Dimensional reduction strategy

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. 

From the figure, it appears that the trajectories reach exactly the same manifold, but it is easy to illustrate that a trajectory always only approaches the low-dimensional surface (or even the equilibrium point), but never reaches it exactly. In principle, time is reversible in a system defined by the system of ODEs (2.9). This means that calculating the trajectory from to to fj, and then continuing the simulation backwards in time from to to, the same concentration set should be recovered. This is impossible if trajectories starting from different initial conditions end up at exactly the same point. However, for most applications, the approximate nature of the slow manifolds is not a barrier to their use in model reduction strategies, since where large separations between timescales exist, the error related to the approximation of the slow manifold should be small. 

In contrast to the three-dimensional growth of the TREN dendrimer 48, we thought it would be interesting to synthesize a linear polymer dendrimer based on pentaethylenehexamine. We used the same strategy described in Section IE. A. The primary Michael addition of acrylonitrile yielded the octanitrile 56 without any problems. Similarly, the DIBAH reduction led to the amine 57 (Scheme 13). Contrary to all expectations, the addition of acrylonitrile failed at this step. In the mass spectra, we... 

SAMs illustrate a strategy for synthesis based on the idea of reduction in dimensionality. The generic idea underlying SAMs is to use a surface, or some other two-dimensional or pseudo two-dimensional system, as a template and to assemble molecules on it in reasonably predictable geometry using appropriate coordination chemistry to connect the surface with the adsorbed molecules. For this strategy to work, one needs ... 

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