Analytical Equation Of State Research Articles

Phase equilibria of binary mixtures with alkanes, ketones, and esters based on the Transferable Anisotropic Mie force field

The transferable anisotropic Mie potential (TAMie) force field is extended to acetates and formates. Intramolecular bending and torsion potentials were obtained from quantum chemical calculations. The parameters defining the Mie pair potential, representing the van der Waals interactions, as well as point charges were optimized. These force field parameters were optimized based on an objective function that considers vapor pressure and liquid density data of several members of one chemical group simultaneously. Like in earlier work, the parameterization is conducted using an analytic equation of state to locally approximate the objective function, ensuring swift convergence of the optimization problem. Acetates, from ethyl ester to n-pentyl ester, have average errors (AAD) of 0.9–6.9% in vapor pressure and of 0.4–0.7% in liquid densities. Formates, from ethyl formate to n-butyl formate have average errors of 2.0–4.2% in vapor pressure and 0.8–1.1% for liquid densities for reduced temperature ranges of T/Tc=0.56 to 0.96, where Tc is the critical temperature of the considered substance. For a binary mixture of ethyl acetate with n-hexane results of the TAMie force field are found in good agreement to experimental data. Because phase equilibrium data for esters with alkanes is scarce, we also investigate mixtures of ketones with alkanes. For mixtures of butanone with n-hexane and butanone with n-heptane, molecular simulations are in satisfying agreement to experimental data when Berthelot-Lorentz combining rules are employed. After adjusting corrections to the Berthelot combining rules the agreement of molecular simulations to experimental data is very good. Additionally to phase equilibria, we calculated excess properties for the mixture of hexane and butanone, which also show very promising results compared to experimental data.

CO2 storage in fractured nanopores underground: Phase behaviour study

In this paper, thermodynamic phase behaviour, implications, and strategies for the CO2 storage process in the fractured tight/shale reservoirs (i.e., nanoproes) with adsorptions are studied. First, an analytical equation of state is modified to calculate the nanoscale phase behaviour by considering the effects of pore radius and molecule–molecule interactions. Second, a new empirical correlation for calculating the adsorption thickness in nanopores is initially developed. The modified equation of state coupled with the new adsorption thickness correlation and fracture geometry equation is used to calculate the phase behaviour of confined pure and mixing CO2 streams in fractured nanopores with adsorptions. Third, the pressure–volume diagrams, pressure–temperature diagrams, and critical properties of 12 pure substances of CO2, N2, and alkanes of C1–10 and 12 binary and ternary CO2-dominated mixtures are studied. The calculated pressures for all cases in nanopores with and without the adsorptions and fractures are reduced with the system volume increases but increased by increasing the system temperature with constant compositions. The pressures in nanopores are always larger than those in bulk phase at small volumes but in good agreement at large volumes. In comparison with the N2 or C1–10, the pure CO2 is more easily transited to be a liquid or supercritical phase. Any additions of contaminations (e.g., N2 or C1–10) into the pure CO2 increase the pressures in the pressure–volume or temperature diagram while decrease the critical properties to different extent, especially in nanopores, wherein the N2 exerts the strongest effect and the effects of the alkanes are weakened with the carbon number increase. Overall, three optimum strategies are determined for CO2 storage projects in the deep tight/shale formations as follows: large pore radii, purified CO2 streams, and low temperatures.

