Constrained and unconstrained Gibbs free energy minimization in reactive systems using genetic algorithm and differential evolution with tabu list

Abstract

Phase equilibrium modeling plays an important role in design, optimization and control of separation processes. The global optimization problem involved in phase equilibrium calculations is very challenging due to the high non-linearity of thermodynamic models especially for multi-component systems subject to chemical reactions. To date, a few attempts have been made in the application of stochastic methods for reactive phase equilibrium calculations compared to those reported for non-reactive systems. In particular, the population-based stochastic methods are known for their good exploration abilities and, when optimal balance between the exploration and exploitation is found, they can be reliable and efficient global optimizers. Genetic algorithms (GAs) and differential evolution with tabu list (DETL) have been very successful for performing phase equilibrium calculations in non-reactive systems. However, there are no previous studies on the performance of both these strategies to solve the Gibbs free energy minimization problem for systems subject to chemical equilibrium. In this study, the constrained and unconstrained Gibbs free energy minimization in reactive systems have been analyzed and used to assess the performance of GA and DETL. Specifically, the numerical performance of these stochastic methods have been tested using both conventional and transformed composition variables as the decision vector for free energy minimization in reactive systems, and their relative strengths are discussed. The results of these strategies are compared with those obtained using SA, which has shown competitive performance in reactive phase equilibrium calculations. To the best of our knowledge, there are no studies in the literature on the comparison of reactive phase equilibrium using both the formulations with stochastic global optimization methods. Our results show that the effectiveness of the stochastic methods tested depends on the stopping criterion, the type of decision variables, and the use of local optimization for intensification stage. Overall, unconstrained Gibbs free energy minimization involving transformed composition variables requires more computational time compared to constrained minimization, and DETL has better performance for both constrained and unconstrained Gibbs free energy minimization in reactive systems.

Introduction

The accurate modeling of phase equilibrium plays a major role in the design, development, operation, optimization and control of chemical processes. For example, phase behavior has significant impact on equipment and energy costs of separation and purification processes in chemical industry. Further, solving phase equilibrium problems is a dominant task in the process simulation software. The development of reliable methods has long been a challenge and is still a research topic of continual interest [1]. The determination of the number of phases, their identity, and composition at equilibrium of multi-component systems is a complex issue and presents several numerical difficulties [1]. Chemical reactions, if present, increase the complexity and dimensionality of phase equilibrium problems, and so phase split calculations in reactive systems are more challenging due to non-linear interactions among phases and reactions [2]. This fact has prompted growing interest in reliable and efficient methods for the simultaneous computation of physical and chemical equilibrium.

The phase distribution and composition at equilibrium of a reactive mixture are determined by the global minimization of Gibbs free energy (G) subject to mass balance and chemical equilibrium constraints [3]. Specifically, the global optimization problem for reactive phase equilibrium calculations follows the form: minimize F_obj(u) subject to h_j(u) = 0 for j = 1, 2, …, m and u ∈ Ω where u is a vector of continuous variables in the domain Ω ∈ ℜⁿ, m is the number of equality constraints related to material balances and chemical equilibrium, and F_obj(u) = G: Ω ⇒ ℜ is a real-valued function. The domain Ω is defined by the upper and lower limits of each decision variable, which are composition variables. This optimization problem can be formulated using either conventional composition variables (i.e., mole fractions or numbers), or transformed composition variables [3], [4], [5]. Based on the problem formulation and the numerical strategy used for this minimization, the methods can be grouped into two main categories: equation-solving methods and direct optimization strategies. In addition, depending on the handling of material balance constraints, these strategies can also be classified as either stoichiometric or non-stoichiometric [3], [6].

