Particle-based definition of the NDF

In the particle-based definition of the NDF, we begin at the microscale and write a dynamic equation for the rate of change of the disperse-phase particle properties at the mesoscale. The simplest system, which we consider first, is a collection of interacting particles in a vacuum wherein the particles interact through collisions and short-range forces. Such a system is referred to as a granular system. We then consider a disperse two-phase system, wherein the particles are dispersed in a fluid. 

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This average property of the distribution is defined through the number density itself and it represents the mean particle size with respect to the number of particles in the system. Of course, other definitions of the mean particle size are possible, as will become clear below. If we define the th moment of the length-based NDF as... 

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