# Investigation of Different Storage Systems for Solar-Driven Organic Rankine Cycle
## 2 Material and Methods
### Mathematical Formulation Part
#### 2.2.3 Storage Modeling  
In this work, three different storage systems were investigated as they are presented in Figure 2. Figure 2a shows a typical storage system with thermal oil as the sensible heat storage material. Figure 2b shows a system with thermal oil and rocks inside the tank in order to increase the tank's thermal capacity. The last examined system (Figure 2c) is the one with phase change material (PCM) inside the tank. There are tubes in the storage tank where the thermal oil flows and exchanges heat with the filling material, which is the PCM. These tubes have fins in order to increase the heat transfer rates between oil and PCM. It is also useful to state that there are different tubes for charging and discharging processes.  
The system with pure thermal oil (Figure 2a) is modeled by using the following formulas. The general energy balance can be written as:  
\[Q_{st}=Q_{u}-Q_{\textit{lrs}}-Q_{\textit{loss}}, \tag{15}\]  
where the stored heat (\(Q_{\textit{stor}}\)) can be written by the energy balance in the fluid-volume:  
\[Q_{\textit{stor}}=\rho_{st}\cdot c_{p,st}\cdot V.\frac{dT_{st}}{dt}. \tag{16}\]The storage tanks' thermal losses (\(Q_{loss}\)) are estimated as follows:  
\[Q_{loss}=A_{T}\cdot U_{T}\cdot(T_{st}-T_{amb}). \tag{17}\]  
For a cubical tank, its outer area (\(A_{T}\)) is calculated as:  
\[A_{T}=6\cdot V^{\frac{3}{2}}. \tag{18}\]  
The system with the thermal oil-rocks can be modeled by using the Equations (15)-(18) but by modifying the density and the specific heat capacity as follows [29; 30]:  
\[\rho_{st}=\varepsilon\cdot\rho_{oil}+(1-\varepsilon)\cdot\rho_{solid}, \tag{19}\]  
\[c_{p,st}=\frac{\varepsilon\cdot\rho_{oil}c_{p,oil}+(1-\varepsilon)\cdot\rho_{ solid}\cdot c_{p,solid}}{\varepsilon\cdot\rho_{oil}+(1-\varepsilon)\cdot\rho_{ solid}}, \tag{20}\]  
where the subscript \"solid\" indicates the rock and the \"oil\" the thermal oil, while the \"st\" is the total system. Moreover, the void fraction (\(\varepsilon\)) is defined as follows:  
\[\varepsilon=\frac{V_{oil}}{V_{oil}+V_{solid}}=\frac{V_{oil}}{V_{st}}. \tag{21}\]  
In this work, the void fraction was selected at 40%, which is a usual value in the literature [31; 32]. In this work, different materials were studied as possible rocks of the storage system. Table 2 includes the examined materials and their properties (density and specific heat capacity) [33]. More specifically, the studied materials were quartzite, basalt, concrete, bricks, and ceramic.  
The last examined storage system is the use of phase change materials. In this work, a proper material with a high melting temperature was selected in order to operate the ORC in high temperatures and to give the possibility for high energetic performance. For this reason, the NaNO\({}_{3}\) material was selected with melting temperature at 308 \({}^{\circ}\)C and latent heat at 174 kJ/kg [34]. In this work, the PCM  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & \(\rho\) **(kg/m\({}^{3}\))** & \(c_{p}\) **(J/kg K)** \\ \hline Quartzite & 2600 & 850 \\ Basalt & 2900 & 900 \\ Concrete & 2200 & 850 \\ Bricks & 3200 & 800 \\ Ceramic & 3550 & 900 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The examined storage materials as the solid material (rocks) [33].  
Figure 2: Storage tank technologies (**a**) Storage tank with pure thermal oil (**b**) Storage tank with rocks and thermal oil (**c**) Storage tank with PCM and finned tubes with thermal oil.  
was assumed to have its melting temperature and its exchanges heating with the thermal oil by using the following equations [35]:  
\[n_{hex,charge}=\frac{T_{c,out}-T_{c,in}}{T_{c,out}-T_{pcm}}, \tag{22}\]  
\[n_{hex,charge}=\frac{T_{c,out}-T_{c,in}}{T_{c,out}-T_{pcm}}. \tag{23}\]  
This work used a finned configuration and so a high efficiency of 90% was assumed for both charging and discharging efficiencies.