Corrosion deposition in deforming anodes of in-ground cathodic protection systems: A three-dimensional transient study

Abstract

Cathodic protection as a complementary technique is widely considered in the industry along with the selection of suitable materials as well as efficient coating considerations. In cathodic protection (CP), anode produces a current output to protect the structure (cathode) and part of the current output is wasted by the grounding system. We developed a 3D, and time-dependent numerical model to simulate the anode corrosion (using arbitrary Lagrangian-Eulerian method), find the deposit formation (by interface tracking level set method), anode mass loss and current wasted by the grounding system. We also defined a CP efficiency as anode mass loss times the current percentage received by the structure. The CP efficiency can be utilized for anode quantity and placement pattern. The results showed when the conductivity of corrosion products (deposit) is lower than that of soil, the current output from the anode decreases over time. When the deposit conductivity is higher than that of soil, the current output from the anode increases in time, reaches a peak, then decreases. This is due to the competition among three factors: anode surface increase, medium conductivity change, and inhibition effect of the deposition layer. For the studied case, almost 54% of the current from the anode is wasted by the grounding system. To illustrate the significance of CP efficiency, this study compares three alternative anode configurations with the original design. It is observed that for a 14.5 kg anode, burying the anodes 2 m deeper enhances the mean potential distribution on the structure by 4%. Additionally, the configuration featuring the 7.7 kg anode, owing to its superior dimensional design, demonstrates improved performance compared to other configurations.

Keywords

In-ground corrosion Cathodic protection Deformed anodes Lagrangian-Eulerian method Level set technique Interface tracking

Introduction

Cathodic Protection (CP) as an electrochemical process is an effective and widely used technique for mitigating the in-ground corrosion of metallic structures. The performance of this method is greatly influenced by anode bed arrangement, electrode surface characteristics, electrolyte properties, and the corrosion deposits. The findings show that the corrosion deposits have different properties and depend on the species present in the medium. The corrosion deposits can be metal oxides,^1–3 metal salts,^4,5 porous layers,^6,7 or dense layers.⁸ The deposits may be impervious/tenacious or porous. Where the corrosion product is impervious, the corrosive medium cannot pass through the deposits, and corrosion rate decreases.^9–11 Whereas for the porous layer, the corrosion deposit doesn't have a protective effect^12–14 and corrosion accelerates.^15–17

Analytical or numerical studies have modeled the corrosion phenomenon, considering the formation of corrosion products (deposits). Yan et al.¹⁸ were among the pioneers who simulated the deposit formation on cathodic-protected steel. They examined the formation of $Mg (OH)_{2}$ and ${CaCO}_{3}$ on steel and magnesium and investigated the changes in the concentration of active ions and their effects on corrosion. Sun et al.¹⁹ used Archie's law²⁰ to find the effective conductivity and performed a simulation to study the effect of deposition porosity and thickness in time. In another study, Sun et al.²¹ examined the effect of corrosion products on anode and cathode surfaces using a time-dependent Eulerian-Lagrangian model. They used the hypothesis of non-corrosive cathode in their model. Also, they assumed that the deposit layer is isotropic. Zhou et al.²² developed an electrochemical deposition method for repairing concrete cracks. In their study, the level set method (LSM) was used to describe the deposit growth. By defining the level set delta function and deposit rate of $Mg (OH)_{2}$ on the cathode surface in the transport equation, the deposit growth and changes in concentration of ions in terms of time and under different voltages were presented. Zhu et al.²³ using an electrochemical deposition method and a continuous damage improvement framework, proposed an approach for repairing concrete. Wang et al.²⁴ arranged a simulation to determine the deposit's effective diffusion coefficient and conductivity, by considering the deposition of $Mg (OH)_{2}$ and using the Bruggeman relationship.²⁵ They investigated temporal and spatial changes on deposition and considered the effect of time on parameters such as corrosion rate, current density, deposit porosity, potential, overpotential, electrolyte conductivity, and concentration of ${Mg}^{2 +}$ and ${OH}^{-}$ ions. In their modeling, they assumed that the deposited layer is anisotropic.²⁴

In all the mentioned articles, the process of deposit formation was time-dependent and dilute solution theory was used. Also, in most of the studies, the deposited layer was considered isotropic, except for Ref. 24. In Refs. 19, 21, and 22, Archie’s law was used to calculate deposit conductivity, while Wang et al.²⁴ used Bruggeman's law for that. In some sources, e.g. Refs. 19, 21, and 24 the cathode was assumed to be non-corrosive also the LSM was used in some, e.g. Refs. 22 –24. All the models were limited to 2D domains and simple geometries. A 3D modeling is needed to simultaneously consider the anode dissolution as well as the deposit formation, and ion inhibition. The 3D model can be used to study the effect of deposit types (porosity and conductivity) on efficiency of cathodic protection and placement of anodes in in-ground cathodic protection systems.

In the literature, the anode and cathode were positioned in close proximity, resulting in greater deposition on the anode and a smaller volume on the cathode. However, in the current model, where the distance between the piles and the anode is approximately 1 m, the majority of the deposit volume is observed on the anode, with only a minimal amount settling on the cathode. Consequently, the primary focus is directed towards the anode. Furthermore, the soil conductivity is characterized by a ratio of 0.01 compared to the brine solution (0.025 S/m versus 2.5 S/m), as outlined in Ref. 21. Consequently, the deposition layer's growth is anticipated to occur at a slower pace.

