Parametric study of the Crossing elongation effect on the mixing performances using short Two-Layer Crossing Channels Micromixer (TLCCM) geometry

Journal Pre-proof Parametric study of the crossing elongation effect on the mixing performances using short Two-Layer Crossing Channels Micromixer (TLCCM) geometry Kouadri Amar, Douroum Embarek, Khelladi Sofiane

PII:

S0263-8762(20)30104-0

DOI:

https://doi.org/10.1016/j.cherd.2020.03.010

Reference:

CHERD 4030

To appear in:

Chemical Engineering Research and Design

Received Date:

26 November 2019

Revised Date:

2 March 2020

Accepted Date:

8 March 2020

Please cite this article as: Amar K, Embarek D, Sofiane K, Parametric study of the crossing elongation effect on the mixing performances using short Two-Layer Crossing Channels Micromixer (TLCCM) geometry, Chemical Engineering Research and Design (2020), doi: https://doi.org/10.1016/j.cherd.2020.03.010

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Parametric study of the crossing elongation effect on the mixing performances using short Two-Layer Crossing Channels Micromixer (TLCCM) geometry Amar Kouadri a, Embarek Douroum b, Sofiane Khelladi c a

LDMM laboratory, Djelfa university, Djelfa 17000, Algeria

b

LMSR laboratory, Sidi Bel Abbes university, Sidi Bel Abbes 22000, Algeria

c

Arts et Metiers Institute of Technology, CNAM, LIFSE, HESAM University, F-75013 Paris, France

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Highlights

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Graphical abstract

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 Numerical investigations of a short TLCCM micromixer with an elongation of the crossing zone were performed.  The chosen micromixer showed an excellent mixing index which exceeds 85.67% for Re = 0.2 and reaches 99.22% from Re = 50.  Our micromixer has a lower pressure drop compared to other geometries studied recently.

ABSTRACT In this study we investigate the mixing performances of a modified micromixer which achieves a very good mixing quality that can be compared to other micromixers proposed recently, our idea proposes a modification of the crossing zone to reduce the unit number. The numerical simulations have been carried out at low Reynolds

numbers using the CFD Fluent code to solve the 3D momentum equations, continuity equation, and the species transport equations. The elongation of the crossing zone is defined by a parameter called aspect ratio (l/W). A parametric study was realized using five values of aspect ratio (l/W) from 0 to 1 in a wide range of Reynolds numbers: from 0.2 to 80. To analyze the obtained results through the numerical simulations, the mass fraction contours, velocity vectors, velocity profiles, and pressure losses were presented in different cross-sectional planes and positions. The selected geometry (with l/W = 1) has excellent mixing performances where the obtained mixing index exceeds 85.67% for Re = 0.2 and reaches 99.22% for Re = 50, it also has a lower pressure drop compared to other geometries studied recently. Therefore, the selected micromixer shows high mixing performances at low Reynolds numbers, so it can be employed to improve fluid mixing in various microfluidic systems. Keywords : Micromixer, TLCCM, CFD, low Reynolds numbers, chaotic advection, mixing index, mass fraction, pressure drops. 1. Introduction

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Micromixers are widely used in various industrial applications, they represent the important components in microfluidic systems and have many uses in the bioengineering fields, biomedical applications and chemical engineering [1]. Mixing in laminar flow and at low Reynolds numbers is commonly used in different fields [2], for example chemical synthesis, chemical reactions, extraction and purification, emulsion processes,

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polymerization, DNA analysis, detection and analysis of chemical or biochemical content.

Micromixers are generally classified into two categories: active and passive micromixers [1–3]. Active micromixers require external energy for the mixing process to enhance the mixing efficiency (dielectrophoretic,

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electrowetting shaking, magneto-hydrodynamic, and ultrasound disturbance) [28]. These types of mixers are more efficient. However, they are more difficult to integrate and expensive than passive types and require more detailed fabrication while mixing in passive micromixers is a consequence of the interaction between flow and

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channel geometry. Passive micromixers have particular importance according to their simple structures and easy manufacturing [26].

