Global Gibbs free energy minimization in reactive systems via harmony search

Transcription

1 Instituto Tecnologico de Aguascalientes From the SelectedWorks of Adrian Bonilla-Petriciolet 2012 Global Gibbs free energy minimization in reactive systems via harmony search Adrian Bonilla-Petriciolet Ma. del Rosario Moreno-Virgen Juan Jose Soto-Bernal Available at:

2 INTERNATIONAL JOURNAL OF CHEMICAL REACTOR ENGINEERING Volume Note S3 Global Gibbs Free Energy Minimization in Reactive Systems via Harmony Search Adrian Bonilla-Petriciolet Ma. del Rosario Moreno-Virgen Juan Jose Soto-Bernal Instituto Tecnológico de Aguascalientes, Instituto Tecnológico de Aguascalientes, moreno Instituto Tecnológico de Aguascalientes, ISSN DOI: / Copyright c 2012 De Gruyter. All rights reserved.

3 Global Gibbs Free Energy Minimization in Reactive Systems via Harmony Search Adrian Bonilla-Petriciolet, Ma. del Rosario Moreno-Virgen, and Juan Jose Soto-Bernal Abstract Phase equilibrium calculations in systems subject to chemical reactions play a major role in the design of reactive separation schemes including chemical reaction engineering. Basically, these calculations involve the global minimization of Gibbs free energy constrained by the material balances and chemical equilibrium restrictions. However, Gibbs free energy function is non-convex, highly non-linear with many decision variables, and may have several local minimums including trivial and nonphysical solutions. In these conditions, conventional numerical methods are not suitable for solving reactive phase equilibrium problems. Recently, there has been a significant and increasing interest in the development of global strategies for reliably solving reactive phase equilibrium problems subject. Harmony search (HS) is a global stochastic optimization method, which has been conceptualized using the musical process of searching for a perfect state of harmony. Until now, HS has been successfully applied to solve various engineering and optimization problems. However, there are few studies concerning the application of this optimization method for chemical engineering calculations. To the best of our knowledge, the performance of HS for solving reactive phase equilibrium problems has not yet been reported. Therefore, this paper introduces the application of HS-based algorithms to the constrained global minimization of Gibbs free energy in reactive systems. Specifically, we have studied the performance of three variants of HS in reactive phase equilibrium calculations. Our results are useful to identify the capabilities and relative strengths of HS with respect to other stochastic optimization methods for the simultaneous calculation of Corresponding author: A. Bonilla-Petriciolet, Instituto Tecnológico de Aguascalientes. Authors acknowledge the financial support provided by CONACYT, DGEST and Instituto Tecnológico de Aguascalientes (Mexico).

4 physical and chemical equilibrium. KEYWORDS: phase equilibrium, chemical equilibrium, global optimization, harmony search

5 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 1 1. INTRODUCTION The simultaneous calculation of physical and chemical equilibrium is a fundamental numerical procedure in process system engineering including chemical reactor engineering (Wakeham and Stateva, 2004; Saboohi and Avami, 2011). These calculations are especially important in the petroleum and chemical engineering industry where separation units are the core of process performance. Classical thermodynamics establishes that the necessary and sufficient condition to achieve a global stable equilibrium in a multiphase and multicomponent system, at constant temperature and pressure, is the global minimization of the Gibbs free energy function G (Rangaiah, 2001; Wakeham and Stateva, 2004; Rossi et al., 2009). If chemical reactions are present, this thermodynamic function is constrained by the mass balances and chemical equilibrium restrictions. Therefore, phase equilibrium calculations for reactive systems can be formulated as global optimization problems. Formally, this constrained minimization problem can be stated as follows (Bonilla-Petriciolet et al., 2011): 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 R n, m is the number of equality constraints related to material balances and chemical equilibrium, and F obj (u) = G : Ω R is a realvalued function. Several studies have shown that the major challenge of solving phase equilibrium problems is that G is generally non-convex (i.e., there are local optimums including trivial and non-physical solutions), highly non-linear with many decision variables especially for multicomponent and multiphase systems, and often has unfavourable attributes such as discontinuity and nondifferentiability (Rangaiah, 2001; Wakeham and Stateva, 2004). Moreover, the problem complexity and dimensionality increase in systems subject to chemical reactions. Consequently, traditional optimization methods are not suitable for solving reactive phase equilibrium problems because they are sensitive to initial guess values and usually converge to local optimums of G. Until now, the amount of literature on the calculation of reactive phase equilibrium using global optimization approaches is rather limited. The applied numerical strategies include both deterministic and stochastic optimization approaches (Zhang et al., 2011). Deterministic global optimization studies have been focused on the application of homotopy continuation methods (Jalali et al., 1999), branch and bound global optimization (McDonald and Floudas, 1996), interval mathematics (Burgos-Solorzano et al., 2004) and linear programming (Rossi et al., 2009). However, some authors have suggested that, although deterministic methods have proven to be promising, some of them are model dependent, may require problem reformulations or significant computational time especially for multicomponent systems (Nichita et al., 2002). On the other hand, there are only a few studies regarding the solution of phase equilibrium problems Published by De Gruyter, 2012

