Macroscale distributions

Depending on the resolution of the mathematical model, different forms of the species conservation equations may be considered in the porous electrodes. For instance, in the multi-scale modehng of Khaleel et al. [18], a mesoscale lattice Boltzmarm model of the electrodes resolves the species transport in the gas, on the surface of the electrode, and through the bulk solid of the electrode. In this model, Eq. (26.1) is solved in three separate domains with corresponding transport properties and source terms. In contrast, in the macroscale distributed electrochemistry model of Ryan et al. [19], the porous medium of the SOFC electrodes is not explicitly resolved but is included in the model via effective properties. In the effective properties model, the diffusion coefficient of Eq. (26.1) is replaced with an effective diffusion coefficient, which is discussed in Section 26.3.3. 

In practice, granular beds comprising a very large number of catalyst pellets are used. It is well known that the efficiency of a catalytic reactor depends crucially on the liquid phase distribution within the catalyst bed [14]. It is likely that the development of hot spots in a catalyst bed is also related to the character of liquid phase distribution. Therefore, it is very important to map the spatial distribution of the liquid phase in a catalytic reactor for various operation regimes. This eventually should lead to the formulation of the mechanisms responsible for the development of critical phenomena on both a micro- and macroscale. 

Specificity of molecular bioactivity and differentially induced defenses are only two examples of factors that can confound the interpretation of patterns at the macroscale. As our knowledge of marine systems continues to expand, the relative abundance of secondary metabolites in different geographic locations may be better understood. However, the literature supports the idea that local pressures and habitat, genetic composition, mode of response and metabolism of the algae play a significant role in shaping distribution patterns of secondary metabolites (e.g. Wright... 

Experimental data are available for large particles at Re greater than that required for wake shedding. Turbulence increases the rate of transfer at all Reynolds numbers. Early experimental work on cylinders (VI) disclosed an effect of turbulence scale with a particular scale being optimal, i.e., for a given turbulence intensity the Nusselt number achieved a maximum value for a certain ratio of scale to diameter. This led to speculation on the existence of a similar effect for spheres. However, more recent work (Rl, R2) has failed to support the existence of an optimal scale for either cylinders or spheres. A weak scale effect has been found for spheres (R2) amounting to less than a 2% increase in Nusselt number as the ratio of sphere diameter to turbulence macroscale increased from zero to five. There has also been some indication (M15, S21) that the spectral distribution of the turbulence affects the transfer rate, but additional data are required to confirm this. The major variable is the intensity of turbulence. Early experimental work has been reviewed by several authors (G3, G4, K3). 

A unit temperature difference between the upper and the other three sides is assumed to achieve macroscale 2D temperature distribution in addition to the microscale one. The problem is solved using Fluent, with the results shown in Fig. 35. Heat fluxes through the right (equal to the left due to symmetry) and bottom sides are shown in Fig. 36. The fluxes from Fluent are averages along... 

In this model, and also in the next one, the interaction is no longer homogeneously distributed. As a consequence, the segregation is not pure microscale segregation, but a combination of micro- and macroscale segregation. 

In the past, the principles described have been implicitly recognized in several attempts to convert monolithic catalysts into catalytic heat exchangers. While the use of millimeter dimensions and nanoporous ceramic supports meets the primary criteria already mentioned, the parallel channel structure of monoliths is not ideally tailored for heat exchanger applications, and complex header structures are required to uniformly distribute and collect reaction medium and coolant to and from the individual channels (Figure 9). The unsatisfactory interface between the milli- and macroscale has been a major weakness of such concepts. 

FIGURE 15 Schematic illustration of the increase in maximum impeller zone macroscale shear rate and a decrease of average impeller zone macroscale shear rate as tank size is increased, illustrating a wider distribution of shear rates in a large tank than in a small tank. The figure is based on a constant power/volume ratio and geometric similarity between the two tanks. 

When the fluid elements pass through the reactor, the exchange of mass between the fluid elements occurs both on a microscale as well as on a macroscale. The mixing process on a macroscale is characterized by the residence-time distribution of the fluid elements. Usually, only the macromixing is considered to have a... 

Roughness developed in a cleaning or etching solution is a result of uneven dissolution across the crystal surface. Many factors in an etching process may contribute to the uneven distribution of the dissolution rate at both micro- and macroscales. In general, any process that causes temporary or permanent surface inhomogeneity will result in preferential dissolution of some areas relative to other areas. 

At the macroscale, only the average fractional saturation of a phase, or 5nw. is usually known (where, for instance, wetting-phase saturation 5w = Kvetting phase/ Fvoid)- However, the spatial distribution of phases at the microscale is important because of its strong effect on macroscopic behavior. A generic rule of thumb is... 

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

For values of this Peclet number well below 1, as encountered in microreactors, a narrow molecular weight distribution can be achieved, while higher values, like those encountered in macroscale reactors, induce a drastic increase in the polydispersity index (Fig. 6.32) [37,48]. Therefore, microreactors can lead to better control over bulk or semi-dilute polymerization processes. 

The reaction environment in a macroscale mbular reactor is nonuniform due to the distribution of residence times and diffusion-limited transfer of heat and mass in the radial direction. This section describes two approaches to improving reactor performance. 

Fig. 18.58a evolve in response to the square patterned region of Fig. 18.57a, while Fig. 18.586 shows the same region for the uniform coverage case. The square macroscale pattern clearly affects temperature uniformity (+88°C) relative to the uniform distribution case (+33°C). This example demonstrates that broad ranges of relevant length scales can be spanned with an approximate analysis and that micro-macroscale coupling can result in significant modification to system thermal response. 

The microscale model Figure 6 is pseudo onedimensional. Because of random distribution of stacks we assume that the macroscale effective diffusion coefficient is given as D =D, /3 where > 11 is the value directly calculated by (23) using the model of Figure 6. 

Note that supersaturations depend both on the macroscale cloud dynamics represented by the cloud updraft velocities (larger updrafts result in higher supersaturations) and on the microphysics (the details of the aerosol size distribution). Cleaner environments, with lower aerosol concentrations, usually result in higher supersaturations. 

---