Geometric arguments

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

Newton made extensive use of analogous geometrical arguments. Late in life, Newton expressed a preference for the geometric style of the Classical Greeks, which he regarded as more rigorous. 

Now Lienard developed a purely geometrical argument that enabled him to bypass, as it were, the analytical theory, which is discussed in Part II of this chapter. His equation is 8... 

First, we will demonstrate from simple geometric arguments that the existence of a strain rate distribution across the tube gives further support to the previously described existence of a master curve (Fig. 42). 

The above geometrical argument may be supplemented by an explicit demonstration that the singularity at the point D is of the focus-focus type (Eq. (7)), when Eq. (51) applies. The procedure is to employ the local approximations... 

The next account starts by summarizing the three-dimensional monodromy of the resonant swing spring. It is then shown how the geometrical arguments of Section IV may be applied to examining the nature of the two-dimensional monodromy within a given polyad. 

Geometrical arguments are used only for the reduced metal to obtain an estimate of cluster size. This is equivalent to the assumption that the unreduced cobalt is interacting with the support or is present as a cobalt aluminate species. Despite the inherent assumptions of the method, the resulting cluster size much more closely approximates the true cluster size in comparison with reported data that erroneously assume that the cobalt is 100% reduced (i.e., excludes the reoxidation step). 

This theorem is a multidimensional generalization of the geometric arguments given previously. By result 1, in searching for a solution, we need only look at vertices. It is thus of interest to know how to characterize vertices in many dimensions algebraically. This information is given by the next result. 

It is readily shown1 bjr a simple geometrical argument l) Ct D. E. Rutherford, Classical Mechanics, (Oliver Uoytl, 1051) J t2. 

By means of geometric arguments, it can be shown that the number of modes n v) per unit volume within a frequency interval d(v) is given by... 

While the situation with respect to simple vinyl polymers is straightforward, the tacticity and geometrical arguments are more complicated for more complex polymers. Here we will only briefly consider this situation. Before we move to an illustration of this let us view two related chloride-containing materials pictured below. We notice that by inserting a methylene between the two chlorine-containing carbons the description of the structure changes from racemic to meso. Thus, there exists difficulty between the historical connection of meso with isotactic and racemic with syndiotactic. 

We have discussed the tanks-in-series model in the sense that the composition in the system was constant over a cross-section. Recently Deans and Lapidus (D12) devised a three-dimensional array of stirred tanks, called a finite-stage model, that was able to take radial as well as axial mixing into account. Because of the symmetry, only a two-dimensional array is needed if the stirred tanks are chosen of different sizes across the radius and are properly weighted. By a geometrical argument. Deans and Lapidus arrived at the following equation for the (i, j) tank ... 

In this section some geometrical factors are discussed, that are observed to influence the adaption of structure types. By analogy and based on similar geometrical arguments prediction of isostructural componnds has been possible and should be further so in most cases. 

A rough estimation of the magnification M can be obtained from a heuristic geometrical argument to yield... 

The relation between shear stress and shear strain can be established based on the relation between normal stress and normal strain. Equations (F.3) and (F.4). Actually, by rotating the coordinate system 45°, it becomes a problem of normal stress and normal strain. Using geometrical arguments, it can be shown that (see, for example, Timishenko and Goodier, 1970) ... 

By simple geometrical arguments and the definition of convolution, we can verify that the convolution of rect(x) with itself yields A(x), which we define as... 

In addition to its simplicity, such a geometric argument can also be used to predict the changes in the structure of the aggregates as variables such as pH, charge, electrolyte concentration, and chain length of the tail are varied. The importance of the relative effects of the head group area and the size of the tails was first emphasized by Tatar (1955), and the details were developed subsequently by Tanford (1980) and others (see Wennerstrom and... 

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