*2.1. Ablation*

The internal energy balance of the solid and pyrolysis gas takes the form of a classic conduction equation combined with a source term that arises from the pyrolysis gas flow:

$$\frac{\partial}{\partial t}(\rho \mathbb{C}\_{\mathcal{P}} T) + \nabla \cdot \left(-\vec{\lambda} \cdot \nabla T\right) + \nabla \cdot \left(\dot{\mathfrak{m}}\_{\mathcal{S}}^{\prime\prime} h\_{\mathcal{S}}\right) = 0 \tag{1}$$

In this equation, *t* denotes time, *ρ* and *Cp* the solid density and specific heat capacity, respectively, *T* the temperature, *λ* ¯¯ the thermal conductivity tensor, *m*˙ *<sup>g</sup>* the area specific pyrolysis gas mass flow rate vector, and *hg* the enthalpy of pyrolysis gas.

If one assumes that a control volume *V* moves at speed *vgr*, through use of the Gauss' theorem, one obtains [25]:

$$\frac{d}{dt} \int \left[ \int\_{V} \rho \mathbb{C}\_{\mathcal{P}} T \, \mathrm{d} \, V + \int\_{A} -\tilde{\lambda} \cdot \nabla T \cdot \mathrm{d} \, \mathbf{A} + \int\_{A} \int\_{\mathcal{S}} \dot{\mathfrak{m}}\_{\mathcal{S}}'' h\_{\mathcal{S}} \cdot \mathrm{d} \, \mathbf{A} - \int\_{A} \int\_{A} \rho \mathbb{C}\_{\mathcal{P}} T \mathfrak{p}\_{\mathcal{S}^{\mathcal{T}}} \cdot \mathrm{d} \, \mathbf{A} = 0. \tag{2}$$

The terms in Equation (2) can be interpreted as follows from left to right: time change in internal energy of the control volume, conductive heat flux, convective heat flux due to pyrolysis gas movement, and convective heat flux due to grid movement.

With reference to Figure 2, the one-dimensional energy balance can be written as:

$$\begin{array}{ll} \frac{1}{\sqrt{2}} \int\_{V} \rho \mathbb{C}\_{p} T dz + \left( \dot{m}\_{\mathcal{S}}^{\prime\prime} \mathbb{h}\_{\mathcal{S}} \right)\_{right} - \left( \dot{m}\_{\mathcal{S}}^{\prime\prime} \mathbb{h}\_{\mathcal{S}} \right)\_{left} & + \left( \lambda \frac{\partial T}{\partial z} \right)\_{right} - \left( \lambda \frac{\partial T}{\partial z} \right)\_{left} \\ & - \left( \rho \mathbb{C}\_{p} T \mathbb{u}\_{\mathcal{S}^{\prime}} \right)\_{right} + \left( \rho \mathbb{C}\_{p} T \mathbb{u}\_{\mathcal{S}^{\prime}} \right)\_{left} = 0, \end{array} \tag{3}$$

where *z* is a stationary coordinate and *ugr* is the grid velocity.

**Figure 2.** Schematic of a one-dimensional control volume describing the coordinates and labeling used in Equation (3).

The rate of decomposition of the material is temperature-dependent and can be described via an Arrhenius equation:

$$\frac{\mathbf{d}\,\rho\_i}{\mathbf{d}\,t} = -c\_i \left(\frac{\rho\_i - \rho\_{c,i}}{\rho\_{v,i} - \rho\_{c,i}}\right)^{n\_{r,i}} \mathbf{e}^{-\frac{A\_i}{T}}.\tag{4}$$

As the material can be made of multiple components, the index *i* indicates the *i*th component with its own decomposition law; *c* is called the pre-exponential factor, *ρ*, *ρv*, and *ρ<sup>c</sup>* the current, virgin, and char density, respectively, *nr* the reaction order, and *Ai* the scaled activation energy. The density of the material is obtained via the weighted sum of each phase:

$$\rho = \sum\_{i=1}^{N\_P} \Gamma\_i \rho\_{i\nu} \tag{5}$$

where *i* represents the considered phase, *Np* is the number of present phases, Γ*<sup>i</sup>* is the volume fraction of the *i*th phase, and *ρ<sup>i</sup>* is the density of the *i*th phase.

The virgin material is generally assumed to be impermeable, thus forcing all gas to leave through the outer surface [26]. All the material that did not remain solid, i.e., transitioning from virgin to char status, is in gaseous form. Thus, for an internal control volume, as depicted in Figure 2, the pyrolysis gas mass flux is related to the decomposition from virgin to charred material, described in Equation (4), via mass conservation:

$$\frac{\partial \dot{m}^{\prime\prime}\_{\mathcal{S}}}{\partial z} = -\frac{\mathbf{d}\,\rho}{\mathbf{d}\,t} = -\sum\_{i} \Gamma\_{i} \frac{\mathbf{d}\,\rho\_{i}}{\mathbf{d}\,t}.\tag{6}$$

The char ablation needs to be considered in the mass conservation when considering a control volume that includes the receding surface. At the surface of the ablative thermal protection system, chemical reactions such as oxidation and nitridation take place [27]. Three components take part in this reaction:


If chemical equilibrium is assumed, the products of the reaction and their associated properties such as enthalpy can be determined. Programs such as Mutation++ [27] or CEA [28] can compute this equilibrium by minimisation of Gibbs energy. As an input, the ratios of the three components listed above are needed. To obtain these, the mass flux of the charred material *m*˙ *<sup>c</sup>* and pyrolysis gas at the wall *m*˙ *<sup>g</sup>*,*<sup>w</sup>* are non-dimensionalised:

$$B'\_c = \frac{\dot{m}\_c^{\prime\prime}}{\rho\_c \mu\_c \mathbf{C}\_M} \tag{7}$$

$$B'\_{\mathbb{S}} = \frac{\dot{m}''\_{\mathbb{S},w}}{\rho\_{\varepsilon}\mu\_{\varepsilon}\mathbb{C}\_{M}},\tag{8}$$

where *ρ<sup>e</sup>* and *ue* are the boundary layer edge gas density and velocity, respectively; *CM* is the local Stanton number for mass transfer. As a result of the surface chemistry calculation, one receives:

$$B'\_{\mathcal{L}} = f\left(T\_{w\prime}, B'\_{\mathcal{X}'}p\right) \tag{9}$$

$$h\_w = f\left(T\_{w\prime}B\_{\mathcal{S}'}'p\right),\tag{10}$$

where *Tw* is the wall temperature, *hw* the enthalpy of the gaseous surface reaction products, and *p* the pressure [29]. The enthalpy flux that is carried away from the material can then be calculated using (*m*˙ *<sup>c</sup>* + *m*˙ *<sup>g</sup>*,*w*)*hw*. The charred material mass flux *m*˙ *<sup>c</sup>* can be connected to the surface recession rate *s*˙ via:

$$
\dot{m}\_c = \rho\_w \dot{s}\_\prime \tag{11}
$$

where *ρ<sup>w</sup>* is the density of the solid material at the wall.
