Empirical functions

The empirical function of temperature is used for the following properties ... 

Droplet Size Distribution. Most sprays comprise a wide range of droplet sizes. Some knowledge of the size distribution is usuaUy required, particularly when evaluating the overaU atomizer performance. The size distribution may be expressed in various ways. Several empirical functions, including the Rosin-Rammler (25) andNukiyama-Tanasawa (26) equations, have been commonly used. 

One other approach is to measure the rate as a function of a specified argument alone, and then differentiate the function with respect to the argument used. The differentiation can be done graphically, or by fitting an empirical function to the data (like a Fourier series) and differentiating this analytically. 

The last remark deserves a more quantitative explanation. Horowitz et al. [64] have shown that useful correlations are obtained between the oxidation voltages of the anion and the averaged Hammett <7 -function or Ingold and Taft s cr-function of the substituents around a boron atom. Both empirical functions reflect the electron-withdrawing ability of the substituent group. Hammett s averaged cr -values are... 

Between absorption edges, the photoelectric (true) mass absorption coefficient r can be expressed as the following approximate empirical function of Z and X ... 

The potential surfaces Eg, Hn, and H22 of the HF molecule are described in Fig. 1.6. These potential surfaces provide an instructive example for further considerations of our semiempirical strategy (Ref. 5). That is, we would like to exploit the fact that Hn and H22 represent the energies of electronic configurations that have clear physical meanings (which can be easily described by empirical functions), to obtain an analytical expression for the off-diagonal matrix element H12. To accomplish this task we represent Hn, H22, and Eg by the analytical functions... 

Fig. 1.—The empirical function expressing the dependence of carbon-carbon interatomic distance on bond character for single bond-double bond resonance.

This equation, with a = 0.25, was based on the observed electric dipole moments of HC1, HBr, and HI. Since then the value of the dipole moment of HF has been determined it is 1.98 D, which corresponds to 47% ionic character, whereas Equation 5 with a = 0.25 gives 59%. It seems justified to formulate an empirical function, based on the values 5, 11, 17, and 47% for the hydrogen halides HI, HBr, HC1, and HF, as calculated from their... 

The determination of values of interatomic distances in molecules has been found to provide much information regarding electronic structure, especially in the case of substances which resonate among two or more valence-bond structures. The interpretation of interatomic distances in terms of the types of bonds involved is made with use of an empirical function formulated originally for single bond-double bond resonance of the carbon-carbon bond.1 There are given in this... 

Recently, the ocean-basin distribution of marine biomass and productivity has been estimated by satellite remote sensing. Ocean color at different wavelengths is determined and used to estimate near-surface phytoplankton chlorophyll concentration. Production is then estimated from chlorophyll using either in situ calibration relationships or from empirical functional algorithms (e.g., Platt and Sathyendranth, 1988 Field et al., 1998). Such studies reveal a tremendous amount of temporal and spatial variability in ocean biological production. 

For this reason it is understandable that numerous empirical functions have been... 

The combinatorial term Pcomb(X,S) in Eq. (11) denotes a usually small correction for size and somewhat also for shape dilferences of the solutes and solvents, which is reasonably well understood from chemical engineering models. It is an empirical function, depending only on the surface areas and volumes of solute and solvent molecules, which are reasonably known from the COSMO cavities. The error arising from the approximations made in the combinatorial term can be expected to be much less than IkJ mol, or 0.17 log units, and hence can be safely neglected in the context of drug solubiUty estimation. 

An adaptive control system can automatically modify its behaviour according to the changes in the system dynamics and disturbances. They are applied especially to systems with non-linear and unsteady characteristics. There are a number of actual adaptive control systems. Programmed or scheduled adaptive control uses an auxiliary measured variable to identify different process phases for which the control parameters can be either programmed or scheduled. The "best" values of these parameters for each process state must be known a priori. Sometimes adaptive controllers are used to optimise two or more process outputs, by measuring the outputs and fitting the data with empirical functions. 

Many different empirical or semi-empirical functions have been suggested to represent the interaction between two spherical particles. The most successful... 

For a flow boiling crisis, an empirical scaling factor was suggested by Stevens and Kirby (1964). It is a graphical correlation developed from uniform-flux round-tube data. They suggested an empirical function ... 

While the relations z = fdetiQ) can be derived on the basis of natural laws, the estimation of an empirical function z = femp(Q) for the purpose of identification and qualitative analysis is mostly carried out by (linear) least squares to fit the observed z-values for a set of pure component standards or a multicomponent standard. On the other hand, empirical relationships z = emp(Q) in the form of tables, atlases and graphs are developed by collection and classification of experimental results. 

Comparison of the relaxation spectra with those relating to the empirical functions [l]-[4] provided us with more insight into the inherent shortcomings of these functions. The analytical expressions for these spectra were derived from the Equations [l]-[4] by a substitution method involving complex algebra (1,... 

If two independent variables are involved in the model, plots such as those shown in Figure 2.5 can be of assistance in this case the second independent variable becomes a parameter that is held constant at various levels. Figure 2.6 shows a variety of nonlinear functions and their associated plots. These plots can assist in selecting relations for nonlinear functions of y versus x. Empirical functions of more than two variables must be built up (or pruned) step by step to avoid including an excessive number of irrelevant variables or missing an important one. Refer to Section 2.4 for suitable procedures. 

---