Macroscale model turbulence

The flow of the continuous phase is considered to be initiated by a balance between the interfacial particle-fluid coupling - and wall friction forces, whereas the fluid phase turbulence damps the macroscale dynamics of the bubble swarms smoothing the velocity - and volume fraction fields. Temporal instabilities induced by the fluid inertia terms create non-homogeneities in the force balances. Unfortunately, proper modeling of turbulence is still one of the main open questions in gas-liquid bubbly flows, and this flow property may significantly affect both the stresses and the bubble dispersion [141]. 

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. 

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 eddy turbulence model, or simply eddy model, assumes that the small-scale eddies control surface renewal and, subsequently, mass transfer. This model acknowledges a scale dependence. Macroscale movements, those represented by the Reynolds number. Re, are assumed to have a small impact on surface renewal, where the Reynolds number is defined as... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

The terms of the form (m/m/) are called the Reynolds stresses. The RANS equations do not consist of a closed set of equations (there are more unknowns than equations), so if the RANS equations are to be solved, the Reynolds stress terms must be modeled somehow. Typically, this modeling is based on experimental measurements. The application of models developed for macroscale flows to turbulent microchannel flows is dependent on the Reynolds stresses being similar for both cases. Recent experimental evidence suggests a strong similarity between turbulence statistics measured in turbulent microchannel flows and turbulence statistics measured in turbulent pipe and channel flows. Thus, the evidence suggests that turbulent models and codes developed to study macroscale turbulent pipe and channel flow should be applicable to the study of turbulent microchaimel flows. 

The hydrodynamic model development for a circulating fluidized bed follows the same approach as bubbling and turbulent beds. In the macroscale, the gas-solid flow is characterized by a coexistence of a bottom dense region and an upper dilute region. The flow in the radial direction can be described by a core-annular structure with a dense particle region close to... 

Other turbulent ingredients are the knowledge of Eulerian and Lagrangian macroscales which can be evaluated by relying on turbulence theory and modelling. However, the state of art in these evaluations is not very satisfactory because poorly known constants are involved in them. Consequently efficient tests of the particulate models must preferably rely on scale experimental determinations. 

For turbulent flows, the collision rate depends on the frequency at which eddies bring drops into contact. Since the drops are usually small compared to the macroscale (Ft d q, as in Section 12-2.3.1), isotropic turbulence theory can be used to model the collision frequency, the force with which two drops collide, and the time that they remain in contact before subsequent eddies carry them apart. These factors depend on the drop size and the magnitude of the energy dissipation rate, which depends on the impeller speed and diameter. For instance, Coulaloglou and Tavlarides (1977) show that for equal drop size and Ft d q, the collision frequency, (d, d), is given by... 

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