Quantifying the non-equilibrium characteristics of heterogeneous gas–solid flow of smooth, inelastic spheres using a computational fluid dynamics–discrete element method

Abstract

Continuum modelling of dense gas–solid flows strongly depends on the constitutive relations used, including the interphase drag force, particle phase stress and the boundary condition for particle–wall interactions. The lack of scale separation is usually claimed to cause the breakdown of the Navier–Stokes (NS) order continuum theory. In this study, computational fluid dynamics–discrete element method (CFD-DEM) simulations of bubbling, turbulent and fast fluidization of smooth, inelastic spheres were conducted to systematically analyse the valid range of NS theory. An entropy-based criterion $I_{s}$ and Knudsen numbers defined using different characteristic length scales ($Kn_{frac}$, $Kn_{gran}$ and $Kn_{vel}$) were quantified. It was found that (i) except at the centre of bubbles where the solid concentration is quite low, NS theory for discrete particles was valid for bubbling fluidization irrespective of the breakdown criterion. Even at the boundary of the bubbles, values of $I_{s}$ and $Kn$ were small; and (ii) the conclusion depended on the criterion used in turbulent and fast fluidization. If $Kn_{frac}$ was used, NS theory would be generally valid. If $I_{s}$ was chosen, NS theory would be still valid but with lower confidence. However, if $Kn_{vel}$ or $Kn_{gran}$ was selected, NS theory broke down. Because $I_{s}$ includes the non-equilibrium effects caused by the gradient of hydrodynamic fields and particle inelasticity, we may conclude that NS theory was valid for all tested cases. This means that the continuum description of discrete particles is not the main source of the breakdown of NS theory.