An Equation of State for High-Temperature Superconductors

In the present paper the expression of cohesive energy and the bulk modulus as a function of volume are formulated for high-T _c copper oxide superconductors. The model employed consists of long-range electrostatic Coulomb interaction and short-range overlap repulsion. The short-range overlap potential is considered in the Born-Landé inverse power form. The model parameters of the Born-Landé model are calculated from the equilibrium condition and data of bulk modulus at room temperature. The computed values of the pressure derivatives of the bulk modulus atP = 0 and the values of the bulk modulus are found to be in very close agreement with experimental values for high-T _c copper oxide and their nonsuperconducting parent compounds. It is also found that the quantityAU/U( V ₀ ) of these compounds increases with increasing hydrostatic pressure.     

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.     

Equation of State for Molten Alkali Metal Alloys

Calculated results of the liquid density of binary molten alloys of Na–K and K–Cs over the whole range of concentrations and that of a ternary molten eutectic of Na–K–Cs from the freezing point up to several hundred degrees above the boiling point are presented. The calculations were performed with the analytical equation of state proposed by Ihm, Song, and Mason, which is based on statistical-mechanical perturbation theory. The second virial coefficients were calculated from the corresponding-states correlation of Mehdipour and Boushehri. Calculation of the other two temperature-dependent parameters was carried out by scaling. The calculated results cover a much wider range of temperatures and are more accurate than those presented in our previous work.     

Prediction of Some Important Regularities Using a Statistical Mechanical Equation of State

Four important known regularities for several pure substances have been described using a statistical mechanical equation of state (EOS) derived by Deiters. The equation of state depends on three temperature-independent parameters, which can be obtained from the critical constants. The studied regularities included: (i) near linearity of the reduced isothermal bulk modulus as a function of reduced pressure, (ii) the common bulk modulus point on the isotherms of the reduced bulk modulus versus reduced density, (iii) near linearity of the Zeno contour of reduced temperature against reduced density from the Boyle temperature to the triple point, and (iv) near linearity of the mean density of a saturated liquid and its equilibrium vapor as a function of temperature, called the law of rectilinear diameter. The results for several fluids have been compared to the available experimental data. The predictions are often satisfactory in the sensitive regions.     

A High-Temperature, Thermal Non-equilibrium Equation of State for Ammonia

The engineering applications of ammonia extend far beyond the pressure and temperature ranges for which thermodynamic models currently exist in the literature. Thus, a thermal non-equilibrium thermochemical model was developed to compute the composition and thermodynamic properties of ammonia for an extended temperature and pressure range that includes ionization reactions. Thermal non-equilibrium between electrons and heavy particles was included and is presented for ratios of 1/2, 1, 2 and 3. The fourteen-equation nonlinear system produced under the assumptions of ideal gas and two-temperature local thermodynamic equilibrium was solved numerically using a Newton-Raphson method. The thermochemical model is verified for both the composition and thermodynamic properties by comparisons to existing thermochemical models in the literature. These comparisons verify the model for the available, yet limited, temperature and density ranges. Analysis of the composition and thermodynamic properties as a function of the independent properties confirms the necessity for such a model as part of rigorous computations with computational fluid dynamics (CFD) or magnetohydrodynamics (MHD) computer codes. The model can be easily cast in tabular form to complement the set of conservation equations utilized by such codes.     

An Extended Equation of State Modeling Method II. Mixtures

This work is the extension of previous work dedicated to pure fluids. The same method is extended to the representation of thermodynamic properties of a mixture through a fundamental equation of state in terms of the Helmholtz energy. The proposed technique exploits the extended corresponding-states concept of distorting the independent variables of a dedicated equation of state for a reference fluid using suitable scale factor functions to adapt the equation to experimental data of a target system. An existing equation of state for the target mixture is used instead of an equation for the reference fluid, completely avoiding the need for a reference fluid. In particular, a Soave–Redlich–Kwong cubic equation with van der Waals mixing rules is chosen. The scale factors, which are functions of temperature, density, and mole fraction of the target mixture, are expressed in the form of a multilayer feedforward neural network, whose coefficients are regressed by minimizing a suitable objective function involving different kinds of mixture thermodynamic data. As a preliminary test, the model is applied to five binary and two ternary haloalkane mixtures, using data generated from existing dedicated equations of state for the selected mixtures. The results show that the method is robust and straightforward for the effective development of a mixture- specific equation of state directly from experimental data.     

