Application of BEM to Design of the Impressed Current

377

Application of BEM to Design of the Impressed

Current Cathodic Protection Systemfor Ship Hull

by Mitsumasa Iwata*, Member, Yi Huang*, Member

Yukio Fujimoto*, Member

Summary

With the aid of the boundary element method, this paper presents an efficient design and evaluation

tool of the impressed-current cathodic protection (ICCP) system for a ship hull. Desirable working state of ICCP system is obtained by the control of anode current, monitoring the potential of reference electrode. Converting impressed-current anodes to concentrated internal point sources, the Poisson's equation becomes available to express electric behavior in cell. In the present approach, not only

potential and current densities on boundary but also the electric current from anode points and the set potential of the reference electrode can be calculated. The proposed methodology is applied to a practical container ship protected by an ICCP system. Based on the results of analysis, rational design of ICCP system for ship hull becomes possible.

1. Introduction

In order to prevent or delay the corrosion failure of

steel structures immersed in sea water, two well-known

cathodic protection (CP) techniques have been widely

used : the sacrificial anode cathodic protection (SACP)

system and the impressed-current cathodic protection

(ICCP)system. In SACP system, a lot of sacrificial

anodes attached on the ship hull surface, especially

around the stern, has a bad influence on the propulsion

of a ship and has to be exchanged within a few years.

Therefore, the trend to apply ICCP system is seen late-

ly, in which a few number of anodes are enough for the

protection. Moreover, the control of ICCP system to keep complete protection is relatively easy.

The electric field around steel structures protected by

these systems can be mathematically analyzed by FEM

or BEM. From the results of analysis, it becomes pos-

sible to design adequate CP system, including the num-

ber and location of anodes and reference points, capac-

ity of a power supply and etc.. Generally speaking, the

numerical analysis methods are more efficient, reliable

and economical tools in comparison with the empirical

approaches.

N. G. Zamanivanalysed electric field around a model

ship with ICCP system by BEM based on Laplace's

Equation. The authors2),3) developed the BEM for the

so-called partial-infinite field problem and succeeded to

evaluate and optimize the SACP system of ships using

the mesh system of only a half of hull surface. In these

calculations, however, mesh systems had to be renewed,

when the locations and the number of anodes were

changed.

In this paper, anodes are converted to concentrated

internal point sources distributed in the electrolyte

domain. On the basis of this conversion, Poisson's equa-

tion becomes available for expression of the field of a

electric cell. Thereupon, the electric field analysis

method based on Poisson's equation is formulated

through BEM. In this method, anodes and reference

electrodes in ICCP system can be put at any desirable

positions without changing the mesh system. The new method is applied to the analysis of a container ship

with ICCP system.

In the design of an ICCP system of a ship hull, it is

often the case that the location and the number of

anodes and reference electrodes controlling electric

current are restrained by the arrangement of ship struc-

ture. Therefore, the adequate protection must be real-

ized by controlling the electric current supplied from

anodes under the condition that the position of anodes

and reference electrodes are given. In the numerical

example, the potential levels of the control points and

the electric currents supplied from anodes are deter-

mined for a container ship under various conditions.

2. Potential Analysis Method Based on Poisson's Equation

2. 1 Basic Equation When point sources generating current with density

Qs are distributed in the homogeneous electrolyte, com-bining the principle of conservation charge and Ohm's law results in the following Poisson's equation4).

* Departement of Naval Architecture and Ocean

Engineering, Hiroshima University.

Received 10th JAN 1992 Read at the Spring meeting 12th, 13th MAY 1992

378 Journal of The Society of Naval Architects of Japan, Vol. 171

( 1 )

where 0 is the electric potential in the electrolyte field,

and p the specific resistance of the electrolyte.

In modeling of electrochemical problems, it is gener-

ally assumed that the electrolyte domain Q is enclosed

by three boundaries : the boundary S, on which electric

potential (ƒÓ) is given as the known value cbo, S2 on which

current density (q) is given as the known value q0 and

Sac the electrode surfaces. Associated boundary condi-

tions are described by the following equation.

