Equation Of State Predictions Research Articles

Predicting solvation energies of free radicals and their mixtures: A robust approach coupling the Peng-Robinson and COSMO-RS models

Detailed kinetic models are valuable tools for a clear understanding of the dynamics of complex reacting systems based on radical-chain mechanisms, such as combustion and oxidation. Gas-phase automatic kinetic generators, such as EXGAS, RMG, MAMOX, NetGen, REACTION, and GENESYS can be used to establish a detailed network of reaction pathways and the rates associated with each reaction. The range of applications of these gas-phase kinetic generators can be extended to liquids through diffusional and solvation corrections directly applied to the gas-phase thermo-kinetic data. In this work, a flexible framework is proposed for the calculation of the solvation correction involving closed-shell (i.e., non-free radical) and open-shell (i.e., free radicals) molecules. This novel model relies on the Peng-Robinson cubic equation of state (EoS) combined with a quantum-based continuum solvation model (COSMO-RS) through an advanced mixing rule. Unlike most predictive equations of state, the proposed model requires only pure compound inputs: critical temperature (Tc,i), critical pressure (Pc,i), acentric factor (ωi), and the screening charge distribution (σ-profile). These inputs were obtained for C/H/O free radicals using group corrections applied to the known values of the associated closed-shell molecules (parent molecules). The resulting EoS is able to provide fast predictions of solvation quantities of closed-shell molecules and C/H/O free radicals with a mean unsigned error of around 0.30 kcal/mol. The great advantage of this method is that it allows high-throughput computation of solvation quantities in pure solvents and mixtures at any temperature (including the supercritical domain), which could enable the simulation of complex oxidation kinetics in a wide range of applications.

Learning thermodynamically constrained equations of state with uncertainty

Numerical simulations of high energy-density experiments require equation of state (EOS) models that relate a material’s thermodynamic state variables—specifically pressure, volume/density, energy, and temperature. EOS models are typically constructed using a semi-empirical parametric methodology, which assumes a physics-informed functional form with many tunable parameters calibrated using experimental/simulation data. Since there are inherent uncertainties in the calibration data (parametric uncertainty) and the assumed functional EOS form (model uncertainty), it is essential to perform uncertainty quantification (UQ) to improve confidence in EOS predictions. Model uncertainty is challenging for UQ studies since it requires exploring the space of all possible physically consistent functional forms. Thus, it is often neglected in favor of parametric uncertainty, which is easier to quantify without violating thermodynamic laws. This work presents a data-driven machine learning approach to constructing EOS models that naturally captures model uncertainty while satisfying the necessary thermodynamic consistency and stability constraints. We propose a novel framework based on physics-informed Gaussian process regression (GPR) that automatically captures total uncertainty in the EOS and can be jointly trained on both simulation and experimental data sources. A GPR model for the shock Hugoniot is derived, and its uncertainties are quantified using the proposed framework. We apply the proposed model to learn the EOS for the diamond solid state of carbon using both density functional theory data and experimental shock Hugoniot data to train the model and show that the prediction uncertainty is reduced by considering thermodynamic constraints.

Residual Entropy Scaling for Long-Chain Linear Alkanes and Isomers of Alkanes

Using residual entropy scaling approaches, transport properties, such as viscosity and thermal conductivity, can be linked to a thermodynamic property, i.e., residual entropy. It has been demonstrated in the literature that these approaches can be successfully used to correlate transport properties in the gas phase as well as the liquid phase over large temperature and density ranges. Recently, Yang et al. [ J. Chem. Eng. Data 2021, 66(3), 1385–1398] proposed a residual entropy scaling approach for the viscosity of 39 refrigerants and their mixtures, and they extended this approach in a subsequent work to 124 fluids [ Int. J. Thermophys. 2022, 43(12), 183]. The method of Yang et al. requires a fluid-specific scaling factor for each fluid. Yang et al. proposed a method for estimating this parameter as being proportional to the residual entropy at the critical point. In this work, it is demonstrated that for hydrocarbons, the fluid-specific scaling factor can be better approximated as a linear function of the longest carbon chain. All linear and branched alkanes for which accurate multiparameter equations of state are available have been considered in this work. Furthermore, the performance of other predictive equations of state, i.e., the Peng–Robinson and Lee–Kesler–Plöcker equations of state, is evaluated.

