Calculation of Compressed Liquid Excess Volumes and Isothermal

18 Calculation of Compressed Liquid Excess Volumes and Isothermal Compressibilities for

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Mixtures of Simple Species S. P. SINGH and R. C. MILLER Department of Mineral Engineering, University of Wyoming, Laramie, WY 82071

Compressed liquid mixture excess volumes (V ) and isothermal compressibilities have been measured recently for binary and ternary mixtures containing the components nitrogen, argon, methane, and ethane. Selected liquid-phase equations of state (and some corresponding-states methods) have been tested using these new data. Simple two-parameter equations of state, based on the model of hard spheres in a uniform potential field, accurately represent V (P, T, x) for these mixtures. Two empirical binary mixing rule deviation parameters were used, but they were highly correlated. Two methods of reducing the formulation to use only one parameter were shown to result in only small reductions in accuracy. The equations of state accurately predicted V values for ternary mixtures and changes in isothermal compressibility on mixing for all mixtures. E

E

E

" e q u a t i o n s of state h a v e b e e n t r a d i t i o n a l l y d e v e l o p e d to d e s c r i b e t h e pressure-molar volume-temperature-composition

(PVTx)

behavior

f o r fluid m i x t u r e s . I f a n e q u a t i o n of state c a n d e s c r i b e t h e V(P,

T, x)

surface f o r a c r y o g e n i c l i q u i d m i x t u r e to a n a c c u r a c y a p p r o a c h i n g 0 . 1 % , t h e n i t is satisfactory f o r a l l c u r r e n t i n d u s t r i a l needs, i n c l u d i n g c u s t o d y transfer c a l c u l a t i o n s . A n a l y t i c a l e q u a t i o n s of this a c c u r a c y h a v e

been

d e v e l o p e d f o r c e r t a i n p u r e cryogens. I t is n e a r l y i m p o s s i b l e to g e n e r a l i z e these c o m p l i c a t e d equations of state to m i x t u r e s , other t h a n b y c o r r e sponding-states t e c h n i q u e s . A c c u r a c y a p p r o a c h i n g t h e r e q u i r e d l e v e l has 0-8412-0500-0/79/33-182-323$05.25/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.

324

EQUATIONS

b e e n o b t a i n e d for c o r r e s p o n d i n g - s t a t e s

OF

STATE

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

l i m i t e d species a n d ranges of c o m p o s i t i o n , as e n c o u n t e r e d i n l i q u e f i e d n a t u r a l gas m i x t u r e s

(1,2,3).

A n a l t e r n a t i v e a p p r o a c h is to use a n e q u a t i o n of state to b e h a v i o r of the excess v o l u m e : V (Ρ,

T, x).

E

describe

E x c e s s v o l u m e is d e f i n e d i n

terms of the m i x t u r e m o l a r v o l u m e ( V ) a n d the c o m p o n e n t m o l a r v o l u m e s at the same pressure a n d t e m p e r a t u r e :

(V ) {


y= — y - 5 > , ν *

(i)

A n e q u a t i o n of state is u s e d to c a l c u l a t e V a n d the V» v a l u e s , a n d the excess v o l u m e is t h e n o b t a i n e d f r o m E q u a t i o n 1. F o r p r a c t i c a l a p p l i c a tions, V

e

f r o m the e q u a t i o n of state m u s t be c o m b i n e d w i t h k n o w n

c o m p o n e n t m o l a r v o l u m e s to o b t a i n the m i x t u r e m o l a r v o l u m e . T h e h o p e is that, d u e to a c a n c e l l a t i o n of errors, m u c h s i m p l e r equations of state c a n be u s e d to a c c u r a t e l y d e s c r i b e V ( Ρ , T, X) t h a n w o u l d be necessary E

to y i e l d d i r e c t l y V(P,

T, x)

to the r e q u i r e d a c c u r a c y . T h e V

e

m e t h o d is

o n l y of p r a c t i c a l use w h e n the t e m p e r a t u r e of the s o l u t i o n is b e l o w the c r i t i c a l t e m p e r a t u r e s of t h e m a j o r c o m p o n e n t s . F r o m p r e v i o u s w o r k (4,5,6)

the f o l l o w i n g c o n c l u s i o n s c a n b e d r a w n

c o n c e r n i n g the use of equations of state to represent V (Ρ,

T, x)

E

for

simple l i q u i d mixtures. T h e L o n g u e t - H i g g i n s a n d W i d o m ( L H W ) twop a r a m e t e r e q u a t i o n of state ( 7 ) , b a s e d o n the m o d e l of h a r d spheres i n a u n i f o r m p o t e n t i a l field, c a n b e u s e d to a c c u r a t e l y d e s c r i b e V (T, E

l o w pressures for m i x t u r e s of n o n p o l a r , n e a r l y s p h e r i c a l species.

x)

at

Where

m o l e c u l a r size differences are not l a r g e , this r e l a t i v e l y s i m p l e e q u a t i o n w o r k s as w e l l or better t h a n other equations tested, i n c l u d i n g those u s i n g m o r e r i g o r o u s representations of the h a r d - s p h e r e m i x t u r e c o m p r e s s i b i l i t y factor. A

o n e - f l u i d t h e o r y is g e n e r a l l y s u p e r i o r to a t w o - f l u i d t h e o r y i n

g e n e r a l i z i n g the L H W e q u a t i o n of state f o r m i x t u r e s . I n the o n e - f l u i d theory the same f o r m of e q u a t i o n is u s e d for t h e m i x t u r e as for the components,

b u t the e q u a t i o n of state p a r a m e t e r s are a s s u m e d to

be

composition dependent. V a n der W a a l s ( V D W ) forms, quadratic i n mole fractions, p r o v e satisfactory to d e s c r i b e this c o m p o s i t i o n d e p e n d e n c e for s i m p l e species. E x c e s s v o l u m e p r e d i c t i o n s are e x t r e m e l y sensitive to the v a l u e u s e d for the u n l i k e - m o l e c u l e size p a r a m e t e r . S m a l l d e v i a t i o n s m u s t b e a l l o w e d f r o m the c o m m o n l y u s e d a r i t h m e t i c m e a n m i x i n g r u l e for size p a r a m e t e r s i f q u a n t i t a t i v e results are to b e o b t a i n e d for V

e

s i m u l t a n e o u s l y w i t h other

m e a s u r a b l e excess t h e r m o d y n a m i c f u n c t i o n s ( G

E

and H

E

). I t s h o u l d b e

n o t e d that s i m i l a r c o n c l u s i o n s h a v e b e e n d r a w n f o r m i x t u r e s of g l o b u l a r molecules

(8).


18.

