Volume-averaged models

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. 

Re3Tiolds decomposition and time averaging were then applied to the instantaneous variables in the volume average model equations. However, it was assumed that none of the densities fluctuate. The terms of fluctuating quantities with order higher than two were considered small compared to those of first and second order and thus neglected. 

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... 

Zero-order Axisymmetric Volume-average Model... 

Resistance function, derived from volume-averaged model [9]... 

Prins-Jansen et al. [39] Volume- averaged model C AC impedance study [40] Comparison with own agglomerate model [41, 42]... 

Subramanian et al. [46] Volume- averaged model C 3 cm half — cell Commented on by Berg and Findlay [47]... 

Four categories of electrode models can be identified from Table 28.3 the spatially lumped model, the thin-film model, the agglomerate model, and the volume-averaged model. Schemes of the basic concepts of these four model categories are depicted in Figure 28.4. Each of the schemes shows an electrode pore, with the gas channels located at the top and the liquid electrolyte, depicted in gray, at the bottom. In some models, electrolyte is also present in the pore. The reaction zones are indicated by a black face (spot, line, or grid structure). Fluxes of mass and ions are indicated by arrows. 

As in the agglomerate model, the governing equations of the volume-averaged model describe the representative concentration, Cj, and the electric potential, here in the form of the overpotential, rj, via spatially distributed partial differential equations (cf, [39]). The component mass balance is identical with Eq. (28.75) ... 

The component mass flux density according to the volume-averaged model can be obtained from the concentration gradient at the boundary towards the gas channel ... 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

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... 

Le Maguer and Yao (1995) presented a physical model of a plant storage tissue based on its cellular structure. The mathematical equivalent of this model was solved using a finite element-based computer method and incorporated shrinkage and different boundary conditions. The concept of volume average was used to express the concentration and absolute pressure in the intracellular volume, which is discontinuous in the tissue, as a... 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

The first term on the right-hand side is the diffusive flux relative to the volume average velocity. The second term represents a contribution due to bulk flow. It should be emphasized here that the separation of the total flux into two contributions is always possible regardless of the actual transport mechanism through the membrane. In other words, Eq. (7) is purely phenomenological and does not require any specific transport model. 

It is extremely difficult to model macroscopic transport of mass, energy, and momentum in porous media commonly encountered in various fields of science and engineering based on microscopic transport models that account for variation of velocity and temperature as well as other quantities of interest past individual solid particles. The basic idea of porous media theory, therefore, is to volume average the quantities of interest and develop field equations based on these average quantities. 

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