The influence of design variables on the objective function
First, we discuss the effects of the three design variables on the mixing efficiency and pressure drop of the mixer at different Res. Three sets of design variables are selected for simulation, which are the minimum, intermediate and maximum values within the range of the design variables. It can be seen from Fig. 3 that when Re = 1 and 10, d/h has the most significant impact on the mixing efficiency, while w/λ has the most prominent impact on the pressure drop. For the mixing efficiency, a/b obtains the optimal value at the intermediate value, while the other two design variables obtain the optimal value at the boundary. For the pressure drop, the three design variables all achieve the optimal value at the intermediate value. But when the mixing efficiency achieves the optimal value, the pressure drop is the worst value. This shows that we can’t find the micromixer with the best comprehensive performance by only relying on experimental analysis. So we use Pareto genetic algorithm to optimize micromixer is a good choice. It can more intelligently weigh the compromise value of the two performances, thereby obtaining a micromixer with better comprehensive performance.
Influence of design variables on objective functions at different Res.
Influence of mixing unit on mixing efficiency
Next, we discuss the influence of the number of mixing units on the mixing efficiency. We perform numerical simulations on the reference design micromixers with different numbers of mixing units to show the influence of the number of mixing units on the mixing performance. It can be seen from Fig. 4 that as the number of mixing units increases, the mixing efficiency at different Res increases as the length of the micromixer increases. When the number of mixing units and the length of the microchannel increase, the diffusion time and the contact area between the fluids increase. This leads to uniform mixing and improves mixing efficiency. When the number of mixing units is six, the mixing efficiency of the micromixer can reach 80%. When we select the number of mixing units and then modify the design variables within the range, it will have a fluctuating influence on the mixing efficiency. Therefore, six mixing units are selected to leave room for the fluctuation of mixing efficiency and to ensure that the optimization effect is excellent. Therefore, the six mixing units are selected to design the micromixer.
The influence of the number of mixing units on mixing efficiency.
Optimization results at different Res
From the above discussion, it can be seen that the mixing efficiency and pressure drop increase with increasing channel length when Re = 1 and 10. Therefore, finding a micromixer that combines high mixing efficiency and low pressure drop is the problem we need to investigate. We use the Pareto genetic algorithm to perform multi-objective optimization of the two objective functions of mixing efficiency and pressure drop. In order to analyze the Pareto optimal solution, we choose five representative Pareto optimal design at each Pareto optimal frontier through K-means clustering. The optimized results obtained are shown in Fig. 5. It can be seen that on OPT-A (Re = 1), the range of mixing efficiency of Pareto optimal frontier obtained by optimization is 75%-100%, and the pressure drop is in the range of 0–100 Pa. Among them, on OPT-B (Re = 10), the range of mixing efficiency of the Pareto optimal frontier is 80%-100%, and the pressure drop ranges between 0–1400 Pa.
The relationship between mixing efficiency and pressure drop on Pareto optimal frontier at different Res.
Five representative OPT results from numerical simulations on OPT-A and OPT-B are listed in Tables 6 and 7, respectively. It is observed that the optimized design variables a/b, w/λ, and d/h are respectively close to the middle, lower and upper limits of the range. Comparing the numerical results of the two tables can show that on OPT-A, compared with the reference design (RD), the mixing efficiency of OPT-1 at the outlet is increased by 20.59%, and the pressure drop is also increased by 223 Pa. On OPT-B, the mixing efficiency at the outlet of OPT-1 is increased by 14.07% and the pressure drop is also increased by 998.75 Pa compared to RD. The mixing efficiency at the outlet of OPT-5 is increased by 5.55% and the pressure drop is also increased by 72.75 Pa. Through analysis and discussion, compared with the RD, when Re = 1 and 10, the mixing efficiency is increased by 20.59% (OPT-1 on OPT-A) and 14.07% (OPT-1 on OPT-B), respectively. In summary, the optimized results are indeed much better than the reference design. We can also make a good trade-off between the two performances of the micromixer in the optimized solution set, so that we can choose the micromixer with higher comprehensive performance.
Mixing efficiency and pressure drop of OPTs and RD
The micromixer optimized by the Pareto genetic algorithm forms an optimal solution set and the K-means clustering algorithm is used to divide the Pareto optimal solution set into five categories. We select five representative design variables. Figure 6a shows the mixing efficiency of the five optimized micromixers at OPT-A and OPT-B obtained by numerical simulation, and the optimized micromixer is compared with the RD. A comparison of the mixing efficiencies of the above six micromixers at different Res shows that the optimized micromixer has a slightly lower mixing efficiency than the RD at the mixing inlet. As the mixing length increases, the mixing efficiency of the optimized micromixer is gradually greater than that of the RD and the mixing efficiency of the optimized micromixer is greater than that of the RD. On OPT-A, the mixing efficiency of OPT-1 closes to complete mixing. The mixing efficiency of OPT-2 is greater than 90%. The mixing efficiency of OPT-4 is slightly lower than that of OPT-3, and the mixing efficiency of these two micromixers are both greater than 80%. Because of the optimized internal structure of the channel, the molecular diffusion capacity is increased and a large amount of chaotic convection is generated. As a result, the comprehensive performance of the micromixer is significantly improved. On OPT-B, the mixing efficiency of OPT-1 and OPT-2 are both close to complete mixing. The mixing efficiency of OPT-3 and OPT-4 is slightly lower than the first two but still greater than 90%. The mixing efficiency of OPT-5 and RD are slightly lower than 90%. Therefore, it can be concluded that when Re increases further, the flow rate of the fluid will increase and strong convection will be formed in the microchannel, so the mixing efficiency of the micromixer begins to increase.
