Corrosion Science 51 (2009) 962–970
Contents lists available at ScienceDirect
Corrosion Science
journal homepage: www.elsevier.com/locate/corsci
On the extension of CP models to address cathodic protection under a
delaminated coating
Kerry N. Allahar, Mark E. Orazem *
Department of Chemical Engineering, University of Florida, Gainesville, FL 32611, USA
a r t i c l e
i n f o
Article history:
Received 29 February 2008
Accepted 27 January 2009
Available online 24 February 2009
Keywords:
Corrosion
Coatings
C. Cathodic protection
a b s t r a c t
Cathodic protection models for pipeline and other metallic structures developed to address the presence
of coating flaws or holidays do not account explicitly for the presence of a delaminated region surrounding a holiday. The potential drop in the delaminated region was obtained by solving the coupled, non-linear, partial differential, governing equations that describe transport and electrochemical reactions within
the delaminated region. The results were used to develop an analytic expression for the potential drop
across the delaminated region as a function of delaminated region length and the resistivity of the bulk
environment exterior to the holiday. Expressions developed using this approach can be utilized by existing cathodic protection models to deliver improved potential distributions along protected metallic surfaces. The simulations were used to test the commonly employed assumption that transport by diffusion
can be neglected in the delaminated region. The influence of diffusion on current was shown to be
significant.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Impressed-current cathodic protection systems are used in conjunction with organic coatings on pipelines and other metallic
structures to provide protection in regions where the coating has
been damaged [1]. These systems mitigate the corrosion at exposed metal surfaces, also known as coating holidays, by driving
cathodic reactions such as oxygen reduction and hydrogen evolution. One criterion for cathodic protection implementation involves the maintenance of the pipe surface at a potential more
negative than 0.850 V relative to a copper/copper-sulphate reference electrode (or 0.773 V(SCE)). Early cathodic protection models and design equations were developed to account for
attenuation to single uniform pipes protected by distant anodes
or ground beds [2,3]. Semi-analytic approaches have been developed for cases where the pipe can still be considered to be uniform,
but a close anode-to-pipe spacing may cause under-protection of
the region of the pipe most distant from the anode while overprotecting the region closest to the anode [4].
The use of boundary element methods to solve cathodic protection problems in infinite media was introduced by Telles et al. [5],
Chuang[6] and Zamani et al. [7] employed boundary elements to
model cathodic protection of ships. Nisancioglu et al. [8–10] and
Santiago and Telles [11] have incorporated time-dependent polar-
* Corresponding author. Tel.: +1 352 392 6207; fax: +1 352 392 9513.
E-mail address: meo@che.ufl.edu (M.E. Orazem).
URL: http://www.che.ufl.edu/faculty/Orazem (M.E. Orazem).
0010-938X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.corsci.2009.01.023
ization curves into boundary element models for cathodic protection. Brichau and Deconinck presented a model using boundary
element and finite element (FEM) methods for buried pipe networks cathodically protected by anode beds [12]. Their method
of surface discretization assumed cylindrical elements (pipe-sections) with a uniform radial current density distribution. The
assumption of a uniform current density distribution, however, is
valid only for a uniformly coated or fully bare pipeline with a distant anode.
The influence of coating holidays was addressed by Kennelley
et al. [13] and Orazem et al. [14] using finite elements. In subsequent work, a more precise boundary-element approach was used
to model the role of coating holidays [15,16] for a single pipeline
protected by sacrificial ribbon anodes such as those used for the
Trans- Alaska pipeline. This model was extended to account for
multiple pipelines, multiple CP systems, bonds, and arbitrary anode placement [17–21]. These models show the degree of protection provided to coating holidays which expose bare steel, but
the models do not address the influence of the delaminated regions
of the coating that often surround holidays.
The delaminated region resembles a crevice in that the electrolyte within is occluded from the exterior bulk environment and
electrochemical reactions at any point on the metal surface are
influenced by local conditions. Detailed reviews of models for such
occluded systems have been presented by Sharland [22] and Turnbull [23]. A model for the steady-state conditions in a disbondment
system under cathodic protection was described by Song et al.
[24,25]. In this work, the influence of a holiday was not considered,
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
concentration gradients were neglected, and the solution potential
was governed by Laplace’s equation. From experimental observations, however, Leng et al. [26,27] and Furbeth and Stratmann
[28,29] concluded that cathodic delamination is driven by diffusion
of cations within a gel-like region which separates the intact coating and the fully delaminated coating. Their interpretation was
supported by simulations developed by Allahar et al. [30] and
Huang et al. [31]. Thus, transport of ionic species may be expected
to play a critical role in the fully delaminated region as well.
