Genetic algorithm to estimate interaction parameters of multicomponent systems for liquid–liquid equilibria

Computers and Chemical Engineering 29 (2005) 1712–1719

Genetic algorithm to estimate interaction parameters of multicomponent systems for liquid–liquid equilibria Manish K. Singh, Tamal Banerjee, A. Khanna ∗ Department of Chemical Engineering, IIT Kanpur, Kanpur-208016, India Received 8 August 2004; received in revised form 5 February 2005; accepted 8 February 2005 Available online 11 March 2005

Abstract Genetic algorithm (GA) has been utilized to estimate the binary interaction parameters from multicomponent liquid–liquid equilibria data; and its applicability for NRTL and UNIQUAC thermodynamic models has been demonstrated. These models being highly non-convex can have several local extrema. GA leads to globally optimum values; it does not require any initial guess but only the upper and lower bounds of the interaction parameters. It has also been shown to perform better than inside variance estimation method (IVEM) and the techniques used in ASPEN, DECHEMA. The objective function and the root-mean-square deviation for these four techniques have been compared for methanol–diphenylamine–cyclohexane system using UNIQUAC. The applicability of GA to six ternary, two quaternary and two quinary systems has also been undertaken. Invariably the rmsd values for GA are better than reported in the literature. © 2005 Published by Elsevier Ltd. Keywords: Liquid–liquid equilibria; Activity coefficient; Parameter estimation; Genetic algorithm

1. Introduction The accurate prediction of thermo-physical properties and equilibrium conditions of chemical systems is one of the most important uses of thermodynamics in chemical engineering. The most common way to do this is to fit the experimental data to a thermodynamic model and then use that model with fitted parameters for predicting properties at other conditions. But quite often the model parameter estimation process is an optimization without a unique result. It is more than likely to run into situations where the optimization problem is non-convex with several local extrema. One such recently developed method—the inside variance estimation method (IVEM) (Vasquez & Whiting 2000) uses the maximum likelihood method for binary interaction parameter regression. This involves the re-computation of the variance for each iteration of the optimization procedure by automatically reweighting the objective function. The focus of this work is on the application of genetic algorithm (GA) to find the global ∗

Corresponding author. Tel.: +91 512 2597117; fax: +91 512 2590104. E-mail address: [email protected] (A. Khanna).

0098-1354/$ – see front matter © 2005 Published by Elsevier Ltd. doi:10.1016/j.compchemeng.2005.02.020

extrema for estimation of interaction parameters. The method can be extended to other fluid phase equilibria situations. Gradientless random searches are simple and reliable for solving complicated non-convex and multimodal non-linear programming optimization problems. Their convergence may be relatively slow (Maria, 2003). Random searches can be grouped in several classes: adaptive random search (Maria, 1998), simulated annealing (Schoen, 1991), GA (Holland, 1992; Goldberg, 1989), evolution algorithm (Schwefel, 1985) and clustering algorithms (Hanagandi & Nikolaou, 1998). Adaptive random search iteratively adapts based on past information of success or failure. Adaptive random searches include several subclasses: (a) with centroid generation (Maria, 1998), (b) Luus Jakola’s class (Luus, 1998), (c) adaptive step size or a combination of these. Simulated annealing methods are based on the Markov chain theory, taking into account the information from the last step. This can accept a detrimental search step with a Boltzmann distribution probability, thus surpassing local optima. GA’s random point generator presents similarities with biogenetic mutation and natural selection. GA is conducted using information from a population of candidate solutions. Crossover

M.K. Singh et al. / Computers and Chemical Engineering 29 (2005) 1712–1719

Nomenclature A F FiP  FiP

g G gmax G GA IVEM m n qk rk R t T W x xˆ z

interaction parameter objective function correlation factor for experimental data, i = no. of component, P = phase correlation factor for predicted data, i = no. of component, P = phase current generation energy parameter maximum number of generation change in Gibbs free energy genetic algorithm inside variation estimation method number of experimental data or tie lines number of components UNIQUAC area parameter UNIQUAC volume parameter raffinate, gas constant number of tries temperature weight mole fraction calculated mole fraction coordination number

