Recent Mixing Rules for Equations of State - ACS Publications

17 Recent Mixing Rules for Equations of State A n Industrial Perspective Thomas W. Copeman and Paul M . Mathias

Downloaded by PURDUE UNIVERSITY on July 10, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch017

Air Products and Chemicals, Inc., Allentown, PA 18105

The number of applications for equations of state capable of describing phase behavior of mixtures with asymmetric interactions is high - and continuing to grow. Gas-processing applications such as enhanced oil recovery and glycol dehydration involve carbon dioxide, water, hydrogen sulfide, and hydrocarbons at high pressure. Organic-chemical and polymer production involve an increasingly wide variety of conditions, not restricted to low pressures (1). Supercritical extraction is actively being applied to the purification of natural products, which involves mixtures of complex polar compounds near the critical point of a gas like carbon dioxide (2). Engineering models to describe phase equilibrium can be divided into two broad categories: equations of state and activity coefficient models. Equations of state have been successfully applied to mixtures of nonpolar and slightly polar compounds at all conditions of engineering interest. These models have been used most extensively by the gas processing industry for the design of various processes (see for example Refs. 3 and 4). C o n v e r s e l y the mathematical f l e x i b i l i t y o f a c t i v i t y c o e f f i c i e n t models has c o n v e n t i o n a l l y been c o n s i d e r e d n e c e s s a r y t o d e s c r i b e systems w h i c h e x h i b i t h i g h l i q u i d - p h a s e n o n i d e a l i t y . These models have been most e x t e n s i v e l y used by the c h e m i c a l s and polymer i n d u s t r i e s f o r d e s i g n o f v a r i o u s p r o c e s s e s . P r o c e s s d e s i g n f o r the p r o d u c t i o n o f p o l y v i n y l a l c o h o l (5) i s one example. The a c t i v i t y c o e f f i c i e n t approach i s adequate a t low reduced temperatures where the l i q u i d phase i s i n c o m p r e s s i b l e and up t o moderate p r e s s u r e s . The use o f d i f f e r e n t models f o r the v a r i o u s phases p r e c l u d e s c o r r e c t d e s c r i p t i o n o f m i x t u r e c r i t i c a l p o i n t s and a d d i t i o n a l problems a r i s e f o r s u p e r c r i t i c a l components (6). The e q u a t i o n - o f - s t a t e approach does not i n h e r e n t l y s u f f e r f r o m these l i m i t a t i o n s and the e x t e n s i o n o f e q u a t i o n s o f s t a t e t o d e s c r i b e asymmetric i n t e r a c t i o n s i s c u r r e n t l y a h i g h l y a c t i v e a r e a o f r e s e a r c h . T h i s paper r e v i e w s r e c e n t developments o f m i x i n g r u l e s f o r e q u a t i o n s o f s t a t e w i t h an e n g i n e e r i n g - d e s i g n p e r s p e c t i v e . 0097-6156/ 86/ 0300-O352$06.00/ 0 © 1986 American Chemical Society

In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

17. C O P E M A N A N D Μ ΑΤΗ IAS

Mixing Rules: An industrial Perspective

353


E v o l u t i o n of Equation of State Mixing Rules C o n s i d e r a b l e e f f o r t has been expended o v e r the p a s t f i f t y y e a r s to d e v e l o p new e q u a t i o n s o f s t a t e f o r pure components. Much l e s s a t t e n t i o n has been g i v e n t o i m p r o v i n g m i x i n g r u l e s . The o n e - f l u i d m i x i n g r u l e s o f van der Waals (7) (known h e r e a f t e r a s the vdW-1 m i x i n g r u l e s ) a r e s t i l l i n w i d e s p r e a d use; the e q u a t i o n s o f s t a t e o f Soave (8) and Peng and R o b i n s o n (9) a r e two such examples. I t s h o u l d be noted t h a t the vdW-1 m i x i n g r u l e s a r e r e a s o n a b l y w e l l based i n b o t h t h e o r y and the t y p i c a l e x c e s s - f u n c t i o n b e h a v i o r of normal f l u i d s . L e l a n d , C h a p p e l e a r , and Gamson (10) have shown t h a t t h e s e m i x i n g r u l e s a r i s e from v e r y r e a s o n a b l e assumptions f o r the j i r a d i a l d i s t r i b u t i o n f u n c t i o n s . ( A l s o see R e f s . _Π and 1 2 ) . Henderson and Leonard ( 1 3 , 14) have shown t h a t t h e s e m i x i n g r u l e s p r o v i d e good agreement w i t h the q u a s i - e x p e r i m e n t a l machines i m u l a t i o n r e s u l t s f o r h a r d - s p h e r e and Lennard-Jones (6:12) m i x t u r e s . F u r t h e r , V i d a l (l_5) has demonstrated t h a t the vdW-1 mixing r u l e s p r e d i c t excess f u n c t i o n s very s i m i l a r t o r e g u l a r s o l u t i o n t h e o r y and thus t h e y s h o u l d p r o v i d e a good d e s c r i p t i o n o f most n o n p o l a r m i x t u r e s . An i m p o r t a n t landmark i n the development o f i n d u s t r i a l l y s i g n i f i c a n t m i x i n g r u l e s was the work o f S t o t l e r and B e n e d i c t (16) who suggested the use o f b i n a r y i n t e r a c t i o n p a r a m e t e r s . They found t h a t the o n e - f l u i d m i x i n g r u l e s suggested f o r the Benedict-WebbR u b i n e q u a t i o n (17, 18) c o u l d be markedly improved by the use o f s m a l l p a i r - d e p e n d e n t c o r r e c t i o n terms. Another i m p o r t a n t e f f o r t was the work o f P r a u s n i t z and Gunn (19) who noted t h a t the i n t e r a c t i o n parameter was not a t o t a l l y e m p i r i c a l c o r r e c t i o n f a c t o r , but was r e l a t e d t o the t h e o r y o f i n t e r m o l e c u l a r f o r c e s . Many e f f o r t s t o improve the vdW-1 m i x i n g r u l e s attempted t o f i n d b e t t e r forms w h i l e r e t a i n i n g the o n e - f l u i d c o n c e p t . Plocker et a l . (20), Radosz e t a l . (21) and Lee e t a l . (22) v a r i e d an exponent i n the m i x i n g r u l e s , which can be r e p r e s e n t e d a s εσ

=

η

m

o

=

Z Z x i X j

E l X i X j

e j i a j i

a j i

where η = m = 3 r e p r e s e n t s the van der Waals o n e - f l u i d m i x i n g

( l )

(2)

rules.

