10
Calculation
of
Three-Phase
Solid-Liquid-Vapor
Equilibrium Using an Equation of State
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DING-YU PENG and DONALD B. ROBINSON Department of Chemical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6
An efficient computation algorithm is proposed for predicting the initial formation of pure solids in hydrocarbon and related systems. In order to use this method, only the density and vapor pressure of the pure solid are required together with the equation-of-state parameters for the solidforming material. The use of the algorithm is illustrated with the Peng-Robinson equation of state for systems containing carbon dioxide as the solid-forming component at cryogenic temperatures. Good agreement was obtained between the predicted results and the experimental literature values. The predicted triple point for pure carbon dioxide was 72.1 psia at —71.2°Fas compared with the literature value of 76.9 psia at -69.9°F. ' " p h e w i d e s p r e a d use of l o w t e m p e r a t u r e f o r the p r o c e s s i n g of n a t u r a l gas has b e e n s t i m u l a t e d b y e c o n o m i c i n c e n t i v e s a n d i t has b e e n m a d e p o s s i b l e b y advances i n m a t e r i a l s a n d process t e c h n o l o g y .
T h e n e e d to
b e a b l e to a c c u r a t e l y d e s c r i b e t h e b e h a v i o r of these h y d r o c a r b o n m i x t u r e s at c r y o g e n i c c o n d i t i o n s is e v i d e n t . I n a d d i t i o n t o t h e u s u a l v a p o r - l i q u i d e q u i l i b r i u m r e l a t i o n s h i p s , a k n o w l e d g e of t h e s o l u b i l i t y of h e a v i e r h y d r o c a r b o n s a n d other p o t e n t i a l s o l i d - f o r m i n g gases s u c h as c a r b o n d i o x i d e o r h y d r o g e n sulfide is necessary f o r t h e safe a n d effective o p e r a t i o n o f p r o c e s s i n g p l a n t s . T h e presence of s o l i d c a r b o n d i o x i d e , f o r e x a m p l e , m a y f o u l heat exchangers o r i n t e r f e r e w i t h t h e o p e r a t i o n of d i s t i l l a t i o n e q u i p m e n t to t h e p o i n t that costly p l a n t s h u t d o w n s a n d m a i n t e n a n c e m a y be required. I n o r d e r t o a v o i d these p o t e n t i a l l y d a m a g i n g c o n d i t i o n s , i t is u s e f u l to b e a b l e to d e l i n e a t e t h e s o l i d - f o r m i n g regions f o r a n y g i v e n p r o c e s s i n g m i x t u r e . A n u m b e r of studies d e a l i n g w i t h t h e c o r r e l a t i o n o f t h e s o l u b i l i t y 0-8412-0500-0/79/33-182-185$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.
186
EQUATIONS O F S T A T E
of solids i n l i q u i d h y d r o c a r b o n s p o r t e d (1,2,3,4,5).
at c r y o g e n i c c o n d i t i o n s h a v e b e e n r e
S o m e of these a r e b a s e d o n t h e S c a t c h a r d - H i l d e -
b r a n d r e g u l a r s o l u t i o n t h e o r y a n d others o n t h e F l o r y - H u g g i n s e q u a t i o n f o r a t h e r m a l m i x t u r e s . R e g a r d l e s s of t h e basis, h o w e v e r , t h e e q u i l i b r i u m c r i t e r i o n i n a l l cases is that t h e free e n e r g y of t h e p u r e s o l i d m u s t e q u a l its p a r t i a l m o l a l free energy i n t h e s o l u t i o n . I n t h e w o r k d e s c r i b e d i n t h i s c h a p t e r , a c o m p u t a t i o n a l g o r i t h m b a s e d o n e q u a t i n g t h e fugacities of t h e pure solid a n d the component
i n t h e s o l u t i o n is p r e s e n t e d f o r u s e i n
p r e d i c t i n g i n c i p i e n t s o l i d f o r m a t i o n of a c o m p o n e n t f r o m a n e q u i l i b r i u m
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vapor or l i q u i d solution. Thermodynamic
Relationships
T h e e q u i l i b r i u m c o n d i t i o n f o r a s o l i d - l i q u i d - v a p o r ( S L V ) system at a specified t e m p e r a t u r e a n d pressure m a y b e w r i t t e n i n terms of t h e fugacities of the s o l i d - f o r m i n g c o m p o n e n t as (1) where f
fc
s
is t h e f u g a c i t y of c o m p o n e n t
k i n t h e s o l i d phase, f
f u g a c i t y of c o m p o n e n t k i n t h e l i q u i d p h a s e , a n d i
Y
k
fc
L
is t h e
is t h a t of t h e same
c o m p o n e n t i n t h e v a p o r p h a s e . I t is u n d e r s t o o d that t h e f u g a c i t i e s of a l l the other c o m p o n e n t s m u s t f o l l o w t h e e q u a t i o n f L
f V
(2)
A l l of these fugacities are e v a l u a t e d at t h e t e m p e r a t u r e a n d pressure o f the system.
