## 3 Modeling of Momentum Source Model
### Constructal Design Application to the Problem  
As previously mentioned, constructal design is the method used to apply the constructal law of design and evolution [19; 20]. The method consists of the investigation of geometry subjected to constraints (global and local), which define the limitations and search space for the variation in geometry given by the degrees of freedom, and the performance indicators, which are used to define the maximization of the internal currents of the flow system. The main steps for the geometric investigation performed here are illustrated in the flowchart in Figure 2. Steps 1 to 3 consists of the definition of the flow system and performance indicators, and steps 4 and 5 define the constraints of the problem, degrees of freedom and the main parameters of the physical problem. At the same time, it is necessary to understand the physical problem and define a method for its solution (numerical in the present work). After the fifth step, it is important to define whether the problem is optimized or not, as well as the number of cases required to investigate the influence of degrees of freedom, constraints and parameters on the performance indicators. It is worth mentioning that constructal design is not an optimization method, but a method for geometrical investigation based on the principle of constraints and objectives [30; 31]. Here, the method is applied to the geometric investigation of only one degree of freedom (\(L/B\)) on the performance of turbulent flow in stirred tanks due to the high computational effort required for each simulated case. The last three steps (6 to 8) consist of defining the number of cases to be studied, performing the numerical simulations for each configuration and investigating the effect of the degrees of freedom on the performance indicators. It is considered here that the problem is subject to two area constraints: the total area in the tank where the baffle can be placed (\(A_{T}\)) and the area occupied by the baffles (\(A_{a}\)). A sketch of the cross-sectional area of the tank and the above-mentioned areas is illustrated in Figure 3. The constraint areas are given by Equations (14) and (15) as follows:  
\[A_{T}=A_{F}-A_{I} \tag{14}\]  
where \(A_{F}\) is the total fluid area of the tank and \(A_{I}\) is the area that can be occupied by the impeller. \[A_{a}=N_{a}\cdot L\cdot B \tag{15}\]  
where \(N_{a}\) is the amount of baffles, \(L\) is the baffle width and \(B\) is the baffle length. The dimensions \(L\) and \(B\) are defined considering the constraints of the area occupied by the baffles. Thus, there is an equation and three variables, and it is necessary to define two degrees of freedom to solve this problem (\(N_{a}\) and \(L/B\)) as defined by the following equations:  
\[B=\left(\frac{A_{T}\cdot\varnothing}{N_{a}\cdot\left(\frac{L}{B}\right)}\right) ^{1/2} \tag{16}\]  
\[L=\left(\frac{L}{B}\right)\cdot B \tag{17}\]  
where \(\varnothing\) is the ratio between the areas occupied by the baffles and the total area where the baffles can be placed, which is given by the following equation:  
\[\varnothing=\frac{A_{a}}{A_{T}} \tag{18}\]  
In addition, the dimensions \(L\) and \(B\) have a limit to vary, i.e., \(B\) can be varied so that its maximum length does not exceed the area occupied by the impeller. The limit of \(L\) is the maximum width before exceeding the tank wall limits. Thus, with the degrees of freedom defined, the geometric evaluation takes place by varying one of them (\(L/B\)) and keeping the other fixed (\(N_{a}\)). The constraints of the area are also fixed. Here, the investigation is limited to \(N_{a}=4\) for the effect of \(L/B\) on the standard deviation of the mixture inside of the tank domain and the homogenization time, which can be minimized. Figure 4a shows the tree of simulations performed from the application of constructal design, Figure 4b illustrates the dimensions of the baffles inserted in the tank, and Table 2 shows the dimensions of baffles used in each case.