Robust and rapid algorithms for solving equations relevant to chemical process simulation will be developed. Singularities frequently occur in chemical process models. These singularities can dramatically alter the performance of equation-solving methods commonly used in chemical process simulation. Many current simulations based on Newton's method fail in singular regions and must be restarted. This study will address the effects of singular points on important equation-solving aspects such as reliability, rates of convergence, and computational efficiency. This work is concerned with the development of a novel and intelligent complex domain methodology for process simulation in singular regions centered around a unified theoretical and computational framework for (1) identifying singular regions, (2) converging to singular points using quadratic approximation, and (3) determining one or more solutions to process model equations. The proposed approach is capable of finding path-disconnected solutions from a given singular point and is computationally efficient. Theorems that provide rigorous theoretical guarantees of a path existence, placing initial values in a given basin of attraction, and global convergence will be proved. The theoretical and algorithmic architectures will be developed in tandem, will address all types of singular points, and will be implemented as a set of portable Fortan 77 programs. Simulation problems such as distillation, reaction engineering, and general process flowsheet examples will be used to verify all theoretical developments. The algorithms developed may have application to improving chemical process simulations.
