Simple Cubic Equations Of State Research Articles

Compression/dilation of condensed matter vis-à-vis an ideal symmetric material

Both the isothermal Helmholtz energy and the adiabatic internal energy of condensed matter exhibit minima at zero pressure. As a consequence, the leading term in a density expansion of these two thermodynamic potentials is quadratic in density displacement. When the quadratic term dominates, the material behaves symmetrically in response to isotropic compressive and tensile forces. From an atomistic viewpoint, compressing or stretching atomic bonds to the same degree in a symmetric material increases a corresponding thermodynamic potential by the same amount. The quadratic term contributes to the pressure as a simple cubic equation of state (EOS). Among 29 metals and 17 inorganic solids surveyed, only metallic gold satisfies this simple EOS to the highest measured compressions (>40%). Other solids such as Pt and NaCl also follow this simple EOS to significant compressions and all materials should follow it at low compressions/dilations. A thermodynamic protocol is proposed to extract the bulk modulus ${B}_{0}$ as well as higher order modulus pressure coefficients,${B}_{1},{B}_{2},\mathrm{and}{B}_{3},$ from least squares smoothed data without taking derivatives or by appealing to an EOS model. Reliable and unbiased experimental values of these higher order modulus coefficients have been obtained for 46 solids. These moduli are strongly correlated and satisfy the relationship ${B}_{0}^{2}{B}_{3}\ensuremath{\simeq}\ensuremath{-}2{B}_{0}{B}_{1}{B}_{2}$. This relation obtains exactly from an adiabatic EOS based on the Mie (m-n) potential and to a high degree of approximation for the well-known Birch-Murnaghan and Vinet EOS. Truncating the thermodynamic expansion at the fifth order term with only the pressure coefficient ${B}_{1}$ as a parameter, this approximate thermodynamic EOS models experimental values of ${B}_{2}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{B}_{3}$ better than either the third order Birch-Murnaghan or Vinet EOS.

Thermodynamic modeling of hydrogen solubility in a series of ionic liquids

In this study, the Perturbed Hard Sphere Chain Equation of State (PHSC EoS) has been employed to predict the hydrogen solubility in a series of ionic liquids. As ionic liquids have no vapor pressure and no critical parameters and as hydrogen is a light molecule which behaves like a perfect gas, simple cubic equations of state cannot be used for modeling of H2 + ionic liquid mixture. Three main parameters of non-cubic PHSC EoS, i.e. (r) the number of segments per molecule, (σ) the distance between two molecules at zero potential and (ε) well-depth of potential between two non-bonded units have been calculated by regression of the experimental Pressure-Volume-Temperature (PVT) data points. The hydrogen solubility decreases with increasing temperature in [BMIM][BF4], [BMIM][C8SO4], [EMIM][EtSO4], [MDEA][Cl], and [N4,1,1,1][Tf2N], but inverse temperature effect is observed in [BMIM], [BMPY], [EMIM] and [HMIM][Tf2N] as well as [BMIM][MeSO4]. The binary interaction parameter which has been obtained from fitting of the equation of state with experimental hydrogen absorption data shows both the usual and the opposite trend of temperature effect as well.

A viscosity model for ionic liquids based on the Eyring's theory and a cubic EoS

A thermodynamic model based on the combined use of the Eyring's activated state theory and a cubic equation of state was developed here to accurately represent the dynamic viscosity of pure ionic liquids (ILs). Within the Eyring's theory, the net viscous flow of a pure IL is assumed to be governed by four main variables: (1) the energy necessary for a molecule to jump from an initial equilibrium position to a new one, (2) the energy necessary to break the molecular bonds to create a hole (vacant sites) of molecular size in the liquid, (3) the availability of the vacant sites, and (4) the frequency or the mean residence time of the jumping molecules. The various activation-state variables were then related to well-known thermodynamic potentials that in turn were estimated from two simple cubic equations of state of the van der Waals type (Soave or Peng-Robinson). The resulting model was successfully validated during the representation of experimental dynamic viscosities of three families of imidazolium-based ILs ([CXmim][BF4], [CXmim][PF6] and [CXmim][Tf2N]), four pyridinium-based ILs ([bmpy][BF4], [empy][EtSO4], [Et2Nic][EtSO4] and [hemmpy][Tf2N]) and two ammonium-based ILs ([cpmam][MeSO4] and [4bam][doc]) within a temperature range varying from 0 to 80 °C and at pressures from 1 up to 3000 bar thus covering a wide viscosity range of 10–19,610 mPa-s.