Our aim in this study is to improve our previously developed 2D time-dependent model to a 3D model and simultaneously model the anode deformation (using arbitrary Lagrangian-Eulerian finite element approach²²) and deposit formation (using LSM²⁶). The developed methodology is applied to model the cathodic protection of a screw-pile. It was shown that a portion of the output current from the anodes is wasted by the grounding system. The wasted portion depends on the number and location of anodes.

In previous works and using 3D simulations, the design of the anode bed and the selection of the optimal mode in the steady-state were investigated^27,28 and in another work, with the development of simulation factors, the process of protecting the structure over 30 years with anode deformation was studied.²⁹ In the end, by using wire anodes with backfill and ribbon mesh, the cathodic protection process of the tank bottom was investigated and economic aspects and key parameters were examined.³⁰

This paper is organized as follows: in the first section, after presenting the considered geometry and expressing the problem hypotheses, the relations governing the simulation model are described i.e. the relations between the electric and concentration fields. Then, by explaining the arbitrary Lagrangian-Eulerian method and LSM for dissolution and deposit formation on the anode, respectively, and defining the boundary conditions, the problem is simulated. In this regard, validation is performed to confirm the simulation. In the second section, the effect of corrosion deposits on cathodic protection system is mentioned. Then changes in current density, deposit formation, and magnesium and hydroxide ion concentrations are presented as 30-year animated albums. The changes in mass of the anode, the output current from the anode, the input current to the steel structure/copper grounding system are expressed. In the third section, in order to find the effect of anode arrangement and anode type from the sizing point of view, several different configurations have been considered. In this regard, cathodic protection efficiency is defined as a parameter for comparing different anode configurations. To understand the usefulness of the cathodic protection efficiency, the preliminary anode configuration is compared with three other configurations, and the most suitable configuration is selected by considering the performance.

Case study and field data

Screw-piles are favored foundations in transmission and distribution lines in wetlands. However, the saturated nature of soils in marshlands renders a high risk of underground corrosion for any steel structure. The design service life for the screw-pile foundation is 50 years; nonetheless, corrosion inspections confirmed 10% material loss in less than 6 years. Sample photos of corrosion damage on the selected foundation are shown in Figure 1. This screw-pile is used to demonstrate our approach for modeling the cathodic protection. Cathodic protection analysis relies on three fundamental points: I. Cathodic protection is applicable when the metal to be protected must be in an electrolyte. Thus, cathodic protection in the atmosphere or air is ineffective. II. Both the anode and cathode must be present in the same electrolyte for cathodic protection to be effective. III. Corrosion occurs at any metal point where the current (loss of electrons) flows out and into the electrolyte. Corrosion created on above the ground (see Figure 1) is atmospheric corrosion, and due to the fact that air is not considered an electrolyte, the cathodic protection method cannot be used and is out of the scope of this article. In these cases, one of the best ways to protect the structure from atmospheric corrosion is to use paint coating in structure in the vicinity of air (waterline).

The foundation design represents a rather complex geometry that demands advanced modeling approaches to find an optimized anode arrangement for cathodic protection of the foundation. Each footing includes eight battered screw-pile and a large vertical pile in the middle. All piles are made of structural steel (ASTM A2527). The buried external surface area of each battered screw pile is 4.2 m² (45.2 ft²), and for the center pile, it is 12.5 m² (134.6 ft²). As such, a total bare surface area of 46.1 m² (496.6 ft²) needs to be protected by cathodic protection. Additionally, a grounding system including a loop and four spikes is attached to the foundations. The grounding loop and spikes are made of copper with a diameter of 3.81 cm (1.5 inches). The buried surface area of the grounding system is 2.49 m² (26.8 ft²). A saturated copper-copper sulfate reference electrode and a calibrated high-impedance multi-meter (Fluke 28 II) were used to measure structure-to-soil potentials, also known as native or equilibrium corrosion potentials. Measurements were performed at different directions and distances around the structure and showed a slight variation of ±20 mV. An averaged native potential of −700 mV vs. copper/ copper sulfate electrode (CSE) is considered for the structure. Soil resistivity tests were performed per ASTM G57 at the footing of the pole using a soil resistivity meter (Fluke 1623 GEO Earth Ground Tester) to characterize the soil service environment. Measurement data are summarized in Table 1. The results confirm a relatively constant soil resistivity at different soil horizons. Sample photos of field measurements are shown in Figure 2.

Table 1.

Soil resistivity measurement data (per ASTM G57).

Pin Spacing
2 ft. (0.61 m)		4 ft. (1.21 m)	6 ft. (1.83 m)	8 ft. (2.44 m)	10 ft. (3.05 m)	12 ft. (3.66 m)
Measured Soil Resistance ( $Ω$ )
13.11	6.29		4.11	2.90	2.24	1.75
Soil Resistivity ( $Ω$ .cm)
5021	4818		4723	4443	4290	4022
Soil Layers Resistivity ( $Ω$ .cm)
0 ft. – 2 ft. (0 m–0.61 m)	2 ft. – 4 ft. (0.61 m–1.21 m)		4 ft. – 6 ft. (1.21 m–1.83 m)	6 ft.– 8 ft. (1.83 m–2.44 m)	8 ft.– 10 ft. (2.44 m–3.05 m)	10 ft.– 12 ft. (3.05 m–3.66 m)
5021	4631		4542	3773	3770	3064

