Phase Equilibria and Fluid Properties in the Chemical Industry. Estimation and Correlation [1ST ed.] 9780841203938, 9780841204485, 0-8412-0393-8

Table of contents :
Title Page ......Page 1
Copyright......Page 2
ACS Symposium Series......Page 3
FOREWORD......Page 4
DEDICATION......Page 5
PdftkEmptyString......Page 0
PREFACE......Page 6
Acknowledgments......Page 8
1 Origin of the Acentric Factor......Page 9
Literature Cited......Page 17
2 State-of-the-Art Review of Phase Equilibria......Page 19
Thermodynamics: Not Magic but a Tool......Page 20
Equations of State for Both Fluid Phases......Page 24
Vapor-Phase Fugacity Coefficients......Page 35
Liquid-Phase Activity Coefficients......Page 43
Local Composition to Describe Nonrandomness......Page 47
Group-Contributions for Activity Coefficients......Page 54
Chemical Theory for Activity Coefficients......Page 57
Supercritical Components in the Liquid Phase......Page 58
Aqueous Solutions of Weak Volatile Electrolytes......Page 59
Conclusion......Page 63
Literature Cited......Page 68
3 Industrial View of the State-of-the-Art in Phase Equilibria......Page 70
HOW Are Phase Equilibria Problems Treated?......Page 72
WHAT Models Are Used And WHY?......Page 77
WHERE Should Future Development Work Be Directed?......Page 90
Nomenclature......Page 92
B. Equations of State......Page 93
E. Enthalpy......Page 94
4 Measurement of Vapor-Liquid Equilibrium......Page 95
Thermodynamic Considerations......Page 96
Isothermal vs. Isobaric Data......Page 97
Direct vs. Indirect Determination of Vapor Compositions......Page 98
Extension of Isothermal VLE Data with Temperature......Page 100
Measurement of Low-Pressure VLE Data......Page 101
Conclusions and Remarks......Page 104
Notation......Page 105
Literature Cited......Page 106
5 Equilibria in Aqueous Electrolyte Systems at High Temperatures and Pressures......Page 107
Critical Curves and Miscibility......Page 109
Dielectric Behavior......Page 114
Ionization and Complex Formation......Page 117
Conclusion......Page 121
References......Page 124
General Principles......Page 126
Single Component, Two Phase Systems Solid/Solid Equilibria in Crystalline Polymers......Page 127
The Glass/Rubbery State "Equilibrium"......Page 131
The Rubbery State/Liquid Equilibrium......Page 132
Polymer Melt/Monomer Liquid(s)......Page 134
Adsorption Equilibria......Page 143
Conclusions......Page 145
Literature Cited......Page 147
Discussion......Page 149
Characteristics of Equations of State Required by Industry......Page 158
Equations of State Adopted by Industry......Page 164
Soave Equation......Page 170
Practical Applications......Page 180
Summary......Page 202
Nomenclature......Page 204
References......Page 205
8 Applications of the Peng-Robinson Equation of State......Page 208
The Peng-Robinson Equation......Page 209
Applications......Page 211
Nomenclature......Page 224
Literature Cited......Page 227
9 Application of Equations of State in Exxon's Production Operations......Page 229
Literature Cited......Page 231
Discussion......Page 232
Conclusions......Page 235
Acknowledgment......Page 238
I. Correlation of Fluid Properties with Analytic ES......Page 239
II. Non-Analytic ES......Page 241
Literature Cited......Page 242
Desired Characteristics of an EOS for VLE Calculations......Page 244
Characterization of Heavy Hydrocarbons (NBP>200F)......Page 246
Literature Cited......Page 248
Displacement Mechanisms in CO2--Reservoir Oil Systems......Page 249
Phase and Flow Behavior Characteristics of CO2--Reservoir Oil Systems......Page 252
Calculations......Page 258
Literature Cited......Page 261
14 A Computer Model for Calculating Physical and Thermodynamic Properties of Synthetic Gas Process Streams......Page 264
Possibilities for New Prediction Methods......Page 266
A New Modified Van der Waals Equation for Both Polar and Non-Polar Compounds and Their Mixtures......Page 267
Nomenclature......Page 289
Appendix......Page 292
Literature Cited......Page 296
Scope......Page 297
Design Basis and Process Description......Page 300
Effects of K-Value Predictions......Page 303
Effects of Thermophysical Property Correlations for Enthalpy and Entropy......Page 305
Conclusions......Page 311
Literature Cited......Page 315
Discussion......Page 317
References......Page 321
16 Prediction of Thermodynamic Properties......Page 324
The Virial Expansion......Page 325
Intermolecular Forces......Page 326
Liquid and Dense Gas Mixtures......Page 327
Advantages and Disadvantages of the Principle of Corresponding States as a Method of Prediction......Page 334
Literature Cited......Page 336
Formal Theory......Page 338
Comparison with Experiment for the Dilute Gas......Page 341
Corresponding States......Page 344
Modification of the Transport Expressions......Page 346
Comparisons with Experimental Data Mixtures......Page 348
Literature Cited......Page 351
18 Polar and Quadrupolar Fluid Mixtures......Page 352
Intermolecular Forces......Page 353
Theory......Page 354
Results......Page 357
References......Page 373
Discussion......Page 375
19 Criteria of Criticality......Page 377
Development of the General Criticality Criteria......Page 379
Alternate Criteria in the U-Representation......Page 384
Alternate Criteria in Other Potential Functions......Page 386
Criteria in Gibbs Free Energy......Page 393
Concluding Remarks......Page 395
Literature Cited......Page 396
What are the Synthetic Fuels?......Page 398
How are Synthetic Fuels Made?......Page 401
New Gasification Processes......Page 414
Shale Oil......Page 417
What are the Data Needs?......Page 421
How Good Do the Data Need to Be?......Page 423
When Are the Data Needed?......Page 424
Literature Cited......Page 425
Discussion......Page 426
References......Page 428
21 A Group Contribution Molecular Model for Liquids and Solutions Composed of the Groups CH3, CH2, OH,and CO......Page 429
Abstract......Page 431
Literature Cited......Page 436

Citation preview

Phase Equilibria and Fluid Properties in the Chemical Industry Estimation and Correlation Truman S. Storvick,

EDITOR

University of Missouri, Columbia

Stanley I. Sandler,

EDITOR

University of Delaware A symposium co-sponsored by the Engineering Foundation, the American Institute of Chemical Engineers, and the National Science Foundation at the Asilomar Conference Grounds Pacific Grove, CA, January 16-21,

ACS

1977

SYMPOSIUM

SERIES

AMERICAN CHEMICAL SOCIETY WASHINGTON, D. C.

1977

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

60

Library of Congress

CIP Data

Phase equilibria and fluid properties in the chemical industry. (ACS symposium series; 60 ISS Includes bibliographical references and index. 1. Phase rule and equilibrium—Congresses. 2. Thermodynamics—Congresses. 3. Liquids—Congresses. I. Storvick, Truman S., 1928. II. Sandler, Stanley I., 1940. III. Engineering Foundation, New York. IV. Series: American Chemical Society. ACS symposium series; 60. QD501.P384 ISBN 0-8412-0393-8

Copyright ©

660.2'9'63 ACSMC8

60 1-537

77-13804 (1977)

1977

American Chemical Society A l l Rights Reserved. N o part of this book may be reproduced or transmitted in any form or by any means—graphic, electronic, including photocopying, recording, taping, or information storage and retrieval systems—without written permission from the American Chemical Society. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. PRINTED IN T H E UNITED STATES OF AMERICA

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

ACS Symposium Series Robert F. Gould,

Editor

Advisory Board D o n a l d G. Crosby Jeremiah P. Freeman E . Desmond G o d d a r d Robert A . Hofstader J o h n L . Margrave N i n a I. M c C l e l l a n d J o h n B. Pfeiffer Joseph V . Rodricks A l a n C . Sartorelli Raymond B. Seymour R o y L . Whistler Aaron W o l d

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

FOREWORD The A C S SYMPOSIUM SERIES was founded in 1974

to provide

a medium for publishing symposia quickly in book form. The format of the SERIES parallels that of the continuing ADVANCES IN CHEMISTRY SERIES except that in order to save time the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form.

As a further

means of saving time, the papers are not edited or reviewed except by the symposium chairman, who becomes editor of the book.

Papers published in the A C S SYMPOSIUM SERIES

are original contributions not published elsewhere in whole or major part and include reports of research as well as reviews since symposia may embrace both types of presentation.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

DEDICATION This

work is dedicated to the memory of three men who contributed

to our understanding of fluid properties. Ping L . Chueh Shell Development C o . Houston, T X Geral M c G i l l University Montreal, Quebec, Canada Thomas M . Reed University of Florida Gainesville, FL Illness and accident cut short their careers in 1976 and have left us with their last contribution.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PREFACE We

had two goals in organizing this conference.

T h e first was to

* * provide a forum for state-of-the-art reviews of an area of chemical engineering often referred to as "thermodynamics and physical properties." T h e reviews should represent the work of both the academic researcher and the industrial practitioner. This we thought was both necessary and timely because there were obvious dislocations between the current needs of the industrial chemical engineer and the research being done at universities, on the one hand, and the slow acceptance of new theoretical tools by the industria Our second objective was, through these reviews and the ensuing discussion, to develop a collection of research objectives for the next decade. W e asked the session reporters to try to identify the important research problems that were suggested in the presentations and discussions of the sessions, as well as to set down their thoughts in this regard. In this way, the major papers in this volume summarize the current state of research and industrial practice, while the reporter's summaries provide a listing of important questions and research areas that need attention now. T h e conference was attended by 135 engineers and scientists from North America, Europe, Asia and Africa.

They represented, in almost

equal numbers, the industrial and academic sectors. Recognized authorities, presently active in physical properties work, were chosen to be speakers, panel members, session reporters and session chairmen. T h e conference was held at the Asilomar Conference Grounds on the Monterey Penninsula of California, the beautiful setting matched by idyllic weather.

W e have tried to give an accurate account of the material

presented at the conference sessions, but the printed word cannot reflect the friendships that were established nor the extent of the academioindustrial dialogue which was initiated. Similarly, the unusual enthusiasm of the conference is not reflected here. Indeed, this enthusiasm was so great that there were six ad-hoc sessions, continuations of scheduled sessions and meetings packed into the four sunny afternoons of the meeting. Many important areas of work were identified as needing further attention during the next decade.

Several obvious to us (in no special

order) are listed below: • It was generally agreed that nine out of 10 requests for data by

xi

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

design engineers were for vapor-liquid equilbrium or mixture enthalpy data. Reduction to field-level practice of either data banks or estimating procedures to supply this information would be very useful. • Significant progress has been made on the group contribution methods for estimating phase equilibrium data. Further development of these procedures is clearly justified. • Perturbation methods based on theory from physics and chemistry, electronic computer simulation studies, and careful comparisons with real fluid behavior are moving quickly toward producing an effective equation of state for liquids. These efforts are in the hands of the theoretician today, but further development and reduction to practice should be explored. • F l u i d transport properties were not the primary concern at this conference, but progress between prediction and experiment for viscosities and thermal conductivities of gaseous mixtures was reported.

Clearly, much work needs to

be done, especially for liquids. • Real difficulties remain when attempts are made to predict, to extrapolate, or even to interpolate data for multicomponent mixtures containing hydrocarbons, alcohols, acids, etc.

Such systems were affection-

ately identified as a "Krolikowski mess" at the conference.

Multicom-

ponent mixtures of this kind may include more than one liquid and/or solid phase and with components that "commit chemistry" as well as physically distribute between the phases are commonly encountered in industrial practice. from

nightmare

to

T h e goal for the future is to reduce these problems headache

proportions

in industrial applications,

though they may continue to remain an enigma for the theoretician. • Cries for more experimental data were often heard. Special needs include high pressure vapor-liquid equilibrium data; data on several properties for mixtures with very light, volatile components in heavy hydrocarbon mixtures; ionic solutions; acid gases in hydrocarbons; and certainly more emphasis on mixtures containing aromatic hydrocarbons. Data with intrinsic value for design work and accurate enough for discriminating theoretical comparisons should have high priority.

Signifi-

cantly, several conferees stated that their primary sources of new experimental data are rapidly shifting to laboratories outside the United States. A n important measure of the success of a conference is its long-term impact.

It remains to be seen whether this conference results in any

permanent interchange of ideas between academic and industrial engineers and whether the ideas expressed influence research in the coming years.

xii

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

A cknou/ledgments This volume is based on the Engineering Foundation Conference, "The Estimation and Correlation of Phase Equilibria and F l u i d Properties in the Chemical Industry," convened at the Asilomar Conference Grounds, Pacific Grove, C A , on Jan. 16-21, 1977. T h e views presented here are not necessarily those of the Engineering Foundation, 345 East 47th St., New York, N.Y. 10017. T h e advice, financial and moral support, and the concern for local arrangements, publicity, registration by Sandford Cole, Harold Commerer, Dean Benson and their staff permitted us to concentrate on the technical aspects of the meeting. Manuscript typing was done by the University of Missouri, Stenographic Services Department. Major funding for the conference by the National Science Foundation was a key ingredien for many American and have been otherwise unable to participate.

T h e interest and support of

Marshall L i h and William Weigand of the National Science Foundation were especially appreciated. The American Institute of Chemical Engineers made important contributions by co-sponsoring and publicizing the conference. W e also thank the members of the Organizing Committee: Stanley Adler, Pullman-Kellogg Co.; Howard Hanley of the National Bureau of Standards; Robert Reid of the Massachusetts Institute of Technology; and L y m a n Yarborough of the Amoco Production C o . They brought focus and structure to the general concept of the conference we brought to them. Finally, and most important we thank the speakers, session reporters, and chairman who d i d their work diligently and in the best scientific tradition; and the conferees for their enthusiastic participation and important discussion contributions that made this conference special. T. S. STORVICK

STANLEY I. SANDLER

University of Missouri—Columbia

University of Delaware

June 1977

xiii

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

1 Origin of the Acentric Factor K E N N E T H S. P I T Z E R

University of California, Berkeley, Calif. 94720

It was a pleasure t Sandler' invitatio this conference by reviewin which led me to propose the acentric factor in 1955. Although I had followed some of the work in which others have used the acentric factor, the preparation of this paper provided the incentive to review these applications more extensively, and I was most pleased to find that so much has been done. I want to acknowledge at once my debt to John Prausnitz for suggestions in this review of recent work as well as in many discussions through the years. Beginning in 1937, I had been very much interested in the thermodynamic properties of various hydrocarbon molecules and hence of those substances in the ideal gas state. This arose out of work with Kemp in 1936 on the entropy of ethane (1) which led to the determination of the potential barrier restricting internal rotation. With the concept of restricted internal rotation and some advances in the pertinent statistical mechanicsitbecame possible to calculate rather accurately the entropies of various light hydrocarbons (2). Fred Rossini and I collaborated in bringing together his heat of formation data and my entropy and enthalpy values to provide a complete coverage of the thermodynamics of these hydrocarbons in the ideal gas state (3). As an aside I cite the recent paper of Scott (4) who presents the best current results on this topic. But real industrial processes often involve liquids or gases at high pressures rather than ideal gases. Hence it was a logical extension of this work on the ideal gases to seek methods of obtaining the differences in properties of real fluids from the respective ideal gases without extensive experimental studies of each substance. My first step in this direction came in 1939 when I was able to provide a rigorous theory of corresponding states (5) on the basis of intermolecular forces for the restricted group of substances, argon, kryptron, xenon, and in good approximation also methane. This pattern of behavior came to be called that of a simple fluid. It is the reference pattern from which the acentric factor measures the departure. Possibly we should recall the key ideas. The 1 In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

