# Radial Turbine Optimization: A 3D Inverse Design Based Rapid Multi-disciplinary Optimization Strategy

3D Inverse Design Turbine August 21, 2024

In the first part of this article, we showed how to perform the initial design of a radial inflow turbine rotor using the 3D inverse design method.

In this second part, we demonstrate the use of Optima feature in TURBOdesign1 to rapidly optimize the initial rotor design to a less 3D blade (not completely radial-filament), and quickly obtain the desired trade-off between its aerodynamic and mechanical performance.

Optimization Workflow

Figure 1 displays the workflow used by the automatic optimization, where the blade loading and stacking parameters are used to generate the blade shape in TURBOdesign1, and then the resulting performance parameters are fed into Optima which basically applies MOGA to drive the solution towards the optimum design.

Figure 1: Workflow used in automatic optimization

Optimization Setup

The optimization process starts by specifying the range of variation for the input parameters shown in Figure 2, which are the streamwise loading parameters as well as the trailing edge work coefficient because it has an influence on the blade wrap where the stresses are maximum. In fact, there is a machine learning based feature that can provide initial range estimates for these input parameters, which helps to explore a large design space but also avoid infeasible solutions.

Also shown are the constraints where, in addition to the overall diffusion ratio, some wrap angle and work related user-defined parameters are also constrained:

• UD_Wrap is the wrap angle difference between 15% span and hub at blade trailing edge
• UD_RMSE calculates the root mean square error of all the wrap angle difference between these various locations which would be zero for a perfectly radial-fibred blade
• UD_work is calculated by the volume flow averaged rVtheta variation between the leading and trailing edges, and is related to the turbine power through Euler’s turbomachinery equation

Finally, for the optimization objectives we want to minimize the profile loss and the leaving kinetic energy in order to maximize the efficiency.

But at the same time, for keeping the blade stresses in check, we also want to minimize some wrap angle parameters like:

• UD_MaxDeviation is the maximum value of all the wrap angle differences used in the calculation of UD_RMSE
• UD_0.9_MaximumBow is the maximum deviation from linear span at 90% of streamwise location near the trailing edge and would be zero for a perfectly radial-filament blade

Figure 2: Radial turbine optimization setup

In this way, using these wrap angle parameters we ensure that blade integrity is not compromised in the process of improving its aerodynamic performance. In fact, this methodology was successfully used in one of ADT’s recent technical papers that was presented at the 15th IMechE turbocharging conference [1].

Optimization Results

Once the optimization run is complete, which is very fast and only takes about 3 to 4 hours on a single core machine, the design candidates can be seen on a scatter plot between any two objectives, in this case the maximum bow on the X axis and the maximum deviation on the Y axis as shown in Figure 3.

Here we have chosen to see the Pareto front of optimum designs from which we can pick and analyse any design of interest. For the present study, we select a design which gives us the smallest maximum deviation and highest turbine work, and so we use this design for further analysis.

Figure 3: Pareto front of optimum radial turbine rotor designs with the selected design

Figure 4 reveals what changed as a result of the optimization. For the streamwise blade loading, compared to the baseline rotor, the optimizer increases the aft-loading at the shroud even further, in order to reduce the profile loss. Regarding the spanwise work (rVt*) distribution, the optimizer recommends leaving a slight swirl at the trailing edge for improving the blade wrap in this area. As a result, the optimized rotor clearly looks very different from the baseline rotor as shown alongside.

Proceeding to some more results, so far as the TURBOdesign1 performance parameters are concerned, there is a substantial improvement in turbine work and also in the various wrap angle-related UDPs as reported in Figure 5. While the initial design had a high lean or wrap at the trailing edge, the optimized design features a much more radial blade which should help reduce the stresses in this area.

Fig 5: Radial turbine rotor optimized TD1 parameters

Comparison with Baseline Rotor (from DOE+RSM optimization)
Previously we generated a highly optimized design using surrogate models involving multiple CFD and FEA runs [2], and we will now use it as the baseline result to see how our rapid optimized design performs compares to that. As illustrated in Figure 6, the wrap angle distribution in the optimized rotor looks quite close to the baseline near the exducer region. Of course, the baseline rotor is purely radial-filament due to the post modification on the 3D geometry.

Figure 6: Figure 6: Radial turbine rotor wrap angle contours comparison

Moving on to the mechanical performance of the optimized rotor, an FEA analysis provides final confirmation that the peak stress in the optimized rotor is contained well within the yield strength of the material, as displayed in Figure 7. In fact, it is quite similar to what we had with the baseline rotor which was radial filament of course. Although there is a small region of high stress on the pressure side trailing edge, but this is mainly caused by the lack of modeling the fillet.

Fig 7: Radial turbine rotor von-Mises stress contours comparison

Finally, what remains to be verified is the aerodynamic performance of the optimized rotor, and so a stage CFD analysis is run to check the actual performance. As Figure 8 shows, ANSYS TurboGrid is used for the fully structured grids of the rotor and the vaned nozzle passages and CFX for the flow analysis. Here are the different CFD settings where these boundary conditions are chosen to match the baseline design from the previous DOE-based study for a fair comparison:

• P01, T01 = 2.2 bar, 403.2 K
• P2 = 1 bar
• N = 50,000 – 90,000 rev/min
• Nozzle-rotor interface = stage (mixing plane)
• Turbulence model = k-omega SST
• Total mesh size ≈ 1 M
• Average blade y+ ≈ 8

Figure 8: Radial turbine rotor CFD setup

Stage CFD is performed with the optimized rotor at five rotational speeds, keeping the nozzle the same as the baseline design. The results are reported in Figure 9, where the efficiency stays very close to what we had with the baseline rotor from the DOE-based optimization, and is shifted slightly to the right. Furthermore, the mass flow rate remains aligned with the baseline value, while being slightly lower only in the high RPM region.

Figure 9: Radial turbine performance maps with optimized rotor

Proceeding to some more results from the design point CFD, Figure 10 shows velocity vector plots in the blade-to-blade view which confirm good flow behaviour throughout the nozzle and the rotor, where the little under-turned region at 90% span could be related to tip leakage effects.

Conclusion

High speed radial turbines typically present a trade-off between the aerodynamic and structural aspects of the rotor. Essentially, using inverse design with special user-defined parameters for wrap angle control, it is possible to quickly optimize the rotor and achieve the optimum blade loading for trade-off in a matter of a few hours and achieve performances at par with more expensive time-consuming CFD and FEA based methods. Moreover, this methodology involves very less computational resources compared to conventional design methods. Our experience has shown that the choice of optimum loading for controlling profile loss or secondary flows has generality and can be applied to other similar applications. For example, we find that for profile loss control, the type of loading that we use for turbines is applicable to all types of turbines, mixed flow and axial and regardless of the turbine speed or size.

References
[1] Zhang, J., Zhang, L., Zangeneh, M., “A 3D inverse design based rapid multi-disciplinary optimization strategy for radial-inflow turbines”, 15th International Conference on Turbochargers and Turbocharging, Institution of Mechanical Engineers, ISBN: 978-1-032-55154-8

[2] Zhang, J., Zangeneh, M., “Multidisciplinary and multi-point optimisation of radial and mixed-inflow turbines for turbochargers using 3D inverse design method”, 14th International Conference on Turbochargers and Turbocharging, Institution of Mechanical Engineers, ISBN: 978-0-367-67645-2

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### Geet Nautiyal

Geet Nautiyal is a Turbomachinery Application Engineer at Advanced Design Technology, focusing on design, marketing and customer support aspects.

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