Effective Molecular Diameters for Fluid Mixtures - Advances in

4 Effective Molecular Diameters for Fluid Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

Mixtures JAMES I. C. CHANG, FRANK S. S. HWU, and THOMAS W. LELAND Department of Chemical Engineering, P.O. Box 1892, Houston, TX 77001

Rice

University,

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix tures based on the hard-sphere expansion conformai solution theory is developed. The method of Verlet and Wets pro duces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH -CO has been carried out successfully. 4

Applications

of Equations

of State

for

Hard

2

Spheres

*~phe d e v e l o p m e n t of a n a c c u r a t e e q u a t i o n of state f o r m i x t u r e s o f h a r d - ' - s p h e r e s ( I ) has l e d to significant i m p r o v e m e n t s i n t h e c o m p u t a t i o n of t h e r m o d y n a m i c p r o p e r t i e s of m i x t u r e s , b o t h t h r o u g h t h e d e v e l o p m e n t of n e w e q u a t i o n s of state ( 2 ) a n d f r o m i m p r o v e d c o n f o r m a i s o l u t i o n techniques ( 3 ) .

P r o p e r t i e s p r e d i c t e d b y these m e t h o d s , h o w e v e r , a r e

e x t r e m e l y sensitive to t h e n u m e r i c a l v a l u e s a s s i g n e d t o t h e m o l e c u l a r 0-8412-0500-0/79/33-182-071$06.25/l © 1979 American Chemical Society In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.

72

EQUATIONS OF STATE

diameters a n d this c h a p t e r reports a n e w p r o c e d u r e for t h e i r d e t e r m i n a t i o n i n a c t u a l r a t h e r t h a n t h e o r e t i c a l fluids. T h e m e t h o d is a p p l i c a b l e to mixtures w h i c h contain both polar and nonpolar components. It

is i m p o r t a n t to r e a l i z e t h a t the diameters n e e d e d for

thermo

d y n a m i c c a l c u l a t i o n s do not necessarily represent a t r u e m i n i m u m a t t a i n a b l e s e p a r a t i o n distance b e t w e e n m o l e c u l e s .

T h e o b j e c t i v e is r a t h e r to

d e t e r m i n e o p t i m a l or "effective" diameters w h i c h g i v e best results w h e n used w i t h a particular method

of d e a l i n g w i t h the c o n t r i b u t i o n s

of

Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

m o l e c u l a r a t t r a c t i o n . I n this c h a p t e r the effective diameters sought are to b e u s e d s p e c i f i c a l l y w i t h t h e h a r d - s p h e r e e x p a n s i o n ( H S E ) c o n f o r m a i s o l u t i o n t h e o r y of M a n s o o r i a n d L e l a n d ( 3 ) .

T h i s t h e o r y generates the

p r o p e r p s e u d o parameters for a p u r e reference

fluid

to b e u s e d i n

p r e d i c t i n g the excess of a n y dimensionless p r o p e r t y of a m i x t u r e o v e r the c a l c u l a t e d v a l u e of this p r o p e r t y f o r a h a r d - s p h e r e m i x t u r e .

The

v a l u e of this excess is o b t a i n e d f r o m a k n o w n v a l u e of this t y p e of excess for a p u r e reference fluid e v a l u a t e d at t e m p e r a t u r e a n d d e n s i t y c o n d i t i o n s m a d e dimensionless w i t h the p s e u d o parameters.

F o r example, i f X M

represents a n y dimensionless p r o p e r t y for a m i x t u r e of η n o n p o l a r c o n stituents at m o l e fractions x

u

x , . . . x -i at t e m p e r a t u r e Τ a n d d e n s i t y 2

n

P, t h e n : ρ, Χι,

Xo,

. . ·

X

n

-1)

= XlISM

{pd \\, Z

+ X R E p ( f c r / c , d*) P

pd 22, · · · %1) 2> · · · %n-l) X

3

X (p3 ). n s

(1)

3

I n E q u a t i o n 1 X H S M is the v a l u e of X for a h a r d - s p h e r e m i x t u r e of d i a m eters du, d 2 . . · etc., a n d X is the v a l u e of X for a p u r e h a r d - s p h e r e fluid w i t h d i a m e t e r d. T h e v a l u e of X s is c a l c u l a t e d f r o m the C a r n a h a n S t a r l i n g ( C S ) e q u a t i o n (4). X F represents the v a l u e of X as o b t a i n e d f r o m a r e d u c e d e q u a t i o n of state for the p u r e reference fluid e v a l u a t e d at Γ a n d ρ m a d e dimensionless b y the p s e u d o parameters c a n d d . 2

I I S

H

R

E

3

S e v e r a l i m p o r t a n t p r i n c i p l e s c o n c e r n i n g t h e d e t e r m i n a t i o n of effective diameters for r e a l fluids c a n be established b y e x a m i n i n g methods of o b t a i n i n g t h e m for t h e o r e t i c a l fluids w i t h exactly k n o w n potentials. Effective

Diameters

for

Perturbation

Theories

A t h e o r e t i c a l basis for the c o m p u t a t i o n of effective h a r d - s p h e r e diameters is n o w w e l l d e v e l o p e d for t h e p e r t u r b a t i o n t h e o r y . Its n u m e r i cal e v a l u a t i o n r e q u i r e s exact k n o w l e d g e of the i n t e r m o l e c u l a r p o t e n t i a l so that the m e t h o d is n o t i m m e d i a t e l y a p p l i c a b l e to r e a l m o l e c u l e s for w h i c h this p o t e n t i a l is u s u a l l y u n k n o w n . D e s p i t e this difficulty, t h e p r i n c i p l e s i n v o l v e d are i m p o r t a n t i n d e s i g n i n g a p r o c e d u r e for r e a l m o l e cules w i t h the H S E theory.

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

4.

CHANG E T AL.

Molecuhr

Diameters

for Fluid

73

Mixtures

T h e p s e u d o p a r a m e t e r s of t h e H S E t h e o r y are d e r i v e d f r o m e q u a t i o n of state e x p a n d e d i n p o w e r s of 1/kT

about a hard-sphere

an fluid,

as is d e v e l o p e d b y t h e p e r t u r b a t i o n theory. C o n s e q u e n t l y , i t is reasonable to e x p e c t that p r o c e d u r e s for d e f i n i n g o p t i m a l d i a m e t e r s for the p e r t u r b a t i o n t h e o r y s h o u l d w o r k w e l l w i t h the H S E p r o c e d u r e .

T h e first p o r t i o n

of this c h a p t e r shows t h a t this is i n d e e d correct. T h e V e r l e t - W e i s (5)

(VW)

m o d i f i c a t i o n of the W e e k s , C h a n d l e r , a n d A n d e r s o n ( W C A )

(6)

p r o c e d u r e w a s u s e d here to d e t e r m i n e d i a m e t e r s i n a m i x t u r e of L e n Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

nard-Jones ( L J ) (12-6)

fluids.

These diameters then were used i n the

H S E p r o c e d u r e to p r e d i c t the m i x t u r e p r o p e r t i e s . T h e V W d i a m e t e r s are defined as :

d = d |\+ B

The d

B

(^j

8vw]

t e r m is the B a r k e r - H e n d e r s o n ( B H ) ( 7 ) d = B

J

(2) diameter:

f°°[l - e ^ ] d r ο

(3)

w h e r e u is the r e p u l s i v e p o r t i o n of the i n t e r m o l e c u l a r p o t e n t i a l 0

and

2σ σ ι

0

(5)

(1-b) b + 1.362 b - 0.8751 b 2

1-

( 4) 17

2

s

where: 6 - | ( p d » ) I 1 -^(pd )

I

3

(6)

T h e s o l u t i o n of these equations for d r e q u i r e s a n i t e r a t i o n , u s u a l l y u s i n g the B H d i a m e t e r as a first t r i a l . T h e B H d i a m e t e r is d e s i g n e d

to i n c l u d e

temperature-dependent,

s o f t - r e p u l s i o n c o n t r i b u t i o n s i n the h a r d - s p h e r e e q u a t i o n of state.

