A comparative study of particle swarm optimization and its variants for phase stability and equilibrium calculations in multicomponent reactive and non-reactive systems

Particle swarm optimization is a novel evolutionary stochastic global optimization method that has gained popularity in the chemical engineering community. This optimization strategy has been successfully used for several applications including thermodynamic calculations. To the best of our knowledge, the performance of PSO in phase stability and equilibrium calculations for both multicomponent reactive and non-reactive mixtures has not yet been reported. This study introduces the application of particle swarm optimization and several of its variants for solving phase stability and equilibrium problems in multicomponent systems with or without chemical equilibrium. The reliability and efficiency of a number of particle swarm optimization algorithms are tested and compared using multicomponent systems with vapor–liquid and liquid–liquid equilibrium. Our results indicate that the classical particle swarm optimization with constant cognitive and social parameters is a reliable method and offers the best performance for global minimization of the tangent plane distance function and the Gibbs energy function in both reactive and non-reactive systems.     

Liquid-liquid phase equilibria of aqueous two-phase systems containing salts and polyethylene glycol

Liquid-liquid phase equilibria of the ternary systems: (a) polyethylene glycol - ammonium sulfate- water and (b) polyethylene glycol - sodium carbonate -water have been determined experimentally at 15°, 25°, 35° and 45°C and for two different molecular weights of the polymer (Avg. M.W. 1000 and 2000). Details of the glass cell and of the equilibration and analytical procedures used are described. Equilibrium data along with phase diagrams are presented. Finally the effect of temperature and of the molecular weight of the polymer are also discussed.     

Prediction of phase equilibria in hydrocarbon + near-critical solvent systems

In this paper the Hole Quasichemical Group-Contribution model (HM) is used to describe phase diagram peculiarities of multicomponent aqueous systems containing hydrocarbons and alcohols. The model correctly describes liquid-liquid (LL), liquid-liquid-vapour (LLV), liquid-vapour (LV) and vapour-vapour equilibria (VVE), and critical and azeotrope curves. The potential use of the HM in the prediction of the solubilities of hydrocarbons in near-critical water and the densities of coexisting phases in the critical region of the system n-hexane-water are presented.     

Prediction of vapor–liquid and liquid–liquid equilibria for polymer systems: Comparison of activity coefficient models

A large number of equations of state and activity coefficient models capable of describing phase equilibria in polymer solutions are available today, but only a few of these models have been applied to different systems. It is therefore useful to investigate the performance of existing thermodynamic models for complex polymer solutions which have not yet been widely studied. The present work studies the application of several activity coefficient models [P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, New York, NY, 1953; T. Oishi, J.M. Prausnitz, Estimation of solvent activities in polymer solutions using a group-contribution method, Ind. Eng. Chem. Process Design Dev. 17 (1978) 333; H.S. Elbro, A. Fredenslund, P. Rasmussen, A new simple equation for the prediction of solvent activities in polymer solutions, Macromolecules 23 (1990) 4707; G.M. Kontogeorgis, A. Fredenslund, D. Tassios, Simple activity coefficient model for the prediction of solvent activities in polymer solutions, Ind. Eng. Chem. Res. 32 (1993) 362; C. Chen, A segment-based local composition model for the Gibbs energy of polymer solutions, Fluid Phase Equilib. 83 (1993) 301; A. Vetere, Rules for predicting vapor–liquid equilibria of amorphous polymer solutions using a modified Flory–Huggins equation, Fluid Phase Equilib. 97 (1994) 43; C. Qian, S.J. Mumby, B.E. Eichinger, Phase diagrams of binary polymer solutions and blends, Macromolecules 24 (1991) 1655; Y.C. Bae, J.J. Shim, D.S. Soane, J.M. Prausnitz, Representation of vapor–liquid and liquid–liquid equilibria for binary systems containing polymers: applicability of an extended Flory–Huggins equation, J. Appl. Polym. Sci. 47 (1993) 1193; G. Bogdanic, J. Vidal, A segmental interaction model for liquid–liquid equilibrium calculations for polymer solutions, Fluid Phase Equilibria 173 (2000) 241] and activity coefficient from equations of state [F. Chen, A. Fredenslund, P. Rasmussen, Group-contribution Flory equation of state for vapor–liquid equilibria en mixtures with polymers, Ind. Eng. Chem. Res. 29 (1990) 875; M.S. High, R.P. Danner, Application of the group contribution lattice—fluids EOS to polymer solutions, AIChE J. 36 (1990) 1625]. The evaluation of these models was carried out both at infinite dilution and at finite concentrations and the results compared to experimental data. Furthermore, liquid–liquid equilibrium predictions for binary polymer solutions using six activity coefficient models are compared in this work. The parameters were estimated for all the models to achieve the best possible representation of the reported experimental equilibrium behavior.     

