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3.2 Operating Point 2: 200 kW, 113 kRPM. Figure 7 shows the design map for the 200 kW, 133 kRPM design point. This operating point is scaled from operating point 1, while maintain- ing a constant specific speed (Ns). As discussed previously, the design constraints relating to the inlet velocity triangle, b4 and a4, are in the same position. Similarly, the specific speed lines are at the same locations. By scaling the speeds between operating points 1 and 2 in accordance with Eq. (8), it is expected that the resulting geometries are identical, but geometrically scaled. Com- paring actual dimensions, rotor inlet radius (r4), and rotor hub radius (r ) shows that the factor of 2 increase in power has 6h pffiffi resulted in a factor 2 1:41 increase in turbine dimensions. Reviewing the feasible design region, some changes relative to operating point 1 can be observed. The top and bottom boundaries are defined by b4 1⁄4 18 deg and b4 1⁄4 42 deg again. And the left limit is also defined by trailing edge natural frequency (xn). How- ever, the right limit is now defined by the lower limit for stator inlet angle a4 1⁄4 66 deg. Consequently, the feasible design space has shifted to the right. This arises as the blades are now larger, thereby moving the limitations for b4 > 0.9 mm away from the area of interest. The more stringent constraint for trailing edge natural frequency arises due to two effects. First, the new excitation frequency for the 200 kW turbine is reduced to fr 1⁄4 20,741 Hz as the rotational speed is lower. Second, the increased blade height at the outlet (b6) leads to reduction in trailing edge stiffness and a lower natural fre- quency. Due to the ð1=b26Þ relationship in Eq. (5), this causes part of the design space previously feasible to be eliminated. This is a clear example of how scaling a turbine affects the feasible design space. These effects are created by different nonlinear scaling relationships for turbine geometry and feasibility criteria. Figure 8 shows the feasible design in relation to specific speed and efficiency. For this higher power setting, the feasible designs have been pushed toward the lower right and thus a higher specific speed (NS). The resulting efficiencies (gts) now lie in the range of 0.70–0.80. Compared to Fig. 6, it is evident that efficiency con- tours are in the same positions and that the reduction in efficiency is caused by loss of feasible design space. For further comparison, two designs are selected. Design B1, a nonfeasible design, is selected to coincide with design 081008-6 / Vol.139,AUGUST2017 Fig. 8 A 200 kW 113 kRPM turbine efficiency contour Fig. 7 Turbine design map at 200 kW and 113 kRPM A (u 1⁄4 0:28 and w 1⁄4 0.82). This allows a direct comparison of two geometrically similar designs. In addition, design B2 is selected at the center of the feasible design space with a combina- tion of u 1⁄4 0:32 and w 1⁄4 0.81. This allows an investigation of how the shift in design space affects performance. 3.3 Operating Point 3: 100 kW, 120 kRPM. Figure 9 shows the available design space for the 100 kW turbine restricted to a lower speed of 120 kRPM. Again, the contour lines of b4 and a4 remain unchanged and angle b4 constitutes the top and bottom boundaries of the feasible design space. The major difference to the faster case (N 1⁄4 160 kRPM) shown in Fig. 5 is that the right hand boundary of the design space given by the blade height con- straint b4 0.9 mm has shifted toward the left eliminating most of the feasible design space. And the blade natural frequency (xn), which now lies at 44,000Hz, has moved away from the design space. This is the case as the smaller blades are significantly stiffer and as the excitation frequency is lower. The left limit of the feasible design space is set by the upper limits of inlet absolute flow angle (a4 1⁄4 78 deg). Considering turbine dimensions, the TransactionsoftheASME 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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