Phase Equilibria and Fluid Properties in the Chemical Industry

4 Measurement of Vapor-Liquid Equilibrium MICHAEL M . ABBOTT

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Rensselaer Polytechnic Institute, Troy, N.Y. 12181

The importance accorded the measurement of vapor-liquid equilibrium (VLE) data needs little elaboration. A glance at the annual indices for the Journal of Chemical and Engineering Data makes the point nicely. For the five years 1971-1975, at least 50 papers present new VLE data, and many of them contain descriptions of new experimental techniques. Perusal of recent volumes of less specialized journals, such as the AIChE Journal or I&EC Fundamentals, reveals a similar proliferation of VLE studies. The necessity for reliable VLE data is apparent. Many separations processes involve the transfer of chemical species between contiguous liquid and vapor phases. Rational design and simulation of these processes requires knowledge of the equilibrium compositions of the phases. Raoult's and Henry's "laws" rarely suffice as quantitative tools for the prediction of equilibrium compositions; precise work demands the availability of either the equilibrium data themselves, or of thermodynamic correlations derived from such data. Specialists in the field tend to concentrate on one of two broad areas: high-pressure VLE, or low-pressure VLE. My major interests are in the latter area, and hence the thrust of my talk will be in this direction: the measurement and reduction of lowpressure VLE data. Low-pressure VLE experimentation differs from high-pressure experimentation on two major counts. First, the problems of equipment design and operation are less formidable than for high-pressure work. Secondly, more effective use can be made of the thermodynamic equilibrium equations, both in the data reduction process and in the design of the experiments themselves. It is this second feature-the strong interplay of theory with experiment--which to me most distinguishes low-pressure from high-pressure VLE work. The usual product of a low-pressure VLE study is an expression for the composition dependence of the excess Gibbs function G for the liquid phase. If experiments are done at several temperatures, E

87 In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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then the temperature dependence o f parameters i n the equation f o r may a l s o be determined. At low p r e s s u r e s , the p r e s s u r e dependence of i s weak, and may u s u a l l y be ignored. C e r t a i n advantages d e r i v e from t h i s k i n d of r e p r e s e n t a t i o n , f o r the d e s c r i p t i o n of VLE i s compact and y e t has thermodynamic s i g n i f i c a n c e . B a c k - c a l c u l a t i o n of e q u i l i b r i u m curves r e q u i r e s values f o r the pure-component vapor p r e s s u r e s , an e x p r e s s i o n f o r G i n terms of x and T, and equations of s t a t e ( u s u a l l y of s i m p l e form) f o r the vapor and l i q u i d phases. Approximations, when they must be made, a r e w e l l - d e f i n e d and o f t e n s u b j e c t to independent verification. Downloaded by LOUISIANA STATE UNIV on May 8, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch004

E

Thermodynamic C o n s i d e r a t i o n s At e q u i l i b r i u m , the f u g a c i t i e s of each component i must be the same i n the l i q u i d and vapor phases:

The vapor-phase f u g a c i t y i s r e l a t e d to the vapor-phase f u g a c i t y c o e f f i c i e n t (ji-^: f±

v

= y** ±

(2)

±

F u g a c i t y c o e f f i c i e n t _^ i s c a l c u l a b l e from an equation of s t a t e f o r the vapor phase; mixing r u l e s must be a v a i l a b l e f o r the equat i o n - o f - s t a t e parameters. The l i q u i d - p h a s e f u g a c i t y i s r e l a t e d to the l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t y^i f . * = x. f°

(3)

Yl

Here, f° i s the s t a n d a r d - s t a t e f u g a c i t y f o r s p e c i e s i . I f the e q u i l i b r i u m temperature i s lower than the c r i t i c a l temperature o f pure i (the u s u a l case f o r low-pressure VLE), the standard s t a t e i s taken as pure l i q u i d i a t the system T and P. Thus Eq. (3) becomes f*

I

= x.y.f. 11 l

(4)

Determination o f f ^ r e q u i r e s the a v a i l a b i l i t y of equations of s t a t e f o r pure vapor and l i q u i d i , and a v a l u e f o r the vapor p r e s s u r e p| of pure i . Often, f-^ i s approximately equal to P ? . Combination of Eqs. ( 1 ) , ( 2 ) , and (4) y i e l d s the form of the e q u i l i b r i u m equation most commonly used i n low-pressure work a t

a t

W i

- W

( 5 )

I t i s through Eq. (5) that low-pressure VLE measurements p r o v i d e the experimental i n p u t r e q u i r e d f o r a q u a n t i t a t i v e thermodynamic

In Phase Equilibria and Fluid Properties in the Chemical Industry; Storvick, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

4.

