Design of micromixers using CFD modelling

Chemical Engineering Science 60 (2005) 2503–2516

www.elsevier.com/locate/ces

Design of micromixers using CFD modelling

J. Aubina,∗, D.F. Fletcherb, C. Xuereba

aLaboratoire de Génie Chimique UMR 5503 CNRS, 5 rue Paulin Talabot BP-1301, Toulouse, Cedex 1 31106, FrancebDepartment of Chemical Engineering, The University of Sydney, Sydney, N.S.W. 2006, Australia

Received in revised form 15 October 2004; accepted 5 November 2004Available online 29 January 2005

Abstract

The effect of various geometrical parameters of a grooved staggered herringbone micromixer on the mixing performance has beeninvestigated using computational fluid dynamics. Mixing quality has been quantified with spatial data statistics, maximum striation thicknessand residence time analyses. The results show that the number of grooves per mixing cycle does not affect the mixing quality in animportant way. On the other hand, a larger groove depth and width allow the maximum striation thickness to be rapidly reduced, withoutincreasing the pressure drop across the mixer. Wide grooves, however, create significant dead zones in the microchannel, whereas deepgrooves improve the spatial mixing quality.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Mixing; Micromixer; Laminar flow; Computational fluid dynamics (CFD); Striation thickness; Spatial analysis

1. Introduction

Micromixers play a significant role in micro chemical pro-cessing and are employed in a multitude of tasks, includ-ing blending, emulsification and suspension, as well as forchemical reaction and also in combination with integratedheat exchangers. Due to the small dimensions of the mi-cro channels, the flow is predominantly laminar and mixingis therefore limited by molecular diffusion. In order to ef-fectively mix in a reasonable time, fluids must be manipu-lated so that the interfacial surface area between the fluidsis increased massively and the diffusional path is decreased,enhancing the molecular diffusion to complete the mixingprocess (Ehrfeld et al., 2000). A number of mechanisms ex-ist for fluid contacting and mixing, including multilamina-tion, splitting and recombination, hydrodynamic focussing,T- and Y-junctions (Ehrfeld et al., 2000), as well as chaoticadvection (Beebe et al., 2001; Lui et al., 2000; Stroocket al., 2002a,b).

Due to the relatively young age of microreactor engi-neering, common design rules for micromixers have not yet

∗ Corresponding author. Tel.: +33 5 34 61 52 43; fax: +33 5 34 61 52 53.E-mail address:[email protected](J. Aubin).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.11.043

been developed. However, one can see that apart from theirminute size, microreactors are merely continuous laminarflow reactors, which suggests that design approaches formixing in micro channels could be dealt with in a similarmanner to that of laminar mixing in macro scale pipe flow.In undisturbed laminar flow, the streamlines run parallel toone another and there is no convective mixing in the radial ortangential directions, which results in a high degree ofspa-tial inhomogeniety. In addition, the parabolic velocity profileof the laminar flow gives rise totemporalinhomogeneities,which translates into wide residence time distributions. Thusin order to disturb the flow and facilitate mixing in laminarpipeline flow, in-line devices or static mixers are insertedinto the channel. The best designs of such devices createa high degree of plug flow, thus decreasing temporal inho-mogeneities, whilst achieving homogenisation in the radialdirection, which increases the spatial homogeneity (Etchellsand Meyer, 2004). The design of micromixers, which arecomparatively similar devices at a much smaller scale, can belooked at in the same manner, whereby the aim is to providesufficient spatial and temporal mixing as fast as possible.

A number of methods are commonly used in order tocharacterise both the spatial and temporal mixing perfor-mance of large-scale static mixing devices, for example:

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2504 J. Aubin et al. / Chemical Engineering Science 60 (2005) 2503–2516

the coefficient of variation and the intensity of segregation(Hobbs and Muzzio, 1997, 1998a,b; Mickaily-Huber et al.,1996; Rauline et al., 2000; Zalc et al., 2002), Poincarésections (Hobbs and Muzzio, 1998a,b), striation thickness(Fourcade et al., 2001) and residence time distributions(Hobbs and Muzzio, 1997; Nauman et al., 2002; Visser etal., 1999). These can be implemented experimentally and/ornumerically—the numerical approaches have been detailedin Aubin et al. (2003).

