<Emphasis Type="Italic">PVT</Emphasis> Relationships of Binary Mixtures of Indole with 2-Methylnaphthalene and Biphenyl at 333.15 K and Pressures up to 270 MPa

PVT relationships of two binary mixtures of indole with 2-methylnaphthalene and with biphenyl have been measured at 333.15 K and at pressures up to 270 MPa or up to near the freezing pressure of each mixture. The compositions in mole fraction of indole were set to be 0.2500, 0.5000, and 0.7500 for both systems. PVT relationships of indole (at 343.15 K and 353.15 K), 2-methylnaphthane (at 333.15 K and 343.15 K), and biphenyl (at 353.15 K and 363.15 K) under pressure and those for the binary mixtures at 0.1 MPa in the temperature range from 293.15 K to 363.15 K were also measured. PVT data were analyzed with the use of the Tait equation and Carnahan–Starling–van der Waals (CS–vdW) equation. It was found that both the equations can be used to represent the experimental PVT relationships for the pure compounds and the binary mixtures with an average absolute deviations of 0.04% for the Tait equation and 0.29% for the CS–vdW equation. As for mixture density calculations with the CS–vdW equation, the effect of mixing rules was investigated.     

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.     

Vapor–Liquid Equilibrium Measurements for Binary Mixtures Containing 1,1,1,2,3,3,3-Heptafluoropropane (HFC-227ea)1

Isothermal vapor–liquid equilibrium data for two binary mixtures of alternative refrigerants were determined by using an apparatus applying recirculating vapor and liquid. The difluoromethane (HFC-32)+1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea) and 1,1,1,2-tetrafluoroethane (HFC-134a)+1,1,1,2,3,3,3-heptafluoropropane (HFC-227ea) systems were studied at 298.15 and 312.65 K. The pressure and vapor and liquid compositions were measured at each temperature. The experimental data were correlated with the Peng–Robinson equation of state using the van der Waals one-fluid mixing rule. Calculated results show that this equation yields good agreement with the experimental data.     

Temperature Dependence of Densities and Excess Molar Volumes of the Ternary Mixture (1-Butanol + Chloroform + Benzene) and its Binary Constituents (1-Butanol + Chloroform and 1-Butanol + Benzene)

Densities ρ of the 1-butanol + chloroform + benzene ternary mixture and the 1-butanol + chloroform and 1-butanol + benzene binaries have been measured at six temperatures (288.15, 293.15, 298.15, 303.15, 308.15, and 313.15) K and atmospheric pressure, using an oscillating U-tube densimeter. From these densities, excess molar volumes (V ^E) were calculated and fitted to the Redlich–Kister equation for all binary mixtures and to the Nagata and Tamura equation for the ternary system. The Radojković et al. equation has been used to predict excess molar volumes of the ternary mixtures. Also, V ^E data of the binary systems were correlated by the van der Waals (vdW1) and Twu–Coon–Bluck–Tilton (TCBT) mixing rules coupled with the Peng–Robinson–Stryjek–Vera (PRSV) equation of state. The prediction and correlation of V ^E data for the ternary system were performed by the same models.     

Numerical simulation of the head-on collision of two equal-sized drops with van der Waals forces

The head-on collision of two equal-sized drops in a hyperbolic flow is investigated numerically. An axisymmetric volume-of-fluid (VOF) method is used to simulate the motion of each drop toward a symmetry plane where it interacts and possibly coalesces with its mirror image. The volume-fraction boundary condition on the symmetry plane is manipulated to numerically control coalescence. Two new numerical methods have been developed to incorporate the van der Waals forces in the Navier–Stokes equations. One method employs a body force computed as the negative gradient of the van der Waals potential. The second method employs the van der Waals forces in terms of a disjoining pressure in the film depending on the film thickness. Results are compared to theory of thin-film rupture. Comparisons of the results obtained by the two methods at various values of the Hamaker constant show that the van der Waals forces calculated from the two methods have qualitatively similar effects on coalescence. A study of the influence of the van der Waals forces on the evolution and rupture of the film separating the drops reveals that the film thins faster under stronger van der Waals forces. Strong van der Waals forces lead to nose rupture, and small van der Waals forces lead to rim rupture. Increasing the Reynolds number causes a greater drop deformation and faster film drainage. Increasing the viscosity ratio slows film drainage, although the effect is small for small viscosity ratio.     

On Failures of van der Waals’ Equation at the Gas–Liquid Critical Point

This comment is in response to a recent “new comment” by Umirzakov on the article “Gibbs density surface of fluid argon: revised critical parameters.” It was incorrectly asserted that van der Waals equation “proves” the existence of a scaling singularity with a divergent isochoric heat capacity (C_v). Van der Waals’ equation, however, is inconsistent with the universal scaling singularity concept; it erroneously predicts, for instance, that C_v is a constant for all fluid states. Van der Waals hypothetical singular critical point is based upon a common misconception that van der Waals equation represents physical reality of fluids. A comparison with experimental properties of argon shows that state functions of van der Waals’ equation fail to describe the thermodynamic properties of low-temperature gases, liquids and of gas–liquid coexistence. The conclusion that there is no “critical point” singularity on Gibbs density surface remains scientifically sound.     

