Equations of State - American Chemical Society

26 Selection and Design of Cubic Equations of State J.-M. Yu, Y. Adachi, and B.C.-Y. Lu

Downloaded by NORTH CAROLINA STATE UNIV on August 22, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch026

Department of Chemical Engineering, University of Ottawa, Ottawa, Ontario K1N 9B4, Canada

Fourteen known cubic equations of the van der Waals type, Ρ = RT/(V-b) - a(T)/(V + ubV + wb ), were evaluated through the calculations of eight properties of normal alkanes. The roles of u and w in these calculations were demonstrated. A new design proce dure was developed and a new relationship between u and w was suggested. 2

2

R e l i a b l e methods f o r p r e d i c t i n g p h y s i c a l p r o p e r t i e s o f pure compo nents and t h e i r m i x t u r e s (such as s i n g l e and two-phase p r o p e r t i e s , and v a p o r - l i q u i d e q u i l i b r i a ) a r e f r e q u e n t l y r e q u i r e d i n process d e s i g n and m a t e r i a l h a n d l i n g . In view of t h e wide range of s t a t e c o n d i t i o n s found i n p r a c t i c a l a p p l i c a t i o n s , and t h e frequent l a c k o f e x p e r i m e n t a l d a t a , c o n s i d e r a b l e a t t e n t i o n has been p a i d t o t h e development of these methods. In p a r t i c u l a r , a number of e q u a t i o n s of s t a t e have been proposed i n t h e l i t e r a t u r e t o meet t h i s demand. The most popular equations of s t a t e a r e c u b i c i n volume ( o r d e n s i t y ) . A g e n e r i c e x p r e s s i o n f o r t h e c u r r e n t l y popular c u b i c equations of s t a t e may be represented i n t h e form of an extended van der Waal s (VDW) e q u a t i o n , Ρ =KL V-b V +ubV+wb a

2

(1) 2

The q u a d r a t i c e x p r e s s i o n i n volume (V +ubV+wb ) r e p l a c e s t h e V term i n t h e denominator of the a t t r a c t i v e term of t h e o r i g i n a l VDW e q u a t i o n . When t h e parameters u and w are a s s i g n e d c e r t a i n p a r t i c u l a r v a l u e s , Equation 1 can be reduced t o t h e o r i g i n a l VDW equation (u=w=0)U), t h e Redlich-Kwong (RK) equation (u=l, w=0)(2) and i t s v a r i o u s m o d i f i e d forms, t h e Peng-Robinson (PR) equation (u=2, w=-l)(_3), t h e Heyen (H) equation (u+w=l)(4), t h e Schmidt-Wenzel (SW) e q u a t i o n (u+w=l)(5), and a number of other e q u a t i o n s . Some general f e a t u r e s of Equation 1 or i t s e q u i v a l e n t e x p r e s s i o n have been d i s cussed by Abbott {6). There i s evidence i n t h e l i t e r a t u r e (7-9) t h a t p r a c t i c a l l y i d e n t i c a l v a p o r - l i q u i d e q u i l i b r i u m (VLE) values ( T - P - c o m p o s i t i o n ) 2

2

0097-6156/ 86/ 0300-0537$06.75/ 0 © 1986 American Chemical Society

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

2


538

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

can be o b t a i n e d from c u b i c e q u a t i o n s of s t a t e c o n t a i n i n g two t o f o u r parameters, and these r e s u l t s are f r e q u e n t l y comparable t o t h o s e o b t a i n e d from more complex e q u a t i o n s of s t a t e . I t has been f u r t h e r demonstrated (10) t h a t by t r e a t i n g the parameter " a " of E q u a t i o n 1 temperature dependent, the VDW e q u a t i o n i s as capable as o t h e r c u b i c e q u a t i o n s f o r c a l c u l a t i n g VLE v a l u e s . On the o t h e r hand, the c a p a b i l i t i e s of the a v a i l a b l e c u b i c e q u a t i o n s f o r r e p r e s e n t i n g v o l u m e t r i c p r o p e r t i e s vary from e q u a t i o n t o e q u a t i o n , e s p e c i a l l y i n the c a l c u l a t i o n of l i q u i d volumes. Kumar and S t a r l i n g (11) suggested t h a t " a t a p a r t i c u l a r t e m p e r a t u r e , a h i g h e r density"~3ependence l e a d s t o a more a c c u r a t e equation of s t a t e " . They i n c r e a s e d the c o m p l e x i t y of Equation 1, and proposed a f i v e - p a r a m e t e r c u b i c e q u a t i o n of s t a t e f o r p r e d i c t i n g l i q u i d d e n s i t i e s and low temperature vapor p r e s s u r e s . One approach i s t h e r e f o r e t o s e p a r a t e the i s s u e s of VLE c a l c u l a t i o n s and v o l u m e t r i c p r e d i c t i o n s i n the a p p l i c a t i o n of e q u a t i o n s of s t a t e . In o t h e r words, d i f f e r e n t equations are used f o r d i f f e r ent p u r p o s e s . Another approach i s t o f u r t h e r improve the p e r f o r m a n ce of the a v a i l a b l e e q u a t i o n s or develop new e q u a t i o n s t o s a t i s f y the requirements of both VLE and v o l u m e t r i c c a l c u l a t i o n s . The t a s k i s a c t u a l l y reduced t o d e v e l o p i n g a s u i t a b l e e q u a t i o n , w i t h a compromise between s i m p l i c i t y and accuracy f o r r e p r e s e n t a t i o n of v o l u m e t r i c d a t a . In both approaches, g u i d e l i n e s f o r s e l e c t i n g the a p p r o p r i a t e e q u a t i o n s are r e q u i r e d . There i s s t i l l a need t o f u r t h e r e v a l u a t e the a v a i l a b l e c u b i c e q u a t i o n s i n a s y s t e m a t i c manner, so t h a t the u t i l i z a t i o n of a c u b i c e q u a t i o n can reach i t s f u l l p o t e n t i a l . The purpose of t h i s study i s t h e r e f o r e t o e v a l u a t e a v a i l a b l e c u b i c equations of s t a t e of the VDW t y p e as r e p r e s e n t e d by Equation 1 t o i d e n t i f y the m e r i t s and l i m i t a t i o n s of these e q u a t i o n s f o r the purpose of s e l e c t i o n , and t o suggest a s u i t a b l e procedure f o r d e s i g n i n g new c u b i c e q u a t i o n s of t h e same t y p e but t a i l o r e d t o s p e c i f i c p u r p o s e s . E v a l u a t i o n o f Cubic Equations

