Equations of State in Engineering and Research - ACS Publications

14

Industrial Experience in Applying the Redlich-Kwong Equation to Vapor-Liquid Equilibria

Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch014

R. D. GRAY, JR. Exxon Research and Engineering Co., Florham Park, NJ 07932

This chapter provides a critical assessment of the strengths and weaknesses of methods which use the Redlich-Kwong equation of state to correlate and predict vapor-liquid equi libria. A brief review is given of well established strengths (effectiveness for high pressure and cryogenic light hydro carbon systems) and weaknesses (inability to represent de tails of density dependence, problems with some versions of the Redlich-Kwong method for supercritical gases, and difficulties with polar compounds). A more detailed dis cussion is given for (cryogenic H -containing systems as well as for certain problems in representing details of paraffinparaffin binaries for critical region behavior or for wide ranges of conditions and difficulties with heavy hydrocarbon systems). Based on this experience, a set of desirable criteria for the next generation of equation-of-state methods is provided. 2

^T^his c h a p t e r d r a w s o n over eight years experience i n u s i n g t h e R e d l i c h K w o n g ( R K ) e q u a t i o n of state to correlate a n d p r e d i c t v a p o r - l i q u i d equilibrium

( V L E ) behavior i n petroleum refining a n d petrochemical

a p p l i c a t i o n s . T h e p e r s p e c t i v e is that of a t h e r m o d y n a m i c d a t a d e v e l o p m e n t a n d i n t e r n a l c o n s u l t i n g g r o u p w h i c h takes r e s p o n s i b i l i t y f o r t h e a c c u r a c y of t h e p r e d i c t i o n s of t h e d a t a m e t h o d s i t r e c o m m e n d s . k i n d of experience often leads one to u n c o v e r weaknesses

This

i n a data

c o r r e l a t i o n . A t t h e same t i m e , one often is f o r c e d to e x t e n d a c o r r e l a t i o n far outside its i n t e n d e d r e g i o n of v a l i d i t y , sometimes u n c o v e r i n g u n e x p e c t e d c a p a b i l i t i e s . G e n e r a l l y , o n e discovers t h i n g s i r i a p p l i c a t i o n e x p e r i ence that are difficult to discover i n a n y other w a y . 0-8412-0500-0/79/33-182-253$05.00/l © 1979 American Chemical Society

In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

254

EQUATIONS

OF

STATE

T h e objective of this c h a p t e r is to s u m m a r i z e the m a j o r patterns of s t r e n g t h a n d weakness w h i c h r e c u r i n the a p p l i c a t i o n of the R K e q u a t i o n of state to p r e d i c t phase b e h a v i o r . well

S o m e of these patterns a r e q u i t e

e s t a b l i s h e d i n p r e v i o u s l i t e r a t u r e , a n d so a r e s u m m a r i z e d

only

briefly. S o m e less w e l l - r e c o g n i z e d strengths a n d weaknesses are e x p l o r e d i n some d e t a i l . concerning

B a s e d o n this experience, some c o n c l u s i o n s a r e d r a w n

d e s i r a b l e features

of t h e next g e n e r a t i o n of m e t h o d s f o r

p r e d i c t i n g p h a s e - e q u i l i b r i u m b e h a v i o r . I t is h o p e d t h a t this synopsis o f i n d u s t r i a l experience w i l l p r o v e u s e f u l n o t o n l y t o t h e m a n y c u r r e n t l y u s i n g R K m e t h o d s , b u t also to those i n t h e a c a d e m i c w o r l d w h o a r e


engaged i n developing n e w methods. Chronology

of RK Application

to VLE

A n a b b r e v i a t e d c h r o n o l o g y of the d e v e l o p m e n t o f R K m e t h o d s is a u s e f u l w a y to b e g i n this d i s c u s s i o n . A l t h o u g h W i l s o n (1) first p r o p o s e d the i n t r o d u c t i o n of a t e m p e r a t u r e - d e p e n d e n t p a r a m e t e r to r e p l a c e one o f t w o constants i n 1964, m u c h of the p o p u l a r i t y of t h e R K m e t h o d stems f r o m the e x t r e m e l y s i m p l e t e m p e r a t u r e d e p e n d e n c e i n t r o d u c e d b y Soave (2)

i n 1972.

I n t h e i n t e r v e n i n g p e r i o d , Joffe a n d Z u d k e v i t c h

(3,4,5)

i n 1969 a n d 1970 a n d C h a n g a n d L u (6) i n 1970 h a d p r o p o s e d m a k i n g b o t h constants t e m p e r a t u r e d e p e n d e n t , i n s p i r e d i n p a r t b y a series o f p a p e r s b y C h u e h a n d P r a u s n i t z (7,8) w h i c h d e m o n s t r a t e d t h a t t h e R K e q u a t i o n c a n b e a d a p t e d to p r e d i c t b o t h v a p o r a n d l i q u i d p r o p e r t i e s . T h e m o r e c o m p l e x J o f f e - Z u d k e v i t c h ( J Z ) m e t h o d w a s n o t as w i d e l y a d o p t e d as the Soave p r o c e d u r e . T w o n o t e w o r t h y recent d e v e l o p m e n t s a r e t h e P e n g a n d R o b i n s o n equation

a n d t h e G r a b o s k i - D a u b e r t v e r s i o n of t h e Soave

(9,10,11)

m e t h o d (12).

T h e P e n g - R o b i n s o n w o r k is p a r t o f a systematic attack

o n phase b e h a v i o r of interest i n gas p r o c e s s i n g , w i t h e s p e c i a l l y i m p r e s s i v e t r e a t m e n t of c r i t i c a l r e g i o n effects, w h i l e t h e G r a b o s k i - D a u b e r t w o r k p r o v i d e s a c o m p r e h e n s i v e a p p l i c a t i o n of t h e Soave m e t h o d t o a l a r g e variety

of hydrocarbon

a n d related

systems

o f interest i n r e f i n i n g

applications. I n this c h a p t e r , most of t h e a p p l i c a t i o n experience is f o r t h e J Z method,

w h i c h has been

used

at E x x o n R e s e a r c h

a n d Engineering

C o m p a n y since before its p u b l i c a t i o n (see A c k n o w l e d g m e n t

Section).

H o w e v e r , most o f t h e c o n c l u s i o n s a p p l y to t h e Soave o r other f o r m s , w i t h m o d i f i c a t i o n s as n o t e d . Summary

of Procedures

Equation

to VLE

to Adapt

the RK

Prediction

T h e R K e q u a t i o n of state, r e l a t i n g pressure Ρ t o m o l a r v o l u m e V and temperature Τ is

P =

RT V

-6

aT'1/2 V(V + b)


(1)

14.

The Redlich-Kwong

G R A Y

255

Equation

I n a d a p t i n g the R K e q u a t i o n for 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 tions, p u r e - c o m p o n e n t p a r a m e t e r s are a d j u s t e d to m a t c h v a p o r a n d l i q u i d f u g a c i t y a l o n g the v a p o r pressure locus.