The object of the present work is to explore the potential drop
within the delaminated region surrounding a coating holiday in
the context of a solution of the coupled equations for conservation
of ionic species, including migration and diffusion, and local electroneutrality. This work is intended to elucidate the role of ionic transport within the region in which the coating is fully delaminated. In
addition, an expression is sought to relate the potential drop in
the delaminated region to parameters which could be accessed by
existing CP models. Such an expression could provide a modified
boundary condition for CP models to allow estimation of CP requirements for the delaminated region adjacent to a coating holiday.
at the common boundary between the 2-D and 1-D delaminated
region domains using continuity of species concentration, solution
potential, and flux of species in the radial direction. Underlined
variables ci and / designate the concentration of a specie i and
the dimensionless solution potential, respectively, in the 1-D domain. No homogeneous reactions were considered. Symbols that
are not underlined correspond to the respective variables in the
2-D domain.
The electrolyte in the exterior of the holiday-crevice domain
was assumed to contain Na+, Cl, OH, and Fe+2 ions. The pH was
neutral, and the concentration of NaCl was used to vary the resistivity of the electrolyte. An electrolyte composed of 103 M NaCl,
107 M OH, and 1015 M Fe+2 corresponded to a resistivity of
7.9 kX cm. The applied potential W represents the potential of
the metal such that the solution potential at the mouth of the holiday had a value of 0 V(SCE).
2.1. Electrochemical reactions
The electrochemical reactions considered were iron dissolution
Fe ! Feþ2 þ 2e
2. Model development
963
ð1Þ
oxygen reduction
A schematic representation of a delaminated region surrounding a coating holiday is given in Fig. 1. A cylindrical geometry
was assumed with an axis of symmetry centered on the coating
holiday. The delaminated region is bounded by the periphery of
the holiday and the intact and un-detached coating. The section
of metal covered by the intact coating was considered to be fully
passivated and did not contribute to the system. The detached
coating was considered impermeable to the passage of current
and oxygen. When the metal surface is under cathodic protection,
current is delivered to the metal surface through the electrolyte in
the holiday and delaminated region. The gap between the metal
surface and the detached coating was assumed uniform across
the length of the delaminated region.
Variable distributions in the direction normal to the metal surface can be considered negligible in the delaminated region far
from the holiday [32,33]. The model is developed to take advantage
of this feature for deeper portions of the delaminated region. The
holiday and the delaminated region was modeled as being a twodimensional (2-D) domain for positions less than 3 cm and as a
one-dimensional (1-D) domain for positions greater than 3 cm.
The cut-off at 3 cm was chosen as being the point at which the
mass-transfer-limited oxygen reduction current could be considered negligible. The domains were solved individually but coupled
axis of symmetry
detached coating
holiday
delaminated region
intact coating
metal
holiday
2-D
model
1-D
model
Fig. 1. Schematic representation in a cylindrical coordinate system for a delaminated region surrounding a coating holiday. The model for the delaminated region
consisted of a 2-D segment for positions less than 3 cm and a 1-D segment for
positions greater than 3 cm.
O2 þ 2H2 O þ 4e ! 4OH
ð2Þ
and hydrogen evolution
2H2 O þ 2e ! H2 þ 2OH
ð3Þ
The current densities for iron dissolution iFe, oxygen reduction iO2 ,
and hydrogen evolution i H2 were given by
iFe ¼ 10ðWUEFe Þ=bFe
1
1
10ðWUEO2 Þ=bO2
iO2 ¼ ilim;O2
ð4Þ
ð5Þ
and
iH2 ¼ 10ðWUEH2 Þ=bH2
ð6Þ
respectively, where anodic reactions have a positive current density
and cathodic current densities have a negative value.
In these equations the interfacial potential V was given by the
potential driving the electrochemical reaction W U where U
was the solution potential of a reference electrode located adjacent
to the metal surface and W the potential of the metal, also referred
to as the applied potential.
The value of the mass-transfer-limited current density ilim;O2
was calculated as a function of position along the metal surface
of the holiday-delaminated region 2-D domain. The contribution
of oxygen reduction for positions along the metal surface of the
1-D delaminated region domain was assumed to be negligible
when compared to the contribution of hydrogen evolution.
