## 2 Material and Methods
### Mathematical Formulation  
In this section, the basic equations about the mathematical modeling of the systems are presented. Firstly, the detailed modeling of the CO\({}_{2}\) cycle is given. The modeling and the description of the absorption chiller are given both in Appendix A.  
The refrigeration production (Q\({}_{e}\)) in the evaporator of the main cycle is calculated as:  
\[\mathrm{Q}_{e}=\mathrm{m}\cdot(\mathrm{h}_{1}-\mathrm{h}_{4}) \tag{1}\]  
The work input in the CO\({}_{2}\)-compressor (W) is given as:  
\[\mathrm{W}=\mathrm{m}\cdot(\mathrm{h}_{2}-\mathrm{h}_{1}) \tag{2}\]  
Figure 4: Pressure-specific enthalpy diagrams: (**a**) for the reference system, and (**b**) for the systems with dedicated subcooling.  
Figure 3: System with dedicated subcooling using an absorption chiller (ACH-SC). The quantity (Q\({}_{e,\mathrm{ach}}\)) is the subcooling heat rate which is equal to the refrigeration load of the absorption chiller.  
The compressor isentropic efficiency (\(\eta_{\rm{is}}\)) is defined as below:  
\[\eta_{\rm{is}}=\frac{\rm{h_{2 is}-h_{1}}}{\rm{h_{2}-h_{1}}} \tag{3}\]  
In this work, the isentropic efficiency is calculated using the pressure ratio (r) according to the next formula [41]:  
\[\eta_{\rm{is}}=0.9343-0.04478\cdot\rm{r} \tag{4}\]  
where the compression pressure ratio (r) is defined as:  
\[\rm{r}=\frac{\rm{P_{high}}}{\rm{P_{low}}} \tag{5}\]  
The process in the throttling valve is assumed to be adiabatic and so the enthalpy is the same between the inlet and the outlet:  
\[\rm{h_{3}}=\rm{h_{4}} \tag{6}\]  
The reference system COP is defined a below:  
\[\rm{COP}=\frac{Q_{e}}{W} \tag{7}\]  
The COP of the system with mechanical subcooling is given as:  
\[\rm{COP}=\frac{Q_{e}}{W+W_{sc}} \tag{8}\]  
The COP of the system with absorption chiller subcooling is given as:  
\[\rm{COP}=\frac{Q_{e}}{W+Q_{g}} \tag{9}\]  
At this point, it has to be said that the secondary refrigeration cycle with R134a is modeled as the CO\({}_{2}\). The modeling of the absorption chiller is not given in detail here, but it follows the modeling of [42; 43; 44].  
An extra equation that has to be written is associated with the heat transfer in the subcooler. The evaporation temperatures of the R134a cycle (\(\rm{T_{e,m}}\)), as well as of the absorption chiller (\(\rm{T_{e,ach}}\)), are selected to be 5 \({}^{\circ}\)C lower than (\(\rm{T_{34}}\)) [45]:  
\[\rm{T_{34}}=\rm{T_{e,m}}+5 \tag{10}\]  
\[\rm{T_{34}}=\rm{T_{e,ach}}+5 \tag{11}\]  
In this work, except for the energy efficiency of the system (or COP), the exergy efficiency is used. This parameter is defined separately for every system as below:  
Exergy efficiency (\(\eta_{\rm{ex}}\)) of the reference system:  
\[\eta_{\rm{ex}}=\frac{Q_{e}\cdot\left(\frac{\rm{T_{0}}}{\rm{T_{e}}}-1\right)} {W} \tag{12}\]  
Exergy efficiency (\(\eta_{\rm{ex}}\)) of the dedicated mechanical subcooling:  
\[\eta_{\rm{ex}}=\frac{Q_{e}\cdot\left(\frac{\rm{T_{0}}}{\rm{T_{e}}}-1\right)} {W+W_{sc}} \tag{13}\]Exergy efficiency (\(\eta_{\text{ex}}\)) of the dedicated absorption chiller subcooling:  
\[\eta_{\text{ex}}=\frac{\text{Q}_{\text{e}}\cdot\left(\frac{\text{T}_{0}}{\text{T }_{\text{e}}}-1\right)}{\text{W}+\text{Q}_{\text{g}}\cdot\left(1-\frac{\text{T}_ {0}}{\text{T}_{\text{g}}}\right)} \tag{14}\]  
The generator temperature (T\({}_{\text{g}}\)) is the used temperature level for the determination of the exergy input in the absorption machine. The generator temperature is the saturation temperature of the strong solution at the high pressure inside the generator device (see Appendix A). All the temperature levels in the Equations (12)-(14) are in Kelvin units. Moreover, the reference temperature (T\({}_{0}\)) is selected at 298.15 K in this work.