The BACK Equation of State and Phase Equilibria in Pure Fluids and

12 The BACK Equation of State and Phase Equilibria in Pure Fluids and Mixtures J. J. SIMNICK, H. M. LIN, and K. CHU CHAO

Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch012

School of Chemical Engineering, Purdue University, West Lafayette, IN 47907

The Boublik-Alder-Chen-Kreglewski augmented hard-core equation is applied to pure fluids and mixtures with special attention to the representation of phase equilibria. Equation constants are determined for twelve substances, and the three constants which are required of nonpolar, nonquantum fluids are correlated with the critical properties and acentric factor. The equation describes mixture-phase equilibria with the introduction of mixing rules and the use of up to two interaction constants for each binary system. V f a n y equations of state h a v e b e e n p r o p o s e d f o r t h e r e p r e s e n t a t i o n of t h e r m o d y n a m i c p r o p e r t i e s of p u r e fluids a n d m i x t u r e s . T h e success of s e v e r a l equations i n t h e q u a n t i t a t i v e d e s c r i p t i o n of some fluid m i x t u r e systems has a d d e d i n c e n t i v e t o t h e f u r t h e r d e v e l o p m e n t of equations of state i n recent years.

T h e n e w equation of K r e g l e w s k i a n d C h e n

(I)

appears p a r t i c u l a r l y a t t r a c t i v e f o r several reasons. I t is h i g h l y a c c u r a t e i n fitting

the PVT b e h a v i o r of a n u m b e r of substances. O n l y a f e w e q u a t i o n

constants a r e r e q u i r e d f o r e a c h substance, a n d these are p r o p e r t i e s of t h e m o l e c u l e s ( or v e r y c l o s e l y r e l a t e d to t h e m ) a b o u t w h i c h m u c h is a l r e a d y k n o w n . I n this w o r k w e a p p l y t h e e q u a t i o n t o p u r e fluids a n d m i x t u r e s of some c o m m o n substances.

S p e c i a l a t t e n t i o n is p a i d t o t h e r e p r e s e n t a -

t i o n of p h a s e e q u i l i b r i a . The

BACK

Equation

T h e B o u b l i k - A l d e r - C h e n - K r e g l e w s k i ( B A C K ) E q u a t i o n is a n a u g m e n t e d h a r d - c o r e e q u a t i o n of t h e f o r m

j f f - z - z

h

+ *

0-8412-0500-0/79/33-182-209$06.25/l © 1979 American Chemical Society

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

(1)

210

EQUATIONS O F

STATE

E q u a t i o n 1 expresses t h a t t h e c o m p r e s s i b i l i t y f a c t o r of a r e a l fluid is the s u m of a r e p u l s i v e t e r m a n d a n a t t r a c t i v e t e r m . C h e n a n d K r e g l e w s k i ( 1 ) suggested u s i n g B o u b l i k ' s h a r d - c o r e e q u a t i o n z

for the repulsive t e r m

h

(2)

a n d to use the p o l y n o m i a l of A l d e r et a l . ( 3 ) f o r t h e a t t r a c t i v e t e r m

z . Thus a

1 + (3« - 2) j + (3α - 3a + l)e

4


-

2

9 h

/ V \

9

N

/V°\

α

2

ξ

3

(

.

M

* - Ç Ç * D « ( è ) (τ)

3

T h e d e n s i t y of the fluid enters i n E q u a t i o n 2 i n the f o r m of ξ d e f i n e d by É=

0.74048 V°/V

(4)

w h e r e V ° is the c l o s e - p a c k e d v o l u m e of t h e m o l e c u l a r h a r d cores.

The

shape of a h a r d c o r e is expressed b y a, w h i c h is d e f i n e d to b e the surface i n t e g r a l of the r a d i u s of c u r v a t u r e d i v i d e d b y three times the m o l e c u l a r volume.

I t is a constant for e a c h m o l e c u l e a n d is e q u a l to u n i t y f o r

spheres b u t greater t h a n one for other c o n v e x bodies. E q u a t i o n 2 reduces to t h e C a r n a h a n - S t a r l i n g (4) T h e constants D

NM

A l d e r et a l . ( 3 )

hard-sphere equation for a =

1.

i n E q u a t i o n 3 originally were determined

by

to fit t h e i r c o m p u t e r - g e n e r a t e d d a t a . C h e n a n d K r e g -

l e w s k i ( I ) r e d e t e r m i n e d the constants b a s e d o n d a t a o n a r g o n . T h e latter set o f constants is u s e d i n this c h a p t e r .

