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A novel passive micromixer with trapezoidal blades for high mixing efficiency at low Reynolds number flow

A novel passive micromixer with trapezoidal blades for high mixing efficiency at low Reynolds number flow
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     Abstract   —In this paper, we propose a novel passive micromixer structure based on the effect of stretching-folding in both vertical and horizontal directions. The channel depth of the micromixer is tightened at two ends each mixing unit. The fluid flows are repeatedly twisted and bent from left to right and vice versa. With this special structure, our proposed micromixer can create vortices and transversal flow even at low Reynolds number. Therefore, it can efficiently mix low speed liquid flows, making it easy to be built into micro-devices. We conduct intensive simulation to evaluate the performance of our proposed mixer by using COMSOL Multiphysics package with Navier-Stokes, convection-diffusion equation and particle tracking method. The simulation results demonstrated that our micromixer may be the first device that is able to operate with high mixing efficiency independent of the Reynolds number in the range of 0.5 to 60. Especially, at very low Reynolds number the mixing efficiency of our proposed micromixer is 220-240% higher compared with those of rhombic mixer with branch channels and pure rhombic mixer.  Keywords—passive micromixer; Reynolds number; trapezoidal blade; mixing efficiency I.   I  NTRODUCTION  The main purpose of the micromixer is to mix small volumes of separate complex reactants or liquids into a homogenous mixture within automatic integrated and microfabricated devices. Micromixer has become an interesting research topic in recent years because of its important applications in chemistry and biology [1]. For predicting the flow characteristics, a dimensionless  parameter called Reynolds number is defined. Reynolds number is the ratio of inertial forces to viscous forces, Re = (  ρuD  H  )/  μ , where  ρ  is the density of fluid, u  is the average velocity,  D  H   is the hydraulic diameter of channel and  μ  is the fluid viscosity. At the microscale, the Reynolds number is small in value and viscous effect is dominant. Consequently, the fluid stream in microchannel is laminar flow without turbulence which is the great challenge for mixing of the fluids. The mixing effect mostly relies on molecular diffusion along the channel which is inefficient compared to convective mixing effect. Therefore, alternative mixing methods such as stirrers and special geometry designs are required to create turbulent flows by enlarging the contact area of different All authors are with Vestfold University College, Postboks 2243, N-3103 Tønsberg, Norway. *Contacting author: Hai Le The is with Vestfold University College, Postboks 2243, N-3103 Tønsberg, Norway (Tel.: ++47(0)41-18-68-21, Email: mixing liquids to increase the mixing efficiency as well as to reduce the mixing time. Numerous micromixers have been  proposed to enhance the mixing performance. They can be categorized as passive micromixers and active micromixers. Active micromixers operate with external energy sources such as electrokinetic [2], gas pressure driving force [3] and ultrasonic vibration [4], making it complicated in fabrication  process and in integration with other microfluidic systems. In contrast, passive micromixers do not use external sources. Thus, they have the advantages such as stable operation, easy integration, low cost in manufacture. Passive mixers can be classified by the arrangement of the mixed phases: lamination mixing, serial lamination, injection, chaotic advection and droplet [5]. In [6], a planar passive micromixer which includes four-rhombus mixer with turning angle 60° and a converging-diverging element at the outlet is proposed to enhance the mixing efficiency. The mixing of fluid flows exploits the splitting and joining of laminar flows. However,  pure rhombic micromixers have low mixing index of 0.5 when Reynolds number (Re) < 60 and dramatically decrease at lower Reynolds number flow. An advanced rhombic micromixer with branch channels is presented in [7] to increase the mixing performance while reducing the pressure drop. Due to the planar vortices created in the intersection of  branch channels, the mixing quality is improved. However, this micromixer only works effectively at large enough