Evaluation of repulsive particle swarm method for phase equilibrium and phase stability problems

Phase equilibrium and stability problems are of crucial importance in simulation, design and optimization of several separation processes. Recently, these problems have been solved using minimization of Gibbs free energy, using global optimization techniques. In this paper, repulsive particle swarm (RPS), a recent global optimization technique is explored for the solution of phase stability and phase equilibrium.     

Calculation of homogeneous azeotropes in reactive and non-reactive mixtures using a stochastic optimization approach

In this paper we introduce an alternative strategy to find homogeneous azeotropes in reactive and non-reactive mixtures which is based upon the Simulated Annealing optimization technique. This stochastic optimization method is used to robustly solve a system of non-linear equations that results from the equalities of the orthogonal derivatives of the Gibbs energy and the Gibbs energy of mixing in the vapor and the liquid phases. For non-reactive systems, this equation system is solved by considering conventional composition variables while for reactive cases we use the transformed composition variables proposed by Ung and Doherty. Numerical performance of our approach is illustrated using several examples previously reported in the literature and results show that it is a suitable and robust strategy for the calculation of homogeneous azeotropes in mixtures with or without chemical reactions.     

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.     

Isobaric vapor–liquid equilibria for the quaternary reactive system: Ethanol + water + ethyl lactate + lactic acid at 101.33 kPa

Isobaric vapor–liquid equilibrium (VLE) data of the reactive quaternary system ethanol (1) + water (2) + ethyl lactate (3) + lactic acid (4) have been determined experimentally. Additionally, the reaction equilibrium constant was calculated for each VLE experimental data. The experimental VLE data were correlated using the UNIQUAC equation to describe the chemical and phase equilibria simultaneously. For some of the non-reactive binary systems, UNIQUAC binary interaction parameters were obtained from the literature. The rest of the binary UNIQUAC parameters were obtained by correlating the experimental quaternary VLE data obtained in this work. A maximum pressure azeotrope at high water concentration for the binary reactive system ethyl lactate + water has been calculated.     

Application of stochastic algorithms for parameter estimation in the liquid–liquid phase equilibrium modeling

In this work the Simulated Annealing (SA) and Particle Swarm Optimization (PSO) algorithms were employed to modeling liquid–liquid phase equilibrium data. For this purpose, some strategies for stochastic algorithms were investigated from common test functions and used in LLE parameter estimation procedure. The strategy used for the flash calculation was based on the isoactivity criteria associated with phase stability test and interpolation function for the initial estimate to improve reliability of phase equilibria calculations. It is shown that both algorithms SA and PSO were capable of estimating the parameters in models describing liquid–liquid phase behavior of binary and multicomponent systems with a good representation of the experimental data.     

Interfacial properties of water + alcohol mixtures

Mixtures of water with alcohol are important in numerous engineering applications. Caused by the polarity of water and alcohol self-association of water and alcohol as well cross-association between water and alcohol appear in such complex mixtures. These features show significant impact on physical and chemical properties, especially phase equilibrium behaviour and hence interfacial properties. The Cahn–Hilliard theory was combined with original statistical associated fluid theory equation of states (SAFT EOS) in order to describe both the phase behaviour and interfacial properties with respect of association. The paper focuses on theoretical investigations of surface tension, density profiles, surface thickness in vapour–liquid or vapour–liquid–liquid equilibrium of mixtures of water with ethanol or 1-butanol. Results of vapour–liquid equilibrium surface tension calculations were compared with experimental data taken from the literature.     

Novel bare-bones particle swarm optimization and its performance for modeling vapor-liquid equilibrium data

Parameter estimation in models plays a significant role in developing mathematical models which are used for understanding and analyzing physical, chemical and biological systems. Parameter estimation problems for vapor-liquid equilibrium (VLE) data modeling are very challenging due to the high non-linearity of thermodynamic models. Recently, stochastic global optimizations and their applications to these problems have attracted greater interest. Of the many stochastic global optimization methods, Bare-bones particle swarm optimization (BBPSO) is attractive since it has no parameters to be tuned by the user. In this study, modifications are introduced to the original BBPSO using mutation and crossover operators of differential evolution algorithm to update certain particles in the population. The performance of the resulting algorithm is tested on 10 benchmark functions and applied to 16 VLE problems. The performance of the proposed BBPSO for VLE modeling is compared with that of other stochastic algorithms, namely, differential evolution (DE), DE with tabu list, genetic algorithm and simulated annealing in order to identify their relative strengths for VLE data modeling. The proposed BBPSO is shown to be better than or comparable to the other stochastic global optimization algorithms tested for parameter estimation in modeling VLE data.     