6 2 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 in reactive systems using stochastic global optimization methods. Results reported in literature indicate that stochastic optimization techniques have often been found to be as reliable and effective as deterministic methods but may offer advantages for phase equilibrium modeling (Rangaiah, 2001; Bonilla-Petriciolet et al., 2011; Zhang et al., 2011). These methods are robust numerical tools that present a reasonable computational effort in the optimization of multivariable functions (generally less time than deterministic approaches); they are applicable to illstructure or unknown structure problems, require only calculations of the objective function and can be used with both symmetric and asymmetric thermodynamic models (Henderson et al., 2001; Rangaiah, 2001; Shahhosseini and Vakili, 2011). To date, the stochastic optimization methods used for the global minimization of G in reactive mixtures include the Random Search (Lee et al., 1999), Simulated Annealing (Bonilla-Petriciolet et al., 2008), Tabu Search (Bonilla-Petriciolet et al., 2010a), Particle Swarm Optimization (Bonilla- Petriciolet and Segovia-Hernández, 2010), Genetic Algorithms and Differential Evolution (Bonilla-Petriciolet et al., 2010b; Bonilla-Petriciolet et al., 2011). These methods usually show a robust performance but, in some difficult problems, they may fail to locate the global optimum. This fact has encouraged the development of alternative optimization strategies to identify a better approach for solving phase equilibrium problems in reactive systems. Recently, new strategies have been proposed in order to improve the drawbacks of classical stochastic optimization methods. These strategies are based on the following approaches: 1) the application of new meta-heuristics, 2) the development of different variants of existing algorithms, and 3) the hybridization of both diversification and intensification components of available stochastic methods. Specifically, Harmony Search (HS) is a nature inspired algorithm, which mimics the improvisation process of music players (Geem et al., 2001). HS is simple in concept and easy in implementation. In addition, the structure of HS favors to combine or hybridize it with other meta-heuristics. It has been successfully applied to various benchmark and real-world optimization problems including traveling salesman problem, parameter optimization of river flood model and design of pipeline network (Lee and Geem, 2005; Geem, 2009). In the field of chemical engineering, there are few studies concerning the application of this global stochastic optimization method (e.g., Fesanghary et al., 2009) and, to the best of our knowledge, the performance of HS for phase equilibrium calculations in reactive systems has not yet been reported. Note that the study and evaluation of the numerical performance of new algorithms in science and engineering problems is an active research area of applied global optimization. Specifically, this type of studies is important to identify the capabilities and limitations of novel techniques for solving global optimization problems

7 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 3 including phase equilibrium calculations. Therefore, this paper introduces the application of HS-based algorithms to perform the constrained global minimization of Gibbs free energy in reactive systems. Particularly, we have studied the performance and capabilities of three variants of HS in reactive phase equilibrium calculations. 2. DESCRIPTION OF HARMONY SEARCH FOR GLOBAL OPTIMIZATION Harmony Search is a music-inspired meta-heuristic algorithm, which has been introduced by Geem et al. (2001). This stochastic optimization method was developed in an analogy with music improvisation process where music players improvise the pitches of their instruments to obtain better harmony (Lee and Geem, 2005). HS combines heuristic rules and randomness to imitate this music improvisation process. A comprehensive explanation of HS is provided by Geem et al. (2001) and a flow chart is given in Figure 1. Define parameters of Harmony Search Initialize harmony memory Improvise a new harmony Add new harmony to harmony memory? Yes Update harmony memory No No Termination criteria satisfied? Yes Local optimization Figure 1. Flowchart of Harmony Search (HS). Published by De Gruyter, 2012

8 4 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 Briefly, HS involves the following parameters: the harmony memory size (HMS), the harmony memory considering rate (HMCR), the pitch adjusting rate (PAR), the bandwidth or step size for variable perturbation during pitch adjustment (bw), and the number of improvisations (NI). The harmony memory is used to store a set of solution vectors for decision variables and it is equivalent to the population of a genetic algorithm or particle swarm optimization. This harmony memory preserves the history of optimization sequence and is useful to identify promising areas for global optimization because good harmonies can be considered as elements of new solution vectors. The initial values of harmony memory are generated from a uniform distribution in the bounds of decision variables u i = lb i + rand (ub i lb i ) i =1,,n opt (1) where lb i and ub i are the lower and upper bounds for each decision variable u i and rand (0, 1) is a random number, respectively. On the other hand, the parameters HMCR and PAR are used to improve the solution vector and to increase the diversity of the search process (Geem et al., 2001; Lee and Geem, 2005). Specifically, a new harmony (i.e., a new solution vector v i ) is generated using these parameters and the following procedures: a) memory consideration, b) pitch adjustment, and c) random selection. These stochastic operators are summarized in the pseudo-code reported in Figure 2a and they are used to perform both diversification and intensification stages in HS. The diversification is controlled by the pitch adjustment and random selection operators, while memory consideration is generally associated to the intensification process. Note that HMCR is used to determine the degree of contribution of harmony memory (i.e., promising solutions) during random search; while PAR and bw are used to perform the additional random perturbation of decision variables when memory consideration is applied. The random selection contributes to explore different regions of objective function and also to increase the diversity of solution vectors. The proper combination of these operators is important to favor the performance of HS in global optimization. Usually, a boundary violation check must be implemented, in particular during pitch adjustment, to verify the feasibility of v; if v is infeasible, a new harmony is randomly generated inside lower and upper bounds of decision variables. In the context of HS, the generation of a new harmony is called improvisation. Finally, to update the harmony memory, the new harmony (v) replaces the worst harmony in harmony memory only if its objective function value is lower than that of the worst harmony. This iterative procedure is repeated until satisfying a proper convergence criterion. Similar to other stochastic methods, the choice of stopping condition can affect the performance of HS. The