I n the Soave m o d i f i c a t i o n o n l y

the R K p a r a m e t e r a is t e m p e r a t u r e d e p e n d e n t , w h i l e for the J Z m o d i f i c a t i o n b o t h a a n d b are t e m p e r a t u r e d e p e n d e n t . U s i n g the n o m e n c l a t u r e i n t r o d u c e d b y C h u e h a n d P r a u s n i t z

(6,7),

p u r e - c o m p o n e n t p a r a m e t e r s are expressed i n terms of dimensionless p r e m u l t i p l i e r s Ω . a n d Ω .. α

&

τ>2Τ>

Ω


at =

p

α4

2.5

(2)

C i

i^aH e r e , Ω .° =

0.42747 . . . a n d Ω . ° =

α

δ

(3)

0.08664 . . . w i l l be u s e d to d e n o t e

the values of the p r e m u l t i p l i e r s u s e d i n the o r i g i n a l R K e q u a t i o n . I n terms of this n o m e n c l a t u r e , the Soave p r e m u l t i p l i e r , Ω . ° ,

gen

σ

e r a l i z e d i n terms of P i t z e r ' s a c e n t r i c f a c t o r ωι a n d the r e d u c e d t e m p e r a t u r e Tr., is Ω

σί

=

Ω ° T

1/2

RI

α

{1 +

(0.48 +

1-574

-

ω ί

0.176

ωί

2

) (1 -

T> ) } R

/2

(4)

2

T h e J o f f e - Z u d k e v i t c h ( R K J Z ) m o d i f i c a t i o n i n c l u d e s v a r i a t i o n s of b o t h Ω . a n d Ω&. w i t h t e m p e r a t u r e to fit l i q u i d d e n s i t y as w e l l as to m a t c h σ

v a p o r - t o - l i q u i d f u g a c i t y a l o n g the v a p o r pressure locus. method

proposed by

Z u d k e v i t c h a n d Joffe

(4)

(The

original

had matched

f u g a c i t y to a g e n e r a l i z e d v a p o r f u g a c i t y c o r r e l a t i o n , b u t the

liquid present

s t u d y f o l l o w s the m e t h o d t h e y a d o p t e d i n a later p u b l i c a t i o n ( 5 ) . )

By

c o m p a r i s o n w i t h the Soave p r o c e d u r e , this m e t h o d loses some a c c u r a c y i n s a t u r a t e d v a p o r densities

( a l t h o u g h this causes v e r y little loss

of

a c c u r a c y i n v a p o r f u g a c i t y ) , w h i l e g r e a t l y i m p r o v i n g the a c c u r a c y

of

saturated l i q u i d density. T h e p r i n c i p a l d i s a d v a n t a g e of

the R K J Z m e t h o d is the

t e m p e r a t u r e d e p e n d e n c e of Ω . a n d Ω .. σ

&

complex

A l t h o u g h H a m a n et a l .

(13)

p r o v i d e d a g e n e r a l i z e d c o r r e l a t i o n for Ω . a n d Ω&., these q u a n t i t i e s are n o t α

g e n e r a l i z e d i n the f o r m d e s c r i b e d here b u t are generated e a c h t i m e t h e y are n e e d e d f r o m l i q u i d densities a n d v a p o r pressures. V a r i o u s sources of l i q u i d d e n s i t y a n d v a p o r pressure are u s e d : versions of the R i e d e l c o r r e lations

(14,15);

A P I 44 A n t o i n e equations

(16);

petroleum

fraction

correlations; a n d c u r v e fits of e x p e r i m e n t a l i n f o r m a t i o n . A n a d v a n t a g e of the J Z p r o c e d u r e is that for s u b c r i t i c a l c o m p o n e n t s , Ω

α ί

and Ω

6 {

are u s e d o n l y as i n t e r m e d i a t e i n t e r n a l v a r i a b l e s . T h a t is,

s p e c i f y i n g v a p o r pressure a n d s a t u r a t e d l i q u i d d e n s i t y p r o v i d e s a l l of the


256

EQUATIONS

OF

STATE

i n f o r m a t i o n n e e d e d to c a l c u l a t e a a n d b; c r i t i c a l p r o p e r t i e s are not n e e d e d to c a l c u l a t e R e d l i c h - K w o n g parameters of h e a v y solvents or p e t r o l e u m fractions. F o r s u p e r c r i t i c a l c o m p o n e n t s , users of the Soave m e t h o d

generally

use E q u a t i o n 4 e x t r a p o l a t e d a b o v e the c r i t i c a l t e m p e r a t u r e , w h e r e a s the R K J Z m e t h o d uses the l i m i t i n g v a l u e of Ω„. a n d Ω . at the c r i t i c a l t e m &

p e r a t u r e . A l t h o u g h the R K J Z p r o c e d u r e i n t r o d u c e s a c o r n e r i n t o the Ω and Ω

&

α

t e m p e r a t u r e d e p e n d e n c e ( a n d thus i n t o the t e m p e r a t u r e d e p e n d

ence of f u g a c i t y ), it does h a v e the a d v a n t a g e of p r e s e r v i n g the p h y s i c a l l y reasonable h i g h - t e m p e r a t u r e l i m i t of the o r i g i n a l R K m e t h o d for s u p e r


c r i t i c a l gases.

T h a t is, the second v i r i a l coefficient tends a s y m p t o t i c a l l y

t o w a r d s b, a s m a l l p o s i t i v e n u m b e r . T h i s l i m i t i n g b e h a v i o r is significant for the v e r y w i d e t e m p e r a t u r e ranges i n r e f i n i n g a p p l i c a t i o n s . M i x i n g rules u s e d i n the present w o r k f o l l o w the u s u a l p r a c t i c e : the o r i g i n a l m o l a r average m i x i n g r u l e for b, w i t h a n adjustable i n t e r a c t i o n p a r a m e t e r ( C ) i n the m i x i n g r u l e for a: i ;

a=

Σ i

Σ

(5)

ViVjOàj

J

where du =

au-

di

(1 -

(aiajV*

C„)

i = j

(6)

i^j

(7)

T h e major a l t e r n a t i v e to E q u a t i o n 7 is to use the p r o c e d u r e s C h u e h a n d P r a u s n i t z to c a l c u l a t e a

ih

p e r a t u r e , pressure, a n d v o l u m e

of

by calculating pseudocritical tem-

i n a n i n t e r m e d i a t e step, w i t h the

fc

{;

p a r a m e t e r i n the c o m b i n i n g r u l e for p s e u d o c r i t i c a l t e m p e r a t u r e p e r f o r m i n g the f u n c t i o n of E q u a t i o n 7. T h i s has the a d v a n t a g e of p r o v i d i n g a c o n n e c t i o n w i t h the c o n s i d e r a b l e l i t e r a t u r e o n kq p a r a m e t e r s .

