Which Data Are Fitted

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

In the method of linear least squares, the algebraic expression to which data are fitted is linear in the least-squares parameter the method can be used for any polynomial. We will, as an example, fit a quadratic equation to a set of experimental data such as that in Table A.l. The extension to polynomials with terms of more or fewer terms will be obvious. 

Solution of these equations gives the expressions to which data are fit. 

Row control valves, such as back pressure valves, will not necessarily be the same nominal size as the pipe in which they are fitted. Manufacturers data for selection of their products is usually very comprehensive, and their guidance should be sought in case of any doubt. 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

Method (b). The (C,t) data are fitted by a fourth degree polynomial on the second graph, from which the derivative is... 

Obviously whenever binding energy data are fit to Eq. (9), the empirically determined values of k and l automatically take into account the atomic relaxation energy, ntr. Equation 9 gives very good correlations for molecules which have similar structures, presumably because of the automatic accounting for R>ntr and the fact that similar molecules have similar R>W values and that - w is therefore absorbed into the constant /. 

In general, data are fit quite well with the model. For example, with only two binary parameters, the average standard deviation of calculated lny versus measured InY of the 50 uni-univalent aqueous single electrolyte systems listed in Table 1 is only 0.009. Although the fit is not as good as the Pitzer equation, which applies only to aqueous electrolyte systems, with two binary parameters and one ternary parameter (Pitzer, (5)), it is quite satisfactory and better than that of Bromley s equation (J). 

It is shown that the properties of fully ionized aqueous electrolyte systems can be represented by relatively simple equations over wide ranges of composition. There are only a few systems for which data are available over the full range to fused salt. A simple equation commonly used for nonelectrolytes fits the measured vapor pressure of water reasonably well and further refinements are clearly possible. Over the somewhat more limited composition range up to saturation of typical salts such as NaCl, the equations representing thermodynamic properties with a Debye-Hiickel term plus second and third virial coefficients are very successful and these coefficients are known for nearly 300 electrolytes at room temperature. These same equations effectively predict the properties of mixed electrolytes. A stringent test is offered by the calculation of the solubility relationships of the system Na-K-Mg-Ca-Cl-SO - O and the calculated results of Harvie and Weare show excellent agreement with experiment. 

Fig. 3 Relative dipole-bound anion formation rates in RET collisions between Rydberg Xe(nf) atoms with (a) adenine (circles) or imidazole (squares) molecules and (b) adenine-imidazole complex produced in a supersonic beam. Experimental data are fitted to curvecrossing model calculations which lead to the experimental determination of EAdS values, equal to 11 meV for adenine, 23 meV for imidazole and 54 meV for adenine-imidazole complex (reproduced by permission of the American Chemical Society).

Various isothermal DSC studies on epoxy-amine cure have been reported in which the kinetic data are fitted to a generalised form of Equation (4-10) ... 

A disadvantage of simple interpretive methods is that the model to which the retention data (or other data) are fit must be fairly accurate. In other words, an interpretive approach may fail if one or more sample components exhibits anomalous retention. Although rare in SFC, such retention behavior is observed occasionally and is difficult to predict intuitively. Note, however, that by anomalous retention we do not mean behavior that is merely unusual, e.g., retention that decreases smoothly with increasing density (at constant temperature). Retention that varies in a regular (continuous) manner, even if unusual, can usually be modeled with a high degree of accuracy (vide infra). 

Figure 16.23. Two examples of neutralino models that provide a good fit to the excess of cosmic ray positrons observed by the HEAT collaboration. The two sets of data points (open and filled squares) are derived from two different instruments flown in 1994-95 and 2000. The lines represent (i) the best expectation we have from models of cosmic ray propagation in the galaxy ( bkg. only fit ), which underestimate the data points above 7 GeV (ii) the effect of adding positrons from neutralino annihilations (lines SUSY component , SUSY+bkg. fit , and bkg. component , the latter being the resulting background component when the data are fitted to the sum of background and neutralino contributions). (Figures from Baltz, Edsjo, Freese, Gondolo(2002)).

Franck-Condon dissociative continuum. At long times (Af = 3500 fs), a sharp photoelectron spectrum of the free NO(A, 3,v) product is seen. The 10.08 eV band shows the decay of the (NO)2 excited state. The 9.66 eV band shows both the decay of (NO)2 and the growth of free NO(A, 3,v) product. It is not possible to fit these via single exponential kinetics. However, these 2D data are fit very accurately at all photoelectron energies and all time delays simultaneously by a two-step sequential model, implying that an initial bright state (NO)2 evolves to an intermediate configuration (NO)2f, which itself subsequently decays to yield free NO(A, 3s) products [138]... 

In this language, the various f s, c s, - s, and r s are fitting parameters written to coincide with expressions used by spectroscopists to summarize their data. Built on the principle of a linear response only to events past, they reflect only the consequence of that principle. They do not become a theory of the response or even an adequate representation of the actual material for which data are being summarized. There has been some unfortunate confusion on this point because people have taken too seriously the language of models used to codify. The goal is to represent data in the most accurate, tractable, convenient form. 

Deactivation parameters obtained by plotting ln[(l — a) a)] versus time are listed in Table XIX for a number of nickel and nickel bimetallic catalysts. The fact that these plots were generally linear confirms that these data are fitted well by this deactivation model. These data, which include initial site densities for sulfur adsorption, deactivation rate constants, and breakthrough times for poisoning by 1-ppm H2S at a space velocity of 3000 hr-1 provide meaningful comparisons of sulfur resistance and catalyst life for both unsupported and supported catalysts. Table XIX shows that the... 

---