Empirical models Butler-Volmer equation

Eor very high exchange current densities (i.e., rapid reactions), a linearized form of Eq. 27 can often be used. For very slow reaction kinetics, either the anodic or cathodic term dominates the kinetics, and so the other term is often ignored, yielded what is known as a Tafel equation for the kinetics. Often, more complicated expressions than that of Eq. 27 are used. For example, if the elementary reaction steps are known, one can write down the individual steps and derive the concentration dependence of the exchange current density and the kinetic equation. Other examples include accounting for surface species adsorption or additional internal or external mass transfer to the reaction site [9]. All of these additional issues are beyond the scope of this chapter, and often an empirically based Butler-Volmer equation is used for modeling the charge transfer in porous electrodes. 

Additional parameters specified in the numerical model include the electrode exchange current densities and several gap electrical contact resistances. These quantities were determined empirically by comparing FLUENT predictions with stack performance data. The FLUENT model uses the electrode exchange current densities to quantify the magnitude of the activation overpotentials via a Butler-Volmer equation [1], A radiation heat transfer boundary condition was applied around the periphery of the model to simulate the thermal conditions of our experimental stack, situated in a high-temperature electrically heated radiant furnace. The edges ofthe numerical model are treated as a small surface in a large enclosure with an effective emissivity of 1.0, subjected to a radiant temperature of 1 103 K, equal to the gas-inlet temperatures. 

As already mentioned above, the derivation of the Butler-Volmer equation, especially the introduction of the transfer factor a, is mostly based on an empirical approach. On the other hand, the model of a transition state (Figs. 7.1 and 7.2) looks similar to the free energy profile derived for adiabatic reactions, i.e. for processes where a strong interaction between electrode and redox species exists (compare with Section 6.3.3). However, it should also be possible to apply the basic Marcus theory (Section 6.1) or the quantum mechanical theory for weak interactions (see Section 6.3.2) to the derivation of a current-potential. According to these models the activation energy is given by (see Eq. 6.10)... 

According to (6.24), the ionic current density in the film varies exponentially with the electric field. Even though this relation has been derived here from a rather simple model, it holds true quite generally. We therefore can also look at equation (6.24) as an empirical equation that describes the relation between the ionic current, the potential and the thickness of solid oxide films, in a similar way as the Butler-Volmer equation describes the relation between current and potential for a metal-electrolyte interface. 

In recent years, many CFD models for SOFC performance have been developed. Some of these models rely on the empirical notion of area-specific resistivity (ASR), not detailing the kinetics of electrochemical reactions (Yakabe et ah, 2001 Xue et ah, 2005). The others utilize the Butler-Volmer equation for the calculation of activation losses (Iwata et ah, 2000 Larrain et ah, 2003 Aguiar et ah, 2004 Yuan and Liu, 2007 Wang et ah, 2007 Ho et ah, 2008 Zhu and Kee, 2008). However, all these models are numerical and they do not give an irrefutable answer to the questions above. 

Regarding catalyst aging. Darling and Meyers have proposed a mechanistic model, based on empirical parameters, of the Pt oxidation/dissolution in a PEMFC cathode, largely cited by experimentalists in subsequent papers. By using classical Butler-Volmer equations written in terms of the CL electrode potential and empirical parameters (e.g. symmetry factors, zero-exchange current and... 

The Butler-Volmer (BV) approximation is the simplest approach to model and capture the essential features of the empirical Tafel equation. It considers an electrochemical half-cell reaction as an activated process, with the forward and backward reaction rates following an Arrhenius type law according to... 

For an ideal monolayer and nj= 1, this expression is reduced to the adsorption isotherm (2.157) derived using Pethica s equation. It can be concluded therefore, that the two approaches lead to similar results. At the same time, Butler s equation (2.7) always leads to a logarithmic form of the equation of state for mixed monolayers, which often disagrees with the experimental results. For these systems, Volmer s or van der Waals equations of state are more appropriate [58, 98]. Therefore the method based on Pethica s equation is advantageous, enabling one to apply semi-empirical model equations of state for mixed monolayers. 

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