3398
Ind. Eng. Chem. Res. 2000, 39, 3398
No Connection between the AREA Criterion and Phase Stability Has Been Established Paul I. Barton* Department of Chemical Engineering and Energy Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Chyi Hwang Department of Chemical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan
Sir: In 19921 the AREA method was proposed for the prediction of phase equilibria in multicomponent mixtures. This is based on the AREA criterion, which maximizes the area between the straight line connecting two points on the Gibbs energy surface and the Gibbs energy surface itself. It was claimed that the two points satisfying this criterion are the true, stable phase split exhibited by a given mixture. In a later paper,2 “rigorous” proofs are presented that claim to demonstrate that the AREA criterion is necessary and sufficient for phase equilibrium in two-component, two-phase mixtures and necessary for phase equilibrium in multicomponent, two-phase mixtures. The purpose of this correspondence is to highlight the logical flaw in these proofs. As a consequence, given the current state of knowledge, no connection between the AREA criterion and phase equilibrium has been established. This obviously brings into question the wisdom of developing algorithms for the prediction of phase equilibrium based on the AREA criterion. It has been known since the time of Gibbs that the stable phase split exhibited by a given mixture is equivalent to the global minimum of the Gibbs free energy at the specified temperature and pressure. Gibbs also derived the well-known equipotentiality criterion2 (eq 4), which is a necessary condition for phase equilibrium. Let us label with A the proposition that a given candidate phase split is stable and denote that the candidate satisfies the equipotentiality criterion as proposition B (i.e., satisfying eq 4 in ref 2). Then, we have
AfB
(1)
It is important to recall the well-known fact that the converse of eq 1 does not hold (e.g., Theorem 4.1.2 plus Corollary in ref 3), because this condition is derived from stationarity of the Gibbs free energy. Let us also denote with C the proposition that the candidate satisfies the AREA criterion. In ref 2 it is shown via the classical necessary conditions for optimality (stationarity) that eq 5 of that paper is a necessary condition for C. Denote with D the proposition that the candidate satisfies eq 5 of ref 2. We therefore have
CfD
(2)
Similarly, the converse does not hold. Recalling that stationarity is only necessary for a maximum or mini* Corresponding author. Phone: +1-617-253-6526. Fax: +1617-258-5042. E-mail:
[email protected].
mum, the “rigorous proof” in fact only establishes
BTD
(3)
whereas in the multicomponent case it can only be demonstrated that
BfD
(4)
Therefore, in both cases we have
AfD
(5)
The flaw in the argument lies in asserting some connection between A (a stable phase split) and C (the AREA criterion) based on demonstrating eqs 2 and 5. By the rules of classical propositional logic, these two relationships give us no information concerning a connection between A and C. All that has been established is that candidates satisfying A and C are both subsets of the set of candidates satisfying D (and B, in the twocomponent, two-phase case). Indeed, in the extreme A and C could be disjoint sets. The desired connection may indeed exist, but the current state of knowledge embodied in the above propositions does not establish any such connection. In addition, it seems unwise to develop algorithms based on criterion D, because in the multicomponent case, it is weaker than the well-known equipotentiality criterion, which was the basis of many early algorithms. Acknowledgment The research of Paul I. Barton was supported by the Engineering Research Program of the Office of Basic Energy Sciences at the United States Department of Energy under Grant DE-FG02-94ER14447. Literature Cited (1) Eubank, P. T.; Elhassan, A. E.; Barrufet, M. A.; Whiting, W. B. Ind. Eng. Chem. Res. 1992, 31, 942-949. (2) Elhassan, A. E.; Tsvetkov, S. G.; Craven, R. J. B.; Stateva, R. P.; Wakeham, W. A. Ind. Eng. Chem. Res. 1998, 37, 14831489. (3) Bazaraa, M. S.; Sherali, H. D.; Shettty, C. M. Nonlinear Programming: Theory and Algorithms, 2nd ed.; John Wiley: New York, 1993.
IE001100G
10.1021/ie001100g CCC: $19.00 © 2000 American Chemical Society Published on Web 08/04/2000
