Mathematical modeling of free radical and olefin polymerization reactors

Mathematical models of reactors for the polymerization of methylmethacrylate (MMA) and propylene have been developed and analyzed in order to better understand the reactor dynamics and to determine conditions for improved operation. The exploration of the effects of mixing and heat transfer in an MMA polymerization reactor system has been conducted by the development of two models--an imperfect mixing model and a detailed model. To model imperfect mixing in polymerization, a reactor configuration using two tanks in parallel was used. Bifurcation diagrams developed using numerical analysis of the model have been drawn with two variable parameters, an exchange ratio, $\sigma$, and a volume ratio, $\kappa$. If both parameters are small, the lower solution branch of the steady state solutions is perturbed in comparison with a simple model which assumes perfect macro-mixing. If $\sigma$ increases ($\kappa$ = 0.1, $\sigma$ = 1.0), the shape of steady state solution curve differs significantly from that of a simple model as the feed temperature decreases. Two correlations for the overall heat transfer coefficient have been used to study the heat transfer. The steady state solutions of mass and energy balances in the reactor depend on the nature of the heat transfer correlation, as does the number of isola branches. The addition of coolant dynamics to the system results in no isola solution branches and no Hopf bifurcations. In olefin polymerization, a particle size distribution (PSD) in the polymerization reactor has been derived using population balances. Three reasonable reaction mechanisms for Ziegler-Natta catalysts, i.e., a simple reaction model, an active site reduction model, and a two sites model, have been used to derive the average number of active sites. It was observed that the PSD depends not only on residence time, but also on the reaction mechanism and that multiple active sites change the PSD slightly. The PSD, however, does not depend on initial catalyst volume. The mass and energy balances in the reactor have been derived to study the effect of PSD for each reaction mechanism on the reactor dynamics. It was observed that the PSD affects both bed height and particle volume. A feasible region for the reactor operation has been calculated using physical constraints. In a nonisothermal polymerization system, the reactor temperature does not change appreciably as catalyst injection rate increases. A unique steady state solution is found in a gas-phase continuous stirred-bed propylene polymerization reactor. The eigenvalues of the system of equations indicate that the steady state is unstable. A comparison with published data allows the observation that the actual reactor dynamics may be readily explained by using only the PSD derived from a simple reaction mechanism