A Helmholtz energy equation of state for calculating the thermodynamic properties of fluid mixtures

A new approach has been developed for calculating the properties of mixtures based on an equation of state explicit in reduced Helmholtz energy. This approach allows for the representation of the thermodynamic properties over a wide range of fluid states and is based on highly accurate equations of state for the pure components combined at the reduced temperature and density of the mixture. The reducing parameters used for temperature and density depend on composition. For simple mixtures (those that closely follow Raoult's law), a very accurate representation of all thermodynamic properties has been achieved with relatively simple functions. For nonideal mixtures, the reducing functions for density and temperature were modified, and a departure function was added to the equation of state. Generally, the model is able to represent liquid and vapor states with uncertainties of 0.1% in density, 1% in heat capacities and 1% in bubble point pressures if experimental data of comparable uncertainties exist. Two applications of the mixture model concepts were developed independently by the authors in the United States and Germany over the same time period. These applications include the development of individual equations for each binary system and a generalization of the model which is valid for a wide variety of mixtures. The individual approaches are presented with an explanation of the similarities and differences. Although the paper focuses mainly on binary systems, some results for ternary mixtures are also presented.     

Density calculation using GMA equation of state considering mixing and combining rules for some liquid mixtures

The density of nine binary and two ternary liquid mixtures at different temperatures, pressures, and compositions has been calculated using a new equation of state considering mean geometry approximation “MGA”. Although the studied mixtures cover the vast variety of mixtures including the inert gases, polar, nonpolar, refrigerant, and strongly hydrogen-bonded systems, the results in prediction of density show good agreement with experiment. The excellent results have been obtained whenever the size and the strength of intermolecular forces of components in a mixture are very similar. Our results show that the effect of size is more important than that of the strength of intermolecular forces. Since the strength of hydrogen bonding in the system of water/methanol is very high, the agreement of calculated densities with the corresponding experimental values is interesting for which the average absolute deviations are better than 1%. To show the ability of this equation of state in prediction of density, the calculated densities of some liquid mixtures have been compared with those of computed from other equations of state.     

Vapor–liquid equilibrium measurement and modeling for the difluoromethane + pentafluoroethane + propane ternary mixture

Vapor–liquid equilibrium data for the difluoromethane (R32) + pentafluoroethane (R125) + propane (R290) ternary mixture were measured at 5 isotherms between 263.15 K and 323.15 K. The measurement was carried out using a circulation-type apparatus recently developed, which was validated with binary mixtures. With binary interaction parameters obtained for the three corresponding binary mixtures, VLE modeling and prediction were performed for the ternary mixture using the Peng–Robinson equation of state with the classical mixing rules and MHV1 mixing rules. Hou's group contribution model for VLE of new refrigerant mixtures was further tested with the experimental data for the ternary system. The predicted pressure and vapor phase composition were compared with experimental ones.     

Thermodynamic consistency of high pressure ternary mixtures containing a compressed gas and solid solutes of different complexity

A thermodynamic consistency test applicable to high pressure binary gas–solid mixtures is extended to ternary mixtures containing a compressed gas and two solid solutes. A high pressure mixture containing carbon dioxide as solvent and two chemically similar solutes (2,3 dimethylnaphthalene and 2,6 dimethylnaphthalene) and a high pressure mixture containing carbon dioxide as solvent and two chemically different solutes (capsaicin and β-carotene), are considered in the study. Several sets of isothermal solubility data for binary and ternary mixtures are considered in the study. The Peng–Robinson equation of state with the mixing rules of Wong and Sandler have been employed for modeling the solubility of the solid in the case of binary mixtures, while the classical van der Waals mixing rules were used for modeling the ternary mixtures containing two solid solutes. Then the proposed thermodynamic consistency test has been applied. The results show that the thermodynamic test for ternary mixtures can be applied with confidence determining consistency or inconsistency of the experimental data used.     

用二元交互作用函数预测高压下多组分体系的气液平衡

用和体系的状态有关且满足不变性条件的二元交互作用函数,结合F函数修改的立方状态--方程FRKS方程,预测高压下多组分体系的气液平衡.选择15个三元体系及其组分二元系来检验方法的可行性,这些体系覆盖了从简单的接近理想溶液行为的体系到高度非理想体系.计算结果表明,该方法不仅能相当精确地关联各种类型二元系的气液平衡,而且能在仅用组分二元系参数的条件下较准确地预测所考察的所有三元体系的气液平衡     

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.     

