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Application of a new computational method to determine

corrosion current density in a cathodically protected

system

I.QAMAR

S. W.HUSAIN

In the prese nt pa per a nu me ric al ap proac h to de termin e the cor ros ion cur ren t de nsi ty of acathodically protected metal is explained. In this method three independent relationshipsare established and numerical evaluation of the unknowns is carried out by using electro-chemical data obtained very close to the protection potential . The results obtained usinghypothetical data are shown to be comparable with another method proposed by Meas andFujioka. The present method, however, has an advantage over this method in that no priorknowledge of the corrosion current density of the unprotected metal (I e) is required.

Be cau se of this res tric tion , it is sh ow n tha t the me tho d of Me as an d Fu jiok a is sen siti ve toslight inaccuracies in the value of Ie used. On the other hand, the present method implicitlycalculates the value of Ie.

Ma nu scr ipt rec eiv ed 9 July 1991; in final form 8 August 1991. T he a ut ho rs a re a t t h eDr A. Q. Khan Research Laboratories, PO Box 502, Rawalpindi, Pakistan.

Iep= Ie +B 1Pep (7)

Equation (6) has three unknowns I e, Bb and B2 In ourproposed method we formulate three independent equations \

where B=(bla +b1e)/blab1e' The total current density willnow be the sum of partial current densities given byequations (3) and (4). Using the definition of corrosioncurrent density Ie (i.e. Ie = 11 = 12 at E = : = Ee), the totalcurrent density can be written as

(6)

(5)

(3)

(4)

11 =2'3I01B(E- E1)

12=-102 exp(- 2'3(E - E2)/b 2c )

I=Ie+2'3I01BP -Ie exp(-2'3P/b 2e )

which can be rewritten as

I=Ie+B 1P-Ie exp(-B 2P)

where the following designations hold

P=E-Ee

B1 =2'3101B

B2 =23/b2c

It is to be noted that the value of the corrosion currentdensity Iep at cathodic protection potential Eep is the

partial current at Pep=Eep - Ee and is given by thefollowing equation

PROPOSED METHODGenerally three polarisation regimes can be defined, aspointed out by Meas and Fujioka,6 which allow us to

simplify equations (I) and (2):regime I: E is close to E 1regime 2: E is close to E2regime 3: E is away from E1 and E2 .

In order to illustrate the application of our proposedmethod, we have developed equations for regime Ionly.The method, however, is general and can be applied toother regimes as well.

For regime I, E - E 1 is very small and exponentials inequation (1) are approximated by the first term of the

series expansion (exp(y) = y) . Also, as E - E2 would be alarge negative quantity, the contribution of the firstexponential in equation (2) can be neglected. Theseapproximations lead to the following equations

INTRODUCTIONThe kinetics of an electrochemical corrosion process whereonly one anodic and one cathodic reaction take place isdescribed by the following equations

11 =1 01 {exp[2'3(E- Ed/b1a ] -exp[ -2' 3(E- E d/ b1e ]}

.. (l)

12 = 102 {exp[23(E - E2)/b 2a] - exp [- 2'3(E - E2)/b 2e ]}

. (2)

where 101 and 102 are the exchange current densities, E1and E2 are the equilibrium potentials and b1 and b2 are

the Tafel slopes for reactionsI

and 2 respectively. Thepartial current density 11 is known as the corrosion currentdensity and determines the rate at which the metal iscorroding at a potential E. The current density measuredexternally in the circuit is the ~um of the two partialcurrent densities and special techniques must be used todetermine the value of the partial current densities. Atequilibrium corrosion potential Ee, the two partial currentdensities are equal and thus the externally measured currentis zero. The corrosion current density Ie at Ee representsthe corrosion rate of a metal in the absence of any

polarisation. The determination of Ie has been dealt withextensively over the years and many methodsl-4 have

been developed. The present. authors have introduced anew computational methodS to calculate I e from electro-chemical data. In the present paper we apply our method

to determine corrosion current densityI

ep for a systemwhich is cathodically protected at a potential Eep Themethod is applied considering the restriction that acomplete electrochemical corrosion experiment is notfeasible for cathodically protected systems since it isundesirable to carry out measurements away from the

protection potential. The problem has also been addressedby Meas and Fujioka.6 Their method (referred to belowas the MF method), however, requires that Ie be known.Our method, on the other hand, does not require a priorknowledge of Ie.

