Mathematical Methods

Abstract

IntroductionIn this chapter some mathematical methods to solve kinetic modeling problems are explained. A very sound basis for this was already laid many years ago by Flory [1]. Here, we want to present modern mathematical tools that have recently been developed through the use of computers. The focus is on the link between kinetic rate data and reactor type on one hand, and distributive properties -in one or more dimensions -on the other. These distributive or microstructural properties are concerned not only with countable quantities, such as the number of monomer units in a polymer molecule, but also with structure in the case of branched polymer molecules. With structure, we discuss the connectivity of branch points and the lengths of the segments between them. In the greater part of the text, reactors are treated in a simplified manner. We consider continuous and batch reactors, but all of them ideally mixed. The effect of incomplete mixing (segregation, macroand micromixing) is addressed in classical textbooks such as Biesenberger [2] and Dotson et al. [3]. Some recent attempts to include the impact of micromixing on distributions are available in the literature [57][58][59], but this field is still in its infancy. Nevertheless, to still provide a sound basis for issues of mixing, we devote one section to the use of computational fluid dynamics in polymer reaction engineering problems.The microstructural properties that we address are chain length, number of monomer units of one kind (copolymer), number of branch points, number of unsaturated bonds, and number of reactive monomer units (end groups in polycondensation). Problems may require solution of one or more of these properties simultaneously. Here, we will denote this as the dimensionality of the problem at hand. For instance, growth in addition polymerization can be described by a simple 1D (chain length) reaction equation and population balance.In contrast, growth in polycondensation of a trifunctional monomer A with a bifunctional monomer B requires a 3D description. The 3D distribution R n; i; k , where subscripts denote chain length, number of A end groups, number of B end groups, respectively, obeys:Note that this describes a reaction between single end groups of two different molecules; end group combinations within one (longer) molecule require a similar approach. These examples illustrate the importance of the dimensionality of the problem at hand. In general, low dimensionality can be dealt with using analytical or differential methods, while higher dimensionality soon requires a Monte Carlo sampling approach. Note that all of these methods, except MC, start with a population balance. Here we will mainly discuss ways to solve such balances of lower or higher dimensionality. This chapter will start with the description of lower-dimensional problems and show to what extent these can be successful. This often involves the reduction of the problem to lower dimensionality, inevitably leading to averaging over one or more dimensions. The most well...

The method of moments used in polymerization reaction engineering for 70 years: An overview, tutorial, and minilibrary

“…Hence, the distribution used there may not be the most appropriate one for the kinetic growth of the polymer chains. Iedema and Kolhapure [ 68 ] compared the solution of PBE by the Galerkin h – p finite element method (FEM) and the solution of the moment equations using the Hulburt and Katz closure for a polymerization with chain transfer to polymer (LCB) in a continuous stirred‐tank reactor (CSTR), the results are shown in Figure 4A. [ 69 ] Increasing the value of the transfer to polymer rate constant ( k tp ) the discrepancy between the solutions becomes more important.…”

“…Hence, the distribution used there may not be the most appropriate one for the kinetic growth of the polymer chains. Iedema and Kolhapure [68] compared the solution of PBE by the Galerkin hp finite element method (FEM) and the F I G U R E 4 (A) Comparison between the method of moments (MoM) solution (with closure method) and the solution of the population balance equations (PBE) (without closure method). Reproduced with permission from Iedema and Kolhapure [69] Copyright © 2008 John Wiley and Sons.…”

Automated reaction generation for polymer networks

“…Efficient tools have been developed describing macromolecular properties in terms of numbers of monomer units, branches, functional groups, etc. ( Dhooge et al, 2016;Iedema and Kolhapure, 2008;Mastan et al, 2015;Mueller et al, 2011;Tobita and Hamielec, 20 0 0 ). A class of polymerization systems exists, however, with more than usually complex chemistry, for which we nevertheless desire to predict macromolecular properties.…”

Modelling and Optimization of Nonlinear Polymerization Processes

“…To evaluate Mw, only one further differential equation for the second moment is needed apart from the mass balances of all species. Unfortunately, in the case of nonlinear processes, the 3 rd moment for example is required for the calculation of the 2 nd , the 4 th for the 3 rd and so on, the so called the closure problem . To circumvent this problem, assumptions on the form of the molecular weight distribution were used (the most common being that by Ref …”