## 1 Introduction  
The inlet temperatures of gas turbine engines are becoming higher in order to obtain better performances. Those high inlet temperatures could be far beyond the temperature limits of current superalloy materials [1]. More coolant and more complex cooling configurations need to be applied, but the performance of the turbine engines could be weakened. Consequently, highly heat-resistant materials, such as ceramic matrix composite (CMC), have attracted a great deal of attentions in recent years [2; 3; 4]. For the application of CMCs for the components with high temperatures, thermal analysis is an important research topic. [PERSON] et al. [5] studied the temperature distribution of a CMC turbine vane with a film-cooling configuration, both experimentally and numerically, for a gas turbine environment. Their results encouraged the potential applications of CMC materials for turbine airfoil. [PERSON] et al. [6] performed a numerical study of the heat transfer process of a nickel-based alloy vane case and a silicon nitride ceramic vane case in which the thermal conductivities were different. The results showed that the thermal conductivity had a significant effect on the internal heat transfer, which further affected the vane's temperature field. However, as a fiber reinforced composite, the thermal conductivity of the CMC material is anisotropic [7; 8] due to the significant differences between the thermal conductivities in the fiber's axial direction and the radial direction and the differences between the thermal conductivities of the fibers and the matrix. [PERSON] et al. [9] investigated the thermal and mechanical performances of the CMC materials under the cyclic thermal stresses of a gas turbine combustion chamber. The anisotropic thermal conductivity (ATC) was used in the analysis, and the in-plane and the through-thickness thermal conductivities were set as 3 W/(m\(\cdot\)K) and 1.8 W/(m\(\cdot\)K), respectively. [PERSON] and [PERSON] et al. [10] studied the thermal analysis method of a CMC turbine vane. In the simulation, the effective ATCs were used to reflect the CMC's anisotropy. The results showed that the directions of the ATCs changed at different locations on the CMC turbine vane, and as a result, the braided direction was changed due to the warped surface. This variation of the ATC affected the heat transfer in the solid region of the CMC turbine vane. [PERSON] et al. [11] carried out a multi-scale thermodynamic analysis for a CMC turbine vane. The in-plane and the through-thickness thermal conductivities were 12.1 W/(m\(\cdot\)K) and 8.83 W/(m\(\cdot\)K), respectively. The variation of the material's ATC local coordinates due to the vane's curved surface was also considered. The research mentioned above indicates that in the thermal analysis of a CMC component with a high temperature, the ATC is an important parameter. However, the ATC varies for different CMC materials, and the internal structure and the percentages of the components with different thermal properties have significant influences on the ATC. For example, [PERSON] et al. [12] studied the thermal conductivity of triaxial braided composites with fibers that changed directions. The results showed that when the direction of the fibers changed from \(\pm\)25\({}^{\circ}\) to \(\pm\)65\({}^{\circ}\), the thermal conductivity in the thickness direction changed from 4.2 W/(m\(\cdot\)K) to 5.15 W/(m\(\cdot\)K). In the work of [PERSON] et al. [13], the thermal conductivities of 3-D braided composites were investigated. The results showed that the volume fraction of the fibers had a considerable effect on the thermal conductivity. When the volume fraction of the fibers increased from 0.25 to 0.55, the thermal conductivity in the braided direction changed from 0.68 W/(m\(\cdot\)K) to 1.3 W/(m\(\cdot\)K). Thus, the prediction method of the braided CMC's ATC is an important aspect in the application of the CMC, which has been widely studied.