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5 Computation of Phase and Chemical Equilibrium: A Review

Downloaded by UNIV OF MISSOURI COLUMBIA on August 24, 2013 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch005

WARREN D. SEIDER, RAJEEV

GAUTAM , 1

and CHARLES W. WHITE, III

Department of Chemical and Biochemical Engineering, University of Pennsylvania, Philadelphia, PA 19104

Figure 1 i l l u s t r a t e s the p h y s i c a l s i t u a t i o n i n which a mix ture o f chemicals i s allowed t o reach e q u i l i b r i u m . The number o f phases, P, and t h e i r compositions are unknown a t e q u i l i b r i u m . U s u a l l y the s t a t e o f the feed and temperature and pressure of the products are known, and the Gibbs f r e e energy i s minimized

G =

S Σ

C P G?n° + Σ Σ j-1 3 3 j-S+1 *=i

G. n 0

3

(1)

%

subject t o mass balance c o n s t r a i n t s , where C i s the number o f chemical s p e c i e s , Ρ i s the number o f mixed phases (vapor, l i q u i d , s o l i d ) , S i s the number o f condensed species (which appear i n only one pure phase, are normally s o l i d , and do not d i s t r i b u t e amongst other phases), n j i s the number o f moles o f compound j i n phase H, G-£ i s the chemical p o t e n t i a l o f compound j i n phase I, given by

G,. = G °

p

f

o

e

+ RT In - i

r

and o f t e n , fj£ = y j j vapor and = X j Y j f j for liquid phases. When the number o f independent chemical r e a c t i o n s equals C - p, where p i s the rank o f the atom matrix (m-j) , Gibbs f r e e energy i s minimized subject t o atom balance c o n s t r a i n t s :

1

Current address: West V i r g i n i a

Union Carbide Corporation, South Charleston,

0-8412-0549-3/80/47-124-115$05.00/0 © 1980 American Chemical Society In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


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COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING

PHASE 1

' PHASE 2 F

'f z

LP

Q PHASE Ρ Figure 1.

Physical situation. Mixture reaches equilibrium at Τ and P. The num ber of phases, P, and their compositions are unknown.

In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

5.

b. = k

117

Phase and Chemical Equilibrium

SEIDER ET AL.

S C P Σ m. n ? + Σ Σ m.,n.„ j=l * j = S l Λ-1 * 3*

k=l,...,E

(2)

3

+

where b i s the number of gram-atoms o f element k, mj i s the number of atoms o f element k i n compound j , and Ε i s the number of elements. A f t e r computing product compositions, the energy balance gives heat duty. Other s p e c i f i c a t i o n s f o r the product, such as pressure and heat duty (e.g., a d i a b a t i c processes) or pressure and entropy (e.g., i s e n t r o p i c processes) a l s o i n v o l v e f i n d i n g the extremum o f a thermodynamic f u n c t i o n . Given pressure and heat duty, the entropy i s maximized


k

S Σ

G°n 3

j=l

C

C Σ

+

D

P Σ

j = s + 1

£ = 1

G

n

] j

(3)

J* DM

subject t o mass balance c o n s t r a i n t s and the energy balance S Q + Η

Ρ h%

Σ

= F

+

C

Σ

j=i

h η

£=i

36

(4)

36

A l t e r n a t i v e l y , Gibbs f r e e energy (1) i s minimized subject t o these c o n s t r a i n t s . Given p r e s s u r e and entropy, the enthalpy i s minimized S Σ

Η =

C G°n 3

j-i

C

Σ

+

3

P Σ

j-s+i £=i

G. n.. ft

3 1

(5)

3 1

subject t o mass balance c o n s t r a i n t s and the entropy balance, which f o r i s e n t r o p i c systems i s S Σ

=

S F

j

=

1

C

s n ] ]

C

Ρ Σ

+ A

s

l

s η

(6)

