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Journal articles on the topic 'Phase equilibria modelling'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Deiters, Ulrich K. "Modelling Supercritical Phase Equilibria: Problems and Pitfalls." Periodica Polytechnica Chemical Engineering 63, no. 2 (March 18, 2019): 261–69. http://dx.doi.org/10.3311/ppch.12757.

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This article addresses some recurring difficulties and problems of computing phase equilibria involving supercritical fluid phases. These difficulties prevent a full automatization of thermodynamic calculations and require human interference. Examples are the wrong interpretation of experimental data, phase inversion phenomena, or overlooking phases. While none of these insights are knew, publications dealing with them are scattered and sometimes hard to obtain. This article gives a short overview over some of the most common difficulties and pitfalls.

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2

Onderka, B., A. Dębski, and W. Gąsior. "Thermodynamic Assessment Of The Bi-In-Zn System." Archives of Metallurgy and Materials 60, no. 2 (June 1, 2015): 567–75. http://dx.doi.org/10.1515/amm-2015-0175.

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Abstract A thermodynamic description of the entire ternary Bi-In-Zn system was obtained by the CALPHAD modelling of the Gibbs energy of the liquid phase. The experimental data on the phase equilibria and the thermodynamic properties published and complemented by the authors’ own experiments were taken into account. In order to verify the phase equilibria in the Bi-In-Zn system, 15 different samples were studied in the temperature range of 300-900 K by the DTA technique during heating and cooling cycles. Coexisting phases and their composition were analyzed by the SEM and EDX techniques for 9 distinct samples after their thermal equilibration at 373 K and 473 K. Assessment and selected phase equilibrium calculations were performed with ThermoCalc and Pandat softwares, and compared with experimental data. The obtained results reproduce well the experimental data on both the phase equilibria and the thermodynamic properties in the optimized system.

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3

Ghazi, Hala, Francois James, and Hélène Mathis. "Vapour-liquid phase transition and metastability." ESAIM: Proceedings and Surveys 66 (2019): 22–41. http://dx.doi.org/10.1051/proc/201966002.

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The paper deals with the modelling of the relaxation processes towards thermodynamic equilibrium in a liquid-vapour isothermal mixture. Focusing on the van der Waals equation of state, we construct a constrained optimization problem using Gibbs' formalism and characterize all possible equilibria: coexistence states, pure phases and metastable states. Coupling with time evolution, we develop a dynamical system whose equilibria coincide with the minimizers of the optimization problem. Eventually we consider the coupling with hydrodynamics and use the dynamical system as a relaxation source terms in an Euler-type system. Numerical results illustrate the ability of the whole model to depict coexistence and metastable states as well.

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4

Shubin, A. B., and K. Y. Shunyaev. "MnO-C Interaction – Thermodynamic Modelling of Phase Equilibria." Defect and Diffusion Forum 273-276 (February 2008): 632–36. http://dx.doi.org/10.4028/www.scientific.net/ddf.273-276.632.

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The interaction between manganese(II) oxide and carbon in powder mixtures may give manganese carbides, condensed(c) and gaseous(g) manganese at different temperatures and pressures. The initial assumptions and results of thermodynamic modelling of this interaction in the temperature range 1600-2100 K at the pressures 0.1-1.0 MPa are presented in this paper.

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5

Lototsky, M. V., V. A. Yartys, V. S. Marinin, and N. M. Lototsky. "Modelling of phase equilibria in metal–hydrogen systems." Journal of Alloys and Compounds 356-357 (August 2003): 27–31. http://dx.doi.org/10.1016/s0925-8388(03)00095-1.

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6

Achar, D. R. G., R. Munoz-Arroyo, L. Singheiser, and W. J. Quadakkers. "Modelling of phase equilibria in MCrAlY coating systems." Surface and Coatings Technology 187, no. 2-3 (October 2004): 272–83. http://dx.doi.org/10.1016/j.surfcoat.2004.02.018.

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7

Teja, Amyn S. "Modelling phase equilibria: thermodynamic background and practical tools." Chemical Engineering Science 48, no. 18 (September 1993): 3321. http://dx.doi.org/10.1016/0009-2509(93)80216-d.

