## 2 Mathematical Modeling
### Geometrical Evaluation  
Geometrical optimization was done by using an association between constructal design and exhaustive search, as explained in recent literature [47]. Figure 2 illustrates the main steps employed for optimization of the problem. Steps 1 to 3 presents the flow system and identifies a performance indicator (here, it is filling time). Steps 4 and 5 show the constraints of the problem, degrees of freedom, and the main parameters of the physical problem. It is also necessary to understand the physical problem and define a method for its solution (numerical solution in this case). With the defining of geometry and understanding of the physical solution method, it is possible to predict the performance indicator (minimization of filling time of resin impregnation along with the porous domain) for each configuration studied. After Step 6, it is necessary to define if the design should be optimized or not. If the problem is not optimized, a geometrical evaluation is performed, seeking to understand the effect of degrees of freedom over performance indicator, simulating the total number of cases defined. However, if the problem is optimized, it is necessary to specify an optimization method. Here, exhaustive search, which consists of evaluating several geometrical possibilities taking into consideration fixed increments of variation for each degree of freedom (as depicted in Figure 3) for the T-shaped case. Figure 1: Computational domain of the problem, which mimics the liquid resin infusion (LRI) process with two empty channels geometries: (**a**) I-shaped, (**b**) T-shaped. Figure 2: Flow chart illustrating the main steps of the geometrical optimization of I and T-shaped empty channels inserted into the rectangular porous plate for resin impregnation along the domain. For complete optimization, Steps 6b to 9b, illustrated in Figure 2, are executed. It should be emphasized that heuristic-based optimization methods, e.g., genetic algorithm and simulated annealing, have also been associated with constructal design [48]. These kinds of tools are more used in software packages such as FLUENT [41] and Magma [5][49]. These methods required a significantly lower number of cases to be solved, but need verification with exhaustive search for suitable reproduction of the effects of DOFs over the performance indicator  
Each case under analysis is submitted to two restrictions, the total area of the computational domain:  
\[A=H\!L \tag{8}\]  
and the area of the empty channel, which for I-shaped and T-shaped channels, respectively, are represented by:  
\[A_{I}=H_{0}L_{0} \tag{9}\]  
\[A_{T}=H_{0}L_{0}+H_{1}L_{1} \tag{10}\]  
The dimensions of the external domain with porous medium are: \(H=L=0.5\) m for all simulations, i.e., the outer area is a geometric constraint of the problem, and \(H/L=1.0\) is taken into account. It should be noted that a spacing of 0.05 m was taken into consideration for the sides of the fibrous reinforcement mold, and an additional height of 0.25 m was inserted to prevent void regions in the locality defined by the dashed line shown in Figure 1a,b, which is the main analysis region of this study. The areas \(A_{I}\) and \(A_{T}\) can be represented by the ratio between the empty channel and plate areas (fraction area), written by:  
\[\phi=\frac{A_{I}}{A}=\frac{A_{T}}{A} \tag{11}\]  
For the I-shaped empty channel, one degree of freedom (\(L_{0}/H_{0}\)) is considered, whereas the T-shaped empty channel has three degrees of freedom (\(L_{0}/H_{0}\), \(H_{0}/H_{1}\), and \(L_{1}/H_{1}\)). To optimize the geometry of the I-shaped channels, a set of 60 simulations were performed with different \(L_{0}/H_{0}\) ratios and different values of \(\phi\), where \(\phi=0.005\); 0.01; 0.03; 0.05, and 0.1. In the case of T-shaped channels, only \(\phi=0.05\) was evaluated, due to the high number of simulations required for each magnitude of \(\phi\). In this specific case, a total of 189 simulations were performed.