Characteristic Ohmic length

Curves showing the cnrrent densities as functions of x are presented for two val-nes of electrode thickness in Fig. 18.5. The parameter L has the dimensions of length it is called the characteristic length of the ohmic process. It corresponds approximately to the depth x at which the local current density has fallen by a factor of e (approximately 2.72). Therefore, this parameter can be nsed as a convenient characteristic of attenuation of the process inside the electrode. 

When only taking into account the concentration polarization in the pores (disregarding ohmic potential gradients), we must use an equation of the type (18.15). Solving this equation for a first-order reaction = nFhjtj leads to equations exactly like (18.18) for the distribution of the process inside the electrode, and like (18.20) for the total current. The rate of attenuation depends on the characteristic length of the diffusion process ... 

Faraday s constant (96,487 C/mol) overpotential total current current density exchange current density ratio of ohmic constriction to inter-facial resistance surface exchange coefficient volume-specific interfacial resistance in a composite thickness utilization length characteristic length of a porous microstructure... 

The Wagner parameter, W, is the ratio of the kinetic resistance to the ohmic resistance. The Wagner parameter is the ratio of the true polarization slope given by the partial derivative, dE /di, evaluated at the overpotential of interest at constant pressure, temperature, and concentration, divided by the characteristic length and the solution resistance (2,40). 

Here, k is the conductivity l is the characteristic length and dr Jd is the slope of the polarization line. A large Wagner number is indicative of a uniform macroscopic current distribution since it corresponds to a large activation resistance (which tends to level off the current) and a small ohmic resistance (which is geometry-dependent and usually causes non-uniformities). 

R (Ohms) is the measured resistance of the sample over its length L (in the direction of a the electric field) and A is the current carrying cross-section area. The dimensions of SR are reported as Ohm-meter (Q-m). A material following Equation 3.1 is said to be Ohmic whereas, a material following a non-liner power law such as V In in the current-voltage characteristic is non-Ohmic. Non-Ohmic behavior has been discussed by Lampert and Mark (1970), Lacharme (1978), and Kingery (1976). 

The electrical behavior of the highly conductive carbon film can be explained by the theory developed by Ramakrishnan et al. [69,80,81], in which a parameter called the characteristic length is important. The characteristic length on the localized side is given by the localization length of the wave function, whereas on the extended side signifies the scale at which the conductance becomes ohmic. 

In all cases, the ohmic resistance scales with tjK, where is a characteristic length, often the electrode size, and k is the electrolyte conductivity. Thus, the Wagner number decreases as the system size increases. Current distributions thus tend to be less uniform with increasing scale. Likewise, the addition of supporting electrolyte may increase k, reducing the ohmic resistance and thus causing a more uniform current distribution. 

Figure 3. Concept of the Wagner s parameter. If X is a characteristic dimension of the cell (anode-cathode) it can be compared to the Wagner s length L. It means that as a function of the relation between X and L (depending of the electrolyte conductivity and the polarization) the ohmic drop between the anode and the cathode will affect the polarization of the anode and cathode.

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