Phase equilibria and fluid properties in the chemical industry: Estimation and correlation

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

Author: Truman S. Storvick | Stanley I. Sandler

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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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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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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.

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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

2

2^8,82]

NUMBER OF CH GROUPS IN SATURATED COMPONENT TOTAL NUMBER CARBON ATOMS IN SATURATED COMPONENT 3

NUMBER REFERS TO BINARY SYSTEMS

-0.03

0

O

AROMATIC COMPONENT:

•

AROMATIC COMPONENT: TOLUENE

0.1

J

u.Z

L

0.3

BENZENE

0.4

0.5

0.6

07

08

r , DEGREE OF BRANCHING Figure 23. Binary parameters for aromatic-saturated hydrocarbon systems (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.

37

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independent of one another (consider the Gibbs-Helmholtz equation) and t h e r e f o r e , i f one b u i l d s separate models f o r these two q u a n t i t i e s one must g i v e a t t e n t i o n to t h e i r mutual interdependence. But fundamentally h and s r e f e r t o r a t h e r d i f f e r e n t p h y s i c a l phenomena and h i s t o r i c a l l y what has happened i s that those who have proposed equat i o n s f o r g have tended, o f t e n s u b c o n s c i o u s l y , t o g i v e primary a t t e n t i o n e i t h e r to h^ o r to s^. For example, the well-known Wohl expansion (10) which dominated phase e q u i l i b r i u m thermodynamics f o r 20 y e a r s , i s p h y s i c a l l y meaningful only f o r h^ (rather than g ) , and the equation of Wilson which has been modified by dozens of authors, was d e r i v e d as a m o d i f i c a t i o n of an equation a p p l i c a b l e to s . C o n s t r u c t i o n of a r i g o r o u s t h e o r e t i c a l equation f o r g which does proper j u s t i c e to both h and s , i s probably beyond our present theoretical capabilities h e f f o r ha bee expended toward e s t a b l i s h i n g an e x p r e s s i o p r a c t i c a l needs, i s a l s o a t l e a s t grounded on some loose t h e o r e t i c a l b a s i s . We must keep reminding o u r s e l v e s that any p r e s e n t l y a v a i l a b l e theory f o r l i q u i d mixtures i s s e v e r e l y r e s t r i c t e d and, when a p p l i e d to r e a l m i x t u r e s , i t should not be taken too s e r i o u s l y . For a l l but the s i m p l e s t molecules, molecular theory of l i q u i d s o l u t i o n s provides a v a l u a b l e guide, u s e f u l f o r i n t e r p o l a t i o n and e x t r a p o l a t i o n , but i t i s not l i k e l y i n the negr f u t u r e t o do much more than t h a t . Equations f o r g which emphasize h and g i v e l i t t l e a t t e n t i o n to s are common and one t h a t i s p a r t i c u l a r l y popular i s the r e g u l a r s o l u t i o n equation of Hildebrand and Scatchard u s i n g s o l u b i l i t y parameters. With a l i t t l e e m p i r i c a l m o d i f i c a t i o n , the r e g u l a r s o l u t i o n equation can be extremely u s e f u l f o r c e r t a i n kinds of m i x t u r e s . An example, shown i n F i g u r e 23, g i v e s a c o r r e l a t i o n e s t a b l i s h e d by Funk (11) f o r mixtures c o n t a i n i n g aromatic hydrocarbons ( i n p a r t i c u l a r benzene or toluene) w i t h s a t u r a t e d hydrocarbons. By i n t r o d u c i n g the b i n a r y parameter &]_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.

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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.

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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.

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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.

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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

62 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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HOW? WHAT AND WHY? WHERE ? Figure 1.

Sequence 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.

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IN C H E M I C A L

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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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Industrial View of Phase Equilibria

MAIN PROGRAM

>

<

MONITOR SUBROUTINE PROPERTY 1

1 V INITIALIZATION SUBROUTINE PROPERTY 2,

METHOD SUBROUTINE PROPERTY 1

<

>

MONITOR SUBROUTINE PROPERTY 2

METHOD SUBROUTINE PROPERTY 2

Figure 2. Structure of the subroutine system

EQUILIBRIUM VAPOR PRES.

I.D. 1

ACT. FUG.

COEF.

* 7

11

19

15

ENTHALPY

COOES

LIQUID

K. E Q.

VAPOR FUG. COEF. 27

23

MIXING

CODES

LIQUID VAPOR 35

31

39

THERM0

— 1 1 1 1 11 1 1 1 1 1 1 1 11 1 1 1 1 11 1 11 1 11 1 11 1 1

11 t

h

s

u

b

i i i i i

2

1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 11

1

1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 11 1 1

THSUB4

111

1

11

11 1 1 1 1 11 1 1 1 1 11 1 11 1 11 1 11 1 1 1 1 1 1 1 11 1 1 1 1

11

1 11 1 11 1 11 1

LIQ

LIQ

ACT. COEF. 19

ACT. COEF. 19

— REGULAR SOLUTION — LIQUID VOLUME

BASIC CODE

1

1

SUBCODES

1 1

— WILSON EQUATION

MODEL

POLYNOMIAL

— UNMODIFIED — LIQUID VOLUME POLYNOMIAL

Figure 3.

Method code structure

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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MOLECULAR WEIGHT NORMAL

BOILING & FREEZING POINTS

CRITICAL CONSTANTS LIQUID DENSITY

AT REFERENCE TEMPERATURE

HEATS OF FORMATION FOR VARIOUS STATES PITZER'S ACENTRI REFRACTIVE INDEX AT REFERENCE TEMPERATURE SOLUBILITY IN VARIOUS SOLVENTS AT REFERENCE TEMPERATURE SURFACE TENSION AT REFERENCE TEMPERATURE Figure 4.

Examples of single-valued properties in the date bank

PERFECT GAS HEAT CAPACITY Cp = A + BT + CT + DT + ET 2

VAPOR

3

4

PRESSURE

In P = A - B/(T + C) + D In T + ET

N

LIQUID VISCOSITY ln^t= A + B/T + C In T LIQUID THERMAL CONDUCTIVITY In k = A + BT + CT 2

Figure 5. Examples of correlations whose constant parameters are in data bank

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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above i t . The f i r s t code i s the b a s i c method code; each method i s assigned a b a s i c method code. The other four codes are method subcodes; the d e f i n i t i o n of the subcodes depends on the b a s i c method code. L e t ' s look a t the 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 codes as an example. A b a s i c method code equal t o 2 designates the Regular S o l u t i o n model. Then, the f i r s t subcode i s the b a s i c method code f o r the l i q u i d volume c o r r e l a t i o n s r e q u i r e d i n t h i s model, and the next code i s the l i q u i d volume method subcode, i f one i s r e q u i r e d . On the other hand, suppose the b a s i c method code f o r 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 i s equal to 4, which denotes the W i l son Equation. Since s e v e r a l m o d i f i c a t i o n s of the Wilson Equation have been programmed i n our system, the f i r s t subcode i n d i c a t e s which v a r i a t i o n should b used Now th d subcod i th b a s i c method code f o r th t h i s model, f o l l o w e d by yrequisit The methods s p e c i f i e d by subcodes a r e r e s t r i c t e d to that p a r t i c u l a r a p p l i c a t i o n . Thus, one l i q u i d volume method can be used i n the a c t i v i t y c o e f f i c i e n t c a l c u l a t i o n s , another f o r the Poynting c o r r e c t i o n , and a t h i r d f o r pipe s i z i n g c a l c u l a t i o n s . I f a subcode i s m i s s i n g , the b a s i c method code f o r that property w i l l be s u b s t i t u t e d . I n the 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 examples, i f the subcode f o r the l i q u i d volume method i s m i s s i n g , the b a s i c method code s p e c i f i e d f o r l i q u i d volume c a l c u l a t i o n s w i l l be used. I f the b a s i c method codes are m i s s i n g , d e f a u l t values a r e assigned. The d e f a u l t method code f o r a property i s the s i m p l e s t model r e q u i r i n g the l e a s t input data, e.g., i d e a l - g a s model f o r vapor f u g a c i t y c o e f f i c i e n t s . "Data Bank. Another p a r t o f the computer system i s a Data Bank which serves as a r e p o s i t o r y f o r pure component data. The Data Bank contains s i n g l e - v a l u e d p r o p e r t i e s o f the type shown i n F i g u r e 4. I t a l s o contains the constant parameters and a p p l i c a b l e ranges f o r property c o r r e l a t i o n s o f the s o r t i l l u s t r a t e d i n F i g u r e 5. Only p r o p e r t i e s r e q u i r e d f o r engineering c a l c u l a t i o n s are s t o r e d i n the Data Bank. A l l o f the values i n the Data Bank are i n t e r n a l l y c o n s i s t e n t ; the vapor pressure c o r r e l a t i o n s do p r e d i c t the normal b o i l i n g p o i n t s and the c r i t i c a l p o i n t s , the enthalpy and entropy values are based on the same reference s t a t e , etc. Every v a l u e i n the Data Bank has a reference number assoc i a t e d w i t h i t . These r e l a t e to a l i s t of references s t o r e d i n a separate computer f i l e . The references a r e comprehensive - every Data Bank v a l u e can be reproduced from the i n f o r m a t i o n given i n the r e f e r e n c e . The references i n c l u d e a l l o f the data sources and the d a t a - r e d u c t i o n methods used i n o b t a i n i n g the Data* Bank values. Main Programs. There a r e a v a r i e t y o f programs which c a l l upon the Engineering Subprogram L i b r a r y and the Data Bank. S e v e r a l of these programs w i l l be b r i e f l y d e s c r i b e d here. VLEFIT i s a f i t t i n g program which uses v a p o r - l i q u i d e q u i l i b r i u m (VLE) data t o determine the best values f o r the a d j u s t a b l e parameters

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INPUT

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

DATA

OPTIONS:

P-T-X P-T-X-Y P-T-Y

P-T-X-AH

M I X

T-X-tf P - T - X - USER

VARIABLE

ALLOWABLE OBJECTIVE FUNCTIONS:

P Y X

Figure 6.

Vapor-liquid equilibrium datafittingprogram (VLEFIT)

ABSORPTION EXTRACTIVE AZEOTROPIC

DISTILLATION DISTILLATION

FRACTIONATION RECTIFICATION STRIPPING LIQUID - LIQUID

EXTRACTION

Figure 7. Scope of the MMSP (Multicomponent Multistage Separation Processes) programs

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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i n VLE models. I t w i l l handle up t o a quaternary system. The input data options and the a l l o w a b l e o b j e c t i v e f u n c t i o n s are shown i n F i g u r e 6. Any combination of i n p u t data and o b j e c t i v e f u n c t i o n s can be used i n a given run, and weighting f a c t o r s can be attached t o both the input data and the o b j e c t i v e f u n c t i o n s . The MMSP programs were developed f o r the s i m u l a t i o n , design, and c o n t r o l of Multicomponent M u l t i s t a g e Separation Processes. These i n c l u d e the operations l i s t e d i n F i g u r e 7. These processes may be c a r r i e d out i n c o n v e n t i o n a l columns (one feed and two prod u c t s ) , i n complex columns ( m u l t i p l e feeds, m u l t i p l e products, and m u l t i p l e i n t e r s t a g e c o o l e r s or h e a t e r s ) , o r i n a s e r i e s of c o n v e n t i o n a l o r complex columns. A u s e f u l d i a g n o s t i c o p t i o n f o r checking VLE models i s a l s o a v a i l a b l e i n the MMSP programs model based b i n a r y parameters. The meters may be very l i m i t e d and/or a t c o n d i t i o n s removed from those of i n t e r e s t . Therefore, i t i s wise to examine the p r e d i c t i o n s of the model f o r the b i n a r y p a i r s before attempting a multicomponent s i m u l a t i o n . This o p t i o n i n MMSP w i l l use the VLE model to produce y-x, T-x-y, P-x-y, a-x, and K-x diagrams f o r the b i n a r y systems. Two of these p l o t s f o r the acetone-water system are shown i n F i g u r e 8. Another program IPES (Integrated Process Engineering System) i s a process s i m u l a t o r . I t has the a b i l i t y of modeling processes of any s i z e , from a s i n g l e d i s t i l l a t i o n column to a complex p l a n t . IPES i s based on a b u i l d i n g b l o c k concept. The process i l l u s t r a t e d i n the flowsheet of F i g u r e 9-A would be modeled i n IPES u s i n g the scheme i n F i g u r e 9-B. Each b l o c k corresponds to a u n i t or o p e r a t i o n i n the a c t u a l process. The b l o c k s i n IPES i n c l u d e mixer-splitters, reactors, flach units, d i s t i l l a t i o n units, e x t r a c t o r s , heat exchangers, compressors, c o n t r o l u n i t s , and economic u n i t s f o r s i z i n g and c o s t i n g process equipment. The method codes d e s c r i b e d e a r l i e r can be s p e c i f i e d on an o v e r a l l b a s i s o r i n d i v i d u a l l y f o r each b l o c k . WHAT Models Are Used And WHY? An e f f i c i e n t computing system i s capable of producing meaningf r e s u l t s only i f a p p r o p r i a t e models are used. Therefore, a t t h i s p o i n t , I would l i k e t o d e s c r i b e the techniques we use f o r VLE, f o r l i q u i d - l i q u i d e q u i l i b r i u m (LLE), and f o r other s e p a r a t i o n processes This w i l l i n c l u d e examples of t y p i c a l systems, the problems they pose, and how we attempt t o cope w i t h them. Vapor-Liquid E q u i l i b r i u m Models. I w i l l begin by d e s c r i b i n g the models a v a i l a b l e i n our system f o r VLE c a l c u l a t i o n s - the f i e l d i n which we have had the most experience, and i n which the most e f f o r t has been expended. The c a l c u l a t i o n s are based on VLE r a t i o s - K v a l u e s . The K v a l u e of a component i n a mixture i s r e l a t e d t o i t s f u g a c i t y i n the l i q u i d phase and i t s f u g a c i t y i n the vapor phase. The equations f o r determining the f u g a c i t i e s are given i n F i g u r e 10. The vapor f u g a c i t y i s expressed i n the u s u a l f a s h i o n i n terms of a f u g a c i t y c o e f f i c i e n t , 0 . There are 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 H A S E EQUILIBRIA

Figure 8.

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Diagnostic options available in MMSP

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

co

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 H A S E EQUILIBRIA

A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

RECYCLE

CONDEN SATE

STEAM BOTTOMS

REC

FEED

HFD

EHXOI HEAT .USTM

EMSOI MIX

RXFD i

PROD ERXOI ROUT EFLOI CFD EDXOI * REAC FLSH COLM HVY

i

CSTM

ECR02I CONT

Figure 9.

LOTS

Process simulation program

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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73

three expressions f o r the l i q u i d f u g a c i t y . The f i r s t i s the t r a d i t i o n a l approach most o f t e n found i n textbooks based on a s t a n dard s t a t e equal t o the t o t a l system p r e s s u r e ; t h i s i s the o p t i o n most f r e q u e n t l y e l e c t e d . I n the second approach, the standard s t a t e i s a f i x e d r e f e r e n c e p r e s s u r e . I n the t h i r d equation, the _ l i q u i d f u g a c i t y i s expressed i n terms o f a f u g a c i t y c o e f f i c i e n t , 0L. A l t e r n a t e l y , the K v a l u e can be c a l c u l a t e d using an e m p i r i c a l c o r r e l a t i o n depending only on pressure and temperature. The f u g a c i t y c o e f f i c i e n t s are determined from equations o f s t a t e . The equations a v a i l a b l e f o r the vapor phase f u g a c i t y coeff i c i e n t s are l i s t e d i n F i g u r e 11. The two B-W-R equations and the Soave equation are a l s o used f o r l i q u i d phase f u g a c i t y c o e f f i c i e n t s . The Soave equation and the Hayden-0 Connell v i r i a l equat i o n are very recent a d d i t i o n are e l i m i n a t e d from c o n s i d e r a t i o n t i o n of the Redlich-Kwong equation and the BWR equations have r e c e i v e d the most usage. The l i q u i d r e f e r e n c e f u g a c i t y and the 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 models are l i s t e d i n F i g u r e s 12 and 13. A t the present time, Henry's Law f o r s u p e r c r i t i c a l components can be used only w i t h the UNIQUAC equation; the unsymmetric convention i s not i n c l u d e d i n the other 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 models. Vapor pressure w i t h a Poynting c o r r e c t i o n i s u s u a l l y used f o r the l i q u i d r e f e r e n c e f u g a c i t y . The Wilson equation and the NTRL equations are the most commonly u t i l i z e d 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 models. UNIQUAC and UNIFAC have j u s t been added to the system, and i f they f u l f i l l our e x p e c t a t i o n s , they w i l l become the most commonly used models. Enthalpy models are a l s o necessary i n VLE design c a l c u l a t i o n s . The procedures f o l l o w e d f o r vapor and l i q u i d enthalpy c a l c u l a t i o n s are s p e c i f i e d i n F i g u r e 14. The equation of s t a t e approach l - ( i ) i s the most popular f o r vapor enthalpy. For l i q u i d enthalpy, the equation of s t a t e approach l - ( i ) , the Yen-Alexander c o r r e l a t i o n l - ( i i ) , and method 2 o f mole f r a c t i o n averaging the pure component e n t h a l p i e s are used w i t h about equal frequency. Method 3 f o r l i q u i d enthalpy i n c l u d e s a heat o f mixing term, A H ^ ; i t i s not used very o f t e n because we have d i s c o v e r e d t h a t heats of mixing p r e d i c t e d by the 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 models are u n r e l i a b l e unless such data were i n c l u d e d i n the f i t t i n g procedure. A l l o f the property monitor subroutines have a group o f codes which a l l o w i n d i v i d u a l s to supply t h e i r own u s e r - s u b r o u t i n they f i n d the a v a i l a b l e methods inadequate. The user-subroutines have f i x e d names and argument l i s t s which are d e s c r i b e d i n the documentation f o r the monitor s u b r o u t i n e s . Why do these l i s t s c o n t a i n so manv models - some mediocre models, i n f a c t . New models are always being added to the s y s tem, but o l d models are never d i s c a r d e d . Once a process has been s u c c e s s f u l l y designed u s i n g a c e r t a i n model, t h a t model must always be a v a i l a b l e f o r f u t u r e process c a l c u l a t i o n s , expansion o f 1

m

x

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

k, = y/XI =

I.

VAPOR:

(II/x,)/

(
f ? = $ ? Y, T > /IT

LIQUID: 1. f' = If =

2.

* ,X i(nr

f

s

L

e

J ^ x j r U e ^ ^

3. I TT = SYSTEM PRESSURE Pref = REFERENCE PRESSURE

II.

In(KIT) = A + B/T + C / T + D/T 2

Figure 10.

IDEAL

3

Vapor-liquid equilibrium equations

( 0 = 1)

REDLICH - K W O N G VIRIAL

(WOHL CORRELATION)

B-W-R PRAUSNITZ-CHUEH

MOD. REDLICH-KWONG

BARNER - ADLER STARLING

M O D . B-W-R

PERSOHN

M O D . REDLICH - K W O N G

SOAVE

MOD. REDLICH-KWONG

HAYDEN-O'CONNELL Figure 11.

VIRIAL Equations 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.

KROLIKOWSKI

f° l °

l

=

Industrial View of Phase Equilibria

VAPOR •>

L

PRESSURE

AG

Vj /RT ( P r . f - P i ) L

C H A O - SEADER C H A O - ROBINSON PERSOHN B-W- R STARLING HENRY'S

MOD. B-W-R LAW FOR

SUPERCRITICAL

COMPONENTS Figure 12.

IDEAL

Liquid reference fugacity models

Or = I)

REDLICH - KISTER HILDEBRAND

REGULAR

(SOLUBILITY

SOLUTION

PARAMETER)

NRTL WILSON MARGULES UNIQUAC -

SYMMETRIC A N D

UNSYMMETRIC

FORMS.

UNIFAC Figure 13. Activity coefficient models

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

VAPOR A N D LIQUID:

1.

H =

H* + -

—

IDEAL G A S ENTHALPY

(H-H)

—

(.)

EQUATION

(ii)

YEN-ALEXANDER

H

2. LIQUID:

H

(H-H*)

OF STATE

Y ( MOLE \ /TEMPERATURE \ 4 - VFRACTION/j VPOLYNOMIAL/j

=

3. H =

H

1 D E A L

SOLN.

+

A H

SOLN.

M

I

X

FUGACITY MODELS

v v (TEMPERATURE\ <") Z-, i V POLYNOMIAL/j 1

X

AH

Figure 14.

M

I

X

—

LIQUID ACTIVITY COEFFICIENT MODELS

Enthalpy models

HYDROGEN

ACETALDEHYDE

METHANE

CROTON ALDEHYDE

ETHYLENE

ETHYL ETHER

ETHANE

ETHANOL

1 - BUTENE

TERT-BUTANOL

n - BUTANE

WATER

2 - METHYLPENTANE

ETHYLENE GLYCOL

n - OCTANE T: - 1 0 ° C — ^ 3 5 0 ^ P: 5—•1100 psia Figure 15.

Typical VLE problem, Example 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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the u n i t , o r the design o f a new u n i t . The e f f o r t r e q u i r e d to redo an acceptable o l d model cannot be j u s t i f i e d . V a p o r - L i q u i d Equilibrium-Sample Systems. I n p r e p a r i n g t h i s p r e s e n t a t i o n , I took an " a t t i t u d e survey" among my c o l l e a g u e s who are u s i n g t h i s VLE c a l c u l a t i o n a l system. The i n d i v i d u a l s i n v o l v e d i n the s i m u l a t i o n of hydrocarbon u n i t s were g e n e r a l l y s a t i s f i e d w i t h the c u r r e n t s t a t e of a f f a i r s . They have years of experience i n making these c a l c u l a t i o n s , the c o r r e l a t i o n s are a p p r o p r i a t e f o r the compounds of i n t e r e s t to them, and they have accumulated c o r r o b o r a t i v e p l a n t data. However, they d i d admit that the f o l l o w i n g s t i l l present a c h a l l e n g e : h i g h pressure c r i t i c a l phenomena, cryogenic s e p a r a t i o n s , very heavy hydrocarbons - t a r s , and the presence of hydrogen i n the system. The i n d i v i d u a l s i n v o l v e i desig 'chemical u n i t had more g r i e v a n c e s . L e t ' 17. I n the f i r s t example, there are 15 compounds - not an unusual number f o r most models. The compounds vary markedly from each other. The person working on t h i s p r o j e c t was u n t i l r e c e n t l y teaching and doing r e s e a r c h a t a u n i v e r s i t y . She claims t h a t what she misses most from the academic l i f e i s the 'nice compounds' worked w i t h there. The model i s expected to work over a broad range of temperatures and p r e s s u r e s ; t h e r e f o r e , a number of the compounds w i l l be s u p e r c r i t i c a l sometimes - but not always. Pure component data are scarce f o r a number of these compounds; e.g. crotonaldehyde. What are the c r i t i c a l p r o p e r t i e s of ethylene g l y c o l — i t decomposes before reaching a c r i t i c a l p o i n t . VLE data are not a v a i l a b l e f o r a l l of the b i n a r y p a i r s ; t h i s i s not s u r p r i s i n g s i n c e there are 105 b i n a r y p a i r s . Some o f the VLE data are cont r a d i c t o r y ; e.g., a c e t a l d e h y d e — e t h a n o l . What equation of s t a t e i s a p p l i c a b l e f o r hydrocarbons, aldehydes, and a l c o h o l s ? How should the l i q u i d enthalpy be t r e a t e d w i t h so many l i g h t compounds? Pure component data which cannot be e a s i l y measured are e s t i m a t e d — i n c l u d i n g the c r i t i c a l p r o p e r t i e s of ethylene g l y c o l . I n many cases, the e s t i m a t i o n techniques are crude, o r they have not been developed f o r the type of compound under c o n s i d e r a t i o n . I f VLE data are m i s s i n g f o r any of the important b i n a r y p a i r s , they are measured. For the VLE model i n t h i s example, the Prausnitz-Chueh Redlich-Kwong equation of s t a t e was s e l e c t e d f o r the vapor phase f u g a c i t y c o e f f i c i e n t s and the Wilson equation f o r the l i q u i d phase a c t i v i t y c o e f f i c i e n t s . Vapor pressure w i t h a Poynting c o r r e c t i o n and a s a t u r a t e d vapor f u g a c i t y c o e f f i c i e n t c o r r e c t i o n was used f o r the l i q u i d r e f e r e n c e f u g a c i t y ; i t was s u i t a b l y e x t r a p o l a t e d i n t o the h y p o t h e t i c a l l i q u i d r e g i o n f o r s u p e r c r i t i c a l components. The W i l s o n equation parameters were determined from the a v a i l a b l e data w i t h the f i t t i n g program VLEFIT. The hope was t h a t i n the f i t t i n g process, the inadequacies o f the model would be absorbed i n the a c t i v i t y c o e f f i c i e n t parameters. The Wilson equation parameters for the b i n a r y p a i r s w i t h no VLE data were estimated by examining the parameters f o r s i m i l a r p a i r s . I t i s hoped t h a t UNIFAC w i l l 1

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

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

ARGON

ACETIC ACID

NITROGEN

WATER

OXYGEN

ACETALDEHYDE

ETHYLENE

ETHYL ACETATE

ETHANE

VINYL ACETATE

CARBON DIOXIDE

GLYCOL DIACETATE

ACROLEIN

VINYL

PROPIONATE

T: 0 ° C - 200 °C P: ^latm. Figure 16.

Typical VLE problem, Example 2

HYDROGEN METHYL

CHLORIDE

CHLORIDE

DIMETHYL ETHER METHANOL WATER SODIUM CHLORIDE SODIUM HYDROXIDE T: 1 0 ° C - 1 3 0 ° C P: 35 - 135 psia Figure 17. Typical VLE problem, Example 3

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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provide us w i t h a more systematic approach f o r t h i s e s t i m a t i o n procedure. F i n a l l y , the equation of s t a t e was a l s o used f o r vapor enthalpy c a l c u l a t i o n s . The l i q u i d enthalpy was c a l c u l a t e d as the mole f r a c t i o n average o f the pure component e n t h a l p i e s ; once again, the h y p o t h e t i c a l l i q u i d s t a t e was used f o r s u p e r c r i t i c a l components. The system i n the second example i s plagued w i t h a l l o f the usual problems and, i n a d d i t i o n , i t contains a c e t i c a c i d . The person who developed a model f o r t h i s system decided that i t was necessary to account f o r the vapor phase a s s o c i a t i o n of a c e t i c a c i d . Thus, a user equation o f s t a t e subroutine was w r i t t e n , wherei n the a s s o c i a t i o n was r i g o r o u s l y t r e a t e d ; then appropriate c o r r e c t i o n f a c t o r s were determined f o r the vapor f u g a c i t y c o e f f i c i e n t and vapor enthalpy of the apparent s p e c i e s The f l e x i b i l i t y i n the computing system mad This i s a h i g h l y n o n i d e a was developed, the subsequent column c a l c u l a t i o n s were very d i f f i c u l t t o converge. This b r i n g s up the question of what i s the b e s t way to i n i t i a l i z e such c a l c u l a t i o n s — t h e optimum method would r e q u i r e no i n p u t from the user and would be quick and f a i l s a f e . I f t h i s system had contained another a s s o c i a t i n g compound l i k e p r o p i o n i c a c i d , o r i f the system pressure had been h i g h e r , the a n a l y s i s would have been complicated by s e v e r a l orders o f magnitude, and the development o f such a custom model might prove to be too time consuming. The "chemical theory" of Nothnagel has been i n c l u d e d i n the new Hayden-0 Connel v i r i a l equation subroutine to provide a standardized method f o r d e a l i n g w i t h such systems. The t h i r d example h i g h l i g h t s another problem. How does one cope w i t h s e v e r a l s a l t s i n an already n o n i d e a l s o l u t i o n ? I n t h i s case, the s a l t e f f e c t s were accounted f o r by modifying the vapor pressure of the s o l v e n t s ; a vapor pressure depression f a c t o r was c a l c u l a t e d as a f u n c t i o n o f s a l t c o n c e n t r a t i o n . We recognize that a b e t t e r technique than t h i s should be a v a i l a b l e i n the computing system. What should i t be? Expensive and/or r a r e s a l t s are being used as c a t a l y s t s i n r e a c t i o n systems. P r e d i c t i n g t h e i r e f f e c t s on the subsequent separations systems and the design of c a t a l y s t recovery systems r e q u i r e good models. Vapor-Liquid E q u i l i b r i u m Plus Chemical Reaction. To conclude t h i s d i s c u s s i o n o f VLE c a l c u l a t i o n s , I d l i k e to mention that we have i n i t i a t e d a program to a l l o w VLE plus chemical r e a c t i o n i n column c a l c u l a t i o n s . A m o d i f i c a t i o n of Frank's dynamic s i m u l a t i o n method (4) i s being used. The i n i t i a l t r i a l s w i t h complex VLE and k i n e t i c models have been moderately s u c c e s s f u l , but f u r t h e r t e s t ing i s r e q u i r e d . Other s i m u l a t i o n techniques a r e a l s o b e i n g investigated. L i q u i d - L i q u i d E q u i l i b r i u m . The next phase e q u i l i b r i u m s i t u a t i o n I would l i k e to d i s c u s s i s LLE. I n our o r g a n i z a t i o n , t h i s f i e l d has r e c e i v e d l e s s a t t e n t i o n than i t deserves. The a v a i l a b l e models a r e shown i n F i g u r e 18. Most of the a c t i v i t y c o e f f i c i e n t models mentioned p r e v i o u s l y are a p p l i c a b l e i n the f i r s t o p t i o n #

1

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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F L U I D PROPERTIES IN

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w i t h the notable e x c e p t i o n of the Wilson equation. The new UNIQUAC model may prove to have r e a l v a l u e i n t h i s a p p l i c a t i o n ; the developers of the model s t a t e that i t gives a good represent a t i o n of LLE f o r a wide v a r i e t y of mixtures. U n t i l r e c e n t l y , our f i t t i n g program d i d not have the f a c i l i t y to d e a l w i t h LLE data. We s t i l l need guidance i n d e c i d i n g what c o n s t i t u t e s a "good" f i t , what the o b j e c t i v e f u n c t i o n f o r f i t t i n g should be, and what i s the proper data to use. The second o p t i o n i n F i g u r e 18 i s based on a f i t of multicomponent experimental data i n the range of i n t e r e s t . In our l i m i t e d experience, we have l e a r n e d the f o l l o w i n g v a l u a b l e l e s s o n s . One cannot expect to model a complex system by c o l l e c t i n g l a r g e amounts of data and attempting to f i t these data by b r u t e f o r c e . Developing the proper e m p i r i c a l equations f o r systems w i t h s e v e r a l c h e m i c a l l i s very d i f f i c u l t . Statisticall t i n g data proved not to be h e l p f u l i n developing such models. The 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 approach f o r p r e d i c t i n g d i s t r i b u t i o n c o e f f i c i e n t s i s v i a b l e , although i t i s s t i l l d i f f i c u l t . I n order to develop an accurate model, both VLE and LLE data must be used i n determining the parameters f o r the 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 correlation. An example of a system which has been s u c c e s s f u l l y modeled u s i n g the NRTL 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 c o r r e l a t i o n i s shown i n F i g u r e 19. F i r s t , a s e t of NRTL parameters was generated by f i t t i n g b i n a r y VLE and LLE data. Then, these parameters were o p t i mized using multicomponent LLE data. C a l c u l a t i o n s u s i n g t h i s model show e x c e l l e n t agreement w i t h p l a n t data. V a p o r - L i q u i d - L i q u i d E q u i l i b r i u m . We have had l i m i t e d e x p e r i ience i n r i g o r o u s three phase e q u i l i b r i u m c a l c u l a t i o n s , vaporl i q u i d - l i q u i d , p r i m a r i l y i n s i n g l e stage f l a s h u n i t s . The implementation of such a three-phase e q u i l i b r i u m model i n column c a l c u l a t i o n i s scheduled i n the f u t u r e . P r e s e n t l y , a method a l s o e x i s t s wherein complete i m m i s c i b i l i t y i n the l i q u i d phase can be s p e c i f i e d between one component and a l l of the other components i n the system; e.g., between water and a s e t of hydrocarbons. The VLE r a t i o s are normalized on an o v e r a l l l i q u i d b a s i s so that the r e s u l t s can be used i n c o n v e n t i o n a l two-phase l i q u i d - v a p o r e q u i l i brium c a l c u l a t i o n s . Other S e p a r a t i o n Processes. The f i n a l type of phase e q u i l i brium I'd l i k e to mention could be t i t l e d unusual s e p a r a t i o n proc e s s e s — a d s o r p t i o n , i o n exchange, membranes. The i n d i v i d u a l s i n v o l v e d i n these areas are d e f i n i t e l y non-computer people. There are no models i n the present computer system which s a t i s f y t h e i r p a r t i c u l a r needs, but more i m p o r t a n t l y , they can't recommend any that should be added to the system. They f i r m l y b e l i e v e t h a t e m p i r i c a l c o r r e l a t i o n s based on experimental data are b e s t . For new systems, they use a s i m i l a r i t y approach based on previous experience. They v o l u n t e e r e d the o p i n i o n that an a n a l y s i s w i t h a

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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1.

DISTRIBUTION COEFFICIENTS ACTIVITY COEFFICIENTS

2.

DISTRIBUTION COEFFICIENTS FROM EMPIRICAL CORRELATIONS OF TEMPERATURE A N D COMPOSITION Figure 18.

FROM

Liquid-liquid equilibrium models

BENZENE

OCTANE

TOLUENE

CYCLOPENTANE

p-XYLENE

CYCLOHEXANE

CUMENE

METHYLCYCLOPENTANE

n-

METHYLCYCLOHEXANE

PENTANE

n-HEXANE

WATER

HEPTANE

TETRAETHYLENE

T: P:

GLYCOL

100°C-160°C 1 -

Figure 19.

15 a t m . Typical LLE problem

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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t h e o r e t i c a l b a s i s might be n i c e , but they didn't r e a l l y expect that i t was p o s s i b l e . Model S e l e c t i o n . As a point of i n t e r e s t , I'd l i k e to remark at t h i s time that the models i n a design system are decided upon by a few i n d i v i d u a l s . I n a l a r g e company, there i s u s u a l l y a technology s e c t i o n r e s p o n s i b l e f o r such matters; i n smaller organi z a t i o n s , the d e c i s i o n s r e s t w i t h one or two persons. For both, i t i s impossible t o review and evaluate everything being published nowadays. Expert c o n s u l t a n t s are employed to give us guidance i n these matters. We tend to concentrate on the areas i n which our d e f i c i e n c i e s are l i m i t i n g the accuracy of design c a l c u l a t i o n s . Our i n t e r e s t w i l l not be aroused by another m o d i f i c a t i o n of the Redlich-Kwong equation, an i n v o l v e d e x p l a n a t i o n of what the Wilson parameters r e a l l y mean, a c o r r e l a t i o n v a l i d f o r the noble gases, or a c o r r e l a t i o methods that are a p p l i c a b l c i a l l y nonhydrocarbons, and that can be used over a broad range of temperatures and pressures. The c o r r e l a t i o n s must be reasona b l y accurate. We favor c o r r e l a t i o n s which are simple i n form and which r e q u i r e e a s i l y a c c e s s i b l e input data. To optimize the e f f i ciency of i t e r a t i v e computer c a l c u l a t i o n s , we p r e f e r c o r r e l a t i o n s which can be solved d i r e c t l y o r which converge r a p i d l y . To a i d us i n the e v a l u a t i o n and p o s s i b l e implementation of a method, we would a p p r e c i a t e d e t a i l s and sample c a l c u l a t i o n s , which could be published o r a v a i l a b l e as supplementary m a t e r i a l . A f t e r a method i s added t o the system, we have no c o n t r o l over how i t w i l l be used. Therefore, we b u i l d checks i n t o the subroutines versus a p p l i c a b l e temperatures, pressures, a c e n t r i c f a c t o r s , e t c . , — i f we've been t o l d what they are. We have a l s o had the u n f o r tunate experience of checking c o r r e l a t i o n s w i t h i n the published a p p l i c a b l e range and d i s c o v e r i n g e r r a t i c behavior: don't promise more than you can d e l i v e r ! How do we know t h a t we have s e l e c t e d the proper models to implement i n our system? We r e c e i v e feedback on the e f f e c t i v e n e s s and on the inadequacies of the models from the process engineers who are u s i n g them. The shortcomings of the system are commented upon v o c i f e r o u s l y ; the v i r t u e s are accepted m a t t e r - o f - f a c t l y . Laboratory and p l a n t t e s t s are o f t e n made to confirm the v a l i d i t y of a process model or to i n d i c a t e areas where d i s c r e p a n c i e s e x i s t . In the l a t t e r case, the model i s reviewed and a p p r o p r i a t e l y adjusted t o e l i m i n a t e the d i f f e r e n c e s . WHERE Should Future Development Work Be Directed? This question produces a v a r i e t y of responses depending upon whom you ask, when you ask, and what i s being worked on a t the time. The two main c a t e g o r i e s are shown i n F i g u r e 20. H i g h l y Nonideal Systems. Highly nonideal systems are of p r i n c i p a l concern; the n o n i d e a l i t y may be caused by the components present and/or by the o p e r a t i n g c o n d i t i o n s . This category i n c l u d e s s u b j e c t s l i k e c r i t i c a l phenomena, e x t r a c t i v e / a z e o t r o p i c d i s t i l l a t i o n , solvent s e l e c t i o n g u i d e l i n e s , s a l t e f f e c t s , high molecular 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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83

HIGHLY NONIDEAL SYSTEMS

EXPERIMENTAL TECHNIQUES AND DATA Figure 20.

Important areas for future development

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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weight polymer systems, and l i q u i d mixtures at h i g h pressures w i t h s u p e r c r i t i c a l components. D e a l i n g w i t h s u p e r c r i t i c a l components i s one of the major problems today; a good general method does not e x i s t . The Henry's Law approach w i t h the unsymmetric convention f o r a c t i v i t y c o e f f i c i e n t s i s not w e l l understood. Using an equat i o n of s t a t e to represent both phases would be s a t i s f a c t o r y , but c u r r e n t l y t h i s technique i s only a p p l i c a b l e to l i g h t hydrocarbon systems. No one expects to e l i m i n a t e the need f o r experimental data. What we do r e q u i r e are c o r r e l a t i n g models t h a t are approp r i a t e f o r h i g h l y n o n i d e a l systems. The acid-gas system i s a case i n p o i n t ; there i s a mass of data, but no model w i t h a f i r m t h e o r e t i c a l b a s i s . We would l i k e to r e l y on the models to a l l o w us to minimize the necessary experimental data. E s t i m a t i o n techniques l i k e UNIFAC are u s e f u l because they permit us to t r e a t the l e s s important component Experimental Technique ous q u a l i t y of some data a l s o pose severe problems; t h i s i s the second area of concern. R e l i a b l e experimental techniques f o r meas u r i n g ' u s e f u l ' e q u i l i b r i u m data q u i c k l y would be i n v a l u a b l e . Recently, there have been many data p u b l i s h e d on the excess prop e r t i e s of m i x t u r e s ; but the u s e f u l n e s s of such data has yet to be d e f i n e d . Many design p r o j e c t s have c l o s e time schedules, and so the p o s s i b l e data c o l l e c t i o n i s s e v e r e l y l i m i t e d because the experimental methods are so time-consuming. There a l s o seems to be a d e c l i n e i n the amount of h i g h q u a l i t y , new data being publ i s h e d ; the same systems a r e d i s c u s s e d over and over again. In approaching new systems, our l i t e r a t u r e surveys tend to l e a d to f o r e i g n p u b l i c a t i o n s more and more f r e q u e n t l y . Has q u a l i t y data c o l l e c t i o n been abandoned a t a p u b l i s h e d l e v e l ? Conclusions In c o n c l u s i o n , I t h i n k i t i s f a i r to say t h a t s i g n i f i c a n t progress has been achieved i n the i n d u s t r i a l treatment of phase e q u i l i b r i a i n the past t w e n t y - f i v e y e a r s . We are t r y i n g to use a t h e o r e t i c a l l y sound b a s i s f o r our c a l c u l a t i o n s , and we have the c a p a b i l i t y of u s i n g s o p h i s t i c a t e d complex models. We do keep t r a c k of new developments i n the f i e l d and implement them, a l b e i t i n a cautious and c o n s e r v a t i v e f a s h i o n . We are s t i l l faced w i t h many unsolved problems. Some of the techniques used are inadequate or i n a p p r o p r i a t e . I n order to e l i minate these d e f i c i e n c i e s , we must be committed to a program of c o n t i n u a l l y upgrading our s k i l l s . H o p e f u l l y , through meetings l i k e t h i s , we can d e f i n e f u t u r e development goals that w i l l be of mutual i n t e r e s t and b e n e f i t to the i n d u s t r i a l and academic communities. Nomenclature A, B, C, D, E - constant

parameters

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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85

Cp - i d e a l gas molar heat c a p a c i t y fG - f u g a c i t y i n the vapor phase mixture fL - f u g a c i t y i n the l i q u i d phase mixture fOL - l i q u i d reference f u g a c i t y H - molar enthalpy H* - i d e a l gas molar enthalpy (H-H*) - molar enthalpy departure frm the i d e a l gas s t a t e A*Wx ~ molar l i q u i d heat o f s o l u t i o n k - l i q u i d thermal c o n d u c t i v i t y K - vapor-liquid equilibrium ratio P - a b s o l u t e pressure Pref " r e f e r e n c e pressure R - gas constant T - a b s o l u t e temperatur V^- - l i q u i d molar volum - l i q u i d p a r t i a l mola x - mole f r a c t i o n i n l i q u i d phase y - mole f r a c t i o n i n vapor phase a ~ relative volatility Y - liquid activity coefficient u^- l i q u i d v i s c o s i t y - vapor phase f u g a c i t y c o e f f i c i e n t 0 - l i q u i d phase f u g a c i t y c o e f f i c i e n t ir - system pressure Subscripts 1

L

1 - property o f component i P £ " property evaluated a t r e f e r e n c e pressure 7T - property evaluated a t system pressure r e

References A. General 1. Hachmuth, K. H., Chem. Eng. Prog., (1952) 48, 523, 570, 617. 2. Pigford, R. L., Ind. Eng. Chem., (1952) 44, 25. 3. Walsh, T. J., Ind. Eng. Chem., (1952) 44, 45. 4. Franks, R. G. E., "Modeling and Simulation in Chemical Engineering,", Wiley-Interscience, New York: 1972. B. Equations of State 5. Redlich-Kwong: Redlich, O. and Kwong, J. N. S., Chem. Rev., (1949) 44, 233. 6. V i r i a l (Wohl Correlation): Wohl, A., Z. Phys. Chem., (1929) B2, 77. 7. B-W-R: Benedict, M., Webb, G. B., and Rubin, L. C., Chem. Engr. Prog., (1951); 47, 419;J. Chem. Physics. (1940) 8, 334;ibid., (1942) 10, 747. 8. Prausnitz-Chueh modified Redlich-Kwong: Chueh, P. L. and Prausnitz, J. M., Ind. Eng. Chem. Fund. (1967) 6, 492.

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

9. Barner-Adler: Barner, H. B. and Adler, S. B., Ind. Eng. Chem. Fund., (1970) 9, 521. 10. Starling modified B-W-R: Starling, K. E., Hydro. Proc., (1971) 50, (3), 101. 11. Persohn modified Redlich-Kwong: Persohn, T. F., Unpublished method used at Union Carbide Corporation. 12. Soave modified Redlich-Kwong: Soave, G., Chem. Engr. Sci., (1972), 27, 1197. 13. Hayden-O'Connell V i r i a l : Hayden, J. G. and O'Connell, J. P., Ind. Eng. Chem. Proc. Des. Dev., (1975) 14, 209. Nothnagel, K. H., Abrams, D. S., and Prausnitz, J. M., Ind. Eng. Chem. Proc. Des. Dev., (1973) 12, 25. C. Liquid Reference Fugacity 14. Chao-Seader: Chao K C and Seader J D. AIChE Jour nal, (1961) 1, 598 15. Chao-Robinson: Robinson, , , Eng Chem. Proc. Des. Dev., (1971) 10, 221. 16. Persohn: Persohn, T. F., Unpublished generalized method used at Union Carbide Corporation. D. Activity Coefficient 17. Redlich-Kister: Smith, B. D., "Design of Equilibrium Stage Processes", Chapter 2, McGraw-Hill, New York, (1963). 18. Hildebrand: Prausnitz, J. M., "Molecular Thermodynamics of Fluid Phase Equilibria", Chapter 7. Prentice-Hall, Englewood C l i f f s , New Jersey; 1969. 19. NRTL: Renon, H. and Prausnitz, J. M., AIChE Journal, (1968) 14, 135. 20. Wilson: Wilson, G. M., J. Am. Chem. Soc., (1964) 86, 135. 21. Margules: Wohl, K., Trans. Am. Inst. Chem. Engrs., (1946) 42, 215. Brown, G. M. and Smiley, H. M., AIChE Journal (1966) 12, 609. 22. UNIQUAC: Abrams, D. S. and Prausnitz, J. M., AIChE Journal, (1975) 21, 116. O'Connell, "Application of the UNIQUAC Equation for Excess Gibbs Energy to Systems Containing Supercritical Hydrogen, Nitrogen and/or Methane", presented at Sixty-Eighth Annual Meeting of the American Institute of Chemical Engineers, November 17, 1975, Los Angeles, California. 23. UNIFAC: Fredenslund, A., Jones, R. L. and Prausnitz, J. M., AIChE Journal, (1975), 21, 1086. E. Enthalpy 24. Yen-Alexander: Yen, L. C. and Alexander, R. E., AIChE Journal, (1965) 11, 334.

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

4 Measurement of Vapor-Liquid Equilibrium MICHAEL M . ABBOTT Rensselaer Polytechnic Institute, Troy, N.Y. 12181

The importance accorded the measurement of vapor-liquid equilibrium (VLE) data needs little elaboration. A glance at the annual indices for the Journal of Chemical and Engineering Data makes the point nicely. For the five years 1971-1975, at least 50 papers present new VLE data, and many of them contain descriptions of new experimental techniques. Perusal of recent volumes of less specialized journals, such as the AIChE Journal or I&EC Fundamentals, reveals a similar proliferation of VLE studies. The necessity for reliable VLE data is apparent. Many separations processes involve the transfer of chemical species between contiguous liquid and vapor phases. Rational design and simulation of these processes requires knowledge of the equilibrium compositions of the phases. Raoult's and Henry's "laws" rarely suffice as quantitative tools for the prediction of equilibrium compositions; precise work demands the availability of either the equilibrium data themselves, or of thermodynamic correlations derived from such data. Specialists in the field tend to concentrate on one of two broad areas: high-pressure VLE, or low-pressure VLE. My major interests are in the latter area, and hence the thrust of my talk will be in this direction: the measurement and reduction of lowpressure VLE data. Low-pressure VLE experimentation differs from high-pressure experimentation on two major counts. First, the problems of equipment design and operation are less formidable than for high-pressure work. Secondly, more effective use can be made of the thermodynamic equilibrium equations, both in the data reduction process and in the design of the experiments themselves. It is this second feature-the strong interplay of theory with experiment--which to me most distinguishes low-pressure from high-pressure VLE work. The usual product of a low-pressure VLE study is an expression for the composition dependence of the excess Gibbs function G for the liquid phase. If experiments are done at several temperatures, E

87 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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A N D F L U I D PROPERTIES

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then the temperature dependence o f parameters i n the equation f o r may a l s o be determined. At low p r e s s u r e s , the p r e s s u r e dependence of i s weak, and may u s u a l l y be ignored. C e r t a i n advantages d e r i v e from t h i s k i n d of r e p r e s e n t a t i o n , f o r the d e s c r i p t i o n of VLE i s compact and y e t has thermodynamic s i g n i f i c a n c e . B a c k - c a l c u l a t i o n of e q u i l i b r i u m curves r e q u i r e s values f o r the pure-component vapor p r e s s u r e s , an e x p r e s s i o n f o r G i n terms of x and T, and equations of s t a t e ( u s u a l l y of s i m p l e form) f o r the vapor and l i q u i d phases. Approximations, when they must be made, a r e w e l l - d e f i n e d and o f t e n s u b j e c t to independent verification. E

Thermodynamic C o n s i d e r a t i o n s At e q u i l i b r i u m , th the same i n the l i q u i d and vapor phases:

The vapor-phase f u g a c i t y i s r e l a t e d to the vapor-phase f u g a c i t y c o e f f i c i e n t (ji-^: f±

v

= y** ±

(2)

±

F u g a c i t y c o e f f i c i e n t _^ i s c a l c u l a b l e from an equation of s t a t e f o r the vapor phase; mixing r u l e s must be a v a i l a b l e f o r the equat i o n - o f - s t a t e parameters. The l i q u i d - p h a s e f u g a c i t y i s r e l a t e d to the l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t y^i f . * = x. f°

(3)

Yl

Here, f° i s the s t a n d a r d - s t a t e f u g a c i t y f o r s p e c i e s i . I f the e q u i l i b r i u m temperature i s lower than the c r i t i c a l temperature o f pure i (the u s u a l case f o r low-pressure VLE), the standard s t a t e i s taken as pure l i q u i d i a t the system T and P. Thus Eq. (3) becomes f*

I

= x.y.f. 11 l

(4)

Determination o f f ^ r e q u i r e s the a v a i l a b i l i t y of equations of s t a t e f o r pure vapor and l i q u i d i , and a v a l u e f o r the vapor p r e s s u r e p| of pure i . Often, f-^ i s approximately equal to P ? . Combination of Eqs. ( 1 ) , ( 2 ) , and (4) y i e l d s the form of the e q u i l i b r i u m equation most commonly used i n low-pressure work a t

a t

W i

- W

( 5 )

I t i s through Eq. (5) that low-pressure VLE measurements p r o v i d e the experimental i n p u t r e q u i r e d f o r a q u a n t i t a t i v e thermodynamic

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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Measurement of Vapor-Liquid

Equilibrium

89

d e s c r i p t i o n of l i q u i d - p h a s e n o n i d e a l i t i e s . The excess Gibbs f u n c t i o n G p l a y s a c e n t r a l r o l e i n e x p e r i mental s o l u t i o n thermodynamics, f o r i t s c a n o n i c a l v a r i a b l e s (T, P, and x) are those most s u s c e p t i b l e t o measurement and c o n t r o l . A c t i v i t y c o e f f i c i e n t s are r e l a t e d to mole-number d e r i v a t i v e s o f the dimensionless excess Gibbs f u n c t i o n g = G /RT: E

E

toY

±

= [Z(ng)/dn ] ±

(6)

T

Thus &nYj[ i s a p a r t i a l molar property w i t h respect to g. Accordi n g l y , we have the a d d i t i o n a l u s e f u l r e l a t i o n s h i p g = Zx Zny ±

(7)

±

C l a s s i c a l thermodynamic f o r the t o t a l d i f f e r e n t i a dg = -

dT + ~ - dP + Zlny.dx R

T

1

(8) 1

RT F F Here, H i s the excess enthalpy (heat o f mixing) and V i s the excess volume (volume change o f m i x i n g ) . Both H and V are subj e c t t o d i r e c t experimental d e t e r m i n a t i o n . Taking the t o t a l d i f f e r e n t i a l o f Eq. ( 7 ) , and comparing the r e s u l t w i t h Eq. ( 8 ) , we o b t a i n the Gibbs-Duhem equation: E

E

H

Zx dlny ±

±

vE

E

= - ^

d

T

+

i f

d

P

(

9

)

RT Equations (1) through ( 9 ) , o r v a r i a t i o n s upon them, c o n s t i tute the u s u a l thermodynamic b a s i s f o r the r e d u c t i o n and i n t e r p r e t a t i o n o f low-pressure VLE data. Isothermal vs. I s o b a r i c Data P r e c i s e d e t e r m i n a t i o n o f G through low-pressure VLE measurements g e n e r a l l y r e q u i r e s the a v a i l a b i l i t y o f data spanning the e n t i r e range o f l i q u i d compositions. The e x p e r i m e n t a l i s t s t i l l has the o p t i o n , however, o f c o l l e c t i n g h i s data e i t h e r a t i s o b a r i c or a t i s o t h e r m a l c o n d i t i o n s . The q u e s t i o n then a r i s e s whether one type of data i s more u s e f u l than the o t h e r . L e t us assume the a v a i l a b i l i t y of two complete s e t s o f e r r o r f r e e VLE measurements. The f i r s t s e t has been taken a t constant pressure, and c o n s i s t s o f values o f T, x, and y. The second i s i s o t h e r m a l , c o n s i s t i n g of values f o r P, x, and y. We wish to reduce both s e t s o f data, so as to o b t a i n values f o r G . For each data s e t , we may c a l c u l a t e p o i n t values o f y from Eq. ( 5 ) : i

Y. = y.<j>.P/x.f.

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

(10)

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The corresponding values f o r g are then computed from Eq. (7): g = Ex.Any. 1 1 I t i s d e s i r a b l e f o r purposes of c o r r e l a t i o n and i n t e r p r e t a t i o n that experimental values of g be at a s i n g l e T and P, w i t h x the only v a r i a b l e . C l e a r l y , t h i s i s not the case f o r e i t h e r of our experiments. C o r r e c t i o n s must be made i n both cases to reduce the values of g to a common b a s i s . According to Eq. ( 8 ) , t h i s r e q u i r e s the use of independently determined values f o r H or V . Thus, f o r i s o b a r i c data, E

,

T

f

H

E

E

g(T5P,x) = g(T,P,x) - /

^ d T

S i m i l a r l y , f o r isotherma p

T

g(T,P ,x) = g(T,P,x) + /

l

V

E

^

dP

T

1

Here, T i s a r e f e r e n c e temperature f o r the i s o b a r i c data, and P a r e f e r e n c e pressure f o r the i s o t h e r m a l data. Even i f the r e q u i r e d values of H or V are a v a i l a b l e — a n d o f t e n they are n o t — t h e above c o r r e c t i o n s complicate the data a n a l y s i s . One n a t u r a l l y asks whether one or the other of the c o r r e c t i o n s might more s a f e l y be ignored. C a l c u l a t i o n s f o r r e a l s y s tems show that the temperature c o r r e c t i o n i s o f t e n s u b s t a n t i a l , whereas the pressure c o r r e c t i o n i s f r e q u e n t l y n e g l i g i b l e ; a t low p r e s s u r e s , g may depend s t r o n g l y on T, but r a r e l y upon P. Thus low-pressure i s o t h e r m a l VLE data are more e a s i l y reduced to u s e f u l form than are i s o b a r i c data; they are to be p r e f e r r e d f o r t h i s reason. Much of the o l d e r low-pressure work i s i s o b a r i c ; the b e s t modern s t u d i e s are i s o t h e r m a l . The argument i s sometimes advanced t h a t i s o b a r i c VLE data are more u s e f u l f o r process design, because s e p a r a t i o n processes are more n e a r l y i s o b a r i c than i s o t h e r m a l . This argument ignores the f a c t t h a t pressure drops i n d i s t i l l a t i o n columns can be substant i a l , and t h a t they are accounted f o r i n modern design procedures. The best thermodynamic t o o l f o r low-pressure d i s t i l l a t i o n column design i s an e x p r e s s i o n f o r G i n terms of T and x, w i t h values f o r the parameters determined from c a r e f u l l y executed i s o t h e r m a l VLE experiments. E

E

E

D i r e c t v s . I n d i r e c t Determination of Vapor Compositions E

We consider the d e t e r m i n a t i o n of G v i a the r e d u c t i o n of lowpressure i s o t h e r m a l P-x-y data. The procedure i s simple and d i r e c t . Values of are computed from Eq. (10), and the corresponding values of g are c a l c u l a t e d from Eq. ( 7 ) ; the values of g are then smoothed w i t h r e s p e c t to l i q u i d composition. The smoothing procedure may be g r a p h i c a l or n u m e r i c a l , but the eventual product i s an

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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a n a l y t i c a l e x p r e s s i o n f o r g as a f u n c t i o n of x. In the above procedure, no use i s made o f the Gibbs-Duhem equation, Eq. ( 9 ) . Because the pressure-dependence o f G i s weak at low p r e s s u r e s , t h i s equation reduces f o r i s o t h e r m a l c o n d i t i o n s to E

Ex d£ny = 0 i

(11)

i

Equation (11) i n f a c t imposes a thermodynamic c o n s t r a i n t upon the l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t s , a c o n s t r a i n t not n e c e s s a r i l y s a t i s f i e d by values of y^ computed v i a Eq. (10) from r e a l (and t h e r e f o r e p o s s i b l y imperfect) data. However, values o f y^ generated from a smoothing equation f o r g v i a Eq. (6) do s a t i s f y Eq. (11). Comparison of generated w i t h experimental values o f y^ cons t i t u t e s an example of c o n s i s t e n c y t e s t . Many f o r low- and high-pressure VLE data. Van Ness e t a l (j.) and C h r i s t i a n s e n and Fredenslund (2) present readable d i s c u s s i o n s o f such t e s t s . Instead o f s e r v i n g as a b a s i s f o r the t e s t i n g of redundant data, the Gibbs-Duhem equation may be used i n q u i t e a d i f f e r e n t manner, one which a i d s the experimenter i n the design of a VLE experiment of minimal complexity. For purposes o f d i s c u s s i o n , we assume i d e a l - g a s behavior f o r the vapor phase, and p r e s s u r e independence o f l i q u i d - p h a s e p r o p e r t i e s . I n t h i s case, Eq. (10) reduces to

and the Gibbs-Duhem equation becomes, f o r a b i n a r y system, x d5,ny + x d£ny = 0 £

1

2

2

(13)

Equations (12) and (13) y i e l d , on combination and s i m p l i f i c a t i o n , d

y

dP

i

y^-y^

P(y -x ) 1

(14)

1

Equation (14) i s a r e s t r i c t e d form o f the b i n a r y c o e x i s t e n c e equat i o n . The important content o f the equation i s as much conceptual as mathematical: i t i l l u s t r a t e s that simultaneous measurement o f P, x, and y i s unnecessary, that vapor compositions can i n p r i n c i p l e be computed from measurements o f j u s t P and x. Once the y a r e determined by i n t e g r a t i o n , values of y^ f o l l o w d i r e c t l y from Eq. (12). Van Ness (3) presents a d e t a i l e d d i s c u s s i o n of the charact e r i s t i c s and a p p l i c a t i o n o f Eq. (14). Reduction o f VLE data v i a the c o e x i s t e n c e equation i s " i n d i r e c t " , i n t h a t i t makes no d i r e c t use o f experimentally-determined vapor compositions. Other i n d i r e c t approaches are p o s s i b l e . One

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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of the most popular i s t h a t f i r s t proposed by Barker (_4 ) . method i s based upon the equation P = Z (x.y.P 11 i I

S a t

70 l

Barker's

(15)

which f o l l o w s from Eq. ( 5 ) .

Quantity $^ i s d e f i n e d by

S a t

*. = * . ( P / f . ) l l l l A c t i v i t y c o e f f i c i e n t s i n Eq. (15) are r e p l a c e d by the expressions Y, = exp [g -

Z x (|f-) i T P k

]

(16)

k

The composition dependence of g i s expressed by an equation of s u i t a b l e form, and values of undetermined parameters i n t h i s equat i o n are found by a r e g r e s s i o n procedure that y i e l d s a b e s t f i t of the P vs. x data. The $^ depend upon the y^, which are not i n i t i a l l y known; thus an i t e r a t i v e procedure i s r e q u i r e d . The major requirement f o r s u c c e s s f u l a p p l i c a t i o n of Barker's method i s the a v a i l a b i l i t y of an a n a l y t i c a l e x p r e s s i o n f o r g t h a t i s capable of producing a f i t to the P-x data to w i t h i n the l i m i t s of e x p e r i mental u n c e r t a i n t y . Abbott and Van Ness (5) and Abbott (6) d i s cuss the development and s e l e c t i o n of such equations. There has been much d i s c u s s i o n of the r e l a t i v e m e r i t s of d i r e c t measurement, as opposed to i n d i r e c t c a l c u l a t i o n , of vapor compositions. Although the d i f f i c u l t y of p r e c i s e measurement of y's i s g e n e r a l l y conceded, there remains a s t r o n g body of o p i n i o n that the redundant i n f o r m a t i o n p r o v i d e d by experimental values of y (however much i n e r r o r they might be) somehow enhances the r e l i a b i l i t y of the eventual c o r r e l a t i o n f o r g. I do not s u b s c r i b e to t h i s view. The d i f f i c u l t y of c a r r y i n g out the necessary c a l c u l a t i o n s by hand c e r t a i n l y a t one time c o n s t i t u t e d a reasonable argument a g a i n s t i n d i r e c t d e t e r m i n a t i o n of y's, but the e l e c t r o n i c computer has removed t h i s computational b a r r i e r . The experimenter's time i s best spent i n a c h i e v i n g accuracy i n the measurement of the minimum number of v a r i a b l e s r e q u i r e d to c h a r a c t e r i z e the system, not i n d e v i s i n g ingenious ways to c o l l e c t redundant data. E x t e n s i o n of Isothermal VLE Data w i t h Temperature According to Eq. ( 8 ) , the excess enthalpy i s p r o p o r t i o n a l to the temperature d e r i v a t i v e of g: H

E

= -RT

2

(|f)

(17)

Equation (17) suggests an a l t e r n a t i v e procedure to the d i r e c t d e t e r m i n a t i o n of i s o t h e r m a l VLE at s e v e r a l temperatures, namely,

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 use o f heat-of-mixing data f o r e x t r a p o l a t i o n o r i n t e r p o l a t i o n of g w i t h T. Suppose f o r example that i s o t h e r m a l data are a v a i l a ble a t a s i n g l e temperature T-^, and t h a t H data are a v a i l a b l e a t two temperatures near T]_. Suppose f u r t h e r that H /RT i s known to vary approximately l i n e a r l y w i t h T: E

E

E

H /RT = a + bT where a and b depend upon composition o n l y . Values o f g a t some other temperature T can then be found by i n t e g r a t i o n o f Eq. (17): 2

g(T ) 2

= gO^)

- a £n ( T ^ )

- b d ^ )

Obvious extensions to t h i s simple example are p o s s i b l e . Thus, the method can be used comprehensive expression be employed, which a l l o w i n c o r p o r a t i o n o f s e v e r a l s e t s o f i s o t h e r mal VLE data and H data. F i n a l l y , C p data can be i n c o r p o r a t e d i n t o the procedure through use o f the thermodynamic equation E

S

E

E

= P,x

( 1 8 )

An elegant example o f the simultaneous use o f i s o t h e r m a l VLE and H data to p r o v i d e a complete and i n t e r n a l l y c o n s i s t e n t s e t o f excess f u n c t i o n s over a wide temperature range i s the recent work of L a r k i n and Pemberton (7) on the ethanol-water system. E

Measurement o f Low-Pressure VLE Data Hala et_ a l (8) c l a s s i f y VLE measurement techniques i n f i v e major groups: d i s t i l l a t i o n methods, c i r c u l a t i o n methods, s t a t i c methods, dew-point/bubble-point methods, and f l o w methods. One could add to t h i s l i s t some r a t h e r s p e c i a l i z e d techniques, b u t i n the main most modern low-pressure VLE work i s done on two major types o f equipment: c i r c u l a t i o n s t i l l s and s t a t i c e q u i l i b r i u m cells. In low-pressure v a p o r - c i r c u l a t i o n s t i l l s , a l i q u i d mixture i s charged to a d i s t i l l i n g f l a s k , and brought to a b o i l . The evolved vapors are condensed e x t e r n a l l y i n t o a r e c e i v e r ; excess condensate r e t u r n s through an overflow tube back i n t o the d i s t i l l i n g f l a s k , where i t mixes w i t h the b o i l i n g l i q u i d . The compositions of the b o i l i n g l i q u i d and the condensate change c o n t i n u o u s l y w i t h time, u n t i l a s t e a d y - s t a t e c o n d i t i o n i s reached. The s t e a d y - s t a t e comp o s i t i o n s of the b o i l i n g l i q u i d and the vapor condensate are taken to be the e q u i l i b r i u m l i q u i d and vapor compositions. Hala et_ a l (8) d i s c u s s a t great l e n g t h the design and p r i n c i ples o f o p e r a t i o n o f c i r c u l a t i o n s t i l l s . The b e t t e r s t i l l s a r e complicated d e v i c e s , which may i n v o l v e c i r c u l a t i o n o f b o i l i n g l i q u i d as w e l l as o f vapor condensate. Operation i s o f t e n i s o b a r i c , w i t h pressure r e g u l a t i o n provided by a manostat. S p e c 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.

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES I N C H E M I C A L INDUSTRY

Vent

Vacuum <

l

I

Constant^T Bath Figure 1.

Schematic diagram of static VLE apparatus

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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care must be taken when withdrawing samples f o r a n a l y s i s , p a r t i c u l a r l y i f the temperature l e v e l i s h i g h and e s p e c i a l l y f o r the vapor condensate, f o r l o s s by v a p o r i z a t i o n o f the l i g h t e r components can be s i g n i f i c a n t . In the s t a t i c method, a l i q u i d mixture i s charged to an evacuated e q u i l i b r i u m c e l l , immersed i n a constant-temperature bath. E q u i l i b r a t i o n o f the phases i s brought about by vigorous s t i r r i n g of the l i q u i d phase o r , i n some designs, by r o c k i n g the c e l l . When the system has come to e q u i l i b r i u m , the pressure i s read from a gauge. Measurement o f the composition of the vapor phase i s the most d i f f i c u l t p a r t o f the s t a t i c method, and probably accounts f o r i t s i n f r e q u e n t use u n t i l r e l a t i v e l y r e c e n t l y . A t low p r e s s u r e s , the t o t a l mass o f vapor i n the c e l l i s s m a l l so the vapor sample must be extremely s m a l l i n orde turbed on withdrawal o sample y avoided e n t i r e l y i f one e s t a b l i s h e s vapor compositions i n d i r e c t l y , by c a l c u l a t i o n . L i q u i d compositions can be determined g r a v i m e t r i c a l l y , and the P-x data reduced by Barker's method, o r by i n t e g r a t i o n o f the c o e x i s t e n c e equation. F i g u r e 1 i s a schematic drawing o f a modern low-pressure s t a t i c apparatus, designed by Gibbs and Van Ness (9) . Pure l i q u i d s 1 and 2 are c o n f i n e d i n c a l i b r a t e d p i s t o n - i n j e c t o r s P I 1 and PI2. The e q u i l i b r i u m c e l l i s immersed i n a constant-temperature bath, and pressures are read from a Texas Instruments fuzedquartz pressure gauge PG. The vapors are i s o l a t e d from the gauge by a d i f f e r e n t i a l pressure i n d i c a t o r DPI; f i n e adjustment o f the i n e r t gas pressure a c t u a l l y measured by the gauge i s p r o v i d e d by a s e n s i t i v e variable-volume device VV. Heater H prevents condens a t i o n o f vapors i n the l i n e connecting the c e l l w i t h the DPI. The c e l l i s evacuated b e f o r e the beginning o f a run. A measured amount o f l i q u i d 2 i s then i n j e c t e d i n t o the c e l l , the magn e t i c s t i r r e r i s a c t i v a t e d , and a f t e r a s h o r t p e r i o d o f time a vapor-pressure reading i s taken. This pressure i s the vapor p r e s sure P | o f pure 2. Small amounts o f l i q u i d 1 are s u c c e s s i v e l y added to the c e l l , and pressure readings are taken a f t e r each i n j e c t i o n . The process i s continued u n t i l the l i q u i d - p h a s e mole f r a c t i o n o f 1 i s about 0.6. The c e l l i s then emptied and evacuated, component 1 i s added t o the c e l l , and s m a l l amounts o f 2 i n j e c t e d u n t i l X2 i s about 0.6; pressure readings are taken a f t e r each i n j e c t i o n . The complete s e t o f pressure-composition measurements f o r both experiments c o n s t i t u t e s the experimental P-x curve f o r the b i n a r y system. The method i s e a s i l y extended t o ternary systems [Abbott e t a l ( 1 0 ) ] ; one merely adds another p i s t o n - i n j e c t o r to accommodate the t h i r d component. The low-pressure s t a t i c method has been developed to a h i g h degree o f s o p h i s t i c a t i o n by the NPL D i v i s i o n o f Chemical Standards group [Pemberton (11); L a r k i n and Pemberton ( 7 ) ] . I n t h e i r work on the ethanol-water system between 30(°C) and 90(°C), they c l a i m an accuracy i n measured vapor pressures o f +0.02% o r +2.6 (Pa), a 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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whichever i s g r e a t e r . The r e l a t i v e s i m p l i c i t y of the method and the ease of o p e r a t i o n of the equipment seem l i k e l y to make the s t a t i c technique a standard procedure f o r d e t e r m i n a t i o n of lowpressure VLE. A c r i t i c a l requirement f o r s u c c e s s f u l use of the s t a t i c technique i s that the l i q u i d s charged to the c e l l be thoroughly degassed. The c e l l i s a c l o s e d system, so even s m a l l amounts of d i s s o l v e d gases can r e s u l t i n erroneous pressure readings. A common—and time-consuming—degassing technique i n v o l v e s the repeated f r e e z i n g and o f f - g a s s i n g of the pure l i q u i d s . We are now experimenting w i t h degassing by d i s t i l l a t i o n ; our p r e l i m i n a r y r e s u l t s c o n f i r m the s u i t a b i l i t y of the method. Conclusions and Remarks (1) The most u s e f u low-pressur constant temperature. I would discourage the r o u t i n e c o l l e c t i o n of low-pressure i s o b a r i c VLE data. T h e i r r e d u c t i o n and i n t e r p r e t a t i o n i s d i f f i c u l t , and there i s nothing to be obtained from them that c a n t be found from a couple of i s o t h e r m a l experiments, o r , p r e f e r a b l y , a s i n g l e i s o t h e r m a l experiment and some heat-of-mixing data. (2) Measurement of redundant data i s o f t e n d i f f i c u l t and f r e quently unnecessary. For example, r i g o r o u s thermodynamic methods permit the d e t e r m i n a t i o n of vapor compositions from low-pressure i s o t h e r m a l P-x data. I f vapor compositions are not the most d i f f i c u l t q u a n t i t i e s to measure, then a l t e r n a t i v e procedures may be devised i n which f o r example P i s c a l c u l a t e d from i s o t h e r m a l x-y measurements, or l i q u i d compositions from i s o t h e r m a l P-y measurements. The p o i n t to be made i s t h i s : thermodynamics can be used e i t h e r f o r c o n s i s t e n c y t e s t i n g , or to l e s s e n the burden on the e x p e r i m e n t a l i s t by reducing the number of v a r i a b l e s he must measure. (3) The pure-component vapor pressures p l a y a key r o l e i n the r e d u c t i o n of low-pressure VLE data. Since they appear i n the e q u i l i b r i u m equations as n o r m a l i z i n g f a c t o r s f o r the a c t i v i t y c o e f f i c i e n t s , they d i r e c t l y a f f e c t the numerical values of G through every data p o i n t . However, vapor pressures are h i g h l y s e n s i t i v e to temperature and to sample p u r i t y . Thus, to guarantee i n t e r n a l c o n s i s t e n c y w i t h the r e s t of the data s e t , they should be measured w i t h the same equipment and on the same l o t s of m a t e r i a l s as are the mixture vapor p r e s s u r e s . This u n f o r t u n a t e l y i s not always done, and one sometimes has to analyze an otherwise s a t i s f a c t o r y s e t of data w i t h f o r e i g n vapor p r e s s u r e s , computed f o r example from an Antoine equation e x t r a c t e d from the l i t e r a t u r e . T h i s can be r i s k y . I would encourage a l l workers i n t h i s f i e l d to measure and r e p o r t pure-component vapor pressures along w i t h t h e i r mixture data. (4) The vapor-phase f u g a c i t y c o r r e c t i o n , although not always l a r g e , i s o f t e n the weakest l i n k i n the d a t a - r e d u c t i o n process. 1

E

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 source o f u n c e r t a i n t y i s u s u a l l y the mixed second v i r i a l coeff i c i e n t B-^j. Experimental data are r a r e l y a v a i l a b l e f o r t h i s q u a n t i t y , and one must r e s o r t to c o r r e l a t i o n s . Good c o r r e l a t i o n s are a v a i l a b l e , b u t t h e i r mixture data base i s s m a l l and needs to be extended. There i s a s p e c i a l need f o r vapor-phase v o l u m e t r i c data on mixtures c o n t a i n i n g p o l a r and hydrogen-bonding s p e c i e s . (5) I t would be u s e f u l i f VLE e x p e r i m e n t a l i s t s could d e f i n e by mutual agreement h a l f a dozen o r so r e f e r e e t e s t systems, as the c a l o r i m e t r y people have done. These would be b i n a r y systems f o r which pure-component vapor pressures were w e l l known, and which had been c a r e f u l l y s t u d i e d by s e v e r a l i n v e s t i g a t o r s on d i f f e r e n t types o f equipment o f proven r e l i a b i l i t y . The systems would be used f o r t e s t i n g new equipment designs and new VLE datar e d u c t i o n methods. (6) There i s a nee p r i s e d of n o n - o f f - t h e - s h e l colleague confided t o me the s u s p i c i o n that many academic s t u d i e s are based p r i m a r i l y on the a v a i l a b i l i t y o f c e r t a i n reagents i n chromatoq u a l i t y grade. This i s not e n t i r e l y an u n f a i r assessment. Whatever the b a s i s f o r c u r r e n t n e g l e c t o f such systems, the need i s there, and must e v e n t u a l l y be met. (7) We r e q u i r e more multicomponent experimentation. The hope remains that theory w i l l e v e n t u a l l y produce methods f o r reasonable e s t i m a t i o n o f multicomponent phase e q u i l i b r i a from data on the c o n s t i t u e n t b i n a r i e s . A s o l i d data base i s needed f o r the development and t e s t i n g o f such t h e o r i e s . (8) I t i s a p p r o p r i a t e to acknowledge the r o l e o f the computer i n modern VLE experimentation. Data r e d u c t i o n i s a k e y — b u t potent i a l l y t e d i o u s — s t e p i n VLE work. The a v a i l a b i l i t y o f high-speed computational techniques has I t h i n k put t h i s c r u c i a l step back i n t o p e r s p e c t i v e : no longer a major h u r d l e between experimentation and theory, but a v a l u a b l e adjunct to both. Acknowledgment I a p p r e c i a t e f i n a n c i a l support provided by N a t i o n a l Foundation Grant No. ENG75-17367.

Science

Notation

n

excess constant-pressure heat c a p a c i t y f u g a c i t y o f pure species i s t a n d a r d - s t a t e f u g a c i t y o f species i f u g a c i t y o f species i i n s o l u t i o n G /RT excess Gibbs f r e e energy excess enthalpy as s u p e r s c r i p t , designates l i q u i d phase t o t a l number o f moles number of moles o f species i

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

98

P H A S E EQUILIBRIA A N D

P Pf R T v V x^ y.£ Y-j_ E

$±

a t

= = = = = = = = = = -

F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

pressure vapor p r e s s u r e of pure i u n i v e r s a l gas constant a b s o l u t e temperature as s u p e r s c r i p t , designates vapor phase excess volume mole f r a c t i o n of s p e c i e s i i n l i q u i d phase mole f r a c t i o n of species i i n vapor phase a c t i v i t y c o e f f i c i e n t of species i f u g a c i t y c o e f f i c i e n t of species i i n s o l u t i o n f a c t o r i n VLE t o t a l - p r e s s u r e e x p r e s s i o n , Eq. (15)

Literature Cited 1. Van Ness, H. C., Byer 19, 238. 2. Christiansen, L. J. and Fredenslund, Aa., AIChE J. (1975), 21, 49. 3. Van Ness, H. C., AIChE. J. (1970), 16, 18. 4. Barker, J. A., Austral. J. Chem. (1953), 6, 207. 5. Abbott, M. M. and Van Ness, H. C., AIChE J. (1975), 21, 62. 6. Abbott, M. M., "Selection of Correlating Expressions for G ", International Symposium on Reduction of Vapor-Liquid Equilibrium Data, U.E.R. Scientifique de Luminy Printing Office, France (1976). 7. Larkin, J. A. and Pemberton, R. C., "Thermodynamic Properties of Mixtures of Water + Ethanol Between 298.15 and 383.15K", NPL Report Chem. 43, National Physical Laboratory, Division of Chemical Standards, Teddington, England, (1976). 8. Hála, E., Pick, J., Fried, V. and Vilím, O., "Vapour-Liquid Equilibrium, Second Edition" Pergamon Press, Oxford, 1967. 9. Gibbs, R. E. and Van Ness, H. C., Ind. Eng. Chem. Fundam. (1972), 11, 410. 10. Abbott, M. M., Floess, J. K., Walsh, J. E., Jr., and Van Ness, H. C., AIChE. J. (1975), 21, 72. 11. Pemberton, R. C. "Thermodynamic Properties of Aqueous Non-electrolyteMixtures. Vapor Pressures for the System Water + Ethanol at 303.15-363.15K Determined by an Accurate Static Method," Communications of 4th International Conference on Chemical Thermodynamics, VI, 137 (1975). E

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

5 Equilibria in Aqueous Electrolyte Systems at High Temperatures and Pressures E. U . FRANCK University of Karlsruhe, Karlsruhe, Germany

Aqueous e l e c t r o l y t e s o l u t i o n s a r e not r e s t r i c t e d to moderate temperatures and low p r e s s u r e s . They can e x i s t and have been i n v e s t i g a t e d to and f a r beyond the c r i t i c a l temperature of pure water and to pressures of s e v e r a l k i l o b a r s . Research on e q u i l i b r i a i n e l e c t r o l y t e s o l u t i o n s i n such a wide range gives v a l u a b l e s c i e n t i f i c 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 and i n t e r i o n i c i n t e r a c t i o n and k i n e t i c phenomena i n dense f l u i d s . P r a c t i c a l a p p l i c a t i o n s of the unusual combinations of p r o p e r t i e s of dense s u p e r c r i t i c a l f l u i d s can a l s o be a n t i c i p a t e d . Water w i l l always be the predominant e l e c t r o l y t i c s o l v e n t . The temperature and d e n s i t y determine i t s p r o p e r t i e s . The r e l a t i o n of these f u n c t i o n s w i t h pressure can be d e r i v e d from s t a t i c e x p e r i mental pvT-determinations which have i n recent years been extended to almost 1 0 0 0 C and 1 0 kbar. A t p r e s e n t , the i n t e r n a t i o n a l l y recognized steam t a b l e s cover the r e g i o n to 8 0 0 C and 1 kbar ( 1 ) . I t can be expected, however, that c r i t i c a l l y compiled t a b l e s f o r a wider range w i l l be a v a i l a b l e to the i n d u s t r i a l user i n the near f u t u r e . F i g . 1 g i v e s a temperature-density diagram (_2,_3jA>JL»_6) f ° water t o 1 0 0 0 C and a d e n s i t y of 1 . 6 g/cm . The c r i t i c a l p o i n t , CP, ( 3 7 4 C , 2 2 1 b a r ) , and t r i p l e p o i n t , TP, a r e i n d i c a t e d on the g a s - l i q u i d c o e x i s t e n c e curve. The p o i n t s on the broken l i n e extending from TP t o the r i g h t denote the t r a n s i t i o n s between d i f f e r e n t h i g h pressure m o d i f i c a t i o n s of i c e . A t pressures above 2 5 kbar, water d e n s i t i e s have been d e r i v e d from shock wave measurements ( 7 ) . A t 5 0 0 C and 1 0 0 0 C, pressures of about 8 and 2 0 kbar are needed t o produce the t r i p l e p o i n t d e n s i t y of l i q u i d water. The r e g i o n to about 5 0 0 C and 5 kbar i s a t present of p a r t i c u l a r i n t e r e s t f o r l a b o r a t o r y experiments, p o s s i b l e i n d u s t r i a l a p p l i c a t i o n and geochemical phenomena. The d i s c u s s i o n of phase e q u i l i b r i a and i o n i c d i s s o c i a t i o n of s a l t s i n water f o r a wide range of temperatures and d e n s i t i e s r e q u i r e s a knowledge of phase diagrams. Only a few s a l t - w a t e r phase diagrams have been e x t e n s i v e l y i n v e s t i g a t e d e x p e r i m e n t a l l y . 3

99

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r

100

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Density, Figure 1.

g/cm

Temperature-density diagram of water. Cross-hatched area is where most of the hydrothermal experiments were made.

Figure 2.

Schematic P-T- and V-x-diagrams of a simple two-component system

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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This a p p l i e s to the sodium c h l o r i d e - w a t e r system which i s an exemplary case. F i g . 2 gives a schematic p-T-diagram o f a two component system w i t h very d i f f e r e n t c r i t i c a l temperatures. One observes a c r i t i c a l curve which i s a p r o j e c t i o n from the t h r e e dimensional pressure-temperature-composition diagram on the p-Tplane. The curve extends u n i n t e r r u p t e d between the c r i t i c a l p o i n t s of the two pure components. The p r o j e c t i o n of a t h r e e phase-surface, S2LG, between a quadruple p o i n t and the t r i p l e p o i n t T2 can a l s o be seen. I t does not i n t e r s e c t the c r i t i c a l curve i n t h i s example. I n the r i g h t p a r t of F i g . 2 we have a P - x - s e c t i o n taken between M and B i n the p r e v i o u s diagram. A t CP the c r i t i c a l curve penetrates the P-x-plane. In F i g . 3, a s e l e c t i o n of experimental isotherms corresponding to the g-£-curves o NaCl-H20 ( 8 ) . A p o r t i o system and i t i s reasonable to assume, t h a t i t i s o f the u n i n t e r rupted type and w i l l e v e n t u a l l y extend t o the c r i t i c a l p o i n t o f pure NaCl, which has not y e t determined but w i l l very probably be above 2000 C. I t f o l l o w s from F i g . 3, that even above the c r i t i c a l temperature of pure water and a t pressures r e a d i l y a v a i l a b l e , homogeneous f l u i d phases w i t h much more than 10 weight percent NaCl can e x i s t over a wide range of c o n d i t i o n s . This i s undoubtedly t r u e f o r other a l k a l i s a l t s . S o l u t i o n s of t h i s type e x i s t as hydrothermal f l u i d s w i t h i n the earth's c r u s t and can p o s s i b l y be used f o r p r a c t i c a l purposes too. Many other aqueous systems, however, e x h i b i t i n t e r r u p t e d c r i t i c a l curves and d i f f e r e n t behav i o r . An example i s the Si02-H20-system ( 9 ) . C r i t i c a l Curves and M i s c i b i l i t y The e l e c t r o l y t i c p r o p e r t i e s o f water can be v a r i e d by expans i o n . S i m i l a r v a r i a t i o n s can be achieved by admixing an i n e r t v o l a t i l e component to s u p e r c r i t i c a l water a t h i g h p r e s s u r e s . Such mixtures can have i n t e r e s t i n g p r o p e r t i e s w i t h o u t adding s a l t . I n recent years v a r i o u s b i n a r y aqueous systems have been i n v e s t i g a t e d by s e v e r a l groups and the two-phase s u r f a c e s determined. I f the nonaqueous component i s nonpolar o r only s l i g h t l y p o l a r , the c r i t i c a l curves are mostly o f the i n t e r r u p t e d type and the upper branch, which begins a t the c r i t i c a l p o i n t of the pure water, proceeds to r a t h e r h i g h p r e s s u r e s . F i g . 4 shows a number o f examples ( 6 ) . These c r i t i c a l curves are a s o r t o f envelope o f the two-phase r e g i o n . A t the h i g h temperature s i d e of the curves, the components are completely m i s c i b l e a t a l l t o t a l d e n s i t i e s . Above 400 C t h e r a r e gases, n i t r o g e n and l i g h t alkanes a l l appear t o be m i s c i b l e w i t h water. Two o f the curves have a d i f f e r e n t c h a r a c t e r than the o t h e r s . Benzene-water has a c r i t i c a l curve w i t h a minimum below 300 C a t about 200 bar. Thus, dense homogeneous mixtures o f water w i t h benzene and other s i m i l a r aromatic compounds can be obtained r e l a t i v e l y e a s i l y (10). The curve f o r H 0-C02 a l s o proceeds t o 2

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

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Weight % Figure 3. Experimental isotherms for the NaCl-H 0 system in a pressure-salt concentration-diagram 2

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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r a t h e r low temperatures, i n d i c a t i n g a c e r t a i n degree of a t t r a c t i v e i n t e r a c t i o n between the two compounds. This does not mean s t o c h i o m e t r i c a s s o c i a t i o n , however, as shown by recent h i g h temperature-high pressure Raman measurements (11). The two-phase boundary s u r f a c e and the pvT-behavior i n the s u p e r c r i t i c a l r e g i o n has been e x t e n s i v e l y i n v e s t i g a t e d (12,13,14). In analogy to " s a l t i n g out" e f f e c t s a t low temperature, one should expect t h a t a d i s s o l v e d s a l t as a t h i r d component a l s o a f f e c t s the m i s c i b i l i t y of C 0 and H2O i n the c r i t i c a l r e g i o n . This i s indeed the case as was shown some years ago (15). Recently extended measurements w i t h the H^O-CC^-NaCl system has been performed at s e v e r a l c o n c e n t r a t i o n s and the r e s u l t s are shown i n F i g . 5. The curves a r e f o r constant w a t e r - s a l t c o n c e n t r a t i o n r a t i o s . The CO2 content v a r i e s . Thi probably a l s o the c r i t i c a by 50C or more (16). Beginning w i t h the c r i t i c a l p o i n t of pure water, a p a r t of the c r i t i c a l curve of the b i n a r y system l^O-NaCl i s a l s o shown. I t appears t h a t by making use o f the c r i t i c a l data f o r the a p p l i e d w a t e r - s a l t r a t i o , a c e r t a i n u n i f i e d , reduced r e p r e s e n t a t i o n of the v a r i o u s curves can be obtained. A more r i g i d t h e o r e t i c a l treatment of a t e r n a r y system of t h i s k i n d does not seem to be p o s s i b l e a t present. The experimental work i s being continued. This system i s c e r t a i n l y one of the most important i n geochemistry. I r r e s p e c t i v e o f the incomplete q u a n t i t a t i v e knowledge of the i n t e r a c t i o n s f o r a t e r n a r y system l i k e t^O-CX^-NaCl one can attempt to make e s t i m a t e s . F i g . 6 g i v e s a t e n t a t i v e t r i a n g u l a r diagram f o r a constant pressures of 1 kbar to the m e l t i n g temperature of pure sodium c h l o r i d e . The i m m i s c i b i l i t y curves of the b i n a r y systems H2O-CO2 and H20-NaCl a r e r e l a t i v e l y w e l l determined by experiments. The lower r i g h t curve i n the l^O-NaCl plane of the p r i s m i s based on the s o l u b i l i t y curve of NaCl i n water a t e q u i l i b r i u m pressures (8) and can be only q u a l i t a t i v e l y c o r r e c t a t 1 kbar. F i g . 6 suggests a t l e a s t , t h a t there i s a r a t h e r extended range of homogeneous f l u i d s i n the H20-C02-NaCl t e r n a r y system a t h i g h pressure between 400 and 600 C. I t i s hoped that c u r r e n t work w i l l permit the c o n s t r u c t i o n of more r e l i a b l e diagrams f o r d i f f e r e n t pressures i n the f u t u r e . Values o f the thermodynamic f u n c t i o n s based on experiments f o r the two b i n a r y systems I^O-NaCl and H2O-CO2 i n the f l u i d onephase r e g i o n a t h i g h temperatures and pressures have not y e t been s u f f i c i e n t l y determined. New work i n t h i s f i e l d i s being done a t present. As one example, F i g . 7 shows p a r t i a l mola volumes of NaCl i n H2O which were c a l c u l a t e d r e c e n t l y (17) from a c r i t i c a l c o m p i l a t i o n o f e x i s t i n g data. F i g . 8 gives excess Gibbs f r e e energy values o f H2O-CO2 mixtures f o r two s u p e r c r i t i c a l temperatures . These are p a r t of the r e s u l t s from c a l c u l a t i o n s now being done at K a r l s r u h e u s i n g both o l d e r and new experimental data and a R e d l i c h - K i s t e r type equation o f s t a t e . 2

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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2000-

200

300

400

500

•T/°C

Figure 5. Phase boundary curves for constant concentrations (isopleths) for the ternary system H 0-C0 -NaCl 2

2

1000 bar T/°C

600

300

co \mm 2

Figure 6. Tentative boundary surface for one-phase fluid conditions at 1000 bar for the ternary system H>0-C0 NaCl 2

H0 2

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

>JNaCI

Equilibria in Aqueous Electrolyte Systems

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Liquid Halite Saturation.

