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.     

New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures

Huron, M.-J. and Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271.Good correlations of vapour-liquid equilibria can be achieved by applying the same two-parameter cubic equation of state to both phases. The results primarily depend on the method used for calculating parameters and, for mixtures, on the mixing rule. True parameters are the covolume b and the energy parameter a/b. For this latter one, deviations from a linear weighting rule are closely connected to the excess free energy at infinite pressure. Thus any mixing rule gives a model for the excess free energy, or any accepted models for this property can be used as mixing rules.From the above, an empirical polynomial mixing rule is used for data smoothing and evaluation, while for practical work a local composition model is used. The mixing rule thus obtained can be reduced to the classical quadratic rule for some easily predicted values of the interaction energies. For highly polar systems, it includes three adjustable parameters. Using literature data, the new mixing rule is applied, in the low and high pressure range, to binary mixtures with one or two polar compounds, giving good data correlation and sometimes avoiding false liquid-liquid immiscibility.     

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.     

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.     

An improved estimation of water-organic liquid interfacial tension based on linear solvation energy relationship approach

An equation based on the linear solvation energy relationship (LSER) was proposed to predict the interfacial tension between organic liquid and water. The equation takes into account five parameters characterizing properties of the organic liquid molecule: excess molar refraction, solute dipolarity/polarizability, effective hydrogen bond acidity, effective hydrogen bond basicity, and the McGowan molar intrinsic volume. The proposed equation provides a better approximation of the interfacial tension than a similar one derived earlier by Freitas et al. (J. Phys. Chem. B 101 (1997, 7488-7493), which is based on seven terms.     

The prediction of isothermal phase equilibria for non-ideal multicomponent mixtures from heats of mixing

A method is presented for predicting both vapor—liquid and liquid—liquid equilibria for multicomponent mixtures using heat of mixing data for the constituent binary pairs together with pure component vapor pressures. Its application to two highly non-ideal hydrocarbon ternary systems is discussed. The parameters of the hybrid local composition model of Renon and Prausnitz, known as the NRTL equation, were evaluated from heat of mixing data for the three binary pairs in each of the two ternary systems. The parameters thus obtained were used in the multicomponent form of the NRTL equation to predict the ternary vapor—liquid equilibrium data for the completely miscible system cyclohexane(1)—n-heptane(2)—touluene(3) and for the partially miscible system acetonitrile(1)—benzene(2)—n-heptane(3) without the need for any ternary or higher order parameters.This method predicted compositions of the single phase region of the partially miscible ternary system with a standard deviation of 10%. It also predicted compositions for the fully miscible system with a standard deviation of 4.6%. Total pressure curves for the partially miscible and miscible ternaries were predicted with standard deviations of 6.6% and 4.5% respectively. Poor predictions of the binodal curve for the partially miscible region were obtained. The method offers a means of predicting the whole range of ternary phase equilibria for miscible systems.     

On phase equilibria,interfacial tension and phase growth in ternary polymer blends

A gradient squared free energy functional of the Landau-Ginzburg type is combined with Flory-Huggins theory to calculate minimum domain sizes, concentration profiles and interfacial tensions in ternary polymer blends. The dynamic equations governing spinodal decomposition are linearized to show that the minimum size for growth is identical to the thermodynamic minimum on phase volume. It is shown that unseparated, third components are enriched at the interface, reduce interfacial tension, increase stability and increase the minimum domain sizes. Enrichment of the third component at the interface causes concentrations at the major components to lie outside their binodal limits at a distance from the interface. Although the effects are most pronounced when the third component is a compatibilizer, the general phenomena remain true even when the third component is relatively incompatible. Generalizations to blends of N components are presented, and a robust method for calculating multicomponent phase diagrams is described.     

Effect of combining rules for cubic equations of state on the prediction of double retrograde vaporization

In this work, the phenomenon of double retrograde vaporization (DRV) is simulated using the Peng–Robinson equation of state with the classical mixing rules and several combining rules for the cross-energy and cross-co-volume parameters. The binary interaction parameters are set equal to zero in all cases, i.e., the calculations are entirely predictive. An interesting conclusion is that the predictions using the classical combining rules (geometric mean rule for a_ij and arithmetic mean rule for b_ij) provide the best agreement with the experimental data for all the systems tested: methane + n-butane, methane + n-pentane, ethane + limonene, and ethane + linalool. Another interesting observation is that several combining rules for b_ij, other than the arithmetic mean rule, predict the existence of three phases in equilibrium in a very narrow temperature range close to the critical temperature of methane in the methane + n-pentane system, even though, literature data indicates that n-hexane is the first n-alkane to present partial liquid phase immiscibility with methane.     