The

W C A p r o c e d u r e also does this a n d f u r t h e r modifies the d i a m e t e r to b e the p r o p e r

value w h e n using a hard-sphere

distribution function i n

p e r t u r b a t i o n terms d u e to a t t r a c t i o n . T h e V W p r o c e d u r e WCA

result for

some of

the l i m i t a t i o n s of

corrects

the P e r c u s - V e v i c k

the (PY)

a p p r o x i m a t i o n a n d expresses the result i n a s i m p l e a l g e b r a i c f o r m . t a b l e of d

B

a n d 8 w values is g i v e n so no i n t e g r a t i o n is r e q u i r e d . V


A

74

EQUATIONS OF STATE

Use of Yerlet—Weis Diameters

in the HSE

Procedure

T a b l e I , i n the c o l u m n h e a d e d H S E - V W , shows the results of u s i n g E q u a t i o n s 2 t h r o u g h 6 to define the d i a m e t e r s w i t h the H S E m e t h o d to c a l c u l a t e the p r o p e r t i e s reference is a p u r e L J

of

a n e q u i m o l a r m i x t u r e of

fluid.

m a c h i n e - c a l c u l a t e d results of S i n g e r a n d S i n g e r (8) v a n der W a a l s ( V D W ) Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

t h e o r y (10)

LJ

fluids.

The

O t h e r c o l u m n s s h o w c o m p a r i s o n w i t h the

one-fluid theory

(9)

in column M C . The

a n d the V D W t w o - f l u i d

are i n c o l u m n s V D W - 1 a n d V D W - 2 . T h e G H B L

gives t h e G r u n d k e , H e n d e r s o n , B a r k e r , L e o n a r d ( G H B L )

column

(11)

pertu-

b a t i o n t h e o r y results w i t h e a c h d i a m e t e r d e t e r m i n e d b y E q u a t i o n 3. T h e c o l u m n h e a d e d H S E uses a n a p p r o x i m a t i o n m a d e o r i g i n a l l y b y M a n s o o r i a n d L e l a n d (3)

t h a t the d i a m e t e r u s e d i n the h a r d

sphere

equations of state is ο σ, the L J σ p a r a m e t e r for e a c h m o l e c u l e m u l t i p l i e d 0

b y a u n i v e r s a l constant f o r c o n f o r m a i r e q u i r e s that d

i}

be replaced by σ

ί;

fluids.

This approximation then

i n the equations d e f i n i n g t h e H S E

p s e u d o p a r a m e t e r s , E q u a t i o n s 10 a n d 11. T h e results i n the H S E c o l u m n use c

0

=

0.98, the v a l u e for L J fluids o b t a i n e d e m p i r i c a l l y b y M a n s o o r i

a n d L e l a n d . T h i s p r o c e d u r e is correct o n l y for a K i h a r a - t y p e p o t e n t i a l a n d i t is not consistent w i t h the L J fluids i n T a b l e I . F u r t h e r m o r e , this causes o n l y the h i g h t e m p e r a t u r e l i m i t of the r e p u l s i o n effects to

be

i n c l u d e d i n the h a r d - s p h e r e c a l c u l a t i o n . Soft r e p u l s i o n s are p r e d i c t e d b y the reference

fluid.

I n c o m p a r i n g the last t w o c o l u m n s , one c a n see t h a t t h e d e n s i t y d e p e n d e n t V W d i a m e t e r s , w h i l e p e r h a p s not the o p t i m a l values f o r the H S E t h e o r y , are a n d better t h a n the constant c a v a l u e a n d a t e m p e r a t u r e 0

T a b l e I. Q j / «12

Property vu/ σ ΐ 2 =

0.810 H /mr E

V

E

0.900

G /mr H /mr E

E

V

E

1.000

G /Nfc7 H*/mT E

7

V

E

1.111

1.235

G /mr //•VNfcT γε G /NkT H /mT V /~NkT E

E

E

E

C a l c u l a t e d Properties

MC

VdW-j

0.305 0.34 -1.29 0.105 0.15 -0.41 -0.005 0.03 0.02 -0.030 -0.05 -0.10 0.040 -0.10 -0.68

0.334 0.35 -1.39 0.114 0.15 -0.54 -0.011

1.06

-0.18 -0.044 -0.089 -0.27 0.014 -0.12 -0.81


4.

CHANG ET AL.

Molecular

Diameters

for Fluid

75

Mixtures

a n d d e n s i t y d e p e n d e n c e is definitely n e e d e d . T h e s e c o l u m n s also

show

the i m p o r t a n c e of d e f i n i n g a d i a m e t e r w h i c h accounts for a l l the r e p u l s i o n effects i n the

fluid.

T a b l e I shows t h a t the H S E t h e o r y is m u c h better t h a n either of t h e V D W theories, w h i c h justifies the d i r e c t c o m p u t a t i o n s of the h a r d - s p h e r e c o n t r i b u t i o n s as o p p o s e d to t h e i r p r e d i c t i o n b y the reference the case w i t h the V D W theories.

fluid

as is

F u r t h e r m o r e , d e r i v a t i o n of t h e H S E

p s e u d o p a r a m e t e r s considers (1/kT)

terms i n the e x p a n s i o n a b o u t h a r d -

2


sphere p r o p e r t i e s , w h e r e a s i n the V D W theories the e x p a n s i o n is t r u n c a t e d after the first o r d e r 1/kT t e r m a n d a l l of t h e r e p u l s i o n p r o p e r t i e s of the fluid are p r e d i c t e d b y the reference the s e c o n d o r d e r (1/kT)

fluid.

T h e i m p o r t a n c e of i n c l u d i n g

terms is i n d i c a t e d b y the excellent results of

2

the H S E - V W m e t h o d for the e n t h a l p y , i n t h a t this p r o p e r t y is m u c h less d e p e n d e n t

o n the

c h o i c e of

the

diameter

a n d the

hard-sphere

contribution. T h e G H B L p e r t u r b a t i o n p r o c e d u r e is r e m a r k a b l y a c c u r a t e a n d the H S E - V W m e t h o d is o n l y s l i g h t l y better i n its o v e r a l l agreement w i t h the m a c h i n e - c a l c u l a t e d results. T h i s c o m p a r i s o n is not c o m p l e t e l y v a l i d i n t h a t the c o n f o r m a i s o l u t i o n t h e o r y uses p u r e c o m p o n e n t

data while i n

the p e r t u r b a t i o n t h e o r y e a c h t e r m is c a l c u l a t e d f r o m m o l e c u l a r p a r a m e t e r s . The HSE Procedure

for

Mixtures

of Polar and Nonpolar

Molecules

O n e of the advantages of the H S E p r o c e d u r e is t h a t c o n t r i b u t i o n from intermolecular attraction can be predicted more accurately if it consists p r i m a r i l y of l o n g - r a n g e i n t e r a c t i o n s , w h i l e v e r y short-range i n t e r of an Equimolar L J Mixture VdW-2

GHBL

HSE

VW-HSE

^22/&i2 = 1-06 0.236 0.24 -1.13 0.073 0.09 -0.47 -0.007 -0.18 -0.006 -0.03 -0.21 0.077 -0.57

0.271 0.27 -1.39 0.095 0.12 -0.48 -0.006 -0.014 -0.08 -0.033 -0.050 -0.12 0.014 -0.073 -0.61

0.280 0.28 -1.27 0.110 0.15 -0.47 0.01 0.02 -0.01 -0.04 -0.07 -0.26 0.019 -0.07 -0.67

0.316 0.34 -1.21 0.119 0.16 -0.39 0.01 0.043 -0.02 -0.018 -0.04 -0.06 0.039 -0.067 -0.55


76

EQUATIONS OF STATE

a c t i o n effects are r e m o v e d a n d e v a l u a t e d i n a separate c o m p u t a t i o n . T h i s is the case for m i x t u r e s c o n t a i n i n g p o l a r m o l e c u l e s w h e r e t h e l o n g e r - r a n g e i n t e r a c t i o n s are s u i t a b l e f o r e x p a n s i o n i n s p h e r i c a l h a r m o n i c s a n d t h e i r m u l t i p o l e coefficients, the d i p o l e a n d q u a d r a p o l e m o m e n t s , are e x p e r i m e n t a l l y accessible. T h i s is not the case f o r the shorter-range a s y m m e t r i c interactions i n v o l v i n g the o v e r l a p coefficient a n d n o n i s o t r o p i c p o l a r i z a bility.

