7 Three-Parameter, Corresponding-States Conformal Solution Mixing Rules for Mixtures
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of Heavy and Light Hydrocarbons T. J. LEE, L. L. LEE, and Κ. E. STARLING School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019
The conformal solution method is used as the basis for developing mixing rules for the characteristic parameters appearing in a three-parameter, corresponding-states corre lation of thermodynamic properties. A logical extension of the van der Waals one-fluid mixing rules from two to three parameters is shown to yield poor predictions of vaporliquid equilibrium for mixtures of paraffin hydrocarbons with highly dissimilar molecular sizes. Therefore, semiempirical rules were developed with improved predictive capability. The average absolute deviations of predicted methane K-values from experimental binary mixture data for methane with heavier normal paraffin hydrocarbons ranging from ethane through normal decane were 11.0% and 4.6%, respectively, using the modified van der Waals one-fluid mixing rules and the semiempirical mixing rules.
U
sing conformai solution theory models for the prediction of mixture thermodynamic
industrial
behavior
calculations.
is b e c o m i n g
T h e attractiveness
increasingly
popular
of t h e c o n f o r m a i
a p p r o a c h stems l a r g e l y f r o m t h e fact that i t i s faster
for
solution
computationally
t h a n p u r e l y t h e o r e t i c a l methods a n d y e t has a sufficiently g o o d basis i n t h e o r y to a l l o w extension to c o m p l e x m o l e c u l a r i n t e r a c t i o n s (e.g., m u l t i p o l e , d i s p e r s i o n , a n d steric effects), w h i c h w o u l d b e difficult u s i n g p u r e l y e m p i r i c a l methods. T h e f o r m u l a t i o n of c o n f o r m a i s o l u t i o n t h e o r y w h i c h has r e c e i v e d t h e w i d e s t use to date is t h e s o - c a l l e d V D W o n e - f l u i d t h e o r y ( I ) . t h e V D W o n e - f l u i d theory a p p l i e s t o m i x t u r e s of s i m i l a r size
Strictly, molecules
0-8412-0500-0/79/33-182-125$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.
126
EQUATIONS O F S T A T E
f o r w h i c h a l l p a i r p o t e n t i a l s c a n b e expressed Φ^α/σα).
i n the f o r m
=
c
Unfortunately, for many industrial mixtures molecular
y
size
differences c a n b e l a r g e a n d o r i e n t a t i o n effects m a k e i m p o r t a n t c o n t r i b u tions to the p a i r p o t e n t i a l s . T h u s , aside f r o m the a p p r o x i m a t i o n s i n h e r e n t to c o n f o r m a i s o l u t i o n t h e o r y , factors w h i c h a d v e r s e l y affect t h e a c c u r a c y of the V D W o n e - f l u i d t h e o r y f o r the c o m p l e x m o l e c u l a r systems e n c o u n t e r e d i n d u s t r i a l l y i n c l u d e u s i n g the t w o p a r a m e t e r
(c^
and σ ) {ί
pair
p o t e n t i a l a n d r e q u i r i n g of s i m i l a r m o l e c u l a r sizes for the m i x t u r e c o m Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007
ponents. Efforts are i n progress at the U n i v e r s i t y of O k l a h o m a to d e v e l o p a m u l t i p a r a m e t e r , corresponding-states f r a m e w o r k for c o r r e l a t i o n of t h e r m o d y n a m i c properties, taking into account the various orientation contribu tions to p a i r i n t e r a c t i o n s (e.g., d i p o l e - d i p o l e ,
quadrupole-quadrupole,
d i p o l e - q u a d r u p o l e , a n d h i g h e r m u l t i p o l e effects as w e l l as d i s p e r s i o n a n d steric effects).
P r e l i m i n a r y research (2)
i n this d i r e c t i o n has i n v o l v e d
l u m p i n g the c o l l e c t i v e effects of o r i e n t a t i o n c o n t r i b u t i o n s i n t o a s i n g l e t e r m i n the p a i r p o t e n t i a l a n d the r e s u l t a n t expressions for t h e t h e r m o dynamic
properties
from
the
Pople
perturbation theory
(3).
