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An Applicability of Cuckoo Search Algorithm for the Prediction of Multicomponent Liquid-Liquid Equilibria for Imidazolium and Phosphonium based Ionic Liquids Anand Bharti, Prerna Prerna, and Tamal Banerjee Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03449 • Publication Date (Web): 19 Nov 2015 Downloaded from http://pubs.acs.org on November 23, 2015

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Industrial & Engineering Chemistry Research

An Applicability of Cuckoo Search Algorithm for the Prediction of Multicomponent Liquid-Liquid Equilibria for Imidazolium and Phosphonium based Ionic Liquids Anand Bhartia, Prerna b, Tamal Banerjeea,*

a

Department of Chemical Engineering, Indian Institute of Technology Guwahati,Assam - 781039, India

b

*

Department of Chemical Engineering, Birla Institute of Technology Mesra, Jharkhand -835215, India

Author to whom all correspondence should be addressed.

E-mail: [email protected] Tel: +91.361.2582266 Fax: +91.361.2582291

ACS Paragon Plus Environment


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ABSTRACT Cuckoo Search (CS), which is nature-inspired stochastic optimization method, has been successfully applied for estimating binary interaction parameters of UNIQUAC and NRTL activity coefficient models for liquid-liquid ternary systems involving twelve Imidazolium and Phosphonium Ionic Liquids (IL).Thirty nine ternary systems, comprising 371 experimental tie-lines were correlated by the UNIQUAC and NRTL model. The results, expressed by deviations between experimental and calculated mole fractions, are very satisfactory, with global deviation value 0.0053 (UNIQUAC) and 0.0072 (NRTL) which is 63% and 45% better than literature reported values. Three quaternary systems and one Quinary system were taken from literature to compare the capability of CS algorithm with GA and PSO algorithms. The global % RMSD value obtained with CS algorithm was ~0.140.85% as compared to ~1.0-2.0% with GA and PSO algorithms. This shows a higher efficiency for the CS algorithm in solving global optimization problems involved in the thermodynamic modeling of multicomponent systems.


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1. Introduction Liquid-liquid extraction is an important separation technology, with a wide range of applications in chemical, petrochemical and pharmaceutical industries. Liquid–liquid Equilibrium (LLE) data of multi-component systems are essential for proper understanding of the extraction process and for the designing and optimization of separation processes. Excess Gibbs free energy models, such as NRTL1and UNIversal QUAsi-Chemical (UNIQUAC)2 models are commonly used to predict the LLE as they provide good agreement with experimental data.3-6 For LLE prediction, each of these models require binary interaction parameters. These parameters are generally estimated from the known experimental LLE data by the optimization of an objective function. Mathematically the aim of optimization is to find the set of inputs that either maximizes or minimizes the output of the objective function. The objective function is highly non-convex having multiple local optima. For Liquid-Liquid phase equilibria, finding the global optimum (reliable interaction parameters) thus becomes a necessary requirement for the correct LLE prediction. Nature inspired metaheuristic algorithms are becoming increasingly popular to solve global optimization problems. They work remarkably efficiently and have many advantages over traditional, deterministic methods and thus have been applied in almost all areas of science, engineering and industry.7 Nature inspired metaheuristic algorithms are broadly classified into Evolutionary Algorithms (Genetic Algorithm (GA), Differential Evolution (DE)), Physical Algorithms (Simulated Annealing (SA), Harmonic Search (HS)), Swarm Intelligence (Particle Swarm optimization (PSO) , Ant Colony Optimization (ACO)), Bioinspired Algorithms (Artificial Immune System) and others (Cuckoo Search (CS) Algorithm, Bat Algorithm).8Genetic Algorithm remains one of the most widely used optimization algorithms in modern nonlinear optimization. It is based on natural selection process that mimics biological evolution. Simulated Annealing is a trajectory based search algorithm



