Abstract
In this paper, we mainly study the effect of the Koch fractal microchannel on the mixing efficiency. Using the Koch fractal principle can effectively change the geometric shape of the microchannel, and increase the convective contact area of the microfluid, enhance chaotic convection. Changing the geometric shape of the microchannel is an effective way to improve the mixing efficiency of the micromixer. So, it is of great significance to study the influence of fractal principle on the mixing performance of microchannel. This paper introduces the design process of the fractal microchannel. The effects of different microchannel heights and different Reynolds (Res) on the mixing efficiency are studied, we also compared the mixing efficiency of the Primary fractal and secondary fractal with different Res. When the microchannel height is 0.5 mm, the mixing efficiency exceeds 90%. In the main section of the Koch fractal channel, the vortex region produced by the fractal microchannel is an important factor to improve the mixing efficiency of the micromixer. As its excellent mixing performance, the micromixers based on the fractal principle will have great potential in chemical engineering and bioengineering.
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1 Introduction
In recent years, with the development of microprocessing technology, lab-on-a-chip techniques are increasingly used in bioengineering, chemical engineering and medical testing (Nestorova et al. 2017; Rahman et al. 2015; Shah et al. 2014; Rasponi et al. 2015; Hellé et al. 2015). As a microdevice, lab-on-a-chip compared with traditional experiments has its unique advantages (Jiang et al. 2011; Chen et al. 2013), it allows miniaturization and integration of microdevices into a system for a wide range of applications (Puttaraksa et al. 2016). The cost of design and manufacturing is very low (Humayun et al. 2017), and only little sample is needed to complete the experiment. It is precisely because of these advantages, lab-on-a-chip attracts more and more attention, and received a lot of research results.
As an important part of microfluidic chips, the mixing efficiency of the micromixer plays a decisive role in the ease of application of the microfluidic chip, the accuracy of the analysis and the efficiency of the analysis. Mixing efficiency is a basic phenomenon involved in instruments such as DNA molecular detection devices. Well-designed micromixers are critical to miniaturization of microfluidic chips (Fan et al. 2017). As mentioned by Suh et al., “in microfluidic applications, mixing has been understood as one of the most fundamental and difficult-to-achieve issues” (Suh and Kang 2010). Micromixers are usually divided into two categories, active and passive. Active micromixers can effectively mix different concentrations of samples with external forces such as electric fields, magnetic fields, mechanical agitation, etc. (Beskok 2008; Wu and Yang 2006; Cao et al. 2015; Fu et al. 2010; Chen et al. 2014). It can improve the mixing efficiency of the mixer, and shorten the reaction time. But, with the addition of power source, the structure of the active mixer becomes more and more complex (Zhou et al. 2015). On the contrary, due to the simple manufacture, ease of control, and low cost of the passive mixer is concern (Tran-Minh et al. 2014). In comparison to the active mixer a passive mixer can achieve a better mixing effect by just changing the structure of the microchannel. Hossain and Kim (2016) studied the parametric investigation on flow structure and mixing in a micromixer with two-layer crossing channels which was reported by Xia et al. (2005). A novel micromixer with gaps and baffles is proposed by Xia et al. (2016), the enhanced mass transfer results from the better synergy between the velocity vector and concentration gradient caused by the combination of gaps and baffles. Cortes-Quiroz et al. (2009) applied the optimization of the geometry of the staggered herringbone micromixer (SHM) to maximize mixing index and minimize pressure drop. A planar rhombic micromixer with a converging–diverging element has been systematically investigated by Chung et al. (2008), the planar rhombic micromixer improves the performance at a smaller footprint and lower Reynolds number. Shih et al. studied a high-efficiency planar micromixer with convection and diffusion mixing over a wide Reynolds number range (Shih and Chung 2008), it achieves nearly uniform mixing in this adaptive-design mixer at Reynold (Re) = 0.1 and Re ≥ 20. The combination of patterning the channel bed and splitting and recombining the streams can be used to make controlled lamination in a 2D channel system (Tofteberg et al. 2010). A numerical study of an electrothermal vortex enhanced micromixer has been done by Cao et al. (2008) using vortices induced by the electrothermal effect to enhance the mixing efficiency. By changing the structure of the microchannel, it is possible to increase the contact area and enhance the chaotic convection of the fluid (Du et al. 2010). Numerical simulations have great potential in the design optimization of micromixers. A series of studies on design (Chen et al. 2012; Chen and Shen 2015), simulation (Chen et al. 2016a; Chen and Shen 2017), optimization (Chen and Li 2016, 2017) and fabrication (Chen et al. 2016b, c, 2017) of micromixer have been done in our work.
In this study, we mainly showed the effect of the Koch fractal structure of the microchannel on the mixing performance of micromixers. Compared the effect of two types of microchannels and different Res on the mixing efficiency. The effect on the mixing efficiency is also obvious when the height of the micromixer is different. The efficient mixing performance with different parameters makes it important to study the fractal of microchannels.
