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Appl. Sci. 2020, 10, 5069 7 of 20 In Soderberg’s model, the losses are modelled as a function of the aspect ratio (H/l), the nominal loss coefficient ε∗ and the blade deflection ε. The nominal loss coefficient is obtained from the empirical correlation: ζ∗ = 0.04 + 0.06 ε 2, (5) 100 where ε is the flow deflection angle, which corresponds to εN = α1 + α2 for the stator and εR = β2 + β3 for the rotor. The nominal loss coefficients are then obtained as follows: 1051/4 ∗ l ζN = Re (1+ζ ) 0.993+0.075H −1 , (6) 1051/4 ∗ l ζR = Re (1+ζ ) 0.975+0.075H −1 , (7) In Ainely and Mathieson’s model, the tip clearance coefficient Yk is defined in terms of pressure drop as a function of the pitch to chord ratio (s/c), average blade angle βm, the radial tip clearance (k), the average blade height (h) and a constant (B). βm = tan−1[(tan β3 − tan β2/2)], (8) CL = 2 (s/c) (tan β2 + tan β3) cos βm , (9) where B equals to 0.5 for radial tip clearance. Considering the manufacturing tolerances and uncertainties in thermal expansion during the operation of micro-scale turbines, the radial tip clearance for the un-shrouded blades is set to 0.1 mm [6]. The tip clearance loss coefficient is then obtained as follows: , (10) The enthalpy loss coefficient for the rotor (λ) is then obtained as a function the rotor blade exit actual relative and isentropic temperatures (T03rel) and (T3′′) respectively, the rotor exit velocity (V3) and the specific heat capacity of the working fluid (cp). λk = Yk (11) (T03,rel/T3′′) 4. Model Validation One of the challenges that face sCO2 turbine design is the lack of experience and published validation data available within the literature. Most of the experimental test rigs available for sCO2 turbomachines have considered small-scale radial turbines (i.e. Sandia national laboratory (SNL), Naval nuclear laboratory (NNL), The Tokyo institute of technology (TIT), Korea atomic energy research institute (KAERI) among others) [37–40]. Furthermore, previous sCO2 turbine designs mostly focus on large-scale turbines, for which the axial turbine is the preferred configuration. To support this study, the design model has been verified against a study conducted for a 100 MW sCO2 axial turbine design presented by Schmitt et al. [41]. In their study, a first-row aerodynamic analysis for a six-stage 100 MW sCO2 Brayton cycle turbine was presented. The mean-line design, integrated with Soderberg’s correlation, was implemented to predict aerodynamic losses. The results of the mean-line design were verified with a 3D simulation using STAR-CCM+; which verified the 1D mean-line design. Using the same design inputs from that study, the model developed in this work, ignoring the tip clearance losses, has been used to design the same turbine and good agreement between the two models is found (Table 2) [41]. Whilst the presented validation provides a preliminary validation for the design model, further studies (i.e., CFD and experimental) are required to ensure the suitability of the implemented kCL 2 cos2β3 Yk = B h s/c cos3β mPDF Image | Mean-Line Design of a Supercritical CO2 Micro Axial Turbine
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