Printer Friendly

UNIQUAC Interaction parameters with closure for imidazolium based ionic liquid systems using genetic algorithm.

INTRODUCTION

The separation of aromatic hydrocarbons from [C.sub.4] to [C.sub.10] aliphatic hydrocarbon mixtures is challenging since these hydrocarbons have boiling points in a close range and several combinations form azeotropes. Ionic liquids (ILs) have been an area of interest both in clean industrial processes, such as liquid media for separation and extraction. Because of its negligible vapour pressure and wide liquidus range, they are now used as an alternative to conventional solvents (Brennecke and Maginn, 2001; Marsh et al., 2004). The liquid-liquid equilibrium (LLE) data gives a fair indication of the separation power of these novel solvents. Data available till now consists of the infinite dilution activity coefficients (David et al., 2003; Heintz et al., 2003) and the liquid-liquid extraction (Letcher and Deenadayalu, 2003a; Selvan et al., 2000). The LLE data are the important ones in deciding their suitability of these solvents for extraction purpose. Till date ternary tie line data are available for aromatic-aliphatic as well as aliphatic-alcohol separation using ionic liquid as a solvent.

To predict the separation, it is necessary to know the liquid-liquid equilibrium data for a particular system. Various activity coefficient models such as Universal Quasi Chemical (UNIQUAC) and Non-Random Two Liquid (NRTL) can be used to predict the LLE. Each of these models requires proper binary interaction parameters that can represent LLE for highly non-ideal liquid mixtures. These parameters are usually estimated from the known experimental LLE data via optimization of a suitable objective function (Kang and Sandler, 1987). The optimization problem can be either the least square objective function minimization or likelihood function maximization. In both cases the objective function is non-linear and non-convex in terms of optimization variables; this possesses several local minima/maxima/saddle points within the specified bounds of the variables. Various local methods have been proposed to efficiently perform the constrained or unconstrained optimization. These use gradient-based approaches like Gauss-Newton (Britt and Luecke, 1973; Fabries and Renon, 1975; Anderson et al., 1978; Schwetlick and Tiller, 1985), Gauss-Marquardt (Valko and Vajda, 1987) or the successive quadratic programming SQP methods (Tjoa and Biegler, 1991, 1992). Alternatively, the possibility of nongradient methods like simplex pattern may be used (Gmehling et al., 1990; Vamos and Hass, 1994). Therefore, it is necessary to apply a technique that results in the global optimization of the variables. Several such techniques are reported in literature, like Interval methods (Gau and Stadtherr, 2000a), Branch and Bound methods (Esposito and Floudas, 1998; Adjiman et al., 1998; Zhu and Inoue, 2001). Moore et al. (1992) and Ratz and Csendes (1995) have both used parameter estimation as an example in demonstrating more general algorithms based on interval methods. Evolutionary algorithms like genetic algorithm (GA) (Singh et al., 2005; Papadopoulos and Linke, 2004) and simulated annealing (SA) (Papadopoulos and Linke, 2004) have been rarely used for the optimization of LLE processes.

The binary interaction parameters for UNIQUAC and NRTL models are dependent on each other following a linear relationship called closure equations (Ahmad and Khanna, 2003; Julia et al., 2005; Sahoo et al., 2006). For a ternary triplet i-j-k the adjustable binary interaction parameters are related as (Hala, 1972):

([[tau].sub.jk] - [[tau].sub.kj]) = ([[tau].sub.ik] - [[tau].sub.ki])-([[tau].sub.ij] - [[tau].sub.ji]) (1)

The number of such relationships for a c component system is 0.5 x c(c - 3) + 1 (Hala, 1972). In the present work, the closure equation concept has been utilized first on the 116 nonionic liquid based systems, and then it has been used on the ionic liquids along with the closure equations.

THEORY AND CALCULATION

Genetic Algorithm

GA was first proposed by Holland (1975) and has been widely used recently. It is a method that searches for the global optima of an objective function through the use of simulated evolution; the survival of the fittest strategy. Unlike most of the optimization methods, GA does not require any initial guess but only the upper and lower bounds of the variables--in our case the interaction parameters, which is a characteristic of interval methods as well (Gau et al., 2000).

GA explores all regions of the solution space and exponentially exploits promising areas through selection, crossover and mutation operations applied to interaction parameters in the population. Float genetic algorithm (FGA) is better than both binary genetic algorithm (BGA) and SA in terms of computational efficiency and solution quality (Houck et al., 1995). FGA has been used for the estimation of interaction parameters. Flow diagram for FGA is given in Figure 1. FGA starts with initial populations of fixed size. Interaction parameters in the initial populations are generated randomly. Normalized geometric ranking, a probabilistic selection method, is used for the selection of populations. This method selects populations for next generation based on their fitness to the objective function. This is the role played by the probability of selecting the best individual (q). Remainder populations are randomly generated. Size of the population remains the same in each generation. In each generation, new populations are generated using genetic operators - crossover (arithmetic, heuristic, and simple) and mutation (boundary, multi-non-uniform, non-uniform, and uniform). Relationships for selection function and operators are given in Appendix A. GA moves from generation to generation until a termination criterion is met. The most frequently used stopping criterion is a specified maximum number of generations ([G.sub.max]) (Singh et al., 2005; Houck et al., 1996). Various input parameters used for selection function and operators are given in Table 1. Finally GA gives best population of interaction parameters which is available at the top of the list as organized by the ranking method from final generation as the required solution. All types of crossover and mutation operators have been used for our estimation. However, one can omit any of the crossover and/or mutation operators.

[FIGURE 1 OMITTED]

Closure Equations

The number of closure equation is one for ternary, three for quaternary and six for quinary system.

Ternary Systems

A ternary system has the following six binary interaction parameters [A.sub.12], [A.sub.21]; [A.sub.13], [A.sub.31]; and [A.sub.23], [A.sub.32]. Closure equation which describes the relationship between these six binary interaction parameters is (Ahmad and Khanna, 2003):

[A.sub.12] - [A.sub.21] + [A.sub.23] - [A.sub.32] + [A.sub.31] - [A.sub.13] = 0 (2)

That is only five out of six binary interaction parameters are independent.

Quaternary Systems

For a quaternary system, the twelve binary interaction parameters [A.sub.12], [A.sub.21]; [A.sub.13], [A.sub.31]; [A.sub.14], [A.sub.41]; [A.sub.23], [A.sub.32]; [A.sub.24], [A.sub.42]; and [A.sub.34], [A.sub.43].

Three closure equations for the quaternary systems are (Ahmad and Khanna, 2003):

[A.sub.12] - [A.sub.21] + [A.sub.23] - [A.sub.32] + [A.sub.31] - [A.sub.13] = 0 (3)

[A.sub.12] - [A.sub.21] + [A.sub.24] - [A.sub.42] + [A.sub.41] - [A.sub.14] = 0 (4)

[A.sub.23] - [A.sub.32] + [A.sub.34] - [A.sub.43] + [A.sub.42] - [A.sub.24] = 0 (5)

That is only nine out of twelve binary interaction parameters are independent.

Quinary Systems

A quinary system has the following twenty binary interaction parameters: [A.sub.12], [A.sub.21]; [A.sub.13], [A.sub.31]; [A.sub.14], [A.sub.41]; [A.sub.15], [A.sub.51]; [A.sub.23], [A.sub.32]; [A.sub.24], [A.sub.42]; [A.sub.25], [A.sub.52]; [A.sub.34], [A.sub.43]; [A.sub.35], [A.sub.53]; and [A.sub.45], [A.sub.54]. Six closure equations for quinary systems are (Ahmad and Khanna, 2003):

[A.sub.12] - [A.sub.21] + [A.sub.23] - [A.sub.32] + [A.sub.31] - [A.sub.13] = 0 (6)

[A.sub.12] - [A.sub.21] + [A.sub.24] - [A.sub.42] + [A.sub.41] - [A.sub.14] = 0 (7)

[A.sub.12] - [A.sub.21] - [A.sub.25] + [A.sub.52] - [A.sub.51] - [A.sub.15] = 0 (8)

[A.sub.23] - [A.sub.32] + [A.sub.34] - [A.sub.43] + [A.sub.42] - [A.sub.24] = 0 (9)

[A.sub.23] - [A.sub.32] + [A.sub.35] - [A.sub.53] + [A.sub.52] - [A.sub.25] = 0 (10)

[A.sub.24] - [A.sub.42] + [A.sub.45] - [A.sub.54] + [A.sub.52] - [A.sub.25] = 0 (11)

That is only fourteen out of twenty binary interaction parameters are independent.

Parameter Estimation Procedure

The calculations of LLE were carried using the UNIQUAC model as given in Appendix B. Binary interaction parameters are usually obtained from experimental LLE data by minimizing a suitable objective function.

Objective Function

The most common objective function is the sum of the square of the error between the experimental and calculated composition of all the components over the entire set of tie lines. As GA is only for maximization, for minimizing the errors between experimental and calculated mole fractions, the objective function can be defined as:

Max. F = - [m.summation over (t=1)] [2.summation over (l=1)] [c.summation over (i=1)] [w.sub.tli] [([x.sub.tli] - [[??].sub.tli]).sup.2]

w.r.t. [A.sub.ij] ; i, j = 1,2.... c and j [not equal to] i (12)

This objective function, with unit weights i.e. [w.sub.tli] = 1, has been used in this work. The goodness of fit is usually measured by root mean square deviation (rmsd) defined as:

rmsd = [(-F/2mc).sup.1/2] (13)

Modified Rachford Rice Algorithm (Seader and Henley, 1998) is used for the calculation of tie lines.

Parameter Elimination

The closure equations have been implemented by the elimination of the parameters, [A.sub.ij]'s equal to the number of closure equations. These eliminated parameters can be obtained by the simultaneous solution of closure equations. There exist several possibilities of parameter elimination. The number of such possibilities for ternary system is 6, for quaternary system is 220, and for quinary system is 38 760. However, all such possibilities are not feasible. A feasible set is such that the rank of the coefficient matrix of the eliminated parameters in the closure equations is equal to the number of independent closure equations. A general expression for the number of feasible sets of parameter elimination for a c component system with n number of closure equation is given as (Ahmad and Khanna, 2003; Hala, 1972):

[c.sup.c-2] x [2.sup.n] where n = 0.5 x c(c-3) + 1 (14)

The following two rules of thumb must be satisfied to obtain the feasible sets of eliminated parameters (Ahmad and Khanna, 2003),

1. Mirror images (i.e., [A.sub.ij] and [A.sub.ji]) cannot be eliminated simultaneously.

2. Same component cannot appear as subscript in all the eliminated parameters.

Thus, number of feasible sets of parameter elimination of ternary system is 6, for quaternary system is 128 and for quinary system is 8000. Summary of parameter elimination scheme is given in Table 2.

RESULTS AND DISCUSSION

Benchmarking

The benchmarking system involves the predictions of LLE for cyclohexane (1) + xylene (2) + sulpholane (3) ternary system at 35[degrees]C.

Lower and Upper Bounds, ([a.sub.i] and [b.sub.i])

The effect of the value of bounds on the interaction parameters and rmsd values are given in Table 3. The rmsd value remains constant after the bound of -1000 and +1000. The lower values of these, estimate some of the interaction parameters at the edge of the boundary. Keeping these in mind we have chosen -1000 and +1000 as the lower and upper bounds for our estimation.

Population Size, P

Effect of population size on absolute objective function value is shown in Figure 2. It is observed that population size of 100 is sufficient to give good set of parameters.

Maximum Number of Generation, [G.sub.max]

Variation of absolute objective function value against generation number is shown in Figure 3. It is observed that 200th generation is sufficient to give a very good set of parameters; further generations only seem to improve marginally upon the solution obtained. Therefore, value of 200 for [G.sub.max] has been used for our estimation.

Comparison of tie line compositions are given in Table 4. The rmsd value thus obtained is 0.006 as compared to 0.011 in the literature. After benchmarking with the above system, GA has been applied and verified on 88 aromatic and 28 hydrogen bonding multi-component systems. The rmsd values obtained using GA have been compared with the rmsd values reported in literature in terms of percentage gain:

[gain.sup.woce.sub.lit] = ([rmsd.sub.lit] - [rmsd.sub.woce]/[rmsd.sub.lit]) x 100 (15)

[gain.sup.woce.sub.woce] = ([rmsd.sub.woce] - [rmsd.sub.wce]/[rmsd.sub.woce]) x 100 (16)

[gain.sup.wce.sub.lit] = ([rmsd.sub.lit] - [rmsd.sub.wce]/[rmsd.sub.lit]) x 100 (17)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Aromatic Extraction Systems

Application of GA has been studied on 88 aromatic extraction systems as listed in Table 5a to 5c along with their references. UNIQUAC volume r and area q parameters have been taken from Aspen Plus (1998).

