Prediction of Binary Critical Loci by Cubic Equations of State

6

Prediction of Binary Critical Loci by Cubic Equations of State 1

2

2

Robert M. Palenchar , Dale D. Erickson , and Thomas W. Leland 1

Olefins Technology Division, Exxon Chemical Company, Raton Rouge, LA 70821 Department of Chemical Engineering, Rice University, Houston, TX 77251

Downloaded by GEORGE MASON UNIV on June 30, 2014 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch006

2

This paper compares reduced forms of five different cubic equations of state of the van der Waals family as to their ability to predict critical loci of binary mixtures. For the calculation of binary critical loci, the constants for each equation were expressed as functions of composition by the same mixing rules which involve two unlike pair coefficients evaluated by fitting to the experimental critical temperature and critical pressure at the same single composition. Only the critical temperature and critical pressure loci of van Konynenburg and Scott's Type I systems are predicted quantitatively over their entire composition range by the equations examined in this way. Critical volumes in all systems and critical temperature and pressure loci in Type II systems are only qualitatively predicted. Unlike pair coefficients evaluated at one point on a critical line in a two-phase region do not produce satisfactory predictions in portions of a critical locus where a third phase appears. There are no major differences in accuracy among any of the equations in predicting Type I critical loci. However, the Teja equation gave best predictions of Type I critical temperatures and pressures. The Peng-Robinson equation gave best predictions of Type I critical volumes. In Class 1, Type II and in all Class 2 systems, the twoconstant Soave and Redlich-Kwong equations generally predict critical temperatures and critical pressures better than the more elaborate three-constant equations and two constant equations whose form has a larger deviation from that of the original van der Waals equation. The accurate prediction of the properties of equilibrium phases and vapor-liquid e q u i l i b r i a i n mixtures near their c r i t i c a l i s an 0097-6156/86/0300-0132$07.00/0 © 1986 American Chemical Society

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


6.

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Prediction of Binary Critical Loci

