Instantaneous, population balance method

Analogous equations can be used with any other instantaneous distribution. This relatively easy integration extends the use of instantaneous distributions to transient reactor operation and considerably broadens the use of this powerful technique. When compared with the method of moments, instantaneous distribution allows for the complete prediction of CLD and CCD instead of only averages when contrasted to the full solution of the population balances, the method of instantaneous distributions provides the same information at a much shorter time using a more elegant solution, allows the modeler to analyze the problem with a simple glance at the equation, and can even be implemented on simple commercial spreadsheets for easy calculation. 

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... 

Section 2.3 describes phenomenological models for polymerization kinetics with coordination catalysts. Molar and population balances will be derived using what we like to call the standard model for olefin polymerization kinetics with coordination catalysts. Also how molecular weight averages can be modeled in batch, semibatch and continuous reactors are shown using the method of moments and the method of instantaneous distributions. Unfortunately, the kinetics of olefin polymerization is complicated by several factors that are not included in the standard model some of these effects and possible solutions and model extensions are mentioned briefly at the end of Section 2.2. 

In the method of moments, only molecular weight averages are calculated and not the complete MWD. For some applications, this simplification is acceptable, but for others it may not be. In this case, the options are solving the discretized population balances or, more elegantly, using the method of instantaneous distributions. Later it will be shown that... 

The mechanism on long-chain branch (LCB) formation with coordination catalysts was discussed briefly in Section 2.2 and illustrated in Figure 2.18. LCB formation with coordination catalysts is nothing more than a copolymerization reaction with macromonomers made in the reactor through -hydride elimination and transfer to monomer reactions for polyethylene, and -methyl elimination for polypropylene (Scheme 2.2). At this point, the population balances could be re-derived to include LCB-formation reactions and solved by the method of moments. For brevity, however, only the final results of this derivation will be shown, leading to an analytical solution for the instantaneous distribution of MWD for chains containing LCB derivation details are available in the literature [45-47]. 

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