Optimum design of cathodic protection using the Boundary

Optimum design of cathodic protection usingthe Boundary Element Method

K. Amaya1 & S. Aoki21 Tokyo Institute of Technology, Ookayama, Meguro, Tokyo, Japan2 Toyo University, Kujirai, Kawagoe, Saitama, Japan

Abstract

Optimization of cathodic protection systems for a pipeline and a ship was per-formed by using the boundary element method (BEM). For a pipeline, the locationand impressed current of electrodes in a cathodic protection system were opti-mized by minimizing the electric power necessary to keep the potential on themetal surface below a critical value. The non-uniformity of soil conductivity andthe resistance of a long pipeline were taken into account. For a ship, an inverseproblem to estimate the potential distribution on the whole surface of the hull fromthe electric potential data measured with several sensors was solved at first. Then,the optimum current to be impressed to each electrode located on the hull wasdetermined by using the results of the inverse analysis.

1 Introduction

Protecting structures, such as a pipeline and a ship, from corrosion is one of themost important problems in engineering [1]. To prevent corrosion of a pipeline, itsouter surface is painted and also cathodic protection is performed by impressingcurrent into surrounding soil from a finite number of electrodes. It is necessary tooptimize the location and current to be impressed to each electrode.

Application of the boundary element method to the optimization has been stud-ied intensively [2]-[6]. Special consideration is necessary for a pipeline. becauseit is so long that the non-uniformity of the electric conductivity of soil must betaken into account, and also the electric resistance of the pipeline can not beneglected [7, 8].

The cathodic protection is performed also for a ship by impressing current intosurrounding sea from several electrodes on its hull. Some sensors for monitoring

© 2005 WIT Press WIT Transactions on Engineering Sciences, Vol 48, www.witpress.com, ISSN 1743-3533 (on-line)

Simulation of Electrochemical Processes 25

Figure 1: Impressed current CP for ship.

potentials are located at several points on the hull. During navigation, it is neces-sary to estimate the potential on the whole surface of hull from the potential datameasured by a small number of sensors and determine the optimal current to beimpressed to each electrode [8, 9].

In this paper, after reviewing the electrochemical aspects of corrosion, a math-ematical model, boundary element formulation, solution methods are presentedwith a few example problems [7]-[9].

2 Electrochemical background

We consider a typical metal M in an electrolyte. Both anodic and cathodic reac-tions occur simultaneously on the metal surface according to:

Anodic reaction :

M → Mn+ + ne− (1)

Cathodic reaction:

12

O2 + H2O + 2e− → 2OH− (2)

Due to the reactions an electric current flow occurs. The electric current densityi versus electrical potential to a reference electrode (e.g., saturated calomel elec-trode) E curve is called a polarization curve. The relationships between the currentdensity i and the potential E for anodic and cathodic reactions are not obtainedindividually but nominal relationship E = f(i) ,i.e., the algebraic sum of the cur-rents for the two reactions versus E curve is measured.

In the natural state, the reactions become in equilibrium, and anodic currentflows from the anode to the cathode. Corrosion rate is proportional to this anodiccurrent density. It is possible to supress the anodic current density by impressingcurrent from external power supply and reducing the potential of the metal to thecritical value Ep. This method is called cathodic protection.


26 Simulation of Electrochemical Processes

3 Ship

It is necessary to estimate the potential on the whole surface of hull from the poten-tial data measured by some sensors on the hull, and determine the optimal currentsto be impressed to each electrode.

3.1 Basic equations

Let us assume that the region occupied with sea water, Ω, is surrounded by thesurface of the hull, Γm, the surfaces of electrodes, Γe (e = 1, 2, . . . , N ; N = totalnumber of electrodes), the surface of sea water, Γn and infinite boundary, Γ∞ asshown in figure 1. The potential, φ, satisfies the following Laplace equation in Ω;

∇2φ = 0 in Ω (3)

and is given at the location of sensors, xs (s = 1, 2, . . . , M ; M = total numberof sensors);

φ = φs at xs (4)

where xs is the location of a sensor and φs is the (known) value of potential mea-sured by the sensor s. The φ is defined as φ = −E, where E is the potential withrespect to some reference electrode, e.g., SCE.

The current density, i (≡ κ∂φ/∂n; κ = conductivity of sea water, ∂/∂n =outward normal derivative) satisfies the following equations;

i = 0 on Γn (5)

i = 0 on Γ∞ (6)

i = ie on Γe (7)

where ie (e = 1, 2, . . . , N ; N = total number of electrodes) is the current densityon the electrode e. Let Q represent a vector, the components of which are givenwith ie.