Conventional anode bed design

This section presents basic calculations to estimate the weight of anodes required for cathodic protection of the considered foundation. Considering the level of soil corrosivity, high-potential magnesium alloy anodes (type M1, per ASTM B843) are considered.³¹ A nominal lifespan of 30 years is assumed for such CP system. A current density of 9 mA/m² (0.84 mA/ft²), a conservative value for current density, is assumed for cathodic protection. This results in an overestimation of anode mass and can be considered a safety design factor to account for transient effects such as anode passivation (formation of deposition on the anode surface). Considering the buried surface area of 46.1 m², the total current for cathodic protection is estimated at 0.415 A. The capacity of the CP system can be calculated from equation (1):

Q_{c p} = I_{c p} \times Life

(1)

Based on equation (1), the required CP system is estimated as 12.45 A-yr. The required mass for anode material can be calculated from equation (2):

m_{a} = Q_{c p} / (Q_{a} \times E \times U)

(2)

where the value of

Q_{a}

is 0.251 A-yr/kg for selected anode material,^32,33 and the values of E and U are 0.5 and 0.85, respectively.^33,34 From equation (2), at least 116.7 kg anode is needed for protecting the structure.

Numerical approach for anode bed design

The calculated anode weight from equation (2) is used to construct a preliminary anode bed for numerical simulations. It is essential to mention that several anode arrangements with different sizes (shapes) for anodes can be considered; however, a balance between installation cost and performance of the cathodic protection system must be considered. The preliminary anode bed and numerical domain are shown in Figure 3, which includes 2 × 4 × 14.5 kg (2 × 4 × 32 lb) cylindrical anodes (diameter: 14 cm, and height: 51.8 cm) symmetrically arranged between screw-pile pairs, so the total mass is 116 kg and approximately equal to the mass calculated using equation (2). Anode pairs are vertically stacked with equal spacing in between, the bottom depth of lower anodes is 2 m. The radial distance of anodes from the vertical screw-pile is 2.32 m.

A hemisphere of soil around the foundation is selected as the main electrolyte domain. An outer shell domain with proper boundary conditions is considered around the main hemisphere domain to take into account the effects of the infinite soil environment.

Numerical model assumptions

Before expressing the main governing equations, the model assumptions are listed as follows:

The electrolyte (soil) temperature is uniform and steady.

Dilute solution theory is considered.

The iron oxidation reaction is ignored as far as the steel is under cathodic protection.¹⁸

The formed deposits are assumed to be $Mg (OH)_{2}$ and porous with a constant porosity.

A homogeneous reaction occurs in the computational domain.

The cathodes are non-corrosive.

The deposition formation is an irreversible reaction.

Current in the soil

For transient problems, the electrical potential for the computational domain of soil and the deposition are as follows²¹:

\frac{\partial}{\partial t} \nabla^{2} φ + \frac{σ}{υ} \nabla^{2} φ = 0

(3)

Also, the current density in the soil (

i

) can be expressed as follows³⁵:

i = σ \nabla φ

(4)

From the electrical point of view, in an electrochemical reaction, the charge (electron) transfer occurs in a much shorter timespan than the ion transfer. The ion transport generates a potential gradient and the potential gradient desires to restore the system to the state of electro-neutrality. The local electro-neutrality assumption ignores the electrostatic interaction between the ions. It can be possible that make-up ions induce electro-neutrality at each time step for each element selected inappropriate value and leads to a negative concentration which is not a reasonable response.²⁴ As shown in Refs. 24, 36 the Poisson's equation should be considered to fix the unreasonable response:

\nabla^{2} φ = - \frac{F \sum_{i} z_{i} c_{i}}{E_{ε}}

(5)

The deposit formation may increase or decrease the electrical resistance and current density on the electrode surface. The ion may transfer through the deposit or the soil; therefore, the effective conductivity

σ_{e f f}

(S / m)

is described as²⁰:

σ_{e f f} = ε_{l} σ + (1 - ε_{l}) σ_{c p}

(6)

where

ε_{l}

and

(1 - ε_{l})

are the soil and deposit volume fractions. In this way,

ε_{l}

as soil volume fraction varies from 1 in the electrolyte (soil) domain to 0 in the corrosion product deposition region. The parameter

σ_{c p}

is the deposit conductivity and is expressed according to Archie’s law as follows²⁰:

σ_{c p} = ξ s_{L}^{m} ε_{P}^{n} σ

(7)

where

ξ

is the intensity of deposition conductivity (1 for normal and 3 for extremely conductive),

s_{L}

is the fluid saturation (how much liquid is in the soil, and its value changes from 0 to 1), m is the cementation exponent and explains the conductivity of the pores (changes from 1.3 to 2.5 for most sedimentary rocks,²¹ close to 2 for sand-stones and 1 for a fully saturated insulating porous matrix and a conducting liquid²²),

ε_{P}

is the deposit porosity, and n is the saturation coefficient and is normally close to 2.²¹ The deposit porosity (

ε_{P}

) is found as¹⁸:

ε_{P} = \frac{A_{s} - A_{d}}{A_{s}}

(8)

As such,

ε_{P} = 1

represents the anode that is free of deposits. Different parameters affect the deposit porosity, e.g. soil composition, pH, metal behavior, and temperature. Indeed, the deposit porosity affects the corrosion rate and deposit structure.

