Large parameter

Assessment starts with a general calculation based on assumed service conditions and identification of the critical parameters and components, with particular regard to risk. Although relatively crude, this ranking of components is probably correct. It is followed by assessment of the actual service conditions experienced - data that should be available for process plant. This can identify the accumulated exposure or dose , or confirm that parameters lie below a threshold level. It can also identify unexpectedly large values, due for example to rapid transient variations, which would not be found in the planned service history. Such large parameters can have an exceptional effect on service lifetime. This is followed by inspection (materials, dimensions, etc.) of the critical components either on the plant or, if allowed, of the dismantled component, to provide a more confident... 

The general features of the theory, expressed in analytical properties which are largely parameter-independent, are thus in accordance with observation, and there seems to be a good case for further investigation of the q and F indices. 

Let us emphasize that as a result of scaling t with the characteristic hydrostatic time to 260 sec long as compared with that of diffusional relaxation time L2/D 10 sec, D emerges in the system (6.3.9)-(6.3.15) as a large parameter. 

Equation (6.2.21) is an ordinary differential equation having large parameter h r). Its solution under certain standard conditions could be obtained by means of the steepest descent (called also VKB) method. 

Equation (8.3.14) is not an asymptotically exact result for the black sphere model due to the superposition approximation used. When deriving (8.3.14), we neglected in (8.3.11) small terms containing functionals I[Z], i.e., those terms which came due to Kirkwood s approximation. However, the study of the immobile particle accumulation under permanent source (Chapter 7) has demonstrated that direct use of the superposition approximation does not reproduce the exact expression for the volume fraction covered by the reaction spheres around B s. The error arises due to the incorrect estimate of the order of three-point density p2,i for a large parameter op at some relative distances ( f — f[ < tq, [r 2 - r[ > ro) the superposition approximation is correct, p2,i oc ct 1, however, it gives a wrong order of magnitude fn, oc Oq2 instead of the exact p2,i oc <7q 1 (if n — r[ < ro, fi — f[ < ro). It was... 

If S -> 0 at constant bs,bg, and V(Nfol P N ol) on the right-hand side of the first equation of eqns. (148) there appears a large parameter 1/ 0, variations in c are determined accurately to the terms of the order by the substance flow... 

Consequently, in this case, the surface is practically free. Let us consider the asymptotes at high (6g - oo) and low (6g -> 0) pressures. If 6g - oo, the equation for cA will have a small parameter l/6g but not for all summands and on the right-hand side of the equation for 9Z there appears a large parameter 6g. Let us write... 

The limiting values of aPo/aPo for arbitrary k and an infinitely large parameter x 00 equal, in the case of linearly polarized Q-absorption of light ... 

Table 3.3. Limiting values of the ratio aPo/aPo °f various rank ground state multipole moments aPo and the population aPo for an infinitely large parameter X — 00...

Successful application of the AOM parametrization scheme for interpretation of the electronic spectroscopy data based on the values extracted from experiment [159] demonstrates that the general parametrization scheme eq. (2.99) implied by the AOM, most probably reflects some general features of the electronic structure of the good fraction of TMCs. However, numerical estimates of its parameters according to formula eq. (2.99) were not particularly successful. As a result the AOM requires for its application large parameter sets (for the cells) specific for each pair of metal - ligand, which makes the parametrization boundless. The AOM parameters remain empirical quantities just as the 10.Dgs were in the original CFT. 

For the system (2.36), in the limit e —> 0, the term (l/sjkfx) becomes indeterminate. For rate-based chemical and physical process models, this allows a physical interpretation in the limit when the large parameters in the rate expressions approach infinity, the fast heat and mass transfer, reactions, etc., approach the quasi-steady-state conditions of phase and/or reaction equilibrium (specified by k(x) = 0). In this case, the rates of the fast phenomena, as given by the explicit rate expressions, become indeterminate (but, generally, remain different from zero i.e., the fast reactions and heat and mass transfer do still occur). 

A large parameter range in which cluster patterns dominated the spatiotemporal dynamics was also observed for the oscillatory oxidation of H2 in the presence of electrosorbing cations and anions [175], Characteristically, the cluster-type patterns were found at lower concentrations of Cu2+ ions compared with the pulse-type motions shown in Fig. 56. Examples of two-phase and three-phase clusters are depicted in Fig. 63. In these figures the homogeneously oscillating mode has been subtracted. 

Similarly to their product, each of the copolymerization parameters can also be studied independently. Only very small or very large parameters will significantly change with temperature. A parameter smaller than 1 will increase with increasing temperature and vice versa. With increasing temperature the parameter value will always approach 1, and thus the tendency to random order will increase, with decreasing temperature the order will tend to alternation. 

Equations 27—29 with two parameters seem to show reasonable correlation. Although Equation 28 would be unacceptable because of unusually large parameter coefficients, the correlations are completely equivalent statistically. As far as compounds used for the analyses are concerned, it is impossible to choose the two significant parameters. In this case, besides a significant correlation between parameters nr and E8, there are mutual relationships among three parameters. Each parameter is expressed as a linear combination of the other two and is not separated from others. 

With simplified chemical kinetics, perturbation methods are attractive for improving understanding and also for seeking quantitative comparisons with experimental results. Two types of perturbation approaches have been developed, Damkohler-number asymptotics and activation-energy asymptotics. In the former the ratio of a diffusion time to a reaction time, one of the similarity groups introduced by Damkohler [174], is treated as a large parameter, and in the latter the ratio of the energy of activation to the thermal... 

In activation-energy asymptotics, E/R T is a large parameter. Equation (89) then suggests that Wj will be largest near the maximum temperature and that Wjr will decrease rapidly, because of the Arrhenius factor, as T decreases appreciably below Therefore values of the reaction rate some distance away from the stoichiometric surface (Z = ZJ will be very small in comparison with the values near Z = Z. This implies that to analyze the effect of w, it is helpful to stretch the Z variable in equation (92) about Z = Z, A stretching of this kind causes the terms involving the highest Z derivative to be dominant, and therefore equation (92) becomes, approximately. 

Ignition processes often are characterized by a gradual increase of temperature that is followed by a rapid increase over a very short time period. This behavior is exhibited in the present problem if a nondimensional measure of the activation energy E is large, as is true in the applications. Let tc denote an ignition time, the time at which the rapid temperature increase occurs a more precise definition of arises in the course of the development. In the present problem, during most of the time that t < tc, the material experiences only inert heat conduction because the heat-release term is exponentially small in the large parameter that measures E. The inert problem, with w = 0, has a known solution that can be derived by Laplace transforms, for example, and that can be written as... 

Since variables Ca and Cb are not separated in (2.2.71), random values Na and Nq are always correlated. On the other hand, this peculiarity of the equation does not permit to solve it exactly and thus asymptotic expansion has to be used. Equation (2.2.71) has no other stationary solution except trivial F(Ca,Cb) = corresponding to P Na,N ) = Sna,qSnb,o- An asymptotic solution (2.2.71) is sought in the F oo limit (system s volume is a large parameter), when one can assume [16], that... 

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