A New Analytical Model for the Prediction of Vapor–Liquid Equilibrium Densities

A new simple, predictive model for estimating both the vapor and the liquid densities of fluids at the vapor–liquid equilibrium is presented. It is based on the symmetry of the derivatives of the two saturation densities with respect to the temperature, which is a consequence of applying the rectilinear diameter law. No adjustable coefficients are involved, and only two parameters—both with certain physical meaning—have to be calculated for each fluid. The method used for these calculations is straightforward, the required inputs being the critical temperature and density, and the value of the vapor and liquid densities at a reference temperature. The results show that the model is accurate for fluids of different kinds as long as the rectilinear diameter law holds, and that, in general, the accuracy is better than that of the most recent model with no adjustable coefficients.     

Saturated Liquid Densities for 33 Binary Refrigerant Mixtures Based on the ISM Equation of State

In this work, the ISM equation of state based on statistical-mechanical perturbation theory has been extended to liquid refrigerant mixtures by using correlations of Boushehri and Mason. Three temperature-dependent parameters are needed to use the equation of state: the second virial coefficient, B₂(T), an effective van der Waals covolume, b(T), and a scaling factor, α (T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, ΔH_vap, and the liquid density at the normal boiling point, ρ_nb. α(T) and b(T) can also be calculated from second virial coefficients by a scaling rule. The theory has considerable predictive power, since it permits the construction of the PVT surface from the heat of vaporization and the liquid density at the normal boiling point. The equation of state was tested on 33 liquid mixtures from 12 refrigerants. The results indicate that the liquid densities can be predicted to at most 2.8% over a wide range of temperatures, 170–369 K.     

Phase Behaviors of Binary Hard-Sphere Mixtures Using Simple Analytical Equations of State

Recently, we have proposed a unified analytical equation of state (EOS) for solid–liquid–vapor states of matter, and have examined the thermodynamic properties of argon, carbon dioxide, and methane, as well as binary mixtures of methane and carbon dioxide. Also it has been demonstrated that the EOS can be applied for the solid–fluid transition of hard spheres, by eliminating the attractive part of the EOS. The present work is an extension of the earlier calculations for identical hard spheres, and here we examine the phase behavior of binary hard-sphere mixtures. The hard-sphere EOS employed in this study is

Tait's Equation of State for Liquid Mixtures

The methods of statistical theory of liquid state are used to validate the well-known Tait's equation of state for liquid mixtures. An expression is derived which relates the coefficients A and B of Tait's equation of state to the parameters of steepness of repulsion forces m and to the thermodynamic properties of the system. P–V–T–x measurements for a water-acetone system are performed to check the theoretical results. The method of molecular dynamics is used to calculate the parameters of steepness of repulsion forces of a water-acetone mixture at different temperatures and concentrations. It is demonstrated that m in the treated ranges of temperature and pressure assumes a constant value of 15. The theoretically obtained coefficient A coincides with the experimentally obtained value within the experimental error, and the coefficient B describes qualitatively correctly the temperature and concentration dependences obtained as a result of P–V–T–x measurements.     

A Thermodynamic Property Model for the Binary Mixture of Methane and Hydrogen Sulfide

A thermodynamic property model with new mixing rules using the Helmholtz free energy is presented for the binary mixture of methane and hydrogen sulfide based on experimental P_ρ T_x data, vapor–liquid equilibrium data, and critical-point properties. The binary mixture of methane and hydrogen sulfide shows vapor–liquid–liquid equilibria and a divergence of the critical curve. The model represents the existing experimental data accurately and describes the complicated behavior of the phase equilibria and the critical curve. The uncertainty in density calculations is estimated to be 2%. The uncertainty in vapor–liquid equilibrium calculations is 0.02 mole fraction in the liquid phase and 0.03 mole fraction in the vapor phase. The model also represents the critical points with an uncertainty of 2% in temperature and 3% in pressure. Graphical and statistical comparisons between experimental data and the available thermodynamic models are discussed     