Mass transfer

The soil near the participant anode surface becomes more alkaline compared to the bulk, combination of ${Mg}^{2 +}$ and ${OH}^{-}$ generates the corrosion product $Mg (OH)_{2}$ which deposits on the anode surface³⁷ as follows:

{Mg}^{2 +} + 2 {OH}^{-} \Rightarrow Mg (OH)_{2}

(9)

This deposit layer forms a physical barrier and hinders the mass transfer of the participant species. It also affects the concentration field, the electrochemical reactions, and development of metal corrosion. The concentration field and electrochemical reactions affect the deposition rate and deposit thickness. The deposit thickness affects the mass transport of ions. In this paper, like our previous studies and to simplify the discussion, the deposit is assumed isotropic with constant porosity. The precipitation reaction of

Mg (OH)_{2}

is assumed to be an irreversible, which initiates when the ion product exceeds

k_{s p}

, the solubility product for

Mg (OH)_{2}

. The rate of reaction for consumption of

{Mg}^{2 +}

(

R_{s, {Mg}^{2 +}}

) and

{OH}^{-}

(

R_{s, {OH}^{-}}

) ions are as follows³⁸:

R_{s, {Mg}^{2 +}} = k (c_{{Mg}^{2 +}} . c_{{OH}^{-}}^{2} - k_{s p}) \times H (Υ)

(10)

R_{s, {OH}^{-}} = 2 k (c_{{Mg}^{2 +}} . c_{{OH}^{-}}^{2} - k_{s p}) \times H (Υ)

(11)

where k is

3.7 \times 10^{- 7} m^{7} / {mol}^{2} s

,¹⁸

c_{{Mg}^{2 +}}

and

c_{{OH}^{-}}

are concentration of

{Mg}^{2 +}

and

{OH}^{-}

ions,

k_{s p}

0.45 {mol}^{3} / m^{9}

,¹⁸

H (Υ)

is the Heaviside step function and

Υ

is the step function variable described as:

Υ = \frac{c_{{Mg}^{2 +}} . c_{{OH}^{-}}^{2}}{k_{s p}} - 1

(12)

The transport mechanism of

{Mg}^{2 +}

ions emerged at the magnesium surface (anode) and

{OH}^{-}

ions emerged at the steel surface (cathode) are considered as diffusion and ion migration. Assuming diluted species³⁹ in the electrolyte, mass transport of the ionic species is³⁵:

N_{i} = - D_{i, e f f} \nabla c_{i} - z_{i} u_{i, e f f} F c_{i} \nabla φ_{l}

(13)

The effective mobility is calculated using the Nernst-Einstein relation as:

u_{i, e f f} = \frac{D_{i, e f f}}{R T}

(14)

The electric field affects the charged species migration and the kinetics of electrochemical reactions (in which some species are generated and then consumed or transported). The interface development owing to the anode dissolution and deposit formation changes the computational domain.

Effective diffusion coefficients are expressed using the soil volume fraction:

D_{i, e f f} = ε_{l} D_{i} + (1 - ε_{l}) D_{i, c p}

(15)

Because of the porous nature of corrosion product deposition, diffusion coefficients should be updated to consider the effect of porosity.

D_{i, c p}

is defined in terms of tortuosity and porosity of corrosion product^18,38 and.⁴⁰

D_{i, c p} = D_{i} (1 - ε_{p}) + \frac{D_{i}}{τ} ε_{p}^{2}

(16)

where

τ

as the tortuosity of corrosion product region is usually described as follows⁴¹:

τ = (L / L_{c})^{2}

(17)

where L is the macroscopic distance between two points, and

L_{c}

is the actual, microscopic, or effective distance of flow through the soil between the same two points (for porous media:

L_{c} > L

τ < 1

). The finding shows that

τ

varies from 0.01 to 0.84 for saturated soils and from 0.025 to 0.57 for unsaturated soils.⁴¹

The material balance equation of species is expressed as:

\frac{\partial c_{i}}{\partial t} + \nabla . N_{i} = R_{i}

(18)

where

R_{i}

(

mol / m^{3} s

) is the reaction rate of ith species corresponding to

{Mg}^{2 +}

and

{OH}^{-}

ions. The reaction source term is defined in terms of the reaction rate for consumption of

{Mg}^{2 +}

and

{OH}^{-}

ions according to:

R_{i} = - R_{s, i} δ

(19)

where

δ

is the level set delta function which describes the deposition reaction at the corrosion product interface.

The formation of deposits

One of the methods of tracking the moving interfaces is to use a LSM.²⁶ This method is a conceptual framework for surface sets and is a tool for numerical analysis of surfaces and shapes; and tracks the moving interfaces or boundaries utilizing a fixed mesh. Also, the LSM makes it easy to follow shapes that change topology. To find the interface with the velocity field $u$ , equation (20) should be solved²⁶:

\frac{\partial ε_{l}}{\partial t} + u . \nabla ε_{l} = γ \nabla . (ε \nabla ε_{l} - ε_{l} (1 - ε_{l}) \frac{\nabla ε_{l}}{| \nabla ε_{l} |})

(20)

where the interface is defined by

ε_{l} = 0.5

. The parameter

γ

determines the amount of reinitialization or stabilization of the level set function. If

γ

is too small, the thickness of the interface might not remain constant which results in intense oscillations in

ε_{l}

due to numerical instabilities. On the other hand, if

γ

is too large, the interface moves incorrectly. One of the suitable values for

γ

is the maximum magnitude of the velocity field

u

. The parameter

ε

determines the thickness of the region where

ε_{l}

varies smoothly from 0 to 1. The

ε

is a constant within each domain and equals the biggest value of the mesh size in the domain.

