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% FARGEPRINTER lpr -Pclj ut.ps (2.etg, rom 217, Organisk kjemiblokk 3)

%\pagestyle{empty}

\title{\huge \textbf{On the Topology of Ternary VLE Diagrams:
Elementary Cells}}

%\title{\huge \textbf{Elementary VLE Cells - A Key to Simple Analysis
%of Ternary Azeotropic Mixtures}}

%\title{\huge \textbf{Elementary Cells of Ternary VLE Diagrams - \\ A
%Key to Simple Azeotropic Distillation Analysis}}

%\title{\huge \textbf{Short Note: \\ Elementary Cells of Vapor-Liquid
%Equilibrium Diagrams in Azeotropic Distillation Analysis}}

%\title{\huge \textbf{A Key to Simple Azeotropic Distillation Analysis:
%Elementary Cells of Ternary VLE Diagrams}}

%\title{\huge \textbf{A Key to Simple Azeotropic Distillation Analysis:
%Elementary VLE Cells}}

\author{\Large E.~K. Hilmen, V.~N. Kiva\thanks
{Laboratory on Mixtures Separation, Karpov Institute of Physical
Chemistry, Moscow 103064, Russia.} 
and
S. Skogestad\thanks{Corresponding author. Tel.:~+47-7359-4154;
fax:~+47-7359-4080; email: skoge@chembio.ntnu.no} 
\\ \\ \normalsize
Department of Chemical Engineering, Norwegian University of Science
and Technology, \\ \normalsize N-7491 Trondheim, Norway}

%\date{}
% $^{**}$
% $\dagger$ 
%\date{\bf Parts presented at ESCAPE-9, Budapest, May 31-June 2, 1999}
\date{\normalsize \bf Submitted to {\em AIChE Journal} \\ 27 Dec. 2000 \\
Revised 06 Oct. 2001}

\begin{document}  
\maketitle
\thispagestyle{empty}

\begin{abstract}
\noi The classification of ternary vapor-liquid equilibrium (VLE)
diagrams is a key to simple azeotropic distillation analysis.  We find
that all ternary mixtures so far reported occurring in nature can be
qualitatively represented by a combination of only four elementary
cells.
This greatly reduces the number of VLE diagram structures that need to
be analyzed in order to reveal the qualitative characteristics of any
ternary azeotropic mixture.
\end{abstract}

\section*{Introduction}
\label{sec:Introduction}

%The concept of elementary cells was proposed by me as "a working tool"
%for my PhD students in 1989 but never had been published specially. 

Ternary vapor-liquid equilibrium (VLE) diagrams provide a graphical
tool to predict qualitatively the feasible separations for
multicomponent azeotropic mixtures before detailed simulation or
experimental study of their distillation. 
The various graphical representations of the VLE (residue curve and
distillation line maps, isotherm map, equilibrium vector field) are
closely related and are equally capable of characterizing the mixture.
In this paper, we consider {\em residue curves}
\cite{Sch01b,Ost02,Zha67,Ser68a,Doh78a}, and the topological
classification of ternary mixtures into 26 diagrams by
\citeasnoun{Ser70a}. Our considerations apply equally well to {\em
distillation lines} \cite{Zha68c,Sti88}.
% Zha75,
We refer to the review of \citeasnoun{Wid96} and to the literature
mentioned for a detailed description of these tools and their
application to distillation column profiles.
%
Our contribution is to propose a set of elementary topological cells
which are constituents of all feasible ternary VLE diagrams.  We show
that the until now reported ternary mixtures include only four
of these elementary cells.

%Residue curves are focused in this paper, but the concept equally
%applies to {\em distillation lines} \cite{Zha68c,Zha75,Sti88} or other
%graphical representations of the VLE of ternary mixtures. 

% (or eqvivalently, distillation line maps, see for example
%\citeasnoun{Sti88})

%This simplified picture gives a clear/perspicuous rendering of the VLE
%diagrams, and are useful in azeotropic distillation analysis and
%presynthesis.
%(phenomena analysis).  
%We believe that the concept of elementary cells is of practical and
%pedagogical importance.
%(that all ternary VLE diagrams are build up by a few elementary cells
%with a few possible line shapes)
%clarifies the presentation
% (lucid/perspicuous/easy-to-follow/surveyable/clearly set out)
%     - -       presentation/description 
%cells = (single bundles of simple distillation residue curves or
%distillation lines)
%that occur among mixtures reported in the literature (real/known
%mixtures)

%The identification of basic structural elements of ternary VLE
%diagrams helps to reduce the diversity of the distillation line maps
%to combinations of only eight elementary topological cells. The
%concept of elementary cells can be useful in analysis of VLE diagrams
%and the corresponding distillation behavior for mixtures of the
%various classes. The concept of basic shapes of the distillation line
%bundle inside the elementary cells helps to distinguish between
%topologically equivalent cells or mixtures of the same topological
%class having different specific characteristics. This is the topic of
%a succeeding paper.
%Various shapes of the simple phase transformation trajectories are
%considered for primary cells and for cells that are parts of complex
%diagrams.

%--------------------------------------------------------------------
\section{Classification of ternary VLE diagrams}
\label{sec:Classification}

%Pure components and azeotropes are singular points of the VLE
%manifold. These are either nodes or saddles of the composition
%triangle (ternary mixtures).

The number of feasible VLE diagram structures is limited by
topological and thermodynamical constraints. 
%Mixtures may be classified according to their topological structure
%given these constraints.
%\citeasnoun{Gur58} was the first to derive the rule of azeotropy and
%classification of ternary mixtures.
%Feasible structures of residue curve maps, isotherm maps and
%equilibrium vector fields for ternary mixtures are 
A complete classification of these feasible structures for ternary
mixtures was given by \citeasnoun{Ser70a} and Serafimov {\em et al.}
\citeyear{Ser71a,Ser73}, and is more recently presented by
\citeasnoun{Ser96a}.
% \cite{Kuc96}, chpt. 10 Serafimov, see p. 596 or \cite{Ser96}
%
Serafimov's classification results in the 26 classes of ternary VLE
diagrams presented in Figure \ref{fig:classification_b_Ser70}.  
%--------------------------------------------------------------------
\begin{figure}[!hb]    
\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Seraf-b-corr.eps,height=18cm}}}
\caption{\em Serafimov's (1970b) topological classification of ternary mixtures.
Structural
characteristics of residue curves (or distillation lines) are shown by
dashed lines with the singular points indicated by $\bullet$ stable
(unstable) node, $\circ$ unstable (stable) node, and $\triangledown$
saddle, and the region boundaries given by bold lines.}
\label{fig:classification_b_Ser70}
\end{figure} 
%qualitative path of the residue curves
%
%The singular points are marked by $\bullet$ stable (unstable) node,
%$\circ$ unstabel (stable) node and $\triangledown$ saddle.  
% 26 classes presented on p. 593 in [Ser96a] 
The classification is based on the simplifying and common assumption
that there exists no more than one binary azeotrope for each binary
pair and no more than one ternary azeotrope.
%(Biazeotropy exists, but is rare.)
% reference!!
%There is no separatrice going from one node to another (no connetion
%N-N) inside a distillation line region. Separatrices only.
Biazeotropy does exist in real mixtures, but it is relatively rare,
so this is not a very restrictive assumption.
The classification of Serafimov considers topological structures and
thus does not distinguish between antipodal diagrams (switching of
minimum- and maximum-boiling azeotropes) since they have the same
topology.  Transition from one antipode to the other can be made by
simply changing the sign of the nodes and inverting the direction of
the arrows of increasing boiling temperature.
%(saddles stays the same).
%Equilateral triangles have the advantage that they can be rotated or
%mirror imaged to "rearrange" the position of the edges singular
%points.

Actually, \citeasnoun{Gur58} was the first to derive the rule of azeotropy and
propose a thermodynamic topological classification of ternary
mixtures.
%(p. 1987-1988).  
However, his classification was incomplete, and \citeasnoun{Ser68b}
revealed four additional feasible structures and established the 26
topological classes presented in Figure
\ref{fig:classification_b_Ser70}.
% not disputed
Later, Serafimov's classification of ternary mixtures was refined by
distinguishing between antipodes inside each structure class, based on
the reasoning that {\em "minimum- and maximum-boiling azeotropes have
dissimilar physical nature and dissimilar behavior during
distillation"} \cite{Zha75}. This refined classification includes a
total of 49 types of ternary VLE diagrams.
%(presented graphically on page 96-98 in the book by Zharov and
%Serafimov, 1975). 
An even more detailed classification is proposed by \citeasnoun{Mat77}
who also rank the components in the order of their boiling temperatures
("light", "intermediate" and "heavy"). 
%The classes are distinguished according to the relative location of
%the binary azeotropic points on the edges of the composition triangle.
This classification includes 113 diagram classes of which 87 are
presented graphically by \citeasnoun{Doh85a}.
%Doherty and Caldarola (1985a) 
Nevertheless, among these 113 classes there are only the 26
topologically distinct structures of Serafimov. 
%
Actually, the classification of Matsuyama and Nishimura adds some
ambiguity as some of their classes with a ternary saddle azeotrope
have two or three possible topological structures.
%However, note that some of Matsuyama and Nishimura's classes where a
%ternary saddle azeotrope is present actually have two or even three
%possible topological structures.  
For example, their classification code 112-S can be either of
Serafimov's class 3.1-3a or 3.1-3b depending on whether there exists a
saddle - saddle separatrix or not. This is also pointed out by
\citeasnoun{Fou91} who recommend an extension of the Matsuyama and
Nishimura's classification code name in these cases.
%(e.g., 112-S, 112-SL and 112-SH or 433-S, 433-SI and 433-SL). 
%Indetermancy of Matsuyama and Nishimura's classification codes.
%However, the terms "light", "intermediate" and "heavy" components have
%limited meaning in general for nonideal and azeotropic mixtures since
%the relative volatility of the components change within the
%composition space. 
%Therefore, the additional complexity in Matsuyama and Nishimura's
%classification does not give any additional benefits.

In this paper, we use Serafimov's \citeyear{Ser70a} classification of
the topological classes and Zharov and Serafimov's \citeyear{Zha75}
refinement of the antipodal structure types (referred to as the
ZS-type).  The relationships between the classifications of
\citeasnoun{Gur58}, \citeasnoun{Ser70a}, \citeasnoun{Zha75} and
\citeasnoun{Mat77} are presented in Table
\ref{tab:correspondence_classifications}.  
%\ref{sec:appendix}
The table is useful when relating publications where the different
classifications are used.
%Serafimov's class 0.0-1 are often refered to as class 0 (zeotropic).

%Finally, we want to emphasize that the topological structure of the
%VLE diagram identifies the class of a given mixture, but not its
%specific characteristic like the shape of the singular and ordinary
%simple phase transformation paths (residue curves and distillation
%lines).
   