P l o c k e r (20) chose n=0.75 and m=3, w h i l e Radosz (21) chose n=-0.25 and m=3. Lee (22) used n=m=4.5. W h i l e some degree o f s u c c e s s i n i m p r o v i n g m i x t u r e p r e d i c t i o n s i s c l a i m e d f o r each o f the methods, t h e i r a p p l i c a t i o n t o h i g h l y asymmetric p o l a r - n o n p o l a r systems i s l i m i t e d . P e r t u r b a t i o n t h e o r y i s u s e f u l as a g u i d e t o the development o f b o t h pure f l u i d and m i x t u r e e q u a t i o n s o f s t a t e f o r e n g i n e e r i n g calculations ( 2 3 ) . Donohue and P r a u s n i t z (24) d e v e l o p e d an equation o f s t a t e f o r mixtures o f molecules w i t h l a r g e s i z e d i f f e r e n c e s based on a p e r t u r b a t i o n e x p a n s i o n f o r s q u a r e - w e l l f l u i d s at low d e n s i t i e s and i d e a s from polymer s o l u t i o n t h e o r y . The e x p a n s i o n was then t r u n c a t e d a f t e r the f o u r t h a t t r a c t i v e term. W h i l e a s i m p l e q u a d r a t i c c o m p o s i t i o n dependence was o b t a i n e d f o r the f i r s t a t t r a c t i v e term, each o f the h i g h e r - o r d e r terms y i e l d s a d d i t i o n a l ( n o n - q u a d r a t i c ) m i x i n g r u l e s . Good r e s u l t s were o b t a i n e d



354

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

f o r m i x t u r e s c o n t a i n i n g a l k a n e s (up t o C 3 0 ) , a r o m a t i c s and l i g h t inorganic gases. W i l s o n ( 2 5 ) has s u g g e s t e d t h a t t h e l i q u i d - p h a s e e x c e s s G i b b s energy d a t a c a n be used t o p r o v i d e i n f o r m a t i o n f o r i m p r o v i n g m i x i n g r u l e s a t h i g h d e n s i t i e s . V i d a l ( 1 5 ) , Huron and V i d a l ( 2 6 ) , Heyen ( 2 7 ) , and Won ( 2 8 ) have f u r t h e r proposed n o n - q u a d r a t i c m i x i n g r u l e s based on t h e b e h a v i o r o f l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t s . These m i x i n g r u l e s do n o t reproduce t h e r e q u i r e d q u a d r a t i c c o m p o s i t i o n dependence o f t h e second v i r i a l c o e f f i c i e n t , as p o i n t e d o u t by Huron and V i d a l ( 2 6 ) and l a t e r by W h i t i n g and P r a u s n i t z ( 2 9 , 30) and Mollerup (31). These i n i t i a l l o c a l - c o m p o s i t i o n e q u a t i o n s o f s t a t e , however, d i d p r o v i d e an i m p o r t a n t advance by d e m o n s t r a t i n g t h a t complex p o l a r - n o n p o l a r systems c o u l d be c o r r e l a t e d w i t h e q u a t i o n s o f s t a t e i n t h e same ( e m p i r i c a l ) manner as w i t h a c t i v i t y - c o e f f i c i e n t models. F o r example, 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 were d e v e l o p e d f o r e t h a n o l - b e n z e n e and a c e t o n e - w a t e r by Heyen ( 2 7 ) and Huron and V i d a l ( 2 6 ) , r e s p e c t i v e l y . To overcome t h e above-mentioned d e f i c i e n c y a t low p r e s s u r e , W h i t i n g and P r a u s n i t z ( 2 9 , 3 0 ) , M o l l e r u p ( 3 1 ) and Won ( 3 2 ) p r o p o s e d d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n m i x i n g r u l e s . A common p r o p o s a l i n a l l t h r e e models i s t h e use o f c o n v e n t i o n a l m i x i n g r u l e s f o r t h e r e p u l s i v e term i n t h e e q u a t i o n o f s t a t e and l o c a l - c o m p o s i t i o n m i x i n g r u l e s w i t h d e n s i t y - d e p e n d e n t Boltzmann f a c t o r s f o r t h e a t t r a c t i v e term. These e q u a t i o n s o f s t a t e p l e a s i n g l y resemble t h e W i l s o n ( 3 3 ) a c t i v i t y - c o e f f i c i e n t model a t h i g h d e n s i t i e s and c o n v e n t i o n a l e q u a t i o n s o f s t a t e a t low d e n s i t i e s . M a t h i a s and Copeman ( 3 4 ) however demonstrated t h a t t h e l o c a l - c o m p o s i t i o n Peng-Robinson e q u a t i o n o f s t a t e d e r i v e d by M o l l e r u p ( 3 1 ) l a c k s adequate p r e d i c t i v e c a p a b i l i t y f o r n o n p o l a r m i x t u r e s w i t h even moderate s i z e differences. The l o c a l - c o m p o s i t i o n e f f e c t i s t o o l a r g e , w h i c h c a n r e s u l t i n g r e a t l y u n d e r p r e d i c t e d f u g a c i t y c o e f f i c i e n t s ( f o r example, 3-4 o r d e r s o f magnitude f o r methane i n d e c a n e ) . S i z a b l e i n t e r a c t i o n parameters along w i t h s i z e parameters (e.g., s u r f a c e area) a r e n e c e s s a r y f o r r e a s o n a b l e p r e d i c t i o n s o f phase e q u i l i b r i u m . The M o l l e r u p ( 3 1 ) Peng-Robinson e q u a t i o n o f s t a t e , a l o n g w i t h t h e Redlich-Kwong and o t h e r a n a l o g s , was c o n c l u d e d t o be t o o u n r e l i a b l e f o r g e n e r a l a p p l i c a t i o n t o i n d u s t r i a l problems. An i m p o r t a n t i d e a has been proposed by D i m i t r e l i s and P r a u s n i t z ( 3 5 ) . They propose t h a t l o c a l - c o m p o s i t i o n e f f e c t s do n o t r e s u l t when j i i n t e r a c t i o n s a r e d i f f e r e n t f r o m i i but r a t h e r when j i a r e d i f f e r e n t f r o m some " i d e a l " c o m b i n a t i o n s o f i i and j j i n t e r a c t i o n s , r e f e r r e d t o as ji°. D i m i t r e l i s and P r a u s n i t z d e f i n e d ji° as an a r i t h m e t i c mean i n t h e i r work. M a t h i a s and Copeman ( 3 4 ) and M o l l e r u p ( 3 6 ) , however, chose t h e g e o m e t r i c mean and d e v e l o p e d a d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n Peng-Robinson e q u a t i o n o f s t a t e t h a t meets an i m p o r t a n t a d d i t i o n a l l i m i t : non-conformal e f f e c t s a r e not n e c e s s a r i l y l a r g e ( o r even n o n - z e r o ) f o r an asymmetric system. T h i s f e a t u r e e n a b l e s t h e model t o r e t a i n t h e good p r e d i c t i o n s o f t h e s t a n d a r d Peng-Robinson e q u a t i o n o f s t a t e f o r n o n p o l a r systems and f a c i l i t a t e s c o r r e l a t i o n o f phase e q u i l i b r i u m o f h i g h l y n o n - i d e a l p o l a r systems. A d d i t i o n a l forms o f t h e e q u a t i o n o f s t a t e were s u g g e s t e d by M a t h i a s and Copeman ( 3 4 ) based on e x p a n s i o n o f t h e Boltzmann f a c t o r s i n the a t t r a c t i v e part w i t h t r u n c a t i o n a f t e r the second term i n t h e s e r i e s . Reduced c o m p u t a t i o n a l time a l o n g w i t h s u r p r i s i n g l y good r e s u l t s f o r b i n a r y h y d r o c a r b o n - w a t e r l i q u i d - l i q u i d e q u i l i b r i a were shown.



17.