133

important unsolved problem. The b a s i c d i f f i c u l t y i s t h a t t h e r e i s at present no accurate non-analytic equation of state for mixtures. The use of t y p i c a l e n g i n e e r i n g t y p e a n a l y t i c a l e q u a t i o n s of s t a t e w i t h e s s e n t i a l l y e m p i r i c a l m i x i n g r u l e s f a i l s i n the n e a r c r i t i c a l region. The n a t u r e of t h i s f a i l u r e i s w e l l u n d e r s t o o d . The d i v e r g e n c e of the p a r t i a l molar volume of a s o l u t e as i t becomes i n f i n i t e l y d i l u t e when the system approaches the s o l v e n t critical c o n d i t i o n s has r e c e n t l y been e x p l a i n e d c l e a r l y by Chang, M o r r i s o n , and L e v e l t Sengers (1). At f i n i t e c o n c e n t r a t i o n s the vapor and l i q u i d phase p r o p e r t i e s p r e d i c t e d by reduced a n a l y t i c a l e q u a t i o n s u s i n g p s e u d o - c r i t i c a l s approach each o t h e r too r a p i d l y as the system approaches i t s c r i t i c a l c o n d i t i o n s . When d e t e r m i n e d from reduced equations i n terms of p s e u d o - c r i t i c a l s t h e s e p r o p e r t i e s tend to predict critical conditions which l i e below the true critical locus. The t r a d i t i o n a l method of d e a l i n g w i t h t h i s problem has been t o s t o p phase e q u i l i b r i u m c a l c u l a t i o n s w e l l below the c r i t i c a l l o c u s and t o make an e m p i r i c a l e x t r a p o l a t i o n t o an e s t i m a t e of the t r u e c r i t i c a l p r o p e r t i e s of the s y s t e m . It may be possible to overcome these problems with c o r r e s p o n d i n g s t a t e s methods w h i c h use a p r o p e r l y s c a l e d r e f e r e n c e fluid equation. Pseudo-critical relations at present cannot a c c u r a t e l y p r e d i c t the second and t h i r d d e r i v a t i v e s o f the Gibbs f r e e energy w i t h r e s p e c t to c o m p o s i t i o n which are needed to d e f i n e the c r i t i c a l of a m i x t u r e . Furthermore, a d d i t i o n a l research is needed to d e t e r m i n e the b e s t way to map the c r i t i c a l r e g i o n of a m i x t u r e on t o the c r i t i c a l r e g i o n of a pure s c a l e d r e f e r e n c e w i t h o u t destroying the effective representation of multicomponent e q u i l i b r i u m phases which i s currently possible below the near c r i t i c a l region. A l t h o u g h r e s e a r c h w i t h t h e s e and o t h e r methods f o r i m p r o v i n g p r e d i c t i o n s i n the c r i t i c a l r e g i o n i s c o n t i n u i n g , a common need with any method under i n v e s t i g a t i o n is a procedure for e s t i m a t i n g the t r u e c r i t i c a l l o c u s of the m i x t u r e . F o r t h i s r e a s o n , t h i s paper examines a f e a t u r e of the c l a s s i c a l van d e r Waals f a m i l y of e q u a t i o n s which e n a b l e s them t o g i v e a v e r y good r e p r e s e n t a t i o n of the c r i t i c a l l o c i of s i m p l e m i x t u r e s by u s i n g properly assigned unlike pair interaction coefficients. This p r o c e d u r e may cause p r o p e r t i e s near the m i x t u r e c r i t i c a l t o be predicted poorly, but the critical locus itself is described remarkably w e l l . D u r i n g the l a s t twenty y e a r s a g r e a t d e a l of work has been done t o s t u d y the p r e d i c t i o n of b i n a r y c r i t i c a l l o c i i n t h i s manner ( 2 ]). U s i n g the o r i g i n a l van der Waals e q u a t i o n , van Konynenburg and Scott (8) q u a l i t a t i v e l y p r e d i c t e d n i n e d i f f e r e n t t y p e s o f b i n a r y critical loci. These d i f f e r e n t t y p e s a r e i d e n t i f i e d by the b e h a v i o r of c r i t i c a l p r o p e r t i e s i n a p r o j e c t i o n of the Ρ , Τ , χ s u r f a c e of the system on t o a p r e s s u r e - t e m p e r a t u r e p l a n e . These t y p e s a r e summarized i n T a b l e I . Thermodynamics of the C r i t i c a l P o i n t The thermodynamic c o n d i t i o n s f o r the e x i s t e n c e of a c r i t i c a l p o i n t were f i r s t d e r i v e d by Gibbs (9) more than a hundred y e a r s a g o . The d e f i n i n g e q u a t i o n s f o r b i n a r y system c r i t i c a l p o i n t s a r e as f o l l o w s :


134

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

(φ

0

Ρ,Τ =

V(V+b)

'

2>5

0.4278 R

2

/P„

Τ

b - 0.0867 R T / P c

c

Soave ( 1 2 ) : Ρ - RT _ a(T) V=b V(V+b) a(T)

f 6 K

= 0.42747 a(T) R

2

2

Τ /P

c

c

b - 0.08664 R T / P a(t) =* {l+m(l-TrO'5 2 c

c

)}

m

- 0.480 + 1 . 5 7 4 ω -

0.176ω

2


0

. )

6.

PALENCHAR


ET AL.

135

Adachi (13): Y

RT l^b

'

a(T) V(V+c)

a(T) -

A

m w

5

2

Z {l+a(l-Tr°- )} R

Q

2

c

b = Β Z

c

R T /P

c

c = C Z

c

R T /P

c

c

c

2

T /P c

a - 0.479817 + 1.55553ω - 0.287787ω 3


A

0

= B (l+C) /(l-B)

c

2

3

Β = 0.260796 - 0.0682692ω - 0.0367338ω

c

2

[{(4/Β)-3}°·5-3]/2

C = Z

;

= [1/(1-B)]-A /(1+C) 0

Peng and Robinson (14): p — _RT _ a(T) " V-b V(V+b)+b(V-b)

( 8 )

r

a(T) = 0.45724 a(T) R b - 0.07780 R T / P c

o(T)

