Wagner Number

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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. 

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. 

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. 

A dimensionless parameter known as the Wagner number is useful for qualitatively predicting whether a current distribution will be uniform or nonuniform (2,40,41). This parameter helps to answer the question, Which current distribution applies to my cell primary, secondary or tertiary ... 

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 resulting Wagner numbers for various analytical expressions are shown... 

Current distribution Overpotential-current relationship E = f(i) Polarization dE/di resistance term Wagner number... 

Recall that the Wagner number depends on the solution conductivity, characteristic length, as well as the interfacial electrode characteristics. A solution has been given for the primary current distribution where the entire interior pipe surface (radius r0) is uniformly cathodically protected to / and the pipe interface is considered to be nonpolarizable (16). The IR drop down the pipe to a distance, L, can be calculated so that the maximum tolerable potential drop from the entrance to the far end is known ... 

Therefore let us instead consider the more practical case of the tertiary current distribution. Based on the dependency of the Wagner number on polarization slope, we would predict that a pipe cathodically protected to a current density near its mass transport limited cathodic current density would have a more uniform current distribution than a pipe operating under charge transfer control. Of course the cathodic current density cannot exceed the mass transport limited value at any location on the pipe, as said in Chapter 4. Consider a tube that is cathodically protected at its entrance with a zinc anode in neutral seawater (4). Since the oxygen reduction reaction is mass transport limited, the Wagner number is large for the cathodically protected pipe (Fig. 12a), and a relatively uniform current distribution is predicted. However, if the solution conductivity is lowered, the current distribution will become less uniform. Finite element calculations and experimental confirmations (Fig. 12b) confirm the qualitative results of the Wagner number (4). 

Wagner number (Wa) — is the dimensionless parameter describing the so-called secondary -> current distribution at an electrode electrolyte interface (-> electrode, -> electrolyte, -> interface) under the conditions when -> overpotential cannot be neglected, but the -> concentration polarization is negligible [i]. This number is defined as... 

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