Finite Element Modeling of Galvanic Corrosion of Metals
by
Megan Elizabeth Turner
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December 2012
© Copyright 2012
by
Megan Turner
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ...................................................................................................... viii
GLOSSARY ..................................................................................................................... ix
LIST OF KEYWORDS ..................................................................................................... x
ACKNOWLEDGMENT .................................................................................................. xi
ABSTRACT .................................................................................................................... xii
1. Introduction.................................................................................................................. 1
1.1
Background ........................................................................................................ 1
1.2
Problem Description........................................................................................... 1
1.3
Prior Work.......................................................................................................... 2
2. Methodology ................................................................................................................ 4
2.1
Theory ................................................................................................................ 4
2.2
Geometry and Boundary Conditions.................................................................. 4
2.3
Corrosion Rate ................................................................................................... 8
3. Results and Discussion ................................................................................................ 9
3.1
3.2
3.3
Model Validation ............................................................................................... 9
3.1.1
Using the Butler-Volmer Relationship for Current Density-Potential ... 9
3.1.2
Using a Basic Geometry Subjected to Polarization ............................. 11
Results for Different Electrode Geometries ..................................................... 13
3.2.1
Electrode Geometry with a 90 Degree Step ......................................... 13
3.2.2
Electrode Geometry with an Elliptical Step ......................................... 15
3.2.3
Axisymmetric Electrode Geometry...................................................... 17
Results for Different Electrolyte Thicknesses Using a Copper-Zinc Galvanic
Couple .............................................................................................................. 19
4. Conclusion ................................................................................................................. 24
iii
5. References.................................................................................................................. 26
6. Appendix A................................................................................................................ 27
7. Appendix B ................................................................................................................ 29
iv
LIST OF TABLES
Table 1: Input Data Used for Model Validation [3] ......................................................... 9
Table 2: Summary of Data for the Different Geometries ............................................... 18
Table 3: Input Data Used for Copper-Zinc Galvanic Couple [3] ................................... 20
Table 4: Properties of Zinc ............................................................................................. 19
Table 5: Summary of Results for the Copper-Zinc Galvanic Couple with an Electrolyte
2mm and 20mm Thick ..................................................................................................... 23
Table 6: Input Data Used for Copper-Zinc Galvanic Couple [3] ................................... 29
Table 7: Summary of Results for the Copper-Zinc Galvanic Couple with an Electrolyte
2mm and 20mm Thick ..................................................................................................... 32
v
LIST OF FIGURES
Figure 1: Coplanar Model Used for Initial Model Validation [3] .................................... 2
Figure 2: Marine Propulsion Unit Analyzed in Reference [6] ......................................... 3
Figure 3: 2D Cartesian Geometry ..................................................................................... 5
Figure 4: COMSOL Results for B-V Current Density Distribution on the Electrode
Surfaces (Surface Plot) .................................................................................................... 10
Figure 5: COMSOL Results for B-V Current Density Distribution and from Reference
[3]..................................................................................................................................... 10
Figure 6: Current Density and Reaction Rate for the Basic Geometry .......................... 11
Figure 7: COMSOL Results for the System Subjected to Polarization (Surface Plot) .. 12
Figure 8:
COMSOL Results for the System Subjected to Polarization and from
Reference [3] ................................................................................................................... 12
Figure 9: Current Density and Reaction Rate for the System Subjected to Polarization 13
Figure 10:
COMSOL Results for the Electrode Geometry with a 90 Degree Step
(Surface Plot) ................................................................................................................... 14
Figure 11: COMSOL Results for the Electrode Geometry with a 90 Degree Step ........ 14
Figure 12: Current Density and Reaction Rate for the Electrode Geometry with a 90
Degree Step ...................................................................................................................... 15
Figure 13:
COMSOL Results for the Electrode Geometry with an Elliptical Step
(Surface Plot) ................................................................................................................... 16
Figure 14: COMSOL Results for the Electrode Geometry with an Elliptical Step ........ 16
Figure 15: Current Density and Reaction Rate for the Electrode Geometry with an
Elliptical Step................................................................................................................... 17
Figure 16: COMSOL Results for the Axisymmetric Electrode Geometry (Surface Plot)
......................................................................................................................................... 17
Figure 17: COMSOL Results for the Axisymmetric Electrode Geometry..................... 18
Figure 18: Current Density and Reaction Rate of the Axisymmetric Electrode Geometry
......................................................................................................................................... 18
Figure 19: Copper-Zinc Galvanic Couple in a Hydrochloric Acid Solution [3] ............ 19
