#
Analysis of Mixing Efficiency in a Stirred Reactor Using Computational Fluid Dynamics^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Geometrical Model (Tank Reactor and Impeller)

^{2}and 0.282244 m

^{2}, respectively, forming a solid wall with 0.9514656 m

^{2}. A non-structured mesh with triangular cells was selected for a good fit with the cylindrical and semispherical regions of the reactor.

^{2}.

## 3. Semi-Planes, Injection and Monitoring Points, and Initial Assumptions

_{1}, Pi

_{4}, Pi

_{8}, Pi

_{14}, and Pi

_{18}) are placed along the shaft from the reactor surface to the blades. The nearest injection points to the wall are Pi

_{6}, Pi

_{12}, Pi

_{16}, Pi

_{21}, and Pi

_{22}. The points in the middle tank position are Pi

_{2}, Pi

_{3}, Pi

_{5}, Pi

_{7}, Pi

_{10}, Pi

_{13}, Pi

_{15}, and Pi

_{17}. Finally, only one single point was placed on the bottom for evaluating hydrodynamic behavior in this zone (Pi

_{20}). Here, the sub-index “i” is used to indicate the injection point.

_{m}) at every time step (t + Δt) during the simulation. Here, the sub-index “m” is used to indicate the monitoring points. Initially, for t = 0, it is assumed the tracer concentration on each monitoring point is equal to zero (C

_{m}= 0). Some of these points were placed in the middle of the reactor such as Pm

_{2}, Pm

_{7}and Pm

_{12}; some others were placed near the walls such as Pm

_{03}to Pm

_{06}and Pm

_{08}to Pm

_{11}. Finally, the rest of these were placed at the bottom such as Pm

_{13}, Pm

_{14}and Pm

_{15}. Here, evaluation is very important because this region is critical for mixing.

## 4. Assumptions

- The impeller shaft is in the middle position of the reactor body for symmetrical conditions. Additionally, no vibration is assumed during rotation; then, the influence of the shaft rotation is not significant. Consequently, the only elements that provide stirring to the bath are the four blades.
- Rugosity on tank walls (cylinder and semispherical) is neglected as a consequence of the fluid displacement being free, and no drag is induced.
- The impeller rotates at 200 radians per minute: approximately 32 RPMs.
- The surface of the liquid lead is flat; this condition is equivalent to assuming a closed reactor on the tank top. The lead surface is discretized with 1512 two-dimensional cells, and the measured area is 0.149812 m
^{2}. - The maximum face area for a cell used for discretization is 1.9733 × 10
^{−3}m^{2}in contrast the minimum cell face area is 2.0324 × 10^{−6}m^{2}. - The temperature of liquid lead is assumed as 327.46 °C (equal to 600.61 K or 621.43 F). This is the lead melting point, and the lead density equal to 10.66 g/cm
^{3}. Then, liquid lead is a heavy incompressible fluid. - During all analyzed cases on simulations, no heat interchange is assumed (isothermal system); then, the lead properties remain constant.
- The tank is considered as isolated, there is no mass interchange, the volume is constant inside, and the tracer replaces a defined lead volume in the 3D mesh.
- The movement of fluid is not a step pulse defined and not a continuous injection; the tracer volume replaced is moved as a consequence of the inertial forces due to the impeller impulse.
- The time step (Δt) used for simulation was 5.208 × 10
^{−3}and represents a rotation of 1° around the tank circumference considering the rotatory speed of the impeller. - There is no volume or mass interchange during the simulation; moreover, the movement of the fluid is due exclusively to the inertial forces of the velocity generated by the impeller movement.

^{3}= 1.25 × 10

^{−4}m

^{3}. Then, the tracer distribution is saved and evaluated dynamically as simulations run. During every simulation, only one single lead volume is replaced on the injection cells by an identical tracer volume at the beginning of the simulation (t = 0). Consequently, the tracer is assumed as ideal; then, an original lead volume is substituted by another volume with the same physical properties. This method is frequently used by authors who work on physical and computer simulation to evaluate fluid flows [1,2,3,4,5,10,11,12,13,16,17,18,30,35,36,37,38,39,40]. This procedure is conducted experimentally using colorants for painting the original fluid in order to follow the fluid path and show streamlines of fluid [5,6,7,13,26,27,28,29,30,39,40,44,45,46]. Nevertheless, in real industrial practices, solid chemicals for cleaning lead are incorporated to the bath tank using a lance inside. The lance can be placed at different positions in the cylindrical body; but, geometrically, it is complicated to inject below the impeller.

