Revision of a multiparameter equation of state to improve the representation in the critical region: application to water

We have developed a crossover formalism for the thermodynamic surface of pure fluids, which can be applied to any multiparameter equation of state. This procedure has been used to incorporate scaling law behavior into a representation of the thermodynamic properties of water and steam developed by Pruss and Wagner (PW EOS) and adopted recently by the International Association for the Properties of Water and Steam. Our revision to this equation retains most of the functional form and coefficients of the PW EOS, but replaces two of the terms with a crossover representation of scaling law behavior. In order to develop this model, we first developed a new crossover formulation for steam in the critical region, and second, we have incorporated universal crossover functions into the original PW EOS. In the modified form, the PW equation of state reproduces the scaling laws down to dimensionless temperatures τ=10⁻⁷. Far from the critical point the equations practically coincide.     

Crossover Peng-Robinson equation of state with introduction of a new expression for the crossover function

In this research, we use the original Peng-Robinson (PR) equation of state (EOS) for pure fluids and develop a crossover cubic equation of state which incorporates the scaling laws asymptotically close to the critical point and it is transformed into the original cubic equation of state far away from the critical point. The modified EOS is transformed to ideal gas EOS in the limit of zero density. A new formulation for the crossover function is introduced in this work. The new crossover function ensures more accurate change from the singular behavior of fluids inside the regular classical behavior outside the critical region. The crossover PR (CPR) EOS is applied to describe thermodynamic properties of pure fluids (normal alkanes from methane to n-hexane, carbon dioxide, hydrogen sulfide and R125). It is shown that over wide ranges of state, the CPR EOS yields the thermodynamic properties of fluids with much more accuracy than the original PR EOS. The CPR EOS is then used for mixtures by introducing mixing rules for the pure component parameters. Higher accuracy is observed in comparison with the classical PR EOS in the mixture critical region.     

A crossover lattice fluid equation of state for pure fluids

A lattice fluid model is one of the most versatile, molecular-based engineering equations of state (EOS) but, in common with all analytic equations of state, the lattice fluid (LF) EOS exhibits classical behaviour in the critical region rather than the non-analytical, singular behaviour seen in real fluids. In this research, we use the LF EOS and develop a crossover lattice fluid (xLF) equation of state near to and far from the critical region which incorporates the scaling laws valid asymptotically close to the critical point while reducing to the original classical LF EOS far from the critical point. We show that, over a wide range of states, the xLF EOS yields the saturated vapour pressure data and the density data with much better accuracy than the classical LF EOS.     

Modification of Simha–Somcynsky equation of state for small and large molecules

The Simha–Somcynsky equation of state (SS EOS) represents the PVT behavior of polymers quite satisfactorily, but cannot be applied to gases at low pressures. This work proposes a modification of the free volume contribution of the SS EOS to allow representation of gaseous state of low molecular-weight substances by introducing the perturbed hard-chain theory of Beret and Prausnitz into the EOS. In addition to this modification, two universal constants are introduced to the free volume term for better representation of properties of low molecular-weight substances. Characteristic parameters in the modified SS EOS were determined for 44 low molecular-weight substances and 64 polymers. The absolute average deviations (AADs) for critical temperature, critical pressure and vapor pressures were 0.86, 2.38 and 2.01%, respectively, while AAD for critical density and saturated liquid density at normal boiling point were somewhat larger, being 20.46 and 5.30%, respectively. The high performance of the original SS EOS for polymer PVT behavior was maintained in the modified EOS with grand AAD of 0.050% for densities.     

Crossover SAFT equation of state for pure supercritical fluids

A crossover statistical associating fluid theory (SAFT) equation of state (EOS) is used to fit the parameters of eight common pure supercritical fluids (water, ammonia, carbon dioxide, R134a, ethane, propane, ethene and propene) and calculate their thermodynamic properties. Over a wide range including the critical region, the EOS reproduces the saturated pressure data with an average absolute deviation (AAD) of about 1% and the saturated densities with an AAD of about 2%. In the one-phase region, the EOS represents the experimental values of pressure with an AAD of about 1–3%. The results are satisfactory.     