of

State

The t e c h n i q u e s used i n the improvement of the o r i g i n a l VDW e q u a t i o n f o r p h y s i c a l p r o p e r t y p r e d i c t i o n s , w i t h o u t changing the e x p r e s s i o n of Equation 1, may be grouped i n t o the f o l l o w i n g t h r e e c a t e g o r i e s : 1. M o d i f i c a t i o n of the e x p r e s s i o n used i n the denominator of the a t t r a c t i v e term. 2 . A p p l i c a t i o n of a v o l u m e - t r a n s l a t i o n t e c h n i q u e t o the o r i g i n a l VDW e q u a t i o n and the e q u a t i o n s of the f i r s t c a t e g o r y . 3. I n t r o d u c t i o n of temperature dependence t o one or more p a r a m e t e r s . The e q u a t i o n s of RK, PR and SW are examples of the f i r s t c a t e g o r y . The C l a u s i u s (C) e q u a t i o n (12) i s an example of the second c a t e g o r y . The Soave form of the RK e q u a t i o n (SRK)(13) and t h e m o d i f i c a t i o n of the RK e q u a t i o n by Hamam et a l . ( H C L ) Q T ) are examples of the t h i r d c a t e g o r y . In a d d i t i o n t o the above mentioned SRK, PR, SW, HCL and H equat i o n s , we have i n c l u d e d i n our c o n s i d e r a t i o n the Harmens-Knapp (HK) e q u a t i o n ( 1 5 ) , the P a t e l - T e j a (PT) e q u a t i o n ( 1 6 ) , the r e s u l t i n g e q u a t i o n from volume t r a n s l a t i o n of the SRK e q u a t i o n (TSRK) proposed by Peneloux et a l . ( 1 7 ) , the f o u r - p a r a m e t e r e q u a t i o n of Adachi et a l . ( A L S ) ( 9 ) , and the CI e q u a t i o n proposed by Peneloux et a l . ( 1 8 ) .



26. YU ET AL.

539

Design of Cubic Equations of State

The o r i g i n a l VDW, RK and C equations are not e v a l u a t e d because t h e i r poor performance on v o l u m e t r i c p r e d i c t i o n s i s known. On the o t h e r hand, the t h r e e - p a r a m e t e r RK e q u a t i o n s suggested by Adachi et a l . (3RK)(19) and by F u l l e r (F) ( 2 0 ) , and the M a r t i n Equation (21) (a v e r s i o n of the C l a u s i u s e q u a t i o n ) i n two m o d i f i e d v e r s i o n s W C (Adachi et a l . (19)) and KMC (Kubic (22)) are c o n s i d e r e d . Thus, a t o t a l of f o u r t e e n c u b i c equations are e v a l u a t e d i n t h i s work. Among these e q u a t i o n s , the RK e q u a t i o n i s one of the most s u c c e s s f u l two-parameter c u b i c e q u a t i o n s . In the SRK e q u a t i o n , Soave made an e f f o r t i n 1972 t o c l o s e l y reproduce vapor p r e s s u r e s of pure compounds by assuming the parameter " a " of the o r i g i n a l e q u a t i o n t o be temperature dependent. This m o d i f i c a t i o n enhanced t h e a p p l i c a b i l i t y of the RK e q u a t i o n f o r c a l c u l a t i n g VLE v a l u e s . Many of the equations which have subsequently appeared i n t h e l i t e r a t u r e have adopted the same or s i m i l a r m o d i f i c a t i o n s . As f a r as the r e p r e s e n t a t i o n of v o l u m e t r i c data f o r pure f l u i d s i s c o n c e r n e d , M a r t i n (21) concludes t h a t the C l a u s i u s - t y p e e q u a t i o n i s the best of the s i m p l e r c u b i c e q u a t i o n s . However, the c a l c u l a t i o n of v o l u m e t r i c p r o p e r t i e s at s a t u r a t i o n c o n d i t i o n s w i t h o u t c o n s i d e r i n g the e q u a l i t y of f u g a c i t y at the same time as a p p l i e d by M a r t i n c o u l d i n t r o d u c e i n t e r n a l i n c o n s i s t e n c y i n the c a l c u l a t e d v a l u e s . As mentioned above, the C l a u s i u s e q u a t i o n can be o b t a i n e d from a volume t r a n s l a t i o n of the VDW e q u a t i o n . I t i s d i f f i c u l t t o envisage t h a t the volume t r a n s l a t e d VDW e q u a t i o n i s s u p e r i o r t o the v o l u m e - t r a n s l a t e d RK e q u a t i o n (TSRK). A comparison of the c a l c u l a t e d r e s u l t s i s d e f i n i t e l y of i n t e r e s t . It should be mentioned t h a t i n the two m o d i f i e d v e r s i o n s of the M a r t i n e q u a t i o n , MMC (19) and KMC ( 2 2 ) , the parameter " a " of Equation 1 was t r e a t e d as temperature dependent. Parameters o t h e r than " a " of Equation 1 have been assumed t o be temperature dependent i n some c u b i c e q u a t i o n s . For example, the H e q u a t i o n c o n t a i n s t h r e e parameters, two of which are assumed t o be temperature dependent. It i s d e s i g n a t e d i n t h i s work as a 3P2T e q u a t i o n . The o t h e r equations are s i m i l a r l y d e s i g n a t e d . The i n v e s t i g a t e d equations of s t a t e are thus grouped i n t o s i x types as shown i n Table I. The r e l a t i o n s h i p s between u and w f o r these e q u a t i o n s , the parameters which are t r e a t e d temperature dependent as w e l l as t h e p r o p e r t i e s (other than T and P ) used t o determine the paramet e r s of these e q u a t i o n s are a l s o presented i n Table I. In the t a b l e , the two-parameter e q u a t i o n proposed by Harmens (HA) (23) was a l s o i n c l u d e d , because i t s u and w r e l a t i o n s h i p formed p a r t o f the b a s i s i n the development of the SW e q u a t i o n . c