T h e dis-

a d v a n t a g e for the R K J Z m e t h o d is that it i m p o r t s c r i t i c a l p r o p e r t i e s i n t o m i x i n g rules even component

t h o u g h they are not r e q u i r e d to define

parameters for s u b c r i t i c a l c o m p o u n d s .

the

F o r systems

purewhere

c r i t i c a l p r o p e r t i e s are w e l l - d e f i n e d , one r e a d i l y c a n t r a n s f o r m f r o m set of m i x i n g rules to the other, as n o t e d b y K a t o , C h u n g , a n d L u Once

procedures

for

calculating pure-component

one

(17).

parameters

and

m i x i n g rules are e s t a b l i s h e d , the c a l c u l a t i o n of c o m p o n e n t f u g a c i t y coefficients φι for b o t h v a p o r a n d l i q u i d phases f o l l o w s s t a n d a r d p r o c e d u r e s (see

e.g.

(4)).

F o r V L E c a l c u l a t i o n s , the d i s t r i b u t i o n of

components

b e t w e e n phases is expressed g e n e r a l l y as the K - v a l u e — t h e v a p o r

mole

f r a c t i o n d i v i d e d b y the l i q u i d m o l e f r a c t i o n — r e l a t e d to f u g a c i t y

coeffi

cients for e a c h c o m p o n e n t

by:

Ki =

φι liquid/φι v a p o r


(8)

14.

The Redlich-Kwong

G R A Y

Generally

Understood

257

Equation

Capabilities

of RK

Methods

F o r r e f i n i n g a n d gas-processing a p p l i c a t i o n s , the b e n c h m a r k g e n e r a l p u r p o s e V L E m e t h o d is that of C h a o a n d Seader (18), w i t h t h e m o d i f i c a t i o n s of G r a y s o n a n d S t r e e d (19).

generally used

B y comparison w i t h

the C h a o - S e a d e r m e t h o d f o r these a p p l i c a t i o n s , either t h e R K J Z o r Soave v e r s i o n has t h e f o l l o w i n g c a p a b i l i t i e s , w h i c h are r e l a t i v e l y w e l l u n d e r stood:

(a)

scope—wider-than

Chao-Seader,

e x t e n d i n g closer

to the

m i x t u r e c r i t i c a l a n d to c r y o g e n i c t e m p e r a t u r e s ; ( b ) a c c u r a c y — g e n e r a l l y better t h a n C h a o - S e a d e r , even w i t h a r o u g h estimate of t h e C

i}


( o r w i t h dj =

0 for light hydrocarbons); a n d ( c )

parameter

flexibility—the

inter-

a c t i o n p a r a m e t e r s p r o v i d e easy adjustment of t h e m e t h o d for specific systems. O n e e x a m p l e w i l l serve to u n d e r s c o r e t h e reason f o r t h e a d v a n t a g e over C h a o - S e a d e r at h i g h pressure. F i g u r e 1 shows t h e c o n v e r g e n c e of R K J Z K - v a l u e s to u n i t y as t h e m i x t u r e c r i t i c a l pressure is a p p r o a c h e d , f o r a temperature a n d composition

o n the m i x t u r e c r i t i c a l locus f o r t h e

m e t h a n e - e t h a n e - b u t a n e t e r n a r y (20).

T h i s m i x t u r e w a s chosen i n o r d e r

to c h e c k R K J Z a p p a r e n t c r i t i c a l pressure vs. t h e 1972 corresponding-states c o r r e l a t i o n of T e j a a n d R o w l i n s o n (21),

w h i c h p r e s u m a b l y has a better

t h e o r e t i c a l basis t h a n t h e R K J Z m e t h o d . I n these c o m p a r i s o n s , t h e T e j a and

R o w l i n s o n c o r r e l a t i o n uses t w o i n t e r a c t i o n p a r a m e t e r s p e r b i n a r y

p a i r , b a s e d p r i m a r i l y o n fits to b i n a r y c r i t i c a l l o c i ; the R K J Z m e t h o d uses dj =

0 for a l l binaries, based o n binary V L E data.

N o t e that t h e R K J Z

method

not only predicts qualitatively the

a p p r o a c h to m i x t u r e c r i t i c a l c o n d i t i o n s ; i t is also q u a n t i t a t i v e l y s u p e r i o r to t h e T e j a a n d R a w l i n s o n p r o c e d u r e i n this instance. T h e a b i l i t y of t h e R K J Z m e t h o d to sense t h e a p p r o a c h to m i x t u r e c r i t i c a l c o n d i t i o n s has b e e n a great a d v a n t a g e i n its a p p l i c a t i o n , b y c o m p a r i s o n w i t h t h e C h a o Seader m e t h o d , w h i c h has a stated l i m i t a t i o n of pressure less t h a n 0.8 times t h e true c r i t i c a l pressure.

Generally

Understood

Limitations

of RK

Limitations of the R K methods assumed)

Methods

w h i c h have been mentioned

i n p r e v i o u s l i t e r a t u r e i n c l u d e : ( a ) p o o r second v i r i a l

(or

coeffi-

c i e n t p r e d i c t i o n , e s p e c i a l l y f o r c o m p o u n d s h a v i n g n o n z e r o a c e n t r i c factors; ( b ) p o o r p r e d i c t i o n of c o m p o n e n t l i q u i d densities ( t h i s is a d i s a d v a n t a g e o n l y of t h e Soave f o r m ; t h e R K J Z m e t h o d is fit to c o m p o n e n t

liquid

d e n s i t i e s ) ; a n d ( c ) i n a b i l i t y to represent a l l P V T p r o p e r t i e s at t h e c o m ponent

c r i t i c a l s i m u l t a n e o u s l y ; t h e Soave f o r m fails to r e p r o d u c e t h e

c r i t i c a l d e n s i t y w h i l e t h e R K J Z f o r m gives n o n z e r o values of a n d (d P/dV ) 2

2

T

at t h e c r i t i c a l p o i n t .


(dP/dV)

T


,β

CATM)

1

4

3

^

Ί

Δ

5

1

G

c

of RKJZ to P using expenmental

PRESSURE

PRESSURE

2

PRES

CALCULATED

Convergence

CRITICAL

ROWLINSON

CRITICAL

0.337

0.470

0.193

MOL F R A C

EXPERIMENTAL TEJA

Figure 1.

-

-

BUTANE

Π

Ά'

ETHANE

Δ

X

METHANE

DEG F

O

178

O

6

1

•

Δ

O


T

A

0

7

1

G>p>

1

8

Γ

9

1

0

14.

