MS 96-011 Abstract

A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem

Ken McKinnon and Marcel Mongeau

Technical Report MS 96-011

This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n_s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m_b independent conserved quantities, where m_b=n_s when no reaction is possible and m_b <= n_e+1 when reaction is possible and n_e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ``reaction tangent-plane criterion'', to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m_b(n_s-1) dimensions) to a finite sequence of local optimization steps in K(n_s-1)-space, K <= m_b, and global optimization steps in (n_s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn-Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.

Key words
Global optimization, Gibbs free energy, chemical and phase equilibrium, non-convex optimization, tangent-plane criterion, convexity.

Postscript MS (310Kb).
Compressed postscript MS (143Kb).
G-Zipped postscript MS (111Kb).
This is a significantly revised version of MS 94-001
This revision was returned to Journal of Global Optimization in Sept 1996.
Related Publications
Technical Report MS 95-001a
Technical Report MS 96-004