Dimensional scaling

Develop device or system designs utilizing the descriptive governing relationships between the dimensionless terms (and subsequently between the variables) that are similar to those of the tested physical model, but are the dimensional scale of the desired device or system. 

S. S. Schiffman, M.L. Reynolds and F.W. Young, Introduction to Multi-dimensional Scaling Theory, Methods and Applications. Academic Press, New York, 1981. 

Modeling relaxation-influenced processes has been the subject of much theoretical work, which provides valuable insight into the physical process of solvent sorption [119], But these models are too complex to be useful in correlating data. However, in cases where the transport exponent is 0.5, it is simple to apply a diffusion analysis to the data. Such an analysis can usually fit such data well with a single parameter and provides dimensional scaling directly, plus the rate constant—the diffusion coefficient—has more intuitive significance than an empirical parameter like k. 

Fabrication issues can arise when thinning the separator material or in trying to ensure high-quality, preferably pinhole-free coverage of dimensionally scaled-down anodes and cathodes. One strategy that can be used to avoid fabricating a separator at all, at least in electrolyzers and fuel cells, takes advantage of decades of work in compact mixed... 

Dimensional scaling theory [109] provides a natural means to examine electron-electron correlation, quantum phase transitions [110], and entanglement. The primary effect of electron correlation in the D 00 limit is to open up the dihedral angles from their Hartree-Fock values [109] of exactly 90°. Angles in the correlated solution are determined by the balance between centrifugal effects, which always favor 90°, and interelectron repulsions, which always favor 180°. Since the electrons are localized at the D 00 limit, one might need to add the first harmonic correction in the 1/D expansion to obtain... 

D. R. Herschbach, J. Avery, and O. Goscinski, Dimensional Scaling in Chemical Physics, Kluwer, Dordrecht, 1993. 

Agrafiotis, D. K., Rassokhin, D. N., and Lobanov, V. S. (2001) Multi-dimensional scaling and visualization of large molecular similarity tables. J. Comput. Chem. 22,... 

Both Mbssbauer spectroscopy and magnetometry are based on the magnetic behaviour of (essentially) iron in a crystal structure, but operate on different dimensional scales. Whereas Mbssbauer spectroscopy yields information about charge and coordination, magnetometric methods are more sensitive to the type of magnetic coupling and to the magnetic domain status of particles. 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

Third, turbulent transport is represented as a succession of simple laminar flows. If the boundary is a solid wall, then one considers that elements of liquid proceed short distances along the wall in laminar motion, after which they dissolve into the bulk and are replaced by other elements, and so on. The path length and initial velocity in the laminar motion are determined by dimensional scaling. For a liquid-fluid interface, a roll cell model is employed for turbulent motion as well as for interfacial turbulence. 

Multi-dimensional scaling with ideal point... 

Each response has its physical sense and its dimension. To join such models, it is first necessary to introduce a non dimensional scale for each response. The scale must be of the same kind for all responses that are generalized. The choice of the scale is not a routine job and it depends on preliminary information we have about the responses and on the precision which is required from the general response. 

After choosing the non dimensional scale for each response, one should define the rules of combining partial responses. A unique rule or algorithm does not exist. 

Assume that a research subject is characterized by n partial responses yu(u=l, 2,..., n) and that each of these responses is measured in N trials. Then the value of the u-responses in the i-th run is yui (i=l, 2,..., N). Each of the given responses yu has its physical interpretation and its dimension. If we introduce the non dimensional scale with only two values 0 and 1, the 0 would correspond to all those values of partial responses that are unsatisfactory by their quality, and the 1 would correspond exactly to those that are satisfactory. The transformed values of partial responses according to the non dimensional scale are marked yui. yui and it is the transformed value of the u response in the i-th trial. After the transformation we obtained non-dimensional partial responses that should now be generalized. Since partial responses take... 

Partial responses transformed into the non dimensional scale are marked du(u=1.2,...,n) and called partial desirability or individual desirability. As shown in Table 2.6 the desirability scale has the range from 0.0 to 1.0. Two characteristic limit values for quality are within this range 0.37 and 0.63. The 0.37 value is approximately l/e=0.36788, where e is the basis of the natural logarithm, and 0.63 is 1-1/e. 

Consider as a length scale parameter, and develop a dimensional scaling argument to conclude that... 

Fig. 2 A dimensionally scaled comparison of a series of PAMAM dendrimers (NH3 core) with a variety of proteins and bioassemblies. (From Ref, 2001 Elsevier Science.)...

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