Mesoscale model fluid velocity

For the present discussion, we do not need to know the exact form of the mesoscale model for the fluid velocity and characteristic properties (e.g. composition and enthalpy). Instead, we will simply assume that and obey ... 

In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

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)). 

It is important to remind the reader that U is the velocity of the fluid phase seen by the particle, U - U is the slip velocity, dp is the particle diameter, and Vf is the kinematic viscosity of the fluid phase. Note that Eq. (5.33) depends on the particle velocity U and is valid in the zero-Stokes-number limit where U = U so that particles follow the fluid. The correlation in Eq. (5.31) is valid only for RCp < 1 and Sc > 200. For larger particle Reynolds numbers the following correlations can be used Sh = 2 -i- 0.724Rep Sc, which is valid for 100 < RCp < 2000, and Sh = 2 -i- 0.425RCp Sc, which is valid for 2000 < RCp <10. Among the other correlations available, it is important to cite the one proposed by Ranz Marshall (1952) for macroparticles Sh = 2.0 -i- O.bReJ Sc. These expressions assume that the fluid velocity U is known. For micron-sized (or smaller) particles moving in turbulent fluids for which only the ensemble-mean fluid velocity (Uf) is known, it is instead better to employ the mesoscale model derived by Armenante Kirwan (1989) Sh = 2.0 -i- 0.52(Re ) Sc, where Re = is the modi-... 

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