## 1 Introduction  
The four-directional carbon/carbon (4D C/C) composites are lightweight and have good thermodynamic properties, since carbon fiber is used as the reinforcement. In addition, 4D C/C composites have good delamination resistance due to the braided structure. It is used in the nozzle of solid rocket motors [1, 2] to face extremely high-temperature (over 2800 K) [3, 4] and the erosion of high speed gas flow at the same time. Therefore, the structure requires high heat transfer efficiency to protect it from failure. Considering the considerable processing cycles and high production expenses, it is necessary to develop a simple numerical model to predict the performances of the material structure to be designed. Analytic-numerical method and finite elements method (FEM) are two common ways for studying the thermal properties of composites. Analytic-numerical method [5, 6, 7] for predicting the thermal conductivities of composites have been discussed in many literatures. [PERSON] et al. [8] proposed a four-phase confocal elliptical cylinder model for predicting the thermal conductivity of coated fiber reinforced composites. [PERSON] and [PERSON] [9] produced and investigated both analytic and numerical models with regular and randomly distributed fibers in matrix material. [PERSON] and [PERSON] proposed two analytic-numerical algorithms for determination of the volume fraction of fibers [10] and the thermal contact resistance between the components [11] in order to obtain a given value of the transverse effective thermal conductivity. [PERSON] et al. [12] employed 1-D inverse heat conduction method for the estimation of thermal contact conductance for metallic contacts. FEM [13; 14; 15] are also widely used to predict the thermal conductivity of composites. [PERSON] [16] and [PERSON] [17] predicted the thermal conductivities of 3D braided carbon fiber composites by finite element analysis. [PERSON] [18; 19] and [PERSON] [20] proposed a finite element model to predict the thermal conductivities of 2D woven fabric and 2D braided C/SiC composites. [PERSON] [21] used multi-size unit cells to predict the effective thermal conductivities of carbon fiber braided composites. The results show that the FEM is effective for the performance calculation of 3D braided composites, and the meso-structure of 3D braided composites can be considered more specifically [22; 23; 24]. Constructing unit cells is a commonly used method to study thermodynamic properties of 4D C/C composites in finite element analysis, due to its periodic characteristics. Normally, the temperature boundary conditions are applied on the unit cells, and the effective thermal conductivities are calculated according to the Fourier law of heat transfer [16; 25; 26]. The preform of 4D C/C composites consists of carbon fiber rods and carbon fiber bundles. The carbon fiber rods are in axial direction, and the carbon fiber bundles are arranged around the carbon fiber rods in a radial plane with an angle of 0\({}^{\circ}\), 120\({}^{\circ}\), and 240\({}^{\circ}\). Then, the preform is impregnated and carbonized to construct the 4D C/C composites [27]. The prediction of the effective thermal conductivities of 4D C/C composites can be carried out on the unit cells at meso-scale, similar to other composites with periodic structure [28; 29]. Carbon fiber rods and carbon fiber bundles can be regarded as transverse isotropic composite materials [30]. In general, the axial thermal conductivities of carbon fiber rods and carbon fiber bundles can be calculated using classical mixture rule, but there is no suitable theoretical formula for the prediction of transverse thermal conductivities. The numerical models are usually used to calculate the transverse thermal conductivities [31; 32]. The thermal conductivities of the unit cells are predicted by the calculated results of the carbon fiber rods, carbon fiber bundles and the matrix, and were used to evaluate the thermal conductivities of the 4D C/C composite.