Mixing Rules for Cubic Equations of State - ACS Symposium Series

15 Mixing Rules for Cubic Equations of State G. Ali Mansoori

Downloaded by PURDUE UNIVERSITY on March 12, 2013 | http://pubs.acs.org Publication Date: March 24, 1986 | doi: 10.1021/bk-1986-0300.ch015

Department of Chemical Engineering, University of Illinois, Chicago, IL 60680

Through the application of conformal solution theory of statistical mechanics a coherent theory for the development of mixing rules is produced. This theory allows us to use different approximations for the mixture radial distribution functions for derivation of a variety of sets of conformal solution mixing rules some of which are density and temperature dependent. The resulting mixing rules are applied to the van der Waals, Redlich-Kwong, and Peng-Robinson equations of state as the three representative cubic equations of state.

T h e r e e x i s t s a wealth of i n f o r m a t i o n in t h e l i t e r a t u r e about cubic equations of s t a t e a p p l i c a b l e t o v a r i e t i e s of f l u i d s of chemical and engineering interestAlthough cubic equations of s t a t e a r e generally e m p i r i c a l m o d i f i c a t i o n s of the v a n d e r Waals equation of s t a t e , they have found widespread applications in p r o c e s s design calculations because of t h e i r s i m p l i c i t y - Extension of their a p p l i c a b i l i t y to m i x t u r e s i s generally acieved by introduction of m i x i n g rules f o r their parameters. Mixing rules are expressions r e l a t i n g p a r a m e t e r s of a m i x t u r e equation of s t a t e t o pure fluid parameters through, usually, some composition dependent expressions. Except f o r the van der Waals equation of s t a t e the m i x i n g r u l e s f o r cubic equations of state are empirical expressions. In t h e p r e s e n t r e p o r t w e introduce a statistical mechanical conformal solution technique through which we can d e r i v e v a r i e t i e s of s e t s of m i x i n g r u l e s a p p l i c a b l e t o cubic equations of s t a t e . This pressure, energy, and c o m p r e s s i b i l i t y equations of s t a t i s t i c a l m e c h a n i c s . In P a r t II o f t h e p r e s e n t r e p o r t w e introduce the conformal s o l u t i o n t h e o r y of polar fluid m i x t u r e s ( 1 ) and i t s r e l a t i o n s h i p t o t h e i d e a of m i x i n g r u l e s . In P a r t III we introduce the concept of the c o n f o r m a l 0097-6156/ 86/ 0300-0314506.00/ 0 © 1986 American Chemical Society

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

Mixing Rules for Cubic Equations of State

MANSOORI

15.

315


s o l u t i o n m i x i n g r u l e s and we p r o d u c e d i f f e r e n t s e t s of m i x i n g r u l e s b a s e d on d i f f e r e n t a p p r o x i m a t i o n s f o r the mixture radial distribution functions. In P a r t IV w e r e v i e w the e x i s t i n g f o r m s of t h e c u b i c e q u a t i o n s of s t a t e f o r m i x t u r e s and the d e f i c i e n c i e s of t h e i r m i x i n g r u l e s and c o m b i n i n g r u l e s . F i n a l l y , i n P a r t IV w e i n t r o d u c e g u i d e l i n e s for t h e use of c o n f o r m a l solution mixing rules and c o m b i n i n g r u l e s in equations of s t a t e and we d e m o n s t r a t e a p p l i c a t i o n of such m i x i n g r u l e s and combining r u l e s f o r three r e p r e s e n t a t i v e cubic equations of s t a t e . II. C o n f o r m a l S o l u t i o n T h e o r y o f M i x t u r e s Conformal solutions refer to substances whose intermolecular potential energy function, are related to each other and t o those of a r e f e r e n c e fluid, designated by s u b - s c r i p t (oo), according to (1,2) *ij = ' i j ' o o f r / h i j " )

usually

(1)

3

For substances whose intermolecular potential energy f u n c t i o n c a n b e r e p r e s e n t e d b y an e q u a t i o n o f t h e f o r m *ij = i j [ ( i j E

L

/ r

>

n

"(l-ij/r) ]

(2)

m

and f o r w h i c h e x p o n e n t s m a n d n a r e t h e s a m e a s f o r t h e r e f e r e n c e s u b s t a n c e , c o n f o r m a l p a r a m e t e r s f^- a n d h|j w i l l be d e f i n e d b y t h e f o l l o w i n g r e l a t i o n s w i t h r e s p e c t t o t h e i n t e r m o l e c u l a r p o t e n t i a l e n e r g y p a r a m e t e r s E j j a n d L^y f..

hj

= F--/F ij oo» I 1

/ l l

n

= (I L --/i / L

h--

ij

^ ij oo^

(3)

}3

Thus t h e c o n f i g u r a t i o n a l thermodynamic properties of a pure s u b s t a n c e of t y p e (a) a r e r e l a t e d to those of t h e reference substance according to the following relations: F

a

( V , T) = f

P (V, a

a a

T) = ( f

F (V/h 0

a a

/h

a a

S ( V , T) = S ( V / h a

c

G ( P , T) = f a

a a

a a

, T/f

)P (V/h 0

,

T/f

a a

a a

a a

) - NkT£nh

, T/f

) + Nkenh

a a

G (Ph

a a

/f

a a

,T/f

a a

H (Ph

a a

/f

a a

, S )

0

a a

a a

a

(4)

a

) a

(5) (6)

a

) - NkTfcnh

a

a

(7)

and H ( P , S) = f a

w h e r e F, pressure,

0

0

(8)

P, S , G, a n d H a r e t h e H e l m h o l t z f r e e energy, entropy, Gibbs free energy, and e n t h a l p y ,


