Diffusion mesoscale

It is meaningful to examine the relation between microscale model, mesoscale model, and micromodel. For reaction kinetics, microscale and mesoscale models adopt the same kinetics that based on element reaction system. For diffusion, mesoscale model embodies two diffusion mechanisms (one for micropores and another for mesopores and macropores), and microscale model considers one diffusion mechanism since it only has micropores. No diffusion was considered within the macropores. It is obvious that the mesoscale model possesses the same theoretical foundation as the microscale model, but its application scope has been enlarged compared to the microscale model. Therefore, it could be reliably used as a tool to derive some parameters, such as effective chemical kinetics and effective diffusion parameters, for macroscale model. In the section following, we discuss the method on how to link the microscale kinetics to the lumped macroscale kinetics via the mesoscale modeling approach. 

Sewell and co workers [145-148] have performed molecular dynamics simulations using the HMX model developed by Smith and Bharadwaj [142] to predict thermophysical and mechanical properties of HMX for use in mesoscale simulations of HMX-containing plastic-bonded explosives. Since much of the information needed for the mesoscale models cannot readily be obtained through experimental measurement, Menikoff and Sewell [145] demonstrate how information on HMX generated through molecular dynamics simulation supplement the available experimental information to provide the necessary data for the mesoscale models. The information generated from molecular dynamics simulations of HMX using the Smith and Bharadwaj model [142] includes shear viscosity, self-diffusion [146] and thermal conductivity [147] of liquid HMX. Sewell et al. have also assessed the validity of the HMX flexible model proposed by Smith and Bharadwaj in molecular dynamics studies of HMX crystalline polymorphs. 

This function accounts for the mesoscale region and comprises most of the listed distribution functions [154]. It includes three empirical parameters, a, pK, and aw. Having ascertained the relationships between these parameters and the properties of anomalous self-diffusion, fractal morphology, and polydispersity of the finite pore-size, physical significance can be assigned to these parameters in the framework of the percolation models [152],... 

FIGURE 1 Principal mechanisms of blending (A) Diffusion, (B) convection, and (C) shear, (A) Schematic of diffusion At the particle scale ieft) At a mesoscale for initial. state (cenier) and after diffusion initiates (right), (B) Schematic of convection At the particle scale (iefiy, at a mesoscale for initial state (center) and after convection initiates (right). (C) Schematic of shear At the particle scale (ieft) at a me.soscale for initial state (center), and after shear initiates (right). 

Mestayer, P.G., and Anquetin, S. (1995) Climatology of cities, In Diffusion and Transport of Pollutants in Atmospheric Mesoscale Flow Fields, Kluwer Publishers, 165-189. 

Figure 5 Multiscale approach to understand rate of CO2 diffusion into and CH4 diffusion out of a structure I hydrate, (left) Molecular simulation for individual hopping rates, (middle) Mesoscale kinetic Monte Carlo simulation of hopping on the hydrate lattice to determine dependence of diffusion constants on vacancy, CO2 and CH4 concentrations, (right) Macroscopic coupled non-linear diffusion equations to describe rate of CO2 infusion and methane displacement. Graph from Stockie. ...

In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

As mentioned above, the mesoscale model can contain stochastic processes, which lead to the diffusion terms appearing in Eq. (5.2). Using A p as an example, we can write... 

As a concrete example of the diffusion coefficients, let us consider a case in which the only mesoscale variables are U and U. Moreover, let us assume that the Lagrangian... 

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