# Design of an Axial Turbine for Highly Downsized Internal Combustion Engines
## 3 Methodology
### Turbine Stator and Rotor Profiling  
The design of the airfoils was performed in the AxSTREAM profiling module. AxSTREAM automatically proposes the optimized profile for the middle section, and starting from that and from the already calculated parameters and flow condition, it proposes an initial airfoil solution for all the other sections. However, many changes to airfoil parameters are performed, mainly on the stagger angle, inlet wedge angle and outlet wedge angle (as they do not affect the optimized camber line shape), in order to have acceptable Mach numbers over the airfoil and respect the Buri criterion (Figure 8). Further, a second optimization process based on the reduction of profile losses with a DoE method was performed with the airfoil optimization tool embedded in the AxSTREAM profiling module, allowing for a semiautomated airfoil shape improvement. The profiling module is also used to control the turbine manufacturability by checking critical areas, such as the size of the throat of the nozzles, or more generally, the overall turbine solidity, in such a way to ensure it can be machined. After the airfoil optimization and manufacturability check, 2D cascade CFD (RANS) analysis was performed at various spanwise sections to control the effectiveness of the generated airfoils before passing the data to the CFD module, which typically requires much longer setup and solving times. Figure 7: Simplified load and bending moment plot schematic of a prismatic leaned blade under centrifugal loads. (M(s): bending moment; q: load; r; turbine radius; s; blade span). This module was also used to control the leaning, sweeping and bowing of stators and rotor blades to obtain further efficiency improvements (Figure 9). This operation was performed infrequently, as the bending stress on the blades increases substantially due to the centrifugal loads, as shown in the following simplified calculations for the simple case of a prismatic leaned blade. As the principle of superposition is valid for this kind of problem, only the centrifugal loads are considered for this demonstration, as the contribution of the aerodynamics and mechanical load does not affect the partial solution and can be added separately. Therefore, defining the specific centrifugal load in cylindrical coordinates as:  
\[q(r)=\rho a^{2}\left(r_{t}-\frac{r}{2}\right) \tag{2}\]  
then considering the bladewise coordinate,  
\[s=r\ \cos(\theta) \tag{3}\]  
the centrifugal load can be decomposed in a bladewise load \(q(s)_{//}\) and a shear load \(q(s)_{\perp}\), as in Figure 8.  
In particular, the presence of the transverse load with centrifugal force, which evidently would not happen with a radially extruded blade, is the factor generating the added bending moment. In fact, considering  
\[q(s)=\rho a^{2}\left(r_{t}-\frac{s}{2\cos(\theta)}\right)\sin(\theta) \tag{4}\]  
Figure 8: Streamwise Mach numbers (blue) and Buri criterion values (green) over a stator profile section. (“comp”: compressible; “incomp”: incompressible; “SS”: surface side; “PS”: pressure side). the shear force is:  
\[T=\int_{0}^{s^{\prime}}q(s)\;ds \tag{5}\]  
and the bending moment is:  
\[M(s)=\int_{0}^{s^{\prime}}T(s)\;ds=\rho\alpha^{2}\sin(\theta)\bigg{(}r_{t}\frac{ s^{2}}{2}-\frac{s^{3}}{12\cos(\theta)}\bigg{)} \tag{6}\]  
with the plot shown in Figure 8 displaying a strong bending load at the blade root. Moreover, despite the fact that bowed, swept or leaned blades often display better performance, they also come with a significantly more challenging manufacturing process, as they inherently have undercut areas requiring additional CNC milling time, precision and capability. The stuck-up airfoil section was also controlled to evaluate the influence of different choices, both from the aerothermal and structural point of view. Centroid axis, leading edge and trailing edge stuck-up types were tested in different combinations both for the stator and rotor, but the preferred one remained the standard centroid stuck-up type.