Fluid-phase momentum transport

For the fluid-phase momentum density, we can postulate a momentum balance of the form + -folUf Uflf) = A. Sf - ,p AfpIp - [Gpip - .p[Slp + .fAf, (4.92) 

The term [Apfjf is the mass-average acceleration of the fluid seen by the particles due in part to momentum transfer between phases, and Af f is due to forces in the fluid phase. The term Gf]f can be evaluated using Eq. (4.80)  

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. 

A consistent model for the fluid acceleration seen by the particles 

The constraint in Eq. (4.101) states that momentum must be conserved during mass transfer between phases, and it can be combined with the constraint in Eq. (4.103). Minimal consistent models for the fluid acceleration terms in Eq. (4.39) can then be written as 

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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). 

In an Interface between pure fluids relcixation processes proceed so fast that, in the absence of temperature and pressure gradients. Interfaces may be considered as being homogeneous and likewise the interfacial tension. We exclude the extremely d5mamic situations considered in sec. 1.14a. Then the shear components of the interfacial tension tensor will also vanish and the normal or symmetric components are, except for the sigh, identical to the Interfacial tension, which is the same everywhere and, hence, no stresses can be built up in the interface. Any motion of, and in, such interfaces is entirely determined by the momentum transport of the adjacent bulk phases. For an illustration see sec. I.6.4d, example 3. 

These relationships are valid for isolated bubbles moving under laminar flow conditions. In the case of turbulent flow, the effect of turbulent eddies impinging on the bubble surface is to increase the drag forces. This is typically accounted for by introducing an effective fluid viscosity (rather than the molecular viscosity of the continuous phase, yUf) defined as pi.eff = Pi + C pts, where ef is the turbulence-dissipation rate in the fluid phase and Cl is a constant that is usually taken equal to 0.02. This effective viscosity, which is used for the calculation of the bubble/particle Reynolds number (Bakker van den Akker, 1994), accounts for the turbulent reduction of slip due to the increased momentum transport around the bubble, which is in turn related to the ratio of bubble size and turbulence length scale. However, the reader is reminded that the mesoscale model does not include macroscale turbulence and, hence, using an effective viscosity that is based on the macroscale turbulence is not appropriate. 

The reservoir simulator developed for this study simulates two-dimensional areal flow of a single-phase, slightly compressible fluid in a closed square reservoir. It is based on the equations developed in previous studies (11,12). The reservoir rock is assumed to be deformable, heterogeneous and anisotropic. The momentum transport equation is... 

The flow in the gas channels and in the porous gas diffusion electrodes is described by the equations for the conservation of momentum and conservation of mass in the gas phase. The solution of these equations results in the velocity and pressure fields in the cell. The Navier-Stokes equations are mostly used for the gas channels while Darcy s law may be used for the gas flow in the GDL, the microporous layer (MPL), and the catalyst layer [147]. Darcy s law describes the flow where the pressure gradient is the major driving force and where it is mostly influenced by the frictional resistance within the pores [145]. Alternatively, the Brinkman equations can be used to compute the fluid velocity and pressure field in porous media. It extends the Darcy law to describe the momentum transport by viscous shear, similar to the Navier-Stokes equations. The velocity and pressure fields are continuous across the interface of the channels and the porous domains. In the presence of a liquid phase in the pore electrolyte, two-phase flow models may be used to account for the interaction between the gas phase and the liquid phase in the pores. When calculating the fluid flow through the inlet and outlet feeders of a large fuel cell stack, the Reynolds-averaged Navier-Stokes (RANS), k-o), or k-e turbulence model equations should be used due to the presence of turbulence. 

Substituting the single particle general property by the variables 1, Cj(t), jcf(t), hi(t) and respectively, and after introducing (2.65), (4.296), (4.300) and (4.301) into (4.299), the continuum transport equations for the solid phase fluid mass, momentum, granular temperature, molecular enthalpy and the species mass can be obtained with some further manipulations. 

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. 

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