SINGH AND

Mixtures

MILLER

of Simple

325

Species

T h e m e t h o d c a n b e e x t e n d e d to i n c l u d e n o n p h e r i c a l , n o n p o l a r species ( s u c h as the l o w e r m o l e c u l a r w e i g h t a l k a n e s ) b y i n t r o d u c t i o n of a t h i r d p a r a m e t e r i n the e q u a t i o n of state, n a m e l y t h e P r i g o g i n e f a c t o r f o r c h a i n type molecules

( 9 ) . T h i s m o d i f i e d h a r d - s p h e r e e q u a t i o n of state a c c u

r a t e l y describes V ( Γ , x) f o r l i q u e f i e d n a t u r a l gas m i x t u r e s at l o w p r e s e

sures.

Ternary a n d higher mixture V

e

values are a c c u r a t e l y p r e d i c t e d

using only binary m i x i n g rule deviation parameters. Significant deviations b e t w e e n

the model predictions a n d experi

m e n t a t i o n o c c u r i f t h e r e d u c e d t e m p e r a t u r e of a c o m p o n e n t present i n a n a p p r e c i a b l e a m o u n t is h i g h e r t h a n a b o u t 0.85. T h e p u r e l i q u i d f o r Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch018

this c o m p o n e n t becomes h i g h l y e x p a n d e d a t h i g h r e d u c e d t e m p e r a t u r e s , m a k i n g the u n i f o r m p o t e n t i a l field m o d e l suspect. A recent e x p e r i m e n t a l s t u d y (10,11) u r e m e n t s to p r o d u c e V (Ρ,

u s e d d i e l e c t r i c constant meas

T, x) f o r some b i n a r y a n d t e r n a r y m i x t u r e s of

e

the c o m p o n e n t s n i t r o g e n , a r g o n , m e t h a n e , t h e r m a l c o m p r e s s i b i l i t i e s [κ(Ρ, Τ, χ)] p r e s s i b i l i t y o n m i x i n g [Δκ (Ρ, Μ

a n d ethane.

Mixture

iso

a n d changes i n i s o t h e r m a l c o m

Τ, χ)] also w e r e d e r i v e d f r o m these results.

T h e p u r p o s e of t h e present w o r k is to test t h e a b i l i t y of v a r i o u s s i m p l e equation-of-state

forms t o fit t h e V (Ρ,

T, x)

e

surfaces f o r t h e b i n a r y

m i x t u r e s . A n i n t e r c o m p a r i s o n also is m a d e w i t h t h e corresponding-states m e t h o d (1,2).

A l l fitting a n d c o m p a r i s o n s are d o n e o n a consistent basis.

C o m p a r i s o n s a r e m a d e also b e t w e e n values f o r t e r n a r y m i x t u r e s a n d Δ κ

Μ

calculated a n d experimental V

E

values f o r a l l m i x t u r e s . A recent

c o r r e s p o n d i n g states c o r r e l a t i o n f o r i s o t h e r m a l c o m p r e s s i b i l i t i e s (12) is tested also. F u r t h e r details c o n c e r n i n g t h e present i n v e s t i g a t i o n c a n b e f o u n d i n R e f e r e n c e 11.

Equations

of State

for

V

E

A l l of t h e equation-of-state forms c o n

Development of Equations.

s i d e r e d a r e b a s e d o n t h e m o d e l of h a r d c o n v e x p a r t i c l e s i n a u n i f o r m p o t e n t i a l field. I n terms of c o m p r e s s i b i l i t y factor the g e n e r a l f o r m i s

VL

_

2

HP

? _

RT

where z

H P

RTV

(o) K

1

is t h e c o m p r e s s i b i l i t y factor f o r t h e h a r d - p a r t i c l e fluid, a n d a

is a p a r a m e t e r c o n s i d e r e d h e r e to b e i n d e p e n d e n t of b o t h t e m p e r a t u r e a n d density. U s i n g the P r i g o g i n e a p p r o a c h f o r c h a i n - t y p e m o l e c u l e s ( 9 ) ,

2

H P

=

H S

C2 ,


(3)

326

EQUATIONS

where z

H S

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 f o r t h e h a r d - s p h e r e

is a p a r a m e t e r i n t r o d u c e d to a c c o u n t b e c o m e r e s t r i c t e d a t h i g h densities. H S

fluid

and c which

F o r liquids w h i c h are not h i g h l y

(14):

VDW z


considered

a r e d u e to V D W , F l o r y ( F L R ) ( 1 3 ) , L H W ( 7 ) , a n d C a r n a h a n

and Starling ( C S )

H

F L R * H S

L H W

where

STATE

f o r degrees o f f r e e d o m

e x p a n d e d , c is t r e a t e d as i n d e p e n d e n t of Τ a n d V . T h e forms for z

OF

y =

2

= , 1 -

S

1

, 4y '

(4)

_L__

=

=

I I S

A

l

+

(5)

\î>

()

V

6

(i - y)

& / 4 V , a n d & is a p a r a m e t e r c o n s i d e r e d

independent of

temperature a n d density. T h e p a r a m e t e r s a a n d b w e r e d e t e r m i n e d s u c h that e a c h e q u a t i o n of state r e p r o d u c e d the e x p e r i m e n t a l c o m p o n e n t V a n d #c values f o r t h e s

8

saturated l i q u i d s at 100 K . V a l u e s of c w e r e t a k e n d i r e c t l y f r o m the w o r k of R o d o s e v i c h (6).

T h e F , V , * , a n d c values w h i c h w e r e chosen s

s

(11)

8

are g i v e n i n T a b l e I , a n d values of a a n d b w h i c h w e r e d e t e r m i n e d a r e given i n T a b l e I I . T h e sensitivity of V

e

c a l c u l a t i o n s t o t h e c values w a s

tested b y some c a l c u l a t i o n s o n the L H W m o d e l u s i n g c = 1.00 f o r ethane. C o r r e s p o n d i n g values f o r a a n d b are 97.63 χ 1 0 J c m 4

cm

3

3

m o l " a n d 108.04 2

mol" . 1

Table I. Component L i q u i d Molar Volumes ( V ) and Isothermal Compressibilities (#c) at 100 Κ and Vapor Pressure ( P ) Which Were Used along with Shape Factors (c or a) to Determine Equation-of-State Parameters a and b s

s

s

Species

PVMPa

V /cm? mol'

Nitrogen Argon Methane Ethane

0.7790 0.3240 0.0345 0.00001

40.586 30.410 36.544 46.886

s

1

κ'/GPa

1

9.547 3.265 1.701 0.604

c

a

1.03 1.00 1.00 1.50

1.08 0.98 1.00 1.22


18.

