## 2 Modeling
### Thermal Storage System  
The basis for investigations on electrical heating within the storage system are transient models for the solid medium and heating wire. The heat balance equation for the cylindrical honeycomb structure (index \(S\)) is based on a 2D porosity model [27], while the open heating wire (index \(P\)) is based on a 0D model due to negligible local temperature gradients. A schematic representation of the thermal energy storage system with the associated relevant variables is shown in Figure 1.  
Figure 1: Channel shaped solid medium with integrated heating wire and surrounding thermal insulation (convective discharging mode with bypass-configuration not illustrated—see [21]). Considering the honeycomb void fraction \(\varepsilon\) and volume \(V\), the axial \(z\) and radial \(r\) temperature profile \(T_{S}\) over time \(t\) is obtained from Equation (1):  
\[(1-\varepsilon)\rho_{S}c_{S}\frac{\partial T_{S}}{\partial t}=\lambda_{S}\frac{ \partial^{2}T_{S}}{\partial z^{2}}+\lambda_{S,r}\left(\frac{\partial^{2}T_{S}} {\partial r^{2}}+\frac{1}{r}\frac{\partial T_{S}}{\partial r}\right)+k_{rad} \frac{O_{S}}{V}\ x\left(T_{P}^{4}-T_{S}^{4}\right) \tag{1}\]  
and the temperature of the heating wire \(T_{P}\) with a volume of \(V_{P}\) at an electrical heating power \(P\) according to Equation (2):  
\[\rho_{P}c_{P}\frac{\partial T_{P}}{\partial t}=k_{rad}\frac{O_{S}}{V_{P}}\ x \left(T_{S}^{4}-T_{P}^{4}\right)+\frac{P}{V_{P}} \tag{2}\]  
Here, the parameter \(x\) (heating wire assignment) was implemented to capture the homogeneously distributed number of honeycomb channels with heating wire in relation to the maximum number of channels. The heat transport between the heating wire and the honeycomb surface \(O_{S}\) takes place via the effective radiation coefficient \(k_{rad}\), which includes radiation and conduction effects. A model-based determination of this coefficient as a function of the relevant influencing variables is described in Sections 2.2 and 3.1. Both parameters were implemented at the first time inside a porous based approach to allow time-efficient simulations of such a storage system without losing geometrical related information regarding the local structure. For the density \(\rho\), the specific heat capacity \(c\) and the thermal conductivities \(\lambda\), temperature-averaged material quantities are used. The calculation of the effective thermal conductivity in radial direction (\(\lambda_{r}\)) is defined according to [28]. This value considers the effect of void fraction inside the channel shaped honeycomb structure leading to a lower radial heat transport. In addition to the transient heat balance equations, axial and radial boundary conditions for the honeycomb structure are needed. These include heat losses to the ambient temperature \(T_{U}\) at the axial frontal areas (Equation (3)) at \(z=0\)\(\&\)\(z=L\) and at the shell surface at \(r=R\) (Equation (4)) as well as symmetrical boundary conditions at \(r=0\) (Equation (5)). \[\lambda_{S}\frac{\partial T_{S}}{\partial z}\bigg{|}_{z=0\ /\ z=L}=k_{Ins,z} \left(T_{U}-T_{S}\right) \tag{3}\]  
\[\lambda_{S,r}\frac{\partial T_{S}}{\partial r}\bigg{|}_{r=R}=k_{Ins,r}\left(T _{U}-T_{S}\right) \tag{4}\]  
\[\left.\frac{\partial T_{S}}{\partial r}\bigg{|}_{r=0}=0\right. \tag{5}\]  
The stated heat transfer coefficients \(k_{Ins,z}\) and \(k_{Ins,r}\) include heat conduction through the thermal insulation (index \(Ins\)) as well as convective heat transfer (\(\alpha_{U}\)) to the environment according to [29].