Equations of State - American Chemical Society

28 New Mixing Rule for Cubic Equations of State for Highly Polar, Asymmetric Systems A. Z. Panagiotopoulos and R. C. Reid Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge,

Downloaded by PURDUE UNIVERSITY on March 25, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch028

M A 02139

A new two-parameter mixing rule for van der Waals-type cubic equations of state was developed by making the normally used single binary interaction parameter k a linear function of composition. A significant improvement was observed in the representation of binary and ternary phase equilibrium data for highly polar and asymmetric systems. Results are presented for systems with water and supercritical fluids at high pressures, as well as for low-pressure non-ideal systems. Ternary phase equilibrium data at high pressures, including LLG three-phase equilibria, were successfully correlated using parameters regressed from binary data only. ij

C u b i c e q u a t i o n s o f s t a t e (EOS) have become i m p o r t a n t t o o l s i n t h e a r e a o f p h a s e e q u i l i b r i u m m o d e l l i n g , e s p e c i a l l y f o r systems a t p r e s s u r e s c l o s e t o o r above t h e c r i t i c a l p r e s s u r e o f one o r more o f the system components. Among t h e more common o f t h e c u r r e n t l y u s e d c u b i c EOS, a r e t h e Soave m o d i f i c a t i o n o f t h e Redlich-Kwong (1) and the Peng-Robinson EOS ( 2 ) . The f u n c t i o n a l form o f b o t h e q u a t i o n s , as w e l l as s e v e r a l o t h e r p r o p o s e d c u b i c s , c a n be r e p r e s e n t e d i n a g e n e r a l manner as shown i n E q u a t i o n 1 (3.) : RT

a m 2 V + uVb

V -b m

(

1

)

2 + wb m m

where u a n d w a r e n u m e r i c a l c o n s t a n t s . T a b l e I l i s t s t h e v a l u e s o f u and w f o r some common EOS. F o r a m i x t u r e , parameters a and b a r e r e l a t e d t o t h e pure component parameters and t h e m i x t u r e c o m p o s i t i o n t h r o u g h a m i x i n g r u l e . E q u a t i o n s 2 and 3 show one common c h o i c e f o r t h e m i x i n g r u l e , the v a n d e r Waals 1 - f l u i d m i x i n g r u l e : m

a

= m b

m

Υ Υ r . ι J -

m

χ. χ. a. . î j i i J

(2)

J

X x. b . h ι ι ι

0097-6156/86/0300-0571 $06.00/0 © 1986 American Chemical Society

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

(3)

572

EQUATIONS OF STATE: THEORIES AND APPLICATIONS


T a b l e I . Parameters

f o r a few c u b i c E q u a t i o n s o f S t a t e

Equation o f State

u

w

van d e r W a l l s (1873)

0

0

Redlich-Kwong (1949)

1

0

Soave (1972)

1

0

Peng-Robinson (1976)

2

-1

The c r o s s - p a r a m e t e r s a ^ a r e r e l a t e d i n t u r n t o t h e pure-compo nent parameters by a "combining r u l e " . E q u a t i o n 4 shows a common form o f t h e combining r u l e f o r a : £ j

7 a. a.

a.. =

(1-k. .)

(4)

I n E q u a t i o n 4, k i s c a l l e d a b i n a r y i n t e r a c t i o n parameter, and was o r i g i n a l l y i n t r o d u c e d so t h a t t h e e q u a t i o n c a n b e t t e r reproduce e x p e r i m e n t a l c o m p o s i t i o n d a t a i n systems t h a t c o n t a i n components o t h e r than t h e l i g h t hydrocarbons. There has been c r i t i c i s m d i r e c t e d toward t h e o v e r s i m p l i c i t y o f the c u b i c e q u a t i o n form, and r i g h t l y so. N e v e r t h e l e s s t h i s r e p r e s e n t a t i o n does d e s c r i b e , a t l e a s t q u a l i t a t i v e l y , a l l the important c h a r a c t e r i s t i c s o f v a p o r - l i q u i d equilibrium behavior. Alternative e q u a t i o n s o f s t a t e have (and a r e ) b e i n g suggested, b u t , t o d a t e , none have been w i d e l y used and t e s t e d . A l s o , t o o o f t e n , a l t e r n a t e EOS a r e s i g n i f i c a n t l y more complex and b r i n g w i t h them a d d i t i o n a l pure-compo nent and m i x t u r e parameters which must be e v a l u a t e d by r e g r e s s i n g experimental data. We f e e l t h a t t h e k e y t o s u c c e s s i n employing t h e c u b i c e q u a t i o n form t o model phase e q u i l i b r i a i s i n t h e c h o i c e o f t h e m i x i n g and combining r u l e s . Our g o a l , then, was t o s e l e c t t h e s i m p l e s t forms, w i t h the fewest i n t e r a c t i o n parameters t h a t c o u l d be used t o c o r r e l a t e complex phase b e h a v i o r i n b o t h n o n - p o l a r and h i g h l y p o l a r systems. t j

