Analysis of Flow and Mixing Characteristics of Planar Asymmetric Split

Article pubs.acs.org/IECR

Analysis of Flow and Mixing Characteristics of Planar Asymmetric Split-and-Recombine (P-SAR) Micromixers with Fan-Shaped Cavities Guodong Xia,* Jian Li, Xinping Tian, and Mingzheng Zhou Key Laboratory of Enhanced Heat Transfer and Energy Conservation, Ministry of Education, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China ABSTRACT: A modified micromixer design based on the concept of multidirectional vortices and unbalanced splitting and recombining of fluid streams is described in this work. The purpose of this study was to demonstrate that the combination of unbalanced inertial collisions, multidirectional vortices, and converging/diverging flow caused by the fan-shaped cavities contributes to the improvement of the mixing effectiveness. By using computational fluid dynamics (CFD), numerous studies of the mixing performance and flow features were completed at moderately low Reynolds numbers ranging from 1 to 80. Experimental analysis of the mass transfer and mixing quality along this modified micromixer was performed to validate the numerical results. The computational and experimental results for the concentration distributions and flow patterns reveal the following trends: (i) The combination of transverse Dean vortices effect (vertical plane) and expansion vortices (horizontal plane) effectively contribute to the considerable improvement of the mixing performance. The experimental results were consistent with the numerical results in this respect. (ii) The geometric parameters and the arrangement of the fan-shaped cavity structure are two important factors affecting the mixing performance. When the fan-shaped cavity channel is 3 times as wide as the major subchannel, the mixing index of this type of micromixer was found to reach around 75% at Reynolds numbers larger than 60. Furthermore, we also analyzed the relation between mixing efficiency and pressure drop simultaneously.

1. INTRODUCTION Microfluidic systems, such as micropumps, microvalves, and micromixers, have received much attention and have played an important role in biological applications, chemosynthesis, and micrototal analysis systems as a result of rapid developments in fluid system miniaturization.1 The mixing of two or more streams is one of the key processes for chemical reactions to occur, so rapid and high-efficiency mixing is a critical issue for microfluidic applications.2,3 Because of the characteristic dimensions of these mixers, the flow in these devices is predominantly laminar rather than turbulent, which limits the volumetric rates of the flow.4 The mixing of two fluid layers is mainly governed by molecular diffusion, determined primarily by the laminar diffusion coefficient. This mixing process is very slow and inefficient compared to convective mixing. Hence, many types of micromixers have been developed to overcome the problem of mixing by reducing the mixing time, the dead volume of the mixing channel, and the pressure drop.5 Micromixers can be classified as active or passive micromixers depending on whether other external energy is provided or the pumping energy is used to achieve mixing.6,7 Active micromixers influence the flow process by accelerating mixing performance using movable elements or external forces or fields. These might include pulse flow mixers,8 thermal bubble mixers,9 incorporation of piezoelectric elements,10 micromachined magnetic stirrers,11 and acoustic mixers.12 They are complicated and expensive to fabricate and are more prone to failure during use than passive micromixers. Conversely, passive micromixers do not have movable elements and require the addition of any energy input apart from the pressure drop needed to drive the flow. Broadly, passive units mix fluids by using a special geometry that creates a specific flow field, and © 2012 American Chemical Society

the mixing behavior relies entirely on molecular diffusion and/ or chaotic advection. Because of the advantages of the simple construction of such micromixer devices, they are easy to fabricate and have been largely integrated into microfluidic systems for various applications. In particular, they are also well-suited for use in some applications involving sensitive solutions, because they do not involve mechanical shearing, high electric fields, or the generation of significant amounts of heat.13 Optimizing special channel geometries to generate chaotic advection or alternating lamellae is an optimal method that is taken for granted.14 Numerous channel configurations that split, stretch, fold, and break the flow have been reported to achieve chaotic advection.15−18 Meanwhile, combining two or more mixing mechanisms to enhance mixing efficiency has become a current approach. Split-and-recombine (SAR) mixing is a passive micromixing method that splits the streams to be mixed into multiple smaller streams and later rearranges them in alternating thin lamellae. Hence, not only are the diffusion distance and mixing time decreased, but the interfacial area is also increased.13 Depending on their configurations, SAR micromixers can be divided into two categories, threedimensional (3D) SAR mixers and planar SAR (P-SAR) mixers. In contrast to 3D SAR mixers, P-SAR mixers are constructed so that all of the axes of the channels are located on the same plane, and the design is therefore simpler and does not require the complex fabrication of three-dimensional Received: Revised: Accepted: Published: 7816

November 14, 2011 May 14, 2012 May 16, 2012 May 16, 2012 dx.doi.org/10.1021/ie2026234 | Ind. Eng. Chem. Res. 2012, 51, 7816−7827

