L. Bortels

The multi-dimensional upwinding method as a new simulation tool for the analysis of multi-ion electrolytes controlled by diffusion, convection and migration. Part 1. Steady state analysis of a parallel plane flow channel

Abstract A new numerical method is presented for the calculation of concentration, potential and current distributions in two-dimensional electrochemical cells controlled by diffusion, convection and migration of ions. The numerical model, for reasons of generality developed for an explicit time-dependent solution, has been made implicit such that it can easily deal with electrochemical cells at steady-state involving multiple ions. The electrolyte solutions are considered to be dilute and at a constant temperature. This new method, the multi-dimensional upwinding method, originates from the field of fluid mechanics. It is an alternative approach to both finite element methods and finite volume methods. In order to evaluate the method, steady-state computations have been performed on two plane, parallel electrodes embedded in the walls of a flow channel. Tertiary current distributions have been calculated using Butler-Volmer polarisation laws and with the applied cell voltage as a driving force for concentration and potential gradients. Electrolytes with two and three ions were considered and the results, in case of an excessive amount of supporting electrolyte, were compared with the Levecque solution. In all cases, the numerical results are found to be in good agreement with analytical or numerical solutions from literature.

Analytical solution for the steady-state diffusion and migration involving multiple reaction ions Application to the identification of Butler-Volmer kinetic parameters for the ferri-/ferrocyanide redox couple

Abstract An analytical solution for the one-dimensional steady-state transport of ions in an electrolyte towards a planar electrode is obtained. This electrolyte contains more than one electroactive species and any number of non-reacting species. The mass and charge transport equations give rise to an implicit form of a set of non-linear algebraic equations which must be solved numerically. The solution is generally applicable and can deal with any kind of overpotential relation at the electrode. The analytical solution is used to determine the mass and charge transport parameters for the Fe(CN) 6 3− 4− redox couple in a KCl solution for two different electrolyte concentrations (0.03 M Fe(CN) 6 3− 4− + 1.0 M KCl and 0.005 M Fe(CN) 6 3− 4− + 0.2 M KCl ) . The agreement between the experimental and analytical current densities is perfect for both electrolyte solutions under investigation. It is shown that, although an excess of supporting electrolyte is added, neglecting migration results in an under/overestimation of the diffusion coefficient of ferri-/ferrocyanide of 3 to 5% for the Fe(CN) 6 3− 4− / KCl ratios investigated. Furthermore, a mathematical background is given for the wide range of values found in the literature for the charge transfer coefficient and the rate constants of the ferri-/ferrocyanide redox system. The same approach can also be useful for other systems, as to parameter identification procedures.

Quasi-one-dimensional steady-state analysis of multi-ion electrochemical systems at a rotating disc electrode controlled by diffusion, migration, convection and homogeneous reactions

Abstract This article presents a new numerical method for the calculation of concentration, potential and current distributions in electrochemical cells controlled by diffusion, convection, migration and homogeneous reactions of ions. A multi-dimensional upwinding method, originating from the field of fluid dynamics, has been adapted in order to solve this non-linear system. The model developed is able to deal with two-dimensional electrochemical cells involving multiple ions. The electrolyte solutions are supposed to be dilute, at steady state and at a constant temperature. Numerical calculations for a quasi-one-dimensional test case at a rotating disc electrode are performed. These results are compared with both experimental and analytical data, the latter based on a chemical-electrochemical reaction model, for the reduction of silver in a nitrate-thiosulphate solution. The numerical data are found to be in good agreement with the experimental and analytical data. The calculation of concentration profiles provides an interesting insight into the degree of dis-equilibrium of a homogeneous reaction in a thin reaction layer near the electrode.

A general applicable model for AC predictive and mitigation techniques for pipeline networks influenced by HV power lines

This paper presents a recently developed simulation software tool for ac predictive and mitigation techniques for pipeline networks influenced by high-voltage (HV) power lines. The software can deal with any configuration (no limitation on number of pipes, transmission lines, bonds, groundings, coating, and soil resistivity) and is very user friendly and robust since a general applicable algorithm is used to calculate the induced electromagnetic force (EMF), eliminating the need for a subdivision of the pipelines in sections parallel or not to the transmission line(s). With the calculated values for the EMF, the induced voltages and currents are then obtained by solving the well-known transmission-line model using a numerical technique that allows to specify the pipeline parameters (diameter, coating, soil resistivity, ...) for each individual section of the pipeline. In this paper, simulation results will be presented and compared with available theoretical test cases. It will be demonstrated that the calculated values for the induced EMF and the induced voltage and current are in perfect agreement with these theoretical test cases. In addition, it will be proven that commonly used formulas for the induced EMF need to be handled with care, especially when the distance between the transmission line and the pipeline becomes bigger.