Reynolds number to create the vortices. In addition, these two micromixers also have the disadvantage of large footprint. Motivated by the above issues, in this paper, we propose a novel micromixer which has trapezoidal blade injecting on two channel sides of the main channel to create chaotic advection. We believe that this special structure significantly improves mixing efficiency. The footprint of the micromixer is reduced to tighten the channel depth, thereby create converging-diverging element on vertical direction. We evaluate the performance of our proposed micromixer using COMSOL software. The rest of this paper is organized as follows. In Section 2, we present the geometrical structure of our proposed micromixer and its advantages in improving mixing efficiency. In Section 3, mathematical models of fluid flows,  particle tracing, and mixing efficiency are studied. In Section 4, intensive simulations are conducted and discussed. Finally, Section 5 concludes the papers. A Novel Passive Micromixer with Trapezoidal Blades for High Mixing Efficiency at Low Reynolds Number Flow Hai Le The*, Hoa Le-Thanh, Nhut Tran-Minh, and Frank Karlsen 2014 Middle East Conference on Biomedical Engineering (MECBME)February 17-20, 2014, Hilton Hotel, Doha, Qatar 978-1-4799-4799-7/14/$31.00 ©2014 IEEE25   II.   G EOMETRICAL S TRUCTURE OF M ICROMIXER   The 3-dimensional structure schematic of our proposed micromixer is presented as in Fig. 1. As can be seen in Fig. 1, our micromixer consists of three inlets, one outlet and four mixing units. The purpose of three inlets is to create the hydrodynamic focusing at their intersection. The sample goes into the middle inlet while the solvent is brought in from the side inlets. Two side inlets act as the sheath flow to reduce the mixing path, enhancing mixing efficiency by diffusion effect. In the mixing channel, the fluid flow becomes narrow  because of the first trapezoidal blade on the right side. Due to the slope angle, the top fluid flow is forced to go down to mix with bottom fluid flow. The fluid flow continues until reaching the second trapezoidal blade on the left side then turns right. Due to the converging-diverging element at the turning position, the velocity in the direction which is  perpendicular to the main channel may increase rapidly. Hence, the mixing efficiency may be greatly enhanced due to the created vortices and transversal flows. The process is repeated until two fluids are completely mixed together and come out at the outlet. The detailed dimensions of the micromixer model are: l in  = 400  μm , w in  = w = 150  μm , l 1  = 330  μm , l 2  = 465  μm , l 3  = 135  μm , l 4  = 130  μm , l 5  = 265  μm , l 6  = 335  μm , w 1  = w 2  = 75  μm , α  = 75°. In our simulation, those dimensional values are fixed while the channel depth d is changed to evaluate influence of the depth over the mixing quality. Figure 1. The geometrical structure of our proposed micromixer III.   M ATHEMATICAL M ODELS FOR E VALUATING THE P ERFORMANCE OF O UR P ROPOSED M ICROMIXER   Our proposed micromixer model is analyzed using COMSOL Multiphysics package which includes models of laminar flow, transport of diluted species for concentration  profile of mixing, and particle tracking for fluid flow. We will  present in detail those mathematical models and mixing efficiency evaluation as follows.  A. Laminar flow and transport of diluted species model The fluid flow is expressed as Navier-Stokes equation for an incompressible flow as:     .0, T  uuuuupt                (1) 0, u    (2) where  ρ  is the density of fluid, u  is the velocity,  μ is the viscosity and  p  is the pressure. In this paper, the 3-dimensional structure model is used to get the results with highest possible accuracy and reliability. The value of fluid density, and viscosity are 1000 kg/m 3 and 10 -3 kg/(m.s) , respectively. For simplicity, we assume that fluid flows are in steady state and there are no slips at micromixer’s walls. The model boundary conditions are same constant average velocities at the three inlets and fixed pressure equal to 0  Pa  at the outlet. The mixing profile of two species is obtained by solving the following convection-diffusion equation using the achieved velocity field from (1). Then we have   0,  Dccu       (3) where  D  presents the diffusion coefficient, and c  denotes the concentration. The value of diffusion coefficient using in this model is fixed at 10 -9 m 2  /s . The fluids entering to the two side inlets are the same species. Their molar concentration values are set to 0. Meanwhile, the fluid entering the middle inlet has molar concentration set to 1. Three inlets are designed to have the same geometric dimension making the volume flow for each inlet equal. Therefore, the total molar concentration value of fluid at the outlet for complete mixing fluid is 0.33.  