Classical Mixing Rules Research Articles

The Effect of Binary Interaction Parameters and Induced Association on the Performance of the PC-SAFT Equation of State: Validation against a Benchmark Database of High-Quality Binary-System Data

The performance of the Perturbed Chain - Statistical Associating Fluid Theory (PC-SAFT) equation of state (EoS) using classical mixing rules, induced-association schemes for mixtures exhibiting cross-association phenomena between two components that do not self-associate, and one regressed temperature-independent binary interaction parameter (kij) per mixture is assessed using a high-quality reference database containing binary-system data for various properties. The binary mixtures that are included in the database are divided into nine categories (Binary Association Codes, BACs) according to their ability to be involved in a hydrogen bond. Grading of the EoS is done by applying an already published procedure with certain modifications that were also published recently. An important feature is the inclusion of a performance indicator, called the “Success Ratio” (SR), that quantifies the percentage of experimental data points which the model can qualitatively reproduce. The original marks of the PC-SAFT EoS when implemented with classical mixing rules, no induced-association scheme, and a kij parameter set equal to zero are recalculated with the modified grading procedure and are directly compared with the new marks obtained in this work. The effect of including induced-association effects in PC-SAFT for mixtures that belong to BAC6, BAC7, and BAC8 (mixtures in which cross-association takes place without both components being self-associating) is also investigated. The effect of regressing cross-association parameters vs regressing kij parameters for mixtures belonging to the BAC6 group (mixtures in which cross-association arises although the two components do not self-associate) is showcased. The final optimized PC-SAFT EoS utilizes classical mixing rules, induced-association effects (using regressed cross-association parameters for some binary systems belonging to BAC6 and a predictive method for BAC7 and BAC8), and one temperature-independent kij parameter per binary mixture (except for binary systems belonging to BAC6 for which cross-association parameters were adjusted). The final average mark of the optimized EoS is 9.3/20.

Validation of Soave–Redlich–Kwong equation of state coupled with a classical mixing rule for sound speed of non-ideal gas mixture of oxygen-hydrogen as liquid rocket propellants

Numerical simulations of inlet jets flow of propellants in the main combustion chamber equipped in a liquid rocket engine have been conducted by compressible CFD technics employing a van der Waals-type equation of state (EOS) and chemical reaction models. Various simulations have been already conducted both for combustion and non-combustion state in the chamber; however, the accuracy of EOS for the mixture of propellant employing a classical mixing rule (CMR), particularly for the sound speed, has not been understood well. Therefore, we have validated the applicability of Soave–Redlich–Kwong (SRK) EOS as one of the van der Waals-type EOSs coupled with the CMR for the sound speed using molecular dynamics (MD) simulations against an imaginary model of oxygen-hydrogen mixture system. We found that SRK EOS coupled with a well-employed CMR can reproduce the sound speed in the MD simulations within a relative error of several percent and that the CMR hardly increases the deviation of the SRK EOS in the single component. The present results suggest that the SRK EOS with the CMR cannot be the main factor to cause critical errors in the CFD based on RANS, which will give reliability to current CFD simulations for internal flows in a combustion chamber of a liquid rocket engine.

Experimental measurements and thermodynamic modeling of hydrate dissociation conditions for CO2 + THF + MgCl2 + water systems

• Experimental hydrate dissociation conditions for (CO2 + THF + MgCl2 + Water) system are reported. • A thermodynamic model has been developed and its results are in acceptable agreement with the experimental data. • The addition of low-medium concentration THF to the system has a promotion effect. • The addition of MgCl2 has an inhibition effect In this study, experimental and thermodynamic modeling on hydrate dissociation conditions of CO 2 + water systems in the absence/presence of MgCl 2 + THF aqueous solutions at various concentrations were investigated. The ternary systems, including different aqueous solutions of tetrahydrofuran and magnesium chloride (0.05 mass fraction of MgCl 2 , 0.05 mass fraction of THF: solution A; 0.15 mass fraction of MgCl 2 , 0.05 mass fraction of THF: solution B; 0.05 mass fraction of MgCl 2 , 0.20 mass fraction of THF: solution C; 0.15 mass fraction of MgCl 2 , 0.20 mass fraction of THF: solution D) were conducted in the temperature range between 280.9 to 292.9 K and pressure ranges 0.92 to 3.89 MPa. An isochoric pressure–search method was applied to undertake all measurements. The results demonstrated that MgCl 2 acts as an inhibitor in all experiments, while THF acts as a promoter. Comparing mentioned data with the same system in the absence of MgCl 2 , a shift is observed toward the left in the equilibrium curve that this kind of shifting left is more observable at high pressures. In addition, a thermodynamic model was developed utilizing the Peng-Robinson equation of state with classical mixing rules and the UNIQUAC activity coefficient model to predict the equilibrium conditions of the aforementioned system. The model results implied a reasonable agreement with the experimental dissociation data with an average absolute deviation of pressure below 0.3%.