C O P E M A N A N D MATH IAS

Mixing Rules: An Industrial Perspective

355

Ludecke and P r a u s n i t z (37) have proposed a s i m i l a r e q u a t i o n o f s t a t e based on the van der Waals form. They chose the M a n s o o r i e t a l . (38) e x p r e s s i o n f o r the h a r d - s p h e r e p a r t and a s i m p l e van d e r Waals e x p r e s s i o n f o r the a t t r a c t i v e p a r t . For m i x t u r e s , the Mansoori e t a l . e x p r e s s i o n i s used f o r the h a r d - s p h e r e c o n t r i b u t i o n , w h i l e f o r the a t t r a c t i v e c o n t r i b u t i o n the c o n v e n t i o n a l d e n s i t y - i n d e p e n d e n t q u a d r a t i c m i x i n g r u l e i s used as a l e a d i n g term and a d e n s i t y - d e p e n d e n t c o r r e c t i o n ( c u b i c i n mole f r a c t i o n ) i s used t o d e s c r i b e " n o n - c e n t r a l " f o r c e s . Good r e s u l t s were o b t a i n e d f o r v a p o r - l i q u i d and l i q u i d - l i q u i d e q u i l i b r i a i n b i n a r y m i x t u r e s c o n t a i n i n g w a t e r , p h e n o l , p y r i d i n e , methanol and h y d r o c a r b o n s . The e q u a t i o n o f s t a t e , however, o v e r - p r e d i c t s the t w o - l i q u i d r e g i o n i n the t e r n a r y systems s t u d i e d . The above approach i s a s i m p l e , y e t p o t e n t i a l l y v e r y e f f e c t i v e , way t o d e s c r i b e p o l a r , asymmetric systems and s h o u l d be d e v e l o p e d f u r t h e r f o r i n d u s t r i a l a p p l i c a t i o n s . A p o s s i b l e way t o a v o i d v i o l a t i n g the q u a d r a t i c dependence o f the second v i r i a l c o e f f i c i e n t and y e t have s e p a r a t e m i x i n g r u l e s f o r h i g h and low d e n s i t i e s has been suggested by L a r s e n and P r a u s n i t z ( 3 9 ) . These r e s e a r c h e r s have d i v i d e d the a t t r a c t i v e c o n t r i b u t i o n t o the H e l m h o l t z energy i n t o a s e c o n d - v i r i a l p o r t i o n and a d e n s e - f l u i d p o r t i o n , t h e r e b y a l l o w i n g an a r b i t r a r y m i x i n g r u l e f o r the d e n s e - f l u i d p a r t . Good r e s u l t s have been o b t a i n e d f o r the methane-water b i n a r y . T h i s work i s i m p o r t a n t s i n c e i t can be extended t o p r o p e r i d e n t i f i c a t i o n o f v a r i o u s t y p e s o f c o n t r i b u t i o n s at the p u r e - f l u i d l e v e l , l e a d i n g t o the use o f t h e o r e t i c a l l y suggested m i x i n g r u l e s f o r each c o n t r i b u t i o n . L i e t a l . (40) have combined the p u r e - f l u i d e q u a t i o n o f s t a t e d e v e l o p e d by Chung e t a l . (41) w i t h l o c a l - c o m p o s i t i o n m i x i n g r u l e s . The l o c a l - c o m p o s i t i o n m i x i n g r u l e s a l o n g w i t h van der Waals o n e - f l u i d m i x i n g r u l e s were e v a l u a t e d a g a i n s t b o t h v a p o r - l i q u i d e q u i l i b r i u m and d e n s i t y d a t a f o r b i n a r y systems. The p r e d i c t i v e c a p a b i l i t y o f the L i e t a l . model s h o u l d be e v a l u a t e d . T a b l e I p r e s e n t s a l i s t o f systems s t u d i e d by v a r i o u s i n v e s t i g a t o r s u s i n g l o c a l - c o m p o s i t i o n m i x i n g r u l e s . The m a j o r i t y o f s t u d i e s have not i n c l u d e d e v a l u a t i o n o f l i q u i d - l i q u i d e q u i l i b r i u m and d e n s i t y p r e d i c t i o n s . I t appears t h a t improved m i x t u r e models w i l l come f r o m the concept o f d e n s i t y - d e p e n d e n t m i x i n g r u l e s . However, i t i s not c l e a r at the p r e s e n t time w h i c h o f the proposed forms i s the " b e s t . " Most appear t o p r o v i d e s i g n i f i c a n t improvement o v e r vdW-1 for correlation o f the phase e q u i l i b r i u m o f b i n a r y systems, i n c l u d i n g complex b e h a v i o r l i k e l i q u i d - p h a s e i m m i s c i b i l i t y . However, v e r y few s t u d i e s have a d d r e s s e d the more d i f f i c u l t problem o f multicomponent predictions. P a r a l l e l work i n computer s i m u l a t i o n and t h e o r e t i c a l s t a t i s t i c a l mechanics i s e x t r e m e l y i m p o r t a n t as i t p r o v i d e s a g u i d e l i n e f o r the development o f e m p i r i c a l models. Approaches t o D e r i v e

Local Composition Mixing

Rules

The e a r l y d e r i v a t i o n s o f l o c a l - c o m p o s i t i o n m i x i n g r u l e s were based on l a t t i c e t h e o r i e s and/or e m p i r i c a l f o r m u l a t i o n s f o r i n t e r n a l , H e l m h o l t z , o r G i b b s energy (33, 42 - 4 4 ) . The d e r i v a t i o n by Kemeny and Rasmussen ( 4 4 ) , based on l a t t i c e p a r t i t i o n f u n c t i o n s


EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

356


Table I.

B i n a r y Systems S t u d i e s w i t h E q u a t i o n o f S t a t e L o c a l Composition Mixing Rules

Investigators

Year

Systems

Type

Huron and V i d a l (25) Heyen (26 ) Won (27) W h i t i n g and P r a u s n i t z (29) Won (31) M a t h i a s and Copeman (33) M o l l e r u p (35) Ludecke and P r a u s n i t z (36) L i e t a l . (39)

1979 1981 1981 1982 1983 1983 1983 1985 1984

1-8 9-11 12, 13 14 5, 15- 18 1, 10, 11, 119-24 1, 5, 20, 25-34 20, 22, 23, 25, 35-41 1, 5, 6, 14, 16, 19, 20, 25 , 29, 31, 42-46

VLE VLE VLE VLE VLE VLE, LLE VLE VLE, LLE VLE, density

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

acetone-water carbon dioxide-ethane ethane-acetone ethane-methyl a c e t a t e methanol-carbon d i o x i d e propane-ethanol methanol-1,2 d i c h l o r o e t h a n e ac e tone-eye1ohexane e t h a n o l benzene butanol-water carbon dioxide-methane carbon dioxide-napthalene ethylene-napthalene water-methane carbon dioxide-n-octane carbon dioxide-n-decane carbon dioxide-butanol water-carbon d i o x i d e methane-n-decane methanol-benzene isobutylene-methanol benzene-water hexane-water

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

1-methylnapthalene-water methanol-water ethanol-water 2-propanol-water acetone-carbon d i o x i d e methano1-n-hexane me t hano1-eye1ohexane e thano1-n-hexane e t hano1-n-hep t ane e thano1-eye1ohexane e thano1-me thy1eye1ohexane pyridine-water pyridine-benzene wa t e r-cyc1ohexane water-heptane water-octane phenol-water propane-water ethane-butane carbon dioxide-benzene ammonia-water e t hano1-n-dec ane ethane-water



17.