## 5. Mathematical Modeling

_{x}, g

_{y}, and g

_{z}will depend on the orientation of gravity with respect to the chosen set of coordinates; for the reactor analyzed, g

_{x}and g

_{y}will be equal to zero; and only g

_{z}participates in the fluid motion. Then, the continuity equation can be written as follows:

_{m}

^{t+Δt}); finally, data are analyzed using Microsoft Excel. It is important to mention that the tank with a mix of lead + tracer can be considered as a multicomponent fluid problem, as represented in Equation (11) [42]. Here, D

_{eff}is the diffusion and turbulent coefficient.

## 6. Computer Simulations

_{11}, Pi

_{18,}and Pi

_{21}, respectively. These monitoring points were placed inside the tank, near and far from the impeller. Some of the curves in these figures show parabolic behavior and others are sinusoidal. Sinusoidal curves are near the impeller; here, the tracer concentration changed at every time step during simulation due to the strong influence of turbulence and vector velocities. In contrast, parabolic behavior is frequently observed near the walls and at the bottom of the tank, where the homogenization of tracer concentration is very slow, and no strong fluctuations are presented because these points are far away from the impeller influence. The tracer begins to be distributed by stirring from values equal to zero (the tracer is absent on monitoring points). Then, as the simulation time advances, there are some periods when the tracer concentration is increased and others when it is decreased. Nevertheless, these fluctuations are reduced as the simulation continues. The reduction on this fluctuation is considered as a parameter to evaluate mixing. Thus, for long times, all the monitoring points tend to adopt the same averaged concentration. This means that the tracer has been homogeneously distributed. There are curves with high tracer saturations on Figure 6a,c; this is not good for mixing; these excesses must be stirred to obtain a homogeneous distribution. Figure 6a shows that only a few curves are parabolic and remain always with a low tracer concentration. Nevertheless, these curves take a lot of time to reach the final average concentration, indicating poor mixing in these regions. Thus, according to the time scales in the horizontal axes, it is possible to appreciate that the best injection point is Pi

_{18}; here, the tank delay is only 450 s. On the other hand, the worst injection point is Pi

_{21}with delays of more than 1400 s, and the tracer distribution injecting on Pi

_{11}is considered as an intermediate behavior.

_{I}). Figure 7 shows these averaged curves to illustrate the general tracer concentration for the best, an intermediate and the worst injection points. Here, it is possible to confirm Pi

_{18}is the best injection point. The curve is lightly sinusoidal, but fluctuations are quickly damped; thus, the curve tends to the final tracer concentration value and no more than 450 s is required for the mixing. This curve is just above the final tracer concentration, and the fluctuations are minors and quickly damped. In contrast, the worst injection point is Pi

_{21}, where there is too much of a time delay because many of the curves are parabolic and distribution is very slow.

_{11}shows a moderate behavior, because it was placed between the shaft and the wall (in a middle distance). Its curve is also sinusoidal, although a high excess of tracer concentration is appreciated at short simulation times. This variation is also quickly stabilized as time passes, but the time required for a good mixing is near 600 s. Moreover, the fluctuations are so high in comparison with those for the injection point Pi

_{18}. This curve is initially significantly above the final tracer concentration value. Then, this instability must be reduced applying additional stirring in comparison with the point Pi

_{18}.

_{21}is the worst injection point. Here, the hydrodynamic behavior is very different. Mixing is delayed due to the tracer being slowly distributed. The time required for mixing considering this injected point is more than 900 s. This curve is always below the final tracer concentration value due to the difficulty for the distribution all around the tank. This injection point is placed near the top corner of the cylindrical section of the tank; it is very far from the impeller influence. Then, the evolution of tracer distribution is very slow. Furthermore, the buoyancy of tracer is critical for the simulation on the semispherical region because it is so difficult to distribute tracers in this region. Moreover, in Figure 7, it can be seen that the final tracer concentration all around the tank for all the injected points tends to a final value which is considered as the moment when the tracer has been homogeneously distributed and no additional stirring is required. Consequently, the best injection point is with the shortest mixing time as indicated.

