Simplification of Packed-Bed Electrode with a Low Conversion

In a special case i.e., low reactant conversion, in which the bulk electrolyte reactant concentration is assumed constant, the coupled Eqs. (156) and (157) can be simplified into a single differential equation (as C = 1). For a cathodic reaction, the charge balance Eq. (106) becomes 

To solve the simple model without decoupling, there are two choices for the boundary condition, at X = 1, on the membrane side. If the dimensionless current I can be measured and the dimensionless potential ° is unknown, we solve the model to obtain ° by using the boundary condition 

If ° can be measured and I is unknown, we solve the model to obtain (dO dX)x =, by using the boundary condition 

The solution and its derivative are in series forms in which the constant root °, namely the value of the potential at X = 0, is undetermined yet. 

For an easier decomposition solution and for faster convergence to the roots, the model equation can be rewritten in a form, which scales the boundary condition values to 1.0. This rearrangement is shown next. 

If can be measnred and / is nnknown, we solve the model to obtain d t ldX)x = i by nsing the bonndary condition 

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