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model of the micromixer, characterized by a planar asymmetric SAR structure with a fan-shaped cavity located in the major subchannel. More specifically, the configuration consists of a Ttype main channel and periodic P-SAR structures arranged at equal distances along the micromixer. Two fluids separately enter through two different inlets and meet at a T-joint where they undergo a certain extent of molecular diffusion. Then, the overall fluid stream flows into the main channel. Molecular diffusion plays a leading role in micromixing behavior at this time. The main channel is split into two subchannels that are unequal in width and that recombine after a certain fixed distance. The key structure, a fan-shaped cavity, is located in the major subchannel. The separated fluid streams flow through semicircular flow paths and then flow together again. Because of the unequal widths, the rate of mass flow in the major subchannel is greater than that in the minor subchannel. As the rate of mass flow increases, so does the inertia of the fluid in the major subchannel. This difference causes an unbalanced collision of the fluid streams. After each collision, the major subchannels change position across the x axis. The downstream fluids in the minor subchannel change the flow direction.21 After flowing through four equal units, the fluid streams exit the microchannel from the outlet and complete the mixing process. For comparison with existing P-SAR micromixers, we used the same series of geometric parameters as Ansari and Kim.22 The width of the main channel, w, was fixed as 0.3 mm, and the height of the micromixer model was fixed at 200 μm. To maintain a constant area of flow, the sum of the widths of the two subchannels was equal to the width of the main channel, that is, w1 + w2 = w. The ratio of the widths of the two subchannels was the same as that optimized in Ansari and Kim’s analysis, namely, w1/w2 = 2.0. The geometrical parameters of the fan-shaped cavity configuration have a significant influence on the mixing characteristics of the mixer. Therefore, three different values of the ratio of the width of the fan-shaped cavity and to that of the major subchannel were investigated, namely, w3/w1 = 2.0, 3.0, 4.0. The axial lengths of the channel (see Figure 1) were fixed at L0 = 1.2 mm, L1 = 0.5 mm, and L2 = 2.95 mm.

geometries. A large number of investigations have been carried out on P-SAR mixers.19−22 In recent studies, curved channels have become widely used in the design of micromixers. When fluids flow through a curved channel, secondary flows are generated in the cross section of the channel as a result from the imbalance between the centrifugal force and radial pressure fields.23 The secondary flows can generate strong curvatureinduced vortices and a complex perturbation at moderately Reynolds and Dean numbers, thereby efficiently enhancing the fluid mixing. Bessoth et al.20 studied a P-SAR micromixer containing 12 equal units both experimentally and numerically. They found that, after the separation of the fluid streams, the lamellar fluids merged into the microchannels and increased the surface areas of the fluid streams. Ansari et al.21 proposed and fabricated a micromixer based on the concept of unbalanced splits and cross-collisions of fluid streams. Different rates of mass flow in two subchannels were found to create an unbalanced collision of the two fluid streams. In the meantime, the curved subchannels generated secondary flows resulting in enhancement of the extent of mixing. In light of the above-mentioned considerations, a modified passive micromixer with a fan-shaped cavity for microchemical applications is proposed in this work. This new structural design was optimized on the basis of the P-SAR micromixer from Ansari and Kim’s work.22 To yield an improved mixing performance, the effects of geometrical parameters on the mixing performance of the micromixer were investigated in simulations at low Reynolds numbers (1−80) in the laminar flow regime. Different channel width ratios and different locations of the fan-shaped cavity structure, taken as the key variables, were analyzed in terms of mixing efficiency. We also investigated the mass transfer and mixing quality in this modified micromixer experimentally to validate our numerical results based on geometrical structures. In addition, the characteristics of the pressure drop were simultaneously analyzed within the same range of Reynolds numbers.

2. PHYSICAL MODEL OF THE MICROMIXER A modified micromixer design was developed based on the concept of multidirectional vortices and unbalanced splitting and recombining of fluid streams. Figure 1 shows a schematic

3. NUMERICAL ANALYSIS 3.1. Computational Procedures. The flow and mixing characteristics of this modified micromixer were analyzed using the commercial computational fluid dynamics (CFD) software FLUENT 6.3. The CFD code was used to solve threedimensional continuity, Navier−Stokes, and convection− diffusion equations by the finite-volume method. The Péclet number, which is defined as the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient, is a dimensionless number that is relevant to the study of transport phenomena in fluid flows. When simulating certain situations with high Péclet numbers, simple computational models can be used. The integration of discrete equations coupled in the pressure−velocity formulation was realized with the implicit algorithm SIMPLEC (semi-implicit method for pressure-linked equationsconsistent) with the QUICK (quadratic upwind interpolation of convective kinematics) scheme (for momentum and species) and the PRESTO (pressure staggering option) scheme (for pressure). The thermal effect of dissolution and the energy equation were ignored because of chemical inertness. No-slip boundary conditions should be considered when the system satisfies the Navier−Stokes

Figure 1. Schematic diagrams of the modified P-SAR micromixer with fan-shaped cavities in the major subchannel. 7817

dx.doi.org/10.1021/ie2026234 | Ind. Eng. Chem. Res. 2012, 51, 7816−7827


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equation, that is, when the Knudsen number is less than 10−3, where the Knudsen number (Kn) is the ratio of the mean free path (λ) of the molecules of a fluid to a characteristic length (L). For our study, no-slip boundary conditions were used for the solid walls because Kn = (5−400) × 10−6 ≪ 10−3. The flow was considered to be steady and laminar and to be unaffected by gravity. The fluids in the micromixer were assumed to be incompressible and Newtonian in the solution of the governing equations. According to these assumptions, the governing equations, based on the continuity equation, Navier−Stokes equation, and convection-diffusion equation, are represented as follows ⎯ =0 (1) ∇·⇀ V ⎯ ·∇⇀ ⎯ = −∇P + μ∇2⇀ ⎯ ρ⇀ V V V

