Mathematical modeling Butler-Volmer equation

The rate of electron transfer and its potential dependence can be described by the Butler-Volmer equation (20) (see Section 2). An electron transfer often initiates a cascade of homogeneous chemical reactions by producing a reactive radical anion/cation. The mechanism can be described mathematically by a rate equation for each species these form part of the electrochemical model. The rate law of the overall sequence is probed by the voltammetric experiment. 

When modeling fuel cells and gas diffusion electrodes, simplifications are almost always inevitable. These have to be handled with care, since they may easily introduce physical inconsistencies that lead to ill-posed problems or inaccuracy in the mathematical model. If the Tafel equations are used as a simplification to the Butler-Volmer equations, then it has to be established that all part of the electrodes are far from equilibrium. If not, then substantial errors may be introduced by such simplifications (Fig. 18.6). The concentration overpotential has to be accurately introduced in the model, since starved part of the electrodes may have small activation overpotentials and therefore be close to equilibrium for the local gas and electrolyte composition. The use of the Tafel equations in combination with a poor description of the concentration overpotential is a common source of inaccuracy in fuel cell modeling. This may also lead to convergence problems, since the model may not follow the conservation laws and therefore become ill-posed. 

Simplified Butler-Volmer Equation 3 Butler-Volmer Equation with Identical Charge Transfer Coefficients-sinh Simplification A very nice simplification can be made to the BV model if the anodic and cathodic charge transfer coefficients at the electrode are equivalent (i.e., Uc = a a). In this case, no approximation is needed, and a new form explicit in r] and mathematically equivalent to the original BV model can be written. This model is valid over all regions of the electrode polarization, as shown in Figure 4.25. 

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