The Redlich-Kwong

G R A Y

259

Equation

O t h e r l i m i t a t i o n s i n c l u d e : ( d ) p o o r l y d e f i n e d basis for e x t r a p o l a t i n g a a n d b parameters above component critical; (e)

i n a b i l i t y to represent

p o l a r / n o n p o l a r systems i n d e t a i l ; e.g., u s i n g a n average C

xj

to represent a

l o w - p r e s s u r e i s o t h e r m for a n a l c o h o l / h y d r o c a r b o n system gives m a x i m u m errors of p e r h a p s a factor of t w o i n K - v a l u e , w h e r e a s a m e t h o d b a s e d o n a c t i v i t y coefficients w o u l d fit the same d a t a w i t h i n a f e w p e r c e n t ; a n d ( f ) a n i n a b i l i t y to represent other d e r i v e d p r o p e r t i e s ( e n t h a l p y , etc.)

for

n o n p o l a r systems to the same degree of a c c u r a c y as V L E .


Some General

Observations

on Redlich—Ktvong

Methods

T h e t h e r m o d y n a m i c s c o m m u n i t y w a s r a t h e r s l o w to a c c e p t the e a r l y m o d i f i e d R K m e t h o d s for V L E p r e d i c t i o n . T h i s w r i t e r , d e s p i t e b e i n g a n interested observer d u r i n g the d e v e l o p m e n t of the R K J Z m e t h o d , o n l y b e c a m e a n enthusiast for the m e t h o d experience.

after c o n s i d e r a b l e

applications

T h e reason for this e a r l y s k e p t i c i s m w a s t h e f e e l i n g t h a t it

w a s a s k i n g too m u c h of the s i m p l e v o l u m e d e p e n d e n c e of the R K e q u a t i o n to represent the f u g a c i t y f u n c t i o n a l i t y i n b o t h phases w i t h

sufficient

accuracy. It is w o r t h w h i l e to ask the q u e s t i o n : " W h y is the R K J Z ( o r the Soave m e t h o d ) better t h a n one w o u l d e x p e c t ? " A n s w e r i n g this q u e s t i o n i n a n y d e p t h is b e y o n d the scope of this c h a p t e r , a l t h o u g h i t is e x p l o r e d specific systems b e l o w .

H o w e v e r , there are t w o g e n e r a l

for

observations

one c a n m a k e . T h e first o b s e r v a t i o n is that, because of c o m p e n s a t i n g errors, t h e v a p o r f u g a c i t y p r e d i c t i o n s of the R K e q u a t i o n are r e l a t i v e l y i n s e n s i t i v e to the adjustment of the constants necessary to fit v a p o r pressure. T h u s , o n c e c o m p o n e n t f u g a c i t y is m a t c h e d a l o n g the v a p o r pressure locus, the effect of pressure a n d t e m p e r a t u r e o n v a p o r f u g a c i t y is r e a s o n a b l y w e l l represented.

F u r t h e r , the effect of pressure o n l i q u i d f u g a c i t y u s u a l l y

does n o t d e p e n d o n h i g h l y a c c u r a t e l i q u i d densities. H o w e v e r , the R K J Z m e t h o d is g e n e r a l l y m o r e a c c u r a t e t h e n the Soave m e t h o d i n r e p r e s e n t i n g the effect of pressure for l i g h t g a s - h e a v y solvent systems because of its better r e p r e s e n t a t i o n of l i q u i d v o l u m e t r i c b e h a v i o r . T h e second o b s e r v a t i o n is t h a t the success of the R K J Z a n d Soave m e t h o d s m a y b e a t t r i b u t e d to t h e m i x i n g rules ( w h i c h are so i m p o r t a n t i n f u g a c i t y p r e d i c t i o n ) . T h e s e m i x i n g rules essentially i n c o r p o r a t e the v a n der W a a l s one-fluid ( V D W - 1 ) sponding-states w o r k ( 2 1 ) .

a p p r o x i m a t i o n f a v o r e d i n recent

corre-

T h a t is, i f one i n t e r p r e t s t h e b p a r a m e t e r

as p r o p o r t i o n a l to the c r i t i c a l v o l u m e V , t h e n the a p a r a m e t e r is p r o p o r c

t i o n a l to T V , a n d the m i x i n g rules for a a n d b are e q u i v a l e n t to d e t e r C

mining T V

C

C

C

and V

c

for the reference substance ( a l t h o u g h i t s h o u l d b e

n o t e d that the v a l u e of b for the R K J Z m e t h o d is m o r e closely p r o p o r -


260

EQUATIONS

OF

STATE

t i o n a l to c r i t i c a l v o l u m e t h a n i t is for the Soave m e t h o d ). I n a sense, one m i g h t r e g a r d the R K m e t h o d s as a g o o d m i x i n g r u l e c o u p l e d w i t h o n l y the most r u d i m e n t a r y reference substance. N e v e r t h e l e s s , the a c c u r a c y of V L E p r e d i c t i o n s w i t h these m o d e l s is q u i t e c o m p e t i t i v e w i t h those of sponding-states

models

incorporating m u c h

more

elaborate

corre-

reference

fluids. Some Surprising

Successes

with

RK

Methods

O u r e a r l y a p p l i c a t i o n experience w i t h the R K J Z m e t h o d , a p a r t f r o m Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch014

h i g h - p r e s s u r e systems, t e n d e d to be for systems w h e r e it w a s the m e t h o d of last r e s o r t — w h e r e n o t h i n g else t h e n a v a i l a b l e i n o u r c o l l e c t i o n computer programs could work.

T h u s , often its successes w e r e

of

doubly

surprising. One

of

the most

s u r p r i s i n g e a r l y successes w a s

with

cryogenic

H o - h y d r o c a r b o n systems. H e r e w h a t w a s so s u r p r i s i n g w a s not just that the m e t h o d c o u l d be m a d e to w o r k , b u t that it w o r k e d so easily once the p r o c e d u r e for i n c o r p o r a t i n g H was adopted. more

2

developed by C h u e h and Prausnitz (7)

Results for H - h y d r o c a r b o n systems h a v e b e e n e x p l o r e d i n 2

d e t a i l i n a recent

study

sufficient to note that a single C

(22); t J

for the present

v a l u e correlates H

2

d i s c u s s i o n i t is and hydrocarbon

K - v a l u e s over s u b s t a n t i a l ranges of t e m p e r a t u r e a n d pressure for c r y o g e n i c systems. A n o t h e r system for w h i c h success is better t h a n m i g h t b e

expected

is the C 0 - m e t h a n e b i n a r y s h o w n i n F i g u r e 2. H e r e v e r y a c c u r a t e d a t a 2

over a w i d e r a n g e of c o n d i t i o n s h a v e r e c e n t l y b e c o m e a v a i l a b l e

(23,24),

for w h i c h Professors K i d n a y a n d K o b a y a s h i a n d t h e i r students

deserve

s p e c i a l praise. T h i s system is v e r y n o n i d e a l ( o w i n g to the C 0

2

quadru-

p o l e ) , a n d is s h o w n o n a v e r y e x p a n d e d scale; r m s errors i n K - v a l u e at e a c h t e m p e r a t u r e , except for the lowest, w e r e less t h a n 3 % , s m o o t h t r e n d i n dj

a n d the

e x h i b i t e d for d a t a f r o m t w o different sources

demon-

strates r e a l l y r e m a r k a b l e consistency of results b e t w e e n t w o laboratories. T h e points below

— 89 ° C are essentially for C 0

p o i n t near infinite d i l u t i o n i n m e t h a n e .