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8

Drápala, Jaromír, Petr Kubíček, J. Vřeštál, and Monika Losertová. "Study of Reaction Diffusivity in the Copper–Indium–Tin Ternary System." Defect and Diffusion Forum 263 (March 2007): 231–36. http://dx.doi.org/10.4028/www.scientific.net/ddf.263.231.

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Interaction of lead-free solders with copper substrate represents an important phenomenon in the issue of reliability of solder joints. New experimental data describing phase equilibria in the Cu-In-Sn system after long-time diffusion annealing at the 400 °C/50 hours, 600 °C/310 hours and 600 °C/48 hours will be presented. The composition of solders was: 100 % Sn, 75 % Sn + 25 % In, 50 % Sn + 50 % In, 25 % Sn + 75 % In, 100 % In. The fast quenching method was employed to freeze thermodynamic equilibrium after annealing, followed by metallography, microhardness measurements, SEM (Scanning Electron Microscope) and WDX (Wave Dispersive X-ray) analysis. New phase equilibrium data, together with the data from literature, represent the best existing experimental description of phase equilibria in the system in question. The obtained experimental results of the phase equilibria were compared with the thermodynamic modelling by the CALPHAD (Calculation of Phase Diagrams) method and with other authors.

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9

Baciocchi, R., A. Chiavola, and R. Gavasci. "Ion exchange equilibria of arsenic in the presence of high sulphate and nitrate concentrations." Water Supply 5, no. 5 (December 1, 2005): 67–74. http://dx.doi.org/10.2166/ws.2005.0040.

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The aim of this work was to develop a quantitative description of the ion exchange equilibria of arsenic on a strong anionic resin, in the presence of nitrates and sulphates. First, the ion exchange equilibrium data of As(V) and NO3− on a strong anionic resin in chloride form were obtained and described with a model based on the mass action law. Namely, assuming ideal behaviour for both solution and resin phase, the thermodynamic constant of the As(V)/Cl− and NO3−/Cl− ion exchange equilibria were estimated by fitting of experimental data. Then, these equilibrium constants were used to predict the ion exchange behaviour of the ternary system As(V)/NO3−/Cl−, providing a rather good agreement with experimental results. The ion exchange equilibria involving sulphate ions were also studied, showing a very high affinity to the resin phase. This behaviour did not allow a quantitative robust modelling of the equilibrium pattern. The results discussed in this paper represent a first step toward the development of a comprehensive modelling of the ion exchange process for the removal of As(V) from surface and groundwater in the presence of competitive, naturally occurring anions.

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10

Yakymchuk, C., M. Brown, C. Clark, F. J. Korhonen, P. M. Piccoli, C. S. Siddoway, R. J. M. Taylor, and J. D. Vervoort. "Decoding polyphase migmatites using geochronology and phase equilibria modelling." Journal of Metamorphic Geology 33, no. 2 (December 29, 2014): 203–30. http://dx.doi.org/10.1111/jmg.12117.

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11

Cassel, E., M. Matt, M. Rogalski, and R. Solimando. "Phase equilibria modelling for binary systems that contain CO2." Fluid Phase Equilibria 134, no. 1-2 (June 1997): 63–75. http://dx.doi.org/10.1016/s0378-3812(97)00050-2.

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12

Das, Susanta K., and Alejandro D. Rey. "Computational modelling of multi-phase equilibria of mesogenic mixtures." Computational Materials Science 29, no. 2 (February 2004): 152–64. http://dx.doi.org/10.1016/j.commatsci.2003.06.007.

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13

Vřesťál, J., J. Pavlů, U. D. Wdowik, and M. Sob. "Modelling of phase equilibria in the Hf-V system below room temperature." Journal of Mining and Metallurgy, Section B: Metallurgy 53, no. 3 (2017): 239–47. http://dx.doi.org/10.2298/jmmb170704032v.