T h e c o n t r i b u t i o n s of these to the p o t e n t i a l are p r o p o r t i o n a l to

i n v e r s e s e p a r a t i o n distances to p o w e r s Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

c o m p a r a b l e w i t h the 1 / r

These

are

d e p e n d e n c e of r e p u l s i o n p o t e n t i a l s a n d

of f r o m

12 to 24.

the

results are s i m p l i f i e d a n d i m p r o v e d i f a n effective d i a m e t e r c a n b e f o u n d to

include

these

short-range

asymmetric

effects

in

the

hard-sphere

calculation. F o r m i x t u r e s c o n t a i n i n g p o l a r m o l e c u l e s the most effective r o u t e to t h e r m o d y n a m i c p r o p e r t i e s is to d e t e r m i n e first the m o l a l r e s i d u a l H e l m h o l t z free-energy f u n c t i o n A, w h e r e :

(7)

T h i s occurs

because the c o n t r i b u t i o n f r o m the a s y m m e t r i c p o r t i o n

t h e potentials is m o r e c o n v e n i e n t l y c a l c u l a t e d for this p r o p e r t y .

of

Other

f u n c t i o n s c a n b e o b t a i n e d f r o m it b y d i f f e r e n t i a t i o n . F o r e x a m p l e , the c o m p r e s s i b i l i t y factor is

(8)

T h e o r i g i n a l d e r i v a t i o n of the H S E p r o c e d u r e e x t e n d e d to i n c l u d e p o l a r c o m p o n e n t s

(3)

c a n easily

b y c o n s i d e r i n g t h a t the

be

excess

over h a r d - s p h e r e p r o p e r t i e s c a n be d i v i d e d i n t o t w o p o r t i o n s f o r m i x t u r e a n d for a p u r e reference

fluid.

the

T h e first i n c l u d e s the c o n t r i b u t i o n

of the s y m m e t r i c excess f r o m a l l n o n p o l a r i n t e r a c t i o n s p l u s t h e l e a d i n g symmetric contributions i n an expansion i n spherical harmonics for each interaction involving a polar molecule.

T h e second p o r t i o n of t h e excess

is the c o n t r i b u t i o n of the a s y m m e t r i c p a r t of a l l i n t e r a c t i o n p o t e n t i a l s . T h e e x p a n s i o n of p o t e n t i a l s i n s p h e r i c a l h a r m o n i c s a n d the r e s u l t i n g c o n t r i b u t i o n to t h e r m o d y n a m i c p r o p e r t i e s w e r e i n i t i a l l y p r e s e n t e d P o p l e (12)

a n d m o r e r e c e n t l y d e v e l o p e d i n f u r t h e r studies (13,14).

by The

e q u a t i o n p r o d u c e d for t h e excess over h a r d - s p h e r e p r o p e r t i e s shows t h a t t h e first-order p e r t u r b a t i o n coefficient (1/kT)

i n v o l v e s o n l y the s y m m e t r i c

p o r t i o n of a l l potentials. T h e a s y m m e t r i c c o n t r i b u t i o n s first a p p e a r i n the coefficient of the second-order

(1/kT)

2

t e r m w h e r e t h e y are w e i g h t e d


4.

CHANG

Molecular

ET AL.

Diameters

for Fluid

77

Mixtures

b y the d i s t r i b u t i o n f u n c t i o n s f o r molecules w h i c h h a v e o n l y s y m m e t r i c potentials.

C o n s e q u e n t l y , the d i s t r i b u t i o n f u n c t i o n s for the m i x t u r e s t i l l

c a n be r e p r e s e n t e d b y the m e a n d e n s i t y a p p r o x i m a t i o n :

(r, T, p, x i , s

2

(9)

· . .Xn-i


T h i s a s s u m p t i o n means, as i n the o r i g i n a l d e r i v a t i o n , t h a t coefficients of l i k e p o w e r s of (1/kT)

w h i c h involve p a i r interactions can be

made

e q u a l i n the m i x t u r e a n d i n the reference b y the p r o p e r c h o i c e of the p s e u d o p a r a m e t e r s . E q u a t i n g these coefficients of (1/kT) in

the

symmetric

potential portion

of

each

excess

and

gives

(1/kT)

2

the

same

p s e u d o p a r a m e t e r s as the o r i g i n a l t h e o r y , E q u a t i o n s 10 a n d 11. a s y m m e t r i c excess has no coefficients of (1/kT) cients of (1/kT)

2

a n d e q u a t i n g the coeffi

that arise f r o m p a i r i n t e r a c t i o n s results i n E q u a t i o n s

12, 13, a n d 14. S o m e coefficients of (1/kT)

terms i n b o t h t h e s y m m e t r i c

2

and

i n the a s y m m e t r i c excesses arise f r o m t h r e e - b o d y

interactions.

the s y m m e t r i c excess these c a n be e q u a t e d i n m i x t u r e a n d introducing

The

three-body

potential parameters, m a k i n g a

In

reference

superposition

approximation for the triplet distribution function, and equating them i n m i x t u r e a n d reference. that the t h r e e - b o d y

E r r o r s are not serious if one s i m p l y assumes

effects i n m i x t u r e a n d reference

are r o u g h l y

com

p a r a b l e w i t h o u t a n e w p a r a m e t e r to force t h e m to b e e q u a l .

I n the

a s y m m e t r i c excess, h o w e v e r ,

i n the

there are t h r e e - b o d y

coefficients

m i x t u r e w h i c h h a v e no c o u n t e r p a r t i n the p u r e a s y m m e t r i c excess.

These

cannot b e a c c o u n t e d for i n the theory. T h e p s e u d o p a r a m e t e r s are as f o l l o w s :

(10)

ΣΪ) i j ^ij^ d{j X

3

X

(ID i

3

T o obtain pseudo critical values V C P a n d T C P replace: d

3

i}

w i t h [ψ

ίΓ

V

ci

+ V ] jr

ci

I

c« withe, V T e T θ ci

ir

cj

β


78

EQUATIONS OF STATE

where 0

and φ

ir

are shape factors w h i c h a l l o w fluid i to b e p r e d i c t e d b y

ίΓ

reference fluid r w h e n t h e t w o fluids are n o n c o n f o r m a l .

Q W Ç Ç * „ ( ^ )


W ) ^ # E E μ* = ά

3

I

Σ Σ χ

ί

J

I

Χ

ί

( 1 2 )

(13)

^ J

;

(14)

( ^ ^

F r o m the p s e u d o c r i t i c a l p a r a m e t e r s the excess r e s i d u a l H e l m h o l t z f u n c t i o n for this m i x t u r e i s :

=

Α™

[Α

( d ,^) P

3

(15)

~ A™( d*)~^ P

od

3

T h e first b r a c k e t e d t e r m is t h e s y m m e t r i c excess w h i c h is p r e d i c t e d e x a c t l y as t h e excess i n E q u a t i o n 1. T h e s e c o n d t e r m is the a s y m m e t r i c excess w h i c h is c a l c u l a t e d d i r e c t l y .

The I ,

a n d Iio f u n c t i o n s

Is,

e

are

integrals i n v o l v i n g h a r d - s p h e r e d i s t r i b u t i o n f u n c t i o n s of t h e t y p e

h

=

f "

g

n

S

{

y

'

p

d

3

)

y

2

' "

d

(16)

y

Padé a p p r o x i m a n t s for_them i n c o n v e n i e n t f o r m are g i v e n b y S t e l l et a l . (14).

T h e jï, //.Q, a n d Q terms are the p s e u d o d i p o l e , d i p o l e - q u a d r a p o l e ,

a n d q u a d r a p o l e m o m e n t s , r e s p e c t i v e l y . T h e y are c o m b i n e d as i n a p u r e c o m p o n e n t to g i v e the a s y m m e t r i c excess of the m i x t u r e . T h e final result f o r the m i x t u r e is

\ f k

w h e r e the A

H

S

M

= A^( , P

x ,x ...,d ,d ...)+ 1

2

1

2

A

EX

(17)

t e r m is the r e s i d u a l H e l m h o l t z free energy f o r the h a r d -

sphere m i x t u r e c a l c u l a t e d f r o m the M C S L e q u a t i o n . T h e a n a l y t i c a l f o r m of this a n d the c a l c u l a t e d results are p r e s e n t e d b y M a n s o o r i et a l . The A

E

X

t e r m is g i v e n b y E q u a t i o n 15.


