2.3.2. Corrugated Core ITPS

The three-dimensional structure of the ITPS sandwich cores is homogenised to allow a reduced treatment. A unit cell of a corrugated sandwich panel and its defining dimensions are sketched in Figure 3. The corrugated sandwich consists of a top face sheet (TFS) with thickness *tT* and a bottom face sheet (BFS) with thickness *tB* separated by the core thickness *tC*. These are connected by webs of thickness *tW* at an angle of corrugation Θ. The voids in-between are filled with a high temperature insulation material. The pattern repeats with multiples of the unit cell length 2*p*.

**Figure 3.** Dimensioned sketch of a corrugated sandwich unit cell.

Effective properties for the core are derived from rule of mixtures. As density is based on the volume fractions of the respective materials, a volume rule of mixtures is chosen for this property. In contrast, heat capacity is defined by the respective mass fractions, which leads to a mass rule of mixtures. The respective areas of web and filling are simply the volumes divided by the core thickness *tC*. As this factor is common to both volumes, there is no difference between areal and volumetric homogenisation for this case. Thus, homogenisation equations for effective specific heat capacity *Cp*,*eff* , effective density *ρ<sup>e</sup> f f* , and effective thermal conductivity *λeff* of the corrugated core are obtained as follows [16]:

$$\mathbb{C}\_{p,eff} = \frac{\mathbb{C}\_{p,W}\rho\_W t\_W + \mathbb{C}\_{p,F}\rho\_F (p\sin\Theta - t\_W)}{p\sin\Theta} \tag{18}$$

$$
\rho\_{\varepsilon}ff = \frac{\rho\_W V\_W + \rho\_F V\_F}{V\_C} = \frac{\rho\_W t\_W + \rho\_F (p \sin \Theta - t\_W)}{p \sin \Theta} \tag{19}
$$

$$
\lambda\_{eff} = \frac{\lambda\_W A\_W + \lambda\_F A\_F}{A\_\mathbb{C}} = \frac{\lambda\_W V\_W + \lambda\_F V\_\mathbb{F}}{V\_\mathbb{C}} = \frac{\lambda\_W t\_W + \lambda\_F (p \sin \Theta - t\_\mathbb{W})}{p \sin \Theta}.\tag{20}
$$

Here, the indices *W* and *F* refer to properties of the web and filling materials, respectively.

#### 2.3.3. Lattice Core ITPS with Embedded PCM

Orthotropic cell geometry leads to an orthotropic effective thermal conductivity tensor of the composite. Several types of lattice cores exist. The ones most investigated in the literature are the cubic ones inspired by Bravais crystals; see Figure 4. Excluding the *bcc* cell, which exhibits an isotropic morphology, cubic arrangements of these cells exhibit orthotropic behaviour.

**Figure 4.** Possible lattice types for cubic unit cells.

The effective thermal conductivity tensor must be written in the form

$$
\tilde{\lambda}\_{eff} = \begin{bmatrix}
\lambda\_{xy} & 0 & 0 \\
0 & \lambda\_{xy} & 0 \\
0 & 0 & \lambda\_z
\end{bmatrix},
\tag{21}
$$

where *x* and *y* are the in-plane coordinates and *z* the out plane one in Figure 4, and the three-dimensional problem can be reduced to a two-dimensional one. The respective values for the thermal conductivity are obtained using a semi-analytical correlation for the definition of the relevant contributions in the equation, as according to Hubert et al. [22].

$$
\lambda\_i = \lambda\_{PCM} \epsilon + \lambda\_s G\_i (1 - \epsilon),
\tag{22}
$$

where *i* indicates a generic principal direction, *λPCM* is the thermal conductivity of the PCM, *λ<sup>s</sup>* is the thermal conductivity of the metallic lattice structure, *Gi* is a dimensionless term that addresses the topology of the cell and is different for different directions, and is the porosity of the lattice structure. The effective density, effective specific heat, and effective latent heat of fusion can be easily obtained via the mixture rule as described in detail by Piacquadio et al. [23].

#### **3. Solver for Ablative TPS and Corrugated Core ITPS**

Using the governing equations described above, a software tool based on the finite volume method (FVM) for the calculation of the thermal response of both ablative and corrugated core ITPS is implemented in Python®. The implementation in Python® allows a simplified connection of the realised solver with different optimisation packages, which allow one to optimise the thermal mass. This way, an easy and accurate comparison of the two options is achievable. For this reason, the tool is named **Hot-St**ructure and **A**blative **R**eaction **Shi**eld **P**rogram (Hot-STARSHIP). Figure 5 shows a flowchart of the program.

**Figure 5.** Flowchart of the developed FVM solver. The dashed line indicates a shortcut for the non-decomposing case.
