# Surface Erosion Analysis for Thermal Insulation Materials of Graphite and Carbon-Carbon Composite
## 2 Materials and Methods
### Theoretical Erosion Model
\[\left(\overline{\rho v}\overline{\mathrm{Y_{i}}}\right)_{\mathrm{g}}+\left( \overline{\rho}D\frac{\partial\overline{\mathrm{Y_{i}}}}{\partial r}\right)_{ \mathrm{g}}=\dot{\omega}_{\mathrm{i}}-\frac{1}{\mathrm{A}}\frac{\mathrm{d}}{ \mathrm{dt}}\iint\limits_{S_{1}}\rho_{\mathrm{i}}^{\prime}\mathrm{ds}+\rho_{c }\dot{r}_{c}\overline{\mathrm{Y_{i}}} \tag{4}\]  
The left side of Equation (4) indicates the convective mass flux and the diffusion mass flux of the species i, and the right side is the rate of generation of species i, the rate of mass accumulation of species i, and mass flux of species i. Here, \(S_{1}\) is the interfacial surface area, Y is the mass fraction, A is the area, \(\rho\) is the density, and \(\dot{r}_{c}\) is the erosion rate. When the temperature is high at the solid-gas interface, the reactant reacts with carbon, resulting in a very low concentration. The mass fraction can be assumed to have no accumulation. So \(\left(\overline{\mathrm{Y_{i}}}\right)\cong 0\) and Equation (4) is reformulated as follows:  
\[\dot{\omega}_{\mathrm{i}}+\rho_{c}\dot{r}_{c}\overline{\mathrm{Y_{i}}}=0 \tag{5}\]  
The reaction between the reactant \(\mathrm{B_{i}}\) and graphite or the C-C composite \(\mathrm{C_{s}}\) is as follows:  
\[\mathrm{\
u_{c,i}C_{s}+\
u_{Bi}B_{i}=\sum_{j}\
u_{j}Prod_{j}} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Surface Reactions** & **A\({}_{\mathrm{i}}\)** & **E\({}_{\mathrm{i}}\) (Kcal/mol)** & \(\beta\) & \(N\) & **Remarks** \\ \hline \(\mathrm{C_{S}+H_{2}O\to CO+H_{2}}\) & \(4.8\times 10^{5}\) (kg/m\({}^{2}\) s atm\({}^{0.5}\)) & 68.8 & 0.0 & 0.5 & \\ \(\mathrm{C_{S}+CO_{2}\to 2 CO}\) & \(9.0\times 10^{3}\) (kg/m\({}^{2}\) s atm\({}^{0.5}\)) & 68.1 & 0.0 & 0.5 & \\ \hline \(\mathrm{C_{S}+OH\to CO+H}\) & \(3.61\times 10^{2}\) (kg K\({}^{0.5}\)/m\({}^{2}\) s atm) & 0.0 & \(-0.5\) & 1.0 & MOS \\  & 100 (kg K\({}^{0.5}\)/m\({}^{2}\) s atm) & 0.0 & \(-0.62\) & 1.0 & MMOS \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Kinetic data for heterogeneous reactions. Consequently, the diffusion-controlled recession rate (\(\overline{r}_{\rm c,d}\)) on the surface of graphite or the C-C composite can be expressed as Equation (7) based on Equations (5) and (6). \[\dot{r}_{\rm c,d}=\frac{1}{\rho_{\rm c}}\sum_{\rm i}\dot{\omega}_{\rm i}\frac{ \
u_{\rm c,i}}{\
u_{\rm B,i}}\frac{\rm MW_{C}}{\rm MW_{B_{i}}} \tag{7}\]  
Here, MW is the molecular weight and \(\
u\) is the stoichiometric coefficient. The mechanical erosion model is expressed by the porosity of the material, the characteristic exhaust velocity, and the experimental constant. The Hopf equation is expressed by the relationship between the friction factor and surface roughness as Equation (8). \[f=\rm C_{1}e^{\rm C_{2}} \tag{8}\]  
Here,\(f\) is the friction coefficient, C is a constant, and e is a dimensionless constant. Equation (8) is again given by Equation (9), taking into account of the porosity that is the measure of the structural resistance of the material, where \(a_{1}\) is a constant and \(P\) is the porosity. \[f=a_{1}(\log P)^{2} \tag{9}\]  
The shear flow of the gas causes shear stress in the wall, and mechanical erosion has a linear relationship with shear stress. The mechanical erosion rate (\(\dot{r}_{m}\)) is expressed as Equation (10), a function of the characteristic exhaust velocity (\(c^{*}\)) and friction coefficient (\(f\)) related to the shear stress, and is a constant. \[\dot{r}_{m}=a_{2}c^{*}f \tag{10}\]  
Substituting Equation (9) into Equation (10), the mechanical erosion rate is summarized as \(\dot{r}_{m}=\rm ac^{*}(\log P)^{2}\) by the relationship between the porosity and characteristic exhaust velocity, where \(\alpha\) is the final constant [5]. The total erosion rate is obtained using the harmonic mean for thermochemical chemical-controlled recession rate (\(\dot{r}_{\rm c,ch}\)), thermochemical diffusion-controlled recession rate (\(\dot{r}_{\rm c,d}\)), and mechanical erosion rate (\(\dot{r}_{\rm m}\)).