Ind. Eng. Chem. Res. 1992, 31, 893-9131
893
Crossing of Valleys, Ridges, and Simple Boundaries by Distillation in
Homogeneous Ternary Mixtures
Endre Rev
Department of Chemical Engineering, Technical University of Budapest, 1521 Budapest, Hungary
Residue curves, simple distillation boundaries, and rectification column liquid composition profiles
in several example ternary mixtures are calculated and plotted, and afterward the empirical results
are theoretically discussed. Valleys and ridges are not the boundaries to simple distillation residue
curves. Valleys and ridges run along paths different from the residue curve trajectories. Valley floors
and top-ridge lines are frequently crossed by residue curves. The simple distillation boundaries are
crossed to a small extent by single-feed rectification profiles, but this crossing is not significant in
the sense that another boundary to rectification profiles exists not far from the simple distillation
boundary. Significant crossing, i.e., elimination of the boundary by rectification, is not theoretically
excluded, and eight maps providing such an opportunity are listed. However, this phenomenon must
be rather rare in practice because of chemical reasons; therefore, excluding them is a reasonable
working hypothesis. Construction of a scalar potential over the tie-line vector field is impossible.
Introduction
This article deals with the valleys and ridges in the
boiling temperature surface over the composition triangle
and with their role in the formation of simple distillation
boundaries. The possibility of crossing the boundaries by
continuous distillation is also discussed as well as the existence of a scalar potential over the vector field represented by the tie lines.
Vapopliquid equilibria of a homogeneous mixture under
given pressure are characterized by a dual pair of vector
to vector mappings between the coexisting x and y compJsitions together with the corresponding scalar function
T = Tb(x)= Td(y)representing the temperature of coexistence, which is the bubble point temperature of x and
the dew point temperature of y. Thus bubble and dew
point temperature functions are equally appropriate for
studying distillation paths in multicomponent systems.
Usually the liquid compositions and the bubble temperature are traced. The coexistence is also denoted by the
inverse vector-vector functions y(x) and x(y).
In the case of simple equilibrium distillation the composition of the vapor which is removed from the x com)
composition,
position liquid is just the y ( ~ equilibrium
and the temperature is T = Tb(x)= Td(y). The evolution
of the liquid composition in the course of the process can
be calculated by integrating a system of ordinary differential equations:
-dx/dt = Y(X) - x
(1)
where t is any arbitrary selected parameter, e..g, time or
volume in the pot. Irrespective of the selection of t , the
evolved x values constitute continuous trajectories or arcs
in their vector space. Since the sum of the mole fractions
must be equal to unity, the trajectories can be represented
by graphs in a closed simplex of a c - 1 dimension Euclidean space: Cfxi = 1.0; 0.0 I x i ( i = 1, 2, ..., c - 1).
Trajectories of ternary mixtures can be represented in a
Gibbs triangle.
A trajectory starting from x, is called the residue curve
or residue line of q. Integration in the opposite direction
(integration with positive sign in eq 1) results in a continuation of the residue line in the opposite direction, and
a full residue line passing through x,takes form. Doherty
and Perkins (1978) proved that such a line is unique to ga
provided that T(x,) is not a singular point. As a consequence, residue lines cannot cross each other. They also
proved, based on thermodynamics, that T(x) is a naturally
occurring Liapunov surface of eq 1 (their "Theorem Four").
Singular points of eq 1 in the simplex correspond to pure
component vertices and azeotrope compositions; here y(x)
-x=o.
Liquid composition profiles of continuous rectification
with theoretical stages are also related to the coexistence
functions but are different from the trajectories of eq 1.
At total reflux the liquid composition x, on stage n is equal
to the Y , + ~ composition of the vapor leaving tray n 1 and
is in equilibrium with the liquid composition x , + ~on tray
n 1:
+
+
xn = Yn+l = ~ ( x n + l )
(2)
Equation 2 can also be solved in both directions to an
infinite number of stages by performing only bubble and
dew point calculations. The composition profiles at total
reflux may be considered as limiting cases of profiles at
finite reflux ratios.
The Gibbs triangle of a ternary system can sometimes
be divided into simple distillation regions (SDR) according
to the end points of the residue lines. Explicitly: Two
points belong to the same SDR if the residue lines passing
through them point to (arrive a t ) the same singular point
of the simplex; otherwise they belong to different regions.
Those residue lines which join singular points are called
separatrices (including also the residue lines which lie on
the binary edges). The SDR boundaries are those sepmatrices which arrive at saddle points. (Saddle points are
singular points possessing both incoming and departing
separatrices. For example, in a ternary system without
ternary azeotrope and with only one binary azeotrope
between the two higher boiliig components, the minimum
boiling azeotrope is a saddle point if there is a separatrix
from the third, lightest pure component to the azeotrope.)
It is also known that composition profiles of continuous
rectification may cross the SDR boundaries. Van Dongen
and Doherty (1985), demonstrating the occurrence of
distillation regions in the case of continuous columns,
presented considerably crossing of an SDR boundary by
the column profiles in their Figure 12.