Molecular thermodynamic model for equilibria in solution IV. Macroscopic partition functions

The molecular ensembles statistically distributed according to internal specific characteristics and distinguished for the different exchanges with the surroundings are represented on the macroscopic scale by appropriate partition functions. The partition function for osmotic non-reacting ensemble is a function of concentration or activity of the ligand and is suited to the definition of thermodynamic potential μ. The partition function for thermal non-reacting ensemble shows the dependence upon the temperature and that for thermo-osmotic non-reacting ensemble shows the dependence upon both concentration and temperature.

The reaction partition function is suited to show the distribution of the different species over the different enthalpy levels of the reacting ensemble. The dispersion of the distributions are represented by second derivatives of the partition function.

The information contained in the entropy axis of the thermodynamic space for reacting ensembles concerning the induced dilution of the bound ligand and final dilution of the free ligand can be spanned to a formation function diagram where free energy of reaction can be graphically represented.     

Molecular thermodynamic model for equilibria in solution: I. Reacting and non-reacting ensembles

The partition functions of solution thermodynamics are mathematical representations of the properties of molecular ensembles statistically distributed according to specific characteristics. The ensembles are classified as non-reacting or reacting. The non-reacting ensembles are characterized by one mean enthalpy level with dispersion around the mean. The reacting ensembles are characterized by two or more distinctly separated enthalpy levels over which the different species are variably distributed, depending on concentration and/or temperature.

The non-reacting ensembles can be distinguished into microcanonical, thermal, osmotic, thermo-osmotic ensembles, depending on the type of exchange with the surroundings which is connected to the fluctuations of the ensemble variables. The reacting ensembles can be distinguished into thermal, osmotic, thermo-osmotic, electrochemical, electro-osmotic, electro-thermal, electro-thermo-osmotic ensembles, depending on the type of reaction and of exchange with the surroundings.     

Sub-solidus phase equilibria in CeO2–SrO system

In this communication, we report on the synthesis and characterization of a series of compounds with the general composition Ce_1−xSr_xO_2−x (0.0≤x≤1.0), to establish a detailed phase relation in the CeO₂–SrO system. The X-ray diffraction (XRD) pattern of the each product was refined to determine the solid solubility and the homogeneity range. The solid solubility limit of SrO in CeO₂ lattice, under the slow cooled conditions, is represented as Ce_0.91Sr_0.09O_1.91 (i.e. 9 mol% of SrO). A careful delineation of the phase boundary revealed that the stoichiometric SrCeO₃, in fact, contains a little amount of CeO₂ also. The mono-phasic compound could be obtained at the nominal composition Sr_0.55Ce_0.45O_1.45. The nominal composition Sr₂CeO₄, under the heat treatment used in the present investigation, was a bi-phasic mixture of SrCeO₃ and SrO. No new ordered phases were obtained in this system.     

