Industrial Uses of Equations of State: A State-of-the ... - ACS Publications

7 Industrial Uses of Equations of State: A State-of-the-Art Review

Downloaded by UNIV OF ROCHESTER on August 26, 2013 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch007

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.

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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 new equations. 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.

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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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 h i s paper was designed to g i v e the indust r i a l p o i n t of view. I n i n d u s t r y we must have a t o o l that works, 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 d i n s e v e r a l ways to compute the component f u g a c i t i e s . I n the one-equation o f s t a t e approach, the same 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 r i t i c a l r e g i o n , hydrogen systems, immisc i b l e l i q u i d s , e t c . , cannot be f u l l y understood. Any or a l l of 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. Most textbooks i n t h e i r coverage o f equations o f s t a t e d e f i n e the c r i t i c a l s t a t e as: (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 Adopted by Industry 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


(4)

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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 numerator 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 ) equation of s t a t e approach, Eq. (3) i s 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


A N D F L U I D PROPERTIES

I N C H E M I C A L INDUSTRY


P H A S E EQUILIBRIA

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


7.

ADLER ET AL.


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 obtained when i t i s present i f the e a r l y 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


160



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

Ethylene

9.24

0.76

K-Value

Methane-Ethylene K

6.69

5.24

L

r

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


7.


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-Ethylene K

5.24

6.00

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



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 equation i s q u i t e accurate. Note a l s o the accuracy of Table VI f o r h i g h pressure v a p o r - l i q u i d e q u i l i b r i u m such 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,


7.

ADLER E T A L .


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

Enthalpy

Vapor Pressure

Methane

0.60%

2.04%

Propane

0.74%

0.88%

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%

Mixture: Density

Enthalpy

K-Value

C-1 K C

3


164


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

0.028 *

He

30.5

C

2

0.044

He 2

Cl He N

Calc'd 37.5 0.029 0.001 30.0 0.045 0.002

l

N

-279.7

38.7

Cl

N

-288.7

2

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)


7.

165


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

0.035

0.013

0.010

0.438

0.292

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)


166

P H A S E EQUILIBRIA AND


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.



7.

ADLER E T A L .


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 the s o l u b i l i t y data i n the r e g r e s s i o n procedures. In a l l cases, the e r r o r i n the i n o r g a n i c K s were reduced 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 . f

=

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 .


168



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



7.

ADLER E T A L .


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 s r e q u i r e d i n a column by over 33%. (This i s assuming an i-C5/n-C5 r e l a t i v e v o l a t i l i t y of a = 1.15 and a p p l y i n g a " r u l e of thumb" that the approximate number 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



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


7.

ADLER ET AL.


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 , and a m u l t i t u d e o f c h o i c e s , i n the s e l e c t i o n o f the thermodynamic r e l a t i o n s h i p s o r e m p i r i c a l equations f o r 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°,

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