Two mechanisms for the particle-particle collisions—with and without frictional sliding collision are considered in this paper to take into account the effect of the particle-particle collision on the motion of solid particles in two-phase turbulent pipe and channel flows. Based on these mechanisms the correlations of the various velocity components of colliding particles are obtained analytically by using an averaging procedure, which takes into account three collision coordinates, two angles and one geometrical distance between the centers of colliding particles. As a result the various stress tensor components are obtained and then introduced in the mass, linear momentum and angular momentum equations of the dispersed phase. They are considered as additional force factors together with the influence of the particle-turbulence interactions, the viscous drag force, the two types of lift force (Magnus and Saffman) and the gravitational force. The approach applies to the particle-particle collisions based both on the average velocity difference between colliding particles and to the turbulent velocity fluctuation of colliding particles. To close the governing equations of the dispersed phase, the pseudoviscosity (and pseudodiffusity) coefficients with collision origin are determined as well, using a Bousinesque-type eddy-viscosity approach. [To close the governing equations of the dispersed phase, the stress tensor components as well as the pseudoviscosity (and pseudodiffusity) coefficients defined with the help of using a Bousinesque-type eddy-viscosity approach are determined here.] In order to obtain these coefficients the time of the inter-particle collision is calculated from the information on the collision process. The model covers the motion and collisions of both polydisperse and monodisperse particulate systems. The model is validated by comparison with the experimental data of Tsuji et al. (1984) for a vertical pipe.

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