The results of parameter estimation along with the corresponding percentage gain values for the ternary systems are given in Table 6. It is seen that the rmsd value corresponding to the parameters with the closure equation taken into account is less than that without closure equations; approximately 21 percent better than without implementation of closure equation and 48 percent better than literature. This clearly means that parameters obtained with the closure equation will predict the LLE more accurately than those obtained without closure equation. While implementing the closure equation for ternary systems all the six possibilities of parameter elimination have been tried. It was observed that the rmsd values change slightly with the eliminated parameters. This is due to the different search path adopted by the optimization procedure for different eliminated parameter to reach the final optimum point. The interaction parameters with closure equation given in Table 6 correspond to the lowest rmsd for the six possibilities. From parameter elimination for ternary systems, it is observed that most of the cases elimination of parameters with only i=non-aromatic-j=aromatic, i=non-aromatic-j=solvent and i=solvent-j=aromatic combination gives better rmsd values than elimination of other binary pairs. Therefore, for quaternary and quinary systems feasible sets involving parameters with only these pairings, termed as 'likely feasible sets' have been considered. The number of such likely feasible sets of parameter elimination is 3 for ternary, 8 for quaternary and 21 for quinary systems.

The results of parameter estimation along with the corresponding percentage gain values for the quaternary and quinary systems are given in Table 7 and Table 8, respectively. It is seen, for quaternary and quinary systems too, the rmsd value corresponding to the parameters with the closure equations taken into account is less than that without closure equations; 23 percent better for quaternary and 18 percent better for quinary systems. This would mean, for quaternary and quinary systems also the parameters that satisfy the closure equations predict the LLE more accurately than those do not satisfy the closure equations.

The choice of the parameters to be eliminated to implement the closure equations plays a very important role in decreasing the rmsd values for quaternary and quinary systems. All likely feasible combinations of parameter elimination have been tried. The rmsd values varied slightly with the eliminated parameters. Reasons explained for the ternary systems apply here also for this variation. Additionally, the larger deviation is due to the higher dimension of optimization problem encountered. The interaction parameters with closure equations given in Table 7 and Table 8 correspond to the lowest rmsd of all likely feasible combination of parameter elimination.

Hydrogen Bonding Systems

Application of GA has also been verified with 28 hydrogen bonding systems as listed in Table 5d and Table 5e along with their references. UNIQUAC volume r and area q parameters have been taken from Aspen Plus (1998). The results of parameter estimation along with the corresponding percentage gain values for the ternary and quaternary systems are given in Table 9 and Table 10, respectively. It has been observed that for these systems also, the rmsd value corresponding to the parameters with the closure equations taken into account is less than that without closure equations; 25 percent better for ternary and 19 percent better for quaternary systems. The overall percentage gain is now 68 percent for ternary and 50 percent for quaternary systems than literature.

Ionic Liquid Based Systems

The ionic liquid based systems considered for parameter estimation are given in Table 5f along with their references. UNIQUAC structural parameters for ILs as estimated from Polarizable Continuum Model (PCM) have been used and are reported in Table 11. From Table 12 it can be said that the prediction with the closure equations is 6 percent better than without closure equations. Thus, it can be concluded that the implementation of closure equations for IL based systems gives a better prediction.

It has been observed that for some cases, rmsd values reported in the literature are better than GA. It has also been observed that for some cases, without implementation of closure equations exhibit better rmsd values than with closure equations. The reason is that GA gives solution near to global optima, so it should be followed by local minimization method to get exact global optima. Inbuilt MatLab routine, fminsearch has been used for local minimization. Table 13 shows the effect of local minimization on rmsd value. However, only marginal improvement in fourth decimal place has been achieved for the four systems considered. It means GA gives solution very close to global optima. This extra analysis is not needed since this improvement in fourth decimal place is not significant with respect to measurement errors.

CONCLUSIONS

A genetic algorithm, which is a structured search-based optimization method, has been applied without and with closure equations to estimate UNIQUAC binary interaction parameters for liquid-liquid extraction systems. Average root mean square deviation value with implementation of closure equations is 55 percent better than literature as compared to 40 percent better for without implementation of closure equations, which is viewed as a potentially useful improvement for the prediction of liquid-liquid equilibria. The effect of using different sequences of genetic operators and their parametric values has not been studied. Genetic algorithm can be applied to predict binary interaction parameters for vapour-liquid systems.
APPENDIX

Appendix A

GA selection function and operators (Houck et al., 1995, 1996)

Functions/Operators Type Relationship

Selection Function Normalized [P.sub.i] =
 geometric q'[(1 - q).sup.r-1];
 ranking q' = q/1 -[(1 - q).sup.P]

Genetic Operators [a.sub.i] and [b.sub.i] = lower
 and upper bound, respectively,
 for each variable i

 [bar.X] and [bar.Y] = two
 m-dimensional row vectors
 denoting individuals (parents)
 from the population

Mutation Uniform [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

 Boundary [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

 Non-uniform [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

 where f (G) G = [([r.sub.2]
 (1 - G/[G.sub.max])).sup.b]

 Multi-non- Applies the non-uniform
 uniform operator to all of the
 variables in the parent
 [bar.X].

Crossover Simple [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

 Arithmetic [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

 Heuristic [MATHEMATICAL EXPRESSION NOT
 REPRODUCIBLE IN ASCII]

Functions/Operators Type Notations

Selection Function Normalized [P.sub.i] = probability of
 geometric selection the ith individual,
 ranking
 q = probability of selecting
 the best individual,

 r = rank of the individual,
 where 1 is the best,

 P = population size

Genetic Operators

Mutation Uniform

 Boundary r = U(0,1)

 Non-uniform [r.sub.1], [r.sub.2] = U(0,1)]

 G = current generation,

 b = shape parameter,

 [G.sub.max] = maximum number
 of generations

 Multi-non-
 uniform

Crossover Simple r = U(1, m)

 Arithmetic r = U(0, 1)

 Heuristic r = U(0, 1)


Appendix B

UNIQUAC activity coefficient model (Prausnitz et al., 1999)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ACKNOWLEDGEMENT

The authors gratefully acknowledge use of the Genetic Algorithm Optimization Toolbox (GAOT) of MatLab, developed by Houck, Joines and Kay of North Carolina State University (Houck et al., 1995).

NOMENCLATURE
A binary interaction parameters ([degrees]K)
a number of binary interaction parameters
[a.sub.i] lower bound
[b.sub.i] upper bound
c total number of components
f number of retries
F objective function
gain percentage gain in rmsd
[G.sub.max] maximum number of generation
[G.sub.n] number of generation
[G.sub.n+1] next generation
lit from literature
m total number of tie lines
P population size
q area parameter
r volume parameter
rmsd root mean square deviation
T absolute temperature ([degrees]K)
x experimental composition (mole fraction)
[??] predicted composition (mole fraction)
wce with closure equations
woce without closure equations
z lattice coordination number

Greek Symbols

[gamma] activity coefficient
[tau] adjustable binary parameters

Subscripts

1, 2, 3, 4, 5 components
i, j, k components
lit from literature
t tie line
woce without closure equations

Superscripts

I, II phases
l phase
wce with closure equations
woce without closure equations


Manuscript received May 12, 2006; revised manuscript received February 25, 2007; accepted for publication March 21, 2007.

REFERENCES

Adjiman, C. S., S. Dallwig, C. A. Floudas and A. Neumaier, "A Global Optimization Method, ?BB, for General Twice-Differentiable Constrained NLPs--I. Theoretical Advances," Computers & Chemical Eng. 22, 1137-1158 (1998).

Ahmad, S. A. and A. Khanna, "Closure Equations in the Estimation of Binary Interaction Parameters," Korean J. Chem. Eng. 20, 736-744 (2003).

Ahmad, S. A., "LLE Experimentation Parameter Estimation and Validation for Aromatic Extraction Systems," PhD Thesis, Indian Institute of Technology Kanpur (2003).

Albert, M., I. Hahnenstein, H. Hasse and G. Maurer, "Vapor-Liquid and Liquid-Liquid Equilibria in Binary and Ternary Mixtures of Water, Methanol, and Methylal," J. Chem. Eng. Data 46, 897-903 (2001).

Al-Qattan, M. A., T. A. Al-Sahhaf and M. A. Fahim, "Liquid-Liquid Equilibria in Some Binary and Ternary Mixtures with Tetraethylene Glycol," J. Chem. Eng. Data 39, 111-113 (1994).

Anderson, T. F., D. S. Abrams and E. A. Grens II, "Evaluation of Parameters for Nonlinear Thermodynamic Models," AIChE J. 24, 20-29 (1978).

Arce, A., M. Blanco and A. Soto, "Quaternary Liquid-Liquid Equilibria of Systems with Two Partially Miscible Solvent Pairs: 1-Octanol + 2-Methoxy-2-Methylpropane + Water + Ethanol at 25[degrees]C," Fluid Phase Equilib. 146, 161-173 (1998).

Arce, A., M. Blanco and A. Soto, "Determination and Correlation of Liquid-Liquid Equilibrium Data for the Quaternary System 1-Octanol + 2-Methoxy-2-Methylbutane + Water + Methanol at 25[degrees]C," Fluid Phase Equilib. 158-160, 949-960 (1999).

Arce, A., O. Rodriguez and A. Soto, "Experimental Determination of Liquid-Liquid Equilibrium Using Ionic Liquids: tert-Amyl Ethyl Ether + Ethanol + 1-Octyl-3-Methylimidazolium Chloride System at 298.15 K," J. Chem Eng. Data 49, 514-517 (2004a).

Arce, A., O. Rodriguez and A. Soto, "Experimental Determination of Liquid-Liquid Equilibrium Using Ionic Liquids: tert-Amyl Ethyl Ether + Ethanol + 1-Octyl-3-Methylimidazolium Chloride System at 298.15 K," Ind. Eng. Chem. Res. 43, 8323-8327 (2004b).

Arce, A., O. Rodriguez and A. Soto, "Purification of Ethyl tert-Butyl Ether from its Mixtures with Ethanol by Using an Ionic Liquid," Chem. Eng. J. 115, 219-223 (2006).

Ashour, I., "Liquid-Liquid Equilibrium for MTBE + Ethanol + Water and MTBE + 1-Hexanol + Water over the Temperature Range of 288.15 to 308.15 K," J. Chem. Eng. Data 50, 113-118 (2005).

Aspen Plus Ver 10.0, Aspen Technology Inc., Cambridge, MA, U.S.A. (1998).

Banerjee, T., M. K. Singh, R. K. Sahoo and A. Khanna, "Volume, Surface and UNIQUAC Interaction Parameters for Imidazolium Based Ionic Liquids via Polarizable Continuum Model," Fluid Phase Equilib. 234, 64-76 (2005).

Banerjee, T., "Ionic Liquids - Phase Equilibria and Thermodynamic Property Predictions using Molecular Modeling and Dynamics, and their Validation with Experiments," PhD Thesis, Indian Institute of Technology Kanpur (2006)

Brennecke, J. F. and E. J. Maginn, "Ionic Liquids: Innovative Fluids for Chemical Processing," AIChE J 47, 2384-2389 (2001).

Britt, H. I. and R. H. Luecke, "The Estimation of Parameters in Nonlinear, Implicit Models," Technometrics 15, 233-247 (1973).

Cassell, G. W., N. Dural and A.W. Westerberg, "Liquid-Liquid Equilibrium of Sulfolane-Benzene-Pentane and Sulfolane-Toluene-Pentane," Ind. Eng. Chem. Res. 28, 1369-1374 (1989a).

Cassell, G. W., M. M. Hassan and A. L. Hines, "Liquid-Liquid Equilibria for the Hexane-Benzene-Dimethyl Sulfoxide Ternary System," J. Chem. Eng. Data 34, 328-331 (1989b).

Cassell, G. W., M. M. Hassan and A. L. Hines, "Correlation of the Phase Equilibrium Data for the Heptane-Toluene-Sulfolane and Heptane-Xylene-Sulfolane Systems," J. Chem. Eng. Data 34, 434-438 (1989c).