S i n c e t h e i r values h a v e

been

r e p o r t e d i n Réf. 1, t h e y w i l l n o t b e r e p e a t e d here. T h e best d e s c r i p t i o n of l i q u i d s a n d c o m p r e s s e d gases r e q u i r e s the m o l e c u l a r h a r d - c o r e v o l u m e to b e a d e c r e a s i n g f u n c t i o n of t e m p e r a t u r e . T h u s C h e n a n d K r e g l e w s k i express V ° b y means of f ° -= V°°

[1 -

C exp (Su°/kT)]

3

(5)

T h e c h a r a c t e r i s t i c energy u i n E q u a t i o n 3 is i n d e p e n d e n t of t e m p e r a ture for spherical molecules.

H o w e v e r , for nonspherical molecules

u

d e p e n d s o n t e m p e r a t u r e a n d C h e n a n d K r e g l e w s k i use

È-T('+*) in which η =

0 f o r spheres, a n d η > 0 f o r a c e n t r i c m o l e c u l e s .


12.

SIMNICK E T A L .

The BACK

Equation

211

of State

F i v e constants m u s t b e k n o w n for e a c h s u b s t a n c e : V°°, a n d C . O f these five o n l y three m u s t b e d e t e r m i n e d f r o m m e n t a l d a t a , a n d these are V ° ° , a, a n d u°/k.

a, w ° / k , fitting

η/k,

experi

Chen and Kreglewski

suggested a s s i g n i n g values to the other t w o constants: C w a s g i v e n t h e same v a l u e , 0.12, f o r a l l n o n p o l a r substances a n d η/k =

0.6 ω T .

Chen

c

a n d K r e g l e w s k i r e p o r t e d v a l u e s of the constants for 11 substances

(I).

T h e a c c u r a c y of the B A C K e q u a t i o n for t h e r e p r e s e n t a t i o n of d a t a is tested w i t h a r g o n . t h a t is o b t a i n e d .

PVT

F i g u r e 1 shows the q u a n t i t a t i v e a g r e e m e n t

C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a s u c h as i n F i g u r e

1, h o w e v e r a c c u r a t e , is nevertheless f r a g m e n t a r y o n a c c o u n t of the l i m i t e d Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch012

a m o u n t of d a t a o n a n y one substance. I n o r d e r to r e v e a l the b e h a v i o r of the e q u a t i o n o v e r a w i d e r a n g e of c o n d i t i o n s , w e c o m p a r e t h e c o m p u t e d c o m p r e s s i b i l i t y of a r g o n w i t h P i t z e r s g e n e r a l i z e d c o r r e l a t i o n f o r s i m p l e

Figure

1. BACK equation and P V T data for argon: (O), Michels (31); (+), Gibbons Correlation (32); ( ), BACK Equation.


data

212 fluids

EQUATIONS O F

of w h i c h a r g o n is one.

F i g u r e 2 shows t h e c o m p a r i s o n .

STATE

I n the

s m a l l i n s e r t i n t h e figure w e s h o w t h e l i m i t i n g b e h a v i o r of t h e gas as Ρ -»

0.

C l e a r l y the s e c o n d

v i r i a l coefficient

is r e p r e s e n t e d

by

the


equation.

Determining

Equation

Constants

E q u a t i o n constants are d e t e r m i n e d for 12 substances i n t h i s c h a p t e r a n d the results are s h o w n i n T a b l e I. T o d e t e r m i n e t h e three constants V ° ° , a, a n d w ° / k f o r a substance, w e use the c r i t i c a l constants, v a p o r pressure, a n d l i q u i d - d e n s i t y d a t a . A n objective f u n c t i o n is d e f i n e d as the s u m of squares of the r e l a t i v e d e v i a tions of those c a l c u l a t e d f r o m e x p e r i m e n t a l v a l u e s . T h e three e q u a t i o n constants are f o u n d w h e n m i n i m i z i n g the o b j e c t i v e f u n c t i o n . F o r use i n


12.

siMNiCK E T A L .

The BACK

T a b l e I.