⇀ ⎯ ·∇C = D∇2 C V

shown in Figure 2. Figure 3 shows the results of the gridindependence test with five different grids wherein the number

(2) (3)

Water and ethanol were selected as the working fluids for mixing, because a number of studies on mixing in the literature have used these fluids. The inlet temperatures were set to a constant value of 293 K in the simulations. The densities of water and ethanol are 9.98 × 102 and 7.89 × 102 kg·m−3, respectively. The diffusivities of both water and ethanol were set at 1.2 × 10−9 m2·s−1. The viscosities of water and ethanol are 1.004 × 10−3 and 1.2 × 10−3 kg·m−1·s−1, respectively. For the boundary conditions, the “velocity-inlet” profile was used for both inlets. To facilitate comparisons, the fluids at the inlets had the same velocity. The outlet boundary was set to zero static pressure. In fluid mechanics, the Reynolds number, which represents the ratio of the importance of inertial effects to the importance of viscous effects in the flow, is the most important dimensionless number. It can be defined as Re = ρud/μ, where u is a characteristic velocity scale, d is a characteristic length scale, ρ is the density of the fluid, and μ is its dynamic viscosity. The Reynolds number was calculated based on the properties of water and the width of the main channel. Six different Reynolds numbers from 1 to 80 were considered for the fluid streams at the inlet, namely, Re = 1, 10, 20, 40, 60, 80. As mentioned above, the diffusivities of water and ethanol were considered constant and equal. The main reason for this choice is that the diffusion coefficient is one of the most obvious influences on mixing effectiveness. Meanwhile, the Péclet number is the product of the Reynolds number and the Schmidt number (i.e., Pe = Re × Sc = ud/Dab, where Dab is the diffusion coefficient of a mixture) and is relevant in the study of transport phenomena in fluid flows. In micromixing applications, when the Schmidt number is fixed, as the Reynolds number increases, so does the Péclet number. The Péclet number is relatively large in the streamwise direction because of the low diffusivity. Under such conditions, the dependence of the flow on the downstream position is diminished, and variables in the flow tend to become one-way properties. The streamwise diffusion can be ignored with respect to advection. In contrast, this dimensionless parameter of fluid streams in the cross-sectional plane is relatively small. The diffusion in the cross-stream directions cannot be neglected. Complex configurations increase the difficulty of mesh generation. Results can be very accurate if the appropriate element and mesh are selected. Therefore, a grid-independence test is necessary to ensure that the numerical simulation approach is not dependent on the grid size. An example of the structured hexahedral grid system employed in this work is

Figure 2. Examples of the structured grid system (w3/w1 = 3, w = 200 μm).

Figure 3. Grid-independence test (w3/w1 = 2, Re = 40).

of nodes varied from 346 × 103 to 1996 × 103, corresponding to w3/w1 = 2.0 and Re = 40. Finally, from the results of the gridindependence test, a grid system with 1549 × 103 nodes was selected as the optimum grid system for further calculations. The simulations were considered to have reached convergence when the relative velocity was below a set tolerance, namely

∑ ∑ |vi(x , y , z) − vi,0(x , y , z)| ≤ 10−6

(4)

3.2. CFD Mixing Quality Definition. The values of concentration on any arbitrary cross-sectional channels of the micromixer, which were obtained from the simulation results, were used to evaluate the overall mixing performance. In previous works in the literature, mixing intensity has been used as the evaluation standard for mixing performance. Typically, the variance of the component concentration on the cross section of the mixing channel is used to represent the mixing intensity, that is, the variance index is equivalent to the mixing intensity index. The calculation is given by 7818


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1 n

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n

∑ (ci − cm)2 i=1

(5)

where n is the number of sampling points on the cross section; cm is the optimal mixing mass fraction, which is 0.5 in this work; and ci is the mass fraction of sampling point i. The characteristic values at the sampling points were obtained by interpolation from adjacent computational grids. σ is the standard deviation of concentration in the cross section at any specific plane perpendicular to the flow direction. To quantitatively characterize the mixing efficiency, the mixing index was defined as 1 minus the relative standard derivation of the concentration at an individual sampling point σ M=1− σmax (6) where σmax is the maximum standard deviation of concentration over the cross section of the channel. M denotes the value of the mixing index in the specific cross section and is in the range from 0 to 1. A greater mixing index indicates a higher mixing quality: A value of 0 means no mixing at all (for which σ = σmax), whereas a value of 1 indicates perfect mixing (for which σ = 0). Ansari and Kim analyzed the mixing characteristics of unbalanced split-and-recombine micromixers with circular or rhombic channels at different channel width ratios and Reynolds numbers both experimentally and numerically.22 They also used water and ethanol as working fluids and compared the mixing efficiencies of two types of micromixers under a uniform profile. Figure 4 shows that the numerical

Figure 5. Predicted profiles of the mass-fraction distribution on the y− z plane and contours of streamlines on the x−y plane (w3/w1 = 2.0, Re = 60).

value as an interface. The cross-sectional views of the channel structure show the distributions of the scalar values in accordance with the color legend.24