This
a p p r o a c h leads to the t h r e e - p a r a m e t e r , corresponding-states
correlation
f r a m e w o r k r e p o r t e d i n recent w o r k
The
(2)
and used herein.
three
c h a r a c t e r i z a t i o n p a r a m e t e r s i n this c o r r e l a t i o n f r a m e w o r k are t h e c h a r a c t e r i s t i c m o l e c u l a r s i z e / s e p a r a t i o n p a r a m e t e r , σ, the c h a r a c t e r i s t i c m o l e c u l a r energy p a r a m e t e r , c, a n d the c h a r a c t e r i s t i c o r i e n t a t i o n p a r a m e t e r , γ. W i t h i n this t h r e e - p a r a m e t e r , corresponding-states f r a m e w o r k i t is p o s s i b l e to d e r i v e , a l o n g the lines of the m e t h o d u s e d b y S m i t h ( 4 ) , a t h r e e p a r a m e t e r , c o n f o r m a i s o l u t i o n m o d e l , w h i c h is p r e s e n t e d i n S e c t i o n 2. I n the d e r i v a t i o n of the t h r e e - p a r a m e t e r c o n f o r m a i s o l u t i o n m o d e l , c e r t a i n p a r a m e t e r s ( exponents ) i n t h e m i x i n g rules f o r the t h r e e c h a r a c t e r i z a t i o n p a r a m e t e r s are a r b i t r a r y . U s i n g the V D W o n e - f l u i d m i x i n g rules f o r the energy a n d separation parameters, along w i t h a m i x i n g rule for
the
o r i e n t a t i o n p a r a m e t e r d e r i v e d a l o n g the lines of t h e V D W o n e - f l u i d t h e o r y y i e l d s the s o - c a l l e d m o d i f i e d V D W o n e - f l u i d m i x i n g rules i n S e c t i o n 3. T h e m e t h o d o l o g y for the t h e r m o d y n a m i c p r o p e r t i e s c a l c u l a t i o n s p r e s e n t e d h e r e i n is p r e s e n t e d i n S e c t i o n 4. I t is s h o w n i n S e c t i o n 5 t h a t u s i n g t h e modified
VDW
o n e - f l u i d m i x i n g rules y i e l d s a c c u r a t e
predictions
of
m i x t u r e t h e r m o d y n a m i c b e h a v i o r for m i x t u r e s of m o l e c u l e s w i t h d i s s i m i l a r i t i e s as great as m e t h a n e a n d p r o p a n e , b u t t h a t the a c c u r a c y
of
p r e d i c t i o n d e c a y s for l a r g e r m o l e c u l a r d i s s i m i l a r i t i e s . I n S e c t i o n 6, the exponents i n t h e m o d i f i e d V D W o n e - f l u i d m i x i n g r u l e s a r e v a r i e d e m p i r i c a l l y ; t h e r e s u l t a n t m i x i n g r u l e s , r e f e r r e d to h e r e i n as s e m i e m p i r i c a l m i x i n g rules, y i e l d s i g n i f i c a n t l y i m p r o v e d p r e d i c t i o n s f o r m i x t u r e s w i t h c o m p o n e n t s as d i s s i m i l a r as m e t h a n e a n d n o r m a l d e c a n e . T h e i m p l i c a tions of these results are d i s c u s s e d i n S e c t i o n 7.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
7.
Conformai
LEE ET AL.
Anisotropic
Fluid
Conformai
Solution
Mixing
Solution
127
Rules
Model
T h e m e t h o d u s e d here for c o n s i d e r i n g c o n f o r m a i s o l u t i o n m o d e l s for fluids w i t h m o l e c u l a r anisotropics is b a s e d o n the m e t h o d u s e d b y S m i t h (4)
for t r e a t i n g i s o t r o p i c o n e - f l u i d c o n f o r m a i s o l u t i o n m e t h o d s as
a class of p e r t u r b a t i o n methods. T h e objective of the m e t h o d is to closely a p p r o x i m a t e the properties of a m i x t u r e b y c a l c u l a t i n g t h e properties of
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a h y p o t h e t i c a l p u r e reference
fluid.
