Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-02-11T12:12:27.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Enskog kinetic theory for monodisperse gas–solid flows

Published online by Cambridge University Press:  27 September 2012

V. Garzó ,
S. Tenneti ,
S. Subramaniam  and
C. M. Hrenya
V. Garzó
Affiliation:
Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain
S. Tenneti
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
S. Subramaniam
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
C. M. Hrenya*
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Enskog kinetic theory is used as a starting point to model a suspension of solid particles in a viscous gas. Unlike previous efforts for similar suspensions, the gas-phase contribution to the instantaneous particle acceleration appearing in the Enskog equation is modelled using a Langevin equation, which can be applied to a wide parameter space (e.g. high Reynolds number). Attention here is limited to low Reynolds number flow, however, in order to assess the influence of the gas phase on the constitutive relations, which was assumed to be negligible in a previous analytical treatment. The Chapman–Enskog method is used to derive the constitutive relations needed for the conservation of mass, momentum and granular energy. The results indicate that the Langevin model for instantaneous gas–solid force matches the form of the previous analytical treatment, indicating the promise of this method for regions of the parameter space outside of those attainable by analytical methods (e.g. higher Reynolds number). The results also indicate that the effect of the gas phase on the constitutive relations for the solid-phase shear viscosity and Dufour coefficient is non-negligible, particularly in relatively dilute systems. Moreover, unlike their granular (no gas phase) counterparts, the shear viscosity in gas–solid systems is found to be zero in the dilute limit and the Dufour coefficient is found to be non-zero in the elastic limit.

JFM classification

Granular media: Granular media Multiphase and Particle-laden Flows: Particle/fluid flow Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 712 , 10 December 2012 , pp. 129 - 168
Copyright
©2012 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbas, M., Climent, E. & Simonin, O. 2009 Shear-induced self-diffusion of inertial particles in a viscous fluid. Phys. Rev. E 79 (3), 036313.Google Scholar
Agrawal, K., Loezos, P. N., Syamlal, M. & Sundaresan, S. 2001 The role of meso-scale structures in rapid gas–solid flows. J. Fluid Mech. 445, 151–185.Google Scholar
Anderson, T. B. & Jackson, R. 1967 A fluid mechanical description of fluidized beds. Ind. Engng Chem. Fundam. 6, 527–539.Google Scholar
Balzer, G., Boelle, A. & Simonin, O. 1995 Eulerian gas–solid flow modelling of dense fluidized beds. In Computational Gas–Solid Flows and Reacting Systems: Theory, Methods and Practice. Fluidization VIII (ed. Large, J. F. & Lagurie, C.). Engineering Foundation.Google Scholar
Brey, J. J., Dufty, J. W. & Santos, A. 1997 Dissipative dynamics for hard spheres. J. Stat. Phys. 87, 1051–1066.CrossRefGoogle Scholar
Brey, J. J., Dufty, J. W., Santos, A. & Kim, C. S. 1998 Hydrodynamics for granular flows at low density. Phys. Rev. E 58, 4638–4653.CrossRefGoogle Scholar
Brilliantov, N. & Pöschel, T. 2004 Kinetic Theory of Granular Gases. Oxford University Press.CrossRefGoogle Scholar
Campbell, C. S. 1990 Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92.CrossRefGoogle Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51, 635–636.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory of Nonuniform Gases. Cambridge University Press.Google Scholar
Clement, C. P., Pacheco-Martínez, H. A., Swift, M. R. & King, P. J. 2010 The water-enhanced Brazil nut effect. Europhys. Lett. 91, 54001.Google Scholar
