9
A New Equation of State
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HIDEZUMI SUGIE Nagoya Institute of Technology, Gokiso, Showa, Nagoya, Japan BENJAMIN C.-Y. LU University of Ottawa, Ottawa, Ontario, Canada
A new pressure-explicit equation of state suitable for calculating gas and liquid properties of nonpolar compounds was proposed. In its development, the conditions at the critical point and the Maxwell relationship at saturation were met, and PVT data of carbon dioxide and Pitzers table were used as guides for evaluating the values of the parameters. Furthermore, the parameters were generalized. Therefore, for pure compounds, only T , P , and ω were required for the calculation. The proposed equation suecessfully predicted the compressibility factors, the liquid fugacity coefficients, and the enthalpy departures for several arbitrarily chosen pure compounds. c
c
Τ Η h e p u r p o s e o f this i n v e s t i g a t i o n is to d e v e l o p a n e w p r e s s u r e - e x p l i c i t e q u a t i o n of state w h i c h : ( 1 ) y i e l d s a c c e p t a b l e
values of Z , a n d c
satisfies t h e u s u a l t w o i n i t i a l pressure—volume d e r i v a t i v e s a t t h e c r i t i c a l p o i n t a n d t h e M a x w e l l r e l a t i o n s h i p at s a t u r a t i o n ( e q u a l f u g a c i t y f o r c o e x i s t i n g l i q u i d a n d v a p o r p h a s e s ) ; ( 2 ) is s u i t a b l e f o r r e p r e s e n t i n g PVT b e h a v i o r of l i q u i d a n d gas phases over a w i d e r a n g e o f t e m p e r a t u r e a n d pressure; a n d ( 3 ) c a n b e i n t e g r a t e d a n d d i f f e r e n t i a t e d easily f o r o b t a i n i n g d e r i v e d t h e r m o d y n a m i c p r o p e r t i e s . I t is a n t i c i p a t e d t h a t t h e p a r a m e t e r s of t h e r e s u l t i n g e q u a t i o n c a n b e g e n e r a l i z e d i n terms o f t h e c r i t i c a l p r o p e r t i e s a n d t h e a c e n t r i c f a c t o r ω. T h e a p p l i c a t i o n is l i m i t e d , h o w e v e r , to p u r e n o n p o l a r
compounds. 0-8412-0500-0/79/33-182-163$05.50/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.
164
EQUATIONS
Development
of
the
Proposed
Equation
of
OF STATE
State
A s u i t a b l e e q u a t i o n of state m u s t satisfy c e r t a i n l i m i t i n g c o n d i t i o n s a n d f o l l o w some g e n e r a l trends.
O n e of the m o r e i m p o r t a n t c o n d i t i o n s
is t h a t the e q u a t i o n of state m u s t r e d u c e at l o w pressures a n d at a l l t e m p e r a t u r e s to the i d e a l gas e q u a t i o n .
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-
p
Hence (1)
=1
v
RT
as Ρ - » ο a n d V —» oo. E x p r e s s i n g Ρ i n terms of a p o w e r series of
1/V,
the f o l l o w i n g e q u a t i o n w a s o b t a i n e d :
p
=
A .
i n w h i c h h m u s t b e e q u a l to x
• A _
ι
ι
l i
(2)
RT.
A n o t h e r c o n d i t i o n is t h a t t h e v o l u m e of a l l gases at h i g h pressures approaches
a l i m i t i n g v a l u e w h i c h is " p r a c t i c a l l y i n d e p e n d e n t
of
the
t e m p e r a t u r e a n d close to 0.26 V " as suggested b y R e d l i c h a n d K w o n g c
(I).
K e e p i n g this i n m i n d , the d e v e l o p m e n t
of t h e n e w e q u a t i o n
of
state b e g a n u s i n g the f o l l o w i n g e x p r e s s i o n :
V
R
T
- b
(3)
+i(T,V)
where
b
— 0.26
V
(4)
c
I f f ( T , V ) c o u l d b e r e p r e s e n t e d b y a g r o u p of terms s u c h as t h a t expressed i n E q u a t i o n 5,
H
T
'
V
)
V\ (V + k x ) - ! (V + k ! ) » . . . . (V + k ) » ~
=
m
+ * "·
( 5 )
the d i f f e r e n t i a t i o n of E q u a t i o n 3 a n d the i n t e g r a t i o n of t h e t h e r m o d y n a m i c expressions for e v a l u a t i n g fugacities a n d e n t h a l p y d e p a r t u r e s , as r e p r e s e n t e d b y E q u a t i o n s 6 a n d 7, w o u l d b e s i m p l i f i e d . (Ζ -
(H*-H°)
T
=
P V - R T -
J
V
[ p - T
1) - y -
(6)
dV
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(7)
9.
