Ken McKinnon and
Marcel Mongeau
Technical Report MS 96-011
Abstract
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.