classification of scalar property isoline diagrams of homogeneous ternary mixtures

ISSN 0040�5795, Theoretical Foundations of Chemical Engineering, 2012, Vol. 46, No. 3, pp. 221–232. © Pleiades Publishing, Ltd., 2012.Original Russian Text © L.A. Serafimov, V.M. Raeva, V.N. Stepanov, 2012, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2012, Vol. 46, No. 3, pp. 267–277.

221

INTRODUCTION

Design and optimization of chemical processesneed knowledge of various properties of phases. Inrecent years, increasing attention has been focused onthe properties of multicomponent mixtures, includingternary mixtures as the simplest representatives ofmulticomponent ones. Works in this field are mostlyexperimental, and one of their important aspects isrepresentation of experimental data.

The thermal, volumetric, and transport propertiesof liquid mixtures are studied most extensively, since itis these properties that should primarily be known forinvestigating and describing mass transfer in systemsconsisting of several phases, such as liquid–vapor, liq�uid–solid, and solid–solid systems. In recent years,the leading scientific publishers have published tens ofstudies of various properties of ternary mixtures. Therehave been both systematic studies of groups of proper�ties of certain mixtures [1–13] and studies of their par�ticular properties [14–25]. In parameter fixing, any ofthese properties is defined by a single number; that is,they all belong to the family of scalar properties (σ),which include intensive and extensive ones [26]. Thelatter are examined as specific properties; in the caseof mass transfer, the number of moles is additionallytaken into account.

In the general case, the number of degrees of free�dom of a phase is determined by n + 2 variables,namely, the number of moles of the phase, tempera�ture, pressure, and the number of components. Theproperties σ of a phase are usually considered for apreset number of components, temperature, and pres�sure. If the phase is in thermodynamic equilibriumwith another phase, the number of degrees of freedomis is n, according to the phase rule; that is, for investi�gating an intensive or specific quantity σ, it is suffi�

cient to specify the composition of the phase and tem�perature (or pressure).

Since there is no general theory of solutions thatcould predict the properties of an n�component mix�ture (n ≥ 3) [27], physical experiments are necessary inthis field. However, these experiments are very labori�ous even for ternary mixtures. Mathematical models ofheterogeneous equilibria based on the conception oflocal compositions make it possible to limit the exper�iment to binary mixtures. With model parameters esti�mated from binary equilibrium data, it is possible tocalculate some excess functions for multicomponentmixtures [28, 29].The adequacy of the resulting math�ematical model to experimental data is verified for anumber of characteristic compositions.

Here, we report a classification of diagrams of sca�lar property isolines for ternary mixtures with differentnumbers of singular points per element of the concen�tration simplex.

RESULTS AND DISCUSSION

It was demonstrated earlier [30–32] that the molarGibbs energy, molar enthalpy, and molar entropy of afirst�order phase transition can be described usinglocal composition equations and the following rela�tionship [33]:

(1)

The results are represented as isopleth diagrams for the

excess functions and [34].

Relationship (1) can be represented in terms of thetrajectories generated by the gradients of these quanti�ties [35]:

(2)

.E E Eg h T sΔ = Δ − Δ

,EgΔ ,EhΔET sΔ

grad grad grad .E E Eg h T sΔ = Δ − Δ

Classification of Scalar Property Isoline Diagrams of Homogeneous Ternary Mixtures

L. A. Serafimov, V. M. Raeva, and V. N. StepanovMoscow State University of Fine Chemical Technologies, pr. Vernadskogo 86, Moscow, 119571 Russia

e�mail: [email protected] July 20, 2011

Abstract—The necessity of creating a classification has been substantiated for scalar property isolines dia�grams of ternary systems having different numbers of singular points per element of the concentration sim�plex. The total number of possible diagrams has been determined, and the topologically feasible diagramstructures containing two singular points per open element of the concentration simplex have been synthe�sized.

DOI: 10.1134/S0040579512030086

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SERAFIMOV et al.

