A New Equation of State - American Chemical Society

9

A New Equation of State

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HIDEZUMI SUGIE Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Japan BENJAMIN C.-Y. LU University of Ottawa, Ottawa, Ontario, Canada

A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only T , P , and ω were required for the calculation. The proposed equation suecessfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. c

c

Τ Η h e p u r p o s e o f this i n v e s t i g a t i o n is to d e v e l o p a n e w p r e s s u r e - e x p l i c i t e q u a t i o n of state w h i c h : ( 1 ) y i e l d s a c c e p t a b l e

values of Z , a n d c

satisfies t h e u s u a l t w o i n i t i a l pressure—volume d e r i v a t i v e s a t t h e c r i t i c a l p o i n t a n d t h e M a x w e l l r e l a t i o n s h i p at s a t u r a t i o n ( e q u a l f u g a c i t y f o r c o e x i s t i n g l i q u i d a n d v a p o r p h a s e s ) ; ( 2 ) is s u i t a b l e f o r r e p r e s e n t i n g PVT b e h a v i o r of l i q u i d a n d gas phases over a w i d e r a n g e o f t e m p e r a t u r e a n d pressure; a n d ( 3 ) c a n b e i n t e g r a t e d a n d d i f f e r e n t i a t e d easily f o r o b t a i n i n g d e r i v e d t h e r m o d y n a m i c p r o p e r t i e s . I t is a n t i c i p a t e d t h a t t h e p a r a m e t e r s of t h e r e s u l t i n g e q u a t i o n c a n b e g e n e r a l i z e d i n terms o f t h e c r i t i c a l p r o p e r t i e s a n d t h e a c e n t r i c f a c t o r ω. T h e a p p l i c a t i o n is l i m i t e d , h o w e v e r , to p u r e n o n p o l a r

compounds. 0-8412-0500-0/79/33-182-163$05.50/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.

164

EQUATIONS

Development

of

the

Proposed

Equation

of

OF STATE

State

A s u i t a b l e e q u a t i o n of state m u s t satisfy c e r t a i n l i m i t i n g c o n d i t i o n s a n d f o l l o w some g e n e r a l trends.

O n e of the m o r e i m p o r t a n t c o n d i t i o n s

is t h a t the e q u a t i o n of state m u s t r e d u c e at l o w pressures a n d at a l l t e m p e r a t u r e s to the i d e a l gas e q u a t i o n .


-

p

Hence (1)

=1

v

RT

as Ρ - » ο a n d V —» oo. E x p r e s s i n g Ρ i n terms of a p o w e r series of

1/V,

the f o l l o w i n g e q u a t i o n w a s o b t a i n e d :

p

=

A .

i n w h i c h h m u s t b e e q u a l to x

• A _

ι

ι

l i

(2)

RT.

A n o t h e r c o n d i t i o n is t h a t t h e v o l u m e of a l l gases at h i g h pressures approaches

a l i m i t i n g v a l u e w h i c h is " p r a c t i c a l l y i n d e p e n d e n t

of

the

t e m p e r a t u r e a n d close to 0.26 V " as suggested b y R e d l i c h a n d K w o n g c

(I).

K e e p i n g this i n m i n d , the d e v e l o p m e n t

of t h e n e w e q u a t i o n

of

state b e g a n u s i n g the f o l l o w i n g e x p r e s s i o n :

V

R

T

- b

(3)

+i(T,V)

where

b

— 0.26

V

(4)

c

I f f ( T , V ) c o u l d b e r e p r e s e n t e d b y a g r o u p of terms s u c h as t h a t expressed i n E q u a t i o n 5,

H

T

'

V

)

V\ (V + k x ) - ! (V + k ! ) » . . . . (V + k ) » ~

=

m

+ * "·

( 5 )

the d i f f e r e n t i a t i o n of E q u a t i o n 3 a n d the i n t e g r a t i o n of t h e t h e r m o d y n a m i c expressions for e v a l u a t i n g fugacities a n d e n t h a l p y d e p a r t u r e s , as r e p r e s e n t e d b y E q u a t i o n s 6 a n d 7, w o u l d b e s i m p l i f i e d . (Ζ -

(H*-H°)

T

=

P V - R T -

J

V

[ p - T

1) - y -

(6)

dV


(7)

9.

suGiE A N D L U

A New

I n E q u a t i o n 5, l q

Equation

165

of State

are constants, w h i l e n

k

m

n

m

0

are either

z e r o or p o s i t i v e integers. A t t h e c r i t i c a l p o i n t , the c r i t i c a l i s o t h e r m shows a p o i n t of i n f l e c t i o n . Hence


(8)

and

m

-

,

T h e r e l a t i o n s h i p b e t w e e n Z a n d ω m a y b e expressed as f o l l o w s : c

Z =

0.291 -

C

0.080ω

(10)

E x p r e s s i n g E q u a t i o n 3 for the c r i t i c a l i s o t h e r m y i e l d s

P = y ^ + H T

(11)

, V )

