Effects of rectangular microchannel aspect ratio on laminar friction constant

Ian Papautskya, Bruce K. Galeb, Swomitra Mohantya, Tim A. Ameelc, and A. Bruno Frazierd

Departments of aBioengineering and cMechanical Engineering, University of Utah bDepartment of Biomedical Engineering, Louisiana Tech University

dSchool of Electrical and Computer Engineering, Georgia Institute of Technology

ABSTRACT

In this paper, the effects of rectangular microchannel aspect ratio on laminar friction constant are described. The behavior of fluids was studied using surface micromachined rectangular metallic pipette arrays. Each array consisted of 5 or 7 pipettes with widths varying from 150 µm to 600 µm and heights ranging from 22.71 µm to 26.35 µm. A downstream port for static pressure measurement was used to eliminate entrance effects. A controllable syringe pump was used to provide flow while a differential pressure transducer was used to record the pressure drop. The experimental data obtained for water for flows at Reynolds numbers below 10 showed an approximate 20% increase in the friction constant for a specified driving potential when compared to macroscale predictions from the classical Navier-Stokes theory. When the experimental data are studied as a function of aspect ratio, a 20% increase in the friction constant is evident at low aspect ratios. A similar increase is shown by the currently available experimental data for low Reynolds number (<100) flows of water.

Keywords: Micromachined pipette arrays, micromachined channels, microscale effects, aspect ratio effects, laminar fluid flow

1. INTRODUCTION Over the past fifteen years, micromachining technology has been used to develop a number of microfluidic systems in silicon, glass, quartz, or plastics. Microchannels and chambers are the essential part of any such system. In addition to connecting different devices, microchannels are also used for reactant delivery, as biochemical reaction chambers, in physical particle separation, in inkjet print heads, and as heat exchangers for cooling computer chips.1-14 Currently, fluid flows in microchannels and micromachined fluid systems (e.g., pumps and valves) are analyzed using the classical Navier-Stokes equations.15,16 However, a number of publications indicate that flow characteristics on the microscale are different from the macroscale and that the Navier-Stokes equations are incapable of explaining the occurring phenomena.17,18 Thus, in order to successfully design and fabricate such microfluidic devices effectively, the fluid flow on the microscale must be understood.

The first studies of fluid flow on the microscale were performed by Wu and Little.19,. They conducted experiments measuring Darcy friction factors for both laminar and turbulent gas flow in microchannels to evaluate the performance of Joule Thompson cryogenic devices. The microchannels were etched in silicon or glass, with hydraulic diameters ranging from 50 µm to 80 µm. The observed Darcy friction factors were larger than predicted by classical macroscale theory.

Later studies by the University of Pennsylvania group used channels approximately 100 µm wide, with heights ranging from 0.5 µm to 50 µm.20,21,22 The channel cross-sections were either trapezoidal or rectangular, with a relative roughness (absolute roughness ε/hydraulic diameter Dh) of approximately 2%. The fluids tested included alcohol, silicon oil, nitrogen, and

Further author information – I.P. (correspondence): E-mail: ian.papautsky@m.cc.utah.edu; Telephone: (801) 581-8611; Fax: (801) 585-5361

helium. These investigations showed a decrease in Darcy friction factors on the order of 10% to 20% when compared to macroscale tubes at dynamically similar conditions.

Experiments conducted by Choi were performed using glass microtubes with diameters ranging from 3 µm to 81 µm.23 This was the first experimental study using microtubes, while most other researchers used channels with bonded glass covers. The tubes were very smooth, with relative roughness of 0.0003. The data showed Darcy friction factors to be lower than those predicted by theory. Yu et al. extended the work of Choi by conducting experimental studies of flow and heat transfer characteristics in microtubes.24 Yu et al. determined the Darcy friction factors of distilled water and nitrogen in tubes with diameters ranging from 19 µm to 102 µm and found the laminar friction constant (Darcy friction factor f times Reynolds number Re) to be approximately 19% less than the theoretical value of 64.

