Power law correlations

The constant may depend on process variables such as temperature, rate of agitation or circulation, presence of impurities, and other variables. If sufficient data are available, such quantities may be separated from the constant by adding more terms ia a power-law correlation. The term is specific to the Operating equipment and generally is not transferrable from one equipment scale to another. The system-specific constants i and j are obtainable from experimental data and may be used ia scaleup, although j may vary considerably with mixing conditions. Illustration of the use of data from a commercial crystallizer to obtain the kinetic parameters i, andy is available (61). 

In the case where no correlations are available (i.e., the application involves an exotic fluid, a non-traditional stirrer or a very small reactor), experimental measurements of kLa must be performed to afford power law correlations valid for very similar reactor, turbines and fluids. Several techniques for kLa determination have been published [56]. 

The cost of the biofuel plants reported in the literature appeared to follow the same general laws as those of chemical and fuels plants [77, 78] irrespective of the technology applied, the plant cost showed a nice power-law correlation (R2 of 0.88) with the overall energy loss of the plant over two orders of magnitude. It correlated much less (R2 of 0.56) with capacity of the plant. 

Such behavior has not been observed. Kurata et al. [ 129] developed an empirically based suggestion after comparing the results available in those days for star molecules. He also assumes power law correlation between the two contraction factors as given by... 

Equation (9) generalizes earlier porosity-Peclet number power-law correlations (Konstandopoulos et al., 2002) obtained at Pe > 0.3 down to the diffusion limited deposition limit. PeQ is a characteristic cross-over Peclet number defining the scale beyond which the convective mechanism will take over the diffusive mechanism of deposition and eK the large Peclet number asymptote of the porosity. K has a dependence on the aggregate size and it is described in a forthcoming publication (Konstandopoulos, 2007). Using Eq. (9) the experimental data of Fig. 9 can be collapsed on a single curve as shown in Fig. 10. 

The DIERS Power Law Scaling Method121 is more robust than those given above and is applicable to fluids which are not Newtonian. The fluid is assumed to obey a power law and the power law parameters are assumed constant along the length of the relief line. Several experiments, at different L/D are required to fit the power law correlation, which may then be used to obtain G. Full details are given in reference... 

Varotsos C. (2004). Power-law correlations in column ozone over Antarctica. International Journal of Remote Sensing, 26, 3333-3342. 

To calculate the shear rate constant, k, a relationship must be established between shear rate and viscosity of a non-Newtonian calibration fluid. A cone-and-plate viscometer is used to determine a correlation between shear rate and viscosity that can be fit to a power law model. The power law correlation is then applied to viscosity data calculated from the impeller viscometer and Eq. 4. The shear rate constant can be calculated as follows ... 

Yi, y2, and y3 are empirical coefficients that were determined by fitting the nonlinear power law correlation Eq. (91) to 484 average Sherwood numbers computed for 121 different pool dimensions and four different sets of hydro-dynamic conditions. The resulting time invariant, average mass transfer correlation for elliptic pools is given by... 

Schuette and McCreery [34] demonstrated that with decreasing wire diameter there was a significant increase in current enhancement and modulation depth. This approached 100% modulation for a wire of diameter, d = 25 pm vibrated at 160 Hz. They showed that in these circumstances, for low Re numbers, the limiting current strictly followed the wire velocity and used [6] an empirical power-law correlation of mass-transfer coefficient to flow velocity /lim = /min(l + A/ cos(ft>.f)f) with s 0.7. They also noted that the frequency and amplitude dependence of the mean current, and the modulation depth, was linked to whether the flow was strictly laminar or not. Flow modelling indicated that for Re > 5 where Re = u dlv, there was separation of the boundary layer at the wire surface, when aid 1. For Re > 40 the flow pattern became very irregular. Under these circumstances, a direct relation between velocity and current should be lost, and they indeed showed that the modulation depth decreased steeply with increase of wire diameter, down to 10% for 0.8 mm diameter wire. 

