Asymptotic theory of two-phase gas-solid flow through a vertical tube at moderate pressure gradient

Based on the equations, constitutive relations and boundary conditions of the kinetic theory of colliding particles in a gas-solid suspension, the approximate theory of the steady, developed vertical flow of a gas-particulate mixture is developed for the case of moderate gas pressure gradient in a vertical tube. The basic equations and boundary conditions show a singular behaviour of the solution of the problem at the wall. The method of matched asymptotic expansions is applied to develop a boundary layer-type theory for the flow parameters of the particulate phase. The basic equations in the bulk flow are reduced to a system of two ordinary integrodifferential equations for the particle-phase concentration and mean kinetic energy of particle velocity fluctuations (particle-phase pseudotemperature). The distributions of the particle concentration and velocity are found in both the bulk and the boundary layer. The solutions shows the bifurcation of flow parameters, and an explicit criterion is derived to identify a range of the given macroscopic parameters corresponding to upward or downward particulate flow. The integrated parameters (total fluxes of the gas and particle phase) are calculated.