In later studies, Wilding et al. analyzed flow of water and various biological fluids in glass-capped silicon microchannels.25 The channels had trapezoid cross-section and varied from 40 µm to 150 µm in width and from 20 µm to 40 µm in height. The data indicated an approximately 50% increase in the Darcy friction factor from the theoretical values. Similar results were obtained by Jiang et al., who studied flow of water through rectangular and trapezoid cross-section channels.16 The microchannels used in this study were formed by etching a silicon substrate and capping it with a glass wafer. The channel dimensions ranged from 35 µm to 110 µm in width and from 13.4 µm to 46 µm in height. Other recent contributions include Arkilic et al.26 who considered the effect of compressibility on microchannel flow and Pong et al.27 and Shih et al.28 who conducted the most detailed studies to date by measuring not only the overall pressure drop and flow rate, but also pressure distribution along the microchannel.

When referring to the friction factor (f), it is a common practice to use f as a function of Reynolds number (Re). The data are also often reported as a normalized friction constant C* that is defined as

ltheoretica

alexperimant

Ref

RefC =* (1)

where f· Re is dependent on the channel cross-section.

The normalized friction constant C* data from the previously mentioned studies are presented in Figure 1. It is clear that the currently available experimental data are inconclusive, as data appears both above and below the theoretical values. Data for each investigator, however, is consistently either greater or less than the theoretical predictions. Some of these anomalous results may be attributed to roughness of the channels and to uncertainty in the determination of the channel dimensions. Part of the discrepancy may also be due to the lack of a well-controlled surface structure, as most channels are fabricated by the bonding of silicon and glass. In addition, while some investigators estimate the entrance effects, in order to truly ascertain the entrance and exit pressures, the measurements must be taken at locations within the channel where the flow is not affected by the entrance and exit.

Several effects, which are normally neglected when considering macroscale flow, may exist on the microscale. The first of these microscale phenomena are the two- and three-dimensional transport effects. As characteristic lengths are reduced to the same order of magnitude as the hydrodynamic boundary layer thickness, momentum transport in directions other than the streamwise direction can increase significantly.29 Another observed microscale effect is that of temperature variations in the transport fluid that can cause a significant variation in fluid properties (e.g., apparent fluid viscosity) throughout a microsystem, invalidating the often-used assumption of constant properties.22

Fluid dynamics in general is an empirical science that relies heavily on experimentation to determine the effects of changes in flow parameters. Unfortunately, until recently the ability to fabricate channels for microfluidic testing has been limited. With the advent of micromachining, a completely new realm of techniques for precise fabrication of microscale channels has emerged. The ability for precision fabrication of channels, though, has not yet resolved the question regarding microfluidic effects. While a significant effort has been vested into determining these microfluidic effects, the data are inconclusive. Whether the difficulty related to collecting and analyzing data of this type is related to difficulty in making measurements at dimensions of this size or other factors, the questions regarding scale effects in microchannel fluid flow have not yet been answered satisfactory. Consequently, much work remains to be completed in this area so that a complete understanding of fluid dynamics in small channels can be realized.

2. THEORY The momentum equation that governs fluid dynamics is one of the most important equations of fluid mechanics. The momentum equation is essentially Newton’s second law applied to a continuum. The momentum equation states that the time rate of change of linear momentum of a continuum is equal the sum of the forces acting on the continuum. Two types of forces are typically present: body forces that act on the bulk material inside the continuum, and surface forces that act at the boundary surface. Using symbolic notation, the differential momentum equation is written as

( ) ôFVVVV

⋅∇++−∇=∇⋅+∂∂

= ρρρ ptDt

D (2)

where D/Dt denotes the material derivative, ρ is the fluid density, V is the fluid velocity vector, t is time, p is pressure, F is the external force vector (e.g., gravity), ττ is the shear stress tensor, and µ is the fluid viscosity. For a Newtonian fluid, the viscous stress is proportional to the rate of shearing strain and may be expressed in terms of velocity gradients and fluid properties. These equations of motion are commonly referred to as the Navier-Stokes equations. The equations are greatly simplified when applied to incompressible flows in which variations in fluid viscosity can be neglected. Under these conditions the equations reduce to

VFV 2∇−+−∇= µρρ p

Dt

D (3)

If additional simplifying assumptions are made, such as fully developed two-dimensional flow and negligible external forces, the Navier-Stokes equations are simplified to (x direction only)

0

0.5

1

1.5

2

2.5

0.001 0.01 0.1 1 10 100 1000 10000

Reynolds Number (Re )

No

rmal

ized

Fri

ctio

n C

on

stan

t (C

*)Wu&Little [19]

Pfahler 1 [22]

Pfahler 2 [22]

Pfahler 3 [23]

Choi [24]

Yu [25]

Wilding [26]

Jiang [16]

Figure 1. Comparison of the currently available experimental data.

