Using the principles of continuum mechanics, a theory is developed for describing quantitatively the sedimentation of small particles in vessels having walls that are inclined to the vertical. The theory assumes that the flow is laminar and that the particle Reynolds number is small, but c0, the concentration in the suspension, and the vessel geometry are left arbitrary. The settling rate S is shown to depend upon two dimensionless groups, in addition to the vessel geometry: a sedimentation Reynolds number R, typically O(1)-O(10); and Λ, the ratio of a sedimentation Grashof number to R, which is typically very large. By means of an asymptotic analysis it is then concluded that, as Λ → ∞ and for a given geometry, S can be predicted from the well-known Ponder-Nakamura-Kuroda formula which was obtained using only kinematic arguments. The present theory also gives an expression for the thickness of the clear-fluid slit that forms underneath the downward-facing segment of the vessel walls, as well as for the velocity profile both in this slit and in the adjoining suspension.
The sedimentation rate and thickness of the clear-fluid slit were also measured in a vessel consisting of two parallel plates under the following set of conditions: c0 ≤ 0·1, R ∼ O(1), O(10)5 ≤ Λ ≤ O(107) and 0° ≤ α ≤ 50°, where α is the angle of inclination. Excellent agreement was obtained with the theoretical predictions. This suggests that the deviations from the Ponder-Nakamura-Kuroda formula reported in the literature are probably due to a flow instability which causes the particles to resuspend and thereby reduces the efficiency of the process.
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