suGiE A N D L U
A New
I n E q u a t i o n 5, l q
Equation
165
of State
are constants, w h i l e n
k
m
n
m
0
are either
z e r o or p o s i t i v e integers. A t t h e c r i t i c a l p o i n t , the c r i t i c a l i s o t h e r m shows a p o i n t of i n f l e c t i o n . Hence
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(8)
and
m
-
,
T h e r e l a t i o n s h i p b e t w e e n Z a n d ω m a y b e expressed as f o l l o w s : c
Z =
0.291 -
C
0.080ω
(10)
E x p r e s s i n g E q u a t i o n 3 for the c r i t i c a l i s o t h e r m y i e l d s
P = y ^ + H T
(11)
, V )
C
w h i c h also m u s t satisfy E q u a t i o n s 8 t h r o u g h 10 at t h e c r i t i c a l p o i n t . C o n s e q u e n t l y , i t w o u l d b e m o r e c o n v e n i e n t for m a t h e m a t i c a l m a n i p u l a t i o n i f f ( T , V) c
w a s expressed i n terms of three constants w h i c h c o u l d b e
d e t e r m i n e d b y E q u a t i o n s 8 t h r o u g h 10. It w a s p r o p o s e d that f ( T , V )
be r e p r e s e n t e d b y three t r u n c a t e d
c
expressions
of
denominator. or
E q u a t i o n 5, w i t h
each
c o n t a i n i n g three terms i n
I n a d d i t i o n , t h e k's w e r e a s s u m e d to b e either —b,
the zero,
Consequently,
U
T
ν
a(T )
λ
,
e
l \ * c , V )
_ _
φ
6
)
n
l
V
n
( y
2
+
b
) n
t
3
C(T ) C
(
y
_
h
) n t f
%
{
y
+
d(T ) (V -
T h e q u a n t i t i e s a(T ), c
b) 7V *(V n
)
%
,
c
^
b
n
+
b) > n
v
a n d d(T )
c(T ), c
, }
were determined from Equations
c
8 t h r o u g h 10. T h e ris w e r e t a k e n to be either z e r o o r p o s i t i v e integers. I n a d d i t i o n , the f o l l o w i n g r e l a t i o n s h i p b e t w e e n the ris m u s t be satisfied: 2 ^ ni + n + 2
n
3
< n + n 4
5
+
< n + n 7
8
+ rig
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(13)
166
EQUATIONS O F S T A T E
H o w e v e r , t h e s e c o n d v i r i a l coefficient d e r i v e d f r o m E q u a t i o n 11 w o u l d n o t b e t e m p e r a t u r e d e p e n d e n t i f ηχ +
n
2
+
n
^
3
3. C o n s e q u e n t l y , i n
E q u a t i o n 13,
ni + n + n = 2 2
(14)
3
and
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4
5
6
7
8
(15)
9
E q u a t i o n 11 t h e n w a s u s e d to fit t h e c r i t i c a l i s o t h e r m (i.e., t h e c u r v e r e l a t i n g Ρ a n d V at T ) o f c a r b o n d i o x i d e , as r e p o r t e d b y M i c h e l s ( 2 ) , c
u s i n g v a r i o u s values of ris u n d e r t h e c o n d i t i o n s o f E q u a t i o n s 14 a n d 15 a n d t h e v a l u e s o f a(T ),
c ( T ) , a n d d(Tc)
c
determined from Equations
c
8 t h r o u g h 10. T h e best fit w a s o b t a i n e d u s i n g t h e f o l l o w i n g v a l u e s : n i =
0; ri2 = 1; n
3
1; n
—
1; n = 2; η
—
4
5
β
—
1; n
7
0; n
—
—
8
0; a n d n = 7. 9
H e n c e , E q u a t i o n 11 took t h e f o l l o w i n g f o r m :
P
R T
=
*
_
7 - b
(r )
fl
c
F ( F + 6)
c(T )
, +
d(T )
e
c
( 7 - 6 ) 7 2
(
y
+
6
(7 + 6)
)
7
(16) T h e first t w o terms o f t h e r i g h t - h a n d side o f E q u a t i o n 16 a r e i n t h e same f o r m as t h e w e l l - k n o w n R e d l i c h - K w o n g ( R K ) e q u a t i o n of state ( I ) . T h e a c e n t r i c f a c t o r o f c a r b o n d i o x i d e is 0.225. I n o r d e r t o e x t e n d t h e a p p l i c a b i l i t y of E q u a t i o n 16 to a w i d e r r a n g e of ω (0 < ω < the
(V +
terms o f E q u a t i o n 16 w e r e
b)
modified
0.5),
as expressed i n
E q u a t i o n 17. RT
€
V - b
_
a(T )
c(T )
c
V ( y + 6i)
d(T )
c
(V -
b) V
2
c
(V + b ) 2
(V +
b) 3
7
(17) where,
b — (0.1181 + 0.4730ω) V ±
c
(18)
b = (0.2117 + 0.1611ω) V
c
(19)
b — (0.2515 + 0.0283ω) V
c
(20)
2
3
T h e ω v a l u e s u s e d i n this s t u d y a r e i d e n t i c a l t o those p r e v i o u s l y r e p o r t e d ( 3 ) . T h e c a l c u l a t e d v a l u e s o f c r i t i c a l i s o t h e r m pressures u s i n g E q u a t i o n 17 a r e c o m p a r e d w i t h P i t z e r s t a b l e i n T a b l e 1 ( 4 ) . T h e average absolute
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
9.
