phase equilibria and fluid properties in the chemical industry estimation and correlation

Phase Equilibria and Fluid Properties in the Chemical Industry Estimation and Correlation January 16-21, 1977 AMERICAN CHEMICAL SOCIETY WASHINGTON, D... Library of Congress CIP Data Ph

Phase Equilibria and Fluid

Properties in the Chemical Industry Estimation and Correlation

January 16-21, 1977

AMERICAN CHEMICAL SOCIETY WASHINGTON, D C 1977

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

Ther-modynamics—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 660.2'9'63 77-13804

ISBN 0-8412-0393-8 ACSMC8 60 1-537 (1977)

Copyright © 1977

American Chemical Society

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PRINTED IN THE UNITED STATES OF AMERICA

ACS Symposium Series

Robert F Gould, Editor

Advisory Board

Donald G Crosby Jeremiah P Freeman

E Desmond Goddard Robert A Hofstader John L Margrave

N i n a I McClelland John B Pfeiffer Joseph V Rodricks

A l a n C Sartorelli Raymond B Seymour Roy L Whistler

A a r o n W o l d

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 sub- mitted 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

DEDICATION

This work is dedicated to the memory of three men who contributed

to our understanding of fluid properties

Ping L Chueh Shell Development Co

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

PREFACE

We had two goals in organizing this conference The 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 proper- ties." The reviews should represent the work of both the academic re- searcher and the industrial practitioner This we thought was both neces- sary 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 discus- sions 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 vide a listing of important questions and research areas that need atten- tion now

pro-The 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 authori- ties, presently active in physical properties work, were chosen to be speakers, panel members, session reporters and session chairmen The conference was held at the Asilomar Conference Grounds on the Monte- rey 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 academio- industrial dialogue which was initiated Similarly, the unusual enthu- siasm 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

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 theo- retician today, but further development and reduction to practice should

conduc-be done, especially for liquids

• Real difficulties remain when attempts are made to predict, to extrapolate, or even to interpolate data for multicomponent mixtures con- taining 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 The goal for the future is to reduce these problems from nightmare to 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 dis- criminating theoretical comparisons should have high priority Signifi- cantly, several conferees stated that their primary sources of new experi- mental 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 engi- neers and whether the ideas expressed influence research in the coming years

xii

A cknou/ledgments

This volume is based on the Engineering Foundation Conference,

"The Estimation and Correlation of Phase Equilibria and Fluid ties 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 The advice, financial and moral support, and the concern for local arrangements, publicity, registration by Sand- ford Cole, Harold Commerer, Dean Benson and their staff permitted us

Proper-to concentrate on the technical aspects of the meeting Manuscript typing was done by the University of Missouri, Stenographic Services Depart- ment

Major funding for the conference by the National Science tion was a key ingredien

Founda-for many American and

have been otherwise unable to participate The 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 tributions by co-sponsoring and publicizing the conference

con-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 Lyman Yarborough of the Amoco Production Co 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 did their work diligently and in the best scientific tradition; and the conferees for their enthusiastic participation and impor- tant discussion contributions that made this conference special

T S STORVICK

University of Missouri—Columbia

June 1977

STANLEY I SANDLER University of Delaware

xiii

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

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 mechanics it became possible to late rather accurately the entropies of various light hydrocarbons (2) Fred Rossini and I collaborated in bringing together his heat

calcu-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

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 g i v e n 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 t h a t 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 s i x t h - p o w e r 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 v e r y 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 a t

l o n g e r d i s t a n c e s Even w i t h o u t 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 s h o r t e r - r a n g e terms, one c o u l d expect t h a t a

u n i v e r s a l f u n c t i o n might be a good a p p r o x i m a t i o n 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 t h a t 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 s h o u l d r e c a l l t h a t 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 e q u a t i o n of s t a t e from t h i s model One c o u l d o n l y show

t h a t i t y i e l d e d c o r r e s p o n d i n

s t a t e i n terms of the reduce

p r e s s u r e

One c o u l d p o s t u l a t e o t h e r models which would y i e l d a c o r r e s

-p o n d i n g - s t a t e s b e h a v i o r but d i f f e r e n t from t h a t of the s i m -p l e f l u i d However, most such m o l e c u l a r 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 e q u a t i o n s Rowlinson (6) found a somewhat more

g e n e r a l case; he showed t h a t f o r c e r t a i n types of a n g u l a r l y dent 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 f u n c -

depen-t i o n s a r o s e

I had observed e m p i r i c a l l y , however, t h a t the f a m i l y r e l a t i o n

