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Ternary critical point determination of experimental demixion curve:

calculation method, relevance and limits

C. Goutaudier

1

, F. Bonnet

2

, R. Tenu

1

, O. Baudouin

2

, and J.J. Counioux

1

1

Laboratoire des Multimatériaux et Interfaces, UMR 5615 Université Lyon 1/CNRS, 69622 Villeurbanne, France

2

Prosim SA, Immeuble Stratège A, 51 rue Ampère, F-31670 Labège, France

Abstract.

In many cases of miscibility gap in ternary systems, one critical point at least, stable or metastable,

can be observed under isobaric and isothermal conditions. The experimental determination of this invariant

point is difficult but its knowledge is essential. The authors propose a method for calculating the composition of

the invariant solution starting from the composition of the liquid phases in equilibrium. The computing method

is based on the barycentric properties of the conjugate solutions (binodal points) and an extension of the straight

diameter method. A systematic study was carried out on a large number of ternary systems involving diverse

constituents (230 sets ternary systems at various temperatures). Thus the results are presented and analyzed by

means of consistency tests.

1 Introduction

According to the Gibbs’ phase rule, the knowledge of all

the invariant transformations permits to predict

qualitatively each of the equilibrium states of a

multiphase system. Hence, the importance of their

determination implies the development of many methods

and techniques applied to their characterization.

However, among invariant equilibria, the critical

phenomena, which we find particularly by studying the

miscibility gap in a multicomponent system, occupy a

special place. This problem can be illustrated by a

demixing zone in the polythermal diagram or, more

simply, in an isotherm of a ternary system (figure 1). The

binodal curves and surfaces are continuous in a

mathematical sense but, thermodynamically separated by

a critical point or line. This isobaric and isothermal

invariant point (κ

T

) or monovariant line (κ−κ

T

) is difficult

to reach by the direct measurement. So, a calculation

method was developed and tested on numerous

isothermal ternary systems with large demixing zone. The

semi-empirical method consists in using the experimental

results for the different tie-lines. A database was

constructed from experimental data taken from the

National Institute of Standards Technology NIST [1].

The determination of each critical point was

conducted in two steps: the first one is based on a

generalization of the straight diameter method and the

second, on the exploitation of the modulus of the

experimental tie-lines.

This contribution presents the original method of

calculation derived from the rectilinear diameter law and

the analysis of results shows its limits and its relevance.

Figure 1. Shell defining the domain of liq-liq equilibrium in

the 3D representation of the ternary system A + B + C under

constant pressure (κ

T

: ternary invariant critical point).

2 Methodology

2.1 Database

A database was built, containing more than 90 ternary

systems and around 230 sets of experimental data at

various temperatures.

Experimental data have been extracted from NIST

database [1]. The selected systems have a wide

miscibility gap with a sufficient number of experimental

Web of Conferences

DOI: 10.1051/

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,

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01046 (2013)

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points. The required data are the compositions of the two

conjugated liquid phases, the temperature and if they are

available the coordinates of the invariant point.

The database is organized in order to use easily the

data to test the method in order to assess its reliability.

2.2 Straight diameter method

Originally, the formulation of the so-called law of

rectilinear diameter for the determination of the critical

volume of substances became in a very important tool for

researchers in the field of critical phenomena [2]. This

semi-empirical method consists in plotting the midpoints

of each tie-line, defining the associated straight line with

the better regression parameters, and then extrapolating

this latter until its intersection with the experimental

binodal curve [3]. However, this graphical method needs

a fairly good symmetry between both sides of the binodal

curve, which provides a good linearity of the midpoints.

In order to apply this property to unsymmetrical curves,

we have performed a barycentric balance of the two

conjugate points. For each tie-line (i), the barycentric

point P

i

satisfies the following relation:

(

)

2i1ii

LλLλ1P +−=

(1)

where (1-λ)

and λ represent respectively the mass

fractions of the liquids L

1i

and L

2i

in equilibrium at the

study temperature. In the pedagogic system A+B+C

(figure 1), the components of the column vectors

i

P

,

1i

L

and

2i

L

are their mass fractions in the solvent A,

the constituent B (x-axis) and the constituent C (y-axis).

Figure 2. Graphical representation of the straight diameter

method in the ternary system A+B+C (isotherm and isobar).

The straight line D

λ

corresponds to the better regression

parameters for λ

min

.

For a series of tie-lines of the same isotherm, an error

function E is then calculated as function of the parameter

λ, which optimal value λ

min

corresponds to the minimum

of E ie the best fitting parameters. The equation of the

corresponding straight line D

λ

may

be written (figure 2):

WC = a WB + b (2)

where W

B

and W

C

are respectively the mass fractions of

the constituents B and C. Obviously a and b are the

calculated parameters of the straight line D

λ

. According

to the rectilinear diameter law, this line will cut the

binodal curve near the critical point.

