Multidimensional Effects

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. 

Temperature Distribution Along the fuel cell channel, the temperature distribution is directly controlled by the heat transfer boundary conditions. For small fuel cell stacks, with no active coolant flow, the external boundary conditions control the temperature distribution and at low current can be considered as uiriform in temperature. For larger stacks with active cooling, the temperature distribution can be engineered to match the desired humidity profile to control flooding and promote longevity by ehmination of dry- and hot-spot locations. In the in-plane direction, temperature variation exists, with more water accumulation under generally colder lands, as discussed. The temperature distribution in the stack can be fairly 

Current Distribution The current distribution in a PEFC can be measured experimentally with a variety of techniques, as discussed in Chapter 9. For larger fuel cell active areas, gradients in temperature, reactant concentration, humidity, and liquid water can drastically alter the local current density and should be considered in any advanced analysis. In order to predict the current distribution in a cell on a qualitative basis, one has to keep in mind the major driving forces that control the current at a given voltage  

Since the temperature in the PEFC typically varies by at most 15°C, the effect of temperature distribution on the reaction kinetics will be relatively small compared to the temperature interaction with the liquid water distribution and relative humidity that controls membrane ionic conductivity. The local relative humidity in the anode phase typically controls the local current density of an underhumidified fuel cell because electro-osmotic drag exacerbates anode dryout, while water generation at the cathode diminishes any electrolyte dryout in the cathode catalyst layer [11]. That is, if the other parameters are constant, the local current distribution can be predicted with a knowledge of the anode in an underhumidified cell [11]. Anode dryout can be the result of the loss of only a few hundredths of a milligram of water per square centimeter active area in the catalyst layer 

Peak height increased with current, net water imbalance 

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Park N S and Waldeck D FI 1989 Implications for multidimensional effects on isomerization dynamics photoisomerization study of 4,4 -dimethylstilbene in / -alkane solvents J. Chem. Phys. 91 943-52... 

The semiclassical theory introduced above can be extended to low vibrationally excited states [32]. The multidimensionality effects are more crucial in this case. As was found before [62, 70], the energy splitting may oscillate or even decrease against vibrational excitation. This cannot be explained at all by the effective ID theory. 

The simplest way to treat the catalyst layers is to assume that they exist only at the interface of the diffusion media with the membrane. Thus, they are infinitely thin, and their structure can be ignored. This approach is used in complete fuel-cell models where the emphasis of the model is not on the catalyst-layer effects but on perhaps the membrane, the water balance, or multidimensional effects. There are different ways to treat the catalyst layer as an interface. 

A more sophisticated and more common treatment of the catalyst layers still models them as interfaces but incorporates kinetic expressions at the interfaces. Hence, it differs from the above approach in not using an overall polarization equation with the results, but using kinetic expressions directly in the simulations at the membrane/diffusion medium interfaces. This allows for the models to account for multidimensional effects, where the current density or potential changes 16,24,46-48,51,52,54,56,60-62,66,80,82,87,107,125 although... 

The purpose of this section is to describe the general results of models that contain more than one of the layers described above. It is beyond the scope of this article to analyze every model and its results in detail, especially since they have already been discussed to a certain extent in section 2. Many of the models make tradeoffs between complexity, dimensionality, and what effects are emphasized and modeled in detail. It is worth noting that those models that employ a CFD approach seem to be the best suited for considering multidimensional effects. In this section, the ways in which the multilayer models are solved and connected are discussed first. Next, some general trends and results are presented. 

The inclusion of multidimensional effects is important to realistically mimic transport in the fuel cell. This is not to say that certain cases and factors cannot be collapsed to lower dimensionality, but one must be aware of higher dimensional effects, lest they become important. 

This review has highlighted the important effects that should be modeled. These include two-phase flow of liquid water and gas in the fuel-cell sandwich, a robust membrane model that accounts for the different membrane transport modes, nonisothermal effects, especially in the directions perpendicular to the sandwich, and multidimensional effects such as changing gas composition along the channel, among others. For any model, a balance must be struck between the complexity required to describe the physical reality and the additional costs of such complexity. In other words, while more complex models more accurately describe the physics of the transport processes, they are more computationally costly and may have so many unknown parameters that their results are not as meaningful. Hopefully, this review has shown and broken down for the reader the vast complexities of transport within polymer-electrolyte fuel cells and the various ways they have been and can be modeled. 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

Beginning in 1980 or earlier, numerous theoretical and experimental investigations were undertaken. Kramers theory was also extended in several ways, such as introducing a frequency-dependent friction to replace a constant one [42] and introducing multidimensional effects [43-46]. Both of these effects caused deviations from Kramers theory, but his work remained the seminal paper. 

Using the techniques discussed in section III and IV we have been able to study the effect of acceleration on ignition of a homogeneous fuel oxydizer mixture. The ability to study multidimensional effects (buoyancy, turbulence etc.) hinges on the use of numerical methods (slow-flow, asymptotic chemistry etc.) which circumvent the time constraints encountered in brute force techniques. These methods go hand in hand with modem fast computers, especially vector machines where judicious programming allows us to attain the actual memory or CPU cycle time. 

