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A theoretical approach for predicting AC-induced corrosion
Corrosion Science, Vol. 36, No. 6, pp. 1039-1046, 1994 Elsevier Science Ltd. Printed in Great Britain 0010-938X/94 $7.00+0.00
Pergamon
0010-938X(93)E0022-J
A THEORETICAL APPROACH FOR PREDICTING AC-INDUCED CORROSION S. B. LALVANIand X. A. LIN Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, U.S.A.
A b s t r a c t - - A model that considers both anodic dissolution and cathodic reduction for corrosion of
materials u n d e r superimposed sinusoidal voltages has been proposed and developed. A closed form solution permits the estimation of the corrosion current and corrosion potential. The corrosion current is found to increase with the peak voltage of the applied signal. For a constant value of the anodic Tafel slope, a decrease in the cathodic Tafel slope results in higher corrosion current. The corrosion potential is found to be a function of the absolute ratio of the anodic Tafel slope to the cathodic Tafel slope (r). For r less than and greater than unity, an increase in the peak voltage results in more active and more noble corrosion potential, respectively, than its D C corrosion potential. A s s u m p t i o n s inherent and the limitations of the model are also discussed in the paper.
INTRODUCTION
As OUR urban areas become increasingly crowded, the problems associated with AC-induced corrosion also become acute. More and more pipes and ducts share utility tunnels and corridor-of-way paths with high voltage conductors. AC signals flowing through conductors induce currents in the metallic structures such as pipes or towers that are in close proximity. The magnitudes of the induced currents or voltages can be quite appreciable and depend upon the proximity of the conductor and structure, the geometry, the material of the structure, conductor signal levels, insulation dielectric strength and other environmental factors. The aim of this paper is to develop a model for the corrosion of materials that have been subjected to AC fields. Corrosion, an electrochemical process, has traditionally been studied as a DC process. Periodic electrochemical processes are difficult to analyse theoretically and hence mathematically. Gellings t was one of the first authors to perform a comprehensive theoretical treatment of the effect of periodic signals on the polarization behavior of metals. At steady state, the anodic and cathodic currents are equal to one another (in magnitude). From the experimentally determined current (DC) vs potential (i vs E) data, and using the Faraday's Law, a relationship between the weight loss of the material and the level of AC signal was determined by Gellings. 1 Chin and Venkatesh e used the above approach to derive an expression for the normalized corrosion density for a corrosion system with identifiable Tafel regions and known values for DC corrosion potential and corrosion current. Bertocci 3 has also investigated the effect of large amplitude perturbations in increasing the corrosion rates of electrical conduit materials. McKubre 4 has analysed the corrosion problem by obtaining Fourier series of harmonic responses and Manuscript received 14 June 1993; in a m e n d e d form 4 October 1993. 1039
1040
S.B. LALVANIand X. A. LIN
developed an on-line technique for monitoring corrosion in cathodically protected systems. This paper describes an analytical solution for the determination of ACinduced corrosion current and corrosion potential. The model presented below assumes activation control. Corrosion of materials in aqueous solutions is modeled by two electrochemical reactions. The anodic dissolution of metal, as well as cathodic reduction (such as hydrogen evolution or reduction of oxygen) are respectively assumed to follow the Tafel equations shown below: E = ma In ia + ca
(1)
E = mc In ic + Cc.
(2)
The corrosion potential, E c.... PC, is defined as the steady-state DC potential at which the anodic and cathodic currents are equal to one another. From equations (1) and (2), the corrosion potential and corrosion current, icorr,DO are given by: Ecorr,D c --
(3)
maCc -- mcCa
m a -- /g/c
'c°Tr°c=exprCc C cme' l exP', c°rr°C mma aCal exprLcorrOC mc
(4)
A V-induced corrosion The potential, E, at the working electrode can be written as the sum of a DC potential, EDC, and the alternating voltage (AV) signal, Ep sin wt, where Ep and w are respectively the peak voltage and frequency of the signal. When E p = 0, it can be shown that E o c is equal to E c.... PC, given by equation (3). Thus, E is written as: E = EDC + E p sin
wt.
