## 2 Mathematical and Numerical Modeling
### Application of Constructal Design to Design of Double Y-Shaped Cavity  
The optimization methodology is oriented by the CD method that defines a hierarchy path between the degrees of freedom to perform the geometrical evaluation, i.e., the CD defines the search space and the order of DOFs investigated. The optimization process starts with the investigation of one degree of freedom while the other is kept at constant magnitudes. In the present work, it is optimized the degrees of freedom \(\alpha\), \(\beta\), \(S_{1}/H_{0}\), and \(H_{0}/L_{0}\), i.e., a four-level optimization. For all optimization levels, the constraints were defined as \(\varphi_{\rm c}=0.01\), \(\varphi_{1}=\varphi_{2}=0.015\). The degrees of freedom \(H_{1}/L_{1}\) and \(H_{2}/L_{2}\), which describe the double Y-shaped cavity branches, were kept with minimal values on the search space, following the recommendations of the best configuration of double T-shaped cavity studied in [6; 17; 18]. For the first level of the optimization process, the cavity branch angles (\(\alpha\) and \(\beta\)) were optimized synchronously with the same values, as if they were just one degree of freedom (\(\alpha=\beta\)). Then, the meta-heuristic is applied to find the optimal geometry of (\(\alpha\))\({}_{\rm o}\) and (\(\beta\))\({}_{\rm o}\) that returns the (\(\theta_{max}\))\({}_{m}\). The search space of angles investigated was \(\alpha=\beta\in[-80^{\circ},80^{\circ}]\) with the discrete variation of \(\Delta\alpha=\Delta\beta=1^{\circ}\). The values of other ratios are fixed, \(H/L=1.0\), \(H_{0}/L_{0}=10\), and \(S_{1}/H_{0}=0.5\).  
The second level of the optimization process searched for the optimal values of \(\alpha\) and \(\beta\) independently (\(\alpha\
eq\beta\)) in the same search space as the previous optimization level. The other geometric parameters and constraints were kept constant during the optimization process and with the same values used in the previous optimization level. Therefore, the optimization method seeks the optimal geometries of (\(\alpha\))\({}_{\rm o}\) and (\(\beta\))\({}_{\rm 20}\) that lead to (\(\theta_{max}\))\({}_{\rm 2m}\). For the third optimization level, the degree of freedom \(S_{1}/H_{0}\) was added to the investigation, so the optimization algorithms seek for (\(S_{1}/H_{0}\))\({}_{\rm o}\), (\(\alpha\))\({}_{\rm 20}\), and (\(\beta\))\({}_{\rm 30}\) that reaches the three times minimized dimensionless maximum excess of temperature, (\(\theta_{max}\))\({}_{\rm 3m}\). The other geometric parameters continued to be kept fixed with the same magnitudes of the previous optimization process, except for \(H_{0}/L_{0}=16\). Finally, the last experiment performs a fourth optimization level, i.e., taking also into account the ratio \(H_{0}/L_{0}\). Therefore, the three degrees of freedom \(\alpha\), \(\beta\), and \(S_{1}/H_{0}\) were optimized for different magnitudes of \(H_{0}/L_{0}\) in the last investigation.  
Considering that the CD method has the main aim to understand the effect of the degrees of freedom over the optimal geometries and performance indicators, the experiments performed here focus on adapting the meta-heuristic to the problem of interest.  
\begin{table}
\begin{tabular}{c c} \hline
**Number of Elements** & \(\theta_{max}\) \\ \hline Present Work & 0.0756 \\ [PERSON] et al. [6] & 0.0756 \\ [PERSON] et al. [10] & 0.0762 \\ [PERSON] et al. [7] & 0.0755 \\ \hline \end{tabular}
\end{table}
Table 2: Validation of the computational method used in present work with references in the literature [6; 7; 10].  
Supposing the algorithm presents an adequate performance to reach the optimal geometries, it will probably be capable of reproducing the correct effects of geometries over the system performance for complete optimization and evaluates all seven degrees of freedom in future studies with a much lower computational effort than that required with the exhaustive search.