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which mimics the annealing process in material processing. Particle Swarm Optimization (PSO) is inspired by the swarm intelligence of fish and birds.9 The application of Nature inspired metaheuristic algorithms for solving phase equilibrium calculations (PEC), phase stability (PS) problems and parameter estimation (PE) has grown considerably in recent years. Srinivas and Rangaiah10 have applied differential evolution with tabu list (DETL) for phase equilibrium (VLE, LLE, VLLE) and parameter estimation problems. It was found that the performance of DETL was better as compared to DE and tabu list (TS). Further Bonilla-Petriciolet et al11 have evaluated the performance of SA, GA, DE, DETL and PSO for modeling several binary VLE data using local composition models and found that DE and DETL perform better than other algorithms in terms of success rate for parameter estimation. Bonilla-Petricioleta and Hernández 12 have also tested different variants of PSO for performing phase stability and equilibrium calculations in both reactive and non-reactive systems. Their results showed that classical PSO with constant cognitive and social parameters was a reliable method and offered the best performance. Zhang et al.13 have also evaluated the performance of three stochastic algorithms, unified bare-bones particle swarm optimization (UBBPSO), integrated differential evolution (IDE) and IDE without tabu list and radius (IDE_N), and compared their performance for phase equilibrium and phase stability problems. Overall, IDE was found to perform better in terms of global success rate and faster convergence rate for the phase equilibrium problems. H. Zhang et al.14 have further applied IDE algorithm for parameter estimation in VLE modeling problems. Compared to other stochastic algorithms such as SA, PSO, DE, and DETL, IDE was found to be reliable (higher global success rate) for solving these modeling problems. Fateen et al.

15

have also evaluated the performance of three global optimization algorithms,

Covariant Matrix Adaptation-Evolution Strategy (CMA-ES), Shuffled Complex Evolution (SCE) and Firefly Algorithm (FA), for phase and chemical equilibrium calculations. The


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phase equilibrium problems include both multi-component systems with and without chemical reactions. FA was found to be the most reliable among the three techniques, whereas CMA-ES can find the global minimum reliably and accurately even with a smaller number of iterations. Fernández-Vargas et al.16 have applied an improved version of Ant colony optimization (ACO) algorithm, named as ACO with feasible region selection (ACOFRS), to perform thermodynamic calculations i.e. parameter estimation, phase equilibrium calculations and phase stability analysis for liquid–liquid and vapor–liquid equilibrium multicomponent systems. It was concluded that ACOFRS is an alternative method for performing global optimization in phase equilibrium calculations of multicomponent systems and it outperformed other stochastic optimization methods such as PSO, DE and GA for solving VLE parameter estimation problems. Apart from the phase equilibrium calculations (PEC) and phase stability (PS) problems, stochastic global optimization methods has also been applied for estimation of binary interaction parameters in multicomponent LLE systems. Singh et al.17 utilized GA to estimate the binary interaction parameters for NRTL and UNIQUAC models in multicomponent LLE systems and demonstrated that their performance was better than inside variance estimation method (IVEM) and the techniques applied in ASPEN and DECHEMA. Sahoo et al.18 calculated the interaction parameters for NRTL model in ternary, quaternary and quinary LLE systems based on GA and showed that the results obtained using GA were better than other techniques in literature. Ferrari et al.19 applied SA and PSO algorithms for parameter estimation of NRTL and UNIQUAC models for binary and multicomponent LLE systems and showed that both algorithms were capable of modeling liquid-liquid equilibrium data. Merzougui et al.20 used a hybrid algorithm i.e. a combination of GA and LevenbergMarquardt (LM) method, for parameter estimation with NRTL and UNIQUAC models for six LLE systems and found good correlation with experimental data. Vatani et al.6 also