Greek letters α non-randomness factor γ activity coefficient θ area fraction τ interaction parameter Φ segment or volume fraction Subscripts i component i ik component and tie line j phase

and mutation operators generate new offsprings, while a fitness function controls the search progress GA’s have been used in diverse areas of chemical engineering: catalytic process (Moros, Kalies, Rex, & Schaffarczyk, 1996), molecular design (Venkatasubramanian, Chan, & Caruthers, 1994), liquid–liquid extraction process (Papadopoulos and Linke, 2004), polymerization reactors (Silva & Biscaia, 2003) and mixed integer non-linear programming process optimisation (Duran & Grossmann, 1986) and (Edwards, Edgar, & Manousiouthakis, 2000). There are several analogies among adaptive random searches, GA and evolution algorithms. GA’s mutation probability corresponds to the offspring mutation and to adaptive random searches’ control of local/global convergence; periodic domain expansion and contraction of adap-

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tive random searches is analogous to the increasing diversity of the GA population by using crowding scheme. A competition–cooperation among ‘families’ in evolution algorithm is equivalent to (a) avoiding the ‘elitism’ induced by the ‘fitness’ function in GA, (b) the continuous switching between local and global search in Adaptive Random Searches, and (c) the multi-start local searches in clustering algorithms (Hanagandi & Nikolaou, 1998).

2. Genetic algorithm To start a GA process, the potential solution is coded as a vector (i.e. a chromosome) of variables (i.e. genes). The operators along with their expressions are shown in Table 1.The GA moves from generation to generation selecting and reproducing parents until a termination criterion is met. The most frequently used stopping criterion is a specified maximum number of generations gmax .

3. Computation of multicomponent liquid–liquid equilibria Multicomponent liquid–liquid equilibria (LLE) compositions may be calculated using any of the thermodynamic activity coefficient models. The calculations of liquid–liquid equilibria were carried using the NRTL and UNIQUAC model (Table 2). At a given temperature and a given total composition, we find two set of compositions (nI1 , nI2 , . . . , nIN ) II II and (nII 1 , n2 , . . . , nN ) for which G for the system is minimum. The function (nI + nII )G = nI GI + nII GII

(1)

may be minimized under the constraints nIi + nII i = ni ,

i = 1, 2, . . . , N

(2)

where ni is the total number of moles of component i, nIi and nII i are the number of moles of component i in phase I and II, and GI and GII are the Gibbs energies of mixing corresponding to nI moles of phase I and nII moles of phase II. This procedure corresponds to the necessary and sufficient condition of equilibrium. In principle, this would be the best method (Novak, Matous, & Pick, 1987; Sørensen, Magnussen, Rasmussen, & Fredenslund, 1979) for regressing thermodynamic parameters; but for practical reasons usually G-minimization is not used. The convergence to solutions represented by local minima is common (Sørensen et al., 1979). To avoid G-minimization, objective functions are commonly defined through the use of least squares or maximum likelihood objective functions for the isoactivity method or minimization of the residuals between experimental and predicted compositions. Typical objective functions used for liquid–liquid equilibria prediction have been discussed by Vasquez and Whiting

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M.K. Singh et al. / Computers and Chemical Engineering 29 (2005) 1712–1719

Table 1 GA operators and their probabilities and relationships (Michalewicz, 1992) Operators

Type

Relationship

Notations

q (1 − q)rj −1 ,

q

q 1−(1−q)P

Selection

Roulette wheel, ranking method

pj =

Crossover

Simple

T T P1 = (p
11 , p12 , . . . p1k , . . . , p1m
) , P2 = (p21 , p22 , . . . , p2k , . . . , p2m ) , p1k if k < r p2k if k < r c1k = ,c = , p2k if k > r 2k p1k if k > r

