High mass transfer rate is a key advantage of microreactors however, under their characteristic laminar flow, it is dominated by

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Over the last two decades, microreactors have been deployed for a large number of applications, from organic reactions [

Due to the importance of mass transfer, the fluid dynamics inside batch [

Different studies have modified the inlet configuration of mixers to enhance mixing efficiency. For example, in contrast to a normal rectangular T-mixer with two opposite inlets (called

In this work, we present a systematic evaluation and comparison of the different widely adopted mixers, T- and Y- mixers with different flow patterns as well as the advanced fluid dynamic guided design of cyclone mixers for enhanced mixing efficiency with mixing potentials one order of magnitude higher than the commercial configurations. Herein, we demonstrate that inlet cross section area and thus, resulting velocity is a key design parameter to predict mixing efficiency.

In order to quantify the mixing efficiency in mixers under steady state, a number of simulations were set-up evaluating the mixing of streams with equal volumetric flowrate. Solutions of 1 mM aqueous Rhodamine B solution and pure water were considered. Three steps were needed to accurately simulate the concentration profiles of Rhodamine B in micromixers [

The first step was to calculate the velocity field by solving Navier-Stokes equation (Eqs. _{1} and _{2} represent the distance between a point and the height or width edge, _{inlet} was the average velocity of the inlet channel. Outlet pressure was set as 0 gauge pressure. In this paper, the Reynolds number is defined in the outlet channel as Eq.

High quality mesh was drawn by the O-grid tool of ICEM software for circular channel. More than 5 million cells were used for all simulations. Second order and second order upwind schemes were used for pressure and momentum discretization, respectively. Least squares cell-based method was employed to evaluate the gradient. The Coupled scheme provided by Fluent was used to solve the Navier-Stokes equation, which iterates the pressure field and velocity field simultaneously. The convergence tolerance was set as 10^{−5}.

Then backward particle tracking was adopted to simulate the convection contributed concentration profile. The method was based on Lagrangian scalar transport equation (Eq. ^{−4}) starting from the sampling points to the nearest plane with a known tracer concentration, i.e. the planes in both inlet channels with 1 mm distance from the centre of the mixer. Then the traveling time, _{A}^{0} of each sampling particle were obtained. Typically, 200*200 sampling points were set in the array.

In the last step, diffusion was estimated by solving Lagrangian scalar transport equations. However, the diffusion rate along the pathline is unknown, which was estimated by the diffusion rate on the cross sectional plane as Eq. _{A} is the concentration of the tracer, _{p,q} in Eq. _{m,n} in Eq.

A linear equation group, describing the concentrations of each sampling points, was formed which was solved in Matlab to obtain the concentration distribution on the cross section (

The method is validated by the experimentally characterised concentration distributions inside a T-mixer by planar laser induced fluorescence [^{−6}.

Mixing index and mixing potential are adopted to quantify mixing efficiency. Mixing index is defined on a cross sectional plane as Eq. ^{2} represents the concentration variance on the cross section, and ^{2}_{max} is the maximum concentration variance. _{A} denotes the concentration of tracer, and _{A,max} represents the maximum concentration of the tracer in the system, i.e. at the inlet. _{p} is the velocity component perpendicular to the cross section, and

Mixing potential is used to quantify the specific contact area between two streams as Eq.

Mesh independency test was carried out to prove that 5 million cells are enough for the simulation as shown in

A systematic analysis and comparison of the mass transfer mechanism and efficiency of a range of widely available mixers was carried out including Y-mixers and T-mixers. Both, square and circular cross sections were considered as both are commercially available. For the T-mixers, two different flow patterns were also considered, namely

The fluid dynamics and mass transfer of each configuration was evaluated using the Navier-Stokes equation available on ANSYS Fluent combined with a backward particle tracking approach as described in the experimental section [^{2}/m^{3} and ~ 1400 m^{2}/m^{3} in the circular and square Y-mixers, respectively. T-mixers with circular and square cross section with ^{2}/m^{3} and 13,020 m^{2}/m^{3} for the circular and square T-mixers respectively with ^{2}/m^{3} to 6200 m^{2}/m^{3} as the Re increases from 10 to 300. Similarly, the mixing potential in the square T-mixer grows from 1400 m^{2}/m^{3} to 5300 m^{2}/m^{3} in the same Re range.

