Frontal processes modelling

The dynamic olefin insertion process has been modeled using various quantum mechanical methods. A concerted four-center mechanism involving a frontal copla-nar attack of the C=C unit on the Zr-H bond of 1 is associated with a low activation energy of 0-15 kcal mol and has been proposed for the reaction of ethylene (Scheme 8-2) [37]. 

Frontal polymerization carried out as described above can be turned into a continuous process. In order to do this, it is necessary to move the newly formed polymer and the reactive mixture in the direction opposite to the direction of spreading of a thermal front at a velocity equal to the velocity of the front development to feed the reactor with a fresh reactive mass.254 Control of the process, choice of process parameters and proper design of the equipment require solving the system of equations modelling the main physical and chemical processes characteristic of frontal reactions. 

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. 

Several theoretical models were constructed to describe the chromatographic process in the frontal 116.191 and the zonal elution mode 20. The conventional method of obtaining the kinetic parameters consists in fitting the model to the experimental breakthrough curves. Another method based on the split-peak effect is a direct measurement of the apparent association rate constant (7,211. Because of the slow adsorption process, a fraction of the solute injected as a pulse into the immunochromatographic column is eluted as a nonretained peak. This behavior is observed at high flow rates, with very short or low-capacity columns 121—251. 

In the fourth step of the building of a mathematical model of a process the assemblage of the parts (if any) is carried out in order to obtain the complete mathematical model of the process. Now the model dimension can be appreciated and a frontal analysis can be made in order to know whether analytical solutions are possible. 

Frontal chromatography can be used in combination with chromatographic models to study mass-transfer and dispersion processes (e.g., the equilibrium dispersive or the transport model of chromatography [7]). 

The accurate determination of the adsorption isotherm parameters of the two enantiomers on a CSP is of fundamental importance to do computer-assisted optimization to scale up the process. Such determinations are usually done with an analytical column and the most traditional method to determine the parameters and saturation capacity is by frontal analysis (see section 3.4.2). The aim of paper III was to investigate the adsorption behavior and the chiral capacity of the newly developed Kromasil CHI-TBB column using a typical model compound. Many of the previous studies from the group have been made on low-capacity protein columns which has revealed interesting information about the separation mechanism [103, 110, 111], For this reason a column really aimed for preparative chiral separations was chosen for investigation [134], As solute the enantiomers of 2-phenylbutyric acid was chosen. 

Finally we pay attention to the ideal frontal TC (cf. Fig. 4.1). The high temperature front of the zone profile is obviously proportional to the adsorption isobar and so, at least for the localized adsorption model, to the adsorption constant. As such, it would obey Eq. 5.14. It holds for the activities which do not appreciably decay in the course of run. As for the shorter-lived nuclides, both the elution and the formally frontal TC result in non-ideal frontal chromatograms. Their shapes are close to what would arise from ideal processing during t . but they are smeared due to the random lifetimes of nuclei. Still the initial part of the thermochromatogram might be useful for evaluation of the required quantity, provided that the statistics of detected decay events is good. 

In the case of a linear isotherm, the solution of the linear ideal model is trivial for the boundary conditions of elution or frontal analysis. This solution is the boundary condition transported along the column at a velocity that is constant. In the case of SMB, the cychc nature of the process makes the solution more complex to derive but also most useful as it predicts with a reasonable precision the concentration profiles and concentration histories. 

Mathematical models of the frontal copolymerization process were developed, studied and compared with experimental data in [67, 90]. An interesting observation was that the propagation speed of the copolymerization wave was not necessarily related to the propagation speeds in the two homopolymerization processes, in which the same two monomers were polymerized separately. For example, the propagation speeds in the homopolymerization processes could be 1 cm/min in each, but in the copolymerization process, the speed could be 0.5 cm/min. Mathematical models of free-radical binary frontal polymerization were presented and studied in [66, 91]. Another model in which two different monomers were present in the system (thiol-ene polymerization) was discussed in [21]. A mathematical model that describes both free-radical binary frontal polymerization and frontal copolymerization was presented in [65]. The paper was devoted to the linear stability analysis of polymerization waves in two monomer systems. It turned out that the dispersion relation for two monomer systems was the same as the dispersion relation for homopolymerization. In fact, this dispersion relation held true for W-monomer systems provided that there is only one reaction front, and the final concentrations of the monomers could be written as a function of the reaction front temperature. 

Frontal polymerization, the process of propagation of a polymerization wave, is important from both fundamental and applied viewpoints. In this chapter, we reviewed theoretical results on the base model of free-radical frontal polymerization. Based on the analogy of the gasless combustion model and using the methods developed in combustion theory, we determined uniformly propagating polymerization waves and discussed their linear and nonlinear stability. 

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