Chen, J., L. Duan, J. Mi, W. Fei and Z. Li, "Liquid-Liquid Equilibria of Multi-Component Systems Including n-Hexane, n-Octane, Benzene, Toluene, Xylene and Sulfolane at 198.15 K and Atmospheric Pressure," Fluid Phase Equilib. 173, 109-119 (2000a).

Chen, J., Z. Li and L. Duan, "Liquid-Liquid Equilibria of Ternary and Quaternary Systems Including Cyclohexane, 1-Heptene, Benzene, Toluene, and Sulfolane at 298.15 K," J. Chem. Eng. Data 45, 689-692 (2000b).

Chen, J., J. Mi, W. Fei and Z. Li, "Liquid-Liquid Equilibria of Quaternary and Quinary Systems Including Sulfolane at 298.15 K," J. Chem. Eng. Data 46, 169-171 (2001).

David, W., T. M. Letcher, D. Ramjugernath and J. D. Raal, "Activity Coefficients of Hydrocarbon Solutes at Infinite Dilution in the Ionic Liquid, 1-Methyl-3-Octyl-Imidazolium Chloride from Gas-Liquid Chromatography," J. Chem. Thermodynamics 35, 1335-1341 (2003).

Deenadayalu, N., C. N. Ngcongo, M. T. Letcher and D. Ramjugernath, "Liquid-Liquid Equilibria for Ternary Mixtures (an Ionic Liquid + Benzene + Heptane or Hexadecane at T) 298.2 K and Atmospheric Pressure," J. Chem. Eng. Data 51, 988-991 (2006).

Doulabi, F. S. M., M. Mohsen-Nia and H. Modarress, "Measurements and Modeling of Quaternary (Liquid + Liquid) Equilibria for Mixtures of (Methanol or Ethanol + Water + Toluene + n-Dodecane)," J. Chem. Thermodynamics 38, 405-412 (2005).

Esposito, W. R. and C. A. Floudas, "Global Optimization in Parameter Estimation of Nonlinear Algebraic Models via the Error-in-Variable Approach," Ind. Eng. Chem. Res. 37, 1841-1858 (1998).

Fabries, J. F. and H. Renon, "Method for Evaluation and Reduction of Vapor-Liquid Equilibrium Data of Binary Mixtures," AIChE J. 21, 735-743 (1975).

Ferreira, P. O., J. B. Ferreira and A. G. Medina, "Liquid-Liquid Equilibria for the System n-Methylpyrrolidone + Toluene + n-Heptane: UNIFAC Interaction Parameters for n-Methylpyrrolidone," Fluid Phase Equilib. 16, 369-379 (1984).

Fre, R. M. D. and L. A. Verhoeye, "Phase Equilibria in Systems Composed of an Aliphatic and an Aromatic Hydrocarbon and Sulfolane," J. appl. Chem. Biotechnol. 26, 469-487 (1976).

Gau, C. and M. A. Stadtherr, "Reliable Nonlinear Parameter Estimation Using Interval Analysis: Error-in-Variable Approach," Computers & Chemical Eng. 24, 631-637 (2000).

Gau, C., J. F. Brennecke and M. A. Stadtherr, "Reliable Nonlinear Parameter Estimation in VLE Modeling," Fluid Phase Equilib. 168, 1-18 (2000).

Gmehling, J., U. Onken and W. Arlt, "Vapor-Liquid Equilibrium Data Collection: Parts 1-8," Chemistry Data Series Vol. I DECHEMA, Frankfurt Main, Germany (1990), pp. 1977-1990.

Hala, E., "Note to Bruin-Prausnitz One-Parameter and Palmer-Smith Two-Parameter Local Composition Equation," Ind. Eng. Chem. Process Des. Dev. 11, 638 (1972).

Hassan, M. S. and M. A. Fahim, "Correlation of Phase Equilibria of Naphtha Reformate with Sulfolane," J. Chem. Eng. Data 33, 162-165 (1988).

Heintz, A., J. K. Lehmann and C. Wertz, "Thermodynamic Properties of Mixtures Containing Ionic Liquids. 3. Liquid-Liquid Equilibria of Binary Mixtures of 1-Ethyl-3-methylimidazolium Bis(trifluoromethylsulfonyl)imide with Propan-1-ol, Butan-1-ol, and Pentan-1-ol," J. Chem. Eng. Data 48, 472-474 (2003).

Holland, J. H., "Adaptation in Natural and Artificial Systems," University of Michigan Press, Ann Arbor (1975).

Houck, C. R., J. A. Joines and M. G. Kay, NCSU-IE TR 95-09, 1995. (URL:www.ie.ncsu.edu/mirage/GAToolBox/goat.)

Houck, C. R., J. A. Joines and M. G. Kay, "Comparison of Genetic Algorithms, Random Restart and Two-Opt Switching for Solving Large Location--Allocation Problems," Computers & Operations Research 23, 587-596 (1996).

Hughes, M. A. and Y. Haoran, "Liquid-Liquid Equilibria for Separation of Toluene from Heptane by Benzyl Alcohol Tri (ethylene glycol) Mixtures," J. Chem. Eng. Data 35, 467-471 (1990).

Julia, J. A., C. R. Barrero, M. E. Corso, M. C. Grande and C. M. Marschoff, "On the Application of the NRTL Method to Ternary (Liquid + Liquid) Equilibria," J. Chem. Thermodynamics 37, 437-443 (2005).

Kang, C. H. and S. I. Sandler, "Phase Behavior of Aqueous Two-Polymer Systems," Fluid Phase Equilib. 38, 245-272 (1987).

Katayama, H., T. Hayakawa and T. Kobayashi, "Liquid-Liquid Equilibria of Three Ternary Systems: 2-Propanone-Glycerol-Methanol, 2-Butanone-Glycerol-Ethanol, and 2-ButanoneGlycerol-2-Propanol in the Range of 283.15 to 303.15 K," Fluid Phase Equilib. 144, 157-167 (1998).

Lee, S. and H. Kim, "Liquid-Liquid Equilibria for the Ternary Systems Sulfolane + Octane + Benzene, Sulfolane + Octane + Toluene, and Sulfolane + Octane + p-Xylene," J. Chem. Eng. Data 40, 499-503 (1995).

Letcher, T. M. and N. Deenadayalu, "Ternary Liquid-Liquid Equilibria for Mixtures of an Alkane + an Aromatic Compound + 1,3-Dimethyl-2-imidazolidinone at 298.15 K and 1 atm," J. Chem. Eng. Data 46, 177-183 (2001).

Letcher, T. M. and P. K. Naicker, "Liquid-Liquid Equilibria for Mixtures or Water + an Alkanol + a Nitrile Compound at T=298.15 K," J. Chem. Eng. Data 46, 1436-1441 (2001).

Letcher, T. M. and N. Deenadayalu, "Ternary Liquid-Liquid Equilibria for Mixtures of 1-Methyl-3-Octyl-Imidazolium Chloride + Benzene + an Alkane at T=298.2 K and 1 atm," J. Chem. Thermodynamics 35, 67-76 (2003).

Letcher, T. M., S. Zondi and P. K. Naicker, "Liquid-Liquid Equilibria for Mixtures of (Furfural + an Aromatic Hydrocarbon + an Alkane) at T = 298.15 K," J. Chem. Eng. Data 48, 23-28 (2003a).

Letcher, T. M., N. Deenadayalu, B. Soko and D. Ramjugernath, "Ternary Liquid-Liquid Equilibria for Mixtures of 1-Methyl-3-Octylimidazolium Chloride + an Alkanol + an Alkane at 298.2 K and 1 bar," J. Chem. Eng. Data 48, 904-907 (2003b).

Letcher, T. M. and P. Reddy, "Ternary Liquid-Liquid Equilibria for Mixtures of 1-Hexyl-3-Methylimidozolium (Tetrafluoroborate or Hexafluorophosphate) + Ethanol + an Alkene at T=298.2 K," Fluid Phase Equilib. 219, 107-112 (2004).

Letcher, T. M. and P. Reddy, "Ternary (Liquid + Liquid) Equilibria for Mixtures of 1-Hexyl-3-Methylimidazolium (Tetrafluoroborate or Hexafluorophosphate) + Benzene + an Alkane at T=298.2 K and p=0.1 MPa," J. Chem. Thermodynamics 37, 415-421 (2005).

Lin, W. C. and N. H. Kao, "Liquid-Liquid Equilibria of Octane + (Benzene or Toluene or m-Xylene) + Sulfolane at 323.15, 348.15, and 373.15 K," J. Chem. Eng. Data 47, 1007-1011 (2002).

Marsh, K. N., J. A. Boxall and R. Lichtenthaler, "Room Temperature Ionic Liquids and Their Mixtures--A Review," Fluid Phase Equilib. 219, 93-98 (2004).

Meindersma, W. G., A. Podt and B. A. Hann, "Ternary Liquid-Liquid Equilibria for Mixtures of Toluene + n-Heptane + an Ionic Liquid," Fluid Phase Equilib. 247 158-168 (2006).

Mohsen-Nia, M., H. Modarress and F. S. M. Doulabi, "Quaternary Liquid-Liquid Equilibria for Systems of {(Water + Methanol or Ethanol) + m-Xylene + n-Dodecane}," Fluid Phase Equilib. 239, 1-7 (2006).

Mondragon-Garduno, M., A. Romero-Martinez and A. Trejao, "Liquid-Liquid Equilibria for Ternary Systems. I. C6-Isomers + Sulfolane + Toluene at 298.15 K," Fluid Phase Equilib. 64, 291-303 (1991).

Moore, R. E., E. Hansen, A. Leclerc, C. A. Floudas and P. M. Pardolos, "Recent Advances in Global Optimization," Princeton Univ. Press, Princeton, NJ (1992), pp. 321.

Morawski, P., T. M. Letcher and P. K. Naicker, "Liquid-Liquid Equilibria for Mixtures of (Furfuryl Alcohol + an Aromatic Hydrocarbon + Alkane) at T=298.15 K," J. Chem. Eng. Data 47, 1453-1456 (2002).

Morawski, P., T. M. Letcher and P. K. Naicker, "Liquid-Liquid Equilibria for Mixtures of (Furfural + a Chlorinated Aromatic Compound + an Alkane) at T = 298.15 K," J. Chem. Eng. Data 48, 822-826 (2003).

Papadopoulos, A. I. and P. Linke, "On the Synthesis and Optimization of Liquid-Liquid Extraction Processes Using Stochastic Search Methods," Computers & Chemical Eng. 28, 2391-2406 (2004).

Prausnitz, J. M., R. N. Lechtenthaler and E. G. Azevedo, "Molecular Thermodynamics of Fluid Phase Equilibria," 3rd ed., Prentice-Hall (1999).

Radwan, G. M., S. A. Al-Muhtaseb and M. A. Fahim, "Liquid-Liquid Equilibria for the Extraction of Aromatics from Naphtha Reformate by Dimethylformamide/Ethylene Glycol Mixed Solvent," Fluid Phase Equilib. 129, 175-186 (1997a).

Radwan, G. M., S. A. Al-Muhtaseb, A. M. Dowaidar and M. A. Fahim, "Extraction of Aromatics from Petroleum Naphtha Reformate by a 1-Cyclohexyl-2-pyrrolidone/Ethylene Carbonate Mixed Solvent," Ind. Eng. Chem. Res. 36, 414-418 (1997b).

Rappel, R., L. M. N. Gois and S. Mattedi, "Liquid-Liquid Equilibria Data for Systems Containing Aromatic + Nonaromatic + Sulfolane at 308.15 and 323.15 K," Fluid Phase Equilib. 202, 263-276 (2002).

Ratz, D. and T. Csendes, "On the Selection of Subdivision Directions in Interval Branch-and-Bound Methods for Global Optimization," J. Global Optim. 7, 183-207 (1995).

Rawat, B. S. and I. B. Gulati, "Liquid-Liquid Equilibria Studies for Separation of Aromatics," J. Appl. Chem. Biotechnol. 26, 425-435 (1976).

Rawat, B. S. and G. Prasad, "Liquid-Liquid Equilibria for Benzene-n-Heptane Systems with Triethylene Glycol, Tetraethylene Glycol, and Sulfolane Containing Water at Elevated Temperatures," J. Chem. Eng. Data 25, 227-230 (1980).

Sahoo, R. K., T. Banerjee, S. A. Ahmad and A. Khanna, "Improved Binary Parameters using GA for Multi-Component Aromatic Extraction: NRTL Model Without and With Closure Equations," Fluid Phase Equilib. 239, 107-119 (2006).