Critical behavior of Lifshitz dilaton black holes

Until now, the critical behavior of Lifshitz black holes, in an extended $P\ensuremath{-}v$ space, has not been studied, because it is impossible to find an analytical equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this paper, we adopt a new approach toward thermodynamic phase space and successfully explore the critical behavior of ($n+1$)-dimensional Lifshitz dilaton black holes. The most important advantage of this approach is that we keep the cosmological constant as a constant without needing to vary it. For this purpose, we write down the equation of state as ${Q}^{s}={Q}^{s}(T,\mathrm{\ensuremath{\Psi}})$, where $\mathrm{\ensuremath{\Psi}}={(\ensuremath{\partial}M/\ensuremath{\partial}{Q}^{s})}_{S,P}$ is the conjugate of ${Q}^{s}$, and construct a Smarr relation based on this new phase space as $M=M(S,{Q}^{s},P)$, where $s=2p/(2p\ensuremath{-}1)$, with $p$ the power of the power-law Maxwell Lagrangian. We justify such a choice mathematically and show that with this new phase space, the system admits the critical behavior and resembles the van der Waals fluid system when the cosmological constant (pressure) is treated as a fixed parameter, while the charge of the system varies. We obtain the Gibbs free energy of the system and find a swallowtail shape in Gibbs diagrams, which represents the first-order phase transition. Finally, we calculate the critical exponents and show that although thermodynamic quantities depend on the metric parameters such as $z$, $p$, and $n$, the critical exponents are the same as the van der Waals fluid-gas system. This alternative viewpoint of the phase space of a Lifshitz dilaton black hole can be understood easily since one can imagine such a change for a given single black hole, i.e., acquiring charge, which induces the phase transition. Our results further support the viewpoint suggested in [A. Dehyadegari, A. Sheykhi, and A. Montakhab, Phys. Lett. B 768, 235 (2017)].

Representative volume elements for the simulation of isotropic composites highly filled with monosized spheres

A method is proposed for generating reliable representative volume elements (RVEs) that allows reducing the statistical analysis required for the simulation of the mechanical behavior of isotropic composites highly filled with monosized spheres. The method combines (i) an algorithm inspired from molecular dynamics and associated with an analytical equation of state, and (ii) a geometrical analysis using the two-point correlation function and a nearest-neighbor distribution function. A restrictive selection process is defined, which leads to microstructures reasonably close to randomness and isotropy. The pertinence of the proposed generation and selection of RVEs is confirmed by the simulation of their elastic behavior with the finite element method. In particular, it is shown how the selection procedure allows reducing the computational effort required to reach reliable elastic moduli by operating on a limited number of suitable RVEs. The results are in good agreement with the generalized self-consistent model and with original experimental data obtained on a composite where an acrylate matrix was reinforced by sifted glass beads.

Analytical equations of state for triangle-well and triangle-shoulder potentials

In this work we apply the Discrete Perturbation Theory (Benavides and Gil-Villegas, 1999) to obtain an equation of state for the cases of attractive and repulsive triangle potentials of variable range. This technique provides an analytical expression for the Helmholtz free-energy explicit in density, temperature and in the parameters that characterize the intermolecular interaction. We present single state thermodynamic properties for both potentials and vapor-liquid diagrams (attractive case) for different potential ranges and compare with available and new simulation data. For the triangle-well potential, this equation is accurate in a wider range interval (ranges >1.2) than previous theoretical results and to our knowledge it is the first analytical equation of state for the triangle-shoulder potential.

Accurate steam-water equation of state for two-phase flow LMNC model with phase transition

• Design of an incomplete equation of state, accurate and specific to the low Mach number regime. • Derivation of a new thermodynamic relation for the compressibility coefficient involving no derivatives (Proposition 3.1). • Fitting procedures ensuring the thermodynamic consistency (Propositions 3.2 and 3.3). • Investigation of the influence of the equation of state: highlight that classic analytical equations of state lack accuracy. • 2D numerical simulations of stiff test cases with phase transition. This paper is concerned with the design of incomplete equations of state for a two-phase flow with phase transition that are specific to the low Mach number regime. It makes use of the fact that the thermodynamic pressure has small variations in this regime. These equations of state supplement the 2D Lmnc model introduced in previous works. Our innovative strategy relies on tabulated values and is proven to satisfy crucial thermodynamic requirements such as positivity, monotonicity and continuity. In particular, saturation values are exact. This procedure is assessed by means of analytical steady solutions and comparisons with standard analytical equations of state, and shows a great improvement in accuracy.

