Dispersivity asymptotic

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be self-sharpening. (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded oed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.)... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

For the correlation bands obtained by a convolution of many shake-up lines, the size-consistency and size-intensivity requirements imply a convergence towards some asymptotic profile, when going to the polymer limit. This must ideally be achieved through a balance of the scaling properties of the individual shake-up lines, and the dispersion of the intensity of Ih lines over a rapidly increasing number of excited states with increasing system size. 

The standard way to answer the above question would be to compute the probability distribution of the parameter and, from it, to compute, for example, the 95% confidence region on the parameter estimate obtained. We would, in other words, find a set of values h such that the probability that we are correct in asserting that the true value 0 of the parameter lies in 7e is 95%. If we assumed that the parameter estimates are at least approximately normally distributed around the true parameter value (which is asymptotically true in the case of least squares under some mild regularity assumptions), then it would be sufficient to know the parameter dispersion (variance-covariance matrix) in order to be able to compute approximate ellipsoidal confidence regions. 

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic... 

Further work on long polyenes, including vibrational distortion, frequency dispersion effects and electron correlation, would be important for evaluating more accurate asymptotic longitudinal polarizabilities and hyperpolarizabilities. 

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

Casimir and Polder also showed that, at very long range (i.e., separations greater than a characteristic distance R of a few hundred angstrom units), the dispersion interaction takes the modified asymptotic form... 

Phenomenological quasiparticle model. Taking into account only the dominant contributions in (7), namely the quasiparticle contributions of the transverse gluons as well as the quark particle-excitations for Nj / 0, we arrive at the quasiparticle model [8], The dispersion relations can be even further simplified by their form at hard momenta, u2 h2 -rnf, where m.t gT are the asymptotic masses. With this approximation of the self-energies, the pressure reads in analogy to the scalar case... 

Here, the densities of the gaseous and solid fuels are denoted by pg and ps respectively and their specific heats by cpg and cps. D and A are the dispersion coefficient and the effective heat conductivity of the bed, respectively. The gas velocity in the pores is indicated by ug. The reaction source term is indicated with R, the enthalpy of reaction with AH, and the mass based stoichiometric coefficient with u. In Ref. [12] an asymptotic solution is found for high activation energies. Since this approximation is not always valid we solved the equations numerically without further approximations. Tables 8.1 and 8.2 give details of the model. 

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

The symbols used are the same as those used in the previous relation. The term /)eq//)0 has been added in eq. (5.454) to account for the asymptotic approach observed in the typical dispersion versus time curves (Forzatti and Lietti, 1999). 

Fig. 9. Reduced Cole-Cole plot for the asymptotic linear array. Dispersion parameters are a = 0.10, / = 0.54...

The ratio, L/D, of length to diameter of a packed tube or vessel has been found to affect the coefficient of heat transfer. This is a dispersion phenomenon in which the Peclet number, uL/Ddisp, is involved, where D Sp is the dispersion coefficient. Some 5000 data points were examined by Schliinder (1978) from this point of view although the effect of L/D is quite pronounced, no dear pattern was deduced. Industrial reactors have LID above 50 or so Eqs. (6) and (7) of Table 17.18 are asymptotic values of the heat transfer coefficient for such situations. They are plotted in Figure 17.36(b). 

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