Dispersion unity

The influence of the vi.scosity ratio 8 on the flow behavior in a capillary was discussed by Rumscheidt and Mason [lOj. They pointed out that when the viscosity ratio is small, the dispersed droplets are drawn out to great lengths but do not burst, and when the viscosity ratio is of the order of unity, the extended droplets break up into smaller droplets. At very high viscosity ratios, the droplets undergo only very limited deformations. This mechanism can explain our observations and supports our theoretical analysis assumptions, summarized previously as points 2, 3, and 4. 

The BiDER/DMSO + LiC104 interface has been studied by impedance and a very well-developed diffuse layer minimum has been observed, with ,niB independent of cyao/75 The capacitance dispersion was no greater than 2 to 3% in the region -1.5 E< -0.3 V (SCE in H20). Linear Parsons-Zobel plots with/pz very close to unity were obtained, Q was independent of cya0j. 

The ionization process can be described as a one-particle event, when only one term dominates in summation (3), yielding a pole strength close unity. In this case, 1 - Fu gives an estimate of the fraction of photoemission intensity dispersed in many-body effects. On the contrary, small pole strengths are indicative (18-20) of a breakdown of the one-particle picture of ionization. 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

In ecent years the utility of extended X-ray absorption fine structure UXAFS) as a probe for the study of catalysts has been clearly demonstrated (1-17). Measurements of EXAFS are particularly valuable for very highly dispersed catalysts. Supported metal systems, in which small metal clusters or crystallites are commonly dispersed on a refractory oxide such as alumina or silica, are good examples of such catalysts. The ratio of surface atoms to total atoms in the metal clusters is generally high and may even approach unity in some cases. 

Taking into consideration the preparation procedures, the Mo content of 2.1Mo/SC, and the Co/Mo atomic ratio of ca. unity at the maximum HDS activity, highly dispersed Co-Mo binary sulfide clusters, possibly COjMOjSx, in the supercage of the NaY zeolite are suggested for catalytically active species. The HDS activity of the CoSx-MoSx/NaY was not changed even after a 20-h treatment at 673 K in a stream of HjS/H (Fig.3), demonstrating a high thermal stability of the active species. 

Dispersion greater than unity calculated assumed 100% dispersion for calculation of turnover frequency. 

We can characterize the mixed systems most easily in terms of the longitudinal dispersion model or in terms of the cascade of stirred tank reactors model. The maximum amount of mixing occurs for the cases where Q)L = oo or n = 1. In general, for reaction orders greater than unity, these models place a lower limit on the conversion that will be obtained in an actual reactor. The applications of these models are treated in Sections 11.2.2 and 11.2.3. 

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

This result compares to a value of 0.666 predicted on the basis of the segregated flow model. Excellent agreement should be obtained for the first-order case if the dispersion parameter gives a good fit of the experimental F(t) curve. Agreement for reaction orders other than unity will not be nearly as good. 

FIGURE 10.5 Estimating xc from saturation transfer. In the dispersion spectrum of a spin label (TEMPO) the ratio of I /I runs from approximately unity in the rigid limit, when the rotation correlation time xc > ICE3, to approximately zero for xc 10 7. 

Hydrogen uptake of reduced catalysts (X) was measured by volumetric method with an AUTOSORB-l-C analyzer (Quantachrome Instruments). Hydrogen adsorption was carried out at 373 K after in situ H2 reduction at 773 K for 6 h in the adsorption cell. The dispersion and particle size of metallic Co were calculated by the following equations, assuming that the stoichiometry for hydrogen adsorption on the metallic site is unity ... 

In this expression A and Q are distance dispersion resulting from electron-vibrational coupling, and frequency tensor (assumed identical in reactant and product states), respectively (work of formation of precursor and successor states is omitted). If we assume that the frequency tensor is diagonal, then we have simply a sum of independent terms for all inner and outer contributing modes. At sufficiently high temperature, the hyperbolic tangents become unity and we obtain the usual (in this approximation) high-temperature expression ... 

The experimental results for dispersion coefficients in gases show that they can be satisfactorily represented as Peclet number expressed as a function of particle Reynolds number, and that similar correlations are obtained, irrespective of the gases used. However, it might be expected that the Schmidt number would be an important variable, but it is not possible to test this hypothesis with gases as the values of Schmidt number are all approximately the same and equal to about unity. 

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