The dynamics of freestanding films: predictions for poly(2-chlorostyrene) based on bulk pressure dependence and thoughtful sample averaging.

In this paper we model the segmental relaxation in poly(2-chlorostyrene) 18 nm freestanding films, using only data on bulk samples to characterize the system, and predict film relaxation times (τ) as a function of temperature that are in semi-quantitative agreement with film data. The ability to translate bulk characterization into film predictions is a direct result of our previous work connecting the effects of free surfaces in films with those of changing pressure in the bulk. Our approach combines the Locally Correlated Lattice (LCL) equation of state for prediction of free volume values (Vfree) at any given density (ρ), which are then used in the Cooperative Free Volume (CFV) rate model to predict τ(T, Vfree). A key feature of this work is that we calculate the locally averaged density profile as a function of distance from the surface, ρav(z), using the CFV-predicted lengthscale, Lcoop(z), over which rearranging molecular segments cooperate. As we have shown in the past, ρav(z) is significantly broader than the localized profile, ρ(z), which translates into a relaxation profile, τ(z), exhibiting a breadth that mirrors experimental and simulated results. In addition, we discuss the importance of averaging the log of position dependent relaxation times across a film sample (〈log τ(z)〉), as opposed to averaging the relaxation times, themselves, in order to best approximate a whole sample-averaged value that can be directly compared to experiment.

Density Characteristics of the CO2–CH4 Binary System: Experimental Data at 313–353 K and 3–18 MPa and Modeling from the PC-SAFT EoS

Accurate density data and a predictive equation of state (EoS) for the CO2–CH4 system are significantly important for the application of natural gas purification and carbon capture, utilization, and storage (CCUS). In this work, densities of the CO2–CH4 binary system with CO2 mole fractions ranging from 0.0998 to 0.8988 were measured at temperatures of 313–353 K and pressures of 3–18 MPa using a magnetic suspension balance, and the compressibility factors and the relative combined standard uncertainties were also calculated correspondingly. The PC-SAFT EoS was employed for the density modeling, and the PR EoS was used for comparison. The measured density data in this work and experimental density data from the literature were used to optimize the binary interaction parameter kij in the PC-SAFT EoS and PR EoS. The predicting abilities of PC-SAFT and PR EoS with the optimized kij were compared to the case without binary interaction (kij = 0). The PC-SAFT EoS with the optimized kij had better performance tha...

Phase boundaries for synthetic light oils and gas condensate mixtures using predictive equations of state with a solid solution model

AbstractPhase boundaries for five synthetic paraffinic mixtures were predicted using a previously developed Gibbs energy minimization algorithm. In this method, the vapour and liquid phases were described by cubic equations of state (EoS). These EoS employed a predictive temperature‐dependent group‐contribution binary interaction parameter model. The solid phase was described using a universal quasi‐chemical activity coefficient (UNIQUAC) solid solution model. Using this algorithm, the phase boundaries were calculated from a temperature‐search strategy. The predicted phase boundaries were found to be in good agreement with experimental phase boundaries for vapour‐liquid (VL), solid‐liquid (SL), solid‐vapour (SV), and solid‐vapour‐liquid (SVL) regions. The model was also used to predict the phase boundaries at low temperatures where no experimental data were available.

