Theoretical Foundations of Chemical Engineering, Vol. 38, No. 1, 2004, pp. 22–30.
Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 38, No. 1, 2004, pp. 24–32.
Original Russian Text Copyright © 2004 by Serafimov, Tatsievskaya, Frolkova.
Extractive Distillation Systems with Separating Agents
Undistributed between Phases
L. A. Serafimov, G. I. Tatsievskaya, and A. K. Frolkova
Lomonosov State Academy of Fine Chemical Technology, pr. Vernadskogo 86, Moscow, 117571 Russia
Received November 20, 2002
Abstract—Extractive heterogeneous systems with nonvolatile agents are considered. It is shown that at sections where the concentration of the nonvolatile agent is constant these systems disobey the Konovalov law.
All extractive heterogeneous systems are divided
into systems with components distributed between
phases and systems with separating agents undistributed between phases. The systems with components
distributed between phases can be subdivided into systems in which the separating agent is the least volatile
component, and systems in which the separating agent
is the most volatile component. Each of these variants
is implemented in its own standard complex of distillation columns (Fig. 1) with a certain flow arrangement
[1, 2]. From systems in which the separating agent is
the least volatile component, one can single out systems
in which the extracting agent is virtually nonvolatile,
i.e., is absent from the vapor phase [1]. A particular case
of extractive distillation with a nonvolatile agent is salt
distillation [3, 4]. In salt distillation, a solution of a salt
or several salts in a mixture to be separated is used
instead of nonvolatile liquid separating agents. The
extraction efficiency of a mixture of separating agents
is often higher than that of a single agent [5]. The listed
processes of separation of mixtures in the presence of a
separating agent essentially constitute a general classification of extractive heterogeneous systems.
Each multicomponent heterogeneous system is
characterized by a set of the relative volatilities αij of
components, which are determined by the distribution
coefficients Ki and Kj of the components between the
phases:
yi x j
K
-.
α ij = ------i = -------xi y j
Kj
(1)
Here, yi, yj, xi, and xj are the mole fractions of the components i and j in the vapor and liquid phases, respectively.
The matrix of the relative volatilities of the components for an n-component system has the form



a = 



α 11 α 12 … α 1n 
α 21 α 22 … α 2n 

… … … … 

α n1 α n2 … α nn 
(2)
For the first time, a matrix of such a type was introduced for ternary systems [6].
(a)
(b)
A
P1
F
P2
F
A
P2
P1
Fig. 1. Flowcharts of (a) extractive and (b) reextractive distillation (A is an extracting agent, and P1 and P2 are product fractions).
0040-5795/04/3801-0022 © 2004 MAIK “Nauka /Interperiodica”
EXTRACTIVE DISTILLATION SYSTEMS WITH SEPARATING AGENTS
If i = j, then αij = 1 and matrix (2) takes the form



a = 



1 α 12 … α 1n 
α 21 1 … α 2n 

… … … … 

α n1 α n2 … 1 
and Eqs. (7) and (8) appear as
α ir x i
y i = -----------------------,
r
∑ (α
(3)
α
------ik- = α ij ,
α jk
α
------ki- = α ji ,
α kj
∑ (α
(4)
n–1
∑ x , Eq. (7) can be rewritten as
i
1
in
We divide the numerator and denominator of the right