In this case, the temperature and pressure are con�stant.

As for diagrams of temperature–pressure isolines,the vector field of these isopleths is related to the vec�tor field of liquid–vapor tie lines by the followingequation [36, 37]:

(3)

where is the functional matrix of the second deriv�atives of the Gibbs function at constant pressure andtemperature. This matrix is positive�definite. At a con�stant pressure, the temperature along a temperaturepressure isoline is constant.

In the general case, the gradient of a scalar propertyσ for a three�component mixture is defined as gradσ =

For an interior singular point, both partial

derivatives are zero, so gradσ = 0, and, accordingly,this point is an extremum or a saddle. For a boundarysingular point, if one of the derivatives is zero and theother one is constant at this point and remains invari�able for an arbitrary long time, then this point ismixed, topological–analytical type. Finally, the fol�lowing situation is possible for ternary mixtures: bothderivatives are constant, so the singular point is topo�logical [38].

The singular points of a scalar property diagramform some frame that is stable in a certain range ofparameters [38]. The corresponding functional matrixof the second derivatives of the gradient is

(4)

The determinant of matrix (4) has real characteristicroots, since the matrix itself is symmetrical. It wasdemonstrated that, owing to its stability, the scalarfield of σ has generalized�saddle or generalized�nodetype singular points [35].

The liquid–vapor tie�line vector field also has sad�dle� and node�type singular points. The relationshipbetween the gradient of equilibrium temperature Тand the tie�line vector field is given by Eq. (3). At thesame time, the differential equation of free open evap�oration trajectories is

(5)

and has saddle� or node�type singular points at = 0.The differential equation of the trajectories generatedby the gradient of equilibrium temperature is

(6)

( ) ( )P, T

grad ,LLVS T−Δ = −ijG Y X

ijG

, .i jx x

∂σ ∂σ

∂ ∂

2 2

2

2 2

2

.i ji

j i j

x xx

x x x

∂ σ ∂ σ

∂ ∂∂

∂ σ ∂ σ

∂ ∂ ∂

ddt

= −

X Y X

−Y X

grad .ddt

= σ

X

Thus, linear similarity between systems (3) and (6) isobserved at each composition point. This is due to thefact that the vector that is opposite in sign to the gradi�ent of equilibrium temperature and the liquid–vaportie line make an acute angle.

The singular points of vector and scalar fields of thesame type are called differently in different areas ofmathematics. Those who consider dynamic systemsdescribed by Eqs. (5) and (6) use the concept of sta�tionary saddle� and node�type singular points [29].Often the singular points at which the gradient is zeroare referred to as stationary [40]. As for isopleths,which are typically considered in differential geome�try, a simple node� or saddle�type singular point iscalled elliptic or hyperbolic point, depending on therun of the isopleths in the vicinity of this point [41, 42].When a gradient vector field in a plane is considered inmathematical analysis, the concept of potential func�tions is used and the singular points of the gradient arecalled critical points [43, 44]. In order to avoid confu�sion, we will establish a correspondence between theterms used for the same types of singular points (Table 1).

The set of characteristic roots (λ) of matrix (4) forternary systems determines the singular point type.For example, the following situations can take placefor the interior singular points defined by Eq. (5):

(1) and —minimum in σ (ellipticminimum, unstable node);

(2) and —maximum in σ (ellipticmaximum, stable node);

(3) and —minimax (saddle);

(4) and —minimax (saddle).For liquid–vapor systems, these singular points are

referred to as analytical extremes.The classification of boundary singular points is

more complicated. The following singular points mayoccur here, depending on the signs of the characteris�tic roots:

(1) and —elliptic minimum (ingoingminimum point, unstable node);

(2) and —elliptic maximum (outgo�ing maximum point, stable node);

(3) and —hyperbolic point (ingoingmaximum point, simple saddle);

(4) and —hyperbolic point (outgoingminimum point, simple saddle).

