5
A Predictive Method for PVT and Phase Behavior of Liquids Containing Supercritical
Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch005
Components PAUL M. MATHIAS and JOHN P. O'CONNELL Department of Chemical Engineering, University of Florida, Gainesville, FL 32611
Concentration derivatives of fugacities are given in terms of statistical-mechanical direct-correlation function integrals which are insensitive to the detailed nature of the intermolecular forces in dense fluids. From this, a simple method is given for the properties of liquids containing supercritical components using only two pure-component parameters and a binary parameter for each pair. With good solvent density data, Henry's constants and easily estimated binary parameter values, we describe high-pressure vapor-liquid equilibrium in such systems as hydrogen-benzene and nitrogenammonia. Also, using a reference-solvent Henrys constant, values can be found for different pure and mixed solvents. These can be used for high-pressure vapor-liquid equilibrium, e.g., for hydrogen in coal oils knowing only hydrogen in quinoline and pure solvent densities.
M
o d e r n c h e m i c a l p r o c e s s i n g d e m a n d s m u c h better estimates o f p h y s i c a l p r o p e r t i e s t h a n i n p r e v i o u s eras because t h e costs of excessive
d e s i g n a r e n o w often too great. H i g h - p r e s s u r e systems a r e of p a r t i c u l a r c o n c e r n , a n d t h e y are b e c o m i n g m o r e p r e v a l e n t as i n c o a l gasification a n d l i q u e f a c t i o n a n d i n F i s c h e r - T r o p s c h syntheses. A w e a k l i n k i n estim a t i o n t e c h n i q u e s f o r these
systems is t h e v a p o r - l i q u i d e q u i l i b r i u m
d i s t r i b u t i o n of c o m p o n e n t s at t e m p e r a t u r e s w e l l a b o v e t h e c r i t i c a l t e m p e r a t u r e of o n e o r m o r e of t h e species p r e s e n t a n d b e l o w t h e c r i t i c a l t e m p e r a t u r e o f t h e others. 0-8412-0500-0/79/33-182-097$05.00/l © 1979 American Chemical Society
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
98
EQUATIONS O F STATE
P r a u s n i t z (1,2)
has d i s c u s s e d this p r o b l e m extensively, b u t t h e
most successful t e c h n i q u e s , w h i c h are b a s e d o n e i t h e r c l o s e d
equations
of state, s u c h as discussed i n this s y m p o s i u m , o r o n d i l u t e l i q u i d s o l u t i o n reference
states s u c h as i n P r a u s n i t z a n d C h u e h ( 3 ) , are l i m i t e d to
systems c o n t a i n i n g n o n p o l a r species o r d i l u t e q u a n t i t i e s o f w e a k l y p o l a r substances.
T h e p u r p o s e of this c h a p t e r is to d e s c r i b e a n o v e l m e t h o d
f o r c a l c u l a t i n g t h e p r o p e r t i e s of l i q u i d s c o n t a i n i n g s u p e r c r i t i c a l c o m ponents w h i c h r e q u i r e s r e l a t i v e l y f e w d a t a a n d is of g e n e r a l a p p l i c a b i l i t y . U s e d w i t h a v a p o r e q u a t i o n of state, t h e v a p o r - l i q u i d e q u i l i b r i u m f o r these systems c a n b e p r e d i c t e d to a h i g h degree of a c c u r a c y e v e n t h o u g h Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch005
the l i q u i d m a y b e 30 m o l % or m o r e of the s u p e r c r i t i c a l species a n d t h e pressure m o r e t h a n 1000 b a r .
General
Expression
T h e basis of the m e t h o d lies i n t h e m o l e c u l a r t h e o r y w h i c h relates integrals
of
the s t a t i s t i c a l - m e c h a n i c a l d i r e c t
correlation function
to
d e r i v a t i v e s of t h e t o t a l pressure a n d the f u g a c i t y of e a c h species
with
respect to t h e c o n c e n t r a t i o n of the species of t h e system (4,5,6).