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performed the liquid–liquid equilibria calculation for 20 different IL based ternary LLE systems by NRTL and two-suffix Margules models with binary interaction parameters calculated using Genetic Algorithm (GA).A harmony search algorithm was used by Merzougui et al.21 to calculate the interaction parameters of NRTL model for twenty ternary liquid–liquid systems. Kabouche et al.22 have used SA, GA, Nelder-Mead Simplex (NMS), SA-NMS (hybrid) and GA-NMS (hybrid) to estimate interactions parameters of NRTL and UNIQUAC models for LLE systems. The hybrid algorithm namely GA-NMS showed the best performance in terms of RMSD with minimum number of iterations among the others. Bonilla-Petriciolet et al.23 have recently analyzed the capabilities of seven stochastic global optimization methods, SA, GA, DE, PSO, HS, DETL and bare bones PSO (BBPSO), to model mean activity coefficients in aqueous solutions of quaternary ammonium salts at 25 oC using the electrolyte NRTL model. The results indicated that SA, DETL and BBPSO offer better performance for solving parameter estimation problems involved in the modeling of thermodynamic properties of ionic liquids (ILs). Besides the well-known methods, the investigations on nature inspired optimization algorithms are still currently under development. Cuckoo Search (CS) is one of the latest nature-inspired metaheuristic algorithms developed by Yang and Deb24. It is a populationbased method which mimics the breeding behaviour such as brood parasitism of certain cuckoo species. This algorithm is enhanced by the so-called Levy flights. Recent studies have shown that CS is potentially far more efficient than other algorithms in many applications. It has been applied in many areas25 which includes applied thermodynamic calculations. Bhargava et al.26 have applied CS algorithm for solving phase stability, phase equilibrium and reactive phase equilibrium problems. They found that CS offers a reliable performance for solving these thermodynamic calculations and is better than other meta-heuristics for phase equilibrium modeling. Fateen and Petriciolet27 have compared the reliability and


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efficiency of eight promising nature inspired metaheuristic algorithms for the solution of nine difficult phase stability and phase equilibrium problems. These algorithms are the cuckoo search (CS), intelligent firefly (IFA), bat (BA), artificial bee colony (ABC), MAKHA, a hybrid between monkey algorithm and krill herd algorithm, covariance matrix adaptation evolution strategy (CMAES), magnetic charged system search (MCSS), and bare bones particle swarm optimization (BBPSO). The results clearly showed that CS is the most reliable of all tested optimization methods as it successfully solved all thermodynamic problems tested in the study. Fateen and Petriciolet28 have also applied Gradient-Based Cuckoo Search (GBCS) algorithm for solving several challenging phase stability problems and analyzed its performance at different numerical effort levels. The GBCS was found to perform better than the original CS algorithm. In comparison with other stochastic optimization, GBCS proved to be the most reliable without any reduction in efficiency. Till date the computation using CS algorithm is scarce for multicomponent phase equilibria problems. In a recent work by Jaime-Leal et al.29, CS algorithm was used to predict the binary phase equilibrium data for aqueous quaternary ammonium IL mixtures. In their work the mean activity coefficients of quaternary ammonium IL’s was predicted using the eNRTL model. The results obtained were very encouraging with a global success rate of ~86% with CS as compared to ~ 77% with other stochastic methods. However computation with ternary, quaternary or quinary systems are not available in the literature. Keeping this limitation in mind, we have attempted to predict the multicomponent LLE data for both IL and organic solvent systems. In our study, binary interaction parameters of UNIQUAC and NRTL models were estimated using CS algorithm for 39 ternary systems which includes 32 IL based systems and 7 organic solvent based systems. The results (root mean square deviation) thus obtained were compared with those reported in the literature. In the concluding section, three quaternary



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systems and one quinary system were selected to compare the performance of CS with GA and PSO algorithms.

2. Theory and Calculation 2.1. Cuckoo Search (CS) Cuckoo search (CS) is a natured-inspired metaheuristic search algorithm, based on the reproduction behaviour of cuckoos, which has been recently developed by Yang and Deb24,30. Specific egg laying and breeding behaviour of some cuckoos species is the basis of this optimization algorithm. The cuckoo is the brood parasite, which never builds their own nests. They lay their eggs in the nest of other species, leaving host species to care for its eggs. If these eggs are discovered by the host bird, it will either throw out the cuckoo egg or abandon the nest to start afresh. Some species of cuckoos have learnt to mimic the colour and pattern of their own eggs so as to match that of their hosts. This reduces the probability of the eggs being abandoned and therefore, increases their re-productivity. To implement these concepts, the CS employ three idealised rules where: (a) eggs are laid one at a time and then dumped randomly in a host nest;(b) the nest having better eggs or solutions are carried over to next generation; and (c) the host bird usually find an alien egg within a probability of unity. However if an alien egg is found, the host bird can either throw the egg away or abandon the nest, and build a completely new nest. The above rules are valid for single objective optimization problem. For multiobjective optimization problems with K different objectives, the first and last rule need modification. Here each cuckoo lays K eggs at a time (Rule a) and thus a new nest with K eggs are built (Rule c). From implementation point of view, each egg in a nest represents a solution, and a cuckoo egg represents a new solution. The aim is to use the potentially better


