Rate ellipsoid

The ellipsoid n xl + n xl -h = 1, called the rate ellipsoid, has the property that the length of the ray parallel to E is r = v /Vo-The Fletcher indicatrix allows the determination of the polarizations and refractive indices of the waves which propagate in a crystal along the normal v. The intersection of the ellipsoid and the plane perpendicular to v which passes through the center is an ellipse. The vector D normal to v lies in this plane. 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

The anisotropy in g(ri, r2) may be determined by the use of akinematic argument. Consider a bulk of particle subjected to a mean shear flow. The radial distribution function, which is spherical in equilibrium, becomes distorted into an ellipsoidal distribution as a result of the presence of the mean shear. Hence, in order for g(n, r2) to exhibit an anisotropy, g (ri, r2) should depend not only on ap, r, and r2 but also on Tc, vi, and v2. For dimensional homogeneity, g can only be a function of ap, k U2i/Tcl/2, and U x/Tc. For a small deformation rate (or when the magnitude of U2i is small relative to Tc1/2), it is assumed that g(ri, r2) takes the form [Jenkins and Savage, 1983]... 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

Experimentally determined rate constants and X values for organic self-exchange reactions, together with estimates of X and r according to a spherical model and an ellipsoidal model... 

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