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357

and t w o - l i q u i d t h e o r y , p r o v i d e d a thorough e x p l a n a t i o n o f the assumptions r e q u i r e d t o o b t a i n the W i l s o n e q u a t i o n . F o l l o w i n g the q u a s i c h e m i c a l approach, the c o m b i n a t o r i a l f a c t o r i n the configurâtional p a r t i t i o n f u n c t i o n i s e x p r e s s e d i n terms o f the count o f c o n t a c t s and a c o r r e c t i o n f a c t o r , denoted by h. To meet the p r o p e r random-mixture l i m i t , an e x p r e s s i o n was d e r i v e d f o r h u s i n g a T a y l o r s e r i e s e x p a n s i o n about complete randomness. The l a t t i c e t h e o r y d e r i v a t i o n c l e a r l y shows t h a t the W i l s o n e q u a t i o n i s most v a l i d f o r near-random m i x t u r e s o f s i m i l a r s i z e d m o l e c u l e s . Kemeny and Rasmussen a l s o p r e s e n t e d an e q u a t i o n f o r the combinatorial f a c t o r which leads to expressions f o r l o c a l area f r a c t i o n s , as used i n the s u c c e s s f u l UNIQUAC a c t i v i t y c o e f f i c i e n t model ( 4 5 ) . L a t t i c e t h e o r i e s have p r o v i d e d l i t t l e f u r t h e r p r o g r e s s i n d e v e l o p i n g improved m i x i n g r u l e s s i n c e the mid 1970*s. More r e c e n t l y , Lee et a l . (46) d e r i v e d g e n e r a l r e l a t i o n s f o r l o c a l c o m p o s i t i o n s i n terms o f r a d i a l d i s t r i b u t i o n f u n c t i o n s and r e l a t e d p o t e n t i a l s o f mean f o r c e t o the W i l s o n f o r m u l a t i o n . T h i s work r e p r e s e n t s a s i g n i f i c a n t c o n t r i b u t i o n , b r i d g i n g the gap between c o n v e n t i o n a l l o c a l - c o m p o s i t i o n m i x i n g r u l e s and r i g o r o u s f l u i d - p h a s e s t a t i s t i c a l mechanics. Mansoori and E l y (47) have f u r t h e r d e r i v e d a u n i f i e d treatment o f the l o c a l - c o m p o s i t i o n concept f o r f l u i d - p h a s e m i x t u r e s . The energy, p r e s s u r e , and c o m p r e s s i b i l i t y e q u a t i o n s were e x p r e s s e d i n terms o f l o c a l p a r t i c l e numbers. For example, the i n t e r n a l energy i s g i v e n by

Ε

= -NkT/2

Σ Σ χ ι

/ " n j i i r ) (dUj ι ( r ) / d r ) d r

(3)

V a r i o u s a p p r o x i m a t i o n s f o r n j i were a n a l y z e d by Mansoori and E l y . T h e i r g e n e r a l procedure i s based on e q u a t i n g the m i x t u r e p o t e n t i a l energy f u n c t i o n and l o c a l p a r t i c l e number t o t h a t o f a h y p o t h e t i c a l pure f l u i d . An advantage o f the Mansoori and E l y approach i s t h a t m i c r o s c o p i c a p p r o x i m a t i o n s t o the r a d i a l d i s t r i b u t i o n f u n c t i o n s can be t r a n s l a t e d i n t o e x p r e s s i o n s f o r l o c a l p a r t i c l e numbers and m i x i n g r u l e s . The r e l a t i o n s , based on the c o m p r e s s i b i l i t y e q u a t i o n , a r e n o v e l and m e r i t f u r t h e r e v a l u a t i o n . One p r o b a b l e d i s a d v a n t a g e f o r i n d u s t r i a l a p p l i c a t i o n i s t h a t the m i x i n g - r u l e e x p r e s s i o n s a r e g e n e r a l l y not e x p l i c i t w i t h r e s p e c t t o the m i x t u r e parameters, r e q u i r i n g s i g n i f i c a n t a d d i t i o n a l c o m p u t a t i o n a l expense. Seaton and G l a n d t (48) have e x p l o r e d l o c a l c o m p o s i t i o n s f o r m i x t u r e s d e s c r i b e d by the a d h e s i v e i n t e r m o l e c u l a r p o t e n t i a l . The unique f e a t u r e o f t h e i r work i s the unambiguous d e f i n i t i o n o f nearest neighbors s i n c e a t t r a c t i v e f o r c e s are extremely short-ranged and n e i g h b o r i n g m o l e c u l e s a r e i n c o n t a c t . The l o c a l p a r t i c l e numbers were shown t o be n j i = i r p i X j io] i / 3

(4)

where the X j ι a r e o b t a i n e d from the s o l u t i o n o f t h r e e q u a d r a t i c a l g e b r a i c e q u a t i o n s f o r a b i n a r y m i x t u r e . The W i l s o n e q u a t i o n was compared t o the a d h e s i v e - p o t e n t i a l model and c o n c l u d e d t o be inaccurate i n d e s c r i b i n g l o c a l compositions f o r mixtures of


358

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

unequal-sized molecules. T h i s c o n c l u s i o n i s s i m i l a r t o t h a t reached by G i e r y c z and N a k a n i s h i (49) and s u g g e s t s t h a t more a t t e n t i o n s h o u l d be g i v e n t o s i z e d i f f e r e n c e s i n l o c a l - c o m p o s i t i o n formulations.


Comparison o f L o c a l C o m p o s i t i o n F o r m u l a t i o n s

t o Computer S i m u l a t i o n

N a k a n i s h i and co-workers have used m o l e c u l a r dynamics and Monte Carlo simulations t o provide u s e f u l information f o r the l o c a l s t r u c t u r e o f v a r i o u s k i n d s o f m i x t u r e s (49 - 52). Insight into l o c a l f l u i d s t r u c t u r e h a s been o b t a i n e d by comparing l o c a l c o m p o s i t i o n s c a l c u l a t e d from computer s i m u l a t i o n s t o l o c a l c o m p o s i t i o n s p r e d i c t e d from s e m i - e m p i r i c a l models ( l i k e W i l s o n , 2 5 ) . The number o f p a r t i c l e s j around a p a r t i c l e i i s d e t e r m i n e d f r o m the s i m u l a t i o n r e s u l t s as R

nji

The

= 4irNj/V J r g j i ( r ) d r 2

l o c a l composition xji

= n i / (E n d

k

(5)

o f j around i i s k i

)

(6)

N a k a n i s h i and co-workers have s e t t h e upper l i m i t o f i n t e g r a t i o n i n E q u a t i o n ( 5 ) t o t h e d i s t a n c e o f t h e f i r s t peak i n t h e r a d i a l d i s t r i b u t i o n f u n c t i o n . Lennard-Jones p o t e n t i a l and L o r e n t z B e r t h e l o t combining r u l e s were chosen f o r c a l c u l a t i o n s on m i x t u r e s w i t h v a r i e d s i z e and energy p a r a m e t e r s . The Lennard-Jones energy parameters were s u b s t i t u t e d f o r t h e p o t e n t i a l s o f mean f o r c e i n t h e local-composition formulations. I n t h e i r most e x t e n s i v e works (49, 5 2 ) , m o l e c u l a r dynamic c a l c u l a t i o n s were made o v e r a range o f o v e r a l l mole f r a c t i o n s and the l o c a l c o m p o s i t i o n s were compared t o t h e W i l s o n (25) and Renon and P r a u s n i t z (42) f o r m u l a t i o n s . The W i l s o n e q u a t i o n was found t o p r e d i c t l o c a l c o m p o s i t i o n s p o o r l y f o r t h e m i x t u r e s d e s c r i b e d above, w h i l e t h e Renon and P r a u s n i t z e q u a t i o n w i t h

2

< 0 >



17.

