Momentum exchange model

Returning to the momentum equations, the third term on the right-hand side of Eqs. (145) and (146) contains the gas-solid momentum-exchange models Likewise, the fourth term on the right-hand side of Eqs. (146) contains the solid-solid momentum-exchange model fsm . Note that because solid-solid interactions conserve momentum, the latter must be defined such that... 

Determination of accurate models for fm and fsm is nontrivial (Drew and Passman, 1999), and no consensus exists on the exact forms needed to describe particular flows. Nevertheless, it is generally acknowledged that the momentum-exchange model must include drag terms with forms similar to... 

E. Other Correlation Models 1. Momentum-Exchange Model... 

A similar equation to that of Eq. (43) was proposed by Bankoff (B6) on the basis of a bubble-flow model for vertical flow. His derivations are discussed in the following section (Section V, B). Finally, it should be mentioned that the momentum exchange model of Levy (L4), and the slip-ratio model of Lottes and Flinn (L7) are more readily applied for the determination of void fractions than for pressure drops. In general, these methods seem to give rather poorer accuracy than those already discussed. 

A spray is a turbulent, two-phase, particle-laden jet with droplet collision, coalescence, evaporation (solidification), and dispersion, as well as heat, mass and momentum exchanges between droplets and gas. In spray modeling, the flow of gas phase is simulated typically by solving a series of conservation equations coupled with the equations for spray process. The governing equations for the gas phase include the equations of mass, momentum and energy... 

A good estimate of air entrainment flow in a spray can be obtained by examining in detail the momentum exchange between the spray and the surrounding air. We present a similar model to the one developed by Heske-stad et al. (1976). In deriving this simple model we assumed the following ... 

Such a model is meaningful only for hard sphere molecules. For real force fields, the molecules are always in a state of collision, i.e., interaction which causes momentum exchange. 

In order to clarify interphase momentum exchange terms, the case of two-phase flow is considered below. Let phase 1 be a continuous phase and phase 2 a dispersed phase. The interphase exchange force exerted in the i direction on the dispersed phase (phase 2) by the continuous phase can be modeled as... 

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... 

When developing models for polydisperse multiphase flows, it is often useful to resort to conditioning on particle size. For example, in gas-solid flows the momentum-exchange terms between the gas phase and a solid particle will depend on the particle size. Thus, the conditional particle velocity given that the particle has internal-coordinate vector will... 

The Brownian force is the well-known force that becomes important in the case of very small particles suspended in a continuous phase. The Brownian force can be defined as the instantaneous momentum exchange due to collisions between the molecules of the continuous phase with a suspended particle. When the particle is small enough to perceive the molecular nature (and motion) of the continuous phase (i.e. when the particle Knudsen number is large enough), it exhibits a random motion, which was observed as early as 2000 years ago by the Roman Lucretius. The Brownian force is typically described as a stochastic process (Gardiner, 2004), and it can be modeled as a Wiener process " ... 

The interfacial momentum exchange terms in the momentum conservation equations for each phase consist of drag and virtual mass force terms. The drag force for gas and liquid is modeled, respectively, as... 

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