EQUATIONS O F STATE: THEORIES A N D APPLICATIONS

316

respectively. A c c o r d i n g to the above equations, a l l the thermodynamic properties of substance (a) c a n be e x p r e s s e d i n t e r m s o f t h e p r o p e r t i e s of a r e f e r e n c e p u r e substance (o) through the conformal p a r a m e t e r s f and h . The c o n f o r m a l s o l u t i o n t r e a t m e n t of f l u i d s c o m p o s e d of p o l a r m o l e c u l e s i s m o r e c o m p l i c a t e d than f o r n o n - p o l a r fluids. This i s m a i n l y due t o e l e c t r o s t a t i c i n t e r a c t i o n s which cause a d e p a r t u r e of the i n t e r m o l e c u l a r potential from spherical symmetry. The e l e c t r o s t a t i c potential between two otherwise neutral molecules arises from permanent a s y m m e t r y in the charge distribution within the molecules. For any p a i r of l o c a l i z e d charge d i s t r i b u t i o n , the mutual e l e c t r o s t a t i c i n t e r a c t i o n energy can be w r i t t e n in t e r m s o f an i n f i n i t e s e r i e s of i n v e r s e powers of s e p a r a t i o n of any t w o p o i n t s . F o r no o v e r l a p b e t w e e n t h e charge distributions the s e r i e s c o n v e r g e s ( l ) . Thus t h e t r u e p a i l — p o t e n t i a l of p o l a r m o l e c u l e s i s o r i e n t a t i o n - d e p e n d e n t and i s t h e s u m o f d i s p e r s i o n f o r c e a s w e l l a s e l e c t r o s t a t i c interactions. In o r d e r t o e x t e n t u t i l i t y of t h e above f o r m u l a t i o n of the c o n f o r m a l s o l u t i o n t h e o r y t o p o l a r f l u i d s we have p r o p o s e d the f o l l o w i n g a n g l e - a v e r a g e d potential function f o r polar molecular interactions which r e p r e s e n t s the f i r s t o r d e r c o n t r i b u t i o n to the anisotropic f o r c e s ( 1 ) a a


a a

* (r,T) = K€ [(o t j

i j

1 J

/ r ) - (o^/r)™]

+ 7|i Mj /[450(kT) r i

4

4

3

- Q| Qj /(1.4kTr 2

-

n

2

1 0

1 2

] -

n

t

2

Hj /(3kTr ) 2

6

(M Qj +vij Q i

2

2

2

) - (oc |ij +oc |i )/r 1

2

j

2

1

1

2

)/(2kTr ) 8

(9)

6

where K = [n/(n-m and w h e r e | i p Q j , a n d OCJ a r e the dipole moment, quadrupole moment, and p o l a r i z a b i l i t y of m o l e c u l e i , r e s p e c t i v e l y . For a polar fluid, whose intermolecular potential energy function c a n be r e p r e s e n t e d by e q . 9 the c o n f o r m a l p a r a m e t e r s f and h a a

a

a

w i l l have the following f o r m s : f

aa = Eaa< > > oo< > >

where

T

r

/ E

T

r

E,j(T,r) = K e A i j

LijCT.r) =

0

h

1 j

a a = a a< > >

(T,r)[H (T,r)] i j

ij[ ij< > >]~ H

T

r

i j

i j

7M

i

4

ii

j

4

r

/ L

oo( > > J T

r

n / m

, / m

HtjCT.r) = [ C ( T , r ) / A ( T , r ) ] Ajj(T,r) = 1 +

T

a

m /

^ n

m )

/[1800(kT) r 3

, 2

- o m

i j

n

Ke ] j j


< > 10

15.

MANSOORI

and


C|j(T r) = 1 + ii Mj /[12lcTr f

1

2

2

6

"" o m

m

1 j

317

K€ j] 1

+ (7/20)Q Q /[kTr ^ " ' " o y ^ C y i

+ (n +

Q

j

(oc ii

j

1

2

1

2

2

j

2

+y

2

+oc |i

j

Q

2

i

j

2

1

)/[8kTr - o 8

2

)/[4r - a 6

m

m

m

1 j

1 J

m

K6| ] j

Ke ] i j

The b a s i c c o n c e p t o f t h e CST of m i x t u r e s i s t h e s a m e a s f o r pure fluids, except that f and h i n e q s . 4 - 8 s h o u l d be a a


replaced

with

parameters, f

x x = x x < ij> f

f

f

and

x x

a a

h

,

x x

the

mixture

conformal

as g i v e n b e l o w h

ij' i x

)

h

xx = xx( h

Uy

h

ij» i>

< >

x

11

Eqs.ll are c a l l e d the conformal solution mixing rules. F u n c t i o n a l f o r m s of t h e s e m i x i n g r u l e s w i l l be d i f f e r e n t f o r different theories of m i x t u r e s as it will be demonstrated l a t e r in this r e p o r t . In t h e f o r m u l a t i o n of a m i x t u r e t h e o r y w e a l s o n e e d t o know t h e c o m b i n i n g r u l e s for unlike-interaction potential parameters which are u s u a l l y e x p r e s s e d by the f o l l o w i n g e x p r e s s i o n s ffj = ( 1 - k i j ) ( f i t f j j >

1 / Z

;

h i j = C l - *tj)[( h

t i

1 / 3

+hjj

1 / 3

)/2]

3

(12)

w h e r e k j j and fcjj a r e a d j u s t a b l e p a r a m e t e r s . III. S t a t i s t i c a l M e c h a n i c a l T h e o r y of l i i x i n a R u l e s The m o s t i m p o r t a n t r e q u i r e m e n t i n t h e d e v e l o p m e n t o f t h e CST o f m i x t u r e s are mixing rules. In t h e d i s c u s s i o n p r e s e n t e d h e r e we have i n t r o d u c e d a new t e c h n i q u e to r e - d e r i v e t h e e x i s t i n g m i x i n g r u l e s and d e r i v e a n u m b e r of new mixing rules some of which are densityand temperature-dependent. A c c o r d i n g to s t a t i s t i c a l mechanics the m a c r o s c o p i c t h e r m o d y n a m i c p r o p e r t i e s of a p u r e f l u i d a r e r e l a t e d to i t s m i c r o s c o p i c m o l e c u l a r c h a r a c t e r i s t i c s by the f o l l o w i n g t h r e e equations (5,4) oo

