Conservation and Constitutive Equations for Fluid–Particle Flow Systems

Abstract

Chapter 1 presented general forms of conservation equations; this chapter presents mass and momentum conservation equations and constitutive and boundary conditions for fluid–particle flow systems. The conservation equations and constitutive relations are general and can be applied to all regimes of fluid–particle flow from a very dilute particle volume fraction to the packed bed regime. The background and fundamentals of the kinetic theory approach for derivation of the conservation and constitutive equations are presented for the regimes when particle collision is dominant. In addition, the kinetic theory approach has been extended to multi-type particulate flow, assuming a non-Maxwellian velocity distribution and energy non-equipartition. Each group or type of particle is represented by a phase, with an average velocity and a granular temperature. Then, the multi-type governing equations are successfully applied to describe numerical simulation of a simple shear flow of a mixture of two particle groups with different properties. In dense granular flows, in addition to the kinetic and collisional stresses (described by the kinetic theory), the frictional stresses must be considered, which, in dense flow of particles, have a dominant effect. Thus, the frictional behavior of granular matter is also discussed based on soil mechanics principles. Finally, the generalized forms of governing equations and constitutive relations for all particle phase flow regimes are presented in Tables 2.1 and 2.2.