9 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 5 stopping criterions commonly used for HS are based on the number of improvisations NI (i.e., the number of function evaluations NFE). After satisfying the stopping condition, the best solution found by HS, which is stored in harmony memory, is expected to be a near global optimum solution. A local optimization technique (e.g., gradient-based method) can be used at the end of global search for efficiently improving the accuracy of the best solution obtained by HS. Herein, it is convenient to remark that stochastic optimization methods may require a significant computational effort to improve the accuracy of global solution because they explore the search space of decision variables by creating random movements instead of determining a logical optimization trajectory. Therefore, this additional intensification step is required for rapid convergence in the final stage of HS. for i = 1 to n opt do for i = 1 to n opt do if rand (0, 1) HMCR then perform if rand (0, 1) HMCR then perform memory consideration memory consideration begin begin v i = u ij where j (1,,HMS) v i = u ij where j (1,,HMS) if rand (0, 1) PAR then perform pitch if rand (0, 1) PAR then perform pitch adjustment adjustment begin begin v i = v i + (0.5 rand) bw i where bw i is the v i = u i,best where best is the index of the best bandwidth (i.e., step size) harmony in the harmony memory end if end if else perform random selection else perform random selection v i = lb i + rand (ub i lb i ) v i = lb i + rand (ub i lb i ) end if end if end for end for a) Harmony Search b) Global-Best Harmony Search Figure 2. Pseudo-code used to improvise a new harmony (i.e., a new solution vector) in harmony search-based optimization methods. As stated, the parameters HMS, HMCR, PAR and bw are important to determine the performance (i.e., reliability and efficiency) of HS in global optimization. For example, some authors have suggested that small values of HMS may lead to the HS to be trapped in local solutions (Mahdavi et al., 2007). However, increasing HMS generally provides better solution vectors but at the expense of more function evaluations. Therefore, the fine tuning of these parameters is very crucial for solving global optimization problems (Mahdavi et al., 2007; Omran and Mahdavi, 2008). Traditionally, fixed values for HS parameters, which can not be changed during new improvisations, are used in Published by De Gruyter, 2012

10 6 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 global optimization (Geem et al., 2001; Geem, 2009). So, this standard version of HS algorithm is used and referred as HSC in this study. Recently, some modifications have been proposed in the literature to improve the convergence performance of the original HS. According to Geem (2009), the variations proposed for HS may involve: a) mechanisms for the proper initialization of HS parameters, b) mechanisms for the dynamic adaptation of HS parameters during optimization sequence, and c) the application of new operators for HS that includes hybrid methods using other meta-heuristics such as Simulated Annealing or Differential Evolution. In this paper, we have studied two typical and promising variants of HS called: the Improved Harmony Search and the Global-Best Harmony Search (Mahdavi et al., 2007; Omran and Mahdavi, 2008). Particularly, the dynamic adaptation of HS parameters is the most common approach to overcome the drawbacks of classical HS. Results reported by Mahdavi et al. (2007) showed that small PAR values with large bw values may affect the performance of HS and increase the calculations needed to find the global optimum. Although, small bw values in final improvisations increase the fine-tuning of solution vectors but, in early iterations, bw should take a bigger value to diversify the solution vectors. Furthermore, large PAR values with small bw values may cause the improvement of best solutions in final improvisations. Based on this fact, Mahdavi et al. (2007) introduced the Improved Harmony Search (IHS), which uses dynamic values of both parameters PAR and bw. Specifically, the dynamic value of PAR changes with improvisation number as follows ( PARmax PARmin PAR k PAR ) 1 = min + * k NI + (2) where PAR min and PAR max are the minimum and maximum pitch adjusting rates, and k is an improvisation counter. On the other hand, the bandwidth for each improvisation is defined as bw k 1 = bwmax exp ( k / NI)ln( bwmin / bwmax ) + ) (3) where bw min and bw max are the minimum and maximum values for bandwidth, respectively. Note that PAR min, PAR max, bw min and bw max are defined by the user and are problem dependent. Mahdavi et al. (2007) showed that this variant of HS has proven to be competitive with respect to other HS algorithms for solving benchmark and some engineering optimization problems. Based on this fact, we have considered IHS for performing the global minimization of G in reactive systems.

