Current primary distribution

Primary current distribution occurs in the absence of electrode reaction, when the surface overpotential is neglected (i.e., the electrode surface is considered to be nonpolarized) and concentration gradients do not exist. 

When an electrochemical reaction occurs, the electrode surface can no longer be considered an equipotential boundary and activation overvoltage has to be overcome to cause further reaction. The shift in electrode potential—the slope of the electrode potential/current density curve becomes significant—required to induce increased current flow gives rise to secondary current distribution. 

When a shift in potential causes significant concentration changes so that mass transport limits to some extent the current flow, tertiary current distribution occurs. Tertiary distribution exists whenever current flows but at relatively low polarization the effect can be ignored. 

Design methods try to include potential variations in the analysis. Local variations in product distribution and current efficiency are deduced and are part of the overall design. These approaches tend to assume an ideal reactor configuration in which the primary current distribution is uniform. This ideal is sometimes approached in practice but some reactor geometries exhibit severe nonuniform distributions, particularly at the higher current densities demanded in industry. It is important to analyze such systems and determine conditions which will minimize these effects. In the following sections we will consider the problem of current distribution and illustrate some calculations with worked examples. 

With no polarization of the electrode and an electrolyte of uniform concentration, we deal only with primary current distribution, which is given by the solution of the Laplacian equation  

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When a battery produces current, the sites of current production are not uniformly distributed on the electrodes (45). The nonuniform current distribution lowers the expected performance from a battery system, and causes excessive heat evolution and low utilization of active materials. Two types of current distribution, primary and secondary, can be distinguished. The primary distribution is related to the current production based on the geometric surface area of the battery constmction. Secondary current distribution is related to current production sites inside the porous electrode itself. Most practical battery constmctions have nonuniform current distribution across the surface of the electrodes. This primary current distribution is governed by geometric factors such as height (or length) of the electrodes, the distance between the electrodes, the resistance of the anode and cathode stmctures by the resistance of the electrolyte and by the polarization resistance or hinderance of the electrode reaction processes. 

Cell geometry, such as tab/terminal positioning and battery configuration, strongly influence primary current distribution. The monopolar constmction is most common. Several electrodes of the same polarity may be connected in parallel to increase capacity. The current production concentrates near the tab connections unless special care is exercised in designing the current collector. Bipolar constmction, wherein the terminal or collector of one cell serves as the anode and cathode of the next cell in pile formation, leads to gready improved uniformity of current distribution. Several representations are available to calculate the current distribution across the geometric electrode surface (46—50). 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

Seconday Current Distribution. When activation overvoltage alone is superimposed on the primary current distribution, the effect of secondary current distribution occurs. High overpotentials would be required for the primary current distribution to be achieved at the edge of the electrode. Because the electrode is essentially unipotential, this requires a redistribution of electrolyte potential. This, ia turn, redistributes the current. Therefore, the result of the influence of the activation overvoltage is that the primary current distribution tends to be evened out. The activation overpotential is exponential with current density. Thus the overall cell voltages are not ohmic, especially at low currents. 

The rotating hemispherical electrode (RHSE) was originally proposed by the author in 1971 as an analytical tool for studying high-rate corrosion and dissolution reactions [13]. Since then, much work has been published in the literature. The RHSE has a uniform primary current distribution, and its surface geometry is not easily deformed by metal deposition and dissolution reactions. These features have made the RHSE a complementary tool to the rotating disk electrode (RDE). 

The primary current distribution is uniform on the hemisphere. Numerical calculations using the potential theory have shown that the current distribution is essentially uniform on the RHSE if the current density is less than 68% of the average limiting current density [47]. 

The dimensionless limiting current density N represents the ratio of ohmic potential drop to the concentration overpotential at the electrode. A large value of N implies that the ohmic resistance tends to be the controlling factor for the current distribution. For small values of N, the concentration overpotential is large and the mass transfer tends to be the rate-limiting step of the overall process. The dimensionless exchange current density J represents the ratio of the ohmic potential drop to the activation overpotential. When both N and J approach infinity, one obtains the geometrically dependent primary current distribution. 

Figure 8. Primary current distribution on the front surface of the electrodes based on Kirkhof s law calculation for three different cell constructions (A) Both connections to the cell are at the top. The higher resistance path at the bottom sections of the electrode reduces the current flow and results in a nonuniform current distribution. (B) All paths have equal resistance, and a uniform current distribution results. (C) The bipolar construction has equal resistance from one end to the other.

This primary current distribution becomes important in practical electrochemical devices, e.g., fuel cells. Here, one uses porous electrodes to try to increase the active electrode area for a unit apparent external area. This seems a good idea at first, but in reality the resistance of the solution down the pores prevents ions produced in the fuel cell reaction from "getting out and often only a small length of the pore in a porous electrodes is active. 

Throwing Power. It has long been observed that plate thickness distribution does not always follow the primary current distribution. 

Current distribution Distribution of reaction rates on an electrode surface. Primary current distribution is calculated by considering only electric field effects both overpotential and concentration gradients are neglected. Secondary current distribution takes both field effects and surface overpotential into account. Tertiary current distribution takes field effects, surface overpotential, and concentration gradients into account. 