4. EXPERIMENTS Micromixers were fabricated from polydimethylsiloxane (PDMS) and bonded to a glass slide using the standard polydimethylsioxane soft-lithography technique.13 The top PDMS layer, including the mixing-channel structures, arranged the fluidic connections. Experimental results were obtained using the top-view technique. The concentration profiles of a labeled solution were measured along the micromixer. Digital images of the flow were obtained using a fluorescence microscope (Nikon Ecliplse 80i) interfaced with a digital charge-coupled device (CCD) camera (Nikon DS-Fi1). The digital camera was used to monitor the mixing process and capture images during experiments. To measure the concentration field, one inlet stream labeled with black ink (diluted to 0.025 g/mL in deionized water) was held at 20 °C. In the second inlet, only deionized (DI) water was introduced. Flow rates were controlled using microsyringe pumps (Harvard Apparatus, PHD2000). The devices were interfaced with the syringe pumps using silicone tubing. L-shaped bends attached to the silicone tubing were inserted into the fluidic connections of the PDMS chips to complete the flow network. Different diameters of the bends and the round holes make a hermetically sealed connection between them. Figure 6 shows a photograph of the fabricated PDMS mixing chip and an optical micrograph of the mixing channels obtained using the top-view technique.

Figure 4. Comparison of calculated mixing indexes between this article and ref 22.

results of our present work remain consistent with those of Ansari and Kim.22 The validity of the numerical analysis performed in this study was further confirmed. Figure 5 presents predicted profiles of the mass-fraction distribution on the y−z plane and contours of streamlines on the x−y plane for w3/w1 = 2.0 and Re = 60. Initially, the mixing channel is filled with red and blue, which indicates that there are two kinds of solutions with scalar values of 1 and 0, respectively. When these solutions come into contact with each other as an interface, the color changes to green, indicating a scalar value of 0.5; color gradation, as illustrated in the color legend, is due to diffusion of the solutions. Green stripes in the designed channel structure show the contour of a 0.5 scalar

5. RESULTS AND DISCUSSION 5.1. Flow Analysis. According to the simulation results, a fan-shaped cavity structure arranged on the major subchannel has a significant effect on the mixing performance of a P-SAR micromixer. The reason for the effect is that enhanced micromixing is achieved by utilizing a synergistic combination of unbalanced inertial collisions, Dean vortices, and expansion 7819



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Figure 6. (a) Photograph of the fabricated PDMS mixing chip. (b) Optical micrograph of mixing channels obtained using the top-view technique.

Figure 7. Captured images demonstrating the effect of flow rates on mixing results at w3/w1 = 3 for Reynolds numbers of black ink and DI water at the inlet of (a) 1 and (b) 80.

counter-rotating vortices driving the fluid from this high-speed core toward the outer wall.4 Analogously to the expansion vortices in the mixing chamber, the Dean vortices can also further accelerate the mixing process. Ultimately, the multivortex phenomenon appears in the chamber of the fan-shaped cavity configuration. The combination of the effects of the transverse Dean vortices (vertical plane) and the expansion vortices (horizontal plane) contributes to a considerable improvement in the mixing performance. The scale and intensity of these vortices directly affect the recirculation and magnitude of the transverse component to the flow velocity. Compared with the prior design of an unbalanced SAR micromixer, the higher mixing effectiveness in the P-SAR mixer with the fan-shaped cavity configuration is more conducive to implementation. Figure 5 shows predicted profiles of mass-fraction distributions on the y−z plane and contours of streamlines on the x−y plane (w3/w1 = 2.0, Re = 60). It can be clearly seen from Figure 5 that expansion vortices in the horizontal plane generate two counter-rotating vortices on both sides of the channel centerline where the axial velocity is a maximum. These vortices drive the fluid streams passing through the abruptcontraction channel path to sweep outward. In the vertical cross-sectional plane, the distribution of the mass-fraction concentration of two miscible liquids also indicates that the counter-rotating vortices, ascribed to the secondary flow, can significantly perturb the laminar flow profile and increase the contact surface area of the fluid streams.

vortices. The combined vortices induced by the complex configurations along the flow path act together to mix coflowing streams. This implies that the transition of flow patterns is more diversified, frequent, and destabilized. Figure 1 shows that abrupt converging and diverging effects induced by the geometry of the fan-shaped cavity appear in the major subchannel configurations. Expansion vortices are generated in the chamber of the fan shaped-cavity because of an abrupt increase in the cross-sectional area. An intensive perturbation, induced by the expansion vortices, enhances the surface area of the fluid streams to complete diffusion and mixing in the flow streams. The geometric characteristics of variable sections improve the mass flow rate in the chamber of the fan-shaped cavity and intensify the unbalanced inertial collisions when the fluids flow into the next unit, thus increasing further perturbations and providing enhanced mixing. On the other hand, when pressure-driven flow passes through a curved channel, a lateral instability is induced by the geometrical curvature. The magnitude of the differential centrifugal forces that act on the fluid at the center and near-wall regions of the curved-channel progressively increases. The centrifugal force rises to a maximum as the fluid passes through a symmetrical curved channel. Two parallel fluid streams entirely switch positions inside a curved duct when its flow through a complete 180° rotation. The secondary flow (called Dean vortices) is generated in the cross-sectional plane of the channel, which is perpendicular to the streamwise direction of the fluids, as a result of these differential forces. The Dean vortices are a pair of 7820



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Figure 8. Comparison between numerical and experimental results for the distribution of the mass fraction of mixing fluids at Re = 1 and 80 (w3/w1 = 3): (a) numerical simulation results on the x−y plane midway along the channel height, (b) experimental optical micrographs captured by a CCD camera, (c) comparison of the simulated mixing indexes with the present experimental data at the outlet at different Reynolds numbers.