Viscosity Modeling of Ionic Liquids Using the Friction Theory and a Simple Cubic Equation of State

In the present work, the well-known friction theory (FT) based on friction concepts of classical mechanics and the van der Waals theory of fluids has been modified to accurately represent the dynamic viscosity of pure ionic liquids in a rather simplified manner. Unlike previous FT applications to pure ionic liquids, the dilute gas limit for viscosity was excluded from the present model since it is negligible for ionic liquids which exhibit extremely low vapor pressures; the friction term thus prevailed for viscosity calculations. The latter was expressed in terms of new temperature-dependent friction coefficients whereas the repulsive and attractive pressure terms were in turn estimated from two simple cubic equations of state (Soave or Peng–Robinson) rather than using sophisticated multiparameter equations of state such as SAFT- or CPA-based expressions previously used by other authors. The resulting model was successfully validated during the representation of experimental dynamic viscosities of three f...

Calculation of critical points from Helmholtz-energy-explicit mixture models

The calculation of the critical points for a mixture of fluids is of practical interest as the calculated critical points can be used to more reliably and efficiently construct phase envelopes. The number of stable critical points found can also provide insight into whether the mixture has an open or closed phase envelope.In this work we have developed a reliable method for determining all the critical points for a mixture that is modeled with Helmholtz-energy-explicit equations of state. This method extends the algorithms developed in the literature for simpler equations of state to these more complex mixture models. These Helmholtz-energy-explicit equations of state could be either multi-fluid models or transformations of simple cubic equations of state to Helmholtz-energy-explicit forms. This algorithm locks onto the first criticality contour (the spinodal) and traces it to high density, thereby locating all relevant critical points. The necessary analytic derivatives of the residual Helmholtz energy, numerically validated values of the derivatives for validation, sample code, and additional figures and information are provided in the supplemental material.

Modeling the Dynamic Viscosity of Ionic Solutions

In the present work, an Eyring-theory model based on concepts of excess Gibbs energy of activation of the viscous flow has been developed for the accurate correlation or prediction of the dynamic viscosity of ionic solutions: inorganic salt (electrolyte) + solvent and organic salt (ionic liquid) + solvent. For the excess Gibbs energy of activation (GEX,≠), both thermal and mechanical contributions to the viscous flow were considered. Accordingly, a thermal GEX,≠ term was described by mixing rules of the Redlich–Kister-type, whereas the mechanical GEX,≠ term was computed from a simple cubic equation of state in an attempt to overall represent the main molecular interactions (between the ionic species and the solvent) affecting viscosity. The resulting model was successfully validated during the representation of experimental dynamic viscosities of various nonaqueous and aqueous ionic solutions within wide ranges of temperature and composition (or salt molality).

Simulating the capture of CO2 from natural gas: New data and improved models for methane + carbon dioxide + methanol

The simulation of carbon capture unit operations often involves predicting the vapor liquid equilibrium (VLE) for mixtures containing polar, non-polar and quadrupolar compounds. In this work, we investigate how well a simple cubic equation of state (EOS) can predict the results of new low temperature, high-pressure VLE measurements of the ternary methane+carbon dioxide+methanol system, which is important to the Rectisol process used for capturing CO2 from natural gas. The ternary p, T, x measurements presented here are the first such data for this system reported in the open literature. First, predictions made with the Peng Robinson (PR) EOS as implemented in a commercial process simulator were compared to binary p, T, x data measured in this work and also taken from the literature. Significant deviations were found between the measured liquid mole fractions and those predicted with the EOS using the default binary interaction parameters (BIP): the relative standard errors were 39%, 77% and 17% for the methane+methanol, methane+carbon dioxide and methanol+carbon dioxide binaries, respectively. Regression of the PR EOS to the binary VLE data by adjusting the BIPs improved the liquid phase mole fraction predictions for the ternary mixture data by a factor of about 2.5 for methane and methanol. However, improvement by a factor of 4.4 in the carbon dioxide liquid mole fraction was achieved by describing the carbon dioxide+methanol binary with an asymmetric composition and temperature dependent mixing rule and tuning the BIPs therein to VLE data for this binary over a wide temperature range. This reduced the standard error in the liquid phase CO2 mole fractions predicted for the ternary mixture using the optimized model by 79% relative to the default model.