ABBOTT

Measurement of Vapor-Liquid

Equilibrium

89

d e s c r i p t i o n of l i q u i d - p h a s e n o n i d e a l i t i e s . The excess Gibbs f u n c t i o n G p l a y s a c e n t r a l r o l e i n e x p e r i mental s o l u t i o n thermodynamics, f o r i t s c a n o n i c a l v a r i a b l e s (T, P, and x) are those most s u s c e p t i b l e t o measurement and c o n t r o l . A c t i v i t y c o e f f i c i e n t s are r e l a t e d to mole-number d e r i v a t i v e s o f the dimensionless excess Gibbs f u n c t i o n g = G /RT: E

E

toY

±

= [Z(ng)/dn ] ±

(6)

T


Thus &nYj[ i s a p a r t i a l molar property w i t h respect to g. Accordi n g l y , we have the a d d i t i o n a l u s e f u l r e l a t i o n s h i p g = Zx Zny ±

(7)

±

C l a s s i c a l thermodynamics provides the f o l l o w i n g e x p r e s s i o n f o r the t o t a l d i f f e r e n t i a l o f g: dg = -

dT + ~ - dP + Zlny.dx R

T

1

(8) 1

RT F F Here, H i s the excess enthalpy (heat o f mixing) and V i s the excess volume (volume change o f m i x i n g ) . Both H and V are subj e c t t o d i r e c t experimental d e t e r m i n a t i o n . Taking the t o t a l d i f f e r e n t i a l o f Eq. ( 7 ) , and comparing the r e s u l t w i t h Eq. ( 8 ) , we o b t a i n the Gibbs-Duhem equation: E

E

H

Zx dlny ±

±

vE

E

= - ^

d

T

+

i f

d

P

(

9

)

RT Equations (1) through ( 9 ) , o r v a r i a t i o n s upon them, c o n s t i tute the u s u a l thermodynamic b a s i s f o r the r e d u c t i o n and i n t e r p r e t a t i o n o f low-pressure VLE data. Isothermal vs. I s o b a r i c Data P r e c i s e d e t e r m i n a t i o n o f G through low-pressure VLE measurements g e n e r a l l y r e q u i r e s the a v a i l a b i l i t y o f data spanning the e n t i r e range o f l i q u i d compositions. The e x p e r i m e n t a l i s t s t i l l has the o p t i o n , however, o f c o l l e c t i n g h i s data e i t h e r a t i s o b a r i c or a t i s o t h e r m a l c o n d i t i o n s . The q u e s t i o n then a r i s e s whether one type of data i s more u s e f u l than the o t h e r . L e t us assume the a v a i l a b i l i t y of two complete s e t s o f e r r o r f r e e VLE measurements. The f i r s t s e t has been taken a t constant pressure, and c o n s i s t s o f values o f T, x, and y. The second i s i s o t h e r m a l , c o n s i s t i n g of values f o r P, x, and y. We wish to reduce both s e t s o f data, so as to o b t a i n values f o r G . For each data s e t , we may c a l c u l a t e p o i n t values o f y from Eq. ( 5 ) : i

Y. = y..P/x.f.


(10)

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The corresponding values f o r g are then computed from Eq. (7): g = Ex.Any. 1 1 I t i s d e s i r a b l e f o r purposes of c o r r e l a t i o n and i n t e r p r e t a t i o n that experimental values of g be at a s i n g l e T and P, w i t h x the only v a r i a b l e . C l e a r l y , t h i s i s not the case f o r e i t h e r of our experiments. C o r r e c t i o n s must be made i n both cases to reduce the values of g to a common b a s i s . According to Eq. ( 8 ) , t h i s r e q u i r e s the use of independently determined values f o r H or V . Thus, f o r i s o b a r i c data, Downloaded by LOUISIANA STATE UNIV on May 8, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch004