13.625 65.751 64.958 65.518 67.740 77.228 88.351 96.556 110.72 54.383 67.013 64.772 79.037

213

of State

E q u a t i o n Constants a

u°/fc,K

1.0004 1.0566 1.0565 1.0498 1.075 1.0720 1.0799 1.0981 1.1349 1.0587 1.0621 1.0583 1.0705

39.171 435.83 432.20 409.59 418.74 468.33 491.00 517.52 558.07 532.12 552.43 522.46 563.23

3

Hydrogen n-Pentane i-Pentane neo-Pentane (Kreglewski) n-Hexane n-Heptane n-Octane n-Decane Benzene Toluene Cyclohexane m-Xylene

BACK

BACK

v°°, cm /mol

Compound

the o b j e c t i v e

Equation

C 0.241 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

0.499 70.72 62.71 51.28 50 90.11 113.77 134.50 181.57 71.50 91.24 70.72 122.54

f u n c t i o n , the c r i t i c a l constants are d e t e r m i n e d f r o m

e q u a t i o n b y s o l v i n g n u m e r i c a l l y the B A C K

the

e q u a t i o n itself a n d

E q u a t i o n s 7 a n d 8,

0

(7)

Τ

T h e v a p o r pressures are c a l c u l a t e d f r o m the B A C K

equation

n u m e r i c a l l y s o l v i n g f o r the v a p o r a n d the l i q u i d densities, p

G

by

and p , L

s i m u l t a n e o u s l y f r o m the t w o f o l l o w i n g equations at a fixed t e m p e r a t u r e = PG

=

(9)

/*L

(10)

PL

w h e r e μ denotes c h e m i c a l p o t e n t i a l a n d Ρ is the pressure. T h e t e m p e r a t u r e r a n g e of the v a p o r - p r e s s u r e a n d l i q u i d - d e n s i t y d a t a u s e d i n the c a l c u l a t i o n s are s h o w n i n T a b l e I I . T h e t e m p e r a t u r e s are c h o s e n to c o v e r the r e d u c e d t e m p e r a t u r e r a n g e of a p p r o x i m a t e l y 0.60 to 1.0 at e v e n intervals. A l s o s h o w n i n T a b l e I I are the r e l a t i v e d e v i a t i o n s of the c a l c u l a t e d v a p o r pressures, a n d saturated l i q u i d a n d v a p o r v o l u m e s . T h e c a l c u l a t e d c r i t i c a l p r o p e r t i e s are g e n e r a l l y i n g o o d with

accepted

experimental values.

a m o u n t s to 1.2%

for T , 3 . 2 % 0

The

average

for P , a n d 4 . 5 % c

absolute

agreement deviation

f o r V . T h e smallest c


214

EQUATIONS O F STATE

Table II. Relative

Temperature Deviations'

1

Vapor


Component n-Pentane i-Pentane neo-Pentane n-Hexane n-Heptane n-Octane n-Decane Benzene Toluene ra-Xylene Cyclohexane

309-455 301-420 282-420 303^93 323-523 333-553 373-603 343-543 353-573 373-603 333-533

AAD°

2.9 0.9 1.5 3.2 3.1 3.3 4.9 1.0

2.6 0.8 1.3 2.8 2.8 3.1 4.5 0.9 1.6 2.4 1.1

1.8

2.7 1.2

(%)

Pressure

rms

b

Range

BIAS

4

-1.6 -0.2 -0.4 -2.1 -1.7 -1.6 -2.3 -0.3 -0.5 -0.5 -0.1

Relative deviations = dev = (experimental value — calculated value)/experi mental value ; N O B = number of observations. The abbreviation r m s = (Σ dev /NOB) 1 /2. β

b

2

deviations are observed for cyclohexane, 0 . 5 % for T , 0 . 6 % for P , a n d c

c

1.3% f o r V . T h e largest d e v i a t i o n s are o b s e r v e d f o r n-decane, 2 . 6 % f o r c

T , 9 . 2 % f o r P , a n d 1 3 . 6 % f o r V . T h e r e is a t e n d e n c y f o r t h e n o r m a l c

c

c

paraffins to s h o w greater d e v i a t i o n s as the c h a i n l e n g t h is i n c r e a s e d . A s a c h e c k o f o u r p r o c e d u r e f o r d e t e r m i n i n g e q u a t i o n constants w e i n c l u d e neo-pentane i n this w o r k f o r w h i c h e q u a t i o n constants h a v e b e e n r e p o r t e d b y C h e n a n d K r e g l e w s k i . T h e set of constants f r o m this w o r k as w e l l as the set b y C h e n a n d K r e g l e w s k i are b o t h p r e s e n t e d i n T a b l e I , a n d t h e y are i n close agreement. b y t h e different d a t a u s e d .