(5)
The time-average current, Tis defined as: 1
(2~/w i dt,
i - (2~/w) ~0
(6)
where 27dw is the period of the sinusoidal signal of magnitude E p sin The average anodic current, Ta, is then given by:
--
l
f2~/w
-
ia
(2~/w) J0
wt.
ia dr.
(7)
From equations (5) and (1), ia=
exp[EDc- Ca + Epsin wt]
(8)
Substitute (8) into (7) i~
~ exp L
ma
J0
ma
l
dt.
The following expansion is useful in evaluating the above intregral:
(9)
Prediction of AC-induced corrosion
1041
oc
exp (zsin wt) = ~" z~ sinn wt n!
(10)
n=0
Applying the result from equation (10) to equation (9), the following expression is obtained for the average current (for proof, see Appendix): ¢c
-ia
:
e(E°~-ca)/ma
~2m.J
IZ
_|
+ 1 . ,,
1
(11)
It can easily be shown that the series represented in equation (11) is convergent by noting that limit lira
SK+ I
K----~~
SK
<
1,
where St( is the Kth term in the series shown in equation (11). Similarly, it can be shown that the time-average cathodic current is given by: ov
ic=e cc"mc[Z, 2mcJl( I2K + '1
(12)
The AV-induced DC corrosion potential, Ecorr,AV, is defined as the steady-state potential when the time-average anodic and cathodic currents are equal to one another. Thus, by equating (11) and (12), and using the result from equation (3), we obtain: Ecorr,A V :
(13)
Ecorr,D C -- a
where
-¢ m.mc t
/ ~ 1 '~''~mcj
_
('~,
a = 0 when Im.I = Imd. The corrosion current, icorr,AV is obtained by substituting equation (13) into either equation (11) or (12) and using the result obtained in equation (4):
I..... DC
LK=I-(-~.)2\2ma ]
+ 1 = e [-("/'~c)l
=, ~ 1
\2mJ
+ 1 .
(15) In equation (14), a is the shift in the potential from the natural corrosion potential
1042
S. B. LALVANIand X. A. LIN 1.0~
0 •
0.9-8
-L00
-200
-300
-q00
-500 ./
\
-600
I
-700
0
200
,
I
,
a
q00
I
I
I
800
B00
t000
Ep (mV)
FIG. 1. Shift in corrosion potential vs peak voltage for ma
=
100 mV/decade and r -< 1.0.
(under no A V conditions) due to the application of A V modulation, i..... DC is the corrosion rate (equation 4) in the absence of any external A V modulation• RESULTS AND DISCUSSION A shift in the corrosion potential from its D C value ( - a ) was estimated using equation (14) as a function of the peak voltage, Ep, the anodic Tafel slope, ma, and the dimensionless absolute ratio of anodic Tafel slope to the cathodic Tafel slope (r -abs (ma/mc)) for r - 1.0 and r -> 1.0. The calculated results are plotted in Figs i and 2. Similarly, the dimensionless corrosion current, icorr,AV/icorr,DC,was estimated using equation (15); and the results obtained are shown in Figs 3 and 4. The data show that for r < 1, an increase in the p e a k voltage results in a decrease in the corrosion potential while the corrosion potentials become more noble for r > 1. Nonetheless, the corrosion current increases rapidly with Ep for all values of r. The corrosion potential and corrosion current are also dependent upon the numerical values of the Tafel slopes. As an illustration, the c o m p u t e d values of the shift in corrosion potential (from its D C value) and the dimensionless corrosion current are shown in Table 1 for Ep of 500 inV. For a constant value of the absolute dimensionless ratio of the anodic to cathodic Tafel slopes (i.e. r), the data show that corrosion current increases sharply with a decrease in the anodic Tafel slope; however, this increase is m o r e pronounced at higher values of r. For r < 1, a decrease in the anodic Tafel slope (ma) results in more negative or active corrosion potentials, while for r = 0, the corrosion potential is equal to its D C value. For r > 1, the corrosion potential becomes more noble with a decrease in ma; however, for large values of r (say 10), the increase in corrosion potential is relatively small. For a given value of the anodic Tafel slope, the corrosion rate, as well as the corrosion potential increase with r, although the corrosion current increases very rapidly, especially at low values of the anodic Tafei slope. The results obtained are qualitatively in agreement with the
-y
Prediction of AC-induced corrosion
960 760
1043
666 566 466 368 o
26O 106 ~
6
i
, 208
i q86
,~
r--at 600
,
i 900
, i886
Ep (mV)
Shift in corrosion potential vs peak voltage for m . = I00 mV/decade and r ~> 1.0.
