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COMPUTER VISUALIZATION OF BINARY, TERNARY, AND QUATERNARY

FLUID-PHASE EQUILIBRIA

Kong S. Tian and Kenneth R. JollsAssisted by Richard J. Campero

Chemical Engineering DepartmentIowa State University

Copyright © 1998

Ported from Unix to Windows byJasper Yen, Hector Perez, and

Walter G. ChapmanChemical Engineering Department

Rice UniversityCopyright © 2000

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.

Reviewed by Science magazine, Oct. 15, 1999, p. 430, “An Eye for the Abstract.”

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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.

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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).

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TABLE OF CONTENTS

INTRODUCTION

OBJECTIVES, THEORY

COMPUTATION, GRAPHICS

BINARY SYSTEMS

TERNARY SYSTEMS

TERNARIES WITH AZEOTROPES

GRAPHICS DETAIL

QUATERNARY SYSTEMS

CONCLUSIONS

REFERENCES

ACKNOWLEDGMENTS

A FINAL WORD

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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 constructBut can show the effect of a second independent variable.

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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.

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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.

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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 three-phase, 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.

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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.

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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.

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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 dew-point 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).

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.

Click on the drawing to maneuver it.

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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

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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 dew-point curve (at D) where the liquid phase disappears completely. Higher cyclohexane compositions at this temperature and pressure yield a vapor phase only.

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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-T-x,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 (maximum-pressure) 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.

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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.

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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

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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.

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A TERNARY COMPOSITION PRISMWe 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 2-methylpentane/ pentane binary face, then the pentane/benzene face, and finally the benzene/2-methylpentane 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.

* 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.

Click on the drawing.

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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.

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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.

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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.

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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.

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SURPRISE!

But there seems to be something strange happening on the red surface. The bubble-point 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 pressure-composition 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.

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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 dew-point 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.

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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 Peng-Robinson equation with common mixing rules the pressure of the three-phase line is predicted to be 8.62 bar.

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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.

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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

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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.

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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.

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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.

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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

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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 acetonitrile-benzene binary mixture and disappears with the addition of the least amount of acetone.

Click on the drawing to maneuver it.

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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 vapor-liquid 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.

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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 phasepink – liquid and vapor in equilibriumgreen – vapor phase

movable diagram

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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.

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BENDING OVER BACKWARDSIf 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.

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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.

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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.

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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

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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 bubble- and 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.

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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 three-dimensional 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.

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THE PLANE FACTSIn 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.

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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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DETAILSAll 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 vapor-liquid equilibrium at these conditions along four of the six edges of the tetrahedron.

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CHANGING THE PRESSUREAs 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.

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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 hyper-dimensional 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.

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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 phase-equilibrium 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.

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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).

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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.

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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.“

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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 BP-Amoco 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.