Parallel plate model

The relative pressure at which the hysteresis loop closes depends upon the nature of the adsorbate [11] being around 0.42 for nitrogen. Low pressure hysteresis is associated with inelastic distortion of the solid [12]. 

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Figure 1-13 displays the experimental dependence of the double-layer capacitance upon the applied potential and electrolyte concentration. As expected for the parallel-plate model, the capacitance is nearly independent of the potential or concentration over several hundreds of millivolts. Nevertheless, a sharp dip in the capacitance is observed (around —0.5 V i.e., the Ep/C) with dilute solutions, reflecting the contribution of the diffuse layer. Comparison of the double layer witii die parallel-plate capacitor is dius most appropriate at high electrolyte concentrations (i.e., when C CH). 

Fig. 6.62. The Helmholtz-Perrin parallel-plate model, (a) A layer of ions on the OHP constitutes the entire excess charge in the solution. (b) The electrical equivalent of such a double layer is a parallel-plate condenser, (c) The corresponding variation of potential is a linear one. (Note The solvation sheaths of the ions and electrode are not shown in this diagram nor in subsequent ones.)...

Going a step further, what does the parallel-plate model of the double layer have to say regarding the capacity of the interface Rearranging Eq. (6.119) in the form of the definition of differential capacity [Eq. (6.97)],... 

Thus, if E and d are taken as constants, the parallel-plate model predicts a constant capacity, i.e., one that does not change with potential. So it appears that the Helmholtz-Perrin model would be quite satisfactory for electrocapillaiy curves that are perfect parabolas (Fig. 6.56). 

After the initial jubilation that the diffuse-layer model has overcome the weakness of constant capacity with change of potential of the parallel-plate model, one has to... 

What happens when the concentration c0 of ions in solution is very large Equations (6.124) and (6.130) indicate that while CG increases with increasing c0, CH remains constant. Thus, as c0 increases, (1/CG) (1/CH), and for all practical purposes, C CH. That is, in sufficiently concentrated solutions, the capacity of the interface is effectively equal to the capacity of the Helmholtz region, Le., of the parallel-plate model. What this means is that most of the solution charge is squeezed onto the Helmholtz plane, or confined in a region vety near this plane. In other words, little charge is scattered diffusely into the solution in the Gouy-Chapman disarray. 

We have seen how the screw extruder pump is synthesized from a simple building block of two parallel plates in relative motion. We have also seen how the analysis of the screw extruder leads in first approximation back to the shallow channel parallel plate model. We carried out the analysis for isothermal flow of a Newtonian fluid, reaching a model (Eq. 6.3-27) that is satisfactory for gaining a deeper insight into the pressurization and flow mechanisms in the screw extruder, and also for first-order approximations of the pumping performance of screw extruders. 

For parallel plates, Co = 1, this equation reduces to Eq. 6.3-19 with pressure drop replacing pressure gradient. Hence, the terms containing Co can be viewed as correction factors of the parallel plate model for tapered geometries. 

A cross section perpendicular to the root of the screw and parallel to the flights is shown in Fig. 6.55, where the other screw across the slit is shown. Clearly, the two screws move at the same velocity in the same direction therefore, they will appear to each other as stationary. Thus, neglecting curvature and flight effects, the model simplifies to a three-parallel plate model, as shown in Fig. 6.56, with the lower and upper plates representing... 

Fig. 6.56 The three-parallel plate model. The upper and lower stationary plates represent the screws, and the slitted midplate is the barrel surface.

Partially Filled Screw Channel The accompanying figure below shows a partially filled screw channel. Using the single-screw Newtonian parallel plate model show that the free boundary profile is given by19... 

Flow in a thin rectangular channel (Figure 4.2), such as that used in field-flow fractionation, can be treated in a manner similar to that used for cylindrical capillary tubes. If the drag at the edges of the channel is neglected (infinite parallel plate model), then the force balance expression (corresponding to Eq. 4.5 for capillary tubes) becomes... 

If we lake median values of e = 7 and d = 0.52 nm, we find from the Helmholtz parallel-plate model C =11.9 uF/cm in reasonable agreement with experiment. A correct calculation will be much more complicated, taking into account, among other things, the possibility that the value of e at the outer Helmholtz plane may already be higher... 

Parallel-plate model (for adsorption isotherm), 332 Parallel-plate model (of the double layer), 188 Partially blocked electrode, 450 Partial surface coverage, 131 Passivation, 506... 

Figure 1. Ionic strength dependence of observed rate constants for ET from two reduced cytochromes (c555 and C551) to oxidized plastocyanin. Solid lines are theoretical fits to the data points using the parallel plate model (only the V term is included).

Figure 4. Ionic strength dependence of the observed rate constant for electron transfer from reduced Fd to oxidized FNR. The dashed line corresponds to a theoretical fit to the higher ionic strength data using the parallel-plate model. The solid line is a smooth curve drawn through the data points.

The double-layer capacitance is taken into account by assuming a simplified Helmholtz parallel plate model (1). On opening the circuit, the potential difference, V, across the double layer must be reduced by diminution of the charge on each plate. For a cathodic reaction, each electron being transferred from the metal to the solution side of the interface effects an elementary act of reaction and reduces the charge, q, on each plate. Consequently the rate of reduction of this charge is equal to the faradaic current, and Eq. (55) follows, y is assumed to differ from rj simply by the value of the reversible potential ... 

To calculate flow through single fractures, the parallel plate model is commonly used ... 

Innes, W.B. (1957). Use of a parallel plate model in calculation of pore size distribution. Anal. Chem., 29, 1069-73. 

Obtaining equivalent expressions for AP for non-Newtonian fluids is difficult and for most rheological models it is not possible to obtain analytical expressions. Good approximate solutions, however, can be obtained by replacing the narrow annulus geometry by a parallel plate model (108, 109), that is, annular flow is represented by slot flow. The... 

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