Over the past few years, a number of studies using differ-ent types of micromixers have been carried out with the aimof characterising the mixer performance using various ex-perimental techniques, such as fluorescent microscopy andspecial chemical reactions, as well as CFD simulations totrace particle trajectories (Beebe et al., 2001; Bertsch et al.,2001; Ehrfeld et al., 1999; Lui et al., 2000; Stroock et al.,2002a,b). As we discussed in a previous paper (Aubin et al.,2003), these methods are indirect and enable mainly qualita-tive comparisons of micromixers but they do not give quan-titative data, such as mixing times or mixing lengths. Fur-thermore, they rarely make use of the characterisation meth-ods that have been developed for macro-scale static mixers.In Aubin et al. (2003), we have used a numerical approachfor quantifying the mixing quality in micromixers, which isbased on a spatial analysis method that has previously beenused for assessing the performance of macro-scale staticmixers. This statistical approach employs quadrat analysisof Lagrangian particle trajectories whereby two-dimensionalplanes in the mixer are divided into equal sized quadratsand the number of particles within each quadrat is recorded.At each plane, the variance of particle distribution with re-spect to an ideally homogenised distribution is calculated.This in turn gives an idea of the channel length necessary toobtain a particular degree of mixing. Although the method-ology described above is useful for quantifying the rate atwhich particles are spread throughout the mixer, it is highlydependent on the size and number of quadrats used, as wellas the number of particles traced (Lui et al., 1994). If thequadrats are too large, the technique will quantify mixingonly on a coarse scale and thus small scale flow patternsthat are smaller than the quadrat size will be filtered out. Onthe other hand, if the quadrats are very small, a very largenumber of particles would be required, which would de-mand an extremely long computational time. InAubin et al.(2003), the micromixer geometries investigated gave strik-ingly different flow patterns and therefore the quadrat sizeand particle number that were employed were sufficient forthe comparison of the two mixers. In the current study, thedifferences between the various micromixer flow patternsare much smaller and a higher degree of resolution is thusrequired for better quantification.

In our previous paper, we characterised the mixing qualityin a diagonally grooved micromixer (DM) and a staggeredherringbone micromixer (SHM) using computational fluiddynamics (CFD). It was shown that the SHM mixes muchmore efficiently and in a shorter time than the DM. In the

current study, we have decided to study numerically the ef-fect of different geometrical parameters of the SHM on themixing quality, with the aim of improving the mixer design.We present an alternative method for quantifying the levelof spatial homogenisation in micromixers, which is basedon a statistical method called nearest-neighbour analysis. Ameasure of the reduction in maximum striation thickness asa function of mixer length is also used to quantify distribu-tive mixing. Temporal mixing is then assessed by the deter-mination of residence time distributions. Lastly, the effect ofthe various geometrical modifications on the pressure dropis examined.

Sections 2–4 describe the various micromixer geometriesinvestigated, as well as the numerical and characterisationmethods employed in this work. A detailed analysis of theresults is then presented in Section 5 and the conclusionsare given in Section 6.

2. Mixer geometry

The mixer geometries used in this study are based on thatproposed byStroock et al. (2002a)and have an off-centredherring bone pattern in the channel floor (Fig. 1), which cre-ates a transverse velocity component in the flow field. Themixer is composed of several mixing cycles, whereby a mix-ing cycle comprises two sequential regions of grooves, i.e.,two half-cycles. The direction of asymmetry of the herring-bones switches with respect to the centreline of the channelevery half cycle. The mixer consists of a rectangular chan-nel (w = 200�m, h = 77�m, L = 0.01 m) with groovesof depth,dg, and width,Wg, which have a wave vector of2�/100�m−1. In this work, the groove depth, the groovewidth and the number of grooves per cycle,Ng, were var-ied in order to understand their effect on the mixing qual-ity. Table 1presents the different cases investigated and thevalues of the geometrical parameters.

