Parabolic model example

In the present example, the total degrees of freedom is ten. The parabolic model that is to be fit. 

This stochastic model of the flow with multiple velocity states cannot be solved with a parabolic model where the diffusion of species cannot depend on the species concentration as has been frequently reported in experimental studies. Indeed, for these more complicated situations, we need a much more complete model for which the evolution of flow inside of system accepts a dependency not only on the actual process state. So we must have a stochastic process with more complex relationships between the elementary states of the investigated process. This is the stochastic model of motion with complete connections. This stochastic model can be explained through the following example we need to design some flowing liquid trajectories inside a regular porous structure as is shown in Fig. 4.33. The porous structure is initially filled with a fluid, which is non-miscible with a second fluid, itself in contact with one surface of the porous body. At the... 

Many data sets can be explained much better with the help of this theoretically derived model than with the empirical parabolic model [23, 175, 345]. Only two examples shall be given here, one (eq. 99) describing the spasmolytic activities of mandelic acid esters (Table 14 eqs. 85 and 86), the other one describing the antifungal activities of aliphatic amines vs. Rhinocladium beurmanni (eq. 100) [345] in the latter case the parabolic model gives r = 0.967, while a combination of the parabolic model with an additional, highly interrelated MW term yielded r = 0.994 [344] (chapter 3.7). 

From the various findings, it seems that a linear-parabolic model would best fit the conversion of carbides changing based on flow rate, temperature, experimental setup, geometry of the carbide, thickness of CDC, and chlorine treatment time. For example, in at least one study, it appeared that a lower synthesis temperature decreased the linear growth region. ... 

In (eco)toxicological QSAR studies, the molecular descriptor of choice is the n-octanol/water partition coefficient (log P), generally used in a simple regression equation. However, sometimes a simple linear regression model is inadequate to model properly the dependence of biological activity (BA) on logF. For example, fish exposed to very hydrophobic chemicals for a limited test duration have insufficient time to achieve a pseudo-steady state partitioning equilibrium between the toxicant concentration in aqueous circumambient phase and the hydrophobic site of action within the fish. Hansch initiated the use of a parabolic model in log P (equation 1) to overcome... 

The results of the comparison are presented in Table 7.17. The steric hindrance increases the activation energy for a thermally neutral reaction by 8.2, %2, and 24 kJ mol when oxygen-centerc nitrogen-craitered, and alkyl radicals are involved respectively. The data obtained with the aid of the parabolic model. The steric effect is manifested similarly also in the reactions of sterically hindered (Ar O) phenoxyl radicals with various substrates. The reactions of the sterically hindered diphenylpicryl radical (DPPH ) and the unhindered diphenylaminyl radical may serve as another example. In the reactions of DPPH- with phenols, the contribution of the steric effect to E ranges fix>m 23 to 30 kJ mof, i.e.., is very considerable. 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

What information could be obtained if all ten experiments were carried out at only three levels of pH Three levels of x, (f= 3) provides sufficient factor combinations for being able to fit a three-parameter model, but leaves no degrees of freedom for estimating lack of fit / - p = 3 - 3 = 0. Because one of our objectives was to determine if a parabolic relationship provides an adequate model for the observed rate, we must be able to estimate the variance due to lack of fit of the model the number of factor combinations (levels of x, in this example) must therefore be greater than three. 

CT-VPP-REDOR) or the pulse duration fp (CT-VPD-REDOR) then produces CT-REDOR curves, from which the second moment may be evaluated with distinctively superior accuracy as compared to the values obtained from a parabolic fit to the conventional REDOR data. When restricting the experiment to short dipolar evolution times, the two-spin approximation may be applied for the data analysis, which proves to be especially attractive for amorphous solids, for which the exact spin geometry is unknovm. The data presented on the model compoimds illustrate the various facets of CT-REDOR NMR spectroscopy. First application examples, namely, the evaluation of the heteronuclear Li-Ti dipolar couplings within the garnet structure of Li5La3Nb20i2, the determi-nation of the intemuclear B- P distance in frustrated Lewis pairs, the analysis of Na- F dipolar interaction in fluormica or Na- P... 

This is the standard dispersion model for P, and the aim of investigating more complex situations has often been to reduce them to this form with D = De, an effective dispersion coefficient that wraps up the complexities of the underlying situation in a single quantity. Whether this is wise is another matter. For example, in a packed bed the flow is obviously very complex, but both theory [4] and experiment can be invoked to justify an effective Peclet number, URIDe, of about 2. The question that hangs over the use of Eq. (33) is that it is a parabolic equation, with infinite signal speed and controversial boundary conditions. 

Numerous other models have been proposed to explain the deviation of dry oxidation from linear-parabolic kinetics. For example, field-assisted oxidant diffiision during the oxidation of metals was proposed by Cabrera and Mott (75) and used by Deal and Grove (69) to explain the results for thin oxides. Ghez and van der Meulen (76) proposed the dissociation of molecular oxygen into atomic oxygen at the Si-Si02 interface and the re-... 

By use of the proper experimental conditions and Ltting the four models described above, it may be possible to arrive at a reasonable mechanistic interpretation of the experimental data. As an example, the crystal growth kinetics of theophylline monohydrate was studied by Rodriguez-Hornedo and Wu (1991). Their conclusion was that the crystal growth of theophylline monohydrate is controlled by a surface reaction mechanism rather than by solute diffusion in the bulk. Further, they found that the data was described by the screw-dislocation model and by the parabolic law, and they concluded that a defect-mediated growth mechanism occurred rather than a surface nucleation mechanism. 

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

See description of this collision model in Sections VII.C.2 and VII.C.3 (on the example of a parabolic potential well). 

The main advantage of the hat-curved potential is that it is possible to narrow the width Avor of the librational absorption band by decreasing the form factor /. Indeed, Avor attains its maximum value when/ = 1. Note that / = 1 is just the case of the hat flat or its simplified variant, the hybrid model, both of which were described in Section IV. The latter was often applied before (VIG) and is characterized by a rather wide absorption band, especially in the case of heavy water. In another extreme case, / — 0, the linewidth Avor becomes very low. When / = 0, we have the case of the parabolic potential well, whose dielectric response was described, for example, in GT and VIG. Thus, when the form factor/of the hat-curved well decreases from 1 to 0, the width Avor decreases from its maximum to some minimum value. 

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