SINGH AND

Mixtures

MILLER

Table I I .

327

Species

Equation-of-State Parameters for Components a/J c m mol" and b/cm mol" 3

2

1

3

VDW

FLR

10" Xa b

11.78 28.74

Nitrogen 21.84 17.68 96.84 55.53

17.95 56.79

17.67 54.29

10" Xa b

11.32 23.64

Argon 20.52 82.03

19.04 50.40

18.65 49.54

10" Xa b

18.77 30.63

Methane 33.50 35.16 109.91 69.11

35.63 70.84

35.16 69.11

10" Xa b

52.34 41.65

Ethane 92.44 113.69 154.16 102.07

114.66 104.61

104.74 103.05

4

4

4


of Simple

4

LHW

CS

18.75 49.21

GIB

A s e c o n d m e t h o d of h a n d l i n g n o n s p h e r i c a l p a r t i c l e s i n t h e same g e n e r a l m o d e l is to use t h e s c a l e d p a r t i c l e t h e o r y result o f G i b b o n s ( G I B ) ( 1 5 ) for z

H P

:

G I B ^- \ ^ f \ 1

A

)

(8)

y

w h e r e A = Sa — 2, a n d Β = 1 + 3 α — 3α. I n this f o r m u l a t i o n α is a 2

shape factor f o r h a r d convex particles.

G r a b o s k i (16) has d e v e l o p e d a

c o r r e l a t i o n f o r a i n terms of t h e P i t z e r a c c e n t r i c factor. V a l u e s l i s t e d i n T a b l e I w e r e t a k e n f r o m this c o r r e l a t i o n . T h e a a n d b v a l u e s i n T a b l e I I w e r e d e t e r m i n e d b y fit of V a n d κ at 100 K , d e s c r i b e d a b o v e f o r t h e s

8

other equations. T h e a b o v e e q u a t i o n s of state w e r e a p p l i e d to m i x t u r e s b y a s s u m i n g a one-fluid theory using V D W relations: a = Σ

Σ

ί? = Σ Σ

x&fiij,

(9) (10)

z&jbij,

C=EEWJ

(U)

}

and « =

Σ Σ

(12)

XiXjotij.

T h e cross p a r a m e t e r s w e r e specified b y ( 6 ) α

υ

=

(a a„) i(

>/»(!-

k„) (

m

,


(13)

328

EQUATIONS OF

STATE

(14) and Cij =

i{c

+ Cjj)

(15)

«t; =

i(aii

+

(16)

H

or otjj)

S o m e c a l c u l a t i o n s also w e r e p e r f o r m e d

on the G I B model using the


m i x i n g r u l e suggested b y G r a b o s k i (16):

(17)

E q u a t i o n s 13 a n d 14 f o l l o w f r o m t h e L o r e n t z - B e r t h e l o t r u l e s , a n a r i t h m e t i c m e a n f o r u n l i k e - m o l e c u l e size parameters a n d a g e o m e t r i c

mean

for u n l i k e - m o l e c u l e energy p a r a m e t e r s , w i t h d e v i a t i o n s a l l o w e d f o r either rule. T h e r e a r e t w o m i x i n g r u l e d e v i a t i o n p a r a m e t e r s (&,·/ a n d ; ) w h i c h y

m u s t b e e v a l u a t e d f o r e a c h p a i r of species i n a m i x t u r e . I n t h e present investigation, only binary mixture V

e

data were used i n the evaluation

of these parameters. Discussion of Results. d e v i a t i o n parameters (k

{j

F o r e a c h b i n a r y system t h e t w o m i x i n g r u l e

a n d ; ) w e r e first d e t e r m i n e d b y least squares i;

fits of t h e V ( F , T, x) d a t a t a b u l a t e d as " h i g h - p r e s s u r e d a t a " i n A p p e n d i x e

Β o f R e f e r e n c e 11. T h e s e d a t a are at temperatures f r o m 91 to 115 K , w i t h pressures f r o m n e a r s a t u r a t i o n t o 50 M P a . T h e o n l y d a t a n o t u s e d w e r e for b i n a r i e s c o n t a i n i n g n i t r o g e n at 115 Κ a n d pressures less t h a n 5.5 M P a . N i t r o g e n at 115 Κ ( r e d u c e d t e m p e r a t u r e of 0.91) is a h i g h l y e x p a n d e d l i q u i d at l o w e r pressures, a n d t h e a s s u m p t i o n of a u n i f o r m p o t e n t i a l field is n o t v a l i d

These

(5,17).

data were

omitted from

all

fitting

work

r e p o r t e d i n this c h a p t e r . P u r e - f l u i d p a r a m e t e r s w e r e t a k e n f r o m T a b l e s I a n d I I , a n d m i x i n g rules 9 - 1 6 w e r e u s e d . T h e r e s u l t i n g b i n a r y d e v i a t i o n parameters a n d standard deviations for V T h e results o f t h e d a t a

fitting

e

d a t a fits a r e l i s t e d i n T a b l e I I I .

s h o w little difference

between the

parameters o r p r e d i c t i o n s of the L H W a n d C S equations. E q u a t i o n s 6 a n d 7 y i e l d n e a r l y t h e same z

H S

f o r t h e r a n g e of y values e n c o u n t e r e d i n this

study. T h e L H W , C S , a n d G I B m o d e l s consistently h a v e t h e lowest s t a n d a r d d e v i a t i o n s ; h o w e v e r , t h e V D W m o d e l is n o t g r e a t l y i n f e r i o r . T h e F L R m o d e l yields significantly higher standard deviations only for N CH

4

and A r + C H . 2

G

2

+

T h e r e is little f r o m w h i c h t o choose b e t w e e n t h e

L H W a n d G I B e q u a t i o n fits. I n fact, c a l c u l a t i o n s u s i n g L H W w i t h c =


18.

SINGH

AND

Table III. Fitting V

e

Mixtures

MILLER

0


CS

+ Argon 0.0072 0.0010 0.028

0.0077 0.0009 0.028

0.0067 0.0015 0.029

0.0591 0.0038 0.044

0.0565 0.0036 0.045

Nitrogen 0.0124 -0.0008 0.032

k j s

-0.0018 0.0044 0.050

Nitrogen 0.0597 0.0004 0.074

k

0.0273 0.0052 0.014

Argon -\- Methane 0.0349 0.0379 0.0037 0.0041 0.013 0.007

0.0373 0.0042 0.007

0.0389 0.0043 0.007

i s

0.0279 0.0098 0.030

Argon -f- Ethane 0.1054 0.1178 0.0113 0.0164 0.062 0.024

0.1176 0.0165 0.025

0.1261 0.0177 0.029

k J s

-0.0425 0.0014 0.027

Methane 0.0069 0.0023 0.027

0.0184 0.0044 0.024

0.0292 0.0055 0.024

-\-

+

Methane 0.0597 0.0036 0.044

Ethane 0.0181 0.0044 0.024

The standard deviations (s) are in c m m o l " .

