Nonlinear Parameter Estimation of e-nrtl Model For Quaternary Ammonium Ionic Liquids Using Cuckoo Search

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1 Instituto Tecnologico de Aguascalientes From the SelectedWorks of Adrian Bonilla-Petriciolet 215 Nonlinear Parameter Estimation of e-nrtl Model For Quaternary Ammonium Ionic Liquids Using Cuckoo Search Jose Enrique Jaime-Leal Adrian Bonilla-Petriciolet Vaibhav Bhargava, Indian Institute of Technology - Kharagpur Seif E.K. Fateen, Cairo University Available at:

2 chemical engineering research and design 9 3 ( ) Contents lists available at ScienceDirect Chemical Engineering Research and Design journal h om epage: Short communication Nonlinear parameter estimation of e-nrtl model for quaternary ammonium ionic liquids using Cuckoo Search J.E. Jaime-Leal a, A. Bonilla-Petriciolet a,, V. Bhargava b, S.E.K. Fateen c a Instituto Tecnológico de Aguascalientes, Aguascalientes, Mexico b Indian Institute of Technology, Kharagpur, India c Cairo University, Giza, Egypt a b s t r a c t This study introduces the bio-inspired computation method namely Cuckoo Search (CS) as a parameter estimation method for modeling the mean activity coefficients of quaternary ammonium aqueous ionic liquids using the e- NRTL model. CS has not been used before to address this particular parameter estimation problem. Our calculations showed that the CS method was robust to perform the data modeling of this thermodynamic property of ionic liquids and that it can offer a global success rate of 9% for solving this challenging thermodynamic problem. CS offers a better performance than those obtained using other stochastic optimization methods such as simulated annealing, differential evolution, genetic algorithm or particle swarm optimization. This study highlights the capabilities of CS for facing challenging global optimization problems involved in the thermodynamic modeling of ionic liquids. We also show that the complexity of parameter estimation problems of ionic liquids appears to be determined by the type of cation and anion involved. Specially, the problems that involve ionic liquids containing [NH + 4 ] and alkylsulfonates ions are more challenging. 214 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Ionic liquids; Cuckoo Search; Parameter estimation; Global optimization; Quaternary ammonium salts; e-nrtl model 1. Introduction Ionic liquids (ILs) are salts that consist of an organic cation and an inorganic or organic anion. They form stable liquids with melting points less than 15 C (Ghatee et al., 213). Room temperature ILs can be synthetized from a wide variety of cation and anion families. The properties of both ions determine the thermodynamic behavior of the ILs. Thus, it is feasible to synthetize ILs with specific physico-chemical properties for a desired application (Lazzus, 212). In particular, the quaternary ammonium-based salts are considered a class of ILs that has a wide spectrum of potential applications in various chemical processes such as bioindustries, petrochemical, electrochemistry, drug processing and environmental engineering (Ma and Hong, 212; Melo et al., 213). Therefore, the accurate modeling of the thermodynamic properties of these ILs is of fundamental (Haghtalab and Mazloumi, 29; Vega et al., 21). In particular, the knowledge of activity coefficients of ILs is required for the analysis and design of separation processes employing phase equilibrium diagrams. The development of modeling tools for reliably representing and predicting the thermodynamic properties of ILs has been recognized as a necessary step to advance on the research of these electrolytes (Vega et al., Corresponding author. Tel.: x127. address: petriciolet@hotmail.com (A. Bonilla-Petriciolet). Available online 2 June / 214 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

