Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method

This paper presents a new method for multiphase equilibria calculation by direct minimization of the Gibbs free energy of multicomponent systems. The methods for multiphase equilibria calculation based on the equality of chemical potentials cannot guarantee the convergence to the correct solution since the problem is non-convex (with several local minima), and they can find only one for a given initial guess. The global optimization methods currently available are generally very expensive. A global optimization method called Tunneling, able to escape from local minima and saddle points is used here, and has shown to be able to find efficiently the global solution for all the hypothetical and real problems tested. The Tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the objective function. In phase two (tunnelization), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The calculation of multiphase equilibria is organized in a stepwise manner, combining phase stability analysis by minimization of the tangent plane distance function with phase splitting calculations. The problems addressed here are the vapor–liquid and liquid–liquid two-phase equilibria, three-phase vapor–liquid–liquid equilibria, and three-phase vapor–liquid–solid equilibria, for a variety of representative systems. The examples show the robustness of the proposed method even in the most difficult situations. The Tunneling method is found to be more efficient than other global optimization methods. The results showed the efficiency and reliability of the novel method for solving the multiphase equilibria and the global stability problems. Although we have used here a cubic equation of state model for Gibbs free energy, any other approach can be used, as the method is model independent.     

A RELIABLE METHOD FOR LIQUID-LIQUID PHASE EQUILIBRIUM CALCULATION AND GLOBAL STABILITY ANALYSIS

A new algorithm to determine liquid-liquid phase equilibrium at the global minimum of the Gibbs free energy is presented, the optimization results are then used by Newton-Raphson method to obtain the final accurale phase equilibrium compositions. The tangent plane distance function criterion is used as the sufficient and necessary condition for the global stability analysis, its global minimum is obtained by the simulated annealing algorithm in order to overcome the multiple local minimum problem. The Gibbs free energy functions for liquid-liquid equilibrium systems are formulated by NRTL or UNIQUAC activity coefficient equation. Aiming to overcome the nonconvex problem of the Gibbs free energy function the simulated annealing algorithm is used for achieving a near true equilibrium solution, this solution then is used as the initial values of the Newton-Raphson method. Our algorithm was implemented and tested for six examples with three components and up to three liquid phases. In all of the examples, results for liquid-liquid equilibrium calculations and global stability analysis were found to be reliable. The main advantage of our algorithm is its simplicity and that it is model independent.     

应用遗传算法求解多相化学反应体系的相稳定性及平衡组成

基于吉布斯自由能最小法原理,研究用遗传算法求解复杂化学体系的平衡问题。将相数和相态未知体系的平衡计算问题分解为两步最优化问题:假定相态数目下的吉布斯自由能最小化问题和相稳定性判据问题。通过序贯加入新相和用遗传算法交替求解两个最优化子问题,可得到最终体系的相数、相态和平衡组成。对合成甲醇和酯化反应两个多相复杂体系进行了的计算,结果验证了本研究方法的有效性。     

Note on flash and distillation systems

Flash and distillation systems evolving on the equilibrium manifold possess stable steady states when the molar flows and the boundary conditions are fixed and a vapor and liquid phase is present at all times. The results extend the range of validity of a local stability theorem for linear, constant molar overflow systems due to Acrivos and Amundson. The approach to stability analysis is motivated by the tangent plane method developed by Gibbs to open systems as it takes advantage of the negative curvature of the entropy surface to construct a Lyapunov function. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3322–3332, 2013     

The Least Dot Products method: A new numerical paradigm for phase stability analysis of thermodynamic mixtures