A A

This i s e q u i v a l e n t t o minimizing Gibbs f r e e energy. Less common s p e c i f i c a t i o n s such as temperature and heat duty o r temperature and entropy are t r e a t e d s i m i l a r l y . S o l u t i o n of the o p t i m i z a t i o n problem r e q u i r e s algorithms t h a t l o c a t e the phase d i s t r i b u t i o n corresponding t o the g l o b a l optimum. C a s t i l l o and Grossmann (1) term t h i s a mixed-integer programming problem and suggest a t r e e to enumerate the p o s s i b l e combinations of phases at e q u i l i b r i u m . Some computational problems are s p e c i f i c t o chemical e q u i l i b -


118

COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING


rium, while others are s p e c i f i c t o phase e q u i l i b r i u m . Hence, we consider techniques f o r chemical e q u i l i b r i u m i n a s i n g l e phase and f o r phase e q u i l i b r i u m without r e a c t i o n p r i o r t o those f o r chemical and phase e q u i l i b r i u m . Before proceeding, two reviews are worthy o f mention. Z e l e z n i k and Gordon (2) and VanZeggeren and Storey {3) concen t r a t e on thermodynamic fundamentals and numerical methods. T h e i r c o n c l u s i o n s should be reassessed i n view o f r e c e n t developments i n numerical a l g o r i t h m s . Chemical

Equilibrium ,

U n t i l the e a r l y 1940 s, with temperature and pressure g i v e n , compositions i n chemical e q u i l i b r i u m were computed manually by s o l v i n g a s e t o f n o n l i n e a r equations:

j

C Σ l

v..G. = 0 D

=

i=l,...,R

(7)

s u b j e c t t o the atom balances i n Equation (2). Equation (7) was w r i t t e n f o r each o f R independent r e a c t i o n s and was obtained by d i f f e r e n t i a t i n g Equation (1): d T7-

where

G = 0

.

1=1,...,R

i s the extent o f r e a c t i o n i and: R Σ

n. = n? + J 3

i

=

1

v..ξ. 13

j=l,...,C

(8)

nj i s the moles o f compound j i n the feed mixture and v j are the s t o i c h i o m e t r i c c o e f f i c i e n t s o f compound j i n r e a c t i o n i given by the atom balances M v = 0. B r i n k l e y ( 4 p o s t u l a t e d c s p e c i e s at e q u i l i b r i u m , p s p e c i e s , r e f e r r e d t o as "components," were s e l e c t e d t o have l i n e a r l y i n dependent formula v e c t o r s , where p i s the rank o f the atom matrix, ( m j ) , and Yj i s the formula v e c t o r f o r the j t h s p e c i e s , [ m j mj2,...mj ]. Given the choice o f components, the s t o i c h i o m e t r i c c o e f f i c i e n t s f o r an independent s e t o f chemical r e a c t i o n s are computed: T

T

k

lf

E

Ρ Σ V..Y. = Y. j=l 3 1

3

where the number o f independent

i=p+l,...,C

(9)

1

r e a c t i o n s , R = C - p, and the


5.

SEiDER ET AL.

119


species on the right-hand-side are r e f e r r e d t o as "derived s p e c i e s . " Compositions a t e q u i l i b r i u m are c a l c u l a t e d using Equa t i o n (7) and conservation of mass f o r each component

G. 1

Ρ Σ v..G. j=l 3 0

= 0

i=p+l,...,C

(10)

,p

(11)

1


C η. + 3

Σ ν..n. = q. i-p+l 1

j=l,

3

where the feed mixture i s d i s t r i b u t e d amongst components, j = the d e r i v e d s p e c i e s , n£, and solved Equation (11) f o r the moles of components, η j . Equations (10) were solved f o r n and con vergence c r i t e r i a checked. For convergence, components need be s e l e c t e d as those species present i n the highest concentration a t e q u i l i b r i u m . When t h i s i s not the case, Equation (11) frequently gives negative values f o r η j . Browne, e t a l . (Ί) suggested the "optimum component" proce dure i n which components are s e l e c t e d as those species expected to be i n the highest concentration a t e q u i l i b r i u m . C r u i s e (8) a l t e r e d the components during i t e r a t i v e c a l c u l a t i o n t o r e f l e c t changes i n composition. Cruise (8) solved Equations (10) and (11) by a d j u s t i n g extents of r e a c t i o n . To prevent divergence, Smith and Missen (9) a l s o improved i n i t i a l estimates by using a l i n e a r programming procedure which neglects the l o g a r i t h m i c terms i n Equation (1). Because o f these problems, most recent methods avoid the d i s t i n c t i o n between components and d e r i v e d species and take the moles of a l l species as i t e r a t i o n v a r i a b l e s . N a p h t a l i (10) solved Equations (7) by o b t a i n i n g c o r r e c t i o n s to extents o f r e a c t i o n from: άζ.