3 chemical engineering research and design 9 3 ( ) ). However, the thermodynamic behavior of ILs is highly non-ideal (Vega et al., 21) and, if an improper numerical framework is used, failures in the modeling of the physicochemical properties of these electrolytes are commonly encountered (Belveze et al., 24; Bonilla-Petriciolet et al., 213). Recently, Vega et al. (21) have reviewed the approaches used to model the phase behavior of ILs. Overall, local composition models and equations of state (EoS) are widely employed to model thermodynamic properties of ILs in process simulators because they provide robust and straightforward calculations for the description of the ILs phase behavior (Vega et al., 21). Particularly, excess Gibbs free energy models (e.g., Wilson, NRTL, UNIQUAC, UNIFAC) and their modified forms are popular for modeling the activity coefficients of electrolyte solutions. These models are commonly used for process design because they provide a good agreement with experimental data and may offer some advantages such as their applicability to model multicomponent systems using the parameters calculated from the nonlinear regression of experimental data of binary systems (Haghtalab and Mazloumi, 29). However, the parameter estimation problem for modeling the mean activity coefficients of quaternary ammonium aqueous electrolytes using excess Gibbs free energy models is a very challenging global optimization problem (Belveze et al., 24; Jaime-Leal and Bonilla-Petriciolet, 28; Bonilla- Petriciolet et al., 213), because these systems may show complex interactions between ion ion, ion molecule and molecule molecule in the electrolyte solution causing a highly non-linear behavior of the thermodynamic model. In fact, previous studies have shown that the nonlinear parameter estimation problem of the mean activity coefficients of ILs using local composition models may imply non-convex objective functions with multiple solutions (i.e., local optimums and saddle points) (Belveze et al., 24; Jaime-Leal and Bonilla- Petriciolet, 28; Bonilla-Petriciolet et al., 213). These studies have highlighted the issue of the proper selection of the numerical method used for data fitting of ILs thermodynamic properties and the necessity of developing more reliable global optimization techniques for nonlinear parameter estimation. Under this scenario, stochastic global optimization methods are promising and offer a versatile performance for addressing this thermodynamic global optimization problem (Behzadi et al., ; Vatani et al., 212; Bonilla-Petriciolet et al., 213). In particular, bio-inspired stochastic methods have become popular in engineering optimization due to their capability for solving challenging optimization problems (Zang et al., 21). In this study, we propose the application of the Cuckoo Search optimization method (Yang and Deb, 29) for data modeling of the mean activity coefficients of quaternary ammonium aqueous electrolytes using e-nrtl. Cuckoo Search is a bio-inspired computation method, which is based on the brood parasitism of cuckoo species (Yang and Deb, 29). This novel metaheuristic has emerged as a robust numerical tool for solving challenging global optimization problems including those related to phase equilibrium calculations (Bhargava et al., 213; Yang and Deb, 214). However, this stochastic method has not been applied in the modeling of ILs thermodynamic properties. Our results showed that Cuckoo Search offers an outstanding performance for modeling this thermodynamic property of the quaternary ammonium aqueous solutions and it is better than other metaheuristics reported in the literature. This bio-inspired method can be considered as alternative optimization tool for the reliable nonlinear regression of this class of experimental thermodynamic data. 2. Methodology 2.1. Database of mean activity coefficients of quaternary ammonium ILs The experimental data of mean activity coefficients of 35 quaternary ammonium ILs in aqueous solution at C have been used to fit the adjustable parameters of e-nrtl model using Cuckoo Search. These ILs include different cations (e.g., [NH 4 + ], [(CH 3 ) 4 N + ], [(C 2 H 5 ) 4 N + ], [(C 3 H 7 ) 4 N + ]) and anions (e.g., [Cl ], [CH 3 SO 3 ], [NO 3 ]). Table 1 provides a general description of ILs and the experimental data used in this study. The pure salts of these ILs have melting points higher than C. Previous studies have reported that the parameter estimation for modeling the activity coefficients of these ILs using e-nrtl model is a challenging global optimization problem (Belveze et al., 24; Jaime-Leal and Bonilla-Petriciolet, 28; Bonilla- Petriciolet et al., 213) Modeling of the mean activity coefficient of ILs using e-nrtl The mean