Surface Tension Prediction of n-Alkanes by a Modified Peng-Robinson Equation of State Using the Density Functional Theory

Through this study, the ability of a modified Peng-Robinson (MPR) equation of state in predicting the surface tension of n-alkanes based on the density functional theory approach was investigated and compared with other studies. The interfacial layer thickness and the density profile were calculated simultaneously at different temperatures from triple point to near critical point using the modified Peng-Robinson equation of state. It was shown that the calculated thickness of interfacial layer increases with decrease in the chain length of n-alkanes molecules and raising of temperature. The surface tension of n-alkanes was calculated using the calculated values of thin layers’ densities. It was shown that the calculated surface tension of n-alkanes decreases with temperature in accordance with the experiment. The average relative error in prediction of the surface tension by the MPR equation of state was in the range of 2.5-6% while it was 4.6-25.2% by the Peng-Robinson equation of state. The validity of the MPR equation of state in the surface tension prediction of n-alkanes containing C1-C10 has been proved by comparing the results of this work with other studies.

A Fluctuation Equation of State for Prediction of High-Pressure Densities of Ionic Liquids

During this work, we demonstrate, for the first time, that the volumetric properties of pure ionic liquids could be truly predicted as a function of temperature from 219 K to 473 K and pressure up to 300 MPa. This has been achieved by using only density and isothermal compressibility data at atmospheric pressure through the Fluctuation Theory-based Tait-like Equation of State (FT-EoS). The experimental density data of 80 different ionic liquids, described in the literature by several research groups as a function of temperature and pressure, was then used to provide comparisons. Excellent predictive capability of FT-EoS was observed with an overall relative absolute average deviation close to 0.14% for the 15,298 data points examined during this work.

E-PPR78: A proper cubic EoS for modelling fluids involved in the design and operation of carbon dioxide capture and storage (CCS) processes

Considering the broad range of operating conditions that typically characterize carbon dioxide capture and storage (CCS) processes, on the one side, and the high number of species that compose the processed streams, on the other, it becomes evident the necessity of disposing of suitable equations of state being able to flexibly represent the complexity of all these thermodynamic systems. To address the growing industrial need for predictive, sufficiently accurate and simple thermodynamic models, the Enhanced-Predictive-PR78 (E-PPR78) equation of state is proposed here as a suitable model for CCS fluids. Currently, the application of the E-PPR78 model is limited to CCS mixtures of CO2, CH4, N2, O2, Ar, H2, H2S, SO2, NO, CO and H2O. The main objective of this work is the extension of the modelling capability of E-PPR78 to mixtures that, besides the mentioned species, could also contain COS, NH3, NO2, N2O4 and N2O. This result is achieved by introducing, with this paper, four new molecular groups for the E-PPR78 model, namely, “COS” (group 28), “NH3” (group 29), “NO2/N2O4” (group 30) and “N2O” (group 31). More specifically, the paper presents, at first, a review of all the available experimental vapor-liquid-equilibrium (VLE) and enthalpy changes on mixing (hM) data for binary systems containing 2 of the following compounds: CO2, CH4, N2, O2, Ar, H2, H2S, COS, SO2, NH3, NO, NO2, N2O4, N2O, CO and H2O. Secondly, the E-PPR78 model is parameterized over these data: not only new group-interaction parameters are determined for the 4 groups added in this study but many parameters are updated over newly available data. Modelling results are analyzed accounting for the complexity of each system, evaluated according to the classification scheme of Van Konynenburg and Scott. Moreover, the capacity of the model to predict ternary data is meticulously discussed. This research thus provides a predictive equation of state, the E-PPR78, being applicable to most of the currently treated CCS fluids, and discusses its strengths and limitations.

A modified group-contribution PC-SAFT equation of state for prediction of phase equilibria

The group contribution (GC) method proposed by Tamouza et al. (Fluid Phase Equilibria 222–223, (2004), 67–76) has been improved to estimate parameters of the Perturbed-Chain statistical associating fluid theory (hereby called mg-SAFT). The improvement is the “exclusion” of the over-accounting of dispersion energy between intra-molecular segments through an empirical correlation parameter. We have treated three hydrocarbon families (n-alkanes, n-alkyl-cycloalkanes and n-alkyl-benzenes); several pure compound properties such as the enthalpy of vaporization, single-phase density, speed of sound, heat capacities are predicted. The results indicate that most pure compound property predictions are satisfactory, mg-SAFT appears to offer a systematic improvement as compared to the original GC-PC-SAFT approach, but no improvement in prediction of speed of sound. Vapor-liquid equilibria of some binary mixtures are purely predicted using the mg-SAFT (kij = 0). Good agreements between the model predictions and the experimental data were found. Comparison with other existing SAFT-based group contribution approach was performed in calculating vapor pressure and saturated liquid density of heavy hydrocarbon; mg-SAFT demonstrated significant improvements in accuracy.

