Momentum density fluid phase

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. 

Remarkably, no additional models are required to obtain consistent convection of the fluid-phase momentum density. Eor consistency, we can require that [Uf p = Uf so that the final term on the right-hand side of Eq. (4.99) is null however, due to conservation of momentum the final term will be cancelled out due to a contribution from (Apf2>i, as will be shown next. 

Microscale fluid turbulence is, by deflnition, present only when the continuous fluid phase is present. The coefficients Bpv describe the interaction of the particle phase with the continuous phase. In contrast, Bpvf models rapid fluctuations in the fluid velocity seen by the particle that are not included in the mesoscale drag term Ap. In the mesoscale particle momentum balance, the term that generates Bpv will depend on the fluid-phase mass density and, hence, will be null when the fluid material density (pf) is null. In any case, Bpv models momentum transfer to/from the particle phase in fluid-particle systems for which the total momentum is conserved (see discussion leading to Eq. (5.17)). 

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). 

In the preceding chapters we first considered the primary forces acting on a fluidized particle in a bed in equilibrium, and then the elastic forces between particles that come into play under non-equilibrium conditions. These two effects provide closure for the particle bed model, formulated in terms of the particle-and fluid-phase conservation equations for mass and momentum. Up to now, applications have focused on the stability of the state of homogeneous particle suspension, in particular for gas-fluidized systems for which the condition that particle density is much greater than fluid density enables the particle-phase equations to be decoupled and treated independently. The analysis has involved solely the linearized forms of these equations, and has led to a stability criterion that broadly characterizes fluidized systems according to three manifestations of the fluidized state always stable - the usual case for liquids always unstable - the usual case for gases and transitional behaviour - involving a switch, at a critical fluid flux, from the stable to the unstable condition. This characterization has... 

The analysis presented in this chapter has been based on the assumption that particle density is substantially greater than fluid density. This was shown in Chapters 11 and 12 to be valid for all cases of gas fluidization, even under very high-pressure conditions only liquid-fluidized beds of low-density particles exhibited differences of any significance in the single- and the two-phase treatments. This justification, however, relates to the linear analysis of small perturbations. It says little concerning the effect of jumps in fluid pressure across the very considerable discontinuities uncovered in the work described in this chapter. The procedure adopted in Chapter 11, of eliminating fluid-pressure terms by combining the particle- and fluid-phase momentum equations, cannot be utilized here as it involves non-linear manipulations, which are not permitted in the analysis of discontinuous functions. 

A momentum balance for the flow of a two-phase fluid through a horizontal pipe and an energy balance may be written in an expanded form of that applicable to single-phase fluid flow. These equations for two-phase flow cannot be used in practice since the individual phase velocities and local densities are not known. Some simplification is possible if it... 

In problems in which the dispersed phase momentum equations can be approximated and reduced to an algebraic relation the mixture model is simpler to solve than the corresponding multi-fluid model, however this model reduction requires several approximate constitutive assumptions so important characteristics of the flow can be lost. Nevertheless the simplicity of this form of the mixture model makes it very useful in many engineering applications. This approximate mixture model formulation is generally expected to provide reasonable predictions for dilute and uniform multiphase flows which are not influenced by any wall effects. In these cases the dispersed phase elements do not significantly affect the momentum and density of the mixture. Such a situation may occur when the dispersed phase elements are very small. There are several concepts available for the purpose of relating the dispersed phase velocity to the mixture velocity, and thereby reducing the dispersed... 

We have considered the situation of only one phase for any mixture composition this means that there is no surface tension and the fluid behavior is completely characterized by the turbulent flow described by the mass and momentum balance equations. To solve these equations, one needs to model the diffusional mixing of the species present in the system and to identify local values of the thermodynamic and transport properties, as considered in Section 3.2. Here we just point out that once the methods for predicting local values of fluid density and viscosity have been worked out, one should be able to integrate Eqs. (10) and (11). 

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. 

Omran et al. have proposed a 3D, single phase steady-state model of a liquid feed DMFC [181]. Their model is implemented into the commercial computational fluid dynamics (CFD) software package FLUENT . The continuity, momentum, and species conservation equations are coupled with mathematical descriptions of the electrochemical kinetics in the anode and cathode channel and MEA. For electrochemical kinetics, the Tafel equation is used at both the anode and cathode sides. Results are validated against DMFC experimental data with reasonable agreement and used to explore the effects of cell temperature, channel depth, and channel width on polarization curve, power density and crossover rate. The results show that the power density peak and crossover increase as the operational temperature increases. It is also shown that the increasing of the channel width improves the cell performance at a methanol concentration below 1 M. 

It is useful to regard the momentum basis functions as states onto which one can project the memory function or the phase space density correlation function. By referring to the notations established in Table 1 and (125), we see that Ip I is the number density state, ip2 the longitudinal current (2-component) state, the energy state, iJ/4 and ij/s the two transverse current (jc- and y-component) states, etc. The elements given in Table 1, /f(a /3), a, /3 4, are those which determine the thermodynamics as well as the basic hydrodynamic structure of the fluid system. Notice that in the limit of ka- O these... 

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