Computing multiple solutions to systems of interlinked separation columns

Globally convergent homotopy continuation methods have been used successfully to find multiple solutions to systems of nonlinear equations used to model multicomponent, multistage separation processes. However, the solutions were achieved by using a multitude of different starting points. This paper describes a procedure for finding all or some solutions of the nonlinear equation system from just one starting point. A reliable algorithm, which utilizes a deflated decomposition technique to overcome the turning-point problem, an efficient procedure to estimate step sizes, and variable mapping functions to prevent failure when computing physical properties, is developed to follow the homotopy path. The Jacobian matrix is retained as a special structure in all calculation steps. The algorithm has found, from one starting point, all the real roots of the interlinked separation system studied by Chavez et al. (1986). Two additional examples, including one optimization problem and one constrained nonlinear equation system, are presented to illustrate other applications of the algorithm.