Mathematical Modeling of Arborescent Polyisobutylene Production in Batch Reactor Using Novel Material Balance and Monte Carlo Methods
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Four mathematical models have been developed to describe copolymerization of inimer and isobutylene via living carbocationic polymerization in batch reactors. In this system, there are six different propagation rate constants that result from two kinds of vinyl groups and three different propagating end groups. The models are developed with the following goals: 1) to account for all six propagation rate constants without making equal-reactivity assumptions; 2) to predict concentration changes of inimer, isobutylene and polymer over time; 3) to predict values of average branching level, number and weight average molecular weight; 4) to predict molecular weight distribution; 5) to estimate model parameters. First, a simplified but lengthy PREDICI model is developed. Due to simplifying assumptions, this model can only be used for systems with low branching levels (i.e., less than 5 branches per molecule, on average), and only four of the six propagation rate constants are included. This model achieves goals 2 to 5 above, with parameters being estimated using low-branching-level data. Next, a traditional Monte Carlo (MC) model is developed. This MC model achieves all of the goals, except parameter estimation, due to the excessive computational effort that would be required. Third, a more advanced MC model is developed using a combination of dynamic material balances and stochastic calculations. With much shorter computational times (by a factor of ~200), this MC model provides information similar to that provided by the traditional MC model, and also provides information about dangling and internal segments in the polymer molecules. However, this model is still not suitable for parameter estimation. Finally, a “parallel” model is developed in PREDICI, which contains three simulation systems that are solved simultaneously. This model achieves goals 1 to 3 and 5, but cannot predict the weight average molecular weight. For the first time, all six propagation rate constants are included in the parameter estimation, resulting in improved fit to the experimental data. Three of the parameter estimates are not significantly different from zero at the 95% confidence level. Additional data, with higher branching levels should be used, in future, to improve the precision of parameter estimates.