*2.2. Energy Equation for ITPS*

For a passive, non-ablative TPS, the energy equation can be written as a special case of the already described one for an ablative material, in which ablation does not take place.

$$\frac{\partial}{\partial t} \left( \rho\_{eff} \mathbb{C}\_{p\_{eff}} T \right) + \nabla \cdot \left( -\vec{\lambda}\_{eff} \cdot \nabla T \right) = 0,\tag{12}$$

where *ρeff* is the effective density of the material, *Cpeff* its effective specific heat capacity, and *λ* ¯¯ *eff* is the effective thermal conductivity tensor. However, a special treatment is needed to describe the behaviour of the ITPS embedding a PCM. This is done by means of the apparent heat capacity formulation [30]. The term corresponding to the latent heat of fusion is included as an additional non-linear term in the definition of the heat capacity of the material:

$$
\rho\_{eff} \mathbf{C}\_{p\_{eff}} = \frac{\partial H}{\partial T} = \mathbf{C}\_{eff} + L \frac{\partial \mathbf{a}\_l}{\partial T},\tag{13}
$$

where *Ceff* is the actual heat capacity, *H* is the enthalpy, *L* is the PCM latent heat of fusion, and *α<sup>l</sup>* is the liquid fraction at the melting front.