Proposed

Method

We p r o p o s e an e m p i r i c a l m o d i f i c a t i o n o f t h e combining r u l e shown i n E q u a t i o n 4. I n o u r approach, we r e l a x t h e assumption k - k. thus i n t r o d u c i n g a second i n t e r a c t i o n parameter p e r b i n a r y : t j

a.. IJ

Ja.

a. [1-k..+(k..-k..)x.] ι

J

1 J 1 J

j i i

J

±

(5) v

'

E q u a t i o n 5 has t h e f o l l o w i n g c h a r a c t e r i s t i c s : I f k.. - k. recovered.

±

, t h e o r i g i n a l m i x i n g r u l e g i v e n by E q u a t i o n 4 i s


28. PANAGIOTOPOULOS A N D REID

Highly Polar, Asymmetric Systems

573

The " e f f e c t i v e " i n t e r a c t i o n parameter between components i and j a p p r o a c h e s k ^ a s x , t h e m o l e f r a c t i o n o f component i , approaches zero. I t a l s o approaches i fx approaches u n i t y . The apparent asymmetry under an i n t e r c h a n g e o f i and j i s c o r r e c t e d by t h e f a c t t h a t b o t h a ^ and a ^ e n t e r i n t h e c a l c u l a t i o n o f t h e m i x t u r e parameter a s y m m e t r i c a l l y . £

L

i

A

m

A p p l i c a t i o n o f t h e r u l e g i v e n by E q u a t i o n 2 f o r t h e c a l c u l a t i o n o f t h e m i x t u r e parameter a r e s u l t s i n a c u b i c e x p r e s s i o n f o r the mole f r a c t i o n dependence o f t h e m i x t u r e parameter a . T h i s i s d i f f e r e n t from t h e case o f the conventional mixing rule (Equation 4 ) , which r e s u l t s i n a q u a d r a t i c e x p r e s s i o n f o r a . m

m


m

U s i n g t h i s m i x i n g r u l e w i t h E q u a t i o n 1 , we f u g a c i t y c o e f f i c i e n t o f a component i n a m i x t u r e a s :

ο A *k

i n

o -

I

N

f

k ~X7P~

b

=

k —

, FV - . RT " > (

0

1

"

i

n

can o b t a i n the

P(V-b ) RT

^

m

+

2 Yx ( a +a

^ ?

i

ik

) - ΥΥχ χ j (ki „ k i '

1

J

i j

-k j J l ) 7 a1 aJ + χk Yx ^ i (k k i -ki k) "y ka a i J i

i

J

k^

ι

k i

i k

/

v

k

i

bk k

b~ 2

in Ju

z

- 4w b R T

2V + b (u-7u -4w) = 2V + b ( u + y u - 4 w )

(6)