The use of chaotic advection which generates secondary flows is an appropriate way to increase mixing

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efficiency, in the case of Newtonian fluids with stationary flows in laminar regime [4], the planar micromixer geometries do not lead to obtaining a homogeneous mixing and consequently, the mass transfer will be ineffective [6, 7, 10], while the use of the chaotic micromixer geometries promote the mass transfer which

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allows improving the mixing performance in a significant way [16, 20, 24]. Many studies have been carried out on the mixing of Newtonian fluids in various types of micromixers. In passive micromixers where the walls are immobile, the mixing of the flowing fluids can be ensured by the key

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phenomenon which is the chaotic advection. The channels geometry makes it possible to disturb the laminar velocity field which generates the compression, stretching and folding of the fluid layers, this deformation leads to a more homogeneous and rapid mixing [4]. Several models of planar micromixers have been designed and studied previously and recently through numerical and experimental works, Soleymani et al. [5] studied the development of vortices in a T-type micromixers where the Reynolds number values are taken between 12 and 240, they examined the effect of the angle junction, the aspect ratio, and the throttling on the mixing efficiency, their results reveal that the mixing quality is improved by the appearance of the vortices which are generated by the increase of the flow rate and also by the geometrical parameters. Different micromixer geometry: curved channel; square-wave channel; and

zig-zag channel, were studied numerically using a CFD code by Houssain et al. [6], they evaluated the mixing efficiency in the Reynolds number range (from 0.267 to 267) and found that the square-wave micromixer gives the best mixing quality while the curved channel shows a low-pressure drop. The mixing efficiency has been analyzed by Ansari et al. [7] for planar split and recombines micromixers with rhombic and circular subchannels where the Reynolds numbers are ranging from 1 to 80. They found that the unbalanced geometries have higher mixing quality compared to the balanced geometries, in particular, the geometries of circular subchannels where the pressure-drop is also low. A planar micromixer with curved C-shaped channel and radial baffles which generate multidirectional vortices has been examined by Tsai et al. [8], the range of Reynolds number is from 0.054 to 81. Their results indicate that the mixing performance of the micromixer with the first baffle related to the internal cylinder and the second related to the external cylinder is better than that of the micromixer with the reverse arrangement of baffles, while the pressure drops are quite strong. A numerical and experimental study on eight planar micromixers have been carried out by Cheri et al. [9] at a range of Reynolds

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number between 0.1 and 40. The results show that the best mixing index is obtained with the micromixer characterized by a round corner rectangular chamber and a straight shape obstacle where the ratio between the mixing index and the pressure drop is maximal. Solehati et al. [10] have realized a numerical study to analyze the mixing performance of microchannel with wavy structure compared to the basic straight microchannel using various Reynolds numbers ranging from 1 to 200. They obtained that microchannel with wavy structure provides

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higher mixing performance than that of basic microchannel due to the occurrence of chaotic flow created by wavy curves structure. A planar modified zigzag micromixer has been numerically studied by Chen and Li [11]

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at a wide range of Reynolds numbers (0.1-100), they found that the inverse flow generated by the zigzag modification allows to enhance significantly the mixing index compared to the zigzag micromixer, however the pressure drop in the modified zigzag micromixer is higher than the zigzag micromixer. A planar micromixer

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with spiral microchannel has been proposed by Vatankhah and Shamloo [12], who conducted a numerical parametric study which investigates the distance center to center of successive channels. They found that decreasing this parameter allows to enhance the mixing index. Borgohain et al. [13] have studied numerically

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using CFD code the effect of the placement of obstacles designed as thin curved ribs on the mixing performance of a cross T-shaped micromixer. Their results show that the inclusion of curved obstacles along the channel provides better geometry that improves the mixing efficiency in the microchannel. The need of generating chaotic advection with average flow regimes has been attracted the interest of several

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researchers with numerical and experimental works to design adequate geometries that can generate threedirectional flows. Among the first works, we indicate the work of Liu et al. [14], they have conducted a

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numerical and experimental study by evaluating the mixing efficiency in the three-dimensional micromixer, the C-shaped micromixer and the straight channel, for a Reynolds range between 6 and 70, they found that the threedimensional micromixer has a mixing index clearly superior to that of the two other micromixers. Beebe et al. [15] carried out a qualitative analysis of the mixing performance of a three-dimensional micromixer by both experimental and numerical study at a Reynolds number range of 1 to 20. The designed geometry is of repetitive "L" shaped units. Their results show that the mixing efficiency of the three-dimensional micromixer is clearly superior to that of the square-wave micromixer. A parametric study of the mixing in a chaotic micromixer consisting of repeating "L" shaped patterns was performed numerically by Ansari et al. [16]. Reynolds numbers rise from 1 to 70. Both the mixing index and the pressure loss values have been limited in optimal intervals of