P–V–T properties of polymer melts based on equation of state and neural network

An accurate and efficient analytical equation of state (EOS) and artificial neural network (ANN) methods are developed for the prediction of volumetric properties of polymer melts. To apply EOS, the second virial coefficients B₂(T), effective van der Waals co-volume, b(T) and correction factor, α(T) were determined. The second virial coefficient was calculated from a two-parameter corresponding states correlation, which is constructed with two constants as scaling parameters, i.e., temperature (T_f) and density at melting (ρ_f) point. The new correlations were used to predict the specific volumes of polypropylene glycol (PPG), polyethylene glycol (PEG), polypropylene (PP), polyvinylchloride (PVC), poly(1-butene)(PB1), poly (?-caprolactone) (PCL), polyethylene (PE) and polyvinylmethylether (PVME) at compressed state in the temperature range of 298.15–634.6 K. The obtained results show that the two models have good agreement with the experimental data with absolute average deviation of 0.28% and 0.39% for ANN and EOS, respectively. The Comparison of the results with ISM model shows that the proposed models represent an efficient method and are more accurate.     

Volumetric profile of polymer melts from the ISM equation of state

Knowledge of the volumetric or pressure–volume–temperature (PVT) profile of molten polymers is important for both engineering and polymer physics. Ihm–Song–Mason (ISM) equation of state (EOS) has been employed to predict the volumetric properties of 12 molten polymers. The significance of the present paper is three temperature-dependent parameters of the ISM EOS to be determined using corresponding states correlations based on the molecular scaling constants, dispersive energy parameters between segments/monomers (ε) and segment diameter (σ) rather than bulk properties, e.g. the liquid density and temperature both at normal boiling point. The ability of the ISM EOS has been evaluated by comparing the results with 1390 literature datapoints for the specific volumes over the temperature range from 293 to 603.5 K and pressure range from 0.1 to 200 MPa. The average absolute deviation (AAD) of the calculated specific volumes from literature data was found to be 0.52%. The isothermal compressibility coefficients, κ_T values of molten polymers have also been predicted using the ISM EOS. From 684 datapoints examined, the AAD of estimated κ_T was equal to 7.55%. Our calculations on the volumetric and thermodynamic properties of studied polymers reproduce the literature data with reasonably good accuracy.     

A nonparametric scaling equation of state for liquids

A new nonparametric scaling equation of state is suggested. The equation correctly describes the p-ρ-T data and heat capacities of liquids close to the critical vaporization points. It was obtained with the use of the S spinodal and the mixing of scaling fields as a first approximation (asymmetric and nonasymptotic terms were ignored). The new equation was used to approximate the data on ⁴He, C₂H₄, and H₂O in the critical region. The results showed that it correctly described the critical behavior of thermodynamic functions, including isochoric heat capacity, not only in the asymptotic but also over a fairly wide density region at the critical point. The suggested equation of state describes the p-ρ-T data with the same error as the Schofield parametric equation of state. The new equation, however, better reproduces the behavior of heat capacities and is much simpler to use. As distinct from the Schofield equation, the new equation, like classic equations of state, allows the spinodal to be determined from the (?p/?v) _T = 0 condition at T < T _c .     

Simple modification of the temperature dependence of the Sanchez–Lacombe equation of state

In this work, the interaction energy term of the Sanchez–Lacombe equation of state (SL EOS) was modified to take into account the temperature dependence of hydrogen bonding and ionic interactions. A simple function was used in the form of the Langmuir equation that reduces to the original SL EOS at high temperature. Comparisons are shown between the ?*-modified SL EOS and the original SL EOS. The ?*-modified SL EOS could represent volumetric data for the group of non-polar fluids, polar fluids and ionic liquids to within an absolute average deviation (AAD) of 0.85%, 0.51%, and 0.054%, respectively whereas, the original Sanchez–Lacombe EOS gave AAD values of 0.99%, 1.2%, and 0.21%, respectively. The ?*-modified SL EOS provides remarkably better PVT representation and can be readily applied to mixtures.     