B. Particle tracing model Particle tracing model gives us illustrative Poincare maps to quickly evaluate the mixing performance of our proposed micromixer. It is a time-dependent particle tracing module in COMSOL package. A Poincare map is a cross-section which is perpendicular to particle trajectories at specific location. The particle trajectories are calculated by using the Newtonian equation as: ()(),  ppd  d mvmFuvdt     (4) where 2 18 d  pp  F d       is the drag force per unit mass, m  p  is mass  per particle  , u  is the achieved velocity field from (1), v is  particle velocity,  ρ  p   is the particle density, d   p   is the particle diameter. We generate 5000 particles per release at each inlet with assumption that the particle density is proportional to the  particle velocity at all inlets. The meshes of blade edges need to be fine enough to ensure that the motions of particles in the structure of micromixer are accurate. Timing step should be small enough to achieve smoothness of particle trajectories. C. Mixing efficiency evaluation The standard deviation of mean concentration value is computed using (5) for outlet surface to analyze the mixing efficiency. If the value of standard deviation is small, there is small difference between the nearby regions that means the mixture is more homogenous and vice versa. The mixing index is expressed as (6). The value of mixing index is 1 for  perfect mixing and is 0 for no mixing.   2 , outlet outlet  outlet  Aoutlet  A ccdAdA        (5) 26    1,  M         (6) where c is the mean concentration value of completely mixing,  A outlet   is the area of outlet cross-section,  M  is the mixing index, respectively. IV.    N UMERICAL R  ESULTS AND D ISCUSSIONS Fig. 2 shows the velocity profile of the micromixer model at Re = 40. As we can see in Fig. 2a, the velocity is much higher when passing the converging-diverging element compared with other region. According to the 3-dimensional structure of our micromixer, fluid flow is tightened and turned from left to right and vice versa, creating transversal flows. The simulation results prove that the proposed mixer can create transversal flows even at low Reynolds number. Fig. 2b  presents the counter-rotating of transversal flow at six different cross-sections which is perpendicular to the main channel. These transversal flows will dramatically improve the mixing efficiency. (a) (b) Figure 2. (a) Velocity field of the proposed micromixer, (b) Transversal components of the flow at different cross-sections Fig. 3 shows the mixing indices as functions of average velocity in main channel when the channel depth increases from 250  μm  to 300  μm  with 50  μm  per each step. The mixing  performance of the micromixer after four mixing units is highest with d = 200  μm . Generally, mixing indices in all three testing cases maintain high value over 0.8 for the whole range of velocity from 0.001 m/s to 0.6 m/s. However, the maximum value of mixing index when d = 200  μm  is higher compared with that when d = 250  μm  or d = 300  μm . Because of the slope angle of the trapezoidal blades, the top fluid is forced to go down to mix with bottom fluid. If the thickness increases, top fluid need to move a longer distance to reach the bottom fluid causing decrease in the mixing efficiency. Figure 3. Mixing indices as functions of total inlet velocity with three different values of channel depth d = 200  μm , d = 250  μm , and d = 300  μm   Fig. 4 shows the impact of the number of mixing units in the geometrical structure of our proposed micromixer on mixing index, with optimal thickness d = 200  μ m  at Re = 40. As we can see in Fig. 4, increasing the number of mixing units results in improving mixing quality. Figure 4. Impact of the number of mixing units on mixing index at Re = 40 Fig. 5 shows mixing indices as functions of Reynolds number for three different passive micromixers: our proposed micromixer, rhombic mixer with branch channels, and pure rhombic mixer, respectively. The mixing index of our  proposed mixer is stable, maintains high values above 0.8 over full Reynolds