The net current density inet for steel was given by the sum of the
current densities due to the individual electrochemical reactions
[34]. Along the metal surface of the holiday and within the 2-D domain of the delaminated region, the value of inet was given by the
sum of iFe, iO2 , and iH2 . Along the metal surface of the 1-D domain of
the delaminated region, the value of inet was given by the sum of
iFe, and iH2 . As shown in the subsequent section, the mass-transfer-limited value of the oxygen reduction current became negligible deep within the delaminated region. The Tafel parameters
bFe, bO2 , and bH2 used in the polarization expressions for iFe, iO2 ,
and iH2 were 62.6, 66.5, and 132 mV/decade, respectively. The equilibrium parameters EFe, EO2 and EH2 were assigned values of 475,
500, and 870 mV(SCE), respectively. These parameters were obtained by fitting polarization curves to experimental data for a slitholiday [34].
964
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
ported the assumption that the influence of oxygen reduction for
positions greater than 3 cm was negligible.
2.2. Oxygen distribution
Under the assumptions that radial symmetry applied and that
convection could be neglected, the mass-transfer-limited current
density for oxygen reduction ilim;O2 could be obtained by solution of
Conservation of species i yields [38]
1 @cO2 @ 2 cO2 @ 2 cO2
þ
¼0
þ
r @r
@r 2
@z2
ð7Þ
The boundary condition at the mouth of the holiday was that the
concentration was equal to the bulk value, i.e., cO2 ¼ cO2 ;1 . No-flux
conditions applied at the axis of symmetry (r = 0) and along the surface of the detached coating, i.e., N O2 n ¼ 0 where N O2 is the flux of
oxygen and n is the unit vector normal to the surface. As the flux
tended toward zero deep within the delaminated region, a no-flux
condition was assumed to apply at the boundary between the 2-D
and 1-D domains. The boundary condition at the metal surface
was cO2 ¼ 0 under the assumption that oxygen was reduced completely at the metal surface.
A finite difference method was employed to solve Eq. (7) subject
to the respective boundary conditions, and the value of ilim;O2 on
the metal surface was calculated using
ilim;O2 ¼ nFDO2
@cO2
@z
ð8Þ
where n = 4 is the number of electrons transferred in the electrochemical oxygen reduction reaction per molecule of oxygen O2
(see Eq. (2)), F is Faraday’s constant, and DO2 is the diffusion coefficient of oxygen.
The distribution of ilim;O2 on the metal surface obtained as a
solution to Laplace’s equation for cO2 is shown in Fig. 2 for a holiday
of radius 0.25 cm, a delaminated region of 1.25 cm, and a gap of
0.05 cm. The value of ilim;O2 was approximately constant at 31 lA/
cm2 along the holiday metal surface and decreased exponentially
along the disbondment metal surface such that at a position of
0.5 cm the value of ilim;O2 was three orders of magnitude less than
at the holiday surface. The distribution of ilim;O2 used in the present
work was consistent with the results of the mathematical model
developed by Chin and Sabde [35,36]. The large exponential decrease in ilim;O2 corresponded to an exponential decrease in the concentration of oxygen within the disbondment. Depletion of oxygen
within the disbondment was observed in both experimental studies [37] and mathematical models [24]. The exponential decrease
in the value of ilim;O2 with distance in the delaminated region sup-
2
10
10
-1
10
-4
10
-7
2.3. Mathematical model
Holiday
Disbondment
@ci
¼ r N i þ Ri
@t
ð9Þ
where the term on the left-hand-side represents the rate of change
of concentration ci with time t (accumulation) and the terms on the
right-hand-side represent the net input due to the flux Ni and the
net rate of production by homogeneous reactions Ri, respectively.
In dilute electrochemical systems Ni is given by the Nernst-Planck
equation[38].
N i ¼ zi ui ci F rU Di rci þ ci v
where U is the local solution potential, ui is the mobility, Di is the
diffusion coefficient, zi is the charge number and v is the mass average velocity of the electrolyte. The terms on the right-hand side of
Eq. (10) represent the contributions by migration, diffusion, and
convection to the flux of a species, respectively.
The local current density is given by
i¼F
X
zi N i
ð11Þ
i
Combination of Eqs. (10) and (11), in the absence of convection,
yields
i ¼ jrU F
X
zi Di rci
-2
2
i lim,O / μA cm
-10
10
-13
0.00
0.25
0.50
0.75
1.00
1.25
r / cm
Fig. 2. Calculated value of absolute value of oxygen reduction current density as a
function of radial position on the metal surface for a holiday of radius 0.25 cm, a
delaminated region of 1.25 cm, and a gap of 0.05 cm. The dashed line at r = 0.25 cm
indicates the boundary between the holiday and the delaminated region.