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solution (cuckoo egg) to replace the not-so-good solution (egg) in the nests. The steps of the algorithm are as follows: Step-1: Initialization of CS algorithm parameters The following parameters are defined: number of nests (n), discovery probability (pa), lower and upper bounds and maximum number of generations/iterations (itermax). Step-2: Generation of initial population of nests of host birds The initial population of the nests are generated randomly between the lower bound and upper bound. Step-3: Generation of new solutions/eggs by Lévy flights In this step, all the host eggs except the best one are replaced by new cuckoo eggs if objective function corresponding to each new cuckoo egg is better than the corresponding existing host egg objective function. Anew solution is generated by Lévy flight as below

(t +1) = y (t ) + α .S.( y (t ) − y (t ) ).(rand ) i i i best

y

(1)

where α is the step size parameter, rand is a random number from a standard normal t distribution, y (t ) is the ith nest current position, ybest is the current best nest and S is a i

random walk based on the Lévy flights.In CS algorithm, Lévy flights are used to explore the unknown, large-scale search space as it is more efficient than the Brownian random walks. The Lévy flight essentially provides a random walk whose random step length is drawn from a Lévy distribution(equation 2).

Lévy u=t −1−β ,(1 < β ≤ 2)

(2)

This has an infinite variance with an infinite mean.In this work, a Lévy flight has been performed for generating new solutionaccording to Mantegna algorithm31. In Mantegna’s algorithm, the step length Sis calculated by



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s= u 1 v β

(3)

where β is considered to be 1.5 andu and v are drawn from normal distributions. It takes the form:

u N (0,σ u2 ),

v N (0,σ v2 ),

(4)

where

σu

Here

  Γ(1 + β )sin(πβ ) 2 =  ( −1) β  2  Γ[(1 + β ) / 2]β 2

     

1

β ,

σv =1

(5)

Γ is the standard Gamma function.

Step-4:Discovery of alien eggs In this step, the alien eggs discovery is performed for each egg by generating a random number rand ∈ [0,1] and comparing it with discovery probability pa such as:

(t +1) (t ) yi ← yi

if rand ≤ pa,

(t +1) (t ) yi ← yi + ζ ( ytj − yt ) if k

rand>pa

(6)

where y tj and y kt are two different solutions selected randomly by random permutation and ζ is a random number drawn from a uniform distribution.

Step-5:Termination criterion The maximum number of iterations or tolerance criterionmay be used as termination criterion of the algorithm. In this work, maximum number of iterations has been used as termination criterion.The pseudo-code and the flow diagram of the CS algorithm are shown in Figure 1 and Figure 2 respectively.


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2.2.UNIQUAC Model and NRTL Model In UNIQUAC model, the activity coefficient, γi, of component ‘i’ in the multicomponent system is given by:

  θ jτ ij   Φi  z  θi  c c Φi c  ln γ i = ln  ∑ x j l j + q 1 − ln ∑ θ jτ ji − ∑ c  + qi ln   + li − j j =1 j =1 ∑ θ τ  x 2 Φ x  i   i i =1    k =1 k kj  (7) with,

τ ij = exp( −

Aij T

), θ i =

qi xi , Φi = ∑ qk xk k

z ri xi , li = ( rk − qk ) + 1 − rk 2 ∑ rk xk k

where z is lattice coordination number (z=10), ri and qi are, respectively the volume and surface area of the pure component i, xi is the mole fraction of component i and τ ij is interaction parameter between component i and j. The structure parameters ‘r’ and ‘q’ for the compounds have been taken from our previous work4,32-35 and literature5,36-38.In NRTL model, the activity coefficient (γi) of component i in the ternary system is given by: c

ln γ i =

∑τ

ji

G ji x j

j =1 c

∑G x

li l

l =1

with,

c   xrτ rj Grj  ∑ c x j Gij  τ ij − r =1c  +∑ c  j =1 Glj xl Glj xl  ∑ ∑   l =1 l =1 

G ji = exp(−α jiτ ji ), τ ji = g ji − gii , α RT

(8)

ji = α ij

where g is an energy parameter characterizing the interaction of species i and j, xi is the mole fraction of component i and α the non-randomness parameter. Although α can be treated as an adjustable parameter, but in this study α was set equal to 0.2 which is a value prescribed for hydrocarbons.