Figure 20: COMSOL Results for the Copper-Zinc Couple with 2mm and 20mm Thick
Electrolyte (Surface Plot) ............................................................................................... 21
vi
Figure 21: COMSOL Results for the Copper-Zinc Couple with 2mm Thick Electrolyte
......................................................................................................................................... 21
Figure 22: COMSOL Results for the Copper-Zinc Couple with 20mm Thick Electrolyte
......................................................................................................................................... 22
Figure 23: Current Density and Corrosion Rate for Copper-Zinc Couple with 2mm
Thick Electrolyte ............................................................................................................. 22
Figure 24: Current Density and Corrosion Rate for Copper-Zinc Couple with 20mm
Thick Electrolyte ............................................................................................................. 23
Figure 25: COMSOL Results for Linear Current Density Distribution on the Electrode
Surfaces (Surface Plot) .................................................................................................... 27
Figure 26: COMSOL Results for Linear Current Density Distribution on the Electrode
Surfaces............................................................................................................................ 28
Figure 27: COMSOL Results for the Copper-Zinc Couple with 2mm and 20mm Thick
Electrolyte (Surface Plot) ............................................................................................... 30
Figure 28: COMSOL Results for the Copper-Zinc Couple with 2mm Thick Electrolyte
......................................................................................................................................... 30
Figure 29: COMSOL Results for the Copper-Zinc Couple with 20mm Thick Electrolyte
......................................................................................................................................... 31
Figure 30: Current Density and Corrosion Rate along the Anode Surface of the CopperZinc Couple with 2mm Thick Electrolyte ....................................................................... 31
Figure 31: Current Density and Corrosion Rate along the Anode Surface of the CopperZinc Couple with 20mm Thick Electrolyte ..................................................................... 32
vii
LIST OF SYMBOLS
𝑖
current density (A m-2)
𝑞
charge density (C m-3)
t
time (s)
E
electric field intensity (N C-1)
𝜙
corrosion potential (V)
σ
conductivity (Ω-1 m-1)
𝑖0
free current density (A m-2)
𝜙0
free corrosion potential (V)
z
number of electrons transferred (unitless)
F
Faraday’s constant (96487 C mol-1)
𝑣
reaction rate (mol m-2 s -1)
CR
corrosion rate (mm year-1)
Mw
molecular weight (kg mol-1)
ρ
density (kg m-3)
𝑒−
electron
aa
anodic transfer coefficient (V)
ac
cathodic transfer coefficient (V)
α
anodic Tafel parameter (V)
β
cathodic Tafel parameter (V)
viii
GLOSSARY
Anode –
The portion of a galvanic couple which sees a decrease in potential along the
electrode surface
Cathode –
The portion of a galvanic couple which sees an increase in potential along the
electrode surface
Electrolyte –
A solution capable of carrying/transferring charge
(Electrochemical) Potential –
Measure of the voltage in a system
Galvanic Corrosion –
“…the corrosion that occurs as a result of one metal being in contact with
another in a conducting environment” [2]
Overpotential –
The difference between the actual potential and the free potential
ix
LIST OF KEYWORDS
COMSOL
Galvanic Corrosion
Electrochemical Potential
Current Density
Corrosion Rate
x
ACKNOWLEDGMENT
I would like to thank the Rensselaer faculty and staff for their commitment to educating
their students. I would especially like to thank Professor Gutierrez-Miravete for the
guidance he provided throughout this project.
xi
ABSTRACT
Corrosion is an ever-present problem in all different environments, particularly in
marine applications. The goal of this project is to develop a finite element model that
can be used with experimental data to characterize the corrosion of a galvanic couple in
an electrolyte. This project first addresses a simple system with coplanar electrodes.
The finite element model is developed using COMSOL Multiphysics Math Module
through
derivation
of
electrochemical kinetics.
equations
describing
corrosion
thermodynamics
and
The model is validated through replication of previously
determined experimental results and then applied to the study of some system
configurations relevant to marine applications.
xii
1. Introduction
1.1 Background
Corrosion is the breakdown of materials, namely metals, through electrochemical
reactions within their environment.
Corrosion is a consideration in virtually all
engineering applications. Each year, industries invest time and money into trying to
curtail the effects of corrosion.
Many different corrosive environments have been
studied and monitored to develop corrosion control methods [1].
There are different types of corrosion, including galvanic corrosion. Galvanic corrosion
is an electrochemical process that occurs when at least two dissimilar metals are in
electrical contact with one another, and are in a conducting environment known as an
electrolyte [2]. Of these dissimilar metals, one acts as the anode and the other acts as the
cathode.
The anode is the portion of the galvanic couple that undergoes material
dissolution due to the chemical properties of the metals and the environment. Galvanic
corrosion is particularly prevalent in marine applications because seawater acts as a
naturally free flowing electrolyte.
1.2 Problem Description
The amount and rate of corrosion can be correlated to the electrochemical potential
distribution within a system. The goal of this project is to develop a finite element
model that can be used in conjunction with experimental data to characterize the
corrosion of a galvanic couple in an electrolyte. There are a number of factors that can
contribute to the amount of corrosion within a system from material factors (e.g.
geometry) to environmental factors (e.g. type of electrolyte present) [1]. To limit the
effects of these factors, this project first utilized a simple model of coplanar electrodes as
shown in Figure 1 [3].
1
Figure 1: Coplanar Model Used for Initial Model Validation [3]
Utilizing a simple geometry allowed the initial focus to be on development of the
boundary conditions rather than generation of geometry representing a specific system.
Reference [3] data was used for finite element model validation.
Following
development and validation of the finite element model, additional galvanic couple
geometries were evaluated. This project is limited to galvanic couples of one anode and
one cathode.
Distributions of electrochemical potential within the electrolyte were
generated to describe the behavior of the system. Current densities and corrosion rates
along the anode surface were generated to quantify to amount of corrosion.
