Equations of State in Engineering and Research - ACS Publications

7 Three-Parameter, Corresponding-States Conformal Solution Mixing Rules for Mixtures

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of Heavy and Light Hydrocarbons T. J. LEE, L. L. LEE, and Κ. E. STARLING School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019

The conformal solution method is used as the basis for developing mixing rules for the characteristic parameters appearing in a three-parameter, corresponding-states corre lation of thermodynamic properties. A logical extension of the van der Waals one-fluid mixing rules from two to three parameters is shown to yield poor predictions of vaporliquid equilibrium for mixtures of paraffin hydrocarbons with highly dissimilar molecular sizes. Therefore, semiempirical rules were developed with improved predictive capability. The average absolute deviations of predicted methane K-values from experimental binary mixture data for methane with heavier normal paraffin hydrocarbons ranging from ethane through normal decane were 11.0% and 4.6%, respectively, using the modified van der Waals one-fluid mixing rules and the semiempirical mixing rules.

U

sing conformai solution theory models for the prediction of mixture thermodynamic

industrial

behavior

calculations.

is b e c o m i n g

T h e attractiveness

increasingly

popular

of t h e c o n f o r m a i

a p p r o a c h stems l a r g e l y f r o m t h e fact that i t i s faster

for

solution

computationally

t h a n p u r e l y t h e o r e t i c a l methods a n d y e t has a sufficiently g o o d basis i n t h e o r y to a l l o w extension to c o m p l e x m o l e c u l a r i n t e r a c t i o n s (e.g., m u l t i p o l e , d i s p e r s i o n , a n d steric effects), w h i c h w o u l d b e difficult u s i n g p u r e l y e m p i r i c a l methods. T h e f o r m u l a t i o n of c o n f o r m a i s o l u t i o n t h e o r y w h i c h has r e c e i v e d t h e w i d e s t use to date is t h e s o - c a l l e d V D W o n e - f l u i d t h e o r y ( I ) . t h e V D W o n e - f l u i d theory a p p l i e s t o m i x t u r e s of s i m i l a r size

Strictly, molecules

0-8412-0500-0/79/33-182-125$05.00/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.

126

EQUATIONS O F S T A T E

f o r w h i c h a l l p a i r p o t e n t i a l s c a n b e expressed Φ^α/σα).

i n the f o r m

=

c

Unfortunately, for many industrial mixtures molecular

y

size

differences c a n b e l a r g e a n d o r i e n t a t i o n effects m a k e i m p o r t a n t c o n t r i b u tions to the p a i r p o t e n t i a l s . T h u s , aside f r o m the a p p r o x i m a t i o n s i n h e r e n t to c o n f o r m a i s o l u t i o n t h e o r y , factors w h i c h a d v e r s e l y affect t h e a c c u r a c y of the V D W o n e - f l u i d t h e o r y f o r the c o m p l e x m o l e c u l a r systems e n c o u n t e r e d i n d u s t r i a l l y i n c l u d e u s i n g the t w o p a r a m e t e r

(c^

and σ ) {ί

pair

p o t e n t i a l a n d r e q u i r i n g of s i m i l a r m o l e c u l a r sizes for the m i x t u r e c o m Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007

ponents. Efforts are i n progress at the U n i v e r s i t y of O k l a h o m a to d e v e l o p a m u l t i p a r a m e t e r , corresponding-states f r a m e w o r k for c o r r e l a t i o n of t h e r m o d y n a m i c properties, taking into account the various orientation contribu tions to p a i r i n t e r a c t i o n s (e.g., d i p o l e - d i p o l e ,

quadrupole-quadrupole,

d i p o l e - q u a d r u p o l e , a n d h i g h e r m u l t i p o l e effects as w e l l as d i s p e r s i o n a n d steric effects).

P r e l i m i n a r y research (2)

i n this d i r e c t i o n has i n v o l v e d

l u m p i n g the c o l l e c t i v e effects of o r i e n t a t i o n c o n t r i b u t i o n s i n t o a s i n g l e t e r m i n the p a i r p o t e n t i a l a n d the r e s u l t a n t expressions for t h e t h e r m o dynamic

properties

from

the

Pople

perturbation theory

(3).