(13)

u = Ujg + 2 n p j 0 ( r ) g ( r ) r d r 0 2

oo

P = pRT + ( 2 / 3 ) n p J r * r ( r ) g ( r ) r d r 0 2

(14)

oo (15) (4TT/RT)J[g(r)-l]r dr 0 w h e r e u i s t h e i n t e r n a l e n e r g y , P i s t h e p r e s s u r e and KJ i s the isothermal compressibility, 0(r) is the pair intermolecular potential energy f u n c t i o n , and g ( r ) i s t h e

k

t

= 1/pRT -

2



318

radial (or pair) distribution function. Eqs.13-15 are commonly called the energy equation, the virial (or pressure) equation, and the c o m p r e s s i b i l i t y equation, respectivelyFor a multicomponent mixture these equations assume the following forms ( 5 - 5 ) oo

u = u

1 g

+ 2np2iSjX Xjj0 j(r)g (r)r dr 0 i

i

(16)

2

1 j

oo

P = pRT + ( 2 / 3 ) n p 2 2 j X x J r ^ ( r ) g 0 ic = ( l / p R D l B l / S i S j X i X j l B ^ j i

i

j

1 j

| J

(r)r dr

(17)

2

(18)


T

In t h e a b o v e e q u a t i o n s s u m m a t i o n s a r e o v e r a l l t h e ( c ) c o m p o n e n t s o f t h e m i x t u r e , Xj a n d xj a r e t h e m o l e f r a c t i o n s , and |B| i s a c x c d e t e r m i n a n t w i t h i t s r e p r e s e n t a t i v e t e r m s i n the f o l l o w i n g f o r m oo

ij

B

=

x

i ij 6

+

x

i jP ij x

G

G

ij

=

4nJ[g j(r)-1 ] r d r 0 2

1

w h e r e 6,-j i s t h e K r o n e e k e r d e l t a , a n d IBIJJ i s t h e c o f a c t o r o f t e r m B y i n d e t e r m i n a n t |B|. E q s . 1 3 - 1 8 c a n b e u s e d i n t h e manner p r e s e n t e d below in o r d e r t o d e r i v e m i x i n g r u l e s b a s e d on d i f f e r e n t m i x t u r e t h e o r y a p p r o x i m a t i o n s : III.1. O n e - F l u i d T h e o r y o f M i x i n g R u l e s : F o r t h e d e v e l o p m e n t of o n e - f l u i d m i x i n g r u l e s we i n t r o d u c e a p s e u d o - p u r e f l u i d which can r e p r e s e n t the configurational p r o p e r t i e s of a m i x t u r e p r o v i d e d that the p s e u d o - p u r e f l u i d and the m i x t u r e m o l e c u l a r i n t e r a c t i o n s o b e y e q . 1. By r e p l a c i n g eq.1 in e q s . 1 3 , 1 4 , 1 7 , a n d 18 a n d t h e n e q u a t i n g c o n f i g u r a t i o n a l i n t e r n a l e n e r g y , p r e s s u r e , and i s o t h e r m a l c o m p r e s s i b i l i t y of t h e p s e u d o - p u r e f l u i d and t h e m i x t u r e w e w i l l o b t a i n t h e following equations f

xx xxôo(y)%o(y)y

f

xx xx/y^ oo(y)goo(y>Y

h

h

2 d

y 2i2jX x f h jj0 =

,

1

2 d

j

i j

1

o o

( y ) g ( y ) y d y (19)

y 2i2jX x f h Jy0' =

1

j

i j

j j

2

i j

o o

(y)g (y)y dy i j

2

(20) {1-4nph J[g x x

0 0

(y)-1]y dy}-1=2i2 x x |B| /|B| 2

j

i

j

1 J

(21)

It s h o u l d b e p o i n t e d o u t t h a t f o r t h e c a s e o f t h e h a r d - s p h e r e fluid eq.19 vanishes, eq.21 remains the same, while eq.20 reduces to the following form h

x x 9 o o < ) = 2 i 2 j X x h g ( 1) 1

i

j

i j

i j


(22)

MANSOORI

15.


Solution of eqs.19-21 should produce the t w o n e c e s s a r y expressions (mixing rules) relating f and h of t h e p s e u d o - p u r e f l u i d t o f j j a n d h|j o f c o m p o n e n t s o f t h e mixture. F o r t h i s p u r p o s e w e s h o u l d u s e an a p p r o x i m a t i o n technique r e l a t i n g t h e r a d i a l d i s t r i b u t i o n f u n c t i o n s (RDF) i n t h e m i x t u r e t o t h e p u r e r e f e r e n c e f l u i d RDF. H o w e v e r , a t a f i r s t g l a n c e i t s e e m s t h a t w e h a v e i n o u r hand t h r e e e q u a t i o n s and t w o unknowns. A s i t w i l l be d e m o n s t r a t e d b e l o w f o r m o s t o f t h e a p p r o x i m a t i o n s o f t h e m i x t u r e RDFs which a r e used here these t h r e e equations produce t w o mixing rules. In t h e p r e v i o u s i n v e s t i g a t i o n s for the d e v e l o p m e n t of m i x i n g r u l e s (5-11) a l l t h e i n v e s t i g a t o r s have used only eq.19 and/or eq.20. Our s t u d i e s i n d i c a t e t h a t w h i l e e q s . 1 9 and 20 a r e e s s e n t i a l i n t h e d e v e l o p m e n t o f m i x i n g r u l e s , eq.21 can add a new dimension which could be s i g n i f i c a n t i n t h e c a l c u l a t i o n o f p r o p e r t i e s o f m i x t u r e s . In what f o l l o w s d i f f e r e n t a p p r o x i m a t i o n s w i l l be used f o r relating gy to g in order to derive different sets of mixing rules. x


319

O

x

x

x

Q

III. 1 . i . Random M i x i n g A p p r o x i m a t i o n ( R M A ) f o r M i x t u r e RDFs: In t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t t h e n o n - s c a l e d RDF o f a l l t h e c o m p o n e n t s o f t h e m i x t u r e a n d t h e i n t e r a c t i o n RDFs a r e i d e n t i c a l (5_), i . e . Q\](r)

=g

2 2

(r) = ... = g^(r)= ...