Salem, A. B. S. H., "Liquid-Liquid Equilibria for the Systems Triethylene Glycol-Toluene-Heptane, Propylene Carbonate-Toluene-Heptane, and Propylene Carbonate-o-XyleneHeptane," Fluid Phase Equilib. 86, 351-361 (1993).

Schwetlick, H. and V. Tiller, "Numerical Methods for Estimating Parameters in Nonlinear Models with Errors in the Variables," Technometrics 27, 17-24 (1985).

Seader, J. D. and E. J. Henley, "Separation Process Principles," John Wiley, New York (1998), pp. 196.

Selvan, M. S., M. D. McKinley, R. H. Dubois and J. L. Atwood, "Liquid-Liquid Equilibria for Toluene + Heptane + 1-Ethyl-3-methylimidazolium Triiodide and Toluene + Heptane + 1-Butyl-3-methylimidazolium Triiodide," J. Chem. Eng. Data 45, 841-845 (2000).

Singh, M. K., T. Banerjee and A. Khanna, "Genetic Algorithm to Estimate Interaction Parameters of Multicomponent Systems for Liquid-Liquid Equilibria," Computers & Chemical Eng. 29, 1712-1719 (2005).

Sorenson, J. M. and W. Arlt, "Liquid-Liquid Equilibrium Data Collection," DECHEMA, Chemistry Data Series V, Part 2 & 3, Frankfurt am Main (1980).

Tjoa, I. B and L. T. Biegler, "Simultaneous Strategies for Data Reconciliation and Gross Error Detection of Nonlinear Systems," Computers & Chemical Eng. 15, 679-690 (1991).

Tjoa, I. B. and L. T. Biegler, "Reduced Successive Quadratic Programming Strategy for Errors-in-Variables Estimation," Computers & Chemical Eng. 16, 523-533 (1992).

Vaidya, P. S. and R. V. Naik, "Liquid-Liquid Equilibria for the Epichlorohydrin + Water + Methanol and Allyl Chloride + Water + Methanol Systems," J. Chem. Eng. Data 48, 1015-1018 (2003).

Valko, P. and S. Vajda, "An Extended Marquardt-Type Procedure for Fitting Error-in-Variables Models," Computers & Chemical Eng. 11, 37-43 (1987).

Vamos, R. J. and C. N. Hass, "Reduction of Ion Exchange Equilibria Data Using an Error in Variables Approach," AIChE J. 40, 556-569 (1994).

Zhu, Y. and K. Inoue, "Calculation of Chemical and Phase Equilibrium Based on Stability Analysis by QBB Algorithm: Application to NRTL Equation," Chemical Engineering Science 56, 6915-6931 (2001).

Ranjan Kumar Sahoo, Tamal Banerjee and Ashok Khanna *

Department of Chemical Engineering, Indian Institute of Technology, Kanpur, INDIA-208016

* Author to whom correspondence may be addressed. E-mail address: akhanna@iitk.ac.in
Table 1. Values of the parameters used in GA Toolbox of MatLab
(Houck et al., 1995, 1996)

Name Value

Population size, P 100
Maximum number of generations, [G.sub.max] 200
Normalized geometric selection
 Probability of selecting the best individual, q 0.08
Uniform mutation, number of times 4
Non-uniform mutation, number of times 4
 Shape parameter 3
Multi-non-uniform mutation, number of times 6
 Shape parameter 3
Boundary mutation, number of times 4
Simple crossover, number of times 4
Arithmetic crossover, number of times 4
Heuristic crossover, number of times 2
 Feasibility check, number of times 3

refer to Appendix A

Table 2. Parameter elimination scheme

Parameter elimination System
procedure
 Ternary Quaternary Quinary

number of components
c 3 4 5

number of binary
interaction parameters
a = 2 x [sup.c][C.sub.2] 6 12 20

number of closure
equations n = 0.5 x
c(c - 3) + 1 1 3 6

number of parameter
elimination possibilities
[sup.a][C.sub.n] 6 220 38 760

number of feasible sets
of parameter elimination
[C.sup.c-2] x [2.sup.n] 6 128 8000

number of likely feasible
sets of parameter
elimination 3 8 21 *

* non-aromatic-non-aromatic-aromatic-aromatic-solvent system

Table 3. Effect of bounds on UNIQUAC interaction parameters and
rmsd values for the system cyclohexane (1)-xylene (2)-sulpholane
(3) at 35[degrees]C

 [A.sub.ij]/K, woce

Bounds [A.sub.12] [A.sub.13] [A.sub.23]
 [A.sub.21] [A.sub.31] [A.sub.32]

[+ or -] 200 -105.67 200.00 200.00
 84.29 200.00 31.12

[+ or -] 500 -20.93 500.00 229.16
 7.78 92.78 14.92

[+ or -] 1000 -233.41 662.70 269.03
 422.52 67.80 9.20

[+ or -] 1500 237.05 356.13 910.79
 -229.09 127.34 -137.64

[+ or -] 2000 -355.88 668.98 562.32
 840.04 35.41 -98.37

Bounds rmsd, Hitting
 woce bound

[+ or -] 200 0.017 Yes

[+ or -] 500 0.007 Yes

[+ or -] 1000 0.006 No

[+ or -] 1500 0.006 No

[+ or -] 2000 0.006 No

Table 4. Comparison of tielines for cyclohexane (1)-xylene
(2)-sulpholane (3) at 35[degrees]C (Rappel et al., 2002)

Experimental

solvent rich phase

[x.sub.1] [x.sub.2] [x.sub.3]

0.0241 0.0394 0.9365
0.0225 0.0568 0.9206
0.0312 0.1050 0.8639
0.0316 0.1694 0.8000
0.0259 0.1018 0.8723

Experimental

non-aromatic rich phase

[x.sub.1] [x.sub.2] [x.sub.3]

0.7230 0.2710 0.0060
0.6456 0.3457 0.0087
0.4303 0.5520 0.0178
0.3154 0.6658 0.0188
0.4540 0.5276 0.0185

Predicted

solvent rich phase

[[??].sub.1] [[??].sub.2] [[??].sub.3]

0.0276 0.0418 0.9305
0.0266 0.0572 0.9162
0.0238 0.1104 0.8658
0.0237 0.1614 0.8149
0.0240 0.1028 0.8732

Predicted

non-aromatic rich phase

[[??].sub.1] [[??].sub.2] [[??].sub.3]

0.7250 0.2704 0.0046
0.6456 0.3472 0.0072
0.4352 0.5440 0.0208
0.3128 0.6558 0.0314
0.4551 0.5257 0.0192

rmsd [lit] = 0.011

rmsd (woce) = 0.006

Table 5. Ternary, quaternary and quinary systems used for parameter
estimation

System no. System name

 (a) Ternary aromatic extraction systems

1 Pentane(1)-Benzene(2)-Sulpholane(3)
2 Pentane(1)-Toluene(2)-Sulpholane(3)
3 Hexane(1)-Benzene(2)-Sulpholane(3)
4 Hexane(1)-Toluene(2)-Sulpholane(3)
5 Hexane(1)-Xylene(2)-Sulpholane(3)
6 Heptane(1)-Benzene(2)-Sulpholane(3)
7 Heptane(1)-Toluene(2)-Sulpholane(3)
8 Heptane(1) + xylene(2) + Sulpholane(3)
9 Octane(1)-Benzene(2)-Sulpholane(3)
10 Octane(1)-Toluene(2)-Sulpholane(3)
11 Octane(1)-Xylene(2)-Sulpholane(3)
12 Cyclohexane(1)-Benzene(2)-Sulpholane(3)
13 Cyclohexane(1) + toluene(2) + sulpholane(3)
14 Cyclohexane(1)-Xylene(2)-Sulpholane(3)
15 Hexane(1)-Benzene(2)-Sulpholane(3)
16 Hexene(1) + toluene(2) + sulpholane(3)
17 Heptene(1) + benzene(2) + sulpholane(3)
18 Heptene(1) + toluene(2) + sulpholane(3)
19 2-Methylpentane(1) + toluene(2) + sulpholane(3)
20 Hexane(1)-Benzene(2)-Dimethyl Sulphoxide(3)
21 Heptane(1) + benzene(2) + dimethyl sulphoxide(3)
22 Heptane(1) + toluene(2) + dimethyl sulphoxide(3)
23 Cyclohexane(1) + toluene(2) + dimethyl sulphoxide(3)
24 Heptane(1)-Ethylbenzene(2)-Ethylene Glycol(3)
25 Heptane(1) + benzene(2) + diethylene glycol(3)
26 Heptane(1) + toluene(2) + diethylene glycol(3)
27 Hexane(1) + benzene(2) + triethylene glycol(3)
28 Heptane(1) + benzene(2) + triethylene glycol(3)
29 Heptane(1) + toluene(2) + triethylene glycol(3)
30 Cyclohexane(1) + benzene(2) + triethylene glycol(3)
31 Decane(1) + benzene(2) + tetraethylene glycol(3)
32 Decane(1)-Ethylbenzene(2)-Tetraethylene Glycol(3)
33 Hexane(1) + benzene(2) + n-methyl pyrrolidone(3)
34 Heptane(1) + benzene(2) + n-methyl pyrrolidone(3)
35 Heptane(1)-Toluene(2)-N-Methyl Pyrrolidone(3)
36 Hexane(1)-Benzene(2)-N-Methylformamide(3)
37 Heptane(1) + benzene(2) + n-methyl formamide(3)
38 Hexane(1) + benzene(2) + dimethyl formamide(3)
39 Heptane(1) + benzene(2) + dimethyl formamide(3)
40 Dodecane(1)-Benzene(2)-Dimethylformamide(3)
41 Dodecane(1)-Ethylbenzene(2)-Dimethylformamide(3)
42 Heptane(1) + toluene(2) + propylene carbonate(3)
43 Heptane(1) + xylene(2) + propylene carbonate(3)
44 Hexane(1) + benzene(2) + furfural(3)
45 Hexane(1) + toluene(2) + furfural(3)
46 Hexane(1) + xylene(2) + furfural(3)
47 Cyclohexane(1) + benzene(2) + furfural(3)
48 Hexane(1) + chlorobenzene(2) + furfural(3)
49 Hexane(1) + 1,2-dichlorobenzene(2) + furfural(3)
50 Hexane(1) + 1,3-dichlorobenzene(2) + furfural(3)
51 Hexane(1) + 1,2,4-trichlorobenzene(2) + furfural(3)
52 Dodecane(1) + chlorobenzene(2) + furfural(3)
53 Hexadecane(1) + chlorobenzene(2) + furfural(3)
54 Hexadecane(1) + 1,2-dichlorobenzene(2) + furfural(3)
55 Hexadecane(1) + 1,3-dichlorobenzene(2) + furfural(3)
56 Hexadecane(1) + 1,2,4-trichlorobenzene(2) +
 furfural(3)
57 Hexane(1) + benzene(2) + furfuryl alcohol(3)
58 Hexane(1) + methylbenzene(2) + furfuryl alcohol(3)
59 Hexane(1) + 1,2-dimethylbenzene(2) + furfuryl
 alcohol(3)
60 Dodecane(1) + benzene(2) + furfuryl alcohol(3)
61 Dodecane(1) + methylbenzene(2) + furfuryl alcohol(3)
62 Dodecane(1) + 1,2-dimethylbenzene(2) + furfuryl
 alcohol(3)
63 Hexadecane(1) + benzene(2) + furfuryl alcohol(3)
64 Hexadecane(1) + methylbenzene(2) + furfuryl
 alcohol(3)
65 Hexadecane(1) + 1,2-dimethylbenzene(2) + furfuryl
 alcohol(3)
66 Hexadecane(1) + benzene(2) + 1,3-dimethyl-2-
 imidazolidinone(3)
67 Hexadecane(1) + toluene(2) + 1,3-dimethyl-2-
 imidazolidinone(3)
68 Hexadecane(1) + xylene(2) + 1,3-dimethyl-2-
 imidazolidinone(3)
69 Hexadecane(1) + mesitylene(1) + 1,3-dimethyl-2-
 imidazolidinone(3)
70 Hexadecane(1) + ethylbenzene(2) + 1,3-dimethyl-2-
 imidazolidinone(3)
71 Heptane(1) + benzene(2) + thiodiglycol(3)
72 Heptane(1) + toluene(2) + thiodiglycol(3)
73 Heptane(1) + ethylbenzene(2) + ethylene carbonate(3)
74 Heptane(1) + toluene(2) + benzyl alcohol(3)
75 Heptane(1) + toluene(2) + 3-methyl sulpholane(3)
76 Heptane(1) + toluene(2) + mercaptoethanol(3)