E

,

T

f

H

g(T5P,x) = g(T,P,x) - / T S i m i l a r l y , f o r i s o t h e r m a l data,

E

^ d T RT p

T

g(T,P ,x) = g(T,P,x) + /

E

l

V

E

^

dP

T

1

Here, T i s a r e f e r e n c e temperature f o r the i s o b a r i c data, and P a r e f e r e n c e pressure f o r the i s o t h e r m a l data. Even i f the r e q u i r e d values of H or V are a v a i l a b l e — a n d o f t e n they are n o t — t h e above c o r r e c t i o n s complicate the data a n a l y s i s . One n a t u r a l l y asks whether one or the other of the c o r r e c t i o n s might more s a f e l y be ignored. C a l c u l a t i o n s f o r r e a l s y s tems show that the temperature c o r r e c t i o n i s o f t e n s u b s t a n t i a l , whereas the pressure c o r r e c t i o n i s f r e q u e n t l y n e g l i g i b l e ; a t low p r e s s u r e s , g may depend s t r o n g l y on T, but r a r e l y upon P. Thus low-pressure i s o t h e r m a l VLE data are more e a s i l y reduced to u s e f u l form than are i s o b a r i c data; they are to be p r e f e r r e d f o r t h i s reason. Much of the o l d e r low-pressure work i s i s o b a r i c ; the b e s t modern s t u d i e s are i s o t h e r m a l . The argument i s sometimes advanced t h a t i s o b a r i c VLE data are more u s e f u l f o r process design, because s e p a r a t i o n processes are more n e a r l y i s o b a r i c than i s o t h e r m a l . This argument ignores the f a c t t h a t pressure drops i n d i s t i l l a t i o n columns can be substant i a l , and t h a t they are accounted f o r i n modern design procedures. The best thermodynamic t o o l f o r low-pressure d i s t i l l a t i o n column design i s an e x p r e s s i o n f o r G i n terms of T and x, w i t h values f o r the parameters determined from c a r e f u l l y executed i s o t h e r m a l VLE experiments. E

E

E

D i r e c t v s . I n d i r e c t Determination of Vapor Compositions E

We consider the d e t e r m i n a t i o n of G v i a the r e d u c t i o n of lowpressure i s o t h e r m a l P-x-y data. The procedure i s simple and d i r e c t . Values of are computed from Eq. (10), and the corresponding values of g are c a l c u l a t e d from Eq. ( 7 ) ; the values of g are then smoothed w i t h r e s p e c t to l i q u i d composition. The smoothing procedure may be g r a p h i c a l or n u m e r i c a l , but the eventual product i s an


4.


ABBOTT

Equilibrium

91

a n a l y t i c a l e x p r e s s i o n f o r g as a f u n c t i o n of x. In the above procedure, no use i s made o f the Gibbs-Duhem equation, Eq. ( 9 ) . Because the pressure-dependence o f G i s weak at low p r e s s u r e s , t h i s equation reduces f o r i s o t h e r m a l c o n d i t i o n s to E

Ex d£ny = 0


i

(11)

i

Equation (11) i n f a c t imposes a thermodynamic c o n s t r a i n t upon the l i q u i d - p h a s e a c t i v i t y c o e f f i c i e n t s , a c o n s t r a i n t not n e c e s s a r i l y s a t i s f i e d by values of y^ computed v i a Eq. (10) from r e a l (and t h e r e f o r e p o s s i b l y imperfect) data. However, values o f y^ generated from a smoothing equation f o r g v i a Eq. (6) do s a t i s f y Eq. (11). Comparison of generated w i t h experimental values o f y^ cons t i t u t e s an example of a popular e x e r c i s e known as a thermodynamic c o n s i s t e n c y t e s t . Many c o n s i s t e n c y t e s t s have been proposed, both f o r low- and high-pressure VLE data. Van Ness e t a l (j.) and C h r i s t i a n s e n and Fredenslund (2) present readable d i s c u s s i o n s o f such t e s t s . Instead o f s e r v i n g as a b a s i s f o r the t e s t i n g of redundant data, the Gibbs-Duhem equation may be used i n q u i t e a d i f f e r e n t manner, one which a i d s the experimenter i n the design of a VLE experiment of minimal complexity. For purposes o f d i s c u s s i o n , we assume i d e a l - g a s behavior f o r the vapor phase, and p r e s s u r e independence o f l i q u i d - p h a s e p r o p e r t i e s . I n t h i s case, Eq. (10) reduces to

and the Gibbs-Duhem equation becomes, f o r a b i n a r y system, x d5,ny + x d£ny = 0 £