Reynolds numbers where the mixing performance is significantly better with a low pressure loss. Nimafar et al. [17] carried out a comparative experimental study of three micromixers H, O and T-micromixer at low Reynolds number (0.08-4.16), they found that the H-micromixer has the best mixing performance compared to the two others. Another comparative numerical investigation has been performed by Alam and Kim [18] to evaluate the mixing performance of four micromixers characterized by circular mixing chambers interconnected by various constriction channels at Reynolds numbers change from 0.1 to 100. They obtained that their proposed micromixer with crossing constriction channels reveals excellent mixing performance at low Reynolds numbers less than 50 with lower pressure drop than the three others. An experimental and numerical investigation has been carried out by The et al. [19] to evaluate the mixing efficiency of 3D micromixer at various Reynolds number from 0.5 to 100, the highest mixing index value obtained in this range was 95% with slightly high pressure drop. Five 3D serpentine micromixers were numerically examined by Lin [20] for a large interval of Reynolds numbers (8 to160), the chaotic advection mechanism as rotation, continuous rotation and 3D stretching

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was identified to know their effect on the mixing performance as well the effect of the addition of grooves and their positions, however, the pressure drop of all micromixers was similar and significantly high. The effects of splitting, recombination and chaotic advection mechanisms were performed by Chen and Shen [21] using two types of micromixers with repeating E-shaped units, their results show that the mixing index has achieved 96% but the pressure drop has quickly increased. Numerical simulations were realized by Ruijin et al. [22] to compare

numbers the Baker micromixer has highest mixing index.

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the mixing efficiency of three micromixers: Baker and F-shaped micromixers, they found that at low Reynolds

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In order to enhance the chaotic advection that improves the mixing quality, another type of micromixers that is characterized by two fluid layers has been examined in very recent works. We note that the first work in this field has been realized by Xia et al. [23], who have studied numerically and experimentally an excellent

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micromixer composed of two-layer crossing channels, the chaotic advection created in this micromixer allows to have a very high mixing index which reaches 96% at low Reynolds numbers. The dominance of this micromixer has been proven compared with other types. Numerical analyses were performed by Hossain and Kim [24] to

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evaluate the mixing performances of a micromixer composed with repeating OH-shaped units in a range of Reynolds number from 0.1 to 120. The studied micromixer reached 88.4% at Re = 30. After that, they carried out a parametric numerical study [25] using the geometry proposed by Xia et al. [23] in a Reynolds number interval change from 0.2 to 40. The results revealed that the mixing index and the pressure drop were

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significantly influenced by the geometric parameters. Experimental and numerical investigations have been carried out by Hossain et al. [26] where the Reynolds number range was (0.2 to 120), high mixing index values

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were obtained: 96% - 99% with lower pressure drop than TLCCM of Xia et al. [23]. Recently, Raza et al. [27] have performed a numerical study to investigate the mixing performance of a short micromixer with repeating OX-shaped unites in the Reynolds number range: 0.1-200, they also studied the effect of the geometric parameters on the mixing performances. Their results revealed that at least 87% of mixing has obtained, while, the pressure drop increased rapidly. Motivated by the work of Hossain et al. [26], our goal is to design a geometry that offers excellent mixing performances with low pressure losses using a short length (the half of that of Hossain et al. [26]). The mixing performances were examined using three dimensional Navier-Stokes equations over Reynolds number interval

of 0.2-80. The effect of the geometrical parameter which is the aspect ratio on the mixing efficiency has been investigated, than the obtained results of the pressure losses were compared with that of recent works.

2. Micromixer designs The basic geometric model is that of Hossain et al. [26]. Our idea is to exploit the elongation of the crossing zone which is defined by an aspect ratio (l/W), where l/W = 0 represents the reference case [26]. The current proposed geometry is composed of two superimposed layers, with two shifted inlets, the inlet 1 is linked to the top layer while the inlet 2 is linked to the bottom layer. Fig. 1 shows a detailed description of the studied micromixer geometry, such as: Fig. 1(a) represents the elongation pattern at the crossing channels of the first period for four tested aspect ratios considered in this study, so the lengthening of the crossing depends on the aspect ratio l/W which takes the values: 0.25, 0.5, 0.75, and 1.

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Whereas, Fig. 1(b) shows a 3D perspective view of the micromixer which corresponds to the case 1/W = 1 as an example. A top view of the geometry is shown in 2D which shows the two inlets and the outlet as well as the four crossings with their plans: crossing inlet, crossing center and crossing outlet. The different dimensions of the current proposed geometry are illustrated in figures. 1(b) and (c), such as the width W, the height of a layer d, the width of the geometry H, the vertical channel length b, the pitch S, and (b+S) the length of one period are

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kept constants, where their values are respectively: 300 𝜇𝑚, 150 𝜇𝑚, 1070 𝜇𝑚, 150 𝜇𝑚, 640 𝜇𝑚, 790 𝜇𝑚. The sectional dimensions of the inlets and outlet are 150 × 300 𝜇𝑚 and 300 × 300 𝜇𝑚, respectively, the hydraulic diameter is defined by 𝐷ℎ =

2𝑑∙𝑊 𝑑+𝑊

. The total length of the geometry with four units is equal to

𝑙/W = 0.25

𝑙/W = 0.5

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𝑙/W = 0

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3750 𝜇𝑚.