In the following we explain our method and show itsutility with an example. The results of our method arecompared with those obtained from the MF method.Figure Irepresents graphically the various terms used inthis paper.

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126 Qamar and Husain Computation of corrosion current in cathodically protected system

log(Current Density)

Schematic representation of various terms used in electrode

kinetics: thick solid curve is experimentally determin'ed

polarisation curve; thin solid lines are Tafel lines; crosses

represent the three measurement points used to determine

corrosion current density at an applied protection potential Ecp

Eep

E 1

102 101 I ep Ie

B 1 from equation (8) using equations (11) and (9), the SS

function takes the form

SS = I(Ij-G(B 2 2 (12)

where G(B2) is another function of B2 only. The value of

B2 for which the SS function is minimum is determined

by solving equation (12) using Newton's method. Once B2is known, Ie is calculated using equation (11), B1 usingequation (9) and finally Icp using equation (7).

It should be noted that the three unknowns Ic,Bb andB2 cou ld be determin ed u sing equ ation (6) alo ne b y

inserting three d ata po ints and solving the resulting

simultaneous equations. However, such a technique will

be highly sensitive to the accuracy of the three data points.

In practice such accuracy can not be achieved. To improve

the confidence in the results, the number of experimental

data points must be increased and consequently the method

should be flexible enough to accommodate all the data

points. This can easily be done using our proposed method.

It is t o b e n oted that the MF method is b ased on a

simultaneous solution of equations (6) and (8) with the

requirement that the value of Ie be known.

to solve for the unknowns. The first relationship is obtained

by defining a least squares function

SS = I{li-[/c+B 1 Pi-Ic exp(-B2 Pi) ] } 2 . (8)

where the summation is carried ou t o ver all the data

points, i.e. i= I to N. The second relation is obtained by

differentiating equation (6) and determining slope at the

cathodic protection potential Pcp

(dI/dP)cp=B1 +B 2Ic exp(-B2Pcp) . (9)

The LHS of equation (9) is calculated as a numerical slope

and depends only upon the experimental points. However,

the RHS is evaluated analytically for a given set of the

three parameters. Therefore the two sides of equation (9)

are determined independently and would be equal when

the choice of the three parameters is correct.

The th ird relatio nship is o btained by integratin g

equation (6)

A= f P N IdP=Ic(PN- Pl)+!Bl(P~- pi)J p1

+Ic/B 2[ exp(-B2P2)- exp(-B2PdJ (10)

Similar to equation (9), th e area A in equation (10),

determined n umerically u sing the trapezo idal rule, is

independent of the integral evaluated for a given set of

parameters.

Equations (8)-(10) are thus the three relationships among

the three parameters Ic' B 1,and B 2' Eliminating B 1 fromequations (9) and (10) gives

Ic=A/F(B2) (11)

where F(B2) is a function of B2 only. Eliminating Ie and

RESULTS AND DISCUSSIONWe use a typical system to illustrate the usefulness of theproposed method. The anodic reaction consists of the

oxidation of Fe to Fe2 + and the cathodic reaction consists

of reduction of H + to H2. The parameters for the anodic

reaction are 7

101 =320 rnA m-2

E1 = -500mV

b1a=b1e=328 mY/decade

The parameters for the cathodic reaction are 7

102

= 14 rnA m - 2

E2=0

b2a=b2e=123 m V/decade

The Ec and Ie satisfying equation (5) are -237 mV and1182 rnA m - 2 respectively.

