| A persistently difficult problem in chemical engineering is the phase and chemical equilibrium problem which is of crucial importance in several process separation applications. These problems are characterized by multiple local solutions that can lead to the erroneous prediction of the number of phases and the distribution of components among them at equilibrium. Difficulties are encountered due the highly nonlinear nature of models used. Therefore, several interesting properties, simplifications and transformations are presented for the NRTL, Wohl's expansion, UNIQUAC, UNIFAC, ASOG and modified Wilson equations for the liquid phase, and the B-truncated virial equation for the vapor phase. It is also shown that the molar Gibbs function calculated using the Wilson equation is a convex function. These manipulations then provide the framework for the application of global optimization for the following two problems: (i) the minimization of the Gibbs free energy, denoted as Problem (G), and (ii) the minimization of the tangent plane distance function, or the tangent plane stability criterion, denoted as Problem (S). It is shown how two global optimization algorithms can be used to obtain global solutions for Problems (G) and (S). For the NRTL, Wohl's expansion and the B-truncated virial equation, the formulations feature a biconvex objective function subject to bilinear constraints. The global optimization (GOP) algorithm of Floudas and Visweswaran (1990, 1993) is used to obtain global solutions for this class of problems. For the UNIQUAC, UNIFAC, ASOG and modified Wilson equations, the objective function is the difference of two convex functions where the concave portion is separable, and a branch and bound algorithm can be used to obtain global solutions for this category of problems. This implies that global solutions can be guaranteed for both Problems (G) and (S) individually. In addition, a combined algorithm employs them in tandem, using (G) to generate candidate equilibrium solutions which can then be verified for thermodynamic stability by solving (S). All of these global optimization approaches have been incorporated into the package GLOPEQ, and computational results for Problems (G) and (S) and the combined algorithm are presented for a wide variety of examples. These clearly demonstrate the effectiveness of the presented approaches in obtaining global solutions to the phase and chemical equilibrium problem. |