S l i g h t differences a p p e a r to b e c a u s e d

Kreglewski a n d C h e n used A P I Research

P r o j e c t 44 tables w h i l e w e u s e d the recent d a t a of D a s ( 6 ) . W e are interested i n u s i n g the B A C K e q u a t i o n f o r h y d r o g e n m i x t u r e s . T h e r e f o r e w e h a v e d e t e r m i n e d e q u a t i o n constants f o r h y d r o g e n , a n d these are i n c l u d e d i n T a b l e I . PVT d a t a ( 7 ) at t e m p e r a t u r e s of 111-2778 Κ a n d pressures u p t o 1020 a t m are u s e d i n this d e t e r m i n a t i o n . N e i t h e r vapor-pressure n o r critical-point data are used to avoid complications o w i n g to q u a n t u m effects.

I t is f o u n d necessary t o a d o p t a n u n u s u a l

v a l u e of the constant C of 0.241. W i t h this C v a l u e the c a l c u l a t e d pressure shows a r e l a t i v e r o o t - m e a n - s q u a r e d d e v i a t i o n of 0 . 5 % a n d a r e l a t i v e bias of less t h a n 0 . 1 % . A s e n s i t i v i t y analysis has b e e n m a d e of the c a l c u l a t e d results to the. values of the e q u a t i o n constants u s i n g b e n z e n e d a t a . T h e most sensitive constant is ( w ° / k ) .

A v a r i a t i o n o f 1 % f r o m its o p t i m a l v a l u e increases

the error of c a l c u l a t e d v a p o r pressures b y 6 % , a n d o f c a l c u l a t e d l i q u i d


12.

siMNiCK E T A L .

The BACK

Equation

215

of State

a n d F i t t i n g of P u r e F l u i d D a t a Relative


Liquid

Deviations'

(%)

1

Volume

Vapor

Volume

rms

AAD

BIAS

rms

AAD

0.3 0.8 1.3 0.7 2.2 6.1 13.7 0.3 1.1 1.3 1.2

0.2 0.8 1.3 0.6 2.0 6.0 13.7 0.2 1.0 1.1 1.1

0.05 0.8 1.3 0.6 2.0 6.0 13.7 -0.1 -0.3 -0.8 1.1

3.7 2.1 1.6 4.6 5.4 7.9

3.0 1.5 1.3 3.5 4.0 5.7

-0.5 -1.1 -0.7 -1.3 -1.1 -1.4

2.4

1.9

-0.2

— — — —

° A A D = Σ Idevll / N O B . "BIAS = 2 dev/NOB.

v o l u m e s b y 1.0%.

Refs.

BIAS

— — — —

8 9 6 10 10 10 11 10 11-16 11,12,16,17 11

— — — —

S u b s e q u e n t l y a is next i n i m p o r t a n c e .

A 1% variation

i n its v a l u e p r o d u c e s a response of a b o u t 3 % i n c a l c u l a t e d v a p o r pressures a n d 1.4%

i n calculated l i q u i d volumes. A 1 % variation i n V ° °

produces

a n a p p r o x i m a t e l y e q u a l p e r c e n t response i n l i q u i d v o l u m e s , b u t o n l y one t e n t h as m u c h r e l a t i v e c h a n g e i n v a p o r pressure. A 1 % c h a n g e i n (77/k) makes a difference of a b o u t 1.5% l i q u i d volumes.

i n v a p o r pressures, b u t o n l y . 3 % i n

B o t h v a p o r pressure a n d l i q u i d v o l u m e are i n s e n s i t i v e

to C .

Correlation

of

the

Equation

V a l u e s of the B A C K substances.

Constants

e q u a t i o n constants

are n o w a v a i l a b l e for

22

S u i t a b l e correlations of the constants c a n a d d g r e a t l y to t h e

usefulness of the e q u a t i o n . W e h a v e f o u n d V ° ° to b e c o r r e l a t e d w i t h V . F i g u r e 3 shows

the

c

result. V°°

=

A s i m p l e p r o p o r t i o n a l i t y exists for most of the substances 0.21 V .

T h e h i g h e r n o r m a l paraffins s t a r t i n g w i t h C

c

8

with

show

a

t e n d e n c y to d e v i a t e f r o m the l i n e a r r e l a t i o n s h i p . T h e i n t e r a c t i o n e n e r g y u°/k

w a s f o u n d b y K r e g l e w s k i a n d C h e n to

b e e q u a l to t h e c r i t i c a l t e m p e r a t u r e for s m a l l m o l e c u l e s . w e s h o w u°/k

as a f u n c t i o n of T . c

In Figure 4

T h e simple equality holds u p

to

p r o p a n e , a b o v e w h i c h the h y d r o c a r b o n s t e n d to s h o w a c u r v e d o w n w a r d . F i g u r e 5 shows

that t h e shape p a r a m e t e r

a correlates

with

a c e n t r i c factor ω.


the



I

CO

Ο

CO

1

Η

ι

M

to ι—» σ>


12.

siMNiCK E T AL.