Février, P., Simonin, O. & Squires, K. D. 2005 Partitioning of particle velocities in gas–solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. J. Fluid. Mech. 533, 1–46.CrossRefGoogle Scholar
Gardiner, C. W. 1985 Handbook of Stochastic Methods, second edition. Springer.Google Scholar
Garzó, V. 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.CrossRefGoogle Scholar
Garzó, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895–5911.CrossRefGoogle ScholarPubMed
Garzó, V., Dufty, J. W. & Hrenya, C. M. 2007a Enskog theory for polydisperse granular mixtures. Part 1. Navier–Stokes order transport. Phys. Rev. E 76, 031303.Google Scholar
Garzó, V., Hrenya, C. M. & Dufty, J. W. 2007b Enskog theory for polydisperse granular mixtures. Part 2. Sonine polynomial approximation. Phys. Rev. E 76, 031304.Google Scholar
Garzó, V. & Santos, A. 2003 Kinetic Theory of Gases in Shear Flows: Nonlinear Transport. Kluwer Academic.Google Scholar
Garzó, V., Santos, A. & Montanero, J. M. 2007c Modified Sonine approximation for the Navier–Stokes transport coefficients of a granular gas. Physica A 376, 94–107.CrossRefGoogle Scholar
Garzó, V., Vega Reyes, F. & Montanero, J. M. 2009 Modified Sonine approximation for granular binary mixtures. J. Fluid Mech. 623, 387–411.CrossRefGoogle Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization. Academic.Google Scholar
Gidaspow, D. & Jiradilok, V. 2009 Computational Techniques: The Multiphase CFD Approach to Fluidization and Green Energy Technologies. Nova.Google Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–293.Google Scholar
Hrenya, C. M., Galvin, J. E. & Wildman, R. D. 2008 Evidence of higher-order effects in thermally driven granular flows. J. Fluid Mech. 598, 429–450.Google Scholar
Idler, V., Sánchez, I., Paredes, R., Gutiérrez, G., Reyes, L. I. & Botet, R. 2009 Three-dimensional simulations of a vertically vibrated granular bed including interstitial air. Phys. Rev. E 79, 051301.Google Scholar
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.Google Scholar
Koch, D. L. 1990 Kinetic theory for a monodisperse gas–solid suspension. Phys. Fluids A 2, 1711–1723.CrossRefGoogle Scholar
Koch, D. L. & Hill, R. J. 2001 Inertial effects in suspensions and porous-media flows. Annu. Rev. Fluid Mech. 33, 619–647.Google Scholar
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229–263.Google Scholar
Lumley, J. L. & Newman, G. R. 1977 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82, 161–178.CrossRefGoogle Scholar
Lun, C. K. K. & Savage, S. B. 2003 Kinetic theory for inertia flows of dilute turbulent gas–solids mixtures. In Granular Gas Dynamics (ed. Pöschel, T. & Brilliantov, N.), p. 263. Springer.Google Scholar
Lutsko, J. 2005 Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models. Phys. Rev. E 72, 021306.Google Scholar
Ma, D. & Ahmadi, G. 1988 A kinetic model for rapid granular flow of nearly elastic particles including interstitial fluid effects. Powder Technol. 56, 191–207.Google Scholar
Möbius, M. E., Cheng, X., Eshuis, P., Karczmar, G. S., Nagel, S. R. & Jaeger, H. M. 2005 Effect of air on granular size separation in a vibrated granular bed. Phys. Rev. E 72, 011304.CrossRefGoogle Scholar
Möbius, M. E., Lauderdale, B. E., Nagel, S. R. & Jaeger, H. M. 2001 Size separation of granular particles. Nature (London) 414, 270.CrossRefGoogle ScholarPubMed
Montanero, J. M. & Santos, A. 2000 Computer simulation of uniformly heated granular fluids. Granul. Matt. 2, 53–64.CrossRefGoogle Scholar
Montanero, J. M., Santos, A. & Garzó, V. 2007 First-order Chapman–Enskog velocity distribution function in a granular gas. Physica A 376, 75–93.CrossRefGoogle Scholar
Naylor, M. A., Swift, M. R. & King, P. J. 2003 Air-driven Brazil nut effect. Phys. Rev. E 68, 012301.Google Scholar