=

P = population size, pj = probability of selecting a individual based on the rank of solution j, q = probability of selecting the best individual = 0.08 P1 , P2 = parents; C1 , C2 = child, pjk = value of the kth variable of jth parent, k = 1, 2, 3, . . ., m, cjk = value of the kth variable of jth child, 1 < r < m, r begin random, Psc = probability of simple crossover = 0.10 r = U(0, 1), pac = probability of arithmetic crossover = 0.10

C1 = (c11 , c12 , . . . , c1k , . . . , c1m )T , C2 = (c21 , c22 , . . . , c2k , . . . , c2m )T

Arithmetic

C1 = rP1 + (1 − r)P2 , C2 = (1 − r)P1 + rP2

Heuristic

C1 = P1 + r(P1 − P2 ), C2 = P1 , feasibility =

Uniform

pjk =





Mutation

if c1k ≥ ak and c1k ≤ bk ∀k otherwise

1 0

ak = lower limit of bound of the kth variable, bk = upper limit of bound of the kth variable, phc = probability of heuristic crossover = 0.10

U(ak , bk ) if k = i if k = i pjk

i is any random variable, 1 ≤ i≤ m, Pum = probability of uniform mutation = 0.05



ak + (pjk − ak )f (g) if r1 < 0.5, k = i bk − (bk − pjk )f (g) if r1 ≥ 0.5, k = i k = i pjk where f (g) = (r2(1−(g/gmax )) )b pjk

Non-uniform

=

g = generation, gmax = maximum generation, b = shape parameter = 2, r1 , r2 = U(0, 1) random from a uniform distribution between 0 and 1, Pbm = probability of non-uniform mutation = 0.05, b = shape factor



ak if r < 0.5, k = i if r ≥ 0.5, k = i bk pjk if k = i Applies the non-uniform operator to all the variables in parent Pk pjk =

Boundary Multi-non-uniform

viation (rmsd) defined as

(2000). For minimizing the residuals between experimental and estimated mole fractions, the objective function is usually defined as min min F (θ) =

m  II  N  k=1 j=I i=1

j

j

pbm = probability of boundary mutation = 0.05 pbm = probability of multi-non-uniform mutation = 0.08

1/2  m  c  2 j j 2  (xik − xˆ ik )  rmsd =  2mc

(4)

k=1 i=1 j=1

j 2

Wik (xik − xˆ ik )

(3) here ‘m’ refers to the no. of tie lines and ‘c’ the no. of components. In the UNIQUAC model, the interaction parameter appears as Aij /T . So instead of regressing for the interaction parameter (Aij ) itself, we have introduced a term Aij = Aij /T and then carried out regression for Aij .

which takes into account all the components. This objective function, with unit weights, has been used in this work. The goodness of fit is usually measured by root mean square deTable 2 UNIQUAC and NRTL Models (Prausnitz, Lechtenthaler, & Azevedo, 1999) UNIQUAC model ln γi = ln Φxii + 2z qi ln Φθii + li −

Φi xi





x l + qi 1 − ln j j j

li = 2z (rk − qk ) + 1 − rk , τij = exp − NRTL model  c

ln γi =

j=1 c

τji Gji xj

k=1

Gki xk

+

c  j=1



Gc ij xj k=1

Gkj xk

Aij T





 

θ τ − j j ji

j

θj τij k

θk τkj

where θi =

qi xi qT ,

qT =

 k

q k xk , Φ i =

ri xi r T , rT

=



r x , k k k

= Aij , z = 10 (coordination number)

 τij −

 c τij Gij xi i=1 where τji = c k=1

Gkj xk

gji −gii RT

= Aji = Aji T or Aji =

gji −gii R ,

Gji = exp(−αji τji ), αji = αij

M.K. Singh et al. / Computers and Chemical Engineering 29 (2005) 1712–1719

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Table 3 List of systems studied at 298.15 K No.

