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98) Here f A [φ(Q)] and f C [φ(Q)] represent the anodic and cathodic polarization curves. Furthermore, n Q is the unit outward normal to the boundary at the point Q under consideration, as displayed in Fig. 20. G. Zamani Figure 20. The domain under consideration and different segments. Additional possible boundary conditions are shown below. These conditions are applicable depending on the problem at hand: ∂φ(Q) = qA Q ∈ ΓA (impressed current condition) , (99) ∂n Q φ(Q) = φA Q ∈ ΓA (nonpolarizable anode) .

For example, if Numerical Modeling of Certain Electrochemical Processes 19 we expect (ds/dt) − φA ≤ , it is straightforward to show that it translates to the following constraints: tε2r 2 − 1 + 2rA + t 2 ε2 r 4 1/ 2 ≤ φA ≤ 0. (63) Naturally, analytical solutions to two-dimensional and threedimensional problems become intractable. VII. 14–16 In the case of cathodic protection and electrodeposition, the governing partial differential equation is very simple, namely, the Laplace equation. These techniques are classified as follows: (a) Finite-difference method (FDM) (b) Finite-element method (FEM) (c) Boundary-element method (BEM) The general description of these methods is provided in the present section.

56) The expression on the right-hand side of (56) can be linearized by taking the first term of the Taylor series expansion in terms of s(ds/dt) − φA . Performing the linearization and some algebra, one arrives at an explicit expression involving ds/dt: r φA ds = . dt 1 + rs (57) Here, the parameter r is given by r = ζ (αA + αc ). The exact solution for (57) can easily be obtained: s(t) = 1 1 −1 + (1 + 2rA + 2r 2 φA t) /2 . r (58) Since s(0) = L the constant A is calculated from A=L+ r 2 L . 2 (59) Combining the results, we have ds −φA , = 1 dt (1 + 2rA + 2r 2 φA t) /2 ⎛ ⎞ r x ⎠, φ (x, t) = φA ⎝1 − 1 1 + 2rA + 2r 2 φA t /2 C1 (t) = − s 1 ds = φA 1 − 1 dt (1 + 2rA + 2r 2 φA t) /2 .