-s h i p of e q u a t i o n -s of -s t a t e wa-s much broader even than would f o l l o w from R o w l i n s o nfs model I t i n c l u d e d 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 m o l e c u l e s such as tetramethylmethane (neopentane), where

no a p p r e c i a b l e a n g u l a r 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 e l o n g a t e d mole-cules su-ch as -carbon d i o x i d e the a n g u l a r 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 i m p o r t a n t as t h a t 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 t h a t the Lennard-Jones 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 m o l e c u l a r c e n t e r s 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 c o r e And I was a b l e to show t h a t one o b t a i n e d i n good a p p r o x i m a t i o n 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 c o r e s of a l l r e a s o n a b l e shapes By a s i n g l e f a m i l y I mean t h a t 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 e q u a t i o n f o r any

p a r t i c u l a r case W h i l e t h i s d i d not prove t h a t a l l of the complete

e q u a t i o n s 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 n u m e r i c a l 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 s t u d e n t s 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 m o l e c u l e s

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 a p p r o x i m a t i o n , they a r e easy to t r e a t t h e o r e t i c a l l y Why a r e n ' t they s i m p l e f l u i d s ? Many t h e o r e t i c a l papers i g n o r e 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 d e p a r t s from the s i m p l e 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

1 PITZER Origin of the Acentric Factor 3

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 o p e r a t e s between each p a r t of the m o l e c u l e r a t h e r than between m o l e c u l a r c e n t e r s Thus the a t t r a c t i v e term i s s t e e p e r 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 m o l e c u l a r

c e n t e r s T h i s i s shown i n F i g u r e 1, taken from my paper (8) i n 1955

We need not b o t h e r 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 d o t t e d and dashed curves f o r the g l o b u l a r m o l e c u l e The impor-

t a n t f e a t u r e i s the narrowness o f 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 m o l e c u l e s of a

p o l a r o r weakly p o l a r m o l e c u l e s 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 t h a t time o n l y t o c o n s i d e r the second v i r i a l c o e f f i -

c e n t The K i h a r a model was used f o r nonpolar m o l e c u l e s of a l l shapes

t u r e which i s a c o n v e n i e n t 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 t h a t the non-polar core m o l e c u l e s f a l l

a c c u r a t e l y on a s i n g l e c u r v e (indeed a s t r a i g h t l i n e ) W h i l e t h e

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 r e a s o n a b l e s t a n d a r d of a c c u r a c y a t t h a t t i m e For son the y v a l u e s o f 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 a r e 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

compari-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 t h a t 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 e x p e r i m e n t a l b a s i s f o r the t h i r d parameter The vapor p r e s s u r e i s the p r o p e r t y 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 o f the p r o p e r t i e s most

( i )

4 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Figure 1 Intermolecular potential for molecules of a simple

fluid, solid line; and for globular molecules such as C(CH 3 ) If>

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

6 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

w i d e l y measured a t 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 d a t a and the a c c u r a c y of the d a t a 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 S i n c e the c r i t i c a l

data have to be known f o r a reduced e q u a t i o n of s t a t e , the reduced

vapor p r e s s u r e near the normal b o i l i n g p o i n t was an easy c h o i c e f o r

the new parameter The a c t u a l d e f i n i t i o n

w i t h Pr the reduced vapor p r e s s u r e a t Tr = 0.700 seemed c o n v e n i e n t ,

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 p r e s s u r e

v a l u e well-removed from the c r i t i c a l p o i n t

Here I s h o u l d note the work of R i e d e l (9) which was s u b s t a n

-t i a l l y s i m u l -t a n e o u s w i -t h mine bu-t 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 p r e s s u r e 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 t o 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 , a l t h o u g h 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 t h e f u l l a r e a 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 t h a t 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

r e v i e w e r , 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

c h o i c e i n i t i a l l y The c o n c e p t u a l 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 m o l e c u l e s f o l l o w a s i m p l e

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 m o l e c u l e S i n c e 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

m o l e c u l e s 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

a p p r o p r i a t e

I t i s assumed t h a t the c o m p r e s s i b i l i t y f a c t o r and o t h e r p r o p e r

-t 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 -t h a -t

The p r e f e r e n c e of Pr over Vr 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 p r e s s u r e 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 d a t a 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

1 PITZER Origin of the Acentric Factor

i Q \

C H 4 C H 4

C 3 H 8 Figure 4 Intermolecular forces operate

between the centers of regions of stantial electron density These centers are the molecular centers for Ar and

sub-CH 2 groups in C 3 H 8 —hence the name acentric factor for the forces arising from points other than molecular centers

C) and H t O(Q)