11 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 7 On the other hand, Omran and Mahdavi (2008) proposed an alternative version of HS called Global-Best Harmony Search (GHS), which is inspired by the concept of swarm intelligence used in Particle Swarm Optimization. This method modifies the pitch-adjustment step of HS to encourage that a new harmony can mimic the best harmony stored in the harmony memory. Results reported for several benchmark optimization problems indicate that GHS may offer a better performance than those reported for HSC and IHS (Omran and Mahdavi, 2008). In general, GHS has the same structure as HS with the exception of pitch adjustment step used in the improvisation of a new harmony. The pseudocode used to improvise a new harmony in GHS is given in Figure 2b. Omran and Mahdavi (2008) have suggested that using a constant value of PAR improves the performance of GHS and this scheme is even better than GHS using a dynamical value of PAR. So, this approach has been used in the present study for GHS. Although these optimization methods (HSC, GHS and IHS) have been successfully applied in different science and engineering fields, their capabilities have not yet been studied in the global minimization of Gibbs free energy function for systems subject to chemical reactions. Therefore, we have used these HS-based optimization methods for performing reactive phase equilibrium calculations. All methods have been implemented in Fortran subroutines that can be employed for solving global optimization problems with continuous variables. These codes are available to interested readers upon request to the corresponding author. In our calculations, the quasi-newton method implemented in the subroutine DBCONF of IMSL library was used for local optimization. This subroutine calculates the gradient via finite differences and approximates the Hessian matrix according to BFGS formula. The default values of DBCONF parameters in the IMSL library were employed. Finally, with respect to the stopping condition of HS, the following criteria have been considered in this study and implemented for HSC, IHS and GHS: 1) a maximum number of successive improvisations (SNI max ) without improvement in the best function value, or 2) a maximum number of improvisations (NI). 3. PROBLEM FORMULATION OF THE CONSTRAINED GIBBS FREE ENERGY MINIMIZATION FOR PHASE EQUILIBRIUM CALCULATIONS IN REACTIVE SYSTEMS 3.1 Objective function As stated, the computation of phase equilibrium in multireactive systems can be performed via the Gibbs free energy minimization subject to constraints obtained from the material balances and chemical equilibrium restrictions. Recently, Bonilla-Petriciolet et al. (2011) have studied and compared the constrained and Published by De Gruyter, 2012

12 8 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 unconstrained Gibbs free energy minimization in reactive systems using genetic algorithms, differential evolution and both conventional and transformed composition variables. This study concluded that the constrained Gibbs free energy minimization using conventional composition variables is more suitable, in terms of computer time and numerical implementation, for reactive phase equilibrium calculations. Based on this fact, we have considered this approach for testing the performance of HS-based methods. The objective function used in this study for phase equilibrium calculations in reactive systems is derived from the relationship between the Gibbs free energy and the transformed Gibbs free energy; a detailed procedure for obtaining this objective function is given by Bonilla-Petriciolet et al. (2011). In summary, for a c multi-component and π multi-phase system, subject to r independent chemical reactions, the objective function for reactive phase equilibrium calculations can be defined as F obj π 1 = Δg ln K N n (4) j = 1 eq ref, j where Δg is the Gibbs free energy of mixing, ln K eq is a row vector of logarithms of chemical equilibrium constants for all r independent chemical reactions, N is an invertible, square matrix formed from the stoichiometric coefficients of a set of reference components chosen from the r reactions, and n ref is a column vector of moles of each of the reference components. Then, Δg is given by π c Δg = n ij ln( x ij γ ij ) (5) j= 1 i= 1 where γ ij and x ij are respectively the activity coefficient and mole fraction of component i in phase j, and n ij is the mole number of component i in phase j. This objective function is defined using reaction equilibrium constants and must be globally minimized subject to the following mass balance restrictions (Bonilla- Petriciolet et al., 2011) j= ( n ij vi N nref, j ) = nif vin nref, F π i = 1,, c r (6) where n i,f is the initial moles of component i in the feed. These mass balance equations can be rearranged to reduce the number of decision variables of the optimization problem and to eliminate equality constraints which are usually

13 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 9 challenging for optimization methods. Thus, Eq. (6) is used to reduce the number of decision variables applying the following expression n iπ π 1 1 = nif i ref F ref 1 v N ( n, n, π ) ( nij vi N nref, j ) i = 1,..., c r (7) j= 1 With this approach, the decision variables are c (π 1) + r mole numbers (n ij ). Then, the global optimization problem can be solved by minimizing F obj (i.e., Eq. 4) with respect to c (π 1) + r decision variables n ij and the remaining c r mole numbers (n iπ ) are determined from Eq. (7) and subject to the inequality constraints n iπ > 0. Formally, the constrained global minimization of F obj can be stated as min n ij F obj ( n ij ) subject to Eq. (6) (8) n ij > 0 i = 1,..., c j = 1,...,π Using this formulation, there is a substantial reduction of the number of decision variables and we can perform reactive phase equilibrium calculations for multicomponent systems via stochastic optimization methods employing a reasonable numerical effort. In constrained optimization problems, the search space consists of both feasible and infeasible points. For reactive phase equilibrium calculations, feasible points satisfy all the mass balance constraints, i.e., Eq. (6), while infeasible points violate at least one of them (i.e., n iπ < 0 where i = 1,, c r). Based on the results reported by Bonilla-Petriciolet et al. (2011), in this study the penalty function method is used to solve the constrained Gibbs free energy minimization in reactive systems via HS. The penalty method is easy to implement and is considered efficient for handling constraints in the stochastic methods. This approach transforms the constrained problem into an unconstrained problem by penalizing infeasible solutions. For handling the constraints in reactive phase equilibrium calculations, an absolute value of constraint violation is multiplied with a high penalty weight and then added to the objective function (i.e., Gibbs free energy function). In case of more than one constraint violation, all Published by De Gruyter, 2012