This work introduces the Least Dot Products (LDP) method, a new algorithm for phase stability analysis of thermodynamic mixtures. Starting with the fact that tangent plane distance function (the objective function used in global stability tests) is, essentially, the dot product between the vector of compositions of a trial phase and the vector that describes the differences among the chemical potential of the components in the phases, our approach tries to obtain the least values of an auxiliary function that represents an appropriate dot product on a unitary sphere of the n-dimensional space, which is a good approximation for the stability test function. In agreement with the foundations of the Gibbs plane tangent criterion, the new algorithm simply tries to find points where the objective function is negative, which are not (necessarily) stationary points or global minima. Thus, our main contribution is not a new method for general nonlinear problems, or a rule-based termination criteria for a classical optimization method. Consequently, if such points do not exist, then LDP method is not capable to recognize the stability condition. To overcome this problem, we develop also a powerful safeguard algorithm, denominated Projected Simulated Annealing (PJSA) algorithm, which is obtained by projecting the Simulated Annealing algorithm onto the unitary sphere.In the present article, firstly we consider liquid–liquid equilibria at low or moderate pressures, where the excess Gibbs energy is described by NRTL or UNIQUAC models. Secondly, we address vapor–liquid equilibria at high pressures with cubic equations of state.To illustrate the performance of the new methodology, we use here 20 systems studied by other authors. Such problems possess 2–12 variables and constitute severe tests for many optimization methods. For some mixtures, we show that LDP method is capable of determining the instability condition using just two iterations.     

Identification of a packed absorption column

A method is presented for the identification of parameters in the partial differential equations describing the dynamic behaviour of a certain class of two phase flow processes. This method is an extension of a procedure for the identification of single phase flow processes which were introduced in a previous paper. The method is applied to the identification of two parameters in the five simultaneous partial differential equations describing the behaviour of a packed absorption column being used to remove carbon dioxide from air with an aqueous solution of monoethanolamine. These parameters were determined from a single experimental step response. The approach employed was to reduce the partial differential equations to two coupled systems of ordinary differential equations by the method of characteristics. Once in this form the identification problem could be formulated as two coupled boundary value problems and solved with the use of quasilinearization.     

基于LCPSO算法的相稳定性判别

相稳定性判别为相平衡计算的基本课题,常采用Gibbs自由能曲面与切平面的距离函数（TPDF）最小化方法求解。对于强非理想体系,或在高压条件下,其TPDF表现出复杂形态,有平凡解和多极值,传统方法难以求得满足约束的全局最小值,从而导致判别失误。粒子群算法（PSO）虽有全局优化性能,但也会陷于局部极小,且缺少约束处理机制。为此,分析了PSO内在蕴含的线性特点,在种群初化、粒子运动等环节提出应对策略,构建线性约束粒子群算法（LCPSO）,确保种群在可行空间内搜索。还增设调变参数、局部加速等措施,以兼顾算法的全面探测和细化挖掘的能力,提高其全局优化效能。经多个实例的测试表明,LCPSO适用面广,既可用于超额自由能、状态方程等各类热力学模型,又能克服混合模型一阶不连续的困难,应用范围从液液相分裂拓展到汽液液相分裂。与确定性全局算法IN/GB相比,LCPSO速率高,效果好,尤对多元体系更具优势。     

FOAM BEHAVIOR IN PARTIALLY MISCIBLE BINARY SYSTEMS

A number of physical properties of the binary system ethylene glycol and ethyl salicylate have been measured recently. Thus measured values of surface tensions and vapor pressures as functions of composition and temperature are available, and from these may be calculated the Gibbs surface excesses of ethyl salicylate in solutions represented in the phase diagram by the one-phase area. The results of these calculations can be expressed as a series of cosorption contours superimposed on the phase diagram, showing maximum surface activity at a point (epicenter) on the co-existence curve. The epicenter would coincide with the consolute point only when the surface tensions of the two pure liquid components are equal: usually they are not equal, so that the epicenter of the contours is displaced on the co-existence curve toward the component of higher surface tension. This work reports the foam stabilities of solutions for which Gibbs surface excess concentrations are known. The foam stability contours (isaphroics) in general follow the cosorption contours, which in turn are related to the Gibbs elasticity. These results therefore demonstrate that the stability of evanescent foams is directly related to the Gibbs elasticity of the liquid surface. The more stable foams of solutions of compositions near that of the epicenter are inhibited by a slight alteration of total composition that will induce phase separation of a dispersed conjugate solution.     