This

a p p r o a c h leads to the t h r e e - p a r a m e t e r , corresponding-states

correlation

f r a m e w o r k r e p o r t e d i n recent w o r k

The

(2)

and used herein.

three

c h a r a c t e r i z a t i o n p a r a m e t e r s i n this c o r r e l a t i o n f r a m e w o r k are t h e c h a r a c t e r i s t i c m o l e c u l a r s i z e / s e p a r a t i o n p a r a m e t e r , σ, the c h a r a c t e r i s t i c m o l e c u l a r energy p a r a m e t e r , c, a n d the c h a r a c t e r i s t i c o r i e n t a t i o n p a r a m e t e r , γ. W i t h i n this t h r e e - p a r a m e t e r , corresponding-states f r a m e w o r k i t is p o s s i b l e to d e r i v e , a l o n g the lines of the m e t h o d u s e d b y S m i t h ( 4 ) , a t h r e e p a r a m e t e r , c o n f o r m a i s o l u t i o n m o d e l , w h i c h is p r e s e n t e d i n S e c t i o n 2. I n the d e r i v a t i o n of the t h r e e - p a r a m e t e r c o n f o r m a i s o l u t i o n m o d e l , c e r t a i n p a r a m e t e r s ( exponents ) i n t h e m i x i n g rules f o r the t h r e e c h a r a c t e r i z a t i o n p a r a m e t e r s are a r b i t r a r y . U s i n g the V D W o n e - f l u i d m i x i n g rules f o r the energy a n d separation parameters, along w i t h a m i x i n g rule for

the

o r i e n t a t i o n p a r a m e t e r d e r i v e d a l o n g the lines of t h e V D W o n e - f l u i d t h e o r y y i e l d s the s o - c a l l e d m o d i f i e d V D W o n e - f l u i d m i x i n g rules i n S e c t i o n 3. T h e m e t h o d o l o g y for the t h e r m o d y n a m i c p r o p e r t i e s c a l c u l a t i o n s p r e s e n t e d h e r e i n is p r e s e n t e d i n S e c t i o n 4. I t is s h o w n i n S e c t i o n 5 t h a t u s i n g t h e modified

VDW

o n e - f l u i d m i x i n g rules y i e l d s a c c u r a t e

predictions

of

m i x t u r e t h e r m o d y n a m i c b e h a v i o r for m i x t u r e s of m o l e c u l e s w i t h d i s s i m i l a r i t i e s as great as m e t h a n e a n d p r o p a n e , b u t t h a t the a c c u r a c y

of

p r e d i c t i o n d e c a y s for l a r g e r m o l e c u l a r d i s s i m i l a r i t i e s . I n S e c t i o n 6, the exponents i n t h e m o d i f i e d V D W o n e - f l u i d m i x i n g r u l e s a r e v a r i e d e m p i r i c a l l y ; t h e r e s u l t a n t m i x i n g r u l e s , r e f e r r e d to h e r e i n as s e m i e m p i r i c a l m i x i n g rules, y i e l d s i g n i f i c a n t l y i m p r o v e d p r e d i c t i o n s f o r m i x t u r e s w i t h c o m p o n e n t s as d i s s i m i l a r as m e t h a n e a n d n o r m a l d e c a n e . T h e i m p l i c a tions of these results are d i s c u s s e d i n S e c t i o n 7.


7.

Conformai

LEE ET AL.

Anisotropic

Fluid

Conformai

Solution

Mixing

Solution

127

Rules

Model

T h e m e t h o d u s e d here for c o n s i d e r i n g c o n f o r m a i s o l u t i o n m o d e l s for fluids w i t h m o l e c u l a r anisotropics is b a s e d o n the m e t h o d u s e d b y S m i t h (4)

for t r e a t i n g i s o t r o p i c o n e - f l u i d c o n f o r m a i s o l u t i o n m e t h o d s as

a class of p e r t u r b a t i o n methods. T h e objective of the m e t h o d is to closely a p p r o x i m a t e the properties of a m i x t u r e b y c a l c u l a t i n g t h e properties of


a h y p o t h e t i c a l p u r e reference

fluid.