Equation of State for Complex Liquid Mixtures from Surface Tension

The present work shows a successful extension of previous studies to molecular liquids for which the second virial coefficients are not known. Recent advances in the statistical mechanical theory of equilibrium fluids can be used to obtain an equation of state (EOS) for compressed normal liquids and molten alkali metals. Three temperature-dependent quantities are needed to use the EOS: the second virial coefficient, B(T), an effective van der Waals covolume, b(T), and a scaling factor, (T). The second virial coefficients are calculated from a correlation that uses the surface tension, _tr, and the liquid density at the triple point. Calculation of (T) and b(T) follows by scaling. Thus, thermodynamic consistency is achieved by use of two scaling parameters (_tr, _tr). The correlations embrace the temperature range T _tr<T<T _c and can be used in a predictive mode. The remaining constant parameter is best found empirically from _tr data for pure dense liquids. The equation of state is tested on 42 liquid mixtures The results indicate that the liquid density at any pressure and temperature can be predicted within about 5%, over the range from T _tr to T _c.     

Equation of State for Normal Liquid Helium-3 from 0.1 to 3.3157 K

An equation of state for normal liquid ³He has been constructed in the form of Helmholtz free energy as a function of independent parameters—temperature, and density. The equation was fitted simultaneously to the collected experimental p-ρ-T, specific heat, sound velocity, isobaric expansion coefficient and isothermal compressibility coefficient from the world’s literature to accuracies comparable with reasonable experimental errors in the measured quantities. Extensive comparisons between the equation of state and experimental data have been made by a set of deviation plots. The state equation is valid in the region for temperatures from 0.1 K to T _c = 3.3157 K, and for pressures from vapor pressures to melting pressures.     

An Extended Equation of State Modeling Method I. Pure Fluids

A new technique is proposed here to represent the thermodynamic surface of a pure fluid in the fundamental Helmholtz energy form. The peculiarity of the present method is the extension of a generic equation of state for the target fluid, which is assumed as the basic equation, through the distortion of its independent variables by individual shape functions, which are represented by a neural network used as function approximator. The basic equation of state for the target fluid can have the simple functional form of a cubic equation, as, for instance, the Soave–Redlich–Kwong equation assumed in the present study. A set of nine fluids including hydrocarbons, haloalkane refrigerants, and strongly polar substances has been considered. For each of them the model has been regressed and then validated against volumetric and caloric properties generated in the vapor, liquid, and supercritical regions from highly accurate dedicated equations of state. In comparison with the underlying cubic equation of state, the prediction accuracy is improved by a factor between 10 and 100, depending on the property and on the region. It has been verified that about 100 density experimental points, together with from 10 to 20 coexistence data, are sufficient to guarantee high prediction accuracy for different thermodynamic properties. The method is a promising modeling technique for the heuristic development of multiparameter dedicated equations of state from experimental data.     

An Analytical Equation of State for Some Saturated Liquid Metals

An analytical equation of state (EoS) is developed for some saturated molten metals. The equation is that of Ihm, Song and Mason in which the three temperature–dependent parameters, second virial coefficient, van der Waals co–volume, and a scaling parameter, are calculated by means of corresponding states correlations. The required characteristic constants are the heat of vaporization and the density at the melting point, H _vap and _m, respectively. The EoS is applied to these liquid metals to calculate the density at temperatures higher than their melting points. The results are fairly consistent with experiment, maximum difference less than ±4%.     

Crossover SAFT Equation of State and Thermodynamic Properties of Propan-1-ol

In this work we have developed a new equation of state (EOS) for propan-1-ol on the basis of the crossover modification (CR) of the statistical-associating-fluid-theory (SAFT) EOS recently developed and applied to n-alkanes. The CR SAFT EOS reproduces the nonanalytical scaling laws in the asymptotic critical region and reduces to the analytical-classical SAFT EOS far away from the critical point. Unlike the previous crossover EOS, the new CR SAFT EOS is based on the parametric sine model for the universal crossover function and is able to represent analytically connected van der Waals loops in the metastable fluid region. The CR SAFT EOS contains 10 system-dependent parameters and allows an accurate representation of the thermodynamic properties of propan-1-ol over a wide range thermodynamic states including the asymptotic singular behavior in the nearest vicinity of the critical point. The EOS was tested against experimental isochoric and isobaric specific heats, speed of sound, PVT, and VLE data in and beyond the critical region. In the one-phase region, the CR SAFT equation represents the experimental values of pressure with an average absolute deviation (AAD) of less than 1% in the critical and supercritical regions and the liquid densities with an AAD of about 1%. A corresponding states principle is used for the extension of the new CR SAFT EOS for propan-1-ol to higher n-alkanols.     