In general, classical strategies for determining the phase equilibrium of non-reactive systems have been extended and applied to systems subject to chemical reactions [3]. Equation-solving methods are based on the solution of non-linear equations obtained from the stationary conditions of the optimization criterion. Local search methods with and without decoupling strategies are frequently used to solve these equations in conjunction with the mass balance and chemical equilibrium restrictions [7]. However, they are prone to severe computational difficulties and may fail to converge to the correct solution when initial estimates are not suitable, especially for non-ideal multi-component and multi-reactive systems [1], [3], [6]. Note that the minimization of G in reactive systems involves many complexities because it is generally non-convex, constrained, highly non-linear with many decision variables, and often has unfavorable attributes such as discontinuity and non-differentiability (e.g., when cubic equations of state or asymmetric models are used for modeling thermodynamic properties). Additional complexities arise near the phase boundaries, in the vicinity of critical points or saturation conditions, and when the same model is used for determining the thermodynamic properties of the mixture [1], [3]. As consequence, G may have several local minima including trivial and non-physical solutions. In these conditions, conventional numerical methods are not suitable for performing reactive phase equilibrium calculations.

On the other hand, a number of optimization strategies for performing the minimization of G in reactive systems have been proposed, and they comprise local and global methods (e.g., [2], [3], [5], [6], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]). The use of Lagrange multipliers is usually the preferred approach for G minimization but its performance is highly dependent on initial estimates of Lagrange multipliers [17]. There has been significant and increasing interest in the development of deterministic and stochastic global strategies for reliably solving reactive phase equilibrium problems. Studies on deterministic reactive phase equilibrium calculations have been focused on the application of the linear programming [9], [21], branch and bound global optimization [11], homotopy continuation methods [13], [15], [20], and interval analysis using an interval-Newton/generalized bisection algorithm [16]. Although these methods have proven to be promising, some of them are model-dependent, may require problem reformulation or significant computational time for multi-component systems [1], [24].

Alternatively, stochastic optimization techniques have often been found to be as reliable and effective as deterministic methods. Further, they offer more advantages for the global optimization of G. These methods are robust, require a reasonable computational effort for the optimization of multivariable functions (generally less time than deterministic approaches), applicable to ill-structure or unknown structure problems, require only objective function calculations and can be used with all thermodynamic models. In fact, it appears that they may fulfill the requirements of an ideal algorithm: reliability, generality and efficiency. To date, a few attempts have been made in the application of stochastic methods for reactive phase equilibrium calculations, compared to those reported for non-reactive systems [14], [19], [22], [23], [25]. Specifically, Lee et al. [14] introduced the application of the random search method of Luus and Jaakola for the global minimization of G using a non-stoichiometric formulation. On the other hand, Bonilla-Petriciolet et al. [19] formulated the unconstrained optimization problem for G minimization using simulated annealing (SA) and transformed composition variables. In another study, particle swarm optimization (PSO) and several of its variants have been applied for reactive phase equilibrium calculations using transformed composition variables [22]. Recently, our group [23] has tested and compared the performance of differential evolution (DE) and tabu search (TS) for the global minimization of G using reaction-invariant composition variables. Finally, Reynolds et al. [25] outlined a general procedure for the global optimization of G in reactive systems using SA and a non-stoichiometric approach. Results of these studies have shown the potential of stochastic optimization solvers for phase equilibrium calculations subject to chemical reactions.

In particular, the population-based stochastic methods are known for their good exploration abilities; when optimal balance between the exploration and exploitation is found, they can be reliable and efficient global optimizers. This is because at each generation/iteration a whole population of potential solutions is improved rather than a single solution. A variety of population-based stochastic methods have been proposed for chemical engineering applications including the modeling of phase equilibrium, e.g. [26], [27], [28], [29]. Specifically, genetic algorithms (GAs) and differential evolution with tabu list (DETL) have been very successful for performing phase equilibrium calculations in non-reactive systems [26], [28]. However, to the best of our knowledge, there are no studies on the performance of both these strategies for G minimization in systems subject to chemical equilibrium. These methods are suitable and promising for overcoming the numerical difficulties of this global optimization problem.

In this study, the constrained and unconstrained Gibbs free energy minimization in reactive systems have been analyzed and used to assess the performance of GA and DETL. Specifically, the numerical performance of these stochastic methods have been tested using both conventional and transformed composition variables as the decision vector for G minimization, and their relative strengths are discussed. The results of GA and DETL are compared with those obtained using SA, which has shown a competitive performance in reactive phase equilibrium calculations [19]. Our results on a variety of reactive systems indicate that DETL is superior to SA and GA for both the constrained and unconstrained Gibbs free energy minimization in reactive systems.