The BACK

Equation

of State

>l '.(>i/n) Figure 4.

Correlation

of u"/k


217



CO

*1

Ο

CO

S

!

w

00

to t—'

12.

siMNiCK E T A L .

Mixing

The BACK

Equation

of State

219

Rules

T h e B A C K e q u a t i o n is e x t e n d e d to m i x t u r e s w i t h t h e i n t r o d u c t i o n of m i x i n g rules f o r the e q u a t i o n constants.

T h e f o l l o w i n g m i x i n g rules

are u s e d i n this w o r k : «. =


V °

M

Σ ^ i

α

= T , T N N i j I

( i l )

ί

I

V °

I

(12)

I

(Ϊ) w h e r e Ν stands f o r m o l e f r a c t i o n a n d s u b s c r i p t m denotes m i x t u r e s . T h e l i n e a r c o m b i n a t i o n of a b y E q u a t i o n 11 w a s u s e d also b y K r e g lewski and C h e n (5).

T h e m i x i n g of V ° a n d (u/k)

13 e l i m i n a t e s t h e separate m i x i n g of T h e cross i n t e r a c t i o n terms Vy

0

u°/k, and ( w / k )

i ;

b y E q u a t i o n s 12 a n d

C , a n d ». w i t h i =^ / t h a t a p p e a r i n

the m i x i n g rules are r e l a t e d to t h e p u r e fluid q u a n t i t i e s b y

(*?°«+^°«)'

fwi+m)

(14)

and

E q u a t i o n s 14 a n d 15 define

a n d #cy as the dimensionless cross-inter

a c t i o n constants. A t the present stage t h e y h a v e to b e d e t e r m i n e d f r o m fitting

mixture data.

Fugacity

and

K-Yalue

T h e f u g a c i t y f of C o m p o n e n t i i n a fluid m i x t u r e is expressed {

con

v e n i e n t l y i n terms of a f u g a c i t y coefficient φι /i

V

=

i l/iP V

(16) (17)


220

EQUATIONS O F S T A T E

Table III. Temperature Range (K)

System


Pressure Range (atm)

Ki

rms°

N

2

+ Ar

140-240

1-26

2.9%

N

a

+ Ci

138-183

6-48

2.6 2.3

Cx + 02

158-283

1.76-68

5.4 4.0

244-394

3-122

9.8 9.5

Ce

310-410

3-17

4.7

ÎIC4

327-394

27-166

5.6 6.4

433-533

19-175

5.6

Ci + nC Ci + H2

n

-j"

4

H 2 -f- CeHe

β

Range of Conditions

The abbreviation rms (root-mean-square deviation) = ( Σ

(

dev /NOB) / , 2

1

2

J^calc — UT P\ ex

) ; N O B = number of observations.

W h e n e q u i l i b r i u m exists b e t w e e n a gas m i x t u r e a n d a l i q u i d s o l u t i o n .

1?-1ff o r a l l i c o m p o n e n t s i n t h e system.

(18)

C o m b i n i n g E q u a t i o n s 1 6 - 1 8 gives

a n expression of t h e K - v a l u e of i,

(19) T h e f u g a c i t y coefficient is d e r i v e d f r o m t h e B A C K e q u a t i o n w i t h t h e use o f t h e m i x i n g rules of E q u a t i o n s 1 1 - 1 5 b y f o l l o w i n g s t a n d a r d p r o c e dures o f c l a s s i c a l t h e r m o d y n a m i c s . T h e r e s u l t is g i v e n b e l o w .

F o r brevity

w e h a v e left o u t t h e s u b s c r i p t m f r o m q u a n t i t i e s t h a t a p p l y to t h e fluid m i x t u r e as a w h o l e ; thus, e.g., ζ =

z. m


12.

SIMNICK E T A L .

The BACK

Equation

221

of State

and Fitting of Mixture D a t a K

2

rms


4.8%

Refi - 7 . 8 χ 10"

4

- 6 . 1 Χ 10"

18

3

1.4 0.8

0.02150 0.02949

0 0.0555

m

8.2 8.4

-0.03114 -0.04119

0 -0.1198

19,®

6.8 7.3

-0.09262 -0.09452

0 -0.02367

21-2

4.1

-0.17019

0

27

9.4 5.7

-0.9886 -0.9027

0 0.5116

29

6.4

-1.0907

0.7475

80

RT In φ = RT In φ ί

4

9

/ u \

RTz + N

RT

/V°\

+ «'?Ϊ"»(ΪΡ)(Τ)

u

^

where

( 2 2 )