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361

i s known t o be wrong at h i g h d e n s i t i e s . However, the r e a s o n a b l e n e s s and the c l a r i t y o f the a s s u m p t i o n s must a l s o be r e c o g n i z e d as we attempt t o d e v e l o p improved m i x i n g r u l e s ( 5 8 ) . The P r e d i c t i v e and C o r r e l a t i v e Power o f the vdW-1 M i x i n g R u l e s . In a d d i t i o n t o t h e i r f o u n d a t i o n i n t h e o r y , the vdW-1 m i x i n g r u l e s have p r o v i d e d good p r e d i c t i v e and c o r r e l a t i v e c a p a b i l i t y . We have a l r e a d y mentioned the work o f Henderson and Leonard ( 1 3 , 14) who found t h a t the vdW-1 m i x i n g r u l e s p r o v i d e d good p r e d i c t i o n s o f m a c h i n e - s i m u l a t i o n r e s u l t s f o r h a r d - s p h e r e and Lennard-Jones m i x t u r e s . We r e i t e r a t e t h a t c o m p a r i s o n w i t h computer s i m u l a t i o n s i s a s t r o n g t e s t o f t h e o r e t i c a l f o r m u l a t i o n s s i n c e the parameters a r e known and cannot be a d j u s t e d . However, t h e s e c o m p a r i s o n s s h o u l d be made i n the d i l u t e r e g i o n not (as i s u s u a l ) f o r e q u i m o l a r m i x t u r e s . S a n d l e r and co-workers have o b t a i n e d m a c h i n e - s i m u l a t i o n r e s u l t s f o r b o t h pure f l u i d s and m i x t u r e s i n t e r a c t i n g w i t h the s q u a r e - w e l l p o t e n t i a l s (56, 5 7 ) , thus e n a b l i n g a t e s t o f the s i m p l e vdW-1 mixing r u l e s . These s i m u l a t i o n s a r e f o r e q u i s i z e d m o l e c u l e s . Therefore the vdW-1 m i x i n g r u l e s reduce t o the s i m p l e f o r m ε

=

Σ

Σ

XiXj

εjι

(10)

The r e s u l t s f o r the vdW-1 p r e d i c t i o n s a r e p r e s e n t e d i n F i g u r e s 1 and 2. F i g u r e 1 shows t h a t i n the case where the temperature i s r e l a t i v e l y h i g h ( n o t e t h a t ε/kTc - 0.8), the vdW-1 mixing r u l e s p r o v i d e e x c e l l e n t p r e d i c t i o n s o f the configurâtional e n e r g y a t a l l c o m p o s i t i o n s and d e n s i t i e s . I n the case where the temperature i s lower ( F i g u r e 2 ) , the vdW-1 m i x i n g r u l e s p r o v i d e good p r e d i c t i o n s at h i g h d e n s i t i e s , and u n d e r p r e d i c t the configurâtional energy a t low d e n s i t i e s . A b e t t e r mixing rule f o r low-density mixtures i s a l o c a l - c o m p o s i t i o n model based on the second v i r i a l c o e f f i c i e n t : Ε

=

Σ

Σ

XiXj

Ε

(ε^;

Τ,ρ)

(il)

E q u a t i o n ( i l ) i s e x a c t a t z e r o d e n s i t y ( 5 5 ) . The dashed l i n e i n F i g u r e 2 shows t h a t the l o w - d e n s i t y m i x i n g r u l e p r o v i d e s improved p r e d i c t i o n s o f the m i x t u r e d a t a . But the model s t i l l u n d e r p r e d i c t s the d a t a . T h i s i s perhaps caused by m i x t u r e c o n d i t i o n s w i t h i n the u n s t a b l e r e g i o n o f the f l u i d , w h i c h i s s u p p o r t e d by the s c a t t e r i n the s i m u l a t i o n d a t a . S e v e r a l i n v e s t i g a t o r s have p o i n t e d out t h a t the s i m p l e vdW-1 m i x i n g r u l e s p r o v i d e good p r e d i c t i o n s f o r systems c o n t a i n i n g e q u i - s i z e d molecules w i t h d i f f e r i n g energies of i n t e r a c t i o n ( f o r example, see F i s c h e r , 5 8 ) . However, the vdW-1 f o r m i s i n a d e q u a t e f o r the more d i f f i c u l t m i x t u r e s c o n t a i n i n g m o l e c u l e s w i t h l a r g e s i z e d i f f e r e n c e s (54, 4 8 ) . I t s h o u l d be noted t h a t i t i s not n e c e s s a r y t o use the vdW-1 a p p r o x i m a t i o n f o r the r e p u l s i v e p a r t o f an e q u a t i o n o f s t a t e s i n c e t h e o r e t i c a l l y - b a s e d , t r a c t a b l e models a r e a v a i l a b l e f o r u n e q u a l - s i z e d h a r d - s p h e r e m i x t u r e s (60, 3 8 ) . Hu e t a l . (53) have used the Mansoori-Carnahan S t a r l i n g - L e i a n d e q u a t i o n (38) f o r the r e p u l s i v e term and o b t a i n e d good agreement w i t h c o m p u t e r - s i m u l a t i o n data.



362

A p p l i c a t i o n s t o Important I n d u s t r i a l E q u a t i o n s o f S t a t e . It i s u s e f u l t o a p p l y t h e r a d i a l - d i s t r i b u t i o n - f u n c t i o n framework f o r l o c a l c o m p o s i t i o n s ( 4 6 , 55) t o t h e i m p o r t a n t i n d u s t r i a l e q u a t i o n s o f s t a t e . T h i s a l l o w s e x a m i n a t i o n o f t h e l o c a l c o m p o s i t i o n s i m p l i e d by the p r a c t i c a l l y - s u c c e s s f u l vdW-1 m i x i n g r u l e s and t h o s e o f t h e a t t e m p t s t o improve them (29 - 3 1 ) . Perhaps more i m p o r t a n t , t h e r e are i n d i c a t i o n s that the s i m p l i f i c a t i o n s inherent i n the p u r e - f l u i d e q u a t i o n s o f s t a t e must be c o n s i d e r e d when a p p l y i n g t h e m i x i n g r u l e s s u g g e s t e d by a r i g o r o u s t h e o r e t i c a l framework. C o n s i d e r t h e i n t e r n a l e n e r g y o b t a i n e d by a p p l y i n g t h e vdW-1 m i x i n g r u l e s t o t h e common c u b i c e q u a t i o n s o f s t a t e 3aji/T Ε

=

-

F

Σ

v


1

(12)

Σ XiXj J

3 1/T

where t h e f u n c t i o n F i s dependent on t h e f o r m chosen f o r t h e a t t r a c t i v e term o f t h e e q u a t i o n o f s t a t e . F o r example: v

van d e r Waals:

F

v

R e d 1 i ch-Kwong:

F

v

Peng-Robinson:

F

v

(13) (14)

1 l n ( l + pb) b

1 4- (1+72) bp 1 + ( W 2 ) bp

In

= 2/2b

(15)

The Redlich-Kwong and Peng-Robinson forms f o r F reduce t o t h e s i m p l e l i n e a r dependence on d e n s i t y ( E q u a t i o n ( 1 3 ) ) i n t h e l i m i t o f low d e n s i t y . (See F i g u r e 3.) Now c o n s i d e r t h e s t a t i s t i c a l m e c h a n i c a l e x p r e s s i o n f o r t h e i n t e r n a l energy o f a system where t h e t o t a l i n t e r m o l e c u l a r p o t e n t i a l i s pairwise additive. v

Ε = ρ/2 Σ Σ X i X j J u j i ( r ) g j i ( r ) 4 t r r d r 2

(16)

As p-»0, t h e r a d i a l d i s t r i b u t i o n f u n c t i o n g j i ( r ) approaches the Boltzmann f a c t o r -Uji(r)/RT

lim p-»0

(17)

gji(r) = e

I f E q u a t i o n ( 1 7 ) i s s u b s t i t u t e d i n t o E q u a t i o n ( 1 6 ) , i t c a n be shown ( 4 6 ) t h a t t h e i n t e r n a l energy a t low d e n s i t i e s i s r e l a t e d t o the t e m p e r a t u r e d e r i v a t i v e o f t h e second v i r i a l c o e f f i c i e n t : 3B2 j ι

(18)

Ε = Rp Σ Σ X i X j 1

J

3 1/T




COPEMAN A N D MATHIAS




F i g u r e 3. D e n s i t y dependence o f a t t r a c t i v e term - Simple equations o f s t a t e


17.