_{18}). Here, colors indicate regions where the tracer has or has not been distributed. Figure 8a was snapped at the initial time (t = 0). Here, the tracer is originally injected. Figure 8b was snapped at 250 s. There are regions with an excess of tracer concentration, but there are some other regions with a poor tracer concentration such as those near the bottom and the highest corner on the cylinder. Figure 8c was taken at 500 s, and it shows a distribution that is nearly homogeneous. There are four zones remaining with lower different tracer concentrations, which are near the cylinder top region, although the profile shows a very similar tracer concentration generally. Figure 8d shows a profile with the same color in the entire reactor. It means that the all the liquid lead and the tracer are perfectly mixed due to stability and homogeneity criterions. Moreover, for all these figures, the color scales are reduced as the mix inside the tank tends to be homogeneous.

_{11}). Figure 9a was also taken at time t = 0; again, it is possible to identify where the injection point is placed. In Figure 9b, the tracer has begun to be distributed on the reactor, but there are regions with a high and a low tracer concentration in the cylindrical section. Nevertheless, big zones with a low tracer concentration are in the reactor bottom. In Figure 9c, high tracer concentrations remain on the cylindrical section near the tank walls. The tracer has been introduced in the reactor bottom, but the profile is still not homogeneous. Finally, Figure 9d shows the progress of the tracer distribution all around the reactor; zones with very high and low tracer concentrations have disappeared, and the profile is certainly more homogeneous. However, this distribution has a low efficiency in contrast with that shown in Figure 8a–d, and longer times for stirring are necessary for a good mixing.

_{21}). Here, the tracer is injected into a point near the reactor top surface and the wall; this is a very isolated region and velocity vectors are weak here, as is shown in Figure 10a. This point is placed far away from the impeller, and the influence of the rotational movement is weak. As a consequence, the tracer distribution is very slow, as shown in Figure 10b–d. Here, the tracer begins to appear in regions of the cylindrical section, but the bottom of the tanks remains without tracer presence. Nevertheless, huge regions without tracer presence can be appreciated in the middle low cylindrical region and the reactor bottom due to the slow diffusion. The final profile on Figure 10d shows a heterogeneous tracer distribution, indicating the mixing efficiency is very poor, as the tracer has not been distributed in the entire reactor. Moreover, Figure 8, Figure 9 and Figure 10 show that the most difficult zone is the semispherical zone. This behavior can be confirmed observing the curves in Figure 6 and Figure 7 for the bottom region.

- The inertial force induced by the impeller begins to break the stationary condition of the liquid lead inside the tank.
- The distribution of tracer inside any tank reactor involves the effect of the fluid displacement, which depends on the forces applied by the impeller movement.
- The tracer initially placed near the impeller is quickly distributed in comparison with the tracer placed near the top or the tank walls.
- The fluid is re-driven as a function of the tank geometry; thus, the hydrodynamics is different when the fluid makes contact with the cylindrical wall than when it takes the flat top or the semispherical bottom.
- The tracer concentration distribution is different on each monitoring point at every time step according with its position on the tank and the injection point analyzed.
- The final tracer concentration at a large time considered for a good mixing was (C
^{tmax}_{tracer}= 0.00135). This means that the tracer concentration is invariable and the total tracer has been homogeneously distributed all around the tank. Then, additional stirring is not necessary.

## 7. Analysis of the Hydrodynamic Behavior

_{18}). Figure 11a–c show the tracer concentration curves for different zones. The curves for the monitoring points near the impeller shaft are displayed in Figure 11a. The strongest influence of the impeller is on the monitoring point (Pm

_{12}). This point is placed very near the injection point; then, it is quickly saturated in excess, showing the highest concentration values. As a consequence, these regions are quickly saturated; thus, only this curve will always remain over the final tracer concentration. Hence, this zone can be considered as a stagnant but also oversaturated region. In other words, a high tracer concentration remains around this zone without being distributed. After this, the tracer concentration is lightly reduced due to a slow distribution. The tracer concentration behavior is a sinusoidal damped curve, and the alternation is evidence of the strong influence provided by the impeller near the injection and monitoring points. This influence is reduced significantly on the point (Pm

_{7}) with an intermediate position on the cylindrical tank. This curve remains without tracers and always is under the final tracer concentration until the homogeneity is reached. But, in the points far away from the impeller such as the monitoring points (Pm

_{1}) and (Pm

_{2}), this influence is very weak. The tracer concentration remains equal to zero, during the initial 25 s; the tracer is slowly dispersed and delays on arriving toward these zones. This behavior can be attributed to the joining between the shaft and the flat liquid surface. Finally, all the curves in Figure 11a are sinusoidal, and the homogenization is reached after 350 s, indicating a good mixing.

