Secondary distribution

both ohmic and kinetics irreversibilities are considered however, mass transport limitations are assumed to be negligible. Since significant mass transport effects are present only when operating close to the limiting current, e.g., in very dilute solutions, the secondary distribution presents a valid approximation for most electrochemical systems. Here, the Laplace equation, (25), is solved subject to the following boundary conditions  

It is significantly more complicated to analytically solve the secondary distribution (than the primary distribution), since the presence of the kinetics overpotential renders the system nonlinear. 

Available analytical solutions include among many others the disk electrode, Wagner s solution of the current distribution at a corner, current distribution within a through hole and a blind via, and within a thin electrolyte layer. Most of the anal5dical solutions employ linearization about some average current density. Computer-implemented numerical solutions of the secondary distribution (example provided in Fig. 2) are readily obtained and do not require linearization. A few examples include modeling the 

The secondary distribution incorporates the effects of the ohmic resistance, which gives the primary distribution its nonuniform characteristics, in combination with the surface resistance associated with the limited reversibility of the electrode kinetics. The latter (with the exception of special cases, involving, e.g., the use of unique additives) leads to a uniform distribution. While it is difficult to derive accurate analytical solutions for the secondary distribution, we can characterize the degree of nonuniformity by evaluating the relative magnitude of the resistances associated with the surface and ohmic dissipative processes. 

Accordingly, the degree of uniformity of the secondary distribution can be characterized in terms of a dimensionless number, named after Carl Wagner, representing the ratio of the surface to the ohmic 

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Commercially pure (< 99.997%) helium is shipped directiy from helium-purification plants located near the natural-gas supply to bulk users and secondary distribution points throughout the world. Commercially pure argon is produced at many large air-separation plants and is transported to bulk users up to several hundred kilometers away by tmck, by railcar, and occasionally by dedicated gas pipeline (see Pipelines). Normally, only cmde grades of neon, krypton, and xenon are produced at air-separation plants. These are shipped to a central purification faciUty from which the pure materials, as well as smaller quantities and special grades of helium and argon, are then distributed. Radon is not distributed commercially. 

The element of p.f. mainly affects the secondary distribution system which serves industries, agriculture, public utilities and domestic loads. Most of them are highly inductive and result in lowering the system p.f. These loads are largely responsible for most of the distribution losses and voltage fluctuations at the consumer end. In developing countries it is estimated that useful power is lost mainly due to transmission and distribution losses. In India, for instance, it is estimated to result in a loss of about 18-20% of the total useful povver, most of which occurs at the secondary distribution attributable to low p.f.s. 

Mehdizadeh et al. exploited the separability of current distribution on different scales to model the macroscopic current distribution on patterns made up of lines or points distributed over a large workpeice [136], They solved the secondary distribution of the superficial current density sup using a boundary condition which captures the density of small features but not their geometry. The boundary condition is based on a smoothly varying parameter representing the Faradaically active fraction of surface area. 

We attribute this distribution to oligomeric MMA which possesses an initiator fragment, (CH3)2(CN)C. This moiety weighs 68 Da, hence we attribute this distribution to a AIBN initiated MMA oligomer which is vinyl-terminated (VIII). The relative ratio (" 30/1) of the primary distribution to the secondary distribution gives a measure of the CoCTC turnover number. 

The secondary current distribution is calculated by including the effects of the ohmic drop in the electrolyte and the effects of sluggish electrode kinetics. While the secondary distribution may be a more realistic approximation, its calculation is more difficult therefore, we need to assess the relative importance of electrode kinetics to determine whether we can neglect them in a simulation. 

In the secondary distribution of the elements in the Upper Mantle and Crust, Goldschmidt believed that during the formation of a crystalline phase the principal factor controlling the behaviour of an element is the size of its ions. From tables of ionic radii one could predict the crystal structures in which a given ion was most likely to occur. This concept has immense qualitative value in crystal chemistry, and has been remarkably successful in view of the assumptions... 

Figure 2.21. Scheme of the various distributions D, and D2 of the polaritons leading to the observed bulk fluorescence. The model of two main distributions accounts for the narrow lines, the satellite broad bands, and their relative intensities. The energy of the main fluorescence lines is given in reciprocal centimeters. The bold arrow represents the relaxation in the excitonic band to states above 0. The primary distribution of excitons ( >,) relaxes by the creation of acoustical phonons (wavy arrow) to the secondary distribution of polaritons (D2) below E0 as well as to other vibrations in the ground state as given by the spectral model85 or the dynamical model (second ref. 87). 

The secondary distribution applies when kinetic limitations cannot be neglected. The solution adjacent to the electrode Cem no longer be considered to be an equipo-tential surface. The condition at the electrode can be replaced by... 

The uncertainty associated with the interpretation of the impedance response can be reduced by using an electrode for which the current and potential distribution is uniform. There are two types of distributions that can be used to guide electrode design. As described in Section 5.6.1, the primary distribution accounts for the influence of Ohmic resistance and mass-transfer-limited distributions account for the role of convective diffusion. The secondary distributions account for the role of kinetic resistance which tends to reduce the nonuniformity seen for a primary distribution. Thus, if the primary distribution is uniform, the secondary... 