Geometries and concentration profiles at 5 mm of different conventional mixers a. Y-mixer with circular cross section, b. Y-mixer with square cross section, c. T-mixer with circular cross section and jet flow, d. T-mixer with square cross section and jet flow, e. T-mixer with circular cross section and cross flow, d. T-mixer with square cross section and cross flow. All mixers have 1 mm of diameter or equivalent, 3 mm inlet branched channel and 10 mm of total length. Conditions: each inlet has equal flowrate with developed velocity profile, diffusion coefficient 2.8*10^{−10} m^{2}/s

It is important to notice that in all cases, diffusion dominated mass transfer increases at low Reynolds numbers due to the increase in residence time at the given position of 5 mm. In these simulations, both inlets have an equal flowrate. However, the concentration profiles can also be affected by the ratio between the flowrates in each inlet [

To better understand the formation of stream engulfment, Fig.

a. Evolution of the concentration distribution profile in a T-mixer with square cross section and jet flow 1 mm of equivalent diameter, 3 mm inlet branched channel and 10 mm of total length, b. Corresponding mixing index and mixing potential as a function of length. c. Evolution of the concentration profile in a cyclone mixer with four inlets. d. Corresponding mixing index and mixing potential as a function of length. Conditions: each inlet of the T mixer has equal flowrate of 1.50*10^{−7} m^{3}/s with developed velocity profile, each inlet of the cyclone mixer has equal flowrate of 5.89*10^{−8} m^{3}/s with developed velocity profile, outlet Re 300, diffusion coefficient 2.8*10^{−10} m^{2}/s

Modifying the inlet configuration by shifting the relative position of the inlet streams enhances the engulfment of the streams. Instead of introducing the streams perpendicular to the mixer (T-mixer), streams can be introduced tangentially in what is normally called cyclone mixers due to their resemblance to the inlet stream of conventional cyclones used for separation of fine particles [

In order to pursue higher mixing efficiency and better understanding, a parametric study of cyclone mixer is carried out, including the effect of Re, inlet area and inlet shape. Figure ^{2} m^{−3} compared to the ~13,020 m^{2} m^{−3} achieved in the square T-mixer with

Concentration profiles at 5 mm of a cyclone mixer with circular cross section and 4 square cross section inlets (0.98 mm * 0.50 mm, Fig. _{z}) and tangential (V_{y}) velocities by the axial velocity at the line z = 5 mm and y = 0 mm. F. Corresponding mixing index ( ^{−10} m^{2}/s

Similarly to above, the rotation of the fluid is created by the inlet configuration of the mixer as evidenced by the normalised velocity profile (Fig.

To investigate the effect of the angular velocity on the mixing efficiency, the configuration of the cyclone mixer can be further optimised by varying the cross section area of the inlet streams, in other words, varying inlet average velocity while keeping the total volumetric flowrate constant. Figure ^{−1}. The origin of the bimodal distribution is believe to be related to the tangential velocity component which substantially increases with the average inlet velocity (Fig.

a. Typical configuration of a circular cyclone mixer (1 mm diameter) with four inlet streams with rectangular cross sections (0.98 mm * 0.50 mm), b. A cyclone mixer with two inlets (0.98 mm * 0.50 mm), c. Corresponding mixing index ( _{x}), axial (V_{z}) and tangential (V_{y}) velocities as a function of cross section inlet area/inlet average velocity, e. Concentration profiles at 5 mm as a function of cross section inlet area/inlet average velocity and f. The concentration profile at 5 mm in the cyclone mixer with two inlets. Conditions: each inlet has equal flowrate of 3.50*10^{−8} m^{3}/s with developed velocity profile Re 180, diffusion coefficient 2.8*10^{−10} m^{2}/s

For further comparison, a cyclone mixer with two inlets (Fig. ^{2} (0.98 mm*0.5 mm*2 inlets). Two different types of comparison can be done with respect to cyclones of four inlets. Keeping constant the inlet area (0.49 mm^{2}, Fig. ^{−1}) shows a similar axial and tangential velocity components in both cases however, the mixing index and mixing potential are almost double in the four inlet system than in that with two inlets. The reason behind this is due to the decrease of the contact area between the streams when the number is decreased.