A modified clausius equation of state for calculation of multicomponent refrigerant vapor-liquid equilibria

Gow, A.S., 1993. A modified Clausius equation of state for calculation of multicomponent refrigerant vapor-liquid equilibria. Fluid Phase Equilibria, 90: 219-249.

A modified Clausius equation of state with a single temperature dependent energy-volume parameter a(T) in the attractive term was designed to describe the vapor pressure vs. temperature relationship of 39 pure refrigerant fluids including elementary cryogenic materials (e.g. He, Ar, N₂, CO₂, CH₄, etc.), chlorofluorocarbons (CFCs), hydrofluorocarbons (HFCs), hydrochlorofluorocarbons (HCFCs), fluorocarbons (FCs), and various other simple cryogenic compounds. The equation developed represents the vapor-liquid coexistence dome, and the superheated vapor compressibility factor and enthalpy for pure refrigerants.

The vapor-liquid equilibrium for refrigerant mixtures is calculated using a “phi-phi” method with “one fluid” van der Waals mixing and combining rules for the equation of state parameters a_M(T), b_M and c_M. A single interaction constant k₁₂ is used to describe non-ideal behavior of each binary. The binary interaction constant, which is a strong function of temperature, and the sign of which signifies the type of deviations from Raoult's law, is obtained by correlating experimental bubble point data for isothermal binary refrigerant mixtures. The proposed equation of state generally describes binary P-x,y data more accurately the higher the temperature for a given system. The method presented is extended to predict vapor-liquid equilibria for the R14-R23-R13 ternary system at 198.75 K using binary interaction constants at this temperature for the three binaries involved.     

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.     

Isothermal vapor–liquid equilibria for mixtures composed of 1,2-dimethoxybenzene, 2-methoxyphenol,and diphenylmethane

Diphenylmethane was found to be a potential entrainer for separating the closely boiling mixtures of 2-methoxyphenol+1,2-dimethoxybenzene via extractive distillation. To gain insight into the capability of this auxiliary agent, isothermal vapor–liquid equilibrium data were measured for the binary and the ternary mixtures containing 2-methoxyphenol, 1,2-dimethoxybenzene, and diphenylmethane at temperatures from 433.15 to 463.15 K. All the binary data passed thermodynamic consistency tests. However, there exhibits a large discrepancy between the experimental values and the predicted results from the UNIFAC model. The new data were correlated with the Wilson, the NRTL, and the UNIQUAC models, respectively. The model parameters determined from the binary data were applied to predict the phase equilibrium behavior of the ternary system.     

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.     

Computing all the azeotropes in refrigerant mixtures through equations of state

Azeotropic mixtures of fluorocarbon (FC) and hydro fluorocarbon (HFC) with hydrocarbons are gaining popularity as drop-in substitutes for CFCs and HCFCs. A method to compute all the azeotropes in a refrigerant mixture through the equation of state approach is described. The method allows prediction of all the azeotropes in a refrigerant mixture and is in close agreement with the experimental data. Both the vapor and the liquid phase non-idealities are incorporated through fugacity coefficients modeled using Peng–Robinson–Stryjek–Vera equation of state with Wong-Sandler and van der Waals mixing rules. Homotopy continuation based methodology guarantees computation of all the solutions of necessary and sufficient condition of azeotropy in multicomponent refrigerant mixtures. The method establishes the pressure dependency of azeotropic composition allowing prediction of bifurcation pressure where refrigerant azeotropes may appear or disappear and predicts azeotropes at elevated pressures. The approach is independent of equation of state and mixing rules but rely on their ability to represent the phase behavior. The approach is tested with R23–R13, propane–R227ea binary mixtures and a ternary mixture of R32–R125–R143a.     

A three-parameter cubic equation of state for prediction of thermodynamic properties of fluids

A new cubic three-parameter equation of state has been proposed for PVT and VLE calculations of simple, high polar and associating fluids. The parameters are temperature dependent in sub-critical region, but temperature independent in super-critical region. The results for 42 simple and 14 associative pure compounds indicate that the calculated saturation properties and volumetric properties over the whole temperature range, up to high pressures, by the proposed equation of state (EOS), were in better agreement with the experimental data, compared with those obtained by the five well-known EOSs (P–R, P–T, Adachi et al., Yu–Lu, and M4). Two derivative properties, molar enthalpy and heat capacity of water and ammonia have been calculated, and demonstrated the thermodynamic consistency of the EOS parameters. Also VLE calculations have been performed for 41 binary mixtures of different type of fluids, including those of interest in petroleum industry. The results indicated the high capability of the proposed EOS for calculating the thermodynamic properties of pure and fluid mixtures.     

