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Appl. Sci. 2020, 10, 5069 17 of 20 Following from an initial estimate for nozzle loss coefficient the isentropic enthalpy (h′ 2 ) can be (A11) calculated: ′ 12 h2=h2− 2λNC2 where 1, 2 and 3 subscripts correspond to stator inlet, rotor inlet and rotor exit conditions respectively. Using the the axial velocity, the area required to pass the specified mass-flow rate can be found. The passage area of the nozzle and rotor at different planes can be obtained as a function of the density ρ and axial velocity component. A= m ̇ (A12) ρCa 2πN b = AN (A14) U The stator and rotor passage losses can be obtained using the enthalpy loss coefficients. These coefficients are expressed as a function of the enthalpy difference between an isentropic and real expansion and the kinetic energy of the flow. The blade height b and mean radius rm can be calculated using the blade speed and rotational speed N: rm= U (A13) h2 − h2′ λN = C2/2 h3 − h3′′ λR = V3/2 W=h03−h01 From the estimated loss coefficients, the total-to-total isentropic efficiency can be determined. V3 C2 T3−1 ηtt=1+ λR2+2λNT2 h0−h0 13 (A15) (A16) (A17) (A18) Following the above blade calculations, detailed dimensions for the rotor radius at the hub and tip can be obtained by applying the free vortex theory. Additionally, the number blades and blade profile can be obtained through estimating the optimum pitch to chord ratio and pitch to throat ratio as a function of the blade angles [28]. To check the design consistency with the permissible level of stress within the rotor blades the centrifugal stress applied on the blade and the gas bending stress can be calculated using the following equations assuming a tapered blade shape: σmax = 34πN2ρbA (A19) σbending=m ̇Ca[tanα2+tanα3]×h× 1 (A20) nR 2 zc3 where ρb is the density of blade material, nR is the number of rotor blades, h is the mean blade height, A is the mean blade area and z is a constant obtained from a graph by Ainley [26].PDF Image | Mean-Line Design of a Supercritical CO2 Micro Axial Turbine
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