# Mixing Performance of a Cross-Channel Split-and-Recombine Micro-Mixer Combined with Mixing Cell

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Cross-Channel Split-and-Recombine (CC-SAR) Micromixer

^{3}, D = 10

^{−10}m

^{2}s

^{−1}and ν = 10

^{−6}m

^{2}s

^{−1}, respectively. This diffusion constant is a typical value for small proteins in an aqueous solution. The Schmidt (Sc) number is 10

^{4}(the ratio of the kinetic viscosity and the mass diffusion of fluid). The Reynolds number is defined as $Re=\frac{\rho {U}_{mean}{d}_{h}}{\mu}$, where $\rho ,{U}_{mean},{d}_{h}\text{}\mathrm{and}\text{}\mu $ denote the density, the mean velocity at the outlet, the hydraulic diameter of main channel, and the viscosity of the fluid, respectively.

## 3. Governing Equations and Computational Procedure

_{ι}and n are mass fraction in the i

^{th}cell and total number of cells, respectively. ξ is specified as 0.5 which indicates equal mixing of the two solutions.

## 4. Validation of the Numerical Study

_{s}, r, and p are the safety factor of the method, grid refinement ratio, and the order of accuracy of the numerical method, respectively. f

_{coarse}and f

_{fine}are the numerical results obtained with a coarse grid and fine grid, respectively. F

_{s}was specified as 1.25 according to the suggestion by Roache [34]. We analyzed the three numerical solutions obtained with the edge size of 4 μm, 5 μm, and 6 μm. The corresponding number of nodes are 4.76 × 10

^{6}, 2.54 × 10

^{6}, and 1.24 × 10

^{6}, respectively. The computed GCI were reduced from 2.4% to 0.94%. Therefore, the edge size of 5 μm is confirmed as small enough to obtain numerical solutions with reasonable accuracy.

## 5. Results and Discussion

_{i}in the plot), and its value means a percentage of DOM with respect to the total amount of DOM. For low Reynolds number flows such as Re = 0.5 and 2, the first and second mixing units perform better than the remaining units. However, the mixing happens more vigorously in the second and third mixing units for high Reynolds number flows such as Re = 20 and 50. One thing to note is that the fourth mixing unit contributes nothing for the Reynolds number of 50. This suggests that the number of mixing units can be optimized in terms of DOM.

_{i}indicates the DOM increment in either the SAR or the mixing cell. Figure 7 confirms that the mixing cell is an excellent mixing device over a wide range of the Reynolds number. For the Reynolds numbers Re = 0.5, 2, and 20, in particular, the mixing cell contributes to mixing more than the SAR. This explains why the present micro-mixer performs better than other SAR based micro-mixers as shown in Figure 5. The SAR becomes dominant for the Reynolds number larger than about 50.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | width of split block of SAR (μm) |

D | diffusion coefficient (m^{2}/s) |

DOM | degree of mixing |

f_{coarse} | numerical solution obtained with coarse grid |

f_{fine} | numerical solution obtained with fine grid |

GCI | grid convergence index |

H | height of baffle (μm) |

n | total number of sampling points at a specific plane |

MI | mixing index |

MEC | mixing energy cost |

$\Delta p$ | pressure load (Pa) |

r | grid refinement ratio |

Re | Reynolds number |

SAR | split and recombine |

U_{mean} | average velocity at the outlet (m/s) |

x, y, z | Cartesian coordinates |

Greek symbols | |

ε | relative error |

μ | fluid viscosity (kg/(ms)) |

ν | fluid kinematic viscosity (m^{2}/s) |

φ | mass fraction |

ρ | fluid density (kg/m^{3}) |

σ | standard deviation |

Subscripts | |

i | sampling point |

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**Figure 2.**Diagram of the SAR examined by Sheu et al. [26].

**Figure 5.**Variation of the degree of mixing (DOM) (

**a**) and pressure load (

**b**) with the volume flow rate.

**Figure 7.**DOM obtained in SAR and mixing cell of each mixing unit: Re = 0.5 (

**a**) Re = 2 (

**b**) Re = 20 (

**c**) Re = 50 (

**d**).

**Figure 11.**Projected streamlines and contours of the concentration distribution on the plane of the channel half.

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**MDPI and ACS Style**

Juraeva, M.; Kang, D.J.
Mixing Performance of a Cross-Channel Split-and-Recombine Micro-Mixer Combined with Mixing Cell. *Micromachines* **2020**, *11*, 685.
https://doi.org/10.3390/mi11070685

**AMA Style**

Juraeva M, Kang DJ.
Mixing Performance of a Cross-Channel Split-and-Recombine Micro-Mixer Combined with Mixing Cell. *Micromachines*. 2020; 11(7):685.
https://doi.org/10.3390/mi11070685

**Chicago/Turabian Style**

Juraeva, Makhsuda, and Dong Jin Kang.
2020. "Mixing Performance of a Cross-Channel Split-and-Recombine Micro-Mixer Combined with Mixing Cell" *Micromachines* 11, no. 7: 685.
https://doi.org/10.3390/mi11070685