T h e c h a r a c t e r i z a t i o n parameters ( i n
this case, i n t e r m o l e c u l a r p o t e n t i a l p a r a m e t e r s ) of t h e reference
fluid
c h o s e n to be functions of c o m p o s i t i o n
a n d the
(i.e., m o l e f r a c t i o n s )
are
c h a r a c t e r i z a t i o n p a r a m e t e r s for the v a r i o u s possible m o l e c u l a r p a i r i n t e r actions ( l i k e - l i k e a n d u n l i k e - u n l i k e ) . I n p r i n c i p l e , a l l m o l e c u l a r a n i s o tropics (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, a n d h i g h e r m u l t i p o l e interactions, as w e l l as o v e r l a p a n d d i s p e r s i o n i n t e r a c tions ) c a n be i n c l u d e d i n the m e t h o d . H e r e , t h e v a r i o u s m o l e c u l a r a n i s o t r o p i c s are l u m p e d i n t o a single t e r m , so that the i n t e r m o l e c u l a r p o t e n t i a l e n e r g y u y ( r i , ωι, ω ) b e t w e e n M o l e c u l e s 1 a n d 2 of Species i a n d / c a n b e 2
2
w r i t t e n i n the f o r m U u ( r i 2 , ωχ, ω ) = €ij φ° (—) \ ϋ/
+ SijCij ψ
σ
I n E q u a t i o n 1, r
ωι, ω ) /
ρ
2
(1)
2
\ ϋ σ
is the v e c t o r d i s p l a c e m e n t of the m o l e c u l a r centers of
12
M o l e c u l e s 1 a n d 2, r
is the scaler separation, r
12
i 2
=
|r |, a n d ωχ a n d ω 12
2
are the E u l e r angles d e s c r i b i n g the orientations of M o l e c u l e s 1 a n d 2. T h e first t e r m o n the r i g h t - h a n d side of E q u a t i o n 1 i n v o l v i n g φ
Ό
is
r e c o g n i z e d as a n i s o t r o p i c p o t e n t i a l f o r m , so that the t e r m i n v o l v i n g φ
ρ
describes a n i s o t r o p i c effects. T h e c h a r a c t e r i z a t i o n p a r a m e t e r s σ , e
ijy
ί;
S
ijy
respectively,
are
c h a r a c t e r i s t i c distance,
energy,
and
and
anisotropic
s t r e n g t h parameters for the i n t e r a c t i o n b e t w e e n m o l e c u l e s of Species i a n d /. T h e extension of the i s o t r o p i c m i x t u r e c o n f o r m a i s o l u t i o n m e t h o d of S m i t h (4)
to the case of a n i s o t r o p i c m o l e c u l a r systems c a n b e
easily i n the f o l l o w i n g m a n n e r . T h e q u a n t i t i e s a
b
b y the relations a
c
ih
=
i ;
δ^/σ^
1 1 1
=
.
ih
δ ί / W ,
{;
and c =
8 % σ^,
where
ν
ν
made
are d e f i n e d
ti
the exponents k, 1, m , p , q , r , u , v , a n d w are left u n s p e c i f i e d at this p o i n t i n the d e v e l o p m e n t .
T h e c o n f i g u r a t i o n a l H e l m h o l t z free energy A for a n
a n i s o t r o p i c m i x t u r e t h e n c a n be
expanded
about the configurational
H e l m h o l t z free energy of a h y p o t h e t i c a l p u r e reference c h a r a c t e r i z a t i o n p a r a m e t e r s δ*, e
Xy
A
=
A
* + ^ Σ Σ ^ ζ , Κ · oa
ι
x
j
and σ
(or a
χ
-
a) x
Xy
+
b
x>
fluid,
A
-
b)
Xy
x
- ^ Σ Σ ^ ^ ϋ au i j
x
χ
x
x
+ - ^ ^ Σ Σ ^ ί Μ ^ υ — °χ) + h i g h e r order terms oc
with
c ),
j
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(2)
128
EQUATIONS O F S T A T E
w h e r e χ* is t h e m o l e f r a c t i o n of the i t h c o m p o n e n t i n t h e m i x t u r e .
