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40 3 Numerical methods The rotor inlet blade height was solved by using the equation t1 = qm,wf . (3.25) ρ1πD1cr1 Diameter ratios D2t/D1 and D2h/D2t were given as inputs, and the rotor outlet diameters at the blade hub and tip as well as the blade height at the rotor outlet were calculated by using the diameter ratios. The axial outlet velocity c2 was solved as c2 = qm,wf . (3.26) ρ2(πr2t − πr2 ) 2h In this thesis, two approaches to design the diverging section of the supersonic stator nozzles were used. Most of the studied stator nozzle diverging section geometries were designed by using the throat width and the exit flow area of the nozzle, based on the result of 1D-turbine design, and the method of characteristics was not used in these designs. One of the studied nozzle diverging section geometries was designed by using a nozzle design tool developed at Politecnico Di Milano employing the method of characteristics, which was extended to real gas flows according to Zucrow and Hoffman (1977). In this design method, the transonic potential equation was solved in the throat section by means of the approximate solution procedure of Sauer (1947), which is applicable to real gas flows without significant modifications. The transonic flow solution provides the initial data curve for the method of characteristic. The expansion through the divergent section to the designed exit pressure is achieved via an initial circular profile followed by the so- called turning region, in which the nozzle upper wall geometry is determined by imposing the conservation of the mass flow at each section. When comparing the nozzle geometries obtained from a design based on a real gas model or ideal gas assumption, a large difference can be observed in the nozzle length and width, especially if the expansion process occurs in a thermodynamic region in which the funda- mental derivative of gas dynamics Γ is always lower than the corresponding perfect gas value (Guardone, 2013). The perfect gas assumption leads to a shorter nozzle and smaller exit flow area when compared to the design based on real gas models. The nozzle length and the nozzle exit flow area are found to increase with the increasing molecular complex- ity of the fluid, due to the non-ideal dependence of the speed of sound on density along the isentropic expansion with organic fluids (Guardone, 2013). The nozzle design method has been used in previous works for designing an organic vapor wind tunnel (Guardone, 2013) and supercritical CO2 nozzles (Guardone, 2010). The nozzle divergent section ge- ometry is designed by giving the total state of the fluid at the turbine inlet and the nozzle throat width as the input values. The design method based on the method of characteris- tics results in a minimum-length nozzle, in which the desired nozzle outlet pressure and uniform Mach number distribution are achieved at the nozzle outlet. The design of the turbine for the experimental ORC setup was an iterative process in which the turbine design, efficiency prediction, and the geometry were changed, basedPDF Image | WORKING FLUID SELECTION AND DESIGN OF SMALL-SCALE WASTE HEAT RECOVERY SYSTEMS BASED ON ORGANIC RANKINE CYCLES
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