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# Parametric investigation on mixing in a micromixer with two-layer crossing channels

- Shakhawat Hossain
^{1}and - Kwang-Yong Kim
^{1}Email author

**Received:**25 February 2016**Accepted:**30 May 2016**Published:**21 June 2016

## Abstract

This work presents a parametric investigation on flow and mixing in a chaotic micromixer consisting of two-layer crossing channels proposed by Xia et al. (Lab Chip 5: 748–755, 2005). The flow and mixing performance were numerically analyzed using commercially available software ANSYS CFX-15.0, which solves the Navier–Stokes and mass conservation equations with a diffusion–convection model in a Reynolds number range from 0.2 to 40. A mixing index based on the variance of the mass fraction of the mixture was employed to evaluate the mixing performance of the micromixer. The flow structure in the channel was also investigated to identify the relationship with mixing performance. The mixing performance and pressure-drop were evaluated with two dimensionless geometric parameters, i.e., ratios of the sub-channel width to the main channel width and the channels depth to the main channel width. The results revealed that the mixing index at the exit of the micromixer increases with increase in the channel depth-to-width ratio, but decreases with increase in the sub-channel width to main channel width ratio. And, it was found that the mixing index could be increased up to 0.90 with variations of the geometric parameters at Re = 0.2, and the pressure drop was very sensitive to the geometric parameters.

## Keywords

- Chaotic micromixer
- Two-layer crossing channels
- Mixing index
- Navier–Stokes equations
- Reynolds number

## Background

The exponential demand for miniaturization in microfluidic applications highlights the significance of understanding the mechanism that controls mixing of fluid species at the microscale stage. The characteristic dimension of a microfluidic device is usually in a range from ten to several hundred micrometer, where the flow becomes laminar due to the low Reynolds number. Thus, the laminar behavior in the devices causes difficulty in mixing of fluids. Due to the low flow velocity, mixing primarily depends on the molecular diffusion between the fluids, which is very slow process. Diffusion mechanism is governed by the Fick’s law, where the mixing mass flux is proportional to the diffusion coefficient and concentration gradient. Microfluidic devises are widely applied for many chemical and biological practices, leading to the new ideas, such as bio-chip (Schwesinger et al. 1996), bio-MEMS (Linder 2001), microreactor (Hardt et al. 2005), and lab-on-a-chip (Erickson 2005). Rapid as well as efficient mixing is very important for almost all chemical and biological analyses. In order to fulfill the demand of efficient and rapid mixing in microchannels, a variety of passive and active micromixers have been developed so far (Nguyen and Wu 2005; Hessel et al. 2005). Active micromixers are coupled with external excitations by electroosmotic, dielectrophoresis, ultrasonic vibration, electrohydrodynamics, magnetic force, etc. (Hessel et al. 2005). Due to the additional parts for the excitations, the design and fabrication of the active micromixers are rather complex and expensive. However, passive micromixers do not use any exterior power source to induce disturbances. Thus, the passive micromixers can be easily incorporated in a composite microfluidic systems (Nguyen and Wu 2005; Hessel et al. 2005) due to the simplicity in their structures. In the lamination micromixer, mixing occurs by successive separation and rejoining of the fluid streams, which increase the interfacial area between the fluids (Gray et al. 1999). Recently, various techniques have been used to promote mixing in passive micromixers, for example, by introducing geometric adaptation in microchannels (Reyes et al. 2002; Manz et al. 1990; Weibel and Whitesides 2006).

Recently, computational fluid dynamics (CFD) has become very popular technique to analyze the mixing and fluid flow in micromixers. In recent years, many investigations have been performed to improve the mixing in a variety of passive micromixers; T-type micromixer (Gobby et al. 2001) and Y-type micromixer (Sahu et al. 2012), serpentine micromixers (Beebe et al. 2001; Hossain et al. 2009; Ansari and Kim 2009; Sahu et al. 2013), split-and-recombinination (SAR) micromixers (Lee and Lee 2008; Viktorov and Nimafar 2013; Nimafar et al. 2012a, b; Hossain and Kim 2014), patterned grooves micromixers (Ansari and Kim 2007; Hossain et al. 2010a, b), etc. It was exposed that shifting the inlet direction in a T-type micromixer did not appreciably improve the mixing performance (Gobby et al. 2001). In planner serpentine micromixer, sharp corner can produce center-rotating vortices at moderate Reynolds numbers (Re > 25), which promotes the agitating process to enhance the mixing performance (Beebe et al. 2001). Unluckily, at low Reynolds numbers, induced vortices in a micromixer decay sooner than they create an opportunity to considerably stir the mixing species.