scans indicating pixel intensity across the microchannel width at the outlet were recorded. The gray-scale values, which were chosen as the measurement parameters, were normalized by the average gray-scale value of the solution at the inlet. Using the normalized gray-scale values instead of ci, mixing percentages were calculated with eqs 5 and 6, where n is the total number of pixels and cm is the average of the normalized gray-scale values. Figure 8c illustrates a comparison of the predictions with the measured mixing indexes at the outlet for different Reynolds numbers from 1 to 80. The predictions agree reasonably well with the experimental data for the mixing index. Furthermore, the dimensionless parameter characterizing flow in curved-channels is the Dean number (De), defined as De = Re(d/R)0.5, where Re is the Reynolds number, d is the width of the curved channel, and R is the radius of curvature. This dimensionless parameter expresses the relative magnitudes of the centrifugal and inertial forces compared to the viscous forces. At any small Dean number, the secondary flow can generate Dean vortices and efficiently enhance the mixing. In addition, the Schmidt number is the dimensionless number defined as Sc = ν/D, which shows the ratio of the viscous effects to the molecular diffusion effects and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. In the present study, the Reynolds number was on the order of 1−80, and the properties of the working fluids were mostly of the same order of the magnitude. The simulations were performed with a Schmidt number of about 1000 (diffusivity D = 1.2 × 10−9 m2·s−1), which shows that viscous diffusion effects were dominant. Therefore, the extent of mixing was a complex and intriguing function of several factors such as the Reynolds number, channel curvature, and the molecular diffusion coefficient. If the Schmidt number is fixed, the main factors impacting the

We experimentally investigated the mass transfer and mixing quality in this modified micromixer, validating our numerical results. The mixing effect of the designed passive micromixer with fan-shaped cavity structures was demonstrated by deionized (DI) water and black ink flowing in a fabricated micromixer (shown in Figure 7). We chose two significant flow rates of inlet fluids, namely, Re = 1 and 80, to evaluate the flow and mixing performance of the modified micromixer. As shown in Figure 7a, when the Reynolds number of inlet was 1, the DI water and black ink basically formed a laminar flow through the entire mixing path. Only slight molecular diffusion appeared on the contact surface of the two laminar flows, even at the outlet of the mixing channel. In contrast, in Figure 7b, the performances of the flow and mixing changed significantly at higher Reynolds number. After passing through the mixer units, the fluid became uniform in color without obvious fluctuations at the outlet, indicating that the fluids were well-mixed. Qualitative and detailed comparisons between the numerical and experimental results for the distribution of the massfraction of mixing can be seen in Figure 8 for two different flow rates of inlet fluids (w3/w1 = 3). For the higher flow rate in Figure 8b, the experimental results remained remarkably consistent with the numerical simulations. The shape and position of the vortex centers, found in the numerical results, basically agree with those in the experiments. In this work, the images of concentration distributions on the horizontal cross section of the flow were obtained by scanning over the horizontal cross sections. Following image capture, the images are analyzed to extract the gray scale value at each pixel to further verify the simulated results. The public-domain Java image processing software ImageJ (developed by the U.S. National Institutes of Health, Washington, DC) was used.25 The image captures at the outlets were chosen as the computing region. From the captured images, gray-scale line 7821



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Figure 9. Distribution of the concentrations at three different cross sections (C1, C2, and C3 labeled in Figure 1) of the third fan-shaped cavity unit for a Schmidt number of 1000 and a radius of channel curvature of 0.55 mm at Re = (a) 20, (b) 40, (c) 60, and (d) 80.

to twist the concentration contours in Figure 9b−d. Likewise, the distribution of concentrations at these different streamwise distances became increasingly sophisticated along the streamwise direction. Compared with the severe perturbations in the fan-shaped cavity configuration, however, the flow streams in the minor subchannel were more stable. This was mainly because of the high width ratio of the two subchannels (w3/w2 = 6.0), which meant that almost only one kind of fluid passed through the minor subchannel for the first two mixing units. It was also found that the locations of the vortices varied directly with changes in the Dean number. Nevertheless, the flow in the cross-sectional channel was symmetrical over the channel midplane. A comparison of the mixing efficiencies of three types of micromixers under the same conditions is shown in Figure 4. By comparison with unbalanced and symmetrical P-SAR micromixers consisting of circular channels, the mixing effectiveness of the P-SAR micromixer with a fan-shaped cavity structure was significantly enhanced. Note that, at Re = 80, the mixing index of this micromixer increased by 42.3% and 311.1% compared to the other two types of mixers for a value of w3/w1 = 3.0. Moreover, the P-SAR micromixer with a fan-shaped cavity structure is superior to both of the other configurations over the entire range of Reynolds numbers. For Figure 10, the effect of the position of the fan-shaped cavity on the mixing efficiency of the device was assessed at a

