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Effects of Surface Roughness in Microchannel Flows


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EFFECTS OF SU RFACE ROUGHNESS IN MICROCHANNEL FLOWS By AMIT S. KULKARNI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Amit S. Kulkarni

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This document is dedicated to my pare nts for their ever-extending support and contribution.

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ACKNOWLEDGMENTS This thesis would not have been possible without help of certain people and hence I would like to use this opportunity to thank them. I would like to thank my parents in the first place for their constant support, encouragement and love all along my way. I would like to express big thanks for Dr. J. N. Chung for allowing me to work on this project under his tutelage and showing confidence in me and giving me insightful talks that went as building blocks in my thesis as well as my masters education. I would also like to express my sincere gratitude towards Dr. William Lear, Dr. Darryl Butt and Dr. Wei Shyy for their time and effort as my committee member. I would also like to thank Dr. Siddarth Thakur for his assistance with regard to development of the code for the current study. I would also like to thank my roommates for their support during my masters studies. Lastly I would also like to acknowledge the financial support from Motorola. iv

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TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii NOMENCLATURE.........................................................................................................xii ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Introduction.............................................................................................................1 1.2 Liquid Flows in Microchannel................................................................................4 1.3 Unique Aspects of Liquids in Microchannels........................................................7 1.4 Commercial Aspects of Microfluidics....................................................................8 1.5 Scientific Aspects of Microfluidics......................................................................11 1.6 Milestones of Microfluidics..................................................................................11 1.6.1 Device Development..................................................................................11 1.6.1.1 Miniaturization approach.................................................................11 1.6.1.2 Exploration of new effects...............................................................12 1.6.1.3 Application developments................................................................12 1.6.2 Technology Development..........................................................................13 2 LITERATURE SURVEY...........................................................................................14 2.1 Introduction...........................................................................................................14 2.2 Wet Bulk Micromachining...................................................................................14 2.2.1 Wet Isotropic and Anisotropic Etching......................................................15 2.2.2 Surface Roughness and Notching...............................................................16 3 GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM......................22 3.1 Introduction...........................................................................................................22 3.2 Transformation to Body-Fitted Coordinates for 2D geometries...........................23 3.3 Discretized Form of Equations.............................................................................26 3.4 SIMPLE Method...................................................................................................29 v

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3.4.1 Staggered Grid:...........................................................................................34 3.4.2 Non-staggered Grids:..................................................................................34 3.5 Momentum Interpolation Technique....................................................................35 3.6 Validation of Computational Model.....................................................................36 4 FLOW CONFIGURATION AND TEST CASES.....................................................40 5 RESULTS AND DISCUSSIONS...............................................................................47 5.1 Introduction...........................................................................................................47 5.2 Pressure Drop........................................................................................................47 5.3 Flow Friction........................................................................................................59 6 CONCLUSION...........................................................................................................74 LIST OF REFERENCES...................................................................................................76 BIOGRAPHICAL SKETCH.............................................................................................80 vi

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LIST OF TABLES Table page 4.1 Geometric parameters of microchannels...............................................................45 5.1 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 1m average surface roughness.....................................................................................51 5.2 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 2m average surface roughness.....................................................................................52 5.3 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 3m average surface roughness.....................................................................................52 5.4 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 4m average surface roughness.....................................................................................53 5.5 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 5m average surface roughness.....................................................................................53 5.6 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 6m average surface roughness.....................................................................................54 vii

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LIST OF FIGURES Figure page 1.1 Scaling of things......................................................................................................2 1.2 Micro heat exchanger constructed from rectangular channels machined in metal.........................................................................................................................5 1.3 Blood sample cartridge using microfluidic channels...............................................6 1.4 Knudsen number regimes........................................................................................7 1.5 Estimated sales of microfluidic components compared to other MEMS devices....................................................................................................................10 2.1 A wet bulk micromachining process is used to craft a membrane with pierzoresistive elements.........................................................................................15 2.2 Roughness caused in microchannels during anisotropic etching process..............16 2.3 Experimentally measured pressure gradient..........................................................19 2.4 A comparison of the measured data of pressure gradient vs. Reynolds number with the predictions of conventional laminar flow theory........................20 2.5 A comparison of the experimental data of pressure gradient vs. Reynolds number with the predictions of Roughness viscosity model.................................21 3.1 Collocated grid and notation for a 2-D grid on a physical plane...........................27 3.2 Collocated grid and notation for a 2-D grid on a Transformed (Computational) plane...........................................................................................28 3.3 The SIMPLE algorithm..........................................................................................33 3.4 Streamlines for the driven cavity problem at Re = 100 on a grid of 100x100.......37 3.5 U-component of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x100.............................................................................................38 3.6 V-component of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x100..............................................................................................38 viii

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3.7 U-component of velocity profile along the vertical centerline for the driven cavity problem at Re = 100 on a grid of 100x100................................................39 5.1 Parallel plates configuration used in the current study..........................................41 4.1 Section of a channel with sawtooth surface roughness. Maximum height of surface roughness is 1m......................................................................................41 4.2 Selection of a channel with sawtooth surface roughness. Maximum height of surface roughness is 2m.......................................................................................42 4.3 Section of a channel with sawtooth surface roughness. Maximum height of surface roughness is 4m......................................................................................42 4.4. Section of a channel with random surface roughness. Average height of surface roughness is 1m.......................................................................................43 4.5. Section of a channel with random surface roughness. Average height of surface roughness is 2m.......................................................................................44 4.6. Profile of random surface roughness. Average height of surface roughness is 6m...................................................................................................................44 4.7. A comparison of pressure drop in Sawtooth 1 channel with 25000 grid points and 50000 grid points..................................................................................45 4.8. A comparison of pressure drop in Sawtooth 2 channel with 25000 grid points and 50000 grid points.................................................................................46 5.1. A comparison of pressure drop in channels with average surface roughness of 1m to that of theoretical pressure drop in a plain channel..............................48 5.2 A comparison of pressure drop in channels with average surface roughness of 2m to that of theoretical pressure drop in a plain channel.............................48 5.3 A comparison of pressure drop in channels with average surface roughness of 3m to that of theoretical pressure drop in a plain channel.............................49 5.4 A comparison of pressure drop in channels with average surface roughness of 4m to that of theoretical pressure drop in a plain channel.............................49 5.5 A comparison of pressure drop in channels with average surface roughness of 5m to that of theoretical pressure drop in a plain channel.............................50 5.6 A comparison of pressure drop in channels with average surface roughness of 6m to that of theoretical pressure drop in a plain channel..............................50 ix