On the phase behaviour of the CO2 + N2O4 system at low temperatures

• VLE data for CO 2 + NO 2 /N 2 O 4 at 253 to 303 K using a static-synthetic apparatus. • Experimental data compared to the Peng–Robinson (PR) with classic mixing rules. • Melting points of N 2 O 4 in CO 2 were measured. • Models used to predict the phase diagram of the binary system. Carbon dioxide capture transportation and storage is one of the technologies that can be employed to reduce CO 2 emissions from power plants. Unfortunately, in the post combustion capture process, CO 2 is not pure and contains impurities like SO 2 , NO x , N 2 , O 2 and Ar for example. In this paper, Vapour-Liquid-Equilibrium (VLE) of a binary system composed of CO 2 and N 2 O 4 /NO 2 have been investigated. The equipment used is based on the “static-synthetic” method with a variable cell to determine the bubble pressure or saturated pressure of the system. The setup was used to obtain bubble point data at four isotherms (253.43, 273.43, 293.43, 303.43) K and pressures up to 7.3 MPa. The accuracies of the measured temperature and pressure were estimated to be 0.03 K and 0.12 kPa, respectively. The Peng–Robinson equation of state (PR78 EoS) is used to represent the isothermal P, x data.

Method for prediction of liquid-vapor critical points in binary mixtures: geometrical-EOS model

A new method for predicting the Liquid- Vapor critical point of binary mixture, is presented, which is based in geometrical distances. Actually, the method is based on the minimization of the distance between the experimental and calculated values of the critical temperatures and critical pressures. The SRK and PR equations of state along with classical mixing rules of van der Waals were used as thermodynamic models to calculate the critical point of a given mixture. The proposed method requires that the mixture parameters a, b, and the covolume ε = b/v of each equation of state be determined at each iteration by solving the resulting cubic equation. For nine binary mixtures containing: hydrocarbon derivatives, carbon dioxide and alcohols are studied. The AARE of the calculated values is about 0.86% for critical temperature and 2.07% for critical pressure. Good agreements are found between the calculated results and experimental data. The technique is a general purpose one and can be applied in connection with other thermodynamic models.

Assessment of the Perturbed Chain-Statistical Associating Fluid Theory Equation of State against a Benchmark Database of High-Quality Binary-System Data

The perturbed chain-statistical associating fluid theory (PC-SAFT) equation of state (EoS) is applied, and its performance is assessed using a reliable free-to-access benchmark thermodynamic database developed by the authors. A total of 200 non-electrolytic binary systems are included in the database, which are divided into nine groups according to the associating character of the components, that is, their ability to be involved in a hydrogen bond. The properties based on which the assessment is performed include two-phase vapor–liquid and liquid–liquid equilibrium, three-phase vapor–liquid–liquid equilibrium, critical point, azeotropic point, enthalpy of mixing, and heat capacity of mixing data. A specific procedure for the calculation of deviations between model predictions and experimental data and for grading the thermodynamic model is applied based on a previously published proposed methodology. The PC-SAFT EoS is implemented with classical mixing rules, no induced association scheme, and no regressed binary interaction parameters. The results are discussed, and specific characteristics of model performance with respect to different families of mixtures are analyzed.

A coupled MD-FE methodology to characterize mechanical interphases in polymeric nanocomposites

This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents.Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters.As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young’s modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson’s ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.

A simple thermodynamic model for the solubility of thermolabile solids in supercritical fluids

An equation-of-state (EoS) scheme to correlate the solubility of solids in supercritical fluids is presented. The solute fugacity coefficient is obtained using the pure-solvent compressibility factor, and empirical solute-to-solvent parameter ratios of cohesion factors and covolumes. The proposed method is simpler than conventional EoS calculations since no iteration is required. We retain the link to classical cubic EoS and mixing rules and showcase the application employing both the Redlich-Kwong and Peng-Robinson EoS. The method uses two adjustable parameters, which are computed from experimental data for several binary systems and used to predict solubilities. The results have been favorably compared to those computed by other methods. The advantage of EoS-based models over empirical ones has been emphasized for cases where the solubilities are extrapolated beyond the range of experimental data. The proposal is advantageous for correlating solubility of thermolabile solids in supercritical fluids since no critical properties of the solute are required.