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365

Comparing E q u a t i o n (18) w i t h the l o w - d e n s i t y l i m i t o f E q u a t i o n (12) ( F - p ) , we o b t a i n an e x p r e s s i o n f o r a j i i n terms o f the second v i r i a l c o e f f i c i e n t . v

3B2 j

a i d

= -RT

J

1/T


-RT

-RT

>2ji

ι d(l/T)

1 / T

=0

r\

-

- B j ι (T = °°) 2

- bj

(19)

i

Thus, as i s w e l l known, a j ι a r i s e s from the a t t r a c t i v e p o r t i o n o f the j i p a i r p o t e n t i a l . E q u a t i o n s (16) and (17) show t h a t , a t low d e n s i t i e s , the s i m p l e e q u a t i o n s q u a l i t a t i v e l y d e s c r i b e the l o c a l c o m p o s i t i o n s r e s u l t i n g from the Boltzmann f a c t o r . We can now get some i d e a o f the h i g h - d e n s i t y r a d i a l d i s t r i b u t i o n f u n c t i o n assumed by the s i m p l e e q u a t i o n s o f s t a t e . The van der Waals e q u a t i o n assumes t h a t t h e r e i s no e f f e c t o f d e n s i t y . The Redlich-Kwong and Peng-Robinson models assume t h a t the l o w - d e n s i t y Boltzmann f a c t o r r e s u l t s i n an o v e r p r e d i c t i o n o f the h i g h - d e n s i t y configurâtional e n e r g y , but t h i s can be c o r r e c t e d by a s i m p l e f u n c t i o n of d e n s i t y , F /p. We emphasize t h a t t h i s i s a nonunique i n t e r p r e t a t i o n . More important t o the d i s c u s s i o n at hand, the vdW-1 m i x i n g r u l e s assume t h a t the l o c a l c o m p o s i t i o n s are independent o f d e n s i t y s i n c e the c o r r e c t i o n f a c t o r i s the same f o r a l l j i p a i r s . T h i s p o i n t has been noted by S a n d l e r (55) who a l s o s t a t e s t h a t t h i s b e h a v i o r i s i n c o n s i s t e n t w i t h the i n v e s t i g a t i o n s o f C h a n d l e r and Weeks (61) and Weeks et a l . (62) who have shown t h a t the s t r u c t u r e o f a h i g h - d e n s i t y f l u i d i s l a r g e l y d e t e r m i n e d by the r e p u l s i v e f o r c e s . However, these s i m p l e e q u a t i o n s o f s t a t e , t o g e t h e r w i t h the vdW-1 m i x i n g r u l e s , have r e s u l t e d i n r e l i a b l e and f r e q u e n t l y a c c u r a t e p r e d i c t i o n s f o r a wide v a r i e t y o f n o n p o l a r m i x t u r e s i n c l u d i n g those whose components have l a r g e d i f f e r e n c e s i n t h e i r intermolecular forces (63). I t i s i m p o r t a n t t o pursue the n a t u r e and impact o f t h e s e d i f f e r e n c e s t o g u i d e improvements t o the vdW-1 m i x i n g r u l e s f o r m i x t u r e s where they a r e o b v i o u s l y i n a d e q u a t e ( e . g . , p o l a r - n o n p o l a r b i n a r i e s l i k e methanol-benzene). We t h e r e f o r e r e v i e w the assumptions l e a d i n g t o E q u a t i o n ( 1 9 ) : a j i i s not s t r i c t l y a l o w - d e n s i t y parameter. I n f a c t , i t i s more c o r r e c t l y a h i g h - d e n s i t y parameter s i n c e i t s v a l u e i s d e t e r m i n e d by f i t t i n g vapor p r e s s u r e o r phase e q u i l i b r i u m d a t a , f o r w h i c h the v a r i a t i o n o f the l i q u i d - p h a s e f u g a c i t i e s u s u a l l y dominate. Further, we a l s o note t h a t the second v i r i a l c o e f f i c i e n t s p r e d i c t e d by the c u b i c e q u a t i o n s are u s u a l l y l e s s n e g a t i v e t h a n the e x p e r i m e n t a l v a l u e s , i n d i c a t i n g t h a t the e f f e c t o f the a t t r a c t i v e f o r c e s i s u n d e r p r e d i c t e d at low d e n s i t i e s . v


EQUATIONS OF


366

STATE: T H E O R I E S A N D

APPLICATIONS

Perhaps a b e t t e r i n t e r p r e t a t i o n o f E q u a t i o n s (16) and (17) i s t h a t the a j i a r i s e s f r o m an e f f e c t i v e p a i r p o t e n t i a l w h i c h p e r m i t s a r e a s o n a b l e d e s c r i p t i o n o f the i n t e r n a l energy at h i g h d e n s i t i e s . However, t h e s e s i m p l e models do not r e v e a l the form they assume f o r m i c r o s c o p i c q u a n t i t i e s , s i n c e we cannot deduce the form o f a f u n c t i o n from a s i n g l e value of i t s i n t e g r a l . T h i s u n c e r t a i n t y i n the j i d i s t r i b u t i o n f u n c t i o n s f o r the vdW-1 case makes an a n a l y s i s o f the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models v e r y tenuous i n d e e d . One argument, o f f e r e d by S a n d l e r and c o - w o r k e r s , s u g g e s t s t h a t the l o c a l c o m p o s i t i o n s caused by the a t t r a c t i v e f o r c e s a r e h i g h e s t at low d e n s i t i e s and d e c r e a s e w i t h d e n s i t y s i n c e h i g h - d e n s i t y s t r u c t u r e i s d e t e r m i n e d by the r e p u l s i v e f o r c e s . A c c o r d i n g t o t h i s argument the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models (29 - 31) a r e q u a l i t a t i v e l y wrong s i n c e t h e y assume t h a t the l o c a l - c o m p o s i t i o n e f f e c t i s z e r o at low d e n s i t i e s and i n c r e a s e s w i t h d e n s i t y . F u r t h e r comments a r e needed t o put t h i s statement i n t o p e r s p e c t i v e . I t s h o u l d f i r s t be n o t e d t h a t the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models do not p r e d i c t random b e h a v i o r at low d e n s i t i e s s i n c e they d e s c r i b e second v i r i a l c o e f f i c i e n t b e h a v i o r c o r r e c t l y ( E q u a t i o n 19). Now the q u e s t i o n i s : I s i t q u a l i t a t i v e l y c o r r e c t t o assume t h a t the l o c a l - c o m p o s i t i o n e f f e c t i n c r e a s e s w i t h density? S a n d l e r ' s s t u d y , c o n s i s t e n t w i t h o t h e r s t u d i e s , shows us t h a t the W i l s o n Boltzmann f a c t o r s do o v e r e s t i m a t e the l o c a l c o m p o s i t i o n e f f e c t at h i g h d e n s i t i e s . The computer s i m u l a t i o n s o f N a k a n i s h i and co-workers have shown t h a t t h e r e i s some l o c a l - c o m p o s i t i o n e f f e c t a t h i g h d e n s i t i e s due t o the a t t r a c t i v e f o r c e s and t h i s c l u s t e r i n g i s a p p r o x i m a t e l y d e s c r i b e d by the NRTL model. Thus t h e r e i s some j u s t i f i c a t i o n f o r the model at h i g h d e n s i t i e s and the d e n s i t y - d e p e n d e n t l o c a l - c o m p o s i t i o n models at l e a s t meets m a c r o s c o p i c boundary c o n d i t i o n s at low and h i g h d e n s i t i e s . We s t r e s s t h a t " l o c a l - c o m p o s i t i o n " i s an ambiguous and i n t e r m e d i a t e q u a n t i t y . I t i s perhaps more a p p r o p r i a t e t o e v a l u a t e models a c c o r d i n g t o t h e i r d e s c r i p t i o n o f more m e a n i n g f u l p r o p e r t i e s l i k e the configurâtional energy. The