ð12Þ
i
where the conductivity j is defined as
j¼
F2 X 2
z Di ci
RT i i
ð13Þ
The ionic current density has contributions from migration im and
diffusion id [39]. These are the first and second terms, respectively,
on the right-hand side of Eq. (12).
Under the assumption of a steady state, conservation of species
yields
0 ¼ Di zi ci r2 / þ Di zi rci r/ þ Di r2 ci
ð14Þ
in which homogeneous reactions and convection were not considered. In Eq. (14), the dimensionless potential / was given by / =
(UF)/(RgT) using the Nernst-Einstein equation Di = uiRgT, where Rg
and T are the molar gas constant and temperature, respectively.
The system was modeled using cylindrical coordinates. The governing equations consisted of four equations of the form of Eq. (14), one
for each specie, and an algebraic expression for the electroneutrality
condition given by
zNaþ cNaþ þ zCl cCl þ zOH cOH þ zFeþ2 cFeþ2 ¼ 0
10
ð10Þ
ð15Þ
Variables ci and U were fixed at the holiday mouth to the bulk conditions ci,1 and U1, respectively. The solution potential U1 = 0 was
used such that the calculated values for U in the model were referenced to a zero value at the bulk boundary position. The condition
of electroneutrality was applied at all boundaries, and a no-flux
condition Ni n = 0 was applied at all insulating boundaries.
Flux conditions were applied at the metal surface. The no-flux
condition was used for the chemically inert species Na+ and Cl.
The flux conditions for OH and Fe+2 were related to the current
densities due to the cathodic (iH2 and iO2 ) and anodic (iFe) electrochemical reactions on the metal surface, respectively, as
zFeþ2 DFeþ2 cFeþ2
@c þ2 iFe
@/
DFeþ2 Fe ¼
@z
@z
2F
ð16Þ
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
and
zOH DOH cOH
@/
@cOH
iO
iH
DOH
¼ 2þ 2
@z
@z
F
F
ð17Þ
When the delaminated region was smaller than 3 cm, no-flux conditions and electroneutrality were applied at the intact boundary.
For delaminated regions larger than 3 cm, the condition on the common boundary between the 2D-delaminated region and 1D-delaminated region regions was a flux condition given by
@/
@ci
zi Di ci
Di
¼ Ni
@r
@r
ð18Þ
where N i was the flux in the r-direction at the boundary in the 1-D
delaminated region. The value of N i was given by
Ni ¼ zi Di ci
@/
@c
Di i
@r
@r
The conservation equation for species i in a dilute environment
given by Eq. (14) was cast to a 1-D domain in cylindrical coordinates to give
ð20Þ
where Ni was the flux of a species i in the r-direction and Si the production of species from electrochemical reactions from the domain
region > r. The term Si = 0 for the chemically inert species Na+ and
Cl that did not participate in electrochemical reactions. For the
reacting species Fe+2 and OH,
SFeþ2 ¼
iFe
2Fg
ð21Þ
iH2
Fg
ð22Þ
and
SOH ¼
4. Results
The model results were used to elucidate the role of ionic transport within the region in which the coating is fully delaminated. In
addition, an expression was developed to relate the potential drop
in the delaminated region to parameters which could be accessed
by existing CP models.
4.1. Contribution of diffusion transport in delaminated regions
2.4. One-dimensional delaminated region
1 dðrN i Þ
þ Si
r
dr
The assumed values of diffusion coefficients were DNaþ ¼ 1:3341
105 cm2s1, D Cl ¼ 2:0344 105 cm2s1, DOH ¼ 5:2458 105
cm2s1, and DFeþ2 ¼ 0:71172 105 cm2 s1 [40]. The assumed value of DO2 ¼ 2:92 105 cm2s1 corresponded to the diffusion
coefficient in dilute NaCl solution [41].
ð19Þ
where ci and /* were the concentration of a species i and the
dimensionless solution potential in the 1-D delaminated region on
the common boundary.
0¼
965
respectively, where g was the gap size of the delaminated region.
The current densities iFe and iH2 at a position r used in calculating
SFeþ2 and SOH were given by Eqs. (4) and (6), respectively, with U
being replaced by U. The condition of electroneutrality was applied
throughout the domain. The values of ci and /* at the boundary between the 2D-delaminated region and 1D-delaminated region
boundary were set to the conditions at the position located midway
on the boundary of 2D-delaminated region, ci and /*. This approach
was justified because, at this location, these values were independent of the direction normal to the metal surface. The no-flux and
electroneutrality conditions were applied at the intact boundary.
