Liquid-liquid equilibria and theta temperatures in homopolymer-solvent solutions from a perturbed hard-sphere-chain equation of state

The perturbed hard-sphere-chain (PHSC) equation of state is used to calculate liquid-liquid equilibria of binary nonpolar solvent/homopolymer systems exhibiting both an upper critical solution temperature (UCST) and a lower critical solution temperature (LCST). Systems studied include polyisobutylene, polyethylene, and polystyrene solutions. Equation-of-state parameters of homopolymers are obtained by regressing the pressure-volume-temperature data of polymer melts. In polymer solutions, however, theory overestimates the equation-of-state effect which causes the LCST at elevated temperature. To correct the overestimated equation-of-state effect, an empirical adjustable parameter is introduced into the perturbation term of the PHSC equation of state. An entropy parameter is also introduced into the Helmholtz energy of the mixture to correlate quantitatively the dependence of critical temperatures on polymer molecular weight. For systems exhibiting a LCST, two adjustable parameters are required to obtain quantitative agreement of theoretical critical temperatures with experiment as a function of polymer molecular weight. For systems exhibiting both an UCST and a LCST, three adjustable parameters may be necessary. The need for so many empirical binary parameters is probably due to the oversimplified perturbation term which is based on the mean-field assumption. © 1996 John Wiley & Sons, Inc.     

Equation for the coexistence curve of alkanes near critical temperature,based on the van der Waals Model

According to the fluctuation theory of phase transitions, a real liquid near the critical point is an ideal gas of the fluctuations of the order parameter, the size of which is determined by the correlation length of the system. We deduce the extended equation of state of liquids near the critical temperature by including the properties of the real van der Waals gas in this model, i.e., taking into account the own volume of the fluctuations of the order parameter and the interaction forces between them. We use this equation to analyze the temperature dependence of the density of a series of alkanes (C _n H_{2n + 2}, n = 1 − 12) along the line of the liquid-gas equilibrium near their critical temperatures. We show that the parameters of this extended equation of the state of substance are linear functions of the compressibility factor of alkanes.     

Effect of electrolytes on the critical temperature of separation and physicochemical properties of binary liquid systems

The effect ions have on the equilibrium and kinetic properties of solutions near the critical temperature of separation is studied. From an analysis of the experimental data obtained in the work and from the literature it is shown that adding ions to a solution increases the correlation length of the system and changes the magnitude of the fluctuation part of the thermodynamic potential and the forces of interaction among the order parameter fluctuations in the vicinity of the critical point. This changes the parameters of the extended equation of state, increases the viscosity of substances, and lowers the coefficient of volume expansion of the system.     

Modification of the predictive capability of the PRSV-2 equation of state for critical volumes of binary mixtures

The predictive capability of the Peng–Robinson–Stryjek–Vera (PRSV-2) equation of state for critical properties of binary mixtures showing continuous critical lines has been investigated. The procedure adopted by Heidemann and Khalil and discussed by Abu-Eishah et al., in a previous paper, has been followed. The effect of using the pure-component parameters of the PRSV-2 equation of state (κ₁, κ₂ and κ₃), new values of κ₁, revised values of κ₁, or giving zero values for these parameters have been investigated. The effect of using zero values or optimized values for the binary interaction parameter on the PRSV-2 predictive capability of critical properties have also been investigated. The standard and the average of the absolute relative deviations in critical properties are included. The predicted critical temperature and pressure for the 20 nonpolar and 18 polar systems studied here agree well with experimental data, and are always better than those predicted by the group-contribution method. A correction has been introduced here to the critical volume predicted by PRSV-2 equation of state that makes the average deviations between the predicted and experimental values very close to or even better than those predicted by the group-contribution method.     

