Dispersion description

For mass transport problems that do not involve molecular diffusion, such as turbulent-flow problems, or problems with highly complicated passages such as packed beds or porous media, a dispersion model is usually used. Here we assume that the controlling mass transport mechanisms are similar in character to the molecular case and, therefore. 

Note that this is the same equation that appears in Table 6.3, except that we define dispersion coefficients for a species in a given direction. For example, Dax is the dispersion coefficient for component A in the X direction. Actually the dispersion model can be derived rigorously by using the concepts of volume averaging developed by Slattery (1972), Whitaker (1962), and Friedman and Ramirez (1977). 

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As a first approach, all the chain segments can be considered to possess the exact same length. Within this mono-disperse description and considering an open uniform chain of n sites, the magnetic susceptibility per site can be calculated (with the notations of Eq. 7) [29] ... 

In this section we consider electromagnetic dispersion forces between macroscopic objects. There are two approaches to this problem in the first, microscopic model, one assumes pairwise additivity of the dispersion attraction between molecules from Eq. VI-15. This is best for surfaces that are near one another. The macroscopic approach considers the objects as continuous media having a dielectric response to electromagnetic radiation that can be measured through spectroscopic evaluation of the material. In this analysis, the retardation of the electromagnetic response from surfaces that are not in close proximity can be addressed. A more detailed derivation of these expressions is given in references such as the treatise by Russel et al. [3] here we limit ourselves to a brief physical description of the phenomenon. 

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

Dispersion forces caimot be explained classically but a semiclassical description is possible. Consider the electronic charge cloud of an atom to be the time average of the motion of its electrons around the nucleus. 

The McMillan-Mayer theory allows us to develop a fomialism similar to that of a dilute interacting fluid for solute dispersed in the solvent provided that a sensible description of W can be given. At the Ihnit of dilution, when intersolute interactions can be neglected, we know that the chemical potential of a can be written as = W (a s) + IcT In where W(a s) is the potential of mean force for the interaction of a solute... 

Wlrile tire Bms fonnula can be used to locate tire spectral position of tire excitonic state, tliere is no equivalent a priori description of the spectral widtli of tliis state. These bandwidtlis have been attributed to a combination of effects, including inlromogeneous broadening arising from size dispersion, optical dephasing from exciton-surface and exciton-phonon scattering, and fast lifetimes resulting from surface localization 1167, 168, 170, 1711. Due to tire complex nature of tliese line shapes, tliere have been few quantitative calculations of absorjDtion spectra. This situation is in contrast witli tliat of metal nanoparticles, where a more quantitative level of prediction is possible. 

An orbital overlap description of electron delocalization mil dimethylallyl cation H2C=CH—C(CH3)2 is given m Figure 10 2 Figure 10 2a shows the rr bond and the vacant p orbital as independent units Figure 10 2b shows how the units can overlap to give an extended rr orbital that encompasses all three carbons This permits the two rr electrons to be delocalized over three carbons and disperses the positive charge... 

The ohmic drop across the electrolyte and the separator can also be calculated from Ohm s law usiag a modified expression for the resistance. When gas bubbles evolve at the electrodes they get dispersed ia and impart a heterogeneous character to the electrolyte. The resulting conductivity characteristics of the medium are different from those of a pure electrolyte. Although there is no exact description of this system, some approximate treatments are available, notably the treatment of Rousar (9), according to which the resistance of the gas—electrolyte mixture, R, is related to the resistance of the pure electrolyte, R ... 

Although evidence exists for both mechanisms of growth rate dispersion, separate mathematical models were developed for incorporating the two mechanisms into descriptions of crystal populations random growth rate fluctuations (36) and growth rate distributions (33,40). Both mechanisms can be included in a population balance to show the relative effects of the two mechanisms on crystal size distributions from batch and continuous crystallizers (41). 

Method of Moments The first step in the analysis of chromatographic systems is often a characterization of the column response to sm l pulse injections of a solute under trace conditions in the Henry s law limit. For such conditions, the statistical moments of the response peak are used to characterize the chromatographic behavior. Such an approach is generally preferable to other descriptions of peak properties which are specific to Gaussian behavior, since the statisfical moments are directly correlated to eqmlibrium and dispersion parameters. Useful references are Schneider and Smith [AJChP J., 14, 762 (1968)], Suzuki and Smith [Chem. Eng. ScL, 26, 221 (1971)], and Carbonell et al. [Chem. Eng. Sci., 9, 115 (1975) 16, 221 (1978)]. 

Spray Drying Detailed descriptions of spray dispersion dryers, together with apphcation, design, and cost information, are given in Sec. 17. Product quality is determined by a number of properties such as particle form, size, flavor, color, and heat stability. Particle size and size distribution, of course, are of greatest interest from the point of view of size enlargement. 

Perrv, S. G., Bums, D. J., Adams, L. A., Paine, R. J., Dennis, M. G., Mills, M. T., Strimaitis, D. G., Yamartino, R. J., and Insley, E. M., "User s Guide to the Complex Terrain Dispersion Model plus Algorithms for Unstable Conditions (CTDMPLUS)," Vol. I "Model Description and User Instructions," EPA/600/8-89/041, U.S. Environmental Protection Agency, Research Triangle Park, NC, 1989. 

London [11] was the first to describe dispersion forces, which were originally termed London s dispersion forces. Subsequently, London s name has been eschewed and replaced by the simpler term dispersion forces. Dispersion forces ensue from charge fluctuations that occur throughout a molecule that arise from electron/nuclei vibrations. They are random in nature and are basically a statistical effect and, because of this, a little difficult to understand. Some years ago Glasstone [12] proffered a simple description of dispersion forces that is as informative now as it was then. He proposed that,... 

This model was later expanded upon by Lifshitz [33], who cast the problem of dispersive forces in terms of the generation of an electromagnetic wave by an instantaneous dipole in one material being absorbed by a neighboring material. In effect, Lifshitz gave the theory of van der Waals interactions an atomic basis. A detailed description of the Lifshitz model is given by Krupp [34]. 

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