Abstract An approximate, analytical treatment is presented for the rectangular annular fin by transforming the complicated modified Bessel equation of zero order into a rudimentary Cauchy-Euler equation. The essential step in the computational procedure revolves around a simple manipulation of the radial coordinate that sets up a variable coefficient in the third term of the modified Bessel equation of zero order. In the third term, the radial variable will be replaced by the mean radius of the inner and outer radius, whereas the radial variable prevails in the first and second terms. This action paves the way to the easier Cauchy-Euler equation. For a collection of rectangular annular fins of interest in engineering applications, approximate, analytical temperature distributions and heat transfer rates (via the fin efficiency) written in terms of two binomials demonstrate excellent quality levels in all cases. Additionally, relative error distributions are presented in detailed manner using as the baseline cases the classical exact, analytical temperature distributions and heat transfer rates expressed in terms of the complicated modified Bessel functions of first and second kind.

中文翻译：

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通过适应柯西-欧拉方程对矩形环形翅片的近似分析程序

摘要 通过将复杂的零阶修正贝塞尔方程转化为基本的柯西-欧拉方程，给出了矩形环形翅片的近似解析处理。计算过程中的基本步骤围绕径向坐标的简单操作，在修正的零阶贝塞尔方程的第三项中设置可变系数。在第三项中，径向变量将被内半径和外半径的平均半径代替，而径向变量在第一项和第二项中占优势。这个动作为更简单的柯西-欧拉方程铺平了道路。对于工程应用中感兴趣的矩形环形翅片的集合，近似，用两个二项式表示的分析温度分布和传热率（通过翅片效率）在所有情况下都显示出优异的质量水平。此外，使用以第一类和第二类复杂的修正贝塞尔函数表示的经典精确分析温度分布和传热速率作为基线情况，以详细方式呈现相对误差分布。