Statistical Associating Fluid Theory Equation Of State Research Articles

Modeling of nitrogen solubility in unsaturated, cyclic, and aromatic hydrocarbons: Deep learning methods and SAFT equation of state

BackgroundAn accurate understanding of the nitrogen solubility in hydrocarbons is crucial for developing enhanced oil recovery (EOR) by nitrogen injection and the design of thermal separation and chemical conversion processes in chemical industries and oil refineries. Unsaturated, cyclic, and aromatic hydrocarbons as important compounds of crude oil are common solvents in the chemical industry that have received less attention compared to normal alkanes. MethodsIn this paper, four deep learning models, including recurrent neural network (RNN), long short-term memory (LSTM), deep belief network (DBN), and convolutional neural network (CNN) were developed for estimating the nitrogen solubility in unsaturated, cyclic, and aromatic hydrocarbons. To this end, 673 experimental nitrogen solubility data for 23 various hydrocarbons were collected from the literature in a wide ranges of operating pressure (0.03–100.1 MPa) and temperature (78–662.8 K). The input parameters to the models were considered critical temperature, critical pressure, and molecular weight of hydrocarbon solvents accompanying operating conditions of temperature and pressure. Also, the performance of deep learning models was compared with Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), and statistical associating fluid theory (SAFT) equations of state (EOSs). Significant findingsThe CNN model is able to estimate the experimental values of nitrogen solubility with a root mean square error (RMSE) of 0.0211. Also, SAFT EOS outperformed the two cubic EOSs. Based on sensitivity analysis, pressure has the greatest impact on nitrogen solubility in unsaturated, cyclic, and aromatic hydrocarbons, followed by temperature and the critical temperature of hydrocarbons. Nitrogen solubility in unsaturated, cyclic, and aromatic hydrocarbons increases with the increase in pressure and temperature, and it lowers by the increase in the critical pressure, critical temperature, and molecular weight of hydrocarbon solvents. Finally, high reliability of the experimental data and statistically high validity of the CNN model were proved by the Leverage approach. The findings of this study can have implications in increasing the efficiency of gas processing units, thermal separation, and chemical conversion processes.

New mathematical modeling of temperature-based properties of ionic liquids mixture: Comparison between semi-empirical equation and equation of state

Ionic liquids are a significant type of organic salts with attractive and superior physicochemical features for employing in a number of chemical processes. Ionic liquids include one large and asymmetric cationic or anionic functional group. The properties of cationic functional group are almost similar and it is main motivation of this study. In the present investigation, we extended various semi-empirical correlations to predict some temperature-based properties of ionic liquids mixture (IL-mixture). These properties are density, viscosity, surface tension and bubble point pressure. In the case of IL-mixture density, modified Rackett equation and Taylor equation with three scenarios, modified Gardas and Coutinho equation were considered and the obtained results were compared with Peng-Robinson equation of state (EoS), Patel-Teja EoS and perturbed-chain statistical associating fluid theory EoS. In the case of IL-mixture viscosity, two types of activation energy concept models were used and also, Lee-Kesler method and modified Brock and Bird were applied to predict IL-mixture bubble point pressure and IL-mixture surface tension, respectively. A used dataset was collected from reliable literature including 2731 data points of bubble pressure (23 binary system), 4643 data points of density (102 binary system), 1011 data points of viscosity (32 binary system) and 391 data points of surface tension (16 binary system). These data relevant to 12 cation-family and 36 solvents. The all of the data was haphazardly separated in two subsets of train and test with a computer program. The obtained results indicated all used semi-empirical correlations gives the best match with corresponding experimental data and the values of statistical parameters demonstrated this hypothesis.