%In most cases, the (topological) structure of the VLE diagram
%for a given mixutre can be uniquely determined based soley on
%information of the existence and boiling temperature of all singular
%points (pure components and azeotropes). It should be noted however,
%that there are some cases of indeterminacy of the diagram structure.

%--------------------------------------------------------------------
%\afterpage{\clearpage}
%\clearpage
\section{Elementary cells}
\label{sec:Elementary_Cells}

There is a great diversity of VLE diagrams for ternary mixtures caused
by the variety in physical properties of the components and their
molecular interaction. As mentioned, if we assume no biazeotropy 
and consider only topological
differences there are 26 distinct types as presented in Figure
\ref{fig:classification_b_Ser70}.  Furthermore, there is a far greater
diversity in possible shapes (geometry) of the simple phase
transformation trajectories such as residue curves and distillation
lines.  It is possible to reduce this complexity to a combination of a
few topological building blocks (``elementary cells'') and some basic
internal structures (shapes of the simple phase transformation
trajectories).
% structural objects
% diversity/complexity
%In this paper simple phase transformation trajectiories is used as a
%common term for residue curves and distillation lines.
We use residue curves to represent the simple phase transformation
trajectories in this paper.

% Kiv76 uses the concept ``cell'' of conjugated tie-lines

We find that among Serafimov's 26 topological classes there are eight
elementary topological cells (denoted I, II, III, IV, II', III', IV'
and V) that constitute all the ternary diagrams, where a cell is
defined as one residue curve region taken with its boundaries
%, where the region is a family of residue curves with common initial
%and final points.
(that is, a subspace of the composition space constrained by residue
curve boundaries, if any, and the composition simplex).
% Elementary topological cell is a ``closure'' of the distillation
% region, i.e. the join of the region and its borders.
From these eight elementary cells we may construct all the 26 diagrams
as shown in Figure \ref{fig:elementary_occurrence_percent}.
%--------------------------------------------------------------------
%\begin{figure}[!ht]    
%\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Seraf-a-corr.eps,height=18cm}}}
%\caption{Elementary cells recurring in Serafimov's classification of
%ternary mixtures.}
%\label{fig:classification_a_Ser70}
%\end{figure} 
%figure=/home/ekh/99-Survey/Figures/Serafimov_26_classes_occur.eps
%figure=/home/ekh/99-Survey/Figures/Serafimov_26_classes_NEW_cells.eps
%--------------------------------------------------------------------
\begin{figure}[!hb]    
\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Seraf-a-occur.eps,height=18cm}}}
\caption{\em Elementary cells (I, II, III, IV, II', III', IV' and V)
within Serafimov's 26 topological classes. Physical occurrence (\%) of
azeotropic mixtures reported in the literature according to Reshetov's
statistics (based on data from 1965 to 1988).}
\label{fig:elementary_occurrence_percent}
\end{figure} 
Each cell has one unstable and one stable node and some set of saddle
points.
%, but a varying number and relative position of the saddles.
%
There are four ``primary'' diagrams where the composition triangle
consists of a single cell (one residue curve region; Serafimov's class
0.0-1, 1.0-1a, 1.0-1b and 2.0-1), and these elementary cells (denoted
I, II, III, IV) are also the only so far reported for naturally
occurring mixtures.
%(I, II, III, IV) 
%The four primary cells are repeated in Figure
%\ref{fig:cells_simplest_classes}.
%Thus, we refer to the four primary cells in Figure
%\ref{fig:cells_simplest_classes}b as generalized elementary
%topological cells.  
Cells II', III' and IV' are modifications of the primary cells II, III
and IV, respectively (with internal nodes), for which there are no
reported physical mixtures. Cell V only occurs as an element in
Serafimov's class 3.1-1b, and also for this class there is so far no
physical mixture reported.

%--------------------------------------------------------------------
%\afterpage{\clearpage} 
%
The four primary diagrams and the corresponding elementary cells I,
II, III, IV are shown in Figure \ref{fig:cells_simplest_classes}.
%--------------------------------------------------------------------
\begin{figure}[!ht] 
%\vspace{0.4cm}   
\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Fig61-corr.eps,height=12.8cm}}}
\caption{\em Generalization of the four elementary topological cells
I, II, III, IV from the diagrams with one residue curve region. Each
cell is characterized by the set and order of the nodes ($N$) and
saddles ($S$) along the contour of its border. The singular points are
also indicated by $\bullet$ stable (unstable) node, $\circ$ unstable
(stable) node, $\triangledown$ saddle. The dashed lines indicate the
qualitative paths of the residue curves.}
%[Fig. 61]
\label{fig:cells_simplest_classes}
\end{figure} 
%--------------------------------------------------------------------
%\begin{figure}[thb]    
%\centerline{\protect{\psfig{figure=/home/ekh/Survey99/Figures/elementray_1234.eps,height=10cm}}}
%\caption{Structure of elementary topological cells}
%\label{fig:elementary_1234}
%\end{figure} 
%
%A cell can be considered as an autonomous topological object with its
%own set and type of singular points and its own topological structure.
%Each elementary cell is characterized with a certain set and order of
%the singular points along the contour of its border.
Each elementary cell is characterized by a certain set and order of
the singular points (nodes and saddles) along the contour of its
border (that is, by its topology).
%, and may also be identified by the shape of the residue curve
%bundle. 
%after the set of nodes ($N$) and saddles ($S$) 
Accordingly we can name them as given in Figure \ref{fig:cells_simplest_classes}.
%\bit
%\item[I.] {\em Node-saddle-node cell:} ``triangular'' cell with two
%nodes and a saddle;
%%The residue curves are either C- or S-shaped
%%depending on the existence of univolatility lines causing inflexion
%%points.
%\item[II.] {\em Node-saddle/saddle-node cell:} ``rhombic'' cell with
%opposite directed nodes and saddles; 
%%The residue curves are either C- or S-shaped, or both.
%\item[III.] {\em Node-saddle-saddle-node cell:} ``trapezoidal'' cell
%with two saddles between the unstable and stable nodes;
%%The residue curves are U-shaped.
%\item[IV.] {\em Node-(saddle/saddle-saddle)-node cell:} ``pentagonal''
%cell with two saddles between the unstable and stable nodes, and a
%saddle. 
%%The residue curves are both C(S)- and U-shaped.  
%\eit
%%they can be named node-saddle-saddle-node ``triangular'' (I),
%%"rhombic" (II), "trapezoidal" (III) and "pentagonal" (IV) cells.

%Recall that the singular points (nodes and saddles) are either pure
%component vertices or azeotropic points.  The borders of the
%elementary cells are made up by residue curve boundaries, if any, and
%parts of the edges of the composition triangle, as easily seen from
%Figure \ref{fig:elementary_occurrence_percent}.
%
%Only saddle separatrices interior to the composition space can be
%residue curve boundaries.  
%
%The borders of the cells for these classes of ternary mixtures are the
%edges (or parts of edges) of the composition triangle, and they are
%linear. It is apparent that any curvilinear figure having the same set
%and order of the singular points will be topologically equivalent to
%its triangular analog.  
%
%In summary, all the 26 ternary VLE diagram classes can be considered
%as combinations of elementary cells. A cell can be considered as an
%autonomous topological object characterized by a certain set and order
%of the singular points along the contour of its border. Primary
%(single) elementary cells and elementary cells incorporated into more
%complex diagrams are topological equivalent if they have the same set
%and order of the singular points along the border. It is shown that
%all probable/common types of ternary VLE diagrams are combinations of
%these four primary elementary topological cells.

%--------------------------------------------------------------------
%\afterpage{\clearpage}
%\subsection{Identifying elementary cells}
%{\em Identifying elementary cells}

%It is important to distinguish between the {\em topological structure}
%of residue curve maps (i.e., the set of singular points and the
%splitting into residue curve regions), the {\em sketch} of the map
%(i.e., the qualitative pattern of the residue curves with
%representation of their shape), and the {\em exact} residue curve map.
%In this paper we only consider structures of the ternary VLE diagrams
%determined by the set and types of the singular points (unstable and
%stable nodes and saddles), the residue curves going along the edges of
%the composition triangle, and the residue curves separatrices interior
%to the composition space.  Recall that a residue curve region is
%defined as a set of liquid compositions that belongs to a bundle of
%residue curves with common initial and final points. The residue curve
%boundaries are the singular residue curves that divide the composition
%space into these regions. 


%----------------------------%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%EXCLUDED text Oct 20, 2000
%By way of example let us consider the diagram of Serafimov's class
%1.0-2 that consists of two distillation regions shown in Figure
%\ref{fig:splitting_class_1-0-2_regions_and_cells}.
%--------------------------------------------------------------------
%\begin{figure}[!ht]    
%\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Fig60.eps,height=3.2cm}}}
%\caption{\em Distillation regions and cells of distillation lines for
%ternary azeotropic mixture of Class 1.0-2: a) splitting into two
%simple distillation regions; b) cells of distillation lines.}
% [Fig. 60]
%\label{fig:splitting_class_1-0-2_regions_and_cells}
%\end{figure} 
%
%The composition triangle in Figure
%\ref{fig:splitting_class_1-0-2_regions_and_cells}a includes: 
%\bit
%
%\item four singular points: components 1 (unstable node), 2 and 3
%(stable nodes), and of the binary azeotrope A (saddle);
%\item the binary residue curves 1-2, 1- 3, A-2, A-3;
%\item the residue curve region boundary 1-A (saddle separatrix); 
%\item two residue curve regions DR-I and DR-II.
%\eit
%
%Distillation region DR-I is the set of liquid compositions located
%inside the triangle 1-A-3, and distillation region DR-II is the set of
%the liquid compositions located inside the triangle 1-A-2.  
%
%These sets are open in the composition space X. Let us consider the
%closure of a distillation region, i.e., the combination of the
%distillation region and its borders. We define this closure as a "cell
%of distillation lines" because the distillation lines are enclosed in
%this cell as illustrated in Figure
%\ref{fig:splitting_class_1-0-2_regions_and_cells}b.  Then the
%composition triangle X can be presented as a set/joint of two cells
%C-I and C-II, i.e., two closed triangles 1-A-2 and 1-A-3 with common
%curvilinear edge 1-A. A cell can be considered as an autonomous
%topological object with its own set and type of singular points and
%its own topological structure. Each diagram class can be considered as
%a certain combination of such cells of distillation lines. By
%definition a cell of distillation lines encloses a single distillation
%region and has a single unstable node and a single stable node but can
%include several saddle points in different combinations. Each
%elementary cell is characterized with a certain set and order of the
%singular points along the contour of its border.
%The primary cell I is a N-S-N cell, II is a N-(S/S)-N cell, III is a
%N-S-S-N cell, and cell IV is a N-(S/S-S)-N cell.