(1).

4.

CHANG ET AL.

Optimal

Molecular

Diameters

for

the

Diameters HSE

for Fluid

Theory

from

79

Mixtures

PVT

Data

W e w i l l n o w discuss the p r o b l e m of d e t e r m i n i n g effective or o p t i m a l d i a m e t e r s for use w i t h the H S E t h e o r y for r e a l fluids w h e n b o t h the f o r m of the i n t e r m o l e c u l a r p o t e n t i a l a n d its p a r a m e t e r s are u n k n o w n a c c u r a t e equations of state w h i c h represent the PVT

but

behavior over an

extensive range are a v a i l a b l e for the p u r e c o m p o n e n t s . For

n o n p o l a r fluids a n d s y m m e t r i c reference

fluids

for p o l a r s u b


stances w e w i l l assume that the u n k n o w n p o t e n t i a l f u n c t i o n for e a c h m a y be m o d e l e d w i t h a s y m m e t r i c a l p o t e n t i a l c o n s i s t i n g of a h a r d - s p h e r e r e p u l s i o n p o t e n t i a l for spheres of d i a m e t e r d p l u s a n excess w h i c h d e p e n d s o n (r/d)

a n d a s i n g l e energy p a r a m e t e r , e, i n the f o r m e f(r/d).

If the

fluids are n o n s p h e r i c a l , c is a n average w h i c h m a y d e p e n d o n t e m p e r a t u r e a n d to some extent o n density. I f the u n k n o w n true p o t e n t i a l i n v o l v e s a soft r e p u l s i o n , d m a y d e p e n d o n b o t h t e m p e r a t u r e a n d d e n s i t y . It is c o n v e n i e n t to relate this c a n d d to c r i t i c a l constants t h r o u g h n e w types of shape factors so that for a n y fluid i: =

&k Oik

(18)

(T )i c

and

b

0.

h a r d spheres of d i a m e t e r

fluid

The Z

as i n d i c a t e d i n E q u a t i o n 20.

The

t e r m is the C S e q u a t i o n

(pd?)

H S

for

d

it

U s i n g E q u a t i o n s 18 a n d 19, a n e q u i v a l e n t f o r m of E q u a t i o n 20 m a y be expressed i n terms of k n o w n c r i t i c a l constants.

Zi = Z™(pb q> V ) Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

k

ik

(21)

-\

Σ

ν et +

Σ Σ

f y V

cj

( 0 P

+aZ-(p,T)]=0.

(32)

I n this case the r i g h t side of E q u a t i o n 27 becomes o n l y Z(p), w h i c h is f u r n i s h e d d i r e c t l y b y t h e e q u a t i o n of state as i n E q u a t i o n 30. T h i s is c a l l e d the h i g h - t e m p e r a t u r e l i m i t a n d at some t e m p e r a t u r e c o n d i t i o n s of interest, e s p e c i a l l y at l o w densities, t h e o p t i m a l diameters a p p r o a c h i t closely. T h e s e diameters are a l w a y s s m a l l e r t h a n t h e h i g h - d e n s i t y l i m i t of B i e n k o w s k i a n d C h a o . T h e s e l i m i t s are v e r y n e a r l y u p p e r a n d l o w e r b o u n d s f o r t h e o p t i m a l diameters a l t h o u g h they d o n o t closely a p p r o a c h the u p p e r b o u n d at a n y c o n c e i v a b l e density of interest. A f e w cases at l o w d e n s i t y s h o w e d t h e optimal diameter very slightly below the h i g h temperature limit. T h e d i s c r e p a n c y is easily w i t h i n t h e e x p e r i m e n t a l u n c e r t a i n t y , h o w e v e r .


84

EQUATIONS OF STATE

Determination

of

Optimal

Diameters

from

Isochores

W e m u s t n o w c o n s i d e r a m o r e g e n e r a l m e t h o d for use w h e n these l i m i t i n g c o n d i t i o n s are n o t a p p l i c a b l e .

Determining Z

H S

( p d ) w i t h the 3

o p t i m a l d i a m e t e r at a g i v e n finite t e m p e r a t u r e a n d d e n s i t y is c a r r i e d out b y c o n s i d e r i n g a l i m i t e d t e m p e r a t u r e r a n g e a l o n g a n isochore given density.

T h i s t e m p e r a t u r e r a n g e is selected to locate t h e

t e m p e r a t u r e as n e a r to the center of the r a n g e as possible.

at the given

Isochores


extrapolate s m o o t h l y into t h e t w o - p h a s e r e g i o n a n d i n t h e l i q u i d p h a s e at l o w e r t e m p e r a t u r e s i n the r a n g e these extrapolations m a y e v e n p r o d u c e n e g a t i v e c o m p r e s s i b i l i t y factors w i t h o u t adverse effects o n the s o l u t i o n for the diameters.

P r o p e r t i e s a l o n g the isochore c a n be o b t a i n e d either

f r o m d i r e c t e x p e r i m e n t a l d a t a o r f r o m a n e q u a t i o n of state w h i c h r e p r e sents i s o c h o r i c b e h a v i o r w e l l .

If s u c h a n e q u a t i o n of state is u s e d , t h e

t e m p e r a t u r e r a n g e selected m u s t b e s h i f t e d to h i g h e r values i f necessary to i n s u r e t h a t (dP/dp)

T

as c a l c u l a t e d b y the e q u a t i o n is p o s i t i v e at e a c h

t e m p e r a t u r e v a l u e w i t h i n the range. T h e w i d t h of the r a n g e is selected i d e a l l y to d e t e r m i n e at a g i v e n t e m p e r a t u r e a n d d e n s i t y , Γ a n d p, the first a n d s e c o n d d e r i v a t i v e s of t h e dimensionless p r o p e r t y w i t h respect to inverse t e m p e r a t u r e s a n d to p r e d i c t t h e p r o p e r t y at e a c h t e m p e r a t u r e i n t h e r a n g e w i t h a n a c c u r a c y w i t h i n its e x p e r i m e n t a l error b y a q u a d r a t i c f u n c t i o n .

F o r example, if

the

c o m p r e s s i b i l i t y factor is b e i n g e v a l u a t e d , the values of ζ at ρ at e a c h p o i n t i n the r a n g e a b o u t Τ are fit b y least squares t o : ζ = α

0

(

Ρ

(33)

) + ? ψ - + ψ -

I n this w o r k a r a n g e w a s selected c o n s i s t i n g of e l e v e n t e m p e r a t u r e s , 10° F apart, i n c l u d i n g t h e g i v e n t e m p e r a t u r e .

I f (dP/dp)

T

is p o s i t i v e at e a c h

t e m p e r a t u r e , the r a n g e consists of five t e m p e r a t u r e s a b o v e a n d five t e m peratures b e l o w the g i v e n v a l u e ; o t h e r w i s e , t h e r a n g e is s h i f t e d u p w a r d so t h a t t h e l o w e s t t e m p e r a t u r e i n the r a n g e is n e a r e r to the g i v e n t e m p e r a ture.

If (dP/dp)

T

is n e g a t i v e at the g i v e n t e m p e r a t u r e t h e m e t h o d is

i n o p e r a b l e at the g i v e n c o n d i t i o n s .

V a r y i n g t h e w i d t h of t h e r a n g e d i d

not affect the results as l o n g as the c o n d i t i o n s d e s c r i b e d f o r i t w e r e met. A t e v e r y d e n s i t y s t u d i e d , E q u a t i o n 33 gave a n excellent r e p r o d u c t i o n of ζ values a l o n g the 1 0 0 ° F r a n g e as defined here. T h e v a l u e of ψ for the q u a d r a t i c fit of the isochore i n this l i m i t e d r a n g e is defined as ψ to i n d i c a t e that i t contains t w o i n v e r s e t e m p e r a t u r e 2

terms. C o n s e q u e n t l y , f r o m E q u a t i o n 3 3 :

* * = - ( ^ + ψ )


(34)

4.

