Macro-simulation-mathematics modeling

Bove R., Lunghi P., Sammes N.M., 2005. SOFC mathematic model for systems simulations. Part One From a micro-detailed to macro-black-box model. International Journal of Hydrogen Energy, 30(2), 181-187. 

These models are not discussed here and the cited papers may be referred to for details of model equations. When macroscale and microscale segregation exist together (bottom left case of Fig. 5.5), none of the cited models are adequate. For such systems, it is necessary to include detailed interaction of fluid mechanics, mixing and reactions in the mathematical model. Various modeling approaches to simulate reactive flow processes with macro- and microscale segregation are discussed briefly below. 

As noted throughout, while the general equations hold, there are many nuances that can go into the expressions. For example, one can consider agglomerate models, coupling of the equations at the macro and mesoscales with phenomena and models at the nano and particle scales, etc. It is clear that reaction within a porous electrode is complex and nonlinear, making mathematical simulation of it very useful. 

A double stochastic model was presented by us. It seems to be mathematically appropriate to describe not only the micro-level stochastic effects but also the macro-level stochastic effects. Using this model the particle system can be well characterized, namely one can give the residence probability of the particles after n mixing steps one can also compute the average residence probability, and on the basis of the model using Monte-Carlo - simulation one can easily approximate these values. Under certain conditions the stationary state of the process can be also determined. 

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