activity coefficient is an important thermodynamic property of ILs that reflects the non-ideality of the electrolyte solution and it can be used for calculating other relevant properties (e.g., osmotic coefficients). In this study, it is assumed that an aqueous electrolyte consists of water, anions and cations (i.e., there is a complete electrolyte dissociation). Then, the mean activity coefficient ± of a neutral electrolyte M v+ X v is defined as (Lin et al., 1993) (v + + v ) ln ± = v + ln + v ln (1) where v = v + + v are the number of moles of cation (c) and anion (a) obtained from the dissociation of one mole of the electrolyte: M v+ X v v + M Z+ + v X Z. In traditional excess Gibbs energy models, the ion activity coefficient ( i ) is the sum of both long- and short-range interaction effects between ions and solvent molecules (Aznar and Telles, 21), which can be represented by ln i = ln sr i + ln lr i (2) where the long-range contribution accounts for the electrostatic interactions between ions and the short-range contribution includes interactions between all species. For the case of ILs, the short-range contribution for the mean activity coefficient can be expressed by the e-nrtl model (Chen et al., 1982), which is a well-known activity coefficient model used for studying the thermodynamic behavior of electrolytes in aqueous solutions over a wide concentration range. This model is considered as one of the simplest local composition models that can use few adjustable parameters for predicting the ILs thermodynamic behavior. In particular,

4 466 chemical engineering research and design 9 3 ( ) Table 1 Description of the parameter estimation problems of mean activity coefficients of quaternary ammonium-based ILs using e-nrtl model. Ionic liquid Experimental data Global optimum parameters of e-nrtl model a Cation Anion m max ndat cas sca F obj [NH + 4 ] [I ] E 4 [C 2 H 5 SO 3 ] E 4 [CH 3 SO 3 ] E 5 [(CH 3 ) 4 N + ] [NO 3 ] E 2 [CH 3 SO 3 ] E 3 [C 2 H 5 SO 3 ] E 4 [Cl ] E 2 [I ] E 4 [Br ] E 3 [(C 2 H 5 ) 4 N + ] [NO 3 ] E 2 [CH 3 SO 3 ] E 4 [C 2 H 5 SO 3 ] E 3 [I ] E 3 [Cl ] E 2 [Br ] E 2 [(C 3 H 7 ) 4 N + ] [I ] E 4 [Cl ] E 1 [Br ] E 1 [(CH 3 ) 3 (C 2 H 4 OH)N + ] [Cl ] E 3 [Br ] E 2 [(CH 3 ) 2 (C 2 H 4 OH)(C 6 H 5 )N + ] [Br ] E 2 [Cl ] E 3 [(CH 3 ) 3 (C 6 H 5 )N + ] [Br ] E 2 [Cl ] E 3 [(C 2 H 4 OH) 4 N + ] [Br ] E 2 [(C 4 H 9 ) 4 N + ] [CH 3 SO 3 ] E 2 [C 2 H 5 SO 3 ] E 2 [Cl ] E 2 [Br ] E 2 [(CH 3 ) 3 N + -CH 2 -CH 2 -(CH 3 ) 3 N + ] [Cl ] E 2 [I ] E 2 [SO 2 4 ] E 1 [SO 3 -C 6 H 4 -SO 3 ] [SO 3 -C 6 H 4 -CH 2 -CH 2 -C 6 H 4 -SO 3 ] E 2 [NH 4 + ] [B 1 H 1 2 ] E 4 a Global optimum solutions taken from Belveze et al. (24) and Bonilla-Petriciolet et al. (213). the expressions of e-nrtl model for the ion activity coefficient contribution due to short-range interactions are given by ln ln e-nrtl c = e-nrtl a = [ ] cas xs 2G cas (x a G cas + x c G cas + x s ) 2 + scaz c x s G sca scaz a x a x s G sca x a + x s G sca (x s G sca + x c ) 2 cas G cas sca Z c [ ] cas xs 2G cas (x a G cas + x c G cas + x s ) 2 + scaz a x s G sca scaz c x c x s G sca x c + x s G sca (x s G sca + x a ) 2 cas G cas sca Z a is employed for the calculation of the long-range contributions of as a function of the ionic strength and is given by (3) (4) where x s, x a and x c is the mole fraction of solvent, anion and cation, respectively. e-nrtl model has two adjustable parameters cas and sca that represent the interaction energies between the solvent and the ions, which are obtained from the fitting of ILs experimental thermodynamic data. Note that is a non-randomness factor and is usually fixed at a value of.2 (Chen et al., 1982). These equations are used in connection with the Pitzer Debye Hückel model to obtain the activity coefficient of an aqueous ion i. In particular, Pitzer Debye Hückel equation ( ) 1/2 [ ] 2Z 2 ln lr i i = A ϕ M s ln(1 + I 1/2 x ) + Z2 i I1/2 x 2I 3/2 x 1 + I 1/2 x and A ϕ = exp[(t )/273.15] (exp[(T )/273.15]) ln(t/273.15) (t ) [T 2 (273.15) 2 ] (273.15/T) (6) (5)