∂∂

+∂∂

=∂∂

2

2

2

2

y

v

x

u

x

pµ (4)

where x and y are defined by a right-handed coordinate system, and u and v are the velocity components of the fluid in x and y directions.

Two non-dimensional parameters often used to represent fluid flow data are the Reynolds number Re and the Darcy friction factor f. The Reynolds number characterizes the significance of internal and viscous effects and is defined as

µ

ρ have DVRe = (5)

where Vave is the average velocity of the fluid, and Dh is the hydraulic diameter. For channels, the hydraulic diameter is given by

PerimeterWetted

AreasectionalCrossDh

−×=

4 (6)

The Darcy friction factor relates roughness effects to pressure drop in pipes and ducts and is defined by the following equation:

2

8

aveVf

ρwô

= (7)

where ττw is the shear stress at the wall. The Darcy-Weisbach equation, Equation 8, is derived by solving the energy and momentum equations for fully developed pipe flow and combining the result with the Darcy friction factor relation. This equation is used to determine the head loss or pressure drop. The Darcy-Weisbach equation holds for both laminar and turbulent flow.

g

V

D

Lfh ave

hf 2

2

= (8)

where L is channel length, and g is the acceleration due to gravity. The laminar solution for head loss may be found by solving the continuity and momentum equations for the Hagen-Poisseuille flow problem. The result is the commonly used laminar Darcy friction factor.

2

2

ave

h

LV

pDf

ρ∆

= (9)

where ∆p is the pressure difference within the channel. Writing the friction factor in terms of the fluid flow rate (Q = Vave A) and rectangular channel height h and width w yields

wh

wh

LQ

pf

+∆

=33

2

4

ρ (10)

Laminar friction constants for fully developed flow may be determined analytically for various aspect ratios (α). The aspect ratio has been defined as either height/width or width/height such that 0 � α � 1. Solutions provided by White 30 for the

friction constant are presented in Table 1. Since the friction constant is only a function of α, the data in Table 1 can be fit with a polynomial equation (with coefficient of determination R2 = 0.9995)31.

f Re = 96(1-1.3553α+1.9467α2-1.7012α3+0.9564α4-0.2537α5) (11)

Laminar friction constants for flows through channels or ducts with non-rectangular cross-sections are given by Shah and London.32

3. FABRICATION Experiments have been conducted in metallic microchannels using water. Arrays of 5 or 7 metallic microchannels were fabricated using surface micromachining fabrication techniques similar to those previously reported.33,34 A schematic diagram of the fabrication process is shown in Figure 2. Sections of microchannels that are extended from a silicon substrate were fabricated on top of 4-6 µm p+ silicon sacrificial membranes created by KOH etching. The bottom wall was formed by micro electroforming nickel. Thick photoresist was deposited and patterned on top of the nickel. The thick photoresist was used to precisely define the inner dimensions of the microchannels and served as a sacrificial layer removed later in the process. The top and side walls were formed by electroforming nickel over a sputtered seed layer. Once the walls were formed, the thick photoresist was removed from the micropipette interior in an acetone bath. Dry etching using SF6 was used to remove the sacrificial membrane and release the microstructures.

Access ports were fabricated by etching through the substrate for backside access. A cross-section schematic of the micromachined pipette input ports and interface is shown in Figure 3. The use of bottom input ports allowed the macroscale interface between microchannels and macro-tubing to be moved from the front of the device to the back, resulting in a more robust interconnect.

Table 1. Laminar Friction Constant for Various Aspect Ratio

Rectangular Channels

αα f Re

0 96

0.05 89.91

0.1 84.68

0.125 82.34

0.167 78.81

0.25 72.93

0.4 65.47

0.5 62.19

0.75 57.89

1 56.91

Figure 2. Schematic diagram of the micromachined pipette fabrication process. (a) Create sacrificial membrane and deposit seed layers; (b) pattern seed layers and electroform the bottom shell; (c) deposit and pattern sacrificial thick photoresist to define pipette inner dimensions; (d) sputter seed layers and electroform top and side walls; (e) remove sacrificial thick photoresist using acetone bath; (f) remove sacrificial silicon membrane using SF6 plasma.