2

w e l l b e l o w its t r i p l e

Although C 0

2

l i q u i d properties

w e r e e x t r a p o l a t e d c a r e f u l l y i n t o this r e g i o n to o b t a i n R K J Z p a r a m e t e r s , the a p p a r e n t S-curve i n dj

m a y be a n a r t i f a c t of the e x t r a p o l a t i o n .

W a t e r - h y d r o c a r b o n systems, s h o w n i n F i g u r e 3, c o m p r i s e

another

class of systems w h i c h , r a t h e r s u r p r i s i n g l y , c a n b e h a n d l e d a c c u r a t e l y e n o u g h for m a n y purposes.

T h i s w o r k w i t h the R K J Z m e t h o d p a r a l l e l s

s i m i l a r studies b y H e i d e m a n n (25)

w i t h the Soave m e t h o d a n d b y P e n g

a n d R o b i n s o n w i t h t h e i r e q u a t i o n (10).

A s i n t h e i r w o r k , o n l y fugacities

i n the h y d r o c a r b o n - r i c h l i q u i d phases are fit b y the m o d e l d i r e c t l y ; i f l i q u i d w a t e r is present, i t is a s s u m e d to b e p u r e , since the Cy

fitting


the

In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979. 2

RKJZ interaction parameters for the methane-C0 binary: (X), Ref. 24; (A), Ref. 23. rms deviation uptol% above minimum.

0.04

0.06

0.08

Figure 2.

Ci j

0.10

0.12


Range


C ij

4.

0.2-

0.3-

0

0.5-

O

Figure 3.

3

O

TYPE

2

AROMATIC

PARAFIN

4

CARBON

HYDROCARBON

9

6

2

NUMBER

1

RKJZ interaction parameters for the H 0-hydrocarbon

o Δ

V/LH L / L

PATA

V/LW

8

8

systems


5!

C5/

O

ce

o

H

α >

W

to

14.

The Redlich-Kwong

G R A Y

Equation

263

w a t e r f u g a c i t y i n the h y d r o c a r b o n - r i c h phases y i e l d s h y d r o c a r b o n s o l u b i l i t i e s i n l i q u i d w a t e r w h i c h are i n error b y m a n y orders of m a g n i t u d e . F i g u r e 3 shows C sources

d a t a vs. c a r b o n n u m b e r b a s e d o n the d a t a f r o m several

i ;

(26-31).

W h a t is s t r i k i n g here is not just that one c a n " f u d g e " the

RKJZ

m e t h o d to i n c l u d e w a t e r b y i n t r o d u c i n g l a r g e C y values of a b o u t

0.4.

M o r e i m p o r t a n t l y , if one looks at C / s f o r a v a r i e t y of systems, t h e y f a l l {

i n t o r e c o g n i z a b l e patterns. N o t e that C^/s for paraffins, t a k e n f r o m v a p o r l i q u i d a n d l i q u i d - l i q u i d d a t a , are r e a s o n a b l y consistent, b u t t h a t c o r r e l a t i n g d a t a for a r o m a t i c s r e q u i r e s s h a r p l y l o w e r Cy's

because of

the


higher water solubility. O n e s h o u l d , of course, be c a u t i o u s i n e x t r a p o l a t i n g p r e d i c t i o n s b a s e d o n C / s f r o m a n a r r o w t e m p e r a t u r e range. {

some t e m p e r a t u r e d e p e n d e n c e i n C

t ;

T h e available data indicate

for w a t e r , b u t the d a t a are not

a c c u r a t e e n o u g h or a v a i l a b l e over a w i d e r a n g e of c o n d i t i o n s to s u p p o r t t e m p e r a t u r e d e p e n d e n c e for most systems. I n g e n e r a l , w i t h i n the sometimes stringent l i m i t a t i o n of t e m p e r a t u r e d e p e n d e n c e of C

i}

compound

one c a n m a p the infinite d i l u t i o n f u g a c i t y of a n y p o l a r

i n t o h y d r o c a r b o n systems.

F u r t h e r , i f the infinite

dilution

b e h a v i o r f o l l o w s k n o w n patterns w i t h h y d r o c a r b o n t y p e , this c a n b e m a d e the basis for a c o r r e l a t i o n of Cq.

T h i s a b i l i t y to i n c o r p o r a t e p o l a r c o m -

p o u n d s over n a r r o w ranges of c o n c e n t r a t i o n is e x t r e m e l y u s e f u l i n r e f i n i n g and hydrocarbon processing applications. Some Unexpected

and/or

Unexplored

Limitations

A l t h o u g h R K m e t h o d s are s u r p r i s i n g l y v e r s a t i l e , t h e y h a v e a v a r i e t y of l i m i t a t i o n s . T h i s d i s c u s s i o n w i l l concentrate o n those w h i c h m i g h t be c o n s i d e r e d u n e x p e c t e d , or w h i c h serve to define the b o u n d a r i e s of the k n o w n r e g i o n of v a l i d i t y . T h e r e is one k i n d of l i m i t a t i o n w h i c h , a l t h o u g h difficult to s u m m a r i z e c o n c i s e l y , s h o u l d be m e n t i o n e d , since i t m i g h t c o m e as a n u n p l e a s a n t surprise to the u n i n i t i a t e d . T h a t is that the R K J Z or Soave m o d e l s c a n n o t represent c e r t a i n details of l i g h t h y d r o c a r b o n

systems at

t e m p e r a t u r e s to a n y t h i n g close to e x p e r i m e n t a l a c c u r a c y .

subambient T h i s k i n d of

l i m i t a t i o n w a s difficult to d i s t i n g u i s h f r o m systematic e x p e r i m e n t a l error before

the

development

of

more

accurate

experimental

techniques,

notably b y K o b a y a s h i and co-workers a n d K a h r e , w h o have measured l o w - t e m p e r a t u r e phase b e h a v i o r for m e t h a n e b i n a r i e s (32,33,34,

35,36).

If one looks at d e v i a t i o n trends for these systems, i t is a p p a r e n t t h a t no adjustment of C

i ;

w i l l a l l o w one to fit the h e a v y - c o m p o n e n t K - v a l u e ,

w h i l e m a i n t a i n i n g reasonable a c c u r a c y for m e t h a n e at t e m p e r a t u r e s less t h a n 50° to 7 5 ° C a b o v e the m e t h a n e c r i t i c a l t e m p e r a t u r e ( t h a t is, b e l o w


264

EQUATIONS

OF

STATE

a b o u t — 5 0 ° C ) at m o d e r a t e to h i g h pressures (say, a b o v e 10 or 20 a t m ) . T h e s e systematic d e v i a t i o n s ( o f 15 to 4 0 % ) seem to i n d i c a t e p r o b l e m s i n c h a r a c t e r i z i n g s i m u l t a n e o u s l y the extremes of m i x i n g effects ( or p o s s i b l y , systematic e x p e r i m e n t a l e r r o r ) i n e x p a n d e d solvents a n d dense v a p o r , i n the r e g i o n w h e r e the lightest c o m p o u n d is o n l y s l i g h t l y s u p e r c r i t i c a l or is subcritical.