2

2

T /P c

c

{1+Κ(1-ΤΓΟ* )}

-

c

5

2

κ - 0.37464 + 1.54226ω - 0.26992ω

2

Teja U 5 ) : RT _ a(T) ~ V-b V(V+b)+c(V-b)

( 9 )

Y

a(T) = Ω α ( Τ ) a b = Ω, R b

Τ

R

2

'

2

T /Pc c

/P c e

c - Ω R Τ /Ρ c c e Ω = 3Z + 3(1-2Z )Ω + Ω a c C D D 2

2

+ 1-3Z^

Κ

c

Ω, equals the smallest positive root of: Ω J + (2-3Z )Ω + 3Z Ω - Ζ 3 - ο b c b c b c b

2

2

Κ

Ω

c

Ζ

= 1 - 3Z

c

c

= 0.329032 - 0.076799ω + 0.0211947ω

α(Τ)

= {1+F(l-Tr°« )}

F -

0.452413 + 1.30982ω - 0.295937ω

5

2

2

2


EQUATIONS OF STATE: THEORIES AND APPLICATIONS

136

loci predicted by various equations using two unlike pair c o e f f i c i e n t s obtained from a single data point are i l l u s t r a t e d i n Figures 1-13. The errors reported i n Table II f o r a l l the c r i t i c a l properties are obtained by the procedure shown below f o r the c r i t i c a l temperature:

|( Average % Relative Absolute Error

)

T

-

- (χ )

S-^S k

Vexpt

3

I

c expt

w

1 Q Q )

3

The subscript j i n Equation (15) indicates represents the t o t a l number of data points.

(15)

a data

point and Ν


Unlike Pair Interaction Coefficients Table I I I shows the values of the unlike pair coefficients ζ and λ i n Equations (10) and (11) obtained by f i t t i n g to the experimentally measured c r i t i c a l temperature and c r i t i c a l pressure i n an equi-molar, or nearly equi-molar, mixture. Because of the empirical nature of the equations of state i n which these c o e f f i c i e n t s are used, their actual relationship to any e f f e c t i v e pair potential i s rather obscure. The name "unlike pair interaction c o e f f i c i e n t s " i s used i n an i d e a l sense only. Furthermore, i t i s important to note that these values apply only to the projected c r i t i c a l l o c i and bear l i t t l e resemblance to their optimal values obtained by f i t t i n g other portions of the P-T-x surface. 1 2

1 2

Even when f i t t i n g c r i t i c a l l o c i alone, a s i g n i f i c a n t l y better correlation could obviously be obtained with any of the equations tested by optimizing either ζ^ when λχ *1·0> or by optimizing both ζ and X^ together, over the entire composition range. This, however, obscures the r e l a t i v e degree of composition dependence of these unlike pair c o e f f i c i e n t s i n the various equations of state and i s less revealing of the shape of the c r i t i c a l locus predicted by the a n a l y t i c a l form of any p a r t i c u l a r equation of state. However, among the equations with the same mixing rules with unlike pair c o e f f i c i e n t s evaluated i n the same way from the same single data point within the same group of data points, the equation which produces the lowest average absolute error f o r the entire group of these data points should produce the best c o r r e l a t i o n when i t s unlike pair c o e f f i c i e n t s are optimized f o r a l l data points i n the group. s

2

1 2

s

e

t

a

t

2

2

In a few cases i t was not possible to obtain an accurate prediction of both the c r i t i c a l pressure and the c r i t i c a l temperature with any combination of ζ χ ^ λχ2 values. These cases are indicated with a * i n Table I I I and the values presented are those giving the best prediction of T and P . a n

2

c

c

The policy of f i t t i n g the unlike pair c o e f f i c i e n t s at a single data point gives information regarding the r e l a t i v e degree of composition independence i n the values of these c o e f f i c i e n t s among the various equations of state tested. Furthermore, this policy also gives information about the shape of the c r i t i c a l loci predicted by the various equations. This i s i l l u s t r a t e d by Figures 10 and 11 where the c r i t i c a l temperatures and c r i t i c a l pressures of


6.


PALENCHAR ET AL.

137

TABLE I van Konynenburg and Scott C l a s s i f i c a t i o n of Binary C r i t i c a l Loci Class 1 Description

Type I.

