**4. Surface ablation using plasma preheating technology**

### **4.1 Theoretical approach**

To effectively couple the solid and the flow-field during ablation, the mass, momentum, energy, and species have to be conserved. The surface mass balance for each species is given in Eq. (1), where *m*\_<sup>0</sup> *<sup>s</sup>* is the mass flux of species s per second determined from the surface thermochemistry, *ρ<sup>s</sup>* is species density in kg/m<sup>3</sup> , *v* ! is the velocity vector representing the mass-averaged velocity leaving the surface, *n*^ is the unit normal vector to the surface but away from the wall, and the last term is the diffusion of species to or from the surface, where *D* is the diffusion coefficient in m2 /s, *Cs* is the species mass fraction [37].

$$
\dot{\boldsymbol{m}}'\_{\boldsymbol{s}} = \rho\_{\boldsymbol{s}} \overrightarrow{\boldsymbol{v}} \,\hat{\boldsymbol{n}} - \rho \boldsymbol{D} \overrightarrow{\nabla} \mathbf{C}\_{\boldsymbol{s}} \hat{\boldsymbol{n}} \tag{1}
$$

The surface energy balance is expressed in Eq. (2), where *q* ! *<sup>w</sup>* contains both the heat conduction and the diffusive chemical heat flux, the first term on the right is

the heat flux conducting energy away from the surface into the body, the second term is the radiation of heat from the surface into the flow, and the last term is the removal of energy from the surface due to mass removal [38].

$$\overrightarrow{q}\_w = -K\_{\text{solid}} \overrightarrow{\nabla} T\_{\text{solid}} \, \hat{n} - \varepsilon \sigma (T\_w - T\_a)^4 - \dot{m}\_w' h\_{o,w} \tag{2}$$

Generally, *Ksolid* is the thermal conductivity of graphite sample in W/(m.K), *Tsolid* is the temperature of graphite at the edges, *ε* is graphite emissivity, *σ* is the Stefan-Boltzmann constant in W/(m<sup>2</sup> .K<sup>4</sup> ), *Tw* is the surface temperature, *Ta* is the surrounding temperature, and *ho*,*<sup>w</sup>* is species enthalpy in J/kg [39]. Unlike the negative heat flux to the wall from the flow in high enthalpy facilities, the present work adopts a positive heat flux from the wall to the flow.