range. Whereas the mixing index of advanced rhombic mixer only reaches 0.8 at Re = 60. When Re < 60, the mixing index of advanced rhombic mixer remarkably decreases. Pure rhombic mixing always has low mixing index which is lower than 0.5. In summary, at very low Reynolds number from 1 to 13, the mixing index of our 27   micromixer is 220-240% higher compared with those of rhombic mixer with branch channels and pure rhombic mixer. Figure 5. The comparison of mixing indices of three micromixers: our  proposed mixer, advanced rhombic mixer with branch channels, and pure rhombic mixer as functions of Reynolds number Fig. 6 shows the concentration profile of whole geometrical structure of our proposed micromixer and cross-sections at different positions along the channel. The cross-sections illustrate the stretching-folding effect of the two different fluids that will be mixed together. This effect improves the mixing efficiency of our micromixer. As we can see in cross-section #7 of Fig. 6b, two fluids are totally mixed together, forming a mixture with the average molar concentration of 0.39 which is close to the perfect concentration of 0.33. (a) (b) Figure 6. (a) Concentration profile image of model micromixer in COMSOL, (b) Cross-section concentration at difference distances along to main channel Fig. 7a shows the particle trajectories along mixing channel. When passing the converging-diverging element, the  particle trajectories of two fluids are twisted and repeatedly  bent from left to right and vice versa, leading to the enhanced mixing efficiency. Fig. 7b shows the Poincare map of different cross-sections as in Fig. 6. The initial particle position of two fluids is set as in cross-section #1 in which red dots are from middle fluid and  blue dots are from side fluid. The Poincare map #7 shows the uniform mixing between red dots and blue dots which means our proposed micromixer has high mixing efficiency. There are some of particles got stuck to the channel walls because each particle has its own mass. Thus, the number of dots in Poincare map #7 is smaller compared to that in the initial Poincare map #1. (a) (b) Figure 7. (a) Particle trajectories along the mixing channel, (b) Poincare map of different Poincare section at Re = 40 V.   C ONCLUSION  In this paper, we propose a novel micromixer with trapezoidal blades. With special structure, our proposed micromixer can create vortices and transversal flows which  provide high mixing efficiency event with low speed liquids. We conduct intensive simulation by using COMSOL Multiphysics package to evaluate the performance of our mixer. The simulation results with different settings show that our proposed micromixer may achieve stable and higher 80% mixing efficiency for a low Reynolds number from 0.5 to 60. Especially, the mixing efficiency of our micromixer is 220 - 240% higher than other micromixer at very low Reynolds number. Moreover, due to small footprint our micromixer can  be integrated into complex microfluidic systems. R  EFERENCES   [1]   Chin-Tsan Wang, Yuh-Chung Hu and Tzu-Yang Hu, "Biophysical micromixer," Sensors Journal, vol. 9, pp. 5379-5389, July 2009. [2]   Samuel Yu, Tae-Joon Jeon, and Sun Min Kim, "Active micromixer using electrokinetic effects in the micro/nanochannel junction," Chemical Engineering Journal  , vol. 197, pp. 289-294, May 2012. [3]   Lung-Ming Fu, Wei-Jhong Ju, Chien-Hsiung Tsai, Hui-Hsiung Hou, Ruey-Jen Yang, and Yao-Nan Wang, "Chaotic vortex micromixer utilizing gas presure driving force," Chemical Engineering Journal  , vol. 241, pp. 1-7, January 2013. [4]   Zhen Yang, Sohei Matsumoto, Hiroshi Goto, Mikio Matsumoto, and Ryutaro Maeda, "Ultrasonic micromixer for microfluidic systems," Sensors and Actuators A: Physical  , vol. 93, pp. 266-272, April 2001. [5]    Nam-Trung Nguyen and Zhigang Wu, "Micromixer - a review,"  Journal of Micromechanics and Microengineering  , vol. 15, no. 2, pp. 1-16, February 2005. [6]   C.K. Chung, T.R. Shih, T.T. Chen, and B.H. Wu, "Mixing behavior of the rhombic micromixers over a wide Reynolds number range using Taguchi method and 3D numerical simulations,"  Biomedical  Microdevices , vol. 10, no. 5, pp. 739-748, October 2008. [7]   C.K. Chang, T.R Shih, and C.K Chung, "Design and fabrication of an advanced rhombic micromixer with branch channels," in  Proc. of IEEE Conference Nano/Micro Engineered and Molecular Systems , pp. 245-248, February 2011. 28
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