## 4 Results and Discussion
One significant problem that has to be considered when designing this volumetric expander for the ORC and the heat pump in the RRTC is that the volumetric machine needs to cover the operating pressure ratio of the system working in heat pump and ORC modes. As illustrated in Figure 3a, assuming that the mass flow rate of the working fluids equals 1 kg/s, the resulting pressure expansion ratio in heat pump mode is almost one-third that in ORC mode (the pressure ratio in heat pump mode ranges between 1.47 and 1.77, whereas in ORC mode it ranges between 3.09 and 3.70). In addition, the applied volumetric expander needs to be fitted into the range of the volumetric ratio based on the obtained results, which are illustrated in Figure 3b. The results show that the range of the volumetric expansion ratio for ORC mode is lower than that for heat pump mode. The volumetric expansion ratio for ORC mode varies in a range of 0.011-0.015, whereas that for heat pump mode varies in a range of 0.33-0.41. Therefore, pressure and volumetric expansion ratio coverage need to be carefully considered in selecting suitable two-phase rotary volumetric expanders. Figure 2: The obtained data from the MATLAB configuration validated using the experimental data from prior work. As illustrated in Figure 3, the obtained results were modeled in different isentropic efficiency conditions. A prior study [48] discussed the empirical method based on Baumann's rule, which bounds the isentropic efficiency of the expander and vapor quality, and is defined as Equation (20):  
\[\eta_{\text{EXR,is,wet}}=\eta_{\text{EXR,is,dry}}(1-\beta_{\text{B}}(1-x)), \tag{20}\]  
Figure 3: The obtained power of the two-phase expander applied in the RRTC in ORC and heat pump modes in (**a**) pressure ratio–expander power diagram and (**b**) volume ratio–expander power diagram. Because the throttle valve is replaced by a volumetric expander in the RRTC heat pump, the power of the expander can be calculated not only for the ORC mode, but also for the heat pump mode. where \(\beta\) is the Baumann coefficient. This empirical method shows that the expander's isentropic efficiency will gradually decrease. Therefore, as illustrated in Figure 1c, the expansion process in heat pump mode may start at the lowest vapor quality near the saturated liquid state (\(0<x<0.2\)). This is the so-called flashing process. The flashing here occurs in the two-phase region, which can be visualized in the \(T\)-s diagram, referring to Figure 1c for the process 2-3 (blue lines). The design of the two-phase volumetric expander may be conducted based on the features of the volumetric machine operating in ORC mode, in which the higher isentropic efficiency of the expander needs to be achieved. In this case, the isentropic efficiency of the expander in heat pump mode will follow. However, it is necessary to predict this efficiency and design the system so that the good _COP_ can be achieved. The _COP_ and ORC efficiency are important assessment parameters that have to be considered in this Carnot battery technology; therefore, the values of these parameters were modeled in the present study. The obtained results are plotted in Figure 4, in a _COP_-ORC efficiency diagram. The results show that increasing the ORC efficiency will decrease _COP_ based on the designed RRTC, as illustrated in Figure 1. Figure 4a,b visualizes the results based on the change in expander isentropic efficiency. As discussed above, the design of the expander may be based on the ORC mode. From Figure 4, it can be seen that changing the isentropic efficiency of the expander for ORC mode significantly influences the overall ORC efficiency. It shows that, with an expander isentropic efficiency of 0.8, the overall ORC efficiency ranges between 0.09 and 0.10, whereas, for an expander isentropic efficiency of 0.7, the ORC efficiency drops to 0.08.