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Mixing performance of a planar micromixer with circular obstructions in a curved microchannel
chemical engineering research and design 9 2 ( 2 0 1 4 ) 423–434
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Mixing performance of a planar micromixer with circular obstructions in a curved microchannel Afroz Alam, Arshad Afzal, Kwang-Yong Kim ∗ Department of Mechanical Engineering, Inha University, Incheon, 402 751, Republic of Korea
a b s t r a c t A numerical investigation of the mixing and fluid flow in a new design of passive micromixer employing several cylindrical obstructions within a curved microchannel is presented in this work. Mixing in the channels is analyzed using Navier–Stokes equations and the diffusion equation between two working fluids (water and ethanol) for Reynolds numbers from 0.1 to 60. The proposed micromixer shows far better mixing performance than a T-micromixer with circular obstructions and a simple curved micromixer. The effects of cross-sectional shape, height, and placement of the obstructions on mixing performance and the pressure drop of the proposed micromixer are evaluated. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micromixer; Circular obstructions; Curved microchannel; Mixing index; Navier–Stokes equations; Reynolds number
1.
Introduction
A wide range of applications for microfluidic systems is being realized in fields such as miniaturized analytical systems for chemistry and biology (e.g., genomic and proteomic analysis), clinical diagnostics, and micro-total analysis systems (TAS) (Sanders and Manz, 2000; Verpoorte, 2002; Reyes et al., 2002; Auroux et al., 2002; Chow, 2002). Many microsystems need to mix two or more fluid streams to fulfill the task. Therefore, the micromixer unit is one of the critical elements of TAS. To obtain fast diagnosis results, efficient and fast mixing of the reagents is needed. Due to low Reynolds numbers, the flows in microchannels are laminar flows, and thus the mixing is dominated by molecular diffusion. If mixing relies only on the diffusion process, considerable time and a long channel length are necessary, thereby resulting in high-pressure drop and high cost. To overcome these difficulties, researchers have tried to obtain efficient and fast mixing even at low Reynolds numbers. To enhance mixing, an active or passive method can be employed. Active mixing is achieved by perturbing the flow field using external sources of energy including electro-kinetic force (Jacobson et al., 1999; Oddy et al., 2001), ultrasonic actuation (Zhu and Kim, 1998; Yang et al., 2001), thermal power (Mao et al., 2002; Tsai and Lin, 2002), and periodic
∗
pressure perturbation (Fujii et al., 2003; Glasgow and Aubry, 2003; Niu and Lee, 2003). Active micromixers have obvious advantages over passive micromixers, but their complexity with respect to integration with microfluidic systems leads researchers to consider alternatives. On the other hand, passive micromixers do not require an external source of energy other than the basic pressure head used to drive the fluid flow. Passive micromixers can be classified into two types of mixing mechanisms: chaotic advection and lamination. Chaotic micromixers enhance mixing with three-dimensional (3-D) channel structures (Liu et al., 2000; Jen et al., 2003; Kim et al., 2004) or geometrical planar shapes (Hong et al., 2001; Wong et al., 2003; Bhagat et al., 2007). The planar designs are easier to fabricate and to integrate with micro-systems, but they need to operate at Re > 100. This results in a higher pressure drop (Wong et al., 2003) or the need for long channels; e.g., longer than 10 mm when working at Re < 1 to achieve higher performance (Bhagat et al., 2007). Mixing in lamination micromixers depends on the molecular diffusion mechanism (Wu and Nguyen, 2005; Ducree et al., 2006). Although these devices can achieve higher mixing at low Reynolds numbers, their fabrications are usually complicated. Micromixers with obstructions use chaotic advection, and have been proposed by many researchers for a straight channel. In many cases, the splits and recombinations are realized
Corresponding author. Tel.: +82 32 872 3096; fax: +82 32 868 1716. E-mail address: [email protected] (K.-Y. Kim). Received 27 October 2012; Received in revised form 12 September 2013; Accepted 16 September 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.09.008
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Nomenclature Ci Cm d1 , d2 H h Lm Le M P Re R1 , R2 Rm Vj Vij W x, y, z
mass fraction at sampling point i optimal mixing mass fraction diameters of the small and large circular obstructions, respectively height of the channel height of the obstruction length of the main channel length of the exit channel mixing index pressure Reynolds number radii of the inner and outer circular walls radius of the centerline in the curved channel average velocity in xj direction velocity of fluid component i in xj direction width of the main channel streamwise, spanwise, and cross-streamwise coordinates, respectively
Greek letters inclination of the obstruction variance ␥ absolute viscosity fluid density i density of fluid component i molecular diffusion coefficient of fluid compoi nent i
by introducing obstructions into the path of the main flow. Lin et al. (2007) studied numerically and experimentally a T-micromixer with J-shaped baffles. Bhagat et al. (2007) proposed a planar passive microfluidic mixer with different shapes of obstructions in a T-channel that is capable of mixing at low Reynolds numbers. Wang et al. (2007) proposed a Y-micromixer with a cylindrical obstruction in order to disrupt the flow. The disruption to the flow field alters the flow direction from one fluid to another. In this way, convection can occur to enhance the mixing. Hossain and Kim (2010) numerically investigated the effects of geometric parameters on the mixing performance of a straight groove micromixer. They found that the number of grooves per cycle increases the mixing index and decreases the pressure drop. In a recent study, Ansari et al. (2010) improved the mixing performance of a simple split and recombination micromixer using unbalanced splits and collisions of fluid streams over Reynolds number range, 10 ≤ Re ≤ 80. Curved micromixers have been studied by several researchers (Hossain et al., 2009; Jiang et al., 2004; Sudarsan and Ugaz, 2006). They indicate that curved micromixers without obstructions are effective only at high Reynolds numbers. Obstructions in straight channels only cause split and recombinations, but obstructions in curved microchannels cause secondary flows at high Reynolds numbers as well as splits and recombinations. Therefore, curved micromixers with obstructions are effective for mixing at both high and low Reynolds numbers. In the present study, a curved micromixer with obstructions is proposed for efficient mixing for a wide range of Reynolds numbers (Re = 0.1–60). Three obstruction shapes were tested by solving 3-D Navier–Stokes equations to
determine the effect of the cross-sectional shape of the obstructions on mixing performance. The effects of height and the placement of obstructions on mixing were also evaluated. The mixing performance of the proposed micromixer was compared to the performance of a T-micromixer with obstructions and a simple curved channel.