suGiE A N D L U
A New
Table I.
Equation
of State
167
Comparison of Calculated Crtical Isotherm Pressures with Pitzer's Table Average
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Factor
β
(ω)
Region'
0.0
V
0.1
V
0.2
V
0.3
V
0.4
V
0.5
V
RK
I I I I
Deviation This
(%) Work
0.2 6.7 0.2 5.4 0.1 4.0 0.0 4.9 0.1 5.2 0.1 7.0
—
I r
(\)
0.2 16.0 0.3 34.0 0.6 58.0 0.8 90.0 1.2 130.0
I
T h e regions v: P
Absolute
^ 1 (5 points) and / : P > 1 (15 points). r
d e v i a t i o n s o b t a i n e d are m u c h s m a l l e r t h a n those o b t a i n e d b y t h e R K e q u a t i o n , e s p e c i a l l y at h i g h e r ω values. A c o m p a r i s o n of the c a l c u l a t e d a n d e x p e r i m e n t a l c o m p r e s s i b i l i t y factors a l o n g the c r i t i c a l i s o t h e r m f o r sulfur dioxide (5)
and carbon dioxide (2)
c r i t i c a l constants of
t h e substances
is s h o w n i n T a b l e I I .
investigated were
K u d c h a d k e r et a l . ( 6 ) a n d M a t h e w s
The
obtained
from
(7).
I n o r d e r to e x p a n d the a p p l i c a b i l i t y of E q u a t i o n 17 to isotherms o t h e r t h a n the c r i t i c a l , it w a s necessary to d e t e r m i n e t h e t e m p e r a t u r e d e p e n d e n c e of α ( Γ ) , c ( T ) , a n d d(T).
Let
a(T) = a f ( r )
(21)
a
c{T)-cf (T)
(22)
d(T)
(23)
0
=dî {T) d
a n d at Γ =
T , f (T ) =
f (T ) =
a(T ), c =
c ( T ) , and d =
d(T ).
c
p
=
RT
7_6
c
a
c
e
c
_
i (T ) a
c
c
aî (T)
1. I n other w o r d s , a —
cl (T)
a
V(V
=
e
H e n c e E q u a t i o n 17 b e c a m e df (T)
e
+ b) x
+
V
2
(V -
b) (V +
d
b) 2
(V +
bs)
7
(24)
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
168
EQUATIONS O F S T A T E
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Table II.