The
f o l l o w i n g m i x i n g r u l e s a n n u l the first o r d e r t e r m s i n t h e e x p a n s i o n i n E q u a t i o n 2, — E
Z i
i
^
W
^
(3)
8x ^W =
( )
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p
« Λ ν
=
4
Z E z i W « ù X i
(5)
w
i
T h e a p p l i c a t i o n of t h e c o n f o r m a i s o l u t i o n m e t h o d i n i n d u s t r i a l c a l c u l a tions r e q u i r e s the use of the a p p r o x i m a t i o n A =
to a v o i d t h e l e n g t h y
A
x
c o m p u t a t i o n r e q u i r e d to c a l c u l a t e t h e h i g h e r o r d e r terms i n E q u a t i o n 2. T h u s , a p r a c t i c a l strategy for c h o o s i n g t h e exponents k, 1, m , p , q , r , u , v , a n d w i n E q u a t i o n s 3, 4, a n d 5 w o u l d be t h r o u g h m i n i m i z a t i o n of t h e difference A — A
x
( a c t u a l l y , d a t a for a l l a v a i l a b l e m i x t u r e t h e r m o d y n a m i c
p r o p e r t i e s c a n be u s e d s i m u l t a n e o u s l y to d e t e r m i n e t h e exponents
by
r e g r e s s i o n ) . H o w e v e r , m o s t a p p l i c a t i o n s of t h e c o n f o r m a i s o l u t i o n m e t h o d h a v e i n v o l v e d t h e use of exponents b a s e d o n m o l e c u l a r t h e o r y a n d so this a p p r o a c h w a s u s e d i n the i n i t i a l phases of t h e p r e s e n t w o r k . Modified
van
der
Waals
One-Fluid
Mixing
Rules
T h e w e l l k n o w n V D W o n e - f l u i d m i x i n g r u l e s for the c h a r a c t e r i z a t i o n parameters σ
χ
a n d c for i s o t r o p i c fluids are x
σ/ — Σ Σ ^ ϋ €σ Χ
3
χ
i
3
— Σ
Σ
i
^
(6)
3
ν
( ) 7
3
T h u s , t h e V D W o n e - f l u i d r u l e s c o r r e s p o n d to t h e use of t h e f o l l o w i n g v a l u e s of the exponents i n E q u a t i o n s 3 a n d 4, k = 0, q — 1, r = mixtures,
3. S m i t h ( 5 )
Equation 6
is
0, m =
3, ρ
=
the
reasonable
theoretical choice
Smith (5)
has s h o w n f o r h a r d - s p h e r e b i n a r y m i x t u r e s t h a t u s i n g t h e 12
=
m i x i n g rules have
1/2(ση +
been
used).
for
σ
arithmetic mean rule, σ
other
most
specifying
χ
(although
0,1 =
has d i s c u s s e d t h e f a c t t h a t f o r h a r d - s p h e r e Also,
^22), t h e s e c o n d - o r d e r t e r m s i n
E q u a t i o n 2 f o r the H e l m h o l t z free e n e r g y p r o b a b l y c a n b e n e g l e c t e d o n l y w h e n σιι a n d σ
22
differ b y less t h a n a b o u t 1 0 % .
F o r i s o t r o p i c fluids, t h e
p e r t u r b a t i o n e x p a n s i o n of the H e l m h o l t z free e n e r g y a b o u t t h a t of
a
h a r d - s p h e r e system leads to E q u a t i o n 7 w h e n t h e m e a n d e n s i t y a p p r o x i mation
is u s e d
for
the
hard-sphere
pair distribution function
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(5).
7.
Conformai
LEE ET AL.
Solution
Mixing
129
Rules
A l t h o u g h t h e V D W o n e - f l u i d m i x i n g rules y i e l d r e a s o n a b l y predictions
of m i x t u r e b e h a v i o r
for m o l e c u l e s
accurate
w h i c h are not
greatly
d i s s i m i l a r , the cases of e v a l u a t i o n of t h e u n l i k e i n t e r a c t i o n p a r a m e t e r s , σ · and c , where i ^ {;
f r o m the d a t a m a y b e
0
compensatory
i n an
empirical way. F o r t h e d e r i v a t i o n of
a m i x i n g r u l e for the a n i s o t r o p i c s t r e n g t h
p a r a m e t e r , δ*, c o n s i d e r the P o p l e e x p a n s i o n
(3)
of t h e H e l m h o l t z free
e n e r g y , A , a b o u t the free e n e r g y , A , of a n i s o t r o p i c fluid reference system
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0
A =
A0 + Αχ +
A2 +
. . .
(8)
w h e r e A^ are the i t h o r d e r terms i n the e x p a n s i o n . T h e i s o t r o p i c r e f e r e n c e s y s t e m p a i r p o t e n t i a l is d e f i n e d as the u n w e i g h t e d average of the a n i s o t r o p i c p a i r p o t e n t i a l i n E q u a t i o n 1, t h a t is,
(%(ri2, ωι,ω ))ω
where the brackets (
)
ω
(9)
2
- • G O -
d e n o t e t h e a n g l e average.
T h u s , Αχ =
0 and
E q u a t i o n 8 is a p e r t u r b a t i o n e x p a n s i o n f o r A i f h i g h e r - o r d e r t e r m s are small.