2.
Micromixer model
Fig. 1 shows schematic diagrams of the curved and straight micromixers with cylindrical obstructions. In both micromixers, the width (W) and height (H) of the main channel are commonly 100 m. Two different fluids enter into the micromixers from different inlets. The dimensions of the micromixers are as follows: inlet channel length, L0 = 0.2 mm; channel length with obstructions, Lm = 3.5 mm; and exit channel length, Le = 1.8 mm. For the curved channel shown in Fig. 1(a), the radius of the inner circular wall is R1 = 0.2 mm and the radius of the outer circular wall is R2 = 0.3 mm. The angular distance between adjacent large and small obstructions is = 15◦ and is uniform throughout the channel. Therefore, there are six large and twelve small cylindrical obstructions in a half-cycle whose diameters (d1 and d2 ) are 0.04 mm and 0.03 mm, respectively. The height of the obstructions (h) varies from 20 to 100 m. Fig. 1(b) shows a schematic diagram of the T-micromixer with cylindrical obstructions. The dimensions of the obstructions are the same as in the curved channel. The axial distance between the adjacent obstructions is equal to Rm . Water and ethanol were selected as the two working fluids for mixing. The properties of water and ethanol were taken at a temperature of 20 ◦ C. The densities of water and ethanol are 998 and 789 kg m−3 , respectively. The viscosities of water and ethanol are 0.9 × 10−3 and 1.2 × 10−3 kg m−1 s−1 , respectively. The analysis was performed for a Reynolds number range from 0.1 to 60. In this work, a parametric study on the geometry of a curved micromixer with obstructions (Fig. 1(a)) was performed to improve the mixing performance with various shapes, heights, and placements of the obstructions.
3.
Numerical analysis
A commercial CFD code, ANSYS CFX-11.0 (2007), was used to analyze the mixing and fluid flow in the micromixer. The code solves the continuity, Navier–Stokes, and species convection–diffusion equations for steady and incompressible flows using the finite volume method. Each fluid component has its own equation for the conservation of mass as follows: ∂(i Vj ) ∂xj Vj =
=−
∂ ( (V − Vj )) ∂xj i ij
(i Vij )
(2)
i (Vij − Vj ) = −
(1)
i ∂i ¯ ∂xj
(3)
where i is the density of fluid component i in the mixture (i.e., the mass of fluid component i per unit volume); Vj is the average velocity field; i (Vij − Vj ) is the relative mass flux; and Vij is the velocity of fluid component i. The differential motion of the individual components in the mixture is accounted for by the relative mass flux term. This term may be modeled in
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Fig. 1 – Schematics of the micromixers; (a) curved and (b) straight micromixers withcylindrical obstructions, and curved micromixers with (c) diamond (left) and hexagonal (right) obstructions.
a number of ways to include the effects of concentration gradients, pressure gradients, etc. The concentration gradient is the primary effect of the possible relative motion of the mixture components. i is the molecular diffusion coefficient of fluid component i. The details of the governing equations and numerical methods can be found in a previous paper (Ansari and Kim, 2009). An unstructured tetrahedral grid system was created for the full model using the commercial software code ANSYS ICEM 11.0. The numerical simulation cannot be free from numerical diffusion errors (Leonard, 1987; Noll, 1993) that arise from the discretization of the convection terms in the Navier–Stokes equations. It is very difficult to overcome these errors; however, they can be minimized by adopting certain techniques (Hardt and Schonfeld, 2003). The velocities at the inlets and zero static pressure at the outlet were specified as boundary conditions. The validation of numerical solutions by comparison with experimental measurements has been reported for a micromixer with a circular chamber in a previous work (Ansari and Kim, 2009) that used the same governing equations and numerical methods used in this study. The solutions were considered to have attained convergence for a root mean square (RMS) residual value of less than 10−6 . In order to quantify mixing efficiency, the variance of species was determined in a cross section of the mixing
Fig. 2 – Grid dependency test for distribution of mixing index along the microchannel (Re = 15, h/H = 1).
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Fig. 3 – Variations of the mixing index at the exit with Reynolds number for three cross-sectional shapes of the obstruction (h/H = 1.0, Â = 15◦ ).
Fig. 4 – Variations in mixing index at the exit of three different microchannels with Reynolds number (h/H = 1.0).
Fig. 5 – Mixing in three different micromixers at Re = 0.1 (h/H = 1.0); (a) Distributions of the mixing index along the micromixer length. (b) Contours of the mass fraction of ethanol on the y–z plane at different positions along the channel.
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channel perpendicular to the main flow. The variance is expressed as follows:
=
1 N
Ci − Cm
2
(4)
where is the variance over the data range, Ci is the mass fraction at sampling point i, Cm is the optimal mixing mass fraction, and N is the number of sampling points (which was taken as 2500 to ensure high accuracy). The sampling points are equidistant on the cross-sectional plane. The values at the sampling points were obtained by interpolation with the values from adjacent computational cells. The mixing efficiency can be evaluated using the following parameter:
M=1−
2 2 max
(5)
where M is the mixing index and max is the maximum variance over the data range. The variance is maximum for completely unmixed fluids and minimum for completely mixed fluids.
4.
427
Results and discussion
A preliminary grid dependency test was performed to ensure that the numerical results would be independent of the grid size. Five different grid systems ranging from 0.63 × 106 to 2.53 × 106 were tested for h/H = 1.0 and Re = 15. The mixing index was calculated along the length of the channel for each grid system. From the results shown in Fig. 2, 1.892 × 106 was determined as the optimum number of grids that was used for further analysis. Mixing and flow field analyses were performed for a curved micromixer with different obstruction shapes for a Reynolds number range from 0.1 to 60, as shown in Fig. 3. The simulations were carried out for three cross-sectional shapes of obstructions: circular, hexagonal, and diamond. To maintain dimensional consistency of the obstructions, each shape was fit to squares of 40 m × 40 m and 30 m × 30 m for the bigger and smaller obstructions patterns, respectively. For each case, the obstruction height was taken as 100 m (equal to the channel height) and the angular distances between obstructions were kept the same, as shown in Fig. 1(a). The length of the channel was 4 mm. In Fig. 3, the circular and
Fig. 6 – Mixings in three different micromixers at Re = 15 (h/H = 1.0); (a) Distributions of the mixing index along the micromixer length. (b) Contours of the mass fraction of ethanol on the y–z plane at different positions along the channel.