1.3 Prior Work
There is a lot of past precedence of scientists taking an interest in galvanic corrosion,
looking to understand what causes it and what limits or accelerates the process.
Numerous studies have been conducted; some take a more global outlook as in
Reference [4], whereas some take a more focused approach as in Reference [5].
The study conducted in Reference [4] looked at many different galvanic couples
commonly used in seawater applications. The study focused on developing reasonable
models for systems experiencing varying periods of exposure to the corrosive
environment. The studies were also conducted under potentiostatic and potentiodynamic
scenarios. The potentiostatic experiment allowed for analysis of the corrosion under the
natural potentials of the system. While the potentiodynamic experiment introduced
2
potential to the electrodes to analyze the impacts of changing the electrode potential on
the corrosion.
Other studies seek to analyze corrosion for a specific system geometry and environment.
Generally these investigations look to develop methods to prevent or minimize corrosion
of that specific system. This is most commonly achieved through a process known as
cathodic protection. This process involves introducing an additional part, known as a
sacrificial anode, to the system. The sacrificial anode is generally another piece of metal
that is introduced into a system so that it will corrode prior to other elements. Reference
[5] looked at developing a method to model cathodic protection of a carbon steel pipe in
seawater using one or two aluminum sacrificial anodes.
Reference [6] looked at
utilizing a sacrificial zinc anode to cathodically protect the marine propulsion unit shown
in Figure 2.
Figure 2: Marine Propulsion Unit Analyzed in Reference [6]
3
2. Methodology
2.1 Theory
The potential distribution within the electrolyte of a galvanic system is fundamentally
based on the continuity equation for conservation of charge [2] in the electrolyte.
−∇. 𝑖 =
In a steady state system,
𝜕𝑞
𝜕𝑡
𝜕𝑞
[1]
𝜕𝑡
= 0 and ∇. 𝑖 = 0
The relationship between the electric field intensity and the electric potential is
𝐸 = −∇𝜙
[2]
𝑖 = 𝜎𝐸
[3]
and Ohm’s Law is
where σ is the conductivity of the electrolyte.
From the above, the continuity equation becomes
∇𝜎∇𝜙 = 0
[4]
and for constant, isotropic conductivity this yields
∇2 𝜙 = 0
[5]
which is Laplace’s Equation for the potential distribution within the electrolyte.
2.2 Geometry and Boundary Conditions
The model geometry must be representative of the system being analyzed and must be
subjected to appropriate boundary conditions.
There are generally three types of
boundary conditions that are considered at the electrode surfaces: linear, exponential
(Tafel) and Butler-Volmer.
This project utilized the finite element program COMSOL Multiphysics to solve
Equation [5] for selected systems. Geometry was developed to represent the electrolyte
of the system being modeled.
During initial model and boundary condition
development, a simple 2D geometry as shown in Figure 1 was used.
4
This initial
coplanar model was created using a 2D Cartesian geometry as shown in Figure 3.
Additional geometries were also created and evaluated as discussed in Sections 3.2 and
3.3.
2
1
3
A
5
B
4
Figure 3: 2D Cartesian Geometry
The main assumption for this system is that there is no current flow normal to the
insulating boundaries [3]. This assumption is in accordance with Equation [1] at a
steady state, because the charge within the system cannot change. This allows for a Zero
Flux boundary condition to be applied to edges 1-2, 2-3, and 3-4, of Figure 3
Edges 1-5 (Electrode A) and 5-4 (Electrode B) are the electrode surfaces. For this
system, Electrode A is the cathode and Electrode B is the anode.
The electrode
boundary conditions were assigned by using a Flux/Source boundary condition to
represent the Butler-Volmer relationship
5
Note: This model is just a generic case to illustrate the methodology, and does not
represent specific materials’ reduction and oxidation reactions.
For instance, the
cathodic reaction could be oxygen reduction (O2 + e- → O2-) or hydrogen evolution (2H+
+ 2e- → H2) [2] and the parameters for those reactions could be inputs to the model for
specific cases.
The Butler-Volmer relationship for current density is based on the identifying the anodic
and cathodic reactions that are taking place on each electrode. In any environment, a
metal is undergoing both an anodic and a cathodic reaction. This characteristic is what
requires the net current density to account for both reactions on each metal, regardless of
if the metal is acting as the anode or the cathode in a galvanic couple. At the equilibrium
potential (zero overpotential) the anodic and cathodic currents are equal; this point is
known as the exchange current. However, when the overpotential is not equal to zero,
the anodic and cathodic currents are different.
Therefore, for a given electrode, the anodic current density is
𝑖𝑎𝑛𝑜𝑑𝑖𝑐 = 𝑖0 exp [a𝑎𝑛𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
]
[6]
while the cathodic current density is
𝑖𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐 = −𝑖0 exp [−a𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
]
[7]
so that net current density is
𝑖𝑛𝑒𝑡 = 𝑖𝑎𝑛𝑜𝑑𝑖𝑐 + 𝑖𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐 =
= 𝑖0 exp [a𝑎𝑛𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
] − 𝑖0 exp [−a𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
][8]
This is known as the Butler-Volmer equation.