Mixing efficiency and pressure drop of OPTs and RD, (a) the mixing efficiency of OPT and RD varies with the length of the channel at different Res, (b) the pressure drop of OPT and RD varies with the length of the channel at different Res.
The pressure drop is additionally one among the vital indicators for testing the performance of the micromixer. The pressure drop is that the pressure distinction between the inlet and outlet of the micromixer. The higher pressure drop in the micromixer represents the higher energy required at the inlet, resulting in a lower safety of the micromixer. Figure 6b shows the pressure drop of the optimized micromixer and RD. It can be seen that as the length of the micromixer increases, the pressure drop also increases. On OPT-A, the pressure drop of OPT-1 is the highest, which is much higher than the pressure drop of the other five micromixers. The pressure drops of the other five micromixers do not differ much from each other and are between 0 and 50 Pa. On OPT-B, the pressure drop of OPT-1 is the largest, which exceeds 1200 Pa. The pressure drop of the OPT-2 is much lower than that of the OPT-1, which is around 1000 Pa. The pressure drop of the other four micromixers are is not very different and is in the range of 400–700 Pa. When Re increases, the increased flow velocity of the two fluids in the micromixer creates a strong chaotic convection, therefore the pressure drop will increase.
Analysis of the concentration field of the optimized micromixer
In order to study the concentration distribution of the micromixer, we intercept the concentration profiles of four important cross-sections. They are the inlet of the micromixer (A-A section), the two important locations of the continuous mixing unit through the fractal structure (B-B and C-C sections), and the outlet of the micromixer (D-D section). We select two Pareto-optimal designs, OPT-1 and OPT-5, at both ends of the Pareto-optimal boundary and use them to compare with RD. By observing the Fig. 7, it can be seen that the contact area caused by the obstruction of the two fluids through the Cantor fractal baffle increases with the length of the channel. However, the distribution of the concentration contours at the outlet of the OPT-1, OPT-5 and RD on OPT-A is significantly different. Compared with OPT-1 on OPT-A, OPT-5 and RD have more layered interfaces in the concentration profile of the two fluids at section B-B, and better mixing has not been achieved. In the C-C section, the concentration of the fluid on OPT-1 has a single color and is almost completely mixed, while the concentration of both OPT-5 and RD gradually spreads and the concentration color is closer to the middle value. By the time the fluid reaches the outlet, OPT-1 has achieved perfect mixing, while OPT-5 and RD have a reduced number of density profile interfaces and fewer color types. Compared to OPT-A, the concentration distribution of the three micromixers on OPT-B is more inhomogeneous. They both exhibit vortex and spiral phenomena that improve the mixing velocity and mixing performance of both fluids. The increase of Re causes the fluid in the microchannel to produce chaotic convection, and causes the fluids of different concentrations to produce large-scale oscillations, which is conducive to the rapid mixing of the fluids. These factors greatly improve the mixing efficiency of the micromixer.
The concentration contours of OPT and RD at different Res.
Analysis of the velocity field of the optimized micromixer
In order to analyze the mixing mechanism, we chose two Pareto optimal designs, namely OPT-1 and OPT-5, at both ends of the Pareto optimal frontier and compared them with RD to compare the velocity fields. Figure 8 shows the variation of the velocity vectors in the microchannel for four different cross sections. At section A-A, the sample fluids of the three types of micromixers have just arrived at the inlet of the microchannel. In the y–z plane, the fluid velocity is faster on the left side of the microchannel. In sections B-B and C-C, after the sample fluid flows through the mixing units with Cantor fractal baffle, the velocity of the fluid gradually changes to the right side of the microchannel. At the section D-D, the velocity direction of the fluid has completely changed from the left side of the microchannel to the right side of the microchannel. The results show that the fractal baffle plays an important role in changing the direction of fluid flow and enhancing convection. Compared to OPT-A, the three micromixers on OPT-B have a more intense fluid flow and a greater variation in flow direction. The fluids produce a wide range of oscillations and have better mixing performance. Finally, it is found that whether it is on OPT-A or OPT-B, the degree of change of the fluid velocity direction in OPT-1 and OPT-2 is greater than that of RD, and both produce more chaotic convection than RD. Therefore, the mixing performance of the optimized micromixer can be better than that of RD.
