Volume 9 Preprint 8
The slope parameter approach to galvanic anode cathodic protection design with application to marine structures
William H. Hartt
Keywords: Cathodic protection, design, Slope Parameter, marine,<br>galvanic anode, Unified Design Equation, mean current density
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Volume 9 Paper 8
The Slope Parameter Approach to Galvanic
Anode Cathodic Protection Design with
Application to Marine Structures
William H. Hartt,
Centre for Marine Materials, Department of Ocean Engineering, Florida
Atlantic university – Sea Tech Campus, Dania Beach, FL 33004 USA,
Both the newly developed Slope Parameter based method and the
conventional approach for marine cathodic protection design that
utilizes three current densities and reviewed. While the former is
shown to be technically more sound because it is first-principles
based, a historical shortfall in both cases has been absence of a
method also based upon first-principle for projecting design current
density. In the present paper, an approach is developed for
determining, either from literature data or from short-term in-situ
exposures, a value for the design mean current density; and it is
demonstrated how doing this facilitates application of the Unified
Keywords: Cathodic protection, design, Slope Parameter, marine,
galvanic anode, Unified Design Equation, mean current density.
This is a preprint of a paper that has been submitted for publication in the Journal of Corrosion Science
and Engineering. It will be reviewed and, subject to the reviewers’ comments, be published online at
http://www.umist.ac.uk/corrosion/jcse in due course. Until such time as it has been fully published it
should not normally be referenced in published work. © UMIST 2004.
For metals and alloys that lack inherent corrosion resistance, corrosion
control in submerged and buried applications is invariably affected by
cathodic protection (cp) or by cp and coatings together. In the latter
case, the role of the coating is often to render cp more effective and
efficient rather than to provide corrosion protection per se.
Significant advances have transpired during the past 50 or so years in
development of both the theory and practice of cp, largely in response
to escalation in petroleum product production and transportation and
to expansion of these activities offshore. Hallmark events regarding
the development and practice of cp in marine applications include:
The classical inception research of Davy1,2,3 nearly 200 years
Use of galvanic anodes on U.S., British, and Canadian Navy
ships during the first half of the 20 th century.4
Recognition of -0.80 V Ag/AgCl as the protection potential
threshold for steels in natural waters.
Recognition of the role of calcareous deposits 5,6 ,7,8 ,9,10,11
formed on cathodic surfaces in sea water as contributing to the
efficiency and effectiveness of cp.
Introduction and refinement of recommended practices for cp
Development of high performance galvanic Al anodes.15,16,17
Incorporation of the “rapid polarization” concept8,9,18,19 into cp
design of offshore structures.
Development of numerical methods for cp design and
Development of mixed-metal oxide (MMO) anodes for
impressed current (ic) cp.
10. Development of the Slope Parameter concept and the Unified
In this paper, the last of these items is critically reviewed in terms of,
first, basic concepts and, second, situations to which the Slope
Parameter has utility. Also, a method is proposed based upon shortterm exposures for determining the mean design current density.
Transients result when a metal is polarized, as in cp, from its steadystate free corrosion potential. Figure 1 exemplifies this as a plot of
potential (c) and current density (ic ) as a function of time subsequent
to exposure of uncoated 200 mm square steel specimens connected to
a galvanic Al anode through a resistor of the indicated size (designated
R x) to quiescent natural sea water. The initial decay in both c and ic
reflects progressive control by oxygen concentration polarization, but
continued decay of these parameters for the R x=55 specimen was in
response to this control being enhanced by calcareous deposit
formation (item 4 above). The data also illustrate the principle of
“rapid polarization” (item 7) in that this same specimen required less
current density for polarization to a more negative potential than for
the R x=110 specimen, in apparent contradiction to the normal
dependence of c upon ic .
Figure 2 schematically illustrates the relationship between c and
current (I c) for such a polarized system upon establishment of a
polarized steady-state. Here, the steel has been polarized from its
corrosion potential (corr(steel)) to a more negative potential c where,
for protection, c is -0/80 VAg/AgCl or more negative (item 3 above) and
the anode to a , the difference between the two potentials being the
product of the anode current output and net circuit resistance (Ia and
R t, respectively). The latter parameter (Ia ·R t) serves as the basis, in
many instances, for cp design as explained below.
