Explain phase stability analysis and the tangent plane criterion.

Phase stability analysis determines whether a mixture is single-phase or will split into multiple phases. Tangent plane criterion: a phase is stable if the tangent plane to Gibbs energy surface at the feed composition lies below all other points on the surface. Mathematically: minimize TPD = sum(y_i*(mu_i(y) - mu_i(feed))) over all trial compositions y. If minimum TPD < 0, phase is unstable and will split; trial composition gives approximate equilibrium composition. Implementation: multiple starting points needed (pure components, random), global optimization or successive substitution. Essential for: reliable flash calculations, identifying three-phase equilibrium, and avoiding false single-phase solutions.