COMPUTER VISUALIZATION OF
BINARY, TERNARY, AND QUATERNARY
FLUID-PHASE EQUILIBRIA
Kong S. Tian and Kenneth R. Jolls
Assisted by Richard J. Campero
Chemical Engineering Department
Iowa State University
Copyright © 1998
Ported from Unix to Windows by
Jasper Yen, Hector Perez, and
Walter G. Chapman
Chemical Engineering Department
Rice University
Copyright © 2000
Reviewed by Science magazine, Oct. 15, 1999, p. 430, “An Eye for the Abstract.”
The 3D graphics applications in this tutorial were written using Open Inventor, an object-oriented 3D
toolkit developed by Silicon Graphics, Inc., and source-licensed to Template Graphics Software, Inc.
1
FOREWORD
This presentation was first given as part of the final examination of Kong S. Tian for
the degree of Master of Science in Chemical Engineering at Iowa State University,
November 1997.
Some of the images shown here were also part of a talk given by Kenneth R. Jolls –
"Fluid-phase equilibria from a chemical process simulator“ – at the annual meeting of
the American Society for Engineering Education, Milwaukee, WI, June 1997.
Details concerning the computation of phase-equilibrium results and the construction
of various graphical elements in this work are contained in Kong Tian's thesis (under
the same title), which may be obtained from Iowa State University through interlibrary
loan.
Mr. Tian is with Cashette, Inc., Fremont, CA.
2
FOREWORD (Continued)
This presentation was ported from Silicon Graphics hardware to a PC/Windows
environment by Jasper Yen, Hector Perez, and Professor Walter G. Chapman of the
Chemical Engineering Department at Rice University. Jasper (a sophomore
engineering student at Rice University) wrote new display routines for each 3D
graphic using C++ and Open Inventor.
Jasper, Hector (a senior engineering student at Rice), and Professor Chapman
created the PowerPoint presentation. This environment allows the addition of display
features that improve readability of the package.
New features that were added to the Phase software by Jasper and Professor
Chapman include cutting planes for the first binary system. This allows the user to
make constant temperature and constant pressure sections through the shaded
bubble and dew surfaces of the binary phase diagram. This drawing appears on the
cover of Introduction to Chemical Engineering Thermodynamics, 6th edition, by
Smith, Van Ness, and Abbott, McGraw-Hill (2001).
3
TABLE OF CONTENTS
INTRODUCTION
GRAPHICS DETAIL
OBJECTIVES, THEORY
QUATERNARY SYSTEMS
COMPUTATION, GRAPHICS
CONCLUSIONS
BINARY SYSTEMS
REFERENCES
TERNARY SYSTEMS
ACKNOWLEDGMENTS
TERNARIES WITH AZEOTROPES
A FINAL WORD
4
INTRODUCTION
• Computer visualization is an important aid to understanding chemical
thermodynamics.
• Phase diagrams show the effects of temperature, pressure, and composition on
the physical behavior of chemical systems.
• Two-dimensional phase diagrams are:
Familiar.
Easy to construct.
But limited to only one independent variable.
• Three-dimensional phase diagrams:
Require more thought.
Are more difficult to construct
But can show the effect of a second independent variable.
5
OBJECTIVES
• In this work we sought to develop ways to obtain phase-equilibrium data in large
quantities from the ASPEN PLUS chemical process simulator and to configure
those data into formats compatible with standard graphical software packages.
• We have also tried to demonstrate how one can visualize the phase behavior of
multicomponent fluid systems (binaries, ternaries, and quaternaries in vapor-liquid
and liquid-liquid equilibrium) by using fixed and animated three-dimensional
drawings.
6
THEORY–1
All phase-equilibrium data in this project were generated by the equation-of-state
method in which the properties of coexisting states are determined from a single
algebraic form. The model chosen here was the Peng-Robinson equation, a cubic
form patterned after the van der Waals equation but incorporating the Pitzer
accentric factor to account for non-spherically symmetric molecules [D.-Y. Peng and
D. B. Robinson, Ind. Eng. Chem. Fundam., 15, 59 (1976)].
Standard mixing rules were used to generate mixture properties, and binary
interaction parameters were employed to bring the predicted equilibria into the best
possible agreement with experimental data.
7
THEORY–2
Solving the Peng-Robinson equation was accomplished by using various routines
within the ASPEN PLUS chemical process simulator, in particular, the FLASH2 block
(for simple VLE), the DECANTER block (for LLE), and the FLASH3 block (for threephase, liquid-liquid-vapor equilibria).
Other ASPEN PLUS utilities (Sensitivity block, Design-Spec, Regression,
Optimization) were used at various times during the project.
Experimental data (for comparison) were obtained from the DECHEMA Chemistry
Data Series and from an American Chemical Society Symposium Series monograph
by L. H. Horsely on the subject of azeotropes.