T h e c h a r a c t e r i z a t i o n parameters ( i n

this case, i n t e r m o l e c u l a r p o t e n t i a l p a r a m e t e r s ) of t h e reference

fluid

c h o s e n to be functions of c o m p o s i t i o n

a n d the

(i.e., m o l e f r a c t i o n s )

are

c h a r a c t e r i z a t i o n p a r a m e t e r s for the v a r i o u s possible m o l e c u l a r p a i r i n t e r actions ( l i k e - l i k e a n d u n l i k e - u n l i k e ) . I n p r i n c i p l e , a l l m o l e c u l a r a n i s o tropics (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, a n d h i g h e r m u l t i p o l e interactions, as w e l l as o v e r l a p a n d d i s p e r s i o n i n t e r a c tions ) c a n be i n c l u d e d i n the m e t h o d . H e r e , t h e v a r i o u s m o l e c u l a r a n i s o t r o p i c s are l u m p e d i n t o a single t e r m , so that the 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 u y ( r i , ωι, ω ) b e t w e e n M o l e c u l e s 1 a n d 2 of Species i a n d / c a n b e 2

2

w r i t t e n i n the f o r m U u ( r i 2 , ωχ, ω ) = €ij φ° (—) \ ϋ/

+ SijCij ψ

σ

I n E q u a t i o n 1, r

ωι, ω ) /

ρ

2

(1)

2

\ ϋ σ

is the v e c t o r d i s p l a c e m e n t of the m o l e c u l a r centers of

12

M o l e c u l e s 1 a n d 2, r

is the scaler separation, r

12

i 2

=

|r |, a n d ωχ a n d ω 12

2

are the E u l e r angles d e s c r i b i n g the orientations of M o l e c u l e s 1 a n d 2. T h e first t e r m o n the r i g h t - h a n d side of E q u a t i o n 1 i n v o l v i n g φ

Ό

is

r e c o g n i z e d as a n i s o t r o p i c p o t e n t i a l f o r m , so that the t e r m i n v o l v i n g φ

ρ

describes a n i s o t r o p i c effects. T h e c h a r a c t e r i z a t i o n p a r a m e t e r s σ , e

ijy

ί;

S

ijy

respectively,

are

c h a r a c t e r i s t i c distance,

energy,

and

and

anisotropic

s t r e n g t h parameters for the i n t e r a c t i o n b e t w e e n m o l e c u l e s of Species i a n d /. T h e extension of the i s o t r o p i c m i x t u r e c o n f o r m a i s o l u t i o n m e t h o d of S m i t h (4)

to the case of a n i s o t r o p i c m o l e c u l a r systems c a n b e

easily i n the f o l l o w i n g m a n n e r . T h e q u a n t i t i e s a

b

b y the relations a

c

ih

=

i ;

δ^/σ^

1 1 1

=

.

ih

δ ί / W ,

{;

and c =

8 % σ^,

where

ν

ν

made

are d e f i n e d

ti

the exponents k, 1, m , p , q , r , u , v , a n d w are left u n s p e c i f i e d at this p o i n t i n the d e v e l o p m e n t .

T h e c o n f i g u r a t i o n a l H e l m h o l t z free energy A for a n

a n i s o t r o p i c m i x t u r e t h e n c a n be

expanded

about the configurational

H e l m h o l t z free energy of a h y p o t h e t i c a l p u r e reference c h a r a c t e r i z a t i o n p a r a m e t e r s δ*, e

Xy

A

=

A

* + ^ Σ Σ ^ ζ , Κ · oa

ι

x

j

and σ

(or a

χ

-

a) x

Xy

+

b

x>

fluid,

A

-

b)

Xy

x

- ^ Σ Σ ^ ^ ϋ au i j

x

χ

x

x

+ - ^ ^ Σ Σ ^ ί Μ ^ υ — °χ) + h i g h e r order terms oc

with

c ),

j


(2)

128

EQUATIONS O F S T A T E

w h e r e χ* is t h e m o l e f r a c t i o n of the i t h c o m p o n e n t i n t h e m i x t u r e .

The

f o l l o w i n g m i x i n g r u l e s a n n u l the first o r d e r t e r m s i n t h e e x p a n s i o n i n E q u a t i o n 2, — E

Z i

i

^

W

^

(3)

