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Mathematics 2020, 9, 50 7 of 30 The detailed geometry of the P3 turbine can be seen in [31]. Table 1 presents the input conditions for the mean-line model, which are used as the input conditions for the current CFD study. Table 2 summarizes the basic equations in ORC and radial inflow turbine applied in the current study. Table 1. Mean-line and computational fluid dynamics (CFD) input conditions. Engine Points Inlet total pressure, (kPa) Inlet total temperature, (K) Pressure ratio, PR (-) Rotational speed, N (rpm) Mass flow rate, (kg/s) Loading coefficient, Flow coefficient P3 900 * 471.5 * 6.9 * 40,000 * 0.8 1.25 0.25 * Input conditions for CFD analysis. Table 2. Basic equations for turbine and ORC applied in the current study. Equation Q =m. (h −h ) in wf out in Equation # (1) (2) (3) (4) (5) (6) Comments Qin is the heat transfer in the evaporator in kW. hout and hin are enthalpies in kJ/kg at exit and inlet of the evaporator. m.w f is the working fluid mass flow rate in kg/s. Wpump is the power consumed by pump in kW. Pout, Pin, ρin and ηpump are outlet pressure, inlet pressure, inlet density and pump efficiency. ∆hact is the actual enthalpy drop within the turbine stage. Wout is the turbine power output in kW. U and Cθ are the tip and tangential velocities. η is the cycle thermal efficiency. ORC Wnet is the cycle net power in kW. m. (Pout −Pin ) Wpump = wf ρinηpump ∆hact =ηts∆his =h01−h05 W o u t = m .w f ∆ h a c t = m .w f ( U 4 C θ 4 − U 5 C θ 5 ) η = Wnet ORC Qin Wnet = Wout − Wpump 3. Generation of 3D Model This section presents the construction of the 3D model of the turbine stage, namely, stator and rotor. The volute is excluded since the losses are extremely small compared to the losses in the stator and rotor. Although the 3D generation of turbine blades and vanes have extensively explained in the literature such as in Aungier [27]. However, these equations are related to ideal gas equations of state. In the current study, real gas properties of the fluid have been accounted for using REFPROP [35]. 3.1. Stator Model The inlet and exit vane angles are 76.335◦ and 66.75◦, respectively, as obtained from the turbine mean-line model [31]. The distribution of thickness is performed according to the procedure proposed by Aungier [27]. The thickness distribution is performed on a parabolic-arc camber-line as outlined in Aungier [27]. The governing equation for a parabolic- arc camber-line is shown in Equation (7). The coordinate (x,y) is shown in Figure 4. g and b are the location and value of maximum camber along the chord, respectively. x2 and x3 are camber-line angles at inlet and exit, respectively. 2 c − 2g (c − 2g)2 2 c2 − 4gc x+ b xy+ 4b2 y−cx− 4b =0 (7)PDF Image | Generation of 3D Turbine Blades for Automotive ORC
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