α

3

1.00 f o r C H 2

3

GIB

LHW

-0.0175 0.0022 0.029

k

2

FLR

k j s

i s

cm

329

Species

Binary Interaction Parameters (ky and j y ) Obtained by D a t a [Reference 11, pp. 166-82] to Equations of State VDW

C H

of Simple

6

0

1

( e q u i v a l e n t to G I B w i t h « =

and C H + C H 4

2

6

1.00) gave fits of t h e A r +

d a t a w i t h s t a n d a r d d e v i a t i o n s of 0.026 a n d 0.023

mol" , respectively. These are almost identical w i t h the values i n 1

T a b l e I I I , i n d i c a t i n g that t h e r e p r e s e n t a t i o n of V ( Ρ , T , x) b y these e

e q u a t i o n s of state is insensitive to c ( o r a ) values. c o n t r a d i c t s p r e v i o u s results ( 6 ) w h e r e b o t h V

e

This conclusion

and G

E

d a t a w e r e fit

s i m u l t a n e o u s l y w i t h t h e same f o r m of L H W e q u a t i o n . U s i n g E q u a t i o n 17 i n s t e a d of E q u a t i o n 16 i n t h e m i x i n g rules f o r t h e G I B m o d e l d i d n o t greatly c h a n g e t h e a b i l i t y of t h e e q u a t i o n to fit t h e V

e

data.

T h e r e w e r e o n l y s l i g h t changes i n d e v i a t i o n p a r a m e t e r s a n d

s t a n d a r d d e v i a t i o n s f o r the N -f- C H a n d A r + C H 2

The

4

a b i l i t y of t h e e q u a t i o n s of state t o fit V

2

e

6

systems.

d a t a does d e p e n d t o

some extent o n t h e c o m p o n e n t a a n d b v a l u e s . I n p r e v i o u s studies ( 5 , 6 ) a a n d b w e r e d e t e r m i n e d f o r the L H W m o d e l b y fitting a s a t u r a t e d l i q u i d m o l a r v o l u m e a n d a heat of v a p o r i z a t i o n f o r e a c h c o m p o n e n t .

Using

these a a n d b v a l u e s i n s t e a d o f t h e ones f r o m T a b l e I I l e d to s t a n d a r d d e v i a t i o n s f o r the b i n a r y V

e

fits w h i c h w e r e l a r g e r t h a n those i n T a b l e I I I

(cf. R e f . J l ) . T h e m a x i m u m increase w a s f r o m 0.024 t o 0.048 c m m o l " 3


1

330

EQUATIONS

OF

STATE

f o r A r + C H . T h e c o m p a r i s o n i n d i c a t e s that p u r e - f l u i d p a r a m e t e r s are 2

6

best d e t e r m i n e d f r o m v o l u m e t r i c p r o p e r t i e s o f t h e l i q u i d state, i f t h e p u r p o s e is t o represent o n l y V

e

data.

Binary Deviation Parameters.

T h e problem with

fitting

only one

t y p e of excess p r o p e r t y is t h a t t h e t w o b i n a r y d e v i a t i o n p a r a m e t e r s a r e h i g h l y c o r r e l a t e d . F i g u r e s 1 a n d 2 s h o w some confidence ellipses f o r t h e N

+ A r and A r +

2

C H 2

e

binary parameters i n the L H W model. A n y

p o i n t i n s i d e a confidence ellipse represents a set of can reproduce

the experimental data within

a n dfavalues w h i c h

the standard

deviation

specified f o r t h e ellipse. T h e s o l i d p o i n t s represent t h e o p t i m u m p a r a m Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch018

eters ( f r o m T a b l e I I I ) , f o r w h i c h t h e s t a n d a r d d e v i a t i o n s are 0.028 c m mol'

1

for N

2

+ A r a n d 0.024 c m m o l " f o r A r + C H . 3

w i d e r a n g e of sets of fa a n d V

e

1

2

6

3

T h e r e is a r a t h e r

values f o r either system w h i c h w i l l fit t h e

d a t a w i t h i n 0.04 c m m o l ' , w h i c h corresponds to a b o u t 0 . 1 % o f t h e 3

1

m i x t u r e m o l a r v o l u m e s . T h i s is a reasonable l e v e l f o r t h e average u n c e r t a i n t y i n these V

e

d a t a o b t a i n e d b y t h e d i e l e c t r i c constant m e t h o d .

j Figure 1. Optimum binary deviation parameters (k and j^) and confidence ellipses for the fit of the LHW equation of state to the N + Ar V data of Ref. 11, pp. 166-168 i}

2

e


18.

SINGH A N D M I L L E R

Mixtures

0.16

1

of

Simple

1

331

Species

1

1

0.14


k

—

\\·\\

0.12

—

_

$»0.04ατΡ mol"' ^ \ \ \\ 5

1

s«0.06 cm mol" " * \ \

1

0.08

1

0.01

1

1

0.02

Q03

j Figure 2. Optimum binary devia tion parameters (k^ and j ) and con fidence ellipses for the fit of the LHW equation of state to the Ar + CH V data of Ref. 11, pp. 1 7 5 179 y

2

e

6

I f the h i g h - p r e s s u r e l i m i t i n g V fixed

independently.

models

e

can be determined, then

can be

W h e n a p p l i e d t o a b i n a r y m i x t u r e , a n y of t h e

except V D W y i e l d t h e f o l l o w i n g relations i n t h e l i m i t as t h e

pressure approaches

infinity:

6 = 4V, (b — Xibn

ρ »

(18) — X2b 2)

^jg^

2

and 2[2V^/x x +(b 1

h 2

2

1

F o r systems s u c h as A r - f - C H 2

V of

e

+

11

(6n ' +

=

6

8

6 2 2

and C H

b )/2]^ 22

1 / 8

4

)

1

+ C H 2

6

}

i t is o b v i o u s f r o m t h e

d a t a that 50 M P a is not h i g h e n o u g h pressure to g i v e a close estimate yEoo

F

o

r

systems ( N + A r , N 2

2

+ C H , a n d A r + C H ) i t is 4

not c l e a r f r o m t h e d a t a alone w h e t h e r o r n o t V E x t r a p o l a t i o n s of t h e V

e

4

e0 0

has b e e n a t t a i n e d .

data have been made using the C S model a n d

t h e p a r a m e t e r s f r o m T a b l e s I , I I , a n d I I I . F i g u r e s 3, 4, a n d 5 s h o w t h e


332

EQUATIONS O F S T A T E


1> -2

10

~ I

100

1000

10000

p/MPa Figure 3. V VS. Ρ for equimohr N - f CH as calculated by the CS equation of state (parameters from Tables /, II, and III) E

2

k

results f o r e q u i m o l a r m i x t u r e s of N CH .

