William D. Fullmer, Christine M. Hrenya, "Continuum prediction of scale-dependent, anisotropic fluctuating kinetic energy in gas-solid flows", Chemical Engineering Science, Volume 186, 2018, Pages 84-87, ISSN 0009-2509, https://doi.org/10.1016/j.ces.2018.04.035.
Abstract: This note highlights the subtle difference between two measures of fluctuating kinetic energy of the particulate phase in gas-solids flow. One measure arises from kinetic theory, here identified as granular temperature (T), and a second measure resulting from a spatial average, here denoted κ. Somewhat surprisingly, continuum models derived from a kinetic theory considering a single, scalar granular temperature, which is further assumed to be isotropic, is still able to predict anisotropic spatially averaged fluctuating kinetic energies. Furthermore, the scale dependence of the spatially averaged can also be extracted. Comparisons of the continuum predictions to recent direct numerical data are striking, particularly for larger averaging volumes.
Keywords: Two-fluid model; Granular temperature; MFiX; GTSH