## 3 Methodology: Thermal Performance Analysis  
Important design parameters of TES are the dynamic behavior and thermal losses. To investigate the slag-based TES in those areas, a simulation model is set up. The basis is set by a continuous solid phase model published by [PERSON] [26] and the following simplifications. Firstly, the solid is considered as a continuous porous medium and not as a medium composed of independent particles. Secondly, the heat transfer in the bed takes place by conduction. The radial heat exchange is neglected. Both phases are modeled separately. The applied energy equations for the fluid phase (Equation (1)) and solid (Equation (2)) side can be written as:  
\[\text{Fluid}:\ \varepsilon\rho/c_{p}\!\!\!/\left(\frac{d\overline{T}}{dt}+v_{ \infty}\frac{d\overline{T}}{d\overline{x}}\right)=\ k_{fx}\frac{d^{2}\overline{ T}}{d\overline{x}^{2}}+h_{sf}a_{p}\!\left(\overline{\theta}-\overline{T} \right)-U_{w}a_{w}\!\left(\overline{T}-\overline{T}_{0}\right) \tag{1}\]  
\[\text{Conduction}\ \ \ \text{Convection}\ \ \ \ \text{Heat loss}\]  
\[\text{Solid}:\ (1-\varepsilon)\rho_{s}c_{p}\frac{d\overline{\theta}}{dt}=\ k_{ xx}\frac{d^{2}\overline{\theta}}{d\overline{x}^{2}}+h_{sf}a_{p}\!\left(\overline{T}- \overline{\theta}\right) \tag{2}\]  
Both equations are structured in the same way. For example, in Equation (1), the sum of rate of change of enthalpy and the net enthalpy of the fluid flow is given by the term on the left side, where \(\varepsilon\) is the porosity, \(\rho\) is the density, \(c_{p}\) is the specific heat and \(v\) the velocity. The axial heat conduction is calculated with the first term on the right-hand side, where \(k_{\text{fx}}\) is the effective thermal conductivity of the fluid. The subsequent term represents the heat exchange by convection between the particles and the fluid. The heat transfer coefficient \(h_{\text{sf}}\) in the convection term is determined according to [PERSON]'s correlation for a bed of particles (Equation (3)) [28]. A correction factor \(f_{\text{a}}\) (Equation (4)) is required, by which the Nusselt number for one particle is multiplied. The reason for this is that the heat transfer coefficient is higher in case of a packed bed than in case of a single particle. With Equations (5) to (7) the Nusselt number of a single particle can be determined.  
\[Nu=f_{a}Nu_{particle} \tag{3}\]  
\[f_{a}=1+1.5(1-\varepsilon) \tag{4}\]  
\[Nu_{particle}=2+\sqrt{Nu_{lam}^{2}+\ Nu_{turb}^{2}} \tag{5}\]  
\[Nu_{particle}=2+\sqrt{Nu_{lam}^{2}+\ Nu_{turb}^{2}} \tag{6}\]  
\[Nu_{turb}=\frac{0.037\ Re^{0.8}\ Pr}{1+2.443\ Re^{-0.1}(Pr^{2/3}-1)} \tag{7}\]  
Two types of thermal losses are taken into account. On the one hand, the heat loss from the storage wall to the external environment, which is calculated with the last term in Equation (1). On the other hand, the radiation losses in the cold and hot distributor section of the TES tank are considered  
Figure 2: Sketch of the lead concept (**a**) and its specifications (**b**) [27].  
by implementing Equation (8) into the thermal model. \(\varepsilon^{*}\) is the degree of emission and \(\sigma\) the Stefan Bolzman constant.  
\[\dot{Q}_{12}=\varepsilon^{*}\sigma\,A\left(T_{1}^{4}-T_{2}^{4}\right) \tag{8}\]  
The lead concept was already examined in detail by [PERSON] et al. in [24]. The results obtained indicated the applicability of steelmaking slag as a TES inventory material. To confirm these outcomes, a model validation and experimental investigations is performed subsequently.