5 chemical engineering research and design 9 3 ( ) where A ϕ, Z i, and M s are the Pitzer Debye Hückel parameter, the ion charge, the closest approach parameter (=14.9) and the solvent molecular weight, respectively. The ionic strength I x is based on a mole fraction basis and defined by I x = 1 2 Z 2 i x i (7) Given a set of experimental data of IL mean activity coefficients ( ±,m ), expressed in molality scale, the nonlinear data regression used for determining the values of e-nrtl model parameters is based on the global minimization of the objective function ndat F obj = [ln ( exp ±,m ) i ln (±,m calc ) i ]2 (8) i=1 where ln( ±,m ) = 1 v [v + ln( cation ) + v ln( anion )] ln(1 +.1M s mv) (9) m is the molality, ndat is the overall number of experimental points used in data regression, exp and calc correspond to the experimental data and the calculated values using the e- NRTL model, respectively. Eq. (8) is a least square formulation widely applied for data modeling of thermodynamic properties of ILs (Chen et al., 1982; Belveze et al., 24). As stated, this parameter estimation approach using e-nrtl model is an optimization problem where the objective function given by Eq. (8) is usually non-convex with the presence of several local minima and saddle points (Belveze et al., 24; Bonilla- Petriciolet et al., 213). Therefore, the application of a global optimization strategy capable of finding the global minimum of Eq. (8) is important in guaranteeing the success of the experimental data regression stage. In this study, the Cuckoo Search method is used to solve this parameter estimation problem Cuckoo Search for parameter estimation of e-nrtl model Cuckoo Search (CS) is a bio-inspired computation method based on the breeding behavior of certain species of cuckoos (Yang and Deb, 29). Some species lay their eggs in the nest of other host birds. If a host bird discovers the alien eggs, they will either throw them away or abandon the nest and build a new one elsewhere. In brief, the metaheuristic of CS for global optimization is based on three rules: (1) each cuckoo lays one egg at a time and dump its egg in a randomly chosen nest; (2) the best nest with high quality (i.e., the best objective function value) of eggs will carry over to the next generations; and (3) the number of available host nests is fixed, and a fraction P a of the n nests are replaced by new nests (with new random solutions). Each egg represents a new solution of the optimization problem and each nest can hold a single egg only. The aim is to use the new and potentially better solutions to replace the worse solutions in the nests. When generating new solutions for Cuckoo i, a Lévy flight is performed. A stochastic equation for random walk is used to represent the search of each cuckoo. The next location of a cuckoo depends on its current location and the transition probability. The step length is randomly drawn for a Lévy distribution, which has an infinite variance with an infinite mean. Some of the new solutions should be generated by a Lévy walk around the best solution obtained so far, which will speed up the local search. Some other solutions Start Generate an initialrandom population of host nests Get a cuckoo randomly by Lévy flights and evaluate its fitness Fi Select a nest j randomly Fi Fj Replace j by the new solution Abandon a fraction Pa of worse nests and build new ones at new locations via Lévy flights Keep the best solutions/nests Is the stopping criterion satisfied? Find the best solution (the best nest) Stop Fig. 1 Flowchart of Cuckoo Search for global optimization. should be generated by far field randomization to avoid being trapped in a local optimum. For illustration, the pseudo code of the CS is given in Fig. 1. As stated in the Introduction, CS has been applied for solving challenging engineering optimization problems including phase stability and phase equilibrium calculations (Bhargava et al., 213; Walton et al., 213; Yang and Deb, 214). Therefore, this study reports the first application of CS for modeling thermodynamic properties of ILs. 3. Results and discussion In this section, we describe several examples for illustrating the use of CS in the calculation of the binary parameters of e-nrtl model from ILs mean activity coefficient data. In all examples, the global optimization has been performed using the next search intervals for e-nrtl model parameters: cas ( 5, 1) and sca ( 1, 5). All examples have been solved using CS with two different stopping conditions: (a) Iter max = the maximum number of iterations and (b) Sc max = the maximum number of iterations without improvement in the best value of the objective function. Based on the results of preliminary calculations, parameter P a of CS was assigned to. and a population size of 2 was used for all calculations performed in this study. All ILs parameter estimation