C

w h i c h also m u s t satisfy E q u a t i o n s 8 t h r o u g h 10 at t h e c r i t i c a l p o i n t . C o n s e q u e n t l y , i t w o u l d b e m o r e c o n v e n i e n t for m a t h e m a t i c a l m a n i p u l a t i o n i f f ( T , V) c

w a s expressed i n terms of three constants w h i c h c o u l d b e

d e t e r m i n e d b y E q u a t i o n s 8 t h r o u g h 10. It w a s p r o p o s e d that f ( T , V )

be r e p r e s e n t e d b y three t r u n c a t e d

c

expressions

of

denominator. or

E q u a t i o n 5, w i t h

each

c o n t a i n i n g three terms i n

I n a d d i t i o n , t h e k's w e r e a s s u m e d to b e either —b,

the zero,

Consequently,

U

T

ν

a(T )

λ

,

e

l \ * c , V )

_ _

φ

6

)

n

l

V

n

( y

2

+

b

) n

t

3

C(T ) C

(

y

_

h

) n t f

%

{

y

+

d(T ) (V -

T h e q u a n t i t i e s a(T ), c

b) 7V *(V n

)

%

,

c

^

b

n

+

b) > n

v

a n d d(T )

c(T ), c

, }

were determined from Equations

c

8 t h r o u g h 10. T h e ris w e r e t a k e n to be either z e r o o r p o s i t i v e integers. I n a d d i t i o n , the f o l l o w i n g r e l a t i o n s h i p b e t w e e n the ris m u s t be satisfied: 2 ^ ni + n + 2

n

3

< n + n 4

5

+

< n + n 7

8

+ rig


(13)

166

EQUATIONS O F S T A T E

H o w e v e r , t h e s e c o n d v i r i a l coefficient d e r i v e d f r o m E q u a t i o n 11 w o u l d n o t b e t e m p e r a t u r e d e p e n d e n t i f ηχ +

n

2

+

n

^

3

3. C o n s e q u e n t l y , i n

E q u a t i o n 13,

ni + n + n = 2 2

(14)

3

and

3 ^ n + n + w < n + n + n 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.ch009

4

5

6

7

8

(15)

9

E q u a t i o n 11 t h e n w a s u s e d to fit t h e c r i t i c a l i s o t h e r m (i.e., t h e c u r v e r e l a t i n g Ρ a n d V at T ) o f c a r b o n d i o x i d e , as r e p o r t e d b y M i c h e l s ( 2 ) , c

u s i n g v a r i o u s values of ris u n d e r t h e c o n d i t i o n s o f E q u a t i o n s 14 a n d 15 a n d t h e v a l u e s o f a(T ),

c ( T ) , a n d d(Tc)

c

determined from Equations

c

8 t h r o u g h 10. T h e best fit w a s o b t a i n e d u s i n g t h e f o l l o w i n g v a l u e s : n i =

0; ri2 = 1; n

3

1; n

—

1; n = 2; η

—

4

5

β

—

1; n

7

0; n

—

—

8

0; a n d n = 7. 9

H e n c e , E q u a t i o n 11 took t h e f o l l o w i n g f o r m :

P

R T

=

*

_

7 - b

(r )

fl

c

F ( F + 6)

c(T )

, +

d(T )

e

c

( 7 - 6 ) 7 2

(

y

+

6

(7 + 6)

)

7

(16) T h e first t w o terms o f t h e r i g h t - h a n d side o f E q u a t i o n 16 a r e i n t h e same f o r m as t h e w e l l - k n o w n R e d l i c h - K w o n g ( R K ) e q u a t i o n of state ( I ) . T h e a c e n t r i c f a c t o r o f c a r b o n d i o x i d e is 0.225. I n o r d e r t o e x t e n d t h e a p p l i c a b i l i t y of E q u a t i o n 16 to a w i d e r r a n g e of ω (0 < ω < the

(V +

terms o f E q u a t i o n 16 w e r e

b)

modified

0.5),

as expressed i n

E q u a t i o n 17. RT

€

V - b

_

a(T )

c(T )

c

V ( y + 6i)

d(T )

c

(V -

b) V

2

c

(V + b ) 2

(V +

b) 3

7

(17) where,

b — (0.1181 + 0.4730ω) V ±

c

(18)

b = (0.2117 + 0.1611ω) V

c

(19)

b — (0.2515 + 0.0283ω) V

c

(20)

2

3

T h e ω v a l u e s u s e d i n this s t u d y a r e i d e n t i c a l t o those p r e v i o u s l y r e p o r t e d ( 3 ) . T h e c a l c u l a t e d v a l u e s o f c r i t i c a l i s o t h e r m pressures u s i n g E q u a t i o n 17 a r e c o m p a r e d w i t h P i t z e r s t a b l e i n T a b l e 1 ( 4 ) . T h e average absolute


9.

suGiE A N D L U

A New

Table I.

