Trajectory tracking kinetics

Step 1 Begin by constructing a trajectory for a PFR from the feed point, continuing to the compleit conversion of methane or chemical equilibrium. Here, the PFR trajectory is computed by solving the kinetic equations for the reactions of Eqs. (6.55)-(6.57) to give the mole numbers of CH4 and CO. This leads to trajectory (1) in Figure 6.12, which tracks the compositiois from the feed point. A, to chemical equilibrium at point B. 

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

Most CFD providers track particles in the reactive flow field by solving the pertinent equations for the trajectory of a sfafisfically significant sample of individual particles that represents a number of the real particles with the same properties. For example, following the Rosin-Rammler size distribution (Figure 6.6), coal particles are tracked using a statistical trajectory model followed by the modeling of the kinetics of devolatilization and subsequent volatile and char combustion as discussed previously in this chapter (Figure 6.9). Models similar to the law presented earlier are used for droplet combustion of atomized fuel oil. 

The optimization provides the amounts of monomers and CTAs in the reactor at any overall conversion. These profiles are independent of the kinetics of the process and can be regarded as master curves. Once the trajectories of the amounts of monomers and CTAs as a function of the conversion are calculated, the implementation of the closed-loop strategy (Figure 6.14) reduces to tracking these profiles. To do so, on-line measurements of the overall conversion and of the free amount of monomers and CTA are necessary. Reaction calorimetry plus state estimation is probably the easiest, cheapest, and most robust option from an industrial perspective. 

The TST is a well-established methodology for the calculation of the kinetics of infrequent events in numerous physical systems. According to the TST method the gas transport mechanism through a dense polymer system is described as a series of activated jumps. For each transition, a reaction trajectory leading from a local energy minimum to another through a saddle point in the configuration space is tracked, and the transition rate constant is evaluated. 

In the simulation not all encounters will result in reaction due to some constraint. The two most common reasons for the encountering pairs to not react are (i) the reactions are partially diffusion controlled and the boundary is not reactive, or (ii) the radical pair is in the wrong spin state to react. In either case a careful treatment for the reflection is required to correctly model the subsequent kinetics, something which is not easily attainable in the IRT framework as the diflusive trajectories are not tracked. This section now presents a new analytical method of finding the distance of the radical pair following an unsuccessful encounter. 

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