Comparison with Reported Data

The convergence plots of CS for few selected imidazolium- and phosphonium-based systems have been shown in Figure 7.5a-d. For imidazolium-based ILs (Figure 7.5a), the objective function value reduces to

UNIQUAC Volume and Surface Area Structural Parameters of Compounds Used in This Work

TABLE 7.8

^a Data from Santiago, R. S. et al., Fluid Phase Equilib., 278, 54-61, 2009. ^b Data from Rabari, D., and Banerjee, T. Ind. Eng. Chem. Res., 53, 18935-18942, 2014. ^c Data from Gonzalez, E. J. et al., J. Chem. Eng. Data, 55, 4931-4936, 2010. ^d Data from Feng, Y. et al., Fluid Phase Equilib, 398, 10-14, 2015. ^e Data from Luo, L. et al., Fluid Phase Equilib., 403, 30-35, 2015.

Other data taken from Banerjee, T. et al. Fluid Phase Equilib., 234, 64-76, 2005.

the order of 10^-3 in approximately 60 iterations with the UNIQUAC model. For phosphonium ILs (Figure 7.5b) with the 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 the NRTL model, imidazolium ILs (Figure 7.5c) used more iterations as compared to phosphonium ILs (Figure 7.5d) to obtain a function value of the order of 10^-3.

TABLE 7.9

Effect of Bounds on RMSD for System 1: [OMIM][Cl] + Ethanol + TAEE Using UNIQUAC Model

TABLE 7.10

Effect of Bounds on RMSD for System 1: [OMIM][Cl] + Ethanol + TAEE Using NRTL Model

Figure 7.6a-d shows the convergence plots of CS for imidazolium and phosphonium ILs with respect to the maximum number of iterations (Iter_max). For each Iter_max, 30 different trials have been carried out and the obtained minimum value of the objective function has been assigned as F_obj. F_obj is the global optimum calculated using CS with Iter_max = 5000. From the plot, it is clear that the performance of CS improves with the increment of Iter_max. 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 was within 10^-7-10^-8 of the global minimum for phosphonium ILs with the UNIQUAC model. On the other hand, CS was able to find the global minimum with a tolerance of 10^-6-10^-7 for imidazo- lium ILs with the UNIQUAC model. Again with the 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 the SR of CS, the following criterion has been considered: F_obj - F_obj < e, where e is tolerance. Tables 7.11 and 7.12 show the percentage SR (%SR) of CS with different tolerance values and stopping

FIGURE 7.5

Convergence plots of CS for selected (a) imidazolium-based ternary systems with the UNIQUAC model (N = 20, Iter _max = 1000) and (b) phosphonium-based ternary systems with the UNIQUAC model (N = 20, Iter _max = 1000). (Continued)

FIGURE 7.5 (Continued)

Convergence plots of CS for selected (c) imidazolium-based ternary systems with the NRTL model (N = 20, Iter _max = 1500) and (d) phosphonium-based ternary systems with the NRTL model (N = 20, Iter _max = 1500).

FIGURE 7.6

Convergence plots of CS for selected (a) imidazolium-based ternary systems (UNIQUAC model)

w.r.t. Iter _max and (b) phosphonium-based ternary systems (UNIQUAC model) w.r.t. Iter _max.

(Continued)

FIGURE 7.6 (Continued)

Convergence plots of CS for selected (c) imidazolium-based ternary systems (NRTL model) w.r.t. Iter _max and (d) phosphonium-based ternary systems (NRTL model) w.r.t. Iter _max.

conditions. It is clear that %SR of CS increased with the increment of Iter_max and decreased with the increment of tolerance values. With the UNIQUAC model, %SR for imidazolium ILs are in the range 23%-87% (Iter_max = 2000), whereas for phosphonium ILs and (Iter_max = 2000) at a tolerance value of 10^-5, >90% is obtained. With the NRTL model having maximum number

TABLE 7.11

Success Performance (%SR) of CS with the UNIQUAC Model for Selected Ternary Systems