i n t e r m o l e c u l a r p o t e n t i a l must be given by a u n i v e r s a l f u n c t i o n w i t h s c a l e f a c t o r s of energy and d i s t a n c e f o r each substance. By then i t was well-known that the dominant a t t r a c t i v e f o r c e f o l l o w e d an i n v e r s e sixth-power p o t e n t i a l f o r a l l of these substances. A l s o the r e p u l s i v e f o r c e s were known to be very sudden. Thus the i n v e r s e s i x t h , power term w i l l dominate the shape of the p o t e n t i a l curve at longer d i s t a n c e s . Even without d e t a i l e d t h e o r e t i c a l reasons f o r exact s i m i l a r i t y of shorter-range terms, one could expect that a u n i v e r s a l f u n c t i o n might be a good approximation. I n a d d i t i o n one assumed s p h e r i c a l symmetry (approximate f o r methane), the v a l i d i t y of c l a s s i c a l s t a t i s t i c a l mechanics, and that the t o t a l energy was determined e n t i r e l y by the v a r i o u s i n t e r m o l e c u l a r d i s t a n c e s . I should r e c a l l that i t was not f e a s i b l e i n 1939 to c a l c u l a t e the a c t u a l equation of s t a t e from t h i s model One could o n l y show that i t y i e l d e d correspondin s t a t e i n terms of the reduce pressure. One could p o s t u l a t e other models which would y i e l d a c o r r e s ponding-states behavior but d i f f e r e n t from that of the simple f l u i d . However, most such molecular models were s p e c i a l and d i d not y i e l d a s i n g l e f a m i l y of equations. Rowlinson (6) found a somewhat more general case; he showed that f o r c e r t a i n types of a n g u l a r l y dependent a t t r a c t i v e f o r c e s the net e f f e c t was a temperature dependent change i n the r e p u l s i v e term. From t h i s a s i n g l e f a m i l y of funct i o n s arose. I had observed e m p i r i c a l l y , however, that the f a m i l y r e l a t i o n ship of equations of s t a t e was much broader even than would f o l l o w from R o w l i n s o n s model. I t included g l o b u l a r and e f f e c t i v e l y s p h e r i c a l molecules such as tetramethylmethane (neopentane), where no a p p r e c i a b l e angular dependence was expected f o r the i n t e r m o l e c u l a r p o t e n t i a l , and f o r elongated molecules such as carbon d i o x i d e the angular dependence of the r e p u l s i v e f o r c e s seemed l i k e l y to be at l e a s t as important as that of the a t t r a c t i v e f o r c e s . Thus the core model of K i h a r a (7) appealed to me; he assumed that the LennardJones 6-12 p o t e n t i a l a p p l i e d to the s h o r t e s t d i s t a n c e between cores i n s t e a d of the d i s t a n c e between molecular centers. He was a b l e to c a l c u l a t e the second v i r i a l c o e f f i c i e n t f o r v a r i o u s shapes of core. And I was a b l e to show that one obtained i n good approximation a s i n g l e f a m i l y of reduced second v i r i a l c o e f f i c i e n t f u n c t i o n s f o r cores of a l l reasonable shapes. By a s i n g l e f a m i l y I mean that one a d d i t i o n a l parameter s u f f i c e d to d e f i n e the equation f o r any p a r t i c u l a r case. While t h i s d i d not prove that a l l of the complete equations of s t a t e would f a l l i n t o a s i n g l e f a m i l y , i t gave me enough encouragement to go ahead w i t h the numerical w o r k — o r more a c c u r a t e l y to persuade s e v e r a l students to undertake the n u m e r i c a l work. Let me emphasize the importance of f i t t i n g g l o b u l a r molecules i n t o the system. I f these molecules are assumed to be s p h e r i c a l i n good approximation, they are easy to t r e a t t h e o r e t i c a l l y . Why aren't they simple f l u i d s ? Many t h e o r e t i c a l papers ignore t h i s q u e s t i o n . In f l u i d p r o p e r t i e s neopentane departs from the simple f l u i d p a t t e r n much more than propane and almost as much as n-butane. But propane f

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

1.

PITZER

3

Origin of the Acentric Factor

i s much l e s s s p h e r i c a l than neopentane. The e x p l a n a t i o n l i e s i n the narrower a t t r a c t i v e p o t e n t i a l w e l l . The i n v e r s e - s i x t h - p o w e r a t t r a c t i v e p o t e n t i a l now operates between each p a r t of the molecule r a t h e r than between molecular c e n t e r s . Thus the a t t r a c t i v e term i s steeper than i n v e r s e s i x t h power i n terms of the d i s t a n c e between molecular c e n t e r s . This i s shown i n F i g u r e 1, taken from my paper (8) i n 1955. We need not bother w i t h the d i f f e r e n c e s between the models y i e l d i n g the dotted and dashed curves f o r the g l o b u l a r molecule. The important f e a t u r e i s the narrowness of the p o t e n t i a l w e l l f o r e i t h e r of these curves as compared t o the s o l i d curve f o r the molecules of a simple f l u i d . I t was easy t o show that the i n t e r m o l e c u l a r p o t e n t i a l curves f o r s p h e r i c a l molecules would y i e l d a s i n g l e f a m i l y of reduced equations of s t a t e . I f one takes th K i h a r model w i t h s p h e r i c a l the the r e l a t i v e core s i z e ca t i o n t o the energy and d i s t a n c equation of s t a t e . With an adequate understanding of g l o b u l a r molecule b e h a v i o r , I then showed as f a r as was f e a s i b l e that the p r o p e r t i e s of other nonp o l a r or weakly p o l a r molecules would f a l l i n t o the same f a m i l y . I t was p r a c t i c a l a t that time only to c o n s i d e r the second v i r i a l c o e f f i cent. The K i h a r a model was used f o r nonpolar molecules of a l l shapes w h i l e R o w l i n s o n s work provided the b a s i s f o r d i s c u s s i o n of p o l a r molecules. F i g u r e 2 shows the reduced second v i r i a l c o e f f i c i e n t f o r s e v e r a l cases. Curves l a b e l e d a / p r e f e r t o s p h e r i c a l - c o r e molecules w i t h a i n d i c a t i n g the core s i z e , c o r r e s p o n d i n g l y £/p i n d i c a t e s a l i n e a r molecule of core l e n g t h £, w h i l e y r e f e r s to a d i p o l a r molecule w i t h y = u /e ^r where u i s the d i p o l e moment. The non-polar p o t e n t i a l i s 1

Q

Q

0

Q

(i) where p i s the s h o r t e s t d i s t a n c e between c o r e s . For the p o l a r molecules I omitted the core, thus p = r . While the curves i n F i g u r e 2 appear t o f a l l i n t o a s i n g l e f a m i l y , t h i s i s i n v e s t i g a t e d more r i g o r o u s l y i n F i g u r e 3 where the reduced second v i r i a l c o e f f i c i e n t a t one reduced temperature i s compared w i t h the same q u a n t i t y a t another temperature. Tg i s the Boyle temperature which i s a convenient r e f e r e n c e temperature f o r second v i r i a l c o e f f i c i e n t s . One sees that the non-polar core molecules f a l l a c c u r a t e l y on a s i n g l e curve (indeed a s t r a i g h t l i n e ) . While the p o l a r molecules d e v i a t e , the d i f f e r e n c e i s o n l y 1% a t y = 0.7 which I took as a reasonable standard of accuracy a t that time. For comparison the y values of c h l o r o f o r m , e t h y l c h l o r i d e , and ammonia are 0.04, 0.16, and 4, r e s p e c t i v e l y . Thus the f i r s t two f a l l w e l l below the 0.7 v a l u e f o r agreement of p o l a r w i t h non-polar e f f e c t s w h i l e ammonia i s beyond that v a l u e . The next q u e s t i o n was the c h o i c e of the experimental b a s i s f o r the t h i r d parameter. The vapor pressure i s the property most s e n s i t i v e t o t h i s t h i r d parameter; a l s o i t i s one of the p r o p e r t i e s most

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

4

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY 1

11

1

1

1 -

r

!

_^*- *^-

if 1

0

\J

0. r/r . 0

Figure 1. Intermolecular potential for molecules of a simple fluid, solid line; and for globular molecules such as C(CH ) dashed lines 3

T

B

If>

/ T .

Figure 2. Reduced second virial coefficients for several models: solid curve, simple fluid; curves labeled by a./p , spherical cores of radius a; curves labeled by l/p , linear cores of length 1; curves labeled by y, molecules with dipoles 0

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

0

Figure 3. Check on family relationship of curves of Figure 2. Comparison of deviations from simple fluid at (T /TJ = 3.5 with that at (T /T) = 2.0 B

B

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

6

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

w i d e l y measured at l e a s t near the normal b o i l i n g p o i n t . Thus both the a v a i l a b i l i t y of data and the accuracy of the data f o r the purpose s t r o n g l y i n d i c a t e d a vapor p r e s s u r e c r i t e r i o n . Since the c r i t i c a l data have to be known f o r a reduced equation of s t a t e , the reduced vapor pressure near the normal b o i l i n g p o i n t was an easy choice f o r the new parameter. The a c t u a l d e f i n i t i o n a) = -£og P

r

- 1.000

(2)

w i t h P the reduced vapor pressure a t T = 0.700 seemed convenient, but the a c t u a l d e t e r m i n a t i o n of a) can be made from any vapor pressure v a l u e well-removed from the c r i t i c a l p o i n t . Here I should note the work of R i e d e l (9) which was substant i a l l y simultaneous w i t h mine but whose f i r s t paper preceded s l i g h t l y . His work was p u r e l y e m p i r i c a l mentary. He chose f o r h i vapor pressure c u r v e , but i n h i s case the d i f f e r e n t i a l s l o p e a t the c r i t i c a l p o i n t . That seemed to me to be l e s s r e l i a b l e and a c c u r a t e , e m p i r i c a l l y , although e q u i v a l e n t o t h e r w i s e . F o r t u n a t e l y R i e d e l and I chose to emphasize d i f f e r e n t p r o p e r t i e s as our r e s p e c t i v e programs proceeded; hence the f u l l area was covered more q u i c k l y w i t h l i t t l e d u p l i c a t i o n of e f f o r t . A l s o I needed a name f o r t h i s new parameter, and that was d i f f i c u l t . The term " a c e n t r i c f a c t o r " was suggested by some f r i e n d l y reviewer, p o s s i b l y by a r e f e r e e ; I had made a l e s s s a t i s f a c t o r y choice i n i t i a l l y . The conceptual b a s i s i s i n d i c a t e d i n F i g u r e 4. The i n t e r m o l e c u l a r f o r c e s between complex molecules f o l l o w a simple e x p r e s s i o n i n terms of the d i s t a n c e s between the v a r i o u s p o r t i o n s of the molecule. Since these f o r c e s between n o n - c e n t r a l p o r t i o n s of the molecules must be c o n s i d e r e d , the term " a c e n t r i c f a c t o r " seemed appropriate. I t i s assumed that the c o m p r e s s i b i l i t y f a c t o r and other propert i e s can be expressed i n power s e r i e s i n the a c e n t r i c f a c t o r and that a linear expression w i l l usually s u f f i c e . r

r

pv = — = z RT

i

z

( 0 )

(0) z

= z -

z

r

r

( 1 )

The preference of P over V as the second independent v a r i a b l e i s p u r e l y e m p i r i c a l ; the c r i t i c a l pressure i s much more a c c u r a t e l y measurable than the c r i t i c a l volume. The e m p i r i c a l e f f e c t i v e n e s s of t h i s system was f i r s t t e s t e d w i t h v o l u m e t r i c data as shown on F i g u r e 5. Here pv/RT a t a p a r t i c u l a r r

r

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

1.

Origin of the Acentric Factor

PITZER

Ar

Ar

i Q \

CH

C

CH

4

4

Figure 4. Intermolecular forces operate between the centers of regions of substantial electron density. These centers are the molecular centers for Ar and

3 8 H

CH groups in C H —hence the name acentric factor for the forces arising from points other than molecular centers. 2

3

8

i.O

0.8-

1.30 .25 1.20

1.15

0.6 PV RT'

' 1.10

0.4

• - 1.05

1

0

#f 1.00

0.2-

A Xe

CH

4

C H HS 2

C(CH ) n-C H, 3

6

4

2

C

3

7

0

l6

2

6 6 H

C0 C H

n-C H H0

4

2

NH

8

3

I

0.1

0.2 CJ.

0.3

0.4

Figure 5. Compressibility factor as a function of acentric factor for reduced pressure 1.6 and reduced temperature shown for each line. Where several substances have approximately the same acentric factor, the individual points are indistinguishable except for n-C H C) and H O(Q). 7

t

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

16

8

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

reduced temperature and pressure i s p l o t t e d a g a i n s t u). The most important r e s u l t appears only by i m p l i c a t i o n ; the r e s u l t s f o r C(CH^)^, n-CifiiQ 6 6> C02 are so n e a r l y equal t h a t they appear as s i n g l e p o i n t s on these p l o t s . Here we have f o u r w i d e l y d i f f e r e n t shapes of molecules which happen to have about the same a c e n t r i c f a c t o r , and they f o l l o w corresponding s t a t e s a c c u r a t e l y among themselves. A l s o to be noted from F i g u r e 5 i s the f a c t that the h i g h l y p o l a r molecules NH3 and H2O depart from the system. Furthermore the dependence on a) i s l i n e a r except f o r the c r i t i c a l r e g i o n . My immediate r e s e a r c h group used g r a p h i c a l methods i n d e a l i n g w i t h the experimental data and r e p o r t e d a l l of our r e s u l t s i n numeric a l t a b l e s (10). At t h a t time the best a n a l y t i c a l equation of s t a t e was t h a t of B e n e d i c t , Webb and Rubin (11) which employed e i g h t parameters and s t i l l f a i l e d to f i t v o l u m e t r i c data w i t h i n experimental accuracy. Bruce Sage suggeste for the normal p a r a f f i n s bot y the a c e n t r i c f a c t o r system. T h i s work (12) was done p r i m a r i l y by J . B. O p f e l l a t C a l Tech. The r e s u l t s showed that the a c e n t r i c f a c t o r system was a great advance over the simple p o s t u l a t e of corresponding s t a t e s , but the f i n a l agreement was i n f e r i o r to that obtained by g r a p h i c a l and numerical methods. Thus we continued w i t h numerical methods f o r the f u g a c i t y , entropy, and enthalpy f u n c t i o n s (13), although we d i d present an e m p i r i c a l equation f o r the second v i r i a l c o e f f i c i e n t (14). This work was done by Bob C u r l ; he d i d an e x c e l l e n t job but found the almost i n t e r m i n a b l e g r a p h i c a l work very tiresome. Thus I was pleased t h a t the B r i t i s h I n s t i t u t i o n of Mechanical Engineers i n c l u d e d C u r l i n the award of t h e i r C l a y t o n P r i z e f o r t h i s work. A f i f t h paper w i t h H u l t g r e n (15) t r e a t e d mixtures on a p s e u d o c r i t i c a l b a s i s , and a s i x t h w i t h Danon (16) r e l a t e d K i h a r a core s i z e s to the acentric factor. N a t u r a l l y , I am v e r y pleased to note t h a t o t h e r s have extended the accuracy and range of our t a b l e s and equations w i t h c o n s i d e r a t i o n of more recent experimental r e s u l t s . Of p a r t i c u l a r l y broad importance i s the 1975 paper by Lee and K e s s l e r (17) which presents both improved t a b l e s and a n a l y t i c a l equations f o r a l l of the major f u n c t i o n s i n c l u d i n g vapor p r e s s u r e s , v o l u m e t r i c p r o p e r t i e s , e n t h a l p i e s , e n t r o p i e s , f u g a c i t i e s , and heat c a p a c i t i e s . Their equation i s an e x t e n s i o n of that of Benedict, Webb, and Rubin now c o n t a i n i n g twelve parameters. They considered more recent experimental data as w e l l as a number of papers which had a l r e a d y extended my e a r l i e r work i n p a r t i c u l a r areas. I r e f e r to t h e i r b i b l i o g r a p h y (17) f o r most of t h i s more d e t a i l e d work, but I do want to note the improved equation of Tsonopoulos (18) f o r the second v i r i a l c o e f f i c i e n t . This equation deals a l s o w i t h e f f e c t s of e l e c t r i c a l p o l a r i t y . In a d d i t i o n to r e f e r e n c e s c i t e d by Lee and K e s s l e r there i s the work Lyckman, E c k e r t , and P r a u s n i t z (19) d e a l i n g w i t h l i q u i d volumes; they found i t necessary to use a q u a d r a t i c e x p r e s s i o n i n u). A l s o Barner and Quinlan (20) t r e a t e d mixtures at high temperatures and p r e s s u r e s , and Chueh and P r a u s n i t z (21) t r e a t e d the c o m p r e s s i b i l i t y C

H

a n d

9

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

1.

