Parallel plate analogy

Fordham, E. J., Bittleston, S. H., and Ahmadi Tehran , M., Viscoplastic flow in centered annuli, pipes and slots, Ind. Eng. Chem. Res. iO(3) 517- 524 (1991). Worth, R. A., Accuracy of the parallel-plate analogy for representation of viscous flow between coaxial cylinders, J. Appl. Polym. Sci. 24 319-328 (1979). McKelvey, J. M., Polymer Processing, Wiley, New York, 1962. 

Worth R A (1979), Accuracy of parallel - plate analogy for representation of viscous flow between cyUnders ,/oMwa/ of Applied Polymer Sciences, 24,319-328. 

A number of more or less equivalent derivations of the electrocapillary Eq. V-49 have been given, and these have been reviewed by Grahame [113]. Lippmann based his derivation on the supposition that the interface was analogous to a parallel-plate condenser, so that the reversible work dG, associated with changes in area and in charge, was given by... 

Figure 18, taken from Ref. 77, describes several models proposed for the Li electrodes in solutions, their equivalent circuit analogs, and the expected impedance spectra (presented as Nyquist plots). Assuming parallel plate geometry for the solid electrolyte interface, as well as knowledge of the surface species involved from spectroscopy (and thus their dielectric constant, which is around 5 for many surface species formed on Li, including R0C02Li, Li2C03, LiF, ROLi, etc. [186]), it is possible to estimate the surface film s thickness from the electrode s capacitance (calculated from the model fitted to the spectra) ...

Lippmann s proof was essentially as follows. The interface was supposed to be analogous to a parallel plate condenser with a difference of potential A between the plates the work required to increase the charge on the plates by dQ is dQ.A. Work is also required to increase the area, owing to the surface tension hence the total reversible work is... 

This, by analogy to charging a parallel plate condenser, is given by ... 

When only the inert electrolyte is present in the polarographic cell a residual current will still flow. This current, which is non-faradaic, is attributable to the formation of an electrical double layer in the solution adjacent to the electrode surface (Fig. 3). At all applied potentials, a current flows to develop this double layer, and the process may be considered analogous to the charging of a parallel plate capacitor. Therefore, the charging current is a capacitance current and varies during the drop lifetime, i.e., with the size of the mercury drop. When the drop surface area is increasing rapidly from the start of the drop lifetime, the capacitance current is a maximum, falling to a minimum near the end of the drop lifetime when the drop size is at a... 

Analogous to circular ducts, the fully developed turbulent Nusselt numbers for uniform wall temperature and uniform wall heat flux boundary conditions in parallel plate ducts are nearly identical for Pr > 0.7 and Re > 105. This is also true for the Nusselt number of turbulent thermally developing flow in a parallel plate duct [147]. 

In the oldest theory of the electrode-electrolyte interface (compact layer theory), the double layer was considered analogous to a parallel plate condenser with a plane of charges on the metal side and a second plane of opposite charges on the solution side (4,5). According to this theory, the capacity of the double layer should be independent of potential which is not observed experimentally. 

We note from these results that the more the molecules are overlapped (the smaller y and z), the higher the binding energy of the pair. We came to a similar conclusion from an analogy with the energy of a charged parallel-plate capacitor. 

Because we have the model of parallel charged layers of different potential we have a direct analogy with a parallel plate condenser and it is therefore perhaps no surprise that a capacitance exists across the double layer. 

There is always a characteristic potential for any electrode/solution interface where there is no net charge on the electrode. This is the potential of zero charge. ( pzc) which for Hg/HaO has a value of about —0.5 V (SCE). At any other potential, charge develops on the electrode and there is an analogy with a parallel plate condenser across the electrical double layer formed at the electrode. A capacitance is created, Cdl, and a capacitive current, Iqp is given by ... 

Mass and heat transfer with laminar flow in parallel-plate and cylindrical geometries is a well-known theoretical problem which has been treated by a large number of Investigators Cl-9). Perhaps the first papers in this area were those by L vtque Cl) and Graetz C2) who studied the heat transfer problem. In the absence of natural convection and for dilute systems, the heat and mass transfer problems are analogous. 

The complex permittivity is given by s (o)= C (o) /Co, analogous to equation (1) for the static case, where Co= A.Bold is the vacuum capacitance of a parallel-plate capacitor ( =area of the plate, So the vacuum permittivity and d the plate separation (A/d) must have dimensions in cm) and C is the complex capacitance of the same capacitor filled with the material under study. Under the influence of a sinusoidal electric field, the complex permittivity relates to the impedance through ... 

In a manner analogous to TMA and DMA, a specimen can be subjected to a constant or oscillating electric field rather than a mechanical stress during measurements. Dipoles in the material will attempt to orient with the electric field, while ions, often present as impurities, will move toward the electrode of opposite polarity. The resulting current flow is similar in nature to the deformation brought about by mechanical tests and represents a measure of the freedom of charge carriers to respond to the applied field. The specimen is usually presented as a thin film between two metal electrodes so as to form a parallel plate capacitor. Two types of test can be performed. 

In additional to cylindrical capillary tubes, another type of capillary with a simple cross-sectional shape is the slit capillary, i.e., a channel formed between two parallel plates. For an electrolyte solution flowing through a slit capillary, it can be shown that, in analogy to Eq. 13,... 

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