FOAM BEHAVIOR IN PARTIALLY MISCIBLE BINARY SYSTEMS

A number of physical properties of the binary system ethylene glycol and ethyl salicylate have been measured recently. Thus measured values of surface tensions and vapor pressures as functions of composition and temperature are available, and from these may be calculated the Gibbs surface excesses of ethyl salicylate in solutions represented in the phase diagram by the one-phase area. The results of these calculations can be expressed as a series of cosorption contours superimposed on the phase diagram, showing maximum surface activity at a point (epicenter) on the co-existence curve. The epicenter would coincide with the consolute point only when the surface tensions of the two pure liquid components are equal: usually they are not equal, so that the epicenter of the contours is displaced on the co-existence curve toward the component of higher surface tension. This work reports the foam stabilities of solutions for which Gibbs surface excess concentrations are known. The foam stability contours (isaphroics) in general follow the cosorption contours, which in turn are related to the Gibbs elasticity. These results therefore demonstrate that the stability of evanescent foams is directly related to the Gibbs elasticity of the liquid surface. The more stable foams of solutions of compositions near that of the epicenter are inhibited by a slight alteration of total composition that will induce phase separation of a dispersed conjugate solution.     

A simple and unified algorithm to solve fluid phase equilibria using either the gamma–phi or the phi–phi approach for binary and ternary mixtures

A new algorithm is proposed for calculating phase equilibria in binary systems at a fixed temperature and pressure. This algorithm is then extended to ternary systems (in which case, the mole fraction of one constituent in a given phase must be fixed in order to satisfy the Gibbs’ phase rule). The algorithm has the advantage of being very simple to implement and insensitive to the procedure used to initialize the unknowns. Most significantly, the algorithm allows the same solution procedure to be used regardless of the thermodynamic approach considered (γ–φ or φ–φ), the type of phase equilibrium (VLE, LLE, etc.) and the existence of singularities (azeotropy, criticality and so on).     

STRATIFIED THREE PHASE FLOW IN PIPES - STABILITY AND TRANSITION

The stability of stratified three phase flow (water-oil-gas) was analyzed using two approaches: A straight forward Kelvin-Helmholtz stability analysis on the two interfaces termed here the “exact” approach and a simplified approach. 1 n the “exact” approach the two interfaces (water-oil and oil-gas) are perturbed, while in the simplified approach the perturbed interface is only the upper oil-gas interface. Both approaches include the viscous Kelvin-Helmholtz analysis in which the shear stresses are laken into account and the inviscid K H analysis where the shear stresses are neglected.

Comparison with some experimental results suggests that the simplified method is a better predictor of the transition from stratified flow than the “exact” approach, suggesting, perhaps, that the stability analysis on the upper interface alone is preferred.     

一种计算分子筛标准Gibbs生成自由能的新方法

引 言 自Mobil公司于20世纪60年代成功合成高硅分子筛β[1]及ZSM-5[2]以来,分子筛的合成及其应用便成为催化领域研究的热点.然而,对于分子筛的合成及其改性研究基本上都是凭着研究者的经验,而缺乏理论的有效指导.研究表明,分子筛的热力学性质(如生成焓、生成自由能等)对于分子筛的合成及其性能(如热稳定性等)的预测具有重要的指导意义.但由于分子筛的结构非常复杂,测定分子筛的热力学性质十分困难,到目前为止,只有少数分子筛的某些热力学参数已用实验测定[3].为此,人们一直在探索用各种方法直接计算或预测分子筛的热力学参数.     

Global solution approaches in equilibrium and stability analysis using homotopy continuation in the complex domain

Recently, significant attention has been shown to physical and chemical equilibrium and stability analysis in the real and complex domains. In this work, a new procedure involving the continuation method in the complex domain using bifurcation theory is propounded. Based on this method, homotopy branches in real and complex space are connected to each other through bifurcation branches. Thus, by just one initial guess, multiple solution branches are found. When calculations are only made in the real domain, multiple solutions are not always found from an arbitrary initial guess. Examples are presented to show the application of the method to nonlinear sets of equations in phase equilibrium, chemical and phase equilibrium, and stability analysis. These types of problems are believed to contain significant nonlinearities in process simulations. The results can be applied to flowsheet calculations.     

乙酸-水-乙酸正丁酯三相体系的热力学分析与共沸精馏过程模拟

乙酸-水-乙酸正丁酯体系的分离是汽-液-液三相共沸精馏过程,相分裂的判断和精馏塔算法的改进是进行该精馏过程模拟的两个主要问题.本文根据现有的基础数据给出了该体系的逸度和活度因子计算模型,然后讨论了Gibbs自由能的变化规律和热力学稳定性条件,给出了该体系相分裂判据,并得到了完整的相分裂区;最后,根据多相精馏的特点对现有的精馏塔算法进行了改进,对该三元非均相共沸精馏过程进行了数值模拟,所得沿塔温度和各组分浓度分布曲线与工业实测数据吻合.     

