Sphere Definition

These businesses within the business, described in Section 6.6, divide the supply chain into market-product-operations combinations. So the charter should list each of the markets, each of the products, and each of the operations that will be incorporated into the supply chain effort. A larger company will have many options to add or delete parts of their business. In the interest of having a manageable effort — that is, a project that is neither too narrow to have an impact nor too big to complete — is likely to speed implementation in focused areas. 

Most companies will identify several spheres. Each of these could become a separate project effort. As spheres are defined, if resources are limited, priorities should establish the sequence for moving ahead. 

---

Alternatives for sphere definition are endless. Extracting ideas for spheres from a complicated functional thicket is daunting. Cross-functional teams, with members who have a big-picture perspective, are often the best way to tackle the task. Table 9.1 provides examples of spheres. Examples 1 through 4 are product-producing spheres. Example 5 is an enable sphere, serving the needs of product-producing spheres with a central sourcing function, like Sphere A. 

Product/customer grouping differing customer needs by group producing candidates for sphere definition. 

QFD ties customer requirements that lead to competitive advantage with the activities and processes that make up the supply chain. Before starting, a QFD user should be aware of the differences that can exist in customer preferences. This could lead to sphere definition early, or at least some candidate spheres. When survey results are known, focused supply chains can cater to the customer requirements addressed by each sphere. QFD leads to a design for the questionnaires and organizes the resulting data to speed decision making. 

Sphere definition Operations supporting flows by customer and product Top down 9... 

For particles of regular geometric shape, values of and ip can be calculated readily from the equivalent sphere definition and by Eq. (6.20), as listed in Table 6.2 for a number of geometric solids. However, the catalyst most commonly used in ammonia synthesis is a crushed and classified material which consists of a mixture of irregular particles having linear dimensions defined by the sieves used in the classification. The surface area of individual catalyst particles is very irregular and complex, and does not permit direct measurement or calculation. Consequently, a nominal sphericity factor of 0.65, as reported in the literature for... 

Solvent-excluded surfaces correlate with the molecular or Connolly surfaces (there is some confusion in the literature). The definition simply proceeds from another point of view. In this c ase, one assumes to be inside a molecaile and examines how the molecule secs the surrounding solvent molecules. The surface where the probe sphere does not intersect the molecular volume is determined. Thus, the SES embodies the solvent-excluded volume, which is the sum of the van der Waals volume and the interstitial (re-entrant) volume (Figures 2-119. 2-120). 

Figure 10.3-54 illustrates how the reaction center is derived from the disconnection of the strategic bond and which additional bond spheres are considered in the definition of the reaction substructure search queiy. 

In the polymer literature each of the five quantities listed above is encountered frequently. Complicating things still further is the fact that a variety of concentration units are used in actual practice. In addition, lUPAC terminology is different from the common names listed above. By way of summary, Table 9.1 lists the common and lUPAC names for these quantities and their definitions. Note that when

[Pg.593]

Heywood [Heywood, Symposium on Paiticle Size Analysis, lust. Chem. Engrs. (1 7), Suppl. 25, 14] recognized that the word shape refers to two distinc t charac teiistics of a particle—form and proportion. The first defines the degree to which the particle approaches a definite form such as cube, tetr edron, or sphere, and the second by the relative proportions of the particle which distinguish one cuboid, tetrahedron, or spheroid from another in the same class. He replaced historical quahtative definitions of shape by numerical shape coefficients. 

Although there are no definitive data on media shape and grinding, ball-milling data indicate that spheres are the most effective shape. [Norris, Trans. Jn.st. Min. Metall, 63(567), 197-209 (1954)]. 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

The prescription proposed in the original Meister-Kroll-Groot [138,139] theory for hard spheres requires the determination of the local density and the averaged density as two independent variational variables by minimizing the grand potential with respect to these variables. The modification introduced by Rickayzen et al. [143,144] arises from another definition of the average density... 