Comparison of Experimental and Calculated Compressibility Factors along the Critical Isotherm Average Absolute Deviation (%)
Component
Region
Number of Data Points
S u l f u r d i o x i d e (δ)
P < l P r > l
16 49
1.46 12.91
0.30 0.89
P
16 8
3.01 10.80
0.89 1.83
r
C a r b o n d i o x i d e (2) r
> i
RK
(\)
This
Work
I n o r d e r to o b t a i n a s u i t a b l e expression f o r ί „ ( Γ ) , E q u a t i o n 2 4 w a s u s e d to o b t a i n the f o l l o w i n g expression f o r t h e s e c o n d v i r i a l coefficient:
Β = Urn (~\ p-*o\dp/
(25)
ai (T)/RT
b
=
a
T
w h i c h t h e n w a s u s e d to fit t h e P i t z e r a n d C u r l c o r r e l a t i o n o f t h e s e c o n d v i r i a l coefficient ( δ ) : BP
C
(0.1445 + 0.073ω) -
RT
(0.330 -
0.46ω)/Τ
Γ
C
-
(0.1385 + 0 . 5 0 ) / Γ
-
0.0073ω/Γ
ω
Γ
-
2
(0.0121 + 0.097ω)/Γ
(0.1711 + 0 . 2 1 4 7 ω ) Τ , + +
(0.2630 + 1.1065ω)/Γ
+ 0.0173ω/7ν when Γ = T , T = c
r
(26)
3
w a s as f o l l o w s :
a
e
Γ
8
T h e expression o b t a i n e d f o r î (T) f (T)
Γ
Γ
(0.8340 +
1.2211ω)
(0.0741 + 0.3120ω)/Γ
Γ
2
(27)
7
1 a n d E q u a t i o n 2 7 r e d u c e s to f ( T ) = 1. 0
c
N e x t , E q u a t i o n 24 w a s r e a r r a n g e d b y s p l i t t i n g t h e t h i r d t e r m o f t h e r i g h t - h a n d side o f t h e e q u a t i o n i n t o t w o terms. H e n c e ,
P
=
RT V - b
aîa(T) ViV + bJ
eîe(T) V (V-b)
+'
2
eî (T) V (V +
dU(T)
e
2
b) 2
(V +
bV 3
(28) where e = b +
(29)
b
2
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
9.
suGiE A N D L U
A New Equation
169
of State
I n o r d e r t o represent P V T d a t a o v e r a w i d e t e m p e r a t u r e r a n g e b y m e a n s of E q u a t i o n 28, i t w a s necessary to m a k e b o t h t h e q u a n t i t y b i n t h e t h i r d 9
t e r m o f t h e r i g h t - h a n d side of t h e e q u a t i o n , a n d t h e q u a n t i t y 6
depend
3
ent o n t e m p e r a t u r e . T h u s , _
p
RT
ai (T)
eU(T)
a
V _ b
ViV+bJ
V (V
ei,(T)
-
2
b')
V (V
di (T) d
+ b)
2
(V +
2
WV
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(30) w h i c h is t h e final expression of t h e p r o p o s e d e q u a t i o n . T h e e v a l u a t i o n of the t e m p e r a t u r e - d e p e n d e n t q u a n t i t i e s f (T),
i (T),
e
f (T),
g
a n d fc ' w a s b a s e d o n : ( 1 ) s a t i s f y i n g t h e M a x w e l l r e l a t i o n s h i p
d
3
at s a t u r a t i o n ( t h e f u g a c i t y of t h e l i q u i d c a l c u l a t e d f r o m E q u a t i o n 30 s h o u l d b e e q u a l to t h e f u g a c i t y of t h e v a p o r c a l c u l a t e d f r o m t h e same equation);
(2)
s a t i s f y i n g t h e g e n e r a l i z e d c o r r e l a t i o n of t h e s a t u r a t e d
l i q u i d v o l u m e p r o p o s e d e a r l i e r b y L u et a l . ( 9 ) ; a n d (3) fitting PVT d a t a as c o r r e l a t e d i n P i t z e r s tables (4) over t h e c o m p l e t e r a n g e of T a n d P . T
T h e f o l l o w i n g set of t e m p e r a t u r e f u n c t i o n s numerous
fitting
trials w e r e
finally
made:
6' = 0.26 Τ · V Γ
b ' =
e
U(T) g
=
+ x /T*
Xl
2
X l
_
2
(31)
c
T
i (T)
i (T)
0
(0.2515 + 0.0283ω) T V
8
-
(32)
C
TV
4
(33)
— 7V -
8
(34)
1
+ xJTf
+ xJT*
α - ζ/T Χ
- 0 . 3 5 8 8 + 0.4982ω + 0.8208ω 0.2993 + 0.3038ω -
0.6829ω
x —
0.9826 -
0.7758ω -
0.0343ω
χ =
0.0883 -
0.0106ω -
0.1054ω
χ =
-0.0114 -
3
4
5
0.0156ω + 0.0018ω
(35)
7
r
2
x _ 2
T
w a s o b t a i n e d after
2
(36)
2
2
2
F o r t h e p u r p o s e of s a t i s f y i n g t h e M a x w e l l r e l a t i o n s h i p m o r e
pre
cisely, a n a d j u s t m e n t w a s m a d e o n t h e t e m p e r a t u r e - d e p e n d e n t f u n c t i o n f (T). a
T h e final expression o b t a i n e d f o r f (T) a
f (Γ)
is as f o l l o w s :
(0.1664 + 0.0043ω) T + (0.8137 - 1.2204ω) (0.2861 + 1 . 1 2 9 7 ω ) / Γ + (0.0666 + 0.2977ω)/Γ + 0.0173ω/Τ r
fl
+
Γ
Γ
Γ
7
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(37)
170
EQUATIONS O F S T A T E
It s h o u l d b e m e n t i o n e d t h a t t h e difference b e t w e e n E q u a t i o n s 3 7 a n d 2 7 is v e r y s m a l l . T h e s e c o n d v i r i a l coefficient c a l c u l a t e d f r o m E q u a t i o n 2 5 u s i n g t h e n e w expression of f ( T ) s t i l l agrees v e r y w e l l w i t h t h e a
correlation of Pitzer a n d C u r l ( E q u a t i o n 2 6 ) . Numerical
Values
of the Parameters
of the Proposed
Equations
A s m e n t i o n e d above, t h e final expression of t h e p r o p o s e d
equation
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is r e p r e s e n t e d b y E q u a t i o n 3 0 : RT
ρ
efe(T)
ai (T) a
V -b
V(V + b )
-
V (V 2
x
.