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Fig. 7 – Velocity vectors on the x–y plane at the mid-height of the channel (Re = 15, h/H = 1.0, z/H = 0.5). hexagonal obstructions show almost the same mixing performance regardless of the Reynolds number, while the diamond obstructions shows far less mixing performance compared to the others except at Reynolds numbers greater than 50. Therefore, the circular obstruction was chosen for further study due to simple geometry and easy fabrication. In comparison with the unbalanced split and recombination micromixer proposed by Ansari et al. (2010) where inertial effect dominates the mixing process for Reynolds number of the order of 100, the present design yielded nearly complete mixing at Re = 30. The mixing performance of the curved micromixer with a circular obstruction was compared to those of the Tmicromixer with a circular obstruction and a simple curved micromixer without obstruction. The height of the obstruction was taken as 100 m for both the curved and T-micromixers. Fig. 4 shows the variation of the mixing index at the exit with the Reynolds number for the three micromixers. The mixing index was evaluated at the exit of the last mixing plane for the following Reynolds numbers: 0.1, 0.5, 1, 5, 10, 15, 30, 45, and 60. The curved micromixer with circular obstructions shows the best mixing performance throughout the Reynolds number range compared to the other two micromixers. At Re = 0.1, the mixing index of the curved micromixer with circular obstructions is 88%, whereas the curved channel without obstruction and the T-micromixer with circular obstructions show a 70% and 82% mixing index, respectively. In Fig. 4, it is found that the mixing indices of these mixers substantially decreased with increased Reynolds number. With an increase in Reynolds number from 0.1 to 5, the mixing index decreased due to the decrease in the residential time for diffusion. However, as the Reynolds number increased beyond Re = 10, the mixing index increased again in all the micromixers. This is because the increase in inertial force caused advection to increase, which produced a disturbance in the flow field and thus enhanced the mixing (ANSYS CFX-11.0, 2007). At Re = 15,
the curved micromixer with obstructions shows a mixing index of 88%, and the T-micromixer with obstructions and the simple curved micromixers show 64% and 28%, respectively. The introduction of circular obstructions in the curved channel significantly improved the mixing performance of the curved channel for most of the Reynolds numbers considered. However, the performance decreased steadily as the Reynolds number increased beyond Re = 5. The curved micromixer with obstructions shows an almost constant mixing index (about 88%) beyond Re = 15. The mixing index for the T-micromixer with obstructions also shows only a small variation in this Reynolds number range. The simple curved channel without obstructions shows a rapid increase in the mixing index as the Reynolds number was increased from 15 to 60. Therefore, this simple curved channel shows a higher mixing performance than the T-micromixer with obstructions for Reynolds numbers greater than 45. This indicates that the channel curvature becomes more effective than the obstructions for the enhancement of mixing at these higher Reynolds numbers. However, the mixing is relatively low compared to the present design for a similar Reynolds number range 10 ≤ Re ≤ 60. Fig. 5(a) shows the development of the mixing index along the micromixer length for three micromixers at Re = 0.1. The mixing indices of the simple curved channel and Tmicromixer with a circular obstruction increased from 0.18 to 0.72 and from 0.22 to 0.84, respectively, through the micromixers. On the other hand, in the curved micromixer with circular obstructions, the mixing index increased from 0.32 to 0.90 monotonically for the same channel length. Fig. 5(b) shows the mass fraction contours of ethanol on the y–z plane at different axial locations for Re = 0.1. In the case of the simple curved channel, two streams flowing in parallel meet each other exactly at the central plane of the channel, and thus the contours show a nearly symmetric distribution at each axial location. The other two micromixers with circular obstructions
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Fig. 8 – Streamlines showing flow patterns in three different micromixers (h/H = 1.0); (a) Re = 0.1 and (b) Re = 15.
clearly show better mixing due to the presence of a split and recombination process that increases the contact surface area for diffusion. Fig. 6(a) shows the development of a mixing index along the micromixer for three micromixers at Re = 15. As the Reynolds number is substantially higher than that of Fig. 5, convection dominates the mixing at this Reynolds number (Re = 15), as shown in Fig. 4. As shown in Fig. 6(a), the qualitative behaviors of the mixing index developments are similar to those shown in Fig. 5(a). However, quantitatively, the simple curved micromixer shows a much lower mixing index distribution than that at Re = 0.1 (Fig. 5(a)). Fig. 6(b) indicates that split and re-combinations due to obstructions in the curved channel promote mixing with a complicated flow field as shown in Fig. 7. The split-and-recombination process
provides multi-lamination, which causes an increased interfacial area for increased diffusion and residence time. The obstruction caused the parabolic velocity distribution to even out and give more time for diffusion at the two interfaces of the two fluids. Due to the asymmetric layout, the obstructions give non-uniform resistance to the flow in the lateral direction. Therefore, the fluids find a path of less resistance through which to flow. This means that part of the fluid flow is distorted and flow is redirected from one component to another; hence, a convective effect occurs. The streamlines plotted in Fig. 8 show flow patterns of the three micromixers at Re = 0.1 and 15. At Re = 0.1, the streamlines were parallel and smooth, but at Re = 15 they became rough in all three micromixers. The two micromixers with
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Fig. 9 – Effect of obstruction placement on mixing index and pressure drop with Reynolds number; (a) Mixing index vs. Reynolds number. (b) Pressure drop vs. Reynolds number (h/H = 1.0, channel length = 4 mm).
obstructions show the split-and-recombination streamlines, which are responsible for the enhancement of mixing compared to a simple curved micromixer. Fig. 9 shows the effects of obstruction placement, the mixing index, and the pressure drop with Reynolds number. With an increase in (and a decrease in the number of obstructions), the mixing index decreased as expected. The pressure drop also increased with a decrease in , as expected. At Re = 60, the pressure drops in straight and curved microchannels with similar hydraulic diameter and total length were 10 kPa and 23 kPa, respectively. Fig. 10 shows the velocity vectors on x–y plane at mid-height of the channel for Re = 60 (h/H = 1.0 and = 45◦ ). Flow separation occurs behind each cylinder
Fig. 10 – Velocity vectors on the x–y plane at the mid-height of the channel (Re = 60, h/H = 1.0, Â = 45◦ ).
