## 3 Uncertainty-Based Optimization Design for the Multilayer TPS of Hypersonic Wing
### Transient Heat Transfer Analysis of the Multilayer TPS over Whole Trajectory  
Based on the FEA model, the transient heat transfer simulation of the multilayer TPS tile over the whole trajectory is conducted to internal heat transfer characteristics. For the heat loads, the heat flux in Figure 5 is imposed on the C/C panel over the trajectory. Meanwhile, the heat radiation is also applied on the C/C panel with an emissivity of 0.8. The other surfaces are set to be adiabatic.  
Based on the law of Fourier heat conduction and energy conversation, the general governing equations of the transient structural heat transfer can be expressed as:  
\[\rho_{s}c_{s}\frac{\partial T_{s}}{\partial t}=\frac{\partial}{\partial x_{i}} \bigg{(}k_{s}\frac{\partial T_{s}}{\partial x_{i}}\bigg{)}+\dot{Q}\ +q_{\Gamma}\quad(i=1,2,3) \tag{16}\]  
where \(t\) is the time, \(T_{s}\) is the structural temperature, \(\rho_{s}\) is the structural density, \(c_{s}\) is the structural specific heat, \(k_{s}\) is the structural thermal conductivity, \(\dot{Q}\) is the volumetric heating source in a solid, and \(q_{\Gamma}\) is the heat flux imposed on the boundary.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Density (g/cm\({}^{3}\))** & **Conductivity [W/(m\(\cdot\)k)]** & **Specific Heat [J/(kg\(\cdot\)C)]** \\ \hline HTTI & 0.40 & 0.072 & 900 \\ MDDI & 0.33 & 0.055 & 870 \\ C/C plate & 1.75 & 8.0 & 94.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Physical properties of the multilayer TPS.  
Figure 7: Illustration of TPS structure for the hypersonic wing. (**a**) Three regions in the hypersonic wing; (**b**) Multilayer TPS; (**c**) Geometry of the multilayer tile; (**d**) FEM model of the multilayer tile.  
Moreover, the outer surface of the C/C panel is set as the second type of thermal boundary condition, and the surface heat radiation effect is considered as follows:  
\[q_{\rm T}=q-\varepsilon\sigma_{\rm s}\left(T_{w}^{4}-T_{0}^{4}\right) \tag{17}\]  
where \(q\), \(\varepsilon\), \(T_{w}\) and \(T_{0}\) are aerodynamic heat flux, surface emissivity, surface temperature and ambient temperature, respectively; \(\sigma_{\rm s}\) denotes the Stefan-Boltzmann constant, whose value is \(5.67\times 10^{-8}\,\rm W/(m^{2}\cdot K^{4})\).  
In this study, the finite element method is used to discretize Equation (16), and the numerical solutions for the nonlinear equilibrium equations have been carried out with the general finite element solver ANSYS Mechanical APDL Version 2022R1. The computational grid is a structured grid generated by ANSYS Mechanical, the analysis element type is set to be SOLID 70, and the mesh number is 160,000.  
The calculated maximum temperature evolution of each layer and the inner surface are shown in Figure 8, where \(T_{1\_\_\_\_\_}\), \(T_{2\_\_\_\_\_}\), \(T_{3\_\_\_\_}\), \(T_{4\_\_\_\_}\) represent the maximum temperature of the C/C panel, HTTI, MTTI and the inner surface, respectively. It can be found that:  
(1) The internal heat transfer characteristics of each layer are different due to the differences of the material properties. For example, the maximum temperature of the C/C panel rises much faster than that of either HTTI or MTTI for the higher heat conductivity.  
(2) As the effects of aerodynamic heating gradually recede over the trajectory, the input heat through the outer surface of the C/C panel is less than the conducted heat inside  
Figure 8: Maximum temperature evolution of each layer and the inner surface.  
material, thus the high temperature area is gradually moved inward from the C/C panel to MTTI, as shown in the temperature nephograms in Figure 8.  
(3) After 1000 s (the end time of the trajectory), though the aerodynamic heating stops, the heat stored in the material continues conducting to the interior. Therefore, the maximum temperature on the inner surface keeps increasing and reaches the peak around 2500 s, which means at least an extra 1500 s analysis time is required to obtain the reasonable results after the trajectory ends.