( 2 )

where n is the unit outward normal of S2 and fac(ƒÓ) is

the function (always non-linear) obtained from experi-

ment as the polarization characteristics of electrodes. In

the calculation model of an ICCP system, it should be

noted that the furface of the impressed-current anode

must not be included in the domain boundary (S= S1

+ S2+ Sac), since it is converted to the internal point

source Qs

Through the weighted residual theorem and Green's

second identity, the boundary value problem described

by Eq. ( 1 ) and Eq. ( 2 ) is converted to the equivalent

boundary integral equation5).

( 3 )

where P and Q denote the source point and field point,

respectively, Op is the potential of the source point, and

cp is the shape coefficient at the source point (P) . cp

depends upon the geometric shape around the source

point (P) and is calculated through the so-called rigid-

body movement method'. In the derivation of Eq. ( 3 ) ,

the fundamental solution, ƒÓ*(P, Q), of Eq. ( 1 ) and its

derivative, q*(P, Q)=(1/ƒÏ)(•ÝƒÓ/•Ýn), about the outward

normal ( n) of the boundary ( S) were taken as the weight-

ed function of the weighted residual theorem.

In the problem of a ship hull floating in sea water, the

electrical potential ƒÓ is defined with reference to the

supposed exterior boundary infinitely faraway from the

structure-electrolyte interface. In the field of electro-

chemistry, however, the electropotential is always

defined on the basis of the potential of a reference

electrode such as the AgCl electrode. For this reason,

another unknown variable ƒÓref has to be introduced into

the formulation2).

Hence, when the non-linear polarization characteris-

tics of electrodes are approximated by the connected

several straight lines, the following equation holds in

each piece of the multi-lines.

( 4 )where q denotes the current density on the electrode

surface, Rp the polarizing resistance of the electrode

surface, ƒÓ0 the electropotential with reference to AgCl

electrode where the current density is equal to zero.

Substituting qs as current density on the anode sur-

face denoted by Sk(= 1,•c, K), where K is the number

of impressed-current anodes, and Qs =0 at any other

points in 9, Eq. ( 3 ) becomes

( 5 )

The function .95*(P, Q)can be approximated to a con-

stant on the anode surface Sk, since Sk is relatively

small. When ƒÓ*k at the central point of Sk is used as the

representive value of ƒÓ*(P, Q), the boundary integral

equation takes the following form,

( 6 )

where Ik =•çsk qs ds is the electric current supplied from k

-th anode.

The above approximations for the impressed-current

anode are made based on the assumption that the anode

size is relatively small. Fig. 1 illustrates the procedure

of the approximation. ( a ) shows the real case of the

anode, in which current is supplied from one surface of

the anode plate ; ( b ) is an approximated case of the

anode, of which the the area is half of ( a ) and current

is supplied from both surfaces ; ( c ) shows that the

anode is taken as one point at the center of the real

anode ( a ). The first approximation from ( a ) to ( b)

makes the integral of the kernel (q*(P, Q)ƒÓ) over the

anode surface Sk disappear, and the second approxima-

tion from ( b ) to ( c ) leads to a simplified expression of

the boundary integral equation as shown in Eq. ( 6 ) .

This process does not affect the electric current supplied

from the impressed-current anode.

In an ICCP system, the current equilibrium principle

described by Eq. ( 7 )should be satisfied,

( 7 )

that is,

( 8 )

2. 2 Numerical method

The protected state of an ICCP system can be

clarified by the Eqs. ( 6 ) and ( 8 ). In the numerical

approach to solve this problem, the structure-electrolyte

interface S is divided into a finite number of triangular

Fig. 1 Approximations of impressed-current anode

Application of BEM to Design of the Impressed Current Cathodic Protection System for Ship Hull 379

and/or rectangular elements, on which the distribution of potential and current density take the linear and/or bilinear form2). Discretized expression of Eqs. ( 6 ) and ( 8) is as follows,

( 9 )

in which, the coefficient matrix Hij and Gij are calcu-

lated by the formulas developed in Ref. 2), while ƒÓ*ik is

determined from the fundamental solution ƒÓ*(P, Q)

relying on the position of node point i and anode k.

