A time-averaged model for gas-solids flow in a one-dimensional vertical channel

In this study, we are interested in deriving time-smoothed governing and constitutive equations for gas-solids flow in moderately dense systems where particle-particle collision is the main energy dissipation mechanism. Results obtained from dynamic simulations of a gas-solids flow in a 1D channel are used to show that it is possible to obtain expressions for the time-averaged constitutive relations based on Taylor series expansion. We demonstrate, by comparing with time-averaged transient results, that the 1st term (or laminar) in the series expressions of most non-linear constitutive relations can yield inaccurate quantitative and qualitative results. This means that steady-state models derived by simply removing the partial time derivative from the governing equations are not suitable for gas-solids flows. This study shows that it was necessary to include many terms of the Taylor series expression of non-linear constitutive relations (such as the granular energy dissipation term) due to large-scale oscillations that were computed for all flow variables at all locations in the 1D domain. In some cases, the Taylor series expansion diverged and the Euler transformation was used to improve the convergence of these series. In this moderately dense flow system, turbulence in the gas-phase was found to be just a reaction to turbulence in the solids phase that resulted from the large-scale motion of solids clusters. This resulted in a negative turbulent gas viscosity computed due to the fact that gas (in the horizontal direction) flows only to occupy regions vacated by clusters of solids. The steady-state results obtained using the time-smoothed gas-solids flow model compared well with the time-averaged results obtained using the transient model for all flow variables.