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Fig. 5 Turbine design map for 100 kW, 160 kRPM operating point Similarly, absolute angle (a4) is defined by the following relationship: 1 w a4 1⁄4tan nu (11) This implies that the range of feasible designs’ limits imposed by a4 and b4 is fixed in the flow and head coefficient state space. To alter these ranges, the meridional velocity ratio (n) has to be altered. The remaining contours give an indication of turbine geometry trends. Inlet blade height (b4) is inversely linked to stator exit angle (a4), showing low angles are desirable to maintain tall blades. A further parameter to consider is exit blade height (b6), which has a significant influence on the rotor blade stiffness which influences blade trailing edge natural frequency (xn). The effect of flow coefficient on geometry can also be explored. Reducing flow coefficient has a strong effect on blade height, leading to increases in both b4 and b6. The resulting changes in flow area particular at the exit are manifested in an increased rotor exit blade angle (b6rms). According to Kopplea [25], the optimum value of b6rms should be in the range 50 deg to 70 deg, though the value of b6rms should not be a feasibility check criterion. Cur- rently, all the feasible designs fall within this range. Similarly, increasing head coefficient has a strong effect on rotor inlet radius (r4) and rotor hub radius at the outlet (r6h). The absolute values of these dimensions, approximately 20 mm and 6 mm, reiterate the compact and highly power-dense nature of these sCO2 turbines. Particular attention should be given to the contours for blade trailing edge frequency (xn), which sets the left limit of the feasi- ble quadrangle. To ensure sufficient separation between the exci- tation frequency, a natural frequency limit of xn 1⁄4 58,666 Hz (2 fr for 160 kRPM) is selected. If rotor excitation to be identi- fied as an issue, the position of this line can be changed by adjust- ing material properties or blade thickness as defined in Eq. (4). The final contours to consider are specific speed (Ns) and effi- ciency (gts), shown in Fig. 6 for clarity. The feasible area lies JournalofTurbomachinery Fig. 6 ing point Turbine efficiency contour for 100 kW, 160 kRPM operat- between Ns 1⁄4 0.28 and 0.32. This is lower than the typically rec- ommended specific speed range. However, as shown by Rohlik [27], high-performance turbines are still feasible in this range by utilizing high-stator exit angles, which is in agreement with the observed trends from Fig. 5. The corresponding contours of total- to-static efficiency show that the feasible design cases have effi- ciencies between 0.74 and 0.82. In theory, more efficient designs could be attained by moving toward the top left of the flow and head coefficient state space. However, as shown in Fig. 5, designs here are not feasible as the geometries require too high inlet stator angles, a4, and due to the rotor dynamic constraint. Using Figs. 5 and 6, it is possible to select an optimum rotor design, while balancing performance, geometric parameters, and separation from feasibility limits. In the current study, for compar- ison with the other design points, the geometry at the center of the feasible design space has been selected. This is marked by A in Fig. 5 and corresponds to the combination of u 1⁄4 0:28 and w 1⁄4 0.82. If the focus was the selection of a high-performance design (high gts), a design closer to the top left of the feasible area should be selected. AUGUST2017,Vol.139 / 081008-5 Downloaded From: http://turbomachinery.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jotuei/936123/ on 04/05/2017 Terms of Use: http://www.asme.org/a bPDF Image | S CO2 Radial Turbine Design as a Function of Turbine Size
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