(23)

When t h i s a p p r o x i m a t i o n i s r e p l a c e d in e q s . 1 9 - 2 1 , eq.21 w i l l v a n i s h a n d e q . 1 9 a n d 20 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xx< > = 2 i S j X x ^ ( r ) r

i

*'xx< > r

=

j

(24)

i j

SiSjXiXjf'ijCr)

(25)

For e x a m p l e , in the case of the L e n n a r d - J o n e s (12-6) intermolecular potential function we will derive the f o l l o w i n g m i x i n g r u l e s (12) f r o m e q s . 1 3 and 14. 'xx^xx f

xx xx h

2

= SiSjXiXjfijhij

4

= SiSjXiXjfjjhjj

2

4

(

2

6

)

(27)

For a h a r d - s p h e r e potential we will d e r i v e only one m i x i n g r u l e t h r o u g h the RMA and that i s d e r i v e d by r e p l a c i n g e q . 2 3 in 2 2 . The r e s u l t i n g m i x i n g r u l e w i l l be

h

x x

, / 3

= ZiZjXiXjh^'S


( 2 8

)


320

l l l . l . i i . Conformal Solution Approximation (CSA) f o r Mixture RPFs: This a p p r o x i m a t i o n technique seems m o r e logical f o r use in the d e v e l o p m e n t of m i x i n g r u l e s than R M A . A c c o r d i n g t o t h i s a p p r o x i m a t i o n t h e s c a l e d RDFs i n a m i x t u r e a r e a l l identical (5J, i-egii(y)=

9zz(y) = - - - = g j j ( y ) = . . .

(29)

When we u s e t h i s a p p r o x i m a t i o n in e q s . 1 9 and 20 t h e y both produce the same m i x i n g r u l e which i s


f

XX XX h

= SlSjXiXjf^hy

(

N o w , b y r e p l a c i n g e q . 2 9 i n 21 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g |B*|/pRTic

Txx

= SiSjX|Xj|B*||j

3

0

)

will

(31)

where |B*| j = x [ 6 j + X j ( h j / h ) ( p R T i c - 1 ) ] . Eq.30 is actually the second van d e r Waals mixing rule which i s well known, but e q . 3 1 i s a new m i x i n g r u l e f o r h which i s replacing the f i r s t van d e r Waals mixing rule. This new mixing rule, in p r i n c i p l e , i s a c o m p o s i t i o n - , t e m p e r a t u r e - , and d e n s i t y - d e p e n d e n t m i x i n g r u l e . This i s because K y w h i c h a p p e a r s i n t h e r i g h t a n d l e f t hand s i d e s o f t h i s equation i s g e n e r a l l y t e m p e r a t u r e - and d e n s i t y - d e p e n d e n t . For example, f o r a binary m i x t u r e eq.31 can be w r i t t e n in t h e f o l l o w i n g f o r m (5.) t

i

1

1

x x

T x x

x

x

x

h

x

x

= {liSjXjXjhjj + x x ( h 1

{1+x

2

1 1

h

2 2

1 2

2

)(pRTic

By u s i n g t h e h a r d - s p h e r e p o t e n t i a l ( b y r e p l a c i n g 22) w e w i l l d e r i v e t h e f o l l o w i n g m i x i n g r u l e

e q . 2 9 in

h

x

x

2 2

1 2

)(pRTK

- 1 ) }/ (31-1)

n

- 2h

T x x

-1)}

l X 2

(h +h

-h

x

T x x

= SiSjXtXjhjj

(32)

This mixing rule i s the f i r s t van d e r Waals m i x i n g rule which, in conjunction with eq.30 is usually used f o r calculation of mixture thermodynamic properties ( 7 , 8 . 1 0 , 1 1 ). It s h o u l d b e p o i n t e d o u t t h a t e q . 3 2 c o n s t i t u t e s another mixing rule f o r hard-sphere m i x t u r e s . A s a result, while the CSA approximation produces two mixing rules f o r potential functions with two parameters, it also produces two mixing rules f o r a h a r d - s p h e r e potential which i s a one-parameter potential function.


MANSOORI

15.


321

MLI.iii, H a r d - S p h e r e Expansion (HSE) A p p r o x i m a t i o n f o r M i x t u r e RDFs: It i s d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d ( x ) can be expanded around t h e h a r d - s p h e r e ( h s ) RDF in t h e f o r m ( 3 ) g

x x

(y)= g

(y)+ (f x

h s

X

/ T

o*^l

+

< xx o*> 92 f

/ T

2

+

—

< > 33

Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e h a r d - s p h e r e m i x t u r e RDFs as t h e f o l l o w i n g


gij= 9 i j

h s

(y)

(fij/T *)g!(y)

+

0

+

(fij/To*) g (y) 2

2

+

—

(

3 4

>

The j u s t i f i c a t i o n b e h i n d t h i s e x p a n s i o n i s g i v e n e l s e w h e r e ( 6 9 ) . Now b y r e p l a c i n g e q s . 3 3 a n d 3 4 i n e i t h e r o f e q s . 1 9 o r 20 w e w i l l b e a b l e t o d e r i v e t h e f o l l o w i n g t w o m i x i n g r u l e s by e q u a t i n g the coefficients of the s e c o n d and t h i r d o r d e r i n v e r s e t e m p e r a t u r e t e r m s of the r e s u l t i n g e x p r e s s i o n . T

f

XX XX = h

SiSjX^jfyhy

' x x ^ x x = 2i2jX x f 2h l

J

1 J

(

3

5

)