Arrays of micromachined pipettes of different dimensions have been fabricated on top of silicon substrates. A summary of pipette dimensions is given in Table 2. The pipettes used in the study were 11.75 mm in length with 2 mm extending from the substrate, while 4 mm separated the pressure measurement ports from the pipette ends. The electroplated nickel walls were 10-20 µm thick. An example of a micromachined pipette array fabricated on top of a silicon substrate is shown in Figure 4.

The surface roughness of the micromachined structures was measured using an Atomic Force Microscope (Digital Instruments Dimension 3000). The RMS roughness of the electroplated nickel that was used to build the channels, measured over the surface of a 25-µm square, was determined to be less than 15 nm resulting in a relative roughness of 0.00028.

4. METHODS

Interfaces machined from acrylic were used to connect the fabricated micromachined pipette arrays (MPA) with 0.794 mm diameter Teflon tubing. The acrylic interfaces were attached to the MPA using a ultra-violet (UV) curable medical device adhesive (Loctite Inc.). Teflon tubing connected the acrylic interfaces to either a pressure source or a pressure transducer. A photograph of an interfaced micromachined pipette array is shown in Figure 5.

The acrylic interface provided ports for the fluid inlet and for downstream static pressure measurement. For the range of experimental flow rates, the entrance length of the developing flow was calculated to be less than 0.5 mm. However, the static pressure measurement port was positioned 4 mm downstream in order to eliminate any possible entrance effects. The extended ends of the pipettes were used as outputs.

Each array consisted of 5 or 7 pipettes. While each pipette had an individual input port, the acrylic interface provided a single pressure source through a manifold. The volume

Figure 3. Schematic diagram of the micromachined pipette array cross-section. The pipette arrays are interfaced from the backside in order to move the bulky interface to the back. Two input ports are used: one as an inlet, and one for downstream static pressure measurements.

Table 2. Summary of the Fabricated Pipette Array Dimensions

Name Height,

µm Width,

µm Length,

mm Aspect Ratio

Cthor

1B 22.71 600 7.75 0.0379 91.33

2A 23.32 600 7.75 0.0389 91.22

2B 24.65 600 7.75 0.0411 90.96

3A 24.94 300 7.75 0.0831 86.39

4A 26.35 150 7.75 0.1757 78.11

Figure 4. Scanning electron microscope images of a typical micromachined pipette array. A. Individual pipettes of the array are 11.75 mm in length, 190 µm in width, and 50 µm in height. Pipettes extend 2 mm of the substrate edge. B. The inner cross-sectional dimensions of pipettes are 150 µm x 22 µm. Structural material is electroformed nickel.

A

B

of the manifold was several orders of magnitude greater than the volume of each pipette, allowing a uniform pressure distribution across input ports of the entire array. Thus, the flow through each pipette was assumed equal.

The connections between the pump, tubing, pressure transducers, and other equipment were of critical importance in the accurate testing of the micromachined pipette arrays. A schematic of the complete setup as well as the packaging connections is shown in Figure 6. A controllable syringe pump (KD Scientific) was used as the flow source with 3, 5, and 10 mL syringes. The pump was connected to the microchannel using 1.588 mm Teflon tubing that was reduced to 0.794 mm Teflon tubing. The Teflon tubing was then connected to the acrylic interface using a PEEK Microtight nut from Upchurch Scientific. Two Microtight nuts were attached to the interface allowing both flow injection and static pressure measurement.

Three different pressure transducers (Omega, PX180 series) were used to measure absolute pressure. Two transducers produced signals between 0 and 100 mV as the output signal across their pressure measurement range with an input voltage of 10 V. One transducer had a range of 689.5 kPa, the other a range of 103.4 kPa. The third transducer produced an output signal between 4 and 20 mA, had a range of 103.4 kPa, and used excitation input of 20 V. The tubing connecting the transducer to the channel was 1.588 mm Teflon tubing that was again reduced to 0.794 mm tubing. The power supply used was an HP 6128C.

The output of the pressure transducers was measured using a HP 3458A multimeter with 8-digit accuracy. The measurements were then sent to a PC where the raw numbers were converted into pressures, friction factors, and Reynolds numbers using LabView software and the equations presented in the theory section.

Prior to each experiment, tubing connecting the interface and transducer was bled in order to eliminate any trapped air bubbles. During pipette dispensing, a water droplet formed at the end of the array. Therefore, to eliminate effects of the

Figure 5. A photograph of the machined acrylic interface attached to a micromachined pipette array (MPA). Teflon tubing (0.794 mm diameter) connects the interface to an external pressure source.