PITZER

Origin of the Acentric Factor

9

of l i q u i d s . Reid and Sherwood (22) g i v e an e x t e n s i v e t a b l e i n c l u d i n g a c e n t r i c f a c t o r s as w e l l as c r i t i c a l constants f o r many substances. On the t h e o r e t i c a l s i d e , one great advance has been i n the development of p e r t u r b a t i o n t h e o r i e s of a g e n e r a l i z e d van der Waals type. Here one assumes t h a t the molecular d i s t r i b u t i o n i s d e t e r mined p r i m a r i l y by r e p u l s i v e f o r c e s which can be approximated by hard cores. Then both the s o f t n e s s of the cores and the a t r a c t i v e f o r c e s a r e t r e a t e d by p e r t u r b a t i o n methods. Barker and Henderson (23) have r e c e n t l y reviewed t h e o r e t i c a l advances i n c l u d i n g t h e i r own outstanding work. Rigby (24) a p p l i e d these modern Van der Waals methods t o n o n - s p h e r i c a l molecules which represent one type of molecules w i t h non-zero a c e n t r i c f a c t o r s . I n a somewhat s i m i l a r manner Beret and P r a u s n i t z (25) developed equations a p p l i c a b l e even to h i g h polymers and r e l a t e d the i n i t i a l departures from simple f l u i d s t o the a c e n t r i c f a c t o r But i n my view the approac f i r s t on g l o b u l a r molecules. These could be modeled by K i h a r a potent i a l s w i t h s p h e r i c a l cores or by other p o t e n t i a l s a l l o w i n g the w e l l to be narrowed. The great advantage would be the r e t e n t i o n of spheri c a l symmetry and i t s t h e o r e t i c a l s i m p l i c i t y . Rogers and P r a u s n i t z (26) made an important beginning i n t h i s area w i t h c a l c u l a t i o n s based on K i h a r a models a p p r o p r i a t e f o r argon, methane, and neopentane w i t h e x c e l l e n t agreement f o r the p r o p e r t i e s s t u d i e d . While they do not d i s c u s s these r e s u l t s i n terms of the a c e n t r i c f a c t o r , the t r a n s formation of s p h e r i c a l core r a d i u s t o a c e n t r i c f a c t o r i s w e l l e s t a b l i s h e d (16, 27), Rogers and P r a u s n i t z were a l s o able to t r e a t mixtures very s u c c e s s f u l l y although those c a l c u l a t i o n s were burdensome even w i t h modern computers. I b e l i e v e f u r t h e r t h e o r e t i c a l work using s p h e r i c a l models f o r g l o b u l a r molecules would be f r u i t f u l . The move to an a n a l y t i c a l equation by Lee and K e s s l e r was undoubtedly a wise one i n view of the marvelous c a p a c i t y of modern computers to d e a l w i t h complex equations. I would expect f u t u r e work to y i e l d s t i l l b e t t e r equations. There remains the q u e s t i o n of the u l t i m a t e accuracy of the a c e n t r i c f a c t o r concept. How a c c u r a t e l y do molecules of d i f f e r e n t shapes but w i t h the same a c e n t r i c f a c t o r r e a l l y f o l l o w corresponding s t a t e s ? Apparently t h i s accuracy i s w i t h i n experimental e r r o r f o r most, i f not a l l , present data. Thus the a c e n t r i c f a c t o r system c e r t a i n l y meets engineering needs, and i t i s p r i m a r i l y a matter of s c i e n t i f i c c u r i o s i t y whether d e v i a t i o n s a r e p r e s e n t l y measurable. I t has been a pleasure to review these aspects of the " a c e n t r i c f a c t o r " w i t h you and I look forward to your d i s c u s s i o n of recent advances i n these and other areas. f

Literature Cited

1.

Kemp, J . D. and P i t z e r , K. S., J . Chem. Phys., (1936) 4, 749; J. Am. Chem. Soc. (1937) 59, 276.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

10

2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

P i t z e r , K. S., J . Chem. Phys., (1937) 5, 469, 473, 752; (1940) 8, 711; Chem. Rev. (1940) 27, 39. Rossini, F. D., P i t z e r , K. S., Arnett, R. L., Braun, R. M. and Pimentel, G. C., "Selected Values of the Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds," Carnegie Press, Pittsburgh (1953). Scott, D. W., J . Chem. Phys. (1974) 60, 3144. P i t z e r , K. S., J . Chem. Phys. (1939) 7, 583. Rowlinson, J . S., Trans. Faraday Soc. (1954) 50, 647; "Liquids and Liquid Mixtures," 2nd ed. Chapter 8, Butterworth, London (1969). Kihara, T., Rev. Mod. Phys. (1953) 25, 831 and papers there cited. P i t z e r , K. S., J . Am. Chem. Soc. (1955) 77, 3427. Riedel, L., Chem. Ing 27, 209, 475; (1956 P i t z e r , K. S., Lippman, D. Z., Curl, J r . , R. F., Huggins, C. M. and Petersen, D. E., J . Am. Chem. Soc. (1955) 77, 3433. Benedict, M., Webb, G. B. and Rubin, L. C., J . Chem. Phys. (1940) 8, 334. Opfell, J . B., Sage, B. H. and P i t z e r , K. S., Ind. Eng. Chem. (1956) 48, 2069. Curl, J r . , R. F., and P i t z e r , K. S., Ind. Eng. Chem. (1958) 50, 265. P i t z e r , K. S. and Curl, J r . , R. F., J . Am. Chem. Soc., (1957) 79, 2369. P i t z e r , K. S. and Hultgren, G. O., J . Am. Chem. Soc. (1958) 80, 4793. Danon, F. and P i t z e r , K. S., J . Chem. Phys. (1962) 36, 425. Lee, B. I . and Kesler, M. G., A.I.Ch.E. Journal (1975) 21, 510. Tsonopoulos, C., A.I.Ch.E. Journal (1974) 20, 263. Lyckman, E. W., Eckert, C. A. and Prausnitz, J . M., Chem. Engr. S c i . (1965) 20, 703. Barner, H. E. and Quinlan, C. W., I . and E.C. Proc. Des. Dev. (1969) 8, 407. Chueh, P. L. and Prausnitz, J . M., A.I.Ch.E. Journal (1969) 15, 471. Reid, R. C. and Sherwood, T. K., "The Properties of Gases and Liquids," 2nd ed., McGraw-Hill Book Co., New York (1966). Barker, J . A. and Henderson, D., Rev. Mod. Phys. (1976) 48, 587. Rigby, M., J . Phys. Chem (1972) 76, 2014. Beret, S. and Prausnitz, J . M., A.I.Ch.E. Journal (1975) 21, 1123. Rogers, B. L. and Prausnitz, J . M., Trans. Faraday Soc. (1971) 67, 3474. Tee, L. S., Gotoh, S., and Stewart, W. E., Ind. Eng. Chem. Fund. (1966) 5, 363.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2 State-of-the-Art Review of Phase Equilibria J. M. PRAUSNITZ University of California, Berkeley, Calif. 94720

I welcome the opportunity to discuss the state of the art for calculating phase equilibria in chemical engineering first, because I consider it a high honor to have been chosen for this important assignment and second, because it may give me a chance to influence the direction of future research in this field. When I mentioned these two reasons to one of my more candid coworkers, he said "What you really mean is, that you enjoy the opportunity to go on an ego trip and that you are glad to have an audience which you can subject to your prejudices." While this restatement of my feelings is needlessly unkind, I must confess that it bears an element of truth. The assignment that Professor Sandler has given me--to review applied phase equilibrium in an hour or two--is totally impossible and it follows that in choosing material for this presentation, I must be highly selective. Since time is limited, I must omit many items which others, in exercising their judgment, might have included. At the outset, therefore, I want to apologize to all in the audience who may feel that some publications, notably their own, have received inadequate attention. While I have tried to be objective and critical in my selection, it is human nature to give preference to that work with which one is most familiar and that, all too often, tends to be one's own. Nevertheless, I shall try to present as balanced a picture as I can. After more than 20 years, I have developed a certain point of view conditioned by my particular experience and I expect that it is pervasive in what I have to say. However, I want very much to assure this audience that I present my point of view without dogmatic intent; it is only a personal statement, a point of departure for what I hope will be vigorous discussion during the days ahead. My aim in attending this conference is the same as yours: at the end of the week I want to be a little wiser than I am now, at the beginning.

11 In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

12

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Thermodynamics:

Not Magic but a Tool

A l l too o f t e n , when I t a l k w i t h chemical engineers from i n d u s t r y who have l i t t l e experience i n thermodynamics, I o b t a i n the impression that they look upon me as a medicine man, a magician who i s supposed to i n c a n t obscure formulas and, i n e f f e c t , produce something out of nothing. This audience knows b e t t e r but n e v e r t h e l e s s , we must remind o u r s e l v e s that thermodynamics i s not magic, that i t i s only a u s e f u l t o o l f o r e f f i c i e n t o r g a n i z a t i o n of knowledge. Thermodynamics alone never t e l l s us the value of a d e s i r e d e q u i l i b r i u m property; i n s t e a d , i t t e l l s us how the d e s i r e d e q u i l i b r i u m property i s r e l a t e d to some other e q u i l i b r i u m property. Thus thermodynamics provides us w i t h a time-saving bookkeeping system: we do not have to measure a l l the e q u i l i b r i u m p r o p e r t i e s ; we measure only some and then we can c a l c u l a t e o t h e r s . Thus, from a tage of thermodynamics i i f we know how the Gibbs energy of mixing v a r i e s w i t h temperature, we need not measure the enthalpy of mixing s i n c e we can c a l c u l a t e i t u s i n g the Gibbs-Helmholtz equation, o r , i n a b i n a r y system, i f we know how the a c t i v i t y c o e f f i c i e n t of one component v a r i e s w i t h compos i t i o n , we can use the Gibbs-Duhem equation to c a l c u l a t e the other. We must keep reminding ourselves and others as to j u s t what thermodynamics can and cannot do. F a l s e expectations o f t e n l e a d t o c o s t l y disappointments. While the l i m i t a t i o n s of c l a s s i c a l thermodynamics a r e c l e a r enough, the p o t e n t i a l l y v a s t p o s s i b i l i t i e s opened by s t a t i s t i c a l thermodynamics a r e s t i l l f a r from r e a l i z e d . J u s t what modern p h y s i c s can do f o r us w i l l be discussed l a t e r i n the week; f o r now, I j u s t want to say that even a t t h i s e a r l y stage, simple molecular ideas can do much to s t r e t c h the range of a p p l i c a t i o n of thermodynamics. When thermodynamics i s coupled w i t h the molecular theory of matter, we can c o n s t r u c t u s e f u l models; w h i l e these only roughly approximate t r u e molecular behavior, they n e v e r t h e l e s s enable us t o i n t e r p o l a t e and e x t r a p o l a t e w i t h some confidence, thereby reducing f u r t h e r the experimental e f f o r t r e q u i r e d f o r r e l i a b l e r e s u l t s . When my n o n t e c h n i c a l f r i e n d s ask me what I , a molecular thermodynamicist do, I answer w i t h a naive but e s s e n t i a l l y accurate analogy: I am a greedy tax c o l l e c t o r . From the s m a l l e s t p o s s i b l e c a p i t a l , I t r y t o e x t r a c t the l a r g e s t p o s s i b l e revenue. Keeping i n mind that thermodynamics i s no more than an e f f i c i ent t o o l f o r o r g a n i z i n g knowledge toward u s e f u l ends, I f i n d t h a t , f o r phase-equalibrium work, thermodynamics provides us w i t h two procedures, as shown i n F i g u r e 1. Our aim i s to c a l c u l a t e f u g a c i t i e s and we can do so e i t h e r using method ( a ) , based e n t i r e l y on an equat i o n of s t a t e a p p l i c a b l e to both phases a and $, or u s i n g method (b), which uses an equation of s t a t e only f o r c a l c u l a t i n g the vaporphase f u g a c i t y and a completely d i f f e r e n t method, expressed by the a c t i v i t y c o e f f i c i e n t , f o r c a l c u l a t i n g condensed-phase f u g a c i t i e s . I now want to examine these two methods because they a r e the ones which have been used i n e s s e n t i a l l y a l l a p p l i e d p h a s e - e q u i l i b r i u m work.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

13

Review of Phase Equilibria

PRAUSNITZ

FOR EVERY COMPONENT i , £ l

= £ l

IN PHASES a AND 0

f = FUGACITY

EITHER

- J n

i

d

V

"

l

n

^1

= MOLES OF i ;

V = TOTAL VOLUME

fl.y.P

f^-Tx.f?

OR

(b)

fY= I

AND

I

y,x = COMPOSITION;

I

'

1

1

1

° = STANDARD STATE

0 = FUGACITY COEFFICIENT (FROM EQUATION OF STATE) T = ACTIVITY COEFFICIENT Figure 1.

Two thermodynamic methods for calculation of fluid-phase equilibria

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

(b)

(a)

METHOD

P - V - T - X DATA ARE SUFFICIENT; IN PRINCIPLE, NO PHASE EQUILIBRIUM DATA NEEDED.

EASILY UTILIZES THEOREM OF CORRESPONDING STATES.

CAN BE APPLIED TO CRITICAL REGION.

SIMPLE LIQUID-MIXTURE MODELS ARE OFTEN SATISFACTORY.

EFFECT OF TEMPERATURE IS IN f , NOT r .

APPLICABLE TO WIDE VARIETY OF MIXTURES, INCLUDING POLYMERS AND ELECTROLYTES.

2.

3.

4.

1.

2.

3.

PRIMARILY

NO STANDARD STATES.

1.

ADVANTAGES

CUMBERSOME FOR COMPONENTS.

2. 3.

NEED SEPARATE METHOD TO FIND v

1.

DIFFICULT TO APPLY IN CRITICAL REGION.

Q

DIFFICULT TO APPLY TO POLAR COMPOUNDS, LARGE MOLECULES, OR ELECTROLYTES.

3.

SUPER-CRITICAL

OFTEN VERY SENSITIVE TO MIXING RULES.

2.

0

NO REALLY GOOD EQUATION OF STATE AVAILABLE FOR ALL DENSITIES

1.

DISADVANTAGES

2.

PRAUSNITZ

Review of Phase Equilibria

15

When encountering a p a r t i c u l a r p h a s e - e q u i l i b r i u m problem, the very f i r s t d e c i s i o n i s to decide which of these methods i s most s u i t a b l e for the p a r t i c u l a r problem. I t i s t h e r e f o r e important t o review the r e l a t i v e advantages and disadvantages of both methods; these are summarized i n F i g u r e 2. The s t a t e of the a r t today i s such that f o r mixtures of simple, or what P i t z e r has c a l l e d "normal" f l u i d s , we can o f t e n c a l c u l a t e v a p o r - l i q u i d e q u i l i b r i a , even at high p r e s s u r e s , w i t h good success u s i n g some e m p i r i c a l equation of s t a t e . However, f o r mixtures i n c l u d i n g one or more s t r o n g l y p o l a r or hydrogen-bonding component, we must r e s o r t t o the use of a c t i v i t y c o e f f i c i e n t s and standardstate fugacities. As i n d i c a t e d i n F i g u r e 2, an equation of s t a t e f o r a l l f l u i d phases has many advantages because one very troublesome f e a t u r e v i z . s p e c i f y i n g a standar troublesome because we f r e q u e n t l y multicomponen mixtures where a t l e a s t one component i s s u p e r c r i t i c a l . I n that event, the choice of a p r o p e r l y defined a c t i v i t y c o e f f i c i e n t and standard s t a t e i n t r o d u c e s formal d i f f i c u l t i e s which are o f t e n mathematically inconvenient and, f o r p r a c t i c a l implementation, r e q u i r e parameters from experimental data that are only r a r e l y available. For l i q u i d - p h a s e m i x t u r e s , p o l a r or nonpolar, i n c l u d i n g polymers and e l e c t r o l y t e s , a t low o r moderate p r e s s u r e s , the a c t i v i t y c o e f f i c i e n t provides the most convenient t o o l we have but our fundamental knowledge about i t i s sparse. Thermodynamics gives us l i t t l e h e l p ; we have three well-known r e l a t i o n s : f i r s t , the Gibbs-Duhem equation which r e l a t e s the a c t i v i t y c o e f f i c i e n t of one component i n a s o l u t i o n t o those of the o t h e r s , second, the Gibbs-Helmholtz equation which r e l a t e s the e f f e c t of temperature on the a c t i v i t y c o e f f i c i e n t to the enthalpy of mixing and f i n a l l y , an equation which r e l a t e s the p a r t i a l molar volume t o the e f f e c t of pressure on the a c t i v i t y c o e f f i c i e n t . These i l l u s t r a t e what I s a i d e a r l i e r , v i z . that c l a s s i c a l thermodynamics i s l i t t l e more than an e f f i c i e n t o r g a n i z a t i o n of knowledge, r e l a t i n g some e q u i l i b r i u m p r o p e r t i e s to o t h e r s , thereby reducing experimental work. But the p r a c t i c a l a p p l i c a t i o n s of these c l a s s i c a l thermodynamic r e l a t i o n s f o r a c t i v i t y c o e f f i c i e n t s are l i m i t e d , i n c o n t r a s t to the more powerful thermodynamic r e l a t i o n s which enable us to c a l c u l a t e f u g a c i t i e s u s i n g only v o l u m e t r i c p r o p e r t i e s . From a s t r i c t l y thermodynamic p o i n t of view, u s i n g an equation of s t a t e i s more e f f i c i e n t than using a c t i v i t y c o e f f i c i e n t s . I f we have an equation of s t a t e a p p l i c a b l e to a l l phases of i n t e r e s t , we can c a l c u l a t e not only the f u g a c i t i e s from v o l u m e t r i c data but a l s o a l l the other c o n f i g u r a t i o n a l p r o p e r t i e s such as the enthalpy, entropy and volume change on mixing. Our i n a b i l i t y to use equations of s t a t e f o r many p r a c t i c a l s i t u a t i o n s f o l l o w s from our inadequate understanding of f l u i d s t r u c t u r e and i n t e r m o l e c u l a r f o r c e s . Only f o r simple s i t u a t i o n s do we have t h e o r e t i c a l i n f o r m a t i o n on s t r u c t u r e and f o r c e s f o r e s t a b l i s h i n g an equation of s t a t e w i t h a t h e o r e t i c a l b a s i s and only f o r the more