eî (T)
dî (T)
g
6')
d
V (V + b ) 2
a
(V +
i> ') 3
(30) in which 6~0.26F 6 i = (0.1181 + 0.4730ο.) V b = (0.2117 + 0.1611*.) F 6' = 0.267Λ°· 7 W = (0.2515 + 0.0283ω) T V aî (T) = ο ι Τ + α + a /T + a / T + a /T ai = a*yiRV a = a*y RT V a = a*y RT W a = a*y RT V a = a*y RT *V 0.1664 - 0.2243ο. 2/ = 0.8137 - 1.2204a> 2/3= 0.2861 + 1.1297ω 2/4 = 0.0666 + 0.2977ω 2/5 = 0.0173ω eî (T) =eT* e — e'ETWc β· — c * / (0.4717 + 0.1611».) «MT) - Λ + g / T + /r + QJT* + g / T ' C
e
2
c
2
ο
r
a
3
2
c
4
2
7
s
c
2
2
c
3
c
4
4
c
5
y
c
3
c
3
5
c
c
e
i
2
e
2
2
2
grj = g = g = g = 3
4
s
X l
x = x = x = x 2
3 t
5
dft(T)
— d—
ff3
4
e*x RT V e*x RT sV* e*x RT V e«x ET F 0.3588 + 0.4982ω + 0.2993 + 0.3038ω 0.9826 - 0.7758ο, 0.0883 - 0.0106α. 0.0114 - 0.0156ω + 3
2
c
c
7
4
5
2
c
3
B
c
8
c
2
c
c
2
0.8208ω 0.6829ω 0.0343ω 0.1054 0.0018ω
2
2 2
ω
2
2
άΓ1Λ d*RT V* 2S
c
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
7
9.
suGiE A N D L U
A New Equation
171
of State
I n t h e a b o v e expressions, V =
Z RT /P
C
C
C
C
Z = 0.291 - 0.080ω C
T h e v a l u e s of a*, c * , a n d d* w e r e d e t e r m i n e d f r o m E q u a t i o n s 8 t h r o u g h 10. T h e values o b t a i n e d f o r s e v e r a l ω values at r e g u l a r i n t e r v a l s are l i s t e d
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b e l o w to serve as e x a m p l e s :
Testing
ω
a»
c*
d*
0.0 0.1 0.2 0.3 0.4 0.5
1.3827 1.4342 1.4872 1.5419 1.5982 1.6561
0.2973 0.2859 0.2761 0.2680 0.2615 0.2564
0.7464 0.7443 0.7443 0.7467 0.7517 0.7594
of the Proposed
Equation
T h e a p p l i c a b i l i t y of t h e p r o p o s e d e q u a t i o n w a s tested i n terms of its p r e d i c t e d values of t h e c o m p r e s s i b i l i t y factors, l i q u i d f u g a c i t y coefficients, a n d i s o t h e r m a l e n t h a l p y departures o f p u r e Compressibility Factors.
compounds.
A t o t a l of 2772 Ζ values of P i t z e r s t a b l e
w a s u s e d to test t h e c a p a b i l i t y of t h e p r o p o s e d e q u a t i o n f o r c a l c u l a t i n g c o m p r e s s i b i l i t y factors of p u r e n o n p o l a r
compounds.