side of Eq. (11) by  1 –
x j and obtain


r+1
n
∑
α in X i
-.
y i = -----------------------------------------r
ir
(12)
– 1) Xi + 1
The physical meaning of this procedure consists in
n
1
∑ (α
j
r+1
the fact that, taking
in x i )
α in x i
-.
y i = -----------------------------------------n–1
∑x
(11)
1
(5)
In the general case, the phase equilibrium can be
described by the equation
α in x i
- , i = 1, 2, 3, …, n – 1, n.
y i = ----------------------(7)
n
Since xn = 1 –
– 1 )x i + 1 –
∑ (α
In the transposed matrix, it is convenient to believe that
the relative volatilities in a certain row are independent;
then, the other rows can be found using expressions (4).
Let the component with respect to which all the volatilities are determined be referred to as the base component. According to expressions (3) and (5), any of the
components of the mixture can be the base component.
Let the component n be the base component. Then, the
relative volatilities constitute a tuple of the form
⟨ α 1n, α 2n, α 3n, …, α n – 2, n, α n – 1, n, 1⟩ .
(6)
∑ (α
ir
1
Expressions (4) imply that matrix (3) of the relative volatilities is degenerate; i.e., its determinant is always
zero. After transposing, the matrix takes the form
1 α 21 … α n1 
α 12 1 … α n2 

… … … … 

α 1n α 2n … 1 
ir x i )
α ir x i
-.
y i = ----------------------------------------------------------n
r
α 12 α 23 α 34 …α n1 = 1.



