Abstract The paper aimed at obtaining an analytical solution for the steady-state heat transfer in a hollow sphere made of functionally graded material. Two-dimensional distribution of temperature is considered to be in both radial and peripheral directions and the conductivity coefficients in both directions are a function of radius. The general boundary conditions for the interior and exterior surface of the sphere are considered so that the resulting solution can be extended to treat a wide range of functional cases. The obtained solutions are in the form of Bessel and Legendre functions. Unknown solution coefficients are achieved by applying boundary conditions and orthogonal Legendre functions' relations. The paper additionally provides results for two practical test cases to assess the robustness of the achieved solution. The influence of material constants and conductivity ratio are investigated in order to shed a light on material selection. The results confirm that the achieved general exact solution is able to adequately calculate the distribution of temperature. The validated findings of the current paper could be considered as a clue for tailoring of functionally graded spheres, like spherical vessels, based on the actual thermal boundary conditions in the manufacture process.

中文翻译：

---

FG 空心球温度分布的二维解析解：一般热边界条件

摘要 本文旨在获得功能梯度材料制成的空心球中稳态传热的解析解。温度的二维分布被认为是在径向和圆周方向上，并且两个方向的电导系数都是半径的函数。考虑了球体的内表面和外表面的一般边界条件，从而可以将所得解决方案扩展到处理各种功能情况。得到的解是 Bessel 和 Legendre 函数的形式。未知解系数是通过应用边界条件和正交勒让德函数关系来实现的。该论文还提供了两个实际测试用例的结果，以评估所实现解决方案的稳健性。为了阐明材料选择，研究了材料常数和电导率比的影响。结果证实，所获得的一般精确解能够充分计算温度分布。当前论文的验证结果可以被视为根据制造过程中的实际热边界条件定制功能梯度球体（如球形容器）的线索。