8x ^W =

( )


p

« Λ ν

=

4

Z E z i W « ù X i

(5)

w

i

T h e a p p l i c a t i o n of t h e c o n f o r m a i s o l u t i o n m e t h o d i n i n d u s t r i a l c a l c u l a tions r e q u i r e s the use of the a p p r o x i m a t i o n A =

to a v o i d t h e l e n g t h y

A

x

c o m p u t a t i o n r e q u i r e d to c a l c u l a t e t h e h i g h e r o r d e r terms i n E q u a t i o n 2. T h u s , a p r a c t i c a l strategy for c h o o s i n g t h e exponents k, 1, m , p , q , r , u , v , a n d w i n E q u a t i o n s 3, 4, a n d 5 w o u l d be t h r o u g h m i n i m i z a t i o n of t h e difference A — A

x

( a c t u a l l y , d a t a for a l l a v a i l a b l e m i x t u r e t h e r m o d y n a m i c

p r o p e r t i e s c a n be u s e d s i m u l t a n e o u s l y to d e t e r m i n e t h e exponents

by

r e g r e s s i o n ) . H o w e v e r , m o s t a p p l i c a t i o n s of t h e c o n f o r m a i s o l u t i o n m e t h o d h a v e i n v o l v e d t h e use of exponents b a s e d o n m o l e c u l a r t h e o r y a n d so this a p p r o a c h w a s u s e d i n the i n i t i a l phases of t h e p r e s e n t w o r k . Modified

van

der

Waals

One-Fluid

Mixing

Rules

T h e w e l l k n o w n V D W o n e - f l u i d m i x i n g r u l e s for the c h a r a c t e r i z a t i o n parameters σ

χ

a n d c for i s o t r o p i c fluids are x

σ/ — Σ Σ ^ ϋ €σ Χ

3

χ

i

3

— Σ

Σ

i

^

(6)

3

ν

( ) 7

3

T h u s , t h e V D W o n e - f l u i d r u l e s c o r r e s p o n d to t h e use of t h e f o l l o w i n g v a l u e s of the exponents i n E q u a t i o n s 3 a n d 4, k = 0, q — 1, r = mixtures,

3. S m i t h ( 5 )

Equation 6

is

0, m =

3, ρ

=

the

reasonable

theoretical choice

Smith (5)

has s h o w n f o r h a r d - s p h e r e b i n a r y m i x t u r e s t h a t u s i n g t h e 12

=

m i x i n g rules have

1/2(ση +

been

used).

for

σ

arithmetic mean rule, σ

other

most

specifying

χ

(although

0,1 =

has d i s c u s s e d t h e f a c t t h a t f o r h a r d - s p h e r e Also,

^22), t h e s e c o n d - o r d e r t e r m s i n

E q u a t i o n 2 f o r the H e l m h o l t z free e n e r g y p r o b a b l y c a n b e n e g l e c t e d o n l y w h e n σιι a n d σ

22

differ b y less t h a n a b o u t 1 0 % .

F o r i s o t r o p i c fluids, t h e

p e r t u r b a t i o n e x p a n s i o n of the H e l m h o l t z free e n e r g y a b o u t t h a t of

a

h a r d - s p h e r e system leads to E q u a t i o n 7 w h e n t h e m e a n d e n s i t y a p p r o x i mation

is u s e d

for

the

hard-sphere

pair distribution function


(5).

7.

Conformai

LEE ET AL.

Solution

Mixing

129

Rules

A l t h o u g h t h e V D W o n e - f l u i d m i x i n g rules y i e l d r e a s o n a b l y predictions

of m i x t u r e b e h a v i o r

for m o l e c u l e s

accurate

w h i c h are not

greatly

d i s s i m i l a r , the cases of e v a l u a t i o n of t h e u n l i k e i n t e r a c t i o n p a r a m e t e r s , σ · and c , where i ^ {;

f r o m the d a t a m a y b e

0

compensatory

i n an

empirical way. F o r t h e d e r i v a t i o n of

a m i x i n g r u l e for the a n i s o t r o p i c s t r e n g t h

p a r a m e t e r , δ*, c o n s i d e r the P o p l e e x p a n s i o n

(3)

of t h e H e l m h o l t z free

e n e r g y , A , a b o u t the free e n e r g y , A , of a n i s o t r o p i c fluid reference system


0

A =

A0 + Αχ +

A2 +

. . .

(8)

w h e r e A^ are the i t h o r d e r terms i n the e x p a n s i o n . T h e i s o t r o p i c r e f e r e n c e s y s t e m p a i r p o t e n t i a l is d e f i n e d as the u n w e i g h t e d average of the a n i s o t r o p i c p a i r p o t e n t i a l i n E q u a t i o n 1, t h a t is,

(%(ri2, ωι,ω ))ω

where the brackets (

)

ω

(9)

2

- • G O -

d e n o t e t h e a n g l e average.