T h e fugacities of c o m p o n e n t
k f o r t h e fluid phases c a n b e
calculated using the rigorous thermodynamic equation dv -
In Ζ
(3)
N o r m a l l y o n e c a n assume t h a t t h e s o l i d phase is n o t a s o l i d s o l u t i o n ; consequently,
t h e s o l i d p h a s e is essentially p u r e c o m p o n e n t
k whose
fugacity follows the equation
(4) T h i s c a n b e w r i t t e n as
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
10.
P E N G A N D ROBINSON
Three-Phase
187
Equilibrium
w h e r e P * is the v a p o r pressure of c o m p o n e n t k at the system t e m p e r a t u r e , k
v
Y
k
is the m o l a r v o l u m e of c o m p o n e n t k i n its v a p o r state, a n d v
is t h e
s
k
m o l a r v o l u m e of s o l i d c o m p o n e n t
k.
I n E q u a t i o n 5, t h e first i n t e g r a l
represents the f u g a c i t y coefficient of the p u r e c o m p o n e n t k at its v a p o r pressure w h i l e t h e second i n t e g r a l accounts for the c o m p r e s s i o n
effect
u p o n the s o l i d at the system pressure. T h i s is greater t h a n t h e s o l i d - v a p o r e q u i l i b r i u m pressure for
this c o m p o n e n t
N e g l e c t i n g the pressure d e p e n d e n c y
at the
system
temperature.
of the m o l a r v o l u m e of s o l i d k,
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we readily obtain
T h e a b o v e e q u a t i o n s i m p l y states t h a t the f u g a c i t y of s o l i d c o m p o n e n t
k
at pressure Ρ equals the f u g a c i t y of p u r e k at its v a p o r pressure m u l t i p l i e d b y a P o y n t i n g c o r r e c t i o n factor. Evaluation of Fugacities Using an Equation of State.
The fugaci-
ties o f the c o m p o n e n t s i n the fluid phases are r e l a t e d to the v o l u m e t r i c a n d phase b e h a v i o r of the m i x t u r e w h i l e the f u g a c i t y of the s o l i d c o m p o n e n t d e p e n d s o n l y o n t h e PVT
r e l a t i o n s h i p of the p u r e
component.
T h e o r e t i c a l l y it is p o s s i b l e to e v a l u a t e the fugacities u s i n g e x p e r i m e n t a l v o l u m e t r i c a n d / o r phase e q u i l i b r i u m d a t a i n c o n j u n c t i o n w i t h E q u a t i o n s 3 a n d 6. H o w e v e r , these d a t a are n o r m a l l y either u n a v a i l a b l e or insuffi c i e n t a n d a n equation-of-state m o d e l has to be u s e d to c o m p u t e
the
fugacities. M a n y equations of state are a v a i l a b l e f o r c a l c u l a t i n g the q u a n t i t i e s i n E q u a t i o n s 3 a n d 6.
W h e n s e l e c t i n g a s u i t a b l e e q u a t i o n of
state,
c o n s i d e r a t i o n s h o u l d be g i v e n to the fact t h a t the c a p a b i l i t y of a n e q u a t i o n of state to a c c u r a t e l y represent the S L V three-phase system hinges o n t h e a b i l i t y of the same e q u a t i o n to d e s c r i b e the s i m p l e r t w o - p h a s e
liquid-
v a p o r system. I n this s t u d y w e h a v e chosen t h e P e n g - R o b i n s o n e q u a t i o n of state ( 6 )
to m o d e l the p h a s e a n d v o l u m e t r i c b e h a v i o r of the
fluid
phases of m i x t u r e . T h e v o l u m e t r i c b e h a v i o r of t h e p u r e c o m p o n e n t i n its v a p o r - p h a s e r e g i o n at t h e s o l i d - f o r m a t i o n t e m p e r a t u r e also
k
was
r e p r e s e n t e d i n terms of this c l o s e d - f o r m e q u a t i o n . T h e P e n g - R o b i n s o n e q u a t i o n has b e e n a p p l i e d successfully to the p r e d i c t i o n of l i q u i d - l i q u i d vapor ( L L V ) equilibrium (7), hydrate formation ( 8 , 9 ) , and mixture c r i t i c a l - p o i n t d e t e r m i n a t i o n ( J O ) i n a d d i t i o n to the u s u a l v a p o r - l i q u i d e q u i l i b r i u m c a l c u l a t i o n s . T h i s e q u a t i o n has t h e f o l l o w i n g f o r m :
P =
RT ν — b
v(v
a(T) + b) + b(v
— b)
In Equations of State in Engineering and Research; Chao, K., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(7)
188
EQUATIONS O F STATE
a n d w h e n t h e c l a s s i c a l d e r i v a t i v e s at t h e c r i t i c a l p o i n t a r e i m p o s e d it yields
= 0.45724^^-
a(T ) e
(8)
b = 0.07780 ^
(9)
*c
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A t temperatures other t h a n the critical point
— ο(Γ ) α(Γ ,«)
a(T)
0
«(7V,
