Micro-scale modelling description

In general, the differential description is useful for processes where there is a wide separation of scales between the smallest macroscopic scales of interest and the microscopic scales associated with the internal structure of the fluid. If the micro-scales were always of molecular magnitude then questions of scale separation would seldom arise. But, in many of the models employed for engineering purposes, the characteristic scales of the internal structure being described are themselves macroscopic in nature. In such situations the desired separation between the calculated and modeled scales is much less clear cut, and one must be careful not to attribute quantitative significance to any predicted solution features with scales comparable to the internal micro-scale. When a continuum description is pushed to far, i.e. applied on scales too small, one can only hope that such inaccuracies are not catastrophic in nature. 

In the macroscopic description of porous media, structural features are hidden in effective parameters, and phenomena that are characteristic of the micro-scale are obscured by rather crude, often semi-empirical, models. Of course, this general deficiency is also true for the macroscopic description of gel drying. However, the transition to the second drying period, and in particular the development of cracks at this moment, is a particular problem, where the necessity to account for pore-scale phenomena becomes even more evident. The purely macroscopic description needs criteria that not only depend on the microstructure itself, but also on how it behaves during drying. 

For PRE this implies the combination of several disciplines such as polymer chemistry, thermodynamics, characterization, modeling, safety, mechanics, physics, and process technology. PRE problems are often of a multi-scale and multifunctional nature to achieve a multi-objective goal. One particular feature of PRE is that the scope ranges from the micro scale on a molecular level up to the macro scale of complete industrial systems. PRE plays a crucial role in the transfer of information across the boundaries of different scale regions and to provide a comprehensive and coherent basis for the description of these processes [19]. 

Multiscale descriptions of particle-droplet interactions in spray processing of composite particles are realized based on Multiphase Computational Fluid Dynamics (M-CFD) models, in which processes such as liquid atomization and particle-droplet mixing spray (macro-scale), particle-droplet collision (mesoscale), and particle penetration into droplet (micro-scale) are taken into account as shown in Fig. 18.52. Thereby, the incorporation efficiency and sticking efficiency of solid particles in matrix particles are correlated with the operatiOTi conditions and material properties. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

Computational fluid dynamics enables us to investigate the time-dependent behavior of what happens inside a reactor with spatial resolution from the micro to the reactor scale. That is to say, CFD in itself allows a multi-scale description of chemical reactors. To this end, for single-phase flow, the space resolution of the CFD model should go down to the scales of the smallest dissipative eddies (Kolmogorov scales) (Pope, 2000), which is inversely proportional to Re-3/4 and of the orders of magnitude of microns to millimeters for typical reactors. On such scales, the Navier-Stokes (NS) equations can be expected to apply directly to predict the hydrodynamics of well-defined system, resolving all the meso-scale structures. That is the merit of the so-called DNS. 

In fact, extremum tendencies expressing the dominant mechanisms in systems like turbulent pipe flow (Li et al, 1999), gas-liquid-solid flow (Liu et al, 2001), granular flow, emulsions, foam drainages, and multiphase micro-/nanoflows also follow similar scenarios of compromising as in gas-solid and gas-liquid systems (Ge et al., 2007), and therefore, stability conditions established on this basis also lead to reasonable descriptions of the meso-scale structures in these systems. We believe that such an EMMS-based methodology accords with the structure of the problems being solved, and hence realize the similarity of the structures between the physical model and the problems. That is the fundamental reason why the EMMS-based multi-scale CFD improves the... 

The study of these systems have become possible thanks to the development of various preparation routes, from sophisticated routes for the preparation of model materials with controlled nanostructures to industrial routes for the production of large quantity of materials. It has benefited as well from the development of new experimental techniques, allowing the properties of matter to be quantitatively examined at the nanometre scale. These include Hall micro-probe [3] or micro-SQUID magnetometry [4], XMCD at synchrotron radiation facilities [5] and scanning probe microscopes [6]. This is not the topic of this chapter to describe in detail these various techniques. They are only quoted in the following sections. The reader may find in the associated references the detailed technical descriptions that he may need. 

In contrast to the nano-scale, where the periodic arrangement of atoms on crystal lattices is well established, and the macro-scale, where a continuous distribution of matter is assumed, adequate quantitative descriptions are notably lacking for structure at the micro- and mesoscales, where properties are described in terms of the behavior of dislocations, material in grains, particles of different phases and the boundaries among them. The traditional means of describing these microstructural attributes with descriptive terms that call to mind familiar shapes fails to provide an adequate quantitative basis for transferring this information to quantitative models. 

In MD the considered microscopic material properties and the underlying constitutive physical equations of state provide a sufficiently detailed and consistent description of the micro mechanical and thermal state of the modeled material to allow for the investigation of the local tool tip/workpiece contact dynamics at the atomic level (Hoover 1991). The description of microscopic material properties considers, e.g., microstmcture, lattice constants and orientation, chemical elements, and the atomic interactions. The following table lists the representation of material properties and physical principles in MD, which have to be described numerically in an efficient way to allow for large-scale systems, i.e., models with hundred thousands, millions, or even billions of particles (Table 1). 

The description is based on the previously defined single-particle (Lagrangian) or one-point (Eulerian) joint velocity-composition (micro-)PDF, /(r,yr). As mentioned in Section 12.4.1, in the one-point description no information on the local velocity and scalar (species concentrations, temperature,. ..) gradients and on the frequency or length scale of the fluctuations is included and the related terms require closure models. The scalar dissipation rate model has to relate the micro-mixing time to the turbulence field (see (12.2-3)), either directly or via a transport equation for the turbulence dissipation rate e. A major advantage is that the reaction rate is a point value and its behavior and mean are described exactly by a one-point PDF, even for arbitrarily complex and nonlinear reaction kinetics. 

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