INTRODUCTION
Determination of net anode resistance is an important component in design of cathodic protection systems, both for marine systems and elsewhere. Approaches that have evolved historically include, 1) empirical curve fitting of test data,1,2 2) derivation of closed form expressions from potential field theory,3,4 & 3) numerical solution of the Laplace equation.5,6 For the ideal case of a single spherical anode in an infinite homogeneous electrolyte of resistivity ñe, an exact solution exists as, where Rra?¨r2 is resistance between the anode surface (sphere radius ra) to a distance in the electrolyte from the sphere center r2. When r2 >> ra, Equation 1 reduces to, For geometries that are more practical from an engineering perspective, simplifying assumptions are necessary in the derivation. Thus, Sunde derived an expression for resistance, Rl, of a single cylinder of radius r and length l as, When l >> r, as is the case for typical standoff galvanic anodes on space-frame structures, this reduces to, which was derived earlier by Dwight, although the modified Dwight equation, .. is more commonly employed in practice. For anodes where r ¡Ö l and non-cylindrical geometries, an equation of the form A ñ K R e · =, (6 has been employed, where K is a coefficient that is a function of l/r and A is anode surface area. With K = 0.315, this expression is known as McCoy?s formula,2 which is widely used to calculate resistance of bracelet anodes on marine pipelines, although it has been reported to be conservative by 26-134 percent.7 , 8 Nisancioglu9 has provided a critical review of the various resistance formulas with discussion of their utility and applicability. Numerical modeling approaches to anode resistance determination has involved Finite Difference Method (FDM), Finite Element Analysis (FEA), and Boundary Element Modeling (BEM). Because of cost and training factors, these methodologies have mostly been utilized in association with unusual and complex geometries.10
Cathodic protection designs involving multiple anodes often require that resistance of an assemblage or array be determined. If the anode are identical and their potential fields are non-overlapping, as is likely to be the case for space frame structures and pipelines with distributed galvanic anodes, then the anodes can be treated as resistors in parallel; and, N R R a
T =, (7 where RT is net resistance, Ra is resistance of an individual anode, and N is the number of anodes.11,12 In other situations, such as retrofits, anodes may be clustered with overlapping fields such that Equation 7 is overly simplistic. Based upon potential field theory, Sunde developed analytical expressions for electrode arrays of simple, periodic geometries.4 Thus, for a one dimensional array of identical, parallel electrodes equally spaced at a distance S, ( ) ( ) ( ) ( )
where Rl, the resistance of a single anode, is given by Equation 3. Likewise, for electrodes equally spaced about the perimeter of a circle of diameter D, While this expression seems of limited applicability for marine cp because of its limitation to circular arrays, Pierson and Hartt13 showed that resistance of a M×N array of conductors, subject to the limitation that N=2a, where a=1,2,3,?, can be calculated from it according to the following sequence:
1. Divide the array into subgroups of four conductors each,
2. Calculate the resistance of each subgroup using Equation 9,
3. Replace the four conductors of each subgroup with a centrally positioned single electrode of equivalent
resistance, and
4. Reapply Equation 9 to the new reduced-N array. The procedure is repeated until only a single conductor remains with diameter of the equivalent electrodes in
