Multidimensional Model Extension

If the nonuniformities in the temperature and species concentrations among the catalyst channels need to be considered, a 2D or 3D modeling approach is required. In this case, the transient energy balance equation of the monolith should be extended to two or three dimensions. The heat conduction equation of the solid phase is formulated in polar coordinates for 2D simulations (13.18) and in Cartesian coordinates for 3D simulations (13.19). 

The other catalyst equations are the same, as described above for the ID model. 

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The computational or analytical prediction of the current, species, and temperature distributions in a fuel cell can be generated by extension of the basic concepts of this text into multidimensional models with various levels of complexity and solved using the tools of computational fluid dynamics (CFD). Models of various complexity abound in the literature, and commercial packages now exist from several CFD software developers for this purpose. 

The introduction and implementation of heteronuclear-based multidimensional techniques have revolutionized the protein NMR field. Large proteins (> 100 residues) are now amenable to detailed NMR studies and structure determination. These techniques, however, necessarily require a scheme by which and isotopes can be incorporated into the protein to yield a uniformly labeled sample. Additional complications, such as extensive covalent post-translational modifications, can seriously limit the ability to efficiently and cost effectively express a protein in isotope enriched media - the c-type cytochromes are an example of such a limitation. In the absence of an effective labeling protocol, one must therefore rely on more traditional proton homonuclear NMR methods. These include two-dimensional (1) and, more recently, three-dimensional H experiments (2,3). Cytochrome c has become a paradigm for protein folding and electron transfer studies because of its stability, solubility and ease of preparation. As a result, several high-resolution X-ray crystal structure models for c-type cytochromes, in both redox states, have emerged. Although only subtle structural differences between redox states have been observed in these... 

A major effort has been put recently in the development of simulations of explosions that go beyond the one-dimensional approximation. This is motivated not only by the difficulty of obtaining successful CCSNe in onedimensional simulations, but also by the mounting observational evidence that SN explosions deviate from spherical symmetry, not to mention the possible connection between the so-called soft long-duration gamma-ray bursts, and grossly asymmetric explosions accompanied with narrow jets of relativistic particles, referred to as JetSNe. The multi-dimensional extension of the simulations opens the potentiality to treat in a proper way different effects that may turn out to be essential in the CCSN or JetSNe process. As briefly reviewed by e.g. [26], they include fluid instabilities, or rotation and magnetic fields on top of the neutrino transport already built into the one-dimensional models. Acoustic power may be another potential trigger of CCSNe [27] (see also [24] for a brief review of multidimensional simulations). 

Extensive development and validation of multidimensional codes has been carried out. The comparison of computed and expoimental results has provided a successful demonstration of the potential of 3D and 2D modelling. Qualitatively good results have been obtained. Quantitative results are reasonable depending on the modelling. 

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... 

Despite its title, and although it contains discussion of relevant numerical techniques, this article is not a comprehensive survey of the numerical methods currently employed in detailed combustion modeling. For that, the reader is referred to the reviews by McDonald (1979) and Oran and Boris (1981). Rather, the aim here is to provide an introduction that will stimulate interest and guide the enthusiastic and persistent amateur. The discussion will center mainly about low-velocity, laminar, premixed flames, which form a substantial group of reactive flow systems with transport. Present computational capabilities virtually dictate that such systems be studied as quasi-one-dimensional flows. We also consider two-dimensional boundary layer flows, in which the variation of properties in the direction of flow is small compared with the variation in the cross-stream direction. The extension of the numerical methods to multidimensional flows is straightforward in principle, but implementation at acceptable cost is much more difficult. 

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