## 3 Finite Element Analyses
### Thermal Conductivities of Carbon Fiber Rods and Carbon Fiber Bundles  
The geometry models of carbon fiber rods and carbon fiber bundles were directly built in ABAQUS. It should be noted that the temperature gradient was prescribed to be 25 K rather than 300 K when calculating the thermal conductivities of carbon fiber rod and carbon fiber bundle. In general, carbon fiber rods and carbon fiber bundles can both be regarded as transverse isotropic composites. The longitudinal thermal conductivity of transverse isotropic composites is usually calculated by classical mixture rule (Parallel Model). However, there is no uniform theoretical formula for the prediction of transverse thermal conductivities. There are quite a few differences in the results of the transverse thermal conductivities calculated by different formulas [43]. In this work, the Series Model, Charles Model [44], Pilling Model [45], and Maxwell Model [46] are used to calculate the transverse thermal conductivities, compared with results calculated by the FEM analysis. The temperature and heat flux distributions of carbon fiber rods and carbon fiber bundles are shown in Figures 9 and 10, respectively. The numerical results are presented in Table 2.  
\[\text{Parallel Model}:\lambda_{yl}=\lambda_{fl}V_{f}+\lambda_{m}(1-V_{f}), \tag{16}\]  
\[\text{Series Model}:\lambda_{yt}=\frac{\lambda_{fl}\lambda_{m}}{\lambda_{m}V_{ f}+\lambda_{fl}(1-V_{f})}, \tag{17}\]  
\[\text{Charles Model}:\lambda_{yt}=\frac{\lambda_{m}(1-V_{f})+\lambda_{fl}(1+V_{ f})}{\lambda_{fl}(1-V_{f})+\lambda_{m}(1+V_{f})}, \tag{18}\]  
\[\text{Pilling Model}:\lambda_{ft}=\frac{\lambda_{m}}{4}\left[\sqrt{(1-V_{f})^{2 }(\frac{\lambda_{ft}}{\lambda_{m}}-1)^{2}+4\frac{\lambda_{ft}}{\lambda_{m}}}-( 1-V_{f})(\frac{\lambda_{ft}}{\lambda_{m}}-1)\right]^{2}, \tag{19}\]  
\[\text{Maxwell Model}:\lambda_{ft}=\lambda_{m}\frac{\lambda_{ft}+2\lambda_{m}+2 V_{f}(\lambda_{ft}-\lambda_{m})}{\lambda_{ft}+2\lambda_{m}-2V_{f}(\lambda_{ft}- \lambda_{m})}, \tag{20}\]  
where \(\lambda_{yl}\), \(\lambda_{yt}\), \(\lambda_{fl}\), and \(\lambda_{ft}\) represent the thermal conductivities in the longitudinal direction of the yarn, in the transverse direction of the yarn, in the longitudinal direction of the fiber, and in the transverse direction of the fiber, respectively, \(V_{f}\) is the fiber volume fraction of yarn, and \(\lambda_{m}\) is the thermal conductivities of the matrix. As shown in Table 2, the calculated results of the parallel model in Equation (15) are close to the results of the numerical model in the longitudinal direction. However, there is a large error in the results calculated by the Series Model in Equation (16) and the Charles Model in Equation (17) in the transverse direction of yarn. The results predicted by Pilling model in Equation (18) are close to the results calculated by the FEM analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material/Heat Treating** & \multicolumn{3}{c}{**Thermal Conductivity/(W\(\cdot\)m\({}^{-1}\)\(\cdot\)K\({}^{-1}\))**} \\
**Temperature** & **Transverse** & **Circumferential** & **Longitudinal** \\ \hline T300 fiber/PAN 2673 K & 0.76 & 0.76 & 76 \\ Resin based Matrix 2673 K & 64.3 & 64.3 & 64.3 \\ Pitch based Matrix 2673 K & 0.64 & 64.3 & 257 \\ Disordered Graphite & 110 & 110 & 110 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermal conductivities of the constituent materials. Figure 10: Temperature and heat flux distributions of the carbon fiber rod: (**a**) temperature, (**b**) heat flux.