Future

vdW-1 m i x i n g r u l e s a r e r e a s o n a b l y based i n t h e o r y and have enabled r e l i a b l e p r e d i c t i o n s i n many n o n p o l a r systems. F u r t h e r work u s i n g vdW-1 m i x i n g r u l e s as a b a s i s f o r e x t e n s i o n appears t o be warranted. P e r t u r b a t i o n theory suggests that separate mixing r u l e s are needed f o r d i f f e r e n t i n t e r m o l e c u l a r e f f e c t s . Approaches based on p e r t u r b a t i o n t h e o r y s h o u l d c o n t r i b u t e t o the c o n t i n u e d development o f new m i x i n g r u l e s . Most l o c a l c o m p o s i t i o n m i x i n g r u l e f o r m u l a t i o n s now resemble W i l s o n ' s e q u a t i o n , w h i c h b o t h o v e r e s t i m a t e s the l o c a l - c o m p o s i t i o n e f f e c t and becomes l e s s a c c u r a t e f o r m i x t u r e s w i t h i n c r e a s i n g s i z e d i f f e r e n c e s . New i d e a s a r e needed t o improve t h e s e d e f i c i e n c i e s . For example, E l y (64) has r e c e n t l y shown f o r s q u a r e - w e l l m i x t u r e s


17.

COPEMAN AND

-βε

j

nji - e


MATHIAS

367

3pj

ι

δ j

i

-

+

Bpj

8pi

Hard

4TT

R

3

(20)

Sphere

which can be s u b s t i t u t e d i n t o the energy e q u a t i o n ( E q u a t i o n 3) t o d e r i v e the m i x t u r e e q u a t i o n o f s t a t e . E l y ' s development i n c o r p o r a t e s hard-sphere s i z e e f f e c t s i n t o the l o c a l - c o m p o s i t i o n f o r m u l a t i o n . F u r t h e r work i n t h i s d i r e c t i o n i s s t r o n g l y encouraged.


Conclusions The c a p a b i l i t y o f e q u a t i o n s o f s t a t e t o d e s c r i b e asymmetric m i x t u r e s has improved s u b s t a n t i a l l y o v e r the p a s t f i v e y e a r s . The o l d r u l e o f thumb t h a t one must use a c t i v i t y c o e f f i c i e n t models f o r m i x t u r e s w i t h p o l a r components i s b e g i n n i n g t o f a d e away. T h i s t r e n d w i l l b e n e f i t c h e m i c a l p r o c e s s d e s i g n and o p t i m i z a t i o n , as more than one approach has been needed, i n some c a s e s , t o s i m u l a t e one e n t i r e process. Equations of s t a t e w i t h l o c a l - c o m p o s i t i o n mixing r u l e s o f f e r s h o r t - t e r m p o t e n t i a l t o d e s c r i b e h i g h l y p o l a r - a s y m m e t r i c systems i n the same ( e m p i r i c a l ) manner as a c t i v i t y c o e f f i c i e n t models. These models now l a r g e l y resemble W i l s o n ' s e q u a t i o n and new i d e a s a r e needed t o d e v e l o p improved f o r m u l a t i o n s . Computer s i m u l a t i o n s t u d i e s a r e b e i n g used t o p r o v i d e i n f o r m a t i o n on l o c a l f l u i d s t r u c t u r e . Some improvements t o m i x i n g r u l e s have been s u g g e s t e d , however, have not y e t been e v a l u a t e d on r e a l systems. F u r t h e r work t o improve m i x i n g r u l e s f o r a p p l i c a t i o n to r e a l systems based on computer s i m u l a t i o n i s b a d l y needed. While comparisons based on l o c a l c o m p o s i t i o n s can p r o v i d e some guidance i n d e v e l o p i n g b e t t e r models, a n a l y s i s o f m a c r o s c o p i c p r o p e r t i e s , such as a c t i v i t y c o e f f i c i e n t s and i n t e r n a l energy, i s recommended t o a v o i d p o t e n t i a l a m b i g u i t i e s and p i t f a l l s . A unified formulation of the energy, p r e s s u r e and c o m p r e s s i b i l i t y e q u a t i o n s i n terms o f l o c a l p a r t i c l e numbers now e x i s t s and can s t r a i g h t f o r w a r d l y be used t o d e v e l o p m i x t u r e e q u a t i o n s o f s t a t e f o r new f o r m u l a t i o n s . Acknowledgments We w i s h t o thank P r o f e s s o r s S. S a n d l e r , J . M. P r a u s n i t z and L. L. Lee and Dr. J . F. E l y f o r s e n d i n g us p r e p r i n t s o f t h e i r u n p u b l i s h e d p a p e r s , and P r o f e s s o r P r a u s n i t z f o r h i s h e l p f u l comments. L i s t o f Symbols a b Β Ε g k η Ν R Τ u

a t t r a c t i v e c o n s t a n t i n van der Waals-based e q u a t i o n o f s t a t e s i z e c o n s t a n t i n van d e r Waals-based e q u a t i o n o f s t a t e second v i r i a l c o e f f i c i e n t i n t e r n a l energy radial distribution function Boltzmann's c o n s t a n t l o c a l p a r t i c l e number number o f m o l e c u l e s l i m i t of integration a b s o l u t e temperature intermolecular pair potential


