Equations of State in Engineering and Research

3 Cubic Equations of State: An Interpretive Review

Downloaded by UNIV OF ARIZONA on November 9, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch003

MICHAEL M. ABBOTT Department of Chemical and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY 12181

All cubic equations of state proposed to date are special cases of a general five-parameter expression. The apparent flexibility of such an expression is partly illusory, however, because of the inherent limitations imposed by its density dependence. Historically, the greatest successes with cubic equations had been achieved with variants of the RedlichKwong equation of state. Recent work, e.g., that of Peng and Robinson, has demonstrated the suitability of other cubic forms, and has inspired renewed efforts to identify the best cubic expression. One must approach such efforts with realistic expectations, and with an understanding of what cubic equations can and cannot do. The

Cubic

Equation

of

State

t h e m a n y p r o p o s e d f o r m s o f t h e a n a l y t i c e q u a t i o n of state, t h e

/^\f

p o l y n o m i a l s i n v o l u m e are o f p a r t i c u l a r p r a c t i c a l i m p o r t a n c e ; e.g., this class i n c l u d e s t r u n c a t i o n s of t h e v i r i a l e q u a t i o n

i n density, the

p r e f e r r e d w o r k i n g forms of t h e v i r i a l e q u a t i o n o f state. finding

E f f i c i e n t root-

techniques are available for the solution of p o l y n o m i a l equations,

a n d m o r e o v e r t h e n u m b e r o f roots is a l w a y s k n o w n . T h e s i m p l e s t u s e f u l p o l y n o m i a l e q u a t i o n of state is o n e t h a t is c u b i c i n m o l a r v o l u m e , f o r s u c h a n expression is c a p a b l e o f y i e l d i n g t h e i d e a l gas e q u a t i o n i n t h e l i m i t as V —> oo, a n d o f r e p r e s e n t i n g b o t h l i q u i d a n d v a p o r - l i k e v o l u m e s f o r sufficiently l o w t e m p e r a t u r e s .

If w e require

that t h e e q u a t i o n b e e x p l i c i t i n pressure, t h e n a l g e b r a i c a r g u m e n t s l e a d us t o a

five-parameter

expression o f t h e f o r m ( I )

P =

RT(V

2

+ aV +

β)

Equations 1155 In16th St. of N.State W.in Engineering and Research; Chao, K., et al.;

Advances American M/Qch i n n t n n inΠChemistry; Ρ ΟΩΛΟΜ

Chemical Society: Washington, DC, 1979.

48

EQUATIONS

O F

STATE

P a r a m e t e r s α, β , λ, μ, a n d ν c a n i n p r i n c i p l e a l l b e f u n c t i o n s of t e m p e r a t u r e Τ a n d , f o r m i x t u r e s , of c o m p o s i t i o n . L i q u i d s are relatively incompressible.

T h e r e f o r e , i f o u r e q u a t i o n is

to represent l i q u i d - l i k e v o l u m e s , i t m u s t generate a steep p o r t i o n o f iso t h e r m f o r a p p r o p r i a t e l y s m a l l values of V . T h i s b e h a v i o r incorporated

r e a d i l y is

b y i n t r o d u c t i o n of a z e r o i n t o t h e d e n o m i n a t o r

e q u a t i o n at V = y

3

of the

b, w h e r e & is a s m a l l , p o s i t i v e n u m b e r :

w + ν

+

2

+

μ

ν

Ε

(ν

b)

-

(ν

2

+ sv + c )


F i n a l l y , w e m a y e l i m i n a t e p a r a m e t e r s a a n d β i n f a v o r of t w o n e w q u a n t i t i e s , Θ a n d η, v i a t h e definitions Θ ΞΞΞΞ RT{8 ?

s

-a)

( j 8 - c ) / ( » - e )

C o m b i n i n g t h e a b o v e equations t h e n y i e l d s P

_

_RT V - b

®(Υ-η) (V -b)(V + 8V + ) 2

P a r a m e t e r s b, δ, c, Θ, a n d η d e p e n d t h e i r n u m e r i c a l values of course

m K

€

'

generally o n Τ a n d composition;

depend

upon

t h e i d e n t i t i e s of t h e

c h e m i c a l species i n t h e system. E q u a t i o n 1 c a n be considered a generalization of the v a n der W a a l s ( V D W ) e q u a t i o n , to w h i c h i t reduces as t h e s i m p l e s t n o n t r i v i a l s p e c i a l case.