Colloidal suspensions are, per definition, mixtures of mesoscopic particles and atomic liquids. What happens if there are several different species of particles mixed in the solvent One can invent several different sorts of mixtures small and large particles, differently charged ones, short and long rods, spheres and rods, and many more. Let us look into the literature. One important question when dealing with systems with several components is whether the species can be mixed or whether there exists a miscibility gap where the components macroscopically phase-separate. 

The SSW s (screened spherical waves) and their accompanying hard core spheres were defined in [3,4] and we assume the reader is familiar with their definition. 

Now, equation 9.27 must of course be modified in the presence of collisions. Since, by definition, the rate of change of f x, v, t) (Px(Pv is equal to the increase in hard-spheres in d xd v as a result of collisions, we have, in general, that... 

It is hardly possible in chemistry to introduce a contrast between elements and simple bodies, as the definition of element since Lavoisier is based on the simple body. It seems to me to be even less apt simply to equate the terms element/atom and simple body/molecule, respectively for apart from the fact that there are simple bodies whose molecules are single atoms, molecules and atoms belong indubitably to one and the same group of scientific concepts, while the essential difference between element and simple body in the Men-deleeffian sense of the words, lies in their belonging to quite different spheres in epistemology ([33], p 57). 

As outlined in Section III.A, knowledge of the molecular wavefunction implies knowledge of the electron distribution. By setting a threshold value for this function, the molecular boundaries can be established, and the path is open to a definition of molecular shape. A quicker, but quite effective, approach to this entity is taken by assuming that each atom in a molecule contributes an electron sphere, and that the overall shape of a molecular object results from interpenetration of these spheres. The necessary radii can be obtained by working backwards from the results of MO calculations21, or from some kind of empirical fitting22. 

It is useful to redefine the characteristic linear dimension L of the spherical particle as its volume per unit surface area. This is, in effect, consistent with the definition of L adopted for the platelet where L is half its thickness. Then, for the sphere ... 

According to Vitanov et a/.,61,151 C,- varies in the order Ag(100) < Ag(lll), i.e., in the reverse order with respect to that of Valette and Hamelin.24 63 67 150 383-390 The order of electrolytically grown planes clashes with the results of quantum-chemical calculations,436 439 as well as with the results of the jellium/hard sphere model for the metal/electro-lyte interface.428 429 435 A comparison of C, values for quasi-perfect Ag planes with the data of real Ag planes shows that for quasi-perfect Ag planes, the values of Cf 0 are remarkably higher than those for real Ag planes. A definite difference between real and quasi-perfect Ag electrodes may be the higher number of defects expected for a real Ag crystal. 15 32 i25 401407 10-416-422 since the defects seem to be the sites of stronger adsorption, one would expect that quasi-perfect surfaces would have a smaller surface activity toward H20 molecules and so lower Cf"0 values. The influence of the surface defects on H20 adsorption at Ag from a gas phase has been demonstrated by Klaua and Madey.445... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

As surface area and pore structure are properties of key importance for any catalyst or support material, we will first describe how these properties can be measured. First, it is useful to draw a clear borderline between roughness and porosity. If most features on a surface are deeper than they are wide, then we call the surface porous (Fig. 5.16). Although it is convenient to think about pores in terms of hollow cylinders, one should realize that pores may have all kinds of shapes. The pore system of zeolites consists of microporous channels and cages, whereas the pores of a silica gel support are formed by the interstices between spheres. Alumina and carbon black, on the other hand, have platelet structures, resulting in slit-shaped pores. All support materials may contain micro, meso and macropores (see text box for definitions). 

Since the parameter Rg is known to be a basic SEC size parameter for random-coil type molecules (Equation 4), and since it is also a well-defined statistical average size parameter, applicable to solutes of any shape including the sphere and rodlike molecules, Rg has been chosen in this work to serve as a common, reduced, solute-size parameter for describing the theory of SEC separation. By definition ... 

---