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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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