R e p r e s e n t a t i o n of P h y s i c a l

c

Properties

The p r o p e r t i e s of ten normal alkanes from methane t o n-decane, o b t a i n e d from g e n e r a l i z e d c o r r e l a t i o n s and t a b u l a t i o n s a v a i l a b l e i n the l i t e r a t u r e were used as the b a s i s f o r comparing the performance of the f o u r t e e n e q u a t i o n s . A t o t a l of e i g h t p r o p e r t i e s were c o n s i d e r e d . The vapor p r e s s u r e s , p , were o b t a i n e d from the g e n e r a l i z e d c o r r e l a t i o n proposed by Gomez-Nieto and Thodos (24) f o r nonpolar substances. The s a t u r a t e d l i q u i d volumes, V * , were o b t a i n e d from a m o d i f i e d R a c k e t t e q u a t i o n using the i n p u t parameters suggested by Spencer and A l d e r ( 2 5 ) . The s a t u r a t e d vapor volumes, V , were o b t a i n e d from the c o r r e l a t i o n of B a r i l e and Thodos ( 2 6 ) . The second v

v


EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

540

Table I

Features of Some Cubic Equations Ρ = RT/(Y-b) - a ( T ) / ( V

TYPE


2P1T

3P1T

4P1T 2P2T 3P2T 3P3T

EOS

u

w

VDW RK SRK PR HA HK SW PT 3RK MMC TSRK CI ALS HCL Η KMC F

0 1 1 2 3 1-w 1-w 1-w

0 0 0 -1 -2

fU) fU)

f(co) f(ω) f() 0 a(T) fU,b) a(T) u /4 a(T) 0 a(T)

p

b^ bj b b

p

v

v

c c

r

V

r

b^ bj

2

2

b

2

p

c

> ν

PV*

bj b

P , Critical P ,V* P\V* P\B, p ,V*,V v

c

bU)

Isotherm

V

b(T) b(T) b(T)

2

j V

v

v

v i r i a l c o e f f i c i e n t s , B, were o b t a i n e d from the c o r r e l a t i o n o f Tsonopoulos (27)« For these f o u r p r o p e r t i e s , p o i n t s were taken at 0.02 i n t e r v a l s i n the T range of 0.5 t o 0.80 and at T equals 0 . 8 5 , 0.90, 0.95 and 0.98 f o r a t o t a l of 20 Τ v a l u e s . The c o r r e l a t i o n o f Lee and K e s l e r (28) was used t o o b t a i n the values f o r l i q u i d c o m p r e s s i b i l i t y T F c t o r ( Z , 0.30 < T < 0 . 9 9 , 0.01 < P < 1 0 . 0 , 315 p o i n t s ) , c o m p r e s s i b i l i t y f a c t o r of vapor ( Z , 0.55 < T < 0 . 9 9 , 0.01 < P < 0 . 8 , 56 p o i n t s ) , c o m p r e s s i b i l i t y f a c t o r of gas above t h e c r i t i c a l temperature ( Z , 1.01 < T < 4 . 0 0 , 0.01 < P < 1 0 . 0 , 240 p o i n t s ) and c o m p r e s s i b i l i t y f a c t o r along the c r i t i c a l isotherm ( Z , T = 1 . 0 , 0.01 < Ρ < 10.0, 15 p o i n t s ) . A summary of the c a l c u l a t i o n r e s u l t s , i n terms of o v e r a l l average a b s o l u t e percent d e v i a t i o n s , i s presented i n Table I I . The r e p o r t e d values may be s l i g h t l y d i f f e r e n t from s i m i l a r c a l c u l a t i o n s a v a i l a b l e i n the l i t e r a t u r e due t o the d i f f e r e n c e i n t h e covered ranges of Tp and P and i n the number of data p o i n t s s e l e c t e d f o r the c a l c u l a t i o n . I t i s w e l l known t h a t a c c u r a t e r e p r e s e n t a t i o n of vapor p r e s s u res i s e s s e n t i a l f o r 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 t i o n s . For t h i s r e a s o n , vapor p r e s s u r e values have been used t o determine t h e values of parameter " a " , or Q (= a Pc/R Τ ) . The g e n e r a l i z e d e x p r e s s i o n s of t h e o r i g i n a l authors f o r " a were used t o c a l c u l a t e p and p r a c t i c a l l y a l l e q u a t i o n s t e s t e d y i e l d e d a c c e p t a b l e r e s u l t s . As f a r as V* i s c o n c e r n e d , the m o d i f i e d M a r t i n (MMC) and the SRK equations y i e l d the l a r g e s t d e v i a t i o n s . The d e v i a t i o n s i n t h e c a l c u l a t e d V v a l u e s f o l l o w c l o s e l y t o those f o r p , and t h e d e v i a t i o n s i n Z v a l u e s f o l l o w c l o s e l y t o those f o r V^. p

r

x

p

p

v

p

r

s u p

f

r

c

r

p

2

2

a

v

v

v

A


Table II

Summary of O v e r a l l Average A b s o l u t e Percent D e v i a t i o n s the C a l c u l a t e d P h y s i c a l P r o p e r t i e s f o r Ten Normal