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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common f l u i d s do we have s u f f i c i e n t experimental i n f o r m a t i o n to e s t a b l i s h r e l i a b l e e m p i r i c a l equations of s t a t e . Thanks to c o r r e s ponding s t a t e s , we can extend the a v a i l a b l e e m p i r i c a l b a s i s to a much wider c l a s s of f l u i d s but again, we are l i m i t e d here because corresponding s t a t e s cannot e a s i l y be extended to p o l a r or hydrogenbonding m a t e r i a l s . Our b i g g e s t b o t t l e n e c k i s that we have not been able to e s t a b l i s h a u s e f u l s t a t i s t i c a l mechanical treatment f o r such f l u i d s nor even to c h a r a c t e r i z e the i n t e r m o l e c u l a r f o r c e s between t h e i r molecules. At l i q u i d - l i k e d e n s i t i e s , the d i p o l e moment i s not good enough and the s t r e n g t h of a hydrogen bond depends not only on p a r t i c u l a r c o n d i t i o n s l i k e d e n s i t y and temperature but, what i s worse, a l s o on the method used to measure i t . L a t e r i n the week, when we d i s c u s s the c o n t r i b u t i o n of theory, we s h a l l h o p e f u l l y r e t u r n to some of these problems Equations of S t a t e f o r Bot Let us now see what k i n d of p r a c t i c a l p h a s e - e q u i l i b r i u m problems we can handle using nothing beyond one of the many c u r r e n t l y a v a i l a b l e equations of s t a t e . For r e l a t i v e l y simple mixtures, e.g., those found i n processing of n a t u r a l gas and l i g h t petroleum f r a c t i o n s , we do w e l l w i t h one of the many m o d i f i c a t i o n s of the BenedictWebb-Rubin equation; i n i t s o r i g i n a l v e r s i o n , t h i s equation had e i g h t constants f o r each f l u i d but i n l a t e r v e r s i o n s t h i s number had i n c r e a s e d , sometimes c o n s i d e r a b l y so. To i l l u s t r a t e , Figure 3 shows c a l c u l a t e d and observed K f a c t o r s f o r methane i n heptane at two temperatures. In these c a l c u l a t i o n s , Orye (1) f o l l o w e d the u s u a l procedure; he assumed a o n e - f l u i d theory, i . e . , he assumed t h a t the equation of s t a t e of the mixture i s the same as that of a pure f l u i d except that the c h a r a c t e r i s t i c constants depend on composition according to some more or l e s s a r b i t r a r y r e l a t i o n s known as mixing r u l e s . Experience has repeatedly shown that at l e a s t one of these mixing r u l e s must c o n t a i n an a d j u s t a b l e b i n a r y constant; i n t h i s case, that constant i s M-^j which was found by f i t t i n g to the b i n a r y data. U n f o r t u n a t e l y , the c a l c u l a t e d r e s u l t s are o f t e n h i g h l y s e n s i t i v e to the mixing r u l e s and to the value of the a d j u s t a b l e parameter. In t h i s case Orye found what many others have a l s o found, v i z . , that the a d j u s t a b l e b i n a r y parameter i s more-or-less i n v a r i a n t w i t h d e n s i t y and composition but o f t e n depends on temperature. Another example, a l s o from Orye, i s given i n F i g u r e 4 f o r the system methane-carbon d i o x i d e at -65°F. The continuous l i n e through the diamonds i s not c a l c u l a t e d but connects the experimental p o i n t s of Donnelly and Katz; the c a l c u l a t e d l i n e s are dashed and the c i r c l e s and t r i a n g l e s i n d i c a t e p a r t i c u l a r c a l c u l a t i o n s , not data. F i r s t we note that the value of M^j has a strong e f f e c t , e s p e c i a l l y on the l i q u i d u s curve; a ten percent change i n M-^j produces a l a r g e e r r o r i n the bubble pressure. When M^j i s adjusted e m p i r i c a l l y to 1.8, much b e t t e r r e s u l t s are achieved but note that Orye r e p o r t s no c a l c u l a t i o n s i n the c r i t i c a l r e g i o n . There are two good reasons f o r t h i s : f i r s t , a l l c l a s s i c a l a n a l y t i c a l equations tend to be poor i n the

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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1200

Figure 3.

Methane-n-heptane (Orye, 1969) • O Kohn (1961); equation

Modified BWR

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

1000

1.0 Mole Fraction Methane

Figure 4. Methane-carbon dioxide (Orye, 1969). Temp., —65°F; 0 Donnelly and Katz (1954); O modified BWR equation, Mn = 1.8; A modified BWR equation, original mixing rule, Mn = 2.0.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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c r i t i c a l r e g i o n and second, computational problems are o f t e n severe because convergence i s hard to achieve. A s i m i l a r s i t u a t i o n i s shown i n F i g u r e 5, taken from S t a r l i n g and Han (2^), who used on 11-constant v e r s i o n of the BWR equation. Again, note t h a t an a d j u s t a b l e b i n a r y constant k^i i s r e q u i r e d . A l s o , note t h a t , c o n t r a r y to u s u a l p r a c t i c e , the l i n e s represent experiment and the p o i n t s represent c a l c u l a t i o n s , suggesting problems i n the c r i t i c a l r e g i o n . I t i s evident t h a t the more constants i n an equation of s t a t e , the more f l e x i b i l i t y i n f i t t i n g experimental data but i t i s a l s o c l e a r t h a t to o b t a i n more c o n s t a n t s , one r e q u i r e s more experimental i n f o r m a t i o n . For example, a twenty-constant equation of s t a t e , e s s e n t i a l l y an e x t e n s i o n of the BWR equation, was proposed by Bender (_3) who a p p l i e d i t to oxygen argon n i t r o g e n and a few l i g h t hydrocarbons. For these f l u i d s accurate r e p r e s e n t a t i o n range. To i l l u s t r a t e one u n u s u a l l y f i n e f e a t u r e of Bender's equat i o n , F i g u r e 6 shows the r e s i d u a l heat c a p a c i t y of propylene f o r s e v e r a l temperatures near the c r i t i c a l temperature, 365 K. This i s a very s e n s i t i v e t e s t and Bender's equation does a remarkable j o b . Bender has a l s o a p p l i e d h i s equation to mixtures of argon, n i t r o g e n , and oxygen, u s e f u l f o r design of a i r - s e p a r a t i o n p l a n t s . For each b i n a r y mixture, Bender r e q u i r e s 3 b i n a r y parameters. With a l l these constants and a l a r g e computer program, Bender can c a l c u l a t e not only accurate v a p o r - l i q u i d e q u i l i b r i a but a l s o heats of mixing as shown i n F i g u r e 7. The heats of mixing here are very s m a l l and agreement between c a l c u l a t i o n and experiment i s e x t r a o r d i n a r y . However, i t i s c l e a r that c a l c u l a t i o n s of t h i s s o r t are r e s t r i c t e d t o those few systems where the molecules are simple and s m a l l , where we have no s i g n i f i c a n t p o l a r i t y , hydrogen bonding or other s p e c i f i c "chemical f o r c e s " and, u n f o r t u n a t e l y , t o those cases where we have l a r g e q u a n t i t i e s of experimental data f o r both pure f l u i d s and f o r b i n a r y m i x t u r e s . I n the process i n d u s t r i e s we r a r e l y meet a l l these necessary c o n d i t i o n s . I f our accuracy requirements are not extremely l a r g e , we can o f t e n o b t a i n good approximations u s i n g c a l c u l a t i o n s based on a simple equation of s t a t e , s i m i l a r i n p r i n c i p l e t o the Van der Waals equation. The s i m p l e s t s u c c e s s f u l v a r i a t i o n of Van der Waals' equation i s t h a t by R e d l i c h and Kwong, proposed i n 1949. That equation, i n t u r n , i s to a p p l i e d thermodynamics what Helen of Troy has been to l i t e r a t u r e ; you r e c a l l t h a t i t was the b e a u t i f u l Helen who i n s p i r e d the l i n e "...the face t h a t launched a thousand s h i p s . " Ten years ago the B e a t l e s turned on an e n t i r e g e n e r a t i o n of teenagers and i n s p i r e d c o u n t l e s s v a r i a t i o n s and e x t e n s i o n s ; s i m i l a r l y , s t a r t i n g about ten years ago, the Redlich-Kwong equation i n i t i a t e d an epoch of i m i t a t i o n unequalled i n the h i s t o r y of a p p l i e d thermodynamics. The number of m o d i f i e d RK equations i s probably c l o s e to a hundred by now and, s i n c e I am amongst f r i e n d s , I must confess to having c o n s t r u c t e d a few myself. A few years ago, there was an a r t i c l e i n Chemical Engineering Science devoted e x c l u s i v e l y to v a r i a t i o n s on

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Figure 5.

Predicted and experimental K-values for the methane-hydrogen sulfide system at 40°F

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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373 15 K

365 15 K

Measurements of Bier et al o

•

348 15 K 365 15 K 373 15 K

X

36815 K 396 15 K

•

42315 K 473 15 K this work

e a. -388 15 K

398 15 K

348

Figure 6. Comparison of the residual isobaric heat capacities of propylene of Bier et al. with those predicted by the equation of state of Bender

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

60

c

t7 * / f /

i 40

20

Figure 7. Molar excess enthalpies of the binary system Ar-0 (Bender). Temp. = 84°K: O exptl, equation of state of Bender; Temp. = K: + exptl, equation of state 2

EXPERIMENTAL (BESSERER A N D ROBINSON TEMP F LIQUID VAPOR 100 • o 220 • •

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Figure 8. Pressure-equilibrium phase composition diagram for isobutane-carbon dioxide system, calculations using Peng-Robinson equation of state In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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the RK equation but i t i s now h o p e l e s s l y out of date and even then, between the time the paper was w r i t t e n and the time i t was p u b l i s h e d , seven new v a r i a t i o n s had appeared. (I get much of t h i s i n f o r m a t i o n d i r e c t l y from Otto R e d l i c h , who keeps a c l o s e eye on " c h i l d r e n " of h i s 1949 a r t i c l e . I n c i d e n t a l l y , I am happy to r e p o r t that Otto, aged 80, i s w e l l , a c t i v e and very pleased about the recent p u b l i c a ­ t i o n of h i s thermodynamics book by E l s e v i e r . Whenever Otto has a need to f e e l young a g a i n , he t a l k s w i t h the i n c r e d i b l e J o e l Hildebrand who, at 95, i s h a l e , h e a r t y , i n good humor and busy w r i t i n g a monograph on transport properties i n l i q u i d s . ) Perhaps the most s u c c e s s f u l v a r i a t i o n on the RK equation i s that proposed by Soave (4) who expresses the RK constant a by an e m p i r i c a l f u n c t i o n of reduced temperature and a c e n t r i c f a c t o r . This e m p i r i c a l f u n c t i o n was determined from vapor-pressure data f o r p a r a f f i n s and t h e r e f o r e , when Soave*s equatio and one a d j u s t a b l e b i n a r t y p i c a l l i g h t - h y d r o g e n m i x t u r e s ; however, i t p r e d i c t s poor l i q u i d d e n s i t i e s . This i l l u s t r a t e s a p o i n t known to a l l workers i n the e q u a t i o n - o f - s t a t e f i e l d ; i t i s not d i f f i c u l t to represent any one thermodynamic property but i t i s d i f f i c u l t , w i t h one equation of s t a t e , to represent them a l l . A comparatively recent v a r i a t i o n on the RK equation was proposed by Peng and Robinson ( 5 ) ; i t i s s i m i l a r to Soave s equation but appears to have b e t t e r behavior i n the c r i t i c a l r e g i o n ; an example i s given i n F i g u r e 8 f o r the isobutane-carbon d i o x i d e system. I n t h i s case the c r i t i c a l r e g i o n i s p r e d i c t e d w e l l and the a d j u s t a b l e b i n a r y parameter i s independent of temperature i n the r e g i o n 100 to 220°F. C a l c u l a t i n g phase e q u i l i b r i a from v o l u m e t r i c data does not n e c e s s a r i l y r e q u i r e an a n a l y t i c a l equation of s t a t e . The v o l u m e t r i c data can be s t o r e d i n t a b u l a r or a n a l y t i c a l form f o r an a r b i t r a r i l y chosen r e f e r e n c e substance and then, u s i n g corresponding s t a t e s , these data can be used to p r e d i c t p r o p e r t i e s of other f l u i d s , i n c l u d i n g m i x t u r e s . This procedure, o f t e n c a l l e d the p s e u d o - c r i t i c a l method o r , i n a more elegant form, the theory of conformai s o l u t i o n s , has been a p p l i e d by numerous authors. Here time permits me t o c a l l a t t e n t i o n to only one example, a p a r t i c u l a r l y u s e f u l one, i n i t i a t e d by Rowlinson and Mollerup and e x t e n s i v e l y developed by Mollerup i n recent years ( 6 ) . Using Goodwin's e x c e l l e n t experimental data f o r methane as a r e f e r e n c e , Mollerup c a l c u l a t e s w i t h h i g h accuracy thermodynamic p r o p e r t i e s of mixtures encountered i n the n a t u r a l - g a s i n d u s t r y . To do so, he uses the o l d Van der Waals mixing r u l e s but he pays very c l o s e a t t e n t i o n to the a l l - i m p o r t a n t b i n a r y constants. F i g u r e 9 shows e x c e l l e n t agreement between c a l c u l a t e d and e x g e r i mental r e s u l t s f o r the system methane-ethane from 130 to 200 K, u s i n g only one temperature-independent b i n a r y constant. Even more impressive i s the e x c e l l e n t r e p r e s e n t a t i o n f o r carbon monoxidemethane shown i n F i g u r e 10 where the c r i t i c a l r e g i o n i s reproduced almost w i t h i n experimental e r r o r . F i n a l l y , F i g u r e 11 shows that the corresponding-states method a l s o g i v e s e x c e l l e n t e n t h a l p i e s of 1

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

METHANE 199.9? K

J

»

J

i_

1

5

10

50

PRE S S U R E . A T M

Figure 9. K-values vs. pressure for methane-ethane mixtures; corresponding-states method of Mollerup and Rowlinson

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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CARBON MONOXIDE

10

•J \9067k

1 2 3

178K

^k

05

01

005

METHANE —

Predicted

O Experimental 05

1

5

10

50

PRESSURE, ATM

Figure 10. K-values vs. pressure for methane-carbon monoxide mixtures (Mollerup, 1975)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

O

X

N

20

Figure 11.