(a)

𝑙/W = 0.75

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(b)

(c)

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3. Numerical analysis

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Fig. 1. The micromixer geometry model: (a) Different geometrical aspect ratios, (b) Proposed micromixer with l/W=1, (c) schematic information of studied micromixer.

3. 1. Governing equations

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In this study, the flow and mixing simulations were performed using Computational Fluids Dynamics (CFD) code ANSYS Fluent 16.0. The governing equations of 3D steady and incompressible flows are the continuity equation (Eq. (1)) and the momentum equation (Eq. (2)) which can be expressed in the following form: ⃗ =0 ∇V 1 ⃗ ∇)V ⃗ = − ∇P + ν∇2 ⃗V (V ρ

(1) (2)

where V, ρ and P represent the velocity, fluid density and static pressure respectively. The species transport equation (Eq. (3)) for a fluid with constant density is a convection-diffusion equation given by:

⃗ ∇ Ci = D∇2 Ci V

(3)

where D is the diffusion coefficient and Ci the concentration. The boundary conditions were a no-slip velocity condition applied on the wall surfaces, a constant uniform velocity imposed at the inlets and zero static pressure at the outlet, In addition, the mass fraction at the inlet 1 equals 0 and equals 1 at the inlet 2. The SIMPLEC (SIMPLE-Consistent) algorithm was used to solve the pressure-velocity coupled equations. In this study, a high resolution second-order upwind scheme was used for the discretization of the convection terms in the momentum equations as well as the species transport equation. To ensure maximum accuracy of the numerical solutions, the governing equations were solved iteratively, the convergence criterion was a root mean square (RMS) residual values for all variables becomes less than 10−6 . In this work, pure water and dye-water were used to simulate the Newtonian fluid flows. The density of water

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used in this work was 1000 kg/m3. The diffusion coefficient of the water was assumed to be 1×10-11 m2/s. Also, by this hypothesis, the mixing is governed mainly by the chaotic advection which is the most dominant mechanism. The Reynolds number is defined as: 𝑅𝑒 =

𝜌𝑉𝐷ℎ 𝜇

(4)

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To quantify and compare the mixing performance of the micromixer, the mixing index MI in each cross-

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sectional plane perpendicular to the flow direction is calculated as follows: σ MI = 1 − σ0

(5)

where σ represents the standard deviation of mass fraction at cross-section defined as: N

1 ∑(ci − c̅)2 N

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σ2 =

(6)

i=1

with N denoting the number of sampling points inside transversal cross-section, ci is the mass fraction at

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sampling point i, and c̅ is the optimal mixing mass fraction of ci. σ20 = c̅(1 − c̅)

(7)

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where σ0 is the maximum standard deviation over the range of data. The value of the standard deviation was taken to be maximum for unmixed fluids (σ = σ0 ), with a mixing index of MI = 0 and minimum for completely

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mixed fluids, with a mixing index of MI = 1. A higher MI value indicates a more homogeneous concentration and a better mixing performance. 4. Results and discussion 4. 1. CFD code validation In order to validate our obtained results by the CFD code, a quantitative comparison has been realized with a recent work of Raza et al. [27], where the mixing index values at the exit plane of the micromixer at different Reynolds number were compared as shown in Fig. 2, it is clear from this figure that our results are very comparative and coincide with those of Raza et al. [27], where the values of the mixing index have the same evolution over a wide range of Reynolds number.

To clarify this comparison, the relative error between our results and those of Raza et al. [27] has been calculated and illustrated in Table 1, this error begins with values in the order of 10% in the diffusion regime in the range (Re = 0.2 ~1) then becomes almost negligible in the convection regime (Re > 1). 1.0

Mixing index (MI)

Raza et al., [27] Present study

0.9

0.8

0.6 0

20

40

60

80

Re Fig. 2. Quantitative validation

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0.7

100

120

MI (Present study) 0.78462038

1

0.875523

0.78691506

10.12057250

10

0.95288

0.896372684

5.93016077

20

0.959056

0.93100916

2.92442151

30

0.983036

0.968076446

1.52177072

40

0.995138

0.979387198

1.58277566

50

0.995379

0.986486769

0.89335128

60

0.998579

0.983267795

1.5332993

70

0.998811

0.990143794

0.86775234

0.999053

0.99186951

0.71902994

0.999526

0.995846213

0.36815322

0.999526

0.995496569

0.40313415

100

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ERROR (%) 17.12162186

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120

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80

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0.1

MI (Raza et al. [27]) 0.946713

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Table 1. Comparative mixing index errors with Raza et al. [27].

