Single first-order model, comparison

The model fit variances are given in Table IV and calculated rate parameters for the two reactions in Table V. In nearly all cases an improvement in model fit was observed. This is typified by the comparison between the fit gg a single first-order and double first-order (DFO) model to the 0 Co data, as shown in Figure 5. Although the DFO model shows considerable improvement over single-site models, residual plots indicate a small systematic deviation at higher concentrations. This is discussed further in the following section. 

Figure 5. Comparison of single first-order and double first-order models to 0co mixing-cell data.

The homogeneous tar conversion is described by an empirical model. The results show that the simple one-tump, single first-order reaction model describes the depletion of the gravimetric tar satisfactory well. The comparison with literature studies shows a good agreement of the determined kinetic parameters. 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

Sobolev [66] hypothesized that the formation of polarization properties of planetary atmospheres takes place in the highest atmospheric layers (where the optical depth does not exceed values of about 1). It is then sufficient to consider only the first-order scattering in the calculation of the second Stokes parameter, Q. Sobolev considered two models of the atmospheric vertical stracture 1) a single-layer model in the form of a semi-infinite cloud layer, and 2) a two-layer model, in which an optically thin gas layer is put on top of the semi-infinite cloud layer. Comparisons of the phase dependence of the degree of linear polarization observed in the visible with calculations for two values of the real part of the refractive index = 1-33 and 1.50) and for varying particle radii showed the best agreement for monodisperse particles with rir= 1.5 and a radius of 1 micrometer. 

Figure 4.5 Simple single-phase model predictions for first-order irreversible catalytic ozone decomposition reaction in comparison with experimental fluidized-bed reactor data of Sun and Grace [44],...

Monte-Carlo calculations provide us with an alternative route to the local properties in the bulk of a liquid which are closer to first principles than the above model although the molgcgle needs to be somewhat simplified. We performed several computations on model liquids in order to evaluate the electric field which a molecule undergoes from the liquid and to compare it with the values predicted by the model. The main result of this comparison is that, due to the error bars of the Monte-Carlo calculation, and to the uncertainties on the dielectric constant of the medium, the model reproduces the electric field fairly well, especially when the charge distribution reduces to a single moment. In turn, noticeable deviations appear between the model and the Monte-Carlo simulation when the charge distribution of the solute is represented by more than one dominant moment (e.g. a dipole and a quadrupole) and when the solvent is represented by point dipoles at the centre of non polarizable molecules. This is easily understandable if one bears in mind that the model replaces this medium by a continuum. Nevertheless these discrepancies are expected to be less important in the case of a real medi] m, due to the molecular polarizabilities which are nonlocal properties. ... 

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