8 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

reduced temperature and p r e s s u r e i s p l o t t e d a g a i n s t u) The most

i m p o r t a n t r e s u l t appears o n l y 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 9 C6H6 > a n d C02 a r e so n e a r l y e q u a l 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

m o l e c u l e s 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 c o r r e s p o n d i n g 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 t h a t the h i g h l y p o l a r

m o l e c u l e s NH3 and H2O d e p a r t from the system Furthermore the dence 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

depen-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 e x p e r i m e n t a l data and r e p o r t e d a l l of our r e s u l t s i n

numeri-c a l t a b l e s ( 1 0 ) At t h a t time the b e s t a n a l y t i numeri-c a l e q u a t i o n 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 p a r a -meters 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 e x p e r i m e n t a l

a c c u r a c y Bruce Sage suggeste

f o r the normal p a r a f f i n s b o t 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 t h a t the a c e n t r i c

f a c t o r system was a g r e a t advance over the s i m p l e p o s t u l a t e of

c o r r e s p o n d i n g s t a t e s , but the f i n a l agreement was i n f e r i o r t o t h a t

o b t a i n e d by g r a p h i c a l and n u m e r i c a l methods

Thus we c o n t i n u e d w i t h n u m e r i c a l methods f o r the f u g a c i t y ,

e n t r o p y , and e n t h a l p y f u n c t i o n s ( 1 3 ) , a l t h o u g h we d i d p r e s e n t an

e m p i r i c a l e q u a t i o n f o r the second v i r i a l c o e f f i c i e n t ( 1 4 ) T h i s

work was done by Bob C u r l ; he d i d an e x c e l l e n t j o b 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 v e r y t i r e s o m e Thus I was

t h i s more d e t a i l e d work, but I do want t o note the improved e q u a t i o n

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 T h i s e q u a t i o n

d e a l s 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 t h e r e 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 t o 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 Q u i n l a n (20) t r e a t e d m i x t u r e s a t h i g h 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

1 PITZER Origin of the Acentric Factor 9

of l i q u i d s R e i d 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 c o n s t a n t s f o r many substances

On t h e t h e o r e t i c a l s i d e , one g r e a t advance has been i n the development o f p e r t u r b a t i o n t h e o r i e s o f a g e n e r a l i z e d van der Waals type Here one assumes t h a t the m o l e c u l a r 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 c o r e s Then b o t h t h e s o f t n e s s of the c o r e s 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

o u t s t a n d i n g work Rigby (24) a p p l i e d these modern Van der W a a l sf methods t o n o n - s p h e r i c a l molecules which r e p r e s e n t one type o f

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 B e r e t and P r a u s n i t z (25) developed e q u a t i o n s 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 t h e i n i t i a l d e p a r t u r e s from s i m p l e

f l u i d s t o t h e 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 m o l e c u l e s These c o u l d be modeled by K i h a r a

poten-t i a l s w i poten-t h s p h e r i c a l cores o r by o poten-t h e r p o poten-t e n poten-t i a l s a l l o w i n g poten-the w e l l

to be narrowed The g r e a t advantage would be t h e r e t e n t i o n of

spher-i c a l symmetry and spher-i t s t h e o r e t spher-i c a l s spher-i m p l spher-i c spher-i t y Rogers and P r a u s n spher-i t z (26) made an i m p o r t a n t b e g i n n i n g 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 t h e p r o p e r t i e s s t u d i e d While they do n o t

d i s c u s s these r e s u l t s i n terms o f the a c e n t r i c f a c t o r , the t r a n s

-f o r m a t i o n o -f 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 a b l e to t r e a t

m i x t u r e s v e r y s u c c e s s f u l l y a l t h o u g h those c a l c u l a t i o n s were some 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

burden-u s i n g s p h e r i c a l models f o r g l o b burden-u l a r molecburden-ules woburden-uld be f r burden-u i t f burden-u l The move t o an a n a l y t i c a l e q u a t i o n by Lee and K e s s l e r was

undoubtedly a w i s e one i n view of the marvelous c a p a c i t y o f modern computers t o d e a l w i t h complex e q u a t i o n s 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 e q u a t i o n s

There remains the q u e s t i o n of the u l t i m a t e accuracy o f t h e

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 t h e 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 c o r r e s p o n d i n g

s t a t e s ? A p p a r e n t l y t h i s a c c u r a c y i s w i t h i n e x p e r i m e n t a l e r r o r f o r most, i f n o t a l l , p r e s e n t d a t a 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 e n g i n e e r i n g 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 p l e a s u r e to r e v i e w these a s p e c t s of the " a c e n t r i c

f a c t o r " w i t h you and I l o o k forward to your d i s c u s s i o n o f r e c e n t advances i n these and o t h e r a r e a s

L i t e r a t u r e C i t e d

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

10 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

2 P i t z e r , K S., J Chem Phys., (1937) 5, 469, 473, 752; (1940)

3 R o s s i n i , F D., P i t z e r , K S., Arnett, R L., Braun, R M and Pimentel, G C., "Selected Values of the P h y s i c a l and Thermo- dynamic Properties of Hydrocarbons and Related Compounds,"

Carnegie Press, Pittsburgh (1953)

4 Scott, D W., J Chem Phys (1974) 60, 3144

5 P i t z e r , K S., J Chem Phys (1939) 7, 583

6 Rowlinson, J S., Trans Faraday Soc (1954) 50, 647; "Liquids and Liquid Mixtures," 2nd ed Chapter 8, Butterworth, London (1969)