1

2

2

(13)

Equations (12) and (13) y i e l d , on combination and s i m p l i f i c a t i o n , d

y

dP

i

y^-y^

P(y -x ) 1

(14)

1

Equation (14) i s a r e s t r i c t e d form o f the b i n a r y c o e x i s t e n c e equat i o n . The important content o f the equation i s as much conceptual as mathematical: i t i l l u s t r a t e s that simultaneous measurement o f P, x, and y i s unnecessary, that vapor compositions can i n p r i n c i p l e be computed from measurements o f j u s t P and x. Once the y a r e determined by i n t e g r a t i o n , values of y^ f o l l o w d i r e c t l y from Eq. (12). Van Ness (3) presents a d e t a i l e d d i s c u s s i o n of the charact e r i s t i c s and a p p l i c a t i o n o f Eq. (14). Reduction o f VLE data v i a the c o e x i s t e n c e equation i s " i n d i r e c t " , i n t h a t i t makes no d i r e c t use o f experimentally-determined vapor compositions. Other i n d i r e c t approaches are p o s s i b l e . One


92

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of the most popular i s t h a t f i r s t proposed by Barker (_4 ) . method i s based upon the equation P = Z (x.y.P 11 i I

S a t

70 l

Barker's

(15)

which f o l l o w s from Eq. ( 5 ) .

Quantity $^ i s d e f i n e d by

S a t

*. = * . ( P / f . ) l l l l Downloaded by LOUISIANA STATE UNIV on May 8, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch004

A c t i v i t y c o e f f i c i e n t s i n Eq. (15) are r e p l a c e d by the expressions Y, = exp [g -

Z x (|f-) y i T P x k

]

(16)

k

The composition dependence of g i s expressed by an equation of s u i t a b l e form, and values of undetermined parameters i n t h i s equat i o n are found by a r e g r e s s i o n procedure that y i e l d s a b e s t f i t of the P vs. x data. The $^ depend upon the y^, which are not i n i t i a l l y known; thus an i t e r a t i v e procedure i s r e q u i r e d . The major requirement f o r s u c c e s s f u l a p p l i c a t i o n of Barker's method i s the a v a i l a b i l i t y of an a n a l y t i c a l e x p r e s s i o n f o r g t h a t i s capable of producing a f i t to the P-x data to w i t h i n the l i m i t s of e x p e r i mental u n c e r t a i n t y . Abbott and Van Ness (5) and Abbott (6) d i s cuss the development and s e l e c t i o n of such equations. There has been much d i s c u s s i o n of the r e l a t i v e m e r i t s of d i r e c t measurement, as opposed to i n d i r e c t c a l c u l a t i o n , of vapor compositions. Although the d i f f i c u l t y of p r e c i s e measurement of y's i s g e n e r a l l y conceded, there remains a s t r o n g body of o p i n i o n that the redundant i n f o r m a t i o n p r o v i d e d by experimental values of y (however much i n e r r o r they might be) somehow enhances the r e l i a b i l i t y of the eventual c o r r e l a t i o n f o r g. I do not s u b s c r i b e to t h i s view. The d i f f i c u l t y of c a r r y i n g out the necessary c a l c u l a t i o n s by hand c e r t a i n l y a t one time c o n s t i t u t e d a reasonable argument a g a i n s t i n d i r e c t d e t e r m i n a t i o n of y's, but the e l e c t r o n i c computer has removed t h i s computational b a r r i e r . The experimenter's time i s best spent i n a c h i e v i n g accuracy i n the measurement of the minimum number of v a r i a b l e s r e q u i r e d to c h a r a c t e r i z e the system, not i n d e v i s i n g ingenious ways to c o l l e c t redundant data. E x t e n s i o n of Isothermal VLE Data w i t h Temperature According to Eq. ( 8 ) , the excess enthalpy i s p r o p o r t i o n a l to the temperature d e r i v a t i v e of g: H

E

= -RT

2

(|f)

(17)

Equation (17) suggests an a l t e r n a t i v e procedure to the d i r e c t d e t e r m i n a t i o n of i s o t h e r m a l VLE at s e v e r a l temperatures, namely,


4.