Numerical Modeling of Transcritical and Supercritical Fuel Injections Using a Multi-Component Two-Phase Flow Model

In the present numerical study, implicit large eddy simulations (LES) of non-reacting multi-components mixing processes of cryogenic nitrogen and n-dodecane fuel injections under transcritical and supercritical conditions are carried out, using a modified reacting flow solver, originally available in the open source software OpenFOAM®. To this end, the Peng-Robinson (PR) cubic equation of state (EOS) is considered and the solver is modified to account for the real-fluid thermodynamics. At high pressure conditions, the variable transport properties such as dynamic viscosity and thermal conductivity are accurately computed using the Chung transport model. To deal with the multicomponent species mixing, molar averaged homogeneous classical mixing rules are used. For the velocity-pressure coupling, a PIMPLE based compressible algorithm is employed. For both cryogenic and non-cryogenic fuel injections, qualitative and quantitative analyses are performed, and the results show significant effects of the chamber pressure on the mixing processes and entrainment rates. The capability of the proposed numerical model to handle multicomponent species mixing with real-fluid thermophysical properties is demonstrated, in both supercritical and transcritical regimes.

Experimental measurement and modelling of vapor-liquid equilibrium for 3,3,3- Trifluoropropene (R1243zf) and trans-1,3,3,3-Tetrafluoropropene (R1234ze(E)) binary system

• Experimental measurement of VLE data for R1243zf + R1234ze(E) by a static analytical method. • The R1243zf + R1234ze(E) system is a zeotropic system within 283.15 to 323.14 K. • Experimental data show good agreement with the predicted results of PR-MC-vdW and PR-MC-MHV2 models. The environmental performance becomes the critical factor of the selection of refrigerant. 3,3,3-trifluoropropene (R1243zf) and trans-1,3,3,3-tetrafluoropropene (R1234ze(E)) are environmental friendly candidates to replace R134a. Whereas, in the consideration of environmental factor, safety, and energetic efficiency at the same time, none of the pure refrigerants has been proved to be a suitable solution. Mixed refrigerants provide an approach to this problem because the characteristics of refrigerant blends can be optimized for particular applications in the way pure component refrigerant alone cannot be. The purpose of this work is to explore an R1243zf/R1234ze(E) blend which can be possibly used as a long-term alternative to replace hydrofluorocarbons (HFCs) and hydrochlorofluorocarbons (HCFCs). We report the measurements of saturated vapor pressures of R1234ze(E) and R1243zf ranging from 273.17 to 353.13 K, and isothermal vapor-liquid equilibrium of R1243zf + R1234ze(E) systems from 283.15 to 323.14 K. The measurements are performed by static-analytic type apparatus coupled with two electromagnetic capillary samplers (ROLSI®, patent of Armines). The experimental data are correlated by the Peng-Robinson (PR) equation of state (EoS) associated with Mathias-Copeman (MC) alpha function and classical mixing rules. For the comparison of conventional mixing rules and excess free energy ( G E ) mixing rules models, classic van der Waals one fluid mixing rules and modified Huron-Vidal second-order (MHV2) mixing rules are employed respectively to the correlation of experimental VLE data. The modeling results are in good agreement with the measured data, while the PRMC-MHV2 model exhibits better performance in VLE prediction.

Vapour-liquid equilibrium data for the carbon dioxide (CO2) + carbon monoxide (CO) system

In carbon capture, utilization and storage (CCUS), a thorough understanding of thermophysical behaviour of the candidate fluid is an essential requirement for accurate design and optimised operation of the processes. In this communication, vapour liquid equilibrium data (VLE) of the binary mixtures of CO + CO2 are presented. A static-analytic method was used to obtain VLE data at six isotherms (253.15, 261.45, 273.00, 283.05, 293.05, 298.15) K and pressures up to 12 MPa. The standard uncertainties of the measured temperature and pressure were estimated to be 0.1 K, 0.005 MPa, respectively. Also, the standard uncertainty of the measured molar composition of each phase is found to be less than 1.1%. The measured experimental results are then compared with some predictive thermodynamic equations of state (EoS) (i.e. the Peng Robinson (PR-78) with classical or Wong-Sandler mixing rules, the GERG, and EoS-CG without and with a specific departure function) and available data in the literature. A sound agreement is observed between the results of this work and some of the VLE data published in the open literature. Furthermore, for all isotherms, the best agreement is observed between experimental results and predicted VLE data from the PR-EoS with the Wong-Sandler mixing rules and the EoS-CG with a specific departure function. However, a significant deviation is found between measured results and VLE data calculated using the GERG-EoS.