3.2 Corrosion rate identification

Let us consider an inverse problem where the potential distribution on the surfaceof hull is estimated from the potential data measured by several sensors. We willuse the boundary element method. If we chose the nodal values of potential ofeach element as the unknown parameters, the number of potential data would betoo small compared with that of unknown parameters. Even if the number of datacould be increased enough to solve the inverse problem, the solution would beabnormally oscillated due to the ill-posedness [11].



Hence, let us make use of the a-priori information that φ and i on Γm must sat-isfy the polarization characteristics. At first, we will approximate the polarizationcurve by a function, e.g., Tafel’s expression [1];

−φ = fm(i; αj) on Γm (8)

where αj (j = 1, 2, . . . , L; L = total number of parameters) is a parameter in thefunction. Then, the several parameters in the function are estimated by an inverseanalysis. After obtaining the polarization characteristics, the potential distributionis easily calculated by a direct analysis.

Following the usual boundary element formulation, we obtain the followingequations from (3) [3];

κ[H ]

φ∞φe

−fm(im; αj)

− [G]

0ie

im

= 0 (9)

By assuming a set of initial values of the unknown parameters, αj , we solve adirect problem, i.e., solve (9), at first and then minimize the following cost func-tion;

cost(αj) =M∑

s=1

(φs(est) − φs)2 (10)

where φs(est) is estimated value of potential at the location of sensor s. Calcula-tions are repeated by modifying the parameters employing a minimizing technique,e.g., the conjugate gradient method, until the value of cost function, cost(αj),becomes less than the measuring error, ε.

Even if the above procedure is followed, the solution for αj sometimes dependson the assumed initial values. By varying initial values widely, we can obtain thepossible solution area, Π. In order to reduce the area, Π, we use the potential dataobtained under different sets of ie (impressed current density on each electrode)as well as the original data. Furthermore, we utilize fuzzy a-priori informationexpressed as membership functions [1]. For example, a-priori information are suchthat if the hull is made of a low alloy steel, then the parameters of polarizationcharacteristics curve must be within some range. Let Fi(i = 1, · · ·n; n = thenumber of a-priori information) denote the fuzzy set expressed by the membershipfunction. The solution can be obtained as the center of gravity of a set, Π ∩ F1 ∩F2 · · · ∩ Fn.

3.3 Optimization of impressed currents

The power necessary for cathodic protection is give by the following equation;

P (Q) =N∑

e=1

∫Γe

φe(Q) + fe(ie(Q))ie(Q)dΓ (11)



where the polarization characteristics of the electrode e is given by a known func-tion, −φ = fe(i), and the components of a vector Q are ie (e = 1, 2, · · · , N). It isnecessary to satisfy the following protection condition on every part of Γm;

φ ≥ φp (i.e., E ≤ Ep) on Γm (12)

where Ep(= −φp) is the critical value of potential for protection.The problem is to find the vector Q which minimizes P in (11) under the condi-

tion (12). It is, however, necessary for φ and i in (11) and (12) to satisfy (9), wherethe αj has already been obtained in the last section. Since αj changes during nav-igation, it is necessary to determine effectively the optimal current impressed toeach electrode, corresponding to the change of αj .

To do this, we combine (11) and (12) by defining a new cost function, P ∗(Q) [8];

P ∗(Q) = P (Q) + ν

K∑k=1

h2k(Q) (13)

Here, ν is a penalty number (large positive number) and the function hk is definedon Γm as

hk = (−φk + φp) u (−φk + φp) (k = 1, 2, . . . , K) (14)

where u(·) is the unit step function, φk is the potential at the nodal point k and K isthe total number of nodes on Γm. If the protection condition (12) is not satisfied,P ∗(Q) takes a large value.

Thus, it becomes possible to determine effectively the optimal currents impressedto each electrode by minimize P ∗(Q). The procedure is as follows;

1. Assume Q.2. Solve (9) for φ and i on Γe by using the polarization characteristics obtained

in the last section.3. Evaluate (13) and modify Q using a minimizing technique, e.g., the conju-

gate gradient method.4. Return to 2. and repeat the procedure until convergence is reached.