2

m

An a l g o r i t h m f o r t h e d e t e r m i n a t i o n o f t h e phase e q u i l i b r i u m p r o p e r t i e s u s i n g t h e e q u a t i o n s d e s c r i b e d above was developed. The main c h a r a c t e r i s t i c s o f t h e method used a r e : P u r e component p a r a m e t e r e v a l u a t i o n : A r e c e n t l y p r o p o s e d technique ( 7 ) was used f o r t h e s u b c r i t i c a l components o f t h e m i x t u r e s under c o n s i d e r a t i o n . T h i s t e c h n i q u e , s i m i l a r t o t h e J o f f e a n d Z u d k e v i t c h m e t h o d (8.) , p r o v i d e s p u r e component p a r a m e t e r s t h a t e x a c t l y r e p r o d u c e vapor p r e s s u r e and l i q u i d volume o f a compound a t a temperature o f i n t e r e s t . The r e a s o n t h i s approach was p r e f e r r e d over t h e c o n v e n t i o n a l a c e n t r i c f a c t o r c o r r e l a t i o n i s t h a t t h e l a t t e r does n o t work w e l l f o r h i g h l y p o l a r o r a s s o c i a t e d c o m p o n e n t s . F o r s u p e r c r i t i c a l components though, t h e normal a c e n t r i c f a c t o r c o r r e l a t i o n was used. Multicomponent e q u i l i b r i a were c a l c u l a t e d as f o l l o w s : We s t a r t by assuming t h a t t h e component and i n t e r a c t i o n parameters a r e known a t a g i v e n temperature, and p o s t u l a t e t h e e x i s t e n c e o f a g i v e n number o f phases (2 o r 3 f o r t h e examples g i v e n b e l o w ) . Then, Newton's method i s a p p l i e d f o r t h e s o l u t i o n o f t h e system o f n o n - l i n e a r e q u a t i o n s g i v e n by E q u a t i o n 7. -

f f ( » f[

where t h e s u p e r s c r i p t s p e c i f i e s component number.

) ,

i « 1,2 ... Ν

(7)

t h e phase and i r e f e r s t o t h e


574


F o r t h e c a l c u l a t i o n s p r e s e n t e d i n t h i s paper, we e l e c t e d t o use the Peng - R o b i n s o n EOS. The v a l u e s o f t h e two b i n a r y i n t e r a c t i o n parameters were r e g r e s s e d from e x p e r i m e n t a l b i n a r y phase e q u i l i b r i u m d a t a on a g i v e n i s o t h e r m . I t was found t h a t a l t h o u g h t h e temperature v a r i a t i o n o f t h e parameters i s weak, i t must be t a k e n i n t o account i f a q u a n t i t a t i v e agreement w i t h e x p e r i m e n t a l d a t a over a wide temper a t u r e range i s d e s i r e d .