3. Numerical method
The linearized governing equations for species mass-transfer
and electroneutrality were discretized using second-order difference approximations. The resulting non-linear systems of equations were solved using an iterative method. The mathematical
model was developed using Compaq Visual Fortran, Version 6.1 with
double precision accuracy. The algorithm is described in reference
[32]. In all the simulations grid spacings of Mr = 0.025 cm and
Mz = 0.0125 cm were used in the radial and axial directions, respectively. The holiday radius and delaminated region gap were fixed at
0.25 cm and 0.05 cm, respectively. The convergence criterion for
the iterative algorithm was 0.01%.
The axial variation of potential and concentration in the delaminated region was negligible, and the dominant component of current density was in the radial direction. The percentage
contribution of the radial diffusion component of current density
to the total radial component, i.e., ir,d/ir, was used as a measure of
the importance of diffusion transport.
The distribution of the radial component of current density in a
1 cm deep delaminated region is shown in Fig. 3(a) with applied
potential as a parameter. The current density had a positive value
in the delaminated region as current was supplied to the metal surface. The variations of ir with position were similar for the three applied potentials. The value of ir decreased with position
approaching a zero-value as the intact boundary was approached.
For a given position in the delaminated region, larger values of ir
were associated with more negative applied potentials. This was
because hydrogen evolution became more significant with more
negative applied potentials. Also, for a given bulk electrolyte resistivity and at a given position in the delaminated region, a change to
a more negative applied potential resulted in increases in the absolute values of ir,d and the contribution due to migration ir,m.
The percentage contribution of ir,d to ir as a function of position in
the delaminated region is shown in Fig. 3(b) with applied potential as
a parameter. For r > 1.225 cm the ir,d-contributions is not included as
ir approached a zero value due the no-flux condition for ci at the coating boundary, r = 1.25 cm. The variations of the contributions of ir,d
(and ir,m) with position were similar for the three applied potentials.
For a given applied potential, the value of the ir,d-contribution decreased with position in the delaminated region and approached an
asymptotic value. The asymptotic values for the applied potentials
were 45 to 52% indicating that the contribution of ir,d to the transport
in the delaminated region was significant for the applied potential
range. At the entry to the delaminated region the ir,d-contribution
was approximately 55% for the three applied potentials.
The bulk electrolyte provided the boundary conditions for the
composition at the mouth of the holiday. The distribution of the radial ionic current density in a 1 cm disbondment is shown in
Fig. 4(a) with bulk resistivity as a parameter. The distributions of
ir were similar for the 79 kX cm and 0.79 kX cm bulk resistivities
were similar to that of 7.9 kX cm bulk resistivity that was discussed. For a given position in the delaminated region, larger values of ir were associated with smaller resistivities.
The percentage contributions of ir,d to the radial current density
as a function of position are shown in Fig. 4(b) with bulk resistivity
as a parameter. For the bulk resistivity of 7.9 kX cm, the ir,d-contribution decreased with position with the overall decrease being
approximately 5%. The ir,d-contributions was uniform for the bulk
resistivities of 0.79 and 79 kX cm for positions greater than
0.29 cm. For positions ranging from 0.25 cm to 0.29 cm, the value
of ir,d decreased by less than 2%. Therefore, the relative contributions of the diffusion current density to the ionic current density
966
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
a
b
30
20
-0.723 V
-1
-2
i r / μA cm
100 i r,d ( i r ) / %
55
-0.823 V
-0.773 V
-0.723 V
10
50
-0.773 V
-0.823 V
45
0
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
r / cm
1.00
1.25
r / cm
Fig. 3. (a) Calculated value of ionic current density and (b) relative diffusion current density a a function of position in the delaminated region with applied potential as a
parameter. The depth of the delamination was 1 cm and the applied potential was 0.773 V(SCE). The dashed line at r = 0.25 cm indicates the boundary between the holiday
and the delaminated region.
a
b
70
-1
0.79 kΩ cm
7.9 kΩ cm
79 kΩ cm
-2
i r / μA cm
100 i r,d ( i r ) / %
0.79 kΩ cm
20
10
60
50
7.9 kΩ cm
40
79 kΩ cm
0
0.25
0.50
0.75
1.00
1.25
0.25
0.50
0.75
1.00
1.25
r / cm
r / cm
Fig. 4. (a) Calculated values of ionic current density and (b) relative diffusion current density as functions of position in the delaminated region with bulk resistivity as a
parameter. The depth of the delamination was 1 cm and the applied potential was 0.773 V(SCE). The dashed line at r = 0.25 cm indicates the boundary between the holiday
and the delaminated region.
was approximately uniform except for a maximum change of 5% at
a bulk resistivity of 7.9 kX cm. For a given position in the delaminated region, larger relative contributions of the diffusion current
were associated with smaller bulk resistivities. This was consistent
with the theory of supporting electrolytes, where the effect of
migration was reduced with decreased resistivity [38].