3.4 Numerical example

We consider a ship which has six electrodes and six sensors as shown in fig-ure 3. At first, a direct analysis was performed for the case where the currentdensities impressed to the electrodes on the left side hull were 0.04, 0.04 and0.4 [A/m2] (from bow to stern), and those on the right side hull were 0.01, 0.01and 0.01 [A/m2] (from bow to stern). It was assumed that the hull was made ofpainted low alloy steel plates and their polarization curves are given by

φ = −0.600 sinh−1(200i) + 0.650 (15)

where the units of φ and i are [V] and [A/m2], respectively.



Figure 2: Fuzzy a-priori information about polarization curve.

140m

11m

Electrode Sensor

746 elements

Figure 3: Locations of electrodes and sensors.

0.765 0.767 0.769 0.771 0.773 0.775 0.777 0.779 0.781 0.783 (V) 0.767 0.770 0.773 0.776 0.779 0.782 0.785 0.788 0.791 0.794 (V)

Figure 4: Calculated potential distri-bution on ship hull.

Figure 5: Potential distribution afteroptimization.

Direct analysis was performed by employing 764 constant elements. The resulton potential distribution is shown in Figure 4. The potential values at the locationof the sensors were rounded off to three significant figures to take account of themeasurement accuracy, and were used as the input data in the following inverseanalysis.

In the inverse analysis, the polarization curves were estimated at first. It wa sassumed that the polarization curve of the painted hull is represented in the fol-



lowing form;φ = −α1 sinh−1(α2i) + α3 (16)

where αj(j = 1, 2, 3) are the parameter to be estimated. We used a-priori infor-mation about polarization characteristics, the membership functions of which wereshown in figure 2.

After estimating these parameters, the potential distribution was calculated bya direct analysis using the estimated polarization characteristics. The estimatedpotential distribution agreed well with the exact solusion which was shown in Fig-ure 4. It is observed in Figure 4 that the protection condition (φ ≥ φp = 0.77V )is not satisfied on the white part of the hull. This is attributable to the impropervalues of impressed currents to the electrodes.

Optimization of the impressed currents was carried out by using the methoddescribed in the preceding chapter. The potential distribution due to the optimizedimpressed current is shown in Figure 5. It is found that the protection condition issatisfied everywhere on the hull.

4 Underground pipeline

4.1 Basic equations

The pipeline is long that the electrode conductivity of soil can not be assumed tobe uniform, but varies gradually as shown in Figure 6 The current density vectori(x) [A/m2] in soil is defined as:

i(x) = −κ(x)gradφ(x) (17)

where κ(x)[Ω−1m−1] is electrical conductivity at a point x in the domain Ω andφ(x) [V] is an electrical potential at x.

The electrical potential within the soil, φ(x), obeys the following governingequation:

−div κ(x)gradφ(x) =me∑i=1

Ieiδ(x − xe) (18)

where δ() the delta function, me the total number of electrodes, Iei the impressedcurrents and xe the location of the ith electrodes. It is assumed that the electrodesare so small compared with the size of Ω to be regarded as points.

The pipeline is so long that its internal resistance can not be neglected, andhence the potential in pipeline φ = m0 depends on location x. The polarizationcurve E = f(i) also depends on location x. Therefore the boundary condition isdescribed as follows:

−φ(x) − φm0(x) = f(i, x) on Γm (19)

The protective condition is written as:

−φ(x) − φm0(x) ≤ Ep(x) (20)



The optimal locations and impressed currents are to be determined in such away that the sum of the necessary power to achieve the complete protection and thecost to lay electrodes underground is minimized. We use the following objectivefunction:

P (Ie, xe) =me∑i=1

(∫Γei

φei + fei(iei)ieidΓ + kzezei

2

)(21)

where Ie, xe are the vectors, the components of which are Iei and xei, respec-tively. The Γei is the boundary which is radius of ε distant from electrode ei (εis very small compared with the size of Γ), φei , iei and fei(iei) are the poten-tial, the current density and the polarization curve of electrode ei respectively, kze

the coefficient of cost to lay the electrode underground, and ze is the depth of theelectrode.

The problem considered here is to find the locations and the impressed currentsof electrode that minimize (21) under the conditions (19) and (20).

