The application of global homotopy continuation methods to chemical process flowsheeting problems

The equation based approach to chemical process flowsheeting requires a method for finding all real roots of general systems of nondifferential algebraic equations with probability one. As an approach to it, the global homotopy continuation methods have been investigated, which find roots by tracking the homotopy paths.; The conventional path tracking algorithms sometimes miss roots owing to jumping from one segment of the global homotopy path to another even if there exist homotopy paths to the roots. A robust path tracking algorithm has been developed which avoids segment jumping by controlling the relative change in the determinant of the augmented Jacobian of the homotopy function.; The starting point criterion proposed by Kuno and Seader (1988) gives restrictions for the starting point for global fixed point homotopy that produce homotopy paths classified into three cases. All real roots are required to lie on a finite length of the real domain homotopy path in two of the cases. However, several additional cases have been found which were not considered in their method. The additional cases indicate that real domain calculations cannot guarantee location of all real roots in any case.; It has been found that in some cases there exist real roots that are unreachable from the starting point even through path following in the complex domain. This shows that the global homotopy continuation methods which use only one starting point do not guarantee to find all real roots.; Although the global homotopy continuation methods are not probability one, case studies show that the Newton homotopy gives wide convergence to physically feasible solutions, and that it can be applied to equation based chemical process flowsheeting to achieve robust equation solving.