Property p

v

Z* Z sup Z B v

z


c

Property p

v

V* V

v

Z* Z sup Z Β v

z

c

541


26. YU ET AL.

NT

SRK

200 200 200 3150 560 2400 150 200

1.51 13.4 1.23 11.2 0.95 2.37 8.50 17.2

TSRK 1.51 3.77 1.20 3.67 0.64 1.35 5.42 16.1

PR 2.55 5.47 2.54 4.96 0.58 1.54 4.68 15.5

SW

HK 1.02 4.37 1.45 4.14 0.55 1.47 4.05 L5.1

1.44 2.78 1.22 2.96 0.38 1.41 4.08 15.6

PT 1.95 2.56 2.01 3.03 0.40 1.28 4.13 15.4

in

3RK

MMC

1.29 4.00 1.08 3.37 0.56 1.36 4.92 15.9

2.33 12.4 2.51 12.6 0.44 1.89 5.24 15.4

CI

ALS

HCL

Η

KMC

F

0.96 3.65 1.26 3.93 0.49 1.30 6.24 15.6

1.42 2.61 1.74 3.13 0.45 1.30 4.00 15.4

1.88 0.65 4.10 2.61 2.24 2.98 8.37 23.2

10.5 2.26 10.3 5.53 2.24 3.72 8.90 16.9

3.56 5.41 4.01 13.5 0.19 2.09 5.32 1.71

1.83 2.01 1.97 5.28 0.99 6.16 12.6 17.4

The d i f f e r e n c e s i n the c a l c u l a t e d Z , Z and Z v a l u e s among t h e 14 e q u a t i o n s are not too s i g n i f i c a n t . The KMC e q u a t i o n g i v e s t h e lowest d e v i a t i o n s i n the c a l c u l a t e d Β v a l u e s , which were used i n the d e t e r m i n a t i o n of the parameters of the e q u a t i o n . A l t h o u g h some of the f i n d i n g s mentioned above c o u l d be e n v i saged from the f e a t u r e s l i s t e d i n Table I, t h e r e are s e v e r a l i n t e r e s t i n g p o i n t s r e v e a l e d by the r e s u l t s of Table I I . 1. The c u r r e n t l y popular c u b i c equations of s t a t e (SRK and PR) do not y i e l d the best r e s u l t s . 2 . The performance of the 3P2T Η e q u a t i o n i s i n f e r i o r t o t h a t of the 3P1T PT e q u a t i o n . Indeed, the o v e r a l l performance of e q u a t i o n s c o n t a i n i n g more than one temperature-dependent parame t e r i s g e n e r a l l y i n f e r i o r , i n d i c a t i n g t h a t these equations are m a i n l y s u i t a b l e f o r r e p r e s e n t i n g the p h y s i c a l p r o p e r t i e s used i n the f o r c e d f i t t i n g p r o c e d u r e . 3. The M a r t i n e q u a t i o n i s not s u i t a b l e f o r VLE and v o l u m e t r i c calculations simultaneously. 4 . A t h r e e - p a r a m e t e r c u b i c e q u a t i o n w i t h o n l y the parameter " a " t r e a t e d temperature dependent i s adequate f o r the purpose of t h i s s t u d y . The performance of the s i x equations of t h i s type (HK, SW, PT, 3RK, TSRK, CI) seems t o be adequate. F u r t h e r examination of the d e v i a t i o n s o b t a i n e d f o r i n d i v i d u a l a l k a n e s (as shown i n Table I I I ) from the 2P1T and 3P1T e q u a t i o n s reveals that: 1. The PR e q u a t i o n y i e l d s l a r g e r d e v i a t i o n s i n p v a l u e s f o r l a r g e r m o l e c u l e s , and the d e v i a t i o n s i n V* i n c r e a s e w i t h i n c r e a s e i n molecular weight. v

s u p

c

v


542


2.

For t h e SRK e q u a t i o n , t h e d e v i a t i o n s i n V * a l s o i n c r e a s e w i t h increase in molecular weight, 3 . A l l 2P1T and 3P1T equations y i e l d l a r g e d e v i a t i o n s i n B, and t h e deviations increase with increase in molecular weight. 4 . The TSRK e q u a t i o n y i e l d s d e v i a t i o n s i n p i d e n t i c a l t o those of the SRK e q u a t i o n , c o n f i r m i n g the f a c t t h a t volume t r a n s l a t i o n does not a f f e c t the p c a l c u l a t i o n s . 5. As f a r as t h e d e v i a t i o n s i n V * and are c o n c e r n e d , the PT e q u a t i o n appears t o be s l i g h t l y s u p e r i o r i n t h e f a m i l y o f e q u a t i o n s (HK SW and PT) which can be r e p r e s e n t e d by t h e u + w = 1 r e l a t i o n s h i p . Among t h e equations r e s u l t i n g from volume t r a n s l a t i o n (MMC, TSRK, and C I ) , t h e performance of the TSRK e q u a t i o n i s b e t t e r because t h e i n d i v i d u a l d e v i a t i o n s are about t h e same f o r t h e t e n a l k a n e s . The d e v i a t i o n s obtained from t h e CI e q u a t i o n tend t o i n c r e a s e w i t h i n c r e a s e i n m o l e c u l a r w e i g h t . A comparison of the SW, PT, TSRK and CI e q u a t i o n s r e v e a l s t h a t e q u a t i o n s w i t h substance-dependent Q values y i e l d b e t t e r r e p r e s e n t a t i o n of these two p r o p e r t i e s . A l l equations appear t o r e p r e s e n t V * and Ζ t o t h e same degree of a c c u r a c y . 6. Although n e a r l y a l l t h e equations y i e l d low d e v i a t i o n s i n t h e c a l c u l a t e d Z and Z v a l u e s , t h e r e i s a tendency toward l a r g e r d e v i a t i o n s at l a r g e r m o l e c u l a r w e i g h t s . 7. The f a m i l y of equations r e p r e s e n t e d by the u + w = 1 r e l a t i o n s h i p y i e l d s lower d e v i a t i o n s i n the c a l c u l a t i o n of Z v a l u e s than t h e e q u a t i o n s o b t a i n e d from t h e v o l u m e - t r a n s l a t i o n method, and the i n d i v i d u a l d e v i a t i o n s are about the same. Thus, t h e r e s u l t s r e p o r t e d i n Tables II and I I I p r o v i d e some guidance i n t h e s e l e c t i o n of e q u a t i o n s among t h e 14 c u b i c e q u a t i o n s f o r r e p r e s e n t i n g p h y s i c a l p r o p e r t i e s of pure normal f l u i d s . The c h o i c e depends on the p r o p e r t i e s t o be emphasized and the m o l e c u l a r weight of t h e substances t o be c o n s i d e r e d . v


v

a c

λ

v

s u p

c

Design o f Cubic E q u a t i o n s o f S t a t e In E q u a t i o n 1, i f u and w are c o n s i d e r e d as c o n s t a n t s f o r a l l substances (such as t h e VDW, RK, SRK, PR and HA e q u a t i o n s ) , t h e r e s u l t a n t e q u a t i o n would be a two-parameter e q u a t i o n . I f u and w are r e l a t e d by an exact mathematic r e l a t i o n s h i p (such as the HK, SW, PT, 3RK, MMC and TSRK e q u a t i o n s ) , Equation 1 would become a t h r e e parameter c u b i c e q u a t i o n . I f u and w are not r e l a t e d t o each o t h e r through an exact mathematic r e l a t i o n s h i p (such as t h e ALS e q u a t i o n ) , E q u a t i o n 1 would y i e l d a f o u r - p a r a m e t e r e q u a t i o n . Although t h e CI e q u a t i o n i n i t s o r i g i n a l form (18) appears t o be a f o u r - p a r a m e t e r e q u a t i o n , t h e u and w r e l a t i o n s h i p of t h i s e q u a t i o n can be a p p r o ximated by the e x p r e s s i o n shown i n Table I. In o t h e r words, i t may be approximates as a v o l u m e - t r a n s l a t e d PR e q u a t i o n . I f we adopt t h e c l a s s i c a l c o n d i t i o n s at the c r i t i c a l p o i n t as two of our c o n s t r a i n t s , (ôP/ôV) = 0 (2) T c