%Let us consider the diagrams that consist of several distillation
%cells. For simplicity we consider only the probable types of the VLE
%diagrams, excluding the improbable classes from
%consideration. Occurrence and probability of the ternary VLE diagram
%classes are discussed in the next section.  
%%(Section \ref{sec:Occurrence}) Table
%%\ref{tab:elementary_topological_cells} 
%The splitting of these diagrams
%into cells of distillation lines are presented in Figure
%\ref{fig:splitting_cells_probable_classes}.
%--------------------------------------------------------------------
%\begin{figure}[!ht]    
%\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Fig62-corr.eps,height=10cm}}}
%\caption{Elementary cells recurring in complex VLE diagrams (with more
%than one simple distillation region) that are known to exist among
%azeotropic mixtures reported in the literature.}
% [Fig. 62]
%\label{fig:splitting_cells_probable_classes}
%\end{figure} 
%%Splitting into cells of distillation lines for the probable VLE
%%diagrams with more than one simple distillation region. Elementary
%%cells indicated by I, II, III and IV.
%One can see that the diagrams in Figure
%\ref{fig:splitting_cells_probable_classes} are certain combinations of
%only the four generalized primary topological cells presented in
%Figure \ref{fig:cells_simplest_classes}.
%
%----------------------------%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There is an important difference between an elementary cell of the
``primary'' diagrams (one residue curve region) and an elementary cell
incorporated into more complex diagrams. If an elementary cell is a
primary diagram, the saddle point is a pure component point. The
borders of the cell are the edges of the composition triangle
(linear). The (stable and unstable) nodes are pure component points or
points of binary azeotropes.  If an elementary cell is a constituent
of a complex diagram, at least one of its saddles is a binary or a
ternary azeotrope, and, therefore, at least one of the borders of the
cell is a residue curve boundary (curved) showed by the solid thick
lines in Figure \ref{fig:elementary_occurrence_percent}.  One of the
nodes can be a point of a ternary azeotrope.  In general, the
composition space is broken into several residue curve regions (cells)
if there are more than one unstable node or more than one stable node.

Despite these differences, a single elementary cell and an elementary
cell incorporated into a complex diagram are topological equivalent.
%same set of singular point types
Note, however, that inside similar topological cells there may be
various shapes of the residue curves (simple phase transformation
trajectories).  
%This topic is described in a succeeding paper on the geometry of VLE
%diagrams.
% \cite{Kiv2000c}.

%---------------------------------------------------------------------  
%\clearpage
%
%In summary, all feasible VLE diagrams are either elementary
%topological cells (one residue curve region), or a combination of two
%or more elementary topological cells. We have only eight elementary
%topological cells, namely: \bit
%\item[I.] {{\em NSN} cell:} ``triangular''
%\item[II.] {{\em N(S/S)N} cell:} "rhombic" 
%\item[III.] {{\em NSSN} cell:} "trapezoidal"
%\item[IV.] {{\em N(S/SS)N} cell:} "pentagonal" 
%\item[II'.] {{\em NS(S/S)N} cell:} "rhombic" with internal nodal point
%\item[III'.] {{\em NS(\underline{SS})N} cell:} ``trapezoidal'' with
%internal nodal point
%\item[IV'.] {{\em NS(S/SS)N} cell:} "pentagonal" with internal nodal
%point
%\item[V.] {{\em N(SS/SS)N} cell:} ``hexagonal'' 
%\eit
%
%The last four cells are not found and are imporbable. Therefore, we
%can restrict our considerations to the diagrams representing or
%including the four elementary cells I, II, III and IV.


%---------------------------------------------------------------------  
%\clearpage
%\newpage
\section{Occurrence in nature}
\label{sec:Occurrence}
%{\bf Occurrence}: reported by Reshetov (based on data from 1965-1988)

Even though all the classified diagrams in Figure
\ref{fig:elementary_occurrence_percent} are topologically and
thermodynamically feasible, their occurrence in nature is limited by
the probability of certain combinations of molecular interactions.  In
this section we present data on the occurrence of the different
classes among the mixtures reported in the literature.  This permits
us to exclude rare or improbable diagram classes from consideration.
%in the investigation of the correlation between VLE diagram structures
%and feasible separations at distillation.
%
%Factors that limit the (natural) occurrence of mixtures
%with certain types of VLE diagrams are revealed.

