A New, Generalized Equation of State Valid Within the Critical Region

Jul 22, 2009 - Chemical Engineering Department, Texas A&M University, College Station, ... 1 Present address: Division of Energy, Arya-Mehr University...
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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

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

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