Hence, for Electrode A, which acts as the cathode, the total current density is
𝑖𝐴 = 𝑖0,(𝐴) [exp (
a𝑎,𝐴 𝑧𝐹(𝜙−𝜙0,(𝐴) )
−a𝑐,𝐴 𝑧𝐹(𝜙−𝜙0,(𝐴) )
𝑅𝑇
𝑅𝑇
) − exp (
for Electrode B, which acts as the anode, the total current density is
6
)]
[9]
𝑖𝐵 = 𝑖0,(𝐵) [exp (
a𝑎,𝐵 𝑧𝐹(𝜙−𝜙0,(𝐵) )
−a𝑐,𝐵 𝑧𝐹(𝜙−𝜙0,(𝐵) )
𝑅𝑇
𝑅𝑇
) − exp (
)] [10]
At low overpotentials, (𝜙 − 𝜙0 ) < 0.01𝑉, the Butler-Volmer equation reduces to the
following linear relationships for current density-potential
Cathodic
𝑖𝐴 =
Anodic
𝑖𝐵 =
𝑖0,(𝐴) 𝑧𝐹(𝜙−𝜙0,(𝐴) )
[11]
𝑅𝑇
𝑖0,(𝐵) 𝑧𝐹(𝜙−𝜙0,(𝐵) )
[12]
𝑅𝑇
On the other hand, at large overpotentials an exponential (Tafel) relationship results.
For anodic polarization (𝜙 − 𝜙0 ) > 0.01𝑉, the Butler-Volmer equation reduces to
𝑖𝑛𝑒𝑡 = 𝑖0 exp [a𝑎𝑛𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
]
[13]
such that the expression for the overpotential is
(𝜙 − 𝜙0 ) = 𝛼 log e
𝑖𝑛𝑒𝑡
[14]
𝑖0
This is known as the anodic Tafel equation.
For cathodic polarization (𝜙 − 𝜙0 ) < −0.01𝑉, the Butler-Volmer equation reduces to
𝑖𝑛𝑒𝑡 = −𝑖0 exp [−a𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐
𝑧𝐹(𝜙−𝜙0 )
𝑅𝑇
]
[15]
such that the expression for the overpotential is
(𝜙 − 𝜙0 ) = 𝛽 log e
𝑖𝑛𝑒𝑡
𝑖0
[16]
This is known as the cathodic Tafel equation.
It should be noted that the input data in Table 1 that was used for model validation refers
to anodic and cathodic Tafel parameters (slopes). Based on Equations [14] and [16], the
relationships that were used to determine the appropriate transfer coefficients are
a𝑎𝑛𝑜𝑑𝑖𝑐 =
𝑅𝑇
𝑧𝐹𝛼
a𝑐𝑎𝑡ℎ𝑜𝑑𝑖𝑐 =
7
𝑅𝑇
𝑧𝐹𝛽
[17]
[18]
Regardless of the relationship used for the current density, the Flux Source boundary
condition reflects the relationship in Equation [19] for the surfaces of Electrode A and
Electrode B respectively
𝜕𝜙
𝜕𝑛
=
𝜕𝜙
=
𝜕𝑦
−𝑖𝐴 𝑜𝑟 𝐵
𝜎
[19]
where y is the distance from the electrode surface. For the system in Figure 1, the
derivative normal to the surface is the same as the derivative with respect to y.
2.3 Corrosion Rate
Based on the current density as derived from Equation [19]
𝜕𝜙
𝜎=𝑖
[20]
𝑖 = 𝑧𝐹𝑣
[21]
𝜕𝑛
and Faraday’s Law
the reaction rate is
𝑣=
𝜎
𝑧𝐹
(
𝜕𝜙
𝜕𝑛
)
[22]
Corrosion rate is then defined as
𝐶𝑅 = 𝑣 𝑀𝑤 /𝜌
[23]
Reaction and corrosion rates are only developed for the anodic surface of each system,
because only the anode corrodes.
8
3. Results and Discussion
3.1 Model Validation
The geometry that is used in the model reflects the configuration shown in Figure 1. The
input data used is shown in Table 1.
Table 1: Input Data Used for Model Validation [3]
Property
Value
Units
αA anodic reaction of metal A
0.05
V
αB anodic reaction of metal B
0.05
V
βA cathodic reaction of metal A
0.05
V
βB cathodic reaction of metal B
0.05
V
σ conductivity of the electrolyte
10
Ω-1m-1
𝑖0,(𝐴) free current density of metal A
1
Am-2
𝑖0,(𝐵) free current density of metal B
1
Am-2
a surface length of metal A
0.01
m
b surface length of metal B
0.01
m
w thickness of the electrolyte
0.01
m
𝜙0,(𝐴) free corrosion potential of metal A
0.5
V
𝜙0,(𝐵) free corrosion potential of metal B
-0.5
V
Tafel Parameters:
3.1.1
Using the Butler-Volmer Relationship for Current Density-Potential
Using Butler-Volmer (B-V) relationships for the current densities at the electrode
surfaces, in accordance with Equations [9] and [10], produced the potential distribution
in the electrolyte shown in Figure 4.
9
Figure 4: COMSOL Results for B-V Current Density Distribution on the Electrode Surfaces
(Surface Plot)
Figure 5 describes the COMSOL results for potential in the electrolyte along the
electrode surfaces and the top of the electrolyte respectively as compared to the
calculations in Reference [3].
