Polymerization processes mathematical modeling

Mathematical modeling is a powerful tool not only for the development of process understanding but also for that of the advanced reactor controls in polymerization processes. The modeling techniques for polymerization processes are reasonably well developed and several commercial simulation packages are available. The modeling of heterogeneous polymerizations such as precipitation polymerization and emulsion polymerization remains a challenge. In the past decade, excellent... 

Obviously, construction of a mathematical model of this process, with our present limited knowledge about some of the critical details of the process, requires good insight and many qualitative judgments to pose a solvable mathematical problem with some claim to realism. For example what dictates the point of phase separation does equilibrium or rate of diffusion govern the monomer partitioning between phase if it is the former what are the partition coefficients for each monomer which polymeric species go to each phase and so on. 

Although the papers represent the whole range of kinds of polymers and processes, there are common themes which reveal the dominant concerns of polymerization reactor engineers. Fully half the papers are concerned rather closely with devising and testing mathematical models which enable process variables to be predicted and controlled very precisely. Such models are increasingly demanded for optimization and com-... 

Since the depolymerization process is the opposite of the polymerization process, the kinetic treatment of the degradation process is, in general, the opposite of that for polymerization. Additional considerations result from the way in which radicals interact with a polymer chain. In addition to the previously described initiation, propagation, branching and termination steps, and their associated rate constants, the kinetic treatment requires that chain transfer processes be included. To do this, a term is added to the mathematical rate function. This term describes the probability of a transfer event as a function of how likely initiation is. Also, since a polymer s chain length will affect the kinetics of its degradation, a kinetic chain length is also included in the model. 

In building mathematical models of product formation in a mold it is possible to treat a polymeric material as motionless (or quasi-solid), because the viscosity grows very rapidly with the formation of a linear or network polymer thus, hydrodynamic phenomena can be neglected. In this situation, the polymerization process itself becomes the most important factor, and it is worth noting that the process occurs in nonisothermal conditions. 

Values of both parameters, p and Mn, are generalized characteristics of any polymerization process therefore, it is reasonable to construct a mathematical model of the kinetics for the parameter P and then calculate Mn fromp with Eq. (2.3). 

Polymerization of lactams in reactive processing proceeds with the involvement of a catalyst and direct or indirect activators. A mathematical model of the process must be a kinetic equation relating the rate of conversion of a monomer to a polymer to the reagent concentrations and temperature. The general form of the model is... 

An important step in the production process is the preparation of a standard specimen. This specimen is used to qualify principle production parameters such as the long-term stability of the reactive mixture, polymerization cycle, and the performance characteristics of the material obtained. Simultaneous determination of the reaction parameters allows us to use mathematical modelling to optimize the reactive processing regime. 

The modeling of a polymerization process is usually understood as formulation of a set of mathematical equations or computer code which are able to produce information on the composition of a reacting mixture. The input parameters are reaction paths and reactivities of functional groups (or sites) at monomeric substrates. The information to be modeled may be the averages of molecular weight, mean square radius of gyration, particle scattering factor, moduli of elasticity, etc. Certain features of polymerizations can also be predicted by the models. 

The analytical predictor, as well as the other dead-time compensation techniques, requires a mathematical model of the process for implementation. The block diagram of the analytical predictor control strategy, applied to the problem of conversion control in an emulsion polymerization, is illustrated in Figure 2(a). In this application, the current measured values of monomer conversion and initiator feed rate are input into the mathematical model which then calculates the value of conversion T units of time in the future assuming no changes in initiator flow or reactor conditions occur during this time. 

Single Phase Polymerization Mathematical Modelling Ziegler-Natta Polymerization Polymerization Processes (Monograph) Emulsion Polymerization Polymerization Reactions and Reactors Continuous Reactors (ed. volume)... 

It was estimated that, if all the Surfmers contributed to stabilization, the surface coverage would be close to 20% at the end of the process. When Surfmer burial is considered, the minimum surface coverage is in the region of 14.7-15.0 % [35]. The authors have also studied the influence of the addition procedure on the evolution of the Surfmer conversion and concluded that, despite the low reactivity due to the presence of the alkenyl double bond, the incorporation could be increased to 72% from the original 58% obtained with a constant feeding rate. A mathematical model able to describe Surfmer polymerization was used in the optimization process [36]. 

A mathematical model of emulsion polymerization as a whole would be too complicated. Therefore Smith and Ewart divided the process into three phases, and appropriately simplified the situation in each of these. The start of polymerization and the monomer-polymer transformation up to the disappearance of micelles were designated as phase I, the subsequent time interval until the complete consumption of monomer droplets as phase II, and the remaining part of polymerization as phase IIIt. 

Based on these kinetic and microscopic observations, olefin polymerization by supported catalysts can be described by a shell by shell fragmentation, which progresses concentrically from the outside to the centre of the support particles, each of which can thus be considered as a discrete microreactor. A comprehensive mathematical model for this complex polymerization process, which includes rate constants for all relevant activation, propagation, transfer and termination steps, serves as the basis for an adequate control of large-scale industrial polymerizations with Si02-supported metallocene catalysts [A. Alex-iadis, C. Andes, D. Ferrari, F. Korber, K. Hauschild, M. Bochmann, G. Fink, Macromol. Mater. Eng. 2004, 289, 457]. 

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