When the trajectories generated by the gradient(Eq. (6)) are considered, the signs of all characteristicroots are inverted.

This work presents an isopleths diagram classifica�tion based on thermodynamical�topological analysis.

Hereafter, we adhere to the terminology used indifferential geometry for the reason that the results ofphysical and computational experiments for three�component mixtures are usually represented as anisopleth diagram for a scalar property σ.

1 0λ < 2 0λ <

1 0λ > 2 0λ >

1 0λ > 2 0λ <

1 0λ < 2 0λ >

Г 0λ < 0Bλ <

Г 0,λ > 0Bλ >

Г 0λ > 0Bλ <

Г 0λ < 0Bλ >

THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING Vol. 46 No. 3 2012

CLASSIFICATION OF SCALAR PROPERTY ISOLINE DIAGRAMS 223

Note that the number of possible diagrams of thescalar fields of different properties is finite and inde�pendent of the property type and is determined by thediagram topology [45]. This means that, for givennumbers of singular points in different elements of theconcentration triangle, it is possible to determine thetotal number of diagrams by relying on relationshipsbetween the numbers of elliptic and hyperbolic points.

We will distinguish between mixtures whose con�centration simplex has no more than one, two, andthree singular points per its element. We will thusobtain three subsets of diagrams. These subsets areactually a single set of scalar property diagrams thatcan turn into one another as the conditions are varied.Taking into account the topological constraints on thepossible numbers of diagram types makes it really pos�sible to compile an atlas of σ isopleth diagrams. Thisatlas could be used at the data processing stage to verifywhether a given diagram obeys the general qualitativerules imposed on all possible diagrams. This is partic�ularly true for diagrams constructed by empiricalmethods. This approach would prevent errors such aswere noted in our earlier publication [46].

Earlier, we investigated the local features of scalarproperty diagrams for mixtures consisting of an arbi�trary number of components [35]. Here, we assumethat any diagram examined is stable in some range ofexternal parameters, implying that it has only simplesingular points.

In the study of the nonlocal features of the scalarfield of any property σ, we will use the Hopf theorem[42], according to which the relationship between sin�gular points with different Poincare indices on asphere is independent of the particular field acting onthis sphere. The algebraic sum of the indices of thesesingular points is equal to the Euler characteristic. Thesign of the Poincare index of a simple singular point is

equal to the sign of the product of the characteristicnumbers of this point. For a one�dimensional sphere(binary mixture), there is a single root, i.e., signi =signλ. For a two�dimensional sphere (which is equal indimensionality to a concentration triangle), signi =sign(λ1λ2). In view of this, all elliptic singular points,irrespective of whether they correspond to a maximum

or minimum have apositive Poincare index. All hyperbolic points

have positive index [44].

In the general case, the Euler characteristic isdefined as

(7)

where R is the genus of the surface and n is the numberof components in the mixture [37]. For a sphere, R =0. According to Eq. (7), the Euler characteristic is Ψ =0 for binary mixtures and Ψ = 2 for ternarymixtures.

Any concentration triangle is equivalent to a semi�sphere [37]. Therefore, each singular point inside aconcentration triangle will be twice represented on asphere (with the coefficient 2) and each boundary,which will be mapped on the equator, will have thecoefficient 1.

Thus, for binary mixtures characterized only byelliptic points (maximum and minimum), we have2E2 + E1 = 0. The general equation reflecting the non�local properties of σ isopleth diagrams of three�com�ponent mixtures is

(8)

with

(9)

( 1 0,λ > )2 0λ > ( 1 0,λ < ),2 0λ <

( 0,iλ <

)0jλ >

1(1 ) 1 ( 1) ,nR −⎡ ⎤Ψ = − + −⎣ ⎦

E H E H3 32( ) 0.Γ Γ− + − =

E H E H E2 2 1.Γ Γ− = − +

Table 1. Singular point designations in different areas of mathematics

Isopleth diagrams Gradient�generated trajectory diagrams

Differential geometry Mathematical analysis Qualitative theory of differential equations

Interior singular points

Elliptic pointsCritical points

Stable and unstable nodes

Hyperbolic points Saddles

Boundary singular points

Elliptic points Ingoing minimum and outgoing maximum points Boundary nodes

Hyperbolic points Outgoing minimum and ingoingmaximum points Boundary saddles

224


SERAFIMOV et al.

