Chapman–Enskog closure approximation in the kinetic theory of dilute turbulent gas-particulate suspensions

The Chapman–Enskog approach is applied to a generalised Fokker–Planck equation for the ensemble-averaged phase-space number density of particles to find a closure approximation for the third-order fluctuating velocity correlations in the particle phase of a turbulent dilute gas-particulate suspension. The resulting closed set of continuum equations is shown to be free of empirical parameters, provided the particles are sufficiently large and the turbulence in the both phases is locally isotropic. Special attention is paid to the case where the particle phase is near equilibrium state. In this case the transport equations for diagonal components of the particle Reynolds stresses reduce to a conservation equation for the turbulent kinetic energy. The effective energy transfer coefficient is calculated and the boundary conditions at the rigid wall are formulated. The resulting system of continuum equations and boundary conditions is analysed for fully developed flow between the vertical plane walls of a dilute but densely loaded gas-particulate suspension. Such a flow models a number of practical applications, e.g. the flow in the riser section of fast (circulating) fluidised bed.