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Centrifugal Turbines for ORC Applications velocity component conserves throughout the expansion, the so-called repeating-stage arrangement is obtained, see, e.g., Fig. 3.2. Repeating stages are usually characterized by similar stator and rotor blades, namely NNN α3 = α1,W3 = C2,W2 = C3. (3.7) Notably, these conclusions rest on assumptions that can only be partly realized in radial machines, since the peripheral speed varies in the stream-wise direction. Proper design rules are therefore to be established if centrifugal turbine arrangements are of interest. Even if the constraints imposed by the repeating stage approach are neglected, radial outflow machines dimensioned on the basis of common axial turbine design criteria, i.e., fulfilling Eq. 3.3, 3.4, 3.5, and 3.6, may show i) different velocity triangles per-stage, notwithstanding the geometrical similarity of the stages, and ii) a characteristic meridional channel shape, which tends to be convergent in the first stages while becoming divergent in the last stages, see, e.g., § and Refs. [11, 32]. A converging-diverging meridional shape of the flow channel is typically an undesirable fea- ture, particularly in case of mini-ORC turbines, since i) the blade height might already be insuf- ficient at the inlet (a full-admission first stage is preferable as far as efficiency is concerned), and ii) large variations of the meridional channel lead to span-wise velocity components which, even though not captured by a mean-line analysis, deteriorate the performance of the turbine. In a radial turbine, contrary to what happens in axial turbines, the stage chord has an influ- ence on the distribution of the stage diameters along the machine, and thus on the work extraction process. This is in turn governed by the variation of peripheral speed, being Dout = Din + b, (3.8) w ∝ U2 = ω2 D2out. (3.9) This feature has an impact also on the variation of the volumetric flow rate, and consequently of the flow passage area Aout needed to accommodate the flow along the expansion. For instance, by applying the continuity equation to the outlet section of the bladed region of a subsonic row, see, e.g., Fig. 3.2, such area may be expressed as m ̇ Aout = Hout o = ρout Vout Nblds ǫ , (3.10) where Hout and o are the blade height and the channel outlet section width (normal to the flow) at the outlet of the channel, respectively, and Vout is the corresponding flow velocity magnitude. Nblds is the number of blades, while ǫ accounts for possible correction factors (e.g., that for blocking effects due to the boundary layer or the blade section). Assuming a rectilinear suction blade end-side, the relation among the outlet section width o and the blade geometric discharge angle BDA is expressed as o = S cos(BDA), where the blade pitch S is evaluated according to the equation S=πD N−1. out blds Finally, in order to express the blade height Hout, Eq. 3.10 may be rearranged into m ̇ Hout = ρout Vout cos(BDA) Dout π ǫ . (3.11) (3.12) (3.13) 65PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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