Table 2. Composition–scalar property diagrams for three�component systems with two singular points per open elementof the concentration simplex

Diagram class

Diagram type E1 H1 E2 H2 E3 H3 M2

1 2 3 4 5 6 7 8 9

3. [2. 0. 0]. 01 0 3 2 0 0 0 2

2 2 1 1 1 0 0 2

3. [2. 1. 0]. 03 1 2 2 1 0 0 3

4 3 0 1 2 0 0 3

3. [2. 1. 1]. 05 0 3 3 1 0 0 4

6 2 1 2 2 0 0 4

3. [2. 2. 0]. 07 0 3 3 1 0 0 4

8 2 1 2 2 0 0 4

3. [2. 2. 1]. 09 1 2 3 2 0 0 5

10 3 0 2 3 0 0 5

3. [2. 2. 2]. 011 0 3 4 2 0 0 6

12 2 1 3 3 0 0 6

3. [2. 0. 0]. 1

13 0 3 1 1 1 0 2

14 2 1 0 2 1 0 2

15 2 1 2 0 0 1 2

3. [2. 1. 0]. 1

16 1 2 1 2 1 0 3

17 3 0 0 3 1 0 3

18 3 0 2 1 0 1 3

19 1 2 3 0 0 1 3

3. [2. 1. 1]. 1

20 0 3 2 2 1 0 4

21 2 1 1 3 1 0 4

22 2 1 3 1 0 1 4

23 0 3 4 0 0 1 4

3. [2. 2. 0]. 1

24 0 3 2 2 1 0 4

25 2 1 1 3 1 0 4

26 2 1 3 1 0 1 4

27 0 3 4 0 0 1 4

3. [2. 2. 1]. 1

28 1 2 2 3 1 0 5

29 3 0 1 4 1 0 5

30 3 0 3 2 0 1 5

31 1 2 4 1 0 1 5

3. [2. 2. 2]. 1

32 0 3 3 3 1 0 6

33 2 1 2 4 1 0 6

34 2 1 4 2 0 1 6

35 0 3 5 1 0 1 6

3. [0. 0. 0]. 2 36 2 1 0 0 1 1 0



Table 2. (Contd.)