FIG. 2.
L6OH8
I
I
21~
4g~
I
I
4 3 2 1666
qP 8 -g <
4 3 2
lee
B
G~8
~
10B~
[p (my) FIG. 3.
Dimensionless corrosion current vs peak voltage for m~, = 100 mV/decade and r -< 1.0.
1044
S . B . LALVANI a n d X. A . LIN
O. 1E8
0.1E7
O. IE6
0.1E5 > <.
O. 1E4
O. 1E3
O.1E2
0.1E1
B
2BE
qgB
Bile
BEE
IEEE
Ep (mV) FIG. 4.
Dimensionless corrosion current vs peak voltage for m~ = 100 mV/decade and r -> 1.0.
observed AC-induced corrosion effects, s-7 It is also observed experimentally that the rate of material corrosion increases exponentially with the peak voltage, s This model predicts that the corrosion current and corrosion potential are independent of the frequency of the applied signal. However, at high excitation voltages and at high frequencies, the overall rate of reaction will be mass-transfer controlled, and hence, the corrosion current observed will be frequency-dependent. Impedance due to the double layer capacitance decreases with frequency, hence the corrosion rate observed will also decrease at high frequency values. A more realistic description of the AC-induced corrosion should also take into account the concentration polarization effects, as well as the double layer capacitance.
TABLE i .
SHIFT IN CORROSION POTENTIAL AND DIMENSIONLESS CORROSION CURRENT AT Ep OF
500 m V
Anodic Tafel slope, m~ (mV/decade)
r = 0.1"
r = 1"
r = 10"
r = 0.1"
100 75 50 25
-294.8 -321.2 -350.3 -381.0
0 0 0 0
398.4 401.1 403.8 406.5
1.4 1.7 2.6 10.5
E ..... AV -- E . . . . . DC ( m Y )
lcorr.AV -- lcorr,DC ( m Y ) r = 1" 2.7 1.2 2.8 4.4
X x x X
101 102 103 10 7
r = 10" 1.5 2.6 9.7 5.0
X × × ×
10 3 10 a 106 1014
* T h e dimensionless absolute ratio of the anodic Tafel slope to cathodic Tafel slope.
Prediction
of AC-induced
corrosion
1045
CONCLUSIONS
1. The average corrosion current, as a result of the imposed alternating volage, is predicted to increase with the peak potential. 2. The shift in the corrosion potential is dependent upon the absolute ratio of the anodic Tafel slope to the cathodic Tafel slope (P-). For the two cases, r < 1 and Y > 1, an increase in the peak potential results respectively in a decrease and increase in the corrosion potential. When Y = 3, there is no change in the corrosion potential with peak voltage. 3. An increase in the numerical value of r, results in more noble corrosion potentials but much higher corrosion currents. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
P. J. GELLINGS, Electrochim. Acta 7, 19 (1962) D.-T. CHIN and S. VENKATESH, J. electrochem. Sot. 126, 1908 (1979). U. BERTOCCI, Corrosion 35,221 (1979). M. C. H. MCKUBRE, On-line Corrosion Monitoring in Cathodically Protected Systems, Final EPRl Report No. CS-5695 on Project 1689-7, May 1988. R. SETHIand D.-T. CHIN, J. electroanal. Chem. 160,79 (1984). S. Z. FERNANDES. S. G. MEHENDALE and S. VENKATACHALAM,J. appl. electrochem. 10,649 (1980). D.-T. CHIN and P. SACHDEV, J. electrochern. Sot. 130, 1714 (1983). M. PAGANO and S. B. LAI.VANI, Corros. Sci. 36, 127 (1994).