A new approach in modelling phase equilibria and gas solubility in electrolyte solutions and its applications to gas hydrates

This paper presents a new predictive model for phase equilibria and gas solubility calculations in the presence of electrolyte solutions. It treats salts as pseudo-components in an equation of state (EoS) by defining the critical properties and acentric factor for each salt. The water–salt, gas–salt and salt–salt binary interaction parameters (BIP) have been determined by using available experimental data on freezing point depression and boiling point elevation as well as gas solubility and salt solubility data in saline solutions.The methodology has been applied in modelling sodium chloride, potassium chloride and their mixtures, as well as solubility of methane and carbon dioxide in aqueous single and mixed electrolyte solutions.The developed model is capable of accurately predicting the phase behaviour, gas hydrate stability zone and potential salt precipitation in single and mixed electrolyte solutions. The model predictions are compared with available independent experimental data, including hydrate inhibition characteristics of single and mixed electrolyte solutions, and good agreement is demonstrated.     

Application of the perturbed Lennard–Jones chain equation of state to solute solubility in supercritical carbon dioxide

The perturbed Lennard–Jones chain (PLJC) equation of state is a thermodynamic model based on the perturbation theory of liquid state. This equation has been shown in the past to be a successful model for phase equilibria calculations of binary and ternary fluid mixtures and polymer solutions. In this work, we employed for the first time the PLJC equation to model the solubility of 39 solids in supercritical carbon dioxide. It was shown that the model achieves good correlation with three temperature independent parameters. A comparison of the PLJC with the commonly used Peng–Robinson equation reveals the PLJC equation gives better correlation to the solubility data than the Peng–Robinson model that utilizes temperature dependent parameters.     

PSRK: A Group Contribution Equation of State Based on UNIFAC

A group contribution equation of state called PSRK (Predictive Soave-Redlich-Kwong) which is based on the Soave-Redlich-Kwong equation (Soave, 1972) has been developed. It uses the UNIFAC method to calculate the mixture parameter a and includes all already existing UNIFAC parameters. This concept makes use of recent developments by Michelsen (1990b) and has the main advantage, that vapor-uquid-equilibria (VLB) can be predicted for a large number of systems without introducing new model parameters that must be fitted to experimental VLB-data. The PSRK equation of state can be used for VLB-predictions over a much larger temperature and pressure range than the UNIFAC γ--approach and is easily extended to mixtures containing supercritical compounds. Additional PSRK parameters, which allow the calculation of gas/gas and gas/alkane phase equilibria, are given in this paper. In addition to those mixtures covered by UNIFAC, phase equilibrium calculations may also include gases like CH₄ C₂H₆, C₃H₆, c₄H₁₀, CO₂, N₂, H₂ and CO.     

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.     

Temperature Dependence of Adsorption of Polymer Mixtures from Solutions

Adsorption from binary (poly(butyl methacrylate)-CCl(4), poly-styrene-CCl(4)) and ternary (poly(butyl methacrylate)-polystyrene-CCl(4)) solutions has been studied at 10, 25, and 60 degrees C. It was found that with increasing temperature the values of adsorption grew due to worsening of the thermodynamic quality of the common solvent. Worsening of the quality of the solvent leads to a decrease in size of the macromolecular coils and to an increase in the critical concentrations of overlapping macromolecules in solution. As a result, the state of macromolecules in solution depends on temperature and determines adsorption values. From the temperature dependence of adsorption using the Clapeyron-Clausius equation, the differential enthalpy of adsorbate (polymer) DeltaH was calculated for each polymer, by adsorption both from binary solution and from the mixture. Determining DeltaH values from the temperature dependence of adsorption allows us to find this value simultaneously for each polymer in the polymer mixture. It was established that transition of a polymer from solution onto the surface leads to an increase in its enthalpy in the case of adsorption from both binary and ternary solutions. DeltaH increases with increasing coverage of the surface. By transition from solution onto the surface, the enthalpy of the adsorbed polymer increases; i.e., the polymer transits in an energetically less favorable state. This effect is more pronounced for adsorption from the mixture, which may be connected with the presence of the second polymer in the adsorption layer. Copyright 2000 Academic Press.     