150 200 Temperature /*C Figure 7. Partial molal volume of NaCl in the liquid phase for vapor-saturated NaCl solutions from 0 wt % to halite saturation between 80 and 325 C. (calculated by J. L. Haas, Jr., U.S. Geological Survey, Reston, Va., 1975)

H2O- CO2 1.5-

G /KJmol" E

20

400 °C 500 °C

40 60 -** X Anol /o ,

c02

Figure 8. Excess free energy for supercritical H 0-C0 mixtures 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.

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D i e l e c t r i c Behavior A q u a n t i t a t i v e d i s c u s s i o n and p r e d i c t i o n of phase e q u i l i b r i a and f l u i d p r o p e r t i e s of e l e c t r o l y t e s o l u t i o n s w i l l o f t e n r e q u i r e the knowledge of the s t a t i c d i e l e c t r i c constant o r " p e r m i t t i v i t y " , e, of the s o l v e n t f l u i d . The methods of p r e d i c t i o n of the d i e l e c t r i c constant of p o l a r f l u i d s f o r a wide range of d e n s i t i e s and temperatures i n c l u d i n g the s u p e r c r i t i c a l r e g i o n a r e s t i l l i n an e a r l y stage. I t has been p o s s i b l e , however, to determine t h i s q u a n t i t y experimentally f o r number of f l u i d s of s m a l l , p o l a r molecules a t sub- and s u p e r c r i t i c a l c o n d i t i o n s . C y l i n d r i c a l autoclaves have been used w i t h b u i l t - i n condensers, the c a p a c i t y of which could be determined and v a r i e d w h i l e f i l l e d w i t h the f l u i d samples at high temperatures an the predominant e l e c t r o l y t i of a number of other l e s s p o l a r f l u i d s a r e a l s o important i n i n d u s t r y and may become more so i n the f u t u r e . Examples a r e methanol, a c e t o n i t r i l e , l i q u i d ammonia and c e r t a i n Freons. F i g . 9 gives experimental r e s u l t s f o r e of m e t h y l e h l o r i d e along the i s o bars (18). As expected, i n c r e a s i n g the temperature and decreasing the pressure lowers the d i e l e c t r i c constant. Since the d i p o l e moment of the i s o l a t e d CH3F molecule of 1.85 debye u n i t s i s almost equal to that of an i s o l a t e d water molecule (1.84 debye u n i t s ) a comparison w i t h water i s p a r t i c u l a r l y i n t e r e s t i n g . The higher e-values of dense water have to be r e l a t e d to i t s s t r u c t u r e caused by hydrogen bonds. The temperature and pressure dependence of the d i e l e c t r i c constant of m e t h y l f l u o r i d e appears to be r a t h e r s i m i l a r to those of s e v e r a l of the more p o l a r Freons and of ammonia. I t may be p o s s i b l e to d e s c r i b e and p r e d i c t q u a n t i t a t i v e l y the d i e l e c t r i c p r o p e r t i e s of such p o l a r , non-hydrogen bonded f l u i d s on the b a s i s of the Debye-Onsager-Frohlich-Kirkwood theory by d e v i s i n g s u i t a b l e expressions f o r the "Kirkwood C o r r e l a t i o n F a c t o r . " For comparison i n F i g . 10 an e-T-diagram f o r HC1 i s given (19). Even i n the h i g h d e n s i t y s u p e r c r i t i c a l r e g i o n e i s much lower than i n CH3F. This mainly due to the lower d i p o l e moment of HC1 (1.05). F i g . 11 gives a temperature-density diagram of water w i t h a number of i s o b a r s . Superimposed on these are curves of constant values of the d i e l e c t r i c constant (20). These l a t t e r ecurves a r e based on an e x t e n s i v e c r i t i c a l survey of 47 papers w i t h experimental data which were p u b l i s h e d between 1920 and 1975. As a r e s u l t of t h i s survey an equation f o r e = f ( t , p ) was d e r i v e d of the form: e

=

1 +

a-^T^^p

+ ( a x " ^ + a^ + a ^ x ) p

2

2

-1 _,_ , 2. 3 + (a_x + a,x + a_x ) p O

D

+ /( a x— 2 +. a T -1 + _l

8

g

/

a

.4

1 Q

)p

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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Equilibria in Aqueous Electrolyte Systems

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0 Figure 11.

A N D F L U I D PROPERTIES

OS

g[g/cm3]

IN C H E M I C A L INDUSTRY

1.0

Isodielectric constant curves on a T-g diagram

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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(T = T/100, T i n K and p i n g/cm^) . This equation represents the e x i s t i n g data. I t i s obvious that even i n water very high d i e l e c t r i c cons t a n t s occur only at r e l a t i v e l y low temperatures and h i g h d e n s i t i e s . There e x i s t s , however, a wide range of s u p e r c r i t i c a l c o n d i t i o n s , where the d i e l e c t r i c behavior i s comparable to that of the more p o l a r o r g a n i c s o l v e n t s which to some extent can act as e l e c t r o l y t i c s o l v e n t s . F i n a l l y , F i g . 12 shows a reduced temperatured e n s i t y diagram of HC1, I^O and CH^CN w i t h e x p e r i m e n t a l l y d e t e r mined e-values o f these three compounds, the d i p o l e moments which n e a r l y i n c r e a s e as 1 : 2 : 4 from HC1 to CH3CN. Such reduced d i a gram may e v e n t u a l l y permit f i r s t estimates of the d i e l e c t r i c prop e r t i e s of other p o l a r f l u i d s . I o n i z a t i o n and Complex Some p o l a r f l u i d s e x h i b i t a s m a l l degree of i o n formation even i n the l i q u i d s t a t e at room temperature and a t o r d i n a r y p r e s sure. This s e l f i o n i z a t i o n can be of c o n s i d e r a b l e i n f l u e n c e on the d i s s o c i a t i o n e q u i l i b r i a of d i s s o l v e d e l e c t r o l y t e s . The phenomena of s o l v o l y s i s or h y d r o l y s i s i n aqueous systems depend on the

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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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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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i o n d i s s o c i a t i o n of the s o l v e n t and may thus be o f i n f l u e n c e on c e r t a i n phase e q u i l i b r i a a l s o . This i s p a r t i c u l a r l y t r u e f o r h i g h temperature aqueous s o l u t i o n s . Examples are the "hydrothermal" f l u i d s of the earth's c r u s t , which are important as n a t u r a l t r a n s port media f o r m i n e r a l s . Hot aqueous f l u i d s o f t h i s k i n d can a l s o occur during e x p l o r a t i o n procedures f o r o i l o r f o r geothermal energy. The i o n product of water, K^, t h a t i s the product of the a c t i v i t i e s o r c o n c e n t r a t i o n s o f h y d r o x y l ions and hydrogen iorts i n mol V i s c l o s e t o 10"-^ mol^ £-2 o m temperature and normal pressure. I t can be expected that t h i s q u a n t i t y w i l l i n c r e a s e a t e l e v a t e d temperatures as w e l l as e l e v a t e d p r e s s u r e s . This has been confirmed i n recent years by many d i f f e r e n t experiments, some of them extending to 100 kbar and almost 1000 C F i g 13 shows the r e s u l t s i n a diagra i n the form o f isotherm log has i n c r e a s e d to -11 and a t 500 C and a t the t r i p l e p o i n t d e n s i t y o f 1 g/cm^, l o g i s -8. This means t h a t one hundredth of a percent o f the water i s d i s s o c i a t e d i n t o i o n s a t these c o n d i t i o n s . At such c o n d i t i o n s even a l k a l i s a l t s o f the s t r o n g m i n e r a l a c i d s can be hydrolyzed to an extent comparable to t h a t of acetates i n aqueous s o l u t i o n s a t o r d i n a r y c o n d i t i o n s . I f one c o n s i d e r s the decrease of the d i e l e c t r i c constant o f water a t h i g h e r temperatures, one c o u l d expect a lowering of the i o n i c d i s s o c i a t i o n of d i s s o l v e d e l e c t r o l y t e s . This i s indeed the case shown by e l e c t r o l y t i c conductance measurements. One example i s shown i n F i g . 14 (21). I t gives i s o b a r s of the s p e c i f i c conductance o f 0.01 m o l a l KC1 s o l u t i o n s to 800 C. Up to 300 C the conductance r i s e s a t a l l pressures because o f decreasing water v i s c o s i t y and i n c r e a s i n g i o n m o b i l i t y . Beyond 300 C, however, the d e c l i n i n g i s o b a r s i n d i c a t e a r e d u c t i o n o f charge c a r r i e r conc e n t r a t i o n caused by i o n a s s o c i a t i o n and decreasing d i s s o c i a t i o n constants. F i g . 14 gives r e s u l t s only f o r a very d i l u t e s a l t s o l u t i o n . However, the hot n a t u r a l b r i n e s which are found i n c e r t a i n p l a c e s o f t e n have very h i g h s a l t c o n c e n t r a t i o n s . The conductance and d i s s o c i a t i o n e q u i l i b r i a i n such s o l u t i o n s approach the behavior of fused s a l t s . This i s demonstrated i n F i g . 15. I t shows experimental values o f the molar conductance o f NaCl i n aqueous s o l u t i o n s a t the s u p e r c r i t i c a l temperatures of 400 and 500 C as a f u n c t i o n o f the mole f r a c t i o n o f the s a l t (22). Measurements could not be made beyond x = 0.1. The molar conductance of the fused s a l t has been determined, however. I t i s not very temperature dependent, so that an e x t r a p o l a t i o n o f the conductance to an assumed supercooled l i q u i d a t 400 C o r 500 C seems to be p o s s i b l e . These values are shown i n F i g . 15. I t i s c l e a r t h a t s o l u t i o n s w i t h s a l t mole f r a c t i o n s o f x = 0.2 and h i g h e r have prop e r t i e s c l o s e r t o the pure fused s a l t than to the d i l u t e s o l u t i o n s as f a r as the conductance i s concerned. A l a r g e f r a c t i o n of the ions i s already i n some way a s s o c i a t e d which must i n f l u e n c e the thermodynamic f u n c t i o n s a c c o r d i n g l y . a t

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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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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 o b s e r v a t i o n that s u p e r c r i t i c a l dense water can i o n i z e d i s s o l v e d s a l t s suggests that one attempt e l e c t r o l y t i c decomposit i o n a t such c o n d i t i o n s . I t i s w e l l known, t h a t an i n c r e a s e i n temperature reduces overvoltages and i t has been shown very e a r l y that a t room temperature w i t h water, h i g h pressures a c t i n the same d i r e c t i o n . Both e f f e c t s are shown by a number of simple curr e n t - v o l t a g e curves f o r d i l u t e , aqueous NaOH s o l u t i o n s i n F i g . 16. I t i s i n t e r e s t i n g t o observe, that the curves i n c r e a s e q u i c k l y a t very low v o l t a g e s i f the temperature i s h i g h and that beyond 400 C the decomposition p o t e n t i a l i s no longer c l e a r l y v i s i b l e . A comp l e t e a n a l y s i s of t h i s behavior has not yet been made. I t i s c e r t a i n , however, t h a t i t i s caused by the combined i n f l u e n c e o f h i g h i o n m o b i l i t i e s , h i g h d i f f u s i o n constants o f n e u t r a l molecules, complete m i s c i b i l i t y o f oxygen and hydrogen i n the aqueous s o l u t i o n and reduced a d s o r p t i o t u r e s . As a consequenc p o l a r i z a b l e above 400 C. Very high c u r r e n t d e n s i t i e s have been obtained a t such c o n d i t i o n s . I n order to consider the p o s s i b i l i t i e s o f t e c h n i c a l a p p l i c a t i o n , a knowledge o f phase boundary s u r faces and c r i t i c a l curves f o r water-oxygen and water-hydrogen systems would be d e s i r a b l e . In the f i e l d s o f power p l a n t o p e r a t i o n and h i g h temperature c o r r o s i o n , i n metallurgy and i n geochemistry, a knowledge o f s t a b i l i t y regions and other e q u i l i b r i u m data f o r heavy metal complexes i n h i g h temperature and s u p e r c r i t i c a l aqueous s o l u t i o n s are o f i n t e r e s t . A t present the r e l e v a n t i n f o r m a t i o n i s s t i l l l i m i t e d . Most o f the r e s u l t s come from geochemical i n v e s t i g a t i o n s (23,24). Spectroscopy i n the v i s i b l e and near u l t r a v i o l e t complex e q u i l i b r i a of h a l i d e s of c o b a l t (25), n i c k e l (26), copper (27), z i n c (28), l e a d and other heavy metals (23). F i g . 17 gives one example w i t h a d s o r p t i o n s p e c t r a f o r c o b a l t c h l o r i d e a t 300 C and 500 C. The e x t i n c t i o n c o e f f i c i e n t a t 20 C and normal pressures i s shown which corresponds to the pink c o l o r e d s o l u t i o n s of b i v a l e n t hexaquo c o b a l t complexes. A t 300 C and the r e l a t i v e l y modest pressure o f 350 bar, the a d s o r p t i o n i s c o n s i d e r a b l y i n c r e a s e d and s h i f t e d t o g r e a t e r wavelengths. The s o l u t i o n s are b l u e , t e t r a h e d r a l f o u r - l i g a n d c o b a l t complexes p r e v a i l . This tendency i s even more pronounced a t 500 C. I n e q u i l i b r i u m , such compressed s u p e r c r i t i c a l s o l u t i o n s c o n t a i n the b i v a l e n t metal ions mainly as complexes w i t h the lower c o o r d i n a t i o n number of f o u r . Analogous behavior has been found f o r n i c k e l and copper. S t a b i l i t y cons t a n t s have been determined, which may be u s e f u l i n the d i s c u s s i o n of c o r r o s i o n products i n s u p e r c r i t i c a l steam and of the composit i o n of hydrothermal f l u i d s . Conclusion As a concluding remark i t may be p o i n t e d out t h a t p r o p e r t i e s of the k i n d discussed above may a l s o be i n v e s t i g a t e d w i t h other c l a s s e s of f l u i d s . F l u i d s a l t s and f l u i d metals can be mentioned as examples. F i g . 18 shows the s p e c i f i c conductance o f mercury

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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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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at s u p e r c r i t i c a l c o n d i t i o n s (29). I f the pressure i s r a i s e d to about 200 b a r s , m e t a l l i c behavior i s found even i n the gas phase. S i m i l a r phenomena and thermodynamic p r o p e r t i e s are being s t u d i e d not o n l y f o r mercury but a l s o f o r some a l k a l i m e t a l s .

Abstract A survey is given of a variety of properties of dense polar fluids and fluid mixtures at sub- and supercritical conditions. Aqueous systems in particular are considered. Critical curves and miscibility, dielectric behavior, ionization and complex formation are discussed and illustrated by examples. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

"Properties of Wate editor, Springer, Heidelberg-New York, 1969. Maier, S., Franck, E. U. Ber. Bunsenges. phys. Chem. (1966) 70, 639. Köster, H., Franck, E. U. Ber. Bunsenges. phys. Chem. (1969) 73, 716. Burnham, C. W.; Holloway, J . R.; Davis, N. F. Am. J . S c i . , (1969) 267 A, 70. "Water at High Temperatures and Pressures", K. Tödheide in "Water" Vol. I, p. 463, F. Franks, editor, Plenum Press, N. Y. London 1972. Franck, E. U. Pure and Applied Chemistry (1974) 38, 449. Rice, M. H . , Walsh, J . M. J . Chem. Physics (1957) 26, 824. Sourirajan, S., Kennedy, G. C. Amer. J . Sci. (1962) 260, 115. Kennedy, G. C.; Wasserburg, G. J.; Heard, H. C.; Newton, R. C. Publ. No. 150, Inst. of Geophysics, UCLA, 1960. Schneider, G. M. in "Topics in Current Chemistry" Vol. 13, p. 559, Springer, Heidelberg, New York, 1970. Kruse, R., Franck, E. U. Ber. Bunsenges, physik. Chem. (1976) 80, 1236. Franck, E. U . , Tödheide, K. Z. Physik. Chem. N. F . , (1959) 22, 232. Tödheide, K., Franck, E. U. Z. Physik. Chem. N. F . , (1963) 37, 387. Greenwood, H. J. Amer. J . Sci. (1969) 267, 191. Takenouchi, S., Kennedy, G. C. Amer. J . Sci. (1964) 262, 1055. Gehrig, M. Thesis, 1975, Inst. for Physical Chemistry, University of Karlsruhe. Haas, J . L., Jr. U. S. Geol. Survey, Open File Report 75675, (October 1975). Reuter, K. Thesis 1974, Inst. for Physical Chemistry, University of Karlsruhe.

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19. Harder, W. Thesis 1972, Inst. for Physical Chemistry, University of Karlsruhe. 20. Uematsu, M.; Harder, W.; Franch, E. U. "The Static Dielectric Constant of Water to 550 C and 5 kbar". Report of the "International Association on the Properties of Steam" (IAPS) Sept. 1976. 21. Quist, A. S.; Marshall, W. L.; Franck, E. U.; v. Osten, W. J. Physical Chemistry, (1970)74,2241. 22. Klostermeier, W. Thesis 1973, Inst. for Physical Chemistry, University of Karlsruhe. 23. Helgeson, H. C.; Kirkham, D. H. Amer. J. Science, (1976) 276, 97. 24. Franck, E. U., J. Solution Chemistry, (1973) 2, 339. 25. Lüdemann, H. D., Franck, E U Ber Bunsenges Physik Chem (1967) 71, 455. 26. Lüdemann, H. D., Franck (1968) 72, 514. 27. Scholz, B.; Lüdemann, H. D.; Franck, E. U. Ber. Bunsenges. Physik. Chem. (1972) 76, 406. 28. Schulz, K. Thesis 1974, Inst. for Physical Chemistry, University of Karlsruhe. 29. Hensel, F., Franck, E. U. "Experimental Thermodynamics" Vol. II, ed.: B. Le Neindre, B. Vodar, p. 975, Butterworths, 1975 (London). 30. Hensel, F. Angewandte Chemie, (1974) 86, 459.

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

6 Polymer Equilibria A. BONDI Shell Development Co., Houston, Tex. 77001

The purpose of the present paper i s to provide a g e n e r a l i z e d view of those phase e q u i l i b r i u m c h a r a c t e r i s t i c s of polymers o r polymer dominated systems which a r e unique to the "polymer w o r l d " . This uniqueness may flow from the h i g h molecular weight of t e c h n i c a l l y important polymers o r from the a s s o c i a t e d high v i s c o s i t y o r from the g l a s s y c o n d i t i o n s which c h a r a c t e r i z e the u s e f u l s o l i d s t a t e o f many members of that c l a s s . Another purpose o f t h i s paper i s to assess the t e c h n i c a l and the economic s i g n i f i c a n c e of ignorance i n t h i s area o f r e s e a r c h , because the purpose o f t h i s conference i s served more by e x h i b i t i n g what we do not know than by the proud d i s p l a y o f a seemingly impregnable, coherent body o f v a l i d a t e d t h e o r e t i c a l understanding. The scope of t h i s p r e s e n t a t i o n i s a sweeping survey w i t h a few in-depth excursions i n t o v a l l e y s of s i g n i f i c a n t ignorance. General P r i n c i p l e s The dominant problem of phase e q u i l i b r i a i n v o l v i n g a polymer phase i s the measurement problem: When has (or can) e q u i l i b r i u m be c a l l e d " e s t a b l i s h e d " . The usual means f o r r a p i d e q u i l i b r a t i o n , f o r c e d c o n v e c t i o n , i s not only d i f f i c u l t t o implement, i t may even be i m p o s s i b l e to do, when the r e q u i r e d energy i n p u t would a c t u a l l y break chemical bonds along the polymer molecule's backbone chain. Another problem i s that o f data c o r r e l a t i o n , o r o f data genera l i z a t i o n by means of w e l l founded theory. A t the rough approximat i o n l e v e l we a r e (thanks t o the work by F l o r y , Huggins, P r i g o g i n e , P r a u s n i t z , and o t h e r s ) i n good shape, and h e r e t o f o r e that has been q u i t e adequate. But we s h a l l see that recent s t u d i e s o f concent r a t e d systems e x h i b i t v a p o r / l i q u i d e q u i l i b r i a which are not even described q u a l i t a t i v e l y by e x i s t i n g theory, assuming that the measurements are r e l i a b l e . In t h i s d i s c u s s i o n we s h a l l take f o r granted that the reader i s f a m i l i a r w i t h the body of theory, according to which polymers 118

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are c h a r a c t e r i z e d by t h e i r molecular weight, t h e i r molecular weight d i s t r i b u t i o n , t h e i r cohesive energy d e n s i t y (or other measure of p a i r p o t e n t i a l between unbonded neighbors) and the f l e x i b i l i t y of c h a i n s , measured e m p i r i c a l l y or c h a r a c t e r i z e d by barr i e r s to i n t e r n a l r o t a t i o n and energies of r o t a t i o n a l i s o m e r i z a t i o n of main c h a i n bonds or bond groupings, the c r o s s l i n k d e n s i t y , i f any, and e l e c t r o s t a t i c charge d e n s i t y , e s p e c i a l l y i n the case of p o l y e l e c t r o l y t e s . The l a t t e r w i l l not be d e a l t w i t h i n t h i s survey. S i n g l e Component, Two Phase Systems S o l i d / S o l i d E q u i l i b r i a i n C r y s t a l l i n e Polymers Polymorphism i s f a crystallin than among c r y s t a l s compose t h i s p a u c i t y i s , of course, the c o n s t r a i n t imposed by t h e i r one dimensional i n f i n i t y upon r o t a t i o n a l d i s o r d e r , the primary cause of polymorphism among organic s o l i d s . The few known cases have been assembled on Table 1. The s i t u a t i o n i s q u i t e d i f f e r e n t , i f we admit a s s o c i a t i o n polymers as l e g i t i m a t e polymers. An outstanding example i s the c l a s s of the a l k a l i and e a r t h a l k a l i metals soaps of long chain f a t t y a c i d s . T h e i r polymorphic behavior i s w e l l e s t a b l i s h e d ( 1 ) , and i s a s c r i b e d to the i n c r e a s i n g m o b i l i t y of the alkane groups w i t h i n c r e a s i n g temperature. T y p i c a l examples are shown on Table 2. Polymeric c r y s t a l l i n e long c h a i n e s t e r s of v i n y l compounds are, of course, the C-C backbone chain e q u i v a l e n t of those association-polymers . S o l i d / L i q u i d E q u i l i b r i a . Few polymer e q u i l i b r i a have been s t u d i e d as thoroughly as those a t the m e l t i n g p o i n t of c r y s t a l l i n e polymers. In l a r g e measure t h i s i n t e r e s t may be caused by the d e s i r e to understand the p e c u l i a r c h a i n f o l d i n g p r o p e n s i t y of many c r y s t a l l i n e polymers. The c u r r e n t l y accepted theory (2) a s s o c i a t e s chain f o l d i n g w i t h an entropy phenomenon, such that chain f o l d i n g should disappear w i t h i n c r e a s i n g temperature; and, indeed, i f one r a i s e s the m e l t i n g temperature high enough by conducting the s o l i d i f i c a t i o n from the melt at high enough h y d r o s t a t i c pressure, chain f o l d i n g can be avoided, and completely s t r a i g h t chain c r y s t a l s are formed without chain f o l d s ( 3 ) . Thus the l i q u i d / s o l i d , pressure/ temperature phase diagram of polymeric c r y s t a l l i n e s o l i d s , i l l u s t r a t e d on F i g u r e 1, i s not q u i t e comparable w i t h those of s i m p l e r compounds, because the two ends of each curve represent m a t e r i a l s i n two q u i t e d i f f e r e n t c r y s t a l morphologies. As t h i s change i s gradual, and r e l a t e d to the molecular weight d i s t r i b u t i o n , the T vs. pressure curves on F i g u r e 1 are without d i s c o n t i n u i t y . A unique aspect of h i g h polymer s o l i d / l i q u i d e q u i l i b r i a i s the o r i e n t a t i o n - i n d u c e d c r y s t a l l i z a t i o n a t temperatures above T (4). The b e s t known i n s t a n c e s of c r y s t a l l i z a t i o n of T > T under m

m

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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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EQUILIBRIA A N D FLUID PROPERTIES IN C H E M I C A L INDUSTRY

TABLE 1 EXAMPLES OF POLYMORPHISM AMONG CRYSTALLINE POLYMERS

3

Crystal Morphology

poly-l-butene

rho ter ortho

o t , c m 135/6 122/4 106

poly-l-pentene

mono port

130 80

2 2

125 75

phex mono phex hex

100 96 141 148

Polymer

poly-4-methylpentene-

poly 1,3 butadiene, trans I II

mono ortho

23 14

tri mono

67/73 62/77

v i n y l c y c l o pentene

tri tet

292 270

poly-p-xylylene, a

mono ortho

375 420

poly a r c y l o n i t r i l e , synd.

hex ortho

317 341

poly v i n y l c h l o r i d e

ortho mono

273 310

poly v i n y l f l u o r i d e

hex ortho

200 230

p o l y 8-amino c a p r y l i c a c i d , a

mono phex

185 220

poly c i s isoprene poly-cyclo-octene,

trans

'From Polymer Handbook, Brandrup, J . and Immergut, E. H., John Wiley, New York, 1975.

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

6. BONDI

Polymer Equilibria

121

TABLE 2 POLYMORPHISM OF ASSOCIATION POLYMERS EXAMPLE:

THE ALKALI SALTS OF LONG CHAIN FATTY ACIDS*

S o l i d / S o l i d T r a n s i t i o n Temperatures and F i n a l M e l t i n g P o i n t s , °C Carbon Number o f F a t t y A c i d : Cation

C^

C

7

C

8

C

9

c

io

K m.p.

Rb m.p.

C

s

m.p.

a)

12

237

L i m.p.

Na m.p.

C

C

14

233

C

16

C

18

223

226

197

191

361

363

360

355

348

329

311

302

283

235

242

243

243

245

246

246

251

255

-

-

-

-

218

218

215

212

208

210

198

189

185

181

179

171

168

165

140

141

138

136

135

113

117

116

>400

>400

>400

>400

>400

395

375

362

348

305

301

291

282

277

273

271

269

267

-

195

170

^400

380

375

358

300

291

284

281

385

370

358

345

295

290

279

273

Demus, D.; and Sackmann, H. From Baum, E.; 37. H a l l e (1970), 19, (5)

Wiss . Z. Univ.

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

122

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES I N C H E M I C A L INDUSTRY

250

200 o o Journal of Macromolecular Science

Figure 1. Phase diagrams of crystalline polymers (6) Figure 1A. Phase diagrams of polyethylene. Melting (solid points) and crystallization (open points) temperatures of: (• O ecc, (| Q) unknown structure

150 -

0

2

4

6

(A)

Figure IB. Pressure dependence of the melting temperature, T , of polyethylene-chain crystals ( ) and foldedchain crystals ( ) m

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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123

Polymer Equilibria

400 T

C

(D) Figure ID. Pressure dependence for samples (a) crystallized at atmospheric pressure, and the measurement pressure

(b) crystallized at

c o n d i t i o n s of t e n s i l e e l o n g a t i o n are those of n a t u r a l rubber and, o c c a s i o n a l l y , of melt s p i n n i n g . S i m i l a r l y , o r i e n t a t i o n - i n d u c e d c r y s t a l l i z a t i o n has been observed i n melt flow, such as i n i n j e c t i o n molding, a t high shear r a t e s . A very high degree of s t e r e o r e g u l a r i t y or " t a c t i c i t y " i s necessary, so that l o n g , u n i n t e r rupted p a r a l l e l a l i g n e d s t r e t c h e s of molecule chains can form s t a b l e c r y s t a l c r y s t a l n u c l e i that w i l l grow r a p i d l y i n t o macroscopic c r y s t a l s . A t h e o r e t i c a l treatment of s t r a i n induced c r y s t a l l i z a t i o n w i t h q u a n t i t a t i v e p r e d i c t i v e power i s s t i l l i n i t s infancy. The r e l a t i o n of T or b e t t e r of AS and AH (and AV ) to molecular s t r u c t u r e of c r y s t a l l i n e polymers has been thoroughly discussed elsewhere (5) and i t s d i s p l a y would go beyond the scope of t h i s review. S u f f i c e i f to say here, that T a t atmospheric pressure i s measured so e a s i l y that i t s e s t i m a t i o n by p r e d i c t i v e methods seems an uneconomical t h i n g to do. The e s t i m a t i o n of the slope of the T vs. P l i n e , (6) the s o - c a l l e d m e l t i n g curve of c r y s t a l l i n e polymers seems no more d i f f i c u l t than of other substances. However, given the high v i s c o s i t y of the polymer melt, e s p e c i a l l y a t high pressures, n u c l e a t i o n may be so r e t a r d e d , to make experimental m e l t i n g p o i n t determination q u i t e u n c e r t a i n . m

m

m

m

m

m

The Glass/Rubbery State " E q u i l i b r i u m " Glassy polymers are of such t e c h n i c a l (and economic) importance that i t would seem pedantic to ignore the very obvious p h y s i c a l change at the g l a s s t r a n s i t i o n temperature. Morphologically, both phases are l i q u i d l i k e , i . e . h i g h l y disordered on an atomic s c a l e . Furthermore the dramatic change i n v i s c o s i t y at the g l a s s t r a n s i t i o n temperature (Tg) i s not much more gradual than i t i s a 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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PHASE

EQUILIBRIA

A N D FLUID

PROPERTIES

IN CHEMICAL

INDUSTRY

T . However, i t s strong dependence upon the c o o l i n g r a t e (Figure 2) c l e a r l y d i f f e r e n t i a t e s T from T . The curve of Tg v s . P (Figure 3) has q u a l i t a t i v e l y the same appearance as the m e l t i n g curve, but the slope i s w e l l a p p r o x i mated, but not p r e c i s e l y d e s c r i b e d as that of a second order t r a n s i t i o n depending on the d i f f e r e n c e s i n the slopes of V v s . T and V vs. P between rubbery and g l a s s y s t a t e . Hence F i g u r e 3 does not q u a l i f y as a genuine phase e q u i l i b r i u m , even i f i t looks and a c t s l i k e one. m

g

m

The Rubbery S t a t e / L i q u i d E q u i l i b r i u m There i s now i n c r e a s i n g evidence f o r the r e a l i t y of the b l i p s i n the d i f f e r e n t i a l thermal a n a l y s i s a t T > Tg, a t a temperature i d e n t i f i e d as T ^ i o r l i q u i d / l i q u i p h y s i c s o f t h i s phenomeno l o g i c a l l y as w e l l as i n terms of molecular motions. According t o some conjectures ( 8 ) T ; Q i s the temperature (range) c h a r a c t e r i z i n g the change from entanglement of neighboring chains to a s t a t e o f f r e e r i n t e r m o l e c u l a r movement. The p a r a l l e l change i n T ^ and i n v i s c o s i t y w i t h moelcular weight (shown on F i g u r e 4 ) i s the source of t h i s c o n j e c t u r e . E x i s t e n c e of a r e l a t i o n between the v i s c o s i t y

V

-10

10

0 T - Tg °

20

°C

Figure 2. Effect of cooling rate f on the observed glass-transition temperature and specific volume according to the theory of Saito et al. The rate 10 C/sec was chosen for the standard for which T = T ° and V = 1.000; dV /dT was assumed independent of cooling rate. (A. Bondi, "Physical Properties of Molecular Crystals, Liquids and Glasses" Wiley, New York, N.Y., 1968) s

g

g

g

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

g

6.

BONDI

125

Polymer Equilibria

T (°C) g

260r

lOOl

1

"

1

1

1

0

2

4

6

8

10

P (Kb) g

Figure 3. Vitrification Phase diagram of polystyrene. Note that vitrification takes place over a pressure range. 4

I

0

Ht

I

I

40 Nt

I

I

l

80 Ar T

b

i

t

120 Kr

!

I

I

1

160 0 40 Xe H t Nc

,in°K

i — I I

80 Ar

I

1

120 Kr

I

160 Xc

T ,in°K b

Figure 4. Logarithm of the solubility coefficients of rare gases in various homopolymers vs. the boiling point, T , of the gas. [Lundstrom, J. E., Bearman, R. J., J. Polym. Sci., Polym. Phys. (1974) 12, 97] PVA = polyvinyl acetate), SR = silicone rubber, SRDC = silicone rubber, SRGE = 5 % phenyl silicone rubber, TREGEM = poly(tetraethyleneglycol dimethacrylate), PE = polyethylene, NRUA = natural rubber. b

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 H A S E EQUILIBRIA AND

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INDUSTRY

of polymer melts and c h a i n entanglement i s f a i r l y w i d e l y accepted

Two Component, Two-Phase System Melt/Gas The p r e d i c t a b i l i t y of the s o l u b i l i t y of gases and vapors i n polymer m e l t i n g i s s t i l l j u s t as e l u s i v e as i t i s f o r simple l i q u i d s . Some hoary g e n e r a l i z a t i o n s about the r e l a t i v e e f f e c t s of the m o l e c u l a r f o r c e constants (e and r ) of gases or of t h e i r c r i t i c a l c o n s t a n t s , and the s o l u b i l i t y parameter of the polymer melt can be used f o r i n t e r p o l a t i o n purposes i n a narrow group of systems. But a good theory, or even a r e l i a b l e engineering c o r r e l a t i o n f o r a wide range of such systems have y e t to be developed. The s o l u b i l i t y of polymer melts i n h i g h l y compressed gases (at T > T and P » P ) t i o n medium f o r polymer c o n d i t i o n s pressure i s a more convenient v a r i a b l e than s o l v e n t composition, and a l e s s d e s t r u c t i v e v a r i a b l e than temperature. However so f a r only e x p l o r a t o r y experiments have been p u b l i s h e d i n t h i s f i e l d (10). Q

c

c

Polymer Melt/Monomer L i q u i d ( s ) The s o l u b i l i t y of polymer melts i n s i n g l e component and i n b i n a r y monomeric l i q u i d mixtures i s so w i d e l y known and f r e q u e n t l y reviewed both w i t h r e s p e c t to p r a c t i c a l a p p l i c a t i o n s , e s p e c i a l l y f o r polymer f r a c t i o n a t i o n , and w i t h r e s p e c t to theory that l i t t l e need be s a i d here. The p r o p e r t i e s of monomer l i q u i d s and of p o l y mer melt which determine t h e i r mutual m i s c i b i l i t y have been assemb l e d on Table 3 and are seen to be q u i t e s i m i l a r to those which determine the i n t e r a c t i o n between gases and polymer melts on F i g ure 4. Four cases are of s p e c i a l i n t e r e s t , the p r e c i p i t a t i o n of a d i s s o l v e d polymer melt by a d d i t i o n of a non-solvent, the separat i o n of c h e m i c a l l y d i f f e r e n t polymers by mixtures of s e v e r a l s o l vents (below t h e i r CST), (11) the d i s s o l u t i o n of a polymer melt by mixture of two non-solvents, and the s w e l l i n g of a c r o s s l i n k e d polymer melt by monomeric s o l v e n t s . The i n c i d e n c e of the f i r s t three cases i s described i n F i g u r e s 5 and 6, r e s p e c t i v e l y , f o r nons p e c i f i c i n t e r a c t i o n s between polymer and s o l v e n t ( s ) . Specific i n t e r a c t i o n s a r e q u a l i t a t i v e l y j u s t as p r e d i c t a b l e as i n the case of s p e c i f i c i n t e r a c t i o n s between monomeric l i q u i d s , say m o l e c u l a r compound formation or acid/base i n t e r a c t i o n . But q u a n t i t a t i v e pred i c t i o n s are even more d i f f i c u l t here than i n the monomer case. Although polymers are commonly mixtures w i t h a molecular weight d i s t r i b u t i o n of f i n i t e w i d t h we t r e a t them as s i n g l e components, or b e t t e r as q u a s i - s i n g l e components. I f we do so, a l l the consequences of the phase r u l e are found to be v a l i d . Moreover,

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.

± .03

Impossible

Too few d a t a so f a r on g l a s / g l a s s phase b o u n d a r i e s

P r e d i c t i o n from Experimental Data

P r e d i c t i o n from E s t i m a t e d Data

F of F u n c t i o n a l Group C o m b i n a t i o n

2

6^;6

Estimated

Data

N o t h i n g , most prop e r t i e s cannot be measured i n t h e glassy state.

2

A v a i l a b l e from Experiment

x

0

g

2

= f(T)±.2

6

S i n g l e phase properties only very approximately.

Phase B o u n d a r i e s No

2

.03

= f(T)

1' log n

n

g

T (1);T (2)

T ( l ) , T (2),

6 (1),6 ( 2 ) , M 6 ;6 D/A c h a r a c t e r i s t i c s

Data

Required

Time and P r o x i m i t y of Tg

Time and P r o x i m i t y o f Tg

Constraints

Glass/Glass Blends

The p r o b l e m i s too new f o r m e a n i n g f u l f o r m u l a t i o n of i t s specific character.

S i z e of I n d i v i d u a l Blocks; P r o x i m i t y to Block: Tg

Phase Diagrams and P r o p e r t i e s o f M i x t u r e s o f Gas, Glass P l a s t i c i z e r Solvents with Block Systems Copolymers

For d i l u t e s o l u t i o n s o f non-polar p o l y m e r s - f a i r . For h i g h l y p o l a r i n t e r a c t i o n s - i r r e v e r s i b l e . For concentrated s o l u t i o n a b s o r p t i o n i s o f t e n domin a t e d by dynamic phenomena. P r e d i c t i o n s not achievable now.

Polymer C o n c e n t r a t i o n Geometry o f A d s o r p t i o n S u r f a c e ; Time

Adsorption Equilibria

TABLE 3. PREDICTABILITY OF TWO VERSUS MULTICOMPONENT - TWO PHASE SYSTEMS

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P H A S E EQUILIBRIA

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

6. BONDI

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129

the i n c i d e n c e and shape o f m i s c i b i l i t y gaps i s p r e d i c t e d q u a l i t a t i v e l y c o r r e c t l y by molecular theory. But no e x i s t i n g theory can p r e d i c t q u a n t i t a t i v e l y the i n t e r e s t i n g m i s c i b i l i t y gap geometries shown on Figures 6 through 8 (12). The s w e l l i n g o f c r o s s l i n k e d polymer melts i s j u s t a l i m i t e d d i s s o l u t i o n . The only d i f f e r e n c e i s t h a t the e f f e c t i v e molecularweight f o r i n t e r a c t i o n w i t h the s o l v e n t i s (M ) t h a t of the chains between c r o s s l i n k s . Hence m i n i m i z a t i o n of s w e l l i n g i s achieved by the choice of polymer and s o l v e n t chemistry that minimizes mutual compatibility. There i s i n c r e a s i n g evidence that the a c t i v i t y of the s o l v e n t such as p l a s t i c i s e r s i n h i g h l y concentrated systems i s not adequately represented by e x i s t i n g f o r m u l a t i o n s , such as the F l o r y Huggins r e l a t i o n s and t h e i In view o f the i n c r e a s i n g l aggregation of polymer molecules i n t o "superaggregates" i n concent r a t e d s o l u t i o n s (14,15) these d e v i a t i o n s a r e not s u r p r i s i n g . "Concentrated' means concentrations i n excess of F a r more work needs t o be done on the systematics of the aggregation cons t a n t s as f u n c t i o n of c o n c e n t r a t i o n , solvent-polymers i n t e r a c t i o n , and temperature before t h e o r i e s o f the e q u i l i b r i a from concentrated polymer s o l u t i o n s can even be formulated. c

1

M e l t / M e l t I n t e r a c t i o n . The s i m p l e s t f o r m u l a t i o n of polymer s o l u b i l i t y i n terms of r e g u l a r s o l u t i o n theory on Table 3 o r F i g ure 4 makes i t very c l e a r that h i g h molecular weight (or volume) of the s o l v e n t must be compensated f o r by s m a l l d i f f e r e n c e s i n t h e i r s o l u b i l i t y parameter, i f mutual c o m p a t i b i l i t y i s to be maintained. When both components of the mixture are h i g h polymers, compatib i l i t y i s commonly achieved only when t h e i r s o l u b i l i t y parameters d i f f e r by l e s s than .05 (cal/cm^)!' 2 i n other words, most polymer melts a r e incompatible w i t h each other. Even d i l u t e s o l u t i o n s of two d i f f e r e n t polymers i n a s i n g l e s o l v e n t w i l l separate i n t o d i l u t e phases. The r a r e exceptions of compatible yet d i f f e r e n t polymers a r e shown on Table 4. Hence most s o - c a l l e d "polyblends" are h i g h l y d i s p e r s e d two phase systems which are prevented from s e p a r a t i n g i n t o l a r g e s c a l e phases by p o t e n t i a l energy b a r r i e r s impeding flow (16) o r by the s k i l l f u l i n t r o d u c t i o n of c r o s s l i n k s (19). #

Glass/Gas (or Vapor). Polymers i n the g l a s s y s t a t e are p l a s t i c i z e d by most monomeric substances w i t h which they a r e compatible. Hence we are concerned here w i t h those s o l u t i o n s o f substances i n the g l a s s f o r which Tg > T ( o b s e r v a t i o n ) . Measurements of the part i a l pressure of the v o l a t i l e s o l u t e over such g l a s s y s o l u t i o n s a r e g e n e r a l l y b e t t e r represented by a combination of a Langmuir i s o therm w i t h Henry's law. The common e x p l a n a t i o n i s that the b u l k o f the v o l a t i l e s o l u t e i s adsorbed on the surfaces of minute voids i n the g l a s s , r a t h e r than being d i s s o l v e d i n the g l a s s (18,19). These

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

130

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES I N C H E M I C A L INDUSTRY

513 37000

503 O

o

2700000

37000 0.1

0. POLYSTYRENE-OCETANE

POLYSTYRENE -CYCL0HEXANE British Polymer Journal

Figure 6. Experimental examples of the various types of miscibility gaps in polymer solutions (12)

British Polymer Journal

Figure 7. Experimental example of the miscibility gap for polyvinyl alcohol-water, a hydrogen bonding system (12)

Figure 8. Lower critical solution temperature observed in glassy state by mechanical relaxation spectroscopy [Akiyama, S., et al, Kob. Roab. (1976) 5, 337.]

0

0.5

WT. FRACTION

PVN

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

6.

BONDI

Polymer Equilibria

131 TABLE 4

COMPATIBLE POLYMER PAIRS*

Polymer 1 Polystyrene

Polymer 2 poly-2-methyIs tyrene

ti

p o l y - m e t h y l v i n y l ether

ii

poly-2,5 d i a l k y l - p h e n y l e n e oxide

it

benzyl-cellulos

poly caprolactam

polyviny v a r i o u s poly ethers

II

nitrocellulose

polyvinylidene fluoride II

polymethyl methacrylate^^ "

ethyl

b u t a d i e n e / a c r y l o n i t r i l e copolymer

polyvinyl chloride II

e t h y l e n e / v i n y l a c e t a t e copolymer

Polymethyl methacrylate ( i s o )

same s y n d i o t a c t i c nitrocellulose

From data by S. Davidson and by D. R. P a u l . ^ C o n f i r m e d by B r i l l o u i n - S c a t t e r i n g , P a t t e r s o n , G. D. e t a l . , Macromol. (1976), 9 603. 9

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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v o i d s are s a i d to be the r e s u l t of the i n a b i l i t y of the g l a s s to acquire i t s e q u i l i b r i u m d e n s i t y i n the face of v i s c o u s r e s i s t a n c e to b u l k flow. The p r a c t i c a l importance of t h i s o b s e r v a t i o n and of the i n c r e a s i n g evidence f o r the o r i e n t a t i o n dependence of gas or vapor s o l u b i l i t y i n g l a s s y systems d e r i v e s from the i n c r e a s i n g l y widespread use of g l a s s y polymers as gas and vapor b a r r i e r f i l m s . Their p e r m e a b i l i t y f o r the gas i s , of course, the product of gas s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t . Given t h i s p r a c t i c a l importance, i t i s o b v i o u s l y awkward that the e q u i l i b r i u m constants of the dual mode s o r p t i o n equations cannot be c o r r e l a t e d w i t h the p r o p e r t i e s of the polymer and of the permeant. Such c o r r e l a t i o n s are reasonably s u c c e s s f u l i n the rubbery s t a t e . Owing to .th of the g l a s s on the i n c i d e n c v o i d s , a r e l i a b l e g e n e r a l i z e d c o r r e l a t i o n f o r p r i o r e s t i m a t i o n s of the three constants f o r gas s o r p t i o n i n g l a s s y polymers may never be p o s s i b l e . Yet more complicated i s the s o r p t i o n e q u i l i b r i u m of more s o l u b l e substances i n g l a s s y polymers. Since they p l a s t i c i z e the g l a s s f a r more than the l e s s s o l u b l e gases do, s o r p t i o n causes d r a s t i c changes i n the polymer, i n c l u d i n g the b u i l d i n g up of s t r e s s e s along the s o r p t i o n f r o n t . S o r p t i o n e q u i l i b r a t i o n under such circumstances i s w i t h a glass only at very low s o l v e n t a c t i v i t i e s , w h i l e at higher s o l v e n t a c t i v i t i e s i t would be w i t h a rubbery substance. The " v i t r i f i c a t i o n c o n c e n t r a t i o n " of the s o l v e n t which separates these two regimes i s , of course, w e l l known from p l a s t i c i z a t i o n experiments, but has r a r e l y been systematized f o r other s o l v e n t s . C r y s t a l / S o l v e n t . Given the d i f f i c u l t y of f i n d i n g s u i t a b l e s o l v e n t s f o r the t e c h n i c a l l y important high m e l t i n g c r y s t a l l i n e polymers, i t i s perhaps not s u r p r i s i n g that only l i t t l e work has been done on the discovery of e u t e c t i c s between c r y s t a l l i n e p o l y mers and s o l v e n t s of n e a r l y s i m i l a r m e l t i n g p o i n t s . These s o l v e n t s then must be completely m i s c i b l e w i t h the polymer i n t h e i r respect i v e melt s t a t e s , and completely i m m i s c i b l e as s o l i d s . Such systems have acquired p r a c t i c a l s i g n i f i c a n c e because of the unique f i b r i l l a r morphology of c r y s t a l l i n e polymer which p r e c i p i t a t e s at or near the e u t e c t i c p o i n t . In the c u r r e n t context i t i s important to note that the e x p e r i m e n t a l l y observed phase diagram, F i g u r e 9, d i f f e r s s u b s t a n t i a l l y from that p r e d i c t e d by the Flory-Huggins theory. At present i t i s not c l e a r whether that d i f f e r e n c e i s r e a l or whether i t i s due to retarded c r y s t a l l i z a t i o n of the p o l y mer (12,20). G l a s s / P l a s t i c i z e r . B a s i c a l l y t h i s system i s j u s t a v a r i a n t of the l i q u i d / l i q u i d system. However, the e f f e c t of phase separat i o n on the g l a s s t r a n s i t i o n temperature i s not only time dependent, as expected, but can a l s o be q u i t e unique. This i s best

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 10. Variation of the DSC glass-transition temperature of bisphenol-A polycarbonate with concentration of plasticizer. Curve A: Pentaerythriol tetra nonanoate; Curve B: Trimellitic acid tridecyloctyl ester T (1) and T (2) of two polymer solutions in equilibrium with each other after storage for one day at room temperature; Curve C: Tritolyl phosphate. g

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i l l u s t r a t e d by the recent thorough s t u d i e s of polycarbonate (21). T y p i c a l l y , i n the case of p a r t i a l m i s c i b i l i t y , there i s j u s t a s e p a r a t i o n i n t o pure p l a s t i c i z e r and a polymer phase w i t h f i x e d p l a s t i c i z e r c o n c e n t r a t i o n and c o n s t a n t , c o r r e s p o n d i n g l y depressed Tg, r e g a r d l e s s of the i n i t i a l mixture composition. In one case, w i t h t r i ( d e c y l o c t y l ) t r i m e l l i t a t e as p l a s t i c i z e r , the polymerp l a s t i c i z e r phase separates a f t e r some time i n t o two polymer phases w i t h w i d e l y d i f f e r i n g p l a s t i c i z e r content and correspond i n g l y two constant T g s across the e n t i r e phase diagram, as shown on F i g u r e 10. That f i g u r e a l s o shows the curve f o r the normal d e p r e s s i o n of Tg w i t h i n c r e a s i n g p l a s t i c i z e r c o n c e n t r a t i o n . None of the otherwise s u c c e s s f u l t h e o r i e s of polymer s o l u b i l i t y behavior p r e d i c t t h i s c u r i o u s phenomenon f

Multi-Component/Multi-Phas Block Copolymers. The c h e m i c a l l y s t a b i l i z e d permanent two phase systems which a r e o b t a i n e d by b l o c k - c o p o l y m e r i z a t i o n of i n c o m p a t i b l e chains e x h i b i t the r a t h e r unexpected gas s o l u b i l i t y c h a r a c t e r i s t i c s shown on F i g u r e 11. T e n t a t i v e l y t h i s p e c u l i a r behavior has been a s c r i b e d to the s p e c i a l p r o p e r t i e s of the " i n t e r f a c i a l " r e g i o n between the two polymer phases (19). I f the r e a l i t y of the s p e c i a l p r o p e r t i e s of such an i n t e r f a c i a l r e g i o n i s v a l i dated by independent experiments, we can speak of a component phase. Given the e a r l y development stage of t h i s f i e l d , a d e f i n i t i v e theory must be based on f a r more experimental work than has been produced so f a r . Co-Extruded F i l m s . The l i m i t e d c o m p a t i b i l i t y of polymers makes i t p o s s i b l e to coextrude an a r b i t r a r y number of d i f f e r e n t polymer f i l m s or f i b e r s , and m a i n t a i n t h e i r separate i d e n t i t y i n d e f i n i t e l y , even though they are i n contact on a molecular s c a l e . Even l a r g e d i f f e r e n c e s i n d e n s i t y are not l i k e l y to l e a d to observable Taylor i n s t a b i l i t i e s over p r a c t i c a l l y s i g n i f i c a n t time i n t e r v a l s a t temperatures at which the f i l m s are e f f e c t i v e l y s o l i d . V i s c o s i t y r e d u c t i o n through composition changes or e l e v a t i o n of the temperature would, of course, a l l o w the development of such buoyancy d r i v e n i n s t a b i l i t i e s . One economic d r i v i n g f o r c e f o r making such m u l t i l a y e r f i l m s i s the need f o r f i l m s of low p e r m e a b i l i t y f o r s e v e r a l d i f f e r e n t gases, which can r a r e l y be achieved by a s i n g l e polymer. The s o l u b i l i t y of d i f f e r e n t gases i n the d i f f e r e n t f i l m s i s independent of each other. Adsorption E q u i l i b r i a The a d s o r p t i o n of polymers out of t h e i r s o l u t i o n s a t l i q u i d / s o l i d , l i q u i d / l i q u i d , and l i q u i d / g a s i n t e r f a c e s i s a t present a very f r u s t r a t i n g s u b j e c t f o r broad g e n e r a l i z a t i o n s . This f r u s t r a t i o n i s caused by the appearance of ever more independent v a r i a b l e s

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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OF NITROGEN THROUGH K RATON 1101

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Figure 11. Solubility (S)ofN and 0 in Kraton Block Copolymer, determined in unsteady state experiments; note the minimum near 30 C (22) 2

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which complicate the s i t u a t i o n . A d s o r p t i o n of non-polar or s l i g h t l y p o l a r polymers out of d i l u t e athermal or near athermal s o l u t i o n on p l a n a r s u r f a c e s can be d e s c r i b e d reasonably q u a n t i t a t i v e l y by the e s t a b l i s h e d techniques of s t a t i s t i c a l mechanics and thermodynamics of polymer s o l u t i o n s (23). The outcome i s t h a t the polymer c o i l attaches to the s u r f a c e at comparatively few s p o t s . Consequently the t h i c k n e s s of the adsorbed l a y e r i s of the same order as the diameter of the random c o i l i n the d i l u t e s o l u t i o n from which i t has been adsorbed. Conversely, very p o l a r polymers, s t r o n g l y adsorbed out of thermal s o l u t i o n s are found to be s t r o n g l y attached i n most unc o i l e d c o n f i g u r a t i o n , i . e . , n e a r l y f l a t on the p l a n a r s u r f a c e , e s p e c i a l l y i f t h e i r molecular weight i s not very high (24). In such cases a s u b s t a n t i a l f r a c t i o n of the adsorbed polymer can be so s t r o n g l y adsorbed to b h e l i c a l configuration i c a l h e l i c e s l y i n g f l a t i n the s o l i d / l i q u i d i n t e r f a c e (25). A d s o r p t i o n of polymers approaches e q u i l i b r i u m very s l o w l y because i n g e n e r a l the lowest molecular weight components are adsorbed f i r s t , even i f l e a s t s t r o n g l y . They are then s l o w l y d i s p l a c e d by the h i g h e r molecular weight components which dominate the a d s o r p t i o n e q u i l i b r i u m , e s p e c i a l l y i n athermal s o l u t i o n s . I f the adsorbent i s porous, the h i g h e r molecular weight components may be excluded on g e o m e t r i c a l grounds, and the steady s t a t e does not represent the thermodynamic e q u i l i b r i u m of the corresponding planar surface. A d s o r p t i o n e q u i l i b r i a f o r polymers out of concentrated s o l u t i o n s as f u n c t i o n of c o n c e n t r a t i o n f r e q u e n t l y e x h i b i t very pronounced maxima ( F i g . 12). These unusual curves can be accounted f o r i f one assumes that the adsorbed s p e c i e s are i n aggregation e q u i l i brium i n the s o l u t i o n , depending upon the amount of s u r f a c e area per u n i t volume of s o l u t i o n . Hence one expects t h a t the a d s o r p t i o n e q u i l i b r i u m out of concentrated polymer s o l u t i o n may not only be approached w i t h " i n f i n i t e " slowness but i s a l s o a f u n c t i o n of the system c h a r a c t e r i s t i c s , and the d e f i n i t i o n of r e p r o d u c i b l e c o n d i t i o n s c o n t a i n s many more v a r i a b l e s than one i s used to from the more common work w i t h d i l u t e s o l u t i o n . This complexity i s p a r t i c u l a r l y awkward when one deals w i t h the important case of competit i v e a d s o r p t i o n of polymers out of concentrated multicomponent s o l u t i o n s , a common phenomenon i n many i n d u s t r i a l processes, such as p a i n t adhesion, c o r r o s i o n p r e v e n t i o n , l u b r i c a t i o n , e s p e c i a l l y wear prevention, etc. Conclusions This quick survey of n o n - e l e c t r o l y t e polymer e q u i l i b r i a demons t r a t e s that phenomenologically, of course, systems c o n t a i n i n g polymers do not d i f f e r from monomeric systems. An important comp l i c a t i o n i s the everpresent p o l y m o l e c u l a r i t y of i n d i v i d u 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.

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C ,g/IO 0

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Figure 12. Adsorption equilibria for polycarbonate (PC) and polystyrene (PS) out of concentrated solution in CC/ at 25° C onto different amounts of Aerosil Silica: (1) (2) (3) (4) (5) g/l. 5 10 20 30 40 A = g (Adsorbate)/g (Adsorbant) (15) 4

polymer s p e c i e s , which s y s t e m a t i c a l l y d i s t o r t s a l l phase diagrams of polymeric systems. But exact r e l a t i o n s between the shape of phase diagrams and the width of molecular weight d i s t r i b u t i o n s have y e t to be developed. S i m i l a r l y , molecular theory s u c c e s s f u l l y d e s c r i b e s one aspect of polymeric systems, namely t h e i r d i l u t e s o l u t i o n behavior, but on o c c a s i o n does not even p r e d i c t q u a l i t a t i v e l y the phase e q u i l i b r i a of concentrated polymer systems, and r a r e l y does so q u a n t i t a t i v e l y . The high v i s c o s i t y of such systems o f t e n prevents or c e r t a i n l y slows the approach to e q u i l i b r i u m , so t h a t the measurement problem can be s u b s t a n t i a l . The g l a s s y "phase" owes i t s separate e x i s t e n c e to the very high v i s c o s i t y . Moreover, the p r a c t i c a l importance of polymers i n the g l a s s y s t a t e has l e d to f a r more s t u d i e s of "phase" e q u i l i b r i a i n v o l v i n g t h i s pseudo s o l i d , than has ever been devoted to monomer i c g l a s s e s . Hence the present survey i n c l u d e s g l a s s y systems as w e l l . Needless to say t h a t a d e f i n i t i v e theory of g l a s s t r a n s i t i o n pseudo phase boundaries has yet to be developed, many attempts i n t h a t d i r e c t i o n not w i t h s t a n d i n g . F i n a l l y , what would be the economic impact of improved t h e o r e t i c a l understanding of polymer e q u i l i b r i a , given t h a t polymer proc e s s i n g and a p p l i c a t i o n s technology have grown f a s t e r than those of any n o v e l m a t e r i a l , except perhaps s o l i d s t a t e e l e c t r o n i c s . I n the absence of m u l t i s t a g e s e p a r a t i o n s processes i n v o l v i n g polymers, one of the most important economic d r i v i n g f o r c e s of i n d u s t r i a l thermodynamic s t u d i e s i s m i s s i n g from t h i s f i e l d . Where, on the other hand, improved t h e o r e t i c a l i n s i g h t leads to q u a l i t a t i v e changes i n

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the understanding of polymer p h y s i c s , new v i s t a s would be opened f o r i m a g i n a t i v e i n v e n t o r s of new compositions of matter or of n o v e l processes. Those developments c o u l d have a f a r g r e a t e r impact than improving the p r e c i s i o n of our estimates of the l o c a t i o n of phase boundaries from molecular parameters. One area promising q u a l i t a t i v e l y n o v e l c o n c l u s i o n s might be a good theory of superaggregate formation i n concentrated polymer s o l u t i o n s .

Abstract The present survey examines the current state of ignorance wherever polymers form part of a phase equilibrium in non-electrolyte systems, whether in or with solids, liquids, gases or surfaces (by adsorption). Overall we note the absence of a reliable means to estimate the effect o of phase equilibria. A by solvent/non-solvent combinations. Among the more significant ignorance islands we find: tensile strain and strain rate induced crystallization; equilibria i n concentrated polymer solutions, the detailed shape of miscibility gaps; solubility of polymers in high pressure gases; vapor-liquid equilibria in block copolymer systems; adsorption of polymers out of their concentrated solutions. Ignorance in these areas is noted by the engineer insofar as he cannot even make crude a priori estimates of these equilibria, and after asking a chemist for a few experimental data points, is hard pressed to put those few data into a correlational framework for expansion into the data network that he commonly needs for process design calculations, or for materials properties estimation over the usual range of the independent variables. The absence of large scale multistage separations processes involving polymers removes the economic incentive which drives phase equilibrium studies i n the world of simple chemical substances. Moreover, given the large number of composition variables available to and exploited by the polymer chemist as well as the multidimensionality of the measurement problem, we can expect the elimination of ignorance islands by the private sector only where continued ignorance is too costly to bear as a source of design and performance uncertainties. We can only hope that important new insights from investigators in academia will encompass one or more ignorance islands incidental to the resolution of more basic problems in polymer physics. Literature Cited 1. Ekwall, Per, Adv. i n Liquid Crystals 1, 1 (1975), Academic Press. 2. Fischer, E. W., Koll. Z. & Polymers (1966), 213, 113, 93; (1969), 231, 472; (1971), 247, 858. 3. Wunderlich, B., et a l . , J. Polymer Sci. (1964), A2, 3694.

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4. Mandelkern, L., "Crystallization of Polymers", McGraw-Hill, New York, 1964. 5. Bondi, A., Chem. Rev. (1965), 67, 565. 6. Bhateja, S. K. and Pae, K. D., J. Macromal. Sci-Revs. Macromol. Chem. (1975), 13, (1). 7. Boyer, R. F., J. Polym. Sci. (1966), C14, 267. 8. Boyer, R. F., Stadhicki, S. J. and Gilham, J. K., J. Appl. Polymer Sci. (1976), 20, 1245. 9. Porter, R. S. and Johnson, J. F., Chem. Rev. 66, (1966) 1 (see also Ref. 8). 10. Ehrlich, P. and Woodbrey, J. C., J. Appl. Polymer Sci. (1969), 13, 117. 11. Kuhn, R., Makromol. Chem. (1976), 177, 1525. 12. Koningsveld, R., Br. Polym. J. (1975), 7, 435. 13. Bonner, D. C., J. C13(2), 263. 14. Lipatov, Y. S. and Sergeeva, L. M., Adv. Coll. and Intf. Sci. (1976), 6, 1. 15. Su, C. S., Patterson, D., and Schreiber, H.P., J. Appl. Polym. Sci. (1976), 20, 1025. 16. Y. Mori and H. Tanzawa, J. Appl. Polym. Sci. (1976), 20, 1775. 17. Sperling, L. H., etal.,Polym. Eng. & Sci. (1972), 12, 101. 18. Vieth, W. R., Frangoulis, C. S. and Rionda, J. A., J. Coll. Intf. Sci. (1966), 22, 454. 19. Koros, W. J., Paul, D. R., and Rocha, A. A., J. Polym. SciPolym. Phys. (1976), 14, 687. 20. Pennings, A. J. and Smith, P., Br. Polym. J. (1976), 7, 460. 21. Onu, A., Legras, R., Mercier, J. P., J. Polym. Sci-Polym. Phys. (1976), 14, 1187. 22. M. I. Ostler, "Structure and Properties of Ordered Block Copolymer Membranes", PhD Thesis, Case-Western Reserve U., 1975. 23. Silberberg, A., Polym. Sci. (1970), Part C30, 393. 24. Fowkes, F. M., Schick, J., Bondi, A., J. Coll. Sci. (1960), 15, 531. 25. Eirich, F. R., etal.,Preprints 48th Nat'l Colloid Symp. U. Texas, June 1974, p. 165/7.

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

Discussion J. P. O'CONNELL

The papers presente both the g e n e r a l i t i e s an t h i s area which i s so e s s e n t i a l i n the design and modeling of chemical processes. I t would be redundant f o r me to repeat what i s s a i d i n them, but i t may be a p p r o p r i a t e f o r me to summarize the impressions that I r e c e i v e d and note some of the more r e l e v a n t d i s c u s s i o n s which f o l l o w e d . F i r s t , I t h i n k i t i s c o r r e c t to say "you ve come a long way, baby!" i n the sense that the thermodynamic and p h y s i c a l b a s i s has been l a i d f o r n e a r l y a l l of the important s i t u a t i o n s encountered, there a r e many c o r r e l a t i o n s which a r e a t l e a s t adequate f o r most p r a c t i c a l purposes, and there a r e many data a v a i l a b l e on systems of importance which a r e c o n t i n u a l l y drawn upon f o r screening purposes and design c a l c u l a t i o n s . For example, there a r e many equations of s t a t e (about which an e x c e l l e n t r e p o r t i s given elsewhere i n t h i s volume) and c o r r e s ponding s t a t e s c o r r e l a t i o n s which a r e used d a i l y w i t h c o n s i d e r a b l e s a t i s f a c t i o n f o r "normal" f l u i d s i n c l u d i n g mixtures (nonpolar and weakly p o l a r - substances). There a r e numerous accurate and f l e x i b l e a c t i v i t y c o e f f i c i e n t c o r r e l a t i o n s which can d e s c r i b e more complex systems provided data a r e a v a i l a b l e to determine t h e i r b i n a r y parameters. I n the absence of t h i s , the methodology of group c o n t r i butions has passed through s u c c e s s i v e generations to a f a i r l y high degree of g e n e r a l i t y and accuracy. There a r e g a s - l i q u i d chromatography f o r i n f i n i t e d i l u t i o n p r o p e r t i e s and other automated apparatuses f o r easy measurement of h i g h l y accurate v a p o r - l i q u i d e q u i l i b r i u m data, p a r t i c u l a r l y a t lower pressures. Computational s o p h i s t i c a t i o n , i n c l u d i n g the f i t t i n g of data and convergence of i t e r a t i v e c a l c u l a t i o n s , has reached impressive h e i g h t s . Although perhaps l e s s w e l l recognized, broadly a p p l i c a b l e t h e o r i e s from s t a t i s t i c a l mechanics and most of the fundamentals of the c r i t i c a l r e g i o n have been worked out so that development toward a p p l i c a t i o n can be s t a r t e d . However, i t i s a l s o t r u e that "you a i n ' t there y e t ! " because there a r e many systems which cannot even be c h a r a c t e r i z e d and T

1

141

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142

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others f o r which present d e s c r i p t i o n s are i n s u f f i c i e n t l y a c c u r a t e , i f not a c t u a l l y erroneous. There have been many " f a i l u r e s " (defined as designs based on p h y s i c a l property i n f o r m a t i o n whose performance d e v i a t e d s i g n i f i c a n t l y from s p e c i f i c a t i o n , f o r worse, or perhaps b e t t e r , i n an economic sense) and p o s s i b l y " d i s a s t e r s " for which no a l t e r n a t i v e methods e x i s t even now. In a d d i t i o n , as time passes our a b i l i t y to a f f o r d l a r g e s a f e t y f a c t o r s w i l l be decreased. For example, how should hydrocarbon f r a c t i o n s r e a l l y be c h a r a c t e r i z e d ? What are the s p e c i e s which a c t u a l l y e x i s t i n systems i n which s t r o n g or even weak chemical r e a c t i o n s can occur? Is there a way of p u t t i n g a "chemist i n the computer" to warn of such phenomena? No equation of s t a t e p r e s e n t l y a v a i l a b l e d e s c r i b e s mixtures of p o l a r and nonpolar substances i n the dense gas or l i q u i d regions (though th for moderately dense systems) G r i f f i t h s (Phys. Revs., 1973) shows the formalisms of the c r i t i c a l s c a l i n g laws f o r m i x t u r e s , these have not been developed f o r use. G e n e r a l i z e d c o r r e l a t i o n s f o r l i q u i d s c o n t a i n i n g s i g n i f i c a n t amounts of i o n i z e d s p e c i e s along w i t h n o n e l e c t r o l y t e s are n o n e x i s t e n t . Most aspects have not been worked out f o r the thermodynamic p r o p e r t i e s and e q u i l i b r i a of systems of macromolecules i n the c r y s t a l l i n e , g l a s s y , or s o l u t i o n form. There i s much u n c e r t a i n t y about the use of d i l u t e s o l u t i o n r e f e r e n c e s t a t e s f o r s u p e r c r i t i c a l components, p a r t i c u l a r l y i n m u l t i s o l u t e , m u l t i s o l v e n t s o l u t i o n s . The p r e d i c t i o n of l i q u i d - l i q u i d phase boundaries and the concomitant v a p o r - l i q u i d d i s t r i b u t i o n c o e f f i c i e n t s i s not on a par w i t h that of l e s s complex systems. I n a d d i t i o n to c o r r e l a t i o n inadequacy, even for those s i t u a t i o n s where equations are a v a i l a b l e , inadequate data bases e x i s t f o r e s t a b l i s h i n g parameters, p a r t i c u l a r l y f o r many group i n t e r a c t i o n s . I t i s accepted t h a t , except f o r accurate p r e d i c t i o n of t e r n a r y and higher c r i t i c a l p o i n t s (such as l i q u i d l i q u i d p l a i t p o i n t s ) , o n l y b i n a r y i n t e r a c t i o n s need to be charact e r i z e d , u s u a l l y by a s i n g l e constant f o r e i t h e r equations of s t a t e or a c t i v i t y c o e f f i c i e n t s . Yet a very l a r g e number of b i n a r i e s have not yet been addressed. To improve upon t h i s s i t u a t i o n , the major source of ideas must come from a p p l y i n g molecular concepts through adopting s t a t i s t i c a l thermodynamics. The e f f o r t s d e s c r i b e d by John P r a u s n i t z and others i n the conference are o n l y beginnings. Yet t h e i r success warrants f u r t h e r i n v e s t i g a t i o n . One aspect of t h e i r use w i l l undoubtedly e s t a b l i s h new c h a r a c t e r i z a t i o n methods r e q u i r i n g data which are d i f f e r e n t from the normal k i n d . F i n a l l y , then " l e t s get moving!" Our speakers exhorted us to concentrate our e f f o r t s on the unsolved problems. Incremental changes i n present c o r r e l a t i o n s (except to c o r r e c t obvious d e f i c i e n c i e s i n form under e x t r a p o l a t i o n ) , and new measurements of systems whose p r o p e r t i e s are s u b j e c t to l i t t l e u n c e r t a i n t y (except f o r checking equipment) are u n i n t e r e s t i n g . The c h a l l e n g e i s to be c r e a t i v e . I b e l i e v e the g r e a t e s t success w i l l occur when communi-

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

O'CONNELL

Discussion

143

c a t i o n and c o o p e r a t i o n , such as at t h i s conference, are e s t a b l i s h e d . In g e n e r a l , the l i m i t i n g f a c t o r f o r new ideas i s the b r i d g e of development. Let us a l l r e s o l v e to b u i l d i t whether we are researchers or p r a c t i t i o n e r s . With a l l the p o s s i b i l i t i e s a v a i l a b l e , l e t i t not be s a i d that a delay or a l a c k of a s o l u t i o n to the t e c h n o l o g i c a l problems of the present and f u t u r e could be l a i d at our f e e t f o r want of adequate d e s c r i p t i o n s of p h y s i c a l p r o p e r t i e s . Comment submitted by J . S.

Rowlinson:

John P r a u s n i t z had mentioned the e x c e l l e n t agreement w i t h experiment that M o l l e r u p had obtained i n the g a s - l i q u i d c r i t i c a l r e g i o n of b i n a r y hydrocarbon m i x t u r e s . M o l l e r u p s r e s u l t s were obtained w i t h a good r e f e r e n c e equation of s t a t e f o r methane (but one which i s c l a s s i c a l i a c c u r a t e l y the known n o n c l a s s i c a namic f u n c t i o n s at the c r i t i c a l p o i n t ) , and w i t h a o n e - f l u i d model based on a mole f r a c t i o n average (or "mole f r a c t i o n based mixing rules"). I t i s , perhaps, worth comment that no improvement would be made by u s i n g a more c o r r e c t r e f e r e n c e equation of s t a t e ( i . e . , one w i t h the c o r r e c t s i n g u l a r i t i e s ) unless the mole f r a c t i o n averaging i s a l s o abandoned. To do t h i s f i r s t s t e p , without the second, means that the s i n g u l a r i t i e s are l o s t i n the mixture c a l c u l a t i o n s — the mixture c r i t i c a l p o i n t i s s t i l l c l a s s i c a l . I f , however, the mole f r a c t i o n average be abandoned a l s o , and r e p l a c e d by mixing r u l e s based on a composition v a r i a t i o n of the a b s o l u t e a c t i v i t y Xj_ = exp ( u i / k T ) , where u i s the chemical p o t e n t i a l , then the s i n g u l a r i t i e s are preserved on going from pure r e f e r e n c e equation of s t a t e to the mixture. T h i s has been shown by Leung and G r i f f i t h s (Phys. Rev.) f o r He - % e m i x t u r e s . I t may be that t h i s composit i o n v a r i a b l e i s worth f u r t h e r study i n the mixing r u l e s . ?

Question by D. T. Binns: In computer c a l c u l a t i o n s f o r d i s t i l l a t i o n or f l o w s h e e t i n g , K-values subroutines may be c a l l e d thousands of times. Routines based on complex equations of s t a t e can make the computation c o s t l y . Is t h i s a problem here? Have you, or anyone e l s e present, any experience of f i t t i n g data, e i t h e r measured or c a l c u l a t e d by an a p p r o p r i a t e equation of s t a t e , to something simple f o r use i n computer programs? For i n s t a n c e , i t should be p o s s i b l e to f i t the two Redlich-Kwong parameters per component i n s t e a d of c a l c u l a t i n g them from c r i t i c a l c o n s t a n t s . I t would a l s o be p o s s i b l e to i n c o r p o r a t e enthalpy data which may be a v a i l a b l e , so h e l p i n g t h e r modynamic c o n s i s t e n c y . Reply by T. K r d l i k o w s k i : There have been some examples, but computer c o s t s are now

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

so

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low that i t i s not worth the t r o u b l e . Some economy can be achieved by c a l c u l a t i n g K-values only every f i f t h i t e r a t i o n . go ahead and use the best equation of s t a t e .

We

Comments by A. Bondi: Regarding the paper presented by J . P r a u s n i t z : 1.

For the advance guess or the c o r r e l a t i o n of a d s o r p t i o n isotherms under s u p e r c r i t i c a l c o n d i t i o n s , I can see no a l t e r n a t i v e to the e s t i m a t i o n of the need p ( f o r the d r i v i n g f o r c e p/p ) v i a the e x t r a p o l a t i o n of the vapor pressure curve beyond T , which you deprecated.

2.

Your e l i p s e s , the r a t h e s t r i k i n evidenc f o th f i n a p p r o p r i a t e data r e d u c t i o exposure. Chemica engineer highly g the computer represent data by q u i t e complex r e g r e s s i o n equat i o n s . But many seem to be f a r l e s s c a r e f u l i n watching f o r a u t o c o r r e l a t i o n among c o e f f i c i e n t s i n these equations, and i n a s c e r t a i n i n g the s i g n i f i c a n c e of each c o e f f i c i e n t than are our c o l l e a g u e s i n economics and elsewhere i n teh b e h a v i o r a l s c i e n c e s . The absence of r e f e r e n c e to a u t o c o r r e l a t i o n and c o e f f i c i e n t q u a l i t y c r i t e r i a from most papers on the m u l t i constant equations of s t a t e i s r a t h e r s e r i o u s evidence f o r t h i s s t a t e of a f f a i r s . Maybe t h i s needs some more emphasis i n the teaching process.

Dr. K r o l i k o w s k i , you and your coworkers' o b s e r v a t i o n d i f f i c u l t i e s i n p h y s i c a l p r o p e r t i e s e s t i m a t i o n f o r two process streams c o n t a i n i n g aldehydes, a l c o h o l ( s ) and water i s not s u r p r i s i n g . Aldehydes r e a c t w i t h water to form hemihydrates by 0

OH /

R -C-H + H 0 « RC o

1

2

\ OH

and they r e a c t w i t h a l c o h o l s to form hemiacetals by 0 1

R -C-H + R 0H 3«R -C-0-R 1 2 1 \ 2 OH o

o

In both cases there i s a steep i n c r e a s e i n v i s c o s i t y , as shown by the examples of my own experience, shown on F i g u r e s 1 and 2, and n a t u r a l l y a s u b s t a n t i a l i n c r e a s e i n d e n s i t y , as shown on F i g u r e s 3 and 4. These r e a c t i o n s are r e v e r s i b l e a n d — a s shown—of l e s s importance a t high than a t low temperature. Moreover, once one has recognized what chemistry i s going on, the mixture p r o p e r t i e s can be estimated w i t h f a i r approximation, as shown by the compari-

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

O'CONNELL

Discussion

100

0.11 0

|

I I I i I I I I 0.2 0.4 0.6 0.8 Mole Fraction, 3-Ethoxy Propionaldehyde = x

I 1.0

Figure 1. Viscosity of mixtures of 3-ethoxy propionaldehyde with water at various temperatures

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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A N D FLUID PROPERTIES IN C H E M I C A L INDUSTRY

100

Figure 2.

Viscosity of mixtures of 3-ethoxy propionaldehyde with ethanol at various temperatures

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

O'CONNELL

147

Discussion

Figure 4. Volume contraction (V ) of mixtures of 3-ethoxy propionaldehyde with ethanol at 0°C compared with deviations of the viscositufrom the ideal mixing rule E

American Chemical Society Library In Phase Equilibria and Fluid Properties in theSt., Chemical 1155 16th N.W.Industry; Storvick, T., et al.; ACS Symposium Series;Washington, American Chemical Society: Washington, DC, 1977. O.C. 20036

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sons of Table 1. A few l i t e r a t u r e data f o r mixtures of acetaldehyde w i t h water show a s i m i l a r behavior. The main p o i n t of these d i s c u s s i o n remarks i s to emphasize the need f o r a very a l e r t chemist on the p r o p e r t i e s e s t i m a t i o n team. In my case i t was a "gut f e e l i n g " when I asked f o r the mixture p r o p e r t i e s t o be measured r a t h e r than s l i p a r o u t i n e e s t i m a t i o n i n t o a process manual. The wise e x p l a n a t i o n of the chemistry came only as we a l l contemplated the unexpected experimental evidence. The p o t e n t i a l c o s t l i n e s s of p r o p e r t y e s t i m a t i o n e r r o r s caused by the unexpected formation of (loose) covalent or molecular compounds among process streams components c a l l s f o r an e a r l y remedy. One of these might be to have the e d i t o r of C.E.P. commission a w e l l known a u t h o r i t y on molecular compounds and r e l a t e d chemistry to w r i t e an a r t i c l e f o r engineers i n c l u d i n g a t a b l e w i t h danger s i g n a l notebooks. These remarkable data presented by Dr. Franck are so d i f f i c u l t to o b t a i n e x p e r i m e n t a l l y t h a t t h e i r e s t i m a t i o n w i t h paper and p e n c i l i s c l e a r l y worthwhile. Can one estimate the d i e l e c t r i c constants i n your s u p e r c r i t i c a l range from a combination of

Table 1 - E s t i m a t i o n of v i s c o s i t y * of Hemihydrate and of Hemiacetal of 3-Ethoxy Propionaldehyde compared w i t h o b s e r v a t i o n on F i g u r e s 1 and 2.

Substance H

t, °c

^est.

^obs.

4

0.22

0.23

0

0.18

0.12

2

OH Et-O-C"" H

OH

2

C

0 1

0 I o"

*By method i n A. Bondi " P h y s i c a l P r o p e r t i e s of Molecular C r y s t a l s , L i q u i d s and G l a s s e s " , Wiley 1968.

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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Discussion

149

d i e l e c t r i c p o l a r i s a b i l i t y , such as modern C l a u s i u s - M o s o t t i d e r i v a t i v e s , w i t h the reduced temperature (T/T )? Can one estimate the e l e c t r i c a l c o n d u c t i v i t y i n your s u p e r c r i t i c a l range from any theory of the Walden product A-n or other s e m i - e m p i r i c a l correlation? Dr. Franck i n d i c a t e d that the C a l u s i u s - M o s o t t i e f f e c t had been t r e a t e d using the Onsager-Kirkwood c o r r e c t i o n s w i t h agreement between experiment and theory that i s only s a t i s f a c t o r y . Chemical r e a c t i o n s commonly occur between water and other molecules. Hydrocarbons s t a r t to r e a c t at 400C so the chemical e q u i l i b r i u m and p h y s i c a l e q u i l i b r i u m are simultaneously observed. S o l i d - l i q u i d melt l i n e s have not been s t u d i e d at the K a r l s r u h e l a b o r a t o r y . Some c a u t i o n must be used at these "superpressures" because molecules deform enough that they change t h e i r chemical i d e n t i t y .

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

7 Industrial Uses of Equations of State: A State-of-the-Art Review STANLEY B. ADLER, CALVIN F. SPENCER, H A L OZKARDESH, and CHIA-MING KUO Pullman Kellogg Co., Houston, TX 77046

The objective of the presentations at this conference, as the authors of this paper understand them, is to document the state of the art in various areas of thermodynamics. In turn the ultimate objective is to enable academia to grasp the needs of industry, while industry becomes better acquainted with the new tools developed in the universities. Those of us from industry on the week's program have been asked to describe, from the industrial side of the fence, the current activities, developments, and applications of particular assigned areas of phase equilibria or physical properties correlation. This paper pertains to equations of state, especially their application in phase equilibrium predictions. It is not the intention of this talk, nor would it be appropriate, to review all the equations of state that have been published. An excellent paper by Tsonopoulos and Prausnitz (1) does make such a review. Instead, this paper will present the applications and limitations of some of the principal equations of state in current use. At the very outset--before even getting to the body of this paper--it is necessary to picture the magnitude of industrial involvement with equations of state. The two lists given in Table I summarize the basic outlines of this involvement. In one category the research aspects of industrial work in equations of state have been assembled. In a second category, the applications in which equations of state are intermixed with thermodynamic principles in day-to-day process and design problems and computations are shown. To put i t another way, the first list deals with sharpening the tools; the second list with using the tools. Characteristics of Equations of State Required by Industry When equations of state are used in industry they must possess two essential characteristics: (1) Versatility and (2) Workability. 150 In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

7.

ADLER E T A L .

Industrial Uses of Equations of State

151

V e r s a t i l i t y . To best d e s c r i b e the v e r s a t i l i t y possessed by equations of s t a t e , a q u o t a t i o n i s taken from P r o f e s s o r Joseph M a r t i n ( 2 ) , a long time i n v e s t i g a t o r i n t h i s f i e l d . "The second reason f o r developing new equations of s t a t e concerns the e x c e p t i o n a l power and u t i l i t y of an equation of s t a t e . When combined w i t h a p p r o p r i a t e thermodynamic r e l a t i o n s , a well-behaved equation can p r e d i c t w i t h h i g h p r e c i s i o n i s o t h e r m a l changes i n heat c a p a c i t y , enthalpy, entropy and f u g a c i t y , vapor pres-

TABLE I Research 1. 2. 3. 4.

5.

6. 7. 8.

9.

Aspects

Develop completely ne Improve e x i s t i n g equations. Test e x i s t i n g ones f o r successes and f a i l u r e s . Study a p p l i c a t i o n to mixtures. a. new models b. new i n t e r a c t i o n constants Extend given equations of s t a t e to other p r o p e r t i e s , modify i f necessary f o r improvement e.g., Redlich-Kwong t o enthalpy [Barner et a l . , (_3) ] Extend to lower and lower temperatures. Get constants f o r more and more substances. Extend u s e f u l n e s s of equations of s t a t e to new a p p l i c a t i o n s . a. a d d i t i o n to m i n i m i z a t i o n of f r e e energy technique b. to s o l u t i o n of freeze-out problems Develop a whole new approach—combining e x i s t i n g equations: one f o r l i q u i d and one f o r the vapor, and even one f o r pure liquid fugacity.

Thermodynamic A p p l i c a t i o n s i n Process Engineering 1. C o m p r e s s i b i l i t y . 2. Enthalph and heats of mixing. 3. Liquid-vapor e q u i l i b r i a : bubble-point, dew-point, f l a s h v a p o r i z a t i o n s ; p r e d i c t i o n s i n the r e t r o g r a d e r e g i o n . 4. L i q u i d - l i q u i d e q u i l i b r i a . 5. L i q u i d - v a p o r - s o l i d e q u i l i b r i a and freeze-out problems. 6. A l l phases w i t h chemical r e a c t i o n e q u i l i b r i a s i m u l t a n e o u s l y . 7. P r e d i c t i o n o f c r i t i c a l p o i n t s and the c r i t i c a l locus of a mixture. 8. P r e d i c t i o n and i n t e r p r e t a t i o n of r e s u l t s f o r thermodynamic anomalies, e.g., m u l t i p l e bubble-points. 9. I n c o r p o r a t i o n of equation of s t a t e as a t o o l i n computerized flow sheet c a l c u l a t i o n s f o r an e n t i r e process. 10. I n c a l c u l a t i o n of data c h a r t s issued company-wide i n Data Books, p a r t i c u l a r l y i n K constant c h a r t s f o r l i q u i d - v a p o r equilibrium.

