Effective-scale Transport Models

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions, 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations 

Both MTPM and DGM share the Darcy s constitutive equation governing the permeation molar flux intensity of the z th component Nf, 

The DGM by Mason and Malinauskas (1983) is the frequently used alternative to effective Fick s diffusion for the calculation of the multicomponent diffusion and convection in the porous media. A number of special features of multicomponent diffusion and convection in the pore space have been outlined by Krishna (1993). 

The microstructure of the multiphase media is often the product of phase transitions, e.g. (i) capillary condensation in the porous media, (ii) phase separation in polymer/polymer and polymer/solvent systems, (iii) nucleation and growth of bubbles in the porous media, (iv) solidification of the melt with a temporal three-phase microstructure (solid, melt, gas), and (v) dissolution, crystallization or precipitation. The subject of our interest is not only the topology of the resulting microstructured media, but also the dynamics of its evolution involving the formation and/or growth of new phases. 

---

The combustion reaction rate is controlled both by the availability of fuel and oxygen kinetic effects (temperature). In full-scale fire modeling, the resolvable length and time scales are usually much larger than those associated with the scales of the chemical combustion reaction, and it is common to assume that the reactions are infinitely fast. The local reaction rate depends on the rate at which oxygen and fuel are transported toward the surface of stoichiometric mixture fraction, shown in Figure 20.2 as a point where both oxygen and fuel mass fractions go to zero. For almost 20 years, the EBU or eddy dissipation models were the standard models used by the combustion CFD community. With the EBU, in its simplest form, the local rate of fuel consumption is calculated as [3] ... 

The effective-scale models have been most often used in the description of transport and reaction processes within the porous structure of catalysts. Such models are based on the introduction of an effective diffusion coefficient De, that is used in the analogy to the Fick s law for the description of diffusion... 

A quantitative description of heterogeneous catalytic reactors for design, scaling-up, control or optimization purposes requires several parameters. Some of them, including the effective diffusivity and some parameters for the transport models and also the intrinsic chemical rate, should be determined in special experiments. 

Lenz C-J, Muller F, Schliinzen KH (2000) The sensitivity of mesoscale chemistry transport model results to boundary values. Environ Monit Assess 65 287-298 L6pez SD, Liipkes C, Schliinzen KH (2005) The effects of different k-e-closures on the results of a micro-scale model for the flow in the obstacle layer. Meteorol Z 14 839-848 Muller F, Schliinzen KH, Schatzmann M (2000) Test of numerical solvers for chemical reaction mechanisms in 3D air quality models. Environ Model Softw 15 639-646 Schliinzen KH (1990) Numerical studies on the inland penetration of sea breeze fronts at a coastline with tidally flooded mudflats. Beitr Phys Atmos 63 243-256 Schliinzen KH, Katzfey JJ (2003) Relevance of subgrid-scale land-use effects for mesoscale models. Tellus 55A 232-246... 

To incorporate coupled effects into regional scale flow and transport, artificial thermal and dense dispersion terms can be introduced into the coupled transport model. The thermal term should be isotropic and the dense term should coincide with gravity acceleration vector. 

---