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5.7 A comparison of pressure drop in channels with sawtooth surface roughness to that of theoretical pressure drop in a plain channel as the roughness is increased................................................................................................................54 5.8 A comparison of pressure drop in channels with random surface roughness to that of theoretical pressure drop in a plain channel as the roughness is increased................................................................................................................55 5.9 Experimentally measured pressure gradient (a) SS and (b) FS microtubes, and comparison with the classical theory..............................................................55 5.10 A comparison of the measured data of pressure gradient vs. Reynolds number with the predictions of conventional laminar flow theory for trapezoidal microchannels.....................................................................................56 5.11 A comparison of pressure drop in rough channels, reduced width channels and smooth channel......................................................................................................58 5.12 A comparison of pressure drop in rough channels, reduced width channels and smooth channel................................................................................................58 5.13 A comparison of pressure drop rough channels, reduced width channels and smooth channel......................................................................................................59 5.14 A comparison of friction factor in channels with average surface roughness of 1m to that of theoretical friction factor in a plain channel..............................60 5.15 A comparison of friction factor in channels with average surface roughness of 2m to that of theoretical friction factor in a plain channel..............................61 5.16 A comparison of friction factor in channels with average surface roughness of 3m to that of theoretical friction factor in a plain channel..............................61 5.17 A comparison of friction factor in channels with average surface roughness of 4m to that of theoretical friction factor in a plain channel..............................62 5.18 A comparison of friction factor in channels with average surface roughness of 5m to that of theoretical friction factor in a plain channel..............................62 5.19 A comparison of friction factor in channels with average surface roughness of 6m to that of theoretical friction factor in a plain channel..............................63 5.20 A comparison of friction factor in channels with random surface roughness to that of theoretical friction factor in a plain channel as the roughness is increased................................................................................................................64 x

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5.21 A comparison of friction factor in channels with sawtooth surface roughness to that of theoretical friction factor in a plain channel as the roughness is increased................................................................................................................64 5.22 Friction factor fexp vs Re for some SS and FS microtubes and comparison with the classical theory.........................................................................................65 5.23 A comparison of f.Re in channels with average surface roughness of 1m to that of theoretical f.Re in a plain channel...............................................................67 5.24 A comparison of f.Re in channels with average surface roughness of 2m to that of theoretical f.Re in a plain channel...............................................................67 5.25 A comparison of f.Re in channels with average surface roughness of 3m to that of theoretical f.Re in a plain channel...............................................................68 5.26 A comparison of f.Re in channels with average surface roughness of 4m to that of theoretical f.Re in a plain channel...............................................................68 5.27 A comparison of f.Re in channels with average surface roughness of 5m to that of theoretical f.Re in a plain channel...............................................................69 5.28 A comparison of f.Re in channels with average surface roughness of 6m to that of theoretical f.Re in a plain channel...............................................................69 5.29 A comparison of friction constant in channels with sawtooth surface roughness to that of theoretical friction constant in a plain channel as the roughness is increased............................................................................................70 5.30 A comparison of friction constant in channels with sawtooth surface roughness to that of theoretical friction constant in a plain channel as the roughness is increased............................................................................................71 xi

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NOMENCLATURE W = Width of the channel L = Length of the channel. x, y = Coordinate directions Re = Reynolds number u = Velocity in x-direction v = Velocity in y-direction u m = Average velocity of fluid. U 0 = Initial velocity of fluid. p = Pressure P = Pressure difference = Viscosity = Density f = Friction factor C f = Coefficient of friction Q = Flow rate h = Average height of surface roughness D h = Hydraulic diameter = Computational coordinate directions J = Jacobian U = Contravariant velocity in x-direction V = Contravariant velocity in y-direction W = Matrix of variables discretized in time E = Matrix of Flux vectors in direction F = Matrix of Flux vectors in direction S = Matrix of Source terms F = Mass flux at the control volume H i = Compact notation for advective and convective flux R p = Stencil error S p = Momentum source term p* = Guessed pressure p = Correction pressure u* = Guessed u-component of velocity u = Correction u-component of velocity v* = Guessed v-component of velocity v = Correction v-component of velocity xii

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EFFECTS OF SURFACE ROUGHNESS IN MICROCHANNEL FLOWS By Amit S. Kulkarni May 2004 Chair: J.N.Chung Major Department: Mechanical and Aerospace Engineering An incompressible 2-D Navier Stokes solver is used to study the effects of surface roughness in microchannel flows. A rough microchannel is generated by selecting the maximum height of the roughness element in the microchannel, and the randomness present in the length of the channel so as to maintain a constant average surface roughness in the channel. Two different types of surface roughness, namely, random surface roughness and sawtooth surface roughness profiles, are generated for the study. The actual geometry is converted into the computational geometry using the transformation of coordinates. The geometry is solved for the specified boundary conditions using body fitted coordinate system. The governing equations are discretized using the non-staggered manner. The governing equations are solved using a SIMPLE method along with the momentum interpolation technique. The study investigates the pressure drop and the friction factor in rough channels. During the first part the pressure drop in the channels is found to be more than the xiii

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pressure drop estimated by the conventional laminar flow theory. The surface roughness is the factor causing the change in pressure drop. The second part of the study investigates the departure of the friction behavior from the classical thermofluid correlations. Although the friction factor in the laminar flow is supposed to be independent of surface roughness, this does not hold true in the case of microchannels. As we increase the surface roughness in the channels, we see more deviation of friction factor from the standard values. These numerical studies are compared with the experimental results done by researchers in this field. The study attempts to set caveats for innovative and inquisitive minds that aspire to study actual flow behavior in microchannels. xiv

