Equilibrium theory of bidensity particle-laden suspensions in thin-film flow down a spiral separator

Spiral gravity separators are designed to separate multi-species slurry components based on differences in density and size. Previous studies [S. Lee et al., Phys. Fluids 26, 043302 (2014); D. Arnold et al., Phys. Fluids 31, 073305 (2019)] have investigated steady-state solutions for mixtures of liquids and single particle species in thin-film flows. However, these models are constrained to single-species systems and cannot describe the dynamics of multi-species separation. In contrast, our analysis extends to mixtures containing two particle species of differing densities, revealing that they undergo radial separation—an essential mechanism for practical applications in separating particles of varying densities. This work models gravity-driven bidensity slurries in a spiral trough by incorporating particle interactions, using empirically derived formulas for particle fluxes from previous bidensity studies on inclined planes [J. T. Wong and A. L. Bertozzi, Phys. D 330, 47–57 (2016)]. Specifically, we study a thin-film bidensity slurry flowing down a rectangular channel helically wound around a vertical axis. Through a thin-film approximation, we derive equilibrium profiles for the concentration of each particle species and the fluid depth. Additionally, we analyze the influence of key design parameters, such as spiral radius and channel width, on particle concentration profiles. Our findings provide valuable insights into optimizing spiral separator designs for enhanced applicability and adaptability.