-two isothermal conditions and by the prediction of the triple-values dew-point curve for the methane-n-butane mixtures. Based on a paper presented at the Research Seminar on “Vapor-Liquid Equilibria in Multicomponent Mixtures” November 2–6, 1975 at Jablonna, Poland.     

Excess Gibbs free energy of butyl acetate with cyclohexane and aromatic hydrocarbons at 308.15K

Molar excess Gibbs free energies of mixing (C ^E ) for butyl acetate+cyclohexane or benzene or toluene or o- or m- or p-xylene were calculated by using Barker’s method from the measured vapor pressure data by static method at 308.15±0.01 K over the entire composition range. The G ^E values for the binary mixtures containing cyclohexane or benzene are positive; while these are negative for toluene, o-, m- and p-xylene system over the whole composition range. The G ^E values of an equimolar mixture for these systems vary in the order: cyclohexane>benzene>o-xylene>m-xylene>p-xylene>toluene. The G ^E values for these systems were also calculated by Sanchez and Lacome theory using the previously published excess enthalpy and excess volume data. It is found that while values of G ^E from Sanchez and Lacombe theory are in reasonably good agreement with those calculated by Barker method for m-xylene and p-xylene mixtures, agreement is very poor for other systems although they predict the sign of G ^E data except in the case of mixtures containing benzene.     

Thermodynamic modeling of vapor-liquid equilibria and excess properties of the binary systems containing diethers and n-alkanes by cubic equation of state

A comparison of the performances of two different approaches of cubic equations of state models, based on a classical van der Waals and mixing rules incorporating theG ^E equation, was carried out for correlation of Vapor-Liquid Equilibria (VLE), H^E and C _P ^E data alone, and simultaneous correlation of VLE+H^E, VLE+C _P ^E , H^E +C _P ^E and VLE+H^E +C _P ^E data for the diethers (1,4-dioxane or 1,3-dioxolane) with n-alkane systems. For all calculations the Peng-Robinson-Stryjek-Vera cubic equation of state (PRSV CEOS) was used. A family of mixing rules for the PRSV CEOS based on the Modified van der Waals one-fluid mixing rule (MvdW1) and two well-known CEOS/G^E mixing rules (MHV1 and MHV2), was considered. The NRTL equation, as the G^E model with linear or reciprocal temperature dependent parameters, was incorporated in the CEOS/G^E models. The results obtained by the CEOS/G^E models exhibit significant improvement in comparison to the MvdW1 models.     

Calculation of complex phase equilibria in the critical region of fluid mixture based on multi-fluid lattice equation of state

Quantitative correlation of critical loci and multiphase behaviors has received considerable attention because the increased industrial importance of processes operating within the high-pressure region such as supercritical fluid extraction. However, in the critical region, classical thermodynamic models such as equations of state (EOS) frequently fail to correlate phase equilibrium properties. Recently, the present authors proposed a new lattice-hole EOS based on the multi-fluid approximation of the nonrandom lattice theory. The model requires only two molecular parameters reflecting size and interaction energy for a pure fluid and one additional interaction parameters for a binary mixture. In this work, the reliable applicability of the EOS was demonstrated to various phase equilibria of complex mixtures in the critical region. Demonstration of the EOS was made to calculate multiphase behaviors such as solid-liquidvapor (SLV) equilibria and critical loci of binary complex mixtures at high pressure. For P-T, P-x, and T-ρ phase diagrams tested, the model agrees well with experimental data. This paper was presented at the 8th APCChE (Asia Pacific Conferation of Chemical Engineering) Congress held at Seoul between August 16 and 19, 1999.     

EXCESS ENTHALPY AND VAPOR-LIQUID EQUILIBRIA WITH THE MHV2 AND SOAVE MIXING RULES

Calculations and predictions of excess enthalpy (H^E) and vapor-liquid equilibrium (VLE) were performed using the Gibbs energy mixing rules MHV2 and a modification of it by Soave. The Soave-Redlich-Kwong equation of state was combined with the UNIQUAC equation. Four sets of parameters estimated in the UNIQUAC model were used for each of seven binary systems: the first estimated from VLE data, the second and the third estimated from H^E data for two versions of the UNIQUAC equation, and the fourth estimated from both H^E and VLE data simultaneously. It was found that H^E calculations can be performed with the mixing rules; the average relative errors fell from around 200% for the conventional mixing rule to around 60% for MHV2 combined with DECHEMA UNIQUAC parameters and was as little as 20% when the UNIQUAC parameters had been estimated from H^E and VLE data simultaneously. However, the approach suffers from the same shortcomings as far as cross-prediction between H^E and VLE data is concerned, as does the UNIQUAC equation used alone. There is a discrepancy between values obtained with the mixing rule and those obtained with the UNIQUAC equation directly. This discrepancy is smaller for the Soave modification of the mixing rule.     

Prediction of High‐Pressure Phase Equilibria using Cubic EOS: What Can Be Learned?

Modern semi‐empirical cubic EOSs are usually attached by complex functionalities, such as Huron‐Vidal (HV)‐type mixing rules. Although this practice improves the flexibility of the models, it also complicates consideration of an overall picture of phase behavior. As a result, G^E‐based equations are usually adjusted to the experimental data by way of a local fit. The latter approach tends to ignore that different regions of the thermodynamic phase space are closely inter‐related. The present study demonstrates that the contribution of (HV)‐type mixing rules to predicting high‐pressure phase equilibria can be quite modest and that in addition they may generate non‐realistic phase diagrams.     

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.