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Mathematics 2020, 9, 50 3 of 30 require a good geometrical representation [29]. Mean-line modelling is yet incapable of capturing the flow behaviour, such as flow separation and shock waves. Thus, 3D simulations are essential [30]. Within the turbine, the rotor is the key component to produce work. Ahead of starting the 3D simulations, the 3D shape of the turbine blades and vanes should be generated. Most of the studies in the literature use commercial software for this task. However, such types of software are expensive and time-consuming. This makes it imperative to develop fast and accurate mathematical modelling for blade generation. As small-scale radial inflow turbines are characterized with small mass flow rates, the turbine usually adopts radial blades at the inlet (zero-degree blade angle). Radial inlet blades are beneficial to avoid bending stresses. To increase the turbine work (enthalpy drop within the rotor), the tangential velocity at the rotor inlet should be increased as explained un the previous study [31]. Using backswept (non-radial) blades results in positive relative angle and consequently, larger enthalpy drop. However, utilizing backswept blades results in higher bending stresses at the rotor leading edge and failure is likely to occur, leading to catastrophic consequences both physically and economically. Therefore, it is very essential to ensure that the turbine rotor can withstand the operating conditions of the flow and have adequate life in service. Therefore, structural evaluation of the turbine stage using FEA, which is widely applied in the analysis of engineering problems [32], is essential. Colonna et al. [33] developed an in-house Euler solver to perform complete CFD simulations for supersonic turbines. They mainly focused on the effects of different equations of state (EoS) for real gases. The results showed a significant deviation when an ideal gas EoS was applied, while the Span–Wagner and Peng–Robinson–Stryjek–Vera equations were very similar. Harinck et al. [34] performed a complete steady-state simulation for their supersonic radial inflow turbine using ANSYS CFX. The property tables were generated using REFPROP [35]. Their results showed that the improved stator model was able to deliver the required tangential velocity components with a Mach number value as high as 2.85. Sauret and Gu [36] presented 3D CFD simulations of ORC radial inflow turbines using commercial software to validate the mean-line model. The results showed good agreement between the mean-line and numerical results. The results also showed the significant effects of the turbine off-design conditions on the ORC system. Uusitalo et al. [37] simulated a high supersonic small-scale ORC turbine stator where a real gas model was implemented in a CFD solver using both the k −ε and k −ωSTT turbulence models. The results showed that Mach number at the stator exit was 2.27 for k −ε solver and 2.31 for k −ω STT. Recently, White [38] performed a full CFD analysis using ANSYS CFX as a validation for his mean- line model of the ORC radial turbine. The agreement between the mean-line and CFD was very good with a deviation of 0.3% in the total to static efficiency. He also validated his model with the CFD by comparing the stagnation and static thermodynamic properties at the inlet and exit of the turbine rotor. The results were very accurate with a maximum deviation of 6.3% in the static pressure at rotor inlet. Similarly, Verma et al. [39] validated his 1D model with a complete CFD simulation using ANSYS CFX, and the deviation between the mean-line and CFD simulation in the turbine efficiency was 0.17%. Their CFD results showed a choking condition at the design point with Mach number of 1.55. Nithesh and Chatterjee [40] presented a study that combines 1-D design and 3D CFD. The numerical CFD analyses were in good agreement with 1-D model. Song et al. [41] studied the performance of radial outflow turbines at off-design conditions using 1D and CFD analyses. Their 1D results correlated well with the CFD results. Daabo et al. [42] presented two design models for axial and radial turbines to be used in solar powered Brayton cycles. Their CFD analyses were validated against experimental work and then used to validate the mean-line models. Dong et al. [43] explored the sensitivity of turbine design to some design parameters such as stator velocity coefficient. The proposed model was followed by a detailed CFD study in which a good agreement was obtained. Sun et al. [44] investigated the effects of non-equilibrium condensation flow on the performance of radial inflow turbines using CFD studies. Wang et al. [45] studied the effects of different zeotropic mixtures on the design of radial inflow turbines. There 1D results were comparedPDF Image | Generation of 3D Turbine Blades for Automotive ORC
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