8
COMPUTATION
All computing was carried out on Project Vincent (DECstation) and Silicon Graphics
IRIS workstations at Iowa State University.
GRAPHICS
The visualization software used in this work was MOVIE.BYU (from Brigham Young
University) and Open Inventor (from Silicon Graphics and from Template Graphics
Software, Inc., for the PC/Windows environment). In this public presentation only
Open Inventor images will be shown. MOVIE.BYU images may be seen in the
papers from The Chemical Engineer and from Physics Today mentioned in the
references.
9
BINARY EQUILIBRIA
Vapor-liquid equilibria are often represented on
two-dimensional plots, such as isothermal
pressure-composition diagrams or isobaric
temperature-composition diagrams. In the first
drawing we see three isothermal P-x,y sections
for the system cyclohexanone-cyclohexane at
the temperatures 450 K, 500 K, and 550 K.
Dew-point curves are drawn with fine lines and
bubble-point curves are shown in bold.
The second drawing shows three isobaric T-x,y
sections for this system taken from the same
region of the phase diagram at 10 bar, 15 bar,
and 20 bar.
10
3D help
Two-dimensional plots are useful for
obtaining numerical results, but taken
individually they do not show the
complete character of binary equilibria.
To convey the three-dimensional nature
of such functions, we should plot them
in pressure-temperature-composition
space.
The bubble-point locus and the dewpoint locus become surfaces in this
space that connect along the sides of
the diagram to form the vapor-pressure
curves and at the upper end to form the
critical curve(s).
Click on the drawing to maneuver it.
This is the complete P-T-x,y diagram for cyclohexanone/cyclohexane. Click on it for the
maneuverable diagram. Then use the Boil and Dew switch buttons to toggle the surfaces
on/off. Using the pointer, click and drag the arrow to move the cutting planes. Note the
connection between the sectioning of the 3-D surfaces and the 2-D plots on the previous
page.
11
The drawing shows isothermal
sections (solid) and isobaric
sections (dashed) through the
bubble- and dew-point surfaces of
this binary system. Bubble-point
curves are red and dew-point
curves are green. Vapor-pressure
curves are white.
White "tie lines" connect the
intersections of isothermal and
isobaric loops and indicate liquid
and vapor states that can coexist in
equilibrium.
The upper three isotherms are for
temperatures greater than the
critical temperature of cyclohexane
and are incomplete near their points
of attachment to the (yellow) critical
curve. This situation will be
mentioned again in a later slide.
* These data were obtained by using the
FLASH2 computing block in ASPEN PLUS
12
INTERPRETING PHASE DIAGRAMS
To understand these drawings it is helpful to think of the experiments needed to produce
the data that they show. Imagine starting with pure cyclohexanone as a liquid at a
particular temperature (say 500 K) and at a pressure somewhat above the white vapor
pressure curve (marked "start"). Then add increasing amounts of cyclohexane at the
same conditions (P,T). The mixture moves along the directed yellow line, remaining a
liquid until the composition crosses the bubble-point curve (at B), after which a vapor
phase appears. Adding still more cyclohexane moves the composition toward the dewpoint curve (at D) where the liquid phase disappears completely. Higher cyclohexane
compositions at this temperature and pressure yield a vapor phase only.
13
BINARY AZEOTROPES
If the two components of a binary
mixture have about the same vapor
pressure and/or the mixture exhibits
sufficiently nonideal behavior, an
azeotrope (a constant-boiling
mixture) is frequently the result.
The following image gives the P-Tx,y phase diagram for the system
cyclopentane-acetone, in which
azeotropic behavior (see the solid
yellow line) persists all the way to
the critical curve. This is a
minimum-boiling (maximumpressure) system. The azeotrope is
shown here at compositions in the
vicinity of 0.5 mole fraction, but at
lower temperatures it moves toward
the cyclopentane side of the
diagram.
14
A TOUCHING SITUATION
The fine structure of the azeotrope
is easier to see if we enlarge the
previous drawing and show a
smaller range of temperature and
pressure. Observe how the order
of volatility changes for
compositions to the left and right of
the azeotropic points. Near the
cyclopentane side the vapor phase
(green) is richer in acetone than
the liquid, but toward the acetone
side the reverse is true.
Note also that the pressure axis on
these binary drawings is nonlinear.
This magnifies the low-pressure
region and makes the conditions in
that range easier to distinguish.
15
3D help
Even more detail can be seen if we rotate the three-dimensional diagram so that we
can view it from any direction. Particularly important are the two-dimensional (flat)
projections from the front, top, and sides.
