Nonuniform scaling

The coordinates, x- and y-axes, are provided with nonuniform scales, denoted by u and v, as follows ... 

We will now show that the disttibution Eq. (14.33) will be a straight line in the coordinate system x-y, whose axes are provided with the nonuniform scales of Eqs. (14.34)-(14.35). To show this we first write Eq. (14,33) in the integrated form... 

The above definition of the external potentials coupled to densities of the nonuniformly scaled electronic charges of subsystems in - (A, A g) again implies for the two extreme imiform scaling points [see Fig. 2 and Eq.(44)] ... 

Examples of nonuniform heating-control problems above 10(X) F (538 C) are (1) nonuniform scale formation with carbon steels, (2) questionable completion of the combustion reaction (pic contact the load surface), (3) sticky scale with resultant rolled-in scale, (4) spotty decarburization of high carbon steels, (5) some stainless steels may not tolerate contact with the reducing atmosphere within the flames, and (6) using impingement heating for steel pieces of heavy cross section could cause formation of reflective scale with resultant reduction of heat transfer. 

The VCD spectra of the two conformers of (2S,3S)-dideuteriobutyro-lactone 20 were calculated using the MFP method and several basis sets. The ab initio SCF/6-31G force field yielded good qualitative agreement between the summed spectra and the experimentally measured one. The appearance of the VCD spectrum and the qualitative agreement with experiment were not improved by nonuniform scaling of the ab initio force field. Simpler theoretical models of calculating VCD spectra did not yield satisfactory results even at the qualitative level. ... 

Detention efficiency. Conversion from the ideal basin sized by detention-time procedures to an actual clarifier requires the inclusion of an efficiency factor to account for the effects of turbulence and nonuniform flow. Efficiencies vaiy greatly, being dependent not only on the relative dimensions of the clarifier and the means of feeding but also on the characteristics of the particles. The cui ve shown in Fig. 18-83 can be used to scale up laboratoiy data in sizing circular clarifiers. The static detention time determined from a test to produce a specific effluent sohds concentration is divided by the efficiency (expressed as a fraction) to determine the nominal detention time, which represents the volume of the clarifier above the settled pulp interface divided by the overflow rate. Different diameter-depth combinations are considered by using the corresponding efficiency factor. In most cases, area may be determined by factors other than the bulksettling rate, such as practical tank-depth limitations. 

Deposit attack occurs when there is nonuniform deposition of a film on a metal surface. The most common form appears as unequal scale deposits in an aqueous environment. Unequal film provides for concentration cells, which degrade the metal by galvanic means. 

Synthetic chemists desire well defined reaction conditions. Process chemists demand them. Nonuniform heating and difficulties with mixing and temperature measurement are technical constraints that initially limited the scale of microwave chemistry with dry media and have not yet been overcome. Poor reproducibility also has been reported, probably resulting from differences in performance and operation of individual domestic microwave ovens [13-15]. Consequently, most, if not all, of the disclosed applications of dry media are laboratory-scale preparations. However, as discussed in other chapters, this does not prevent their being interesting and useful. 

Fig. 28. Wall temperature maps for 1-hole particles (a) without wall conduction (b) with wall conduction, on the same temperature scale as (a) (c) with wall conduction, on a temperature scale chosen to show nonuniform temperature features.

A further problem of the size press operation is that of films splitting. At the exit of the nip, the film sometimes splits into two layers with one part following the paper and the other remaining on the roll surface. The problem manifests itself as small scale nonuniformities on the paper surface caused by threads of the size solution being drawn out and deposited unevenly on the surface to give a kind of orange peel effect. 

Cox chart chem A straight-line graph of the logarithm of vapor pressure against a special nonuniform temperature scale vapor pressure-temperature lines for many substances intersect at a common point on the Cox chart. kaks, chart cp See chemically pure. 

As noted in Sect. 10.1, heterogeneities play a dominant role in the migration of contaminants in the subsurface. Nonuniform, preferential patterns of flow and transport are ubiquitous. It is important to recognize that, at the field scale, contaminant movement generally is very difficult to anticipate. In natural soils and aquifer materials, macropores, soil cracks and aggregates, fissures, solution channels, root paths, and wormholes, as well as variable mineral composition (e.g., clay aggregates... 

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