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fixed χ; (b) Variation of the kinematic efficiency with ρr for different χ and fixed βout. Since ξr depends on the fluid dynamic and geometric conditions that are different in each row, it is not possible to choose a single value of ρr for all rows that optimizes ηis,i. This justifies the choice of a velocity triangles optimization based on the selection of the ρr that maximizes the ηkin,i. Int. J. Turbomach. Propuls. Power 2020, 5, 19 9 of 17 As shown in Figure 10, for a fixed geometry (fixed χ and βout) and by varying the velocity, and consequently ξr, the maximum of the isentropic efficiency is close to the range of the maxima of the kinematic efficiency. This supports the choice of an optimization procedure based on the velocity of the kinematic efficiency. This supports the choice of an optimization procedure based on the triangles. velocity triangles. Figure 10. Comparing the variation of ηkin and ηis with ρr for fixed χ and βout and different flow velocity. Figure 10. Comparing the variation of ηkin and ηis with ρr for fixed χ and βout and different flow velocity. A MATLAB code was written to obtain a complete mean line analysis, and then the flow in the A MATLAB code was written to obtain a complete mean line analysis, and then the flow in the resulting geometry was simulated via a CFD analysis to validate the model. The inputs of the algorithm resulting geometry was simulated via a CFD analysis to validate the model. The inputs of the are shown in Table 7. algorithm are shown in Table 7. m. [kg/s] pin [Pa] pout [Pa] Tin [K] Working Fluid Mass Flow Rate Inlet pressure Outlet pressure Inlet temperature τall [MPa] ωmax [rpm] δb [–] lmin [m] bmin [m] βout [deg] Allowable Stress at the Shaft 80 Maximum speed 4400 Blade encumbrance 0.85 Minimum blade height 0.01 Radial chord 0.01 Minimum relative outlet angle 18◦ Table 7. Inputs and constraints of the algorithm. By setting a trial rotational speed ω = ωmax the shaft diameter can be found that sets a lower limit to the inlet radius at the entrance of the first row. Thence we obtain a matrix of all possible combinations of rin and βin that satisfy the continuity equation for all the l > lmin. The abovementioned matrix leads to another matrix of all possible values of ηkin. By taking the maximum value we can also find the optimal values of ρr and χ. With all these parameters in place, it is now possible to calculate the loss coefficient ξr and the enthalpy drop for the first stage (∆hi). n◦ rows i = 1 ∆hi = ∆htot, (13) If (13) is not satisfied it is possible to design additional rows and if an exact match is not achieved, a modified rotational speed ω = ωmax − ∆ω is selected and the process is iteratively repeated, as shown in Figure 11. Since the maximum efficiency is reached for b = 0 and βout = 0 and since these conditions are clearly unphysical, a minimum value must be arbitrarily specified. In addition, the change of the cross-sectional area along the radius requires the designer to specify a minimum value of the axial blade length at the inlet: ξ·m. hmin = wf , (14) 2πωrin2ρ tan β0δb where the leakage factor ξ was assumed here to be 0.98.PDF Image | Optimal Design of a Ljungstrom Turbine for ORC Power
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