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Int. J. Turbomach. Propuls. Power 2020, 5, 19 15 of 17 4. Conclusions To the best of our knowledge, there is not a commercially available Ljungström blade design software, and therefore the present study proposes the use of the kinematic efficiency as a new relevant design parameter in the Ljungström design. However, as discussed in Section 2, such a choice opens up a lot of points worthy of further discussion and investigation. The main reason is that this procedure does not take into account the centrifugal forces (absent in axial flows), and therefore the “optimality” of the blade profile proposed by the analytical model must be verified by an experimental validation. In the case studied here, this validation was obtained by a series of 2D time-dependent and by a steady state 3D CFD simulations. Though a full picture of the time evolution of the 3D flow is not available, nevertheless, some useful considerations can be drawn. As clearly shown in Table 9, the axial blade length is a crucial parameter in Ljungström design. By looking at Equation (17), it is clear that once all parameters, except for density, are fixed, in fact the inlet radius of each row is determined by the assumption of a fixed radius ratio, while the flow coefficient is fixed by the assumption of a constant outlet angle. In light of the above, it is also clear that the axial blade length depends on the mass flow rate and on the enthalpy drop. Following Shepherd and Kearton, it was assumed that all blades, except those in the innermost ring, have the same cross-section and they are all set similarly in the rotor. Some authors question the adequacy of these assumptions, used in the past for the design of large scale steam-powered ROT, since they lead to geometrical optimization instead of maximizing the isentropic efficiency [7]. According to these authors, if one removes the assumptions of constant βout and χ, it is possible to exploit two additional degrees of freedom for the axial blade length Equation (17). As a consequence, the formula of the kinematic efficiency changes its structure so that a multi-dimensional optimization is required. Some existing designs split the turbine into a high pressure and a low pressure stage to avoid the reduction in axial blade length in the downstream blade crowns of the turbine. This alternative solution consists of a small reduction in the mass flow rate at the inlet (first stage) and an additional injection in the low pressure wheel; however, this is not in contrast with our model, although further investigations could be useful. Clearly all the proposed changes should be analyzed in more depth because none of the above is supported by calculations. The method we propose is based on kinematic efficiency maximization, while isentropic efficiency determines the turbine performance. Even though ηkin does not take into account fluid dynamic losses, it is maximized with respect to ρr, the ratio between inlet and outlet radii. The latter also plays a major role in the axial blade length Equation (17). In the end, the shorter the blade length, the larger the wall friction losses and the lower the isentropic efficiency. Author Contributions: Conceptualization, U.C. and E.S.; Formal analysis, U.C.; Investigation, U.C.; Methodology, U.C.; Supervision, E.S.; Validation, E.S.; Writing—original draft, U.C.; Writing—review and editing, E.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.PDF Image | Optimal Design of a Ljungstrom Turbine for ORC Power
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