Ind. Eng. Chem. Res. 1988,27, 1728-1732
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(15) Melikhov, I. V.; Kelebeev, A. S. Kristallographia 1979,24,410. (16) Melikhov, I. V.; Kelebeev, A. S.; BaEiE, S. J . Colloid Interface Sci. 1986, 112, 54. (17) Serebryakov, Ju. A.; Khamskii, E. V. Kristallographia 1970,15, 1226. (18) Sohnel, 0.;Mullin, J. W. Cryst. Res. Technol. 1987, 22, 555. (19) von Smoluchowski, M. Z. Phys. Chem. 1917,92, 129. (20) Nielsen, A. E.; Sohnel, 0. J. Crystal Growth 1971, 11, 233. (21) Handbook of Chemistry and Physics; Weast, R. C., Ed.; CRC: Cleveland, OH, 1974-1975; p 8-142. (22) Linke, W. F. Solubilities of Inorganic and Metal-organic Compounds; Van Nostrand: New York, 1958. (23) Falkenhagen, H. Theor. Elektrolyte 1971, 289. (24) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959; p 145.
(25) Landolt-Bornstein Zahlenwerte und Funktionen, II. band, 7 . Teil; Springer-Verlag: Berlin, 1960; p 259. (26) International Critical Tables; McGraw-Hill: New York, 1929; Vol. 6, p 233. (27) Sohnel, 0.; HandGiovl, M. Cryst. Res. Technol. 1984,19,477. (28) Brown, D. J.; Felton, P. G. Chem. Eng. Res. Des. 1985,63, 125. (29) Nancollas, G. H.; Purdie, N. Q.Reu. 1969, 18, 1. (30) Sohnel, 0.; Mullin, J. W. J. Crystal Growth 1978, 44, 377. (31) Fiiredi-Milhofer, H.; MarkoviE, M.; Komunjer, Lj.; PugariE, B.; BabiE-IvanEiE, V. Croat. Chem. Acta 1977, 50, 139. (32) Tomazic, B.; Mohanty, R.; Tadros, M.; Estrin, J. J . Cryst. Growth 1986, 75, 339. Receiued for review November 30, 1987 Accepted May 9, 1988
Reduction Theorem for Phase Equilibrium Problems Eric M. Hendriks K o n i n k l i j k e l S h e l l Laboratorium, A m s t e r d a m ( S h e l l Research B.V.), P.O.Box 3003, 1003 A A A m s t e r d a m , T h e N e t h e r l a n d s
If the excess Gibbs free-energy function for n-component mixtures depends on composition only through a limited number, K , of linear functions (scalar products), then the set of two-phase equilibrium equations and the equations of the stability test can be reduced to a set of only K + 1 equations; the Newton-Raphson correction equations can be reduced from a system of n linear equations to one of size, K 1,and the spinodal curve can be evaluated from the criterion of positive semidefiniteness for a ( K 1)-dimensional quadratic form. Applications include examples ranging from mixtures of hydrocarbons with L non-hydrocarbons ( K = 2L 2 ) to polymer mixtures. In a number of special cases, the simplications have been exploited by other authors, among other things t o save computer time and storage. T h e present work shows the mathematical structure behind these examples and generalizes these to a well-defined class of models.
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1. Introduction The aim of this paper is t o present a general mathematical theorem in phase equilibrium equations. It shows, how for a well-defined class of thermodynamic models, the dimensionality of various phase equilibrium problems can be reduced. For this reason, it has been called the reduction theorem. For a number of special cases, the possibility of a reduction has been noted by other authors, who have used it among other things to save computer time and storage for multicomponent mixtures and to derive a simple criterion for the limit of stability, the spinodal curve, in polymer physics. This paper shows that the simplifications are due to the same underlying mathematical structure. So it unifies results in different fields. The conditions of the theorem are easy to verify. They are often fulfilled, a t least approximately. The theorem may then serve as a basis for a perturbation expansion. The reduction in storage and computer time, such as has been achieved by other authors, is due to the algebraic structure leading to the theorem and is due to the large number of components and small number of parameters. If these conditions are fulfilled for a specific problem/ model, then we expect that computer savings are possible. After introduction of some basic concepts used in the evaluation of phase equilibria in section 2, we shall formulate the theorem in section 3, prove it in section 4, and give a number of examples in section 5. Finally, some conclusions will be given in the last section. 2. Basic Concepts
In phase equilibrium problems, the thermodynamic model is usually stated by specifying the molar Gibbs free-energy function of a homogeneous phase, directly or 0888-5885/88/2627-1728$01.50/0
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indirectly (through an equation of state, for example) as a function of composition 2,pressure p , and temperature T. Quite generally, it is of the form n
G = RT(Cxi log xi i=l
+ g($,p,T))
(1)
for a mixture of n components. When an equation of state is used, G is determined only up to a temperature-dependent but otherwise constant term, which is of no importance here. The free energy (GbJ of a multiphase system is simply the sum of the free energies of the various phases. The equilibrium situation a t fixed temperature and pressure corresponds to the global minimum of G,,, in thermodynamic state space. Metastable states correspond to local minima. A necessary condition for equilibrium is that the chemical potentials of the various species be equal in all phases. For a two-phase equilibrium pi' = pi"
