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

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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 cutoff wavenumber from which the length scale is estimated. Finally, the model results are compared to experimental data.

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Abbreviations

\(A_1\) to \(A_9\) :

Constants

\(a\) :

Constant

\(B\) :

Constant

\(B_1\) to \(B_5\) :

Constants

\(b\) :

Constant

\(C_1\) to \(C_5\) :

Constants

\(c\) :

Granular compactness

\(D\) :

Tube diameter (m)

\(d\) :

Grain diameter (m)

\(g\) :

Gravity acceleration (\(\mathrm{m/s}^2\))

\(H\) :

Humidity index

\(k\) :

Wavenumber (\(\mathrm{m}^{-1}\))

\(P\) :

Pressure (Pa)

\(P_{\mathrm{atm}}\) :

Atmospheric pressure (Pa)

\(R\) :

Tube radius (m)

\(v_s\) :

Velocity of individual grains (m/s)

\(W\) :

Grain flow rate (kg/s)

\(z\) :

Vertical coordinate (m)

\(\kappa \) :

Redirection coefficient

\(\gamma \) :

Ratio of specific heats

\(\lambda \) :

Wavelength of the plugs (m)

\(\mu _a\) :

Dynamic viscosity of air (Pa s)

\(\mu _s\) :

Friction coefficient between grains

\(\rho _{s}\) :

Specific mass of each grain (\(\mathrm{kg/m}^3\))

\(\omega _r\) :

Angular frequency (rad/s)

\(\omega _i\) :

Growth rate (\(\mathrm{s}^{-1}\))

\(\sigma _{\mathrm{zr}}\) :

Stress between the tube wall and the grains (\(\mathrm{N/m}^2\))

\(\sigma _{\mathrm{zz}}\) :

Vertical stress operating on the grains (\(\mathrm{N/m}^2\))

\(a\) :

Relative to air

\(s\) :

Relative to grains

\(0\) :

Relative to the basic state

\(\tilde{}\) :

Relative to the perturbation

\(\hat{}\) :

Relative to the amplitude of perturbations

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Acknowledgments

Erick de Moraes Franklin is grateful to FAPESP (Grant no. 2012/19562-6) and to FAEPEX/UNICAMP (conv. 519.292, projects AP0008/2013 and 0201/14). Carlos Alvarez Zambrano is grateful to SENESCYT. The authors thank Rodolfo M. Tomazela for the help with the experimental device.

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Correspondence to Erick de Moraes Franklin.

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Technical Editor: Francisco Ricardo Cunha.

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Franklin, E.d.M., Zambrano, C.A. Length scale of density waves in the gravitational flow of fine grains in pipes. J Braz. Soc. Mech. Sci. Eng. 37, 1507–1513 (2015). https://doi.org/10.1007/s40430-014-0291-3

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  • DOI: https://doi.org/10.1007/s40430-014-0291-3

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