A new two-parameter cubic equation of state for predicting phase behavior of pure compounds and mixtures

In this work, a new two-parameter cubic equation of state is presented based on perturbation theory for predicting phase behavior of pure compounds and of hydrocarbons and non-hydrocarbons. The parameters of the new cubic equation of state are obtained as functions of reduced temperature and acentric factor. The average deviations of the predicted vapor pressure, liquid density and vapor volume for 40 pure compounds are 1.116, 5.696 and 3.083%, respectively. Also the enthalpy and entropy of vaporization are calculated by using the new equation of state. The average deviations of the predicted enthalpy and entropy of vaporization are 2.393 and 2.358%, respectively. The capability of the proposed equation of state for predicting some other thermodynamic properties such as compressibility, second virial coefficient, sound velocity in gases and heat capacity of gases are given, too. The comparisons between the experimental data and the results of the new equation of state show the accuracy of the proposed equation with respect to commonly used equations of state, i.e. PR and SRK. The zeno line has been calculated using the new equation of state and the obtained result compared with quantities in the literatures. Bubble pressure and mole fraction of vapor for 16 binary mixtures are calculated. Averages deviations for bubble pressure and mole fraction of vapor are 9.380 and 2.735%, respectively.     

Modelling investigation on the thermal conductivity of pure liquid,vapour, and supercritical refrigerants and their mixtures by using Heyen EOS

This work presents a literature survey of the available data regarding the thermal conductivity of refrigerants. About 31 pure refrigerants that contain 7127 data points are selected for the temperature range of 91.35–580.00 K, a pressure range of (0.000111-500) bar, and thermal conductivity range of (0.007–0.27) W m^?1 K^?1 containing liquid, vapour, and supercritical phases. Seven binary and three ternary mixtures are also collected both in liquid and vapour phases with an overall of 803 data points. Based on the similarity between the pressure-volume-temperature and Tλ (thermal conductivity) P diagrams, the thermal conductivity model based on Heyen equation of state has been developed for pure refrigerants and their mixtures. The genetic algorithm is used to determine the adjustable parameters of the model. The calculation results prove that this proposed model can reproduce and predict thermal conductivity of refrigerants with good accuracy (overall AAD = 6.85% for pure compounds, AAD = 6.14% for binary mixtures and AAD = 9.32% for ternary mixtures).     

Isobaric vapor–liquid equilibrium for methyltrichlorosilane–dimethyldichlorosilane–benzene system

Isobaric vapor–liquid equilibrium (VLE) for the system methyltrichlorosilane–dimethyldichlorosilane–benzene and that of the three binary systems were measured with a new pump-ebulliometer at the pressure of 101.325 kPa. These binary compositions of the equilibrium vapor were calculated according to the Q function of molar excess Gibbs energy by the indirect method and the resulted VLE data agreed well with the thermodynamic consistency. Moreover, the experimental data were correlated with the Wilson, NRTL, Margules and van Laar equations by means of the least-squares fit, the acquired optimal interaction parameters were fitted to experimental vapor–liquid equilibrium data for binary systems. The binary parameters of Wilson equation were also used to calculate the bubble point temperature and the vapor phase composition for the ternary mixtures without any additional adjustment. The predicted vapor–liquid equilibrium for the ternary system accorded well with the experimental results. The separation factor of methyltrichlorosilane against dimethyldichlorosilane in benzene was also reported. The VLE of binary and multilateral systems provided essential theory for the production of the halogenated silane.     