A crossover SAFT-VR equation of state for pure fluids: preliminary results for light hydrocarbons

SAFT is perhaps the most versatile, fundamentally, based engineering equation of state in use today. However, in common with all analytic equations of state, SAFT exhibits classical behavior in the critical region rather than the non-analytical, singular behavior seen in real fluids. Recently, so-called crossover equations of state have been developed which solve this shortcoming by incorporating the scaling laws valid asymptotically close to the critical point while reducing to the original classical equation of state far from the critical point. We have combined the SAFT-VR equation of state with an analytical crossover technique to obtain the SAFT-VRX equation of state. The SAFT-VRX approach combines the accurate low temperature behavior of SAFT-VR with a precise representation of the critical region. Preliminary results are presented for hydrocarbon systems which illustrate the accuracy of the SAFT-VRX approach over the entire fluid phase region.     

Thermodynamic properties of synthetic natural gases: Part 4. Dew point curves of synthetic natural gases and their mixtures with water: measurement and correlation

Experimental measurements of dew points for five synthetic natural gases (SNG)+water mixtures were carried out between 2.1×10⁵ and 73.2×10⁵ Pa in the temperature range from 224.3 to 270.2 K. The experimental results were analysed in terms of both an equation of state (EOS) model and an excess function–EOS method, which reproduced the experimental data with an AAD from 2.1 to 3.4 K and from 1.9 to 3.0 K, respectively.     

Dew points of binary carbon dioxide + water and ternary carbon dioxide + water + methanol mixtures: Measurement and modelling

Dew points for four carbon dioxide + water mixtures between 1.2×10⁵ and 41.1×10⁵ Pa in the temperature range from 251.9 to 288.2 K, and eight carbon dioxide + water + methanol mixtures between 1.2×10⁵ and 43.5×10⁵ Pa and temperatures from 246.0 to 289.0 K were experimentally determined. The experimental results obtained on the binary and ternary systems were analysed in terms of a predictive excess function–equation of state (EF–EOS) method, which reproduced the experimental dew point temperature data with absolute average deviation (AAD) between 0.8 and 1.8 K for the systems with water, and from 0.0 to 2.7 K for the systems with water and methanol. The experimental results obtained for carbon dioxide + water mixtures, with molar fraction of water lower than 0.00174, at pressure values higher than 5×10⁵ Pa were also compared to a predictive equation of state model. It reproduced experimental dew point temperature data with AAD between 0.2 and 0.6 K.     

A crossover cubic equation of state near to and far from the critical region

In this research, we use the Patel–Teja (PT) cubic equation of state [N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473.] and develop a crossover cubic model near to and far from the critical region, which incorporates the scaling laws asymptotically close to the critical point and it transformed into original classical cubic equations of state far away from the critical point. This equation of state is used to calculate thermodynamic properties of pure systems (carbon dioxide, normal alkanes from methane to heptane). We show that, over a wide range of states, the equation of state yields the saturated vapour pressure data and the saturated density data with a much better accuracy than the original PT equation of state.     

Density measurements of liquid 2-propanol at temperatures between (280 and 393) K and at pressures up to 10 MPa

Liquid densities for 2-propanol have been measured at T = (280, 300, 325, 350, 375, and 393) K from about atmospheric pressure up to 10 MPa using a vibrating tube densimeter. The period of vibration has been converted into density using the Forced Path Mechanical Calibration method. The R134a has been used as reference fluid for T ? 350 K and water for T > 350 K. The uncertainty of the measurements is lower than ±0.05%. The measured liquid densities have been correlated with a Starling BWR equation with an overall AAD of 0.025%. The same BWR equation agrees within an AAD lower than 0.2% with the experimental values available in the literature over the same temperature and pressure range.     

A new coordination number model by using the local composition theory

A new coordination number model for square-well (SW) fluid is proposed in this work. It is based on the local composition theory and starts from the point that total site numbers around one molecule can vary with reduced density and reduced temperature. The total site numbers are correlated with reduced density and reduced temperature by using local composition theory and Monte Carlo (MC) simulation results. This new model agrees well with the MC simulation results for the pure SW fluids at the low-density limit and in the high-density region. In addition, it can predict the transition to the face-centered cubic (fcc) fluid property which is shown in Young and Alder’s [J. Chem. Phys. 73 (1980) 2430] phase diagram. An equation of state (EOS) is derived from this new coordination number model and gives satisfactory compressibility factors for SW fluids.     