6 468 chemical engineering research and design 9 3 ( ) Table 2 Results of the parameter estimation of mean activity coefficients of tetramethylammonium salts in water at C using e-nrtl model and Cuckoo Search. Ionic liquid SR(%) for CS a SR(%) for CS + SQP a Iter max Sc max Iter max Sc max Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U [(CH 3 ) 4 N + ][NO 3 ] 28.9 ± ± ± ± [(CH 3 ) 4 N + ][CH 3 SO 3 ] 6.1 ± ±. 4.3 ± ±. [(CH 3 ) 4 N + ][C 2 H 5 SO 3 ] 8.2 ± ± ± ± [(CH 3 ) 4 N + ][Cl ] 3.9 ± ± ± ± [(CH 3 ) 4 N + ][I ] 15. ± ± ± ± [(CH 3 ) 4 N + ][Br ] 3.8 ± ± ± ± a Mean = mean SR, = standard deviation of SR, SR L = minimum SR and SR U = maximum SR. problems and CS algorithm were coded in MATLAB 211a. In addition, we have considered the application of SQP (Sequential Quadratic Programming) algorithm of MATLAB, with default parameter settings of function fmincon and tolerances values of 1 8, as a local intensification strategy to improve the performance of CS. An overview of SQP is provided by Fletcher (198). Success rate (SR) of CS has been calculated for each IL parameter estimation problem using numerical trials with random initial values of e-nrtl model parameters. For the calculation of SR, we have considered the following criterion: F * obj F obj,stoc ε where F * obj is the known solution of the parameter estimation problem (i.e., the global optimum), F obj,stoc is the calculated value by CS for the given stopping condition and ε = 1 6 is the precision for the solution found. For selected ILs, we have provided a detailed summary of the CS performance in parameter estimation. Example 1. Tetramethylammonium ILs In this example, we have considered 6 tetramethylammonium ILs for testing the performance of CS in e-nrtl parameter estimation. These ILs contain halides, nitrates and alkylsulfonates as anions. Results of CS for solving the parameter estimation problems are reported in Table 2 using both stopping conditions and the local optimization method. In particular, these calculations were performed using different values of the stopping conditions, i.e.: (a) Iter max : 1 15 and (b) Sc max : 1 5, respectively. Overall, the SR of CS for performing the e-nrtl parameter estimation ranged from to 84% and from 7 to % without and with the local optimization method using Iter max. On the other hand, the reliability of CS ranged from to 59% and from to % using CS with the local optimization strategy and Sc max. For both stopping conditions, the SR of CS using the local optimization method is higher than that obtained with the stochastic method alone. However, it is interesting to remark that CS is reliable to find the global optimum solution with a high precision, without using the local optimization SQP, if a proper number of function evaluations is used (i.e., Iter max or Sc max ). For illustration, Fig. 2 shows the convergence profile of Cuckoo Search without SQP during the parameter estimation of e-nrtl model for some ILs. It is clear that the performance of CS increased with the number of function evaluations (i.e., with increments of both Iter max and Sc max ) and may reach a high precision in the solution obtained. In particular, CS is very reliable for finding the global solution of the parameter estimation problem for ILs with halides even without using the local intensification strategy. In these electrolytes, CS may show a SR > 8% with a reduced numerical effort especially using SQP. For example, in ILs [(CH 3 ) 4 N + ][Cl ] and [(CH 3 ) 4 N + ][Br ], CS showed a SR of 8% using Iter max >, while CS + SQP showed the same SR but at Iter max >. In contrast, CS showed a worse 1.E E+ 1.E E-1 + Serie1 [( CH3) 4 N ][ C2H5SO3 ] + Serie2 [( CH3) 4 N ][ I ] 1.E-2 1.E-3 1.E-4 F * obj Fobj,stoc 1.E-2 1.E-3 1.E-5 1.E-6 1.E-7 1.E-8 1.E-4 1.E-5 1.E-9 1.E-1 1.E-11 + Serie1 [( C2H5) 4 N ][ CH 3SO3 ] 1.E-12 Serie2 + [( C2H5) 4 N ][ I ] 1.E-13 Iter max of Cuckoo Search Fig. 2 Convergence profiles of Cuckoo Search in the parameter estimation of e-nrtl model using selected ionic liquids.