Predicting the phase behavior of fluorinated organic molecules using the GC-SAFT-VR equation of state

Fluorinated molecules such as perfluoroalkanes (PFA), perfluoroalkylalkanes (PFAA), fluoroalkanes, and hydrofluoroethers (HFE) possess attractive physical properties that have resulted in their use in a wide range of applications. However, while there is an abundance of thermophysical data for hydrocarbons in the literature, only limited studies have been performed that report the properties of the corresponding fluorinated species. Predictive approaches are therefore needed to accurately and reliably determine the physical properties of these molecules. The statistical associating fluid theory (SAFT) is a commonly used molecular-based equation of state that in its various forms has been applied to study a wide range of fluid systems. In recent work, several group contribution (GC) SAFT equations of state have been proposed, such as the GC-SAFT-VR equation that combines the SAFT equation for potentials of variable range (VR) with a group contribution approach that uniquely allows for the description of hetero-segmented chains. The GC-SAFT-VR equation has been shown to provide an excellent description of the phase behavior of pure associating and non-associating fluids and their mixtures, with a minimal reliance on fitting the model parameters to experimental data. Specifically, parameters for key functional groups (such as CH3, CH2, CH, CH2=CH, C=O, C6H5, ether and ester, OH, NH2, CH=O, COOH) have been obtained by fitting to experimental vapor pressure and saturated liquid density data for selected low molecular weight fluids and then used to predict the phase behavior of pure fluids and their mixtures without further adjusting the group parameters. To expand upon this effort, here we report parameters for the CF3, CF2, CF, CH2F, CHF2, and CHF functional groups and their cross interactions. The theoretical predictions are compared with experimental data for pure PFAs, PFAAs, and HFEs, as well as binary mixtures of alkanes, alkenes, PFAs, PFAAs, and CO2 in order to test the transferability of the new group parameters. The GC-SAFT-VR approach is found to accurately predict the phase behavior of the systems studied.

Four phase hydrate equilibria of methane and carbon dioxide with heavy hydrate former compounds: Experimental measurements and thermodynamic modeling

We experimentally investigated the hydrate dissociation condition for four phase hydrate (H)-aqueous liquid (LAq)-hydrocarbon rich liquid (LHC)-vapor (V) for the ternary systems of help gas-heavy hydrate former-water. Methane and carbon dioxide are known as help gases and benzene and cyclohexane are considered as heavy hydrate formers. The experimental data were generated using an isochoric pressure-search method. Two different equations of state (EOS) were employed to study the equilibrium phase behavior of ternary four phase systems. The EOSs considered are Valderama-Patel-Teja EOS combined with non-density dependent mixing rule (VPT+NDD) and Statistical Associating Fluid Theory EOS proposed by Huang and Radosz (SAFT-HR). The required binary interaction parameters (BIP) were obtained using vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) data. The hydrate phase was modeled by the modification of the solid solution theory of van der Waals and Platteeuw. To obtain reliable results, distortion of cages due to occupation of large molecules was considered. The Kihara parameters of cyclohexane were adjusted to hydrate dissociation data. Model calculations for hydrate forming conditions were found to be in satisfactory agreement with the newly reported data in this work and literature data.

Prediction of thermodynamic properties of sodium dodecyl sulfate aqueous solutions through the hetero-SAFT equation of state

A special version of statistical associating fluid theory (SAFT), the so-called hetero-segmented SAFT equation of state, has been extended to calculate the thermodynamic properties of sodium dodecyl sulfate aqueous solutions. The predicted properties were included the vapor pressure as well as second thermodynamic properties, such as speed of sound. Evaluation of the model has been done through comparison of the calculated vapor pressure with experimental data in the range of 298.15–313.15 K. The hetero-SAFT model is found to be able to correctly describe the thermodynamic derivative properties as well as pVT and VLE properties of ionic surfactant solutions.

Study of Protein Partitioning in Polymer-Salt Aqueous Two-Phase Systems Using Electrolyte-SAFT (E-SAFT) Equation of State

The statistical associating fluid theory (SAFT) has been applied to correlate partition coefficients of proteins in polymer-salt aqueous two-phase systems. Accordingly, interactions among the association on the partition coefficients of proteins have been thoroughly studied by the SAFT equation of state. The parameters of the SAFT equation of state (EoS) for pure water were obtained from our previous work. Additionally, the parameters of SAFT for poly (ethylene glycol) with diverse molecular weights were estimated using densities of aqueous polymer solutions. Finally, partition coefficients of proteins in aqueous two-phase systems were investigated. The restrictive primitive mean spherical approximation (RP-MSA) has been used for long-range interactions whereas the parameters of SAFT for proteins were obtained, primarily by utilizing the electrolyte-SAFT (E-SAFT). The results showed that the E-SAFT is capable of accurately correlate partition coefficients of proteins in aqueous two-phase systems.