(ô P/ôV2) 2

T c

=0

%

(3)

a two-parameter e q u a t i o n y i e l d s a constant c r i t i c a l c o m p r e s s i b i l i t y f a c t o r , ζ ; and a t h r e e - , or f o u r - p a r a m e t e r e q u a t i o n may y i e l d substance-dependent ζ v a l u e s . 0

Γ



6

1.86 0.87 1.03 0.90 1.43 0.56 1.57 0.97 1.09 2.04

10

C C C C^ C C C C C C

1.64 1.12 1.42 1.31 1.49 1.08 1.63 0.85 1.71 2.81

SRK

3.99 7.08 8.92 10.2 12.4 15.1 16.2 18.5 19.5 21.8

10

9

8

7

6

5

3

2

x

9

8

7

6

5

4

3

2

C, C C C C C C C C C

10

9

8

7

d C, C, C* Cc C C C C C

Property

2.11 3.09 2.97 2.53 1.08 2.05 1.42 2.11 3.35 4.70

9.25 7.04 5.83 4.97 3.55 2.28 2.95 4.99 5.85 7.96

1.15 2.51 2.61 2.42 0.94 2.35 1.28 2.19 4.16 5.94

PR

1.36 1.10 1.19 1.01 1.40 1.11 1.96 1.93 1.60 1.87

5.11 4.97 4.96 5.13 4.89 4.00 4.26 3.57 3.74 3.05

1.00 0.67 0.85 0.75 1.00 0.68 1.41 1.37 1.10 1.42

HK

3RK

1.32 2.73 2.72 2.35 0.64 1.64 1.08 1.58 2.10 3.36

1.60 1.01 1.21 1.04 1.54 0.88 1.73 0.95 1.11 1.80

1.64 1.12 1.42 1.31 1.49 1.08 1.63 0.85 1.71 2.81

T

TSRK (0.50 < r

2.06 1.60 1.55 1.55 2.80 1.90 2.86 2.38 2.89 3.66

v

MMC

Vapor Pressure, p

PT

3.50 3.10 2.88 2.83 2.56 2.36 2.26 2.32 2.30 2.42

4.13 4.60 4.49 3.93 3.62 4.22 3.32 3.33 4.00 4.35

5.39 5.22 6.24 9.49 16.0 13.2 13.4 15.1 17.1 23.0

0.49 1.60 1.73 1.52 0.39 1.22 0.38 1.25 1.78 1.87

1.51 2.85 2.77 2.26 0.77 1.53 1.47 2.03 2.12 2.80

1.80 0.86 0.98 0.67 1.54 0.52 1.72 1.15 0.57 0.95

2.42 1.86 1.76 1.74 3.11 1.92 3.09 2.58 2.88 3.77

Saturated Vapor Volume, V

3.56 3.28 2.97 2.50 2.37 3.21 2.34 2.83 2.25 2.46 v

1.91 1.04 1.24 0.96 1.33 0.50 1.42 0.68 1.00 1.93

p

p

(0.50 < T

3.69 3.69 3.68 3.67 3.69 3.74 3.75 3.74 4.04 4.05

CI

0.56 2.20 1.85 1.27 0.90 1.12 1.26 1.61 1.62 1.89

ALS

2.82 3.09 2.92 2.41 2.32 3.17 2.26 2.48 2.30 2.36

1.47 1.07 1.18 1.13 0.89 1.19 1.30 1.63 1.22 1.52

1.02 2.42 1.98 1.44 1.23 1.44 1.69 2.12 1.91 2.19

< 0.98)

2.33 2.14 2.47 2.55 3.45 4.66 4.46 5.12 4.34 4.98

< 0.98)

0.97 0.66 0.78 0.79 0.69 0.96 1.13 1.18 1.02 1.41

< 0.98)

Saturated Liquid Volume, V* (0.50 < T

0.37 1.73 1.88 1.69 Ό.39 J O.39 1.57 0.54 1.48 2.19 2.59

SW

page

0.58 0.76 1.20 1.27 1.76 3.02 2.49 3.23 2.63 3.18

1.55 1.04 1.42 1.44 1.10 1.60 0.90 1.38 3.19 4.72

F

C o n t i n u e d on n e x t

3.92 3.12 2.70 3.14 3.92 4.29 5.98 6.94 9.26 10.8

1.85 3.61 4.30 4.55 5.28 4.04 3.46 1.50 1.85 5.15

KMC

2.46 1.52 1.87 1.71 1.30 1.60 1.11 1.16 2.92 4.02

4.14 3.95 5.83 8.16 10.8 11.5 11.0 12.9 15.6 19.4

1.42 1.15 1.49 0.84 2.18 1.53 2.71 3.74 4.17 3.38

3.44 2.32 4.02 5.93 6.70 10.3 11.5 15.0 20.0 25.3

H

2.17 4.46 5.36 5.57 6.31 4.56 3.72 1.62 1.80 4.55

2.41 2.01 2.32 2.83 4.93 3.37 4.22 4.47 5.89 8.58

0.97 0.32 0.16 0.71 0.75 0.31 0.43 0.60 0.85 1.36

0.89 0.64 0.70 0.76 1.78 0.76 0.99 1.08 3.82 7.42

HCL

Table III Comparison of Average Absolute Percentage Deviations Obtained from 14 Cubic Equations of State for Ten Normal Alkanes



