The primary distribution

When all kinds of overpotentials can be neglected, we say that the current distribution is a primary distribution. 

The solution adjacent to the electrodes is an equipo-tential surface. The boundary conditions are constant and the Laplace equation has simple, classic mixed boundary conditions (Neumann and Dirichlet, see section 2.3). 

In some cases, analytic solutions exist. Powerful techniques to find them are conformal mapping and separation of the variables. These solutions are also very important to check numerical methods. 

The conductivity being constant throughout the solution (mostly a realistic assumption), the primary distribution depends only on geometrical factors. 

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When a battery produces current, the sites of current production are not uniformly distributed on the electrodes (45). The nonuniform current distribution lowers the expected performance from a battery system, and causes excessive heat evolution and low utilization of active materials. Two types of current distribution, primary and secondary, can be distinguished. The primary distribution is related to the current production based on the geometric surface area of the battery constmction. Secondary current distribution is related to current production sites inside the porous electrode itself. Most practical battery constmctions have nonuniform current distribution across the surface of the electrodes. This primary current distribution is governed by geometric factors such as height (or length) of the electrodes, the distance between the electrodes, the resistance of the anode and cathode stmctures by the resistance of the electrolyte and by the polarization resistance or hinderance of the electrode reaction processes. 

Let us consider the primary distribution network of Example 23.2 as shown in Figure 24.25(a) leeding an LT load ol 29.4 MW at 0.98 p.1. through a 33/0.4 kV transformer. The following line parameters have been considered ... 

For ease of calculation, let us consider the impedance of the transformer as its leakage reactance, ignoring resistance and draw an equivalent circuit diagram as in Figures 24.25 (b) and (c) Assuming the length of the primary distribution line to be 15 km, the total line parameters will become... 

The primary distribution of protection current density (see Section 2.2.5) for a given geometry and driving voltage, can be seen as follows ... 

Different locations of parent elements. and Th are generally located in minor phases within host rocks. Due to different U/Th ratios in these phases, recoil from the two chains may be affected by different surrounding matrix characteristics or mineral sizes. Not only might the primary distribution of U and Th be different, but earlier weathering or alteration may also have redistributed U and Th. This is discussed further below. 

Figure 4.17 General phenonenaloglcal retention model for a solute that participates in a secondary chemical equilibrium in liquid chromatography. A - solute, X - equilibrant, AX analyte-equilibrant coeplex, Kjq - secondary chemical equilibrium constant, and and are the primary distribution constants for A and AX, respectively, between the mobile and stationary phases.

Equation (7.20) is general, although the expression for Te itself depends on the primary distribution, the gel fraction, and the relative rates of random cross-linking and chain scission. 

Determination of the sol fraction together with the primary distribution thus provides a method for calculation of the crosslinking density q. The gel fraction increases usually rather sharply as soon as the gel point is passed. For example, in a network composed of monodisperse primary chains with 2.5 crosslinks per primary molecule, 90% of the polymer is included in the gel fraction. 

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. 

There is a tendency for Vss and Vc to correlate one with another, which implies that the volume of distribution is predominantly determined by distribution in the vascular and interstitial space as well as unspecific protein binding in these distribution spaces. The distribution rate is inversely correlated with molecular size and is similar to that of inert polysaccharides, suggesting that passive diffusion through aqueous channels is the primary distribution mechanism [57]. 

Goldschmidt s ideas on the primary distribution of the elements in the Earth have not been seriously challenged (see, however, Bums and Fyfe, 1966a). From studies of minerals in meteorites and phases from blast furnaces, Golschmidt classified the elements as siderophilic if they are inert (relative to iron) and enter the metallic phase, chalcophilic if they are concentrated in sulphides, lithophilic if they are concentrated in silicates and atmophilic if they are gaseous and are present in the atmosphere. Those elements enriched in organisms were also classed as biophilic. 

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). 

Under the assumption that the concentrations are uniform within the electrolyte, potential is governed by Laplace s equation (5.52). Under these conditions, the passage of current through the system is controlled by the Ohmic resistance to passage of current through the electrolyte and by the resistance associated with reaction kinetics. The primary distribution applies in the limit that the Ohmic resistance dominates and kinetic limitations can be neglected. The solution adjacent to the electrode can then be considered to be an equipotential surface with value o- The boundary condition for insulating surfaces is that the current density is equal to zero. 