T
a = 



(10)
1
Apparently,
1
α ji = ------ ,
α ij
23
(8)
– 1 )x i + 1
1
Let us assume that, in an n-component heterogeneous system, the first r components are volatile and
the other, (n – r) components are nonvolatile; i.e., the
concentrations of the latter components in the vapor
phase are zero. Then, we have the following tuple of the
relative volatilities:
⟨ α 1r, α 2r, α 3r, …, α r – 1, r, 1, 0, 0, …, 0, 0⟩ ,
(9)
∑x
j
= const and assuming all the
r+1
concentrations of the nonvolatile components to be
constant, one can obtain a linear section of the concentration simplex of the liquid phase. The dimension of this
section coincides with the dimension of the concentration simplex of the vapor phase, which is by (n – r) lower
than the dimension of the concentration simplex of the
liquid phase and is n – 1 – (n – r) = r – 1. Equation (12)
is actually an equation in terms of the relative concentrations Xi (i = 1, 2, …, r) of all the volatile components,
since, for any of them, we have
xi
xi
-.
(13)
- = ---------X i = -------------------r
n
1–
∑x
j
r+1
∑x
i
1
The processes of equilibrium open evaporation and
distillation in concentration simplexes are represented
by trajectories, which are certain sequences of points
representing concentrations. If a process is characterized by a vector field of liquid–vapor equilibrium tie
lines, then such fields form a certain finite set of diagram structures, which differ in the number, type, and
relative positions of singular points. These structures
were classified earlier [7–9].
The addition of nonvolative agents leads to a diagram of a new structure. Let us call the initial diagram
the base diagram, and let the diagram of the new structure be referred to as the derivative diagram. The decisive role in the formation of the new structure is played
by the base diagram. Furthermore, it was found [10]
that the relationship between the base and derivative
diagrams is ambiguous. This is because mixtures containing nonvolative agents do not form n-component
azeotropes. Therefore, azeotropes that can be formed
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SERAFIMOV et al.
stant. We constructed [2] such sections for a case where
the base mixture is binary and the system contains a
single nonvolatile component. A similar section is presented in Fig. 2. It is seen that the set of simplexes of
dimension (r – 1) consists of a set of sections, for each
of which the concentration of the nonvolatile component is constant. We also demonstrated [2] that the trajectories of extractive distillation with a nonvolatile
agent lie completely within one of the sections of the
triangle.
3 (A)
1''
2''
1'
2'
1
2
Fig. 2. Section of the concentration simplex of a ternary system containing one nonvolatile extracting component A (the
arrowhead lines are the tie-line vector field of one of the sections).
by the initial (base) mixture only change their type. The
emergence of new azeotropes is impossible here.
In n-component systems with components distributed between phases, the processes of open equilibrium
evaporation and distillation are represented by points
within a concentration simplex of dimension (n – 1). If,
in a system, r components are distributed between the
phases and (n – r) components are located only within
the liquid phase, then the processes of open equilibrium
evaporation and actual distillation are modeled by
points within a derivative concentration simplex of
dimension (n – 1) and fractionation processes are represented by points within a concentration simplex of
dimension (r – 1) (r < n), which is a section of the derivative diagram.
Previously [2], we showed that to make the dimensions of the simplexes of the vapor and liquid phases
consistent it is necessary to construct a section of the
simplex of the liquid phase such that the concentrations
of the nonvolatile components at this section are con-
For a ternary mixture, the tuple of the relative volatilities has the form
⟨ α 12, 1, 0⟩ .
Let us consider more complex cases. For a quaternary mixture, the following tuples of the relative volatilities are possible:
⟨ α 13, α 23, 1, 0⟩ ,
⟨ α 12, 1, 0, 0⟩ .
These cases are illustrated in Fig. 3. The first tuple is
similar to that of a ternary liquid system, with the difference that sections at which the concentration of the
nonvolatile component is constant are triangles here
(Fig. 3a). The second tuple corresponds to a system
with two nonvolatile components. The construction of
sections of dimension 1 is shown in Fig. 3b.
For a quinary system, the following tuples of the relative volatilities are possible:
⟨ α 14, α 24, α 34, 1, 0⟩ ,
⟨ α 13, α 23, 1, 0, 0⟩ ,
⟨ α 12, 1, 0, 0, 0⟩ .
(a)
1
(b)
1
1'
1'
1''
1''
(A)
3
2
3'
2'
2''
3''
4
(A)
3
2''
O
4
2'
2
(A)
Fig. 3. Sections of the concentration simplex (tetrahedron) of a quaternary system containing (a) one and (b) two nonvolatile extracting agents A.
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25
(b)
1
(a)
1
1'
2
3
3
2
1'
3'
2'
2'
3'
4
4'
5
(A)
(A)
4
O
5
(A)
(c)
1
1'
(A) 3
2
2'
O
(A)
(A)
4
5
Fig. 4. Sections of the concentration simplex (pentatope) of a quinary system containing (a) one, (b) two, and (c) three nonvolatile
extracting agents A.
Figure 4 shows the corresponding sections. In the general case, the number ρ of possible tuples is
ρ = r – 1 = n – 2.
(14)
The table presents the number of possible tuples of
the relative volatilities and their relationship with the
number of nonvolatile components in multicomponent
(n = 3–10) systems.
As already noted, because of the nonlinearity of the
vector fields under consideration, separation methods
using several nonvolatile components can be expected
to be more efficient than methods using a single separating agent. Moreover, the field of search for the most
optimum extracting agents is much wider here.
Making the dimensions of the simplexes of the
vapor and liquid phases consistent is a justified rather
than artificial step. This means that the sum of the concentrations of nonvolatile components remains virtually constant along the height of each section of a distillation column, varying insignificantly only because
of the redistribution of the amount of the liquid and
vapor phases due to the difference between the heats of
evaporation of the base components [2].
Let us derive equations describing heterogeneous
systems with a nonvolatile agent.
Let us consider a two-phase n-component heterogeneous system in which r components (1, 2, 3, …, r) are
Number of possible tuples of relative volatilities in multicomponent systems
n–r
ρ
n
1
2
3
4
5
6
7
3
1
4
1
1
5
1
1
1
6
1
1
1
1
7
1
1
1
1
1
8
1
1
1
1
1
1
9
1
1
1
1
1
1
1
10
1
1
1
1
1
1
1
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SERAFIMOV et al.
distributed between the phases and (n – r) components
are undistributed.
At constant pressure, the vapor phase of this system
is described by the zero-potential equation
y 1 dµ 1 + y 2 dµ 2 + … + y r – 1 dµ r – 1
+ y r dµ r + ∆s v dT = 0,
Here, ∆sv and ∆sl are the entropies of the vapor and liquid phases, respectively; T is temperature; and µ is the
chemical potential of a component.
Let us transpose the second and subsequent terms
from the left to the right sides of Eqs. (15) and (16):
y 1 dµ 1 + y 2 dµ 2
(15)
= –y 3 dµ 3 – … – y r – 1 dµ r – 1 – y r dµ r – ∆s v dT ,
and the liquid phase is described by the equation
x 1 dµ 1 + x 2 dµ 2 + … + x r – 1 dµ r – 1
+ x r dµ r + … + x n dµ n + ∆s l dT = 0.
x 1 dµ 1 + x 2 dµ 2 = – x 3 dµ 3 – … – x r – 1 dµ r – 1
– x r dµ r – … – x n dµ n – ∆s l dT .
(16)
Using the Sarrus rule [11], we obtain
( –y 3 dµ 3 – … – y r – 1 dµ r – 1 – y r dµ r – ∆s v dT )
y2
( –x 3 dµ 3 – … – x r – 1 dµ r – 1 – x r dµ r – … – x n dµ n – ∆s l dT ) x 2
dµ 1 = --------------------------------------------------------------------------------------------------------------------------------------------------------- .
y1 x1
y2 x2
Expanding the determinants in the numerator and denominator gives
x 2 ( –y 3 dµ 3 – … – y r dµ r – ∆s v dT ) + y 2 ( x 3 dµ 3 + … + x r dµ r + … + x n dµ n + ∆s l dT )
-,
dµ 1 = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------x1 x2 ( K 1 – K 2 )
or
( –y 3 dµ 3 – … – y r dµ r – ∆s v dT ) + K 2 ( x 3 dµ 3 + … + x r dµ r + … + x n dµ n + ∆s l dT )
-.
dµ 1 = -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------x1 ( K 1 – K 2 )
Combining the terms with the factors dµi and dT in the
numerators, we obtain
1
dµ 1 = – --------------------------------- x 3 ( K 3 – K 2 )dµ 3 + …
x 1 K 2 ( α 12 – 1 )
∫
+ x r – 1 ( K r – 1 – K 2 )dµ r – 1 + x r ( K r – K 2 )dµ r
n
–…–
∑ x* K dµ + ( ∆s
i
2
i
v
– K 2 ∆s l )dT .
r+1
Here, x *i (i = r + 1, r + 2, …, n) is the concentration of
the nonvolatile component in the liquid phase.
Ultimately, we have
x 1 ( α 12 – 1 )dµ 1 + x 3 ( α 32 – 1 )dµ 3 + …
+ x r – 1 ( α r – 1, 2 – 1 )dµ r – 1 + x r ( α r, 2 – 1 )dµ r
= x *r + 1 dµ r + 1 + … + x *n – 1 dµ n – 1
∆s
+ x *n dµ n –  --------v- – ∆s l dT .
 K2