The prediction of isobaric phase equilibria for non-ideal multicomponent mixtures from heat of mixing data

A method for predicting isobaric binary and ternary vapor—liquid equilibrium data using only isothermal binary heat of mixing data and pure component vapor pressure data is presented. Three binary and two ternary hydrocarbon liquid mixtures were studied. The method consists of evaluating the parameters of the NRTL equation from isothermal heat of mixing data for the constituent binary pairs. These parameters are then used in the multicomponent NRTL equation to compute isobaric vapor—liquid equilibrium data for the ternary mixture. No ternary or higher order interaction terms are needed in the ternary calculations because of the nature of the NRTL equation. NRTL parameters derived from heat of mixing data at one temperature can be used to predict vapor—liquid equilibrium data at other temperatures up to the boiling temperature of the liquid mixture.For the systems studied this method predicted the composition of the vapor phase with a standard deviation ranging from 1–8% for the binary systems and from 4–12% for the ternary systems.     

Effects of ion solvation on phase equilibrium and interfacial tension of liquid mixtures

We study the bulk thermodynamics and interfacial properties of electrolyte solution mixtures by accounting for electrostatic interaction, ion solvation, and inhomogeneity in the dielectric medium in the mean-field framework. Difference in the solvation energy between the cations and anions is shown to give rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The ion solvation affects the phase equilibrium of the solvent mixture, depending on the dielectric constants of the solvents, reflecting the competition between the solvation energy and translation entropy of the ions. Miscibility is decreased if both solvents have low dielectric constants and is enhanced if both solvents have high dielectric constant. At the mean-field level, the ion distribution near the interface is determined by two competing effects: accumulation in the electrostatic double layer and depletion in a diffuse interface. The interfacial tension shows a nonmonotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations and decreases approximately as the square root of the salt concentration for dilute solutions, reaching a minimum near 1 mM. We also find that, for a fixed cation type, the interfacial tension decreases as the size of anion increases. These results offer qualitative explanations within one unified framework for the long-known concentration and ion size effects on the interfacial tension of electrolyte solutions.     

Solid and liquid phase equilibria and solid-hydrate formation in binary mixtures of water with amines

Solid and liquid phase diagrams have been constructed for {water+triethylamine,or+N,N-dimethylformamide(DMF) or+N,N-dimethlacetamide (DMA)} Solid-hydrates form with the empirical formulae N(C2H5)3 3H2O,DMF 3H2O,DMF 2H2O,DMA 3H2O and (DMA)2 3H2O.All are congruently melting except the first which melts incongruently.The solid-hydrate formation is attributed to hydrogen bond.The results are compared with the references     

A combined determination of phase diagrams of asymmetric binary mixtures by equations of state and transitiometry

Transitiometric investigations of the pure compounds tetracosane and anhydrous caffeine as well as of the mixtures (methane+tetracosane$\text{methane}+\text{tetracosane}$) and (carbon dioxide+caffeine$\text{carbon dioxide}+\text{caffeine}$) are reported for pressures up to 180 MPa. The results are compared with calculations from equations of state; the selection of reference data and the fitting of parameters is explicitly discussed. It is demonstrated how the calculations can aid the interpretation of transitiometric signals, and how the combination of transitiometry and thermodynamic modelling can be used to construct and understand high-pressure phase diagrams of asymmetric mixtures.     

Effect of three-body interactions on the vapor-liquid phase equilibria of binary fluid mixtures

Gibbs-Duhem Monte Carlo simulations are reported for the vapor-liquid phase coexistence of binary argon+krypton mixtures at different temperatures. The calculations employ accurate two-body potentials in addition to contributions from three-body dispersion interactions resulting from third-order triple-dipole interactions. A comparison is made with experiment that illustrates the role of three-body interactions on the phase envelope. In all cases the simulations represent genuine predictions with input parameters obtained independently from sources other than phase equilibria data. Two-body interactions alone are insufficient to adequately describe vapor-liquid coexistence. In contrast, the addition of three-body interactions results in very good agreement with experiment. In addition to the exact calculation of three-body interactions, calculations are reported with an approximate formula for three-body interactions, which also yields good results.     

Analytical applications of liquid-liquid phase equilibria: analysis of binary mixtures of chemically similar components

A new, simple and rapid method based on the principle of liquid-liquid phase equilibria has been developed for the analysis of binary mixtures of chemically similar organic compounds. The method does not require elaborate instrumentation and can be used to analyse mixtures of members of homologous series. The application of the method has been illustrated by analysing binary mixtures of n-hexane and n-octane; the maximum uncertainty in this analysis is ~2%.     

