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Dh 1⁄4 C26 (13) exit 2 3.5.3 Tip Clearance Loss. To calculate tip clearance losses, a bespoke model was developed to suit the high tip clearance ratios (up to 10%) experienced with small turbines. The model is a poly- nomial response surface fit to experimental data presented by Futral and Holeski [32], optimized for the high clearance range. The resulting relationship is 3.5.7 Loss Discussion. Several of the above loss models including their empirical components have been developed for the use with air or light gases as the working fluid. In the absence of substantial sCO2 radial turbine data, these models are the best starting point for preliminary analysis. However, consideration must be given for their appropriateness both for the small size of the turbines and the high density of the fluid. Of the significant loss models, both exit energy loss and incident loss are related to kinetic energy in the nonrotating environment. It is not expected that these losses are affected by scale or fluid density. Passage losses, however, are a complex combination of both viscous losses within the passage and losses due to secondary flows induced due to Coriolis and centripetal forces. As such, pas- sage shape and fluid density strongly affect these losses. The suit- ability of CETI loss model for small air turbines was demonstrated by Futral and Wasserbauer [38]. He validated the model with a 116.6mm diameter turbine operating at a specific speed (Ns) of 0.29. This gives confidence in the model to correctly capture the passage shapes of the turbines currently explored. How- ever, no comparison data are available for secondary flow structures and how these are affected by the high fluid densities of sCO2. Hence, some uncertainty is associated with this loss prediction. Tip clearance losses are a manifestation of how the desired flow pattern is affected by the tip gaps and the associated mass flow that is lost to high-speed jets between adjacent rotor passages. As such, rotor geometry and size will be the major drive for variation. The turbine used for the development of the response surface model had a diameter of 152.9 mm and a specific speed of 0.91 [32]. As the large specific speed difference results in highly differ- ent passage aspect ratios, some uncertainty in the model is expected. Finally, windage losses are caused by the well-studied phenom- enon of fluid being entrained by a moving disk. This phenomenon has been well characterized in regards to Reynolds number (Re). As the rotational Reynolds numbers of the test turbines are within the existing ranges of experimental data, no additional errors are expected. 3.5.8 Options for Loss Reduction. The loss model formula- tions also allow a deduction about possible routes to improve effi- ciency. For example, from the tip clearance loss model in Eq. (14), it is clear that radial clearance at the turbine outlet is a domi- nant factor and requires close control. Similarly, to minimize pas- sage loss designs with tall blades and short passages (low ðLh =Dh Þ), a throat close to the rotor inlet is desired. This can be achieved through the use of higher specific speed designs that create a throat closer to the rotor inlet. Incident losses can be minimized by reduc- ing the relative flow angle between the incoming flow and the rotor blades at the inlet. However, to effectively achieve this, actual flow directions should be obtained using higher order methods that account for the flow development in the vaneless gap. The rotor exit losses, which are high for these small turbines, can be reduced through the use of efficient diffusers. An alterna- tive approach would be to increase the exit flow area while reduc- ing the outlet hub radius. Such a design change would maintain the appropriate exit velocity triangles, without the need for high blade exit angles (b6), leading to a reduction of both W6rms and C6. Finally, an initially counterintuitive trend is seen for windage losses. Here, design C shows the largest loss, despite having a rotor radius (r4) and speed comparable to designs B1 and B2. This is a consequence of windage loss being primarily dependent on rotor speed and rotor radius. Thus for design C, which has a lower output power, windage has a much higher impact on efficiency. This highlights that there should be a strong push to minimize rotor radius in order to control this loss. 4 Discussion The current work uses an enhanced mean-line design code, which has been adapted for the simulation of sCO2 turbines in the AUGUST2017,Vol.139 / 081008-9 Dhtip 1⁄4 Dh0 ð0:09678 A 1:69997 Rþ0:096844 R2 0:03379 A RÞ 1 100 (14) Here, A and R are the axial and radial clearance ratio in percent- age. The agreement between the response surface and the experi- mental data was confirmed using hypothesis testing, showing an agreement in excess of 99.99%. For the current design space analysis, the values for A and R are set to 9% and 4% in accordance with recommendations by Glass- man [33] or the minimum permissible tip clearance. Considering a minimum tip clearance of 0.1 mm, this results in a constant loss of 7.4% once an inlet blade height of 1.11 mm is exceeded. 3.5.4 Incidence Loss. The incidence loss is modeled as a com- plete conversion of relative tangential velocity into internal energy of the working fluid [17,34] at the rotor inlet W2 Dhincidence 1⁄4 h4 2 (15) 3.5.5 Windage Loss. Windage losses arise due to flow being entrained on the rear surface of the rotors disk in the cavity between the rotor rear-face and the turbine housing. This loss is calculated based on the empirical correlations for torque acting on rotating disks, developed by Daily and Nece [35] Mr1⁄41Cmqx2r45 (16) 4 where q is the average fluid density in the rear cavity, and Cm is a flow regime dependent torque coefficient defined as C 1⁄4 m 0:1 3:7 eb r4 Re0:5 re 0:2 0:102 b for1103PDF Image | S CO2 Radial Turbine Design as a Function of Turbine Size
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