(17)
In Eq. (17), the left side characterizes the volatile
components, and the right side describes the nonvolatile agents and contains the entropy term.
The dimension of the base simplex is (r – 1) and
coincides with the dimension of the simplex of the
vapor phase. This simplex has r vertices, which correspond to the volatile components. The dimension of the
simplex corresponding to the nonvolatile components
is (n – r – 1). The dimension of the derivative concentration simplex of the entire multicomponent system is
(n – 1). Both the base simplex and the simplex representing the nonvolatile components lie in the boundary
space of the derivative simplex.
In the simplex representing the nonvolatile components at certain concentrations, there is a point corresponding to the composition of the mixture of these
components. Connecting this point by edges to the vertices of the base simplex gives a pyramid, whose base
is the base simplex. If the overall composition and total
amount of the nonvolatile components are fixed, we
have one of many sections of the constructed r-edge
pyramid. The dimension of the section of the pyramid
and the number r of vertices coincide with the respective characteristics of the base simplex. A feature of this
geometric construction is the fact that, if the base mixture can form an r-component azeotrope, there is a unit
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α line of multiplicity (r – 1) within the pyramid. The
dimension of this line is unity. The isothermal-isobaric
surfaces within the pyramid are represented by some
hypersurfaces, whose dimension coincides with the
dimension of the base simplex.
The unit α line of multiplicity (r – 1) either ends at
a point of the simplex of the nonvolatile components or
degenerates on one of the faces of dimension (r – 2) of
the derivative simplex. Therefore, the α line intersects
either all the isothermal-isobaric hypersurfaces or some
of them.
Suppose a representative point moves over one of
the isothermal-isobaric hypersurfaces (dT = 0). The
expression (∆sv – K2∆sl) under conditions that are far
from critical is always more than zero. Then, the
entropy term in Eq. (17) is zero. If the isothermal-isobaric hypersurface and the unit α line of multiplicity
(r – 1) intersect, then we have
ence of the heats of evaporation of the volatile components.
Let us derive an equation for the zero potential in
terms of the relative fractions of the volatile components, which are expressed through the absolute fractions as
xi
xi
-.
X i = ---------- = ---------------------n
r
∑x
∑
X 1 dµ 1 + X 2 dµ 2 + … + X r – 1 dµ r – 1 + X r dµ r
1–
(18)
i
i
n
i
(19)
i
r+1
i
Since
r–1
∑X
i
= 1 – Xr,
1
Eq. (19) can be written as
X 1 ( dµ 1 – dµ r ) + X 2 ( dµ 2 – dµ r ) + …
+ X r – 1 ( dµ r – 1 – dµ r ) + dµ r
∆s l
- dT +
+ ---------------------n
= dΨ*. Then, according to
r+1
Eq. (18) at constant temperature along the unit α line,
this quantity attains an extremum. The dimension of the
hypersurface Ψ* is (n – r). The dimension of the isothermal-isobaric hypersurface is (n – 2). If Ψ* = const,
then, along the unit α line of multiplicity (r – 1), we
have dT = 0. This means that the hypersurface at which
Ψ* = const and the isothermal-isobaric hypersurface
touch each other at the point where the unit α line of
multiplicity (r – 1) intersect them. Similar results were
obtained by another method for a particular case of systems containing a single nonvolatile component [10].
If the components r + 1, r + 2, …, n are nonvolatile,
their concentrations during extractive distillation are
constant within a section of the column. The column
contains two sections (Fig. 1), the first of which is
installed between the points of introduction of the
extracting agents and the initial mixture, and the second
(stripping) section is mounted between the point of feed
of the initial mixture and the point of removal of the distillation residue. Thus, the trajectory of extractive distillation in each section is at a section of dimension (r – 1).
Insignificant deviations may be caused by the differ-
∑ x*
x i*
- dµ * = 0.
∑ -------------1 – x*
r+1
n
∑ x* dµ
i
We divide Eq. (16) for the zero potential of the liquid
n