 (b) Quaternary aromatic extraction systems

77 Hexane(1) + benzene(2) + xylene(3) + sulpholane(4)
78 Heptane(1) + benzene(2) + toluene(3) + sulpholane(4)
79 Octane(1) + toluene(2) + xylene(3) + sulpholane(4)
80 Hexane(1) + heptane(2) + toluene(3) + sulpholane(4)
81 Hexane(1) + octane(2) + benzene(3) + sulpholane(4)
82 Heptane(1) + octane(2) + xylene(3) + sulpholane(4)
83 Cyclohexane(1) + heptene(2) + benzene(3) +
 sulpholane(4)
84 Cyclohexane(1) + heptene(2) + toluene(3) +
 sulpholane(4)
85 Heptane(1) + toluene(2) + triethylene glycol(3) +
 benzyl alcohol(4)

 (c) Quinary aromatic extraction systems

86 Hexane(1) + heptane(2) + toluene(3) + xylene(4) +
 sulpholane(5)
87 Hexane(1) + octane(2) + benzene(3) + toluene(4) +
 sulpholane(5)
88 Heptane(1) + octane(2) + benzene(3) + xylene(4) +
 sulpholane(5)

 (d) Ternary hydrogen bonding systems

89 2-Propanone(1) + glycerol(2) + methanol(3)
90 2-Butanone(1) + glycerol(2) + ethanol(3)
91 2-Butanone(1) + glycerol(2) + 2-propanone(3)
92 Methyl tert-butyl ether(1) + ethanol(2) + water(3)
93 Methyl tert-butyl ether(1) + 1-hexanol(2) + water(3)
94 Epichlorohydrin(1) + water(2) + methanol(3)
95 Allyl chloride(1) + water(2) + methanol(3)
96 Water(1) + methanol(2) + methylal(3)
97 Water(1) + methanol(2) + 1,4-dicyanobutane(3)
98 Water(1) + ethanol(2) + 1,4-dicyanobutane(3)
99 Water(1) + 1-propanol(2) + 1,4-dicyanobutane(3)
100 Water(1) + methanol(2) + butanenitrile(3)
101 Water(1) + ethanol(2) + butanenitrile(3)
102 Water(1) + 1-propanol(2) + butanenitrile(3)
103 Water(1) + methanol(2) + benzonitrile(3)
104 Water(1) + ethanol(2) + benzonitrile(3)
105 Water(1) + 1-propanol(2) + benzonitrile(3)
106 2-Methoxy-2-methyl propane(1) + ethanol(2) +
 water(3)
107 2-Methoxy-2-methyl propane(1) + 1-octanol(2) +
 water(3)
108 1-Octanol(1) + ethanol(2) + water(3)

 (e) Quaternary hydrogen bonding systems

109 Water(1) + methanol(2) + xylene(3) + dodecane(4)
110 Water(1) + ethanol(2) + xylene(3) + dodecane(4)
111 Methanol(1) + ethanol(2) + xylene(3) + dodecane(4)
112 Water(1) + methanol(2) + toluene(3) + dodecane(4)
113 Water(1) + ethanol(2) + toluene(3) + dodecane(4)
114 Methanol(1) + ethanol(2) + toluene(3) + dodecane(4)
115 1-Octanol(1) + 2-methoxy-2-methyl propane(2) +
 water(3) + ethanol(4)
116 1-Octanol(1) + 2-methoxy-2-methyl buta ne(2) +
 water(3) + methanol(4)

 (f) Ternary ionic liquid based systems

117 [Hmim][[PF.sub.6]] (1) + Benzene(2) + Heptane(3)
118 [Hmim][[BF.sub.4]](1) + Benzene(2) + Heptane(3)
119 [Hmim][[BF.sub.4]](1) + Benzene(2) + Dodecane(3)
120 [Hmim][[BF.sub.4]](1) + Benzene(2) + Hexadecane(3)
121 [Hmim][[PF.sub.6]] (1) + Benzene(2) + Dodecane(3)
122 [Hmim][[PF.sub.6]] (1) + Benzene(2) + Hexadecane(3)
123 [Omim][CI](1) + Ethanol(2) + Hexadecane(3)
124 [Omim][CI](1) + Ethanol(2) + tert-Amyl ethyl
 ether (3)
125 [Hmim][[BF.sub.4]](1) + Ethanol(2) + Heptene(3)
126 [Hmim][[BF.sub.4]] (1) + Ethanol(2) + Hexene(3)
127 [Hmim][[PF.sub.6]] (1) + Ethanol(2) + Hexene(3)
128 [Hmim][[PF.sub.6]] (1) + Ethanol(2) + Heptene(3)
129 [Bmim][[CF.sub.3]S[O.sub.3]](1) + Ethanol(2) +
 tert-Amyl ethyl ether (3)
130 [Bmim][([CF.sub.3]S[O.sub.3]](1) + Ethanol(2) +
 Ethyl-tert-Butyl Ether(3)
131 [Omim][CI](1) + Methanol(2) + Hexadecane(3)
132 [Omim][CI] (1) + Benzene(2) + Heptane(3)
133 [Omim][CI] (1) + Benzene(2) + Dodecane(3)
134 [Omim][CI] (1) + Benzene(2) + Hexadecane(3)
135 [Emim][OcS[O.sub.4]] (1) + Benzene (2) + Heptane(3)
136 [Emim][OcS[O.sub.4]] (1) + Benzene(2) +
 Hexadecane(3)
137 [Omim][MDeg] (1) + Benzene (2) + Heptane(3)
138 [Omim][MDeg] (1) + Benzene (2) + Hexadecane(3)
139 [Mmim][MeS[O.sub.4]] (1) + Toluene (2) + Heptane(3)
140 [Emim][EthS[O.sub.4]] (1) + Toluene (2) + Heptane(3)
141 [Bmim][[EthS[O.sub.4]] (1) + Toulene (2) +
 Heptane(3)
142 [Emim][EthS[O.sub.4]] (1) + Ethanol(2) + Hexene(3)
143 [Emim][EthS[O.sub.4]] (1) + Ethanol(2) + Heptene(3)
144 [1,2-Emim][EthS[O.sub.4]] (1) + Ethanol(2) +
 Hexene(3)
145 [1,2-Emim][EthS[O.sub.4]] (1) + Ethanol(2) +
 Heptene(3)

System no. Temp., [degrees]C

 (a) Ternary aromatic extraction systems

1 17.0, 25.0, 50.0
2 17.0, 25.0, 50.0
3 25.0, 30.0, 50.0, 75.0, 100.0
4 25.0, 35.0, 50.0
5 25.0, 35.0, 50.0
6 25.0, 110.0
7 25.0, 30.0, 50.0, 75.0, 100.0
8 25.0
9 25.0, 35.0, 45.0, 100.0
10 25.0, 50.0, 75.0
11 25.0, 30.0, 45.0, 50.0
12 25.0, 75.0, 100.0
13 25.0
14 35.0, 50.0
15 25.0, 50.0, 100.0
16 25.0
17 25.0
18 25.0
19 25.0
20 10.0, 25.0, 50.0
21 20.5
22 25.0
23 20.0
24 20.0, 30.0, 50.0
25 50.0
26 25.0
27 20.0
28 20.0
29 25.0
30 20.0
31 29.0
32 25.5, 39.0, 50.0
33 25.0
34 25.0
35 15.0, 25.0, 40.0
36 20.0, 25.0
37 20.0
38 25.0
39 20.0
40 20.0, 30.0, 40.0
41 20.0, 30.0, 50.0
42 25.0
43 25.0
44 25.0
45 25.0
46 25.0
47 25.0
48 25.0
49 25.0
50 25.0
51 25.0
52 25.0
53 25.0
54 25.0
55 25.0
56 25.0
57 25.0
58 25.0
59 25.0
60 25.0
61 25.0
62 25.0
63 25.0
64 25.0
65 25.0
66 25.0
67 25.0
68 25.0
69 25.0
70 25.0
71 50.0
72 50.0
73 40.0
74 25.0
75 25.0
76 25.0

 (b) Quaternary aromatic extraction systems

77 25.0
78 25.0
79 25.0
80 25.0
81 25.0
82 25.0
83 25.0
84 25.0
85 25.0, 40.0

 (c) Quinary aromatic extraction systems

86 25.0
87 25.0
88 25.0

 (d) Ternary hydrogen bonding systems

89 20.0
90 20.0
91 20.0
92 25.0
93 25.0
94 0.0
95 0.0
96 20.0
97 25.0
98 25.0
99 25.0
100 25.0
101 25.0
102 25.0
103 25.0
104 25.0
105 25.0
106 25.0
107 25.0
108 25.0

 (e) Quaternary hydrogen bonding systems

109 25.0
110 25.0
111 25.0
112 25.0
113 25.0
114 25.0
115 25.0
116 25.0

 (f) Ternary ionic liquid based systems

117 25.0
118 25.0
119 25.0
120 25.0
121 25.0
122 25.0
123 25.0
124 25.0
125 25.0
126 25.0
127 25.0
128 25.0
129 25.0
130 25.0
131 25.0
132 25.0
133 25.0
134 25.0
135 25.0
136 25.0
137 25.0
138 25.0
139 25.0
140 25.0
141 25.0
142 40.15
143 40.15
144 40.15
145 40.15

System no. Reference

 (a) Ternary aromatic extraction systems

1 (Cassell et al., 1989a)
2 (Cassell et al., 1989a)
3 (Fre and Verhoeye, 1976, Hassan and Fahim, 1988)
4 (Chen et al., 2000a, Rappel et al., 2002)
5 (Chen et al., 2000a, Rappel et al., 2002)
6 (Rawat and Gulati, 1976, Rawat and Prasad, 1980)
7 (Fre and Verhoeye, 1976, Hassan and Fahim, 1988)
8 (Cassell et al., 1989c)
9 (Lee and Kim, 1995, Lin and Kao, 2002)
10 (Lee and Kim, 1995, Lin and Kao, 2002)
11 (Hassan and Fahim, 1988, Lee and Kim, 1995, Lin
 and Kao, 2002)
12 (Fre and Verhoeye, 1976)
13 (Mondragon-Garduho et al., 1991)
14 (Rappel et al., 2002)
15 (Fre and Verhoeye, 1976)
16 (Mondragon-Garduho et al., 1991)
17 (Chen et al., 2000b)
18 (Chen et al., 2000b)
19 (Mondragon-Garduho et al., 1991)
20 (Cassell et al., 1989b)
21 (Sorenson and Arlt, 1980)
22 (Sorenson and Arlt, 1980)
23 (Sorenson and Arlt, 1980)
24 (Radwan et al., 1997a)
25 (Sorenson and Arlt, 1980)
26 (Sorenson and Arlt, 1980)
27 (Sorenson and Arlt, 1980)
28 (Sorenson and Arlt, 1980)
29 (Hughes and Haoran, 1990)
30 (Sorenson and Arlt, 1980)
31 (Al-Qattan et al., 1994)
32 (Al-Qattan et al., 1994)
33 (Ahmad, 2003)
34 (Sorenson and Arlt, 1980)
35 (Ferreira et al., 1984)
36 (Ahmad, 2003, Sorenson and Arlt, 1980)
37 (Sorenson and Arlt, 1980)
38 (Ahmad, 2003)
39 (Sorenson and Arlt, 1980)
40 (Radwan et al., 1997a)
41 (Radwan et al., 1997a)
42 (Salem, 1993)
43 (Salem, 1993)
44 (Letcher et al., 2003a)
45 (Letcher et al., 2003a)
46 (Letcher et al., 2003a)
47 (Sorenson and Arlt, 1980)
48 (Morawski et al., 2003)
49 (Morawski et al., 2003)
50 (Morawski et al., 2003)
51 (Morawski et al., 2003)
52 (Morawski et al., 2003)
53 (Morawski et al., 2003)
54 (Morawski et al., 2003)
55 (Morawski et al., 2003)
56 (Morawski et al., 2003)
57 (Morawski et al., 2002)
58 (Morawski et al., 2002)
59 (Morawski et al., 2002)
60 (Morawski et al., 2002)
61 (Morawski et al., 2002)
62 (Morawski et al., 2002)
63 (Morawski et al., 2002)
64 (Morawski et al., 2002)
65 (Morawski et al., 2002)
66 (Letcher and Deenadayalu, 2001)
67 (Letcher and Deenadayalu, 2001)
68 (Letcher and Deenadayalu, 2001)
69 (Letcher and Deenadayalu, 2001)
70 (Letcher and Deenadayalu, 2001)
71 (Rawat and Gulati, 1976)
72 (Rawat and Gulati, 1976)
73 (Radwan et al., 1997b)
74 (Hughes and Haoran, 1990)
75 (Rawat and Gulati, 1976)
76 (Rawat and Gulati, 1976)