This chapter concerns the structures and propagation velocities of the deflagration waves defined in Chapter 2. Deflagrations, or laminar flames, constitute the central problem of combustion theory in at least two respects. First, the earliest combustion problem to require the simultaneous consideration of transport phenomena and of chemical kinetics was the deflagration problem. Second, knowledge of the concepts developed and results obtained in laminar-flame theory is essential for many other studies in combustion. Attention here is restricted to the steadily propagating, planar laminar flame. Time-dependent and multidimensional effects are considered in Chapter 9. 

Aller (1984) created a mechanistic model for the multi-dimensional transport of dissolved pore-water species by animals. He observed that ammonia profiles caused by sulfate reduction in the top-ten-centimeter layer of Long Island Sound sediments could not be interpreted by onedimensional diffusion (Equation (3)). The multidimensional effects of irrigation were reproduced mathematically by characterizing the top layer of... 

Consequently, the proposed model allows the necessary information regarding the electrolyte-metal electrode interface and about the character of the electronic conductivity in solid electrolytes to be obtained. To an extent, this is additionally reflected by the broad range of theoretical studies currently published in the scientific media and is inconsistent with some of the research outcomes relative to both physical chemistry of phenomena on the electrolyte-electrode interfaces and their structures. Partially, this is due to relative simplifications of the models, which do not take into account multidimensional effects, convective transport within interfaces, and thermal diffusion owing to the temperature gradients. An opportunity may exist in the further development of a number of the specific mathematical and numerical models of solid electrolyte gas sensors matched to their specific applications however, this must be balanced with the resistance of sensor manufacturers to carry out numerous numbers of tests for verification and validation of these models in addition to the technological improvements. 

An attempt has been made in this chapter to introduce the reader to gas radiation, which is vitally important for current technological and environmental problems. However, because of the complexity of the subject, some of its aspects—such as thick gases near boundaries, intermediate optical thicknesses, spectral effects, weighted nongray-ness, multidimensional effects, and scattering—are left untreated. Readers interested in these topics should consult the references of this chapter and the current literature. 

Unlike conventionally used volume-averaged equations, equation 20 does reduce to the Brinkman equation 5 when the inertial effects are negligible and to Darcy s law, equation 3, when both inertial and multidimensional effects are negligible. 

Still based on the first approach, Liu et al. (32) studied the wall effects while excluding the multidimensional effects. By heuristic arguments, they derived wall effect correction factors for both viscous... 

Multidimensional Effects. In the previous section, we studied the wall effect on the shear factor. To give a full account of the wall effects, we now look at the no-slip flow effect posed by the containing wall (multidimensional effect) on the total pressure drop. For simplicity, let us rewrite the normalized pressure drop factor, fv, based on the permeability of the medium rather than the particle diameter,... 

Let Cmd be a coefficient for multidimensional effect on the normalized pressure factor,... 

Figure 12 shows the multidimensional effect on the normalized pressure drop factor from both the exact solution, equation 121, and its approximation, equation 123. It can be observed that equation 123 gives a fairly good approximation to the exact solution. The multidimensional effect is significant when the normalized bed radius is small, say, F1/2D/2 < 100. When the normalized bed radius is large, Crrul - 1. 

From equation 124, we observe that the coefficient of multidimensional effect is also related to the ratio of the superficial velocity near the center line to the average superficial velocity, that is,... 

Hence, we expect that the larger the multidimensional effect is, the larger the deviation of the center line velocity from the average velocity. In the extreme case where F = 0, or the medium is infinitely permeable, the velocity ratio reaches 2, whereas the coefficient of the multidimensional effect is infinite. 

The coefficient for multidimensional effect, equation 121 or 123, is strictly speaking valid for Darcy s flow only. However, it is possible to estimate the multidimensional effect. Because the multidimensional effect is only significant near the wall, the shear factor used for evaluating the multidimensional coefficient may take an average value in the boundary layer. Let... 

When the porous medium is contained in a cylindrical bed, the multidimensional effect is given by equation 127. When the pressure drop is needed, there is no need to solve the governing equation numerically. Equations 106 and 128 can be used for the estimation of pressure drop as well as scaling-up of a packed bed. 

The function p7( ) is a symmetric bell-shaped curve centered at T, and pT( ) is narrower when the effective potential barrier is wider. For an ideal dynamical bottleneck kt is unity deviations from unity indicate that recrossing or other multidimensional effects are important. 

The value of the transmission coefficient kt is shown for each feature in Table 2. (The value of kt for the last feature is greater than 1 because it includes contributions from higher energy transition states that have not been included in the fit.) Many of the values of the transmission coefficients are very close to unity, suggesting that these features correspond to quantized transition states that are nearly ideal dynamical bottlenecks to the reactive flux. Several of the values of kt deviate from unity this could be the result of the assumption of parabolic effective potential barriers or from recrossing or other multidimensional effects. 

There have been extensive experiments on the isomerization of stilbene and its derivatives (125, 126), none of which to date have received any serious theoretical numerical analysis. In this experiment, changes in pressure have caused changes of orders of magnitude in the rate. It is time that a serious molecular dynamics study of this system be undertaken. This might also be useful in understanding the importance of multidimensional effects, since typically, in an isomerization of a big molecule, one could expect that more than one angle participates strongly in the isomerization process. 

Each new topic contained most of the complications present in the previously discussed topics. The basic ingredients in the studies presented are reasonable representations of both the flow field and the chemical kinetics. The complexity arose because we considered multidimensional effects and expanded the range of scales considered. In general, the current level of understanding of the details of the interactions between the chemistry and flow is less accurate as the flow and the chemical reactions become more and... 

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