Length scales disparate

Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models. 

In this article we propose a homogenized form of the modified convection-diffusion equations to describe contaminant transport in expansive clays char-acaterized by three disparate length scales and two levels of porosity. The microscale consists of macromolecular structures saturated by an electrolye... 

Consider an example from nucleation and growth of thin films. At least three length scales can be identified, namely, (a) the fluid phase where the continuum approximation is often valid (that may not be the case in micro- and nanodevices), (b) the intermediate scale of the fluid/film interface where a discrete, particle model may be needed, and (c) the atomistic/QM scale of relevance to surface processes. Surface processes may include adsorption, desorption, surface reaction, and surface diffusion. Aside from the disparity of length scales, the time scales of various processes differ dramatically, ranging from picosecond chemistry to seconds or hours for slow growth processes (Raimondeau and Vlachos, 2002a, b). 

We saw in the last section that disparate length scales and time scales exist for turbulent flows. Various time scales also are associated with the... 

Fig. 3. Panorama of plasma etching using silicon etching with chlorine as an example. This figure also shows the disparate length scales involved from the reactor, to the sheath, to the microfeature, to the atomic scale. Cl radicals and CIJ ions are generated in the plasma by electron impact of gas molecules (a). Ions accelerate in the sheath and bombard the wafer along the vertical direction (b), thereby inducing anisotropic etching of microscopic features to yield SiCU, a volatile product (c). Ion bombardment creates a modified layer at the surface where Cl is mixed within the Si lattice (d).

It is instructive to first consider the forward problem from the reactor to the feature, ignoring the coupling back into the reactor. Because of the disparity of the length scales involved, it helps to break down the problem into several pieces. An approach proposed by Economou and Alkire [89] is shown in Fig. 38. The near wafer space is separated into two regions. Region I and Region II. Region I contains the... 

Phase separation in macroscale equipment either uses density differences between the two fluids to drive the separation, as in settlers, or these differences play an important role in the technical layout of the separator, e.g. in distillation towers. In macroscopic two-phase flow, length scales vary between the size of the apparatus and the interface-dictated Laplace length scale /o/(g AQ)) of entrained bubbles or drops. The former is often on the order of meters, whereas the latter is on the order of millimeters. This significant disparity in length scales makes it virtually impossible to separate macroscopic two-phase flows in a single step. 

The disparate time and length scales that control heterogeneous catalytic processes make it essentially impossible to arrive at a single method to treat the complex structural behavior, reactivity and dynamics. Instead, a hierarchy of methods have been developed which can can be used to model different time and length scales. Molecular modeling of catalysis covers a broad spectrum of different methods but can be roughly categorized into either quantum-mechanical methods which track the electronic structure or molecular simulations which track the atomic structme (see the Appendix). 

FIGURE 3.5 Gases, excluded volume effects, and probability. The upper and lower diagrams illustrate N particles in volume V on disparate length scales. Two molecules such as indicated by the arrows cannot occupy the same imaginary cells at the same time. The probability of molecules occupying neighboring cells scales in the manner of the second pressure term of the van der Waals equation. 

Going from planar to porous electrode introduces another length scale, the electrode thickness. In the case of a PEM fuel cell catalyst layer, the thickness lies in the range of IcL — 5-10 pm. The objective of porous electrode theory is to describe distributions of electrostatic potentials, concentrations of reactant and product species, and rates of electrochemical reactions at this scale. An accurate description of a potential distribution that accounts explicitly for the potential drop at the metal/electrolyte interface would require spatial resolution in the order of 1 A. This resolution is hardly feasible (and in most cases not necessary) in electrode modeling because of the huge disparity of length scales. The simplified description of a porous electrode as an effective medium with two continuous potential distributions for the metal and electrolyte phases appears to be a consistent and practicable option for modeling these stmctures. 

Multiple length scales and time scales involved in electrochemical and mechanical phenomena seriously complicate the analysis of battery cells and materials. Special care has to be taken to devise a framework to incorporate the disparate length and time scales in the modeUng of Li-ion batteries. 

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