T h u s , Αχ =

0 and

E q u a t i o n 8 is a p e r t u r b a t i o n e x p a n s i o n f o r A i f h i g h e r - o r d e r t e r m s are small.

F o r s m a l l a n i s o t r o p i c s , t r u n c a t i o n at A

2

is a c c u r a t e , w h i l e

l a r g e anisotropies t h e use of t h e Padé a p p r o x i m a n t u s e d b y S t e l l

A — A

+

0

(1-A /A ) a

2

y i e l d s g o o d results. H e r e i n the t r u n c a t i o n at A order term A

2

for

(6)

2

w i l l be used. T h e second-

is g i v e n b y t h e r e l a t i o n

A2 — ^

E

E

^

W

f

d

r

i

d

r

2

»Qii°

(10)

w h e r e ρ is the m o l e c u l e n u m b e r d e n s i t y , Γ is absolute t e m p e r a t u r e , k is B o l t z m a n n ' s constant, r i a n d r

2

a r e t h e p o s i t i o n vectors of M o l e c u l e s 1

a n d 2 , a n d g ° is t h e i s o t r o p i c p a i r d i s t r i b u t i o n f u n c t i o n . F o r t h e case i n y

w h i c h φ / c a n b e w r i t t e n as t h e p r o d u c t f u n c t i o n {

A

2

^)l>(«i,«2)

(ID

/ à r u ' r u ' V t f a s ' i D * ) .

(12)

becomes A

2

= -

η ^ Σ Σ ^ Α ν ^

8


130

EQUATIONS

where r *

F o r e x a m p l e , i f t h e p e r t u r b a t i o n c o n t r i b u t i o n to

— r /a .

12

12

OF STATE

12

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

the

p e r t u r b a t i o n c o n t r i b u t i o n c o u l d b e a p p r o x i m a t e d b y t h e f o l l o w i n g expres s i o n , o w i n g to P o p l e

(3),

*w4>if> — so t h a t F Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007

2

(σ< ·Α )

=

{j

0i and 0

;

1 2

[3 c o s 0! -

(jr^J

12

1 2

3 cos θ -

2

and D =

[3 c o s 0 2

2

-

X

2]

2

3 cos

θ

2

2

-

(13) 2],

where

are t h e p o l a r angles of o r i e n t a t i o n of M o l e c u l e s 1 a n d 2.

To

o b t a i n the expression f o r δ^ t h e f o l l o w i n g a p p r o x i m a t i o n is i n t r o d u c e d ,

g i j

where r * =

° ( ^ ' j ^ ' ™ ' *

r/a

Xy

ρ* =

1

'

ρσ , Τ* = χ

3

* · - ) -

f

kT/e .

T h e approximation i n E q u a

x

o

°

(

*

r

^

T

#

)

( 1 4 )

t i o n 14 is s i m i l a r to, b u t m o r e stringent t h a n , 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 . W i t h the a s s u m p t i o n i n E q u a t i o n 14, A

becomes

2

A,=

~ ^ f f i T Z ^ W ^ ' /

drV"**0.°

(15)

I t is t h e n l o g i c a l to choose the f o l l o w i n g m i x i n g r u l e f o r the a n i s o t r o p i c s t r e n g t h p a r a m e t e r ( o v e r l a p p a r a m e t e r i n t h e specific e x a m p l e ) δ*, — Σ Σ ^ Α Λ ί Α ϋ i

(16)

8

3

T h i s m i x i n g r u l e c o r r e s p o n d s to t h e use of t h e f o l l o w i n g values of exponents i n E q u a t i o n 5, u = free energy, A * =

2, ν =

2, w =

w h e r e Ν is t h e n u m b e r of m o l e c u l e s

A/NkT,

the

3. T h e r e d u c e d H e l m h o l t z then

takes the f o r m A* w h e r e ρ* — ρσ \ Γ * = χ

A * G

kT/t

x>

δ , ΤΓ (D ) 2

(17)

p*JJ(T*)

2

2

a n d J is t h e i n t e g r a l x

J.—f

(18)

dr*r* F g ° 2

2

x

N o t e t h a t A * is of t h e f o r m A* = A * 0

(19)

