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Unconstrained Gibbs Free Energy Minimization for Phase Equilibrium Calculations in Nonreactive Systems, Using an Improved Cuckoo Search Algorithm Seif-Eddeen K. Fateen Department of Chemical Engineering, Cairo University, Giza, Egypt

Adrián Bonilla-Petriciolet* Department of Chemical Engineering, Instituto Tecnológico de Aguascalientes, Aguascalientes, México ABSTRACT: This study introduces a strategy to improve the effectiveness of Cuckoo Search (CS) algorithm for the unconstrained Gibbs free energy minimization in phase equilibrium calculations of nonreactive systems. Specifically, the gradient information of the unconstrained Gibbs free energy function, which is readily available, is used to enhance the balance between diversification and intensification stages of the CS algorithm for phase-split calculations in multicomponent systems. The results showed that it is feasible to improve the numerical performance of the CS algorithm using the gradient information of the Gibbs free energy function; this improved method provides better results for phase equilibrium calculations in nonreactive systems with insignificant additional computational effort. This gradient-based Cuckoo Search (GBCS) algorithm outperformed the conventional CS algorithm, in terms of its reliability and efficiency in solving phase equilibrium problems, especially for multicomponent systems.

1. INTRODUCTION The minimization of the Gibbs free energy for the prediction of phase behavior of a multicomponent mixture is one of the most important and challenging global optimization problems, in the context of applied thermodynamics for chemical engineering. This optimization problem can be solved using both constrained and unconstrained formulations,1,2 where the objective is to determine the number, type, and composition of the phases at equilibrium under the given operating conditions. Optimization methods for minimization of the Gibbs free energy in nonreactive systems employ local or global solving strategies.1,2 They include the accelerated successive substitution method, Nelson’s method, the linearly constrained minimization method, homotopy continuation algorithms, and deterministic and stochastic global optimization strategies.1−7 Overall, several optimization strategies do not provide a guarantee of obtaining the global minimum of Gibbs free energy function, because of its multivariable, nonlinear, and nonconvex nature.1−3 Considering these challenging characteristics, it is easy to understand the difficulty of finding a global minimum of the Gibbs free energy in phase equilibrium calculations (PEC). In particular, phase-equilibrium problems have been successfully solved using optimization metaheurisitics.2,8−17 These methods combine heuristics in high-level frameworks to explore effectively the search space for finding the global optimum solution. They offer several advantages for performing the global minimization of Gibbs free energy, because they are not problem-specific, may use a long-/short-time form of memory to guide the search for the global optimization avoiding the convergence to local optima, and may use domain-specific knowledge of the problem to explore the best promising areas.2 These optimization strategies have been tested and evaluated on © 2014 American Chemical Society

several phase equilibrium problems with vapor−liquid equilibrium (VLE), liquid−liquid equilibrium (LLE), and vapor− liquid−liquid equilibrium (VLLE).8−17 Metaheuristics used for PEC include the most important and popular single-point and population-based methods: Simulated Annealing, Genetic Algorithms, Tabu Search, Differential Evolution, Particle Swarm Optimization, Firefly Algorithm, Ant Colony Optimization, Cuckoo Search, and other algorithms.8−17 In particular, Bhargava et al. 16 found out that Cuckoo Search (CS) is one of the most reliable metaheuristics for the unconstrained Gibbs free energy minimization in PEC of nonreactive systems. Herein, it is important to remark that the numerical performance and convergence behavior of optimization metaheuristics are determined by both the intensification and diversification strategies, which guide and modify the numerical operations to efficiently explore the search space of decision variables in order to find (near-)global optimal solutions. Specifically, the diversification (exploration) stage is devoted to exploring the search space, and for the computation of the objective function value of very different points in the search domain to find the promising area. On the other hand, the intensification (exploitation) stage investigates the promising area to locate, as near as possible, the global optimum solution that the algorithm is looking for. Although in recent years, there have been significant advantages in the application of metaheuristics to perform the Gibbs free energy minimization for PEC, finding a good balance between diversification and Received: Revised: Accepted: Published: 10826