In o rd er to test o ur method and compare i t with the

MF method, we use hypothetical data generated using

equation (5) with the above parameters. In practice such

data would be obtained experimentally. The data basically

consist of three current values I(P cp) , I (P1), and I(P 3).Note that only three data points have been used in order

to have a valid comparison with the MF method. However,

we recommend the use o f as many exp erimental data

points as possible to increase the reliability of the results.

Additionally, it is desirable to keep PI and P3 close to

Pcp, i.e. P d should be sma ll. On the other ha nd theinstrumental resolution imposes certain restrictions as to

how small a value ofP d can be applied. We have therefore

Table 1 Comparison of calculated corrosion current density lep at various cathodic protection

potentials Ecp

lcp, rnA m-2

Appliedpotential,mV

-260

-300

-400

-490

*True value = 1182

Presentmethod

107889945154

MF

method

1078898450

52

Actual partialcurrent densityusing equation (3)

107889844 9

45

Ic from the presentmethod*, rnA m-2

11821182.11821182

British Corrosion Journal 1992 Vol. 2 7 N O.2

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Q amar and Husain Computat ion of c orrosion c urre nt in c athodical ly protec te d sy stem 127

Table 2 Effect of 1;" inaccuracy in Ie on the computed

value of corrosion current density Iep using MF

method

Applied lcp, rnA m-2

potential,

mV Exact Computed

-260 1078 1093

-300 898 921-400 449 528-490 45 334

taken P d as 1 m V. A comparison of the results obt~ined

using our method and the MF method is shown in Table 1.

It is seen that the values o f Icp obtained using the two

metho ds are q uite close to each other as well as to the

true current obtained using equation (3).

The proposed method has also been tested in the other

regimes of applied potential. The results obtained are

similar to those presented here and will not be discussed

for the sake of brevity. It is interesting to note that, contrary

to the a ssumption of Meas a nd Fujioka that Ie is

generally known and Tafel constants are not known, it is

generally believed that the reverse is true. In fact the well

known linear polarisation method2 used to calculate leis

based on the assumption that Tafel constants are generally

known. In addition, an accurate determination of leis

generally a difficult task. We have -therefore tested the

sensitivity of the MF method by introducing a certain

degree of inaccuracy in the true value of Ie. The results

thus obtained are shown in Table 2. A large error in Iep is

observed at potentials near E1 whereas the error decreases

as the protection potential approaches corrosion potential

Ee The solution is therefore more sensitive to the value of

Ie used in a region where cathodic protection is likely to

be used, i.e. away from the corrosion potential.

Although implicit in the above discussion, it cannot be

overemphasised that our method generates the value of Ie

(the corrosion current density of the freely corroding metal)

the determination of which in itself is a major problem in

electrochemical corrosion studies. As shown in the last

column of Table 1, the values of Ie computed from the

data at various potentials are the same as the exact value

of Ie.

CONCLUSIONSThe nu merical approach u sed in the p resent p ap er to

determine corrosion current density of a cathodically

protected metal provides a reliable method for a wide

range of applied potentials. The results are comparable to

those obtained by another method developed by Meas and

Fujioka.6 Th e main advantage o f ou r metho d ov er the

MF method is that it does not require a prior knowledge

of Ie , the corrosion current density of the freely corroding

metal. In fact our method generates the value of I e , the

determination of which is generally a problem.

REFERENCES1. J. TAFEL: Z. Phys. Chem., 1905, 50, 641.2. M. STERN and A. L. GEARY: J. Electrochem. Soc., 1957, 104, 56.

3. J. JANKOWSKI and R. JUCHNIEWICZ: Corros. Sci., 1980, 20, 841.4. N. D. GREENE and R. H. GANDHI: Mater. Perform., 1982, 21, 34.5. I. QAMAR and s. W. HUSAIN: Br. Corros. J., 1990,25,202.6. Y. MEAS and J. FUJIOKA: Carras. Sci., 1990, 30, 929.

7. M. POURBAIX: 'Lectures on electrochemical corrosion'; 1973,New York, Plenum Press.

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