# Detailed Design and Aerodynamic Performance Analysis of a Radial-Inflow Turbine
## 3 Preliminary Design
### Rotor Preliminary Design  
The velocity diagram is illustrated in Figure 3. In this figure, \"C\" and \"W\" are absolute and relative velocity, respectively. The blade and flow angle are computed related to the tangential direction.  
In rotor design, it is assumed that the inlet blade angle is constrained to be 90\({}^{\circ}\). Since the turbine is designed for optimum performance, the exit absolute tangential velocity is considered to be zero. On the other hand, it is a reasonable assumption that the absolute Mach number at rotor exit has a relative low value in a well-designed turbine. By these assumptions, the rotor tip radius and rotor inlet tangential velocity can be computed by the following equations and using Equations (4)-(6).  
\[U_{4}=\
u_{ts}C_{0 ts}\;\;\mbox{where}\;\;C_{0 ts}=\sqrt{2\Delta H_{id}} \tag{4}\]  
\[r_{4}=U_{4}/\omega \tag{5}\]  
\[C_{\theta 4}=U_{4}\eta_{ts}/\left(2\
u_{ts}^{2}\right) \tag{6}\]  
[PERSON] [4] recommended rotor inlet absolute flow angle as a function specific speed presented in Equation (7). [PERSON] [9] suggested Equation (8) to estimate the number of rotor blades based on Jamieson expression.  
\[\alpha_{4}=10.8+14.2n_{s}^{2} \tag{7}\]  
Figure 3: Velocity diagram in a rotor blade.  
Figure 2: Design station for calculating geometric and aerodynamic parameters.  
\[N_{R}=\frac{\pi}{30}(110-\alpha_{2})\tan\alpha_{2} \tag{8}\]  
[PERSON] [7] provided the useful default relations to compute the hub radius, the ratio of meridional velocity at rotor exit to rotor inlet meridional velocity and the rotor axial length (Equations (9)-(11)). He also suggested the default equations to estimate the rotor blade thickness at the inlet and discharge stations. However, in this work, the rotor blade thickness was considered constant at all points of the rotor blade to simplify the final machining process. Though the hub radius, the blade thickness and the axial length are influenced by aerodynamic and mechanical considerations, it is more effective to use the default value for these parameters in the preliminary design. These parameters can be optimized according to aerodynamic and structural analysis.  
\[r_{h5}=0.185r_{4} \tag{9}\]  
\[C_{m5}/C_{m4}=1+5(b_{4}/r_{4})^{2} \tag{10}\]  
\[\Delta z_{R}=1.5(r_{s5}-r_{h5}) \tag{11}\]  
A well-designed rotor should satisfy a rotor exit relative Mach number less than unity. By this criteria, the mean blade pitch at the rotor exit and the rotor mean throat width could be computed by Equations (12) and (13).  
\[s_{5}=2\pi r_{5}/N_{R} \tag{12}\]  
\[o_{5}=s_{5}C_{m5}/W_{5} \tag{13}\]  
In some other research, the hub and shroud contours were assumed to be elliptical and circular arcs, respectively [9; 12]. However, in this work, the hub contour is considered the largest circular arc, which is compatible with the rotor dimensions to minimize passage curvature effects. Since the value of the rotor axial length was less than the value of the rotor tip ratio, a linear segment is added at the rotor inlet (Figure 4). The rotor shroud contour is defined by a power law relation to obtain reasonable passage area distribution, which is shown in Equations (14) and (15) [7].  
\[r_{s}=r_{s5}+(r_{4}-r_{s5})\xi^{n} \tag{14}\]  
\[\zeta=(z_{s}-z_{5})/(\Delta z_{R}-b_{4}) \tag{15}\]  
The value of \(n\) should be selected in the range of 2 through 9 so that the mean passage area should be average of the rotor area at inlet and exit stations. In this study, this parameter is considered to be equal to 7.  
Figure 4: Rotor hub contour.  
Two different types of rotor blade usually are used in RI turbines: Straight-line element (three dimensional) and radial element blades. Straight-line element blades are constructed by connecting corresponding blade camber-line. These blades are easy to design and manufacture and usually provide more efficient blades than radial element blades. Here the rotor blades are designed based on straight-line element presented in [7,21].