Figure 3 shows steady-state c -ic data for individual steel specimens
that were polarized in natural sea water by a galvanic Al anode but
with an external resistor of the indicated size (R x=75 to 5,750 ) in
the circuit between anode and cathode.26 If a potential scan were
Potential, V (Ag/AgCl)
Rx = 55 Ohms
Rx = 110 Ohms
Exposure Time, hours
Current Density, mA/m2
Rx = 55 Ohms
Rx = 110 Ohms
Exposure Time, hours
Figure 1: Plots of (a) potential and (b) current density versus exposure
tine for steel-Al anode couples in natural sea water.
- Potential +
1 O H O 2e 2OH
H O e H OH
Figure 2: Schematic illustration of the polarized steady-state of steel in
natural water upon cathodic polarization by a galvanic Al anode.
Potential , VSCE
Current Density, mA/m 2
Figure 3: Steady state potential-current density data for steel
specimens connected through different size resistors to aluminium
performed on a particular specimen from c to corr(steel) subsequent
to the polarized steady-state being achieved, a cathodic curve as in
Figure 2 would result. However, the steady-state c-ic trend for
multiple specimens that encompass a range of steady-state c values
assumes a sigmoidal shape where ic is maximum near -0.80 V Ag/AgCl
and minimal near -1.00 VAg/AgCl . Such behaviour is a consequence of
calcareous deposits that form in the range of the latter potential being
particularly protective. These data illustrate that, while protection is
achieved upon polarization to -0.80 V Ag/AgCl, the ic to affect this is
approximately three times greater than at -1.00 VAg/AgCl .
Consequently, the latter value is typically adapted as the cp target for
offshore structural steel installations.
Figure 426 expands upon Figure 3 and shows the c-ic decay trend for
typical individual specimens, each with a unique Rx, from early
exposure to steady-state (or nearly so). Data at specific times (24,
120, 460, and 3200 hours) have been connected, thereby defining the
cathodic polarization curve at these times. This reveals the transition
from a cathodic polarization curve characterized by classical oxygen
concentration polarization (24 hours) to the long-term sigmoidal one
that reflects various degrees of calcareous deposit formation (3200
Rx , O hms
Current Density, mA/m2
Figure 4: Potential-current density decay trends at four different times
from experiments involving different Rx.
Likewise, Figure 5 shows a similar plot to the one in Figure 4 for
SS316; 27 and Figure 6 illustrates the long-term, steady-state results
alone. These reveal that, while the peak and minimum ic for the SS
and carbon steel are approximately the same, these tend to occur at a
more negative potential for the former. This may reflect reduced
catalytic efficiency for the SS such that greater polarization was
required to affect the same ic and calcareous deposit formation rate.
9 6 0h rs
2 2 0 0h rs
4 8 hrs
2 4 0d y s
1 0 0 h rs
48 0 hrs
3 6h rs
1 4 hrs
2 7 h rs
C u rr e n t D e n s it y , m A / m
Figure 5: Potential-current density decay trends at four different times
for experiments involving different R x.
Galvanic Anode CP Design for Offshore Structures
Early cp design procedures for offshore structures specified a
single current density (the mean, im , which is analogous to I c in Figure
2); and the corresponding required number of anodes, Nm , to provide
this current density for the design life, T, was calculated from a
modified form of Faraday’s law,
C u r re n t D e n s it y , m A / m
Figure 6: Steady state c-ic data for SS specimens connected through
different size resistors to aluminium anodes.
N m c
Ac = structure surface area,
C = anode current capacity,
w = weight of an individual anode, and
u = anode utilization factor.
However, with recognition of the rapid polarization phenomenon, the
design method transitioned to one based on not only im but also an
initial (io ) and final (if) current density, where the former is relatively
large in order to affect rapid polarization and the latter is of sufficient
magnitude to ensure repolarization should calcareous deposits
become storm disrupted in late life. These current densities are a
function of water temperature and movement and, hence, vary
spatially. Tables 1 and 2 list values of io and if and of i m, respectively,
according to one recommended practice. 13
Table 1: Design values for i o and if.
Table 2: Listing of i m values for offshore structures in different
By this method, the design approach involves, first, calculation of Ia
based upon assuming a value for the voltage drop in Figure 2 for both
the initial and final conditions and assuming Rt ≈Ra , where Ra is
resistance of an individual anode. Thus,
The N corresponding to the initial and final conditions (No and Nf,
respectively) is then determined from the expression,
N c c ,
where i c is successively set equal to io and if. In doing this,
consideration is given to the fact that c and ic decrease and Ra
increases as the exposure progresses. Ideally, the three N values
would be the same; however, this is seldom the case since the method
is an algorithm rather than being first-principles based.