An Equation of State for 1,1,1-Trifluoroethane (R-143a)

A fundamental equation of state has been developed for 1,1,1-trifluoroethane (R-143a) using the dimensionless Helmholtz energy. The experimental thermodynamic property data, which cover temperatures from the triple point (161 K) to 433 K and pressures up to 35 MPa, are used to develop the present equation. These data are represented by the present equation within their reported experimental uncertainties: ±0.1% in density for both vapor and liquid phase P––T data, ±1% in isochoric specific heat capacities, and ±0.02% in the vapor phase speed-of-sound data. The extended range of validity of the present model covers temperatures from 160 to 650 K and pressures up to 50 MPa as verified by the thermodynamic behavior of the isobaric heat-capacity values over the entire fluid phase.     

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 mercury

This paper presents a procedure for predicting the equation of state of mercury, by including mercury in the scope of a new statistical mechanical equation of state that is known for normal fluids. The scaling constants are the latent heat of vaporization and the density at the melting temperature, which are related to the cohesive energy density. Since experimental data for the second virial coefficient of mercury are scarce, a corresponding-states correlation of normal fluids is used to calculate theB(T) of mercury. The free parameter of the ISM equation, λ, compensates for the uncertainties inB(T). Also, we can predict the values of two temperature-dependent parameters, α(T) andb(T), with satisfactory accuracy from a knowledge of ΔH _vap andp _m, without knowing any details of the intermolecular potentials. While the values ofB(T) are scarce for mercury and the vapor pressure of this metal at low temperatures is very small, an equation of state for mercury from two scaling parameters (ΔH _vap,p _m) predicts the density of Hg from the melting point up to 100° above the boiling temperature to within 5%.     

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%.     

Equation of state for compressed liquids and their mixtures from the cohesive energy density

A procedure is presented, based on statistical-mechanical theory, for predicting the equation of state of compressed normal liquids and their mixtures from two scaling constants that are available from measurements at ordinary pressures and temperatures. The theoretical equation of state is that of Ihm, Song, and Mason, and the two constants are the enthalpy of vaporization and the liquid density at the triple point, which are related to the cohesive energy density of regular solution theory. The procedure is tested on a number of substances ranging in complexity from Ar and CO₂ to n-heptane and toluene. The results indicate that the liquid density at any pressure and temperature can be predicted within about 5%, over the range from T _tp to T _c and up to the freezing line. Possible methods of determining the scaling constants are discussed, as well as other possible choices for scaling constants.Paper dedicated to Professor Joseph Kestin.     

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%.     

An analytical equation of state for molten alkali metals

This paper brings the molten alkali metals into the scope of a new statistical mechanical equation of state that is known to satisfy normal fluids over the whole range. As for normal fluids, the latent heat of vaporization and density at freezing temperature are the only inputs (scaling factors). The correspondingstates correlation of normal fluids is used to calculate the second virial coefficient,B ₂(T), of alkali metals, which is scarce experimentally and its calculation is complicated by dimer formation. Calculations of the other two temperature-dependent constants,(T) andb(T), follow by scaling. The virial coefficients of alkali metals cannot be expected to obey a law of corresponding states for normal fluids. The fact that two potentials are involved may be the reason for this. Thus, alkali metals have the characteristics of interacting through singlet and triple potentials so that the treatment by a single potential here is fortuitous. The adjustable parameter of the equation of state,, compensates for the uncertainties inB ₂(T). The procedure used to calculate the density of liquids Li through Cs from the freezing line up to several hundred degrees above the boiling temperatures. The results are within 5 %.     