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CHAPTER 1 INTRODUCTION 1.1 Introduction Tool making has always differentiated our species from all others on earth. Aerodynamically correct spears were carved by Homo sapiens close to 40,000 years ago. Man builds size consistent with his size, typically in the range of two orders of magnitude larger or smaller than himself as shown in Fig 1.1. Humans have always striven to explore, build and control the extremes of length and time scales. The Great Pyramid of Khufu was originally 147m high when completed around 2600 B.C., while the Empire State Building is 449m high, constructed during 1950. At the other end of the spectrum of man-made artifacts, a dime is less than 2cm in diameter. Watchmakers have practiced the art of miniaturization since the 13 th century. The invention of the microscope in the 17 th century opened the way for direct observation of microbes and plant and animal cells. Smaller things were man-made in the later half of 20 th century. The transistor-invented in 1947, in todays integrated circuits has a size or 0.18m in production and approaches 10nm in research laboratories using electron beams. Manufacturing processes that can create extremely small machines have been developed in recent years (Angell et al. 1983; Gabriel et al. 1988, 1992; Ashley, 1996; Amato, 1998; Knight 1999; Chalmers, 2001). Electrostatic, magnetic, electromagnetic, pneumatic and thermal actuators, motors, valves, gears etc less than size of 100m in size 1

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2 Fig 1.1 Scaling of things, in meters. Lower scale continues in the upper bar from left to right. have been fabricated. These have been used as sensors for pressure, temperature, mass flow, velocity, sound and chemical composition; as actuators for linear and angular motions and also part of complex systems such as robots, micro heat engines, and micro heat pumps. Integrated microfluidic systems with a complex network of fluidic channels are routinely used for chemical and biological analysis and sensing. They have generated a considerable activity, at economic and scientific levels, and their importance in our everyday life, is expected to considerably increase over the next few years. The rapid development of microelectronics and molecular biology has simulated an increasing interest in miniaturization characterized by flow and heat transfer in confined tiny geometries.

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3 Micorelectromechanical systems (MEMS) refer to devices that have a characteristic length of less than 1mm but more than 1m, that combine electrical and mechanical components that are fabricated using integrated circuit batch-processing technologies. Current manufacturing techniques for MEMS include surface silicon micromachining; bulk silicon micromachining; lithography, electrodeposition and plastic molding, and electrodischarge machining (EDM). As shown in Figure 1.1, MEMS are more than four order magnitudes larger than hydrogen atom, but about four order magnitude smaller than the traditional man-made artifacts. Microdevices can have characteristic lengths smaller than the diameter of a human hair. Some of the MicroElectroMechanical Systems (MEMS) devices used for momentum and energy transfer have characteristic lengths of microns. Microfluidics is about flows of liquids and gases, single or multiphase, through microdevices fabricated by MEMS. It is mainly fostered by the development of lab-on a chip devices, i.e. systems able to perform in impressive number of tasks on a small chip, such as mixing, separating, analyzing, detecting molecules. In the next few years, a variety of such systems, designed to identify and analyze DNA from a drop of blood will be made available. Also in context of propulsion of miniaturized rockets, for space applications, microfluidics arose into a new discipline. MEMS are finding increased applications in a variety of industrial and medical fields, with a potential market in billions of dollars. Accelerometers for automobile airbags, keyless entry systems, dense arrays of micromirrors for high definition optical displays, scanning electron microscope tip to image single atoms, micro-heat exchangers for cooling electronic circuits, reactors for separating biological cells, blood analyzers

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4 and pressure sensors for catheter tips are but a few use. Microducts are used in infrared detectors, diode lasers, miniature gas chromatographs and high frequency fluidic control systems. Micropumps are used for ink-jet printing, environmental testing and electronic cooling. Potential medical applications for small pumps include controlled delivery and monitoring of minute amounts of medication, manufacturing of nanoliters of chemicals and development of artificial pancreas. 1.2 Liquid Flows in Microchannel Nominally, microchannels can be defined as channels whose dimensions are less than 1mm and greater than 1m. Above 1m the flow exhibits behavior that is same as macroscopic flows. Below 1m the flow is characterized as nanoscopic. Currently, most microchannels fall into the range of 30 to 300m. Microchannels can be fabricated in many materials-glass, polymers, silicon metals using various processes including surface micromachining, bulk micromachining, molding, embossing and conventional machining with microcutters. Microchannels offer advantages due to their high surface-to-volume ratio and their small volumes. The large surface to volume ration leads to a high rate of heat and mass transfer, making microdevices excellent tools for compact heat exchangers. For example, the device in Fig 1.2 is a cross-flow heat exchanger constructed from a stack of 50 14mm-14mm foils, each containing 34 200m wide x 100m deep channels machined into the 200 m thick stainless steel foils by the process of direct high precision mechanical micromachining (Brander et al., 2000; Schaller et al., 1999). The direction of the flow in adjacent foils is alternated 90, and the foils are attached by the means of diffusion bonding to create a stack of cross-flow heat exchangers capable of transferring 10kW at a temperature difference of 80K using water flowing at 750kg/hr. The

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5 impressive large of heat transfer is accomplished mainly by the large surface area covered by the interior of the microchannel: approximately 3600mm 2 packed into a 14mm cube. A second example of the application of microchannels is in the area of microelectromechanical systems (MEMS) devices for biological and chemical analyses. The primary advantages of microscale devices in these applications are the good match with the scale of biological structures and potential for placing multiple functions for chemical analysis on a small are. Microchannels are used to transport biological materials such as (in order of size) proteins, DNA, cells and embryos or to transport chemical samples. Flow in biological devices and chemical analysis microdevices are usually much slower than those in heat transfer and chemical reactor microdevices. Fig 1.2 Micro heat exchanger constructed from rectangular channels machined in metal. (The MEMS Handbook (2002))

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6 Fig 1.3 Blood sample cartridge using microfluidic channels (The MEMS Handbook (2002)) Knudsen Number: It is defined as the ratio of the mean free path over the characteristic geometry length or a length over which very large variations of macroscopic quantity make take place. It is given by the following formula LKn where = mean free path L = characteristic length According to the Knudsen number the flow regimes can be divided into various regimes. These are: continuum, slip, transition and free-molecular flow regimes. Discrete particle or molecular based model is the Boltzmann equation. The continuum based models are the Navier-Stokes equations. Euler equations correspond to inviscid continuum limit which shows a singular limit since the fluid is assumed to be inviscid