222

EQUATIONS O F

STATE

dV° dN'i

(23)

j

ρ

(24)


Θ

1η φ =

(α

2

-

=

\

^

V°

1) 1η ( ! - £ )

+

( α + 3α)£ 2

(1 Μ

+ Σ Some Binary

(25)

dNij

+

ζ

-3α£

2

-ξ)

2

(26)

— 1 — 1η ζ

Mixtures

U s i n g the B A C K e q u a t i o n , w e h a v e s t u d i e d t h e p h a s e e q u i l i b r i u m of s e v e r a l b i n a r y m i x t u r e s for w h i c h e x p e r i m e n t a l d a t a are a v a i l a b l e o v e r a n e x t e n d e d r a n g e of c o n d i t i o n s .

T a b l e I I I presents the m i x t u r e systems,

the t e m p e r a t u r e a n d pressure ranges of t h e d a t a , t h e o v e r a l l

fitting

of

K - v a l u e s b y the B A C K e q u a t i o n , a n d the i n t e r a c t i o n constants ν a n d κ. The

BACK

equation

constants

for

the p u r e

components

have

been

r e p o r t e d either i n T a b l e I or i n R e f . J . T h e i n t e r a c t i o n constants ν a n d κ i n T a b l e I I I are d e t e r m i n e d f o r e a c h b i n a r y system b y fitting the e x p e r i m e n t a l K-values

of b o t h c o m p o n e n t s i n

t h e least s q u a r e sense for the r e l a t i v e d e v i a t i o n s . T h e s i m p l e s t m i x t u r e s i n T a b l e I I I are the t w o b n i a r y systems nitrogen w i t h argon and w i t h methane.

of

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

w i t h l i t e r a t u r e d a t a are s h o w n i n F i g u r e s 6 a n d 7. T h e m o l e c u l e s are a l l q u i t e s m a l l a n d s i m i l a r i n i n t e r a c t i o n energies. T h e i n t e r a c t i o n p a r a m e t e r s have small values. F o r nitrogen + zero. F o r nitrogen + m a k e s l i t t l e difference

a r g o n , b o t h p a r a m e t e r s are p r a c t i c a l l y

m e t h a n e , κ has a s m a l l b u t s i g n i f i c a n t v a l u e . B u t i t to t h e h i g h a c c u r a c y i f ν is set e q u a l to

F i g u r e 7 shows t h a t q u a n t i t a t i v e a g r e e m e n t

is o b t a i n e d

even

zero.

i n the

retrograde region. F o u r b i n a r y systems of m e t h a n e are i n c l u d e d i n T a b l e I I I . T h e use of a zero v a l u e of ν is tested o n three of t h e m a n d f o u n d to g i v e t h e same results as the best n o n z e r o v a l u e . I t appears t h a t ν =

these systems, a n d o n l y one i n t e r a c t i o n constant, κ, needs to b e


about

0 for a l l deter-

12.

SIMNICK E T A L .

The BACK

Equation

223

of State

m i n e d for e a c h of these p a i r s . I n F i g u r e s 8 a n d 9 w e c o m p a r e c a l c u l a t e d K - v a l u e s w i t h e x p e r i m e n t a l d a t a for m e t h a n e +

ethane a n d m e t h a n e

+

η-butane, r e s p e c t i v e l y . T w o b i n a r y m i x t u r e s of h y d r o g e n h a v e b e e n s t u d i e d . A l a r g e p o s i t i v e v a l u e of ν a n d a s u b s t a n t i a l n e g a t i v e v a l u e of κ are o b t a i n e d f o r systems.

Having ν =

0 gives

d e f i n i t e l y i n f e r i o r results.

The

both BACK

e q u a t i o n gives a n excellent r e p r e s e n t a t i o n of t h e t w o systems w i t h t h e


use of b o t h i n t e r a c t i o n constants, as s h o w n i n F i g u r e s 10 a n d 11.

70

80

90

100

110 Τ,

120

130

140

150

Κ

Figure 6. Experimental and BACK-predicted K-values for N + Ar: (O), Wilson et al (IS); ( ), BACK Equation; κ = - 7 . 8 Χ ΙΟ ; ρ = - 6 . 1 X 2

4

JO" . 3


EQUATIONS O F

STATE


224

Figure 7. Experimental and BACK-predicted K-values for N + CH : ( O , • , Δ , Ο λ experimental; ( ), BACK Equation; κ = 0.0215; ν = 0. s


h

siMNiCK E T A L .