Comparison between the artificial neural network system and SAFT equation in obtaining vapor pressure and liquid density of pure alcohols

Vapor pressure and liquid density of 20 pure alcohols were correlated using an artificial neural network (ANN) system and statistical associating fluid theory (SAFT) equation of state. The SAFT equation has five adjustable parameters as temperature-independent segment diameter, square-well energy, number of segment per chain, association energy and association volume. These parameters can be obtained by a non-linear regression method using the experimental vapor pressure and liquid density data. In continue, the vapor pressure and liquid densities of pure alcohols were estimated by using an artificial neural network (ANN) system. In the neural network system, it is assumed that thermodynamic properties of pure alcohols depend on temperature, critical properties and acentric factor. The best network topology was obtained as (4-10-2). The weights connection and biases were obtained using batch back propagation (BBP) method for 611 experimental data points. The average absolute deviation percent (ADD%) for vapor pressure of pure alcohols for ANN system and SAFT equation of state are 3.593% and 3.378%, respectively. Also, the average absolute deviation percent (ADD%) for liquid density of pure alcohols for ANN system and SAFT equation of state are 0.792% and 1.367%, respectively. The results emphasized that the artificial neural network can more accurately predict thermophysical properties of pure alcohols than the SAFT equation of state.

Prediction of vapor–liquid equilibrium and PVTx properties of geological fluid system with SAFT-LJ EOS including multi-polar contribution. Part I: Application to H 2O–CO 2 system

Molecular based equations of state (EOS) are attractive because they can take into account the energetic contribution of the main types of molecular interactions. This study models vapor–liquid equilibrium (VLE) and PVTx properties of the H 2O–CO 2 binary system using a Lennard-Jones (LJ) referenced SAFT (Statistical Associating Fluid Theory) EOS. The improved SAFT-LJ EOS is defined in terms of the residual molar Helmholtz energy, which is a sum of four terms representing the contributions from LJ segment–segment interactions, chain-forming among the LJ segments, short-range associations and long-range multi-polar interactions. CO 2 is modeled as a linear chain molecule with a constant quadrupole moment, and H 2O is modeled as a spherical molecule with four association sites and a dipole moment. The multi-polar contribution to Helmholtz energy, including the dipole–dipole, dipole–quadrupole, and quadrupole–quadrupole contribution for H 2O–CO 2 system, is calculated using the theory of Gubbins and Twu (1978). Six parameters for pure H 2O and four parameters for pure CO 2 are needed in our model. The Van der Waals one-fluid mixing rule is used to calculate the Lennard-Jones energy parameter and volume parameter for the mixture. Two or three binary parameters are needed for CO 2–H 2O mixtures, which are evaluated from phase equilibrium data of the binary system. Comparison with the experimental data shows that our model represents the PVT properties of CO 2 better than other SAFT EOS without a quadrupole contribution. For the CO 2–H 2O system, our model agrees well with the vapor–liquid equilibrium data from 323–623 K. The average relative deviation for CO 2 solubility (expressed in mole fraction) in water is within 6%. Our model can also predict the PVTx properties of CO 2–H 2O mixtures up to 1073 K and 3000 bar. The good performance of this model indicates that: (1) taking account of the multi-polar contribution explicitly improves the agreement of calculated properties with experimental data at high temperatures and high pressures, (2) the molecular-based EOS with just a few parameters fit to data in the sub-critical region can predict the thermodynamic properties of fluids over a wide range of P– T conditions.