A common feature of the simple distillation process and
the single-feed rectification process is the monotonous
change of T with time ( t ) and stages ( n ) , respectively.
(Attention: a column with multiple feeds is not strictly
subjected to this rule.) This feature is expressed by the
theorem that T(x) is a Liapunov surface. Azeotrope compositions can be found by experimentally studying the T(x)
surface. It is a generally accepted opinon among chemical
engineers that isothermal lines in the triangular diagram
Q88~-5S85/92/2631-0893$03.Q0/Q
0 1992 American Chemical Society
894 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
give useful information on the possible distillation pathways or profiles. For example, Tamir and Wisniak (1978)
reported an excellent interpolation function to fit measured
temperature data for multicomponent mixtures. They
claimed that their equation is useful in obtaining saturation isotherms, and the figures made of them “indicate
possible paths in multicomponent distillation”. It is also
well-known that valley floors and top-ridge lines in the
T(x) surface may be utilized to identify ternary azeotrope
points; see, e.g., Swietoslawski (1963).
Problems
It is again a widespread opinon that valleys and ridges
cannot be crossed and they are the boundaries to the
distillation regions. Such statements occur at least in the
Russian literature (see, e.g., Bushmakin and Kish (1957))
but are also supported by an unfortunate mistake in the
fundamental article of Doherty and Perkins (1978). (In
the Explanations section later in the present article a lively
description of the valley and ridge situations is provided
in order to clear up the picture, and the most probable
motive of misinterpreting the role of valleys and ridges is
pointed out.)
The possibility of crossing the SDR boundaries by rectification is inconsistent with the above opinion. But why
the SDR boundaries can be crossed by rectification is
sufficiently and clearly explained by Doherty and Perkins
(1978). It is because the tie lines are always tangent to the
residue lines at the liquid point and the boundaries are
usually curved (with exception of the binary lines).
Analyzing the phenomena of ”distillation anomaly” reported by Ewell and Welch (1945) and the comment made
on it by Swietoslawski (1963),Doherty and Perkins (1978)
concluded that “we do not feel that ridge projections are
appropriate candidates for distillation boundaries”.
However, they claimed in their “Corollary two” that ”valley
floors must begin and end at pure component or azeotropic
boiling points. They cannot arbitrarily start and finish at
any other temperaturecomposition point. Similarly ridges
in the temperature surface giue rise to stable separatrices
which must also begin and end at different singular
points.” Moreover, they gave a method to determine the
valley and ridge projections and added: “Therefore, it is
possible to locate the separating manifolds (i.e., higher
dimensional separatrices) .,. which are precisely the distillation boundaries for the mixture.” Here they obviously
considered valleys and ridges as borders.
On the basis of their “Corollarytwo” they concluded that
“Swietoslawski’s claims about ridge lines are most certainly
incorrect since they effectively contradict the Second Law
of Thermodynamics.”
There is some confusion in the above opinions. If rectification profiles cross the boundaries and the valley and
ridge projections are exactly the separatrices, then these
valleys and ridges are crossed. If they are crossed, how
and why do they give rise to boundaries? The statement
that valleys and ridges are crossed and they constitute
SDR boundaries, thus SDR boundaries are crossed, is
theoretically not conflicting. But it is also a question of
whether, how, and to what extent the boundaries are
crossed. On the other hand, Swietoslawski’s claims are well
established by experimental results of independent researchers. He claimed: “Numerous experiments carried
out by our group have shown that the top-ridge line
reaches a point lying on the triangle side ... In some cases,
the top-ridge line disappeared earlier.” See section 72 and
figures X.1, X.2, X.3, X.5, X.13, X.14, and X.22 in his book.
Having accepted that the SDR boundaries are crossed
by column profiles, another question emerges about how
Table I. Pure Material Data
1
2
3
4
5
6
7
8
9
10
11
component
name
acetone
methyl acetate
chloroform
methanol
tetrachloromethane
ethyl acetate
benzene
cyclohexane
2-propanol
heptane
toluene
Antoine coefficients
A
B
C
formula
C3H60 7.117 14 1210.595 229.664
C3HB02 7.065 24 1157.630 219.726
CHC13 6.95465 1170.966 226.232
CH40 8.080 97 1582.271 239.726
CC14
6.840 83 1177.910 220.576
C4H802 7.101 79
C,jH6
6.87987
C&2
6.851 46
C3H80 8.878 29
C7H16 6.893 86
C7HB
6.95087
1244.951
1196.760
1206.470
2010.330
1264.370
1342.310
VL
74.05
79.84
80.67
40.73
97.09
217.881 98.49
219.161 89.41
223.136 108.75
252.636 76.92
216.640 147.47
219.187 106.85
significant this crossing is. Doherty and Caldarola (1985)
claimed “... it is known that sometimes a continuous
distillation profile will actually cross a simple distillation
boundary in regions of high curvature. However, it seems
that a reasonable working assumption is the following:
Material balance lines joining distillate, feed, and bottoms
compositions in a continuous distillation are forbidden to
cross simple distillation region boundaries ...”. This latter
assumption is questioned by Laroche et al. (1990), who
stated that this criterion “rejects many candidate entrainers which actually make separation feasible”,therefore
the crossing of the boundaries is of industrial importance.