x *i  and retain only
phase by the difference  1 –


r+1
such terms of the left side that characterize the volatile
components:
and ultimately obtain
Let us denote
∑ x*
r+1
∆s l
- dT +
+ ---------------------n
A comparison of Eqs. (16) and (18) shows that the
equation of the zero potential of the liquid phase when
T = const, P = const, and all the relative volatilities of
all the initial (volatile) components are unity is decomposed into two equations whose right sides are zero.
This means that the zero potential of the liquid phase in
this case is the sum of the zero potential of the volatile
components and the zero potential of the nonvolatile
components.
1–
i
1
α 12 = α 32 = α 42 = α r – 1, 2 = α r, 2 = 1
x *r + 1 dµ r + 1 + … + x *n – 1 dµ n – 1 + x *n dµ n = 0.
27
1–
∑ x*
i
n
x *i
∑ ----------------------- dµ* = 0.
1 – ∑ x*
(20)
i
n
r+1
i
r+1
r+1
A similar equation for the zero potential of the vapor
phase can be obtained by eliminating yr from Eq. (15):
y 1 ( dµ 1 – dµ r ) + y 2 ( dµ 2 – dµ r ) + …
+ y r – 1 ( dµ r – 1 – dµ r ) + dµ r + ∆s v dT = 0.
(21)
The phases are in equilibrium; therefore, dµli = dµvi,
i = 1, 2, 3, …, r. Subtracting Eq. (20) from Eq. (21)
gives
( y 1 – X 1 ) ( dµ 1 – dµ r ) + ( y 2 – X 2 ) ( dµ 2 – dµ r )
+ ( y 3 – X 3 ) ( dµ 3 – dµ r ) + …
+ ( y r – 1 – X r – 1 ) ( dµ r – 1 – dµ r )
(22)


