## 2 Material and Methods
### Followed Methodology  
In this work, the three systems were examined for different operating conditions. The refrigeration production was produced in four different temperatures (T\({}_{\text{e}}\)), from \(-35\)\({}^{\circ}\)C up to \(5\)\({}^{\circ}\)C, with a step of \(10\)\({}^{\circ}\)C, while the heat rejection temperature (T\({}_{\text{c}}\)) was studied from \(35\)\({}^{\circ}\)C up to \(50\)\({}^{\circ}\)C, with a step of \(5\)\({}^{\circ}\)C. Totally, 20 different operating scenarios were investigated in this work. In every case, the system was optimized properly in order to found the optimum design. The optimization criterion was the exergy efficiency in every case. For the reference system (Ref) and the system of mechanical subcooling (M-SC), the optimization with the exergy efficiency leads to the same results with the optimization with the COP as the optimization criterion. On the other hand, for the system with the absorption chiller, the optimization with the exergy efficiency did not give exactly the same results with the optimization based on maximum COP. However, the selection of the exergy efficiency as the optimization criterion was based on the greater suitability of this criterion in order to evaluate, properly, the temperature level of the generator temperature (T\({}_{\text{g}}\)). The COP evaluates the (W) and the (Q\({}_{\text{g}}\)) as the same quantities, and the higher exergy potential of the work, compared to the heat source, was not taken into consideration. During the optimization procedure, the high-pressure level in the gas cooler was an important optimization variable. This variable was optimized by using the following equivalent optimization parameter. This parameter is called \"pressure ratio\" and it is the ratio of the maximum pressure (\(\text{p}_{\text{high}}\)) to the critical pressure of the CO\({}_{2}\) (\(\text{p}_{\text{crit}}\)) which is 73.77 bar. \[\text{a}=\frac{\text{P}_{\text{high}}}{\text{P}_{\text{crit}}} \tag{15}\]  
For the system with subcooling, the evaporator temperature in the secondary cycle (T\({}_{\text{e,m}}\)) and in the absorption chiller (T\({}_{\text{e,ach}}\)) were also optimization variables. Moreover, the generator temperature was an extra optimization variable in the system with the absorption chiller. The optimization was conducted using the conjugate directions method or \"Powell's method\", which is included in the simulation tool EES [40]. The relative convergence tolerance was chosen at \(10^{-8}\) and the maximum number of iterations (function calls) at 3000. Practically, this method is an iterative method, which assumes some initial values for the optimization variables and it tries to find the optimum values by using the gradient values of the objective function. Moreover, it has to be said that some reasonable assumptions were made in this work. These assumptions are given below [25; 27; 46; 47] and they regard the main cycle, as well as the subcooling cycles. Some of them have been given in the mathematical formulation of the paper, but they are listed also below in order to be all presented together. * There is no pressure drop in the devices. * All the systems are in steady-state conditions. * There is no superheating in the evaporator outlet for the main cycle, as well as for the subcooling cycles. The expansion in the throttling valves is adiabatic, which means that the inlet enthalpy is equal to the outlet enthalpy because the process is adiabatic without any work production/consumption (first thermodynamic law). This assumption is applied in the main cycle, as well as in the subcooling cycles. The heat exchanger effectiveness in the absorption chiller is 70%. The cooling capacity is 100 kW in all the cases. The heat rejection temperature is the same in all cases. More specifically this parameter (T\({}_{\text{c}}\)) regards the gas cooler outlet temperature, the condenser of the M-SC, the condenser of the ACH-SC, and the absorber of the ACH-SC.