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2.3. Liquid-Liquid Equilibria Modeling The thermodynamic equilibrium condition for multicomponent liquid-liquid system can be described by the following expression:

( i = 1, 2, 3,....)

γ iI xiI = γ iII xiII

(9)

where γ i , the activity coefficient of component i in a phase ( I or II) is predicted using the NRTL model. x iI and xiII

represents the mole fraction of component i in phase I and II

respectively. The compositions of the extract and raffinate phases are calculated using a flash algorithm( Figure 3) as described by the modified Rashford-Rice Algorithm39. The optimum binary interaction parameters are those which minimize the difference between the experimental and calculated compositions and are given by the relation m

c

II

(

Fobj = ∑∑∑ xikl − xˆikl k =1 i =1 l = I

)

2

(10)

The RMSD values, which provide a measure of the accuracy of the correlations, were calculated according to the following expression: 1/ 2

 Fobj  RMSD =    2 mc 

 m c II ( x l − ˆx l )2  ik ik  = ∑ ∑ ∑  k =1 i =1 l = I 2 mc   

1/ 2

(11)

where, m refers the number of tie lines and c refers to the number of components. Here xikl and xˆ ikl are the experimental and predicted values of mole fraction for component i for the kth tie line in phase l, respectively. Figure 3 shows the flow diagram of the total algorithm used in this work for the calculation of binary interaction parameters.


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3. Results and Discussions Thirty two IL and seven organic based ternary systems were selected from literature to test our approach with CS. They were correlated with both UNIQUAC and NRTL models with interaction parameters estimated by the CS algorithm. The systems used in this work are given in Table 1a and Table 1b respectively. The structure parameters ‘r’ and ‘q’ for all compounds studied in this work have been reported in Table 2a and Table 2b. System no 1,[OMIM][Cl] + Ethanol + TAEE has been used for benchmarking study.

3.1. Effect of Bounds Similar to other metaheuristic algorithms, lower and upper bounds are also important factors in CS algorithm for solving optimization problems. A check has been done to observe how close the obtained interaction parameters lie within the defined bounds. If any of the interaction parameters are close to the boundary, the bound has to be refined to a domain such that it covers the entire set of interaction parameters. The effect of the value of bounds on RMSD values have been considered and shown in Table 3a for UNIQUAC and Table 3b for NRTL model for system 1.A population size of 20 was used for the benchmarking system. Maximum number of iterations of 2000 was used as the stopping condition.Based on the recommendations of Yang and Deb24, pa of 0.25 was used. For each bound, 30 numerical trials have been carried out with random initial values of interaction parameters. It can be inferred from Table 3a and Table 3b that the RMSD values are minimum at a bound of -1000 to +1000 for UNIQUAC model and -100 to +100 for NRTL model. All the parameters were found to lie within the same range.Keeping this in mind, we have chosen the lower and upper bounds for estimation of interaction parameters for all the systems.



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3.2. Maximum Number of Iterations and Population size The effect of the maximum number of iterations and population size on RMSD values have been considered and shown in Figure 4a and Figure 4b. For UNIQUAC model, population size of 20 and maximum iteration of 1000 is sufficient to give a very good RMSD value.A further increase in population size and iteration was not able to improve the solution (Figure 4a).For NRTL model, population size of 20 and maximum iteration of 1500 was ableto give a very high RMSD value (Figure 4b). Keeping this in mind, we have chosen 20 as the population size for all systems. In a similar manner, 1000 and 1500 were chosen as the maximum number of iterations for UNIQUAC and NRTL model respectively. After the benchmarking with the above system, CS has been applied on the other ternary systems. For each system, 30 trials has been carried out with random initial values of Population and the lowest RMSD along with the corresponding interaction parameters were then selected as the final result. 3.3.Comparison with Reported Data The convergence plots of CS for few selected imidazolium and phosphonium based systems have been shown in Figure 5 a-d. For imidazolium based ILs (Figure 5a), the objective function value reduces to the order of 10-3 in approximately 60 iterations with UNIQUAC model. For phosphonium ILs (Figure 5b) with UNIQUAC model, the objective function value reduces to the order of