Diagram class

Diagram type E1 H1 E2 H2 E3 H3 M2

1 2 3 4 5 6 7 8 9

3. [1. 0. 0]. 237 1 2 1 0 1 1 1

38 3 0 0 1 1 1 1

3. [1. 1. 0]. 2

39 0 3 2 0 1 1 2

40 2 1 1 1 1 1 2

41 0 3 0 2 2 0 2

3. [2. 0. 0]. 2

42 0 3 2 0 1 1 2

43 2 1 1 1 1 1 2

44 0 3 0 2 2 0 2

3. [1. 1. 1]. 2

45 1 2 2 1 1 1 3

46 3 0 1 2 1 1 3

47 3 0 3 0 0 2 3

48 1 2 0 3 2 0 3

3. [2. 1. 0]. 2

49 3 0 1 2 1 1 3

50 1 2 2 1 1 1 3

51 3 0 3 0 0 2 3

52 1 2 0 3 2 0 3

3. [2. 1. 1]. 2

53 0 3 3 1 1 1 4

54 2 1 2 2 1 1 4

55 2 1 4 0 0 2 4

56 0 3 1 3 2 0 4

57 2 1 0 4 2 0 4

3. [2. 2. 0]. 2

58 0 3 3 1 1 1 4

59 2 1 2 2 1 1 4

60 2 1 4 0 0 2 4

61 0 3 1 3 2 0 4

62 2 1 0 4 2 0 4

3. [2. 2. 1]. 2

63 1 2 3 2 1 1 5

64 3 0 2 3 1 1 5

65 1 2 5 0 0 2 5

66 3 0 4 1 0 2 5

67 1 2 1 4 2 0 5

68 3 0 0 5 2 0 5

3. [2. 2. 2]. 2

69 0 3 4 2 1 1 6

70 2 1 3 3 1 1 6

71 0 3 6 0 0 2 6

72 2 1 5 1 0 2 6

73 0 3 2 4 2 0 6

74 2 1 1 5 2 0 6

226


SERAFIMOV et al.

The H1 points are not involved in Eqs. (8) and (9),because they acquire a zero Poincare index on beingmapped onto a sphere.

Let the number of singular points inside a concen�tration triangle be М3 and let the number singularpoints on the triangle edges (which are treated as opensets) be М2. Equation (8) will then be representable as

(10)

The possible solutions of Eq. (10) provide a basisfor classification of σ diagrams for ternary mixtures.This classification is based on the numbers of ellipticand hyperbolic singular points belonging to differentelements of the concentration triangle and, accord�ingly, representing different numbers of components.

On being mapped onto a sphere, the singular pointslocated on the edges of the concentration triangle become

elliptic ( → ) or hyperbolic ( → ).The classification suggested here is an equivalence

classification [47], since it regards a scalar property σ.It is based on the topological image of a diagram thatis formed by different types of singular points locatedin different boundary elements of the concentrationtriangle and inside this triangle. The vertices of theconcentration triangle (1, 2, 3) and its edges are con�sidered to be distinguishable. The total number ofbinary singular points is =

The class of a diagram is coded by a set of numbers.For example, in a class 3.[2.2.1.].2 ternary diagram,

EH H3 2 1

3 22

1 2 .2

M M+ +− = +

E(2)2 E(3)

2 H(2)2 H (3)

2

2M 12 13 23.M M M+ +

edges 1�2 and 1�3 have two binary singular pointseach, edge 2�3 has one singular point, and two singularpoints are inside the concentration triangle. The num�bers subsequent to this designation specify the mixturetype, which is determined by the relationship betweenthe numbers of elliptic and hyperbolic singular points.Class 3.[2.2.1.].2 includes six types. For example, atype 3.[2.2.1.].2�63 diagram, as distinct from the otherdiagrams of this class, has three elliptic binary singularpoints and two hyperbolic ones (Table 2). Thus, thisclassification is detailed up to diagram type. Subtypes,which differ in the mutual arrangement of elliptic andhyperbolic singular points, are taken into account hereonly for systems that have no more than one singularpoint per element of the concentration triangle,because their number in the other cases is vary large.

For binary diagrams as such, there are four classes,according to the conventional classification, which are1.0, 1.1, 1.2, and1.3, and their number is equal to thenumber of types. If a binary diagram is considered as aconstituent of a concentration triangle, the number oftypes will increase to five. The composition–scalarproperty diagrams of binary mixtures have been inves�tigated in detail [33, 35, 46, 49, 50].

The σ isopleth diagrams of ternary mixtures thathave no more than one singular point per element(vertex, edge, inner region) of the concentration sim�plex are shown in Fig. 1 [51]. There are 7 classes and14 types in this case. However, for given numbers ofelliptic and hyperbolic points, these points can be dif�ferently arranged in the concentration triangle. This

Table 3. Classes of composition–scalar property diagrams having no more than three singular points per open element ofthe concentration triangle

