Discrete phases numerical techniques

We will not attempt here to give a detailed explanation of the numerical aspects (fundamentals of discretization, error estimates, and error control) of CFD since a number of excellent texts are available in the literature that deal in depth with this matter (Fletcher, 1988a,b Hirsch, 1988, 1990). First some general aspects of the numerical techniques used for solving fluid flow problems are discussed and, subsequently, a distinction is made between single-phase flows and... 

This method has been devised as an effective numerical technique of computational fluid dynamics. The basic variables are the time-dependent probability distributions 7 (x, t) of a velocity class a on a lattice site x. This probability distribution is then updated in discrete time steps using a deterministic local rule. A careful choice of the lattice and the set of velocity vectors minimizes the effects of lattice anisotropy. This scheme has recently been applied to study the formation of lamellar phases in amphiphilic systems [92, 93]. 

Equations (10-98) through (10 100) constitute 7+1 governing equations for 7+1 variables Xj (/= ,...-/) andp/. They can be solved numerically, for example, by a discretization technique where a set of coupled differential equations is replaced by a set of NxM finite difference equations on a grid consisting of M mesh points. The necessary boundary conditions can be established by requiring the reaction equilibrium (i.e.. Equation (10 99)) and the sum of the mole fractions equal to one (i.e.. Equation (10 100) at the membrane interface and equality of the pressure at the membrane interface and the pressure in the adjacent gas phase. Additional boundary conditions can be obtained from mass balances coupling the molar fluxes from the gas phase to the membrane interface with those at the interface. Details can be found elsewhere [Sloot et al.. 1990]. 

There are two main approaches for the numerical simulation of the gas-solid flow 1) Eulerian framework for the gas phase and Lagrangian framework for the dispersed phase (E-L) and 2) Eulerian framework for all phases (E-E). In the E-L approach, trajectories of dispersed phase particles are calculated by solving Newton s second law of motion for each dispersed particle, and the motion of the continuous phase (gas phase) is modeled using an Eulerian framework with the coupling of the particle-gas interaction force. This approach is also referred to as the distinct element method or discrete particle method when applied to a granular system. The fluid forces acting upon particles would include the drag force, lift force, virtual mass force, and Basset history force.Moreover, particle-wall and particle-particle collision models (such as hard sphere model, soft sphere model, or Monte Carlo techniques) are commonly employed for this approach. In the E-E approach, the particle cloud is treated as a continuum. Local mean... 

Direct numerical simulations (DNS) At the most detailed level of description, the gas flow field is modeled at scales smaller than the size of the solid particles. The interaction of the gas phase with the solid phase is incorporated by imposing no-slip boundary conditions at the surface of the solid particles. This model thus allows one to measure the effective momentum exchange between the two phases, which is a key input in aU the higher scale models. Many different types of DNS models exist, such as the lattice Boltzmann model (Ladd, 1994 Ladd and Verberg, 2001) or immersed boundary techniques (Peskin (2002), UMmann (2005)). The goal of these simulations is to construct drag laws for dense gas—solid systems, which are used in the discrete particle type models. 

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