10-3 in approximately 100, 250 and 350 iterations

respectively for [Phosph], [DCA] and [DEC] anions. With NRTL model, imidazolium ILs (Figure 5c)used more number of iterations as compared to phosphonium ILs (Figure 5d) in order to obtain a function value with order 10-3. Figure 6 a-d shows the convergence plots of CS for imidazolium and phosphonium ILs with respect to maximum number of iterations (Itermax). For each itermax, 30 different trials has been carried out and the obtained minimum value of objective function has been


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assigned as Fobj. F*obj is the global optimum calculated using CS with itermax = 5000. From the plot, it is clear that the performance of CS improves with increment of Itermax. In particular, CS is very reliable for finding the global solution with high precision for phosphonium ILs as compared to imidazolium ILs. This is due to the fact that the solution were within 10-7-10-8 of the global minimum for phosphonium ILs with UNIQUAC model. On the other hand, CS was able to find the global minimum with a tolerance of 10-6-10-7 for imidazolium ILs with UNIQUAC model. Again with NRTL model, CS could converge to the global minimum within a tolerance of 10-3-10-5 for imidazolium ILs and 10-6-10-7 for phosphonium ILs. For the calculation of success rate (SR) of CS, following criterion has been * ≤ ε where ε is tolerance. Table 4 a-b shows the %SR of CS with considered: Fobj − Fobj

different tolerance values and stopping conditions. It is clear that %SR of CS increased with increments of Itermax and decreased with increment of tolerance values. With UNIQUAC model, %SR for imidazolium ILs are in the range of 23-87% (itermax=2000) whereas for phosphonium ILs >90% (itermax=2000) at tolerance value of 10-5 is obtained. With NRTL model having maximum number of iterations as 2000, success rate in % for imidazolium ILs are in the range of 0-3.3% whereas for phosphonium ILs they are in the range of 3.3-23.3% with a tolerance value of 10-5. This implies that CS gave a lower order of success in the parameter estimation of imidazolium ILs especially with NRTL model. Thus CS can be recommended tool for imidazolium IL’s as less computational effort is required with NRTL model. The RMSD values calculated using CS algorithm have been compared with the RMSD values reported in literature and tabulated in Table 5a-5d. For IL based liquid-liquid ternary systems, global RMSD value with CS are 0.0056 (Table 5a) and 0.0076 (Table 5c) for UNIQUAC and NRTL model respectively. This is 64% and 45% better than literature



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reported global value of 0.0156 and 0.0139 for 305 tie-lines. For organic solvent based liquid-liquid ternary systems, global RMSD value with CS is 0.0036 (Table 5b) and 0.0048 (Table 5d) for UNIQUAC and NRTL model respectively. This again is 53% and 43% better than literature reported global value of 0.0077 and 0.0085 for 66 tie-lines.The overall RMSD values with CS were 0.0053 and 0.0072 for UNIQUAC and NRTL model respectively. This included 371 tie lines. So in a nutshell this again is 63% and 45% better than literature reported global value of 0.0145 and 0.0131. These results are extremely satisfactory when compared to the deviation reported by Santiago et al.5 and Aznar3. It is also about the same order of magnitude as reported by Vatani et al. 6. Santiago et al.5 further correlated LLE data of fifty ternary systems involving twelve different IL’s, comprising 408 experimental tie-lines by UNIQUAC model. They obtained a global deviation of 1.75%. Aznar3 correlated the LLE data for 24 IL based ternary systems (184 tie-lines) by NRTL model and obtained global root mean square (RMS) deviations of 1.4%. Aznar3 and Santiago et al.5 estimated the interaction parameters using the Simplex method. Vatani et al.6 performed the liquid–liquid equilibria calculation for 20 different IL based ternary systems by NRTL model with binary interaction parameters predicted using Genetic Algorithm (GA). The overall RMSD value was 0.39% for 169 tielines. In order to confirm our capability with CS, a comparison with GA and PSO algorithms was made for quaternary and quinary system.Three quaternary systems, Ethanol + Water + Pentane + Hexane; Ethanol + Water + Pentane + Cyclohexane; Ethanol + Water + Hexane + Cyclohexane and one quinary system, Ethanol + Water + Pentane + Hexane + Cyclohexane, as studied by Khansary and Sani52 were selected for the same. The experimental data for the above mentioned systems has been taken from Huang et al.53.


