## 3 Numerical Methodology
### Operating Conditions and Parameters Definition  
In the simulation, there were two kinds of numerical models, i.e., the RVE model and the full-size model. The RVE model was a minimal periodic element in the 2.5-D braided CMC, while the full-size model was based on the sample's real size for the 2.5-D braided CMC.  
Figure 4: A schematic diagram of the principal direction of the thermal conductivity (PDTC).  
Simultaneously, to study the influence of the sample thickness on the Effective Thermal Conductivity (ETC) and the critical thickness that was suitable for the RVE model, full-size models with different thickness of the \(z\)-axis were built. The thicknesses \(\mathrm{L_{Z}}\) were set as 2.8 mm, 4.0 mm, 7.6 mm, 11.2 mm, 14.8 mm, and 18.4 mm, as shown in Figure 6. The boundary conditions and the mesh generation strategies were the same for the model shown in Figure 5b. The corresponding numbers of the meshes were 798,492, 1,297,051, 2,785,965, 4,264,916, 5,465,846, and 6,760,725, respectively.  
Additionally, to make the results and conclusions more useful, in the investigation of the thermal conductivities with increasing thickness, the absolute value ratio, n (as shown in Equation (5)), and the level of anisotropy, \(r\) (as shown in Equation (6)), of the thermal conductivities of the fiber bundles were changed.  
\[\mathrm{n}=\frac{k_{\zeta}}{9.66}=\frac{k_{\eta}}{1.48}=\frac{k_{\
u}}{1.48} \tag{5}\]  
\[r=\frac{k_{\zeta}}{k_{\eta}}=\frac{k_{\zeta}}{k_{\
u}} \tag{6}\]  
Figure 5: The mesh and boundary conditions.  
Figure 6: Full-size models with different thicknesses.  
where \(k_{\zeta}\), \(k_{\eta}\), and \(k_{\
u}\) are the thermal conductivities along the \(\zeta\), \(\eta\), and \(\
u\) directions and their values are 9.66 W/(m\(\cdot\)K), 1.48 W/(m\(\cdot\)K), and 1.48 W/(m\(\cdot\)K), respectively, of the sample in Figure 1.  
The operating conditions of the absolute value ratio n and the level of anisotropy \(r\) were set according to the data in Reference [28], as shown in Table 1. The ATC of the sample shown in Figure 1a was set as the basic Case 1. For Cases 2-5, the values of \(k_{\zeta}\), \(k_{\eta}\), and \(k_{\
u}\) were increased at the same time to improve the absolute value ratio n and the level of anisotropy r was kept constant. For Cases 6-9, the \(k_{\eta}\) and \(k_{\
u}\) were kept constant and the \(k_{\zeta}\) was increased to improve the level of anisotropy \(r\). Correspondingly, the absolute value n did not exist. The corresponding \(k_{\zeta}\), \(k_{\eta}\), and \(k_{\
u}\) for each case are also given in Table 1.  
In the analysis of the results, the parameter \(F_{f}\) that represents the volume fraction of the fibers is defined as follows:  
\[F_{f}=\frac{V_{f}}{V_{t}} \tag{7}\]  
where \(V_{f}\) is the volume of fibers and \(V_{t}\) is the total volume of the model.  
Additionally, although the constant temperature boundary conditions in different models were the same, it should be noted that the thickness changed, so the temperature and the heat flux in different models varied. To compare the temperature and the heat flux in different models fairly, the relative temperature fluctuations \(\delta_{T}\) and the relative heat flux fluctuations \(\delta_{bf}\) are defined:  
\[\delta_{T}=\frac{T_{i}-T_{avg}}{10\left[\text{K}\right]} \tag{8}\]  
\[\delta_{bf}=\frac{hf_{i}-hf_{avg}}{hf_{avg}} \tag{9}\]  
where \(T_{i}\) is the local temperature, \(T_{avg}\) is the average temperature, and 10 K is the difference between the two constant temperature boundary conditions. The \(hf_{i}\) term is the local heat flux, and \(hf_{avg}\) is the average heat flux.  
When the relative deviation between the ETCs based on the RVE model and the full-size model was smaller than 5%, the corresponding thickness of the full-size model was defined as the \"critical thickness\", L\({}_{Z}\). For different ATC cases in Table 1, the critical thickness L\({}_{Z}\) which was suitable for the RVE model was investigated in this paper.