## 3 Thermodynamic Analysis
### Combined ORC-Single Evaporator VCR Cycle Equations  
Exergy of a point can be expressed as:  
\[\mathrm{e_{i}}=(\mathrm{h_{i}}\mathrm{-h_{0}})\mathrm{-T_{0}}\mathrm{\cdot(s_{i} \mathrm{-s_{0}})} \tag{1}\]  
Flow exergy of any point in the system can be expressed as:  
\[\mathrm{\dot{E}_{i}}=\mathrm{\dot{m_{i}}\mathrm{\cdot e_{i}}} \tag{2}\]  
Thermal and exergy efficiency calculations for combined ORC-single evaporator VCR cycle can be seen from below mentioned equations.  
Coefficient of performance of refrigeration cycle:  
\[\mathrm{COP_{VCR}}=\frac{\mathrm{\dot{Q}_{evap}}}{\mathrm{\dot{W}_{comp}}} \tag{3}\]  
Thermal efficiency of ORC:  
\[\mathrm{\eta_{IORC}}=\frac{\mathrm{\dot{W}_{turb}}}{\mathrm{\dot{Q}_{boil}}+ \mathrm{\dot{W}_{pump}}} \tag{4}\]  
Coefficient of performance of combined system:  
\[\mathrm{COP_{sys}}\mathrm{=\eta_{IORC}\mathrm{-COP_{VCR}}} \tag{5}\]  
Carnot efficiency of ORC:  
\[\mathrm{\eta_{Carnot,ORC}}\mathrm{=1}\mathrm{-\frac{T_{L,ORC}}{T_{H,ORC}}=1 \mathrm{-\frac{T_{0}}{T_{H,ORC}}}} \tag{6}\]  
Exergy efficiency of ORC:  
\[\mathrm{\eta_{lex,ORC}}=\frac{\mathrm{\eta_{IORC}}}{\mathrm{\eta_{Carnot,ORC}}} \tag{7}\]  
Carnot coefficient of performance:  
\[\mathrm{COP_{Carnot,VCR}}=\frac{T_{L,VCR}}{T_{H,VCR}\mathrm{-T_{L,VCR}}=\frac{ T_{L,VCR}}{T_{0}\mathrm{-T_{L,VCR}}}} \tag{8}\]  
Exergy efficiency of refrigeration cycle:  
\[\mathrm{\eta_{ex,VCR}}=\frac{\mathrm{COP_{VCR}}}{\mathrm{COP_{Carnot,VCR}}}= \frac{\mathrm{\dot{Q}_{evap}\mathrm{\cdot(\frac{T_{0}}{T_{L,VCR}}\mathrm{-1})} }}{\mathrm{\dot{W}_{comp}}} \tag{9}\]  
Exergy efficiency of combined system:  
\[\mathrm{\eta_{ex,sys}}\mathrm{=\eta_{ex,ORC}\mathrm{\cdot\eta_{ex,VCR}}} \tag{10}\]  
Energy and exergy equations related to components in refrigeration cycle of combined system are given in Table 4.  
Energy and exergy equations related to components in the organic Rankine cycle of combined system are given in Table 5.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Component** & **Energy and Energy Equations** \\ \hline \multirow{6}{*}{COMPRESSOR} & \(\hat{W}_{\text{comp}}=\hat{m}_{\text{VCR}}(\text{h}_{2}\text{-h}_{1})\) \\  & \(\hat{\text{E}}_{\text{dest,comp}}=\hat{\text{E}}_{1}-\hat{\text{E}}_{2}+\hat{W}_{ \text{comp}}\) \\  & \(\text{EDR}_{\text{comp}}=\hat{\text{E}}_{\text{dest,comp}}/\hat{\text{E}}_{ \text{dest,total}}\) \\  & \(\eta_{\text{ll,comp}}=\frac{\hat{\text{E}}_{\text{E}}-\hat{\text{E}}_{1}}{ \hat{\text{W}}_{\text{comp}}}\) \\  & \(\text{v}_{\text{comp}}=\frac{\hat{\text{E}}_{\text{E}}}{\hat{\text{W}}_{\text{ comp}}+\hat{\text{E}}_{1}}\) \\ \hline \multirow{6}{*}{VCR CONDENESR} & \(\hat{\text{Q}}_{\text{cond,VCR}}=\hat{m}_{\text{VCR}}(\text{h}_{2}\text{-h}_{3})\) \\  & \(\hat{\text{E}}_{\text{dest,cond,VCR}}=\hat{\text{E}}_{2}-\hat{\text{E}}_{3}( -1-\frac{\hat{\text{E}}_{1}}{\hat{\text{T}}_{\text{VCR}}})\hat{\text{Q}}_{ \text{cond,VCR}}\) \\  & \(\text{EDR}_{\text{cond,VCR}}=\hat{\text{E}}_{\text{dest,cond,VCR}}/\hat{\text{E}}_{ \text{dest,total}}\) \\  & \(\eta_{\text{ll,cond,VCR}}=\frac{\hat{\text{E}}_{\text{cond,VCR}}}{\hat{\text{E}}_{ \text{dest,$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ $\text{$\text{$\text{$\text{$\text{$\text{$\text{$}}}}}}}$}$}}}}}}}=0\) \\  & \(\hat{\text{E}}_{\text{Q},\text{cond,VCR}}=(1-\frac{\hat{\text{E}}_{\text{T}}}{ \hat{\text{T}}_{\text{VCR}}})\hat{\text{Q}}_{\text{cond,VCR}}=0\) \\  & \(0\) (\(\text{T}_{\text{H},\text{VCR}}=\text{T}_{0}\)) \\  & \(\text{v}_{\text{cond,VCR}}=0\) \\ \hline \multirow{6}{*}{EXPANSION VALVE} & \(\hat{\text{E}}_{\text{dest,v}}=\hat{\text{E}}_{\text{dest,v}}\) \\  & \(\text{EDR}_{\text{ev}}=\hat{\text{E}}_{\text{dest,v}}/\hat{\text{E}}_{\text{dest, total}}\) \\  & \(\text{v}_{\text{ev}}=0\) \\ \hline \multirow{6}{*}{EVAPORATOR} & \(\hat{\text{Q}}_{\text{evap}}=\hat{\text{m}}_{\text{VCR}}(\text{h}_{2}\text{-h}_{ 4})\) \\  & \(\hat{\text{E}}_{\text{dest,evap}}=\hat{\text{E}}_{4}-\hat{\text{E}}_{1}-(\frac{ \hat{\text{T}}}{\hat{\text{T}}_{\text{VCR}}-1})\hat{\text{Q}}_{\text{evap}}\) \\  & \(\text{EDR}_{\text{evap}}=\hat{\text{E}}_{