1-D model

In this paper, we first briefly describe both the single-channel 1-D model and the more comprehensive 3-D model, with particular emphasis on the comparison of the features included and their capabilities/limitations. We then discuss some examples of model applications to illustrate how the monolith models can be used to provide guidance in emission control system design and implementation. This will be followed by brief discussion of future research needs and directions in catalytic converter modeling, including the development of elementary reaction step-based kinetic models. 

The basic scheme for the numerical solution is the same as that used for the 1 -D model, except that in this case the solid temperature field used to solve the DAE system for each monolith channel must be calculated from the three-dimensional solid-phase energy balance equation. The three-dimensional energy balance equation can be solved by a nonlinear finite element solver (such as ABAQUS) for the solid-phase temperature field while a nonlinear finite difference solver for the DAE system calculates the gas-phase temperature and... 

For examples of such correlations in 1-D models, see Chapter 4 in Ben-Naim (1992). 

Figure 1. Fuel-cell schematic showing the different model dimensionalities. 0-D modeis are simpie equations and are not shown, the 1-D models comprise the sandwich (z direction), the 2-D modeis comprise the 1-D sandwich and either of the two other coordinate directions x or y), and the 3-D models comprise aii three coordinate directions.

The next two major models were those by Fuller and Newman and Nguyen and White,who both examined flow effects along the channel. These models allowed for a more detailed description of water management and the effect of dry gas feeds and temperature gradients. Throughout the next few years, several more 0-D models and 1-D models were generated. Also, some simulations examined... 

More complicated expressions than those above can be used in the 0-D models, but these usually stem from a more complicated analysis. For example, the equation used by Ticianelli and co-workers comes from analysis of the catalyst layer as a flooded agglomerate. In the same fashion, eq 21 can be embedded and used to describe the polarization behavior within a much more complicated model. For example, the models of Springer et al. " and Weber and Newman " use a similar expression to eq 21, but they use a complicated 1-D model to determine the parameters such as ium and R. Another example is the model of Newman,who uses eq 22 and takes into account reactant-gas depletion down the gas channels by, in essence, having a limiting current density that depends on the hydrogen utilization. All of these types of models, which use a single equation to describe the polarization behavior within a more complicated model, are discussed in the context of the more complicated model. 

Although the fuel-cell sandwich is the heart of a fuel cell, there are important effects that are not found when only a 1-D model is used. These effects basically arise from the fact that a fuel cell is in reality a 3-D structure, as shown in Figures 1 and 5. Many models explore these effects and are discussed in this section. These models always include the fuelcell sandwich as one of the dimensions. First, the 2-D models are examined and then the 3-D ones. Because there is another review in this issue that focuses on these models in more depth, the discussion below is shortened. 

The general results of the 3-D models are more-or-less a superposition of the 2-D models discussed above. Furthermore, most of the 3-D models do not show significant changes in the 1-D sandwich in a local region. In other words, a pseudo-3-D approach would be valid in which the 1-D model is run at points in a 2-D mesh wherein both the channel and rib effects can easily be incorporated. Another pseudo-3-D approach is where the 2-D rib models are used and then moved along the channel, similar to the cases of the pseudo-2-D models described above. This latter approach is similar to that by Baker and Darling. In their model, they uncouple the different directions such that there is a 1-D model in the gas channel and multiple 2-D rib models. However, they neither treat the membrane nor have liquid water. In all, the use of CFD means that it is not significantly more complicated to run a complete 3-D model in all domains. 

The model presented here is a combination of a 1-D model for gas channels and a 3-Dmodel for solid and porous regions. 

The 1-D concentric cylinder models described above have been extended to fiber-reinforced ceramics by Kervadec and Chermant,28,29 Adami,30 and Wu and Holmes 31 these analyses are similar in basic concept to the previous modeling efforts for metal matrix composites, but they incorporate the time-dependent nature of both fiber and matrix creep and, in some cases, interface creep. Further extension of the 1-D model to multiaxial stress states was made by Meyer et a/.,32-34 Wang et al.,35 and Wang and Chou.36 In the work by Meyer et al., 1-D fiber-composites under off-axis loading (with the loading direction at an angle to fiber axis) were analyzed with the... 

Application of the 1-D Model Transient Creep and Stress Redistribution... 

Above 40 km altitude the observed concentration of Os is greater than that calculated with a faster rate for reaction 32, and a low (O2), the agreement is significantly worsened in this 1-D model. 

Thermal modelling is a powerful way to predict the performance and the temperature response of MHP/ mHP. Unfortunately most of developed thermal models are 1-D models and empirical correlations are employed to determine fraction factor of vapour flow. 

Tikhonov s theorem of uniqueness for a one-dimensional (1-D) model (Tikhonov... 

In the special case of a 1-D model, the WKBJ approximation of Green s function takes the form (Bleistein et al., 2001, p. 69)... 

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