6 A New, Generalized Equation of State Valid Within the Critical Region MOHAMMED S. NEHZAT , KENNETH R. HALL, and PHILIP T. EUBANK 1
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2
Chemical Engineering Department, Texas A&M University, College Station, TX 77843
An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon cor relation of the two leading terms: the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data.
' " p h e c r i t i c a l r e g i o n p r o v i d e s a severe test f o r t h e a p p l i c a b i l i t y o f a n y e q u a t i o n of state, a n d often just as severe a test f o r t h e p a t i e n c e of the c o r r e l a t o r . T h e reason is t h e almost p a t h o l o g i c a l b e h a v i o r o f
fluids
as t h e y a p p r o a c h t h e i r c r i t i c a l p o i n t s . M a n y t h e r m o d y n a m i c p r o p e r t i e s e i t h e r b e c o m e zero o r else d i v e r g e to i n f i n i t y a t t h e c r i t i c a l p o i n t . T h i s s t u d y is a n o t h e r r a t h e r successful a t t e m p t to correlate t h e fluid properties i n the critical region. W e have chosen an isochoric equation of state w i t h constant c u r v a t u r e t o represent these p r o p e r t i e s a n d w e Present address: Division of Energy, Arya-Mehr University of Technology, Isfahan, Iran. * Author to whom correspondence should be addressed. 1
0-8412-0500-0/79/33-182-109$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.
110
EQUATIONS OF
h a v e i m p o s e d some of the ideas f r o m the s c a l i n g hypothesis.
STATE
Because
this is a c o r r e l a t i o n , w e h a v e a l l o w e d d a t a to influence t h e m o d e l .
When
the d a t a d i d not p r o v i d e c o n c l u s i v e g u i d a n c e , w e chose to m a k e the c o r r e l a t i o n i n t e r n a l l y consistent. W e selected ethylene a n d w a t e r as test substances for the c o r r e l a t i o n . T h e s e t w o c o m p o u n d s are i m p o r t a n t c o m m e r c i a l substances a n d t h e y are also i n t e r e s t i n g f r o m a scientific v i e w p o i n t . I n a d d i t i o n , the c r i t i c a l r e g i o n correlations for these c o m p o u n d s w e r e i n r a t h e r p o o r agreement w i t h the
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data.
Previous
Work
A n d r e w s ( J ) first d i s c o v e r e d the c r i t i c a l p o i n t of a S h o r t l y thereafter i n 1873, V a n d e r W a a l s (2)
fluid
i n 1869.
p r e s e n t e d his d i s s e r t a t i o n ,
" O n the C o n t i n u i t y of the G a s a n d L i q u i d State." T h i s a n d later w o r k i n the f o l l o w i n g t w e n t y years p r o v i d e d the c l a s s i c a l t h e o r y of the c r i t i c a l r e g i o n for
fluids.
H o w e v e r , V e r s c h a f f e l t i n the e a r l y 1900's f o u n d
the
c r i t i c a l exponents β a n d δ to b e a b o u t 0.35 a n d 4.26, r e s p e c t i v e l y , c o m p a r e d w i t h the c l a s s i c a l values of 1 / 2 a n d 3. T h e surface t e n s i o n e x p o nent also w a s f o u n d to be near 1.25 i n s t e a d of the c l a s s i c a l v a l u e of 3 / 2 . A n excellent d e t a i l e d h i s t o r i c a l r e v i e w of this p e r i o d has b e e n g i v e n b y L e v e l t Sengers
(3).
I n 1965, W i d o m (4)
p r o p o s e d a n o n c l a s s i c a l m o d e l for the c r i t i c a l
r e g i o n , the s c a l i n g hypothesis. T h i s m o d e l w a s r e m a r k a b l y successful a n d s p a w n e d a t r e m e n d o u s n u m b e r of papers b o t h a p p l y i n g a n d r e f i n i n g the m o d e l . B o o k s b y S t a n l e y ( 5 ) a n d M a (6)
together w i t h the c o m p r e h e n
sive r e v i e w b y L e v e l t Sengers, G r e e r , a n d Sengers ( 7 ) p r o v i d e the neces sary b a c k g r o u n d m a t e r i a l . T h e c r i t i c a l r e g i o n d e s c r i p t i o n b y the s c a l i n g m o d e l w a s so successful that n o serious c h a l l e n g e s arose for 10 years. B e c a u s e of its m a t h e m a t i c a l c o m p l e x i t y a n d l a c k of s i m p l e t r a c t a b i l i t y i n terms of m e a s u r 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.e., t h e heat of v a p o r i z a t i o n ) , s c a l e d equations of state are p o p u l a r w i t h f e w e x p e r i m e n t a l t h e r m o d y n a m i c i s t s a n d p r a c t i c i n g engineers. correctness,
most
simply do
W h i l e some q u e s t i o n its
not u n d e r s t a n d this
theoretical physicists a n d mathematicians.
field
dominated
D e s p i t e the efforts
of
by the
E q u a t i o n of State S e c t i o n of the N B S i n W a s h i n g t o n to p o p u l a r i z e the subject t h r o u g h u s e f u l a p p l i c a t i o n s , f e w engineers c a n use a n y of the results save s u c h p r i n t e d t h e r m o d y n a m i c p r o p e r t y tables as those for steam ( S ) .