F o r s m a l l a n i s o t r o p i c s , t r u n c a t i o n at A
2
is a c c u r a t e , w h i l e
l a r g e anisotropies t h e use of t h e Padé a p p r o x i m a n t u s e d b y S t e l l
A — A
+
0
(1-A /A ) a
2
y i e l d s g o o d results. H e r e i n the t r u n c a t i o n at A order term A
2
for
(6)
2
w i l l be used. T h e second-
is g i v e n b y t h e r e l a t i o n
A2 — ^
E
E
^
W
f
d
r
i
d
r
2
»Qii°
(10)
w h e r e ρ is the m o l e c u l e n u m b e r d e n s i t y , Γ is absolute t e m p e r a t u r e , k is B o l t z m a n n ' s constant, r i a n d r
2
a r e t h e p o s i t i o n vectors of M o l e c u l e s 1
a n d 2 , a n d g ° is t h e i s o t r o p i c p a i r d i s t r i b u t i o n f u n c t i o n . F o r t h e case i n y
w h i c h φ / c a n b e w r i t t e n as t h e p r o d u c t f u n c t i o n {
A
2
^)l>(«i,«2)
(ID
/ à r u ' r u ' V t f a s ' i D * ) .
(12)
becomes A
2
= -
η ^ Σ Σ ^ Α ν ^
8
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
130
EQUATIONS
where r *
F o r e x a m p l e , i f t h e p e r t u r b a t i o n c o n t r i b u t i o n to
— r /a .
12
12
OF STATE
12
t h e p a i r p o t e n t i a l w e r e the o v e r l a p p o t e n t i a l f o r l i n e a r m o l e c u l e s ,
the
p e r t u r b a t i o n c o n t r i b u t i o n c o u l d b e a p p r o x i m a t e d b y t h e f o l l o w i n g expres s i o n , o w i n g to P o p l e
(3),
*w4>if> — so t h a t F Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007
2
(σ< ·Α )
=
{j
0i and 0
;
1 2
[3 c o s 0! -
(jr^J
12
1 2
3 cos θ -
2
and D =
[3 c o s 0 2
2
-
X
2]
2
3 cos
θ
2
2
-
(13) 2],
where
are t h e p o l a r angles of o r i e n t a t i o n of M o l e c u l e s 1 a n d 2.
To
o b t a i n the expression f o r δ^ t h e f o l l o w i n g a p p r o x i m a t i o n is i n t r o d u c e d ,
g i j
where r * =
° ( ^ ' j ^ ' ™ ' *
r/a
Xy
ρ* =
1
'
ρσ , Τ* = χ
3
* · - ) -
f
kT/e .
T h e approximation i n E q u a
x
o
°
(
*
r
^
T
#
)
( 1 4 )
t i o n 14 is s i m i l a r to, b u t m o r e stringent t h a n , the m e a n d e n s i t y a p p r o x i m a t i o n . W i t h the a s s u m p t i o n i n E q u a t i o n 14, A
becomes
2
A,=
~ ^ f f i T Z ^ W ^ ' /
drV"**0.°
(15)
I t is t h e n l o g i c a l to choose the f o l l o w i n g m i x i n g r u l e f o r the a n i s o t r o p i c s t r e n g t h p a r a m e t e r ( o v e r l a p p a r a m e t e r i n t h e specific e x a m p l e ) δ*, — Σ Σ ^ Α Λ ί Α ϋ i
(16)
8
3
T h i s m i x i n g r u l e c o r r e s p o n d s to t h e use of t h e f o l l o w i n g values of exponents i n E q u a t i o n 5, u = free energy, A * =
2, ν =
2, w =
w h e r e Ν is t h e n u m b e r of m o l e c u l e s
A/NkT,
the
3. T h e r e d u c e d H e l m h o l t z then
takes the f o r m A* w h e r e ρ* — ρσ \ Γ * = χ
A * G
kT/t
x>
δ , ΤΓ (D ) 2
(17)
p*JJ(T*)
2
2
a n d J is t h e i n t e g r a l x
J.—f
(18)
dr*r* F g ° 2
2
x
N o t e t h a t A * is of t h e f o r m A* = A * 0
(19)
+ s f*(T*, *) 2
x
P
T h i s result is i d e n t i c a l to t h e expression w h i c h is o b t a i n e d f r o m t h e p e r t u r b a t i o n e x p a n s i o n of A f o r a p u r e fluid. T h u s , r e f e r r i n g to E q u a t i o n 2, t h e first-order c o n f o r m a i s o l u t i o n r e l a t i o n for a n i s o t r o p i c fluids is ^ * ( Γ , ρ, { σ „ } ,
{8
(21)
*)
Ρ
€ , a n d δ* g i v e n i n E q u a t i o n s X
6, 7, a n d 16. T h e equation-of-state expression f o r t h e absolute pressure Ρ is o b Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007
t a i n e d f r o m E q u a t i o n 19 u s i n g t h e t h e r m o d y n a m i c r e l a t i o n
pkT
Y
dp*
/ ν,
(22) τ
t h e r e s u l t a n t expression f o r t h e c o m p r e s s i b i l i t y f a c t o r Ζ = P/pkT is Ζ = Z + 8.*Zi
(23)
0
where
η
\
Op
/ Ν,
(24) Τ
' ' - φ ) , . , - Λ Μ = ^ ) 1 . Calculation
of Thermodynamic
( 2 5 )
Properties
F o r t h e c a l c u l 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 , E q u a t i o n 2 3 w a s used i n a n empirical manner.