O n c e this effect w a s r e c o g n i z e d as a source of c o n f u s i o n i n

c o r r e l a t i n g e x p e r i m e n t a l d a t a , it w a s not v e r y i m p o r t a n t i n the a p p l i c a t i o n e x p e r i e n c e r e p o r t e d here. T h e Soave m e t h o d is s l i g h t l y better t h a n the R K J Z m e t h o d i n r e p r e s e n t i n g this r e g i o n , a n d it appears that the P e n g R o b i n s o n m e t h o d is s u b s t a n t i a l l y better.


M o s t other w o r k e r s t r e a t i n g these d a t a h a v e a s s u m e d t h a t these d i s c r e p a n c i e s are s i m p l y manifestations of the c o m m o n p r o b l e m of syst e m a t i c e x p e r i m e n t a l error for h e a v y - c o m p o n e n t older

e x p e r i m e n t a l d a t a for

light hydrocarbons.

K - v a l u e f o u n d i n the F u r t h e r m o r e , these

d e v i a t i o n s are m a s k e d b y t h e c o m m o n p r a c t i c e of r e p o r t i n g d e v i a t i o n s i n b u b b l e - p o i n t pressure a n d i n absolute differences i n v a p o r m o l e f r a c t i o n , w h i c h are b o t h r e l a t i v e l y i n s e n s i t i v e to d e v i a t i o n s i n K-value.

heavy-component

N e v e r t h e l e s s , i f the h i g h p r e c i s i o n of the n e w e r e x p e r i m e n t a l

results is to be b e l i e v e d , there are systematic errors i n these

methods

w h i c h m i g h t b e i m p o r t a n t i n some a p p l i c a t i o n s . A r e l a t e d l i m i t a t i o n of the R K J Z or Soave m e t h o d s , w h i c h a g a i n is difficult to q u a n t i f y , is also w o r t h m e n t i o n i n g because i t is not o b v i o u s f r o m p u b l i s h e d results. T h e a c c u r a c y of these m e t h o d s deteriorates i f one attempts to fit too w i d e a r a n g e of c o n d i t i o n s e v e n for n o r m a l

fluids.

T h i s effect is often m a s k e d b y the g e n e r a l l y spotty q u a l i t y of the e x p e r i m e n t a l d a t a base, b u t one g e n e r a l l y c a n d i s c e r n s u b s t a n t i a l systematic d e v i a t i o n trends b y c a r e f u l e x a m i n a t i o n of a c c u r a t e d a t a . T h u s , u l t i m a t e l y the s i m p l i f i e d v o l u m e d e p e n d e n c e of the R K e q u a t i o n does p l a c e l i m i t a tions o n its a c c u r a c y .

C o n s e q u e n t l y , one s h o u l d k e e p i n p e r s p e c t i v e the

c l a i m s of Z u d k e v i t c h a n d Joffe ( 3 ) , to represent h y d r o c a r b o n

systems

b a s e d o n C,-/s d e t e r m i n e d f r o m one or t w o d a t a p o i n t s , or of Soave to represent h y d r o c a r b o n systems w i t h dj

=

(2)

0. T h e s e c l a i m s are q u i t e

t r u e i n the context i n w h i c h t h e y w e r e m a d e , d e m o n s t r a t i n g the g e n e r a l i t y of the R K methods. N e v e r t h e l e s s , w h e n h i g h e s t a c c u r a c y is r e q u i r e d , one s h o u l d be r e c o n c i l e d to different C i / s for different regions. A n o t h e r l i m i t a t i o n i n a r e g i o n of a p p a r e n t s t r e n g t h is i n the r e p r e sentation of c r i t i c a l - r e g i o n effects at v e r y h i g h pressures, s u c h as those o c c u r r i n g i n m i x t u r e s of m e t h a n e w i t h m o d e r a t e l y h e a v y h y d r o c a r b o n s . A n extreme e x a m p l e is s h o w n i n F i g u r e 4, w h i c h shows c a l c u l a t e d a n d e x p e r i m e n t a l phase b o u n d a r i e s at 1 0 0 ° F for the p s e u d o b i n a r y m i x t u r e of m e t h a n e w i t h K e n s o l - 1 6 , a n a r r o w - b o i l i n g o i l w i t h c a r b o n n u m b e r i n the C i 5 to C (37).

2 0

r a n g e . T h i s m i x t u r e w a s s t u d i e d i n 1950 b y R z a s a a n d K a t z

T h e c a l c u l a t e d - p h a s e envelopes

s h o w the extreme s e n s i t i v i t y of




1

οι

K)

δ*

ta -Ci

OS

ο

Ci"

g-

> ni

Ο

h-


0

-

I

%

•

&

A

l

Λ

TYPE

ι

5

ι

ι

ι

2

b /b,

1

CONSTANT

tz

TYPE B

•

•

ι DATA

HENRY'S

ι

I

I

w

Ο

10

•

Ο

1

10

+

t

1 1 1 1

1

RKJZ interaction parameters for methane-C

l

• •

Ο

I

Ν

Ρ

SOLVENT

t

Figure 5.

I

1

1

ι

1

ι

1

hydrocarbons

15


ι

I

1

•

ο

1

20

-

14.

The Redlich-Kwong

G R A Y

Equation

267

m i x t u r e c r i t i c a l pressure ( c a l c u l a t e d b y the R K J Z m e t h o d ) to C \ is

Cij =

for

0, the c a l c u l a t e d c r i t i c a l pressure is w e l l b e l o w the e x p e r i m e n t a l

v a l u e of a b o u t 12000 p s i a , w h i l e for C

i y

=

0.05, it is f a r a b o v e i t . T h i s

demonstrates that extreme care is necessary to represent c r i t i c a l - r e g i o n effects for m i x t u r e s w i t h l a r g e differences i n m o l e c u l a r size. I f m o l e c u l a r - s i z e differences are too l a r g e , i t is n o t o n l y the c r i t i c a l r e g i o n w h i c h s h o u l d be of c o n c e r n .