The graphics software used for these drawings, Open Inventor, permits rotation,
zooming, and sectioning interactively. The model may be moved in any direction
using the mouse, but precise positioning is more easily achieved by turning the
vertical and horizontal thumb-wheels marked Rotx and Roty. Remove the
perspective by clicking on the bottom icon to produce the two-dimensional P-x,y
view, then Rotx (down) to get the T-x,y view. Zoom to fill the screen and observe
the slight movement of the azeotropic line toward the cyclopentane (left) side of the
diagram as the temperature decreases.
movable diagram
16
TERNARY SYSTEMS
Fluid-phase equilibrium diagrams for ternary mixtures are four-dimensional, so one
variable must be fixed in order to have a three-dimensional figure to plot. For most
of our examples we have chosen to fix the temperature, and thus in those cases we
show isothermal pressure-composition diagrams or "prisms."
Ternary compositions are conveniently shown on a triangular plot with two
independent compositions and the third determined by difference.
The drawings show triangular plotting areas in Cartesian (right-angle) and
equilateral form.
17
3D help
A TERNARY COMPOSITION PRISM
We can plot a single dependent variable (for
example, the pressure) in the prism space above
a ternary composition triangle, thus letting us
construct isothermal, pressure-composition
surfaces for both dew-point and bubble-point
states. Such a display is shown in the figure on
the right for the ternary system benzene, normal
pentane, and 2-methylpentane. Red represents
bubble-point states, green represents dew-point
states, and (faint) white tie lines connect three
pairs of coexisting equilibrium phases at a
common pressure of 0.3 bar. The bubble-point
surface is drawn in transparent red so that the
dew-point surface and the tie lines can be seen
beneath it.
More details are visible if we look at the movable
ternary prism. Click on the image and then add a
slight tilt by moving Rotx (down). Next rotate the
model using Roty (left) to see, first, the 2methylpentane/ pentane binary face, then the
pentane/benzene face, and finally the benzene/2methylpentane face. Look between the two
surfaces to see the three white tie lines, each at a
pressure of 0.3 bar. Note how the ternary
surfaces converge to the binary P-x,y curves for
this temperature as the various compositions go to
zero at the faces of the prism.
Click on the drawing.
* It may be easier to distinguish these features in
the movable 3-D view if you switch the dew- and
bubble-point surfaces alternately from transparent
to opaque by clicking on the dew and bubble
buttons.
18
INTERSECTIONS
Open Inventor graphics software
can perform sectioning and thus
detect the intersections between
surfaces in three-dimensional
space. In this drawing an isobaric
plane is added to the prism so as
to show, by its intersections with
the bubble- and dew-point
surfaces, bubble- and dew-point
curves for the ternary mixture
corresponding to the temperature
of the prism and the specific
pressure of the plane.
First, this is done with the static
composition prism of the previous
image. The plane is drawn at a
pressure of 0.5 bar.
19
3D help
ANIMATION
Now we move the plane up and down interactively, which causes its intersections
with the equilibrium surfaces to move up and down along with the varying pressure.
By viewing the prism from above, at the same time as we view it from the side, the
pressure dependencies of the bubble- and dew-point curves, for the fixed
temperature of the prism, are more easily observed. Create the movable diagram.
The right hand drawing is a two-dimensional image taken from above the cutting
plane. Use the pick arrow to drag the slider, change the pressure, and raise and
lower the plane.
movable diagram
Color in these drawings makes them not only more interesting to look at but also
easier to interpret quantitatively. In the 2-D view, moving diagonally upward from left
to right, the three colors denote first the liquid phase, then the liquid-vapor
region, and finally the vapor phase. Moving the isobaric plane to simulate
pressure changes gives us an idea of how the system behaves and where the
equilibria lie.
Computer graphics gives us access to a wide variety of visualizing attributes that can
both enhance the appearance of an image and also make it more useful as a source
of physical information.
20
3D help
AN ISOBARIC PRISM
The same ternary mixture of benzene,
normal pentane, and 2-methylpentane can
be shown also in a constant-pressure,
temperature-composition prism. The upper
surface (transparent green) now represents
dew-point states and the lower surface
(opaque red) gives bubble-point states.
Here, we show a fixed isothermal
intersecting plane. Click on the drawing for
a movable pair of images. The plane can
then be raised and lowered so as to show
the influence of temperature on vapor and
liquid compositions at a given constant
pressure. As you study the 2-D view,
moving right to left, remember that you are
seeing, first, the plane itself and, then, the
plane through the transparent dew-point
surface, and finally the bubble-point surface
through the dew-point surface.
Click on the drawing.
21
LIQUID IMMISCIBILITY
Now we move to a different ternary
system and construct the same kind of
isothermal, pressure-composition
prism as we drew before. The
components are normal pentane,
nitrobenzene, and isobutane. First, we
obtain the data in the same way as
before - by using the FLASH2 block in
ASPEN PLUS on the assumption that
we will find nothing but the usual
vapor-liquid equilibria.
The results appear in the drawing to
the right. Because nitrobenzene is so
much less volatile than either
isobutane or normal pentane ( by two
orders of magnitude), the dew-point
vapor (the green surface) contains
very little nitrobenzene except at the
lowest pressure shown.
22
SURPRISE!