The distribution of the radial ionic current density ir in the disbondment is shown in Fig. 5(a) with disbondment length as a
parameter. The variations of ir with position were similar for the
different disbondment lengths. The value of ir decreased with position and approached a zero value as the disbondment tip was approached. For a given position in the disbondment that was
common between two distributions, a larger value of ir was associ-
b
60
3 cm
2 cm
1 cm
-1
40
100 i r,d ( i r ) / %
i r / μA cm
-2
a
ated with the longer delaminated region. This was because a larger
cathodic current on the metal surface was associated with the
longer delaminated region.
The relative contribution of ir,d to ir in the delaminated region are
shown in Fig. 5(b) with delaminated region length as a parameter.
The variation of the ir,d contribution for the 2 and 3 cm disbondments were similar to the corresponding variations for the 1 cm
delaminated region. For a position common between two distributions, a smaller value of the ir,d-contribution was associated with the
longer delaminated region. This demonstrated that the contribution
of diffusion in the delaminated region was reduced with delaminated region length. However, the ir,d-contribution was still significant,
especially over the first 1 cm into the delaminated region.
20
1 cm
50
2 cm
3 cm
40
0
0
1
2
r / cm
3
0
1
r / cm
2
3
Fig. 5. (a) Calculated values of the ionic current density and (b) relative diffusion current density as functions of position in the delaminated region with delaminated region
length as a parameter. The dashed line at r = 0.25 cm indicates the boundary between the holiday and the delaminated region.
967
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
4.2. Potential distribution in the delaminated region
Results from the model were used to develop an expression for
the potential drop as a function of delaminated region length and
bulk resistivity for a given applied potential. The concentrations
of the bulk NaCl used were approximately 0.01, 0.005, 0.001,
0.0005, 0.0002, and 0.0001 M. These corresponded to bulk resistivities of 0.79 1.58, 7.9, 15.8, 39.5, and 79 kX cm, respectively. The
applied potential used in the simulations was 0.773 V(SCE), similar to a criterion used for the implementation of impressed-current cathodic protection.
The distributions of solution potential and local electrolyte
resistivity, expressed as a percentage of the bulk resistivity, are
shown in Fig. 6(a) and (b), respectively, with delaminated region
length as a parameter for a bulk electrolyte resistivity of 0.79 kX
cm. The distributions of solution potential were similar for the different lengths, with a more negative value of the potential at the
innermost part of the delaminated region, the intact boundary,
associated with a longer delaminated region. The distributions of
local electrolyte resistivity were also similar with a smaller local
electrolyte resistivity at the intact boundary associated with a
longer delaminated region.
The model results were used to calculate the potential drop
across the delaminated region and the resistivity at the boundary
between the delaminated and intact regions. These values are
presented in Fig. 7(a) and (b), respectively, as functions of delamination depth with bulk resistivity as a parameter. For a given
bulk resistivity, larger solution potential drops and smaller local
electrolyte resistivities were associated with longer delaminated
regions. For a given delaminated region length, larger solution potential drops were associated with larger bulk resistivities. The
change in the local electrolyte resistivity was larger for larger
bulk electrolyte resistivities for a given delaminated region
length.
The potential drop across the delaminated region is seen to be a
function of the delaminated region length and bulk resistivity. A
design equation was developed for conditions of a 0.5 cm holiday
radius, 0.05 cm gap size, 0.05 cm coating thickness, and a metal potential of 0.773 VSCE relative to a zero value of solution potential
at the mouth of the holiday.
The equation of a distribution was assumed of the form
MU ¼ a1 þ
a2
1 þ expðða3 r d Þ=a4 Þ
ð23Þ
where a1, a2, a3, and a4 were design parameters. The values of these
design parameters were obtained by fitting by inspection the distribution to the model-calculated values. The values of the design
parameters are given in Table 1.
a
0
The value of MU as a function of delaminated region length with
bulk resistivity as a parameter calculated using these design
parameters and Eq. (23) are shown in Fig. 8 as dashed lines. There
was a good correlation between the model-calculated (symbols)
and equation-calculated (dashed lines) values for MU.
