Fitting data

The reactor model was fitted to the bench-scale plant data. The set of parameters was optimized with the Levenberg-Marquardt algorithm by minimizing the sum of squared residuals (SSR) between experimental data and model predictions. 

Parameter estimations were assessed by computing individual confidence intervals based on t tests. 

FIGU RE 8.8 Comparison between experimental kinetic data and model predictions. (—) 5%. 

The point of a transient kinetic experiment is to establish the reaction pathway by examination of the concentration dependence of the rate of reaction. It is the goal of proper data analysis to ferret out the best fit to the data to establish the reaction mechanism containing the minimal number of steps. I will describe two 

The conventional data analysis involves the fitting of data to an equation describing the time dependence of the reaction, leading to the best estimates for the constants defining the equations. Analytical solutions to most simple reaction sequences can be obtained (7, 5, 63). Solutions of differential equations describing the series of first-order (or pseudo-first-order) reactions will always be a sum of exponential terms [Eq. (22)]. Thus for a single exponential, the fitting process provides the amplitude (A), the rate of reaction (X), and the end point (C) 

For more complex kinetics, the data can be fit to an equation including two or more exponential terms. A double-exponential equation is defined by 

The distinction between a single- and double-exponential fit is sometimes obvious, but in other cases, in which the difference in rates of reaction between the two phases is small (less than fourfold), it can be difficult to resolve the two rates. In these cases, the evaluation of the goodness of fit and the justification for including a second exponential term cannot be based solely on a reduction in 

Presteady-state burst kinetics can be fit to an equation of the form 

To use classical integration techniques, we will first generate a table of discrete points where we evaluate the function seperated by equal intervals of Ax = 0.25 (Table 7.4). 

Let us first use the trapezoidal rule to calculate the above integral 

Had we used a higher precision for the Gauss points, the quadrature would have rendered a solution even closer to the exact integral. 

Data fitting or modeling of data is often required to find a numerical representation for a set of data points. In polymer processing, we often want to tit complex models, such 

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Thermodynamic consistency requites 5 1 = q 2y but this requirement can cause difficulties when attempts ate made to correlate data for sorbates of very different molecular size. For such systems it is common practice to ignore this requirement, thereby introducing an additional model parameter. This facihtates data fitting but it must be recognized that the equations ate then being used purely as a convenient empirical form with no theoretical foundation. 

E] Smooth pipe data. Data fits within 4% except at Nsc> 20,000, where experimental data is underpredicted. 

The best fit, as measured by statistics, was achieved by one participant in the International Workshop on Kinetic Model Development (1989), who completely ignored all kinetic formalities and fitted the data by a third order spline function. While the data fit well, his model didn t predict temperature runaway at all. Many other formal models made qualitatively correct runaway predictions, some even very close when compared to the simulation using the true kinetics. 

Figure 4 Data plus Herstions 1,2, and 7 in regression analysis (data fit) for the optical coating glass/TI02/Ag/TI02-...

G. Vizkelethy. NticL Instr. Meth. B45,1, 1990. Description of the program SENRAS. used in fitting NRA spectra includes examples of data fitting. 

There are relatively few kinetic data on the Friedel-Crafts reaction. Alkylation of benzene or toluene with methyl bromide or ethyl bromide with gallium bromide as catalyst is first-order in each reactant and in catalyst. With aluminum bromide as catalyst, the rate of reaction changes with time, apparently because of heterogeneity of the reaction mixture. The initial rate data fit the kinetic expression ... 

The activation data collected by Zavitsas et al. at 30 and 57°C was extrapolated to 70-90°C by Ferrero and Panetti and tested against experiment. The data fit well throughout the temperature range. Fig. 9 and Table 6 show these activation data for each species involved in the methylolation process. 

Equation 1-111 is known as the Lineweaver-Burk or reciprocal plot. If the data fit this model, a plot of l/V versus 1/Cg will he linear with a slope K /V x intercept l/V x-... 

Pauli spin susceptibility for the aligned CNTs has been measured and it is reported that the aligned CNTs are also metallic or semimetallic [30]. The temperature dependence of gn and gx s plotted in Fig. 5(a). Both values increase with decreasing temperature down to 40 K. A similar increase is observed for graphite. The g-value dependence on the angle 0 at 300 K is shown in Fig. 5(b) (inset). The g-value varies between gn = 2.0137 and gx= 2.0103 while the direction of magnetic fields changes from parallel to perpendicular to the tubes. These observed data fit well as... 

The situation is more eomplieated when the set of reasonable eontributing struetures are not all equivalent. Examine the geometry and atomie eharges forphenoxide anion. Do these data fit any one of the possible resonanee struetures (draw all reasonable possibilities), or is a eombination of two or more resonance contributors necessary ... 

