Boundary layers overlap layer

Figure 36. Defect concentration and conductance effects for three different thicknesses Li L2 Lj. The mesoscale effect on defect concentration (l.h.s.) discussed in the text, when L < 4J, is also mirrored in the dependence of the conductance on thickness (r.h.s.). If the boundary layers overlap , the interfacial effect previously hidden in the intercept is now resolved. It is presupposed that surface concentration and Debye length do not depend on L. (Both can be violated, c , at sufficiently small L because of interaction effects and exhaustibility of bulk concentrations.)36 94 (Reprinted from J. Maier, Defect chemistry and ion transport in nanostructured materials. Part II. Aspects of nanoionics. Solid State Ionics, 157, 327-334. Copyright 2003 with permission from Elsevier.)...

In addition to the particle size and the particle number fluctuation in the measurement volume, boundary layer, overlapping, and other effects, as mentioned earlier, have an impact on the fluctuating transmission signals as well. Therefore, the determination of the particle sizes from the transmission signals requires knowledge of quantitative correlation between these factors and the transmission signal. 

From the results obtained in [344] it follows that the composites with PMF are more likely to develop a secondary network and a considerable deformation is needed to break it. As the authors of [344] note, at low frequencies the Gr(to) relationship for Specimens Nos. 4 and 5 (Table 16) has the form typical of a viscoelastic body. This kind of behavior has been attributed to the formation of the spatial skeleton of filler owing to the overlap of the thin boundary layers of polymer. The authors also note that only plastic deformations occurred in shear flow. 

In nature, it is likely to encounter convective dissolution of many crystals. In this case, if their boundary layers do not overlap and the flow velocity fields do not overlap, each crystal may be viewed as dissolving individually without interacting with other crystals. However, if their boundary layers overlap or their flow velocity fields overlap, the above treatment would not be accurate. Furthermore, when there are many crystals, the whole parcel of crystal-containing fluid may sink or rise (large-scale convection), leading to completely different fluid dynamics. Such problems remain to be solved. 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

When the flow is constrained to a channel bounded on both faces, then at a sufficient distance from the entry point, xe, the two hydrodynamic boundary layers associated with each wall overlap and the fluid velocity varies parabolically with position across the channel. 

Kato and Wen (5) found, for the case of packed beds,that there was a dependency of the Sherwood and Nusselt numbers with the ratio dp/L. They proposed that the fall of the heat and mass transfer coefficients at low Reynolds numbers is due to an overlapping of the boundary layers surrounding the particles which produces a reduction of the available effective area for transfer of mass and heat. Nelson and Galloway W proposed a new model in terms of the Frossling number, to explain the fall of the heat and mass transfer coefficients in the zone of low Reynolds numbers. 

Finally, Kato and WenO) have proposed that the drastic fall observed for heat and mass transfer coefficients in the zone of low Reynolds numbers is due to an overlapping of the boundary layers surrounding the solid. 

This overlapping will in fact reduce the available area for heat and mass transfer. During the present work, some boundary layer thicknesses were estimated for the experimental conditions of this work. As a result, the boundary layers only overlap for Reynolds numbers below 0.826. For the case of Reynolds numbers of 1.74 and 3.05 using the particle diameter of 0.035 cm., the boundary layers do not overlap.Table III shows some of the values obtained.Clearly, this effect cannot explain completely the low heat and mass transfer coefficients at low Reynolds numbers. 

A turbulent flow is characterised by velocity fluctuations which overlap the main flow. The disturbed flow is basically three-dimensional and unsteady. At sufficiently high Reynolds numbers, the boundary layer is also no longer laminar but turbulent, such that the velocities, temperatures and concentrations all vary locally at a fixed position, as Fig. 3.14 shows for a velocity component wt. At every position it can be formed as the sum of a time-mean value (T here is the integration time)... 

Intuitively one could imagine that the boundary layer as a whole can be characterized in terms of the boundary layer thickness and related dimensionless groups. However, experimental data reveals that the laminar shear is dominant near the wall (i.e., in the inner wall layer), and turbulent shear dominates in the outer wall layer. There is also an intermediate region, called the overlap wall region, where both laminar and turbulent shear are important. 

The constant A cannot be determined from the boundary condition at the wall but must be obtained from the matching requirement that (4-27) reduce to the form of the core solution (4-17) in the region of overlap between the boundary layer and the interior region. Now, any arbitrarily large, but finite, value of Y will fall within the boundary-layer domain on the other hand, the corresponding value of y can be made arbitrarily small in the asymptotic limit R0J - oo. Thus the condition of matching is often expressed in the form... 

In considering the matching condition (5-211) it is necessary to either express the core solution in terms of the boundary-layer variables or else express the boundary-layer solution in terms of the core variables. Because both approximations to the solution are valid in the region of overlap where the matching condition applies, either choice is acceptable. However, for present purposes, it is more convenient to introduce the boundary-layer variables into the core solution. 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

Perhaps the most realistic model is the random pore model of Bhatia and Perlmutter (1980 1981a, b 1983), which assumes that the actual reaction surface of the reacting solid B is the result of the random overlapping of a set of cylindrical pores. Surface development as envisaged in this model is illustrated in Figure 11.12. The first step in model development is therefore the calculation of the actual reaction surface, based on which the conversion-time relationship is established in terms of the intrinsic structural properties of the solid. In the absence of intraparticle and boundary layer resistances, the following relationship is obtained ... 

The comer region, where boundary layers from adjacent walls overlap, must be acceptably small. This requirement establishes an upper bound on... 

Immediately outside the comer region where mass transfer boundary layers from adjacent walls do not overlap or influence each other, the molar density of reactant A at the catalytic surface must be acceptably close to Ca. inlet- This requirement establishes an upper bound on Zsiart-... 

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