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7 and non-conducting. Euler flow corresponds to Kn = 0.0. Navier-Stokes equations with slip boundary conditions are used for Slip flow regime. Kn = 0.00011 0.1 0.01 0.001100 10 Continum flow Free-molecule flow Transition regime Slip-flow regime Fig 1.4 Knudsen number regimes. (The MEMS Handbook (2002)) Since the density of liquids is 1000 times the density of gases, Kn doesnt play an important role. The flow is in the conti nuum regime. For example, water has lattice spacing of 0.3 nm. In a 1m gap and 50 m diameter channel Knudsen number is 3x10-4 and 6x10-4, respectively, which are well within the continuum flow regime. 1.3 Unique Aspects of Liquids in Microchannels Flow in microscale devices differ from the macroscopic counterparts for two reasons: 1. The small scale makes molecular effects such as wall slip more important. 2. Small scale amplifies the magnitudes of certain ordinary continuum effects to extreme levels. Consider, strain rate and shear rate which scale in proportion to the velocity scale, Us, and inverse proportion to the length scale, Ls. Thus, 100 mm/s flow in a 10 m channel experiences a shear rate of order of 104 s-1. Acceleration scales are also similarly enhanced. The effect is even more dramatic if one tries to maintain the same volume flux while scaling down. The flux scales as Q ~ UsL2 s, so at constant flux Q ~ L-2 s, and both shear and acceleration go as L-3 s. Fluids that are Newtonian at ordinary rates of shear and

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8 extension can become Non-Newtonian at very high rates. The pressure gradient becomes especially large in small cross-section channels. For fixed volume flux, the pressure gradient increases as L -4 s Electrokinetic effects occur at the interface between liquids and solids such as glass due to chemical interaction. The result is electrically charged double layer that indicates a charge distribution in a very thin layer of fluid close to the wall. Application of an electric field to this layer creates a body force capable of moving the fluid as if it were slipping over the wall. Molecular effects in liquids are difficult to predict because the transport theory is less well developed than the kinetic theory of gases. For this reason, studies of liquid microflows in which molecular effects play a role are much more convincing if done experimentally. Liquids are essentially incompressible. Consequently, the density of a liquid in microchannel flow remains very nearly constant as a function of distance along the channel, despite the very large pressure gradients that characterize the microscale flow. This behavior greatly simplifies the analysis of liquid flows relative to gas flows wherein the large pressure drop in a channel leads to large expansion and large changes of thermal heat capacity. Liquids in contact with solids or gases have surface tension in the interface. At the microscale, the surface tension force becomes one of the most important forces, far exceeding body forces such as gravity and electrostatic fields. 1.4 Commercial Aspects of Microfluidics With the recent achievement in the Human Genome Project and the huge potential of biotechnology, microfluidic devices promise to be a big commercial success.

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9 Microfluidic devices are tools that enable novel applications unrealizable with conventional equipment. The apparent interest and participation of the industry in microfluidics research and development show the commercial values of devices for practical applications. With this commercial potential, microfluidics is poised to become the most dynamic segment of the MEMS technology thrust. From its beginnings with the now-traditional microfluidic devices, such as inkjet print heads and pressure sensors, a much broader microfluidic market is now emerging. Fig 1.4 shows the estimated sales of microfluidic devices in comparison to other MEMS devices. The estimation assumes an exponential growth curve based on the survey data of 1996[System Planning Corporation, 1999) The projection considers four types of microfluidic devices: fluid control devices, gas and fluid measurement devices, medical testing devices, and miscellaneous devices such as implantable drug pumps. The curve in the fig shows that microfluidic devices sales exceed all other application areas, even the emerging radio frequency MEMS devices (RF-MEMS). Commercial interests are focused on plastic microfabrication for single-use disposable microfluidic devices. The major application applications of microfluidics are medical diagnostics, genetic sequencing, chemistry production, drug discovery and proteomics. Microfluidics can have a revolutionizing impact on chemical analysis and synthesis, similar to the impact of integrated circuits on computers and electronics. Microfluidic devices could change the way instrument companies do business. Instead of selling a few expensive systems, companies could have a mass market of cheap, disposable drug dispensers available for everyone will secure a huge market similar to that of computers today.

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10 As computing power has improved from generation to generation by higher operation frequency as well as parallel architecture, in the same way, microfluidics revolutionizes chemical screening power. Furthermore, microfluidics will allow the pharmaceutical industry to screen combinatorial libraries with high throughput-not previously possible with manual, bench-top experiments. Fast analysis is enabled by the smaller quantities of materials in assays. Massively parallel analysis on the same microfluidic chip allows higher screening throughput. Microfluidic assay can have several hundred to several hundred thousand parallel processes. The high performance is extremely important for DNA-based diagnostics in pharmaceutical and health care applications. Fig 1.5 Estimated sales of microfluidic components compared to other MEMS devices. (Fund. and App. of Microfluidics (2002))

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11 1.5 Scientific Aspects of Microfluidics Scientists from almost all traditional engineering and science disciplines have begun pursuing mircrofluidics research, making it a truly multidisciplinary field representative of the new economy of the 21 st century. Electrical and mechanical engineers contribute novel enabling technologies to microfluidics. Fluid mechanics researchers are interested in the new fluid phenomena possible at the microscale. In contrast to the continuum based hypotheses of conventional macroscale flows, flows physics in microfluidic devices is governed by a transitional regime between the continuum and molecular-dominated regimes. Besides new analytical and computational models, microfluidics has enabled a new class of fluid measurements of microscale flows using in situ microinstruments. Life scientists and chemists also find in microfluidics novel, useful tools. Mircofluidic tools allow them to explore new effects not possible in traditional devices. These new effects, new chemical reactions, and new microinsturments lead to new applications in chemistry and bioengineering. These reasons explain the enormous interest of research disciplines in microfluidics. 1.6 Milestones of Microfluidics There are two major aspects considered as the milestones of microfluidics: the applications-driven development of devices and development of fabrication technologies. 1.6.1 Device Development 1.6.1.1 Miniaturization approach With silicon micromachining as the enabling technology, researchers have been developing silicon microfluidic devices. The first approach of making miniaturized devices was shirking down conventional principles. This approach is representative of the research conducted in the 1980s through the mid-1990s. In this phase of microfluidics