Scores of s p e c i a l i z a t i o n s of E q u a t i o n 1 h a v e b e e n p r o p o s e d

since

v a n d e r W a a l s ' t i m e ; a f e w of t h e m are l i s t e d i n T a b l e I a n d c a t e g o r i z e d a c c o r d i n g to t h e values ( o r types of f u n c t i o n s ) a s s u m e d f o r p a r a m e t e r s Θ, η, δ, a n d c S i g n i f i c a n t l y , a l l of t h e m o d e r n equations h a v e a t e m p e r a t u r e - d e p e n d e n t Θ. A l s o , most c u b i c equations i n c o r p o r a t e t h e constraint η =

b, a n d i n a d d i t i o n h a v e zero values f o r c e r t a i n p a r a m e t e r s

and/or

specified r e l a t i o n s h i p s a m o n g some of the p a r a m e t e r s b, η, δ, a n d c. Table I.

C l a s s i f i c a t i o n o f Some C u b i c E q u a t i o n s o f S t a t e

Equation v a n der W a a l s (1873) B e r t h e l o t (1900) C l a u s i u s (1880) R e d l i c h - K w o n g (1949) W i l s o n " (1964) P e n g - R o b i n s o n (1976) L e e - E r b a r - E d m i s t e r (1973)

Θ

V

ο α/Τ α/Τ a/T ®w(T)

b b b b b b

i/2

ΘΡΚ(Γ) ® L E E ( Τ)

S

€

0 0 2c b b 2b b

0 0 c 0 0 -b 0

"Similarly, Barner et al. (1966) and Soave (1972).

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

2

2

3. The

Cubic

A B B O T T

Virial

Form

Equations

49

of State

of the Cubic

Equation

T h e p r e d i c t i v e c a p a b i l i t i e s o f a n e q u a t i o n o f state a t l o w densities m a y b e tested c o n v e n i e n t l y b y c o m p a r i n g e x p e r i m e n t a l values o f t h e v i r i a l coefficients against those i m p l i e d b y t h e ( e m p i r i c a l ) e q u a t i o n of state. A l t e r n a t i v e l y , r e a l i s t i c , l o w - d e n s i t y b e h a v i o r m a y b e b u i l t i n t o a n e m p i r i c a l e q u a t i o n o f state b y d i r e c t i n c o r p o r a t i o n o f expressions for one or m o r e of the v i r i a l coefficients.

T h u s i t is u s e f u l t o express E q u a t i o n 1

i n the v i r i a l f o r m


Z—1

+ B/V + C/V + D/V* 2

+

. . .

H e r e Β is the s e c o n d v i r i a l coefficient, C is the t h i r d , etc. F o r a m i x t u r e c o n t a i n i n g specified substances, the v i r i a l coefficients d e p e n d o n Τ a n d composition

only;

moreover,

the composition

d e p e n d e n c e of m i x t u r e

v i r i a l coefficients is k n o w n . T h e Ζ - e x p l i c i t f o r m of E q u a t i o n 1 is V

(&/RT)

V - b

V (V — η)

(V -b)(V

2

+ SV + «)

K

E x p a n d i n g t h e right side of E q u a t i o n 2 i n inverse p o w e r s

>

of V a n d

c o m p a r i n g the r e s u l t w i t h the v i r i a l e q u a t i o n , w e find t h a t

B

=

c

=

D

=

b

~ W f b 2

6

3

(

-w +w

-w

+

w

+ {

s

+

a

)

( 3 b )

> ~~ir

r

3

}

< δ2

£

+ ^

< > 3c

etc. S i n c e the v i r i a l coefficients d e p e n d o n Τ a n d c o m p o s i t i o n o n l y , the e q u a tion-of-state p a r a m e t e r s c a n d e p e n d a t most o n Τ a n d c o m p o s i t i o n , as a l r e a d y n o t e d . T h e s e c o n d v i r i a l coefficient Β is the o n l y one for w h i c h a d e c e n t d a t a base a n d r e l i a b l e e s t i m a t i o n p r o c e d u r e s

are available;

a c c o r d i n g t o E q u a t i o n 3a, v a l u e s f o r Β ( a s i m p l i e d b y o u r e q u a t i o n o f state) are d e t e r m i n e d c o m p l e t e l y b y s p e c i f i c a t i o n of p a r a m e t e r s b a n d Θ. Unconstrained

y

Dimensionless

Forms

of the Cubic

Equation

A p p l i c a t i o n of E q u a t i o n 1 or 2 requires the availability of numerical values for the equation-of-state p a r a m e t e r s . R e s t r i c t i n g ourselves f o r the m o m e n t t o systems c o n t a i n i n g a single c o m p o n e n t , a n d a n t i c i p a t i n g the


EQUATIONS

O F

S T A T E

e v e n t u a l use of corresponding-states c o n c e p t s , w e i n t r o d u c e a set

of

dimensionless p a r a m e t e r s v i a the definitions ^

(RT /P )b

(4a)

δ

=

(RT /P )t

(4b)

V =

(RT /P )^

(4c)

[R T /P )Î

(4d)

(R T /P )$

(4e)