7 chemical engineering research and design 9 3 ( ) Table 3 Results of the parameter estimation of mean activity coefficients of tetraethylammonium salts in water at C using e-nrtl model and Cuckoo Search. Ionic liquid SR(%) for CS a SR(%) for CS + SQP a Iter max Sc max Iter max Sc max Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U [(C 2 H 5 ) 4 N + ][NO 3 ] 43.7 ± ± ± ± [(C 2 H 5 ) 4 N + ][CH 3 SO 3 ] 14.2 ± ± ± ± [(C 2 H 5 ) 4 N + ][C 2 H 5 SO 3 ] 16.1 ± ± ± ± [(C 2 H 5 ) 4 N + ][I ] 43.4 ± ± ± ± [(C 2 H 5 ) 4 N + ][Cl ] 31.2 ± ± ± ± [(C 2 H 5 ) 4 N + ][Br ] 37.1 ± ± ± ± a Mean = mean SR; = standard deviation of SR; SR L = minimum SR and SR U = maximum SR. performance in the parameter estimation of alkylsulfonatesbased ILs. In these electrolytes, CS may fail several times to find the global optimum solution for the defined solution precision ε, see results reported in Table 2. Alkylsylfonates-based ILs appear to be the most challenging parameter estimation problems of tested tetramethylammonium ILs. Example 2. Tetraethlyammonium ILs Table 3 contains the SR of CS for data regression of the mean activity coefficients of ILs with tetraethlyammounim cations. In this example, we have also used the stopping conditions given for Example 1. CS offers the best performance to find the e-nrtl model parameters for ILs with anions [NO 3 ], [Cl ], [I ] and [Br ] with and without the use of SQP. For electrolytes [(C 2 H 5 ) 4 N + ][NO 3 ] and [(C 2 H 5 ) 4 N + ][I ], CS may find the global optimum parameters of e-nrtl model without using the local intensification strategy but using a proper numerical effort (i.e., number of function evaluations). However, the use of SQP improves significantly the SR especially at early iterations. Again, CS may fail to locate the global optimum solution of parameter estimation problems for IL with anions [CH 3 SO 3 ] and [C 2 H 5 SO 3 ] using both stopping conditions and the local optimization method. With respect to the choice of the stopping condition, both CS and CS + SQP showed higher SRs using Iter max as convergence criterion than those obtained with Sc max. In general, the local intensification stage using SQP has more impact on CS for solving the e-nrtl parameter estimation problems when low values of Iter max or Sc max are used. These results are consistent with other thermodynamic calculations using stochastic optimization methods, e.g. Bonilla-Petriciolet et al. (213) and Bhargava et al. (213). Finally, Fig. 2 reports the CS convergence profiles for selected ILs and these results confirmed that CS is a robust optimization strategy for solving e-nrtl parameter estimation problems with proper values of Iter max or Sc max. This metaheuristic is capable of finding global optimum solutions with high precision even when these parameter estimation problems may have objective functions with several local and saddle points (Belveze et al., 24; Bonilla-Petriciolet et al., 213). The performance of CS and CS + SQP in the parameter estimation of remaining ILs is reported in Table 4, while Fig. 3 shows the global SR of CS and CS + SQP for data regression of the mean activity coefficients of all ILs and using both convergence conditions. Overall, global SR of CS ranged from to 81% without SQP, while CS + SQP showed a SR up to 88% at tested stopping conditions. The impact of the local optimization strategy is more significant at low levels of both Iter max and Sc max. Figs. 4 and 5 show the numerical performance of CS as function of the cation and anion type of ILs. This analysis indicates that the complexity of parameter estimation problem is given by: (a) ILs with cation: [(C 4 H 9 ) 4 N + ] < [(C 3 H 7 ) 4 N + ] < [(C 2 H 5 ) 4 N + ] [(CH 3 ) 4 N + ] < [NH 4 + ]; and (b) ILs with anion: [I ] < [Br ] [Cl ] < [C 2 H 5 SO 3 ] < [C 2 H 5 SO 3 ], respectively. Results confirmed that ILs with [NH 4 + ] and alkylsulfonates ions involve the more challenging parameter estimation problems. In these electrolytes, CS may fail to locate the global solution of parameter estimation problem even using high values of either Iter max or Sc max. In contrast, CS is successful for reliable finding the global optimum values of cas and sca for electrolytes with halides and nitrates cations. As stated, Belveze et al. (24) have shown that the presence of multiple local minima in the objective function used for e-nrtl parameter estimation is not uncommon. Specially, [NH 4 + ]- and alkylsulfonates-based ILs may show local Global SR, % a) b) Cuckoo Search+SQP Cuckoo Search Iter max Sc max Fig. 3 Global success rate of Cuckoo Search in the parameter estimation of mean activity coefficients of tetramethylammonium salts in water at C using e-nrtl model. Stopping condition: (a) Iter max and (b) Sc max.