APPENDIX Integration by parts gives us, 2i/l+ 2nln, sin”-’ sin” wt dt = !0 I0 = sin”-’
wt{sin wt dt}
wt(-cos
wtIw)J(:.71W+
(cos wtlw) (n - 1) sirY2
wt cos wt. w dt
0 =(n-1)
Therefore,
the following
Z;r/M 2n/w sin”-2 wf dt - (n - 1) sin” wt dr. 0 (1
recursion
formula
is derived
ZIT/W’ n _ sin” wt dr = i0 n Repeatedly
integrating
1 2;rtw sin”-’
wt dt.
!0
by parts the right hand side of the above equation,
we obtain
wtdt=..’
in-l - w n
n-3
-
3
n-2
4 2 i ,)
- -1 2n sin” t dr,
In-l -__. w n
n-3 ____.
-. -
n - even
n-odd
where r = wt. The integral term in equations (a) and (b) is, respectively, equal to 2~ and zero. In the formulation shown in the text, even and odd numbers of n are substituted by 2K and 2K - 1, respectively. To simplify the expression in (a), we rewrite it by multiplying the denominator to both numerator and denominator and replacing n by 2K:
(a)
(b)
1046
S.B. LALVAN1and X. A. L1N 2K-1 2K
3 l_2K(2K-1)(2K-2)(2K-3)...4.3.2.1 [2K. ( 2 K - 2)...4.2] 2 4 2
2K-3 2K- 2
(2K)! {2KK . ( K - I ) . . - 2 . 1 }
(2K)! 2 22K(K!)2
Thus, equations (a) and (b) appear [2ar (2K)!
2.n:/wsin" wtdt = { w 22K(K!) 2'
n =2K
(c)
n=2K+l
(d)
f
0
k
0
,
where K = 0, 1, 2 . . . .
Substituting equation (10) to equation (9) results in the following: -
w
[EDc-ca]I2:~/WzEnsinnwt
ia = ~ exp [ ~ J
,,
Vm:n!
dt
n =l)
-~exp
t
m~
1 ~=~m~n!)o
~-~ exp t
m,,
~ . a m,,2K (2K).T ] K=o
K=o (K!): \2m~J
~K-,
(K!)2 ~2m,,]
w 22K(K!) 2
Pergamon
0010-938X(93)E0022-J
A THEORETICAL APPROACH FOR PREDICTING AC-INDUCED CORROSION S. B. LALVANIand X. A. LIN Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, U.S.A.
A b s t r a c t - - A model that considers both anodic dissolution and cathodic reduction for corrosion of
materials u n d e r superimposed sinusoidal voltages has been proposed and developed. A closed form solution permits the estimation of the corrosion current and corrosion potential. The corrosion current is found to increase with the peak voltage of the applied signal. For a constant value of the anodic Tafel slope, a decrease in the cathodic Tafel slope results in higher corrosion current. The corrosion potential is found to be a function of the absolute ratio of the anodic Tafel slope to the cathodic Tafel slope (r). For r less than and greater than unity, an increase in the peak voltage results in more active and more noble corrosion potential, respectively, than its D C corrosion potential. A s s u m p t i o n s inherent and the limitations of the model are also discussed in the paper.