Correlation for the Carbon Dioxide and Water Mixture Based on The Lemmon–Jacobsen Mixture Model and the Peng–Robinson Equation of State

Models representing the thermodynamic behavior of the CO₂–H₂O mixture have been developed. The single-phase model is based upon the thermodynamic property mixture model proposed by Lemmon and Jacobsen. The model represents the single-phase vapor states over the temperature range of 323–1074 K, up to a pressure of 100 MPa over the entire composition range. The experimental data used to develop these formulations include pressure–density–temperature-composition, second virial coefficients, and excess enthalpy. A nonlinear regression algorithm was used to determine the various adjustable parameters of the model. The model can be used to compute density values of the mixture to within ±0.1%. Due to a lack of single-phase liquid data for the mixture, the Peng–Robinson equation of state (PREOS) was used to predict the vapor–liquid equilibrium (VLE) properties of the mixture. Comparisons of values computed from the Peng–Robinson VLE predictions using standard binary interaction parameters to experimental data are presented to verify the accuracy of this calculation. The VLE calculation is shown to be accurate to within ±3 K in temperature over a temperature range of 323–624 K up to 20 MPa. The accuracy from 20 to 100 MPa is ±3 K up to ±30 K in temperature, being worse for higher pressures. Bubble-point mole fractions can be determined within ±0.05 for CO₂.     

Relationship Between Interatomic Separation and Pressure for Ionic Solids at High Temperatures

The equation of state derived from the Anderson approximation and the theory of interionic potentials based on the quantum mechanical form of the overlap repulsive energy are used to investigate the relationship between interatomic separation and pressure for three ionic solids (LiF, NaF, and CsC1) at high temperatures. The values of van der Waals dipole–dipole and dipole–quadrupole energies are also included in the model. The results obtained are in agreement with the available experimental data.     

Equation of state for a weakly ionized gas-discharge plasma

A modification is proposed for the van der Waals equation for a real gas, which transforms it into an equation of state for a thermally nonequilibrium weakly ionized gas-discharge plasma. This modification reflects the existence of collective interactions between neutral particles in the gas-discharge plasma, which generates quasistructural formations impeding the compression and expansion of the plasma. This equation completely describes the characteristics of the dynamic processes in the gas-discharge plasma, provides a quantitative explanation of these processes, and reveals the essential physical features of this phenomenon. Pis’ma Zh. Tekh. Fiz. 23, 81–88 (July 26, 1997)     

Dynamic van der waals interaction between atom and cold wall

A dynamic potential of the van der Waals interaction between a neutral atom (sodium) and a metal (gold) wall is calculated for the atom moving parallel to the surface at a velocity within 0.5–5 Bohr units. The results show that the absolute value of the interaction potential at small (0.3–3 nm) distances from the wall significantly decreases as compared to the static van der Waals potential. As the velocity decreases and/or the distance from the wall increases, the interaction potential asymptotically tends to the static values.     

State equation of mineral sands

The pressure-volume analogy between compressible fluids and macroscopic sand bodies (Ivsic et al. in Phys A, 277:47–61, 2000) is further extended using quantitative determination of corresponding empirical constants based on adapted van der Waals state equation. The isothermal constants obtained by interpretation of triaxial sand tests at so called “critical state of sand” are clearly related to the universal ideal gas properties and molar properties of mineral sands. The corresponding constants for sand and gases or other volatile liquids have the same order of magnitude. The apparent bulk repulsion/attraction effects in sand bodies are also discussed.     

The Effect of Equations of State on Simulation of Convective Flow and Heat Transfer in Near-Critical Liquids

Continuous media obeying an equation of state obtained within the mean-field model are treated in the vicinity of the critical point. This equation is an asymptotic approximation of an arbitrary two-parameter equation of state in which the first and the second derivatives of pressure with respect to density are zero and some other derivatives exist and are bounded. The thermodynamic relations, stratification, and characteristic time scales are obtained as functions of the coefficients in the equation of state. A numerical simulation is performed of convective flow and heat transfer in a square domain with side heat input; in so doing, approximate van der Waals and Redlich–Quong equations and an experimental relation are used as the equation of state. It is found that the simulation results agree qualitatively for all three cases.     

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.     

Development of a New Alpha Function for the Peng–Robinson Equation of State: Comparative Study of Alpha Function Models for Pure Gases (Natural Gas Components) and Water-Gas Systems

Numerous modifications have been suggested for the temperature dependence of the attractive term of the Peng–Robinson equation of state (PR-EOS), through the alpha function. In this work, a new alpha function combining both exponential and polynomial forms is proposed. Pure-compound vapor pressures for different molecular species were fitted and compared using different alpha functions including the Mathias–Copeman and Trebble–Bishnoi alpha functions. The new alpha function allows significant improvements of pure compound vapor pressure predictions (about 1.2% absolute average percent deviations) for all the systems considered, starting from a reduced temperature of 0.4. In addition, a generalization of the classical Mathias–Copeman alpha function was proposed as a function of the acentric factor. These alpha functions were used for VLE calculations on water+various gases including gaseous hydrocarbons. A general procedure is presented to fit experimental VLE data. The corresponding thermodynamic approach is based on the Peng–Robinson equation of state with the above cited alpha functions. It includes the classical mixing rules for the vapor phase and a Henry's law approach to treat the aqueous phase.     