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Phase transformation from orthorhombic HfV2 structure to cubic C15 Laves phase structure, which occurs during heating at about 114 K, is well known. In this contribution, a thermodynamic description of this phenomenon is provided supported by ab initio calculations. We utilize the third generation of thermodynamic database extending the Scientific Group Thermodata Europe (SGTE) unary data to zero Kelvin and demonstrate that it may be also applied to intermetallic phases. The data from a recent thermodynamic assessment of the Hf-V system (valid for temperatures above 298.15 K) were used and extended to zero Kelvin by the same method as it was used for unary data. Under the assumption of validity of harmonic approximation and electronic contribution to the heat capacity, the thermodynamics of C15 and orthorhombic phase were described. With the help of ab initio approach, we demonstrate that the HfV2 orthorhombic phase and C15 Laves phase are mechanically stable at 0 K and thanks to entropy stabilization they are in equilibrium with pure element phases in the temperature region of structural change.

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14

Yakymchuk, Chris. "Applying Phase Equilibria Modelling to Metamorphic and Geological Processes: Recent Developments and Future Potential." Geoscience Canada 44, no. 1 (April 20, 2017): 27. http://dx.doi.org/10.12789/geocanj.2017.44.114.

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Phase equilibria modelling has played a key role in enhancing our understanding of metamorphic processes. An important breakthrough in the last three decades has been the ability to construct phase diagrams by integrating internally consistent datasets of the thermodynamic properties of minerals, fluids and melts with activity–composition models for mixed phases that calculate end-member activities from end-member proportions. A major advance in applying phase equilibria modelling to natural rocks is using isochemical phase diagrams to explore the phase assemblages and reaction sequences applicable for a particular sample. The chemical systems used for modelling phase equilibria are continually evolving to provide closer approximations to the natural compositions of rocks and allow wider varieties of compositions to be modelled. Phase diagrams are now routinely applied to metasedimentary rocks, metabasites and intermediate to felsic intrusive rocks and more recently to ultramafic rocks and meteorites. While the principal application of these phase diagrams is quantifying the pressure and temperature evolution of metamorphic rocks, workers are now applying them to other fields across the geosciences. For example, phase equilibria modelling of hydrothermal alteration and the metamorphism of hydrothermally altered rocks can be used to determine ‘alteration vectors’ to hydrothermal mineral deposits. Combining the results of phase equilibria of rock-forming minerals with solubility equations of accessory minerals has provided new insights into the geological significance of U–Pb ages of accessory minerals commonly used in geochronology (e.g. zircon and monazite). Rheological models based on the results of phase equilibria modelling can be used to evaluate how the strength of the crust and mantle can change through metamorphic and metasomatic processes, which has implications for a range of orogenic processes, including the localization of earthquakes. Finally, phase equilibria modelling of fluid generation and consumption during metamorphism can be used to explore links between metamorphism and global geochemical cycles of carbon and sulphur, which may provide new insights into the secular change of the lithosphere, hydrosphere and atmosphere.RÉSUMÉLa modélisation des équilibres de phases a joué un rôle clé dans l’amélioration de notre compréhension des processus métamorphiques. Une percée importante au cours des trois dernières décennies a été la capacité de construire des diagrammes de phase en y intégrant des ensembles de données cohérentes des propriétés thermodynamiques des minéraux, des fluides et des bains magmatiques avec des modèles d'activité-composition pour des phases mixtes qui déduisent l’activité des membres extrêmes à partir des proportions des membres extrêmes. Une avancée majeure dans l'application de la modélisation d'équilibre de phase aux roches naturelles consiste à utiliser des diagrammes de phases isochimiques pour étudier les assemblages de phase et les séquences de réaction applicables pour un échantillon particulier. Les systèmes chimiques utilisés pour la modélisation des équilibres de phase évoluent continuellement pour fournir des approximations plus proches des compositions naturelles des roches et permettent de modéliser de plus grandes variétés de compositions. Les diagrammes de phase sont maintenant appliqués de façon routinière aux roches métasédimentaires, aux métabasites et aux roches intrusives intermédiaires à felsiques et plus récemment aux roches ultramafiques et aux météorites. Bien que l'application principale de ces diagrammes de phase consiste à quantifier l'évolution de la pression et de la température des roches métamorphiques, les utilisateurs les appliquent maintenant à d'autres spécialités des géosciences. Par exemple, la modélisation des équilibres de phase de l'altération hydrothermale et du métamorphisme des roches d’altération hydrothermale peut être utilisée pour déterminer les « vecteurs d'altération » des gisements minéraux hydrothermaux. La combinaison des résultats des équilibres de phase des minéraux constitutifs des roches avec des équations de solubilité des minéraux accessoires a permis d’en savoir davantage sur la signification géologique des âges U–Pb des minéraux accessoires couramment utilisés en géochronologie (par exemple zircon et monazite). Les modèles rhéologiques basés sur les résultats de la modélisation des équilibres de phase peuvent être utilisés pour évaluer comment la résistance de la croûte et du manteau peut changer à travers des processus métamorphiques et métasomatiques, ce qui a des implications sur une gamme de processus orogéniques, y compris la localisation des séismes. Enfin, la modélisation des équilibres de phase de la génération et de l’absorption des fluides pendant le métamorphisme peut être utilisée pour explorer les liens entre le métamorphisme et les cycles géochimiques globaux du carbone et du soufre, ce qui peut fournir de nouvelles perspectives sur le changement séculaire de la lithosphère, de l'hydrosphère et de l'atmosphère.

