Vapor–liquid equilibria of copper using hybrid Monte Carlo Wang—Landau simulations

Because of the large temperatures and pressures involved, the experimental determination of the vapor–liquid equilibria and of the critical properties of metals is fraught with difficulties. We show in this work how we determine these properties for a metal using hybrid Monte Carlo Wang–Landau simulations in the isothermal–isobaric ensemble on the example of copper. We use a many-body potential, known as the quantum corrected Sutton–Chen embedded atom model, to model the interactions between Cu atoms. We obtain the following estimates for the critical temperature _Tc=5696±50$T_c=5696\pm 50$ K, the critical density _ρc=1.80±0.03${\rho }_c=1.80\pm 0.03$ g/cm³, and the critical pressure _Pc=1141±100$P_c=1141\pm 100$ bar. Our results lie within the range of values found in experiments for the critical temperature (between 5140 K and 7696 K), for the critical pressure (between 420 bar and 5829 bar) and for the critical density (1.9 g cm⁻³).     

Effect of nonvolatile solute on vapor–liquid equilibrium at fixed liquid composition

The influence of some nonvolatile solutes on boiling points of two azeotropic mixtures (1-propanol–water and methanol–tetrahydrofuran systems) was determined by means of isobaric vapor–liquid equilibrium experiments. A basic thermodynamic equation of nonvolatile solute effect on vapor–liquid equilibrium at fixed liquid composition was derived. Based on the theoretical analysis about the equation, two criterions of universal significance were obtained: (1) when a little nonvolatile solute dissolves in a binary liquid mixture with constant composition, if the vapor composition of less volatile component is increased the boiling point must be elevated at given pressure or the vapor pressure must be depressed at given temperature; (2) when a little nonvolatile solute dissolves in an azeotropic mixture, any kind of nonvolatile solute always causes the elevation of boiling point at given pressure or the depression of vapor pressure at given temperature irrespective of the variation of vapor composition. Verifying through the experiments of this paper and lot of the relevant experimental data in the literature, all of the experimental results were in agreement with both criterions without exception.     

Critical parameters of liquid–vapor equilibrium and its geometric interpretation

Expressions are derived for the Gibbs and free energies of a liquid–vapor system. The critical parameters of the liquid are determined, and the character of the relationship between them is found. The temperature dependence of the vaporization heat is found. The analytical expression of the empirical Tait rule is substantiated. General patterns of subcritical and critical equilibria were revealed. A geometric definition of a critical state is proposed, and the existence of two critical states is proven.     

Soft-SAFT modeling of vapor–liquid equilibria of nitriles and their mixtures

Nitriles are strong polar compounds showing a highly non-ideal behavior, which makes them challenging systems from a modeling point of view; in spite of this, accurate predictions for the vapor–liquid equilibria of these systems are needed, as some of them, like acetonitrile (CH₃CN) and propionitrile (C₂H₅CN), play an important role as organic solvents in several industrial processes. This work deals with the calculation of the vapor–liquid equilibria (VLE) of nitriles and their mixtures by using the crossover soft-SAFT Equation of State (EoS). Both polar and associating interactions are taken into account in a single association term in the crossover soft-SAFT equation, while the crossover term allows for accurate calculations both far from and close to the critical point. Molecular parameters for acetonitrile, propionitrile and n-butyronitrile (C₃H₇CN) are regressed from experimental data. Their transferability is tested by the calculation of the VLE of heavier linear nitriles, namely, valeronitrile (C₄H₉CN) and hexanonitrile (C₅H₁₁CN), not included in the fitting procedure. Crossover soft-SAFT results are in excellent agreement with experimental data for the whole range of thermodynamic conditions investigated, proving the robustness of the approach. Parameters transferability has also been used to describe the isomers n-butyronitrile and i-butyronitrile. Finally, the nitriles soft-SAFT model is further tested in VLE calculation of mixtures with benzene, carbon tetrachloride and carbon dioxide, which proved to be satisfactory as well.     

Isobaric vapor–liquid–liquid equilibrium and vapor–liquid equilibrium for the system water–ethanol-1,4-dimethylbenzene at 101.3 kPa

An all-glass, dynamic recirculating still equipped with an ultrasonic homogenizer has been used to determine vapor–liquid (VLE) and vapor–liquid–liquid (VLLE) equilibria. Consistent data have been obtained for the ternary water + ethanol + p-xylene system at 101.3 kPa for temperatures in the range of 351.16–365.40 K. Experimental results have been used to check the accuracy of the UNIFAC, UNIQUAC and NRTL models in the liquid–liquid region of importance in the dehydration of ethanol by azeotropic distillation.     

Prediction of vapor–liquid equilibria properties of several HFC binary refrigerant mixtures

New correlations have been developed to estimate saturated vapor pressures of eight HFC binary refrigerant mixtures, namely HFC125/134a, HFC125/143a, HFC134a/236fa, HFC134a/245fa, HFC143a/134a, HFC143a/152a, HFC32/125, and HFC32/134a. In this prediction method, the saturated vapor pressures of mixtures can be calculated by the thermoproperties of pure components, without any adjustable parameters determined by experimental data. The overall average absolute deviation of pressures is <1% compared with experimental data.     

