Improving Convergence Behavior of Nonlinear Equation Systems in Intensified Process Models by Decomposition Methods

Abstract

Modelling and simulating intensified processes can be a rather demanding task, since convergence problems occur frequently and force the numerical algorithm to abort before a solution for the related equation system is found. These systems are mostly large-scaled, they contain a high number of nonlinear terms and their equations are strongly coupled through certain variables. A promising way to eliminate the convergence problems is presented here in the form of two decomposition methods alias Dulmage-Mendelsohn decomposition and bordered block transformation that arise from linear algebra. They divide the overall equation system into lower dimensional subsystems which can be solved separately in sequence. This is said to reduce the numerical costs and improve the condition in most cases. To check these assertions the decomposition methods were applied on the equation system of a reactive distillation column. Subsequently the original and the decomposed versions were iterated by a simultaneous solver. Their performance during iteration concerning condition, numerical costs and convergence were compared and validated by the condition number, the function counts and the number of successful iterations starting from systematically chosen initial values out of the solution environment. Based on the results we discuss use cases for the decomposed and the original versions and close this work by presenting some ideas about how the efficiency of the decomposition methods could be further improved.