CHANG ET AL.

Molecular

Diameters

for Fluid

85

Mixtures

W e c a n find the t e m p e r a t u r e d e p e n d e n c e t h a t is n o t a c c o u n t e d f o r b y t h e q u a d r a t i c fit b y c o m p a r i n g ψ w i t h 2

the m a x i m u m p o s s i b l e

range

a n d assuming that ψ

describes

Λ

of temperatures.

T h e difference

is d e

fined as

(ψ* - φ )

δ =

(35)

2


since t h e H S E p s e u d o c r i t i c a l values f o r the excess over t h e h a r d - s p h e r e b e h a v i o r w e r e d e r i v e d b y c o n s i d e r i n g o n l y terms i n ( 1 / T ) a n d ( 1 / T ) i n its e x p a n s i o n .

2

F u r t h e r m o r e , these terms i n v o l v e d o n l y p a i r w i s e c o n

t r i b u t i o n s f r o m t h e a t t r a c t i v e p o r t i o n of 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 . C o n s e q u e n t l y , at c o n d i t i o n s w h e r e t h e coefficients a\ a n d a i n E q u a t i o n 2

34 c o n t a i n p r e d o m i n a t l y a t t r a c t i v e c o n t r i b u t i o n s of this t y p e , the aZ~(p, T) t e r m i n E q u a t i o n 2 7 contains n o t r i p l e t p o t e n t i a l effects of o r d e r C o n s e q u e n t l y , i t is e n t i r e l y i n c l u d e d a m o n g (1/T)

3

i n t h e ψ expression.

(1/T) . 2

t h e interactions of

order

I f a n e x p a n s i o n of Z (p, T ) i n p o w e r s o f +

χ

1 / T gives coefficients of ( 1 / T ) a n d ( 1 / T )

2

w h i c h are n e g l i g i b l e i n c o m

p a r i s o n w i t h t h e a t t r a c t i v e c o n t r i b u t i o n s to these terms, t h e i r presence w i l l n o t a p p r e c i a b l y affect t h e Γ p s e u d o a t t r a c t i o n p a r a m e t e r p r e d i c t e d b y t h e H S E theory.

C o n s e q u e n t l y , t h e o n l y soft-sphere

contributions

w h i c h need to be i n c l u d e d i n the hard-sphere term b y adjusting the d i a m e t e r are those i n terms of o r d e r ( 1 / T )

3

and higher i n the

term.

I n this case, t h e difference b e t w e e n ψ» a n d ψ represents a l l terms o f 2

h i g h e r o r d e r t h a n 1 / T w h i c h n e e d to b e c o m b i n e d i n t o t h e h a r d - s p h e r e 3

result.

T h i s difference

terms of o r d e r

(1/T)

defines t h e δ p a r a m e t e r i n E q u a t i o n 35. I f a l l 3

were

zero, t h e best Z (pd! ) v a l u e at these HS

3

conditions w o u l d be the a leading term i n E q u a t i o n 33 for the quadratic Q

fit. W i t h corrections for the h i g h e r o r d e r terms, t h e best Z ( p d ) v a l u e is HS

Z

H S

3

(,d ) — ( a - S ) 8

(36)

0

T h e n e g a t i v e s i g n is t h e result of t h e c o n t r i b u t i o n to ζ as g i v e n b y —φ as d e f i n e d i n E q u a t i o n s 26 a n d 34. E q u a t i o n 36 m a y b e r e g a r d e d as t h e best a p p r o x i m a t i o n to E q u a t i o n 2 7 u n d e r these c o n d i t i o n s . T h e l i m i t s of v a l i d i t y of E q u a t i o n 36 are i n d i c a t e d b y t h e m a g n i t u d e a n d s i g n of t h e a (p) t e r m i n E q u a t i o n 34. T h e assumptions l e a d i n g t o 2

E q u a t i o n 36 b e c o m e i n v a l i d at h i g h densities. b e l o w pV ~ 0.6, δ ^ c

0, a n d a ~ 0

A t the lowest

densities

Z(p), t h e h i g h - t e m p e r a t u r e l i m i t of

the e q u a t i o n of state i n E q u a t i o n 30. T h e a (p) i n E q u a t i o n 34 is s m a l l 2

a n d negative. A s densities increase a b o v e pV =

0.6 t h e absolute v a l u e

of a begins to increase w h i l e i t is s t i l l n e g a t i v e .

P r e s u m a b l y this means

c

2

an a t t r a c t i o n c o n t r i b u t i o n is b e i n g r e p r e s e n t e d .

P o s i t i v e c o n t r i b u t i o n s of


86

EQUATIONS OF

STATE

t h e soft r e p u l s i o n are a p p a r e n t l y s t i l l n e g l i g i b l e a n d E q u a t i o n 36, w h i c h r e q u i r e s this, is s t i l l v a l i d . T h i s causes the v a l u e of δ i n E q u a t i o n 35 to be n e g a t i v e a n d Z As

i n E q u a t i o n 36 increases.

H S

d e n s i t y increases

decrease.

f u r t h e r t h e absolute

v a l u e of

a

2

begins

A l t h o u g h i t s t i l l r e m a i n s n e g a t i v e at this p o i n t , the

to

absolute

v a l u e of δ is a m a x i m u m . T h e a t e r m , w h i c h is b e c o m i n g less n e g a t i v e 2

i n this w a y , is c o n s i d e r e d to b e a l t e r e d b y the onset of the p o s i t i v e c o n t r i b u t i o n s of soft r e p u l s i o n w h i c h at these densities begins to affect the coefficient of ( 1 / T ) . Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

2

T h i s m a x i m u m s i n |δ| a n d \a \ o c c u r at a r e d u c e d 2

d e n s i t y of a b o u t 1.6 T h e r e d u c e d d e n s i t y of 1.6 is c o n s i d e r e d to be the u p p e r l i m i t of the v a l i d i t y of E q u a t i o n 36.

A t densities h i g h e r t h a n this |δ| a n d \a \ 2

decrease r a p i d l y a n d a itself e v e n t u a l l y becomes p o s i t i v e , i n t e r p r e t e d as 2

its d o m i n a t i o n b y p o s i t i v e soft-repulsion effects.

Diameters from Equation

36 g i v e p o o r results i n this r e g i o n . T h e r e is no w a y t h a t these soft effects c a n be separated f r o m a t t r a c t i o n effects a n d the o p t i m a l d i a m e t e r c a n not be c a l c u l a t e d . T h e diameters c a n b e p r e d i c t e d o n c e m o r e at v e r y h i g h densities where a

2

has b e c o m e v e r y l a r g e a n d p o s i t i v e , i n d i c a t i n g d o m i n a n c e

the second-order t e r m b y s o f t - r e p u l s i o n effects.

three b o d y c o n t r i b u t i o n s to this t e r m are no l o n g e r

negligible.

o p t i m a l diameters t h e n are o b t a i n e d b y p l a c i n g a l l of the (1/T)

2

of

It is also v e r y l i k e l y that The

second-order

t e r m i n the h a r d - s p h e r e e q u a t i o n since it is n o w r e p u l s i o n d o m i

n a t e d . T h e t e m p e r a t u r e r a n g e u s e d for the q u a d r a t i c fit is r e d u c e d f r o m 100° to 5 0 ° F w i t h the g i v e n t e m p e r a t u r e near the center of this shorter range. T h e objective is n o w to o b t a i n a n accurate r e p r e s e n t a t i o n of e a c h ζ v a l u e i n the r a n g e b y a least-squares fit of the l i n e a r r e l a t i o n :

z =

+

a/

(37)

^f

T h e ai t e r m is a l w a y s n e g a t i v e at r e a d i l y accessible densities. S i n c e n o n e of the n e g a t i v e 1 / T d e p e n d e n c e s h o u l d a p p e a r i n t h e h a r d - s p h e r e e q u a t i o n a n d E q u a t i o n 37 represents the ζ values a c c u r a t e l y i n the shorter r a n g e , the best Z

H S

result i s : Z

H S

(38)

(pd ) = O o ' 3

E q u a t i o n 38 t h e n is s o l v e d for t h e d i a m e t e r . T h i s l i n e a r fit m e t h o d gives excellent results for the o p t i m a l d i a m e t e r at r e d u c e d densities of

about

2.4 a n d h i g h e r . T h e r e d u c e d d e n s i t y r e g i o n b e t w e e n 1.6 a n d 2.4 is thus a n i n d e t e r m i n a n t r e g i o n . A s a first a p p r o x i m a t i o n , t h e best Z

H S

( p d ) values i n this 3

r e g i o n w e r e a s s u m e d to be g i v e n b y a spline-fit i n t e r p o l a t i o n b e t w e e n


4.