7 Kihara, T., Rev Mod Phys (1953) 25, 831 and papers there

11 Benedict, M., Webb, G B and Rubin, L C., J Chem Phys

16 Danon, F and P i t z e r , K S., J Chem Phys (1962) 36, 425

17 Lee, B I and Kesler, M G., A.I.Ch.E Journal (1975) 21, 510

18 Tsonopoulos, C., A.I.Ch.E Journal (1974) 20, 263

19 Lyckman, E W., Eckert, C A and Prausnitz, J M., Chem Engr

22 Reid, R C and Sherwood, T K., "The Properties of Gases and

L i q u i d s , " 2nd ed., McGraw-Hill Book Co., New York (1966)

23 Barker, J A and Henderson, D., Rev Mod Phys (1976) 48, 587

24 Rigby, M., J Phys Chem (1972) 76, 2014

25 Beret, S and Prausnitz, J M., A.I.Ch.E Journal (1975) 21,

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 i t 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 exer- cising 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, a l l 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 i t 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

12 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Thermodynamics: Not Magic b u t a T o o l

A l l t o o o f t e n , when I t a l k w i t h c h e m i c a l engineers from i n d u s t r y who have l i t t l e e x p e r i e n c e i n thermodynamics, I o b t a i n the i m p r e s s i o n

t h a t they l o o k upon me as a medicine man, a m a g i c i a n 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 o f

n o t h i n g T h i s audience knows b e t t e r b u t n e v e r t h e l e s s , we must remind

o u r s e l v e s t h a t thermodynamics i s not magic, t h a t i t i s o n l y 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 v a l u e o f a d e s i r e d e q u i l i b r i u m p r o p e r t y ; i n s t e a d ,

i t t e l l s us how t h e d e s i r e d e q u i l i b r i u m p r o p e r t y i s r e l a t e d t o some

o t h e r e q u i l i b r i u m p r o p e r t y Thus thermodynamics p r o v i d e s us w i t h a

t i m e - s a v i n g bookkeeping system: we do n o t have t o 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 o n l y 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 m i x i n g v a r i e s w i t h temperature,

we need n o t measure the e n t h a l p y of m i x i n g 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 e q u a t i o n , 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 compo-

s i t i o n , we can use the Gibbs-Duhem e q u a t i o n t o c a l c u l a t e the o t h e r

We must keep reminding o u r s e l v e s and o t h e r s as t o j u s t what dynamics can and cannot do F a l s e e x p e c t a t i o n s o f t e n l e a d t o c o s t l y

thermo-d i s a p p o i n t m e n t s

W h i l e t h e l i m i t a t i o n s o f 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 d i s c u s s e d l a t e r i n the week; f o r now,

I j u s t want t o say t h a t even a t t h i s e a r l y s t a g e , simple m o l e c u l a r

i d e a s can do much t o s t r e t c h the range o f a p p l i c a t i o n of

thermo-dynamics When thermodynamics i s coupled w i t h the m o l e c u l a r theory

of m a t t e r , we can c o n s t r u c t u s e f u l models; w h i l e these o n l y r o u g h l y approximate t r u e m o l e c u l a r b e h a v i o r , 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 c o n f i d e n c e , thereby r e d u c i n g

f u r t h e r t h e e x p e r i m e n t a l 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 m o l e c u l a r thermodynamicist

do, I answer w i t h a n a i v e b u t e s s e n t i a l l y a c c u r a t e analogy: I am a greedy t a x 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 t h e l a r g e s t p o s s i b l e revenue

Keeping i n mind t h a t 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 p h a s e - e q u a l i b r i u m work, thermodynamics p r o v i d e s us w i t h two

p r o c e d u r e s , as shown i n F i g u r e 1 Our aim i s t o 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 u s i n g method ( a ) , based e n t i r e l y on an equa-

t i o n o f s t a t e a p p l i c a b l e t o both phases a and $, o r u s i n g method

( b ) , w h i c h uses an e q u a t i o n o f s t a t e o n l y f o r c a l c u l a t i n g the phase f u g a c i t y and a c o m p l e t e l y d i f f e r e n t method, expressed by the

vapor-a c t i v i t y c o e f f i c i e n t , f o r c vapor-a l c u l vapor-a t i n g condensed-phvapor-ase f u g vapor-a c i t i e s

I now want t o 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

2 PRAUSNITZ Review of Phase Equilibria 13

FOR EVERY COMPONENT i , IN PHASES a AND 0

y,x = COMPOSITION; ° = STANDARD STATE

0 = FUGACITY COEFFICIENT (FROM EQUATION OF STATE)

T = ACTIVITY COEFFICIENT

- J d "l ^1 TOTAL VOLUME

Figure 1 Two thermodynamic methods for calculation of fluid-phase equilibria

2 PRAUSNITZ Review of Phase Equilibria 15

When e n c o u n t e r i n g 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 v e r y

f i r s t d e c i s i o n i s to d e c i d e which of these methods i s most s u i t a b l e

f o r 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 r e v i e w the

r e l a t i v e advantages and d i s a d v a n t a g e s of both methods; these a r e summarized i n F i g u r e 2