2 Theoretical background
2.1 Koch fractal principle
Figure 1 shows the design principle of the Koch fractal microchannel, the length of the primary fractal a = 1.8 mm, the angle of the two sides E, F are 60° and 120°. The angle between the four polylines adjacent to “a” is 60°. The straight lines of the upper and lower borders of the channels are parallel to each other. The side length of secondary fractal b = a/3 = 400 μm is obtained based on the primary fractal. a and b are a unit of the primary and secondary fractal microchannels, respectively. The geometric structure of the Koch fractal micromixer with two inlets and one outlet is designed by the CAD software. Figure 2 shows the whole structure of the Koch fractal microchannel based on the micromixer. The total length of the two inlets is 4.3 mm. The main channel consists of two Koch fractal units of length 5.4 mm. The outlet length and width are 4.2 mm and 0.3 mm respectively. We define the height of the microchannel as H. It is set to 0.1, 0.3 and 0.5 mm for this study.
2.2 Governing equations
The following Navier–Stokes (N–S) equations are used for the fluid dynamic analysis (Wong et al. 2004):
where u is the velocity vector, \( \uprho \) is the fluid density, \( \upeta \) is the dynamic viscosity and p is the pressure. Equation (1) is the equation of momentum balance. Equation (2) is the equation of continuity for incompressible fluids.
The following convective diffusion equation is used for concentration analysis:
where c is the concentration, D is diffusion coefficient and u is the velocity vector.
The Re of square microchannel is calculated as follows (Capretto et al. 2011):
where Re is the Reynolds number, \( D_{d} \) is the equivalent diameter, A is the height, B is the width, V is the velocity and \( \nu \) is the kinematic viscosity. The velocity of the fluid for each simulation has been shown in Table 1. These velocities are calculated from Eqs. 4 and 5.
Mixing efficiency of the sample can be calculated by the formula as follow.
where M is the mixing efficiency, N is the total number of sampling points, \( c_{i} \) and \( \bar{c} \) are normalized concentration and expected normalized concentration, respectively. Mixing efficiency ranges from 0 (0%, not mixing) to 1 (100%, full mixed).
3 Results and discussion
3.1 Mixing efficiency at different heights
In order to studied the effect of different heights on the mixing efficiency, we have defined the height of the microchannels as 0.1, 0.3, 0.5 mm, and study the mixing efficiency of secondary classification with different Re 0.1, 0.5, 1, 5, 10, 50 and 100.
In Fig. 3, we show the positions of section A and section B of the primary and the secondary fractal micromixers and compare the secondary fractal sections. Figure 4 shows the simulation results of different heights and different Res. With Re increasing, laminar flow is gradually broken in each micromixer. It is a meaningful principle that a high Re can encourage the vortical flow, which produces secondary flow to enhance mixing. But, at high Re 100, when the height of the micromixer is increased, the change in the laminar flow state is noticeable. It can be seen clearly from Fig. 6, at different Res, the curves of concentration gradients are nonlinear. As the microchannel height decreases, the mixing efficiency at the same Re is gradually reduced. When Re is 0.1 and 100, the secondary fractal can almost achieve a complete mixing state. When Re is 1 and the channel height is 0.1 mm the mixing effect is not good enough (Fig. 5).
3.2 Comparison of the two kinds of Koch fractal micromixers on the mixing efficiency
Based on the above study, we chose the micromixers with a microchannel height of 0.5 mm for a comparative study on the mixing efficiency of two structures of fractal. As shown in Fig. 3, we compared the mixing effect of the two mixers in sections A and B. Two micromixers are simulated and the simulation results are shown in Fig. 7, the mixing index at the exit of the micromixers is shown in Fig. 8.
In Fig. 7, when Re is less than 10, the laminar flow states of the two fractal microchannels are similar in section A and B. The structure at the turn affects the intensity of vortex and the mixing performance at high Re 100, the laminar flow state disappears. As shown in Fig. 6 that the fractal structure can achieve better mixing efficiency at Re 0.1 and 100. As shown in Fig. 8 the mixing indices of both the fractal micromixers are almost beyond 90%. It is also shown that the secondary fractal has an advantage over the primary fractal micromixer in mixing efficiency at various Re.
3.3 Flow analysis of fractal section
In view of above analysis, at low Re and high Re should draw our attention. Figures 9 and 12 have shown the mixing performance of the micromixers at two Re, namely Re 0.1, and Re 100. That Figs. 9 and 10 present primary fractal micromixers at two Re, and Figs. 11 and 12 present secondary fractal, micromixers at two Re. The color of the streamlines in Figs. 9, 10, 11 and 12 denotes the concentration distribution and the color of arrows in the section A and B represents the velocity magnitude.
