Square and Rectangular UMEs

The location of a planar rectangular electrode of width w and length / in a Cartesian coordinate system is shown in Fig. 12.11a. The origin of the coordinate system is at the centre of the electrode. For normalisation we make use of the same equations as for the UMBE. 

The right-hand side of Eq. (12.89f) contains a factor 4 which does not appear in the definition of the dimensionless time for the UMDE, Eq. (12.16). This is because space is normalized by the half width of the electtode, w/2, while for the UMDE the radius a is used. The diffusion equation in three dimensions is then 

Tmax is the duration of the experiment in dimensionless units, see Eq.(12.89f). The parameter L is the dimensionless half-length of the band and is given by the ratio l/w. The boundary conditions for a simple potential step experiment to the diffusion limit involving one electroactive species have been given before for an 

Equation (12.94a) is the initial condition—at the beginning of the computation (or experiment) the concentration in the whole domain is equal to the bulk concentration. Equation (12.94b) is the boundary condition at the electrode, where the electroactive species reacts instantly, so that the concentration is zero. Equations (12.94c) and (12.94d) are the no-flux conditions at the insulating plane surrounding the electrode. Equations (12.94e) and (12.94f) are symmetry conditions along the XZ and YZ plane, respectively. Finally, the last three equations are the far field conditions where C remains at its normalised bulk value. 

For the current calculation, integration has to be performed along the x direction (0 X w/2) and along the y direction (0 y 1/2). This is expressed by the double integral 

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