At the first attempt, the optimization model was run through 126 generations to produce 1,259,049 designs. Because of the strict constraints on the output parameters, only 188,340 of the designs (about 15%) were feasible. Figure 7 clearly demonstrates the superiority of the computerized optimization over the manual design. Figure 8 shows the Pareto frontier of the best designs from the first optimization, which generated a 1.9% performance gain in efficiency and 9 degrees of the exit angle. The Pareto graph shows a few discontinuities that clearly demonstrate the nonlinear nature of this problem and the advantage of using a genetic algorithm in this case. Despite the success of the first phase of the turbine design, a more rigorous analysis of the optimized designs presented in Figure 8 revealed unexpectedly large angles of attack (DBeta1) between the blades and the gas flow (up to 20 degrees).

#### Fig. 10 - Pareto designs (second optimization)

The problem, here, is that the turbine loss model has not been validated at such angles, so another limiting condition was added in the form |DBeta1|≤10deg. A further 220,440 configurations were calculated using the Non-dominated sorting genetic algorithm II (NSGA-II). According to mF [4], NSGA-II implements the crowding distance approach which guarantees the diversity and spread of the solutions on the Pareto front by estimating the density of the solutions in the objective space and guiding the selection process towards a uniform spread. The points belonging to the same front are sorted such that a higher ranking is given to the points located in the less populated regions of the front. Since this sorting demands additional calculations, we used the NSGA-II algorithm because of the smaller population. No advantage over the MOGA-II algorithm was observed.

After the addition of a constraint on the blade’s angle of attack, only 23,030 feasible designs were obtained (about 10%) from this second optimization. Figure 9 shows that the best efficiency reached was 85.4%, which reduced to 85.2% when the exit angle was above 80 degrees (generally, the exit angle should be close to 90 degrees). Figure 10 shows the Pareto frontier of the best designs from the second optimization. A prime number of vanes (23) and blades (31) was obtained, with a potential performance gain of up to 1.5% in efficiency and 6 degrees of the exit angle. To maximize the efficiency of the optimized turbine, another 546,000 designs were calculated under the following conditions:

- Vane number ZN=23 (primary number);
- Blade number ZM=29 & 31 (primary numbers);
- Exit angle limited by 80 degrees;
- Maximum efficiency was the only target.

These additional conditions reduced the number of design variables and objectives, making it easier to find a solution. As a result, the turbine’s efficiency was improved up to 85.5% (see Figure 11). As one can see, the main gain in the turbine’s efficiency was already achieved after 20-30,000 designs, while a very moderate, but clear improvement was obtained after that. As already mentioned, 33 input variables were used in the design of the small axial-flow turbine. Obviously, not all these parameters equally affecting the turbine’s performance. A sensitivity analysis was performed using the mF tool [4] to detect the most important input variables. This enabled the exclusion of certain variables from the optimization, reducing the required computational effort. This analysis is particularly useful to better understand the physical model. In particular, sensitivity analysis functions can be used in the Response surface method (RSM) training process. Figure 12 illustrates some of the sensitivity analysis results. The following 7 of the 33 parameters were responsible for 80% of the efficiency variation:

- BETA1K – blade inlet angle
- BETA2K – blade exit angle
- RNTIP – stator tip radius
- LW2AD – relative rotor exits Mach number
- L1AD – absolute stator exit Mach number
- ALFA1K – stator exit angle

The second section from the turbine disk, next to the hub, was found to be the most important section for maximum efficiency. It is to be noted that the huge amount of data generated through the optimization process becomes difficult to treat: we found that the mF operations in the working table became slow at about 500,000 designs. Large tables cannot be treated by MS Excel due to its limit of 1,048,576 rows and 32,000 graph points. The optimization process was therefore split to cater for these limitations, and only the feasible designs were treated in post-processing analysis. As a result of the optimization, the turbine exit angle was improved by 2÷6 degrees, while the efficiency was increased by 1.5%. One may question how meaningful this result is. Visual inspection of Figure 7 demonstrates a sufficient improvement in optimized efficiency compared to the manual design that was based on a few dozen attempts by a turbine design expert. However, some quantitative estimation of the performance improvement gained through the optimization is desirable. To this end, we used the classical Smith chart [5], which is a map that describes the empirical correlations between the efficiency of the state-of-theart axial turbine stages, and the loading and flow coefficients, and which is widely accepted as feasible for use during preliminary turbine design. According to the Smith chart, the turbine efficiency achieved with the manual design was about 4% below the maximum efficiency of the axial turbine under consideration. This meant that the efficiency improvement of 1.5% achieved with design optimization had closed about 40% of the potential gain in turbine efficiency. Other measures, such as the advanced blade geometries, hub and tip contouring, abradable seals, stacking, and 3D blade design can close the remaining gap.