%Studies of the published data on the VLE of azeotropic mixtures show
%that many of the predicted diagram structures are not reported to the
%present day.  
\citeasnoun{Ser68b} analyzed the occurrence of different types of VLE
diagrams among 418 reported (experimental) data on ternary azeotropic
mixtures. \citeasnoun{Res98} made a similar study for 1609 ternary
mixtures (in which 1365 are azeotropic) based on thermodynamic data
that were published during the period from 1965 to 1988.
% experimental data for ternary mixtures
%Reshetov's statistics present literatury (experimental) data on
%ternary mixtures only.  However, the original references usually don't
%include the determination of the structure. it can be incomplete, and
%so on. Therefore Reshetov made the treatment of these data to
%determine the class/type of the mixture. Of course, sometimes his
%determination can be erroneous.
%
To the best of our knowledge we have not found any other publications
that address this issue.
%During last 15 years Reshetov collected literature data on VLE for his
%own database ("bank of data") and made an analysis of these
%data. There is a source of his statistics.
%This statistics is unpublished, and you can made a reference on it as
%"a private communication" or include this statistics in your work with
%reference that these data were collected and analized by Dr. Reshetov,
%as you like.
The occurrence of the various classes as reported by Serafimov and
Reshetov are given in Table \ref{tab:occurence_of_classes_published}.
%---------------------------------------------------------------------  
%\afterpage{\clearpage}
%\clearpage
%\newpage
\begin{table}[!hb]  
\vspace{-0.4cm}
\caption{\em Occurrence of ternary VLE diagram structures found in 
published mixture data}
\label{tab:occurence_of_classes_published} 
\vspace{-0.6cm}
\bc
%  \renewcommand{\baselinestretch}{0.92}  
  {\small
\begin{tabular}{|c|c|c|c|c|}  
% \hhline{#=}
\multicolumn{5}{c}{} \\ \hline 
                  & Occurrence   &         &            & Occurrence \\ 
Serafimov's class & Serafimov    & ZS-type      & Set of azeotropes $^a$ & Reshetov  \\
                  & (until 1968) &    &            & (1965-1988) \\ \hline 
 0.0-1  & -                 & 1   & zeotropic          & 244 \\ \hline 
 1.0-1a & 13                & 3a  & min                & 283 \\ 
        &                   & 7a  & max                & 12 \\  
 1.0-1b & 2                 & 3b  & min                &  4 \\ 
        &                   & 7b  & max                &  1 \\  
 1.0-2  & 20                & 4   & min                & 95 \\ 
        &                   & 8   & max                & 21 \\ \hline 
 1.1-1a & None              & 2a  & min + min Az$_3$   & None \\
        &                   & 6a  & max + max Az$_3$   & None \\ 
 1.1-1b & None              & 2b  & min + min Az$_3$   & None \\
        &                   & 6b  & max + max Az$_3$     & None \\  
 1.1-2  & 7                 & 5   & min + S$_3$          & 8 \\
        &                   & 9   & max + S$_3$          & 8 \\ \hline 
 2.0-1  & 1                 & 15  & min + max            & 9 \\ 
 2.0-2a & None              & 17  & min + max            & 2 \\
        &                   & 18  & max + min            & 3 \\ 
 2.0-2b & 77                & 11a & min + min            & 280 \\
        &                   & 21a & max + max            & 6 \\ 
 2.0-2c & 2                 & 11b & min + min            & 10 \\
        &                   & 21b & max + max            & 2 \\ \hline 
 2.1-1  & None              & 13  & min + max + min Az$_3$ & None \\
        &                   & 14  & min + max + max Az$_3$ & None \\ 
 2.1-2a & None              & 16a & min + max + min Az$_3$ & None \\
        &                   & 16b & min + max + max Az$_3$ & None \\  
 2.1-2b & 3                 & 10  & min + min + min Az$_3$ & 55 \\
        &                   & 20  & max + max + max Az$_3$ & None \\ 
 2.1-3a & 14                & 19  & min + max + S$_3$     & 37 \\ 
 2.1-3b & 5                 & 12  & min + min + S$_3$     & 2 \\
        &                   & 22  & max + max + S$_3$     & 1 \\ \hline 
 3.0-1a & None              & 29  & min + min + max        & None \\
        &                   & 33  & max + max + min        & None \\ 
 3.0-1b & None              & 28  & min + min + max        & 9 \\
        &                   & 34  & max + max + min        & 3 \\ 
 3.0-2  & 85                & 24  & min + min + min        & 114 \\
        &                   & 37  & max + max + max        & None \\ \hline 
 3.1-1a & 3                 & 27b & min + min + max + min Az$_3$ & None \\
        &                   & 32b & max + max + min + max Az$_3$ & None\\ 
 3.1-1b & None              & 26  & min + min + max + min Az$_3$ & None \\
        &                   & 31  & max + max + min + max Az$_3$ & None \\ 
 3.1-1c & None              & 27a & min + min + max + max Az$_3$ & None \\
        &                   & 32a & max + max + min + min Az$_3$ & None \\ 
 3.1-2  & 171               & 23  & min + min + min + min Az$_3$ & 355 \\
        &                   & 36  & max + max + max + max Az$_3$ & None \\ 
 3.1-3a & None              & 25a & min + min + min + S$_3$   & None \\
        &                   & 38a & max + max + max + S$_3$   & None \\ 
 3.1-3b & None              & 25b & min + min + min + S$_3$   & None \\
        &                   & 38b & max + max + max + S$_3$   & None \\ 
 3.1-4  & 15                & 30  & min + min + max + S$_3$   & 41 \\
        &                   & 35  & max + max + min + S$_3$   & 4 \\ \hline 
\multicolumn{5}{l}{\vspace{-0.3cm} }\\
%\multicolumn{5}{l}{\footnotesize $^a$ ZS-type refers to the refined
%classification on antipodal structures by \citeasnoun{Zha75}.} \\
%\multicolumn{5}{l}{\vspace{-0.3cm} }\\
\multicolumn{5}{l}{\footnotesize $^a$ min - minimum boiling binary azeotrope; max - maximum boiling binary azeotrope;} \\
\multicolumn{5}{l}{\vspace{-0.4cm} }\\
\multicolumn{5}{l}{\footnotesize \hspace{0.1cm} Az$_3$ - ternary node azeotrope; S$_3$ - ternary
   saddle azeotrope.} \\
\end{tabular} 
} % end small
\ec
\end{table}   
%The ZS-type of antipodal structure refers to the refined
%classification by \citeasnoun{Zha75}.
%
(Reshetov's statistics are also presented in Figure
\ref{fig:elementary_occurrence_percent}).  Note that
\citeasnoun{Res98} found that the three mixtures reported for class
3.1-1a in the statistics by \citeasnoun{Ser68b} actually was of
another structure class. 
%(result of an erroneous determination of the ternary azeotrope from the ternary
%mixtures' experimental data).
% incorrect 
%, giving the occurrence of the 15 azeotropic classes that are found
%among the real mixtures reported.
From Reshetov's data we see that only 16 of Serafimov's 26 classes are
reported to occur in nature. If we also differentiate between minimum-
and maximum-boiling azeotropes (ZS-type), then we find that 27 of the
48 classes are reported. 
%
%Recurrence of the elementary cells of different type in more complex
%diagrams is presented in Table \ref{tab:recurrence_cells}.
%---------------------------------------------------------------------  
%\begin{table}[!ht]  
%\caption{Recurrence of elementary cells in Serafimov's topological
%classes of ternary VLE diagrams}
%\label{tab:recurrence_cells} 
%\vspace{-0.6cm}
%\bc
%\begin{tabular}{|c|c|}  
%\multicolumn{2}{c}{} \\ \hline 
%Cell &  Serafimov's classes that include this cell \\ \hline  
%I & 0, 1.0-2, 1.1-2, 2.0-2a, 2.0-2b,
%2.0-2c, 2.1-2a, 2.1-3a, \\
%& 2.1-3b, 3.0-1b, 3.0-2, 3.1-1c, 3.1-3a, 3.1-3b, 3.1-4 \\ 
%II & 1.0-1a, 2.0-2b, 2.1-2b, 2.1-3a,
%3.0-2, 3.1-1b, 3.1-1b, 3.1-3b, 3.1-2, 3.1-4 \\ 
%III & 1.0-1b, 2.0-2a, 2.0-2c, 2.1-3b,
%3.0-1a, 3.1-1a, 3.1-3a \\ 
%IV & 2.0-1, 2.1-2b, 3.0-1b \\ 
%II' & 1.1-1a \\ 
%III' & 1.1-1b, 2.1-2a, 3.1-1a \\ 
%IV' & 2.1-1, 3.1-1c \\ 
%V & 3.1-1b \\ \hline
%\end{tabular} 
%\ec
%\end{table}  
%
%---------------------------------------------------------------------  
%\begin{table}[!ht]  
%\caption{Recurrence of elementary cells in ternary mixtures classes}
%\label{tab:recurrence_cells_classifications} 
%\vspace{-0.6cm}
%\bc
%{\small
%\begin{tabular}{||l|c|c||}  
%\multicolumn{3}{c}{} \\ \hline \hline
%                & Serafimov (1970) & Cells recurring \\ \hline 
%No azeotropes   & 0.0-1  & I \\ \hline 
%One binary azeotrope & 1.0-1a & II \\ \cline{2-3}
%                & 1.0-1b & III \\ \cline{2-3} 
%                & 1.0-2  & I, I \\ \hline 
%One binary and one ternary azeotrope  & 1.1-a  & II' \\ \cline{2-3} 
%                & 1.1-1b & III' \\ \cline{2-3} 
%                & 1.1-2  & I, I, I, I \\ \hline 
%Two binary azeotropes & 2.0-1  & IV \\ \cline{2-3} 
%                & 2.0-2a & I, III \\ \cline{2-3} 
%                & 2.0-2b & I, II \\ \cline{2-3} 
%                & 2.0-2c & I, III \\ \hline 
%Two binary and one ternary azeotrope & 2.1-1  & IV' \\ \cline{2-3} 
%                & 2.1-2a & I, III' \\ \cline{2-3} 
%                & 2.1-2b & II, IV \\ \cline{2-3} 
%                & 2.1-3a & I, I, I, II \\ \cline{2-3} 
%                & 2.1-3b & I, I, I, II \\ \hline 
%Three binary azeotropes & 3.0-1a & III, III \\ \cline{2-3} 
%                & 3.0-1b & I, IV \\ \cline{2-3} 
%                & 3.0-2  & I, I, II \\ \hline 
%Three binary and one ternary azeotrope & 3.1-1a & III, III' \\ \cline{2-3} 
%                & 3.1-1b & II, V \\ \cline{2-3} 
%                & 3.1-1c & I, IV' \\ \cline{2-3} 
%                & 3.1-2  & II, II, II \\ \cline{2-3} 
%                & 3.1-3a & I, I, III, III \\ \cline{2-3} 
%                & 3.1-3b & I, I, I, II \\ \cline{2-3} 
%                & 3.1-4  & I, I, II, II \\ \hline \hline
%\end{tabular} 
%}
%\ec
%\end{table}   
%
The distribution reported in these studies does not necessarily
reflect the real occurrence in nature. The azeotropic data selection
is small and occasional. Moreover, the distribution can be distorted
compared to the unknown natural distribution since the published
mixture data are results of deliberate searches for entrainers for
specific industrial separation problems.  Nevertheless, these data are
interesting and can be used for some deductions:
% indications/pointing hand
%Of course, these statistics cannot present unknown "occurence of
%mixtures in Nature" because (i) the selections are too small, and (ii)
%the distribution in published data can be essentially deformed as
%compare to unknown "Natural distribution" due of purposeful searching
%of the entrainers for some industrial problems, i.e., purposeful
%searching of certain types of mixtures entrainer-base binary.
%Nevertheless, these statictics give a good foundation for some speculations.
%
%Both statistics show the same trend. For example, 
%From Reshetov's statistics we can make some important (useful) conclusions:
%Percentage of occurence out of the total azeotropic mixtures data
%analyzed. It indicates the commoness of the various classes in
%nature. It is important to stress that this no more than a "pointing 
%hand" on which types that are more important to focus etc.
\bit
\item Serafimov's class 3.1-2 with three minimum-boiling binary
azeotropes and one minimum-boiling ternary azeotrope has the largest
number of reported mixtures.  About 26 \% of the 1365 ternary
azeotropic mixtures in the study by Reshetov are of this class.
%, and 41 \% of 418 mixtures in the study by Serafimov 
%From this we conclude that a VLE diagram structure of Class 3.1-2 and
%ZS-type 23 () is probably one of the most common thermodynamic
%behaviors among ternary azeotropic mixtures.

\item Elementary cells I and II cover more than 90 \% of all the
reported ternary azeotropic mixtures.  The three most common
structures are Serafimov's classes 1.0-1a (21.6 \%), 2.0-2b (21.0 \%)
and 3.1-2 (26.0 \%).  Among these three classes only cells I and II
occur.
% about 92.9 \% Reshetov's statistics
% about 97.4 \% Serafimov's statistics

\item Ternary azeotropes are common in nature. About 38\% 
of the reported ternary mixtures have a ternary azeotrope. These are
Serafimov's classes 1.1-2, 2.1-2b, 2.1-3a, 2.1-3b, 3.1-2 and 3.1-4.
% 51.4 \% in Serafimov's statistics

%\item Less than 6\% of the reported mixtures include cells with
%possible MSS (string of saddle points, cell III, III', IV and IV').
\eit

%It is important to recognize that these numbers are only indications
%of the relative commonness of the different classes. 
For each of the reported classes, one may find a great number of
(similar) mixtures in nature. However, for those classes with no
reported mixtures, and with a structure that requires a set of
molecular interactions that are unlikely, we expect that few if any
real mixtures will be found.  The experimental data for ternary
azeotropic mixtures used for the above statistics is rather limited.
However, the experimental data for binary azeotropic pairs is
considerably more extensive. For example, \citeasnoun{Gme94} have
compiled data for 18 800 binary systems (involving about 1 700
components). From this we can estimate the behavior of a large number
of ternary mixtures combinations from a VLE model, but to our
knowledge no systematic effort has yet been done.
%This has been done by Morari and coworkers?
%
%For example, the mixture acetone, heptane and benzene is of
%Serafimov's class 1.0-1b. We can search other similar components to
%check whether also these from mixtures of this class. 

It is well-known that binary maximum-boiling azeotropes are less
abundant than minimum-boiling azeotropes. According to
\citeasnoun{Lec49} the ratio of minimum-boiling versus maximum-boiling
azeotropes that occurs in nature is about 9 to 1.
%\cite{Doh93}
The statistics of Reshetov confirm this heuristic rule.  Thus, the
more maximum-boiling binary azeotropes that are included in the
ternary mixture, the less is the probability of its occurrence.  This
is clearly demonstrated in Table
\ref{tab:occurence_of_classes_published}.  In particular, no ternary
mixture with three binary maximum-boiling azeotropes has been found
among Reshetov's selection of 1365 mixtures.
Ternary maximum-boiling azeotropes are also very rare.
%Binary maximum-boiling
%azeotropy is relative rare but still an occurring phenomena,
% Baburina et al. (1988) referred to in Bek96b possibly non-existing,
% see p. 4269
%--------------------------------------------------------------------
%\begin{figure}[!ht]    
%\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Fig33.eps,height=10cm}}}
%\caption{Occurrence of various combination of binary azeotropes in
%ternary mixtures.}
% [Fig. 33]
%\label{fig:occurence_various_binary}
%\end{figure} 
%
As a result, even for the topological structures where the existence is
beyond question, the occurrence of antipodes with maximum-boiling
azeotropes is much less than that of antipodes with minimum-boiling
azeotropes. 
%given in Table \ref{tab:occurence_antipodes_common_classes}.
%---------------------------------------------------------------------  
%\begin{table}[!ht]  
%\caption{Occurrence of antipodes of the most common topological
%classes of ternary VLE diagrams}
%\label{tab:occurence_antipodes_common_classes} 
%\vspace{-0.6cm}
%\bc
%\begin{tabular}{||c|cc||}  
%\multicolumn{3}{c}{} \\ \hline  \hline 
%Serafimov's & \multicolumn{2}{c||}{Reshetov's Statistics (1965-1988)} \\ 
%Class              & min. azeotrope(s) & max. azeotrope(s) \\ \hline \hline 
%1.0-1a             & 283               & 12                \\
%1.0-2              & 95                & 21                \\
%2.0-2b             & 280               & 6                 \\
%2.1-2b             & 55                & None              \\ \hline  \hline 
%\end{tabular} 
%\ec
%\end{table}  
%
%A more detailed evaluation of the occurrence and probability of the
%various ternary VLE diagrams structures are given in
%\citeasnoun{Kiv2000a}.
%Low probability does not mean that these structures do not exist, but
%the information can be used as an Ockham's Razor in analysis of VLE
%and azeotropic distillation. Consider the most probable structures. 
%
%One can see that the diagrams listed in Table
%\ref{tab:most_common_types},
%(1.0-2, 1.1-2, 2.0-2a, 2.0-2b, 2.0-2c, 2.1-2b, 2.1-3a, 2.1-3b, 3.0-1b,
%3.0-2, 3.1-2, 3.1-3b, 3.1-4) 
%are combinations of only the four primary topological cells presented
%in Figure \ref{fig:cells_simplest_classes}.