(i = 1, ..., n )
In addition, the conditions of material balance have to be satisfied: r$
+ n?I = niF
(i =
1,
..., n)
(3)
in which ni7 (y = I, 11, and F) are mole numbers. The superscript F denotes feed, i.e., the mixture as a whole. These equations are easily generalized to situations with more phases. After the solution has been found, it still has to be tested, whether or not it corresponds to a global minimum of Gb, (stability test). An important symmetric matrix, related to the above phase equilibrium equations, is AijT
= (8pi/hj)T
0 1988 American Chemical Society
(7 = I, 11)
(4)
Ind. Eng. Chem. Res., Vol. 27, No. 9, 1988 1729 If the differentiations are performed at constant pressure, the elements of A are second derivatives of the Gibbs free-energy function, which are used in numerical algorithms to solve the phase equilibrium equations (2) and (3). For instance, if the standard Newton-Raphson method is used, a t each iteration a linear set of equations
Aa = 3 with A = A' + A" (5) must be solved. This set occurs also as the correction equations when a perturbation expansion is used (Hendriks, 1988). If the differentiations in (4)are performed at constant volume, the matrix elements will have different values, corresponding to the second derivatives of the Helmholtz free energy. This matrix provides a local criterion for metastability: if it is positive semidefinite, i.e., if all its eigenvalues are nonnegative, then the corresponding phase is metastable. The border between metastable and unstable regions in thermodynamic state space, the spinodal curve, is therefore characterized by the vanishing of the smallest eigenvalue of A. For models without pressure dependence, the matrix of second derivatives of the Helmholtz energy cannot be used. In that case, the criterion for metastability follows from the positive semidefiniteness of the matrix of second derivatives of the Gibbs free energy with respect to mole fractions, where one of these mole fractions has been expressed in the others, before differentiation (see the end of section 4). The condition for absolute stability has been analyzed in detail by Baker et al. (1982). A criterion that follows from this analysis has been used by Michelsen (1982). It can be formulated as follows: if for all the solutions (2t,A) of the equatiori pi(2')
- pi(?') = A
3. Formulation of the Theorem In this section we formulate the reduction theorem. If the Gibbs free-energy function is of the form of (1) with g ( 2 , p , T )depending on composition through a set of K independent linear functions, Bk,K < R , i.e.,
(aE nK)
n
(k = 1,
..a,
( n d/dni),T ~ = n
(nT d / a n T ) p , T+ ( d / a d p , T -
C(xj d / d x j ) , , T (9)
j=l
-
which follows from the formal transformation of variables [nj) (r+,x,) with r+ Enj,leads to the following formula for the chemical potential: K pi
= log x i
+ l=O ChlPli
(10)
xEl
Blhl;and in which hl = dh/dP1for 1 = 1,...,K ho = h 1 for i = 1, ..., n. In terms of K factors, defined in the usual ways as Ki = x ? I / x ? , (2) and (3) can therefore be rewritten as Poi
with n
Ki = exp(-(CAhiPii)) 1=0
E,
(12)
in which the symbol A denotes the difference between the two phases (AQ = Q" - Q') and 4 = nT1/r+ is the fraction of the mixture in phase I. We introduce a set of generalized K factors, ~1
( I = 1, ...,K )
= B)'/B)
(13)
and define, for convenience,
(7)
where
Bk = E P k j x ,
4. Proof of the Theorem Let us proceed to the proof of the theorem, starting with the first part, concerning the reduction of the phase equilibrium equations. If the conditions of the theorem, (71, hold, then application of the identity,
(6)
in which y represents the phase under consideration and t a trial phase, the resulting A is positive, and then the phase y is absolutely stable. This is a global criterion. To check it is difficult and time consuming.
g ( t , p , T )= h(B,p,T)
theorem holds also in cases where the variable characterizing species assumes continuous values, so that the summation in (8) must be interpreted as an integral. It could be a moment of a mole weight distribution function, for example. The theorem can also be extended to polymer mixtures, where the Gibbs free-energy function is written in a form different from (1)(van Dijk and Hendriks, 1988) with the ideal mixing term replaced by the Flory-Huggins expression.
KO
= 1;
Poi
=1
(i = 1, ..., n)
(14)
A set of equations involving K I can now be derived by multiplication with Pli of (11)on the one hand, and (3) on the other hand, followed by summation over i. This yields
j=l
Pkj
is independent of composition; rank (B) = K ) (8)
then the following statements hold: (1) The corresponding set of phase equilibrium equations, (2) and (3), and also the equation of the stability test, (6), can be reduced to a set of K + 1 simultaneous nonlinear equations. (2) The system of II linear equations, ( 5 ) , can be reduced to an equivalent set of K + 1 linear equations. (3) The spinodal curve can be calculated from the criterion of positive semidefiniteness for a ( K + 1)dimensional quadratic form; it depends on composition only through the quantities Y k l = Cix,Bki@lcand Bk. From the proof, which will be given in the next section, the explicit form of the reduced set of equations follows. Especially when K
