## 4 Results and Discussion  
Different types of volumetric expanders can be potentially applied in the considered system, such as gerotor, scroll, screw, piston, and multi-vane expanders [43, 44]. Each type of expander has its own operating characteristics, geometry, and design [45, 46]. A previous study [16] examined and discussed the possibility of applying volumetric expanders in different operating conditions, including two-phase expansion. This investigation shows that, compared to reciprocating volumetric expanders (i.e., piston expanders), some rotary volumetric expanders (e.g., multi-vane machines) tend to work under two-phase conditions without any serious issues. Therefore, the rotary type of two-phase volumetric expanders may be suitable for application in a reversible Rankine-based Carnot battery. When designing the volumetric expanders, two important parameters, namely, pressure and volumetric expansion ratio, must be considered. These parameters can be expressed as described in Equations (18) and (19):  
\[\delta_{\text{P}}=\frac{p_{\text{in}}}{p_{\text{out}}}=\frac{p_{2}}{p_{3}}. \tag{18}\]  
\[\delta_{\text{V}}=\frac{\dot{V}_{\text{in}}}{\dot{V}_{\text{out}}}=\frac{ \dot{m}_{\text{in}}/\rho_{\text{in}}}{\dot{m}_{\text{out}}/\rho_{\text{out}}} =\frac{\rho_{\text{out}}}{\rho_{\text{in}}}=\frac{\rho_{3}}{\rho_{2}}, \tag{19}\]  
where subscripts \(p\) and \(V\) represent the pressure and volume, respectively. Moreover, \(p\), \(\dot{V}\), and \(\rho\) describe pressure, volumetric flow rate, and density, respectively. The MATLAB configuration model was validated using experimental data from a previous study [47] using a multi-vane expander, and both modeling and experimental results are plotted in Figure 2. The validated results show that the MATLAB configuration model has a 1-3% relative error. Some parameters, such as heat losses, internal friction, and other characteristics that were not considered in the modeling, may result in inaccuracy. In this case, only isentropic expander efficiency was established in the configured model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **ORC Mode** & **Heat Pump Mode** \\ \hline HE1 pinch point (K) & 5 & 5 \\ HE2 pinch point (K) & 5 & 5 \\ HE1 secondary fluid temperature (K) & 348.15—348.15 + 10 & 348.15—348.15 + 10 \\  & (from storage) & (to storage) \\  & 288.15 & 338.15 \\ HE2 secondary fluid temperature (K) & (from air) & (from waste heat) \\ Simulation temperature step (K) & 1 & 1 \\ Isentropic efficiency of expander (-) & 0.8 and 0.7 & \(\eta_{\text{EX,is}}\) < 0.8 \\ Isentropic efficiency of a pressure & 0.8 (a liquid pump) & 0.8 (a compressor) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The reversible Rankine-based thermodynamic cycle (RRTC) setup applied in this study. Using the data from the MATLAB model appears to be a good approach for obtaining the expected result during the design stage because the relative error was observed to be less than 10%. The following section presents and discusses the modeling data based on the working fluid thermal properties concerning the power of the expander if it runs in both modes (i.e., heat pump and ORC modes). The obtained modeling results shown in pressure ratio-power and volumetric ratio-power diagrams are presented in Figure 3. This figure shows that, with the assumption of a mass flow rate at 1 kg/s for both operating modes, the generated power varies as a function of pressure and volumetric expansion ratios. Figure 3 illustrates the variation in the two-phase volumetric expander power operating in ORC and heat pump modes. The output power of this expander for ORC mode is around 23-43 kW, whereas for heat pump mode it is around 1.4-5.5 kW. The result shows a big difference between the obtained expander output power for ORC and heat pump modes. In heat pump mode, the expander generates less power than in ORC mode. This is because the change in the specific enthalpy of the working fluid is a function of the temperature range, which, for the heat pump and the ORC, is also different.