2.3 Modulus method

The second step of calculation is based on a particular

property of the critical points: at this point indeed, the

internodal length becomes zero. Since each tie-line can be

identified by its intersection with the straight line D

λ

previously obtained by the diameter method, the length of

the tie-line can be considered as a function of the abscissa

or the ordinate of this point. As the variation law is

unknown, it was assumed that the raising of the

internodal distance to the α

th

power (L

α

) linearly varies

with the mass fractions W

B

or W

C

of the constituents B

and C. At each tie-line (i), a couple of values (W

Bi

, W

Ci

)

can be calculated by using the equation (2).

For the series of experimental tie-lines, an error

function E is then calculated as function of the parameter

α, which optimal value α

min

corresponds to the common

minimum of the E function, with W

B

or W

C

ordinates. In

the figure 3, we have plotted the modulus L of this length

at the power α

min

as a function of W

B

or W

C

. On a very

short distance, we have then extrapolated the straight

lines obtained up to the modulus zero. So this

extrapolation leads to the coordinates of the critical point

κ

T

.

Figure 3. Power α of the tie-line’s internodal distance L as a

function of the mass percentage of component B. Extrapolation

to zero length corresponds to one of the coordinates of the

critical point κ.

3 Results

The proposed method for the determination of the critical

point has been tested on 230 experimental ternary

systems having a liquid-liquid equilibrium area. It gives

consistent results in most cases, provided to have a

sufficient number of tie-lines.

Two error functions have been tested in order to find

the most suitable regression parameters for each step of

the calculation (λ and α). The first one was the classical

01046-p.2

39

th

JEEP – 19

th

- 21

st

March 2013 – Nancy

correlation coefficient R. The second one was defined as

follow:

(

)

(

)

∑

−−=

i

BcalcBi

α

calc

α

i

ww

LL

N

E

1

or

(

)

(

)

∑

−−

i

CcalcCi

α

calc

α

i

ww

LL

N

1

(3)

where N is the number of experimental tie-lines.

As an example, figure 4 compares the changes of the

error functions calculated from the system 2-

Propanol+Toluene+Water. The 13 experimental tie lines

were determined by Washburn at 298K under

atmospheric pressure using refractive indices

measurements [4]. In this figure the variations are plotted

in normalized arbitrary unit and lead to the quite similar

minimum value.

Figure 4. Comparison of the changes of the error functions

calculated from the system 2-Propanol+Toluene+Water. (R and

E values are plotted in normalized arbitrary unit. Experimental

data from [4]).

In these conditions, the optimal fitting parameters for the

studied case are λ = 0.527 and α = 1.837.

Then in order to assess the reliability of the

calculation method, the coordinates of the critical point

are compared with those available in the literature or

calculated by classical method using binary interaction

parameters. Typical values are given in table 1 for the

system 2-Propanol + Toluene + Water at 298K and are in

a very good agreement.

Table 1. Coordinates of the critical point κ

T

calculated from

several methods (in molar fraction ; T=298K).

Method

2-

Propanol

Toluene

Water

NRTL 0.29 0.06 0.65

UNIQUAC 0.30 0.075 0.625

This work 0.289 0.058 0.653

In contrast, the method does not work in a number of

cases. Often critical analysis of the experimental results

provides the solution easily, as shown in figure 5 for the

ternary system Benzene + Hexane + Perfluorohexane. It

should check the experimental tie-lines in order to

remove aberrant lines intersecting.

Figure 5. An example of ternary system for which the method

of rectilinear diameter cannot work [1].

4 Conclusion

The knowledge of the ternary critical point of demixion

areas is essential for any definition of an operation unit

based on liquid-liquid extraction. The original calculation

method was assessed on a large number of experimental

data. The method derives from the rectilinear diameter

law on which a barycentric balance is applied. The

quality of the conjugated experimental points is the

dominant parameter.

References

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Kazakov, J.W. Magee, I.M. Abdulagatov, K.

Kroenlein, C.A. Diaz-Tovar, J.W. Kang, R. Gani,

NIST ThermoData Engine Version 7.0, Database

#103B, NIST Thermodynamics Research Center

(2011)

2. L.P. Cailletet, E. Mathias, C. R. Séances Acad. Sci.

102 1002 (1886); J. Phys. Théor. Appl. 5 549 (1886)

3. S. Reif-Acherman, Quim. Nova. 33 2003 (2010)

4. E.R. Washburn, A.E. Beguin, J. Am. Chem. Soc. 62

579 (1940)

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