Phase equilibria in binary mixtures of ethane and hexadecane

This paper reports experimental results of a study of the phase behaviour of binary mixtures of ethane + hexadecane. In the near-critical region of ethane liquid + vapour and solid hexadecane + liquid two-phase boundaries have been measured. Also the three-phase equilibrium solid hexadecane + liquid + vapour has been determined experimentally. The experimental data cover the complete mole fraction range. Pressures up to 18 MPa were applied and the investigation was performed in a temperature region from about 260 K up to 450 K.     

Chemical potentials and phase equilibria of Lennard-Jones mixtures: a self-consistent integral equation approach

We explore the vapor-liquid phase behavior of binary mixtures of Lennard-Jones-type molecules where one component is supercritical, given the system temperature. We apply the self-consistency approach to the Ornstein-Zernike integral equations to obtain the correlation functions. The consistency checks include not only thermodynamic consistencies (pressure consistency and Gibbs-Duhem consistency), but also pointwise consistencies, such as the zero-separation theorems on the cavity functions. The consistencies are enforced via the bridge functions in the closure which contain adjustable parameters. The full solution requires the values of not only the monomer chemical potentials, but also the dimer chemical potentials present in the zero-separation theorems. These are evaluated by the direct chemical-potential formula [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] that does not require temperature nor density integration. In order to assess the integral equation accuracy, molecular-dynamics simulations are carried out alongside the states studied. The integral equation results compare well with simulation data. In phase calculations, it is important to have pressure consistency and valid chemical potentials, since the matching of phase boundaries requires the equality of the pressures and chemical potentials of both the liquid and vapor phases. The mixtures studied are methane-type and pentane-type molecules, both characterized by effective Lennard-Jones potentials. Calculations on one isotherm show that the integral equation approach yields valid answers as compared with the experimental data of Sage and Lacey. To study vapor-liquid phase behavior, it is necessary to use consistent theories; any inconsistencies, especially in pressure, will vitiate the phase boundary calculations.     

Examining the effect of binary interaction parameters on VLE modelling using cubic equations of state

Vapor-liquid equilibrium (VLE) data are important in the optimization of thermodynamic cycles. As energy concerns continue to grow, improving the efficiencies of power and refrigeration cycles is increasingly important. Numerical simulations using empirical equations of state provide an excellent alternative to time consuming experimental measurement of VLE data. However, it is important to understand the limitations of using correlative equations for data prediction. In this study, a water-ethanol mixture is simulated with various VLE models. Non-optimal binary interaction parameters are considered and model accuracy is evaluated in terms of average absolute percent deviation (%AAD) between simulated and experimental bubble and dew point pressures. For this system, it is found that as the correlative accuracy of a model increases, the predictive ability decreases. Specifically, the temperature dependence of the binary interaction parameters is shown to be an important consideration for the water-ethanol system when more complex combining rules are implemented.     

Calculation and prediction of fluid phase equilibria from an equation of state

Liquids or compressed gases consisting of light molecules show deviations from classical mechanics, which are caused by the discontinuity of energy levels. From the assumption that each molecule is confined to a cell with a size depending on the free volume, a quantum correction is derived which extends any van der Waals type equation of state to quantum gases. The correction is applied to a semiempirical equation of state developed by the author. The extended equation yields reasonable critical compressibility factors and gives a better representation of PVT data than the uncorrected equation. Furthermore high pressure phase equilibria in mixtures containing helium and hydrogen have been calculated. Again the agreement with experimental data is improved; the adjustable binary interaction parameters have values close to the Berthelot-Lorentz rules and are less temperature dependent.     

表面活性剂与油相动态界面张力影响因素研究

对不同类型表面活性剂烷基糖苷(APG1214)、咪唑啉(IAS)、十二烷基苯磺酸钠(SDBS)、烷基酚醚羧酸盐(ss-231)的油水动态界面张力进行了研究。在60℃,5 000 r·min-1条件下,考察了表面活性剂的浓度、表面活性剂的结构、正构烷烃碳数以及原油中活性物质对形成低界面张力影响。实验结果表明:表面活性剂亲水基的亲水性越强,亲水基之间排斥力越小,使得在油水界面排布的密度越大,降低界面张力的效果会更好;当表面活性剂疏水碳链与烷烃碳链相似时,降低界面张力的效果会更明显;无碱体系中原油中的活性物质可在油水界面上形成粘弹性界面膜,这种界面膜的形成减少了表面活性剂分子在界面的吸附,使界面张力升高。