April 22, 2014 June 15, 2014 June 17, 2014 June 17, 2014 dx.doi.org/10.1021/ie5016574 | Ind. Eng. Chem. Res. 2014, 53, 10826−10834

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intensification stages of current metaheuristics is still a challenge, especially for multicomponent systems. The lack of the proper balance of these stages may lead to a poor convergence performance and low algorithm reliability. This problem prevails even for the most promising metaheuristics such as Cuckoo Search or Bare-Bones Particle Swarm Optimization, which may show some numerical disadvantages for PEC.13,16 The desirable scenario for the application and implementation of a metaheuristics for the global minimization of the Gibbs free energy function in a process simulator is that the optimization method can quickly identify regions in the search space with high-quality solutions, without wasting too much computational time in unpromising regions. In particular, a recent study 18 showed that CS is a reliable stochastic optimization method for solving the unconstrained Gibbs minimization problem and it may offer a better performance than other state-of-the-art stochastic methods. However, it also showed that there is a need to improve the efficiency of the CS algorithm, especially for multicomponent systems. Based on these facts, this study is focused on the development of a strategy to improve the effectiveness of the CS algorithm for the unconstrained Gibbs free energy minimization in PEC of nonreactive systems. This improvement strategy is based on the use of the gradient information of the unconstrained Gibbs free energy function to enhance the balance between diversification and intensification stages of the CS algorithm for PEC. One of the most commonly used and wellstudied concepts in numerical optimization is the gradient. The gradient at any point in the decision space indicates the direction in the search space along which the function to be optimized improves the most.19 Therefore, this direction, which is derived from the objective function gradient, can be incorporated into the metaheuristics to improve their numerical performance. Our results showed that it is feasible to improve the performance of the CS algorithm using the gradient information of the Gibbs free energy function. This improved method provides better results for PEC in nonreactive systems without adding a computational complexity to the optimization algorithm. In summary, this study introduces a modification to the CS algorithm, which maintains the reliability of the algorithm but significantly enhances its efficiency in the unconstrained Gibbs free energy minimization for PEC in multicomponent systems.

∑ nij = zinF

⎛ Δμij ⎞ ∑ ∑ nij⎜⎜ ⎟⎟ ⎝ RT ⎠ j=1 i=1 π

(2)

0 ≤ nij ≤ zinF

i = 1 , ..., c ; j = 1 , ..., π

(3)

where nij and Δμij/RT are the mole numbers and chemical potentials of species i in phase j, respectively; c and π are the numbers of components and phases at equilibrium, respectively; zi is the mole fraction of component i in the feed; and nF is the total moles in the feed. Decision variables nij for this constrained optimization problem are c·π. The unconstrained minimization of the Gibbs free energy function can be performed using alternative variables χij instead of nij as decision variables.8 For multiphase nonreactive systems, new decision variables χij ∈ (0,1) can be defined and employed by using the following expressions for the mass balances: ni1 = χi1 zinF

i = 1 , ..., c

(4)

j−1

nij = χij (zinF −

∑ nim)

i = 1 , ..., c ; j = 2 , ..., π − 1

m=1

(5) π−1

niπ = zinF −

∑ nim

i = 1 , ..., c (6)

m=1

where the number of decision variables χij is c(π − 1) for the unconstrained Gibbs free energy minimization of nonreactive systems. In most of the reported studies, authors assumed that the number and type of phases at equilibrium are known; such problems are also refereed as phase split calculations. In this study, the same assumption is made. For example, for a twophase split problem, the objective function becomes Δg = RT