One continuous gas-liquid c r i t i c a l l i n e connecting the two pure component c r i t i c a l points CI and C2. CI represents the c r i t i c a l of the component with the lowest c r i t i c a l temperature and C2 i s the c r i t i c a l of the other component.

I- A. The same as I, with the addition of a negative azeotrope.


II.

Two c r i t i c a l lines: One i s a vapor-liquid c r i t i c a l l i n e between the pure component c r i t i c a l s CI and C2. The second i s a l i q u i d - l i q u i d c r i t i c a l l i n e representing the merger of two liquids with limited m i s c i b i l i t y i n the presence of a s o l i d phase. Because of the limited m i s c i b i l i t y , this c r i t i c a l l i n e persists to Immeasurably high pressures so that i t s o r i g i n may be regarded as occurring at an i n f i n i t e l y large pressure. From this i n f i n i t e pressure this liquid-liquid critical line connects at lower pressure with an upper c r i t i c a l end point (UCEP), representing the high pressure termination point of a three-phase liquid-liquid-solid line. At this termination point the two l i q u i d phases become i d e n t i c a l . The o r i g i n of this l i q u i d - l i q u i d l i n e at an i n f i n i t e c r i t i c a l pressure i s designated with the symbol C by van Konynenburg and Scott (8). m

I I - A.

Type

The same azeotrope.

as

II,

but

with

the

addition

of

a

positive

Class 2 Description

III-HA. Two c r i t i c a l l i n e s , but no continuous c r i t i c a l l i n e runs between CI and C2. The f i r s t of the c r i t i c a l l i n e s i s a vaporliquid critical line from CI to the high temperature termination point (UCEP) of a liquid-liquid-gas three phase l i n e where the l i q u i d richest i n the component with the lowest c r i t i c a l temperature and the gas phase become i d e n t i c a l . The other critical line connects one of two possible low temperature origins to the c r i t i c a l C2 of the component with the higher c r i t i c a l temperature. The portion of this second c r i t i c a l l i n e near i t s low temperature o r i g i n i s a l i q u i d l i q u i d c r i t i c a l line which changes to a vapor-liquid c r i t i c a l l i n e before approaching C2. The two points of o r i g i n for this second c r i t i c a l l i n e identify two sub-types of Type III-HA defined as IIla-HA and IIIb-HA as follows: Continued on next page


138

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

Table I Continued IIIa-HA. In Type IIIa-HA the two liquids which merge along the low temperature portion of the second c r i t i c a l l i n e have limited m i s c i b i l i t y so that their merger requires increasingly higher c r i t i c a l pressures as the c r i t i c a l temperature i s lowered. The low temperature o r i g i n of this l i q u i d - l i q u i d c r i t i c a l l i n e i n this case may be regarded as occurring at an i n f i n i t e c r i t i c a l pressure and i s given the symbol C by van Konynenburg and Scott (8).


m

IIIb-HA. The second low temperature point of o r i g i n for the c r i t i c a l l i n e to C2 occurs at the high pressure termination point, or UCEP, of a three phase l i q u i d - l i q u i d - s o l i d l i n e at a point where the two l i q u i d phases become i d e n t i c a l i n the presence of the s o l i d . Type IIIb-HA was not considered by van Konynenburg and Scott because i t s o r i g i n at the UCEP of a three phase l i q u i d - l i q u i d - s o l i d l i n e cannot be represented by the o r i g i n a l van der Waals equation. In Type IIIa-HA or Type IIIb-HA the entire three phase l i q u i d liquid-gas l i n e l i e s at pressures above the vapor pressure curves of both of the two pure components. This condition represents what i s called "hetero-azeotropic" behavior and i s designated by the symbol "HA" i n the c l a s s i f i c a t i o n of this type of c r i t i c a l behavior. I I I . The same as III-HA except that the three phase l i q u i d - l i q u i d gas l i n e ending with the UCEP l i e s between the vapor pressure curves of the two pure components. Like Type III-HA, Type III may be subdivided into two sub-types I l i a and I l l b , depending upon the o r i g i n of the c r i t i c a l l i n e to C 2 . Type I l i a originates at and Type I l l b originates at the UCEP terminating a three phase l i q u i d - l i q u i d - s o l i d l i n e . Because the three phase liquid-liquid-gas l i n e does not l i e outside the two pure component vapor pressure curves, there i s no heteroazeotropic behavior i n Type I l i a or Type I l l b systems. IV.