Utilizing the polarization characteristics defined by

Eq. ( 4 ) , and replacing the elements of {} related to

the control point by the electric current vector (Ik), Eq.

( 9 ) can be rearranged into a standard and closed sys-

tem. The unknown vector of the system includes the

potential ƒÓ, ƒÓref and the electric current {Ik}. However,

the potential at the control point is eliminated. Eq. ( 9 )

is solved by an iteration approach named as the piece-

wise quasi-linear method6) such that the final solution

satisfies the non-linear polarization characteristics.

With the aid of the numerical method developed

herein, the desirable working state of an ICCP system

for ship hull surface will be achieved under several

conditions through the iteration algorism as shown in

Fig. 2.

2.3 Verification of the present method

The accuracy of the present method is investigated

through a simple ICCP system. Fig. 3 is a thin plate

moving on the surface of sea water with the speed of 23

kt. The surface of the plate is coated by Tar-epoxy

paint and it is in the state of Fig. 3. The specific resis-

tance of sea water is 0.250ƒ¶ •Em. The polarization

characteristics shown in Fig. 7 ( b), which will be discus-

sed in next chapter, is used for this model. On the

plate-seawater interface, impressed-current anode is

arranged near one corner. The 6m •~ 6m region around

the anode is an insulated wall. Through the analysis by

the proposed method, it is made clear that electric

current with the value of 13.5 A is necessary to main-

tain the protection level of -800 m V for this system.

This model is also analysed by the same method

developed by N. G. Zamani", changing the anode size to

0.3 m •~ 0.3m, 0.3m •~ 0.6m and 1.0 m •~ 1.0 m respectively.

As the current density on the anode surface, the value of

(13.5A/Anode area) is given for each condition.

The potential values were calculated at the ten points

shown in Fig. 3 by the present and Zamani's methods.

Table 1 compares the results. A good agreement of the

potential can be seen between both methods. However,

a little difference is seen at the two points near the

anode. In practical engineering, this region is always

coated perfectly to be an insulated wall. Therefore, the

present method can be expected to be applicable to the

design and evaluation of ICCP system in practical

engineering.

Fig. 2 Flow chart for design procedure

Fig. 3 Thin plate model used for verification

Table 1 Comparison of calculated potential values

(mV) between two methods (Current from

anode is 13.5A)


3. Numerical example

In order to demonstrate the applicability of the

proposed method, a practical container ship protected by an ICCP system is taken up, and the capacity of

power sources of the system and potential values of control points required to protect the ship body surface were investigated under various conditions. When steel in sea water is polarized below -780m V vs.AgCl by a cathodic protection method, it is in a protected state. However, the potential value of -800m V vs. AgCl is empirically established as the target value required for complete protection.

3. 1 Condition of calculation ( 1 ) Principal dimensions and outline of the system

The principal dimensions of the container ship taken

up here are Gross Tonnage of 72,206t, width(B)of B ) of 32m,

fully loaded draught ( d ) of 13m, total length of 267m

and the length between vertical lines (Lpp) of 250m. The

wetted area of the hull surface in full and light condi-

tions are 12,500m2 and 8,800m2, respectively. This ship

has two bronze propellers and their total surface area

are 60.0m2.

The hull surface is coated by the Tar-epoxy paint

with 0.25mm thickness, and also protected by the ICCP

system outlined in Fig. 4. Three anodes, Anode( 1),

Anode( 2 ) and Anode( 3 ) , are mounted on each side of

ship hull. In order to relieve the influence of overprotec-

tion, the area of 7.2m •~ 6.0m around each anode on the

hull surface is more perfectly painted as shown in Fig.

5. The perfectly painted part is regarded as the insulat-

ed wall in this calculation. Three reference electrodes,

R. E. ( 1 ) , R. E. ( 2 ) and R. E. ( 3 ), are mounted on ship

body. R. E. ( 1 ) controls the total current of Anode( 1)

and Anode( 2), and R. E. ( 3 ) controls that of Anode(

3). The fundamental solutions ƒÓ*(P, Q)and q*(P, Q)

which satisfy the conditions of the symmetry plan of the

ship and sea surface boundary were developed in Ref.