Van Dongen and Doherty (1984) recognized this problem
and concluded practically to the same solution which is
also one of our conclusions: the T(x) surface is not a
potential function over the tie-line vectors.
The common factor in the above doubts is the relative
situation of ridges and valleys with respect to the boundaries in the composition triangle, as well as the distinction
between the SDR and the rectification profile boundary
lines. These questions are of great importance in process
design practice. In the course of searching for answem and
explanations another question also emerged whether the
T(x) surface can be utilized for assigning the distillation
pathways or the construction of a better “potential
surface”, instead, is possible.
Study
To answer the above questions, several ternary systems
were studied and the results evaluated, as a first step
before theoretical analysis. The problems being of a
qualitative nature, the crucial point is not the accuracy but
the general features. As a consequence, in this study the
only calculations performed were based on a well-established model with reliable parameters. For the sake of
simplicity all calculations were made using the two-parameter Wilson model (Wilson, 1964) at atmospheric
pressure. The model is described in and all the parameter
data are taken from Gmehling and Onken (1982). The
materials and the model parameters that are applied in
the present article are listed in Table I and Table 11.
Each system was analyzed as follows:
1. The T(x)temperatures were calculated in many
points of a dense grid in the composition triangle. Then
minima and maxima were searched for along sections
parallel to the edges in the three possible directions. In
case of necessity the number of sections per edge were
adapted to the resulted range of data. In some cases other
directions were also scanned. The extrema1 data were
collected and plotted.
2. Several residue curvea, starting from the tight vicinity
of the singular points, were computed and the trajectories
plotted in order to determine the loci and the shape of the
separatrices. The method suggested by Doherty and
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 896
ACETONE
a
Table 11. Wilson Interactivity Parameters (Species from
Table I)
CHLOROFORM
BENZENE
ACETONE
b
i
i
1
1
1
1
2
2
3
3
3
4
4
4
4
5
5
7
7
8
10
3
4
7
Aij
-112.4539
-124.9332
650.7928
1016.1121
183.6587
-125.6860
-361.7944
-397.1513
-161.8065
1018.4621
2959.3110
3215.5675
1743.2189
280.3865
140.6542
110.0965
345.9681
257.6138
17.5185
8
3
4
4
6
7
6
8
10
11
8
9
8
9
9
11
a
A,
-309.6043
551.4545
-264.9213
305.4414
-497.9734
891.1353
1694.0241
-63.5668
49.6010
-225.1660
600.1533
732.1596
295.8779
-285.2382
1230.7604
146.4216
880.7213
1515.9240
209.9140
TOLUENE
0%
0
4
61
am 0.10
to.1.c
O.M
0.30
0.40
0.m
0.50
0.70
0.80
0.90
CHLOROFORM
1.00
BENZENE
ACETONE
C
QSJ
HEPTANE
,
M E T HA NOL
TOLUENE
b
I
C H L O R O 2 J
FORM
050
r'
0 40
J \
0 50
d
{/
0 60
0 70
4
-BENZENE
Figure 1. Example 1. (a) Map. (b) Column profiles crossing the
ridge and the separatrix. (c) Enlargement of part of (b).
Perkins (1978) was not applied.
3. In some cases rectification column profiles, at total
reflux, were calculated to check if and how far the profiles
cross the SDR boundary. The profile calculations were
started from over or from the tight vicinity of the separatrix and evolved up to 200 stages in both directions (this
proved practically to be infinite number of stages).
Some of the maps thus obtained are shown in Figures
1-7.
The first system shown is acetone/chloroform/benzene,
which exhibits a negative binary azeotrope on the acetone/chloroform edge. In Figure l a a dashed-dotted,
070
HEPTANE
59053'C
080
0 90
100
METHANOL
Figure 2. Example 2. (a) Map with rectification profiles. (b)
Enlargement.
almost straight line denotes a ridge starting from the
maximum boiling binary azeotrope and pointing to about
52% benzene in the chloroform/benzene edge. No extremum exists in this point in the T function along the
chloroform/benzene line. The temperature increases
monotonously from chloroform to benzene. The temperature surface is smooth around the point of arrival. Even
896 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
ETHVL
CYCLOHEXANE
a
n.
METHANOL
ACETATE
\
I
.n
ACETONE
METHANOL
b
CYCLOHEXANE
CHLOROFORM
Figure 5. Example 5.
METHANOL
\
METHANOL~23-
,
,
,
,
\
,
,
,
,
,
I
~
'
ACETONE
0
do
0.42
0.44
0.16
0 40
0.50
Figure 3. Example 3. (a) Partial map with valley lines and residue
curves. (b)Enlargement of the central part of (a).
2 -PROPANOL
CHLOROFORM
METHYL ACETATE
Figure 6. Example 6.
