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Model Development

The present model has been obtained by considering a mixture of N solid phases (N is the number of solid phases); each phase is composed of smooth inelastic hard spheres, and it was also assumed the collision of particles is the dominant interaction force between two particles. The assumption of hard spheres suggests that the collisions are almost instantaneous, so that binary collisions may safely be assumed. Each particulate phase i contains particles of mass mi and diameter di that collide with each other in the phase i. The collisions/interaction between particle i and other particles of different phases occurs at the interface between phase i and the other particulate phases. Each particle in the phase i moving with instantaneous velocity ci is subject to an external force Fiext. At any time t, the probable number of particles per unit of volume, dr, with velocity varying between ci and ci + d ci, is the product of the single velocity distribution function f] (c, r, t) and the variation of the velocity d ci

Hence, the mean value of any property of phase i, ^i( ci) is defined as

Thus, the equation of change for the particle property of phase i may be expressed as


У2 {yci) |p is defined as the difference between the postcollisional and the


precollisional properties of particle i due to all possible collisions with all the particles in the mixture. The collision of particle i with the other particles in the same phase results in the constitutive relation for each phase, and the interaction of phase i with other phases results in interfacial forces between particulates phases.


The average of (iyci) |p is defined as


f2 = /2(?i, ri; cp, rp) is the complete pair distribution function defined as the probability of finding, at time t, two particles i and p, such that they are centered on ri and rp and have velocities within the range ci, ci + dci and cp, cp + dcp

Following Jenkins and Savage (1983), the assumption of chaos along with the consideration of the correlation function allows us to write the pair distribution function as the product of the single velocity distributions,/ and fp, weighted by the spatial pair distribution function at contact gip(?i, ep),


cip = ci — cp is the relative instantaneous velocity and dip = kip is the unit vector connecting the centers of the two particles, located at ri and rp, respectively, and directed from i to p. In the remaining text, we consider kip = k.

N l '

The collisional rate of production per unit of volume, < igi — щi > |p, was

p=1 V

evaluated by Jenkins and Mancini (1989) as a sum of a symmetric (Ycip) and antisymmetric (xcip) terms


Here Xdp and ycip are the collisional fluxes and sources, respectively. Substituting Eqs. (2.74) and (2.75) into the equation of change, Eq. (2.71), the continuity, momentum, and fluctuating energy equations were obtained for pt equal to mi, mici, and |miC2, respectively.

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