The Conduction Layer Model

FIGURE 4.26 The conduction layer model (a) conduction layer growth on plates and wall for a cavity without interior solids (6) similar growth for a cavity with an interior solid (c) conduction layer model applied to a horizontal cavity having 6 = 0 and D L. 

When 0 = 180°, the hot, light fluid lies above the cold, heavy fluid, so the stationary fluid layer (in which there is no fluid motion) is inherently stable, and Nu = 1 for all Ra. (In terms of the conduction layer model, for 0 = 180° both the conduction layers are infinite, so the conduction layers always overlap, and Nu = 1.)... 

Equation 4.78 with kx and k2 having values appropriate to water at moderate temperatures (Pr = 6) is plotted in Fig. 4.27, together with relevant data for water. Figure 4.28 shows a plot of Eq. 4.78 for various values of Pr, covering only those ranges in Ra at which the equation has been tested. Also plotted are the predictions of the conduction layer model given by Eq. 4.77. This model is seen to be correct only in the limit of small Ra (Ra < 1708) and large Ra... 

FIGURE 4.28 Plot of Eq. 4.78 for various values of Prandtl numbers, describing the heat transfer across a horizontal cavity with DIL a 10 and heating from below also shown is the heat transfer predicted by the conduction layer model (Eq. 4.77). 

Figure 4.31 shows a plot of Eq. 4.89 for a circular cylinder cavity with perfectly conducting walls and various values of D/L. As is clear from the graph, the Nusselt number rises very steeply with Ra after initiation of convection, and very rapidly approaches the value of Nu for the horizontally extensive cavity. This behavior is consistent with the conduction layer model at high Ra, the conduction layers on the walls at the sides are so thin that they have no effect on the heat transfer at sufficiently low Ra, they are so thick that they overlap (even though those on the horizontal plates do not), so that their presence governs the condition for a stationary fluid. 

Figure 4.366 shows conduction layers applied to each cylinder. Both conduction layer thicknesses are shown as being much less than the spacing L so that they do not touch—this situation will always occur if Ra is large enough to make A, + A-,model predicts that, provided the conduction layers do not touch, Nu will be independent of E. In fact, this is what is... 

Cylinders With Horizontal Axes. The conduction layer model has been shown to accurately predict the heat transfer for horizontal cylinders [164, 223], but because of the need to iteratively solve for the central region temperature it does not yield an explicit expression for Nu. However, by making additional approximations Raithby and Hollands [223] were able to derive an explicit relation for the heat transfer when the (assumed laminar) conduction layers do not overlap ... 

Other 3D Enclosures With Interior Solids. Warrington and Powe [278] showed that so far as the heat transfer is concerned, cubes and stubby cylinders behave similarly to equivalent spheres of the same volume. This appears to be the case for both the inner and outer body shape. So Eqs, 4.121,4.124, and 4.128 appear to be applicable to other inner and outer body shapes as well, it being understood that D0 = (6V0/7t)l/3 and D, = (6 Vz/Jt)1 3, where Va and U, are the inner and outer body volumes, respectively. Sparrow and Charmichi [258], using stubby cylinders for the inner and outer body shapes, confirmed the conduction layer model prediction that the heat transfer is independent of eccentricity E when Ra (based on inner cylinder diameter) is greater than about 1500. 

Furthermore, we will take all other properties as constant and independent of temperature. Due to the high temperatures expected, these assumptions will not lead to accurate quantitative results unless we ultimately make some adjustments later. However, the solution to this stagnant layer with only pure conduction diffusion will display the correct features of a diffusion flame. Aspects of the solution can be taken as a guide and to give insight into the dynamics and interaction of fluid transport and combustion, even in complex turbulent unsteady flows. Incidentally, the conservation of momentum is implicitly used in the stagnant layer model since ... 

It is worth comparing these locally obtained values with the effective conductivity creff of the same sample measured in a conventional setup. A measurement with macrosopic electrodes yields one semicircle in the complex impedance plane and an effective conductivity of 42 10 9 ft 1 cm-1. According to the brick layer model for... 

A remarkable difference in the Pt spectra of oxide- and carbon-supported platinum is especially clear for the 2.5-nm sample the fuel cell material shows much less intensity at the bulk resonance position (1.138 G/kHz). A similar difference is shown by the spectrum for the 2.0-nm sample. In terms of the NMR layer model, this comparison means that the healing length is larger in the carbon-supported material. It is not clear whether this result is related to the conducting nature of the carrier or to the presence of the electrolyte comparisons between wet and dry samples are needed. 

The microscopic characteristics of a real adsorbate layer have been considered [33, 34] by separating the x, y and z components of the electric field at the interface (Fig.4), and applying the Lorentz oscillator model to microscopically represent the adsorbate in the three-layer model. For the case of external reflection at a vacuum/semi-conducting, where Ss is real (no absorption) and isotropic, we can write ... 