4.2 Resistance of pipeline

We discretize the pipeline to n nodes. Because of the preservation of the electricalcharge, in a jth element of pipeline, we assume the following equation :

im0jSp − im0j+1Sp = Sjij + ij+1

2+ cj(Iek

) (22)

where Sp is the cross section area of pipeline, Sj is the longitudinal area of the jthelement, ij and im0j are the radial and the axial current density of the jth noderespectively. The function cj(Iek

)(k = 1 ∼ me) is defined as the following:

cj(Iek) =

Iek, if the electrode ek connects

the jth element

0, if the electrode ek does not connect

the jth element

(23)

The relations between the radial current density and the axial current density ofwhole elements are expressed as the following combined form:

[A]im0 = [B]i + Ie (24)

where i and im0 are the vector of the radial and the axial current densityrespectively, and Ie is the vector which consists of cj(Iek

).Following equation can be assumed by the Ohm’s law.

im0 = −1ρ

dφm0

dx(25)

where ρ [Ω · m] is the resistivity of the internal pipeline.



In each element of the pipeline, assuming the distribution of the axial currentdensity im0 is linear, the axial potential of the surface pipeline is expressed as:

φm0j = −ρ

∫ xj

xj−1

(x − xj−1

xj − xj−1im0j +

xj − x

xj − xj−1im0j−1

)dx + φm0j−1 (26)

where xj is the location of jth node. Accordingly,

φm0j − φm0j−1 = −ρ

2(xj − xj−1)(im0j + im0j−1) (27)

The base potential is the potential on the nth node.

φm0n = 0.0 (28)

The relations between the axial current density and the internal potential ofpipeline are expressed as:

[C]φm0 = [D]im0 (29)

where φm0 the vector of the internal potential of pipeline.We obtain the relation between the radial current density and the internal poten-

tial of pipeline by (24) and (29), as follows:

φm0 = [E]i + [F ]Ie (30)

4.3 Boundary element formulation

Let us assume the conductivity for the corrosion domain is κ(x) = Kx + K0

where K and K0 are constants and x is the x-component of x. The boundaryintegral equation is expressed as:

c(y)φ(y) =∫

Γ

φ∗(x, y)i(x) − i∗(x, y)φ(y)dΓ

+me∑i=1

Ieiφ∗(xe , y) (31)

where Γ is the boundary of the domain under consideration, x = (x, y, z)T is theposition vector of observation point, and y = (xs, ys, zs)T is the position vectorof source point.

In this case, the fundamental solution is given by the following equation [10]:

φ∗(x, y) =1

2πKr√

r2 + 4(x + K0/K)(xs + K0/K)(32)

i∗(x, y) is expressed as:

i∗(x, y) = κ(x)∂φ∗(x, y)

∂n(33)

where n is the component of the unit outward normal vector at y ∈ Γ and ∂∂n

denotes the partial derivative along the same normal vector.



Location x [m]

Counterelectrode

Soil

Air

Pipeline

Current Ω

Con

duct

ivity

[m

]

Ω-1

-1

Location x [m]

Con

duct

ivity

[m

]Ω-1

-1

0.02

0.01

Electrode

Pipeline

0 5000 10000 15000

Ω 1 Ω 2 Ω 3

Interface

Figure 6: Cathodic protection systemfor pipeline.

Figure 7: Example problem impressedcurrent CP for pipeline.

0 2500 5000 7500 10000 1250015000-5000-2500

02500

5000-5000

-4000

-3000-2000

-1000

0

Pipeline

y[m]

z[m

]

x[m]

Interface

DepthLengthRadius

Resistivity

2.0[m]15000[m]

0.3[m]1.4x10 [ m]-7 Ω

0.5

1

1.5

2

2.5

3

3.5

0 2500 5000 7500 10000 12500 15000

InitialOptimized

Protective potential

Pot

entia

lφ−

φ m0

[V]

x[m]

Figure 8: Boundary element mesh forunderground pipeline.

Fig u r e 9 : Po ten tial d istr ibu tio n on pipeline.

By applying the proposed method in the above section, we can optimize thelocation of electrodes and impressed current to each electrode.

4.4 Numerical example

An example is shown in Figure 7 in which a pipeline is laid in soil with non-uniform electric conductivity. Let x = (x, y, z)T . The design variables are thelocations xei, xei and the currentsIei of electrode (i = 1, 2, 3. i.e., N = 3), assum-ing the electrode locations yei = 0.0.

The whole domain is divided into three subdomains Ω1, Ω2, Ω3. The electricconductivities in these sub-domains are shown in Figure 7, and the boundary ele-ment mesh is shown in Figure 8.

The data in Table 1 was used for the initial design. The electric power was265.0[W]. Other input data should be referred to [7, 8].

Figure 9 shows the distribution of potential before and after optimization. Itis found that the pipeline was over-protected before optimization, while after theoptimization the whole pipeline is economically protected. Table 2 shows the opti-mization results. The electric power was reduced from 265.0 [W] to 83.1 [W].