The corrosion of anode includes two steps: (i) magnesium ions dissolve from the anode surface into the soil near the anode (equation (21)) and (ii) the magnesium ions react with hydroxide ions and form the deposit on the anode surface (equation (22)). In other words, as the anode surface is deformed due to dissolution, the $Mg (OH)_{2}$ corrosion product deposits on the anode surfaces. The velocity field used in the transport equation for level set variable needs to consider both deposition velocity (i.e. $u_{d e p}$ ) and deformation/dissolution velocity ( $u_{d i s}$ ). The magnesium (anode) dissolution velocity in normal direction is defined as follows:

u_{d i s} = \frac{i_{l o c}}{2 F} \frac{M_{M g}}{ρ_{M g}}

(21)

The corrosion product deposition velocity in normal direction is defined as follows:

u_{d e p} = R_{s, {Mg}^{2 +}} \frac{M_{Mg {(OH)}_{2}}}{ρ_{Mg {(OH)}_{2}}}

(22)

The deposition of corrosion products on the anode surface can significantly affect the electrochemical behavior of the anode at the anode-soil interface. In this study, arbitrary Lagrangian-Eulerian (ALE) approach is used to track the anode interface location. For this method, combination of spatial and material coordinate systems is used, respectively associated with Eulerian and Lagrangian formulations. The Eulerian approach studies the physical parameters at fixed positions in space and the Lagrangian traces the moving particles. The ALE has the advantage of both and shows a great ability to model the moving interfaces.²⁴ For the transient case in this work, the mesh displacement of the computational grid is governed by the following equations²¹:

\frac{\partial^{2}}{\partial X^{2}} \frac{\partial x}{\partial t} + \frac{\partial^{2}}{\partial Y^{2}} \frac{\partial x}{\partial t} + \frac{\partial^{2}}{\partial Z^{2}} \frac{\partial x}{\partial t} = 0

(23a)

\frac{\partial^{2}}{\partial X^{2}} \frac{\partial y}{\partial t} + \frac{\partial^{2}}{\partial Y^{2}} \frac{\partial y}{\partial t} + \frac{\partial^{2}}{\partial Z^{2}} \frac{\partial y}{\partial t} = 0

(23b)

\frac{\partial^{2}}{\partial X^{2}} \frac{\partial z}{\partial t} + \frac{\partial^{2}}{\partial Y^{2}} \frac{\partial z}{\partial t} + \frac{\partial^{2}}{\partial Z^{2}} \frac{\partial z}{\partial t} = 0

(23c)

Boundary conditions

The Tafel equations are used as boundary conditions for anodes and cathodes as follows³⁵:

i_{a} (φ) = i_{0, a} \times 10^{\frac{η_{a (φ)}}{A_{a}}} and i_{c} (φ) = i_{0, c} \times 10^{\frac{η_{c (φ)}}{A_{c}}}

(24)

and

η

as the overpotential is described as³⁵:

η_{a} = φ_{a} - φ_{s} - E_{e q, a} and η_{c} = φ_{c} - φ_{s} - E_{e q, c}

(25)

The values of Tafel parameters depend on electrode material and soil properties. In order to show the influence of the concentration parameters in simulations, the boundary condition at the anode surface is²¹:

- N_{{Mg}^{2 +}} . \hat{n} = R_{{Mg}^{2 +}} = \frac{i_{a} (φ) . ν_{{Mg}^{2 +}}}{n_{{Mg}^{2 +}} F}

(26a)

The boundary condition at the steel (cathode) surface is²¹:

- N_{{OH}^{-}} . \hat{n} = R_{{OH}^{-}} = \frac{i_{c, s} (φ) . ν_{{OH}^{-}}}{n_{{OH}^{-}} F}

(26b)

All other boundaries are insulated as follows²¹:

- N_{i} . \hat{n} = 0

(26c)

where

ν_{{Mg}^{2 +}}

and

ν_{{OH}^{-}}

are, the stoichiometric coefficients and

n_{{Mg}^{2 +}}

and

n_{{OH}^{-}}

are the chemical reaction coefficients of

{Mg}^{2 +}

and

{OH}^{-}

ions, respectively. The initial conditions and concentration at the ground surface are set as follows:

c_{{Mg}^{2 +}} = 0 mol / m^{3}

(27a)

c_{{OH}^{-}} = 1 \times 10^{- 3} mol / m^{3}

(27b)

Information about other parameters is presented in Table 2.

Table 2.

Simulation parameters.

Parameter	Value	Ref.	Parameter	Value
$D_{{Mg}^{2 +}}$	$7 \times 10^{- 11} [m^{2} / s]$	⁴²	$E_{e q, a}$	−1.75 [V_CSE]
$D_{O H^{-}}$	$5.27 \times 10^{- 9} [m^{2} / s]$	⁴³	$E_{e q, c, s t}$	−0.7 [V_CSE]
R	8.3145 [ $J / mol K$ ]	—	$E_{e q, c, c o p}$	−0.35 [V_CSE]
T	298.45 [K]	—	$i_{0, a}$	0.1 [A/m²]
$τ$	0.8	⁴¹	$i_{0, c, s t}$	0.001 [A/m²]
$s_{L}$	1	²⁰	$i_{0, c, c o p}$	0.001 [A/m²]
m	1	²⁰	$A_{a}$	0.05 [V_CSE]
n	2	²⁰	$A_{c, s t}$	−0.16 [V_CSE]
$M_{Mg}$	0.025 [ $kg / mol$ ]	²⁹	$A_{c, c o p}$	−0.15 [V_CSE]
$M_{Mg {(OH)}_{2}}$	0.058 [ $kg / mol$ ]	¹⁹	$n_{{Mg}^{2 +}}$	+ 2
$ρ_{Mg}$	1820 [ $kg / m^{3}$ ]	²⁹	$n_{{OH}^{-}}$	+ 4
$ρ_{Mg {(OH)}_{2}}$	2450 [ $kg / m^{3}$ ]	¹⁹	$ν_{{Mg}^{2 +}}$	−1
			$ν_{{OH}^{-}}$	+ 4