Estimation of gas solubility in petroleum fractions using PR-UMR and group contributions methods

Solubility of N2, CO2 and CH4 in petroleum fluids is estimated using the Peng-Robison cubic equation of state and three different mixing rules: the classical mixing rule of van der Waals (vdW), the modified Huron-Vidal of order one (MHV1) mixing rule and the universal mixing rule (UMR). The petroleum fluids were split out in pseudocomponents using a mass distribution procedure. Then, to determine equation-of-state parameters, pseudocomponents were characterized by a group contribution method, where the Gibbs free energy is minimized. In the proposed methodology, mixing-rule parameters for predicting gas solubility from binary mixtures of identifiable components were used in unidentifiable petroleum fluids. Twenty-two systems, which include bitumens, heavy oils, refinery cuts and coal liquids, were considered in order to assess the methodology. Good agreement between experimental data and predicted values were obtained for the UMR and vdW mixing rules, but significant deviations are obtained with the MHV1 mixing rule. The total absolute average deviations are 13%, 15% and 30% for the UMR, vdW and MHV1 mixing rules, respectively.

Vapor-liquid equilibrium measurements for 5 binary mixtures involving HFO-1336mzz(E) at temperatures from 313 to 353 K and pressures up to 2.735 MPa

Performances of systems using refrigerants are highly dependent on the selection of the working fluids. In order to guide this choice, the thermophysical properties of these fluids need to be correctly correlated. Experimental data are required for models to work; The knowledge of thermophysical properties of working fluids is required for designing ORC and High Pump (HP) systems. Isothermal vapor-liquid equilibrium (VLE) data for the HFO-1336mzz(E) (trans-1,1,1,4,4,4-hexafluoro-2-butene) binary mixture with HFC-290 (Propane), HFC-134a (1,1,1,2-tetrafluoroethane), HFC-152a (1,1-difluoroethane), HFC-227ea (1,1,1,2,3,3,3-heptafluoropropane) and HFO-1234ze(E) (trans-1,3,3,3-tetrafluoropropene) were measured at 313 K, 333 K and 353 K. The experimental technique used is based on static analytic methods. All of the data (P-x-y) were well correlated with the Peng-Robinson equation of state, associated with a Mathias-Copeman alpha function and classical mixing rules. The results of such modeling are in good agreement with the measured data.

Rheology of non-Newtonian liquid mixtures and the role of molecular chain length

We have examined mixtures of toluene, a Newtonian liquid, with several non-Newtonian liquids (i.e. surfactants) across the full composition range, from pure toluene to pure surfactant, for the purpose of exploring their rheological properties. Surfactants of similar viscosities but significantly different molecular weights were chosen, in order to verify the role of the molecular chain length. We discovered that the classical mixing rule, based on excess activation energy, succeeds for mixtures with shorter molecular chain, but fails for mixtures with longer ones. We suggest here a hypothesis which explains this difference as the result of expanding-collapsing of flexible long-chain molecules in shear flow. In order to support this hypothesis, we applied longitudinal rheology measurements which reproduce such oscillatory effects under the controlled conditions of an ultrasound longitudinal wave. This method confirms that there are qualitative differences in the rheology of these liquids mixtures with short and long molecular chains.

Effect of Microgeometry on Modeling Accuracy of Fluid-Saturated Rock Using Dielectric Permittivity

A common practice for estimating subsurface constituents from remote sensing methods is to use analytical effective medium models relating effective dielectric permittivity to properties of the targeted region. These models suggest that the effective permittivity depends on the volumetric fraction and phase property of each constituent of the composite. Most effective medium models are based on idealized approximations of the composite’s geometry. Some studies have shown that the analytical mixing rule may underestimate or overestimate the effective property when there are geometrical variations. In this paper, we use numerical experiments to compute the effective dielectric permittivity of composites with different microgeometries having varying amounts of internal interfaces. By comparing the numerical results with the classic analytical mixing rules, we quantify the discrepancy with a diagram that indicates the high deviation region. The study is carried out by using various fluid–solid permittivity contrasts to suit a wide range of applications.