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sure, l a t e n t heat of v a p o r i z a t i o n , a c t i v i t y c o e f f i c i e n t s , and v a p o r - l i q u i d e q u i l i b r i a i n mixtures, not to mention the a s s i s t a n c e i t o f f e r s i n t r a n s p o r t prop e r t y c o r r e l a t i o n s . U n f o r t u n a t e l y , even though the u s e f u l a p p l i c a t i o n s of an equation of s t a t e are so e x t e n s i v e and a t t r a c t i v e , the development of a high performance equation proves to be so i n v o l v e d that to date no one has come c l o s e to d i s c o v e r i n g a s i n g l e r e l a t i o n which i s t r u l y good over a wide range of density." As M a r t i n p o i n t e d out, the equation of s t a t e i s a powerful t o o l . Working w i t h i t to s o l v e problems i s an e x c i t i n g occupation. N e v e r t h e l e s s , b e f o r e g e t t i n g i n t o the e x c i t i n g s i d e of the p i c t u r e , i t i s necessary to p a i n t the p r o s a i c , more mundane s i d e . The r e a son f o r doing so i s that t r i a l p o i n t of view. I n that i s f l e x i b l e , t h a t can reach a s o l u t i o n without having a computer f a i l u r e a f t e r the c a l c u l a t i o n has proceeded as f a r as the mid-tray i n a one hundred t r a y d i s t i l l a t i o n tower. The word f l e x i b l e , mentioned i n the previous sentence, covers a myriad of a t t r i butes. The equation of s t a t e , as M a r t i n s a i d , must be s u i t a b l e f o r a l l the thermodynamic and p h y s i c a l p r o p e r t i e s — n o t j u s t enthalpy or c o m p r e s s i b i l i t y as many p u b l i s h e d equations of s t a t e are. As an i n i t i a l requirement i t should be a b l e to generate these p r o p e r t i e s f o r vapors, both f o r pure components and mixtures. 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 demand, as an a d d i t i o n a l requirement, a p p l i c a t i o n to l i q u i d s and l i q u i d m i x t u r e s . As an added r e q u i r e ment i t would be expedient to have i t apply to petroleum f r a c t i o n s which are a continuum of many components that cannot i n d i v i d u a l l y be d e f i n e d . This i s t r u l y a s k i n g a l o t , y e t a l l the requirements are i n t e g r a l p a r t s of a t y p i c a l process design c a l c u l a t i o n . Some of the shortcomings of p u b l i s h e d expressions which l a c k such f l e x i b i l i t y are i l l u s t r a t e d below. These examples are based on s t u d i e s performed by the T e c h n i c a l Data Group a t Pullman K e l l o g g over the l a s t f i f t e e n years. Some years ago Barner et a l . (3) developed a m o d i f i c a t i o n of the Redlich-Kwong equation of s t a t e f o r a p p l i c a t i o n i n enthalpy c a l c u l a t i o n s . The o r i g i n a l Redlich-Kwong equation, intended f o r compressib i l i t y , was being m i s a p p l i e d by a c l i e n t f o r enthalpy computations. Doing so, the c l i e n t disagreed w i t h a s p e c i f i e d duty on a flowsheet. The Redlich-Kwong equation was m o d i f i e d to represent enthalpy by f i t t i n g i t s constants to the C u r l - P i t z e r enthalpy t a b l e s . The end r e s u l t : i t helped the c l i e n t , but the r e v i s e d equation f a i l e d to represent f u g a c i t y adequately. S i m i l a r l y , i n another p u b l i c a t i o n Barner and A d l e r (4) r e v i s e d the J o f f e equation to represent q u i t e a number of vapor p r o p e r t i e s . I t was, as expected, an u t t e r f a i l u r e f o r r e p r e s e n t i n g l i q u i d s , y e t w e l l a p p l i c a b l e to the gaseous mixtures of those components f o r which BWR constants were not a v a i l a b l e and where i t would be i m p r a c t i c a l to determine them, p a r t i c u l a r l y i f the components are not o f t e n encountered.

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As an i l l u s t r a t i o n o f an equation o f s t a t e that w i l l not be a p p l i c a b l e to m i x t u r e s , consider that the usual mixing r u l e s r e q u i r e t a k i n g square r o o t s , cube r o o t s , or the l i k e . I f the cons t a n t s are n e g a t i v e f o r one o r more of the components i n the mixt u r e , extension to mixtures i s not f e a s i b l e . The most important i n d u s t r i a l a p p l i c a t i o n o f equations o f s t a t e i s , and w i l l continue to be, i n phase e q u i l i b r i u m p r e d i c t i o n s . A l l such c a l c u l a t i o n s are based on the e q u i l i b r i u m c r i t e r i o n : f

v

= h

= h

(

1

)

where, f = component f u g a c i t y V, L, S = vapor, l i q u i d , and s o l i d phases, r e s p e c t i v e l y . Equations of s t a t e are a p p l i e ponent f u g a c i t i e s . I n th equation i s used t o get the f u g a c i t y o f each component i n a l l the c o e x i s t i n g phases. In a second approach, one equation o f s t a t e i s used t o get vapor f u g a c i t y , w h i l e d i f f e r e n t equation(s) are employed f o r the l i q u i d o r s o l i d phases. These approaches are explained i n greater d e t a i l i n the next s e c t i o n o f the paper. Although most e q u i l i b r i u m c a l c u l a t i o n s i n v o l v e a s i n g l e vapor and l i q u i d phase o n l y , a number o f e q u i l i b r i u m phase combinations such as l i q u i d - l i q u i d - v a p o r , l i q u i d - s o l i d w i t h or w i t h o u t a vapor phase, and v a p o r - s o l i d phases are encountered i n i n d u s t r y . Furthermore, any o f these p h y s i c a l e q u i l i b r i u m c o n d i t i o n s may a l s o r e q u i r e computations o f the chemical r e a c t i o n e q u i l i b r i a . For example, consider the f o l l o w i n g r e a l i s t i c i n q u i r y from an engineer designing a p i l o t p l a n t . The chemists completed t h e i r bench-scale work f o r the h y d r o l y s i s o f propylene to make an a l c o h o l and an ether. They reported the optimum temperature to be 240F and the pressure to be 500 p s i a . The chemical engineer wanted to know how many phases the r e a c t o r e f f l u e n t would have a t these c o n d i t i o n s : a l l vapor, a l i q u i d phase w i t h the a l c o h o l product and condensed steam i n e q u i l i brium w i t h the vapor, o r perhaps three phases i n v o l v i n g the unreacted, condensed propylene as w e l l . This complex e q u i l i b r i u m problem i n v o l v i n g both p h y s i c a l and chemical e q u i l i b r i u m w i t h n o n i d e a l l i q u i d s and a n o n i d e a l vapor best represents what chemical process engineering i s a l l about. Problems l i k e these can be solved by a p p l y i n g the p r i n c i p l e of the m i n i m i z a t i o n o f f r e e energy ( 5 ) . I n t h i s approach the f r e e energy i s r e l a t e d t o f u g a c i t y o f each component i n each phase, whereby the f u g a c i t i e s are c a l c u l a t e d from equations o f s t a t e . No other known approach t o the s o l u t i o n of such problems i s r e a l l y s a t i s f a c t o r y . The f a c t that t h i s approach i s a l s o d i r e c t and easy to v i s u a l i z e as one looks a t successive computer p r i n t - o u t s o f the i t e r a t i o n s to the f i n a l s o l u t i o n makes i t a l l the more the one to be recommended. Furthermore, the ether p r o p e r t i e s , enthalpy being the most important, are computed on a b a s i s c o n s i s t e n t w i t h the e q u i l i b r i a , by means of the same equation of s t a t e . I t i s f o r

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t h i s s o r t of problem that the v e r s a t i l i t y of the equation of s t a t e r e a l l y pays o f f . W o r k a b i l i t y . Even i f the equation i s a p p l i c a b l e to l i q u i d s and vapors, pure components and mixtures, p h y s i c a l p r o p e r t i e s , thermodynamic p r o p e r t i e s , and e q u i l i b r i a , there i s s t i l l the quest i o n of w o r k a b i l i t y . In simple terms, can i t accomplish what i t proposes to do? Very f r e q u e n t l y a new thermodynamic r e l a t i o n s h i p i s c a r e f u l l y d e r i v e d , t e s t e d on a few systems, and p u b l i s h e d e i t h e r i n the open l i t e r a t u r e or i n a company engineering r e p o r t . I t may look prom i s i n g , but u n t i l i t i s t e s t e d on a wide v a r i e t y of systems, and a wide range of temperatures and pressures and other c o n d i t i o n s , i t s r e a l m e r i t s or shortcomings i n areas such as handling c l o s e b o i l i n g components, the c i b l e l i q u i d s , e t c . , canno these areas may be o u t s i d e the a p p l i c a b i l i t y of the new r e l a t i o n ship. Some years ago a research chemist t r i e d to develop a thermodynamic consistency t e s t f o r hydrogen and helium systems where one component i s above i t s c r i t i c a l temperature, and where a vapor pressure and t h e r e f o r e , the a c t i v i t y c o e f f i c i e n t , cannot be obtained. His r e l a t i o n s h i p r e q u i r e d the use of a g e n e r a l i z e d c h a r t to get a p r o p e r t y — o f course i t was before the days of computers. When attempts were made to use the g e n e r a l i z e d c h a r t , i t was found that near the c r i t i c a l r e g i o n an accurate property value could not be read, c e r t a i n l y not accurate enough to t e s t c o n s i s t e n c y w i t h o u t adding as much e r r o r as would be measured i n the consistency t e s t . Unless one takes the time to s u f f i c i e n t l y t e s t , one does not see the problems that may be encountered. A second c o n s i d e r a t i o n , which i s c e r t i a n l y very important to i n d u s t r y , i s w o r k a b i l i t y without the i n t e r v e n t i o n of man, as w i t h a computer. Not only w i l l the equation of s t a t e be used i n an i t e r a t i v e f a s h i o n to converge to a s o l u t i o n , but a l s o on s u c c e s s i v e stages i n equipment design, to produce a whole s e r i e s of r e s u l t s . With a l l these repeated uses, i t must lend i t s e l f to ease of s o l u tion with scarcely a f a i l u r e . This e n t i r e p r e s e n t a t i o n c o u l d be devoted to the problem of g e t t i n g some equations of s t a t e to converge on the c o r r e c t r o o t . Some of the problems are l i s t e d here: 1. In a p p l i c a t i o n s of equations of s t a t e to v a p o r - l i q u i d e q u i l i b r i a , a v o i d convergence to a t r i v i a l s o l u t i o n , part i c u l a r l y a l l the K values being 1.0 (where K = y/x i s the r a t i o of the mole f r a c t i o n s i n vapor and l i q u i d , respectively). 2. The s e l e c t i o n of the c o r r e c t r o o t f o r the vapor d e n s i t y and then the l i q u i d d e n s i t y , when the two phases are i n equilibrium. 3. T r e a t i n g convergence to negative r o o t s . Some of the cookbook r u l e s to e x t r i c a t e the computer from t h i s s i t u a t i o n

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simply l e a d t o a negative c o m p r e s s i b i l i t y again a few trials later. Besides p r o v i d i n g f e a s i b l e computer routes to handle the three aforementioned problems, s p e c i a l care must be taken to p r e vent the computer f l a s h a l g o r i t h m from o s c i l l a t i n g back and f o r t h i n K constant ( 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 because the p r e s c r i b e d f l a s h c o n d i t i o n s o f temperature and pressure do not f a l l i n the two-phase r e g i o n . I t would be presumptuous to assume t h a t a l l f l a s h c o n d i t i o n s are r e a l i s t i c ; engineers do, and w i l l continue to, submit u n r e a l i s t i c temperature and p r e s s u r e s , i n the range o f subcooled l i q u i d s and superheated vapors. Proper c r i t e r i a must be s e l e c t e d f o r e s t a b l i s h i n g whether two phases f o r a g i v e n mixture r e a l l y e x i s t a t the s e t c o n d i t i o n s . Frequently the d e f i n i t i o n o f the c r i t i c a l p o i n t o f a pure compound i s o f t e n erroneously used equations o f s t a t e d e f i n (2)

= 0

In simple terms, the c r i t i c a l isotherm has a p o i n t of i n f l e c t i o n at the c r i t i c a l p o i n t . By s t u d y i n g the behavior o f an isotherm p r e d i c t e d from an equation o f s t a t e , i t i s p o s s i b l e to e s t a b l i s h whether two phases e x i s t . For example, i f the i s o t h e r m a l second d e r i v a t i v e changes s i g n over a range o f pressures and volumes, a vapor and l i q u i d are present. This t e s t u n f o r t u n a t e l y a l s o works for mixtures a t c o n d i t i o n s f a r removed from the c r i t i c a l r e g i o n , but breaks down i n the c r i t i c a l and r e t r o g r a d e r e g i o n . Such a t e s t i s v a l i d f o r pure components o n l y . For m i x t u r e s — a n d K cons t a n t s a r i s e only f o r m i x t u r e s — t h e thermodynamics are f a r more complicated, and Eq. (2) should never be a p p l i e d . I t should be added t h a t problems i n t h i s area are o f t e n complicated by the f a c t that many o f the e x i s t i n g equations o f s t a t e are not v a l i d i n the c r i t i c a l r e g i o n and o f t e n e x h i b i t unusual behavior i n t h a t p o r t i o n of the phase envelope. Proper c r i t e r i a f o r r a p i d l y i d e n t i f y i n g u n r e a l i s t i c f l a s h c o n d i t i o n s are only developed a f t e r s u f f i c i e n t experience i s gained working w i t h the r e s p e c t i v e equation of s t a t e and p r o p e r l y a n a l y z i n g f l a s h r e s u l t s , both i t s successes and f a i l ures . As p o i n t e d out i n the preceding paragraph, some equations, by t h e i r very nature, complicate matters. As an a d d i t i o n a l example, consider the s p e c i f i c l i m i t a t i o n s o f the Chao-Seader c o r r e l a t i o n and others l i k e i t , which a l l o w only about 20% methane i n the l i q u i d . I n a t y p i c a l f l a s h problem i t i s c o n v e n t i o n a l f o r a f i r s t set of K constants to be f u r n i s h e d e i t h e r by the engineer o r i n i t i a l i z e d by the f l a s h program. These K s immediately l e a d to vapor and l i q u i d compositions. With these compositions the component f u g a c i t i e s and 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 are c a l c u l a t e d , which i n t u r n l e a d to a seemingly b e t t e r s e t o f K s . I f the f i r s t f

T

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l i q u i d c o m p o s i t i o n — a d m i t t e d l y f a r removed from the c o r r e c t o n e — contains 80% methane i n the l i q u i d , the c o r r e l a t i o n ' s range of a p p l i c a b i l i t y has been exceeded. These K s are so f a r removed from the r e a l s e t that the r e s u l t i n g second t r i a l may be worse than the f i r s t , and convergence never obtained. Thus, even though the mixture composition answer may be w e l l w i t h i n the range of the c o r r e l a t i o n , the i n i t i a l or even the i n t e r m e d i a t e t r i a l s may exceed that range, and lead to erroneous t r i a l values from which the computer s o l u t i o n a l g o r i t h m w i l l never recover. Summarizing, i t takes a f l e x i b i l e equation of s t a t e and the proper combination of thermodynamics, common sense, p a t i e n c e and programming f i n e s s e to u t i l i z e equations of s t a t e e f f e c t i v e l y i n computerized v a p o r - l i q u i d e q u i l i b r i u m and other design c a l c u l a t i o n s . T

Equations of S t a t e Adopte As s t a t e d e a r l i e r , two-phase l i q u i d - v a p o r e q u i l i b r i u m i s the predominant problem i n i n d u s t r i a l e q u i l i b r i u m c a l c u l a t i o n s . To i n c o r p o r a t e equations of s t a t e , Eq. (1) i s o f t e n expanded and r e w r i t t e n as: o x. • y • f = y. • . • ir (3) l 'Li Li i i where, f o r each component i , X i = mole f r a c t i o n i n the l i q u i d phase T

T

;

YL^ o ^Li

Y

= l i q u i d phase a c t i v i t y c o e f f i c i e n t =

P

u r e

liquid

fugacity

y_^ = mole f r a c t i o n i n the vapor phase 71

(j>_^ = vapor phase f u g a c i t y c o e f f i c i e n t fy./y^ " 7T = system pressure When the s i n g l e equation of s t a t e approach i s used, Eq. (3) reduces t o :

XiTT K. 1

hi

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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where i s the l i q u i d - v a p o r e q u i l i b r i u m constant o f component i . The same equation o f s t a t e i s then employed to get both the numerato r and denominator i n t h i s e x p r e s s i o n using standard thermodynamic r e l a t i o n s h i p s . The work o f Benedict, Webb and Rubin ( 6 ) , o f S t a r l i n g ( 7 ) , and the s e r i e s o f Exxon papers ( L i n e t a l . , ( 8 ) ; L i n and H o p k e , ( _ 9 ) ) — a l l on v a r i o u s forms-of the BWR, Soave, and PengRobinson equations o f s t a t e are examples o f the use o f one equation of s t a t e t o perform the whole c a l c u l a t i o n . I t should be added that the o r i g i n a l developments i n t h i s area t r e a t e d the f L i / i term i n the numerator as a separate e n t i t y and m u l t i p l i e d the f i n a l answer by 1/tt f o r consistency. Most contemporary approaches r e l a t e both numerator and denominator t o equations o f s t a t e v i a the f u g a c i t y c o e f f i c i e n t route, the only d i f f e r e n c e i n l i q u i d and vapor being the d e n s i t y and the equation constants obtained from the r e s p e c t i v e mixing r u l e s . In the two ( m u l t i ) equatio r e s t a t e d as: Y fO L. L. l x (5) i x. l x

This s o - c a l l e d two equation o f s t a t e method o f t e n r e q u i r e s three equations: one f o r the n o n i d e a l i t y o f the l i q u i d , one f o r the n o n i d e a l i t y o f the vapor, and one f o r the standard s t a t e l i q u i d f u g a c i t y to which the a c t i v i t y c o e f f i c i e n t must be r e f e r r e d . These three r e l a t i o n s h i p s determine r e s p e c t i v e l y , YL> and f L i n Eq. (5). The Chao-Seader c o r r e l a t i o n and i t s many m o d i f i c a t i o n s , and the Chueh-Prausnitz c o r r e l a t i o n are examples o f t h i s approach. I n the Chao-Seader c o r r e l a t i o n f ^ and ir a r e combined as a pure l i q u i d f u g a c i t y c o e f f i c i e n t , v, so that Eq. (5) has three d i s t i n c t p a r t s — e a c h r e q u i r i n g a unique equation, as j u s t described. Obviously i t i s d e s i r a b l e t o use a s i n g l e equation o f s t a t e f o r the whole computation. I t i s s i m p l e r , more c o n s i s t e n t , and l e s s work f o r both the engineer and computer programmer. However, the hard, c o l d f a c t i s that no one equation o f s t a t e can meet the v e r s a t i l i t y requirements s t a t e d e a r l i e r . M o d i f i c a t i o n s o f S t a r l i n g ' s BWR-11 equation are e x c e l l e n t i n the cryogenic r e g i o n f o r p r e d i c t i n g both VLE, and thermal and p h y s i c a l p r o p e r t i e s . However, the equation i s not a p p l i c a b l e i n many p e t r o c h e m i c a l processing c a l c u l a t i o n s because o f l i m i t a t i o n s caused by the a v a i l a b i l i t y of constants f o r a number o f needed components. The v a r i o u s m o d i f i c a t i o n s of the Chao-Seader c o r r e l a t i o n are u s e f u l f o r p r e d i c t i n g vaporl i q u i d e q u i l i b r i u m i n a wide v a r i e t y o f process s.treams; however, t h i s c o r r e l a t i o n cannot be used t o p r e d i c t thermal o r p h y s i c a l prop e r t i e s . The Soave and Peng-Robinson equations are steps i n the r i g h t d i r e c t i o n , but f u r t h e r work i s needed to f i l l the needs of industry. A few of the equations o f s t a t e that have widespread use i n i n d u s t r i a l VLE c a l c u l a t i o n s are discussed below. The comments, which deal mainly f o r the aforementioned category 1, "The Research 0

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

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

TEMPERATURE, °F Figure 1. Ethylene — enthalpy of saturated liquid

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

7.

ADLER ET AL.

Industrial Uses of Equations of State

159

11

Aspects , i n Table I , are based on personal experience and i n f o r m a t i o n obtained from other i n d u s t r i a l sources. Again no attempt has been made t o p r o v i d e a t e c h n i c a l t r e a t i s e on each equation employed by i n d u s t r y . Benedict-Webb-Rubin Equation. The o r i g i n a l form o f the BWR equation contained e i g h t constants. I t was intended p r i m a r i l y f o r c o m p r e s s i b i l i t y , v a p o r - l i q u i d e q u l i b r i u m and enthalpy, although the r e l a t i o n s h i p s f o r the other thermodynamic f u n c t i o n s were publ i s h e d . I t was used f o r the phase e q u i l i b r i a o f hydrocarbon systems i n the temperature range of 26 F t o 400 F and the pressure range of 65 t o 2000 p s i a . The e r r o r i n the K constants was reported as 3.4% f o r t h i s range o f c o n d i t i o n s . I t should be emphas i z e d that hydrogen was not present i n the systems s t u d i e d and good r e s u l t s cannot be obtaine BWR form i s used. The equation a p p l i e s e q u a l l y w e l l t o l i q u i d s and vapors; the f i r s t equation of s t a t e known by the authors to apply to both phases, and, almost f o r t y years l a t e r , i s s t i l l one of the most capable f o r doing so. Because i t s a n a l y t i c a l form p e r m i t t e d a p p l i c a t i o n to m i x t u r e s , and allowed the r e q u i r e d mathematical operations f o r o b t a i n i n g a l l the thermodynamic p r o p e r t i e s i n a thermodynamically c o n s i s t e n t manner, the BWR equation a t t r a c t e d widespread i n d u s t r i a l a t t e n t i o n . I f components were e i t h e r above t h e i r normal b o i l i n g p o i n t s or present i n s m a l l amounts, the mixture enthalpy p r e d i c t i o n s were e x c e l l e n t . A t lower temperatures, enthalpy r e s u l t s were sometimes found to be so poor that a c o l d stream i n a heat exchanger could be computed to e x i t a t a temperature lower than i t went i n — " a negative s p e c i f i c heat '! (See F i g u r e 1) This problem was e l i m i n ated by p e r m i t t i n g the constant C t o change w i t h temperature (Barner and A d l e r , ( 1 1 ) ) . This i n f a c t , added two more constants. S t a r l i n g (7) added an eleventh constant and i n c l u d e d a b i n a r y i n t e r a c t i o n c o e f f i c i e n t i n the mixing r u l e s which extended the range of a p p l i c a b i l i t y o f the equation. With the greater number of constants and the e l i m i n a t i o n o f low temperature negative s p e c i f i c heats, came new problems. Cons t a n t s could no longer be obtained from pure component p r o p e r t i e s . Experimental e q u i l i b r i u m data, d e n s i t y data, and enthalpy data f o r mixtures had to be i n c l u d e d i n the r e g r e s s i o n procedure f o r o b t a i n i n g the constants. To develop a more u s e f u l s e t o f constants f o r a p a r t i c u l a r component, as many mixtures as p o s s i b l e c o n t a i n i n g that component should be i n c l u d e d . Experience has shown that two s e t s of almost i d e n t i c a l cons t a n t s can g i v e d i v e r s e r e s u l t s when compared to experimental measurements o f K's and v i c e v e r s a . As shown i n Table I I , s m a l l d i f ferences i n only f i v e of the eleven BWR constants can l e a d to s i g n i f i c a n t changes i n the p r e d i c t e d K s . The o p p o s i t e trend i s observed i n Table I I I f o r a t h i r d s e t o f constants. Although four of the c o n s t a n t s , B , D , E , and d, d i f f e r a p p r e c i a b l y i n magnitude 1

Q

T

Q

Q

Q

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

160

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

TABLE I I COMPARISON OF RESULTS OF TWO SIMILAR SETS OF BWR-11 CONSTANTS FOR ETHYLENE

% Deviation From Experimental Data Property

System

Set 1

Set 2

Density

Ethylene

0.62

0.72

Vapor Pressure

Ethylen

9.24

0.76

K-Value

Methane-Ethylen l K

Ethane-Ethylene

C2 K 2 K 2

10.70

5.98

0.90

0.88

1.58

0.84

9.60

3.97

10.65

15.26

C

C

Hydrogen-Ethylene K

BWR- •11 Constants f o r C

2

(Ethylene)

Set 1

Set 2

Difference

,-10 C xlO

.180628

.183101

1.4%

Y x l Or

1

.227978

.228344

0.2

b x l Or

1

.257883

.265697

3.0

a x l Or

5

.156497

.158859

1.5

.604935

.604567

Q

-0.0006

E , c, d a r e i d e n t i c a l o

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

7.

Industrial Uses of Equations of State

ADLER E T A L .

161

TABLE I I I COMPARISON OF RESULTS OF TWO DIFFERENT SETS OF BWR-11 CONSTANTS FOR ETHYLENE

% Deviation From Experimental Data System

Property

Set 2

Set 3

Density

Ethylene

0.72

0.63

Vapor Pressure

Ethylene

0.76

1.70

K-Value

Methane-Ethylen 5.98

7.76

0.88

1.24

0.84

2.01

3.97

4.19

15.26

14.96

C

E thane-E thy1ene

2

K

C

2 Hydro gen-Ethyl ene KiH C

2

2

BWR-11 Constants f o r C,

B o D xlO" o d xlO" E

xlO"

1 1

6

1 1

Set 2

Set 3

Difference

.593445

.445599

-24.9%

.821161

.745192

9.3

.845194

1.067700

26.3

.263014

.404265

53.4

:A , C , a, b, c, y, a are w i t h i n 5% o o

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

162

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

the p r e d i c t i o n s are of s i m i l a r accuracy. A f i n a l example i s given i n Table IV f o r the methane-propane system. The S t a r l i n g r e s u l t s are based on the g e n e r a l i z e d form of the BWR-11 equation. The improvement gained by using Exxon's constants, which were obtained from more s o p h i s t i c a t e d r e g r e s s i o n techniques and a w i d e r , m u l t i property range o f input data, i s evident. I t should a l s o be mentioned that the BWR-11 equation causes some problems i n the convergence of computer f l a s h programs that might not be observed f o r some of the s i m p l e r equations of s t a t e . To some degree these problems have been d e t a i l e d i n the e a r l i e r s e c t i o n on w o r k a b i l i t y . Once these problems are s o l v e d , the r e s u l t s obtained are q u i t e worthwhile. For one such example see Table V which gives the r e s u l t s f o r a mixture c a l c u l a t i o n i n the cryogenic r e g i o n , one of the areas where BWR equatio accuracy of Table VI f o as occur i n petroleum r e s e r v o i r work. Besides Pullman K e l l o g g , o r g a n i z a t i o n s such as Exxon, Chicago Bridge and I r o n Works, Northern N a t u r a l Gas Corp., Union Carbide and U n i v e r s i t y o f Oklahoma a l s o use and probably p r e f e r the BWR equation of s t a t e . As a f i n a l comment on the BWR equation, we quote Hopke and L i n (12) on the problem of the determination of the BWR constants: In our r e g r e s s i o n work, we found that attempts to f i t pure component and b i n a r y mixture data alone w i l l not y i e l d a unique s e t of o p t i m a l BWR s parameters f o r a given component. On the c o n t r a r y , many w i d e l y d i f f e r e n t parameter s e t s can g i v e about the same f i t to the data. Moreover, the d i f f e r e n t parameter sets w i l l y i e l d d i f f e r e n t r e s u l t s when used to p r e d i c t thermodynamic p r o p e r t i e s a t c o n d i t i o n s o u t s i d e of the temperature and p r e s s u r e range of the data base used t o determine the parameter s e t s . A l s o , e x t r a p o l a t i o n o f l i g h t hydrocarbon parameters to o b t a i n e s t i mates of parameters f o r h e a v i e r hydrocarbons i s i m p o s s i b l e unless a unique s e t of optimal parameters i s o b t a i n e d . " " i n t h i s work we developed a new r e g r e s s i o n procedure to o b t a i n a unique s e t of o p t i m a l parameters f o r i s o butane, normal butane, iso-pentane, normal pentane and carbon d i o x i d e . This new procedure, which i s d e s c r i b e d i n t h i s paper, i n v o l v e s using multicomponent K-value data to determine one of the eleven BWR pure component parameters, and then r e g r e s s i n g on the remaining ten parameters u s i n g d e n s i t y , enthalpy, vapor p r e s s u r e and K-value data f o r pure components and b i n a r y m i x t u r e s . I n c l u d i n g these multicomponent data has the e f f e c t o f extending the temperature range of the data base to lower temperatures." !l

T

Soave Equation

the

The Soave equation (13) i s one of the many m o d i f i c a t i o n s of o r i g i n a l Redlich-Kwong equation o f s t a t e . As shown below,

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

7.

ADLER E T A L .

Industrial Uses of Equations of State

163

TABLE IV COMPARISON OF BWR-11 RESULTS: EXXON'S 1974 CONSTANTS STARLING'S GENERALIZED CONSTANTS

Absolute Average D e v i a t i o n Exxon (1974)

Property

Starling

Pure Component: Density

Methane

0.60%

2.04%

Enthalpy

Methane

0.78 B t u / l b

1.84 B t u / l b

Propane

1.40 B t u / l b

1.31 B t u / l b

Methane Propane

0.63% 0.94%

1.23% 2.43%

MethanePropane

0.76%

1.87%

MethanePropane

1.30 Btu/lb

3.64 Btu/lb

K

1.94%

9.58%

4.41%

13.74%

Vapor Pressure Mixture: Density

Enthalpy

K-Value

C-1 K C

3

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

164

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES IN C H E M I C A L INDUSTRY

TABLE V CRYOGENIC PHASE EQUILIBRIA PREDICTED BY BWR-11 He-N -C System a t 2000 p s i a

K = y/x Temp., °F -306.7

Component He N

-297.7

He N C

-288.7

2

2

38.7 0.028

30.5 0.044

He 2

Cl He N

Calc'd 37.5 0.029

30.0 0.045 0.002

l

N

-279.7

Expt'l**

2

Cl

24.3

24.5

0.068

0.067

0.007

0.004

19.5

20.2

0.10

0.097

0.012

0.008

* Indeterminate Smoothed Experimental Data by Boone, DeVaney, and Stroud, Bur. of Mines, RI-6178 (1963)

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

7.

165

Industrial Uses of Equations of State

ADLER ET AL.

TABLE V I PHASE EQUILIBRIA AT RESERVOIR CONDITIONS BY BWR-11

T = 120F

P = 3566 p s i a

L i q u i d (x) Component

T

Expt l

Cal'd

K = y/x

Vapo r (y) T

Expt l

Cal d

f

Expt'l

Cal'd

Cl

0.528

0.511

0.907

0.913

1.72

1.78

c

0.045

0.045

0.038

0.037

0.838

0.832

C3

0.029

0.031

0.016

0.016

0.557

0.514

c

0.029 0.026

0.030

0.007

0.005

0.272

0.163

0.032

0.038

0.006

0.004

0.198

0.093

0.312

0.311

0.014

0.016

0.044

0.052

2

4

C

5

C6

c

7

+

T = 200 F

P = 4957 p s i a

l

0.594

0.593

0.883

0.862

1.49

1.45

C2 C

0.041

0.041

0.040

0.039

0.966

0.938

3

0.024

0.026

0.019

0.018

0.786

0.711

C

4

0.023

0.027

0.014

0.014

0.631

0.517

0.018

0.023

0.009

0.008

0.489

0.372

0.022

0.029

0.009

0.008

0.411

0.271

0.262 0.278 T = 200F

0.026

0.092

0.195

Cl

0.679

0.727

0.840

0.815

1.24

1.12

c

2

0.040

0.040

0.038

0.039

0.959

0.970

C

3

0.021

0.022

0.018

0.020

0.874

0.886

C4

0.020

0.020

0.015

0.016

0.751

0.795

C5

0.015

0.015

0.020

0.011

0.648

0.709

0.018

0.019

0.010

0.012

0.540

0.634

0.208

0.157

0.059

0.088

0.284

0.563

C

C

5 C6 C

c

7+

6

Mixture C r i t i c a l : E x p t ' l Data:

T = 128F c

P

p

c

=

0.051 6740 p s i a

= 3450 p s i a

Roland, C , IEC, 37_, 930 (1945)

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

166

P H A S E EQUILIBRIA AND

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

Soave has i n t r o d u c e d a temperature-dependent a t t r a c t i v e f o r c e term i n t o the equation: RT_ _ a(T) V-b V(V + b)

. K

a(T) i s c a l c u l a t e d from the f o l l o w i n g e x p r e s s i o n : R2T 2 a(T) = 0.42747 { — ~ } a(T)

J

(7)

where, a(T)

0 , 5

= 1.0 + (0.480 + 1.574co

2

- 0.176a) ) (1 - T **) .

(8)

For m i x t u r e s , a =

(I x

b = Z x

±

a

±

)

b .

(9) (10)

For u n l i k e i n t e r a c t i n g p a i r s , a.. = (1 - K . . ) ( a . a . ) ^ !J i j i 3 where K^j = 0.0 f o r hydrocarbon-hydrocarbon interactions. The comments i n the next few paragraphs are based on the work done a t Pennsylvania S t a t e U n i v e r s i t y , where, under the auspices of the API Subcommittee on T e c h n i c a l Data, the API T e c h n i c a l Data Book (14) was r e v i s e d . The Penn State group presented s i x r e p o r t s (15) on v a p o r - l i q u i d e q u i l i b r i u m . A f t e r thoroughly checking t h e i r e q u i l i b r i u m data s e t f o r thermodynamic c o n s i s t e n c y , they t e s t e d a number of v a p o r - l i q u i d e q u i l i b r i u m p r e d i c t i o n methods w i t h the b i n a r y hydrocarbon-hydrocarbon p o r t i o n of the data s e t and concluded t h a t the Soave equation and Peng-Robinson equation (16) were the most accurate and the g e n e r a l i z e d of the a v a i l a b l e methods. Additional testing w i t h a l a r g e amount of t e r n a r y and some higher-order hydrocarbon mixture K data i n d i c a t e d that the use of the Soave equation gave the best r e s u l t s . With respect to i n d u s t r i a l a p p l i c a t i o n , the f o l l o w i n g three questions r e g a r d i n g the Soave equation are p e r t i n e n t : 1. I s i t usable i n the cryogenic region? 2. Does i t apply to mixtures c o n t a i n i n g i n o r g a n i c gases? 3. What problems do petroleum f r a c t i o n s present? With regard to p o i n t one, the Penn S t a t e group found t h a t the o r i g i n a l , unmodified Soave equation cannot approach the accuracy of the BWR equation i n the cryogenic r e g i o n . I n a d d i t i o n , they i n t e n d to recommend the BWR f o r cryogenic systems i n the r e v i s e d v a p o r - l i q u i d e q u i l i b r i u m chapter of the Data Book. R e f e r ence w i l l probably be made to S t a r l i n g ' s 1 1 - c o e f f i c i e n t BWR work.

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

7.

ADLER E T A L .

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167

The o r i g i n a l Soave equation i s not a p p l i c a b l e to mixtures c o n t a i n i n g hydrogen. As shown i n one API r e p o r t on v a p o r - l i q u i d e q u i l i b r i u m (15), t h i s f a i l u r e occurs because a t high reduced temperatures a(T) approaches zero and the a t t r a c t i v e term i n the Soave equation vanishes. Because i n p r a c t i c a l s i t u a t i o n s hydrogen K values are r e q u i r e d a t l a r g e reduced temperatures, the Penn State group i n t r o d u c e d a r e v i s e d a-expression f o r hydrogen. The improvement was t h r e e - f o l d w i t h regard to hydrogen K value p r e d i c t i o n , and the r e v i s e d expression was shown to be e x c e l l e n t f o r both b i n a r y and multicomponent mixtures c o n t a i n i n g hydrogen. The Penn S t a t e group developed p a i r i n t e r a c t i o n terms, j , f o r mixtures c o n t a i n i n g i n o r g a n i c gases. These values were obtained from a l a r g e s e t of b i n a r y e q u i l i b r i u m data and some s o l u b i l i t y data. The v a p o r - l i q u i d e q u i l i b r i u m data were however given much more weight than th dures. In a l l cases, th by a f a c t o r of four w i t h these m o d i f i c a t i o n s . These i n t e r a c t i o n c o e f f i c i e n t s were found to be r e l a t i v e l y independent o f both temperature and pressure, and could be g e n e r a l i z e d as a f u n c t i o n of the absolute s o l u b i l i t y parameter d i f f e r e n c e between the i n t e r a c t i n g p a i r s . ^ i ^ i n o r g . "~ ^HCI• Unique expressions are r e q u i r e d f o r CO2 HC p a i r s , H2§-HC p a i r s , and N2-HC p a i r s . I n a l l other cases, i n c l u d i n g H2, K^. = 0.0. Multicomponent system c a l c u l a t i o n s based on these i n t e r a c t i o n parameters were found to agree w e l l w i t h the experimental data. The Soave equation, w i t h the m o d i f i c a t i o n s made by the Penn State group does very w e l l f o r systems that are o f i n t e r e s t t o a wide c r o s s - s e c t i o n o f the chemical i n d u s t r y . Another p o i n t i n favor of the Soave equation i s that no a d d i t i o n a l a d j u s t a b l e parameters are r e q u i r e d beyond T , p , to, and 6 f o r each compound. For companies using the Chao-Seader method, where 6 i s already a v a i l a b l e i n computer s t o r a g e , t r a n s i t i o n to Soave i s c e r t a i n l y advantageous. The Penn S t a t e group has not answered p o i n t three i n any o f t h e i r r e p o r t s . However, t h i s p o i n t was d i s c u s s e d a t the 1976 API Sub-committee meeting i n Washington, and i t was f e l t that petroleum f r a c t i o n s probably would not cause any p a r t i c u l a r problems w i t h regard t o the s o l u b i l i t y parameter c o r r e l a t i o n needed i n the presence of i n o r g a n i c gases. N e v e r t h e l e s s , i t i s b e l i e v e d t h a t each company must draw t h e i r own c o n c l u s i o n i n t h i s area. T e s t i n g a few t y p i c a l r e f i n e r y type mixtures would s u f f i c e . =

c

c

Chao-Seader C o r r e l a t i o n . Reference was made e a r l i e r to the w e l l known and much used Chao-Seader c o r r e l a t i o n f o r the p r e d i c t i o n of v a p o r - l i q u i d e q u i l i b r i u m f o r p r i n c i p a l l y hydrogen-hydrocarbon systems w i t h s m a l l amounts o f CO2, H2S, 02> N2, e t c . The h e a r t of the c o r r e l a t i o n c o n s i s t s of s e v e r a l equations to represent l i q u i d f u g a c i t y . The other two c o n s t i t u e n t s , the Scatchard-Hildebrand equation f o r a c t i v i t y c o e f f i c i e n t s and the Redlich-Kwong equation f o r the vapor-phase n o n i d e a l i t y , were already w e l l e s t a b l i s h e d .

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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A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

10.0

Figure 2.

Comparison of Chao-Seader and Grayson-Streed liquid fugacity coefficient correction terms

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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169

Because o f i t s s i m p l i c i t y and g e n e r a l i t y , the o r i g i n a l c o r r e l a t i o n and i t s m o d i f i c a t i o n s by Cavett (17) and Grayson and Streed (18) have found wide a p p l i c a t i o n s i n petroleum and gas p r o c e s s i n g indust r i e s . However, a r t i c l e s , too many to enumerate, have p o i n t e d out the weaknesses i n the c o r r e l a t i o n . Some o f our o b s e r v a t i o n s are l i s t e d below: 1. The l i q u i d f u g a c i t y c o e f f i c i e n t e x p r e s s i o n does not reproduce pure component vapor pressures a t s a t u r a t i o n c o n d i t i o n s . This reason alone makes the c o r r e l a t i o n unsafe f o r the design of s e p a r a t i o n s o f c l o s e - b o i l i n g components. For such designs as the s e p a r a t i o n o f i s o pentane from normal pentane i n a g a s o l i n e i s o m e r i z a t i o n u n i t , a 5% e r r o r i n i-C5 vapor pressure can change the number o f t r a y r e q u i r e d i colum b 33% (Thi i s assuming an and a p p l y i n g a " r u l e of thumb that the approximate num ber o f t h e o r e t i c a l p l a t e s r e q u i r e d f o r a given s e p a r a t i o n of f i x e d r e f l u x r a t i o i s i n v e r s e l y p r o p e r t i o n a l to ( a - 1 ) ) . 2. I n the o r i g i n a l and Cavett v e r s i o n s o f the c o r r e l a t i o n , the r e a l - f l u i d c o r r e c t i o n term, i n the l i q u i d fugac i t y equation, v = v ( ^ ) v ( l ) , leads to p r o g r e s s i v e l y lower hydrocarbon K s w i t h i n c r e a s i n g temperature a t component reduced temperatures g r e a t e r than one. This occurs because the numerical value o f v ( l ) drops s u b s t a n t i a l l y at these c o n d i t i o n s as shown i n F i g u r e 2. One of the Grayson and Streed m o d i f i c a t i o n s on the o r i g i n a l ChaoSeader c o r r e l a t i o n was to f i x v ( l ) a t i t s c a l c u l a t e d value at T = 1.0 f o r a l l temperatures above the c r i t i c a l temperature o f a component, which i n c r e a s e d the a p p l i c a b i l i t y of the c o r r e l a t i o n from 500 F to 800 F. 3. I n g e n e r a l , w i t h the o r i g i n a l and the m o d i f i e d forms o f the Chao-Seader c o r r e l a t i o n , the p r e d i c t e d e q u i l i b r i u m K values o f the l i g h t components (those w i t h K s g r e a t e r than 1.0) are o f t e n too h i g h and those of the heavy components (species w i t h K s l e s s than 1.0) are too low, making the r e l a t i v e v o l a t i l i t y too l a r g e and the designs unsafe i n c e r t a i n a p p l i c a t i o n s . More s p e c i f i c a l l y , K s f o r components w i t h reduced temperatures greater than 0.9 a r e u s u a l l y overestimated, and K's f o r components w i t h temperatures l e s s than 0.9 are u s u a l l y underestimated. As shown i n F i g u r e 3 f o r the Grayson-Streed c o r r e l a t i o n these overand underestimations may reach i n t o l e r a b l e l e v e l s as the system pressure i n c r e a s e s . A c l a s s i c example of the problem o f underestimation of heavy component K values can be seen i n F i g u r e 4, which i s taken from a paper by Robinson and Chao (19). R e f e r r i n g to t h i s f i g u r e , they comment: "The 0 F temperature corresponds to T = 0.472 f o r n-heptane; and T = 0.411 a t -60 F. The comparison appears to be t y p i c a l of the heavy substances a t low T . The c a l c u l a t e d K values are g e n e r a l l y lower than the a v a i l a b l e a >

T

R

f

T

f

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.

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Figure 3. Percent deviation in Grayson-Streed K-value predictions vs. reduced pressure at different reduced temperatures for paraffin-paraffin binary systems

100

1000 PRESSURE, psia

Figure 4. n-Heptane in binary mixtures with methane

• DATA (Chang et ai. 1966) —

ROB INSON-CHAO CORRELATION

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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171

experimental data, even a t the lowest pressure that i s reported i n the experimental work, i . e . 100 p s i a . The cause of the d e v i a t i o n remains unresolved a t t h i s time." M u l t i e q u a t i o n Approach to V a p o r - L i q u i d E q u i l i b r i a . The c o r r e l a t i o n s mentioned e a r l i e r were developed s p e c i f i c a l l y f o r hydrocarbon systems and, i n g e n e r a l , a r e not a p p l i c a b l e to systems c o n t a i n ing p o l a r and a s s o c i a t i n g components. The v a p o r - l i q u i d e q u i l i b r i u m c o r r e l a t i o n s f o r systems w i t h such components are best handled w i t h a m u l t i - e q u a t i o n o f s t a t e procedure u s i n g Eq. ( 5 ) . This method i s a l s o used i n developing v a p o r - l i q u i d e q u i l i b r i u m c o r r e l a t i o n s f o r the design o f s e p a r a t i o n u n i t s f o r c l o s e - b o i l i n g hydrocarbons. The s e p a r a t i o n of the vapor and l i q u i d f u g a c i t i e s and the a c t i v i t y c o e f f i c i e n t s i n the fundamental e q u i l i b r i u m r e l a t i o n s h i p a l l o w great f l e x i b i l i t y , an t i o n o f the thermodynami e s t i m a t i o n o f each of these q u a n t i t i e s . For the vapor f u g a c i t y c o e f f i c i e n t any of the equations o f s t a t e mentioned e a r l i e r o r some o t h e r , such as the v i r i a l equation, may be used. I n the l a t t e r case, the v i r i a l c o e f f i c i e n t s may be determined e x p e r i m e n t a l l y o r estimated u s i n g three- o r four-parameter g e n e r a l i z e d c o r r e l a t i o n s . The standard s t a t e f u g a c i t y of the pure l i q u i d , o f course, i s estimated from the exact thermodynamic r e l a t i o n s h i p : rO

f

L

O

= p

.O

r

(j) exp {

V

(TT R

T

- p°) -, >

/

r, s 0

(12)

where p°, <j)°, and V a r e , r e s p e c t i v e l y , the vapor pressure, f u g a c i t y c o e f f i c i e n t a t the standard s t a t e , and the p a r t i a l molar l i q u i d volume of the component, the l a t t e r u s u a l l y taken to be equal to the s a t u r a t e d l i q u i d molar volume o f the pure component. When any o f the components i s above i t s c r i t i c a l temperature, the pure l i q u i d fugac i t y may be estimated (a) by equation(s) w i t h l i m i t e d e x t r a p o l a t i o n of vapor p r e s s u r e , (b) from a g e n e r a l i z e d e m p i r i c a l equation such as Chao-Seader (or Grayson-Streed), o r (c) by b a c k - c a l c u l a t i o n from experimental v a p o r - l i q u i d e q u i l i b r i u m data. A l t e r n a t i v e l y , one can use Henry s constants w i t h unsymmetric convention f o r a c t i v i t y coeff i c i e n t s , thereby e l i m i n a t i n g the need f o r h y p o t h e t i c a l l i q u i d fugac i t i e s f o r pure components, as i n the Chueh-Prausnitz c o r r e l a t i o n . The e m p i r i c a l equations proposed f o r the c o r r e l a t i o n of the a c t i v i t y c o e f f i c i e n t s a r e c e r t a i n l y d i v e r s e . I n a d d i t i o n to the c l a s s i c Margules and Van Laar equations, there are l o c a l composition equations such as Wilson, nonrandom t w o - l i q u i d (NRTL), H e i l , UNIQUAC, and many others i n both c l a s s e s . Each equation used, whether f o r <J)°, fg , o r y, has i t s p a r t i c u l a r advantages and disadvantages, and l i m i t a t i o n s on i t s range o f a p p l i c a b i l i t y ; these and other f a c t o r s i n f l u e n c e the s e l e c t i o n o f the equations, o r p a r t i c u l a r combination of equations used. F o r example, i n many K e l l o g g design a p p l i c a t i o n s the f o u r - s u f f i x Margules form o f the Wohl equation f o r a c t i v i t y c o e f f i c i e n t s i s T

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

172

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F L U I D PROPERTIES IN

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INDUSTRY

p r e f e r r e d because of our e x t e n s i v e l i b r a r y of constants developed through past experience, f a m i l i a r i t y w i t h i t s behavior, the f l e x i b i l i t y o f f e r e d by the a d d i t i o n of constants, and i t s a p p l i c a b i l i t y to i m m i s c i b l e systems as w e l l as m i s c i b l e ones. The multicomponent form of t h i s equation, which may be seen i n some p u b l i c a t i o n s occupying h a l f of the page, can be reduced to two terms as shown below: n n n n n n n log y. = 4 Z Z Z x.x. Xp3. « - 3 Z Z Z Z x.x.x. x«3 . ., « j - i k - i £=1 ^ i«i j - i k=l £-1 ^ (13) 1

1

l

j

U

1

1

l j k £

i s

where n i s the number of components and $ i j k c r e l a t e d to the b i n a r y Wohl c o n s t a n t s , A s and D s, and the t e r n a r y constant, C*, depending on the values of the i n t e g e r s i , j , k, and £, (Adler et a l . , ( 2 0 ) ) . L i s t e d below are som s t a t e approach was used i n c o r r e l a t i n g v a p o r - l i q u i d e q u i l i b r i u m data needed a t Pullman K e l l o g g e i t h e r f o r process development s t u d i e s or f o r the design of commercial s c a l e p l a n t s . They have been chosen to show the wide spectrum of components. In a l l cases the Wohl equation was used f o r the a c t i v i t y coefficients. T

T

High Pressure A p p l i c a t i o n s : Isopropanol - I s o p r o p y l Ether - Water - Propylene H2 - N2 - CH4 - A or He H - N - CO - C H Ethane - Ethylene Propane - Propylene Propane - Propylene - Propadiene - Methylacetylene C h l o r i n a t e d Hydrocarbons - HC1 2

2

2

6

Low Pressure A p p l i c a t i o n s : Oxygenated Hydrocarbons HCN - Water H e r b i c i d e P l a n t Data Table V I I shows r e s u l t s f o r the f i r s t system, i s o p r o p a n o l i s o p r o p y l ether - water - propylene, i n which the experimental compositions i n each of the three phases are compared w i t h the values p r e d i c t e d by the method j u s t d e s c r i b e d . A m o d i f i e d R e d l i c h Kwong equation of s t a t e f o r vapor f u g a c i t y , Chao-Seader equation w i t h a d j u s t e d parameters f o r l i q u i d f u g a c i t y , and the Wohl equation for the a c t i v i t y c o e f f i c i e n t s were used. The p r e d i c t i o n s were based only on data f o r b i n a r y systems. Practical Applications In the previous s e c t i o n the use of equations of s t a t e 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 p r e d i c t i o n s was emphasized. This p o r t i o n of the paper expands on t h i s t o p i c w i t h some s p e c i f i c v a p o r - l i q u i d e q u i l i b r i u m example c a l c u l a t i o n s , and a l s o deals w i t h a number of

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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173

TABLE V I I PROPYLENE - IPA

IPE - WATER THREE-PHASE EQUILIBRIA 220 F, 600 p s i a

Hydrocarbon Liquid-Phase Composition (Mole F r a c t i o n ) Compound

Exp.

Propylene

0.767

0.750

IPE

0.069

0.067

Calc.

IPA Water

Aqueous Liquid-Phase Composition (Mole F r a c t i o n ) Exp.

Calc.

Propylene

0.002

0.003

IPE

0.0002

0.0002

IPA

0.033

0.038

Water

0.965

0.959

Vapor-Phase Composition (Mole F r a c t i o n ) Exp.

Calc.

Propylene

0.922

0.910

IPE

0.021

0.021

IPA

0.032

0.035

Water

0.025

0.033

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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EQUILIBRIA

AND

FLUID PROPERTIES

IN

CHEMICAL

INDUSTRY

other i n d u s t r i a l a p p l i c a t i o n s of equations of s t a t e . Most of these a p p l i c a t i o n s f a l l i n t o category two of Table I. The d i s c u s s i o n on f r e e z e - o u t problems, however, overlaps category one, The Research Aspects of I n d u s t r i a l Work i n Equations of S t a t e . Obviously every i n d u s t r i a l a p p l i c a t i o n of equations of s t a t e can not be covered i n t h i s s e c t i o n of the paper. Thus, f o r examp l e , c a l c u l a t i o n of Joule-Thompson c o e f f i c i e n t s , heats of mixing, and the v e l o c i t y of sound, and o b t a i n i n g v a r i o u s other parameters r e q u i r e d i n compressor c a l c u l a t i o n s , which are an i n t e g r a l p a r t of i n d u s t r i a l design, w i l l not be d i s c u s s e d . C r i t i c a l P r o p e r t i e s From Equations of S t a t e . In i n d u s t r i a l work c a l c u l a t i n g the c r i t i c a l locus of a mixture i s f r e q u e n t l y necessary. Because t h i s i s done at perhaps 0.05 mole f r a c t i o n i n t e r v a l s f o r a b i n a r y system l a t i o n demands as d i r e c p o s s i b l e . This precludes s o l v i n g simultaneous equations, such as s e t t i n g the second and t h i r d d e r i v a t i v e s of the f r e e energy w i t h r e s p e c t to composition equal to zero as i n d i c a t e d below: {^f>

= 0

(14)

= 0.

(15)

T,P

A} T,p

This i s the c l a s s i c a l thermodynamic approach to s o l v i n g t h i s problem. Rather, an equation of s t a t e and s e m i e m p i r i c a l c o r r e l a t i o n s are used; e.g. the Chueh-Prausnitz c o r r e l a t i o n scheme (10). In t h i s approach the c r i t i c a l pressure of the mixture i s obtained i n d i r e c t l y from a m o d i f i e d v e r s i o n of the Redlich-Kwong-Chueh equat i o n of s t a t e a f t e r T and V have been obtained d i r e c t l y from q u a d r a t i c mixing r u l e s which employ the r e s p e c t i v e Chueh-Prausnitz i n t e r a c t i o n parameters, T j ^ ^ 1 2 ^ p l based on t h i s method i s given i n F i g u r e 5 f o r the ethane-n-heptane b i n a r y mixture. Agreement i s r a t h e r good except i n the immediate v i c i n i t y of the maximum c r i t i c a l pressure of the mixture. A member of the T e c h n i c a l Data Group at Pullman K e l l o g g has done some promising work on the p r e d i c t i o n of the c r i t i c a l propert i e s of mixtures (Spencer ( 2 1 ) ) . In h i s recommended procedure the true c r i t i c a l pressure i s obtained from the Redlich-Ngo equation of s t a t e which i s r e s t a t e d below i n a form v a l i d a t the c r i t i c a l p o i n t only: 3Z RT a P = cm cm _ m (16) cm ( _b ) T h V (V + b ) ' cm m cm cm cm m c m

c m

a n

v

#

n

e

x

a

m

e

K

V

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

7.

ADLER ET AL.

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175

I WO

Figure 5. Comparison of calculated and experimental critical pressures for the ethanen-heptane system

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

176

PHASE

EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

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where, Z

= Z x. Z . 1 ci

cm

(17)

and ^ and b are c a l c u l a t e d from Eqs. (9) and (10), r e s p e c t i v e l y . The t r u e c r i t i c a l temperature, T , and true c r i t i c a l volume, V , are obtained d i r e c t l y from c o r r e l a t i o n s f o r these p r o p e r t i e s that were developed i n the study. The optimum values of a^j ( i ^ j ) , which were obtained by r e g r e s s i o n of the a v a i l a b l e experimental b i n a r y mixture c r i t i c a l pressure data, have been c o r r e l a t e d as a g e n e r a l i z e d f u n c t i o n of a s i z e parameter, the molecular weight r a t i o , (MWR) and shape parameter, and the moment of i n e r t i a r a t i o m

cm

(MIR)

c m

.

a

1 2

=

[A

+

B (MWR

A unique s e t of c o e f f i c i e n t s i s r e q u i r e d f o r both b i n a r y methane and methane-free i n t e r a c t i o n s . The o v e r a l l c r i t i c a l pressure r e s u l t s obtained from the modif i e d Redlich-Ngo Procedure are g i v e n i n Table V I I I . The Chueh-Prausn i t z entry was obtained u s i n g the o r i g i n a l (published) v e r s i o n of that c o r r e l a t i o n (10). The K r e g l e w s k i equation, a s e m i e m p i r i c a l d i r e c t p r e d i c t i o n method, i s the recommended c o r r e l a t i o n f o r the c r i t i c a l pressure i n the API T e c h n i c a l Data Book. As i s c l e a r l y e v i d e n t , the Redlich-Ngo procedure i s s u p e r i o r to the other methods. Although p r e l i m i n a r y r e s u l t s have i n d i c a t e d that t h i s technique i s e x c e l l e n t f o r multicomponent m i x t u r e s , a d d i t i o n a l study i s needed, p a r t i c u l a r l y f o r mixtures c o n t a i n i n g h e a v i e r components such as petroleum f r a c t i o n s , b e f o r e t h i s c o r r e l a t i o n can be i n c o r p o r a t e d i n everyday i n d u s t r i a l design c a l c u l a t i o n s . Volumetric P r o p e r t i e s From an Equation of S t a t e . I n g e n e r a l , most equations of s t a t e g i v e r e l a t i v e l y good p r e d i c t i o n s of v o l u m e t r i c p r o p e r t i e s a t h i g h temperature and low pressure. However, near the s a t u r a t i o n envelope and e s p e c i a l l y i n the c r i t i c a l r e g i o n , v o l u m e t r i c p r e d i c t i o n s based on equations of s t a t e are poor, p a r t i c u l a r l y f o r the s a t u r a t e d l i q u i d . Therefore, w i t h the exception of 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 , where i n t e r n a l l y c o n s i s tent l i q u i d d e n s i t i e s are needed to c a l c u l a t e the l i q u i d f u g a c i t y , e m p i r i c a l l i q u i d d e n s i t y c o r r e l a t i o n s are normally used i n indust r i a l design c a l c u l a t i o n s . N e v e r t h e l e s s , w i t h more s o p h i s t i c a t e d equations of s t a t e , reasonable p r e d i c t i o n s of the s a t u r a t e d l i q u i d d e n s i t y can be obtained. Consider the r e s u l t s given i n Table IX. A number of e m p i r i c a l l i q u i d d e n s i t y c o r r e l a t i o n s and a l s o Pullman K e l l o g g s v e r s i o n of the BWR-11 equation of s t a t e have been evaluated w i t h a set of LNG bubble-point d e n s i t y data. [The most accurate LNG data g e n e r a l l y a v a i l a b l e are those of Klosek & McKinley (22), Shana a (23), M i l l e r & Rodesevitch (24), and M i l l e r (25)]. f

f

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

\DLER ET AL.

Industrial Uses of Equations of State

TABLE V I I I EVALUATION OF EQUATIONS FOR PREDICTING CRITICAL PRESSURE

Ave. Dev., P s i a Methane Methane-Free

Method M o d i f i e d Redlich-Ngo

90.0

11.1

Chueh-Prausnitz

278.0

28.6

K r e g l e w s k i (API)

411.3

18.8

TABLE IX RESULTS OF EVALUATION OF EQUATIONS FOR PREDICTING THE BUBBLE POINT DENSITY OF LNG

Correlation

Percent Avg. D e v i a t i o n *

BWR-11

2.81

Watson

3.33

Yu-Lu

0.29

Spencer-Danner

0.54

Chiu-Hsi-Lu # 1

0.58

Chiu-Hsi-Lu # 2

0.26

* Basis - 130 p o i n t s

(20 mixtures)

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.

UNSTABILIZED

LIGHT DISTILLATE

3

DESULFURIZED PRODUCT

Figure 6. Distillate hydroclesulfurization

IIO°F 940 psia

I

2

H S OFF GAS

STRIPPER

-*-H.P. PURGE

HIGH PRESSURE SEPARATOR^

HYDROGEN RECYCLE

REACTOR

7.

ADLER ET AL.

Industrial Uses of Equations of State

179

As i s e v i d e n t , the BWR-11 i s not as accurate as most of the e m p i r i c a l c o r r e l a t i o n s (exception Watson (26)) t h a t have been derived s p e c i f i c a l l y f o r p r e d i c t i n g l i q u i d density. C a l c u l a t i o n s i n v o l v i n g the design of storage areas, and e s p e c i a l l y custody t r a n s f e r d e f i n i t e l y r e q u i r e the more accurate e m p i r i c a l equations. Liquid-Vapor E q u i l i b r i u m (Design Case S t u d i e s ) . As mentioned e a r l i e r , equations o f s t a t e c e r t a i n l y have u t i l i t y i n i n d u s t r i a l design s t u d i e s r e q u i r i n g a knowledge o f the phase behavior. This s e c t i o n deals w i t h some c h a l l e n g i n g l i q u i d - v a p o r e q u i l i b r i u m design c a l c u l a t i o n s , where experience as w e l l as a workable and f l e x i b l e equation of s t a t e a r e r e q u i r e d . The f i r s t example deals w i t h a very unusual design s i t u a t i o n that has occurred twice during the l a s t year, m u l t i p l e b u b b l e - p o i n t s . In the f i r s t occurrenc designed. A simple flowshee o i l i s preheated i n a furnace and c a t a l y t i c a l l y r e a c t e d w i t h hydrogen to convert the hydrocarbon mercaptans to H 2 S , which i s subsequently s t r i p p e d o f f . A process engineer d i d a computer c a l c u l a t i o n i n v o l v i n g heating of a bubble-point l i q u i d (from the bottom of the high-pressure separator) from 110 F a t a design pressure of 940 p s i a , to feed the H2S s t r i p p i n g column, and then f l a s h i n g the stream. The comput e r output i n d i c a t e d that the stream was a l l l i q u i d ; t h a t i s , even though i t was a t i t s bubble-point i n i t i a l l y and was then heated to a 500 F higher temperature i t was s t i l l l i q u i d . The composition of the bubble-point l i q u i d i s given i n the following table. HYDRODESULFURIZER HIGH PRESSURE SEPARATOR EFFLUENT

Mole Percent

Mole Percent 2

3.2

300 NBP

0.36

HS

5.6

345 NBP

0.37

Cl

4.62

460 NBP

6.34

c

0.47

565 NBP

12.50

C3 C

0.57

645 NBP

16.28

0.46

705 NBP

16.37

100 NBP

0.05

760 NBP

15.04

155 NBP

0.28

805 NBP

9.82

245 NBP

0.54

855 NBP

7.14

H

2

2

4

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

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

1000 940 Psia 900--

/

LIQUID

/

IN C H E M I C A L INDUSTRY

T

800--

7 0 0 - - uT

O CALCULATED CHUEH-PRAUSNITZ CRITICAL POINT

LIQUID AND VAPOR 600- - o-

500 • -

400-

- GRAYSON-STREED PREDICTIONS - CHAO-SEADER PREDICTIONS • PROJECTED CRITICAL REGION BEHAVIOR TEMPERATURE,°F -+—I —I 600 800 200 400

1400

Figure 7. Pressure-temperature diagram for hydrodesulfurization effluent stream

700 NICHOL'S ET AL (1957) GRAYSON-STREED PREDICTIONS 600 'p CALC'D CRITICAL PT. * NICHOL'S EST. CRITICAL 500 - -

400-VAPOR 300--

200 100

Figure 8.

200

300

400

500

600

700

Pressure-temperature diagram for a 4.0 mol % H -96.0 mol % n-hexane system 2

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

7.

ADLER E T A L .

Industrial Uses of Equations of State

181

Although the mixture c o n s i s t s o f about 80% very heavy o i l s , the l i g h t components s t i l l i n f l u e n c e mixture behavior. I n f a c t , a very s m a l l amount of hydrogen can have a l a r g e e f f e c t on the phase envelope. The P-T diagram shown i n F i g u r e 7 i l l u s t r a t e s the s i t u a t i o n . The s o l i d bubble-point and dew-point l i n e s are based on the Grayson-Streed m o d i f i c a t i o n of the Chao-Seader c o r r e l a t i o n and the dashed bubble-point l i n e i s p r e d i c t e d from the Chao-Seader c o r r e l a t i o n . N o t i c e the very exaggerated minimum i n the p r e d i c t e d bubble-point curve. Indeed, the engineer was performing h i s c a l c u l a t i o n s i n a s i n g l e phase r e g i o n , and the computer was c o r r e c t ; i n f a c t , i f the stream was heated 200 F h i g h e r , o r t o about 810 F, a second bubble-point would have been reached. This behavior perhaps can be l a b e l e d r e t r o g r a d e h i g h e r than the p r e d i c t e However, i t i s a s s u r e d l y d i f f e r e n t from what one normally p e r c e i v e s to be i s o b a r i c r e t r o g r a d e v a p o r i z a t i o n ( i . e . , two bubble-points a t a f i x e d pressure above the c r i t i c a l pressure separated by a twophase r e g i o n ) . Because t h i s type o f behavior i s seldom d i s c u s s e d i n the l i t e r a t u r e , and a l s o because i t was our f i r s t exposure to the problem i n some time, an attempt was made to l o c a t e some experimental data to c o n f i r m the p r e d i c t e d e q u i l i b r i u m b e h a v i o r , and a l s o to convince the process engineer that the s i t u a t i o n could occur. Data f o r a 4% H i n hexane system are given i n F i g u r e 8. The phase envelope represented by the s o l i d l i n e s i s p r e d i c t e d from the Grayson-Streed c o r r e l a t i o n . The dashed l i n e represents the experimental phase envelope. Again n o t i c e the minimum i n the bubble-point curve, and that as many as three b u b b l e - p o i n t s , A, B, and C, can occur a t constant pressure above the true c r i t i c a l pressure of the mixture. An a d d i t i o n a l s e t o f experimental data are given i n F i g u r e 9 f o r v a r y i n g amounts of hydrogen i n a p e t r o leum naphtha. A t the lowest c o n c e n t r a t i o n of hydrogen the minimum i n the bubble-point curve extends below the c r i t i c a l p o i n t of the mixture. At the medium c o n c e n t r a t i o n , the phenomena occurs only at pressures h i g h e r than the c r i t i c a l p o i n t . A t the h i g h e s t conc e n t r a t i o n the phenomenon was not observed. A f t e r r e v i e w i n g other data, i t was concluded t h a t m u l t i p l e bubble p o i n t behavior may occur when hydrogen (or more g e n e r a l , any s l i g h t l y s o l u b l e gas) i s present i n the mixture a t about 7 mole percent o r l e s s . The BWR-11 equation of s t a t e , and the Chao-Seader, and GraysonStreed computational procedures were shown to be adequate f o r pred i c t i n g t h i s type of behavior. However, without some e x t e r n a l g u i dance, a commercial f l a s h r o u t i n e based on one of these c o r r e l a t i o n s w i l l not generate m u l t i p l e b u b b l e - p o i n t s . For example, cons i d e r the general case shown i n F i g u r e 10. A t the design pressure P P", i f the moles i n the vapor phase t o the t o t a l moles i n feed i s s e t equal to 0.0, i n d i c a t i n g a bubble-point c a l c u l a t i o n , a computer f l a s h program w i l l converge t o only one o f the bubblep o i n t s (A, B, C). For i n s t a n c e , i f the temperature i s i n i t i a l i z e d 2

f

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

182

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

100-

TIMPIRATUftl.*r Figure 9. Hydrogen-petroleum naphtha pressure-temperature diagram

LIQUID

LIQUIDVAPOR

Figure 10. Pressure-temperature diagram for a mixture having a minimum in the bubble point

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

7.

ADLER E T A L .

Industrial Uses of Equations of State

183

at F the f l a s h would probably converge t o bubble-point A. L i k e w i s e , i f the i n i t i a l guess was a t p o i n t E, the bubble-point a t B would be obtained. I n g e n e r a l , the process engineer i s not aware o f t h i s behavior and could be m i s l e d by the r e s u l t s . Therefore, c e r t a i n checks should be b u i l t i n t o a f l a s h program to a l l o w f o r the presence o f m u l t i p l e b u b b l e - p o i n t s . F o r i n s t a n c e , i f the design pressure exceeds the t r u e c r i t i c a l pressure, o r i f a s m a l l amount of hydrogen o r other s l i g h t l y s o l u b l e gas i s present i n the mixt u r e , the f l a s h v a p o r i z a t i o n c a l c u l a t i o n could be programmed so that a f t e r convergence i s obtained w i t h the i n i t i a l temperature guess, a d d i t i o n a l bubble-point c a l c u l a t i o n s are a u t o m a t i c a l l y attempted u s i n g a temperature guess twice the i n i t i a l temperature or even using the c r i t i c a l temperature i t s e l f . Using such a procedure, a l l three, and f o r some systems four bubble-points could be p r e d i c t e d . The Chao-Seade never converge to p o i n t be d e s c r i b e d l a t e r ) and i s an i n h e r e n t weakness o f both methods. For some m i x t u r e s , i f the BWR-11 o r BWR-8 equation i s used i n the f l a s h c a l c u l a t i o n , the bubble-point a t C can be obtained. However, i n some case s t u d i e s , i f the i n i t i a l temperature guess was D, the bubble-point a t B would be obtained r a t h e r than C. I n these cases, going one step f u r t h e r , even i f the i n i t i a l temperature guess was temperature C the program would converge on bubble-point B. This could be a problem i n the convergence p o r t i o n o f the f l a s h r o u t i n e i t s e l f o r perhaps a general s t a b i l i t y problem w i t h the equation of state. As mentioned e a r l i e r t h i s phenomenon occurred twice w i t h i n the past year, and the second time was c e r t a i n l y e a s i e r to understand. The second case i s shown i n F i g u r e 11. A process engineer was t r y i n g to confirm a 10 degree d i f f e r e n c e between the p r e d i c t e d bubble-point temperature A and a c l i e n t ' s value a t p o i n t B f o r a given design pressure ( s o l i d l i n e ) . The system i s the bottoms o f a n i t r o g e n wash tower which contains n i t r o g e n , methane, and a s m a l l amount of hydrogen. A s e r i e s o f computer f l a s h c a l c u l a t i o n s between A and B and then a t temperatures higher than B were made. For these c a l c u l a t i o n s the message "subcooled l i q u i d " was p r i n t e d out. By i n c r e a s i n g the temperature i n l a r g e increments a two phase r e g i o n was l o c a t e d a t p o i n t E. Because o f the aforementioned work i n t h i s area, m u l t i p l e bubble-point phenomena were suspected, and the phase envelope f o r the mixture was generated using the BWR-11 equation o f s t a t e . Because ( i ) the BWR-11 equation i s , i n g e n e r a l , e x c e l l e n t f o r a l l three components i n the mixture, ( i i ) our p r e d i c t e d c r i t i c a l p o i n t a t D f e l l i n l i n e w i t h the envelope, and ( i i i ) o f the experience gained from F i g u r e s 7 to 10, i t was f e l t t h a t a t r u e p i c t u r e of the n i t r o g e n mixture behavior was obtained. The second example deals w i t h a dew-point c a l c u l a t i o n f o r a t y p i c a l n a t u r a l gas mixture based on the BWR-11 equation of s t a t e . As shown i n F i g u r e 12, a t a design pressure o f 620 p s i a the s o l u t i o n a l g o r i t h m i n i t i a l l y diverged but f i n a l l y approached a s o l u t i o n i n the r e g i o n of the t r u e c r i t i c a l point of the mixture, -100 F . However, complete convergence was not obtained u n t i l 22 F, 122 F beyond the t r u e c r i t i c a l temperature of the mixture! I t ' s c e r t a i n l y In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

184

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

l \ I \ I \

1000" 900-.5

H -N -C, 2

X

s

2

BWR-11 PREDICTIONS

PREDICTED \CRITICAL POINT

800-

QL

uf or

SUBCOOLED LIQUID

700-

A

co 600-• CO _ LU

y

X SUPER\ HEATED

\A B

H

g

500400300-400

-350

-300

-250

-200

-150

TEMPERATURE, °F Figure 11.

Nitrogen wash tower bottoms phase diagram

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

-100

7.

ADLER ET AL.

Industrial Uses of Equations of State

185

p o s s i b l e to have a s o l u t i o n a t 22 F, because the r e t r o g r a d e bulge i n a n a t u r a l gas can extend a p p r e c i a b l y beyond the t r u e c r i t i c a l p o i n t of the mixture. For a s o l u t i o n c l o s e to the t r u e c r i t i c a l p o i n t as might be i n d i c a t e d by the r e s u l t s , the design pressure must exceed the t r u e c r i t i c a l pressure of the mixture, which i n t h i s case was 882 p s i a . F i g u r e 13 shows the experiment phase behavior f o r t h i s mixt u r e . +22 F i s an upper r e t r o g r a d e dew-point. I f the design pressure was lowered a second dew-point could a l s o be l o c a t e d a t a temperature of +22 F. This i s , of course, the lower dew-point pressure. The dew-point of a n a t u r a l gas i s very s e n s i t i v e to the heavy ends c h a r a c t e r i z a t i o n . Consider a n a t u r a l gas stream w i t h the f o l l o w i n g composition: Componen N

0.69

2

Ci

950.0

C

92.0

2

C3

C

12.0 +

4

<10.0

The d i f f e r e n t c h a r a c t e r i z a t i o n s o f the heavy ends, which make up l e s s than 1% o f the e n t i r e m i x t u r e , were considered. In Case 1 normal butane was s e l e c t e d t o represent the heavy ends, i n Case 2 normal pentane, and i n Case 3 normal hexane. The phase envelopes for the three cases were p r e d i c t e d w i t h the BWR-11 equation o f s t a t e . The r e s u l t s are given i n F i g u r e 14. The bubble-point curves almost c o i n c i d e , and the t r u e c r i t i c a l p o i n t s are not very s e n s i t i v e to composition. However, as shown, the dew-point i s g r e a t l y a f f e c t e d by the c h a r a c t e r i z a t i o n of the heavy ends compos i t i o n , and there i s about a 70 F d i f f e r e n c e i n the maximum dew-points of the three m i x t u r e s . Therefore, when working w i t h a n a t u r a l gas type system, i f the breakdown o f the C+ f r a c t i o n i n t o a d d i t i o n a l compounds i s a v a i l a b l e , i t should be used, e s p e c i a l l y i n f l a s h c a l c u l a t i o n s , to get an accurate r e p r e s e n t a t i o n of the phase behavior. The l a s t example deals w i t h some troublesome f l a s h c a l c u l a t i o n s which were based on the Chao-Seader c o r r e l a t i o n . In a p a r t i c u l a r design study a dew-point c a l c u l a t i o n and bubble-point c a l c u l a t i o n f a i l e d to converge f o r a C - C 5 system a t a pressure o f 1015 p s i a . I n other words, a pressure o f 1015 p s i a apparently exceeded the border curve o f the mixture. Some ethane systems do have two-phase regions above 1000 p s i a . However, w i t h out c o n s i d e r i n g the mixture composition o r without c a l c u l a t i n g the c r i t i c a l p r o p e r t i e s , i t i s d i f f i c u l t to p r o v i d e a reasonable e x p l a n a t i o n of these f l a s h c a l c u l a t i o n f a i l u r e s . The composition and p r e d i c t e d c r i t i c a l p r o p e r t i e s o f the mixture are given below: 2

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

186

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

T E M P E R A T U R E , °F

Figure 13.

Figure 14.

Phase diagram for a typical natural gas

Pressure-temperature diagram for natural gases with a varying heavy concentration

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

7.

ADLER E T A L .

Industrial Uses of Equations of State

C -C 2

5

187

System Mole F r a c t i o n

Ethane Propylene Propane Isobutane Butane Isobutylene Isopentane T p

,0004 .1269 .0758 .3786 .1107 .3013 .0063 = 276.05 F

c m

cm

=

584.75 p s i a

The mixture i s e s s e n t i a l l s i z e d i f f e r e n c e i n the ture should be c l o s e - b o i l i n g and a l s o have a c r i t i c a l p o i n t l o c a t e d between the cricondenbar and cricondentherm of the phase envelope. The p r e d i c t e d c r i t i c a l p r o p e r t i e s a r e based on a m o d i f i e d ChuehP r a u s n i t z technique. Based on the composition o f the m i x t u r e , i t i s u n l i k e l y t h a t the two-phase r e g i o n w i l l extend a p p r e c i a b l y beyond the t r u e c r i t i c a l p o i n t . Thus, c o n s i d e r i n g the p r e d i c t e d c r i t i c a l p r e s s u r e , a reasonable estimate of the cricondenbar would be about 675 p s i a . Therefore, q u a l i t a t i v e l y , i t was concluded t h a t the design pressure of 1015 p s i a f a r exceeded the two-phase r e g i o n and that the process engineer was attempting these f l a s h e s i n a homogenous l i q u i d r e g i o n . To s u b s t a n t i a t e these c o n c l u s i o n s , a bubble-point and dewp o i n t curve f o r the mixture were p r e d i c t e d using the Chao-Seader c o r r e l a t i o n . The r e s u l t s a r e given i n F i g u r e 15. The dashed l i n e to the l e f t i s the p r e d i c t e d bubble-point curve. The dashed l i n e to the r i g h t i s the p r e d i c t e d dew-point curve. The s m a l l t r i a n g l e represents the p r e d i c t e d c r i t i c a l p o i n t of the mixture. I t appears that the envelope should c l o s e i n the area of the t r u e c r i t i c a l p o i n t . The s o l i d l i n e i n the upper r i g h t hand p o r t i o n of the graph represents the attempted design pressure, c e r t a i n l y f a r above the two-phase r e g i o n . In t h i s study, attempts were made a t c a l c u l a t i n g dew-points and bubble-points a t 50 p s i a i n t e r v a l s , from 550 p s i a to the design pressure o f 1000 p s i a . A t most p r e s s u r e s , convergence was not achieved. However, a t a pressure o f 900 p s i a s o l u t i o n s were obtained f o r both the bubble-point and dew-point. These u n r e a l i s t i c roots a r e an unfortunate r e s u l t of Chao-Seader type p r e d i c t i o n s i f the l i m i t s o f the c o r r e l a t i o n , T < 0.97, are not observed. This i s a problem w i t h the c o r r e l a t i o n , r a t h e r than o f the f l a s h a l g o r i t h m . (A more extreme s i t u a t i o n w i l l be d i s c u s s e d s h o r t l y . ) The end r e s u l t may have been t h a t the engineer, r e a l i z i n g that the design pressure of 1015 p s i a was too h i g h , lowered h i s c o n d i t i o n s to about 900 p s i a and unknowingly accepted the r e s u l t s . To a v o i d these problems, one should remember t h a t f o r t h i s type of mixture R

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

188

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

1100DESIGN

PRESSURE

I000-900--

XX

800-700-600--

A // // // // It

if) CL

LlT

tr

T p

c

c

=276.05 =584.75

(/) if)

UJ

QC CL

^/

/ JP

200--

v

A

CHAO SEADER PREDICTIONS COMPUTER GENERATED RESULTS PREDICTED CRITICAL POINT

TEMPERATURE, °F

100

200 Figure 15.

300

400

Pressure-temperature diagram for a C -C system 2

5

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

7.

ADLER ET AL.

Industrial Uses of Equations of State

189

there i s no way a bubble-point can e x i s t a t a temperature g r e a t e r than the t r u e mixture c r i t i c a l temperature. The observations o f the previous problem were based on pred i c t e d p r o p e r t i e s , f o r both the phase envelope and the c r i t i c a l p o i n t . To j u s t i f y these c o n c l u s i o n s , a s i m i l a r mixture was s t u d i e d , but one f o r which the experimental c r i t i c a l r e g i o n phase envelope and c r i t i c a l p o i n t data were a v a i l a b l e . The composition and c r i t i c a l p r o p e r t i e s o f the mixture are given below: C ~C

System Mole F r a c t i o n

Pentane Hexan Heptan Octane Nonane T

cm P c m

.246

.178 .162 5 1 4

6 F

= = 448.6 p s i a

At about the equimolar composition o f the m i x t u r e , i t i s u n l i k e l y that the bubble-point curve w i l l e x i s t a t temperatures g r e a t e r than the t r u e c r i t i c a l temperature of the mixture. The Chao-Seader generated phase envelope f o r t h i s mixture i s given i n F i g u r e 16. The dashed l i n e represents the c r i t i c a l r e g i o n phase envelope, t h a t i s , the experimental phase envelope. The remainder of the phase envelope c o i n c i d e d r e l a t i v e l y w e l l w i t h the p r e d i c t e d v a l u e s , and i s not i n c l u d e d i n the diagram. Although the c r i t i c a l r e g i o n c l e a r l y terminates a t a pressure of about 450 p s i a , bubble-point and dew-point p r e d i c t i o n s were obtained up to 600 p s i a b e f o r e the f l a s h program f i n a l l y f a i l e d to converge. This behavior, which has been l a b e l e d the " f u n n e l i n g e f f e c t " , i s s p e c i f i c to Chao-Seader type c o r r e l a t i o n s and i n d i c a t e s these c o r r e l a t i o n s need improvement. Most o f these erroneous r e s u l t s can be r e j e c t e d by comparison w i t h r e s u l t s f o r s i m i l a r mixtures and w i t h the p r e d i c t e d c r i t i c a l p r o p e r t i e s . The f u n n e l i n g e f f e c t does not occur, a t l e a s t f o r the cases t h a t have been cons i d e r e d , when a s i n g l e equation o f s t a t e approach i s used i n a f l a s h program. Freeze-out o f a S o l i d from a L i q u i d S o l u t i o n . Thermodynamic textbooks and j o u r n a l a r t i c l e s , f o r the most p a r t , g i v e g r e a t e s t coverage to the phase e q u i l i b r i a o f f l u i d phases but, w i t h few exceptions, l i t t l e t o s o l i d s . However, p r e d i c t i o n of s o l i d f l u i d e q u i l i b r i a i s an important i n d u s t r i a l problem. This i s espec i a l l y true f o r process designs r e l a t e d t o n a t u r a l gas and cryogenic p l a n t s , where the s o l u b i l i t y l i m i t s a t the i n c i p i e n t s o l i d phase s e p a r a t i o n of, f o r example, water, C 0 and heavy hydrocarbons must be a c c u r a t e l y known to prevent c o s t l y p l a n t shutdowns w i t h 2

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

P H A S E EQUILIBRIA

400

450

Figure 16.

H -210

Figure 17.

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

1 -190

500

550

600

Pressure-temperature diagram for a C -C system 5

1 -170

1

1

-150 -130 TEMPERATURE, °F

1

1

-110

-90

9

-70

Estimation of freeze-out temperatures by intersection of fugacity curves

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

7.

Industrial Uses of Equations of State

ADLER E T A L .

