Particle mass disperse-phase momentum

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Likewise, the disperse-phase momentum is related to the particle mass by... 

Consider again a system wherein all particles have the same volume and mass, and the disperse-phase momentum density is gp = ppOp. The transport equation for the disperse-phase momentum density for this case is ... 

Based on the above-mentioned assumptions, the mass, momentum and energy balance equations for the gas and the dispersed phases in two-dimensional, two-phase flow were developed [14], In order to solve the mass, momentum and energy balance equations, several complimentary equations, definitions and empirical correlations were required. These were presented by [14], In order to obtain the water vapor distribution the gas phase the water vapor diffusion equation was added. During the second drying period, the model assumed that the particle consists of a dry crust surrounding a wet core. Hence, the particle is characterized by two temperatures i.e., the particle s crust and core temperatures. Furthermore, it was assumed that the heat transfer from the particle s cmst to the gas phase is equal to that transferred from the wet core to the gas phase i.e., heat and mass cannot be accumulated in the particle cmst, since all the heat and the mass is transferred by diffusion through the cmst from the wet core to the surrounding gas. Based on this assumption, additional heat balance equation was written. 

Scmi the net source due to dispersed phase particles (Eq. (4.11)). Fd, Fi and Fvm are drag, lift and virtual mass forces (Section 4.2.1). It must be noted that Eq. (7.9) assumes that the volume-averaged momentum transfer (from the dispersed phase)... 

This formulation is particularly convenient when Euler-Lagrange simulations are used to approximate the disperse multiphase flow in terms of a fimte sample of particles. As discussed in Sections 5.2 and 5.3, although some of the mesoscale variables are intensive (i.e. independent of the particle mass), it is usually best to start with a conserved extensive variable (e.g. particle mass or particle momentum) when deriving the single-particle models. For example, in Chapter 4 we found that must have at least one component, corresponding to the fluid mass seen by a particle, in order to describe cases in which the disperse-phase volume fraction is not constant. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

Below, we will discuss two important examples of zeroth-order point processes as seen from the perspective of the particle-phase NDF. However, some zeroth-order point processes such as the formation of the disperse phase from the fluid phase are accompanied by a change of state in the fluid phase (i.e. the total mass and momentum of the two phases are conserved). Thus, seen in the perspective of the particle and fluid phase, the overall process requires a decrease in the mass of the fluid phase equal to the mass of the formed particle, which is represented by the term 5m in Eq. (4.68) on page 119. In other words, the term 5m in Eq. (4.77) representing the rate of addition of mass to the particle phase from the fluid phase must follow from the source term for the zeroth-order point process for formation of the disperse phase. Using the properties of delta functions, we can formally write the source term in Eq. (5.1) for a zeroth-order point process as... 

As explained throughout the book, disperse multiphase systems are characterized by multiple phases, with one phase continuous and the others dispersed (i.e. in the form of distinct particles, droplets, or bubbles). The term polydisperse is used in this context to specify that the relevant properties characterizing the elements of the disperse phases, such as mass, momentum, or energy, change from element to element, generating what are commonly called distributions. Typical distributions, which are often used as characteristic signatures of multiphase systems, are, for example, a crystal-size distribution (CSD), a particle-size distribution (PSD), and a particle-velocity distribution. 

The physical picture used in the CFD approach considers the primary fluid phase as a continuum. The time-averaged Navier-Stokes equations are then solved for this continuum. The secondary phase (dispersed phase) is treated by following it (drops/ bubbles or solid particles) through the flow field determined in the previous step. Mass, momentum, and energy exchange can occur between the two phases involved. 

The second great limitation of CFD is dispersed, multiphase flows. Multiphase flows are common in industry, and consequently their simulation is of great interest. Like turbulent flows, multiphase flows (which may also be turbulent in one or more phases) are solutions to the equations of motion, and direct numerical simulation has been applied to them (Miller and Bellan, 2000). However, practical multiphase flow problems require a modeling approach. The models, however, tend to ignore or at best simplify many of the important details of the flow, such as droplet or particle shape and their impact on interphase mass, energy, and momentum transport, the impact of deformation rate on droplet breakup and coalescence, and the formation of macroscopic structures within the dispersed phase (Sundaresan et al., 1998). 

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... 

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