Modelling asymptotic approach

The proposed model of the structure of oxyfluoride melts corresponds with the conductivity results shown in Fig. 69. The specific conductivity of the melt drops abruptly and asymptotically approaches a constant value with the increase in tantalum oxide concentration. This can be regarded as an additional indication of the formation of oxyfluorotantale-associated polyanions, which leads to a decrease in the volume in which light ions, such as potassium and fluorine, can move. The formation of the polyanions can be presented as follows ... 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

At low 7t, the denominator simplifies to unity in each case and both models are linear in n. For sufficiently high n, the parenthesis in the denominator approaches Ketv, the initial rate for the dual-site model then approaches zero, and that of the single-site model approaches a constant value. Thus the plot of the experimental data will indicate that the dual-site model is preferable if a maximum exists in the data, or that the single-site model is preferable if a horizontal high-pressure asymptote exists. Hence, for the data of Franckaerts and Froment (FI) shown in Fig. 2, the dual-site model is preferred over the single-site model. 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

It was found that over a narrow free energy range rapidly increases with asymptotically approaching a limiting value smaller than . Although the observed behavior remains in accordance with the Marcus model prediction, it was, somewhat unexpectedly, found that not the electron annihilation energetics is a only factor... 

In contrast, the asymptotic approach puts minimal strain on the computer but demands more of the modeller. The convergence of the computed solutions is usually easy to test with respect to spatial and temporal resolution, but situations exist where reducing the timestep can make an asymptotic treatment of a "stiff" phenomenon less accurate rather than more accurate. This follows because the disparity of time scales between fast and slow phenomena is often exploited in the asymptotic approach rather than tolerated. Furthermore, the non-convergence of any particular solution is often easier to spot in timestep splitting with asymptotics because the manner of degradation is usually catastrophic. In kinetics calculations, lack of conservation of mass or atoms signals inaccuracy rather clearly. 

The exact solutions are not valid if any of the model inputs differ from the distribution type that is the basis for the method. For example, the summation of lognormal distributions is not identically normal, and the product of normal distributions is not identically lognormal. However, the Central Limit Theorem implies that the summation of many independent distributions, each of which contributes only a small amount to the variance of the sum, will asymptotically approach normality. Similarly, the product of many independent distributions, each of which has a small variance relative to that of the product, asymptotically approaches lognormality. 

The Langmuir-adsorption model predicts an asymptotic approach to monolayer surface coverage as adsorbate partial pressure approaches saturation this is the Type-I isotherm of Figure 5.7. The Langmuir model, though proven for many ultraclean, well-ordered surfaces interacting with small-molecule adsorbates, is oversimplified for many real-world systems. Nonetheless, it is the foundation upon which much of adsorption theory is built and as such provides a useful con-... 

In some cases, adsorption of analyte can be followed by a chemical reaction. The Langmuir-Hinshelwood (LH) and power-law models have been used successfully in describing the kinetics of a broad range of gas-solid reaction systems [105,106]. The LH model, developed to describe interactions between dissimilar adsorbates in the context of heterogeneous catalysis [107], assumes that gas adsorption follows a Langmuir isotherm and that the adsorbates are sufficiently mobile so that they equilibrate with one another on the surface on a time scale that is rapid compared to desorpticm. The power-law model assumes a Fre-undlich adsorption isotherm. Bodi models assume that the surface reaction is first-order with respect to the reactant gas, and that surface coverage asymptotically approaches a mmiolayer widi increasing gas concentration. 

Lippincott and Schroeder (1242) have discussed the relation between Av, and R in terms of a covalent description of the H bond (see Section 8.2.5). Their model leads to an asymptotic approach of Av toward zero for large R. This prediction is reasonable, and whether the covalent description is correct or not it points the obvious fault of the proposed linear relation between Ay, and R. Certainly, as the distance becomes large the H bond perturbation must become unimportant and Ay, must approach zero in a continuous fashion. The study by Glemser and Hartert (782) shows that the linear relation between Ay,... 

Graphical methods provide a first step toward interpretation and evaluation of impedance data. An outline of graphical methods is presented in Chapter 16 for simple reactive and blocking circuits. The same concepts are applied here for systems that are more typical of practical applications. The graphical techniques presented in this chapter do not depend on any specific model. The approaches, therefore, can provide a qualitative interpretation. Surprisingly, even in the absence of specific models, values of such physically meaningful parameters as the double-layer capacitance can be obtained from high- or low-frequency asymptotes. 

For example, if the reaction controlling the sorption of each molecule of a contaminant is identical and the capacity of a sorbent for these molecules is operationally limitless, a linear isotherm relationship is prescribed in which the sorbed-phase concentration is a constant proportion of the solution-phase concentration. When the sorption reactions are identical but sorption capacity is limited, an asymptotic approach to a maximum sorbed-phase concentration might be expected. These two limiting-condition models have been described and compared with others for description of the sorption of hydrophobic contaminants on a variety of natural soils, sediments, and suspended solids... 

A nonlinear local isotherm model is clearly required for description of sorption reactions between the TCB and the shale isolate. A variety of conceptual and empirical models for representing nonlinear sorption equilibria, exists (2). The Langmuir model is one of the ideal limiting-condition-type models cited earlier. It is predicated on a uniform surface affinity for the solute and prescribes a nonlinear asymptotic approach to some maximum sorption capacity. 

The available continuum models for dispersed multi-phase flows thus follow one of two asymptotic approaches. The dilute phase approach is formulated based on the continuum mechanical principles in terms of the local conservation equations for each of the phases. A macroscopic model is then obtained by averaging the local equations based on an appropriate averaging procedure. In the dense phase approach, on the other hand, a kinetic theory description is adopted for the dispersed particulate phase (granular material), whereas an averaged continuum model formulation is adopted for the interstitial phase. 

Exponential Function In some cases of disease progression, such as recovery from an injury or some other temporary disease state, the model should be able to describe the improvement over time. In such cases, recovery can be approximated by an exponential function parameterized for the baseline status So and the rate constant of recovery kprog. The exponential function has the property of asymptotically approaching 0 and so is best used in situations where the severity scores have a minimum value of 0 or, in the case of some biomarkers, do not occur in the nondiseased state. 

Thus, the question of central concern raised in our contribution has been the macroscopic formulation of EET and its relation to the experimental observable of excimer fluorescence in a time-resolved experiment. EET has been discussed, hers, as a dispersive, i.e., time-depen-dent process in deterministic monomer-excimer models which had been the subject of a detailed kinetic analysis in recent work (3 8, 4.S.). With the use of rate function k(t) (Equation 4) it is natural to yield typical non-exponential intensity-time profiles, either in form of an asymptotic approach (Equations 5,6), or in closed form analytical solutions (Equations 7,8). The physios emer-... 

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