Polydisperse suspensions with particles of a finite number N of size classes have been widely studied in laboratory experiments. However, in most real-world applications the particle sizes are distributed continuously. In this paper, a well-studied one-dimensional kinematic model for batch sedimentation of polydisperse suspensions of small equal-density spheres is extended to suspensions with a continuous particle size distribution. For this purpose, the phase density function Φ = Φ(t, x, ξ), where ξ ∈ [0, 1] is the normalized squared size of the particles, is introduced, whose integral with respect to ξ on an interval [ξ1, ξ2] is equivalent to the volume fraction at (t, x) occupied by particles of that size range. Combining the Masliyah–Lockett–Bassoon (MLB) model for the solid-fluid relative velocity for each solids species with the concept of phase density function yields a scalar, first-order equation for Φ, namely the equation of the generalized kinetic theory. Three numerical schemes for the solution of this equation are introduced, and a numerical example and an L1 error study show that one of these schemes introduces less numerical diffusion and less spurious oscillations near discontinuities than the others. Several numerical examples illustrate the simulated behavior of this kind of suspensions. Numerical results also illustrate the solution of an eigenvalue problem associated with the equation of the generalized kinetic theory.
Monte Carlo simulation for soot dynamics