Uncertainty analysis applied to thermodynamic models and process design—2. Binary mixtures

A simple procedure is proposed to evaluate the uncertainty on binary interaction parameters propagated from the uncertainties present in the physical properties and equation parameters applied for their calculation using a simple cubic equation of state. A useful database containing 87 binary mixtures present in natural gas processing was constructed through critical collection of available experimental vapour–liquid equilibrium (VLE) data and their associated uncertainties.A thermodynamic consistency test was performed on each isothermal dataset to investigate the quality of the associated VLE data. Upon acceptance of the VLE data based on its quality and consistency, binary interaction parameters and associated uncertainties were determined using a combination of nonlinear regression and Monte Carlo simulation, taking into consideration the uncertainties of pure components, equation of state parameters, and VLE data. The Monte Carlo simulation was also used for the error propagation to estimate the uncertainty on the calculated VLE. Sample calculations were presented illustrating the effect of uncertainties on the PXY and TXY diagrams of ethane/propane and methane/hydrogen sulfide binary mixtures. The required minimum number of stages for a simplified de-ethanizer was calculated taking into account uncertainties of basic input parameters. In addition the effect of uncertainties on the position of calculated cricondenbar and cricondentherm was investigated.

Simple cubic equation of state applied to hard-sphere, Lennard-Jones fluids, simple fluids and solids

Based on the expansion and extension of the virial equation of state (EOS) of hard-sphere fluids solved by the Percus–Yevick integration equation, a universal cubic (UC) EOS is developed. The UC EOS is applied to model hard-sphere and Lennard-Jones (LJ) fluids, simple Ar and N2 liquids at low temperatures, and supercritical Ar and N2 fluids at high temperatures, as well as ten solids, respectively. The three parameters are determined for the hard-sphere fluid by fitting molecular dynamics (MD) simulation data of the third to eighth virial coefficients in the literature; for other fluids by fitting isothermal compression data; and for solids by using the Einstein model. The results show that the UC EOS gives better results than the Carnahan–Starling EOS for compressibility of hard-sphere fluids. The Helmholtz free energy and internal energy for LJ fluids are predicted and compared with MD simulation data. The calculated pressures for simple Ar and N2 liquids are compared with experimental data. The agreement is fairly good. Eight three-parameter EOSs are applied to describe isothermals of ten typical solids. It is shown that the UC EOS gives the best precision with correct behavior at high-pressure limitation. The UC EOS considering thermal effects is used to analytically evaluate the isobaric thermal expansivity and isothermal compressibility coefficients. The results are in good agreement with experimental data.

Predicting the Viscosity of Asymmetric Hydrocarbon Mixtures with the Expanded Fluid Viscosity Correlation

Mass-based mixing rules are proposed for the Expanded Fluid viscosity correlation that are suitable for asymmetric mixtures and replace the original volumetric mixing rules. The Expanded Fluid correlation provides viscosity values as a function of fluid density and characterizes each pure compound with three fluid-specific parameters: c2, ρso and c3, when using experimental densities and two parameters, c2, ρso, when using a cubic equation of state. The proposed set of mixing rules predicts the viscosity of over 90 binary mixtures with overall average absolute relative deviations (AARD) of 2.9% and 7.8% using measured densities and densities estimated from an equation of state, respectively. To improve the predictions in equation of state applications, a binary interaction parameter is introduced to the mixing rules, compensating for inaccuracies associated with density predictions from simple cubic equations of state. Using fitted interaction parameters, the overall AARD is 3.6%. The binary interaction p...

Modeling of Highly Nonideal Systems: 2. Prediction of High Pressure Phase Equilibria with the Group Contribution NRTL-PR EoS

The purpose of this study is to propose a simple cubic equation of state for the prediction of phase equilibria in highly non ideal systems. The NRTL-PR equation of state is based on the association of the Peng−Robinson EoS and the generalized NRTL model developed in the first part of this work. This model, which derives from the lattice two-fluid theory, was specially developed to take account for the differences in size and shape between mixture components. The main advantage of the resulting NRTL-PR equation of state is to yield a simple and predictive model, depending on very few interaction parameters. In addition, it also allows satisfactory predictions of excess enthalpies, liquid−vapor and liquid−liquid equilibria occurring in mixtures formed by very asymmetric compounds, such as hydrocarbons, water, and ethylene glycols.