2 - PROPANOL
,DO
(00
0
,90
t
CYCLOHEXANE
f
*
+
063
070
om
-
BENZENE
Figure 4. Example 4.
in a finite distance from the edge, e.g., 5 % acetone, or
farther, say 20% acetone, there is no extremum temperature along a section parallel to the chloroform/benzene
edge. The ridge is perceptible only along the sections
parallel to the acetone/chloroform edge.
The dashed lines represent residue curves. All the
residue curves in this system run into pure benzene. The
two displayed residue curves are near enough the binary
saddle point to merge practically into one common line
after surpassing 30% benzene. This common line is continued in the reverse direction by a dotted line till the
binary azeotrope. This line starting from the azeotrope
and ending at pure benzene is a separatrix (though it is
76
co3
010
020
03c
CYCLOHEXANE
GLO
OM
090
100
CAR EON
TETRACHLORIDE
Figure 7. Example 7.
not an SDR boundary). The residue curve starting from
the vicinity of chloroform and near the chloroform/acetone
edge does cross the ridge line. The ridge itself does not
behave as a boundary.
In Figure l b only the ridge, the separatrix, and four
rectification profiles at total reflux are shown. All the
rectification profiles cross both the ridge and the separatrix. The distillate composition approaches pure chloro-
Ind. Eng. Chem. Res., Vol. 31, No. 3,1992 897
form for an infinite number of stages while the bottom
approaches pure benzene. The profiles run along and near
the separatrix, and they cross it at the points denoted by
numbers 1-4. A small part of the triangle is enlarged in
Figure IC. It can be seen that the crossings occur with
finite angles.
Two minimum-boiling binary azeotropes occur in the
system of methanol/heptane/toluene shown in Figure 2a.
The two azeotropes are not connected by a valley. One
valley starting from the methanol/ toluene binary minimum moves almost along the binary edge with roughly
20% slope, and at about [0.35,0.55,0.1] it turns right by
90° and runs almost straight to about 75% toluene and
0% heptane. Another valley starting from the methanol/toluene azeotrope runs almost straight along the edge
with a very small slope to about 99% methanol and 1%
heptane. Both valley floors are represented by dashed
lines. There is a boundary separatrix, represented by a
dotted line; it joins the two binary azeotropes. The smooth,
curved full lines are residue curves; the broken lines are
rectification profiles.
An interesting region is enlarged in Figure 2b. Observe
the shape of the profiles. The full points near the residue
curves play a double role. When such a point represents
liquid composition, then the tie line joining it to the
equilibrium vapor composition is tangent to the residue
m e . When the same point represents vapor composition,
then the tie line joining it to the equilibrium liquid composition is tangent to another residue curue passing
through the liquid composition point far from this residue
curve. This is the reason why these profiles are so heavily
broken.
The residue curves and column profiles cross the valley
floors. The profiles which cross the boundary are not
plotted (but exist).
There are three minimum-boiling binary azeotropes and
a minimum-boiling ternary azeotrope in the system of
acetone/methanol/cyclohexane;see Figure 3a. The valley
floors, represented by full lines, start from the binary
azeotropic points and arrive at neutral binary points denoted here by squares, and all cross at the ternary minimum. There are three SDR boundaries which are exactly
the separatrices; they join the ternary minimum to the
three binary minima. Only one of the SDR boundaries is
plotted, by a dotted line, from the ternary minimum to the
acetone/cyclohexane minimum. One of the other SDR
boundaries, that which joins the ternary minimum to the
acetone/methanol one, is only indicated by the residue
curves pointing to different stable nodal points.
The residue curves, represented here by dashed lines,
cross the valleys. The SDR boundaries are obviously
different from the valley lines, but one could ask if the
boundary is started with a slope tangent to one of the
valley lines at the ternary minimum or not. To answer this
question, the central part of the triangle is enlarged and
shown in Figure 3b. The valley lines do not exactly cross
in one point due to the inaccuracy in collecting the data
on the loci of the lines. However, they are accurate enough
to cross near the azeotrope designated here by a full point.
The slope of the separatrix is apparently different from
those of the valley floors.
A similar map is shown in Figure 4 (benzene/cyclohexane/2-propanol). One of the residue curves (full lines)
pointing to pure cyclohexane crosses two valleys. The
boundary imagined between the two plotted residue curves
crosses a valley.
The system chloroform/methanol/ethyl acetate is shown
in Figure 5. One maximum-boiling and two minimum-
boiling binary extrema are present. Correspondingly, two
dashed valley floor lines and a ridge projection are seen.
These three avoid each other and arrive somewhere at the
binary edges. On the other hand, an SDR boundary separatrix joins the two binary minima. The two region attractors are the pure methanol and the maximum-boiling
binary azeotrope.
A bipositivenegative saddle ternary azeotrope is shown
in the system methyl acetate/chloroform/methanol;see
Figure 6. The valley floor lines (dashed) and the top-ridge
line (dotted) behave as usual and meet at a single crossing
point which is a saddle. The separatrices are not plotted
but only indicated by the full line residue curves. The
residue curves starting from the vicinity of the chloroform/methanol positive azeotrope cross both a valley and
a ridge. The ridge runs along the separatrix in a long
section making it easy to confuse them.
