A review of two-dimensional (2D) analytical models of skin and proximity effects in large industrial furnaces with three electrodes arranged in an equilateral triangle is given. The models cover three different cases: one electrode only, three electrodes where two are approximated by line currents, and induced shell currents where all electrodes are approximated by line currents. The first two models show how the skin and proximity effects depend on electrode material properties and size, and the distance between the electrodes. The third model shows how the strength of the induced shell currents will depend on electrode position and furnace size. These models are compared to numerical studies including distributed electrodes and shell currents. The analytical models are accurate when induced shell currents can be disregarded. However, strong shell currents may have a significant impact on the current distribution within the electrodes. This electrode-shell proximity effect competes with the electrode-electrode proximity effect. Finally, the 2D models have been compared with three-dimensional (3D) case studies of large industrial furnaces. In 3D, the shell currents are significantly smaller than what are predicted by the 2D models, but they are sufficiently strong to cause a significant correction of the electrode current density.

中文翻译：

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电极和炉壳中的趋肤效应和邻近效应

给出了三个电极排列成等边三角形的大型工业炉中趋肤效应和邻近效应的二维 (2D) 分析模型的综述。这些模型涵盖三种不同的情况：只有一个电极，三个电极，其中两个电极近似于线电流，以及感应壳电流，其中所有电极近似于线电流。前两个模型显示了皮肤和邻近效应如何取决于电极材料特性和尺寸以及电极之间的距离。第三个模型显示感应壳电流的强度将如何取决于电极位置和熔炉尺寸。这些模型与包括分布式电极和壳电流在内的数值研究进行了比较。当可以忽略感应壳电流时，分析模型是准确的。然而，强壳电流可能对电极内的电流分布产生重大影响。这种电极-壳邻近效应与电极-电极邻近效应相互竞争。最后，将 2D 模型与大型工业炉的三维 (3D) 案例研究进行了比较。在 3D 中，壳电流明显小于 2D 模型预测的电流，但它们足够强以导致电极电流密度的显着修正。