Abstract
When a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.
Highlights
Shallow granular free-surface flows are one of the most common particle transport mechanisms in both our natural environment and industry
Hazardous geophysical mass flows such as snow avalanches (Savage & Hutter 1989; Ancey 2012), debris flows (Iverson 1997), rockfalls, pyroclastic flows (Branney & Kokelaar 1992) and lahars (Vallance 2000) all fall into this category, while in industry examples include chute flows (Pouliquen 1999a) as well as thin fluid-like avalanches that develop in the free-surface layers of heaps (Williams 1968), silos (Schulze 2008) and rotating tumblers (Gray & Hutter 1997; Hill et al 1999)
It is precisely these granular avalanches that are efficient at segregating particles by size (Middleton 1970; Savage & Lun 1988) and density differences between the particles (Drahun & Bridgwater 1983)
Summary
Shallow granular free-surface flows are one of the most common particle transport mechanisms in both our natural environment and industry. The first term ensures that particle percolation is driven by intrinsic rather than partial pressure gradients, the second provides a linear resistance to motion and the final term models diffusive mixing of the particles This form of the interaction drag is simple and allows an explicit formula for the normal velocity of species A and B to be derived. Dividing (2.25) and (2.26) by the intrinsic density, ρν∗, yields the segregation equations In each of these equations, the first term on the left-hand side describes the rate of change of the concentration φν with time, the second describes the transport of φν due to the bulk flow field, the third is due to segregation (with a typical φaφb structure) and the term on the right-hand side accounts for diffusive remixing of the particles.
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