In
e q u a t i o n f o r m these are
(1)
^(τ;Τ )άν^-0^(Τ )/ )£
)Ε
Ρ
and
dP/RTl d
Pj
where
=
P i
N
\
T
t
P
k
^
=
x , x is t h e m o l e f r a c t i o n , iP
(2)
Σ Χ Λ Ι - C , )
is t h e f u g a c i t y , a n d c
t
y
is t h e
d i r e c t c o r r e l a t i o n f u n c t i o n . T h e essence of o u r m e t h o d is to express t h e d i r e c t c o r r e l a t i o n f u n c t i o n i n t e g r a l s , C y , i n terms of r e d u c e d t e m p e r a t u r e a n d d e n s i t y , i n t e g r a t e E q u a t i o n 2 f r o m a s u i t a b l e reference state T , P , r
p , x to t h e final state Γ, P , x* to solve f o r the final d e n s i t y , p , a n d t h e n r
r
f
f
integrate E q u a t i o n 1 f o r a l l species i n t h e system to o b t a i n t h e final state fugacity,
f r o m t h e reference
state f u g a c i t y (fi/Xi) .
U s i n g a single
r
d u m m y i n t e g r a t i n g v a r i a b l e , t, t h e equations b e c o m e In p = In x{ + In ( ? , / * , ) ' + In (pVp ) r
£
[Σ
W
- p% ) r
-
T)/P
]
D
T
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(
3
)
5.
MATHiAS AND O'CONNELL
Phase Behavior
of
99
Liquids
and
^
-
1
w
-
- / ; [ Σ ^ . 4 > ] *}
{-
A w
(«
w h e r e , i n the integrals
P =
P
' +
(6)
Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch005
T o c o m p l e t e t h e c a l c u l a t i o n , the m o d e l f o r t h e C
i ;
( w h i c h is a f u n c
t i o n of T , p, a n d χ t h r o u g h Τ , T , a n d ^ defined b e l o w ) needs to b e i ;
specified, t h e c h o i c e of reference state m a d e , a n d t h e i n p u t p a r a m e t e r and
p r o p e r t y values n e e d to b e g i v e n . E a c h of these s h a l l b e d e s c r i b e d
b e l o w f o r t h e representative cases w h i c h w e h a v e e x a m i n e d . Direct
Correlation
Function
Integrals
W e use a p e r t u r b a t i o n t h e o r y as t h e basis f o r a corresponding-states expression f o r t h e c o r r e l a t i o n f u n c t i o n i n t e g r a l s , t h e major p o r t i o n b e i n g that f r o m r i g i d spheres a n d a m i n o r c o n t r i b u t i o n f r o m a n a d d e d v i r i a l coefficient
t e r m . ( T h i s c o n c e p t h a d b e e n chosen
second
previously b y
B i e n k o w s k i et a l . ( 7 ) i n another context w h i l e a p p l i c a t i o n of t h e p r i n c i p l e of c o r r e s p o n d i n g states f o r t h e c o r r e l a t i o n f u n c t i o n integrals h a d b e e n examined b y Gubbins a n d O'Connell (8).)
- C V ' & f l
Ο (Τ,Τ ρ) υ
φ
- 2 V » [-iîiL P
-
i ;
yÇ-
(£?)] (7)
S p e c i f i c a l l y , w e h a v e u s e d t h e C a r n a h a n - S t a r l i n g E q u a t i o n (9) f o r r i g i d spheres. E q u a t i o n s 1 a n d 9-18 o f R e f . 9 w e r e u s e d t o o b t a i n E q u a t i o n 8 here. -
Ç0
=
(σ, + σ , ) / ( 1 -
ίβ) + [3σ*Μ
8
+ 3fj (σ#,,)*(σ, + ) + 9(σ,σ&) » / (1 - &)
4
2
+
6ί ( σ ·) 2
σ ί
;
2
In (1 -
h)
2
3
(8)
3
+ [6 + & ( —15 +
(σ, + σι) & [ 6 + £ , ( - 1 5 + 1%) ] / & +
(
+ & (σ,σ,) V (1 - ξ )
Χ {9£ (σ, + σι) + 6 W ;
+
2
+ fo W l / ( 1 -
σ}
-
(σ, + σ , ) + σ σ,]
9&)]/&
2
21 + &(26 - 14&) ) ] / & ( 1 - f » ) } 3
&){£,
-
(σ, + σ,)& + & σ