191

plugged p i p e l i n e s o r heat exchanger u n i t s due to e x c e s s i v e accum u l a t i o n of s o l i d s formed by f r e e z e - o u t s . Equations o f s t a t e can be used f o r determining the e q u i l i b r i a i n s o l i d - l i q u i d systems. The s t a r t i n g p o i n t i s the o b s e r v a t i o n that a t the p o i n t where the f i r s t minute s o l i d p a r t i c l e p r e c i p i t a t e s out, the composition of the l i q u i d i s not a l t e r e d from i t s i n i t i a l composition. Therefore, the f u g a c i t y o f the l i q u i d s o l u t i o n can be c a l c u l a t e d a t s u c c e s s i v e temperatures u n t i l a temperature i s reached where the computed f u g a c i t y of the p r e c i p i t a t i n g component i n the l i q u i d equals that of the pure s o l i d . The fugac i t y of the pure s o l i d must f i r s t be computed by a p p l i c a t i o n o f the same equation of s t a t e to the s o l i d i n s i m i l a r s o l u t i o n s , f o r which experimental s o l i d - l i q u i d o r s o l i d - l i q u i d - v a p o r data a r e a v a i l a b l e . When a p p r o p r i a t e , the two-equation o f s t a t e approach can, o f course, be a p p l i e d An a p p l i c a t i o n i s i l l u s t r a t e temperature a t which s o l i d carbon d i o x i d e would s t a r t to form i n a condensed n a t u r a l gas system c o n t a i n i n g 5% d i s s o l v e d carbon d i o x i d e at a pressure o f 440 p s i a . I n one case, as i n d i c a t e d by the dashed l i n e s , the 11-constant BWR equation o f s t a t e i s used to s o l v e the problem. I n the o t h e r , a two-equation approach i n which the 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 a r e c o r r e l a t e d by the Wohl equation i s used. For BWR a p p l i c a t i o n , curve CC represents the f u g a c i t y of pure s o l i d carbon d i o x i d e as b a c k - c a l c u l a t e d from experimental data compiled by Kurata (29) on the s o l u b i l i t y o f s o l i d carbon d i o x i d e i n l i q u i d hydrocarbons. Curve FF r e l a t e s the computed f u g a c i t y of a 0.05 mole f r a c t i o n C 0 l i q u i d s o l u t i o n over a temperature range where f r e e z i n g might be expected. A t the p o i n t of i n t e r s e c t i o n , the f u g a c i t y o f the component i n l i q u i d s o l u t i o n equals the f u g a c i t y o f the pure s o l i d ; t h e r e f o r e , p r e c i p i t a t i o n o c c u r s . The temperature of the i n t e r s e c t i o n i s the s o l u t i o n to the problem. In the two-equation of s t a t e approach, the Redlich-Kwong-Chueh equation of s t a t e was used f o r vapor-phase f u g a c i t i e s and the Wohl equation f o r 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 i n the r e d u c t i o n of the experimental v a p o r - l i q u i d and v a p o r - l i q u i d - s o l i d e q u i l i b r i u m data. The a c t i v i t y c o e f f i c i e n t s so obtained were subsequently c o r r e l a t e d as a f u n c t i o n of temperature. Curves C C' and F F have the same s i g n i f i c a n c e as curves CC and FF. While n e i t h e r of the "pure s o l i d " f u g a c i t y curves i n F i g u r e 17 may represent the r e a l v a l u e s , each i s c o n s i s t e n t w i t h the equations used i n the data r e d u c t i o n and, theref o r e , r e l i a b l e f o r p r e d i c t i o n of the needed i n f o r m a t i o n as long as t h i s c o n s i s t e n c y i s maintained. As shown i n F i g u r e 17, the pred i c t e d f r e e z i n g temperature f o r the problem s t a t e d e a r l i e r i s -136 F by BWR-11 a p p l i c a t i o n , and 0138 F by the two-equation of s t a t e approach. The g r a p h i c a l technique of i n t e r s e c t i n g curves i s , o f course, only one o f the v a r i e t y o f ways t h i s problem can be s o l v e d . Once the fundamental thermodynamic parameters a r e e s t a b l i s h e d , computer programs can be used to o b t a i n the answers. As mentioned e a r l i e r , the m i n i m i z a t i o n o f the f r e e energy i s the u n i v e r s a l method o f s o l v i n g chemical and phase e q u i l i b r i u m problems. A computer program developed to handle multicomponent2

f

f

1

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

192

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

multiphase systems can be designed not only to determine the f r e e z e - o u t temperatures, b u t , f o r a g i v e n m i x t u r e , one can f o l l o w complete changes i n the e q u i l i b r i u m phases as temperature and pressure c o n d i t i o n s change. An i n t e r e s t i n g method from a fundamental p o i n t of view i s the g r a p h i c a l approach to f r e e energy m i n i m i z a t i o n . Since a t e q u i l i b r i u m , i . e . when t o t a l f r e e energy of the system i s minimum, the chemical p o t e n t i a l ( p a r t i a l m o l a l f r e e energy) of a component i s the same i n a l l phases, a common tangent l i n e to the f r e e energy curves drawn f o r each phase would e s t a b l i s h the e q u i l i b r i u m concent r a t i o n s . A common example f o r t h i s i s given i n most textbooks f o r l i q u i d - l i q u i d e q u i l i b r i a (King, (30); P r a u s n i t z , (31)). An a p p l i c a t i o n of t h i s p r i n c i p l e f o r s o l i d - l i q u i d e q u i l i b r i a i s shown i n F i g u r e 18 f o r the carbon dioxide-methane system a t -100 F. The f r e e energie values which a r e r e l a t e follows: Pure s o l i d C 0 phase; G 0

= G° + RT I n

c

L i q u i d phase; G = Z x. G? L i i

+ AG

T

(19)

M

(20)

where the standard r e f e r e n c e s t a t e i s taken as the pure subcooled l i q u i d a t the system temperature. The f r e e energy of m i x i n g , AG^, i s r e l a t e d to the a c t i v i t y c o e f f i c i e n t s by AG

M

= RT Z x. I n x. y. l i i

(21)

From Eqs. (19), (20), and (21), the r e l a t i v e m o l a l f r e e energies may be d e f i n e d as _

M

[

]

R T s o l i d CO

"

FG

G

G

^

°]

F

RT

"

l

n

y~

(22) L

[^Hn . . , = — RT l i q u i d J

E

X ±

G

]

= £ x. l n x. y. i l 'i

RT

(23)

The r e l a t i v e p a r t i a l m o l a l f r e e energy of carbon d i o x i d e i n the l i q u i d s o l u t i o n f o r a g i v e n c o n c e n t r a t i o n may be o b t a i n e d by drawing a tangent to the r e l a t i v e f r e e energy of mixing curve a t t h a t c o n c e n t r a t i o n (Lewis and R a n d a l l , (32)). The i n t e r c e p t o f t h i s tangent on t h e o r d i n a t e , X Q l-°> relative partial m o l a l f r e e energy of carbon d i o x i d e i n the l i q u i d and should have the same n u m e r i c a l v a l u e f o r carbon d i o x i d e i n a l l e q u i l i b r i u m phases. I n v e r s e l y , a tangent l i n e drawn from the p o i n t correspondi n g to the r e l a t i v e m o l a l f r e e energy of CO2 i n the s o l i d phase (x = 1.0 f o r pure C 0 ) , would touch the l i q u i d r e l a t i v e f r e e energy curve a t a c o n c e n t r a t i o n which i s i n e q u i l i b r i u m w i t h the pure =

C

i s

t

h

e

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.

7.

ADLER E T A L .

Industrial Uses of Equations of State

Figure 18. Estimation of the solubility of solid carbon dioxide in methane by the tangent method

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

193

194

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

s o l i d carbon d i o x i d e . The (fs/fj^) value f o r t h i s example i s taken from a p l o t given by Myers and P r a u s n i t z (33), and the a c t i v i t y c o e f f i c i e n t s a r e computed using a t w o - s u f f i x Margules equation. S o l u b i l i t y . Another important a p p l i c a t i o n of equations of s t a t e i n chemical process design i s i n converting s o l u b i l i t y data to v a p o r - l i q u i d e q u i l i b r i u m constants. The only e q u i l i b r i u m data that a r e a v a i l a b l e f o r such gaseous components as H 2 , O 2 , CH4, CO, CO2, H 2 S , the a c e t y l e n e s , e t c . , i n such s o l v e n t s as water, methanol, DMF, f o r example, a r e i n the form of s o l u b i l i t y data which may be found i n S e i d e l l (35) i n the form of Bunsen and Ostwald c o e f f i c i e n t s . A simple way to deal w i t h the low pressure s o l u b i l i t y data, which are u s u a l l y a t atmospheric pressure i s to assume the vapor phase to be an i d e a l ga 1.0, <(>£ = 1.0, thereby reducin (5) 1 0 K

=

±iH X

=

Y f

L° _L_

(24)

TT

from which a c t i v i t y c o e f f i c i e n t s can be computed and c o r r e l a t e d . When warranted, some improvement can be made by a l l o w i n g f o r the vapor pressure of the solvent so that f o r the s o l u t e gas y = p/fr. Further d e t a i l s and refinements on t h i s point a r e given by F r i e n d and Adler (34). Once the a c t i v i t y c o e f f i c i e n t s are e s t a b l i s h e d , the K s a t h i g h e r pressures a r e c a l c u l a t e d using Eq. (5) and an equation of s t a t e to c a l c u l a t e <j> as a f u n c t i o n of temperature, pressure and vapor composition, the l a t t e r being e s t a b l i s h e d by a t r i a l and e r r o r procedure. Whether or not the f u g a c i t y c o r r e c t i o n i s a l s o i n c l u d e d i n the r e d u c t i o n of the s o l u b i l i t y data f o r more p r e c i s e a c t i v i t y c o e f f i c i e n t s , * and whether or not these a c t i v i t y c o e f f i c i e n t s are developed and c o r r e l a t e d w i t h the standard s t a t e of Henry's constants or h y p o t h e t i c a l l i q u i d f u g a c i t i e s , i t i s obvious that equat i o n s of s t a t e again make p o s s i b l e an important chemical engineering computation. !

Summary This paper has d e a l t w i t h the c h a r a c t e r i s t i c s of equations of s t a t e r e q u i r e d by i n d u s t r y , has discussed a number of equations that are used i n i n d u s t r i a l 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 , and has covered a number of everyday and sometimes unusual p r a c t i c a l a p p l i c a t i o n s of equations of s t a t e . I n a l l three areas an attempt was made to analyze the shortcomings, d e f i c i e n c i e s , and handicaps of s p e c i f i c equations of s t a t e as w e l l as equations of s t a t e i n general, from an i n d u s t r i a l viewpoint. I t i s hoped that some of the m a t e r i a l discussed i n t h i s paper w i l l prove advantageous i n f u t u r e equation of s t a t e development work.

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

7.

ADLER E T A L .

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195

I t i s r e a l i z e d that the demands o f i n d u s t r y may be steep and, i n some cases, unreasonable; however, every p u b l i s h e d equation o f s t a t e has some l i m i t a t i o n s . The a v a i l a b l e equations o f s t a t e were developed and o r i g i n a l l y a p p l i e d i n r e l a t i o n to c e r t a i n systems, and t h e i r g r e a t e s t value i s i n the i n t e r p o l a t i o n and e x t r a p o l a t i o n of those systems. I n many cases without m o d i f i c a t i o n , a p p l i c a t i o n of the equation of s t a t e beyond the o r i g i n a l parameters i s i n v a l i d . Perhaps, the development o f a w h o l l y f l e x i b l e and workable equation of s t a t e , as d e f i n e d e a r l i e r , i s an u n a t t a i n a b l e g o a l . However, to be more o p t i m i s t i c , the authors b e l i e v e that t h i s challenge can be met. What then has to be done to make the equation of s t a t e a f l e x i b l e , i n c l u s i v e , s u c c e s s f u l t o o l ? F i r s t l y , a general purpose, three-constant, three-parameter equation i s needed I t s use should be u n r e s t r i c t e d as to syste u t i l i z e d i n these development y d i f f i c u l t t o estimate i n t h e i r own r i g h t . These parameters should be developed from m u l t i p r o p e r t y ( v o l u m e t r i c , enthalpy, v a p o r - l i q u i d e q u i l i b r i u m ) pure component and d e f i n e d mixture data. Extension to p o l a r and a s s o c i a t e d substances i s needed and e v e n t u a l l y must i n c l u d e t e s t i n g the c o r r e l a t i o n a g a i n s t a l l a v a i l a b l e r e l i a b l e data. I f i n t e r a c t i o n parameters are r e q u i r e d i n the treatment o f mixtures, an attempt should be made a t o b t a i n i n g temperature and p r e s s u r e independent values which can be g e n e r a l i z e d as a f u n c t i o n of a convenient size-shape o r p o l a r i t y parameter. F i n a l l y , the new developments should aim a t e l i m i n a t i n g the shortcomings o f the e x i s t i n g equations o f s t a t e enumerated e a r l i e r . As a concluding thought, i f a f l e x i b l e and t r u l y workable equat i o n o f s t a t e i s developed, there remains a f i n a l f a c t o r which i s s p e c i f i c to i n d u s t r y . For example, some t w e n t y - f i v e years a g o — a n d that i s before the a v a i l a b i l i t y of c o m p u t e r s — t h e s e n i o r author o f t h i s paper was developing a s e t of c h a r t s f o r the enthalpy of petroleum f r a c t i o n s . I n a d d i t i o n t o the u s u a l parameters o f temperature and pressure, these c h a r t s a l s o were f u n c t i o n s o f g r a v i t y , of the slope of t h e i r d i s t i l l a t i o n curve and of a c h a r a c t e r i z i n g f a c t o r p e c u l i a r to petroleum f r a c t i o n s . With a l l these v a r i a b l e s , months of hand c a l c u l a t i o n s were e n t a i l e d . During t h i s work i t was d i s c o u r a g i n g to hear a furnace design engineer comment that the charts would never be used, because the heat of c r a c k i n g c o r r e l a t i o n s used w i t h the petroleum f r a c t i o n enthalpy curves were d e r i v e d from the o l d enthalpy c h a r t s that would be r e p l a c e d . Of course, the heat of c r a c k i n g c o r r e l a t i o n could a l s o be m o d i f i e d . So i t i s w i t h any new c o r r e l a t i o n produced today. I f i t i s i n t r o d u c e d , every c h a r t , graph, c o r r e l a t i o n and a n a l y t i c a l e x p r e s s i o n i n a computer program that was d e r i v e d from the proceeding c o r r e l a t i o n must be r e v i s e d . This can best be seen by l o o k i n g a t the i n g r e d i e n t s of the Chao-Seader c o r r e l a t i o n . I f the Redlich-Kwong equation of s t a t e i s r e p l a c e d w i t h i t s Chueh m o d i f i c a t i o n , the l i q u i d - f u g a c i t y equat i o n , which o r i g i n a l l y took up the s l a c k i n developing the o v e r a l l c o r r e l a t i o n from experimental data, a l s o must be r e v i s e d . C o n s i d e r i n g

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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A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

that i t i s based on more than 3000 experimental p o i n t s and cons i s t s of about 20 constants, t h i s i s a b i g task. And w i l l the o v e r a l l r e s u l t be b e t t e r ? I n summary, when a new p i e c e of work i s introduced i n the l i t e r a t u r e no matter how good i t i s , i t can not immediately be put i n t o use. Acknowledgment Acknowledgment i s made to other members of our s t a f f who a s s i s t e d i n the p r e p a r a t i o n of t h i s paper; K a t h l e e n Cowan, Denise Hanley, J a n i c e Lambrix, and Rosalyn Quick. Nomenclature =

a A A

Constant Constant i n Eq Constant i n BWR-11 Eq. (Table I D Constant B = Constant i n Eq. (18) B = Constant i n BWR-11 Eq. (Table I D c Constant i n BWR-11 Eq. (Table I D C Constant i n Eq. (18) Constant i n BWR-11 Eq. (Table I D co 8 = Constant i n BWR-11 Eq. (Table I D D Constant i n BWR-11 Eq. (Table I D E° Constnat i n BWR-11 Eq. (Table I D ? = F u g a c i t y ; f,, component f u g a c i t y ; f°, standard s t a t e f u g a c i t y o r f u g a c i t y of pure component G = Gibbs molar f r e e energy; G°, Gibbs molar f r e e energy of pure component a t standard s t a t e K = E q u i l i b r i u m constant, y/x K^j = Binary i n t e r a c t i o n c o e f f i c i e n t i n E q . ( l l ) n = Number o f components, used i n Eq.(13) P,p = System p r e s s u r e P , P = C r i t i c a l pressure P = True c r i t i c a l p r e s s u r e of mixture p" = P a r t i a l p r e s s u r e o f component i n m i x t u r e p° = Vapor p r e s s u r e o f pure component P = Reduced p r e s s u r e , P/P R = Gas constant T = Temperature T = C r i t i c a l temperature T = True c r i t i c a l temperature of m i x t u r e T = Reduced temperature T/T V = Molar volume V = P a r t i a l molar l i q u i d volume of component V = True c r i t i c a l molar volume of mixture x = Mole f r a c t i o n of component i n l i q u i d phase y = Mole f r a c t i o n of component i n vapor phase ci C r i t i c a l c o m p r e s s i b i l i t y f a c t o r of component i cm C r i t i c a l c o m p r e s s i b i l i t y f a c t o r of mixture = = =

Q

C

C

c m

r

c

c

c m

r

c

c m

z

=

z

=

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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197

Greek a = Constant used i n BWR-11 Eq. (Table I I ) and parameter i n Eqs. (6) and (7) 3 = C o e f f i c i e n t i n Wohl Eq. (13) Y = Constant used i n BWR-11 Eq. (Table I I ) Y = A c t i v i t y c o e f f i c i e n t o f a component i n l i q u i d 6 = S o l u b i l i t y parameter v w j = simple f l u i d l i q u i d f u g a c i t y c o e f f i c i e n t v ' ' = L i q u i d f u g a c i t y c o e f f i c i e n t c o r r e c t i o n term TT = System pressure x = Chueh-Prausnitz i n t e r a c t i o n parameter <j> = Vapor phase f u g a c i t y c o e f f i c i e n t <j)° = F u g a c i t y c o e f f i c i e n t a t standard s t a t e u) = A c e n t r i c f a c t o 1

Subscripts c = i,j,k,l= L = m = S = Y =

C r i t i c a l point Components i n a mixture L i q u i d phase Mixture Solid Vapor

References 1. Tsonopoulos, C. and Prausnitz, J. M., Cryogenics, (1969), 9, 315. 2. Martin, J. J., Appl. Thermodyn. Symp., (1967), 59, 34. 3. Barner, H. E., Pigford, R. L., and Schreiner, W. C., Proc. Am. Petrol. Inst., Vol. 46 (III), (1966), 244. 4. Barner, H. E. and Adler, S. B., Ind. Eng. Chem. Fundamentals, (1970), 9, 521. 5. Dluzniewski, J. H. and Adler, S. B., I. Chem. E. Symp. Ser., (1972), No. 35, 4:21. 6. Benedict, M., Webb, G. B., and Rubin, L. C., Chem. Eng. Progr., (1951), 47, 449. 7. Starling, K. E., Fluid Thermodynamic Properties for Light Petroleum Systems, p. 221, Gulf Publishing Company, Houston, Texas (1973). 8. Lin, C-J., etal.,Can. J. Chem. Eng., (1972), 50(10), 644. 9. Lin, C-J. and Hopke, S. W., A.I.Ch.E. Symp. Ser., (1974), 70 (140), 37. 10. Chueh, P. L. and Prausnitz, J. M., A.I.Ch.E.J., (1967), 13, 1107. 11. Barner, H. E., and Adler, S. B., Hydrocarbon Process, (1968) 37 (10), 150. 12. Hopke, S. W. and Lin, C-J., Application of the BWRS Equation to Natural Gas Systems, Presented at the 76th National A.I.Ch. E. Meeting, Tulsa, Oklahoma, March 10-13, 1974.

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

13. Soave, G., Chem. Eng. Sci., (1972), 27, 1197. 14. American Petroleum Institute, Technical Data Book--Petroleum Refining, 2nd Edition, Chap. 4, 7, & 8, Washington, D.C., 1977. 15. American Petroleum Institute, Report Nos. API-1-74, API-4-74, API-3-75, API-4-75, API-1-76, and API-3-76, The Pennsylvania State University, University Park, Pennsylvania (Private Communications). 16. Peng, D-Y. and Robinson, D. B., Ind. Eng. Chem. Fundamentals, (1976), 15, 59. 17. Cavett, R. H., Proc. Am. Petrol. Inst., Vol. 42 (III), (1962), 351. 18. Grayson, H. G., The Influence of Differences in Phase Equilibria Data on Design Presented at the Midyear Meeting of the American Petroleum' California, May 14 19. Robinson, R. L . , Jr. and Chao, K. C., Ind. Eng. Chem. Process Design Develop., (1971), 10, 221. 20. Adler, S. B., Ozkardesh, H., and Schreiner, W. C., Hydrocarbon Process., (1968), 47(4), 145. 21. Spencer, C. F., A Semi-Empirical Study of the True Critical Properties of Defined Mixtures, Ph.D. Thesis, Pennsylvania State University (August, 1975). 22. Klosek, J. and McKinley, C., Proceedings, First International Conference on LNG, Chicago, Illinois (1968). 23. Shana'a, M. Y. and Canfield, F. B., Trans. Faraday Soc., (1968), 64, 2281. 24. Miller, R. C. and Rodesevitch, J. B., A.I.Ch.E.J., (1973), 18, 728. 25. Miller, R. C., Chem. Eng., (1974), 81 (23), 134. 26. Watson, K. M., Ind. Eng. Chem., (1943) 35, 398. 27. Nichols, W. B., Reamer, H. H., and Sage, B. H., A.I.Ch.E.J., (1957), 3, 262. 28. Kay, W. B., Chem. Rev., (1941), 29, 501. 29. Kurata, F., Solubility of Solid Carbon Dioxide in Pure Light Hydrocarbons and Mixtures of Light Hydrocarbons, Research Report RR-10, Gas Processors Association, Tulsa, OK (1974). 30. King, M. B., Phase Equilibrium in Mixtures, p. 65, Pergamon Press, Inc., New York, New York (1969). 31. Prausnitz, J. M., Molecular Thermodynamics of Fluid-Phase Equilibria, p. 234, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1969). 32. Lewis, G. N. and Randall, M., Thermodynamics, 2nd Edition, Revised by K. S. Pitzer and L. Brewer, p. 207, McGraw Hill Book Co., Inc., New York, New York (1961). 33. Meyers, A. L. and Prausnitz, J. M., Ind. Eng. Chem. Fundamentals, (1965) 4, 209. 34. Friend, L. and Adler, S. B., Chem. Eng. Progr., (1957) 53, 452.

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

7.

ADLER E T A L .

Industrial Uses of Equations of State

35. Seidell, A., Solubilities of Inorganic and Metal Organic Compounds, 4th Edition, Vol. 1 & 2, D. Van Nostrand Company, Inc., New York, New York (1958).

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

199

8 Applications of the Peng-Robinson Equation of State DONALD B. ROBINSON, DING-YU PENG, and HENG-JOO NG University of Alberta, Edmonton, Alberta, Canada

The recently proposed Peng-Robinson equation of state (1) incorporates the best features of the Soave (2) treatment of the Redlich-Kwong (3) equation into a new model which has some signi ficant advantages over earlier two-parameter equation of state models. The purpose of this contribution is to indicate how the equation performs when it is used for calculating fluid thermo dynamic properties for systems of industrial interest. The flexibility of the equation and the generality of the situations for which it can be used to give answers of acceptable reliability at a reasonable cost are illustrated through example calculations of vapor pressure, density, vapor-liquid equilibrium, critical properties, three phase L L G equilibrium for systems containing water, and HLG, HL L G, and HL L equilibrium in hydrate forming systems. The equation shows its best advantages in any situation involving liquid density calculations and in situations near the critical region, but it is usually better than other two para meter models in all regions. In developing the new equation certain criteria were estab lished at the outset. In order for the equation to be acceptable the following conditions had to be met. 1

1

1

2

2

1

2

a. The equation was to be a two-constant equation not higher than cubic in volume. b. The model should result in significantly improved perfor mance in the vicinity of the critical point, in particular with Z and liquid density calculations. c. The constants in the equation should be expressable in terms of P , T and ω. d. The mixing rules for evaluating mixture constants should not contain more than one fitted binary interaction para meter, and if possible this parameter should be temper ature, pressure, and composition independent. c

c

c

200

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

8.

Peng-Robinson Equation of State

ROBINSON E T A L .

e.

201

The equation should be s u f f i c i e n t l y general i n i t s a p p l i c a b i l i t y so that the one equation could be used t o handle a l l f l u i d property c a l c u l a t i o n s f o r systems normally encountered i n the p r o d u c t i o n , t r a n s p o r t a t i o n , or p r o c e s s i n g or n a t u r a l gas, condensate, or other r e l a t e d hydrocarbon mixtures.

The Peng-Robinson Equation The d e t a i l s of the development of the Peng-Robinson (PR) equat i o n are given i n the o r i g i n a l paper ( 1 ) . The f i n a l r e s u l t s are summarized here f o r convenience. The equation has the form:

p

=

_RT

a(T

v - b

v ( v + b) + b(v - b)

In t h i s equation: RT b = 0.07780 a(T) = a ( T ) c

c • a(T ,a>) R

2 2 R T a(T ) = 0.45724 „ c P c C

Z = 0.307 c a^(T ,co) = 1 + K ( 1 - T S R

R

K = 0.37464 + 1.54226a) - 0.26992a)

2

The values of b and a ( T ) are obtained by equating the f i r s t and second d e r i v a t i v e of pressure w i t h respect t o volume t o zero along the c r i t i c a l isotherm at the c r i t i c a l p o i n t , s o l v i n g the two equations simultaneously f o r a and b, and then u s i n g the equation of s t a t e t o s o l v e f o r Z . The v a l u e of K f o r each pure component was obtained by l i n e a r i z i n g the or2(T ,a)) V S T ^ r e l a t i o n s h i p between the normal b o i l i n g point and the c r i t i c a l p o i n t . This i s s l i g h t l y d i f f e r e n t that the method of Soave (2) which assumed a l i n e a r r e l a t i o n s h i p from T = 1.0 to T = 0.7. The i n f l u e n c e of a) and K was obtained by a l e a s t squares f i t of the data f o r a l l components of i n t e r e s t . c

c

R

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.

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TABLE 1.

Comparison of Vapor Pressure P r e d i c t i o n s (Reprinted w i t h Permission from Ind. Eng. Chem. Fundamentals, 15, 293 (1976a). Copyright by The American Chemical S o c i e t y . ) R e l a t i v e E r r o r , %. No. of Data Points

Component Methane

28

Ethane

27

Propane

31

Isobutane n-Butane

AAD SRK

BIAS PR

1 .44

SRK

0.66

RMS PR

SRK

PR

0 .47

0.38

1. 57

0. 77

-0.34

0.95

0.38

0 .70

0.34

-0 .10

27

1 .06

0.32

0 .82

0.16

1. 18

0.34

28

0 .75

0.37

0 .47

-0.22

0.86

0.42

Isopentane

15

0 .46

0.54

0 .17

-0.53

0.,49

0. 60

n-Pentane

30

0 .92

0.58

0 .50

-0.29

1.02

0. 66

n-Hexane

29

1 .55

0.90

1 .31

0.37

1. 75

1.06

n-Heptane

18

1 .51

0.79

1 .48

0.63

1.88

1. 04

n-Octane

16

1 .99

1.04

1 .97

1.02

2. 24

1. 26

Nitrogen

17

0 .56

0.31

0 .00

-0.02

0.75

0.37

Carbon Dioxide

30

0 .53

0.62

0 .50

-0.49

0.63

0.71

Hydrogen S u l f i d e

30

0 .66

0.96

0 .34

0.42

1.00

1. 48

1 .01

0.60

0 .68

0.11

1.19

0.73

Average, %

N l a

AAD = ± = L

I d.i

1

BIAS

1=1

RMS

d. = e r r o r f o r each p o i n t l

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

8.

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203

The mixing r u l e s which are recommended are as f o l l o w s : b

= m

a

m

a.

(1 - 6. .) a.' in i

In the equation f o r a y , i s a f i t t e d parameter p r e f e r a b l y obtained by o p t i m i z i n g the p r e d i c t i o n of b i n a r y bubble p o i n t p r e s sures over a reasonable rang Applications Vapor Pressure. One of the important c o n s i d e r a t i o n s f o r any equation o f s t a t e that i s t o be used 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 i s whether or not i t can a c c u r a t e l y p r e d i c t the vapor pressure of pure substances. Table 1 shows a comparison between the PR equation and the Soave-Redlich-Kwong (SRK) equation f o r p r e d i c t i n g the vapor pressure of ten p a r a f f i n hydrocarbons and three commonly encountered non-hydrocarbons. I t w i l l be noted that the use of the PR equation has improved the RMS r e l a t i v e e r r o r by a f a c t o r of about 40 percent over the SRK p r e d i c t i o n s u s i n g the same data. F u r t h e r , i t i s seen that the SRK p r e d i c t i o n s are biased high i n every case but one and that PR p r e d i c t i o n s are evenly s p l i t between p o s i t i v e and negative departures w i t h a r e s u l t i n g o v e r a l l p o s i t i v e b i a s that i s only 16 percent of the value obtained u s i n g the SRK equation. D e n s i t i e s . I t i s w e l l known that the SRK equation tends to p r e d i c t l i q u i d m o l a l volumes that are too h i g h , and that t h i s i s p a r t i c u l a r l y n o t i c e a b l e i n the v i c i n i t y of the c r i t i c a l p o i n t . The f a c t t h a t the PR equation gives a u n i v e r s a l c r i t i c a l c o m p r e s s i b i l i t y f a c t o r of 0.307 compared t o 0.333 f o r the SRK equation has improved the a b i l i t y of the PR equation t o p r e d i c t l i q u i d d e n s i t i e s . F i g u r e 1 i l l u s t r a t e s the performance of the two equations f o r p r e d i c t i n g the m o l a l volume of s a t u r a t e d l i q u i d s and vapors f o r pure n-pentane. A t reduced temperatures above about 0.8, the average e r r o r i n l i q u i d d e n s i t y has been reduced by a f a c t o r o f about 4 by using the new equation. At lower reduced temperatures, the p r e d i c t i o n s by the new equation are b e t t e r by a f a c t o r of about 2. Both equations g i v e acceptable p r e d i c t i o n s of the vapor d e n s i t y . The a b i l i t y o f the new equation t o p r e d i c t the s p e c i f i c volume of s a t u r a t e d l i q u i d s i n multicomponent systems i s c l e a r l y i l l u s t r a t e d i n F i g u r e s 2 and 3. F i g u r e 2 shows the c a l c u l a t e d l i q u i d volume percent i n the r e t r o g r a d e r e g i o n f o r a 6 component p a r a f f i n hydrocarbon mixture c o n t a i n i n g components from methane through n-decane. F i g u r e 3 shows the same k i n d o f i n f o r m a t i o n f o r a 9 component system

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A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

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

Peng-Robinson Equation of State

ROBINSON ET AL.

0

5

10 VOLUME

15

20

25

PERCENT LIQUID

Figure 3. Volumetric behavior of nine-component condensate-type fluid containing hydrocarbons, nitrogen, carbon dioxide, and hydrogen sulfide at 250°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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A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

• YARBOROUGH, 1972 I —PR PREDICTION

O

<

ZD

Figure 4. Experimental and PR predicted equilibrium ratios for a ninecomponent system containing hydrocarbons, nitrogen, carbon dioxide, and hydrogen sulfide at 100°F

10* 10 PRESSURE, PSIA

Figure 5. Experimental and SRK predicted equilibrium ratios for a nine-component system containing hydrocarbons, nitrogen, carbon dioxide, and water at 100° F

10* io PRESSURE, PSIA

J

3

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

8.

ROBINSON E T A L .

Peng-Robinson Equation of State

207

c o n t a i n i n g N2, CO2 and H2S i n a d d i t i o n to the 6 hydrocarbons. The g r e a t l y improved performance of the PR equation i s obvious, p a r t i c u l a r l y i n the upper r e t r o g r a d e r e g i o n , where an improvement from about 20 t o over 100 percent r e s u l t s . The a b i l i t y of the equation to p r e d i c t upper dew p o i n t s f o r these systems i s a l s o e v i dent. I t should perhaps be emphasized that none of the experimental data on the mixtures was used i n e v a l u a t i n g the parameters f o r making the p r e d i c t i o n s . V a p o r - L i q u i d E q u i l i b r i u m . One of the advantages of u s i n g a simple two-constant cubic equation of s t a t e i s the r e l a t i v e s i m p l i c i t y w i t h which they may be used to perform 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 . The time-consuming i t e r a t i v e procedures r e q u i r e d by m u l t i c o n s t a n t equations of s t a t e f o r c a l c u l a t i n g the c o e x i s t i n g phase d e n s i t i e s can be avoided b o b t a i n i n th analyticall from the cubic equation. The a b i l i t y of the PR equation to p r e d i c t e q u i l i b r i u m phase compositions and phase envelopes has been evaluated f o r a v a r i e t y of systems, i n c l u d i n g those c o n t a i n i n g both hydrocarbon and nonhydrocarbon components. Examples of some of these are i l l u s t r a t e d i n F i g u r e s 4, 5, 6, and 7. F i g u r e s 4 and 5 show comparisons between e q u i l i b r i u m r a t i o s determined e x p e r i m e n t a l l y and those c a l c u l a t e d by the PR and SRK equations f o r a nine component system c o n t a i n i n g hydrocarbons from methane to n-decane, together w i t h n i t r o g e n , carbon d i o x i d e , and hydrogen s u l f i d e . The composition of t h i s mixture i s given i n Table 2. An i n s p e c t i o n of the f i g u r e s shows t h a t i n general the agreement between the experimental data and both p r e d i c t i o n s i s good. However, a more d e t a i l e d a n a l y s i s shows that the SRK p r e d i c t i o n would y i e l d a somewhat higher convergence pressure f o r the system than that p r e d i c t e d by the PR equation. Although i t cannot be seen from F i g u r e s 4 and 5, the K - f a c t o r s converge to u n i t y f o r the PR p r e d i c t i o n a t 3400 p s i a and by the SRK p r e d i c t i o n a t 3700 p s i a . The u n d e s i r a b l e consequences of t h i s may be explained as f o l l o w s . Each e q u i l i b r i u m r a t i o curve f o r the heavier components passes through a minimum at a c e r t a i n o p e r a t i n g pressure. For these curves t o converge to u n i t y a t a lower pressure means that the e q u i l i b r i u m r a t i o s must i n c r e a s e more r a p i d l y as pressure increases from the value a t the minimum to the value a t the convergence pressure. By the same reasoning, the r a t i o s f o r the l i g h t e r components, i n p a r t i c u l a r n i t r o g e n and methane, must decrease f a s t e r i n the same pressure i n t e r v a l . However, i t i s the e q u i l i b r i u m r a t i o s f o r the heavier components that c o n t r o l the dew p o i n t pressures. Thus i f the PR equation p r e d i c t e d the dew p o i n t pressure f o r a p a r t i c u l a r system i n the upper retrograde r e g i o n , the SRK equation would have t o go to a higher pressure t o f i n d the same set of e q u i l i b r i u m r a t i o s and consequently i t would p r e d i c t a higher dew p o i n t pressure. The e r r o r s r e s u l t i n g from t h i s c h a r a c t e r i s t i c are c l e a r l y seen i n Figures 2 and 3. F i g u r e 6 shows the experimental and p r e d i c t e d phase envelope and c r i t i c a l p o i n t f o r a four-component l e a n gas mixture. Although the agreement along the bubble p o i n t locus i s good, the agreement

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

Figure 6. Experimental and predicted phase envelope and critical point for lean gas mixture (data from Ref. 16)

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

Peng-Robinson Equation of State

ROBINSON ET AL.

Table 2 Composition of Sour Gas M i x t u r e Used f o r E q u i l i b r i u m R a t i o Comparisons Yarborough (15) Component N

2

C

l

Mole F r a c t i o n 0.0299 0.7180

C0

0.0302

o

0.0456 HS

0.0377

2

0.0251

nC

0.0533

r

0.0377 nC

0.0300

10

400

PR C A L C U L A T E D • BUBBLE POINTS o D E W POINTS • CRITICAL POINTS

-150

-100

TEMPERATURE

-50

°F

Figure 7. Predicted phase envelopes and critical point for selected lean gas mixtures containing varying butanes, pentanes, and heptane

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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along the dew p o i n t locus does not appear to be as good. However, measurements of t h i s type are exceedingly d i f f i c u l t to make and the pressures of i n i t i a l l i q u i d f o r m a t i o n may be open to some u n c e r t a i n t y . I n i t i a l l i q u i d f o r m a t i o n at -120 and -140 F as c a l c u l a t e d from the GPA Engineering Data Book (Revised 1976) agrees w i t h the p r e d i c t i o n s . As a f u r t h e r assessment of the d i s c r e p a n c y , f l a s h c a l c u l a t i o n s were c a r r i e d out a t each of the experimental dew p o i n t s , and the r e s u l t s i n d i c a t e d that the system was more than 99.9 volume percent vapor i n each case. Thus i t becomes a q u e s t i o n of whether or not one can a c t u a l l y observe something l e s s than 0.1 volume percent l i q u i d . F i g u r e 7 i l l u s t r a t e s the response of the equation to s l i g h t changes i n the c o n c e n t r a t i o n of components h e a v i e r than propane i n a multi-component m i x t u r e . The three systems shown each c o n t a i n a s m a l l amount of n i t r o g e n amount of methane, ethane of butanes, pentanes, and heptane. The compositions of the three mixtures together w i t h the c a l c u l a t e d c r i t i c a l p r o p e r t i e s are shown i n Table 3. I t w i l l be noted that the phase envelopes f o r mixtures A and C are t y p i c a l , but that the phase envelope f o r mixture B e x h i b i t s a double r e t r o g r a d e r e g i o n . A d e t a i l of t h i s r e g i o n and the c r i t i c a l p o i n t are shown i n F i g u r e 7. The authors are unaware of other r e p o r t s of a double r e t r o g r a d e r e g i o n i n multicomponent systems, although the phenomenon i s w e l l known i n b i n a r y systems (Chen et a l , (4, 5 ) ) . C r i t i c a l P r o p e r t i e s . A method f o r c a l c u l a t i n g the c r i t i c a l p r o p e r t i e s of multicomponent systems using an equation of s t a t e has r e c e n t l y been developed by Peng and Robinson ( 6 ) . The method i s based on the r i g o r o u s thermodynamic c r i t e r i o n f o r the c r i t i c a l s t a t e enunciated by J . W i l l a r d Gibbs (7) and uses the PR equation of s t a t e to d e s c r i b e the f l u i d p r o p e r t i e s . I n t h i s study, the p r e d i c t e d and experimental c r i t i c a l temperature and pressure were compared f o r 32 mixtures c o n t a i n i n g from 3 to 12 components. The systems included p a r a f f i n hydrocarbons from methane to n-decane, n i t r o g e n , carbon d i o x i d e , and hydrogen s u l f i d e . The average a b s o l u t e e r r o r i n pressure was 25 p s i a or 2.33% and the average a r i t h m e t i c e r r o r was +5.7 p s i a or 0.13%. For temperature the corresponding e r r o r s were 7.7 R or 1.31% and +7.0 °R or 1.14%. The range of c r i t i c a l p r e s sures v a r i e d by a f a c t o r of 4.43 and the range of c r i t i c a l temperatures by a f a c t o r of 2.88. F i g u r e 6 i l l u s t r a t e s a t y p i c a l comparison between the p r e d i c t e d and experimental c r i t i c a l p o i n t f o r a four-component system and F i g u r e 7 and Table 3 i l l u s t r a t e the c a l c u l a t e d c r i t i c a l p r o p e r t i e s for systems c o n t a i n i n g up to 10 components. Water-Hydrocarbon Systems. The a p p l i c a t i o n of the PR equation to two and three-phase e q u i l i b r i u m c a l c u l a t i o n s f o r systems c o n t a i n i n g water has r e c e n t l y been i l l u s t r a t e d by Peng and Robinson ( 8 ) . As i n the case of other hydrocarbon-non-hydrocarbon m i x t u r e s , one f i t t e d b i n a r y i n t e r a c t i o n parameter f o r water w i t h each of the hydrocarbons i s r e q u i r e d . These parameters were obtained from experimental data a v a i l a b l e i n the l i t e r a t u r e on each of the water-hydrocarbon b i n a r i e s .

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

8.

ROBINSON E T A L .

211

Peng-Robinson Equation of State

Table 3 S e l e c t e d Mixtures Used f o r Phase Envelope and C r i t i c a l P o i n t C a l c u l a t i o n s

Component Nitrogen Carbon D i o x i d e Methane

Mixture A Mole % 1.67 0.20 97.21

Mixture C Mole %

Mixture B Mole % 1.76 0.17 97.18

1.27 0.34 97.57 0.78

Ethane Propane Isobutane

0.805

0.78

n-Betane Isopentane n-Pentane

0.015

0.016 0.002 0.001

— — —

0.003

0.02

n-Heptane C r i t i c a l Pressure, psia C r i t i c a l Temperature, °F

— — — 706.4

706.6

707.2

-113.0

-112.7

-113.2

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

212

P H A S E EQUILIBRIA A N D F L U I D

PROPERTIES I N C H E M I C A L

INDUSTRY

DATA OF M c K E T T A & KATZ IO-'

-

O

(1948)

100°F

A

160°P

a

220°F

V

280°F PREDICTION

10 -2 A_AAA_4L

10 -3 z o

n

500

0

OGE)

1000

PRESSURE,

1500

2000

2500

psia Canadian Journal of Chemical Engineering

Figure 8. Water content of hydrogen-rich liquid and vapor in a three-phase system containing water, methane, and n-butane

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

8.

ROBINSON ET AL.

Peng-Robinson Equation of State

213

The three phase program was then evaluated by comparing p r e d i c t e d and experimental r e s u l t s f o r the methane-n-butane-water and n-butane-1 butene-water systems as reported by McKetta and Katz (9) and Wehe and McKetta (10). Figures 8 and 9 i l l u s t r a t e the type of agreement that was obtained f o r the water content and the hydrocarbon d i s t r i b u t i o n i n both the hydrocarbon l i q u i d and vapor phases f o r the methane n-butanewater system. I t can be seen from F i g u r e 8 that the p r e d i c t i o n s reproduce the water content very w e l l a t a l l temperatures. F i g u r e 9 shows that the agreement between experimental and p r e d i c t e d hydrocarbon c o n c e n t r a t i o n s i n the vapor and l i q u i d phases i s good a t 100 F, although the agreements does not seem t o be as good a t 2220 F. The experimental data f o r 220 F may be open to q u e s t i o n s i n c e the c r i t i c a l pressure f o r methane-n-butan about 1350 p s i a by Robert these authors on the methane-n-butane system a t 220 F are i n c l u d e d for comparison. I t seems d o u b t f u l that the presence of water i n t h i s system would i n c r e a s e the c r i t i c a l p r e s s u r e to about 1550 p s i a as i n d i c a t e d by the three component d a t a . In view of t h i s , the p r e d i c t e d r e s u l t s a r e thought t o be j u s t as good a t 220 F as a t 100 F. The programs used to make the above p r e d i c t i o n have been s u c c e s s f u l l y used f o r three phase f l a s h , bubble p o i n t , and hydrocarbon and/or water dew p o i n t c a l c u l a t i o n s . G e n e r a l l y the c a l c u l a t i o n s converge r a p i d l y and normally r e q u i r e only about 20 i t e r a t i o n s . The programs p r e d i c t hydrocarbon c o n c e n t r a t i o n s i n the aqueous l i q u i d phase which are s e v e r a l orders of magnitude lower than the reported experimental d a t a . I n order t o o b t a i n a good p r e d i c t i o n of these very d i l u t e c o n c e n t r a t i o n s a d i f f e r e n t and p o s s i b l y temperature dependent 6 j j may be r e q u i r e d . An a d d i t i o n a l example of the a b i l i t y of the PR equation t o p r e d i c t the water content of gases s a t u r a t e d w i t h water i s shown i n F i g u r e 10. The comparisons shown here are f o r the PR p r e d i c t i o n s and the GPA E n g i n e e r i n g Data Book (Revised 1976) v a l u e s f o r t h e water content of n a t u r a l gases. I t w i l l be noted that the GPA data and the p r e d i c t i o n s are p r a c t i c a l l y c o i n c i d e n t over the e n t i r e range of pressures and temperatures. Hydrates. Programs have been developed f o r p r e d i c t i n g hydrate forming c o n d i t i o n s f o r the HLiG, H L i L G , and H I ^ I ^ e q u i l i b i a . These use the approach developed by P a r r i s h and P r a u s n i t z (12) but w i t h the PR equation used throughout f o r a l l f l u i d p r o p e r t y c a l c u l a t i o n s . D e t a i l s of the method have been reported by Ng and Robinson (13). T y p i c a l r e s u l t s f o r the HLiG e q u i l i b r i u m are shown f o r the carbon dioxide-propane-water system and the methane-isobutane water system i n F i g u r e 11. The p r e d i c t e d and experimental pressures are compared a t the e x p e r i m e n t a l l y determined hydrate temperature. The mixtures of carbon d i o x i d e and propane i n c l u d e d c o n c e n t r a t i o n s from 6 t o 84 mole percent propane on a w a t e r - f r e e b a s i s i n the gas phase and the mixtures o f methane and isobutane included c o n c e n t r a t i o n s from 0.4 t o 63.6 mole percent isobutane on the same b a s i s . I t w i l l be seen that the p r e d i c t e d and experimental r e s u l t s compare f a v o r a b l y 2

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

214

P H A S E EQUILIBRIA

0

0.2

METHANE

A N D F L U I D PROPERTIES

0.4

0.6

I N C H E M I C A L INDUSTRY

0.8

1.0

AAOLE FRACTION, DRY BASIS

Figure 9. Hydrocarbon distribution in the hydrocarbon-rich liquid and vapor in three-phase system containing water, methane, and n- butane

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

Peng-Robinson Equation of State

ROBINSON E T A L .

~

i

•

60

i

i

i

i

•

PR PREDICTED GPA DATA BOOK (1976)

100

140

180

TEMPERATURE

220

260

°F

Figure 10. Predicted equilibrium water content of natural gas in contact with water liquid

Figure 11. Comparison of experimental and predicted hydrate formation pressures in the HL G region (data from Refs. 17,18) t

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

216

P H A S E EQUILIBRIA AND

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

throughout, although the methane—isobutane values are biased s l i g h t l y high. F i g u r e 12 shows a comparison between p r e d i c t e d and experimental H L 1 L 2 G data f o r the methane-isobutane-water system. None of the four-phase data were used i n e s t a b l i s h i n g the parameters used i n making the p r e d i c t i o n s . Recently, Ng and Robinson (14) have extended the methods used above f o r the HLiG and H L 1 L 2 G e q u i l i b r i a to the H L i L r e g i o n . This was accomplished by developing a method f o r p r e d i c t i n g the slope of the H L 1 L 2 locus i n multicomponent systems. By combining t h i s w i t h the p r e d i c t e d H L 1 L 2 G p o i n t f o r the system i t i s p o s s i b l e to c a l c u l a t e the hydrate forming temperature i n the H L i L r e g i o n at any pressure. F i g u r e 13 shows a compariso hydrate forming c o n d i t i o n l i q u i d m i x t u r e s . The composition of the mixtures i s given i n Table 4. These i n c l u d e a mixture of e s s e n t i a l l y p a r a f f i n hydrocarbons, a mixture c o n t a i n i n g a s i g n i f i c a n t amount of carbon d i o x i d e , and a mixture c o n t a i n i n g a wide range of molecular weights. The mixture c o n t a i n i n g the carbon d i o x i d e shows a d i f f e r e n c e between the p r e d i c t e d and experimental hydrate formation temperature of about 2 F although the slopes of the p r e d i c t e d H L i L l o c i i agree very w e l l w i t h those of the experimental data i n every case. The discrepancy between the experimental and p r e d i c t e d h y d r a t i o n temperatures probably r e s u l t s from a combination of s m a l l e r r o r s i n both the bubble p o i n t and hydrate programs. 2

2

2

Future Work Undef ined F r a c t i o n s . The example a p p l i c a t i o n s i l l u s t r a t e d above show the a b i l i t y of the PR equation to make acceptable f l u i d property p r e d i c t i o n s f o r a wide v a r i e t y of s i t u a t i o n s when the composition of the system i s f u l l y d e f i n e d . At the present time, a s u i t a b l e method has not been developed f o r c h a r a c t e r i z i n g the c r i t i c a l p r o p e r t i e s and a c e n t r i c f a c t o r f o r an undefined f r a c t i o n such as C y or i t s e q u i v a l e n t . Work i s c u r r e n t l y i n progress on t h i s problem. The importance of being a b l e to handle t h i s k i n d of s i t u a t i o n f o r indust r i a l systems i n v o l v i n g petroleum f r a c t i o n s can r e a d i l y be a p p r e c i a t e d . Water Content of Systems Containing CO? and H2S. Frequently r e s e r v o i r f l u i d s and petroleum f r a c t i o n s c o n t a i n s i g n i f i c a n t concent r a t i o n s of CO2 and H2S. When these f l u i d s are i n the presence of l i q u i d water and an e q u i l i b r i u m gas and/or l i q u i d phase i t i s of i n t e r e s t to be a b l e to c a l c u l a t e the water content of the gas and hydrocarbon l i q u i d . A s u i t a b l e scheme f o r making these p r e d i c t i o n s i s c u r r e n t l y being developed. P r e l i m i n a r y i n d i c a t i o n s are t h a t the 6 y values f o r water w i t h CO2 or H2S w i l l have to be temperature dependent. +

Nomenclature a b

-

a t t r a c t i o n parameter i n PR equation Van der Waals covolume In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

ROBINSON ET AL.

Peng-Robinson Equation of State

Figure 12. Comparison of experimental and predicted four-phase Hhth^G equilibrium in the methane-isobutane-water system

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

218

PHASE

EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L

INDUSTRY

NG AND ROBINSON (1976) NG AND ROBINSON (1976) VERMA (1974) PR PREDICTED PREDICTED HL,L G 2

Figure 13. Comparison of experimental and predicted hydrate-forming conditions in selected liquid mixtures in the region

50 TEMPERATURE

60 °F.

Table 4 Composition of Mixtures Used f o r P r e d i c t i n g Hydrate Formation i n the H^L^ Region

Component M Methane Ethane Propane Isobutane Nitrogen Carbon Dioxide n-Decane

Mixture I Mole % 2.2 30.6 50.8 16.2 0.2

c

Mixture I I Mole %

C

Mixture I I I Mole % 14.5

17.0 38.6 18.9

27.1

25.5 —

58.4

Ng and Robinson (13) Verma (16)

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

8.

G H Ll L P R T v x Z 2

ROBINSON ET AL.

" -

-

Peng-Robinson Equation of State

219

gas phase s o l i d hydrate phase w a t e r - r i c h l i q u i d phase l i q u i d phase r i c h i n hydrate former(s) pressure gas constant temperature molar volume mole f r a c t i o n compressibility factor

Greek L e t t e r s a 6 y a)

-

scaling factor i i n t e r a c t i o n paramete c h a r a c t e r i s t i c constant used f o r PR parameters acentric factor

Subscripts c i,j m -

c r i t i c a l property component i d e n t i f i c a t i o n mixture p r o p e r t y

Literature Cited

1. Peng, D.-Y., and Robinson, D. B., Ind. Eng. Chem. Fundamentals, (1976) 15, 59. 2. Soave, G., Chem. Eng. Sci. (1972) 27, 1197. 3. Redlich, O., and Kwong, J. N. S., Chem. Rev., (1949) 44, 233. 4. Chen, R. J., Chappelear, P. S., and Kobayashi, R., J. Chem. Eng. Data, (1974) 19, 53. 5. Chen, R. J., Chappelear, and Kobayashi, R., Chem. Eng. Data, (1974) 19, 58. 6. Peng, D.-Y, and Robinson, D. B., "A rigorous Method for Predicting the Critical Properties of Multicomponent Systems Using an Equation of State, " AIChE Journal, (1977) 23, 137. 7. Gibbs, J. Willard, "On the Equilibrium of Heterogeneous Substances," (October 1876 - May 1877). Collected Works, Vol. 1, p. 55, Yale University Press, New Haven, Connecticut, 1928. 8. Peng, D.-Y., and Robinson, D. B., "Two- and Three Phase Equilibrium Calculations for Systems Containing Water," Can. Journal of Chem. Eng., (1976) 54, 595. 9. McKetta, J. J., and Katz, D. L., Ind. Eng. Chem., (1948) 40, 853. 10. Wehe, A. H., and McKetta, J. J., J. Chem. Eng. Data, (1961) 6, 167. 11. Roberts, L. R., Wang, R. H., Azarnoosh, A., and McKetta, J. J., J. Chem. Eng. Data, (1962) 7, 484. 12. Parrish, W. R., and Prausnitz, J. M., Ind. Eng. Chem. Process Des. Develop., (1972) 11, 26.

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

13. Ng, H.-J., and Robinson, D. B., Ind. Eng. Chem. Fundamentals (1976) 15, 293. 14. Ng, H.-J., and Robinson, D. B., "Hydrate Formation in Condensed Systems," AIChE Journal, (1977), In Press. 15. Yarborough, L., and J. Chem. Eng. Data, (1972) 17, 129. 16. Gonzales, M. H., and Lee, A. L., J. Chem. Eng. Data, (1968) 40, 853. 17. Robinson, D. B., and Mehta, B. R., and J. Can. Pet. Tech., (1971) 10, 33. 18. Wu, B.-J., Robinson, D. B., and Ng, H.-J., J. Chem. Thermo., (1976) 8, 461.

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

9 Application of Equations of State in Exxon's Production Operations S. W. HOPKE Exxon Production Research Co., Houston, TX 77001

Equations of state tion of phase behavior in field separation facilities, pipelines, gas plants and condensate or volatile oil reservoirs. These simulation programs are executed scores of times each working day. There are several state equation methods available to our engineers. One of them, the BWRS equation, (1 - 3), is more accurate than the others and is used for almost all of our simulations that employ state equations. The BWRS equation is Starling's modification of the BenedictWebb-Rubin equation of state. It contains eleven adjustable pure component parameters plus a binary interaction parameter for each component pair. Thus, a typical 20 component mixture would be characterized by 220 pure component parameters and 180 different binary interaction parameters--a total of 400 constants. Exxon's set of constants were determined by multi-property regression, a procedure in which parameters are adjusted until available data on density, enthalpy, vapor pressure, K-values, sonic velocity, and specific heats are all matched simultaneously. The large number of constants to be determined requires that these data be accurate and that they cover a wide range of conditions. Nearly 20,000 data points were used to determine our set of constants. The large amount of data required limits the components that can be handled to the relatively few for which such data exist and may also place a practical limit on the number of parameters desirable in an equation of state. To be useful for typical petroleum production applications, a state equation must be able to predict the phase behavior of mixtures containing hydrocarbon fractions. The BWRS equation can do this for mixtures containing fractions with molecular weights as high as 500. The BWRS parameters for these fractions are obtained from correlations of the parameters, made dimensionless by dividing by appropriate powers of the critical temperature and critical volume, with the acentric factor. Accurate predicted K-values for light components in oils result from correlations of binary interaction parameters. Reliable simulations of absorber plant 221

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

222

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

performance are p o s s i b l e even when only the molecular weight and d e n s i t y o f the a b s o r p t i o n o i l s are known. In condensate and v o l a t i l e o i l r e s e r v o i r s , i n h i g h pressure s e p a r a t i o n f a c i l i t i e s , and i n p i p e l i n e s a s s o c i a t e d w i t h p r o d u c t i o n of r e s e r v o i r s i n remote l o c a t i o n s , phase behavior i s dominated by hydrocarbons h e a v i e r than C ^ Q . Since the c o n c e n t r a t i o n s and prop e r t i e s o f these heavy hydrocarbon f r a c t i o n s are not u s u a l l y known, r e l i a b l e phase behavior p r e d i c t i o n s f o r these produced f l u i d s a r e i m p o s s i b l e unless they are based upon experimental phase behavior data f o r the f l u i d i n q u e s t i o n . Consequently, to p r e d i c t the phase behavior o f these systems, we take the f o l l o w i n g approach. First, the composition o f the f l u i d i s measured t o C10+ and the molecular weight and d e n s i t y determined f o r each o f the C5 t o C10+ f r a c t i o n s . The C^o"" f r a c t i o n i s the the C I Q , C15, C 2 0 and i n t e r a c t i o n parameters of these heavy hydrocarbon f r a c t i o n s w i t h other components are adjusted u n t i l the model matches experimental phase behavior data f o r the f l u i d . The minimum phase behavior data r e q u i r e d i s the volume percent l i q u i d a t s e v e r a l pressures and at a constant temperature. Because of the l a r g e number o f constants i n v o l v e d , the BWRS method r e q u i r e s s l i g h t l y longer computation times than s i m p l e r , l e s s accurate methods. Considerable a t t e n t i o n to improving comput a t i o n e f f i c i e n c y has enabled us to use the BWRS equation i n t r a y t o - t r a y column c a l c u l a t i o n s and i n a one-dimensional c o m p o s i t i o n a l r e s e r v o i r s i m u l a t o r , a p p l i c a t i o n s which i n v o l v e l a r g e numbers o f s t a t e equation c a l c u l a t i o n s . While the BWRS equation o f s t a t e i s adequate f o r most o f our a p p l i c a t i o n s , there are problem areas. Some low temperature proc e s s i n g c o n d i t i o n s approach the c r i t i c a l r e g i o n , where c a l c u l a t i o n s l o s e accuracy and are d i f f i c u l t to converge. To date, we have not s y s t e m a t i c a l l y evaluated the accuracy near mixture c r i t i c a l p o i n t s nor s t u d i e d ways t o improve convergence i n t h i s r e g i o n . We have not had great success modeling water w i t h the BWRS equation. Water i s present i n most p r o d u c t i o n systems and we have had t o use separate c o r r e l a t i o n s f o r water behavior. Binary i n t e r a c t i o n parameters, e s p e c i a l l y f o r component p a i r s that g r e a t l y d i f f e r i n molecular s i z e , tend to be temperature dependent. This i n d i c a t e s t h a t the mixing r u l e s can be improved. In the f u t u r e , we p l a n t o continue i n c o r p o r a t i n g new data i n t o our c o r r e l a t i o n s as they become a v a i l a b l e . We a l s o await w i t h i n t e r e s t s t u d i e s c u r r e n t l y underway i n the u n i v e r s i t i e s t h a t might l e a d to f a s t , g e n e r a l i z e d equations o f s t a t e t h a t can handle water and p o l a r compounds w h i l e r e t a i n i n g adequate accuracy f o r e n g i neering a p p l i c a t i o n s . 1

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

9.

HOPKE

Equations of State in Exxon's Operations

223

Literature Cited 1. Lin, C. J. and Hopke, S. W., AIChE Symposium Series (1974), 70, No. 140, 37. 2. Hopke, S. W. and Lin C. J. presented at 76th National AIChE meeting, Tulsa, Oklahoma (1974). 3. Hopke, S. W. and C. J. Lin, Proc. 53rd Gas Processors Assn. Annual Conv. (1974) p. 63.

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

10 Phase Equilibria from Equations of State T. E. DAUBERT Pennsylvania State University, University Park, PA 16802

As I am last to make opening remarks I have decided to take a slightly different approach to the problem. The previous six speakers have alluded to the meaning and application of various equations--their advantages, disadvantages and accuracy. As co-director of the API Technical Data Book project my major interest is in ascertaining what equation is the most generally interpolatable and extrapolatable with reasonable accuracy over a wide range rather than what is the most accurate equation within a limited range of applicability. My major premise is: No equation of state is necessarily better than the data used to validate it. In order to be generally valid any equation must be tested over a wide range of molecular type and weight of components, a wide range of compositions, and the entire temperature and pressure span. We've evaluated many of the equations available and conclude that a decision as to the universally "best" possible equation cannot be made with current data. In addition, I personally would suggest that until more data are available further refinements of existing base equations may be empty exercises. Thus, what I would like to do is show some examples of the data that are available and then point out where the largest gaps exist and what must be done aboutit.My discussion will be limited to hydrocarbon systems and systems of hydrocarbons and the industrially important gases--H, N2, HS, CO, and CO. It will become apparent that the easy data already have been taken and the difficult data are yet to be determined. 2

2

2

Discussion Table 1 shows the data which have been gathered for our work from a comprehensive survey of the literature. All binary data were rigorously tested for thermodynamic consistency using the method of Van Ness and co-workers (jL, 2) using the general coexistence equation reduced to constant temperature or constant pressure according to the data source. It is readily apparent that less than 224

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. Nonhydrocarbon

Nonhydrocarbon Hydrogen Nitrogen Hydrogen + Hydrogen

Nonhydrocarbon Hydrogen Hydrogen S u l f i d e Carbon Dioxide Carbon Monoxide Nitrogen

Quaternary Hydrocarbon Hydrocarbon - Water Petroleum F r a c t i o n s

Ternary Hydrocarbon -

Binary Hydrocarbon -

Binary Hydrocarbon Ternary Hydrocarbon Quaternary Hydrocarbon F i v e and S i x Component Hydrocarbon

Type System

Sulfide

Types and Amounts of VLE Data A v a i l a b l e

Table 1

209 97 54

750 350 700 240 450

2836 670 134 60

430 240 323 156 184

1850

P o i n t s of Data Available "Consistent"

226

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L

INDUSTRY

t w o - t h i r d s of the b i n a r y data were c o n s i s t e n t . Thus, only the c o n s i s t e n t p o i n t s were used f o r data e v a l u a t i o n . The method was extended t o t e r n a r y systems. However, s i n c e necessary parameters f o r t e s t i n g o f t e n were not a v a i l a b l e f o r many data s e t s , a l l a v a i l a b l e data which were not o b v i o u s l y i n e r r o r were used f o r testing. From t h i s t a b l e i t i s apparent that data on t e r n a r y hydrocarbon-non-hydrcarbon systems are i n short supply as are data on a l l four and h i g h e r component systems. However, the former need i s the most c r i t i c a l f o r t e s t i n g purposes as e s t i m a t i n g equations which are a c c u r a t e f o r t e r n a r i e s , i n g e n e r a l , are accurate f o r higher multicomponent systems. The t a b l e a l s o shows no e n t r i e s f o r hydrocarbon-water o r petroleum f r a c t i o n systems. Such systems w i l l be d i s c u s s e d l a t e r Table 2 breaks dow Table 2 C o n s i s t e n t B i n a r y Hydrocarbon VLE Data

System Types A

Number o f Points

Carbon Number Range

B

B

A

r s

Paraffin - Paraffin

727

c

Paraffin - Olefin

160

c -c

Olefin - Olefin

c

2

?

C

2~ 10

C

C

2- 6

C

A 6" 7

C

29

C

P a r a f f i n - Naphthene

133

c

r 8

C

P a r a f f i n - Aromatic

450

c

r s

VS

O l e f i n - Aromatic

128

V 8

V 8

Naphthene - Aromatic

160

V 8

C

6" 8

C

C

7

C

7- 10

Naphthene - Naphthene

3

Aromatic - Aromatic

60

Total

1850

3 c

c

C

C

6

V 9 C

C

C

C

C

molecular type systems and carbon number range. Of systems of r e a l i n t e r e s t , date i s very sparse f o r naphthene-naphthene and aromaticaromatic systems. Data are n o n e x i s t e n t above C^Q f o r any of the system types.

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

10.

Phase Equilibria

DAUBERT

227

from Equations of State

Table 3 shows the d i s t r i b u t i o n of pressure ranges of the data Table 3 Pressure Range of B i n a r y Hydrocarbon Pressure Range (psia) P < 14.7 14.7

<

p <_ 100

100 £ p £ 1000 1000

<_

Data

Percent of Data P o i n t s 37 6 44

p

f o r the t o t a l b i n a r y hydrocarbon data s e t . Few data e x i s t i n the important range of 1 - 7 atm. o r above 65 atm. Table 4 l i s t s the c o n s i s t e n t b i n a r y hydrocarbon-nonhydrocarbon VLE data. Data are a g a i n e s s e n t i a l l y n o n e x i s t e n t above hydrocarbons c o n t a i n i n g ten carbon atoms. Aromatic data i s e s s e n t i a l l y l i m i t e d to benzene and naphthenic data are almost absent. Table 5 summarizes data f o r multicomponent systems. For hydrocarbons, the same d e f i c i e n c i e s e x i s t as were present f o r b i n a r i e s — only worse. Aromatic and naphthenic data are n o n e x i s t e n t . For nonhydrocarbon-hydrocarbon systems no data are a v a i l a b l e f o r CO o r CO2 and other data are extremely sketchy. Two areas l i s t e d i n Table 1 were not d i s c u s s e d — h y d r o c a r b o n water b i n a r i e s and petroleum f r a c t i o n s . For the former systems only e i g h t systems of experimental VLE data ( e t h y l e n e , propylene, 1-butene, 1-hexene, n-hexane, cyclohexane, benzene, and n-nonane) were a v a i l a b l e . Use o f s o l u b i l i t y data and c a l c u l a t e d VLE data added only f i v e more systems (propane, propyne, cyclopropane, nbutane, and 1,3-butadiene). For petroleum f r a c t i o n systems only three sources and seven systems have been c h a r a c t e r i z e d w e l l enough to use i n a n a l y t i c a l c o r r e l a t i o n s to be t e s t e d . These systems i n c l u d e three naphtha-fuel o i l systems, two hydrogen-hydrocrackate f r a c t i o n s , and two hydrogen-hydrogen s u l f i d e - h y d r o c r a c k a t e f r a c t i o n s . No reasonable work can be done without a d d i t i o n a l data. Conclusions The major c o n c l u s i o n of t h i s survey i s o b v i o u s — m o r e data are necessary i n order to v a l i d a t e an extended range, completely g e n e r a l equation of s t a t e . P a r t i c u l a r l y important areas f o r which new data must be taken or p r o p r i e t a r y data must be r e l e a s e d are l i s t e d below i n no p a r t i c u l a r order as p r i o r i t i e s depend on the users needs. 1. All systems w i t h hydrocarbon components above

C . 10

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

228

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Table 4 C o n s i s t e n t B i n a r y Hydrocarbon - Nonhydrocarbon VLE Data

Nonhydrocarbon

Hydrocarbon Type and Carbon Number Range

Hydrogen

P a r a f f i n s (C^ O l e f i n s ( C -* 0

c ) 12

Approximate Number of Data Points 280 60

3>

C

Naphthenes (C^ + C ) ?

50

Aromati 430 Hydrogen S u l f i d e

P a r a f f i n s (C

n

c

io>

220 13

Naphthene (C^)

7

Aromatic (Cg)

240 Carbon D i o x i d e

P a r a f f i n s (C.

282 C

10> 24

Olefin (C ) 3

Naphthene (C >

8

Aromatic (C^)

9

3

323 Carbon Monoxide

Paraffins

V

105 51

Aromatic (C^)

156 Nitrogen

P a r a f f i n s (CAromatic (C^)

C

10>

168 16 184

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

10.

Phase Equilibria

DAUBERT

229

from Equations of State

Table 5 3, 4, 5, and 6 Component Systems

Type System

No. of Systems

No. of Data Points

12 1 2

435 13 67

4

15

1

53

Hydrocarbon Ternary Hydrocarbon Paraffin (Ci - C ) P a r a f f i n - O l e f i n (Ci - C3 P a r a f f i n - Naphthene - O l e f i n (C5 - C7) M i s c e l l a n e o u s System w i t P a r a f f i n s O l e f i n s , Acetylene 1 0

Quaternary Hydrocarbon P a r a f f i n ( C - Cy) P a r a f f i n - O l e f i n - D i o l e f i n - Acetylene ±

(c ) 3

P a r a f f i n - O l e f i n - D i o l e f i n - Acetylene (C ) 5

F i v e and S i x 5 Paraffin 6 Paraffin 6 Paraffin

1

Carbon Atom Hydrocarbon (C-^ - C^Q) (C^ - C ) - O l e f i n (C4)

66 134

28 17 15

1 0

60

Nonhydrocarbon Ternary Nonhydrocarbon Hydrogen - Methane - Ethane Hydrogen - Methane - Propane Hydrogen - Methane - Ethylene N i t r o g e n - Methane - Ethane N i t r o g e n - Methane - Hexane Hydrogen - Hydrogen S u l f i d e - C^ aromatic or p a r a f f i n o r naphthene Quaternary Nonhydrocarbon Hydrocarbon N i t r o g e n - Methane Ethane Hydrogen - Benzene - Cyclohexane - Hexane Hydrogen - Hydrogen S u l f i d e - Methane Isopropylcyclohexane

82 30 97 44 53 54

1 1

360

7 36

1 52

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

230

P H A S E EQUILIBRIA

2. 3. 4. 5. 6.

A N D F L U I D PROPERTIES

IN C H E M I C A L

INDUSTRY

Aromatic and naphthenic b i n a r y and t e r n a r y hydrocarbon systems. Nonhydrocarbon-naphthenic and aromatic b i n a r y and t e r n a r y systems, e s p e c i a l l y hydrogen-aromatic-naphthenic systems. Water-hydrocarbon b i n a r y and t e r n a r y systems of a l l types. Water-hydrocarbon-nonhydrocarbon t e r n a r y systems. Petroleum f r a c t i o n and petroleum fraction-nonhydrocarbon systems.

Progress i n o b t a i n i n g new data i s e s s e n t i a l i f the users of equations of s t a t e ever hope to adopt a r e l i a b l e , dependable, and accurate p r e d i c t o r .

References Cited 1. Van Ness, H. C., "Classical Thermodynamics of Non-Electrolyte Solutions," Pergamon Press, Oxford (1964). 2. Van Ness, H. C., Byer, S. M., Gibbs, R. E. AIChE Journal, (1973) 2, 238. Acknowled gment The R e f i n i n g Department of the American Petroleum I n s t i t u t e i s g r a t e f u l l y acknowledged f o r f i n a n c i a l support of t h i s work.

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

11 Equations of State in the Vapor-Liquid Critical Region P. T. EUBANK Texas A&M University, College Station, TX 77843

Equations of stat analytic and those that are not. Analytic equations of the form P(ρ,T,[Z]) cannot provide an accurate description of thermodynamic properties in the critical region whether for the pure components or their mixtures. Scaled ES are non-analytic in the usual P (ρ,T) coordinates but assume analyticity inμ(ρ,T)for pure components. The choice of variables for a scaled ES for a mixture is not welldefined although Leung and Griffiths (1) have used P(T,[μ]) with success on the He-He system. Phase diagrams are simplier in such coordinates as the bubble-point surface and dew-point surface col lapse into a single sheet. Analytical equations operating outside the critical region generally correlate P-ρ-T-[Z] data with greater difficulty as one of the components is exchanged for a compound of greater acentricity or, particularly, polarity. Disclaimers usually accompany classical equations to discourage use with highly polar compounds. Steam is a good example--both the ES and correlation procedures used to produce steam tables differ considerably from those used with hydrocarbons. Scaled ES operating in the critical region, where intermolecular forces are not so important, do not incur additional difficulty with highly polar compounds. i

3

i

4

i

I. Correlation of Fluid Properties with Analytic ES Angus (2) has recently surveyed modern ES including 1. virial-type 2. extended BWR including Strobridge, Bender, Gosman-McCartyHust, and Stewart-Jacobsen 3. Helmholtz free energy equations as used by Pollak and by Keenan-Keyes-Hill-Moore for water/steam 4. orthogonal polynomials as in the NEL steam tables of 1964 5. spline functions as used by Schot in 1969 for steam 6. the Goodwin ES (non-analytic) 7. the Kazavchinskii ES. 231

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

232

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

The form of the ES p l u s the number and type of c o n s t r a i n t s to be p l a c e d upon i t s constants depends on the compound and the r e g i o n of reduced pressure and temperature over which the equation i s to be used. Thirst, we w i l l examine compounds t h a t are not h i g h l y p o l a r — i . e . , d i p o l e moment <0.5D. A. Non-Polar Compounds. I f we w i s h to f i t the data f o r n i t r o g e n ( T = 126.26K) above 200K, a Bender (3) ES, c

(P/pRT) = 1 + Bp + C p where B = n

2

+ Dp

1

- n /T 2

- n /T 3

C = n

6

+ n /T

+ n

D = n

9

+

E = n F =

n

0

?

+ Ep

4

+ Fp

- n./T 4

3

5

2

2

n

p2

+ (G + H p ) p e ~ " 2 0 ,

- n /T 5

(1)

4

c

n /T 1Q

+

n

2

Q

3

n /T 12

n /T 13

G = n /T

3

H = n /T

3

u

1 ?

+ n /T

4

+

+ n /T

4

+ n^/T ,

1 5

l g

n^/T

5

5

could be used w i t h only the f o l l o w i n g c o n s t r a i n t s : 1. the equation f o r B c o n t a i n i n g f i v e constants should accura t e l y reproduce the l i t e r a t u r e second v i r i a l c o e f f i c i e n t s over the d e s i r e d temperature range. 2. the constants i n the equation f o r C may a l s o be c o n s t r a i n e d to f i t any C (T) data and/or to p r o v i d e a C vs T curve i n agreement w i t h the Lennard-Jones p a i r - p o t e n t i a l ( 4 ) . 3. a t h i g h d e n s i t i e s the Bender equation could be f u r t h e r cons t r a i n e d to reproduce a hard-sphere ES. However, i f we wish to f i t both the l i q u i d and vapor PVT s u r face f o r n i t r o g e n a t temperatures above 80K, then a d d i t i o n a l cons t r a i n t s must be p l a c e d on the Bender c o n s t a n t s . As most of the p r o p e r t i e s i n the two-phase r e g i o n are f u n c t i o n s only of temperat u r e , are r e l a t e d by the Clapeyron e q u a t i o n , and o f t e n r e p r e s e n t measurements ( i . e . , heats of v a p o r i z a t i o n ) independent of the single-phase d e n s i t y data, i t i s g e n e r a l l y b e s t to f i x the c o e x i s tence dome f i r s t and l a t e r c o n s t r a i n the PVT s u r f a c e to c o n s i s t e n t l y pass through i t . D o u s l i n and coworkers (5) p r o v i d e e x c e l l e n t examples as to the d e t e r m i n a t i o n of accurate and c o n s i s t e n t two-phase p r o p e r t i e s . The c r i t i c a l c o n s t a n t s , the vapor p r e s s u r e curve, and the s a t u r a t i o n d e n s i t i e s are found g r a p h i c a l l y from the two-phase data. The heat of v a p o r i z a t i o n i s c a l c u l a t e d from the Clapeyron equation and checked f o r power-law b e h a v i o r a g a i n s t the temperature v a r i a b l e

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

11.

Equations of State in Vapor-Liquid

EUBANK

Critical Region

233

T = 1 as d i s c u s s e d by H a l l and Eubank ( 6 ) . This reference a l s o provides a method f o r accurate determination o f the vapor pressure slope a t the CP, *c =- r < V

d T

]

R l

T R

( 2 )

= i ,

which ranges from about f o u r f o r ^He to over e i g h t f o r water. The f o l l o w i n g a d d i t i o n a l c o n s t r a i n t s may now be placed on our Bender ES constants: 4. slope of c r i t i c a l i s o c h o r e a t CP equal to ty . 5. O P/8p ) = 0 a t CP, w i t h m = 1, 2, 3, 4. 6. obey Maxwell's equal area r u l e on each s u b c r i t i c a l i s o therm spaced 10K. At t h i s p o i n t there i s the danger t h a t our twenty-constant Bender equation has bee no f r e e constants l e f t der (2) g e n e r a l l y used c o n s t r a i n t s 1, 4, 5 (m = 1,2) and 6 (about 5 isotherms) which leaves seven f r e e constants. c

m

m

T c

_B. P o l a r Compounds. The same procedures can be a p p l i e d to h i g h l y p o l a r compounds except that i t i s d i f f i c u l t to a c c u r a t e l y represent the single-phase d e n s i t y data without a l a r g e number of unconstrained constants. For the water example, the P o l l a k equat i o n uses 40 c o n s t a n t s , Keenan-Keyes-Hill-Moore used 50 and Jusa used over 100 as d i d Schot i n her b i c u b i c s p l i n e f i t . The major d i f f i c u l t y concerning ES f o r p o l a r compounds i s the l a c k o f a c c u r a t e , comprehensive data o r , i n most cases, the l a c k of any data. Steam i s probably the only h i g h l y p o l a r compound f o r which there i s a l a r g e body of data f o r a v a r i e t y o f p r o p e r t i e s such as vapor pressure, c r i t i c a l c o n s t a n t s , s a t u r a t i o n d e n s i t i e s , heats o f v a p o r i z a t i o n , single-phase d e n s i t i e s i n both l i q u i d and vapor, c a l o r i m e t r i c , Joule-Thornson, s o n i c v e l o c i t i e s and heat capac i t y measurements (7_,jB,9) . Recent data (10) and c o m p i l a t i o n work (11) have improved the p i c t u r e f o r ammonia. There i s a l s o a r e a sonable body o f data f o r methanol but i t i s not a l t o g e t h e r thermodynamically c o n s i s t e n t (12). I n c o n s i s t e n c i e s are more l i k e l y to occur between sets o f PVT data from d i f f e r e n t l a b o r a t o r i e s due p r i m a r i l y to s u r f a c e e f f e c t s — p h y s i c a l a d s o r p t i o n and chemical r e a c t i o n . L i k e w i s e , c a l c u l a t e d enthalpy d e v i a t i o n s w i t h pressure a t constant temperature from d e n s i t y measurements are not so l i k e l y to agree w i t h the c a l o r i m e t r i c data as f o r non-polar compounds. II.

Non-Analytic ES

The reader may note t h a t there i s now a tendency to use some of the r e s u l t s c o n s i s t e n t w i t h s c a l i n g theory and the data i n conj u n c t i o n w i t h an assumed a n a l y t i c ES. Examples are (1) a nona n a l y t i c vapor pressure equation such as that of Goodwin o r of Wagner and (2) m = 3,4 i n c o n s t r a i n t 5, above. While t h e o r e t i c a l l y i n c o n s i s t e n t , an a n a l y t i c ES so c o n s t r a i n e d g e n e r a l l y provides a

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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b e t t e r approximation i n the c r i t i c a l r e g i o n although i n c o r r e c t a t the CP. Two d i f f i c u l t i e s that a r i s e i n using n o n - a n a l y t i c ES a r e that (1) they a r e not e a s i l y t r a c t a b l e and (2) they are not d e s i r able o u t s i d e the c r i t i c a l r e g i o n . By t r a c t a b l e , we mean the ease w i t h which one can d e r i v e an equation f o r a measureable q u a n t i t y , such as the heat of v a p o r i z a t i o n , from a reasonably simple s c a l e d ES, such as that due to V i c e n t i n i - M i s s o n i , L e v e l t Sengers and Green (13), i n the form o f u(p,T). One o f t e n d e r i v e s an expression i n v o l v i n g a power s e r i e s as i n Ref. 6 f o r the vapor pressure. Attempts to blend a n a l y t i c and n o n - a n a l y t i c ES along an a r b i t r a r y c r i t i c a l r e g i o n face the u s u a l d i f f i c u l t y o f b l e n d i n g d i f f e r ent a n a l y t i c e q u a t i o n s — d i s c o n t i n u i t i e s a r e l i k e l y to occur i n the second d e r i v a t i v e s of the PVT s u r f a c e ( i . e . , the heat c a p a c i t y ) plus the d i s s i m i l a r i t y of the form of the two equations A successf u l example i s the b l e n d i n m e t r i c s c a l e d ES by Chapel carbon d i o x i d e . One a l t e r n a t e procedure suggested by L e v e l t Sengers i s to use the a n a l y t i c ES as the background term and add on the necessary n o n - a n a l y t i c a l terms i n a manner such that they only cont r i b u t e i n the c r i t i c a l r e g i o n . A second a l t e r n a t i v e i s j u s t the r e v e r s e — t a k e the CP as reference and add a n a l y t i c terms to the s c a l e d ES as one moves away from the CP.

Literature Cited 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Leung, S. S. and Griffiths, R. B., Phys. Rev. A (1973), 8, 2670. Angus, S., "Guide to the Correlation of Experimental Thermodynamic Data on Fluids", IUPAC Thermodynamic Tables Project Centre, Department of Chemistry, Imperial College, London. (May 1975). Bender, E., Kaltetechnik-Klimatisierung 23, 258 (Sept. 1971). Mason, E. A. and Spurling, T. H., "The Virial Equation of State", p. 13, Pergamon Press, 1969. Douslin, D. R. and Harrison, R. H., J. Chem. Thermodynamics (1973), 5, 491. Hall, K. R. and Eubank, P. T., I&EC Fundam. (1976), 15, 323. Keenan, J. H., Keyes, F. G., Hill, P. G., and Moore, J. G., "Steam Tables", Wiley and Sons, Inc., New York, 1969. Levelt Sengers, J. M. H. and Greer, S. C., Int. J. Heat and Mass Transfer (1972), 15, 1865. Baehr, H. D. and Schomaecker, H., Forsch. Ingenieurw. (1975), 41 (2), 43. Baehr, H. D., Garnjost, H., and Pollak, R., J. Chem. Thermodynamics (1976), 8, 114. Haar, L. and Gallagher, J. S., "Thermodynamic Properties of Ammonia", Nat. Bur. of Stds., Washington, D. C. (to be published). Eubank, P. T., Chem. Engr. Sym. Ser. (1970), 66 (98), 16.

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

11.

EUBANK

Equations of State in Vapor-Liquid

Critical Region

235

13. Vicentini-Missoni, M., Levelt Sengers, J. M. H., and Green, M. S., J. Res. Natl. Bur. Stand. (1969), 73A, 563. 14. Chapela, G. A. and Rowlinson, J. S., J. Chem. Soc. (Faraday Trans.I) (1974), 70, 584.

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

12 On the Development of an Equation of State for Vapor-Liquid Equilibrium Calculations M. G. KESLER and B. I. L E E Mobil Oil Corp., Princeton, NJ 08540

Mobil's VLE and relate use in the modeling of refinery units, as well as petrochemical plants. Because of the diversity and overlap of these needs, we favor the use of an equation of state (EOS) for most thermodynamic calculations in support of process modeling. Our recent experience in the development of t Lee-Kesler generalized correlation and in the use of a modified Soave-Redlich-Kwong equation (SRK) give us reason to believe that t above goal is attainable, at least to meet current needs. We would like to outline a few criteria that might be helpful in guiding future work in this area. Desired Characteristics of an EOS for VLE Calculations An important requirement of an EOS, to fill the needs of the petroleum industry, is that i t apply to a wide range of hydrocarbon from methane,to components boiling above 1500F and to wide boiling mixtures of these components, including also H , H O N , O , CO2, C and H S. The EOS would need to cover temperature ranges of -250 to 1000F and pressures from vacuum to several thousand psi, correspond to reduced temperatures (T ) of 0.3 to 30 and to reduced pressures (P ) of 0.01 to 10. As a corollary of the above, the EOS should conform with the principle of corresponding states, that is, i t should be generaliza The advantages of a generalizable form are in the ease of developi an EOS and of extending its application. A generalized form of th coefficients should also help the development of mixing rules. It is worth noting that the BWR EOS shows excessive and erroneous dep dence on composition that can be traced to the specificity of the constants. A good starting point for the development of an EOS for VLE application is reasonable representation of the compressibility, this connection we have plotted in Figure 1, representation of the compressibility factor of n-octane for several isotherms, using th Lee-Kesler, Soave and Peng-Robinson (P-R) EOS's U , 2, 3). As is 2

2

2

2

2

r

r

236

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

KESLER

AND L E E

Vapor-Liquid

Figure 1.

Equilibrium

Calculations

Compressibility factor comparison

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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seen from the f i g u r e , the Soave r e s u l t s show s i g n i f i c a n t departure from the Lee-Kesler, p a r t i c u l a r l y near the c r i t i c a l and i n the l i q u i d r e g i o n . P-R shows much c l o s e r agreement w i t h the Lee-Kesler except near^the c r i t i c a l , where the P-R gives a unique Z of 0.307. Since the Z of substances of i n t e r e s t to the petroleum i n d u s t r y v a r i e s from about 0.2 to 0.3, t h i s i s a severe shortcoming of P-R and of other modified R-K EOS's. The Soave and P-R m o d i f i c a t i o n s of the R-K EOS have remarkably improved VLE r e p r e s e n t a t i o n . Nevertheless, we b e l i e v e that the above weaknesses lead to thermodynamic i n c o n s i s t e n c i e s , hence to d i f f i c u l t i e s of e x t r a p o l a t i o n and r e p r e s e n t a t i o n of thermal p r o p e r t i e s . We favor a three-parameter EOS, approaching the complexity of BWR, that would represent c o m p r e s s i b i l i t y w i t h greater accuracy than modified R-K EOS's do. c

Q

C h a r a c t e r i z a t i o n of Heavy Hydrocarbons (NBP>200F) Accurate r e p r e s e n t a t i o n of VLE i n petroleum p r o c e s s i n g by an EOS depends on the accuracy of input data that c h a r a c t e r i z e s the component or n a r r o w - b o i l i n g f r a c t i o n . I n t h i s connection i t would be h e l p f u l to review current methods of c h a r a c t e r i z i n g a t y p i c a l crude. The s t a r t i n g p o i n t f o r that i s a crude assay shown i n F i g u r e 2. The crude, b o i l i n g between 100 and 1000F, i s d i v i d e d i n t o f r a c t i o n s b o i l i n g w i t h i n a range o f , say, 25F. Each f r a c t i o n i s c h a r a c t e r i z e d by a b o i l i n g p o i n t (NBP) and a s p e c i f i c g r a v i t y (SG). Current pract i c e i s to d e r i v e from these two p r o p e r t i e s , the other p r o p e r t i e s needed f o r the EOS such as P , T , W, MW, e t c . The current s t a t e - o f t h e - a r t based on the corresponding s t a t e s p r i n c i p l e can be represented by: c

c

( C o r r e l a t i o n s based on NBP

and

SG)

f ( T , P , a)) = f [ T (NBP,SG),P (NBP,SG),O)(NBP,SG)]=0(NBP,SG) c c c c

(1)

We b e l i e v e that the above mapping can be improved, p a r t i c u l a r l y f o r hydrocarbons d e r i v e d from c o a l , s h a l e , e t c . , by i n t r o d u c i n g a t h i r d parameter, such as v i s c o s i t y . The proposed can be represented by: ( C o r r e l a t i o n s based on NBP, f(T ,P ,o)) c

c

SG, and

y)

= f [T (NBP,SG,y)^ (NBP,SG,y),co(NBP,SG,y) ]=0(NBP,SG,u) c

c

(2)

Two

of the reasons f o r the suggested r e l a t i o n (2) are:

1.

The i d e a l - v a p o r s p e c i f i c heats of naphthenes d i f f e r s i g n i f i c a n t l y from the s p e c i f i c heats of a mixture of n - p a r a f f i n s and aromatics w i t h the same NBP and SG as those of the naphthenes (4). The v i s c o s i t i e s of petroleum f r a c t i o n s w i t h the same NBP and SG are d i f f e r e n t from cracked than f o r v i r g i n stocks ( 5 ) . In f a c t ,

2.

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

12.

KESLER

AND L E E

Vapor-Liquid

Figure 2.

Equilibrium Calculations

Typical crude assay

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

v i s c o s i t y i s suggested as a t h i r d parameter because i t r e f l e c t s w e l l s t r u c t u r a l d i f f e r e n c e s between substances. In connection w i t h the crude assay, shown i n F i g u r e 2, r e l a t i o n (2) i m p l i e s that v i s c o s i t y would be measured simultaneously w i t h the NBP and SG, as a f u n c t i o n of the percentage d i s t i l l e d o f f the crude, hence as a f u n c t i o n of temperature.

Literature Cited 1. Lee, B. I. and Kesler, M. G., AIChE J., (1975) 21, 510. 2. Soave, G., Chem. Eng. Sci., (1972) 27, 1197. 3. Peng, D. and Robinson 15, 59. 4. American Petroleum Institute Research Project 44, Texas A & M University. 5. Watson, K. M., Wien, J. L. and Murphy, G. B., Ind. Eng. Chem. (1936) 28, 605.