Consequently, the highest of the three N’s is selected, meaning that
over-design results for the other two criteria. This over-design has
been projected to be by approximately 32 percent for Gulf of Mexico
The Slope Parameter Concept for CP Design and Analysis
The Slope Parameter, S, analytically represents the c-i c decay path
along which polarization occurs. Thus, from a modified representation
of Ohm’s law, 28
c = (Rt·Ac)·ic + a ,
a linear interdependence between c and ic is projected provided Rt·A c
and a are constant with time, as is generally the case for steels in
natural waters The Slope Parameter then is defined by the expression,
S R t
Ac c .
Figure 7 shows a c-ic plot for the Rx = 55 and 110 specimens in
Figure 1 and indicates a linear trend in both cases. The slopes
Potential, V (Ag/AgCl)
Rx = 55 Ohms
Rx = 110 Ohms
Current Density, mA/m2
Figure 7: Plot of c versus ic for cathodically polarized steel specimens
in natural sea water.
measured from the graph are 2.3 and 4.3 ∙
m2 for Rx = 55 and 110 ,
respectively, whereas the corresponding calculated values (Equation 5),
assuming R x=Rt, are 2.2 and 2.4 ∙
m2, thus indicating good agreement
between the two. The data for different specimens in Figures 4 and 5
also illustrate an approximately linear c -i c decay with time.
The Unified Design Equation
It can be assumed that multiple, identical galvanic anodes upon space
frame structures act as resistors in parallel. Thus,
Rt a .
Combining Equations 1, 5, and 6 then yields,
which is termed the Unified Design Equation. Because all terms on the
right side are design choices, the process is reduced to anode
dimensioning such that the requisite value for R a∙
w is realized.
Equation 7 is first-principles based and incorporates both io and im ,
the former implicitly within S (Equation 5) and the latter explicitly.
Current Density to Affect CP
While the criterion for cp is in terms of potential (item 3 above), design
is accomplished via current density (Tables 1 and 2). However, with
due respect to design ic values, there is no first-principles method for
projecting these. In this regard, Figure 8 shows a compilation of ic
versus time data from the literature 29 for various structures and
exposure panels worldwide and reveals a power law relationship
beyond about 100 hours between these two parameter as,
ic 10 c
where c and d are constants. Table 3 lists values for these two
parameters considering that the warm and cold water data conform to
distinct populations. Likewise, by integrating this expression over the
design life, and dividing by that time, im is determined as,
where k is a multiple of the standard deviation, , for the warm or cold
water data. Values for are 0.23 for warm water and 0.23 for cold.
Current Density, mA/m^2
Figure 8: Cathodic protection current density as a function of
exposure time for offshore structures and test panels.
Table 3: Values for the Equation 8 constants.
Application of the Slope Parameter in CP Design
In a design situation for which relevant data from prior exposures does
not exist, test panels similar to the specimens from which the data in
Figures 1 and 7 were developed and with appropriately sized resistors
(R x) can be exposed at the site of interest and c and ic recorded as a
function of time. The exposures need only continue until it can be
assured that the knee on the sigmoidal c -i c curve has been missed
and an appropriate value for S thereby assessed.
Selection of an appropriate im is more complex since data scatter in
Figure 8, even after partitioning according to warm versus cold water,
can approach an order of magnitude. However, provided exposure
conditions are such that a stable film forms, results for individual
experiments have proven to exhibit less scatter than in Figure 8, as
shown by Figures 1 and 7 for laboratory specimens and by Figure 930
for a 410 day panel exposed in deep water Gulf of Mexico. The values
for c and d (Equation 8) and for are then determined and an
appropriate k is selected according to the desired conservatism. The
corresponding im is then calculated from Equation 9. This method
avoids any need for extending the test panel exposure to long-term.
Based upon this i m and the corresponding S (see above), an optimized
design for the galvanic anode cp system can be accomplished based
upon Equation 7.
Current Density, mA/m2
Figure 8: Current density versus time data for a steel panel
cathodically polarized by an Al anode in deep water Gulf of Mexico.
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