Predicting mixture phase equilibria and critical behavior using the SAFT-VRX approach

The SAFT-VRX equation of state combines the SAFT-VR equation with a crossover function that smoothly transforms the classical equation into a nonanalytical form close to the critical point. By a combinination of the accuracy of the SAFT-VR approach away from the critical region with the asymptotic scaling behavior seen at the critical point of real fluids, the SAFT-VRX equation can accurately describe the global fluid phase diagram. In previous work, we demonstrated that the SAFT-VRX equation very accurately describes the pvT and phase behavior of both nonassociating and associating pure fluids, with a minimum of fitting to experimental data. Here, we present a generalized SAFT-VRX equation of state for binary mixtures that is found to accurately predict the vapor-liquid equilibrium and pvT behavior of the systems studied. In particular, we examine binary mixtures of n-alkanes and carbon dioxide + n-alkanes. The SAFT-VRX equation accurately describes not only the gas-liquid critical locus for these systems but also the vapor-liquid equilibrium phase diagrams and thermal properties in single-phase regions.     

Quintic equation of state for pure substances in sub- and supercritical range

A new quintic equation of state (EOS) for pure substances and mixtures is proposed. The equation is based on critical parameters and one saturation point. The proposed Q5EOS is a generalisation of many cubic equations of state. Equation Q5 has five parameters, four of which are temperature-independent. The temperature-dependent parameter a is expressed by a relation based on a simple power function. Parameters defining this function can be calculated from saturation data, Boyle temperature and supercritical data.     

Common compression factor and bulk modulus quiescence points of liquid cesium

In this paper, a recently developed analytical equation of state (EOS) is used to investigate the bulk modulus of compressed liquid cesium and to locate common compression factor and the bulk modulus quiescence point(s). This EOS is applied quick well to Na and Rb far from T_c. Bulk modulus of liquid cesium have two quiescence points, a sharp one in the range 1100–1500 K and a diffused one in the range 1600–1900 K. Therefore, two types of liquid cesium metal may be identified with characteristic structure and interaction potential energy. It is a constant independent of temperature, however, some residual change is seen due to the change in the values of integral of pair correlation function as temperature is increased. Furthermore, it is related to the shape of the unit cell and the atomic size at equilibrium. Observation of distinct liquid in the metal–nonmetal transition range is compared with NMR studies and molecular dynamic results.     

Equation of state for molten alkali metal alloys

Results on the density of binary and ternary alkali metal alloys of Cs–K, Na–K, Na–K–Cs, at temperatures from the freezing point up to several hundred degrees above the boiling point are presented. The theoretical equation of state is that of Ihm, Song, and Mason. The second virial coefficients, B(T), are calculated by using the corresponding states correlation of Boushehri and Mason. Calculation of the other two temperature-dependent parameters, α(T) and b(T), are performed by scaling rules with the latent heat of vaporization and the freezing point density as scaling constants. The results are within 5%.     

A new equation of state for metal alloys

A perturbed hard-trimer (PHT) equation of state (EOS) has been employed to model volumetric properties of molten metals and their alloys considering a trimer expression obtained from the statistical associating fluid theory. The van der Waals dispersion forces were applied as perturbation term. Two parameters appeared in the PHT EOS, reflecting the dispersive energy among trimers, ε and the hard-core diameter σ were determined based on the molecular scaling parameters. The performance of the proposed PHT EOS has been evaluated by predicting the saturated and isobaric densities of 16 molten metals including alkali metals, alkali earth and refractory metals over the temperature range within 234–7400 K and pressures up to 436 MPa. From 677 data points examined, the average absolute deviation (AAD) of the predicted densities from the experimental ones was found to be 1.60%. Furthermore, the estimated uncertainties of predicted densities of alloys were ±3.00%.