The V

4

2

+

CH ,C H + 4

4

values w e r e c a l c u l a t e d f r o m t h e j

e0 0

12

using E q u a t i o n 2 0 . These

figures

C H , and A r + 2

6

values f r o m T a b l e I I I

i n d i c a t e that, e v e n f o r pressures as

h i g h as 1 0 , 0 0 0 M P a , t h e h i g h - p r e s s u r e l i m i t ( V

e 00

) is y e t to b e a t t a i n e d .

A pressure of 1 0 , 0 0 0 M P a is c o n s i d e r a b l y h i g h e r t h a n t h e s o l i d i f y i n g pressures of these m i x t u r e s . I f t h e m o d e l b e h a v i o r is q u a l i t a t i v e l y correct, it appears that d i r e c t d e t e r m i n a t i o n of V

e0 0

is not possible. T h u s , j

12

can

not be e v a l u a t e d i n this f a s h i o n . T h e r e are a n u m b e r of other m e t h o d s w h i c h c a n b e u s e d to u n c o u p l e the ; ' i a n d k

p a r a m e t e r s . O n e of t h e best w a y s w o u l d b e to s i m u l t a n e

12

2

o u s l y fit G , H , a n d V E

E

e

d a t a . T h i s w o r k has n o t p r o c e e d e d t o t h a t stage

as y e t . T h e first m e t h o d t r i e d to u n c o u p l e j

12

to zero.

and k

12

T h i s is n o t e q u i v a l e n t t o setting ;

1 2

w a s to set V

e q u a l t o zero.

e00

equal

T h e result

f r o m E q u a t i o n s 1 4 a n d 2 0 is to use a n a r i t h m e t i c m e a n r u l e d i r e c t l y f o r the cross b p a r a m e t e r , bi = i(b 2

n

+ b ) 22


(21)

18.

SINGH A N D MILLER

Mixtures

ι

ι ι

11

V

0

of Simple

ι

E o e

1

1

1

1

1

1

1

=Q007cm mor 3

333

Species

1 1

' ''

1

l

-0.2

ο

-

ε Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch018

ΊΞ

>

-0.4 _

\οοκ/

//

_\08κ//

-

ΙΙ5Κ/ -0.6

II

I

I I I 10

I

M i l 100

I

I I I I 1000

I

I I I

10000

ρ/ΜΡα Figure 4. V VS. Ρ for equimolar CU + CH as calculated by the CS equation of state (parameters from Tables I, II, and III) E

h

2

6

T h e m i x t u r e b b e c o m e s the m o l e f r a c t i o n average of c o m p o n e n t v a l u e s :

b — xb 1

11

(22)

+ xb 2

22

T h e remaining binary deviation parameter k

12

fitting

the 5 0 M P a V

e

d a t a of R e f e r e n c e 11.

was determined b y again

These k

12

v a l u e s , the c o r r e

s p o n d i n g / i 2 values f r o m E q u a t i o n 20, a n d t h e s t a n d a r d d e v i a t i o n s for the V

e

d a t a fit are l i s t e d i n T a b l e I V f o r the L H W ( V

=0)

e 0 0

model.

B y c o m p a r i s o n w i t h the T a b l e I I I L H W values (also l i s t e d ) , o n l y N

2

+

CH

4

and A r +

C H 2

6

for

are the s t a n d a r d d e v i a t i o n s s i g n i f i c a n t l y

i n c r e a s e d for this m e t h o d . A s e c o n d m e t h o d u s e d to u n c o u p l e / Ί a n d k 2

son a n d H i z a parameters k

12

(18)

w a s to use the R o b i n -

Î2

e m p i r i c a l o b s e r v a t i o n that k

a n d ;Ί

i2

2

=

The binary

6j . 12

w e r e d e t e r m i n e d for the L H W m o d e l b y least

squares fit of the same V

e

d a t a , subject to t h e c o n s t r a i n t k

=

12

6/Ί . 2

The

r e s u l t i n g p a r a m e t e r s a n d s t a n d a r d d e v i a t i o n s for the fits are l i s t e d i n T a b l e I V as t h e L H W ( J t

12

=

6j ) 12

model.

The A r +

C H 2

6

d a t a are


334

EQUATIONS OF

STATE

Ο


Ε "Ε

0 1

ο

I

10

100

1000

10000

ρ/ΜΡα Figure 5. V VS. Ρ for equimolar Ar + C H as calculated by the CS equation of state (parameters from Tables I, II, and III) E

4

m o r e closely fit t h a n b y t h e V slightly worse.

— 0 method, but the N

E0 0

2

+

C H

4

fit is

F o r the other systems t h e d e v i a t i o n s a r e a g a i n n e a r l y t h e

same as the T a b l e I I I results. E i t h e r o f t h e a b o v e m e t h o d s i s u s e f u l i n r e d u c i n g t h e n u m b e r of b i n a r y p a r a m e t e r s w h i c h m u s t b e fit t o e x p e r i m e n t a l V

e

data from t w o

to o n e . I f o t h e r excess p r o p e r t y d a t a a r e c o n s i d e r e d , a r e - e v a l u a t i o n o f these m e t h o d s w i l l b e necessary. Excess volumes were calculated

Predictions for T e r n a r y Mixtures.

f r o m t h e equations o f state f o r t h e n e a r l y e q u i m o l a r t e r n a r y m i x t u r e s o f N

2

+ Ar-f C H

and A r +

4

C H

4

+

C H 2

for w h i c h data are given i n

6

A p p e n d i x Β o f R e f e r e n c e 11. T h e root m e a n square d e v i a t i o n s the experimental a n d calculated V

e

between

values a r e g i v e n i n T a b l e V . O n l y

component a n d binary parameters from Tables I , I I , a n d I I I were used i n these c a l c u l a t i o n s . T h e d e v i a t i o n s f o r t h e t e r n a r y m i x t u r e s a r e o n t h e same o r d e r as those f o r t h e c o n s t i t u e n t b i n a r i e s . T h e s e s i m p l e equations o f state c a n predict ternary V

e

values f o r m i x t u r e s of N , A r , C H , a n d C H 2

4

2

r e q u i r i n g any ternary parameters i n the formulation.


6

without

18.