14 10 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 constraint violations are first multiplied with the penalty weight, and all of them are added to the objective function value. This penalty function is given by F r F = F obj obj if n ij > 0 i = 1,..., c + p otherwise, j = 1,..., π, (9) where p is the penalty term whose value is positive. In phase equilibrium calculations, the infeasible solutions (i.e., n iπ < 0) imply that the Gibbs free energy function of phase π can not be determined due to the logarithmic terms of the activity or fugacity coefficients. Results reported by Bonilla-Petriciolet et al. (2011) indicated that a suitable penalty term for handling infeasible solutions in the constrained Gibbs free energy minimization for systems subject to chemical reactions is given by n unf n i i= 1 p = 10 π (10) where n iπ is obtained from Eq. (7) and n unf is the number of infeasible mole numbers (i.e., n iπ < 0 where i = 1,..., c r). So, we have used this formulation for testing the performance of HS-based methods in phase equilibrium calculations of reactive systems. 3.2 Description of reactive phase equilibrium problems To attain a better understanding of the performance of HS-based methods, we have used several reactive phase equilibrium problems reported in literature as cases of study (Bonilla-Petriciolet and Segovia-Hernández, 2010; Bonilla- Petriciolet et al., 2011). In summary, these problems include multicomponent reactive systems with vapor-liquid and liquid-liquid equilibrium (i.e., symmetric and asymmetric models). Initial feed composition, operating conditions, thermodynamic models and global solutions of these problems are reported in Tables 1 and 2 and a detailed description of all problems can be found in Bonilla- Petriciolet and Segovia-Hernández (2010) and Bonilla-Petriciolet et al. (2011). These reactive systems are a suitable suite of well-known test problems, which have been used to evaluate the performance of other stochastic optimization methods (e.g., Simulated Annealing, Particle Swarm Optimization, Genetic Algorithm, and Differential Evolution). In fact, they have different degrees of difficulty and features, and we consider that are useful to systematically test the performance of HS-based methods.

15 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems RESULTS AND DISCUSSION 4.1 Parameter tuning of HS-based methods Literature indicates that the fine-tuning of algorithm parameters is required to improve the performance of a stochastic method for a particular problem. Therefore, the key parameters of HSC, GHS and IHS have been tuned by finding the global minimum of problems No. 1 and 4. Following previous studies (e.g., Bonilla-Petriciolet and Segovia-Hernández, 2010; Bonilla-Petriciolet et al., 2011), parameter tuning has been performed by varying one parameter at a time while the rest was fixed at nominal values. These nominal values were defined using results reported in the literature and our preliminary calculations. For parameter tuning, each HS method was run 100 times on every problem using different random number seeds and initial guesses for each set of parameter values. The suggested parameter values for all stochastic methods are reported in Table 3. For all calculations performed using HS-based methods, we have defined lb i = 10-5 and ub i = ; while HMS = 10n opt where n opt is the number of decision variables used in reactive phase equilibrium calculations. In general, our results indicate that suggested values are a reasonable compromise between numerical effort and reliability of tested stochastic methods. All our calculations were performed on an Intel Pentium M 1.73 GHz processor with 504 MB of RAM. 4.2 Performance of HSC, GHS and IHS in the constrained minimization of Gibbs free energy of reactive systems In this section, we report and compare the performance of HSC, IHS and GHS for reactive phase equilibrium calculations. These methods are evaluated by considering both the ability of the stochastic method to find the global minimum of Eq. (4) and the numerical effort that it required. Each test problem has been solved 100 times using HS methods, each time with a different random number seed such that the initial values of decision variables and random operators are different in all numerical trials. Figures 3 and 4 show the performance of stochastic methods with respect to the relative errors for the solutions obtained in all calculations performed at different levels of computational efficiency, which are obtained by changing the stopping conditions NI and SNI max. As stated, the stopping conditions NI and SNI max contribute to the trade-off between efficiency and reliability of HS. Therefore, the performance of three methods is illustrated by changing these stopping conditions. This approach is adopted in the present study because generally no correlation can be established a priori between an optimization problem and the required numerical effort for finding the global Published by De Gruyter, 2012

16 12 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 optimum. So, the proper stopping condition has to be determined by a sensitive analysis. It is convenient to remark that these results correspond to the implementation of stochastic methods without the local optimization strategy. Table 1. Examples selected for the constrained minimization of Gibbs free energy in reactive systems using HS-based methods. No. System Feed conditions Thermodynamic models 1 A 1 + A 2 A 3 + A 4 (1) Ethanol (2) Acetic acid (3) Ethyl acetate (4) Water n F = (0.5, 0.5, 0.0, 0.0) at 355 K and kpa NRTL model and ideal gas. K eq,1 = A 1 + A 2 A 3, and A 4 as an inert component. (1) Isobutene (2) Methanol (3) Methyl ter-butyl ether (4) n-butane 3 A 1 + A 2 + 2A 3 2A 4 (1) 2-Methyl-1-butene (2) 2-Methyl-2-butene (3) Methanol (4) Tert-amyl methyl ether 4 A 1 + A 2 A 3 + A 4 (1) Acetic acid (2) n-butanol (3) Water (4) n-butyl acetate n F = (0.3, 0.3, 0.0, 0.4) at K and kpa n F = (0.354, 0.183, 0.463, 0.0) at 335 K and kpa n F = (0.3, 0.4, 0.3, 0.0) at K and kpa Wilson model and ideal gas. Δ / R = T G o rxs T lnt 0 ΔGrxs ln Keq,1 = where T is in K. RT Wilson model and ideal gas / T K eq, 1 = e where T is in K. UNIQUAC model. 450 ln K, 1 = where T is in K. eq T 5 A 1 + A 2 A 3 n F = (0.6, 0.4, 0.0) Margules solution model. E g = 3.6x1 x x1 x x2 x3 RgT K eq,1 = A 1 + A 2 + 2A 3 2A 4 with n F = (0.1, 0.15, 0.7, 0.0, Wilson model and ideal gas. A 5 as inert component 0.05) at 335 K and / T K eq, 1 = e (1) 2-Methyl-1-butene kpa where T is in K. (2) 2-Methyl-2-butene (3) Methanol (4) Tert-amyl methyl ether (5) n-pentane 7 A 1 + A 2 A 3 n F = (0.52, 0.48, 0.0) at K and kpa 8 A 1 + A 2 A 3 + A 4 n F = (0.048, 0.5, 0.452, 0.0) at 360 K and kpa Margules solution model. K eq,1 = 3.5 NRTL model. K eq,1 = 4.0

