## 2 Turbine Design Methodology  
sCO\({}_{2}\) condensation (transcritical) power cycles were firstly proposed by [PERSON] and [PERSON] in 1968 where efficiencies up to 50 % can be achieved [16; 17].However, to achieve condensation within the cycle it is necessary to lower the temperature of the CO\({}_{2}\) below its critical temperature (31.1 \({}^{\circ}\)C). Unfortunately, achieving this within CSP plants, which are typically in dry arid regions with high solar irradiation, requires water-cooling which is usually not feasible in these sites. To overcome this issue, it has been proposed to raise the critical point of the working fluid by doping CO\({}_{2}\) with another fluid. For example, mixing CO\({}_{2}\) with various additives, such as C\({}_{3}\)H\({}_{8}\), C\({}_{4}\)H\({}_{8}\), C\({}_{4}\)H\({}_{10}\), C\({}_{5}\)H\({}_{10}\), C\({}_{5}\)H\({}_{12}\) and C\({}_{6}\)H\({}_{6}\), has been shown to increase the critical temperature to up to 60 \({}^{\circ}\)C [18]. Moreover, the application of sCO\({}_{2}\) blends in CSP plants have been previously studied [18] and have been shown to enhance efficiency of a Brayton power cycle by 3-4% compared to pure sCO\({}_{2}\). Therefore, sCO\({}_{2}\) blends have been proposed instead of pure sCO\({}_{2}\) to increase the critical temperature of the working fluid [19; 20]. The presented design methodology has been developed as part of a preliminary study related to the Horizon 2020 SCARABEUS project [21], and thus the methodology is capable of designing turbines intended for sCO\({}_{2}\) blends. However, for simplicity at this stage, pure sCO\({}_{2}\) will be considered as the working fluid. Considering a blend merely changes the inputs into the equation of state used to predict thermodynamic properties and thus the method can be readily extended to blends without any change to the analysis. It should be noted that the turbine design is likely to be sensitive to the chosen blend, but such investigations are left for a future work. The turbine design is part of a condensation sCO\({}_{2}\) cycle, as shown in Figures 1 and 2, with a net power output of 100 kW. The turbine inlet parameters are obtained from a thermodynamic cycle analysis and are selected as a compromise between cycle efficiency, component life, complexity and feasibility [11]. Accordingly, the turbine inlet temperature is limited to 650 \({}^{\circ}\)C. The turbine inlet pressure is limited to 17 MPa with an expansion ratio of 3 to reduce the power block weight, size and price [22]. The cycle is designed assuming a pump inlet temperature of 20 \({}^{\circ}\)C, pressure of 6 MPa, compressor isentropic efficiency of 70%, turbine isentropic efficiency of 80% and recuperator effectiveness of 90%. For the selected parameters, the cycle has a thermal efficiency of 32.8% with a mass-flow rate of 0.65 kg/s. To design the turbine, a mean-line turbine design approach is adopted to provide a fast and accurate estimation of the turbine geometry and the expected isentropic efficiency. This has been developed in MATLAB and coupled with NIST REFPROP [23] to account for the sCO\({}_{2}\) fluid behaviour. The model is also capable of operating with Simulis, which enables novel fluid blends to be considered [24]. Within the model the steady-state mass, energy and momentum equations are solved to obtain the geometric parameters of the turbine. In this section, the turbine mean-line design is developed along with a parametric study to examine the effect of changing various design parameters on the turbine performance, at various rotational speeds, and assess blades pressure and centrifugal loads. The designs were evaluated for turbine inlet conditions of 923 K, 170 bar, expansion ratio of 3. For brevity, a full description of the model is not presented here, but can instead be found in Appendix A. Instead, a brief overview of the process is provided here.