Equation

of State

167

Comparison of Calculated Crtical Isotherm Pressures with Pitzer's Table Average


Factor

β

(ω)

Region'

0.0

V

0.1

V

0.2

V

0.3

V

0.4

V

0.5

V

RK

I I I I

Deviation This

(%) Work

0.2 6.7 0.2 5.4 0.1 4.0 0.0 4.9 0.1 5.2 0.1 7.0

—

I r

(\)

0.2 16.0 0.3 34.0 0.6 58.0 0.8 90.0 1.2 130.0

I

T h e regions v: P

Absolute

^ 1 (5 points) and / : P > 1 (15 points). r

d e v i a t i o n s o b t a i n e d are m u c h s m a l l e r t h a n those o b t a i n e d b y t h e R K e q u a t i o n , e s p e c i a l l y at h i g h e r ω values. A c o m p a r i s o n of the c a l c u l a t e d a n d e x p e r i m e n t a l c o m p r e s s i b i l i t y factors a l o n g the c r i t i c a l i s o t h e r m f o r sulfur dioxide (5)

and carbon dioxide (2)

c r i t i c a l constants of

t h e substances

is s h o w n i n T a b l e I I .

investigated were

K u d c h a d k e r et a l . ( 6 ) a n d M a t h e w s

The

obtained

from

(7).

I n o r d e r to e x p a n d the a p p l i c a b i l i t y of E q u a t i o n 17 to isotherms o t h e r t h a n the c r i t i c a l , it w a s necessary to d e t e r m i n e t h e t e m p e r a t u r e d e p e n d e n c e of α ( Γ ) , c ( T ) , a n d d(T).

Let

a(T) = a f ( r )

(21)

a

c{T)-cf (T)

(22)

d(T)

(23)

0

=dî {T) d

a n d at Γ =

T , f (T ) =

f (T ) =

a(T ), c =

c ( T ) , and d =

d(T ).

c

p

=

RT

7_6

c

a

c

e

c

_

i (T ) a

c

c

aî (T)

1. I n other w o r d s , a —

cl (T)

a

V(V

=

e

H e n c e E q u a t i o n 17 b e c a m e df (T)

e

+ b) x

+

V

2

(V -

b) (V +

d

b) 2

(V +

bs)

7

(24)


168



Table II.

Comparison of Experimental and Calculated Compressibility Factors along the Critical Isotherm Average Absolute Deviation (%)

Component

Region

Number of Data Points

S u l f u r d i o x i d e (δ)

P < l P r > l

16 49

1.46 12.91

0.30 0.89

P

16 8

3.01 10.80

0.89 1.83

r

C a r b o n d i o x i d e (2) r

> i

RK

(\)

This

Work

I n o r d e r to o b t a i n a s u i t a b l e expression f o r ί „ ( Γ ) , E q u a t i o n 2 4 w a s u s e d to o b t a i n the f o l l o w i n g expression f o r t h e s e c o n d v i r i a l coefficient:

Β = Urn (~\ p-*o\dp/

(25)

ai (T)/RT

b

=

a

T

w h i c h t h e n w a s u s e d to fit t h e P i t z e r a n d C u r l c o r r e l a t i o n o f t h e s e c o n d v i r i a l coefficient ( δ ) : BP

C

(0.1445 + 0.073ω) -

RT

(0.330 -

0.46ω)/Τ

Γ

C

-

(0.1385 + 0 . 5 0 ) / Γ

-

0.0073ω/Γ

ω

Γ

-

2

(0.0121 + 0.097ω)/Γ

(0.1711 + 0 . 2 1 4 7 ω ) Τ , + +

(0.2630 + 1.1065ω)/Γ

+ 0.0173ω/7ν when Γ = T , T = c

r

(26)

3

w a s as f o l l o w s :

a

e

Γ

8

T h e expression o b t a i n e d f o r î (T) f (T)

Γ

Γ

(0.8340 +

1.2211ω)

(0.0741 + 0.3120ω)/Γ

Γ

2

(27)

7

1 a n d E q u a t i o n 2 7 r e d u c e s to f ( T ) = 1. 0

c

N e x t , E q u a t i o n 24 w a s r e a r r a n g e d b y s p l i t t i n g t h e t h i r d t e r m o f t h e r i g h t - h a n d side o f t h e e q u a t i o n i n t o t w o terms. H e n c e ,

P

=

RT V - b

aîa(T) ViV + bJ

eîe(T) V (V-b)

+'

2

eî (T) V (V +

dU(T)

e

2

b) 2

(V +

bV 3

(28) where e = b +

(29)

b

2


9.

suGiE A N D L U

A New Equation

169

of State

I n o r d e r t o represent P V T d a t a o v e r a w i d e t e m p e r a t u r e r a n g e b y m e a n s of E q u a t i o n 28, i t w a s necessary to m a k e b o t h t h e q u a n t i t y b i n t h e t h i r d 9

t e r m o f t h e r i g h t - h a n d side of t h e e q u a t i o n , a n d t h e q u a n t i t y 6

depend

3

ent o n t e m p e r a t u r e . T h u s , _

p

RT

ai (T)

eU(T)

a

V _ b

ViV+bJ

V (V

ei,(T)

-

2

b')

V (V

di (T) d

+ b)