The BACK

Equation

225

of State


12.

0.031

1

I

2

I

3

Figure 8. Expenmental ethane: ( O , • , Δ , Ο »

I

I

4

I

I

I I I

5 6 7 8910

p,atm

and BACK-predicted λ experimental; ( -0.0311; v = 0.

χ

I

20

1

30

1

40

1—ι 60

ι

ι

80

ι

I

100

K-values for methane + ), BACK Equation; κ =


EQUATIONS O F


226


STATE


12.

The BACK

SIMNICK E T A L .

0.011 10

1

1

20

30

1

Equation

1—

50

227

of State

:—I

1

70

100

p, atm

200

300

1

1

500

Figure 10a. Comparison of K-values of hydrogen in H + η-butane with BACK equation: (O, • , Δ , < 0 > +), experimental; ( ), BACK Equa tion; κ = -0.9027; ν = 0.5116. 2



228

EQUATIONS

1 I

10

ι

20

ι

30

ι

ι

50

ι

i i i i

70

100

p, atm

1

1

200

1

OF STATE

1—ι

400

600

Figure 10b. Comparison of K-values of η-butane in H + η-butane with BACK equation: (O, • , Δ , Ο λ experimental; ( ), BACK Equation; κ = ^0.9027; v = 0.5116. 2


SIMNICK E T A L .

The BACK

Equation

of State


12.


229



to

CO

CO

*1

Ο

co

H Ο

>

M

fO

ο

12.

siMNiCK E T A L .

Discussion

and

The BACK

Equation

231

of State

Conclusion

E v e n t h o u g h the B A C K e q u a t i o n constants are m o l e c u l a r p r o p e r t i e s a n d are k n o w n f o r some m o l e c u l e s , w e p r e f e r to treat t h e m as e m p i r i c a l constants d e t e r m i n e d for the best

fitting

of t h e r m o d y n a m i c d a t a .

We

d o so because m o l e c u l a r p r o p e r t i e s g e n e r a l l y are n o t k n o w n w i t h a c c u r a c y . T h e constants thus o b t a i n e d correlate w i t h p r o p e r t i e s of t h e

fluid

t h a t are r e l a t e d c l o s e l y to m o l e c u l a r p r o p e r t i e s . T h e B A C K e q u a t i o n is c a p a b l e of a c c u r a t e l y d e s c r i b i n g t h e p h a s e e q u i l i b r i a of some p u r e fluids a n d m i x t u r e s . T h e a c c u r a c y appears t h e Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch012

best for fluids of g l o b u l a r m o l e c u l e s , b u t not q u i t e as g o o d for l o n g - c h a i n molecules.

N o r m a l octane seems to be s i g n i f i c a n t l y less w e l l fitted t h a n

the shorter chains a n d n - d e c a n e is e v e n w o r s e .

However,

even

the

r e l a t i v e l y p o o r a c c u r a c y here appears to b e s u p e r i o r to t h a t a t t a i n e d b y a n y other equations of state w h e n a p p l i e d to m i x t u r e s . Glossary

of

Symbols

c

= constant i n E q u a t i o n 5

K

= e q u i l i b r i u m ratio =

M

= i n d e x for E q u a t i o n 3

u n i v e r s a l constants i n E q u a t i o n 3 y/x

N

= index for E q u a t i o n 3

N

= mole fraction

R

= u n i v e r s a l gas constant

T

= absolute t e m p e r a t u r e

= volume fugacity fk — B o l t z m a n n ' s constant

V

P

= pressure

u

= interaction energy

χ = liquid-phase mole fraction y ζ

= vapor-phase mole fraction = compressibility

Greek Letters a =

s p h e r o c y l i n d e r constant i n E q u a t i o n 2

η =

constant f o r i n t e r a c t i o n energy t e m p e r a t u r e d e p e n d e n c e , Equation 6

κ=

interaction energy m i x i n g parameter

μ=

chemical potential

ν=

interaction volume-mixing parameter

φ=

f u g a c i t y coefficient


232

EQUATIONS O F

£=

reduced volume, Equation 4

ρ=

density

ω

=

STATE

acentric factor

Subscripts c =


j\ =

critical property component indices

G =

gas- or v a p o r - p h a s e p r o p e r t y

L =

liquid-phase property

m =

mixture property

r =

reduced property

Superscripts V = v a p o r - or gas-phase p r o p e r t y L = liquid-phase property h =

h a r d sphere

a == a t t r a c t i v e ° =

m o l e c u l a r p r o p e r t y , as i n V ° ,

u°/k

Acknowledgment F u n d s for this r e s e a r c h w e r e s u p p l i e d b y the E l e c t r i c P o w e r R e s e a r c h Institute t h r o u g h r e s e a r c h project R P - 3 6 7 .

