microsystems and microstructures 155€¦ · microsystems and microstructures 159 3, model...

Microsystems and Microstructures 155

Fluid modelling in micro flow channels

M.J. Harper,* C.M. Turner," J.E.A. Shaw"

"Company Research Laboratory, British Nuclear Fuels pic,

Springfields Works, Salwick, Preston, PR4 OXJ, UK

Central Research Laboratories Limited, Dawley Road,

Abstract

Over the last few years there has been an increased level of interest in thedevelopment of miniaturised systems which require liquids to flow throughnarrow channels, often only a few microns across. Whilst much has beenwritten on the design and fabrication of pumps, valves, flow channels and ontheir integration into useful systems, relatively little work has been done onmodelling fluid flow behaviour on this scale. As micro-fluidic systems becomemore complex, the ability to model such flows will be seen as increasinglyvaluable.

This paper begins by reviewing the limited amount of previous fluidmodelling work which has been done on the micro-scale. The approach toComputational Fluid Dynamics modelling adopted by the authors and their useof the CFDS-FLOW3D package for this work are then described. Importantassumptions and boundary conditions adopted in the analysis are outlined anda comparison given between code predictions and experimental results forfluid flows through a number of different channels of the order of tens ofmicrons in width, showing good agreement. A more complex example in theuse of CFDS-FLOW3D as a design tool for a micro-fluidic manifold structureis then given which, when compared with experimental observations, againshows that the methods employed give a valuable insight into flow behaviourson this scale. Finally, areas where future work is needed to improve micro-fluidic modelling are summarised.

Transactions on the Built Environment vol 12, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509

156 Microsystems and Microstructures

1. Introduction

With the rapidly expanding growth in the use of microfluidic devices, there isan increased requirement for experimental time scales and costs to be reducedthrough the development of a predictive design capability. An important factorin this development is the effective predictive modelling of fluid motion andchemical reactions within micro flow channels. To gain confidence in thesemodelling predictions for flows on this scale, the models must be firstvalidated through rigorous comparison with experimentation.

Up until now, little work has been carried out to this end, although someilluminating developments have been established by Zemel et. al. [1,2,3] whodemonstrated, using particular organic liquids, that for flows in channels withwidths of down to 0.1 |Lim, observations were within a 5% agreement of thetheoretical solution of the Navier Stokes equations, whilst flows in channelssmaller than this showed significant deviations from Navier Stokespredictions. Makihara et. al. [4] also demonstrated that the Navier Stokesequations may be used for analysis of highly viscous liquid flow inmicrocapillary channels. Some computational fluid dynamic modelling hasbeen used by Ohki et. al. [5] in the characterisation of micro sheath flow.These are the most important results suggesting that flows on the micron scalefall within the continuum representation of liquids, as may be intuitivelyexpected, since 0.1 |im is still a very large length scale compared with thoseassociated with the molecular interactions of the fluid.

Beyond this, very little experimental work has been carried out to validatetheoretical mathematical modelling. In the 60s, Eringen [6,7] and co-workersdeveloped an extensive representation of polar fluid motion on the micronscale which predicted a small but significant deviation from that given by theNavier Stokes equations, although as far as the present authors are aware thiswork was never validated experimentally. Additionally, the motion of microgas flows has shown a significant deviation from the Navier Stokes equations(Beskok et. al. [8]).

The purpose of this paper is to validate further the use of mathematicalmodelling of liquids in micro flow channels, particularly using computationalfluid dynamics (CFD) described here in detail, and so prepare the initialgroundwork required for a reliable predictive design capability to be used forthe optimisation of microfluidic devices.

2. Micro Fluid Modelling Using Computational Fluid

Dynamics (CFD)

The development of CFD mathematical models for fluid motion on the macroscale and the numerical solution of such models using high poweredcomputers has been used with great success over the past 10-15 years so theextension of these well established techniques into micro fluidics is a logical



step forward. Once validated experimentally, such mathematical modelling hasbeen shown to allow full non-intrusive investigation of the flow without costlyand lengthy experimentation, so that only the final optimised design need bepassed to the fabrication stage, thus establishing modelling as a predictivedesign tool, cutting experimentation time and fabrication costs. Othermodelling benefits include the shedding of light onto the fundamental physicaland chemical processes at work within fluids. For instance, if the physics andchemistry of a particular process are not understood, then models can be builtassuming different mechanisms in isolation from the experimentation, andsuch models, when compared with the experimental data, may highlight aparticular model as most accurately representing the physics.