2

(V +

2

WV


(30) w h i c h is t h e final expression of t h e p r o p o s e d e q u a t i o n . T h e e v a l u a t i o n of the t e m p e r a t u r e - d e p e n d e n t q u a n t i t i e s f (T),

i (T),

e

f (T),

g

a n d fc ' w a s b a s e d o n : ( 1 ) s a t i s f y i n g t h e M a x w e l l r e l a t i o n s h i p

d

3

at s a t u r a t i o n ( t h e f u g a c i t y of t h e l i q u i d c a l c u l a t e d f r o m E q u a t i o n 30 s h o u l d b e e q u a l to t h e f u g a c i t y of t h e v a p o r c a l c u l a t e d f r o m t h e same equation);

(2)

s a t i s f y i n g t h e g e n e r a l i z e d c o r r e l a t i o n of t h e s a t u r a t e d

l i q u i d v o l u m e p r o p o s e d e a r l i e r b y L u et a l . ( 9 ) ; a n d (3) fitting PVT d a t a as c o r r e l a t e d i n P i t z e r s tables (4) over t h e c o m p l e t e r a n g e of T a n d P . T

T h e f o l l o w i n g set of t e m p e r a t u r e f u n c t i o n s numerous

fitting

trials w e r e

finally

made:

6' = 0.26 Τ · V Γ

b ' =

e

U(T) g

=

+ x /T*

Xl

2

X l

_

2

(31)

c

T

i (T)

i (T)

0

(0.2515 + 0.0283ω) T V

8

-

(32)

C

TV

4

(33)

— 7V -

8

(34)

1

+ xJTf

+ xJT*

α - ζ/T Χ

- 0 . 3 5 8 8 + 0.4982ω + 0.8208ω 0.2993 + 0.3038ω -

0.6829ω

x —

0.9826 -

0.7758ω -

0.0343ω

χ =

0.0883 -

0.0106ω -

0.1054ω

χ =

-0.0114 -

3

4

5

0.0156ω + 0.0018ω

(35)

7

r

2

x _ 2

T

w a s o b t a i n e d after

2

(36)

2

2

2

F o r t h e p u r p o s e of s a t i s f y i n g t h e M a x w e l l r e l a t i o n s h i p m o r e

pre

cisely, a n a d j u s t m e n t w a s m a d e o n t h e t e m p e r a t u r e - d e p e n d e n t f u n c t i o n f (T). a

T h e final expression o b t a i n e d f o r f (T) a

f (Γ)

is as f o l l o w s :

(0.1664 + 0.0043ω) T + (0.8137 - 1.2204ω) (0.2861 + 1 . 1 2 9 7 ω ) / Γ + (0.0666 + 0.2977ω)/Γ + 0.0173ω/Τ r

fl

+

Γ

Γ

Γ

7


(37)

170


It s h o u l d b e m e n t i o n e d t h a t t h e difference b e t w e e n E q u a t i o n s 3 7 a n d 2 7 is v e r y s m a l l . T h e s e c o n d v i r i a l coefficient c a l c u l a t e d f r o m E q u a t i o n 2 5 u s i n g t h e n e w expression of f ( T ) s t i l l agrees v e r y w e l l w i t h t h e a

correlation of Pitzer a n d C u r l ( E q u a t i o n 2 6 ) . Numerical

Values

of the Parameters

of the Proposed

Equations

A s m e n t i o n e d above, t h e final expression of t h e p r o p o s e d

equation


is r e p r e s e n t e d b y E q u a t i o n 3 0 : RT

ρ

efe(T)

ai (T) a

V -b

V(V + b )

-

V (V 2

x

.

eî (T)

dî (T)

g

6')

d

V (V + b ) 2

a

(V +

i> ') 3

(30) in which 6~0.26F 6 i = (0.1181 + 0.4730ο.) V b = (0.2117 + 0.1611*.) F 6' = 0.267Λ°· 7 W = (0.2515 + 0.0283ω) T V aî (T) = ο ι Τ + α + a /T + a / T + a /T ai = a*yiRV a = a*y RT V a = a*y RT W a = a*y RT V a = a*y RT *V 0.1664 - 0.2243ο. 2/ = 0.8137 - 1.2204a> 2/3= 0.2861 + 1.1297ω 2/4 = 0.0666 + 0.2977ω 2/5 = 0.0173ω eî (T) =eT* e — e'ETWc β· — c * / (0.4717 + 0.1611».) «MT) - Λ + g / T + /r + QJT* + g / T ' C

e

2

c

2

ο

r

a

3

2

c

4

2

7

s

c

2

2

c

3

c

4

4

c

5

y

c

3

c

3

5

c

c

e

i

2

e

2

2

2

grj = g = g = g = 3

4

s

X l

x = x = x = x 2

3 t

5

dft(T)

— d—

ff3

4

e*x RT V e*x RT sV* e*x RT V e«x ET F 0.3588 + 0.4982ω + 0.2993 + 0.3038ω 0.9826 - 0.7758ο, 0.0883 - 0.0106α. 0.0114 - 0.0156ω + 3