Model validation

The validation examined the effect of bimetallic corrosion between a magnesium alloy and mild steel in contact with brine solution. In this regard, the developed model in this study is validated against the experimental finding using SVET (Scanning vibrating electrode technique) of a bimetallic corrosion in,⁴⁴ and a numerical result in,²¹ see Figure 4, where current density distributions are shown in the magnesium-steel coupled surface at a time interval of 1000 s. Negative and positive values determine the cathodic reduction and anodic dissolution currents, accordingly. The maximum anodic current density in,^21,44 and this study are: 84.1 A/m², 81.6 A/m² and 80.7 A/m², accordingly. Hence, our numerical results are in excellent agreement with experimental results.

Results and discussions

To accurately model the electrochemical process of cathodic protection for steel structure, a nonlinear set of governing equations needs to be solved numerically. In this study, we utilized COMSOL MULTIPHYSICS (V 5.2), a finite element PDE solver, to impressively solve these equations and draw the distributions of potential and current density within the computational domain.

In order to create a suitable mesh, considering the importance of dissolution and formation of deposition on the anode, we employed an extra fine element size in the meshing process. The resulting meshes comprise 86,618 mesh vertices, 446,343 tetrahedron elements, 49,384 triangle elements, 7268 edge elements, and 315 vertex elements, with an average element quality of 0.8468. With the completion of this meshing process, now we discuss about the deposition formation process and its simulation.

The $Mg (OH)_{2}$ deposits formed on the anode surface are porous. So, the electrical resistance is not uniform along the anode surface, which can cause a different current density distribution along the anode surface. Also, the deposit affects the mass transfer of ions. In Figure 5, the current density distribution on anode 5 (see Figure 3) is depicted over 30 years, in the presence and absence of deposition as an animated album.

The corrosion deposition may increase or decrease the electrical resistance at the anode-soil interface depending on the deposit conductivity relative to soil conductivity (see Figure 5). The deposit formation also affects the ion transport and inhibits the ion transfer which should result in slower electrochemical reactions.

When the deposition conductivity is lower than the soil conductivity (e.g. Figure 5(b)), formation of deposit layer increases the electrical resistance at the anode-soil interface; as well as ion transport inhibition. As such, the current output from the anode constantly reduces. However, when the deposit conductivity is higher than that of soil (e.g. Figure 5(c)), as the anode dissolves, the medium near the anode becomes more conductive, and as shown in Figure 5(c) the current density increases in time.

Further investigations show that the current density of the case in Figure 5(c) increases in time, reaches a peak then decreases, see Figure 6(a). In other words, the increase in anode corrosion rate (more current output) means the cathodic protection is higher. So, when the deposit conductivity is higher than the electrolyte conductivity, the cathodic protection is more effective at first, and after passing the peak current value, the corrosion rate and thus the cathodic protection process decreases. This phenomenon is explained by the competition between the expanding anodic surface and the inhibiting effect of the deposit layer. Thus, the growing deposit layer can inhibit the current density, while the enlarging of the electrode area over time leads to an increase in the current. Initially, the thin deposit layer has little protective effect and thus the expanding electrode area dominates the competition, causing the current to increase with time. Over time, the deposit layer becomes increasingly thick and forms an effective protective barrier to inhibit the corrosion current density. In this condition, the expansion of the electrode area cannot neutralize the inhibiting effect of the growing deposit layer, and as a result, the current decreases with increasing time.

It is worth noting that neglecting the effect of deposition in the simulations may result in over-predicting or under-predicting the anode corrosion, depending on the corrosion deposition conductivity, see Figure 6(b).

Using Figure 6(a) and (b), one may compare the output current from anode and anode mass loss after 30 years for three cases: $σ_{c p} = 1.62 σ$ (corrosion product conductivity is greater than that of soil by 1.62 times), no deposition (or conductivity of corrosion product is equal to that of soil) and $σ_{c p} = 0.56 σ$ (corrosion product conductivity is less than that of soil by 0.56 times). The current output and mass loss values for the cases after 30 years are: “0.38 A, 8 kg”, “0.31 A, 6.6 kg”, and “0.25 A, 5.5 kg”, accordingly.

As expected current output from anodes is divided between the steel structure and the copper grounding system. In the following, the distribution of the CP currents between the steel structure and the copper grounding system is shown in Figure 6(c) and (d), respectively. Simulation results show that around 54% of the CP current is received by the grounding system, and only about 46% is used to protect the main structure, despite a significant portion of steel structure.

An animated trend of deposit formation on anode-soil interface is shown over 30 years in Figure 7. The formation of deposit begins from the two ends of the anodes and then covers the whole anode surface.

