Turbulent flow modeling Hinze-Tchen model

A variety of statistical models are available for predictions of multiphase turbulent flows [85]. A large number of the application oriented investigations are based on the Eulerian description utilizing turbulence closures for both the dispersed and the carrier phases. The closure schemes for the carrier phase are mostly limited to Boussinesq type approximations in conjunction with modified forms of the conventional k-e model [87]. The models for the dispersed phase are typically via the Hinze-Tchen algebraic relation [88] which relates the eddy viscosity of the dispersed phase to that of the carrier phase. While the simplicity of this model has promoted its use, its nonuniversality has been widely recognized [88]. 

Here we present the derivation of the transport coefficients due to diffusion of discrete particles in a homogeneous turbulent flow. Although the Hinze-Tchen model was developed in a more general form for general particle-fluid multiphase flows, we introduce this model only for the cases of gas-solid flows. Some assumptions for this model are the following ... 

For the Reynolds number range typical of drag reduction (Re 105), / is about 0.02 from the Moody chart (see Fig. 11.7). The typical turbulent intensity of gas in a pipe flow is about 5 percent. Using the Hinze-Tchen model (see 5.3.4.1), the ratio of the velocity fluctuation of the particles to that of the gas may be given by Eq. (5.196) as... 

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