The kinetic equation for gas-particle flow

As a second example, we consider the kinetic equation (KE) for monodisperse, isothermal solid particles suspended in a constant-density gas phase. For clarity, we assume that the particle material density is significantly larger than that of the gas so that only the fluid drag and buoyancy terms are needed to account for momentum exchange between the two phases (Maxey Riley, 1983). In this example, the particles are large enough to have finite inertia and thus they evolve with a velocity that can be quite different than that of the gas phase. 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are 

In fhese expressions, gx is the x-component of the gravity force, pp is the particle density, and pg is the gas-phase pressure. Tp and Tc are characteristic time scales for drag and collisions, respectively, t/g is the velocity of the gas phase, and eq is the equilibrium 

The disperse-phase kinetic equation is coupled to the continuity and momentum equations for the continuous phase, which are given, respectively, by 

The phase-density ratio is defined by f i = Pp/pg and, since cpi 1, the buoyancy term is negligible. The two new dimensionless numbers generated in this process are the phase-velocity ratio 03 = t/p/t/, and the disperse-phase Knudsen number KUp = uItJL. In addition, the dimensionless form ofEqs. (1.10), 

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