Numerical simulation of transient current responses in diluted electrochemical ionic systems

A versatile model for the simulation of electrochemical processes at a rotating disc electrode is presented. This model is based on the dilute solution ion model, in combination with an adequate description of electrode reaction kinetics. A brief introduction is given to the theoretical aspects of the model and the numerical solution technique being used to solve this model for the rotating disc electrode (RDE). A multiple ion electrolyte system with an excess of supporting electrolyte is considered. For several reaction conditions (reversible up to irreversible reactions) results of steady-state and linear sweep voltammetry simulations are presented and are compared with analytical and literature data. Based on computations of the current response, while sinusoidal varying potentials with different frequencies are applied, electrochemical impedances are also derived and compared with analytical results.

Numerical steady state analysis of current density distributions in axisymmetrical systems for multi-ion electrolytes: application to the rotating disc electrode

The multidimensional upwinding method (MDUM) is used for the numerical calculation of concentration, potential and current density distributions at a rotating disc electrode controlled by diffusion, convection, migration and homogeneous reactions of multiple ions. MDUM, previously applied for two-dimensional applications, is now extended to axisymmetrical problems. The electrolyte solutions are supposed to be dilute, at steady state and at a constant temperature. Numerical calculations are performed for two electrochemical systems. In the first system, silver is reduced from a nitrate + thiosulphate solution. This system is controlled by diffusion, convection and a chemical-electrochemical reaction, while migration has a minor influence. In the second system, copper is reduced from a copper sulphate + sulphuric acid solution. Since the amount of sulphuric acid is not excessive, migration has a significant influence on the current density distribution at the electrode. For the copper system, numerical data obtained from MDUM are compared with both experimental results and numerical data from the literature and are found to be in very good agreement. MDUM provides reliable results for both systems without making any simplifications of the governing transport equations for dilute solutions and with a reasonable computational effort.

Analytical solution for the steady-state diffusion and migration. Application to the identification of Butler-Volmer electrode reaction parameters

Abstract An analytical solution for the one-dimensional steady-state transport of ions in an electrolyte between two planar electrodes has been obtained. This electrolyte contains one electroactive species and any number of non-reacting species. The mass and charge transport equations give rise to an implicit form of a set of non-linear algebraic equations which must be solved numerically. It has been shown that the same set of equations, with only a very small modification, can easily be used to solve the stagnant boundary layer problem. The solution is generally applicable and can deal with any kind of over potential relation at both anode and cathode. The analytical solution for the stagnant boundary layer has been used to determine the diffusion coefficient for the reacting ion and the kinetic parameters in the Butler-Volmer overpotential relation for the electrodeposition of copper from a 0.01 M CuSO 4 + 0.1 M H 2 SO 4 solution. The resulting parameters are in good agreement with the values found in the literature. Analytical results obtained with these parameters match very well with the experimental data for current densities ranging from secondary up to limiting current values and for different values of the rotation speed (100, 500 and 1000 rev min −1 ). Also, it has been shown that neglecting migration can lead to an overestimation of the diffusion coefficient of about 15%.

Three-Dimensional Boundary Element Method and Finite Element Method Simulations Applied to Stray Current Interference Problems. A Unique Coupling Mechanism That Takes the Best of Both Methods

Abstract In this paper it will be demonstrated how a boundary element method (BEM) model based on so-called pipe elements and a finite element method (FEM) model that is limited in space can be cou...

Numerical solution of electro-osmotic flow in a flow field effect transistor

Abstract Recently, a general applicable numerical model for the simulation of steady electro-osmotic flow in rectangular micro-channels has been presented and evaluated based on available analytical solutions [Anal. Chem. 74 (19) (2002) 4919]. This model combines the steady multi-ion transport model and the Navier–Stokes equations, and has been implemented in a finite element code. The model includes the ion distribution in the Helmholtz double layer and considers only one single electrical potential field variable throughout the domain. In this paper, the model is further evaluated through comparison with experimental data obtained in a so-called flow field effect transistor or FlowFET [Science 286 (1999) 942]. It has been found that the numerical solutions are in very good agreement with the experimental data and with the commonly used slip boundary condition model.

Numerical 3-D Simulation of a Cathodic Protection System for a Buried Pipe Segment Surrounded by a Load Relieving U-Shaped Vault

Abstract A three-dimensional, multi-domain boundary element method approach was used to determine the performances of a cathodic protection system applied to a buried pipeline of limited longitudinal dimension, surrounded by a U-shaped vault of infinite resistivity (insulator). A Laplace model describing the potential problem was solved in combination with nonlinear boundary conditions that hold for the polarization behavior of the coated pipeline and coating holidays. The efficiency of this cathodic protection system was examined as a function of the soil conductivity, the position of the defects, and the presence of the U-shaped vault. The objective of these simulations was the quantitative determination of the reducing effect of this structure on the protection level of the outer pipe surface, as a result of its spatial obstruction for the protective currents that originate from the far-field anode beds.