Results P o l a r - S u p e r c r i t i c a l F l u i d Systems. As s u g g e s t e d e a r l i e r , t h e p r i n c i p a l m o t i v a t i o n b e h i n d t h e development o f t h e new m i x i n g r u l e , h a s b e e n t h e r e p r e s e n t a t i o n o f phase e q u i l i b r i u m i n systems t h a t c o n t a i n a s u p e r c r i t i c a l component and one o r more h i g h l y p o l a r compounds, such as water. The inadequacy o f t h e c o n v e n t i o n a l method f o r such systems i s demonstrated i n F i g u r e 1, t h a t shows t h e p r e d i c t e d c o m p o s i t i o n o f t h e two phases f o r t h e system c a r b o n d i o x i d e - water a t 323 K, f o r b o t h the one-parameter and t h e two-parameter m i x i n g r u l e s . Parameter k was f i t t e d t o t h e s u p e r c r i t i c a l f l u i d phase c o n c e n t r a t i o n o f water (Y2) f o r b o t h models. Parameter k f o r t h e two parameter model was f i t t e d t o t h e l i q u i d phase c o m p o s i t i o n o f C 0 ( X I ) . I t i s c l e a r t h a t when t h e a d j u s t a b l e parameter i n t h e s i n g l e - p a r a m e t e r c o r r e l a t i o n i s f i t t e d t o t h e c o m p o s i t i o n o f one phase, t h e r e s u l t s f o r t h e o t h e r p h a s e a r e v e r y p o o r . I n c o n t r a s t , t h e two-parameter c o r r e l a t i o n p r e d i c t s t h e c o m p o s i t i o n o f b o t h phases e s s e n t i a l l y w i t h i n e x p e r i m e n t a l a c c u r a c y t h r o u g h o u t a p r e s s u r e range from a t m o s p h e r i c t o 1000 b a r . An i m p o r t a n t f e a t u r e o f t h e new method i s t h a t t h e two p a r a meters a r e e s s e n t i a l l y u n c o r r e l a t e d i n many c a s e s , as shown i n t h e p r e v i o u s example, i n w h i c h t h e parameters were d e t e r m i n e d from d a t a i n d i f f e r e n t phases. T h i s i s g e n e r a l l y t r u e o n l y f o r systems i n w h i c h t h e c o m p o s i t i o n s o f t h e c o e x i s t i n g phases a r e v e r y d i f f e r e n t . I n t h i s r e s p e c t , t h e proposed method i s s i m i l a r t o a p r e v i o u s l y s u g g e s t e d t e c h n i q u e f o r h i g h l y asymmetric systems (9) i n w h i c h a d i f f e r e n t i n t e r a c t i o n parameter i s used f o r each phase. The advantage o f t h e p r o p o s e d method i s b e s t seen i n systems t h a t approach a c r i t i c a l p o i n t ( f o r example a l i q u i d - l i q u i d phase ' s p l i t ) , and t h e r e f o r e cannot be a d e q u a t e l y m o d e l l e d i f a d i f f e r e n t e q u a t i o n i s used f o r d i f f e r e n t phases. I n F i g u r e 2, a comparison i s made between model p r e d i c t i o n s and the e x p e r i m e n t a l l y o b s e r v e d b e h a v i o r f o r t h e C 0 - water system over a temperature range from 298 Κ t o 348 Κ (pure C0 i s s u b c r i t i c a l below 304.2 K) . V e r y good agreement i s o b t a i n e d f o r a l l p r e s s u r e s f o r t h e c o m p o s i t i o n o f b o t h phases. An i n t e r e s t i n g q u e s t i o n a r i s e s i f we c o n s i d e r t h e l o w e r p r e s s u r e range examined. A t those c o n d i t i o n s , t h e d e n s i t y o f t h e gas phase i s low and d e v i a t i o n s from i d e a l i t y a r e s m a l l . One o f t h e reasons f o r the c o m p l e x i t y o f p r e v i o u s l y proposed m o d i f i c a t i o n s o f t h e m i x i n g r u l e s f o r c u b i c EOS (4,5,6) i s t h e r e q u i r e m e n t t h a t t h e m i x i n g r u l e s h o u l d reduce t o a q u a d r a t i c form f o r a m i x t u r e second v i r i a l c o e f f i c i e n t B a t l o w d e n s i t i e s , whereas t h e o b s e r v e d b e h a v i o r a t t h e h i g h d e n s i t y l i m i t c a n o n l y be m o d e l l e d w i t h a h i g h e r o r d e r m i x i n g r u l e . F o r a c u b i c EOS o f t h e type shown i n E q u a t i o n 1, t h e e x p r e s s i o n 1 2

2 1

2

2

2

m



28.

PANAGIOTOPOULOS AND REID

-1.0

Η

•

Wîebe and Gaddy

+

Coan and King

— -1.5

Η

-2.0

Η


PR

575

(1941)

(1971)

EOS

CM >CP

ο

k21 tr-

"2.5

k12 -

a -0.198

k21 -

+0.160

h0.160

Η 200

400

600

Pressure (bar) F i g u r e 1. E x p e r i m e n t a l and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - water a t 323 Κ. X I i s t h e mole f r a c t i o n o f C0 i n t h e l i q u i d phase and Y2 t h e mole f r a c t i o n o f water i n t h e f l u i d phase. 2


576


f o r t h e m i x t u r e second v i r i a l c o e f f i c i e n t i s a Β m

b m

m

(9)