INTRODUCTION
As OUR urban areas become increasingly crowded, the problems associated with AC-induced corrosion also become acute. More and more pipes and ducts share utility tunnels and corridor-of-way paths with high voltage conductors. AC signals flowing through conductors induce currents in the metallic structures such as pipes or towers that are in close proximity. The magnitudes of the induced currents or voltages can be quite appreciable and depend upon the proximity of the conductor and structure, the geometry, the material of the structure, conductor signal levels, insulation dielectric strength and other environmental factors. The aim of this paper is to develop a model for the corrosion of materials that have been subjected to AC fields. Corrosion, an electrochemical process, has traditionally been studied as a DC process. Periodic electrochemical processes are difficult to analyse theoretically and hence mathematically. Gellings t was one of the first authors to perform a comprehensive theoretical treatment of the effect of periodic signals on the polarization behavior of metals. At steady state, the anodic and cathodic currents are equal to one another (in magnitude). From the experimentally determined current (DC) vs potential (i vs E) data, and using the Faraday's Law, a relationship between the weight loss of the material and the level of AC signal was determined by Gellings. 1 Chin and Venkatesh e used the above approach to derive an expression for the normalized corrosion density for a corrosion system with identifiable Tafel regions and known values for DC corrosion potential and corrosion current. Bertocci 3 has also investigated the effect of large amplitude perturbations in increasing the corrosion rates of electrical conduit materials. McKubre 4 has analysed the corrosion problem by obtaining Fourier series of harmonic responses and Manuscript received 14 June 1993; in a m e n d e d form 4 October 1993. 1039
1040
S.B. LALVANIand X. A. LIN
developed an on-line technique for monitoring corrosion in cathodically protected systems. This paper describes an analytical solution for the determination of ACinduced corrosion current and corrosion potential. The model presented below assumes activation control. Corrosion of materials in aqueous solutions is modeled by two electrochemical reactions. The anodic dissolution of metal, as well as cathodic reduction (such as hydrogen evolution or reduction of oxygen) are respectively assumed to follow the Tafel equations shown below: E = ma In ia + ca
(1)
E = mc In ic + Cc.
(2)
The corrosion potential, E c.... PC, is defined as the steady-state DC potential at which the anodic and cathodic currents are equal to one another. From equations (1) and (2), the corrosion potential and corrosion current, icorr,DO are given by: Ecorr,D c --
(3)
maCc -- mcCa
m a -- /g/c
'c°Tr°c=exprCc C cme' l exP', c°rr°C mma aCal exprLcorrOC mc
(4)
A V-induced corrosion The potential, E, at the working electrode can be written as the sum of a DC potential, EDC, and the alternating voltage (AV) signal, Ep sin wt, where Ep and w are respectively the peak voltage and frequency of the signal. When E p = 0, it can be shown that E o c is equal to E c.... PC, given by equation (3). Thus, E is written as: E = EDC + E p sin
wt.
(5)
The time-average current, Tis defined as: 1
(2~/w i dt,
i - (2~/w) ~0
(6)
where 27dw is the period of the sinusoidal signal of magnitude E p sin The average anodic current, Ta, is then given by:
--
l
f2~/w
-
ia
(2~/w) J0
wt.
ia dr.
(7)
From equations (5) and (1), ia=
exp[EDc- Ca + Epsin wt]
(8)
Substitute (8) into (7) i~
~ exp L
ma
J0
ma
l
dt.
The following expansion is useful in evaluating the above intregral:
(9)
Prediction of AC-induced corrosion
1041
oc
exp (zsin wt) = ~" z~ sinn wt n!
(10)
n=0
Applying the result from equation (10) to equation (9), the following expression is obtained for the average current (for proof, see Appendix): ¢c
-ia
:
e(E°~-ca)/ma
~2m.J
IZ
_|
+ 1 . ,,
1
(11)
It can easily be shown that the series represented in equation (11) is convergent by noting that limit lira
SK+ I
K----~~
SK
<
1,
where St( is the Kth term in the series shown in equation (11). Similarly, it can be shown that the time-average cathodic current is given by: ov
ic=e cc"mc[Z, 2mcJl( I2K + '1
(12)
The AV-induced DC corrosion potential, Ecorr,AV, is defined as the steady-state potential when the time-average anodic and cathodic currents are equal to one another. Thus, by equating (11) and (12), and using the result from equation (3), we obtain: Ecorr,A V :
(13)
Ecorr,D C -- a
where
-¢ m.mc t
/ ~ 1 '~''~mcj
_
('~,
a = 0 when Im.I = Imd. The corrosion current, icorr,AV is obtained by substituting equation (13) into either equation (11) or (12) and using the result obtained in equation (4):
I..... DC
LK=I-(-~.)2\2ma ]
+ 1 = e [-("/'~c)l
=, ~ 1
\2mJ
+ 1 .
(15) In equation (14), a is the shift in the potential from the natural corrosion potential
1042
S. B. LALVANIand X. A. LIN 1.0~
0 •
0.9-8
-L00
-200
-300
-q00
-500 ./
\
-600
I
-700
0
200
,
I
,
a
q00
I
I
I
800
B00
t000
Ep (mV)
FIG. 1. Shift in corrosion potential vs peak voltage for ma
=
100 mV/decade and r -< 1.0.