3. Numerical methods

The numerical simulation of the flow and mixing inthe micromixers has been performed using ANSYS-CFX5(ANSYS, 2003). This is a general purpose commercial CFDpackage that solves the Navier–Stokes equations using afinite volume method via a coupled solver. The analysisprocedure has been carried out in two steps. Firstly, thevelocity and pressure fields in the mixer are solved. Thesevalues are then used to calculate particle trajectories withinthe flow field.

3.1. Flow computation

A mesh composed of approximately 1 330 000 hexahe-dral elements (1 500 000 nodes) was used. A preliminarygrid convergence study was carried out in order to verify

J. Aubin et al. / Chemical Engineering Science 60 (2005) 2503–2516 2505

half-cyclehalf-cycle

αh

w

h

2π/q

Fig. 1. Geometry of the staggered herringbone micromixer (Stroocket al., 2002a).

Table 1Geometrical parameters of the different micromixers investigated

Case Groove depth, Groove width, Number ofdg (h) Wg (�m) grooves per cycle

Reference 0.23 50 20(Stroock et al., 2002a)1 (dg = 0.30 h) 0.30 50 202 (dg = 0.35 h) 0.35 50 203 (Wg = 25�m) 0.23 25 204 (Wg = 75�m) 0.23 75 205 (Ng = 10) 0.23 50 106 (Ng = 30) 0.23 50 30

that the solution is grid independent. The boundary condi-tion at the channel inlet was a uniform velocity profile withvx = 0.02 ms−1 andvy = vz = 0 ms−1. This corresponds toa laminar flow regime with a Reynolds number(Re) ∼= 2.At the outlet, a constant pressure condition (P = 0) was im-posed and no-slip boundary conditions were applied at allwalls. Water at 25◦C and 1 atm was used as the operatingfluid. The CFX5.6 solver was used to solve the steady-statemomentum and continuity equations for the fluid flow in themicromixers. The advection terms in each equation were dis-cretised using a bounded second order differencing schemeto minimise the effects of numerical diffusion. Simulationswere typically considered converged when the normalisedresiduals for the velocities fell below 1× 10−6.

3.2. Particle tracking

In this study, a 50–50% mixture of mass-less fluid parti-cles in the flow is followed using a Lagrangian particle track-ing method in order to quantify the mixing quality. Initially,2480 evenly distributed particles are place on the right-handside of the mixer inlet. This approach avoids the introductionof numerical diffusion which results if a scalar is tracked,which confuses the mixing behaviour. The movement of theparticle tracers in the flow is determined by integration thevector equation of motion for each particle:

dxdt

= v(x). (1)

In order to obtain a sufficient degree of accuracy when inte-grating the equation of motion, a fourth order Runge–Kuttascheme with adaptive step size has been employed. Fur-

thermore, a restitution coefficient of unity is applied to themicrochannel walls. This avoids particle trajectories beingtrapped near the walls where the local velocity is close tozero (less than 2% of particles are stopped between the inletand the outlet of the mixer).

4. Characterisation methods

4.1. Spatial mixing

Spatial mixing has been characterised by two methods.The first is a variation of nearest-neighbour analysis, whichis used in spatial data statistics and measures the distancebetween a particular coordinate and a tracer particle. Thesecond method measures the reduction in the maximum stri-ation thickness at different positions along the microchannel.

4.1.1. Analysis using point-event distancesNearest-neighbour analysis, like quadrant analysis, is

used in spatial data statistics to study the spatial distributionof point patterns on a plane (Diggle, 1983). The nearest-neighbour method differs from quadrant analysis in that itis based on the distance between each point and its nearest-neighbour and that it does not require the studied regionto be divided into quadrants. Several different nearest-neighbour methods exist; some are based on the mean dis-tance between the two nearest particle locations comparedwith a randomly distributed set of particle locations, oth-ers use the distance between a randomly chosen point andthe nearest particle position (point-event distance). Suchanalysis methods have been widely used in ecology (e.g.,the study of the distribution of trees) and biology (e.g., thestudy of the distribution of cell nuclei). Concerning otherscientific disciplines however, it appears that spatial dataanalysis has been employed only occasionally in materialsscience to study the distribution of microporosity in alloymaterials (e.g.,Anson and Gruzleski, 1999).