Type

Name of the system

Reference

1 2 3 4 5 6 7 8 9 10

Ternary Ternary Ternary Ternary Ternary Ternary Quaternary Quaternary Quinary Quinary

Methanol (1) + diphenylamine (2) + cyclohexane (3) Chloroform (1)–propanoic acid (2)–water (3) Chloroform (1)–2-propanone (2)–water (3) Methanol (1)–propanol (2)–hexane (3) Water (1)–methanol (2)–1,4-dicyanobutane (3) Water (1)–octane (2)–diethylene glycol monobutylether (3) Hexane (1) + heptane (2) + toluene (3) + sulfolane (4) Hexane (1) + octane (2) + m-xylene (3) + sulfolane (4) Hexane (1) + octane (2) + benzene (3) + xylene (4) + sulfolane (5) Hexane (1) + heptane (2) + toluene (3) + xylene (4) + sulfolane (5)

Sørensen & Arlt (1980) Sørensen & Arlt (1980) Sørensen & Arlt (1980) Sørensen & Arlt (1980) Letcher & Naicker (2001) Liu, Chiou, & Chen (2002) Chen, Mi, & Li (2001) Chen et al. (2001) Chen et al. (2001) Chen et al. (2001)

Table 4 UNIQUAC parameters for methanol (1) + diphenylamine (2) + cyclohexane (3) Approach

A12 , A21

A13 , A31

A23 , A32

Objective function value

rmsd

rmsd deviation w.r.t GA (%)

ASPENa DECHEMAb,c IVEMa GA (first trial) GA (second trial) GA (third trial)

−663.62, −260.69 103.66, −179.61 82.25, −245.58 440.68, −202.20 −489.27, −220.99 −746.03, −320.42

17.76, 663.73 14.02, 704.77 32.73, 603.95 10.23, 662.70 −5.99, 655.62 −10.72, 661.50

−223.76, −691.49 −108.50, −75.54 −83.10, −164.61 −155.55, 388.01 −398.24, 140.48 −210.23, −830.13

0.0053 0.0032 0.0018 0.0008 0.0010 0.0009

0.0121 0.0094 0.0071 0.0047 0.0047 0.0043

157.4 100 51.06 0.0 0.0 0.0

% Deviation = (rmsd(reported) − rmsd(predicted by GA)/rmsd(predicted by GA) × 100. a Vasquez and Whiting (2000). b Sørensen and Arlt (1980). c Sergeeva et al. (1971).

Now if the typical values of Aij are of the order of 1000, this redefinition would bring it down to an order of 10 and help in better regularization of the problem. The tie lines are calculated by a LLE separation calculation using the modified Rachford Rice algorithm (Seader & Henley 1998).

4. Results and discussion Applications of GA has been studied on 10 systems as listed in Table 3.The benchmarking system involves the liquid–liquid equilibria predictions for the ternary system—methanol (1) + diphenylamine (2) + cyclohexane (3) using the UNIQUAC model. Experimental equilibrium data for this system were taken from Sørensen and Arlt

(1980), originally reported by Sergeeva, Eskaraeva, Usmanova, and Glybovskaya (1971). Fig. 1 presents the experimental tie-lines and tie-lines for parameters estimated using GA, IVEM, ASPEN, DECHEMA. The parameter sets are from the ASPEN Plus regression package and for DECHEMA it was reported as published in DECHEMA DATA Series (Sørensen & Arlt, 1980). The values of the parameters are not unique, for the same objective function there can be many sets of parameters, this explains the mutual difference among the parameters reported in this work and those reported in the literature. It is also observed that, in comparison to GA, other techniques produce poor predictions of the slopes in the experimental tie lines. GA also shows an improved predicted phase envelope, and good agreement with the experimental data (making it more reliable for process design and simulation). The values for four sets of binary in-