Solubility Of Amino Acids In Water And Aqueous Solutions By the Statistical Associating Fluid Theory

The method previously developed is applied to determine the parameters of the statistical associating fluid theory (SAFT) equation of state for amino acids. With the parameters determined, the SAFT equation of state is applied to model the amino acid solubility in water with very good precision, including l-tyrosine, l-leucine, l-aspartic acid, l-tryptophan, l-glutamic acid, l-alanine, dl-alanine, dl-valine, l-phenylalanine, dl-serine, l-proline, l-serine, and glycine. On the basis of the results of binary systems of amino acid/water, the solubility of l-alanine and l-leucine in water at high pressures up to 3500 bar is simulated; the amino acid solubilities in an aqueous solution is modeled, including systems dl-alanine/dl-valine/H2O at 25 °C, dl-alanine/dl-serine/H2O at 25 °C, l-glutamic acid/l-aspartic acid/H2O at 25, 40, and 60 °C, l-serine/l-aspartic acid/H2O at 25, 40, and 60 °C, and l-serine/l-glutamic acid/H2O at 25, 40, and 60 °C, and good agreement between modeling results and data are obtained.

Effect of alcohol on the solubility of amino acid in water

In a previous work, the parameters of the statistical associating fluid theory (SAFT) equation of state for amino acids were determined by using the method developed. The solubility of amino acids in water was modeled. In this work, the SAFT equation of state has been applied to describe the solubility of amino acids in aqueous alcohol solutions. The systems include dl-alanine/ethanol/water, glycine/ethanol/water, dl-valine/ethanol/water, dl-serine/ethanol/water, glycine/1-propanol/water, glycine/2-propanol/water, l-alanine/2-propanol/water, l-leucine/ethanol/water. Binary interaction parameters between amino acid and alcohol are needed by the SAFT model to get good modeling results.

Vapor−Liquid Equilibria Simulation and an Equation of State Contribution for Dipole−Quadrupole Interactions

A systematic investigation on vapor-liquid equilibria (VLEs) of dipolar and quadrupolar fluids is carried out by molecular simulation to develop a new Helmholtz energy contribution for equations of state (EOSs). Twelve two-center Lennard-Jones plus point dipole and point quadrupole model fluids (2CLJDQ) are studied for different reduced dipolar moments micro*2=6 or 12, reduced quadrupolar moments Q*2=2 or 4 and reduced elongations L*=0, 0.505, or 1. Temperatures cover a wide range from about 55% to 95% of the critical temperature of each fluid. The NpT+test particle method is used for the calculation of vapor pressure, saturated densities, and saturated enthalpies. Critical data and the acentric factor are obtained from fits to the simulation data. On the basis of this data, an EOS contribution for the dipole-quadrupole cross-interactions of nonspherical molecules is developed. The expression is based on a third-order perturbation theory, and the model constants are adjusted to the present 2CLJDQ simulation results. When applied to mixtures, the model is found to be in excellent agreement with results from simulation and experiment. The new EOS contribution is also compatible with segment-based EOS, such as the various forms of the statistical associating fluid theory EOS.

Renormalization group theory for fluids including critical region. II. Binary mixtures

Abstract In our previous work [J. Mi, C. Zhong, Yi.-G. Li, J. Chen, Chem. Phys., 305 (2004) 37–45], an equation of state (EOS) based on the combination of renormalization group theory (RG) and the statistical associating fluid theory (SAFT) was proposed for describing pure fluid thermodynamic properties both inside and outside critical region, which was extended to binary mixtures in this work. A variety of binary systems were considered in this work, including nonpolar/nonpolar, nonpolar/polar, nonpolar/associating and associating/associating mixtures. Two adjustable parameters are required by the new EOS for each binary system, which are obtained by fitting the vapor–liquid equilibria (VLE) data at one temperature. The calculated results show that the new EOS gives satisfactory predictions for critical properties as well as the VLE at other temperatures, both inside and outside critical region. This work demonstrates that RG theory is a very useful tool for accurately describing fluid properties inside critical region, and a combination of it with SAFT EOS can lead to a new EOS possessing the advantages of both theories, applicable to the whole phase equilibrium region of binary mixtures.