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15

Rodríguez-Palmeiro, Iago, Oscar Rodríguez, Ana Soto, and Christoph Held. "Measurement and PC-SAFT modelling of three-phase behaviour." Physical Chemistry Chemical Physics 17, no. 3 (2015): 1800–1810. http://dx.doi.org/10.1039/c4cp04336g.

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Liquid–liquid–liquid phase equilibria of ternary systems (water + surfactant ionic liquid + n-dodecane) were measured experimentally. Phase behaviour was accurately predicted by PC-SAFT using only pure-component parameters.

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16

Durelle, Jeremy, Jesse R. Vanderveen, and Philip G. Jessop. "Modelling the behaviour of switchable-hydrophilicity solvents." Phys. Chem. Chem. Phys. 16, no. 11 (2014): 5270–75. http://dx.doi.org/10.1039/c3cp55391d.

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17

Hoffmeister, Hans. "Heterogeneous phase equilibria as a basis for crevice corrosion modelling." Steel Research 70, no. 12 (December 1999): 519–23. http://dx.doi.org/10.1002/srin.199905678.

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18

Anderko, Andrzej. "Equation-of-state methods for the modelling of phase equilibria." Fluid Phase Equilibria 61, no. 1-2 (January 1990): 145–225. http://dx.doi.org/10.1016/0378-3812(90)90011-b.

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19

Acharya, S., and J. P. Hajra. "Thermodynamic modelling of phase equilibria in Al-Ga-P-As system." Bulletin of Materials Science 28, no. 2 (April 2005): 179–85. http://dx.doi.org/10.1007/bf02704238.

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20

Bogatu, Cristina, Anca Duţă, Theo W. de Loos, and Dan Geană. "Modelling fluid phase equilibria in the binary system trifluoromethane + 1-phenylpropane." Fluid Phase Equilibria 428 (November 2016): 190–202. http://dx.doi.org/10.1016/j.fluid.2016.06.027.

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21

Gürth, M., A. Grytsiv, J. Vrestal, V. V. Romaka, G. Giester, E. Bauer, and P. Rogl. "On the constitution and thermodynamic modelling of the system Ti–Ni–Sn." RSC Advances 5, no. 112 (2015): 92270–91. http://dx.doi.org/10.1039/c5ra16074j.

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22

Zhao, Baojun, Kun Su, and Xiaodong Ma. "Experimental Determination of Phase Equilibria in the Na2O-SiO2-WO3 System." Metals 11, no. 12 (December 13, 2021): 2014. http://dx.doi.org/10.3390/met11122014.

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The present study investigated phase equilibria in the Na2O-SiO2-WO3 system experimentally using high-temperature equilibration, quenching, and electron probe X-ray microanalysis (EPMA). New thermodynamic information on the Na2O-SiO2-WO3 system was derived based on the newly obtained experimental results and data from the literature. The primary phase fields of sodium metasilicate, sodium disilicate, and tridymite were determined along with the isotherms at 1073, 1173, and 1273 K. The solubilities of WO3 in SiO2, Na2Si2O5, and Na2SiO3, and the solubility of SiO2 in Na2WO4 were accurately measured using EPMA. Comparisons between the existing and newly constructed phase diagram were carried out and the differences are discussed. The phase equilibrium data will be beneficial to the future development of sustainable tungsten industries and thermodynamic modelling in WO3 related systems.