Density and Surface Tension of Electromagnetically Levitated Cu–Co–Fe Alloys

The density and surface tension of Cu–Co–Fe alloys have been measured using the non-contact technique of electromagnetic levitation. At temperatures above and below the liquidus point, the density and surface tension are linear functions of temperature. The experimental density results can be predicted by means of the regular solution model from the binary phases alone, i.e., no assumption about ternary interactions needs to be made. The surface tension values are in good agreement with numerical solutions of the Butler equation if the known Gibbs excess energies are used. It is found that, for Cu–Co–Fe, it is possible to predict the surface tension from the binary systems as well. In addition to this, the surface tension is insensitive to substitution of the two transition metals, Co and Fe.     

Density and Thermal Expansion of Liquid Ag–Cu and Ag–Au Alloys

The densities of liquid Cu–Ag and Ag–Au alloys were measured using the technique of electromagnetic levitation. This technique involves producing shadow images of the sample from which the volume is calculated by an image processing algorithm. The density and thermal expansion of several alloys and the pure elements copper, gold, and silver are measured at temperatures above their melting points. In addition, they were investigated as a function of either the copper or gold concentration. It was found from data analysis that the densities can be derived from a linear combination of the molar volumes of the elements and that thermodynamic excess quantities are negligibly small.     

Equation of State for Nonpolar Fluid Mixtures: Prediction from Boiling Point Constants

Our previous corresponding-states correlation for the second virial coefficient of nonpolar fluids, based on the normal boiling point parameters, has been employed to predict the equation of state of nonpolar fluid mixtures. The analytical equation of state is that of Ihm, Song, and Mason, which requires three temperature-dependent parameters, i.e., the second virial coefficient, a scaling constant for softness of repulsive forces, and a van der Waals covolume. In the previous work, we showed that the temperature-dependent parameters could be calculated by knowing the boiling point constants. In this work, it is shown that using a simple geometric mean for the boiling point temperature and an arithmetic mean for the liquid density at the normal boiling point is sufficient to determine the temperature-dependent parameters for mixtures. The equation of state has been utilized to calculate the liquid density of several nonpolar fluid mixtures. The agreement with experiment is good.     

Equations of state for hard-sphere fluids

Equations of state and contact values of hard-sphere radial distribution functions (rdf's) which are given by a linear combination of the Percus— Yevick and scaled-particle virial expressions are considered. In the one-component case the mixing coefficient() is, in general, a function of the volume fraction. In mixtures the coefficient( _i ,d _i ), in general, depends upon the volume fraction _i , and diameterd _i , of each species,i andj. For the contact valuesY _ij of the rdf's, the mixing coefficients _ij ( _k ) also depend on speciesi andj. Density expansions for the exact for the one-component hard-sphere fluid are obtained and compared with several approximations made in earlier works and in our own work, as well as with simulations. For a mixture, it turns out that one cannot obtain the exact fourth virial coefficient by using a linear combination of the Percus-Yevick and scaled-particle virial expressions forY _ij unless one allows _ij to depend on mole fractionsx _i even at the zeroth order of its density expansion. We also find that _ij must depend on particle speciesi andj in order to satisfy the exact limits obtained earlier by Sung and Stell. A new equation of state for the binary hard-sphere mixture which satisfies all the exact limits we have considered is suggested.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

The ISM Equation of State Applied to Refrigerants

In this work, we apply an equation of state based on statistical–mechanical perturbation theory to liquid refrigerants and their mixtures. Three temperature-dependent parameters are needed to use the equation of state: the second virial coefficient, B ₂(T), an effective van der Waals covolume, b(T), and a scaling factor, (T). The second virial coefficients are calculated from a correlation based on the heat of vaporization, H _vap, and the liquid density at the freezing point, _fp. (T) and b(T) can also be calculated from the second virial coefficient by a scaling rule. Based on the theory, these two temperature-dependent parameters depend only on the repulsive branch of the potential function, and therefore, by our procedure, can be found from H _vap and _fp. The theory has considerable predictive power, since it permits the construction of the p–v–T surface from the heat of vaporization plus the triple-point density. The equation of state is tested for pure, two- and three-component liquid refrigerant mixtures.     