The design parameters are shown as functions of the bulk resistivity in Fig. 9. To include an explicit dependence of Eq. (23) on
resistivity, the expression
ak ¼ pk lnðq1 Þ þ qk
was fitted to each parameter where pk and qk were fitting parameters
for the design parameters ak. The values of the fitting parameters,
obtained using Microsoft EXCEL, were p1 = 2.1715 mV, q1 =
4.1785 mV, p2 = 10.857 mV, q2 = 52.559 mV, p3 = 1.32 46 cm,
q3 = 5.8378 cm, p4 = 0.1954 cm, and q4 = 2.6706 cm.
The expression for the solution potential drop in the delaminated region was
MUðr d ; q1 Þ ¼ a1 ðq1 Þ þ
ð25Þ
5. Discussion
Many mathematical models of pitting and parallel-sided systems have been presented in the literature that calculate the steady-state and transient conditions within the occluded domain.
These include the hemispherical pitting system [42–46]. the parallel-sided delaminated region with an active tip and passive walls
[47–53] and the parallel-sided delaminated region with active
walls [54,47,53,55–59]. Very few models presented in the literature, however, address the coupling between the holiday and a delaminated region.
The assumptions used in constructing the models presented in
literature for holiday/delaminated-region systems and delaminated-region systems were aimed at reducing the complicity of the
system. In one model for a circular delaminated region surrounding a holiday, only oxygen reduction was considered significant
and the diffusion coefficients of the species were equated to eliminate the coupling of the governing equations for concentration
and solution potential [35,36]. In the transient model for a rectangular holiday with a delaminated region adjacent to one side, elec-
100
1 cm
3 cm
80
100 ρ / ρ∞
Φ / mV
a2 ðq1 Þ
1 þ expða3 ðq1 Þ rd Þ=a4 ðq1 Þ1
where the design parameters were functions of bulk resistivity in
kX cm given by Eq. (24). The value of the MU as a function of delaminated region length is shown in Fig. 8 with bulk electrolyte resistivity as a parameter. The values of model-calculated MU are shown
together with the fits of Eq. (23) and the design-equation-calculated
values of Eq. (25). There was agreement between the values of MU
calculated using the design equation and the model.
b
1 cm
ð24Þ
5 cm
-20
8 cm
10 cm
-40
60
3 cm
40
5 cm
8 cm
20
10 cm
0
0
2
4
6
r / cm
8
10
0
2
4
6
8
10
r / cm
Fig. 6. Calculated (a) solution potential and (b) local electrolyte resistivity as functions of position on the metal surface with delaminated region length as a parameter for
bulk electrolyte resistivity of 0.79 kX cm. The dashed line at r = 0.25 cm indicates the boundary between the holiday and the delaminated region.
968
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
a
b
0
100
0.79 kΩ cm
80
ΔΦ / mV
100 ρtip / ρ∞
0.79 kΩ cm
-20
-40
7.9 k Ω cm
-60
7.9 kΩ cm
40
20
79 k Ω cm
-80
60
79 k Ω cm
0
-100
0
2
4
6
8
10
0
2
4
6
8
10
r d / cm
r d / cm
Fig. 7. Calculated values for (a) solution potential and (b) resistivity at the intact boundary as functions of delaminated region length with bulk resistivity as a parameter. The
trends associated with the values are given by the dashed lines.
Table 1
Design parameter values associated with design equation for the solution potential
drop.
Bulk resistivity (kX cm)
Design parameters
a1 (mV)
a2 (mV)
a3 (cm)
a4 (cm)
0.79
1.58
7.9
15.8
39.5
79
4
5
8
10
13
14
50
59
75
85
95
100
6.2
5.5
3
2.2
1
0.1
2.8
2.6
2.1
2.0
2.0
1.9
0
0.79 k Ω cm
ΔΦ / mV
-20
-40
7.9 k Ω cm
-60
79 k Ω cm
-80
-100
0
2
4
6
8
10
r d / cm
Fig. 8. Calculated solution potential drop as a function of delaminated region length
with bulk resistivity as a parameter. The symbols represent values calculated using
the mathematical model, the dashed line represent the fit of Eq. (23), and the solid
line represent the values using design equation given by Eq. (25).
troneutrality was not maintained explicitly but was satisfied by
adjusting the concentration of a specie such as Na+ that was not involved in homogeneous or electrochemical reactions [60].
Models of a delaminated region adjacent to a holiday have been
presented where the influence of the holiday was not considered. A
method involving control volumes was developed for a transient
model for the conditions of the one-dimensional delaminated region system [61]. Song et al. assumed the concentration gradients
in the delaminated region were negligible and the solution potential was governed by Laplace’s equation [24,25]. Another model
was presented where the holiday was considered and a twodimensional solution was obtained [62,63].