But there seems to be something strange happening on the red surface. The bubblepoint curve on the pentane-nitrobenzene face of the ternary prism has a wave in it, while
the previous binary bubble-point curves that we have seen were all monotonic. In the
wavy region tie lines intersect the bubble-point curve three times and pass outside the
closed loop. How can that be?
Let's examine this further using ASPEN PLUS: precisely what does the pressurecomposition diagram for the pentane-nitrobenzene binary mixture look like at this
temperature? The P-x,y diagram generated by FLASH2 is shown on the next page.
23
HILLS AND VALLEYS
Thermodynamic stability theory tells
us that such behavior cannot occur.
A continuous binary bubble-point
curve with these characteristics is
forbidden by the Second Law of
Thermodynamics. Instead, this
system will split at a unique
pressure into two immiscible phases
in liquid-liquid equilibrium. One
phase will be rich in pentane and
one rich in nitrobenzene. And
because of the location of the dewpoint curve, there will also be a third
equilibrium phase consisting of a
pentane-rich vapor.
The FLASH2 block in ASPEN PLUS cannot make this distinction by itself. We have to run
the simulation, apply sound thermodynamic reasoning to the results, and then make any
further adjustments that are needed.
24
THREE-PHASE EQUILIBRIUM
To solve the problem, we use
another ASPEN PLUS
computational block, FLASH3,
which anticipates a vapor phase
plus two liquid phases from a
flash operation. We specify a
binary feed stream of
intermediate stoichiometry, fix the
outlet temperature of the flash
block to be that of the isothermal
prism (398.2 K), and extract just
enough heat from the flash
process to cause three phases to
form. The results for the normal
pentane-nitrobenzene binary
system are shown at the left.
The horizontal line is called a three-phase tie line, and it shows the compositions of the
three fluid phases that can coexist in equilibrium at this temperature. Using the PengRobinson equation with common mixing rules the pressure of the three-phase line is
predicted to be 8.62 bar.
25
By adding increasing amounts of
isobutane to the binary mixture and
repeating the procedure with FLASH3,
we are now able to describe the
complete three-phase equilibrium region
for the isothermal ternary system. In the
drawing on the right each pair of yellow
tie lines connects two immiscible liquid
states (liquids 1 and 2) to an equilibrium
vapor state (just visible under the
transparent bubble-point surface). The
original "wave" has been cut out of the
red surface and replaced by the
saturated liquid curves and a group of
these tie lines, each representing a
distinct three-phase pressure.
The three-phase region closes in a
critical point (shown by the symbol "C" in
the drawing), but FLASH3 cannot
determine that point, and we terminate
the region in a final tie line close to (but
slightly short of) the actual critical
condition.
26
3D help
Rotating the diagram and zooming in to look at these states in greater detail may
help clarify the nature of the equilibria, particularly in the three-phase region.
Note especially that each of the connected pairs of tie lines (LLE, VLE) corresponds
to a particular three-phase pressure and that this pressure rises monotonically with
isobutane concentration. For a ternary system having three phases in equilibrium,
this behavior is predicted by the Gibbs Phase Rule:
f=c-p+2=3-3+2=2
The two degrees of freedom in this case are the temperature (a fixed variable for the
entire drawing) and the changing isobutane concentration.
movable drawing
27
To help define the two-phase liquid
region – what we often refer to as a
"miscibility gap" – let us first construct
some constant-pressure sections (again
in yellow) in the liquid phase(s) above
the bubble-point surface. Each section
shows saturated liquid curves for liquids
1 and 2 and a group of tie lines, each of
which connects an equilibrium pair of
states between the two. The
DECANTER block in ASPEN PLUS was
used for these liquid-liquid equilibria
(LLE) determinations.
The two-phase liquid region extends to
very high pressures, but the ability of the
Peng-Robinson equation to represent
those states becomes poorer as the
pressure rises. To obtain better
accuracy, one would have to use a more
complex equation of state and/or
different mixing rules to represent the
actual fluid-phase mixture more closely.
28
CRITICAL POINTS
The isobaric saturation curves for the two
immiscible liquids meet in a line of LLE
critical points. In the laboratory we could
find such points by changing the
compositions of the two liquids (at
particular pressures) so that they move
toward one another in the phase
diagram. If this were done carefully, the
two immiscible phases would coalesce
into a single phase at an exact critical
point for each pressure.
Phase-equilibrium algorithms, such as the
ones in ASPEN PLUS, cannot do this –
either they produce distinct equilibrium
phases or they fail. To determine critical
points computationally (and thus construct
a critical line), we must use a different
technique that applies the critical criteria
derived from stability reasoning. One
such method was developed by
Heidemann and Khalil [AICHE J, 26, 5,
769 (1980)]. That scheme is used here.