T h e c a l c u l a t e d values o b t a i n e d f r o m
E q u a t i o n 3 0 , together
with
those o b t a i n e d f r o m t h e equations of R e d l i c h et a l . ( 1 0 ) , E d m i s t e r et a l . (11), R e d l i c h a n d K w o n g (1), a n d S u g i e et a l . (12) a r e c o m p a r e d w i t h the Ζ values of P i t z e r s w o r k i n T a b l e I I I . T h e p r o p o s e d e q u a t i o n p r o v i d e s t h e smallest s t a n d a r d d e v i a t i o n .
Table III. Comparison of Calculated Compressibility Factors w i t h P i t z e r s Table by Various Methods for a T o t a l of 2772 D a t a Points at 22 T and 21 P Conditions" T
r
Standard R e f . 10
R e f . 11 Réf. 1 R e f . 12 This work β
Deviation
0.013 0.025 0.054 0.024 0.0108
0.8 < T < 4, 02 < P < 9 ; ω = 0.0-0.5. t
r
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
172
EQUATIONS O F S T A T E
A c o m p a r i s o n of the c a l c u l a t e d a n d e x p e r i m e n t a l Ζ values of p r o p a n e (13)
a n d s u l f u r d i o x i d e ( 5 ) , i n the gas r e g i o n is s h o w n i n T a b l e I V . I n
a d d i t i o n , t h e c a l c u l a t e d results o b t a i n e d f r o m the equations of R e d l i c h a n d K w o n g ( I ) , R e d l i c h a n d D u n l o p (14), et a l . (12)
G r a y et a l . ( 1 5 ) , a n d S u g i e
also are i n c l u d e d i n T a b l e I V . T h e p r o p o s e d e q u a t i o n y i e l d s
the best results.
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Table IV.
Comparison of Experimental and Calculated Ζ Values for Propane and Sulfur Dioxide Average
τ
F(psia)
Ref. 14
Ref. 1 Propane
220°F 340°F 460°F
20-6000 20-6000 20-6000
Overall
Absolute
10-300 10-300 10-300
Overall
(%)
Ref. 15
Ref. 12
This Work
(13)
6.3 1.7 1.0
5.2 1.5 0.4
1.9 1.9 4.2
1.0 1.2 1.8
0.80 1.47 1.67
3.0
2.4
2.7
1.3
1.31
Sulfur dioxide 157.5°C 200°C 250 ° C
Deviation
(5)
13.5 4.9 2.2
6.4 3.3 1.0
3.3 0.9 3.3
0.8 2.0 1.2
0.77 1.02 1.32
6.9
3.6
2.5
1.3
1.04
I n a d d i t i o n , t h e c a l c u l a t e d a n d t h e e x p e r i m e n t a l Ζ v a l u e s of h y d r o g e n sulfide ( 1 6 ) ,
i n the regions i n c l u d i n g gas, v a p o r , a n d l i q u i d , are
c o m p a r e d i n T a b l e V . T h e c a l c u l a t e d results o b t a i n e d f r o m the e q u a t i o n s of R e d l i c h a n d K w o n g (1 ) a n d S u g i e et a l . (12)
also are i n c l u d e d i n t h i s
t a b l e f o r c o m p a r i s o n . A g a i n , t h e p r o p o s e d e q u a t i o n y i e l d s the best results.
Table V .
Comparison of Experimental and Calculated Ζ Values for Hydrogen Sulfide (16) Average
T
Number Points
r
0.744 0.834 0.893 1.012 1.102 1.191 Overall
0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7 0.01-7.7
34 34 34 34 34 34
of
Absolute
Deviation
(%)
Ref. 1
Ref. 12
This Work
3.64 3.94 4.00 3.84 2.54 1.69 3.27
1.53 1.52 1.40 0.55 0.88 0.66 1.09
1.37 1.01 0.84 0.53 0.66 0.85 0.88
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
9.
SUGIE A N D L U Fugacities.
A New
Equation
173
of State
T h e f u g a c i t y coefficient, φ, w h i c h is e q u a l t o t h e r a t i o
of f u g a c i t y to pressure, c a n b e c a l c u l a t e d f r o m E q u a t i o n 38. dV ( Z - l ) ~ r -
v
/
^ z - i - m z - m V^ T J _
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ele(T)
V-b'
|_6
RTb'
Γ_1_
RTb
lb
b
x
(38)
Vj
V
11
V + b
m
2
v
V +
11
V
ei,(T) 2
+' ^RTbi - m
_
dU(T) 6RT(V
V j
2
+
b ')« 3
T h e a b o v e e q u a t i o n w a s u s e d to o b t a i n φ v a l u e s , at r e g u l a r i n t e r v a l s of T
r
from T
T
=
0.5 to T
=
r
butane, a n d n-pentane.
