Mathematical Modelling of Electrochemical Machining Processes

The solution to free boundary problems arising from electrochemical machining processes is considered. A mathematical description of the machining process is presented with particular consideration given to its electrochemical properties and to the appropriateness of its treatment as a potential problem with Dirichlet boundary conditions. A number of different machining configurations are considered, for which the evolution of the workpiece is determined. In all but the simplest of cases the use of numerical techniques is necessary to include geometric and electrochemical effects upon machining rates. The adoption of these techniques allows realistic problems to be considered when the surface geometries are irregular and the associated boundary conditions are nonlinear. In the one-dimensional case, the smoothing of surface irregularities is examined by a perturbation procedure and previous work is extended to include higher order terms and a more general description of metal dissolution. The numerical solution of Laplace's Equation in two-dimensional regions is considered by using a boundary integral technique. This formulation determines electric potentials and fields throughout the machining domain solely through the use of the boundary conditions and a description of the surface geometry. Irregular boundary geometries are considered and nonlinear boundary conditions are applied through the adoption of a suitable iterative procedure. The evolution of the workpiece with time is readily calculated, rather than merely a final steady state configuration. The presence of overpotential effects upon workpiece evolution is also considered, together with the effects of insulated portions of the electrode. Electrochemical effects immediately adjacent to a metallic surface have been analysed in some detail. A singular perturbation analysis of the Nernst-Planck equations is performed with the inclusion of curvature effects. An analysis is made of the case of a simple, symmetric electrolyte and is extended to include the more general case of an electrolyte containing several species of differing valencies. A complete description of the current and species distributions throughout the computational domain is derived. Within the boundary layer analysis it is demonstrated that surface curvature effects have the greatest influence on the solution.