Figure 7 shows the interesting ternary system of tetrachloromethane/cyclohexane/2-propanol.Only two binary
minima and a ternary saddle are present. The valley floors
can be easily confused with the separatrices (this happened, e.g., to Yuan and Lu (1963), also acknowledged by
Tamir and Wisniak (1978)). The ridge runs with considerable deviation from the other two separatrices. The new
fact is that the starting point of the ridge coincides with
a pure component vertex. This is a special case due to the
particular behavior of the tetrachloromethane/cyclohexane
binary system and implies interesting consequences, as
discussed in the next section.
The studied examples exhibit the following common
features.
1. Each binary positive or negative azeotropic point is
a starting point of a valley or a ridge, respectively.
2. The valley and ridge lines arrive at a binary point
on one of the edges of the triangle. There is no extremum
at the point of arrival.
3. The starting points of the valley floors and top-ridge
lines are the binary azeotrope points except in the case of
the carbon tetrachloride/cyclohexane/2-propanolsystem.
In that case the starting point of the ridge is one of the
end points of the edge. (Later we will see that it is not
an exception but a special case.)
4. If a ternary azeotrope exists in the system, then the
valley floors and/or the ridge lines all cross there.
5. The valley floors and top-ridge lines are vigorously
crossed by both the residue curves and the rectification
profiles.
6. The separatrices are not identical to the valley or
ridge lines, and apparently there is no other relation between them than that they both start from the singular
points.
7. The rectification profiles cross the SDR boundaries
in such a manner that one side is far from the boundary
and points to a nodal point while the other side runs near
the separatrix and points to that nodal point which the
crossed separatrix arrives at or is started from.
Explanations
Crossing the Valleys and Ridges. In binary mixtures
an azeotropic point is always a locus of a peak or a pit in
the T ( x )function. Azeotropic points cannot be crossed
by simple distillation or by rectification becaw that would
alter the direction of temperature change. This experience
is generalized, by mistake, in the opinion that ridges and
valleys cannot be crossed. This mistake might have been
supported by the fact that the temperature along a valley
or a ridge changes monotonously (neglecting the case of
ternary saddle points). Our ordinary experience that to
cross a valley it would be necessary first to descend and
898 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
F i r e 8. View of an inclined valley: v, valley; m. minimum; I,
crossing of the vaUey by the level line and by the BC face of the ABC
triangle.
then to ascend is a simple delusion. This is highlighted
in Figure 8.
Imagine you are walking downhill a steep slope just in
the trace line of a valley designated by the letter -v” in
Figure 8. Another person arrives from right or left, perpendicular to your direction, and he or she would first
climb down and then climb up crossing your trace. His
or her path could be B-m-A or A-m-B where m is the
point of crossing and there is a minimum of T (surface
height over sea level) along the A-m-B line in point m.
Turn from this direction by a small angle, e.g., lo, and you
shall easily c l i i b out of the valley, meanwhile unceasingly
descending. Such a line could be perhaps that along Cx-B. You could perhaps observe a second other person
who crosses the valley, in point x, while maintaining the
same height above sea level. His or her trace line assigns
the borders of possible directions suitable for climbing out
of the valley with monotonous ascending. (The same
picture applies in the opposite direction and also in the
case of crossing a ridge.)
The direction of progress is,of course, not up to you but
is assigned by the tie lines. We always have to move in
the direction of y(x) - x or x - y(x). If the direction of
the tie lines were the same as that of the temperature
gradient, then crossing the valley would be impossible
because the gradient is always tangent to the valley floor
projection. Fortunately (or unfortunately) nothing forces
the tie linea M behave in such manner. The temperature
surface is not a scalar potential function of the vector field
y ( x ) - x but only a Liapunou function.
Valley a n d Ridge Endings on t h e Binary Edges.
The same, above-outlined picture explains how a valley
or a ridge can reach a binary point of the triangle without
pointing to a stationary point. Draw an imaginary triangle
on the horizontal projection under the hillside in such a
manner that one of the edges is just a line crossing the
projection of the valley in the way described above. Draw
a vertical prism over the triangle and mark the lines of
intersection of the faces of the prism with the surface. You
shall be given a curved triangle like that marked by the
A, B, and C vertices in Figure 8. No m i n i u m exists along
the B C edge or along a section parallel with the B C edge
near it. The ualley is perceived only in other crossing
directions. Naturally, thii picture works only with inclined
valleys.
Though there is no extremum of T i n the end point,
there is usually an extremum in the first derivative of T
along the edge at that point (point x in the given example).
The derivative does not reach zero and does not change
sign but runs through a minimum.
In contrast to the general opinion, ending of a valley or
a ridge on a binary edge is not the exception but the
general rule. No general chemicophysicalrelation is known
that would force the lines to tend toward extrema or toward pure component vertices. Among the ternary system
analyzed is the tetrachloromethane/cyclohexane/2propanol, but even in that m e the ridge does not end but
starts at a vertex according to our terms. (See below in
this section.)