29
LOOKING AROUND CORNERS AND UNDER THINGS
The previous image is now repeated as a movable model (next page) so that it can
be inspected from all angles. You may find it interesting to turn, tilt, and zoom the
drawing so that you are looking at the light-blue LLE critical line from the direction of
the isobutane vertex but beneath the red bubble-point surface. The critical line
protrudes downward into the VLE space, while (for equilibrium states) it should
really stop at the intersection with the 3-phase line.
The program that determines the LLE critical states doesn't know whether a given
point is above or below the 3-phase region, so like the wave in the bubble-point
surface, we must cut off the critical line using sound thermodynamic reasoning.
30
3D help
GETTING TO THE BOTTOM OF IT ALL
Phase equilibrium results when the governing thermodynamic potential [here the
Gibbs (or free) energy] attains a minimum value for coexisting phases rather than for
a single phase alone. This tendency is a consequence of the entropy maximum
principle and one of the foundation ideas in classical thermodynamics. At pressures
above the three-phase line (and for compositions inside the miscibility gap)
coexisting liquid states give the lowest value, but for conditions below the line a
coexisting liquid and vapor produce the minimum. At a precise three-phase point all
three fluid states can coexist, and equilibrium is guaranteed by a uniform chemical
potential (composition derivative of the Gibbs energy) for each component across all
phases. These ideas are expressed clearly in the text by Modell and Reid (now
Modell and Tester) cited in the references.
movable diagram
31
3D help
AZEOTROPES IN TERNARY SYSTEMS
Ternary systems also form
azeotropes when their pure
components are of a similar
volatility or mix with significant
nonideality. One such example is
the system acetonitrile, benzene,
acetone. The first two
components form a binary
azeotrope at 333.2 K, while the
two binaries with acetone are
"relatively ideal.“
Rotate and zoom the movable
model so that you can look
between the red and green
surfaces. Note that the azeotrope
occurs only in the acetonitrilebenzene binary mixture and
disappears with the addition of the
least amount of acetone.
Click on the drawing to maneuver it.
32
RISING ALL AROUND
If we replace acetone in the previous system with ethanol, all three binary pairs form
maximum-pressure azeotropes, and the full ternary mixture becomes similarly
azeotropic at a central composition. The following two fixed images show different
views of the isothermal, pressure-composition prism for this new, interesting vaporliquid equilibrium situation. Note that the ternary azeotrope occurs at the highest
pressure generated by the system at this temperature.
As in the earlier drawings, the bubble-point surface is rendered in transparent red
and the dew-point surface in opaque green.
33
34
3D help
As before, we can pass a movable, isobaric plane through this model to show the
behavior of the bubble- and dew-point curves with varying pressure. For any given
pressure the two curves are connected by tie lines, although none have been drawn
in these figures. Remember to enable the movable plane by clicking on the arrow
symbol in the upper right-hand corner.
Note in particular how the separate bubble- and dew-point sections become joined
as the pressure rises. At a pressure just below the maximum point (where the
ternary azeotrope occurs), the final connection is made, and the bubble-point (outer)
and dew-point (inner) curves each become closed loops with the azeotropic
composition in the center.
With both surfaces clicked as transparent, the colors seen in the 2-D view represent
the following:
blue –
liquid phase
pink –
liquid and vapor in equilibrium
green – vapor phase
movable diagram
35
3D help
LOOKING THE OTHER WAY
Holding the pressure constant instead
of the temperature gives an inverted
geometry that shows this system to
have both binary and ternary
azeotropes that are "minimum-boiling."
We draw the fixed image for a
pressure of 0.6 bar. Click as before to
display a movable pair of drawings that
show an isothermal intersecting plane
and the corresponding bubble-point (T,
X1, X2) and dew-point (T,Y1,Y2) curves
in 3-D and in 2-D projection.
In this case the curves become
connected as the plane (and thus the
temperature) moves down. By carefully
moving the slider you can detect the
minimum by noting the point at which
the curves disappear from the 2-D
drawing – at approximately 326.7 K.
Click on the drawing to maneuver it.
36
BENDING OVER BACKWARDS
If a ternary system contains binary azeotropes of opposite sign, the resulting phase diagram
can be even more complicated and show what is often called a saddle azeotrope. The
system acetone, chloroform, ethanol behaves in this way, and the bubble- and dew-point
surfaces wind around each other to form saddle shapes that touch at a single, interior
azeotropic point. Again we show alternate views of the isothermal, pressure-composition
prism in these two fixed drawings. The ternary azeotrope is visible in both but is marked only
in the one on the left.
Click on the drawing to maneuver it.
3D help
37
EXTREMUM POINTS - OPTIMIZATION
The interior azeotropic points in the previous ternary systems were found by the
Optimization routine in ASPEN PLUS. In the first examples (with all positive
azeotropes), the global maximum in the pressure or minimum in the temperature
was sought. A similar result could also be obtained by seeking a minimum in the
pressure (or temperature) difference between the bubble- and dew-point functions,
i.e., by finding the point where the surfaces touch after excluding the sides and
edges of the prism from the search region.