Chemical equilibrium in a heterogeneous fluid phase system: thermodynamic regularities and topology of phase diagrams

Singularities of the systems with simultaneous phase and chemical equilibria were examined. The thermodynamic regularities were revealed and topology of phase diagrams was presented for multicomponent heterogeneous systems including two or several equilibrium liquid phases (solution layering). The thermodynamic conditions for displacement of phase and chemical equilibrium are considered.     

Isobaric (vapour + liquid) equilibria for sulfolane with toluene, ethylbenzene, and isopropylbenzene at 101.33 kPa

Isobaric (vapour + liquid) equilibrium data have been measured for the (toluene + sulfolane), (ethylbenzene + sulfolane), and (isopropylbenzene + sulfolane) binary systems with a modified Rose-Williams still at 101.33 kPa. The experimental data of binary systems were well correlated by the non-random two-liquid (NRTL) and universal quasi-chemical (UNIQUAC) activity coefficient models for the liquid phase. All the experimental results passed the thermodynamic consistency test by the Herington method. Furthermore, the model UNIFAC (Do) group contribution method was used. Sulfolane is treated as a group (TMS), the new group interaction parameters for CH₂–TMS, ACH–TMS and ACCH₂–TMS were regressed from the VLE data of (toluene + sulfolane) and (ethylbenzene + sulfolane) binary systems. Then these group interaction parameters were used to estimate phase equilibrium data of the (isopropylbenzene + sulfolane) binary system. The results showed that the estimated data were in good agreement with the experimental values. The maximum and average absolute deviations of the temperature were 4.50 K and 2.39 K, respectively. The maximum and average absolute deviations for the vapour phase compositions of isopropylbenzene were 0.0237 and 0.0137, respectively.     

The phase envelopes of alternative solvents (ionic liquid,CO2) and building blocks of biomass origin (lactic acid,propionic acid)

Solid–liquid, liquid–liquid and vapour–liquid equilibrium measurements for binary and ternary systems containing building blocks of biomass origin such as propionic acid, lactic acid and alternative solvents like carbon dioxide and 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ionic liquid have been carried out at 313.15 K. The binary solid–liquid and liquid–liquid equilibrium measurements were performed at ambient pressure. The vapour–liquid equilibrium was studied in the range of pressure from 3.54 to 12 MPa while ternary systems were examined at 9, 10 and 12 MPa. The samples from the coexisting phases were taken and the compositions of both liquid and vapour phases were determined experimentally. The three-phase system was observed for lactic acid + ionic liquid + CO₂ as well. The achieved results were correlated using the Peng–Robinson equation of state with the Mathias–Klotz–Prausnitz mixing rule. The set of interaction parameters for the employed equations of state and the mixing rule for the investigated systems were obtained.     

Efficient methods to study phase equilibria in multinary aqueous systems

We report the results of developing efficient universal isothermal methods for the determination of compositions of all phases (several solid phases and a liquid) involved in invariant equilibria in multicomponent aqueous systems using chemical analysis (a combination method) or not (an optimized sections method) to determine the composition of an equilibrium liquid phase.     

Solution of the isochoric–isoenergetic flash problem by direct entropy maximization

A new algorithm to find the phase equilibrium conditions in systems with specified values of internal energy, volume, and number of moles of each component present (isochoric–isoenergetic flash problem) is proposed. The core of the procedure consists of maximizing the system entropy by iterating on the values, in each phase, of internal energy, volume, and number of moles of each component. Analytical expressions for the physical properties and derivatives required by the calculations were generated by computer algebra. The algorithm tests for the possible need to add or remove phases during the course of iterations. The paper discusses possible numerical difficulties during application of the procedure and how to overcome them. The algorithm has shown to be robust and capable of solving multiphase equilibrium problems, avoiding trivial solutions.     

