In an electrolysis tank where electrorefining or electrowinning is performed, differences in the local density of the electrolyte due to variations in ion concentration bring about natural convection near each electrode. This natural convection affects the supply of metal ions from the bulk solution to the cathode surface and the movement of ionic species generated at the anode surface. This paper describes calculation methods that are based on boundary layer theory and have been proposed in the past to analyze natural convection caused by the ion concentration distribution in an electrolyte. The boundary layer theory makes it possible to understand the diffusion-limited current density at the cathode by linking it to the concentration of metal ions in the electrolytic bath and the physical properties of the electrolyte. First, we introduce a method of converting the boundary layer equations from partial differential equations into ordinary differential equations using similarity variables, and then derive theoretical formula for the flow velocity of natural convection and the diffusion-limited current density at the cathode with the help of numerical calculations. Furthermore, we derive a formula for estimating the diffusion-limiting current density from the size of the electrode and the physical properties of the electrolyte using dimensional analysis, which is widely used in fluid mechanics. Next, we will explain the von Kármán-Pohlhausen integration method as another approach for analyzing the flow velocity of natural convection and current density distribution. Experimental methods for observing the boundary layer and a calculation model improved for application to electrolysis at current densities below the diffusion limit are also presented.

In recent years, there has been an increase in cases where new tunnels are excavated parallel to existing tunnels along highways and major national roads. Although separation distances are secured in the planning and design phases to avoid interference between the tunnels, there can still be impacts on existing tunnels in poor ground conditions. In areas with poor ground stability, the excavation of a new tunnel can still lead to unanticipated impacts on the structural integrity of nearby, pre-existing tunnels. Previous research has documented measurement results and numerical analyses of factors such as existing tunnel lining stress and the behavior of the intermediate ground. However, few studies have sufficiently compared and discussed measurement results after the completion of new tunnel excavation with predicted analysis results. This study aims to analyze measurement data from an actual parallel tunnel construction project and examine the behavior of both the existing and new tunnels through numerical back-analyses based on these measurements, assessing the validity of these numerical predictions. From the measurements of ground displacement, lining stress, and crack displacement in the existing tunnel, it was observed that the loosening loads initially acting during the excavation of the existing tunnel were reactivated by the excavation of the new tunnel, indicating the effects of close-proximity construction. The back-analysis results showed discrepancies between measured and predicted values of crown settlement and inner displacement when using a standard stress release rate for the ground properties and lateral earth pressure coefficient of the existing tunnel in the new tunnel excavation analysis. Adjusting the stress release rate brought the displacement predictions closer to the measured values, and aligning the displacements allowed the lining stress to be reasonably predicted. Consequently, this approach to back-analysis demonstrates that it is possible to estimate the impact on the existing tunnel during new tunnel excavation.