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4. 2. Grid independency test It is important to choose an adequate mesh in our calculations, an unstructured mesh has been generated with tetrahedral elements (see figure 3 (a)), this choice is related to the complex shape of the current proposed geometry which presents sharp angles. In order to study the sensitivity of the results to the mesh, several meshes were tested for the current proposed geometry of l/W = 0.5 and with Reynolds number Re = 30. For that, four meshes were conceived with an increasing number of cells using the following values: 222917, 429113, 643203, and 1083357 cells as shown in Fig. 3 (b).

(a) 222 917 Cells 429 113 Cells 643 203 Cells 1 083 357 Cells

0.6

0.4

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Mixing index (MI)

0.8

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1.0

0.0 2

4

6 x/W

8

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12

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0.2

(b)

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Fig. 3. Sensibility of the results to the mesh, (a) Capture of the tetrahedral grid, (b) Evolutions of mixing index along the mixing channels. From the figure 3 (b), we see that there is a remarkable difference between the three first meshes which becomes very weak between the two last meshes. The relative errors of the mixing index in different planes between the

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last mesh and the others have been calculated, where the error between the third and fourth mesh is 0.94587% at the exit which considered negligible, so through the results of the sensitivity test, a grid with 643203 cells

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corresponds to a cell of 16 μm in size is sufficiently refined to find accurate results with a reduced computation time.

4. 3. The mixing performance of studied micromixers In order to choose a suitable geometry with a half-length of that of Hossain et al. [26] , which consists of four units. Four aspect ratios l/W = 0.25; 0.5; 0.75 and 1 were considered, the obtained results through the numerical simulations are presented in terms of the mixing index and the mass fraction in the exit plane, Fig. 4 (a) shows the evolution of the mixing index in the exit plane on the Reynolds range (0.2-80) for the considered geometries.

Fig. 4 (a) shows clearly the superiority of the geometry characterized by aspect ratio of 1/W = 1 over the whole range of the Reynolds numbers, where the mixing index starts with a value close to 85% at very low Reynolds numbers (0.2-5), then reaches 90% from Re = 20, and continues its progression until to 98% at Reynolds numbers more than 40. The results obtained previously (Fig. 4 (a)) are supported by the mass fraction distribution at the exit plane for the four geometries (see Fig. 4 (b)). It’s clear that the mixing quality depends to the Reynolds number and essentially to the aspect ratio l/W, at low Reynolds numbers (0.2 -5), all the geometries have the same mixing quality, and from Re = 15, The mixing quality enhanced with increasing elongation until to l/W = 1 where the mixing quality is excellent, this is explained by the homogeneous distribution of the mass fraction, so the aspect ratio of l/W = 1 is sufficient to achieve excellent mixing performance with less losses, which prevents us for

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don't exceeding the value of l/W = 1.

1.0

l/W=0.25 l/W=0.5 l/W=0.75 l/W=1

0.8

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0.7 0.6 0.5 0.4 20

40

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0

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Mixing index (MI)

0.9

60

80

Re

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(b) Fig. 4. (a) Variation of the mixing index at the outlet with Reynolds numbers for different aspect ratio, (b) Mass fraction distribution at the outlet for different aspect ratio.

Numerical simulations were carried out for three geometries: our chosen geometry (l/W = 1), the geometry of l/W = 0 with four units and the geometry of l/W = 0 with nine units. We indicate that the first two geometries of four units have the same length which represents the half of that of nine units. Fig. 5 illustrates the evolution of the mixing index for all geometries in different Reynolds numbers. On the Reynolds number range (0.2 to 20), the geometry with l/W = 0 of nine units has slightly higher mixing index values than the current proposed geometry with a difference of 5 to 10%, but from Re = 30, the current proposed geometry is more better, so an elongation of l/W = 1 and with only four units we can arrive to have an excellent mixing quality compared to the previously studied geometries.

0.9

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Mixing index (MI)

1.0

0.8

l/W=0 (Four units) l/W=1 (Four units) l/W=0 (Nine units)

0.6 10

20

30

40 Re

50

60

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0.7

70

80

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Fig. 5. Comparison between our chosen geometry (l/W=1) and the geometries used by Hossain et al. [26] (l/W=0).