Phase equilibria in the Ag2Te-ZnTe and Ag2Te-Zn systems

The state diagrams (T-x) of the systems Ag₂Te-ZnTe(I) and Ag₂Te-Zn(II) are offered on the ground of data obtained by differential thermal analysis, X-ray phase analysis, microstructural analysis and measurements of the density and the microhardness of samples synthesized. The systems studied are quasibinary sections of the ternary system Ag-Zn-Te. System I is characterized by two eutectic and three eutectoidal non-variant equilibria as well as by an intermediate compound Ag₂ZnTe₂, which melts congruently at 880°C. The latter exists in the range from 120 to 880°C in two polymorphic modifications (T_ʅ→β=515°C). System II is characterized by one eutectic, two eutectoidal and one peritectic nonvariant equilibria, boundary solid solutions on the ground of Ag₂Te and Zn and one intermediate phase of the composition Ag₄Zn₃Te₂, which melts congruently at 880°C.     

Phase equilibria in the CdI2-Ag2Se system

The phase diagram of the system CdI₂-Ag₂Se is studied by means of X-ray diffraction, differential thermal analysis and measurements of the density of the material. The unit cell parameters of the intermediate phase 2CdI₂·3Ag₂Se were determined a = 0.6387 Å, b = 4.311 Å, c = 4.044 Å; α = 113.72°, β = 90.27° and γ = 94.85°. The intermediate phase 2CdI₂·3Ag₂Se has a polymorphic transition at 125 °C. It melts incongruently at 660 °C.     

pePC-SAFT: Modeling of polyelectrolyte systems 2. Aqueous two-phase systems

This work considers aqueous two-phase systems (ATPS) containing one polymer-polyelectrolyte as well as one salt. To model the liquid-liquid equilibria (LLE) of these systems, the recently presented model pePC-SAFT has been employed. ATPS containing poly(acrylic acid) of different degrees of neutralization or poly(vinyl pyrrolidone), respectively, were considered. The binary interaction parameters used between water-poly(acrylic acid) and water-poly(vinyl pyrrolidone) were adjusted to vapor-liquid equilibrium (VLE) data of these systems. ATPS consisting of poly(vinyl pyrrolidone)-water-sodium sulfate were predicted as function of temperature as well as of molar mass of the polymer. For poly(acrylic acid) systems, ATPS were predicted as function of charge density (degree of neutralization) for different types of salt. For these calculations, the polyelectrolyte model parameters were determined from the non-charged polymer whereas the effect of increasing charge density has been purely predicted by the model. Using this approach, it is possible to predict the shrinking of the liquid-liquid equilibrium region with increasing charging of the polyelectrolyte.     

Phase equilibria and crystal structure of the complex oxides in the Ln-Ba-Co-O (Ln=Nd, Sm) systems

The phase equilibria in the Ln-Ba-Co-O (Ln=Nd, Sm) systems were systematically studied at 1100 °C in air. The homogeneity ranges and crystal structure of the solid solutions: Ln_2−xBa_xO_3−δ (0<x≤0.1 for Ln=Nd and 0<x≤0.3 for Ln=Sm), Nd_3−yBa_yCo₂O₇ (0.70≤y≤0.80), BaCo_1−zSm_zO_3−δ (0.1≤z≤0.2) were determined by X-ray diffraction of quenched samples. The values of oxygen content (5+δ) for slowly cooled LnBaCo₂O_5+δ (Ln=Nd, Sm) samples were estimated as 5.73 for Ln=Nd, and 5.60 for Ln=Sm. The unit cell parameters were refined using Rietveld full-profile analysis. It was shown that NdBaCo₂O_5.73 possesses tetragonal structure and SmBaCo₂O_5.60 - orthorhombic structure. The projections of isothermal-isobaric phase diagrams for the Ln-Ba-Co-O (Ln=Nd, Sm) systems to the compositional triangle of metallic components were presented.     

Modelling of phase equilibria for associating mixtures using an equation of state

In the present work, the group contribution with association equation of state (GCA-EoS) is extended to represent phase equilibria in mixtures containing acids, esters, and ketones, with water, alcohols, and any number of inert components. Association effects are represented by a group-contribution approach. Self- and cross-association between the associating groups present in these mixtures are considered. The GCA-EoS model is compared to the group-contribution method MHV2, which does not take into account explicitly association effects. The results obtained with the GCA-EoS model are, in general, more accurate when compared to the ones achieved by the MHV2 equation with less number of parameters. Model predictions are presented for binary self- and cross-associating mixtures.     

