Journal of Global Optimization 5: 205-251, 1994. 205 @ 1994 KluwerAcademic Publishers. Printed in the Netherlands. Decomposition Based and Branch and Bound Global Optimization Approaches for the Phase Equilibrium Problem CONOR M. McDONALD and CHRISTODOULOS A. FLOUDAS* Department of Chemical Engineering, Princeton University, Princeton, N.J. 08544-5263, U.S.A. (Received: 19 October 1993; accepted: 15 March 1994) Abstract. An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee e-global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNiversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A numberof interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to obtain e-global solutions in this case. For the UNIQUAC equation, the new properties show how the objective function can be transformed into the difference of two convex functions (i.e. a D.C. programming problem is obtained), where the concave portion is separable. A branch and bound algorithm based on that of Falk and Soland [6] is used to guarantee convergence to an e-global solution. Examples are presented which demonstrate the performance of both algorithms. Key words: Global optimization, phase equilibrium, biconvex and DC programming problems. 1. Introduction A crucial step in the design of any separation process is the ability to predict the behavior of the fluids, when there may be several fluid phases and components that may or may not be reacting. For many separations processes, the assumption that the fluids are in equilibrium is made. The goal is to effectively model these processes over a potentially wide range of operating conditions. Such models can yield complex and nonlinear expressions with resultant difficulties in obtaining the solutions that actually describe the process. For the phase and chemical equilibrium problem there have been essentially two basic approaches. The first of these is equation based, and is not considered in this work. A useful reference in this area is the book of Smith and Missen [21]. An increasingly popular approach is to explicitly minimize the thermodynamic function that describes the equilibrium condition. In the context of this work, this function will be the Gibbs free energy, and a global minimum implies that the Author for correspondence