The s t a t e o f the a r t today i s such t h a t f o r m i x t u r e s o f s i m p l e ,

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 a t h i g h 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 e q u a t i o n of s t a t e However, f o r m i x t u r e s

i n c l u d i n g one o r more s t r o n g l y p o l a r o r 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 s t a n d a r d

-s t a t e f u g a c i t i e -s

As i n d i c a t e d i n F i g u r e 2, an e q u a t i o n of s t a t e f o r a l l f l u i d phases has many advantages because one v e r y troublesome f e a t u r e

v i z s p e c i f y i n g a s t a n d a r

troublesome because we f r e q u e n t l y multicomponen

m i x t u r e s 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 t h a t event, the c h o i c e o f a p r o p e r l y d e f i n e d a c t i v i t y c o e f f i c i e n t and

c i e n t p r o v i d e s the most convenient t o o l we have but our fundamental knowledge about i t i s s p a r s e Thermodynamics g i v e s us l i t t l e h e l p ;

we have t h r e e well-known r e l a t i o n s : f i r s t , the Gibbs-Duhem e q u a t i o n 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 e q u a t i o n which r e l a t e s the e f f e c t o f temperature on the a c t i v i t y c o e f f i c i e n t

to the e n t h a l p y o f m i x i n g and f i n a l l y , an e q u a t i o n 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 o f p r e s s u r e 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 t h a t

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 -t o o -t h e r s , thereby r e d u c i n g e x p e r i m e n t a l work But the p r a c t i c a l a p p l i c a t i o n s

we can c a l c u l a t e not o n l y the f u g a c i t i e s from v o l u m e t r i c d a t a but

a l s o a l l the o t h e r 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 e n t h a l p y , entropy and volume change on m i x i n g

Our i n a b i l i t y to use e q u a t i o n s 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 u n d e r s t a n d i n g 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 s i m p l e 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 e q u a t i o n 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 o n l y f o r the more

16 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

common f l u i d s do we have s u f f i c i e n t e x p e r i m e n t a l 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 w i d e r c l a s s of f l u i d s but a g a i n , we are l i m i t e d here because

-c o r r e s p o n d i n g s t a t e s -cannot e a s i l y be extended to p o l a r or bonding 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 t h a t we have not been

hydrogen-a b l e to e s t hydrogen-a b l i s h hydrogen-a u s e f u l s t hydrogen-a t i s t i c hydrogen-a l m e c h hydrogen-a n i c hydrogen-a l trehydrogen-atment 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 m o l e c u l e s 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 o n l y 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 b u t , 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 t h e o r y , 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

-e i g h t c o n s t a n t s 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 numb-er 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 , F i g u r e 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 a t 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 t h e o r y , i e , he assumed t h a t the

e q u a t i o n of s t a t e of the m i x t u r e i s the same as t h a t of a pure

f l u i d except t h a t the c h a r a c t e r i s t i c c o n s t a n t s depend on c o m p o s i t i o n

a c c o r d i n g 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 m i x i n g

r u l e s E x p e r i e n c e has r e p e a t e d l y shown t h a t a t l e a s t one of these

m i x i n g 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 c o n s t a n t ; i n t h i s case, t h a t c o n s t a n t i s M-^j which was found by f i t t i n g to the b i n a r y

d a t a 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 a r e o f t e n h i g h l y s e n s i

t i v e to the m i x i n g r u l e s and to the v a l u e of the a d j u s t a b l e p a r a meter I n t h i s case Orye found what many o t h e r s have a l s o found,

-v i z , t h a t the a d j u s t a b l e b i n a r y parameter i s m o r e - o r - l e s s i n -v a r i a n t

w i t h d e n s i t y and c o m p o s i t i o n but o f t e n depends on temperature

Another example, a l s o from Orye, i s g i v e n i n F i g u r e 4 f o r the system methane-carbon d i o x i d e a t -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 e x p e r i m e n t a l p o i n t s of

D o n n e l l y and K a t z ; the c a l c u l a t e d l i n e s a r e 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 d a t a F i r s t we note t h a t the v a l u e of M^j has a s t r o n g 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 c u r v e ; a t e n percent change i n M-^j produces a l a r g e e r r o r

i n the bubble p r e s s u r e When M^j i s a d j u s t e d 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 t h a t 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 a r e 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

2 PKAUSNITZ Review of Phase Equilibria 17

1200

Figure 3 Methane-n-heptane (Orye, 1969) • O Kohn (1961); Modified BWR

equation

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

2 PRAUSNITZ Review of Phase Equilibria 19

c r i t i c a l r e g i o n and second, c o m p u t a t i o n a l problems a r e o f t e n severe because convergence i s hard to a c h i e v e

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, t a k e n 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 e q u a t i o n