n


x *i
∆s l 
---------------------- dµ *i = 0.
- dT –
+  ∆s v – ---------------------n
n


r+1

1–
x i*
1–
x *i 


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At constant pressure, we have
∂ ( µi – µn )
- dT +
dµ i – dµ n = ----------------------∂T
r–1
= –
∑
1
r–1
∂ ( µi – µn )
-d X
∑ ----------------------∂X
i
i
1
r–1 r–1
∂S
--------dT +
∂ Xi
K *i manifolds of dimension (r – 2) or the point of intersection of α manifolds of the same dimension.
(23)
∂ g
---------2 d X k .
∂ Xk
k = 1i = 1
2
∑∑
Substituting Eq. (23) into Eq. (22) yields
∆s l
-–
– ∆s v – ---------------------n
r–1
1–
1
∑ x*
∂s l
∑ ( y – X ) ------∂X
i
i
dT
+
n–1
i
dT =
i
(24)
r+1
n
∑
x *i
---------------------- dµ i* =
n
r+1
1–
r–1 r–1
∂ g
---------2 ( y i – X i )d X k .
∂ Xk
k = 1i = 1
∑∑
∑ x*
i
Equation (24) is the van der Waals–Storonkin equation at P = const for a case where an n-component system contains an arbitrary number r of volatile components and (n – r) nonvolatile components.
Let us consider the behavior of the system over an
isothermal-isobaric hypersurface (dT = 0). At the point
where the component (r – 1) meets the condition yi = Xi,
we obtain
x *i
∑ ----------------------- dµ* = 0.
1 – ∑ x*
i
n
(25)
r+1
i
r+1
This means (see Eq. (18)) that
dΨ*
---------------------- = 0.
n
1–
∑ x*
r+1
Let us prove that, for all the volatile components, the
condition yi = Xi is equivalent to the condition αir = 1.
Indeed, if K *i = 1, then K *r = 1; therefore, K *i /K *r = 1.
However, K *i /K *r = Ki /Kr and, hence, αir = Ki /Kr = 1.
In the simplex of the relative concentrations of the
volatile components at constant concentrations of the
nonvolatile components, the point of intersection of the
n
hypersurface
∑ x* = const (i = r + 1, r + 2, …, n) and
i
r+1
∂T
-d X ,
∑ ------∂X
(26)
i
i
1
2
r+1
n
Note that all studies of azeotropy assumed that the
temperature differential at an azeotropic point is zero.
Among such studies are many investigations using the
van der Waals–Storonkin equation. This condition is
necessary but not sufficient. As we showed [12] by the
example of ternary systems, the temperature differential can be zero in two cases. If the total differential for
the systems under investigation is represented as
the unit α line of multiplicity (n – 2) in these coordinates is a point of r-component azeotrope. The point
can be determined as the point of intersection of unit
then, in the first case, all the partial derivatives are zero:
∂T/∂Xi = 0, which is characteristic of the point of extremum of a scalar temperature field. In the second case,
the scalar product of orthogonal vectors—the temperature gradient vector (∂T/∂X1, ∂T/∂X2, …, ∂T/∂Xr – 1) and
the composition change vector (dX1, dX2, …, dXr – 1)—
is zero, since the motion of the representative point over
the isothermal-isobaric hypersurface is considered.
Here, ∂T/∂Xi ≠ 0; however, dT = 0.
Thus, the fact that the differentials of temperature
and the function Ψ* along the unit α line are zero
means that the partial derivatives ∂T/∂Xi and ∂Ψ*/∂Xi
are nonzero, and that the surfaces over which T = const
and Ψ* = const touch each other at a so-called
pseudoazeotropic point. This point in the coordinates of
the relative concentrations of the volatile components
substantially differs from an ordinary azeotropic point.
At the latter point, if the concentrations of all the components in the vapor and the liquid are respectively
equal, then the condition ∂T/∂xi = 0 is necessarily met.
In the case under consideration, let us write, in vector form, the equation that is derived from Eq. (24) by
a previously published method [12]. Let us first denote
the entropy term as the scalar function
∆s l
-–
∆S l v = ∆s v – ---------------------n
1–
∑ x*
r–1
∂s l
∑ ( y – X ) ------∂X
i
i
i
1
i
r+1
and use Ψ* as a scalar function of the relative concentrations of the volatile components. These manipulations give an analogue of the modified form of the van
der Waals–Storonkin equation [12] in vector form
under isobaric conditions:
THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING
Vol. 38
No. 1
2004
EXTRACTIVE DISTILLATION SYSTEMS WITH SEPARATING AGENTS
∂Ψ*
----------∂ X1
∂T
--------∂ X1
∂Ψ*
∂T
------------------1
---------------------–
∆S
∂
X
∂
X2
lv
2
n
1–
x*
…
…
i
r+1
∂Ψ*
∂T
--------------------------∂ Xr – 1
∂ Xr – 1
Suppose the temperature gradient is zero; then,
∂Ψ*
----------∂ X1
∂Ψ*
----------1
---------------------∂ X2
n
1–
x *i …
∑
 g
g 12
11

