Current distribution calculation methods

Fig. 11. A comparison of calculated and experimentally measured current distributions for the pattern shown in Fig. 10. The six resist-defined features are numbered from left to right. For each feature, three bars are shown the left bar shows the experimental data obtained by Rosset et al. [44, 45], while the center and right bars correspond to current distributions calculated by two different methods. (Figure reprinted and caption paraphrased from West et al. [43] by permission of the publisher. The Electrochemical Society, In ).

In contrast to a direct injection of dc or ac currents in the sample to be tested, the induction of eddy currents by an external excitation coil generates a locally limited current distribution. Since no electrical connection to the sample is required, eddy current NDE is easier to use from a practical point of view, however, the choice of the optimum measurement parameters, like e.g. the excitation frequency, is more critical. Furthermore, the calculation of the current flow in the sample from the measured field distribution tends to be more difficult than in case of a direct current injection. A homogenous field distribution produced by e.g. direct current injection or a sheet inducer [1] allows one to estimate more easily the defect geometry. However, for the detection of technically relevant cracks, these methods do not seem to be easily applicable and sensitive enough, especially in the case of deep lying and small cracks. 

Although there have been many published reports of chemical shift calculations neglecting electron correlation, it is well established that electron correlation is important for carbenium ions, triple bonds, and other systems sensitive to correlation. Currently, Gauss MP2 method (distributed as part of ACES II) (85) is the most extensively validated methodology for carbenium ion chemical shift calculations. 

With this method, control equipment is simple and inexpensive and it is straightforward to calculate the charge delivered. However, if high currents are used, problems are encountered with batteries which have porous electrode structures because of non-uniform current distribution, and severe gassing may occur towards the end of the charge. At low current,... 

In the case of solid electrolytes, such a calibration is usually impossible. The configuration of measuring cells should be selected to provide uniform current distribution or to enable use of a definite solution of differential Ohm s law for the conductivity calculations [ii-iv]. The conductivity values are typically verified comparing the data on samples with different geometry and/or electrode arrangement, or using alternative measurement methods. 

The calculation methods for pore distribution in the microporous domain are still the subject of numerous disputes with various opposing schools of thought , particularly with regard to the nature of the adsorbed phase in micropores. In fact, the adsorbate-adsorbent interactions in these types of solids are such that the adsorbate no longer has the properties of the liquid phase, particularly in terms of density, rendering the capillary condensation theory and Kelvin s equation inadequate. The micropore domain (0.1 to several nm) corresponds to molecular sizes and is thus especially important for current preoccupations (zeolites, new specialised aluminas). Unfortunately, current routine techniques are insufficient to cover this domain both in terms of the accuracy of measurement (very low pressure and temperature gas-solid isotherms) and their geometrical interpretation (insufficiency of semi-empirical models such as BET, BJH, Horvath-Kawazoe, Dubinin Radushkevich. etc.). 

Perhaps the first numerical investigation of lithographically patterned electrodeposition was published by Alkire et al. [46]. In this work, the finite-element method was used to calculate the secondary current distribution at an electrode patterned with negligibly thin insulating stripes. (This is classified as a secondary current distribution problem because surface overpotential effects are included but concentration effects are not.) Growth of the electrodeposit was simulated in a series of pseudosteady time steps, where each node on the electrode boundary was moved at each... 

In order to develop eddy current measurement techniques for CFRP it is necessary to understand the effect of the anisotropic resistance on the eddy currents. A method was developed which enables the visualisation of eddy currents in CFRP. In this method the z-component of the magnetic field was measured using a receiver coil. From the two-dimensional magnetic field distribution the current distribution can be calculated. 

Kawamoto (2) developed a two-dimensional model that is based on a double iterative boundary element method. The numerical method calculates the secondary current distribution and the current distribution within anisotropic resistive electrodes. However, the model assumes only the initial current distribution and does not take into account the effect of the growing deposit. Matlosz et al. (3) developed a theoretical model that predicts the current distribution in the presence of Butler-Volmer kinetics, the current distribution within a resistive electrode and the effect of the growing metal. Vallotton et al. (4) compared their numerical simulations with experimental data taken during lead electrodeposition on a Ni-P substrate and found limitations to the applicability of the model that were attributed to mass transfer effects. 