Ternary systems Sr-{Ni,Cu}-Si: Phase equilibria and crystal structure of ternary phases

Phase relations were established in the Sr-poor part of the ternary systems Sr-Ni-Si (900 °C) and Sr-Cu-Si (800 °C) by light optical microscopy, electron probe microanalysis and X-ray diffraction on as cast and annealed alloys. Two new ternary compounds SrNiSi₃ (BaNiSn₃-type) and SrNi_9−xSi_4+x (own-type) were found in the Sr-Ni-Si system along with previously reported Sr(Ni_xSi_1−x)₂ (AlB₂-type). The crystal structure of SrNi_9−xSi_4+x (own-type, x=2.7, a=0.78998(3), c=1.1337(2) nm; space group P4/nbm) was determined from X-ray single crystal counter to be a low symmetry derivative of the cubic, parent NaZn₁₃-type. At higher Si-content X-ray Rietveld refinements reveal the formation of a vacant site (□) corresponding to a formula SrNi_5.5Si_6.5□_1.0. Phase equilibria in the Sr-Cu-Si system are characterized by the compounds SrCu_2−xSi_2+x (ThCr₂Si₂-type), Sr(Cu_xSi_1−x)₂ (AlB₂-type), SrCu_9−xSi_4+x (0≤x≤1.0; CeNi_8.5Si_4.5-type) and SrCu_13−xSi_x (4≤x≤1.8; NaZn₁₃-type). The latter two structure types appear within a continuous solid solution. Neither a type-I nor a type-IX clathrate compound was encountered in the Sr-{Cu,Ni}-Si systems.Structural details are furthermore given for about 14 new ternary compounds from related alloy systems with Ba.     

Phase equilibria and excess properties for binary systems in reactive distillation processes Part I. Methyl acetate synthesis

Isothermal vapor–liquid equilibrium (VLE) and excess enthalpy (H^E) data were measured for binary systems required for the design of reactive distillation processes for the methyl acetate production. The isothermal P–x data were measured with the help of a computer-operated static apparatus. A commercial isothermal flow calorimeter was used for the determination of the heats of mixing. Temperature-dependent interaction parameters for the UNIQUAC model were fitted simultaneously to the experimental data from this work and other authors.     

Making equation of state models predictive: Part 2: An improved PCP-SAFT equation of state

We use the PCP-SAFT equation of state, which is of the Van der Waals type and has a sound physical basis, to predict mixture properties, such as vapor–liquid and liquid–liquid equilibria, as well as excess enthalpies. We use molecular properties, such as dipole moment, quadrupole moment, polarizability and dispersion interaction coefficients, that have been determined quantum mechanically in Part I of this publication and adjust the remaining three pure compound parameters to pure compound data. We finally present a new combination rule for the dispersion energy parameter ? that is based on the quantum mechanically determined data. The predictions based on quantum mechanically determined pure compound properties along with the new combination rule show an improved performance compared to the original PCP-SAFT combination rule.     

Acid-base equilibria of some pyridinol derivatives in binary water/organic solvent systems

Summary pK _a values of 2-hydroxy 3-pyridinol (HHP), 2-mercapto 3-pyridinol (MHP), and 2-carboxy-3-pyridinol (CHP) were determined by potentiometric titration in water/organic solvent mixtures containing 20 mole% of organic solvent at 25±0.1 °C and 0.1M ionic strength (KNO₃) applying an empiricalpH correction for mixed aqueous solvents. The influence of the organic solvents on the dissociation constants and tautomeric equilibria of the pyridinol derivatives is discussed. The effect of the molecular structure of the compounds onpK _a is also explained. Titrations of a mixture of two weak diprotic acids (HHP andCHP) in a water/dimethylsulphoxide medium containing 20 mole% organic solvent at constant ionic strength were evaluated using theGran method.