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

13 Oil Recovery by C O Injection 2

RALPH SIMON Chevron Oil Field Research Co., LaHabra, CA 90631

In the U.S. ( e x c l u d i n g Alaskan North Slope) =420 b i l l i o n b a r r e l s of o i l have been d i s c o v e r e d ( F i g u r e 1 ) . Of t h i s amount 110 b i l l i o n have a l r e a d y been produced and an a d d i t i o n a l 30 b i l l i o n can be economically recovered a t present p r i c e s as primary o r secondary p r o d u c t i o n . Of the ^280 b i l l i o n unrecoverable by primary and secondary methods o n l y a f r a c t i o n can be recovered. P u b l i s h e d s t u d i e s suggest t h a t CO2 might recover 5 t o 10 b i l l i o n b a r r e l s . T h i s amount depends on economic parameters, t h e primary one being t h e p r i c e of crude o i l . The s u i t a b i l i t y of a s p e c i f i c r e s e r v o i r f o r CO2 i n j e c t i o n can be estimated from phase behavior measurements, p h y s i c a l p r o p e r t y c a l c u l a t i o n s , and displacement t e s t s i n v a r i o u s porous media. R e s e r v o i r s t h a t a r e candidates f o r CO2 i n j e c t i o n g e n e r a l l y have pressures exceeding 1500 p s i a (100 atmospheres) and crude o i l s w i t h s p e c i f i c g r a v i t y <0.87 (>30° A P I ) . In 1976 t h e r e a r e 6 commercial CO2 i n j e c t i o n p r o j e c t s ( 1 ) , and more a r e planned. I t ' s d o u b t f u l t h a t enough g o o d - q u a l i t y n a t u r a l l y o c c u r r i n g CO2 can be found f o r a l l of them. Displacement

Mechanisms i n C O 2 — R e s e r v o i r O i l Systems

The p h y s i c a l phenomena t h a t occur d u r i n g CO2 i n j e c t i o n can be e x p l a i n e d w i t h t h e a i d of a s e r i e s of s i m p l i f i e d network model drawings o f a porous medium. F i g u r e 2 r e p r e s e n t s a v i r g i n r e s e r v o i r c o n t a i n i n g o i l as t h e continuous phase and connate water i n i s o l a t e d c l u s t e r s of pores. F i g u r e 3 shows the r e s e r v o i r a f t e r a w a t e r f l o o d , i n d i c a t i n g the r e s i d u a l o i l not d i s p l a c e d by the water and the continuous water phase r e a c h i n g from i n l e t t o o u t l e t . F i g u r e 4 i l l u s t r a t e s the i n v a s i o n of CO2 f o l l o w i n g the w a t e r f l o o d . The advancing CO2 d i s s o l v e s i n the r e s i d u a l r e s e r v o i r o i l , causing i t t o s w e l l , decreasing i t s v i s c o s i t y , and v a p o r i z i n g the more v o l a t i l e components i n the o i l . The advancing vapor phase i s a l s o a b l e t o enter and d i s p l a c e pores c o n t a i n i n g r e s i d u a l o i l 241

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 H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

BILLION BARRELS US DISCOVERIES*

420

PRODUCED TO DATE

110

PRODUCIBLE WITH EXISTING PROCESSES AND ECONOMICS

30

FROM COn

5-10

EXCLUDING ALASKAN NORTH SLOPE

Figure 1

0

WATER

•

OIL

•

INJECTED WATER

ED c o

Figure 2.

(INTERSTITIAL)

2

Oil (continuous) with connate water

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

13.

SIMON

Oil Recovery by C0

2

243

Injection

0

WATER (INTERSTITIAL)

•

OIL

•

INJECTED WATER

•

Figure 3.

co

2

Oil after waterflood

0

WATER (INTERSTITIAL)

•

OIL

•

INJECTED WATER

E3 c o

Figure 4.

2

Oil with C0 injection 2

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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because of low i n t e r f a c i a l t e n s i o n between the advancing gas and r e s i d u a l l i q u i d . The combination of these phenomena enables CO2 to d i s p l a c e o i l i n e i t h e r secondary or t e r t i a r y c o n d i t i o n s . Phase and Flow Behavior C h a r a c t e r i s t i c s of C02 — R e s e r v o i r O i l Systems The mechanisms described i n the previous s e c t i o n are expressed q u a n t i t a t i v e l y i n Figures 5 through 10. Figure 5 shows CO2 s o l u b i l i t y i n o i l as a f u n c t i o n of pressure and temperature. Note that at t y p i c a l r e s e r v o i r c o n d i t i o n s e.g. 2,000 p s i a (135 atmospheres), and 170°F (77°C) the s o l u t i o n cont a i n s 65 mol percent CO2. At these c o n d i t i o n s the d e n s i t y of the

Figure 5.

C0

2

solubility in oil

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

13.

SIMON

Oil Recovery by C0

2

Injection

245

CO2 and some r e s e r v o i r o i l s a r e n e a r l y equal, thus m i n i m i z i n g CO2 gravity override. F i g u r e 6 shows o i l phase v o l u m e t r i c expansion. With 65 mol percent CO2 i n s o l u t i o n the s w e l l i n g can be 25 percent. As the o i l s w e l l s i t d i s p l a c e s adjacent o i l toward the producing w e l l s . F i g u r e 7 d i s p l a y s the v i s c o s i t y of C02~crude o i l m i x t u r e s . The data show that d i s s o l v e d CO2 can reduce v i s c o s i t y as much as 100 f o l d . This improves the m o b i l i t y r a t i o and decreases the tendency of CO2 to f i n g e r i n the h i g h p e r m e a b i l i t y paths. A comp l e t e d i s c u s s i o n of F i g u r e s 5, 6, and 7 and r e l a t e d measurements i s i n Reference 2. F i g u r e 8 shows the two phase boundary and c r i t i c a l p o i n t f o r a C 0 2 - r e s e r v o i r o i l system and notes t h a t a s o l i d p r e c i p i t a t e d a t

1.40

-

0

.20

X co

40

.60

2

Figure 6. Swelling factor vs. mol fraction C0

2

*vol @ safn press, ir temp vol @ 1 atm ir temp

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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A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

1.0 | -

0

1000

2000

SATURATION PRESSURE, PSIA

Figure 7.

Viscosity of C0 -crude oil mixtures at 120°F 2

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

13.

SIMON

Oil Recovery

Figure 8.

by CO

$

Injection

Pressure-composition diagram at 130°F: Oil A

American Chemical Society Library 1155 16th St., N.W. In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; Washington, D.C. 20036 ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

247

248

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CO2 c o n c e n t r a t i o n s >60 mol percent. C r i t i c a l p o i n t s of systems s t u d i e d ranged from 60 mol percent CO2 and 2570 p s i a (175 atmospheres) to 75 mol percent and 4890 p s i a (330 atmospheres). F i g u r e 9 p r o v i d e s "K" data f o r one C T ^ - r e s e r v o i r o i l system at one p r e s s u r e . For systems near the c r i t i c a l p o i n t , the K s approach 1.0 and a s i g n i f i c a n t amount o f o i l v a p o r i z e s . F i g u r e 10 i n d i c a t e s how i n t e r f a c i a l t e n s i o n between gas and l i q u i d phases v a r i e s w i t h CO2 c o n c e n t r a t i o n and p r e s s u r e , approaching zero near the c r i t i c a l p o i n t . The flow behavior of C 0 2 ~ r e s e r v o i r o i l systems i n porous media i s s t u d i e d by performing displacement t e s t s i n both bead-packed s l i m tubes (length/diameter >100) and c o n s o l i d a t e d sandstone and f

Figure 9.

KP vs. F plot for 55 mol % C0 , oil at 130°F 2

45 mol % reservoir

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

13.

SIMON

Oil Recovery by C0

2

249

Injection

δ s ο «s

to ©ι Ο

υ s S

Ι

to v. ce

ο

Ί to H to to .g

to Ο ο »-< to κ.

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

250

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

limestone cores. The tubes are used to determine the optimum pressure f o r o i l displacement (Figure 11 ( 3 ) ) ; the cores to measure o i l recovery versus pore volumes i n j e c t e d . F i g u r e 12 i n d i c a t e s the p o t e n t i a l o f CO2 to recover o i l from a p r e v i o u s l y waterflooded reservoir. Calculations The phase and flow behavior data d e s c r i b e d above are the b a s i s f o r c a l c u l a t i o n s used to evaluate new p r o j e c t s , design approved ones, and operate them e f f i c i e n t l y . These c a l c u l a t i o n s a r e done p r i n c i p a l l y w i t h a Compositional S i m u l a t o r . The main c a l c u l a t i o n

1001-

O AT71°F 80

AT C0 BREAKTHROUGH 2

Q LU CC LU >

60

o o

40

20 -

0 1200

1600 2000 FLOOD PRESSURE (PSIG)

Figure 11. Oil recovered from C0

2

2400

floods of 48-ft long sand pack

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

to ox

Figure 12. Oil recovery vs. HCPVI In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES

IN C H E M I C A L INDUSTRY

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

13.

SIMON

Oil Recovery by C0

2

Injection

253

i s a performance p r e d i c t i o n t h a t shows recovery versus time f o r a s p e c i f i e d number and l o c a t i o n o f w e l l s , i n j e c t i o n r a t e , r e s e r v o i r pressure and temperature, e t c . Compositional S i m u l a t o r c a l c u l a t i o n s a r e a l s o used to h e l p understand the e f f e c t s o f mass t r a n s f e r as CO2 advances through the r e s e r v o i r and d i s p l a c e s o i l . Two examples: F i g u r e 13 shows the C2+ i n the gas phase versus time and d i s tance. Note how the gas phase e n r i c h e d w i t h 20 mol percent C2+ moves through the r e s e r v o i r . F i g u r e 14 d i s p l a y s i n t e r f a c i a l t e n s i o n versus time and d i s tance. Here a low ( e s s e n t i a l l y zero) i n t e r f a c i a l t e n s i o n f r o n t i s shown moving through the r e s e r v o i r . Under these c o n d i t i o n s CO2 provides a m i s c i b l e displacement. Research In recent years there have been s i g n i f i c a n t advances i n o b t a i n ing data on C 0 2 - r e s e r v o i r o i l systems. However, more data a r e needed i n order to e v a l u a t e , design, and operate i n d i v i d u a l CO2 i n j e c t i o n p r o j e c t s w i t h o u t f i r s t making e x t e n s i v e , c o s t l y , and time-consuming experimental s t u d i e s . A d d i t i o n a l i n f o r m a t i o n i s needed p r i m a r i l y i n two c a t e g o r i e s : 1. P r e d i c t i n g the p r o p e r t i e s o f two-phase m i x t u r e s — s p e c i f i c a l l y , more r e l i a b l e K c o r r e l a t i o n s and i n t e r f a c i a l t e n s i o n equations. 2. Understanding mass t r a n s f e r and p r e d i c t i n g i t s e f f e c t versus time and d i s t a n c e as a CO2 f r o n t moves through the r e s e r v o i r .

Abstract Various authors have estimated that five to ten billion barrels of U.S. crude oil (excluding Alaskan North Slope) may be potentially recoverable by CO injection. This recovery will result when CO dissolves in the oil, swellsit,reduces its viscosity and lowers interfacial tension between the injected gas and the residual oil thus enhancing oil displacement. Also CO vaporizes the volatile part of the oil, carrying it from the reservoir in the gas phase. Physical property correlations and displacement tests with CO -reservoir oil systems are needed to calculate the effect of CO injection. The calculations are used in evaluating new projects, designing approved ones, and operating them efficiently. Research is needed to improve the reliability of present calculation methods, particularly for predicting the properties of two-phase mixtures. 2

2

2

2

2

Literature Cited (1)

O i l & Gas Journal, April 5 (1976).

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

254

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

to to ΰ ."co

to

I CO

•2 "co

s to

to to S

8

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

13.

SIMON

Oil Recovery by C0

2

Injection

(2) Simon, R., and Graue, D. J., J. Pet. Tech., (1965). (3) Holm, L. W., and Josendahl, V. Α., J. Pet. Tech., (1974).

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

255

14 A Computer Model for Calculating Physical and Thermodynamic Properties of Synthetic Gas Process Streams GRANT M. WILSON and MARK D. WEINER Brigham Young University, Provo, UT 84602

This paper describes our progress to date on development of a computer model based on a thermodynamically-consistent correlation that will accurately and reliably predict enthalpies, entropies, densities and ultimately K-values for synthetic gas systems over the range of typical synthetic gas processing conditions. A review of various gasification processes is given in Table 1. This table shows that process operating temperatures range from 550º to 3000ºF. The gasification of liquid hydrocarbons is done at temperatures ranging from 550ºF to 1000ºF, while coal gasification processes operate at temperatures from 500ºF to 3000ºF. Gasification pressures range from ambient to 1500 psia, and may be extended to 4000 psia. The principal chemical reactions of these processes can be characterized by the following equations. C Η + H O -> H + CO

(1)

CO + H 0 -> H + C0

(2)

x

y

2

2

CO +

2

2

3H

2

2

->

CH

4

+

HO 2

(3)

In some cases hydrogen is produced by reactions 1 and 2, and then the feedstock is hydrogenated to produce gas and oil. High operating temperatures are achieved either by pre-heating the reactants or by partial combustion using oxygen or air. Reactor products and by-products therefore consist of the following compounds. Hydrogen Carbon monoxide Carbon dioxide

Argon Hydrogen sulfide Carbonyl sulfide 256

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.

1700-3000 500-1870

l i q u i d hyd.

l i q u i d hyd.

l i q u i d hyd.

coal

coal

coal

coal

coal

coal

c o a l o r coke

c o a l or coke

coal

coal

Gasynthan

JGC MethaneR i c h Gas (MRG)

Shell Gasification

Agglomerating Gasification

Bi-Gas

CO^ Acceptor

Coal S o l u t i o n Gasification

Hydrane

Lurgi-SNG/Coal

Molten S a l t C o a l Gasification

Koppers-Totzek

U-Gas

Winkler G a s i f i c a t i o n

Burner

l i q u i d hyd.

CRG Hydrogasification

2000

700-1900

2900

1700

600-1850

1450-1800

not given

1600-2100

not given

not given

550-750

600-1000

Feedstock

Process (5) Temp. °F

not given

300-350

not given

1200

1500

1000

not g i v e n

150

1000-1500

100-400

not given

not g i v e n

300-500

375

Pressure, p s i a

Table 1, Process C o n d i t i o n s of Various G a s i f i c a t i o n Methods

3

3

4

H /CO

2

H /C0

2

H /C0

SNG, 950 B t u / f t

3

SNG, 950 + B t u / f t

95% CH

3

SNG, 1000 + B t u / f t

SNG, 950 + B t u / f t

SNG, 900 B t u / f t

2

H /C0

2

98% CH. 4 50% H , 45% CO

97% CH. 4 98% CH. 4

Product

3

1

I—

S3

to

<S

-4

CO

Co

< > s 9°

ο

Co

Ο a

S"

(S

S*

CO

M

w

α

>

F ο

nix

258

P H A S E EQUILIBRIA A N D

Water Methane Nitrogen Oxygen

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

Ammonia Light hydrocarbons Heavy hydrocarbons

The prediction and correlation of energy requirements, heat exchanger duty, chemical equilibria, and separation equilibria requires a knowledge of the enthalpy, compressibility, fugacity, and vapor-liquid equilibrium properties of these compounds and their mixtures. Existing prediction methods apply primarily to hydrocarbon mixtures at relatively low temperatures and their application to mixtures containing significant concentrations of water, CO , H S, and COS at high temperature i questionable Thus accurat prediction methods are neede methods or by development of new methods. A discussion of various alternatives for development of a new prediction method is given in the next section of this report. 2

2

Possibilities for New Prediction Methods Equation-of-state methods appear to be the most likely candidates for reliable accurate data. They are capable of predicting enthalpy, entropy, density, fugacity, vapor-liquid, and liquid-liquid data from one equation in regions where both low and high densities are encountered. Rapid computation requires that the equation be simple yet accurate without computational difficulties in areas surrounding the c r i t i c a l point. Possible candidates for this purpose are the following. 1. Modify existing correlations based on the Redlich-Kwong equation of state. a) Mark V (P-V-T, Inc., Houston, Texas) b) Soave (1) method 2. Adapt the BWR equation 3. Develop new equations of state Correlations based on the Redlich-Kwong equation of state have a "built i n " volume-dependence limitation. The equation has the right form for non-polar compounds but not for polar compounds. For example, i f one correlates the solubility of CO2 in water at 77 F; then one would expect that the correlation should also predict the solubility of water in CO2. This is not the case, since the predicted solubility of water in CO2 at liquid-liquid saturation i s 2.1 mole percent instead of the measured value of 0.25 mole percent (2). This represents a factor-of-eight prediction error. No amount of change of the temperature parameters w i l l correct this error because i t is related to the volume dependence of the equation. Since water is a principal component of synthetic gas processes, an accurate prediction of this mixture seems essential. The BWR equation has not been adequately tested for predicting

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

14.

WILSON

AND

Synthetic Gas Process Streams

WEINER

259

the p r o p e r t i e s of p o l a r and non-polar compounds. The author has some doubts whether i t would be s u i t a b l e . A second disadvantage i s t h a t the BWR equation has so many constants t h a t i t takes about f i v e times the computer time r e q u i r e d by the Redlich-Kwong equation. What i s needed then i s a new equation comparable i n comput a t i o n speed to e x i s t i n g Redlich-Kwong c o r r e l a t i o n s , but w i t h the c a p a b i l i t y of c o r r e l a t i n g the p r o p e r t i e s of both p o l a r and nonp o l a r mixtures. A new m o d i f i e d Van der Waals equation which meets these requirements i s d e s c r i b e d i n the next s e c t i o n of t h i s r e p o r t . A New M o d i f i e d Van der Waals Equation f o r Both P o l a r and Non-Polar Compounds and T h e i r M i x t u r e s Van Laar (3>4) used the Van der Waals equation to d e r i v e the now w i d e l y used Van Laa c i e n t s i n n o n - i d e a l mixtures

ln

Y

£ n

l

b l

2

+

(

^2 = x

V

x b/

(

= x.

2

Tx b/ V

l b l

<>

2

where b = Van der Waals b

a

a

= 1_ RT

1

a

l' 2

=

V

a

_^1 _ 2 2 b b

k

;

1

a

n

2

^

e rW a a

^

s

a

As d e r i v e d , the equation gave only approximate r e s u l t s ; but i t was found that by making the b s and a s e m p i r i c a l parameters a l a r g e number of mixtures could be a c c u r a t e l y c o r r e l a t e d . Of course, by t h i s method the parameters l o s e t h e i r p h y s i c a l s i g n i f i c a n c e as parameters i n the Van der Waals equation. N e v e r t h e l e s s , the equat i o n has proved to be a very u s e f u l equation, and i s now more a p p r o p r i a t e l y w r i t t e n i n the f o l l o w i n g form. T

T

x S l n

l n

Y

Y

(

l- x

2 -

I x s / 1 12 S

l S l

(x

B

2

T x s / 2 12 S

l S l

B

2

where S, B = e m p i r i c a l

3

<>

parameters.

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

4

<>

260

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

I f the o r i g i n a l Van Laar equation came from the Van der Waals equation, then what equation of s t a t e corresponds to the e m p i r i c a l Van Laar equation? I f t h i s equation of s t a t e were known, one would have an equation of s t a t e w i t h e m p i r i c a l parameters to a d j u s t f o r a s s y m e t r i c n o n - i d e a l behavior i n mixtures. The authors b e l i e v e t h a t such an equation i s d e r i v a b l e by assuming t h a t v o i d spaces i n a f l u i d can be considered as a d d i t i o n a l component of a mixture. When t h i s method i s used w i t h the e m p i r i c a l Van Laar equation, then the f o l l o w i n g m o d i f i e d Van der Waals (M-VDW) equation i s produced. (See the appendix for details.) S x S (A

V Z Z x .1 k

PV _ _ V V - b

/RT)

(

where b = molecular-volume parameter analogous to Van der Waals b A j k = energy parameter analogous to Van der Waals jk S = symmetry parameter from the e m p i r i c a l Van Laar equation B = Zxibi i S = ExiSi i A

This equation reduces to the form of the Van der Waals equation when S equals b. Thus the model i s i n agreement w i t h the o r i g i n a l Van Laar equation when S = b; t h i s confirms the method by which i t was d e r i v e d . The parameters Sj and Ajj/RT have been assumed to be temperature dependent as f o l l o w s . Tc. £

n

(

=

yv

C

(

j ~T

1 )

( 6 )

Tc. S

A

j j j

/

R

T

=

A

+

3

("T ) 1

Tc. +

Y

("T ) 1

+

Tc. 6

(

"T

1

)

+

Tc. . A

( 7 )

00

Equation (6) i s s i g n i f i c a n t because i t shows that as T -> then Sj b j . T h i s means that the equation approaches the Van der Waals form a t h i g h temperatures. The parameter S can be considered to be the e f f e c t i v e volume of a molecule r e s u l t i n g from i n t e r a c t i o n f o r c e s between the molecules. The t r a j e c t o r y of a molecule passing a c e n t r a l molecule i s changed as a r e s u l t of molecular i n t e r a c t i o n s , and passing molecules thus c o l l i d e more f r e q u e n t l y w i t h the c e n t r a l molecule than would otherwise be expected. When t h i s happens, the e f f e c t i v e s i z e of the molecule i s l a r g e r than i t s a c t u a l s i z e . At h i g h temperatures, the molecular i n t e r a c t i o n s

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

14.

WILSON

AND

WEINER

Synthetic Gas Process Streams

261

become s m a l l compared to the k i n e t i c energy of the molecules and the t r a j e c t o r y of a molecule i s n e g l i g i b l y a l t e r e d . When t h i s happens, the e f f e c t i v e s i z e of a molecule i s the same as the a c t u a l s i z e of the molecule. The parameter A j j has the u n i t s of energy per u n i t volume and i s analogous to the square of the s o l u b i l i t y parameter i n the Scatchard-Hildebrand equation. Numerous t e s t s of the ScatchardH i l d e r b r a n d equation have shown that there i s s i g n i f i c a n c e to the s o l u b i l i t y parameter. P r e v i o u s l y , there has been no way to i n t e r p r e t t h i s type of parameter from an equation of s t a t e . This new M-VDW equation could provide t h i s opportunity. P r e l i m i n a r y s t u d i e s show that the M-VDW equation can be adjusted to a c c u r a t e l y p r e d i c t the p h y s i c a l p r o p e r t i e s of water, hydrogen, carbon d i o x i d e and methane Comparisons w i t h e x p e r i mental data are given i comparisons between experimenta volume, vapor pressure, and saturated vapor c o m p r e s s i b i l i t y f a c t o r data f o r these compounds. These are p r e l i m i n a r y r e s u l t s , nevert h e l e s s the agreement between experimental and p r e d i c t e d data from the M-VDW equation are q u i t e good. With some exceptions, l i q u i d molar volumes are p r e d i c t e d w i t h i n about ±2%. This could be improved by assuming a s m a l l temperature dependence of the b parameter. Vapor pressure data appear to be p r e d i c t e d to b e t t e r than ±1%, and saturated vapor c o m p r e s s i b i l i t y f a c t o r s are accur a t e l y p r e d i c t e d except at c o n d i t i o n s c l o s e to the c r i t i c a l temperature where t h i s property changes q u i t e d r a s t i c a l l y w i t h only s m a l l changes i n temperature. The p r e d i c t i o n of these p r o p e r t i e s presumably can be improved by a d j u s t i n g e i t h e r the A or S parameters. One important c o n c l u s i o n from these comparisons i s that water p r o p e r t i e s can be p r e d i c t e d w i t h accuracy comparable to other non-polar compounds. This r e s u l t i s important i n d e t e r mining the s u i t a b i l i t y of the M-VDW f o r c o r r e l a t i n g and p r e d i c t i n g the p r o p e r t i e s of mixtures c o n t a i n i n g p o l a r and non-polar compounds. Tables 2 to 5 compare p r o p e r t i e s at temperatures below the c r i t i c a l temperature of the compounds. Tables 6 to 9 extend the comparisons to temperatures above the c r i t i c a l temperatures. The t a b l e s compare experimental and p r e d i c t e d c o m p r e s s i b i l i t y f a c t o r data. D e v i a t i o n s between experimental and p r e d i c t e d data are s m a l l except i n regions near the c r i t i c a l p o i n t . A l l equations of s t a t e tend to d e v i a t e i n t h i s r e g i o n , and the authors b e l i e v e the accuracy i s comparable to p r e d i c t i o n s from other equations of state. From past experience, the authors have found that i f other p h y s i c a l p r o p e r t i e s such as vapor pressure and c o m p r e s s i b i l i t y f a c t o r are a c c u r a t e l y p r e d i c t e d ; then enthalpy w i l l be a c c u r a t e l y p r e d i c t e d . This was found to be so i n t h i s case as i s shown i n Tables 10 to 15. Tables 20, 11, and 12 compare published and c a l c u l a t e d enthalpy data i n the s a t u r a t i o n regions f o r water, CO2

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.

.4231

.4252

600

-0.49

1543

680.8

.3623

.3675

500

-1.41

.3280

.3358

400

67.01 247.3

-1.97

.3082

.3144

300

11.53

.949

Exp^

-2.32

-0.23

.2993

.3000

200

%Diff

4.13

M-VDW

.3026

.2906

100°F

Temperature

P

Ex <*>

L i q u i d Molar Volume f t 3 / l b --mole

0.29 1.66

.9012 .8143 .6914

.8986 .8010 .6468

1.12 0.80 -0.32

250.07 686.26

6.90

-0.01 .9573 .9574

0.67

67.46

1538.07

-0.09

.9872

.9881

-0.30

11.50

%Diff

-0.09

M-VDW

.9978

Exp<*>

.9987

%Diff

Saturated Vapor C o m p r e s s i b i l i t y Factor

-0.11

.948

M-VDW

Vapor Pressure psia

Table 2, WATER, Comparison of P r e d i c t e d and Experimental L i q u i d Molar Volume, Vapor Pressure, and Saturated Vapor C o m p r e s s i b i l i t y Factor

3

I—I

a >

g

O X W

W H W

§

a

S

1

*i r

a

> >

S

F

w

W

to to

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

.4059

.4326

.4727

.5400

.6902

.4278

.4521

.4876

.5454

.6920

-430.87°F

-423.67

-416.47

-409.27

-402.07

smperature

M-VDW

3

-0.26

-0.99

-3.06

-4.31

-5.12

%Diff

L i q u i d Molar Volume ft /lb-mole

160.67

83.69

37.41

13.07

2.966

P

Ex ^

161.45

84.36

37.50

13.04

2.966

M-VDW

0.00

%Diff

0.49

0.80

0.24

-0.23

Vapor Pressure psia

1.23 5.52

.7164 .5446

.7077 .5161

0.31

.8304

.8278

-0.04

-0.17

%Diff

.9124

.9662

M-VDW

.9128

.9678

Exp^

Saturated Vapor Compressibility Factor

Table 3, NORMAL HYDROGEN, Comparison of P r e d i c t e d and Experimental L i q u i d Molar Volume, Vapor P r e s s u r e , and Saturated Vapor C o m p r e s s i b i l i t y Factor

3

85

3

o o Co

C/3

w

s

o

>

O

I—I

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

.5985

.6324

.6914

.7860

-40

0

40

(8)

-69.6°F

Temperature

EX£

.7949

.6916

.6253

.5887

M-VDW

3

1.13

0.03

-1.12

-.164

%Diff

L i q u i d Molar Volume ft /lb-mole (8)

567.3

305.8

145.87

75.15

Exp

561.9

308.2

147.2

74.7

M-VDW

%Diff

-0.95

0.78

0.91

-0.60

Vapor Pressure psia

.6714

.7923

.8713

.9142

Exp,

(8)

.6796

.7865

.8698

.9175

M-VDW

1.,22

-0,.73

-0.,17

0..36

%Diff

Saturated Vapor Compressibility Factor

Table 4, CARBON DIOXIDE, Comparison of P r e d i c t e d and Experimental L i q u i d Molar Volume, Vapor Pressure,and Saturated C o m p r e s s i b i l i t y F a c t o r

Hi

§ o <=n! e

•

g

O X W

§

M

H

M

§

a

w a > >

— ii

I—I

c

w

to

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

-0.56 -0.61 -0.70 0.36

.8933 .8142 .7082 .5564

.898 .819 .713 .554

0.23 1.22 1.65 1.43

151.83 301.91 534.54

150.0 297.0

0.52 2.65 5.37

.7144

.8135

1.0142

.7107

.7925

.9625

-190

-160

-130

527.0

64.65

64.5

-0.56

.6524

0.21 .9483

.946

0.20

%Diff

-0.69

21.56

.6561

21.71

-220

-250

M-VDW

.9817

0.00

-1.01

.5839

-280°F

ExpW

Saturated Vapor C o m p r e s s i b i l i t y Factor

.980

%Diff

4.90

M-VDW

.6098

4.90

ExpW

.6160

%Diff

-0.75

M-VDW

Vapor Pressure psia

.5795

nperature

Exp®

3

L i q u i d Molar Volume ft /lb-mole

Table 5, METHANE, Comparison of P r e d i c t e d and Experimental L i q u i d Molar Volume, Vapor P r e s s u r e , and Saturated Vapor C o m p r e s s i b i l i t y F a c t o r

ss

o

a

C/3

3

B

a

>

i

3 p

266

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

Table 6, WATER, Comparison of P r e d i c t e d and Experimental C o m p r e s s i b i l i t y Data i n Super Heat Region Data Obtained From (6)

700°F

a

1200°F

)

EXP

M-VDW

%DIFF

500

.9866

.9865

-0.01

-0.09

1000

.9727

.9733

0.06

.7341

1.86

2000

.9445

.9452

0.07

.6103

.6400

4.87

2500

.9302

.9315

0.14

3000

.4274

.5000

16.99

3000

.9159

.9179

0.22

4000

.1662

.1582

-4.81

4000

.8870

.8910

0.45

5000

.1940

.1834

-5.46

5000

.8579

.8644

0.76

PSIA

EXP

M-VDW

%DIFF

PSIA

500

.9443

.9422

-0.22

1000

.8809

.8801

1600

.7915

.7969

2000

.7207

2500

1600°F

900°F PSIA

EXP

M-VDW

%DIFF

500

.9953

.9946

-0.07

-0.09

1000

.9901

.9894

-0.07

.8996

0.00

1600

.9841

.9832

-0.09

.8723

.8732

0.10

2000

.9800

.9792

-0.08

2500

.8366

.8395

0.35

2500

.9750

.9743

-0.07

3000

.7998

.8035

0.46

3000

.9699

.9695

-0.04

4000

.7221

.7289

0.94

4000

.9596

.9587

-0.09

5000

.6397

.6488

1.42

5000

.9493

.9492

-0.01

EXP

M-VDW

%DIFF

PSIA

500

.9703

.9704

0.01

1000

.9390

.9382

1600

.8996

2000

a)

T h i s i s s l i g h t l y below the c r i t i c a l temperature of 706 F.

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

14.

WILSON

267

Synthetic Gas Process Streams

AND WEINER

Table 7, NORMAL HYDROGEN, Comparison of P r e d i c t e d and Experimental C o m p r e s s i b i l i t y Data i n Super Heat Region Data Obtained From (7)

-398 .47°F PSIA

a

-351 .67°F

)

PSIA

EXP

M-VDW

%DIFF

0.22

19.24

.9941

.9948

0.07

2.53

183.11

.9462

.9514

0.55

.3872

-1.05

876.18

.8488

.8582

1.11

.6302

-3.22

1540.14

.9552

.9582

0.31

EXP

M-VDW

%DIFF

17.75

.9708

.9729

150.93

.6882

.7056

221.91

.2890

.2784

443.52

.3913

833.26

.6512

-189 .67°F

-391 .27°F PSIA

EXP

M-VDW

%DIFF

EXP

M-VDW

%DIFF

19.94

.9757

.9777

0.20

80.90

1.003

1.0028

-0.02

159.45

.7803

.7935

1.69

1385.83

1.074

1.0668

-0.67

235.87

.6415

.6635

3.43

2192.64

1.132

1.1221

-0.87

368.28

.4292

.4360

1.58

3640.20

1.254

1.2398

-1.13

612.24

.5167

.5122

-0.87

5894.57

1.462

1.4552

-0.47

PSIA

80 .33°F

-380 ,47°F PSIA

M-VDW

%DIFF

1.007

1.0059

-0.11

3311.01

1.140

1.1325

-0.66

2.44

6126.76

1.266

1.2596

-0.51

.63.26

2.46

9005.71

1.396

1.3989

0.21

.6347

0.51 12526.87

1.553

1.5748

1.40

EXP

M-VDW

%DIFF

14.04

.9887

.9897

163.27

.8623

310.38

PSIA

EXP

0.10

162.39

.8726

1.19

.7287

.7465

467.33

.6174

747.00

.6315

a)

T h i s i s s l i g h t l y above the c r i t i c a l p o i n t a t -399.93°F.

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

268

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

Table 8, CARBON DIOXIDE, Comparison of P r e d i c t e d and Experimental C o m p r e s s i b i l i t y Data i n Super Heat Region Data Obtained From (8)

400°F

100 PSIA

EXP

M-VDW

%DIFF

200

.9885

.9886

0.01

2.90

400

.9763

.9775

0.12

.7080

5.75

800

.9534

.9547

0.14

.5921

2.28

1000

.9427

.9439

0.13

EXP

M-VDW

%DIFF

PSIA

200

.9260

.9386

1.36

400

.8471

.8717

600

.7585

.7961

800

.6695

1000

.5789

600°F

140 °F PSIA

EXP

M-VDW

%DIFF

200

.9955

.9966

0.11

3.31

400

.9912

.9934

0.22

.8477

5.40

600

.9865

.9904

0.40

.7337

.7882

7.43

800

.9822

.9875

0.54

.6853

.7230

5.50

1000

.9777

.9848

0.73

EXP

M-VDW

%DIFF

PSIA

200

.9406

.9522

1.23

400

.8727

.9016

600

.8043

800 1000

1000°F

240 °F PSIA

M-VDW

%DIFF

200

.9992 1.0020

0.28

0.23

400

.9986 1.0042

0.56

.9175

0.33

600

.9979 1.0063

0.84

.8859

.8891

0.36

800

.9972 1.0085

1.13

.8564

.8605

0.48

1000

.9964 1.0108

1.45

EXP

M-VDW

%DIFF

PSIA

200

.9718

.9737

0.20

400

.9433

.9455

600

.9145

800 1000

a)

EXP

This i s s l i g h t l y above the c r i t i c a l temperature at 88°F.

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

14.

WILSON

AND WEINER

Synthetic

Gas Process

269

Streams

Table 9, METHANE, Comparison of P r e d i c t e d and Experimental C o m p r e s s i b i l i t y Data i n Super Heat Region Data O b t a ined From (9) -100°F PSIA

500°F EXP

M-VDW

%DIFF

100

1.0016

1.0000

-0.16

0.37

300

1.0023

1.0030

0.07

.6820

1.44

600

1.0047

1.0064

0.17

.3217

.3045

-5.35

1000

1.0095

1.0115

0.20

2000

.4646

.4350

PSIA

EXP

M-VDW

%DIFF

PSIA

EXP

M-VDW

100

.9694

.9706

0.12

100

1.0016

1.0021

0.05

300

.9026

.9061

0.39

300

1.0062

1.0065

0.03

600

.7924

.8009

1.07

600

1.0126

1.0131

0.05

1000

.6250

.6378

2.05

1000

1.0215

1.0222

0.07

2000

.5206

.4931

-5.28

2000

1.0440

1.0438

-0.02

EXP

M-VDW

100

.9567

.9567

300

.8588

.8620

600

.6723

1000

%DIFF

PSIA

60°F

1000°F %DIFF

1500°F PSIA

EXP

M-VDW

%DIFF

PSIA

EXP

M-VDW

%DIFF -0.05

100

.9808

.9818

0.10

100

1.0023

1.0018

300

.9405

.9439

0.36

300

1.0062

1.0056

-0.06

600

.8788

.8867

0.90

600

1.0124

1.0112

-0.12

1000

.7974

.8097

1.54

1000

1.0206

1.0188

-0.18

2000

.6777

.6696

-1.20

2000

1.0420

1.0371

-0.47

100°F PSIA 100

2000°F

EXP

M-VDW

%DIFF

PSIA

.9906

.9915

0.09

100

M-VDW

%DIFF

1.0022

1.0012

-0.10 -0.19

EXP

300

.9711

.9752

0.42

300

1.0057

1.0038

600

.9440

.9502

0.66

600

1.0115

1.0076

-0.39

1000

.9108

.9200

1.01

1000

1.0192

1.0192

-0.64

2000

.8584

.8639

0.64

2000

1.0378

1.0260

-1.14

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

270

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Table 10 Comparison of C a l c u l a t e d Water Enthalpy Data w i t h L i t e r a t u r e Data i n theS a t u r a t i o n Region (10) Temp °F

Pressure PSIA

Enthalpy, BTU/LB

Calc 32.02 213.03

.089 a

40.8

LIQUID Diff Lit 0

VAPOR Calc

Lit

Diff

40.8

1080.5

1075.5

5.0

15.0

181.

320.28

90.0

292.2

290.7

1.5

1191.8

1185.3

6.5

377.53

190.0

354.7

350.9

3.8

1204.9

1197.6

7.3

414.25

290.0

395.9

390.6

5.3

1210.8

1202.6

8.2

444.60

400.0

430.6

424.2

6.4

1213.8

1204.6

9.2

503.08

700.0

500.0

491.6

8.4

1214.4

1201.8

12.6

567.19

1200.0

581.4

571.9

9.5

1205.3

1184.8

20.5

613.13

1700.0

645.1

636.5

8.6

1190.5

1158.6

31.9

662.11

2400.00

722.0

719.0

3.0

1163.2

1103.7

59.5

a)

Enthalpy base adjusted to f i t the l i q u i d enthalpy a t 213.03 F

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

14.

WILSON

AND WEINER

271

Synthetic Gas Process Streams

Table 11 Comparison of C a l c u l a t e d Carbon D i o x i d e Enthalpy Data w i t h Literature"'" Data i n the S a t u r a t i o n Region (11) Temp °

Pressure

F

P

S

I

„ , ., , , Enthalpy, BTU/LB T

A

Calc

mTT/TT

LIQUID Lit Diff

Calc

VAPOR Lit

Diff

-4. 6 -1.8

139.6

137.3

2.3

-69.9

75.1

-50.0

118.3

-6.4

145.9

0.0

0.0

—

140.1

137.9

2.2

-20.0

215.0

12.2

9. 2

3.0

140.8

138.7

2.1

0.0

305.8

23.8

18. 8

5.0

140.9

138.9

2.0

20.0

421.8

35.3

29. 6

5.7

140.2

138.5

1.7

40.0

567.3

47.0

41. 8

5.2

138.5

136.8

1.7

60.0

747.4

59.5

55. 7

3.8

135.5

132.2

3.3

80.0

969.3

73.9

74. 0 -0.1

129.8

119.0

10.8

-40.0

a

a)

-19.

Enthalpy base adjusted to f i t the l i q u i d enthalpy a t -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.

272

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Table 12 Comparison of C a l c u l a t e d Methane Enthalpy Data w i t h L i t e r a t u r e Data i n the S a t u r a t i o n Region (9) Temp o/

Pressure p

s

i

„ , , „ Enthalpy, BTU/LB n m T / T

A

Calc -280

4. 90

-260

LIQUID Diff Lit

-1934. 0 -1934.0

VAPOR Calc

Lit

Diff

—

-1706.1

-1705. 8

-0.3

13. 8

-1918

14. 7

-1917. 0 -1917.0

—

-1697.4

-1697. 5

0.1

-235

39. 0

-1897. 4 -1896.4 -1.0

-1689.2

-1689. 2

—

-210

87. 6

-1875. 6 -1875.2 -0.4

-1682.8

-1683. 0

0.2

-185

169. 7

-1852. 2 -1851.2 -1.0

-1679.5

-1679. 2

-0.3

-160

297. 0

-1825. 9 -1826.2

0.3

-1680.3

-1679. 2

-1.1

-135

482. 0

-1797. 0 -1795.3

1.3

-1688.3

-1686. 8

-1.5

673. 1

-1760. 1 -1730.0--30.1

-1708.6

-1730. 0

21.4

-258.,68

-116.,5

b

a

a)

Enthalpy base a d j u s t e d to f i t the l i q u i d enthalpy at -258.68°F.

b)

C r i t i c a l p o i n t of methane.

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

14.

WILSON

A N D WEINER

Synthetic Gas Process Streams

273

Table 13 Comparison of C a l c u l a t e d Water Enthalpy Data w i t h L i t e r a t u r e Data i n the Superheat Region (10) Temp °F 750

a

Pressure PSIA

Calc

1000

1361.8

1358.7

3.1

1500

1332.4

1328.0

4.4

2000

1299.6

1292.6

7.0

2500

1262.6

1250.6

12.0

3500

1158.9

1127.1

31.8

4000

933.8

1007.4

-73.6

6000

819.4

822.9

-3.5

8000

796.1

796.6

-0.5

10000

783.8

783.8

—

15000

769.6

769.7

0.1

250

1806.4

1800.2

6.2

500

1803.2

1796.9

6.3

Enthalpy, BTU/LB Lit

Diff

3000

1500

a)

750

1799.9

1793.6

6.3

1000

1796.7

1790.3

6.4

1500

1790.2

1783.7

6.5

2000

1783.8

1777.1

6.7

2500

1777.1

1770.4

6.7

3000

1770.6

1763.8

6.8

3500

1764.1

1757.2

6.9

4000

1757.6

1750.6

7.0

6000

1731.8

1724.2

7.6

8000

1706.4

1698.1

8.3

10000

1681.7

1672.8

8.9

15000

1624.6

1615.9

8.7

b

S l i g h t l y above c r i t i c a l temperature of water at 706 F.

b) The e r r o r of 6 BTU/lb i s an e r r o r i n the i d e a l gas enthalpy at 1500 F r e f e r r e d t o the l i q u i d a t 213 F. This e r r o r i s caused by a s m a l l e r r o r i n the i d e a l gas heat c a p a c i t y of water. This problem w i l l be c o r r e c t e d . In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

274

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

Table 14 Comprison of C a l c u l a t e d Carbon Dioxide Enthalpy w i t h L i t e r a t u r e Data i n the Superheat Region ( i i ) Temp °F •20

Pressure PSIA

Calc

Enthalpy, BTU/LB Lit

Diff

1.0

152.5

151.5

1.0

10.0

152.1

151.1

1.0

25.0

151.4

150.4

1.0

50.0

150.2

149.0

1.2

200.0

141.9

139.9

2.0

1.0

100.0

300

1000

219.6

218.8

0.8

10.0

219.5

218.8

0.7

25.0

219.3

218.7

0.6

50.0

218.9

218.4

0.5

100.0

218.3

217.8

0.5

200.0

216.9

216.4

0.5

400.0

214.2

214.0

0.2

600.0

211.4

211.4

0

1000.0

205.7

205.9

-0.2

1.0

399.9

399.0

0.9

10.0

399.9

399.0

0.9

25.0

399.8

399.0

0.8

50.0

399.8

398.9

0.9

100.0

399.6

398.9

0.7

200.0

399.3

398.8

0.5

400.0

398.7

398.4

0.3

600.0

398.1

398.0

0.1

397.0

397.2

-0.2

1000.0

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

14.

WILSON

A N D WEINER

Synthetic Gas Process Streams

275

Table 15 Comparison of C a l c u l a t e d Methane Enthalpy Data .th L i t e r a t u r e Data i n theSuperheat Region Temp

-60

Pressure PSIA

Calc

Enthalpy, BTU/LB Lit

Diff

1.0

-1595.7

-1595.6

-0.1

25.0

-1596.9

-1596.8

-0.1

50.0

-1598.2

-1597.9

-0.3

400.0

-1617.3

-1617.0

-0.3

600.0

-1630.0

-1629.8

-0.2

1000.0

-1662.0

-1661.9

-0.1

2000.0

-1727.3

-1720.8

-6.5

1.0

-1110.0

-1107.8

-2.2

25.0

-1110.1

-1107.9

-2.2

50.0

-1110.3

-1108.0

-2.3

100.0

-1110.4

-1108.3

-2.1

200.0

-1110.8

-1108.9

-1.9

400.0

-1111.5

-1109.8

-1.7

600.0

-1112.3

-1110.3

-2.0

1000.0

-1113.7

-1112.6

-1.1

2000.0

-1117.0

-1116.3

-0.7

1.0

518.5

518.6

-0.1

50.0

518.4

518.6

-0.2

100.0

518.7

518.8

-0.1

200.0

518.8

519.2

-0.4

400.0

519.0

519.8

-0.8

600.0

519.2

520.4

-1.2

1000.0

519.7

521.8

-2.1

2000.0

521.2

525.3

-4.1

100.0 200.0

700

2200

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

276

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

Table 16, WATER, Comparison of P r e d i c t e d and Experimental Second V i r i a l C o e f f i c i e n t s

T°K

M-VDW

G. S. K e l l

(12) Literature — M. P. V u k a l o v i c h

353.16

-561.4

-844.4

373.16

-465.75

-453.6

423.16

-308.59

-326

-283.3

473.16

-217.35

-209

-196.1

523.16

-160.18

573.16

-122.2

-117.

623.16

-95.67

-92.38

-90.2

673.16

-76.53

-73.26

-72.4

723.16

-62.24

-59.36

-60.6

773.16

-51.29

-50.4

823.16

-42.72

-42.0

923.16

-30.33

-29.4

1073.16

-18.75

-17.0

1173.16

-13.68

-11.6

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.

-316.61 -296.24 -223.46 -174.05 -151.70 -100.80 -69.99 -58.74 -49.35 -34.54 -14.64 -1.82 7.15 13.81 21.10

1. 2. 3. 4. 5. 6. 7.

E. R. K. A. K. M. B.

-46.3 -29.1

-147.4 -100.7 -69.5

1*

G. Butcher (12) S. Dadson (12, 13) E. MacCormack (L2, 13) Perez Masia (12, 13) Schafer (12, 13) P. V u k a l o v i c h (12_, 13) L. T u r l i n g t o n (12)

Primary Data Source:

203.83 209.03 233.34 258.15 273.15 323.15 373.15 398.15 423.15 473.15 573.15 673.15 773.15 873.15 1023.15

M-VDW

-70..2 -58,.4

-147. ,4

2

-50.59 -34.08 -13.58 -1.58 6.05 12.11

4

-142 -104, .3 -73,.9 -59..4 -52,.6

Literature

-156.36 -102.63 -71.85

3 -330 -302 -210

5

Table 17, CARBON DIOXIDE, Comparison of P r e d i c t e d and Experimental Second V i r i a l

-103.1 -73.1 -61.7 -52.5 -36.8 -15.9 -3.4 4.0 10.4 15.8

6

Coefficients

-173. ,5 -149. ,3 -104. ,5

7

278

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Table 18, HYDROGEN, Comparison of P r e d i c t e d and Curve F i t Experimental Second V i r i a l C o e f f i c i e n t s

T^K

M-VDW

15 25 50 100 200 300 400

-239.2 -106.0 -30.3 -1.6 9.7 13.0

Literature

(12) —

±5 ±3 ±2 ±1 ±0.5 ±0.5

-230 -111 -35 -1.9 11.3 14.8

Table 19, METHANE, Comparison of P r e d i c t e d and Curve F i t Experimental Second V i r i a l C o e f f i c i e n t s

T^K

M-VDW

110 150 200 250 300 400 500 600

-415.5 -204.8 -107.3 -62.7 -37.7 -11.1 2.6 11.0

(12) Literature— -344 -191 -107 -67 -42 -15.5 -0.5 8.5

±10 ±6 ±2 ±1 ±1 ±1 ±1 ±1

Table 20, P r e d i c t e d S o l u b i l i t y of Water i n Carbon Dioxide a t L i q u i d - L i q u i d S a t u r a t i o n from the M-VDW and Mark V Equations of State a t 25 C Solubility Mole % Measured (2)

Error %

0.25

Predicted

b

M-VDW

0.20

Predicted

b

Mark V

2.1

20 840

b) The CO2-H2O i n t e r a c t i o n c o e f f i c i e n t was adjusted t o f i t the measured s o l u b i l i t y of CO2 i n H2O which i s 2.55 mole % as reported by F r a n c i s . In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

14.

WILSON

>^

00 vO

O O

vO

O O

ON

d

rH rH

O

00

rH vO 00 00 00
OO

00 vO O 00

CN rH ON 00 rH

o

CN rH

vO

rH

00 rH

O

CN 00
O 00 vO CN 00 O
O

X>

00 LO
ON

u •H rH •H

279

Synthetic Gas Process Streams

AND WEINER

R>. OO

ON

•

O

O

rH

o

O

I

rH 00 VO VO ON O rH O rH rH

O

• • • rH rH rH

rH rH rH

•H CO CO CU

U P,

00 rH CN vO O N rH rH 00 vO 00 O N

VO

e o

M

CN rH vO 00

ON

rH vO rH O N O 0O O rH rH < F vO

IN

rH vO OO O rH CN rH rH rH

OO

CO

ON

ON

O

OO

ON

0O

N sj- 1^ (N 00 rH CN O

M

O o o

rH

O I

O rH CN

O

00 00 CM CM rH 00 O N

O O O

vO

O

O

rH

ON

CN

O

OO

rH 00 rH vO O N CO M vO

o

ON

00

O

O

O

rH CD PN G O CU O

•H

J-I rH CU A X

O

O

IN

CN

o o d

CN

vO CN vO M

OO vO 00 I

w A

O

CM rH OO rH O VO 00

crj < r 00 0) 4-1 A

•H

CO

0

CM CM O N 00 OO 00 O N

I

M

ON

M ON

vO

CM O

00 CM «N

O

o O o

o o O

rH

rH rH rH

rH rH rH

CO 00 00 00 M CM •sD .CM rH I N 00 00 O N O N O N

rH vO OO CN VO O N CO O rH 0O

o

J-I

rH

CU CO J-I

RN

CU

J-I

4-» O

CD

O

a, & CU

rH O vO 00 OO O

J-I

CU

CO •H J-L

CM vO IN OO OO

fx. ON

ON

O

O

ON

00 00

IN

rH

O

M

O rH vO VO

o o o o rH rH rH rH

H

00 rH O rH rH

crj 4J a. 6 crj o crj - crj

O

rH vO CN rH

vO 0O rH 00

O

d o

o d

rH CN

o o d

CO

ON!

00 rH

CN rH VO CN O rH CN

rH rH rH

rH rH rH

00 0O rH M

R» •
O

d d d

O

M

rH CN

rH

M

CN

O O

O

vO

CM Q O

CJ> I

PS W H

CN I N I N rH vO O ON 00 vO

ON

ON

ON

CO

ON

00

00

ON

ON

OO

vO 00

ON

ON

ON

O O O O

00 ON

rH

O O O O

ON

O O

O

O

O

O

O

CM CU

rH rO

crj EH

rH

rH rH CM O CO vO 00

ON CO

ON

ON

ON

ON

rH

Si

M

ON

P«4 O

CNO O U

^5 O

CM CN CM CM <J- C M O OO O C M
H

H

U O 0O

00 vO ON

O M ON

M M ON

o CM CM CN CN st (N O IN 00

O

CM s t rH rH rH

ON

M OO ON

O

ON

ON

ON

ON

ON

ON

CN

i P^

o

CM CN CM CN O O CN

O

CJ

M

vO rH rH O

O

pc<

o

CN CN CM

CN N
rH

O

00

CM O

CM

o

CN s j rH rH rH

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

280

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

and methane r e s p e c t i v e l y . Water d e v i a t e s on the average about 7 BTU/lb i n both the l i q u i d and vapor r e g i o n s except at h i g h pressures i n the vapor r e g i o n . This b i a s could be c o r r e c t e d by a d j u s t i n g the enthalpy base, and the e r r o r a t high pressures i n the vapor could be minimized by simultaneous f i t t i n g of m u l t i p l e p r o p e r t i e s . We cons i d e r the agreement here to be q u i t e good c o n s i d e r i n g that no enthalpy data have yet been used i n developing the model. Enthalpy data on CO2 i n Table 11 show s i m i l a r behavior except t h a t a s m a l l e r adjustment needs to be made. Data on methane i n Table 12 show almost p e r f e c t agreement except at the c r i t i c a l p o i n t where the computation method does not prove r e l i a b l e . Superheat data i n Tables 13, 14, and 15 a l s o show good agreement. Data on water i n Table 13 d e v i a t e near the c r i t i c a l p o i n t of water. Thus at 750 F and 4000 p s i a the l a r g e s higher and lower pressure there appears to be a constant b i a s i n the c a l c u l a t e d enthalpy of water, and t h i s e r r o r can be c o r r e c t e d by use of a b e t t e r i d e a l - g a s heat c a p a c i t y equation. Data i n Table 14 on CO2 agree very w e l l w i t h p u b l i s h e d data. Data on methane i n Table 15 a l s o show good agreement except at 700 F where d e v i a t i o n s of about -2 BTU/lb occur. This e r r o r i s probably due to i d e a l - g a s enthalpy e r r o r , and can be corrected. We suspect some v a r i a n c e between c a l c u l a t e d and measured data i s due to i n c o n s i s t e n c i e s between v a r i o u s s e t s of data. This i s shown i n Tables 16 to 19 where c a l c u l a t e d second v i r i a l c o e f f i c i e n t s are compared w i t h l i t e r a t u r e data from v a r i o u s sources on water, CO2, hydrogen, and methane r e s p e c t i v e l y . C a l c u l a t e d data i n a l l four t a b l e s appear to agree w i t h the experimental data almost w i t h i n the s c a t t e r of the data. This demonstrates two t h i n g s . One i s that the comparison depends on whose data you compare w i t h and the other i s that p r e d i c t a b i l i t y almost w i t h i n experimental accuracy has been made without f i t t i n g v i r i a l c o e f f i c i e n t data d i r e c t l y f o r both p o l a r and non-polar compounds. One problem i n p r e d i c t i n g values a t 3000 F i s the l a c k of experimental data. This means that t h e o r e t i c a l models f o r the second v i r i a l c o e f f i c i e n t w i l l have to be used i n order to extend to these h i g h e r temperatures. P r e d i c t e d data f o r mixtures of carbon d i o x i d e and water have been made at low temperature and at h i g h temperatures. These comp a r i s o n s are given i n Tables 20 and 2 1 . Table 20 compares the p r e d i c t e d s o l u b i l i t y of water i n l i q u i d carbon d i o x i d e a t l i q u i d l i q u i d s a t u r a t i o n at 25 C from the M-VDW and Mark V equations of s t a t e . This comparison shows a very l a r g e d i f f e r e n c e between the two equations of s t a t e . The M-VDW equation p r e d i c t s the s o l u b i l i t y w i t h i n 20% w h i l e the Mark V i s i n e r r o r by a f a c t o r of e i g h t . Because of the a d j u s t a b l e symmetry parameter, the M-VDW equation proves to be s i g n i f i c a n t l y s u p e r i o r to the Mark V. Presumably the Soave equation would g i v e r e s u l t s comparable to the Mark V because they both assume the Redlich-Kwong volume dependence. Table 21 compares experimental and p r e d i c t e d c o m p r e s s i b i l i t y data on C02~water mixtures at h i g h temperature from the M-VDW

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

14.

WILSON

AND

WEINER

Synthetic Gas Process Streams

281

equation of s t a t e . This comparison shows good agreement w i t h measured data where d e v i a t i o n s on the order of ±1% depending on the temperature and p r e s s u r e . Parameters used i n p r e d i c t i n g data given i n Tables 2 to 21 are summarized i n Table 22. Summary and Future Work The r e s u l t s of t h i s study show the f o l l o w i n g . 1. The M-VDW equation can be used to c o r r e l a t e both lowtemperature and high-temperature p h y s i c a l p r o p e r t y data. 2. P o l a r compounds such as water can be c o r r e l a t e d w i t h accuracy comparable to non-polar compounds. 3. Accurate l i q u i d - l i q u i the M-VDW equatio 4. The a s s y m e t r i c a c t i v i t y c o e f f i c i e n t behavior of n o n - i d e a l mixtures i s p r e d i c t a b l e . 5. This new equation should be comparable i n computation speed to e x i s t i n g Redlich-Kwong m o d i f i c a t i o n s such as Mark V o r Soave. Although not presented here, i t can be shown that the M-VDW parameters can be c a l c u l a t e d from group c o n t r i b u t i o n s . T h i s makes p o s s i b l e the c a l c u l a t i o n of heavy hydrocarbon parameters w i t h o u t a knowledge of the c r i t i c a l c o n s t a n t s . By t h i s method, p a r a f f i n , napthene, and aromatic s o l v e n t e f f e c t s can be taken i n t o account without knowing the a c t u a l compounds present i n a g i v e n hydrocarbon fraction. Work i s i n progress to g e n e r a l i z e the parameters i n the M-VDW model so that they w i l l be p r e d i c t a b l e from the a c e n t r i c f a c t o r and perhaps a p o l a r parameter. A l s o work i s planned to extend the model f o r p r e d i c t i n g enthalpy, heat c a p a c i t y , entropy, d e n s i t y , and phase behavior ( i n c l u d i n g l i q u i d - l i q u i d - v a p o r phase behavior) of water, hydrogen, carbon d i o x i d e , carbon monoxide, n i t r o g e n , methane, and other l i g h t hydrocarbons f o r temperatures from 0 F to 3000 F and pressures from atmospheric to 4000 p s i a . Subsequent to t h i s , the c o r r e l a t i o n w i l l be extended, a c c o r d i n g to present p l a n s , t o i n c l u d e systems c o n t a i n i n g i n t e r m e d i a t e and heavy o i l s and t a r s of the types encountered i n s y n t h e t i c gas p r o c e s s i n g . The model i s expected to take i n t o account the aromatic, naphthenic, and p a r a f i n i c nature of these o i l s and t a r s . U l t i m a t e l y i t i s planned to a l s o extend the M-VDW model to i n c l u d e low-temperature p r o c e s s i n g f o r hydrogen p u r i f i c a t i o n . Nomenclature Symbol a. l

Definition Van der Waals a

a!

d e r i v e d from a. and b., Eqns. 1-2

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.

-1.56706 6.05555 -2.09764

-0.256358

1.71075

2.49569

-0.374112 0

0.149

1.05

a

3

Y

6

a

B

C

0.372

0

1.12534

0.54

0.013

673.08

343.2

0.44

0.414

0

-0.0876456

0.676355

3.56806

-0.704135

Derived Constants

0.231

1072.81

547.57

0.348

3206.2

Pressure P s i a

Methane

C r i t i c a l Constants

Carbon D i o x i d e

A c e n t r i c Factor

1165.1

Temperature °R

Water

0.268

0.301

0.195429

-1.08136

1.87177

2.46398

0.00595060

-0.214

190.75

59.74

Hydrogen

Table 22, C r i t i c a l and Derived Constants Used i n Computer Program

14.

WILSON AND WEINER

Symbol a. , 1 J

Synthetic Gas Process Streams

283

Definition derived from B.: ^, A-^, and A ^ , Eqn. a-6

a^. ^

a^j i n t e r a c t i o n parameter when i = v o i d spaces

a

a y when both i and j = v o i d spaces

R H

A

Helmholtz f r e e energy

A^

I d e a l Helmholtz f r e e energy of mixin

A°

Helmholt component

A

M-VDW parameter i n Eqn. 5

A_„

i n t e r a c t i o n parameter defined by Eqn. a-4 b = Z.x.b. i l l b. = Van der Waals volume l

b b.

I

b

n

B

volume of "one mole" of v o i d spaces Van Laar parameter i n Eqn. a-1

B!. 1 J

Analogous Van Laar parameter i n Helmholtz f r e e energy Eqn., a-3

C

Temperature parameter f o r Sj i n Eqn. 6

G

Gibbs f r e e energy

E G .1

Excess Gibbs f r e e energy of mixing I d e a l Gibbs f r e e energy of mixing

G^

Free energy of pur component

n^

moles of v o i d spaces g i v e n by Eqn. a-7

n^

moles of component i

P

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

284

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

Symbol S.

IN C H E M I C A L INDUSTRY

Definition Van Laar parameter

1

Van Laar S^ f o r v o i d spaces S

S = . x.S. 1 i i

T

Absolute temperature

T c V

C r i t i c a l temperature (absolute)

Z

Volume

V mole f r a c t i o n of component i compressibility

factor

Parameters f o r c a l c u l a t i n g A. . i n Eqn. 7 33

Activity coefficient

Appendix D e r i v a t i o n of M-VDW Model The excess f r e e energy of mixing g i v e n by the Van Laar equation i s given as f o l l o w s . G Z E.Z.n.S.n.S.B.. — = .n.^ny. = - I - J - I - I - J - J - I J RT p

(a-1)

Wk

where B-j^ = 0. The t o t a l f r e e energy i s g i v e n as the f r e e energy of the components p l u s the i d e a l f r e e energy of mixing p l u s the excess f r e e energy of mixing as f o l l o w s .

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

14.

WILSON

AND

285

Synthetic Gas Process Streams

WEINER

I f the same a n a l y t i c a l form i s assumed f o r the Helmholtz f r e e energy, one o b t a i n s the f o l l o w i n g . A If

1

Z o , A , Z £ n_.S n S^B! . i i i RT J V V A

=

n

J

A

8

J

1

,

g

1 3

( a

"

o x

3 )

Wk where the prime i s used to show that the B j j parameters have d i f f e r e n t v a l u e s than B i j . The pure component f r e e energy A^ can be defined i n terms of parameters and A j ^ as f o l l o w s . 1

A° = S.A. . l l i i

(a-4)

This d e f i n i t i o n can be following.

1

A —

A , Z .Z,n,S, n.S .a. . =— + iJ ±k j j ij

, (a-5)

Wk where A

a. . = B! . + i i iJ iJ

+

A

lj

(a-6)

Y-^-

In the next step, we assume that v o i d spaces i n a f l u i d can be considered as an a d d i t i o n a l component i n c a l c u l a t i n g the Helmholtz f r e e energy. This i s done by assuming a molar volume f o r v o i d spaces so that the number of moles of v o i d spaces can be c a l c u l a t e d as f o l l o w s .

n

V

u

H

Z

n

b

= ~ k k k T H

(a-7)

b

where b^ = molar volume of molecules excluding b

v o i d spaces

= molar volume of v o i d spaces

S u b s t i t u t i o n i n t o Equation (a-5) g i v e the f o l l o w i n g .

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

286

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

1

A

A =— +

RT

RT

—

V ;— Z n

~

b

S

k k

V

+2(v

r n

H

v v

n . S. n.S.a. . k k 3 3 13

S

E k k

H

b

-Wk> a v

2

g + (v

2

-Wk> (b

2a

HH

V The

(a-8)

f r e e energy of a v o i d space i s zero, so a

HH

S

0

( a

"

9 )

A l s o a t i n f i n t e volume, the f r e e energy equals the f r e e energy of an i d e a l gas a t u n i t pressure: |_,

. !a 2

V=°°

?

y

j

H

.

< s

_

1 0 )

H

\ or

A?=2b S.a . H

(a-11)

H

When Equation (a-9) i s s u b s t i t u t e d i n t o (a-8) and the equation i s rearranged a l g e b r a i c a l l y we o b t a i n the f o l l o w i n g . S

H Iln.S.n.S.a. . + 2 (V-Ek. IL b. ) — Z n . S . a . A , , 1 1 3 3 13 He k k'b 3 3 Hj A_ 12 M3 RT S u

A

-A RT

1

=

+

( -i2) a

rl In the Van Laar equation, as f o l l o w s .

f

one of the S s i s a r b i t r a r y ; so we d e f i n e

(a-13) With t h i s change our f i n a l Helmholtz f r e e energy equation i s as follows.

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

14.

WILSON

A

A

IZn.S.n.S.a. . + 2(V-£, n. b. ) — Zn.S.a . ia J 3 !J k k k b j j Hj 1

1

1

— =— + RT RT

287

Synthetic Gas Process Streams

A N D WEINER

u

1

2

_ , ? k

k

(a-14) K

._ _ k^

J

k

The equation of s t a t e i s d e r i v e d from the d e r i v a t i v e of A w i t h respect to volume as f o l l o w s .

P

(

( a

" - W>T,n

"

1 5 )

The r e s u l t i n g equation i „

5

RT

. I /

D

I.Z.n.S.n.S.A../RT

T ;

<• 3V

T,n

( V - I ^ l ^ +

Z ^ S J

where A., /RT i s d e f i n e d as f o l l o w s . lk

a

^

+

= " i3

a

+

Hi

a

( a

Hj

"

1 ? )

The d e r i v a t i v e o f the i d e a l Helmholtz f r e e energy of mixing w i t h respect t o volume i s assumed t o given by the V-Z.n.b. term i n the Van der Waals equations; o r

v

Z

MVRT) y

8V T , n

=

n

i i

(a v

V-E.n.b. i l l

- ) 18

'

Thus our f i n a l equation, c a l l e d the M o d i f i e d Van der Waals equation, i s as f o l l o w s . E.n. E.E.n.S.n.S.(A../RT) -L = L L _ 3 .1 ,1 *3 RT V-Zn.b. (V + E n S Zn£) l 1

1

1

2

k

or

Z = ^~ V-b

V

1

k

J _

k

(a-19)

k

S.x.S. (A. ./RT) J J ^ (V + S-b)

X

This equation i s the same as Equation

(a-20)

(5) i n t h i s paper.

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

P H A S E EQUILIBRIA

288

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

This equation i s unique because the mixture r u l e s a r e already d e f i n e d . A l s o Van Laar parameters d e r i v e d from f i t t i n g b i n a r y and multicomponent v a p o r - l i q u i d e q u i l i b r i u m data r e l a t e d i r e c t l y to parameters i n the equation of s t a t e . By simple s u b s t i t u t i o n i t can be shown that A j / R T i n Equation (a-17) i s r e l a t e d to Bj— i n Equation (a-3) by the equation: k

iJr

= - V

^

k

( a

-

2 1 )

This s u b s t i t u t i o n i s not q u i t e as simple as i t appears because the Bjk parameters a r e defined as Van Laar i n t e r a c t i o n parameters when no v o i d spaces a r e present t h i i d i f f e r e n t fro th l defi nition. Although not shown here, i t can be shown that the S^, and A..^ parameters can be c a l c u l a t e d from group c o n t r i b u t i o n s as i n the c a l c u l a t i o n of a c t i v i t y c o e f f i c i e n t s from group c o n t r i b u t i o n s .

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Soave, G., Chem. Eng. Sci. (1977) 27, 1197. Francis, A., J. Phys. Chem. (1954) 58, 1099. van Laar, Z. Physik. Chem. (1910) 72, 723. Prausnitz, J. M., "Molecular Thermodynamics of Fluid Phase Equilibria," Prentice Hall, Englewood C l i f f s , New Jersey, 1969, pp. 264-269. "Gas Processing Handbook" Section in Hydrocarbon Processing (1975) 54, No. 4, 112-125. Perry, J. H., "Chemical Engineer's Handbook," 4th Edition, McGraw-Hill, New York, 1968. NBS Technical Note 120 (November 1968). Sweigert, R. L., Weber, P. and Allen, R. L., Ind. & Eng. Chem. (1946) 38, 185-200. Conjar, L. N. and Mannuig, F. S., "Thermodynamic Properties and Reduced Correlations for Gases," Gulf Publishing Co., Houston, 1967. Meyer, C. A., McClintock, R. B., Silvestri, G. J. and Spencer, R. C., Jr., Editors, "Thermodynamic and Transport Properties of Steam," ASME, United Engr. Center, New York, New York, 1967. "ASHRHE Thermodynamic Properties of Refriguants," Am. Soc. of Heating and Refrigerating and Air Conditioning Engineers, Inc., New York, 1969. Dymond, J. H. and Smith, E. B., "The Virial Coefficients of Gases--A Critical Computation," Clarendon Press, Oxford, 1969. Vukalovich, M. P. and Altunin, V. V., "Thermophysical Properties of Carbon Dioxide," Collets Publishing Ltd., London and Wellingborough, 1969. Greenwood, H. J., Am. J. Sci., Schairer (1969) 267-A, 191-208.

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

15 Thermophysical Properties: Their Effect on Cryogenic Gas Processing D. G. ELLIOT, P. S. CHAPPELEAR, R. J . J. CHEN, and R. L . MCKEE McDermott Hudson Engineering, Houston, TX 77036

The changing economi industry has precipitated the design and construction of gas pro cessing plants to recover 80 percent or more of the contained ethane. Typically, high ethane recoveries are attained by employing a cryogenic process utilizing a turboexpander. Historically, the availability of fundamental thermophysical property data required to design gas processing plants has lagged behind the design and construction of such facilities (1). The cryogenic process is no exception. Due largely to the efforts of the Gas Processors Association, pertinent experimental data has been taken and correlated by several methods. These thermophysical property correlations are generally available to the gas processing industry. It is the authors' intent to illustrate the relationship between process design and accurate prediction of K-values, enthalpy, entropy, and CO solubility. 2

Scope In 1970, White et al (2) demonstrated the importance of accurate K-value correlations in cryogenic plant design by comparing predicted product recoveries for a typical cryogenic plant using several correlations available at that time. The various K-value correlations gave predicted recoveries for ethane from 25.7% to 45.0% and for propane from 66.7% to 90.7%. Since that time much progress has occurred in correlations. Existing correlations have been refined and new correlations have been introduced. It is the authors intent to demonstrate the importance of accurate thermophysical properties on plant design and to compare the properties predicted by generally available correlations. This has been done by comparing predictions both from K-value correlations and enthalpy/entropy correlations for the major components in a typical cryogenic process. In addition, the predicted CO concentrations near the conditions of CO solids formation, a condition limiting ethane recovery in many plants, are presented. The correlations examined, listed in Table I, are of three 1

2

2

289

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

290

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

TABLE I CORRELATIONS Description

Symbol Empirical:

GPA CONV

Reference

K-values from Hadden convergence pressure concept (1972 e d i t i o n ) w i t h CH4, C2H5, 3 3 from b i n a r y data a t low temperatures. K-values (convergence) computer c a l c u l a t e d from GPA C o e f f i c i e n t fit C

(4)

H

HUDSON

McDermott Hudson p r o p r i e t a r y c o r r e l a t i o n f o r K-values of CH4, C 2 H 6 , and C 3 H 8 . Uses GPA CONV f o r h e a v i e r components .

MARK V

Wilson m o d i f i c a t i o n of the Redlich-Kwong equation of s t a t e ; computer program marketed by PVT, I n c .

Equation of State: van der Waals

BWR

(6)

LEE

Lee e t a l . c o r r e l a t i o n as programmed i n the GPA computer package K & H.

SOAVE

Soave m o d i f i c a t i o n of the (8) Redlich-Kwong equation of s t a t e , as programmed i n the GPA computer package K & H.

P-R

Peng-Robinson m o d i f i c a t i o n (9) of the Redlich-Kwong equat i o n of s t a t e . Values computer c a l c u l a t e d from c o r r e l a t i o n by P-R d i s t r i b u t e d by GPA.

SH BWR

Starling-Han m o d i f i c a t i o n of the Benedict-WebbRubin equation of s t a t e as programmed i n the GPA computer package K & H.

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

(Z)

(10) (11)

15.

ELLIOT E T A L .

Cryogenic Gas Processing

291

TABLE I (Continued)

Iipe

Symbol

Description

Conformal S o l u t i o n :

K VAL

1972 program f o r K-value by CHEMSHARE f o r h i g h methane content streams. Based on b i n a r y K data f o r t e n r e f e r e n c e systems.

PROP-75

Property-75 R i c e U n i v e r s i t y shape f a c t o r c o r r e s ponding s t a t e s program f o r

Reference

(12)

r e f e r e n c e . 1975 r e v i s i o n of computer program d i s t r i b u t e d by GPA. K DELTA

T. W. Leland corresponding (13) states c o r r e l a t i o n using data f o r b i n a r y systems as r e f e r e n c e . Computer program marketed by S i m u l a t i o n Science Inc. ( S S I ) .

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

292

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

types: e m p i r i c a l , equation of s t a t e , and conformal s o l u t i o n . E m p i r i c a l c o r r e l a t i o n s are graphs of experimental data p l o t t e d a g a i n s t a c o r r e l a t i n g parameter. Equation of s t a t e c o r r e l a t i o n s a r e d i v i d e d i n t o two types: one i s s i m i l a r to the van der Waals equation w i t h a s m a l l number of parameters; the other i s based on the m u l t i p l e cons t a n t BWR equation. A l l equation of s t a t e c o r r e l a t i o n s i n v o l v e the computation of the parameters f o r the equation of s t a t e which w i l l represent the multicomponent mixture. These computations a r e based on parameters determined f o r the pure components, combining r u l e s , and p o s s i b l y b i n a r y i n t r a c t i o n parameters. The conformal s o l u t i o n or corresponding s t a t e s c o r r e l a t i o n computes pseudo-reduced c o n d i t i o n s f o r the mixture to conform to the r e f e r e n c e substance. The r e f e r e n c e substance may be a pure component or a m i x t u r e ; i t s p r o p e r t i e format. The computer programs used i n t h i s study a r e c u r r e n t v e r s i o n s . Other programming of the same equations may g i v e s l i g h t l y d i f f e r e n t r e s u l t s . The a b b r e v i a t i o n s l i s t e d i n Table I are used throughout the t e x t . Design B a s i s and Process D e s c r i p t i o n P r o p e r t i e s p r e d i c t e d by these c o r r e l a t i o n s determine not only the economic f e a s i b i l i t y of gas p r o c e s s i n g i n s t a l l a t i o n s ; they a l s o d i c t a t e the d e s i g n ( s i z e ) of i t s major components. These components are u s u a l l y the compressor, turboexpander, heat exchangers, e x t e r n a l r e f r i g e r a t i o n ( i f any), and demethanizer. A t y p i c a l arrangement f o r a cryogenic gas p l a n t of these components i s shown i n F i g u r e 1. The f o l l o w i n g process c o n d i t i o n s are s e l e c t e d : 1. 2. 3. 4. 5. 6. 7. 8.

I n l e t f l o w r a t e of 70 MMscfd I n l e t gas pressure of 250 p s i a I n l e t gas temperature of 120°F High pressure separator a t -80°F and 800 p s i a Expander o u t l e t pressure a t 200 p s i a Expander i s e n t r o p i c e f f i c i e n c y a t 80% (commonly c a l l e d adiabatic) Compressor i s e n t r o p i c e f f i c i e n c y at 75% (commonly c a l l e d adiabatic) I n l e t gas composition i n Table I I .

In t h i s t y p i c a l p l a n t , the i n l e t gas i s compressed from 250 p s i a to 815 p s i a . I t i s then cooled by an a i r c o o l e r to 120°F f o l l o w e d by exchanging heat w i t h the demethanizer r e b o i l e r and the p l a n t r e s i d u e gas. Any a d d i t i o n a l r e f r i g e r a t i o n r e q u i r e d to c o o l the i n l e t gas to -80 F b e f o r e e n t e r i n g the h i g h pressure separator i s f u r n i s h e d from an e x t e r n a l source. The l i q u i d separated i n the h i g h pressure separator i s f e d to the demethanizer where the methane i s s t r i p p e d out. Energy i s recovered from the h i g h pressure separator vapor by reducing the p r e s s u r e through a turboexpander. The expander o u t l e t i s much c o l d e r and p a r t i a l l y condensed. This energy i s converted to

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

15.

293

Cryogenic Gas Processing

ELLIOT E T AL.

TABLE I I - COMPOSITIONS USED IN CALCULATIONS

Component

I n l e t Gas Mol % Mol/Day

Residue Gas Mol % Mol/Day

L i q u i d Product Mol % Mol/Day 0

Nitrogen

0.40

738

0.44

738

Carbon D i o x i d e

0.52

959

0.30

498

2.91

461

Methane

90.16

166,308

98.52

166,124

1.16

184

Ethane

4.69

8,651

0.70

1,181

47.15

7,470

Propane

1.85

3,412

0.04

73

21.07

3,339

Isobutane

0.79

Normal Butane

0.51

941

5.94

941

Isopentane

0.27

498

3.14

498

Normal Pentane

0.18

332

2.10

332

Hexane P l u s *

0.63

1,162

7.33

1,162

100.00

15,844

100.00 *A11 c a l c u l a t i o n s

184,458 100.00

168,614

used n-Heptane f o r the Hexane P l u s f r a c t i o n .

RESIDUE GAS -167"F 200 PSIA

(-B0T

800 PSIAQ

HP. SEPARATOR <

PRODUCT 24*F

Figure 1.

LhQ

M GAL PUMP "T5AY~

Schematic for typical turboexpander gas processing plant

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.

(Mbtu/hr)

24,099

29,421

Total Liquids

Cooling Duty

20,290 3,154 2,011

3.954 0.673 1.367 0.331 0.131 0.0614 0.0437 0.0161 0.0173 0.0086

Methane Ethane Propane

L i q u i d Condensed (Mol/Day)

Nitrogen Carbon Dioxide Methane Ethane Propane Isobutane Normal Butane Isopentane Normal Pentane Hexane Plus

K-Value at -80F, 800 p s i a

K-VALUE CORRELATION

24,550

42,333

30,804 4,704 2,490

3.954 0.572 1.310 0.250 0.110 0.0614 0.0437 0.0161 0.0173 0.0086

GPA CONV 1050 PSIA HUDSON

24,459

41,429

29,919 4,427 2,619

3.616 0.543 1.321 0.277 0.0889 0.0400 0.0287 0.0131 0.0096 0.0012

P-R

24,409

40,157

28,646 4,390 2,619

3.644 0.574 1.337 0.269 0.0834 0.0366 0.0261 0.00114 0.00838 0.00092

SOAVE

23,564

22,283

12,635 3,136 2,306

5.262 0.598 1.672 0.241 0.0665 0.0262 0.0176 0.00686 0.00497 0.00052

SH BWR

26,447

83,061

68,692 6,345 2,951

2.723 0.414 1.161 0.298 0.127 0.0889 0.0475 0.0366 0.0193 0.00363

LEE

24,275

35,969

25,511 3,615 2,472

3.374 0.706 1.336 0.336 0.0932 0.0386 0.0271 0.0097 0.0064 0.0003

K VAL

TABLE I I I EFFECT OF K-VALUE CORRELATION IN HIGH PRESSURE SEPARATOR (PROPERTY 75 FOR THERMODYNAMIC PROPERTIES)

24,330

35,600

24,588 4,095 2,564

4.019 0.645 1.378 0.266 0.0797 0.0392 0.0295 0.0098 0.00628 0.000302

K DELTA

24,455

41,023

29,458 4,482 2,619

3.512 0.599 1.327 0.267 0.0876 0.0407 0.0293 0.0142 0.0107 0.0018

MARK V

15.

ELLIOT ET A L .

Cryogenic

Gas

295

Processing

u s e f u l work by t h e turboexpander d r i v e n compressor which compresses the r e s i d u e gas to the d e l i v e r y pressure. The turboexpander o u t l e t stream i s then fed to a low temperature separator (normally i n the top of the demethanizer) where condensed l i q u i d i s used to r e f l u x the tower. The c o l d vapor goes overhead w i t h the demethanizer r e s i d u e gas. E f f e c t s of K-Value P r e d i c t i o n s High pressure separator and turboexpander designs a r e h i g h l y dependent on v a p o r - l i q u i d e q u i l i b r i u m K-values. V a r i a t i o n i n K-values i s examined by s e l e c t i n g a common b a s i s (Property-75) f o r enthalpy/entropy c a l c u l a t i o n s . High Pressure Separator The compressed i n l e t gas a f t e r being cooled to 120°F by an a i exchanger system. For th enter t h i s system a t 120°F and 810 p s i a and i s cooled to -80°F a t 800 p s i a , the c o n d i t i o n s of the high pressure s e p a r a t o r . At the f i x e d c o n d i t i o n s of the high pressure s e p a r a t o r , the l i q u i d knockdown and i t s composition a r e o n l y dependent on the K-value c o r r e l a t i o n . The r e s u l t s of the h i g h pressure separator f l a s h c a l c u l a t i o n s are t a b u l a t e d i n Table I I I . For ease i n comparing r e s u l t s , K-values p r e d i c t e d by the Hudson c o r r e l a t i o n as d e f i n e d by Table I a r e used as a common b a s i s . The range of K-values and amount of l i q u i d knock-down i s summarized as f o l l o w s :

Methane K-value Ethane K-value Propane K-value L i q u i d Condensed

Deviation -11% to -4% to -40% t o -47% t o

from 28% 34% 19% 96%

Hudson (1.161 t o 1.672) (0.241 to 0.336) (0.0665 to 0.131) (22,283 to 83,061 mols/day)

The l a r g e v a r i a t i o n i n the amount of l i q u i d condensed i s mainly caused by the v a r i a t i o n i n the p r e d i c t e d methane K-value. This v a r i a t i o n becomes i n c r e a s i n g l y pronounced as t h e c o n d i t i o n s of the gas approach the c r i t i c a l p o i n t where most of the c o r r e l a t i o n s a r e inadequate or f a i l a l t o g e t h e r . U n f o r t u n a t e l y , t h i s c o n d i t i o n occurs f r e q u e n t l y i n gas p l a n t design. I n such cases, one should be extremely c a u t i o u s i n u s i n g these c o r r e l a t i o n s . Turboexpander. Vapor from the h i g h pressure separator feeds the expander. Table IV was generated using the d i f f e r e n t K-value c o r r e l a t i o n s w i t h t h e f o l l o w i n g common b a s i s : Expander I n l e t Temperature -80 F Expander I n l e t Pressure 800 p s i a Expander Discharge Pressure 200 p s i a Expander I s e n t r o p i c E f f i c i e n c y 80% The gas flow r a t e was determined from the high pressure separator f l a s h c a l c u l a t i o n s . Otherwise, the expander i n l e t gas would not be at i t s dewpoint f o r the c o r r e l a t i o n being used. This f a c t precludes

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.

1,130

1,052

1,051

143,029

8 .674 0 .1654 1 .283 0 .0421 0 .0032 0 .0005 0 .0002

9.068 0. 332 1. 268 0. 0446 0. 0026 0. 0006 0. 0003

9 .189 0 .229 1 .335 0 .0524 0 .0039 0 .0006 0 .0003

155,037 142,125

-166.2

P-R

-166.1

HUDSON

-165.1

CONV

1,061

144,301

8. 539 0. 186 1. 290 0. 0406 0. 0029 0. 0004 0. 0005

-167.1

SOAVE

1,195

162,175

8.,491 0.,229 1. 316 0,.0431 0. 0045 0. 0009 0. 0005

-165.8

SH BWR

K VAL

8. 337 0. 299 1. 263 0. 0556 0. 0042 0. 0006 0. 0003

752

1,088

101,397 148,489

12:.28 0. 0131 1. 241 0. 0424 0. 0042 0. 0009 0. 0004

-164.4 -163.2

LEE

1,153

148,858

9.417 0 .271 1 .251 0 .0455 0 .0041 0 .00051 0 .0002

-166.4

K DELTA

1,054

143,435

7..735 0,.219 1,.283 0,.0383 0,.0033 0..0006 0.,0003

-166.5

IffARK V

*K-values f o r methane, ethane and propane a r e obtained from i n t e r p o l a t i o n of b i n a r y c h a r t and i n f i n i t e d i l u t i o n c h a r t i n GPA 1972 Handbook. K-values f o r the r e s t of components are obtained by convergence method u s i n g 800 p s i convergence pressure.

Expander Horsepower

(Mol/Day)

I n l e t Gas Flow Rate

Expander Discharge Temperature, °F K-Value a t 200 p s i a Nitrogen Carbon D i o x i d e Methane Ethane Propane Isobutane Normal Butane

K-Value C o r r e l a t i o n s

GPA*

TABLE IV EFFECT OF K-VALUE CORRELATIONS ON EXPANDER (PROPERTY 75 FOR THERMODYNAMIC PROPERTIES)

15.

ELLIOT E T A L .

Cryogenic Gas Processing

297

the i s o l a t i o n of the expander c a l c u l a t i o n from the h i g h pressure separator c a l c u l a t i o n when comparing o v e r a l l ethane recovery. Again, w i t h the Hudson c o r r e l a t i o n as the common b a s i s , the v a r i a t i o n i n p r e d i c t e d K-values and r e c o v e r i e s a r e summarized as follows:

Methane K-value Ethane K-value Propane K-value L i q u i d Condensed

Deviation -2.1% to -14% to 11% t o -36% to

from Hudson 5.3% (1.241 to 1.335) 25% (0.0382 to 0.0556) 73% (0.0029 to 0.0045) 21% (13,576 to 25,584 mols/day)

The t o t a l e f f e c t of the h i g h pressure separator and the turboexpander f l a s h c a l c u l a t i o n s i s shown by the estimated ethane recovery i n Table IV whic propane recovery f o r a l l The methane K-value shows very good agreement f o r a l l the c o r r e l a t i o n s , but the ethane and propane K-values e x h i b i t a l a r g e v a r i a t i o n . V a r i a t i o n i n the amount o f l i q u i d condensed a t the expander o u t l e t and the expander horsepower a r e d i r e c t l y r e l a t e d t o the amount of vapor separated a t h i g h pressure separator. The magnitude of t h i s v a r i a t i o n can a f f e c t the expander design q u i t e s i g n i f i c a n t l y . The l i q u i d s condensed by the expansion a r e l i s t e d i n Table V along w i t h the t o t a l l i q u i d s . The t o t a l l i q u i d condensed i s the sum of expander o u t l e t l i q u i d and h i g h pressure separator l i q u i d . Since the ethane recovery i s d i r e c t l y p r o p o r t i o n a l t o the t o t a l l i q u i d s condensed, the wide v a r i a t i o n i s s i g n i f i c a n t . A l s o , the t o t a l amount of l i q u i d s has a l a r g e e f f e c t on the design of the demethanizer. I n the f i n a l a n a l y s i s , the most important number i n the p l a n t i s product r a t e . Here, the c a l c u l a t e d o v e r a l l v a r i a t i o n i n ethane recovery i s 6.3%. This corresponds t o 5,520 g a l l o n s per day of ethane. I f one assumes ethane i s worth $0.15 per g a l l o n , the incremental product revenue i s $828 per day, l e s s gas shrinkage of $176 per day based on gas a t $0.85 per MMscf. The r e s u l t i n g net income i s $652 per day o r more than $200,000 per year. Obviously, t h i s v a r i a t i o n i n a n t i c i p a t e d revenue i s s i g n i f i c a n t enough t o a f f e c t investment d e c i s i o n s . Moreover, h i s t o r y has shown that i t i s not unusual f o r a c o m p e t i t i v e b i d t o be awarded on the b a s i s of 1% higher guaranteed ethane recovery. E f f e c t s of Thermophysical Property C o r r e l a t i o n s f o r Enthalpy and Entropy To examine enthalpy e f f e c t s of the v a r i o u s c o r r e l a t i o n s , K-values from the Soave c o r r e l a t i o n were s e l e c t e d as a common b a s i s . Compari s o n s a r e presented here i n terms of the segments of the design c a l c u l a t i o n s f o r an expander p l a n t i n the order of the f l o w shown on F i g u r e 1. Comparison of P r e d i c t i o n s f o r Compressors. There a r e two compression schemes a v a i l a b l e f o r the design of a cryogenic

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.