As shown in Figs. 9 and 11, the mixing is limited by molecular diffusion and the chaotic advection is ineffective at Re 0.1. So, the mixing at low Re is dominated by the residence time and depends on the total path of the flow. The secondary fractal channel significantly increases the total path of the micromixer compared to the primary fractal. This undoubtedly contributes to increase in the residence time of the sample within the microchannel, and improves the mixing efficiency of the secondary fractal micromixer. Comparing sections in Figs. 10 and 12, there are obvious chaotic convection phenomena in section A and B. However, the strong chaotic region in Fig. 12 is significantly weaker than Fig. 10. But, due to the impact of the fractal structure, the fluid continuously changes the flow in the secondary fractal microchannels, and a weakly chaotic region of the arrow appears in Fig. 12. These two important factors lead to the result that the mixing efficiency of secondary fractal microchannel is better than the primary fractal microchannel.
4 Conclusions
In this paper, we proposed a novel design for 3D micromixers based on Koch fractal principle. Lots of intensive numerical simulations were conducted to compare primary fractal with secondary fractal micromixer. By comparing the mixing performance of two micromixers, it can be concluded as follows:
-
1.
With increase in the height of the microchannel, the mixing efficiency of micromixers based on Koch fractal principle is getting better. Due to the impact of the fractal structure, the mixing efficiency of three different heights of secondary fractal mixers at each Re is high, beyond 80%. Especially at the height 0.5 mm, the mixing efficiency is beyond 90%.
-
2.
The mixing effect of secondary fractal micromixer is better than that primary fractal micromixer. Both the fractal structures can achieve more than 90% mixing efficiency at a height of 0.5 mm.
-
3.
Fractal structure not only extends the total path of the flow at low Re but also enhances the chaotic advection in fractal microchannel at high Re. It is significant for the mixing performance of passive micromixers. Fractal micromixer will have potential application value in field such as chemical industry and biochemistry due to its outstanding mixing performance.
References
Beskok A (2008) An electroosmotically stirred continuous micro mixer. In: ASME 2008 6th international conference on nanochannels, microchannels, and minichannels. American Society of Mechanical Engineers, pp 1655–1659
Cao J, Cheng P, Hong FJ (2008) A numerical study of an electrothermal vortex enhanced micromixer. Microfluid Nanofluid 5(1):13–21
Cao Q, Han X, Li L (2015) An active microfluidic mixer utilizing a hybrid gradient magnetic field. Int J Appl Electromagnet Mech 47(3):583–592
Capretto L, Cheng W, Hill M, Zhang X (2011) Micromixing within microfluidic devices. In: Microfluidics. Springer, Berlin, pp 27–68
Chen X, Li T (2016) A novel design for passive misscromixers based on topology optimization method. Biomed Microdevice 18(4):57
Chen X, Li T (2017) A novel passive micromixer designed by applying an optimization algorithm to the zigzag microchannel. Chem Eng J 313:1406–1414
Chen X, Shen J (2015) Simulation in system-level based on model order reduction for a square-wave micromixer. Int J Nonlinear Sci Numer Simul 16(7–8):307–314
Chen X, Shen J (2017) Design and simulation of a chaotic micromixer with diamond-like micropillar based on artificial neural network. Int J Chem React Eng 15(2):1617–1628
Chen X, Liu C, Xu Z, Liu J, Du L (2012) Macro-micro modeling design in system-level and experiment for a micromixer. Anal Methods 4(8):2334–2340
Chen X, Liu C, Xu Z, Pan Y, Liu J, Du L (2013) An effective PDMS microfluidic chip for chemiluminescence detection of cobalt(II) in water. Microsyst Technol 19(1):99–103
Chen CY, Lin CY, Hu YT (2014) Inducing 3D vortical flow patterns with 2D asymmetric actuation of artificial cilia for high-performance active micromixing. Exp Fluids 55(7):1765
Chen X, Li T, Zeng H, Hu Z, Fu B (2016a) Numerical and experimental investigation on micromixers with serpentine microchannels. Int J Heat Mass Transf 98:131–140
Chen X, Shen J, Zhou M (2016b) Rapid fabrication of a four-layer PMMA-based microfluidic chip using CO2-laser micromachining and thermal bonding. J Micromech Microeng 26(10):107001
Chen X, Li T, Shen J (2016c) CO2 laser ablation of microchannel on PMMA substrate for effective fabrication of microfluidic chips. Int Polym Proc 31(2):233–238