10

10

9

8

7

6

5

4

3

2

C, C C C C C C C C C

10

9

8

7

Cg C C C C

C5

C, CI C, C*

υ

7

8

c c

IÏ

2

1

c c

Property

0.88 0.71 0.60 0.52 0.45 0.43 0.45 0.49 0.57 0.67

8.35 6.64 5.46 4.52 3.52 3.03 2.96 3.52 4.97 6.65

PR

SW

PT

3RK l

MMC

3.16 3.11 3.09 3.06 3.02 2.99 2.92 2.94 2.98 3.09

3.78 3.34 3.14 3.01 2.91 2.88 2.95 3.25 3.88 4.54

0.44 0.48 0.51 0.53 0.55 0.56 0.58 0.59 0.61 0.62

0.17 0.23 0.28 0.32 0.37 0.40 0.45 0.49 0.52 0.56

0.20 0.27 0.31 0.35 0.39 0.42 0.46 0.49 0.52 0.55

0.20 0.31 0.39 0.46 0.55 0.61 0.68 0.74 0.80 0.84

0.18 0.25 0.31 0.36 0.42 0.47 0.53 0.58 0.62 0.66

v

3.72 5.82 7.69 9.40 11.6 13.3 15.6 17.5 19.7 21.9

1.14 1.55 1.83 2.05 2.32 2.51 2.75 2.94 3.17 3.39

2.13 1.66 1.48 1.37 1.31 1.31 1.36 1.44 1.57 1.73

ALS

HCL

0.18 0.29 0.38 0.47 0.60 0.68 0.80 0.89 1.00 1.10

p

3.58 3.43 3.40 3.38 3.43 3.46 3.81 3.85 4.08 4.28

3.25 3.17 3.20 3.34 3.75 3.96 4.47 4.59 4.69 4.90

3.64 3.29 3.17 3.11 3.06 3.03 3.00 2.99 2.99 2.99

P

p

3.60 2.26 2.23 2.76 3.09 3.09 2.77 2.26 1.83 2.26

< 0.99, 0.01 < r < 10.0)

CI

1.15 1.26 1.33 1.39 1.46 1.51 1.57 1.62 1.68 1.73

1.71 1.36 1.18 1.09 1.08 1.16 1.35 1.52 1.73 1.92

0.83 0.87 0.92 1.01 1.14 1.27 1.44 1.59 1.78 1.96

1.16 1.20 1.20 1.20 1.23 1.27 1.36 1.48 1.67 1.89

0.98 1.08 1.18 1.32 1.57 1.84 2.21 2.52 2.89 3.25

0.98 1.06 1.11 1.17 1.24 1.32 1.44 1.55 1.71 1.89

sup

0.88 0.89 0.97 1.07 1.22 1.33 1.48 1.60 1.73 1.88

0.84 0.86 0.93 1.01 1.15 1.27 1.45 1.62 1.82 2.03

r

0.39 0.38 0.39 0.40 0.43 0.45 0.48 0.50 0.53 0.56

0.76 1.20 1.50 1.78 2.12 2.39 2.72 3.00 3.32 3.63

1.94 2.00 2.05 2.11 2.18 2.24 2.33 2.41 2.50 2.60

5.09 5.01 5.00 5.02 5.08 5.21 5.44 5.78 6.42 7.25

H

p

0.10 0.13 0.16 0.18 0.20 0.21 0.22 0.23 0.23 0.24

5.85 7.57 9.07 10.5 12.4 13.9 16.0 17.8 20.1 22.3

KMC

1.05 1.52 1.92 2.30 2.79 3.17 3.65 4.03 4.45 4.88

3.17 3.21 3.29 3.37 3.53 3.68 3.89 4.09 4.35 4.62

1.22 1.23 1.29 1.49 1.79 2.08 2.45 2.76 3.15 3.54

(1.01 < T < 4.00, 0.01 < P < 10.0)

0.58 0.49 0.43 0.39 0.40 0.42 0.47 0.51 0.56 0.63

Compressibility Factor of Vapour, Z (0.55 < T < 0.99, 0.01 < P < 0.8)

4.21 4.53 4.65 4.66 4.55 4.37 4.12 3.74 3.41 3.18

3.44 3.02 2.93 2.90 2.86 2.86 2.85 2.87 2.91 3.00

Γ

TSRK

Liquid Compressibility Factor, l (0.30 < τ

HK

Compressibility Factor of Gas above the Critical Temperature, Z

0.19 0.39 0.56 0.71 0.89 1.04 1.21 1.35 1.52 1.67

3.67 5.30 6.83 8.27 10.2 11.7 13.7 15.5 17.6 19.6

SRK

Table III Continued


4.80 5.15 5.41 5.67 6.00 6.27 6.62 6.91 7.24 7.57

0.95 0.95 0.95 0.95 0.96 0.98 1.01 1.03 1.06 1.08

5.12 5.19 5.13 5.09 5.12 5.17 5.36 5.40 5.57 5.74

F

Ο

£

"O

g ? ζ

>

00

=

S

χ m

^ § m

^1

§ Ο

^ >

3


0

C*

Cq

c'

7

C* C* C* C

Co

C, C,

9

8

7

6

5

3

c c c" c c c c c

2

Property

7.73 11.8 14.0 15.7 17.5 18.8 20.2 21.2 22.2 23.1

4.33 5.30 6.19 7.02 8.07 8.89 9.93 10.8 11.8 12.7

SRK

HK

SW

PT

3RK

MMC

TSRK c

T

CI

ALS

r

HCL

H

7.83 10.9 12.6 14.0 15.6 16.7 18.0 18.9 19.9 20.7

4.44 4.21 4.18 4.21 4.26 4.46 4.72 4.94 5.43 5.99

7.16 10.6 12.4 13.8 15.5 16.4 17.5 18.4 19.2 19.9

3.31 3.50 3.64 3.77 3.94 4.12 4.36 4.47 4.59 4.70 4.12 3.70 3.79 4.04 3.93 4.03 4.34 4.24 4.61 4.49

3.99 4.35 4.46 4.68 4.81 4.94 5.34 5.49 5.47 5.68

3.84 4.11 4.34 4.65 5.01 5.29 5.66 6.06 6.50 6.98 4.12 4.48 4.74 4.98 5.28 5.53 5.84 6.10 6.40 6.69

7.81 11.2 13.0 14.3 15.8 16.8 18.0 18.8 19.6 20.3

7.63 11.1 12.9 14.2 15.7 16.7 17.8 18.6 19.5 20.2

7.77 11.4 13.3 14.7 16.2 17.3 18.4 19.1 20.0 20.6

7.49 11.1 12.9 14.2 15.5 16.7 18.0 18.7 19.5 19.9

7.59 11.4 13.3 14.8 16.4 17.5 18.6 19.5 20.4 21.2

3.30 3.46 3.74 3.71 4.00 3.98 4.31 4.45 4.37 4.67

7.26 10.8 12.7 14.2 15.7 16.7 17.9 18.8 19.7 20.4

< 0.98)

