Disperse-phase momentum transport

Note that, in general, the transport and source terms in Eq. (4.85) will not be in closed form. Erom the definition in Eq. (4.84), we can identify three limiting cases. 

If the particle material density (pp) is constant, then ppUp = pporpllv. 

These cases would suggest that the best averaged velocity to use depends on which internal coordinates are held constant. However, because conservation of momentum is of fundamental importance, it is always best to use the mass-average velocity in Eq. (4.85). For this reason, Eq. (4.85) should always be included in the set of moment-transport equations used to model the disperse phase. The disperse-phase momentum source terms appearing on the right-hand side of Eq. (4.85) are defined as follows. The first term. 

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  

For multicomponent particles, all components in a particle have the same velocity and thus Up can be found from Eq. (4.85). The only difference from the single-component case is that the terms for mass transfer and nucleation may be different for each component (e.g. just one component may change phases), in which case the right-hand side of Eq. (4.85) would include sums over the contributions for each component. 

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Adding together Eqs. (4.71) and (4.72) yields a realizability constraint for the velocity fields, namely Vx Uyoi = 0, where Uyoi = apUp + afUf. As mentioned earlier, this constraint must be incorporated into the conditional source terms in the disperse-phase momentum transport equation. Note that, in general, Uyoi t Umix unless the fluid and the particles have the same material density. 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

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. 

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... 

Thus, Eq. (1.28) reduces to the transport equation for the disperse-phase mean momentum ... 

Other Considerations. The presence of a third phase can affect liquid-liquid dispersion and coalescence. Fine solids have little effect on drop dispersion but often affect coalescence. Gas bubbles affect dispersion by reducing the effective continnons phase viscosity and lead to a loss in momentum transport, hence dispersion capability. Tiny gas bubbles reduce probability of coalescence by interfering with film drainage rates between colliding drops. This subject is complex and is best stndied experimentally at different scales. 

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). 

The governing momentum equations of the continuous phase are solved numerically after discretization in space and time. In almost aU multiphase flows, the coupling between the continuous and dispersed phase dominates the transport equations of the continuous phase. When trying to solve these equations, it is therefore useful to locally Hnearize the interphase momentum force density as follows ... 

In the bubble column the velocity profile of recirculating liquid is shown in Fig. 27, where the momentum of the mixed gas and liquid phases diffuses radially, controlled by the turbulent kinematic viscosity Pf When I/l = 0 (essentially no liquid feed), there is still an intense recirculation flow inside the column. If a tracer solution is introduced at a given cross section of the column, the solution diffuses radially with the radial diffusion coefficient Er and axially with the axial diffusion coefficient E. At the same time the tracer solution is transported axially Iby the recirculating liquid flow. Thus, the tracer material disperses axially by virtue of both the axial diffusivity and the combined effect of radial diffusion and the radial velocity profile. 

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