Figure 5: COMSOL Results for B-V Current Density Distribution and from Reference [3]
10
The COMSOL results correlate qualitatively and quantitatively well to the results from
Reference [3]. The potential varies from -0.2293V to 0.2293V along the surface of the
electrode, and varies from approximately -0.11V to 0.11V along the top of the
electrolyte, both of which agree with the calculated data in Reference [3]. The good
correlation suggests that the finite element model produced reliable results.
Figure 6 shows the current density along the entire electrode surface, and the associated
reaction rate along the anodic surface. The maximum current density is approximately
4500A/m2 and the approximate maximum reaction rate is 0.024mol/m2s.
Figure 6: Current Density and Reaction Rate for the Basic Geometry
The reaction rate is being reported for this system because this is a generic case and
there are no specific values for molecular weight (Mw) or density (ρ).
Appendix A describes the initial attempt at model validation using the linear current
density-potential expressions. However, the results obtained using the linear expressions
did not correlate quantitatively well with the experimental results because this system
has a large overpotential, for which the linear approximation is not valid.
3.1.2
Using a Basic Geometry Subjected to Polarization
In addition to the input data provided in Table 1, further model validation was completed
using the same system with an applied polarization of 4.5V. The resulting potential
distribution in the electrolyte is shown in Figure 7. Figure 8 describes the potential in
the electrolyte along the electrode surfaces. The same B-V boundary conditions were
11
utilized on the electrode surfaces as in Section 3.1.1. In order to apply the polarization,
the Zero Flux condition on edge 2-3 of Figure 3 was replaced with a Dirichlet Boundary
Condition. The Dirichlet Boundary Condition allowed the edge to be held at a constant
value, in this case 4.5V. The potential along the top of the electrolyte is not shown for
this case, because it will always be equal to 4.5V due to the polarization of the system.
Figure 7: COMSOL Results for the System Subjected to Polarization (Surface Plot)
The COMSOL results and the calculated results from Reference [3] correlate
qualitatively and quantitatively well as shown in Figure 8. The potential along the
electrode surface varies from -0.0772V to approximately 0.9V, which agrees with the
experimental data.
Figure 8: COMSOL Results for the System Subjected to Polarization and from Reference [3]
12
The good correlation further validates that the finite element model appropriately
characterizes the system.
Figure 9 shows the current density along the entire electrode surface, and the associated
reaction rate along the anodic surface. The maximum current density is approximately
17500A/m2 and the approximate maximum reaction rate is 0.09mol/m2s. These values
are much higher than those for the un-polarized system. By polarizing the system, the
change in potential normal to the surface is greater for this system, thereby increasing
both the current density and reaction rate.
Figure 9: Current Density and Reaction Rate for the System Subjected to Polarization
3.2 Results for Different Electrode Geometries
Following validation of the finite element model, additional cases were evaluated using
the same boundary conditions used in Section 3.1.1 and the input data from Table 1, but
for different electrode geometries. The following geometries were developed to explore
the impacts of variations in geometry of a galvanic couple on potential distribution
within the electrolyte, as well as current density along the entire electrode surface and
reaction rate along the anodic surface. Current density and associated reaction rate were
developed in accordance with Equations [21] and [23] respectively.
3.2.1
Electrode Geometry with a 90 Degree Step
The following geometry depicts two non-coplanar electrodes.
Electrode A extends
further into the electrolyte at a 90 degree angle to the surface of Electrode B. This
13
geometry could be representative of an area where two plates are not flush to one
another.
Figure 10 shows the potential distribution in the electrolyte that was produced. Figure
11 describes the potential in the electrolyte along the electrode surfaces and the top of
the electrolyte respectively. The potential varies from -0.2082V to 0.2782V. This
system shows an increase in the range of the potential within the electrolyte, and a shift
to more positive potentials.
Figure 10: COMSOL Results for the Electrode Geometry with a 90 Degree Step (Surface Plot)
Figure 11: COMSOL Results for the Electrode Geometry with a 90 Degree Step
14
Figure 12 shows the current density along the entire electrode surface, and the associated
reaction rate along the anodic surface. The maximum current density is approximately
8500A/m2 and the approximate reaction rate is 0.045mol/m2s. These values are higher
than those for the coplanar electrode geometry. By increasing the electrode surface area,
more charge can be transferred to the electrolyte, which creates a larger gradient along
the electrode surfaces. Per Equations [21] and [22], current density and reaction rate are
directly related to the potential gradient normal to the electrode surface.
Figure 12: Current Density and Reaction Rate for the Electrode Geometry with a 90 Degree Step
3.2.2
Electrode Geometry with an Elliptical Step
The following geometry depicts two non-coplanar electrodes. Electrode A extends
further into the electrolyte elliptically from the surface of Electrode B. This geometry
could be representative of an area where one plate has been machined to meet its
abutting plate.
Figure 13 shows the potential distribution in the electrolyte that was produced. Figure
14 describes the potential in the electrolyte along the electrode surfaces and the top of
the electrolyte respectively. The potential varies from -0.2164V to 0.2533V. This
system produces results similar to the electrode geometry with a 90 degree step.