Single and Mixed Gas Adsorption Equilibria of Carbon Dioxide/Methane on Activated Carbon

Single gas adsorption isotherms of methane and carbon dioxide on micro-porous Norit RB1 activated carbon were determined in a gravimetric analyser in the temperature range of 292 to 349 K and pressures to 0.8 Mpa. Furthermore binary isotherms of carbon dioxide and methane mixtures were determined at 292 K and pressures up to 0.65 MPa. Adsorbed phase compositions were determined from the gravimetric data by the rigorous thermodynamic method of Van Ness.These experimental binary equilibrium data were compared with equilibrium data calculated by the Ideal Adsorbed Solution (IAS) model. Only moderate agreement could be obtained.Finally, activity coefficients, accounting for the non-ideality of the adsorbate mixture, were calculated from the experimental data. The Wilson equation, derived for bulk solutions, was fitted on these activity data and the Wilson interaction parameters were determined. The Wilson equation proved to correlate the experimental data reasonably. However, the Wilson interaction parameters are not only completely different from those found for bulk solutions, but also the physical interpretation of these parameter values is completely lacking.It is concluded that new solution models should be developed encompassing both non-ideal solution behaviour and surface heterogeneity.     

A perturbed hard-sphere-chain equation of state for mixtures: Prediction from critical point constants

A modified perturbed hard-sphere-chain equation of state by Eslami [H. Eslami, Fluid Phase Equilibr. 216 (2004) 21-26], is extended to mixtures. The resulting equation of state for mixtures consists of two temperature-dependent parameters as well as an additional parameter, reflecting the segment size for pure components. The temperature-dependent parameters of the equation of state are correlated as universal functions of the reduced temperature. It is shown that knowing just the critical constants of pure components is sufficient to calculate the temperature-dependent parameters. The equation of state for mixtures is checked against the experimental pressure-volume-temperature data for a large number of mixtures, having varieties of molecular sizes and shapes. It is shown that no interaction parameter is needed to describe the behavior of fluid mixtures. Among about 3500 data points for mixtures, the average absolute deviation, compared to the experimental data, is about 0.93%.     

变阱宽方阱链流体状态方程应用于CO2-物理吸收溶剂系统

采用变阱宽方阱链流体（SWCF-VR）状态方程关联了CO2在几种常规物理吸收溶剂中的溶解度数据,得到了二元交互作用参数,建立了二元交互作用参数与温度的关联式．结果表明,采用一个二元交互作用参数,SWCF．VR方程均能很好地描述CO2在常规流体中的溶解度,尤其能满意再现甲醇低温洗工艺中CO2-甲醇的相行为．利用建立的二元交互作用参数与温度的关联式,可将SWCF-VR状态方程拓展应用于预测二元系统气液两相的密度以及CO2-物理吸收溶剂多元系统的气液平衡．     

A new cubic equation of state for simple fluids: pure and mixture

A two-parameter cubic equation of state is developed. Both parameters are taken temperature dependent. Methods are also suggested to calculate the attraction parameter and the co-volume parameter of this new equation of state. For calculating the thermodynamic properties of a pure compound, this equation of state requires the critical temperature, the critical pressure and the Pitzer’s acentric factor of the component. Using this equation of state, the vapor pressure of pure compounds, especially near the critical point, and the bubble point pressure of binary mixtures are calculated accurately. The saturated liquid density of pure compounds and binary mixtures are also calculated quite accurately. The average of absolute deviations of the predicted vapor pressure, vapor volume and saturated liquid density of pure compounds are 1.18, 1.77 and 2.42%, respectively. Comparisons with other cubic equations of state for predicting some thermodynamic properties including second virial coefficients and thermal properties are given. Moreover, the capability of this equation of state for predicting the molar heat capacity of gases at constant pressure and the sound velocity in gases are also illustrated.     

Equation of state for square-well chain molecules with variable range II. Extension to mixtures

An equation of state (EOS) developed in our previous work for square-well chain molecules with variable range is further extended to the mixtures of non-associating fluids. The volumetric properties of binary mixtures for small molecules as well as polymer blends can well be predicted without using adjustable parameter. With one temperature-independent binary interaction parameter, satisfactory correlations for experimental vapor–liquid equilibria (VLE) data of binary normal fluid mixtures at low and elevated pressures are obtained. In addition, VLE of n-alkane mixtures and nitrogen + n-alkane mixtures at high pressures are well predicted using this EOS. The phase behavior calculations on polymer mixture solutions are also investigated using one-fluid mixing rule. The equilibrium pressure and solubility of gas in polymer are evaluated with a single adjustable parameter and good results are obtained. The calculated results for gas + polymer systems are compared with those from other equations of state.     