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23

James, François, Mauricio Sepúlveda, and Patrick Valentin. "Statistical Thermodynamics Models for Multicomponent Isothermal Diphasic Equilibria." Mathematical Models and Methods in Applied Sciences 07, no. 01 (February 1997): 1–29. http://dx.doi.org/10.1142/s0218202597000025.

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We propose in this paper a whole family of models for isothermal diphasic equilibrium, which generalize the classical Langmuir isotherm. The main tool to obtain these models is a fine modeling of each phase, which states various constraints on the equilibrium. By writing down the Gibbs conditions of thermodynamical equilibrium for both phases, we are led to a constrained minimization problem, which is solved through the Lagrange multipliers. If one of the phases is an ideal solution, we can solve explicitly the equations, and obtain an analytic model. In the most general case, we have implicit formulas, and the models are computed numerically. The models of multicomponent isotherm we obtain in this paper are designed for chromatography, but can be adapted mutatis mutandis to other cases.

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24

Davies, R. H., J. A. Gisby, A. Dinsdale, M. Tyrer, C. S. Walker, and F. P. Glasser. "Thermodynamic modelling of phase equilibria in cement systems: multiple sublattice model for solids in equilibrium with non-ideal aqueous phase." Advances in Applied Ceramics 113, no. 8 (September 23, 2014): 509–16. http://dx.doi.org/10.1179/1743676114y.0000000205.

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25

MEDRAJ, M., R. HAMMOND, W. T. THOMPSON, and R. A. L. DREW. "PHASE EQUILIBRIA, THERMODYNAMIC MODELLING AND NEUTRON DIFFRACTION OF THE AlN-Al2O3-Y2O3SYSTEM." Canadian Metallurgical Quarterly 42, no. 4 (January 2003): 495–507. http://dx.doi.org/10.1179/cmq.2003.42.4.495.

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26

Guevara, V. E., and M. J. Caddick. "Shooting at a moving target: phase equilibria modelling of high-temperature metamorphism." Journal of Metamorphic Geology 34, no. 3 (February 10, 2016): 209–35. http://dx.doi.org/10.1111/jmg.12179.

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27

Garrido, Nuno M., Georgios K. Folas, and Georgios M. Kontogeorgis. "Modelling of phase equilibria of glycol ethers mixtures using an association model." Fluid Phase Equilibria 273, no. 1-2 (November 2008): 11–20. http://dx.doi.org/10.1016/j.fluid.2008.08.006.

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28

Li, Jun, Zhigang Zhang, Xiaorong Luo, and Xiaochun Li. "Modelling of phase equilibria in CH4–C2H6–C3H8–nC4H10–NaCl–H2O systems." Applied Geochemistry 56 (May 2015): 23–36. http://dx.doi.org/10.1016/j.apgeochem.2015.02.006.

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29

Ferreira, Olga, Esteban A. Brignole, and Eugénia A. Macedo. "Modelling of phase equilibria for associating mixtures using an equation of state." Journal of Chemical Thermodynamics 36, no. 12 (December 2004): 1105–17. http://dx.doi.org/10.1016/j.jct.2004.07.001.

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30

Gabas, N., and C. Laguerie. "Modelling of liquid-solid phase equilibria with UNIFAC. Application to sugar systems." Computers & Chemical Engineering 17 (January 1993): S239—S244. http://dx.doi.org/10.1016/0098-1354(93)80236-g.

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31

Gabas, N. "Modelling of liquid-solid phase equilibria with UNIFAC. Application to sugar systems." Computers & Chemical Engineering 17, no. 1 (1993): S239—S244. http://dx.doi.org/10.1016/0098-1354(93)85037-m.

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32

Mathis, Hélène. "A thermodynamically consistent model of a liquid-vapor fluid with a gas." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 1 (January 2019): 63–84. http://dx.doi.org/10.1051/m2an/2018044.

