Modeling at the particle scale

In this study, we have shown how gas-liquid flow through a random packing may be represented by a percolation process. The main concepts of percolation theory allow us to account for the random nature of the packing and to derive a theoretical expression of the liquid flow distribution at the bed scale. This flow distribution allows us to establish an averaging formula between the particle and bed scales. Using this formula, we propose the bed scale modelling of some transport processes previously modelled at the particle scale. 

An "entropy analysis of the liquid flow through the percolation structures allow us to derive a theoretical expression of flow distribution. This expression may be used as the basis of averaging formula of various hydrodynamic mecanisms. The resulting models involve both parameters characterizing the mechanism modeled at the particle scale and a parameter defining the effective solid wettability, i.e. the minimum liquid velocity u. The various models analysed in this paper and compared with experiments yield logical variations of the parameter u with the operating conditions (solid wettability, liquid viscosity). 

The bed scale corresponds to the whole bed or to a volume containing a large number of particles. That is the level at which we want to derive models for the investigated transport processes. However these processes are generally ruled by gas-liquid-solid interactions occurring at the particle scale. That is the reason why it is necessary to model these processes at the particle scale. The change of scale or volume averaging between both levels is ruled by the percolation process, i.e., by the velocity distribu-... 

Let us then consider a suspension of identical, neutrally buoyant solid spheres of radius a. We are interested in circumstances in which the length scale of the suspension at the particle scale (that is, the particle radius) is very small compared with the characteristic dimension L of the flow domain so that the suspension can be modeled as a continuum with properties that differ from the suspending fluid because of the presence of the particles. Our goal is to obtain an a priori prediction of the macroscopic rheological properties when the suspension is extremely dilute, a problem first considered by Einstein (1905) as part of... 

In such a case, the various heat and mass balances could be described at the bed scale by diffusion-like equations. It is indeed the approach in many models (e.g. axial and radial dispersions). However, if cne locks the bed closer,e.g. at the particle scale as in the close-up of figure U, the process representation is completely different. The liquid flow is distributed in different channels according to the local geometrical features of the packing. For example, the number of channels entering and leaving the cell represented by the close-up in figure 4, depends on these features and varies randomly form point to point. 

In a kinetic investigation, the rate-determining step and, hence, the functional form of the rate model are not known a priori also unknown are the rate constants and adsorption equilibrium coefficients. Hence, the aim of data procurement and correlation is both model discrimination and parameter estimation which are completed in tandem [17]. The critical problem at this point is to obtain reliable experimental data from which kinetic models that reflect steady-state chemical activity can be extracted and evaluated. In order to measure correctly the rates of chemical events only, (i) external and internal mass and heat transport resistances at the particle scale have to be eliminated,... 

For the heterogeneous catalytic process to be effective, the reactants present in the surrounding fluid phase must be transported to the surfece of the solid catalyst, and after the reaction, the products formed must be carried back from the surface to the bulk fluid. The path of the physical rate processes at the particle scale is divided into two parts, as depicted in the 7-step sequence of the continuous reaction model used in microkinetic analysis ... 

In contrast with the previous two sections, where the models focused on the segregation dynamics at the particle-scale level, in this section, we focus on a model developed for the evolution of macro-scale segregation pattern. 

Chemical dynamics and modeling were identified as important research frontiers in Chapter 4. They are critically important to the materials discussed in this chapter as well. At the molecular scale, important areas of investigation include studies of statistical mechaiucs, molecular and particle dynamics, dependence of molecular motion on intermolecular and interfacial forces, and kinetics of chemical processes and phase changes. 

Once the analytical method is validated for accuracy at the laboratory scale, it can be used to obtain extensive information on process performance (blend homogeneity, granulation particle size distribution, and moisture content) under various conditions (blender speed, mixing time, drying air temperature, humidity, volume, etc.). Statistical models can then be used to relate the observable variables to other performance attributes (e.g., tablet hardness, content uniformity, and dissolution) in order to determine ranges of measured values that are predictive of acceptable performance. 

In the preceeding section, we have shown how fluid flows through a packed bed may be observed at various levels, leading to completely different interpretations. Actually, when modelling a transport process, it is always necessary to consider at least two observation levels the bed scale and the particle scale. 

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