Mixing of two uid streams in a microchannel using theTaylor-Aris dispersion eect

Hengzi Wang, Pio Iovenitti, Erol Harvey and Syed Masood

Industrial Research Institute Swinburne,Swinburne University of Technology, PO Box 218,

Hawthorn,Victoria 3122, Australia

ABSTRACT

Taylor-Aris dispersion is an unwanted eect in some applications, such as chromatography, because of its rapiddispersion along channel axis direction to cause the diculties to separation. However, this eect can be usedin solving mixing problems. In this presentation, the authors studied mixing of two streams of food dye solutionin a microchannel. The two liquids sandwiched in the axis direction, and due to non-slip boundary condition,the slugs of liquids stretched and thus increase their interfacial area. This phenomena was rstly studied byTaylor about the dispersion of solute in a circular capillary, and then improved by Aris. Numerical analysis wasapplied to design mixing section in microchannels, and simplied experiments were conducted to illustrate theconcept.

Keywords: Taylor-Aris dispersion, Microuidics, microchannel, MEMS, micromixer, passive mixing

1. INTRODUCTION

It is well known that the Reynolds number is low in typical microuidic channels, and the ow is laminar undernormal conditions, especially for liquids. Therefore, the mixing of two uid streams in a microchannel reliesmainly on diusion. For a typical microuidic device, the length scale is too large for a rapid diusion andtoo small to include mechanical agitation. It is possible to achieve this by dynamic mixing with the assistanceof externally forced mass transport, i.e. Yang et al.1 used ultrasonic waves to enhance mixing, and Knight2

described fast mixing by forming and controlling nanoscale, submerged uids jets. However, it is much easierto control mixing in a laminar ow using a static micromixer. The principle of a static micromixer can becategorized into four types. (a). T-shape micromixer. The T-mixer simply combines two or more uid streams,which ow parallel to each other in the microchannel, and mixing relies purely on molecular diusion.36 TheT-mixer normally has a small channel width of the order of tens of microns and sucient channel length. (b).Geometrically splitting and recombining substreams. In this way, large contact surfaces and small diusionpaths are generated (Ehrfeld et al.,7 Schwesinger, et al.,8 Koch et al.9, 10). (c). Chaotic Mixer. Stroock etal.11 presented this mixer to stretch and fold the streams by rotating the streamline inside the microchannel.The circulation of the streamlines was due to the transverse pressure component created by the anisotropicresistance to viscous ow. Johnson et al.12 reported similar structures machined by Excimer laser to achieverapid mixing. However, the obvious drawback for these mixers is the dead volume created by these structures.(d). Altering ow direction laterally. This method tries to create stirring (convective) eects by forcing oneuid stream into another (Liu et al.,13 and He et al.14). Lius serpentine-shaped micromixer was more favorableto high Reynolds numbers ( 70), and Hes in-situ micromixer could alter the ow direction, shrink the channeland divide the main stream into substreams. However, Hes mixer is not likely to be used in a pressure-drivenenvironment, due to the large pressure drop in its packed columns. Taylor and Aris15, 16 described the dispersionof solute in a circular pipe along the longitudes (axial) direction rather than transverse direction, like manyT-type microchannel do. There were also extensive discussions about the Taylor-Aris dispersion in a rectangular

Send correspondence to Hengzi Wang: E-mail: hwang@swin.edu.au, Telephone: 61 3 9214 4332, Fax: 61 3 9214 5050

Biomedical Applications of Micro- and Nanoengineering, Dan V. Nicolau, Abraham P. Lee, Editors, Proceedings of SPIE Vol. 4937 (2002) 2002 SPIE 0277-786X/02/$15.00

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channel.17, 18 From these discussions, it is generally accepted that the apparent diusion coecient D (Taylorsdispersivity) in a rectangular channel can be described in equation (1),

D = D +1210

u2W 2

Df(

h

W) (1)

where D is the molecular diusion coecient, W is the width of the channel, u is the mean velocity in thechannel, h is the height of the channel and f is the geometric function of a rectangular channel, decided bythe height to width aspect ratio h/W . It is surprisingly that the geometric function f is not unity,18 instead,f = 7.95 while h/w 0. For a Taylor dispersivity D, the length between the dispersed leading and trailingedges of slug, lm, shall be estimated by equation (2) developed by Einstein when he studied Brownian motion,19

lm =2dt (2)

Taylor dispersivity D is much large than the molecular diusivity D, therefore, for a xed width d of a channel,the mixing time t is reduced.

2. EXPERIMENTAL AND NUMERICAL DESIGNS

2.1. Numerical setupWhile experiments can illustrate the dispersion a slug of solute in a microchannel and reveal that it is quickerthan pure diusion, numerical simulations can explain some details better and also save time. A straight channelwas modelled numerically using the computational uid dynamic (CFD) software package MemCFD r v2001.3dfrom Coventor. Flow visualization was performed with Coventor visualizer.20 MemCFD r uses a nite volumemethod to solve the Navier-Stokes equations. The channels were meshed into 8-nodes hexahedral elements. Thesimulation was run as transient (steady for studying obstacles in a microchannel), laminar, Newtonian, with 2uids to evaluate the dispersion of one uid in another. The inlets were assigned ow rate boundary condition,and all the channel walls were assigned wall boundary condition (velocity components: vi = 0, i = 1 3). Fluidproperties were calculated and consistent with the uids used in the experiments. In this case, low concentrationof food coloring dyes in water solutions were used.The length, width and height of the channels were 2mm, 200 m and 100m respectively. Simulations wereperformed on Win NT4.0 with Pentium III 800MHz CPU and 512MB memory.