15
Figure 13: COMSOL Results for the Electrode Geometry with an Elliptical Step (Surface Plot)
Figure 14: COMSOL Results for the Electrode Geometry with an Elliptical Step
Figure 15 shows the current density along the entire electrode surface, and the associated
corrosion rate along the anodic surface. The maximum current density is approximately
8500A/m2 and the approximate maximum reaction rate is 0.045mol/m2s. The surface
area of the electrode with the elliptical step is slightly less than the surface area of the 90
degree step, which is why the values are slightly less.
16
Figure 15: Current Density and Reaction Rate for the Electrode Geometry with an Elliptical Step
3.2.3
Axisymmetric Electrode Geometry
The next case assumed a 2D axisymmetric geometry.
This geometry could be
representative of a pipe made of two different metals.
Figure 16 shows the potential distribution in the electrolyte that was produced. Figure
17 describes the potential in the electrolyte along the electrode surfaces and the top of
the electrolyte respectively. The potential distribution for this electrode geometry is the
same as with the coplanar geometry.
Figure 16: COMSOL Results for the Axisymmetric Electrode Geometry (Surface Plot)
17
Figure 17: COMSOL Results for the Axisymmetric Electrode Geometry
Figure 18 shows the current density along the entire electode surface, and the associated
reaction rate on the anodic surface. Since this system had the same potential distribution
as the coplanar geometry, it follows that the current density and the reaction rate would
be the same as well.
Figure 18: Current Density and Reaction Rate of the Axisymmetric Electrode Geometry
Table 2 provides a summary of the data discussed above.
Table 2: Summary of Data for the Different Geometries
Coplanar
Polarization
90 Degree Step
Elliptical Step
Axisymmetric
Max. Potential (V)
0.2293
4.5
0.2782
0.2533
0.2293
Min. Potential (V)
-0.2293
-0.0772
-0.2082
-0.2164
-0.2293
Range (V)
0.4586
4.5772
0.4864
0.4697
0.4586
4500
17500
8500
8500
4500
0.024
0.09
0.045
0.045
0.024
Max. Current
Density (A/m2)*
Max. Reaction
Rate (mol/m2 s)*
*Maximum current densities and corrosion rates are approximate
18
3.3 Results for Different Electrolyte Thicknesses Using a Copper-Zinc
Galvanic Couple
In this section computed results are shown for the copper-zinc galvanic couple in Figure
19 [3]. The inputs used for this system are included in Table 4. This case is included
because it is a real situation with experimental data, rather than a generic system.
Figure 19: Copper-Zinc Galvanic Couple in a Hydrochloric Acid Solution [3]
For this system, the anodic reaction is:
𝑍𝑛 → 𝑍𝑛+2 + 2𝑒 −
[24]
Table 3 provides data needed to support calculation of the corrosion rate of a zinc anode.
Table 3: Properties of Zinc
Property
z number of electrons transferred
Mw molecular weight
ρ density
19
Value
Units
2
--
65.409 x 10-3
kg mol-1
5610
kg m-3
Table 4: Input Data Used for Copper-Zinc Galvanic Couple [3]
Property
Value
Units
aa,Cu anodic transfer coefficient of Copper*
0.025
V
aa,Zn anodic transfer coefficient of Zinc*
0.025
V
ac,Cu cathodic transfer coefficient of Copper*
0.025
V
ac,Zn cathodic transfer coefficient of Zinc*
0.025
V
σ conductivity of HCl solution
0.42
Ω-1m-1
𝑖0(𝐶𝑢) free current density of Copper
1
Am-2
𝑖0(𝑍𝑛) free current density of Zinc
1
Am-2
0.0075
m
0.02
m
0.002 and 0.02
m
𝜙𝐶𝑢 free corrosion potential of Copper
-0.845
V
𝜙𝑍𝑛 free corrosion potential of Zinc
-0.985
V
a surface length of Copper
b surface length of Zinc
w thickness of the HCl solution
*Note: These values are not from Reference [3], they have been chosen based on
standard values rather than the experimental data
Figure 20 shows the potential distribution produced in the 2mm and 20mm thick
electrolyte. Figure 28 describes the potential in the electrolyte along the electrode
surfaces and along the top of the electrolyte respectively for the system with a 2mm
thick electrolyte, while Figure 29 provides the same information for the 20mm thick
electrolyte.
20
Figure 20: COMSOL Results for the Copper-Zinc Couple with 2mm and 20mm Thick Electrolyte
(Surface Plot)
Both systems have approximately the same range of potentials -0.9615V to -0.9249V for
a 2mm electrolyte and -0.9506V to -0.9383V for a 20mm electrolyte. However, the
distributions shown in Figure 21 and Figure 22 indicate that even though the range of
potentials is approximately the same for both of the electrolyte thicknesses, the
distribution of the potential within the electrolyte is very different. As expected, there
are more significant differences in the potentials at the top of the electrolyte for each
thickness.
