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In the design and optimization of separation systems based on alternative solvents, accurate models are a key. Furthermore, the reliable computation of phase and chemical equilibria from these models is a critical need. However, it is well known that standard computational methods are subject to failure, thus leading to potentially incorrect results even when the underlying models are accurate. We are developing a new, completely reliable technique, based on interval mathematics, for solving such problems. This will be applied, for example, to design the metal chelate/CO2 extraction processes discussed above.
The determination of phase stability, i.e., whether or not a given phase will split into multiple phases, is the key step in phase equilibrium calculations. The phase stability problem is frequently formulated in term of the tangent plane condition. Minima in the tangent plane distance are sought, usually by solving a system of nonlinear equations for the stationary points. If any of these yield a negative tangent plane distance, indicating that the tangent plane intersects (or lies above) the Gibbs energy or mixing surface, the phase is unstable. The difficulty lies in that, in general, given any arbitrary equation of state or activity coefficient model, most computational methods cannot find with complete certainty all the stationary points, and thus there is no guarantee that the phase stability problem has been correctly solved.
Standard methods for solving the phase stability problem typically rely on the use of multiple initial guesses, carefully chosen in an attempt to locate all stationary points in the tangent plane distance function. However, these methods offer no guarantee that the global minimum in the tangent plane distance has been found. Because of the difficulties that thus arise, there has been significant recent interest in the development of more reliable methods for solving the phase stability problem. Among these are the methods of Eubank et al., who use exhaustive search on a grid, Sun and Seider, who use a homotopy-continuation approach, and Wasylkiewicz et al., who use topological considerations. McDonald and Floudas show that for certain activity coefficient models, the phase stability problem can be reformulated to make it amenable to solution by powerful global optimization techniques, which provide a mathematical guarantee of reliability. However, in general there appears to remain a need for an efficient general-purpose method that can perform phase stability calculations with mathematical certainty for any arbitrary equation of state or activity coefficient model.
An alternative approach, based on interval mathematics, that satisfies these needs was originally suggested by Stadtherr et al., who applied it in connection with activity coefficient models, as later don also by McKinnon et al. Recently, we have extended this method to vapor/liquid equilibrium problems modeled using equations of state, including the Van der Waals, Peng-Robinson, and Soave-Redlich-Kwong equations. In our work is this area, we plan to continue the development of this method and extend its application to solid/fluid equilibrium problems such as the metal chelate/CO2 extraction processes discussed above, and to other phase and reaction equilibrium problems involving environmentally benign solvents. Instructors can contact Professor Stadtherr for this information.