2

c

c

7

4

5

2

c

3

B

c

8

c

2

c

c

2

0.8208ω 0.6829ω 0.0343ω 0.1054 0.0018ω

2

2 2

ω

2

2

άΓ1Λ d*RT V* 2S

c


7

9.

suGiE A N D L U

A New Equation

171

of State

I n t h e a b o v e expressions, V =

Z RT /P

C

C

C

C

Z = 0.291 - 0.080ω C

T h e v a l u e s of a*, c * , a n d d* w e r e d e t e r m i n e d f r o m E q u a t i o n s 8 t h r o u g h 10. T h e values o b t a i n e d f o r s e v e r a l ω values at r e g u l a r i n t e r v a l s are l i s t e d


b e l o w to serve as e x a m p l e s :

Testing

ω

a»

c*

d*

0.0 0.1 0.2 0.3 0.4 0.5

1.3827 1.4342 1.4872 1.5419 1.5982 1.6561

0.2973 0.2859 0.2761 0.2680 0.2615 0.2564

0.7464 0.7443 0.7443 0.7467 0.7517 0.7594

of the Proposed

Equation

T h e a p p l i c a b i l i t y of t h e p r o p o s e d e q u a t i o n w a s tested i n terms of its p r e d i c t e d values of t h e c o m p r e s s i b i l i t y factors, l i q u i d f u g a c i t y coefficients, a n d i s o t h e r m a l e n t h a l p y departures o f p u r e Compressibility Factors.

compounds.

A t o t a l of 2772 Ζ values of P i t z e r s t a b l e

w a s u s e d to test t h e c a p a b i l i t y of t h e p r o p o s e d e q u a t i o n f o r c a l c u l a t i n g c o m p r e s s i b i l i t y factors of p u r e n o n p o l a r

compounds.

T h e c a l c u l a t e d values o b t a i n e d f r o m

E q u a t i o n 3 0 , together

with

those o b t a i n e d f r o m t h e equations of R e d l i c h et a l . ( 1 0 ) , E d m i s t e r et a l . (11), R e d l i c h a n d K w o n g (1), a n d S u g i e et a l . (12) a r e c o m p a r e d w i t h the Ζ values of P i t z e r s w o r k i n T a b l e I I I . T h e p r o p o s e d e q u a t i o n p r o v i d e s t h e smallest s t a n d a r d d e v i a t i o n .

Table III. Comparison of Calculated Compressibility Factors w i t h P i t z e r s Table by Various Methods for a T o t a l of 2772 D a t a Points at 22 T and 21 P Conditions" T

r

Standard R e f . 10

R e f . 11 Réf. 1 R e f . 12 This work β

Deviation

0.013 0.025 0.054 0.024 0.0108

0.8 < T < 4, 02 < P < 9 ; ω = 0.0-0.5. t

r


172


A c o m p a r i s o n of the c a l c u l a t e d a n d e x p e r i m e n t a l Ζ values of p r o p a n e (13)

a n d s u l f u r d i o x i d e ( 5 ) , i n the gas r e g i o n is s h o w n i n T a b l e I V . I n

a d d i t i o n , t h e c a l c u l a t e d results o b t a i n e d f r o m the equations of R e d l i c h a n d K w o n g ( I ) , R e d l i c h a n d D u n l o p (14), et a l . (12)

G r a y et a l . ( 1 5 ) , a n d S u g i e

also are i n c l u d e d i n T a b l e I V . T h e p r o p o s e d e q u a t i o n y i e l d s

the best results.


Table IV.

Comparison of Experimental and Calculated Ζ Values for Propane and Sulfur Dioxide Average

τ

F(psia)

Ref. 14

Ref. 1 Propane

220°F 340°F 460°F

20-6000 20-6000 20-6000

Overall

Absolute

10-300 10-300 10-300

Overall

(%)

Ref. 15

Ref. 12

This Work

(13)

6.3 1.7 1.0

5.2 1.5 0.4

1.9 1.9 4.2

1.0 1.2 1.8

0.80 1.47 1.67

3.0

2.4

2.7

1.3

1.31

Sulfur dioxide 157.5°C 200°C 250 ° C

Deviation

(5)

13.5 4.9 2.2

6.4 3.3 1.0

3.3 0.9 3.3

0.8 2.0 1.2

0.77 1.02 1.32

6.9

3.6

2.5

1.3

1.04

I n a d d i t i o n , t h e c a l c u l a t e d a n d t h e e x p e r i m e n t a l Ζ v a l u e s of h y d r o g e n sulfide ( 1 6 ) ,

i n the regions i n c l u d i n g gas, v a p o r , a n d l i q u i d , are

c o m p a r e d i n T a b l e V . T h e c a l c u l a t e d results o b t a i n e d f r o m the e q u a t i o n s of R e d l i c h a n d K w o n g (1 ) a n d S u g i e et a l . (12)

also are i n c l u d e d i n t h i s

t a b l e f o r c o m p a r i s o n . A g a i n , t h e p r o p o s e d e q u a t i o n y i e l d s the best results.