PressureTransducer

Computer/Signal

Processing

Meter

Power Supply

Acrylic Interface

MicromachinedPipette Array

Teflon Tubing

PUMP

Figure 6. A schematic diagram of the system setup and components showing flow from a syringe pump to the micromachined pipette array, pressure measurement using a downstream static port, and a sequence for signal processing and analysis.

Static Port

MPA

Input Port

droplet surface tension, the pipette arrays were submerged just below the water surface.

5. RESULTS AND DISCUSSION The experimental data were obtained for water flows at room temperature with Re spanning three orders of magnitude from 0.001 to 10. The experimental data in the terms of normalized friction constant C* as a function of Re are presented in Figure 7. Each data point is the statistical average of at least ten values, with vertical bars indicating the range of the deviation.

An increase in the friction constant compared to the macroscale laminar flow predictions at the same Re is shown in Figure 7. The average C* for the experimental data is approximately 1.1956, a nearly 20% increase from the theoretical value of one. The uncertainty analysis indicates the average uncertainty for f and Re to be 7.77 % and 1.51 % respectively at low pressures and 5.61% and 0.13% at high pressures. The greatest contribution to the uncertainty is related to the channel height measurement.

An important effect that is readily apparent in Figure 7 is the reduction in friction constant as the pressure, and thus the flow rate and the Reynolds number, increases. This may be a microscale effect. A possible explanation suggested earlier35 was the compliance of the plastic syringe used in the experiments. In the experiments described in this paper, a glass syringe was used, eliminating the possibility of compliance. Another possible explanation is the change in channel height or width that may occur at higher pressures by either elongation of the channel sidewalls, deflection of the channel walls, or their combination. Since the Darcy friction factor varies approximately as the third power of the microchannel height (f = O(h3)), as seen from Equation 10, height measurements are very critical and any change in channel dimensions during experiments will significantly affect the results..

Using the classical stress-strain relationship, the elongation of the channel sidewalls is estimated to be approximately 1 nm for the maximum fluid pressure.36 This value is four orders of magnitude less than the microchannel height and, therefore, this effect may be neglected.

The microchannels had a 10-µm thick nickel top and bottom walls, with the bottom wall electroplated on top of a 400-µm thick silicon substrate. Consequently, the bottom wall can be assumed rigid with no deflection. The sidewalls are 20 µm

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.001 0.01 0.1 1 10

Reynolds Number (Re)

No

rmal

ized

Fri

ctio

n C

on

stan

t (C

*)

MPA 1B

MPA 2A

MPA 2B

MPA 3A

MPA 4A

Figure 7. The normalized friction constant C* as a function of Reynolds number Re. The uncertainties of the friction factor f and Reynolds number Re are 7.77 % and 1.51 % respectively at low pressures and 5.61% and 0.13% at high pressures.

high and 20 µm thick, formed by the overlap of the top and bottom shells. Thus, it is unlikely that any deflection would occur at the channel sidewalls. The expected deflection of the microchannel walls can then be estimated by modeling the top microchannel wall as a membrane that is fixed along the long edges and is simply supported by the short edges with a uniformly-distributed load. The membrane deflection for such a structural design is shown in Equation 12 where ∆h is the maximum membrane deflection, α is a constant related to geometry, P is pressure, w is channel width, E is Young’s modulus of the membrane, and t is the membrane thickness.

3

4

Et

Pwh

α=∆ (12)

Given that the Young’s modulus of nickel is 207 GPa and α = 0.0016 for this design36, the maximum membrane deflection is estimated to be approximately 0.3 µm for a 10-µm thick wall in a 20-µm high microchannel, an approximate 1.5% change. Thus, deflection of the top wall will result in an approximate 4.6% change in the Darcy friction factor. This change, however, is too small to explain the 10-15% drop off in the normalized friction constant at higher Reynolds numbers.

Writing the Darcy friction factor in terms of maximum membrane deflection yields Equation 13

2

3

)1(

8

αµ +∆

=⋅wh

LQ

pRef (13)

As Equation 13 shows, the friction factor varies as the third order of membrane deflection. Thus, a partial explanation of the trend shown in Figure 7 is a result of the fact that f was calculated assuming no deflection, and some deflection is expected as pressure and Re are increased. The increased cross-sectional area produces a reduction in f. In fact, if the top walls of microchannels did deflect during tests, then the reported friction factors are lower then the actual values and thus must be increased appropriately.

