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former ensure that the turbine can be manufactured and operated within the constraint of today’s technologies, while the latter ensures that the geometries generate appropriate flow features and efficiency. 2.2.1 Manufacturing Criteria. The manufacturing criteria include manufacturing limitations, structural constraints, and vibration constraints. These arise both due to tooling limitations and operational effects, for example, thermal expansion that affects the actual operating geometry. Manufacturing limitations are provided by De Miranda Ventura [19] and are based on radial inflow turbines used in turbochargers. The minimum inlet radius is set to 10 mm to ensure that sufficient machining resolution is available to generate the turbine geometry. The next constraint is tip clearance to blade height ratio, which is realized as a minimum blade height. For the size of turbines under consideration, operat- ing clearances are limited to approximately 0.1 mm due to manu- facturing tolerances and uncertainties in thermal expansion during operation. Consequently, to maintain appropriate tip clearances ratios of less than 10%, the inlet blade height (b4) is limited to 0.9 mm. A structural constraint is implemented by comparing the rotor elastic stress (rr) caused by centrifugal loading to the mate- rial yield stress (rY). Rotor stress (rr) is calculated by the equation from Marscher [23] rr 1⁄40:3qmU42 (3) To allow for uncertainties and variations, a conservative limit of 0.9 rY has been selected. For the current study, INCONEL IN718 is used as the rotor material due to its good performance when exposed to high thermal and corrosion resistance. IN718 is one of the number of materials implemented in TOPGEN. In addition, the rotor blades of radial turbine are exposed to additional dynamic stresses that arise from the aerodynamic exci- tation of blade and disk modes. A vibration constraint is imple- mented to prevent rotor excitations that can lead to blade damage and fatigue failure. The primary cause of vibrations is the interac- tion between nozzle guide vanes and rotor blades, which result in excitation frequency of NðRPMÞ Z fr1⁄4 sðHzÞ (4) 60 The vibration constraint is set by comparing the excitation fre- quency (fr) to the natural frequency of the rotor trailing edge (xn), which is considered as the most flexible part of the turbine and prone to excitations. The natural frequency is calculated using the model presented by Blevins and Plunkett [24] Fig. 2 trailing edge thickness 6:94 xn 1⁄42pb26 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E t2 12qð12Þ (5) and relative flow angle (b4). As reported by Koppela [25] and pre- sented by Woolley and Hatton [26] for optimum performance, the inlet absolute angle (a4) should lie between 68deg and 76deg, while the relative angle (b4) should be considered within the range 40deg to 20deg. To allow investigation of designs at the edges of the feasible space, these ranges have been increased by 6 2 deg, respectively. For the geometry constraints, according to the research of Roh- lik [27], the rotor inlet radius to rotor outlet tip radius ratio ðr4=r6tÞ should not be less than 1.42 and the outlet hub to tip radius ratio ðr6h=r6tÞ should not be less than 0.4. Regarding the operational constraints, TOPGEN discards cases with a total-to-static efficiency (gts) below 50%. The resulting cri- teria used for the TOPGEN feasibility check are summarized in Table 1, and the definitions for rotor geometric parameters are shown in Fig. 3. 2.3 Selection of Test Cases. Due to current limitations of available test facilities, it is envisaged that the next generation of turbines will be designed in the 100–200 kW range, which is the focus of the current study. Design know-how from these turbines allows read-across to larger megawatt scale designs. As the operating conditions for the cycle are fixed by the con- centrated solar thermal application (see Table 2), the first variable to be selected is rotational speed (N). Speed is limited by rotor materials (see feasibility checks) and the bearing design choices. Gas foil bearings permit operation at speeds up to 350 kRPM [28]. However, for the current study, oil-lubricated rolling ele- ment bearings are considered only as they are more robust for early development. The results of a survey of commercially available bearings by Swann [29] highlight the relationship between bearing diameter and maximum bearing speed in Fig. 4. Reviewing these data, and based on recommendations from Swann, two turbine design speeds have been selected for the 100 kW design study. The pre- ferred speed of 160 kRPM and a more conservative speed of 120 kRPM allow for operational margins in relation to shaft diameter The schematic of the rotor blade plate thickness and Furthermore, in the current designs, the actual trailing edge thickness is tapered (tt 1⁄4 ðt=2Þ), as shown in Fig. 2, to reduce the trailing edge loss while maintaining trailing edge stiffness. Typically, a factor 4 is applied to separate the excitation and response frequencies [13]. However, for the current design study, this factor has been reduced to 2 for the preliminary design pro- cess. It is expected that approaches, such as filleting the blade root radius, trailing edge cutting, and 3D geometry optimization, all of which are highly efficient for small rotors, can be employed to re- establish a larger frequency separation during detailed design. 2.2.2 Guideline Performance Criteria. The guideline per- formance criteria consist of flow feature constraints, geometric constraints, and operational constraints. They are recommended values from literature, which have been proven to deliver high- performance turbines. The flow feature constraints are settings to obtain the optimum range for the rotor inlet absolute angle (a4) JournalofTurbomachinery Table 1 66–78 deg 40 deg to 20 deg <1 1.42 0.4 0.9 mm TOPGEN feasibility check criteria Parameters a4 b4 M4 r4/r6t r6h/r6t b4 Constraints Parameters Constraints r4 10 mm vi 1.0% rr <0.9 rY gts 50% fr 2 xn AUGUST2017,Vol.139 / 081008-3 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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