The crevice is considered one-dimensional, but the ereviee profile may be arbitrary. The crevice and the free surface around it is treated as a galvanic element. The potential variation and current distribution is calculated numerically by the finite difference method (FDM). This is based on the eleetrolyte in the crevice being divided into finite resistors, as shown in Figure 7.19. 

In connection with modem offshore oil and gas activities, numerical methods have been applied which make it possible to carry out more accurate and extensive calculations of potential variation and current distribution, in which the total system... 

The boundary element method has, up to now, scarcely been used to calculate current distributions. 

To calculate current distributions in electrochemical systems, the indirect methods are less attractive as supplementary calculations are needed to obtain U and U on the boundaries. On the other hand the RBEM can be a good alternative provided the distance d is well chosen [ 83]. 

Nevertheless, when singularities occur, it is sometimes observed that calculated primary current distributions, obtained with the combination of analytical and numerical integration, are worse than the less accurate classical integration method where only points belonging... 

Y. Nishiki et al. [ 77] used the finite element method with linear elements to calculate the primary current distribution for a series of values of the parameters (d /p, 2 b/p, w/p, P-1/P2) of "the unit-cell... 

Applying the Boundary Element Method to Electrochemical Calculations of Primary Current Distribution". 

Calculation of Current Distribution and Electrode Shape Change by the Boundary Element Method. ... 

Calculation of the Current Distribution at a Cathode by the Method of Straight Lines . Translated from Elektrokhimiya, Vol. 1, No. 2,... 

In 1964., Klingert et al. [ 60], as well as Fleck et al. [ J , 2] outlined the first computer programs for calculating current distributions by the finite difference method. 

The nature of current distribution influences the shape generation. The recession takes place in the direction of current density and the amount of recession depends on the magnitude of current density which can be explained by Eqn (3.5). Current distribution is calculated for a given time step by numerical solution of Laplace equation with nonlinear boundary conditions. Finite element method and boundary element method have been used for simulation of shape evolution during EMM. The new shape is obtained from the immediate previous shape by displacing the boundary proportional to the magnitude and in the direction of current density. The results of these simulation techniques agreed with the experimental results [6]. 

One example is the maximum entropy formalism (MEF), which assiunes the final distribution is that which maximizes the entropy production. A detailed discussion is given in [24]. In general, the method is able to calculate the correct shape of the distribution. However, experimental results are needed to obtain the magnitude. At the current time, no method is predictive without some experimental input [4]. 

The current bypass in the multiple-channel RuOj/YSZ cell was estimated by the following indirect method. Two series of current-voltage curves were determined under the same conditions, one before and another after deposition of the catalyst. Measurements were made at various gas compositions and at different temperatures, the same in both cases. It was assumed that coating the support with the catalyst does not change the current distribution in the solid electrolyte, but simply opens new parallel conduction pathways. The current bypass was then calculated from the currents measured at a same cell potential. 

Because of the complex geometries and the nonlinear boundary conditions involved in the current distribution problems, there are few analytical solutions. The primary concurrent distribution profiles for various geometries have been calculated and tabulated in an excellent series of papers by Kojima [8,9] and Klingert et al. [10]. Prentice and Tobias [11] reviewed current distribution problems solved by numerical methods in the literature. The finite difference method and the finite element method are widely used for determining current distribution profiles. 

Cathode Corrosion. One of the problems encountered with MDC-29 diaphragm cells is the severe cathode corrosion at the cefi liquor/hydrogen interface near the cathode screen, which prevents deposition of good diaphragms. The primary current distribution profiles were calculated [15] using the finite difference method and the results... 

Design methods try to include potential variations in the analysis. Local variations in product distribution and current efficiency are deduced and are part of the overall design. These approaches tend to assume an ideal reactor configuration in which the primary current distribution is uniform. This ideal is sometimes approached in practice but some reactor geometries exhibit severe nonuniform distributions, particularly at the higher current densities demanded in industry. It is important to analyze such systems and determine conditions which will minimize these effects. In the following sections we will consider the problem of current distribution and illustrate some calculations with worked examples. 

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