## 2 Modeling
### Effective Radiation Coefficient  
The electrical heating of the storage system is based on a heating wire passing the honeycomb channels several times and heats the solid via thermal radiation and conduction. A central value for this system is given by the effective radiation coefficient \(k_{rad}\), which includes both heat transport mechanism. A consideration of the individual heat transport mechanisms in \(k_{rad}\) is necessary, since a detailed geometric representation of the honeycomb structure is not covered by the computationally efficient porosity model. For its determination as a function of the relevant influencing variables, a geometry-resolved model of the honeycomb structure is required serving as a basis for a novel parametrizable formulation of the effective radiation coefficient \(k_{rad}\). A schematic representation of the detailed model is shown in Figure 2.  
The detailed model-divided into three zones-is based on symmetrical section of a single honeycomb channel with integrated heating wire (zone I), the surrounding solid wall of the honeycomb (zone II) and the honeycomb structure without heating wires (zone III). The corresponding radii of the individual zones can be determined from geometric variables related to the honeycomb structure and the heating wire assignment \(x\). For a honeycomb with a void fraction \(\varepsilon\) and a specific surface \(a_{V}\), the radius \(r_{0}\) results from Equation (12):  
\[r_{0}=\frac{2}{a_{V}}\varepsilon \tag{12}\]  
the radius \(r_{1}\) from Equation (13):  
\[r_{1}=\frac{2}{a_{V}}\sqrt{\varepsilon} \tag{13}\]  
and the radius \(r_{2}\) from Equation (14):  
\[r_{2}=\frac{2}{a_{V}}\sqrt{\frac{\varepsilon}{x}} \tag{14}\]  
Within the solid honeycomb wall (zone II), the radial transient heat balance is based on Equation (15):  
\[\rho_{S}c_{S}\frac{\partial T_{S,1}}{\partial t}=\lambda_{S}\bigg{(}\frac{ \partial^{2}T_{S,1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial T_{S,1}}{ \partial r}\bigg{)} \tag{15}\]  
and for the surrounding porous honeycomb structure (zone III) on Equation (16):  
\[(1-\varepsilon)\rho_{S}c_{S}\frac{\partial T_{S,2}}{\partial t}=\lambda_{S,r }\bigg{(}\frac{\partial^{2}T_{S,2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial T _{S,2}}{\partial r}\bigg{)} \tag{16}\]  
The boundary conditions include for the solid honeycomb wall (Zone II) heat transport by radiation at the heating wire temperature \(T_{P}\) according to Equation (17), for the porous honeycomb structure (Zone III) adiabatic conditions according to Equation (18) and for the interface between the two zones heat conduction according to Equation (19). \[\lambda_{S}\frac{\partial T_{S,1}}{\partial r}\bigg{|}_{r=r_{0}}=C_{rad}\ \bigg{(}T_{P}^{4}-T_{S,1}^{4}\bigg{)} \tag{17}\]  
\[\left.\frac{\partial T_{S,2}}{\partial r}\right|_{r=r_{2}}=0 \tag{18}\]  
\[\lambda_{S}\frac{\partial T_{S,1}}{\partial r}\bigg{|}_{r=r_{1}}=\left. \lambda_{S,r}\frac{\partial T_{S,2}}{\partial r}\right|_{r=r_{1}} \tag{19}\]  
The pure radiation parameter \(C_{rad}\) in Equation (17) is defined for a configuration with concentric cylinders. With regard to the honeycomb surface at \(r=r_{0}\), the radiation parameter \(C_{rad}\) is calculated as described in Equation (20):  
\[C_{rad}=\sigma\frac{\beta_{S}\ \beta_{P}\ \varphi_{SP}}{1-(1-\beta_{S})\ (1-\beta_{P})\ \varphi_{SP}\ \varphi_{PS}} \tag{20}\]  
\[r_{0}=\frac{2}{a_{V}}\varepsilon \tag{12}\]0 Heat transport model: radiation between heating wire and honeycomb (zone I), heat conduction in the surrounding solid wall (zone II) and the porous structure (zone III) of the honeycomb. with \(\beta\) as emission coefficients, \(\sigma\) as the Stefan-Boltzmann constant, \(\varphi_{SP}\) and \(\varphi_{PS}\) as geometry-dependent parameters [31] referring to the honeycomb (Index \(S\)) and heating wire dimension (Index \(P\)), respectively.