⎛ Δμ ⎞ ∑ niα⎜ iα ⎟ + ⎝ RT ⎠ i=1 c

c

⎛ Δμiβ ⎞ ⎟⎟ ⎝ RT ⎠

∑ niβ⎜⎜ i=1

(7)

where the subscripts α and β refer to the two phases at equilibrium. The decision variables χiα for a two-phase split problem are related to the mass balances as follows: niα = χiα zinF

i = 1 , ..., c

niβ = zinF − χiα zinF

2. THE UNCONSTRAINED GIBBS FREE ENERGY MINIMIZATION FOR PHASE EQUILIBRIUM CALCULATIONS IN NONREACTIVE SYSTEMS In a phase equilibrium problem, a mixture of substances at a given temperature (T), pressure (P), and total molar amount may separate into two or more phases. The global minimization of the Gibbs free energy must be performed for calculating the equilibrium state of a mixture.1,2 This approach was introduced by White et al.20 and, to date, many optimization strategies have been studied for the global minimization of this thermodynamic function.1−3 For a stable equilibrium at given T and P, the Gibbs free energy of mixing is the thermodynamic objective function, which must be at the global minimum. This optimization problem can be defined as g − g0 Δg min = = RT RT

i = 1 , ..., c

j=1

(8)

i = 1 , ..., c

(9)

On the other hand, the gradient of the unconstrained Gibbs free energy function can be easily derived as follows: 1 ⎛ ∂Δg ⎞ ⎜⎜ ⎟⎟ = RT ⎝ ∂χkα ⎠ T

c

π

+

∑ ∑ nij j=1 i=1

π

c

⎛ Δμij ⎞ ∂nij ⎟⎟ ⎝ RT ⎠ ∂χkα

∑ ∑ ⎜⎜ j=1 i=1

∂(Δμij /RT ) ∂χkα

k = 1 , ..., c (10)

The derivative of the number of moles niα and niβ, with respect to the decision variables χiα, is ⎧ zinF i = k ∂niα =⎨ ∂χkα i≠k ⎩0 ⎪

⎪

c

∂niβ

(1)

∂χkα

subject to 10827

⎧−zinF i = k =⎨ i≠k ⎩0

(11)

⎪

⎪

(12)

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Figure 1. Pseudocode of the gradient-based cuckoo search (GBCS) algorithm used for the unconstrained minimization of the Gibbs free energy in nonreactive systems.

abandoned. Therefore, the numerical implementation of CS for global optimization is based on the following rules:21−23 (1) Each cuckoo lays an egg in a random nest. The egg represents a set of solutions for the optimization problem. (2) The best eggs (i.e., solutions) are contained in a fraction of the nests and are carried over to the next iteration. (3) The number of nests does not change. A host bird can find an alien egg with a specified probability pa ∈ [0,1]. If an alien egg is found, the host can abandon the nest or discard the egg, and then build a new nest elsewhere. This condition is implemented in CS with the assumption that a fraction pa of nh nests is replaced by new nests. The use of Lévy flights in the CS algorithm makes its local and global search stages effective. A Lévy flight is a trajectory that consists of taking successive random steps. This sequence of sudden jumps, which are chosen from a power-law tail probability density function, is a characteristic of the Lévy flight, which is considered as the optimum random search pattern. To generate a new egg, a Lévy flight is performed using the coordinates of a randomly selected egg. This step can be represented by

The second term of eq 10 is identical to zero, based on the Gibbs−Duhem equation. Thus, the gradient of the unconstrained objective function of a two-phase split problem (e.g., eq 7), is analytically obtained as ⎡ Δμ Δμkβ ⎤ 1 ∂Δg ⎥ = zknF ⎢ kα − RT ∂χkα RT ⎦ ⎣ RT

k = 1 ,..., c (13)

It is important to remark that the evaluation of eq 13 (i.e., the gradient of Δg/RT) does not involve any derivatives of the chemical potential function and can be easily implemented independent of the thermodynamic model used (i.e., EoS or local composition model). Finally, the computational cost associated with the calculation of this gradient is insignificant, from a practical point of view, since the chemical potentials are calculated in the course of the calculation of the objective function. We will illustrate that the use of the gradient of the Gibbs free energy function may increase the effectiveness of metaheuristics for PEC, because it provides the direction of maximum improvement to search for the optimum of the objective function. Based on this fact, in the following section, we describe the approach used to improve the numerical performance of Cuckoo Search for the unconstrained minimization of Gibbs free energy in two-phase equilibrium calculations of nonreactive systems.