%\clearpage
% also \citeasnoun{Poe96} 
\citeasnoun{Poe94} propose to reduce the number of ternary VLE
diagrams by considering only "physically meaningful" structures. Their
list includes 19 of Serafimov's 26 classes,
%(0.0-1, 1.0-1a, 1.0-1b, 1.0-2, 1.1-2, 2.0-1, 2.0-2a, 2.0-2b, 2.0-2c,
%2.1-2b, 2.1-3a, 2.1-3b, 3.0-1a, 3.-1b, 3.0-2, 3.1-1b, 3.1-2, 3.1-3a,
%3.1-4).
excluding 1.1-1a, 1.1-1b, 2.1-1, 2.1-2a, 3.1-1a, 3.1-1c, and 3.1-3b.
%In our opinion, 
However, it is impossible in principle to state that some classes of
ternary mixtures cannot exist in nature or are ``physically
meaningless'', because all the structures in Figure
\ref{fig:elementary_occurrence_percent}
%predicted by the rule of azeotropy 
are thermodynamically and topologically feasible. We can only discuss
the {\em probability} of the existence of some types of the VLE
diagram structures.

%We can also predict ternary mixtures behavior from VLE model based on
%experimental data for the binary pairs.  

%The determination of azeotropic behavior of a ternary mixture from
%ternary VLE data can lead to very poor results because the number of
%data points for the ternary mixutre is usually limited and only a part
%of the composition space is covered. Usually, the use of n=binary
%azeotropic data to correlate 


%--------------------------------------------------------------------
%\afterpage{\clearpage}
\clearpage
\section{Use of elementary cells}
%Examples 

%perspicuous 
The concept of elementary topological cells is a simplification which
primarily is made to reduce the number of structures of ternary VLE
diagrams. It is useful for preliminary analysis of azeotropic
distillation (presynthesis).  Based on the knowledge of the distillation
behavior of azeotropic mixtures of the primary diagrams elementary
cells in Figure \ref{fig:cells_simplest_classes}, or combinations of
these, we also have information about what behavior to expect from
other more complex mixtures.
%Nevertheless, using the concept of elementary cells we can at least
%say something about the possible behavior of other mixtures with VLE
%structures that constitute the same elementary cells.
It is important to recognize that each real mixture has its own
specific thermodynamic characteristics and should therefore be
analyzed in detail in a later step of the separation synthesis. 
%The specific features may form obstacles as well as openings for the
%successful separation of the mixture in question.
   
The concept of elementary cells is even more important when using the
classification of Matsuyama and Nishimura with its less surveyable 113 classes.
%Multiple steady states means that a system with as many parameters
%specified as its degrees of freedom may exhibit different solutions at
%steady state.  

The elementary cell concept may be extended to mixtures with more than three
components, but grahical visualization is then more difficult. Of course, for
a multicomponent mixture one can analyze each ternary subsystem.
For example, each of the four triangles of
a four-component tethrahedon is one of the 26 ternary classes and (for
the real mixtures reported so far) four elementary ternary cells.


%--------------------------------------------------------------------
%\clearpage
\subsection{The album}

%The thermodynamical behavior of nonideal and azeotropic mixtures is
%highly system specific, but the structural variance is limited. This
%makes it possible to investigate classes of mixtures and develop
%generalized separation schemes for each.
% \cite{Hau98}, see page 20 

%numerous possible diagrams when the internal structures (geometry) are
%taken into account makes this an almost impossible task.
%including subtypes (or `isomers') with deformation of the distillation
%lines due to the presence of univolatility lines etc., 

%Only possible to analyze a few main archetypes of ternary
%diagrams. Correct selection of these archetypes is very important.


One goal underlying the classification of ternary mixtures is to have
a complete album of possible VLE diagram structures with their
corresponding scheme of separation by distillation.
%giving the exact product areas
But this is not established knowledge. Prediction of feasible
distillation product compositions for even some of the simplest
diagram structures is still under development. Furthermore, methods or
separation schemes to separate all classes of (ternary) azeotropic
mixtures are not established.  One reason is that there are many
possible structures with deformation of the simple phase
transformation paths due to regions with different volatility order
within the composition space, making this an almost impossible task.
Instead, we propose to consider selected VLE diagrams and specific
mixtures of these diagrams in detail: \ben

\item Zeotropic mixture, ideal and nonideal
with univolatility line(s) \\ (Serafimov's class 0.0-1: cell I);
% with C- and S-shape

\item Mixture with one separatrix (one binary azeotrope) \\
(Serafimov's class 1.0-2: combination of two cell I's);
%, illustrated in Figure \ref{fig:Class_1_0-2};
% with C/S-shape

%\item Mixture without separatrix, but with one binary azeotrope
%\\ (Serafimov's class 1.0-1a: cell II);


\item Mixture without separatrix, but with one binary azeotrope
\\ (Serafimov's class 1.0-1a: cell II and 1.0-1b: cell III);
%, U-shape

\item Mixture without separatrix, but with two binary azeotropes
nodes \\ (Serafimov's class 2.0-1: cell IV). 
%, C/U-shape
\een

An illustration of a ternary mixture with one binary saddle azeotrope
(Class 1.0-2) consisting of two elementary cell I's is given in Figure
\ref{fig:Class_1_0-2}.
%--------------------------------------------------------------------
\begin{figure}[!ht]    
\centerline{\protect{\psfig{figure=/home/ekh/99-Methods/Figures/Class_1_0-2.eps,height=4.2cm}}}
\caption{\em Ternary azeotropic mixture with one binary saddle
azeotrope (Class 1.0-2 consisting of two Cell I's). Residue curves;
unidistribution (solid) and univolatility (dash-dotted) lines are
given.}
\vspace{-0.1cm}
\label{fig:Class_1_0-2}
\end{figure} 
%
The left cell has ``C-shaped'' residue curves, and the right cell has
``S-shaped'' residue curves (inflection) caused by the univolatility
line $\alpha_{23}$. Although both cells are of type I, this difference
in shape may have a large effect on the actual separation process. For
details on the internal structure (``shape'') of ternary VLE diagrams,
the reader is referred to \citeasnoun{Hil2000} and \citeasnoun{Res99}.

%Further, by using the concept of elementary topological cells, we can
%make at least some deductions about the behavior of other mixtures
%with VLE diagrams that include combinations of these elementary cells.
%Let us consider some uses of elementary cells.

%--------------------------------------------------------------------
%\clearpage
\subsection{Multiple steady states}
%{\em Easy prediction of MSS:}

%Method of predicting MSS in distillation based on analysis of the
%structure of distillation diagrams \citePon96}.

%> It will be very interesting to apply our approach of elementary
%> cells to something useful. May be we can use it for critical
%> analysis of Guettinger's concepts?
%
%Yes, to include something like this is apropriate for the paper on
%elementary cells (last chapter of survey) that we probably will
%publish separately. I agree, U-cells and multiplicity is a good
%example. To illustrate that from this simplified view we can derive a
%lot of qualitative information (even if it is a too simplified and not
%complete). This is the main point of using the concept of elementary
%cells (and I am sure that other researchers must have thought about it).

%Azeotropic mixtures are known to sometimes result in distillation
%multiplicities. 

As an example of the use of elementary cells we may consider the
possibility for multiple steady states in (homogeneous) azeotropic
distillation.  Such multiplicities may lead to problems in column
operation and control, as well as problems in column design and
simulation.  When two or more multiple steady states exists for the
same inputs it is possible that, for some disturbance, the column
profile jumps from the desirable (in terms of product specifications)
to an undesirable steady state.  However, such catastrophic jumps may
be avoided by proper control of the column, and the separation schemes
for such mixtures may well be feasible and economical.
%
%The existence of MSS may also lead to a higher convergence failure
%rate in the steady-state simulation of a distillaiton column
%\cite{Bek96b}.  The convergenced solution the simulation package
%arrives at depends on the choice of initial conditions.  
%
%%Potential problems in the operation of multiplicity columns have been
%%studied by \citeasnoun{Bek96b}. 
%
%With the computation of only one of the solutions lurks the
%possibility of disregarding other eligible and possibly attractive
%solutions. This may result in misleading conclusions and decisions
%regarding the separation under consideration \cite{Gut98}. 
%
%The $\infty/\infty$ analysis (see, for example \citeasnoun{Bek96b}) is
%able to find all possible solutions and provides useful initial
%profiles for steady-state simulators in order to calculate the desired
%solution.
% Aspen Plus not able to compute MSS
%
The possibility of multiple steady states at infinite efficiency of
the distillation (infinite reflux, infinite theoretical equilibrium
trays, and $D/F$ from 0 to 1) was first noted by \citeasnoun{Bal70}.
They considered a mixture of Serafimov's class 1.0-1b (primary diagram
of elementary cell III with U-shaped residue curves) with a binary
maximum-boiling azeotrope (ZS-type 7b), and found that for these
mixtures it is feasible to have multiple products for the same value
of the parameter $D/F$.  Later, \citeasnoun{Pet71} analyzed this issue
in more detail and included bifurcation analysis. This analysis is also
included in the textbook by \citeasnoun{Pet83}, where Petlyuk writes
{\em ``the existence of more than one saddle along a distillation line
for sharp (infinite reflux, infinite column) separation going from an
unstable node to a stable node leads to multiplicity of the separation
products for the same feed composition and the same value of parameter
$D/F$''}.
% (excessive subregion of distillation) 
The geometrical considerations given by Petlyuk and coworkers are also
found in \citeasnoun{Ser71a}.  This is a sufficient, but not a
necessary condition for multiplicity of the
mixture. \citeasnoun{Bek93} give a similar condition, roughly that the
{\em existence of two or more neighboring saddles} may lead to output
multiplicity.
% (referred to as inherent multiplicity)
% REFERENCE ! maybe Bek93?
%Mixtures with such {\em inherent} structural properties are 
%
%The elementary cells III, III', IV and IV' satisfy this geometrical condition.
%cells III, III', IV and IV' with a string of saddle points
% with the characteristic U-shaped residue curves
\citeasnoun{Bek93} also identify other structural characteristics
that may induce multiple steady states, such as highly curved
distillation boundaries (``pseudosaddles''), as for the mixture of
acetone-chloroform-methanol.