producing two counter-rotating vortices. The flow separation was first observed at Re = 15. However, the vortex motion was not sufficiently developed. Increase in Reynolds number led to augmentation of the vortex motion. It is expected that the vortex formation led to enhanced energy dissipation, and thereby resulted in higher pressure drop. It can be seen that for the most efficient design (obstructions with circular shape, h/H = 1.0, and = 15◦ ), mixing index approaches the asymptotic value of 0.88 at Re = 15. And, the pressure drop is in the acceptable range for 0.1 ≤ Re ≤ 30 for microfluidic applications. In conclusion, the optimum operation range of the micromixer is 0.1 ≤ Re ≤ 30 in terms of mixing performance and pressure drop. It is noted that with an increase in Reynolds number, the mixing index for all the cases for approaches the same value. Fig. 11 shows the contours of the mass fraction of ethanol on the x–y plane at the middle height of the channel for different numbers of obstructions at Re = 0.1 and 15. The fact that the mixing is enhanced as the number of obstructions increases (Fig. 9(a)) is confirmed visually in this figure. At Re = 0.1, it is shown in Fig. 11(a) that the mixing steadily improved along the channel length in all cases. However, at Re = 15, oscillation of the mass fraction is observed along the channel length, especially in the cases with a smaller number of obstructions. Fig. 12 shows the effects of obstruction height on mixing efficiency with Reynolds number. The obstruction was placed at the base of the x–y plane. The obstruction height was varied from 20 to the full channel height of 100 m, and was fixed at 150 . Fig. 11 clearly shows that with increased obstruction height, the mixing index generally increases. However, as an exception, the mixing index for h/H = 1.0 becomes slightly lower than the index for h/H = 0.4–0.8 at Reynolds numbers greater than 15. The mixing indices for h/H = 0.4–0.8 are nearly the same beyond Re = 30 regardless of Reynolds number. At full channel height, the micromixer works like a split-andrecombine micromixer as the inlet stream splits into four sub-streams and then recombines into laminae. Better mixing is achieved using this configuration due to an increase in the interfacial area of the fluid stream that facilitates faster diffusion. And, it is also noted that the obstructions of full channel height can be patterned with a single lithography step.
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Fig. 11 – Mass fraction distributions of ethanol on x–y plane at the middle of the channel height vs. number of obstructions (h/H = 1.0); (a) Re = 0.1 and (b) Re = 15. Fig. 13 shows velocity vector plots on the y–z planes along the channel length for a fixed obstruction height, h = 60 m at three different Reynolds numbers, Re = 0.1, 15, and 30. It can be clearly seen that the flow is periodic in the micromixer at the Reynolds numbers tested in the present work. Fig. 14 shows velocity vector plots on the y–z planes P-1 and P2 for different obstruction heights and Reynolds numbers. Considering the periodicity of the flow, the flow patterns were compared only at the cross-sectional planes P-1 and P-2. The planes are located at first peak and crest of the channel, respectively. At Re = 0.1, as the height of the obstructions varied from 20 to 100 m (full channel height), the strength of the transverse flow started to increase and became maximum at a height of 100 m. At Re = 15, due to increased inertia force compared to Re = 0.1, there is a sign of the start of the formation of secondary flows with the increase in obstruction height. However, when the obstruction heights approach the channel height, the secondary flows disappear.
Fig. 12 – Effect of the obstruction height on mixing index with Reynolds number.
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Fig. 13 – Velocity vector plots on y–z planes along the channel length; (a) Re = 0.1, (b) Re = 15, and (c) Re = 30.
Fig. 14 – Velocity vector plots on y–z planes P-1 and P-2 for different obstruction heights; (left) Re = 0.1, (middle) Re = 15, and (right) Re = 30.
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At Re = 30, two counter-rotating vortices start to form at the center of the plane at the peak positions and near the walls at the crest positions for h = 20 and 40 m. However, at h = 60 m, only one vortex formed at the peak position and the vortices disappeared when the height of the obstruction reached beyond 60 m. The strength of the transverse flow increased with increased obstruction height.
5.
Conclusion
A planar curved laminating passive micromixer that is capable of mixing at low Reynolds numbers was proposed in this work. Detailed study of the effects of geometry and placement of obstructions on mixing was carried out using Navier–Stokes analysis. The study was performed using a Reynolds number range from 0.1 to 60. First, the effect of the cross-sectional shape of the obstructions on mixing was investigated. Second, the mixing performances of the T-micromixer with obstructions and a simple curved micromixer without obstructions were compared to that of the proposed micromixer. Finally, the effects of the obstruction height and Reynolds number on mixing performance were evaluated. Micromixers with circular and hexagonal obstructions show almost the same mixing performances regardless of the Reynolds number, while micromixer with diamond obstructions shows far less mixing performance compared to the others except at Reynolds numbers greater than 50. The curved channel with circular obstructions shows much higher mixing performance than those of the T-micromixer with obstructions and a simple curved micromixer without obstructions throughout the Reynolds number range considered. The proposed micromixer with circular obstructions shows the best mixing index at the exit of the micromixer, 88% among the tested micromixers, at Reynolds numbers of 0.1 and greater than 15. The minimum mixing index, 72%, was seen at Re = 5. The pressure drop increased with an increased number of obstructions. However, this increase in pressure drop is not substantial, and can be an acceptable tradeoff for improved mixing. The mixing generally increased as the height and number of the obstructions increased. However, both of these effects disappeared as the Reynolds number approached 60, where the formation of secondary flows due to channel curvature dominated the other mechanisms due to obstructions. The proposed micromixer design is planar, and hence requires only simple fabrication.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant No. 2009-0083510 funded by the Korean government (MSIP) through Multi-phenomena CFD Engineering Research Center.