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12 development, a number of silicon microvalves, micropumps, and microflow sensors were developed and investigated (Shoji S. et al. 1994). Two general observations of scaling laws can be made in this development stage: the power limit and the size limit of the devices. Assuming that the energy density of actuators is independent of their size, scaling down the size will decrease the power of the device by the length scale cubed. This means we cannot expect micropumps and microvalves to deliver the same power level as conventional devices. The surface-to-volume ratio varies as the inverse of the length scale. Large surface area means large viscous forces, which in turn requires powerful actuators to be overcome. Often integrated microactuators cannot deliver enough power, force, or displacement to drive a microfluidic device, so an external actuator is the only option for microvalves and micropumps. The use of external actuators limits the size of those microfluidic devices, which range from several millimeters to several centimeters. 1.6.1.2 Exploration of new effects Since the mid-1990s, development has shifted to the exploration of new actuating schemes of microfluidics. Because of the power and size constraints research efforts have concentrated on actuators with no moving parts and non-mechanical pumping principles. Electrokinetic pumping, surface tension-driven flows, electromagnetic forces and acoustic streaming are effects that usually have no impact at macroscopic length scales. However, at the microscale they offer particular advantages over mechanical principles. 1.6.1.3 Application developments Concurrent with the exploration of the new effects, microfluidics today is looking for further application fields beyond conventional fields such as flow control, chemical analysis, biomedical diagnostics and drug discovery. New applications utilizing

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13 microfluidics for distributed energy supply, distributed thermal management and chemical production are promising. 1.6.2 Technology Development Similar to trends in device development, the technology of making microfluidic devices has also seen a paradigm shift. Since mid-1990, with chemists joining the field, microfabrication technology has been moving to plastic micromachining. With the philosophy of functionality above miniaturization and simplicity above complexity, microfluidic devices have been kept simple, sometimes only with a passive system of microchannels. The actuating and sensing devices are not necessarily integrated into microdevices. These microdevices are incorporated as replaceable elements in bench top and handheld tools. Batch fabrication of plastic devices is possible with many replication and forming techniques. The master for replication can be fabricated with traditional silicon-based micromachining technologies. Complex based microfluidic devices based on plastic microfabrication could be expected in the near future with further achievements of plastic-based microelectronics.

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CHAPTER 2 LITERATURE SURVEY 2.1 Introduction Rough walls exist in all flow systems, where they may lead either to deterioration or improvement of the desired functionality. Wall roughness can be increased to promote mixing of the fluid, or reduced to eliminate flow disturbances. The related problem of the laminar-turbulent transition over a rough wall is one of the classical problems in fluid mechanics that has so far defied all analytical efforts. Recently, the effects of surface roughness became of interest from the point of view of passive/active flow control strategies, where once is interested in determining the smallest possible surface modification that may induce the largest possible changes in flow field. 2.2 Wet Bulk Micromachining In wet bulk micromachining, features are sculpted in the bulk of materials such as silicon, quartz, SiC, GaAs, InP, Ge and glass by orientation-dependent (anisotropic) and/or by orientation-dependent (isotropic) wet etchants. The technology employs pools as tools instead of plasma. A vast majority of micromachining work is based on single crystal silicon. These tools are used to fabricate microstructures either in parallel or serial processes. The principle commercial Si micromachining tools used today are the well-established wet bulk micromachining. A typical structure fashioned in a bulk micromachining process is shown in Fig 2.1. Despite all the emerging technologies, Si wet bulk micromachining, being the best characterized micromachining tool, remains the 14

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15 most popular in industry. Two types of etching used for wet bulk micromachining are wet isotropic etching and anisotropic etching which are discussed as below. Fig 2.1 A wet bulk micromachining process is used to craft a membrane with pierzoresistive elements. (The MEMS Handbook (2002)). 2.2.1 Wet Isotropic and Anisotropic Etching Wet etching of Si is used mainly for cleaning, shaping, polishing and characterizing structural and compositional features. Wet etching is also faster as compared to typical dry etching. Modification of wet etchant and/or temperature can alter the selectivity to silicon dopant concentration and type especially when using alkaline etchants, to crystallographic orientation. Isotropic etchants, also polishing etchants, etch in all crystallographic directions at the same rate; they usually are acidic, and lead to rounded isotropic features in single crystalline Si. They are usually used at room temperatures or slightly above (< 50C). Anisotropic etchants, etches away crystalline silicon at different rates depending on the orientation of exposed crystal plane. Typically the pH stays above 12, while more

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16 eleva ted temperatures are used for these slower type etchants (> 50C). These etchants are reaction rate limited. When carried out properly, anisotropic etching results in geometric shapes bounded by perfectly defined crystallographic planes. 2.2.2 Surface Roughness and Notching Anisotropic etchants frequently leave too rough surface behind, and a use of ughness also referred to as notching or pillowge ch is ds the process control of MEMS and micro-fluidic devices. It has been found that for fluid flow in mic isotropic etch is required to touch-up. Ro ing, results when centers of exposed areas etch with a seemingly lower averaspeed compared with the borders of areas. This difference can be as 1 to 2m, whiquiet considerable if one is etching 10 to 20m thick structures. Fig 2.2shows these kinof roughness in the channels. Fig 2.2 Roughness caused in microchannels during anisotropic etching process. (The MEMS Handbook (2002)). The liquid flow characteristics in microchannels are important in the design and rochannels, the flow behavior often deviates significantly from the predictions of conventional theories of fluid mechanics.