The velocity vector diagram of OPT and RD at different Res.
The concentration surface of the micromixer is at different Res
In order to investigate more clearly the effect of Cantor fractal baffle parameters on the mixing performance of the micromixer, OPT-1, OPT-5 and RD are selected for comparison. Figure 9 shows the concentration surface of the three types of micromixers at different Res. When the fluid just enters the microchannel, the two different fluids present a symmetrical distribution in the microchannel. Once the fluid enters the mixing zone through the Cantor baffle, the two fluids change considerably within the microchannel. At the same time, the concentration of the fluid starts to change in a wavy pattern and is distributed at the top and bottom of the mixing zone on both sides. This phenomenon helps to expand the contact area between the two fluids in the microchannel and increases the molecular diffusion capacity.
Mixing performance of OPT and RD at different Res, (a) OPT-1, (b) OPT-5, (c) RD.
On the OPT-A, the OPT-1 has the fastest mixing velocity. As the fluid flows through the four fractal baffles, the fluid is almost completely mixed. When the fluid reaches the outlet, the two fluids are perfectly mixed. The mixing performance of OPT-5 is better than that of OPT-1. Due to the parametric design of the fractal baffle, the OPT-5 achieves basic mixing at the outlet of the channel. Finally, the mixing performance of the RD is far inferior to that of the optimized micromixer. On OPT-B, OPT-1 achieves perfect mixing at the outlet of the channel, and both OPT-1 and OPT-5 can outperform RD in mixing. When Re increases, it helps to drive more chaotic convection between the two fluids, which in turn increases the mixing efficiency. The stronger rotation of the velocity streamline near the baffle makes the mixing performance better. Through the above study, it is found that the mixing performance of the micromixer optimized by Pareto genetic algorithm is better than that of RD.
The concentration cut line of the micromixer is at different Res
We study the concentration curve of the micromixer along the length of the channel, and discuss the influence of the Cantor fractal baffle on the concentration distribution of the micromixer. We chose OPT-1, OPT-5 and RD for research. On the y–z coordinate system, we collect the concentration data in the microchannel. The coordinate origin (0, 0) of the inlet of the left channel is set as the center. We take three cross-sections in the microchannel. Their coordinates on the Y-axis are 0.1 mm, 0.0 mm, and − 0.1 mm, respectively, and their coordinates on the Z-axis are all 0.1 mm. The Fig. 10 shows the concentration convergence curve at different positions at different Res. It can be seen that the X-axis concentration fluctuates greatly due to the influence of the Cantor fractal baffle.
Concentration curves of OPT and RD along the length of the microchannel at different Res, (a) OPT-1, (b) OPT-5, (c) RD.
On OPT-A, the concentration of OPT-1 at the inlet remains constant and there is little mixing. As the length of the channel increases, the two fluids begin to mix and the concentration curve begins to fluctuate dramatically. When it reaches the outlet of the micromixer, it is essentially perfect and the concentration is close to 0.5 mol/m3 at this moment. On OPT-5, the two fluids begin to mix as the length of the channel increases. The concentration curve fluctuation is not as severe as OPT-1 and the frequency is also low. When they reach the outlet of the micromixer, the mixing of the two fluids is not completed. The concentration does not reach 0.5 mol/m3. However, as the length of the channel increases, the concentration curve of the two fluids gradually approaches 0.5 mol/m3, but the concentration is relatively far from complete mixing when they reach the outlet.
On OPT-B, as the length of the channel increases, the fluid in OPT-1 gradually mixes and the concentration curve approaches 0.5 mol/m3 at the outlet. However, the fluctuations in the concentration curve are not as severe as at low Re. The fluid in OPT-5 oscillates in the channel under the influence of the Cantor fractal baffle. Because of the short residence time of the two fluids in the microchannel and the incomplete mixing of the fluids, the three concentration curves do not converge completely to 0.5 mol/m3 at the outlet of the micromixer. From the above analysis, it can be seen that the chaotic convection caused by the fractal baffle to the fluid flow has far-reaching significance for improving the mixing efficiency.
Application of multi-objective genetic algorithm in different Res
In order to demonstrate the applicability of the Pareto genetic algorithm for multi-objective optimization at different Res, we select RD to calculate its mixing efficiency and pressure drop at Re = 1, 10, 25, 50, and 100. Figure 11 shows the variation of mixing efficiency and pressure drop of RD with increasing Re. Through research, it is found that at different Res, the pressure drop at the outlet of RD increases as the mixing efficiency increases. Their relationship is proportional. We are looking for a micromixer with high mixing efficiency and low pressure drop, so this means that the two objective functions of mixing efficiency and pressure drop are in a competitive relationship at any Res. As long as the objective functions are in competition, they can be optimized using the Pareto genetic algorithm, so the multi-objective optimization approach is applicable to different Res.
Mixing efficiency and pressure drop of RD at different Res.