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Chen, S. S.; Kreglewski, A. Ber. Bunsenges. 1977, 81(10), 1048. Boublik, T. J. Chem. Phys. 1975, 63(9), 4048. Alder, B. J.; Young, D. Α.; Mark, M. A. J. Chem. Phys. 1972, 56(6), 3013. Carnahan, N . F.; Starling, Κ. E. J. Chem. Phys. 1969, 51(6), 1184. Kreglewski, Α.; Chen, S. S. J. Chim. Phys. 1978, 75(4), 347. Das, T. R.; Reed, C. O.; Eubank, P. T. J. Chem. Eng. Data 1977, 22(1), 16. McCarty, R. D. "Hydrogen Technological Survey—Thermophysical Prop erties"; NASA Spec. Publ. 1975, 3089. Das, T. R.; Reed, C. O., Jr.; Eubank, P. T. J. Chem. Eng. Data, 1977, 22(1), 3. Ibid, p. 9. Young, S., "Pysico-Chemical Constants of Pure Organic Compounds," 2nd ed.; Timmermanns, J., Ed.; Elsevier: Amsterdam, 1950. Zwolinski, B. J., Ed. "Selected Values of Physical and Thermodynamic Properties of Hydrocarbons"; 1953-1977 API Research Project 44, Texas A&M University. Francis, A. Ind. Eng. Chem. 1957, 49(10), 1779.



12. SIMNICK ET AL.

The BACK Equation of State

233

13. Krase, N. W.; Goodman, J. B. Ind. Eng. Chem. 1930, 22(13), 13. 14. Zmaczynski, M . A. "Physico-Chemical Constants of Pure Organic Com pounds," 2nd ed.; Timmermanns, J., Ed.; Elsevier: Amsterdam, 1950; p. 151. 15. Griswold, J.; Andrea, D.; Klein, V. A. "Physico-Chemical Constants of Pure Organic Compounds," 2nd ed.; Timmermanns, J., Ed.; Elsevier: Amster dam, 1965; Vol. 2, p. 99. 16. School of Chemical Engineering, Purdue University, IN, unpublished data. 17. Glaser, F.; Ruland, H. Chem. Ing. Tech. 1957, 29, 772. 18. Wilson, G. M.; Silverberg, P. M.; Zellner, M . G. Adv. Cryog. Eng. 1965, 10, 192. 19. Wichterle, I.; Kobayashi, R. J. Chem. Eng. Data 1972, 17(1), 9. 20. Price, R. Α.; Kobayashi, R. J. Chem. Eng. Data 1959, 4(1), 40. 21. Elliot, D. G.; Chen, R. J. J.; Chappelear, P. S.; Kobayashi, R. J. Chem. Eng. Data 1974, 19(1), 71. 22. Karhre, L. C. J. Chem. Eng. Data 1974, 19(1), 67. 23. Sage, Β. H.; Hicks, B. L.; Lacey, W. N . Ind. Eng. Chem. 1940, 32(3), 1085. 24. Weise, H. C.; Jacobs, J.; Sage, Β. H. J. Chem. Eng. Data 1970, 15(1), 1970. 25. Roberts, L. R.; Wang, R. H.; Azarnoosh, Α.; McKetta, J. J. J. Chem. Eng. Data 1962, 7(4), 484. 26. Sage, B. H.; Budenholzer, R. Α.; Lacey, W. N . Ind. Eng. Chem. 1940, 32(9), 1262. 27. Gunn, D. D.; McKetta, J. J.; Ata, N. AIChE J. 1974, 20(2), 347. 28. GPA Tech. Publ. TP-4, "Low Temperature Data from Rice University for Vapor-Liquid and P-V-T Behavior," April, 1974. 29. Klink, A. E.; Cheh, H. Y.; Amick, Ε. H. AIChE J. 1975, 21 (6), 1142. 30. Connolly, J. F. J. Chem. Phys. 1962, 36(11), 2897. 31. Michels,A.;Levelt, J. M.; DeGraff, W. Physica 1958, XXIV, 659. 32. Gibbons, R. M.; Kuebler, G. P. "Research on Materials Essential to Cryocooler Technology—Thermophysical and Transport Properties of Argon, Neon, Nitrogen, and Helium-4"; 1968, AFML-TR-68-370. RECEIVED September 5, 1978.


The BACK Equation of State and Phase Equilibria in Pure Fluids and

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