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4. 4. The transition from diffusion to convection regime

It’s obvious to distinguish two flow regimes in the laminar regime, these regimes are the diffusion regime for very low Reynolds numbers and the convection regime for Reynolds little high. To clarify the transition between

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the two regimes in the proposed Reynolds number range, the results obtained were represented by the mass fraction contours in the different cross sectional planes P 1-P5 (see Fig. 1(c)). From Fig. 6, the flow behavior

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presented by the mass fraction contours shows that the fluid layers for Re = 0.2 and Re = 5 evolve in an identical manner along the crossing planes, and from the third crossing plane the quality of the mixing begins to develop which is interpreted by the phenomenon of molecular diffusion. Whereas from the Re = 15, the fluid layers undergo deformations translated by the rotation, the compression and the stretching, this phenomenon represents well the chaotic advection. To distinguish the diffusion and convection mechanisms, it is important to represent the mass fraction distribution of the three planes of each crossing (Crossing inlet, Crossing center and Crossing outlet). For Re = 0.2 (see Fig. 7(a)) the behavior of the fluid layers in each cross undergo relatively a small evolution in terms of deformation where the mixing has been improved by molecular diffusion. And according to the Fig. 7(b), it can be seen that from the mass fraction distribution at each crossing plane, the positioning of the

fluid layers undergoes a remarkable deformation, especially between the crossing inlet and its center where the fluids make a rotation of 70 ° then a rotation of 50 ° in the outlet plane of the first crossing. Thus in the second crossing the fluids have the same behavior and continue their rotations with the similar angles, but we can see precisely an enlargement of the separation line which becomes a mixing zone in the form of two attached cells which develops in both planes, the crossing center and outlet planes such that these cells detach and increase the mixing zone and continue to occupy an important zone especially in the outlet plane of the second crossing, this development translates clearly the sudden increase in the mixing index from the outlet of the second crossing. Overall, the two fluids make three rotations between the first and the fourth crossing, such as in two successive crossings and precisely in similar planes the fluids undergo a rotation of 2π, which significantly improves the mixing quality at the same time. Using this geometrical configuration (l/W = 1), the elongation of the crossing has provided a sufficient and optimal contact zone which makes it possible to give a lot of rotations of the fluid

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layers, which are interpreted by the appearance of the chaotic advection which improves the mixing quality.

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Fig. 6. Mass fraction distribution at cross-sectional planes P1-P5 for different Reynolds numbers with l/W=1

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(a)

(b)

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Fig. 7. Mass fraction distribution at different cross-sectional planes with l/W=1: (a) Re = 0.2 (b) Re = 15.

4. 5. Hydrodynamic behavior and flow structure at cross-sectional planes Fig. 8 shows the flow structure in our chosen micromixer (l/W=1), the velocity vectors are plotted in the four

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crossings. At very low Reynolds number (Re = 0.2), the flow structure in the three crossing planes is similar for the four crossings, at the crossing inlet, the two fluids enter through the diagonally symmetrical corners, than, it can be seen in the crossing center planes that along the crossing the flow becomes unidirectional where there is no secondary flow, after that, the fluids quit through the other two corners (Fig. 8-a). With Re = 15(Fig. 8-b), the

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flow structure is changed importantly, this is clearly shown in the crossing inlet plane where we find two opposing flows up and down which develops by creating a vortex in the crossing center plane which revolves

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around the geometry axis. Subsequently, the fluids quit through the two diagonal corners in the same manner as the case Re = 0.2, this structure shows the rotational movement that it undergoes the two fluids during their passage through the elongation. Whereas, the mixing at low Reynolds number (Re = 0.2) is ensured by the molecular diffusion.

(b)

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Fig. 8. Velocity vector plot on y-z plane at different cross-sectional planes with l/W=1: (a) Re = 0.2 (b) Re = 15 .

