Pseudo-continuum model

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

Two-phase continuum models, in which the solid particles and their associated stagnant fillets of fluid are regarded as a continuous pseudo-solid phase, are to be preferred to the more traditional cell model of a particle bathed in fluid, which does not allow conduction from particle to particle. In previous studies, such models have been simplified by considering a one-dimensional model only (25). This study considers the full equations, which are, in dimensionless form ... 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

The dynamics of the pseudo-gel state are described by a continuum model in which cooperative diffusion of the network chains occur (117). A simple single-exponential relaxation law is obtained with characteristic time given by ... 

According to the classification given by Froment and Bishoff," the continuum models can be divided in two categories pseudo-homogeneous and heterogeneous models. 

Whether a pseudo-homogeneous or a heterogeneous continuum model must be used for reactor scale-up depends on the relative importance of the transport resistances. The gas phase resistance on the gas/liquid interface (as already mentioned) usually can be neglected, but this is not always the case with the other transport resistances as demonstrated experimentally by several authors [46,52]. 

Quasi-continuum models Of these, the quasi-continuum model is the most common. Here, the solid-fluid system is considered as a single pseudo-homogeneous phase with properties of its own. These properties, for example, diffusivity, thermal conductivity, and heat transfer coefficient, are not true thermodynamic properties but are termed as effective properties that depend on the properties of the gas and solid components of the pseudo-phase. Unlike in simple homogeneous systems, these properties are anisotropic, that is, they have different values in the radial and axial directions. KuUcami and Doraiswamy (1980) have compiled all the equations for predicting these effective properties. Both radial and axial gradients can be accounted for in this model, as well as the fact that the system is really heterogeneous and hence involves transport effects both within the particles and between the particles and the flowing fluid. 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

Under a certain set of assumptions, continuum mechanics yields governing equations, (35) and (39), that are identical to those proposed by the SAFT group. The predictions of this model were reviewed earlier detailed comparisons with experimental data for capacity loss will not be reproduced here but may be found in their original publication. In essence, we find that the predictions of the SAFT model are consistent with rigorous continuum mechanics under the assumption of pseudo-steady-state SEI growth limited by SEI electronic conductivity. 

A preliminary modelling analysis involved the parametric study of a multi-tubular externally-cooled fixed-bed reactor for a generic selective oxidation process, where the catalyst load consisted of cylindrical honeycomb monoliths with washcoated square chaimels, made of highly conductive supports. In this early work, the attention was focused on the effect of catalyst design. Simulation results were generated by a steady-state, pseudo-continuous 2D monolithic reactor model, where the catalyst is regarded as a continuum consisting of a static, thermally connected solid phase... 

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