_{11}shows a similar behavior than the point Pm

_{12}. The tracer concentration is quickly increased. Although the tracer concentration value is considerably more minor than in point (Pm

_{12}), the sinusoidal behavior is also similar. Nevertheless, this point reaches the final tracer concentration value faster than the point Pm

_{12}due to its fluctuations being lower. The points in the middle of tank body such as the Pm

_{8}, Pm

_{9}and Pm

_{10}also show a sinusoidal behavior, but these points remain briefly with a tracer concentration equal to zero; then, they begin to increase its tracer concentration. These curves always remain below the final tracer concentration value until reached. Curves for the points in the faraway positions such as Pm

_{3}, Pm

_{4}and Pm

_{5}also remain below the final tracer concentration during the first 120 s and are damped curves due to the slow mixing process. Finally, the tracer concentration in the lowest region of the semispherical reactor has longer times without change (C

_{tracer}= 0), as is shown in Figure 11c. The curves for the monitoring points Pm

_{14}and Pm

_{15}are parabolic due to the tracer being slowly dispersed here. Nevertheless, the curve for the point Pm

_{13}is sinusoidal due to this point being the nearest to the impeller. These three curves also remain always below the final tracer concentration value: evidence again that the bottom is the most complicated region for distribution.

## 8. Validation of Hydrodynamics

^{−7}m

^{3}, hexahedral cell 9.6594 × 10

^{−7}m

^{3}, and honeycomb cell 9.71766 × 10

^{−7}m

^{3}, respectively; these are similar volumes which were selected to measure the efficacy of meshes.

_{i=21}, P

_{i=18}, and P

_{i=11}). Here, all the solutions tend to the final tracer concentration; then, it is possible to affirm the model works and results are reliable. Nevertheless, at the beginning of the simulation, there are notorious differences regarding approaching and variations. Simulations for the injection point P

_{i=21}is with a parabolic form due to the final tracer being slowly distributed from points near the top. Consequently, it takes longer for the tracer delay to be distributed. In contrast, simulations for the points P

_{i=18}and P

_{i=11}are with a sinusoidal form due to there being a dynamic tracer interchange as a result of the impeller movement, but the tracer is quickly distributed. Then, it is also possible to say all meshes tend to final tracer concentration, but the tetragonal mesh has reduced fluctuations. Additionally, the transitory states are the most complicated to calculate.

_{i=1}, and P

_{i=9}on every monitoring point is shown in Figure 13a,b. Here, again, all curves tend to the final tracer concentration value, evidencing that this value can be considered as a feasible criterion for a homogeneous distribution. Some points near the impeller tend to the final tracer concentration, but other delays are longer. Then, the difference on tracer concentration is a measure of the hydrodynamic difficulty to distribute the tracer on every tank zone. Here, again, it is possible to appreciate that the tracer concentration variations are minor on all monitoring points for the injection point P

_{i=9}. It is possible to appreciate that monitoring points near the impeller tends to the final tracer concentration quickly, and that monitoring points placed near the injection points are with a notorious tracer excess. In contrast, monitoring points on the semispherical zone have a tracer deficit.

