Dispersion operator

Table 7-4 shows flow patterns and applications of some commercially available impellers. Generally, the axial flow pattern is most suitable for flow sensitive operation such as blending, heat transfer, and solids suspension, while the radial flow pattern is ideal for dispersion operations that require higher shear levels than are provided by axial flow impellers. Myers et al. [5] have described a selection of impellers with applications. Further details on selection are provided by Uhl and Gray [6], Gates et al. [7], Hicks et al. [8] and Dickey [9]. 

This is formally exact and can be practically implemented in the spectral domain without further approximations, but sometimes a finite number of series expansion terms is used to fit the linear chromatic dispersion of a medium or of a waveguide. What we understand under NEE in the following assumes an exact treatment of the dispersion operator. 

Fig. 13.1. Supercontinuum generated in a femtosecond pulse propagating in air. Full curve was obtained from the full UPPE simulation while the other curves correspond to NEE equation simulated with two and three terms included in the dispersion operator...

It is observed, even in the partial solution to the problem, that realistic models of the droplet coalescence and breakage processes as discussed in Section V,D,2 have yet to be employed. A parallel development has occurred. The work is currently at the point where the realistic model of the droplet dynamics can be applied to the pertinent problems of extent of reaction and solute depletion in dispersions. The success of this effort would permit the researcher and designer to predict dispersed-phase reactor performance from fundamental properties of the dispersion, operation conditions of the vessel, and knowledge of the intrinsic kinetics. 

In suspending solids, the size and the surface area of the solid particles exposed to the liquid are fixed, as is the total volume of suspended solids. In gas-liquid or liquid-liquid dispersion operations, by contrast, the size of the bubbles or drops and the total interfacial area between the dispersed and continuous phases vary with conditions and degree of agitation. New area must constantly be created against the force of the interfacial tension. Drops and bubbles are continually coalescing and being redispersed. In most gas-dispersion operations, bubbles rise through the liquid pool and escape from the surface and must be replaced by new ones. 

A new chapter on membrane separations has been added, and the order of the chapters on multicomponent distillation, extraction, drying, and crystallization has been made more logical. The discussion of particulate solids has been shortened and two former chapters on properties and handling of solids and of solids mixing have been combined into one. New material has been added on flow measurement, dispersion operations, supercritical extraction, pressure-swing adsorption, crystallization techniques, crossflow filtration, sedimentation, and many other topics. The treatment of dimensional analysis has been condensed and moved from the appendixes to Chapter 1. 

In recent years the in-line static mixer has been adopted for a large number of blending and dispersing operations. Many in-line mixers are now commercially available. The geometrical details of these devices differ greatly but the principles of operation are basically the same in each case. 

For mixing operations which require the continuous production of finely dispersed solids, emulsions, stable foams, etc., the in-line dynamic mixer in one of its several forms could be used. These usually consist of a rotor which spins at high speed inside a casing and the feed materials are pumped continuously to the unit. Inside the casing the fluid is subjected to extremely high shear forces which are required for the dispersing operation, see Figure 7.10. 

Although the scientific principles behind this simple example of practical technology are easily understood, it illustrates the benefits that can be realized by considering the blending process as a dispersion operation that may be followed, if necessary, by an operation to retard the rate at which the ingredients of the blend demix. In special cases, of course, the latter operation may be rendered unnecessary by the selection of blend ingredients that are miscible in the first instance. 

The effect of mixing (time and intensity) on the rheology of the pigment slurry is summarized by Robinson et al. (1997). Breakdown of the dispersion agglomerates requires imposition of shear on the slurry. Consequently, the impeller tip speed is critical for disperser operation and mixer design. As the maximum shear rate [ranging from 100 to 10" s in dispersion operations (Makinen, 1999)] determines the ultimate size of the particles, extending the dispersion time cannot compensate for an inadequate shear level. 

For miniaturized chemical analysis, there are often samples of given size but not fixed permanent flow rate, where two samples A and B are required to be mixed with each other. A ring-shaped micromixer as shown in Figure 4.8 can be used for this situation, where a peristaltic pump drives the flow around a ring. The Taylor dispersion operates along the ring, and after some rotations, the two spots A and B mix completely. The speed of rotation of the fluids is U, and the transverse distance, that is, the height of the channel, is b. The effective diffusivity is... 

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