Characterization of ferrite magnetic nanoparticle modified polymeric composites by modeling

Abstract This study aims to develop a computational method for characterizing the magnetic and dielectric properties of ferrite nanoparticle modified polymeric nanocomposites for mobile device applications. The spinel structured nickel zinc ferrite (Ni0.5Zn0.5Fe2O4) nanoparticles were synthesized using chemical thermal treatment. The nanocomposites were fabricated using epoxy matrix dispersed with the nanoparticles through ultrasonic agitation mixing. The effects of the ferrite nanoparticles’ shape, concentration, and distribution on the magneto-dielectric properties of the nanocomposites were systematically investigated by computer modeling. The isothermal magnetization curves of the nanocomposites have been experimentally measured for validating the numerical prediction. The effective relative permittivity and permeability of the nanocomposites predicted by numerical method were compared with the results of analytical methods (classical mixing rules of Maxwell-Garnett rule and Bruggeman rule).

Experimental Phase Equilibrium Data for Rotenone in Supercritical Carbon Dioxide

In this paper, we report data for the phase transition of the binary [rotenone (1) + CO2 (2)] system obtained visually using the static synthetic indirect method and a variable-volume cell at temperatures between 303 and 343 K and pressures near 20 MPa. The molar fraction varied from 1.0 × 10–5 to 2.75 × 10–5. For a molar fraction of rotenone greater than 2.75 × 10–5, the phase transition data could not be measured at pressures of 30 MPa because of the experimental limitations of the equipment. The pressure limitation arose because of the difficulty in solubilizing the system where the solubilization occurs close to or above the limits of the experimental apparatus (30 MPa). The transition of the system was identified as the solid–fluid type, and the phase transition profiles are similar. The Peng–Robinson equation of state (PR-EoS) with the classical mixing rule was used to correlate the experimental data to theory, and excellent agreement was found between the experimental and calculated values.

<i>Ab-initio</i> validation of a cubic equation of state with classical mixing rule for oxygen-hydrogen mixture system focusing on its intermolecular interaction between the different kinds

<i>Ab-initio</i> validation of a cubic equation of state with classical mixing rule for oxygen-hydrogen mixture system focusing on its intermolecular interaction between the different kinds

Peng–Robinson Equation of State Model for Polycyclic Aromatic Hydrocarbons and Long-Chain Hydrocarbons Solubilities in Supercritical Fluids. Correlations Based on Solute Molecular Properties

The Peng–Robinson equation of state (EOS) with the classical van der Waals quadratic mixing rules is used in this study to calculate the solid solubilities of polycyclic aromatic and long chained hydrocarbons in supercritical carbon dioxide, ethane, and ethylene. The effect of the binary interaction parameters kij (for parameter a) and lij (for parameter b) in the van der Waals mixing rules, on the pressure dependence of the solute solubility in the supercritical fluid is examined. The addition of a third parameter, λ2, that modifies the pure component covolume parameter b of the solute, significantly enhanced the ability of the EOS to fit the experimental data. Reasonable optimally fitted binary interaction parameters kij and lij and pure component parameter λ2 were obtained. The EOS parameters were correlated with molecular properties and can be used as estimates for predicting solubilities when experimental data is missing. A phase stability test is proposed to guarantee that the stable root is always ...

Solubility of oleic acid, triacylglycerol and their mixtures in supercritical carbon dioxide and thermodynamic modeling of phase equilibrium

To evaluate the selectivity of supercritical carbon dioxide (scCO2) as a solvent to deacidify vegetable oils, the solubility of oleic acid and of a mixture of oleic acid and refined sunflower oil in scCO2 was measured. Sunflower oil was adopted as a pseudo-pure component and the phase equilibrium of ternary and binary systems were thermodynamically modeled by the Peng-Robinson equation of state with classical mixing rule. The values of the estimated and/or adopted critical properties in modeling thermodynamics were shown to be of great influence on the quality of data correlation. Although correlating data from binary systems is possible, the modeling was incapable of satisfactorily predicting the behavior of the phase equilibrium of the ternary system. The large deviations observed may derive from the presence of specific intermolecular interactions between the components of these systems that were not considered in this type of modeling.