For the 2mm thick electrolyte, the potentials at the top surface of the
electrolyte vary from approximately -0.968V to -0.926V, whereas for the 20mm thick
electrolyte, they vary from approximately -0.948V to -0.946V
Figure 21: COMSOL Results for the Copper-Zinc Couple with 2mm Thick Electrolyte
21
Figure 22: COMSOL Results for the Copper-Zinc Couple with 20mm Thick Electrolyte
Figure 23 shows the current density along the anode surface, and the associated
corrosion rate for the 2mm thick electrolyte, while Figure 24 provides the same
information for the 20mm thick electrolyte.
For the 2mm thick electrolyte, the
approximate maximum current density is 0.2A/m2 and the approximate maximum
corrosion rate is 0.4mm/year. Similarly, for the 20mm thick electrolyte, the approximate
maximum current density is 0.18A/m2 and the approximate maximum corrosion rate is
0.345mm/year
Figure 23: Current Density and Corrosion Rate for Copper-Zinc Couple with 2mm Thick
Electrolyte
22
Figure 24: Current Density and Corrosion Rate for Copper-Zinc Couple with 20mm Thick
Electrolyte
Table 5: Summary of Results for the Copper-Zinc Galvanic Couple with an Electrolyte 2mm and
20mm Thick
2mm Thick Electrolyte
20mm Thick Electrolyte
Max. Potential (V)
-0.9249
-0.9398
Min. Potential (V)
-0.9615
-0.9506
Range (V)
0.0366
0.0108
Max. Current Density (A/m2)*
0.2
0.18
Max. Corrosion Rate (mm/year)*
0.4
0.345
*Maximum current densities and corrosion rates are approximate
Appendix B describes this system using linear expressions for the current density. Due
to the relatively small overpotential in this system, linear conditions were a reasonable
system to review prior to implementing the Butler-Volmer conditions.
23
4. Conclusion
Each of the geometries that were investigated in Section 3.2 utilized the same system
inputs. In doing so, the results that were generated could be evaluated with respect to
the different geometries. Each of the geometries demonstrates the same qualitative trend
of potential in the electrolyte along the electrode surfaces, while the same overall
magnitude of potential and zero potential at the intersection of the two electrodes are
also seen. Table 2 summarizes the maximum and minimum potentials that were seen in
the electrolyte for each of the systems in Section 3.2.
The geometry with the 90 degree step saw the greatest range of potentials in the
electrolyte, followed by the geometry with the elliptical step. These ranges were also
shifted more positive than for the coplanar electrode geometry. This increase in range of
potential is due to the greater electrode surface area, which allows for more transfer of
charge from the electrode to the electrolyte. The shift to more positive potentials is
because the increase in surface area was only made to the cathode.
The axisymmetric
model yielded the same results as the coplanar 2D Cartestian model developed during
finite element model validation. As expected, when all else is equal, the greater the
electrode surface area, the greater the potential change in the electrolyte.
Table 2 also summarizes the approximate maximum current density along the entire
electrode surface, and maximum reaction rate along the anode surface for each of the
systems in Section 3.2. The impacts of electrode geometry can also been seen with
current densities and reaction rates. As with the potential distributions, the coplanar
systems (Cartesian and axisymmetic) exhibit the same results for current density and for
reaction rate. The same correlation is seen for the systems with the 90 degree step and
the elliptical step. These results were expected since both current density and reaction
rate are based on potential distribution at the electrode surface.
Table 5 summarizes the maximum and minimum potentials that were seen in the
electrolyte for the copper-zinc galvanic couple for both a 2mm and 20mm thick
24
electrolyte. Table 5 also summarizes the approximate maximum current densities and
maximum corrosion rates along the anode surface.
The data in Table 5 would indicate that the thickness of the electrolyte did not have a
significant impact on the potential distribution within the electrolyte. However, Figure
20 demonstrates that even if the range of potentials is approximately the same for both of
the electrolyte thicknesses, the distribution of the potential within the electrolyte is very
different. There is a lot less variation of the potential at the top of the electrolyte in the
system with the thicker electrolyte. This result is in keeping with physical principles.
The same trend can be seen with processes such as heat transfer and diffusion; the
greater the distance from an input, the less impact is seen from the input. This same
rationale also explains the similar results for potential distribution 1mm from the
electrode surfaces.
The similarities in the maximum current densities and maximum corrosion rates were
also expected. These values are dependent on the behavior of potential at and near the
electrode surface and as such are minimally impacted by changing the thickness of the
electrolyte.
25
5. References
[1]
Zhang, X. G. (2011). Galvanic Corrosion. In Uhlig's Corrosion Handbook (Third
ed., pp. 123-143). John Wiley & Sons, Inc.
[2]
Oldfield, J. W. (1988). Electrochemical Theory of Galvanic Corrosion. In H. P.
Hack (Ed.), Galvanic Corrosion, ASTM STP 978 (pp. 5-22). Philadelphia, PA:
American Society for Testing and Materials.
[3]
Doig, P., & Flewitt, P. E. (1979). A Finite Difference Numerical Analysis of
Galvanic Corrosion for Semi-Infinite Linear Coplanar Electrodes. Journal of The
Electrochemical Society , 126 (12), 2057-2063.
[4]
Hack, H. P., & Scully, J. R. (1986). Galvanic Corrosion Prediction Using Longand Short-Term Polarization Curves. Corrosion , 42 (2), 79-90.
