Scaling indices

The total gain in weight over the seven 6-h cycles is designated the scaling index and this value is plotted against test temperature for a series of steels of varying chromium content in Fig. 7.28. These were laboratory-produced... 

Fig. 7.28 Scaling index versus temperature for several chromium steels...

Fig. 7.29 Temperature to give a scaling index of 100 versus silicon content (10-89-13-14% Cr)...

The theory of models requires that in scale-up from a model (index m) to industrial scale (index x), not only must the geometric similarity be ensured but also all dimensionless numbers describing the problem must retain the same numerical values (rii = idem). This means that in scale-up of boats or ships, for example, the dimensionless numbers governing the hydrodynamics here... 

Other indices proposed that have tried to overcome the basic limitations of LSI include the Stiff and Davies Saturation Index, Larson and Buswell Index, Puckorius (Practical) Scale Index, Oddo-Tomson Index, and Larson-Skold Corrosivity Index, some of which are discussed briefly here. 

SDSI = pHactUai pCa pTA K 4.3.3.2 Puckorius (Practical) Scale Index... 

The method used is outlined in Ref. CE9 (Chapter 4). The capital cost estimate is adjusted to the correct plant capacity by applying a scaling index. The appropriate scaling index for nitric acid plants is 0.6 (Ref. CE9 Table 19, p.184). Therefore ... 

Figure 10.6 Projected scaling indexes of the Dow Water and Process Solutions ROSA design program.

Using DEA, we have established that there are statistical processes for which 8 = H and statistical processes for which 8 H, both of which scale. However, there is a third class of processes for which the scaling index is a function of the Hurst exponent, but the relation is not one of their being equal. This third class is the Levy random walk process (Levy diffusion) introduced by Shlesinger et al. [65] in their discussion of the application of Levy statistics to the understanding of turbulent fluid flow. 

The Levy-walk intermittent noise is characterized by an uncorrelated waiting-time sequence and a Levy or an inverse power-law waiting-time distribution function such as given by Eq. (97) with 1 < a < 2. This interval for the scaling index insures that although the second moment diverges, the first moment is finite. The presence of a Levy-walk process in a given time series can be detected by means of the asymptotic relation Eq. (99), which we refer to as... 

It should be stressed that even though the second moment for a Levy walk is finite, the scaling index obtained from the second moment does not give the correct scaling properties of the time series. This is a word of caution regarding the application of FVSMs to determine the scaling properties of a time series. Even when the second moment has the form... 

Inserting Eq. (109) into Eq. (108) yields the equation for the scaling index ... 

The scaling index or fractal dimension marks the system s response and can be used as an indicator of the system s state of health. Since the fractal dimension is also a measure of the level of complexity, the change in dimension with disease suggests a new definition of disease as a loss of complexity, rather than the loss of regularity [56], This observation was first made by Goldberger and West, see, for example, Ref. [110]. 

Hawthorn, D., 1992, The scaling index is dead, long live the scaling equation, in Bohnet, M., Bott, T.R., Karabelas, A.J., Pilavachi, P.A., Semeria, R. and Vidil, R. eds. Fouling Mechanisms Theoretical and Practical Aspects. Editions Europeenes Thermique et Industrie, Paris. 

Neonates, and especially preterm infants, are a population at risk of suffering the consequences of iodine deficiency because of the impact of neonatal hypothyroxinemia on brain development. We evaluate the possible association between mental development scores at different ages and iodine intake during the neonatal period. Sixty-seven preterm infants were subdivided into GA groups for data analysis. The mental development scores reported here are those of the Brunet-Lezine scale index for children (0-24 months of age). The children were tested at 6, 9, 12, 18 and 24 months of postnatal age, and results were corrected for their GA. The test assesses P, motor abilities and postural... 

The spectral dimension d is a true property of a fractal and is determined only by its connectivity. It differs from the mass scaling index [see Equation (11.1)] or fractal dimension df and from the scaling index of the diffusion constant 8 by the fact that it does not depend on the way in which a fractal has been inserted into the Euclidean space with dimension d. The dependence of d and 8 on df is described by Equation (11.3). 

Attempts to resolve this contradiction have been undertaken [68]. Maritan and Stella [69] calculated the fractal dimensions for 50 proteins using two methods. In one method, fractal dimension is a scaling index for the contour length of a macromolecule with respect to the distance between its ends (it is equal to 1.19-1.82), while in the other method, the fractal dimension is a scaling index for the total mass with respect to the length, which is equal to 1.62-2.24. The calculations were performed using X-ray diffraction crystallographic parameters of proteins. 

In Eqs. (4) and (5) z is not strictly determined at the scaling approach the efiective coordination number of ( -measured lattice, parameters y and y are assiuned as imiversal scaling indexes. 

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