Xit + 1 = Xit + α ⊕ Levy( ́ λ)

(14)

where ⊕ denotes entry-wise multiplication, α is the step size, and Lévy (λ) is the Lévy distribution. The condition for displacing an egg to the new position is that the value of the objective function is found better than another randomly selected egg. The step size α, which depends on the scale of the optimization problem, controls the scale of random search. A fraction (1 − pa) of the nests selected at random is abandoned and replaced by new ones at new locations via local random walks:

3. STRATEGY TO IMPROVE THE BALANCE OF DIVERSIFICATION AND INTENSIFICATION STAGES OF CUCKOO SEARCH FOR THE UNCONSTRAINED MINIMIZATION OF THE GIBBS FREE ENERGY In this study, we have used the Cuckoo Search (CS)21−23 as an optimization algorithm for the unconstrained minimization of the Gibbs free energy for PEC in nonreactive systems. The use of CS and its variants for solving multimodal optimization problems is gaining wide popularity, because of its ease of use and effective performance.21−25 This metaheuristic emulates the brood parasitism behavior of the cuckoo birds, which lay their eggs in the nests of other species. This nature parasitism behavior implies that the cuckoo eggs evolved to mimic the egg appearance of local host birds to avoid being discovered and

Xit + 1 = Xit + α(X jt − Xkt )

Xtj

(15)

Xtk

where and are two different solutions selected randomly by random permutation and α is a random number drawn from a uniform distribution. The only parameter to be tuned is the fraction of nests to be abandoned (1 − pa). However, this value is not critical for the optimization algorithm and Yang and Deb21 suggested using a value of pa = 0.25. 10828

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Table 1. Description, Thermodynamic Models, and Feed Conditions of Selected Phase Equilibrium Problems No.

system

feed conditions

thermodynamic models

1 2 3

n-butyl acetate + water toluene + water + aniline N 2 + C1 + C2

nF = (0.5, 0.5) at 298 K and 101.325 kPa nF = (0.29989, 0.20006, 0.50005) at 298 K and 101.325 kPa nF = (0.3, 0.1, 0.6) at 270 K and 7600 kPa

4

C1 + H2S

nF = (0.9813, 0.0187) at 190 K and 4053 kPa

5

C2 + C3 + C4 + C5 + C6

nF = (0.401, 0.293, 0.199, 0.0707, 0.0363) at 390 K and 5583 kPa

6

C1 + C2 + C3 + C4 + C5 + C6 + C7−16 nF = (0.7212, 0.09205, 0.04455, 0.03123, 0.01273, 0.01361, 0.07215, 0.01248) at 353 K and 38500 kPa + C17+ C1 + C2 + C3 + iC4 + C4 + iC5 + C5 + nF = (0.614, 0.10259, 0.04985, 0.008989, 0.02116, 0.00722, 0.01187, 0.01435, C6 + iC15 0.16998) at 314 K and 2010.288 kPa C1 + C2 + C3 + C4 + C5 + C6 + C7 + nF = (0.6436, 0.0752, 0.0474, 0.0412, 0.0297, 0.0138, 0.0303, 0.0371, 0.0415, C8 + C9 + C10 0.0402) at 435.35 K and 19150 kPa

7 8

Our previous study24 has indicated that the convergence performance of CS can be improved via the modification of the local random walk in which a fraction (1 − pa) of the nests is replaced. In particular, the gradient of the objective function can be employed to determine the step direction and magnitude for generating new solutions.24 Using the gradient information, new nests are generated randomly from the worse nests but in the direction of the minimum as seen from the point of view of the old nests. Therefore, the new solutions can be obtained using stepi = α(X jt − Xkt )

(16)

⎛ stepi ⎞ Xit + 1 = Xit + stepi ⊗ sign⎜ − ⎟ ⎝ dfi ⎠

(17)

NRTL model NRTL model SRK EoS with classical mixing rules SRK EoS with classical mixing rules SRK EoS with classical mixing rules SRK EoS with classical mixing rules SRK EoS with classical mixing rules SRK EoS with classical mixing rules

global optimum −0.0201983 −0.3529567 −0.5477911 −0.0198922 −1.1836523 −0.8387826 −0.7697724 −1.1211758

Herein, it is convenient to remark that a comparison between a set of promising stochastic methods for solving phase equilibrium problems has been recently published.18 Its outcome clearly showed that CS is a reliable method for solving phase stability and phase equilibrium problems and it suggested further research to be directed toward the improvement of the efficiency of CS, especially for multicomponent systems. Thus, in this study, we limited the comparison of the results between the original CS algorithm and the developed modified version of CS, which made use of the readily available gradient of the unconstrained Gibbs free energy function, to probe the improvements in algorithm efficiency.

