Recently, application of numerical simulation methods such as boundary element method (BEM) or finite element method (FEM) to the electrochemistry problems has been studied intensively, and these methods have become powerful tools. With the progress of numerical simulation, “Inverse Problems” have received a lot of attention. Inverse Problem is a research area dealing with finding out unknown information from external or indirect observation through the model of the system.
The background of inverse problems is the modeling and simulation of natural phenomena. When observations are taken of these phenomena, the observation data are used to infer knowledge about physical states. One of the most spectacular success in the field of inverse problems was the invention of an inversion algorithm for Computed Tomography. Not only the computed tomography in medical area, applications of inverse problems arise in many fields of engineering, too. In the fields of electrochemical engineering, there are many problems which can have a benefit from inverse analysis approach.
Our research focuses on the techniques for monitoring the accurate current density distribution and concentration distribution on the surface of the electrode during the polarization characteristic measurement. We have developed a novel technique to identify current density and concentration distribution just on the electrode surface from the data of total current, bulk concentration and bulk flow rate. In order to measure total currents, we use electrochemical flow cell that can control bulk flow rate.
In this inverse analysis, the concentration distribution on the electrode surface, i.e., boundary conditions of boundary value problem with advection diffusion equation are not fully given, but the total currents and bulk concentration are obtained by practical measurement to compensate for lack of unknown boundary conditions. The solution would be abnormally unstable due to the ill posedness. Hence, let us make use of the a-priori information that concentration and its flux proportional to the current density on the electrode must satisfy the polarization characteristics.
we will approximate the polarization characteristics by a function of the concentration and its flux, e.g., piecewise bi-linear function with several number of parameters. Firstly, the velocity field of the flow cell is analyzed with finite volume method as shown at top-right in the figure. The velocity filed is described with Navier-Stokes equation. The figure represents the procedure of the whole inverse analysis. Assuming parameters of polarization characteristics, the direct analysis is performed and total currents is calculated. Minimizing the residual between the measured and the calculated total currents with nonlinear least square method, the parameters of polarization characteristics can be obtained. After obtaining the polarization characteristics, the concentration and the current density distribution on the electrode are easily calculated by the direct analysis.
Our primary aim is to verify the applicability of the inverse analysis in the estimation of the polarization characteristics. In order to concentrate on the evaluation of process, we have created necessary measured data through a regular/direct numerical simulation instead of carrying out real experimental measurements. Such a procedure allows the present technique to be isolated from other factors such as unexpected measurement or simulation model error. Up on this, to accommodate an important aspect of experimental measurements, a small perturbation/error is added to the calculated results. These values are assumed as the measured data and used in the inverse analysis to determine the polarization characteristics. The final identified results by the proposed technique are in good agreement with the results computed by the numerical simulation. In order to demonstrate the effectiveness of this technique, the identification using real experimental data is also performed.