RT

The proposed m i x i n g r u l e does n o t obey t h e r e q u i r e m e n t f o r a q u a d r a t i c dependence o f B , s i n c e t h e dependence o f a on t h e mole f r a c t i o n s o f t h e components o f a m i x t u r e i s c u b i c . Because o f t h i s , i t was s u r p r i s i n g a t f i r s t t h a t such a good agreement was found a t the l o w p r e s s u r e range between model p r e d i c t i o n s and e x p e r i m e n t a l results. To compare t h e performance o f t h e p r o p o s e d m i x i n g r u l e f o r the m i x t u r e v o l u m e t r i c p r o p e r t i e s , we c a l c u l a t e d t h e m i x t u r e second v i r i a l c o e f f i c i e n t a t 323 Κ u s i n g E q u a t i o n 7 w i t h t h e v a l u e s o f k and k c a l c u l a t e d e a r l i e r , as w e l l as u s i n g a common i n t e r a c t i o n parameter e q u a l t o t h e i r a r i t h m e t i c average. I n t h e l a t t e r c a s e , t h e m i x i n g r u l e r e s u l t s i n a q u a d r a t i c f u n c t i o n a l i t y f o r B . As c a n be seen from F i g u r e 3, t h e d i f f e r e n c e i n t h e c a l c u l a t e d dependence o f B on t h e m i x t u r e c o m p o s i t i o n between t h e two cases i s i n d e e d s m a l l . A d d i t i o n a l examples o f t h e model performance f o r p o l a r - super c r i t i c a l f l u i d systems a r e shown i n F i g u r e s 4 and 5 f o r t h e c a r b o n d i o x i d e - e t h a n o l and t h e c a r b o n d i o x i d e - acetone systems. A g a i n , agreement between model p r e d i c t i o n s and experiment i s e x c e l l e n t , even quite c l o s e t o mixture c r i t i c a l p o i n t s . I t i s i n t e r e s t i n g t o note, t h a t f o r t h e system c a r b o n d i o x i d e - a c e t o n e , t h e o p t i m a l v a l u e s o f the two parameters a r e q u i t e c l o s e t o each o t h e r . U s i n g t h e conven t i o n a l m i x i n g r u l e f o r t h i s p a r t i c u l a r system would have r e s u l t e d i n a l m o s t as good agreement between experiment and p r e d i c t i o n as f o r the two - parameter c o r r e l a t i o n . I t i s e n c o u r a g i n g t o n o t e t h a t t h e proposed c o r r e l a t i o n n a t u r a l l y leads i t s e l f t o the e l i m i n a t i o n o f the second parameter i f t h e systems b e i n g m o d e l l e d a r e r e l a t i v e l y s i m p l e . m

m

1 2


2 1

m

m

Low P r e s s u r e VLE. Up t o t h i s p o i n t , we have f o c u s e d o u r d i s c u s s i o n on s y s t e m s a t r e l a t i v e l y h i g h p r e s s u r e s . F o r such systems, t h e e q u a t i o n o f s t a t e approach has a l r e a d y been t e s t e d e x t e n s i v e l y . I f we a r e i n t e r e s t e d i n m o d e l l i n g t e r n a r y phase b e h a v i o r , we a l s o need i n t e r a c t i o n parameters between t h e l e s s v o l a t i l e components o f a m i x t u r e t h a t do n o t form two-phase systems a t h i g h p r e s s u r e s . The e a s i e s t way t o o b t a i n such parameters i s t o u t i l i z e VLE d a t a a t low pressures that are widely available. As a t e s t o f t h e a p p l i c a b i l i t y o f t h e method f o r l o w p r e s s u r e s y s t e m s , we c o r r e l a t e d i s o t h e r m a l VLE d a t a f o r t h e b i n a r y system e t h a n o l - water a t a range o f temperatures ( 1 8 ) . The r e s u l t s a r e p l o t t e d i n F i g u r e 6. Agreement i s a g a i n v e r y good, even c l o s e t o t h e a z e o t r o p i c r e g i o n , and t h e a c c u r a c y o f t h e p r e d i c t i o n s i s c l e a r l y comparable t o t h e a c c u r a c y o f excess Gibbs Free Energy models w i t h the same number o f a d j u s t a b l e parameters. T e r n a r y Systems One o f t h e most s t r i n g e n t t e s t s f o r a proposed c o r r e l a t i o n i s i t s a b i l i t y t o p r e d i c t t e r n a r y b e h a v i o r when o n l y t h e b i n a r y b e h a v i o r i s known. I n F i g u r e 7 t h e model p r e d i c t i o n s f o r t h e t e r n a r y system c a r b o n d i o x i d e - e t h a n o l - water a r e p r e s e n t e d . The m o d e l p r e d i c t i o n s a r e based s o l e l y on v a l u e s o f t h e i n t e r a c t i o n parameters r e g r e s s e d from b i n a r y d a t a a t t h e same temperature ( t h e v a l u e s o f t h e pure component and m i x t u r e parameters u s e d a r e a l s o shown on t h e same f i g u r e ) .