Application of SAFT‐VR Equation of State for Prediction of Thermophysical Properties of Sugar Solutions

AbstractThe thermodynamic properties of sugar solutions, which are important solutions in food industries, have been calculated. Because of the large molecules and hydrogen bonding in the water/sugar solutions, the statistical association fluid theory of variable ranged (SAFT‐VR) equation of state was used to calculate the thermodynamic properties such as water activity, solubility, vapor pressure and boiling temperature of the aqueous sugar solutions, including glucose, sucrose and fructose. The SAFT‐VR parameters were calculated using vapor pressure and density data for water and water activity data for sugars. The total average absolute deviation percentage (AAD%) for all the calculated properties was 1.93%. The results showed that the model can accurately predict the properties of the sugar solution.Practical ApplicationsSugar solutions have important roles and uses in different industries such as food production, food preservation industries, chemical feed stock, pharmaceutical industries, metabolic reactions, industrial operations and chemical processes, such as crystallization, evaporation and selective separation. For example, water activity in a solution can be used to stop deteriorative reactions. The density of the solution is important in separation processes and mixing equipment. Solubility of sugar in water is another important property in food preservation and processes such as gasification and separation. The boiling temperature of the solution is an important thermodynamic property in different industrial processes such as purification and separation processes.

High-pressure solubility of light gases in heavy n-alkanes from a predictive equation of state: Incorporating Henry's law constant into binary interaction parameter

Using fugacity coefficient of a cubic equation of state, Henry's law constant of a solute in a solvent is incorporated into binary interaction parameter of the classical attractive parameter mixing rule. The developed equation is a function of temperature. The binary interaction parameter is evaluated by pure component critical properties and acentric factors of the solute and the solvent and the Henry's law constant of the solute in the solvent. The developed model accurately describes the solubility of gases including methane, ethane, nitrogen, carbon dioxide and hydrogen sulphide in heavy n-alkanes from low to high pressure for wide range of temperature. The solubility of methane and carbon dioxide in water is also predicted adequately.

Evaluation of various types' equations of state for prediction of rate of double gas hydrate formation based on Kashchiev model in flow loop apparatus

This paper compares the effects of using various types of equations of state such as Peng Robinson (PR), Soave Redlich Kwong (SRK), Esmaeilzadeh – Roshanfeker (ER), Patel Teja (PT) and Valderrama–Patel–Teja (VPT) on the rate of hydrate formation based on the Kashchiev model for gaseous mixtures with experimental data points obtained in a mini flow loop apparatus in presence of the kinetic inhibitors (such as modified starch and l-tyrosine) at various pressures (1–4 MPa) and specified temperature (277.15 K). For this purpose, a laboratory mini flow loop apparatus was set up to measure the rate of gas hydrate formation when a hydrate forming mixture is contacted with water in the presence or absence of dissolved inhibitor under suitable temperature and pressure conditions. In each experiment, a water blend saturated with pure gas is circulated up to a required pressure. Pressure is maintained at a constant value during experimental runs by means of the required gas make-up. The total average absolute deviation was found to be 18.7%, 19.8%, 19.2%, 20.2% and 20.7% of the PR, ER, SRK, VPT and PT equations of state for calculating of gas consumption for gaseous mixtures during gas hydrate formation (in the presence and absence of kinetic gas hydrate inhibitor), respectively. Comparison results between the calculated and experimental data points of gas consumption were obtained in a flow loop indicate that the PR and ER equations of state have lower errors than the SRK, VPT and PT equations of state for this model.