ι

= -AG.dX ι

i=l,...,R

(12)

where dX i s the s t e p - s i z e and C AG. = 1

Σ V..G. j=l

1 D

3

He showed t h a t p o s i t i v e dX give d£ t h a t reduce Gibbs f r e e energy. This method i s analogous t o t h a t of steepest descent, a f i r s t order method f o r minimization o f Gibbs f r e e energy. Ma and Shipman (11) used N a p h t a l i s method t o estimate compositions a t e q u i l i b r i u m and the Newton-Raphson method t o achieve convergence. Several n o n l i n e a r programming methods have been a p p l i e d t o 1


120

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APPLICATIONS T O C H E M I C A L

ENGINEERING


minimize Gibbs f r e e energy. But these are normally formulated t o determine compositions i n chemical and phase e q u i l i b r i u m and are presented i n the s e c t i o n "Chemical and Phase E q u i l i b r i u m . " Heidemann (12) observes t h a t "the chemical r e a c t i o n e q u i l i b rium problem i n a homogeneous phase i s known t o have an unique s o l u t i o n except when the thermodynamic model o f the phase can e x h i b i t d i f f u s i o n a l i n s t a b i l i t y . " Hence, f o r chemical e q u i l i b r i u m i n a s i n g l e phase, l o c a l minima i n Gibbs f r e e energy do not occur and the search i s s i m p l i f i e d . Phase E q u i l i b r i u m Algorithms f o r computation o f compositions i n vapor and l i q u i d phases a t e q u i l i b r i u m u s u a l l y solve the n o n l i n e a r a l g e b r a i c equations G. 3

= G. 3-

V

j=l,...,C

(13)

j=l,...,C

(14)

i n the form y. = k.{T,p,x,y}x.

together with mass balances and the energy balance. Equations (13) are obtained by d i f f e r e n t i a t i n g Equation (1) f o r S = 0 and £ = 2 with d n j = -dnjL. With temperature and pressure given, the f l a s h equation: V

C

ζ. (1 - k.) -, ττ- = 0 . . 1 + o(k. - 1) 3=1 3 Σ

3

1

(15)

/Ί

can be solved f o r α (moles o f vapor/moles o f feed) using simple r o o t - f i n d i n g procedures. For non-ideal s o l u t i o n s k j i s a f u n c t i o n o f l i q u i d composi t i o n and a. Hence, Equation (15) must be s o l v e d i t e r a t i v e l y with m a t e r i a l balance equations and computation times i n c r e a s e . Boston and B r i t t (13) present an algorithm, given the s t a t e of the feed and s i x p r a c t i c a l sets o f s p e c i f i c a t i o n s shown i n F i g u r e 2. They d e f i n e v o l a t i l i t y parameters

u. = I n b 3

where k

1

j=l,...,C

(16)

K

i s the "reference e q u i l i b r i u m r a t i o " d e f i n e d by: C In k b

=

Σ

j=l

w. In k. 3

3


(17)


5.


SEIDER ET AL.

Specified

Calculated

II

α, P ο,τ

y, Ρ

III

Q,V

IV

ν, τ

I

Figure 2.

121

V

V, Ρ

VI

Τ, Ρ

τ,ν LP P,Q Tj Q

ν, ο

Specification by Boston and Britt (13) for vapor-liquid equilibrium


COMPUTER

122

APPLICATIONS

TO CHEMICAL

ENGINEERING

Wj are weighting f u n c t i o n s computed as a f u n c t i o n o f V/L, y, and k, as d e s c r i b e d i n t h e i r paper. Furthermore, a second r e l a t i o n f o r the reference e q u i l i b r i u m r a t i o i s d e f i n e d as:


in

- A + B

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