8 47 chemical engineering research and design 9 3 ( ) Table 4 Performance of Cuckoo Search in the parameter estimation of mean activity coefficients of quaternary ammonium-based ILs using e-nrtl model. Ionic liquid a SR (%) for CS b SR (%) for CS + SQP b Iter Scmax Iter Scmax Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U Mean ± SR L SR U [NH + 4 ] [I ] 46.6 ± ± ± ± [NH + 4 ][C 2 H 5 SO 3 ] 46.8 ± ± ± ± [NH + 4 ][CH 3 SO 3 ] 5.3 ± ± ± [(C 3 H 7 ) 4 N + ][I ] 34.2 ± ± ± ± [(C 3 H 7 ) 4 N + ][Cl ] 31.1 ± ± ± ± [(C 3 H 7 ) 4 N + ][Br ] 37.6 ± ± ± ± [(C 4 H 9 ) 4 N + ][CH 3 SO 3 ] 49.9 ± ± ± ± [(C 4 H 9 ) 4 N + ][C 2 H 5 SO 3 ] 51.6 ± ± ± ± [(C 4 H 9 ) 4 N + ][Cl ] 46.7 ± ± ± ± [(C 4 H 9 ) 4 N + ][Br ] 46.1 ± ± ± ± [M 2+ ][Cl ] ± ± ± ± [M 2+ ][I ] ± ± ± ± [M 2+ ][SO 2 4 ] 19. ± ± ± ± [M 2+ ][SO 3 -C 6 H 4 -SO 3 ] 5.8 ± ± ± ± [M 2+ ][SO - 3 -C 6 H 4 -CH 2 -CH 2 -C 6 H 4 -SO 3 ] 36.9 ± ± ± ± [NH + 4 ] 2 [B 1 H 2 1 ] 47.3 ± ± ± ± [(CH 3 ) 3 (C 2 H 4 OH)N + ][Cl ] 19. ± ± ± ± [(CH 3 ) 3 (C 2 H 4 OH)N + ][Br ] 34.3 ± ± ± ± [(CH 3 ) 2 (C 2 H 4 OH)(C 6 H 5 )N + ][Br ] 46.7 ± ± ± ± [(CH 3 ) 2 (C 2 H 4 OH)(C 6 H 5 )N + ][Cl ] 44.7 ± ± ± ± [(CH 3 ) 3 (C 6 H 5 )N + ][Br ] 46.7 ± ± ± ± [(CH 3 ) 3 (C 6 H 5 )N + ][Cl ] 42.1 ± ± ± ± [(C 2 H 4 OH) 4 N + ][Br ] 42.3 ± ± ± ± a [M 2+ ] = [(CH 3 ) 3 N + -CH 2 -CH 2 -(CH 3 ) 3 N + ]. b Mean = mean SR, = standard deviation of SR, SR L = minimum SR and SR U = maximum SR. a) a) Global SR, % b) Global SR, % b) Iter max Iter max Fig. 4 Effect of the cation type on the global success rate of Cuckoo Search for the parameter estimation of mean activity coefficients of tetramethylammonium salts in water at C using e-nrtl model. (a) Cuckoo Search and (b) Cuckoo Search + SQP. Fig. 5 Effect of the anion type on the global success rate of Cuckoo Search for the parameter estimation of mean activity coefficients of tetramethylammonium salts in water at C using e-nrtl model. (a) Cuckoo Search and (b) Cuckoo Search + SQP.