In this study, we have carried out a statistical analysis us-ing an adapted point-event distance methodology. Instead ofchoosing random point coordinates and calculating their dis-tance to the nearest particle tracer positions, we have used aset of uniformly spaced points that represent a perfectly ho-mogeneous mixture. In a two-dimensional plane, the mostuniform pattern for a set of points occurs in a regular hexag-onal tessellation with a point at the centre of each hexagon,Figs. 2and3.

In this work, we have used 2480 tracer particles to analysethe flow. To obtain the most uniform distribution of thisnumber of points, we arrange them as an 80× 31 array intheY–Z plane. Considering theY–Z dimensions of the microchannel, this gives an inter-point distance ofdy = 2.50�min the Y-direction anddyz = 2.78�m in the direction oftheY–Z vector. An approximation of the mean inter-pointdistance of this uniform array of points can be calculated asthe mean distance between all points in a regular hexagonal


10

7

6

5

4

3

2

1

02

Y (m)

Z (

m)

x 10−5

x 10−4

Fig. 2. Uniform pattern for a set of points occurring in a regular hexagonaltessellation.

dY

dY dY

dY

dYZ

dYZdYZ

dYZ

dYZdYZ

dYZdYZ

Z

Y

Fig. 3. Hexagonal tessellation for the calculation of the mean inter-pointdistance.

tessellation with a point at the centre (Eq. (2),Fig. 3).

dR = 4dY + 8dYZ. (2)

For different cross-sectional planes along the micro mixerchannel, point-particle location distances,di , are determinedby calculating the distance between each particular locationof the evenly distributed points,Pi(yi, zi), and the locationof the nearest particle tracer,Pt(yt , zt ):

di =√

(yi − yt )2 + (zi − zt )

2. (3)

From this the spread of the point-particle location distancesfrom the expected mean distance of the uniform point patterncan be determined using

var= 1

N − 1

N∑i=1

(di − dR)2. (4)

A ‘well mixed’ criterion has been set such that for any point-particle location distance,di , which is less than the expectedmean distance,dR, a value equal todR is assigned todi .This means that variance value of 0 corresponds to a uniformspatial distribution of tracer particles.

4.1.2. Striation thicknessThe maximum striation thickness at different positions

along the micromixer has been measured at the mid-height ofthe channel. To do this, it has been assumed that all particleswhich have aZ-coordinate between 35 and 40�m are at themid height of the channel. The distance in theY-direction,�y, between two consecutive particles is then calculated.The distance�y is then tested by fixing a threshold value,�yT , in order to determine whether two consecutive particlesare part of the same striation or not. In this study, the valueof �yT has been set to 5�m, which is the same resolutionas in theZ-direction. The following test is then executed:

�y��yT : Consecutive particles belong to the same

striation, f (x) = 1,

�y >�yT : Consecutive particles belong to different

striations, f (x) = 0. (5)

Fig. 4(a) shows the functionf (x) for the reference case at thepositionX = 0.04 cm along the micromixer as an example.It can be seen that this method for determining the thick-ness of striations correctly depicts the corresponding parti-cle tracer pattern that is observed inFig. 4(b). The data fromthe functionf (x) are then used to calculate the maximumstriation thickness at any position along the micromixer.