The nature of the flow regime is determined from the magnitude of a scalar,known as the Reynolds Number, which is a non-dimensional value derivedfrom the equations of conservation of mass and momentum:

^ (1)

where p is the fluid density , U is the velocity, d is a length scale characteristicof the flow such as the channel diameter, and |H is the fluid viscosity. On the

macro scale the fluid is considered to be laminar for Re < 2000-3000 whilst onthe micro scale the transition to turbulence is not well established, but is stillconsidered to be laminar for Re < -300 (Pfahler et.al. [1]). All flowsconsidered in this paper have a Reynolds Number of less than 1.0, placingthem well within the laminar regime. This is an important advantage aslaminar flows are easier to model accurately than turbulent flows.

The motion of the fluid in a laminar regime is essentially modelled usingequations (2) and (3) for the conservation of mass and momentumrespectively:

at

(2)

(3)

where p is the pressure and g is the gravitational constantSolution methods for these equations described in this paper are for use

with the commercial CFD package, CFDS-FLOW3D [9]. This has been usedfor its wide, well validated modelling capability, fast efficient solutionalgorithms and ease of use. Formulating and solving the microfluidic problemsdiscussed here using CFDS-FLOW3D consist of three main parts:



(i) Building the geometry within which the fluidic problem is to be solvedThe flow domain is divided into cuboidal cells which completely fill thedomain within which the relevant equations of conservation of fluidmomentum and mass are solved discretely. Inlets, outlets and any internalsolid regions are specified. Consideration is made to ensure that the grid cellsdo not have large aspect ratios and that areas where there are large gradients ofa variable have a finer grid resolution to capture all important fluidic features.This grid adjustment often requires some prior insight into the location ofimportant features existing in the flow and the physical/chemical processespresent.

(ii) Defining the flow problem This consists of describing the physics of thefluid computationally, including relevant modelling assumptions for theincompressibility of the fluid, Newtonian shear stress, the neglect of viscousdissipation and the laminar nature of the flow. Initial and boundary conditions,fluid parameters and program control specifying the numerical algorithms andtolerances are also included at this point. These mathematical equations arethen solved iteratively on a discrete cell by cell basis until the mass residualtolerance, which is effectively the error is less than a prescribed small amount,at which point the problem is accepted as having converged to a uniquesolution.

(iii) Analysing the solution data Output data is usually given in the form ofnumerical fields describing the values of the various variables throughout thedomain. This output data may be then fed through post processing softwarewhere the raw data can be viewed graphically in three dimensions with coloursrepresenting the varying magnitudes of the variable under consideration. Thisallows the data to be fully interrogated to ensure that the solution is of asensible form, and that full detailed information about the flow can beextracted from the data, much of which would be missed by simply lookingthrough pages of numerical data points.

There are certain limitations in using CFD to model fluids.1. Complicated fluidic structures may necessitate dividing the flow domain

into too many cells, thus making the computations to obtain the solution tothe equations of fluid motion within each cell prohibitively expensive inCPU time. This results in most fluidic modelling of this kind being limitedto discrete fluidic units, rather than entire fluidic systems. However, thereis nothing to prevent the solution for the outflow of one unit being used asthe inflow condition on a subsequent unit, eventually to build up piecewise fluidic systems.

2. There are still many physical and chemical processes which have not beenincorporated as models within commercial CFD software e.g. certaincomplex chemical reactions or electro-magnetic coupling.

3. Some physical insight is required to model all flow features accurately.



3, Model Validation

Simple micro fluidic channels and manifolds have been fabricated, and flowswithin them investigated, to establish the validity of using CFD as a techniquefor modelling flows on the micron scale. The channel dimensions weremeasured using an Alpha-Step 200 from Tencor Instruments.

