A previously developed first principles based equation for potential attenuation along marine pipelines and risers with multiple, equally spaced, identical galvanic anodes was modified to accommodate an error in its derivation. Both the original and modified equations account for the electrolyte (anode), coating, polarization, and metallic path resistances and, as such, represent improvements over previously available representations. Coordinate Mapping Based Finite Difference Method (CoMB-FDM) solutions are presented for different pipe and cathodic protection conditions; and to the extent possible, the accuracy of these is verified.
External corrosion of marine pipelines and risers is normally controlled using a combination of galvanic anode cathodic protection (cp) and coatings, where the purpose of the latter is to render the former more practical and cost effective. The cp protection criterion for these systems invariably is based upon maintaining a cathode potential of -0.80 VAg/AgCl or more negative at the mid-anode spacing. However, the one-dimensional nature of pipelines renders potential modeling and cp design more complex than for space-frame structures. Recently, Pierson et al.1 derived a first principles based potential attenuation equation for pipelines protected by multiple, identical, spherical, equally spaced galvanic anodes that incorporates the electrolyte (anode), coating, polarization, and metallic path resistance terms. The equation was considered an improvement over previously available analytical expressions,2,3 because these disregard the electrolyte resistance term and over Boundary Element Modeling (BEM) which does not include metallic path resistance. The Pierson et al. governing equation is, ( ) ( ) ( ) z U z U z E '
Where Ec(z) is the polarized pipe potential at distance z from an anode and
Um(z) and Ue(z) are potentials in the metallic pipe and electrolyte, respectively, at z.
Steady-state was assumed, and Ec(z) was considered to be a linear function of cathodic current density, ic(z), according to,
where á is the polarization resistance for the cathodic process (normally O2 reduction) and ã is a coating quality factor (ratio of total-to-bare surface area). Figure 1 graphically illustrates the polarization behavior projected by Equation 2. As in the work of Morgan2 and Uhlig,3 ) z ( U"m was expressed as,
where Rm is pipe resistance per unit length and rp is the pipe outer radius.
In evaluating the electrolyte resistance term, Pierson et al. expressed the potential difference between the anode and the cathode, ( ) z Ue . , as: ( ) ( ) ( ) z R z I z U e a e · = . , (4)
where Ia(z) is the current remaining in the electrolyte at z, ( ) ( ) ?ç · · · · · ?ß
L z c p a dt t i r z I ð 2 2 , (5) and ( ) z Re the resistance between the anode and the cathode at z (L in Equation 5 is the half anode spacing).
From Equations 1-5 an expression for potential attenuation along a pipeline was derived as?.