A g a i n , note t h a t an a d j u s t a b l e b i n a r y c o n s t a n t 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 t o u s u a l p r a c t i c e , the l i n e s r e p r e s e n t experiment and the p o i n t s r e p r e s e n t c a l c u l a t i o n s , s u g g e s t i n g problems

i n the c r i t i c a l r e g i o n

I t i s e v i d e n t t h a t the more c o n s t a n t s i n an e q u a t i o n 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 e x p e r i m e n t a l d a t a but i t i s a l s o

c l e a r t h a t t o 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 e x p e r i m e n t a l

i n f o r m a t i o n For example, a twenty-constant e q u a t i o n 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 o f the BWR e q u a t i o n , was proposed by Bender (_3) who a p p l i e d i t t o oxygen argon n i t r o g e n and a few l i g h t hydro-carbons For t h e s e f l u i d s

a c c u r a t e 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 o f Bender's

equa-t i o n , F i g u r e 6 shows equa-the r e s i d u a l heaequa-t c a p a c i equa-t y o f p r o p y l e n e f o r

s e v e r a l temperatures near the c r i t i c a l temperature, 365 K T h i s i s

a v e r y s e n s i t i v e t e s t and Bender's e q u a t i o n does a remarkable j o b Bender has a l s o a p p l i e d h i s e q u a t i o n to m i x t u r e s of argon, n i t r o g e n , and oxygen, u s e f u l f o r d e s i g n 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 m i x t u r e , Bender r e q u i r e s 3 b i n a r y parameters With a l l these

c o n s t a n t s and a l a r g e computer program, Bender can c a l c u l a t e not

o n l y a c c u r a t e 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 o f m i x i n g as shown i n F i g u r e 7 The heats of m i x i n g here a r e v e r y 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 t h a t c a l c u l a t i o n s of t h i s s o r t a r e

r e s t r i c t e d t o those few systems where the m o l e c u l e s a r e s i m p l e 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 o r

o t h e r s p e c i f i c " c h e m i c a l 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 e x p e r i m e n t a l d a t a 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 p r o c e s s i n d u s t r i e s we

r a r e l y meet a l l these n e c e s s a r y c o n d i t i o n s

I f our a c c u r a c y r e q u i r e m e n t s a r e not e x t r e m e l y l a r g e , we can

o f t e n o b t a i n good a p p r o x i m a t i o n s u s i n g c a l c u l a t i o n s based on a

s i m p l e e q u a t i o n o f 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

e q u a t i o n 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'

e q u a t i o n i s t h a t by R e d l i c h and Kwong, proposed i n 1949 That

e q u a t i o n , i n t u r n , i s to a p p l i e d thermodynamics what Helen o f Troy has been t o 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 f a c e t h a t launched a thousand s h i p s " Ten y e a r s ago the B e a t l e s t u r n e d 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 t e n y e a r s ago, the Redlich-Kwong e q u a t i o n i n i t i a t e d an epoch

of i m i t a t i o n u n e q u a l l e d i n the h i s t o r y of a p p l i e d thermodynamics The number o f m o d i f i e d RK e q u a t i o n s i s p r o b a b l y 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 c o n f e s s t o h a v i n g

c o n s t r u c t e d a few m y s e l f A few y e a r s ago, t h e r e was an a r t i c l e i n Chemical E n g i n e e r i n g S c i e n c e devoted e x c l u s i v e l y to v a r i a t i o n s on

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Figure 5 Predicted and experimental K-values for the methane-hydrogen

sul-fide system at 40°F

PRAUSNITZ Review of Phase Equilibria

10

365 15 K

e a

373 15 K Measurements of Bier et al

-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

22 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Figure 7 Molar excess enthalpies of

the binary system Ar-0 2 (Bender)

Temp = 84°K: O exptl, equation

of state of Bender; Temp = K: +

exptl, equation of state

//

//

1200

EXPERIMENTAL (BESSERER AND ROBINSON 1973) TEMP # F LIQUID VAPOR

Figure 8 Pressure-equilibrium phase composition diagram for

isobutane-carbon dioxide system, calculations using

Peng-Robin-son equation of state

2 P R A U S N i T z Review of Phase Equilibria 23

the RK e q u a t i o n but i t i s now h o p e l e s s l y out o f 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 t o r e p o r t t h a t O t t o , aged 80, i s w e l l , a c t i v e and v e r y p l e a s e d about the r e c e n t p u b l i c a ­

t i o n o f 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 H i l d e b r a n d 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

t r a n s p o r t p r o p e r t i e s 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 e q u a t i o n i s t h a t proposed by Soave (4) who expresses the RK c o n s t a n t a by an e m p i r i c a l

f u n c t i o n o f reduced temperature and a c e n t r i c f a c t o r T h i s e m p i r i c a l

f u n c t i o n was determined from v a p o r - p r e s s u r e d a t a 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 e q u a t i o