Scalar Potential Surface and Contour Lines. In
their “Theorem Four“ Doherty and Perkina (1978) exactly
proved that “The temperature surface is a naturally occurring Liapunov surface for the ordinary differential
equations” (eq 1). In their “Corollary two” they wrote: “It
follows from Theorem Four that an (inclined) valley...”;
see above. The key of the confusion is that a Liapunov
function is not necessarily a scalar potential function. If
it were, the tie lines would point in the direction of the
gradient of T. However, this is not the case. Evidently,
the reasoning of their ‘Corollary two” is defective. Fortunately they also recognized the error and corrected it in
the article of Van Dongen and Doherty (1984).
The question emerges naturally whether a potential
surface can somehow be constructed over the vector field
y(x) - x since in that case the contour lines would give
sufficient information about the possible directions of
movement in the composition triangle. Really, all the
equilibrium calculations are based on the general thermodynamic potential functions free energy, free enthalpy,
etc.
However, the potential functions of thermodynamics are
valid over a domain of system states and not over the
vector field represented by the tie lines. The answer to
the above question is “no”, because no thermodynamic
relation farces the tie-line uector field to satisfy the
necessary condition that the rotation (curl) of the vector
field should uanish aver the whole domain; i.e., the relation
a ~ v ~-( xi)
~ )- aCvj(x) axj
(3)
axi
for all i, j pairs is not an identity. Or, if thexe is a potential
function V over the composition triangle, then it must also
be valid over the edges. It is really possible (and easy) to
construct V over an edge:
V(X) = V(0)
+ fCv(x)
- x ) Clx
=
1
V(0)+ ~ ‘ Y C X )dr - 2 (4)
The integration constant V(0)may be chosen arbitrarily
for a binary mixture but may not be chosen independently
for the closed cycle of edges in a composition triangle. For
if the potential increment from pure component i to pure
component j is denoted by V,, then the overall increment
of the cycle must be equal to zero:
v, + v,, + v,,
=0
(5)
If follows that the sum of integrals under the equilibrium
cwvea in the McCabe-Thiele diagrams (in the appropriate
order and direction) must be equal to 1
Suppose now
that one fmds three components A, B, and C satisfying eq
5. Dropping one of them, e.g., C, and selecting another
one, D, with different properties will usually alter the
values of the integrals so that eq 5 is violated. Equation
5 is not generally satisfied.
Valleys and Ridges from the Corner. In a “normal”
case a ridge or a valley should start from an azeotropic
point. Having accepted this point of view, the situation
seen in Figure 7 is an “abnormal” one. The ridge starting
from the pure cyclohexane vertex approaches the cyclohexane/tetrachloromethaneedge, and one could think that
the ridge starts from an internal point of that edge very
Ind. Eng. Chem. Res., Vol. 31,No.3, 1992 899
I
Y
saddle
point
stable
nodal point
0.1
b.1
Figure 9. Saddle point (a) and stable nodal point (b).
A
D
G
F
Figure 10. Minimal pattern of the significant crossing.
near the pure component. One can find conflicting references in the data collection of Horsley (1973) for the
binary system in question about whether that mixture
exhibits azeotropic features.
A simple solution of this dilemma is that there is a
tangential azeotrope in this binary system at the cyclohexane end. This decision cannot be made on the basis
of the equilibrium curve plotted in Gmehling and Onken
(1982) because of the small relative volatility. The existence of a tangential azeotrope is in accordance with the
rule that the ridge should start from an azeotropic point.
Verification of the existence of tangential azeotrope in
a binary system generally with small relative volatility may
involve serious practical difficulties. Detecting the existence of a ridge or a valley starting from a corner of a
ternary system might be sometimes easier. This phenomenon provides a new method to reveal the presence
of a tangential azeotrope.
Rectification Profiles Crossing the Boundaries.
The rectification profiles easily cross the separatrices because the separatrices (being residue curves) follow the
tie-line directions continuously while the rectification
profiles move along the tie lines in discrete steps and
change the role the points play from liquid to vapor composition and vice versa (see also Figure 2b). This is clearly
explained by Doherty and Perkins (1978).
Only those separatrices are boundaries which arrive at
saddle points. The residue curves approaching the saddle
point from different sides of the separatrix are deflected
into different directions (see Figure 9a), and therefore they
arrive at different stable nodal points. If a separatrix
arrives at a stable nodal point, as in Figure 9b, the residue
curves are attracted to the same point and distillation
regions are not formed.
While from the viewpoint of SDR the nature of a sepmatrix depends only on the point of arrival, both ends of
the separatrix are signifcant in characterizing the behavior
of the rectification profiles. The distillate end is attracted
by local minima, and the bottoms end is attracted by local
maxima; therefore any nodal point is a stable attractor for
the profiles, while the saddle points are deflectors.