SINGH AND MILLER

Mixtures of Simple

335

Species

Table I V . Binary Interaction Parameters (kij and and Standard Deviations ( s / c m mol" ) between Model Calculations and the Experimental V D a t a from Reference 1 1 , pp. 1 6 6 — 8 2 3

1

e


LHW (Table

III)

LHW (v*- — 0)

LHW (kii=6j )

h j s

0.0072 0.0010 0.028

Nitrogen 0.0099 0.0004 0.028

fc

0.0597 0.0036 0.044

Nitrogen 0.0717 0.0013 0.053

0.0379 0.0041 0.007

Argon 0.0435 0.0032 0.011

fc

0.1178 0.0164 0.024

Argon + 0.1348 0.0144 0.042

k

0.0181 0.0044 0.024

Methane 0.0198 0.0042 0.025

j s

fc j s

j s

i s

+

+

-\-

MOLi

MOL1

ti

Argon 0.0067 0.0011 0.028

0.0007 0.0043 0.022

— — —

Methane 0.0414 0.0069 0.062

0.0340 0.0117 0.027

0.032 0.015 0.044

Methane 0.0313 0.0052 0.011

0.0250 0.0165 0.015

— — —

Ethane 0.1062 0.0177 0.032

0.0542 0.0319 0.020

— — —

-f- Ethane 0.0231 0.0038 0.026

-0.0092 0.0048 0.025

-0.007 0.004 0.026

Table V . Root Mean Square Deviations ( c m mol" ) between Model Predictions and the Experimental V D a t a for Nearly Equimolar T e r n a r y Mixtures from Rerefence 11, pp. 1 8 3 — 1 8 4 3

1

e

System N + Ar + C H Ar + C H + C H 2

4

4

2

6

VDW

FLR

LHW

0.039 0.034

0.054 0.044

0.029 0.022

CS 0.029 0.022


336

EQUATIONS

OF

STATE

Predictions of Change in Isothermal Compressibility on Mixing.

Λ

q u a n t i t y closely r e l a t e d to the c h a n g e i n excess v o l u m e w i t h pressure at constant t e m p e r a t u r e a n d c o m p o s i t i o n

is the c h a n g e

i n isothermal

compressibility on mixing:

Δκ

Μ

—

κ

- Σ

(23)

I n this e q u a t i o n #c is the m i x t u r e i s o t h e r m a l c o m p r e s s i b i l i t y , a n d the x

{

a n d Ki are the c o m p o n e n t m o l e fractions a n d i s o t h e r m a l c o m p r e s s i b i l i t i e s Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch018

at the same t e m p e r a t u r e a n d pressure as the m i x t u r e .

V a l u e s of

Δκ

Μ

w e r e d e r i v e d f r o m V ( Ρ , T, x) i n R e f e r e n c e 11. T h i s q u a n t i t y w a s q u i t e e

l a r g e for a l l the systems s t u d i e d (10,11).

Whereas the m a x i m u m V

o b s e r v e d w a s o n the o r d e r of 1 0 % of the m i x t u r e V , t h e m a x i m u m Δ κ

ρ/ΜΡα Figure 6. Change in isothermal compressibility on mixing (Δκ ) vs. Ρ for a nearly equimolar mixture of N + CH at 108 K: (·), Ref. 11, p. 170; (—), VDW; (——), FLR; ( LHW. Μ

2

h


E

Μ


18.

SINGH A N D M I L L E R

Mixtures

of Simple

337

Species

p/MPa Figure 7. Change in isothermal compressibility on mixing (Δκ ) vs. Ρ for a nearly equimolar mixture of Ar + C H at 115 K: (·), Ref. 11, p. 176; (—), VDW; ( FLR; (—),LHW. Μ

2

6

w a s a b o u t 1 5 0 % of the m i x t u r e κ. T h e m i x t u r e s w e r e a l w a y s less c o m p r e s s i b l e t h a n a m o l e - f r a c t i o n average of c o m p o n e n t

κ values w o u l d {

indicate. Comparisons have been made between Δ κ experimental V

e

d a t a a n d p r e d i c t i o n s of the equations of state. F i g u r e s

6, 7, a n d 8 are p l o t s of Δ κ N

2

+

values d e r i v e d f r o m the

Μ

C H , Ar +

C H

4

2

6

Μ

vs. Ρ for some n e a r l y e q u i m o l a r m i x t u r e s of

and A r +

CH

4

+

C H . 2

6

T h e d a t a are f r o m

A p p e n d i x Β of R e f e r e n c e 11. T h e curves w e r e d r a w n b y u s i n g t h e V D W , F L R , a n d L H W equations w i t h p a r a m e t e r s f r o m T a b l e s I , I I , a n d I I I . P r e d i c t i o n s of Δ κ

Μ

b y a l l m o d e l s a p p e a r to be i n f a i r l y g o o d agree

m e n t w i t h the e x p e r i m e n t a l d a t a . M a x i m u m d i s c r e p a n c i e s are o b s e r v e d f o r the F L R e q u a t i o n at the lowest pressures. T h e V D W p r e d i c t i o n s are in

s l i g h t l y better o v e r a l l agreement

w i t h e x p e r i m e n t t h a n the L H W

m o d e l , b u t b o t h are q u i t e satisfactory w i t h average d e v i a t i o n s o n t h e o r d e r of 0.1 G P a " . 1

T h e C S a n d G I B models predict Δ κ

Μ

values v e r y

close to those of the L H W m o d e l .


338

EQUATIONS

OF

STATE


0

20

60

40

p/MPa Figure 8. Change in isothermal compressibility on mixing (Δκ ) vs. Ρ for a nearly equimolar mixture of Ar + CH + C H at 115 K: (·), Ref. 11, p. 184; (—), VDW; ( ), FLR; ( LHW. Μ

k

Corresponding-States

2

6

Calculations

Extended Corresponding States.

A n e x t e n d e d c o r r e s p o n d i n g states

m e t h o d has r e c e n t l y b e e n d e v e l o p e d b y M o l l e r u p a n d R o w l i n s o n

(1,2,3)

for a p p l i c a t i o n to l i q u e f i e d n a t u r a l gas a n d other s i m p l e l i q u i d m i x t u r e s . M e t h a n e is u s e d as a reference substance w i t h p r o p e r t i e s f o r this taken from G o o d w i n

(19).

fluid

S i z e a n d e n e r g y - r e d u c i n g p a r a m e t e r s are

t a k e n to b e t e m p e r a t u r e d e p e n d e n t b y use of t h e shape factors d e v e l o p e d b y L e a c h et a l . (20).