To show the intensity of the secondary flow in the elongation and its utility in the mixing quality enhancement, the velocity profiles were plotted in the three planes of the fourth crossing as an example because we find that the velocity profiles in all crossing planes have an identical shape. From Fig. 9(a), the radial velocity at the crossing inlet is great at the top and bottom and characterized by a symmetrical profile related to the center where the velocity is nul, then decreases in the other crossing planes. While in Fig. 9(b), the tangential velocity at the crossing inlet is very low, this is due to the dominance of the radial flow, and then takes values identical to those of the radial velocity in the crossing center, this similarity is due to the homogeneous vortex created within the elongation. At the crossing outlet, the tangential velocity takes values opposite compared to that of the center, this is due to the entanglement of the streamlines as seen previously. Therefore, the elongation gives a sufficient space to the fluids to create intense secondary flows accompanied by vortex which directly affect the mixing efficiency. 0.20

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Crossing inlet Crossing center Crossing outlet

0.15 0.10

0.00

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Vy (m/s)

0.05

-0.05

-0.15

0.00006

0.00012

0.00018

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-0.20 0.00000

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-0.10

0.00024

0.00030

Z-coordinate (m)

(a)

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0.08 0.06

Crossing inlet Crossing center Crossing outlet

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0.04

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Vz (m/s)

0.02 0.00

-0.02 -0.04 -0.06

-0.08 -0.000150

-0.000075

0.000000

0.000075

0.000150

Y-coordinate (m)

(b) Fig. 9. Velocity profiles at the 4th crossing with l/W=1 and Re = 15, (a) Radial velocity profiles (b) Tangential velocity profiles.

To show the effect of the elongation on the secondary flows generated during the passage of the fluids through the geometry channels, the streamlines are presented in figure 10-a for l/W = 0 and figure 10-b for l/W = 1 at Re = 15 . It is obvious that the geometry with l/W = 1 allows to create entanglements of the streamlines in the crossings and also vortices in the sides of the vertical channels which favor chaotic advection. Although in the geometry with l/W = 0, the vortices created in the vertical channels are less intense and the passage in the crossings takes place without entanglement, the secondary flows which generate entanglements and vortices in the current proposed geometry (l/W = 1) contribute strongly to the improvement of the mixing compared with

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the basic geometry (l/W = 0).

(a)

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(b)

Fig. 10. Streamlines colored by mass fraction at Re = 15 (a) l/W=0 (b) l/W=1 4. 6. The pressure drops:

Another important characteristic of the flow in our proposed micromixer (l/W=1) represents the pumping power

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required to move the fluids. To express this importance, the pressure difference between the two inlets and the outlet of the mixing channel was calculated as a difference between area-weighted averages of total pressure. In

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Fig. 11(a), we compare the results of the current proposed geometry of 3.75 mm with those of a recent geometries of Hossain et al. [24] and Raza et al. [27], with the same length of 2.975 mm in an interval of Reynolds number (0.2 – 80). We see that the current proposed micromixer presents the lowest pressure losses

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than the two other micromixers on all used Reynolds numbers. 100

Present study (l/W=1) Hossain et al., [24] Raza et al., [27]

60

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Pressure drop (kPa)

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80

40

20

0 0

20

40 Re

(a)

60

80

1200 l/W = 1 l/W = 0.75 l/W = 0.5 l/W = 0.25 l/W = 0

Friction coefficient, f

1000 800 600 400 200 0 15

30

45

60

75

90

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0

Re

(b) Fig. 11. Evolutions of the pressure drops and the friction coefficient with the Reynolds number,

(a) Comparison of the pressure drop between the current proposed micromixer and those proposed by Hossain et

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al. [26] and Raza et al. [27], (b) Evolution of the friction coefficient for different geometries. The local friction coefficient is expressed in terms of pressure drop as follows: ∆𝑃 𝐷ℎ 1 2 𝐿 𝜌𝑈 2 𝑚

(8)

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𝑓=

Where ΔP is the pressure difference between the inlet and outlet micromixer sections, while Dh is the hydraulic

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diameter of the inlet micromixer section and Um is the mean flow velocity. Figure 11(b) shows the evolution of the friction coefficient as a function of Reynolds number for the five aspect ratios (l/W = 0; 0.25; 0.5; 0.75 and 1). It can be seen that the friction coefficient decreases with the increase of

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the Reynolds number and its values are almost identical for all considered geometries. Therefore, it is deduced that the elongation of the crossing does not have a considerable effect on the friction coefficient. The mixing performance of other typical micromixers contains Newtonian fluids, with certain involving two

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layers with OH-shaped units (Hossain S et al. [24]), X-shaped units (Hossain S et al. [26]), and recently OXshaped units (Raza W et al. [27]). A quantitative comparison has been carried out to compare the mixing index

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of the current proposed micromixer with those of previous and recent studies on various micromixers as shown in Table 2. It’s indicated that the mixing index can be reach to 99% only for micromixers characterized by two layers under certain Reynolds number. However, the present micromixer with a short length had high values of mixing index which varied from 0.8567 to 0.9985 at very low Reynolds numbers, which clearly shows the superiority of the current proposed micromixer.