4. RESULTS AND DISCUSSION In this study, the two optimization algorithms (CS and GBSC) and the thermodynamic models were coded in MATLAB. For the comparison of two methods, we have used pa = 0.25 and nh = 10nvar, where nvar is the number of decision variables. We studied eight PEC problems, whose details can be found in Table 1. These problems are multimodal with the number of decision variables ranging from 2 to 10. Note that these phase equilibrium problems have been used in previous studies for testing other global optimization strategies, including classical metaheuristics.8,10,12,13,15,16 Each problem was solved 100 times independently with a different random number seed, for a robust performance analysis. GBCS and CS were evaluated according to the reliability and efficiency for finding the global optimum of the unconstrained Gibbs free energy function. The efficiency is determined by recording the number of function evaluations (NFE) for each optimization algorithm, where a low value of NFE means a higher efficiency. Note that NFE is an unbiased indicator of the computational costs required by a certain algorithm and is independent of the host hardware. On the other hand, the reliability was measured by the success rate (SR) at certain numerical effort. The success rate is defined as the ratio of number of runs in which the global minimum was attained within a tolerance at this numerical effort to the total number of runs. In addition, we also report a plot of the average best value against NFE. The best values are averaged over all the runs and plotted against NFE, which was calculated at each iteration. Since the NFE needed for each iteration differ among the metaheuristics, the plot of average best value against NFE is a useful indication of reliability versus efficiency of the optimization method. Finally, the performance profiles26 for the reliability and efficiency

where the sign function obtains the sign of its argument and dfi is the gradient of the objective function at each decision variable, ∂f/∂xi. Note that if the objective function gradient is negative, the step direction is made positive; otherwise, the step direction is made negative. It is important to remark that traditional metaheuristics do not utilize the gradient information in their diversification and intensification schemes. On the other hand, eqs 16 and 17 do not alter the structure of the optimization algorithm and no additional parameters are needed to implement them. In a previous study,16 CS algorithm has proven its superiority over other algorithms, in terms of reliability (i.e., the ability to find the global minimum), for PEC. However, its degree of diversification and intensification is far from optimal, especially in challenging PEC. Note that, in the vicinity of the global minimum of a phase equilibrium problem, the original CS is advantageous over other methods, because of its use of the local random search.18 Preliminary calculations performed using CS with eqs 16 and 17, instead of eq 15 in different phase equilibrium problems suggested that it is more convenient to use an adaptive scheme for the generation of new solutions, which may combine the gradient information and the local random walk. Thus, we propose that eqs 16 and 17 are used if the gradient of the unconstrained Gibbs free energy function was >10−3; otherwise, the traditional local random search is applied in the CS algorithm. Pseudocode for this improved algorithm (gradient-based cuckoo search, GBCS) is given in Figure 1, which has been coded in MATLAB. In the following section, we will compare the convergence performance of both CS and GBCS in selected PEC. 10829

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Figure 2. Evolution of the mean best function value with NFE for CS and GBCS in the unconstrained Gibbs free energy minimization of nonreactive systems

metrics of both CS and GBCS have been calculated using the following equations: rpsm =

ρs (ς) =

t psm min{t psm: 1 ≤ sm ≤ ns}

1 size{p ∈ ℜ: rpsm ≤ ς} np

(19)

where ns is the number of metaheuristics (i.e., 2) to be tested, np is the number of problems used in this comparison, tpsm is the value