Identifying Pure-Component Parameters of an Analytic Equation of State Using Experimental Surface Tension or Molecular Simulations with a Transferable Force Field

A meaningful parametrization of an analytic equation of state can be obtained by adjusting the pure-component parameters to the vapor-pressure data and liquid-density data of the considered pure substance. For substances with no vapor-pressure data available, the parametrization of an equation of state is more subtle. In this study, we investigate two different routes to retrieve pure-component parameters for the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) for such cases. In the first route, classical density-functional theory is applied, and PCP-SAFT parameters are adjusted to experimental liquid density as well as to data of surface tensions. The second approach uses results for liquid density as well as enthalpies obtained from molecular-dynamics simulations using transferable force fields. It is shown that through this route, pure-component parameters for polymeric compounds with ‘vanishing’ vapor pressure can be obtained without experimental data as input. We assess the approaches for several n-alkanes, 1-alkenes, and ethers as training substances. For both routes, rather satisfactory results are obtained for the correlated thermodynamic properties as well as for the predicted ones.

Lagrangian theory of structure formation in relativistic cosmology. V. Irrotational fluids

We extend the general relativistic Lagrangian perturbation theory, recently developed for the formation of cosmic structures in a dust continuum, to the case of model universes containing a single fluid with a single-valued analytic equation of state. Using a coframe-based perturbation approach, we investigate evolution equations for structure formation in pressure-supported irrotational fluids that generate their rest-frame spacetime foliation. We provide master equations to first order for the evolution of the trace and traceless parts of barotropic perturbations that evolve in the perturbed space, where the latter describes the propagation of gravitational waves in the fluid. We illustrate the trace evolution for a linear equation of state and for a model equation of state describing isotropic velocity dispersion, and we discuss differences to the dust matter model, to the Newtonian case, and to standard perturbation approaches.

ISM equation of state and PVTx profile of polymer solutions

ABSTRACTAn analytical equation of state (EoS) has been previously employed to predict the volumetric profile of several molten polymers. The appearing temperature-dependent parameters in the EoS were determined using corresponding states correlations based on the molecular scaling constants, dispersive energy parameters between segments/monomers (ε) and segment diameter (σ). In this study, that EoS was extended to several copolymer melts for predicting their specific volumes and some mechanical coefficients. The ability of the EoS has been evaluated by comparing the results with 738 literature data points for the specific volumes of four copolymer melts over the temperature range from 422.35 to 506 K and pressures up to 200 MPa. The average absolute deviation (AAD) of the calculated specific volumes from literature data was found to be 1.12%. The EoS was also extended to predict PVTx behaviour of seven polymer solutions involving molecular solvents such as water, anisole, 1-octanol and propane by the use of some operating interaction parameter between polymer-solvent molecules. Our calculations on the PVTx profile of studied solutions predicted 1212 experimental data points over the pressure range within 0.1–50 MPa with the AAD equal to 0.96%.

Analytic Equation of State for the Square-well Plus Sutherland Fluid from Perturbation Theory

Analytic expressions were derived for the compressibility factor and residual internal energy of the square-well plus Sutherland fluid. In this derivation, we used the second order Barker-Henderson perturbation theory based on the macroscopic compressibility approximation together with an analytical expression for radial distribution function of the reference hard sphere fluid. These properties are expressed in terms of density, temperature, and the potential parameters. Derived equations successfully applied to hard-core Lennard-Jones fluid. It is found that the agreement between theory and simulation is quite good for both the compressibility factor and the residual internal energy for a wide range of densities and temperature.

Pure Substance and Mixture Viscosities Based on Entropy Scaling and an Analytic Equation of State

This study proposes a simple model for viscosities, based on entropy scaling, for real substances and mixtures. The residual entropy is calculated with the perturbed chain polar statistical associating fluid theory (PCP-SAFT). The model requires two or three pure component parameters, noting, however, that an entirely predictive group contribution approach as proposed in our previous work [Loetgering-Lin O.; Gross J. Ind. Eng. Chem. Res. 2015, 54, 7942–7952] gives also very good results. Overall, 140 real substances are considered with relative mean deviations from experimental data of about 5% (without excluding “outliers”). We performed molecular simulations for mixtures of simple model fluid in order to determine a suitable mixture model. A completely predictive approach for viscosities of real mixtures is thereby obtained. The model is evaluated for 566 mixtures with about 34,500 experimental data points of various complexity (i.e., nearly ideal systems as well as highly asymmetric mixtures). Mixtures...

