Phase equilibrium calculations using a modified form of the complete local concentration model

FINIHInlIIW[ [OIIILlllIA ELSEVIER

•uid Phase Equilibria 135 (1997) 209-226

Phase equilibrium calculations using a modified form of the complete local concentration model Isamu Nagata *, Kazuhiro Tamura Department of Chemist O' and Chemical Engineering, DiL'ision of Physical Sciences, Kanazawa University, 40-20 Kodatsuno 2-chome, Kanazawa, lshikawa 920, Japan

Received 26 February 1996; accepted 12 March 1997

Abstract

The workability of the modified complete local composition model of Wang-Chao, which includes a non-randomness parameter and two energy parameters per binary systems, is studied in the reproduction of binary coexistence curves over a wide temperature range in binary vapor-liquid equilibria and in the correlation of ternary and quaternary liquid-liquid equilibria using additional ternary and quaternary parameters. © 1997 Elsevier Science B.V. Keywords: Activity coefficient; Application; Gibbs energy; Liquid-liquid equilibria; Vapor-liquid equilibria

1. I n t r o d u c t i o n

Phase equilibrium calculations are fundamentally important for separation process design. Many excess Gibbs free energy models have been proposed to reproduce phase equilibria in binary and multicomponent systems for strongly non-ideal mixtures [1 ], but these models cannot always represent accurately the phase equilibria, especially liquid-liquid equilibria in multicomponent systems. Wang and Chao [2] showed how well their complete local composition model (CLC) could reproduce binary vapor-liquid equilibria (VLE) and predict ternary liquid-liquid equilibria (LLE) from binary information. However, our preliminary study showed that the CLC model did not work well for many ternary LLE systems other than those described by these authors and good representation of VLE for methanol + aromatic hydrocarbon systems cannot be obtained. In this paper we present the workability of a modified CLC model in the reproduction of binary coexistence curves over a wide temperature range, in binary vapor-liquid equilibria and in the correlation of ternary and quaternary liquid-liquid equilibria using additional ternary and quaternary parameters. * Corresponding author. E-mail: [email protected]; fax: + 81-76-234-4829. 0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 ( 9 7 ) 0 0 0 8 2 - 4

210

L Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

2. Solution model

To represent liquid-liquid immiscibility phenomena as well as vapor-liquid equilibria, Wang and Chao [2] proposed the complete local composition model by including an enthalpic contribution term expressed by an equation similar to the NRTL equation in addition to an entropic contribution term given by the Wilson equation. The molar excess Gibbs free energy of binary mixtures is expressed by

Z

gE=-~[X, X21(gzI--g,,)+X2X,e(g,e--g22)]+RT

(

Xlln

(D'I @22) +xzln X1 X-~

(1)

where Z is the coordination number set equal to 6; x i, the stoichiometric mole fraction of component i, and gij, the molecular interaction energy between the i - j pair. The local mole and volume fractions, x21 and @91, are given by x 2 exp( - o~2j g 2 J R T )

x2, =

X 1 exp( -- a21 g l l / R T )

nt- y 2 exp(

- oe2, g ~ , / R V )

(2)

x 2 v} exp( - oee,/RT) =

x I v~ exp( - ee21g, I / R T ) + x 2 u} exp( - a21 g 2 , / R T )

(3)

where u/L is the liquid molar volume of component i, and c~ij ( = aji) is the non-randomness parameter, similar to that used in the NRTL equation, as suggested by Zou and Prausnitz [3] in simultaneous fit of infinite-dilution activity coefficients and miscibility limits for 2-butanone + water and 2-pentanol + water at 70°C. Setting a~2 = c~21 = 1 in Eqs. (2) and (3) reduces to the original CLC model. The alternative expressions Xl2 and qbl2 could also be given by Eqs. (2) and (3) by interchanging the subscripts 1 and 2. The activity coefficients for components 1 and 2 are obtained from Eq. (1) by partial differentiation with respect to the number of moles of component i ln-/l=~

1 (z)[ 2 ] ~- x 2 1 ( g 2 , - g ~ , ) + x 2 x , _ 2 - - ( g 1 2 - g 2 2 ) xl

-ln(x,+A,2x2)+x

Ai 2 2 xi+A,2x2

A2t A21x,+x2

(4)

where Ai2 is defined as A,z = __--g-exp|Pl [

R'T

= _.-~exp| vl" L

- T

(5)

and In Y2, whose expression is given by Wang and Chao [2], can be obtained by interchanging the subscripts 1 and 2. The original model assumes the only binary parameters due to the two-body interaction contribution. In a ternary mixture both two-body interactions and three-body interactions are present among different interacting molecules. The original model can take into account the three-body interactions in a ternary mixture. It is possible to include six ternary parameters in two terms: three parameters in the enthalpic term and three parameters in the entropic term [4,5]. Then six ternary parameters may be necessary in these two terms or six parameters may be too many. A minimum number of three

I. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

211

adjustable ternary parameters may be preferable. We preliminarily studied two cases incorporating three adjustable ternary parameters into the model: I, only the term due to differences in interaction energies includes three ternary parameters; II, only the term due to the entropy effect has three parameters. Case II gave a slightly better description of ternary LLE than those obtained by using case I. In this paper we present calculated results based on case II. In a ternary mixture, we assume that Eq. (1) can be extended by introducing three adjustable ternary parameters A23~, A132, and AI23 into the logarithmic term of the Wilson-type equations as described previously [5]. The correction term reduces to zero whenever the ternary system degenerates to a binary system. The excess Gibbs free energy of ternary systems is expressed by 3

3

= i

. x i In

EG~ix* k

E. a j i x j + -2 E E. a j k i X j X k j k j

(6)

where Gji = exp( - aji~ji ) gji -- gii

•j,-

(7)

aji

R-----T-- r

(s)

and the ternary parameters A j k i = Akj i # O, and Ag i = A j j j = Aji i = 0. Similarly, the excess Gibbs free energy of quaternary systems involving a minimum set of three ternary parameters A231, A132, and AI23 per each ternary and four quaternary parameters A2341, A1342, A1243, and A1234 is given by 4

gE --

Y'~ T j i G j i x j =

RT

Xi

E

i

4

J4

E

[ 4

1 4

4

~.x / l n . 1 Y ' ~ A j i x J + 2 ~ - ' ' Y'~Ajkixjxk" Gk i Xk

j

k

j

k

1 4

+6EEE /

4

4

k

j

A j , ti xj x k X I )

(9)

where the quaternary parameters Aj~ti = A / , ~ = Akjti = A k u i = At/,i = Atkji # 0 and Ajjji = Akkji = Ak/ki = 0. Further details of incorporating the additional ternary and quaternary parameters into the entropic term of the CLC model are given by Nagata and Watanabe [6]. The activity coefficient of component i for ternary and quaternary mixtures can be similarly obtained by differentiating Eqs. (6) and (9) with respect to the number of moles of component i. The activity coefficient of component 1 for a quaternary mixture is expressed as

In Yl =

--a--

E+oj, +

q-

~

~-.-----

' E. oj, x+

Tli --

4

E+oj +

212

I. Nagata, K. Tamura/ Fluid Phase Equilibria 135 (1997) 209-226

+ 1 - ln(x I + A12x 2 ÷ AI3X 3 + Al4X 4 + A 2 3 1 x 2 x 3 + A 2 4 1 x 2 x 4 + A 3 4 1 x 3 x 4 1 - A 2 3 1 x 2 x 3 - A241XzX 4 - A 3 4 1 x 3 x 4 - 2 A 2 3 4 1 x 2 x 3 x 4 -}-A2341x2x3x4) - X l

4 EAliXi i

[

+ A231x2x3 -~- Az41X2X 4 + A 3 4 1 x 3 x 4 + A 2 3 4 1 x 2 x 3 x 4

xz [ az, + a,32x3(1 - x,) + a142 x4(1 - Xl) - a 3 4 2 x 3 x 4 -[- a1342x3x4(1 - 2x,) 1

4

EA2ixi i

-k- Aj32 XlX 3 + AI42XlX4 ~- A 3 4 2 x 3 x 4 + A 1 3 4 2 x l x 3 x 4

A31 -k- A,23x2(1 - X l ) -[- A143x4(1 - x , ) -- A243x2x 4 + A1243x2x4(1 -- 2xt) --

X 3

4

EA3ixi i

x4{

+ A i 2 3 x l x 2 + Ai43XlX4 + A243x2x4 + A 1 2 4 3 x l x 2 x 4

A41-[- A 1 2 4 x 2 ( 1 - x 1 ) - b A 1 3 4 x 3 ( 1 - x 1 ) - A 2 3 4 x 2 x 3 -[- A I 2 3 4 x 2 x 3 ( 1 4 E A 4 i x i + Ai24xi y 2 Jr- A134XlX 3 .-+-A 2 3 4 x 2 x 3 -.]- A 1 2 3 4 x l x 2 x 3 i

2Xl)

(1o) The expression for In Y2 is obtained by changing the subscripts 1 to 2, 2 to 3, 3 to 4, and 4 to 1. Similarly, the expressions In Y3 and In ")/4 are given by cyclic advancement of the subscripts.