Indeed, Figure 7 shows a plot of volume fraction of deposition over 30 years. The region with volume fraction 1 represents the corrosion product deposition and value 0.5 represents the position of corrosion product interface. The simulation results show when the deposition conductivity $σ_{c p}$ is lower than the soil conductivity $σ$ , formation of deposit layer increases the electrical resistance at the anode-electrolyte interface; and current output from the anode reduces (see Figure 7(a)). This corresponds to less cathodic protection on the cathode side, i.e. the structure. However, when the deposition conductivity $σ_{c p}$ is higher than the electrolyte conductivity $σ$ , formation of deposit layer decreases the electrical resistance and current output from the anode increases (Figure 7(b)).

Due to the competition between the transport process and the electrochemical reaction, as the electrons are free to leave the anode, ${Mg}^{2 +}$ concentration on the anode increases in time (see Figure 8(a)), due to the generation of ${Mg}^{2 +}$ from the Mg dissolution. However, ${OH}^{-}$ concentration decreases in time (see Figure 8(b)), as the ${OH}^{-}$ ions are hindered by the deposits and fresh ${OH}^{-}$ ions cannot move towards the anode and the existing ${OH}^{-}$ ions are consumed by ${Mg}^{2 +}$ to form the deposits. As shown in Figure (8), the super-saturation of $Mg (OH)_{2}$ is controlled by the concentration of ${Mg}^{2 +}$ ions, and the higher concentration of ${Mg}^{2 +}$ ions on the anode surface is the driving force for the formation of $Mg (OH)_{2}$ .

Figure 9 shows the distribution of potential and cathodic protection current density distribution on the main structure after one year, 15 years and 30 years from the protection. The results confirm a decrease in protection level. Based on the NACE SP0169 standard,⁴⁵ middle and bottom sections of structure depict low protection levels compared to the top section. The reason is that the focus of the anode arrangement is in the upper part of the structure, and most of the protection current is consumed by the copper grounding system.

As explained in Figures 3, 6(c) and (d) almost 54% of the output current from the anode is wasted by the grounding system, and only about 46% is used to protect the structure. The designer goal is to protect the buried structure from corrosion, not the grounding system. The number of anodes, their location, and cost (anode and installation) are the most essential parameters of the cathodic protection system, and in practice, these parameters should be optimized. So here we define a cathodic protection efficiency as follows:

e f f i c i e n c y = anode mass loss fraction \times current fraction received by the steel structure

(28)

To better understand this efficiency, the original design is compared with 3 other artificial cases. These configurations are shown in Figure 10.

Figure 11 shows cathodic protection current density and potential distribution on the steel structure after 1 year and 30 years. As can be seen, the 3^rd configuration shows a better performance in terms of the protection potential range. The reason is that the 7.7 kg sacrificial anode has the highest efficiency due to the optimal design of the anode size and protects the structure better, while in the configuration 1 and 2 due to the use of 14.5 kg anodes, we will have a much lower efficiency, while the effect Burial of the anodes at different heights has also been shown. For a better understanding, refer to Figure 12 and Table 3.

Table 3.

The data of different configurations where $σ_{c p} = 1.13 σ$ .

Configuration num.	Main	1^nd	2^nd	3^rd
Number and mass of anodes	2 × 4 × 14.5 kg = 116 kg	1 × 8 × 14.5 kg = 116 kg	2 × 4 × 14.5 kg = 116 kg	2 × 8 × 7.7 kg = 123.2 kg
Anode mass loss after 30 years (%)	47.3	50.3	49	79
Current received by the steel structure from anodes (%) *	46	53	57	66
Cathodic protection efficiency after 30 years	0.473 × 0.46 = 0.22	0.503 × 0.53 = 0.27	0.49 × 0.57 = 0.28	0.79 × 0.66 = 0.52
Material cost ($)⁴⁶	8 × 750 = 6000	8 × 750 = 6000	8 × 750 = 6000	16 × 425 = 6800
Drilling cost ($)⁴⁷	2500	2800	3600	5600
Total cost ($)	8500	8800	9600	12400

*The remaining current is received by the grounding system

As shown in Figure 10, the anode mass in all the configurations is somewhat equal. After 30 years, in the 3^rd configuration, the mean potential (829.1 mV) and current density (6.96 mA/m²) on the surface of the structure are much higher than other configurations (785 mV–3.89 mA/m² in 1^st configuration and 793.2 mV–4.07 mA/m² in 2^nd configuration), which results in much higher mass dissolution compared to other configurations (79% mass loss), see Figure 12. Also, the efficiency of the 3^rd configuration is higher (see Table 3). As shown in Table 3, in the 1^st and 2^nd configurations, 53% and 57% of the current is received by the structure, respectively while in the 3^rd configuration 66% of the current was received by the structure. Comparing configurations 1 and 2 (14.5 kg), despite the higher mass loss of the anodes in configuration 1, in configuration 2, because the anodes are buried deeper, the bottom of the structure is better protected, although the cost of drilling and installation is higher in the 2^nd configuration.

Overall, the comparison indicates that from the protection perspective, the 3^rd configuration displays relatively better performance than other configurations but at a higher cost. Thus, trading between cost and level of protection is the most important challenge and creates a condition to perform an industrial optimization in future, of course, if the structure is close to the protection range or on the threshold of losing the protection range.

The information in Table 3 shows the inefficiency of the 14.5 kg anode. In this regard, the current output in each anode and the overall current have been investigated. Using the empirical tests around the magnesium anode,^48–50 the relation 29 is presented for the output current from every anode as follows:

I_{Mg} = C_{e} F^{'} Y^{'} / ρ^{'}

(29)

Table 4 (a to c) shows the parameters affecting the current output and describe the values related to

F^{'}

Y^{'}

and

C_{e}

parameters obtained based on experimental values.