N o t e i n F i g u r e 4 that the C

i ;

which

represents the c r i t i c a l p o i n t w o u l d not represent the d e w - p o i n t l i n e v e r y w e l l . M o r e s t r i k i n g is F i g u r e 5, i n w h i c h t h e C i / s necessary to correlate methane-heavy hydrocarbon behavior (recently measured by Prausnitz


a n d c o - w o r k e r s (38,39, b /bi 2

40) ) are p l o t t e d vs. the r a t i o of R K J Z p a r a m e t e r s

( w h e r e 2 designates the solvent a n d 1 the m e t h a n e ) ; this r a t i o is

essentially e q u i v a l e n t to the r a t i o of c r i t i c a l v o l u m e s . N o t e the d i v e r g e n t trends i n C

{j

necessary to correlate these h i g h l y a s y m m e t r i c systems: as

solvent m o l e c u l a r w e i g h t increases, d e c r e a s i n g C i / s are n e e d e d to c o r r e late H e n r y ' s constant for m e t h a n e ; b u t i n c r e a s i n g C i / s are n e e d e d to correlate s e c o n d v i r i a l cross-coefficients

( a n d thus d e w - p o i n t c o m p o s i t i o n

at h i g h p r e s s u r e ) . T h e s e trends are for the R K J Z m e t h o d ; s i m i l a r t r e n d s ( a l t h o u g h c o n s i d e r a b l y d i s p l a c e d ) o c c u r f o r the Soave v e r s i o n .

Clearly

the trends d e m o n s t r a t e d i n F i g u r e 5 i m p o s e l i m i t a t i o n s o n the use of R K m e t h o d s for a s y m m e t r i c systems at h i g h pressures. F o r m a n y a p p l i c a t i o n s , the m e t h o d w i l l be satisfactory p r o v i d e d one u n d e r s t a n d s the n a t u r e of the l i m i t a t i o n s . F i n a l l y , i t is w o r t h w h i l e n o t i n g t h a t t h e strengths of the

RKJZ

m e t h o d — i t s c a p a b i l i t y to represent c r i t i c a l - r e g i o n b e h a v i o r , as w e l l as nonideal m i x i n g — c a n i n practice impose limitations. A s noted by Deiters a n d S c h n e i d e r (41),

this c a p a b i l i t y makes i t p o s s i b l e to represent p h a s e

b e h a v i o r of r e m a r k a b l y c o m p l e x t o p o l o g y , e s p e c i a l l y for h i g h - p r e s s u r e systems. T h i s c o m p l e x phase b e h a v i o r is a l w a y s p o t e n t i a l l y present, e v e n t h o u g h flash or d i s t i l l a t i o n a l g o r i t h m s d o not a c c o u n t for i t . C o n s e q u e n t l y , a l g o r i t h m s w h i c h w o r k w e l l for the C h a o - S e a d e r m e t h o d ( w h i c h r e p r e sents n o n i d e a l m i x i n g b u t not c r i t i c a l p h e n o m e n a ) or one of the m e t h o d s b a s e d o n the B W R e q u a t i o n of state ( w h i c h represents c r i t i c a l p h e n o m e n a b u t is not u s e d u s u a l l y for systems e x h i b i t i n g n o n i d e a l m i x i n g ) , m a y develop new pathology w h e n used w i t h a R K method.

In practice,

m a i n t a i n i n g a R K m e t h o d c a n b e e x p e c t e d to r e q u i r e m o r e

advanced

expertise i n b o t h p r o g r a m m i n g a n d a p p l i e d t h e r m o d y n a m i c s . The

Next

Generation

of

Equation-of-State

Methods

A l o g i c a l c o n c l u s i o n for this c h a p t e r is, b a s e d o n t h i s a p p l i c a t i o n s experience, to reflect o n w h a t features i n a n e w g e n e r a t i o n of equations of state w o u l d represent d e s i r a b l e i m p r o v e m e n t s over the R K m e t h o d s for


268

EQUATIONS

phase-equilibrium calculations.

OF

STATE

S u c h reflections are of g e n e r a l interest

since the strengths a n d weaknesses of the R K m e t h o d s seem to b e s h a r e d b y a n y of the c o r r e s p o n d i n g states m e t h o d s u s i n g the V D W - 1 a p p r o x i m a t i o n . H e r e is a p e r s o n a l , b u t b y no means o r i g i n a l , list of features: ( a ) the a c c u r a c y ( a n d

flexibility)

of R K J Z or Soave m e t h o d s i n r e -

g i o n w h e r e these m e t h o d s are satisfactory; ( b ) g o o d r e p r e s e n t a t i o n of d e n s i t y of b o t h phases, i n c l u d i n g g o o d r e p r e s e n t a t i o n of second v i r i a l coefficients; ( c ) s i m u l t a n e o u s r e p r e s e n t a t i o n of p u r e - c o m p o n e n t ties ( , Pc

(dP/dT)

C9

T , F ); c

c

( d ) w e l l - d e f i n e d s u p e r c r i t i c a l e x t r a p o l a t i o n of Downloaded by CORNELL UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch014

critical proper-

temperature-depend-

ent p a r a m e t e r s ; ( e ) w e l l - u n d e r s t o o d e x t r a p o l a t i o n to h i g h m o l e c u l a r w e i g h t s ; (f) independently

adjustable

infinite

d i l u t i o n fugacities

for

each

component i n a binary; ( g ) g r o u p c o n t r i b u t i o n features b u i l t into p u r e - c o m p o n e n t

and/or

mixture parameters; a n d ( h ) s p e c i a l m o d i f i c a t i o n s for p o l a r c o m p o u n d s i n a l l of the a b o v e . M a n y of these features are a l r e a d y i n some of the e m e r g i n g m e t h o d s , and

a l l are at least u n d e r s t u d y s o m e w h e r e .

ambitious enough

No

one

to i n c l u d e a l l i n a single m e t h o d .

has y e t Obviously,

cannot expect a n y m o d i f i c a t i o n s of the s i m p l e R K m e t h o d s to

been one

combine

a l l of these features, a l t h o u g h some of t h e m c a n be i n t r o d u c e d b y v a r i o u s a d d - o n artifices. T h u s , one c a n expect the R K m e t h o d s e v e n t u a l l y to b e l a r g e l y s u p p l a n t e d i n a p p l i c a t i o n w o r k w h e r e these features are i m p o r t a n t b y the m o r e elaborate m e t h o d s n o w u n d e r d e v e l o p m e n t . It is o u t s i d e the scope of this c h a p t e r to assess the p o t e n t i a l of these e m e r g i n g methods, except to c o m m e n t that for those of us i n t e r e s t e d i n a w i d e r a n g e of m o l e c u l a r size, t h e p e r t u r b e d h a r d - c h a i n m o d e l of D o n o h u e a n d P r a u s n i t z (42)

appears to c o m e closest to c o m b i n i n g a l l the

features of interest. R e g a r d l e s s of w h i c h of the n e w m e t h o d s u l t i m a t e l y find w i d e use i n i n d u s t r i a l a p p l i c a t i o n s , the process of s e l e c t i n g , a d a p t i n g , a n d testing t h e m for this p u r p o s e w i l l take years. D u r i n g this p e r i o d , t h e R K m e t h o d s w i l l p r o v i d e the b e n c h m a r k b y w h i c h the e m e r g i n g m e t h o d s are j u d g e d .