+ s f*(T*, *) 2

x

P

T h i s result is i d e n t i c a l to t h e expression w h i c h is o b t a i n e d f r o m t h e p e r t u r b a t i o n e x p a n s i o n of A f o r a p u r e fluid. T h u s , r e f e r r i n g to E q u a t i o n 2, t h e first-order c o n f o r m a i s o l u t i o n r e l a t i o n for a n i s o t r o p i c fluids is ^ * ( Γ , ρ, { σ „ } ,

{8

(21)

*)

Ρ

€ , a n d δ* g i v e n i n E q u a t i o n s X

6, 7, a n d 16. T h e equation-of-state expression f o r t h e absolute pressure Ρ is o b Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007

t a i n e d f r o m E q u a t i o n 19 u s i n g t h e t h e r m o d y n a m i c r e l a t i o n

pkT

Y

dp*

/ ν,

(22) τ

t h e r e s u l t a n t expression f o r t h e c o m p r e s s i b i l i t y f a c t o r Ζ = P/pkT is Ζ = Z + 8.*Zi

(23)

0

where

η

\

Op

/ Ν,

(24) Τ

' ' - φ ) , . , - Λ Μ = ^ ) 1 . Calculation

of Thermodynamic

( 2 5 )

Properties

F o r t h e c a l c u l 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 , E q u a t i o n 2 3 w a s used i n a n empirical manner.

O n l y d a t a f o r n o n p o l a r n o r m a l paraffin

h y d r o c a r b o n systems w e r e u s e d i n t h e c o r r e l a t i o n d e v e l o p m e n t so t h a t as a n a p p r o x i m a t i o n , t h e P i t z e r a c e n t r i c factor, ω, c o u l d b e t a k e n as a n estimate of t h e c o l l e c t i v e s t r e n g t h of m o l e c u l a r anisotropics (i.e., δ ω).

2

=

B e c a u s e t h e use of t h e resultant c o r r e l a t i o n f o r p o l a r systems w a s

a n t i c i p a t e d , t h e p a r a m e t e r y ( γ = δ ) , r e f e r r e d to h e r e i n as t h e o r i e n t a t i o n 2

p a r a m e t e r , w a s u s e d i n s t e a d of t h e a c e n t r i c f a c t o r fluids).

(γ ^

ω for other

T h e e q u a t i o n of state i n E q u a t i o n 2 3 t h e n takes t h e f o r m Ζ(Τ·,

Λ

γ) = Ζ AT*,

ρ*) + γΖΛΤ*,

w h e r e Ζ is t h e c o m p r e s s i b i l i t y f a c t o r a n d Z

0

(26)

ρ*)

a n d Z i are f u n c t i o n s of t h e

r e d u c e d t e m p e r a t u r e T * = kT/e a n d r e d u c e d d e n s i t y ρ * = ρσ . 3

T h e equation-of-state

f o r m u s e d h e r e i n is t h e m o d i f i e d

Benedict-

W e b b - R u b i n ( M B W R ) e q u a t i o n as g i v e n b y H a n a n d S t a r l i n g ( 7 ) . I t is r e f o r m u l a t e d i n t o t h e f o r m of E q u a t i o n 26 b y expressing t h e constants appearing linearly i n the equation into t w o parts—one isotropic part a n d one a n i s o t r o p i c p a r t , Bi =

ai + ybi


(27)

132

α

Λ

EQUATIONS O F

STATE

b e i n g the i s o t r o p i c p a r t a n d bi b e i n g t h e a n i s o t r o p i c p a r t , w h e r e as

noted above, γ ~

δ is a n o r i e n t a t i o n p a r a m e t e r a c c o u n t i n g f o r t h e n o n 2

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

Therefore,

t h e M B W R e q u a t i o n c o r r e s p o n d i n g to E q u a t i o n 26 assumes the f o r m l + f[Bx

Ζ _ +

* [B

P

2

-

5


+

- B T*' B T*~i

-

X

2

B T*~

3

3

- B T*- ]

Q

Bsp* T*~* 2

9

+ ρ** [B T*-

2

10

+ B T*-* 7

[ (1 + B

4 P

X

* ) exp ( - V 2

2

-

BiiT*" ]

+

B T*' ]

)

5

12

2

]

(28)

w h e r e fo i n E q u a t i o n 27 is z e r o to i n s u r e l i n e a r i t y of Ζ i n γ, ρ* is t h e 4

r e d u c e d d e n s i t y , ρ* = kT/e.