3. Calculation procedure

3.1. Binary liquid-liquid equilibria The thermodynamic relation for liquid-liquid equilibria for component i between two liquid phases, I and II, is given by ( Xi~/i) I : ( Xia/i) lI

(11)

Y'~x~ = 1 and Y'~x~l = 1 i i

(12)

with

To represent binary coexistence-curves over a wide temperature range, we assume a12 = c~2~ = 1 and quadratic temperature dependence of the two energy parameters:

a12 = Al2 + BlzT + C12T 2

(13)

a21 =A21 -}- B 2 1 T + C21 T2

(14)

Model parameter estimating procedure is the same as previously described by Nagata et al. [7]. The

I. Nagata, K. Tamura/ Fluid PhaseEquilibria 135 (1997)209-226

213

coefficients Ajj B/j and Cij of energy parameters agj in Eqs. (13) and (14) are obtained by minimizing the following objective function using the Marquardt method [8]. Initial values of the parameters Aij, Bij and Cij used for the Marquardt minimization are estimated from the temperature dependence for each set of the energy parameters at each given temperature. The parameter determination is performed by the least-squares minimization with the following objective function using the simplex algorithm modified as Nelder and Mead [9]: Sl = E l l X i , e x p T i I _ x l I i

i,expTi I,] < 10-

,0

i = 1,2

(15)

The summation is performed for all original experimental data. The phase concentrations xl,ca~c~ and Xlxa~c are calculated with given model parameters by the use of a Newton-type algorithm on the isoactivity criterion: 82 l II TiII ----0 (16) Xi,calc Vi I - - X i,calc The objective function used for the Marquardt algorithm is expressed by the sum of the squares of the difference between experimental and calculated compositions of component 1 in the two phases: l

l

Y'-(Xl,exp--Xl,calc J

$3=

)2

I1

+ Y'~(X,.e,p-J

x u 12 1.calc,

(17)

3.2. Binary vapor-liquid equilibria Binary VLE calculations are performed using the thermodynamic equations

P&iYi = TixiPiSqb; exp[ vL( P - - Pi')/RT]

(18)

P In chi = ( 2 Y'~ yj B,j - Y'~ Y'. Yi Yi Bij ) RT

(19)

j

i j

where P is the total pressure, 4) the fugacity coefficient at P and T (q 5s, that at P~ and T); y, the vapor-phase mole fraction; PS, the pure-component vapor pressure taken from the original references, B the second virial coefficient calculated from the Hayden-O'Connell method [10], and v L the pure-liquid molar volume calculated by a modified Rackett equation [11]. For carboxylic acid-containing systems, the chemical theory based on the dimerization of the acid [12] is used to calculate ~b: = --exp

(20)

yi

where zi is the true vapor-phase mole fraction of species i, which differs from the stoichiometric mole fraction y~, and BsV~ is the free contribution to the second virial coefficient of component i (Hayden and O'Connell, [10]). An optimum set of the model energy parameters is obtained by minimizing the objective function El =

~i { ( Pi'calc - - Pi,exp ) 2 •

--¢Tp2

(Ti'calc-- Ti'exp)2 q- (Xi'calc--Xi'exp)2 2 q2 O"T

(Yx

--O'v2

(21)

where the subscripts calc and exp indicate, respectively, the most probable calculated value corresponding to each measured point and the experimental value. The standard deviations in the

214

L Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

experimental values are taken as: o-p = 1 m m H g , o-T = 0.05 K, o-x = 0.001 and o~,,= 0.003. More details of the calculations are available everywhere [13].

3.3. Ternary liquid-liquid equilibria The prediction of ternary L L E based on only binary parameters is not always good. So we introduce a m i n i m u m number of three ternary parameters to obtain good correlation of ternary LLE. The ternary parameters can be obtained by minimizing the following objective function with the simplex algorithm

3 2 (xi;kxx p _ xijk.c,,,c)-

min Y'~ Y'~

F2=

i where values.

i=

j

1,2,3 ( c o m p o n e n t s ) ,

A

experimental

similar

(22)

6M

method

j = 1,2 ( p h a s e s ) ,

is u s e d

to

obtain

k = 1,2 . . . . M ( t i e - l i n e s ) , a n d r a i n m e a n s the

quaternary

parameters

in fitting

the

minimum model

to

results.