 (b) Quaternary aromatic extraction systems

77 (Chen et al., 2000a)
78 (Chen et al., 2001)
79 (Chen et al., 2000a)
80 (Chen et al., 2001)
81 (Chen et al., 2000a)
82 (Chen et al., 2001)
83 (Chen et al., 2000b)
84 (Chen et al., 2000b)
85 (Hughes and Haoran, 1990)

 (c) Quinary aromatic extraction systems

86 (Chen et al., 2001)
87 (Chen et al., 2000a)
88 (Chen et al., 2001)

 (d) Ternary hydrogen bonding systems

89 (Katayama et al., 1998)
90 (Katayama et al., 1998)
91 (Katayama et al., 1998)
92 (Ashour, 2005)
93 (Ashour, 2005)
94 (Vaidya and Naik, 2003)
95 (Vaidya and Naik, 2003)
96 (Albert et al., 2001)
97 (Letcher and Naicker, 2001)
98 (Letcher and Naicker, 2001)
99 (Letcher and Naicker, 2001)
100 (Letcher and Naicker, 2001)
101 (Letcher and Naicker, 2001)
102 (Letcher and Naicker, 2001)
103 (Letcher and Naicker, 2001)
104 (Letcher and Naicker, 2001)
105 (Letcher and Naicker, 2001)
106 (Arce et al., 1998)
107 (Arce et al., 1998)
108 (Arce et al., 1998)

 (e) Quaternary hydrogen bonding systems

109 (Mohsen-Nia et al., 2006)
110 (Mohsen-Nia et al., 2006)
111 (Mohsen-Nia et al., 2006)
112 (Doulabi et al., 2005)
113 (Doulabi et al., 2005)
114 (Doulabi et al., 2005)
115 (Arce et al., 1998)
116 (Arce et al., 1999)

 (f) Ternary ionic liquid based systems

117 (Letcher and Reddy, 2005)
118 (Letcher and Reddy, 2005)
119 (Letcher and Reddy, 2005)
120 (Letcher and Reddy, 2005)
121 (Letcher and Reddy, 2005)
122 (Letcher and Reddy, 2005)
123 (Letcher et al., 2003c)
124 (Arce et al., 2004a)
125 (Letcher and Reddy, 2004)
126 (Letcher and Reddy, 2004)
127 (Letcher and Reddy, 2004)
128 (Letcher and Reddy, 2004)
129 (Arce et al., 2004b)
130 (Arce et al., 2006)
131 (Letcher et al., 2003b)
132 (Letcher and Deenadayalu, 2003)
133 (Letcher and Deenadayalu, 2003)
134 (Letcher and Deenadayalu, 2003)
135 (Deenadayalu et al., 2006)
136 (Deenadayalu et al., 2006)
137 (Deenadayalu et al., 2006)
138 (Deenadayalu et al., 2006)
139 (Meindersma et al., 2006)
140 (Meindersma et al., 2006)
141 (Meindersma et al., 2006)
142 (Banerjee, 2006)
143 (Banerjee, 2006)
144 (Banerjee, 2006)
145 (Banerjee, 2006)

Table 6. Binary interaction parameters for ternary aromatic
extraction systems

 Binary interaction parameters
 with closure equations,
 [degrees]K
System Temp.,
no. [degrees]C [A.sub.12] [A.sub.21]

1 17.0 319.87 -16.19
 25.0 272.41 -133.51
 50.0 -14.37 -22.37
2 17.0 309.35 -174.20
 25.0 105.54 -101.67
 50.0 406.48 -214.90
3 25.0 299.75 -81.21
 30.0 905.56 -50.68
 50.0 463.66 -268.49
 75.0 288.04 -243.42
 100.0 362.39 -101.39
4 25.0 241.62 -151.67
 35.0 301.77 -210.31
 50.0 -328.17 153.53
5 25.0 140.31 -187.94
 35.0 -191.48 158.71
 50.0 -10.72 -74.13
6 25.0 414.62 -240.71
 110.0 421.53 -146.68
7 25.0 223.47 -109.66
 30.0 702.28 -152.71
 50.0 111.49 -106.09
 75.0 308.47 -233.72
 100.0 426.60 -264.65
8 25.0 -46.23 23.47
9 25.0 522.15 -49.81
 35.0 386.10 -12.71
 45.0 203.31 -112.49
 100.0 441.07 -228.79
10 25.0 -136.24 523.44
 50.0 300.64 -237.15
 75.0 -65.16 1.26
11 25.0 -189.22 206.87
 30.0 127.64 317.93
 45.0 -34.69 -29.11
 50.0 -189.22 206.87
12 25.0 51.34 -35.42
 75.0 730.89 -333.79
 100.0 293.99 -313.01
13 25.0 940.22 143.73
14 25.0 105.59 -125.36
 50.0 380.39 -257.82
15 25.0 225.42 -191.02
 50.0 401.38 -196.08
 100.0 379.82 -203.92
16 25.0 270.71 -165.62
17 25.0 629.93 -318.68
18 25.0 569.50 -296.72
19 25.0 28.17 -63.35
20 10.0 127.64 -13.75
 25.0 122.15 -86.79
 50.0 356.98 -164.27
21 25.0 242.95 -136.94
22 25.0 5.89 9.21
23 25.0 400.90 -234.81
24 25.0 76.34 167.60
 30.0 -118.39 222.34
 40.0 -167.55 308.37
25 25.0 -74.13 146.31
26 25.0 -29.54 47.86
27 25.0 23.42 18.77
28 25.0 165.80 -95.55
29 25.0 40.44 35.17
30 25.0 172.11 -77.95
31 25.0 -188.46 537.50
32 25.0 -14.87 322.58
 39.0 -50.64 -90.25
 50.0 88.16 -35.18
33 25.0 402.53 12.81
34 25.0 163.31 -137.62
35 15.0 676.91 -170.08
 25.0 577.84 -162.56
 40.0 -323.12 -37.84
36 25.0 179.48 -138.68
37 25.0 29.39 -76.27
38 25.0 111.00 -81.27
39 25.0 -202.71 -107.39
40 25.0 682.61 233.79
 30.0 -149.33 349.98
 40.0 922.88 -502.29
41 25.0 480.52 -119.93
 30.0 145.27 -354.74
 50.0 343.47 -85.80
42 25.0 220.33 -164.20
43 25.0 -144.30 100.68
44 25.0 555.74 108.02
45 25.0 423.57 -181.82
46 25.0 282.98 -174.89
47 25.0 45.95 -119.47
48 25.0 533.91 -263.76
49 25.0 -163.95 -128.17
50 25.0 820.60 -77.85
51 25.0 166.60 -97.50
52 25.0 853.98 -96.47
53 25.0 255.73 -152.94
54 25.0 96.01 -309.60
55 25.0 129.65 -131.96
56 25.0 192.09 -45.66
57 25.0 839.73 -304.08
58 25.0 713.05 -274.33
59 25.0 426.32 -239.76
60 25.0 430.67 -204.02
61 25.0 513.07 -251.34
62 25.0 28.36 45.02
63 25.0 421.97 -191.39
64 25.0 465.79 -251.04
65 25.0 -20.47 78.39
66 25.0 660.48 -404.78
67 25.0 779.16 -322.69
68 25.0 77.59 -118.98
69 25.0 485.92 -255.70
70 25.0 868.87 -359.30
71 25.0 -201.27 518.31
72 25.0 67.62 -63.50
73 25.0 -218.39 133.92
74 25.0 -104.72 -278.86
75 25.0 -128.37 725.69
76 25.0 203.37 -117.48
average

 Binary interaction parameters with
 closure equations, [degrees]K
System
no. [A.sub.13] [A.sub.31] [A.sub.23]

1 311.90 285.91 5.29
 571.79 273.82 -0.11
 456.24 146.03 187.82
2 618.56 123.38 75.36
 515.34 86.98 187.13
 670.58 198.82 21.72
3 367.10 182.57 16.75
 367.13 99.55 13.16
 690.30 112.75 -36.17
 661.99 13.07 95.27
 393.99 113.41 49.70
4 637.25 73.91 152.60
 697.68 60.41 165.66
 517.67 61.70 697.79
5 803.23 36.84 377.60
 353.69 98.75 530.05
 531.68 76.47 360.05
6 744.69 170.01 -7.01
 405.71 78.50 14.89
7 473.44 174.80 62.87
 311.33 117.66 -155.37
 479.87 51.00 182.02
 650.05 11.46 113.05
 570.18 40.27 -5.56
8 359.16 119.76 253.67
9 415.12 114.80 91.31
 336.19 200.29 69.69
 528.51 61.14 144.60
 529.93 110.82 -29.69
10 250.90 169.34 671.31
 706.43 135.17 79.52
 461.96 48.13 336.30
11 374.46 89.48 533.62
 197.47 223.97 334.58
 412.92 46.23 258.94
 374.46 89.48 533.62
12 359.73 105.84 131.85
 670.77 49.95 -145.16
 551.58 21.09 -43.00
13 468.68 157.30 29.74
14 666.94 65.81 311.64
 935.42 12.56 270.38
15 508.71 35.69 58.87
 439.29 58.82 -24.46
 441.44 18.19 7.26
16 542.49 127.67 56.15
17 706.53 48.77 -104.00
18 856.38 -6.35 55.43
19 443.10 103.67 174.73
20 507.34 119.00 238.97
 642.98 91.70 240.94
 578.78 90.10 63.93
21 584.89 85.95 130.90
22 498.89 57.27 361.11
23 834.54 58.86 147.19
24 491.68 131.78 564.01
 364.45 225.71 560.90
 372.49 129.29 794.16
25 221.93 294.97 190.58
26 429.12 77.34 411.74
27 364.02 93.20 239.52
28 493.93 60.19 179.18
29 331.53 107.76 224.21
30 363.34 115.95 91.23
31 -166.45 637.33 61.85
32 -95.56 600.55 -27.97
 14.25 302.79 -90.00
 -16.26 381.14 -141.00
33 229.64 19.17 121.81
34 264.26 6.44 -58.41
35 255.68 13.27 -83.86
 223.40 19.77 -84.55
 206.78 9.41 46.04
36 819.03 19.96 387.11
37 932.09 -30.03 673.43
38 535.74 28.44 267.67
39 346.87 19.52 54.62
40 402.59 -10.42 575.06
 399.61 -20.11 694.54
 419.53 -4.45 -445.78
41 520.76 -107.57 193.06
 739.53 -36.37 7.74
 379.22 -6.50 114.93
42 748.88 9.36 279.74
43 630.33 19.42 662.12
44 278.05 58.98 211.31
45 403.81 53.77 -37.06
46 399.36 40.64 14.80
47 331.96 31.78 47.92
48 529.15 25.66 -93.56
49 343.67 48.23 34.23
50 361.25 53.99 -24.93
51 327.77 62.71 41.97
52 505.19 -107.38 190.23
53 466.55 -46.51 99.45
54 403.24 109.33 -210.76
55 556.58 -98.39 262.33
56 361.49 13.54 150.91
57 618.19 21.29 -176.27
58 591.81 14.32 -122.13
59 547.60 -0.65 4.12
60 466.52 -39.78 10.52
61 773.08 -114.64 131.58
62 454.05 -101.08 468.64
63 593.66 -33.25 64.47
64 825.71 -69.58 145.27
65 516.96 -108.03 534.30
66 364.90 84.56 -336.29
67 440.07 -57.93 -180.64
68 333.00 -35.17 102.47
69 420.06 -24.11 -98.38
70 477.34 -37.46 -243.16
71 201.50 195.73 719.29
72 508.68 92.56 325.53
73 564.53 762.06 174.42
74 297.15 -37.97 -126.45
75 215.30 45.63 925.42
76 510.09 28.21 192.62
average

System Binary
no. interaction
 parameters
 with closure
 equations,
 [degrees]K
 rmsd rmsd
 [A.sub.32] [(wce).sup.c] (woce)