If either end of a separatrix or boundary is a nodal
point, then that node will attract one of the end points
of the rectification profile which crosses that separatrix;
therefore the profile will not deviate from the separatrix
to a great distance. In this case there exists another
boundary, i.e., a boundary to the rectification profiles, and
this crossing cannot be practically utilized to bridge the
region boundary. Significant crossing of the separatrix
in both directions may therefore be expected only i f the
separatrix joins two saddle points. In that case the column profiles would run as indicated by the dashed line in
Figure 10.
The distillate is attracted by the unstable nodal point
A, the bottom is attracted by the stable nodal point G, and
the profile is deflected into the appropriate directions at
the saddle points B and F. Note that all six nodes have
role in forming the pattern. Lumping A with D or lumping
A
A
B
C
a.)
b.1
Figure 11. Two nisp candidates for significant crossing.
A
B
D
C
Figure 12. Ternary saddle azeotrope with three binary minima.
C with G would lead to the formation of directed cycles,
which is forbidden to occur in residue curve maps. The
arrows A to D or C to G may not be reversed for the same
reason. The other nodes have been already discussed,
while the other directed edges not yet discussed are significant in maintaining the characteristics (being saddles
and stable or unstable nodal points) of the nodes. Another
possible rectification profile path could be C-B-F-D instead of A-B-F-G. The actual pattern depends on the
curvature of the B-F boundary (i.e., which side is convex).
Existence of such patterns cannot be excluded merely
by topology. Two theoretical cases are shown in Figure
11. In Figure l l a a possible rectification profile crossing
a boundary is A-D-F-B and another is E-D-F-C depending on the curvature of the D-F boundary. In Figure
l l b the possible paths are A-D-E-B and C-D-E-F. At
least one naturally occurring example is known for the case
of two azeotropes in a single binary mixture (the system
benzene/perfluorobenzene,reported by Chinikamala et al.
(1973));however, such a ternary system is not yet known
and must be very rare. Another example, the system dinitrogen pentoxidelwater, reported by Lloyd and Wyatt
(1955),is problematic to interpret because of the influence
of explicit chemical equilibrium in the system.
After restricting the spectrum of possible systems to the
cases of at most one binary azeotrope per edge and at most
one ternary azeotrope, the systems remaining in the scope
are the 1125,1219, 2119, 3349,3439 and 4335 maps according to the systematization of Matsuyama and Nishimura (1977). One of them (system 112s) is shown in
900 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992
A
a.1
D
C
G
b. 1
C.)
d.)
Figure 13. The four possible allocations of the vertices of Figure 10 if there is no ternary azeotrope in the system.
Figure 12. The possible rectification profile paths are
D-S-%A, D-S-E-C, D-S-F-A, and D-S-F-B according
to the starting points and the curvatures. However, all are
very unlikely to occur in nature. If three species repel each
other in such a way that three binary minima occur, then
it is rather unlikely that they form a saddle ternary
azeotrope. The same applies to the cases of three binary
maxima. Kogan (1971)excluded this possibility, saying
that in order to form a saddle at least one valley and one
ridge must meet and in that case at least’one minimum
and one maximum binary azeotrope should exist in the
system. However, against our empirical results that the
valleys and ridges start from a singular point, the opposite
case cannot be excluded theoretically. It is theoretically
possible that some ternary interactions alter the behavior
of the pairs in the ternary composition region of interest.
A ridge or a valley might emerge just in an internal point
(see also the note of Swietoslawski above on the disappearance [fade out] of valleys or ridges inside the triangle).
Of course, in these cases the type of the ternary singular
point is not an elementary one in the terms of mathematical topology.
Suppose the “abnormal” cases discussed above are excluded. The six singular nodes of Figure 10 are to be
settled in the triangle. Both B and F must be binary or
ternary saddles. A ternary saddle divides the domain into
four regions in the same triangle shape, and the six singular
nodes would have to be settled in one of these smaller
triangles with the saddle ternary azeotrope being on one
of the edges or in one of the vertices of the smaller triangle.
A vertex of a triangle cannot play the role of B or F
because, though it can be a saddle, it can be a saddle only
in such a form that one of the edges is arriving at and the
other is departing from it and all the residue curves arriving from other directions are deflected. If B and F are
both binary saddles (or we consider the shape formed by
dividing the triangle diagram into smaller triangles), then
they straddle one of the vertices and either A and D or C
and G have to be settled on that vertex and on one of the
two edges incident to that vertex (see Figure 13.) In either
case two binary azeotropes would sit on one edge what is
against the presumptions.
We may conclude that the working assumption given
by Doherty and Caldarola (1985) is well established and
is acceptable. It should, however, be pressed that this
assumption is only based on analyzing the case of single-feed rectification.
On the other hand, quantitatively small effects may
imply significant change in the final decisions, and thus
the suggestions of Larwhe et al. (1990) ought to form basis
of a more detailed analysis. Especially the case of multiple-feed rectification is not so strictly constrained as the
single-feed case, and the continuous mixing along the
column (extractive or absorptive distillation) might lead
to a radically different shape of the composition profiles.