4.2. Temporal mixing

4.2.1. Residence time distribution (RTD)The RTD for the fluid flowing through the various mi-

cromixer geometries was calculated by determining theparticle trajectories as described in paragraph 3.2, and byrecording the particle residence times after 1 cm of mi-cromixer length. This allows the calculation of a residencetime distribution,E(t), as described byFolger (1992):

E(t) = �N

N· 1

�t, (6)

where�N is the number of particles that have a residencetime in the mixer between timet andt+�t andN is the totalnumber of particles released at the entry of the microchannel.The cumulative residence time distribution,F(T ) can thenbe calculated as

F(T ) =∑t �T

E(t).�t. (7)

The residence times of the particles were then normalised bythe expected mean residence time,�=V/Q, in the mixer toobtain the normalised residence time�. The residence timedistribution and the cumulative distribution as a function of� are then expressed as

E(�) = �.E(t), (8)

F(�) =∑

��

E(�).��. (9)


Fig. 4. Determination of the maximum striation thickness for the reference case atX = 0.4 cm: (a) function f(x); (b) corresponding particle tracer patternwith observed striation thicknesses.

5. Results and discussion

5.1. Visualisation of mixing

In order to visualise the mixing phenomena in the variousmicromixer geometries, the particle locations at differentaxial positions along each mixer are plotted, as shown inFigs. 5–7.A detailed description of the mechanism of mixingin this type of micromixer has already been given inAubinet al. (2003)and will therefore not be repeated here.

Fig. 5compares the effect of groove depth on the mixingpatterns in the micro channel. Several observations are notedas the groove depth increases (fromFig. 5(a)–(c)). In thefirst part of the microchannel, the flux of particles from theright-hand side to the left-hand side increases, this in turnfavours the exchange of particles between the left-hand andright-hand sides of the channel. Consequently, the rate ofdecrease of the maximum striation thickness increases as thegroove depth increases and spatial homogenisation of theparticle tracers occurs more quickly.

The groove width has an important effect on the spatialmixing quality as can be seen inFig. 6 . If the grooves arevery narrow (Fig. 6(b)), the exchange of fluid between theleft-hand and right-hand sides of the channel is very poorbecause the amount of fluid that enters the grooves and at-tains a transverse velocity component is reduced. The result-ing mixing pattern is very orderly, with a finger-like struc-ture being created at the end of each mixing cycle. Whenthe grooves are wide compared with the reference case, asshown inFig. 6(c), the mixing pattern is very irregular. Theexchange of fluid between both sides of the mixer is im-portant and the rate of reduction of the largest striation in-creases considerably. However, it is also noted that for thecase of wide grooves, the distribution of particles is muchless uniform than in the reference case. The particles tend togravitate towards the centre of the channel, leaving the upperand lower corners void of particles. This suggests the pos-sible presence of dead zones, which lack circulation. Sucha hypothesis can be verified using the RTD analysis.

Fig. 7 shows the effect of the number of grooves per cy-cle on the mixing quality. When the number of grooves percycle is halved with respect to the reference case, short fine

finger-like striations are formed at the end of each cycle. Thefineness of these striations would clearly facilitate moleculardiffusion. Nevertheless, the largest striation at the centrelineof the channel is not that much different from that of thereference case geometry. When the number of grooves percycle is greater than that of the reference case, the finger-like structures are relatively thick. However, they have timeto be folded upon themselves, which then decreases the stri-ation thickness. Nevertheless, after 1 cm of mixer length,the maximum striation thickness appears to be more or lesssimilar to that present in the reference geometry.

5.2. Quantification of spatial mixing

In order to evaluate the homogeneity of the mixing pat-terns presented in Section 5.1, the spread of the particle pat-terns from the ideal uniformly distributed pattern was quan-tified by calculating the variance as described in Section4.1.1. At each position along the micromixer, the variance,var, is normalised by the variance at the entry of the chan-nel, var0. A normalised variance value of 0 corresponds to auniform spatial distribution of tracer particles and thus goodmixing, whilst a value of 1 indicates that no spatial mixinghas occurred compared with the initial state of the system.