=

0.44

40 60 P R E S S U R E , ATM

80

100

Excess enthalpy of methane-nitrogen mixtures at 201.2°K (Mollerup, 1975)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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mixing f o r gaseous mixtures at h i g h p r e s s u r e s , c o r r e c t l y r e p r o ducing the observed maxima when the excess enthalpy i s p l o t t e d a g a i n s t pressure. These few i l l u s t r a t i o n s should be s u f f i c i e n t to o u t l i n e our present p o s i t i o n w i t h respect to phase e q u i l i b r i u m c a l c u l a t i o n s using an equation of s t a t e f o r both phases. I have e a r l i e r pointed out some of the advantages of t h i s type of c a l c u l a t i o n but I want now to add one more: i f we can c o n s t r u c t an equation of s t a t e a p p l i c a b l e to normal f l u i d s and t h e i r b i n a r y m i x t u r e s , then we need not worry about how to c a l c u l a t e e q u i l i b r i a i n t e r n a r y (or h i g h e r ) m i x t u r e s . For mixtures of normal f l u i d s , pure-component parameters and b i n a r y parameters are almost always s u f f i c i e n t f o r c a l c u l a t i n g e q u i l i b r i a i n multicomponent m i x t u r e s . For multicomponent mixtures of normal f l u i d s , the o n e - f l u i d theory i s u s u a l l y s a t i s f a c t o r y u s i n g only pure-component and b i n a r tremendous importance i n mixtures are much more common than b i n a r i e s . P r e d i c t i n g m u l t i component e q u i l i b r i a u s i n g only pure-component and b i n a r y data i s perhaps one of the g r e a t e s t triumphs of a p p l i e d thermodynamics. Having p r a i s e d the uses o f equations of s t a t e , I must a l s o p o i n t out t h e i r contemporary l i m i t a t i o n s which f o l l o w from our i n a b i l i t y to w r i t e s e n s i b l e equations of s t a t e f o r molecules t h a t are very l a r g e o r very p o l a r , or both. That i s where the f r o n t i e r l i e s . I see l i t t l e p o i n t i n pursuing f u r t h e r the obsession of modif y i n g the Redlich-Kwong equation. We must i n t r o d u c e some new p h y s i c s i n t o our b a s i c n o t i o n s of how to c o n s t r u c t an equation of s t a t e and there we must r e l y on suggestions s u p p l i e d by t h e o r e t i c a l p h y s i c i s t s and chemists. U n f o r t u n a t e l y most of these are "argon people" although, I am happy t o say, i n the l a s t few years a few brave t h e o r i s t s have s t a r t e d to t a c k l e n i t r o g e n . Some computer-type t h e o r i s t s have spent a l o t of time on water and on p r o t e i n s but these h i g h l y complicated s t u d i e s are s t i l l f a r removed from e n g i neering a p p l i c a t i o n s . N e v e r t h e l e s s , there are some new t h e o r e t i c a l ideas which could be used i n f o r m u l a t i n g new equations of s t a t e s u i t a b l e f o r those f l u i d s that cannot now be d e s c r i b e d by the u s u a l equations of s t a t e . Not t h i s morning, but perhaps l a t e r i n t h i s conference, I hope to have an o p p o r t u n i t y to say a few words about that. Vapor-Phase F u g a c i t y C o e f f i c i e n t s I now t u r n to what I have e a r l i e r c a l l e d Method ( b ) , that i s , f u g a c i t y c o e f f i c i e n t s f o r the vapor phase only and a c t i v i t y c o e f f i c i e n t s f o r a l l condensed phases. Method (b) i s used whenever we d e a l w i t h mixtures c o n t a i n i n g molecules t h a t are l a r g e or p o l a r or hydrogen-bonded or e l s e when a l l components are s u b c r i t i c a l and the pressure i s low. At modest vapor d e n s i t i e s , our most u s e f u l t o o l f o r vapor-phase f u g a c i t y c o e f f i c i e n t s i s the v i r i a l equation of s t a t e truncated a f t e r the second term. For r e a l f l u i d s , much i s known about second v i r i a l

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY BP

c

_

j(o)

-

J (T )

j ( l ) +

R

J (T ) R

B = SECOND VIRIAL COEFFICIENT; P

- CRITICAL PRESSURE;

c (j) « ACENTRIC FACTOR

j

AND f

T

R

j(2)

J (T )

+

R

=* T/T

c

T = CRITICAL TEMPERATURE c

ARE KNOWN FUNCTIONS SIMILAR TO THOSE

FIRST PROPOSED BY PITZER AND CURL.

TSONOPOULOS PROPOSES

FOR NONPOLAR FLUID FOR POLAR (NONHYDROGEN-BONDED) FLUIDS

a * 0

BUT

b * 0.

Figure 12. Correlation of second virial coefficients (Tsonopoulos)

0

100

200

REDUCED DIPOLE MOMENT, / /

300 R

Figure 13. Dependence of a on reduced dipole moment for nonhydrogen bonding compounds (Tsonopoulos, 1974)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

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29

c o e f f i c i e n t s ; l i t t l e i s known about t h i r d v i r i a l c o e f f i c i e n t s and n e a r l y nothing i s known about higher v i r i a l c o e f f i c i e n t s . Therefore, a p p l i c a t i o n i s l i m i t e d to moderate d e n s i t i e s , t y p i c a l l y d e n s i t i e s up to about 1/2 the c r i t i c a l . There are two major advantages of the t h e o r e t i c a l l y - d e r i v e d v i r i a l equation: f i r s t , the v i r i a l c o e f f i c i e n t s can be q u a n t i t a t i v e l y r e l a t e d to the i n t e r m o l e c u l a r f o r c e s and second, extension to mixtures r e q u i r e s no a d d i t i o n a l assumptions. For e n g i n e e r i n g , the f i r s t advantage i s important because i t enables us to i n t e r p r e t , c o r r e l a t e and m e a n i n g f u l l y e x t r a p o l a t e l i m i t e d v i r i a l c o e f f i c i e n t data, and the second i s important because we do not have to guess a t a r b i t r a r y mixing r u l e s f o r expressing the composition dependence of the v i r i a l c o e f f i c i e n t s . Any standard thermodynamics t e x t t e l l s us how t o c a l c u l a t e f u g a c i t y c o e f f i c i e n t s from the v i r i a l equation of s t a t e The most important problem i s to estimat we can u t i l i z e an extende by the c o r r e l a t i o n of Tsonopoulos (7) shown i n F i g u r e 12. The f i r s t term on the r i g h t holds f o r simple f l u i d s ; the second term c o r r e c t s for a c e n t r i c i t y and the t h i r d term c o r r e c t s f o r p o l a r i t y and hydrogen bonding. The constants a and b cannot be completely g e n e r a l i z e d but good estimates are o f t e n p o s s i b l e by observing trends w i t h i n chemical f a m i l i e s . F i g u r e 13 shows r e s u l t s f o r constant a p l o t t e d a g a i n s t a dimensionless d i p o l e moment; s i n c e p o l a r i t y i n c r e a s e s a t t r a c t i v e f o r c e s , we f i n d , as shown, that constant a becomes more negative as the reduced d i p o l e moment r i s e s , g i v i n g a more negative second v i r i a l c o e f f i c i e n t . F i g u r e 14 gives some r e s u l t s f o r constant b f o r a l c o h o l s , again p l o t t e d a g a i n s t reduced d i p o l e moment. Note t h a t the p o s i t i o n of the OH r a d i c a l has a n o t i c e a b l e e f f e c t . For these f l u i d s cons t a n t a i s s l i g h t l y p o s i t i v e because the hydrogen-bonding nature expressed by constant b dominates, e s p e c i a l l y at lower temperatures. To estimate c r o s s v i r i a l c o e f f i c i e n t s B ] ^ ' one must make some assumptions about the i n t e r m o l e c u l a r f o r c e s between molecules 1 and 2 and then s u i t a b l y average the molecular parameters appearing i n the c o r r e l a t i o n . Only f o r simple cases can any general r u l e s be used; whenever we have p o l a r components, we must l o o k c a r e f u l l y a t the molecular s t r u c t u r e and use judgment which, u l t i m a t e l y , i s based on experience. The v i r i a l equation i s u s e f u l f o r many cases but, when there i s strong a s s o c i a t i o n i n the vapor phase, the t h e o r e t i c a l b a s i s of the v i r i a l equation i s not v a l i d and we must r e s o r t to what i s commonly c a l l e d a "chemical treatment", u t i l i z i n g a chemical e q u i l i b r i u m constant f o r d i m e r i z a t i o n . D i m e r i z a t i o n i n the vapor phase i s e s p e c i a l l y important f o r organic a c i d s and even a t low p r e s s u r e s , the vapor-phase f u g a c i t y c o e f f i c i e n t s of mixtures c o n t a i n i n g one (or more) o r g a n i c a c i d are s i g n i f i c a n t l y removed from u n i t y . F i g u r e 15 shows some r e s u l t s based on the c o r r e l a t i o n of Hayden and O C o n n e l l (8) c a l c u l a t e d by Tom Anderson f o r the system p r o p i o n i c a c i d methyl i s o b u t y l ketone a t 1 atm along the v a p o r - l i q u i d s a t u r a t i o n curve. When the mole f r a c t i o n of a c i d i s very low, the f u g a c i t y c o e f f i c i e n t s a r f

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

30

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY i

1

1

1

r Ethanol

0.06-

2-Propanol \ ^

tert-Butanol

Methanol

?

2-Butanol s ^ V / ^ Isobutanol 1-Propanol 1-Butanol b = 0.00908 +0.0006957

f

J 0

10

( 2 )

= 0.0878/T

6

- b/T

R

R

8

L

20

30

40

50

60

REDUCED DIPOLE MOMENT, / /

Figure 14.

R

^ )

70

80

90

100

R

Dependence of b on reduced dipole moment for alcohols (Tsonopoulos, 1974)

Dew-Point Temperature, °C 1 —

ient at Satui

o o

M4.9 2.0

124.2 i

129.3 1

!33.'2 1

136.9 i

-

1.5

1.0

'

-

*2

o

Coeff

140.0

-

0.7

-

o 0.5 o o» if

i i

0.3 0

0.2

1 0.4

1 0.6

i 0.8

1.0

Vapor-Phase Mole Fraction Propionic Acid Figure 15.

Fugacity coefficients for saturated propionic acid (1)—methyl isobutyl ketone (2) at 1 atm

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

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Review of Phase Equilibria

31

near u n i t y because d i m e r i z a t i o n between a c i d molecules i s n e g l i g i b l e . As the mole f r a c t i o n of a c i d r i s e s , d i m e r i z a t i o n becomes i n c r e a s i n g l y l i k e l y and t h e r e f o r e , on the r i g h t s i d e of the diagram, the f u g a c i t y c o e f f i c i e n t s of both components are w e l l removed from u n i t y even though the temperature i s reasonably high (140C) and the pressure i s only 1 atm. At high p r e s s u r e s , where the v i r i a l equation i s no longer u s e f u l , e m p i r i c a l equations must be used t o c a l c u l a t e f u g a c i t y c o e f f i c i e n t s . However, contrary to Method ( a ) , the equation of s t a t e now need not hold f o r both the vapor phase and the l i q u i d phase; v a l i d i t y i n the vapor phase i s s u f f i c i e n t . To i l l u s t r a t e , I now show some r e s u l t s using an equation d e v e l oped by de S a n t i s and Breedveld (9) f o r gases a t h i g h pressures c o n t a i n i n g water as one of the components As i n d i c a t e d i n Figure 16 the equation i s l i k e tha f o r c e constant a i s d i v i d e For mixtures of water w i t h nonpolar components, the c r o s s - c o e f f i c i e n t a-]^ i s found from the geometric-mean assumption, but i n t h i s assumpt i o n only the nonpolar p a r t of constant a f o r water i s used because the nonaqueous component i s nonpolar. Figure 17 shows that the modified RK equation gives good f u g a c i t y c o e f f i c i e n t s f o r aqueous water but t h i s i s hardly s u r p r i s i n g s i n c e the constants a and b were determined from steam-table data. More g r a t i f y i n g are the r e s u l t s shown i n Figure 18 which show that c a l c u l a t e d v o l u m e t r i c p r o p e r t i e s a t high pressures are i n e x c e l l e n t agreement w i t h e x p e r i ment f o r gaseous mixtures of water and argon. The equation of de S a n t i s and Breedveld has r e c e n t l y been a p p l i e d by Heidemann to the problem of w e t - a i r o x i d a t i o n . When vapor-phase f u g a c i t y c o e f f i c i e n t s are c a l c u l a t e d from t h i s equation of s t a t e , and l i q u i d - p h a s e f u g a c i t i e s are c a l c u l a t e d from the p r o p e r t i e s of pure water c o r r e c t e d f o r s o l u b i l i t y of gases i n the water, i t i s p o s s i b l e to c a l c u l a t e the saturated water content and other e q u i l i b r i u m prope r t i e s of combustion gases. Figure 19 shows the saturated water content i n n i t r o g e n and F i g u r e 20 shows how that water content i s enhanced when CO2 i s present i n the gas phase; r e s u l t s are shown f o r two molar compositions (dry b a s i s ) : 20% CO2, 80% N2 and 13% CO2, 87% N2. E s p e c i a l l y at moderate temperatures, the pressure of CO2 s u b s t a n t i a l l y r a i s e s the saturated water content. Figure 21 shows enthalpy c a l c u l a t i o n s , again based on the equation of s t a t e of de S a n t i s and B r e e d v e l t , u s e f u l f o r designing a w e t - a i r o x i d a t i o n process. In v a p o r - l i q u i d e q u i l i b r i a according to Method (b), f u g a c i t y c o e f f i c i e n t s c o n s t i t u t e only h a l f the s t o r y , u s u a l l y (but not always. ) the l e s s important h a l f , w h i l e i n l i q u i d - l i q u i d e q u i l i b r i a f u g a c i t y c o e f f i c i e n t s p l a y no r o l e at a l l . We now must t u r n our a t t e n t i o n to the l a s t and i n some respects the most d i f f i c u l t t o p i c , v i z . , the activity coefficient. 1

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY P

a(T)

RT

=

" ^

B

' T

(WATER) "

'

u

1 / 2

cm3 6

(o)

a(T)

v(v+b)

a

ra

/ °i

(1) a(T)

+

(NONPOLAR)

e

/TABULATED VALUES OBTAINED\ I FROM STEAM TABLES. J

(POLAR)

FOR BINARY MIXTURES CONTAINING WATER(1) AND NONPOLAR GAS(2)

b

=

y

l

b

l

+

a = Y& 1

a

Y

B

2 2

+ y

1

(o) = (a a )

1 2

x

1

/

2

2

(o) TO FIND a

x

, USE B

1 2

(SECOND VIRIAL COEFFICIENT) DATA FOR

MIXTURES OF WATER WITH N , Ar, CH , ETC. 2

4

Figure 16. Vapor-phase equation of state for mixtures containing water (de Santis and Breedveld)

Figure 17. Fugacity coefficients for gaseous water. Calculations using equation of state proposed by de Santis et al.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PRAUSNITZ

Review of Phase Equilibria

Figure 19. Water content of nitrogen; comparison with experiment (Heidemann)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

0

100

Figure 20.

200 TEMPERATURE ° C

Effect of C0 /N

Figure 21.

2

2

300

ratio on saturated water content (Heidemann)

Enthalpy of saturated combustion gases (Heidemann)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

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35

Liquid-Phase A c t i v i t y C o e f f i c i e n t s In p r i n c i p l e , everybody knows that an a c t i v i t y c o e f f i c i e n t has no s i g n i f i c a n c e unless there i s a c l e a r d e f i n i t i o n of the standard s t a t e to which i t r e f e r s . In p r a c t i c e , however, there i s a l l too o f t e n a tendency to n e g l e c t p r e c i s e s p e c i f i c a t i o n of the standard s t a t e and i n some cases f a i l u r e t o g i v e t h i s exact s p e c i f i c a t i o n can lead t o s e r i o u s d i f f i c u l t i e s . This problem i s e s p e c i a l l y important when we consider s u p e r c r i t i c a l components or e l e c t r o l y t e s i n l i q u i d mixtures and, a l i t t l e l a t e r , I s h a l l have a few comments on t h a t s i t u a t i o n . But f o r now, l e t us consider mixtures of t y p i c a l n o n e l e c t r o l y t e l i q u i d s a t a temperature where every component can e x i s t as a pure l i q u i d . I n that event, the s t a n d a r d - s t a t e f u g a c i t y i s the f u g a c i t y of the pure l i q u i d a t system temperature and pressure and that f u g a c i t y i s determine pressure. F i g u r e 22 reviews some well-known r e l a t i o n s between a c t i v i t y c o e f f i c i e n t s and excess f u n c t i o n s . A l l t h i s i s s t r i c t l y c l a s s i c a l thermodynamics and the e n t i r e aim here i s that of c l a s s i c a l thermodynamics; v i z . , to organize our knowledge of e q u i l i b r i u m p r o p e r t i e s i n an e f f i c i e n t way so t h a t , by r e l a t i n g v a r i o u s q u a n t i t i e s to one another, we can minimize the amount of experimental e f f o r t r e q u i r e d f o r engineering design. There are three noteworthy f e a t u r e s i n F i g u r e 22: 1. The excess f u n c t i o n s used here are i n excess of those which apply to a p a r t i c u l a r k i n d of i d e a l s o l u t i o n v i z . that ( e s s e n t i a l l y ) given by Raoult's law. This choice of i d e a l i t y i s a r b i t r a r y and f o r some s i t u a t i o n s a d i f f e r e n t d e f i n i t i o n of i d e a l s o l u t i o n may be more s u i t a b l e . F u r t h e r , choosing ( e s s e n t i a l l y ) R a o u l t s law as our d e f i n i t i o n of an i d e a l s o l u t i o n , we are n a t u r a l l y l e d to the use of mole f r a c t i o n x as our c h o i c e of composition v a r i a b l e . That i s not n e c e s s a r i l y the best c h o i c e and there are s e v e r a l cases ( n o t a b l y , polymer s o l u t i o n s and s o l u t i o n s of e l e c t r o l y t e s ) where other measures of composition are much more convenient. 2. Equation (2) can be d e r i v e d from Equation (1) only i f we use the Gibbs-Duhem equation. Therefore, i f we organize our e x p e r i mental i n f o r m a t i o n a c c o r d i n g t o the scheme suggested by F i g u r e 22, we assure that the f i n a l r e s u l t s obey at l e a s t a c e r t a i n degree of thermodynamic c o n s i s t e n c y . 3. The excess Gibbs energy g^ i s a combination of two terms as shown i n Equation ( 3 ) . When we t r y t o c o n s t r u c t models f o r g^, we o f t e n do so d i r e c t l y but, i f we want our model to have p h y s i c a l s i g n i f i c a n c e we should i n s t e a d make models f o r h^ and s^ because these are the p h y s i c a l l y s i g n i f i c a n t q u a n t i t i e s t h a t can be r e l a t e d to molecular behavior; g^ i s only an o p e r a t i o n a l combination of them. The excess enthalpy i s concerned p r i m a r i l y w i t h e n e r g e t i c i n t e r a c t i o n s between molecules w h i l e excess entropy i s concerned p r i m a r i l y w i t h the s t r u c t u r e of the s o l u t i o n , i . e . , the s p a t i a l arrangements of the molecules which leads us to concepts l i k e randomness and segreg a t i o n . U n f o r t u n a t e l y , excess enthalpy and excess entropy are not f

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

36

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

(1)

x

E

T

±

(2)

(USES RAOULT S LAW FOR IDEALITY)

g = RT Y, ± l i

1

lnT

= ACTIVITY COEFFICIENT;

RT I n * = f T 6 n

E\ g

)

(BASED ON GIBBS-DUHEM EQUATION)

= TOTAL NO MOLES;

(3)

g

x = MOLE FRACTION

n

±

= NO MOLES OF COMPONENT i

Ts

E

DETERMINED PRIMARILY BY INTERMOLECULAR FORCES

DETERMINED PRIMARILY BY MOLECULAR STRUCTURE (SIZE, SHAPE, POSITION, DEGREES OF FREEDOM)

Figure 22.