4. Calculated results

4.1. Binary liquid-liquid equilibria Table 1 gives the temperature-dependent parameters and absolute arithmetic mean deviations between experimental data and calculated valued for mutual solubility of eight typical binary systems,

Table 1 Calculated results for mutual solubility data of binary systems Type a

System (1 + 2 )

Temp. range

Average abs.

Parameters

/°C

dev./mol%

A 12

B12/K

CI 2 / K 2

A21

B21/K

C21/K 2

408.1763 -459.9395 - 1631.9892 868.7613 1663.9957 -2042.1467 1836.4345 -2828.3686 5660.2618 - 6285.4359 400.3713 -2873.3566 2532.0035 - 1550.3920 2606.4492 - 3633.4241

5.6984 2.7429 12.4736 -4.7980 - 9.5373 15.6368 - 8.3533 14.8686 - 29.3836 34.1578 -0.0527 13.4740 - 15.8045 7.8874 - 11.3088 17.5069

0.0083 -0.0050 -0.0212 0.0071 0.0147 -0.0286 0.0129 -0.0204 0.0424 - 0.0478 0.0018 -0.0166 0.0298 - 0.0085 0.0149 -0.0211

I

1-Butanol + water

53.33-127.46

0.32

I

Furfural+cyclohexane

16.30-66.30

1.01

I

Nitroethane + n-hexane

2.90-29.30

1.29

II

l-Propoxy-2-propanol + water

34.50-171.70

1.78

II

2-Butoxyethanol + water

49.10-128.00

0.55

II

Nicotine+water

61.50-233.00

0.96

III

Dipropylamine+water

-4.80-74.80

0.57

III

l-Methylpiperidine + water

48.30-236.00

1.93

Reference [14] [15] [16] [17] [18] [19] [20] [21]

I, System with an upper critical solution temperature; II, System with both upper and lower critical solution temperatures; III, System with a lower critical solution temperature.

1. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

215

(e)

(A) 413.1

40315 393.15

393.1

383 15 373 15

373.1 363.15 74 353.15

353.1

E

34315 333,1

333.1

323.1 f

313.1

I

I

t

t

313.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Mole fraction of l-bu~anol 293.1 0.05

0.1

Mole fraction

0.]5

0.25

0.2

of 2-butoxyethanol

(D)

(c) 523.15[

353 15

503151

343.15

48311

333.15

483.

323.15

443.1

313.15

423.1!

%

303.15 403.15 293.15 383.15 283.1 :

363.15

273.1~

343.15 323.15

t

~

O.OS 0.1

t

I

I

I

I

O.IS

0.2

0.25

0.3

0.35

Mole fraction

Of nicotine

263.15 0.4

[

i

i

I

i

i

i

0.1

0.2

0.3

0.4

O.S

0.6

0.7

0.8

Mole fraction of diDropylamine

Fig. I. Calculated coexistence curves for four binary systems: O, experimental; - - , calculated. (A) l-butanol + water [14]; (B) 2-butoxyethanol + water [l 8]; (C) nicotine + water [19]; (D), dipropylamine+ water [20]. and Fig. 1 compares the calculated values with the experimental results for typical 4 systems. It can be seen that the model with quadratic temperature dependence of binary energy parameters can successfully reproduce the binary coexistence curves over wide temperature ranges for nicotine + water and 2-butoxyethanol + water mixtures having both upper and lower critical solution temperatures.

4.2. Binary uapor-liquid equilibria The binary energy parameters obtained from binary phase equilibrium data reduction, together with the root-mean-square deviations between the experimental and calculated values are presented in Table 2. The values of nonrandomness parameter o~12 is generally set as 1 except for a few systems. The original CLC model shows erroneously phase separation for the mixtures of methanol + toluene and water + 1-propanol. The present modification by using the nonrandomness parameter (3{12 is able to avoid it. To correct the original CLC model and to represent the binary VLE correctly, we use the

216

1.Nagata,K. Tamura/ FluidPhaseEquilibria135(1997)209-226

~

~

II

.

~

~

.

~

~

~

w

.

I

~

~

?

~

'-

w

.

-

~- -

.

~

~

~

+

+

+

+

+

+

+

+

I

~

~

o +

N~'o Z~

~

a.