ABBOTT


Equilibrium

93

the use o f heat-of-mixing data f o r e x t r a p o l a t i o n o r i n t e r p o l a t i o n of g w i t h T. Suppose f o r example that i s o t h e r m a l data are a v a i l a ble a t a s i n g l e temperature T-^, and t h a t H data are a v a i l a b l e a t two temperatures near T]_. Suppose f u r t h e r that H /RT i s known to vary approximately l i n e a r l y w i t h T: E

E

E

H /RT = a + bT where a and b depend upon composition o n l y . Values o f g a t some other temperature T can then be found by i n t e g r a t i o n o f Eq. (17): Downloaded by LOUISIANA STATE UNIV on May 8, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0060.ch004

2

g(T ) 2

= gO^)

- a £n ( T ^ )

- b d ^ )

Obvious extensions to t h i s simple example are p o s s i b l e . Thus, the method can be used f o r computation o f i s o b a r i c VLE data. More comprehensive expressions f o r the temperature dependence o f H can be employed, which a l l o w i n c o r p o r a t i o n o f s e v e r a l s e t s o f i s o t h e r mal VLE data and H data. F i n a l l y , C p data can be i n c o r p o r a t e d i n t o the procedure through use o f the thermodynamic equation E

E

S

E

E

= P,x

( 1 8 )

An elegant example o f the simultaneous use o f i s o t h e r m a l VLE and H data to p r o v i d e a complete and i n t e r n a l l y c o n s i s t e n t s e t o f excess f u n c t i o n s over a wide temperature range i s the recent work of L a r k i n and Pemberton (7) on the ethanol-water system. E

Measurement o f Low-Pressure VLE Data Hala et_ a l (8) c l a s s i f y VLE measurement techniques i n f i v e major groups: d i s t i l l a t i o n methods, c i r c u l a t i o n methods, s t a t i c methods, dew-point/bubble-point methods, and f l o w methods. One could add to t h i s l i s t some r a t h e r s p e c i a l i z e d techniques, b u t i n the main most modern low-pressure VLE work i s done on two major types o f equipment: c i r c u l a t i o n s t i l l s and s t a t i c e q u i l i b r i u m cells. In low-pressure v a p o r - c i r c u l a t i o n s t i l l s , a l i q u i d mixture i s charged to a d i s t i l l i n g f l a s k , and brought to a b o i l . The evolved vapors are condensed e x t e r n a l l y i n t o a r e c e i v e r ; excess condensate r e t u r n s through an overflow tube back i n t o the d i s t i l l i n g f l a s k , where i t mixes w i t h the b o i l i n g l i q u i d . The compositions of the b o i l i n g l i q u i d and the condensate change c o n t i n u o u s l y w i t h time, u n t i l a s t e a d y - s t a t e c o n d i t i o n i s reached. The s t e a d y - s t a t e comp o s i t i o n s of the b o i l i n g l i q u i d and the vapor condensate are taken to be the e q u i l i b r i u m l i q u i d and vapor compositions. Hala et_ a l (8) d i s c u s s a t great l e n g t h the design and p r i n c i ples o f o p e r a t i o n o f c i r c u l a t i o n s t i l l s . The b e t t e r s t i l l s a r e complicated d e v i c e s , which may i n v o l v e c i r c u l a t i o n o f b o i l i n g l i q u i d as w e l l as o f vapor condensate. Operation i s o f t e n i s o b a r i c , w i t h pressure r e g u l a t i o n provided by a manostat. S p e c i a l


P H A S E EQUILIBRIA

A N D FLUID PROPERTIES I N C H E M I C A L INDUSTRY

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