2 results
Enhanced sedimentation in narrow tilted channels
- Eric Herbolzheimer, Andreas Acrivos
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- Journal:
- Journal of Fluid Mechanics / Volume 108 / July 1981
- Published online by Cambridge University Press:
- 20 April 2006, pp. 485-499
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The analysis of Acrivos & Herbolzheimer (1979) is extended to describe the sedimentation of dilute suspensions in tilted two-dimensional channels in which the spacing between the plates is small compared with their length. The theory assumes that the flow is laminar and that the suspension consists of monodisperse spherical beads having small particle Reynolds number. Expressions for the flow fields in the clearfluid region and in the suspension, as well as for the location of the interface separating these two regions, are obtained asymptotically in the limit of Λ [Gt ] 1 with $R\Lambda^{-\frac{1}{3}}\ll 1$, where R and A are as defined in the previous work. The present analysis differs from that given earlier in that the aspect ratio, i.e. the ratio of the height of the suspension to the channel width, is now taken to be O(A1/3) rather than O(1) as was the case before. Under these conditions, the solution of the time-dependent equations leads to the surprising prediction that the clear-fluid layer which forms beneath the downward-facing plate attains a steady shape only along the lower portion of the channel while, in contrast, its thickness increases with time for locations along the channel that are above some critical point. Because of this transient behaviour, the well-known Ponder-Nakamura-Kuroda (PNK) formula overestimates the rate at which the top of the suspension region falls with time; however, the PNK results for the volumetric settling rate still hold under the conditions considered in this paper. It is shown that this discontinuity in the interface shape can be suppressed in continuous settling systems but only if the feed and withdrawal locations are chosen properly.
Batch sedimentation experiments were conducted in a channel with parallel flat walls under the following sets of conditions: H0/b ≈ 90, 5° ≤ θ ≤ 45°, 0·01 ≤ c0 ≤ 0·025, 1·7 × 107 < Λ < 3·5 × 107, and 1·8 < R < 2·1, where θ is the angle of inclination of the vessel from the vertical, and c0 is the initial volume fraction of solids in the suspension. The experimental observations were found to be in excellent agreement with the theoretical predictions.
Enhanced sedimentation in settling tanks with inclined walls
- Andreas Acrivos, Eric Herbolzheimer
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- Journal:
- Journal of Fluid Mechanics / Volume 92 / Issue 3 / 12 June 1979
- Published online by Cambridge University Press:
- 19 April 2006, pp. 435-457
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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.