Again the so-called “abnormal” cases cannot be excluded
exactly and may result in preferable structures in particular cases.
Conclusions
Ternary vapor-liquid equilibrium systems with homogeneous liquid phase were studied. Valleys and ridges in
the temperature surface were compared to the situation
of simple distillation residue curves and region separatxices.
The possibility of crossing the valleys and ridges by residue
curves or rectification composition profiles and the possibility of crossing the separatrices by rectification profiles
were studied. All the study was made by calculations and
theoretical analysis. The following conclusions can be
drawn.
1. Although the temperature surface is a Liapunov
surface of the residue curves, it is not a scalar potential
function over the tie lines; therefore the valleys and ridges
can be crossed along the tie lines with monotonously altering temperature. A particular case of this kind of
crossing is the crossing of the valleys and ridges along the
binary edges. Generally the valleys and ridges arrive not
at pure component vertices but at binary points where
there is not extremum in the temperature along that edge.
These results are consonant with the conclusions of Van
Dongen and Doherty (1984).
2. Not only is the temperature surface not a scalar
potential function over the tie line vector field, but such
a potential cannot even be constructed.
3. Valleys and ridges start from azeotropic points in the
*normaln case. (“Abnormal” cases, when the starting
points are inside the triangle, are not excluded theoretically.) A particular subcase within this normal case is the
starting from a pure component edge. The presence of
such a line indicates the existence of a binary tangential
azeotrope. This phenomenon might be utilized in verifying
the presence of the tangential azeotrope.
4. Valley and ridge projections in the composition triangle are different from the simple distillation separatrices
and region boundaries. The valleys and ridges are crossed
by both simple distillation and rectification because the
tie lines are not tangent to the valley or ridge lines. Even
if there are three binary azeotropes in the system, it is
possible (and usual or normal) that all the valleys and
ridges avoid each other and no one couple of them are
joined by a valley or ridge. The valleys and/or ridges meet
only if there is a ternary azeotrope; in that case they meet
at the point of azeotropy.
5. Both stable and unstable nodes are attractors to one
end of a distillation column (at an infinite number of
trays). Since one end of a simple distillation separatrix
is incident with a nodal point in the normal case, generally
one end of a rectification profile crossing that separatrix
Ind. Eng. Chem. Res. 1992,31, 901-908
is attracted to that nodal point and only the other end can
depart from the separatrix line to a significant distance.
(This other end not only can but will depart because a
residue line joining two nodal points is not a separatrix.)
In order to depart from the crossed separatrix significantly in both directions, forming of such a pattern is
necessary that the crossed separatrix join two saddle points
as in Figure 10. The existence of such a pattern is not
theoretically excluded, and some candidate combinations
are presented in this article. However, occurrence of such
a pattern in nature must be at least rather rare (if anyhow
possible), and therefore the working assumption of Doherty
and Caldarola (1985) based on the rareness of such cases
is well established.
Nomenclature
A to G and S = singular pointa in the composition triangle
t = time or any arbitrarily selected parameter
T = temperature of phase equilibrium
V = scalar potential
V , = liquid volume parameter in the Wilson model
x = liquid composition (mole fraction)
y = vapor composition (mole fraction)
Indices
b = bubble point
d = dew point
i and j = components
n = stage number
Abbreviations
SDR = simple distillation region
Literature Cited
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Received for review May 1, 1991
Revised manuscript received September 4, 1991
Accepted November 14,1991
Modeling Zero- Gravity Distillation
Eduardo
A.Rarnirez-GonzBlez*
Departamento de Zngenieda Qdmica, Centro de Znuestigacidn e n Quimica Aplicada, Apartado Postal 379,
Saltillo, Coahuila 25100, Mexico
Carlos M a r t i n e z a n d J e s h Alvarez
Departamento de Zngenierh de Procesps e Hidrbulica, Universidad Autdnoma Metropolitana- Unidad
Zztapalapa, Avenida Michoacdn y Purisima, Colonia Vicentina, Apartado Postal 55-534, Mixico, D.F. 09340,
Mexico
Zero-gravity distillation (ZGD) is a recently reported operation, where capillarity is the basic driving
mechanism. The operation name is due to the fact that gravity forces do not act on the separation
process. Previous laboratory-scale experimental work has demonstrated operation feasability. In
this work, ZGD is conceptualized in terms of first principles. The resulting model consists of two
partial differential equations for liquid and vapor compositions coupled to algebraic equations for
gas-liquid interface equilibrium, mass flux,and bubble-point conditions. The model exhibits the
interplay of capillarity, mass transfer, and phase equilibrium. With a standard finite-element
technique, the partial differential equations are discretized, and the resulting algebraic system is
solved with a standard method. Good agreement with reported experimental data is obtained.
1. Introduction
Because of ita impact on plant energy requirements,
distillation is a current subject of research along three main
directions: (i) energy integration schemes, (ii) improved
design of distillation apparatus, and (iii) new ways of
carrying out separations. Integration schemes (Andreco-
0888-588519212631-0901$03.00/0 0 1992 American Chemical Society