For the saddle azeotrope only the latter method could be used.
38
A WORD ABOUT THE GRAPHICS
The surfaces seen in these drawings were produced from large arrays of ASPEN
PLUS-generated data points that were later connected into triangular graphical
elements. It may be interesting to look at these connected arrays of points in the
absence of the smooth shading effect created by the graphics software.
The following three static images show these arrays for the bubble and dew-point
surfaces (first separately and then together) for the maximum-pressure ternary
azeotrope seen in the earlier slide.
39
40
41
42
3D help
POINTS INTO POLYGONS
The following movable image of the dew-point surface can be zoomed and rotated
so that the fine detail of the data-point interconnections can be studied. Points were
obtained from the simulator in a repetitive manner by fixing one concentration,
incrementing a second, and using the FLASH2 block to determine the pressure for
the dew-point and bubble-point functions separately at the fixed temperature of the
prism. A triangulation algorithm then generated the spatial connections between the
computed points. Click on the lower right-hand icon to change from points to
triangles.
You may choose to look either at the triangulated surface or at the actual data points
themselves. Use Rotx (or the hand cursor) to view the drawing directly from above,
then zoom to see the distribution of composition data.
movable diagram
43
QUATERNARY SYSTEMS
Four-component systems are
essentially at the limit of our ability to
show fluid-phase equilibria through
fixed computer graphics. Representing
such systems completely requires a
five-dimensional display. To get a
three-dimensional structure for plotting,
we usually fix the temperature and
pressure and show the remaining
composition variable in a tetrahedral
space. All that can be usefully drawn
are the two sets of quaternary
compositions that make up the bubbleand dew-point surfaces for the specific
temperature and pressure chosen.
To understand the tetrahedral diagram
look first at these figures that represent
the quaternary system A,B,C,D. Then
continue reading for more details and a
pictorial drawing of a quaternary
composition point.
44
The figure on the left has three
identical, mutually perpendicular,
triangular faces (I,II,III) and a fourth
face of a different size that angles
across the Cartesian coordinate
system. Compositions in this "right"
tetrahedron are given by the XB, XC,
XD values of any interior point, with XA
determined by difference.
In the figure on the right the B and C
vertices have been repositioned so as
to make all of the triangular faces
equal in area. This is the threedimensional equivalent of what was
done on page 17 with the ternary
composition triangle to change it from
right to equilateral. In the equilateral
tetrahedron, compositions are
measured as distances along vectors
perpendicular to each face and
passing through the opposite vertex.
Taken as fractional distances through
the tetrahedron, these four quantities
always sum to unity.
45
THE PLANE FACTS
In the drawing on the right we show this
pictorially. The equilateral tetrahedron is
cut by three oversized, colored planes of
constant composition (Xj=0.2). Each plane
is perpendicular to the vector (not drawn)
along which its composition is measured.
The planes intersect at the interior point
marked, and that point represents a
quaternary mixture of composition
XB = XC = XD = 0.2, XA = 0.4, at the fixed
temperature and pressure of the diagram.
Since all three spatial dimensions are used
to show the composition of the system,
there is nothing left to represent any
additional thermodynamic properties. Thus
we can show only whether a point
corresponds to a homogeneous mixture (by
coloring it) or a multiphase region ( by
leaving it blank.) In the following
quaternary examples we color only those
mixture compositions that are precisely
bubble- or dew-point states.
46
A SIMPLE QUATERNARY
The system benzene, acetone, normal pentane, normal butane forms a quaternary
mixture at 298.2 K, 0.5 bar, in which there are neither azeotropes nor immiscible
liquids. To obtain VLE data for this system, we fix the temperature and two
compositions and program ASPEN PLUS (using Design-Spec) to find the value of
the third composition that gives the fixed pressure of the tetrahedron, first for
bubble-point states and then for dew-point states.
The fixed pressure that we choose must lie between the vapor pressure of normal
butane (the most volatile component) and that of benzene (the least volatile
component) at the fixed temperature of the plot.
The results are shown in the next image.
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DETAILS
All points in the tetrahedron below
the four-sided, red bubble-point
surface represent homogeneous
liquid states; all points above the
four sided, green dew-point
surface represent homogeneous
vapor states. The dashed white
tie lines connect saturated liquid
and vapor quaternary
compositions that can coexist in
equilibrium at the temperature and
pressure of the diagram. The
dashed blue tie lines connect
binary states that are in vaporliquid equilibrium at these
conditions along four of the six
edges of the tetrahedron.
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CHANGING THE PRESSURE
As the pressure changes within
the stated limits, the bubble-point
and dew-point surfaces move
through the tetrahedron so as to
show how the compositions of the
equilibrium phases depend
(isothermally) upon the pressure.