(i) ExperimentationFabrication The micro channels and manifolds were fabricated using acombination of standard micro-engineering wet etch techniques and glass tosilicon anodic bonding.

Mass Flow Data The experimental mass flow data was determined byflowing de-ionised water through the channels under a known pressure, for aset time and the outlet flow of the devices was collected. Hydrostatic pressurewas used to control the inlet pressure and the outlet pressure was atatmosphere, giving an easily measurable pressure difference. Pressuredifferences in the tubing used to connect the device is neglected as virtually allthe pressure drop will occur in the micro-channels. A calibrated balanceaccurate to lOOjug was used to measure the outlet mass. Data collection wasautomated by a 386 PC, allowing readings to be taken over several hours. Thedevice outlet flowed into a container on the balance, giving continuous massmeasurements. This container was sealed to minimise loses due toevaporation. The temperature was measured and used with published data [10]to calculate the fluid viscosity.

Flow Visualisation The fluid flow patterns were experimentally highlightedusing latex beads in the fluid and image processing techniques to obtain snap-shots of the latex bead motion. Near neutral density Sjurn latex beads wereincluded in the de-ionised water and flowed through the micro-manifolddevices under a known hydrostatic pressure. The fluid flow was viewed undera microscope and recorded on video at 25 frames/s. The video images werethen captured onto a PC using a Fast Electronic Movie Machine Pro™ videocapture card. Individual frames were then removed from the video sequenceusing Microsoft Video Edit™ to give a set of four consecutive frames. ZSoftPublishers Paintbrush™ was used to provide image subtraction, allowing thestill background to be removed, highlighting the trajectory of the particles.

(ii) CFD ResultsMicrofluidic channels The CFD geometry used for the model recreatedexactly the geometry used in the experimental channels, allowing directcomparison. The liquid properties of the de-ionised water were incorporatedinto the model, as were the pressure inlet and outlet boundary conditions andinitial conditions. The model was then solved for the steady state situation.



Three particular channel geometries with the same mean path lengths werecompared. The geometries of the three channels are shown in figure 1. Thecross sections of all three were the same due to their similar etchedconstruction. The same pressure drop was the driving force in all three caseswhich produces an average flow velocity of around 0.5 mm/s so that theReynolds Number which was approximately 0.1 falls well within the laminarregime.

The CFD calculated velocity profiles of all three channels showed theexpected fully developed Poisseuille flow profiles, with the highest flow ratethrough the straight channel followed by the curved channel followed by thesquare shaped channel. The comparison of CFD modelled flow rates toexperimental flow rates is shown in figure 2, with errors of less than 5% in allcases.

• Experiment D CFD Model

Moss FlowRote

(a) (b) (c)

Figure 1Micro Channel Geometry:(a) Straight Channel, (b) CurvedChannel, (c) Square Channel(All dimensions in jiim)

SquareChannel

Figure 2 Channel GeometryMass Flow Rate ComparisonBetween Experiment& CFD Model

Micro manifolds In the same way as for the micro channels, CFD modelswere built to represent exactly the geometry and physics of the real deviceshown in figure 3 with the SEM image of the manifold shown in figure 4 (Alldimensions are in microns). The deep area shown on the SEM is where theinlet syringe is inserted. The rest of the manifold is 25 )Lim deep. Again theflow was driven by a known pressure drop forcing the flow to enter throughthe multiple 30 micron channels and exit through the central 50 micron widechannel leading from the central triangular area.



14(H

Figure 3Manifold Geometry

Figure 4SEM Image of Manifold

The velocity distribution within the manifold was calculated fromphotographic images shown in figure 5 which illustrate a series of foursuperimposed snapshots at known time steps. The distance between thesuccessive locations of the latex balls can be easily measured from thephotographs allowing direct measurement of each ball's velocity. Assumingthat the latex balls do not affect the flow, but are simply swept along with it,these results may be used as flow velocity validation data for the CFD model.

