The geometry of separation boundaries: I. Basic theory and numerical support

Separation has always been an important task in the chemical industry. Even with the recent shift in emphasis and efforts to unify the physical and biological sciences, separation will remain an important workhorse in production and product/process design. The theory of simple distillation and residue curve maps has a long history—roughly 100 years—and has made a significant impact in the way many new separation processes are synthesized and designed. The key synthesis/design concept in this approach centers on understanding the ways in which constant boiling mixtures (or azeotropes) define separation boundaries. It is now well established that these azeotropes (or eutectics in melt or fractional crystallization and other solid–liquid separations) often define curved separation boundaries that place limitations on the degree of separation that can be achieved. Despite this, there remains no clear and exact understanding of separation boundaries and no straightforward way of accurately computing them in practice. A geometric methodology is presented that shows that exact separation boundaries can be defined through the use of differential geometry and dynamical systems theory and formulated as a constrained global optimization problem. Our novel approach is based on the observation that, for ternary homogeneous liquids, separation boundaries correspond to local maxima in the line integral within a given separation region. In addition, it is shown that these local maxima in the line integral correspond to one-sided cusps and that global optimization is absolutely necessary, given that several local maxima can exist within any separation region. Several numerical examples are presented that show that the proposed geometric approach can accurately find separation boundaries for homogeneous and heterogeneous mixtures. Finally, it is shown that our new methodology is very general and readily extends to mixtures with four or more components, reactive separations, nonequilibrium models, and other processes such as crystallization and vapor degreasing. © 2005 American Institute of Chemical Engineers AIChE J, 52: 582–594, 2006