Investigation of Volumetric Properties of Some Glycol Ethers Using a Simple Equation of State

In this work, a simple equation of state (EoS) has been used to predict the density and other thermodynamic properties such as the isobaric expansion coefficient, α _P , the isothermal compressibility, κ _T , and the internal pressure, P _i , of six glycol ethers including diethylene glycol monobutyl ether (DEGBE), propylene glycol propyl ether (PGPE), diethylene glycol monomethyl ether (DEGME), diethylene glycol monoethyl ether (DEGEE), triethylene glycol dimethyl ether (TriEGDME), and tetraethylene glycol dimethyl ether (TEGDME) at different temperatures and pressures. A comparison with literature experimental data has been made. Additionally, statistical parameters between experimental and calculated densities for the GMA EoS and four other EoSs (Soave–Redlich–Kwong, Peng–Robinson, Soave–Redlich–Kwong with volume translation, and Patel–Teja) indicate the superiority of the GMA EoS.     

Extension of the Neoclassical Theory of Capillarity to Advanced Cubic Equations of State

The neoclassical Redlich–Kwong (RK) theory of capillarity is extended to the Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR) equations of state. Use of the SRK and PR fluid models results in poorer predictions of interfacial tension compared to the RK model because the RK overpredicts vapor densities to a greater extent than SRK or PR, reducing the corresponding RK interfacial tension predictions to be in better agreement with accepted values. The limits of the theory applied to cubic equations are reached by proposing modified SRK and PR fluid models based on a known interfacial tension datum and knowledge of the fluid molecular structure. These modified fluid models provide improved accuracy in interfacial tension predictions of 6% (SRK) and 10% (PR) for the fluid set in this study when compared to applying the RK model (17%). These modified fluid models also provide improved predictions of bulk liquid density, but sacrifice accuracy in pressure and vapor density predictions.     

Three-dimensional analysis of the spontaneous instability for soft thin viscoelastic films

Based on the continuum mechanics and the bifurcation theory, a three-dimensional theoretical model for a soft thin viscoelastic film bonded to a rigid substrate is investigated. Considering the competition among van der Waals interaction potential energy, strain energy, and surface energy, three-dimensional governing equations of the spontaneous instability are derived, and the analytical results of time-dependent critical conditions are obtained. Furthermore, the phase diagram of instability due to van der Waals interaction and variation of the dimensionless characteristic wavenumber and the critical stiffness of interaction with the critical time are discussed.     

Modeling and thermodynamic consistency of solubility data of refrigerants in ionic liquids

Phase equilibrium data of temperature, pressure and solubility (T-P-x) of hydrofluorocarbon-type refrigerants in different types of ionic liquids over wide ranges of temperatures and pressures are modeled and tested for thermodynamic consistency. Experimental data taken from the literature for nineteen binary mixtures refrigerant + ionic liquids with a total of forty eight isothermal data sets are considered in the study. The modified Peng–Robinson equation of state proposed by Kwak and Mansoori is used for correlating the P-T-x data and a flexible thermodynamic consistency method is applied to analyze the data. Modeling is found acceptable in all cases, meaning that deviations in correlating the data are low, proving at the same time the claimed flexibility of a well-founded model that uses simple van der Waals mixing rules. Thirty eight data sets resulted to be thermodynamically consistent, nine were found to be not-fully consistent and only one set was found to be thermodynamically inconsistent.     

The Effects of Different Property Models in a Computational Fluid Dynamics Simulation of a Reciprocating Compressor

Computational fluid dynamics was applied to model a simple reciprocating compressor using R134a (1,1,1,2-tetrafluoroethane) as the working fluid. The sensitivity of the compressor model to various property models was quantitatively assessed by calculating the work required to carry out several compression cycles. The ideal gas equation, a virial equation using only the second virial coefficient, and the Peng–Robinson equation were compared to a reference-quality Helmholtz energy equation of state. Significant errors, up to 12% in the density of the outflowing gas, can result from the use of the ideal gas model. The Peng–Robinson equation resulted in density errors up to 6.3%. The virial equation gave values closest to those calculated using the Helmholtz energy equation of state, with errors in density up to 4.7%. The results also show that an increase in accuracy in work and mass flow calculations achieved by using the Helmholtz energy equation of state is obtainable without an impractical increase in computation time.     

An equation of state for pure fluids describing the critical region

Density fluctuations of a pure nuid are treated by a cell model, in which the fluid is divided into cells containing different numbers of particles. A probability function for the particle number is derived. This function, after convolution with a classical (mean field) equation of state, leads to an improved equation of state which is valid in the critical region. The equation of state is analytical, hence not exact in the immediate vicinity of the critical point. As an example, the convolution is applied to the Carnahan-Starling/van der Waals equation of state; the resulting equation of state is used to correlate thermodynamic properties of several simple fluids.