17 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 13 In particular, these figures show the percent of solutions (F f, %) found by HS methods, from all calculations performed, with relative errors less than or equal to e; this relative error (e) is defined as HS opt obj Fobj HS Fobj F e = (11) opt F obj HS F obj where is the known global optimum for each tested problem and is the best objective function value recorded in the harmony memory of HSC, GHS and IHS, respectively. Therefore, our results provide a general perspective of the solution quality obtained by HS-based methods. Table 2. Global minimum of the reactive examples studied for the constrained global optimization of Gibbs free energy. No. Equilibrium 1 Global minimum of F obj Decision variables, n opt 1 Vapor-Liquid , β i,1 for i =1,2,3 2 Vapor-Liquid , β i,1 for i =1,2,4 3 Vapor-Liquid , β i,1 for i =1,2,3 4 Liquid-Liquid , β i,1 for i =1,2,3 5 Liquid-Liquid , β i,1 for i =1,2 6 Vapor-Liquid , β i,1 for i =1,2,3,5 7 Liquid-Liquid , β i,1 for i =1,2 8 Liquid-Liquid , β i,1 for i =1,2,3 1 Global solution of all problems can be found in Bonilla-Petriciolet et al. (2011). Figures 3 and 4 indicate that both the reliability and the solution quality obtained by HSC, GHS and IHS in the constrained minimization of Gibbs free energy are dependent on the stopping condition and, as a consequence, are related to the numerical effort. Note that increasing the value of both stopping conditions (i.e., NI or SNI max ) improves the performance of three methods for reactive phase equilibrium calculations. But, the reliability of HSC, GHS and IHS is generally better using stopping condition NI compared to SNI max (see results of Figures 3 and 4). On the other hand, it appears that the precision of solutions found by GHS without local optimization is better than those obtained by HSC and IHS for both stopping conditions. With illustrative purposes, Figure 5 provides the convergence histories of opt HS the norm of Fobj Fopt for all methods in the constrained global minimization of G of examples No. 1, 6 and 8. This norm is based on the average (over 100 runs) Published by De Gruyter, 2012

18 14 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 of the best objective function HS F opt recorded in the harmony memory at different improvisations (i.e., NFE). In general, the mean objective function value of best harmony (i.e., solution vector) obtained by GHS and IHS is usually lower than that achieved by HSC in reactive equilibrium calculations. As expected, the objective function value of the best solution obtained by all HS-based methods is nearer to the global minimum as NI increases. However, GHS and IHS can find solution vectors closer to the global minimum solution and they outperformed the classical HS. Moreover, it appears that the convergence curves of GHS and IHS are faster than that of HSC using NI as stopping condition. In fact, both variants of HS can be more effective for finding the promising area of the global optimum. Table 3. Suggested values of parameters in HSC, IHS and GHS for performing the constrained global minimization of Gibbs free energy in reactive systems. Method Parameter Suggested value HSC HMCR 0.5 PAR 0.25 Bw ub i lb i GHS HMCR 0.5 PAR 0.75 Bw ub i lb i IHS HMCR 0.5 PAR min 0.5 PAR max 0.95 bw min bw max ub i lb i These results are in agreement with the observations reported by Mahdavi et al. (2007) and Omran and Mahdavi (2008). In particular, these authors have indicated that the modifications of traditional HS may allow performing global optimization more reliably. In contrast to classical HS, it appears that GHS and IHS can achieve a better tradeoff between the capabilities to explore the search space and to exploit the experience accumulated during the optimization sequence. Overall, our numerical experience in reactive phase equilibrium calculations suggests that GHS may offer the better performance for solving highdimensional problems (i.e., multicomponent and multireactive systems). For the sake of brevity, the performance results of three HS methods implemented with the local optimization method are summarized as the global