Thermodynamic model for the formation of solid paraffinic compounds in petroleum

A new model for the formation of solid compounds in petroleum is proposed. The model is based on the assumption that the solid precipitates are composed of pure immiscible heavy hydrocarbons. The weight of solid waxes is calculated using the direct numerical minimization of the Gibbs free energy of a two-phase multicomponent liquid solution-solid precipitate system. The hydrocarbon is precipitated from the solution only when the thermodynamic condition accounting for the formation of the solid phase for this component is met. The simulated results are compared with the experimental data obtained for artificial solutions and natural oils. The approach proposed in the present paper generalizes the models existing in this area.     

Application of the bio-inspired Krill Herd optimization technique to phase equilibrium calculations

The Krill Herd optimization technique, which is based on the simulated herding behaviour of the krill crustacean, is applied to calculations involving phase equilibrium and phase stability, as the application of this emerging technique is extremely limited in the literature. In this work, the Krill Herd algorithm (KH)¹ and the modified Lévy-flight Krill Herd algorithm (LKH)² has been applied to phase stability (PS)³ and phase equilibrium calculations in non-reactive (PE)⁴ and reactive (rPE)⁵ systems, where global minimization of the total Gibbs energy is necessary. Several phase stability and phase equilibrium systems were considered for the analysis of the performance of the technique that includes both vapour and liquid phase conditions.The Krill Herd algorithm was found to reliably determine the desired global optima in PS, PE and rPE problems with generally higher success rates and lower computing time requirements than previously applied metaheuristic techniques such as those involving swarm intelligence and genetic and evolutionary algorithms.     

Incremental and simultaneous identification of reaction kinetics: methods and comparison

Short cut procedures such as the differential method are still popular for the identification of reaction kinetics. Despite their lack of statistical rigor their computational efficiency makes them a valuable tool for initial analysis. A new incremental approach is introduced here to extend the differential method. This identification procedure is split in a sequence of inverse problems, thereby reducing uncertainty and computational complexity. Since error propagation remains critical for the success of such a stepwise approach the influence of error propagation has to be analyzed. Consequently, the performance of the incremental strategy is compared to that of the simultaneous approach which exhibits optimal statistical properties. A pedagogical example is considered in detail to extract general conclusions.     

Computation of phase and chemical equilibria

A new method is proposed for the computation of simultaneous phase and chemical equilibria by minimization of the Gibbs energy. A non-linear programming approach is used which enables one to determine the number and identity of phases at equilibrium, as well as the compositions of each component in each phase. Any thermodynamic model which predicts fugacities can be incorporated easily in this method. Numerical results of various systems that involve gases, liquids and solids are presented.     

Customized minimization techniques for phase equilibrium computations in reservoir simulation

This paper discusses a minimization algorithm for the solution of the Gibbs free energy minimization problem involving at most two phases. There are three major aspects to this paper. The first is the selection of descent directions and step lengths in order to handle the poorly scaled phase equilibrium problems associated with mixtures near bubble points and critical points. The second is the prevention of convergence to trivial solutions in both the Gibbs free energy minimization problem for two-phase mixtures and the associated phase stability test for single-phase mixtures. The third is the use of effective convergence criteria to obtain either the desired level of accuracy in the solution or the maximum accuracy allowed by the problem and the computer.     

Application of headspace analysis to the study of aroma compounds-lipids interactions

Taking into account interactions between aroma compounds and food components is necessary to better manage the flavoring of food products. These interactions occur at a molecular level and reflect changes, at a macroscopic level, in thermodynamic equilibria, such as solubility or volatility. The rate of transfer of an aroma compound from the liquid to the vapor phase can be affected as well. The behavior of aroma compounds in water and lipid solutions was studied in two complementary ways, a thermodynamic and a kinetic approach (head-space analysis). The transfer rate of volatiles at the liquid-water interface does not only depend on the hydrophobicity of the aroma compounds. Vapor-liquid partition and activity coefficients show the presence of solute-solvent interactions. The Gibbs free energy values indicate their physicochemical nature.