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Chapter 3 This last variable greatly affects the meridional channel shape, which may be characterized by the so-called flaring angle δ, i.e., the ratio between the blade height at the outlet and at the inlet of a given row. Assuming a constant span among subsequent rows, the blade height at every row inlet may be specified as equal to that at the outlet of the previous row, i.e., for every row ‘i’ apart for the first stator (for which i=1), the equality Hi = Hi-1 holds. This leads to the following expression for in out the flaring angle of the ith row H i ρi-1 Vi-1 cosBDAi-1 D i δi≈ out∝ out · out · · in . Hin ρi Vi cos BDAi Din +b out out ABCD (3.14) Terms A and B in Eq. (3.14) are determined by the expansion process across the machine, i.e., by the distribution among the stages of the loading and of the reaction degree R. If simplifying assumptions are adopted, by assigning, for instance, common values to every stage as in Eqs. 3.6 and 3.5, both A and B may be considered approximately the same for all the stages. Referring to Fig. 3.2, and using V to indicate the velocity magnitude of the flow discharged by a generic row, it can be noted that Vi = Ci and Vi-1 = Wi-1 if the ith row is a stator. Conversely, Vi = Wi out out out out out out and Vi-1 = Ci-1 if the ith row is a rotor. This implies that, if the reaction degree in around 0.5, out out B ≈ 1, being Ci = Wi-1 for a stator, and Wi = Ci-1 for a rotor. Hence, the influence of term out out out out B on δ vanishes. In radial-outflow turbines B tends to be lower than one as a consequence of the increasing peripheral speed that, in turn, increases also the rotor outlet velocity. This fact can be handily explained by resorting to the energy balance in the relative frame of reference, which states the conservation of rothalpy rt across the rotating cascade, i.e. referring to Fig. 3.2 which leads to rt=h2+W2 −U2 =h3+W3 −U3 , 22 22 U2 W=Wi =Vi =2(rt−h+3). 2 (3.15) (3.16) 22 22 3 out out 3 Compared to an axial stage having the same features, i.e., the same stator outlet velocity Vi-1, out terms of Eq. (3.14). Regarding term C, assuming that all the blades have the same value of BDA according to Eq. 3.3, the equality C = 1 is obtained. The meridional channel shape becomes therefore primarily a function of terms A and D. The quantity represented by term D, i.e., the row diameters ratio, is plotted against the row inlet diameter in Fig. 3.3, considering different values of the radial chord. It can thus be seen how, for radial stages placed at progressively larger diameters, D tends to unity, i.e., its potential influence on δ vanishes. In other words, as expected, centrifugal stages characterized by small values of the ratio b/Din tend to behave like axial stages, with the chord size having little influence in determining the meridional channel shape, which is rather determined by the expansion process, i.e., by term A in Eq. (3.14). This is the case of the ROTs described in Ref. [28], and proposed for Rankine cycle power systems working with potassium and ranging in size from 20 up to 200 MWM. In the case small of ROT’s, however, the low mass flow rate leads to comparatively large b/Din ratios, due pri- marily to the need of obtaining acceptable blade height at the turbine inlet. The design is therefore inlet rothalpy and expansion ratio (which leads to a similar value of static outlet enthalpy h3), a centrifugal stage is affected by a greater W due to the larger kinetic term U2/2, In these conditions 3 B has an adverse effect on the flaring angle, and this can be compensated only by acting on the other 66 3PDF Image | New Concepts FOR Organic Rankine Cycle Power Systems
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