## 4 Discussion of Results
### Recommended Operating Conditions  
COP values offered by both cycle configurations are merged in Figure 12, where the COP value at each evaporating and environment level corresponds to the best performing system. As it can be observed, the cascade system gets over the MS solution at high environment temperatures and low evaporating levels. In fact, the environment temperature for a given evaporating level that defines the border of both systems is expressed by Equation (11), which was fitted from the results of the models. At environment temperatures above the value given by Equation (11), the cascade solution operates with highest COP. Also, the optimum modes of operation of the MS cycle are depicted in Figure 12. The operating conditions between an environment temperature of 25.3 \({}^{\circ}\)C and that defined by Equation (11) will be in transcritical conditions, whereas all lower environment levels the best performing cycle will be in subcritical. As it is observed in Figure 12, the environmental conditions at which the plant would be operated in transcritical are very narrow, which means that the correct design of the first CO\({}_{2}\) heat exchanger would be as condenser. In an attempt to summarize all the results of Figure 12, the COP dependence of both cycles versus the temperature difference between the cold and hot sources, Equation (12), is presented in Figure 13. Data used in Figure 13 correspond to all the  
Figure 11: Ratio of compressor’s power consumptions of the cascade cycle.  
calculated points to represent Figure 12. It can be observed that the MS cycle offers highest COP values at reduced temperature lifts and the cascade the other way round. The limit is at a temperature lift of 28.5 K approximately. However, it is important to note that the difference between the COP values of the MS cycle regards the cascade are higher at low temperature lifts that the difference between them at high temperature lifts. Those COP differences will condition the operation of the system along different environment temperatures, therefore a climatic evaluation would be needed to compare both cycles. That is discussed in Section 4.2.  
\[\mathrm{T_{env}}=25.95+0.4\mathrm{T_{o}} \tag{11}\]  
\[\mathrm{\Delta T}=\mathrm{T_{env}}-\mathrm{T_{o}} \tag{12}\]  
Figure 12: Best performing cycle for the different operating conditions.  
Figure 13: COP dependence on cold and hot sink temperature lift. MS and cascade cycles.  
Although in Figure 12 it seems that a smooth transition between the MS and the cascade cycle would be possible, it will only happen when the cycle is sized to operate in both configurations. To illustrate this reasoning, the compressor's displacements for the low and high temperature cycles for both configurations are presented in Figure 14. Those data correspond to the displacements for a refrigeration plant with 50 kW of cooling capacity designed for an environment temperature of 30 \({}^{\circ}\)C. It can be observed that the differences of the CO\({}_{2}\) compressor are not much significant between both cycles' solutions, but the compressor of the cascade cycle would be up to 300% higher than that needed for the MS cycle. If the plant is sized to be operated as MS cycle, its operation as cascade would not be possible because of the compressors, and, although not evaluated, because of the size of the subcooler and the gas-cooler.