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Centrifugal Turbines for ORC Applications hence, may induce a diffusion effect due to the increase of passage area in the flow direction. The minimum blade height Hmin constitutes a critical parameter as the size of the machine is reduced and, in the present case, it is assigned a value of 10 mm. The flaring angle δ, i.e. the angle between the end-wall contour and the radial direction, is constrained in the typical range adopted for axial machines of ±30 ◦ [5, 23]. The lower constraint on the inlet diameter Din may be tentatively de- termined in order to be compatible with the minimum blade height, keeping however in mind that, owing to the possibility of a converging meridional channel, this does not necessarily occur in the first stage, see §3.4. Regarding the 6-stage machine, the design assumption of repeating stages (§3.4) is adopted in order to illustrate its consequences on the result. Coherently, the main variables of the problem, i.e. the pressure drops, the degrees of reaction, the radial chords, and the blades outlet geometric angles, are assumed to be the same for all the stages. In particular, the blades angles, which are given values typical for turbine cascades, are opposite in sign between stators and rotors. The global total-to- static pressure ratio is evenly distributed among the stages, i.e. (pT,in/pS,out)stg = (pT,in/pS,out)1/Nstgs . The reaction degree is left to vary between 0.4 and 0.6, in a region of high stage-performance. This allows also to split almost equally the expansion ratio between the stator and the rotor, thus limit- ing the maximum Mach number within the stage. The constraints imposed on the radial chord are selected in order to preserve acceptable blade aspect ratios and to limit the turbine dimension, i.e. its maximum diameter. In the 3-stage machine, due to the increased stage-loading, all the preceding simplifications are removed in order to limit the maximum Mach number to 1.4 and to respect the constraint on the maximum flaring angle. In particular, the pressure drops, the degrees of reaction, the radial chords, and the blades outlet geometric angles are allowed to assume values differing among the stages, according to the novel design procedure introduced at the end of §3.4. 3.5.3 Results: Transonic Turbine The main features of the optimized 6-stage turbine are shown in Tab. 3.3. The optimal velocity triangles are shown in Fig. 3.5a and, as expected (§3.4), there is no similarity among them, notwith- standing the design assumption of repeating-stages. Another expected feature of the machine is the shape of the meridional channel, see Fig. 3.5b, which appears to be slightly convergent in the first stages, while in the last one the flaring angle reaches the prescribed upper limit of 30 ◦. This trend is strictly combined to the other quantities of the problem, and more insight may be gained by considering again Eq. (3.14). An increase in the flow passage area is required along the stream-wise direction, in order to accommodate for the considerable growth of the volumetric flow rate as the expansion process proceeds. Being the blades angle and the radial chord values constrained to be the same in this case, the only free variable that can be exploited to this end is the flaring angle. In the first stages, where the increase in passage area determined by the diameter increase is comparatively large, the meridional channel tends to be convergent. On the contrary, an increasing divergence is needed in the subsequent stages. The adoption of a comparably large number of stages reduces the blade aerodynamic load- ing, which is an interesting quantity for underlying differences and analogies with respect to the axial architecture. In axial machines, this quantity is normally proportional to the stage specific work w. Conversely, for radial outflow configurations, the reduction of the the aerodynamic load- 71PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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