## 2 Thermodynamic Analysis of the Solar ORC System with Zeotropic Mixtures
### Thermodynamic Model and Boundary Conditions  
A low-temperature solar-driven ORC system was simulated to provide the output parameters of the ORC using different zeotropic mixtures as WFs. The calculations are carried out with and without the regenerator to analyse the effect of its performance on every working fluid mixture. The ORC system performances are computed in respect of expressing the energy balance in every single component formerly defined, the thermal and electrical powers, efficiencies, and the cycle efficiency. The next hypotheses have been done in this study:  
* Steady state conditions are presumed. * The ORC system is powered by the net thermal power generated by a solar source in which the pump energy consumption and other inherent losses of the solar energy system loop are pre-accounted for. It is the case for the back-up as well. * Kinetic energy and potential energies of operating modes are neglected. * Pressure drops and heat losses are disregarded in the heat exchangers, condenser, evaporator, and pipes. * The flow directions of WFs in the heat exchangers, condenser, evaporators, and regenerator are counter-current. * The formulations of the zeotropic mixtures do not vary during the ORC mode. The inputs and outputs of the ORC cycle are provided in Tables 1 and 2, respectively. The equations developed to calculate the thermodynamic performance of each component of the ORC cycle with and without regeneration, based on Figures 1 and 2 and Tables 1 and 2, are given below:  
\[p_{1}=p_{6}=p_{6^{\prime}}=p_{7}=p_{6a} \tag{1}\]  
\[s_{1}=s_{2} \tag{2}\]  
\[p_{2}=p_{3}=p_{4}=p_{5}=p_{2a} \tag{3}\]  
\[s_{5}=s_{6^{\prime}} \tag{4}\]  
\[\eta_{E}=\frac{h_{5}-h_{6}}{h_{5}-h_{6}^{\prime}}=0.8 \tag{5}\]  
Solar collector-evaporator heat inlet without regenerator:  
\[Q_{E}=h_{5}-h_{2} \tag{6}\]  
Solar collector-evaporator heat inlet with regenerator:  
\[Q_{E}=h_{5}-h_{2a} \tag{7}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Input Description** & **Parameters** & **System Unit** \\ \hline WF#1 name & & - \\ WF#2 name & & - \\ WF#1 mass fraction & & - \\ WF#2 mass fraction & & - \\ Dead-state temperature & \(T_{0}\) & \({}^{\circ}\)C \\ Lowest temperature/condensation bubble point & \(t_{1}\) & \({}^{\circ}\)C \\ Evaporation dew-point temperature & \(t_{4}\) & \({}^{\circ}\)C \\ Highest temperature & \(t_{5}\) & \({}^{\circ}\)C \\ Mass flow rate & \(\dot{m}\) & kg/s \\ Expander efficiency & \(\eta_{I}\) & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Inputs of the ORC system. Expansion work outlet:  
\[W_{s}=h_{5}-h_{6} \tag{8}\]  
Heat transfer in regenerator:  
\[Q_{1}=h_{6}-h_{6a}=h_{2a}-h_{2} \tag{9}\]  
Condenser heat rejected without regenerator:  
\[Q_{C}=h_{6}-h_{1} \tag{10}\]  
Condenser heat rejected with regenerator:  
\[Q_{C}=h_{6a}-h_{1} \tag{11}\]  
Pump work inlet:  
\[W_{p}=h_{2}-h_{1} \tag{12}\]  
Net power outlet:  
\[W_{N}=W_{s}-W_{p} \tag{13}\]  
The thermodynamic efficiency of the ORC cycle is computed by means of the next equation:  
\[\eta_{I}=\frac{W_{N}}{Q_{E}}=\frac{W_{S}-W_{P}}{Q_{E}} \tag{14}\]  
Carnot cycle efficiency:  
\[\eta_{C}=1-\frac{T_{1}}{T_{5}} \tag{15}\]  
The second law efficiency is estimated based on a work by [PERSON] et al. (2018) on thermal analysis on zerotropic mixtures for ORC. The exergy of the absorbed heat is calculated using the following equation:  
\[\dot{E}_{Q}=\ \dot{\mathrm{Q}}_{\mathrm{eva}}\left(1-\frac{T_{0}}{T_{m,eva}}\right) \tag{16}\]  
where \(T_{0}\) is the dead-state temperature, \(\dot{\mathrm{Q}}_{\mathrm{eva}}\) is the heat absorbed by the evaporator and \(T_{m,eva}\) is the mean temperature of the evaporator. The latter is calculated as follows:  
\[W_{s}=h_{5}-h_{6} \tag{8}\]0  
where \(T_{h,in}\) and \(T_{h,out}\) are the inlet and outlet temperature of the working fluid, respectively. The second law efficiency is calculated as follows:  
\[W_{s}=h_{5}-h_{6} \tag{8}\]1  
where \(W_{N}\) and \(\dot{E}_{Q}\) are the net power output and the exergy of the absorbed heat, respectively.