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 T h i s i l l u s t r a t e s a p o i n t known t o 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 t o r e p r e s e n t any one thermodynamic p r o p e r t y but i t i s d i f f i c u l t , w i t h one e q u a t i o n of

s t a t e , t o r e p r e s e n t them a l l

A c o m p a r a t i v e l y r e c e n t v a r i a t i o n on the RK e q u a t i o n was proposed

by Peng and Robinson ( 5 ) ; i t i s s i m i l a r to S o a v e1s e q u a t i o n but appears to have b e t t e r b e h a v i o r i n the c r i t i c a l r e g i o n ; an example

i s g i v e n i n F i g u r e 8 f o r the i s o b u t a n e - c a r b o n 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 t o 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 e q u a t i o n 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 o r 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 c o r r e s p o n d i n g 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 o f o t h e r f l u i d s ,

i n c l u d i n g m i x t u r e s T h i s 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 e l e g a n t form, the t h e o r y of conformai s o l u t i o n s , has been a p p l i e d by numerous a u t h o r s Here time p e r m i t s me t o c a l l

a t t e n t i o n to o n l y 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 M o l l e r u p and e x t e n s i v e l y developed by M o l l e r u p i n

r e c e n t y e a r s ( 6 ) U s i n g Goodwin's e x c e l l e n t e x p e r i m e n t a l data f o r methane as a r e f e r e n c e , M o l l e r u p 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 o f m i x t u r e s 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 m i x i n g r u l e s but

he pays v e r y 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 c o n s t a n t s

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 t o 200 K,

-u s i n g o n l y one temperat-ure-independent b i n a r y c o n s t a n t Even more

i m p r e s s i v e 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

monoxide-methane 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 e x p e r i m e n t a l e r r o r F i n a l l y , F i g u r e 11 shows t h a t the c o r r e s p o n d i n g - s t a t e s 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

24 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

Figure 9 K-values vs pressure for methane-ethane mixtures;

correspond-ing-states method of Mollerup and Rowlinson

2 PRAUSNITZ Review of Phase Equilibria 25

26 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

2 PRAUSNITZ Review of Phase Equilibria 27

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 t o 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 m i x t u r e s of normal f l u i d s , pure-component parameters and b i n a r y parameters a r e 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 m i x t u r e s

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

o n l y pure-component and b i n a r

tremendous importance i n

m i x t u r e s a r e 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 o n l y 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 e q u a t i o n s o f 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 e q u a t i o n s of s t a t e f o r m o l e c u l e s t h a t

a r e v e r y l a r g e o r v e r y p o l a r , o r b o t h 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 p u r s u i n g f u r t h e r the o b s e s s i o n of

modi-f y i n g the Redlich-Kwong e q u a t i o n 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 o f how to c o n s t r u c t an e q u a t i o n o f

s t a t e and t h e r e we must r e l y on s u g g e s t i o n s 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 c h e m i s t s U n f o r t u n a t e l y most of these a r e "argon

p e o p l e " a l t h o u g h , I am happy t o say, i n the l a s t few y e a r s a few brave t h e o r i s t s have s t a r t e d t o 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 o f time on water and on p r o t e i n s but these h i g h l y c o m p l i c a t e d s t u d i e s a r e s t i l l f a r removed from e n g i -

n e e r i n g a p p l i c a t i o n s N e v e r t h e l e s s , t h e r e are some new t h e o r e t i c a l

i d e a s which c o u l d be used i n f o r m u l a t i n g new e q u a t i o n s of s t a t e

s u i t a b l e f o r those f l u i d s t h a t cannot now be d e s c r i b e d by the u s u a l

e q u a t i o n s o f s t a t e Not t h i s morning, but perhaps l a t e r i n t h i s

c o n f e r e n c e , I hope to have an o p p o r t u n i t y t o say a few words about

t h a t

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 ) , t h a t 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 o n l y 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 m i x t u r e s c o n t a i n i n g molecules t h a t a r e l a r g e o r p o l a r o r gen-bonded o r e l s e when a l l components a r e s u b c r i t i c a l and the

hydro-p r e s s u r e 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 e q u a t i o n o f s t a t e t r u n c a t e d 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

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

BP c _ j ( o ) j ( l ) j ( 2 )

- J ( T R ) + J ( T R ) + J ( T R )

B = SECOND VIRIAL COEFFICIENT; T R =* T / T c

P - CRITICAL PRESSURE; T = CRITICAL TEMPERATURE

c c

(j) « ACENTRIC FACTOR

j AND f 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)

0 100 200 300

REDUCED DIPOLE MOMENT, / / R

Figure 13 Dependence of a on reduced dipole moment for nonhydrogen

bonding compounds (Tsonopoulos, 1974)

2 PRAUSNITZ Review of Phase Equilibria 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 n o t h i n g i s known about h i g h e r v i r i a l c o e f f i c i e n t s T h e r e f o r e ,

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 a r e two major advantages o f 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 e q u a t i o n : 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 t o the i n t e r m o l e c u l a r f o r c e s and second,

