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Appl. Sci. 2020, 10, 4168 17 of 26 3 at 23,550 rpm and the lowest with Πts = 7 at 29,268 rpm. The maximum Mach number at stator outlet is lower for higher ω (as deduced from Figure 10) meaning that the entropy rise in the stator exit channel is also lower, hence the higher efficiencies. 96 94 92 90 ηtt [%] 88 86 84 82 80 (a) 19268 rpm 21409 rpm 23550 rpm ηts [%] 88 86 84 82 80 78 76 74 72 DP 19268 rpm 21409 rpm 23550 rpm DP 70 234567234567 Πts [-] (b) Πts [-] 130 Figure 11. Variation of efficiencies Rwainthgetoftarel-qtuoi-rsetdatic pressure ratio at three rotational speeds: 120 DP power output (a) Total-to-total efficiency, (b) Total-to-static efficiency. 100 90 outletattheoff-designconditions. Theflowangleatrotorinlet21(40β9rp)mplaysaroleindetermining 70 4 110 Table 10 shows the change in relative flow angle at rotor inlet and absolute flow angle at rotor Ẇ [MW] 80 19268 rpm incidence losses. The recommended values of rotor incidence angles for a radial turbine are in the ◦◦ 50 range of −40 –−20 [36]; incidence angles around the suggested values are only obtained in two cases 40 60 23550 rpm (−37.01◦ & −44.19◦), all which co3r0respond to pressure ratios below choking conditions and those cases 234567 correspond to relatively high efficiencies. Ideally, the closer the absolute flow angle at rotor outlet is to (c) zero, the lower the kinetic energy loss will be; therefore the lower the difference between ηtt and ηts is. Values of α5 ≈ 0◦ are attained at higher rpms because of the lower Mach number at stator outlet/rotor inlet which in turn leads to a lower relative Mach at rotor outlet and thus can have lower absolute flow angles for similar pressure ratios at different speeds. Table 10. Variation of relative flow angle at rotor inlet and absolute flow angle at rotor outlet at off-design conditions. Πts 234567 β4 [◦] Πts [-] 19,268 rpm 21,409 rpm 23,550 rpm 19,268 rpm 21,409 rpm 23,550 rpm −1.07 43.97 49.88 −37.01 −44.19 40.92 −58.45 0.47 26.21 α5 [◦] 56.5 36.8 15.95 63.06 -27.85 26.09 69.81 48.48 34.49 51.10 51.28 44.21 45.29 33.57 36.22 0.02 −10.64 9.56 −2.35 18.65 5.87 51.25 45.65 37.29 −17.22 −10.27 −3.04 The total-to-static pressure ratio and rotational speed of the most and least efficient points are used in 3D CFD to visualise the flow behaviour for each case. Figure 12 shows the relative Mach number distribution for both cases at mid-span. One can note the absence of the shockwave and supersonic flows at the stator outlet channel for the conditions with high rpm which leads to much improved efficiency values (ηtt = 82.40% ηts = 72.39%). The right-hand-side figure (b) also shows that the inefficient turbine (ηtt = 65.19% ηts = 57.22%) exhibits loss features that include the presence of a vortex, incidence losses (β4 = 35.44◦) and suction side separation or flow reversal as explained in Section 3.3.2. Whereas, the left-hand-side figure (a) shows smooth flow acceleration and deceleration as would be ideal in a radial turbine case with proper rotor incidence of β4 = −12.93◦. The aim would be to design a turbine that coincides with such flow behaviour, however, operating conditionsPDF Image | Radial Turbine Design for a Utility-Scale Supercritical CO2 Power
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