## 5 Conclusions  
This paper discusses the result of a modeling analysis of the possible application of two-phase volumetric expanders for a novel reversible Rankine-based cycle for Carnot batteries. The results of this study show a significant difference in the values of the assessment parameters, such as pressure ratio, volumetric expansion ratio, and output power for the same volumetric expander working in ORC and heat pump modes. These parameters have to be considered together during the design and selection of two-phase volumetric expanders that can operate in both ORC and heat pump modes. To implement a single expander to generate power from ORC mode and reduce the power consumption of heat pump mode (by installing an expander instead of a throttle valve), it is suggested to design the expander based on the values of the assessment parameters obtained for ORC mode, and then fit the design in a way that covers the pressure and volumetric expansion ratio needed for the system operation in heat pump mode.  
Figure 5: The variation in \(\zeta\) for different values of \(T_{\text{HE1}}\) temperature and different expander isentropic efficiency.  
The obtained result of the RRTC modeling simulation using propane as the working fluid shown in the power-pressure ratio diagram illustrated in Figure 3a shows that, if one expander is applied for both cycles (ORC and heat pump modes), this expander needs to cover the pressure ratio between 1.4 (minimal) and 3.7 (maximal). In addition, this device is also required to cover the volumetric ratio between 0.01 (minimal) and 0.41 (maximal), as depicted in the power-volumetric ratio diagram (Figure 3b). Therefore, it is suggested to design the expander in the RRTC mentioned above having a pressure ratio of 2.55 \(\pm\) 1.15 and a volumetric ratio of 0.21 \(\pm\) 0.105. It may be possible to obtain these design parameters if rotary-type volumetric machines, such as multi-vane expanders, are applied in the RRTC system. Using these parameters, performance of Carnot battery technology in the range of 0.50-0.98 may be achieved with an HE1 temperature in ORC mode of 343.15-353.15 K.  
Conceptualization, methodology, software, validation, formal analysis, investigation, visualization, writing--original draft preparation, S.D.; writing--review and editing, S.D., P.K., A.R.I; supervision, P.K., A.R.I.; All authors have read and agreed to the published version of the manuscript.  
This work was supported by NAWA STER Program Internationalization of Wroclaw University of Science and Technology Doctoral School.  
Not applicable.  
The authors would like to thank [PERSON] and [PERSON] for the invitation to publish this article.  
The authors declare no conflict of interest.  
The specific heat capacity (J/kg\(\cdot\)K)  
The specific enthalpy (J/kg)  
The specific work of expander (kJ/kg)  
Mass flow rate (kg/s)  
Power (Watt)  
Pressure (Pa)  
Temperature (K)  
Heat transfer rate (Watt)  
Volumetric flow rate (m\({}^{3}\)/s)  
Vapor quality (\(-\))  
Greek letters:  
Baumann coefficient  
Ratio (\(-\))  
Efficiency (\(-\))  
Performance of Carnot battery (\(-\))  
Density (kg/m\({}^{3}\))  
Subscripts:  
1,2,3,4 Process in the cycle (blue color for heat pump mode, red color for ORC mode)  
B
Baumann rule  
CPR  
Compressor  
ExR  
Expander  
GM  
Reversible motor-generator  
HE  
Heat exchanger  
in  
Input  
is  
Isentropic process  
1 Liquid  
LP  
Liquid pump  
out  
Output