Any s t a n d a r d 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 e q u a t i o n o f s t a t e The most

c h e m i c a l 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 c o n s t a n t a p l o t t e d

a g a i n s t a d i m e n s i o n l e s s 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, t h a t c o n s t a n t a becomes more

n e g a t i v e as the reduced d i p o l e moment r i s e s , g i v i n g a more n e g a t i v e 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 g i v e s some r e s u l t s f o r c o n s t a n t b f o r a l c o h o l s ,

a g a i n 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

con-s t a n t a i con-s con-s l i g h t l y p o con-s i t i v e becaucon-se the hydrogen-bonding n a t u r e expressed by c o n s t a n t b dominates, e s p e c i a l l y a t lower temperatures

To e s t i m a t e 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 m o l e c u l a r parameters a p p e a r i n g i n

the c o r r e l a t i o n Only f o r s i m p l e cases can any g e n e r a l 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 m o l e c u l a r 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 e x p e r i e n c e

The v i r i a l e q u a t i o n i s u s e f u l f o r many cases b u t , when t h e r e i s

s t r o n g 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 o f the

v i r i a l e q u a t i o n 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 " c h e m i c a l t r e a t m e n t " , u t i l i z i n g a c h e m i c a l e q u i l i b r i u m

c o n s t a n t 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 o r g a n i c a c i d s and even a t low p r e s s u r e s , t h e vapor-phase f u g a c i t y c o e f f i c i e n t s o f m i x t u r e s c o n t a i n i n g one (or

more) o r g a n i c a c i d a r e 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 o f Hayden and

OfC 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 a l o n g the v a p o r - l i q u i d s a t u r a t i o n c u r v e When the mole f r a c t i o n of a c i d i s v e r y low, the f u g a c i t y c o e f f i c i e n t s a r

30 PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

0 0 6

-i 1 1 1 r

2-Propanol tert-Butanol \ ^

Ethanol

2-Butanol s ^ V / ^ Isobutanol

Methanol

?

1-Propanol 1-Butanol

b = 0.00908 +0.0006957 ^ R )

f ( 2 ) = 0 0 8 7 8 / T R 6 - b / T R 8

J L

0 10 20 30 40 50 60 70 80 90 100

REDUCED DIPOLE MOMENT, / / R

Figure 14 Dependence of b on reduced dipole moment for alcohols (Tsonopoulos, 1974)

Dew-Point Temperature, °C M4.9 124.2 129.3 !33.'2 136.9 140.0 1—

Figure 15 Fugacity coefficients for saturated propionic acid

(1)—methyl isobutyl ketone (2) at 1 atm

2 PRAUSNITZ Review of Phase Equilibria 31

near u n i t y because d i m e r i z a t i o n between a c i d m o l e c u l e s 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 o f both components are w e l l removed from u n i t y even though the temperature i s r e a s o n a b l y h i g h (140C) and the p r e s s u r e i s

o n l y 1 atm

At h i g h p r e s s u r e s , where the v i r i a l e q u a t i o n i s no l o n g e r u s e f u l ,

e m p i r i c a l e q u a t i o n s 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, c o n t r a r y to Method ( a ) , the e q u a t i o n of s t a t e now need not

h o l d 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 u s i n g an e q u a t i o n d e v e l oped by de S a n t i s and B r e e d v e l d (9) f o r gases a t h i g h p r e s s u r e s

-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 F i g u r e 16the e q u a t i o n i s l i k e t h a

r e s u l t s shown i n F i g u r e 18 which show t h a t 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 h i g h p r e s s u r e s 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 m i x t u r e s of water and argon

-The e q u a t i o n of de S a n t i s and B r e e d v e l d 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 e q u a t i o n 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 o f gases i n the w a t e r , i t i s p o s s i b l e

to c a l c u l a t e the s a t u r a t e d water content and o t h e r e q u i l i b r i u m

prop-e r t i prop-e s o f combustion gasprop-es F i g u r prop-e 19 shows thprop-e s a t u r a t prop-e d watprop-er

c o n t e n t i n n i t r o g e n and F i g u r e 20 shows how t h a t water c o n t e n t i s enhanced when CO2 i s p r e s e n t i n the gas phase; r e s u l t s are shown

f o r two molar c o m p o s i t i o n s (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 a t moderate temperatures, the p r e s s u r e o f CO2

s u b s t a n t i a l l y r a i s e s the s a t u r a t e d water c o n t e n t F i g u r e 21 shows

PHASE EQUILIBRIA AND FLUID PROPERTIES IN CHEMICAL INDUSTRY

" ^ ' T 1 / 2 v ( v + b )

B (WATER) " u ' 6 c m 3/r a° ie

(o) ( 1 ) a(T) a + a(T)

(NONPOLAR) (POLAR)

/TABULATED VALUES OBTAINED\

I FROM STEAM TABLES. J

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

TO FIND a x , USE B 1 2 (SECOND VIRIAL COEFFICIENT) DATA FOR

MIXTURES OF WATER WITH N 2 , A r , CH 4 , ETC

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