Excess functions and activity coefficients

0.02 9 - (*i V

x v)

E

2

2

agreement between c a l c u l a t e d and experimental r e s u l t s i s much improved and, as shown here f o r a l i m i t e d c l a s s of mixtures, i t i s possible to correlate molecular s t r u c t u r e . The parameter Z^_2 ^ H i a b s o l u t e v a l u e but i t has a pronounced e f f e c t , as i n d i c a t e d i n F i g u r e 24. The b a s i c i d e a of s o l u b i l i t y parameter was f i r s t d e s c r i b e d by Hildebrand about 50 years ago. I t was b a r e l y known to chemical e n g i neers u n t i l about 20 years ago but s i n c e then i t has been both used and abused e x t e n s i v e l y f o r a v a r i e t y of purposes, both l e g i t i m a t e and otherwise, f a r beyond J o e l H i l d e b r a n d s w i l d e s t dreams. I t shows up i n the p a i n t and v a r n i s h i n d u s t r y , i n m e t a l l u r g y , p h y s i o l o g y , c o l l o i d chemistry and pharmacology and r e c e n t l y I have seen i t m u t i l a t e d i n a magazine a r t i c l e on " s c i e n t i f i c " a s t r o l o g y . Before l e a v i n g the s o l u b i l i t y parameter, I want to p o i n t out one use which, w h i l e not new, has perhaps not r e c e i v e d as much a t t e n t i o n as i t deserves. I r e f e r t o the use of the s o l u b i l i t y parameter f o r d e s c r i b i n g the s o l v e n t power of a dense gas, w i t h p a r t i c u l a r r e f e r e n c e to high-pressure gas e x t r a c t i o n , c e r t a i n l y not a n o v e l process but one which i s r e c e i v i n g renewed a t t e n t i o n i n c o a l l i q u e f a c t i o n and i n food processing. E

E

E

E

E

E

E

E

E

E

w

s s

m

a

i

t

n

n

!

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY 1 o

0

1

1

r

0.6

0.8

EXPERIMENTAL

0.2 MOLE

0.4 FRACTION

1.0

BENZENE

Figure 24. Experimental and calculated excess Gibbs energies for two binaries containing benzene at 50°C (Funk)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

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Review of Phase Equilibria

Using the well-known t a b l e s by P i t z e r , i t i s e a s i l y p o s s i b l e to c o n s t r u c t a g e n e r a l i z e d s o l u b i l i t y parameter diagram as shown i n F i g u r e 25. I n t h i s p a r t i c u l a r case the a c e n t r i c f a c t o r 0.075 i s c l o s e to that of ethylene because when t h i s chart was prepared over ten years ago, a p p l i c a t i o n was d i r e c t e d a t i n t e r p r e t i n g s o l u b i l i t y data f o r naphthalene i n ethylene at high pressure. J u s t to o r i e n t o u r s e l v e s , when the temperature i s around 20 C and the pressure i s about 400 atm, the s o l u b i l i t y parameter i s i n the r e g i o n 5 o r 6 (cal/cm^)!/^, only about one o r two u n i t s lower than that of a l i q u i d p a r a f f i n l i k e hexane. When s o l u b i l i t y data i n compressed ethylene are used w i t h the Scatchard-Hildebrand equation to back-out a s o l u b i l i t y parameter f o r l i q u i d naphthalene, we f i n d r e s u l t s shown i n Figure 26. The remarkable f e a t u r e s of t h i s f i g u r e are f i r s t , that the s o l u b i l i t y parameter obtained i s i n good agreement w i t h what one would o b t a i n by e x t r a p o l a t i n l i q u i d naphthalene to temperature that the backed-out s o l u b i l i t y parameter i s n e a r l y constant w i t h pressure. This i n d i c a t e s that the Hildebrand r e g u l a r s o l u t i o n equat i o n i s u s e f u l f o r mixtures of nonpolar f l u i d s r e g a r d l e s s of whether these are l i q u i d s or gases, provided only that the d e n s i t y i s s u f f i ciently large. F i n a l l y , a u s e f u l f e a t u r e of the s o l u b i l i t y parameter i s shown i n F i g u r e 27 f o r n i t r o g e n . S i m i l a r diagrams can be constructed f o r any f l u i d ; n i t r o g e n i s here shown only as an example. Note that the s o l u b i l i t y parameter i s s t r o n g l y s e n s i t i v e to both pressure and temperature i n the c r i t i c a l r e g i o n . H i g h l y s e l e c t i v e e x t r a c t i o n can t h e r e f o r e be c a r r i e d out by s m a l l changes i n temperature and pressure. F u r t h e r , such s m a l l changes can be e x p l o i t e d f o r e f f i c i e n t s o l v e n t r e g e n e r a t i o n i n continuous s e p a r a t i o n processes. L o c a l Composition to Describe Nonrandomness For mixtures c o n t a i n i n g p o l a r and hydrogen-bonded l i q u i d s , equations f o r g which emphasize h ( r a t h e r than s ) tend to be u n s a t i s f a c t o r y because i n t h e i r b a s i c f o r m u l a t i o n such equations give l i t t l e a t t e n t i o n to the d i f f i c u l t problem of nonrandomness. In a r e g u l a r s o l u t i o n , the molecules are " c o l o r - b l i n d " which means that they arrange themselves i n a manner d i c t a t e d only by the r e l a t i v e amounts of the d i f f e r e n t molecules that are present. It i s easily p o s s i b l e to add a c o r r e c t i o n which takes i n t o account the e f f e c t of molecular s i z e , as given by the Flory-Huggins expression. (Size c o r r e c t i o n s are e s s e n t i a l f o r polymer s o l u t i o n s . ) V a r i a t i o n s on that expression [e.g., Staverman or Tompa (12)] can a l s o account, i n p a r t , f o r d i f f e r e n c e s i n molecular shape. But, f o r s t r o n g l y i n t e r a c t i n g molecules, r e g a r d l e s s of s i z e and shape, there are l a r g e d e v i a t i o n s from random mixing; such molecules are f a r from " c o l o r - b l i n d " because t h e i r choice of neighbors i s h e a v i l y i n f l u e n c e d by d i f f e r e n c e s i n i n t e r m o l e c u l a r f o r c e s . An i n t u i t i v e idea toward d e s c r i b i n g t h i s i n f l u e n c e was introduced by Wilson w i t h h i s n o t i o n of l o c a l compos i t i o n , shown s c h e m a t i c a l l y i n F i g u r e 28 (13). Viewed m i c r o s c o p i c a l l y , E

E

E

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

40

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

I

Figure 25.

2

3

4

5

6

7

8

9

10

Solubility parameters for dense gases with an acentric factor of 0.075

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

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41

Review of Phase Equilibria

200

220

240

280

Figure 26. Solubility parameter of naphthalene calculated from solubility data in gaseous ethylene

TEMPERATURE, *K

Figure 27. Solubility parameter for nitrogen

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

42

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

15 of type 1

Overall mole fractions: Xi = x — \ Local mole fractions: 2

*

2 1

_ Molecules of 2 about a central molecule 1 ~~ Total molecules about a central molecule 1 1, as shown

*2i+*n

=

*12+*22

= 1

X21

~ f

Figure 28. Local compositions and the concept of local mole fractions (Cukor)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

2.

Review of Phase Equilibria

PRAUSNITZ

43

a s o l u t i o n i s not homogeneous because molecules have d e f i n i t e p r e f e r ences i n choosing t h e i r immediate environment, l e a d i n g to very s m a l l r e g i o n s , sometimes c a l l e d domains, which d i f f e r i n composition. There i s no obvious way to r e l a t e l o c a l composition to o v e r a l l ( s t o i c h i o m e t r i c ) composition but the use of Boltzmann f a c t o r s provides us w i t h one reasonable method f o r doing so. I n the l a s t ten y e a r s , Wilson's equation f o r g has enjoyed much p o p u l a r i t y and, f o r c e r t a i n m i x t u r e s , notably alcohol-hydrocarbon s o l u t i o n s , i t i s remarkably good. U n f o r t u n a t e l y , i t has one major f l a w : i t i s not a p p l i c a b l e to part i a l l y m i s c i b l e mixtures and, as a r e s u l t , there has been a f l u r r y of a c t i v i t y to extend and modify Wilson's equation; new m o d i f i c a t i o n s appear almost monthly. Time does not permit me to d i s c u s s any of these m o d i f i c a t i o n s but I want to p o i n t out an important development which only r e c e n t l y has become i n c r e a s i n g l y evident. In many cases the c h o i c e of model f o r g f o r data r e d u c t i o n ; that meters from l i m i t e d experimental data i s o f t e n more important than d e t a i l s w i t h i n the model. When reducing experimental data t o o b t a i n model parameters, a t t e n t i o n must be given to the e f f e c t of experimental e r r o r s . Not a l l experimental measurements are e q u a l l y v a l u a b l e and t h e r e f o r e , a proper s t r a t e g y f o r weighting i n d i v i d u a l experimental p o i n t s i s needed to o b t a i n "best" parameters (14, 15). Any r e a l i s t i c s t r a t e g y shows a t once that f o r any given set of b i n a r y data, there i s no unique s e t of model parameters. I n a t y p i c a l case, there are many s e t s of parameters which are e q u a l l y good, as i l l u s t r a t e d i n Figure 29 prepared by Tom Anderson f o r the system ethanol-cyclohexane. I n t h i s p a r t i c u l a r case, the model used was Wilson's but that i s not important here. The important message i s t h a t any p o i n t i n the e l l i p s e s shown can represent the experimental data e q u a l l y w e l l ; there i s no s t a t i s t i c a l s i g n i f i c a n c e i n p r e f e r r i n g one p o i n t over another. The e l l i p t i c a l r e s u l t s shown i n F i g u r e 29 are t y p i c a l ; the parameters have a tendency to be c o r r e l a t e d but, without a d d i t i o n a l i n f o r m a t i o n , i t i s not p o s s i b l e to say which p o i n t w i t h i n the e l l i p s e i s the b e s t . Since the two e l l i p s e s do not o v e r l a p , we are j u s t i f i e d i n a s s i g n i n g a temperature dependence to the parameters. When we do so, we o b t a i n the pressure-composition diagrams shown i n F i g u r e 30. I n t h i s case we assumed that the temperature dependence of one parameter i s p a r a l l e l to that of the other ( i . e . , we used t h r e e , not f o u r , a d j u s t a b l e parameters s i n c e b has no s u b s c r i p t s ) but that procedure i s not always s u c c e s s f u l ; f r e q u e n t l y f o u r parameters are r e q u i r e d . On the other hand, i f the two e l l i p s e s i n F i g u r e 29 had a r e g i o n of o v e r l a p , there would be no good reason to use temperature-dependent parameters; two temperature-independent parameters would be sufficient. Another example prepared by Tom Anderson i s shown i n F i g u r e 31 f o r the system butanol-water; i n t h i s case the UNIQUAC model was used r a t h e r than Wilson's because we are concerned w i t h v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a . The l e f t s i d e shows t h a t when vaporl i q u i d and l i q u i d - l i q u i d e q u i l i b r i a are reduced s e p a r a t e l y , we o b t a i n E

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

In

ACS

Phase

Symposium

— X22 i

* A

2

- in

* Xi2

2

i

= a « + b/T = a + b/T

• Data of Scatchard and Satkiewicz, 1964

Figure 30. Vapor-liquid equilibrium using Wilsons equation for ethanol(l)-cyclohexane(2)

| *

tg

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

300

~

1

\

Figure 31.

0

1

1 9

, cal/mole

-300

1

1

0

1

FROM VAPOR-LIQUID/ AND LIQUID-LIQUID DATA 20-118°C

1

UNIQ UAC parameters for butanol(l)-water(2) (—99% Confidence elipses)

Au

300

1

\ \ \ ^

FROM LIQUID-LIQUID

FROM VAPOR-LIQUID/ DATA 93-118 C

r\

NT"

1

>

300

46

PHASE

EQUILIBRIA

AND

FLUID

PROPERTIES

IN

CHEMICAL

INDUSTRY

two e l l i p s e s w i t h no o v e r l a p . When both sets of data are reduced s i m u l t a n e o u s l y , we o b t a i n the e l l i p s e shown on the r i g h t . F i g u r e 32 compares c a l c u l a t e d w i t h experimental r e s u l t s using temperatureindependent parameters. Agreement i s f a i r but not as good as we would l i k e i t to be. Further a n a l y s i s shows that s i g n i f i c a n t l y b e t t e r agreement cannot be obtained by a l l o w i n g the parameters to vary l i n e a r l y w i t h 1/T. For t r u l y s a t i s f a c t o r y agreement i t i s necessary f o r t h i s system to a s s i g n a quadratic dependence on 1/T. Once we have obtained good d e s c r i p t i o n s of b i n a r y l i q u i d mixt u r e s , we can o f t e n p r e d i c t the p r o p e r t i e s of multicomponent l i q u i d mixtures using only b i n a r y data. This procedure saves much e x p e r i mental e f f o r t and i t i s usually- s u c c e s s f u l f o r multicomponent vaporl i q u i d e q u i l i b r i a but o f t e n i t i s not f o r multicomponent l i q u i d liquid equilibria. Group-Contributions

for Activit

The v a r i e t y of equations based on the l o c a l composition concept has given us an improved t o o l f o r handling s t r o n g l y n o n i d e a l s o l u t i o n s but, perhaps more important, these equations have s t i m u l a t e d another development which, i n my view, i s p a r t i c u l a r l y promising f o r chemical engineering a p p l i c a t i o n . I r e f e r to the g r o u p - c o n t r i b u t i o n method f o r e s t i m a t i n g a c t i v i t y c o e f f i c i e n t s , a technique where a c t i v i t y c o e f f i c i e n t s can be c a l c u l a t e d from a t a b l e of groupi n t e r a c t i o n parameters. The fundamental i d e a , d a t i n g back over 50 years to Langmuir, i s that i n a l i q u i d s o l u t i o n of polyatomic molec u l e s , i t i s not the i n t e r a c t i o n s of molecules, but the i n t e r a c t i o n s of f u n c t i o n a l groups comprising the molecules (e.g., CH^, 2 > > e t c . ) which are important; F i g u r e 33 i l l u s t r a t e s the general i d e a . About 15 years ago, Deal, Derr and Wilson developed the ASOG g r o u p - c o n t r i b u t i o n method based on W i l s o n s equation where the important composition v a r i a b l e s are not the mole f r a c t i o n s of the components but the mole f r a c t i o n s of the f u n c t i o n a l groups (16, 17). In chemical technology the number of d i f f e r e n t f u n c t i o n a l groups i s much s m a l l e r than the number of molecular s p e c i e s ; t h e r e f o r e , the g r o u p - c o n t r i b u t i o n method provides a very powerful scale-up t o o l . With a r e l a t i v e l y s m a l l data base to c h a r a c t e r i z e group i n t e r a c t i o n s , i t i s p o s s i b l e to p r e d i c t a c t i v i t y c o e f f i c i e n t s f o r a very l a r g e number of systems, i n c l u d i n g those f o r which no experimental data are a v a i l a b l e . There i s no time now to d i s c u s s g r o u p - c o n t r i b u t i o n methods; we s h a l l have an o p p o r t u n i t y l a t e r t h i s week to go i n t o some d e t a i l s . Here I j u s t want to mention that some of the d i f f i c u l t i e s and l i m i t a t i o n s of the ASOG method have been overcome by a s i m i l a r method, c a l l e d UNIFAC (18), based on the UNIQUAC equation. Very r e c e n t l y , Fredenslund and Rasmussen i n Denmark and Gmehling and Onken i n Germany have s i g n i f i c a n t l y extended the e a r l i e r UNIFAC work; i n a p u b l i c a t i o n now i n p r e s s , the UNIFAC data base has been much enlarged and, t h e r e f o r e , the range of a p p l i c a t i o n i s now much i n c r e a s e d . The l a t e P r o f e s s o r R a t c l i f f at M c G i l l has a l s o developed a g r o u p - c o n t r i b u t i o n method and, d u r i n g the l a s t few years, P r o f e s s o r s Chao and N0

1

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

C 0 0 H

2.