368 V X β δ ε μ ρ


σ


volume f l u i d - p h a s e mole f r a c t i o n 1/KT Kronecker d e l t a energy parameter i n p o t e n t i a l chemical p o t e n t i a l density hard-sphere diameter

function

Literature Cited 1. Sandler, S. I. Proceedings of the National Science Foundation Workshop, "Thermodynamic Needs for the Decade Ahead - Theory and Experiment," Washington, D. C., October 28-29, 1983. 2. Ely, J. F.; Baker J. K. "A Review of Supercritical Extraction;" NBS Technical Note 1070, 1983. 3. Chappelear, P. S.; Chen, R. J.; Elliot, D. E. Hydrocarbon Processing, 1977, 215-217. 4. George, B. A. Proceedings of the Sixty-first Annual Convention, Gas Processors Association, Dallas, March 15-17, 1982. 5. Miller, E. J.; Geist, J. M. Presented at the Joint Meeting of the Chemical Industry Engineering Society of China and the AIChE, Beijing, China, 1982. 6. Abrams, D. S.; Seneci, F.; Chueh, P. L.; and Prausnitz, J. M. Ind. Eng. Chem. Fundam., 1975, 14:52-54.31 7. van der Waals, J. D. Z. Phys. Chem., 1890, 5:133-173. 8. Soave, G. Chem. Eng. Sci., 1972, 27:1197. 9. Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam., 1976, 15:59-64. 10. Leland, T. W.; Chappelear, P. S.; Gamson, B. W. AIChEJ, 1962, 8:482-489. 11. Reid, R. C.; Leland, T. W. AIChEJ, 1965, 11:228-237. 12. Leland, T. W.; Chappelear, P. S. Ind. Eng. Chem., 1968, 60(7):15-43. 13. Henderson, D.; Leonard, P. J. Proc. Natl. Acad. Sciences, 1970, 67:1818-1823 14. Henderson, D.; Leonard, P. J. Proc. Natl. Acad. Sciences, 1971, 68:632-635 15. Vidal, J. Chem. Eng. Sci., 1978, 33:787-791. 16. Stotler, H. H.; Benedict, M. Chem. Eng. Prog. Symp. Ser., No. 6, 1953, 49:25-36. 17. Benedict, M.; Webb, G. B.; and Rubin, L. C. J. Chem. Phys., 1940, 8:334. 18. Benedict, M.; Webb, G. B.; and Rubin, L. C. Chem-Eng. Prog., 1951, 47:419. 19. Prausnitz, J. M.; Gunn, R. D. AIChEJ, 1958, 4:430-435. 20. Plocker, U.; Knapp, H.; Prausnitz, J. M. Ind. Eng. Chem. Proc. Des. Dev., 1978, 17:324. 21. Radosz, M.; Lin, H. M.; and Chao, K. C. Ind. Eng. Chem. Proc. Des. Dev., 1982, 21:653-658. 22. Lee, T. J.; Lee, L. L.; Starling, Κ. E. "Equations of State in Engineering and Research," 1979; Meeting of the American Chemical Society, September 11-14, 1978, 125-141. 23. Henderson, D. "Equations of State in Engineering and Reasearch," 1979, Meeting of the American Chemical Society, September 11-14, 1978, 1-30.



17. COPEMAN AND MATHIAS

Mixing Rules: An industrial Perspective

24. Donohue, M. D.; Prausnitz, J. M. AIChEJ, 1978, 24:849-860. 25. Wilson, G. M. Contribution No. 29, 1972, Center for Thermochemical Studies, Brigham Young University, March. 26. Huron, M. J.; Vidal, J. Fluid Phase Equilibria, 1979, 3:255-271. 27. Heyen, G. Proceedings of the 2nd World Congress of Chemical Engineering, October 4-9, 1981, 41-46. 28. Won, K. W. Presented at the Annual AIChE Meeting, New Orleans, November 8-12, 1981. 29. Whiting, W. B.; Prausnitz, J. M. Paper presented at Spring National Meeting, American Institute Chemical Engineers, Houston, Texas, April 5-9, 1981. 30. Whiting, W. B.; Prausnitz, H. M. Fluid Phase Equilibria, 1982, 9:119-147. 31. Mollerup, J. Fluid Phase Equilibria, 1981, 7:121-138. 32. Won, K. W. Fluid Phase Equilibria, 1983, 10:191-210. 33. Wilson, G. M. J. Am. Chem. Soc., 1964, 86:127-130. 34. Mathias, P. M.; Copeman, T. W. Presented at the Third International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, Callaway Gardens, Georgia, April 10-15, 1983. Also published in Fluid Phase Equilibria, 13:91-108. 35. Dimitrelis, D.; Prausnitz, J. M. Presented at Fall Annual Meeting, American Institute of Chemical Engineering, Los Angeles, CA, Nov. 14-19, 1982. 36. Mollerup, J. Fluid Phase Equilibria, 1983, 15:189-207 37. Ludecke, D.; Prausnitz, J. M. Fluid Phase Equilibria, 1985, 22:1-19. 38. Mansoori, G. Α.; Carnahan, N. F.; Starling, Κ. E.; Leland, T. W. J. Chem. Phys., 1971, 54:1523-1525. 39. Larsen, E. R.; Prausnitz, J. M. AIChEJ, 1984, 30:732-738. 40. Li, M. H.; Chung, F. T.; Lee, L. L.; Starling, Κ. E. Presented at the Fall Annual Meeting, American Institute of Chemical Engineers, San Francisco, CA, November 25-30, 1984. 41. Chung, T. H.; Khan, M. M.; Lee, L. L.; Starling, Κ. E. Fluid Phase Equilibria, 1984, 17:351. 42. Renon, H.; Prausnitz, J. M. AIChEJ, 1968, 14:135-144. 43. McDermott, C.; Ashton, N. Fluid Phase Equilibria, 1977, 1:33-35. 44. Kemeny, S.; Rasmussen, P. Fluid Phase Equilibria, 1981, 7:197-203. 45. Abrams, D. S.; Prausnitz, J. M. AIChEJ 1975, 21:116-128. 46. Lee, L. L.; Chung, R. H.; and Starling, Κ. E. Fluid Phase Equilibria, 1983, 12:105-124. 47. Mansoori, G. Α.; Leland, Jr., T. W. J. Chem. Soc., Faraday Trans. II, 1972, 68:320-344. 48. Seaton, Ν. Α.; Glandt, E. D. Presented at the Fall Annual Meeting, American Institute of Chemical Engineers, San Francisco, CA, November 25-30, 1984. 49. Gierycz, P.; Nakanishi, K. Fluid Phase Equilibria, 1984, 16:255-273. 50. Nakanishi, K.; Tanaka, H. Fluid Phase Equilibria, 1983, 13:371-380. 51. Nakanishi, K.; Taukabo, K. J. Chem. Phys., 1979, 70:5848-5850.


369


370

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

52. Gierycz, P.; Tanaka, H.; Nakanishi, K. Fluid Phase Equilibria, 1984, 16:241-253. 53. Hu, Y.; Ludecke, D.; Prausnitz, J. M. Fluid Phase Equilibria, 1984, 17:217-241. 54. Shing, K. S.; Gubbins K. E. Mol. Phys., 1983, 49:1121-1138. 55. Sandler, S. I. Fluid Phase Equilibria, 1985, 19:233-257. 56. Lee, K. H.; Lombardo, M.; Sandler, S. I. "The Generalized van der Waal's Partition Function II. Application to Square-Well Fluid," Fluid Phase Equilibria, 1985, in press. 57. Lee, Κ. H.; Sandler, S. I.; Patel, N. C. "The Generalized van der Waal's Partition Function III. Local-Composition Models for a Mixture of Equal Size Square-Well Molecules," Fluid Phase Equilibria, 1985, in press. 58. Mansoori, G. Α.; Ely, J. F. "Statistical Mechanical Theory of Local Compositions," submitted to Fluid Phase Equilibria, 1984, 59. Fischer, J. Fluid Phase Equilibria, 1983, 10:1-7. 60. Lebowitz, J. L.; Rowlinson, J. S. J. Chem. Phys., 1964, 41:133-138. 61. Chandler, D.; Weeks, J. D. Phys. Rev. Lettrs., 1970, 24:849 62. Weeks, J. D.; Chandler, D.; Anderson, H. C. J. Chem. Phys., 1971, 54:5237. 63. Oellrich, L.; Plocker, V.; Prausnitz, J. M.; Knapp, H. International Chemical Engineering, 1981, 21:1-16. 64. Ely, J. F., 1985, to be submitted for publication. RECEIVED November

5, 1985


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