(36)

1 J

These m i x i n g r u l e s a r e used f o r c a l c u l a t i o n of e x c e s s p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e m i x t u r e (13) at t h e s a m e t h e r m o d y n a m i c c o n d i t i o n s ( 9 ) . A p p l i c a t i o n o f t h e HSE a p p r o x i m a t i o n in eq.21 w i l l not produce any additional mixing rule. I l l . l . i v . D e n s i t y E x p a n s i o n (DEX) A p p r o x i m a t i o n f o r M i x t u r e R D F s : i t h a s b e e n d e m o n s t r a t e d t h a t t h e RDF o f a p u r e f l u i d can b e e x p a n d e d a r o u n d t h e d i l u t e g a s RDF, e x p [ - 0 ( r ) / k T ] , i n the f o r m (14) g

x x

( y ) = [1 + F

x x

( y ) ] exp[-0 (r)/kT]

(37)

x x

Let us a l s o a s s u m e that we could make a s i m i l a r e x p a n s i o n f o r RDFs i n a m i x t u r e a r o u n d t h e d i l u t e g a s m i x t u r e RDFs a s the f o l l o w i n g g i j ( y ) = [1 + F

( y ) ] exp[-0 (r)/kT]

x x

(38)

1 j

Now b y r e p l a c i n g e q s . 3 7 a n d 3 8 i n e q . 1 9 a n d a f t e r a n u m b e r of a l g e b r a i c m a n i p u l a t i o n s w e w i l l d e r i v e t h e f o l l o w i n g mixing rule fxx xx=2 2 x x f h

1

j

1

j

1 j

h {1-(f /f i j

i j

x x

-1)[u-u

+T(C -C v

v i g

1 g

)/kT

)/(u-u

1 g

)]}


(39)


322


The l a t t e r m i x i n g r u l e c a n b e u s e d , j o i n e d w i t h a n o t h e r m i x i n g r u l e , f o r c a l c u l a t i o n of m i x t u r e p r o p e r t i e s . Similar a p p r o x i m a t i o n s c a n be u s e d i n o r d e r t o d e r i v e o t h e r m i x i n g r u l e s f r o m the v i r i a l and c o m p r e s s i b i l i t y e q u a t i o n s . III.2. M u l t i - F l u i d T h e o r y o f M i x i n g R u l e s : The b a s i c a s s u m p t i o n i n d e v e l o p i n g t h e m u l t i - f l u i d m i x i n g r u l e s i s the same as the o n e - f l u i d a p p r o a c h e x c e p t that in this case we w i l l s e a r c h f o r a h y p o t h e t i c a l multicomponent ideal m i x t u r e which could r e p r e s e n t the configurational p r o p e r t i e s of a multicomponent r e a l m i x t u r e , both w i t h the s a m e number of c o m p o n e n t s and at the same t h e r m o d y n a m i c conditions. In t h i s c a s e e q s . 1 9 - 2 1 w i l l be r e p l a c e d b y t h e f o l l o w i n g s e t of e q u a t i o n s f

xi xiôo(y)9oo(y^V dy=2jXjf h jj0

f

h

x 1

2

h

x 1

Jy0-

o o

(y)g

o o

i j

1

(y)y dy=2jX f h 2

j

i j

1 j

o o

(y)g j(y)y dy

Jy^'

o o

(40)

2

i

(y)g (y)y dy i j

2

{l-4nph J[g (y)-1]y2dy}-1=2 X3|B| /|B| x i

0 0

j

i j

(41) (42)

E x p r e s s i o n s f o r Bjj and G|j w i l l b e t h e s a m e a s i n e q . 1 8 . In the c a s e of the h a r d - s p h e r e f l u i d e q . 4 0 w i l l r e d u c e to the following form h

xigoo

eq.41

h s

( )=2jX h g 1

j

i j

j j

h s

(1),

(40-1)

w i l l v a n i s h and e q . 4 3 w i l l r e m a i n t h e s a m e .

Ml.2.i. A v e r a g e Potential Model ( A P M ) f o r M i x t u r e RDFs: t h i s a p p r o x i m a t i o n i t i s a s s u m e d t h a t (5_), g ( r ) = [ g ^ r ) + gjj(r)]/2 l j

Q (r) * gjj(r)

In (43)

t 1

When this a p p r o x i m a t i o n i s r e p l a c e d in e q s . 4 0 - 4 2 , eq.42 w i l l v a n i s h a n d e q . 4 0 and 41 w i l l p r o d u c e t h e f o l l o w i n g m i x i n g rules *xi< > = 2 j X j 0 ( r )

(44)

*'xi(r) = SjXj^'i^r)

(45)

r

i j

For e x a m p l e , in the c a s e of t h e intermolecular potential function f o l l o w i n g m i x i n g rules(JL2) . f

h

2

= 7-x-f-h-

Lennard-Jones (12-6) we w i l l derive the

2


(

4

6

)

15.