The elementary cells that may lead to output multiplicity are III,
III', IV and IV'. From this we can predict the {\em possibility} of
multiple steady states (sufficient condition) for any given mixture
that is caused by these structural characteristics.
% but rather defined as the closure of the composition trajectories of 
% the given distillation process constrained by the distillation
% boundaries, if any, and the composition simplex.
%For mixture classes with MSS the effect of the (separation scheme)
%design on control and operation can be significant.
%effect of equipment, unstable operation
%
%According to Morari et al. existence of MSS in distillation is
%common. They report to have found 24 mixture systems of Serafimov's
%class 1.0-1b through database search.
%
% Morari et al necessary and sufficient geometrical condition for the
% existence of MSS: two or more neighbouring saddles
% highly curved boundaries
%
From Figure \ref{fig:elementary_occurrence_percent} we see that 14 of
the 26 diagrams include these elementary cells (1.0-1b, 1.1-1a,
1.1-1b, 2.0-1, 2.0-2a, 2.0-2c, 2.1-1, 2.1-2a, 2.1-2b, 3.0-1b, 3.1-1a,
3.1-1b, 3.1-1c, 3.1-3a).  From the statistics by Reshetov we find that
about 7 \% of the reported VLE diagram structures include these cells.
% (14 diagrams)
% Morari: 24 of M&N 113 classes have MSS (30 %)
% opposed to the claim by \citeasnoun{Gut98}.
%For example, mixtures of Serafimov's class 2.0-1, that is, the primary
%diagram of cell IV, have possible MSS in distillation.
%It is the characteristic U-shaped residue curves that may lead to
%MSS. 
Serafimov's class 2.1-2b, which includes cell IV, is relatively common
(4 \%), and this class may give multiplicities for feeds in the region
with U-shaped residue curves.
%Examples of mixtures of this type is ethanol, 2-butanone and octane,
%and hexane, acetone and ethanol.
% needs to check
%The simple phase transformation trajectories of these mixtures have
%C-, S- and U-shape.
%Among the five most common ternary mixture classes, none has the
%possibility of MSS due to the shape of the lines (no series of two or
%more saddles in a row). 
%
%If two or more saddles are present in the mixtures under
%consideration, one should check whether MSS might be a problem.
%

The significance of multiplicities for column operation have been
studied by Morari and coworkers
\cite{Lar92a,Lar92b,Bek93,Bek96b,Gut96b,Gut97}.
%Here we refer to output multiplicities where there are multiple
%outputs (different profiles, product compositions) for given column
%design and operating conditions.
%
% see Bekiaris and Morari (1996b), p. 4269
Mixtures with one binary minimum-boiling azeotrope of Serafimov's
class 1.0-1b (primary diagram of cell III with U-shaped residue
curves) has been the focus,
%(Matsuyama and Nishimura's class 001 and 003), 
in particular the mixture of acetone - heptane - benzene.  However,
from the occurrence statistics we see that Serafimov's class 1.0-1b is
not very common.
%(output multiplicities) 
%Multiplicities of this type are caused by the VLE as well as the
%complex recycle structure between the stages \cite{Gut97}.
%(type II in Guttinger 1997) 
%\cite{Lee99}.
%$\infty/\infty$-analysis - simulations and (industrial plant)
%experiments. 
%They studied the mixture of methanol (1) - methyl butyrate (2) -
%toluene (3), and the mixture of acetone - heptane - benzene.  
%These mixtures has a VLE topology of Serafimov's class 1.0-1b with
%U-shaped distillation lines (or, equivalently, residue curves). This
%corresponds to Matsuyama and Nishimura's class 001.
% according to Guttinger thesis this is a very frequent type of
% thermodynamical system (class), 4300 examples in database search.

\citeasnoun{Bek96b} found multiple steady states for the mixture
ethanol - ethyl propanoate - toluene (Serafimov's class 2.0-2a,
ZS-type 18, which includes cells I and III) for a specific feed
region.
%(Matsuyama and Nishimura's class 401)
The specific feed region corresponds exactly to elementary cell III.
%(that is, the residue curve region with two neighboring saddles and
%U-shaped residue curves). 
%In general, any azeotropic mixture with a VLE diagram structure that
%includes cell III has the possibility of multiplicities.
%MSS in distillation for feeds located in the composition region of
%this cell.
% Serafimov's class 2.0-2a consists of the two elementary cells I and III. 
% ethanol, ethyl propanoate and toluene.
%This mixtures has a VLE topology of Serafimov's class 2.0-2c with two
%simple distillation regions one with C-shaped residue curves (cell I)
%and the other with U-shaped residue curves (cell III). This
%corresponds to Matsuyama and Nishimura's class 201.  
%\citeasnoun{Gut98} refer to this example as {\em ``classes with MSS
%for some feeds''}.  


%---
%One of the main advantages of Serafimov's classification is that it is
%surveyable.  It is easy to identify recurring cells within the 26
%classes. All the classifications differentiate on topology, but only
%Serafimov's classification is true to this basic idea.
%
%Matsuyama and Nishimura's classification is unnecessary complex and
%leads to an overwhelming number of diagrams to relate. For example,
%Matsuyama and Nishimura's class 222-m and 444-M are just antipodes of
%Serafimov's class 3.1-2.
%For example, Serafimov's class 3.0-1b has 12 Matsuyama and Nishimura
%diagrams all with identical topology.
%
%Both 020 and 400 of Matsuyama and Nishimura's classification are of
%Serafimov's class 1.0-2.

%In addition, the use of right-angled triangles (Rozeboom triangles)
%instead of equilateral triangles (Gibbs triangles) in
%\citeasnoun{Doh85a} presentation of the ternary VLE diagrams makes it
%(a bit) more difficult to identify/relate the diagrams with identical
%topology but of different Matsuyama and Nishimura's class because the
%structure gets asymmetrical.  Table
%\ref{tab:correspondence_two_classifications} may be a useful tool to
%relate the different classifications.

%Table should probably be made ``opposite'': Listed after Matsuyama and
%Nishimura's class as given in their article and give the corresponding
%Serafimov' class.

%--------------------------------------------------------------------
\section{Pseudo-component subsystem}
% An related idea/approach

%This statement is closely related to the idea behind 
%The main ideal behind the elementary cells concept is that we are not
%``blind'' dealing with a complex azeotropic ternary mixture that
%splits into several distillation regions.
%(such as acetone-chloroform-methanol).  
%Insights gained from analyzing ideal ternary mixtures and azeotropic
%mixtures of the primary diagrams (elementary topological cells I, II,
%III and IV) can be applied in preliminary analysis of the distillation
%behavior of any real mixture.
%, considering each subsytem (cells).

An idea related to the concept of elementary cells is the concept of
pseudo-components.  This approach is known to many engineers working
with azeotropic distillation.
%, for example, when reliable data for the azeotropic mixture in
%question are not available, or, to overcome the modeling difficulties
%of proper prediction of the nonideal thermodynamic behavior of
%azeotropic mixtures. 
\citeasnoun{Vog99} proposes to analyze real azeotropic mixtures as
subsystems approximated by zeotropic mixtures where each azeotrope is
represented by a pseudo-component. A residue curve region with $k$
singular points (pure component and azeotropes) is thus represented as
a $k$-component zeotropic mixture by assuming constant relative
volatilities between the real components and the pseudo-components
(azeotropes). {\em Ideal distillation lines} are calculated for each
subsystem.  However, this strong simplification has major
pitfalls. The ideal distillation lines (or residue curves) diverge
from the real (exact) ones as azeotropic mixtures necessarily have
univolatility lines that deform the simple phase transformation
trajectories and cause S-, $\Omega$-, and even more complex shapes
(internal structures) of the distillation lines (residue curves).  For
example, a ternary azeotropic mixture of Serafimov's class 1.0-2
(Figure \ref{fig:Class_1_0-2}) may be considered to be a quaternary
system with the binary saddle azeotrope as an intermediate boiling
pseudo-component. From this, for both of the cells I in the diagram,
C-shaped ideal distillation lines are calculated.  However, this is
not true for the real mixture. Furthermore, the approach results in
straight line distillation boundaries which in general are curved
\cite{Sch02}.
%Even though the approach is shown to work considerable well for some
%example mixtures, this approach is not adequate for real azeotropic
%mixtures in general. In fact, the opposite would be to expect. Each
%azeotrope causes the existence of at least one univolatility line in
%the composition interior. Mixtures that form azeotropes usually have a
%very complex internal structure (geometry of the residue curves).

% what about class 1.0-1b one-region U-shape? how to model
% illustrated in Figure 8

We argue that when only the {\em qualitative} shape of the curves are
needed, one can sketch the residue curve map (by hand) rather than
calculate the exact, but incorrect, ideal subsystems map based on
the constant relative volatility assumption.
%This gives more useful information than the
%``pseudocomponent''-approach. 
%If the only thermodynamical data available is the information about
%the existence of azeotropes and the boiling temperature of the
%azeotropes and pure components (and maybe also the azeotrope
%compositions), we only have topological information and thus only the
%topological structure of the azeotropic mixture in question can be
%determined (we can, in most cases, decide the type of the residue
%curve singular points from this information). From the structure
%determination we can identify the topological class and elementary
%cells that constitutes this class, and from their structual properties
%we can have some ideas of what distillation behavior (peculiarities)
%to expect from the mixture (such as possibilities of MSS in
%distillation).
For example, we know that for mixtures of Serafimov's class 1.0-2
(Figure \ref{fig:Class_1_0-2}) with one binary azeotrope saddle and a
separatrix that splits the composition space into two cells I, one of
these cells must have a univolatility line extending from the
azeotrope and to one of the binary edges resulting in an inflection
point of the residue curves (S-shape).
%That is, the internal structure (geometry) differs form that of an
%ideal subsystem mixture.
%
%Often, it is useful to calculate only the residue cruve boundaries
%exact and sketch the other (ordinary) residue curves by hand based on
%information about unidistribution and univolatility maps.
%
%Therefore, in this paper we propose to assume that an elementray cell
%incorporated into a complex diagram behave similarily to an elementray
%cell of the primary diagram, rather than to assume ideal constant
%relative volatility within each cell (mixture subsystem).
%, that is, that the cell behave like a ideal ternary- or quaternary
%mixture (because this approach excludes the important deformation of
%the residue curve paths due to the presence of univolatility lines).
%This is also a strong simplification, however still it incorporated
%some of the inherent nonideality of azeotropic mixtures.

\citeasnoun{Vog99} emphasizes that his main point is not to
approximate the distillation lines of real mixtures by the
distillation lines of ideal systems, but to show that the distillation
behavior of real (zeotropic and azeotropic) mixtures is not
fundamentally different from the distillation behavior of ideal
systems, and that, therefore, the large body of knowledge developed
from the theory of multicomponent distillation may be applied to
better understand the distillation of real multicomponent mixtures.
This is in line with the idea behind elementary cells presented in
this paper.