van Noije, T. P. C. & Ernst, M. H. 1998 Velocity distributions in homogeneous granular fluids: the free and heated case. Granul. Matt. 1, 57–64.Google Scholar
Pannala, S., Syamlal, M. & O’Brien, T. J. 2011 Computational Gas–Solids Flows and Reacting Systems: Theory, Methods and Practice. doi:10.4018/978-1-61520-651-3.CrossRefGoogle Scholar
Pöschel, T. & Brilliantov, N. 2006 Breakdown of the Sonine expansion for the velocity distribution of granular gases. Europhys. Lett. 74, 424–430.Google Scholar
Rericha, E. C., Bizon, C., Shattuck, M. D. & Swinney, H. L. 2002 Shocks in supersonic sand. Phys. Rev. Lett. 88, 014302.Google Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidisation. Part 1. Trans. Inst. Chem. Engrs Lond. 32, 35.Google Scholar
Sánchez, P., Swift, M. R. & King, P. J. 2004 Stripe formation in granular mixtures due to the differential influence of drag. Phys. Rev. Lett. 93, 184302.Google Scholar
Sangani, A. S., Mo, G., Tsao, H.-K. & Koch, D. L. 1996 Simple shear flows of dense gas–solid suspensions at finite Stokes numbers. J. Fluid Mech. 313, 309–341.Google Scholar
Santos, V., Garzó, V. & Dufty, J. W. 2004 Inherent rheology of a granular fluid in uniform shear flow. Phys. Rev. E 69, 061303.CrossRefGoogle ScholarPubMed
Santos, A. & Montanero, J. M. 2009 The second and third Sonine coefficients of a freely cooling granular gas revisited. Granul. Matt. 11, 157–168.CrossRefGoogle Scholar
Simonin, O., Zaichik, L. I., Alipchenkov, V. M. & Février, P. 2006 Connection between two statistical approaches for the modelling of particle velocity and concentration distributions in turbulent flow: the mesoscopic Eulerian formalism and the two-point probability density function method. Phys. Fluids 18 (12), 125107.CrossRefGoogle Scholar
Sinclair, J. L. & Jackson, R. 1989 Gas-particle flow in a vertical pipe with particle–particle interactions. AIChE J. 35, 1473–1486.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.Google Scholar
Tenneti, S., Fox, R. O. & Subramaniam, S. 2010 a Instantaneous particle acceleration model for gas–solid suspensions at moderate Reynolds numbers. In 7th International Conference on Multiphase Flow, Tampa, Florida. http://ufdc.ufl.edu/UF00102023/00189.Google Scholar
Tenneti, S., Garg, R., Hrenya, C. M., Fox, R. O. & Subramaniam, S. 2010b Direct numerical simulation of gas–solid suspensions at moderate Reynolds number: quantifying the coupling between hydrodynamic forces and particle velocity fluctuations. Powder Technol. 203, 57–69.CrossRefGoogle Scholar
Tsao, H.-K. & Koch, D. L. 1995 Simple shear flows of dilute gas–solid suspensions. J. Fluid Mech. 296, 211–245.Google Scholar
Wen, C. Y. & Yu, Y. H. 1966 Mechanics of fluidization. Chem. Engng Prog. Symp. Ser. 62, 100–111.Google Scholar
Wildman, R., Martin, T. W., Huntley, J. M., Jenkins, J. T., Viswanathan, H., Fen, X. & Parker, D. J. 2008 Experimental investigation and kinetic-theory-based model of a rapid granular shear flow. J. Fluid Mech. 602, 63–79.CrossRefGoogle Scholar
Wylie, J. J., Zhang, Q., Xu, H. Y. & Sun, X. X. 2008 Drag-induced particle segregation with vibrating boundaries. Europhys. Lett. 81, 54001.Google Scholar
Xu, Y. & Subramaniam, S. 2006 A multiscale model for dilute turbulent gas-particle flows based on the equilibration of energy concept. Phys. Fluids 18, 033301.CrossRefGoogle Scholar
Yan, X., Shi, Q., Hou, M., Lu, K. & Chan, C. K. 2003 Effects of air on the segregation of particles in a shaken granular bed. Phys. Rev. Lett. 91, 014302.CrossRefGoogle Scholar
Zaichik, L. I., Simonin, O. & Alipchenkov, V. M. 2009 An Eulerian approach for large eddy simulation of particle transport in turbulent flows. J. Turbul. 10 (4), 1–21.Google Scholar
Zeilstra, M. A., van der Hoef, M. A. & Kuipers, J. A. M. 2008 Simulations of density segregation in vibrated beds. Phys. Rev. E 77, 031309.Google Scholar