intensity of the multivortices are the Reynolds number and the radius of curvature. Figure 9 shows the distributions of the concentration in three different cross sections (C1, C2, and C3 labeled in Figure 1) of the third fan-shaped cavity unit at several different flow Reynolds numbers for a Schmidt number of 1000 and a channel radius of curvature of 0.55 mm. As previously mentioned, the secondary flow consisted of a pair of counterrotating vortices positioned symmetrically above and below the horizontal midplane. The direction of the secondary flow was perpendicular to the flow direction of the fluids. The two vortex cores were symmetrical to the horizontal plane near the midpoint of the channel height, and symmetrical transverse flows developed in the channel. At moderately low Reynolds numbers in Figure 9a, insufficient transverse flows existed in the curved channels because the centrifugal effects were not strong enough to significantly perturb the laminar flow profile. The curvature-induced vortices dissipated very rapidly under the viscous action, and the mixing attributed to the secondary flow was not notable. As the Dean number increased, the crossstream velocity component, ascribed to the Dean vortices, acted to transport fluid from the inner wall of the channel radially toward the outer wall, and the fluid pattern obviously changed. For Reynolds number values of 40, 60, and 80, Dean vortices were visible in the cross-sectional channel of the fan-shaped cavity. It can be seen that the effect of the secondary flow was 7822



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Figure 10. Variations of the Reynolds numbers with the mixing index in three different arrangements of the fan-shaped cavity at different Reynolds numbers.

certain range of Reynolds numbers. In this figure, the outlet mixing index is plotted as a function of Reynolds number. It is apparent that the mixing indexes of three different geometric structures increased as the Reynolds number increased. When the location of the fan-shaped cavity structure was on one side of the x axis, the computation results indicated that the mixing index rose gradually at relatively low flow rates, as seen in Figure 10. Beyond Re = 60, the mixing index maintained a stable trend and reached a maximum value, around 53%. It also can be seen that the changing tendency of the function curve of the Reynolds number and mixing index remained consistent, not only in the P-SAR micromixer with the fan-shaped cavity structure but also in the unbalanced circular ones. The mixing index was a minimum at Re = 10. On the contrary, the same function curve maintained a stable increase when the fanshaped cavities were arranged at only one side of the main channel. No marked turning point appeared, unlike for the relevant curves in the other two types of structures. This is because this layout of fan-shaped cavities increases the fluid path and mixing time to enhance the mixing performance, but only up to a certain degree. On the other hand, when the location of the fan-shaped cavity was shifted about the x axis (as shown in Figure 1), the mixing index rose dramatically and quickly exceeded approximately 75% at Re = 80. For the cases of two different locations of the fan-shaped cavities, Figure 11 shows contour plots of the mass fraction of water on the x−y plane in the cross section passing through the half-height of the channels at three different Reynolds numbers. Although the same phenomenon of expansion vortices appears in both different structures, significant differences in the intensity of the unbalanced collision of fluid streams were observed in the two different configurations. The distribution of the mass fraction at the outlet region in the design where the fan-shaped cavities were located on only one side of the x axis did not appear to change markedly at different Reynolds numbers. This phenomenon is consistent with the change of the mixing index in Figure 10. High mixing performance appeared only in fan-shaped cavities, similarly to a relatively closed mixing chamber. As the Reynolds number increased, the mixing index at the outlet gradually rose until maintaining a certain degree of mixing index, not following the changing of Reynolds numbers. This is because the fluids from inlets 1 and 2 flow through this structure only along their own respective sides and do not cross each other. In the case where the

Figure 11. Mass-fraction distributions of water on the x−y plane for different positions of the fan-shaped cavity structure in a cross section passing through the half-height of channels at different Reynolds numbers: (a,c,e) fan-shaped cavity structure on one side of the x axis (i.e., one side location) for Re = (a) 20, (c) 60, and (e) 80; (b,d,f) fanshaped cavity structure shifted about the axis (i.e., different side location) for Re = (b) 20, (d) 60, and (f) 80.

positions of fan-shaped cavities were shifted, the mixer showed obviously greater mixing performance at all Reynolds numbers. However, in the case where the positions of the fan-shaped cavity structure were located on one side or different sides, the mixers still showed better mixing performance than an unbalanced split-and-recombine micromixer at all Reynolds numbers. In summary, it is concluded that the geometrical parameters and arrangement of fan-shaped cavity structure are two important factors affecting the mixing performance of this passive micromixer. 5.2. Mixing Analysis. The flow analysis within the fanshaped cavity structure showed that the interaction of the expansion vortices (generated in the horizontal plane) and the Dean vortices (generated in the vertical plane) has a significant influence on the mixing performance of the fluids in the channels. This micromixer enhances severe perturbations in two different planes that are mutually perpendicular, significantly promoting the mixing performance. The high mixing indexes obtained in a P-SAR micromixer with a fan-shaped cavity are a synergistic combined effect of Dean vortices in the bending channel, unbalanced inertial collisions in the crosschannel region, and expansion vortices in the abrupt increase and decrease of the cross-sectional area. Compared with other similar T-type structural micromixers,15,18,26 the design idea of 7823



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Figure 12. Concentration distributions in three different micromixer structures: (a) balanced circular ring channel, (b) unbalanced circular ring channel, (c) fan-shaped cavity subchannel.