Fig. 8shows the effect of the different geometrical param-eters on the normalised variance at different positions alongthe micromixer. InFig. 8(a), it can be seen that the spatialmixing quality improves as the groove depth increases. Atapproximately positionX=0.85 cm, the variation in the nor-malised variance appears to plateau off for the three casesand thus the spatial pattern of the tracer varies very little.This suggests that from this position onwards, the striationpatterns are being modified much more slowly by advectionand that the final phase of mixing will be limited by molec-ular diffusion. It can be seen inFig. 8(b) that quantitativelythe groove width has a strong effect on the homogenisation.When the grooves are very narrow, spatial mixing is veryslow and at least 2 cm of mixer length would be needed toobtain the same quality of mixing achieved in only 1 cm ofthe reference case mixer geometry. The curve correspond-ing to the wide grooved micromixer is quite unique. The


(b) (c)

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Fig. 5. Effect of the groove depth,dg , on mixing patterns: (a) reference case (dg = 0.23 h); (b) dg = 0.30 h; (c) dg = 0.35 h.

mixing quality appears to evolve in a similar manner to thereference case mixer until approximatelyX=0.45. After thispoint, the tracer tends to accumulate near the centre of thechannel creating empty zones in the channel corners, thuscausing the normalised variance to plateau off. The numberof grooves per cycle does not appear to have a very impor-tant effect on the spatial distribution of the tracer, as shownin Fig. 8(c). It is however noticeable that towards the endof the mixer, the curves for casesNg = 10 andNg = 30 are

still varying considerably compared with the reference case.This suggests that advection is still playing an important rolein the modification of the flow structures.

5.3. Striation thickness

The reduction in maximum striation thickness as a func-tion of the micromixer length for the various geometries


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Fig. 5. (continued).

is shown in the semi-log plots given inFig. 9. The dottedline indicates a striation thickness of 10�m, which has beenfixed as a reference. The time for a striation to mix by puremolecular diffusion can be estimated by

tdiff = s2

D, (10)

wheres is the striation thickness andD is the molecular dif-fusion. For a striation of width 10�m and a typical molecu-lar diffusion in liquids of 2×10−9 m2/s, the mixing time by

diffusion is approximately 0.05 s. With the flow rate studiedhere, this is less than one tenth of the mean residence timein the mixer and in this time, the fluid will be convectedonly about 0.1 cm. It can thus be assumed that a sufficientlevel of mixing is obtained in the mixers when the maximumstriation thickness is 10�m or less.

Fig. 9(a) compares the effect of different groove depthson the maximum striation thickness. As observed in the par-ticle patterns inFig. 5, the rate of reduction of the maximumstriation thickness is proportional, but not linearly, to the


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Fig. 6. Effect of the groove width,Wg , on mixing patterns: (a) reference case (Wg = 50�m); (b) Wg = 25�m; (c) Wg = 75�m.

groove depth. The results suggest that a maximum groovedepth exists whereby the decrease in the maximum stria-tion thickness is no longer significantly improved. Quanti-tatively, a striation thickness of 10�m is obtained after ap-proximately 0.84 cm of mixer whendg =0.35 h and 0.92 cmwhendg =0.30 h. This means that one can expect very goodmixing in both geometries with just 1 cm of mixer length. Inorder to attain a striation thickness of 10�m in the referencecase mixer, however, a much longer micromixer is required,around 1.2 cm and one would expect complete mixing within1.3 cm of mixer length. The graphs inFig. 9(b), which com-pare the effect of groove width on the striation thickness,are remarkably different, as suggested by the particle pat-

terns inFig. 6. Firstly, it can be seen that the maximum stri-ation thicknesses for the three cases do not decrease in anexponential manner, as is expected for typical chaotic flow(Ottino, 1989). This is most apparent for the geometry withwide grooves. Here, the rate of reduction of the maximumstriation thickness greatly increases along the length of themicromixer. In this geometry, a striation thickness of 10�mis obtained after approximately 0.75 cm of mixer, which ismuch less than the length required by the reference case ge-ometry. For the case with narrow grooves, a relatively longmixer is needed to attain the same quality of mixing. Theeffect of the number of grooves per cycle on the striationthickness is compared inFig. 9(c). The results quantitatively


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x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567

x 10−5

x 10−4

−10

0

1

1

2

2

34567


show the striation thickness does not depend on this geomet-rical parameter, which confirms the qualitative observationsmade inFig. 7.