19 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 15 success rate (GSR). GSR is defined as the average successes rate on the collection of reactive phase equilibrium problems tested: GSR = 1 N prob N prob i= 1 SR i (12) where SR i is the success rate on the i th problem. SR is measured in terms of number of times the algorithm located the global minimum out of 100 trials and is also reported as percentage. We have also determined the computational efficiency, which is measured in terms of average number of function evaluations. The average NFE was calculated using successful trials only. A trial is considered successful if the global optimum is obtained with an absolute error of 10-5 opt or lower in the objective function value, i.e. F F HS Note that NFE includes both the function calls for evaluating the objective function using the stochastic method (NFE stoc ) and the function calls for the local optimization (NFE loc ). In general, GHS and IHS can achieve high GSR values (i.e., GSR > 80%), and its performance is usually better than that of HSC using either NI or SNI max as the stopping condition, see results reported in Table 4. GSR of the methods is affected when the stopping conditions are limited to low values (i.e., early function evaluations). However, the intensification stage using the local optimization method plays a major role for improving numerical performance of HS-based methods especially at early improvisations. Note that the intensification step is needed for efficiently improving the accuracy of final solutions once the solutions from harmony memory are clustered around the global optimum especially in problems with several decision variables (i.e., multicomponent reactive systems). In general, GSR of GHS is better than that obtained for IHS and HSC using NI as the stopping condition especially at low improvisations. But, as NFE increases, IHS may offer a better performance. If SNI max is used as the convergence criterion, the results indicate that GHS can achieve a slightly higher GSR than those obtained for both HSC and IHS. On the other hand, GHS, HSC and IHS may fail to find the global minimum of the Gibbs free energy function in some test problems specially using few NFE. Specifically, the three stochastic methods each followed by the local optimization method, show high reliability (SR 90%) for reactive problems No. 1-3, 6, and 8, irrespective of the stopping criterion used (i.e., NI or SNI max ). In fact, the global optimum is found even using low values of these stopping conditions. However, HS methods failed several times to find the global optimum of G in reactive problems No. 4, 5 and 7. In particular, example No. 4 is very challenging and useful for testing the performance of new global optimization obj obj Published by De Gruyter, 2012

20 16 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 a) e b) F f, % c) NI/HMS Figure 3. Percent of solutions based on relative errors of a) HSC, b) GHS and c) IHS in the constrained minimization of Gibbs free energy of reactive systems using NI as stopping condition.

21 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 17 a) e b) F f, % c) SNI max /(n opt HMS) Figure 4. Percent of solutions based on relative errors of a) HSC, b) GHS and c) IHS in the constrained minimization of Gibbs free energy of reactive systems using SNI max as stopping condition. Published by De Gruyter, 2012

22 18 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 a) 1.E+00 HSC GHS IHS 1.E-01 1.E-02 b) 1.E-03 1.E+00 1.E-01 1.E-02 1.E-03 opt obj F F HS opt c) 1.E-04 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E NI/HMS Figure 5. Convergence profiles of HSC, GHS and IHS for the constrained minimization of Gibbs free energy for selected reactive systems: a) No. 1, b) No. 6 and c) No. 8.

23 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 19 Table 4. Global success rate (GSR) of HSC, GHS and IHS for the constrained global minimization of Gibbs free energy in reactive systems. GSR (%) for NI/HMS SNI max /(n opt HMS) Method HSC GHS IHS strategies due to the presence of comparable minima. All HS methods showed a SR < 10% in this reactive systems. It is expected that the performance of the optimization methods for solving constrained problems is related to the capability of each method for handling constraints and searching the feasible region of the total search space. Results of this study show that GHS and IHS appear to be more effective to escape from the infeasible region and to exploit the search space. HSC may fail more frequently to identify feasible optimal solutions in the constrained minimization of G for reactive systems. These failures may be due to the presence of local optima and/or flat objective function near the global solution in some reactive problems. In addition, the feasible region is very restricted for some examples. In summary, the performance of HS-based methods is dependent on the problem under study and, as a consequence, the numerical effort for escaping from local minima depends on the problem characteristics. Overall, the reliability of HS methods is better using NI compared to that of SNI max. This may be because of slow convergence and/or more function evaluations are required for escaping from both the infeasible space and local minimum region reached in the initial improvisations. As indicated, the maximum Table 5. NFE of HSC, GHS and IHS each followed by a local optimization method for the constrained minimization of Gibbs free energy in reactive systems. NFE for NI/HMS SNI max /(n opt HMS) Method n var HSC 4 2,158 4,150 10,144 20,139 30,136 40,131 60,129 1,858 3,793 5,296 GHS 2,154 4,153 10,143 20,133 30,129 40,130 60,125 2,346 5,245 9,236 HIS 2,156 4,154 10,147 20,136 30,126 40,119 60,113 1,784 3,321 4,955 HSC 5 3,189 5,669 13,088 25,604 38,030 50,468 75,426 3,780 6,206 10,438 GHS 3,129 5,586 12,921 25,341 37,784 50,277 75,227 7,017 15,774 27,412 HIS 3,180 5,648 13,114 25,473 37,907 50,355 75,290 3,259 5,648 8,604 HSC 6 3,855 6,453 15,318 30,240 45,220 60,207 90,204 5,116 10,040 15,020 GHS 3,300 6,238 15,220 30,214 45,213 60,212 90,207 8,735 17,746 28,312 HIS 3,546 6,381 15,249 30,206 45,209 60,208 90,207 4,462 8,917 13,606 Published by De Gruyter, 2012

24 20 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 number of function evaluations to find the global optimum can not be judged a priori for an arbitrary function. In some global optimization problems, this may lead to unnecessary function calls when the minimum is reached long before the maximum number of improvisations, thus increasing computational effort (Bonilla-Petriciolet et al., 2011). For any global optimization problem, it is expected that all stochastic methods can converge to a global minimum as NFE (Blum and Roli, 2003). Therefore, to avoid the premature convergence and reduce computational effort, it is better to employ both NI and SNI max with suitable values as the stopping conditions (i.e., the algorithm terminates after satisfying either NI or SNI max ). So, for solving difficult global optimization problems, the performance of HS-based methods could be improved using larger NI or SNI max but at the expense of significant computational effort. Note that the computational effort of the HS stochastic methods increases with NI and SNI max (see Table 5). In the tests performed on the reactive problems, numerical effort of HSC, GHS and IHS is almost the same when NI is used as the stopping criterion. However, NFE required by IHS is generally less than that of both HSC and GHS if SNI max is used alone as convergence condition. Finally, the performance of HS-based method has been compared with results reported for other stochastic optimization methods (Bonilla-Petriciolet et al., 2011). In particular, Bonilla-Petriciolet et al. (2011) studied the performance of Genetic Algorithms (GA), Simulated Annealing (SA) and Differential Evolution with tabu list (DETL) for solving the constrained GA IHS GSR, % Iter of GA or NI/HMS of IHS Figure 6. GSR of Genetic Algorithm (GA) and Improved Harmony Search (IHS) for the constrained minimization of Gibbs free energy of reactive systems.