Thermodynamic modeling of phase equilibrium for gas hydrate in single and mixed refrigerants by using sPC-SAFT equation of state

This paper demonstrates the ability of simplified PC-SAFT equation of state for prediction of hydrate dissociation pressure of refrigerants namely (CFC13, HCFC22, HFC23, HFC32, HFC125, HCFC141b, HFC134a, HFC152a, HFC143a and R744) and their mixtures at different temperatures and concentrations. The presented model is based on water chemical potential difference in the liquid and hydrate phases. The Van der Waals and Platteeuw model was also applied to illustrate the chemical potential of water in hydrate phase. The simplified Perturbed-Chain Statistical Associating Fluid Theory (sPC-SAFT) equation of state was used to determine the fugacity of each component in vapor and liquid phases. Moreover, the temperature dependent binary interaction parameters (kij) were adjusted, taking advantage of the well-described phase equilibria. The Kihara potential parameters were generated based on the dissociation conditions of single refrigerants hydrate to use for prediction of mixture gas hydrate dissociation conditions. The agreement between the experimental and predicted pressure is acceptable in the presented model; the average absolute deviation of model is between (0.06–3.80) for single systems and (1.90–4.99) for mixed systems.

Effect of Isomeric Structures of Branched Cyclic Hydrocarbons on Densities and Equation of State Predictions at Elevated Temperatures and Pressures

The cis and trans conformation of a branched cyclic hydrocarbon affects the packing and, hence, the density, exhibited by that compound. Reported here are density data for branched cyclohexane (C6) compounds including methylcyclohexane, ethylcyclohexane (ethylcC6), cis-1,2-dimethylcyclohexane (cis-1,2), cis-1,4-dimethylcyclohexane (cis-1,4), and trans-1,4-dimethylcyclohexane (trans-1,4) determined at temperatures up to 525 K and pressures up to 275 MPa. Of the four branched C6 isomers, cis-1,2 exhibits the largest densities and the smallest densities are exhibited by trans-1,4. The densities are modeled with the Peng-Robinson (PR) equation of state (EoS), the high-temperature, high-pressure, volume-translated (HTHP VT) PREoS, and the perturbed chain, statistical associating fluid theory (PC-SAFT) EoS. Model calculations highlight the capability of these equations to account for the different densities observed for the four isomers investigated in this study. The HTHP VT-PREoS provides modest improvements over the PREoS, but neither cubic EoS is capable of accounting for the effect of isomer structural differences on the observed densities. The PC-SAFT EoS, with pure component parameters from the literature or from a group contribution method, provides improved density predictions relative to those obtained with the PREoS or HTHP VT-PREoS. However, the PC-SAFT EoS, with either set of parameters, also cannot fully account for the effect of the C6 isomer structure on the resultant density.

Prediction of sour gas compressibility factor using an intelligent approach

Compressibility factor (z-factor) values of natural gasses are essential in most petroleum and chemical engineering calculations. The most common sources of z-factor values are laboratory experiments, empirical correlations and equations of state methods. Necessity arises when there is no available experimental data for the required composition, pressure and temperature conditions. Introduced here is a technique to predict z-factor values of natural gasses, sour reservoir gasses and pure substances. In this communication, a novel mathematical-based approach was proposed to develop reliable model for prediction of compressibility factor of sour and natural gas. A robust soft computing approach namely least square support vector machine (LSSVM) modeling optimized with coupled simulated annealing (CSA) optimization tool was proposed. To evaluate the performance and accuracy of this model, statistical and graphical error analyses have been used simultaneously. Moreover, comparative studies have been conducted between this model and nine empirical correlations and equations of state. The obtained results demonstrated that the proposed CSA-LSSVM model is more robust, reliable and efficient than the existing correlations and equations of state for prediction of z-factor of sour and natural gasses.