O n l y d a t a f o r n o n p o l a r n o r m a l paraffin
h y d r o c a r b o n systems w e r e u s e d i n t h e c o r r e l a t i o n d e v e l o p m e n t so t h a t as a n a p p r o x i m a t i o n , t h e P i t z e r a c e n t r i c factor, ω, c o u l d b e t a k e n as a n estimate of t h e c o l l e c t i v e s t r e n g t h of m o l e c u l a r anisotropics (i.e., δ ω).
2
=
B e c a u s e t h e use of t h e resultant c o r r e l a t i o n f o r p o l a r systems w a s
a n t i c i p a t e d , t h e p a r a m e t e r y ( γ = δ ) , r e f e r r e d to h e r e i n as t h e o r i e n t a t i o n 2
p a r a m e t e r , w a s u s e d i n s t e a d of t h e a c e n t r i c f a c t o r fluids).
(γ ^
ω for other
T h e e q u a t i o n of state i n E q u a t i o n 2 3 t h e n takes t h e f o r m Ζ(Τ·,
Λ
γ) = Ζ AT*,
ρ*) + γΖΛΤ*,
w h e r e Ζ 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 a n d Z
0
(26)
ρ*)
a n d Z i are f u n c t i o n s of t h e
r e d u c e d t e m p e r a t u r e T * = kT/e a n d r e d u c e d d e n s i t y ρ * = ρσ . 3
T h e equation-of-state
f o r m u s e d h e r e i n is t h e m o d i f i e d
Benedict-
W e b b - R u b i n ( M B W R ) e q u a t i o n as g i v e n b y H a n a n d S t a r l i n g ( 7 ) . I t is r e f o r m u l a t e d i n t o t h e f o r m of E q u a t i o n 26 b y expressing t h e constants appearing linearly i n the equation into t w o parts—one isotropic part a n d one a n i s o t r o p i c p a r t , Bi =
ai + ybi
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(27)
132
α
Λ
EQUATIONS O F
STATE
b e i n g the i s o t r o p i c p a r t a n d bi b e i n g t h e a n i s o t r o p i c p a r t , w h e r e as
noted above, γ ~
δ is a n o r i e n t a t i o n p a r a m e t e r a c c o u n t i n g f o r t h e n o n 2
s p h e r i c i t y of the m o l e c u l e - p a i r p o t e n t i a l s u n d e r c o n s i d e r a t i o n .
Therefore,
t h e M B W R e q u a t i o n c o r r e s p o n d i n g to E q u a t i o n 26 assumes the f o r m l + f[Bx
Ζ _ +
* [B
P
2
-
5
Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007
+
- B T*' B T*~i
-
X
2
B T*~
3
3
- B T*- ]
Q
Bsp* T*~* 2
9
+ ρ** [B T*-
2
10
+ B T*-* 7
[ (1 + B
4 P
X
* ) exp ( - V 2
2
-
BiiT*" ]
+
B T*' ]
)
5
12
2
]
(28)
w h e r e fo i n E q u a t i o n 27 is z e r o to i n s u r e l i n e a r i t y of Ζ i n γ, ρ* is t h e 4
r e d u c e d d e n s i t y , ρ* = kT/e.