(under no A V conditions) due to the application of A V modulation, i..... DC is the corrosion rate (equation 4) in the absence of any external A V modulation• RESULTS AND DISCUSSION A shift in the corrosion potential from its D C value ( - a ) was estimated using equation (14) as a function of the peak voltage, Ep, the anodic Tafel slope, ma, and the dimensionless absolute ratio of anodic Tafel slope to the cathodic Tafel slope (r -abs (ma/mc)) for r - 1.0 and r -> 1.0. The calculated results are plotted in Figs i and 2. Similarly, the dimensionless corrosion current, icorr,AV/icorr,DC,was estimated using equation (15); and the results obtained are shown in Figs 3 and 4. The data show that for r < 1, an increase in the p e a k voltage results in a decrease in the corrosion potential while the corrosion potentials become more noble for r > 1. Nonetheless, the corrosion current increases rapidly with Ep for all values of r. The corrosion potential and corrosion current are also dependent upon the numerical values of the Tafel slopes. As an illustration, the c o m p u t e d values of the shift in corrosion potential (from its D C value) and the dimensionless corrosion current are shown in Table 1 for Ep of 500 inV. For a constant value of the absolute dimensionless ratio of the anodic to cathodic Tafel slopes (i.e. r), the data show that corrosion current increases sharply with a decrease in the anodic Tafel slope; however, this increase is m o r e pronounced at higher values of r. For r < 1, a decrease in the anodic Tafel slope (ma) results in more negative or active corrosion potentials, while for r = 0, the corrosion potential is equal to its D C value. For r > 1, the corrosion potential becomes more noble with a decrease in ma; however, for large values of r (say 10), the increase in corrosion potential is relatively small. For a given value of the anodic Tafel slope, the corrosion rate, as well as the corrosion potential increase with r, although the corrosion current increases very rapidly, especially at low values of the anodic Tafei slope. The results obtained are qualitatively in agreement with the
-y
Prediction of AC-induced corrosion
960 760
1043
666 566 466 368 o
26O 106 ~
6
i
, 208
i q86
,~
r--at 600
,
i 900
, i886
Ep (mV)
Shift in corrosion potential vs peak voltage for m . = I00 mV/decade and r ~> 1.0.
FIG. 2.
L6OH8
I
I
21~
4g~
I
I
4 3 2 1666
qP 8 -g <
4 3 2
lee
B
G~8
~
10B~
[p (my) FIG. 3.
Dimensionless corrosion current vs peak voltage for m~, = 100 mV/decade and r -< 1.0.
1044
S . B . LALVANI a n d X. A . LIN
O. 1E8
0.1E7
O. IE6
0.1E5 > <.
O. 1E4
O. 1E3
O.1E2
0.1E1
B
2BE
qgB
Bile
BEE
IEEE
Ep (mV) FIG. 4.
Dimensionless corrosion current vs peak voltage for m~ = 100 mV/decade and r -> 1.0.
observed AC-induced corrosion effects, s-7 It is also observed experimentally that the rate of material corrosion increases exponentially with the peak voltage, s This model predicts that the corrosion current and corrosion potential are independent of the frequency of the applied signal. However, at high excitation voltages and at high frequencies, the overall rate of reaction will be mass-transfer controlled, and hence, the corrosion current observed will be frequency-dependent. Impedance due to the double layer capacitance decreases with frequency, hence the corrosion rate observed will also decrease at high frequency values. A more realistic description of the AC-induced corrosion should also take into account the concentration polarization effects, as well as the double layer capacitance.
TABLE i .