_{tracer}

^{Δt}− C

_{tracer}

^{Δt−1})

_{i=21}, P

_{i=1}, P

_{i=2}and P

_{i=6}are with similar behaviors and are the most complicated for distribution; these points are near the upper wall and top of the tank. The tracer placed in these points initially is very far from the impeller influence. Velocity vectors are weak, and the inertial forces of the fluid move the tracer so slowly. An additional handicap against tracer distribution is the fact that properties of the tracer and melting lead are the same (ρ

_{tracer}= ρ

_{lead}); consequently, the influence of the buoyancy forces makes difficult to drive the tracer to the bottom of the tank. Industrially, this can be a serious problem if the density of the reacts added for the cleaning lead is equal to or less than lead density (ρ

_{tracer}< ρ

_{lead}). Then, the industrial suggestion was to place the reacts near the bottom tank. Moreover, the vector in the semispherical bottom impulses up the fluid. In the same way, the injection points P

_{i=18}, P

_{i=19}, P

_{i=20}and P

_{i=22}form a common area; these are the best injection areas due to the influence of the impeller being near. Then, the place tracer can be strongly impulsive, while tracer concentration on the semispherical zone and near the impeller influence denotes a very different hydrodynamic behavior, as can be seen in Figure 15a,b.

- Curves of tracer concentrations tend to approach the final tracer concentration faster; thus, mixing time is reduced.
- Fluctuation of the tracer concentration curve is minor.
- Tracer in excess is minor, and it is quickly distributed.

_{tracer}≠ ρ

_{lead}) and modify the tracer volume at the initial condition to know if there is enough for the cleaning process or if the tracer volume can contribute to reduce the mixing time. Another option to explore is to simulate different tank configurations and different impeller configurations.

_{i=21}, P

_{i=18}and P

_{i=11}are employed to test the mesh influence. Fine mesh is expressed as a percentage of nodes listed in Table 4. Thus, a 200% mesh has double the nodes for discretization. Here, it is evident that the times vary with wider meshes, but its influence is reduced as the mesh became smaller.

## 9. Influence of the Impeller Speed

## 10. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Geometrical model of the tank reactor; two-dimensional views (side left, frontal and upper).

**Figure 3.**Control semi-plane showing the position of every analyzed injection point (points placed to the right of the central line of the batch reactor).

**Figure 4.**Control semi-plane showing the monitoring points (points placed to the right of the central line of the batch reactor).

**Figure 5.**Velocity vector distribution (

**a**) all around the entire tank and (

**b**,

**c**) on the blades of the impeller.

**Figure 6.**Tracer concentration curves for all the monitoring points (

**a**) corresponding to the injected point (Pi

_{11}); (

**b**) corresponding to the injected point (Pi

_{18}); and (

**c**) corresponding to the injected point (Pi

_{21}).

**Figure 7.**Tracer concentration for the best, intermediate, and worst injection points (Pi

_{18}, Pi

_{11}, Pi

_{21}, respectively); curves for averaged behavior.

**Figure 8.**Tracer concentration profiles at different simulation times for the injection point (Pi

_{18}). (

**a**) t = 0 s, (

**b**) t = 250 s, (

**c**) t = 500 s (

**d**) t = 750 s.

**Figure 9.**Tracer concentration profiles at different simulation times for the injection point (Pi

_{11}). (

**a**) t = 0 s, (

**b**) t = 250 s, (

**c**) t = 500 s, (

**d**) t = 750 s.

**Figure 10.**Tracer concentration profiles at different simulation times for the injection point (Pi

_{21}). (

**a**) t = 0 s, (

**b**) t = 250 s, (

**c**) t = 500 s, (

**d**) t = 750 s.

**Figure 11.**Tracer concentration curves on different regions inside the tank for monitoring points: (

**a**) near the body impeller, (

**b**) near the tank walls, and (

**c**) in the bottom of the tank.

**Figure 12.**Comparison between different kinds of meshes used for simulation of the mixing process for (

**a**) injection point (P

_{i=21}), (

**b**) injection point (P

_{i=18}), the best injection point, (

**c**) injection point (P

_{i=11}).

**Figure 13.**Tracer concentration on every monitoring point (

**a**) for injection point p

_{i=1}and (

**b**) for injection point p

_{i=9}.

**Figure 14.**Error as a function of the tracer concentration on every monitoring point (

**a**) for injection point P

_{i=1}and (

**b**) for injection point P

_{i=9}.

**Figure 16.**Comparison of results between computer simulation and physical modeling (

**a**) for the injection point P

_{i=21}, (

**b**)for the injection point P

_{i=18}, and (

**c**) for the injection point P

_{i=11}.

**Figure 17.**Tracer concentration curves for different impeller rotate speed corresponding to (

**a**) the best injection point (P

_{i18}), (

**b**) an intermediate injection point (P

_{i11}), and (

**c**) the worst injection point (P

_{i21}).