Considering that the parameters for the MNDO/d method for all first row elements (which are present in most of the training set of compounds) are identical to MNDO, the improvement by addition of d-functions is quite impressive. It should also be noted that MNDO/d only contains 15 parameters, compared to 18 for PM3, and that some of the 15 parameters are taken from atomic data (analogously to the MNDO/AMl parameterization), and not used in the molecular data fitting as in PM3. 

The isotherms for the two enantiomers of phenylalanine anilide were measured at 40, 50, 60 and 70 C, and the data fitted to each of the models given in Equations (1-3) [42]. The isotherms obtained by fitting the data to the Langmuir equation were of a quality inferior to the other two. Fittings of the data to the Freundlich and to the bi-Langmuir equations were both good. A comparison of the residuals revealed that the different isotherms of d-PA were best fitted to a bi-Langmuir model, while the... 

FIGURE 6.6 Schilcl regression for pirenzepine antagonism of rat tracheal responses to carbachol. (a) Dose-response curves to carbachol in the absence (open circles, n = 20) and presence of pirenzepine 300 nM (filled squares, n = 4), 1 jjM (open diamonds, n=4), 3j.lM (filled inverted triangles, n = 6), and 10j.iM (open triangles, n = 6). Data fit to functions of constant maximum and slope, (b) Schild plot for antagonism shown in panel A. Ordinates Log (DR-1) values. Abscissae logarithms of molar concentrations of pirenzepine. Dotted line shows best line linear plot. Slope = 1.1 + 0.2 95% confidence limits = 0.9 to 1.15. Solid line is the best fit line with linear slope. pKB = 6.92. Redrawn from [5],... 

FIGURE 6.17 Fitting of data to models, (a) Concentration response curves obtained to an agonist in the absence (circles) and presence of an antagonist at concentrations 3 jiM (triangles) and 30 j.lM (diamonds), (b) Data fit to model for insurmountable orthosteric antagonism (Equation 6.31) with Emax = 1, Ka = 1 pM, t = 30 and KB = 1 pM. 

FIGURE 10.5 Full agonist potency ratios, (a) Data fit to individual three-parameter logistic functions. Potency ratios are not independent of level of response. At 20%, PR = 2.4 at 50%, PR = 4.1 and at 80%, PR = 6.9. (b) Curves refit to logistic with common maximum asymptote and slope. PR = 4.1. The fit to common slope and maximum is not statistically significant from individual fit. 

The same conclusion can be drawn from another statistical test for model comparison namely, through the use of Aikake s information criteria (AIC) calculations. This is often preferred, especially for automated data fitting, since it is more simple than F tests and can be used with a wider variety of models. In this test, the data is fit to the various models and the SSq determined. The AIC value is then calculated with the following formula... 

FIGURE 11.16 Control dose-response curve and curve obtained in the presence of a low concentration of antagonist. Panel a data points. Panel b data fit to a single dose-response curve. SSqs = 0.0377. Panel c data fit to two parallel dose-response curves of common maximum. SSqc = 0.0172. Calculation of F indicates that a statistically significant improvement in the fit was obtained by using the complex model (two curves F = 4.17, df=7, 9). Therefore, the data indicate that the antagonist had an effect at this concentration. 

FIGURE 11.18 Asymmetrical dose-response curves, (a) Dose-response data fit to a symmetrical Hill equation with n = 0.65 and EC50 = 2.2 (solid line) or n= 1, EC50 = 2 (dotted line). It can be seen that neither symmetrical curve fits the data adequately, (b) Data fit to the Gompertz function with m = 0.55 and EC50= 1.9. 

The reason for the success of this type of data fitting is that for moderately large barriers it becomes unimportant whether escape for the image potential is treated within the framework of the Onsager or the RS model. An indication that the Onsager description is, nevertheless, more appropriate is the intersection of j(/ ) curves for variable temperature at high electric fields. This is a characteristic feature of Onsager processes 24J. 

The plotting of Dixon plot and its slope re-plot (see 5.9.5.9) is a commonly used graphical method for verification of kinetics mechanisms in a particular enzymatic reaction.9 The proposed kinetic mechanism for the system is valid if the experimental data fit the rate equation given by (5.9.4.4). In this attempt, different sets of experimental data for kinetic resolution of racemic ibuprofen ester by immobilised lipase in EMR were fitted into the rate equation of (5.7.5.6). The Dixon plot is presented in Figure 5.22. 

Thus a lineal- relation between In (CA/CA0) and the reactor length should exist if the model accurately describes the immobilised cell reactor. The experimental data fitting the model was discussed earlier. 

These data are plotted in Figure 10-3 about the Gaussian curve for which the standard deviation is the square root of the mean. The data of Rutherford and Geiger, which were obtained by counting alpha-particles, are plotted about the same curve. In the figure, both sets of data fit the Gaussian about equally.well. 

The vacuum decomposition [1106] of nickel maleate at 543—583 K was predominantly a deceleratory process. Following an initial surface reaction, data fitted the kinetic expression... 

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