Random-grain model

Frieden (1971,1972) attacked this problem by seeking the solution that has the greatest probability of occurrence. The lower-bound condition [d(x) positive] was a natural consequence of the physical random-grain model that he used. This work was unique in that separate solutions were found for o(x) and the noise. In its simplest form, the original method required the data to be all positive. Even though o(x) satisfies this requirement, i(x) often does not, because of noise excursions that, in spectra, may be found in the base-line region. A bias term was added to the data to ensure positivity. 

Fig. 5.13. Relaxation time r3 plotted vs. temperature for the coarse-grained model of PE with N = 20, using the random hopping algorithm (upper set of data) or the slithering snake algorithm (lower set of data), respectively. The time r3 is of the same order as the Rouse relaxation time of the chains, and is defined in terms of a crossing criterion for the mean-square displacements [41], g3(t = r3) = g2(t = r3) [See Eqs. (5.2) and 5.3)]. From [32]...

The simplest models are those in which the internal structure of a pellet is not considered, and its behavior as a whole is modeled. These are normally called the macroscopic or basic models. In other models, the behavior of the distinctive elements of a pellet, such as the grain, micrograin, or the pore, constitutes the central feature such models account for structural changes during reaction. The so-called random pore models are the most common. 

This is best understood intially by considering the process of diffusion. Ghromatographic peaks represent chemical species that have been concentrated in space and time and the process of diffusion will immediately disperse them in space as a function of time. The conceptual basis of diffusion lies in the concept of the random walk model, wherein particles/molecules in suspension or solution are being jostled continuously by collisions with other particles or molecules. This is also referred to as Brownian motion, and is readily apparent when observing small particles with a microscope, such as some pollen grains, that seem to be in constant and random motion as they gradually spread out from any center of concentration. 

In contrast, random displacements of individual sites of a chain (or a few neighboring sites), when feasible, can be a valuable tool (see Fig. 1). For this approach to be applicable, the chain backbone cannot have rigid constraints (e.g., rigid bonds and bond angles). It is particularly effective for coarse-grained models that allow wide fluctuations of individual sites around their bonded neighbors. Relevant examples are the bond-fluctuation model and certain bead-spring models such as that employed by Binder... 

Fig. 24 Schematic pictures of flow-induced crystallization from the polymer melt (a) for the random coil model (b) using the folded-chain fringed-micellar grain model...

As mentioned before, the carbonation of CaO is a typical non-catalytic, gas-solid reaction. As snch, it has been extensively modelled using either random pore or grain models. The random pore model was developed and first applied by Bhatia and Perlmntter [57] to model the sulphation of lime and subsequently extended by Sun et al. [65]. The pores were assumed as an assembly of randomly oriented cylinders of uniform diameter, which initially overlapped (Fig. 6.17). The initial increase in the reaction rate was attributed to the growth of the surface area of the CaO-CaCOs interface, which is, however, overshadowed in later stages by the intersection of the growing surfaces, leading subsequently to a decrease in the reaction rate. 

Hereby, the first term is the conservative force on particle i due to interaction with other particles and the additional terms represent interactions with the continuum medium. R is the Langevin random force and is the friction coefficient. In DPD simulation, the polymer chain is represented by a coarse-grain model in which a coarse-grain particle (bead) represents a group of atoms. The total acting force on a DPD particle i is defined as ... 

Different models for the description of the solid structure (grain model, random pore model,. ..)... 

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