Modeling Navier-Stokes Solvers

In the Lagrangian frame, droplet trajectories in the spray may be calculated using Thomas 2-D equations of motion for a sphere 5791 or the simplified forms)154 1561 The gas velocity distribution in the spray can be determined by either numerical modeling or direct experimental measurements. Using the uncoupled solution approach, many CFD software packages or Navier-Stokes solvers can be used to calculate the gas velocity distribution for various process parameters and atomizer geometries/configurations. On the other hand, somesimple expressions for the gas velocity distribution can be derived from... 

The core of any CFD model is its Navier-Stokes solver. The numerical solution of these equations is considered by many to be a mature field, because it has been practiced for over 30 years, but the nature of turbulence is still one of the unsolved problems of physics. All current solvers are based on the approximations that have their effects on the applicability of the solver and the accuracy of the results—also in the fire simulations. The aspects of turbulence modeling are discussed in the next section. 

Only a few LES simulations have been reported describing the turbulent flow in single phase stirred tanks (e.g., [20, 77, 18]). The lattice-Boltzmann method is used in the more recent publications since this scheme is considered to be an efficient Navier-Stokes solver. Nevertheless, the computational requirements of these models are still prohibitive, therefore the application of this approach is restricted to academic research. No direct simulations of these vessels have been performed yet. 

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 last section described a variety of physical effects that may need to be taken into account in the modeling of flow and heat transfer in microdevices. In addition to the importance of including the correct physical models, it is also very import to address numerical solution accuracy, as some of the phenomena require extremely accurate or different numerical methods to capture them correcfly. Here we split the discussion into two sections the first deals with Navier-Stokes solvers and the second introduces novel, physics-specific methods. 

Navier-Stokes solver. Nevertheless, the computational requirements of these models are still prohibitive, therefore the application of this approach is restricted to academic research. No direct simulations of these vessels have been performed yet. 

Momentum conservation requires that an equal and opposite force be applied to the fluid. Both discrete and continuous degrees of freedom are subject to Langevin noise in order to balance the frictional and viscous losses, and thereby keep the temperature constant. The algorithm can be applied to any Navier-Stokes solver, not just to LB models. For this reason, we will discuss the coupling within a (continuum) Navier-Stokes framework, with a general equation of state p p). We use the abbreviations for the viscosity tensor (46), and... 

Since the computational model is time independent, a stationary solver can be used. The continuity equation together with the momentum equations is solved separately from the species equations to smoothly achieve converged solution. Returned values from the continuity and Navier-Stokes equations are then automatically called by the chosen solver and used in the convection-diffusion equation. 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

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