28.



577

Mole fraction C02 F i g u r e 2. E x p e r i m e n t a l and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - water a t s e v e r a l t e m p e r a t u r e s . E x p e r i m e n t a l d a t a a r e from r e f e r e n c e s 12-15.

θ η ? ο

- 5 -

ιο

-10 -

Ε

-40

Η 0

,

, 0.2

,

, 0.4

1

1 0.6

1

1 0.8

1

1

Mole Fraction C02 F i g u r e 3. P r e d i c t e d second v i r i a l c o e f f i c i e n t v e r s u s c o m p o s i t i o n f o r t h e c a r b o n d i o x i d e - water system a t 323 K.


1


578


120

0

0.2

0.4

0.6

0.8

Mole fraction C 0 2 F i g u r e 4. Experimental (10) and p r e d i c t e d phase b e h a v i o r f o r t h e system c a r b o n d i o x i d e - e t h a n o l .

0

0.2

0.4

0.6

equilibrium

0.8

Mole Fraction Carbon Dioxide F i g u r e 5. Experimental (16) and p r e d i c t e d phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - a c e t o n e a t 313 K.


1



579


28.

Figure 6. Experimental (18) and p r e d i c t e d b e h a v i o r f o r the system e t h a n o l - water.

phase

equilibrium



580


As c a n b e s e e n f r o m F i g u r e 7, t h e a c c u r a c y o f t h e m o d e l p r e d i c t i o n s a t t h e low e t h a n o l c o n c e n t r a t i o n range i s v e r y good. The c a l c u l a t e d t i e - l i n e s d e v i a t e from t h e e x p e r i m e n t a l d a t a as t h e p l a i t p o i n t i s approached, b u t s t i l l t h e c o r r e c t q u a l i t a t i v e b e h a v i o r i s p r e d i c t e d . One more i l l u s t r a t i v e example o f t h e c a p a b i l i t i e s o f the model i s g i v e n i n F i g u r e 8 f o r t h e t e r n a r y system carbon d i o x i d e - acetone - w a t e r a t 313 K. The most prominent c h a r a c t e r i s t i c o f t h e model p r e d i c t i o n s f o r t h e system i s an e x t e n s i v e three-phase r e g i o n . A t t h i s temperature, t h e s o l u b i l i t y o f carbon d i o x i d e i n t h e acetone phase i s v e r y h i g h even a t moderate p r e s s u r e s . T h i s c a n be seen a l r e a d y i n F i g u r e 8a t h a t corresponds t o a p r e s s u r e o f 20 b a r . As p r e s s u r e i s i n c r e a s e d t o approx. 22 b a r ( F i g u r e 8b), t h e l i q u i d phase undergoes a phase s p l i t , i n t o a lower l i q u i d phase r i c h i n w a t e r and a m i d d l e l i q u i d phase r i c h i n acetone, w h i c h c o e x i s t w i t h a f l u i d phase r i c h i n C 0 . The t h r e e phase r e g i o n i n c r e a s e s i n s i z e as p r e s s u r e i s i n c r e a s e d ( F i g u r e s 8c, 8d) and t h e n g r a d u a l l y s h r i n k s a g a i n ( F i g u r e 8e) . A t approx. 82 b a r (not shown on F i g u r e 8 ) , t h e system passes through a c r i t i c a l s t a t e a g a i n , and t h e m i d d l e l i q u i d phase becomes i d e n t i c a l t o t h e s u p e r c r i t i c a l phase. A t even h i g h e r p r e s s u r e s , o n l y two phases c o e x i s t , and t h e e f f e c t o f p r e s s u r e on t h e c o m p o s i t i o n o f t h e phases i s much s m a l l e r ( F i g u r e 8 f ) . T h i s c o m p l e x b e h a v i o r was o b s e r v e d e a r l i e r f o r t h e system e t h y l e n e - acetone - water by E l g i n and W e i n s t o c k ( 1 1 ) . T h e i r " Type 2 " q u a l i t a t i v e phase diagrams b e a r a s t r i k i n g resemblance t o our m o d e l p r e d i c t i o n s . The m o d e l p r e d i c t i o n s a r e f u l l y s u p p o r t e d by e x p e r i m e n t a l e v i d e n c e from o u r l a b o r a t o r y as i n d i c a t e d on F i g u r e s 8c, 8d and 8e, i n which t h e measured c o m p o s i t i o n s o f t h e t h r e e phases a r e shown i n a d d i t i o n t o t h e model r e s u l t s . The c a l c u l a t e d phase c o m p o s i t i o n s a r e n o t i n e x a c t agreement w i t h t h e e x p e r i m e n t a l d a t a , b u t t h e c o r r e c t p r e d i c t i o n o f t h e appearance and d i s a p p e a r a n c e o f t h e t h i r d phase s t r o n g l y i m p l i e s t h a t t h e model c a p t u r e s t h e s u b s t a n t i a l f e a t u r e s o f t h e p h y s i c a l r e a l i t y . A more complete p r e s e n t a t i o n o f t h e p e r t i n e n t e x p e r i m e n t a l r e s u l t s and comparison w i t h model p r e d i c t i o n s i s g i v e n elsewhere ( 1 0 ) . 2

