Efficient Numerical Integration of Stiff Differential Equations in Polymerisation Reaction Engineering: Computational Aspects and Applications Iv´ an Zapata-Gonz´ alez, 1, 2 Enrique Sald´ ıvar-Guerra, 1 * Antonio Flores-Tlacuahuac, 3 Eduardo Vivaldo-Lima 4 and Jos´ e Ortiz-Cisneros 2 1. Centro de Investigaci´ on en Qu´ ımica Aplicada (CIQA), Blvd. Enrique Reyna 140, Saltillo, Coahuila 25253, Mexico 2. Facultad de Ciencias Qu´ ımicas, Universidad Aut´ onoma de Coahuila, Blvd. Venustiano Carranza, Col. Rep´ ublica 25280, Saltillo, Coahuila, Mexico 3. Departamento de Ingenier´ ıa Qu´ ımica y Ciencias Qu´ ımicas, Universidad Iberoamericana, Santa F´ e, Prol. Paseo de la Reforma 880, 01210 M´ exico, DF, Mexico 4. Facultad de Qu´ ımica, Departamento de Ingenier´ ıa Qu´ ımica, Universidad Nacional Aut´ onoma de M´ exico, 04510 M´ exico, DF, Mexico The modelling of the full molecular weight distribution in addition polymerisation gives rise to very large dimension (10 3 –10 6 ) systems of ordinary differential equations, often exhibiting serious stiffness issues. This article summarises a methodology recently implemented by our group, in which the QSSA is applied on the fast dynamic species in order to reduce the stiffness, and then the remaining equations are solved by computationally inexpensive explicit algorithms (such as Euler). Specific features of the methodology are illustrated by application to the academically and industrially relevant systems of controlled radical polymerisation (RAFT and NMP cases) and coordination catalysis polymerisation. Keywords: quasi-steady-state, molecular weight distribution, NMP, RAFT, coordination polymerisation INTRODUCTION S tiff differential equations appear in several problems in chemical engineering, such as those in the following fields: analysis of reaction kinetics of systems (Gelinas, 1972; Shieh et al., 1988) involving very reactive intermediate chemical species (e.g. combustion and thermal cracking), modelling and simula- tion of atmospheric chemistry problems (Verwer et al., 1996), fluid turbulence research (Moin and Mahesh, 1998), dynamic optimisation of processing systems (Biegler, 1984), simulation of biomedical systems (Hines, 1984), optimal design of experiments (Bauer et al., 2000), molecular dynamics calculations (Baldridge et al., 1989), and in the simulation of stochastic biochemical sys- tems (Li et al., 2008), just to mention a few fields (Shampine and Gear, 1979; Shampine, 1985; Cash, 2003). The stiffness of these systems arise from the different dynamics exhibited by ‘slow’ and ‘fast’ species (reactive intermediates); however, there seems to be no systematic numerical treatment of these problems that exploit the different dynamics present in the system. Stiffness issues can also appear not just from process dynamic behaviour but also because of numerically ill-conditioned systems. Recently (Biegler et al., 2007), there has been interest in the simulation and optimi- sation of distributed parameter systems. In these systems, stiffness can also arise because of an improper spatial discretisation when ∗ Author to whom correspondence may be addressed. E-mail address: esaldivar@ciqa.mx Can. J. Chem. Eng. 90:804–823, 2012 © 2012 Canadian Society for Chemical Engineering DOI 10.1002/cjce.21656 Published online 17 February 2012 in Wiley Online Library (wileyonlinelibrary.com). | 804 | THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING | | VOLUME 90, AUGUST 2012 |