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Yang, Z., Matsumoto, S., Goto, H., Matsumoto, M., Maeda, R., 2001. Ultrasonic micromixer for microfluidic systems. Sens. Actuators A 93, 266–272. Zhu, X., Kim, E.S., 1998. Microfluidic motion generation with acoustic waves. Sens. Actuators A 66, 355–360.
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Mixing performance of a planar micromixer with circular obstructions in a curved microchannel Afroz Alam, Arshad Afzal, Kwang-Yong Kim ∗ Department of Mechanical Engineering, Inha University, Incheon, 402 751, Republic of Korea
a b s t r a c t A numerical investigation of the mixing and fluid flow in a new design of passive micromixer employing several cylindrical obstructions within a curved microchannel is presented in this work. Mixing in the channels is analyzed using Navier–Stokes equations and the diffusion equation between two working fluids (water and ethanol) for Reynolds numbers from 0.1 to 60. The proposed micromixer shows far better mixing performance than a T-micromixer with circular obstructions and a simple curved micromixer. The effects of cross-sectional shape, height, and placement of the obstructions on mixing performance and the pressure drop of the proposed micromixer are evaluated. © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micromixer; Circular obstructions; Curved microchannel; Mixing index; Navier–Stokes equations; Reynolds number
1.
Introduction
A wide range of applications for microfluidic systems is being realized in fields such as miniaturized analytical systems for chemistry and biology (e.g., genomic and proteomic analysis), clinical diagnostics, and micro-total analysis systems (TAS) (Sanders and Manz, 2000; Verpoorte, 2002; Reyes et al., 2002; Auroux et al., 2002; Chow, 2002). Many microsystems need to mix two or more fluid streams to fulfill the task. Therefore, the micromixer unit is one of the critical elements of TAS. To obtain fast diagnosis results, efficient and fast mixing of the reagents is needed. Due to low Reynolds numbers, the flows in microchannels are laminar flows, and thus the mixing is dominated by molecular diffusion. If mixing relies only on the diffusion process, considerable time and a long channel length are necessary, thereby resulting in high-pressure drop and high cost. To overcome these difficulties, researchers have tried to obtain efficient and fast mixing even at low Reynolds numbers. To enhance mixing, an active or passive method can be employed. Active mixing is achieved by perturbing the flow field using external sources of energy including electro-kinetic force (Jacobson et al., 1999; Oddy et al., 2001), ultrasonic actuation (Zhu and Kim, 1998; Yang et al., 2001), thermal power (Mao et al., 2002; Tsai and Lin, 2002), and periodic
∗
pressure perturbation (Fujii et al., 2003; Glasgow and Aubry, 2003; Niu and Lee, 2003). Active micromixers have obvious advantages over passive micromixers, but their complexity with respect to integration with microfluidic systems leads researchers to consider alternatives. On the other hand, passive micromixers do not require an external source of energy other than the basic pressure head used to drive the fluid flow. Passive micromixers can be classified into two types of mixing mechanisms: chaotic advection and lamination. Chaotic micromixers enhance mixing with three-dimensional (3-D) channel structures (Liu et al., 2000; Jen et al., 2003; Kim et al., 2004) or geometrical planar shapes (Hong et al., 2001; Wong et al., 2003; Bhagat et al., 2007). The planar designs are easier to fabricate and to integrate with micro-systems, but they need to operate at Re > 100. This results in a higher pressure drop (Wong et al., 2003) or the need for long channels; e.g., longer than 10 mm when working at Re < 1 to achieve higher performance (Bhagat et al., 2007). Mixing in lamination micromixers depends on the molecular diffusion mechanism (Wu and Nguyen, 2005; Ducree et al., 2006). Although these devices can achieve higher mixing at low Reynolds numbers, their fabrications are usually complicated. Micromixers with obstructions use chaotic advection, and have been proposed by many researchers for a straight channel. In many cases, the splits and recombinations are realized
Corresponding author. Tel.: +82 32 872 3096; fax: +82 32 868 1716. E-mail address: [email protected] (K.-Y. Kim). Received 27 October 2012; Received in revised form 12 September 2013; Accepted 16 September 2013 0263-8762/$ – see front matter © 2013 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cherd.2013.09.008
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Nomenclature Ci Cm d1 , d2 H h Lm Le M P Re R1 , R2 Rm Vj Vij W x, y, z
mass fraction at sampling point i optimal mixing mass fraction diameters of the small and large circular obstructions, respectively height of the channel height of the obstruction length of the main channel length of the exit channel mixing index pressure Reynolds number radii of the inner and outer circular walls radius of the centerline in the curved channel average velocity in xj direction velocity of fluid component i in xj direction width of the main channel streamwise, spanwise, and cross-streamwise coordinates, respectively
Greek letters inclination of the obstruction variance ␥ absolute viscosity fluid density i density of fluid component i molecular diffusion coefficient of fluid compoi nent i
by introducing obstructions into the path of the main flow. Lin et al. (2007) studied numerically and experimentally a T-micromixer with J-shaped baffles. Bhagat et al. (2007) proposed a planar passive microfluidic mixer with different shapes of obstructions in a T-channel that is capable of mixing at low Reynolds numbers. Wang et al. (2007) proposed a Y-micromixer with a cylindrical obstruction in order to disrupt the flow. The disruption to the flow field alters the flow direction from one fluid to another. In this way, convection can occur to enhance the mixing. Hossain and Kim (2010) numerically investigated the effects of geometric parameters on the mixing performance of a straight groove micromixer. They found that the number of grooves per cycle increases the mixing index and decreases the pressure drop. In a recent study, Ansari et al. (2010) improved the mixing performance of a simple split and recombination micromixer using unbalanced splits and collisions of fluid streams over Reynolds number range, 10 ≤ Re ≤ 80. Curved micromixers have been studied by several researchers (Hossain et al., 2009; Jiang et al., 2004; Sudarsan and Ugaz, 2006). They indicate that curved micromixers without obstructions are effective only at high Reynolds numbers. Obstructions in straight channels only cause split and recombinations, but obstructions in curved microchannels cause secondary flows at high Reynolds numbers as well as splits and recombinations. Therefore, curved micromixers with obstructions are effective for mixing at both high and low Reynolds numbers. In the present study, a curved micromixer with obstructions is proposed for efficient mixing for a wide range of Reynolds numbers (Re = 0.1–60). Three obstruction shapes were tested by solving 3-D Navier–Stokes equations to
determine the effect of the cross-sectional shape of the obstructions on mixing performance. The effects of height and the placement of obstructions on mixing were also evaluated. The mixing performance of the proposed micromixer was compared to the performance of a T-micromixer with obstructions and a simple curved channel.