Micromixers TLCCM

Fluids

Lt (mm)

Re

MI

Authors

Year

Glycerol & liquid food dye

12

0.01–60

0.99

Xia et al. [23]

2005

3D « L-shaped units »

Water & Ethanol

12.9

1–70

0.62–0.82

Ansari et al. [16]

2009

Two layers with OHshaped units

Water & Ethanol

4.35

0.1–120

0.46–0.88

Hossain et al. [24]

2015

3D « E-shaped units »

Water & ink in water

9.7

0.1–100

0.73–0.96

Chen et al. [21]

2017

TLCCM (l/W=0), nine X-shaped units

Water & dye

7.5

0.2–120

0.96–0.99

Hossain et al. [26]

2017

Two layers with OXshaped units

Water & dye

2.975

0.1–120

0.87–0.99

Raza et al. [27]

2018

3D « Unbalanced Split and Recombine »

Water & dye

7.9

0.1–120

0.28−0.99

Raza et al. [28]

2019

TLCCM (l/W=1)

Water & dye

3.75

0.2–80

0.85–0.99

Present study

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Table 2. Comparison of the mixing index between the current proposed micromixer and previous studied

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micromixers.

5. Conclusions

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Numerical simulations of the mixing of Newtonian fluids were performed in a chaotic modified two-layer crossing channels micromixer. The numerical results were in good agreement with the recent numerical data.

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The proposed micromixer with four mixing units had at least 85% of mixing index with Re = 0.2 and exceeds 90% for Re = 20 and reaches more than 99.22 % from Re = 50. A parametric numerical study was carried out for four aspect ratios (l/W = 0.25-1), the quantitative and qualitative analysis of the results shows that the geometry with l/W = 1 shows an excellent mixing index in a wide range of Reynolds numbers (0.2-80). The mixing

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performance of the chosen geometry through this study was compared with those of the two geometries without elongation of four and nine units, this comparison shows the dominance of the current proposed micromixer compared to the others. The presentation of the mass fractions along the geometry in cross-sectional planes for

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different Reynolds numbers allows us to distinguish the transition from the diffusion regime to the convection regime, and also to understand the mixing mechanism using the chaotic advection. The presentation of the vectors and velocity profiles in cross-sectional planes of the elongated crossing zones makes it possible to

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illustrate the structure of the induced flows which shows the generated secondary flows and their influence on the mixing performances. The chosen micromixer in this work through a parametric study shows the superiority of the current proposed micromixer in terms of mixing quality and the lowest pressure losses compared to the other recently examined micromixers. Overall, the micromixer studied was very efficient for all values of Reynolds number considered in this study which leads us to suggest this proposed modified geometry to have a fast mixing at low Reynolds numbers. However, the micromixer proposed represents a preliminary contribution to the design of an optimal micromixer which requires support by experiments.

Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] C.Y. Lee, L.M. Fu, Recent advances and applications of micromixers, Sensors and Actuators B. 259 (2018) 677–702. [2] C.Y. Lee, W.T. Wang, C.C. Liu, L.M. Fu, Passive mixers in microfluidic systems: A review, Chem. Eng. J. 288 (2016) 146–160. [3] A. Ghanem, T. Lemenand, D.D. Valle, H. Peerhossaini, Static mixers: Mechanisms, applications, and characterization methods – A review, Chem. Eng. Res. Des. 92 (2014) 205–228.

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[7] M.A. Ansari, K.Y. Kim, Mixing performance of unbalanced split and recombine micomixers with circular and rhombic sub-channels, Chem. Eng. J. 162 (2010) 760–767.

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[9] M.S. Cheri, H. Latifi, M.S. Moghaddam, H. Shahraki, Simulation and experimental investigation of planar micromixers with short-mixing-length, Chem. Eng. J. 234 (2013) 247–255. [10] N. Solehati, J. Bae, A.P. Sasmito, Numerical investigation of mixing performance in microchannel Tjunction with wavy structure, Computers & Fluids. 96 (2014) 10–19.

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[23] H.M. Xia, S.Y.M. Wan, C. Shu, Y.T. Chew, Chaotic micromixers using two-layer crossing channels to exhibit fast mixing at low Reynolds numbers, Lab Chip 5. (2005) 748–755. [24] S. Hossain, K.Y. Kim, Mixing analysis in a three-dimensional serpentine split-and-recombine micromixer, Chem. Eng. Res. Des. 100 (2015) 95–103.

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[27] W. Raza, S. Hossain, K.Y. Kim, Effective mixing in a short serpentine split-and-recombination micromixer, Sensors and Actuators B: Chemical. 258 (2018) 381–392.

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