~

-

+++~ +

+

+-

~

~

~

._-+

+

~

~ + =

~ - ~- ~ +~ +~

+ ~ ~ ~

~ +t~ + ~~ ~.~- +~ +

~

1. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

0

0

0

0

0

0

I

0

0

0

0

0

0

0

0

II

0

0

0

0

0

0

0

~

0

0

0

~

.

0

0

0

~

~

0

0

.

0

0

0

0

0

0

0

0

.

0

0

0

.

0

0

0

0

0

0

.

~

0

.

~

0

0

~

0

0

217

0

.

.,--,

.~..=

~ + + o ~ ~

+ ~= =

+

~ ++

+

~

~+

~ ~ =o= o

.~ ~ . ~~

~

+++++~++++

218

1. Nagata, K. Tumura / Fluid Phase Equilibria 135 (1997) 209-226

+ ¢q

~

a

~

-

~

Z

~

Z

~

+

¢q

+ m

z II

I I I

II

!

I

I~II

I

[

[

I

dM

I 1 [ I

I

II

d

d d dd

d d d d

d d d

dd

d ~d

2 2

~ 0 00~

+ + + ~

z + + -° ~ ÷ ~

+ ~

+

+ +~~ + + + + + + ~ ° + +

+~Z ~ ~ + + +

~'U'~

~

Z

1. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209 226

I

I

I

I

~

I

I

t

I

I

~ _ ~ ~ I

I

~

I

I

I

I

I

I

:~

~o~ I

~ ' ~

I

I

I

I

>~--~

=~'~

+

+

-F +

+

+++

+ + + +

+++

o

219

1. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

220

values of ~12 reduction.

=

1.3 for methanol + toluene, and 2.3 for water + l-propanol, for the binary VLE data

4.3. Ternary liquid-liquid equilibria In ternary systems we studied two common types. In type I, only one binaries is partially miscible; in type II, two binaries are partially miscible and the third binary is completely miscible. Table 3 gives the calculated results of ternary liquid-liquid equilibria for 35 systems including methanol, water, acetic acid, nitromethane, and acetonitrile were studied in this work. The average root-mean square deviations between the experimental and calculated results for the 35 systems were 0.87% and those of comparable 10 ternary systems were 0.91 mol.% for the present modification and 1.83 mol.% for the NRTL model with three ternary parameters [4]. From these calculated results we observed the following: (1) For type II systems whose solubility lines are nearly straight the model can predict well the experimental data. (2) For most systems of type I the original model predict a rough estimate of two phase regions are always larger than the experimental ones. (3) For type I systems containing methanol and hydrocarbons the ternary predictions using the original CLC model are grossly in error. However, the present modification including additional ternary parameters gives improved calculated results. Salazar-Sotelo et al. [92] proposed a method to represent with the NRTL model the ternary LLE system for acetonitrile + benzene + n-heptane at 45°C using only the binary parameters obtained from simultaneous correlation of the VLE of binary systems making up the acetonitrile + benzene + nheptane system, mutual solubilities for acetonitrile + n-heptane and some ternary tie-lines for the acetonitrile + benzene + n-heptane system. It is to be noted that the binary NRTL parameters for the acetonitrile + n-heptane system given in Ref. [92] should be read as c~3 = 1626.0cal mol-1 (instead of 626.0calmol-~), c31 = 1381.2calmol-1 and a13 = 0.393, because the data cannot be reproduced using the binary parameters listed in Ref. [92]. The root-mean-square deviation between experimental ternary tie-line results for nine data points and calculated with the binary parameters are obtained: 0.54mo1.% and 0.65 mol.% by the present method with the NRTL and modified CLC model; and 1.09 mol.% by the method of Salazar-Sotelo et al. [92] using the NRTL model. The present method shows better representation for the ternary LLE of the acetonitrile + benzene + n-heptane system at 45°C compared with the NRTL model. In the quaternary LLE system formed by components A, B, C and D, if we treat independently the constituting ternary A + B + C and A + B + D systems according to the method of Salazar-Sotelo et al., two sets of binary parameters for the A + B system may be obtained. In calculation of the quaternary LLE for the A + B + C + D system, we must use a single set of binary parameters for the A + B system. To obtain such a common set of parameters for the A + B system, we should readjust all constituent binary parameters for any quaternary (or more than quaternary) system. Our method does not need such calculations and the binary parameters used for any ternary system are entirely independent of ternary parameters. Furthermore the method of Salazar-Sotolo et al. provides only predicted results for quaternary or multicomponent systems. Our method can correlate quaternary (or multicomponent) LLE more accurately by introducing quaternary (or multicomponent) parameters.