5.4. Residence time distributions

The cumulative residence time distributions are presentedin Fig. 10. It can be seen fromFigs. 10(a) and (c) that nei-ther the groove depth nor the number of grooves per cycleaffect the residence time distributions significantly. Never-theless, it is noticed that the RTDs in these mixer geometriesdo deviate from the typical Poiseuille flow distribution and

tend towards a plug flow distribution, which means that thegrooves present in channels act as flow inverters and equalisethe residence times as macro-scale static mixers do. Thecurves inFig. 10(b) are however quite different from oneanother. When the grooves are narrow, they have little effecton the flow and the RTD is very close to that of Poiseuilleflow. For wide grooves, however, the RTD is much closer tothat of typical plug flow. As seen in the particle pattern plotsin Fig. 6, the tracer in the wide groove geometry moves tothe centre of the microchannel, leaving zones void of parti-cles in the upper and lower corners. This means that parti-cles move with much similar velocities as they are avoidingthe low velocities close to the channel walls, thus the tracer


X =

0.1

cm

X

= 0

.2 c

m

X =

0.3

cm

X

= 0

.4 c

m

X =

0.5

cm

x 10

−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

(a) (b) (c)

Fig. 7. Effect of the number of grooves,Ng , per cycle on mixing patterns: (a) reference case (Ng = 20); (b) Ng = 10; (c) Ng = 30.

RTD tends towards a plug flow RTD. In order to verify theexistence of possible dead zones in this micromixer geome-try, the mean residence time of the tracer has been comparedwith the expected mean residence time (�=V/Q), as shownin Table 2. The mean residence time in significantly lower(15%) than the expected mean residence time, whereas forthe other mixer geometries there is generally not more the5% difference. This means that for the wide groove geom-

etry, the effective volume of the micromixer is decreased,which confirms the presence of dead zones in the channelcorners. It should be stressed that the mass-less tracer usedhere does not take into account any mixing via moleculardiffusion. For the simulation of a scalar concentration or theexperimental mixing of a dye, however, one would expectthe tracer to mix into these dead zones via molecular diffu-sion but very slowly compared with the rest of the channel.


X =

0.6

cm

X

= 0

.7 c

m

X =

0.8

cm

X

= 0

.9 c

m

X =

1.0

cm

x 10

−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

x 10−5

x 10−4

7 6 5 4 3 2 1 0

–1

0 1 2

(a) (b) (c)


5.5. Pressure drop

The pressure drop,�P , across the various micromixersis presented inTable 3. It is observed that the groove depthand the number of grooves per cycle have almost no effecton the pressure drop across the micromixer: deviations fromthe pressure drop across the reference case geometry are lessthan 1%. The width of the grooves, however, has a small

effect on the pressure drop: an increase in groove widthdecreases�P .

6. Conclusions

CFD simulations have been performed in order to inves-tigate the effect of various geometrical parameters in a stag-


0.001

0.010

0.100

1.000

0.001

0.010

0.100

1.000

0.001

0.010

0.100

1.000

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

X (cm) X (cm)

X (cm)

Reference case (dg = 0.23h)dg = 0.30hdg = 0.35h

(a) (b)

(c)

Var

/Var

o

Var

/Var

o

Var

/Var

o

Reference case (Wg = 50 µm)Wg = 25 µmWg = 75 µm

Reference case (Ng = 20)Ng = 10Ng = 30

Fig. 8. Effect of geometrical parameters on the normalised variance: (a) effect of groove depth; (b) effect of groove width; (c) effect of number ofgrooves per cycle.