1 315.36 0.007 0.009
 107.83 0.005 0.006
 -114.39 0.005 0.005
2 63.73 0.006 0.007
 -34.02 0.004 0.004
 171.34 0.009 0.007
3 213.18 0.006 0.006
 701.82 0.007 0.009
 118.43 0.004 0.005
 -22.19 0.007 0.008
 232.90 0.008 0.006
4 -17.45 0.003 0.005
 40.47 0.010 0.010
 -239.88 0.005 0.005
5 -60.54 0.004 0.004
 -75.08 0.004 0.004
 -31.75 0.002 0.004
6 73.64 0.005 0.005
 255.89 0.005 0.008
7 97.36 0.006 0.009
 505.95 0.002 0.002
 -29.27 0.003 0.004
 16.65 0.002 0.003
 155.78 0.004 0.004
8 -55.42 0.007 0.008
9 362.95 0.003 0.004
 332.60 0.005 0.007
 -6.97 0.004 0.006
 221.06 0.006 0.006
10 -69.92 0.005 0.007
 46.05 0.006 0.007
 -143.95 0.004 0.004
11 -147.45 0.005 0.006
 170.79 0.008 0.009
 -113.33 0.004 0.004
 -147.45 0.005 0.006
12 -35.28 0.005 0.006
 298.70 0.006 0.007
 33.51 0.003 0.004
13 514.85 0.007 0.008
14 -58.53 0.005 0.006
 -14.27 0.005 0.006
15 2.29 0.003 0.005
 192.53 0.005 0.007
 167.75 0.004 0.005
16 77.66 0.006 0.007
17 186.85 0.006 0.009
18 58.92 0.005 0.009
19 -73.18 0.006 0.009
20 -7.98 0.006 0.009
 -101.40 0.003 0.004
 96.50 0.004 0.003
21 11.76 0.006 0.008
22 -83.83 0.004 0.005
23 7.21 0.003 0.004
24 112.84 0.005 0.005
 81.43 0.005 0.005
 75.04 0.003 0.004
25 43.18 0.006 0.007
26 -17.43 0.002 0.002
27 -26.65 0.003 0.007
28 6.79 0.002 0.003
29 5.72 0.002 0.006
30 93.90 0.007 0.008
31 139.67 0.007 0.009
32 330.69 0.004 0.006
 238.15 0.009 0.009
 379.74 0.01 0.011
33 301.06 0.003 0.004
34 -15.28 0.004 0.007
35 520.72 0.002 0.003
 452.22 0.002 0.004
 -436.61 0.002 0.003
36 -93.80 0.003 0.004
37 -183.03 0.005 0.005
38 -47.36 0.003 0.006
39 -368.05 0.001 0.004
40 610.87 0.003 0.007
 -224.49 0.003 0.006
 555.41 0.007 0.009
41 165.18 0.010 0.008
 -268.15 0.008 0.009
 158.48 0.005 0.008
42 -75.25 0.001 0.004
43 -193.77 0.003 0.003
44 439.96 0.005 0.007
45 218.29 0.003 0.007
46 113.96 0.003 0.004
47 -86.85 0.001 0.004
48 200.61 0.005 0.007
49 -296.99 0.006 0.007
50 566.27 0.006 0.006
51 41.01 0.004 0.007
52 528.12 0.008 0.008
53 -4.93 0.003 0.006
54 -99.06 0.007 0.007
55 -131.04 0.010 0.011
56 40.71 0.007 0.007
57 370.64 0.005 0.008
58 287.75 0.007 0.012
59 121.95 0.004 0.007
60 138.91 0.011 0.011
61 8.28 0.008 0.010
62 -103.15 0.015 0.016
63 50.91 0.003 0.011
64 -33.19 0.005 0.009
65 -189.55 0.008 0.008
66 448.63 0.005 0.007
67 423.21 0.008 0.008
68 -69.13 0.007 0.008
69 199.07 0.005 0.008
70 470.21 0.007 0.007
71 -6.06 0.004 0.004
72 40.53 0.002 0.002
73 19.64 0.002 0.002
74 -287.43 0.004 0.006
75 -98.31 0.009 0.010
76 31.59 0.004 0.008
average 0.005 0.006

System rmsd [gain.sup.
no. [[lit].sup.d] woce.sub.lit]
 (%)

1
2
3
4
 0.020 50.00
 0.011 54.55
5 0.012 66.67
 0.012 66.67
 0.018 77.78
6
7
8
9 0.004 0.00
 0.009 22.22
 0.006 0.00
 0.008 25.00
10 0.002 -250.00
 0.007 0.00
 0.005 20.00
11 0.006 0.00
 0.005 20.00
 0.006 0.00
12
13
14 0.011 45.45
 0.016 62.50
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33 0.006 33.33
34
35 0.004 25.00
 0.004 0.00
 0.014 78.57
36 0.012 66.67
37
38 0.007 14.29
39
40
41
42
43
44 0.008 12.50
45 0.008 12.50
46 0.007 42.86
47
48 0.011 36.36
49 0.010 30.00
50 0.012 50.00
51 0.016 56.25
52 0.011 27.27
53 0.012 50.00
54 0.013 46.15
55 0.013 15.38
56 0.011 36.36
57 0.020 60.00
58 0.020 40.00
59 0.020 65.00
60 0.030 63.33
61 0.020 50.00
62 0.030 46.67
63 0.030 63.33
64 0.030 70.00
65 0.030 73.33
66 0.012 41.67
67 0.016 50.00
68 0.016 50.00
69 0.014 42.86
70 0.007 0.00
71
72
73
74
75
76
average 0.013 32.87

System [gain.sup. [gain.sup.
no. wce.sub.woce] wce.sub.lit]
 (%) (%)

1 30.11
 16.67
 11.11
2 18.57
 0.00
 -25.00
3 0.00
 22.22
 20.00
 12.50
 -33.33
4 40.00
 0.00 50.00
 0.00 54.55
5 0.00 66.67
 0.00 66.67
 50.00 88.89
6 0.00
 37.50
7 33.33
 0.00
 25.00
 33.33
 0.00
8 12.50
9 25.00 25.00
 28.57 44.44
 33.33 33.33
 0.00 25.00
10 28.57 -150.00
 14.29 14.29
 0.00 20.00
11 16.67 16.67
 11.11
 0.00 20.00
 16.67 16.67
12 16.67
 14.29
 25.00
13 12.50
14 16.67 54.55
 16.67 68.75
15 40.00
 28.57
 20.00
16 14.29
17 33.33
18 44.44
19 33.33
20 40.43
 25.00
 -46.67
21 25.00
22 20.00
23 25.00
24 0.00
 0.00
 8.33
25 14.29
26 0.00
27 57.14
28 33.33
29 66.67
30 12.50
31 22.22
32 33.33
 -1.14
 9.26
33 25.00 50.00
34 42.86
35 33.33 50.00
 50.00 50.00
 33.33 85.71
36 25.00 75.00
37 0.00
38 50.00 57.14
39 75.00
40 57.14
 46.77
 22.83
41 -25.00
 20.21
 36.25
42 75.00
43 0.00
44 28.57 37.50
45 57.14 62.50
46 25.00 57.14
47 75.00
48 28.57 54.55
49 14.29 40.00
50 0.00 50.00
51 42.86 75.00
52 0.00 27.27
53 50.00 75.00
54 0.00 46.15
55 9.09 23.08
56 0.00 36.36
57 37.50 75.00
58 41.67 65.00
59 42.86 80.00
60 0.00 63.33
61 20.00 60.00
62 6.25 50.00
63 72.73 90.00
64 44.44 83.33
65 0.00 73.33
66 28.57 58.33
67 0.00 50.00
68 12.50 56.25
69 37.50 64.29
70 0.00 0.00
71 0.00
72 0.00
73 0.00
74 33.33
75 10.00
76 50.00
average 20.97 47.69

Table 7. Binary interaction parameters for quaternary aromatic
extraction systems

 Binary interaction parameters
 with closure equations,
 [degrees]K

System Temp., [A.sub.12] [A.sub.13]
no. [degrees]C [A.sub.21] [A.sub.31]

 372.37 -951.30
77 25.0
 -122.67 601.11
 135.00 146.11
78 25.0
 -126.09 -121.47
 -145.41 90.89
79 25.0
 384.02 236.51
 -275.32 812.38
80 25.0
 -148.72 -326.96
 182.18 342.63
81 25.0
 -85.23 259.29
 229.39 -86.89
82 25.0
 -503.81 983.40
 -109.07 367.92
83 25.0
 -769.98 -854.59
 -209.67 288.51
84 25.0
 -834.01 -880.05
 330.10 800.34
85 25.0
 10.77 225.77
 836.96 812.09
 40.0
 -145.33 320.97

average

 Binary interaction parameters
 with closure equations, [degrees]K

System [A.sub.14] [A.sub.23] [A.sub.24]
no. [A.sub.41] [A.sub.32] [A.sub.42]

 358.44 -871.18 -44.35
77
 173.98 726.92 266.23
 664.41 -218.29 225.75
78
 43.42 354.11 -134.16
 353.15 969.22 526.89
79
 275.59 -108.39 -80.09
 419.48 3.00 399.09
80
 114.38 -108.18 -32.62
 434.16 423.31 245.71
81
 88.59 130.75 167.54
 156.53 10.88 156.53
82
 170.25 569.13 903.45
 683.13 248.22 419.32
83
 -470.84 -313.38 -73.74
 586.64 226.83 441.32
84
 -546.81 -293.96 -67.79
 331.17 363.21 297.55
85
 -57.83 107.97 227.88
 -409.48 -341.99 269.19

 -912.76 149.18 748.20

average

 Binary
 interaction
 parameters
 with closure
 equations,
 [degrees]K

System [A.sub.34] rmsd
no. [A.sub.43] (woce)

 547.78
77 0.008
 -739.73
 729.77
78 0.009
 -202.54
 -40.16
79 0.008
 430.47
 260.87
80 0.006
 -59.66
 173.05
81 0.005
 387.44
 81.61
82 0.005
 270.27
 -7.99
83 0.003
 60.55
 57.27
84 0.004
 68.95
 249.91
85 0.005
 435.48
 -177.86
 0.006
 -190.02
average 0.006

System rmsd [gain.sup.
no. (wce) wce.sub.woce]
 (%)

77 0.011 27.27

78 0.007 -28.57

79 0.008 0.00

80 0.009 33.33

81 0.009 44.44

82 0.007 28.57

83 0.008 62.50

84 0.008 50.00

85 0.006 16.67

 0.006 0.00

average 0.008 23.42

Table 8. Binary interaction parameters for quinary aromatic
extraction systems

 Binary interaction parameters with
 closure equations, [degrees]K

System no. [A.sub.12] [A.sub.21] [A.sub.31]
 [A.sub.13] [A.sub.23] [A.sub.32]
 [A.sub.14] [A.sub.24] [A.sub.33]
 [A.sub.15] [A.sub.25] [A.sub.34]

 -951.56 -118.86 949.22
 -174.61 -145.73 145.39
86
 -82.30 -139.07 181.79
 6.25 174.70 728.78
 542.64 50.25 660.14
 348.79 120.82 924.56
87
 13.05 -77.73 142.55
 151.71 49.33 444.86
 -91.60 -803.49 -629.77
 854.41 646.35 -125.94
88
 262.38 107.78 186.11
 662.87 274.65 31.09
average

 Binary interaction parameters
 with closure equations,
 [degrees]K

System no. [A.sub.41] [A.sub.51] rmsd
 [A.sub.42] [A.sub.52] (woce)
 [A.sub.43] [A.sub.53]
 [A.sub.44] [A.sub.54]

 949.54 367.70
 60.06 -296.56
86 0.013
 89.80 -33.60
 650.93 -19.46
 468.56 330.50
 870.17 720.51
87 0.012
 286.71 312.30
 356.02 79.30
 -731.26 -246.86
 -173.97 76.81
88 0.007
 676.65 605.54
 51.18 135.09
average 0.011

System no. rmsd [gain.sup.
 (wce) wce.sub.woce]
 (%)

86 0.019 31.58

87 0.011 -9.09

88 0.010 30.00

average 0.013 17.50

Table 9. Binary interaction parameters for ternary hydrogen
bonding systems

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.12] [A.sub.21] [A.sub.13]

89 254.09 155.76 -756.29
90 413.12 114.95 486.30
91 284.50 172.68 982.83
92 369.81 -149.79 920.24
93 361.54 -89.57 911.75
94 939.42 493.37 624.79
95 990.55 539.77 579.74
96 -127.00 939.97 183.33
97 519.77 -21.51 203.82
98 95.08 27.36 192.89
99 174.49 72.13 192.23
100 566.71 -58.58 441.00
101 201.03 48.10 389.24
102 253.58 212.04 311.04
103 135.60 39.44 525.06
104 392.37 189.13 496.44
105 243.18 237.55 559.95
106 555.97 -219.24 866.61
107 382.60 38.66 905.92
108 255.72 -58.36 553.09
average