Conclusions A new t w o - p a r a m e t e r m i x i n g r u l e i s p r o p o s e d f o r use i n c u b i c E q u a t i o n s o f S t a t e and i s shown t o be e s p e c i a l l y u s e f u l i n c o r r e l a t i n g t h e phase e q u i l i b r i u m b e h a v i o r i n h i g h l y p o l a r systems t h a t c a n n o t b e c o r r e c t l y r e p r e s e n t e d by a c o n v e n t i o n a l one-parameter m i x i n g r u l e . The i n t r o d u c t i o n o f E q u a t i o n 5, i s a t t h i s p o i n t a p u r e l y e m p i r i c a l c o r r e c t i o n t o t h e o r i g i n a l form o f t h e m i x i n g r u l e . The m o d i f i c a t i o n i s r e l a t e d , b u t i s n o t d i r e c t l y d e r i v e d from, the i d e a o f l o c a l c o m p o s i t i o n s , t h a t has been shown i n t h e p a s t t o r e s u l t i n improved r e p r e s e n t a t i o n o f t h e phase e q u i l i b r i u m b e h a v i o r i n h i g h l y p o l a r and asymmetric m i x t u r e s . Among t h e a d v a n t a g e s o f t h e p r o p o s e d m i x i n g r u l e , a r e t h e relative simplicity of the r e s u l t i n g expressions f o r the derived thermodynamic p r o p e r t i e s , t o g e t h e r w i t h t h e f a c t t h a t t h e model c a n be e a s i l y reduced t o t h e c o n v e n t i o n a l one-parameter m i x i n g r u l e f o r w h i c h a s u b s t a n t i a l amount o f r e g r e s s e d i n t e r a c t i o n parameters e x i s t s . The model i s shown t o a c c u r a t e l y reproduce d a t a f o r b o t h h i g h


28.

PANAGIOTOPOULOS A N D REID


581


Ethanol (2)

Water (3)

C02C1) Τ = 313 Κ . Ρ = 103 ba

F i g u r e 7. T e r n a r y phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - e t h a n o l - water ( e x p e r i m e n t a l d a t a from 12,12.) .

Wator

P - 20.• bar

Ρ - 40. 1 bar

C

0

W

a

t

e

r

2

^

Water

Ρ - 22. 3 bar

C

m

5 7 m l

b

a

r

W

o

t

e

2

C p

0

0 2

*

Q t e r

r

Ρ - 30.0 bar

Ρ - 100 bar

F i g u r e 8. T e r n a r y phase e q u i l i b r i u m b e h a v i o r f o r t h e system c a r b o n d i o x i d e - acetone - water a t 313 Κ ( e x p e r i m e n t a l d a t a (10) f o r t h e t h r e e - p h a s e e q u i l i b r i u m c o m p o s i t i o n s i n c , d and e a r e shown as f i l l e d t r i a n g l e s ) .



