The use of the Kramers-Kronig (KK) relations to evaluate the consistency of impedance data has been limited by the fact that the experimental frequency domain is necessarily finite. Current algorithms do not distinguish between the residual errors caused by a frequency domain that is too narrow and discrepancies caused by a system which does not satisfy the constraints of the KK equations. A new technique is presented which circumvents the limitation of applying the KK relations to impedance data which truncate in the capacitive region. The proposed algorithm calculates impedance values below the lowest experimental frequency which "force" the data set to satisfy the KK equations. Internally consistent data sets yield low-frequency impedance values which are continuous at the lowest measured experimental frequency. A discontinuity between the calculated low-frequency values and the experimental data indicates inconsistency which cannot be attributed to the finite experimental frequency domain.

Mathematical models were developed to predict cathodic protection (CP) requirements for coated pipelines protected by parallel anodes. This work was motivated by the need to estimate current and potential distribution on a pipe when anodes are placed nearby or when discrete coating holidays expose bare steel. The mathematical model solves Laplace's equation for potential with boundary conditions appropriate for the pipe being protected, the anode, and any region through which current does not pass. The current density on bare steel was assumed to be composed of contributions from corrosion, reduction of dissolved oxygen, and evolution of hydrogen. Kinetic parameters were obtained from independent experiments. The anode was assumed to have a constant potential, and current was allowed to flow through the coating under the assumption that the coating is a highresistance ionic conductor. A boundary element technique coupled with Newton-Raphson iteration was used to solve the governing equations for two-(2-D) and three-dimensional (3-D) configurations. Results showed good agreement with experimental values and can be used to assess viability of CP designs. KEY WORDS: anodes, boundary elements, cathodic protection, coatings, films and film formation, holidays, modeling, pipelines and line pipe steel, polarization curves 0010-9312/97/000063/$5.00+$0.50/0 © 1997, NACE International FIGURE 8. CP current delivered to a coated pipe with a 5.67-cm (2.25-in.)-diam circular holiday calculated by a 2-D BEM model. Polarization resistance is negligible compared to the ohmic resistance at high electrolyte resistivities.

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