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.31] [A.sub.23] [A.sub.32]

89 -371.79 -808.44 -325.61
90 -304.56 294.57 -198.11
91 -148.46 825.43 -194.04
92 308.91 218.74 127.01
93 333.40 456.86 329.62
94 -123.43 297.74 -4.43
95 -25.05 248.06 94.06
96 466.18 910.39 126.27
97 581.14 -137.35 781.26
98 602.94 -162.50 315.27
99 601.88 -208.20 303.81
100 766.97 -158.62 792.64
101 808.77 -194.05 378.42
102 836.89 -152.48 414.91
103 921.89 -37.81 455.18
104 957.41 -84.48 579.73
105 932.06 -92.04 285.72
106 398.97 -3.02 304.56
107 319.60 560.56 318.18
108 355.38 140.24 256.61
average

System rmsd rmsd rmsd
no. (wce) (woce) [lit]

89 0.006 0.005
90 0.004 0.006
91 0.006 0.008
92 0.003 0.007
93 0.002 0.003
94 0.006 0.008 0.014
95 0.008 0.007 0.034
96 0.001 0.002
97 0.002 0.003 0.011
98 0.002 0.004 0.007
99 0.001 0.002 0.013
100 0.002 0.003 0.014
101 0.004 0.006 0.014
102 0.002 0.003 0.008
103 0.004 0.006 0.010
104 0.003 0.003 0.014
105 0.002 0.003 0.007
106 0.004 0.003 0.005
107 0.004 0.005 0.005
108 0.002 0.003 0.021
average 0.003 0.005 0.013

System [gains.sup. [gains.sup. [gain.sup.
no. woce.sub.lit] wce.sub.woce] wce.sub.lit]
 (%) (%) (%)

89 -20.00
90 33.33
91 25.00
92 57.14
93 33.33
94 42.86 25.00 57.14
95 79.41 -14.29 76.47
96 50.00
97 72.73 33.33 81.82
98 42.86 50.00 71.43
99 84.62 50.00 92.31
100 78.57 33.33 85.71
101 57.14 33.33 71.43
102 62.50 33.33 75.00
103 40.00 33.33 60.00
104 78.57 0.00 78.57
105 57.14 33.33 71.43
106 40.00 -33.33 20.00
107 0.00 20.00 20.00
108 85.71 33.33 90.48
average 58.72 25.47 67.99

Table 10. Binary interaction parameters for quaternary hydrogen
bonding systems

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.12] [A.sub.13] [A.sub.14]
 [A.sub.21] [A.sub.31] [A.sub.41]

 -618.19 878.71 805.27
109
 -913.69 699.99 904.18
 -722.43 358.82 604.64
110
 419.59 986.45 842.39
 557.96 336.32 329.52
111
 335.10 347.72 598.95
 -800.21 -518.21 602.46
112
 5.25 756.00 968.50
 -372.85 388.63 697.55
113
 781.83 922.79 982.67
 -145.52 -320.28 -107.67
114
 383.53 474.27 888.41
115 -60.10 601.45 356.54
115
 162.75 216.53 65.02
 202.54 601.44 198.02
116
 223.59 353.95 -2.70
average

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.23] [A.sub.24] [A.sub.34]
 [A.sub.32] [A.sub.42] [A.sub.43]

 -474.38 -161.15 -687.71
109
 -357.61 233.26 -410.08
 390.77 623.24 560.36
110
 -123.63 -281.03 170.49
 -29.76 -8.53 -114.30
111
 204.50 483.76 143.74
 -438.48 254.74 188.71
112
 30.27 -184.69 -719.47
 676.61 865.53 432.57
113
 56.08 -4.04 183.53
 -48.06 -32.56 -187.40
114
 217.44 434.47 14.14
115 812.39 494.83 358.44
115
 204.62 -19.53 451.85
 751.89 239.31 14.01
116
 483.35 17.54 60.79
average

System rmsd rmsd rmsd
no. (wce) (woce) [lit]

109 0.016 0.018 0.022
110 0.009 0.020 0.026
111 0.014 0.016 0.021
112 0.015 0.019 0.022
113 0.012 0.024 0.030
114 0.010 0.017 0.033
115 0.006 0.005 0.018
116 0.006 0.005 0.011

average 0.011 0.016 0.023

System [gain.sup. [gain.sup. [gain.sup.
no. woce.sub.lit] wce.sub.woce] wce.sub.lit]
 (%) (%) (%)

109 18.18 11.11 27.27
110 23.08 55.00 65.38
111 23.81 12.50 33.33
112 13.64 21.05 31.82
113 20.00 50.00 60.00
114 48.48 41.18 69.70
115 72.22 -20.00 66.67
116 54.55 -20.00 45.45

average 34.25 18.86 49.95

Table 11. UNIQUAC volume and area parameters (Banerjee et al.,
2005, 2006)

Ionic liquid r q

[[C.sub.1]mim][Cl] 6.34 4.27
[[C.sub.2]mim][Cl] 7.19 4.88
[[C.sub.3]mim][Cl] 8.00 5.40
[[C.sub.4]mim][Cl] 8.74 5.76
[[C.sub.5]mim][Cl] 9.60 6.42
[[C.sub.6]mim][Cl] 10.43 6.87
[[C.sub.7]mim][Cl] 11.20 7.41
[[C.sub.8]mim][Cl] 11.99 7.89

[[C.sub.1]mim][[BF.sub.4]] 7.47 4.85
[[C.sub.2]mim][[BF.sub.4]] 8.38 5.33
[[C.sub.3]mim][[BF.sub.4]] 9.25 5.95
[[C.sub.4]mim][[BF.sub.4]] 10.06 6.37
[[C.sub.5]mim][[BF.sub.4]] 10.84 7.04
[[C.sub.6]mim][[BF.sub.4]] 11.66 7.39
[[C.sub.7]mim][[BF.sub.4]] 12.62 8.10
[[C.sub.8]mim][[BF.sub.4]] 13.19 8.36

[[C.sub.1]mim][[PF.sub.6]] 8.59 5.54
[[C.sub.2]mim][[PF.sub.6]] 9.43 5.96
[[C.sub.3]mim][[PF.sub.6]] 10.19 6.53
[[C.sub.4]mim][[PF.sub.6]] 11.03 6.96
[[C.sub.5]mim][[PF.sub.6]] 11.89 7.59
[[C.sub.6]mim][[PF.sub.6]] 12.87 8.17
[[C.sub.7]mim][[PF.sub.6]] 13.90 8.79
[[C.sub.8]mim][[PF.sub.6]] 14.23 8.94

[[C.sub.4]mim][[CF.sub.3]S[O.sub.3] 12.46 7.52
[[C.sub.1]mim][[CH.sub.3]S[O.sub.4]] 8.77 5.75
[[C.sub.4]mim][[CH.sub.3]S[O.sub.4] 11.16 7.13
[[C.sub.2]mim][[C.sub.2][H.sub.5]S[O.sub.4] 10.38 6.76
[[C.sub.2]mim][[C.sub.8][H.sub.17]S[O.sub.4] 14.89 9.49
[[C.sub.2]dmim][[C.sub.2][H.sub.5]S[O.sub.4] 11.55 6.84
[[C.sub.8]mim][DEG *] 19.28 11.83
[BmPy][[BF.sub.4]] 10.69 6.86

* diethyleneglycolmonomethylethylsulphate

Table 12. Binary interaction parameters for ternary ionic
liquid systems

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.12] [A.sub.21] [A.sub.13]

117 -35.70 132.74 140.59
118 9.20 137.23 8.96
119 290.82 -131.54 336.36
120 184.35 59.69 206.44
121 355.48 -72.68 221.03
122 -86.64 223.55 84.25
123 779.67 99.70 562.86
124 878.39 -300.67 452.04
125 559.07 -39.59 405.73
126 660.55 -10.29 259.87
127 206.18 -30.19 88.68
128 222.56 -22.06 182.49
129 585.08 -191.93 249.86
130 25.30 64.04 42.62
131 -24.22 57.69 258.50
132 284.83 50.51 406.98
133 48.45 139.62 459.10
134 327.08 438.13 322.45
135 499.98 0.45 151.28
136 442.20 32.11 490.05
137 391.50 437.10 351.60
138 125.43 29.29 354.78
139 479.63 486.10 144.23
140 491.98 285.45 102.17
141 474.70 74.14 277.55
142 586.36 38.75 72.88
143 22.72 44.16 -28.55
144 -23.46 751.48 72.14
145 593.04 743.24 -83.62
average

System Binary interaction parameters with
no. closure equations, [degrees]K

 [A.sub.31] [A.sub.23] [A.sub.32]

117 112.05 162.70 -34.28
118 235.74 26.11 124.86
119 -87.12 62.63 61.52
120 701.84 -93.85 526.20
121 415.54 -101.58 521.09
122 408.67 67.77 82.00
123 -135.49 39.15 20.78
124 -75.05 -345.52 306.45
125 -45.17 54.39 202.15
126 13.31 15.00 439.27
127 142.30 -46.10 243.89
128 48.02 28.22 138.37
129 23.19 -192.88 357.46
130 69.88 464.83 453.33
131 456.68 124.76 241.03
132 65.61 136.48 29.44
133 497.98 110.90 58.60
134 482.35 102.99 151.84
135 293.43 147.61 789.28
136 403.30 363.65 686.99
137 183.13 143.84 -70.23
138 232.74 246.81 220.92
139 109.11 158.72 117.13
140 21.62 98.15 224.13
141 4.08 128.57 255.66
142 126.34 114.71 715.78
143 71.36 259.61 338.08
144 789.48 204.95 147.36
145 695.96 179.17 808.55
average

System rmsd rmsd
no. (wce) (woce)

117 0.010 0.012
118 0.014 0.018
119 0.013 0.014
120 0.012 0.014
121 0.016 0.017
122 0.008 0.009
123 0.008 0.007
124 0.008 0.006
125 0.007 0.009
126 0.006 0.006
127 0.009 0.014
128 0.009 0.012
129 0.004 0.004
130 0.013 0.006
131 0.002 0.003
132 0.017 0.015
133 0.009 0.015
134 0.007 0.016
135 0.015 0.009
136 0.007 0.006
137 0.009 0.006
138 0.009 0.005
139 0.005 0.007
140 0.006 0.008
141 0.004 0.009
142 0.017 0.008
143 0.004 0.013
144 0.004 0.005
145 0.012 0.016
average 0.009 0.010

System rmsd # [gain.sup.
no. [lit] wce.sub.woce]
 (%)

117 0.015 16.67
118 0.018 22.22
119 0.006 7.14
120 0.016 14.29
121 0.004 5.88
122 0.004 11.11
123 0.004 -14.29
124 0.007 -33.33
125 0.009 22.22
126 0.005 0.00
127 0.022 35.71
128 0.037 25.00
129 0.003 0.00
130 0.005 13.33
131 0.000 -33.33
132 0.004 -13.33
133 0.002 40.00
134 0.000 56.25
135 -66.67
136 -16.67
137 -50.00
138 -80.00
139 0.0084 28.57
140 0.0091 25.00
141 0.0148 55.56
142 -41.67
143 69.23
144 20.00
145 25.00
average 0.008 6.41

# Corresponding NRTL rmsd value from literature

Table 13. Effect of local minimization on rmsd values

System no. Temp., rmsd
 [degrees]C (wce)

1 25.0 0.0053
80 25.0 0.0059
103 25.0 0.0040
117 25.0 0.0097

System no. rmsd Difference
 (after local
 minimization)

1 0.0051 0.0002
80 0.0058 0.0001
103 0.0039 0.0001
117 0.0096 0.0001
COPYRIGHT 2007 Chemical Institute of Canada
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2007 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Sahoo, Ranjan Kumar; Banerjee, Tamal; Khanna, Ashok
Publication:Canadian Journal of Chemical Engineering
Date:Dec 1, 2007
Words:15000
Previous Article:Gas holdup and solid-liquid mass transfer in Newtonian and non-Newtonian fluids in bubble columns.
Next Article:New attraction term for the Soave-Redlich-Kwong equation of state.

Terms of use | Privacy policy | Copyright © 2019 Farlex, Inc. | Feedback | For webmasters