Modeling volume-averaged formulation

Following the steps for formulation of a CFD model introduced earlier, we begin by determining the set of state variables needed to describe the flow. Because the density is constant and we are only interested in the mixing properties of the flow, we can replace the chemical species and temperature by a single inert scalar field (x, t), known as the mixture fraction (Fox, 2003). If we take = 0 everywhere in the reactor at time t — 0 and set / = 1 in the first inlet stream, then the value of (x, t) tells us what fraction of the fluid located at point x at time t originated at the first inlet stream. If we denote the inlet volumetric flow rates by qi and q2, respectively, for the two inlets, at steady state the volume-average mixture fraction in the reactor will be... 

The model can be classified as a homogeneous mixture model which is formulated directly on the averaging scales (i.e., the control volume coincides with the averaging volume). It is further assumed that the relative velocity in the interface grid cells is zero (v = 0), and that all the scales of turbulence are resolved (i.e., in this respect this homogeneous model formulation resembles a direct numerical simulation). 

To save space the governing time averaged equations for each of the primary variables are not listed here as their mathematical form coincides with the volume averaged model formulation given in sect 3.4.1. Nevertheless, it is important to note that the physical interpretations of the mean quantities and the temporal covariance terms differ from their spatial counterparts. Furthermore, the conventional constitutive equations for the unknown terms are discussed in chap 5, and the same modeling closures are normally adopted for any model formulation even though their physical interpretation differ. 

As for the collision density in the macroscopic model formulation, the average collision frequency of fluid particles is usually described assuming that the mechanisms of collision is analogous to collisions between molecules as in the kinetic theory of gases. The volume average coalescence frequency, ac d d, Y), can thus be defined as the product of an effective swept volume rate hc d d, Y) and the coalescence probability, pc d d, Y) (e.g., [16, 92, 114, 39, 46, 118]) ... 

In what follows the magnetoviscosity phenomenon is analyzed by formulating the local ferrohydrodynamic model, the upscaled volume-average model in porous media with the closure problem, and solution and discussion of a simplified zero-order steady-state isothermal incompressible axisymmetric model for non-Darcy-Forchheimer flow of a Newtonian ferrofluid in a porous medium of the... 

At a higher level, the flow field is modeled at a scale much larger than the size of the particles, and the fluid velocity and pressure are obtained by solving the volume-averaged Navier-Stokes equations. The particle particle interactions (particle wall as well) are formulated with the so-called discrete particle models (DPMs), which are based on the schemes that are traditionally used in molecular dynamics simulations, with the addition of dissipation of mechanical energy. 

In the continuum (Euler-Euler)-type formulation, the gas, liquid, and solid phases are assumed to be continuum and the volume-averaged mass and momentum equations (see Table 6.10) are solved for each phase separately to predict the pressure, phase holdup, and phase velocity distributions. As a result of time and volume averaging, additional terms appear in the momentum conservation equations. These additional terms need closure models and such unclosed terms are highlighted in Table 6.10. 

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