M i x t u r e p r o p e r t i e s are c a l c u l a t e d b y u s i n g V D W

o n e - f l u i d e q u a t i o n s , s i m i l a r i n f o r m to E q u a t i o n s 9, 10, 1 1 , a n d 12. T w o b i n a r y i n t e r a c t i o n p a r a m e t e r s are u s e d w h i c h are e q u i v a l e n t to /y a n d i n E q u a t i o n s 13 a n d 14. A l l c a l c u l a t i o n s r e p o r t e d here w e r e m a d e u s i n g a computer program obtained directly from M o l l e r u p

(21).

T h e t w o b i n a r y p a r a m e t e r s w e r e d e t e r m i n e d b y least squares fit of the b i n a r y V

e

d a t a ( 0 - 5 0 M P a ) f r o m A p p e n d i x Β of R e f e r e n c e 11, e x a c t l y

as w a s d o n e f o r t h e equations of state d i s c u s s e d a b o v e . B i n a r y p a r a m e t e r s a n d s t a n d a r d d e v i a t i o n s f o r t h e fits are c o m p a r e d w i t h those f r o m T a b l e I I I f o r the L H W m o d e l i n T a b l e I V . M O L 1 is t h e e x t e n d e d c o r r e s p o n d i n g


18.

SINGH

Mixtures

AND MILLER

of Simple

states w i t h o p t i m i z e d b i n a r y parameters f r o m the V b i n a r y p a r a m e t e r s as d e t e r m i n e d b y M o l l e r u p (21) fitting

liquefied natural

gas

and

339

Species

petroleum

e

fits, a n d M O L 2 uses from simultaneously

gas

densities

and

phase

e q u i l i b r i a d a t a . M o l l e r u p d i d not r e p o r t b i n a r y parameters for systems containing argon.

F r o m the values i n T a b l e I V it is o b v i o u s t h a t the

e x t e n d e d c o r r e s p o n d i n g states are c a p a b l e of d e s c r i b i n g the V (Ρ, T, e

surfaces as a c c u r a t e l y as the best of the s i m p l e equation-of-state

x)

methods.

I n fact, there is a slight i m p r o v e m e n t for the n i t r o g e n - c o n t a i n i n g m i x t u r e s . It is e n c o u r a g i n g that the p a r a m e t e r s o p t i m i z e d to the V

e

d a t a are not

g r e a t l y different t h a n those d e t e r m i n e d b y s i m u l t a n e o u s fit of densities Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch018

a n d phase e q u i l i b r i a for N

2

+

C H and C H 4

4

+

C H . 2

6

F i g u r e s 9 a n d 10 s h o w c o m p a r i s o n s of L H W a n d M O L 1 c a l c u l a t i o n s w i t h experimental V and

Ar +

C H . 2

6

e

values for n e a r l y e q u i m o l a r m i x t u r e s of N

2

+

CH

4

T h e t w o c a l c u l a t i o n a l m e t h o d s y i e l d curves s l i g h t l y

different i n shape, b u t they b o t h fit the d a t a a b o u t e q u a l l y w e l l . V a l u e s of Δ κ

Μ

h a v e not b e e n c a l c u l a t e d u s i n g e x t e n d e d c o r r e s p o n d i n g

states. S i n c e g o o d V of Δ κ

Μ

E

V S . F curves at constant Τ a n d χ are o b t a i n e d , values

s h o u l d be i n reasonable a g r e e m e n t w i t h e x p e r i m e n t a t i o n .

0

20

40

60

p/MPa Figure 9. V VS. Ρ for a nearly equimolar mixture of N + CH^ at 108 K: (·), Ref. 11, p. 170; (—), MOLl;( ),LHW. E

2



340

EQUATIONS

OF

STATE

p/MPa Figure 10. V VS. Ρ for a nearly equimolar mixture of Ar + C H at 115 K: (·), Ref. 11, p. 176; (—), MOLl;( ),LHW. E

2

Corresponding-States B r e l v i a n d O ' C o n n e l l (12)

6

Correlation for Isothermal

Compressibility.

d e v e l o p e d a corresponding-states

correlation

for l i q u i d i s o t h e r m a l c o m p r e s s i b i l i t i e s b a s e d o n a f o r m suggested statistical t h e r m o d y n a m i c s .

A

volume, and critical volume

knowledge

of

the

temperature,

by

molar

are r e q u i r e d to c a l c u l a t e t h e i s o t h e r m a l

c o m p r e s s i b i l i t y at a n y g i v e n state for s i m p l e l i q u i d s . a p p l i e d the c o r r e l a t i o n to m i x t u r e s (22);

These

authors

h o w e v e r , there w e r e n o d a t a

a v a i l a b l e for s i m p l e l i q u i d m i x t u r e s . B a s e d o n d a t a for m o r e m i x t u r e s near r o o m t e m p e r a t u r e , they c o n c l u d e d

complex

that m o l e - f r a c t i o n

or

v o l u m e - f r a c t i o n averages of c o m p o n e n t values y i e l d close a p p r o x i m a t i o n s to m i x t u r e i s o t h e r m a l c o m p r e s s i b i l i t i e s . R e c e n t d a t a (10,11)

for s i m p l e

l i q u i d m i x t u r e s at l o w t e m p e r a t u r e s do not substantiate this c o n c l u s i o n . Extremely

large

deviations

from

simple

component

averages

were

o b s e r v e d for b i n a r y a n d t e r n a r y m i x t u r e s of N , A r , C H , a n d C H . 2

4

2

6

I n this w o r k c o m p o n e n t c r i t i c a l v o l u m e s w e r e t a k e n f r o m R e i d et a l . (23).

T h e y are g i v e n i n T a b l e V I , a l o n g w i t h r o o t m e a n square d e v i a

tions b e t w e e n the c o r r e l a t i o n p r e d i c t i o n s a n d the i s o t h e r m a l c o m p r e s s i b i l i t i e s t a b u l a t e d i n A p p e n d i x A of R e f e r e n c e MPa).

For A r , C H , and C H 4

2

G

11

(pressures u p to 50

the c o r r e l a t i o n is q u i t e a c c u r a t e , g i v i n g

i s o t h e r m a l c o m p r e s s i b i l i t i e s w i t h average d e v i a t i o n s w i t h i n 5 % . tions f o r N

2

are o n this o r d e r for 91 a n d 100 K , b u t t h e y


Devia become

18.

SINGH AND MILLER

Mixtures

of Simple

341

Species

Table V I . Component Critical Volumes ( V ) from Reference 23 and Root Mean Square Deviations (s) between the Βrelvi—O'Connell Correlation (12) and Component Isothermal Compressibilities from Reference 11, pp. 124-128 c

Species N Ar CH CoH

Y /cm

s/GPa

1

89.5 74.9 99.0 148.0

2

4


mol'

3

c

6

1.00 0.09 0.08 0.02

p r o g r e s s i v e l y l a r g e r at 108 a n d 115 K .