Figure 5Photographic snapshots of latex balls

The CFD solution (figure 6) shows contours of speed. Superimposed uponthese are mathematically calculated particle tracks effectively following thestreamlines within the flow.



Rreall: Port id* Track:/ g Contours of Speed (H/

Figure 6CFD Solution: Contours of Speed (m/s) with superimposed streamlines

These qualitatively compare very well and show the proportional speedincrease at the constriction.

4. Future work

Micro fluidic modelling is still in its infancy requiring further work to place iton an equal footing with its well established larger scale counterpart. The mostimportant difference between modelling macro and micro scale flows is thedominance of interfacial and wall adhesion surface tension effects on themicro scale. These effects need to be included into any CFD model used inorder to represent the micro flow accurately. Another important logisticalimprovement is in accurate data collection techniques, particularly in obtainingmore accurate distributions of velocity, acceleration and turbulence intensity.

In addition to the improvements to the modelling, there are a number ofapplication areas requiring model development and validation. Theseapplication areas include the optimal design of heat transfer units frommicroengineered devices using micro fluidic cooling systems, particularlyapplicable in the microelectronics industry; precise control of chemicalreactions which are inherently safer and more controllable; micro gas flows,although these have been shown to exhibit slip at the wall (Beskok et. al. [8]);micro switching systems; micro mixing; and the useful miniaturisation of thewealth of existing macro fluidic systems, whether for optimal control, safety orcompactness.



5. Conclusion

CFD modelling of flows within microfluidic devices has been further validatedthrough direct quantitative and qualitative comparison with experimentation.Results suggest that CFD is a valid technique to model accurately the fluiddynamics within complex micro geometries, for channels with dimensionsdown to 25 |Lim, and that this effective mathematical modelling of micro flowsprovides a cheap and fast predictive tool for the design of microfluidic devices.

Acknowledgements

The authors acknowledge the assistance of their colleagues who have assistedin the fabrication of the devices used in this work and who have participated indiscussions of the results.

References

1. Pfabler, J., Harley, J., Bau, H. & Zemel, J. Liquid Transport in Micron andSubmicron Channels, Sensors and Actuators, 1990, A21-A23, pp. 431-434.

2. Bau, H., Harley, J., Hava, S., Pfabler, J., Toroa, A., Wilding, P., Urbanek, W.,Wang, T.K., Weisberg, A. & Zemel, J. Energy and Mass Transport in MesoscaleStructures, Proceedings of the 8th Symp. on Energy Engineering Sciences: Micro/Macro Studies of Multiphase Media, pp. 71-78, 1991.

3. Urbanek, W., Zemel, J.N. & Bau, H.H. An Investigation of the TemperatureDependence of Poiseuille Numbers in MicroChannel Flow, J. Micromech.MfcmcMg., 1993, 3, pp. 206-208.

4. Makihara, M., Sasakura, K. & Nagayama, A. The Flow of Liquids inMicrocapillary Tubes - Review of the Applicability of the Navier StokesEquations, Seimitsu Kogakkai-Shi (Precision Engineering Institute Journal)1993, 59/3/93, pp.31-36 (399-404).

5. Ohki, H., Miyake, R. & Yamazaki, I. Three-Dimensional Flow Characteristics ofMicro Flow Chamber, in FLUCOM '91, pp. 125-129, Proceedings of the 3rd Int.Symp. on Fluids Control, Measurement and Visualisation, 1991.

6. Eringen, A., Simple Microfluids, Int. J. Eng. Sci., 1964, 2, pp. 205-217.7. Eringen, A., Theory of Micropolar Fluids, J. Math. Mech., 1966, 16, pp. 1-18.8. Beskok, A. & Karniadakis, G.E. Simulation of Slip-Flows in Complex Micro-

Geometries, Micromechanical Systems, ASME, 1992, 40, pp. 355-370.9. CFDS-FLOW3D User Guide, AEA Technology, CFDS, 8.19 Harwell

Laboratory, Oxon, 1994.10. Anderson, H.L., editor-in-chief, A Physicist's Desk Reference: The second

edition of physics vade mecum, 2nd. ed., pp. 200, 1989.


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