PRAUSNiTz

Review

of Phase

Equilibria

47

Figure 32. Temperature-equilibrium phase composition diagram for butanol(l)— water(2) system. Calculations are based on UNIQUAC equation with temperature-independent parameters.

American Chemical Society Library 16thinSt., N.W. Industry; Storvick, T., et al.; In Phase Equilibria and Fluid1155 Properties the Chemical ACS Symposium Series; American Chemical Washington, D . C . Society: 20036Washington, DC, 1977.

48

P H A S E EQUILIBRIA

E.G.

A N D FLUID PROPERTIES

(1)

(2)

ACETONE

TOLUENE

IN CHEMICAL

(CH)

in

Y

i

=

In

Y

i

(COMBINATORIAL)

C

F (x,

F (X , K

MOLE

a,

Q/

R, a ,

FRACTION

GROUP

Figure 33.

• *n y .

R

C-Qh

(RESIDUAL)

R »

GROUP VOLUME

X »

MOLE FRACTION

Q

C

R

(

INDUSTRY

MN

T)

OF GROUP

INTERACTION

K;

PARAMETER

Group contributions to activity coefficients y and y t

2

Q2 ( - )

301 OA

r = 0.502 b = 2.62

-0.6

0.2

0.4

( - ) (cal/cc) 0.6

0.8

1.0

Figure 34. Activity coefficients for ethanol (A)-triethy Limine (B) system at 34.85°C (Nitta and Katayama, 1973)

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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Greenkorn at Purdue have s t a r t e d to work i n t h i s area. The groupc o n t r i b u t i o n method n e c e s s a r i l y provides only an approximation but f o r many a p p l i c a t i o n s that i s s u f f i c i e n t . For p r a c t i c a l - m i n d e d chemical engineers t h i s new r e s e a r c h i n a p p l i e d thermodynamics r e p r e sents perhaps the most e x c i t i n g development s i n c e P i t z e r ' s a c e n t r i c factor. Chemical Theory f o r A c t i v i t y C o e f f i c i e n t s While the l o c a l composition concept has been h i g h l y u s e f u l f o r s t r o n g l y n o n i d e a l m i x t u r e s , i t i s a l s o p o s s i b l e to represent data f o r such s o l u t i o n s by assuming t h a t molecules a s s o c i a t e o r s o l v a t e t o form new molecules. I t f o l l o w s from t h i s viewpoint that a b i n a r y mixture of A and B i s r e a l l y not a b i n a r y m i x t u r e but i n s t e a d a multicomponent mixture mers, a l s o polymers A2, A c o n t a i n i n g both A and B i n v a r i o u s p o s s i b l e s t o i c h i o m e t r i c p r o p o r t i o n s . D e v i a t i o n s from i d e a l behavior are then explained quant i t a t i v e l y by a s s i g n i n g e q u i l i b r i u m constants to each of the postul a t e d chemical e q u i l i b r i a . This i s a Pandora's box because, i f we assume a s u f f i c i e n t number of e q u i l i b r i a , a d j u s t the s t o i c h i o m e t r y of the polymers and copolymers and a l s o a d j u s t the e q u i l i b r i u m c o n s t a n t s , we can o b v i o u s l y f i t anything. N e v e r t h e l e s s , the chemical method makes sense provided we have independent chemical i n f o r m a t i o n (e.g., s p e c t r o s c o p i c data) which a l l o w s us to make s e n s i b l e a p r i o r i statements concerning what chemical species are present. For example, we know that a c e t i c a c i d forms dimers, t h a t a l c o h o l s polymerize to dimers, t r i m e r s , e t c . and that chloroform and acetone are l i n k e d through a hydrogen bond. Thus an " e n l i g h t e n e d " chemical theory can o f t e n be used to represent experimental data w i t h only a few parameters where a s t r i c t l y e m p i r i c a l equation r e q u i r e s many more parameters t o g i v e the same f i t . The l i t e r a t u r e i s r i c h i n examples of t h i s s o r t ; a recent one by N i t t a and Katayama (19) i s given i n F i g u r e 34 which shows a c t i v i t y c o e f f i c i e n t s f o r the system e t h a n o l t r i e t h y l a m i n e . Here A stands f o r a l c o h o l and B f o r amine. S u b s c r i p t C denotes chemical c o n t r i b u t i o n ; i n a d d i t i o n to the chemical e f f e c t s , there are p h y s i c a l f o r c e s between the " t r u e " molecules and these are taken i n t o account through the parameter b which has u n i t s of energy d e n s i t y . E q u i l i b r i u m constant K^, f o r continuous p o l y m e r i z a t i o n of e t h a n o l , i s obtained independently from alcohol-hydrocarbon mixture data. Parameter r i s the r a t i o of the e q u i l i b r i u m constant f o r A + B"Z5£AB t o K^. The e x c e l l e n t f i t i s , t h e r e f o r e , obtained w i t h two a d j u s t a b l e parameters, b and r . While chemical t h e o r i e s are o f t e n u s e f u l f o r d e s c r i b i n g s t r o n g l y n o n i d e a l l i q u i d m i x t u r e s , they are n e c e s s a r i l y s p e c i f i c , l i m i t e d to a p a r t i c u l a r type of s o l u t i o n . I t i s d i f f i c u l t to c o n s t r u c t a general theory, a p p l i c a b l e to a wide v a r i e t y of components, without i n t r o d u c i n g complicated a l g e b r a and, what i s worse, a p r o h i b i t i v e l y l a r g e number of parameters. This d i f f i c u l t y a l s o makes chemical theory i m p r a c t i c a l f o r multicomponent mixtures and indeed, w h i l e the

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l i t e r a t u r e i s r i c h w i t h a p p l i c a t i o n of chemical theory to b i n a r i e s , there are few a r t i c l e s which apply chemical theory to t e r n a r y (or higher) m i x t u r e s . For p r a c t i c a l chemical e n g i n e e r i n g , t h e r e f o r e , the chemical theory of l i q u i d s o l u t i o n s has l i m i t e d u t i l i t y . S u p e r c r i t i c a l Components i n the L i q u i d Phase I have p r e v i o u s l y s t r e s s e d the d i f f i c u l t y of standard s t a t e s when we d e a l w i t h s u p e r c r i t i c a l components. For these components, e.g., methane or n i t r o g e n , a t o r d i n a r y temperatures, i t has been common p r a c t i c e to i g n o r e the problem simply by e x t r a p o l a t i n g purel i q u i d f u g a c i t i e s to temperatures above the c r i t i c a l . This i s convenient but u l t i m a t e l y u n s a t i s f a c t o r y because there i s no unambiguous way to perform an e x t r a p o l a t i o n f o r a h y p o t h e t i c a l q u a n t i t y . The most commo p l o t of the f u g a c i t y versu Experience has shown t h a t at temperatures f a r above the c r i t i c a l , t h i s i s a bad assumption but r e g a r d l e s s of what shape the p l o t i s assumed to be, on semilog paper, the t h i c k n e s s of the p e n c i l can a l r e a d y make a s i g n i f i c a n t d i f f e r e n c e . The only p o s s i b l e s a t i s f a c t o r y procedure f o r proper use of a c t i v i t y c o e f f i c i e n t s of s u p e r c r i t i c a l components i s to use Henry's constants as the s t a n d a r d - s t a t e f u g a c i t y . Henry's constants are not h y p o t h e t i c a l but are e x p e r i m e n t a l l y a c c e s s i b l e ; a l s o , at l e a s t i n p r i n c i p l e , they can be c a l c u l a t e d from an equation of s t a t e . Remarkably l i t t l e a t t e n t i o n has been given to the formal thermodynamics of l i q u i d mixtures c o n t a i n i n g s u p e r c r i t i c a l components. Using Henry's constants i n t r o d u c e s a v a r i e t y of problems but they are by no means insurmountable. Y e t , chemical engineers have stubbornly r e s i s t e d u s i n g Henry's constants f o r s t a n d a r d - s t a t e f u g a c i t i e s ; whenever I have t r i e d to i n t e r e s t my i n d u s t r i a l colleagues i n t h i s p o s s i b i l i t y I f e l t l i k e a gun-control e n t h u s i a s t t a l k i n g to the N a t i o n a l R i f l e A s s o c i a t i o n . As long as the s o l u t i o n i s d i l u t e , Henry's constant i s s u f f i c i e n t but as the c o n c e n t r a t i o n of s o l u t e r i s e s , unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s must be introduced and at present we have l i t t l e experience w i t h these. While b i n a r y mixtures can be handled w i t h r e l a t i v e ease, major formal d i f f i c u l t i e s a r i s e when we go to multicomponent mixtures because, u n f o r t u n a t e l y , Henry's constant depends on both s o l u t e and s o l v e n t and, t h e r e f o r e , when we have s e v e r a l s o l v e n t s present, we must be very c a r e f u l to d e f i n e our s t a n dard s t a t e s and corresponding a c t i v i t y c o e f f i c i e n t s i n a thermodynamically c o n s i s t e n t way. About ten years ago the l a t e Ping Chueh and I wrote a monograph on the use of unsymmetric a c t i v i t y c o e f f i c i e n t s f o r c a l c u l a t i n g K f a c t o r s i n hydrocarbon and n a t u r a l - g a s m i x t u r e s , but i t never caught on. About f i v e years ago I was window shopping i n the Time Square s e c t i o n of New York and to my amazement I saw a copy of our monograph on a t a b l e i n a used book s t o r e , completely surrounded by books on pornography. I t appeared that my colleagues were t r y i n g to t e l l me something.

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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However, times change and now that pornography i s accepted w i t h l i t t l e o p p o s i t i o n , maybe Henry's constants f o r standard-state fugac i t i e s can be accepted too. John O'Connell a t F l o r i d a has been working on t h i s and i n Figure 35 we see some r e s u l t s f o r excess Henry's constants f o r ethylene, carbon d i o x i d e and carbon monoxide i n b i n a r y s o l v e n t mixtures (20). To a f i r s t approximation, the l o g a r i t h m of Henry's constant f o r a gas i n a mixed s o l v e n t i s given by a simple m o l e - f r a c t i o n average; F i g u r e 35 shows d e v i a t i o n s from that f i r s t approximation. F i g u r e 36 presents another example, given by N i t t a and Katayama (21); i t shows Henry's constant f o r n i t r o g e n i n mixtures of n-propanol and iso-octane. Two p l o t s a r e shown, one against mole f r a c t i o n and the other a g a i n s t volume f r a c t i o n of the s o l v e n t mixture. Since i s o octane i s a much l a r g e r molecule than propanol i t i s not s u r p r i s i n g that the v o l u m e - f r a c t i o n p l o f r a c t i o n p l o t , but, n e v e r t h e l e s s s t r a i g h t - l i n e behavior. Katayama a p p l i e s h i s chemical model f o r e x p l a i n i n g the d e v i a t i o n w i t h r e s u l t s shown i n Figure 37. One c o n t r i b u t i o n of the F l o r y Huggins type c o r r e c t s f o r s i z e d i f f e r e n c e s , another (chemical) c o n t r i b u t i o n c o r r e c t s f o r a s s o c i a t i o n of a l c o h o l molecules and f i n a l l y , a physical contribution corrects for differences i n intermolecular f o r c e s . The sum of the c o r r e c t i o n s gives good agreement w i t h e x p e r i ment. Since the c o r r e c t i o n s f o r s i z e and a s s o c i a t i o n were c a l c u l a t e d from other data, only one a d j u s t a b l e parameter was used i n preparing the f i n a l p l o t . Aqueous S o l u t i o n s of Weak V o l a t i l e E l e c t r o l y t e s I have i n d i c a t e d e a r l i e r that the chemical theory of l i q u i d mixtures presents some d i f f i c u l t i e s and that the use of Henry's constants a l s o gives us headaches. However, when we come to s o l u t i o n s of v o l a t i l e e l e c t r o l y t e s we are r e a l l y i n a bad way because now we must use not only the awkward chemical theory but i n a d d i t i o n , those unpleasant Henry's constants. We have no r e a l choice here because i n d i l u t e aqueous s o l u t i o n , weak v o l a t i l e e l e c t r o l y t e s (e.g., ammonia, hydrogen s u l f i d e , s u l f u r d i o x i d e ) d i s s o c i a t e i n t o ions and thus there i s r e a l chemistry going on which we cannot ignore. Further, s i n c e ions a r e n o n v o l a t i l e , we must use unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s ; the f u g a c i t y of a pure v o l a t i l e e l e c t r o l y t e l i q u i d which i s not i o n i z e d doesn't t e l l us anything that would be u s e f u l f o r a d i l u t e aqueous s o l u t i o n where the s o l u t e i s , a t l e a s t i n p a r t , i n i o n i c form. The s i t u a t i o n we must d e s c r i b e i s shown s c h e m a t i c a l l y i n Figure 38. The h o r i z o n t a l e q u i l i b r i u m i s chemical, c h a r a c t e r i z e d essent i a l l y by a d i s s o c i a t i o n constant, and the v e r t i c a l e q u i l i b r i u m i s p h y s i c a l , c h a r a c t e r i z e d e s s e n t i a l l y by Henry's constant. Detailed development toward q u a n t i t a t i v e r e s u l t s a l s o r e q u i r e s unsymmetrically normalized a c t i v i t y c o e f f i c i e n t s , i . e . , those a c t i v i t y c o e f f i c i e n t s which go t o u n i t y not as the composition approaches the pure s o l v e n t ,

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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Deviation of Henrys constant from that in an ideal solution (O'Connell)

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Figure 36. Henry's constants for nitrogen in n-propanol (A)-isooctane (B) mixture vs. mole fraction and volume fraction (Katayama et al., 1973)

Figure 37. Experimental and calculated In K-values vs. volume fraction for n-propanol (A)-isooctane (B) mixtures (Katayama et al, 1973) lnic = ln H ,

N2 m

— %®j In H ^ j

j

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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P

T

Vapor Phase

Mol ecular Elec trolyte

Mol ocular Elec rolyte ~

Figure 38. Vapor-liquid equilibrium in a single-solute system

(1)

^ Ions

Liquid Phase

MASS BALANCE m

A "

m

a

\

+

(m

+

+

m

-

)

m f MOLALITY SUBSCRIPT A = STOICHIOMETRIC SUBSCRIPT a - MOLECULAR

(2)

DISSOCIATION

K -

EQUILIBRIUM

V -

T ^ l

(3)

AS m ^ O

ELECTRONEUTRALITY m

(4)

ACTIVITY

+

*m

VAPOR-LIQUID EQUILIBRIA

^ a

P

~ a a m

T

H ( P C

>

H = HENRY'S CONSTANT PC - POYNTING CORRECTION (P = VAPOR-PHASE FUGACITY COEFFICIENT

Figure 39.