ρσ , 3
a n d T * is t h e r e d u c e d t e m p e r a t u r e , T *
T h e c h a r a c t e r i s t i c m o l e c u l a r d i s t a n c e p a r a m e t e r , σ, a n d
=*
energy
p a r a m e t e r , c, w e r e e s t i m a t e d f r o m the c r i t i c a l constants u s i n g the r e l a t i o n s ,
0.3189
(29)
Pc _ c
—
kT
c
(30)
1.2593
w h e r e k is t h e B o l t z m a n n constant. P e r t i n e n t relations f o r other t h e r m o d y n a m i c p r o p e r t i e s h a v e b e e n p r e s e n t e d elsewhere
(2).
E q u a t i o n s 29
a n d 30 are b a s e d o n t h e r e l a t i o n s h i p s of t h e L e n n a r d - J o n e s
(12-6)
p o t e n t i a l p a r a m e t e r s for a r g o n to the a r g o n c r i t i c a l constants. T h e use of E q u a t i o n s 29 a n d 30 i n the M B W R e q u a t i o n of state g i v e n i n E q u a t i o n s 27 a n d 28 w o r k s w e l l for p u r e n o r m a l paraffin h y d r o c a r b o n s . v e r s a l constants α a n d bi, i = έ
1, . . . 12 (fo
4
=
s i m u l t a n e o u s l y u s i n g d e n s i t y , v a p o r pressure, a n d e n t h a l p y data for
methane
The uni
0) were determined b y departure
t h r o u g h n o r m a l d e c a n e i n m u l t i p r o p e r t y analysis.
A v e r a g e absolute d e v i a t i o n s of p r e d i c t e d f r o m e x p e r i m e n t a l p r o p e r t i e s w e r e 1.00%
f o r d e n s i t y , 1.13 B t u / l b for e n t h a l p y a n d 0 . 8 5 %
Table I .
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane
for v a p o r
Generalization Parameters of Pure Materials to Be Used with Generalized Equation of State Critical Temp. (°F)
Critical Density (lb-mol/cu
Molecular Weight ft)
-116.43 90.03 206.13 305.67 385.42 453.45 512.85 563.79 610.50 651.90
0.6274 0.4218 0.3121 0.2448 0.2007 0.1696 0.1465 0.1284 0.1150 0.1037
16.042 30.068 44.094 58.12 72.146 86.172 100.198 114.224 128.24 142.276
Orientation Parameter (y) 0.01289 0.09623 0.1538 0.1991 0.2530 0.3054 0.3499 0.4004 0.4463 0.4880
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
7.
LEE E T AL.
pressure.
Conformai
T h u s , the
Solution
Mixing
133
Rules
m u l t i p a r a m e t e r , corresponding-states
correlation
f r a m e w o r k p r o v i d e d b y the p e r t u r b a t i o n e q u a t i o n f o r m i n E q u a t i o n 26 a n d the r e s u l t a n t g e n e r a l i z e d M B W R e q u a t i o n i n E q u a t i o n 28 y i e l d s g o o d results for the p u r e n o r m a l paraffin h y d r o c a r b o n s .
V a l u e s of the c r i t i c a l
constants a n d o r i e n t a t i o n p a r a m e t e r s u s e d i n this w o r k are g i v e n i n T a b l e I , w h i l e the values of the constants
a n d h i i n E q u a t i o n 2 7 are g i v e n
in Table II.
Downloaded by NORTH CAROLINA STATE UNIV on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch007
Table II.
Generalized Parameters Used in the M B W R Equation B, =
ParameteT i 1 2 3 4 5 6 7 8 9 10 11 12 Use
of
the
The
Modified
van
modified V D W
a, + y b i
ai
b,
1.45907 4.98813 2.20704 4.86121 4.59311 5.06707 11.4871 9.22469 0.094624 1.48858 0.015273 3.51486
0.32872 -2.64399 11.3293
der
Waals
2.79979 10.3901 10.3730 20.5388 2.76010 -3.11349 0.18915 0.94260
One-Fluid
Rules
o n e - f l u i d m i x i n g rules f o r
σ, χ
e
x
and S
in
x
E q u a t i o n s 6, 7, a n d 16 w e r e u s e d to d e t e r m i n e the a b i l i t y of this f o r m u l a t i o n of the c o n f o r m a i s o l u t i o n m o d e l f o r p r e d i c t i n g m i x t u r e b e h a v i o r . T h e f o l l o w i n g relations w e r e u s e d f o r u
ih