Applications of polydisperse sedimentation models

This paper reviews some recent advances in mathematical models for the sedimentation of polydisperse suspensions. Several early models relate the settling velocity to the solids concentration for a monodisperse suspension. Batchelor's theory for dilute suspensions predicts the settling velocity in the presence of other spheres that differ in size or density. However, this theory is based on the questionable assumption that identical spheres have identical velocities, and leads to significantly differing results for spheres that differ only slightly in size or density. Since Batchelor's analysis cannot be extended to concentrated suspensions, one needs to revert to semi-empirical equations and computational results. A rational model developed from the basic balance equations of continuum mechanics is the Masliyah-Lockett-Bassoon (MLB) model. A useful tool for evaluating polydisperse hindered settling models in general is a stability analysis. Basically, a model should reflect that, for polydisperse suspensions of equal-density spheres, instabilities such as blobs or fingers during separation are never observed. These structures do not form if the model equations are hyperbolic. The MLB model provably has this property, in contrast to certain extrapolations of the Batchelor model. The sedimentation process of a suspension can be simulated by either solving the conservation equations numerically by using a sophisticated scheme for conservation laws, or by using a particle-based method. Numerical examples illustrating both methodologies are presented, with an emphasis on fluidization problems.