## 5 Results and Discussion
### Influence of Different Geometric Configurations on the Fluid Dynamic Behavior and Mixture Time  
In order to explore the developed model, some new theoretical recommendations were proposed for the influence of the dimensions of the baffles after applying constructal design [19; 30]. The studied cases have the same dimensions for the tank previously shown in Figures 1 and 3. The only difference here is the shape of the baffles. The degree of freedom \(N_{a}\) (number of baffles) remains fixed (\(N_{a}=4\)), while the degree of freedom \(L/B\) (corresponding to the size of the baffles) varies. Thus, seven different dimensions for the baffles were studied in addition to the unbaffled tank shown in Figure 4.  
Figure 14: Results for turbulence kinetic energy profiles obtained with the present code and those predicted by [PERSON] et al. [11]. Based on the results presented in Table 4 and in Figures 16 and 17, it can be seen that the average turbulence kinetic energy is the highest for the unbaffled tank, and as the \(L/B\) ratio decreases, this average also decreases. This decrease occurs until \(L/B=0.08\), then the average turbulence kinetic energy starts to increase, i.e., the case where \(L/B=0.08\) presents the lowest value for the turbulence kinetic energy. It is worth mentioning that, in spite of high amounts of turbulence kinetic energy, the unbaffled tank and the lowest ratio of \(L/B\) presented obvious dead spot regions. For the unbaffled tank, they were at the periphery, whereas they were behind the baffles when \(L/B=0.02\). This behavior can impair the mixture in the whole domain. Figure 18 shows the fields of turbulent viscosity for the same configurations presented in Figure 17. The results for turbulent viscosity corroborate the findings for turbulence kinetic energy, such that poor distribution of the turbulent viscosity occurred at the periphery of the unbaffled tank even though the highest magnitudes occurred in the central region. Moreover, the most distributed fields associated with high magnitudes of turbulent viscosity were noticed for the case when \(L/B=1.0\), while at the lowest value for \(L/B\), large dead spots were caused by the intrusion of the baffles. To evaluate the influence of the baffles on the mixture time, numerical simulations were performed using the species transport model. A tracer with the same fluid-dynamic properties as the main fluid was inserted into the flow, as shown the Figure 19, and the standard deviation for the mass fraction of the tracer as a function of time was monitored in the tank domain. Table 5 presents the maximum values for the tracer mass fraction as a function of time in the tank domain. The relative difference for the tracer mass fraction at each instant of time was calculated in order to obtain the instant at which the mass fraction changes were no longer relevant. Thus, when the values for the tracer mass fraction from one time step to the next showed a relative difference smaller than 5%, the variation was considered no longer relevant, and the tracer had achieved an equilibrium in the system. As shown in Table 5, the case that required the longest time, 6.5 s, to reach equilibrium in the system was when \(L/B=0.05\). The fastest time, 4.0 s, occurred when \(L/B=0.14\) and \(L/B=0.02\). Thus, all simulations modeling the behavior of the tracer in the system flow were run for a maximum time of \(t=7.0\) s. The result with the smallest standard deviation in the system was considered to represent the best mixture, as shown in Figure 20.  
Figure 17: Distribution of the fields of turbulence kinetic energy in the baffled tank for different ratios of \(L/B\) compared to the unbaffled tank. It was observed that all cases initially had similarly large magnitudes for the standard deviation, since the tracer mass fraction kept changing as the system was being homogenized.