Notes: SYS-2: [BMIM][TfO] + Ethanol + TAEE; SYS-20: [BMIM][BF4] + Benzene + Heptane; SYS-30: [BMIM][Tf₂N] + Ethanol + Water; SYS-22: [TDTHP][DCA] + Butanol + Water; SYS-23: [TDTHP][DEC] + Butanol + Water; SYS-25: [TDTHP] [Phosph] + Butanol + Water.

of iterations as 2000, %SR for imidazolium ILs are in the range 0%-3.3%, whereas for phosphonium ILs they are in the range 3.3%-23.3% with a tolerance value of 10^-5. This implies that CS gave a lower order of success in the PE of imidazolium ILs, especially with the NRTL model. Thus, CS can be a recommended tool for imidazolium as well as phosphonium ILs with UNIQUAC model.

Tables 7.13 and 7.14 show the %SR of CS as a function of the cation- and anion-type ILs. With the UNIQUAC model, %SR of imidazolium IL with [BF₄] anion is greater than [PF₆] anion (Iter_max = 2000) at a tolerance value of 10^-5, while for phosphonium ILs, the different anion types (DCA/DEC/ Phosph) do not make much effect on %SR of CS. With same [TFO] anion, %SR of [EMIM] IL is greater than [BMIM] IL at a tolerance value of 10^-5.

The RMSD values calculated using the CS algorithm have been compared with the RMSD values reported in the literature and tabulated in Tables 7.15 through 7.18. For IL-based liquid-liquid ternary systems, the global RMSD values with CS are 0.0056 (Table 7.15) and 0.0076 (Table 7.17) for the UNIQUAC and the NRTL models, respectively. This is 64% and 45% better than the global values of 0.0156 and 0.0139 reported in the literature for 305 tie lines. For organic-solvent-based liquid-liquid ternary systems, the

TABLE 7.12

Success Performance (%SR) of CS with the NRTL Model for Selected Ternary Systems

Notes: SYS-2: [BMIM][TfO] + Ethanol + TAEE; SYS-20: [BMIM][BF4] + Benzene + Heptane; SYS-30: [BMIM][Tf₂N] + Ethanol + Water; SYS-22: [TDTHP][DCA] + Butanol + Water; SYS-23: [TDTHP][DEC] + Butanol + Water; SYS-25: [TDTHP] [Phosph] + Butanol + Water.

global RMSD values with CS are 0.0036 (Table 7.16) and 0.0048 (Table 7.18) for the UNIQUAC and the NRTL models, respectively. These again are 53% and 43% better than global value of 0.0077 and 0.0085 reported in the literature for 66 tie lines. The overall RMSD values with CS are 0.0053 and 0.0072 for the UNIQUAC and the NRTL models, respectively, for 371 tie lines. So in a nutshell, these are 63% and 45% better than the global values of 0.0145 and 0.0131 reported in the literature.

These results are extremely satisfactory when compared to the deviation reported by Santiago et al. (2009) and Aznar (2007). They are also about the same order of magnitude as reported by Vatani et al. (2012). Santiago et al. further correlated the LLE data of 50 ternary systems involving 12 different ILs, comprising 408 experimental tie lines, using the UNIQUAC model with a global deviation of 1.75%. Aznar correlated the LLE data for 24 IL-based ternary systems (184 tie lines) using the NRTL model and found global RMSD of 1.4%. Both Aznar (2007) and Santiago et al. (2009) estimated the interaction parameters using the Simplex method. Vatani et al. (2012) performed the LLE calculation for 20 different IL-based ternary systems using the NRTL model with binary interaction parameters calculated using the GA. The overall RMSD value for 169 tie lines was 0.39%.

TABLE 7.13

Effect of the Anion Type on the Success Performance (%SR) of CS with the UNIQUAC Model for Selected Ternary Systems

Notes: SYS-18: [HMIM][BF4] + Benzene + dodecane; SYS-19: [HMIM][PF6] + Benzene + dodecane; SYS-22: [TDTHP] [DCA] + Butanol + Water; SYS-23: [TDTHP][DEC] + Butanol + Water; SYS-25: [TDTHP][Phosph] + Butanol + Water.