CHANG E T AL.

p o i n t s at p

R

Molecular

Diameters

> 2.4 a n d those at p

R

for Fluid

87

Mixtures

< 1.6. F o r l i q u i d s at l o w temperatures

t h e i n d e t e r m i n a n t r e g i o n is l e n g t h e n e d b e c a u s e t h e l i q u i d n o l o n g e r c a n b e e x t r a p o l a t e d to r e d u c e d densities n e a r 1.6 because of the s t a b i l i t y l i m i t . L o w d e n s i t y values at p

R

< 0.6 are s t i l l o b t a i n a b l e b y e q u a t i n g Z

H S

(pd ) 3

to t h e h i g h - t e m p e r a t u r e l i m i t of t h e e q u a t i o n of state. T h e b e h a v i o r of t h e q u a d r a t i c a n d l i n e a r fit m e t h o d s is s h o w n i n F i g u r e 1.

T h e i n t e r p r e t a t i o n of t h e a

coefficient b e h a v i o r i n terms of

2

s o f t - r e p u l s i o n effects i n the q u a d r a t i c fit a (p)

assumes t h a t t h e d a t a fitted


z

6.0|

1

1

/>v Figure 1.

Computation

of Z

HS

-j

Γ

c

from BWR-S

equation (CH

k

at T / T = c


1.0)

88

EQUATIONS O F STATE

to t h e isochores gives a v a l i d s e c o n d d e r i v a t i v e w i t h respect t o 1 / Γ . T h i s m a y n o t b e t h e case f o r values g e n e r a t e d

by the B W R - S

R e g a r d l e s s of this, t h e q u a d r a t i c fit m e t h o d b e l o w p = R

fit m e t h o d a t p

R

equation.

1.6 a n d t h e l i n e a r

> 2.4 g i v e excellent results f o r t h e o p t i m a l d i a m e t e r s .

This was checked

b y increasing a n d decreasing Z

H S

(pd ) 3

about the

p r e d i c t e d v a l u e a n d n o t i n g t h e effect o n t h e p r e d i c t e d m i x t u r e p r o p e r t i e s . T h e weakest prediction for Z

H S

( p d ) is i n t h e spline-fit r e g i o n . A t y p i c a l 3


result o f this test o n t h e p r e d i c t e d v a l u e is s h o w n i n T a b l e II.

Table II. Effect of Z on Calculated ζ i n the Incalculable Region for 2 H S

H S e

HS

%

Z

5.31 1.77

2.73 3.39 (α -8) 0

Error

=3.48-> 0.29 0 -0.99 -2.23

3.68" 3.74 ' 3.94 4.21 50% C H ; 50% C H . Τ = 160°F; Ρ = 5000 psia. * B y spline fit between Z „ at p > 2.4 and Z at p < 1.6. Predicted by interpolation to 0% error.

α

4

3

8

s

H S

R

R

c

Application

of the

Method

A l t h o u g h o n l y c o m p r e s s i b i l i t y factor c a l c u l a t i o n s are u s e d example

i n t h e e x p l a n a t i o n of t h e m e t h o d , o t h e r p r o p e r t i e s

as a n can be

p r e d i c t e d e q u a l l y w e l l . B e c a u s e of t h e t e m p e r a t u r e a n d d e n s i t y d e p e n d ence of t h e diameters a n d shape factors n e e d e d to relate t h e m t o c r i t i c a l constants i t is best to d e t e r m i n e separate values of t h e m f o r e a c h c o m ponent.

T h r e e basic d i m e n s i o n l e s s p r o p e r t i e s

should be determined.

T h e s e are t h e ones best s u i t e d to t h e use of t h e H S E m e t h o d w i t h a n e q u a t i o n of state i n terms of t e m p e r a t u r e a n d d e n s i t y .

These are the

c o m p r e s s i b i l i t y factor, z; t h e i n t e r n a l e n e r g y d e v i a t i o n ( U * —

U)/RT;

a n d a dimensionless f u g a c i t y r a t i o , l n ( f / p R T ) . A l l o t h e r d e s i r e d p r o p e r ties c a n b e o b t a i n e d f r o m t h e m . T h e l n ( f / p R T ) similarly.

a n d ζ are calculated

T h e c o m p u t a t i o n scheme is o u t l i n e d as s h o w n i n T a b l e III.

Table III. Shape Factors and Diameters for Nonconformal Fluids with U n k n o w n Potentials Example for Compressibility Factor 1. Calculate Z™{pr°)

di and d =Z

i

H S

from:

r

(p °) i


4.

Molecular

CHANG E T A L .

Diameters

Table III.

for Fluid

89

Mixtures

Continued

o b t a i n e d f r o m p u r e - c o m p o n e n t equations of state b y m e t h o d r e p o r t e d i n t h i s c h a p t e r (22). C o n d i t i o n s a t w h i c h shape factors are e v a l u a t e d (28) : p VCP m

Pr

(V ) c

p

m


and

,

ο pi° =

pmVCP (V )^i c

r

= mixture density S o l v e for φ a t t r i a l V C P . έ

T o force c o n f o r m a l i t y betwen C o m p o n e n t i a n d Reference r: 1 + V+ V

2

Î

S o l v e for η S o l v e f o r d: 1/3

1/3

and

di =

Calculate

0 :

d

r

Np ° r

2.

ir

to force c o n f o r m a l i t y between C o m p o n e n t i a n d Reference r: Ζ,ίρΛΓ, )

Z (pr°,

0

T °)

r

Î

τ

f r o m e q u a t i o n of state o r k n o w n v a l u e s f o r pure p o l a r fluid

from equation of state for the pure nonpolar reference fluid

3

d?

I« ( di ) 0

Pl

3

Z

+

r

]

E X A

î asymmetric excess calculated below

(kT-ψ}

Ti° = t e m p e r a t u r e a t w h i c h ^ m u s t be e v a l u a t e d i n a m i x t u r e (28). U s i n g p r e v i o u s l y c a l c u l a t e d d solve for Ti° : h

Calculate 0 at trial T C P value from T ° : ir

T

C

p

(TJidir v*c/»"*r

{

a —n d

T *° r

r

=

Tu

(T )r e

N o w c a l c u l a t e new T C P a n d V C P a n d repeat u n t i l t h e y a r e constant.


90

EQUATIONS O F STATE

F o r (U* — U)/RT

n o h a r d - s p h e r e p r o p e r t y c a l c u l a t i o n s are m a d e a n d

the a t e r m of t h e q u a d r a t i c fit a l o n g t h e c o m p r e s s i b i l i t y f a c t o r isochores 0

c a n be e q u a t e d to Z

H S

(pd ). 3

T h i s is t h e n s o l v e d for t h e d i a m e t e r u s e d

i n the p s e u d o p a r a m e t e r c o m p u t a t i o n s . C a l c u l a t e d results f o r m e t h a n e a n d p r o p a n e o b t a i n e d b y H w u (22) w i t h a n ethane reference are p r e s e n t e d i n T a b l e I V . T h e B W R - S e q u a


t i o n is u s e d f o r a l l p u r e c o m p o n e n t s . Table I V . P

T h e agreement is g e n e r a l l y excellent.

Compressibility Factors—Density"

Exp"

Cale.

HSE

(psia)

(lb-mol/ff?)

(lb-mol/ft )

200 400 800 1000 3000 5000 7000 9000

0.0322 0.0692 0.1648 0.2252 0.6760 0.7955 0.8636 0.9091

0.0321 0.0691 0.1666 0.2333 0.6778 0.7955 0.8631 0.9101

Cale.

Exp"

HSE

ζ

ζ

0.9351 0.8692 0.7313 0.6670 0.6676 0.9451 1.2189 1.4886

0.9369 0.8696 0.7228 0.6448 0.6645 0.9446 1.2191 1.4866

3

*Sage and Lacey {23).