4.4. Quaternary liquid-liquid equilibria Table 4 shows the calculated results for six quaternary systems, indicating that the quaternary parameters are generally necessary for good correlation of LLE. The ternary parameters obtained from

I. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 2 0 9 - 2 2 6

Z =

-e +

0

t"xl

•~

221

+

E

£'4

~

0

< o a a ~

• II

II

11 II

II

II

g~¢

II

II

II II

II

e~ 0

II

~ v v ,~ v ~,,v--<- ~ ~-,<- ~

-

e~

g

II

II

~

-

II

II

~

-

II

~

II

-

~

II II -

II II

II

II

,~ "7

,', ~

~

~'-

/

,-a ©

d

~

.=

~5

=-~

E

~ l.=,

=

6 0,1

~

~1

~

t",l

~

(",1

N

~

~

~

~

"~,

0%

I

~+e+~, ~

r¢3

,.,

+

~"

t'M

~

+

"~ +

+=+~= 0

~ e.,

~

g

~

+~:+

[-.,r,..)

[....

~

gg2N~

°

222

I. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

the ternary completely miscible VLE systems constituting the quaternary LLE systems set to zero, because the ternary parameters are not necessary in representing the ternary VLE with good accuracy. Comparison between the present model and the modified NRTL equation is given for two quaternary

(.,a}

0.4

Methanol

0.2

0.8

0.6

n-Octane

(B)

V

0.2

0.4

0,4 0.6 Molefraction Water

0.2 r,

0.8

Cyclohexane

0.8 /'x

\ 0.6 0.8 .8" ~

1

-Butanol

V

0.2

~-V"

0.4

V

V

0.6

0.8

0.2

2-Butanone

Molefraction Fig. 2. Calculated liquid-liquid equilibria for ternary systems constituting quaternary systems at 298.15K. - - - , experimental tie-line, - - , calculated with binary and ternary parameters. (A) methanol + benzene 4- cyclohexane + noctane [83]; (B) water + ethanol + 2-butanone + l -butanol [90].

1. Nagata, K. Tamura / Fluid Phase Equilibria 135 (1997) 209-226

223

systems, showing that both models with ternary and quaternary parameters give good agreement with the experimental results. Fig. 2 shows that the present model including three ternary parameters and four quaternary parameters successfully reproduces the experimental quaternary results.

5. Conclusion The modified CLC model as proposed in this paper shows good performance in the accurate representation of binary, ternary and quaternary LLE. For the reproduction of ternary and quaternary liquid-liquid equilibrium data, additional ternary and quaternary parameters should be included in the model.

6. List of symbols A j i , Bji, Cji aji Bii

FI

g g j, M N P Pi s

R S I, S 2 , S 3

T Xi Xji

Yi Z Zi

coefficients of Eqs. (13) and (14) binary energy parameter as defined by (gji - g i i ) / R second virial coefficient for i - j pair free contribution to second virial coefficient of component i objective function as defined by Eq. (21) objective function as defined by Eq. (22) coefficient as defined by e x p ( - OtjiTji) excess Gibbs free energy molecular interaction energy between j - i pair number of tie-lines number of data points total pressure vapor pressure of pure component i universal gas constant objective functions for calculation procedure absolute temperature molar volume of pure liquid i liquid-phase mole fraction of component i local mole fraction of molecule j about a central molecule i stoichiometric vapor-phase mole fraction of component i coordination number true vapor-phase mole fraction of component i

6.1. Greek letters o~ji

Yi

% Aji

non-randomness parameter activity coefficient of component i binary parameter as defined by Eq. (8) Wilson equation like parameter

L Nagata, K. Tamura/lquid Phase Equilibria 135 (1997) 209-226

224

Ajki Ajkli O'p, O'T, O'x, O~v

%

ternary parameter quaternary parameter standard deviations in pressure, temperature, liquid-phase mole fraction, and vapor-phase mole fraction local volume fraction of molecule j about a central molecule i fugacity coefficient of component i at P and T fugacity coefficient of pure component i at p.s and T

6.2. Subscripts calc exp

i,j,k,l 1,2,3,4

calculated experimental components components

6.2.1. Superscripts excess quantity liquid phases

E

I, II

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