In this section we plan to discuss the dynamic versus the thermodynamic approach to noncanonical equilibrium. We plan to show that the thermodynamic method, in the form of an information approach, does not yield satisfactory result. The dynamic approach is more promising than the information approach, even if it involves difficult conceptual problems that have not yet been solved. We argue, however, that the strange response of fluctuations with inverse power law correlation to perturbation, illustrated in Section VII, might turn out to be useful in settling these difficult conceptual problems. 

Table 4-4 Power Law Correlation Between G and G", and Frequency (rad s ), and the Cross Over Frequency for Solutions of LBG (natural pH and ionic strength)...

At the long time extreme, the autocorrelation function can again be expanded in a Taylor series to yield the long-time memory indicated by the inverse power-law correlation function... 

For contact nucleation, the most common type of secondary nucleation, the combination of potential causes usually calls for a power law correlation of the type... 

Jullien, R. and Botet, R., Aggregation and Fractal Aggregates, World Scientific, Singapore, 1987. Berry, M.V. and Percival, I.C., Optics of fractal clusters such as smoke. Opt. Acta, 33, 577, 1986. Ereltoft, T., Kjems, J.K., and Sinha, S.K., Power-law correlations and finite-size effects in silica particle aggregates studied by small-angle neutron scattering, Phys. Rev. B, 33, 269, 1986. 

A somewhat simpler, but not fundamental, approach is given in Ref. 376. The mass of data for various types of roughness (Fig. 11.8) is now so large that straightforward power-law correlations for Nusselt number and friction factor can be generated using large computers and statistical analysis software. The power-law correlations developed in this manner are... 

The offset strip fin (Fig. 11,13d) is widely used in compact heat exchangers. Once again, the data are so numerous that it is possible to develop power-law correlations for heat transfer factor and friction factor [377] ... 

These equations are based on experimental data for 18 different offset strip fin geometries, and they represent the data continuously in the laminar, transition, and turbulent flow regions, as shown in Fig. 11.13c. The development of accurate power-law correlations for a variety of enhancement configurations is possible when large databases are available. 

The cost of more complex equipment items, as reactors, furnaces, dryers, or filters can be estimated for preliminary design by means of a global quotation called the capacity ratio method. The cost is expressed by a power-law correlation as ... 

Mass action forms of rate correlation, often referred to as power law correlations, are widely applied, particularly for reactions in homogeneous phases. Table 1.1 gives a representative selection of correlations for various reactions. It is seen in several of the examples that the reaction orders are not those to be expected on the basis of the stoichiometric coefficients. Since these orders are normally established on the basis of experimental observation, we may consider them correct as far as the outside world is concerned, and the fact that they do not correspond to stoichiometry is a sure indication that the reaction is not proceeding the way we have written it on paper. Thus we will maintain a further distinction between the elementary steps of a reaction and the overall reaction under consideration. The direct application of the law of mass action where the orders and the stoichiometric coefficients correspond will normally pertain only to the elementary steps of a reaction, as will the dependence of rate on temperature to be discussed in the next section. Also,... 

In Table 3.4 is a summary of some eomparisons provided by Weller on the basis of this argument. The power-law correlations generally provide an adequate representation of the rate data, although they are not (nor do they propose to be) strongly based on any fundamental consideration of nonideal surfaces. In fact, the adequacy of both power-law and LH equations can be explained, in part, by the fact that the general LH form,... 

Freltoft, T., Kjems, J. K. and Sinha, S. K. (1986). Power-law correlations and finite-size effects in sUica particle aggregates studied by small-angle neutron scattering. Phys. Rev. B, 33, 269-275. 

Sinha, S.K., Fretoft, T. and Kjems, J.K. (1984). Observations of power law correlations in silica particle aggregates by small-angle neutron scattering. In Kinetics of Aggregation and Gelation, Family, F. and Landau, D.P. (eds). North-Holland, New York, pp. 87-90. 

Note the presence of two screening lengths and is of macromolecular dimension and is simply related to the radius of gyration = Kg/y/l. p, on the other hand, is short range at high density and is determined from the core conditioa Note also that at high densities there is a wide intermediate region of intersite separation, r where power law correlations occur. 

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