Macroscopic flow model

Coimectivity is a term that describes the arrangement and number of pore coimections. For monosize pores, coimectivity is the average number of pores per junction. The term represents a macroscopic measure of the number of pores at a junction. Connectivity correlates with permeability, but caimot be used alone to predict permeability except in certain limiting cases. Difficulties in conceptual simplifications result from replacing the real porous medium with macroscopic parameters that are averages and that relate to some idealized model of the medium. Tortuosity and connectivity are different features of the pore structure and are useful to interpret macroscopic flow properties, such as permeability, capillary pressure and dispersion. 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

Methane can be oxidatively coupled to ethylene with very high yield using the novel gas recycle electrocatalytic or catalytic reactor separator. The ethylene yield is up to 85% for batch operation and up to 50% for continuous flow operation. These promising results, which stem from the novel reactor design and from the adsorptive properties of the molecular sieve material, can be rationalized in terms of a simple macroscopic kinetic model. Such simplified models may be useful for scale up purposes. For practical applications it would be desirable to reduce the recycle ratio p to lower values (e.g. 5-8). This requires a single-pass C2 yield of the order of 15-20%. The Sr-doped La203... 

Ideal flow models contain inherent assumptions about mixing behavior. In BMF, it is assumed that all fluid elements interact and mix completely at both the macroscopic and microscopic levels. In PF, microscopic interactions occur completely in any plane perpendicular to the direction of flow, but not at all in the axial direction. Fluid elements at different axial positions retain their identities as they progress through the vessel, such that a fluid element at one axial position never interacts with a fluid element at another position. 

On the basis of the considered macroscopic flow pattern, the dominant circulation flows (/ c and Fc/2) subdivide the reactor into three parallel levels, where each level is then divided into Nc/3 equally sized compartments of equal volume Vc = Vr/Nc. Every compartment is modeled as a nonstationary ideal continuous stirred tank reactor, with a main inlet and outlet flow, which connects the given compartment with adjacent compartments on the same level, and secondary exchange flow rates accounting for the turbulent mixing with adjacent compartments laying on the upper and/or lower level (Fig. 7.3). 

The "laminar" macroscopic flow equations contain phenomenological terms which represent averages over the macroscopic dynamics to include the effects of turbulence. Examples of these terms are eddy viscosity and diffusivity coefficients and average chemical heat release terms which appear as sources in the macroscopic flow equations. Besides providing these phenomenological terms, the turbulence model must use the information provided by the large scale flow dynamics self-consistently to determine the energy which drives the turbulence. The model must be able to follow reactive interfaces on the macroscopic scale. 

A. Lagrangian Framework. An ideal subgrid model should be constructed on a Lagrangian hydrodynamics framework moving with the macroscopic flow. This requirement reduces purely numerical diffusion to zero so that realistic turbulence and molecular mixing phenomena will not be masked by non-physical numerical smoothing. This requirement also removes the possibility of masking purely local fluctuations by truncation errors from the numerical representation of macroscopic convective derivatives. 

Despite the fact that there is some progress in modeling the macroscopic flow structure of slurry reactors, a number of microscopic phenomena are very difficult to capture in macroscopic flow simulation models such as the possible accumulation of solid particles near the gas-liquid interface, which significantly affects the mass transfer characteristics of the slurry system (Beenackers and van Swaaij, 1993). 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

It has been shown that the ILG model can adequately simulate well-established macroscopic flow regimes, like Poiseuille s law (Di Pietro et al., 1994). The ILG model was also applied to study evaporation in porous media (Pot, 1994), infiltration in two-dimensional saturated and non-saturated porous media with macropores (Di Pietro, 1996) and water storage in roughed infiltrating and non-infiltrating surfaces (Garcia Sanchez et al., 1996). 

Heterogeneously catalyzed reactions. Macroscopic fluid models are combined with microscopic transport models in the catalyst particles to describe how concentration changes with time and position in a catalytic reactor. Special considerations must be given to the selection of experimental temperature and catalyst particle size to minimize (and hopefully eliminate) internal transport limitations on the catalytic reaction rate. The next requirement is that the flow pattern in the reactor Is accurately represented by the well-mixed or plug-flow assumption. The subsequent discussion applies to gas-phase reactants. 

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