N i t r o g e n is h i g h l y c o m p r e s s i b l e

at 115 K . T h e average d e v i a t i o n c o u l d be r e d u c e d s o m e w h a t b y u s i n g a slightly higher V

c

v a l u e for n i t r o g e n .

T h r e e m e t h o d s w e r e tested for c a l c u l a t i n g m i x t u r e i s o t h e r m a l c o m pressibilities. T h e first u s e d a m o l e f r a c t i o n average of c o m p o n e n t v a l u e s :

= Σ

κ Experimental component

(24)

values w e r e u s e d r a t h e r t h a n c o r r e l a t i o n p r e

dictions. T h e second and third methods were based on using a one-fluid m i x t u r e theory to a p p l y the B r e l v i - O ' C o n n e l l corresponding-states

corre

l a t i o n to m i x t u r e s : < -

V

Σ

(25)

V

Σ

cij

C r o s s parameters w e r e c a l c u l a t e d f r o m

V< = [i(V i}

+ V ) / S

1/s

Ci

w i t h the d e v i a t i o n p a r a m e t e r (; ) i;

d + in)I .

i )

3

26

either t a k e n as zero or o p t i m i z e d to

fit b i n a r y m i x t u r e d a t a . In

Table V I I a comparison

is m a d e

between

root

mean

d e v i a t i o n s f r o m e x p e r i m e n t a l b i n a r y d a t a f o r the three methods.

square Experi

mental mixture isothermal compressibilities were taken from A p p e n d i x Β of R e f e r e n c e 11.

C o m p o n e n t values for use i n E q u a t i o n 24 c a m e f r o m

A p p e n d i x A of the same reference.

U s e of κ values f r o m the c o r r e l a t i o n {

w o u l d h a v e y i e l d e d s l i g h t l y h i g h e r d e v i a t i o n s for the systems c o n t a i n ing

N . 2

It is g e n e r a l l y m u c h better to use the E q u a t i o n 25 a p p r o a c h t h a n the m o l e - f r a c t i o n average

of the c o m p o n e n t

values to o b t a i n m i x t u r e

i s o t h e r m a l c o m p r e s s i b i l i t i e s . V e r y little is g a i n e d b y t r y i n g to o p t i m i z e the d e v i a t i o n parameters to b i n a r y d a t a . It w o u l d a p p e a r that /y = a satisfactory a p p r o x i m a t i o n i n this m e t h o d .


0 is

342

EQUATIONS

OF

STATE

Table VII. Root Mean Square Deviations ( G P a ) between Calculated Mixture Isothermal Compressibilities and Experimental Values from Reference 1 1 , pp. 166—184, and Optimum Deviation Parameters (;) for Equation 26 -1

RMS Deviations int κ Using Equations 25 and 26 with System N N Ar Ar 2


2

CH4

+ Ar + C H + C H + C H

(k

4

4

2

-4-

e

C 2 H 6

N + Ar + C H Ar + C H + C H 2

4

4

2

e

1.20 2.27 0.13 0.61 0.19 1.68 0.53

= 0.46 0.44 0.13 0.03 0.06 0.41 0.05

0)

(Optimum 0.39 0.43 0.07 0.01 0.03

Optimum \) n

Values 0.0059 0.0035 0.0052 0.0030 -0.0050

— —

— —

Acknowledgment T h e authors are g r a t e f u l to the U . S . N a t i o n a l S c i e n c e F o u n d a t i o n f o r financial s u p p o r t of this i n v e s t i g a t i o n .

Literature Cited 1. Mollerup, J.; Rowlinson, J. S. Chem. Eng. Sci.1974, 29, 1373. 2. Mollerup, J. Adv. Cryog. Eng. 1975, 20, 172. 3. Haynes, W. M.; Hiza, M. J.; McCarty, R. D. Pap.—Int. Conf. Liquified Nat. Gas 1977, 2, paper 11. 4. Liu, Y.-P.; Miller, R. C. J. Chem. Thermodyn. 1972, 4, 85. 5. Massengill, D. R.; Miller, R. C. J. Chem. Thermodyn. 1973, 5, 207. 6. Rodosevich, J. B.; Miller, R. C. AIChE J. 1973, 19, 729. 7. Longuet-Higgins, H. C.; Widom, B. Mol. Phys. 1964, 8, 549. 8. Herring, W. Α.; Winnick, J. J. Chem. Thermodyn. 1974, 6, 957. 9. Prigogine, I.; Trappeniers, N.; Mathot, V. Faraday Discuss. Chem. Soc. 1953, 15, 93. 10. Singh, S. P.; Miller, R. C. J. Chem. Thermodyn. 1978, 10, 747. 11. Singh, S. P. "Dielectric Constants, Excess Volumes and Compressibilities for Liquid Mixtures of Nitrogen, Argon, Methane and Ethane from 91 to 115 Κ at Pressures to 50 MPa," Ph.D. Thesis, University of Wyoming, Laramie, 1978. 12. Brelvi, S. W.; O'Connell, J. P. AIChE J. 1972, 18, 1239. 13. Flory, P. J. J. Am. Chem. Soc. 1965, 87, 1833. 14. Carnahan, N. F.; Starling, Κ. E. AIChE J. 1972, 18, 1184. 15. Gibbons, R. M. Mol. Phys. 1970, 18, 809. 16. Graboski, M . "A Generalized Hard Sphere Equation of State for VaporLiquid Equilibria Prediction," Ph.D. Thesis, Pennsylvania State Uni versity, University Park, 1977. 17. Miller, R. C. J. Chem. Phys. 1971, 55, 1613. 18. Robinson, R. L.; Hiza, M. J. Adv. Cryog. Eng. 1975, 20, 218. 19. Goodwin, R. D. Natl. Bur. Stand. (U. S.), Tech. Note 1974, No. 653. 20. Leach, J. W.; Chappelear, P. S.; Leland, T. W. Proc. Am. Pet. Inst. 1966, 46, 223.p


18. SINGH AND MILLER Mixtures of Simple Species 343

21. Mollerup, J. "The Computer Program LNG PROPERTY for the Calculation of the Thermodynamic Properties of Natural Gas and Petroleum Gas Mixtures," Instituttet for Kemiteknik, Danmarks Tekniske Hojskole, Denmark, 1977. 22. Brelvi, S. W.; O'Connell, J. P. AIChE J. 1975, 21, 1024. 23. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. "The Properties of Gases and Liquids," 3rd ed.; McGraw-Hill: New York, 1977.


RECEIVED August 15, 1978.


Calculation of Compressed Liquid Excess Volumes and Isothermal

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