T h e p o o r e r results at 1000 p s i a a p p a r e n t l y are d u e to a weakness i n t h e B W R - S reference e q u a t i o n at this p o i n t . I t c o i n c i d e s w i t h t h e m i n i m u m of t h e c u r v e of ζ vs. P .

_

T a b l e V shows the c o m p u t a t i o n s b y H w u f o r Η * — Η i n a m e t h a n e p r o p a n e m i x t u r e i n c o m p a r i s o n w i t h t h e p r e d i c t i o n s of M o l l e r u p 18) u s i n g t h e V D W o n e - f l u i d t h e o r y w i t h shape factors.

(16,17,

T h e improve

m e n t of t h e H S E m e t h o d is v e r y slight. T h e t h e o r e t i c a l advantages o f the H S E m e t h o d f o r e n t h a l p y c a l c u l a t i o n s m a y b e offset here b y u s i n g a g e n e r a l l y p o o r e r reference e q u a t i o n of state t h a n that u s e d b y M o l l e r u p . T a b l e V I presents p r e l i m i n a r y c a l c u l a t i o n s b y C h a n g (24) p o l a r - n o n p o l a r m i x t u r e . T h e h i g h e s t pressures m a y b e i n v a l i d

for a because

they w e r e m a d e before t h e m e t h o d f o r e v a l u a t i n g t h e o p t i m a l d i a m e t e r s was developed.

T h e s e c o m p u t a t i o n s use t h e h i g h - t e m p e r a t u r e l i m i t of

the B W R - S e q u a t i o n f o r Z

H S

( p d ) to o b t a i n t h e d i a m e t e r . I t w a s h o p e d 3

that c o m p a r i s o n w i t h t h e B W R - S e q u a t i o n w o u l d s h o w a m o r e d i s t i n c t a d v a n t a g e of t h e t h e o r e t i c a l c o m p o s i t i o n d e p e n d e n c e of the H S E m e t h o d . I n fact, t h e t w o m e t h o d s

g i v e a b o u t t h e same results.

Probably no

c o n c l u s i o n a b o u t this c a n b e d r a w n f r o m t h e c o m p a r i s o n because t h e constants d e t e r m i n e d b y H o p k e a n d L i n (25)

for t h e B W R - S e q u a t i o n

w e r e o b t a i n e d b y fitting t h e e q u a t i o n to this b i n a r y . T h e results are g i v e n i n the c o l u m n headed B W R S E .

T h e test of t h e i m p r o v e d c o m p o s i t i o n


4.

CHANG E T AL.

Molecular

Table V .


τ (Of)

e 5 e

Diameters

for Fluid

E n t h a l p y Calculations of Ρ

Expt."

91

Mixtures (H — H*)

M-R

HSE

c

( psia)

(Btu/lb)

(Btu/lb)

(Btu/lb)

100

250 1250 1750

-20.4 -142.4 -143.6

-18.5 -142.4 -143.1

-19.3 -144.0 -144.4

50

750 1250 1750

-157.1 -156.1 -155.5

-154.9 -154.8 -154.3

-156.8 -156.3 -156.0

0

750 1250 1750

-169.3 -168.5 -167.1

-167.1 -166.2 -165.1

-168.3 -167.4 -166.5

-50

750 1250 1750

-180.7 -179.3 -178.0

-178.6 -177.2 -175.8

-178.4 -177.1 -176.0

23.4% C H , 76.6% C H . Yesavage-Powers (24). Mollerup-Rowlinson (16,17,18). 4

3

8

Table V I . Calculated T h e r m o d y n a m i c Properties for Methane—Carbon D i o x i d e M i x t u r e at 100°F P

Ζ

(psia)

Expt. (23)

BWR-SE Xj

200 600 800 1000 2000 3000 4000 5000 7000 9000

0.9512 0.8347 0.7830 0.7160 0.4438 0.4958 0.5921 0.6947 0.8982 1.1012

β

0.9606 0.8584 0.8360 0.7921 0.6100 0.6055 0.7755

HSE =

0.9491 0.8748 0.8305 0.7855 0.6056 0.6031 0.7678

=

(A -

A*;

(A -

A*

RT

RT

BWR-SE

HSE

-0.0507 -0.1670 -0.2357 -0.3150 -0.8184 -0.9831 -1.0336 -1.0561 -1.0722 -1.0724

-0.0503 -0.1653 -0.2331 -0.3116 -0.8080 -0.9613 -1.0153 -1.0389 -1.0556 -1.0557

-0.0469 -0.1309 -0.1807 -0.2308 -0.5268 -0.6928 -0.7838

-0.0408 -0.1303 -0.1791 -0.2336 -0.5302 -0.6870 -0.7747

0.2035 ° 0.9500 0.8443 0.7836 0.7165 0.4518 0.5033 0.5992 0.6999 0.9087 1.1058

0.9489 0.8400 0.7787 0.7121 0.4458 0.4942 0.5910 0.6941 0.9005 1.1025 X/

200 600 800 1000 2000 3000 5000

Ζ

Ζ

04055 0.9595 0.8771 0.8336 0.7873 0.6067 0.6067 0.7748

The χι = mole fraction methane.


92

EQUATIONS OF STATE

d e p e n d e n c e w i l l h a v e to a w a i t c a l c u l a t i o n s w h e n a t h i r d c o m p o n e n t

is

a d d e d , m a k i n g a m i x t u r e not i n c l u d e d i n the fitting of the constants. It is e n c o u r a g i n g that a t h e o r e t i c a l l y b a s e d m e t h o d for a m i x t u r e of a n o n p o l a r s y m m e t r i c p o t e n t i a l fluid, m e t h a n e , a n d a s t r o n g q u a d r a p o l e

fluid,

C0 , 2

p r o d u c e s s u c h g o o d results. T h e n o n p o l a r reference fluid w a s m e t h a n e . Conclusions


I n s u m m a r y the results s h o w that i t is i n d e e d p o s s i b l e to e x t e n d the H S E m e t h o d successfully to m i x t u r e s c o n t a i n i n g p o l a r m o l e c u l e s .

Meth

ods h a v e b e e n d e v e l o p e d to o b t a i n effective diameters a n d shape factors w h i c h are o p t i m a l for use w i t h the H S E theory. A l t h o u g h the d e t e r m i n a t i o n of d i a m e t e r s for fluids w i t h u n k n o w n p o t e n t i a l f u n c t i o n s w i t h these m e t h o d s is not possible at a l l densities, e n o u g h c a l c u l a t i o n s c a n b e m a d e to a l l o w a c o r r e l a t i o n b y fitting the results to the V W equations f o r the o p t i m a l d i a m e t e r w i t h the p e r t u r b a t i o n theory.

T h e success of t h e V W

d i a m e t e r s for the H S E t h e o r y w a s c o n f i r m e d . T h e results o b t a i n e d encourage f u t u r e s t u d y a n d illustrate the p o w e r of c o n f o r m a i s o l u t i o n methods. It is reasonable to expect t h a t the excellent a c c u r a c y o b t a i n e d b y M o l l e r u p (16,17,18)

w i t h the V D W

one-fluid

t h e o r y for n a t u r a l gas m i x t u r e s c a n be e x p e c t e d w i t h t h e H S E t h e o r y for p o l a r m i x t u r e s a n d other systems i n w h i c h there are large d i s s i m i l a r i ties b e t w e e n the c o m p o n e n t s a n d the reference

fluid.

Nomenclature English a

u

Letters a = coefficients of 1 / T a n d ( 1 / T ) e q u a t i o n of state 2

2

i n a n e x p a n s i o n of a n

a = u n i v e r s a l p r o p o r t i o n a l i t y constant b e t w e e n c a n d T fluids c o n f o r m a i w i t h fluid k fc

c

for a l l

A = dimensionless r e s i d u a l H e l m h o l t z free e n e r g y f u n c t i o n A = m o l a l H e l m h o l t z free e n e r g y at a g i v e n Τ a n d Ρ A * = m o l a l H e l m h o l t z free e n e r g y at Τ a n d Ρ i f the fluid o b e y e d t h e p e r f e c t gas l a w b

fc

= u n i v e r s a l p r o p o r t i o n a l i t y constant b e t w e e n σ a n d V fluids c o n f o r m a i w i t h fluid k 3

c

for a l l

b , B = constants i n the B W R S t a r l i n g e q u a t i o n of state 0

0

d = effective h a r d - s p h e r e d i a m e t e r d = p s e u d o p a r a m e t e r u s e d to f o r m t h e r e d u c e d d e n s i t y i n p r e d i c t i n g the m o l e c u l a r a t t r a c t i o n c o n t r i b u t i o n t o a m i x t u r e property 3

f =

fugacity


4.