Transferability of cross-interaction pair potentials: Vapor-liquid phase equilibria of n-alkane/nitrogen mixtures using the TAMie force field

The van der Waals contribution of force fields designed for calculating phase equilibria and thermodynamic properties are usually adjusted to experimental data of pure components. Using these force fields for mixtures by applying combining rules for the cross-wise interaction potentials does not always lead to satisfactory results. This study considers the cross-wise van der Waals energy parameters εαβ among individual pairs of (united-)atom groups as adjustable (but subsequently transferable) parameters. The cross-energy parameters are simultaneously adjusted to an experimental training set of several mixtures. An analytic equation of state, the Perturbed-Chain Polar Statistical Associating Fluid Theory (PCP-SAFT), is used for ensuring very swift convergence of the optimization procedure. As an example, we consider the phase equilibria of n-alkane/nitrogen mixtures. The cross-energy parameters (-CH3 to N and -CH2- to N) are adjusted to three mixtures as the training set. We explore the transferability of the so-obtained force field for varying temperatures and for mixtures that were not considered in the training set. Phase equilibria of nine n-alkane/nitrogen mixtures (each at various temperatures) are determined using Monte Carlo simulations in the grand canonical ensemble. Results for vapor-liquid equilibrium calculations show very good agreement with experimental data.

Determination of structure factor and effective intermolecular parameters of liquid alkali metals using mean spherical approximation theory

The structural properties of liquid cesium (Cs) were investigated in the temperature range of 400–2000 K by application of the effective hard-core attractive Yukawa (H-Y) potential function to the mean spherical approximation (MSA) theory. An analytical equation of state was applied to calculate the molecular parameters of a potential function at any thermodynamic state. The proposed approach was used to predict the behavior of the structure factor, S(k), at a wide range of wave vectors (k). The results obtained were compared with the available experimental measurements, which revealed a good agreement over the whole liquid range for Cs. The structure factor at a long wavelength limit, S(0), was also calculated and compared with the experimental data, which showed the accuracy of the Ornstein-Zernike (OZ) behavior of S(k) at the thermodynamic states below the boiling temperature (Tb). For the case of Cs, the presented intermolecular potential model was very much responsive to the thermodynamic states and led to the indication of the two distinct isotherm ranges T <Tb and T >Tb. Two analytical expressions were obtained for the temperature dependency of the potential well-depth (Keff) over the temperature ranges of T <Tb and T >Tb. Furthermore, the proposed approach showed several metal-nonmetal transitions for Cs at about 500, 1100, 1350, and 1650 K. Using the Keff relations for the other liquid alkali metals Na, K, and Rb provided the S(k) results that were more accurate in comparison with the experimental data.

Modeling mass transfer and reaction of dilute solutes in a ternary phase system by the lattice Boltzmann method.

In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

Analytical Equations of State of Natural Gas and Their Role in Mathematical Modeling

Analytical Equations of State of Natural Gas and Their Role in Mathematical Modeling

Predicting PR EOS binary interaction parameter using readily available molecular properties

The proper design, operation, simulation and optimization of numerous chemical processes require the knowledge of phase behavior of coexisting phases over wide ranges of pressure and temperature. A convenient method for describing equilibrium phase behavior is use of analytic equations of state. The most widely used equations of state (EOS) are the cubic van der Waals type equations such as the Soave–Redlich–Kwong (SRK) and Peng–Robinson (PR) EOS. These equations utilize one or two empirical binary interaction parameters (BIPs) for each binary pair in a mixture. These BIPs can have a significant effect on the predicted properties of mixtures and are generally required for accurate predictions.In a recent work, we developed generalized models to estimate the BIPs in the PR EOS for non-ideal, low-pressure systems using a quantitative structure–property relationship (QSPR) modeling approach. In a departure from our earlier work, we present in this study a different modeling approach which utilizes only readily available molecular properties of the components involved in a mixture. This greatly simplifies application of our new models, since they do not require elaborate computational methods for generation of molecular descriptors.To develop the present generalized model, a VLE database consisting of 1010 binary systems comprised of non-ideal, low-pressure binary systems as well as asymmetric, high-pressure binary systems was utilized. Using this database, the BIPs for the PR EOS were first determined by regressing the VLE experimental data for each system. Then, the BIPs were generalized in terms of molecular properties of components in a mixture. Results indicate that the generalized model provides bubble-point pressure predictions with an average absolute deviation (AAD) of 9% when compared with an AAD of 4.7% through direct regressions of the BIPs. Further, the model developed in this study was compared to our previous model by predicting bubble-point pressures of non-ideal, low-pressure binary systems. The predictions from the model developed in this study were comparable to our previous results (9.2% compared to 8.3% AAD), thus demonstrating the usefulness of the simpler approach presented in this work.