%This subsystem thinking presented by \citeasnoun{Vog99} is in the same
%line of thoughts as the concept of elementary cells presented in this
%paper.

%--------------------------------------------------------------------
%\section{Discussion}
%
%{\bf On occurrence}
%
%Given a ternary mixture where all three binary pairs form azeotropes
%(ie., mixture of class 3), the probability that the mixture forms a
%ternary azeotrope is high. If it is found that no ternary azeotrope
%exists (at the given pressure or temperature), then the topological
%structure of class 3.0-2 is the most probable.
%
%The authors wish for an updated list of the natural occurrence of the
%various ternary mixture classes based on eg. the azeotropic data
%collected in \citeasnoun{Gme94} (Lecat 1918, Lecat 1949, Ogorodnikov
%et al. 1971, Horsley 1973 and the Dortmund Data Bank, Gmehling et
%al. 1977). This azeotropic data book reports to comprise more than 18
%800 systems involving approximately 1 700 compounds, in which
%approximately 47\% of the systems show azeotropic behavior.
%(ie., 8 836 binary and ternary azeotropic systems).?
%
%``Theoretical'' compounds that form the ternary mixtures that are not
%reported in Reshetov statistics based on UNIFAQ model. 
% DECHEMA database
%
%More than 90\% of the binary azeotropic data show positive azeotropy
%(ie., minimum boiling azeotropes) \cite{Gme94}. More than 80\% of
%these are homoazeotropic.
%
%{\bf Further work: internal structure - geometry}
%
%Inside each topological cell there may be various shapes of the simple
%phase transformation trajectories (residue curve and distillation line
%paths).  This topic is described in a succeeding paper on geometry of
%VLE diagrams \cite{Kiv2000c}.

%--------------------------------------------------------------------
%\clearpage
\section{Conclusion}

The concept of elementary topological cells of ternary VLE diagrams is
a key to a simple and surveyable azeotropic distillation analysis.
Any real ternary mixture can be qualitatively represented by a
combination of only four elementary cells.  This greatly reduces the
number of VLE diagram structures that need to be analyzed in order to
reveal the qualitative characteristics of any ternary azeotropic
mixture.  
% added 05 oct 01

%Residue curves are focused in this paper, but the concept equally
%applies to {\em distillation lines} \cite{Zha68c,Zha75,Sti88} or other
%graphical representations of the VLE of ternary mixtures. We refer to
%the review of \citeasnoun{Wid96} and to the literature mentioned for a
%detailed description of these tools and their application to column
%profiles.

%An elementary topological cell may also be defined as the closure of
%the composition trajectories of the given distillation process
%constrained by the distillation boundaries, if any, and the
%composition simplex.
%This is useful in the prediction of MSS
%However, we only talk about the topology.

%Feasible structures of VLE diagrams of ternary (zeotropic and
%azeotropic) mixtures are limited by rigorous thermodynamical and
%topological constraints and are classified into 26 distinct
%structures. Eight structural objects, named elementary cells, are
%identified as constituent of the VLE diagram structures. 

%An elementary cell can be considered as an autonomous topological
%object characterized by a certain set and order of the singular points
%along the contour of its border. Primary (single) elementary cells and
%elementary cells incorporated into more complex diagrams are
%topological equivalent if they have the same set and order of the
%singular points along the border. It is shown that only the four
%primary topological cells recurs in the diagrams that represent the
%VLE structure of real mixtures reported in the literature.
% Only need to know four

%The main idea behind the concept of elementary cells is that from this
%simplified view we can derive a lot of qualitative information (and
%the authors as sure that other researchers within this field make use
%of the same line of thoughts).
%%(even if it is too simplified and not complete). 

%All types of the classified diagrams are topologically and
%thermodynamically feasible, but their occurrence in nature is
%determined by the probability of certain combinations of molecular
%interactions.  Data on the natural occurrence of various mixture
%classes permit to exclude rare or improbable diagrams from
%consideration in the investigation of the correlation between VLE
%diagram structures and feasible separations upon distillation.
  
%All ternary VLE diagrams are build up by a few elementary cells with a
%few possible line shapes.  In this paper we reveal the set of eight
%basic topological elements as single bundles of simple distillation
%residue curves or distillation lines. These cells constitute the more
%complex diagrams.  We show that the naturally occurring types of
%ternary VLE diagram structures include only four of these elementary
%cells.

%The concept simplifies the conception of the VLE diagrams and makes a
%surveyable rendering that we believe is of practical and, in
%particular, pedagogical importance.  Examples of its use in
%distillation analysis and presynthesis for mixtures of selected
%classes are given.
%The concept that the distillation line (residue curve) maps are
%represented by combinations of elementary topological cells 
%It is important to stress that this is a strong simplification .

%\citeasnoun{Ser71a} give a complete classification of possible
%topological structures of VLE diagrams for ternary (zeotropic and
%azeotropic) mixtures.  Within the 26 classified VLE diagrams one can
%identify four {\em elementary cells} that recurs in the VLE diagram of
%any real ternary mixture.  The VLE diagrams may be
%further categorized according to the shape of the distillation lines
%or, equivalently, the shape of the simple distillation residue curves.
% \cite{Kiv99}.
% ref{sec:c3:VLE_diagrams}
%All other more complex classes of the {\small VLE} diagrams
%consists of such elementary cells. This simplifies the analysis.  For
%example, the most common ternary azeotropic mixtures occurring in
%nature is of class 3.1-2 [in Serafimov's thesis 1968 \cite{Kiv99}]
%with three binary azeotropes, one ternary azeotrope and three
%distillation regions, and they seem very complex to analyze in terms
%of feasible distillation separations.
%An other illustrative example is mixtures of class 2.1-2b which is 
%also quite common in nature. This diagram consists of two 
%{\em S/S-cells} and one {\em U-cell} and seems very complex to 
%analyze in terms of feasible distillation separations.
%The idea is that by knowing only a few (simplified) structural
%elements of the VLE diagrams we also have information about any
%complex VLE diagram that consists of these basic elements or cells.
%%the elementary cells recurs in the complex VLE diagrams. 
%Thus, we can reduce the number of diagram structures to be analyzed in
%order to reveal the qualitative characteristics of any azeotropic mixture.
%% regularities/
%We can make use of this simplifying concept to organize our analysis
%of distillation of various types of ternary mixtures.

%We show that consideration of elementary cells is an useful approach
%that can give information about topological structure of residue curve
%(distillation line) map. This approach is particular useful when we
%try to find some general predictions of the distillation behavior of
%the mixtures of different types. Nevertheless, it is necessary to
%analyze the the exact distillation lines, or, at least the structure
%of unidistribution and univolatility lines for each specific mixture
%in question.

%Finally, we want to emphasize that the topological structure of the
%VLE diagram identifies the class of a given mixture, but not its
%specific characteristic like the shape of the singular and ordinary
%simple phase transformation paths (residue curves and distillation
%lines).
   
%The concept of elementary topological cells is an abstraction of the
%structural properties of VLE diagrams. 
%that greatly reduces the number
%structures that need to be analyzed in order to reveal the qualitative
%characteristics of any ternary azeotropic mixture. 

%Statistics on the physical existence (occurrence) of
%azeotropic mixtures are also included.

%The idea is that by knowing the elementary cells we also have
%information about any complex VLE diagram. 

%Any real ternary mixture can be qualitatively represented by a
%combination of only four elementary topological cells.

%--------------------------------------------------------------------
%\clearpage
\section*{Acknowledgments}
%\addcontentsline{toc}{section}{Introduction} 

Dr.~S.A. Reshetov at the Karpov Institute of Physical Chemistry in
Moscow is greatfully acknowledged for providing the statistics on the
occurrence of Serafimov's classes for reported ternary azeotropic
mixtures.

%---------------------------------------------------------------------  
%\clearpage
%\secapp{4A}{}
%Ternary VLE diagram classifications relationship
%\label{sec:appendix}
%\vspace{0.2cm}
%---------------------------------------------------------------------  
\begin{table}[!b]
%\vspace{-1.4cm}
\caption{\em Relationship between different classifications of
ternary VLE diagrams}
\label{tab:correspondence_classifications} 
\vspace{-0.8cm}
\bc
%  \renewcommand{\baselinestretch}{0.92}  
%  {\footnotesize
  {\small
\begin{tabular}{|l|c|c|c|c|}  
\multicolumn{5}{c}{} \\ \hline 
& Gurikov (1958) & Serafimov (1970) & ZS-type$^a$ & Matsuyama
\& Nishimura (1977) \\ \hline
%& 22 classes   & 26 classes & 49 classes & 113 classes \\ \hline
%& Gurikov & Serafimov & ZS-type$^a$ & Matsuyama and Nishimura \\ 
%& (1958)  & (1970) & (1975) & (1977) \\ \hline
Zeotropic        &  1    & 0.0-1  & 1   & 000 \\ \hline 
One binary & 4a & 1.0-1a & 3a  & 100 \\ 
azeotrope     &    &        & 7a  & 030 \\ \cline{2-5}
    & 4b & 1.0-1b & 3b  & 001 \\
    &    &        & 7b  & 003 \\ \cline{2-5}
    & 3  & 1.0-2  & 4   & 020 \\
    &    &        & 8   & 400 \\ \hline 
One binary & 2a & 1.1-1a & 2a  & 200-m \\
and one &    &        & 6a  & 040-M \\ \cline{2-5}
ternary      & 2b & 1.1-1b & 2b  & 002-m \\
azeotrope     &    &        & 6b  & 004-M \\ \cline{2-5}
    & 5  & 1.1-2  & 5   & 010-S \\
    &    &        & 9   & 300-S \\ \hline 
Two binary & 9 & 2.0-1  & 15  & 031, 103, 130 \\ \cline{2-5}
azeotropes     & 8a & 2.0-2a & 17  & 023, 320 \\
    &    &        & 18  & 401, 410 \\ \cline{2-5}
    & 8c & 2.0-2b & 11a & 102, 120, 021 \\
    &    &        & 21a & 043, 430, 403 \\ \cline{2-5}
    & 8b & 2.0-2c & 11b & 201, 210, 012 \\
    &    &        & 21b & 034, 304, 340 \\ \hline 
Two binary & 7 & 2.1-1  & 13  & 032-m, 230-m, 203-m \\
and one &   &        & 14  & 041-M, 140-M, 104-M \\ \cline{2-5}
ternary  & - & 2.1-2a & 16a & 420-m, 402-m  \\
azeotrope     &   &        & 16b & 024-m, 420-M  \\ \cline{2-5}
    & 6 & 2.1-2b & 10  & 022-m, 220-m, 202-m \\
    &   &        & 20  & 044-M, 440-M, 404-M \\ \cline{2-5}
    & 10a & 2.1-3a & 19  & 013-S, 310-S, 301-S \\ \cline{2-5}
    & 10b & 2.1-3b & 12  & 011-S, 110-S, 101-S \\
    &     &        & 22  & 033-S, 330-S, 303-S \\ \hline 
Three binary & 14a & 3.0-1a & 29  & 411 \\
azeotropes    &     &        & 33  & 323 \\ \cline{2-5}
    & 14b & 3.0-1b & 28  & 123, 321, 132, 213, 312, 231 \\
    &     &        & 34  & 413, 314, 431, 341, 134, 143 \\ \cline{2-5}
    & 13  & 3.0-2  & 24  & 212, 122, 221 \\
    &     &        & 37  & 434, 344, 443 \\ \hline 
Three binary & - & 3.1-1a & 27b & 421-m, 412-m \\
and one    &   &        & 32b & 423-M, 324-M \\ \cline{2-5}
ternary    & 12 & 3.1-1b & 26  & 232-m, 223-m, 322-m \\
azeotrope      &    &        & 31  & 414-M, 441-M, 144-M \\ \cline{2-5}
    & -  & 3.1-1c & 27a & 142-M, 241-M, 124-M, 214-M, 421-M, 412-M \\
    &    &        & 32a & 423-m, 324-m, 432-m, 342-m, 234-m, 243-m \\ \cline{2-5}
    & 11 & 3.1-2  & 23  & 222-m \\
    &    &        & 36  & 444-M \\ \cline{2-5}
    & 15 & 3.1-3a & 25a & 121-S, 112-S, 211-S \\
    &    &        & 38a & 343-S, 334-S, 433-S \\ \cline{2-5}
    & -  & 3.1-3b & 25b & 121-S, 112-S  \\
    &    &        & 38b & 343-S, 334-S, 433-S \\ \cline{2-5}
    & 16 & 3.1-4  & 30  & 131-S, 113-S, 311-S \\
    &    &        & 35  & 133-S, 313-S, 331-S \\ \hline 
%Total & 22 & 26 & 49 & 113 \\ \hline 
\multicolumn{5}{c}{\vspace{-0.2cm} } \\ 
\multicolumn{5}{l}{$^a$ ZS-type refers to the refined
classification of the 49 antipodal structures by \citeasnoun{Zha75}, p. 96-98}
\end{tabular} 
} % end small
\ec
\vspace{-0.4cm}
\end{table}   

