A comparison of the pseudo-steady-state and shrinking-core model for the reduction of titanium dioxide to titanium

A multistage shrinking-core model is proposed for the electrodeoxidation of titanium dioxide to titanium. This takes place through a series of steps from TiO₂ to Ti₃O₅ to Ti₂O₃ to TiO to Ti. Ideally, the model would incorporate a number of shells of the above lower oxide phases with the shrinking core of TiO₂ in the center but this would be mathematically intractable. A simpler method would be to use the shrinking-core model for each of the individual reductions. Taking the experimental parameters and diffusion coefficient of oxygen in the different phases into account, an analytical solution is developed for the transient differential equation. The first ten eigenvalues are taken into account for the computation of the series solution. This is then compared with a solution based on a pseudo-steady-state approximation to the transient equations. Based on the results, for higher values (>0.01) of dimensionless applied current density, I _d , both the solutions disagree in terms of the time it takes for the core to shrink completely. The difference appears to be decreasing with lower values of I _d but for larger values of I _d the pseudo-steady-state approximation fails to yield results close to the analytical solution.