Concerning the two-layer model, the thickness and properties of each layer depend on the nature of the electrolyte and the anodisation conditions. For the application, a permanent control of thickness and electrical properties is necessary. In the present chapter, electrochemical impedance spectroscopy (EIS) was used to study the film properties. The EIS measurements can provide accurate information on the dielectric properties and the thickness of the barrier layer [13-14]. The porous layer cannot be studied by impedance measurements because of the high conductivity of the electrolyte in the pores [15]. The total thickness of the aluminium oxide films was determined by scanning electron microscopy. The thickness of the single layers was then calculated. The information on the film properties was confirmed by electrical characterisation performed on metal/insulator/metal (MIM) structures. 

Fig. 4. The dead-layer model for analyzing changes in PL intensity for two states, a) and b). As indicated in the figure, state a) corresponds to the PL intensity in a N2 ambient and state b) corresponds to the PL intensity in the presence of a gaseous amine. The symbols CB and VB represent the solid s conduction and valence band edges, respectively. For each state, the PL intensity is proportional to the amount of incident light (intensity Iq absorptivity a ) absorbed beyond the nonemissive layer whose thickness is D. The ratio of the two PL Intensities leads to eq. 1.

The concept of surrounding the surfaces by a layer of stationary fluid, called the conduction layer, is useful for the present enclosure problem as well as for the external and open cavity problems. Unless the conduction layer thickness is greater than the cavity dimensions, a central region is produced (Fig. 4.26a and b), which experience has shown takes up a nearly uniform temperature this region can therefore be modeled as isothermal. Once the thicknesses of the conduction layers have been specified, finding the heat transfer and the temperature Tcr of this central region is a relatively straightforward heat conduction problem. 

As yet, the conduction layer approach has only been tested quantitatively on those problems in which the influence of the side walls is unimportant. Even for these problems the model has met with only mixed success in closely predicting the heat transfer. However, it does predict the correct trends and the correct asymptotes, it is useful in correlating experimental data, and does afford a simple physical understanding to problems that, when viewed from a different perspective, often appear very complex. The practitioner may find it useful for problems for which there is insufficient information from other sources. 

The conductivity of a fictitious borehole solution is related with the parameters of the three-layered model as ... 

The latter expression implies that the largest signals are obtained in the case of the depletion layer model. The effect of the surface chemistry on the resistance changes becomes weaker where the conduction moves into the accumulation layer. 

It is interesting to compare the series model (Figure 4.1.2a) with the corresponding parallel model (Figure 4.1.2b), in which the layers are stacked across the electrodes. For the parallel layer model, the complex conductivity follows a linear mixing rule... 

The behavior of this circuit differs qualitatively from the previous one, because conductances gi, g2 and capacitances Ci, C2 are in parallel. Thus the circuit is equivalent to that of Figure 4.1.3b, which shows only one relaxation. For the microstructure of Figure 4.12b the individual relaxations cannot be resolved by any method, graphical, CNLS, or other. Although at first glance this model would appear to be as plausible as the series layer model, it fails to describe the behavior of grain boundaries in ceramics. 

A salient point in this work is that, by specifying a variation in conductivity with distance from the grain boundary core, conductivity profiles in both orientations can be integrated and a phenomenological description obtained that is consistent with the brick-layer model. On one hand, this is encouraging, because it shows... 

Many additional studies have been conducted with the boundary layer model by taking into account the variation of physical properties with composition (or temperature) and by relaxing the assumption that Vy = 0 at y = 0 when mass transfer is occurring. Under conditions of high mass transfer rates one finds that mass transfer to the plate decreases the thickness of the mass transfer boundary layer while a mass flux away from the wall increases the boundary layer thickness The analogous problem of uniform flux at the plate has also been solved. Skelland describes a number of additional mass transfer boundary layer problems such as developing hydrodynamic and mass transfin- profiles in the entrance region of parallel flat plates and round tubes. 

Results showed that the textile-based sensor behavior is close to the expected one and already at this stage the stmcture might be used to indicate the presence of a pressure. The deviation was mainly due to the lateral movement of the conductive layers. As the distance decreases the rigidity of the spacer stmcture produces a shear force, which makes the conductive layers move laterally so that the overlapping area is no longer constant. Future work should aim to resolve the unwanted lateral motion of the conductive layers as well as to make a precise model of the partially filled capacitor in order to predict the effective permittivity of the dielectric. If fliese issues are taken care of the stmcture will also be suitable for making absolute measurements of either distance or pressure. 

What happens when the cychc cluster is increased Depending on its shape and size different sets of fe-points are reproduced, but in the EHT matrix elements the number of interactions included (interaction radius) increases as the periodically reproduced atomic sites distance is defined by the translation vector of a cyclic cluster as a whole. It is important to reproduce in the cyclic-cluster calculations the states defining the bandgap. As the overlap matrix elements decay exponentially with the interatomic distance one obtains the convergence of results with increasing cyclic cluster. Of course, this convergence is slower the more diffuse are the AOs in the basis. Prom band-structure calculations it is known that for BNhex in the one-layer model the top of the valence band and the bottom of the conduction band are at the point P of the BZ reproduced in the cyclic cluster considered. 

The triple layer model does not significantly improve the description of acid-base properties of minerals when compared to a Stem model. The Stem model has fewer adjustable parameters (acid-base properties). For electrokinetic data, a detailed analysis should be conducted. 

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