Class Number of types Class Number of types Class Number of types

3.[3.0.0.].0 2 3.[3.3.1.].1 4 3.[2.0.0.].3 3

3.[3.1.0.].0 2 3.[3.3.2.].1 4 3.[2.1.0.].3 4

3.[3.1.1.].0 2 3.[3.3.3.].1 4 3.[2.1.1.].3 5

3.[3.2.0.].0 2 3.[3.0.0.].2 4 3.[2.2.0.].3 5

3.[3.2 1.].0 2 3.[3.1.0.].2 5 3.[2.2.1.].3 6

3.[3.2.2.].0 2 3.[3.1.1.].2 6 3.[2.2.2.].3 7

3.[3.3.0.].0 2 3.[3.2.0.].2 6 3.[3.0.0.].3 4

3.[3.3.1.].0 2 3.[3.2.1.].2 6 3.[3.1.0.].3 5

3.[3.3.2.].0 2 3.[3.2.2.].2 6 3.[3.1.1.].3 6

3.[3.3.3.].0 2 3.[3.3.0.].2 6 3.[3.2.0.].3 6

3.[3.0.0.].1 4 3.[3.3.1.].2 6 3.[3.2.1.].3 7

3.[3.1.0.].1 4 3.[3.3.2.].2 6 3.[3.2.2.].3 8

3.[3.1.1.].1 4 3.[3.3.3.].2 6 3.[3.3.0.].3 7

3.[3.2.0.].1 4 3.[0.0.0.].3 1 3.[3.3.1.].3 7

3.[3.2.1.].1 4 3.[1.0.0.].3 2 3.[3.3.2.].3 8

3.[3.2.2.].1 4 3.[1.1.0.].3 3 3.[3.3.3.].3 8

3.[3.3.0.].1 4 3.[1.1.1.].3 4



circumstance is responsible for the existence of sub�types, which are designated by a letter added to thenumerical code of the diagram (Fig. 1). In the subsetconsidered, the number of diagrams with unorientedtrajectories is 26.

Orientation is determined by the run of the trajec�tories generated by the gradient whose direction coin�

cides with the direction in which the scalar field of σincreases. If the vertices are replaced with troughs, wewill obtain oppositely oriented diagrams (Fig. 2). If thelatter are taken into consideration, there will be49 composition–property diagrams, because threediagrams among the reoriented ones will be identical(interconvertible by rotation or inversion of the con�

E1

E1 E1 E1 E1

E1

E2 E2

E1 E1

H2 E3

H2

3.1.1–1a3.1.0–23.1.0–1b3.1.0–1a3.0.0–1

E3 H2

E1 E1 E1

E1E2 E2 H2 E2 E2

E1H2

E1

H3

3.2.0–2b3.2.0–2a3.2.0–13.1.1–23.1.1–1bE2

3.2.1–3a3.2.1–2b3.2.1–2a3.2.1–13.2.0–2c

3.3.1–1a3.3.0–23.3.0–1b3.3.0–1a3.2.1–3b

3.3.1–3b3.3.1–3а3.3.1–23.3.1–1с3.3.1–1b

3.3.1–4

E1

E2

E1

H2E2 H2

E3

H2H2

E1 E1

E3 E3

H2 H2

E1

E1

E1

H3

E2 E2

E1

E2 E2 E2H2

E1 E1 E1

H2 H2

E1 E1

E2

H3

E2

E1 E1

E1

E3

E2

H2E2E2H2

E1 E1

E2 H2

H2H2 H2 E2

H3

E1

E1 E1 E1 E1

E1 E1

E1E1

E3

H2 H2

H3

E2 E2 E2 E2

H3

H2

E1

E2 E2

E2

E3 H2H2

H2

Fig. 1. Composition–scalar property diagrams of three�component systems having no more than one singular point per open setof the concentration triangle: E = elliptic point and H = hyperbolic point.

228


SERAFIMOV et al.

centration triangle) [37]. This subset of diagrams alsohas been investigated in detail [52].

Things are different with the subsets of scalar isop�leth diagrams for ternary mixtures having no morethan two or three singular points per diagram element.In this case, the number of mixture subtypes is much

larger [53]. It is, therefore, reasonable at this stage tolimit classification to diagram types and to determinethe diagram subtype for each particular mixture ofinterest. An analysis of the complete subsets of dia�grams with several singular points per diagram elementgoes beyond the limits of a regular scientific article.