ρσ , 3

a n d T * is t h e r e d u c e d t e m p e r a t u r e , T *

T h e c h a r a c t e r i s t i c m o l e c u l a r d i s t a n c e p a r a m e t e r , σ, a n d

=*

energy

p a r a m e t e r , c, w e r e e s t i m a t e d f r o m the c r i t i c a l constants u s i n g the r e l a t i o n s ,

0.3189

(29)

Pc _ c

—

kT

c

(30)

1.2593

w h e r e k is t h e B o l t z m a n n constant. P e r t i n e n t relations f o r other t h e r m o d y n a m i c p r o p e r t i e s h a v e b e e n p r e s e n t e d elsewhere

(2).

E q u a t i o n s 29

a n d 30 are b a s e d o n t h e r e l a t i o n s h i p s of t h e L e n n a r d - J o n e s

(12-6)

p o t e n t i a l p a r a m e t e r s for a r g o n to the a r g o n c r i t i c a l constants. T h e use of E q u a t i o n s 29 a n d 30 i n the M B W R e q u a t i o n of state g i v e n i n E q u a t i o n s 27 a n d 28 w o r k s w e l l for p u r e n o r m a l paraffin h y d r o c a r b o n s . v e r s a l constants α a n d bi, i = έ

1, . . . 12 (fo

4

=

s i m u l t a n e o u s l y u s i n g d e n s i t y , v a p o r pressure, a n d e n t h a l p y data for

methane

The uni

0) were determined b y departure

t h r o u g h n o r m a l d e c a n e i n m u l t i p r o p e r t y analysis.

A v e r a g e absolute d e v i a t i o n s of p r e d i c t e d f r o m e x p e r i m e n t a l p r o p e r t i e s w e r e 1.00%

f o r d e n s i t y , 1.13 B t u / l b for e n t h a l p y a n d 0 . 8 5 %

Table I .

Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane

for v a p o r

Generalization Parameters of Pure Materials to Be Used with Generalized Equation of State Critical Temp. (°F)

Critical Density (lb-mol/cu

Molecular Weight ft)

-116.43 90.03 206.13 305.67 385.42 453.45 512.85 563.79 610.50 651.90

0.6274 0.4218 0.3121 0.2448 0.2007 0.1696 0.1465 0.1284 0.1150 0.1037

16.042 30.068 44.094 58.12 72.146 86.172 100.198 114.224 128.24 142.276

Orientation Parameter (y) 0.01289 0.09623 0.1538 0.1991 0.2530 0.3054 0.3499 0.4004 0.4463 0.4880


7.

LEE E T AL.

pressure.

Conformai

T h u s , the

Solution

Mixing

133

Rules

m u l t i p a r a m e t e r , corresponding-states

correlation

f r a m e w o r k p r o v i d e d b y the p e r t u r b a t i o n e q u a t i o n f o r m i n E q u a t i o n 26 a n d the r e s u l t a n t g e n e r a l i z e d M B W R e q u a t i o n i n E q u a t i o n 28 y i e l d s g o o d results for the p u r e n o r m a l paraffin h y d r o c a r b o n s .

V a l u e s of the c r i t i c a l

constants a n d o r i e n t a t i o n p a r a m e t e r s u s e d i n this w o r k are g i v e n i n T a b l e I , w h i l e the values of the constants

a n d h i i n E q u a t i o n 2 7 are g i v e n

in Table II.


Table II.

Generalized Parameters Used in the M B W R Equation B, =

ParameteT i 1 2 3 4 5 6 7 8 9 10 11 12 Use

of

the

The

Modified

van

modified V D W

a, + y b i

ai

b,

1.45907 4.98813 2.20704 4.86121 4.59311 5.06707 11.4871 9.22469 0.094624 1.48858 0.015273 3.51486

0.32872 -2.64399 11.3293

der

Waals

2.79979 10.3901 10.3730 20.5388 2.76010 -3.11349 0.18915 0.94260

One-Fluid

Rules

o n e - f l u i d m i x i n g rules f o r

σ, χ

e

x

and S

in

x

E q u a t i o n s 6, 7, a n d 16 w e r e u s e d to d e t e r m i n e the a b i l i t y of this f o r m u l a t i o n of the c o n f o r m a i s o l u t i o n m o d e l f o r p r e d i c t i n g m i x t u r e b e h a v i o r . T h e f o l l o w i n g relations w e r e u s e d f o r u

ih

Equations of State in Engineering and Research - ACS Publications

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