By looking at an animated series
of such drawings, one might get a
sense of the four-dimensional
pressure-composition functions
[P=f(XA,XB,XC)T, DP or BP] that
characterize these equilibria.
As the pressure falls to the vapor
pressure of acetone (0.29 bar) the
red and green surfaces join at the
acetone vertex, and at still lower
pressures they retract from that
point toward the benzene vertex in
the form of separate, three-sided
surfaces, with binary VLE
remaining along only three of the
six edges.
49
As the pressure rises to 0.68 bar a
similar situation develops at the
pentane vertex. The four-sided
surfaces join at that point and then
retract toward the butane vertex
as separate three-sided surfaces.
By viewing a time series of these
isothermal tetrahedrons, each
drawn for a different pressure, this
complicated movement could be
understood more easily.
Animation introduces time as an
independent display variable that
can portray an additional physical
quantity (here the pressure). With
sufficient data, computing power,
and memory, this effect could help
us visualize these movements and
let us comprehend the hyperdimensional relationships that they
represent.
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A COMPLEX QUATERNARY
For the final example, we choose a quaternary system in which three of the six
binary sub-mixtures have maximum-pressure azeotropes and the remaining binaries
are relative ideal. Acetonitrile, benzene, ethanol, and acetone form such a system
at 333.2 K.
Three of the four ternary sub-mixtures have a single binary azeotrope (as in an
earlier drawing), and we show these ternaries in the next slide. Each isothermal
prism is intersected by a plane at a pressure of 0.6 bar.
51
52
The remaining ternary contains all three binary azeotropes and gives a phase
diagram similar to the maximum-pressure (minimum temperature) model shown
earlier in this presentation. As in the previous screen, the prism is intersected by a
plane at 0.6 bar, lower than any of the azeotropic pressures.
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GETTING IT ALL TOGETHER
Phase-equilibrium information from
the ternary sub-mixtures provides
the boundaries for ASPEN'S
quaternary calculations, which in
turn give the complete bubble-point
and dew-point surfaces within the
tetrahedron. Because of the three
binary azeotropes, there are also
three separate pairs of bubble- and
dew-point surfaces.
The complete tetrahedron with the
three pairs of vapor-liquid
equilibrium surfaces is shown here,
again with bubble-point states red
and dew-point states green.
Dashed white tie lines identify
quaternary VLE pairs for each
section, and dashed blue tie lines
show binary equilibrium states
along the edges.
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BRINGING PRESSURE TO BEAR
Now we raise the pressure until the
first binary (acetonitrile-ethanol)
becomes precisely azeotropic and
redraw the diagram. The phaseequilibrium surfaces move inward and
connection is made at the azeotropic
point along the acetonitrile-ethanol
edge of the tetrahedron.
This connection implies the same
physical situation that we saw (in
principle) on pages 14 and 15 and
(directly) on page 52 – the vapor and
liquid phases of the acetonitrile-ethanol
binary [within the ternary (within the
quaternary)] have the same
composition (about 58 mole percent
ethanol) at 333.2 K and 0.638 bar.
Understanding multi-component phase
relationships is often made simpler by
knowing the phase behavior of the
various sub-mixtures of the system.
55
With further increases in pressure, all of the bubble-point surfaces ultimately
connect (as do all of the dew-point surfaces), and the ensemble moves through the
tetrahedron toward the acetone vertex. At pressures greater than the vapor
pressure of acetone (at this temperature), all VLE disappears and the interior of the
tetrahedron represents homogeneous liquid states only.
A single isothermal-isobaric tetrahedron gives very limited information about this
complex quaternary system. But with time representing either temperature or
pressure in an animated sequence of such drawings, a more complete
understanding of these higher dimensional equilibria could be obtained.
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CONCLUSIONS
 In this project we have applied high-performance computer graphics to
the representation of complex, multicomponent, fluid-phase equilibria
as predicted from a cubic equation of state.
 We have proved that such data can be generated in the large quantities
needed for computer visualization by using the ASPEN PLUS chemical
process simulator for repetitive calculations.
 We have shown that complex equilibria are best interpreted through an
understanding of their lower dimensional components.
 We have suggested that time can be used to model an independent
variable in an animated display of a hyperdimensional thermodynamic
function.
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OTHER EXAMPLES
There are many other kinds of phase behavior to be shown through computer
visualization. We have chosen examples that merely dramatize the rich variety of
fluid-phase relationships that one encounters in chemical thermodynamics.
At the low-temperature end are the many types of fluid-solid equilibria that are often
of interest to materials scientists. At the other extreme are the various classes of
fluid-phase critical behavior described, for binary systems, by van Konynenburg and
Scott [Philos. Trans. R. Soc. London, 298, 495 (1980)]. The latter subject was
explored using an equation-of-state method and vector-graphics visualization in a
project at Cornell University by Charos et al. [ Fluid Phase Equilibria, 23, 59
(1985)].