9 chemical engineering research and design 9 3 ( ) optimums and a global optimum with values of the objective function very similar (i.e., the difference between the objective function values of local and global optimum is <1 3 ). These comparable solutions are stronger attractors for the stochastic optimization method. For example, the parameter estimation problem of [NH 4 + ][C 2 H 5 SO 3 ] has 3 saddle points and one local optimum, which has a difference of <1 4 with respect to the objective function value of the global optimum. Therefore, CS converged several times to the local optimum. This appears to be the main reason for the lowest SR of CS and other stochastic optimization methods for solving these parameter estimation problems. Our numerical experience indicated that this failure is common for several stochastic optimization methods, e.g. Bonilla-Petriciolet et al. (213). In fact, the development of novel and robust diversification and intensification strategies that can overcome these pitfalls commonly encountered in real-life application problems, even for low-dimension problems, is an open topic for research using metaheuristics. CS + SQP is a reliable strategy for performing data regression of ILs mean activity coefficients using a reasonable numerical effort (i.e., low values of Iter max and Sc max ). However, our results indicate that Iter max is the best convergence criterion for solving this type of parameter estimation problems using CS. In particular, a convergence criterion of Iter max = appears to be proper for performing the data regression of electrolyte activity coefficients using e-nrtl model and CS + SQP. Note that we have solved the ILs parameter estimation problems using SQP only and our results indicated that the SR of this local method is almost zero in the most challenging problems. As expected, this local method is more efficient than CS but less reliable for solving these parameter estimation problems. Similar findings have been reported by Belveze et al. (24) using the simplex pattern search routine of MATLAB. Our results suggest that the performance of CS is mainly dependent on the ion type of the ionic liquid. It appears that the objective function used for parameter estimation of e-nrtl model is more complex for ILs with [NH 4 + ] and alkylsulfonates ions. This result could suggest a relationship between the structure and properties of the ionic liquids and the complexity of the global optimization problem to be solved for e-nrtl parameter estimation. Overall, the e-nrtl model is able to capture the non-ideal phase behavior of ILs analyzed in this study. The behavior of experimental activity coefficients of tested ILS is different and some of these compounds may show micelle formation in solutions (Belveze et al., 24). In particular, e-nrtl showed some limitations for accurately representing the experimental data in ILs that could form micelles in solution and/or at high ILs concentrations due to the incomplete salt dissociation (Belveze et al., 24). For these ILs, the value of the objective function could be very sensitive to the parameters of thermodynamic model and it may vary in several orders of magnitude. Therefore, this feature can be identified as a factor that makes the e-nrtl parameter estimation as a challenging global optimization problem. Further studies are required to determine if this relationship prevails for other ILs and other thermodynamic properties. For illustration, we have compared the performance of CS with those reported for other stochastic optimization methods (Bonilla-Petriciolet et al., 213). Fig. 6 shows the global SR of different stochastic optimization methods (e.g., Harmony Search, Simulated Annealing, Differential Evolution, Particle Swarm Optimization) using the same set of ILs tested in this study and e-nrtl model. This comparison has been performed using the same numerical effort (i.e., number of function Fig. 6 Performance of several stochastic optimization methods for the parameter estimation of mean activity coefficients of tetramethylammonium salts in water at C using e-nrtl model. evaluations) for all metaheuristics. Other stochastic methods may reach a maximum success rate of 77%, while CS showed the highest reliability (SR 86%) for solving these parameter estimation problems. It is clear that CS + SQP outperformed other metaheuristics in the data regression of mean activity coefficients of ILs when the number of function evaluations (NFE) is 2. In particular, SA offered the best performance at early function evaluations (i.e., <). Our numerical experience indicated that population-based stochastic methods may require slightly more NFE for identifying the promising area of the global optimum in comparison to the point-topoint methods. This could be the reason that SA outperformed CS and other population-based methods for ILs parameter estimation at low values of NFE. In summary, we can conclude that CS is a robust optimization method for determining the e-nrtl model parameters for predicting and representing the mean activity coefficients of ILs. 4. Conclusions This study provides an overview of the capabilities of Cuckoo Search for modeling the mean activity coefficients of ionic liquids. This paper introduces the first application of Cuckoo Search to solve the global optimization problem of modeling the activity coefficients of a set of quaternary ammonium salts using the e-nrtl model. Results showed that, depending on the type of quaternary ammonium IL, the complexity of the global optimization problem may significantly increase. Specially, the modeling of activity coefficients of ILs with [NH 4 + ] and alkylsulfonates ions appears to be the most challenging parameter estimation problems. On the other hand, Cuckoo Search outperformed other stochastic optimization methods used for e-nrtl parameter estimation and can be considered as the best option for solving this challenging global optimization problem. However, this metaheuristic may fail to solve the parameter estimation problem if the local optimum has an objective function with comparable value to the global optimum especially using a low numerical effort. Further studies should be focused on the performance improvement of this bio-inspired optimization method using low values of convergence criterion and for optimization problems with local and global optimums with comparable objective function values. This numerical aspect is relevant to enhance the application of this and other metaheuristics for solving global optimization problems in thermodynamics and related fields.

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