# Surface Erosion Analysis for Thermal Insulation Materials of Graphite and Carbon-Carbon Composite
## 2 Materials and Methods  
### Theoretical Erosion Model  
Figure 1 shows the physical and chemical processes of rocket nozzle erosion in which convection and radiation heat transfer occurs through combustion gas, and conduction heat transfer occurs in the nozzle. A heterogeneous reaction occurs at the nozzle surface and gas-phase chemical species diffusion, pyrolysis, and evaporation occur. Finally, by combustion gas flow, surface deviation occurs and the nozzle surface is eroded. Thermochemical erosion can be divided into a chemical-controlled recession process and a diffusion-controlled recession process; the effect depends on the surface temperature [6]. Surface temperature, surface pressure, reactant and product concentration, and density are the most influential factors for thermochemical erosion. Density of material has a large effect on erosion, and the higher the density, the more resistant to erosion. As the combustion chamber pressure increases, the density of the gas increases and the erosion increases [7]. Additionally, the higher the temperature, the greater the effect of erosion, which is due more to the diffusion-controlled recession process than the chemical-controlled recession process [4]. In the effects of chemical species, erosion is proportional to the concentration of H\({}_{2}\)O, CO\({}_{2}\), and OH, and inversely proportional to the concentration of H\({}_{2}\) by the reaction of H\({}_{2}\)O and CO\({}_{2}\), which are the main causes of erosion, wherein the C-C composite or graphite produce CO [8; 9]. Mechanical erosion is dominated mainly by the characteristic exhaust velocity, the friction coefficient and porosity. The characteristic exhaust velocity is a measure of combustion performance and depends on chamber pressure, nozzle cross-sectional area, and mass flow rate [10]. Figure 1: Physical and chemical processes associated with rocket nozzle erosion. The coefficient of friction is the shear stress at the surface and the porosity is the measure of the structural resistance [6]. The chemical-controlled recession process in the thermochemical erosion can be described in Equation (1) as a heterogeneous chemical reaction, where H\({}_{2}\)O, CO\({}_{2}\), and OH are the main oxide and H\({}_{2}\), CO, and H are the main products [11]. \[ \begin{array}{rcl}\mathrm{C_{s}+H_{2}O}&\rightarrow&\mathrm{CO+H_{2}}\\ \mathrm{C_{s}+CO_{2}}&\rightarrow&2\mathrm{CO}\\ \mathrm{C_{s}+OH}&\rightarrow&\mathrm{CO+H}\end{array}  \tag{1}\]  
Here, \(\mathrm{C_{s}}\) is the carbon of the material surface and other species such as O, O\({}_{2}\), and NO are ignored because they have little effect on the chemical reaction. The reaction rate constant (\(\mathrm{K_{i}}\)) for species i is expressed by the Arrhenius equation as follows:  
\[\mathrm{K_{i}}=\mathrm{A_{s,i}T_{i}^{\beta}\exp(-\frac{E_{a,s,i}}{R_{u}T_{s}})} \tag{2}\]  
Here, A is the pre-exponential factor, \(\beta\) is the temperature exponent, E is the activation energy, \(\mathrm{R_{u}}\) is the universal gas constant, and T is the surface temperature. The chemical dominant erosion rate at the surface is expressed as Equation (3) [12]. \[\dot{r}_{c,ch}=\sum\mathrm{K_{i}P_{i}^{n_{i}}/\rho_{c}} \tag{3}\]  
Here, \(\mathrm{P_{i}}\) is the pressure of i chemical species and the reaction rate data for the heterogeneous reaction are shown in Table 1[13]. \(\rho_{c}\) is density of graphite or carbon-carbon composite. The diffusion-controlled recession process in thermochemical erosion can be expressed as a surface mass balance (SMB) [14] at the solid-gas interface, as illustrated in Figure 2 and Equation (4).