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A Parametric Study of Mixing in a Granular Flow a Biaxial Spherical Tumbler

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Dynamical Systems: Modelling (DSTA 2015)

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Abstract

We report on a computational parameter space study of mixing protocols for a half-full biaxial spherical granular tumbler. The quality of mixing is quantified via the intensity of segregation (concentration variance) and computed as a function of three system parameters: angles of rotation about each tumbler axis and the flowing layer depth. Only the symmetric case is considered in which the flowing layer depth is the same for each rotation. We also consider the dependence on \(\bar{R}\), which parametrizes the concentric spheroids (“shells”) that comprise the volume of the tumbler. The intensity of segregation is computed over 100 periods of the mixing protocol for each choice of parameters. Each curve is classified via a time constant, \(\tau \), and an asymptotic mixing value, bias. We find that most choices of angles and most shells throughout the tumbler volume mix well, with mixing near the center of the tumbler being consistently faster (small \(\tau \)) and more complete (small bias). We conclude with examples and discussion of the pathological mixing behaviors of the outliers in the so-called \(\tau \)-bias scatterplots.

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Notes

  1. 1.

    In an experiment, the flowing layer is at an angle with respect to the horizontal, while, in the model, the coordinate system is rotated backwards by this fixed angle. Also, we assume that the effects of side walls and transport in the transverse direction are negligible.

  2. 2.

    In an experiment, several intermediate steps are performed between the rotations [8], however, as far as the mathematical analysis is concerned, the rotations are completely independent.

  3. 3.

    In Figs. 2, 4, 5 and 6 below, the quality of this fit is given as two numbers in brackets in the legend of the rightmost plot. The first number is the coefficient of determination \(R^2\); the second number is the root-mean-squared error in the fit.

  4. 4.

    Though these numbers are for a quasi-2D circular tumbler, they are relevant here too thanks to the geometric similarity assumption used to construct the 3D continuum model. That is to say, if we suppose the quasi-2D tumbler of Jain et al. [24] is the \(x=0\) (or \(z=0\)) cut through the 3D spherical tumbler considered here, then \(\varepsilon _z\) (or \(\varepsilon _x\)) herein is precisely \(\delta _0/R\) in [24].

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Acknowledgments

I.C.C. was supported, in part, by a Walter P. Murphy Fellowship from the Robert R. McCormick School of Engineering and Applied Science and by US National Science Foundation grant CMMI-1000469 at Northwestern and by the LANL/LDRD Program through a Feynman Distinguished Fellowship at Los Alamos National Laboratory, which is operated by Los Alamos National Security, L.L.C. for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396. We thank Stephen Wiggins for suggesting the \(\tau \)-bias scatterplots and useful discussions.

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Authors and Affiliations

  1. Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, 87545, USA

    Ivan C. Christov

  2. Department of Chemical and Biological Engineering, Department of Mechanical Engineering and The Northwestern Institute on Complex Systems (NICO), Northwestern University, Evanston, 60208, USA

    Richard M. Lueptow & Julio M. Ottino

  3. Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK

    Rob Sturman

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  1. Ivan C. Christov
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  2. Richard M. Lueptow
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  3. Julio M. Ottino
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  4. Rob Sturman
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Correspondence to Ivan C. Christov .

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Editors and Affiliations

  1. Dept. of Autom., Biomech. and Mechatr., Łódź University of Technology, Łódź, Poland

    Jan Awrejcewicz

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Christov, I.C., Lueptow, R.M., Ottino, J.M., Sturman, R. (2016). A Parametric Study of Mixing in a Granular Flow a Biaxial Spherical Tumbler. In: Awrejcewicz, J. (eds) Dynamical Systems: Modelling. DSTA 2015. Springer Proceedings in Mathematics & Statistics, vol 181. Springer, Cham. https://doi.org/10.1007/978-3-319-42402-6_13

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