82.3

86.5

354

63,435

47,184 7,840 3,394

86.1

355

62,125

45,819 7,803 3,394

20,696

15,900 3,376 775

P-R

86.3

337

60,355

44,067 7,802 3,394

20,198

15,421 3,412 775

SOAVE

84.5

88.4

503

96,637

47,148

272

79,501 8,153 3,394

13,576

10,809 1,808 443

LEE

31,210 7,581 3,394

24,865

18,575 4,445 1,088

SH BWR

82.1

86.1

338

59,838

60,520

446

43,661 7,788 3,394

24,238

19,073 3,693 830

K DELTA

44,602 7,544 3,394

24,551

19,091 3,929 922

K VAL

87.4

319

61,590

45,248 7,876 3,394

20,567

15,790 3,394 775

MARK V

''Estimated Ethane Stripped = T o t a l mol of methane condensed x mol r a t i o of ethane to methane i n the expander o u t l e t vapor x i n e f f i c i e n c y of the column (1.10 i s used).

Estimated Ethane Recovery, Percent

387

55,005

Total Liquids

^Estimated Ethane Stripped

39,105 7,507 3,394

Methane Ethane Propane

(Mol/Day)

21,102

25,584

Total Liquids

Total Liquids

16,380 3,136 904

HUDSON

18,815 4,353 1,383

GPA* CONV

Methane Ethane Propane

Turboexpander L i q u i d s Condensed (Mol/Day)

K-Value C o r r e l a t i o n s

EFFECT OF K-VALUE CORRELATION ON TURBOEXPANDER LIQUIDS AND TOTAL RECOVERY (PROPERTY 75 FOR THERMODYNAMIC PROPERTIES)

TABLE V

15.

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ET AL.

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299

turboexpander p l a n t . Since f r e e pressure drop i s u s u a l l y not a v a i l a b l e t o supply the energy of s e p a r a t i o n , e i t h e r compression of the feed gas o r recompression of the r e s i d u e gas must be used (or some combination o f the two). S e l e c t i o n i s dependent upon the feed gas pressure l e v e l . U s u a l l y , precompression i s used f o r low pressure feed and recompression i s used f o r h i g h pressure feed. Precompressor. The example presented here i n F i g u r e 1 uses precompression. Since compressors are normally the most expensive pieces o f equipment i n the e n t i r e p l a n t , the process i s u s u a l l y designed t o f i t the c l o s e s t compressor frame s i z e a v a i l a b l e . The design b a s i s i s : Flow Rate: Gas Composition:

70 MMscfd I n l e t gas as shown i n

Suction Pressure: S u c t i o n Temperature: Horsepower A v a i l a b l e : Isentropic Efficiency:

120°F 6,000 hp s i t e r a t e d 75%

Compressor discharge c o n d i t i o n s a r e t a b u l a t e d i n Table V I . The v a r i a t i o n i n d i s c h a r g e pressure i s about 12 p s i pressure from 809.5 to 821.5 p s i a . Although a higher d i s c h a r g e pressure r e s u l t s i n g r e a t e r recovery, the 12 p s i i s not a s i g n i f i c a n t d i f f e r e n c e . Even a hand c a l c u l a t i o n by the k-method (3) y i e l d e d similar results. The v a r i a t i o n i n d i s c h a r g e temperature i s 6.8°F. For an a i r c o o l e r c o o l i n g the compressed gas t o 120°F, t h i s r e s u l t s i n a design duty v a r i a t i o n up t o 3.5% depending on which c o r r e l a t i o n i s chosen. Recompressor. Although not shown here, a s i m i l a r comparison can be made f o r a case u s i n g recompression by choosing an estimated r e s i d u e gas composition. The design b a s i s i s : Flow Rate: Gas Composition: Suction Pressure S u c t i o n Temperature: Horsepower A v a i l a b l e : Isentropic Efficiency:

70 MMscfd Residue Gas as shown i n Table I I 250 p s i a 120°F 6,000 hp s i t e r a t e d 75%

C a l c u l a t e d d i s c h a r g e c o n d i t i o n s are t a b u l a t e d i n Table V I . A l l the c o r r e l a t i o n s agree reasonably w e l l i n p r e d i c t i n g discharge pressure and temperature. I n l e t Gas C o o l i n g . For t h i s comparison, the i n l e t gas c o n d i t i o n s a r e s e t as 120°F and 810 p s i a . The h i g h pressure separator c o n d i t i o n s are s e t as -80°F and 800 p s i a . C a l c u l a t e d c o o l i n g d u t i e s a r e g i v e n i n Table V I I . The v a r i a t i o n i n i n l e t gas c o o l i n g duty i s -0.9% t o 1.8% of the Property-75 c a l c u l a t i o n . ( A l l comparisons i n t h i s s e c t i o n are made w i t h the Property-75 r e s u l t s ) .

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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EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

TABLE VI CALCULATED COMPRESSOR DISCHARGE CONDITIONS INLET: 70 MMSCFD AT 120°F 250 PSIA 6000 HP AT 75% EFFICIENCY I n l e t Gas Compression PSIA °F

Residue Gas Compression °F PSIA

PROP-75

811.8

319.6

792.2

335. 8

P-R

821.5

320.8

799.6

337 2

SOAVE

809.7

LEE

809.5

315.5

794.7

332. 8

SH BWR

813.8

316.7

798.6

334. 9

GPA-k

817.3

314.0

798.7

331. 8

TABLE V I I - COMPARISON OF ENTHALPY AND ENTROPY CORRELATIONS (K-VALUES FROM GPA K&H SOAVE) SOAVE Feed Gas C o o l i n g Duty, MBtu/Hr

P-R

LEE

PROP-75

24,727

24,203

24,858

24,409

Expander Horsepower

1,050

1,018

942

1,061

Expander Discharge Temperature, F

-167.8

-168.2

-167.0

-167.1

R e b o i l e r Duty, MBtu/Hr

5,727

5,858

5,662

5,223

Residue Heating Duty, MBtu/Hr

17,937

17,512

18,394

18,057

1,063

833

802

1,129

R e f r i g e r a t i o n Duty, MBtu/Hr

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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Expander. A l l gas q u a n t i t i e s and compositions a r e h e l d constant by u s i n g a f i x e d K-value c o r r e l a t i o n s i n c e the expansion process i s i s e n t r o p i c . The e f f e c t of the enthalpy-entropy c o r r e l a t i o n s appears i n the c a l c u l a t e d horsepower and f i n a l temperature. The r e s u l t s i n Table V I I show from -1.0% t o -11.2% d e v i a t i o n i n horsepower and from +0.1°F t o -1.1°F i n expander d i s c h a r g e temperature. I f the same b a s i s f o r ethane s t r i p p e d as Table V i s used, the estimated ethane recovery i s : LEE PR0P-75 SOAVE P-R

86.0% @ 86.1% @ 87.0% @ 87.5% @

-167.0°F -167.1°F -167.8°F -168.2°F

This v a r i a t i o n i s p r i m a r i l t h i s narrow temperature range. The methane K-values a r e 1.260 t o 1.290; the ethane K-values a r e 0.0398 to 0.0406. Demethanizer R e b o i l e r . A f i x e d recovery was assumed by s e t t i n g compositions of the r e s i d u e gas and product as g i v e n i n Table I I . The feed t o the demethanizer i s s e t by the c a l c u l a t i o n s from steps B and C above w i t h a minor v a r i a t i o n i n the feed compos i t i o n . With the column o p e r a t i n g a t 200 p s i a , the r e s i d u e gas at a dew p o i n t of -167°F, and the product a t a bubble p o i n t of 24.1°F, enthalpy c a l c u l a t i o n s were made around the column f o r the v a r i o u s c o r r e l a t i o n s . The r e b o i l e r duty was c a l c u l a t e d as f o l l o w s : R e b o i l e r Duty = Enthalpy of the Residue 4- Enthalpy of the Product - Enthalpy of the Feed The c a l c u l a t e d r e b o i l e r duty v a r i e s from +8.4% t o 12.2%. A higher r e b o i l e r duty s i g n i f i e s a higher c o o l i n g duty a v a i l a b l e f o r the feed gas, which would r e s u l t i n a higher recovery. Residue Gas Heating. R e s u l t s of the c a l c u l a t i o n s f o r h e a t i n g the r e s i d u e gas from -167°F to 110°F a r e a l s o g i v e n i n Table V I I . This i s the a v a i l a b l e duty f o r c h i l l i n g the i n l e t gas i n the gasgas exchanger. The v a r i a t i o n here i s s m a l l , from -3.0% to +1.9%. This i s not s u r p r i s i n g s i n c e the r e s i d u e gas i s mostly methane and only s e n s i b l e heat i s i n v o l v e d . E x t e r n a l R e f r i g e r a t i o n . The e x t e r n a l r e f r i g e r a t i o n duty r e q u i r e d f o r the p l a n t i s c a l c u l a t e d from an o v e r a l l heat balance of the e n t i r e p l a n t . R e f r i g e r a t i o n Duty = Feed Gas C o o l i n g Duty - R e b o i l e r Duty - Residue Gas Heating Duty The r e s u l t s l i s t e d i n Table V I I vary from -29.0% t o -5.8%. The major c o n t r i b u t o r t o these d i f f e r e n c e s i s the r e b o i l e r duty. The c a l c u l a t e d low temperature (expander o u t l e t ) c o n d i t i o n s are markedly different.

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

302

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

to β

to Ο υ "ο CO

M—«

Ο

Ο CO

-S

1 to

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

15.

ELLIOT E T A L .

Cryogenic Gas Processing

303

S o l i d s Formation by Carbon D i o x i d e The p o s s i b i l i t y f o r freeze-up e x i s t s when carbon d i o x i d e i s present i n the feed gas t o an expander p l a n t . For the c a l c u l a t i o n s a feed composition of 0.52 mol% carbon d i o x i d e i s used. A t f i x e d c o n d i t i o n s , v a r i a t i o n s i n the c a l c u l a t i o n s f o r the carbon d i o x i d e c o n c e n t r a t i o n a r i s e s o l e l y from K-value c o r r e l a t i o n s . The r e s u l t s from S e c t i o n I I I of t h i s paper, where thermodynamic p r o p e r t i e s a r e f i x e d by the Property-75 c o r r e l a t i o n and compositions a r e c a l c u l a t e d f o r v a r i o u s K-value c o r r e l a t i o n s , a r e s e l e c t e d as the b a s i s f o r the comparison. Any freeze-up w i l l occur i n the demethanizer. The feeds t o t h i s column a r e the expander o u t l e t t o the top of the column and the l i q u i d from the h i g h pressur tray. A detailed tray-to-tra t r a y s u s i n g the Soave K-value c o r r e l a t i o n . A t t r a y c o n d i t i o n s , the l i m i t i n g CO2 content i n the l i q u i d phase was found from Figure 2. These r e s u l t s , presented i n Table V I I I , show the top three t r a y s of the column a r e c l o s e t o freeze-up c o n d i t i o n s . F l a s h c a l c u l a t i o n s were made a t the same pressure and temperature as the Soave c a l c u l a t i o n s f o r the top three t r a y s f o r a l l of the K-value c o r r e l a t i o n s . Feed composition on each tray f l a s h was set by the t o t a l composition from the t r a y - t o - c a l c u l a t i o n . The amount of CO2 was adjusted by the r a t i o of CO2 knockdown i n the expander o u t l e t f o r the K-value c o r r e l a t i o n t o the Soave case. The r e s u l t s i n Table IX show that the Peng-Robinson c o r r e l a t i o n p r e d i c t s that t r a y s 2 and 3 w i l l be only 0.21 t o 0.25 mol% below freeze-up c o n d i t i o n s . This i s too c l o s e f o r o p e r a t i n g c o n t r o l . A p l a n t design based on the Peng-Robinson c o r r e l a t i o n without t r e a t i n g f a c i l i t i e s would n e c e s s i t a t e higher o p e r a t i n g pressure f o r the demethanizer and a higher temperature and probably a lower recovery, i n order t o avoid s o l i d s formation. Other c o r r e l a t i o n s such as K VAL and K DELTA would p r e d i c t s a f e o p e r a t i o n a t 200 p s i a . The carbon d i o x i d e K-value has a d e f i n i t e e f f e c t on p l a n t operat i o n and recovery, since the carbon d i o x i d e f r e e z i n g c o n d i t i o n s must be avoided. No data e x i s t i n the r e g i o n of i n t e r e s t f o r the t e r n a r y methane-ethane-carbon d i o x i d e system; such data would be u s e f u l f o r e v a l u a t i o n of these K-value c o r r e l a t i o n s . Conclusions The e f f e c t s of c o r r e l a t i o n s on turboexpander p l a n t design have been examined from two views: K-values and enthalpy/entropy. A summary of the r e s u l t s i s given i n F i g u r e 3 f o r three p o r t i o n s of the p l a n t . D e t a i l e d r e s u l t s a r e given i n Tables I I I through IX. K-Value C o r r e l a t i o n s . The K-values used have been shown t o have a s i g n i f i c a n t e f f e c t on p l a n t d e s i g n , more, i n f a c t ; than enthalpy/entropy. The t o t a l v a l u e of the plant i s expressed i n the t o t a l ethane recovery; t h i s v a r i e d from 82.1% t o 88.4%. The d i f f e r ence i s e q u i v a l e n t t o more than $200,000 per year i n a n t i c i p a t e d revenue. 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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A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

TABLE V I I I TRAY-TO-TRAY DEMETHANIZER SOAVE K-VALUES PROP-75 THERMOPHYSICAL PROPERTIES 200 PSIA

Carbon D i o x i d e mol % Vapor Liquid

Limiting Solubility mol % CO2 L i q u i d

Tray Number

Temperature F

1

-166

0.32

1.72

2.45

2

-164

0.43

2.23

2.62

3

-158

0.62

2.85

3.27

Feed 4

-126

1.20

2.52

8.5

5

-121

1.80

3.38

6

-104

4.19

5.46

7

-66

10.5

7.29

8

-21

15.5

6.21

11.3

2.92

Reboiler

24.1

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

15.

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Cryogenic Gas Processing

305

TABLE IX SOLIDS FORMATION DEMETHANIZER PROP-75 THERMOPHYSICAL PROPERTIES 200 PSIA

K-Value Correlation

Tray No^. 1 -166 F mol % C0„ C0 L i q . 2

Tray No. 2 — . o— 164^ mol % C0 L i q . "CO. 2

Tray No^. 3 158"F mol % C0 L i q . CO, 2

GPA CONV

0. 261

P-R

0.166

1. 82

0.171

2. 37

0.,195

3.,06

SOAVE

0.187

1. 75

0.193

2. 26

0.,220

2,.85

SH BWR

0. 223

1. 39

0. 229

1. 83

0.,262

2,.30

MARK V

0. 219

1. 55

0. 225

2. 00

0.,257

2,.54

K VAL

0.28

1. 01

0. 294

1. 44

0.,335

1..96

K DELTA

0. 271

1. 26

0. 291

1. 57

0,,338

2,.17

Figure 2

2. 45

2. 62

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

3,.27

306

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Figure 3. Phase envelope for feed gas with three process paths: Variations in predictions for different enthalpy correlations at a fixed K-value correlation denoted by ®; variations in predictions for different K-value correlations at a fixed enthalpy correlation indicated by @. Approximate critical conditions are indicated by an X.

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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ELLIOT E T A L .

Cryogenic Gas Processing

307

The h i g h pressure c o o l i n g duty v a r i e s from - 4 . 0 % t o 7 . 7 % of the s e l e c t e d base. This 1 2 % d i f f e r e n c e a f f e c t s s i z i n g of heat exchangers, amount of e x t e r n a l r e f r i g e r a t i o n , and s i z e of expander. The expander horsepower v a r i e s from 7 5 2 to 1 1 5 3 . This c a l c u l a t i o n i s coupled w i t h the h i g h pressure separator c a l c u l a t i o n . I f i n a c t u a l p l a n t o p e r a t i o n s the design expander discharge temperature c a l c u l a t e d i s not reached, the t o t a l recovery w i l l not meet design. The d e s i g n of the demethanizer tower, the r e b o i l e r , and the r e f r i g e r a t i o n requirement i s a f u n c t i o n of the t o t a l l i q u i d s condensed. The l a r g e s t c a l c u l a t e d v a l u e i s t w i c e the s m a l l e s t . Freeze-up due to carbon d i o x i d e s o l i d formation can occur i n the c o l d p o r t i o n s of the process. The CO^ K-value i s important a t c o n d i t i o n s approaching the formation of s o l i d s , but no t e r n a r y experimental data now e x i s lations . Enthalpy/Entropy C o r r e l a t i o n s . The most expensive item i n a turboexpander p l a n t i s the compressor which i s designed from enthalpy/ entropy c a l c u l a t i o n s . F o r a f i x e d horsepower the r e s u l t s are almost the same f o r P r o p e r t y - 7 5 , Peng-Robinson, GPA K&H Soave, GPA K&H Lee, GPA K&H S t a r l i n g - H a n BWR, and GPA-k method. The l a r g e s t v a r i a t i o n p r e d i c t e d i s i n the discharge temperature (maximum d i f f e r e n c e s of 6 . 8 ° and 5 . 4 ° F ) which would a f f e c t the discharge c o o l e r s i z e s (assuming 120°F) by 3 % . The e f f e c t on the h i g h pressure c o o l i n g duty and the expander horsepower i s r a t h e r s m a l l f o r the d i f f e r e n t H & S c o r r e l a t i o n s as shown i n F i g u r e 3 . The major e f f e c t i s seen i n the c a l c u l a t e d demethanizer r e b o i l e r duty, which v a r i e s from 8 . 4 % t o 1 2 . 2 % . This i s l a r g e l y a r e s u l t of the l a r g e d i f f e r e n c e s i n c a l c u l a t e d enthalpy a t the expander o u t l e t c o n d i t i o n s . I f the enthalpy balance i s i n e r r o r , s u f f i c i e n t r e f r i g e r a t i o n w i l l not be a v a i l a b l e and the expected recovery w i l l not be obtained. Acknowledgment The authors thank P a t r i c i a J . C a s t i l l o , C h a r l o t t e L. Eberwein, and Deborah L. Woods f o r t h e i r e f f o r t s i n the p r e p a r a t i o n of the manuscript. A l s o , the authors are indebted to Dr. J . R a n d a l l Johnson f o r h i s h e l p f u l comments.

Literature Cited 1. 2. 3.

E l l i o t t , D. G., McKee, R. L., and White, W. E., Gas Processors Assn. Proc. (1976) 55, 8 1 . White, W. E., Wilson, G., and Kobayashi, R., Nat. Gas Processors Assn. Proc. (1970) 49, 126. GPSA Engineering Data Book, Ninth Edition, Gas Processors Assn., Tulsa, Okla., 1972, Section 5, "Centrifugal Compressors".

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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4. GPSA Engineering Data Book, Ninth Edition, Gas Processors Assn., Tulsa, Okla., 1972, Section 18, "Equilibrium Ratios". 5. Wilson, G., Adv. Cryog. Eng. (1964) 9, 168. 6. Redlich, O., and Kwong, I.N.S., Chem. Rev. (1949) 44, 233. 7. Lee, B. L., and Edmister, W. C., Nat. Gas Processors Assn. Proc. (1971) 50, 56. 8. Soave, G., Chem. Eng. Sci. (1976) 15, 59. 9. Peng, D.-Y., and Robinson, D. B., Ind. Eng. Chem. Fundamentals (1976) 15, 59. 10. Starling, K. E., and Han, M. S., Hyd. Proc. (May 1972) 51 (5), 129. 11. Benedict, M., Webb, G. B., and Rubin, L. C., J. Chem. Phys. (1940) 8, 334. 12. Leach, J. W., Chappelear (1968) 14, 568. 13. Chapela-Castañares, B. A., and Leland, T. W., AIChE Sym. Series (1975) 70, No. 140, 48.

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

Discussion LYMAN YARBOROUGH

Comments: Two constant equations of s t a t e can be f i t t e d to s p e c i f i c f l u i d p r o p e r t i e s i n the two-phase r e g i o n up to and i n c l u d i n g the c r i t i c a l . However, i f t h i s i s done, c o n s i d e r a b l e accuracy i s l o s t when c a l c u l a t i n g other f l u i d p r o p e r t i e s not used i n the f i t t i n g process; i . e . , i f pure component vapor pressures and l i q u i d phase d e n s i t i e s up to the c r i t i c a l p o i n t and e q u a l i t y of vapor and l i q u i d phase fugac i t i e s are used to determine the two c o n s t a n t s , then p r o p e r t i e s such as enthalpy c o r r e c t i o n and the d e r i v a t i e s of (3P/8v^) a t or near the c r i t i c a l p o i n t w i l l be i n c o r r e c t . These equations of s t a t e do not have a s u f f i c i e n t number of independent v a r i a b l e s to f i t a l l properties correctly. Too many equations of s t a t e and m o d i f i c a t i o n s of equations of s t a t e are being used. The number of equations should be reduced by s e l e c t i n g the best ones, and those equations of s t a t e should be f u r t h e r developed. Q:

Because of environmental r e g u l a t i o n s , the c o n c e n t r a t i o n of d i s s o l v e d components i n water i s of importance. Can the s o l u b i l i t y of d i s s o l v e d gaseous and l i q u i d components i n water be estimated by equations of s t a t e ?

A:

P r e s e n t l y a v a i l a b l e equations of s t a t e can be used to accur a t e l y c a l c u l a t e the water content of gas and hydrocarbon l i q u i d phases but not the gaseous or l i q u i d hydrocarbon content of the water phase. Equations of s t a t e can p r e d i c t o n l y q u a l i t a t i v e r e s u l t s f o r the water phase.

Q:

At cryogenic temperatures some n a t u r a l gas mixtures can form two l i q u i d phases, and l i q u i d - l i q u i d - v a p o r phase e q u i l i b r i a occurs. Can t h i s behavior be c a l c u l a t e d ?

A:

Yes, e q u i l i b r i u m f o r three-phase systems can be c a l c u l a t e d f o r 309

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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non-aqueous systems; but l i t t l e data are a v a i l a b l e to compare a g a i n s t and good i n i t i a l values are r e q u i r e d to s t a r t the calculation. Comments: C a l c u l a t i o n s employing f r e e energy m i n i m i z a t i o n o f t e n show l e s s s e n s i t i v i t y to s t a r t i n g v a l u e s than other methods. A d d i t i o n a l e q u i l i b r i u m data on l i q u i d - l i q u i d - v a p o r phase systems are needed. Q:

How i s the a c e n t r i c f a c t o r c a l c u l a t e d f o r petroleum f r a c t i o n s ? What k i n d of values are used i n equations of s t a t e ?

A:

The a c e n t r i c f a c t o r c a l c u l a t e d normal b o i l i n g p o i n t f o r the f r a c t i o n matches the one measured e x p e r i m e n t a l l y . Values of one and greater have been obtained i n t h i s manner.

Comments: The a c e n t r i c f a c t o r was developed as a p e r t u r b a t i o n from the s p h e r i c a l molecules and was never intended f o r use w i t h l a r g e molecules. An a l t e r n a t i v e could be a parameter employed by Prigogine. A l a r g e a c e n t r i c f a c t o r doesn't make sense p h y s i c a l l y but i s used to c o r r e l a t e vapor pressures f o r the petroleum f r a c t i o n s . Perhaps a vapor pressure c o r r e l a t i o n f o r petroleum f r a c t i o n s could supplant the use of a c e n t r i c f a c t o r . The Lee-Kesler method f o r c o r r e l a t i n g gas phase p r o p e r t i e s ( r e f e r ence equation and corresponding s t a t e s approach) has been extended to v a p o r - l i q u i d e q u i l i b r i a c a l c u l a t i o n s . The c a l c u l a t i o n of accurate phase e q u i l i b r i a r e s u l t s i s more d i f f i c u l t than f o r the p r o p e r t i e s of s i n g l e phase f l u i d s . For h i g h l y asymmetric mixtures (small and l a r g e m o l e c u l e s ) , recent s t u d i e s have shown that mixing r u l e s which provide parameters i n t e r mediate between those of Van der Waal's method and Kay's method should be used. This may be h e l p f u l i n the corresponding s t a t e s approach. I t i s not r e a l l y p o s s i b l e to represent mixture behavior by a pseudo pure component approach. P o s s i b l y should separate c o r r e l a t i o n equation i n t o r e p u l s i v e and a t t r a c t i v e p o t e n t i a l s . The r e p u l s i v e p o t e n t i a l could be represented by the hard core p o t e n t i a l and only the a t t r a c t i v e p o t e n t i a l c o r r e l a t e d by the p s e u d o c r i t i c a l approach.

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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DisCUSSion

Comment by H. T. Davis: In the emerging theory of i n t e r f a c i a l thermodynamics, the p r o p e r t i e s of the thermodynamic f u n c t i o n s (PVT or f r e e energy q u a n t i t i e s ) a t d e n s i t i e s corresponding to thermodynamically unstable s t a t e s p l a y a c r u c i a l r o l e . For example, the i n t e r f a c i a l t e n s i o n i s computed from a formula c a l l i n g f o r Helmholtz f r e e energy data a t every d e n s i t y (composition) between the compositions of the b u l k phases i n e q u i l i b r i u m . A p a r t i c u l a r i m p l i c a t i o n of the theory i s that Method (a) of P r o f e s s o r P r a u s n i t z s c l a s s i f i c a t i o n i s u s e f u l f o r i n t e r f a c i a l c a l c u l a t i o n s of v a p o r - l i q u i d systems, but Method (b) i s not. For those i n t e r e s t e d i n the i n t e r f a c i a l theory to which I r e f e r , the f o l l o w i n g papers and references t h e r e i n may be u s e f u l : f

1. 2.

Bongiorno, V., and Bongiorno, V., S c r i v e n and I n t e r f a c e Science.

Comment by John P. O'Connell: There a r e a number of aspects of improving equations of the "van der Waal" type (e.g., R.K.) which d i d not appear i n the presentat i o n s and d i s c u s s i o n s , three of which I want to address. (I also f e e l t h a t present knowledge of the c r i t i c a l s t a t e s c a l i n g laws should be i n c l u d e d i n f u t u r e developments.) The form of the equation i s

Z = £

= f(v/b) - ^ g ( v / b )

where a, b can be f u n c t i o n s of T, x_« Excluded Volume The f ( v / b ) represents excluded volume e f f e c t s of the l i q u i d s by a r i g i d - b o d y formula, the van der Waals form v/Z x.b. l l f(v/b) =

v/Z x.b. -1 l l

i s not a good one. A b e t t e r one i s that of Mansoori, et a l . (1) known as the "Carnahan-Starling" form 2

3

f ( v / b ) = [1 + v/b + ( v / b ) - ( v / b ) ] / ( l - v / b )

3

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

312

2.

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES

IN C H E M I C A L INDUSTRY

Second V i r i a l C o e f f i c i e n t s These equations y i e l d

expressions

3(T,x) = EZx.x.3..(T) = b(T,x) - a(T,x)/RT i] Since h i g h l y accurate forms f o r are known, The T,_x behavior of a and b should be chosen a p p r o p r i a t e l y . This can be q u i t e important even f o r the l i q u i d s i n c e second v i r i a l c o n t r i b u t i o n s to the i n t e g r a l s f o r f u g a c i t y and enthalpy a r e not n e g l i g i b l e . 1

3.

J

1 J

Thermal Pressure C o e f f i c i e n t and C o m p r e s s i b i l i t y of L i q u i d s For l i q u i d s where T < T , v < v /2

] and ~ \ _ ar 3T Jj v x T av J I Tx

J

y

over a wide range of T,v. Thus, s e t t i n g y \

,2 ^ ^ j = 0 we f i n d a r e l a t i o n f o r the 3T / v,x V

R

temperature v a r i a t i o n of a and b

T

K

2

df _ a_ dg\/9"b 9b \ df Rv db 3T jx

.o f d ^ l / B a A

2

[

d

b

"

R

H

3

T

;x

2

d^l/8b\ d

b

JWx"

g / 9 a\ R

AaT

2

;,

I f a has a T dependence which y i e l d s a second d e r i v a t i v e , b must a l s o depend on T. Since l i q u i d r e s u l t s depend s t r o n g l y on the value of b, t h i s could be s i g n i f i c a n t . B r e l v i (_2,3) showed that t h i s r e s u l t could be g e n e r a l i z e d to

£n

\

+

D(v*/v)

=ln

i ^\ 1+

" " * A + B(v

+ c(v

/v)2 +

3

c c where v* = v f o r nonpolar substances and v * < v f o r p o l a r substances. For t h i s 5% c o r r e l a t i o n of K-p, the substances included Ar, a l c o h o l s , hydrocarbons from CH^ to those w i t h MW > 400, Amines, e t c . Gubbins (3) showed how the g e n e r a l i z a t i o n to a l l f l u i d s should be expected (though not why the T dependence i s absent). I t was shown that f o r T > T a 2-parameter corresponding s t a t e s p l o t f o r K-j-RT/v of H2O and Ar°is s u c c e s s f u l over wide ranges of T/T* and

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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v/v* w i t h the values of T* and v * corresponding to i n t e r m o l e c u l a r f o r c e parameters of the s p h e r i c a l l y symmetric f o r c e s . These should be r e l a t e d t o the a and b parameters of the equation of s t a t e s i n c e c o m p r e s s i b i l i t y i s e a s i l y measured.

References 1. Mansoori, G. A., Carnahan, N. F., Starling, K. E., and Leland, T. W., J. Chem. Phys. (1971) 54, 1523. 2. Brelvi, S. W. and O'Connell, J. P., AIChE J., (1972) 18, 1239; (1975) 21, 171, 1024. 3. Gubbins, K. E. and O'Connell, J. P., J. Chem. Phys. (1974) 60, 3449. Q: Approximately what cell was the solid precipitate, and what were the API gravities of the two oils tested? A: The solid precipitate varied from 2% to 5% of the total fluid, and the gravity for theoilstudied at 130F was 40 API while the gravity for theoilstudied at 255F was 26 API. Q: What is the minimum value of the interfacial tension that can be measured using the pendant drop apparatus? Has this been checked against other experimental techniques? A: Experience has shown the pendant drop and spinning drop techniques agree down to interfacial tensions of 0.01 dynes/cm. The spinning drop apparatus is not applicable at pressures up to 2000 psia. Comment: The Weinaug and Katz correlation was based on surface tension and is not designed to correlate interfacial tension. Q: Does water affect the process of CO displacing oil? 2

A: No, the water is nearly immiscible in both phases. The interfacial tension between the C02~water and water-oil phases i s high. Water is often injected alternately with the CO2 for mobility control. Q: Is i t possible to use any activity coefficient model and derive a consistent equation of state in a manner similar to that presented? A: No. One must be careful of the reference state. In the derivation shown in the paper, the reference state is the pure component.

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 H A S E EQUILIBRIA AND

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Q:

What parameters are r e q u i r e d to c h a r a c t e r i z e the heavy f r a c t i o n s which a r e expected to be r i c h i n aromatic hydrocarbons?

A:

I don't know.

Q:

This problem of c h a r a c t e r i z a t i o n i s s t i l l being s t u d i e d by both API and GPA f o r petroleum base f l u i d s . At l e a s t one e a s i l y measured parameter i s r e q u i r e d i n a d d i t i o n to normal b o i l i n g p o i n t and s p e c i f i c g r a v i t y .

Comments: Some Russian a r t i c l e s have suggested r e f r a c t i v e index as a charact e r i z i n g parameter. R e f r a c t i v e index c o r r e l a t e y y specifi g r a v i t y to be considered as another independent parameter. A parameter suggested by P r i g o g i n e f o r l a r g e molecules combines the d e n s i t y , the c o e f f i c i e n t of thermal expansion, and the i s o t h e r m a l c o m p r e s s i b i l i t y of the f l u i d . This might be u s e f u l f o r l a r g e petroleum molecules a l s o . Q:

What i s the s i g n i f i c a n c e of the assumed hole i n the l i q u i d s t a t e i n your development?

A:

I t i s r e a l l y j u s t an a d j u s t a b l e parameter, a s o r t of f r e e volume.

Q:

I n the d e r i v a t i o n , what assumptions were made concerning the s i z e of the holes?

A:

No assumptions were necessary because the hole volume c a n c e l l e d out.

Q:

When comparing the phase e q u i l i b r i a p r e d i c t i o n methods you used a g a i n s t experimental data, which one appears to be best?

A:

The Soave Redlich-Kwong marketed by GPA. However, one c o r r e l a t i o n may be good i n a c e r t a i n r e g i o n but not as good i n another r e g i o n .

Q:

Why show comparisons w i t h a l l these c o r r e l a t i o n s ? Why not throw out some that are known not to be a p p l i c a b l e ?

A:

A l l c o r r e l a t i o n s used i n process s i m u l a t o r s which are a v a i l able on a time-sharing b a s i s , or which are marketed by GPA, were t e s t e d to show which are poor and which are good f o r t h i s type of p l a n t . This a l l o w s the engineer to see why some c o r r e l a t i o n s should not be used.

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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315

Comments: Some of the two constant equations of s t a t e p r e d i c t good phase e q u i l i b r i a r e s u l t s a t cryogenic temperatures. Experimental data measured on an LNG p l a n t were used to compare a g a i n s t p r e d i c t i o n s from both the Soave Redlich-Kwong and the Peng-Robinson equations of s t a t e marketed through GPA, and both c a l c u l a t i o n methods compared w e l l w i t h the data. Phase e q u i l i b r i a c a l c u l a t i o n s from both an Orye-type BWR and the Soave Redlich-Kwong equation of s t a t e were compared w i t h data from an LNG p l a n t . Both c o r r e l a t i o n s showed good comparisons w i t h the p l a n t data and experimental data measured by P-V-T, I n c . The e n t h a l p i e s p r e d i c t e d u s i n g the Soave Redlich-Kwong equation of s t a t e were poorer than thos correlation. The c a l c u l a t e d ethane recovery i s very s e n s i t i v e to the method used t o c h a r a c t e r i z e the heptanes plus f r a c t i o n . I f t h i s f r a c t i o n contained p r i m a r i l y naphthenic and aromatic components, the c a l c u l a t e d ethane recovery could be s e v e r a l percent d i f f e r e n t than i f the f r a c t i o n contained only n - p a r a f f i n hydrocarbons. The charact e r i z a t i o n of the heavy f r a c t i o n i s important even when that f r a c t i o n c o n t s i s t s of l e s s than 0.5 mol % of the i n l e t feed.

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

16 Prediction of Thermodynamic Properties JOHN S. ROWLINSON Physical Chemistry Laboratory, Oxford, England

No chemical engineer has ever r e l i e d on d i r e c t l y measured values f o r a l l the thermodynamic p r o p e r t i e s he uses. I t i s one of the great v i r t u e s of c l a s s i c a l thermodynamics (indeed, i t has been s a i d that i t i s i t s only v i r t u e (j )) that i t a l l o w s us to c a l c u l a t e one p h y s i c a l or chemical property from another, f o r example, a l a t e n t heat of evaporation from the change of vapour pressure w i t h temperature. Less obvious, but almost e q u a l l y secure c a l c u l a t i o n s can be made by f i t t i n g measured values t o an e m p i r i c a l equation and c a l c u l a t i n g other p r o p e r t i e s by a subsequent manipul a t i o n of that equation. Thus we can f i t the pressure of a gas to a Redlich-Kwong (RK) or Benedict-Webb-Rubin (BWR) equation, from which the other p r o p e r t i e s such as changes of enthalpy and entropy are then d e r i v e d . S i m i l a r l y we f i t a c t i v i t y c o e f f i c i e n t s to W i l son's equation and c a l c u l a t e K-values. None of t h i s , necessary and u s e f u l though i t i s , i s what i s meant by p r e d i c t i o n , f o r t h a t term i s u s u a l l y reserved f o r c a l c u l a t i o n s which go beyond the comfortable s e c u r i t y of c l a s s i c a l thermodynamics (2,3). They may s t i l l be e m p i r i c a l , f o r example, the e x t r a p o l a t i o n of a BWR equation to a range remote from t h a t of the experimental evidence used to determine i t s parameters, or the e s t i m a t i o n of BWR parameters f o r a mixture from those f o r the pure components. I t i s , however, becoming more common to r e s t r i c t the word p r e d i c t i o n to methods of c a l c u l a t i o n which a r e based, a t l e a s t i n p a r t , on the two t h e o r e t i c a l d i s c i p l i n e s of quantum and s t a t i s t i c a l mechanics, and i t i s t h a t f i e l d which i s the s u b j e c t of t h i s review. The d i s t i n c t i o n between e m p i r i c i s m and theory i s not sharp, nor i s i t unchanging. Thus before 1938-39 we should have s a i d that the p r i n c i p l e of corresponding s t a t e s was e m p i r i c a l , but a f t e r the work of de Boer and of P i t z e r (_4,5_,6) we could see i t s b a s i s i n s t a t i s t i c a l mechanics, and i t now ranks as a t h e o r e t i c a l l y - b a s e d p r i n c i p l e . Moreover i t was only a f t e r i t s t h e o r e t i c a l b a s i s , and hence i t s t h e o r e t i c a l l i m i t a t i o n s , had become evident that we were L

316

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16.

ROWLINSON

Prediction of Thermodynamic Properties

317

able t o d i s c u s s w i t h success s y s t e m a t i c departures from the p r i n c i p l e . These departures are now handled by means of the a c e n t r i c f a c t o r , as P r o f e s s o r P i t z e r has d e s c r i b e d i n the opening paper of t h i s meeting. The sequence of e m p i r i c i s m , f o l l o w e d by theory, f o l l o w e d by an e x t e n s i o n which i s e m p i r i c a l i n form but which i s guided by theory, i s one which i s , I b e l i e v e , l i k e l y t o be f o l lowed i n c r e a s i n g l y i n the development of methods of p r e d i c t i o n of v a l u e t o the chemical engineer. There i s no need to s t r e s s the confidence we a l l f e e l i n c a l c u l a t i o n s t h a t are s e c u r e l y founded on c l a s s i c a l thermodynamics, but i t i s important to emphasize t h a t those based on s t a t i s t i c a l mechanics are not i n h e r e n t l y any l e s s secure. This s c i e n c e i s now about 100 years o l d , i f we reckon Maxwell, Boltzmann and Gibbs t o be i t s founders, and that i s ample time f o r any f a u l t s i n the foundations t o have r e v e a l e l a t i o n s may be more s p e c u l a t i v have i n t r o d u c e d , but the e x i s t e n c e of these approximations i s always e v i d e n t , even i f t h e i r consequences are not f u l l y known. When the approximations are n e g l i g i b l e our confidence i n the c a l c u l a t i o n s should be high (_2,3) . The D i l u t e Gas S t a t i s t i c a l c a l c u l a t i o n s are most accurate i n the c a l c u l a t i o n s of the p r o p e r t i e s of the d i l u t e or p e r f e c t gas, where they provide the l i n k between q u a n t a l and s p e c t r o s c o p i c determinations of molecular energy l e v e l s , on the one hand, and molar thermodynamic p r o p e r t i e s on the o t h e r . Even twenty years ago (7) i t was accepted that s t a t i s t i c a l c a l c u l a t i o n s were more accurate than experimental measurements f o r the heat c a p a c i t i e s , energies and e n t r o p i e s of such simple gases as A r , N 2 , H 2 , CO, C02 and H2O; indeed i t would now be hard t o f i n d a recent measurement of these p r o p e r t i e s . The case f o r s t a t i s t i c a l c a l c u l a t i o n , w h i l e s t i l l s t r o n g , i s not so overwhelming once we go to more complicated molecules. The l i m i t i n g f a c t o r i s the a b i l i t y of the s p e c t r o s c o p i s t t o a s s i g n energy l e v e l s to molecules, an a b i l i t y which depends on the symmetry and r i d i g i t y of the molecule. I f these d e s i r a b l e p r o p e r t i e s are absent, as i n the a l c o h o l s , f o r example, then h i s job i s d i f f i c u l t , but even here h i s accuracy i s now not n e c e s s a r i l y worse than that of a l l but the very best measurements. The p o s i t i o n has improved c o n s i d e r a b l y over that d e s c r i b e d by Reid and Sherwood (8) ten years ago, as i s shown by the methods and r e s u l t s f o r o r g a n i c molecules d e s c r i b e d i n a review by F r a n k i s s and Green ( 9 ) . The d i l u t e gas, i t s m i x t u r e s , and chemical r e a c t i o n s o c c u r r i n g t h e r e i n , can now be regarded as a solved problem. The V i r i a l Expansion When we go t o the next l e v e l of d i f f i c u l t y , the imperfect gas, then the claims made f o r s t a t i s t i c a l mechanics, although

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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s t i l l s t r o n g , must be moderated. We note, f i r s t , that i t t e l l s us the a p p r o p r i a t e form of the equation of s t a t e , namely that the compression f a c t o r Z = pV/nRT, has a power s e r i e s expansion i n the molar d e n s i t y n/V, where n i s the amount of substance. This expansion i s convergent at the low d e n s i t i e s and divergent a t high. We do not know the l i m i t of convergence but i t must be below the c r i t i c a l d e n s i t y at temperatures at and below T . Perhaps f o r t u n a t e l y , the mathematical range of convergence i s unimportant i n p r a c t i c e s i n c e the expansion i s useless when i t needs to be taken to more than two or three terms. More important i s that s t a t i s t i c a l mechanics does t e l l us the composition dependence of each of the v i r i a l c o e f f i c i e n t s of t h i s expansion. The n v i r i a l c o e f f i c i e n t i s determined by the f o r c e s between a group of n molecules, and so i s a polynomial of degree n i n the mole f r a c t i o n s y^ c

t n

B = E I y. y. B. . ±

j

1

J

=

(B..

B..),

(1)

ij Ji

1J

which i s a r e s u l t we owe dynamics .

to s t a t i s t i c a l not c l a s s i c a l thermo-

Intermolecular Forces When we come 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 s of pure and mixed gases we come to one of the p r i n c i p a l d i f f i c u l t i e s of a l l methods of p r e d i c t i o n based on s t a t i s t i c a l mechanics - what do we know of i n t e r m o l e c u l a r f o r c e s ? Here c a u t i o n i s needed, f o r twenty years ago we thought that we knew more about them than we did. The Lennard-Jones p o t e n t i a l i s u(r) = 4 e [ ( a / r )

1 2

6

- (a/r) ]

(2)

where e i s the depth a t the minimum and a, the c o l l i s i o n diameter: i . e . u(a) = 0. Because of some c a n c e l l a t i o n of e r r o r s , which need not be discussed here, and under the i n f l u e n c e of an important book by H i r s c h f e l d e r , C u r t i s s and B i r d (10), i t was thought that the Lennard-Jones p o t e n t i a l was an accurate r e p r e s e n t a t i o n of the f o r c e s between simple molecules such as Ar, N£, O2, CH4, e t c . The accepted value of e/k f o r the Ar-Ar i n t e r a c t i o n was 120K; we know now that the p o t e n t i a l has a d i f f e r e n t f u n c t i o n a l form and that e/k i s 141 + 2 K, Moreover i t i s now only f o r the i n e r t gases (11) and f o r some of t h e i r mixtures that we are c o n f i d e n t that we know e, a, and the f u n c t i o n a l form of u ( r ) . Nevertheless t h i s s a l u t a r y shock to our confidence should not lead us i n t o the opposite e r r o r , s i n c e f o r many p a i r s of molecules, l i k e or u n l i k e , we s t i l l know much about the s t r e n g t h of the i n t e r m o l e c u l a r p o t e n t i a l r e l a t i v e to that of another p a i r , and such r e l a t i v e r a t h e r than absolute values o f t e n s u f f i c e f o r p r e d i c t i o n . F o r t u n a t e l y the second v i r i a l c o e f f i c i e n t and many other thermodynamic p r o p e r t i e s of gases and l i q u i d s are not very sens i t i v e to the exact form of the p o t e n t i a l (indeed our e r r o r w i t h the

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

16.

ROWLINSON

Prediction of Thermodynamic Properties

319

Lennard-Jones p o t e n t i a l could not have a r i s e n i f they were), and so p r e d i c t i o n s can be b e t t e r than the p o t e n t i a l s they depend on. Thus the e d i t i o n s of the American Petroleum I n s t i t u t e Research P r o j e c t 44 p u b l i s h e d s i n c e the 1950 s have r e l i e d on the LennardJones p o t e n t i a l f o r the c a l c u l a t i o n of the p r o p e r t i e s of methane to 1500 K and 100 atm. The temperature i s 1000 K higher than any measurements of B but I t h i n k that the r e s u l t s are more r e l i a b l e than any that could be obtained by the e x t r a p o l a t i o n of an e m p i r i c a l equation of s t a t e , however complicated, that had been f i t t e d to the p-V-T p r o p e r t i e s a t low temperatures. When we come to mixtures even r e l a t i v e knowledge of the s t r e n g t h of i n t e r m o l e c u l a r f o r c e s i s harder to o b t a i n . I n pract i c e we have to b a c k - c a l c u l a t e from one or more observed p r o p e r t i e s i n order to o b t a i n the strengths of the u n l i k e f o r c e s which we then use to c a l c u l a t e anothe Thus i n one c a l c u l a t i o n a ^12 f ° system CO2 + N2 was used to a i d the c a l c u l a t i o n of the v i r i a l c o e f f i c i e n t s B12 " ^112 * so to p r e d i c t the s o l u b i l i t y of s o l i d CO2 i n compressed a i r (12). More u s u a l i s the use of one thermodynamic property (B12, T as a f u n c t i o n of y, etc.) to c a l c u l a t e another (K-values, enthalpy of l i q u i d mixtures, e t c . ) . In each case, however, we need an experimental measurement on which to base our estimate of the s t r e n g t h of the 1-2 f o r c e s i f we a r e t o have a secure base f o r our c a l c u l a t i o n s . Intermolecular f o r c e s a r e , to a good approximation, f o r c e s between p a i r s of molecules, and so experimental evidence on the p r o p e r t i e s of b i n a r y mixtures w i l l always be the foundation of s a t i s f a c t o r y methods of p r e d i c t i o n . F o r t u n a t e l y there a r e now r a p i d , indeed almost autom a t i c , methods of measuring heats of mixing and vapour pressures at l e a s t f o r b i n a r y mixtures a t and near room temperatures. It is only the need f o r data a t inconvenient pressures and temperatures and, above a l l , the need f o r the p r o p e r t i e s of multi-component mixtures, that j u s t i f i e s the great e f f o r t put i n t o the development of methods of p r e d i c t i o n . f

rt

n

e

a n a

anc

c

L i q u i d and Dense Gas Mixtures In using the v i r i a l equation of s t a t e our problems are those of knowing enough of the i n t e r m o l e c u l a r f o r c e s ; the s t a t i s t i c a l mechanics we a r e u s i n g i s e s s e n t i a l l y exact. When we come to dense gases and l i q u i d s the i n t e r m o l e c u l a r f o r c e problem i s s t i l l w i t h us but we have now the a d d i t i o n a l problem of having to i n t r o duce approximations i n the s t a t i s t i c a l mechanics. This i s not the p l a c e to attempt t o review progress i n the theory of l i q u i d s and l i q u i d mixtures (13, 14, 15, 16), although t h i s has been g r a t i f y i n g l y r a p i d i n the l a s t decade, but simply to s t a t e that the methods of p r e d i c t i o n of v a l u e to engineers a r e s t i l l based on the s t a t e of the s t a t i s t i c a l a r t of about ten years ago. No doubt the more recent work on theory w i l l , i n time, be used, but t h i s has not yet happened to any a p p r e c i a b l e extent. This s e c t i o n of t h i s review i f t h e r e f o r e r e s t r i c t e d t o methods based on the p r i n c i p l e

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

320

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

of corresponding s t a t e s , on i t s e x t e n s i o n by means of the a c e n t r i c f a c t o r to the l e s s simple f l u i d s , and to mixtures by the best p s e u d o - c r i t i c a l approximation a v a i l a b l e to us. The p r i n c i p l e was f i r s t used to r e l a t e the p-V-T p r o p e r t i e s of one pure substance t o those of another by w r i t i n g : P.(V,T) = ( h / f ) [ P ( V / h , T / f . . ) ] ± i

l i

o

(3)

i i

where f . . and h.. a r e the r a t i o s of the c r i t i c a l c o n s t a n t s : XI xx C

C

f.. = T. /T ° and h.. = V.°/V (4) XX X O XX X o The s u b s c r i p t s a r e doubled because the s i z e of the parameters f and h depends on the s t r e n g t h s of the p o t e n t i a l energies u ^ i ( r ) and u ( r ) between two o. O Q

f.. = e. ./e h. . = a. /o (5) XX XX OO XX XX oo These equations enable the (unknown) pressure of substance i to be c a l c u l a t e d from the (presumed known) pressure of substance o a t a d i f f e r e n t volume and temperature. A more g e n e r a l l y u s e f u l equat i o n i s that r e l a t i n g the c o n f i g u r a t i o n a l p a r t s of the Helmholtz f r e e energy (17) 3

3

A.(V,T) = f..[A (V/h.., T / f . . ) ] - nRT£nh.. X XX o xx' XX XX from which equation (3) f o l l o w s a t once, s i n c e : P = - OA/3V)

(6)

r

Equations (3) to ( 6 ) , although s i m p l e , a r e not accurate except f o r c l o s e l y r e l a t e d p a i r s of substances, f o r example Kr from Ar, or i-C4Hl0 from n-C4Hio that i s , p a i r s f o r which u ( r ) can be expected t o have g e o m e t r i c a l l y s i m i l a r shapes. They have the consequence that f o r both substances, i and o, the reduced vapour pressure ( P / P ) i s the same f u n c t i o n of the reduced temperature (T/Tc), and that both substances have the same v a l u e f o r the c r i t i c a l compression r a t i o Zc = pcv /nRT . I f choose argon as a r e f e r e n c e substance, s u b s c r i p t o, then ( P / P ) i s 0.100 a t (T/T ) =0.7, and Z = 0.293. The same r a t i o s , namely 0.100 and 0.293, a r e found f o r k r y p t o n and xenon, but lower v a l u e s f o r a l l other substances other than the " q u a n t a l " f l u i d s , hydrogen, helium, and neon. Moreover, the departures of other substances from the reduced behavior of argon are not random but can be w e l l c o r r e l a t e d by a t h i r d parameter i n a d d i t i o n to f and h, which i s a measure of the i n c r e a s i n g departure of u ( r ) from the s p h e r i c a l nonpolar form i t has f o r argon. I f , f o l l o w i n g P i t z e r (18, 19, 20), we d e f i n e an a c e n t r i c f a c t o r , a), by: a

c

c

c

w

Q

e

C

0

c

C

Q

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

16.

Prediction of Thermodynamic Properties

ROWLINSON

a)

Q

321

C

= -1.000 - l o g ( P / P ) 1 0

i

(7)

i

then a) i s zero f o r A r , Kr, and Xe and takes the f o l l o w i n g values for some other substances: Substance

U)

0.021

°2

0.040 0.013

CH, 4 C

2 6

C

3 8

C

6 6

0.105

H

H

0.215

H

0.348

H 0 2

The change of Z° from the v a l u e of 0.293 i s found to be r e l a t e d to the v a l u e of oo: Z

c

= 0.293/(1 + 0.375 a>) .

(8)

E i t h e r w o r (Z - 0.293) can be used as a t h i r d parameter w i t h which to extend the p r i n c i p l e of corresponding s t a t e s to substances which depart from s t r i c t agreement w i t h equations (3) t o ( 6 ) . The f i r s t was used by P i t z e r and h i s c o l l e a g u e s , (18-20) and the second by Hougen, Watson, and Ragatz (21). The f i r s t , has the p r a c t i c a l advantage that the vapour pressure a t (T/T ) = 0.7 i s known f o r most substances s i n c e t h i s temperature i s not f a r above the normal b o i l i n g p o i n t . P i t z e r wrote, f o r any thermodynamic f u n c t i o n Y: c

C

C

Y.(V/V.°, T/T. ) = Y (V/V °, T/T ) + 1 1 l o o ' o u> Y '(V/V °, T/T °) o o o

(9)

where Yo i s the same f u n c t i o n as f o r argon and Y i s an e m p i r i c a l l y determined c o r r e c t i o n f u n c t i o n . A more convenient way of expressing the same r e s u l t was devised by Leland and h i s c o l l e a g u e s (22, 23) who introduced what they c a l l e d shape f a c t o r s , 0 and <(>. These are f u n c t i o n s of the reduced d e n s i t y and temperature which are d e f i n e d as f o l l o w s . Let us suppose t h a t , f o r a p a i r of substances, i and o, which do not n e c e s s a r i l y conform mutually t o a p r i n c i p l e of corresponding s t a t e s , we use (1) and (4) to d e f i n e f u n c t i o n s f -it. . and h... I t can be shown (17, 24) t h a t these -ii — — Q

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

322

P H A S E EQUILIBRIA AND

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

equations can be s a t i s f i e d simultaneously and c o n s i s t e n t l y only by one p a i r of v a l u e s of and h-^ at each V and T of the substance i. C l e a r l y i f i and o do conform to the p r i n c i p l e ( i . e . i f co-j^ = 0 ) ) then f and h, so d e f i n e d , would be constants given by equat i o n ( 4 ) . I f i and o do not conform, then they are s l o w l y v a r y i n g f u n c t i o n s of V and T. Leland t h e r e f o r e w r i t e s : o o

f..

11,00

C

C

(V/V. , T/T.°) = (T.°/T °) x 6.. 1

1

1 0

(V/V.°, T/T. )

11,00

1

1

(10) C

h.. (V/V. , T/T.°) = (V.°/V °) x (V/V.°, T/T.°) 11,00 1 1 1 o ' n,oo 1 1 Y

where 0 and cj) are the shape f a c t o r s of substance i , w i t h r e f e r e n c e to substance o. They can be expressed: 3..

11,00

= 1 + (a).. X

11

0

00

'

(ID |>..

11,00

C

= 1 + ( ( * ) . . - a) ) x F^(V/V , 11 00 (j) '

T/T°)

!

where w i t h i n the ambit of P i t z e r s f o r m u l a t i o n , F 0 and F ^ are u n i v e r s a l f u n c t i o n s , determinable e m p i r i c a l l y by comparing a few f l u i d s w i t h , say, argon (25, 26). With t h i s apparatus we are equipped to c a l c u l a t e the c o n f i g u r a t i o n a l f r e e energy, and hence the other thermodynamic p r o p e r t i e s , f o r a s i n g l e component, i , from a knowledge of A (V,T) f o r argon (or other r e f e r e n c e substance of our c h o i c e ) , from the u n i v e r s a l f u n c t i o n s F Q and F^, and the constants V-^ , T^ , and u±±* The l i m i t a t i o n of the method i s that the departure of i from the reduced p r o p e r t i e s of o should be s u f f i c i e n t l y s m a l l and r e g u l a r to be d e s c r i b e d by the one parameter 00. In p r a c t i c e t h i s means that i f our r e f e r e n c e substance i s one f o r which w i s a p p r o x i mately zero, we are r e s t r i c t e d to substances f o r which w i s l e s s than about 0.25. This i n c l u d e s most simple molecules except NH3 and H2O, and the lower hydrocarbons, but excludes such t e c h n i c a l l y important substances as the a l c o h o l s and the lower amines, ammonia and water. Many attempts have been made to extend the r e c i p e above to mixtures by i n t r o d u c i n g parameters f and h ( o r , what i s equivalent, T and V ) which are averaged parameters a s c r i b e d to a s i n g l e h y p o t h e t i c a l substance, s u b s c r i p t x, which i s chosen to represent the m i x t u r e . That i s , we w r i t e f o r the c o n f i g u r a t i o n a l f r e e energy of a mixture of C components: Q

0

x

0

x

c

x

x

A . _(V,T) = A (V,T,) + I n. RT£nx. mixt x .,1 1 N

,

(12)

/

1=1

A (V,T) = f X

.A X

where Z n. = n and

(V/h , T/f ) - n RT£nh O

c Z

i=i

X

X

(13) X

x. = 1. 1

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

(14)

16.

Prediction of Thermodynamic Properties

ROWLINSON

323

The parameters f and h c l e a r l y depend on a l l molecular i n t e r a c t i o n s i n the m i x t u r e , that i s , on both those between l i k e molecules, f i i and h i i , and those between u n l i k e molecules, f j j and h y . Since i n t e r m o l e c u l a r f o r c e s are, a t l e a s t approximately, p a i r - w i s e a d d i t i v e , we do not need to i n t r o d u c e three-body parameters, f i j k > e t c . The most s u c c e s s f u l r e c i p e f o r combining the parameters, and the one f o r which there are the best t h e o r e t i c a l arguments (27) i s that u s u a l l y c a l l e d the van der Waals approximation: x

f h

c I

=

x

I x.x.f..h..

i=l j=l h

= x

c c Z E x.x.h... i• = li j• = -il i J J

(15)

a

J

These equations, together w i t h equations (12) t o (14), are a s o l u t i o n t o the problem of c a l c u l a t i n g the thermodynamic prope r t i e s of a multi-component m i x t u r e , provided: (1) That the change of f and h w i t h reduced volume and temperature can again be i n c o r p o r a t e d i n t o shape f a c t o r s , as f o r pure substances i n equation (10) and (11). (2) That we can a s s i g n values to the cross-parameters f i j and h y w i t h i not equal to j . The f i r s t problem has been s o l v e d , but needs care i f t h e r modynamic c o n s i s t e n c y i s t o be preserved (17, 22-26). The second i s u s u a l l y solved by w r i t i n g : f

f

ij = V

( 1 6 )

i i V "

h.. = n..(%h./

/3 +

^h..

1 / 3

)

3

(17)

u>. . = h (u). . + a>. . ) . ij i i 33

(18)

I f nothing i s known of the b i n a r y system formed from species i and j then i t i s u s u a l t o adopt the L o r e n t z - B e r t h e l o t assumption:

= n . = 1.

(19)

±

Measurements of almost any thermodynamic property of the b i n a r y mixture allows a t l e a s t one of these parameters to be determined more p r e c i s e l y . I t i s u s u a l t o r e t a i n rii-j 1 and t o use the b i n a r y measurements to estimate • T y p i c a l values a r e : (28-31) =

J

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

324

P H A S E EQUILIBRIA AND

Substances Ar +

0

F L U I D PROPERTIES IN

5

2

0.99 0.99

N

2

+ CO

N

2

+

CH

0.97

4

CO + CH. 4 CH + C H

6

0.99

CH

8

0.97

4

4

2

+ C H 3

0.99

^3^£I

^2^2

C Hg + n 3

C0

o

C0

0

2

2

C

0.99

H

4 10

+

CH, 4

0.94

+

C H. 2 6

0.91

Q

CH, + CF, 4 4 C

H

6 6

C H E M I C A L INDUSTRY

+

c

"

0.92 C

0.97

H

6 12

Such f i g u r e s are s u b j e c t to an u n c e r t a i n t y of at l e a s t 0.01, s i n c e the theory does not e x a c t l y correspond w i t h r e a l i t y and so d i f f e r e n t p r o p e r t i e s of the b i n a r y mixture l e a d to s l i g h t l y d i f f e r e n t values of The p r o p e r t i e s most commonly used to estimate 5 are the second v i r i a l c o e f f i c i e n t and T as f u n c t i o n s of x, and the excess Gibbs f r e e energy and enthalpy. There i s now abundant evidence that 5 i s u s u a l l y s i g n i f i c a n t l y l e s s than u n i t y , p a r t i c u l a r l y i n mixtures of c h e m i c a l l y d i f f e r e n t type. The use of values d e t e r mined from b i n a r y mixtures leads to more accurate p r e d i c t i o n s f o r multi-component mixtures than the u n i v e r s a l use of equation (19). The p r i n c i p l e of corresponding s t a t e s , extended as above to mixtures of a c e n t r i c molecules, has been a p p l i e d to the c a l c u l a t i o n of many of the p r o p e r t i e s needed f o r the design of s e p a r a t i o n equipment. The examples reviewed b r i e f l y here are taken from our own work on cryogenic f l u i d s , l i q u i f i e d n a t u r a l gas (LNG), mixtures of hydrocarbons, and mixtures of carbon d i o x i d e w i t h hydrocarbons. In a l l t h i s work methane was used as the r e f e r e n c e substance. c

Liquid-Vapour E q u i l i b r i u m . For mixtures of Ar + N2 + O2 the method d e s c r i b e d above y i e l d s e x c e l l e n t K-values (28). S i m i l a r r e s u l t s were obtained by Mollerup and Fredenslund f o r Ar + N2> using a method almost i d e n t i c a l w i t h that d e s c r i b e d here (32). The p r i n c i p a l e r r o r i n the work on the ternary system arose i n the pure

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

16.

Prediction of Thermodynamic Properties

ROWLINSON

325

components. P i t z e r * s a c e n t r i c f a c t o r , although i t provides an exact f i t to the vapour pressure a t (T/T ) = 0 . 7 , leads to s m a l l e r r o r s i n the vapour pressures a t other temperatures. The b o i l i n g p o i n t o f each component i s almost c e r t a i n l y one of the experimental f a c t s known before the c a l c u l a t i o n s a r e s t a r t e d f o r the multi-component mixture, and so i t i s n a t u r a l to modify the procedure above i n such a way as to use t h i s i n f o r m a t i o n . This can be done by t r e a t i n g w as a f l o a t i n g parameter which, a t each pressure, i s chosen a f r e s h so as to reproduce e x a c t l y the b o i l i n g p o i n t o f each component. I f t h i s i s done, (29) then the l i q u i d vapour e q u i l i b r i u m of Ar + N2 + O2 i s reproduced w i t h a mean e r r o r o f 0.14% which i s almost w i t h i n experimental e r r o r , and e x c e l l e n t r e p r e s e n t a t i o n s a r e obtained o f the systems: C3H8 + C3H5: C3H3 + n - C H : C2Hfc + C^: and c

4

C0

2

+ n-C4H

1 0

10

(29).

Compression f a c t o r For one-phase systems comparison of theory w i t h experiment i s hindered by the l a c k of good experimental r e s u l t s f o r m u l t i component systems. However, Z i s given (29) t o 0.1 - 1.0% f o r pressures up t o 200 t o 300 bar f o r a i r , f o r CH4 + H2S, and f o r CH4 + CO2. The enthalpy of a i r i s e q u a l l y w e l l p r e d i c t e d (29). A z e o t r o p i c L i n e s . The c a l c u l a t i o n of l i q u i d - v a p o u r e q u i l i b r i u m i n c l u d e s , i n p r i n c i p l e , the d e t e r m i n a t i o n of the c o n d i t i o n s f o r azeotropy. N e v e r t h e l e s s , i n p r a c t i c e these a r e best formulated and t e s t e d f o r i n a separate c a l c u l a t i o n (30). Values o f £^2 l than u n i t y lead c o r r e c t l y to the appearance of azeotropes a t h i g h reduced temperatures f o r the systems: CO2 + C2H5: CO2 + C2H4: and e s s

C02 + C H2 2

(30).

C r i t i c a l L i n e s . The c a l c u l a t i o n of the thermodynamic c o n d i t i o n s f o r c r i t i c a l p o i n t s i n even a b i n a r y system i s a severe t e s t of any method of p r e d i c t i o n , because o f the s u r p r i s i n g v a r i e t y o f c r i t i c a l l i n e s t h a t can occur ( 2 ) . The Redlich-Kwong equation has been used by chemical engineers (33, 34, 35) to represent the s i m p l e s t c l a s s of b i n a r y system, when there i s but one c r i t i c a l l i n e i n p-T-x space. This j o i n s the g a s - l i q u i d c r i t i c a l p o i n t of the two pure components. The treatment described above can r e p r e sent, q u a n t i t a t i v e l y f o r simple systems, and q u a l i t a t i v e l y f o r more complex systems, the c r i t i c a l l i n e s of many b i n a r y mixtures (30). I n the more complex of these, the l i n e s r e p r e s e n t i n g l i q u i d l i q u i d c r i t i c a l s t a t e s i n t r u d e i n t o the g a s - l i q u i d c r i t i c a l r e g i o n , g i v i n g r i s e to a t o p o l o g i c a l l y wide v a r i e t y of behavior. Density o f LNG. An accurate knowledge of the d e n s i t y of LNG i s needed f o r i t s metering and pumping. The d e n s i t i e s of some n a t u r a l and some " s y n t h e t i c " mixtures have been measured but i t i s i m p o s s i b l e t o cover a l l compositions that might be encountered i n p r a c t i c e . The problem of p r e d i c t i n g d e n s i t i e s i n such a system i s one that i s i d e a l l y s u i t e d to the method d e s c r i b e d above.

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

326

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F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

The r e s u l t s obtained were i n c o r p o r a t e d i n a program which s p e c i f i e d f i r s t whether the c o n d i t i o n s are those o f : (1) the compressed l i q u i d , (2) the l i q u i d a t a bubble p o i n t of s p e c i f i e d temperature, (3) or the l i q u i d at a bubble p o i n t of s p e c i f i e d pressure. The program then y i e l d e d the l i q u i d d e n s i t y w i t h an accuracy u s u a l l y b e t t e r than 0.2%, and, f o r c o n d i t i o n s (2) and ( 3 ) , the composition, p r e s s u r e , (or temperature) and d e n s i t y of the vapour phase (31). This program was prepared by Mollerup (36) and was issued by him to many i n t e r e s t e d i n t e s t i n g t h i s method of p r e d i c t i o n . S a v i l l e and h i s colleagues (37) l a t e r modified and shortened programs of t h i s k i n d , f i r s t by a l g e b r a i c improvements which removed one l a y e r of i t e r a t i o n , secondly by u s i n g more e f f i c i e n t programming, and t h i r d l y by u s i n g Stewart and Jacobsen s equation f o r n i t r o g e n as the r e f e r e n c e equation (38). (Vera and P r a u s n i t z (39) had e a r l i e r p o i n t e d out the advantage they chose S t r o b r i d g e ' s t i o n f o r methane.) I t i s d i f f i c u l t to compare r a t e s of computation, but i t may be u s e f u l to estimate c o s t s . At the standard r a t e s quoted by commercial bureaux i n B r i t a i n , S a v i l l e estimates that the c a l c u l a t i o n of the e q u i l i b r i u m s t a t e s of l i q u i d and vapour i n a multi-component mixture and the t a b u l a t i o n of t h e i r e n t h a l p i e s , d e n s i t i e s and f u g a c i t i e s , c o s t s now 3 cents per p o i n t . (Within the u n i v e r s i t y the r a t e s are only 2% of t h i s , but these costs are not economically r e a l i s t i c . ) A sum of 3 cents i s , of course, n e g l i g i b l e f o r a s m a l l number of c a l c u l a t i o n s . I t i s a p p r e c i a b l e , but not p r o h i b i t i v e , i f the program i s to be run many times as, f o r example, i n an i t e r a t i v e design c a l c u l a t i o n f o r a d i s t i l l a t i o n o i s t r i p p i n g column. One p o i n t which i s worth emphasizing here i s the need i n such c a l c u l a t i o n s f o r h i g h accuracy i n " d i f f i c u l t " regions of the phase diagram. Separation processes may account f o r h a l f the c a p i t a l cost of a modern p l a n t , and the c a p i t a l cost of s e p a r a t i o n v a r i e s roughly as (£n a ) - l , where a i s the v o l a t i l i t y r a t i o . Hence high accuracy i s needed i n those r e g i o n s , o f t e n near i n f i n i t e d i l u t i o n or azeotropes, where a i s c l o s e to u n i t y . The t e s t i n g of methods of p r e d i c t i o n does not u s u a l l y pay enough a t t e n t i o n to t h i s p o i n t . Moreover the cost of e r r o r i n these f i e l d s make i t almost c e r t a i n that the design i s t e s t e d e x p e r i m e n t a l l y before a f i n a l d e c i s i o n i s taken. There are d e f i n i t e l i m i t s to what p r e d i c t i v e methods can be expected to do. f

Advantages and Disadvantages of the P r i n c i p l e of States as a Method of P r e d i c t i o n

Corresponding

The method d e s c r i b e d above i s t r u l y p r e d i c t i v e i n that i t uses thermodynamic i n f o r m a t i o n from one- and, i f p o s s i b l e , two-component systems to c a l c u l a t e the same or d i f f e r e n t thermodynamic p r o p e r t i e s of multi-component mixtures. There are other methods which have been used f o r the same purpose of which one of the most powerful i s the g e n e r a l i z a t i o n to mixtures of the Benedict-Webb-Rubin equation,

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

16.

ROWLINSON

327

Prediction of Thermodynamic Properties

and of equations d e r i v e d from i t . For each substance we must determine a set of eight or more parameters from the p-V-T p r o p e r t i e s . I f we use the BWR equation f o r mixtures by w r i t i n g for each of i t s e i g h t parameters X an equation of the form: X

1 / n

1 / n

E x.X. where (n = 1,2, o r 3) (20) i=l then the equation has p r e d i c t i v e power—we o b t a i n the p r o p e r t i e s of the mixture from those of the pure components. But such use, although o f t e n of v a l u e , s u f f e r s from the same l o s s of accuracy as the use of the p r i n c i p l e of corresponding s t a t e s w i t h a l l K . = 1. I f we are t o get the best use out of equations such as the BWR then i t i s necessar become themselves f u n c t i o n duce parameters which represent the a c t u a l behaviour of the r e l e v a n t b i n a r y system (40, 41). We have then a f l e x i b l e and a c c u r a t e method of p r e d i c t i n g the p r o p e r t i e s of multi-component m i x t u r e s . When comparative t e s t s have been made, the accuracy of the BWR equation and of the p r i n c i p l e of corresponding s t a t e s has been about the same (29), but the extended BWR equations are more a c c u r a t e f o r the systems to which they have been f i t t e d . The use of the p r i n c i p l e of corresponding s t a t e s i s now l i t t l e more expensive i n computing time. The advantages of the p r i n c i p l e a r e , f i r s t , that i t i s more c l o s e l y t i e d t o theory and so can more s a f e l y t o e x t r a p o l a t e d to new r e g i o n s of pressure and temperature. I t s second advantage i s that i t i s much more e a s i l y extended t o take i n new components. A l l we need are T i , V-^ , u±± and estimates of C y f o r the b i n a r y systems formed from the new and each o f the o l d components. I f the l a t t e r are not known then they can be put equal t o u n i t o r , b e t t e r , guessed by analogy w i t h c h e m i c a l l y s i m i l a r systems. To extend the BWR, or Stewart and J a c o b s e n ^ equation, or Bender s equation (41) to a d d i t i o n a l components r e q u i r e s a l a r g e body of p-V-T and, p r e f e r a b l y , c a l o r i m e t r i c i n f o r m a t i o n on the new components, and a program to f i t between e i g h t and 20 parameters. A s i m i l a r new f i t must be made f o r each of the b i n a r y systems i n v o l v i n g the new component i f the best accuracy i s to be achieved. The g r e a t e s t l i m i t a t i o n of both methods i s t h e i r r e s t r i c t i o n to comparatively simple molecules. The problem of p r e d i c t i n g w i t h such h i g h accuracy the p r o p e r t i e s of multi-component systems cont a i n i n g ammonia, water, a l c o h o l s , phenols, amines, e t c . i s s t i l l s u b s t a n t i a l l y unsolved. I t i s hard to say how r a p i d progress w i l l be here. Undoubtedly there are r e g u l a r i t i e s i n the behaviour o f such s e r i e s of compounds as the a l c o h o l s , but f o r two reasons I do not want t o s p e c u l a t e f u r t h e r on how mixtures c o n t a i n i n g h i g h l y p o l a r substances can be handled. The f i r s t i s t h a t , a t l e a s t i n the f i r s t i n s t a n c e , such methods are l i k e l y t o be more e m p i r i c a l than those I have d e s c r i b e d , perhaps of the k i n d c a l l e d "molecular thermodynamics" by P r a u s n i t z (42), and the second i s that I do not x

=

±

c

0

9

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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P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

know how f a r companies and other agencies have gone i n developing such methods i n schemes of p r e d i c t i o n that a r e w h o l l y or p a r t l y unpublished.

Abstract If the prediction of thermodynamic properties is to be soundly based then it requires an understanding of the properties of molecules and the forces between them, and of the methods of statistical mechanics. This paper reviews the extent of our knowledge in these fields and of our ability to make useful predictions. Particular emphasis is placed on methods that derive from the principle of corresponding states. Literature Cited 1. McGlashan, M. L., "Chemical Thermodynamics", Vol. 1, Chap. 1. Specialist Report, Chemical Society, London, 1973. 2. Bett, K. E., Rowlinson, J. S., and Saville, G., "Thermodynamics for Chemical Engineers", Chapt. 9, M.I.T. Press, Cambridge, Mass., 1975. 3. Reed, T. M. and Gubbins, K. E., "Applied Statistical Mechanics", McGraw-Hill, New York, 1973. 4. de Boer, J. and Michels, A., Physica, (1938) 5, 945. 5. Pitzer, K. S., J. Chem. Phys., (1939) 7, 583. 6. Guggenheim, E. A., J. Chem. Phys., (1945) 13, 253. 7. "Tables of Thermal Properties of Gases", National Bureau of Standards Circular 564, Washington, D.C., 1955. 8. Reid, R. C. and Sherwood, T. K., "The Properties of Gases and Liquids: Their Estimation and Correlation", Chapt. 5, 2nd edn., McGraw-Hill, New York, (1966). 9. Frankiss, S. G. and Green, J. H. S., "Chemical Thermodynamics", Vol. 1, Chap. 8, Specialist Report, Chemical Society, London, (1973). 10. Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B., "Molecular Theory of Gases and Liquids", John Wiley, New York (1954). 11. Smith, E. B., Physica, (1974), 73, 211. 12. Rowlinson, J. S., Chem. Ind. (1961), 929. 13. Smith, W. R., "Statistical Mechanics", Vol. 1, Chap. 2, Specialist Reports, Chemical Society, London, 1973. 14. McDonald, I. R., Vol. 1, Chap. 3, Ibid. 15. Gray, C. G., "Statistical Mechanics", Vol. 2, Chap. 5, Specialist Reports, Chemical Society, London, 1975. 16. Hansen, J. P. and McDonald, I. R., "Theory of Simple Liquids", Academic Press, London, 1976. 17. Rowlinson, J. S. and Watson, I. D., Chem. Eng. Sci., (1969) 24, 1565. 18. Pitzer, K. S. and Curl, R. F., "Thermodynamic and Transport Properties of Fluids", p. 1, London, Institution of Mechanical Engineers, 1958.

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

16.

ROWLINSON

Prediction of Thermodynamic Properties

329

19. Edmister, W. C., Petrol. Refiner, (1958) 37, 173. 20. See reference 8, Appendix A. 21. Hougen, O. A., Watson, K. M., and Ragatz, R. A., "Chemical Process Principles: Part II, Thermodynamics", p. 569, 2nd ed., John Wiley, New York, 1959. 22. Leland, T. W., Chappelear, P. S., and Gamson, B. W., AICHE J. (1962) 8, 482. 23. Reid, R. C. and Leland, T. W., AICHE J. (1965) 11, 228. 24. Canfield, F. B. and Gunning, A. J., Chem. Eng. Sci., (1971) 26, 1139. 25. Leach, J. W., Chappelear, P. S. and Leland, T. W., Proc. Am. Petrol. Inst., (1966) 46, 223. 26. Leach, J. W., Chappelear, P. S. and Leland, T. W., AICHE J. (1968) 14, 568. 27. Leland, T. W., Rowlinson Faraday Soc., (1968) , 28. Watson, I. D. and Rowlinson, J. S., Chem. Eng. Sci. (1969) 24, 1575. 29. Gunning, A. J. and Rowlinson, J. S., Chem. Eng. Sci. (1973) 28, 529. 30. Teja, A. S. and Rowlinson, J. S., Chem. Eng. Sci. (1973) 28, 521. 31. Mollerup, J. and Rowlinson, J. S., Chem. Eng. Sci. (1974) 29, 1373. 32. Mollerup, J. and Fredenslund, A., Chem. Eng. Sci. (1973) 28, 1295. 33. Joffe, J. and Zudkevitch, D., Chem. Eng. Prog. Symp. Ser., No. 81, (1967) 63, 43. 34. Spear, R. R., Robinson, R. L., and Chao, K. C., Ind. Eng. Chem. Fundam. (1969) 8, 2. 35. Hissong, D. W. and Kay, W. B., AICHE J. (1970) 16, 580. 36. Mollerup, J., Program ICLNG. Instituttet for Kemiteknik, Danmarks Tekniske Højskole, Lyngby, Denmark. 37. Saville, G., Department of Chemical Engineering, Imperial College of Science and Technology, London, England, private communication. 38. Stewart, R. B. and Jacobsen, R. T., Center for Applied Thermodynamic Studies, College of Engineering, Idaho University, Moscow, Idaho, private communication. 39. Vera, J. H. and Prausnitz, J. M., Chem. Eng. Sci. (1971) 26, 1772. 40. See examples and papers cited in, S. K. Sood and G. G. Haselden, AICHE J. (1970) 16, 891. 41. Bender, E., The Calculation of Phase Equilibrium from a Thermal Equation of State, applied to the Pure Fluids Argon, Oxygen, and Nitrogen, and their Mixtures, Müller, Karlsruhe, 1973. 42. Prausnitz, J. M., "Molecular Thermodynamics of Fluid-Phase Equilibria", Prentice-Hall, Englewood C l i f f s , N.J., 1969.

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

17 Can Theory Contribute: Transport Properties HOWARD J . M . HANLEY National Bureau of Standards, Boulder, CO 80302

In t h i s l e c t u r e we comment b r i e f l y o ho molecula t h e o r y that i s s t a t i s t i c a l mechanics and k i n e t i c t h e o r y — c a n d e s c r i b e t r a n p o r t c o e f f i c i e n t s , e.g., the v i s c o s i t y (n)» thermal c o n d u c t i v i t y (X), and the d i f f u s i o n (D), of f l u i d s such as argon, n i t r o g e n , oxygen, carbon d i o x i d e , and the simple hydrocarbons. U n f o r t u n a t e l y i t turns out that s t a t i s t i c a l mechanics i s not always s a t i s f a c t o r y i n t h i s context because of the s e r i o u s conceptu< and mathematical d i f f i c u l t i e s i n d e s c r i b i n g a f l u i d i n n o n e q u i l i brium. And the problems a r e not only t h e o r e t i c a l : an assessment oi any theory from the viewpoint of the engineer has been, and i s , handicapped by l a c k of q u a n t i t a t i v e data. R e l i a b l e data f o r pure f l u i d s over a wide temperature and pressure range were not a v a i l a b l e u n t i l about ten years ago, w i t h only l i m i t e d e x c e p t i o n s : few data a r e y e t a v a i l a b l e f o r mixtures i n the l i q u i d . So, although i n p r i n c i p l e theory can c o n t r i b u t e , i n p r a c t i c e i t s c o n t r i b u t i o n i s not always obvious. One should d i s t i n g u i s h between formal s t a t i s t i c a l mechanics anc approximate t h e o r i e s , based on s t a t i s t i c a l mechanics, which bypass i n some way the d e t a i l e d problems of the molecular theory f o r a f l u : i n n o n e q u i l i b r i u m . Such t h e o r i e s w i l l not be discussed here ( 1 ) , w i t h the e x c e p t i o n of corresponding s t a t e s . Formal Theory One can f o r convenience say that formal s t u d i e s of t r a n s p o r t phenomena i n f l u i d s a r e based on two p r i n c i p l e approaches. The more g e n e r a l examines how a system responds to a f o r c e or g r a d i e n t : the t r a n s p o r t c o e f f i c i e n t s a r e a q u a n t i t a t i v e measure of the response, and can be expressed i n terms of the decay of f l u c t u a t i o n s induced by the f o r c e . I t can be shown that any t r a n s p o r t c o e f f i c i e n t , L, f o r a f l u i d which i s not too f a r from e q u i l i b r i u m can be w r i t t e n as an i n t e g r a l o f the corresponding c o r r e l a t i o n f u n c t i o n :

330

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

17.

L

W

331

Properties

Transport

HANLEY

(1)

°<0 (o)-J (x)]>dT L

L

T

where J ( o ) and J ( ) i s the f l u x a s s o c i a t e d w i t h L at a r b i t r a r y time o and at time T , r e s p e c t i v e l y . For example the s i m p l e s t e x p r e s s i o n i s that f o r the s e l f - d i f f u s i o n c o e f f i c i e n t , ^ l L

L

D

D

:

A ( O ) - ( T ) S dx

u

(2)

?

*

OO

*

where v i s the v e l o c i t y of a molecule. S i m i l a r l y f o r the v i s c o s i t y , and the other t r a n s p o r t p r o p e r t i e s . The expressions are standard (1). The a l t e r n a t i v e approac gas, i . e . , a gas f o r which one only has to c o n s i d e r b i n a r y molecular c o l l i s i o n s . This procedure i s c l e a r l y f a r more r e s t r i c t e d than the c o r r e l a t i o n f u n c t i o n route but the corresponding t r a n s p o r t express i o n s are standard and p r a c t i c a l . For example, the Chapman-Enskog s o l u t i o n (3) of the Boltzmann equation g i v e s the d i l u t e gas viscos i t y , n , and the other c o e f f i c i e n t s , e.g.,

= n

o

5_ 16

1/2

(7TmkT) 2 (2,2)* 7TO ft '

m K

}

n

1

where m i s the molecular mass, k i s Boltzmann s constant, a i s a s i z e parameter c h a r a c t e r i s t i c of the i n t e r m o l e c u l a r p a i r p o t e n t i a l , <j>. ft(2,2)*—{-he c o l l i s i o n i n t e g r a l — d e p i c t s the dynamics of a b i n a r y c o l l i s i o n and i s a f u n c t i o n of <j>. The r e l a t i o n s h i p between the g e n e r a l c o r r e l a t i o n f u n c t i o n approach and the d i l u t e gas should, i n p r i n c i p l e , be c l e a r - c u t . In f a c t , Equation (3) can be d e r i v e d from the corresponding equation (2). Many authors have attempted to g e n e r a l i z e the Boltzmann equation f o r a l l d e n s i t i e s ; a l t e r n a t i v e l y , many authors have attempted to w r i t e the f l u x e s of Equation (1) i n terms of the dynamics of molec u l a r c o l l i s i o n s . But these are d i f f i c u l t problems. A s i m p l i s t i c e x p l a n a t i o n of the d i f f i c u l t i e s , which perhaps should be compared to those encountered f o r a f l u i d i n e q u i l i b r i u m , i s as f o l l o w s . In e q u i l i b r i u m , i t turns out one i s concerned w i t h molecular c o n f i g u r a t i o n s over d i s t a n c e s of the order of the range of the i n t e r m o l e c u l a r p o t e n t i a l (see F i g u r e l a ) . In n o n e q u i l i b r i u m , however, the c h a r a c t e r i s t i c d i s t a n c e s are of the order of the mean f r e e path, or g r e a t e r (see F i g u r e l b ) . A d e s c r i p t i o n and understanding of the dynamics of the molecular motion i n the f l u i d r e q u i r e s c e r t a i n i n t e g r a l s to be evaluated over such d i s t a n c e s . They have not been s o l v e d f o r a r e a l i s t i c p o t e n t i a l , other than f o r the d i l u t e gas. The t h r u s t of k i n e t i c theory i n the l a s t few years has been to

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 H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Figure 1. (a) General form of the intermolecular pair potential, , as a function of separation, r. The parameter v, defined by (r) = 0, corresponds to the molecular diameter: e, is the maximum value of . (b) Schematic of a possible sequence of collisions between three molecules, 1,2, and 3, for three times t = 0, t', t".

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

17.

HANLEY

Transport Properties

333

t r y t o understand the nature of the dynamical events i n a n o n e q u i l i brium system and, i n p a r t i c u l a r , t o e x p l a i n the decay of t h e c o r r e l a t i o n f u n c t i o n w i t h time. Comparison w i t h Experiment f o r the D i l u t e Gas S u b s t a n t i a l progress has been made r e c e n t l y i n a p p l y i n g the k i n e t i c theory d i l u t e gas e x p r e s s i o n s , both f o r the t r a n s p o r t and e q u i l i b r i u m p r o p e r t i e s . For example, we now have a good grasp on how a model i n t e r m o l e c u l a r p o t e n t i a l can be used to r e l a t e theory to data. A l s o simple n o n s p h e r i c a l molecules can be considered systematically. Some r e s u l t s f o r carbon d i o x i d e (4) are presented here g r a p h i c a l l y to i l l u s t r a t e the type of agreement one can expect between k i n e t i c theory and experiment In our work w i t h carbo c u l a r p o t e n t i a l i s of the form 4>(re e 4>) = <> ( ) + 4> (re-e,*) l z s ns 1z

(4)

r

and that the molecules i n t e r a c t a c c o r d i n g to the c o o r d i n a t e system shown i n Figure 2. <j> i s the s p h e r i c a l , angle independent c o n t r i b u t i o n g i v e n by the m-6-8 p o t e n t i a l s

6

m * (r) s

=e

(?) - - v i v

8 - -

(f) ''-"(f) <j<»>

8>

where cf>(a) = 0, (j) ( r i ) = - e and d = r i / a . The parameters, m and y, represent the s t r e n g t h of the r e p u l s i v e term and an i n v e r s e e i g h t a t t r a c t i v e term, r e s p e c t i v e l y . <J> i s a f u n c t i o n of a quadrupole moment, the p o l a r i z a b i l i t y and the r e l a t i v e angles of orientation. Comprisons between theory and experiment f o r the v i s c o s i t y , thermal c o n d u c t i v i t y , e q u i l i b r i u m second v i r i a l c o e f f i c i e n t , and the d i e l e c t r i c second v i r i a l c o e f f i c i e n t a r e shown i n Figures 3 - 6 . One observes that a wide range of independent thermophysical p r o p e r t i e s have been f i t t e d q u i t e w e l l . I t should be s t r e s s e d that the p o t e n t i a l parameters a r e the same f o r a l l p r o p e r t i e s , and that the v a l u e s of quadrupole moment and the p o l a r i z a b i l i t y were taken from independent experiments and were not t r e a t e d as a d j u s t a b l e parameters. I t should a l s o be s t r e s s e d that the f i t s depend on a proper c h o i c e of the s p h e r i c a l p a r t of the p o t e n t i a l , <|>. Models f o r <|> s i m p l e r than Equation (5) are inadequate. R e s u l t s s i m i l a r to those of Figures 3-6 a r e a v a i l a b l e f o r other simple polyatomic molecules ( F 2 , N 2 , CH4, e t c . ) the r a r e m

n

m

n

ns

s

s

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

334

P H A S E EQUILIBRIA

1

CL X

O

A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

-1

1

1

1

r

Viscosity of CO2

o o oo

-2

o J

400

L

800

J

I

L

1200

1600

TEMPERATURE, K Figure 3. Viscosity of C0 : Comparison between values determined from Equation 3, using the potential (4), and experiment (details in Ref. 4) 2

CL

X

.