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REFERENCES
Examples of other computer-generated thermodynamic drawings may be found in the following
references:
Jolls, K. R., and G. P. Willers, Cryogenics, 18, 6, 329 (1978).
Jolls, K., J. Burnet, and J.T. Haseman, Chem. Engr. Educ., XVII, 112 (1983).
Jolls, K., "Research as an Influence on Teaching," J. Chem. Ed., 61, 5, 393 (1984).
Walas, S.M., Phase Equilibria in Chemical Engineering, Butterworth, 4, 5, 116 (1985).
Jolls, K. R., Chemical Engineering Progress, 85, 2 (1989).
Jolls, K. R., "Gibbs and the Art of Thermodynamics," Proceedings of the Gibbs Symposium,
D. Caldi, G. Mostow, eds., Amer. Math. Soc., 293 (1990).
Jolls, K. R., M. C. Schmitz, and D. C. Coy, "Seeing is believing: a new look at an old subject,"
The Chemical Engineer, May 30, 42 (1991).
More on the next page
59
Schmitz, M. C., "Visualizing thermodynamic concepts through high-performance computer
graphics," M.S. thesis, Dept. of Chem. Engr., Iowa State University (1991).
Jolls, K. R., and D. C. Coy, "Gibbs's models visualized," Physics Today, 96, (March 1992).
Coy, D. C., "Visualizing thermodynamic stability and phase equilibrium through computer
graphics," Ph.D. dissertation, Dept. of Chem. Engr., Iowa State Univ. (1993).
Modell, M., and J. Tester, Thermodynamics and Its Applications, 3rd edition, Prentice Hall,
129, 216, 218 (1997).
Kyle, B., Chemical and Process Thermodynamics, 3rd edition (PVT graphics tutorials on the
enclosed CD-ROM), Prentice Hall, (1998).
Jolls, K., "Visualization in Classical Thermodynamics," Proceedings, Annual Meeting of A.B.E.T.,
San Diego (October 1996).
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ACKNOWLEDGMENTS
Kong Tian would like to express his gratitude to his major professor, Dr. Kenneth R. Jolls, for
patiently guiding and constantly encouraging him throughout this research.
He is grateful to the members of his thesis committee - Professors Hugo Franzen, Glenn
Schrader, and Dennis Vigil, all of Iowa State University - for their thoughtful attention.
He would like also to thank Professor Dean Ulrichson for his assistance with various aspects of
using ASPEN PLUS, and Professor Judy Vance and doctoral candidate Perry Miller from the ISU
Mechanical Engineering Department for their advice on using Open Inventor and the Silicon
Graphics utility “Showcase.”
Special thanks go to chemical engineering doctoral graduate Richard Campero who worked on
the animation programming and final organization of the computer graphics. Dr. Campero is with
Westvaco Corporation in Covington, Virginia.
More on the next page
61
Angela Lair, BSChE 1998 (ISU), created the hyperscripts that determine the sequence of the
tutorial and provide interconnections between the text slides and the still and movable images.
Ms. Lair is with Procter and Gamble in Cincinnati, Ohio.
Lee Teras, BSChE 1998 (ISU Honors Program), was responsible for the isobaric ternary
diagrams and also helped modify and expand Kong Tian's original “Showcase” presentation. Mr.
Teras is also with Procter and Gamble in Cincinnati.
ASPEN PLUS is a product of Aspen Technology, Inc., Cambridge, MA.
The Silicon Graphics workstations used in this project were obtained through a gift to Iowa State
University from electrical engineering alumnus Edward R. McCracken, formerly CEO of Silicon
Graphics, Inc., and currently Chairman of its Board of Directors.
Past financial support for Dr. Jolls’ visualization research has come from The National Science
Foundation, the University of California at Berkeley, the Camille and Henry Dreyfus Foundation,
The CACHE Corporation, Union Carbide Corporation, and Iowa State University.
Support for the creation of this tutorial was provided by the General Electric Foundation under a
grant entitled "Improving Instruction in Thermodynamics and Related Courses through Scientific
Visualization.“
More on the next page
62
Support to Professor Chapman at Rice University for porting of the visualization software to the
PC/Windows environment and for enhancements to the presentation has been provided by an
Innovative Teaching Grant from The George R. Brown Foundation and by a grant from the BPAmoco Foundation.
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A FINAL WORD
In the spring of 1873 J. Willard Gibbs published the first of his three great papers,
"Graphical Methods in the Thermodynamics of Fluids." The first sentence of that
paper has provided continuing motivation for the work described here and in earlier
publications from this laboratory:
"Although geometrical representations of propositions
in the thermodynamics of fluids are in general use and
have done good service in disseminating clear notions in
this science, yet they have by no means received the
extension in respect to variety and generality of which
they are capable.”
Gibbs published these words in the Transactions of the Connecticut Academy of
Arts and Sciences in his second year as Professor of Mathematical Physics at Yale
University.
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