# 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 *m _{i}* and diameter

*d*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 c

_{i }_{i}is subject to an external force

*F*At any time t, the probable number of particles per unit of volume,

_{iext}.*dr,*with velocity varying between

*c*and

_{i}*c*+ d

_{i}*c*is the product of the single velocity distribution function f

_{i},^{]}

*(c, r,*t) and the variation of the velocity d

*c*

_{i}

Hence, the mean value of any property of phase i, ^_{i}( c_{i}) is defined as

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

N

У2 *{y**_{ci}*) |

_{p}is defined as the difference between the postcollisional and the

p=^{1}

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.

*^{N}*|

The average of (*iy**_{ci}*) |

_{p}is defined as

p=1

f2 = /2(?i, r_{i}; c_{p}, r_{p}) 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 r_{i} and r_{p} and have velocities within the range *c*_{i}*, c**_{i}* +

*dc*

*and c*

_{i}_{p}, c

_{p}+

*dc*

_{p}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 *f***p***,* weighted by the spatial pair distribution function at contact *g _{ip}(?i,* e

_{p}),

where

*c*_{ip} *= c**_{i}* — c

_{p}is the relative instantaneous velocity and

*d*

_{ip}*= k*

*is the unit vector connecting the centers of the two particles, located at r*

_{ip}_{i}and r

_{p}, respectively, and directed from i to p. In the remaining text, we consider

*k*

_{ip}*= k.*

**N l ' **

The collisional rate of production per unit of volume, < *ig*_{i} *— щ*_{i} *>* |_{p}, was

p=1 ^{V}

evaluated by Jenkins and Mancini (1989) as a sum of a symmetric (Y_{cip}) and antisymmetric *(x _{c}i_{p})* terms

where

Here *Xdp* and y_{cip} 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 *p _{t}* equal to

*m*, and |m

_{i}, m_{i}c_{i}_{i}C

^{2}, respectively.