Given the importance of the channel dimensions measurement, there is a possibility that the currently available experimental data contains slight errors that become magnified during the data analysis process. The earlier papers by the University of Pennsylvania group,20,21 for example, specify that the channel height measurements were within ±20% accuracy, which would indicate that the friction factor might vary by as much as 73%. Such variation in the friction factor might reposition the data for flow of nitrogen and n-propanol from the current location of approximately 20% below the theoretical values (Figure 1).

The microchannel inner height was measured using a KLA Tencor P10 profilometer to high accuracy (±1 nm) during the fabrication process following the sacrificial thick photoresist deposition. The channel height was assumed to remain unchanged throughout the remaining fabrication steps. During the wet etching (acetone) of the sacrificial thick photoresist, however, the top walls could deflect inward due to the strong capillary forces.37 Also, it is possible that some residual sacrificial photoresist remained inside the microchannels. Presence of such obstructions inside the channel would significantly reduce the channel’s hydraulic diameter. To ensure that it was not the case, the top shell of the microchannels was removed using needle-nose tweezers and the inside visually inspected to reveal clean surfaces.

Several other measurements are critical to the accurate determination of friction factors, in addition to the microchannel height. The length and width of the channel were measured to high accuracy (±1 nm) using a profilometer (KLA Tencor P10). The temperature was measured using a thermometer with accuracy to 1/100 °C. The temperature was used to calculate the density and viscosity of the DI water used in the experiments. The temperature was converted to a density and viscosity using a curve fit to standard data. Fluid flow rates were set on the pump that measured flow rates with accuracy to 1/100 mL/hr. Error in flow rates could come from a number of areas including the velocity of the pump and syringe diameter measurements especially at the higher Reynolds numbers, though both are expected to be highly accurate.

The experimental data in the form of the friction constant C as a function of the aspect ratio α, as well as the currently available data from several other investigators is shown in Figure 8. The black solid line represents Equation 11. It is clear that the majority of the available experimental data are above the theoretical predictions, especially at the lower aspect ratios. Overall, the data may be separated into three groups. First, the data at aspect ratios of 0.1 or less were collected using

microchannels with heights of approximately 20 to 30 µm and at Reynolds numbers less than 10. The second set, for 0.2 < α < 0.5, was gathered using microchannels with heights in the 20 to 40 µm range and Reynolds numbers of up to 100. The third set of data, for α > 0.9, was collected in microchannels with heights approximately 100 µm and Reynolds numbers up to 1000.

The larger microchannels, with characteristic dimensions on the order of 100 µm, produce function constants very close to the theoretical predictions. It is possible that at such dimensions, the microscale effects are minimal and are not readily observable. It is also possible that flow at high Reynolds numbers may minimize the microscale effects. The data for α < 0.5, obtained using microchannels with heights of 20-40 µm, result in friction constants approximately 20% above the predicted values. At these smaller microchannel heights, the microscale effects are more evident. However, it is also possible that this apparent microscale effect is due to the decrease in Reynolds number.

The data presented in this paper suggest the presence of microscale effects at low aspect ratios and low Reynolds number flows, though it is not clear which is the contributing and dominating factor. It is possible that both conditions play a role. Experiments must be conducted using rigid microchannels of various aspect ratios within the same range of Reynolds numbers in order to more definitely determine the effects of rectangular microchannel aspect ratios on laminar friction constant.

6. CONCLUSIONS Micromachining technologies have been used as a method of batch fabrication of metallic micromachined pipette arrays. These arrays were used to study fluid behavior on the microscale. The experimental results indicate an approximate 20% increase in the laminar friction constant for a specified driving potential compared to macroscale predictions from the classical Navier-Stokes theory. The experimental data also indicate that microscale effects are present at low microchannel aspect ratios and at low Reynolds number flows. Further studies will determine which of the contributing factors dominate the microscale flow characteristics in rectangular microchannels. More data must be collected to make a firm conclusion.

0

20

40

60

80

100

120

140

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Aspect Ratio (αα )

Fri

ctio

n C

on

stan

t (C

= f

Re)

Gale [38]

Jiang [16]

Papautsky

Wu [39]Wilding [26]

Theory

Stanley [40]

Figure 8. Experimental data presented as the friction coefficient C = f Re as a function of aspect ratio α.

