We propose a reinterpretation of the Gibbs tangent plane stability criterion for phase equilibria [1] via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding Lagrangian dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. Based on this interpretation we propose a very simple algorithm for the calculation of phase equilibria. We discuss its relation with existing algorithms and present case studies based on the NRTL and UNIQUAC activity coefficient models.

The focus of this presentation is the calculation of a stable phase split for multi-component mixtures at constant temperature and pressure. Given an overall composition we are interested in obtaining a stable state comprising one or more phases. Stability in thermodynamics is typically established based on extremum principles which offer one of the alternative formulations of the second law of thermodynamics. The general statement of a (primal) extremum principle is that a state is stable if and only if among all admissible states it is an extremum (minimum or maximum) of an appropriately selected physical property. The set of admissible states is limited by considerations such as material balances. Which physical property is maximized or minimized depends on the constraints considered. At constant pressure and temperature, Gibbs free energy tends to a minimum.

Based on this extremum principle we formulate the minimization of Gibbs free energy subject to material balance as a mathematical program and then consider its Lagrangian dual. Duality is a fundamental mathematical concept used in a variety of subdisciplines such as topology, functional analysis, algebra and mathematical programming. Often it is simpler to prove properties of a primal problem by considering its dual problem. In convex programs satisfying a constraint qualification, the optimal solution values of the primal and dual programs are equal, while in nonconvex programs a duality gap typically exists. In this latter case, the optimal solution value of the dual is a strict lower bound to the optimal solution value of the primal. We demonstrate that the stable phase split is the solution of the dual problem. Only in the absence of duality gap does the solution coincide with the solution of the primal problem; this case can correspond to a single phase being stable (no guarantee of uniqueness). This observation not only allows a reinterpretation of the wellknown stability criterion of the Gibbs tangent plane, but also is to our best knowledge the first description of a dual extremum principle.

Our treatment does not require any assumptions on differentiability of the Gibbs free energy, which by itself is a significant advance. Moreover, the number of phases is not restricted to a finite number and the presence of all species in all phases is not required. We only require continuity of the Gibbs free energy and that each species is present in the overall composition. While the dual problem is always convex [2, p. 486], it is potentially very hard to solve. The particular dual considered here has a specific structure which suggests a very simple procedure for the solution of phase equilibria based on the algorithm by Blankenship and Falk [3] for semi-infinite programs and the algorithm by Falk and Hoffman [4] for min-max problems. Similarly to the methods by Michelsen [5, 6] and by McDonald and Floudas [7, 8, 9] our proposal is a two-stage process. In the first stage the slope of a candidate hyperplane is obtained through a linear program. In the second stage the maximal interception of this hyperplane is obtained through the global solution of a nonlinear program, which is equivalent to the tangent distant minimization of the existing methods. Our proposal does not require differentiability and can use any global optimization algorithm.

We present simple case studies based on the NRTL and UNIQUAC activity coefficient models. Finally, we discuss extensions of the observed result to phase-equilibrium for reacting systems and systems where a different thermophysical property model is used for different phases [10].

References

[1] L. E. Baker, A. C. Pierce, and K. D. Luks. Gibbs energy analysis of phase equilibria. Soc. Petrol. Engrs. J., 22:731-742, 1982.

[2] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, Massachusetts, Second edition, 1999.

[3] J. W. Blankenship and J. E. Falk. Infinitely constrained optimization problems. Journal of Optimization Theory and Applications, 19(2):261-281, 1976.

[4] J. E. Falk and K. Hoffman. A nonconvex max-min problem. Naval Research Logistics, 24(3):441-450, 1977.

[5] M. L. Michelsen. The isothermal flash problem 1. Stability. Fluid Phase Equilibria, 9(1):1-19, 1982.

[6] M. L. Michelsen. The isothermal flash problem 2. Phase-split calculation. Fluid Phase Equilibria, 9(1):21-40, 1982.

[7] C. M. McDonald and C. A. Floudas. Decomposition based and branch-and-bound global optimization approaches for the phase-equilibrium problem. Journal of Global Optimization, 5(3):205-251, 1994.

[8] C. M. McDonald and C. A. Floudas. Global optimization and analysis for the Gibbs free-energy function using the UNIFAC, Wilson, and ASOG equations. Industrial & Engineering Chemistry Research, 34(5):1674-1687, 1995.

[9] C. M. McDonald and C. A. Floudas. Global optimization for the phase-stability problem. AIChE Journal, 41(7):1798-1814, 1995.

[10] J. V. Smith, R. W. Missen, and W. R. Smith. General optimality criteria for multiphase multireaction chemical-equilibrium. AIChE Journal, 39(4):707-710, 1993.