C

C

c

C

2

€ Ξ Ξ

®= Downloaded by UNIV OF ARIZONA on November 9, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch003

c

C

2

2

2

2

C

H e r e T a n d P are the c r i t i c a l t e m p e r a t u r e a n d pressure. T h e d i m e n s i o n c

c

less p a r a m e t e r s [ d e s i g n a t e d b y a c i r c u m f l e x ( ) ] A

d e p e n d at m o s t o n

r e d u c e d t e m p e r a t u r e T a n d o n t h e i d e n t i t y of t h e substance c o n s i d e r e d . r

I f the t w o - p a r a m e t e r t h e o r e m of c o r r e s p o n d i n g states a p p l i e d , these p a r a m e t e r s w o u l d be i n d e p e n d e n t of c h e m i c a l species. E q u a t i o n s 1 a n d 2 c a n be p u t i n t o r e d u c e d f o r m b y u s i n g E q u a t i o n 4 a n d the a d d i t i o n a l definitions

(5a)

Pr^P/P

c

T = T/T

c

(5b)

Vτ ΞΞΞ V/V

c

(5c)

r

(5d)

t = P V /RT c

Parameter f

c

Q

c

c

is a n a p p a r e n t c r i t i c a l c o m p r e s s i b i l i t y factor. W e g i v e i t a

s p e c i a l s y m b o l to p r e c l u d e its g e n e r a l i d e n t i f i c a t i o n w i t h t h e e x p e r i m e n t a l c r i t i c a l c o m p r e s s i b i l i t y factor Z , to w h i c h , i n p r a c t i c a l a p p l i c a t i o n s of c

c u b i c equations of state, i t is often a s s u m e d not e q u a l . I n s u c h cases, the r e d u c e d v o l u m e V

r

is defined a l w a y s w i t h respect to a

hypothetical) critical volume V

c

(possibly

d e f i n e d i n terms of P , T , a n d £ v i a c

c

c

E q u a t i o n 5. C o m b i n i n g Equations 1 and 2 w i t h Equations 4 and 5 then yields the e q u i v a l e n t expressions

and


3.

Cubic

A B B O T T

Equations

51

of State

T h e expressions f o r the v i r i a l coefficients m a y b e w r i t t e n also i n d i m e n sionless f o r m . D e f i n i n g dimensionless v i r i a l coefficients b y %=

(8a)

BP RT c

e

d=CP /R T 2

c

2

(8b)

2

c

Ù = DP /R T 3

3

C

(8c)

3

C


etc., w e c a n w r i t e E q u a t i o n 3 as

(9a) ^

(9b) (9c)

Incorporation

of

the

Classical

Critical

Constraints

N u m e r i c a l values for the equation-of-state p a r a m e t e r s m a y b e establ i s h e d i n m a n y w a y s , a n d , since no e q u a t i o n of state is perfect, different values are o b t a i n e d d e p e n d i n g u p o n t h e m e t h o d s u s e d .

O n e class of

m e t h o d s , w h i c h w e c a n c a l l " b r u t e - f o r c e " m e t h o d s , i n v o l v e s t h e use of n o n l i n e a r regression t e c h n i q u e s to d e t e r m i n e b y analysis of e x p e r i m e n t a l d a t a the best values of the p a r a m e t e r s f o r r e p r e s e n t a t i o n o f a p a r t i c u l a r p r o p e r t y ( o r p r o p e r t i e s ) o v e r a l i m i t e d r a n g e of t e m p e r a t u r e a n d pressure. T h e equations w h i c h result c a n b e q u i t e p r e c i s e f o r t h e i r i n t e n d e d p u r pose. A n o t h e r m u c h o l d e r a p p r o a c h is to i m p o s e a f e w selected m a t h e m a t i c a l or n u m e r i c a l constraints o n the e q u a t i o n of state a n d to d e t e r m i n e n u m e r i c a l values f o r the p a r a m e t e r s b y s o l v i n g the r e s u l t i n g system of equations.

These techniques, w h i c h w e c a n c a l l "algebraic"

methods,

l e a d to e q u a t i o n s of state g e n e r a l l y less p r e c i s e t h a n those r e s u l t i n g f r o m the b r u t e - f o r c e a p p r o a c h , b u t also often less l i k e l y to generate a b s u r d results outside the i n t e n d e d r a n g e of a p p l i c a t i o n of the e q u a t i o n . M a r t i n (2)

discusses a n d a p p l i e s s o m e of the features of r e a l - f l u i d

b e h a v i o r w h i c h l e n d themselves to c o n v e n i e n t expression as a l g e b r a i c constraints for equation-of-state studies.

T h e t w o constraints p r o b a b l y

most often u s e d are t h e c l a s s i c a l c o n d i t i o n s (θΡ/θν) ,„ = Γ

(3*P/dV*) ,

T ct

=

0

(10)

0

(ID


52

EQUATIONS

O F

S T A T E

w h i c h assert t h a t the c r i t i c a l i s o t h e r m has a p o i n t of h o r i z o n t a l i n f l e c t i o n at the c r i t i c a l state ( c r ) .

