Fig. 9.7 Reaction probability calculated using the mean reaction time and compared with random flights simulations using an outer elastic boundary and inner absorptive boundary with an encounter radius of 3 A. a n = 0.0005 A ps and b n = 0.001 A ps . Here MC refers to random flights simulation |

The independent reaction time (1RT) model was introduced as a shortcut Monte Carlo simulation of pairwise reaction times without explicit reference to diffusive trajectories (Clifford et al, 1982b). At first, the initial positions of the reactive species (any number and kind) are simulated by convolving from a given (usually gaussian) distribution using random numbers. These are examined for immediate reaction—that is, whether any interparticle separation is within the respective reaction radius. If so, such particles are removed and the reactions are recorded as static reactions. [Pg.222]

Fig. 167. Output of the simulation program SIMxNNy44/45. The program produces the distribution, the first peak (Fp), the median reaction time (Med), the internal meancycEN, and the externally observable (med-lin)/ET.Only the peaks of the xl ly distribution are simulated, the intermediate values are produced by a small random factor which has to be added ( z). A comma means 50ms on the x-axis. The times of Con, Lin etc. are given in millisecond, too |

With models for catalyst decay and effectiveness now in hand, the simulation of lignin liquefaction could be achieved given the initial lignin structure (as described earlier) and model compound reaction pathways and kinetics, both thermal and catalytic. Construction of a random polymer, as outlined earlier, began the simulation. This structural information combined with the simulated process conditions to allow calculation of the reaction rate constants, selectivities and associated transition probabilities. The largest rate constant then specified the upper limit of the reaction time step size. [Pg.249]

Analytical treatment of the diffusion-reaction problem in a many-body system composed of Coulombically interacting particles poses a very complex problem. Except for some approximate treatments, most theoretical treatments of the multipair effects have been performed by computer simulations. In the most direct approach, random trajectories and reactions of several ion pairs were followed by a Monte Carlo simulation [18]. In another approach [19], the approximate Independent Reaction Times (IRT) technique was used, in which an actual reaction time in a cluster of ions was assumed to be the smallest one selected from the set of reaction times associated with each independent ion pair. [Pg.268]

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. [Pg.353]

The simulation of lignin liquefaction combined a stochastic interpretation of depolymerization kinetics with models for catalyst deactivation and polymer diffusion. The stochastic model was based on discrete mathematics, which allowed the transformations of a system between its discrete states to be chronicled by comparing random numbers to transition probabilities. The transition probability was dependent on both the time interval of reaction and a global reaction rate constant. McDermott s ( analysis of the random reaction trajectory of the linear polymer shown in Figure 6 permits illustration. [Pg.247]

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