Three c r i t i c a l l i n e s : A vapor-liquid c r i t i c a l l i n e runs from CI to the high temperature UCEP termination of a l i q u i d - l i q u i d gas three phase l i n e where a l i q u i d and the gas phase become identical. In this case, however, this liquid-liquid-gas line i s discontinuous. As the temperature i s lowered, i t ends at a LCEP where the two l i q u i d phases merge. At temperatures below this LCEP there i s a short range where the l i q u i d s are completely miscible and only a l i q u i d and gas phase are present. At s t i l l lower temperatures, there i s a second UCEP where two l i q u i d phases are again c r i t i c a l . Below this second, or low temperature, UCEP the liquid-liquid-gas three phase l i n e continues with decreasing temperature u n t i l i t ends with the formation of a s o l i d phase at a quadruple point. A second c r i t i c a l l i n e runs from the LCEP to C 2 . Near the LCEP i t i s a l i q u i d - l i q u i d c r i t i c a l l i n e and i n aproaching C2 i t changes to a vapor-liquid c r i t i c a l l i n e .


6.


PALENCHAR ET AL.

139

Table I Continued The third c r i t i c a l l i n e i s a l i q u i d - l i q u i d c r i t i c a l locus between the second or low temperature UCEP and a C point at an immeasurably high pressure. m


V.

Two c r i t i c a l l i n e s : In this system there i s a single three phase liquid-liquid-gas l i n e ending at a UCEP where one of the l i q u i d s and the gas phase become i d e n t i c a l . As the temperature i s lowered, this three phase l i n e stops at a LCEP where the two l i q u i d phases merge. One of the c r i t i c a l lines i s a gas-liquid c r i t i c a l locus from CI to the UCEP. The second c r i t i c a l l i n e runs from the LCEP to C2 and consists of a l i q u i d - l i q u i d c r i t i c a l l i n e near the LCEP which changes to a vapor-liquid c r i t i c a l l i n e when approaching C2.

V-A. The same as azeotrope.

Type

V

but with

the addition

of a negative

Class 3 The van Konynenburg and Scott c l a s s i f i c a t i o n includes a Class 3 behavior which i s exhibited by very complex mixtures with strong s p e c i f i c interactions, usually involving hydrogen bonds, between the components. These systems have LCSTs where there i s a minimum i n a T-x coexistence curve. Systems i n this class cannot be represented by equations of state of the van der Waals family.

y

-

\

~

\

/

\

ce Q. (I)

α.

\ »

TJ5J

EXPERIMENTAL TEJA EQUATION PEN6 AND ROBINSON

70·

TSO

Si

eSi

δϊ~

TEMPERATURE (R)

Figure 1.

C r i t i c a l l o c i of propane-hexane

system.


EQUATIONS 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

\

/


\

tu oc V) 2CA ·

»

%T% ïSi

4··

EXPERIMENTAL TEJA EQUATION PEN6 ANO ROBINSON

Sî

sot

880

TEMPERATURE (R)

Figure 2. C r i t i c a l l o c i of m ethane-car bon dioxide system.

V en

en ο en S. tu m oc a.

»

"Soô

EXPERIMENTAL TEJA EQUATION AOACHI EQUATION 800

680

700

TEMPERATURE (R)

Figure 3.

C r i t i c a l l o c i of butane-carbon dioxide system.


6.


PALENCHAR ET AL.

» 0

*-*

141

EXPERIMENTAL (CURVE #1) EXPERIMENTAL (CURVE #2) SOAVE EQUATION

ο


α. 3

CL

usually decreasing i t s value. Unfortunately, when the calculated and experimental Pc values are brought into good agreement by this process, the error i n the calculated Tc i s made larger than the i n i t i a l r e s u l t obtained with both X and ζ equal to 1.0 . 1 2


=

a n

Prediction of Binary Critical Loci by Cubic Equations of State

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