Molecular dynamics simulation of the equilibrium liquid–vapor interphase with solidification

The equilibrium structure of the finite, interphase interfacial region that exists between a liquid film and a bulk vapor is resolved by molecular dynamics simulation. Argon systems are considered for a temperature range that extends below the melting point. Physically consistent procedures are developed to define the boundaries between the interphase and the liquid and vapor phases. The procedures involve counting of neighboring molecules and comparing the results with boundary criteria that permit the boundaries to be precisely established. Two-dimensional radial distribution functions at the liquid and vapor boundaries and within the interphase region demonstrate the physical consistency of the boundary criteria and the state of transition within the region. The method developed for interphase boundary definitions can be extended to nonequilibrium systems. Spatial profiles of macroscopic properties across the interphase region are presented. A number of interfacial thermodynamic properties and profile curve-fit parameters are tabulated, including evaporation/condensation coefficients determined from molecular flux statistics. The evaporation/condensation coefficients away from the melting point compare more favorably with transition state theory than those of previous simulations. Near the melting point, transition theory approximations are less valid and the present results differ from the theory. The effects of film substrate wetting on evaporation/condensation coefficients are also presented.     

Thermodynamics of the multicomponent vapor–liquid equilibrium under capillary pressure difference

We discuss the two-phase multicomponent equilibrium, provided that the phase pressures are different due to the action of capillary forces. We prove the two general properties of such an equilibrium, which have previously been known for a single-component case, however, to the best of our knowledge, not for the multicomponent mixtures. The importance is emphasized on the space of the intensive variables P, T and μⁱ, where the laws of capillary equilibrium have a simple geometrical interpretation. We formulate thermodynamic problems specific to such an equilibrium, and outline changes to be introduced to common algorithms of flash calculations in order to solve these problems. Sample calculations show large variation of the capillary properties of the mixture in the very neighborhood of the phase envelope and the restrictive role of the spinodal surface as a boundary for possible equilibrium states with different pressures.     

Molecular thermodynamics of salt effect in vapor–liquid equilibrium of ethanol–water systems

Scaled particle theory was used to derive a general expression for the salt effect parameter, K, of isobaric vapor–liquid equilibrium for ethanol–water-1-1 type electrolytic systems, which appears in the Furter equation. This expression was essentially a sum of two terms: 1, the hard sphere interaction term calculated by Masterton–Lee's equation, 2, the soft sphere interaction term calculated by Y. Hu's molecular thermodynamical model, in which the diameters of nacked ions were replaced by that of solvated ions, the solvation coefficients (i.e., in the radio of the latter to the former) were taken to be adjustable parameters, their magnitude implies the ionic solvation rules. A correlation equation for the local dielectrical constant around central ions with liquid concentration was obtained by mapping out experimental points. The calculated salt effect parameters of 9 ethanol–water-1–1 type electrolytic systems were in good agreement with the literature values within the wide range of liquid concentration.     

Determination of vapor—liquid equilibrium diagrams of multicomponent systems

A method for the determination of vapor–liquid phase diagram structure of five-component systems based on the analysis of types and Poincare indexes of singular points of the geometric scan and full structure of the concentration simplex is proposed. Validity of the proposed method was demonstrated by vapor–liquid equilibrium modeling in five-component mixtures: ethanol + water + toluene + butanol + chlorbenzene and acetone + chloroform + ethanol + cyclohexane + water.     

Evaluation of stochastic global optimization methods for modeling vapor–liquid equilibrium data

Parameter estimation for vapor–liquid equilibrium (VLE) data modeling plays an important role in design, optimization and control of separation units. This optimization problem is very challenging due to the high non-linearity of thermodynamic models. Recently, several stochastic optimization methods such as Differential Evolution with Tabu List (DETL) and Particle Swarm Optimization (PSO) have evolved as alternative and reliable strategies for solving global optimization problems including parameter estimation in thermodynamic models. However, these methods have not been applied and compared with respect to other stochastic strategies such as Simulated Annealing (SA), Differential Evolution (DE) and Genetic Algorithm (GA) in the context of parameter estimation for VLE data modeling. Therefore, in this study several stochastic optimization methods are applied to solve parameter estimation problems for VLE modeling using both the classical least squares and maximum likelihood approaches. Specifically, we have tested and compared the reliability and efficiency of SA, GA, DE, DETL and PSO for modeling several binary VLE data using local composition models. These methods were also tested on benchmark problems for global optimization. Our results show that the effectiveness of these stochastic methods varies significantly between the different tested problems and also depends on the stopping criterion especially for SA, GA and PSO. Overall, DE and DETL have better performance for solving the parameter estimation problems in VLE data modeling.     

