Heat transfer in granular materials: effects of nonlinear heat conduction and viscous dissipation

In this paper, we study the heat transfer in a one-dimensional fully developed flow of granular materials down a heated inclined plane. For the heat flux vector, we use a recently derived constitutive equation that reflects the dependence of the heat flux vector on the temperature gradient, the density gradient, and the velocity gradient in an appropriate frame invariant formulation. We use two different boundary conditions at the inclined surface: a constant temperature boundary condition and an adiabatic condition. A parametric study is performed to examine the effects of the material dimensionless parameters. The derived governing equations are coupled nonlinear second-order ordinary differential equations, which are solved numerically, and the results are shown for the temperature, volume fraction, and velocity profiles. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Kolumban Hutter,et al.  A priori bounds and structural stability for double-diffusive convection incorporating the Soret effect , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[2]  J. Duran,et al.  Sands, Powders, and Grains , 2000 .

[3]  Mehrdad Massoudi,et al.  On the flow of granular materials with variable material properties , 2001 .

[4]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[5]  Liqiu Wang.  Vector-field theory of heat flux in convective heat transfer , 2001 .

[6]  T. N. Narasimhan,et al.  Fourier’s heat conduction equation: History, influence, and connections , 1999, Journal of Earth System Science.

[7]  Jovo P. Jarić,et al.  FOURIER'S LAW OF HEAT CONDUCTION IN A NONLINEAR FLUID , 1999 .

[8]  H. Jaeger,et al.  Granular solids, liquids, and gases , 1996 .

[9]  Hans J. Herrmann,et al.  Modeling granular media on the computer , 1998 .

[10]  Brian Straughan.  A note on convection with nonlinear heat flux , 2007 .

[11]  S. Savage,et al.  The Mechanics of Rapid Granular Flows , 1984 .

[12]  M. Massoudi,et al.  A method for measuring the material moduli of granular materials: Flow in an orthogonal rheometer , 1990 .

[13]  Mehrdad Massoudi,et al.  A continuum model for granular materials: Considering dilatancy and the Mohr-Coulomb criterion , 2001 .

[14]  R. Soto,et al.  Departure from Fourier's Law for Fluidized Granular Media , 1999 .

[15]  Bernard D. Coleman,et al.  Thermodynamics and departures from Fourier's law of heat conduction , 1963 .

[16]  George Gabriel Stokes,et al.  Mathematical and Physical Papers vol.1: On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids , 2009 .

[17]  T. Fisher,et al.  Models for metal hydride particle shape, packing, and heat transfer , 2012, 1205.1073.

[18]  H. Jakobsen.  Flows of Granular Materials , 2014 .

[19]  J. J. Fourier,et al.  The Analytical Theory of Heat , 1878, Nature.

[20]  Ingo Müller,et al.  On the entropy inequality , 1967 .

[21]  R. Rivlin,et al.  Stress-Deformation Relations for Isotropic Materials , 1955 .

[22]  I. F. Collins,et al.  Elastic/plastic models for soils and sands , 2005 .

[23]  R. L. Braun,et al.  Stress calculations for assemblies of inelastic speres in uniform shear , 1986 .

[24]  J. Oldroyd.  On the formulation of rheological equations of state , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[25]  V. A. Cimmelli,et al.  On the Thermodynamics of Korteweg Fluids with Heat Conduction and Viscosity , 2011 .

[26]  Mehrdad Massoudi,et al.  On the heat flux vector for flowing granular materials—part II: derivation and special cases , 2006 .

[27]  T. Y. Na,et al.  Computational methods in engineering boundary value problems , 1979 .

[28]  I. Liu.  On Fourier's law of heat conduction , 1990 .

[29]  Mehrdad Massoudi,et al.  Fully developed flow of granular materials down a heated inclined plane , 1994 .

[30]  Dimitrios Kolymbas,et al.  An outline of hypoplasticity , 1991, Archive of Applied Mechanics.

[31]  J. E. Dunn,et al.  On the Thermomechanics of Interstitial Working , 1985 .

[32]  P. G. de Gennes,et al.  Reflections on the mechanics of granular matter , 1998 .

[33]  S. Elaskar,et al.  Constitutive relations for compressible granular materials using non-Newtonian fluid mechanics , 1998 .

[34]  J. Maxwell,et al.  On Stresses in Rarified Gases Arising from Inequalities of Temperature , 2022 .

[35]  Mehrdad Massoudi,et al.  On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background , 2006 .

[36]  Mehrdad Massoudi,et al.  Conservation laws and constitutive relations for density-gradient-dependent viscous fluids , 2005 .

[37]  M. Massoudi.  Mathematical Modelling of Granular Materials , 2010 .

[38]  Joseph J. McCarthy,et al.  Mixing and heat conduction in rotating tumblers , 2010 .