2.
Micromixer model
Fig. 1 shows schematic diagrams of the curved and straight micromixers with cylindrical obstructions. In both micromixers, the width (W) and height (H) of the main channel are commonly 100 m. Two different fluids enter into the micromixers from different inlets. The dimensions of the micromixers are as follows: inlet channel length, L0 = 0.2 mm; channel length with obstructions, Lm = 3.5 mm; and exit channel length, Le = 1.8 mm. For the curved channel shown in Fig. 1(a), the radius of the inner circular wall is R1 = 0.2 mm and the radius of the outer circular wall is R2 = 0.3 mm. The angular distance between adjacent large and small obstructions is = 15◦ and is uniform throughout the channel. Therefore, there are six large and twelve small cylindrical obstructions in a half-cycle whose diameters (d1 and d2 ) are 0.04 mm and 0.03 mm, respectively. The height of the obstructions (h) varies from 20 to 100 m. Fig. 1(b) shows a schematic diagram of the T-micromixer with cylindrical obstructions. The dimensions of the obstructions are the same as in the curved channel. The axial distance between the adjacent obstructions is equal to Rm . Water and ethanol were selected as the two working fluids for mixing. The properties of water and ethanol were taken at a temperature of 20 ◦ C. The densities of water and ethanol are 998 and 789 kg m−3 , respectively. The viscosities of water and ethanol are 0.9 × 10−3 and 1.2 × 10−3 kg m−1 s−1 , respectively. The analysis was performed for a Reynolds number range from 0.1 to 60. In this work, a parametric study on the geometry of a curved micromixer with obstructions (Fig. 1(a)) was performed to improve the mixing performance with various shapes, heights, and placements of the obstructions.
3.
Numerical analysis
A commercial CFD code, ANSYS CFX-11.0 (2007), was used to analyze the mixing and fluid flow in the micromixer. The code solves the continuity, Navier–Stokes, and species convection–diffusion equations for steady and incompressible flows using the finite volume method. Each fluid component has its own equation for the conservation of mass as follows: ∂(i Vj ) ∂xj Vj =
=−
∂ ( (V − Vj )) ∂xj i ij
(i Vij )
(2)
i (Vij − Vj ) = −
(1)
i ∂i ¯ ∂xj
(3)
where i is the density of fluid component i in the mixture (i.e., the mass of fluid component i per unit volume); Vj is the average velocity field; i (Vij − Vj ) is the relative mass flux; and Vij is the velocity of fluid component i. The differential motion of the individual components in the mixture is accounted for by the relative mass flux term. This term may be modeled in
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Fig. 1 – Schematics of the micromixers; (a) curved and (b) straight micromixers withcylindrical obstructions, and curved micromixers with (c) diamond (left) and hexagonal (right) obstructions.
a number of ways to include the effects of concentration gradients, pressure gradients, etc. The concentration gradient is the primary effect of the possible relative motion of the mixture components. i is the molecular diffusion coefficient of fluid component i. The details of the governing equations and numerical methods can be found in a previous paper (Ansari and Kim, 2009). An unstructured tetrahedral grid system was created for the full model using the commercial software code ANSYS ICEM 11.0. The numerical simulation cannot be free from numerical diffusion errors (Leonard, 1987; Noll, 1993) that arise from the discretization of the convection terms in the Navier–Stokes equations. It is very difficult to overcome these errors; however, they can be minimized by adopting certain techniques (Hardt and Schonfeld, 2003). The velocities at the inlets and zero static pressure at the outlet were specified as boundary conditions. The validation of numerical solutions by comparison with experimental measurements has been reported for a micromixer with a circular chamber in a previous work (Ansari and Kim, 2009) that used the same governing equations and numerical methods used in this study. The solutions were considered to have attained convergence for a root mean square (RMS) residual value of less than 10−6 . In order to quantify mixing efficiency, the variance of species was determined in a cross section of the mixing
Fig. 2 – Grid dependency test for distribution of mixing index along the microchannel (Re = 15, h/H = 1).
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Fig. 3 – Variations of the mixing index at the exit with Reynolds number for three cross-sectional shapes of the obstruction (h/H = 1.0, Â = 15◦ ).
Fig. 4 – Variations in mixing index at the exit of three different microchannels with Reynolds number (h/H = 1.0).
Fig. 5 – Mixing in three different micromixers at Re = 0.1 (h/H = 1.0); (a) Distributions of the mixing index along the micromixer length. (b) Contours of the mass fraction of ethanol on the y–z plane at different positions along the channel.
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channel perpendicular to the main flow. The variance is expressed as follows:
=
1 N
Ci − Cm
2
(4)
where is the variance over the data range, Ci is the mass fraction at sampling point i, Cm is the optimal mixing mass fraction, and N is the number of sampling points (which was taken as 2500 to ensure high accuracy). The sampling points are equidistant on the cross-sectional plane. The values at the sampling points were obtained by interpolation with the values from adjacent computational cells. The mixing efficiency can be evaluated using the following parameter:
M=1−
2 2 max
(5)
where M is the mixing index and max is the maximum variance over the data range. The variance is maximum for completely unmixed fluids and minimum for completely mixed fluids.
4.
427
Results and discussion
A preliminary grid dependency test was performed to ensure that the numerical results would be independent of the grid size. Five different grid systems ranging from 0.63 × 106 to 2.53 × 106 were tested for h/H = 1.0 and Re = 15. The mixing index was calculated along the length of the channel for each grid system. From the results shown in Fig. 2, 1.892 × 106 was determined as the optimum number of grids that was used for further analysis. Mixing and flow field analyses were performed for a curved micromixer with different obstruction shapes for a Reynolds number range from 0.1 to 60, as shown in Fig. 3. The simulations were carried out for three cross-sectional shapes of obstructions: circular, hexagonal, and diamond. To maintain dimensional consistency of the obstructions, each shape was fit to squares of 40 m × 40 m and 30 m × 30 m for the bigger and smaller obstructions patterns, respectively. For each case, the obstruction height was taken as 100 m (equal to the channel height) and the angular distances between obstructions were kept the same, as shown in Fig. 1(a). The length of the channel was 4 mm. In Fig. 3, the circular and
Fig. 6 – Mixings in three different micromixers at Re = 15 (h/H = 1.0); (a) Distributions of the mixing index along the micromixer length. (b) Contours of the mass fraction of ethanol on the y–z plane at different positions along the channel.