(18) 10830

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Table 2. Minimum NFE for the Average Best Value To Reach a Tolerance Value (ε) from the Known Global Minimum Using CS and GBCS in the Unconstrained Minimization of Gibbs Free Energy Minimum Number of Function Evaluations, NFEa tolerance, ε

a

PEC No. 1

PEC No. 2

PEC No. 3

1 × 10−3 1 × 10−4 1 × 10−5 1 × 10−6 1 × 10−7

140 540 980 1540 2260

930 1770 2730 5130 9810

390 1410 2670 4170 6030

1 × 10−3 1 × 10−4 1 × 10−5 1 × 10−6 1 × 10−7

100 580 900 1300 1900

1230 2250 3150 4650 9390

150 750 1830 3330 5250

PEC No. 4 CS Method 60 60 1060 1980 2860 GBCS Method 60 60 940 2460 3580

PEC No. 5

PEC No. 6

PEC No. 7

PEC No. 8

1250 2650 3850 5450 13650

4560 14160 29360 51280 75280

11970 24570 39690 57870 77490

10100 20300 33300 51900 80100

550 1150 2350 4150 10450

1040 4080 16400 39920 65840

2970 6030 11610 21690 36270

1700 4500 10900 26500 51100

Boldface numbers represent the more efficient algorithm.

Table 3. Success Rate of CS and GBCS in the Unconstrained Minimization of the Gibbs Free Energy at Different Iterations Success Rate (%)a iteration

a

PEC No. 1

PEC No. 2

PEC No. 3

10 25 50 100 200 300 400

2 31 95 100 100 100 100

0 1 9 91 99 100 100

0 0 16 100 100 100 100

10 25 50 100 200 300 400

2 51 100 100 100 100 100

0 0 3 95 98 100 100

0 0 40 100 100 100 100

PEC No. 4 CS Method 2 26 79 100 100 100 100 GBCS Method 5 23 87 100 100 100 100

PEC No. 5

PEC No. 6

PEC No. 7

PEC No. 8

9 53 85 100 100 100 100

0 0 0 0 0 32 100

0 0 0 0 0 18 100

0 0 0 0 1 100 100

0 4 93 100 100 100 100

0 0 0 0 6 99 100

0 0 0 7 100 100 100

0 0 0 2 100 100 100

Boldface numbers represent the more efficient algorithm.

of the performance metric for problem p and metaheuristic sm, rpsm is the performance ratio used to compare the performance on problem p by metaheuristic sm with the best performance by any metaheuristic on this problem; and ρs(ς) is the fraction of the total number of problems for which metaheuristic sm has a performance ratio rpsm within a factor of ς of the best possible ratio. Note that rpsm = 1 for the metaheuristic that performs the best on a specific problem p. The reliability of the metaheuristic in accurately finding the global minimum of the unconstrained Gibbs free energy function is considered as the principal goal; hence, the metric used to obtain the reliability performance profile is defined as t psm =

Δg Δg * − RTcalc RT

the evaluation of the efficiency of both CS and GBCS in obtaining the global minimum of the unconstrained Gibbs free energy. This metric is the minimum number of NFE needed to reach within 10−5 of the global minimum of Δg/RT. This efficiency performance profile compares how fast CS and GBCS can find the global minimum with a tolerance level of 10−5 and it is useful to identify the metaheuristic that reaches the solution fastest for the phase equilibrium problems tested. The performance results of both CS and GBCS are presented in three different ways. For each problem, the mean best values are plotted versus NFE for each of the eight phase equilibrium problems. These plots are reported in Figure 2. The minimum NFE required to reach a certain tolerance from the known global minimum for each problem were calculated and presented in Table 2, while the reliability (i.e., success rates) of both metaheuristics for solving PEC are reported in Table 3 and Figure 3, which shows the success rate at different number of iterations. For the purpose of this study, the success rate is defined as the percentage of runs that converged to within 10−6 of the known global minimum. Finally, the performance profiles of CS and GBCS for the reliability and efficiency metrics are

(20)

where f * and fcalc is the known global minimum and the mean value of the unconstrained Gibbs free energy function (i.e., f = Δg/RT) calculated by the metaheuristic over several runs. This reliability performance profile compares how accurately CS and GBCS can find the global optimum value of Δg/RT, relative to each other. We have also used another performance metric for 10831

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Figure 3. Global success rates (GSR, %) of the CS and GBCS algorithms in the unconstrained minimization of the Gibbs free energy for the phase equilibrium calculations (PEC) of nonreactive systems.