<

i i i i Thermal Conductivity of C0

i 2

i r (Corrected)

8

o

?

a.

0

x Q>

o 2

-

8

200

400

600

800

1000

1200

TEMPERATURE, K Figure 4. Thermal conductivity of C0 : comparison between kinetic theory and experiment (details in Ref. 4) 2

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

17.

HANLEY

200

335

Transport Properties

400

60

TEMPERATURE, K Figure 5. Second virial coefficient of C0 predicted from statistical mechanics using the potential of Equation 4 (Ref. 4) 2

300

200

400

500

TEMPERATURE, K Figure 6. Dielectric virial, B , of C0 . This is defined by the expansion 2

D

€

, o - == A + B D

DP

+ . . .

where e is the dielectric constant, is the density (comparison between theory and experiment, Ref. 4) P

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

336

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

gases, and t h e i r m i x t u r e s . On the whole, t h e r e f o r e , the a p p l i c a t i o n of k i n e t i c theory to the simple d i l u t e gas or gas mixture i s s a t i s f a c t o r y . Of course, the g e n e r a l l y good agreement between experiment and c a l c u l a t i o n can break down f o r more complex molecules. C l e a r l y , the p o t e n t i a l of Equation (4) can be too simple, and there are some approximations i n e v a l u a t i n g the k i n e t i c theory expressions which are not v a l i d i f one has to consider a very n o n s p h e r i c a l molecule, such as benzene 05). A d i s c u s s i o n i s given i n references (4) and ( 5 ) . Comparison w i t h Experiment f o r the Dense Gas and L i q u i d : Computer S i m u l a t i o n The computer has l e d to s i g n i f i c a n t c o n t r i b u t i o n s i n our understanding of l i q u i d s . I t has played l e s s of a d i r e c t r o l e i n c a l c u l a t i n g property data. N e v e r t h e l e s s c o r r e c t to imply t h a t r i g o r o u t r a n s p o r t c o e f f i c i e n t s are r e s t r i c t e d to the d i l u t e gas because Equation (2) can be evaluated n u m e r i c a l l y i f the f l u i d i s simulated on the computer. For example, i t i s r e l a t i v e l y s t r a i g h t f o r w a r d to c a l c u l a t e the s e l f - d i f f u s i o n c o e f f i c i e n t v i a Equation ( 3 ) . In f a c t , Figure 7 shows the r e s u l t s obtained by us (6) f o r methane, assuming that methane i n t e r a c t s w i t h the m-6-8 p o t e n t i a l of Equation ( 5 ) . Agreement between the computer c a l c u l a t i o n and experiment i s good. Whether, however, one can conclude that computer s i m u l a t i o n i s a p r a c t i c a l t o o l f o r c a l c u l a t i n g t r a n s p o r t c o e f f i c i e n t s i n general i s open to q u e s t i o n . I t i s more d i f f i c u l t to o b t a i n the v i s c o s i t y and thermal c o n d u c t i v i t y to w i t h i n a reasonable e r r o r than i t i s to d e t e r min and s e l f - d i f f u s i o n , and the l a t t e r i s not a t r i v i a l c a l c u l a t i o n . * Corresponding

States

The c o n t r i b u t i o n of s t a t i s t i c a l mechanics i s not, of course, l i m i t e d to the k i n d of r e s u l t s shown i n F i g u r e s 3-7. Several approximate t h e o r i e s have been proposed which are based on s t a t i s t i c a l mechanics but which r e q u i r e some assumption, or assumptions, to avoid a d e t a i l e d d e s c r i p t i o n of a f l u i d i n n o n e q u i l i b r i u m . A good example i s the Enskog theory and i t s m o d i f i c a t i o n s (1). As remarked, however, such t h e o r i e s w i l l not be discussed here w i t h one e x c e p t i o n , namely the theory of corresponding s t a t e s . One s p e c i f i c a p p l i c a t i o n i s o u t l i n e d as f o l l o w s .

* However, a d i f f e r e n t approach has been introduced r e c e n t l y by Hoover (_7), and by other authors, which could e v e n t u a l l y make computer s i m u l a t i o n a v i a b l e t o o l . Rather than evaluate the c o r r e l a t i o n funct i o n , i t i s suggested that a l a b o r a t o r y experiment, which would lead to the t r a n s p o r t c o e f f i c i e n t , be simulated. For example, a flow system can be set up on the computer to represent a shear f l u x produced by a v e l o c i t y g r a d i e n t . Numerical e v a l u a t i o n of the f l u x f o r a given gradient g i v e s the v i s c o s i t y c o e f f i c i 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.

17.

337

Transport Properties

HANLEY

DIFFUSION OF METHANE Density (/>] Temperature lO^DIcalc) lOjoDlexp) kg/ms K kg/m kg/ms 2.2 1.6 121.5 440.9 3

440.9

151.4

2.5

2.7

347.4

172.8

5.7

6.0

258.7

202.8

10.1

10.2

192.4

213.0

9.4

10.6

133.6

233.0

12.7

12.0 Molecular Physics

Figure 7. Table comparing experimental diffusion coefficients for methane with values calculated from Equation 3 using computer simulation (6)

We have developed (8) a corresponding s t a t e s procedure the object of which i s t o p r e d i c t the v i s c o s i t y and thermal conduct i v i t y c o e f f i c i e n t of a pure f l u i d o r a mixture from thermodynamic (PVT) data. The procedure i n c o r p o r a t e s the extended corresponding s t a t e s approach t o the thermodynamic p r o p e r t i e s of nonconformal f l u i d s introduced by Leland, and by Rowlinson and t h e i r co-workers (9). The thermodynamic p r o p e r t i e s of a f l u i d , a, can be r e l a t e d t o the p r o p e r t i e s of a reference f l u i d , o, v i a ( f o r example) the functions f and h where aa,o aa,o f

C

aa, o

C

= ( T /T ) 6 ; aa o aa, o

h

C

C

(p /p ) cj) (6) o aa aa, o

aa, o

The shape f a c t o r s , 0 and cf), are weak f u n c t i o n s of temperature (T) and d e n s i t y (p) and can be regarded as c h a r a c t e r i s t i c of the P i t z e r a c e n t r i c f a c t o r s of f l u i d s a and o, r e s p e c t i v e l y . The s u p e r s c r i p t c denotes the c r i t i c a l p o i n t value. Applying extended corresponding s t a t e s ideas to the t r a n s p o r t p r o p e r t i e s , one has f o r the v i s c o s i t y and thermal c o n d u c t i v i t y c o e f f i c i e n t s of f l u i d , a, a t a given d e n s i t y and temperature w i t h respect to the e q u i v a l e n t c o e f f i c i e n t of f l u i d , o (10) n

n (P,T) = n (ph , T/f ) FH ' o aa, o ' aa, o a, o and

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

(7)

338

P H A S E EQUILIBRIA

X (p,T) = A

a

A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

A

o

(ph , T/f ) FH aa, o aa, o a, o

(8)

where FH

n

FH

A

= (M /M ) a o

a, o

= (M /M ) o a

a, o

1

/

2

1

/

2

2 / 3

1

/

2

h" f aa, o aa, o 2 / 3

1

h~ aa, o

/

(9)

2

f aa, o

(10)

For a mixture x, one has n (p,T) = n (Ph , T/f ' ) F H x o x, o x, o x, o n

(11)

w i t h a corresponding equatio A (p g w i t h the Van der Waals o n e - f l u i d approximation (11) are x

h

x, o

= I I x x h a 3 a$, o

x, o

h = £ £ x x f „ h „ x, o a 3 a3, o a3, o a p

n

n

(12) f

0

with f

a3, o

= E (f f a3 aa, o 33, o

and

(13) = V

a3, o

/1 a3 ^2

1/3 aa, o

1 1/3 \ 2 33, o ) n

where £ £ and i p ^ a r e the b i n a r y i n t e r a c t i o n parameters. We use the e x p r e s s i o n M = E x M f o r the mixing r u l e f o r the molecular weight a

a

x

a

a

[but see Mo and Subbins (11)]. I t i s important to note that Equations (7) and ( 8 ) , and the e q u i v a l e n t equations f o r a mixture can f a i l to p r e d i c t c o r r e c t l y the t r a n s p o r t p r o p e r t i e s of a nonconformal f l u i d i f the corresponding s t a t e s parameters a r e estimated from thermodynamic data. The equat i o n s can break down, i n p a r t i c u l a r , i f p/p > 1. c

M o d i f i c a t i o n of the Transport Expressions I t i s proposed t h a t the p r e d i c t i v e c a p a b i l i t y of the extended corresponding s t a t e s approach to t r a n s p o r t phenomena would be improved s i g n i f i c a n t l y i f one c o n s i d e r s equations of the form ( f o r the v i s c o s i t y c o e f f i c i e n t , f o r example) n (p,T) = n (ph , T/f ) FH X (p,T). ql o aa, o ' aa, o a, o a, o 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.

(14)

17.

339

Transport Properties

HANLEY

A c o r r e c t i o n f a c t o r , x \ i s introduced which should be u n i t y i f f l u i d s a and o f o l l o w simple two-parameter corresponding s t a t e s , but should be a s t r o n g f u n c t i o n of d e n s i t y and a weak f u n c t i o n of temperature otherwise. However, X^ should n e v e r t h e l e s s approach u n i t y as the d e n s i t y approaches zero. In l i k e manner, we i n t r o d u c e X^ ( P » ) f ° mixture, and X^ Q(p,T) and X (p,T) f o r the thermal c o n d u c t i v i t y c o e f f i c i e n t of the pure f l u i d and mixture, r e s p e c t i v e l y . We have suggested expressions f o r X^ and X^. Based on the behavior of the t r a n s p o r t c o e f f i c i e n t s according to the M o d i f i e d Enskog Theory ( 1 ) , one can d e r i v e f o r the pure f l u i d , (and s i m i l a r l y f o r the mixture) T

rt

n

e

Q

x

X

n

a, o

(p,T)

Q

= Q G ^a, o

Q i s f u n c t i o n of p and T d e f i n e d as a, o [1

Q x

a,

1/exp

C

= [(bp) /(bp) ] " L

o

N

K

/

3

]

(16)

a o

where b i s a term given by b = B + TdB/dT, w i t h B the e q u i l i b r i u m second v i r i a l c o e f f i c i e n t . G^ and G are r a t i o s g i v e n , respect i v e l y by Gg^ = []Jj/[]8 and* 'The bracket expressions are 0

a

0

0

n

[ ] = [1/bpx + 0.8 + 0.761 bpx] (17) []

X

=

[1/bpx + 1.2 +

0.755

bp] X

with PR

\3T j

P

f o r f l u i d s a o r o. R i s the gas constant. I t should be noted that the c a l c u l a t i o n of the c o r r e c t i o n f a c t o r s X and X^ r e q u i r e only thermodynamic (PVT) i n f o r m a t i o n f o r f l u i d a or x w i t h respect to the r e f e r e n c e f l u i d ; f o r example, b Equation (16) can be obtained from the second v i r i a l c o e f f i c i e n t of the r e f e r e n c e f l u i d , B , a c c o r d i n g t o the e x p r e s s i o n n

a

Q

(h b (T) = h B + T d a aa, o o

a a

^ ° dT T

B ) °

(19)

and s i m i l a r l y f o r the term bpx Equation (18).

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

P H A S E EQUILIBRIA AND

340

F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

Comparisons w i t h Experimental Data M i x t u r e s Apart from the d i l u t e and moderately dense gas, v i s c o s i t y and thermal c o n d u c t i v i t y data f o r nonconformal mixtures are scarce and t h e i r r e l i a b i l i t y are o f t e n d i f f i c u l t to assesss. The m o d i f i e d equations, however, were checked as f a r as p o s s i b l e over a wide range of experimental c o n d i t i o n s . Methane was chosen as the r e f e r ence f l u i d [equation of s t a t e data from Goodwin (12), t r a n s p o r t data from the c o r r e l a t i o n of Hanley, McCarty and Haynes (13)]. Table 1 g i v e s t y p i c a l and r e p r e s e n t a t i v e r e s u l t s f o r the methanepropane system. The data (estimated accuracy ^ 3%) are those of Huang, S w i f t and Kurata (14). [Values from Equation (11) are given i n parentheses and the agreement between c a l c u l a t i o n and experiment i s seen to be poor.] Table 2 shows a compariso values f o r an n a t u r a l ga the f o l l o w i n g composition: CH4, 91.5%; C H , 1.8%; C Hg, 0.8%; 4 10> °-6%; 2 ' 5%; others 0.3%. We cannot e v a l u a t e the accuracy of the data but the p r e d i c t i o n appears reasonable. R e s u l t s f o r a nitrogen-carbon d i o x i d e mixture are shown i n Table 3: data from r e f e r e n c e (15). Note that n e i t h e r n i t r o g e n , or carbon d i o x i d e i s used as the r e f e r e n c e f l u i d . Table 4 compares our p r e d i c t i o n w i t h thermal c o n d u c t i v i t y data of a nitrogen-ethane mixture from Gilmore and Comings (16). Values from the unmodified equation are given i n parentheses. Finally, Table 5 shows the r e s u l t s f o r a methane-n-butane mixture (17). 2

C

H

6

3

N

TABLE 1.

VISCOSITY OF A METHANE (50%) c - PROPANE (50%) MIXTURE AT 153.15 K. ( T ffc 319 K, f\^7.7 mol/fc) c

P

p

D(calc)

n(exp) -1

mol/£

yg cm

-1 s

18.69

2780

2614

(2286)*

18.78

2860

2710

(2360)

18.96

3030

2878

(2509)

19.12

3200

3002

(2659)

19.28

3380

3232

(2807)

19.42

3550

3407

(2952)

^ R e s u l t s from the extended corresponding s t a t e s equation (11).

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

17.

HANLEY

Transport Properties

341

TABLE 2. NATURAL GAS VISCOSITY AT 273 K n(exp)

P

n(calc) -1 -1

atm

yg cm

s

20.

109.3

110.0

60.

132.2

123.6

100.

160.2

145.8

200.

244.

300.

313.7

298.8

400

372.3

345.0

TABLE 3. NITROGEN-CARBON DIOXIDE VISCOSITY

Comparison between p r e d i c t e d v i s c o s i t y c o e f f i c i e n t s f o r a n i t r o g e n (62%) - carbon d i o x i d e (38%) mixture a t 289 K.

P

p

n(exp)

n(calc) -1

atm

mol/£

yg cm

-2 s

20.

0.9

167.0

167.0

60.

2.8

179.5

186.5

100.

4.9

200.5

213.6

120.

7.0

212.5

229.1

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

342

PHASE

TABLE 4.

EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

THERMAL CONDUCTIVITY OF A NITROGEN (59.8%) - ETHANE (40.2%) MIXTURE AT 348.16 K ( T £ 260 K, p £ 12.0 mol/A) c

p

INDUSTRY

c

X(exp)

mol/p

A(calc) mW m ^ K ^

7.34

46.0

46.11

( 61.5)*

10.16

55.2

54.69

( 72.65)

13.75

70.3

69.47

( 92.4)

16.43

85.4

84.4

^ R e s u l t s from the extended corresponding s t a t e s equation c o r r e s ponding to equation (11).

TABLE 5. THERMAL CONDUCTIVITY OF A METHANE (39.4%) - n-BUTANE MIXTURE (60.6%) a t 277.6 K ( T ^ 390 K, p ^ 6.25 mol/£) c

p

c

A(exp)

mol/£

n(calc) mW m ^ K ^

12.26

99.00

95.33

12.49

101.46

99.63

13.01

113.39

109.99

13.25

114.90

115.05

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

17.

HANLEY

Transport Properties

343

The r e p r e s e n t a t i v e r e s u l t s reported here i n d i c a t e our procedure i s s a t i s f a c t o r y . I n other words i t appears, given the l i m i t e d data a v a i l a b l e a t t h i s time, that t r a n s p o r t p r o p e r t i e s can be p r e d i c t e d s a t i s f a c t o r i l y v i a thermodynamic data. Hence only one s e t of parameters (9,cj),i|; £,etc.) a r e r e q u i r e d t o f i t both thermodynamic and transport properties. a

Abstract A very brief outline of theoretical calculations of the transport coefficients is given. A method to predict the viscosity and thermal conductivity coefficients of pure fluids and mixtures is presented. Literature Cited 1. Hanley, H. J. M., McCarty, R. D. and Cohen, E. G. D., Physics (1972) 60, 322. 2. McQuarrie, D. G., "Statistical Mechanics" (Harper and Row, N.Y., 1976). 3. Chapman, S. and Cowling, T. G., "The Mathematical Theory of Non-Uniform Gases," (Cambridge Univ. Press, London, 1970). 4. Ely, J. F. and Hanley, H. J. M., Molecular Physics (1975) 30, 565. 5. Evans, D. J., Ph.D. Thesis, Australian National University, Canberra, Australia (1975). 6. Hanley, H. J. M. and Watts, R. O., Molecular Physics (1975) 29, 1907. 7. Ashurst, W. T. and Hoover, W. G., AIChE Journal (1975) 21, 410. 8. Hanley, H. J. M., Cryogenics (1976) 16, 643. 9. Rowlinson, J. S. and Watson, I. D., Chem. Eng. Sci. (1969) 24, 1565. 10. Haile, J. M., Mo, K. C. and Gubbins, K. E., "Advances i n Cryogenic Eng." Vol. 21, Ed. K. D. Timmerhaus and D. H. Weitzel (Plenum Press, N.Y., 1975) 501. 11. Mo, K. C. and Gubbins, K. E., Chem. Eng. Commun. (1974) 1, 281. 12. Goodwin, R. D., Nat. Bur. Stand. (U.S.), Tech. Note No. 653 (1974). 13. Hanley, H. J. M., McCarty, R. D. and Haynes, W. M., Cryogenics (1975) 15, 413; J. Phys. Chem. Ref. Data (In press). 14. Huang, E. T. S., Swift, G. W. and Kurata, F., AIChE J. (1967) 13, 846. 15. Golubev, I. F., "Viscosity of Gases and Gas Mixtures," Israel Program for Scientific Translations (Jerusalem, 1970). 16. Gilmore, T. F. and Comings, W. E., AIChE Journal (1966) 12, 1172. 17. Carmichael, L. T., Jacobs, J. and Sage, B. H., J. Chem. Eng. Data (1968) 13, 489.

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

18 Polar and Quadrupolar Fluid Mixtures K. E . GUBBINS and C. H . TWU Cornell University, Ithaca, NY 14853

Phase e q u i l i b r i u m c a l c u l a t i o n empirical or t h e o r e t i c a one uses an e m p i r i c a l equation o f s t a t e f o r one o r more of the phases i n v o l v e d ; f o r the l i q u i d phase one o f the e m p i r i c a l equat i o n s f o r the excess Gibbs energy (Wilson, van Laar, etc.) i s u s u a l l y used. The s e m i e m p i r i c a l approach gives good r e s u l t s provided that one has a s i g n i f i c a n t amount o f experimental data a v a i l a b l e f o r the mixture. However, such methods are b e t t e r s u i t e d t o i n t e r p o l a t i o n o f e x i s t i n g data than to e x t r a p o l a t i o n o r p r e d i c t i o n . T h e o r e t i c a l methods are based i n s t a t i s t i c a l thermodynamics, r e q u i r e l e s s mixture data, and should be more r e l i a b l e f o r p r e d i c t i o n . T h e o r e t i c a l l y - b a s e d methods that have found extensive use by chemical engineers i n c l u d e r e g u l a r s o l u t i o n theory ( 1 ) , corresponding s t a t e s methods (conformal s o l u t i o n theory) (2-4), and p e r t u r b a t i o n expansions based on a hard sphere f l u i d as reference system (3,_4) • mixtures of simple nonpolar molecules these methods g i v e good r e s u l t s , e s p e c i a l l y the c o r r e sponding s t a t e s and p e r t u r b a t i o n expansion t h e o r i e s ( 3 ) . However, a l l three t h e o r i e s are based on the assumption that the molecules are s p h e r i c a l , w i t h i n t e r m o l e c u l a r f o r c e s that are a f u n c t i o n only of the i n t e r m o l e c u l a r s e p a r a t i o n . This assumption i s s t r i c t l y v a l i d only f o r mixtures o f the i n e r t gases (Ar, Kr, Xe) and f o r c e r t a i n fused s a l t s and l i q u i d metals. I n s p i t e o f t h i s r e s t r i c t i o n the corresponding s t a t e s and p e r t u r b a t i o n methods have been a p p l i e d w i t h success to mixtures i n which the i n t e r m o l e c u l a r f o r c e s depend on the molecular o r i e n t a t i o n s , e.g., mixtures cont a i n i n g 02, N , l i g h t hydrocarbons, e t c . (see r e f . 4 f o r review o f a p p l i c a t i o n s up t o 1973). The extension to weakly n o n s p h e r i c a l molecules can be accomplished, f o r example, by i n t r o d u c i n g shape f a c t o r s as suggested by Leland and h i s colleagues ( 5 ) . These methods have been e x t e n s i v e l y e x p l o i t e d f o r both thermodynamic (2) and t r a n s p o r t (6) p r o p e r t i e s . They are p r e d i c t i v e only i f equat i o n s are a v a i l a b l e f o r the composition dependence o f the shape f a c t o r s . The e x i s t i n g methods f o r doing t h i s work w e l l f o r r e l a t i v e l y simple mixtures, but break down when c o n s t i t u e n t s w i t h F

o

r

2

344

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

18.

345

Fluid Mixtures

GUBBINS A N D T W U

s t r o n g l y orientation-dependent forces are present (e.g., s t r o n g l y p o l a r o r quadrupolar c o n s t i t u e n t s ) . In t h i s paper we review a r e c e n t l y developed t h e o r e t i c a l method f o r phase e q u i l i b r i u m c a l c u l a t i o n which e x p l i c i t l y accounts f o r s t r o n g l y orientation-dependent f o r c e s . These a n i s o t r o p i c forces are taken i n t o account through a p e r t u r b a t i o n scheme i n which the reference f l u i d i s composed o f simple s p h e r i c a l molec u l e s ; i n p r a c t i c e , the known p r o p e r t i e s o f argon, o r those o f a Lennard-Jones f l u i d simulated on the computer, may be used. Such a p e r t u r b a t i o n scheme was f i r s t suggested by Pople (_7) more than twenty years ago, but was not immediately used f o r l i q u i d phase c a l c u l a t i o n s because the reference f l u i d p r o p e r t i e s were not s u f f i c i e n t l y w e l l known. Since 1972 the theory has been extended and improved, and i t s s u c c e s s f u l a p p l i c a t i o n to l i q u i d s o f s t r o n g l y p o l a r o r quadrupolar molecule The most s u c c e s s f u the Theory S e c t i o n . I n the S e c t i o n on R e s u l t s the theory i s used to c l a s s i f y mixture phase diagrams i n terms o f the i n t e r m o l e c u l a r forces i n v o l v e d , and a l s o to p r e d i c t v a p o r - l i q u i d e q u i l i b r i a f o r s e v e r a l b i n a r y and ternary mixtures. Intermolecular

Forces

The i n t e r m o l e c u l a r p a i r p o t e n t i a l u between two a x i a l l y symm e t r i c molecules o f components a and $ can be w r i t t e n as a sum o f parts u*

6

( r

W

)

-

(r)

u ^

+

+

t

(re^) <

r

W >

( r Q ^ )

+

+

U

ind

(

r

e

iV>

(1

>

where u ^ ^ , u d i , u , and U i j are m u l t i p o l a r ( e l e c t r o s t a t i c ) , a n i s o t r o p i c d i s p e r s i o n , a n i s o t r o p i c charge o v e r l a p , and i n d u c t i o n terms, r e s p e c t i v e l y . Here r i s the i n t e r m o l e c u l a r s e p a r a t i o n , (9ll) are the p o l a r angles g i v i n g the o r i e n t a t i o n of molecule i , the r_ d i r e c t i o n being taken as the p o l a r a x i s , and c|> = <j>i - 2« We take the i s o t r o p i c c e n t r a l p o t e n t i a l u ( r ) to be an (n,6) model (8), s

o v

n c

Q

where eapN a g and n ^ are p o t e n t i a l parameters. The m u l t i p o l a r , d i s p e r s i o n , overlap and i n d u c t i o n p o t e n t i a l s are u s u a l l y a p p r o x i mated by the f i r s t few terms i n a s p h e r i c a l harmonic expansion.* a

a

+In p r a c t i c e not a l l o f these c o n t r i b u t i o n s are e q u a l l y important, and some may be neglected f o r p a r t i c u l a r f l u i d s . I n c a l c u l a t i n g thermodynamic p r o p e r t i e s f o r the f l u i d s considered i n t h i s review, the m u l t i p o l a r forces make the l a r g e s t a n i s o t r o p i c c o n t r i b u t i o n .

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

P H A S E EQUILIBRIA A N D

346

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

Equations f o r these terms are given i n s e v e r a l reviews (7-10). The l e a d i n g m u l t i p o l e term i n u i i s the d i p o l e - d i p o l e p o t e n t i a l i n the case of p o l a r f l u i d s , and the quadrupole-quadrupole potent i a l f o r f l u i d s of l i n e a r symmetrical molecules (C0 , N , e t c . ) . These are given by m u

t

2

u

yy

=

—

( S

r

3 Q U

QQ

C

Q

a B

7T~ 4r^

=

S

1 2

n ( 1

"

5 c

"

2 C

C

1 2

,2 l "

5 C

2

}

( 3 )

_ 2

2 +

2 1 7 C

C

1

2^.2 1 2

+

2

2 S

S

2 2 C

where s^ = s i n 0-^, c-^ = cos 6^, c = cos <|>, and y and Q are the d i p o l e and quadrupole moments. Experimental values of y and Q have been t a b u l a t e d by Stogryn and Stogryn (11). The a n i s o t r o p i c o v e r l a p p o t e n t i a l may be approximated f o r symmetrical l i n e a r molecules ( C 0 , N , B r , e t c . ) by (7) 2

aB U

ov

. ^ / a 3 \ l 2 .a3 , *z(—} ° l "a3( r ) 2

2

2 ,

a

Q

=

4

6

e

C

2

+

3

C

2

2

a

"

0

.

...

2 )

( 5 )

w h i l e f o r unsymmetrical ymmetrical l i n e a r molecules (HC1, NO, N ) , e t c . ) one has 2

aB , [ a 3 ] 1 2 . aS, = 4 e V j 6_ (c_ ov a3 \ r / 1 1 a

u

0

(6)

c ) 2

where n i n Eq. (2) i s taken to be 12, and where 6± and 6 are dimensionless o v e r l a p parameters t h a t must l i e w i t h i n the ranges -0.5 £ 61 <_ 0.5 and -0.25 <_ 62 £ 0.5. The a n i s o t r o p i c d i s p e r s i o n p o t e n t i a l i s given to a good approximation by the London express i o n (7,9,10). 2

Theory In thermodynamic p e r t u r b a t i o n theory the p r o p e r t i e s of the r e a l system, i n which the p a i r p o t e n t i a l i s u $ , are expanded about the values f o r a r e f e r e n c e system, i n which the p a i r potent i a l i s ug£. Here we take the r e f e r e n c e p o t e n t i a l to be the (n,6) p o t e n t i a l , so t h a t the a n i s o t r o p i c p a r t s of the p o t e n t i a l i n Eq. (1) are the p e r t u r b a t i o n . Expanding the Helmholtz f r e e energy A i n powers of the p e r t u r b i n g p o t e n t i a l about the v a l u e A f o r the r e f e r e n c e system gives a

Q

A = AQ

+ A

where A , A 3 , 2

2

+ A3

+ . . .

. . . are the second-, t h i r d - o r d e r , e t c .

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

(7)

18.

347

Fluid Mixtures

Twu

GUBBINS AND

p e r t u r b a t i o n terms (the f i r s t - o r d e r term v a n i s h e s ) . General expressions f o r A and A3 f o r an a r b i t r a r y i n t e r m o l e c u l a r potent i a l have been worked out (12,13). Equations f o r the other t h e r modynamic p r o p e r t i e s (pressure, i n t e r n a l energy, e t c . ) may be obtained by a p p l y i n g the u s u a l thermodynamic i d e n t i t i e s to ( 7 ) . When (7) i s terminated at the t h i r d - o r d e r term i t i s found to g i v e good r e s u l t s f o r moderately p o l a r f l u i d s (e.g., HC1) but to f a i l f o r strong d i p o l e s (e.g., H 0, NH3, acetone). This i s shown for a l i q u i d s t a t e c o n d i t i o n i n F i g u r e 1 . + S i m i l a r r e s u l t s are found f o r quadrupole f o r c e s . The slow convergence of (7) f o r s t r o n g m u l t i p o l e strengths l e d S t e l l et_ a l . (14) to suggest the f o l l o w i n g simple Pade approximation f o r the f r e e energy. 2

2

A

o

+

A

2 1 2J

Eq. (8) i s found to g i v e e x c e l l e n t r e s u l t s , even f o r the s t r o n g e s t d i p o l e or quadrupole moments observed i n nature (Figure 1 ) . F o r the a n i s o t r o p i c overlap p o t e n t i a l of Eq. ( 5 ) , the Pade approximat i o n gives good r e s u l t s f o r -0.2 £ 6 _ + 0 . 3 . This range of 6 i n c l u d e s molecules such as HC1, C0 , ethane, e t h y l e n e , methanol, etc., f o r which 6 i s 0.2 or l e s s . The expressions f o r A and A3 i n v o l v e the s t a t e v a r i a b l e s , i n t e r m o l e c u l a r p o t e n t i a l parameters, and c e r t a i n i n t e g r a l s J and K f o r the r e f e r e n c e f l u i d . Thus, i f the only a n i s o t r o p i c p a r t of the p o t e n t i a l i n (1) i s the d i p o l e - d i p o l e term of Eq. ( 3 ) , then 2

2

2

2

2

2TTNP T

A A

2

=

32 A

3

t.

1 A

2 2

" 3kT a s V s l ^ F ) } V e 7 1 3

TT*

135

,14TK1/2 (~T~)

aB

xx x

r

/ Q

0

a 6

.

( 9 )

yy

2

p N —o

(kT) I

5

T

J

(, l(

1

\ 2 2 2 ^aSy )ju y y K (10) G

^°a VW

a

6

a3y p = N/V i s where N i s Avogadro's number, Boltzmann constant, T i s temperature, and t i o n of component a . The summations are the mixture. J and are i n t e g r a l s

3

Y VW

number d e n s i t y , k i s X g = N /N i s mole f r a c over the components o f over the two- and t h r e e A

U U

+For HC1 y*0.8, w h i l e f o r acetone and H 0 y* i s about 1 . 2 and 2-3, r e s p e c t i v e l y ; f o r N and CO2 the reduced quadrupole moment Q* i s about 0.5 and 0.9, r e s p e c t i v e l y . Here y* = y / ( e a ^ ) l / 2 and Q* = A / ( e ) ^ . The i n f l u e n c e of p o l a r or quadrupolar f o r c e s on the thermodynamic p r o p e r t i e s i s determined by y* and Q*, r a t h e r than by y and Q themselves. Thus the e f f e c t s of these f o r c e s are g r e a t e s t f o r s m a l l molecules w i t h r e l a t i v e l y l a r g e y or Q v a l u e s . 2

2

5

a

American Chemical Society Library

In Phase Equilibria and Fluid Properties the Chemical Industry; Storvick, T., et al.; 1155 in16th St., N.W. ACS Symposium Series; American ChemicalD.C. Society: Washington, DC, 1977. Washington, 20036

348

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY

Kor Q*

-14 L

8 -.2 I

-.1 I

.1

0

.2 I

I

.3 I

.4 I

- 6.0 -

- 6.5 -

\x \ \ \ \ \ \ \ \

-7.0

-

P* = 0.85 T =1:29 #

-7.5 -

u Ne c

I

-8.0 -

(b)

Figure 1. Comparison of the perturbation expansion to third order ( ), Equation 7, with the Fade approximant to the series ( ), Equation 8, with molecular dynamics computer simulation data (15, 16). U is the configurational contribution to the internal energy. The potential is of the form u + u , where u is the Lennard-]ones (12,6) potential and u , is given by either Equation 3 or 4 in Figure (a), and is the anisotropic overlap model of Equation 5 in Figure (b). Here = fi/(e
0

a

0

a

/A*

3

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

18.

Fluid Mixtures

GUBBINS A N D T W U

349

molecule c o r r e l a t i o n f u n c t i o n s f o r the reference f l u i d . These i n t e g r a l s have been c a l c u l a t e d f o r the case n = 12 from molecular dynamics r e s u l t s f o r a pure Lennard-Jones (12,6) f l u i d (12,13, 17) ,and are given as a f u n c t i o n of reduced d e n s i t y p* = po~^ and temperature T* = kT/e by 4

J

= -0.488498 p *

In T* + 0.863195 p *

2

+ 0.761344 p *

2

W

xln T* -0.750086 p * -0.218562 In T* -0.538463 In Kyuy = -1.050534 p * x£

n

2

In T* + 1.747476 p *

2

(11)

+ 1.769366 p*

T*-l.999227 p *

The equations f o r A and A3 f o r other types o f a n i s o t r o p i c f o r c e s are l i s t e d elsewhere (12,13,17). In order to make mixture c a l c u l a t i o n s using the above equat i o n s , methods f o r c a l c u l a t i n g A , J°[jj, and K y j ^ (together w i t h J and K i n t e g r a l s f o r any other p a r t s of the a n i s o t r o p i c p o t e n t i a l ) a r e necessary. Accurate methods f o r these c a l c u l a t i o n s have been desc r i b e d by Twu et_ a l . (13,17). The thermodynamic p r o p e r t i e s o f the r e f e r e n c e mixture were r e l a t e d t o those of a pure, conformal r e f e r ence f l u i d by u s i n g the van der Waals 1 form o f the corresponding s t a t e s theory (2,3,13). The p r o p e r t i e s of t h i s pure reference f l u i d were obtained from experimental data f o r argon, by using the multiparameter equations o f Gosman e t a l . (18) i n reduced form. M i x t u r e values o f J $ and K $Y were a l s o r e l a t e d to pure f l u i d values using the van der Waals 1 theory (17). 2

Q

a

a

Results In t h i s s e c t i o n we g i v e as examples two uses o f the theory o u t l i n e d above: (a) the c l a s s i f i c a t i o n o f b i n a r y phase diagrams, c r i t i c a l behavior, e t c . i n terms of the i n t e r m o l e c u l a r f o r c e s i n v o l v e d , and (b) comparison w i t h experiment f o r some b i n a r y and t e r n a r y systems. C l a s s i f i c a t i o n o f Binary Phase Diagrams. F i g u r e 2 shows a c l a s s i f i c a t i o n o f b i n a r y phase diagrams suggested by S c o t t and Van Konynenburg (19), w i t h examples o f each c l a s s . This c l a s s i f i c a t i o n i s based on the presence o r absence of three-phase l i n e s and the way c r i t i c a l l i n e s connect w i t h these; t h i s i s best seen on a P,T p r o j e c t i o n . I n c l a s s e s I , I I and VI the two components have s i m i l a r c r i t i c a l temperatures, and the g a s - l i q u i d c r i t i c a l l i n e passes c o n t i n u o u s l y between the pure component c r i t i c a l p o i n t s ; c l a s s I I and V I mixtures d i f f e r from c l a s s I i n that they are more n o n i d e a l and show l i q u i d - l i q u i d i m m i s c i b i l i t y . Class I I behavior i s common, whereas c l a s s V I , i n which c l o s e d s o l u b i l i t y

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.

Figure 2. Classification of binary fluid phase diagrams (19) with examples of each class. Vertical and horizontal axes are pressure and temperature, respectively. Solid lines labelled 1 and 2 are vapor pressure curves for the pure components; solid lines labelled LLG are three-phase (liquid-liquidr-gas) lines; dashed lines are critical loci.

18.

GUBBINS

Fluid Mixtures

AND T W U

351

loops occur, a r i s e s i n only a r e l a t i v e l y few cases ( u s u a l l y water mixed w i t h a l c o h o l s o r amines). Mixtures of c l a s s e s I I I , IV and V are u s u a l l y composed of components w i t h w i d e l y d i f f e r e n t c r i t i c a l temperatures (Tj/T^ _> 2 ) , and the g a s - l i q u i d c r i t i c a l curve does not pass c o n t i n u o u s l y from one pure component to the other (e.g., because the l i q u i d - l i q u i d i m m i s c i b i l i t y r e g i o n extends to that of the g a s - l i q u i d c r i t i c a l c u r v e ) . Included i n c l a s s I I I are systems that e x h i b i t "gas-gas" i m m i s c i b i l i t y . These s i x c l a s s e s may be f u r t h e r s u b d i v i d e d , according to whether o r not azeotropes a r e formed, e t c . D e t a i l e d d i s c u s s i o n of the e x p e r i mental behavior of b i n a r y systems i s given i n s e v e r a l reviews (20-22). In t h i s s e c t i o n we r e p o r t c a l c u l a t i o n s f o r f l u i d s i n which one o f the c o n s t i t u e n t potential, 1 2

u (r) = 4 [ ( ^ ) Q

£

6

- (^) ]

(13)

w h i l e the other c o n s t i t u e n t i n t e r a c t s w i t h a p o t e n t i a l u + u ^ i t - , where u i s again the (12,6) model and u ^ i s e i t h e r the d i p o l e d i p o l e o r quadrupole-quadrupole p o t e n t i a l (Eq. (3) o r ( 4 ) ) . I n p a r t i c u l a r , we i n v e s t i g a t e the r e l a t i o n s h i p between the i n t e r m o l e c u l a r f o r c e s and t h e r e s u l t i n g c l a s s of b i n a r y phase diagram. V a p o r - l i q u i d , l i q u i d - l i q u i d , and "gas-gas" e q u i l i b r i u m s u r f a c e s , three phase l i n e s , c r i t i c a l l i n e s , and a z e o t r o p i c l i n e s a r e c a l c u l a t e d . For a more d e t a i l e d d i s c u s s i o n o f the methods and r e s u l t s , the papers o f Twu e t aJL. (M,1Z>23) should be c o n s u l t e d . The u n l i k e p a i r parameters a g and e ^ are r e l a t e d to the l i k e p a i r parameters by the u s u a l combining r u l e s , Q

Q

m u

a

a

E

?

aB = « 3

( e

t

( 1 5 )

aa

where n £ and £ g a r e c l o s e to u n i t y . Figures 3 to 5 i l l u s t r a t e the e f f e c t s o f adding a d i p o l e d i p o l e o r quadrupole-quadrupole i n t e r a c t i o n (Eq. (3) o r (4)) to one o f the components i n a b i n a r y Lennard-Jones mixture. In this case the Lennard-Jones parameters a r e chosen so that the r e f e r e n c e system approximates an argon-krypton mixture (24): a

W

a

-

k

3

°ArAr " '

1

1

4 0 5

'

9

1

8

K

W

-

k

3

°KrKr = "

1

6

6 3 3

7

- °

?

ArKr = ° -

V

K

r =

9 8 9

^

The argon-krypton r e f e r e n c e system i s a c l a s s I system w i t h no azeotrope,and argon i s the more v o l a t i l e component. I f a d i p o l e moment i s now added t o the argon component, so t h a t the m i x t u r e

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

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

Figure S. Effect of dipole moment fi * on gas-liquid critical lines ( ) and on pure component vapor pressures ( ) for class I and II systems. Reference system is an argon—krypton mixture. A

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

GUBBINS A N D

TWU

Fluid Mixtures

Figure 4. Class II behavior for a polar-nonpolar mixture (argonkrypton reference system) shown as a P,T projection

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.

K

K

1

2

Figure 5. P,T projection for quadrupolar-nonpolar mixtures (QA* = 0) showing class III behavior. The L L G line shown is for the case Q * = 1-0. For higher Q * values this line lies closer to the vapor pressure curve for pure argon.

T, K

CO

18.

GUBBINS

AND T W U

Fluid Mixtures

355

becomes a polar/nonpolar one, the e f f e c t i s to make the A r component l e s s v o l a t i l e r e l a t i v e to the K r component. F i g u r e 3 shows the e f f e c t o f the d i p o l e moment on the g a s - l i q u i d c r i t i c a l curve and on the pure component vapor p r e s s u r e s . A t a v a l u e o f y^ o f 1.6 the v o l a t i l i t y o f the two components i s about the same and the system has a p o s i t i v e azeotrope. D e s p i t e the s i m i l a r i t y i n the v o l a t i l i t i e s and b o i l i n g p o i n t s o f the two components, t h e i r i n t e r m o l e c u l a r f o r c e s are very d i f f e r e n t , and the system i s o f c l a s s I I , showing l i q u i d - l i q u i d i m m i s c i b i l i t y a t lower temperat u r e s . F i g u r e 4 shows the three-phase l i n e , the a z e o t r o p i c l i n e , and the l i q u i d - l i q u i d c r i t i c a l l i n e f o r t h i s case. I f the v a l u e of i s f u r t h e r i n c r e a s e d the Kr component becomes the more v o l a t i l e one, and f o r s u f f i c i e n t l y l a r g e y^ the system passes to c l a s s I I I . S i m i l a r r e s u l t s are found when a quadrupole i s added t o Ar i n s t e a d of a d i p o l e (17,23) I f the d i p o l e i s adde component, l e a v i n g the A r i n t e r a c t i o n unchanged, the e f f e c t i s to i n c r e a s e the r e l a t i v e v o l a t i l i t y , and the system soon changes from c l a s s I to IV, and then q u i c k l y to I I I (Table 1 ) . This b e h a v i o r i s shown f o r the case of quadrupole i n t e r a c t i o n s i n F i g u r e 5. For Q£ > -1.7 the s l o p e of the c r i t i c a l curve becomes p o s i t i v e a t h i g h p r e s s u r e s , so that the c r i t i c a l temperature exceeds that of e i t h e r pure component a t s u f f i c i e n t l y h i g h pressures. Such systems e x h i b i t "gasgas i m m i s c i b i l i t y (20,21,25). Table 1 summarizes the c l a s s e s of phase b e h a v i o r found f o r these polar/nonpolar systems, using an argon-krypton r e f e r e n c e system, and compares i t w i t h the b e h a v i o r f o r s i m p l e nonpolar Lennard-Jones systems. An important d i f f e r e n c e between the two types o f systems i s that the Lennard-Jones mixtures do not form azeotropes, and appear t o e x h i b i t c l a s s I I b e h a v i o r only when the components have very d i f f e r e n t vapor pressures and c r i t i c a l temperatures ( T ^ / T > 2 ) . I n p r a c t i c e , the l i q u i d ranges o f the two components would not o v e r l a p i n such cases, so that l i q u i d l i q u i d i m m i s c i b i l i t y (and hence c l a s s I I b e h a v i o r ) would not be observed i n Lennard-Jones mixtures (the only e x c e p t i o n t o t h i s statement seems to be when the u n l i k e p a i r i n t e r a c t i o n i s improbab l y weak). Thus, the use of t h e o r i e s based on the Lennard-Jones or o t h e r i s o t r o p i c p o t e n t i a l models cannot be expected to g i v e good r e s u l t s f o r systems of c l a s s I I , and w i l l probably g i v e poor r e s u l t s f o r most systems of c l a s s e s I I I , IV and V a l s o . The polar/nonpolar mixtures s t u d i e d here e x h i b i t four of the s i x c l a s s e s of behavior shown i n F i g u r e 2. C l a s s V i s presumably present a l s o , but i s i n d i s t i n g u i s h a b l e from c l a s s IV because the l o c a t i o n of the s o l i d - f l u i d boundary i s not c a l c u l a t e d . For p o l a r / n o n p o l a r mixtures i n which the r e f e r e n c e system i s a weakly n o n i d e a l Lennard-Jones m i x t u r e , i n c r e a s i n g the d i p o l e moment o f the p o l a r molecule causes a continuous t r a n s i t i o n among the classes, 11

c

c

a

I -> I I -> (V) •> (IV) -> I I I

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

356

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

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

O

18.

GUBBINS

Fluid Mixtures

AND T W U

357

Classes IV and V are observed only f o r c e r t a i n v a l u e s o f the Lennard-Jones parameters. The same sequence i s found f o r quadr u p o l a r / n o n p o l a r m i x t u r e s . The p o t e n t i a l models considered here f a i l to account f o r the c l a s s V I systems, i . e . , those w i t h low temperature lower c r i t i c a l s o l u t i o n p o i n t s . Such behavior i s b e l i e v e d to a r i s e from s t r o n g u n l i k e p a i r f o r c e s , and presumably r e q u i r e s d i f f e r e n t p o t e n t i a l models than those used here. I t i s i n t e r e s t i n g to note that i t i s p o s s i b l e t o observe a t r i c r i t i c a l p o i n t * i n b i n a r y p o l a r / n o n p o l a r (or quadrupolar/nonp o l a r ) systems o f the type considered above. Thus i f the p o l a r component i s a and y i s i n c r e a s e d , the t r i c r i t i c a l p o i n t i s observed as an i n t e r m e d i a r y stage i n the t r a n s i t i o n from c l a s s I I to c l a s s I I I b e h a v i o r . This i s shown i n F i g u r e 6, the t r i c r i t i c a l p o i n t o c c u r r i n g where the v a p o r - l i q u i d c r i t i c a l curve, l i q u i d l i q u i d c r i t i c a l curve, an should be noted that th b i n a r y mixture does not v i o l a t e the phase r u l e , s i n c e y a c t s as an a d d i t i o n a l degree o f freedom). a

a

Comparison w i t h Experiment. We c o n s i d e r systems i n v o l v i n g the p o l a r c o n s t i t u e n t s HC1 and HBr, and the quadrupolar c o n s t i t u ents CO2, ethane, ethylene and a c e t y l e n e , together w i t h the nonp o l a r monatomic f l u i d xenon. F o r the l i k e p a i r i n t e r a c t i o n the i n t e r m o l e c u l a r p o t e n t i a l models used were: Xe:

u ( r ) = u (n,6)

HC1:

u(r6 0 <J>) = 1 z

u

(16)

(n,6) + u - (yy + yQ + QQ) o mult

+ u + u.. ov dis HBr:

u(r0 6 c))) = 1 Z 1

9

u

CO 2' 2 2 ' 2 4 ' 2 6 C

H

C

H

C

H

(17) (n,6) + u _ (yy + yQ + QQ) o mult :

u ( r 9

e 1

( j ) ) 2

=

u

n

0

(18)

6

^ » )

+ u _ (QQ) + u + u,. mult ov d i s

(19)

x

where u i s the (n,6) p o t e n t i a l , yy i s d i p o l e - d i p o l e , yQ i s d i p o l e quadrupole, and QQ i s quadrupole-quadrupole i n t e r a c t i o n , u ^ is the London model f o r a n i s o t r o p i c d i s p e r s i o n , and u i s g i v e n by Eq. (6) f o r HC1 and by (5) f o r the other cases. The u n l i k e p a i r p o t e n t i a l model f o r CT^/ethane, GX^/ethylene, CO?/acetylene, and ethane/ethylene was Eq. (19). For Xe/HCl and Xe/HBr the models Q

i g

Q v

+

A normal ( b i ) c r i t i c a l p o i n t occurs when two phases ( g a s - l i q u i d , l i q u i d - l i q u i d , o r "gas-gas")become i n d i s t i n g u i s h a b l e , i . e . , t h e i r i n t e n s i v e p r o p e r t i e s become i d e n t i c a l . A t r i c r i t i c a l p o i n t occurs when three phases become i d e n t i c a l ; f o r the b i n a r y systems considered here, one o f these phases i s gaseous and the other two are l i q u i d .

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

358

PHASE

EQUILIBRIA

A N D FLUID PROPERTIES

IN C H E M I C A L

INDUSTRY

5

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

18.

GUBBINS

Fluid Mixtures

AND T W U

359

Xe/HCl

u(rGfi<j)) = u (n,6) + u. , + u 12 o ind ov

Xe/HBr

u(r0 9 (j)) = u (n,6) + u. , 1 z o ma 1

+ u,. dis

(20) (21)

o

For the Xe/HCl case the o v e r l a p model corresponding to Eq. (6) i s

u

ov

= 4e

4^F ? 5

6 c

°

s e

i

< 2 2 >

and 0 i i s the p o l a r angle f o r the HC£ molecule. The term U i j i n (20) and (21) i n c l u d e s r u p o l e , quadrupole-induce rupole terms. The above p o t e n t i a l models were a r r i v e d a t by c o n s i d e r i n g a l l p o s s i b l e terms i n Eq. (1) i n each case, and o m i t t i n g terms t h a t were found to be n e g l i g i b l e . M u l t i p o l e moments and p o l a r i s a b i l i t i e s were a v a i l a b l e from independent experimental measurements. For the l i k e - p a i r i n t e r a c t i o n the remaining parameters (e, a, n, 6]_, o r 62) were determined as those g i v i n g the best f i t t o the s a t u r a t e d l i q u i d d e n s i t y and vapor pressure data (13). The l i k e p a i r parameters are shown i n Table 2. For the u n l i k e - p a i r i n t e r a c t i o n s one must a l s o know the parameters e £ , c ? £ , n £ and where i i s 1 o r 2. The l a s t two parameters were estimated from the combining r u l e s n c

a

n a =
6?

3

a a BB Q 0

) a

a

a

1 / 2

(23) 3

= 1/2 ( S ? + 6^ )

(24)

w h i l e e g and o ^ are g i v e n by (14) and (15). The q u a n t i t i e s n g and were taken t o be the values g i v i n g the b e s t f i t t o v a p o r - l i q u i d e q u i l i b r i u m data f o r the b i n a r y mixture a t a s i n g l e temperature. The u n l i k e - p a i r parameters are given i n Table 3. F i g u r e s 7 and 8 show a comparison o f theory and experimental data o f Calado €it a l . (27,28) 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 i n the systems Xe/HCl and Xe/HBr. F i g u r e 9 shows a comparison f o r the excess volume f o r Xe/HBr ( s i m i l a r agreement i s obtained f o r Xe/ HC1). These systems are h i g h l y n o n i d e a l , and both the p o l a r and quadrupolar f o r c e s are important. The Xe/HCl system has a p o s i t i v e azeotrope which i s w e l l reproduced by the theory, and appare n t l y e x h i b i t s l i q u i d - l i q u i d i m m i s c i b i l i t y a t low temperatures (27). The theory p r e d i c t s the v a p o r - l i q u i d compositions and p r e s sures w e l l a t each temperature, the maximum e r r o r being 0.1 b a r , and a l s o reproduces the excess volume. I t i s i n t e r e s t i n g to note that the composition curve f o r Xe/HBr i s c u b i c . Such behavior a

a

a

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.

2

C H

2

6

C2H2 C2H4

Xe HCl HBr C0

(a) (b) (c) (d)

Substance

(K)

3.961 3.670 3.790 3.687 3.901 4.097 4.324

n 12 9 12 16 16 13 13

( a )

0.10

01

J

-0.1 +0.3 0.1 -0.2

02 4.10

a(A )

3

0Q ( b )

0.257 0.270 0.158 0.132

1.07 (c) 0.788(c)

1 8

y x 10 esu cm

3.80 4.0 -4.30 5.01 3.85 -0.65

2 6

2

(c) (c) (c) (d) (d) (c)

Q x 10 esu cm

From s a t u r a t e d l i q u i d density and pressure. g I s o t r o p i c p o l a r i z a b i l i t y , from Reed and Gubbins. From Stogryn and S t o g r y n . H From S p u r l i n g and Mason. ^ K = ( a i r a _)/(an+ 2a_i_), where anand aj_ are the p o l a r i z a b i l i t i e s p a r a l l e l and p e r p e n d i c u l a r to the symmetry a x i s , r e s p e c t i v e l y .

231.52 153.23 248.47 244.31 253.66 229.32 238.99

e/k

( a )

POTENTIAL PARAMETERS

Table 2

18.

GUBBINS A N D

TWU

Fluid Mixtures

Figure 9. Excess molar volumes for Xe-HBr from theory (lines) and experiment (28) (points)

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

361

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

362

INDUSTRY

Table 3 VALUES OF THE CROSS-INTERACTION PARAMETERS

6

HCl/Xe HBr/Xe C0 /C H 2

2

6

CO2/C2H4 CO2/C2H2

C H /C H6 2

2

2

aS 1

System 1.0050 0.9996 1.0000 1.0169 1.0000 1.0000

1.0042 0.9880 0.9635 0.9038 0.8380 1.0197

10.3923 12.0000 14.4222 14.4222 16.0000 13.0000

+0.10 -0.15 0.00 +0.10 -0.05

i s common f o r mixtures c o n t a i n i n g p o l a r c o n s t i t u e n t s , b u t i s not observed i n simple nonpolar mixtures (20). F i g u r e 10 shows t h e o r e t i c a l p r e d i c t i o n s f o r the system CO2/ ethane over a wide range o f temperatures, pressures and composit i o n s . The system has a p o s i t i v e azeotrope over the whole o f the l i q u i d range. E x c e l l e n t agreement i s obtained between theory and experiment f o r the v a p o r - l i q u i d e q u i l i b r i a a t a l l temperatures, although s m a l l d i s c r e p a n c i e s appear i n the c r i t i c a l l i n e . C a l c u l a t e d a z e o t r o p i c compositions agree w i t h experimental values w i t h i n 1% over the e n t i r e l i q u i d range; c a l c u l a t e d a z e o t r o p i c p r e s sures agree w i t h i n 0.1 bar. F i g u r e 11 shows the p r e d i c t e d vaporl i q u i d e q u i l i b r i u m curves f o r CX^/acetylene. This system i s o f p a r t i c u l a r i n t e r e s t because i t forms a n e g a t i v e azeotrope; according to the theory t h i s azeotrope i s bounded above. The n e g a t i v e azeotrope i n t h i s system i s b e l i e v e d to a r i s e because the quadr u p o l e moments o f C 0 and a c e t y l e n e are of o p p o s i t e s i g n (20). The t h e o r e t i c a l p r e d i c t i o n s are i n good agreement w i t h the very meager experimental data f o r t h i s system. These are l i m i t e d t o a measurement o f the b o i l i n g p o i n t of the azeotrope a t 1 b a r , (32) and to measurements o f the c r i t i c a l p o i n t f o r an equimolar mixture (29,33). For the systems CT^/ethylene and ethane/ethylene, the agreement between theory and experiment i s s i m i l a r to t h a t f o r C02/ethane. F i g u r e 12 shows a t e s t o f the theory f o r the t e r n a r y system C02/ethane/ethylene. Such a t e s t i s o f p a r t i c u l a r i n t e r e s t , s i n c e no new p o t e n t i a l parameters need be i n t r o d u c e d . For temperatures ranging from 263 t o 293K, and pressures from 20 t o 60 b a r , the mean d e v i a t i o n between theory and experiment was o n l y 0.003 i n mole f r a c t i o n and 0.16 bar i n p r e s s u r e . The r e s u l t s f o r the t e r nary system are as accurate as those f o r the b i n a r y systems. Thus the b i n a r y systems are adequate b u i l d i n g b l o c k s f o r c a l c u l a t i n g the p r o p e r t i e s of the multicomponent systems. There i s no need to invoke any c h a r a c t e r i s t i c a l l y three-component parameter. 2

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

18.

GUBBINS

AND T W U

Fluid Mixtures

363

80

/ / / / /

0.

0.2

0.4 x

0.6

co 'yco 2

0.8

1.

2

Figure 10. Vapor-liquid equilibria and critical line for C0 -ethane from theory (lines) and experiment (29-31,) (points) 2

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

364

PHASE

EQUILIBRIA A N D F L U I D

PROPERTIES IN C H E M I C A L

INDUSTRY

32

T = 188.1 K

Figure 11.

Calculated vapor-liquid equilibria for COo-acetylene

CO

32 atm

C H 2

6

Figure 12. Vapor—liquid equilibria for the system C0 -ethane— ethylene at 263° K from theory (lines) and experiment (34) (points) 2

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

18.

GUBBINS

AND TWU

Fluid Mixtures

365

Acknowledgments We a r e g r a t e f u l to the N a t i o n a l Science Foundation, the Ameri c a n Gas A s s o c i a t i o n , and the Petroleum Research Fund (administered by the American Chemical S o c i e t y ) f o r f i n a n c i a l support o f t h i s work. I t i s a p l e a s u r e to thank L. A. K. S t a v e l e y , J . C. G. Calado, A. Fredenslund and J . M o l l e r u p f o r sending papers p r i o r to p u b l i c a t i o n , and B. Widom f o r a h e l p f u l d i s c u s s i o n .

Abstract A method based on thermodynamic perturbation theory is described which allows strong directional intermolecular forces to be taken into account whe This is applied to the cal loci for mixtures containing polar or quadrupolar constituents. Two applications of the theory are then considered. In the first, the relation between intermolecular forces and the type of phase behavior is explored for binary mixtures in which one component is either polar or quadrupolar. Such systems are shown to give rise to five of the six classes of binary phase diagrams found in nature. The second application involves comparison of theory and experiment for binary and ternary mixtures. References 1. Hildebrand, J. H., Prausnitz, J. M., and Scott, R. L., "Regular and Related Solutions," Van Nostrand Reinhold Co., New York 1970. 2. J. S. Rowlinson, this volume. 3. Smith, W. R., "Statistical Mechanics, Vol. 1, Specialist Periodical Report," ed. K. Singer, Chemical Society, London (1973). 4. Gubbins, K. E., AIChE Journal, (1973) 19, 684. 5. T. W. Leland, P. S. Chappelear and B. W. Gamson, A.I.Ch.E. Journal, (1962), 8,482; J. W. Leach, P. S. Chappelear and T. W. Leland, Proc. Am. Petrol. Inst., (1966), 46, 223. 6. K. C. Mo and K. E. Gubbins, Molec. Phys., (1976),31,825; J. M. Haile, K. C. Mo and K. E. Gubbins, Adv. Cryogen. Engng., (1976), 21,501; S. Murad and K. E. Gubbins, Chem. Eng. Science, (1977), 32, 499. 7. For a review see Egelstaff, P. A., Gray, C. G., and Gubbins, K. E., "Molecular Structure and Properties," International Review of Science. Physical Chemistry, Series 2, Volume 2, ed. A. D. Buckingham, Butterworths, London 1975. 8. Reed, T. M., and Gubbins, K. E., "Applied Statistical Mechanics," McGraw Hill, New York 1973. 9. Gray, C. G., and Van Kranendonk, J., Canad. J. Phys., (1966) 44 2411.

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

366

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

10. Armstrong, R. L., Blumenfeld, S. M. and Gray, C. G., Canad. J. Phys. (1968) 46, 1331. 11. Stogryn, D. E. and Stogryn, A. P., Molec. Phys. (1966) 11, 371. 12. Flytzani-Stephanopoulos, M., Gubbins, K. E., and Gray, C. G., Molec. Phys., (1975) 30, 1649. 13. Twu, C. H., Ph.D. Dissertation, University of Florida (1976). 14. S t e l l , G., Rasaiah, J. C., and Narang, H., Molec. Phys., (1974) 27, 1393. 15. Wang, S. S., Egelstaff, P. A., Gray, C. G., and Gubbins, K. E., Chem. Phys. Lett., (1974) 24, 453. 16. Haile, J. M., Ph.D. Dissertation, University of Florida (1976). 17. Twu, C. H., Gubbins (1976) 64, 5168. 18. Gosman, A. L., McCarty, R. D., and Hust, J. D., Nat. Stand. Ref. Data Serv. Natl. Bur. Stand. 27 (1969). 19. Scott, R. L., and Van Konynenburg, P. H., Discuss. Faraday Soc., (1970) 49, 87. 20. Rowlinson, J. S., "Liquids and Liquid Mixtures," Butterworths, London, 2nd edition 1969. 21. Schneider, G. M., Adv. Chem. Phys., (1970) 17, 1. 22. Scott, R. L., Ber. Bunsenges. Phys. Chem. (1972) 76, 296. 23. Twu, C. H., and Gubbins, K. E., to be published. 24. McDonald, I. R., "Statistical Mechanics, Vol. 1 Specialist Periodical Report," ed. K. Singer, Chemical Society, London (1973). 25. Streett, W. B., Canad. J. Chem. Eng., (1974) 52, 92. 26. Spurling, T. H., and Mason, E. A., J. Chem. Phys., (1967) 46, 322. 27. Calado, J. C. G., Kozdon, A. F., Morris, P. J., da Ponte, N., Staveley, L. A. K., and Woolf, L. A., J. Chem. Soc., Faraday Trans. I, (1975) 71, 1372. 28. Calado, J. C. G., and Staveley, L. A. K., private communication. 29. Kuenen, J. P., Phil. Mag. (1897) 44, 174. 30. Fredenslund, A., and Mollerup, J., J. Chem. Soc. Faraday Trans. I., (1974) 70, 1653. 31. Mollerup, J., J. Chem. Soc. Faraday Trans. I, (1975) 71, 2351. 32. Clark, A. M., and Din, F., Trans. Faraday Soc., (1950) 46, 901. 33. Dewar, J., Proc. Roy. Soc. London, (1880) 30, 542. 34. Fredenslund, A., Mollerup, J., and Hall, K. R., J. Chem. Eng. Data, (1976), 21, 301.

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

Discussion TRUMAN S. STORVICK

Progress on the t h e o r e t i c a moderately dense and dens l i m i t e d by the l a c k of an equation of s t a t e f o r dense f l u i d phases. The past decade has seen r a p i d progress on t h i s problem using e l e c t r o n i c computers to o b t a i n ensemble average p r o p e r t i e s f o r f l u i d systems modelled w i t h simple p a r t i c l e s . The e q u i l i b r i u m and n o n - e q u i l i b r i u m s t a t i s t i c a l mechanical and k i n e t i c theory r e l a t i o n s h i p s admit the d i r e c t c a l c u l a t i o n of the f l u i d p r o p e r t i e s without i d e n t i f y i n g the mathematical form of the equation of s t a t e . This a p p l i c a t i o n of computers has r e v o l u t i o n i z e d the f l u i d p r o p e r t i e s s t u d i e s i n ways as profound as t h e i r a p p l i c a t i o n to process design problems. The determination of dense f l u i d p r o p e r t i e s from ab i n i t i o quantum mechanical c a l c u l a t i o n s s t i l l appears to be some time from p r a c t i c a l completion. Molecular dynamics and Monte C a r l o c a l c u l a t i o n s on r i g i d body motions w i t h simple i n t e r a c t i n g f o r c e s have q u a l i t a t i v e l y produced a l l of the e s s e n t i a l f e a t u r e s of f l u i d systems and q u a n t i t a t i v e agreement f o r the thermodynamic p r o p e r t i e s of simple pure f l u i d s and t h e i r mixtures. These c a l c u l a t i o n s form the b a s i s upon which p e r t u r b a t i o n methods can be used to o b t a i n p r o p e r t i e s f o r polyatomic and p o l a r f l u i d systems. A l l t h i s work has provided i n s i g h t f o r the development of the p r i n c i p l e of corresponding s t a t e methods that d e s c r i b e the p r o p e r t i e s of l a r g e r molecules. The p e r t u r b a t i o n methods were discussed i n d e t a i l and a p p l i c a t i o n s to polar-nonpolar mixtures described the p o t e n t i a l power of these techniques. The energy and d i s t a n c e s c a l i n g parameters for each component are obtained from pure component vapor pressure data. D i p o l e or quadrupole moments are obtained from independent measurements. An energy i n t e r a c t i o n parameter i s evaluated from v a p o r - l i q u i d e q u i l i b r i u m data at one temperature. An e f f e c t i v e equation of s t a t e f o r these mixtures i s obtained from the formalizm using t h i s s m a l l set of data. More work must be done to improve the accuracy of the c a l c u l a t i o n s to provide design data but i t c l e a r l y shows promise and continued e f f o r t should be p r o d u c t i v e . 367

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

368

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

The c a l c u l a t i o n of the t r a n s p o r t p r o p e r t i e s of moderately dense f l u i d s u s i n g k i n e t i c theory and an e q u i l i b r i u m equation of s t a t e shows remarkable agreement between p r e d i c t i o n s and e x p e r i mental measurements. This success suggests that the t h e o r e t i c a l s t r u c t u r e r e q u i r e d to o b t a i n the f u n c t i o n a l forms that produced these r e s u l t s should be s t u d i e d . The v e r i f i c a t i o n of t h i s procedure f o r a wider range of c o n d i t i o n s and mixtures could p r o v i d e a way to o b t a i n t r a n s p o r t p r o p e r t i e s of f l u i d s . The q u e s t i o n and answer s e s s i o n focused on problems t h a t must be t r e a t e d i n order to make the t h e o r e t i c a l advances a v a i l a b l e to the design people. I t takes great a t t e n t i o n to the d e t a i l e d s t r u c ture of the theory to understand what the e s s e n t i a l f e a t u r e s of the models are and what experimental i n f o r m a t i o n must be s u p p l i e d to make q u a n t i t a t i v e c a l c u l a t i o n s There are some s p e c i a l area 1.

2.

3.

4.

Experimental data to t e s t t h e o r e t i c a l procedures i s o f t e n not a v a i l a b l e . Thermodynamic and t r a n s p o r t data on mixtures cont a i n i n g hydrogen and a c i d gases (HCl, NH3, SO2, etc.) would be very u s e f u l . Dense gas data are g e n e r a l l y not a v a i l a b l e . Accurate measurements over wide ranges of temperature and d e n s i t y are e s p e c i a l l y u s e f u l . Requests f o r data from f i e l d engineers to company data centers are c u r r e n t l y d i s t r i b u t e d about 70% phase e q u i l i b r i u m , 25% enthalpy and 5% a l l other data. Design d e c i s i o n s are made on the b a s i s of the d i s t r i b u t i o n c o e f f i c i e n t s between phases and on the process energy requirements. The requests r e f l e c t t h i s demand. Procedures to a c c u r a t e l y p r e d i c t these data would be w i d e l y adopted and are i n great demand. Polyatomic molecules are d i f f i c u l t to t r e a t because the s i z e , shape, f l e x i b i l i t y , and charge d i s t r i b u t i o n are a l l important f a c t o r s i n the f l u i d equation of s t a t e . Complex molecules do not s a t i s f y random mixing c r i t e r i a and the energy and entropy f u n c t i o n s must c o n t a i n proper accounting of these non-random effects. The extended p r i n c i p l e of corresponding s t a t e methods are being f u r t h e r developed. Wider experience w i t h t h i s approach might produce e a r l i e r adoption of new methods of p r e d i c t i o n because most p r a c t i c i n g engineers have a b e t t e r i n t u i t i v e understanding of these methods.

The r a p i d development of the theory of dense f l u i d systems w i l l r e q u i r e a number of workers who can transform these r e s u l t s i n t o p r a c t i c a l p r o p e r t i e s p r e d i c t i o n schemes. This i s o f t e n a d i f f i c u l t step to accomplish but i t i s e s s e n t i a l i f we are to reduce the time between the t h e o r e t i c a l d e s c r i p t i o n of f l u i d behavior and the use of t h i s understanding to help make design d e c i s i o n s .

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

19 Criteria of Criticality MICHAEL M O D E L L Massachusetts Institute of Technology, Cambridge, MA 02139

Over a century ago, Gibbs (1) developed the mathematical c r i t e r i a of c r i t i c a l i t y . He defined the c r i t i c a l phase as the t e r m i n a l s t a t e on the b i n o d a l s u r f a c e and reasoned that i t has one l e s s degree of freedom than the b i n o d a l s u r f a c e ( i . e . , f = n-1). He then developed two equations as the c r i t e r i a of c r i t i c a l i t y ; these two equations, when imposed upon the Fundamental Equation, reduce the degrees of freedom from n+1 t o n-1. Gibbs presented three a l t e r n a t e sets of the two c r i t i c a l i t y criteria. I n terms of the v a r i a b l e s e t T , V , y i , . . . , y _ i , N , they a r e : n

n

= 0

(1)

n-1

= 0 n-1

369

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

(2)

P H A S E EQUILIBRIA

370

AND FLUID

PROPERTIES IN C H E M I C A L

INDUSTRY

To ensure s t a b i l i t y of the c r i t i c a l phase, Gibbs noted that the f o l l o w i n g r e l a t i o n must a l s o be s a t i s f i e d :

(3)

> 0 T,V,y ,...,y 1

n-1

Gibbs noted that Eq. (1) may lead to an indeterminant form. When t h i s occurs, he suggested permutting the component s u b s c r i p t s or using "another d i f f e r e n t i a l c o e f f i c i e n t of the same general form." He noted that Eq. (1) i s the c r i t e r i o n of the l i m i t of s t a b i l i t y ( i . e . , the s p i n o d a l s u r f a c e ) . I n an e a r l i e r d i s c u s s i o n on that s u b j e c t , Gibbs showed that a l l of the f o l l o w i n g p a r t i a l d e r i v a t i v e s v a n i s h on the s p i n o d a l s u r f a c e

(») V-'

.f-h)

V, ...,y V

V

n

/

1 ,v,u ,...,p T

2

V n

n /

T,V,p ,..., 1

V l

(4) Thus, any one of these forms could be used i n place of Eq. ( 1 ) , w i t h corresponding changes made i n Eqs. (2) and ( 3 ) . As an a l t e r n a t i v e to Eqs. (1) and ( 2 ) , Gibbs suggests " f o r a p e r f e c t l y r i g o r o u s method there i s an advantage i n the use of S^V, N^,...,N as independent v a r i a b l e s . " I n t h i s v a r i a b l e s e t the c r i t i c a l i t y c o n d i t i o n s become: n

(5)

R , = 0 n+1

(6) SS

where

R

s

N S n

N

N Nn 1

N

i

N

i i

(7)

n+1

SN

N N n n

1 n

and S i s the determinant formed by r e p l a c i n g one of the rows of R

n 1 ? n lJ h

+

(R

+

(I

W

R

N

< n l> +

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

19.

Criteria of Criticality

MODELL

SS

371

s

N S n

N

N HL n 1

N

i

N

i i

e.g., S

(8) SN

( R

N.N

N N , n n-1

1 n-1

n-1 ( R

n l> +

( R

n l> +

n+l\

where the n o t a t i o n i s = i n the d i f f e r e n t i a t i o n ar

O R

N

+

1

/

9

N

L

Gibbs s t a t e s that Eqs. (7) and

)

S , V , N N L

n (8) " w i l l hold true of every c r i t i c a l phase w i t h o u t e x c e p t i o n , " implying that t h i s s e t of c r i t i c a l i t y c r i t e r i a i s more s t r i n g e n t than Eqs. (1) and ( 2 ) . Gibbs presents a t h i r d a l t e r n a t i v e s e t w i t h T,P, N . ,N as independent v a r i a b l e s : ^ n

N

N

2 1

N n-1 -N-1

N

N N

n-1 2

N

i i

N

12

N

2

2

0 (9)

N

and

C

=

i

N N 2 n-1

1 n-1i N

n-1 n-1

0

(10)*

where C i s the determinant formed by r e p l a c i n g one of the rows of the B-determinant by B B„ ... B. . Gibbs does not s p e c i f y i f N N N 1 2 n-1 Eqs. (9) and (10) are t r u e w i t h o u t e x c e p t i o n or i f there a r e r e s t r i c t i o n s t o t h e i r use. T

T

N

Development of the General C r i t i c a l i t y C r i t e r i a Each of the c r i t i c a l i t y c r i t e r i a presented by Gibbs can be shown *Gibbs uses the symbols U and V f o r the B- and C-determinants. We use B and C here t o avoid c o n f u s i o n w i t h i n t e r n a l energy and volume.

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

372

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

to be s p e c i a l cases of more general forms. The development of the general form f o l l o w s from the unique f e a t u r e of the c r i t i c a l phase which d i s t i n g u i s h e s i t from other phases i n the v i c i n i t y ; namely, the c r i t i c a l phase i s the s t a b l e c o n d i t i o n on the s p i n o d a l s u r f a c e . This statement i s e q u i v a l e n t t o Gibbs argument that the c r i t i c a l phase l i e s on the b i n o d a l surface and, t h e r e f o r e , i s s t a b l e ( w i t h respect to continuous changes) and a l s o s a t i s f i e s the c o n d i t i o n of the l i m i t of s t a b i l i t y (which d e f i n e s the s p i n o d a l curve). S t a r t i n g w i t h the Fundamental Equation i n the i n t e r n a l energy representation, U = f (S,V,N,...,N ) u l n

(11)

we can t e s t the s t a b i l i t y of a substance by expanding U i n a Taylor s e r i e s and examining th h o l d i n g the t o t a l entropy AU = 6U + ~j 6 U + ~r 6 U + ... 2

3

(12)

For e q u i l i b r i u m to e x i s t , 6U = 0

(13)

and f o r the system to be s t a b l e ,

where 6 U i s the lowest order non-vanishing v a r i a t i o n . The s p i n o d a l surface represents the c o n d i t i o n s where the second-order v a r i a t i o n f i r s t l o s e s the p o s i t i v e , d e f i n i t e character of the s t a b l e , s i n g l e phase (3). The second-order v a r i a t i o n can be expressed as

2

ss^> n 2 £ j=i n n

j = l k=l

+ 2U (6S)(6V) gv

(U

SN. J

6

^

+ U

VN. J

+ U

(

)2

w^

6V)6N. J (15)

IT „ 6N. 6N N.N, j k J

k

or, i n c l o s e d form, as a sum of squares, u

u

ss sv

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

Criteria of Criticality

MODELL

373

SS

'sv

SN,

VS

vv

VN,

N V

N

N

11 6Z

2 3

+

'SS "VS

U

U

°SV

'SS

U

VS

l S

N l

N S n

V

VV

U

\

SN,

V

N

N V n

J

SN

VN

1 n

N

i i

N n 1

N N n n 6Z

'SS

J

U

VS

N

n-1

_S

U

'SV

SN.

SN

VV

VN,

VN

N V

N

N

,V n-1

U

N

n-1

n-1

i i

N N 1 n-1

-Nn-1 1

N

N

n+2

n-1

n

N

n-1

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

(16)

374

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L

INDUSTRY

Using the shorthand n o t a t i o n , A

u

i - ss SS

i+2

sv

SS

SV

VS

vv

N

i

SN.

NV

s

N

N.V

N.S

N.N, l 1

1

1

N- N. 1 l

N

i i

N.N.

Eq. (16) takes the form,

9

9

6^U = A , 6 Z /

n

+

+

Z

2

A

i

9

6Z.

(17)*

The l a s t term i n Eqs. (16) or (17) can be shown to be zero as a v i r t u e of the f a c t that only n+1 i n t e n s i v e v a r i a b l e s are independent f o r a s i n g l e phase. The proof f o l l o w s . Let us expand each of the n+2 i n t e n s i v e v a r i a b l e s T, P, u-^,..., u as f u n c t i o n s of S^,V,N^,. .. ,N . n

dT =

n

(ff) ^V,N

dS +

dV + ^S,N

Z 1=1

(fjU 8 N

iS,V,N[i]

d N 1

*For the form of the 6Zj v a r i a t i o n s , see ( 3 ) ; s i n c e they always appear as squared terms, we need not consider them f u r t h e r here.

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

19.

MODELL

or

dT = U

Criteria of Criticality

dS + U d V + I

s g

U

gv

375

i=l dP = U

d

dS + U

v s

d i+

"i-V

Vl

U

% = U

N

d

n-1 l

S

d S

S

V N

dN.

(19)

i

+

d N

1

= N

(18)

(20)

V - ' i V- i

1

d

±

1

dV + I U i=l

w

dN

g N

+

*

+

U

D

N

n

1=1

V I n-1

H

V

d

1

d V

+

n

Z

+

1

?

1

° H ,N. l i = l n-1 1 dN

1

U i=l n

dN.

(

2

1

)

(22)

1

Since the n+2 d i f f e r e n t i a l s of T,P,ui_,. . . , u are r e l a t e d by the Gibbs-Duhem equation, any one from t h i s set can be expressed as a f u n c t i o n of the other n+1. Thus, i f P,y]_,...,y are h e l d constant ( i . e . , dP = d u i = ... = d y = 0 ) , then T must a l s o be constant ( i . e . , dT = 0 ) . I t f o l l o w s that the determinant of the m a t r i x of c o e f f i c i e n t s i n Eqs. (18) to (22) must be zero. Since t h i s d e t e r minant i s A +2 i t f o l l o w s that A 2 0. Therefore, the general form of Eq. (17) i s n

n

n

=

n

S

n+

2 2 6 U = A 6Z + Z n

Z

+

1

Z

1

1

2

A 2j

2 6Z.

Z

(23)

Vl

The l i m i t of s t a b i l i t y i s d e f i n e d as the c o n d i t i o n under which 6^U_ l o s e s i t s p o s i t i v e , d e f i n i t e c h a r a c t e r . That i s , one of the c o e f f i c i e n t s i n Eq. (23) vanishes. As has been shown p r e v i o u s l y (_3), when the s p i n o d a l s u r f a c e i s approached from a s t a b l e s i n g l e phase r e g i o n , the c o e f f i c i e n t of 6 Z i i s among the f i r s t to reach zero ( i . e . , i f another c o e f f i c i e n t a l s o v a n i s h e s , i t does so simultaneously w i t h the c o e f f i c i e n t of 6 Z ^ ) . Therefore, on the s p i n o d a l s u r f a c e n +

n +

A , = 0 n+1

(24)

This equation i s one of the c r i t e r i a of c r i t i c a l i t y . I t i s equival e n t to reducing the degrees of freedom from n+1 f o r a s t a b l e s i n g l e phase to n on the s p i n o d a l s u r f a c e . Since A +i i s the determinant n

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

376

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L

INDUSTRY

of the m a t r i x of c o e f f i c i e n t s of Eqs. (18) to (21) a t constant N , the requirement of A +i = 0 i s e q u i v a l e n t to dT = dP = dy^ = ... = d y _ i = 0. That i s , i f n v a r i a b l e s from the set T,P,y-[, . . . , y - l are held constant, the remaining v a r i a b l e w i l l a l s o be constant. n

n

n

n

The second c r i t e r i o n of c r i t i c a l i t y i s that the d i f f e r e n t i a l of A 2 must v a n i s h f o r a l l p o s s i b l e v a r i a t i o n s . Thus, n+

dA

dN.=0 l S,V,N [ i ]

n+1

(25) A l t e r n a t i v e l y , Eq. (25) ca equations from the set.Eqs. (18) to (21) a t constant . Choosing dT = dP = dy^ = ... = dy _2 = 0 from t h i s s e t , the second c r i t e r i a of c r i t i c a l i t y becomes: R

n

SS

VS

u SN,

vv

VN,

VN

N

N-N . 1 n-1

N S

N

( A

S n-2 0

n-2

n l>

( A

+

u SN n-1

u sv

n l\ +

N

1 1

N

( A

N

N. n-2 1 0

( A

n l>

n-1

N n-2 n-1 9

n l> +

+

n-1 (26) A l t e r n a t e C r i t e r i a i n the U-Representation The two c r i t e r i a of c r i t i c a l i t y developed i n the l a s t s e c t i o n are s i m i l a r t o , but no i d e n t i c a l w i t h Eqs. (5) and (6) which were s t a t e d by Gibbs. We s h a l l now show that the two are equivalent and that they are p a r t i c u l a r s e t s from a more g e n e r a l form of the c r i t e r i a . We began the d e r i v a t i o n of Eq. (24) by expanding U i n terms of S,V,N ,...N Eq. (15), and then closed the sum of squares, Eq. (16). In the process, we maintained the o r d e r i n g of the independent v a r i a b l e s as S_,V,Nj_, . . ,N . I f we had chosen t h e ^ r d e r as S_,Ni, . . .N ,V, then the numerator of the c o e f f i c i e n t of &Z +i, which was A +i, would have been R + i , as expressed by Eq. ( 7 ) . Thus, Eqs. (5) and (24) d i f f e r only i n the o r d e r i n g of v a r i a b l e s i n c l o s i n g the sum of squares and, hence, are of equal v a l i d i t y . The same statement a l s o a p p l i e s to 1

n

n

n

n

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.

19.

Criteria of Criticality

MODELL

377

Eqs. (6) and (26). I t f o l l o w s immediately that there a r e n+2 e q u i v a l e n t forms of Eqs. (5) or (24) and (6) or (26), each formed by o m i t t i n g one of the n+2 v a r i a b l e s

, x

S_,V,NT_, . .. ,N . N

The general form can be s t a t e d i n terms of y =f (z^, . . ., z ) where y(°) i s U and z ^ , . . . , z i s any o r d e r i n g of the n+2 v a r i a b l e s jS,V,N ... ,N ( i . e . , m=n+2). The general form of Eq. (23) becomes m

m

L5

N

2

6 y

= V

( o )

1

6Z

1 1

n+1 V. JL + E •

9

V

-

i

(27)

6Z/ J

J =2 j-1 ( o )

( o )

y 22

21

f

where V, =

2k

y

(28)

k

'kl and y <

o )

13

=

2

8 y

( o )

'k2

'kk

/3z.3z.. i J

In t h i s terminology, the two g e n e r a l c r i t e r i a which apply without e x c e p t i o n , a r e :

of c r i t i c a l i t y ,

( 0 )

( o )

'll

y 12

'21

y 22

yK n + l ) y

y

( 0 ) y

2(n+l)

y

(n+1)(n+1)

y

n+1

y

and

y

(n+l)l (n+l)2 ,
'12 ,(o)

y

( 0 )

y

l(n+l)

y

2(n+l)

y

n(n+l)

22

(29)

f

= 0

"n+1 n2

f

( P

n+l\

P

< n+l\

P

< n+l>n+l

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

(30)

378

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

A l t e r n a t e C r i t e r i a i n Other P o t e n t i a l Functions In a d d i t i o n to the c r i t e r i a of c r i t i c a l i t y i n the U-represent a t i o n , Gibbs presented two a l t e r n a t i v e s e t s ; namely Eqs. (1) and (2) and Eqs. (9) and (10). These are two of many sets which we s h a l l show d e r i v e from r e p r e s e n t a t i o n s of the Fundamental Equation i n a l t e r n a t e p o t e n t i a l f u n c t i o n s . We s h a l l use the methodology of Legendre transformations to s h i f t from one set of independent v a r i a b l e s to another.* I t has been shown that the common p o t e n t i a l f u n c t i o n s of enthalpy, H, Helmholtz f r e e energy, A, and Gibbs f r e e energy, G_, can be viewed as Legendre transforms of U_ i n the f o l l o w i n g independent v a r i a b l e s e t s : H = f^P,^,...,^

A = f (T,V,N ...,N )

(32)

G = f (T,P,N ,...,N )

(33)

A

r

G

n

1

n

The f u n c t i o n s f j j , f ^ and fQ are e n t i r e l y e q u i v a l e n t to £\j i n Eq. (11). That i s , the i n f o r m a t i o n content of the Fundamental Equation, Eq. (11), i s maintained i n the Legendre t r a n s f o r m a t i o n . For multicomponent substances, a d d i t i o n a l p o t e n t i a l f u n c t i o n s may be defined wherein one or more mole numbers, N^, i s transformed to i t s conjugate c o o r d i n a t e , u^. F o l l o w i n g the n o t a t i o n of Beegle, et a l . , ( 3 ) , we s h a l l c a l l these the prime p o t e n t i a l f u n c t i o n s , wherein the number of primes i n d i c a t e the number of Nj_ that have been so transformed. That i s , U

?

(34)

= f (S,V,y ,N ,...,N ) IJI

U" = f

u l t

1

2

n

( S . V . y ^ , N ,...,N ) 3

H' = f ( S , P , u , N , . . . , N ) Hl

1

2

1

2

1

2

(37)

n

G' = f , ( T , P , y , N , . . . , N ) G

(36)

n

A' = f ( T , V , y , N , . . . , N ) Af

(35)

n

(38)

n

*Refer to M o d e l l and Reid (_2) or Beegle, et_ a_l. (3) f o r a d e s c r i p t i o n of the Legendre technique and f o r a more d e t a i l e d e x p l a n a t i o n of terminology.

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

19.