7.59 11.1 12.9 14.3 15.7 16.9 18.0 19.0 20.0 20.7

p

3.15 3.92 5.06 5.77 6.50 6.96 7.40 7.67 7.89 8.05

Second Virial Coefficient, Β (0.50 < T

3.59 3.72 Ï.12 4.23 4.32 4.21 4.09 4.12 4.19 4.26

8.54 14.7 18.1 20.7 23.5 25.5 27.8 29.4 31.1 32.7

2.92 5.48 6.45 7.19 6.19 8.96 9.91 10.7 11.5 12.4

13.4 13.7 14.6 15.5 16.5 17.4 18.3 19.0 19.8 20.5

7.27 7.77 8.19 8.49 8.85 9.12 9.45 9.68 9.98 10.2

Compressibility Factor along the Critical Isotherm, Z ( r = 1.0, 0.01 < P < 10.0)

PR


1.09 0.77 0.87 1.00 1.20 1.43 1.88 2.34 2.93 3.56

3.81 4.10 4.33 4.61 4.99 5.30 5.70 6.20 6.78 7.34

KMC

8.81 12.6 14.6 16.0 17.7 18.8 20.0 20.8 21.7 22.5

9.80 10.7 11.3 11.8 12.4 12.9 13.5 14.0 14.5 15.0

F

546


Should the parameters of a two-parameter e q u a t i o n be t r e a t e d temperature dependent up t o the c r i t i c a l p o i n t by a f o r c e d f i t t i n g p r o c e d u r e , the s e l e c t i o n of p and V would be d e s i r a b l e f o r t h i s purpose. However, such an equation can no l o n g e r s a t i s f y both Equations 2 and 3. In o t h e r words, the r e p r e s e n t a t i o n of the c r i t i c a l i s o t h e r m i n the c r i t i c a l region cannot be s a t i s f a c t o r y . Although a 3P2T e q u a t i o n (having the f l e x i b i l i t y of o b t a i n i n g substance-dependent ζ v a l u e s ) might overcome t h i s d i f f i c u l t y , two such e q u a t i o n s were e v a l u a t e d i n t h i s work, Η and KMC, and both e q u a t i o n s y i e l d e d poorer o v e r a l l performance than the e v a l u a t e d 3P1T equations. Hence, a f u r t h e r examination of the l i m i t a t i o n s and b e h a v i o r of 3P1T c u b i c e q u a t i o n s of s t a t e would be u s e f u l i n t h e d e s i g n of new c u b i c e q u a t i o n s . A l l c u b i c equations of s t a t e s u f f e r c e r t a i n s h o r t c o m i n g s . As d i s c u s s e d above, no e q u a t i o n e v a l u a t e d c o u l d s a t i s f a c t o r i l y r e p r e sent a l l the e i g h t p h y s i c a l p r o p e r t i e s s i m u l t a n e o u s l y . I f one f o r c e s an e q u a t i o n t o p r o v i d e a c c e p t a b l e Β v a l u e s , such as the KMC e q u a t i o n , l a r g e r d e v i a t i o n s would occur i n o t h e r p r e d i c t i o n s , such as the V* and Z* v a l u e s . On the o t h e r hand, i f one i g n o r e s the r e p r e s e n t a t i o n of B, t h e r e i s a chance of o b t a i n i n g an e q u a t i o n which can s i m u l t a n e o u s l y p r o v i d e p r e d i c t i o n s of the o t h e r seven p h y s i c a l p r o p e r t i e s w i t h reasonable r e s u l t s . The parameter v a l u e s of " a " capable of p r e d i c t i n g vapor p r e s s u r e s of pure substances would i n general be s u i t a b l e f o r 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 t i o n s . Because some a p p r o x i m a t i o n i s i n v o l v e d i n c o r r e l a t i n g Q by means of a temperature f u n c t i o n , some d e v i a t i o n s occur i n t h e calculated p values. T h i s shortcoming i s not due t o the form of the c u b i c e q u a t i o n of s t a t e , but due t o the temperature f u n c t i o n itself. The d e v i a t i o n s i n p due t o t h i s inadequacy are shown i n Table III. T h i s inadequacy i s a l s o r e f l e c t e d i n the c a l c u l a t e d V v a l u e s . A p l o t of p and V d e v i a t i o n s as a f u n c t i o n of T would r e v e a l t h a t one d e v i a t i o n curve i s m i r r o r image of the o t h e r a l o n g t h e z e r o - d e v i a t i o n l i n e f o r the 2P1T and 3P1T e q u a t i o n s . S i n c e the d e v i a t i o n s i n the c a l c u l a t e d Z and Z v a l u e s are p r a c t i c a l l y the same among the 2P1T and 3P1T equations and these d e v i a t i o n s are of t h e l e a s t concern of t h i s s t u d y , the f o l l o w i n g d i s c u s s i o n i s m a i n l y c e n t e r e d on the d e v i a t i o n s of V * , Z* and Z P . To i l l u s t r a t e the drawbacks of e q u a t i o n s c o n t a i n i n g two temp e r a t u r e - d e p e n d e n t parameters, the d e v i a t i o n curves o b t a i n e d f o r the HCL and Η e q u a t i o n s are p l o t t e d as a f u n c t i o n of T i n F i g u r e 1. The c o r r e l a t i o n equations o r i g i n a l l y proposed ( 4 , 14) were used t o generate the curves f o r n-hexane. As both equations used p and V f o r d e t e r m i n i n g t h e i r parameters, they y i e l d low d e v i a t i o n s i n t h e s e two p r o p e r t i e s , even though the accuracy of the r e p r e s e n t a t i o n was l o s t somewhat i n the c o r r e l a t i o n s . However, the d e v i a t i o n s o b t a i n e d i n the V v a l u e s are too l a r g e , due t o the s h i f t of e r r o r s through the l i m i t a t i o n s of a c u b i c e q u a t i o n . The r e s u l t s o b t a i n e d from a m o d i f i e d PR e q u a t i o n w i t h both parameters a and b s i m u l t a n e o u s l y f i t t e d t o p and V* of n-hexane are a l s o i n c l u d e d i n F i g u r e 1 f o r c o m p a r i s o n . The s h i f t of e r r o r s t o V i s a l s o very e v i d e n t . The d i f f i c u l t y encountered by equations c o n t a i n i n g two t e m p e r a t u r e dependent parameters f o r r e p r e s e n t i n g the c r i t i c a l region i s d e p i c t e d v

