Front development in superimposed processes

The theoretical approach to modelling frontal polymerization is based on the well developed theory of the combustion of condensed materials.255 6 The main assumptions made in this approach are the following the temperature distribution is one-dimensional die development of the reaction front is described by the energy balance equation, including inherent heat sources, with appropriate boundary and initial conditions. Wave processes in stationary and cyclical phenomena which can be treated by this method, have been investigated in great detail. These include flame spreading, diffusion processes, and other physical systems with various inherent sources. 

In some cases, the heat source can include both the heat of polymerization and the heat output of crystallization of the newly formed products. This is the case in anionic activated e-caprolactam polymerization. This dual heat source must be included in the energy balance equation. As was discussed above, the temperature dependence of the crystallization rate is somewhat complicated. Nevertheless, the propagation of the heat wave is analogous to other well-known cases of wave propagation from consecutive reactions. 

Theoretical modelling and analysis of the results for the superimposed processes of polymerization and crystallization was carried out for wave propagation in anionic activated reaction of e-caprolactam polymerization.258 In the steady situation, the process is described by the system of differential equations  

Solution of the system of Eqs (4.25) - (4.29) gives the velocity of wave propagation. Numerical values of the constants in the kinetic equations for polymerization and crystallization were found from a standard calorimetric experiment. Then we can find the solution and compare it with experimental data,as is done in Fig. 4.41. The results are quite satisfactory. This means that the theoretical model of front polymerization (with crystallization of the newly synthesized product) correctly represents main features of the process and can be used for plant design. 

It can be shown that three different modes of front propagation during the formation of polycaproamide can be observed, depending on the relationship between the process parameters. In the first mode, which was found experimentally, the zones of polymerization and crystallization coincide. In the second mode these zones are separated in space. The third mode, which was predicted theoretically, is characterized by a non-monotonic distribution of the degree of crystallinity. However, it is not clear whether this situation can actually be observed in anionic e-caprolactam polymerization because even slight variations in parameters transfers the system into another regime. 

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