[5]
Yan, J. F., Pakalapati, S. N., Nguyen, T. V., & White, R. E. (1992).
Mathematical Modeling of Cathodic Protection Using the Boundary Element
Method with a Nonlinear Polarization Curve. J. Electromchem. Soc. , 139 (7),
1932-1936.
[6]
Astley, D. J. (1988). Use of the Microcomputer for Calculation of the
Distribution of Galvanic Corrosion and Cathodic Protection in Seawater
Systems. In H. P. Hack (Ed.), Galvanic Corrosion, ASTM STP 978 (pp. 53-78).
Philadelphia, PA: American Society of Testing and Materials.
26
6. Appendix A
Using a Basic Geometry with Linear Expressions for Current Density
Using linear relationships for the current densities at the electrode surfaces, in
accordance with Equations [11] and [12], produced the potential distribution in the
electrolyte shown in Figure 25. Figure 26 describes the potential in the electrolyte along
the electrode surfaces and the top of the electrolyte respectively.
Figure 25: COMSOL Results for Linear Current Density Distribution on the Electrode Surfaces
(Surface Plot)
The COMSOL results in Figure 26 correlate qualitatively well to the results from
Reference [3], as shown in Figure 5. However, there are discrepancies in the values for
the potential within the system. This linear relationship is only appropriate when there is
a very small overpotential in the system (generally <0.01V), which is not applicable in
this case.
distribution.
The linear approximation gives conservative results for the potential
This system is more appropriately represented by Butler-Volmer
relationships for the current density.
27
Figure 26: COMSOL Results for Linear Current Density Distribution on the Electrode Surfaces
28
7. Appendix B
Results for Different Electrolyte Thicknesses Using a Copper-Zinc Galvanic Couple
Using Linear Expressions for Current Density
The copper-zinc galvanic couple that was evaluated is representative of Figure 19, and
the inputs that were used for this system are included in Table 6.
Table 6: Input Data Used for Copper-Zinc Galvanic Couple [3]
Property
Value
Units
0.42
Ω-1m-1
𝑖0(𝐶𝑢) free current density of Copper
1
Am-2
𝑖0(𝑍𝑛) free current density of Zinc
1
Am-2
0.0075
m
0.02
m
0.002 and 0.02
m
𝜙𝐶𝑢 free corrosion potential of Copper
-0.845
V
𝜙𝑍𝑛 free corrosion potential of Zinc
-0.985
V
σ conductivity of HCl solution
a surface length of Copper
b surface length of Zinc
w thickness of the HCl solution
Linear relationships were used for the current density along the electrode surfaces, in
accordance with Equations [11] and [12]. The maximum overpotential for this system is
0.14V, which is greater than 0.01V, but still relatively small making it a reasonable
approximation. Figure 27 shows the potential distribution produced in the 2mm and
20mm thick electrolyte. Figure 28 describes the potential in the electrolyte along the
electrode surfaces and along the top of the electrolyte respectively for the system with a
2mm thick electrolyte, while Figure 29 provides the same information for the 20mm
thick electrolyte.
29
Figure 27: COMSOL Results for the Copper-Zinc Couple with 2mm and 20mm Thick Electrolyte
(Surface Plot)
Both systems have approximately the same range of potentials -0.9846V to -0.8594V for
a 2mm electrolyte and -0.9745V to -0.8864V for a 20mm electrolyte. However, the
distributions shown in Figure 28 and Figure 29 indicate the total range of potential is not
enough information to characterize the whole system. As expected, there are more
significant differences in the potentials at the top of the electrolyte for each thickness.
For the 2mm thick electrolyte, the potentials at the top surface of the electrolyte vary
from approximately -0.985V to -0.873V, whereas for the 20mm thick electrolyte, they
vary from approximately -0.938V to -0.855V
Figure 28: COMSOL Results for the Copper-Zinc Couple with 2mm Thick Electrolyte
30
Figure 29: COMSOL Results for the Copper-Zinc Couple with 20mm Thick Electrolyte
Figure 30 shows the current density along the anode surface, and the associated
corrosion rate for the 2mm thick electrolyte, while Figure 31 provides the same
information for the 20mm thick electrolyte.
For the 2mm thick electrolyte, the
approximate maximum current density is 5 A/m2 and the approximate maximum
corrosion rate is 10 mm/year. Similarly, for the 20mm thick electrolyte, the approximate
maximum current density is 4.5 A/m2 and the approximate maximum corrosion rate is
8.5 mm/year
Figure 30: Current Density and Corrosion Rate along the Anode Surface of the Copper-Zinc
Couple with 2mm Thick Electrolyte
31
Figure 31: Current Density and Corrosion Rate along the Anode Surface of the Copper-Zinc
Couple with 20mm Thick Electrolyte
Table 7: Summary of Results for the Copper-Zinc Galvanic Couple with an Electrolyte 2mm and
20mm Thick
2mm Thick Electrolyte
20mm Thick Electrolyte
Max. Potential (V)
-0.8594
-0.8864
Min. Potential (V)
-0.9846
-0.9745
Range (V)
0.1252
0.0881
Max. Current Density (A/m2)*
5
4.5
Max. Corrosion Rate (mm/year)*
10
8.5
*Maximum current densities and corrosion rates are approximate
32