*2.3. Material Properties*

In this section, the treatment for the material properties of each considered concept is described.

#### 2.3.1. Ablative Material

Difficulties in the determination of the local material properties arise from the thermal properties of the ablative TPS that depend on the degree of char *β*. As the heat shield material decomposes, the material properties, such as thermal conductivity or heat capacity, change. One commonly used approach in literature [31,32] to model this change is to prescribe fully virgin and fully charred material properties and interpolate based on the weight fraction of virgin and charred material.

The extent of the decomposition reaction *β* can be calculated through

$$
\beta = \frac{\rho\_v - \rho}{\rho\_v - \rho\_c},
\tag{14}
$$

where *v* refers to the virgin and *c* to the charred material; *β* is therefore 0 when the whole material is virgin and 1 for a fully charred state. Because of the assumption that a denser material contributes to the material properties to a higher degree, the weight fraction *wv* of virgin material is introduced:

$$w\_{\upsilon} = \frac{\rho\_{\upsilon}}{\rho\_{\upsilon} - \rho\_{\varepsilon}} \left( 1 - \frac{\rho\_{\varepsilon}}{\rho} \right) = \frac{\rho\_{\upsilon}}{\rho} (1 - \beta). \tag{15}$$

The char weight fraction *wc* is then

$$w\_c = 1 - w\_v = \frac{\rho\_c}{\rho} \beta. \tag{16}$$

The heat capacity *cp* for instance is computed using:

$$w\_p(T, w\_\upsilon) = w\_\upsilon \mathcal{C}\_{p,\upsilon}(T) + w\_\varepsilon \mathcal{C}\_{p,\varepsilon}(T) = w\_\upsilon \mathcal{C}\_{p,\upsilon}(T) + (1 - w\_\upsilon)\mathcal{C}\_{p,\varepsilon}(T). \tag{17}$$