Measurements and calculations of hydrocarbon mixtures liquid density by simple cubic equations of state

The continuing importance of pressure-volume-temperature (PVT) property calculations of reservoir fluids mixtures in design of equipments used for storage, transportation and separation processes need considerable experimental and theoretical attempts to present a simple, accurate and theoretical based equation of state to replace the existing empirical based equations of state. The first author and his co-workers previously presented two parameters cubic theoretical based equation of state (MMM). This equation was used in PVT and vapour-liquid-equilibrium (VLE) calculations of pure and fluid mixtures. The comparisons for the different pure and mixture fluids calculations indicated the advantages of this equation of state over the well-known empirical-based two parameters cubic equation of state (RK EOS). In this study, PVT data for binary, ternary, quaternary and multicomponent oil mixtures at the different temperature and ambient pressure are measured, and reported. The measured PVT data were compared with those obtained by using the simple cubic equations of state. The calculated results by MMM EOS are compared with SRK and PR EOSs which are the commonly used three parameters cubic EOSs in the oil and gas calculations. The comparisons indicate that, although the SRK and PR EOSs needs an additional parameter (acentric factor, ω), the MMM equation of state is more accurate than both SRK and PR equations for the PVT calculations of oil mixtures.

Modeling solubilities of subcritical and supercritical fluids in polymers with cubic and non-cubic equations of state

The original statistical associating fluid theory (CK-SAFT), Perturbed-Chain SAFT (PC-SAFT), Sanchez–Lacombe (SL) and Sako–Wu–Prausnitz (SWP) equations of state (EoS) are used to model the solubilities of subcritical and supercritical fluids such as N 2, C 2H 4, C 4H 10, iso-C 4H 10, CO 2 in molten, thermally softened or semicrystalline polymers. Classical van der Waals one-fluid mixing rule with only one adjustable binary parameter is used for all the models. Modeling results are compared with the experimental data for 21 binaries including 78 systems published in open literature. Good agreement can be obtained with the PC-SAFT and SL EoS where the polymers are molten or thermally softened. Moreover, SL EoS gives a reasonable representation of the sorption data for semicrystalline polymers but the results are less satisfactory when compared with amorphous polymer systems. The performance of the CK-SAFT is least satisfactory in most case. Furthermore, SWP EoS provides an encouraging results although it is only a simple cubic equation of state.

Eyring-Theory-Based Model To Estimate Crude Oil Viscosity at Reservoir Conditions

The well-known absolute reaction rate theory of Eyring was successfully extended to the accurate estimation of crude oil viscosities over a wide pressure range: from dead to live oil conditions. Two simple cubic equations of state (Soave or Peng−Robinson) were incorporated into the viscosity model to calculate all of the thermodynamic properties required within the Eyring’s modeling framework. Satisfactory density and viscosity representations of various crude oils (with low-to-moderate molar masses) were achieved by using two tuning parameters valid over a wide pressure range.

Structure-Composition Relationships in Ternary Solvents Containing Methylbenzoate

Thermophysical properties of the hexane+1-chlorohexane (or hexanoic acid or diisopropylether)+methylbenzoate ternary systems and their binary constituents are reported at 298.15 K and 0.1 MPa over the whole composition range. The properties and the optimized geometry of the gas-phase components were appraised from the density functional theory. To find out the causal link between the thermophysical measurements and the molecular level features, the derived mixing and excess functions of the ternary systems were looked into according to the scaled particle and Kirkwood-Buff analyses. The hydrogen bonding and dipole interactions along with the geometry effects brought about by the very different size and shape of the components give rise to complex mixed structures. Application of semiempirical models and use of simple cubic equations of state combined with a one-parameter van der Waals mixing rule has led to prediction of the ternary properties with variable degree of precision.

Correlation and prediction of interface tension for fluid mixtures: An approach based on cubic equations of state with the wong-sandler mixing rule

Concentration profiles, interface thickness, and interface tensions have been calculated for mixtures applying the gradient theory to the Peng-Robinson equation of state. The approach is based on an accurate local fit of vapor-liquid equilibrium (VLE) data, and, for this purpose, the flexibility of the original Wong-Sandler mixing rule has been taken into account. Correlation and prediction capabilities of experimental interfacial tension data are analyzed for the quadratic mixing rule and the present approach. The method, which is discussed in detail in this work, provides an improved scheme for calculating interfacial properties, both for polar and nonpolar mixtures. According to results, a better correlation and prediction of the interfacial tensions can be obtained from an adequate interpolation of the VLE, using simple cubic equations of state. Examples are presented for binary and ternary mixtures.

