FEM (Finite Element Method) Simulation of the Three-Dimensional Boundary Layer Close to a Rotating Semi-Spherical Electrode in Electrochemical Cells

Abstract

Iron rotating disk electrodes are widely in applied electrochemistry due to the fact of requiring an easily built experimental setup and to the existence of a simple semi-analytical similarity solution for the steady hydrodynamic equations describing the flow close to the electrode. Based on this solution the current in the cell is theoretically evaluated and compared to the experimental results. A caveat of such experimental apparatus results from the fact that, particularly at high current regimes, dissolution of the iron electrode in the 1M H2SO4 solution of the electrolyte endamages the geometry of the electrode surface and the electrode no longer can be used. An alternative to overcome the problem consists in using rotating semi-spherical electrodes, which keep the geometry in the dissolution process. However, no similarity solution exists for the complete hydrodynamic equations exits for this case. First, a boundary layer approximation is required to describe the flow in the neighborhood of the electrode and second, a solution is found, in terms of a power series of the polar angle, multiplying functions describing the dependency of the velocity componentes on the radial coordinate. In this work we briefly review the principles of the power series solution for the steady solution of the boundary layer developed by constant viscosity electrolytes close to rotating semi-spherical electrodes and propose a numerical Finite Element (FEM) procedure to obtain the velocity profiles for polar angles (southern direction) ranging 0, angle, pi/2. Spatial discretization of the diffusive and pressure terms is performed by the Galerkin method, and of the material derivative, through the Semi-lagrangian method. The numerical code, developed in C++, uncouples the velocity and pressure through the discrete projection method. Conjugate gradient method is used to solve the velocity whereas pressure is solved with the generalized minimal residual method (GMRES). The results obtained are compared with the semi-analytical solution obtained by the power series method. The effect of boundary conditions imposed in the simulations at the equator plane is analyzed and compared to the conditions assumed in the semi-analytical solution.