7. ACKNOWLEDGEMENTS The authors would like to acknowledge Jeff Griffin at Clariant Corp. for supplying AZ4620 photoresist and Olivia Wang for help with data analysis. This material is based upon work supported by a University of Utah Technology Innovation Grant.

8. REFERENCES 1. R. Srinivasan, I.-M. Hsing, J. Ryley, M. P. Harold, K. F. Jensen, and M. A. Schmidt, "Micromachined chemical reactors

for surface catalyzed oxidation reactions," in IEEE Solid-State Sensor and Actuator Workshop, Hilton Head, SC, June 2-6, 1996.

2. D. J. Harrison, Z. Fan, K. Fluri, and K. Seiler, "Integrated electrophoresis systems for biochemical analysis," in IEEE Solid-State Sensor and Actuator Workshop, Hilton Head, SC, June 13-16, 1994.

3. S. C. Terry, J. H. Jerman, and J. B. Angell, “A gas chromatographic air analyzer fabricated on a silicone wafer,” IEEE Trans. Electron. Devices, vol. ED-26, p. 1880, 1979.

4. R. R. Reston and E. S. Kolesar, “Silicon-micromachined gas chromatography system used to separate and detect ammonia and nitrogen dioxide--part I: design, fabrication, and integration of the gas chromatography system,” IEEE J. Microelectromech. Sys., vol. 3, p. 134, 1994.

5. G. Ocvirk, E. Verpoorte, A. Manz, H. M. Widmer, “Integration of a micro liquid chromatography onto a silicon chip,” in Proc. Tranducers 95, Stockholm, Sweden, June 25-29, 1995.

6. A. Manz, “Miniaturized chemical analysis systems based on electroosmotic flow,” in IEEE Micro Electro Mechanical Systems Conf., Nagoya, Japan, Jan. 26-30, 1997.

7. C. S. Effenhauser, A. Manz, and M. H. Widmer, “Glass chips for high-speed capillary electrophoresis separation with submicrometer plate heights,” Anal. Chem., vol. 65, p. 2637, 1993.

8. S. C. Jacobsen, A. W. Moore, and J. M. Ramsey, “Fused quartz substrates for microchip electrophoresis,” Anal. Chem., vol. 67, p. 2059, 1995.

9. B. K. Gale, A. B. Frazier, and K. Caldwell, “Micromachined electrical field-flow fractionation (µ-EFFF) system,” in IEEE Micro Electro Mechanical Systems Conf., Nagoya, Japan, Jan. 26-30, 1997.

10. R. C. Anderson, G. J. Bogdan, and R. J. Lipshutz, “Miniaturized genetic-analysis system,” in IEEE Solid-State Sensor and Actuator Workshop, Hilton Head, SC, June 2-6, 1996.

11. J. Chen and K. D. Wise, "High-resolution silicon monolithic nozzle array for inkjet printing," in Proc. Transducers 95, Stockholm, Sweden, June 25-29, 1995.

12. M. B. Bowers and I. Mudawar, "Two-phase electronic cooling using mini-channel and micro-channel heat sinks: part 1—design criteria and heat diffusion constraints," Trans. ASME, vol. 116, p. 290, 1994.

13. N. C. J. Chen, D. K. Felde, and G. L. Yoder, "Thermal analysis of two-phase microchannel cooling," in ASME Micro Electro Mechanical Systems Conf., Atlanta, GA, Nov. 17-22, 1996.

14. Tuckerman, D. B. and Pease, R. F. W., “High-performance heat sinking for VLSI,” IEEE Electron Dev. Lett., Vol. EDL-2, No. 5, pp. 126-129, 1981.

15. J. P. Brody and P. Yager, "Low Reynolds number micro-fluidic devices," in Proc. Solid-State Sensor and Actuator Workshop, Hilton Head, SC, June 2-6, pp. 105-108, 1996.

16. X. N. Jiang, Z. Y. Zhou, J. Yao, Y. Li, and X. Y. Ye, "Micro-fluid flow in microchannel," in Proc. Transducers 95, Stockholm, Sweden, June 25-29, pp. 317-320, 1995.

17. X. F. Peng and B. X. Wang, "Liquid flow and heat transfer in microchannels with/without phase change," in Proc. 10th International Heat Transfer Conference, Brighton, UK, Aug. 14-18, pp. 159-177, 1994.

18. D. B. Holmes and J. R. Vermeulen, "Velocity profiles in ducts with rectangular cross sections," Chem. Engng. Sci., Vol. 23, pp. 717-722, 1968.