Liquid–liquid equilibrium calculations for methanol–gasoline blends using continuous thermodynamics

This work presents the application of continuous thermodynamics to investigate the limited miscibility of methanol–gasoline blends. To predict the liquid–liquid equilibrium of these systems, the Gaussian distribution function was used to represent the composition of paraffins in the gasoline. The naphthenes and aromatics were represented by model compounds. A model has been developed using three different continuous versions of the UNIFAC model. Methanol is an associating component, and association affects phase equilibria. Therefore, the CONTAS (continuous thermodynamics of associating systems) model based on the Flory–Huggins equation, for multicomponent methanol–gasoline blends has also been investigated. The predicted results including the cloud point curve, shadow curve and phase separation data have been compared with experimental data and good agreement was found for the two UNIFAC and CONTAS models.     

Liquid–liquid phase behaviour,liquid–liquid density,and interfacial tension of multicomponent systems at 298 K

Experimental liquid–liquid phase diagrams are presented for the multicomponent systems isooctane–benzene–(80 mass% methanol + 20 mass% water)–5 mass% isobutyl alcohol (2-methyl-1-propanol) and isooctane–benzene–(80 mass% methanol + 20 mass% water)–15 mass% isobutyl alcohol, at 298.15 K. The density and interfacial tension of conjugate phases of concentration located in the isothermal binodal have been determined at 298.15 K for the partially miscible systems: isooctane–benzene–methanol, isooctane–benzene–(80 mass% methanol + 20 mass% water), isooctane–benzene–(80 mass% methanol + 20 mass% water)–5 mass% isobutyl alcohol, and isooctane–benzene–(80 mass% methanol + 20 mass% water)–15 mass% isobutyl alcohol. The experimental tie-line data define the binodal or coexistence curve of the two studied multicomponent systems and depending on the initial isobutyl alcohol concentration the liquid–liquid phase diagram is either of type II, with low alcohol concentration, or type I, with high concentration of alcohol, which is a clear indication that the solubility of the partially miscible systems is greatly enhanced via the co-solvency phenomenon. It is observed that both the density of each conjugate phase and the interfacial tension of each tie-line are valuable indicators of the degree of solubility of the multicomponent systems. Furthermore the experimental tie-lines data were correlated with the NRTL and UNIQUAC solution models with satisfactory quantitative results.     

On the phase equilibrium Stockmayer fluids

The fluid phase equilibrium of the Stockmayer fluid is investigated using a thermodynamic perturbation theory approach. The reference and the perturbation potential are the Lennard–Jones potential and the dipolar–dipolar interactions, respectively. They are assumed to be represented by the modified Benedict–Webb–Rubin equation of state [J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78 (1993) 591–618] and the Padé approximant [G. Stell, J.C. Rasaiah, H. Narang, Mol. Phys. 27 (1974) 1393–1414], respectively. The asymmetry found in an analogous study [M.E. van Leeuwen, B. Smit, E.M. Hendriks, Mol. Phys. 78 (1993) 271–283] based on the BWR equation of state [J.J. Nicolas, K.E. Gubbins, W.B. Streett, D.J. Tildesley, Mol. Phys. 37 (1979) 1429–1454] is now not observed on the vapour–liquid equilibrium coexistence curves of Stockmayer fluids with dipolar strength of μ^*2 = 1, 2, 3, and 4. Results agree with computer simulations for dipolar strength of μ^*2 = 1; however as strength dipole increases, liquid densities are over-estimated.     

Evaluation of stochastic global optimization methods for modeling vapor–liquid equilibrium data

Parameter estimation for vapor–liquid equilibrium (VLE) data modeling plays an important role in design, optimization and control of separation units. This optimization problem is very challenging due to the high non-linearity of thermodynamic models. Recently, several stochastic optimization methods such as Differential Evolution with Tabu List (DETL) and Particle Swarm Optimization (PSO) have evolved as alternative and reliable strategies for solving global optimization problems including parameter estimation in thermodynamic models. However, these methods have not been applied and compared with respect to other stochastic strategies such as Simulated Annealing (SA), Differential Evolution (DE) and Genetic Algorithm (GA) in the context of parameter estimation for VLE data modeling. Therefore, in this study several stochastic optimization methods are applied to solve parameter estimation problems for VLE modeling using both the classical least squares and maximum likelihood approaches. Specifically, we have tested and compared the reliability and efficiency of SA, GA, DE, DETL and PSO for modeling several binary VLE data using local composition models. These methods were also tested on benchmark problems for global optimization. Our results show that the effectiveness of these stochastic methods varies significantly between the different tested problems and also depends on the stopping criterion especially for SA, GA and PSO. Overall, DE and DETL have better performance for solving the parameter estimation problems in VLE data modeling.     