Wagner number Dimensionless ratio of polarization resistance to electrolyte resistance. A low value is characteristic of a primary current distribution a high value corresponds to a secondary current distribution. 

Primary current distribution. The current distribution is governed solely by the electric field. No other effects are considered. 

The primary current distribution represents the distribution resulting solely from resistance to current flow in the electrolyte. Since temperature and concentration variations as well as overpotential are neglected, this type of current distribution is usually easy to calculate. 

If the assumptions inherent in the primary current distribution model are reasonable for the system being considered, then a simulation of the system behavior is relatively straightforward. 

FIGURE 7 Primary current distribution on a wavy electrode. Resistance to current flow is represented schematically by the size of the resistors. 

FIGURE 10 Current distribution on a disk electrode. The primary current distribution approaches infinity at the junction of the electrode and the coplanar insulator. The secondary current distribution is more uniform. Average current density is /aVg and the electrode radius r0. 

The primary potential distribution is, by definition, uniform adjacent to the electrode surface, but the current distribution is highly nonuniform (Fig. 10). It is a general characteristic of the primary current distribution that the current density is infinite at the intersection of an electrode and a coplanar insulator. This condition obtains at the periphery of the disk electrode, and the current density becomes infinite at that point. Additional resistance due to kinetic limitations invariably reduces the nonuniformity of the current distribution. In this system the current distribution becomes more uniform as the Wagner number increases. Theoretically, the current distribution is totally uniform as the Wagner number approaches infinity. 

Channel flow between plane parallel electrodes is shown in Fig. 11. This geometry is similar to that of the disk in that an electrode and an insulator intersect in the same plane. Because of many geometric similarities, the general characteristics of the primary and secondary current distributions are similar. At the edges the local current density is infinite for the primary current distribution (Fig. 12). Increasing the kinetic limitations tends to even out the current distribution. The significant contrasts appear in a comparison of the tertiary current distributions. In channel flow, the fluid flows across the electrode rather than normal to it. Consequently, the electrode is no... 

Figure 3 Sinusoidal two-electrode cell with insulating boundaries. In the primary current distribution, all current flow occurs at the peak in the sinusoid. For increasing ratios of RrJR[>. the current uniformity increases.

Isopotential lines are parallel to the electrode surfaces for what is known as the primary current distribution (no interfacial electrode polarization, or zero polarization resistance). Said another way, the solution adjacent to an electrode surface is an equipotential surface (1). This primary current distribution applies to the case of extremely fast electrochemical reactions (e.g., nonpolar-izable electrode reactions). This current distribution situation is only of interest to the corrosion engineer in cases where high current densities might be flowing (i.e., in relatively nonpolarizable cells). 

Figure 4 Primary current distribution rules applied to the case of a flush mounted electrode with an insulating edge of various angles.

The primary current distribution The current distribution is determined solely by the potential field in the solution. Hence the solution conductivity and geometry are the only factors considered, and the potential across the electrochemical interface is assumed to be negligible, such as when the electrode reactions are extremely fast (i.e., a nonpolarizable electrode). 

The Wagner parameter may be thought of as the ratio of the kinetic resistance to the ohmic resistance. Hence when the Wagner number approaches numbers less than one, the ohmic component dominates the current distribution characteristics, and when it is much larger than one, the kinetic component dominates. In practice, the primary current distribution is said to exist when W < 0.1, and the secondary current distribution exists if W > 10 (6). The Wagner parameter is the ratio of the true polarization slope, dE ue/di (evaluated at the overpotential of interest) divided by the characteristic length and the solution resistance (1,6). 

The corrosion engineer can use this information in the following way. If the primary current distribution applies (W < 0.1), then current distributions are likely to be nonuniform unless one of the ideal cell geometries leading to uniform primary current distributions (discussed in Table 2) is used. In the former case, errors in polarization resistance and kinetic parameters are likely. In the latter case, rjapp must still be corrected for iRa, using the relationships given in Eq. (2) but the value of ViR will be the same at all positions along the electrode surface. 

The rotating disk electrode will have a uniform tertiary current distribution but an extremely nonuniform primary current distribution with the current density at the electrode edge approaching infinity (8-12). For a disk electrode of radius r0, embedded in an infinite insulating plane with the counterelectrode far away, the primary current distribution is given by... 

It is clear from the analysis of a disk electrode that when the electrode and insulator reside in the same plane (i.e., a flush mounted disk or plate), the primary current distribution is infinite at the electrode edge (13). Alternatively, the primary current distribution at the edge is zero if the angle between the insulator and the edge is less than 90°. The primary current distribution is not zero or infinite only when the angle is exactly 90°. This is shown in Fig. 4. 

It is common in corrosion laboratories and in field corrosion monitoring probes to immerse two vertical rods parallel to one another in an electrolyte. In the lab, one of the rods consists of a high-density graphite counterelectrode while the other is a working electrode. A reference electrode may be placed in between the two rods. In the field, polarization resistance or electrochemical noise measurements are often made between two nominally identical rods that both consist of the material of interest. The primary current distribution is nonuniform with respect to circumferential position about each electrode when the distance between the two rods is small in comparison to the radius of the rod, Fig. 10a (16). Again, the value of Ra varies from where the rods face each other to where they... 

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