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This work is devoted to the consistent modeling of a three-phase mixture of a gas, a liquid and its vapor. Since the gas and the vapor are miscible, the mixture is subjected to a non-symmetric constraint on the volume. Adopting the Gibbs formalism, the study of the extensive equilibrium entropy of the system allows to recover the Dalton’s law between the two gaseous phases. In addition, we distinguish whether phase transition occurs or not between the liquid and its vapor. The thermodynamical equilibria are described both in extensive and intensive variables. In the latter case, we focus on the geometrical properties of equilibrium entropy. The consistent characterization of the thermodynamics of the three-phase mixture is used to introduce two Homogeneous Equilibrium Models (HEM) depending on mass transfer is taking into account or not. Hyperbolicity is investigated while analyzing the entropy structure of the systems. Finally we propose two Homogeneous Relaxation Models (HRM) for the three-phase mixtures with and without phase transition. Supplementary equations on mass, volume and energy fractions are considered with appropriate source terms which model the relaxation towards the thermodynamical equilibrium, in agreement with entropy growth criterion.

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33

Korneev, Ivan A., Vladimir V. Semenov, and Tatiana E. Vadivasova. "Synchronization of Periodic Self-Oscillators Interacting via Memristor-Based Coupling." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2050096. http://dx.doi.org/10.1142/s0218127420500960.

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A model of two self-sustained oscillators interacting through memristive coupling is studied. The memristive coupling is realized by using a cubic memristor model. Numerical simulation is combined with theoretical analysis by means of quasi-harmonic reduction. It is shown that the specifics of the memristor nonlinearity results in the appearance of infinitely many equilibrium points which form a line of equilibria in the phase space of the system under study. It is established that the possibility to observe the effect of phase locking in the considered system depends on both parameter values and initial conditions. Consequently, the boundaries of the synchronization region are determined by the initial conditions. It is demonstrated that introducing or adding a small term into the memristor state equation gives rise to the disappearance of the line of equilibria and eliminates the dependence of synchronization on the initial conditions.

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34

Guo, Feng. "Global Analysis of a Liénard System with Quadratic Damping." Discrete Dynamics in Nature and Society 2018 (September 2, 2018): 1–9. http://dx.doi.org/10.1155/2018/1249620.

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In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

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35

Dubessy, Jean, Régis Thiery, and Martin Canals. "Modelling of phase equilibria involving mixed gas clathrates: Application to the determination of molar volume of the vapour phase and salinity of the aqueous solution in fluid inclusions." European Journal of Mineralogy 4, no. 5 (October 14, 1992): 873–84. http://dx.doi.org/10.1127/ejm/4/5/0873.

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36

Eltekova, N. A., and Yu A. Eltekov. "Adsorption Equilibria in Carbon/Hydrocarbon/Water Systems." Adsorption Science & Technology 15, no. 2 (January 1997): 109–14. http://dx.doi.org/10.1177/026361749701500204.

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Two equations derived from the theory of volume filling of micropores (the DR and DS equations) were used for the description of the liquid-phase adsorption of benzene, toluene and p-xylene from aqueous solution on to two commercial microporous activated carbons. Comparison of the experimental and calculated adsorption isotherms showed good accord between the theoretical and experimental values. The results of this work indicate that the parameters of these equations can be used for modelling a water purification process based on the activated carbons.

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37

Bostan, Mihaï, and José Antonio Carrillo. "Fluid models with phase transition for kinetic equations in swarming." Mathematical Models and Methods in Applied Sciences 30, no. 10 (August 7, 2020): 2023–65. http://dx.doi.org/10.1142/s0218202520400163.

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We concentrate on kinetic models for swarming with individuals interacting through self-propelling and friction forces, alignment and noise. We assume that the velocity of each individual relaxes to the mean velocity. In our present case, the equilibria depend on the density and the orientation of the mean velocity, whereas the mean speed is not anymore a free parameter and a phase transition occurs in the homogeneous kinetic equation. We analyze the profile of equilibria for general potentials identifying a family of potentials leading to phase transitions. Finally, we derive the fluid equations when the interaction frequency becomes very large.

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38

Rybka, Piotr. "Dynamical modelling of phase transitions by means of viscoelasticity in many dimensions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 121, no. 1-2 (1992): 101–38. http://dx.doi.org/10.1017/s0308210500014177.