Table 5 Comparison of tie lines for methanol (1)–diphenylamine (2)–cyclohexane (3) (Sergeeva et al., 1971) Methanol rich phase

Cyclohexane rich phase

x1(expt)

x1(pred)

x2(expt)

x2(pred)

x3(expt)

x3(pred)

x1(expt)

x1(pred)

x2(expt)

x2(pred)

x3(expt)

x3(pred)

0.820 0.800 0.789 0.776 0.762 0.749 0.736 0.712

0.824 0.801 0.788 0.774 0.758 0.744 0.732 0.709

0.000 0.005 0.008 0.011 0.015 0.018 0.020 0.024

0.000 0.006 0.010 0.012 0.015 0.017 0.017 0.019

0.180 0.195 0.203 0.213 0.223 0.233 0.244 0.264

0.176 0.193 0.202 0.214 0.227 0.238 0.250 0.272

0.075 0.113 0.145 0.184 0.213 0.247 0.295 0.361

0.086 0.116 0.144 0.179 0.207 0.242 0.289 0.356

0.000 0.003 0.006 0.009 0.012 0.016 0.018 0.022

0.000 0.002 0.004 0.008 0.012 0.016 0.020 0.026

0.925 0.884 0.849 0.807 0.775 0.737 0.687 0.617

0.914 0.882 0.852 0.813 0.781 0.741 0.691 0.617

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M.K. Singh et al. / Computers and Chemical Engineering 29 (2005) 1712–1719

Table 6 Interaction parameter dependence on the bounds for methanol (1)–diphenylamine (2)–cyclohexane (3) Bounds

UNIQUAC parameters

± 200 ± 500 ± 1000 ± 2000 ± 5000 ± 10000

A12 , A21

A13 , A31

A23 , A32

−30.82, 200.00 499.54, 168.65 −648.97, −187.14 226.84, 154.72 −3639.40, −78.88 −118.83, 9908.50

200.00, 200.00 81.18, 500.00 7.83, 686.79 8.68, 679.68 5.61, 675.04 17.17, 694.28

200.00, 100.70 500.00, 2.07 −199.78, −527.54 68.91, 2.07 244.43, 3835.70 −338.50, 1192.60

rmsd

Hitting bound

0.0722 0.0146 0.0042 0.0043 0.0042 0.0050

Yes Yes No No No No

Table 7 Correlation matrix for predicted and experimental tie lines of methanol (1)–diphenylamine (2)–cyclohexane (3) Extract phase F1E F1E  F1E F2E  F2E F3E  F3E F1R  F1R F2R  F2R F3R  F3R

Raffinate phase

 F1E

1.0 0.999 −0.975 −0.996 −0.996 −0.998 −0.995 −0.990 0.997 −0.990 0.996 0.991

0.999 1.0 −0.969 −0.994 −0.999 −0.999 −0.997 −0.993 −0.996 −0.992 0.997 0.993

 F2E

F2E −0.975 −0.969 1.0 0.988 0.962 0.966 0.952 0.937 0.974 0.942 −0.954 −0.938

 F3E

F3E

−0.996 −0.994 0.988 1.0 0.991 0.991 0.986 0.978 0.996 0.982 −0.987 −0.978

−0.998 −0.999 0.962 0.991 1.0 0.999 0.998 0.996 0.994 0.993 −0.998 −0.996

−0.998 −0.999 0.966 0.991 0.999 1.0 0.998 0.996 0.995 0.994 −0.998 −0.996

 F1R

F1R −0.995 −0.997 0.952 0.986 0.998 0.998 1.0 0.998 0.993 0.995 −0.998 −0.999

 F2R

F2R

−0.990 −0.993 0.937 0.978 0.996 0.996 0.998 1.0 0.988 0.996 −0.998 −0.999

−0.997 −0.996 0.974 0.996 0.994 0.995 0.993 0.988 1.0 0.993 −0.994 −0.989

−0.990 −0.992 0.942 0.982 0.993 0.994 0.995 0.996 0.993 1.0 −0.995 −0.997

F3R 0.996 0.997 −0.954 −0.987 −0.998 −0.998 −0.999 −0.998 −0.994 −0.995 1.0 0.998

 F3R

0.991 0.993 −0.938 −0.978 −0.996 −0.996 −0.998 −0.999 −0.989 −0.997 0.998 1.0

Table 8 Interaction parameters for ternary systems at 298.15 K System no. of Table 3