The present model calculated the steady-state electrolyte composition in the holiday/crevice system while accounting for the
presence of a holiday, transport by diffusion and migration, and
relevant cathodic and anodic electrochemical reactions. The model
was used to demonstrate that the influence of diffusion was significant for the ±50 mV range about the 0.730 V criterion, for the
bulk resistivity ranging from 0.79 to 79 and for any delaminated
region length. The influence of diffusion was seen to be the greatest
at the entry of the delaminated region from the holiday. These results demonstrated that the commonly used assumption that concentration gradients can be neglected in the delaminated region is
not valid. The interpolation formula presented as Eq. (25) provides
an approach for representing the simulation results while accounting for the resistivity of the environment exterior to the holiday
and the length of the delaminated coating. The expression used requires assumption of a fixed delamination gap thickness, a fixed
holiday size, and a fixed applied potential. These limitations can
be alleviated by using the model and the approach outlined to develop analogous expressions.
Future extensions to this work should account for the permeability of the coating to dissolved gases and ionic species, homogeneous reactions such as dissociation of water and ferrous ion
hydrolysis, pH-dependence of electrochemical reactions, and use
of concentrated rather than dilute solution expressions for transport phenomena. Inclusion of these phenomena may be expected
to yield competing influences on the estimated variation of potential within the delaminated region. Local increases in pH will result
in reduced electrical current associated with the hydrogen evolution reaction [55]; whereas, transport of oxygen through a permeable coating will increase the oxygen reduction current. Rouw
observes, however, that the barrier property of organic coatings
alone is not sufficient to prevent corrosion and that the most significant contribution of a coating to the mitigation of corrosion
arises from the reduction of the rates of transport of ionic species
[64]. Inclusion of coating permeability to ionic species will tend
to reduce the variation of potential associated with large concentration gradients. It should be noted that the species included in
the present model were the dominant species needed for calculation of electrolyte resistivity. Addition of H+ and Fe(OH)+ concentrations as variables would not contribute significantly to the
resistivity. Thus, inclusion of H+ and the water dissociation reaction
would improve the pH calculation in the delaminated region, but
would have minimal impact on the electrolyte resistivity.
While, as described above, refinements to account for homogeneous reactions and coating permeability are desirable, the present
work serves the useful purposes of illustrating the importance of
changes in local composition within the delaminated region and
969
K.N. Allahar, M.E. Orazem / Corrosion Science 51 (2009) 962–970
a
15
b
a2 / mV
a1 / mV
-60
10
-80
5
-100
0
20
40
60
0
80
20
c
60
80
60
80
d
6
2.5
4
a4 / cm
a3 / cm
40
ρ∞ / kΩ cm
ρ∞ / kΩ cm
2
2.0
0
0
20
40
60
80
0
ρ∞ / kΩ cm
20
40
ρ∞ / kΩ cm
Fig. 9. Parameters for solution potential drop equation as functions of bulk electrolyte resistivity. (a) a1, (b) a2, (c) a3, and (d) a4.
of providing a means for modifying existing CP programs to estimate the potential underneath delaminated regions.
6. Conclusions
A mathematical model was presented for the steady-state conditions in a delaminated region surrounding a circular holiday on a
metal surface under cathodic protection. Oxygen reduction and
distributions normal to the metal surface were found to be negligible for delaminated regions greater than 3 cm. The results demonstrate that the commonly employed assumption that
concentration gradients are negligible is not valid within the delaminated region. The results from the model were used to construct
an expression for the potential drop in a holiday-delaminated region system as a function of bulk resistivity and delaminated region length for a given applied potential. The expression can also
be used to estimate the potential drop from the bulk to the holiday
metal surface in the absence of a surrounding delaminated region.
This expression can be used in conjunction with existing CP models
to refine estimates for the potential requirement of an impressedcurrent system. The use of the equation avoids the coupling of a
full-numerical model solution of the holiday-delaminated region
system to a CP model. The development of a transient model for
the pseudo steady-state electrochemistry of a crevice surrounding
a holiday that incorporates a pH-dependent hydrogen evolution
reaction, water dissociation explicitly, and coating permeability
to oxygen and ionic species is recommended. The results associated with this model could be used to construction analogous
expressions for the potential drop similar to the approach outlined
in the present work.
Acknowledgements
This work was supported by the Gas Research Institute through
Grant 5097-260-378. The authors acknowledge the contribution of
Dr. N. Sridhar at the Center for Nuclear Waste Regulatory Analysis,
Southwest Research Institute, San-Antonio, Texas.
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