2) . Therefore, only half of the hull surface under water

line was discretized and analysed. The mesh systems for

both full and light load conditions are shown in Fig. 6.

( 2 ) Polarization characteristics

In numerical analysis of cathodic protection prob-

lems, the influence of electrode speed moving in electro-

lyte is taken in consideration through selection of the

suitable polarization curve. Fig. 7 shows the polariza-

tion characteristics used in these calculations. In this

figure, ( a ) shows the polarization characteristics of

rotating bronze propeller, and ( C ) shows the polariza-

tion curve of the bare steel at the speed of 23kt which

will be used in the case that a part of paint coatings on

Fig. 4 Outline of ICCP system for container ship

Fig. 5 Painting procedure around anode


(a) Bronze propeller (b) Painted steel (c) Bare steel

the hull surface is stripped. These two polarization

curves were estimated from Ref. 7).

Because the real polarization curve of ship hull cov-

ered with Tar-epoxy paint and moving at the speed of

23kt was not found, it was estimated from the test

results of the container ship described above. Under the

conditions shown in Table 2, following items were

measured, the total electric current supplied from two

anodes, the electric current applied into a propeller and

the electric potential values at three reference elec-

trodes. The polarization characteristics which satisfies

the measured data can be estimated through an iter-

ative backward approach. As the result, the polariza-tion characteristics shown in Fig. 7 ( b ) was obtained for the Tar-epoxy painted ship hull at the speed of 23kt.

The calculation was carried out for the following two operating states. ( 1 ) Electric current is supplied from a single anode,

Anode ( 1). ( 2 ) Electric current is supplied from two anodes,

Anode ( 1 ) and Anode ( 2). In the case ( 2 ) , the electric current ratio of two anodes was assumed.

The electric behavior calculated for the above two states are shown in Figs. 8 and 9. The both figures show that the calculated results can be almost adjusted to all of the measured values. So that, the polarization char-acteristics shown in Fig. 7( b ) can be regarded as the

proper one for this container ship at the speed of 23kt. It is seen from Fig. 9 that the electric current of Anode

Fig. 6 Mesh system, position of anodes and control

points

Fig. 7 Polarization characteristics used in calculation

Table 2 Test condition


( 2 ) is considerably lower than that of Anode ( 1 ). This phenomenon should be remarked in the design of anodes capacity.

The polarization curve of the bare steel shown in Fig. 7( c ) will be used in the case that a part of paint coat-ings on the hull surface is stripped.

3. 2 Results of calculation ( 1 ) Influence of cargo weight

The electric behavior around this container ship was measured only for the light load condition. For design of ICCP system, however, that in full load condition is required. So, the analysis was carried out for full load ship under the same condition as shown in Fig. 8. The result is shown in Fig. 10. The potential distribution in full load condition is almost same as that in the light load condition, while the electric current into the surface of full load ship is larger than that into the light load ship. This change of electric current is correspond-ing to the increment of the immersed area of hull surface.

( 2 ) Effect of insulated propeller When the propellers are rotaing, they are sometimes

insulated from hull through oil film between a shaft and a bush of propeller, and then the burden of ICCP system will be lightened. In this condition, the current is sup-

plied only from Anode ( 1 ). Electric fields around the ship was calculated under full and light load conditions. The results of calculation are shown in Fig. 11 and Fig. 12. Because the whole surface of hull is in a completely

protected state, it can be concluded that the control potential of -850mV vs. AgCl given to R. E. ( 1 ) is a suitable set point. The electric currents supplied into hull surface shown in Figs. 11 and 12 are almost agree with those into hulls of the full and light ship with earthed propeller shown in Figs. 8 and 10, respectively. ( 3 ) Influence of a stripped part of paint coatings

The ICCP system was investigated under the condi-tion that paint coatings were partially stripped near the bow of the ship. It is assumed that the immersed area of the paint stripped part is 1,226m2 and 684m2 for full and light load conditions, respectively, and this corresponds

Fig. 8 Comparison of measured and calculated electric

behavior (Anode( 1) is working)

Fig. 9 Comparison of measured and calculated electric

behavior (Anode ( 1 ) and Anode ( 2 ) are work-

ing)

Fig. 10 Influence of cargo weight on electric behavior


to about 10% of the whole wetted area of hull surface.