Table V .

Comparison of Experimental and Calculated Ζ Values for Hydrogen Sulfide (16) Average

T

Number Points

r

0.744 0.834 0.893 1.012 1.102 1.191 Overall

0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7

34 34 34 34 34 34

of

Absolute

Deviation

(%)

Ref. 1

Ref. 12

This Work

3.64 3.94 4.00 3.84 2.54 1.69 3.27

1.53 1.52 1.40 0.55 0.88 0.66 1.09

1.37 1.01 0.84 0.53 0.66 0.85 0.88


9.

SUGIE A N D L U Fugacities.

A New

Equation

173

of State

T h e f u g a c i t y coefficient, φ, w h i c h is e q u a l t o t h e r a t i o

of f u g a c i t y to pressure, c a n b e c a l c u l a t e d f r o m E q u a t i o n 38. dV ( Z - l ) ~ r -

v

/

^ z - i - m z - m V^ T J _


ele(T)

V-b'

|_6

RTb'

Γ_1_

RTb

lb

b

x

(38)

Vj

V

11

V + b

m

2

v

V +

11

V

ei,(T) 2

+' ^RTbi - m

_

dU(T) 6RT(V

V j

2

+

b ')« 3

T h e a b o v e e q u a t i o n w a s u s e d to o b t a i n φ v a l u e s , at r e g u l a r i n t e r v a l s of T

r

from T

T

=

0.5 to T

=

r

butane, a n d n-pentane.

1, f o r l i q u i d m e t h a n e , ethane, p r o p a n e , n -

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

of t h e a v a i l a b l e t a b u l a t i o n s i n t h e l i t e r a t u r e (17,18,20)

i n Figures 1

t h r o u g h 5. E x c e l l e n t a g r e e m e n t w a s o b t a i n e d . I n a d d i t i o n , φ v a l u e s w e r e c a l c u l a t e d u s i n g E q u a t i o n 38 f o r s a m e five c o m p o u n d s b u t at l o w e r t e m p e r a t u r e s ( T s i m i l a r c o m p a r i s o n is s h o w n i n F i g u r e s 6 a n d 7.

r

=

the

0.4 a n d 0 . 3 ) .

G o o d agreement

A

gen

e r a l l y is o b t a i n e d w i t h the e x c e p t i o n of m e t h a n e . T h e isothermal enthalpy depar

Isothermal Enthalpy Departures.

tures f r o m the i d e a l - g a s state w e r e c a l c u l a t e d f o r six p u r e , s a t u r a t e d l i q u i d s ( m e t h a n e , ethane, p r o p a n e , η-butane, i - b u t a n e , a n d u s i n g E q u a t i o n 39.

n-pentane)

T h e p r o p o s e d e q u a t i o n w a s , of course, u s e d i n its

derivation ( H * - H ° ) , - P V - R T bRT

[ - (-ff) ~] p

J

V

F

d

+

b)

l n

— γ —

0.2ef.(D

Γ 2

b'

I b'

b

η l b

2

3

V-b'

[ ψ

_ Z s

b Τ'yν Γ

(v +

2

Γ 1

2

V ( V - b ' )

T

df (T)V

g

ψ

eî.{T)

a

ef (T)

-

dV

aî (T) 7 + b i

V - b

v(v

T

. 2

m

,

, +

V V+b

2

Fx

,

Τ&x " In

y +

h

1

Ί

11 T J _1_

V-b' V

m

+ ^

V

+

,

Π

"·"

Vj

_

+

b' J

V F

4

6(7 +

df (T)W (V + b/y

6 ') 3

e

t


(39)


EQUATIONS O F


STATE

suGiE A N D L U

A New Equation

of State


9.


175

EQUATIONS O F


176


STATE

suGiE A N D L U

A New Equation

of State


9.


177

EQUATIONS O F


178


STATE



h-· -α CO

Ci"

S"

ο"

a

•ο

Ci

ϋ

>

ο Β

C O


EQUATIONS O F


STATE

9.

suGiE A N D L U

A New

Equation

181

of State

in which


Fi -

af„(T) -

F

2

= ef.(T)

F

3

= βί,(Γ) -

F

4

-

df (D d

-

-

Τ (

θ

α

ί

Τ ( ^ γ

^

1

Γ

)

)

2

=

) -

Τ

Γ [

) = a + 2 a / T + ZaJT*

σ α Α

*Γ

;

3

+

8α /Τ'

5β/Γ*

0i + 3 g / T + 704/r 2

) = 2.8 d / Γ · 1

2

6

+ +

50 /Γ* 8g /T' 3

5

8

T h e values o b t a i n e d f r o m E q u a t i o n 39 a n d several other (11,12,19,21,22,23,24,25)

5

are c o m p a r e d

methods

w i t h the available experi

m e n t a l values i n the l i t e r a t u r e (26,27,28,29).

T h e c o m p a r i s o n , expressed

i n terms of average absolute d e v i a t i o n s , is p r e s e n t e d i n T a b l e V I . H o w ever, the results o b t a i n e d b y the p r o p o s e d e q u a t i o n are o n l y f a i r .