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17 There has been a lot of research work done in the areas, of heat transfer and fluid flow through microchannels. Wu and Little (1983) measured the friction factors for the flow y ss crochannels had the dimeney the nels, namely 0.76 and 1.09 mm wanging from 0.133 to 0.367 mm and width to height ratios from 0.333 of gases in miniature channels. The test channels were etched in glass and silicon with hydraulic diameters ranging from 55.81 to 83.08 m. The tests involved both laminar and turbulent flow regimes. They found that the friction factors of both flow regimes in these channels were larger than predictions form the correlations for the macroscale pipes. The transition to turbulent flow was found to be as low as 350. Theattributed these deviations to relatively high surface roughness, asymmetric roughneand uncertainty in the determination of channel dimensions. Harley and Bau (1989) measured the friction factors in microchannels with trapezoidal and rectangular cross sections. The trapezoidal mi sions of 33m (depth), 111m (top width), and 63m (bottom width), and a rectangular channel had the dimensions of 100m in depth, and 50m in width. Thfound that the product of f.Re ranged from 49 for the rectangular channel to 512 fortrapezoidal channel in contrast to the classical value of 48. Adams et al. (1998) conducted single phase flow studies in microchannels using water as working fluid. Two diameters of circular microchan ere used for investigation. It was found that the Nusselt numbers are larger than those in macrochannels. Peng and Peterson (1996) investigated water flows in rectangular microchannels with hydraulic diameters r to 1. Their experimental results indicated that the flow transition occurs at Reynolds number 200-700. This transitional Re decreases as the size of the microchannel

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18 decreases. The flow friction behaviors of both the laminar and turbulent flows weto depart from classical correlations. The friction factors were either large or small than the predictions of the classical theories. The geometrical parameters, such as hydraulic diameter and aspect ratio, were found to have important effects on flow. Yu et al. (1995) studied the fluid flow and heat transfer characteristics of dry nitrogen gas and water in microtubes with diameters of 19, 52 and 102 m re found The Reynolds numbss for he was found that when microtube diameters are belowce tum and heat transfer was done by Mala and Li (1999) and Qu et al. (2000nels r er in their study ranges from 250 to over 20,000. The average relative roughne53 m microtube was measured and the value is approximately 0.0003. The flow friction results indicate that for laminar flow, in microtubes, the value of the product f.Re, is between 49.35 and 51.56, instead of 64. Lim et al. (2000) conducted experimental study on water flow in microtubes. Tdiameters range from 49.3 to 701.9 m. It 300 m, the f.Re decreases (departs from fully developed f.Re of 64) as the diameter decreases. However one of the very few works done in understanding the effect of surfaroughness on momen ). They investigated flow in trapezoidal silicon microchannels. A significant difference between the experimental data and the theoretical predictions was found. Experimental results indicated that pressure gradient and flow friction in microchanare higher that those given by conventional laminar flow theory. The measured highepressure gradient and flow friction was attributed to the surface roughness of microchannels.

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19 Mala and Li (1999) measured pressure gradients of water flow in microtinner diameter ra ubes with nging from 50.0 to 254.0 m. The Reynolds number was up to 2100. They ace found for larger microtubes with inner diameter above 150 m, the experimental results were in rough agreement with conventional theory. For smaller microtubes, the pressure gradients are up to 35% higher than these predicted by the conventional theoryAs the Reynolds number increased the difference between the experimental and the conventional results also increased. They attributed these effects to the change in flow mode from laminar to turbulent at low Reynolds number, or due to the effects of surfroughness. Fig 2.3 Experimentally measured pressure gradient (a) SS and (b) FS microtubes, and comparison with the classical theory. (Mala, Li, Int. J. heat and Fluid flow (1999) 142-148)

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20 Fig. 2.4. A comparison of the measured data of pressure gradient vs. Reynolds number with the predictions of conventional laminar flow theory. (a) (1) dh = 51.3 mm; (2) dh= 62.3 mm ;( 3) dh = 64.9 mm (Qu et al. Int. J. of Heat and Mass Transfer 43 (2000) 353-364) As seen in the above Fig 2.4 theoretical curves fall below the experimental curves, which mean that at any given flow rate, a higher pressure gradient is required to force the liquid to flow through those microchannels than the predictions of the conventional laminar flow theory. They proposed a Roughness-Viscosity model to interpret the experimental data more correctly. According to this concept the value of roughness viscosity R should have a higher value near the wall and gradually diminish as the distance from the wall increases. Roughness viscosity R should also increase as Re increases.

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21 Fig. 2.5. A comparison of the experimental data of pressure gradient vs. Reynolds number with the predictions of Roughness viscosity model. (a) (1) dh = 51.3 mm; (2) dh= 62.3 mm ;( 3) dh = 64.9 mm (Qu et al. Int. J. of Heat and Mass Transfer 43 (2000) 353-364) However, not a lot of work is done in computational modeling the microchannels with surface roughness. In the following work, we try to model the surface roughness in the microchannels and model the flow field in the microchannels to study the effect of the surface roughness on the pressure drop and the friction factor in the microchannel.

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CHAPTER 3 GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM 3.1 Introduction Simulation of full Navier-Stokes equations for different fluid flows arises in various engineering problems. Various different algorithms have been proposed and developed by various researchers. But an approach that is fully robust from the point of view of numerical and modeling accuracy as well as efficiency has yet to be developed. Existing algorithms for Navier-Stokes equations can be generally classified as density-based methods and pressure-based methods. For both of these methods, the velocity field is normally obtained using the momentum equations. Density based methods are employed for compressible flows where the continuity equation is used to obtain the density of the fluid, while pressure information is obtained using the equation of state. The system of equations is solved simultaneously. These methods can be extended with modification to the low Mach number regime where the flows are incompressible where the density has no role to play in determining, the pressure field. (Fletcher 1988, Hirsch 1990). Pressure based methods (Patankar 1980, Shyy 1994) are developed for the incompressible flow regime. These obtain the pressure field via a pressure correction equation which is formulated by manipulating the continuity and momentum equations. The solution procedure is conventionally sequential in nature, and hence, can more easily accommodate a varying number of equations depending on the physics of the problem involved, without necessity of reformulating the entire algorithm. These methods can be 22

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23 extended to compressible flows by taking in account the dependence of density on pressure, via the equation of state. The discretization schemes used in the present algorithm have been developed primarily for incompressible flows and are differentiated by the geometric shape of the interpolation function used to estimate the fluxes on a control volume interface. The first-order upwind, central difference, second-order upwind (Warming & Beam 1976, Shyy et al. 1992, Thakur & Shyy 1993) and QUICK (Quadratic Upwind Interpolation for Convective Kinematics) are the examples of these kind of schemes. The continuity and the momentum equation in the Cartesian coordinates can be written as follows 0yvxut 3.1 yuyxuxxpyuvxuutu 3.2 yvyxvxypyvvxuvtv 3.3 3.2 Transformation to Body-Fitted Coordinates for 2D geometries The above form of Navier-Stokes equations is for Cartesian coordinates (x, y). For arbitrarily shaped geometries generalized body-fitted coordinates are employed, denoted by (, ) where = (x,y), = (x,y). The transformation of the physical domain (x, y) to the computational domain (, ) is achieved by transformation metrics which are related to the physical coordinates as follows (Anderson et. al 1984).

