The Stacking Model

Properties and Application. The two independent statistical distributions of the two-phase stacking model are the distributions of amorphous and crystalline thicknesses, hi x) and h2 x). Both distributions are homologous. The stacking model is commutative and consistent. If the structural entity (i.e., the stack as a whole) is found to show medium or even long-ranging order, the lattice model and its variants should be tested, in addition. As a result the structure and its evolution mechanism may more clearly be discriminated. 

Model Construction. In the stacking model alternating amorphous and crystalline layers are stacked. Likewise the combined thicknesses in the convolution polynomial are generated by alternating convolution from the independent distributions hi =h -khi, h4 = hi h, and/15 =hi h2. In general it follows 

If structure visualization by means of the IDF or CDF has shown that h and h2 can be modeled by Gaussians, all the combined thickness distributions are Gaussians as well. Each normalized Gaussian is completely described by mean di and standard deviation oi, and Eq. (8.101) is reduced to a relation 

The identification of the crystalline thickness may be possible from the result of the fit. As a frequent result of a model fit, the relative standard deviations of the fundamental domain thickness distributions, Ci/r/i and 02/ 2, differ considerably. In many cases the broader distribution can be attributed to the amorphous (or soft) phase. Even higher significance of the assignment can be achieved if the material is studied in time-resolved SAXS experiments during processing (under thermal load, mechanical load). Thus it is not always necessary to resort to secondary methods in order to resolve the ambiguity inherent to Babinet s theorem. 

Continued to infinity, the series of cluster weight parameters is intrinsically computed from a general weight parameter (number of lamellae in the sample) and the coupling parameter c. 

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The stacking model (Fig. 8.42c) does not carry this inconsistency [128,229], It cannot be discriminated from the lattice models if the polydispersity is strong. For small polydispersity even the lattice models make physical sense, because then the mutual penetration is negligible. Computation and fitting of stacking and lattice models are described in Sects. 8.7.3.4 and 8.133. 

In analogy to the treatment of the stacking model Jo (s) = 0 is valid, if the structural entities are embedded in matrix material. Compact material, again, may require a correction because of the merging of particles from abutting structural entities... 

As has already been mentioned in the discussion of the stacking model, such equations are particularly useful for the analysis of nanostructured material with weak disorder in order both to assess the perfection of the material and to discriminate among lattice and stacking models (cf. Sect. 8.8.3). 

As mentioned above, stack models are useful for analyzing full system performance including perhaps auxiliary components in the system such as compressors. In terms of equations, almost all of the models use simple global balances and equations because single cells are not the focus of the models thus, they use equations similar to eqs 21 and 78. In terms of other equations, normally they use typical flow and heat balances as well as the appropriate current and voltage relations, which take into account how the cells are connected together. The stack models can be separated into two categories, those that consider the stack and those that consider... 

Couple the stack model with a system model. 

The fuel cell module takes a hydrogen inlet flow rate and a requested power, then determines if sufficient power can be supplied. The stack model uses a simple map of efficiency versus power. This data is read from an input file to allow the fuel cell to be calibrated to real performance data. If sufficient... 

In Section 9.2 below, a summary of the nomenclature used in the chapter is given. In Section 9.3, a summary of fuel cell stack geometry, and a discussion of the dimensional reductions used in the model is given. In Section 9.4, the model of 1-D MEA transport is presented, followed by Section 9.5 on the model of channel flow for a unit cell and Section 9.6 on the electrical and thermal coupling in a stack environment. In Section 9.7, a summary of the stack model is given followed by its discretization. In Section 9.8, the iterative solution strategy for the discrete system is presented, followed by sample computational results in Section 9.9. The current state of stack modeling in this framework and future directions are summarized in the final section. 

Unit cells (up to 100 or more) are placed in series into fuel cell stacks. The anode plate of one cell is placed next to the cathode plate of the next (so that their voltages will add in the stack). The combined plates are called bipolar plates. The reduced dimensional geometry of the stack model is shown in Figure 9.3. 

At this level of the model, consider to be given the local current density i, the anode and cathode catalyst temperatures (da and 6c), the anode and cathode channel vapor concentrations (c and Cc), the anode channel hydrogen concentration (ch), and the cathode channel oxygen concentration (Co)-To be determined from the MEA model are the local cell voltage v, the diffusive water flux through the membrane / from cathode to anode, and the heat generated due to membrane resistance and cathode overpotential losses. As mentioned above, in this simplified model, the membrane resistivity is taken to be constant. Other resistances (except to in-plane currents in the bipolar plates in the stack model) are neglected. 

The stack model is now considered with M=13 cells. All cells except the center cell 1 are run at base conditions. The center cell is run with the anomalous cathode stoich Sc = 1.3. This anomaly could arise, for example. 

As became obvious in the previous section, a detailed description of a complete fuel cell is computationally very demanding. The stack models thatt are discussed in this section are on a higher abstraction level. They serve mainly as one component of a complete fuel-cell system. The discussion of system simulation is beyond the scope of this chapter, and at this point only the characteristics of the stack models are mentioned. 

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