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it is now possible to calculate the loss coefficient ξr and the enthalpy drop for the first stage (∆hi). ∑° ∆h = ∆h , (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 Int. J. Turbomach. Propuls. Power 2020, 5, 19 10 of 17 in Figure 11. Figure 11. Schematic representation of the design algorithm. Figure 11. Schematic representation of the design algorithm. The above equation has two degrees of freedom. By fixing a proper range of values for β0 and rin Since the maximum efficiency is reached for b = 0 and βout = 0 and since these conditions are all possible combinations that meet the equation can be calculated. These ranges are constrained by clearly unphysical, a minimum value must be arbitrarily specified. In addition, the change of the two different conditions. β0 must be limited in order to obtain wout,1 > win,1: cross-sectional area along the radius requires the designer to specify a minimum value of the axial blade length at the inlet: 1 0 < β < cos−1χρ ∙ , (15) 0r 222 h =1+ρ−2ρcosβ, +χρ (14) r r out r where the leakage factor ξ was assumed here to be 0.98. while rin for the first row must be larger than the shaft radius that was calculated with the following formula: dshaft≥K3 16P, (16) πω where K = 1.3 is a safety factor, P [MPa] the power at the shaft and ω [rpm] its rotational speed [17]. Subsequently the code designs the first row using the inputs provided by the optimization function. It starts from the velocity triangles, calculates the isentropic heat drop using the loss coefficients and finally derives the thermodynamic quantities at the outlet of the row. The quantities thus derived are used as inputs for the following stage, leading to an iterative process. The proposed algorithm leads to a choice in the number of stages as a function of the maximum angular speed. As shown in Table 8 and, as expected, the χ parameter is strictly correlated to the efficiency of the turbine. A closer look to these results reveals that the less efficient thermodynamic cycles presents the higher isentropic efficiencies. The Ljungström turbine performs better with low enthalpy drop and high-volume flow rates (higher specific speed ns). Moreover, since the cross-sectional area depends on the volume flow rate variation, not all the selected fluids lead to geometrical characteristics, such as ξ·m. hmin = wf , (17) 2π·ω·rin2·ρ· tan β0.δbPDF Image | Optimal Design of a Ljungstrom Turbine for ORC Power
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