i–j

1

1–2 1–3 2–3 rmsd

2

1–2 1–3 2–3 rmsd

3

UNIQUAC

NRTL

Predicted (GA)

Reported (reference of Table 3)

Predicted (GA)

Reported (reference of Table 3)

Aij

Aji

Aij

Aji

Aij

Aji

Aij

−746.03 −10.72 −210.23 From GA

−340.42 661.50 −830.13 0.0043

103.66 14.02 −108.50 Lit.

−179.61 704.77 −75.54 0.0094

−489.27 −5.99 −398.24 From GA

−220.99 655.62 140.48 0.0047

NA

−95.66 17.50 17.32 From GA

336.25 −47.71 345.99 0.0084

50.26 149.54 21.19 Lit.

−27.50 30.98 −2.45 0.0103

690.68 25.79 484.20 From GA

−21.18 46.39 422.61 0.0032

217.00 233.74 57.36 Lit.

−105.93 265.54 8.89 0.0036

1–2 1–3 2–3 rmsd

−47.74 65.28 28.12 From GA

20.65 34.53 −13.54 0.0055

13.48 125.21 37.60 Lit.

−21.49 37.44 −6.73 0.0097

636.38 37.66 −19.63 From GA

226.18 203.51 177.16 0.0048

−0.61 169.81 4.90 Lit.

−37.90 134.32 63.65 0.0110

4

1–2 1–3 2–3 rmsd

32.07 0.07 −2.78 From GA

20.06 77.51 601.23 0.0567

3.56 7.02 −13.01 Lit.

−45.10 65.41 −7.64 0.0859

209.43 570.68 323.98 From GA

−61.41 90.40 306.40 0.0560

28.19 85.22 −106.96 Lit.

5

1–2 1–3 2–3 rmsd

−74.80 15.52 105.54 From GA

79.00 32.87 16.48 0.0088

−39.33 3.01 8.30 Lit.

−39.09 54.13 31.99 0.0160

632.87 704.99 23.13 From GA

575.61 86.26 692.18 0.0039

858.91 −387.06 1392.59 206.28 205.92 272.43 Lit. 0.0110

6

1–2 1–3 2–3 rmsd

119.79 −46.34 43.03 From GA

120.23 42.77 −26.22 0.0015

26.99 15.06 27.89 Lit.

295.58 −23.78 −10.84 0.0015

524.61 701.09 674.01 From GA

10.16 294.03 111.99 0.0076

NA

Aji

−162.56 32.87 −40.98 0.1080

0.0098 0.0086

0.0048 0.0069

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1.56, 2.02 −0.95, −0.46 0.19, −0.71 −0.79, 0.09 0.39, −0.69 0.17, 1.54 2.39, 1.87 −0.41, −0.78 1.01, −0.28 1.43, 2.40 0.24, 0.48 −0.89, 0.03 0.49, −0.93 0.96, 1.98 1.04, 0.14 0.85, −0.64 Aji = Aji T . a