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24 222112111ffffJyxyx 3.4 where, f 11 = y f 12 = x f 21 = y f 22 = x and J is the Jacobian of the transformation given by, J = x y x y Each term in the equations (Eq. 3.1-3.3) is transformed to the (, ) coordinate system. The resulting governing equations in generalized body-fitted coordinates are presented in the complete form as follows: Continuity: 0)(VUt 3.5 U momentum equation: uquqJuquqJpfpfVuUutU32212111)()()( 3.6

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25 V momentum equation: 12221223()()()VuUvVvfpfpqqtJuuqqJ u 3.7 where, 221yxq 222yxq yyxxq3 and U, V are the components of the contravariant velocity (which is the scalar product of the velocity vector and the area vector at a control volume interface). vxuyU uyvxV The contravariant velocity components can be interpreted as the volume flux normal to control volume interfaces; specifically, U is the local volume flux along the coordinate, V along the coordinate. In a compact form, the governing equations can be written as follows SFEWt 3.8 where, vuW 3.9 and the flux and source vectors are given by

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26 11122123UuuUufpqqJuuVvfpqqJ E 3.10 21232223VuuVufpqqJuuVvfpqqJ F 3.11 vuSS0S 3.12 3.3 Discretized Form of Equations The governing equations presented in the previous section are discretized on a structured grid. The velocity components and the scalar variables like pressure are located on the grid in a non-staggered manner as shown in Fig 3.1. For the flow domain, the positive direction (increasing index i) is denoted as the east direction and the negative direction (decreasing index i) as the west. Similarly, the north and south directions are along the positive (increasing j) and negative (decreasing j) directions, respectively. The index notation for momentum control volumes is illustrated using a u-component control volume. The u-component associated with a representative grid point (i, j) is labeled E (index i + 1) and EE (index i + 2). Similarly, the first and second neighbors along the west, north, and south directions are labeled, respectively, as W and WW, N and NN, S and SS. The east face is also denoted as i + interface indicating that

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27 it lies between the us located at (i, j) and ) (i+1, j). Similarly, the west, north, south faces are also denoted as i j + j respectively. Fig 3.1 Collocated grid and notation for a 2-D grid on a physical plane. The governing equations written in the generalized body fitted coordinates are integrated over the control volume whose dimensions in the computational domain are given by x The discretized form of the governing equations is finally obtained by choosing the dimensions of the control volume as = 1, = 1. The discretized form of the continuity and momentum equations are as presented below. The discretized form of the continuity equation can be written as: 0nsewVU 3.13 where,

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28 weew etc. We denote the mass flux at each control volume face by F and rewrite the continuity equation as 0nsewFF 3.14 Fig 3.2 Collocated grid and notation for a 2-D grid on a Transformed (Computational) plane. The discretized form of the momentum equations can be obtained in a similar manner. The details of the momentum equations are as follows.

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29 U momentum equation nsewnsewnsewuquqJuquqJpfpfVuUu32212111 3.15 V momentum equation nsewnsewnsewuquqJuquqJpfpfVvUv32221222 3.16 The two terms on the left hand side of the equation (write equation number), are the convective fluxes at the control volume faces; the first two terms on the right hand side are the pressure fluxes and the last two terms are the diffusion fluxes. The standard central difference operator is employed for the pressure and the diffusive fluxes. 3.4 SIMPLE Method Implicit methods are preferred for steady and slow transient flows, because they have less stringent time steps restrictions as compared the explicit schemes. Many solutions methods for steady incompressible flows use a pressure (or pressure-correction) equation to enforce mass conservation at each time step or otherwise known as outer iteration for steady solvers. The acronym SIMPLE stands for Semi-Implicit Method for Pressure Linked Equations. The algorithm was originally put forward by Patankar and Splading and is essentially a guess-and-correct procedure for the calculation of pressure. Until the early 1980s the SIMPLE family of methods was generally only employed on staggered grids.

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30 However later several investigators (Rhie et al. 1981) reported success in implementing pressure-correction schemes on a regular grid. During the current investigation we use SIMPLE method on non-staggered grids. The disadvantages mentioned above in using non-staggered grid are taken care by using the Rhie-Chow Momentum interpolation technique. Discretized U-momentum and V-momentum equations are written in following form P PnbnbWPPauauppAS P 3.17 P PnbnbSPPavavppAS P 3.18 where, S p :is the source momentum term A p : cell face area of the u-control volume. To initiate the SIMPLE calculation process a pressure field p* is guessed. Discretized momentum equations Eqn. 3.17 and Eqn. 318 are solved using the guessed pressure field to yield velocity components u* and v* as follows: **** P PnbnbWPPauauppAS P 3.19 **** P PnbnbSPPavavppAS P 3.20 Now we define the correction p as the difference between the correct pressure field p and the guessed pressure field p*, so that p pp 3.21 Similarly we define velocity corrections u and v to relate the correct u and v velocities to the guessed solutions u* and v* *uuu 3.22 *vvv 3.23 Substitution of correct pressure field p into momentum equations yields the correct velocity field (u,v). Discretized U-momentum and V-momentum equations Eq. (3.17) and Eq. (3.18) link the correct velocity fields with the correct pressure field.