Transferable Anisotropic United-Atom Force Field Based on the Mie Potential for Phase Equilibria: Aldehydes, Ketones, and Small Cyclic Alkanes

The Transferable Anisotropic Mie potential (TAMie) is extended to ketones, aldehyhdes, and small cyclic alkanes. Parameters defining the Mie pair potential, representing van der Waals interactions, as well as partial point charges were treated as adjustable. These adjustable parameters were identified by minimizing deviation of calculated vapor pressure data and liquid density data to experimental data. The optimization procedure was supported by an analytic equation of state, according to Hemmen and Gross [J. Phys. Chem. B, 2015, 119, 11695−11707]. The optimal point charges for aldehydes and ketones were found by varying them along a predefined grid of values. Results for phase equilibria of pure substances show good agreement with experimental data. The absolute average deviations for aldehydes and ketones in the reduced temperature range TR = T/Texpcrit = 0.55–0.96 are lower than 3.4%. For cyclohexane and cyclopentane the deviations are 0.6%.

An effective series expansion to the equation of state of unitary Fermi gases

Using universal properties and a basic statistical mechanical approach, we propose a general equation of state for unitary Fermi gases. The universal equation of state is written as a series solution to a self consistent integral equation where the general solution is a linear combination of Fermi functions. First, by truncating our series solution to four terms with already known exact theoretical inputs at limiting cases, namely the first three virial coefficients and using the Bertsch parameter as a free parameter, we find a good agreement with experimental measurements in the entire temperature region in the normal state. This analytical equation of state agrees with experimental data up to the fugacity z = 18, which is a vast improvement over the other analytical equations of state available where the agreements is only up to . Second, by truncating our series solution to four terms again using first four virial coefficients, we find the Bertsch parameter , which is in good agreement with the direct experimental measurement of . This second form of equation of state shows a good agreement with self-consistent T-matrix calculations in the normal phase.

Equation of state for polymers based on crystallization data

A simplified procedure with minimum input information is presented to predict an analytical equation of state for crystallizable polymer melts from enthalpy of fusion and crystalline density as scaling constants. The second virial coefficients are calculated from a two-parameter corresponding states correlation obtained from two constants which are heat of fusion, ΔHfus, and the molar density at the crystalline point, ρcrys. This new correlation has been applied to the Ihm-Song-Mason equation of state to predict the volumetric behavior of some polymer melts at compressed state with temperature range from 311.5K to 557.25K and pressures up to 200MPa. For 924 data points, the overall average absolute relative deviation (AARD) of pVT properties was found to be 0.46%. The isothermal compressibility coefficients and liquid bulk modulus have also been predicted using mTM EOS. From 840 data points examined, the AARD of both isothermal thermodynamic properties was equal to 9.71×10−2%. Also the enthalpy of fusion (ΔHfus) and entropy of fusion (ΔSfus) of three polymer melts are predicted using the mTM EOS. Our calculations on the volumetric and thermodynamic properties of studied polymers reproduce the literature data with reasonably good accuracy.

An analytical equation of state for ammonia at high temperatures and high pressures

An analytical equation of state is presented for ammonia to describe its thermodynamic properties over wide ranges of temperatures and pressures up to 3000K and 20GPa, respectively. The equation of state contains well-defined parameters, two of which have been slightly adjusted around their global values based on a limited set of experimental data. The properties of ammonia predicted by the new equation of state are in very good agreement with the experimental data and the first-principles molecular dynamics simulation results over a broad range of temperatures and pressures. The new equation of state is expected to supplement molecular simulations at conditions where experimental data are not yet available.