this modified micromixer is innovative in terms of the mixing mechanism, which occurs in different planes. Figure 12 shows concentration contours at Re = 80 in three different structures, namely, a balanced circular ring channel, an unbalanced circular ring channel, and a fan-shaped cavity subchannel. One can see, first, that the fluids of different concentrations in the balanced circular ring microchannel flow side by side even at high Reynolds number, and no recirculation or chaotic flow is generated. The fluids pass through the semicircular microchannels, except for a certain amount of mixing occurring at the entrance of T-junction and the cross-channel region. The mixing is mainly dependent on the molecular diffusion under these conditions. In contrast, some different phenomena appear in the unbalanced circular ring channel and fan-shaped cavity subchannel. The advection of mixing coflowing streams, ascribed to the unbalanced configurations, promotes enhanced mixing. The distribution of the concentration becomes complex, especially in the fanshaped cavity subchannel. A strong motion of fluids exists in the fan-shaped cavity structure, provided perturbing and shifting the interface of the fluid streams. This feature does not appear in the first two structures. It clearly shows that, after a certain number of mixing units, the concentration of fluids in Figure 12c (i.e., fan-shaped cavity subchannel) acquires the optimal mixing efficiency, whereas the worst result appears in Figure 12a. The main mixing mechanism for passive micromixers is to increase the contact area, fluid path, and mixing time of different miscible liquids. Thus, the aim of optimizing the geometric parameters of the channel is to improve the intensity of combined vortices and achieve highly efficient and rapid mixing. Figure 13 displays a comparison between the mixing index and the Reynolds number at various values of the ratio

Figure 13. Comparison between mixing index and Reynolds number at various values of w3/w1.

w3/w1 along this micromixer. The mixing performance was analyzed for three different channel width ratios (w3/w1 = 2.0, 3.0, 4.0). The results showed that, when the Reynolds number was appropriately small, such as ranging from 1 to 10, the variation trends of the Reynolds number and mixing index were just the opposite: M decreased as Reynolds number increased. The extent of mixing was invariant with the channel width ratio because of the dominant position of molecular diffusion at low Reynolds numbers. When the Reynolds number was on the order of 40 or larger, ranging from 40 to 80, the different channel width ratios had a substantial influence on the mixing index. The micromixer acquired the optimum value of M for a value of w3/w1 = 3.0 in a higher range of Reynolds numbers, while the relation between Reynolds number and the optimum mixing index shows a regular variation. This corresponds to an alternating change trend: For Re = 40, the higher mixing 7824



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efficiency was achieved for a value of w3/w1 = 2.0; for Re = 60 or 80, the optimum mixing effect was obtained for w3/w1 = 3.0. Compared with the value for the unbalanced SAR micromixer designed and simulated by Ansari and Kim,22 the mixing index of this modified micromixer improved by 28.8%, 42.3%, and 36.5% under the same physical conditions at a high range of Reynolds numbers. Meanwhile, the pressure drop of the fluids in the channels increased with increasing Reynolds number. That sets higher requirements on the assembly of microfluidic chips and experimental conditions. 5.3. Pressure Drop Analysis. Pressure drop is a term used to describe the decrease in pressure from one point in a pipe or tube to another point downstream and is another index used to evaluate the micromixing process, as it influences the mixing performance of the micromixer. Pressure losses of fluid in the microchannel consist of linear pressure losses and local pressure losses.27 For a preliminary structural design, it is necessary to take the channel pressure drop into account. Additional complex configurations certainly increase the pressure drop, as well as difficulties in device fabrication. Micromixers with higher pressure drops require greater driving forces. This not only increase the power required to drive the device in microfluidic chips, but is also adverse to their integration. Therefore, analyses on the the pressure drop characteristics were carried out. The apparent flow friction factor was used in this work, and the mixing indexes as a function of the friction factor are plotted for different geometric parameters. The overall apparent friction factor, defined in eq 7, incorporates both the pressure drop caused by the higher wall shear stress due to the higher velocity gradient normal to the wall and that caused by the momentum flux variation due to the change of the velocity field from a uniform profile at the inlet to a specific profile somewhere in the duct. Almost all of the analyses available in the literature consider a uniform velocity condition at the inlet. To account for the developing region, the pressure drop equations are presented in terms of an apparent friction factor.28 The friction factor depends on the flow conditions, the flow-channel geometry, and the surface conditions f=

ρD h Δp 2LG

2

=

Figure 14. Comparison of Reynolds numbers with pressure drop for various values of w3/w1 at the outlet.

width ratios have almost no influence on the pressure drop in the channel. Moreover, the pressure drop in the fan-shaped cavity structure channels is less than that in unbalanced or balanced circular ring channels. The fluid in the fan-shaped cavity flows so slowly that a laminar stagnation zone forms. The stagnation zone makes the main flow slip over the fan-shaped cavities, which reduces the pressure drop. When the working fluid enters the expansion and contraction regions, jet and throttling effects occur that interrupt the hydraulic boundary layer and contribute to the chaotic mixing of fluid and increase the specific surface area of the channel, which enhances the mixing performance. The numerical and experimental analyses were performed for a certain range of Reynolds numbers below 80 in our study. When the Reynolds number is smaller, slipping over the reentrant cavities is dominant, and when the Reynolds number is larger, jet and throttling effects prevail, so as to form a negative pressure gradient in the fan-shaped cavity. This is the main reason why the new micromixer shows a low pressure drop in Figure 14. Because of the high specific surface areas, several issues need to be noticed when evaluating pressure drops in microfluidic flows. The inlet and outlet losses and fully developed frictional losses need to be quantified for microfluidic channels. The channel dimensions and the flow-channel geometry have a major effect on the friction factor calculations. As the friction factor changes, so does the mixing index. The relation between the mixing index and the overall apparent friction factor is plotted for different geometric parameters in Figure 15. For the conditions of three different structural micromixers, the value of M decreases first and then increases gradually as the value of f continues to increase. The overall apparent friction factor f incorporates both the pressure drop and the momentum flux variation. For a given friction factor, the mixing index acquired in the P-SAR micromixer with a fan-shaped cavity structure was higher than that in the other two traditional structures. Increasing the channel width ratios of the fan-shaped cavities, however, did not cause a prominent change of the mixing indexes. When w3/w1 varies in the range of 2.0−4.0, the micromixer can achieve a higher mixing index within a certain range of values of the friction factor. The greater the pressure drop, the higher the driving forces needed. For a value of Re = 80, these three types of micromixers, with high mixing efficiency and low pressure drop, meet the technical requirements of microfluidic chips. As the Reynolds number increased

