Intermediate Stage Model

It is rather unfortunate that of ail the sintering stages the most important is also the most difficult to model. For example, any intermediate-stage model that does not take into account the details of particle packing has very limited validity. A cursory examination of the results shown in Fig. 10.10 should make this point amply clear. 

Similar assumptions are made for the final sintering stage, so that nondensifying mechanisms can be neglected, which is similar to that in the intermediate stage model [27, 38]. The procedure to derive the sintering equations for the intermediate... 

This concept can also be applied to the intermediate stage of sintering. Coble s intermediate stage model also took the surface area of the pore into account. The kinetic equations based on the two models are, in fact, the same except for the numerical constants. 

The material transport by diffusion through a thin liquid film at the contact area is similar to that in Coble s intermediate stage model.Therefore,... 

In a sense the formation of t) -H2 complexes can be thought of as an intermediate stage in the oxidative addition of H2 to form two M-H bonds and, as such, the complexes might serve as a model for this process and for catalytic hydrogenation reactions by metal hydrides. Indeed, intermediate cases between and... 

When a number of competing reactions are involved in a process, and/or when the desired product is obtained at an intermediate stage of a reaction, it is important to keep the residence-time distribution in a reactor as narrow as possible. Usually, a broadening of the residence-time distribution results in a decrease in selectivity for the desired product. Hence, in addition to the pressure drop, the width of the residence-time distribution is an important figure characterizing the performance of a reactor. In order to estimate the axial dispersion in the fixed-bed reactor, the model of Doraiswamy and Sharma was used [117]. This model proposes a relationship between the dispersive Peclet number ... 

The model of clusters or ensembles of sites and bonds (secondary supramolecular structure), whose size and structure are determined on the scale of a process under consideration. At this level, the local values of coordination numbers of the lattices of pores and particles, that is, number of bonds per one site, morphology of clusters, etc. are important. Examples of the problems at this level are capillary condensation or, in a general case, distribution of the condensed phase, entered into the porous space with limited filling of the pore volume, intermediate stages of sintering, drying, etc. 

Figure 1. Three isotope plot of O isotopes in Allende inclusions. Deviations are plotted in 6 units which are %o deviations relative to the terrestrial SMOW standard. In a two stage model, normal inclusions had initially a composition close to point A and exchanged with a reservoir poorer in O in the region of point D (Clayton et al. 1973). FUN inclusions underwent an intermediate step along a fractionation line between point A and point C. Then each inclusion exchanged with the same O poor reservoir D (Clayton and Mayeda 1977).

This is namely the main reason why fnmolecular models (e.g., Brusselator [16]) being often physically non-transparent but having only two intermediate stages have attracted such great attention in synergetic studies. 

Section we show that presence of two such intermediate stages is more than enough for the self-organization manifestation. Lotka [22] was the first to demonstrate theoretically that the concentration oscillations could be in principle described in terms of a simplest kinetic scheme based on the law of mass action [4], Its scheme given by (2.1.21) is similar to that of the Lotka-Volterra model, equation (2.1.27). The only difference is the mechanism of creation of particles A unlike the reproduction by division, E + A - 2A, due to the autocatalysis, a simpler reproduction law E —> A with a constant birth rate of A s holds here. Note that analogous mechanism was studied by us above for the A + B — B and A + B — 0 reactions (Chapter 7). 

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