Figure 5. Efficiency performance profiles of the CS and GBCS algorithms in the unconstrained minimization of the Gibbs free energy for the phase equilibrium calculations (PEC) of nonreactive systems.

shown in Figures 4 and 5, respectively. A brief discussion of the results follows.

2c. The savings in computational effort ranges from 60% at the 10−3 level to 13% at the 10−7 level, with respect to the results obtained with CS. Note that GBCS showed 100% reliability at 100 iterations. PEC No. 4 is a binary system with VLE and the performance results indicated that GBCS did not outperform CS, in terms of numerical effort. In particular, at the 10−7 level, GBCS needed 25% more computational effort, as depicted in Table 2. PEC No. 5 is a five-component system with VLE. As expected, both CS and GBCS required significantly more NFE than the previously discussed phase equilibrium problems to be able to converge to the global minimum of the unconstrained Gibbs free energy function, as shown in Figure 2e. In this case, the use of GBCS resulted in a reduction of the computational effort ranging from 56% at the 10−3 level to 24% at the 10−7 level, and there is an improvement in algorithm reliability (see results reported in Tables 2 and 3). On the other hand, PEC No. 6 is an eighthydrocarbon mixture with VLE, which required more NFE to be solved by both metaheuristics. However, GBCS outperformed CS, as in the case of PEC No. 5, as illustrated in Figure 2f. The reduction in computational effort of GBCS ranged from 77% at the 10−3 level to 13% at the 10−7 level, and this improved method may offer 100% reliability for PEC at ∼300 iterations (see results reported in Table 3). PEC No. 7 is a nine-component mixture that shows a VLE, which is modeled using the Soave−Redlich−Kwong equation of state (SRK EoS). The reduction of computational effort at all levels when using GBCS were more than 53% and 75% when compared with the original CS, as shown in Table 2 and Figure 2g. Since this is a difficult phase split calculation, this significant reduction in NFE translates to a considerable difference in computational time. In addition, the reliability of GBCS is 100% at 200 iterations, while CS did not converge to the global optimum of Δg/RT at the same numerical effort. This same convergence pattern is repeated for PEC No. 8, which is reported in Figure 2h. This mixture has 10 components and shows a VLE. In this case, the reduction in computational effort for GBCS ranged from 83% at the 10−3 level to 26% at the 10−7 level, as depicted in Table 2. Again, GBCS exhibited a success rate of 100% in PEC at 200 iterations, while CS failed to find the global optimum solution. Figure 3 shows the global success rate of both CS and GBCS for tested phase equilibrium problems. It is clear

Figure 4. Reliability performance profiles of the CS and GBCS algorithms in the unconstrained minimization of the Gibbs free energy for the phase equilibrium calculations (PEC) of nonreactive systems.

PEC No. 1 is a two-component liquid−liquid equilibrium (LLE) problem that is relatively easy to solve. The two algorithms were able to solve the problem satisfactorily to the level of 10−7 distance from the global minimum of Δg/RT within ∼2000 NFE, as indicated in Table 2. GBCS was slightly better than CS in performing PEC, achieving a 100% success rate at 50 iterations, but the savings in computational time is not significant, as depicted in Figure 2a. PEC No. 2 is also an LLE problem for a ternary system and the convergence pattern to the global minimum of the two algorithms is similar to that for PEC No. 1. Both algorithms performed the unconstrained minimization of the Gibbs free energy satisfactory with no significant savings in computational time for GBCS, as indicated by the minimum NFE required to reach a certain distance from the global minimum as shown in Table 2. GBCS and CS reached a 100% success rate at iterations ≥300 (see Table 3). GBCS converged to the global minimum of Δg/RT for PEC No. 3, which is a ternary system with vapor−liquid equilibrium (VLE), in a significantly less NFE, as shown in Table 2 and Figure 10832