Aqueous solutions of weak volatile electrolytes

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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but i n s t e a d , as the aqueous s o l u t i o n becomes i n f i n i t e l y d i l u t e . F i g u r e 39 i n d i c a t e s the four c o n d i t i o n s t h a t must be s a t i s f i e d . For most s o l u t e s of i n t e r e s t , chemical e q u i l i b r i u m constant K i s known as a f u n c t i o n of temperature; the major d i f f i c u l t y l i e s ^ i n c a l c u l a t i n g H and y*. F i g u r e 40 shows two equations f o r y . Both s t a r t out w i t h the Debye-Hdckel term which depends p r i m a r i l y on i o n i c s t r e n g t h but t h a t term alone i s a p p l i c a b l e only to very d i l u t e s o l u t i o n s . Guggenheim adds an e s s e n t i a l l y e m p i r i c a l f i r s t - o r d e r c o r r e c t i o n and t h i s i s s u f f i c i e n t f o r i o n i c strengths to about 1 or 2 molar. For more concentrated s o l u t i o n s , P i t z e r has proposed a semit h e o r e t i c a l equation which, however, has many parameters and a l l of these depend on temperature (22). Time does not permit a d e t a i l e d d i s c u s s i o n but, i n view of the importance of these s o l u t i o n s i n chemical e n g i n e e r i n g l e t me q u i c k l y show a few r e s u l t s . F i g u r experimental data i n the regio aqueou reduced to y i e l d Henry's constants (the i n t e r c e p t ) and Guggenheim's c o e f f i c i e n t 3 (the s l o p e ) . Note t h a t the a b s c i s s a g i v e s the molecular m o l a l i t y of ammonia, not the t o t a l m o l a l i t y ; t h e r e f o r e , F i g u r e 41 i m p l i c i t l y i n c l u d e s the e f f e c t of i o n i z a t i o n as d e t e r mined by the independently-measured chemical d i s s o c i a t i o n constant. A s i m i l a r a n a l y s i s was made f o r s o l u t i o n s of CO2 i n water. F i g u r e 42 g i v e s p a r t i a l pressures f o r the t e r n a r y system ammoniacarbon d i o x i d e when the t o t a l ammonia m o l a l i t y i s 0.128; these r e s u l t s were p r e d i c t e d u s i n g only b i n a r y data; no t e r n a r y data were used. I n t h i s example the s o l u t i o n i s d i l u t e and Guggenheim's equat i o n i s adequate; f o r higher c o n c e n t r a t i o n s , P i t z e r ' s equation i s r e q u i r e d as shown i n F i g u r e 43 based on very recent (and as yet unpublished) work by Renon and coworkers. The l i n e on the r i g h t i s the same as the one shown i n the p r e v i o u s f i g u r e ; the m o l a l i t y i s low. The l i n e on the l e f t i s a t higher ammonia c o n c e n t r a t i o n and, as we proceed to higher r a t i o s of carbon d i o x i d e t o ammonia, the t o t a l m o l a l i t y goes w e l l above 2. We see that Edwards' l i n e , based on Guggenheim's equation, i s s a t i s f a c t o r y at f i r s t but shows i n c r e a s i n g d e v i a t i o n s as the t o t a l m o l a l i t y r i s e s . The r e s u l t s shown here are again based on b i n a r y data alone. A t 20°C, experimental data are r e l a t i v e l y p l e n t i f u l and i t was p o s s i b l e to evaluate a l l the parameters i n P i t z e r ' s equation but at higher temperatures, where good data are s c a r c e , i t i s not easy to use P i t z e r ' s equation u n t i l some r e l i a b l e method can be found to estimate how temperature a f f e c t s the parameters. F i n a l l y , I should mention t h a t the c a l c u l a t i o n s shown here are based on simultaneous s o l u t i o n of 14 equations. A good computer program i s an a b s o l u t e n e c e s s i t y . Conclusion I have t r i e d t h i s morning to present a survey of the present s t a t u s of a p p l i e d p h a s e - e q u i l i b r i u m thermodynamics. In one sense, the survey i s much too long because I am sure your p a t i e n c e has been pushed w e l l beyond i t s e l a s t i c l i m i t . In another sense, i t i s much

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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GUGGENHEIM

In

Ve.

r.

4-

1+/T WHERE:

.m. *J J

z = CHARGE A = KNOWN CONSTANT I = IONIC STRENGTH = -

) z.m. J J

2 V

PITZER

In

Az

r.

2

l57T

+

ZE i c

j k

m

J

m

+

!

l

n

(

1

+

b

/

r

)

k

WHERE: a=2 AND b = l . 2 IF

i AND j ARE MOLECULAR SPECIES,

0.

Figure 40. Activity coefficients in electrolyte solutions

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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Figure 42. Vapor-liquid equilibria at 20°C for ammonia-carbon dioxide-water containing excess ammonia

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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Figure 43.

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Calculated and experimental partial pressures of CO at 20°C for the C0 -H 0 system: Effect of total concentration (Renon et al.) 2

2

2

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

NH 3

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too s h o r t because I have had t o omit many worthwhile c o n t r i b u t i o n s . I a l r e a d y f e a r the hurt and i n s u l t e d looks that I am l i k e l y to r e c e i v e from some of you f o r the r e s t of the week, i f not l o n g e r ! Let me q u i c k l y summarize what to me are the main i m p l i c a t i o n s of t h i s f r a n k l y p e r s o n a l i z e d survey. F i r s t , those of us who are i n the u n i v e r s i t i e s must get over our a r g o n - f i x a t i o n and s t a r t t h i n k i n g b o l d l y about molecules that are l a r g e , n o n - s p h e r i c a l , p o l a r and hydrogen bonded. In other words, l e t us pay more a t t e n t i o n to the r e a l w o r l d . For chemical engineers i t i s b e t t e r , I t h i n k , t o s o l v e approximately new and r e a l problems than to improve m a r g i n a l l y s o l u t i o n s to o l d problems. I am hopeful that t h i s conference w i l l c o n t r i b u t e toward that end. Second, we must stop the game of composing v a r i a t i o n s on o l d themes. The Redlich-Kwong equation the BWR equation the Wilson equation, a l l these represen we should not honor them regard them as great monuments, i n s p i r i n g us toward t a c k l i n g new frontiers. Where, then, are these new f r o n t i e r s which demand our a t t e n t i o n ? I can here mention only s i x that I f i n d p a r t i c u l a r l y c h a l l e n g i n g i n t e l l e c t u a l l y and i n d u s t r i a l l y important: 1. C o n s t r u c t i o n of approximate, but p h y s i c a l l y s e n s i b l e , equat i o n s of s t a t e a p p l i c a b l e to complex molecules i n both gaseous and l i q u i d phases. 2. Vapor-phase experimental work (PVT and g a s - s a t u r a t i o n measurements) to provide fundamental i n f o r m a t i o n on i n t e r m o l e c u l a r f o r c e s i n asymmetric b i n a r y m i x t u r e s , i . e . , those mixtures where the two components are s t r o n g l y d i f f e r e n t , e i t h e r i n molecular s i z e , o r p o l a r i t y , or both. 3. V a p o r - l i q u i d e q u i l i b r i u m experiments on mixtures of complex molecules, i n c l u d i n g p o l y n u c l e a r aromatics, polymers and h i g h l y p o l a r s o l v e n t s such as g l y c o l s , p h e n o l i c s and other "nasty" l i q u i d s . The systems water-ethanol and benzene-cyclohexane have each been s t u d i e d about 50 times. Enough of t h a t . L e t ' s measure e q u i l i b r i a i n systems where we cannot now estimate the r e s u l t s w i t h i n even an order of magnitude. 4. More a t t e n t i o n must be given to data r e d u c t i o n methods. Some data are c l e a r l y more v a l u a b l e than others and we must i n c o r porate t h a t d i s t i n c t i o n i n t o our experimental p l a n s . 5. We can u s u a l l y do a p r e t t y good job c a l c u l a t i n g vaporl i q u i d e q u i l i b r i a f o r multicomponent mixtures of t y p i c a l n o n e l e c t r o l y t e s . However, f o r multicomponent l i q u i d - l i q u i d e q u i l i b r i a the s i t u a t i o n i s much l e s s f a v o r a b l e and we should g i v e more a t t e n t i o n to those e q u i l i b r i a . 6. F i n a l l y , l e t us l e a r n to use more the powerful methods of s t a t i s t i c a l mechanics; l e t us overcome our f e a r of p a r t i t i o n funct i o n s and l e t us not h e s i t a t e to i n t r o d u c e some e n l i g h t e n e d empiricism into t h e i r construction. This assembly of over 100 s c i e n t i s t s and engineers represents a wide v a r i e t y of knowledge, i n t e r e s t s and experiences. Our meeting

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here p r o v i d e s us w i t h a unique, unprecedented o p p o r t u n i t y to exchange views, s t i m u l a t i n g us a l l to new achievements. I have presented t h i s r a p i d overview of previous accomplishments w i t h the c o n v i c t i o n that these accomplishments must serve us not as ground f o r s e l f - c o n g r a t u l a t i o n but as a firmament on which to b u i l d toward a b r i g h t e r f u t u r e . My f e e l i n g — a n d I hope i t i s yours, t o o — m u s t be that of the French philosopher who s a i d , "From the a l t a r s of the past l e t us c a r r y the f i r e , not the ashes." Acknowledgment For f i n a n c i a l support extending over many y e a r s , the author i s g r a t e f u l to the N a t i o n a l Science Foundation, the Gas Processors A s s o c i a t i o n , the American Petroleum I n s t i t u t e the Donors of the Petroleum Research Fund (administere S o c i e t y ) , Union Carbide C o r p o r a t i o S p e c i a l thanks a r e due to Mr. Thomas F. Anderson f o r e x t e n s i v e assistance i n preparing t h i s report.

Literature Cited 1. Orye, R. V., Ind. Eng. Chem. Process Des. Dev., (1969) 8, 579. 2. Starling, K. E., and Han, M. S. Hydrocarbon Processing, (1972), 51, 107. 3. Bender, E., Cryogenics, (1973) 13, 11; (1975) 15, 667. 4. Soave, G., Chem. Eng. Sci., (1972) 27, 1197. 5. Peng, D., and Robinson, D. B., Ind. Eng. Chem. Fundam., (1976) 15, 59. 6. Mollerup, J., and Rowlinson, J. S., Chem. Eng. Sci., (1974) 29, 1373; Mollerup, J., Advan. Cryog. Eng., (1975) 20, 172. 7. Tsonopoulos, C., A.I.Ch.E. Journal, (1974) 20, 263. 8. Hayden, J. G., and O'Connell, J. P., Ind. Eng. Chem. Process Des. Dev., (1975) 14, 209. 9. de Santis, R., Breedveld, G. J. F., and Prausnitz, J. M., Ind. Eng. Chem. Process Des. Dev., (1974) 13, 374. 10. Wohl, K., Trans. A.I.Ch.E., (1946) 42, 215. 11. Funk, E. W., and Prausnitz, J. M., Ind. Eng. Chem., (1970) 62, 8. 12. Tompa, H., Trans. Faraday Soc., (1952) 48, 363; Staverman, A. J., Rec. Trav. Chim. Pays-bas, (1950) 69, 163; Donohue, M. D., and Prausnitz, J. M., Can. J. Chemistry, (1975) 53, 1586. 13. Cukor, P. M., and Prausnitz, J. M., Intl. Chem. Eng. Symp. Ser. No. 32 (Instn. Chem. Engrs., London) 3:88 (1969). 14. Anderson, T. F., Abrams, D. S., Grens, E. A., and Prausnitz, J. M., paper presented at the 69th Annual A.I.Ch.E. Meeting, Chicago, I l l i n o i s , 1976; submitted to A.I.Ch.E. Journal. 15. Fabries, J., and Renon, H., A.I.Ch.E. Journal, (1975) 21, 735. 16. Derr, E. L., and Deal, C. H., Intl. Chem. Eng. Symp. Ser. No. 32 (Instn. Chem. Engrs., London) 3:40 (1969).

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17. Wilson, G. M . , and Deal, C. H., Ind. Eng. Chem. Fundam., (1962) 1, 20. 18. Fredenslund, Aa., Jones, R. L., and Prausnitz, J . M., A.I.Ch.E. Journal, (1975) 21, 1086; Fredenslund, Aa., Michelsen, M. L . , and Prausnitz, J . M., Chem. Eng. Progr., (1976) 72, 67; Fredenslund, Aa., Gmehling, J., Michelsen, M. L., Rasmussen, P. and Prausnitz, J . M., Ind. Eng. Chem. Process Des. Dev. (in press). 19. Nitta, T., and Katayama, T., J . Chem. Eng. Japan, (1973) 6, 1. 20. Orye, R. V . , Ind. Eng. Chem. Process Des. Dev., (1969) 8, 579. 21. Nitta, T., and Katayama, T., J . Chem. Eng. Japan (1973) 6, 1. 22. Pitzer, K. S., and Kim, J . J., J . Amer. Chem. Soc. (1974) 96, 5701. 23. Edwards, T. J., Newman J., and Prausnitz J M. A.I.Ch.E Journal (1975) 21,

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

3 Industrial View of the State-of-the-Art in Phase Equilibria T. S. KROLIKOWSKI Union Carbide Corp., Chemicals and Plastics Div., S. Charleston, W.Va. 25303

Twenty-five years ago, in 1952, there was a series of articles in Chemical Engineering Progress (1) entitled: "Industrial Viewpoints on Separation Processes". In the section on phase equilibrium data, it was noted that "The complete representation of such data for mixtures containing more than three components becomes impractically complex". Simplified calculations for multicomponent systems were recommended, and i f the predicted values did not agree with experimental data, a system of minor correction factors should be devised. In the same year, one of the annual review articles in Industrial and Engineering Chemistry (2) mentioned that the BWR equation of state seemed to provide the most accurate method thus far developed for estimating K-factors for hydrocarbon systems. Use of the equation was deemed tedious, and a procedure for using the equation in a simplified form suitable for rapid equilibrium calculations was to be presented. Charts based on the procedure were available from the M. W. Kellogg Co., New York. Another review article (3) observed that automatic computers have entered the field of ditillation calculations. The author remarks: "The difficulty is that the machines cannot evaluate the errors in the assumptions set up by the operator, and therefore the value of the numbers produced by the machine gives a false impression of accuracy". That statement is as valid today as i t was twenty-five years ago. On the other hand, the industrial approach to phase equilibria has changed over the years. In this state-of-the-art report, I will describe our present practices and concerns. This presentation will be subdivided according to the sequence presented in Figure 1 - HOW, WHAT and WHY, WHERE. HOW are phase equilibria problems treated in an industrial situation? WHAT methods and correlations are used, and WHY are these techniques used? WHERE should future development work be directed? Of necessity, the conditions described here are based

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HOW? WHAT AND WHY? WHERE ? Figure 1.

Sequence of presentation

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p r i n c i p a l l y on my own work environment. They may not be u n i v e r s a l l y t r u e , but they a r e c e r t a i n l y r e p r e s e n t a t i v e of the c u r r e n t s t a t e of a f f a i r s i n i n d u s t r y . HOW Are Phase E q u i l i b r i a Problems Treated? The t e c h n i c a l p o p u l a t i o n i n i n d u s t r y can be d i v i d e d i n t o the computer people and the non-computer people. The non-computer people tend to use simple c o r r e l a t i o n s , g e n e r a l i z e d models and graphs, and estimates based on t h e i r experience or i n t u i t i o n . The computer people, who are i n the m a j o r i t y , have the c a p a b i l i t y of developing very complex models. S e v e r a l years ago a t Union Carbide, an e f f o r t was i n i t i a t e d to improve the e f f e c t i v e n e s s of the computer system used by these i n d i v i d u a l s f o r design purposes I would l i k e to spend some time d e s c r i b i n Subprogram L i b r a r y syste Engineering Subprogram L i b r a r y . The l i b r a r y c o n s i s t s of computer subroutines which have been w r i t t e n to perform engineering process and design c a l c u l a t i o n s and to s o l v e v a r i o u s types of mathematical problems. The subroutines a r e w r i t t e n i n accordance w i t h a standard format, and they use c o n s i s t e n t technology. They are f u l l y documented i n a manual so that they can be e a s i l y used by other programmers. There a r e three kinds of thermodynamic and p h y s i c a l property subroutines: monitor s u b r o u t i n e s , method s u b r o u t i n e s , and i n i t i a l i zation subroutines. Their i n t e r - r e l a t i o n s h i p i s i l l u s t r a t e d i n F i g u r e 2. There i s a monitor subroutine f o r every p r o p e r t y ; vapor molar volume, vapor f u g a c i t y c o e f f i c i e n t s , l i q u i d a c t i v i t y c o e f f i c i e n t s , e t c . Suppose a main program needs the value of property 1. I t w i l l c a l l the monitor subroutine f o r property 1 and supply a computation method code. The monitor subroutine checks the code and c a l l s the a p p r o p r i a t e method subroutine to perform the c a l c u l a t i o n s of property 1. I f the method subroutine r e q u i r e s the value of property 2, i t c a l l s the monitor subroutine f o r property 2 and so f o r t h . The monitor subroutines a l l o w us t o w r i t e very general main programs w i t h a v a r i e t y of options f o r c a l c u l a t i n g thermodynamic and p h y s i c a l p r o p e r t i e s . I t i s a l s o very easy to add a new method to the system; one simply i n c l u d e s a new c a l l statement i n the monitor s u b r o u t i n e . The main program does not have to be reprogrammed when adding a new method. The t h i r d type of subroutine i s the i n i t i a l i z a t i o n subroutine which c a l c u l a t e s the constant parameters a s s o c i a t e d w i t h a method s u b r o u t i n e ; f o r i n s t a n c e , the Redlich-Kwong equation of s t a t e c o n s t a n t s . The main program c a l l s the r e q u i r e d i n i t i a l i z a t i o n subroutines once f o r any given s e t of components. Code S t r u c t u r e . As mentioned e a r l i e r , a monitor subroutine checks a method code to determine the c a l c u l a t i o n a l method. A f l e x i b l e scheme has been developed f o r s p e c i f y i n g method codes. The codes r e q u i r e d f o r v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s a r e shown i n F i g u r e 3. Each column represents a s e t of f i v e codes t r a n s m i t t e d to a monitor subroutine f o r the property designated

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