Criteria of

MODELL

379

Criticality f

Note that A, H and U a r e a l l s i n g l e v a r i a b l e Legendre t r a n s forms; G, A , H and U" are double transforms; and G , A", H" and U" are t r i p l e transforms. D e f i n i n g E,± as the conjugate coordinate of z., 1

1

f

f

1

3v

( 0 )

the k - v a r i a b l e Legendre transform i s

^• «i

vvi

f

(40)

Ck In terms of the y transform, the c r i t i c a l i t y c r i t e r i a can be s i m p l i f i e d by reducing the V- determinant forms i n Eq. (27) to s i n g l e p a r t i a l d e r i v a t i v e s . I t has been shown (3) that v

y

(k-1) kk

J

\ _

(41)

V

S u b s t i t u t i n g Eq. (41) i n t o Eq. (27), .2 (o) _ (o) 2 6y - y 6Z n

1

+ y

(1)

1 2

6Z

2 £

+ y

(2) 3 3

6Z

2

(n) + ... + ^(n+1)(n+1)

6 Z

3

2 n+1

(42) Instead of Eq. (29), we o b t a i n f o r the c r i t e r i o n of the s p i n o d a l surface:

y

z \ (n) (n+1)(n+1)

. _2 (n) /d y \ v

=

=0

(«)

The second c r i t e r i o n of c r i t i c a l i t y becomes

y

(n) (n+l)(n+l)(n+l)

3

=

(

n

)

P 7 \ ( 3 ) \ n+1 /

=0 t

(44)

Z

^l'-"' n' n+2 To ensure s t a b i l i t y

of the c r i t i c a l phase, we f u r t h e r r e q u i r e

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

380

v y

P H A S E EQUILIBRIA

( >

A N D F L U I D PROPERTIES

\

=/^-I

n

IN C H E M I C A L INDUSTRY

>o

(45)

( . 4 1 V n+1 I

(n+l)(n+1)(n+1)(n+1)

3z

^l'-

,,,

Z

^n' n+2

Using the G-prime n o t a t i o n , (n)

y

m

£

(n-2)»

=

f ( T ) P ) y i )

...

) 1 J n

_

_

2 ) N n

(46)

i j N n )

and G.(n-2)'

(n) n+1

C~ \

\ 3N _

/

( 4 7 )

"n-1

Thus, Eqs. (43) to (45) become:

y

9 y

(n) n+l)(n+l)

/ n-l \ V n-l/

=

0

( 4 g )

S N

T,P,y ,..., ,N 1

V 2

n

L 0

0

(49)

> 0

(50)

n

( n )

y (n+l)(n+1)(n+1) y

and y

( n )

y

(n+1) (n+1) (n+1) (n+1) T

'

P

^ l

V 2 - \

I f we choose to work i n the A-prime system, we have

y

(n) _ (n-1)' A

=

f(T>

v,u ,...,u _ ,N ) 1

n

1

(51)

n

Two e q u i v a l e n t c r i t e r i a a r e obtained because z + i could be taken as V or N , the two e x t e n s i v e v a r i a b l e s of U which a r e not transformed to o b t a i n A^ " ) '. Therefore, n

n

11

y

(n) (n+l)

=

1

.(n-1) \

1

_

\ 3V

^

\ j

=

_

p

( 5 2 ) C 5 2 )

L

,

V

1

P

n - 1 '

n

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

19.

Criteria of Criticality

MODELL

«

< n

i y

-(4~,

)

- u

Equations (43) t o (45) then take

GO T , y (ft

(53)

e i t h e r of two e q u i v a l e n t forms:

V

l

,N

n

T,u ,..., ,N 1

1

r

n

(54) ,...,

1

V l

"(y)T , u , . . . , u _ , N °

381

1

' " \3N /

n

1

> 0

(56)

n

=0

(57)

ft)

(58)

We have now reached another one o f our o b j e c t i v e s : to show that the Gibbs c r i t e r i a of Eqs. (1) t o (3) a r e a s p e c i a l case of a broader g e n e r a l i t y . Equations (57) t o (59) a r e i d e n t i c a l t o Eqs. (1) t o ( 3 ) . By our development h e r e i n , we a l s o see that Eqs. (1) t o (3) a r e but one of many e q u i v a l e n t sets which o b t a i n from the n - v a r i a b l e Legendre transform. The p a r t i c u l a r s e t f o r Eqs. (1) to (3) d e r i v e s from the A . ( ~ l ) ' system. Other e q u i v a l e n t f i r s t c r i t i c a l i t y c r i t e r i a f o r b i n a r y , ternary and quaternary systems a r e given i n Table I . n

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

382

P H A S E EQUILIBRIA

u

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

o

cd

II

e u Q)

4J

LP)

cd c r

o II

rH •H rH

Cd O •rl 4-»

•H H

P.

O

PM

M

cdd J-I

II

co

H

II

II

II

CO 25

o

o

o

O

CO 25

CM Pr-> rH

CN rH L_J

II

CO

25

> J

CN 3-

H

rH

CN P.

II

CO 3.

»»

H

CO

25

V

3

o

II

25 CM O

>^

u

cd a

•H

H

II

CN CO CO

w

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

19.

Criteria of Criticality

MODELL

383

The second c r i t i c a l i t y c r i t e r i a f o l l o w d i r e c t l y by a second differentiation. Gibbs noted t h a t Eq. (1) may, i n some i n s t a n c e s , be i n d e t e r minate, w h i l e Eq. (5) holds t r u e without e x c e p t i o n . Eq. (1) i s a s p e c i a l case of Eq. (43) w h i l e Eq. (5) i s a s p e c i a l case of Eq. ( 2 9 ) . The d i f f e r e n c e between the two c r i t e r i a can be seen by comparing Eqs. (27) and (42). The c o e f f i c i e n t of the l a s t d i f f e r e n t i a l term i n each of the equations d e f i n e s the f i r s t c r i t i c a l i t y c r i t e r i o n . They a r e e q u i v a l e n t ; i . e . , n

y

( ) (n+1) (n+1)

^n+1 — n

=

, ,

n

v

( 6 0 )

I f , under some circumstances Eq. (29) w i l l be s a t i s f i e Thus, Eq. (29) i s the more general c r i t e r i o n . On the other hand, when Eq. (43) i s indeterminate, Gibbs s t a t e s that we could always a l t e r the o r d e r i n g of the independent v a r i a b l e s to remove the indeterminate c o n d i t i o n . This procedure would be equiv a l e n t to choosing another form of (n) , such as i l l u s t r a t e d i n Table I . (n+1)(n+1) ,. As an a l t e r n a t i v e to r e o r d e r i n g v a r i a b l e s when y, .-. , i s . , . _ , . (n+1)(n+1) indeterminate, the f o l l o w i n g procedure can be used. A p p l y i n g Eq. (60) s u c c e s s i v e l y n times, y

J

= y y

n+1

(

n

V

)

(n+l)(n+l) n+1 v

(

=

n

)

(n

V ""

V

y

n

y

(n+l)(n+l) n n

i;)

w

n

V

n-1

(

o r

p n

+i=TT

( 6 1 )

k=l Thus, P + i i s equal to the product of the c o e f f i c i e n t s i n Eq. (42). n

When y / ? . i s indeterminate ( i . e . , V = 0 ) , then y ^ ^ should (n+1) (n+1) r -n be zero or indeterminate; i f y i s a l s o indeterminate, then ( -2) y, A, , should be zero or indeterminate; e t c . I t then f o l l o w s • (n-l)(n-l) that when y ( + ] ^ ) ( 3 ) indeterminate, we could choose p r o g r e s s i v e l y n

n

l W

N

J

n

n

7

N

( n )

i s

n

n+

s m a l l e r values of Y^""^ u n t i l we o b t a i n one which i s zero.

In this

manner, we can s a t i s f y the f i r s t c r i t i c a l i t y c r i t e r i o n without reordering v a r i a b l e s . Let us examine the i m p l i c a t i o n s of these procedures i n ternary and b i n a r y systems. For a ternary system, Eq. (42) i s .2 (o) (o) „ 2 ^ (1) _ 2 ^ (2) 2 _,_ (3) 2 6 y = y 6Z + y ^ 6Z + y ^ 6Z + y (62) x

n

7

±

X 7

£

3

4 4

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

384

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES IN C H E M I C A L INDUSTRY

In terms of the normal o r d e r i n g of v a r i a b l e s f o r U [see Eq. ( 1 1 ) ] , Eq. (62) can be w r i t t e n i n terms of p o t e n t i a l f u n c t i o n s as « U =U 2

G

6Z

S

2

+^

±

2

6Z

+

2

6Z

+ G ' ^

2 3

(63)

6Z?

where G = f ( T , P , N i , N , N ) and G = f ( T , P , u i , N 2 , N ) . I f we a r e d e a l i n g w i t h a s p e c i a l case i n which and P4 v a n i s h simultaneously, then G N - J N ^ = P3/P2 0 " N2N2 ^ 4 ^ 3 indeterminate. A simple r e o r d e r i n g of v a r i a b l e s t o , e.g., G_ = f (T,P,N2,Nj,N ) and G = f(T,P,U2> 1*N3) should r e s u l t i n a f i n i t e v a l u e of G ^ N o * v a l u e of G ' N ^ - L * l i l y > we could t e s t the f i r s t c r i t i c a l i t y c r i t e r i o n by using Eq. (61): T

2

3

=

a n a

3

G?

=

i

s

!

3

N

a

A

p . k v

O

\

R

=

O) Y

4 4

G

' N

e

(2)

N

G

2

N

r

n

a

t

v

c

a

z

e

r

o

e

(1)

3 3

Y

2

t

n

2 2

Y

I

N

y

\

I

l l

'

V °SS

(

6

5

)

(2) (3) That i s , we can t e s t f o r y ^ when i s indeterminate. As pointed out by Heidemann ( 4 ) , i n t e s t i n g the s t a b i l i t y of l i q u i d - l i q u i d c o e x i s t i n g phases i n a t e r n a r y system, there a r e c o n d i t i o n s w i t h i n the unstable r e g i o n where y^y vanishes ( i . e . , where yffl i s n e g a t i v e ) . Thus, when t e s t i n g transforms of lower order than y^ ^\ / > s t be sure that we have approached the c o n d i t i o n uAcfer wRicn y ( ) i s indeterminate from a r e g i o n of proven stability. ( >< ) For a b i n a r y system, Eqs. (42), (62) and (63) are truncated a f t e r the t h i r d term. I f we a r e d e a l i n g w i t h the s p e c i a l case i n which #2 " ^3 v a n i s h s i m u l t a n e o u s l y , then Ayy = ^2/^1 0 * ^NiNi P3/P2 i s indeterminate. T h i s s p e c i a l case i s known to occur when a b i n a r y mixture e x h i b i t s a z e o t r o p i c behavior i n the c r i t i c a l r e g i o n ; that i s , when the locus of a z e o t r o p i c c o n d i t i o n s i n t e r s e c t s the l o c u s of c r i t i c a l c o n d i t i o n s ( 5 ) . For such a system, the r e o r d e r i n g of v a r i a b l e s might take the f o l l o w i n g form. S t a r t i n g w i t h U = f(^,N^,N2,V), we would have v ( D = f (T,Ni,N ,V) and y ( ) = f (T,y]_,N2,V) . The truncated b i n a r y form of Eq. (62) would then become n

w

e

m u

n

l N /

y

n + 1

1 1 N

n + 1

a n a

=

a n c

=

2

2

2

« £ =U

s s

^

+

6Z

2 2

+ A ' ^

6Z

2 3

(66)

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

19.

385

Criteria of Criticality

MODELL

I f t h i s form i s s u c c e s s f u l i n removing the indeterminate c o n d i t i o n , then we should f i n d A ^ ^ ^ non-vanishing and A ^ ^ ®** I f i t i s u n s u c c e s s f u l , then another choice of reordered v a r i a b l e s should be pursued. As an a l t e r n a t i v e to r e o r d e r i n g the independent v a r i a b l e s , we can use Eq. (61) f o r the b i n a r y : 1

V

m

y

3

°

r

(2) (1) (o) 33 22 l l y

"\

3

V

=

( 6 7 )

y

V

^

U

K

'

SS

(

6

8

)

Thus, when a z e o t r o p i c behavio we could t e s t Ayy ^ d G ^ N y produc two to determine the v a n i s h i n g of £ 3 . This l a t t e r procedure was a p p l i e d s u c c e s s f u l l y by Teja and K r o p h o l l e r ( 6 ) . a

C r i t e r i a i n Gibbs Free Energy The t h i r d c r i t e r i a put f o r t h by Gibbs, Eqs. (9) and (10), a r e the forms f r e q u e n t l y quoted i n the c u r r e n t l i t e r a t u r e . These c r i t e r i a , which were presented by Gibbs f o r G = f G ( T , P , N ] _ , . . . , N ) , a r e the s p e c i a l case of y ( ) = f ? 2 > Z 3 > • • • > Z n + 2 ) • The form o f f e r e d by Gibbs can be developed d i r e c t l y from the procedure of the l a s t s e c t i o n . Using Eq. (61) t o evaluate P » N

2

2

V

=

(1) (o) 22 l l y

2

y

(

K

6

9

)

J

Eq. (61) can be r e w r i t t e n as

2

k=3

*Note that f o r a pure m a t e r i a l , the f i r s t c r i t i c a l i t y c r i t e r i o n i s y^l) = 0 where y(Jp = Ayy or A ^ N O ) • A r e o r d e r i n g of v a r i a b l e s does not remove the c r i t e r i o n of mechanical i n s t a b i l i t y . On the other hand, f o r the b i n a r y , Ayy may v a n i s h w h i l e some other form of 22^ ( " * > N j N i ) should not v a n i s h . Thus, to t e s t mechanical s t a b i l i t y of a B i n a r y , one would have to evaluate Ayy. However, i n the general case the r e g i o n of mechanical i n s t a b i l i t y w i l l l i e w i t h i n the r e g i o n of m a t e r i a l i n s t a b i l i t y and, thus, the c o n d i t i o n of mechanical i n s t a b i l i t y w i l l be academic. I t i s always the v a n i s h i n g of the l a s t term i n Eq. ( 2 7 ) , or i t s e q u i v a l e n t , Eq. ( 4 2 ) , that d e f i n e s the l i m i t of i n t r i n s i c s t a b i l i t y . y

e

g

A

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

386

P H A S E EQUILIBRIA

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Each term i n the product of Eq. (70) can be expressed i n terms of d e r i v a t i v e s of y ( w by u s i n g the step-down operator [Eq. (20) of Beegle, et a l . , ( 3 ) ] :

Y

(k-1) kk

m y

(k-2) _ kk

y

(k-2) k(k-l)

2

(k-2) (k-l)(k-l)

/ y / y

K

'

± J

Repeated use of the step-down operator leads to the f o l l o w i n g equation: (2) 33

y

34

y

34

y

44

v y

( 2 )

3k

y

J> 2

y

v y

( 2 )

4k

3k

y

( 2 )

v kk y

(k-D kk

(72) v 33

(2) 3(k-l)

( 2 )

y

v y

( 2 )

34

( 2 )

v 3(k-l) y

y

34

v y

( 2 )

44

( 2 )

v 4(k-l) y

y

..(2) *4(k-l)

..(2) (k-1)(k-1) y

S u b s t i t u t i n g Eq. (72) f o r each of the terms i n the product of Eq. (70), we o b t a i n : (2) ^33 n+1

=

( 2 )

y 34

^3k

^44

'4k

y

^34

(73)

( 2 )

( 2 )

y 3(n+l)

y4 ( n + l ) y

v

( 2 )

y

y

( n + l ) (n+1) Eq. (73) i s the determinant of Eq. (9) when y i s taken as G_. f o r a quarternary system (n=4), Eq. (73) i s ,(2)

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

Thus,

19.

387

Criteria of Criticality

MODELL

(74)

G,

G,

G,

(75)

G,

G,

G

N N 3

3

From t h i s a n a l y s i s , i t should be c l e a r that the c r i t e r i o n o f f e r e d by Gibbs i n Eq. (9) i s e q u i v a l e n t to the requirement that the product of Eq. (70) should v a n i s h . That i s , f o r a quarternary system, f o r example,

= 0

(76)

F o l l o w i n g the d i s c u s s i o n i n the l a s t s e c t i o n , t h i s form may be indeterminant i f y ^ v a n i s h e s . Thus, f o r a system i n which an azeotrope i n t e r s e c t s the c r i t i c a l l o c u s , we would expect Eq. (76) to be indeterminant. I n such cases, the a l t e r n a t e procedures outl i n e d i n the preceding s e c t i o n should be f o l l o w e d . Concluding Remarks In t h i s paper, we have developed the c r i t e r i a of c r i t i c a l i t y f o r multicomponent, c l a s s i c a l systems. The development u t i l i z e d Legendre transform theory described i n the f i r s t two papers of t h i s set (Beegle, et_ al., 1974a,b). I t was shown that the c r i t e r i a can be expressed i n terms of (n+1)-order determinants i n v o l v i n g second order p a r t i a l d e r i v a t i v e s of U [Eqs. (29) and ( 3 0 ) ] , s i n g l e second order p a r t i a l d e r i v a t i v e s of n - v a r i a b l e Legendre transforms of U [Eqs. (43) and ( 4 4 ) ] , or determinants of t w o - v a r i a b l e Legendre transforms [Eq.(73)]. The a l t e r n a t e s e t s presented by Gibbs a r e s p e c i a l cases of the g e n e r a l c r i t e r i a presented h e r e i n .

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

388

P H A S E EQUILIBRIA AND

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

Abstract The general criteria of criticality for a classical thermodynamic system are developed in terms of the internal energy representation of the Fundamental Equation, U, (S,V,N,...,N). Alternative criteria in other variable sets are derived using Legendre transformations. The criteria of criticality as stated by Gibbs are shown to be special cases of the general criteria. The development utilizes extensive potential functions with mole number variables rather than mole fractions. 1

n

Notation A = t o t a l Helmholtz f r e e energy A-£_|_2 determinant d e f i n e d above Eq. (17) B = determinant defined i n Eq. (9) 8 = determinant defined i n Eq. (26) #k = determinant defined i n Eq. (28) E_j_2. determinant d e f i n e d i n Eq. (30) G_ = t o t a l Gibbs f r e e energy H = t o t a l enthalpy Nj = moles of component j n = number of components i n mixture P = pressure R +1 = determinant d e f i n e d i n Eq. (7) S = determinant d e f i n e d i n Eq. (8) Si = t o t a l entropy T = temperature IJ = t o t a l i n t e r n a l energy V_ = t o t a l volume y(k) k - v a r i a b l e Legendre transform Zk = e x t e n s i v e parameter d e f i n e d i n Eq. z^ = e x t e n s i v e independent v a r i a b l e =

=

n

n

=

(16)

Greek L e t t e r s Uj £j

= chemical p o t e n t i a l of component j = i n t e n s i v e independent v a r i a b l e that i s the conjugate coordinate of Z j

Literature Cited 1. Gibbs, J. W., "On the Equilibrium of Heterogeneous Substances," Trans. Conn. Acad. I I I , 108 (1876). 2. Modell, M., and R. C. Reid, Thermodynamics and Its Applications,

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

19.

MODELL

Criteria of Criticality

389

Ch. 5, 7, Prentice-Hill, Englewood C l i f f s , N. J. (1974). 3. Beegle, B. L., M. Modell and R. C. Reid: (a) "Legendre Transformations and Their Application in Thermodynamics," AIChE J., 20, 1194 (1974); (b) "Thermodynamic Stability Criterion for Pure Substances and Mixtures," Ibid., 1200 (1974). 4. Heidemann, R. A., "The Criteria for Thermodynamic Stability," AIChE J., 21, 824 (1975). 5. Rowlinson, J. S., Liquids and Liquid Mixtures, 2nd Ed., Ch. 6. Butterworth, London (1969). 6. Teja, A. S., and H. W. Kropholler, "Critical States of Mixtures in which Azeotropic Behaviour Persists in the Critical Region," Chem. Engng Sci., 30, 435 (1975).

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

20 Thermodynamic Data Needs in the Synthetic Fuels Industry HOWARD G. HIPKIN Research and Engineering, Bechtel Corp., San Francisco, CA 94119

The petroleum industry most of the energy used society produced by product over the past 40 years, a broad base of thermodynamic data for the hydrocarbons it processes. Even though the petroleum industry is mature, the data development and correlation effort has not slacked off, and indeed has accelerated in the last two decades. A similar, but more proprietary, effort has been carried on by the chemical industry for nonhydrocarbons. In view of these long-lived programs, the question arises,--does the new synthetic fuels industry, which promises to become important in the last quarter of the century, need specific data programs or are the present data systems adequate for its needs? This paper looks at that question and attempts to outline areas where work is needed. As in the petroleum industry, the need is for thermodynamic properties of a small number of pure compounds, and for mathematical procedures to predict the properties of mixtures of those compounds. What are the S y n t h e t i c Fuels? The s y n t h e t i c f u e l s comprise a spectrum of gaseous, l i q u i d , and s o l i d f u e l s . Gaseous f u e l s i n c l u d e h i g h - , medium, and lowheating values gas from c o a l , s i m i l a r gases from f e r m e n t a t i o n or p y r o l y s i s of biomass, low-heating v a l u e gas from r e t o r t i n g shale o i l , and hydrogen. L i q u i f i e d n a t u r a l gas, w h i l e not a s y n t h e t i c , i s i n c l u d e d here because i t promises to be an important answer to the near-term energy shortage, and because the data needs f o r LNG are s p e c i f i c . High-Btu gas from c o a l i s methane, made by r e a c t i n g carbon monoxide and hydrogen. I t can r e p l a c e n a t u r a l gas i n a l l i t s a p p l i c a t i o n s without any equipment changes. I t s heat of combustion i s about 950 Btu per standard cubic f o o t . Medium-Btu gas i s the carbon monoxide and hydrogen from which high-Btu gas i s made. I t s heat of combustion i s about 300 Btu per standard cubic f o o t . Since methanation i s n o t r e q u i r e d , i t i s cheaper to produce than high-Btu 390

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

20.

HIPKIN

Thermodynamic Data Needs

391

gas; but i s not i n t e r c h a n g e a b l e w i t h n a t u r a l gas, and i t i s more expensive to t r a n s p o r t because of i t s l a r g e r volume. Low-Btu gas i s carbon monoxide and hydrogen d i l u t e d w i t h atmospheric n i t r o g e n . I t i s made by u s i n g a i r i n s t e a d of oxygen i n the c o a l g a s i f i e r . I t i s the cheapest gas to make from c o a l , but i t s heat of combustion i s only 150 Btu per standard cubic f o o t , and the v e r y l a r g e volume r e q u i r e d prevents i t s t r a n s p o r t a t i o n much away from the g a s i f i e r . The major use f o r low-Btu gas i s expected to be f o r combined c y c l e power g e n e r a t i o n (gas t u r b i n e f o l l o w e d by steam t u r b i n e ) . Gas made by p y r o l y s i s of biomass, or of m u n i c i p a l s o l i d waste, resembles c o a l s y n t e h s i s gas i n composition, and can be produced i n the same h e a t i n g v a l u e s . Gas made by anaerobic f e r m e n t a t i o n of biomass i s predominantly methane and carbon d i o x i d e . When the c a r bon d i o x i d e i s removed, i t i s a replacement f o r n a t u r a l gas. Depending on the metho s i t i o n and q u a n t i t y i s mad burned as i n - p l a n t f u e l , but i t could be processed f o r syngas. An i n t e r e s t i n g new development i s the p r o d u c t i o n of high-Btu gas by h y d r o t r e a t i n g o i l s h a l e . This p r o c e s s , under development by the I n s t i t u t e of Gas Technology, p r o v i d e s more complete u t i l i z a t i o n of the organic carbon content than simple r e t o r t i n g . Hydrogen, from decomposition of water, may become the dominant gaseous f u e l w e l l a f t e r the turn of the century. This concept i s the b a s i s f o r the s o - c a l l e d "hydrogen economy", which v i s u a l i z e s a v i r t u a l l y i n e x h a u s t i b l e supply of n o n - p o l l u t i n g gaseous energy from t h i s source. The hydrogen economy s u f f e r s from three problems; the high-energy, high-temperature heat source r e q u i r e d , the r e l a t i v e l y low c o n v e r s i o n e f f i c i e n c y of that energy, and the d i f f i c u l t y of s t o r i n g hydrogen. Since t h i s form of s y n t h e t i c f u e l i s u n l i k e l y to be important i n the next 20 or 30 y e a r s , i t w i l l not be c o n s i dered f u r t h e r i n t h i s paper. L i q u i f i e d n a t u r a l gas has been produced i n commercial quant i t i e s f o r a number of years i n the Middle East and A l a s k a . A number of p r o j e c t s f o r l i q u i f y i n g waste n a t u r a l gas i n the Middle East, Indonesia, and A l a s k a , s h i p p i n g the LNG to the U n i t e d S t a t e s , and r e g a s i f y i n g i t to augment our d i m i n i s h i n g s u p p l i e s a r e under a c t i v e development a t present. LNG appears to be the cheapest way to supply gaseous f u e l t o the American market. The energy r e q u i r e d to l i q u e f y and t r a n s p o r t the gas i s l e s s than 15% of the heating v a l u e of the i n i t i a l gas; as 'a r e s u l t , the energy e f f i c i e n c y of t h i s process i s h i g h e r than that of any of the other processes under consideration. L i q u i d s y n t h e t i c f u e l s i n c l u d e s h a l e o i l and i t s v a r i o u s f r a c t i o n s ; c o a l l i q u i d s by cooking, by hydrogenation, and by F i s c h e r tropsch s y n t h e s i s ; methyl f u e l from c o a l or from overseas n a t u r a l gas; and l i q u i d products from biomass, probably by g a s i f i c a t i o n and Fischer-Tropsch s y n t h e s i s . O i l shale i s one of the most p l e n t i f u l resources i n the U n i t e d S t a t e s . I t i s w e l l known that the r e s e r v e s of o i l shale i n the Piceance Creek B a s i n of Colorado and Utah exceed a l l the known

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

392

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

C H E M I C A L INDUSTRY

petroleum reserves i n the world. I t i s l e s s w e l l known that the reserves of o i l shale i n the deeper and leaner d e p o s i t s of the eastern s t a t e s are even l a r g e r (1) . Shale o i l i s produced by r e t o r t i n g o i l s h a l e , e i t h e r i n - s i t u , or on the s u r f a c e a f t e r mini n g . The v i s c o u s shale o i l r e q u i r e s a l i g h t hydrogenation to remove s u l f u r , n i t r o g e n , and oxygen, and then can be r e f i n e d l i k e a crude o i l . Simple p r y o l y s i s of c o a l produces l i g h t o i l s and t a r s , both predominantly aromatic i n c h a r a c t e r , together w i t h some gas and a r e s i d u e of s o l i d coke ( u s u a l l y high i n s u l f u r ) . The l i q u i d y i e l d can be i n c r e a s e d by hydrogenating the c o a l under pressure. The hydrogenation a l s o reduces the s u l f u r content of the s o l i d r e s i due, and e l i m i n a t e s the n i t r o g e n content as ammonia. An a l t e r n a t e p r o c e s s i n g scheme i s to g a s i f y the c o a l w i t h steam and oxygen to make hydrogen and carbon monoxide reacted to a spectrum o This mixture i s then r e f i n e d to make v a r i o u s grades of f u e l s and chemicals. At present, only the S a s o l p l a n t i n South A f r i c a uses t h i s Fischer-Tropsch process commercially. A s p e c i a l case of Fischer-Tropsch i s the p r o d u c t i o n of methanol, e i t h e r from g a s i f i e d c o a l or from reformed n a t u r a l gas (overseas). Crude methanol can be used as a f u e l i n gas t u r b i n e s , i n d u s t r i a l b o i l e r s , f u e l c e l l s , and i n t e r n a l combustion engines. I t represents a commercially a v a i l a b l e way to convert c o a l to a l i q u i d f u e l , and has the advantage (as compared to conventional c o a l l i q u i d s ) of e l i m i n a t i n g the r e f i n i n g s t e p . A number of p r o p r i e t a r y processes produce l i q u i d products by p y r o l y s i s of biomass. A l t e r n a t i v e l y , the biomass can be g a s i f i e d and reacted to l i q u i d products by F i s c h e r - T r o p s c h . S o l i d f u e l s are, i n most cases, the u n d e s i r a b l e residues from c o a l processes. Obviously, i f a s o l i d f u e l i s needed, c o a l i t s e l f would be the cheapest m a t e r i a l . The r e s i d u e from most g a s i f i c a t i o n and l i q u i f i c a t i o n processes i s high i n ash and u s u a l l y contains a h i g h e r percentage of s u l f u r than the o r i g i n a l c o a l . I f the combust i b l e content of the r e s i d u e i s low, i t i s d i s c a r d e d . I f , as i n coke, i t i s too l a r g e to be economically d i s c a r d e d , the s o l i d can be g a s i f i e d to produce the hydrogen r e q u i r e d f o r the process. An exception i s Solvent Refined C o a l , a process which produces a l o w - s u l f u r s o l i d f u e l w i t h a m e l t i n g p o i n t of about 350°F. The process a l s o produced a h i g h - s u l f u r , carbonaceous r e s i d u e . The s o l i d r e s i d u e l e f t from r e t o r t i n g o i l s h a l e c o n s i s t s of l a r g e amounts of rock w i t h enough carbon to support combustion. The carbon i s u s u a l l y burned to supply the heat r e q u i r e d f o r r e t o r t i n g . Under some circumstances, t h i s m a t e r i a l could be a s o l i d f u e l f o r other purposes. Most of these processes have a common c h a r a c t e r i s t i c — t h e y are i n c r e a s i n g the hydrogen content of a raw f u e l t h a t i s hydrogend e f i c i e n t compared to petroleum f u e l s . There are three ways i n which t h i s i s done:

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

20.

HIPKIN

Thermodynamic Data Needs

393

The hydrogenation processes add hydrogen d i r e c t l y C + 2H = CH, 2 4 The g a s i f i c a t i o n processes add hydrogen from water. The oxygen i n the water i s r e j e c t e d t o the atmosphere as carbon d i o x i d e . A s i m p l i f i e d r e p r e s e n t a t i o n o f the c o a l - t o - s y n t h e t i c n a t u r a l gas processes i s 2C + 2H 0 = CH, + C 0 . 2 4 2 o

o

I t can be seen that about h a l f the carbon i n the c o a l i s r e j e c t e d i n order t o upgrade the other h a l f . The simple p y r o l y s i residue and a high-hydroge low-hydrogen f u e l . The coking o f c o a l can be represented i n an o v e r s i m p l i f i e d way as 10.71

(CH

n

= C,H, + 4.71C.

The good y i e l d of benzene i n d i c a t e d by t h i s equation i s deceptive, because the c o a l a l s o contains oxygen, n i t r o gen, and s u l f u r , and these elements are d r i v e n o f f as water, ammonia, and hydrogen s u l f i d e , thus reducing the hydrogen a v a i l a b l e f o r forming hydrocarbons. I t i s not p o s s i b l e to f o l l o w the technology o f a l l the synt h e t i c f u e l processes i n a s i n g l e paper. Since c o a l g a s i f i c a t i o n f o r the production o f high-BTU gas i s the process on which most e f f o r t i s c u r r e n t l y concentrated, and s i n c e c o a l g a s i f i c a t i o n i s the f i r s t step i n s e v e r a l l i q u i f i c a t i o n processes, and s i n c e c o a l or char g a s i f i c a t i o n i s the most l i k e l y way o f producing hydrogen f o r a v a r i e t y o f other processes, the v a r i o u s c o a l g a s i f i c a t i o n processes w i l l be f o l l o w e d i n some d e t a i l . I n a d d i t i o n , t h i s paper takes a s h o r t e r look a t i n - s i t u shale r e t o r t i n g . How

are S y n t h e t i c Fuels Made?

This look a t s y n t h e t i c f u e l s i s r e s t r i c t e d t o c o a l g a s i f i c a t i o n , as mentioned above. The present generation of commercial g a s i f i c a t i o n processes i n c l u d e s three, a l l German, and a l l designed f o r chemical feedstock, r a t h e r than f o r f u e l . L u r g i . The L u r g i g a s i f i e r has had more i n s t a l l a t i o n s than any other, p o s s i b l y because i t has been the only pressure g a s i f i e r (up to 500 p s i ) . I t i s the g a s i f i e r used i n s e v e r a l of the c o a l g a s i f i c a t i o n f a c i l i t i e s proposed f o r American SNG p r o d u c t i o n . I t i s a f i x e d bed g a s i f i e r , i n which c o a l i s fed i n a t the top through l o c k hoppers, and ash i s removed a t the bottom, a l s o through l o c k hoppers. Steam and oxygen enter a t the bottom and pass up counter-

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

394

P H A S E EQUILIBRIA

A N D FLUID PROPERTIES

Figure 1.

IN C H E M I C A L INDUSTRY

Lurgi reactor

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

20.

HIPKIN

Thermodynamic Data Needs

395

current t o the c o a l . The s y n t h e s i s gas formed a t the bottom o f the r e a c t o r hydrogenates the c o a l i n the middle s e c t i o n s , then cokes and preheats the incoming c o a l . Because o f the c o u n t e r c u r r e n t o p e r a t i o n , the gas i s cooled t o about 1000°F b e f o r e i t leaves the r e a c t o r . The L u r g i r e a c t o r i s shown s c h e m a t i c a l l y i n F i g u r e 1. Since the c o a l i s heated g r a d u a l l y as i t passes s l o w l y down through the bed, the L u r g i r e a c t o r produces l i g h t o i l s , t a r s , and phenols, as w e l l as the u s u a l i m p u r i t i e s of hydrogen s u l f i d e and ammonia. A t y p i c a l L u r g i gas from bituminous c o a l has the f o l l o w ing dry composition: C0

2

CO

H S, 2

, 4 N , Ar, etc. 2

29 volume% 20

2

P u r i f i c a t i o n o f the gas from any c o a l g a s i f i e r i s a major task f o r s e v e r a l reasons: The gas c o n t a i n s up t o 50% o f unreacted water which, when condensed, i s contaminated w i t h s o l i d f l y a s h and carbon, d i s s o l v e d hydrogen s u l f i d e , ammonia, and c a r bon d i o x i d e , p o s s i b l y w i t h phenols. The water must be t r e a t e d t o remove these contaminants. The f i n e l y d i v i d e d s o l i d s , some o f them sub-micron i n s i z e , are p a r t i c u l a r l y d i f f i c u l t t o remove and, when removed, a r e a d i s p o s a l problem. I t i s always necessary t o remove hydrogen s u l f i d e , and u s u a l l y necessary t o remove carbon d i o x i d e . Any c l e a n up process which removes both a c i d gases d e l i v e r s a stream which i s too d i l u t e i n H2S to be handled i n a Claus p l a n t f o r elemental s u l f u r p r o d u c t i o n . As a r e s u l t , the gas i s u s u a l l y processed t w i c e , once f o r hydrogen s u l f i d e and l a t e r f o r carbon d i o x i d e , a t increased cost. The v a r i o u s p u r i f i c a t i o n steps o f t e n r e q u i r e the gas to be heated and cooled s e v e r a l times i n s u c c e s s i o n . Good heat economy r e q u i r e s fancy heat exchange between these streams, a t i n c r e a s e d c a p i t a l c o s t . With the L u r g i g a s i f i e r , the problem i s f u r t h e r complicated by the n e c e s s i t y to remove o i l s ( l i g h t e r than w a t e r ) , t a r s (heavier than w a t e r ) , phenols and cyanides ( d i s s o l v e d i n w a t e r ) , and other s u l f u r compounds (mercaptans, carbonyl s u l f i d e , and carbon d i s u l f i d e ) As a r e s u l t , the L u r g i gas p u r i f i c a t i o n s e c t i o n i s complicated and correspondingly expensive. F i g u r e s 2 through 8 show the p u r i f i c a -

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

Figure 2. Gasification

B.F. WATER

LOCK GAS

COMPRESSOR

LOCK GAS

GAS LIQUOR

TARRY GAS LIQUOR

CRUDE GAS

CRUDE GAS FILLING GAS

COAL

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

20.

HIPKIN

Thermodynamic Data Needs

397

t i o n scheme used by L u r g i f o r the proposed ANG Goal C a s i f i c a t i o n Company P l a n t ( 2 ) . The gas l e a v i n g the r e a c t o r i s quenched through a v e n t u r i n o z z l e w i t h a m i x t u r e of r e c i r c u l a t e d water, t a r , and l i g h t o i l t o about 800°F. The gas i s then cooled by p a s s i n g up through the tubes o f a v e r t i c a l heat exchanger which generates 100 p s i steam on the s h e l l s i d e . ( I t i s worth n o t i n g , i n p a s s i n g , that c o a l g a s i f i e r s consume l a r g e amounts o f steam,—something over two tons of steam f o r each ton of c o a l . ) The condensate from t h i s exchanger, a m i x t u r e o f t a r , o i l , and water, washes the tubes i n c o u n t e r c u r rent flow and keeps them from f o u l i n g . P a r t o f the condensate i s r e c y c l e d t o the v e n t u r i scrubber (Figure 2 ) . The gas from t h i s exchanger i s s p l i t i n t o two streams. One stream goes to a second c o u n t e r c u r r e n t v e r t i c a l heat exchanger which generates 60 p s i steam where the carbon monoxid v e r t e d t o carbon d i o x i d e , i n order to a d j u s t the r a t i o of carbon oxides to hydrogen f o r subsequent methanation (Figure 3 ) . The second stream bypasses the s h i f t c o n v e r t e r s , and goes to a s e r i e s of four v e r t i c a l exchangers which generate 60 p s i steam and 20 p s i steam, preheat b o i l e r feedwater, and f i n a l l y c o o l the gas w i t h c o o l i n g tower water. The gas from the s h i f t c o n v e r t e r s i s cooled through two b o i l e r feedwater exchangers i n s e r i e s , i s cooled i n an a i r c o o l e r , and f i n a l l y , i s cooled a g a i n s t c o o l i n g tower water, and i s compressed t o j o i n the bypassed stream. I n t h i s p r o c e s s i n g scheme, the gas i s c o u n t e r c u r r e n t l y scrubbed w i t h mixed-phase condensate f i v e times a t s u c c e s s i v e l y lower temperatures. This repeated c o n t a c t i n g , together w i t h the v e n t u r i scrub a t the react o r o u t l e t , removes almost a l l the p a r t i c u l a t e s i n the raw gas from the g a s i f i e r , and does i t i n a way that keeps the exchanger tubes from f o u l i n g w i t h t a r . The condensates from the h i g h temp e r a t u r e exchangers go to " t a r r y gas l i q u o r " treatment, and the condensate from the low temperature exchangers go to " o i l y gas l i q u o r " treatment (Figure 4 ) . At t h i s p o i n t , the gas i s e s s e n t i a l l y a t ambient temperature, and the condensate contains a l l the t a r and p a r t i c u l a t e s , a l l the ammonia and phenol, most o f the water and o i l , and some o f the hydrogen s u l f i d e and carbon d i o x i d e . The gas i s r e f r i g e r a t e d and t r e a t e d i n a R e c t i s o l u n i t t o remove hydrogen s u l f i d e . Rectisol i s L u r g i s tradename f o r a p r o p r i e t a r y process u s i n g r e f r i g e r a t e d methanol as a s o l v e n t . Methanol i s an e x c e l l e n t s o l v e n t f o r hydrogen s u l f i d e , c a r b o n y l s u l f i d e , carbon d i s u l f i d e , carbon d i o x i d e , hydrocarbons, and water. I t d i s s o l v e s hydrogen s u l f i d e pref e r e n t i a l l y to carbon d i o x i d e , and i t i s p o s s i b l e to p r o v i d e a concentrated hydrogen s u l f i d e stream from a R e c t i s o l u n i t . I n the design shown i n F i g u r e 5, the hydrogen s u l f i d e , w i t h some carbon d i o x i d e , i s removed f i r s t , then the gas a f t e r methanation i s scrubbed to remove the remaining carbon d i o x i d e . L i g h t o i l , hydrogen cyanide, and water are a l l recovered i n the R e c t i s o l u n i t . 1

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

Figure 3.

Shift conversion

J

I

TARRY GAS LIQUOR

BFW.

'

60 PSIG ^ STEAM

—>h—^—^

FIRST SHIFT REACTOR

SECOND SHIFT REACTOR

CONVERTED GAS WASTE HEAT EXCHANGER CRUDE GAS

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.

PSIG

H E A T

S T E A M

60

G A S

BFW

G A S

E X C H A N G E R

C O N V E R T E D

G A S

L I Q U O R

BFW

E X C H A N G E R

W A S T E

G A S

C O N V E R T E D

T A R R Y

C R U D E

T

1

G A S

E X C H A N G E R

C O N V E R T E D

BFW

{

H E A T J

E X C H A N G E R

W A S T E

BFW

60 PSIG S T E A M

T.

Zk

B F W

P R E H E A T

G A S

C O O L E R

Gas cooling

AIR

G A S

C O O L E R

C O N V E R T E D

E X C H A N G E R

Figure 4.

LP

AIR

C R U D E

i

G A S

G A S

CW

C O O L E R

F I N A L

CW

CW

C O O L E R

F I N A L

G A S C O M P R E S S O R

C O N V E R T E D B O O S T E R

y

G A S

G A S

A N D

G A S

G A S

L I Q U O R

O I L Y

C R U D E

C O N V E R T E D

F I L L I N G

L O C K

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

OFF GAS TO

TREATED WATER

HCN LIQUOR

WASH WATER

CRUDE AND CONVERTED GAS

SYNGAS TO METHANATION

S RECOVERY

Figure 5.

Rectisol unit

IMPURE WATER

METHANATED GAS

SNG

TO S. RECOVERY

EXPANSION GAS

20.

Thermodynamic Data Needs

HIPKIN

401

The condensates are combined, dropped i n p r e s s u r e to f l a s h o f f d i s s o l v e d gases, and separated by s e t t l i n g i n t o t a r (heavier than w a t e r ) , t a r o i l ( l i g h t e r than w a t e r ) , and contaminated water ( F i g ure 6 ) . The water i s t r e a t e d i n another L u r g i p r o p r i e t a r y p r o c e s s , the Phenosolvan process, to remove phenols and ammonia (Figure 7). The v a r i o u s s u l f u r streams are processed through a Claus p l a n t f o r treatment o f the more concentrated hydrogen s u l f i d e streams. In the Claus p l a n t , p a r t o f the H2S i s burned t o S 0 , which i s r e a c t e d w i t h the r e s i d u a l to make elemental s u l f u r a c c o r d i n g t o the r e a c t i o n 2

S

°2

+

2 H

2

S

=

3 S

+

2 H

2°

Since the t a i l gas from the Claus p l a n t s t i l l contains more s u l f u r than EPA r e g u l a t i o n s p e r m i t p r i e t a r y IFP process f o The low s u l f u r c o n c e n t r a t i o n s are processed i n a S t r e t f o r d u n i t , which a l s o produces elemental s u l f u r . A l l these s u l f u r removal f a c i l i t i e s are shown i n F i g u r e 8. I t can be seen that c l e a n i n g up the crude s y n t h e s i s gas from a L u r g i g a s i f i e r i s a complicated o p e r a t i o n . P a r t o r a l l o f the cost o f t h i s cleanup i s o f f s e t by the v a l u e o f the recovered byproducts,—ammonia, phenols, hydrogen cyanide, aroma t i c s , t a r o i l , t a r , and s u l f u r . The gas cleanup i n a W i n k l e r o r Koopers-Totzek g a s i f i e r i s considerably simpler. W i n k l e r . The Winkler r e a c t o r i s a f l u i d i z e d bed, as shown i n F i g u r e 9. The c o a l i s p u l v e r i z e d b e f o r e b e i n g fed to the bed, and the f l u i d i z i n g medium i s steam and oxygen, o r steam and a i r . The r e a c t o r has no i n t e r n a l moving p a r t s , i s q u i t e s i m p l e , and has a w e l l - e s t a b l i s h e d r e p u t a t i o n f o r r e l i a b i l i t y . The r e a c t o r must operate i n the non-slagging mode to keep the bed f l u i d i z e d , and so i s l i m i t e d to c o a l s w i t h reasonably h i g h ash f u s i o n temp e r a t u r e s . Since the o p e r a t i o n i s not c o u n t e r c u r r e n t , the gas would leave the r e a c t o r a t r e a c t i o n temperature, but an i n t e r n a l heat exchanger i n the gas space above the bed removes some heat and drops the gas temperature. The c h i e f disadvantage o f the Winkler r e a c t o r i s i t s atmospheric p r e s s u r e o p e r a t i o n , b u t the l i c e n s o r i s working on a pressure m o d i f i c a t i o n . Because the c o a l i s brought r a p i d l y to r e a c t i o n temperature and the p y r o l y s i s products are exposed to the f u l l temperature, the Winkler r e a c t o r produces no t a r s o r phenols, and very l i t t l e methane. A t y p i c a l gas composition from bituminous c o a l i s (dry and s u l f u r - f r e e ) : CO£

21 volume %

CO

33

H

41

2

CH

4

3

N , Ar, etc.

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.

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

LIQUOR

LIQUOR

LIQUOR

GAS

TARRY

GAS

GAS

OILY

WASH WATER

LIQUOR

LIQUOR

DUST

UNIT

SEPARATOR

REMOVAL

TAR

GAS

VESSEL

EXPANSION

GAS

LIQUOR

GAS

Figure 6. Gas liquor separation

TANK

LIQUOR

STORAGE

LIQUOR

SEPARATOR

FINAL GAS

SEPARATOR

GAS

4j~LoiL

—,

NH-J S C R U B B E R

GAS

LIQUOR

• EXPANSION GAS

20.

HIPKIN

Thermodynamic Data Needs

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

403

S T R E T F O R D UNIT

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

20.

HiPKiN

Thermodynamic Data Needs

Figure 9. Winkler gasifier

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

405

406

P H A S E EQUILIBRIA A N D

F L U I D PROPERTIES IN

CHEMICAL

INDUSTRY

Most of the ash leaves the r e a c t o r w i t h the gas, so that p a r t i c u l a t e removal i s a more d i f f i c u l t problem than w i t h the L u r g i reactor. Koppers-Totzek. In t h i s r e a c t o r , shown i n F i g u r e 10, f i n e l y powdered c o a l i s e n t r a i n e d w i t h the oxygen i n t o a burner s u r rounded by steam j e t s . The r e a c t i o n temperature i s about 2700°F, no t a r s , phenols, a r o m a t i c s , or ammonia are produced. The react i o n temperature i s above the f u s i o n p o i n t of the ash, and about h a l f the ash i s tapped o f f as a molten s l a g ; the r e s t i s c a r r i e d over w i t h the gas. A t y p i c a l dry and s u l f u r - f r e e gas composition from bituminous c o a l i s C0

H

2

2

CH. 4 N , Ar, etc. 2

12 volume %

33 0.2 1.5

The h i g h e r r e a c t i o n temperature r e s u l t s i n more CO and l e s s C 0 and CH4, as compared to the L u r g i or Winker r e a c t o r s . The K-T i s an atmospheric pressure r e a c t o r , but S h e l l i s working w i t h Koppers to develop a p r e s s u r i z e d model. The gas from the r e a c t o r i s quenched enough to s o l i d i f y the molten ash d r o p l e t s b e f o r e they reach the waste-heat b o i l e r , mounted d i r e c t l y above the g a s i f i e r . A spray washer cools the gas from 500°F to about 95°F a f t e r the b o i l e r . At t h i s temperature, the gas goes through two Thiessen d i s i n t e g r a t o r s , which mechanic a l l y a g i t a t e the gas w i t h water to remove more p a r t i c u l a t e s . A demister f o l l o w s the d i s i n t e g r a t o r s and, i f the gas i s to be compressed, the demister i s f o l l o w e d by e l e c t r o s t a t i c p r e c i p i t a t o r s . The gas i s then t r e a t e d to remove s u l f u r , u s i n g any of a number of d i f f e r e n t processes, i n c l u d i n g dry i r o n o x i d e , S u l f i n o l , or R e c t i s o l . I t i s then s h i f t converted and carbon d i o x i d e i s removed (Figure 11). Obviously, the gas p u r i f i c a t i o n r e q u i r e d f o r W i n k l e r or Koppers-Totzek i s s i m p l e r than that f o r L u r g i , although the problem of p a r t i c u l a t e removal i s worse. The condensates from these processes must s t i l l be t r e a t e d to remove contaminants. 2

New G a s i f i c a t i o n Processes There are about a dozen g a s i f i c a t i o n processes under development i n t h i s country and i n Europe. A number of them have reached the stage of l a r g e p i l o t p l a n t s . The one that i s probably c l o s e s t to c o m m e r c i a l i z a t i o n i s the Texaco p a r t i a l o x i d a t i o n process. Texaco and S h e l l have both l i c e n s e d p a r t i a l o x i d a t i o n processes f o r use w i t h a v a r i e t y of petroleum feeds s i n c e the l a t e 1940 s, and over 250 g a s i f i e r s have been i n s t a l l e d , l a r g e l y f o r making f

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

Figure 10. Koppers-Totzek gasifier In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

408

P H A S E EQUILIBRIA

Figure 11.

A N D F L U I D PROPERTIES

IN C H E M I C A L INDUSTRY

Koppers-Totzek gas cleanup system

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

20.

Thermodynamic Data Needs

HIPKIN

409

hydrogen as ammonia p l a n t feed. The process has been p a r t i c u l a r l y u s e f u l f o r g a s i f y i n g h i g h s u l f u r , h i g h m e t a l l i c content r e s i d s . R e c e n t l y , Texaco has t e s t e d through the p i l o t p l a n t stage, and has announced the a v a i l a b i l i t y f o r l i c e n s i n g o f a m o d i f i c a t i o n to gasify coal (3). The other processes under development, the main ones being Hygas, C0 -Acceptor, Synthane, Bi-Gas, COED, COGAS, Atgas-Patgas, Molten S a l t , B&W, and Exxon C a t a l y t i c , are designed w i t h three main o b j e c t i v e s i n mind: 2

to operate under pressure to c r e a t e modules l a r g e enough to supply gas f o r s y n t h e t i c f u e l s p r o d u c t i o n (a moderate s i z e s y n t h e t i c gas gasifiers, o to maximize p r o d u c t i o n o f methane, and to minimize p r o d u c t i o n of other byproducts, such as o i l s , t a r , phenols, hydrogen cyanide, e t c . The advantages of pressure o p e r a t i o n are that i t e l i m i n a t e s the need t o compress the gas as much, and that i t i n c r e a s e s the amount of methane i n the product from the g a s i f i e r , thus reducing the subsequent l o a d on the methanation f a c i l i t i e s . The advantages o f pressure o p e r a t i o n are most pronounced f o r p r o d u c t i o n o f high-BTU gas, but are a l s o c o n s i d e r a b l e f o r p r o d u c t i o n o f methanol, ammonia, and hydrogen f o r subsequent hydrogenation. The e q u i l i b r i u m o f the methane-forming r e a c t i o n s i s b e t t e r under p r e s s u r e , but o f more importance i s the f a c t that the heat l o a d (that i s , the amount o f oxygen required) i s reduced i f methane i s made d i r e c t l y r a t h e r than carbon monoxide and hydrogen. Shale O i l A r i c h o i l s h a l e from the Piceance Creek B a s i n runs 30 o r 35 g a l l o n s per ton ( F i s h e r a s s a y ) , and l i e s , g e n e r a l l y , deep i n the f o r m a t i o n , — f r o m 200 to 2000 f e e t deep. To supply a s m a l l r e f i n e r y of 100,000 b a r r e l s per day c a p a c i t y , 140,000 tons per day of t h i s r i c h s h a l e must be mined. This i s a l a r g e o p e r a t i o n , much b i g g e r than the l a r g e s t underground c o a l mines i n the country, and l a r g e r than any s i n g l e open-pit o p e r a t i o n i n the western s t a t e s . The only l a r g e r o p e r a t i o n on t h i s c o n t i n e n t i s the Syncrude t a r sands proj e c t i n northern A l b e r t a , a t 250,000 tons per day. Furthermore, the spent s h a l e from such a mine amounts t o about 125,000 tons per day, and i s a f i n e , dry dust occupying more volume than the o r i g i n a l o i l s h a l e . This dust has to be disposed of somehow, and t o keep i t from blowing a l l over the S t a t e o f Colorado, i t has to be soaked w i t h w a t e r , — i n an a r i d country. Since 90% o f the s h a l e i n the B a s i n l i e s below a reasonable l e v e l f o r s t r i p mining, and s i n c e shaft-and-tunnel mining i s c o n s i d e r a b l y more expensive than

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

410

P H A S E EQUILIBRIA A N D F L U I D PROPERTIES I N C H E M I C A L INDUSTRY

OIL RECOVERY

AIR MAKE-UP COMPRESSOR

OiL SHALE RUBBLE BARRIER

5.o.^iVvfo,

'^M 'g§o c

OIL SUMP AND PUMP

Figure 12.

Oxy oil shale process retort operation

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

20.

HIPKIN

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411

open-pit mining, there i s a r e a l i n c e n t i v e f o r an i n - s i t u r e t o r t ing o p e r a t i o n which e l i m i n a t e s the n e c e s s i t y to mine the s h a l e a t all. A number o f concepts f o r i n - s i t u r e t o r t i n g have been proposed. The best developed concept i s shown i n F i g u r e 12. A chamber i s mined out, e i t h e r i n the rock above o r below the formation, o r i n the upper o r lower o i l s h a l e s t r a t a , o r both, and s h a f t s are bored i n t o the s h a l e . Large amounts of c o n v e n t i o n a l e x p l o s i v e s a r e detonated throughout the s h a l e to break i t up. The formation i s thus rendered porous enough to permit the flow o f gas through i t , w h i l e the i n t a c t surrounding s h a l e w a l l s c o n f i n e the gas and pyrol y s i s products. Each such r e t o r t i s , t y p i c a l l y , 300 x 300 x 500 feet. In one concept, a i r i s pumped down through the formation w h i l e the upper s u r f a c e gas o r propane burners. Onc s e l f - s u s t a i n i n g . The hot gases i n f r o n t of the flame heat the shale to r e t o r t i n g temperature and p y r o l y z e i t , l e a v i n g a carbonaceous r e s i d u e which supports combustion when the flame reaches i t . The hot spent s h a l e behind the flame f r o n t preheats the a i r , and the c o l d u n r e t o r t e d s h a l e i n f r o n t of the r e t o r t i n g zone cools the combustion gases and (at l e a s t i n i t i a l l y ) serves to condense the l i q u i d products, which are c o l l e c t e d a t the bottom of the r e t o r t and pumped to the s u r f a c e . An i d e a l i z e d temperature g r a d i e n t f o r an a c t i v e r e t o r t i s shown i n F i g u r e 13. To m a i n t a i n continuous prod u c t i o n , s e v e r a l r e t o r t s are i n d i f f e r e n t stages o f development a t any given time, w i t h one being mined, another being r u b b l i z e d , and a t h i r d i n o p e r a t i o n . I n p r a c t i c e , the hot gases from a r e t o r t i n which the flame f r o n t i s approaching the bottom would most l i k e l y be d i r e c t e d up through a second completed r e t o r t w a i t i n g to be f i r e d , so that the l i q u i d products can be condensed without u s i n g c o o l i n g water. The problem w i t h t h i s scheme i s that the gas from such a r e t o r t has a low h e a t i n g v a l u e , — p r o b a b l y around 50 BTU per c u b i c f o o t , and p o s s i b l y approaching the lower flammable l i m i t . There i s an enormous amount o f t h i s gas, and i t cannot be d i s c a r d e d because: i t i s contaminated w i t h hydrogen s u l f i d e and ammonia i t represents an a p p r e c i a b l e f r a c t i o n of the h e a t i n g value o f the o i l s h a l e . For example, i n a goods i z e d o p e r a t i o n , the f u e l o i l e q u i v a l e n t o f the r e t o r t gas could be 20,000 b a r r e l s per day. To i n c r e a s e the h e a t i n g value o f the gas, and to reduce i t s volume, the gas can be r e c i r c u l a t e d from the r e t o r t through a heater and back t o the r e t o r t . I n t h i s case, no a c t u a l combustion occurs i n the r e t o r t , and the carbon r e s i d u e i s l e f t behind. P a r t o f the gas make i s used to f i r e the heater; the remainder i s a v a i l a b l e f o r steam g e n e r a t i o n , o r f o r s a l e . Obviously, the a d d i t i o n of the heaters i n c r e a s e s the c a p i t a l c o s t , but the i n c r e a s e d cost may be o f f s e t by the i n c r e a s e d v a l u e o f the h i g h BTU gas. A l l of the gas

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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RUBBLE P R E H E A T Z O N E

I

I

I

1

I

TEMPERATURE-

Figure 13.

Idealized temperature gradient for in-situ retorting

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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must be t r e a t e d f o r s u l f u r and ammonia removal; and the water produced from the r e t o r t must be t r e a t e d before discharge. The rock m a t r i x o f western o i l shale i s l a r g e l y carbonate, and during r e t o r t i n g i t i s p a r t l y decomposed. I n some cases, i t may be necessary t o scrub the carbon d i o x i d e from the gas to i n c r e a s e i t s h e a t i n g v a l u e . The base rock of eastern o i l shales is s i l i c a t e . The shale o i l produced i s a viscous m a t e r i a l that r e q u i r e d hydrogenation t o e l i m i n a t e s u l f u r , n i t r o g e n , and oxygen, to reduce i t s v i s c o s i t y , and t o improve i t s s t a b i l i t y . The hydrogenation w i l l be done a t the p r o d u c t i o n s i t e to improve the p r o p e r t i e s of the shale o i l before p i p e l i n i n g to a r e f i n e r y . Hydrogenated shale o i l i s a reasonable s u b s t i t u t e f o r crude o i l , and s t i l l r e q u i r e s refining. What are the Data Needs Three areas where more thermodynamic data are needed are f r e e energies and e n t h a l p i e s of formation; h i g h temperature and high pressure r e g i o n , p a r t i c u l a r l y w i t h p o l a r mixtures; and the cryogenic r e g i o n f o r c e r t a i n mixtures. Free Energies and E n t h a l p i e s o f Formation. Coal i s the b a s i s f o r s e v e r a l of the s y n f u e l s , and probably w i l l be the most important one d o m e s t i c a l l y . Coals vary t r e m e n d o u s l y , — f r o m l i g n i t e to a n t h r a c i t e ; and i n a l l o f i t s forms c o a l i s more r e a c t i v e than g r a p h i t e , which i s the thermodynamic standard s t a t e f o r carbon. In view of the very l a r g e d i f f e r e n c e s i n r e a c t i v i t y , the i n d u s t r y w i l l need b e t t e r heats and f r e e energies o f formation, probably as d i f f e r e n c e s between g r a p h i t e and carbon i n the c o a l , to p r e d i c t r e l i a b l e chemical e q u i l i b r i a and heat requirements. A f i r s t step i s a t a b u l a t i o n o f such data f o r a representat i v e v a r i e t y o f c o a l s . A second step i s the c o r r e l a t i o n o f these data a g a i n s t some e a s i l y measured p r o p e r t i e s o f c o a l , so that the f r e e energy and enthalpy of formation can be p r e d i c t e d f o r a new c o a l from some simple l a b o r a t o r y measurements. The same type of formation i s needed f o r s h a l e s , s i n c e i t can be a n t i c i p a t e d that shales from v a r i o u s sources w i l l vary c o n s i derably. The problem here i s complicated by the need to d i s t i n guish between organic and i n o r g a n i c carbon i n s h a l e s . A f u r t h e r c o m p l i c a t i o n , f o r both coals and s h a l e s , i s that the r e a c t i v i t y of the carbon v a r i e s as g a s i f i c a t i o n proceeds and that f o r accurate work the p r e d i c t i o n of t h i s v a r i a t i o n i s necessary. High-Temperature, High-Pressure Region. The g a s i f i c a t i o n processes are d i s t i n g u i s h e d from the more f a m i l i a r petroleum r e f i n i n g processes by much higher temperatures, and by the f a c t that mixtures c o n t a i n i n g p o l a r compounds are i n v o l v e d . Processing pressures are no higher than many encountered i n petroleum work, but the d i f f e r e n t types o f compounds handled add an e x t r a complexi t y t o the high pressure o p e r a t i o n s .

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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Very few r e f i n e r y operations operate about 800°F. The c o a l g a s i f i c a t i o n processes, on the other hand, s t a r t a t about 1300°F and some operate up to n e a r l y 3000°F. Many o f the new processes operate a t pressures of about 1000 p s i a . R e l i a b l e zero pressure e n t h a l p i e s up to these temperatures a r e a v a i l a b l e f o r most of the m a t e r i a l s found i n g a s i f i e r s , but the mixing r u l e s and pressure c o r r e c t i o n s which the trade has been using f o r nonpolar mixtures are probably inadequate. Another c o m p l i c a t i o n i s that the systems found i n g a s i f i e r s are r e a c t i n g systems a t high temperatures. I n a d d i t i o n to the heterogeneous r e a c t i o n s , the homogeneous gas phase has many p o s s i b l e r e a c t i o n s such as: CO + H 0 = C 0 + H 2

CH + H 0 = C 4

2

2NH = N 3

2

+ 3H

2

and these r e a c t i o n s a r e c a t a l y z e d by the ash components and a r e promoted by the l a r g e area s o l i d surfaces of the coke. For some designs, i t may be necessary to p r e d i c t the chemical e q u i l i b r i a i n these systems. V a p o r - l i q u i d e q u i l i b r i u m p r e d i c t i o n s i n these systems are p a r t i c u l a r l y i n t e r e s t i n g , and w i l l r e q u i r e some d i f f e r e n t techniques than the ones that the petroleum i n d u s t r y has used. At the high pressures of the second generation g a s i f i e r s , water s t a r t s to condense as the gas i s cooled to about 500°F, and continues to condense down to the lowest temperature, probably around 100 F. The condensate contains a p p r e c i a b l e q u a n t i t i e s of hydrogen s u l f i d e and carbon d i o x i d e , probably a l l of the ammonia, p o s s i b l y phenols and cyanides. Depending on the type of g a s i f i e r , a hydrocarbon phase may a l s o c o n d e n s e , — r a n g i n g from t a r to benzene. I n some u n l i k e l y circumstances, two hydrocarbon phases may separate, one l i g h t e r and one h e a v i e r than water. The mutual s o l u b i l i t y of the phases i s a f f e c t e d by the d i s s o l v e d components, a l l of which a r e s o l u b l e i n a l l the phases. And the e q u i l i b r i u m c a l c u l a t i o n s are f u r t h e r complicated by r e a c t i o n s i n the l i q u i d water phase, between the a c i d i c and b a s i c components. The L u r g i gas cleanup system, shown i n Figures 2, 3, and 4, i s a good example of the problems i n v o l v e d . Each of the f i v e counter-current exchangers represents a s e r i e s of complicated, simultaneous e q u i l i b r i u m and heat t r a n s f e r c a l c u l a t i o n s f o r a p o l a r mixture, w i t h a three- and p o s s i b l y four-phase system. ( I f the s o l i d f i n e s are considered, the system i s f o u r , p o s s i b l y f i v e phases; but the s o l i d phase, which most l i k e l y stays w i t h the t a r , i s g e n e r a l l y neglected.) N e i t h e r the a v a i l a b l e enthalpy data, nor the a v a i l a b l e e q u i l i b r i u m c o r r e l a t i o n s , a r e r e a l l y adequate f o r such m i x t u r e s , and the problems would be worse i f the pressures were h i g h e r , as they may be i n the f u t u r e . This i s not to say that

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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L u r g i cannot design such a system; o b v i o u s l y they have, the p l a n t s so designed do work. E i t h e r L u r g i uses p r o p r i e t a r y data, o r they design from past experience; i n e i t h e r case, t h e i r techniques a r e not a v a i l a b l e to the t e c h n i c a l p u b l i c . Nor i s i t known how much s a f e t y i s b u i l t i n t o t h e i r designs. Cryogenic Region. A t the low end of the temperature s c a l e , there are other data needs. Here, the designer needs r e l i a b l e data f o r a s m a l l number o f simple systems. And he needs data on s o l i d - l i q u i d phase r e l a t i o n s f o r a few m a t e r i a l s , notably carbon d i o x i d e , hydrogen s u l f i d e , and the higher hydrocarbons. The c h i e f area where these data (or c o r r e l a t i o n s ) are r e q u i r e d has been f o r l i q u i f i e d n a t u r a l gas p l a n t s . I t would appear t h a t the a v a i l a b l e data should be adequate s i n c e there i s a l o t o f i n f o r m a t i o n on the l i g h t hydrocarbons; w i l l be u n p l e a s a n t l y s u r p r i s e of methane gas a t low temperatures v a r i e s between v a r i o u s p r e d i c t i o n methods. He w i l l a l s o f i n d that i t i s d i f f i c u l t to p r e d i c t where carbon d i o x i d e c r y s t a l l i z e s out of the l i q u i d hydrocarbon phase as n a t u r a l gas i s cooled. Since deep r e f r i g e r a t i o n i s expensive, cryogenic p r o c e s s i n g r e q u i r e s accurate c o r r e l a t i o n s . Exxon has r e c e n t l y p u b l i s h e d a paper on t h e i r new c a t a l y t i c g a s i f i c a t i o n process, i n which c o a l impregnated w i t h potassium carbonate i s g a s i f i e d w i t h steam and oxygen a t a low temperature (1200-1300°F) and a pressure of about 500 p s i ( 4 ) . Under these c o n d i t i o n s , the p r o d u c t i o n o f methane i s maximized, and the overa l l r e a c t i o n i s almost i s e n t h a l p i c . The carbon d i o x i d e and methane are separated from the carbon monoxide and hydrogen, which are r e c y c l e d to the r e a c t o r . The s e p a r a t i o n of methane from carbon monoxide and hydrogen i s cryogenic and, f o r good economy, r e q u i r e s r e l i a b l e enthalpy and e q u i l i b r i u m data. Almost any cryogenic s e p a r a t i o n design becomes, a t some p o i n t , a balance between heat exchange costs and compression c o s t s . And the optimum (minimum t o t a l cost) u s u a l l y occurs a t exchanger temperature d i f f e r e n c e s of only a few degrees, sometimes only a f r a c t i o n o f a degree. To design exchangers to these c l o s e approaches r e q u i r e s very accurate e q u i l i b r i a data, and good e n t h a l py data. I f the p l a n t uses a mixed r e f r i g e r a n t , the need i s more pronounced. How

Good Do the Data Need to Be?

For the high temperature p o l a r mixtures a t h i g h p r e s s u r e s , the need i s p r i m a r i l y f o r p r e d i c t i o n methods that can be f a i r l y sloppy, but even more important i s some knowledge o f the degree of u n c e r t a i n t y i n the method. The design engineer faced w i t h c a l c u l a t i n g the enthalpy o f a mixture o f CO, CO2, H , and CH4 w i t h 50% steam a t 1000 p s i a and 1500°F w i l l probably take the four gases at zero pressure and 1500°F, use some g e n e r a l i z e d pressure c o r r e c t i o n to r a i s e the mixture to 500 p s i a , and add the enthalpy o f 2

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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pure steam a t 500 p s i a and 1500°F. Or, he may take a l l f i v e components at zero pressure (1 p s i a f o r the steam because he i s u s i n g steam t a b l e s ) and 1500°F, and use the g e n e r a l i z e d pressure c o r r e c t i o n to r a i s e the mixture to 1000 p s i a . I f he does both, he w i l l have two d i f f e r e n t answers. I f he uses d i f f e r e n t pressure c o r r e c t i o n s , he w i l l have more d i f f e r e n t answers. I f he i s knowl e d g a b l e , he w i l l worry a l i t t l e about the f a c t that the pressure c o r r e c t i o n i s based on l i g h t hydrocarbons, or on a i r . I f he checks, he w i l l f i n d t h a t the e f f e c t of pressure on the enthalpy of steam, from h i s steam t a b l e s , i s badly p r e d i c t e d by the genera l i z e d c o r r e l a t i o n . What he needs i s some method which t e l l s him that h i s c a l c u l a t e d enthalpy has a reasonable p r o b a b i l i t y of being o f f by, say, + 10%. Given that u n c e r t a i n t y , he can design enough s a f e t y f a c t o r i n t o the u n i t to take care of i t H i s problem occurs when he b e l i e v e o f f by 10%. L a t e r , as the s y n f u e l i n d u s t r y becomes more s o p h i s t i c a t e d , b e t t e r accuracy w i l l be needed. The heat flows around a l a r g e g a s i f i e r p l a n t are immense, and much of t h a t heat i s s u p p l i e d from expensive oxygen r e a c t i n g w i t h the c o a l . A high-BTU syngas p l a n t producing 250 m i l l i o n standard c u b i c f e e t per day of gas a l s o produces about 15,000 tons per day of carbon d i o x i d e , e q u i v a l e n t to a heat of combustion of 5 b i l l i o n BTU per hour. To a v o i d l a r g e and expensive s a f e t y f a c t o r s i n the design of such p l a n t s , a c c u r ate enthalpy methods w i l l be needed. For cryogenic s e p a r a t i o n s , i n d u s t r y already has the a p p r o x i mate methods. What i s needed now i s accurate data and methods to reduce the cost of unnecessary s a f e t y f a c t o r s on expensive deep r e f r i g e r a t i o n . The i n d u s t r y a l s o needs techniques to p r e d i c t s o l i d phase formation. When Are the Data Needed? The petroleum i n d u s t r y and the n a t u r a l gas p r o c e s s i n g indust r y operated f o r about 40 years b e f o r e any r e a l attempt to develop data and c o r r e l a t i o n s was made. They used crude approximations, such as Raoult's and Dalton's Laws, because they were good enough for the p r o c e s s i n g that the i n d u s t r y was doing a t t h a t time. As p r o c e s s i n g became more complex, b e t t e r data were needed and were d e v e l o p e d , — a trend that s t i l l continues. The s y n t h e t i c f u e l i n d u s t r y cannot undergo a comparable i n c u b a t i o n p e r i o d f o r i t s data requirements f o r two main reasons: 1. The s y n t h e t i c f u e l s are intended to augment petroleum f u e l s that are p r e s e n t l y produced by s o p h i s t i c a t e d proc e s s i n g techniques based on adequate thermodynamic data. 2. S y n t h e t i c f u e l p l a n t s w i l l be horrendously expensive, and there i s a l a r g e economic i n c e n t i v e to provide data good enough to e l i m i n a t e expensive s a f e t y f a c t o r s from the design.

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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Literature Cited 1. Musser, William N., and John H. Humphrey, "In-Situ Combustion of Michigan Oil Shale: Current Field Studies", Eleventh Intersociety Energy Conversion Engineering Conference, page 341 (1976). 2. "Joint Application of Michigan-Wisconsin Pipeline Company and ANG Coal Gasification Company for Certificates of Public Convenience and Necessity", Docket No. CP75-278 before the Federal Power Commission, Volume 1 (1975). 3. Crouch, W. G., and R. D. Klapatch, "Solids Gasification for Gas Turbine Fuel: 100 and 300 BTU Gas", Eleventh Intersociety Energy Conversion Engineering Conference, page 268 (1976). 4. Epperly, W. R., an tion for SNG Production", Eleventh Intersociety Energy Conversion Engineering Conference, page 249 (1976).

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

Discussion C. A. ECKERT

Two papers given i encountered by s c i e n t i s t under unusual c o n d i t i o n s . Mike M o d e l l s paper s t r e s s e d the d i f f i c u l t i e s thermodynamicists have i n d e a l i n g w i t h multicomponent systems i n the c r i t i c a l r e g i o n . He reviewed the c r i t e r i a f o r c r i t i c a l i t y i n multicomponent systems and developed the use of Legendre transformations to f i n d a s t a b l e p o i n t on the s p i n o d a l s u r f a c e . Much of the d i s c u s s i o n f o l l o w i n g the paper centered about whether one could use the same type of approach to determine the b i n o d a l s u r f a c e — t h a t i s , i n a p r a c t i c a l sense, f i n d the compos i t i o n of c o e x i s t i n g phases. Some comments on t h i s problem were as follows: H. Ted Davis, U n i v e r s i t y of Minnesota, "The s p i n o d a l c o n d i t i o n s are u s e f u l i n c o n s t r u c t i n g the b i n a r y and ternary phase d i a grams. M e i j e r i n g (_1) has used the s p i n o d a l e x t e n s i v e l y to l o c a t e c r i t i c a l p o i n t s f o r r e g u l a r s o l u t i o n s . Once the c r i t i c a l p o i n t i s l o c a t e d , then a simple numerical technique can be used to march along a b i n o d a l to c o n s t r u c t the b i n o d a l curve. Overlapping b i n o d a l s can then be used to l o c a t e three phases i n e q u i l i b r i u m , where such e x i s t . Such a process i s being c a r r i e d out f o r l i q u i d phase diagrams by J e f f K o l s t a d working w i t h C. E. S c r i v e n and me at Minnesota." John S. Rowlinson, U n i v e r s i t y of Oxford, United Kingdom, " I would l i k e to make two p o i n t s : I. By c o n c e n t r a t i n g on the s p i n o d a l s u r f a c e , as Modell and Reid's elegant transformations do, one runs the r i s k of overl o o k i n g other kinds of behavior on c r i t i c a l s u r f a c e s . For example, the s p i n o d a l curve may be o u t s i d e one of the b i n o d a l curves. This does happen w i t h the ternary diagram f o r which G i s q u a d r a t i c i n composition, a system which was analyzed f u l l y by M e i j e r i n g (_1). Here, there a r e three t r i c r i t i c a l p o i n t s , and i t i s the presence of these, which a r e s i n g u l a r i t i e s not envisaged i n G i b b s treatment, which would i n v a l i d a t e the use of the s p i n o d a l alone as a s o l e c r i t e r i o n of c r i t i c a l behavior. 2. The Gibbs (and r e l a t e d ) treatments assume that the extenE

f

418

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

ECKERT

419

Discussion

s i v e thermodynamic f u n c t i o n s form an a n a l y t i c s u r f a c e U = f(V,S,N^) around the c r i t i c a l p o i n t . This assumption c o n f l i c t s w i t h the known e x i s t e n c e of "weak" s i n g u l a r i t i e s at t h i s p o i n t (e.g., Cy oo) . In a mixture these s i n g u l a r i t i e s can g i v e r i s e to complic a t i o n s not, I t h i n k , encompassed i n M o d e l l s treatment. Thus, R. B. G r i f f i t h s and Wheeler (_2) have suggested that there are "anomalies" on a g a s - l i q u i d c r i t i c a l curve f o r a b i n a r y mixture, not only at the c r i t i c a l azeotrope but a l s o at p o i n t s where the c r i t i c a l curve passes through the extremum w i t h respect to changes of pressure or temperature." The second paper of the evening given by Howard H i p k i n of B e c h t e l Corporation complimented the f i r s t i n that i t s t r e s s e d , from a much more p r a c t i c a l p o i n t of view, s p e c i f i c needs that w i l l be encountered i n the near f u t u r e by i n d u s t r i a l designers d e a l i n g w i t h methods f o r energy what s y n t h e t i c f u e l s migh what we know about such processes now p r e d i c t e d what they indeed might be. He s t r e s s e d the need f o r heats of formation and Gibbs energies of formations at higher temperatures, e s p e c i a l l y f o r c o a l and o i l s h a l e ; f o r high temperature and pressure data e s p e c i a l l y for p o l a r mixtures; and the need f o r c a l c u l a t i o n a l methods f o r handling such data. As one example he held f o r t h the spectre to an a n a l y s t of a five-phase system emerging from a L u r g i g a s i f i e r . Rather extensive d i s c u s s i o n i n v o l v i n g a number of i n d i v i d u a l s f o l l o w e d d e a l i n g w i t h the imminent needs f o r new energy sources w i t h s m a l l l i k e l i h o o d of i t being s a t i s f i e d by n u c l e a r , s o l a r , or geothermal power. However, severe problems e x i s t i n the u t i l i z a t i o n of c o a l or o i l shale i n terms of high c a p i t a l requirements coupled w i t h the environmental r e s t r i c t i o n s on emissions from such p l a n t s . One q u i t e c o n s i d e r a b l e dilemma that becomes apparent i s that the production of energy from s y n t e h t i c f u e l s w i t h our current technology would only be p r a c t i c a l at c u r r e n t c a p i t a l c o s t s i f the s e l l i n g p r i c e of energy went up. However, i t i s q u i t e evident that c a p i t a l costs are l i n k e d to energy c o s t s . Thus, the consensus of the d i s c u s s i o n was that b e t t e r data are needed at higher temperatures and pressures, and i n the i n i t i a l stages, even r a t h e r "sloppy" data would be more u s e f u l than what i s now a v a i l a b l e . This w i l l r e q u i r e b e t t e r m a t e r i a l s and may lead to new techniques such as, f o r example, s u p e r c r i t i c a l s e p a r a t i o n processes and higher temperature processes. C e r t a i n l y the methods we now have f o r e s t i m a t i n g such p r o p e r t i e s w i l l prove inadequate, and new and b e t t e r methods w i l l c e r t a i n l y be r e q u i r e d . As i l l u s t r a t e d so g r a p h i c a l l y by the f i r s t t a l k of the evening these w i l l undoubtedly be more d i f f i c u l t to develop and apply than current methods. However, the general concensus was that i t i s q u i t e c l e a r that the energy c r i s i s w i l l soon be upon us i n a much more s e r i o u s sense than the general p u b l i c a p p r e c i a t e s , and we as s c i e n t i s t s and engineers must begin now to seek s o l u t i o n s i n terms of new data at more extreme c o n d i t i o n s and the thermodynamic framework w i t h i n which to apply them. 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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References 1. Meijering, J. L., Phillips Res. Rep. (1950) 5, 333; (1951) 6, 183. 2. Griffiths, R. B. and Wheeler, J. C. Phys. Rev. (1970) 2, 1047.

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

21 A Group Contribution Molecular Model for Liquids and Solutions Composed of the Groups and

CH , 3

CH , 2

OH,

CO

T. NITTA, E. A. TUREK, R. A. GREENKORN, and K. C. CHAO Purdue University, West Lafayette, IN 47907 Group i n t e r a c t i o n model the d e s c r i p t i o n of a c t i v i t t i o n s . Notable i n t h i s development are the p i o n e e r i n g work by P i e r o t t i , Deal and Derr (]L), Wilson and Deal (2), and subsequent c o n t r i b u t i o n s by S c h e l l e r (3), R a t c l i f f and Chao ( 4 ) , Derr and Deal ( 5 ) , and Fredenslund, Jones, and P r a u s n i t z ( 6 ) . N i t t a e t . a l . (7) extended the group i n t e r a c t i o n model to thermodynamic p r o p e r t i e s of pure p o l a r and non-polar l i q u i d s and t h e i r s o l u t i o n s , i n c l u d i n g energy of v a p o r i z a t i o n , pvT r e l a t i o n s , excess p r o p e r t i e s and a c t i v i t y c o e f f i c i e n t s . The model i s based on the c e l l theory w i t h a c e l l p a r t i t i o n f u n c t i o n d e r i v e d from the C a r n a h a n - S t a r l i n g equation of s t a t e f o r hard spheres. The l a t t i c e energy i s made up of group i n t e r a c t i o n c o n t r i b u t i o n s . An important advantage of the model by N i t t a e t . a l . i s i t s a p p l i c a b i l i t y over a wide temperature range. The same group parameters used i n the same equations have been found to g i v e good r e s u l t s at c o n d i t i o n s f o r which the c e l l model i s known to be a p p l i c a b l e — w h e r e the l i q u i d i s not "expanded", the reduced d e n s i t y i s g r e a t e r than two and the temperature i s not much above the normal b o i l i n g p o i n t . I t i s not necessary to have d i f f e r e n t s e t s of group parameter values f o r d i f f e r e n t temperatures. N i t t a e t . a l . (7) presented the p r o p e r t i e s of the groups CH3, CH2, OH, and CO and t h e i r i n t e r a c t i o n s . Comparisons of the model and experimental data were made f o r a number of pure l i q u i d s and their solutions. A d d i t i o n a l comparisons of s o l u t i o n p r o p e r t i e s w i t h the model c a l c u l a t e d values are presented here to cover the gamut of mixtures made up of the given groups from the non-polar/non-polar, through non-polar/polar, to p o l a r / p o l a r . F i g u r e 1 shows the p r e d i c t e d a c t i v i t y c o e f f i c i e n t s of n-hexane i n s o l u t i o n w i t h n-dodecane compared to experimental data by Broensted and Koefoed ( 8 ) . The same agreement i s obtained between our model and experimental data from the same source on the mixtures of other n-alkanes. F i g u r e 2 shows the p r e d i c t e d a c t i v i t y c o e f f i c i e n t s i n the system ethanol/n-octane at 75C i n comparison w i t h experimental 421

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

21.

NITTA E T A L .

Group Contribution Molecular Model

423

data by Boublikova and Lu ( 9 ) . The agreement i s the same as that p r e v i o u s l y reported by N i t t a e t . a l . (7) f o r the same system a t 45C w i t h data from the same source. There i s a remarkable change i n a c t i v i t y c o e f f i c i e n t s i n t h i s system i n the temperature i n t e r v a l of 30C. Thus the i n f i n i t e d i l u t i o n v a l u e of e t h a n o l i s reduced by a f a c t o r of about two w h i l e that of n-octane i s reduced by only about 10% w i t h t h i s temperature i n c r e a s e . This remarkable change i s q u a n t i t a t i v e l y described by the model. F i g u r e 3 shows the p r e d i c t e d a c t i v i t y c o e f f i c i e n t s i n n-butanol/n-heptane a t 50C i n comparison w i t h the experimental data of A r i s t o v i c h e t . a l . (10). The agreement that i s obtained here f o r the h i g h a l c o h o l i s about the same as the p r e v i o u s l y reported r e s u l t s (_7) f o r the lower a l c o h o l s . F i g u r e 4 shows the p r e d i c t e d excess enthalpy of n-butanol/nheptane a t 15 and 55C i Nguyen and R a t c l i f f (11) and d e v i a t i o n s up to 30 cal/g-mole are observed f o r some compositions. F i g u r e 5 through 8 show the a c t i v i t y c o e f f i c i e n t s i n the system n-hexane/2-propanone a t four temperatures 45, 20, -5, and -25C. Experimental data are from Schaefer and R a i l (12) a t 45 and - 2 0 C , and from R a i l and Schaefer (JL3) a t 20 and -25C. The v a r i a t i o n of the a c t i v i t y c o e f f i c i e n t s w i t h temperature appears to be quant i t a t i v e l y d e s c r i b e d by our model. The 45C isotherm was used i n the development of the p r o p e r t i e s of the CO group and the model i s t h e r e f o r e i n a sense f i t t e d to t h i s isotherm. But the other isotherms were not used i n the development of the model, and the c a l c u l a t i o n s f o r them are of a p r e d i c t i v e nature. F i g u r e 9 shows the a c t i v i t y c o e f f i c i e n t s i n the system 2-propanal/n-hexanol (14). The molecular i n t e r a c t i o n s i n t h i s p o l a r / p o l a r mixture i s complex l e a d i n g to an apparent maximum i n the f i g u r e . The e x i s t e n c e of the maximum i s c o r r e c t l y p r e d i c t e d by our model, but the c a l c u l a t e d v a l u e s seem to vary too r a p i d l y at s m a l l c o n c e n t r a t i o n s of acetone. There a l s o seems to be c o n s i d e r a b l e u n c e r t a i n t y and s c a t t e r i n g of the experimental data i n the same range. Acknowledgment This work was supported by N a t i o n a l Science Foundation through grants GK-16573 and ENG76-09190. D. W. Arnold a s s i s t e d i n the calculations.

Abstract The group contribution molecular model for liquids and solutions developed by Nitta et.al.is applied to properties of liquid solutions made up of the groups CH , CH , OH, and CO, and the results are compared with experimental data. A wide range of molecular species in mixtures is included over a wide temperature range. 3

2

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

21.

NITTA ET A L .

Group Contribution Molecular Model

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.2 .4 .6 MOLE FRRCTION N-HEXRNE

Figure 7. Activity coefficients in n-hexane-2-propanone at 5°C

n

r

~i

r

PREDICTED

cr CD

_l

Figure 8. Activity coefficients in n-hexane—2-propanone at 20°C

L_

.2 .4 .6 .8 MOLE FRRCTION N-HEXRNE

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

KO

21.

NITTA E T

AL.

Group Contribution Molecular Model

Figure 9. Activity coefficients in 2-propanone— n-hexanol 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.

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Literature Cited 1. Pierotti, G. J., Deal, C. H., and Derr, E. L., Ind. Eng. Chem. (1959) 51, 95. 2. Wilson, G. M., and Deal, C. H., Ind. Eng. Chem. Fundamen., (1962) 1, 20. 3. Scheller, W. A., Ind. Eng. Chem. Fundamen. (1965)4,459. 4. Ratcliff, G. A., and Chao, K. C., Canad. J. Chem. Eng. (1969) 47, 148. 5. Derr, E. L. and Deal, C. H., Distillation 1969, Sec. 3, p. 37, Brighton, England: Intern. Conf. Distillation, Sept. 1969. 6. Fredenslund, Aa., Jones, R. L., and Prausnitz, J. M., AIChE J. (1975) 21, 1086. 7. Nitta, T., Turek, E. A., Greenkorn, R. A., and Chao, K. C., AIChE J. (1977) 23, 8. Broensted, J. N., an (1946) 22, No. 17, 1. 9. Boublikova, L., and Lu, B. C. -Y., J. Appl. Chem. (1969) 19, 89. 10. Aristovich, V. Y., Morachevskii, A. G. and Sabylin, I. I., J. Appl. Chem. USSR (1965) 38, 2633. 11. Nguyen, T. H. and Ratcliff, G. A., J. Chem. Eng. Data (1975) 20, 252. 12. Schaefer, K. and Rall, W., Z. Elektrochem. (1958) 62, 1090. 13. Rall, W., and Schaefer, K., Z. Elektrochem. (1959) 63, 1019. 14. Rao, P. R., Chiranjivi, C., and Dasarao, C. J., J. Appl. Chem. (1967) 17, 118.

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