## 2 Refrigeration Cycles, Models and Assumptions
### Calculation Models and Assumptions  
We performed the analysis of the MS and the cascade cycles using simplified but realistic models, which assumptions are detailed then. CO\({}_{2}\) compressor for both configurations is modeled using the overall efficiency as a linear relation with the compression ratio, as detailed by Equation (5). We fitted this relation using experimental data of a semi-hermetic single-stage CO\({}_{2}\) compressor able to operate in subcritical or transcritical [23]. \[\eta_{\rm G,CO_{2}}=0.7359-0.0517{\rm t}_{\rm CO_{2}} \tag{5}\]  
Figure 4: Pressure-enthalpy diagram of the low temperature cycle (CO\({}_{2}\)) of the cascade. For either the MS and cascade configurations, an approach temperature in gas-cooler of 5 K regards the environment temperature and 10 K of superheating degree in evaporator are chosen. For the MS configuration, when working in transcritical conditions, the high-pressure is established by the back-pressure. The tunable parameters are the high-pressure and the subcooling degree in subcooler (SUB = T\({}_{3}-\) T\({}_{4}\)). Both parameters are optimized to obtain the best performing conditions. When working in subcritical, high pressure is computed as saturation temperature of CO\({}_{2}\) at the environment temperature plus a temperature difference in condenser of 5 K, to maintain the same reference level as in transcritical. The exit of the condenser is considered in saturation. Only the subcooling degree in the subcooler is free, it being optimized in the calculations. For the cascade configuration, the tunable variable is the temperature of the intermediate level, being the CO\({}_{2}\) condensing temperature taken as reference and optimized in the calculations. In this case, the exit condition of CO\({}_{2}\) of the cascade heat exchanger is considered in saturation. For both cycles, the lamination processes are assumed isenthalpic and pressure losses and heat transfer to the environment in the lines are neglected. Regarding the secondary refrigerant, R1234 yf is selected for the MS cycle and for the high-temperature cycle. This HFO is one of the new generation of refrigerants introduced to the market with the aim of substitute the R134, being an alternative with low GWP but light inflammable (A2L), that can perform as drop-in replacement. [PERSON] et al. (2015) find out that this drop-in allows increasing the cooling capacity, being a refrigerant suitable for new plants and plants that are already working. The overall efficiency of the compressor is also adjusted as a linear relation with the compression ratio, as detailed by Equation (6), in this case fitted from experimental data of a semi-hermetic compressor ([PERSON], 2016). \[\eta_{\text{G,R1234 yf}}=0.9721-0.0533\text{t}_{\text{R1234 yf}} \tag{6}\]  
The high-temperature cycle or dedicated mechanical subcooling cycle, is thermally linked to the CO\({}_{2}\) cycle using two different approaches: when working as condenser in the cascade configuration, the evaporation temperature of R1234 yf is considered to be 5 K below the CO\({}_{2}\) condensing temperature ([PERSON], 2016), thus being optimized during the calculation. On the other hand, when this cycle operates as mechanical subcooler, its evaporation temperature is computed considering a thermal effectiveness of the subcooler of 60%, Equation (7), being this effectiveness the average value measured in ([PERSON], 2016). This temperature is indirectly optimized by tuning of the optimum subcooling degree in the CO\({}_{2}\) cycle. \[\text{T}_{\text{o,R1234 yf,MS}}=\text{T}_{3}-\frac{\text{SUB}}{\varepsilon}= \text{T}_{3}-\frac{\text{T}_{3}-\text{T}_{4}}{\varepsilon} \tag{7}\]  
For this cycle, a degree of superheat in the evaporator of 5 K is chosen. The exit of the condenser is in saturation and the expansion process is isenthalpic. Also, pressure losses and heat transfer to the environment in pipes are neglected.