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Fig. 7 – Velocity vectors on the x–y plane at the mid-height of the channel (Re = 15, h/H = 1.0, z/H = 0.5). hexagonal obstructions show almost the same mixing performance regardless of the Reynolds number, while the diamond obstructions shows far less mixing performance compared to the others except at Reynolds numbers greater than 50. Therefore, the circular obstruction was chosen for further study due to simple geometry and easy fabrication. In comparison with the unbalanced split and recombination micromixer proposed by Ansari et al. (2010) where inertial effect dominates the mixing process for Reynolds number of the order of 100, the present design yielded nearly complete mixing at Re = 30. The mixing performance of the curved micromixer with a circular obstruction was compared to those of the Tmicromixer with a circular obstruction and a simple curved micromixer without obstruction. The height of the obstruction was taken as 100 m for both the curved and T-micromixers. Fig. 4 shows the variation of the mixing index at the exit with the Reynolds number for the three micromixers. The mixing index was evaluated at the exit of the last mixing plane for the following Reynolds numbers: 0.1, 0.5, 1, 5, 10, 15, 30, 45, and 60. The curved micromixer with circular obstructions shows the best mixing performance throughout the Reynolds number range compared to the other two micromixers. At Re = 0.1, the mixing index of the curved micromixer with circular obstructions is 88%, whereas the curved channel without obstruction and the T-micromixer with circular obstructions show a 70% and 82% mixing index, respectively. In Fig. 4, it is found that the mixing indices of these mixers substantially decreased with increased Reynolds number. With an increase in Reynolds number from 0.1 to 5, the mixing index decreased due to the decrease in the residential time for diffusion. However, as the Reynolds number increased beyond Re = 10, the mixing index increased again in all the micromixers. This is because the increase in inertial force caused advection to increase, which produced a disturbance in the flow field and thus enhanced the mixing (ANSYS CFX-11.0, 2007). At Re = 15,
the curved micromixer with obstructions shows a mixing index of 88%, and the T-micromixer with obstructions and the simple curved micromixers show 64% and 28%, respectively. The introduction of circular obstructions in the curved channel significantly improved the mixing performance of the curved channel for most of the Reynolds numbers considered. However, the performance decreased steadily as the Reynolds number increased beyond Re = 5. The curved micromixer with obstructions shows an almost constant mixing index (about 88%) beyond Re = 15. The mixing index for the T-micromixer with obstructions also shows only a small variation in this Reynolds number range. The simple curved channel without obstructions shows a rapid increase in the mixing index as the Reynolds number was increased from 15 to 60. Therefore, this simple curved channel shows a higher mixing performance than the T-micromixer with obstructions for Reynolds numbers greater than 45. This indicates that the channel curvature becomes more effective than the obstructions for the enhancement of mixing at these higher Reynolds numbers. However, the mixing is relatively low compared to the present design for a similar Reynolds number range 10 ≤ Re ≤ 60. Fig. 5(a) shows the development of the mixing index along the micromixer length for three micromixers at Re = 0.1. The mixing indices of the simple curved channel and Tmicromixer with a circular obstruction increased from 0.18 to 0.72 and from 0.22 to 0.84, respectively, through the micromixers. On the other hand, in the curved micromixer with circular obstructions, the mixing index increased from 0.32 to 0.90 monotonically for the same channel length. Fig. 5(b) shows the mass fraction contours of ethanol on the y–z plane at different axial locations for Re = 0.1. In the case of the simple curved channel, two streams flowing in parallel meet each other exactly at the central plane of the channel, and thus the contours show a nearly symmetric distribution at each axial location. The other two micromixers with circular obstructions
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Fig. 8 – Streamlines showing flow patterns in three different micromixers (h/H = 1.0); (a) Re = 0.1 and (b) Re = 15.
clearly show better mixing due to the presence of a split and recombination process that increases the contact surface area for diffusion. Fig. 6(a) shows the development of a mixing index along the micromixer for three micromixers at Re = 15. As the Reynolds number is substantially higher than that of Fig. 5, convection dominates the mixing at this Reynolds number (Re = 15), as shown in Fig. 4. As shown in Fig. 6(a), the qualitative behaviors of the mixing index developments are similar to those shown in Fig. 5(a). However, quantitatively, the simple curved micromixer shows a much lower mixing index distribution than that at Re = 0.1 (Fig. 5(a)). Fig. 6(b) indicates that split and re-combinations due to obstructions in the curved channel promote mixing with a complicated flow field as shown in Fig. 7. The split-and-recombination process
provides multi-lamination, which causes an increased interfacial area for increased diffusion and residence time. The obstruction caused the parabolic velocity distribution to even out and give more time for diffusion at the two interfaces of the two fluids. Due to the asymmetric layout, the obstructions give non-uniform resistance to the flow in the lateral direction. Therefore, the fluids find a path of less resistance through which to flow. This means that part of the fluid flow is distorted and flow is redirected from one component to another; hence, a convective effect occurs. The streamlines plotted in Fig. 8 show flow patterns of the three micromixers at Re = 0.1 and 15. At Re = 0.1, the streamlines were parallel and smooth, but at Re = 15 they became rough in all three micromixers. The two micromixers with
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Fig. 9 – Effect of obstruction placement on mixing index and pressure drop with Reynolds number; (a) Mixing index vs. Reynolds number. (b) Pressure drop vs. Reynolds number (h/H = 1.0, channel length = 4 mm).