Discussion

and

Conclusion

T h i s i n v e s t i g a t i o n f o l l o w s o u r efforts p r e v i o u s l y m a d e o n the m o d i f i c a t i o n of the R K e q u a t i o n of state (3,30). T h e r e p u l s i v e t e r m of t h e R K e q u a t i o n w a s r e t a i n e d w i t h the a n t i c i p a t i o n that the o r i g i n a l terms w o u l d b e p r e s e r v e d as p a r t of the n e w e q u a t i o n . T h i s p r a c t i c e m a y b e subject to modifications i n f u t u r e endeavors. T h e r e p u l s i v e t e r m m a y be r e p l a c e d b y a m o r e s u i t a b l e t e r m s u c h as t h a t p r o p o s e d b y C a r n a h a n a n d S t a r l i n g (31) . T h e proposed

e q u a t i o n w a s n o t c o m p a r e d w i t h a n y of t h e

more

r e c e n t c u b i c e q u a t i o n s , s u c h as the Soave m o d i f i c a t i o n of t h e R K e q u a t i o n (32) , the P e n g a n d R o b i n s o n e q u a t i o n (33), (34),

and the F u l l e r equation

because a l l of these equations do n o t y i e l d a c c e p t a b l e values of Z . c

Some of t h e ω values u s e d i n this s t u d y differ s l i g h t l y f r o m suggested b y Passut a n d D a n n e r ( 3 5 ) .

those

H o w e v e r , these s m a l l differences

h a r d l y affected t h e c a l c u l a t e d results. I n c o n c l u s i o n , a n e w p r e s s u r e - e x p l i c i t e q u a t i o n of state has b e e n successfully d e v e l o p e d as i n t e n d e d . I t is s u i t a b l e f o r r e p r e s e n t i n g PVT b e h a v i o r of l i q u i d a n d gas phases over a w i d e r a n g e of t e m p e r a t u r e a n d pressure for p u r e , n o n p o l a r c o m p o u n d s . F u r t h e r m o r e , t h e p a r a m e t e r s of the p r o p o s e d e q u a t i o n are g e n e r a l i z e d i n terms of t h e c r i t i c a l p r o p e r ties a n d the a c e n t r i c factor.


182

EQUATIONS O F


Table V I .

Comparison of Experimental and Calculated

Component

TV

Pr

M e t h a n e (26) E t h a n e (27) Propane (26,27,28) i V - B u t a n e (27) i s o - B u t a n e (27) J V - P e n t a n e (29)

0.757-0.990 0.654-0.982 0.660-0.983 0.653-0.967 0.757-0.990 0.662-0.970

0.172-0.945 0.045-0.894 0.041-0.891 0.032-0.794 0.034-0.872 0.032-0.802

No. Data

Overall

of

a,c,d

=

Symbols q u a n t i t i e s r e p r e s e n t e d b y E q u a t i o n s 2 1 , 22, a n d 23, respectively

a * , c*, d* = q u a n t i t i e s d e t e r m i n e d f r o m E q u a t i o n s 8 , 9 , a n d 10 Β = s e c o n d v i r i a l coefficient b = p a r a m e t e r of E q u a t i o n 3 hi, &2> &3 — p a r a m e t e r s of E q u a t i o n 17 9

&' = p a r a m e t e r s of E q u a t i o n 30 3

e = q u a n t i t y d e f i n e d b y E q u a t i o n 29 e* = c * / ( 0 . 4 7 1 7 + F χ,...

4

0

e n t h a l p y at i d e a l gas state

== e n t h a l p y at pressure Ρ

H

p

k

0.1611ω)

F = f u n c t i o n s of E q u a t i o n 39 H° =

hi...

5 7 13 7 7 14 53

Glossary

V

STATE

h = parameters of E q u a t i o n 1 n

... k

m

rii.. . n

— constants of E q u a t i o n 5 = z e r o or p o s i t i v e i n t e g e r

m

Ρ =

pressure

P — c r i t i c a l pressure c

R = gas constant Τ = T = c

temperature critical temperature

T = reduced temperature r

V == v o l u m e V

c

— critical volume

Xi. . . Xs = f u n c t i o n s r e p r e s e n t e d b y E q u a t i o n 36 Ζ =

compressibility factor

Z = c o m p r e s s i b i l i t y f a c t o r at the c r i t i c a l p o i n t c


of Pts.