The numbers of classes and types were determined,using Eq. (10), for σ isopleth diagrams having severalbinary or ternary singular points. For mixtures withtwo or fewer singular points per diagram element, wehave 22 classes, which include 74 types of diagramswith unoriented trajectories (Table 2). Figure 3 pre�sents examples of the run of σ = const isopleths for thissubset. For mixtures with three or fewer singular pointsper element of the concentration triangle, the numberof classes is 50 and the number of types is 223 (Table 3).Figure 4 presents several examples of isopleth dia�grams from different classes of this subset.

Note that the presence of two singular points in thecomposition–property relationships is not a rare casefor binary homogeneous mixtures. The concentrationdependences of excess properties of some binary solu�tions have three singular points [54–58]. Further�

(а)(b)

(c) (d)

E2 E1

H2

E1

H3

H2H2E2

E3

E1

E3

E2

H3

E1

E1

E2

H3

E2

H3

E1 E2 E1

E1

H2

E3

H2

E3

H2

Fig. 3. Scalar property isopleth diagrams of class 3.[1.1.1].2 three�component mixtures: (a) type 45, (b) type 46; (c) type 47, and(d) type 48.

E1

H2

E2

E3

H2

E1

E3

H2H2

E2

Fig. 2. Topographic representation of composition–prop�erty diagrams of ternary mixtures. The minima and max�ima of the scalar property are indicated by light and darkpoints.



more, there are scalar property isoline diagrams of ter�nary mixtures that have several interior singular points[59–61].

In any subset, the diagrams that are in conflict withEqs. (9) and (10) should be rejected as nonexistentones [46]. This approach has been implemented in theHYSIS program package applied to the fairly compre�hensively studied diagram subset in which there is nomore than one singular point per element of the con�centration triangle. The three subsets of scalar isoplethdiagrams of ternary homogeneous mixtures constitute

a single set in which there can be diagram redistribu�tion among the subsets under variation of externalconditions. The way in which these diagrams undergointerconversions, according to bifurcation theory [62],remain unclear.

CONCLUSIONS

The classification suggested in this work covers allpossible composition–property diagrams of homoge�neous ternary mixtures. This classification is of impor�

(а) (b)

(c) (d)

E1H2

E1

E2

H3

E2

H3

E2E1 E1

H2E3

H2

E2 H2 E2H2

E2

E2

E2 E2

E3

H2

E2 E2

E2

H3

(e) (f)

E3

E1

E3

H2

H2

H2 H2

E3

H2

H3

H3 H3

E2

E2

E2

E2

E2

E2 E2 E2

Fig. 4. Scalar property isopleth diagrams of three�component mixtures with three singular points per element of the concentra�tion triangle: (a) 3.[3.0.0.].0, (b) 3.[3.1.1.].0, (c) 3.[3.2.1.].2, (d) 3.[2.1.1.].3, (e) 3.[2.2.0.].3, and (f) 3.[3.3.3.].3.

230


SERAFIMOV et al.

tance both for further development of liquid solutiontheory and for solving various engineering problems. Itis expected to be helpful in the development andimprovement of methods of calculation of thermody�namic properties of ternary systems from limitedexperimental data arrays or only from data known forthe binary constituents.

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research, project no. 10�08�00785a).

NOTATION

—excess molar Gibbs energy, J/mol;

—excess molar enthalpy, J/mol;i—Poincare index of a singular point;k—number of components;R—genus of the surface;

—molar entropy of the liquid–vapor transi�tion, J/(mol K);

—excess molar entropy, J/(mol K);Т—temperature, K;х—composition of the liquid phase in mole fractions;λ—characteristic root;σ—scalar property;Ψ—Euler characteristic.

SUBSCRIPTS AND SUPERSCRIPTS

1, 2, 3—components;Γ—boundary point.

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classification of scalar property isoline diagrams of homogeneous ternary mixtures

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