A r e q u i r e m e n t e q u i v a l e n t to E q u a t i o n s 10 a n d

11 is that E q u a t i o n 6, w h e n e x p a n d e d a n d s p e c i a l i z e d to the c r i t i c a l state (T

r

=

P = r

1 ), has three e q u a l roots; t h a t is, i t m u s t b e of the f o r m

~ D

(V

v

3

=

-

V*

3F

r

2

+ 3V

-

r

1=

0

(12)

T h u s w e o b t a i n , after some a l g e b r a i c m a n i p u l a t i o n s , t h e three e q u a t i o n s 8 = Downloaded by UNIV OF ARIZONA on November 9, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch003

C

6 + C

1 -3£ +

fr_fc'-&

(13)

c

- & c + 3f

Θ

c

(14)

2

+ l)g

(15)

0

S i n c e for a p u r e m a t e r i a l a l l p a r a m e t e r s c a n i n p r i n c i p l e b e f u n c t i o n s of Τ , I h a v e u s e d the s u b s c r i p t c to designate a v a l u e f o r a p a r a m e t e r Γ

e v a l u a t e d at T =

1.

r

I f ζc is r e g a r d e d as a n e m p i r i c a l equation-of-state p a r a m e t e r , different i n g e n e r a l f r o m the e x p e r i m e n t a l c r i t i c a l c o m p r e s s i b i l i t y factor Z , t h e n c

E q u a t i o n s 13, 14, a n d 15 constitute a system of three equations i n six unknowns.

T h u s , three a d d i t i o n a l constraints m u s t b e i m p o s e d just to

m a k e the system d e t e r m i n a t e for the single t e m p e r a t u r e T

r

=

1. I f £ is c

i d e n t i f i e d w i t h Z , t h e n the n u m b e r of u n k n o w n s is r e d u c e d b y one, b u t c

w e are s t i l l left w i t h t w o degrees of f r e e d o m .

A s suggested b y t h e f o r m

of E q u a t i o n s 13, 14, a n d ^ l S ^ w h e n one deals w i t h t h e m o r e g e n e r a l case (£

C

=7^ Z c ) , parameters b

; " , , j F = 0.873 F (exp)

Ic, Vc = 0.150 For case H a , t = 0.075 l i b , R = 0.100 I I c , £ = 0.125 c

e

e

1

r

T

r

r

effects of the c h a n g e i n £ c a n b e seen b y c o m p a r i n g entries f o r C a s e l i b c

w i t h C a s e l a (t

c

=

0.100) o r C a s e l i e w i t h C a s e l b (% = Q

0.125). T h e

n e t effect o n t h e r e p r e s e n t a t i o n of t h e i s o t h e r m is to p r o d u c e p r e d i c t e d pressures greater t h a n t h e e x p e r i m e n t a l v a l u e s f o r r e d u c e d densities u p to a b o u t 0.4, b u t also to i n t r o d u c e a h o r i z o n t a l shift i n the^ c u r v e at s u p e r c r i t i c a l densities, thus y i e l d i n g f o r a p p r o p r i a t e v a l u e s of b

c

case, f o r b

c

( i n this

s o m e w h a t less t h a n 0.100) a " c o m p r o m i s e fit" o f t h e steep

p o r t i o n of t h e i s o t h e r m . T h i s , i n fact, is e x a c t l y w h a t o n e observes w i t h t h e R K e q u a t i o n , a n d i t is o n e o f t h e reasons f o r t h e success of t h a t expression a n d its n u m e r o u s p r o g e n y . T h e p r i c e o n e p a y s , of course, is a less-than-perfect p i c t u r e of t h e c r i t i c a l r e g i o n ( i n t h e p r e s e n t case t h e c r i t i c a l v o l u m e is 1 5 % t o o l a r g e ) , b u t a n a n a l y t i c e q u a t i o n of state is i n c a p a b l e of p r e c i s e d e s c r i p t i o n o f c r i t i c a l p h e n o m e n a a n y h o w .

More

serious is t h e d a m a g e d o n e w i t h respect t o r e p r e s e n t a t i o n of s a t u r a t e d l i q u i d v o l u m e s , f o r unless b is c o n s i d e r e d a b i z a r r e f u n c t i o n of T , t h e r

effect o f t h e l a r g e V

c

is felt i n t h e l i q u i d p h a s e d o w n t o t h e l o w e s t

reduced temperatures. S c h e m a t i c illustrations o f some o f t h e effects just d e s c r i b e d a r e s h o w n i n F i g u r e 1.


56

EQUATIONS

O F

STATE


a

V (exp) r

b

ι

·—· 1

?

Equations of State in Engineering and Research

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