Extended equation for the coexistence curve of molecular liquids in the vicinity of a critical point

An extended equation of state for substances along a phase interface, based on the Van der Waals model of a gas with fluctuations of the order parameter is confirmed by experimental data along the coexistence curve of a wide class of homogeneous and inhomogeneous molecular liquids under the Earth’s field of gravity. It is shown that the parameters of the extended equation for the coexistence curve of homogeneous and spatially inhomogeneous molecular liquids are linear functions of the compressibility factor. This allows us to predict the parameters of the equations of state of molecular liquids that are difficult to investigate experimentally.     

Phase separation of polymer solutions. The calculation of the cloudpoint curve with a concentration and temperature-dependent free energy correction parameter

The free enthalpy correction parameter g in the Flory-Huggins equation for the Gibbs free enthalpy of mixing in polymer solutions is considered generally as a concentration- and temperature-independent parameter. It has been extended here with linear concentration- and temperature-dependent terms. With these parameters, six different types of cloudpoint curves can be predicted. Using the experimental cloudpoint curve for solutions of poly(2,6-dimethyl-1,4-phenylene oxide) in toluene up to about 70 per cent by weight of polymer, a set of g-parameters is obtained, accounting for concentration and temperature dependence. With the parameters thus obtained, the melting point curve has been calculated which agrees very well with the experimental melting points for this system.     

Vapor–liquid equilibrium in the n-butane + methanol system,measurement and modeling from 323.2 to 443.2 K

New experimental vapor–liquid equilibrium (VLE) data for the n-butane + methanol binary system are reported over a wide temperature range from 323.2 to 443.2 K and pressures up to 5.4 MPa. A static–analytic apparatus, taking advantage of two pneumatic capillary samplers, was used. The phase equilibrium data generated in this work are in relatively good agreement with previous data reported in the literature. Three different thermodynamic models have been used to represent the new experimental data. The first model is the cubic-based Peng–Robinson equation of state (EoS) combined with the Wong–Sandler mixing rules. The two other models are the non-cubic SAFT-VR and PC-SAFT equations of state. Temperature-dependent binary interaction parameters have been adjusted to the new data. The three models accurately represent the new experimental data, but deviations are seen to increase at low temperature. A similar evolution of the binary parameters with respect to temperature is observed for the three models. In particular a discontinuity is observed for the k_ij values at temperatures close to the critical point of butane, indicating the effects of fluctuations on the phase equilibria close to critical points.     

A semi-empirical equation of state applied to vapor-liquid equilibria calculations

A five-parameter equation of state is proposed to calculate the vapor-liquid equilibria of compounds in binary and multicomponent mixtures. This equation is closely related to a previous equation of state proposed by the author, the main modification being in the entropic term where the parameter m assumes a constant value for all compounds. It is shown that the van der Waals conditions at the critical point and the Morbidelli-Carra' algorithm enable the calculation of three other constants. Rules are given to calculate the remaining constant K which pertains to the enthalpic term. The proposed method only requires knowledge of the critical constants and of the normal boiling temperature as input parameters. A wide application of the new equation to both polar and non-polar binary systems indicates the following: the proposed method is predictive for ideal or nearly ideal mixtures; the correlation of mixtures of hydrocarbons having very different molar volumes can be obtained by optimizing only the binary interaction parameter linked to the enthalpic term; the new equation also correlates well with strongly non-ideal systems which exhibit a miscibility gap; the prediction of multicomponent vapor-liquid equilibria from the binary data alone is also reliable for both polar and non-polar mixtures.     

An equation of state for hydrochloric acid solutions

Gibbons, R.M. and Laughton, A.P., 1984. An equation of state for hydrochloric acid solutions. Fluid Phase Equilibria, 18:61–68.A modified Redlich-Kwong-Soave (RKS) equation is presented for hydrochloric acid solutions. It is shown that the equation with one binary parameter predicts the azeotropic behaviour of the system accurately and the heats of solution within 10%. These results suggest that the new equation can be applied to a wide range of polar systems.