An effective calculation procedure for two-phase equilibria in multireaction systems

In this work, we present a method for the calculation of two-phase equilibria in multireaction systems. The procedure uses a transformed composition variable in the orthogonal derivatives of the Gibbs function and the tangent plane equation to form a system of non-linear equations. We solve this system with a Newton–Raphson method and our initialization procedure uses results from the reactive stability analysis and the reactive equal area rule. With this initialization strategy, convergence occurs with only a few iterations in the numerical method. Several examples with multiple chemical reactions demonstrate the performance of our approach.     

Phase equilibria of mixed carbon dioxide and tetrahydrofuran hydrates in sodium chloride aqueous solutions

In this work, the competing effects of sodium chloride (NaCl) and tetrahydrofuran (THF) on carbon dioxide hydrate formation are investigated through phase equilibrium measurements. The phase behaviour in the hydrate forming region for the binary system carbon dioxide–water, the ternary systems carbon dioxide–tetrahydrofuran–water and ternary carbon dioxide–sodium chloride–water and, in addition, the quaternary system carbon dioxide–tetrahydrofuran–water–sodium chloride are determined experimentally, using a Cailletet apparatus. All measurements are made in a temperature and pressure region of 275–290 K and 0.5–7.0 MPa, respectively. In these ranges, three different hydrate equilibrium curves are measured namely: H-L_W-V, H-L_W-L_V-V and H-L_W-L_V. The formation of an organic-rich liquid phase in the systems due to a liquid–liquid two-phase split between water and tetrahydrofuran when pressurized with carbon dioxide causes the occurrence of an upper quadruple point (Q₂) to evolve into a four-phase H-L_W-L_V-V equilibrium line. The presence of sodium chloride in the quaternary system enhances the split between the two liquids due to the salting-out effect. It was found that the hydrate promoting effect of tetrahydrofuran is able to suppress the inhibiting effect of sodium chloride especially at lower concentration of sodium chloride.     

Application of the CPA equation of state to reservoir fluids in presence of water and polar chemicals

The complex phase equilibrium between reservoir fluids and associating compounds like water, methanol and glycols has become more and more important as the increasing global energy demand pushes the oil industry to target reservoirs with extreme or complicated conditions, such as deep or offshore reservoirs. Conventional equation of state (EoS) with classical mixing rules cannot satisfactorily predict or even correlate the phase equilibrium of those systems. A promising model for such systems is the Cubic-Plus-Association (CPA) EoS, which has been successfully applied to well-defined systems containing associating compounds. In this work, a set of correlations was proposed to calculate the CPA model parameters for the narrow cuts in ill-defined C₇₊ fractions. The correlations were then combined with either the characterization method of Pedersen et al. or that of Whitson et al. to extend CPA to reservoir fluids in presence of water and polar chemical such as methanol and monoethylene glycol. With a minimum number of adjustable parameters from binary pairs, satisfactory results have been obtained for different types of phase equilibria in reservoir fluid systems and several relevant model multicomponent systems. In addition, modeling of mutual solubility between light hydrocarbons and water is also addressed.     

Modeling of three-phase vapor–liquid–liquid equilibria for a natural-gas system rich in nitrogen with the SRK and PC-SAFT EoS

In this work, we present the modeling of three-phase vapor–liquid–liquid equilibria for a mixture of natural gas (Hogback gas) containing high concentrations in nitrogen (51.8 mol%) with the SRK and PC-SAFT equations of state. The interest of studying this mixture is due to the experimental evidence of the occurrence of multiple equilibrium liquid phases for this mixture over certain ranges of temperature and pressure. The calculation of the multiphase equilibria was carried out by using an efficient numerical procedure based on the minimization of the system Gibbs energy and thermodynamic stability tests to find the most stable state of the system. The results of the calculated vapor–liquid–liquid equilibria (VLLE) show that the PC-SAFT equation of state predicts satisfactorily the phase behavior that experimentally exhibits this mixture, whereas the SRK equation of state predicts a three-phase region wider than the experimentally observed. The two-phase boundary for this mixture was also calculated through flash calculations, and the results showed that this mixture does not present any gas-liquid critical point.