Equations of State for Technical Applications. I. Simultaneously Optimized Functional Forms for Nonpolar and Polar Fluids

New functional forms for multiparameter equations of state have been developed for non- and weakly polar fluids and for polar fluids. The resulting functional forms, which were established with an optimization algorithm which considers data sets for different fluids simultaneously, are suitable as a basis for equations of state for a broad variety of fluids. With regard to the achieved accuracy, the functional forms were designed to fulfill typical demands of advanced technical application. They are numerically very stable, and their substance-specific coefficients can easily be fitted to restricted data sets. In this way, a fast extension of the group of fluids for which accurate empirical equations of state are available becomes possible. This article deals with characteristic features of the new class of simultaneously optimized equations of state. Shortcomings of existing multiparameter equations of state widely used in technical applications are briefly discussed, and demands on the new class of equations of state are formulated. Substance specific parameters and detailed comparisons are given in subsequent articles for the non- and weakly polar fluids (methane, ethane, propane, isobutane, n-butane, n-pentane, n-hexane, n-heptane, n-octane, argon, oxygen, nitrogen, ethylene, cyclohexane, and sulfur hexafluoride) and for the polar fluids (trichlorofluoromethane (CFC-11), dichlorodifluoromethane (CFC-12), chlorodifluoromethane (HCFC-22), difluoromethane (HFC-32), 1,1,2-trichlorotrifluoroethane (CFC-113), 2,2-dichloro-1,1,1-trifluoroethane (HCFC-123), pentafluoroethane (HFC-125), 1,1,1,2-tetrafluoroethane (HFC-134a), 1,1,1-trifluoroethane (HFC-143a), 1,1-difluoroethane (HFC-152a), carbon dioxide, and ammonia) considered to date.     

An Analytical Equation of State for Alkaline Earth Metals

In this work, an analytical equation of state based on statistical mechanical perturbation theory, which was initially developed for normal fluids and can be applied to predict the P–V–T data for saturated liquid alkaline earth metals, is presented. The equation of state is that of Ihm, Song, and Mason, and the temperature-dependent parameters of the equation of state are calculated from a corresponding-states correlation as functions of the reduced temperature. Two scaling constants are sufficient for this purpose, the surface tension and the liquid density at the melting point. The equation of state is used to predict the saturated liquid density of molten alkaline earth metals from the melting point up to 2000 K, for which experimental data exist, within an accuracy of 5%.     

Equations of State for the Ozone-Safe Refrigerants R32 and R125

Equations of state for gaseous and liquid difluoromethane (R32) and pentafluoroethane (R125) were developed. The coefficients of the equations were determined using experimental density data and heat capacities c_v and c_s. The equations satisfy Maxwell's rule. The equations describe the thermodynamic properties of R32 and R125 at temperatures from 140 to 433 K and from 178 to 480 K, respectively, and at pressures up to 70 MPa within the experimental uncertainties. In particular, the root-mean-square deviations of the calculated values of density from the most reliable experimental data are equal to 0.10% for R32 and 0.12% for R125.     

Vapor–Liquid Equilibria for the 1,1,1-Trifluoroethane (HFC-143a)+1,1,1,2-Tetrafluoroethane (HFC-134a) System

A vapor–liquid equilibrium apparatus has been developed and used to obtain data for the binary HFC-143a+HFC-134a system. Fifty-four equilibrium data are obtained for the HFC-143a+HFC-134a system over the temperature range from 263.15 to 313.15 K at 10 K intervals. The experimental data were correlated with the Carnahan–Starling–De Santis (CSD) and Peng–Robinson (PR) equations of state. Based upon the present data, the binary interaction parameters for the CSD and PR equations of state were calculated for six isotherms for the HFC-143a+HFC-134a system. The binary interaction parameters for both equations of state were fitted by a linear equation as a function of temperature. The present data were in good agreement with the calculated results from the CSD equation of state, and the deviations were less than 1.0% with the exception of two points.