582

p r e s s u r e p o l a r - s u p e r c r i t i c a l f l u i d , and l o w p r e s s u r e p o l a r - p o l a r b i n a r y phase e q u i l i b r i u m . I n addition, predictions f o r ternary systems based on c o e f f i c i e n t s r e g r e s s e d from b i n a r y d a t a o n l y , a r e q u a l i t a t i v e l y c o r r e c t f o r the systems s t u d i e d . One p o s s i b l e weakness o f t h e p r o p o s e d model, i s t h a t i t does not reproduce t h e c o r r e c t f u n c t i o n a l dependence o f a m i x t u r e second v i r i a l c o e f f i c i e n t on the c o m p o s i t i o n .


Acknowledgments F i n a n c i a l s u p p o r t f o r t h i s r e s e a r c h was p r o v i d e d b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n (CPE-8318494). One o f t h e a u t h o r s (AZP) was a l s o s u p p o r t e d by a f e l l o w s h i p from H a l c o n I n c . We g r a t e f u l l y acknowledge both sponsors.

Literature Cited 1. Soave, G., Chem. Eng. Sci. 1972, 27, 1197-1203. 2. Peng, D.-Y. and Robinson, D.B., Ind. Eng. Chem. Fundam. 1976, 15(1), 59-64. 3. Schmidt G. and Wenzel Η., Chem. Eng. Sci., 1980, 35, 1503-1512. 4. Whiting, W.B. and Prauznitz, J.M., Fluid Phase Equil. 1982, 9, 119-147. 5. Mollerup, J., Fluid Phase Equil. 1981, 7, 121-138. 6. Mathias, P.M. and Copeman, T.W., Fluid Phase Equil. 1983, 13, 91-108. 7. Panagiotopoulos, A.Z. and Kumar, S.K., Fluid Phase Equil. 1985, 22, 77-88. 8. Joffe, J. and Zudkevitch, D., Ind. Eng. Chem. Fundam. 1970, 9(4), 545-548. 9. Robinson, D.B., Peng, D.-Y. and Chung C.Y.-K., " The Development of the Peng - Robinson Equation and its Application to Phase Equilibrium in a System containing Methanol", paper presented at the Annual AIChE meeting in San Fransisco, CA, November 1984. See also Peng, D.-Y. and Robinson, D. Β., ACS Symposium Series, 1980, 133, 393-414. 10. Panagiotopoulos, A.Z. and Reid, R.C., 1985, " High Pressure Phase Equilibria in Ternary Fluid Mixtures with a Supercritical Fluid ", ACS Division of Fuel Chemistry Preprints, vol.30 No 3, 46-56. 11. Elgin, J.C. and Weinstock, J.J., J. Chem. Eng. Data 1959, 4(1), 3-12. 12. Wiebe, R. and Gaddy, V.L., J. Am. Chem. Soc. 1941, 63,475-77 and Wiebe, R., Chem. Rev. 1941, 29, 475-81. 13. Coan, C.R. and King, A.P., J. Am. Chem. Soc. 1971, 93, 1857-62. 14. Matous, J. et al., Coll. Czech. Chem. Commun. 1969, 34, 3982-85. 15. Zawisza, A. et al., J. Chem. Eng. Data 1981, 26, 388-391. 16. Katayama, T., et al., J. Chem. Eng. Jpn. 1975, 8(2) , 89-92. 17. Kuk, M.S. and Montagna, J.C., Chapter 4 in Paulaitis et al. (ed.), " Chemical Engineering at Supercritical Fluid Conditions ", Ann Arbor Science Publishers, 1983. 18. Pemberton, R.C. and Mash, C.J., J. Chem. Thermodynamics 1978, 10, 867-888. RECEIVED November 5, 1985


Equations of State - American Chemical Society

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