2.2. Experimental Setups

Plane of uniformity

& Mask

Lens Array Homogenizer

Input Beam248 nm (KrF) or 193nm (ArF)

Projection Lens

Workpiece step motion

Figure 1. Excimer micromachining

The microuidic channel was fabricated in the microtechnology laboratory in the Industrial Research Insti-tute of Swinburne (Hawthorn, AU). An Excimer Laser (ExiTech) was used to create the channels. The standard

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Figure 2. Section of Y-channel machined in Polycarbonate by Excimer Laser

1sec

2sec

5sec

8sec

10sec

12sec

15sec

18sec

20sec

Figure 3. Taylor dispersion along a straight channel: numerical simulation for various time intervals

operation procedures can be found in early papers of the laboratory.21 The UV laser beam, with a wavelengthof 248nm, projected through the designed features on a chrome-on-quartz mask, and then the laser beam isfocused to pattern the feature on the substrate (Figure 1 ). The Excimer laser machined structures have adepth resolution of the order of 0.1m and spatial resolutions of the order of 1m or better.

Figure 2 shows the Excimer laser ablated Y-channel on a polycarbonate substrate. Then, the channel waspackaged in a lamination process, a thin PET foil coated with a melting adhesive layer was pressed by a heatedlamination roller onto the structure as the lid. Melinex r Polyester lm type 301 (Dupont Teijin lms), 30min thickness was used in our fabrication, and the lamination equipment was MEGA dry lm laminator model305 (Mega Electronics).The aqueous solutions, one was mixture of water, 2.4% food dye E102 and E122(Yellow), and another one wasmixture of water and 2.1% food dye E133 (blue)(Queen Fine Foods Pty Ltd, Australia), were introduced to thechannel by capillary eects, and the solute plug was pumped into the channel by a syringe. Flow rates weremeasured by weighing the uid collected at the outlet of the channel. Viscosity of the food dye/water solutionwas estimated approximately the same as water. Diusion coecient was calculated according to the chemicalgroups.22

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500m 500m

Figure 4. The solute dispersed at, (a) time t1 and (b) time t2

3. RESULTS AND DISCUSSION

Analytical results can evaluate Taylors dispersivity for rectangular channels with dierent depth to width aspectratio.17 The transient behavior of most practical applications are much easier to interpret through numericalsimulations. In Figure 3, a time series of mass fraction of uid 1 dispersed in uid 2 for a time period of 20 sec-onds. To calibrate with the conditions for analytical interpretation, the Reynolds number was low (Re = 0.005)for diusion to sample all the streamlines.16 It is quite clear that the length of the dispersed uid is longerthan a pure diusion, because of the convective spreading of uid physically. As Taylors dispersivity coecientis a function of mean velocity, D = f(u), it is quite intuitive to have a large ow rate for a quick mixing. Thisis dierent from Taylors original study of dispersion of soluble matter in solvent owing slowly.16 However,in most practical applications, it is not necessary to have a slow motion. As a matter of fact, the velocity isusually higher and has a parabolic shape. While velocity becomes larger, there will be less time for the diusionto sample all the streamlines. In Figure 4, the green region was the mixed uids between uid 1 (yellow) and 2(blue). And there were two transition regions between the yellow/green and green/blue, where the uids beganto diuse into each other. The dispersed solute was initially a narrow parabolic shape at time t1, and thendeveloped into a wider parabolic shape at time t2 in a short distance. Although there is lack of quantitativemeasurement for evaluating the mixed volume of the two uids, the enlarged area of mixed green region andlengthened parabolic shape indicated the enhance of mixing. Nevertheless, numerical simulation agreed withexperiments qualitatively and can be used to access the quality of the mixing. More structured mesh withhigher density of elements can reduce numerical diusion and improve the accuracy, whenever computer re-source allows.It was eective to inject one slug into the main stream of uids for certain analysis. It is ideal to create a seriesof sandwiched slugs of uids proportional to each other according to specication of microuidic mixers. To doit, a dynamic system is required, i.e. a membrane switch or a periodic micropump.The concept of Taylor-Ariss dispersion can be shifted to a more broad way of interpretation, such as usingobstacles in the channel to disrupt ow and create transverse component of ow,23 which is brought into themain stream in the longitudinal direction. It suggested that convection happened in the both the longitudinaldirection and transverse direction. It is dierent from the conventional Taylor-Aris dispersion which requireslow motion and limited to one-dimension, the ow pattern in a microchannel with obstacles was not exactlyslug-like and can be two-dimension or three-dimension.

4. CONCLUSION AND FUTURE WORK

Simple microchannels were fabricated and dispersion of slugs of solute was experimentally and numericallyinvestigated. It is dicult to apply dynamic systems to create sandwiched slugs in the channel. Alternatively,obstacles can be a passive approach to create convective eects, which in principle is similar to Taylor-Arisdispersion.

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Optimising various geometric parameter of obstacles were conducted to create more eective convection, andwill be reported elsewhere. In the meantime, it will be a promising solution to microuidic application byapplying dynamic systems to introduce sandwiched slugs to create complete mixing.

ACKNOWLEDGMENTS

The authors wish to express their gratitude to the Cooperative Research Centre (CRC) for Microtechnology forthe support in this research project. Dr. Rowan Deam also provided very helpful discussions.

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