Differential geometry based homotopy continuation with applications to separation process models

Chemical engineering problems often lead to the search for the solution of a set of highly non-linear algebraic equations. Equations of these type often exhibit multiple solutions. Continuation algorithms are the only assured technique of indicating this multiplicity and finding each of the solutions. Unfortunately the increased robustness of continuation is accompanied by increased computational load. In this work the efficiency of the prediction and step-length procedures of continuation algorithms is improved by incorporation of local differential geometry. The unit tangent, curvature and principle unit normal of the solution path Frenet frame are rigorously calculated at each continuation step. The curvature provides a natural, geometric means of step-length determination while the tangent, unit normal, and curvature provide improved prediction accuracy. The new algorithm requires less continuation steps, less Newton corrections, and encounters fewer poorly chosen step sizes than standard continuation algorithms. CPU time requirements are reduced for most problems of small dimensionality (N ;Further utility of the algorithm is demonstrated by using it to investigate heat integrated ethanol dehydration flowsheets. An azeotropic two column sequence for ethanol dehydration is presented which requires less equipment than extractive distillation and requires less energy than conventional two column configurations.;When continuation is applied to separation problems with complex thermodynamic subroutines and many equilibrium stages, evaluation of the system equations represents the major computational load of the algorithm. For these type of problems a continuation algorithm was developed which minimized function evaluations while approximating the differential geometry of the homotopy path. Three approximations were employed: (1) calculation of the tangent vector from the most recent corrector Jacobian, (2) calculation of the curvature and principle unit normal from a quadratic spline fit to the homotopy path, and (3) employment of quasi-Newton updates during the correction process. The resulting algorithm reduces function evaluations and CPU time requirements for the solution of complex separation problems.