## 3 A Novel Model of the Reversible Rankine-Based Thermodynamic Cycle (RRTC)
### The Mathematical Description of RRTC in Heat Pump Mode  
The process of the RRTC in heat pump mode starts at HE1, whose power is directly related to the heat transfer rate needed to heat/cool the space. In the mathematical description, it can be calculated as an expression of enthalpy changes, in the same was as in Equation (2) according to energy balance. However, it has to be noted that the processoperates in the opposite direction. This means that using the same heat exchanger, in heat pump mode, the HE1 behaves as a condenser instead of an evaporator. Therefore, a special design of HE1 has to be implemented. The condensation process in HE1 starts from the superheated state and proceeds to the liquid state (\(x=0\)), which is referred to the states numbered 1 and 2 (blue color) in Figure 1b,c. Therefore, the process of HE1 in heat pump mode can be expressed as a function of enthalpy changes, described as Equation (12):  
\[\dot{m}_{\text{out}}\Delta h_{\text{out}}=\ \dot{m}_{\text{wf}}\left((\left.c_{ \text{p,wf}}\right|_{T_{2}}^{T_{\text{sat,v}}}\Delta T_{2\to\text{sat,v}})+ \Delta h_{\text{sat,v}\to 1}\right), \tag{12}\]  
where the subscript v represents the vapor.  
Once the working fluid reaches a saturated liquid state (\(x=0\)), which is indicated by state 2 (blue color) in Figure 1b,c, the working fluid needs to be expanded. In small heat pumps, the throttle valves were often used as the expansion devices. Because this novel RRTC uses an innovative expansion device, the expander is used to expand the working fluid; therefore, no throttle valve is installed. It should be noted that the use of expansion devices for expanding saturated liquids is a realistic assumption; similar devices are used in flash cycles [38]. The expansion process of the RRTC in heat pump mode can be described as in Equations (4) and (5).  
The following process is the evaporation during which the working fluid changes the phase from two-phase (\(0<x<1\)) to saturated vapor (\(x=1\)) or superheated state, as presented in Figure 1b,c, indicated with the state 3 and 4 (blue color). Although the same heat exchanger (HE2) can be utilized in the RRTC operating in heat pump mode, the process has a different direction, as shown in Figure 1b,c. The evaporation process in HE2 can be expressed as Equation (7) for the enthalpy change. Therefore, the process of HE2 in heat pump mode can be described as Equation (13):  
\[\dot{m}_{\text{in}}\Delta h_{\text{in}}=\ \dot{m}_{\text{wf}}\Delta h_{3\to 4}. \tag{13}\]  
The possible application of plate heat exchangers as a reversible device (condenser and evaporator) has been discussed elsewhere [39]. After the previous step, the working fluid is ready to be pressurized with the help of a compressor. Typically, the compressor process is similar to using a pump to increase the working fluid's pressure, but in the vapor/gas phase. The compressor power and isentropic efficiency can be expressed using the change in the specific enthalpy of the working fluid, as described in Equations (14) and (15):  
\[P_{\text{CPR}}=\ \dot{m}_{\text{wf}}\Delta h_{4\to 1_{\text{is}}}/\eta_{\text{CPR, is}}, \tag{14}\]  
\[\eta_{\text{CPR,is}}=\ \frac{\Delta h_{4\to 1_{\text{is}}}}{\Delta h_{4\to 1}}. \tag{15}\]  
Therefore, the coefficient of performance of the RRTC operating in heat pump mode can be expressed as Equation (16):  
\[COP_{\text{RRTC,heat pump}}=\frac{P_{\text{CPR}}-P_{\text{EXR}}}{\bar{Q}_{ \text{HE1}}}. \tag{16}\]