SHIFT IN CORROSION POTENTIAL AND DIMENSIONLESS CORROSION CURRENT AT Ep OF
500 m V
Anodic Tafel slope, m~ (mV/decade)
r = 0.1"
r = 1"
r = 10"
r = 0.1"
100 75 50 25
-294.8 -321.2 -350.3 -381.0
0 0 0 0
398.4 401.1 403.8 406.5
1.4 1.7 2.6 10.5
E ..... AV -- E . . . . . DC ( m Y )
lcorr.AV -- lcorr,DC ( m Y ) r = 1" 2.7 1.2 2.8 4.4
X x x X
101 102 103 10 7
r = 10" 1.5 2.6 9.7 5.0
X × × ×
10 3 10 a 106 1014
* T h e dimensionless absolute ratio of the anodic Tafel slope to cathodic Tafel slope.
Prediction
of AC-induced
corrosion
1045
CONCLUSIONS
1. The average corrosion current, as a result of the imposed alternating volage, is predicted to increase with the peak potential. 2. The shift in the corrosion potential is dependent upon the absolute ratio of the anodic Tafel slope to the cathodic Tafel slope (P-). For the two cases, r < 1 and Y > 1, an increase in the peak potential results respectively in a decrease and increase in the corrosion potential. When Y = 3, there is no change in the corrosion potential with peak voltage. 3. An increase in the numerical value of r, results in more noble corrosion potentials but much higher corrosion currents. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
P. J. GELLINGS, Electrochim. Acta 7, 19 (1962) D.-T. CHIN and S. VENKATESH, J. electrochem. Sot. 126, 1908 (1979). U. BERTOCCI, Corrosion 35,221 (1979). M. C. H. MCKUBRE, On-line Corrosion Monitoring in Cathodically Protected Systems, Final EPRl Report No. CS-5695 on Project 1689-7, May 1988. R. SETHIand D.-T. CHIN, J. electroanal. Chem. 160,79 (1984). S. Z. FERNANDES. S. G. MEHENDALE and S. VENKATACHALAM,J. appl. electrochem. 10,649 (1980). D.-T. CHIN and P. SACHDEV, J. electrochern. Sot. 130, 1714 (1983). M. PAGANO and S. B. LAI.VANI, Corros. Sci. 36, 127 (1994).
APPENDIX Integration by parts gives us, 2i/l+ 2nln, sin”-’ sin” wt dt = !0 I0 = sin”-’
wt{sin wt dt}
wt(-cos
wtIw)J(:.71W+
(cos wtlw) (n - 1) sirY2
wt cos wt. w dt
0 =(n-1)
Therefore,
the following
Z;r/M 2n/w sin”-2 wf dt - (n - 1) sin” wt dr. 0 (1
recursion
formula
is derived
ZIT/W’ n _ sin” wt dr = i0 n Repeatedly
integrating
1 2;rtw sin”-’
wt dt.
!0
by parts the right hand side of the above equation,
we obtain
wtdt=..’
in-l - w n
n-3
-
3
n-2
4 2 i ,)
- -1 2n sin” t dr,
In-l -__. w n
n-3 ____.
-. -
n - even
n-odd
where r = wt. The integral term in equations (a) and (b) is, respectively, equal to 2~ and zero. In the formulation shown in the text, even and odd numbers of n are substituted by 2K and 2K - 1, respectively. To simplify the expression in (a), we rewrite it by multiplying the denominator to both numerator and denominator and replacing n by 2K:
(a)
(b)
1046
S.B. LALVAN1and X. A. L1N 2K-1 2K
3 l_2K(2K-1)(2K-2)(2K-3)...4.3.2.1 [2K. ( 2 K - 2)...4.2] 2 4 2
2K-3 2K- 2
(2K)! {2KK . ( K - I ) . . - 2 . 1 }
(2K)! 2 22K(K!)2
Thus, equations (a) and (b) appear [2ar (2K)!
2.n:/wsin" wtdt = { w 22K(K!) 2'
n =2K
(c)
n=2K+l
(d)
f
0
k
0
,
where K = 0, 1, 2 . . . .
Substituting equation (10) to equation (9) results in the following: -
w
[EDc-ca]I2:~/WzEnsinnwt
ia = ~ exp [ ~ J
,,
Vm:n!
dt
n =l)
-~exp
t
m~
1 ~=~m~n!)o
~-~ exp t
m,,
~ . a m,,2K (2K).T ] K=o
K=o (K!): \2m~J
~K-,
(K!)2 ~2m,,]
w 22K(K!) 2