## 3 Modelling
### Solid Media Thermal Energy Storage Systems  
For modelling the discharge procedure of a solid media thermal energy storage system, a one-dimensional cylindrical heterogeneous porous continuum approach is used, providing the axial temperature characteristics for the gaseous (\(F\)) and the solid phase (\(S\)) in time (\(t\)) and space (\(z\)) [28]. Under ideal assumptions with neglections of thermal losses and of the axial heat conduction term due to low heat conductivities for the fluid and solid phase compared to the convective part, the thermal energy balances can be simplified to  
\[\varepsilon\rho_{F}c_{F}\Bigg{(}\frac{\partial T_{F}}{\partial t}+w\frac{ \partial T_{F}}{\partial z}\Bigg{)}=ka_{V}(T_{S}-T_{F}) \tag{1}\]  
\[(1-\varepsilon)\rho_{S}c_{S}\frac{\partial T_{S}}{\partial t}=ka_{V}(T_{F}-T_ {S}) \tag{2}\]  
where the fluid and solid medium temperature are represented by \(T_{F}\) and \(T_{S}\), the fluid velocity by \(w\), the void fraction by \(\varepsilon\) and the specific heat transfer surface by \(a_{V}\). The total regenerator heat transfer coefficient \(k\) is calculated according to Hausen [29; 30] including in a compact way the temporal averaged heat resistance inside the solid phase as well as the heat transfer coefficient of the fluid for the investigated channel-shaped bricks [29]. Additionally, temperature dependent material propertiessuch as the density \(\rho\) and specific heat capacity \(c_{P}\) are needed, given by the selected solid medium as well as by a real gas model for air [31].  
After defining the inlet boundary conditions for the fluid phase, the adiabatic conditions for the solid phase  
\[T_{F}|_{z=0}=T_{U} \tag{3}\]  
\[\frac{\partial T_{S}}{\partial z}\bigg{|}_{z=0\ \&\ z=L}=\ 0 \tag{4}\]  
and by using a central and a backward finite-difference-method in space for the solid and for the fluid phase, the resulting set of differential algebraic equations are solved in time with a commercial simulation tool (Matlab). Mesh studies were performed in order to allow a deviation of less than \(10^{-4}\) relating to the fluid outlet temperature leading to an axial discretization of at least 50 nodes.  
For qualifying the thermal efficiency of different channel-shaped geometric storage medium solutions, a central value for such types of thermal energy storages is introduced in Equation (5). This quantity--the thermal utilization coefficient (\(\eta_{stor}\))--represents the ratio of effective to maximal possible discharged thermal energy and is used in the following for comparing different geometric solutions.  
\[\eta_{stor}=\frac{c_{S}T_{PH}-\int_{0}^{1}[c_{S}T_{S}(z,t=\tau)]\frac{dz}{L}}{ c_{S}T_{PH}-c_{S}T_{U}} \tag{5}\]  
Aside of the described modelling details, a simulation control procedure is integrated inside the unsteady simulation runs to divide the inlet mass flow in a bypass and a storage medium passing part. In an iteratively operating way, the bypass flow during discharging period at ambient condition and the storage medium exiting flow at elevated temperature is controlled in order to fulfil the specifications of a constant mixed outlet temperature. A maximum deviation inside the iteratively simulation runs of less than \(10^{-4}\) relating to the constant mixed outlet temperature was allowed.