To further demonstrate the determining effect of the average velocity on the rotation of the fluid in cyclone mixers and consequently in the mixing efficiency, the height/width ratio of the inlet cross section was varied by keeping the cross section area constant. Figure

a. Concentration profiles at 5 mm of 4 inlet cyclone mixers and b. Corresponding mixing index and mixing potential as a function of the height/width ratio of the rectangular inlet cross section. Conditions: each inlet has equal flowrate of 3.50*10^{−9} m^{3}/s with developed velocity profile, total cross section inlet area 0.49 mm^{2}, inlet velocity 0.0716 m/s, Re 180, diffusion coefficient 2.8*10^{−10} m^{2}/s

To identify the key design parameter affecting mixing efficiency, all the studies presented above with varying Reynolds numbers, inlet areas and average inlet velocities for the cyclone mixers with four inlets were compared simultaneously. A clear direct relationship is presented between both mixing index and mixing potential with respect to average inlet velocity as shown in Fig.

Relationship between inlet average velocity and mixing efficiency expressed as a. mixing index (by changing ^{−10} m^{2}/s

These results provide the foundation for the extrapolation of the trends to other mixers with different dimensions. To demonstrate this, the cyclone mixer in Fig. ^{2}m^{−3} to 1432.3 m^{2}m^{−3}) as the diameters expands ten times from 1 mm to 10 mm. Despite longer time to reach the same mixedness at longer length scale of the mixer, larger mixer provide higher throughput and lower pressure drop. As the geometry expands 10 times, the cross section area increases 100 times, and the volume of the mixer increases 1000 times. The average velocity drops 10 times to keep the same Re, leading to 10 times volumetric flowrate.

Normalised concentration distribution profiles of cyclone mixers with four inlet streams with square cross sections a) cross section at 500 μm, mixer diameter of 100 μm, b) cross section at 5 mm, mixer diameter of 1 mm, and c) cross section at 50 mm, mixer diameter of 10 mm. Conditions: Re 180, 20 °C aqueous solution, developed inlet velocity profile, diffusion coefficient of Rhodamine B 2.8*10^{−10} m^{2}/s

The mixing efficiency in T-, Y- and cyclone mixers is systematically investigated by a novel method based on backward particle tracking, which eliminates numerical diffusion. The method is firstly validated by published experimentally characterized concentration profiles, showing higher accuracy than the default method in commercial software. The cross section shape of T-mixers with a jet flow is a key design parameter to enhance mixing in the inlet section of flow systems. Square channels lead to flow engulfment at high Reynolds number of 300 while circular channels do not. T-mixers with a cross flow always lead to flow engulfment event at relatively low Re of ~100. Y-mixers do not promote convection and the inlet section is solely dominated by diffusion in the investigated range. Relative position of the inlet streams is a key design parameter to promote mixing. Having the inlet streams tangential (cyclone mixers) instead of opposite (T-mixers) to each other, promotes the rotation of the fluid and thus, convection. We also demonstrate that the inlet average velocity is the main design parameter determining mixing efficiency in cyclone mixers. The higher the inlet average velocity, the more pronounced the rotation of the fluid and thus, higher the mixing efficiency. The cyclone mixer with the smallest inlet area presents the highest mixing efficiency among all investigated mixers. The geometry of the inlet configuration, mainly cross section inlet area, has a mild effect on the homogeneity of the fluid rotation.

The authors greatly acknowledge the financial support from UK Engineering and Physical Science and Research Council (grant number EP/L020443/2) and YG thanks the Chinese Scholarship Council and The Cambridge Trust for his studentship.

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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