Polyolefins are commercially produced in continuous reactors that have a broad residence time distribution (RTD). Most of these polymers are made with heterogeneous catalysts that also have a particle size distribution (PSD). These are totally segregated systems, in which the catalyst/polymer particle can be seen as a microreactor operated in semibatch mode, where the reagents (olefins, hydrogen, etc.) are fed continuously to the catalyst/polymer particle, but no polymer particle can leave. The reactor RTD has a large influence on the PSD of the polymer particles leaving the reactor, as well as in polymer microstructure and properties, polymerization yield, and composition of reactor blends. This article proposes a Monte Carlo model that can describe how particle RTD in a single or a series of reactors can affect the PSD of polymer particles made under a variety of operation conditions. It is believed that this is the most flexible model ever proposed to model this phenomenon, and can be easily modified to track all properties of interest during polyolefin production in continuous reactors with heterogeneous catalysts.

Polymer reactor blends, such as bimodal polyethylene or high‐impact polypropylene, are usually produced in multistep processes using two or more reactors in series. Since the polymer particles are subject to reactor residence time distributions (RTD) during the polymerizations, the fractions of the polymer populations made in each reactor will vary from particle to particle. It is shown in the previous publications in this series that reactor RTD has a marked effect on the particle size distribution and on the packing density of polyolefin particles. In this article, the versatile Monte Carlo model is extended to demonstrate how reactor RTD affects particle composition and molecular weight distributions of polyolefin reactor blends made in multistep processes. Increasing the number of reactors in series favors the homogeneity of the product. Moreover, the average fraction of the different polymer populations in the particles depends strongly on the mean reactor residence time and polymerization kinetics.

The sphere packing problem consists on arranging spheres, of different or same radii, on a 3D container without overlapping them. The ratio of the volume occupied by the spheres to the total volume of the container is called packing density. The goal of the sphere packing problem is to find the arrangement that maximizes the packing density. When the spheres are packed randomly, the arrangement is referred to as a random loose packing. If the container is shaken afterward, the arrangement is called random close packing.Several approaches exist to model random packing. Two of them are the collective rearrangement [2][3][4][5] and the sequential deposition [6][7][8][9][10][11][12] methods. In the collective rearrangement method, the spheres are randomly placed in a container and allowed to overlap. In a subsequent step, the spheres are moved and/or reduced in size until the overlap drops to an acceptable threshold. In the sequential deposition method, on the other hand, the spheres are dropped one by one in a container following the gravitational law until they find a stable position. This approach is also known as drop and roll algorithm.In this work, we have implemented a drop and roll algorithm as described by Zhou et al. [7] The algorithm consists in the following steps:• An incoming sphere, is randomly positioned at the top of the container. • The incoming sphere is dropped in small increments down the container. • The dropping stops once the incoming sphere touches the floor of the container or hits another sphere. • If the incoming sphere touches the floor of the container, its position is considered stable, and the algorithm starts positioning a new incoming sphere. • If the incoming sphere touches another sphere, it rolls on the stationary sphere until it hits the floor (and becomes stable), or hits a second sphere. • If the incoming sphere hits a second stationary sphere, it rolls again until it either becomes stable by hitting the floor or a third sphere. • When the incoming sphere hits a third sphere, the stability criteria is evaluated to determine if the incoming sphere becomes stable. If not, the incoming sphere will continue rolling but now on the pair of spheres, or on the sphere with the steepest contact angle. Monte Carlo ModelThe residence time distribution of reactors used to produce polyolefins commercially has a significant impact on many of their properties. This effect often goes unnoticed, since the concept of reactor residence time is often poorly understood even among chemical engineers. In the first part of this series of articles, it is shown how the reactor residence time distribution has a marked impact on the particle size distribution of polyolefin particles using a new Monte Carlo model. In this article, the Monte Carlo model is combined with a drop and roll algorithm to predict how reactor residence time distribution affects the packing density of the polyolefin particles.

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