CHANG ET AL.

g

Molecular

Diameters

for Fluid

93

Mixtures

= 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 for a n if p a i r i n a m i x t u r e w i t h other constituents gREF = 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 for a p a i r i n a p u r e reference fluid i ;

M

gHs _

r a

d i i d i s t r i b u t i o n f u n c t i o n for a p a i r of h a r d spheres i n a a

hard-sphere

fluid

H = molal enthalpy k = B o l t z m a n n ' s constant Downloaded by NORTH CAROLINA STATE UNIV on December 5, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch004

Ν = Avogadro's number Ρ =

pressure

Q =

quadrapole moment

r =

s e p a r a t i o n distance b e t w e e n a p a i r of m o l e c u l a r centers

R = gas constant Τ = T

c

temperature

= critical temperature

u = intermolecular pair potential T7 = m o l a l i n t e r n a l e n e r g y V = V

c

volume

= critical volume

x = m o l e f r a c t i o n of constituent i i n a m i x t u r e {

X = any dimensionless t h e r m o d y n a m i c p r o p e r t y of a

fluid

ζ = c o m p r e s s i b i l i t y factor Ζ = c o m p r e s s i b i l i t y factor of a h a r d - s p h e r e Z

+

fluid

= c o n t r i b u t i o n of soft r e p u l s i o n to a n e q u a t i o n of state expressed i n terms of the c o m p r e s s i b i l i t y f a c t o r

Z " = c o n t r i b u t i o n of i n t e r m o l e c u l a r a t t r a c t i o n to a n e q u a t i o n of state expressed i n terms of the c o m p r e s s i b i l i t y f a c t o r Greek

Letters a = f r a c t i o n of the a t t r a c t i v e c o n t r i b u t i o n to e x p a n d e d equations of state d u e to terms of order ( 1 / T ) a n d h i g h e r a n d threeb o d y interactions of order ( 1 / T ) a n d h i g h e r 3

2

/? =

l/kT

δ = p a r a m e t e r u s e d i n o b t a i n i n g the effective h a r d - s p h e r e d i a m e t e r f r o m isochores 8

V W

= p a r a m e t e r i n the V e r l e i t - W e i s e q u a t i o n for the effective hard-sphere diameter

c = L e n n a r d - J o n e s p a r a m e t e r f o r the a l g e b r a i c m i n i m u m i n the pair potential 7 = p s e u d o p a r a m e t e r u s e d to f o r m t h e r e d u c e d t e m p e r a t u r e i n p r e d i c t i n g the m o l e c u l a r a t t r a c t i o n c o n t r i b u t i o n to m i x t u r e properties. μ, = d i p o l e m o m e n t π = 3.141516


94

EQUATIONS O F STATE

η = dimensionless d e n s i t y i n t h e C a r n a h a n - S t a r l i n g e q u a t i o n φ · = shape f a c t o r coefficient of V to m a k e f l u i d i c o n f o r m a i w i t h fluid / ρ = density ί;

P r

c i

= reduced density, Ρ V

c

σ = L e n n a r d - J o n e s p a r a m e t e r f o r t h e finite s e p a r a t i o n distance at w h i c h the i n t e r m o l e c u l a r p o t e n t i a l is z e r o $ij = shape factor coefficient of T to m a k e fluid i c o n f o r m a i w i t h a reference fluid /. ( F o r a c o m m o n reference t h e s e c o n d s u b s c r i p t is sometimes o m i t t e d . ) $ij = coefficient o f t h e B e r t h e l o t c o m b i n i n g r u l e f o r u n l i k e ij p a i r i n t e r a c t i o n potentials φ = t e m p e r a t u r e d e p e n d e n t p o r t i o n of a d i m e n s i o n l e s s e q u a t i o n of state


ci

Acknowledgment T h e t h e o r e t i c a l studies c o n c e r n i n g t h e test of t h e H S E m e t h o d w i t h m i x t u r e s of L J m o l e c u l e s a n d t h e w o r k i n v o l v e d i n e x t e n d i n g t h e m e t h o d to p o l a r m o l e c u l e s w e r e s u p p o r t e d b y t h e N a t i o n a l Science F o u n d a t i o n . T h e w o r k o n evaluating molecular diameters from P V T data a n d from pure-component

equations of state w a s s u p p o r t e d b y t h e G a s R e s e a r c h

Institute.

Literature Cited 1. Mansoori, G. Α.; Carnahan, N . F.; Starling, Κ. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. 2. Bienkowski, P. R.; Deneholz, H. S.; Chao, K. C. AIChE J. 1973, 19, 167. 3. Mansoori, G. Α.; Leland, T. W. J. Chem.Soc.,Faraday Trans. II 1972, 68, 320. 4. Carnahan, N . F.; Starling, Κ. E. J. Chem. Phys. 1969, 51, 635. 5. Verlet, L.; Weis, J. J. Phys. Rev. A 1972, 5, 939. 6. Weeks, J. D.; Chandler, D.; Andersen, H . C. J. Chem. Phys. 1971, 54, 5237. 7. Barker, J. Α.; Henderson, D. J. Chem. Phys. 1967, 47, 4714. 8. Singer, J. V. L.; Singer, K. Mol. Phys. 1972, 24, 357. 9. Leland, T. W.; Rowlinson, J. S.; Sather, G. A. Trans. Faraday Soc. 1968, 64, 1447. 10. Leland, T. W.; Rowlinson, J. S.; Watson, I. D. Trans. Faraday Soc. 1969, 65, 2034. 11. Grundke, E. W.; Henderson, D.; Barker, J. Α.; Leonard, P. J. Mol Phys. 1973, 25, 883. 12. Pople, J. A. Proc. R. Soc. London, Ser. A 1954, 221, 508. 13. Mo, K. C.; Gubbins, Κ. E. J. Chem. Phys. 1975, 63, 1490. 14. Stell, G.; Rasaiah, J.; Narang, H. Mol. Phys. 1972, 23, 393. 15. Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969, 24, 1565. 16. Mollerup, J.; Rowlinson, J. S. Chem. Eng. Sci. 1974, 29, 1373. 17. Mollerup, J. Adv. Cryog. Eng. 1975, 20, 172. 18. Ibid., 1978, 23, 550.



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19. Hsu, R. "Thermodynamic Properties of Mixtures by the Hard Sphere Expansion Theory," M.S. Thesis, Rice University, 1977. 20. Starling, Κ. E. Hydrocarbon Process. 1972, 50. 21. Ibid., 1973, 51. 22. Hwu, F. S. S. M.S. Thesis, Rice University, 1978. 23. Sage, B. H.; Lacey, W. H. "A.P.I. Project 37"; American Petroleum Insti tute: New York, 1950. 24. Chang, J. I. C. "Improvement of the Hard Sphere Expansion Conformal Solution Theory," M.S. Thesis, Rice University, 1978. 25. Lin, C. J.; Hopke, S. W. "Application of the BWRS Equation to Natural Gas Systems," presented at National A.I.Ch.E. Meeting, Tulsa, Okla homa, March 1974. 26. Reamer, H . H.; Olds, R. H.; Sage, Β. H.; Lacey, W. N . Ind. Eng. Chem. 1944, 88. 27. Bienkowski, P. R.; Chao, K. C. J. Chem. Phys. 1975, 63, 4217. 28. Leach, J. W.; Chappelear, P. S.; Leland, T. W. Proc. Am. Pet. Inst., Sec. 3 1966, 46, 223. 29. Leach, J. W.; Chappelear, P. S.; Leland, T. W. AIChE J. 1968, 14, 568. RECEIVED September 21, 1978.


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