A n o t h e r c o m p l i c a t i o n is the necessity to b l e n d t h e s c a l e d
e q u a t i o n of state w i t h a s e c o n d , a n a l y t i c a l e q u a t i o n of state v a l i d o u t s i d e the critical region ( 9 ) ;
w i t h i n the c r i t i c a l r e g i o n , the a n a l y t i c a l terms
are r e f e r r e d to as the " b a c k g r o u n d . "
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
6.
ΝΕΗΖΑτ E T A L .
A New,
Generalized
Equation
111
of State
T h e r m o d y n a m i c a l l y consistent, n o n a n a l y t i c a l , e m p i r i c a l equations of state i n d u c e d f r o m
e x p e r i m e n t a l measurements
c a n a v o i d the
above
difficulties. S i n c e 1965, at least t w o laboratories a c t i v e l y w e r e d e v e l o p i n g i s o c h o r i c equations of state ( R e f s . 10,11).
T h e s e w o r k e r s h a d the benefit
of the s c a l i n g w o r k a n d i n c l u d e d n o n c l a s s i c a l b e h a v i o r i n the c r i t i c a l r e g i o n for t h e i r equations. T h e e q u a t i o n p r e s e n t e d i n t h i s c h a p t e r arose f r o m u t i l i z i n g the same basic strategy. I n 1976, H a l l a n d E u b a n k ( 12,13)
p u b l i s h e d two papers w h i c h have
d i r e c t b e a r i n g u p o n the present e q u a t i o n of state.
I n t h e first p a p e r ,
t h e y n o t e d the r e c t i l i n e a r b e h a v i o r for the m e a n of the v a p o r a n d l i q u i d Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch006
i s o c h o r i c slopes i s s u i n g f r o m the same p o i n t o n the v a p o r pressure c u r v e n e a r the c r i t i c a l p o i n t a n d the p o w e r l a w b e h a v i o r for the difference i n these slopes.
T h e s e c o n d p a p e r p r e s e n t e d a n e m p i r i c a l d e s c r i p t i o n of
the c r i t i c a l r e g i o n w h i c h g e n e r a l l y a g r e e d w i t h the s c a l i n g m o d e l b u t d i f f e r e d i n one s i g n i f i c a n t w a y — t h e c u r v a t u r e of the v a p o r pressure c u r v e .
Equation
of State
T h e basic f u n c t i o n of a n i s o c h o r i c e q u a t i o n of state is to
describe
isochores as t h e y issue f r o m the v a p o r pressure c u r v e . F i g u r e 1 i l l u s t r a t e s
Temperature Figure 1. Coexistence curve with isochores and the isochonc slope, ψ . At points A and Β the curvature of the isochores is zero as well as along the locus indicated by the dotted line (the locus of the isochonc heat capacity extrema). 0
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
112
EQUATIONS O F S T A T E
the g e n e r a l l y r e g u l a r b e h a v i o r o f these curves o n a P - T d i a g r a m . T h e b a s i c f o r m u l a t i o n is a T a y l o r s series e x p a n s i o n a b o u t t h e v a p o r p r e s s u r e :
p=Pff+
(lf)J
{
-
t
t
+i(fï)J
° î
w
-
t
^
+
•• •
(1) w h e r e Ρ is pressure, Τ is t e m p e r a t u r e , ρ is d e n s i t y , a n d s u b s c r i p t σ denotes a s a t u r a t i o n v a l u e (e.g., Ρ is t h e v a p o r p r e s s u r e ) .
I n the near-critical
σ
r e g i o n , w h i c h w e a r b i t r a r i l y s h a l l define as w i t h i n |1 — T \ ^ 0.01 a n d Downloaded by UNIV OF ARIZONA on December 6, 2012 | http://pubs.acs.org Publication Date: December 1, 1979 | doi: 10.1021/ba-1979-0182.ch006
R
|1 ~~ pie I ^ 0.3, E q u a t i o n 1 represents t h e isochores
adequately
when
t r u n c a t e d after t h e second-order t e r m . F i n a l l y , w e w r i t e t h e e q u a t i o n of state i n r e d u c e d f o r m f o r i n c r e a s e d n u m e r i c a l t r a c t a b i l i t y : ρ_ P
Ρσ
=
P
c
To/ap\ c
+
Ι
P\dTj
( τ - τ
\
ρ
)
σ
T
2
( τ - τ
2 P \dT ) \
+
c
ι
/d P\
2
c
T
σ
2
C
ρ
σ
)
2
Τ
2
σ
(2) o r w i t h t h e u s u a l definitions of r e d u c e d v a r i a b l e s :
P
" =
P
- + (!£)„L