A


0

a

v

v

v

v

v

p

c

v

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D e v i a t i o n Curves o b t a i n e d from e q u a t i o n s two temperature-dependent p a r a m e t e r s .

containing

American Chemical Society Library 1155 16th Si» N.W. Washington, D.C. 20038 In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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

548

APPLICATIONS

i n F i g u r e 2 , u s i n g the e x p r e s s i o n i n terms of a(=a/a ) and p ( = b / b ) , and p l o t t e d i n terms of T . The shape of the two c u r v e s , p a r t i c u l a r l y near the c r i t i c a l p o i n t , prevents simple c o r r e l a t i o n of the parameters i n terms of T . The i n f l u e n c e of values of u and w on the r e p r e s e n t a t i o n of V v a l u e s by means of the 2P1T e q u a t i o n s (SRK, PR and HA) i s i l l u s t r a t e d i n F i g u r e 3. The t h r e e e q u a t i o n s belong t o the u + w = 1 f a m i l y and t h e i r " a " parameters were determined by f i t t i n g vapor p r e s s u r e s . In F i g u r e 3, the c a l c u l a t e d d e v i a t i o n s i n V and V f o r methane, n-hexane and n-decane are p l o t t e d . None of the t h r e e e q u a t i o n s y i e l d e d r e a l l y s a t i s f a c t o r y r e s u l t s . I t i s of i n t e r e s t t o note t h a t the general p a t t e r n s of e r r o r d i s t r i b u t i o n are about the same. It i s i m p o s s i b l e t o o b t a i n low d e v i a t i o n s f o r substances w i t h d i f f e r e n t m o l e c u l a r weights from two-parameter e q u a t i o n s by s i m p l y a d j u s t i n g the u and w v a l u e s . T h i s c o n f i r m s the c o n c l u s i o n reached above t h a t t h r e e - p a r a m e t e r e q u a t i o n s are d e s i r a b l e . The d e v i a t i o n contours of V o b t a i n e d from E q u a t i o n 1 are p l o t t e d on u-w diagrams i n F i g u r e s 4 and 5 f o r methane, n-hexane and n-decane. The numbers i n d i c a t e d on the diagrams r e f e r t o the number of carbons of the a l k a n e s , and the dots r e p r e s e n t the minimum d e v i a t i o n s . The temperature range covered i n F i g u r e 4 (0.5 < Tr < 0.98) i s l a r g e r than t h a t i n F i g u r e 5 (0.5 < Tr < 0 . 8 5 ) . On the o t h e r hand, the c o n t o u r s i n F i g u r e 4 r e p r e s e n t average a b s o l u t e d e v i a t i o n s of 2.5% i n V , w h i l e i n F i g u r e 5, 1%. As p o i n t e d out by Schmidt and Wenzel (J5), the f o l l o w i n g c o n s t r a i n t s must be s a t i s f i e d t o a v o i d the c o n d i t i o n t h a t V + ubV + w b = 0 f o r V > b, c

c

p

f

A


A

v

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A

2

2

w > -1-u

f o r u > -2

(4a)

w > u /4

f o r u < -2

(4b)

2

The n o n e x i s t i n g area i n d i c a t e d i n F i g u r e s 4 and 5 r e p r e s e n t s t h e area excluded by Equation 4. A comparison of the two diagrams i n d i c a t e s t h a t by narrowing the temperature range, the r e l a t i v e p o s i t i o n s of the two groups of d e v i a t i o n contours s h i f t towards l a r g e r v a l u e s of w and s m a l l e r v a l u e of u. In phase e q u i l i b r i u m c a l c u l a t i o n s , i t would be d e s i r a b l e t o have more v o l a t i l e components w e l l r e p r e s e n t e d at h i g h e r T v a l u e s , and l e s s v o l a t i l e components w e l l r e p r e s e n t e d at lower T values. The d e v i a t i o n contours f o r Z i n the temperature and p r e s s u r e ranges of 0.30 < T < 0.99 and 0.01 < P < 10.0 are p l o t t e d on t h e u-w diagram i n F i g u r e 6. The contours r e p r e s e n t 4% average a b s o l u t e d e v i a t i o n s . S i m i l a r l y , the d e v i a t i o n contours f o r Z P i n t h e temperature and p r e s s u r e ranges of 1.01 < T < 4.00 and 0.01 < P < 10.0 are p l o t t e d i n F i g u r e 7. The contours r e p r e s e n t C and C , and the average a b s o l u t e d e v i a t i o n s vary from 1.5 t o 2.5%. When the d e v i a t i o n contours of F i g u r e s 4-7 are superimposed, t h e o v e r l a p p i n g areas f o r the alkanes are p l o t t e d on the u - w diagram i n F i g u r e 8. The d e v i a t i o n contour of V f o r C is also i n c l u d e d i n the f i g u r e f o r r e f e r e n c e . Any l i n e or curve p a s s i n g through these contours w i l l y i e l d the f o l l o w i n g average a b s o l u t e d e v i a t i o n s f o r the alkanes i n the t h r e e p h y s i c a l p r o p e r t i e s : V : 1% (0.5 < T < 0 . 8 5 ) ; 2.5% (0.5 < T < 0.98) p

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V a r i a t i o n of α and β w i t h T PR e q u a t i o n

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550

EQUATIONS OF STATE: THEORIES A N D APPLICATIONS

24.


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1 1 Methane RK PR Ha 1

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I l l u s t r a t i o n s of the i n f l u e n c e of u and w v a l u e s on t h e r e p r e s e n t a t i o n of V and V by means of 2P1T equations. SRK ( u = l , w=0), — PR (u=2, w = - l ) , - - - - - HA (u=3, w=-2). A

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F i g u r e 3c I l l u s t r a t i o n s o f t h e i n f l u e n c e o f u and w v a l u e s on t h e r e p r e s e n t a t i o n o f V by means o f 2PIT e q u a t i o n s . SRK (u=1, w=0), PR (u=2, w=-1), - - - HA (u=3, w=-2).



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D e v i a t i o n c o n t o u r s of V* i n the t e m p e r a t u r e range 0.5 < T < 0 . 9 8 . Curves r e p r e s e n t d e v i a t i o n s of 2.5% i n V * . r


of

554

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

1

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1

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APPLICATIONS

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Equations of State - American Chemical Society

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