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For each system, 20 trials has been carried out and the lowest RMSD along with the corresponding interaction parameters were selected as the final result and reported in Table 6 and Table 7. For quaternary system, CS gave RMSD (%) values in the range of 0.140.44% against ~1.0% for GA and PSO (Table 6). Whereas for quinary system, CS gave RMSD (%) values ~0.85 % against ~2.0% for GA and PSO (Table 7). This clearly shows CS algorithm is a reliable tool to correlate LLE data and even in some cases, it is superior to GA and PSO algorithms. In summary based on %SR and RMSD analyses, it can be concluded that CS algorithm gave good result in terms of both success rate and number of iterations especially for UNIQUAC model. LLE systems containing imidazolium ILs appears to be challenging for parameter estimation for higher precision. Further studies should be focused on the performance improvement of CS algorithm to get result with good precision using low numerical effort. Thus it can be concluded that the UNIQUAC and NRTL models with interaction parameters estimated by CS algorithm, was able to correlate the LLE data successfully. 4. Conclusions Liquid-liquid equilibria data for thirty nine ternary systems including thirty two IL based systems were correlated by the UNIQUAC and NRTL models. The binary interaction parameters were estimated using the Cuckoo Search algorithm. It has been found that population size of 20 is sufficient to satisfactorily predict the LLE with high accuracy.%SR analysis suggested that the performance of CS mainly depends on the type of ILs and LLE systems. Specially, ternary systems containing imidazolium ILs appeared to be most challenging for parameter estimation with NRTL model. RMSD results are extremely encouraging, with deviations in phase compositions 0.0053 (UNIQUAC) and 0.0072 (NRTL). This is 63% and 45% better than literature reported global value of 0.0145 and



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0.0131 for 371 tie-lines. Three ternary systems and one quinary system were further selected for comparison of CS with GA and PSO algorithms. Global %RMSD value obtained with CS algorithm was ~0.14-0.85% as compared to ~1.0-2.0% with GA and PSO. This shows CS algorithm is a reliable tool for the estimation of optimum values of binary interaction parameters for IL based LLE systems. Nomenclature

y

ith nest current position

α

step size parameter

(t ) i

Rand random number S

Step length

Γ

Standard Gamma function

pa

Discovery probability

ζ

Random number

γi

Activity coefficient of component i

xi

Mole fraction of component i

Φ

Segment fraction in UNIQUAC model

θ

Area fraction in UNIQUAC model

z

Lattice coordination number (z=10)

r

Volume parameter in UNIQUAC parameter

q

Surface parameter in UNIQUAC parameter

c

Number of components in LLE system

τ ij

Interaction parameter between component i and j in UNIQUAC/NRTL model

Aij

Energy parameter between component i and j in UNIQUAC model

α

NRTL non-randomness parameter.

m

Number of tie lines

xikl

Experimental values of mole fraction for component i for the kth tie line in phase l

ˆxikl

Predicted values of mole fraction for component i for the kth tie line in phase l

Fobj

Objective function

G ji / g ji Interaction energy parameter.


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begin Objective function f ( x), x = ( x1 ,...., xd )T Generate initial population of n host nests xi (i = 1, 2,..., n) while (t< MaxGeneration) or (stop criterion) Get a cuckoo randomly by Levy flights evaluate its quality/fitness Fi Choose a nest among n (say, j) randomly if(Fi

Applicability of Cuckoo Search Algorithm for the ... - ACS Publications

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