Electrode volume-averaged models

Four categories of electrode models can be identified from Table 28.3 the spatially lumped model, the thin-film model, the agglomerate model, and the volume-averaged model. Schemes of the basic concepts of these four model categories are depicted in Figure 28.4. Each of the schemes shows an electrode pore, with the gas channels located at the top and the liquid electrolyte, depicted in gray, at the bottom. In some models, electrolyte is also present in the pore. The reaction zones are indicated by a black face (spot, line, or grid structure). Fluxes of mass and ions are indicated by arrows. 

There is an inherent coupling of the behavior of the micro-scale variables to the behavior of macro-scale variables. This in itself presents complications when simrrlating these models. A few researchers have tried to address this problem of couphng of scales in these models. The solid state concentration term defined by the micro scale diffusion equation need to be coupled with the governing equations for the macro-scale to predict electrochemical behavior. Wang and co-workers used volume averaged equations and a parabolic profile approximation for solid-phase concentration. Subramanian et al. developed approximations assuming that the solid-state concentration inside the spherical electrode particle can be expressed as a polynomial in the spatial direction. 

The fourth type of electrode models [39, 46] is based on the volume-averaging approach for porous media [51-55]. As indicated in Figure 28.4, it is similar to the agglomerate model, but no macropores are considered. Hence the main direction of mass transport in these micropores is along the depth of the electrode. 

Recently, pore network modeling has been applied to simulate the accumulation of liquid water saturation within the porous electrodes of polymer electrolyte membrane fuel cells (PEMFCs). The impetus for this effort is the understanding that liquid water must reside in what would otherwise be reactant diffusion pathways. It therefore becomes important to be able to describe the effect that saturation levels have on reactant diffusion. Equally important is the understanding of how the properties of porous materials affect local saturation levels. This requirement is in contrast to most continuum modeling of the PEMFC, where porous materials are treated with volume-averaged properties. For example, the relationship between bulk liquid saturation and capillary pressures foimd through packed sand and other soil studies are often employed in continuum models. ... 

Here a typical treatment in the model using a volume-averaging method is described to illustrate how this approach works and what sensible solntions in the computation can be derived from a model with temporal and spatial resolntions. Let Nj be the average flux of species i in the pore solution when averaged over the cross-sectional area of the electrode. Thus, for a plane surface, of normal unit vector n cutting the porous solid, n-N represents the amount of species i crossing this plane in the solution phase, but referred to the projected area of the entire plane rather than to the area of an individual phase. The superficial current density I2 in the pore phase is due to the movanent of charged solutes ... 

Detailed derivations of the porous electrode model and the volume averaging approach for battery modeling can be found in Refs [14,42, 52], respectively. 

The volume-averaged equations for transport of species and charge described earlier apply to the RFB computational model involving porous electrodes and... 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

Most of the models available in the literature are axial symmetric. A second simplification refers to the discretization adopted for the electrodes and electrolyte. Some of the models consider the cathode, electrolyte and anode as a whole and adopt an axial discretization. Electronic/ionic resistivity is computed as the average value of the single resistivites, calculated at the local temperature (Campanari and Iora, 2004). Using this approach means to simplify the solution of mass transfer in the porous media and the conservation of current. Authors have shown that about 200 elements are sufficient to describe the behaviour of a cell 1.5 m long using a finite volume approach (Campanari and Iora, 2004). 

Neutral PU electrodes are also called indifferent or reference electrodes. A PU electrode may be more or less indifferent depending on its area and position with respect to signal sources. The ideal unipolar models presuppose an enormous spherical neutral electrode, infinitely far away and at zero potential. In a limited tissue volume such as the human body, the ideal indifferent electrode is not feasible, and we use flie concept of quality of an indifferent electrode. A practical approach is to take the average potential of several electrodes (e.g., Wilson s central terminal in ECG). It is not ideal to connect electrodes of different potentials resulting in exchange current flow between them, but each electrode may first be connected to a buffer amplifier. 

In this section we describe the equations required to simulate the electrochemical performance of porous electrodes with concentrated electrolytes. Extensions to this basic model are presented in Section 4. The basis of porous electrode theory and concentrated solution theory has been reviewed by Newman and Tiedemann [1]. In porous electrode theory, the exact positions and shapes of aU the particles and pores in the electrode are not specified. Instead, properties are averaged over a volume small with respect to the overall dimensions of the electrode but large with respect to the pore structure. The electrode is viewed as a superposition of active material, filler, and electrolyte, and these phases coexist at every point in the model. Particles of the active material generally can be treated as spheres. The electrode phase is coupled to the electrolyte phase via mass balances and via the reaction rate, which depends on the potential difference between the phases. AU phases are considered to be electrically neutral, which assumes that the volume of the double layer is smaU relative to the pore volume. Where pUcable, we also indicate boundary conditions that would be used if a Uthium foil electrode were used in place of a negative insertion electrode. 

---