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SynopsisWe study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

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39

Chen, Yuqiu, Xiaodong Liang, John M. Woodley, and Georgios M. Kontogeorgis. "Modelling study on phase equilibria behavior of ionic liquid-based aqueous biphasic systems." Chemical Engineering Science 247 (January 2022): 116904. http://dx.doi.org/10.1016/j.ces.2021.116904.

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40

INDARES, A., R. W. WHITE, and R. POWELL. "Phase equilibria modelling of kyanite-bearing anatectic paragneisses from the central Grenville Province." Journal of Metamorphic Geology 26, no. 8 (October 2008): 815–36. http://dx.doi.org/10.1111/j.1525-1314.2008.00788.x.

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41

Al Ghafri, Saif ZS, and J. P. Martin Trusler. "Phase equilibria of (Methylbenzene + Carbon dioxide + Methane) at elevated pressure: Experiment and modelling." Journal of Supercritical Fluids 145 (March 2019): 1–9. http://dx.doi.org/10.1016/j.supflu.2018.11.012.

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42

Perederic, Olivia A., Larissa P. Cunico, Sawitree Kalakul, Bent Sarup, John M. Woodley, Georgios M. Kontogeorgis, and Rafiqul Gani. "Systematic identification method for data analysis and phase equilibria modelling for lipids systems." Journal of Chemical Thermodynamics 121 (June 2018): 153–69. http://dx.doi.org/10.1016/j.jct.2018.02.007.

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43

Alavianmehr, M. M., F. Akbari, and R. Behjatmanesh-Ardakani. "Modelling phase equilibria of pure ionic liquids from a new equation of state." Ionics 22, no. 12 (July 14, 2016): 2447–59. http://dx.doi.org/10.1007/s11581-016-1769-z.

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44

Kamdjeu Kengne, L., Z. Tabekoueng Njitacke, J. R. Mboupda Pone, and H. T. Kamdem Tagne. "The Effects of a Constant Excitation Force on the Dynamics of an Infinite-Equilibrium Chaotic System Without Linear Terms: Analysis, Control and Circuit Simulation." International Journal of Bifurcation and Chaos 30, no. 15 (December 9, 2020): 2050234. http://dx.doi.org/10.1142/s021812742050234x.

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Abstract:

In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite-equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.

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Gourgouillon, Didier, and Manuel Nunes da Ponte. "High pressure phase equilibria for poly(ethylene glycol)s + CO2: experimental results and modelling." Physical Chemistry Chemical Physics 1, no. 23 (1999): 5369–75. http://dx.doi.org/10.1039/a906927e.

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Tork, Thomas, Gabriele Sadowski, Wolfgang Arlt, André de Haan, and Gerard Krooshof. "Modelling of high-pressure phase equilibria using the Sako–Wu–Prausnitz equation of state." Fluid Phase Equilibria 163, no. 1 (September 1999): 61–77. http://dx.doi.org/10.1016/s0378-3812(99)00219-8.

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Tork, Thomas, Gabriele Sadowski, Wolfgang Arlt, André de Haan, and Gerard Krooshof. "Modelling of high-pressure phase equilibria using the Sako–Wu–Prausnitz equation of state." Fluid Phase Equilibria 163, no. 1 (September 1999): 79–98. http://dx.doi.org/10.1016/s0378-3812(99)00220-4.

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Fourie, Frederick Christiaan van Niekerk, Cara Elsbeth Schwarz, and Johannes Hendrik Knoetze. "CO2 + n-dodecane + 1-decanol: High pressure experimental phase equilibria data and thermodynamic modelling." Journal of Supercritical Fluids 151 (September 2019): 49–62. http://dx.doi.org/10.1016/j.supflu.2019.04.019.

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dos Ramos, María Carolina, Felipe J. Blas, and Amparo Galindo. "Modelling the phase equilibria and excess properties of the water + carbon dioxide binary mixture." Fluid Phase Equilibria 261, no. 1-2 (December 2007): 359–65. http://dx.doi.org/10.1016/j.fluid.2007.07.012.

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Vital, J., V. Almeida, and L. S. Lobo. "Kinetic modelling with phase equilibria of α-pinene hydration to terpine in isobutanol—water." Chemical Engineering Journal 50, no. 3 (December 1992): 115–21. http://dx.doi.org/10.1016/0300-9467(92)80018-6.

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