1, f o r l i q u i d m e t h a n e , ethane, p r o p a n e , n -
T h e c a l c u l a t e d v a l u e s are c o m p a r e d w i t h s o m e
of t h e a v a i l a b l e t a b u l a t i o n s i n t h e l i t e r a t u r e (17,18,20)
i n Figures 1
t h r o u g h 5. E x c e l l e n t a g r e e m e n t w a s o b t a i n e d . I n a d d i t i o n , φ v a l u e s w e r e c a l c u l a t e d u s i n g E q u a t i o n 38 f o r s a m e five c o m p o u n d s b u t at l o w e r t e m p e r a t u r e s ( T s i m i l a r c o m p a r i s o n is s h o w n i n F i g u r e s 6 a n d 7.
r
=
the
0.4 a n d 0 . 3 ) .
G o o d agreement
A
gen
e r a l l y is o b t a i n e d w i t h the e x c e p t i o n of m e t h a n e . T h e isothermal enthalpy depar
Isothermal Enthalpy Departures.
tures f r o m the i d e a l - g a s state w e r e c a l c u l a t e d f o r six p u r e , s a t u r a t e d l i q u i d s ( m e t h a n e , ethane, p r o p a n e , η-butane, i - b u t a n e , a n d u s i n g E q u a t i o n 39.
n-pentane)
T h e p r o p o s e d e q u a t i o n w a s , of course, u s e d i n its
derivation ( H * - H ° ) , - P V - R T bRT
[ - (-ff) ~] p
J
V
F
d
+
b)
l n
— γ —
0.2ef.(D
Γ 2
b'
I b'
b
η l b
2
3
V-b'
[ ψ
_ Z s
b Τ'yν Γ
(v +
2
Γ 1
2
V ( V - b ' )
T
df (T)V
g
ψ
eî.{T)
a
ef (T)
-
dV
aî (T) 7 + b i
V - b
v(v
T
. 2
m
,
, +
V V+b
2
Fx
,
Τ&x " In
y +
h
1
Ί
11 T J _1_
V-b' V
m
+ ^
V
+
,
Π
"·"
Vj
_
+
b' J
V F
4
6(7 +
df (T)W (V + b/y
6 ') 3
e
t
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(39)
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EQUATIONS O F
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
STATE
suGiE A N D L U
A New Equation
of State
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9.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
175
EQUATIONS O F
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176
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
STATE
suGiE A N D L U
A New Equation
of State
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9.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
177
EQUATIONS O F
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178
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
STATE
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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h-· -α CO
Ci"
S"
ο"
a
•ο
Ci
ϋ
>
ο Β
C O
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EQUATIONS O F
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
STATE
9.
suGiE A N D L U
A New
Equation
181
of State
in which
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Fi -
af„(T) -
F
2
= ef.(T)
F
3
= βί,(Γ) -
F
4
-
df (D d
-
-
Τ (
θ
α
ί
Τ ( ^ γ
^
1
Γ
)
)
2
=
) -
Τ
Γ [
) = a + 2 a / T + ZaJT*
σ α Α
*Γ
;
3
+
8α /Τ'
5β/Γ*
0i + 3 g / T + 704/r 2
) = 2.8 d / Γ · 1
2
6
+ +
50 /Γ* 8g /T' 3
5
8
T h e values o b t a i n e d f r o m E q u a t i o n 39 a n d several other (11,12,19,21,22,23,24,25)
5
are c o m p a r e d
methods
w i t h the available experi
m e n t a l values i n the l i t e r a t u r e (26,27,28,29).
T h e c o m p a r i s o n , expressed
i n terms of average absolute d e v i a t i o n s , is p r e s e n t e d i n T a b l e V I . H o w ever, the results o b t a i n e d b y the p r o p o s e d e q u a t i o n are o n l y f a i r .
Discussion
and
Conclusion
T h i s i n v e s t i g a t i o n f o l l o w s o u r efforts p r e v i o u s l y m a d e o n the m o d i f i c a t i o n of the R K e q u a t i o n of state (3,30). T h e r e p u l s i v e t e r m of t h e R K e q u a t i o n w a s r e t a i n e d w i t h the a n t i c i p a t i o n that the o r i g i n a l terms w o u l d b e p r e s e r v e d as p a r t of the n e w e q u a t i o n . T h i s p r a c t i c e m a y b e subject to modifications i n f u t u r e endeavors. T h e r e p u l s i v e t e r m m a y be r e p l a c e d b y a m o r e s u i t a b l e t e r m s u c h as t h a t p r o p o s e d b y C a r n a h a n a n d S t a r l i n g (31) . T h e proposed
e q u a t i o n w a s n o t c o m p a r e d w i t h a n y of t h e
more
r e c e n t c u b i c e q u a t i o n s , s u c h as the Soave m o d i f i c a t i o n of t h e R K e q u a t i o n (32) , the P e n g a n d R o b i n s o n e q u a t i o n (33), (34),
and the F u l l e r equation
because a l l of these equations do n o t y i e l d a c c e p t a b l e values of Z . c
Some of t h e ω values u s e d i n this s t u d y differ s l i g h t l y f r o m suggested b y Passut a n d D a n n e r ( 3 5 ) .
