Macrohomogeneous model equations

At macroscopic level, the overall relations between structure and performance are strongly affected by the formation of liquid water. Solution of such a model that accounts for these effects provides full relations among structure, properties, and performance, which in turn allow predicting architectures of materials and operating conditions that optimize fuel cell operation. For stationary operation at the macroscopic device level, one can establish material balance equations on the basis of fundamental conservation laws. The general ingredients of a so-called "macrohomogeneous model" of catalyst layer operation include ... 

With the evaluated site coverage and pore blockage correlations for the effective ECA and oxygen diffusivity, respectively, and the intrinsic active area available from the reconstructed CL microstructure, the electrochemistry coupled species and charge transport equations can be solved with different liquid water saturation levels within the 1-D macrohomogeneous modeling framework,25,27 and the cathode overpotential, q can be estimated. 

In this chapter the scope of our discussion was restricted by the macrohomogeneous model of CL performance and its derivatives. The first numerical macrohomogeneous models of CCL for a PEM fuel cell were developed by Springer and Gottesfeld (1991) and by Bernard and Verbrugge (1991). These models included the diffusion equation for oxygen transport, the Tafel law for the rate of ORR and Ohm s law for the proton transport in the electrolyte phase. A similar approach was then used by Perry, Newman and Cairns (Perry et al., 1998) and by Eikerling and Kornyshev (1998) for combined numerical and analytical studies. 

Pores with radius r < are filled with liquid water, while pores with r > are filled with gas. In an operating fuel cell, s depends on the pore size distribution, wettability distribution, and the distributions of pressures, i.e., and pK The pressure distributions are coupled to stationary fluxes of species as well as to rates of current generation and evaporation via the set of flux and conservation equations that will be presented in the section Macrohomogeneous Model with Constant Properties. When it becomes necessary to distinguish hydrophilic and hydrophobic pores in CCLs (Kusoglu et al., 2012), the liquid saturation is given by Equation 3.101. 

The effects of porous structure and liquid water accumulation on steady-state performance of conventional CCLs were explored in Eikerling (2006) and Liu and Eikerling (2008). In these modeling works, uniform wetting angle was assumed in secondary pores, with a value 0 < 90°. The full set of equations presented in the section Macrohomogeneous Model with Constant Properties are solved with the following boundary conditions ... 

The macroscale model is almost identical to the MHM discussed in the section Macrohomogeneous Model with Constant Properties. In the electrochemical source term of the MHM Equation 4.5, a spatial variation in the agglomerate effectiveness factor must be accounted for... 

In this section, the reactions and general equations for the catalyst layers are presented first. Next, the models are examined starting with the interface models, then the microscopic ones, and finally the simple and embedded macrohomogeneous ones. Finally, at the end of this section, a discussion about the treatment of flooding is presented. 

The above discussion outlines the modeling methodology and equations for understanding porous electrodes from a macrohomogeneous view. 

Simplified models that do not make a priori assumptions about one or more dominant resistances are often of the 1-D macrohomogeneous type. The 1-D assumption is similar to that in mass transfer-based models. The assumption of macrohomogeneity, based on work by Newman and Tobias [50], has proven useful in battery and fuel cell electrode modelling. It implies that the microstructure of the electrode is homogeneous at the level of the continuum equations governing mass transfer, heat transfer, and current conduction in the electrode (Eqs. (l)-(7) and (33)-(37)). This type of model can exploit solutions available in chemical reaction engineering practice and has been elaborated by several researchers in that field [51-55],... 

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