Prediction of ternary vapor–liquid equilibria for 33 systems by molecular simulation

A set of molecular models for 78 pure substances from prior work is taken as a basis for systematically studying vapor–liquid equilibria (VLE) of ternary systems. All 33 ternary mixtures of these 78 components for which experimental VLE data are available are studied by molecular simulation. The mixture models are based on the modified Lorentz–Berthelot combining rule that contains one binary interaction parameter which was adjusted to a single experimental binary vapor pressure of each binary subsystem in prior work. No adjustment to ternary data is carried out. The predictions from the molecular models of the 33 ternary mixtures are compared to the available experimental data. In almost all cases, the molecular models give excellent predictions of the ternary mixture properties.     

Isothermal vapor–liquid equilibria and excess molar volumes for the ternary mixtures containing 2-methyl pyrazine

Isothermal vapor–liquid equilibrium (VLE) data at 353.15 K and excess molar volumes (V^E) at 298.15 K are reported for the binary systems of ethyl acetate (EA)+cyclohexane and EA+n-hexane and also for the ternary systems of EA+cyclohexane+2-methyl pyrazine (2MP) and EA+n-hexane+2MP. The experimental binary VLE data were correlated with common g^E model equations. The correlated Wilson parameters of the constituent binary systems were used to calculate the phase behavior of the ternary mixtures. The calculated ternary VLE data using Wilson parameters were compared with experimental ternary data. The experimental excess molar volumes were correlated with the Redlich–Kister equation for the binary mixtures, and Cibulka’s equation for the ternary mixtures.     

Isobaric vapor–liquid equilibrium of systems containing triethyl orthoformate at 101.3 kPa

Isobaric vapor–liquid equilibrium data of ethanol(1)-triethyl orthoformate(2), benzene(1)-triethyl orthoformate(2) and ethanol(1)-benzene(2)-triethyl orthoformate(3) were measured at 101.3 kPa and under a wide range of temperatures (349–420 K), using the Rose–Williams still modified by the authors. The experimental data of binary systems were tested for thermodynamic consistency with the method of Fredenslund and coworkers and correlated satisfactorily with SRK equation and PR equation of state. The VLE data of ethanol(1)-benzene(2)-triethyl orthoformate(3) ternary system were tested with the method of McDermont–Ellis and were predicted with the parameters of SRK and PR equation of state obtained from binary systems.     

The influence of the temperature on the liquid–liquid equilibrium of the ternary system 1-pentanol–ethanol–water

Liquid–liquid equilibrium data for the ternary system 1-pentanol–ethanol–water have been determined experimentally at 25, 50, 85 and 95°C. These results have been correlated simultaneously by the uniquac method obtaining two sets of interaction parameters: one of them independent of the temperature and the other with a linear dependence. Both sets of parameters fit the experimental results well.     

Experimental investigation of the vapor–liquid equilibrium at 313.15 K of the ternary system tert-amylmethyl ether (TAME)+n-heptane+methanol

Experimental isothermal P–x data at 313.15 K for the ternary system (tert-amylmethyl ether (TAME)+n-heptane+methanol) and for one of the unmeasured constituent binary systems, (tert-amylmethyl ether (TAME)+methanol) are reported. Data reduction by Barker's method provides correlations for g^E using the Margules equation for the binary systems and the Wohl expansion for the ternary system. Wilson, NRTL and UNIQUAC models have been applied successfully to both the binary and the ternary systems. The presence of azeotropes in the ternary system and constituent binaries are studied as well as the presence of immiscible zones.     

Liquid–liquid phase equilibrium in ternary immiscible Al–Bi–Sn melts

In this paper, the liquid-phase separation of ternary immiscible Al–Bi–Sn melts was studied with resistivity and thermal analysis methods at different temperatures. The resistivity–temperature curves appear anomalous and abrupt change as rising temperature, corresponding to the distinctive and low peak of melting process in the differential scanning calorimetry (DSC) curves, indicative of the occurrence of the liquid-phase separation. The anomalous behaviour of the resistivity temperature dependence is attributable to concentration–concentration fluctuations. The microheterogeneity–microhomogeneity transformation causes large fluctuations in concentration, which make the randomness and chaos of the atoms larger, leading to the greater impediment to electron movement and the sharp rise of resistivity. The addition of tin to the Al–Bi immiscible alloys decreases the monotectic reaction. It is concluded that concentration–concentration fluctuations are responsible for the anomalous behaviour of resistivity and DSC methods.     

Correlation and prediction of salt effects on vapor–liquid equilibrium in alcohol–water–salt systems

A modification of the solvation model of Ohe is proposed for the calculation of vapor–liquid equilibria (VLE) in alcohol–water–salt systems. The modified method employs the Bromley equation to calculate the activity of water in salt solutions, and a one-parameter empirical expression to calculate the activity of the alcohol. The single parameter is obtained by fitting ternary alcohol–water–salt data. The method is simple to use and does not require data on the vapor-pressures of alcohol–salt mixtures that are seldom available in the literature. Experimental data for 17 salts in 36 alcohol–water–salt systems, covering a temperature range from 298 to 375 K, and salt concentrations up to about 8 m, were correlated using the new approach. In all, 69 data sets and 1045 data points were correlated satisfactorily. The method was also used to predict VLE in four ternary alcohol–alcohol–salt systems and one quaternary alcohol–alcohol–water–salt system with satisfactory results.