those
H o w e v e r , these s m a l l differences
h a r d l y affected t h e c a l c u l a t e d results. I n c o n c l u s i o n , a n e w p r e s s u r e - e x p l i c i t e q u a t i o n of state has b e e n successfully d e v e l o p e d as i n t e n d e d . I t is s u i t a b l e f o r r e p r e s e n t i n g PVT b e h a v i o r of l i q u i d a n d gas phases over a w i d e r a n g e of t e m p e r a t u r e a n d pressure for p u r e , n o n p o l a r c o m p o u n d s . F u r t h e r m o r e , t h e p a r a m e t e r s of the p r o p o s e d e q u a t i o n are g e n e r a l i z e d i n terms of t h e c r i t i c a l p r o p e r ties a n d the a c e n t r i c factor.
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
182
EQUATIONS O F
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Table V I .
Comparison of Experimental and Calculated
Component
TV
Pr
M e t h a n e (26) E t h a n e (27) Propane (26,27,28) i V - B u t a n e (27) i s o - B u t a n e (27) J V - P e n t a n e (29)
0.757-0.990 0.654-0.982 0.660-0.983 0.653-0.967 0.757-0.990 0.662-0.970
0.172-0.945 0.045-0.894 0.041-0.891 0.032-0.794 0.034-0.872 0.032-0.802
No. Data
Overall
of
a,c,d
=
Symbols q u a n t i t i e s r e p r e s e n t e d b y E q u a t i o n s 2 1 , 22, a n d 23, respectively
a * , c*, d* = q u a n t i t i e s d e t e r m i n e d f r o m E q u a t i o n s 8 , 9 , a n d 10 Β = s e c o n d v i r i a l coefficient b = p a r a m e t e r of E q u a t i o n 3 hi, &2> &3 — p a r a m e t e r s of E q u a t i o n 17 9
&' = p a r a m e t e r s of E q u a t i o n 30 3
e = q u a n t i t y d e f i n e d b y E q u a t i o n 29 e* = c * / ( 0 . 4 7 1 7 + F χ,...
4
0
e n t h a l p y at i d e a l gas state
== e n t h a l p y at pressure Ρ
H
p
k
0.1611ω)
F = f u n c t i o n s of E q u a t i o n 39 H° =
hi...
5 7 13 7 7 14 53
Glossary
V
STATE
h = parameters of E q u a t i o n 1 n
... k
m
rii.. . n
— constants of E q u a t i o n 5 = z e r o or p o s i t i v e i n t e g e r
m
Ρ =
pressure
P — c r i t i c a l pressure c
R = gas constant Τ = T = c
temperature critical temperature
T = reduced temperature r
V == v o l u m e V
c
— critical volume
Xi. . . Xs = f u n c t i o n s r e p r e s e n t e d b y E q u a t i o n 36 Ζ =
compressibility factor
Z = c o m p r e s s i b i l i t y f a c t o r at the c r i t i c a l p o i n t c
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
of Pts.
9.
suGiE A N D L U
( H ° — H )t
A New Equation
Values for Pure Saturated Liquids
p
Average
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183
of State
Absolute
Deviation
(Btu/lb)
Ref. 19
Ref. 21
Ref. 22
Ref. 23
Refs. 24,25
Ref. 11
Ref. 12
This Work
3.7 1.7 1.2 2.2 2.3 1.9 2.2
9.8 2.2 3.5 3.4 6.1 4.6
10.0 8.1 3.4 9.4 13.6 11.7
2.6 5.0 4.6 4.0 4.4 4.8
7.0 5.1 2.7 4.4
32.1 9.3 5.1 4.6
3.1
3.4
3.0 1.9 1.5 2.1 2.5 1.5
4.7 4.3 4.6 4.7 5.5 5.0
4.9
9.4
4.2
4.5
11.0
2.1
4.8
—
—
Greek Letters ρ = density φ = f u g a c i t y coefficient ω
= acentric factor
Literature Cited
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