## 5 Results
### 3D Optimization Evaluation  
Once a 2D optimized design was found, and after the revolution and the addition of the second part of the whole mixer, the 3D optimization method was computed and evaluated.  
Unlike in other methods, the changes derived from an adjoint method are really small between iterations, as it is constantly adjusting itself, so it is quite difficult to see them in a simple view. These changes are expected to be monotonous and asymptotic.  
Figure 11: Flow field comparison between the initial (**left**) and the optimum (**right**) cases.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **2D Ini** & **2D Optim** & **Corresponded 3D** \\ \hline \(\hat{m}\)(kg/s - rad) & 0.942 & 0.944 & 0.94 \\ \(BPR(-)\) & 3.49 & 2.64 & 2.66 \\ \(\Delta_{p}\)(Pa) & 270 & 229 & 229 \\ \(UI_{u}(-)\) & 0.93 & 0.96 & 0.98 \\ \(M1_{\Phi}(-)\) & 0.0311 & 0.0309 & 0.0310 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Surface average results of the main variables.  
In Figure 12, the results of the cost function are summarized. The left top figure shows the minimization of the cost function itself. It is shown that the main correction was performed on the first iteration of the adjoint method, and after each iteration, the corrections were smaller than the previous one until the process converged to a final value.  
The right top shows the evolution of the BPR. Like the cost function, the main change appeared in the first iteration and then it converged until the final value. It is worth mentioning that the BPR evolution, according to the objective value, was favourable to the study, because it reached the objective value at the same time as the cost function was minimized. On the other hand, by observing the UI and the mass flow of the case, the tendency was exactly the opposite. As the cost function decreased, both the UI and the mass flow moved away from the objective value. The UI decreased its value and so did the mass flow. Nevertheless, as the weight for each of the values was quite restrictive, the final mass flow and the objective mass flow differed by just 0.3%.  
Once the cost function was analysed, the influence in the pressure drop had to be checked in order to see if the method worked. In Figure 13, the minimization of the pressure drop is shown and, following the same tendencies as those of the rest of the variables of the cost function, the value decreased (mainly in the first iterations) and converged to a final value of 402 Pa, which meant a decrease of approximately 10% with respect to the initial value.  
Figure 12: Cost function evaluation.  
Together with the evaluation of the method, an analysis of the final geometry solution was conducted. As mentioned earlier, by analysing the differences of the main scalar contours between the original 3D geometry and the optimized one, no remarkable differences could be easily observed, although the pressure drop was minimized. In order to support the final results, an analysis of the total pressure stage by stage along the longitudinal axis of the mixer was performed, included in Figure 14, observing that the decrease in the pressure drop was mainly developed around the initial body of the mixer, where the main parts eligible for their optimization were defined, whereas in the bottom part, as already concluded, the solution was almost perfect, and no remarkable differences were observed.  
Together with the averaged values, the velocity contours are included in Figure 15 in order to check the mentioned differences. Nevertheless, the nature of the adjoint method, which acts by only changing small details of the geometry, and the already well-designed mixer, which leads to small changes in the values, resulted in velocity contours where the differences were almost not noticeable.  
Figure 14: Pressure drop by stages along the \(x\) axis in the mixer.  
Figure 13: Pressure drop minimization.