obstructions show the split-and-recombination streamlines, which are responsible for the enhancement of mixing compared to a simple curved micromixer. Fig. 9 shows the effects of obstruction placement, the mixing index, and the pressure drop with Reynolds number. With an increase in (and a decrease in the number of obstructions), the mixing index decreased as expected. The pressure drop also increased with a decrease in , as expected. At Re = 60, the pressure drops in straight and curved microchannels with similar hydraulic diameter and total length were 10 kPa and 23 kPa, respectively. Fig. 10 shows the velocity vectors on x–y plane at mid-height of the channel for Re = 60 (h/H = 1.0 and = 45◦ ). Flow separation occurs behind each cylinder
Fig. 10 – Velocity vectors on the x–y plane at the mid-height of the channel (Re = 60, h/H = 1.0, Â = 45◦ ).
producing two counter-rotating vortices. The flow separation was first observed at Re = 15. However, the vortex motion was not sufficiently developed. Increase in Reynolds number led to augmentation of the vortex motion. It is expected that the vortex formation led to enhanced energy dissipation, and thereby resulted in higher pressure drop. It can be seen that for the most efficient design (obstructions with circular shape, h/H = 1.0, and = 15◦ ), mixing index approaches the asymptotic value of 0.88 at Re = 15. And, the pressure drop is in the acceptable range for 0.1 ≤ Re ≤ 30 for microfluidic applications. In conclusion, the optimum operation range of the micromixer is 0.1 ≤ Re ≤ 30 in terms of mixing performance and pressure drop. It is noted that with an increase in Reynolds number, the mixing index for all the cases for approaches the same value. Fig. 11 shows the contours of the mass fraction of ethanol on the x–y plane at the middle height of the channel for different numbers of obstructions at Re = 0.1 and 15. The fact that the mixing is enhanced as the number of obstructions increases (Fig. 9(a)) is confirmed visually in this figure. At Re = 0.1, it is shown in Fig. 11(a) that the mixing steadily improved along the channel length in all cases. However, at Re = 15, oscillation of the mass fraction is observed along the channel length, especially in the cases with a smaller number of obstructions. Fig. 12 shows the effects of obstruction height on mixing efficiency with Reynolds number. The obstruction was placed at the base of the x–y plane. The obstruction height was varied from 20 to the full channel height of 100 m, and was fixed at 150 . Fig. 11 clearly shows that with increased obstruction height, the mixing index generally increases. However, as an exception, the mixing index for h/H = 1.0 becomes slightly lower than the index for h/H = 0.4–0.8 at Reynolds numbers greater than 15. The mixing indices for h/H = 0.4–0.8 are nearly the same beyond Re = 30 regardless of Reynolds number. At full channel height, the micromixer works like a split-andrecombine micromixer as the inlet stream splits into four sub-streams and then recombines into laminae. Better mixing is achieved using this configuration due to an increase in the interfacial area of the fluid stream that facilitates faster diffusion. And, it is also noted that the obstructions of full channel height can be patterned with a single lithography step.
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Fig. 11 – Mass fraction distributions of ethanol on x–y plane at the middle of the channel height vs. number of obstructions (h/H = 1.0); (a) Re = 0.1 and (b) Re = 15. Fig. 13 shows velocity vector plots on the y–z planes along the channel length for a fixed obstruction height, h = 60 m at three different Reynolds numbers, Re = 0.1, 15, and 30. It can be clearly seen that the flow is periodic in the micromixer at the Reynolds numbers tested in the present work. Fig. 14 shows velocity vector plots on the y–z planes P-1 and P2 for different obstruction heights and Reynolds numbers. Considering the periodicity of the flow, the flow patterns were compared only at the cross-sectional planes P-1 and P-2. The planes are located at first peak and crest of the channel, respectively. At Re = 0.1, as the height of the obstructions varied from 20 to 100 m (full channel height), the strength of the transverse flow started to increase and became maximum at a height of 100 m. At Re = 15, due to increased inertia force compared to Re = 0.1, there is a sign of the start of the formation of secondary flows with the increase in obstruction height. However, when the obstruction heights approach the channel height, the secondary flows disappear.
Fig. 12 – Effect of the obstruction height on mixing index with Reynolds number.
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Fig. 13 – Velocity vector plots on y–z planes along the channel length; (a) Re = 0.1, (b) Re = 15, and (c) Re = 30.
Fig. 14 – Velocity vector plots on y–z planes P-1 and P-2 for different obstruction heights; (left) Re = 0.1, (middle) Re = 15, and (right) Re = 30.
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At Re = 30, two counter-rotating vortices start to form at the center of the plane at the peak positions and near the walls at the crest positions for h = 20 and 40 m. However, at h = 60 m, only one vortex formed at the peak position and the vortices disappeared when the height of the obstruction reached beyond 60 m. The strength of the transverse flow increased with increased obstruction height.
5.
Conclusion
A planar curved laminating passive micromixer that is capable of mixing at low Reynolds numbers was proposed in this work. Detailed study of the effects of geometry and placement of obstructions on mixing was carried out using Navier–Stokes analysis. The study was performed using a Reynolds number range from 0.1 to 60. First, the effect of the cross-sectional shape of the obstructions on mixing was investigated. Second, the mixing performances of the T-micromixer with obstructions and a simple curved micromixer without obstructions were compared to that of the proposed micromixer. Finally, the effects of the obstruction height and Reynolds number on mixing performance were evaluated. Micromixers with circular and hexagonal obstructions show almost the same mixing performances regardless of the Reynolds number, while micromixer with diamond obstructions shows far less mixing performance compared to the others except at Reynolds numbers greater than 50. The curved channel with circular obstructions shows much higher mixing performance than those of the T-micromixer with obstructions and a simple curved micromixer without obstructions throughout the Reynolds number range considered. The proposed micromixer with circular obstructions shows the best mixing index at the exit of the micromixer, 88% among the tested micromixers, at Reynolds numbers of 0.1 and greater than 15. The minimum mixing index, 72%, was seen at Re = 5. The pressure drop increased with an increased number of obstructions. However, this increase in pressure drop is not substantial, and can be an acceptable tradeoff for improved mixing. The mixing generally increased as the height and number of the obstructions increased. However, both of these effects disappeared as the Reynolds number approached 60, where the formation of secondary flows due to channel curvature dominated the other mechanisms due to obstructions. The proposed micromixer design is planar, and hence requires only simple fabrication.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant No. 2009-0083510 funded by the Korean government (MSIP) through Multi-phenomena CFD Engineering Research Center.
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