A2–4 , A4–2 A2–3 , A3–2 A1–5 , A5–1 A1–4 , A4–1 A1–3 , A3–1 A1–2 , A2–1

−0.82, 2.39 −0.89, 0.28 −1.19, −1.16 −0.09, 1.60 Quinary 9 10

Quaternary 7 8

1.52, −0.46 2.14, 2.36

0.29, 0.33 1.56, 0.83

1.77, 2.35 0.15, 0.46

0.24, 0.79 0.68, 0.32

1.31, 0.38 0.92, 2.37

0.21, 0.47 0.37, 1.68

A1–5 , A5–2

A3–4 , A4–3

A3–5 , A5–3

A4–5 , A5–4

0.0047 0.0102

0.0041 0.0036

rmsd (from GA) A3–4 , A4–3 A2–4 , A4–2 A2–3 , A3–2 A1–4 , A4–1 A1–3 , A3–1 A1–2 , A2–1

NRTL parameters (Ai–j , Aj–I )a

System no. as in Table 3

Table 9 Interaction parameters for quaternary and quinary systems at 298.15 K

teraction parameters involved in this benchmarking system are shown in Table 4. In order to validate GA we have used three different trials using the same objective function; invariably all the trials had approximately the same rmsd’s and objective function. The final values of the objective functions for the four different methods are also compared and the GA estimation has been found to be better by an order of magnitude. With each new generation the estimates get closer to the experimental data and by the end of 200th generation a very good set of parameters is found; further generations only seem to improve marginally upon the solution obtained. Therefore value of 200 for gmax has been used for further study. The experimental and predicted data of obtained methanol (1) + diphenylamine (2) + cyclohexane (3) system has been shown in Table 5. The effect of the value of bounds on the interaction parameters has been demonstrated for it. It can be inferred that the rmsd’s are minimum at a bound of −1000 to +1000.The lower bounds gave values that hit either of the extreme (Table 6). Keeping this in mind we have chosen this bound for our estimation. The goodness of the fit has been estimated by the rmsd values and the quality of estimation has been statistically tested by the correlation matrix, which is symmetric in nature (Table 7). After benchmarking with the above system, GA has been studied and verified for five ternary systems using both UNIQUAC and NRTL models; and both the interaction parameters (literature as well as GA) are shown in Table 8. Finally, multicomponent systems—two quaternary and two quinary systems have been also handled by GA and the rmsd’s are compared with literature in Table 9. All comparisons indicate a better rmsd value. For the quaternary and quinary systems, GA was run to fit the experimental data with the NRTL model. The GA algorithm does not require an initial guess and the parameters generated are entirely new and not reproducible In Table 9 the parameters reported are not predictive i.e. the values of the parameters obtained from ternary systems have not been included for the estimation of parameters for higher order systems. These can be used predictively in GA if we freeze their values and estimate only the remaining parameters. For e.g. in a quaternary system comprising of four components (A–B–C–D) it has 12 parameters. Out of these 12 parameters, 6 can be fixed by freezing the parameter sets of any of the ternary systems (i.e. A–B–C or B–C–D or A–B–D, etc.) and evaluating the other 6 by GA. On the same lines for a quinary system (A–B–C–D–E) which has 20 parameters one can also use the parameters obtained from ternary or quaternary systems. By taking the parameters of ternary systems (A–B–C or B–C–D) one has to evaluate 14 parameters (20 − 6), similarly by taking the parameters of quaternary systems (A–B–C–D or B–C–D–E, etc.) we have to evaluate only 8 parameters (20 − 12). We have not adopted this technique in our work. Our method is not termed predictive in the above sense.

rmsd (reference of Table 3)

M.K. Singh et al. / Computers and Chemical Engineering 29 (2005) 1712–1719

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Fig. 1. Experimental (Sergeeva et al., 1971) and predicted tie lines for the ternary system methanol (1)–diphenylamine (2)–cyclohexane (3) at 298.15 K.

5. Conclusions An approach based on non-traditional optimization (genetic algorithm) for regressing binary interaction parameters in thermodynamic models has been presented. The method is stochastic and population based and does not require a good initial guess. The method produces better results than traditional techniques, which often overlook the optimum found by this method. The improvement on the regression results obtained using the approach allows the thermodynamic model to be more reliable for prediction and simulation. The effect of using different genetic operators’ probabilities has not been studied.

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