Position (m) | Points | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Pi_{1} | Pi_{2} | Pi_{3} | Pi_{4} | Pi_{5} | Pi_{6} | Pi_{7} | Pi_{8} | Pi_{9} | Pi_{10} | Pi_{11} | |

x | 0.050 | 0.110 | 0.110 | 0.050 | 0.110 | 0.190 | 0.110 | 0.050 | 0.080 | 0.110 | 0.150 |

y | 0.440 | 0.440 | 0.385 | 0.330 | 0.330 | 0.330 | 0.275 | 0.220 | 0.022 | 0.220 | 0.220 |

Pi_{12} | Pi_{13} | Pi_{14} | Pi_{15} | Pi_{16} | Pi_{17} | Pi_{18} | Pi_{19} | Pi_{20} | Pi_{21} | Pi_{22} | |

x | 0.190 | 0.110 | 0.050 | 0.110 | 0.190 | 0.110 | 0.050 | 0.110 | 0.110 | 0.190 | 0.190 |

y | 0.220 | 0.165 | 0.110 | 0.110 | 0.110 | 0.055 | 0.000 | 0.000 | −0.110 | 0.440 | 0.000 |

Position (m) | Points | |||||||
---|---|---|---|---|---|---|---|---|

Pm_{1} | Pm_{2} | Pm_{3} | Pm_{4} | Pm_{5} | Pm_{6} | Pm_{7} | Pm_{8} | |

x | −0.200 | −0.200 | −0.200 | −0.200 | −0.200 | −0.200 | −0.200 | −0.200 |

y | 0.450 | 0.400 | 0.350 | 0.300 | 0.230 | 0.150 | 0.100 | 0.000 |

Pm_{9} | Pm_{10} | Pm_{11} | Pm_{12} | Pm_{13} | Pm_{14} | Pm_{15} | ||

x | −0.110 | −0.110 | −0.110 | −0.110 | −0.060 | −0.050 | 0.000 | |

y | 0.450 | 0.230 | 0.000 | −0.110 | −0.150 | −0.165 | −0.180 |

Area (m^{2}) | Volume (m^{3}) | Cells 2D Triangular | Cells 3D Tetragonal | Cells 3D Hexahedron | Cells 3D Honey Comb | |
---|---|---|---|---|---|---|

Tank | ||||||

Cylindrical | 0.66922 | 0.0716019649 | ||||

Semispherical | 0.28445 | 0.0250888176 | ||||

total | 0.951465 | 0.0966907825 | 1062 | |||

shaft | 0.147618 | 16,890 | ||||

impeller | 0.00129032 | 657 | ||||

Lead volume (m^{3}) | ||||||

Cylindrical | 0.0716019649 | 210,235 | 99,923 | |||

Semispherical | 0.0250888176 | |||||

total | 0.0966907825 | 99,923 | 100,100 | 99,500 |

**Table 4.**Times required for the final tracer concentration as a function of the fine mesh employed on the points P

_{i=21}, P

_{i=18}and P

_{i=11}.

Fine Mesh (%) | Time (s) for the Final Tracer Concentration | ||
---|---|---|---|

Pi_{18} | Pi_{11} | Pi_{21} | |

60 | 480 | 680 | 1015 |

80 | 475 | 635 | 945 |

Original | 450 | 600 | 900 |

150 | 445 | 590 | 888 |

200 | 447 | 595 | 895 |

250 | 448 | 591 | 898 |

300 | 450 | 597 | 901 |

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**MDPI and ACS Style**

Ramírez-López, A.
Analysis of Mixing Efficiency in a Stirred Reactor Using Computational Fluid Dynamics. *Symmetry* **2024**, *16*, 237.
https://doi.org/10.3390/sym16020237

**AMA Style**

Ramírez-López A.
Analysis of Mixing Efficiency in a Stirred Reactor Using Computational Fluid Dynamics. *Symmetry*. 2024; 16(2):237.
https://doi.org/10.3390/sym16020237

**Chicago/Turabian Style**

Ramírez-López, Adan.
2024. "Analysis of Mixing Efficiency in a Stirred Reactor Using Computational Fluid Dynamics" *Symmetry* 16, no. 2: 237.
https://doi.org/10.3390/sym16020237