2D h Δp ρLum 2

(7)

where Δp is the pressure drop between the inlet and the outlet of the channel; Dh is the hydraulic diameter for the chosen rectangular microchannel, expressed as Dh = 2wh/(w + h); h is the depth of the microchannel; L is the channel length; and G is the mass flux, which is given by G = ρum, where um is the mean longitudinal velocity. The increased mixing surface areas provided by the fanshaped cavity structures result in higher mixing performance, but also increase the frictional area to cause a higher pressure drop. Although the flow paths are generally short, the pressure gradients generated in microchannels are extremely high. Figure 14 shows the pressure drops, Δp, in channels as a function of Reynolds number for various values of the channel width ratio w3/w1 at the outlet. For the three different channel width ratios, the pressure drop Δp exhibits an approximately linear increase as the Reynolds number increases. This is in line with the theory that linear energy losses are directly proportional to the first power of velocity. Meanwhile, it can be seen that the trends in the pressure drops in the three different mixing channels exhibit no marked differences. In particular, for the P-SAR micromixer with a fan-shaped cavity structure, the different 7825



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ACKNOWLEDGMENTS

The work was financed by the National Natural Science Foundation of China (No. 51176002) and the National Basic Research Program of China (2011CB710704). The authors also acknowledge support for this project from the Research Fund for the Doctoral Program of Higher Education (20111103110009).

■ Figure 15. Relevant functions between f and M for various channel width ratios w3/w1.

further, it was almost impossible to attain such a large driving force and was also not suitable for microfluidic chip applications. Therefore, these three types of micromixers are optimally used in the low range of Reynolds numbers (i.e., 1− 80).

6. CONCLUSIONS Three-dimensional CFD simulations and experimental analysis of a modified designed passive micromixer with fan-shaped cavity structures, based on the principles of multidirectional vortices and unbalanced splitting and recombining of fluid streams, have been performed. Different parameters of the geometrical structure for a certain range of Reynolds numbers (1−80) were investigated, aimed at achieving optimal mixing performance. Furthermore, we also simultaneously analyzed the relation between mixing efficiency and pressure drop. Enhanced micromixing was achieved by utilizing a synergistic combination of unbalanced inertial collisions, Dean vortices, and expansion vortices. The combined vortices induced by the complex configurations along the flow path act together to mix coflowing streams. Multidirectional vortices appear in the chamber of the fan-shaped cavity configuration at high Reynolds numbers. Compared with the earlier design of the unbalanced SAR micromixer, the higher mixing effectiveness in the P-SAR mixer with the fan-shaped cavity configuration is more conducive to implementation. The experimental results were consistent with the numerical simulations. The geometric parameters and the arrangement of the fan-shaped cavity structure are two important factors affecting the mixing performance of this passive micromixer. When the width ratio w3/w1 was 3.0, the mixing index of this type micromixer reached around 75% at Re values larger than 60. For the three different channel width ratios, the pressure drop Δp exhibited an approximately linear increase with increasing Reynolds number. These planar passive micromixers are likely to be optimally used in a range of low Reynolds numbers (i.e., 1−80) to meet the requirements of microfluidic chips.

■

NOMENCLATURE ci = mass fraction of species concentration at sampling point (node) i cm = the optimal mixing mass fraction of species concentration d = width of the curved channel, m Dab = diffusion coefficient of a mixture, m−2·s−1 De = Dean number Dh = hydraulic diameter of the microchannel, m f = overall apparent friction factor G = mass flux, kg·m−2·s−1 h = depth of the microchannel, m Kn = Knudsen number L = characteristic length, m M = mixing index n = number of sampling points on the cross section Pe = Péclet number Q = volume flow rate of the mixer, m3·s−1 R = radius of curvature, m Re = Reynolds number Sc = Schmidt number u = characteristic velocity, m·s−1 um = mean longitudinal velocity, m·s−1 w = width of main channel, m w1 = width of major subchannel, m w2 = width of minor subchannel, m w3 = width of fan-shaped cavity channel, m Δp = pressure drop between the inlet and outlet of the channel, kg·m−1·s−2

Greek Letters

■

ρ = density, kg·m−3 ν = kinematic viscosity, m2·s−1 μ = dynamic viscosity, kg·m−1·s−1 σ = standard deviation of species concentration λ = mean free path length of the molecules

REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +86-10-67391985-8301. Fax: +86-10-67392774. E-mail: [email protected]. Notes

The authors declare no competing financial interest. 7826



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Analysis of Flow and Mixing Characteristics of Planar Asymmetric Split

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