1

10

100

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

X (cm)

X (cm)

Reference case (dg = 0.23h)dg = 0.30hdg = 0.35h

(a)

1

10

100

Reference case (Ng = 20)Ng = 10Ng = 30

(c)

(b)

Str

iatio

n th

ickn

ess

(µm

)

1

10

100

0 0.2 0.4 0.6 0.8 1

X (cm)

Reference case (Wg = 50 µm)Wg = 25 µmWg = 75 µmS

tria

tion

thic

knes

s (µ

m)

Str

iatio

n th

ickn

ess

(µm

)

Fig. 9. Evolution of the maximum striation thickness: (a) effect of groove depth; (b) effect of groove width; (c) effect of number of grooves per cycle.

gered herringbone micromixer and to hence indicate howthe design of such micromixers can be modified to improvethe mixing quality. Qualitative and quantitative analyses ofthe spatial mixing quality show that the groove depth andthe groove width have the greatest influence on homogeni-sation. On the other hand, the number of grooves per cy-cle does not affect the mixing quality. As the groove depthincreases, homogenisation increases. If the groove width isvery narrow, the flow in the microchannel is hardly affectedby the transversal velocity component and spatial mixing isvery poor. For wide grooves however, homogenisation doesnot necessarily improve since the tracer tends to accumu-

late near the centre of the mixer leaving dead zones in thecorners of the channel. A measure of the reduction of max-imum striation thickness has also allowed the quantificationof spatial mixing. It was found that for grooves deeper than30% of the channel height, mixing is greatly improved. Forthe operating conditions studied here, complete mixing canoccur within 1 cm of the microchannel. The wide groovemixer was also found to very efficient in decreasing the stria-tion thickness. On the other hand, the number of grooves percycle does not modify the reduction in striation thickness.The temporal mixing quality was investigated using cumu-lative residence time distributions. The results show that the


1.0

0.0

0.2

0.4

0.6

0.8

0 1 2 3 4 0 1 2 3 4

0 1 2 3 4

Reference case (dg = 0.23h)dg = 0.30hdg = 0.35hPoiseuille FlowPlug Flow

Reference case (Wg = 50 µm)Wg = 25 µmWg = 75 µmPoiseuille FlowPlug Flow

(a)

(c)

(b)

Reference case (Ng = 20)Ng = 10Ng = 30Poiseuille FlowPlug Flow

F (Θ

)

1.0

0.0

0.2

0.4

0.6

0.8

F (Θ

)

1.0

0.0

0.2

0.4

0.6

0.8

F (Θ

)

Θ Θ

Θ

Fig. 10. Effect of geometrical parameters on the cumulative residence time distributions: (a) effect of groove depth; (b) effect of groove width; (c)effectof number of grooves per cycle.

Table 2Mean residence time,tm, compared with the expected mean residencetime, �, for the various micromixer geometries

Case Expected mean Mean residenceresidence time,�(s) time, tm(s)

Reference (Stroock et al., 2002a) 0.551 0.5211 (dg = 0.30 h) 0.565 0.5612 (dg = 0.35 h) 0.575 0.5683 (Wg = 25�m) 0.524 0.5094 (Wg = 75�m) 0.571 0.4865 (Ng = 10) 0.547 0.5006 (Ng = 30) 0.522 0.517

Table 3Pressure drop across the various micromixer geometries

Case Pressure drop,�P (Pa)

Reference (Stroock et al., 2002a) 428.51 (dg = 0.30 h) 426.82 (dg = 0.35 h) 425.83 (Wg = 25�m) 454.84 (Wg = 75�m) 393.25 (Ng = 10) 433.36 (Ng = 30) 430.5

grooves present on the microchannel floor allow flow in-version and equalisation of residence times, compared withtypical Poiseuille flow. It is also confirmed that the widegrooves give rise to important dead zones in the microchan-nel volume. Lastly, a comparison of the pressure drop acrossthe micromixers shows that the groove depth and the num-ber of grooves per cycle do not affect this parameter.

Thus, in order to improve the mixing quality in staggeredherringbone micromixers, one is advised to use a mixer with

deep grooves, having a depth of 30–40% of the channelheight. This promotes both spatial homogenisation and thereduction of striation thickness, without increasing the pres-sure drop across the mixer. A groove width of 50�m (widergrooves increase dead zones in the mixer) and 20 groovesper cycle appear to be adequate, as proposed originally byStroock et al. (2002a).

Future work will be directed toward the experimental in-vestigation of micromixers using visualisation techniquesand micro-Particle Image Velocimetry, with the aim of pro-viding detailed validation data.

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Design of micromixers using CFD modelling

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