Chen X, Li T, Fu B (2017) Surface roughness study on microchannels of CO2 laser fabricating PMMA-based microfluidic chip. Surf Rev Lett 24(02):1750017
Chung CK, Shih TR, Chen TC, Wu BH (2008) Mixing behavior of the rhombic micromixers over a wide Reynolds number range using Taguchi method and 3D numerical simulations. Biomed Microdevice 10(5):739–748
Cortes-Quiroz CA, Zangeneh M, Goto A (2009) On multi-objective optimization of geometry of staggered herringbone micromixer. Microfluid Nanofluid 7(1):29–43
Du Y, Zhang Z, Yim CH, Lin M, Cao X (2010) A simplified design of the staggered herringbone micromixer for practical applications. Biomicrofluidics 4(2):024105
Fan LL, Zhu XL, Zhao H, Zhe J, Zhao L (2017) Rapid microfluidic mixer utilizing sharp corner structures. Microfluid Nanofluid 21(3):36
Fu LM, Tsai CH, Leong KP, Wen CY (2010) Rapid micromixer via ferrofluids. Phys Proc 9:270–273
Hellé G, Roberston S, Cavadias S, Mariet C, Cote G (2015) Toward numerical prototyping of labs-on-chip: modeling for liquid–liquid microfluidic devices for radionuclide extraction. Microfluid Nanofluid 19(5):1245–1257
Hossain S, Kim KY (2016) Parametric investigation on mixing in a micromixer with two-layer crossing channels. SpringerPlus 5(1):794
Humayun Q, Hashim U, Ruzaidi CM, Noriman NZ (2017) A strategy for design and fabrication of low cost microchannel for future reproductivity of bio/chemical lab-on-chip application. In: AIP conference Proceedings, vol 1808(1). AIP Publishing, p 020022
Jiang H, Weng X, Li D (2011) Microfluidic whole-blood immunoassays. Microfluid Nanofluid 10(5):941–964
Nestorova GG, Hasenstein K, Nguyen N, DeCoster MA, Crews ND (2017) Lab-on-a-chip mRNA purification and reverse transcription via a solid-phase gene extraction technique. Lab Chip 17(6):1128–1136
Puttaraksa N, Whitlow HJ, Napari M, Meriläinen L, Gilbert L (2016) Development of a microfluidic design for an automatic lab-on-chip operation. Microfluid Nanofluid 20(10):142
Rahman MM, Ali ME, Hamid SBA, Bhassu S, Mustafa S, Al Amin M, Razzak MA (2015) Lab-on-a-chip PCR-RFLP assay for the detection of canine DNA in burger formulations. Food Anal Methods 8(6):1598–1606
Rasponi M, Gazaneo A, Bonomi A, Ghiglietti A, Occhetta P, Fiore GB, Redaelli A (2015) Lab-on-chip for testing myelotoxic effect of drugs and chemicals. Microfluid Nanofluid 19(4):935–940
Shah P, Zhu X, Chen C, Hu Y, Li CZ (2014) Lab-on-chip device for single cell trapping and analysis. Biomed Microdevice 16(1):35–41
Shih TR, Chung CK (2008) A high-efficiency planar micromixer with convection and diffusion mixing over a wide Reynolds number range. Microfluid Nanofluid 5(2):175–183
Suh YK, Kang S (2010) A review on mixing in microfluidics. Micromachines 1(3):82–111
Tofteberg T, Skolimowski M, Andreassen E et al (2010) A novel passive micromixer: lamination in a planar channel system. Microfluid Nanofluid 8(2):209–215
Tran-Minh N, Frank K, Tao D (2014) A simple and low cost micromixer for laminar blood mixing: design, optimization, and analysis. Biomed Inf Technol 404:91–104
Wong SH, Ward MCL, Wharton CW (2004) Micro T-mixer as a rapid mixing micromixer. Sens Actuators B Chem 100(3):359–379
Wu CH, Yang RJ (2006) Improving the mixing performance of side channel type micromixers using an optimal voltage control model. Biomed Microdevice 8(2):119–131
Xia HM, Wan SYM, Shu C, Chew YT (2005) Chaotic micromixers using two-layer crossing channels to exhibit fast mixing at low Reynolds numbers. Lab Chip 5(7):748–755
Xia GD, Li YF, Wang J et al (2016) Numerical and experimental analyses of planar micromixer with gaps and baffles based on field synergy principle. Int Commun Heat Mass Transf 71:188–196
Zhou B, Xu W, Syed AA, Chau Y, Chen L, Chew B, Xiao X (2015) Design and fabrication of magnetically functionalized flexible micropillar arrays for rapid and controllable microfluidic mixing. Lab Chip 15(9):2125–2132
Acknowledgements
This work was supported by National Natural Science Foundation of China (51405214), The Key Project of Department of Education of Liaoning Province (JZL201715401).
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Chen, X., Zhang, S. 3D micromixers based on Koch fractal principle. Microsyst Technol 24, 2627–2636 (2018). https://doi.org/10.1007/s00542-017-3637-9
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DOI: https://doi.org/10.1007/s00542-017-3637-9