The new experimental data and a new thermodynamic model based on group contribution for correlation liquid–liquid equilibria in aqueous two-phase systems of PEG and (K2HPO4 or Na2SO4)

The new experimental data of liquid–liquid equilibria for aqueous two-phase systems PEG–K₂HPO₄–water and PEG–Na₂SO₄–water are presented. The effects of pH and molecular weight of polyethylene glycol were investigated and the tie lines with binodal curves for both systems are shown. A new thermodynamic model based on group contribution has been proposed for studying the phase behavior of aqueous two-phase polymer–salt systems. The assumptions of NRTL-NRF model and the activity coefficient equation of UNIQUAC-NRF model have been used for the groups. In this new model, UNIFAC-NRF, the nonrandom state of groups were selected as a reference state. The binary interaction parameters were adjusted using the data of binary salt–water systems and the ternary systems were correlated with only six binary adjustable parameters. The Debye–Huckel equation based on Fowller–Guggenheim equation was used to calculate the long range electrostatic interaction of the ions. The UNIFAC-NRF model was applied to correlate the experimental data of aqueous two-phase systems: PEG–K₂HPO₄–water and PEG–Na₂SO₄–water for two different molecular weight of PEG at different pH. The results of the new model showed that it can be used to correlate the LLE in aqueous solution of polymer–salt very well.     

Bubble pressures and saturated liquid molar volumes of trichlorofluoromethane—chlorodifluoromethane mixtures. Representation of refrigerant-mixtures vapor—liquid equilibrium data by a modified form of the Peng—Robinson equation of state

Experimental vapor—liquid equilibrium data and saturated liquid molar volumes of chlorodifluoromethane—trichlorofluoromethane binary mixtures have been obtained at four temperatures (298.15, 323.15, 348.15 and 373.15 K) using apparatus described previously.The experimental vapor—liquid equilibria are represented well by a modified form of the Peng—Robinson equation of state with one interaction parameter, but the mean deviation between the calculated and experimental densities is 5%.Vapor—liquid data for binary refrigerant mixtures from the literature are treated using the modified form of the Peng—Robinson equation of state with one adjusted interaction parameter in the mixing rule for a. The representation is fair and is not improved by introducing an additional parameter in the mixing rule for b.     

Prediction of excess thermodynamic functions and activity coefficients of some polymeric liquid mixtures using a new equation of state

In this work, the excess thermodynamic properties, namely excess molar Gibbs energy, excess molar enthalpy, excess molar entropy, excess molar internal energy, and excess molar Helmholtz energy for four polymer mixtures and blends at different temperatures, pressures, and compositions have been calculated using the GMA equation of state. We have also calculated the activity coefficient for these polymeric mixtures using the GMA equation of state. The values of statistical parameters between experimental and calculated properties show the ability of this equation of state in reproducing and predicting the excess thermodynamic functions and activity coefficients for studied polymeric mixtures.     

Semi-ideal solution theory. 2. Extension to conductivity of mixed electrolyte solutions

Conductivities were measured for the ternary systems NaCl-LaCl(3)-H(2)O and KCl-CdCl(2)-H(2)O and their binary subsystems NaCl-H(2)O, KCl-H(2)O, CdCl(2)-H(2)O, and LaCl(3)-H(2)O at 298.15 K. The semi-ideal solution theory for thermodynamic properties of aqueous solutions of electrolyte mixtures was used together with the Eyring absolute rate theory to study conductivity of mixed electrolyte solutions. A novel simple equation for prediction of the conductivity of mixed electrolyte solutions in terms of the data of their binary solutions was established. The measured conductivities and those reported in literature were used to test the newly established equation and the generalized Young's rule for conductivity of mixed electrolyte solutions. The comparison results show that the deviation of a ternary solution from the new conductivity equation is closely related to its isopiestic behavior and that the deviations are often within experimental uncertainty if the examined system obeys the linear isopiestic relation. While larger deviations are found in the system with large ion pairing effect, the predictions can be considerably improved by using the parameters calculated from its isopiestic results. These results imply that the previous formulation of the thermodynamic properties of aqueous solutions of electrolyte mixtures has a counterpart for transport properties.     

Thermodynamic consistency of experimental VLE data for asymmetric binary mixtures at high pressures

A thermodynamic consistency of isothermal vapor–liquid equilibrium data for 9 non-polar and 8 polar binary asymmetric mixtures at high pressures has been evaluated. A method based on the isothermal Gibbs–Duhem equation was used for the test of thermodynamic consistency using a Φ–Φ approach. The Peng–Robinson equation of state coupled with the Wong–Sandler mixing rules were used for modeling the vapor–liquid equilibrium (VLE) within the thermodynamic consistency test. The VLE parameters calculations for asymmetric mixtures at high pressures were highly dependent on bubble pressure calculation, making more convenient to eliminate the data points yielding the highest deviations in pressure. However the results of the thermodynamic consistencies test of experimental data for many cases were found not fully consistent. As a result, the strategies for solving these problems were discussed in detailed.