Reliable Computation of Phase Stability and Equilibrium from the SAFT Equation of State

In recent years, molecularly based equations of state, as typified by the SAFT (statistical associating fluid theory) approach, have become increasingly popular tools for the modeling of phase behavior. However, whether this, or even a much simpler model, is used, the reliable calculation of phase behavior from a given model can be a very challenging computational problem. A new methodology is described that is the first completely reliable technique for computing phase stability and equilibrium from the SAFT model. The method is based on interval analysis, in particular an interval Newton/generalized bisection algorithm, that provides a mathematical and computational guarantee of reliability and is demonstrated using nonassociating, self-associating, and cross-associating systems. New techniques are presented that can also be exploited when conventional point-valued solution methods are used. These include the use of a volume-based problem formulation, in which the core thermodynamic function for phase equilibrium at constant temperature and pressure is the Helmholtz energy, and an approach for dealing with the internal iteration needed when there are association effects. This approach provides for direct, as opposed to iterative, determination of the derivatives of the internal variables.

Phase behavior of isotactic polypropylene/C4-solvents at high pressure. Experimental data and SAFT modeling

The phase behavior of two types of polypropylene (PP) (commercial Ziegler–Natta PP and a metallocenic PP) in n-butane and in 1-butene is investigated in a high-pressure variable-volume view cell. For these systems, three types of phase equilibrium are observed: vapor–liquid, liquid–liquid and vapor–liquid–liquid. The statistical associating fluid theory equation of state (SAFT-EOS) is used to model the experimental data through bubble-point calculations. A simplified strategy to calculate the three-phase equilibrium in polymeric solutions via SAFT-EOS is presented. Results show a good agreement between the SAFT model and experimental data when a binary interaction parameter is employed to fit the experimental values.

Phase Behavior and Modeling of Supercritical Carbon Dioxide−Organic Acid Mixtures

Pressure−composition isotherms are obtained for binary mixtures of carbon dioxide with formic acid, acetic acid, butyric acid, valeric acid, caproic acid, and caprylic acid at temperatures of 35.0−120.0 °C and pressures up to 250 bar. The accuracy of the experimental apparatus was tested by comparing the measured phase equilibrium data of the carbon dioxide−acetic acid system at 40.0 and 60.0 °C with those of Laugier et al. These six carbon dioxide−polar solute systems exhibit type I phase behavior, which is characterized by an uninterrupted critical mixture curve that has a maximum in pressure. In each system, the mixture critical point increases as the temperatures increases, and also the mixture critical pressure increases as the molecular weight increases. On the contrary, the carbon dioxide−formic acid system shows a higher mixture critical pressure compared with those of the other systems. The experimental data are modeled using both the statistical associating fluid theory (SAFT) and the Peng−Robinson (P−R) equation of state. The SAFT equation of state reasonably models the pressure−composition isotherms for these six systems only if two temperature-independent mixture parameters are used for each system. The P−R equation of state calculated the phase behavior with one or two temperature-independent mixture parameters.

Development of the semi-empirical equation of state for square-well chain fluid based on the Statistical Associating Fluid Theory (SAFT)

A semi-empirical equation of state for the freely jointed square-well chain fluid is developed. This equation of state is based on Wertheim’s thermodynamic perturbation theory (TPT) and the statistical associating fluid theory (SAFT). The compressibility factor and radial distribution function of square-well monomer are obtained from Monte Carlo simulations. These results are correlated using density expansion. In developing the equation of state the exact analytical expressions are adopted for the second and third virial coefficients for the compressibility factor and the first two terms of the radial distribution function, while the higher order coefficients are determined from regression using the simulation data. In the limit of infinite temperature, the present equation of state and the expression for the radial distribution function are represented by the Carnahan-Starling equation of state. This semi-empirical equation of state gives at least comparable accuracy with other empirical equation of state for the square-well monomer fluid. With the new SAFT equation of state from the accurate expressions for the monomer reference and covalent terms, we compare the prediction of the equation of state to the simulation results for the compressibility factor and radial distribution function of the square-well monomer and chain fluids. The predicted compressibility factors for square well chains are found to be in a good agreement with simulation data. The high accuracy of the present equation of state is ascribed to the fact that rigorous simulation results for the reference fluid are used, especially at low temperatures and low densities.