19. P. Wu and W. A. Little, “Measurement of friction factors for the flow of gases in very fine channels used for microminiature Joule-Thomson refrigerators,” Cryogenics, No. 5, pp. 273-277, 1983.

20. J. Harley, J. Pfahler, H. Bau, J. N. Zemel, “Transport processes in micron and submicron channels,” Proc. ASME Heat Transport Processes, Vol. HTD-116, pp. 1-5, 1989.

21. J. Pfahler, J. Harley, and H. Bau, “Liquid transport in micron and submicron channels,” Sensors and Actuators, Vol. A21-23, pp. 431-434, 1990.

22. J. Pfahler, J. Harley, H. Bau, and J. Zemel, “Gas and liquid flow in small channels,” Proc. ASME Micromech. Sensors, Actuators, and Systems, DSC-Vol. 32, pp. 49-60, 1991.

23. S. B. Choi, R. F. Barron, and R. O. Warrington, “Fluid flow and heat transfer in microtubes,” Proc. ASME, DSC-Vol. 32, pp. 123-134, 1991.

24. D. Yu, R. Warrington, R. Barron, and T. Ameel, “An experimental and theoretical investigation of fluid flow and heat transfer in microtubes,” Proc. 4th ASME/JSME Thermal Engineering Joint Conf., March, Maui, Hawaii, 1995.

25. P. Wilding, M. A. Shoffner, and L. J. Kircka., "Manipulation and flow of biological fluids in straight channels micromachined in silicon," Clin. Chem., Vol. 40, pp. 43-47, 1994.

26. E. B. Arkilic, M. A. Schmidt, and K. S. Breuer, “Gaseous slip flow in long microchannels,” IEEE J. Microelectromech. Sys., Vol. 6, No. 2, pp. 167-178, 1997.

27. K.-C. Pong, C.-M. Ho, J. Liu, and Y.-C. Tai, “Non-linear pressure distribution in uniform microchannels,” Proc. ASME, FED-Vol. 197, pp. 51-56, 1994.

28. J. C. Shih, C.-M. Ho, J. Liu, and Y.-C. Tai, “Monatomic and polyatomic gas flow through uniform microchannels,” Proc. ASME, DSC-Vol. 59, pp. 197-203, 1996.

29. S. W. Ma and F. M. Gerner, “Forced convection heat transfer from microstructures,” J. Heat Transfer, vol. 115, pp. 872-880, 1993

30. F. M. White, Fluid Mechanics, 3rd ed., New York, NY: McGraw Hill, 1994.

31. J. P. Hartnett and M. Kostic, “Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts,” Advances in Heat Transfer, vol. 19, pp. 247-356, 1989

32. R. K. Shah and A. L. London, Laminar Flow Forced Convection in Ducts, Academic Press, New York, 1978.

33. I. Papautsky, J. Brazzle, H. Swerdlow, and A. B. Frazier, "A low temperature, IC compatible process for fabricating surface micromachined metallic microchannels," IEEE J. Microelectromech. Sys., Vol. 7, No. 2, pp. 267-273, 1998.

34. I. Papautsky, J. Brazzle, H. Swerdlow, and A. B. Frazier, "Micromachined pipette arrays," in Proc. EMBS 97, Chicago, IL, Oct. 30 - Nov. 2, pp. 2281-2284, 1997.

35. I. Papautsky, J. Brazzle, T. Ameel, and A. B. Frazier, “Laminar fluid behavior in microchannels using micropolar fluid theory,” Sensors and Actuators A, vol. 73, pp. 101-108, 1999.

36. R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th ed., Mc-Graw-Hill, New York, 1982.

37 . C. H. Mastrangelo and C. H. Hsu, “Mechanical stability and adhesion of microstructures under capillary forces—part 1: basic theory,” J. Microelectromech. Syst., vol. 2, no. 1, pp. 33-43, 1993.

38. B. K. Gale, Ph. D. Dissertation, University of Utah, 1999.

39. S. Wu, J. Mai, Y. Zohar, Y. C. Tai, and C. M. Ho, “A suspended microchannel with integrated temperature sensors for high-pressure flow studies,” in IEEE Micro Electro Mechanical Systems Conf., Heidelberg, Germany, Jan. 25-29, 1998.

40. R. S. Stanley, Ph. D. Dissertation, Louisiana Tech University, 1997.