25 Bonilla-Petriciolet et al.: Gibbs Energy Minimization in Reactive Systems 21 minimization of Gibbs free energy in reactive systems. This comparison indicates that the performance of HS-based methods is better than that obtained for GA, and is competitive with respect to DETL and SA. For example, Figure 6 shows the GSR of GA and IHS in the Gibbs free energy minimization of all reactive systems used in this study. It is clear that IHS may offer a better performance than that obtained using GA. Overall, compared to other stochastic methods, HS imposes fewer mathematical requirements and its implementation is straightforward. Based on these facts, HS is among the best algorithms and can be considered an alternative method for solving reactive phase equilibrium problems in chemical reaction engineering. In particular, the variants of HS are more attractive and suitable for performing phase equilibrium calculations in reactive systems. These variants enhance the performance of classical HS for solving challenging global optimization problems. 5. CONCLUSIONS In this study, we have compared the performance of classical harmony search and some of its variants for the constrained minimization of Gibbs free energy in systems subject to chemical reactions. Our results indicate that HS-based optimization algorithms are capable of handling the difficult characteristics of this constrained global optimization problem. In particular, the Global-Best Harmony Search and Improved Harmony Search are better than classical Harmony Search in terms of reliability for the constrained minimization of Gibbs free energy in reactive systems. Therefore, our results indicate that these stochastic methods are suitable and alternative global optimization strategies for phase equilibrium calculations in reactive systems. REFERENCES Blum C., Roli A., Metaheuristics in combinatorial optimization: overview and conceptual comparison, ACM Computing Surveys, 2003, 35, 3, Bonilla-Petriciolet A., Bravo-Sanchez U.I., Castillo-Borja F., Frausto-Hernandez S., Segovia-Hernandez J.G., Gibbs energy minimization using simulated annealing for two-phase equilibrium calculations in reactive systems, Chemical Biochemical Engineering Quarterly, 2008, 22, 3, Bonilla-Petriciolet A., Rangaiah G.P., Segovia-Hernandez J.G., Jaime Leal J.E., Phase stability and equilibrium calculations in reactive systems using differential evolution and tabu search, in: G.P. Rangaiah (Ed.), Stochastic Global Optimization: Techniques and Applications in Chemical Engineering, World Scientific, Singapore, 2010a, pp Published by De Gruyter, 2012

26 22 International Journal of Chemical Reactor Engineering Vol. 10 [2012], Note S3 Bonilla-Petriciolet A., Segovia-Hernández J.G., A comparative study of particle swarm optimization and its variants for phase stability and equilibrium calculations in multicomponent reactive and non-reactive systems, Fluid Phase Equilibria, 2010b, 289, 2, Bonilla-Petriciolet A., Rangaiah G.P., Segovia-Hernández J.G., Constrained and unconstrained Gibbs free energy minimization in reactive systems using genetic algorithm and differential evolution with tabu list, Fluid Phase Equilibria, 2011, 300, 1-2, Burgos-Solorzano G.I., Brennecke J.F., Stadtherr M.A., Validated computing approach for high-pressure chemical and multiphase equilibrium Fluid Phase Equilibria, 2004, 219, 2, Fesanghary M., Damangir E., Soleimani I., Design optimization of shell and tube heat exchangers using global sensitivity analysis and harmony search algorithm, Applied Thermal Engineering, 2009, 29, 5-6, Geem Z.W., Kim J.H., Loganathan G.V., A new heuristic optimization algorithm: harmony search, Simulation, 2001, 76, 2, Geem Z.W., Music-inspired harmony search algorithm theory and applications, 2009, First Ed. Springer, United Sates. Henderson N., de Oliveira J.R., Amaral Souto H.P., Pitanga R., Modeling and analysis of the isothermal flash problem and its calculation with the simulated annealing algorithm, Industrial Engineering Chemistry Research, 2001, 40, 25, Jalali F., Seader J.D., Homotopy continuation method in multi-phase multireaction equilibrium systems, Computers Chemical Engineering, 1999, 23, 9, Lee Y.P., Rangaiah G.P., Luus R., Phase and chemical equilibrium calculations by direct search optimization, Computers Chemical Engineering, 1999, 23, 9, Lee K.S., Geem Z.W., A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice, Computer Methods in Applied Mechanics and Engineering, 2005, 194, 36-38, Mahdavi M., Fesanghary M., Damangir E., An improved harmony search algorithm for solving optimization problems, Applied Mathematics and Computation, 2007, 188, 2, McDonald C.M., Floudas C.A., GLOPEQ: A new computational tool for the phase and chemical equilibrium, Computers Chemical Engineering, 1997, 21, 1, Nichita D.V., Gomez S., Luna E., Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method, Computers Chemical Engineering, 2002, 26, 12,