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■

that GBCS offers a better performance than CS at early iterations, especially for multicomponent systems. Figure 2 shows that, for the phase-split problems with a lesser number of components (i.e., decision variables), both CS and GBCS behaved similarly. However, for problems with a greater number of components (i.e., c ≥ 5), which are relatively more difficult to solve, GBCS significantly outperformed CS in all phase equilibrium problems. This numerical performance is illustrated in the reliability performance profile given in Figure 4 and the efficiency performance profile of Figure 5. Both performance profiles show that GBCS is more effective than CS for the unconstrained Gibbs free energy minimization, especially in multicomponent systems. In summary, our results showed that the incorporation of the gradient of the unconstrained Gibbs free energy function is useful to improve the tradeoff of diversification and intensification stages of the CS algorithm for phase equilibrium calculations in nonreactive systems especially for systems with several components. These results are relevant for compositional reservoir modeling and chemical process simulators, where a great number of phase equilibrium calculations is required for multicomponent systems and the computational time is critical and may represent up to 50% of the simulation total CPU time.27,28 It is clear that this context may limit the application and use of global optimization algorithms, including metaheuristics. Therefore, our study is a step forward toward speeding up the convergence performance of CS without compromising its reliability for the global minimization of the Gibbs free energy, thus enhancing its incorporation and use in process simulators for modeling multicomponent systems.

REFERENCES

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5. CONCLUSIONS In this study, we made use of the gradient of the unconstrained Gibbs free energy function to improve the performance of the Cuckoo Search (CS) algorithm for the solution of phase equilibrium problems. The improved optimization algorithm was evaluated solving difficult phase equilibrium problems. Its performance, in comparison with the original CS algorithm, has been analyzed. The gradient-based Cuckoo Search (GBCS) algorithm outperformed the CS algorithm, in terms of its reliability and efficiency in solving phase equilibrium problems, especially for multicomponent systems. Since CS is a reliable stochastic optimization method for solving phase equilibrium problems,18 the modified version of CS, proposed in this study, made use of the readily available gradient of the Gibbs function to enhance the performance of CS further, in performing phase equilibrium calculations, especially in multicomponent systems. Thus, GBCS is an improved stochastic method for solving this challenging thermodynamic problem. In summary, this study provides insights on the utility of the gradient information to improve the performance of metaheuristics for solving global optimization problems related to the phase equilibrium modeling.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Notes

The authors declare no competing financial interest. 10833

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(21) Yang, X. S.; Deb, S. Cuckoo search via Lévy flights. In Proceedings of World Congress on Nature & Biologically Inspired Computing (NABIC 2009), December 9−11, 2009, Coimbatore, India; IEEE Publications: New York, 2009; pp 210−214 (ISBN 978-1-4244-5053-4). (22) Yang, X. S.; Deb, S. Engineering Optimisation by Cuckoo Search. Int. J. Math. Modell. Numer. Optim. 2010, 1, 330−343. (23) Yang, X. S.; Deb, S. Cuckoo search: Recent advances and applications. Neural Comput. Appl. 2014, 24, 169−174. (24) Fateen, S. E. K.; Bonilla-Petriciolet, A. Gradient-Based Cuckoo Search for global optimization. Math. Probl. Eng. 2014, 2014, Article ID 493740. (25) Fister, I, Jr.; Fister, D.; Fister, I. A comprehensive review of cuckoo search: Variants and hybrids. Int. J. Math. Modell. Numer. Optim. 2013, 4, 387−409. (26) Dolan, E. D.; More, J. J. Benchmarking optimization software with performance profiles. Math. Progr., Ser. A 2002, 91, 201−213. (27) Petitfrere, M.; Nichita, D. V. Robust and efficient trust-region based stability analysis and multiphase flash calculations. Fluid Phase Equilib. 2014, 362, 51−68. (28) Gaganis, V.; Varotsis, N. An integrated approach for rapid phase behavior calculations in compositional modeling. J. Pet. Sci. Eng. 2014, 118, 74−87.

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