Table 4.

(a) Anode shape correction ( $F^{'}$ ),⁵¹ (b) Driving voltage correction ( $Y^{'}$ ),⁵⁰ and (c) Coating effectiveness ( $C_{e}$ ).⁵⁰

(a)
Anode weight kg (lb.)	Type (D × L) (standard anode-cm)	$F^{'}$
4.08 (9)	9 × 35	0.76
7.70 (17)	9 × 65	1.00
14.5 (32)	14 × 52	1.14
21.75 (48)	14 × 82	1.31

(b)
Structural potential (V vs. CSE)	Standard Mg anode	High potential Mg anode
−0.70	1.21	2.14
−0.80	1.07	1.36
−0.85	1.00	1.29
−0.90	0.93	1.21
−1.00	0.79	1.07
−1.10	0.64	0.93
−1.20	0.50	0.79

(c)
Type of anode	Standard Mg anode	High potential Mg anode
Structure with good coating	96000	120000
Structure with a weak or bare coating	120000	150000

Also Dwight's relationship and the output current from each anode are presented as equation (30):

R = \frac{0.005 ρ^{'}}{π L} [L n (\frac{8 L}{d}) - 1], I = \frac{Δ V}{R}

(30)

The experimental relationship 29 (Tefankjian equation) shows far more logical results than Dwight's relationship. Table 5 shows the comparison of the results between the equations of Dwight, Tefankjian as well as Comsol software in terms of current output of each anode.

Table 5.

Comparing the results of output current values from 7.7 and 14.5 kg anodes using equations 29, 30 and Comsol software.

	Equation (29) (mA)	Equation (30) (mA)	Comsol (Averagely) (mA)
$I_{7.7 kg}$	38.7	30.2	39.6
$I_{14.5 kg}$	44.2	30.7	45.2

The summary of calculations is presented in Table 6 for configurations 2 and 3 with identical anode placement. As can be seen despite the anodes having nearly equal mass, the 7.7 kg anodes have 42.9% more current output than the 14.5 kg anodes and provide much better potential distribution and higher efficiency (see Figure 11). Meanwhile, the 14.5 kg anodes do not even supply the current required for protection. Also, considering the 30-year-old design, it was expected that a large percentage of the anode would be consumed to determine the efficiency of the anode, but in the anode of 14.5 kg, about 51% of the mass of the anode remains and only 57% of the output current is directed to the structure and 43% goes to the adjacent structure (see Table 3). Also, optimization and experience show that as the ratio of anode length to diameter increases, current output and anode efficiency increase. Also, the 14.5 kg anodes have an inefficient design due to their low current output compared to their large mass, and to solve the shortcoming of this design, they must be used together with backfill to fix the small current output defect. With the descriptions provided, it can be concluded that the 7.7 kg anode has a suitable design in terms of dimensions and due to its effective current output and suitable efficiency, is more practical in industries.

Table 6.

Specifications related to current output, total weight and dimensional ratio for Config. 2 and 3.

Configuration	Total weight (kg)	Current output of each anode (mA)	Total current output (mA)	The required current of CP (mA)	$\frac{L}{D}$
2 (14.5 kg)	116	44.2	354	415	3.70
3 (7.70 kg)	123.2	38.7	620	415	7.14

In general, our model thoroughly explores the dynamics of current output from dissolved magnesium sacrificial anodes, encompassing concentration, electrical, and electrochemical fields, while incorporating the influence of deposition through the level set method. The investigation spans a 30-year timeframe, addressing critical factors such as anode burial depth and the utilization of anodes. Specifically, we carefully examine the impact on the current directed towards the protected steel structure and the consequential waste current to the ground system. This model can be applied to diverse CP situations, providing valuable insights into anode corrosion, deposit formation, and CP efficiency across varying conditions.

Conclusion

In this study, we conducted a comprehensive three-dimensional and time-dependent numerical simulation to model cathodic protection for in-ground structures. The concentration, electrical, and electrochemical fields, along with their governing relationships, were thoroughly explained to describe the impact of deposition on the current output of magnesium sacrificial anodes. Our findings revealed that if the conductivity of the deposition is lower than the soil conductivity, the output current diminishes over time. Conversely, when the deposition conductivity surpasses the soil conductivity, the output current experiences an initial increase, reaches a peak, and subsequently decreases. Three key factors influencing this process include an increase in anode surface area, alterations in medium conductivity, and the inhibitory effect of the deposition layer. The primary goal of our research is to protect steel structures, recognizing that a significant portion of the anode's output current is lost to the grounding system. In the original cathodic protection system, 54% of the anode current is directed towards the grounding system. Furthermore, we explored the impact of anode burial depth on the current directed towards the steel structure, emphasizing the need to reach a balance between burial depth and increased costs. In a comparative analysis, we evaluated the current output of two anodes weighing 7.7 and 14.5 kg. The results indicated that the 7.7 kg anode exhibits a more effective current output for structural protection, attributed to its superior dimensional design. Our assessment introduced a cathodic protection efficiency parameter, calculated as the anode mass loss fraction multiplied by the current fraction received by the steel structure. To evaluate the significance of this parameter, we compared the original configuration with three alternative configurations, each possessing a relatively equal total anode mass.

Footnotes

Acknowledgments

AM would like to acknowledge the HCT interdisciplinary grant for generous financial support.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

M. J. Shirshahi

Abraham Mansouri

Seyed Farshid Chini

List of symbols Letter

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