He4 State Equation Below 0.8 K

This work develops the Helmholtz potential A(ρ, T) for He⁴ below 0.8 K. Superfluid terms, related to temperature and momentum gradients, are neglected with negligible loss of accuracy in the derived state properties (specific heats, first sound velocity, expansivity, compressibility, etc.). Retained terms are directly related to a bulk fluid compressibility plus phonon and roton excitations in this quantum fluid. The bulk fluid compressibility is found from the empirical equation c₁³ ≈ c₁₀³ + b; P, where c₁ is the velocity of first sound, P is the pressure, and c₁₀ and b are constants; this empirical equation is found to apply also to other helium temperature ranges and to other fluids. The phonon excitations lead to a single temperature-dependent term in A(ρ ,T) up to 0.3 K, with only two more terms added up to 0.8 K. The roton potential, negligible below about 0.3 K, is a single term first derived 60 years ago but little used in more recent work. The final A(ρ ,T) is shown to fit available experimental specific heat data to about ±2% or better. The magnitude of the pressure-independent Gruneisen parameter below 0.3 K is typical of highly compressed normal liquids. Extension of the equation above 0.8 K is hampered by lack of data between 0.8 and 1.2 K     

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.     

Direct Prediction of Cricondentherm and Cricondenbar Coordinates of Natural Gas Mixtures using Cubic Equation of State

A numerical algorithm is presented for direct calculation of the cricondenbar and cricondentherm coordinates of natural gas mixtures of known composition based on the Michelsen method. In the course of determination of these coordinates, the equilibrium mole fractions at these points are also calculated. In this algorithm, the property of the distance from the free energy surfaces to a tangent plane in equilibrium condition is added to saturation calculation as an additional criterion. An equation of state (EoS) was needed to calculate all required properties. Therefore, the algorithm was tested with Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), and modified Nasrifar-Moshfeghian (MNM) equations of state. For different EoSs, the impact of the binary interaction coefficient (k _ij) was studied. The impact of initial guesses for temperature and pressure was also studied. The convergence speed and the accuracy of the results of this new algorithm were compared with experimental data and the results obtained from other methods and simulation softwares such as Hysys, Aspen Plus, and EzThermo.     

Common intersection point independent of mole fraction: A new regularity

A new regularity has been found for compressed liquid mixtures, namely, a common intersection point for the isotherms of the reduced bulk modulus of a compressed liquid mixture as a function of composition. The regularity holds over a specific range of densities from a particular liquid mixture. The regularity has been tested for a LJ (12,6) mixture, Ar+Kr, Kr+Xe, and CO₂+C₂H₆ based on equations of state derived from statistical mechanics and it is valid close to within experimental accuracy.     

Equation of State for Substances that Satisfy the Corresponding States Law

Universal expressions for the free energy of substances that satisfy the law of corresponding states are obtained. The equation of state is constructed using an interpolation between the high temperature Debye approximation and the ideal-gas one. Missing data needed for calculations have been found from the comparison of calculated and tabulated values of entropy and pressure along the phase coexistence curves. The thermodynamics of corresponding states, constructed in this way, allows one to find the triple- and critical-point parameters, as well as the phase coexistence curves with an accuracy that does not exceed the accuracy of the law of corresponding states itself.     

Thermophysical properties from the equation of state of Mason and co-workers

The theory gives formulas for calculating the three temperature-dependent parameters of the equation of state from the intermolecular potential. But the second virial coefficient also serves to predict the entire equation of state in terms of two scaling parameters and, hence, a number of other thermodynamic properties including the Joule-Thomson inversion curve, bulk modulus. secant bulk modulus, and inverse isobaric expansivity among others. Agreement with experimental data is quite good. Paper dedicated to Professor Edward A. Mason.