Analytical solutions to the granular temperature profiles and effective particle-phase viscosities for single-bubble motion in a fluidized bed

We have considered asymptotic solutions at large bubble Reynolds number to the energy transport equation describing the spatial variation of the “granular temperature” in the flow field around a single bubble in a gas fluidized bed. The granular temperature represents the mean peculiar kinetic energy of the particle phase, and is necessary in a “complete” description of the fluid-particle transport phenomena. In the present case of single-bubble motion, the energy source is from the particle-phase viscous dissipation term. At large bubble Reynolds numbers, viscous effects are confined to a thin boundary-layer region, as in the case of a gas bubble in a liquid. It is demonstrated that the Prandtl number for this problem is O(1), or that the momentum and “thermal” particle-phase boundary layers are of the same order of magnitude. The thermal boundary-layer equation for the granular temperature is subsequently derived and solved analytically. It is assumed that outside the thermal boundary layer, where viscous effects are negligible, the granular temperature is zero. From the mathematical solutions to the energy transport equation associated with single-bubble motion, including viscous dissipation terms and utilizing relationships from the kinetic theory of dense gases, we have given analytical expressions for the effective particle-phase shear viscosity and the effective particle-phase hydrostatic pressure, the two properties which characterize the kinetic component of the particle-phase pressure tensor. The theoretical results show remarkably good agreement with reported experimental measurements of the above properties.