18 Calculation of Compressed Liquid Excess Volumes and Isothermal Compressibilities for
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
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 ) {
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
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).
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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 ,
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(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
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
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
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
,
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(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
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
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 =
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
18.
SINGH
AND
Table III. Fitting V
e
Mixtures
MILLER
0
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
18.
SINGH A N D M I L L E R
Mixtures
0.16
1
of
Simple
1
331
Species
1
1
0.14
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
332
EQUATIONS O F S T A T 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.ch018
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
334
EQUATIONS OF
STATE
Ο
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
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.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
E
Μ
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
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 .
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
338
EQUATIONS
OF
STATE
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
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
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 .
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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
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
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
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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.
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
RECEIVED August 15, 1978.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.