%---------------------------------------------------------------------  
%\begin{table}[!ht]
%\vspace{-0.6cm}
%\caption{Correspondence between different classifications of ternary
%VLE diagrams}
%\label{tab:correspondence_classifications} 
%\vspace{-0.6cm}
%\bc
%{\small
%\begin{tabular}{|l|c|c|c|}  
%\multicolumn{4}{c}{} \\ \hline 
%& Serafimov (1970) & ZS-type $^a$ (1975) & Matsuyama and Nishimura (1977) \\ \hline
%Zeotropic        & 0.0-1  & 1   & 000 \\ \hline 
%One binary azeotrope & 1.0-1a & 3a  & 100 \\
%    &        & 7a  & 030 \\ \cline{2-4}
%    & 1.0-1b & 3b  & 001 \\
%    &        & 7b  & 003 \\ \cline{2-4}
%    & 1.0-2  & 4   & 020 \\
%    &        & 8   & 400 \\ \hline 
%One binary and one & 1.1-a  & 2a  & 200-m \\
%ternary azeotrope  &        & 6a  & 040-M \\ \cline{2-4}
%    & 1.1-1b & 2b  & 002-m \\
%    &        & 6b  & 004-M \\ \cline{2-4}
%    & 1.1-2  & 5   & 010-S \\
%    &        & 9   & 300-S \\ \hline 
%Two binary azeotropes & 2.0-1  & 15  & 031, 103, 130 \\ \cline{2-4}
%    & 2.0-2a & 17  & 023, 320 \\
%    &        & 18  & 401, 410 \\ \cline{2-4}
%    & 2.0-2b & 11a & 102, 120, 021 \\
%    &        & 21a & 043, 430, 403 \\ \cline{2-4}
%    & 2.0-2c & 11b & 201, 210, 012 \\
%    &        & 21b & 034, 304, 340 \\ \hline 
%Two binary and one & 2.1-1  & 13  & 032-m, 230-m, 203-m \\
%ternary azeotrope  &        & 14  & 041-M, 140-M, 104-M \\ \cline{2-4}
%    & 2.1-2a & 16a & 420-m, 402-m  \\
%    &        & 16b & 024-m, 420-M  \\ \cline{2-4}
%    & 2.1-2b & 10  & 022-m, 220-m, 202-m \\
%    &        & 20  & 044-M, 440-M, 404-M \\ \cline{2-4}
%    & 2.1-3a & 19  & 013-S, 310-S, 301-S \\ \cline{2-4}
%    & 2.1-3b & 12  & 011-S, 110-S, 101-S \\
%    &        & 22  & 033-S, 330-S, 303-S \\ \hline 
%Three binary azeotropes & 3.0-1a & 29  & 411 \\
%    &        & 33  & 323 \\ \cline{2-4}
%    & 3.0-1b & 28  & 123, 321, 132, 213, 312, 231 \\
%    &        & 34  & 413, 314, 431, 341, 134, 143 \\ \cline{2-4}
%    & 3.0-2  & 24  & 212, 122, 221 \\
%    &        & 37  & 434, 344, 443 \\ \hline 
%Three binary and one & 3.1-1a & 27b & 421-m, 412-m \\
%ternary azeotrope    &        & 32b & 423-M, 324-M \\ \cline{2-4}
%    & 3.1-1b & 26  & 232-m, 223-m, 322-m \\
%    &        & 31  & 414-M, 441-M, 144-M \\ \cline{2-4}
%    & 3.1-1c & 27a & 142-M, 241-M, 124-M, 214-M, 421-M, 412-M \\
%    &        & 32a & 423-m, 324-m, 432-m, 342-m, 234-m, 243-m \\ \cline{2-4}
%    & 3.1-2  & 23  & 222-m \\
%    &        & 36  & 444-M \\ \cline{2-4}
%    & 3.1-3a & 25a & 121-S, 112-S, 211-S \\
%    &        & 38a & 343-S, 334-S, 433-S \\ \cline{2-4}
%    & 3.1-3b & 25b & 121-S, 112-S, 241-S?  \\
%    &        & 38b & 343-S, 334-S, 433-S \\ \cline{2-4}
%    & 3.1-4  & 30  & 131-S, 113-S, 311-S \\
%    &        & 35  & 133-S, 313-S, 331-S \\ \hline 
%\multicolumn{4}{c}{\vspace{-0.3cm} } \\ 
%\multicolumn{4}{l}{$^a$ ZS-type refers to the refined
%classification on antipodal structures by \citeasnoun{Zha75}}
%\end{tabular} 
%} % end small
%\ec
%\end{table}   


%--------------------------------------------------------------------
%\begin{figure}[!ht]    
%\centerline{\protect{\psfig{figure=/home/ekh/99-Survey/Figures/Seraf-a-occur.eps,height=18cm}}}
%\caption{Occurrence of the topological classes among azeotropic
%mixtures reported in the literature according to Reshetov's statistics
%(based on data from 1965 to 1988).}
%\label{fig:occurence_percent}
%\end{figure} 
%\clearpage
%\afterpage{\clearpage}
%

%---------------------------------------------------------------------  
%\afterpage{\clearpage}
%\begin{table}[!ht]  
%\caption{Correspondence between the classifications by Serafimov
%(1970{\em b}) and Matsuyama and Nishimura (1977) on topologically
%feasible VLE diagram structures for ternary mixtures (without
%biazeotropy).}
%\label{tab:correspondence_two_classifications} 
%\vspace{-0.4cm}
%\bc
%{\small
%\begin{tabular}{|l|c|c|}  
%\multicolumn{3}{c}{} \\ \hline 
%                & Serafimov (1970) & Matsuyama \& Nishimura (1977) \\ \hline 
%No azeotropes   & 0.0-1  & 000 \\ \hline  
%One binary azeotrope & 1.0-1a & 100, 030 \\ 
%                & 1.0-1b & 001, 003 \\ 
%                & 1.0-2  & 020, 400 \\ \hline 
%One binary and one ternary azeotrope  & 1.1-a  & 200-m, 040-M \\ 
%                & 1.1-1b & 020-m, 004-M \\ 
%                & 1.1-2  & 010-S, 300-S \\ \hline 
%Two binary azeotropes & 2.0-1  & 031, 103, 130 \\ 
%                & 2.0-2a & 023, 320, 401, 410 \\
%                & 2.0-2b & 102, 120, 021, 043, 430, 403 \\ 
%                & 2.0-2c & 201, 210, 012, 034, 304, 340 \\ \hline 
%Two binary and one ternary azeotrope & 2.1-1  & 032-m, 230-m, 203-m, 041-M, 140-M, 104-M \\% 
%                & 2.1-2a & 420-m, 402-m, 024-m, 420-M  \\ 
%                & 2.1-2b & 022-m, 220-m, 202-m, 044-M, 440-M, 404-M \\ 
%                & 2.1-3a & 013-S, 310-S, 301-S \\ 
%                & 2.1-3b & 011-S, 110-S, 101-S, 033-S, 330-S, 303-S \\ \hline  
%Three binary azeotropes & 3.0-1a & 411, 323 \\ 
%                & 3.0-1b & 123, 321, 132, 213, 312, 231 \\
%                &        & 413, 314, 431, 341, 134, 143 \\ 
%                & 3.0-2  & 212, 122, 221, 434, 344, 443 \\ \hline 
%Three binary and one ternary azeotrope & 3.1-1a & 421-m, 412-m, 423-M, 324-M \\ 
%                & 3.1-1b & 232-m, 223-m, 322-m, 414-M, 441-M, 144-M \\ 
%                & 3.1-1c & 142-M, 241-M, 124-M, 214-M, 421-M, 412-M \\
%                &        & 423-m, 324-m, 432-m, 342-m, 234-m, 243-m \\ 
%                & 3.1-2  & 222-m, 444-M  \\ 
%                & 3.1-3a & 121-S, 112-S, 211-S, 343-S, 334-S, 433-S \\ 
%                & 3.1-3b & 121-S, 112-S, 241-S, 343-S, 334-S, 433-S \\ 
%                & 3.1-4  & 131-S, 113-S, 311-S, 133-S, 313-S, 331-S \\ \hline
%\end{tabular} 
%} % end small
%\ec
%\end{table}   


%---------------------------------------------------------------------
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