A new equation of state for molten alkali metal alloys

A new proposed equation of state (EoS) for pure liquid alkali metals has been extended to predict the volumetric and thermodynamic properties of alkali metal alloys at different temperatures, pressures, and compositions. The new equation which is valid over the whole liquid range is based on a suggested potential function and the interactions of nearest neighbors are according to the characteristics of the soft repulsive interaction in dense and large attractive interaction in expanded liquid alkali metals. The composition dependencies of the parameters of the equation of state are assumed as quadratic functions of mole fraction considering the mean geometry approximation “MGA”. In this paper, the calculated results of liquid density and other thermodynamic properties such as isobaric expansion coefficient, αP, isothermal compressibility, κT, and internal pressure, Pi, of binary molten alloys of Na–K and Cs–K from the freezing point up to several hundred degrees above the boiling point are presented. The results show good agreement between the density values obtained from this equation and the experimental and literature data. To show the ability of this EoS in prediction of density of alkali metal alloys, the results have been compared with some other equations. Although there is no corresponding literature data to compare the results obtained for derived thermodynamic properties, the variations of these properties with temperature and composition show proper trends for these alloys.

Application of PHSC equation of state in prediction of gas hydrate formation condition

In this work the modified perturbed hard sphere chain equation of state (PHSC), coupled with Van der waals Platteuw model, has been used for prediction of gas hydrate formation (dissociation) conditions. The PHSC EOS has been modified using association and electrostatic contributions. In order to evaluate the capability of the PHSC EOS, the hydrate formation conditions of various pure as well as mixed gases in the presence and absence of the thermodynamic inhibitors such as methanol, ethanol, mono ethylene glycol(MEG), NaCl, KCl and CaCl2 have been studied. In the absence of inhibitors, the pure gas hydrate formation conditions have been predicted within 3.4% absolute average relative deviation. The hydrate formation condition of gas mixtures have been predicted within AARD 4.6%. In the presence of organic inhibitors and single electrolyte 6.1% and 5.8% AARD has been observed respectively. A total of 102 systems have been studied; our results showed that the PHSC EOS is capable of prediction of hydrate formation condition with reasonable absolute average relative deviation (AARD) from experimental data.

A new regularity and an equation of state for alkali metals over the whole liquid range

In this paper, we have derived an equation of state (EoS) for liquid alkali metals based on a suggested potential function over the whole liquid range including metal–nonmetal transition. This EoS is based on the interaction of nearest neighbors according to the characteristics of the soft repulsive interaction in dense and large attractive interaction in expanded liquid alkali metals. PVT data of dense liquid alkali metals obey the equation (2Z−1)Vm=A+Bρ over the whole range of liquid densities, where Z, Vm, and ρ are compression factor, molar volume, and molar density, respectively. The intercept A and slope B of this equation are related to the attractive and repulsive parts of suggested potential function, respectively, and both depend on temperature. The new EoS has been used to calculate the molar density and other thermodynamic properties such as isobaric expansion coefficient, αP; isothermal compressibility, κT; and internal pressure, Pi; of liquid alkali metals at any temperature and pressure over the whole liquid range. The results show good agreement between the values obtained using this equation and the experimental and literature data. To show the ability of this EoS in prediction of thermodynamic properties of liquid alkali metals over the whole liquid range, the results have been compared with some other equations.

Multiphase equilibria of waxy systems with predictive equations of state and a solid solution model

A thermodynamic model is developed to predict the vapor–liquid, solid–liquid, solid–vapor and solid–liquid–vapor equilibria of systems containing methane with waxy components. The model takes the advantages of equations of state to describe the vapor and liquid phases. The solid phase is described by Coutinho's activity coefficient model, i.e., the solid phase is assumed to be orthorhombic solid solution. Impact of pressure on the solid phase is included by coupling solid–liquid Clapeyron relationship with Poynting correction factor. While no adjustable parameter is used, the model satisfactorily predicts the phase boundaries of the synthetic and real systems resembling gas condensate mixtures for wide ranges of pressure and temperature.