Volume-Averaged Formulation

For example, the following conservation equation is a typical volume-averaged equation used in a macro-homogeneous model, which represents charge balance in the electrolyte  

Equation 25.14 can be used to determine the electrical potential in the electrolyte phase. Each term in this equation has been volume averaged, where the last term is a source term describing the volumetric electrochemical reaction. Here, a is the specific interfacial area described as the interfacial area per unit of electrode volume. The effective conductivities and x are used to account for the actual tortuous path length of charge transport and local porosity. Two empirical relations are often used in the porous electrode theory, namely  

The multidimensional volume-averaged formulation in the context of a typical Li-ion cell is briefly reviewed below [33, 35]. 

---

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... 

Today there are between 45 and 50 plastic materials. Physical volume, averaging a 13% per year growth rate for the past ten years, reached an estimated 16 billion pounds in 1968, higher than that of any metal except iron and steel, and approaching the total for non-ferrous metals. The number of formulas, grades, and types of these materials is greatly expanded by the use of plasticizers, fillers, and polymerization alternatives. All of these formulations are presumably different from one another and offer the user a broad material selection to fit his property and cost requirements. 

The volume averaged force formulation then 3delds ... 

A simplified formulation of the volume averaged mechanical energy equation supposedly valid for dispersed flows only can be expressed as ... 

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. 

The time after volume averaging procedure can be applied under a unified set of conditions denoting the sum of the two sub-sets of requirements formulated in sects 3.4.1 and 3.4.2 for the pure volume and time averaging procedures to handle the scale disparity in a proper manner. 

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]) ... 

Local Volume Averaging. The local volume-averaging treatment leading to the coupling between the energy equation for each phase is formulated by Carbonell and Whitaker [81] and is given in Zanotti and Carbonell [82], Levee and Carbonell [83], and Quintard et al. [84]. Their development for the transient heat transfer with a steady flow is reviewed here. Some of the features of their treatment are discussed first. 

But this is not all, the same diffusion coefficient D may be used if we use molar quantities in formulation of Pick law for this binary concentration diffusion, cf. (4.562) below. Specifically, using the molar diffusion flow of constituent 1 defined as the corresponding (specific) diffusion flow given above divided by the molar mass Ml of the first constituent (molar quantities are denoted, in addition, by apostrophe e.g. the molar diffusion flow relatively to volume average velocity as... 

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... 

The system stress over the simulation cube after each distortion increment is obtained as a volume average of all atomic site stress tensors that, in turn, is based on the generalized formulation of Theodorou and Suter (1986) of the classical Born and Huang (1954) operations, which now include both torque and force interactions between atoms, giving the atomic site stress-tensor element for the th atom as... 

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. 

---