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Appl. Sci. 2020, 10, 5069 6 of 20 In principle, the optimal axial turbine efficiency occurs within a certain range of specific speed, typically 0.1 < Ns < 1.0 [29]. For the specified specific speed range, the rotational speed ranges varies between 64 and 640 kRPM. Running the turbine at higher specific speed imposes high level of stresses on the rotor blades and could also result in rotordynamic instabilities. Consequently, in the present study the turbine design will be evaluated at three different shaft speeds, namely 150k, 200k and 250k RPM, corresponding to specific speed ranging from 0.23 to 0.39 rad, to achieve an overall turbine efficiency greater than 80%. The shaft speed is determined using Equation (4), by assuming that the specific speed is within the optimal range for axial turbines: Ns = ωQ ̇ , (4) ∆hos3/4 In Equation (4), ω is the rotor rotational speed in rad/s, Q ̇ is the volumetric flow rate at the rotor outlet in m3/s and ∆hos is the isentropic enthalpy drop across the turbine, in J/kg. To examine the effect of the various design parameters on the turbine performance and the feasibility of the design, the flow coefficient and loading coefficients have been varied over the range of 0.2 to 1 and 0.8 and 3 respectively [25]. Also, the the degree of reaction is varied from from 0–0.5 [26]. Then, the radius for root, mean, tip profiles are obtained using the free vortex design equations. Table 1 reports the range of values selected for the various design parameters. Table 1. Input design specifications. Design Parameter Turbine inlet temperature Expansion ratio [-] Rotational Speed [kRPM] Degree of Reaction [-] Ni-Cr-Co density [kg/m3] 3. Loss Modelling Value 650oC [11] 3.0 150–250 0.0–0.5 [26] 8000 Design Parameter Turbine inlet pressure [MPa] Net power output [kW] Flow coefficient [-] Loading Coefficient [-] Inconel 718 equivalent stress [MPa] Value 17 [22] 100 0.2–1.0 [25] 0.8–3.0 [25] 303 at 1073 K [27] To predict axial turbine performance, various loss models have been previously introduced starting from Soderberg [30], and then Ainely and Mathieson [31]. This has been followed by modifications presented by Dunham and Came [32], Craig and Cox [33], Kacker and Okapuu [34], and finally off-design correlations proposed by Moustapha et al. [35]. Soderberg’s loss model accounts for the effect of profile and secondary flow losses, while tip clearance and trailing edge losses are ignored. Profile losses are calculated as a function of the flow deflection while the secondary losses are interpreted as function of the aspect ratio neglecting the effect of inlet boundary layer and blade geometry. While Soderberg’s model is considered to be an oversimplified model where the effect of Mach number (Ma) and fluid non-dimensional parameters are neglected, it is considered to be satisfactory for preliminary design phase as it allows the loss within the stator and rotor to be estimated based on the amount of expansion that occurs within each passage [36].In the presented design framework, velocities approach the sonic speed and therefore Soderberg model is considered to be more accurate for estimating the flow losses as the correlations were derived based on high Mach number data. In the absence of a tip clearance loss correlation in this model, the Ainley and Mathieson correlation has been used. It is worth mentioning that a comparative study was made between using Ainley and Mathieson correlations only and the combination of the two classes, and the same trends were observed with a deviation in the design point efficiency of approximately 1.65%. Equations (5)–(7) represent the loss coefficients predicted using Soderberg model and Equations (10)–(11) represent the tip clearance loss coefficient estimated using the Ainely and Mathieson model.PDF Image | Mean-Line Design of a Supercritical CO2 Micro Axial Turbine
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