A staggered herringbone grooves micromixers use a herringbone grooves pattern on the bottom of the channel, to induce lateral transports and thus to generate chaotic flows (Stroock et al. 2002a, b) to enhance the mixing performance. Hong et al. (2004) proposed a passive micromixer that employs the “Coanda effect” using two-dimensional Tesla structures to generate transverse dispersion. Tesla structure divides the fluid stream into the sub-streams, which recombine later. The splitting and recombining mechanism can create chaotic advection and improve the mixing, significantly. A parametric study of a modified Tesla structure was conducted by Hossain et al. (2010a, b) for a wide Reynolds number range from 0.05 to 40 with two geometric parameters. A staggered overlapping crisscross micromixer based on chaotic mixing principles was designed and fabricated by Wang and Yang (2006). Their numerical and experimental results show that the micromixer can generate chaotic flows to stretch and fold the fluid streams rapidly. Xia et al. (2005) proposed micromixers based on chaotic mixing principles, and performed numerical and experimental investigations of the micromixers. At very low Reynolds numbers, the proposed micromixers can manipulate the flow by splitting-and-recombining and stretching-and-folding which generate chaotic advection, and thus, significantly enhance the mixing. As the generation of chaotic advection does not depend on the inertial forces of fluids, the proposed micromixers worked well especially at low Reynolds number (Re = 0.2).

In the present work, a parametric study of the micromixer proposed by Xia et al. (2005) has been performed to systematically investigate the performance of the micromixer, which shows incredibly high mixing performance at very low Reynolds number. The sub-channel width, main channel width and channel depth were selected as the geometric parameters to be tested, and the mixing index was used as the performance parameter for mixing. The mixing index was evaluated using three-dimensional Navier–Stokes and mass conservation equations with a diffusion–convection model.

## Micromixer model

_{1}+ d

_{2}, where d

_{1}and d

_{2}are the depths of the bottom and top channels, respectively. The reference values of Wc and D, are same as 0.3 mm, and the number of rhombic units is eleven. The symbols Lc and Ls indicate the lengths of connecting channels shown in Fig. 1.

Geometric parameters and their ranges

Design variable | Lower limit | Upper limit | Reference value (Xia et al. 2005) |
---|---|---|---|

D/H | 0.18 | 0.37 | 0.28 |

Wc/H | 0.18 | 0.33 | 0.28 |

## Numerical formulation

**ν**correspond to the velocity, density, and kinematic viscosity of the fluid mixture, respectively. To examine the mixing mechanism, water at 25 °C and a water-dye solution were used as the working fluids. For each fluid component with constant viscosity and density, the mass transport equation of advection–diffusion type (Bird et al. 1960), is formulated as follows;

*C*

_{ i }specify the diffusivity coefficient and concentration of the fluid component, respectively. For modeling of diffusive mixing, the scalar transport equation was used by many researchers for different micromixers (Chung et al. 2008; Cortes-Quiroz et al. 2010; Hinsmann et al. 2001; Afzal and Kim 2015).

To solve the above equations, the following boundary conditions were considered. Pure water at 25 °C was introduced at the Inlet 1 and the water-dye solution enters at Inlet 2. Constant velocity was specified at the inlets, while zero static pressure was assigned at the outlet. No-slip condition was employed at the walls. Physical properties of the water used in this work were same as those in the previous work (Xia et al. 2005). Diffusivity coefficient of the water-dye mixture was 1 × 10^{−11} m^{2}/s. And, the density (*ρ*) and dynamic viscosity (*µ*) of water were 997 kg/m^{3} and 8.8 × 10–3 kg/m-s, respectively (Kirby 2010).

To conduct the numerical analysis, an unstructured tetrahedral grid system was created using ANSYS ICEM 15.0. The numerical diffusion error is generally induced due to discretization of convection terms in the Navier–Stokes equations. Higher-order upwind schemes can be used to minimize the numerical diffusion (Hardt and Schöndfeld 2003). To discretize the advection terms of the governing equations, a high-resolution scheme of second-order approximation was utilized in this work. By the aid of an automatic correction algorithm (Ansari and Kim 2007), the high-resolution scheme minimized the numerical discretization errors. The measurement for the convergence was the root mean square (RMS) residual value of 10^{−7}.

