Kinetic theory for a monodisperse gas–solid suspension

The fluid-dynamic and solid-body interactions among a suspension of perfectly elastic particles settling in a viscous gas are studied. The Reynolds number of the particles, Re≡ρfUa/μ, is small but their Stokes number St≡mŪ/(6πμa2) is large, indicating that particle inertia and viscous forces in the fluid are important. Here, ρf is the density of the fluid, m is the mass of a particle, Ū is the average velocity of the particles, a is their radius, and μ is the fluid viscosity. Equations for the particle velocity distribution and averages of the fluid and particle velocities are derived. For very large Stokes numbers, St≫φ−3/2, where φ is the particle volume fraction, solid-body collisions lead to a nearly Maxwellian velocity distribution. On the other hand, at smaller Stokes numbers, St≪φ−3/2, fluid-dynamic interactions play a more important role in determining the particle velocity distribution and the distribution is not Maxwellian. The amount of energy contained in the particle velocity fluctuations is determined by a balance involving fluid-dynamic interactions, even in the case where the solid-body collisions lead to a nearly Maxwellian velocity distribution. The viscous interactions are found to be similar to those in a fixed bed at high Stokes numbers, St≫φ−3/4, where the particles do not respond quickly to changes in the local fluid velocity. A stability analysis of the averaged equations indicates that the suspension is unstable to particle density waves for St≫φ−3/2. The inertia of the particles is destabilizing, while the energy introduced into the fluctuating motions of the particles by viscous flow interactions is stabilizing.