The method of moments is a simple, efficient method simulating polymerization processes. Its use is said to be limited in nonlinear free radical polymerizations with branching or crosslinking due to the assumptions needed. Here, moment equations are derived without assuming steady state, one radical per molecule, or a statistical distribution of connections. Equations are valid up to the gel point. The bulk solution is formally identical to the pseudo kinetic approach by Tobita and Hamielec if moments of dead polymer are replaced by the sum of dead and life polymers. The method relies on analytical solutions of the moments of the molecular weight distributions (MWD) of instantaneous primary chains. In emulsion polymerization compartmentalization of radicals complicates the calculation. An alternative approximation of these MWDs is presented. The present extension allows nonlinear free radical polymerization to be readily included in the computer based design and optimization of polymerization processes and to check more detailed calculations of the MWD.
A kinetic model for the dynamic simulation of Di Metal Cyanide (DMC) catalyzed alkoxylation is proposed. The model covers the characteristic “catch‐up” kinetics, favoring the growth of small polymers, and high molecular weight tailing (HMWT), which broadens the distribution toward large polymers. Computations are based on the direct numerical integration of fundamental reaction kinetics of a living polymerization with chain length dependent reaction coefficients. It is observed that “catch‐up” kinetics can be modeled by molecular weight (MW) dependent propagation and activation constants. The formation of HMWT can be described by MW dependent segmental growth. Thereby, the segmental growth is influenced by chain length dependent deactivation constants and effects of polymer diffusion. Simulations of (semi‐)batch operations return an appropriate reproduction of measured polymer distributions. Processes with continuous addition of starters (CAOS) and single stage continuous (CSTR) processes are compared to (semi‐)batch operations. Kinetics without “catch‐up” mechanism have a strongly broadened MWD in CAOS/CSTR processes, compared to (semi‐)batch operations. Also, the concentration of short chained polymers, which act as catalyst poison, is increased. In contrast, CAOS/CSTR simulations with “catch‐up” mechanism result in a MWD comparable to results from (semi‐)batch operations and there is no accumulation of short chained polymers.
Summary
Reactions such as branching and crosslinking leading to nonlinear polymers play an important role in many polymerization productions, but they increase the sensitivity of the processes and increase the requirements for their design and control. It is therefore desirable to have an accurate and simple means to model these processes. However, the simplest method for the description of the polymers: the method of moments was said to fail in the nonlinear case because of the so called closure problem. It will be shown here how to extend the method of moments to nonlinear processes in the pregel region − except scission − and thus provide a simple means for the simulation and optimization of the main parameters of a polymer distribution. Furthermore, the assumption of one radical per polymer molecule is no longer needed. The formulas allow average polymer properties to be easily included in flow sheet simulations or computational fluid dynamics codes and allow checking other more elaborate codes for the description of polymer distributions. Examples will be given for free radical polymerizations. An approximation in case of scission will be discussed together with an application to reversible polycondensation.
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