MANSOORI


323

'xihxi = SjXjfjjh^

(47)

4

For a h a r d - s p h e r e rule hx i

1

/

= 2jXjhij

3

p o t e n t i a l we w i l l d e r i v e only one m i x i n g

(*»>

1 / 3

Ml.2.ii. Multi-fluid CSA Approximation for Mixture A c c o r d i n g to this approximation t h e s c a l e d RDFs m i x t u r e a r e r e l a t e d as the following


gij(y) = tgij(y) + gjj(y)]/2

RDFs: in a

g (r) * gjj(y)

(49)

t 1

W h e n w e u s e t h i s a p p r o x i m a t i o n i n e q s . 4 0 a n d 41 t h e y b o t h produce the same m i x i n g rule which is f -h =

Y x f - h -

( °) 5

N o w , b y r e p l a c i n g e q . 4 9 i n 42 an a d d i t i o n a l m i x i n g r u l e be p r o d u c e d w h i c h i s t h e f o l l o w i n g

will

|B*|/ RTK

(51)

P

= SjXjlB*^

T x 1

where |B*|

tj

= x {6 i

i j

+ (x h /2)[(pRTK j

1 j

T x i

-1)/h +(pRTic j-1)/h j]} x 1

T x

X

Eq.50 i s actually the second van d e r Waals m u l t i - f l u i d m i x i n g r u l e , but eq.51 i s a new m i x i n g r u l e f o r h j . By using the h a r d - s p h e r e p o t e n t i a l (by r e p l a c i n g e q . 4 9 in 4 0 - 1 ) we will derive the following mixing rule X

h

= ZjXjhfj

x j

(52)

This mixing rule is the f i r s t m u l t i - f l u i d m i x i n g rule which, in conjunction with eq.50 for calculation of m i x t u r e thermodynamic should be p o i n t e d out that eq.52 c o n s t i t u t e s rule for hard-sphere mixtures. I l l . 2 . i i i . M u l t i - F l u i d HSE M i x i n g R u l e s : as t h e o n e - f l u i d c a s e w e c a n d e r i v e rules f

van der Waals i s usually used properties. It another mixing

In a s i m i l a r m a n n e r the following mixing

h •= Y x f - h -

(53)

f' x i - "h x i• = Y xJ ifj hi j 2

These

2

mixing

rules

(

are

used

for

calculation

of


5

4

excess

)


324

p r o p e r t i e s of a m i x t u r e o v e r the h a r d - s p h e r e

mixture.

III. 2 . i v , M u l t i - F l u i d DEX M i x i n g R u l e s : In a s i m i l a r m a n n e r a s the o n e - f l u i d case we can d e r i v e the f o l l o w i n g m i x i n g rule f

xi xi h

=

SjXjfijh^d-Cf^/fxi-DEupUig)/^ +T(C


This mixing rule can be used, rule, f o r calculation of m i x t u r e

v 1

-C

v 1 g

)/(u u

joined with properties.

r

1 g

)]}

another

(55) mixing

IV. A p p l i c a t i o n o f M i x i n g R u l e s f o r C u b i c E o u a t i o n s o f S t a t e In o r d e r t o a p p l y t h e v a r i e t i e s o f t h e c o n f o r m a l solution mixing rules which a r e introduced here f o r cubic and other equations of s t a t e t h e f o l l o w i n g c o n s i d e r a t i o n s should be taken into account: (i) Conformal solution mixing rules a r e f o r the molecular conformal volume parameter, h, and the m o l e c u l a r c o n f o r m a l e n e r g y p a r a m e t e r , f. (ii) Conformal solution mixing rules a r e applicable for c o n s t a n t s o f an e q u a t i o n o f s t a t e o n l y . B e f o r e u s i n g a s e t o f m i x i n g r u l e s f o r an e q u a t i o n o f s t a t e one h a s t o e x p r e s s t h e p a r a m e t e r s of t h e equation of s t a t e with r e s p e c t t o the m o l e c u l a r c o n f o r m a l p a r a m e t e r s h and f. This w i l l then make i t p o s s i b l e t o w r i t e t h e c o m b i n i n g r u l e s a n d m i x i n g r u l e s f o r t h e e q u a t i o n o f s t a t e . In w h a t f o l l o w s m i x i n g r u l e s and combining rules f o r three representative cubic e q u a t i o n s of s t a t e a r e d e r i v e d and t a b u l a t e d . IV.1. M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e : The v a n d e r W a a l s e q u a t i o n o f s t a t e ( 1 5 ) c a n b e w r i t t e n i n the f o l l o w i n g f o r m Z = Pv/RT = v / ( v - b ) - a/vRT

(56)

P a r a m e t e r b of this equation of s t a t e i s p r o p o r t i o n a l to m o l e c u l a r v o l u m e ( b « h ) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume)(molecular energy (aocfh). Then, in order to apply the mixing rules introduced in this r e p o r t f o r the van d e r W a a l s equation of s t a t e we must r e p l a c e h with b and f w i t h a/b i n a l l t h e m i x i n g r u l e s . In T a b l e I m i x i n g r u l e s f o r t h e v a n d e r W a a l s e q u a t i o n o f s t a t e b a s e d on d i f f e r e n t t h e o r i e s of m i x t u r e s a r e r e p o r t e d . The combining rules f o r a ^ and b y (i*j) of this equation of s t a t e , c o n s i s t e n t w i t h eqs.12 w i l l be


MANSOORI

15.


325

T a b l e I: M i x i n g R u l e s f o r t h e v a n d e r W a a l s E q u a t i o n o f S t a t e One-Fluid

Mixing

Rules

a RMA

[2i2jX Xja jb j]^/2 i

i

i

2 2jX x a b 3]1/2

/ [

i

i

j

i j

i j

Theory{

^ , Theory{

vdW

^ i ^ i ^ ^ j ^ i j l i i . ^ i ^ j ^ i i îJ_

]

a = SiSiXiXiQii


b = SiSjXiXjbij

HSE

a = SiSjXiXja^j

Theory{

b =

[2i2jX Xja j] /2 2jX Xja j /b j i

a = [a DEX

v d W

1

2

i

+(b/vRT)

2

i

i

i

2i2jX Xja j /b i

2

i

]/[ 1+a

i j

v

d

W

/vRT]

Theory{ b = SiSjXiXjbij a = SjSjXjXjaij

CSA

Theory{ 1+A

-|B*|/ ZiSjXiXjlB*^;