SAFT equation of state for fluid mixtures of hard chain copolymers

We present the SAFT (statistical associating fluid theory) equation of state for mixtures consisting of heteronuclear hard chain molecules using Wertheim's first-order perturbation theory extended to monodisperse mixtures. The reference fluid is a hard sphere mixture. Since only pure fluid simulation results are available, comparisons of theoretical results with molecular dynamics simulation data are made for the compressibility factor of pure hard chain copolymer systems, namely, block, alternate and random copolymers. Our comparisons show a very good performance of the equation of state. The comparisons of our theory with other equations of state for heteronuclear hard chain fluids show that the SAFT equation of state is almost equivalent to the generalized Flory dimer theory, and it represents a significant improvement over the generalized Flory, Chiew-Percus-Yevick and the modified form of Chiew-Percus-Yevick theories.

The use of molecular orbital calculations to describe the phase behavior of hydrogen-bonding mixtures

In previous studies we have used Hartree-Fock theory and density functional theory to compute the enthalpy and entropy changes of dimerization for methanol and a number of small carboxylic acids. We have shown that by using these results in a physical equation of state, the statistical associating fluid theory (SAFT), we are able to model the phase behavior of these pure hydrogen-bonding compounds with a reduction in the number of adjustable parameters; in this study, we use the pure-component parameters derived from the results of our molecular orbital calculations to describe the phase behavior of mixtures containing one associating and one nonassociating compound, again using the SAFT equation of state. We show that the use of the pure-component SAFT parameters derived from our quantum-mechanical calculations results in correlations of mixture VLE data with no loss of accuracy, and frequently with improved accuracy, compared to the original parameters reported for use with the SAFT model.

Strengths and limitations of SAFT for calculating polar copolymer-solvent phase behavior

Statistical associating fluid theory (SAFT) is used to calculate the cloud-point behavior of poly(ethylene-co-methyl acrylate) (EMA) copolymers (0–41 mol % methyl acrylate) in ethane, propane, butane, ethylene, propylene, 1-butene, chlorodifluoromethane, and dimethyl ether at temperatures to 250°C and pressures to 2,600 bar. Poor agreement is obtained between calculated and experimental data if the pure component EMA parameters used in SAFT are calculated using mixing rules that average polyethylene (PE) and poly(methyl acrylate) (PMA) parameters. Therefore, two of the three pure component parameters for all of the EMA copolymers are fixed to the values reported for PE and the third parameter, uo/k, for the copolymer containing 31 mol % methyl acrylate (EMA31) is determined by fitting the EMA31-butane cloud-point curve. The value for (uo/k)PMA is then obtained using a mixing rule and the values of uo/k for all of the EMA copolymers are calculated. A good fit of all of the copolymer–solvent cloud-point curves is obtained using a temperature-independent mixture parameter, kij. With this method of calculation it is possible to correlate cloud-point data with the SAFT equation of state if a small amount of experimental data are available. © 1996 John Wiley & Sons, Inc.

(Methanol + ethene): phase behavior and modeling with the SAFT equation of state

Pressure (composition) isotherms for (methanol + ethene) were obtained at the temperatures (303.65, 333.15, 382.35, and 431.95) K. The critical-mixture pressures from these isotherms are in good agreement with the critical-mixture results obtained by Brunner (J. Chem. Thermodynamics1985, 17, 871). The experimental results are modeled with the statistical associating-fluid theory (SAFT), which explicitly accounts for methanol-methanol association and for ethene-methanol complex formation. The SAFT model, with a single temperature-independent mixture parameter fitted to one isotherm, correctly predicts the type of phase diagram exhibited by (methanol + ethene). However, calculated critical pressures are as much as 4 MPa greater than observed values. The Peng-Robinson equation of state is not capable of even qualitatively predicting the correct phase behavior with two temperature-independent mixture parameters, because it cannot account for either methanol-methanol self-association or ethene-methanol complexing.