*N*is the number of sampling points within the plane,

*c*

_{ m }and

*c*

_{ i }signify the optimal mass fraction and mass fraction at sampling point

*i*, respectively. At any cross-sectional plane, optimal mass fraction (

*c*

_{ m }) is 0.5. To investigate the mixing performance of the micromixer quantitatively, the mixing index at a specific cross-sectional plane is defined using the variance (Kockmann et al. 2003) as:

## Results and discussion

^{5}to 1.44 × 10

^{6}as shown in Fig. 2. The test was performed at Re = 40 for the reference design. The mixing index was computed along the length of the micromixer. From the test results, the grid system with 1.21 × 10

^{6}nodes was selected for further analysis. Present computational results were compared with the previous results (Xia et al. 2005) quantitatively and qualitatively as shown in Fig. 3. Figure 3a represents the distributions of the variance of the mass fraction along the channel length at Reynolds number 0.2. The variance of the mass fraction varies from 0 (for complete mixing) to 0.5 (for incomplete mixing). The variance decreases gradually along the channel in both cases, and the present numerical results show good agreements with the previous numerical results. In Fig. 3b, c, distributions of the dye mass fraction are presented. In the present study, the x–y planes (Fig. 3b) are taken at the middle depths of the top and bottom channels, whereas the previous work (Xia et al. 2005) did not described the exact locations of the planes. Distributions of the dye mass fraction on the y–z planes have been compared along the channel length as shown in Fig. 3c. Qualitatively, both the figures (Fig. 3b and c) represent good agreements between these two numerical results.

Comparison of present results on mixing index with those of previous studies

Sr. no | Investigators | Geometrical shapes | Design parameter | Hydraulic Dia. (mm) | Reynolds number | Mixing length (mm) | Mixing index (Mo) variations in the tested range |
---|---|---|---|---|---|---|---|

1 | Ansari and Kim (2009) | 3D serpentine | straight channel length to the channel width (s/d) | 0.15 | 1.0 | 12 | 0.23–0.25 |

2 | Hossain and Kim (2014) | Unbalanced three split and recombine | rhombic angle (θ) | 0.15 | 0.1 | 4.0 | 0.23–0.31 |

3 | Hossain and Kim (2015) | 3D serpentine split and recombine | sub-channel width to the main channel width (w/W) | 0.13 | 0.1 | 2.1 | 0.42–0.43 |

4 | Alam and Kim (2012) | Curved microchannel with rectangular grooves | groove depth to the main channel width (d/W) | 0.1 | 0.5 | 2.66 | 0.20–0.21 |

5 | Present study | Two-layer crossing microchannels | channels depth to the main channel width (D/H) | 0.2 | 0.2 | 7.2 | 0.89–0.99 |

^{6}. The mixing index increases as Pe increases up to 4.0 × 10

^{6}, but further increase in Pe does not change the mixing index. This indicates that the effect of molecular diffusion on the mixing of fluids in this micromixer is negligible for Péclet numbers larger than 4.0 × 10

^{6}.

## Conclusions

This work presents a parametric investigation on flow structure and mixing in a micromixer with two-layer crossing channels which was reported by Xia et al. (2005). The flow and mixing performance were numerically analyzed using ANSYS-CFX inbuilt diffusion–convection model. Present numerical results for mixing agree quantitatively and qualitatively well with the previous numerical results. The mixing index along with pressure-drop have been examined in terms of two geometric parameters related to sub-channel width and depth, i.e., Wc/H and D/H, respectively, at various Reynolds numbers in a range from 0.2 to 40. The mixing index at the exit of the micromixer increases with the increase in D/H, and becomes more sensitive to D/H as Reynolds number decreases. At Re = 40, the effect of D/H on the mixing index is almost negligible, while at Re = 0.2 with the increase in D/H from 0.18 to 0.37, the mixing index at the exit of the micromixer increases from 0.89 to 0.99, which is quite good achievement in mixing at this low Reynolds number. A quantitative comparison among the mixing indexes at low Reynolds numbers in various micromixers indicates that the present micromixer produces the extraordinarily high mixing index at Re = 0.2 compared to the previous micromixers. Velocity vectors at Re = 0.2 show that increase in D/H strengthens the ‘‘saddle-shaped’’ flow pattern. The mixing index at the exit of the micromixer generally decreases with increase in Wc/H, and the decreasing rate of the mixing index with Wc/H becomes larger as D/H and Reynolds number decrease. At Re = 0.2, the maximum relative reductions in pressure drop are about 70 and 75 %, respectively, with the increase in Wc/H and D/H in their tested ranges. This study is expected to provide preliminary input for the design of an optimum micromixer.

## Declarations

### Authors’ contributions

Shakhawat Hossain performed mainly the calculation and analysis of the results, and Kwang-Yong Kim contributed to the idea of this research, discussion of the results, and writing. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Funding

This work was supported by Inha University research grant.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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