X X

A PuMl t i -TFhl u e iodr y {M i x i n g M b

B*

i j

=x (6 i

i j

+x A J

x x

b j/b) i

Rules

= SjXjaij vdW

Theory{ b

=

SjXjbij

= SjXjay HSE

Theory{ b

a

DEX

i

=

la j

=

A

X

X

I

V C

i

j

j

i

2

i

j V Y " K i / v R T ) SjXjajj^/b^j]/[ 1 +a i

bi

= SjXjbij

b

d W

V

/

v

R

T

^

ai

= SjXjaij

Theory{ 1+A

X

j

Theory{

CSA

A

[SjX a j]2/2 x a j /b j

=

PRTK

= pRTK

T

T

x

x

x

i

-1 -l

x i

« | B * | / 2jXj|B*lij;

B*i =x [6 j+x b j(A i/bi+A j/b )/2] j

i

i

= [2a(v-b)^-RTb(2v-b)]/[RTv

j

i

X

X

^-2a(v-b)^l

= [2ai(v-bi) -RTbi(2v-bi)]/[RTv -2ai(v-bi) l 2

2

2


j


326

a

ij = < - ij) ij( ii jj/t> 1

k

b

a

a

1 i

b )

1 / 2

;

)/2]

3

j j

(57) b

t j

= (1-£

1 j

)[(b

1 1

1 / 3

+b

j j

1 / 3


IV.2. M i x i n g R u l e s f o r the R e d l i c h - K w o n g E q u a t i o n of S t a t e : The R e d l i c h - K w o n g equation of s t a t e (16) which i s an e m p i r i c a l m o d i f i c a t i o n o f t h e v a n d e r W a a l s e q u a t i o n c a n be w r i t t e n in the f o l l o w i n g f o r m

Z = Pv/RT = v / ( v - b ) -

a/[RT

3 / 2

(v+b)]

(58)

P a r a m e t e r b of t h i s e q u a t i o n o f s t a t e i s p r o p o r t i o n a l t o m o l e c u l a r v o l u m e (boch) and p a r a m e t e r a i s p r o p o r t i o n a l t o (molecular volume) (molecular energy) or (ai2jX x a j(a/b)^3( _ i

DEX

j

i

1

.

/ 2 i 2 j X

/

[ ( a i j

4/3

X j a i j

p2/3(

b i

b / a

b i r

l/3

)273_|]^

Theory{ b = SiSjXjXjbij _ _ _ _ _ _ _ ^

CSA

Theory{ 1+A =|B*|/ S i S j X i X j l B * ^ ;

B*ij=

X X

Multi-Fluid

Mixing ai -

APM

XiCeij+xjAxxbij/b)

Rules [2jX a j

1 j

2/3b

1 j

4/3]5/2

/ 2 j X j a i j

2/3

b l j

10/3

Theorvl

,

ai =

[2jX

vdW Theory{

J

J

j

a

J

i

j

2/3

b

i

j

1/3]3/2

/

[

2

j

X

j

b

|

j

]

l/2

1 J

bi = SjXjbij

_______ HSE

Theory{

^

^ ^ ^ . z / S b y ^ S j Z ^ x j ^ ^ / S b y - l / S

=

ai = _ > j X a i j ( a / b ) ^ 3 { - ( j

DEX

1

[

a i j

/

)2"/3(

b i j

/

b i

a i

)2/3_

1 ] C i

}

Theory{ bi = SjXjbij

CSA

Theory{ 1+__

C Ci

x i

= |B*|/ _>jXj|B*lij ;

B*ij= x ( 6 j + x b j [ A i

i

j

i

x i

/b +A I

x j

/bj])

= (3/2)(a/bR)T "V/Untv/Cv+b)]-l/2; = C3/2)(ai/biR)T- £n[v/(v+b i ) ] - l / 2 3 / 2

A

X

A

X

X

I

= =

PRTK PRTK

T

x

T x 1

x

-1

-1

=-l+RT /2( 2_ 2)2/[ 3

=

v

b

-l+RT3/2( 2v

b l

2)2/[

R T

R T

{ ( v

{ ( v

v + b

))

v + b i

2_ (

)}2-

a

a i

2 v

+b)(v-b)

2

1

(2v+bi)(v-bi) ]


2


328

and T and P a r e the c r i t i c a l point t e m p e r a t u r e and p r e s s u r e , r e s p e c t i v e l y ; and u> i s t h e a c e n t r i c f a c t o r . In order to utilize the statistical mechanical mixing rules f o r the P e n g - R o b i n s o n equation of s t a t e we must f i r s t s e p a r a t e t h e r m o d y n a m i c v a r i a b l e s f r o m constants of this equation of s t a t e . F o r t h i s p u r p o s e we m a y w r i t e t h i s e q u a t i o n of s t a t e in t h e f o l l o w i n g f o r m c

c

Z = Pv/RT = v / ( v - b ) -

[(A/RT+C-2(AC/RT)

1 / 2

]/


[(v+b)+(b/v)(v-b)]

(61)

w h e r e A = a ( l + 8 ) and C = a 6 / R T . This new f o r m of the P e n g - R o b i n s o n equation of s t a t e i n d i c a t e s that t h e r e e x i s t three independent constant p a r a m e t e r s in this equation w h i c h a r e A , b, a n d C. P a r a m e t e r s b and C a r e p r o p o r t i o n a l t o t h e m o l e c u l a r v o l u m e (boch a n d C « h ) w h i l e p a r a m e t e r A i s proportional to (molecular volume)(molecular energy) or (Aocfh). B a s e d on d i f f e r e n t t h e o r i e s o f m i x t u r e s m i x i n g r u l e s f o r this new f o r m of t h e Peng-Robinson equation of s t a t e a r e r e p o r t e d i n T a b l e III. The combining r u l e s f o r t h e unlike i n t e r a c t i o n p a r a m e t e r s of this equation of s t a t e a r e as t h e f o l l o w i n g 2

c

A

ij = ^- ij) ij( ii jj k

b

A

b

ij = (1-îj)[( ii

c

ij =

b

A

c

/ b

1 / 3 + b

2

c

(62)

ii jj>

jj

b

1 / 3

)/2]

3

^ - ^ n i c ^ ^ c ^ ^ / z ] ^

(63)

(64)

Similar procedures t o those d e m o n s t r a t e d above can be used f o r d e r i v a t i o n of conformal solution mixing r u l e s f o r o t h e r cubic equations of s t a t e .