Glossary

of

Symbols

a,b = p a r a m e t e r s i n R e d l i c h - K w o n g e q u a t i o n of state, E q u a t i o n 1 dij = i n t e r a c t i o n p a r a m e t e r u s e d i n c a l c u l a t i n g m i x t u r e p a r a m e t e r i n Equation 5 Cq = 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 defined b y E q u a t i o n 7


14.

The Redlich-Kwong

G R A Y

Equation

269

Ki = ratio of v a p o r m o l e f r a c t i o n to l i q u i d m o l e f r a c t i o n of C o m p o n e n t i for vapor and liquid i n equilibrium F = system pressure P = c r i t i c a l pressure of C o m p o n e n t i R = gas constant Τ = system t e m p e r a t u r e V = system v o l u m e V = critical volume T . = c r i t i c a l t e m p e r a t u r e of C o m p o n e n t i T . = T/T . = r e d u c e d t e m p e r a t u r e of C o m p o n e n t i il = p r e m u l t i p l i e r to d e t e r m i n e R e d l i c h - K w o n g a p a r a m e t e r Ω . = p r e m u l t i p l i e r to d e t e r m i n e R e d l i c h - K w o n g b p a r a m e t e r . = a c e n t r i c factor for C o m p o n e n t i $ = f u g a c i t y coefficient of C o m p o n e n t i i n m i x t u r e p = critical density Ci

c

c

c

r

ai

6


ω

{

c

Acknowledgments T h e author thanks E x x o n Research & E n g i n e e r i n g C o m p a n y for p e r m i s s i o n to p u b l i s h this w o r k .

J . Joffe a n d D . Z u d k e v i t c h

the o r i g i n a l v e r s i o n of t h e R K J Z p r o g r a m s .

developed

G . A . L y o n c o d e d most of

the adaptations u s e d i n this w o r k , a n d Η . E . N e w m a n s u p p l i e d t h e computer graphics. vided

M . S. G r a b o s k i a n d T . E . D a u b e r t generously

information concerning

pro

t h e i r v e r s i o n of t h e Soave m e t h o d i n

a d v a n c e of p u b l i c a t i o n .

Literature Cited 1. Wilson, G. M . Adv. Cryog. Eng. 1964, 9, 168. 2. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. 3. Zudkevitch, D.; Joffe, J., paper presented at the New Orleans meeting of AIChE, March 1969. 4. Zudkevitch, D.; Joffe, J. AIChE J. 1970, 16(1), 112. 5. Joffie, J.; Schneder, G. M.; Zudkevitch, D. AIChE J. 1970, 16(3), 496. 6. Chang, S. D.; Lu, B.C.-Y. Can. J. Chem. Eng. 1970, 46, 21. 7. Chueh, P. L.; Prausnitz, J. M. Ind. Eng. Chem., Fundam. 1967, 6, 492. 8. Chueh, P. L.; Prausnitz, J. M. AIChE J. 1967, 13, 1099. 9. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem., Fundam. 1976, 15, 59. 10. Peng, D.-Y.; Robinson, D. B. Can. J. Chem. Eng. 1976, 54, 595. 11. Peng, D.-Y.; Robinson, D. B. AIChE J. 1977, 23, 137. 12. Graboski, M. S.; Daubert, T. E. Ind. Eng. Chem., Process Des. Dev. 1978, 17, 443. 13. Haman, S. E. M.; Chung, W. K.; Elshaxal, I. M.; Lu, Β. C.-Y. Ind. Eng. Chem., Process Des. Dev. 1977, 16, 51. 14. Riedel, L. Chem.-Ing.-Tech. 1954, 26, 83. 15. Riedel, L. Chem.-Ing.-Tech. 1954, 26, 259. 16. Am. Pet. Inst. Proj. 44, loose leaf sheets extant 1977. 17. Kato, M.; Chung, W.K.;Lu, Β. C.-Y. Chem. Eng. Sci. 1976, 31, 733. 18. Chao, K. C.; Seader, J. D. AIChE J. 1961, 7, 598. 19. Grayson, H . G.; Streed, C. W. World Pet. Congr. Proceedings, 6th 1963, Sec.III,234. 20. Cota, H. M.; Thodos, G. J. Chem. Eng. Data 1962, 7, 62.


270

EQUATIONS OF STATE

21. Teja, A. S.; Rowlinson, J. S. Chem. Eng. Sci. 1973, 28, 529. 22. Gray, R. D., paper presented at the New York meeting of AIChE, 1977. 23. Davalos, J.; Anderson, W. R.; Phelps, R. E.; Kidnay, A. J. J. Chem. Eng. Data 1976, 21, 81. 24. Mraw, S. C.; Hwang, S. C.; Kobayashi, R. J. Chem. Eng. Data 1978, 23, 135. 25. Heidemann, R. A. AIChE J. 1974, 20, 847. 26. Olds, R. H.; Sage, B. H.; Lacey, W. N . Ind. Eng. Chem. 1942, 34(10), 1223. 27. Rigby, M.; Prausnitz, J. M. J. Phys. Chem. 1968, 72, 330. 28. Coan, C.R.;King, A. D. J. Am. Chem. Soc. 1971, 93(8), 1857. 29. Sage, Β. H.; Lacey, W. N . "Some Properties of Lighter Hydrocarbons in H S, and CO ," Monograph on API Research Project 32, 1955. 30. Kobayashi,R.;Katz, D. L. Ind. Eng. Chem. 1953, 45, 440. 31. Polak, J.; Lu, Β. C.-Y. Can. J. Chem. Eng. 1973, 51, 4018. 32. Wichterle, I.; Kobayashi, R. J. Chem. Eng. Data 1972, 17, 4. 33. Kahre, L. C. J. Chem. Eng. Data. 1974, 19, 67. 34. Elliot, D. G.; Chen, R. J. J.; Chappelear, P. S.; Kobayashi, R. J. Chem. Eng. Data 1974, 19, 71. 35. Chu, I.-C.; Chu, T.-C.; Chen, R. J. J.; Chappelear, P. S.; Kobayashi, R. J. Chem. Eng. Data 1976, 21, 41. 36. Kahre, L. C. J. Chem. Eng. Data 1975, 20, 363. 37. Rzasa, M . J.; Katz, D. L. Trans. Am. Inst. Min., Metall. Pet. Eng. 1950, 189, 119. 38. Chappelow, C.C.;Prausnitz, J. M. AIChE J. 1974, 20, 1097. 39. Cukor, P. M.; Prausnitz, J. M. J. Phys. Chem. 1972, 76, 598. 40. Kaul, Β. K.; Prausnitz, J. M. AIChE J. 1978, 24, 223. 41. Deiters, U.; Schneider, G. M. Ber. Bunsenges. Phys. Chem. 1976, 80, 1316. 42. Donohue, M. M.; Prausnitz, J. M. AIChE J. 1978, 24, 849. RECEIVED October 5, 1978.


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2


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