9.

suGiE A N D L U

( H ° — H )t

A New Equation

Values for Pure Saturated Liquids

p

Average


183

of State

Absolute

Deviation

(Btu/lb)

Ref. 19

Ref. 21

Ref. 22

Ref. 23

Refs. 24,25

Ref. 11

Ref. 12

This Work

3.7 1.7 1.2 2.2 2.3 1.9 2.2

9.8 2.2 3.5 3.4 6.1 4.6

10.0 8.1 3.4 9.4 13.6 11.7

2.6 5.0 4.6 4.0 4.4 4.8

7.0 5.1 2.7 4.4

32.1 9.3 5.1 4.6

3.1

3.4

3.0 1.9 1.5 2.1 2.5 1.5

4.7 4.3 4.6 4.7 5.5 5.0

4.9

9.4

4.2

4.5

11.0

2.1

4.8

—

—

Greek Letters ρ = density φ = f u g a c i t y coefficient ω

= acentric factor

Literature Cited

1. Redlich, O.; Kwong, J. N. S. Chem. Rev. 1949, 44, 233. 2. Michels, C. "Some Physical Properties of Compressed Carbon Dioxide"; University of Amsterdam: Amsterdam, 1937. 3. Sugie, H.; Lu, Β. C.-Y. Ind. Eng. Chem., Fundam. 1970, 9, 428. 4. Pitzer, K. S.; Lippman, D. Z.; Curl, R. F., Jr.; Huggins, C. M.; Petersen, D. E. J. Am. Chem. Soc. 1955, 77, 3433. 5. Kang, T. L.; Hirth, L. T.; Kobe, Κ. Α.; McKetta, J. J. J. Chem. Eng. Data 1961, 6, 220. 6. Kudchadker, A. P.; Alani, G. H.; Zwolinski. Chem. Rev. 1968, 68, 659. 7. Mathews, J. F. Chem. Rev. 1972, 72, 71. 8. Pitzer, K. S.; Curl, R. F., Jr. J. Am. Chem. Soc. 1957, 79, 2369. 9. Lu, Β. C.-Y.; Ruether, J. Α.; Hsi, C.; Chiu, C. H . J. Chem. Eng. Data 1973, 18, 241. 10. Redlich, O.; Ackerman, F. J.; Gunn, R. D.; Jacobson, M . ; Lau, S. Ind. Eng. Chem., Fundam. 1965, 4, 369. 11. Edmister, W. C.; Vairogs, J.; Klekers, A. J. AIChE J. 1968, 14, 479. 12. Sugie, H.; Ono, K.; Hiraide, M.; Lu, Β. C.-Y., presented at the 7th Annual Meeting of Society of Chemical Engineers, Japan, 1973. 13. Johnston, H. L.; White, D. Trans. ASME 1950, 72, 785. 14. Redlich, O.; Dunlop, A. K. Chem. Eng. Prog., Symp. Ser. 1963, 59(44), 95. 15. Gray, R. S., Jr.; Rent, N. H.; Zudkevitch, D. AIChE J. 1970, 16, 991. 16. Reamer, H . H.; Sage, B. H.; Lacey, W. N . Ind. Eng. Chem. 1950, 42, 140. 17. Chao, K. C.; Greenkorn, R. Α.; Olabisi, O.; Hensel, Β. H . AIChE J. 1971, 17, 353. 18. Carruth, G. F.; Kobayashi, R. Ind. Eng. Chem., Fundam. 1972, 11, 509. 19. Lee, B.-I.; Edmister, W. C. Ind. Eng. Chem., Fundam. 1971, 10, 229.



184

EQUATIONS OF STATE

20. Lewis, G. N.; Randall, M. "Thermodynamics," 2nd ed.; Revised by Pitzer, K. S., Brewer, L.; McGraw-Hill: New York, 1961. 21. Stevens, W. F.; Thodos, G. AIChE J. 1963, 9, 293. 22. Yen, L. C.; Alexander, R. E. AIChE J. 1965,11,334. 23. Erbar, J. H . ; Persyn, C. L.; Edmister, W. C., Proceedings of the 43rd Annual Convention Natural Gas Processors Association, March 1964. 24. Benedict, M.; Webb, G. B.; Rubin, L. C. J. Chem. Phys. 1940, 8, 334. 25. Benedict, M.; Webb, G. B.; Rubin, L. C. J. Chem. Phys. 1942, 10, 747. 26. Jones, M . L., Jr.; Mage, D. T.; Faulkner, R. C., Jr.; Katz, D. L. Chem. Eng. Prog., Symp. Ser. 1963, 59(44), 52. 27. Canjar, L . N.; Manning, F. S. "Thermochemical Properties and Reduced Correlations for Gases"; Gulf Publishing Co. : Houston, 1967. 28. Yesavage, V. R., Ph.D. Thesis, University of Michigan, Ann Arbor, 1969. 29. Brydon, J. W.; Walen, N.; Canjar, L . N . Chem. Eng. Prog., Symp. Ser. 1953, 49(7), 151. 30. Sugie, H.; Lu, Β. C.-Y. AIChE J. 1971, 17, 1068. 31. Carnahan, N. F.; Starling, Κ. E. J. Chem. Phys. 1969, 51, 1184. 32. Soave, G. Chem. Eng. Sci. 1972, 27, 1197. 33. Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem., Fundam. 1976, 15, 59. 34. Fuller, G. G. lnd. Eng. Chem., Fundam. 1976, 15, 254. 35. Passut, C. Α.; Daner, R. P. Ind. Eng. Chem., Process Des. Dev. 1973, 12, 365. RECEIVED October 8, 1978.


A New Equation of State - American Chemical Society

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