Some electrochemical systems can be described as blocking electrodes for which no Faradaic reaction can occur. At steady state, the current density for such a system must be equal to zero. The transient response of a blocking electrode is due to the charging of the double layer. At short times or high frequency, the interfacial impedance tends toward zero, and the solution adjacent to Ihe electrode can then be considered to be an equipotential surface. The short-time or high-frequency current distribution, therefore, follows the primary distribution described in the... 

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... 

As has been previously noted, the high volatility of Hg over a wide range of geological environments has led many authors to speculate on the possibility of broad primary dispersion halos surrounding sulphide deposits. Until about thirty years ago the only published case-history studies of the primary distribution of Hg around ore deposits were to be found in the Soviet literature. The most often-quoted example is that of Ozerova (1962, 1971), who studied the dispersion of Hg around Hg deposits as well as those containing Hg as a trace component. Of these the most pertinent to our considerations are the deposits of South Fergana, Uzbekistan. 

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). 

When the primary distribution does not illustrate the current or electric potential distribution well, an additional resistance, that is, the charge transfer electrode resistance, has to be considered. In such cases, we need to account for the electrode kinetics, and the secondary current and potential distributions emerge from the models. For industrial purposes the porous or tortuous electrocatalyst has to be considered as a dynamic system. This means that its porosity shape and density besides the surface roughness and the real geometric area changes all the time. This point makes us think that it... 

For the surface growth and the subsequent variation in the boundary layer, the physical change of the electrocatalyst can be easily evaluated with the help of the current distributions. In general, the primary distribution can be used as Ohm s law ... 

The most widely used parenteral administration avenues are intravenous (iv), intramuscular (im), and subcutaneous (sc). In addition, there are several minor applications (e.g. intraarterial). Application of a protein drug by the different main parenteral administration routes may have profound effects on the pharmacological performances. When the drug is administered iv, it is immediately available for action in the circulation, while drugs administered im or sc need more time to reach the blood (depot effect), and consequently the pharmacokinetic (PK) profiles could be different. Besides the PK, the route of administration may have influence on the primary distribution of the drug. For example, when administered sc, smaller and hydrophiUic proteins tend to enter the venous system, while larger and/or more hydrophobic proteins tend to... 

Both equations describe what we have been calling secondary equilibrium effects to the primary distribution equilibria, which, in this case, is the distribution of neutral SO2 molecules between the aqueous phase and the HS described as follows ... 

In spite of its limitations for complex systems, the Wagner number gives a good qualitative idea of the current distribution in an electrochemical cell. Indeed, when the Wagner number is small (W << 1), the influence of overpotentials can be neglected and the current distribution is nearly the primary distribution. For larger values (- 0.01 < W < 100), the... 

Fig. 1.22 The current distribution around a cathode placed in front of an anode. The primary distribution presents a singularity at the free end. This singularity disappears with increasing Wagner number (1 Wc=0.25 2 Wc=1.03, 3 Wc=5.88, Wa=0).

The unknown constants are associated with nodes. One can add to these trial functions others which are part of a known series approximation to the exact solution. This is particular useful to describe singularities accurately [ 37]. However, due to the nonlinear boundary conditions we are dealing with, adding these terms is not advised. Indeed, although the primary distribution would be more accurate, the secondary and tertiary would not since there is no singularity any more. 

Preliminarily we remark that the number of iterations necessary to converge, depends highly on the first guess. Although a better guess could be made based on approximated Wagner numbers obtained from the primary distribution, in all the subsequent calculations the first guess was the primary distribution itself. 

To check the primary distribution as well as to get a feeling for the influence of overpotentials, the geometry of fig. 3.21 is chosen. This geometry presents a zero, a uniform and an infinite current density. An analytic expression for the primary distribution was found by means of a Schwarz-Chris-toffel transformation and is briefly outlined in appendix A. 1.2. 

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,). 

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