The analysis was carried out under the conditions that

electric current are supplied from Anode ( 1 ) and Anode

( 3 ) and the propellers are earthed to the ship body. The

assumed conditions are considered to be very severe for

the ICCP system.

The longitudinal potential profiles and the electric

current are shown in Fig. 13 and Fig. 14. As a detail

expression, the potential contours on the hull surface

are shown in Fig. 15. The calculated potential on whole

surface of the hull is 10mV lower than the critical

value, -780mV vs.AgCl, necessary for corrosion protec-

tion of steel. Moreover, the potential is maintained

lower that the target value, -800mV vs. AgCl, for the

whole surface of each hull, except for a small region

near the bow in the full load condition. It becomes clear

from this result that the set potential of -850m V vs.

AgCl for R. E. ( 1 ) and -820m V vs. AgCl for R. E. ( 3 )

Fig. 11 Influence of insulated propeller on electric

behavior in light load condition

Fig. 12 Influence of insulated propeller on electric

behavior in full load condition

Fig. 13 Influence of stripped part of coatings on elec-

tric behavior in light load condition

Fig. 14 Influence of stripped part of coatings on elec-

tric behavior in full load condition


are suitable for the complete protection.

The electric current from Anode ( 3 ) in the full load

condition becomes quite high owing to the stripped part,

but it is below the capacity of the mounted power

supply. Therefore, it is concluded that the capacity of

this ICCP system is appropriate for the container ship.

4. Conclusions

For the purpose of the rational design of the impressed-current cathodic protection (ICCP) system, the impressed-current anode is approximated to a concentrated internal point source, and an efficient numeri-cal procedure by BEM has been developed from Poisson's Equation. Moreover, the proposed method has been applied to a container ship with ICCP system. From the results of calculations, it is made clear that the potential distributions on whole surface of ship hull can be estimated and the capacity of ICCP system and set points of reference electrodes required for the com-

plete protection of ship hull can be predicted accurately under various conditions.

Acknowledgement

The authors wish to appreciate Mr. S. Tamari, and

Mr. I. Noguchi, The Nippon Corrosion Engineering Co.,

Ltd., for their help in offering the indispensable data for

calculations by the authors and many helpful recom-

mendations.

References

1) Zamani, N. G.: Boundary Element Simulation of the Cathodic Protection System in a Prototype Ship, Applied Mathematics and Computation, Vol. 26, Noumber 2, (1988), (Reprint).

2) Huang, Y., Iwata, M., Jin, Z. L.: Numerical Analysis of Electropotential Distribution on Sur- face of Marine Structure under Cathodic Protec- tion (Application of Three Dimensional BEM) , J. of SNAJ, Vol. 168, (1990), pp. 589-596.

3) Iwata, M., Fujimoto, Y., Huang, Y., Ogawa, K., Matsushita, K. : Numerical Analysis of Electropotential Distribution on Surface of Ship Body under Cathodic Protection, J. of West- Japan Society of Naval Architects, No. 82, (1991) , pp. 279-286.

4) Helle, H. P. E.: The Electrochemical Potential Distribution around Ships, Trans. RINA, Vol. 123, (1981) , pp. 253-263.

5) Brebbia, C. A.: The Boundary Element Method for Engineering, Pentech Press, London, (1978).

6) Nagai, K., Iwata, M. and Ogawa, K.: Numerical Analysis of Potential Distribution in Electrolyte under Cathodic Protection (Part 2 Application of 3-D FEM), J. of the West-Japan Society of Naval Architects, p. 114, (Mar. 1987).

7) Investigation on the Protection of Corrosion of Ship ; The 50th Research Committee, The Report of the Shipbuilding Research Association of Japan, No. 46, p. 10, (1964).

Fig. 15 Potential contours on hull surface

Documents

Application of BEM to Design of the Impressed Current