Solubilities and Diffusivities of Carbon Dioxide in Ionic Liquids: [bmim][PF6] and [bmim][BF4

Gaseous carbon dioxide (CO2) absorption measurements in l-n-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF6]) and l-n-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF4]) were made using a commercial gravimetric microbalance at temperatures of 283.15, 298.15, 323.15, and 348.15 K and at pressures under 2 MPa. Gas solubilities were determined from absorption saturation (equilibrium) data at each fixed temperature and pressure. The experimental solubility data of CO2 in ionic liquids have been successfully correlated for the first time using a simple cubic equation of state (EOS). Some phase behaviors of CO2 and ionic liquid mixtures as well as Henry's law constants and the volume change of solutions are predicted well with the present EOS model. From time-dependent absorption data, binary diffusion coefficients of CO2 in the ionic liquids have been obtained for the first time by the use of a simple diffusion model. Magnitudes in the observed diffusion coefficients are 10-10 to 10-11 m2 s-1, which are about 10−100 times lower than typical values, found in various organic liquids.

Friction theory prediction of crude oil viscosity at reservoir conditions based on dead oil properties

The general one-parameter friction theory ( f-theory) models have been further extended to the prediction of the viscosity of real “live” reservoir fluids based on viscosity measurements of the “dead” oil and the compositional information of the live fluid. This work representation of the viscosity of real fluids is obtained by a simple one-parameter tuning of a linear equation derived from a general one-parameter f-theory model. Further, this is achieved using simple cubic equations of state (EOS), such as the Peng–Robinson (PR) EOS or the Soave–Redlich–Kwong (SRK) EOS, which are commonly used within the oil industry. In sake of completeness, this work also presents a simple characterization procedure which is based on compositional information of an oil sample. This procedure provides a method for characterizing an oil into a number of compound groups along with the critical constants and parameters that are required by an EOS. The resulting EOS characterized fluid correctly reproduces properties such as the saturation pressure and the liquid phase density, from reservoir conditions to low pressure conditions. The viscosity modeling approach along with the characterization method presented in this work provide a complete self-consistent procedure for the viscosity and density modeling of reservoir fluids.

Critical properties ( pc, Tc, and ρc) and phase equilibria of binary mixtures of CO 2, CHF 3, CH 2F 2, and SF 6

Measurements of the critical properties p c, T c, and ρ c of the pure substances CO 2, CHF 3, CH 2F 2, SF 6 and their binary mixtures were performed by observing the critical opalescence in a static equilibrium cell. The experimental uncertainties of the critical properties are ±6 kPa, ±60 mK, and ±2% for the determination of the density. The composition was determined by gas-chromatographic analysis within ±0.3 mol%. The experimental results for the binary mixtures are described by several simple cubic equations of state with van der Waals mixing and combining rules, using only the experimental critical data of the pure components and those of an approximately equimolar mixture to fit the binary interaction parameters. The capability of the cubic equations of state with these binary interaction parameters not only to describe the critical loci but also to describe VLE data is demonstrated by additional VLE measurements for the binary systems (CHF 3+SF 6) and (SF 6+CH 2F 2). Furthermore, VLE data published by other authors for some of the systems investigated are considered.

One parameter friction theory models for viscosity

In this work the friction theory ( f-theory) for viscosity modeling is used in conjunction with the SRK, PR and PRSV cubic equations of state in order to develop three one parameter general models for viscosity prediction. The models are considered one parameter models because they only require a characteristic critical velocity, which is a parameter normally not tabulated. The models use these rather simple cubic equations of state as a basis to obtain accurate modeling of the viscosity of fluids for wide ranges of temperature and pressure. The general models presented in this work are based on the viscosity behavior of n-alkanes from methane to n-octadecane. Although best performance is obtained for the considered n-alkanes, a good model performance is also obtained for other systems. Thus, recommended characteristic critical viscosity values for several systems are also reported in this work. However, in the case of n-alkanes, an empirical equation for the characteristic critical viscosity is provided so that no additional parameters are required. In addition, with the use of simple mixing rules, the viscosity of several binary to quaternary n-alkane mixtures can also be predicted with a satisfactory accuracy.