Table 1: Initial design.

xe[m] ze[m] Ie[A]

1 2.5×103 -55.0 10.0

2 7.5×103 -55.0 10.0

3 12.5×103 -55.0 10.0

Table 2: Optimal design.

xe[m] ze[m] Ie[A]

1 2.88×103 -141.4 8.74

2 7.39×103 -107.5 5.00

3 13.36×103 -73.7 2.56

5 Conclusions

Optimization of the locations and impressed currents of electrodes in a cathodicprotection system for a long pipeline buried in soil with non-uniform electric con-ductivity was performed by minimizing the cost to lay the electrode undergroundplus the electric power necessary to keep the potential on the metal surface belowa critical value. In the optimization, the multi-region boundary element method(BEM) was applied, and the fundamental solution suitable to the linear change ofthe electric conductivity was used. The resistance of a long pipeline was also takeninto account.

An inverse problem accompanied with performing optimal cathodic protectionfor a ship was solved, i.e., the potential distribution on the hull was estimatedfrom the potential data at several fixed sensors. To remove the ill-posedness of thisinverse problem, the polarization characteristics of the painted hull were estimatedat first and then the potential distribution was obtained by solving a direct problemby BEM using the estimated polarization characteristics.

Furthermore, in the estimation of polarization characteristics fuzzy a-prioriinformation was used. By using this result and the above-mentioned optimizationmethod, the optimal current to be impressed from each electrode was determined.

References

[1] Fontana,M.G.& Greene,N.D., Corrosion Engineering, Tokyo, Japan:McGraw-Hill,1978.

[2] Strommen,R., Keim,W., Finnegan,J.& Mehdizadeh,P., Advances in offshorecathodic protection modeling using the Boundary Element Method, Corro-sion’86, Paper 45, (1986), p.45/1.

[3] Adey,R.A., Brebbia,C.A.& Niku,S.M., Application of Boundary ElementsIn Corrosion Engineering,in Topics in Boundary Element Research VII, ed.C.A.Brebbia Berlin, Germany: Springer-Verlag,pp.34-64, 1990.

[4] DeGiorgi,V.G., Kee,A., Lucas,K.E. & Thomas,E.D., Examination of Mod-eling Assumptions for Impressed Current Cathodic Protection , Proc. ofCORROSION 99 Research Topical Symposium, Cathodic Protection , USA,NACE International, pp.1-16, 1999.



[5] Riemer D.P. & Orazem,M.E., Models for Cathodic Protection of MultiplePipelines with Coating Holidays , Proc. of CORROSION 99 Research Topi-cal Symposium, Cathodic Protection , USA, NACE International, pp.66-82,1999.

[6] Deconinck,J., Bortels L., Nelissen,G. & Dam,A., Pynacker,A., Simmulationand Evaluation of a Cathodic Protection System of Buried PipelinesunderStray Current Influences , Proc. of CORROSION 99 Research Topical Sym-posium, Cathodic Protection, USA, NACE International, pp.83-89, 1999.

[7] Takazawa,H., Amaya,K. & Aoki,S., Optimum Design of Cathodic Protectionfor Long Pipelines, Proc. of Pressure Vessel and Piping 98, PVP.370, NewYork, USA, ASME, pp.63-69, 1998.

[8] Aoki,S., Amaya.K. & Miyasaka, M., Boundary Elemnet Analysis ofCathodic Protec tion for Complicated Structures, Proc. of CORROSION 99Research Topical Symposium, Cathodic Protection: Modeling and Experi-maent, USA, NACE International, pp.45-65, 1999.

[9] Aoki,S., Amaya,K. & Gouka,K., Optimal Cathodic Protection for Ship,Proc.of Boundary Elemen Technology XI, Southampton, England, ComputationalMechanics Pu blications, pp.345-356, 1996.

[10] Amaya,K., Aoki,S., Takahashi, T., Uragou, M. & Nishikawa, A., B.E. Anal-ysis of G alvanic Corrosion of Long Structure Buried in Soil with GradientElectrical Conductivity, Boundary Element 19, Southampton, England, Com-putational Mechanics Pub., pp.789-798, 1997.

[11] Aoki,S. & Amaya,K., Boundary Element Analysis of Inverse Problems inCorrosion Engineering, in Boundary Integral Formulations for Inverse Anal-ysis, ed. L.C.Wrobel and D.B.Ingham, London, England, ComputationalMechanics Publications, pp.151-169, 1997.



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Optimum design of cathodic protection using the Boundary