15.

MANSOORI


T a b l e HI: One-Fluid

329

M i x i n g Rules f o r the P e n a - R o b i n s o n Eg. of S t a t e Mixing Rules

A = [SiSjXiXjAijb^/^^jXiXjA^byS]!/? RMA Theory{ b = t Z i S j X i X j A i j b i j f / S i S j X i X j A i j b i j l J ^ C = [_> iSjXiXjAijCij^/SiSjXiXjAijCij]' ^ 7

A = 2i2jXiXjAjj _Ei_EjXiXjbij vdW Theory! b = 2i_£jXjXjCij C =


A HSE Theory!

2i_!iXiXj Ajj - CSiSjXiXjAij^/SiSjXiXjA^/bij = [S i_>j x i Xj Aj j ] ^ /2 i2j x i Xj Ai j - /C i j C =

DEX Theory!

A - SiSjXiXjAijd-lCAij/bijXb/A)-!!^ = Si-SjX-jXjb-jj C SiSjXjXjCij

=

b

b

=

CSA Theory!

A = SiSiXjXjAjj xx=lB*I/ 2i_>jX |B*hj; C = 2i_!jXiXjCij 1+A

B* =x (6 j+x A b /b);

lXj

ij

i

i

j

xx

ij

Multi-Fluid Mixing Rules A APM Theory! b - [SjXjAijbijJ/^jXjAijbij]Ml C = C__ x A jC 3/2: x A C ]1/2 j

i

ij

j

j

ij

ij

SjXjAtj SjXjbij ZjXjCij

A vdW Theory! b C HSE Theory!

j

Ai = SjXjAij i 2j j ij] /_>jXjAij /bij Ci [ZjXjAylZ/SjXjAijZ/Cy

b

=

[

x

A

2

2

Ai = 2jXjAij{1-[CAij/bij)(b/A)-1]^ i) DEX Theory! i = 2 j j i j Ci - SjXjCij b

x

b

Ai = SjXjAij CSA Theory { 1+A = |B*[V __jX |B*| ; Ci = SjXjCij xi

j

B* =Xi[6ij+(x b /2)(A /bi+A /bj)]

ij

ij

j

ij

xi

A = P R T K - 1 = -HRT/{RTv v-b)^-2Av^/( v ^ + b ^ } A = p R T < - l = -HRT/{RTv /( v - b i ) - 2 A i V / ( v + b i ) } I = {[A-Y(ACRT)]/(2bRTv 2)}en[CY+b-b^2)/(v+b+bv 2)] + ^(ACRT)/!2[^(ACRT)-A]) Ci= {[Ai-v^(AiCiRT)]/(2biRTV2)}£n[(v+b bjV2)/(v+b i+bjV-)] +V(AiCiRT)/{2[V(A iCiRT)-Ail} x x

x i

T x x

2

T x i

2

3

r

2

2

2

r

r


xj


330


Acknowledgment: The a u t h o r t h a n k s P r o f e s s o r C a r o l H a l l f o r h e r h e l p f u l c o m m e n t s and c o r r e c t i o n s . This r e s e a r c h i s s u p p o r t e d b y t h e U.S. D e p a r t m e n t of E n e r g y G r a n t N o . DE-FG02-84ER13229.

Literature Cited 1. Massih, A.R.; Mansoori, G.A. Fluid Phase Equilibria 1983, 10, 57. 2. Brown, W.B. Proc. Roy. Soc. London Series A, 1957, 240;Phil. Trans. Roy. Soc. London Series A, 1957, 250. 3. Hill, T.L. "Statistical Mechanics" McGraw-Hill, New York, N.Y. 1956. 4. Kirkwood, J.G.; Buff, F. J. Chem. Phys. 1951, 19, 774. 5. Mansoori, G.A.; Ely, J. F. Fluid Phase Equilibria 1985, 22, 253. 6. Lan, S.S.; Mansoori, G.A. Int. J. Eng. Science 1977, 15, 323. 7. Leach, J.W.; Chappelear, P.S.; Leland, T.W. AIChE J. 1968, 14, 568; Proc. Am. Petrol. Inst. Series III, 1966, 46, 223. 8. Leland, T.W. Adv. Cryogenic Eng. 1976, 21, 466. 9. Mansoori, G.A.; Leland, T.W. J. Chem. Soc., Faraday Trans.II 1972, 68, 320. 10. Mansoori, G.A. J. Chem. Phys. 1972, 57, 198. 11. Rowlinson, J.S.; Swinton, F.L. "Liquids and Liquid Mixtures" 3rd Ed., Butterworths, Wolborn, Mass. 1982. 12. Scott, R.L. J. Chem. Phys. 1956, 25, 193. 13. Mansoori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W. J. Chem. Phys. 1971, 54, 1523. 14. Mansoori, G.A.; Ely, J.F. J. Chem. Phys. 1985, 82, 406. 15. Van der Waals, J.D. "Over de continuiteit van den gasen vloeistoftoestand" Leiden, 1873. 16. Redlich. O.; Kwong, J.N.S. Chem. Rev. 1949, 44, 233. 17. Peng, D.Y.; Robinson, D.B. Ind. Eng. Chem. Fundam. 1976, 15, 59. RECEIVED November

5, 1985


Mixing Rules for Cubic Equations of State - ACS Symposium Series

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