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Säure-Basen-Gleichgewichte einiger Pyridinolderivate in binären Systemen aus Wasser und organischen Lösungsmitteln

Zusammenfassung DiepK _a-Werte von 2-Hydroxy-3-pyridinol (HHP), 2-Mercapto-3-pyriodinol (MHP) und 2-Carboxy-3-pyridinol (CHP) wurden durch potentiometrische Titration in wäßrigen Systemen mit 20 mol% organischem Lösungsmittelanteil bei 25±0.1 °C und einer lonenstärke von 0.1M KNO₃ unter Anwendung einer empirischenpH-Korrektur für Lösungsmittelgemische bestimmt. Der Einfluß der organischen Lösungsmittel auf die Dissoziationskonstanten und die tautomeren Gleichgewichte der untersuchten Verbindungen und der Einfluß der molekularen Strukturen auf diepK _a-Werte werden diskutiert. Die Titration eines Gemisches von zwei schwachen zweibasigen Säuren (HHP undCHP) in Wasser/Dimethylsulfoxid bei konstanter Ionenstärke wurde mit Hilfe derGranschen Methode ausgewertet.

     

Automated calculation of complete Pxy and Txy diagrams for binary systems

An algorithm for the calculation of global phase equilibrium diagrams has been recently developed [M. Cismondi, M.L. Michelsen, Global phase equilibrium calculations: critical lines, critical end points and liquid–liquid–vapour equilibrium in binary mixtures, J. Supercrit. Fluids 39 (2007) 287–295]. It integrates the calculation of critical lines, liquid–liquid–vapour (LLV) lines and critical end points, and was implemented in the software program GPEC: global phase equilibrium calculations [M. Cismondi, D.N. Nuñez, M.S. Zabaloy, E.A. Brignole, M.L. Michelsen, J.M. Mollerup, GPEC: a program for global phase equilibrium calculations in binary systems, in: Proceedings of the CD-ROM EQUIFASE 2006, Morelia, Michoacán, Mexico, October 21–25, 2006; www.gpec.plapiqui.edu.ar]. In this work we present the methods and computational strategy for the automated calculation of complete Pxy and Txy diagrams in binary systems. Being constructed from the points given by the global phase diagram at a specified temperature or pressure, their calculation does not require the implementation of stability analysis.     

An activity coefficient model for predicting salt effects on vapor–liquid equilibria of mixed solvent systems

In the present study, an activity coefficient model, based on the concept of local volume fractions and the Gibbs–Helmholtz relation, has been developed. Some modifications were made from Tan–Wilson model (1987) and TK–Wilson model (1975) to represent activity coefficients in mixed solvent–electrolyte systems. The proposed model contains two groups of binary interaction parameters. One group for solvent–solvent interaction parameters corresponds to that given by the TK–Wilson model (1975) in salt-free systems. The other group of salt–solvent interaction parameters can be calculated either from vapor pressure or bubble temperature data in binary salt–solvent systems. It is shown that the present model can also be used to describe liquid–liquid equilibria. No ternary parameter is required to predict the salt effects on the vapor–liquid equilibria (VLE) of mixed solvent systems. By examining 643 sets of VLE data, the calculated results show that the prediction by the present model is as good as that by the Tan–Wilson model (1987), with an overall mean deviation of vapor phase composition of 1.76% and that of the bubble temperature of 0.74 K.     

Retention in field-flow fractionation with secondary chemical equilibria

Summary The application range of field-flow fractionation (FFF) can be extended to low molecular weight solutes, as demonstrated a few years ago by Berthod et al., by taking profit of secondary chemical equilibria (SCE) occurring between the bulk carrier and a retained carrier component. The theory of solute retention in this SCE-FFF method is developed for any value of the solute distribution coefficient and of the retention ratio of the retained carrier component, provided that the Brownian mode of retention applies for this component and that the flow velocity profile is parabolic. This removes some of the limitations of the model previously developed by Berthod and Armstrong and sheds light on the potentialities of the SCE-FFF method for physico-chemical studies about secondary chemical equilibria in colloidal systems. Remaining assumptions in the model are discussed.Dedicated to Professor Leslie S. Ettre on the occasion of his 70th birthday.