A characteristic of the primary distribution, in general, is that it is less uniform than the secondary distribution for a given electrode geometry and the electrochemical cell device. There is only one exception that arises from the concentric cylindrical electrode system depicted in Figure 13.2a, where both the primary and the secondary current distributions are uniform in the case of the forced convective hydrodynamics (rotating electrodes). 

In spite of considering simple geometries to derive the electric potential and the current potential that were analytically solved from the Laplace equation, the secondary distributions always requires a numerical integration because of the current-electric potential-dependent boundary conditions. For example, in the case of the electrochemical reactor shown in Figure 13.5, the numerical solution for the secondary distribution of the current is generally presented as a plot of jx/j vs. 

FIGURE 13.6 Current distribution on a plane electrode catalyst for primary distribution (dashed line), secondary distribution (dotted line), and tertiary distribution (continuous line). 

V/hen only activation overpotentials are important, the obtained current distribution is called the secondary distribution. The potential difference across the interface depends on the local current density. Therefore, the solution near the electrodes is no longer an equi-potential surface. Since higher current densities involve larger overpotentials (passivation excluded), the activation overpotentials will tend to make the current distribution more uniform. Also the total current will decrease. This can easily be understood by means of fig. 1.16. 

Figure 1.17 gives an example of a secondary distribution in comparison with a primary distribution. 

Further, primary current distributions are compared with analytic solutions and some other calculated examples are discussed. To check results of secondary distributions, a copper electrorefining cell was built and quantitative data were obtained. 

Fig. 3.5e Analytical integration method secondary distribution due to a small overpotential, the oscillation vanishes. The distribution agrees with that of figure 3.5c.

Secondary distribution Applied voltage (V) P (0cm) Overvoltage Butler-Volmer on cathode equation with Wagner number... 

Therefore, for what concerns the primary distribution, we may conclude that the precision obtained with the BEM is better than we could ever measure and that precision will be as good or even better for secondary distributions because of the smoothing effect of overvoltages (see also section 3.. 2.4,). 

In the table 3.5 the obtained results are compared. Unfortunately no secondary distributions were calculated by Y. Nishiki. 

The resulting primary and secondary distributions a-long the anode are given in fig. 3.26b. 

A better primary distribution could be obtained with a finer element distribution but the secondary distribution would not change much. 

Whereas calculated primary distributions can easily be evaluated with analytic solutions, this is not true for real non-linear secondary distributions and we decided to make experiments for that purpose. 

It was observed that primary distributions can be calculated with a precision of a few percent (on a computer with 6 significant numbers) and that secondary distributions are certainly as precise. 

For 0 < W <00, the same arguments exist and we may conclude that for secondary distributions and starting from a singularity, the electrode profile will grow in such a way that the angle between electrode and adjacent insulator becomes and stays right. In practice this is also confirmed by measurements (see fig. -i.26 to 4..29). 

Fig. 4.15b Anodic, levelling under secondary distribution. Steady state gap 0.41 mm.

It was derived that, starting from a singularity, for secondary distributions the incident angle between an electrode and adjacent insulator becomes right. This was simulated by the program and confirmed by measurements. 

As an example, Fig. 2.9 is a sketch of the various current distributions for a parallel-plate cell with electrodes of length L and of infinite width, and with fully developed laminar flow. The primary current distribution shows the current to be uniform over most of the electrode but with a considerable edge effect at x = 0 and jc = L the current goes to a very high value at these ends. The secondary distribution is similar but is even closer to the ideal while the limiting tertiary distribution shows that in these conditions the current density drops sharply along the electrode. 

The need for a cationization reagent in MALDI analysis of polymers can also create some complications in mass spectral interpretation [42, 56]. For example, the spectrum in Figure 8.3b shows a secondary distribution of lower intensity in addition to the principal distribution. This secondary distribution could be due to cation adduction with different ionic species and/or the presence of other polymeric species with different end-group structures. In this case, the secondary distribution has oligomer mass shifts of -1-22.4 Da from the nearest oligomer of lower mass in the principal distribution. This is consistent with the generation of salt cluster complexes, similar to what has been observed in the ESI of polystyrene [57]. For polyisoprene, it takes the form of [polyisoprene-i-Cu(copper retinoate)]L This amounts to an actual mass shift of -1-363.0 Da with respect to the principal distribution of [polyisoprene-tCujL This is consistent with the observed mass shift of 22.4 Da plus six repeat units of 64.2 Da. 

In general, interfacial impedance is partly capacitative as well as resistive in nature. At high frequencies, the capacitance short-circuits the interface, and the primary distribution is observed for the ac part of the current. As the frequency is lowered, the interface impedance increases, causing a changeover to the secondary distribution. Of necessity, this effect leads to a frequency dependence of the equivalent circuit parameters which describe the system. Of course, if the primary distribution is uniform, there will be no frequency dispersion arising from this source. 

Aaserud et al. reported an on-line coupling of SEC to Fourier transform mass spectrometry (FT-MS) using a modified commercial electron spray ionization (ESI) interface (Fig. 29) [169]. They analyzed a glycidyl methacrylate/butyl methacrylate copolymer with a broad molecular weight distribution, where fractionation and high resolving power were required for adequate characterization. The SEC/ESI/FT-MS also allowed for an unequivocal end-group determination and characterization of a secondary distribution due to the formation of cyclic reaction products. 

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