Length scale of density waves in the gravitational flow of fine grains in pipes

Abstract

Gravitational flow of grains in pipes is frequently encountered in industry. When the grains and pipes are size-constrained, granular flow may result in density waves consisting of alternate high-and low-compactness regions. This paper discusses the length scale of density waves that appear when fine grains fall vertically in pipes. A one-dimensional model and a linear stability analysis of the model are presented. The analysis suggests the presence of long-wavelength instability for the most unstable mode, moreover, a cut-off wavenumber from which the length scale is estimated. Finally, the model results are compared to experimental data.

Intermittent gravity-driven flow of grains through narrow pipes

“…These experiments and those of Franklin and Alvarez [14] showed that the plug sizes are 3 < λ/D < 11, which is in perfect agreement with the proposed model. However, as only one tube diameter was employed, we then compare these results with the previously published results.…”

“…Under these conditions, there is a redirection of forces within the plug, and the Janssen effect [23] is expected. Franklin and Alvarez [14], using the Janssen's theory, calculated the mean size of a compact plug and reported the length of plugs was in the range 3 < λ/D < 11, which is in accordance with the results of the present study.…”

“…where ρ s is the specific mass of each grain, g is the gravitational acceleration, P is the air pressure, σ zz is the vertical stress operating on the grains, and σ zr is the stress between the tube wall and the grains. We use here the closure of σ zz and σ zr proposed by Franklin and Alvarez [14]. The first one is to take into account the redirection of forces through a constant coefficient (dimensionless) [1] κ: σ zr = µ s κσ zz , where µ s ≈ tan(32 o ) is the friction coefficient between the grains, and the grains and the pipe walls.…”

“…Recently, Franklin and Alvarez [14] presented a linear stability analysis and experimental results for the vertical chute of grains in a narrow pipe. They found a dimensional dispersion relation to be solved numerically, and the analysis was limited to some small ranges of grains and pipes.…”

“…The experiments were performed in a 1 m long glass tube of 3 mm internal diameter aligned vertically, and the grains consisted of glass beads of specific mass ρ s = 2500 kg/m 3 divided in two different populations: grains with diameter within 212 µm ≤ d ≤ 300 µm and within 106 µm ≤ d ≤ 212 µm. Franklin and Alvarez [14] reported the existence of granular plugs with length in the range 3 < λ/D < 11, where λ is the plug length.…”