 g 21
g 22
= 
 …
…

 g r – 1, 1 g r – 1, 2
… g 1, r – 1
… g 2, r – 1
…
…
… g r – 1, r – 1
∑
(27)

y1 – X 1


y2 – X 2


…

 P, T y r – 1 – X r – 1
r+1
 g
g 12
11

 g 21
g 22
= 
 …
…

 g r – 1, 1 g r – 1, 2
or, in abridged form,
∑
… g 1, r – 1 
… g 2, r – 1 

…
… 

… g r – 1, r – 1 
(29)
y1 – X 1
y2 – X 2
P, T
…
yr – 1 – X r – 1
1
---------------------- gradΨ* = G ( y – X ).
n
1–
x *i
(29‡)
∑ x*
i
r+1
r+1
∂ gl
are the second derivatives of the
where gij = -------------∂ Xi X j
Gibbs g potential with respect to the concentrations of
the volatile components.
2
Suppose the magnitude of the liquid–vapor tie-line
vector is zero. Then, according to Eq. (27), we have
∂Ψ*
----------∂ X1
REFERENCES
(28)
∑
or
1
---------------------- gradΨ* – ∆S l v gradT = 0.
n
This is a condition for a local extremum of the scalar
temperature field. Thus, at ordinary azeotropic points,
the temperature gradient and the tie-line vector are zero
at the same point, whereas for systems with nonvolatile
components, these vectors are zero at different points.
Consequently, the systems considered in the coordinates of the relative concentrations of the volatile components at a section of the derivative simplex disobey
the known Konovalov law.
ACKNOWLEDGMENTS
This work was supported by the Russian Foundation
for Basic Research (project no. 02-03-33104).
∂T
--------∂ X1
∂Ψ*
∂T
------------------1
---------------------–
∆S
∂
X
∂
X2 = 0
lv
2
n
1–
x *i …
…
r+1
∂Ψ*
∂T
--------------------------∂ Xr – 1
∂ Xr – 1
1–
∂Ψ*
-------------∂ Xr – 1
or
1
---------------------- gradΨ* – ∆S l v gradT = G ( y – X ), (27‡)
n
1–
29
(28‡)
∑ x*
i
r+1
Although the difference of the gradients multiplied by
scalar factors is zero, each of them is nonzero. This is
the condition for a pseudoazeotropic point.
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No. 1
2004
30
SERAFIMOV et al.
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THEORETICAL FOUNDATIONS OF CHEMICAL ENGINEERING
Vol. 38
No. 1
2004