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31 Subtraction of Eq. (3.19) and Eq. (20) from Eq. (3.17) and Eq. (3.18) respectively gives ***()() P PPnbnbnbWWPPPauuauuppppA 3.24 ***()() P PPnbnbnbSSPPPavvavvppppA 3.25 Using the correction formulae Eq. (3.21-3.23) the Eq. (3.24) and Eq. (3.25) may be written as follows: P PnbnbWPauauppA P 3.26 P PnbnbSPavavppA P 3.27 At this point an approximation is introduced nbnbau and nbnbau are dropped to simplify equations (3.26) and (3.27) for velocity corrections. Omission of these terms is the main approximation of the SIMPLE algorithm. We obtain P PWPudpp 3.28 P PSPvdpp 3.29 where P P P Ada Equations (3.28) and (3.29) describe the corrections to be applied to velocities through equations (3.22) and (3.23) which gives P WPuudpp 3.30 P SPvvdpp 3.31 The velocity field obtained above is subjected to the constraint that it should satisfy continuity equation. Continuity equation is the discretized form is as shown below. 0EPNPuAuAvAvA 3.32 Substitution of the corrected velocities Eq.(3.30) and Eq. (3.31) into discretized continuity equation (3.32) gives the equation for pressure correction p. The pressure correction equation in the symbolic form is as shown below P PEEWWNNSSapapapapapb P 3.33

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32 where, P EWNSaaaaab P are the coefficients obtained from the continuity equation. P b is the imbalance arising from the incorrect velocity field u*, v*. By solving Eq. (3.33) the pressure correction p can be obtained at all points. Once the pressure correction field is known the correct pressure field is obtained from Eq. (3.21) and velocity components through correction formulas Eq.(3.30) and Eq.(3.31). The omission of terms such as nbnbau and nbnbau does not affect the final solution because the pressure correction and velocity corrections will be zero in a converged solution giving p* = p, u* = u, v* = v.

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33 Fig 3.3 The SIMPLE algorithm **** P PnbnbWPPPauauppAS **** P PnbnbSPPPavavppAS PPEEWWNNSSPapapapapapb P WPuudpp P SPvvdpp *pppppp START STEP 1: Solve the discretized momentum equations Set STEP 2: Solve pressure correction equation p* = p; u* = u; v* = v STEP 3: Correct pressure and velocities No Convergence Yes STOP Above we present an algorithm of the SIMPLE method which is employed in this code.

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34 3.4.1 Staggered Grid: The main idea of the staggered grid is to evaluate scalar variables, such as pressure, density, temperature etc., at ordinary nodal points but to calculate velocity components on staggered grids centered around the cell faces. Advantages of the staggered grid: 1. Checkered board problem is solved. 2. Divergence condition is satisfied. 3. Compact stencil is used for the Pressure Poison equation 4. Strong coupling between p, and u,v. However, the disadvantages of staggered grid are: There is more computational complexity especially for the non-Cartesian, non-orthogonal meshes (especially 3D meshes) 1. More memory is required for storage of variables. 2. Difficult and inefficient for multi-grid solvers. 3.4.2 Non-staggered Grids: In non-staggered grid we have all the variables p, u, v are solved at same point. This retains a strong coupling between p and u, v. The checkered board pattern of the pressure field that results when using a Non-staggered grid without certain modifications to the original staggered grid scheme is highlighted. The necessary remedies must be applied to the non-staggered scheme to overcome the checkered board pressure field. In the approach presented here all quantities are solved and stored at the element centroid. The face values of the velocity components have to be calculated from these element based values. This leads to the need to employ an alternative interpolation

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35 method, which does not suffer from the checkerboard effect. The Rhie Chow interpolation method offers one approach which satisfies these requirements We need to use the Rhie-Chow momentum-interpolation technique to satisfy the continuity and using the non-staggered grid. 3.5 Momentum Interpolation Technique Consider a control volume as shown in the figure3.1, We write the u momentum equation in the symbolic form as follows ppWEMMMppxPbuAuA, pppppxPHuA' where, buAHWEMMMp,' ppppppxPAHu where, pppAHH' Now, we write u-momentum equation at face e in the same form. eepeeexPAHu 7.1 where, EpeHHH21 xPPxppEe But,

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36 ppppppxPAuH EEpEEExPAuH So the equation becomes, xPPAxPPAxPPAAuuupEEEpEWppppEpEpEpppEpe2121 The 1 st term in the above equation is the linear interpolation of the velocity, while the term in [] is the pressure smoothening term. It is of the following form, WpEEEpPPPPA334 which is 3 rd order pressure distribution. When we use it in the momentum equation it is 4 th order dissipation. This 4 th order dissipation is added the continuity equation. The cell face values of velocities computed as indicated above are then used to compute the continuity terms. Several investigators have compared the accuracy and computational efficiency of the non-staggered and staggered grid versions of SIMPLE family of methods. Among these studies are the works of Burns et al. (1986), Peric et al. (1988) and Malaaen (1992). Generally the accuracy and convergence rate of both formulations have been found comparable. The difference between the two results has been less than estimated numerical error in the calculations of the either scheme. 3.6 Validation of Computational Model The standard lid-driven cavity flow problem is presented as a test case, in order to validate the code. The problem has been extensively used to asses various codes and schemes by several researchers, and serves as a useful test case owing to substantial

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37 skewness of the streamlines of the flow relative to the grid employed for numerical simulation. The streamline contours obtained for Reynolds number of 100 on a grid of 100x100 using the SIMPLE method described above are plotted in Fig 3.1 after the steady state has reached. Fig 3.2 illustrates the u velocity profiles plotted along the center line of the cavity for 100x100 grid at Reynolds number of 100. The well known results of Ghia et al. (1982) have been used as a benchmark to assess the performance of the code. Fig 3.4 Streamlines for the driven cavity problem at Re = 100 on a grid of 100x100

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38 Fig 3.5 U-component of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x100 Fig 3.6 V-component of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x100

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39 u comp of velocity00.10.20.30.40.50.60.70.80.91-0.4-0.200.20.40.60.811.2u comp of velocityy 100x100 Ghia Fig 3.7 U-component of velocity profile along the vertical centerline for the driven cavity problem at Re = 100 on a grid of 100x100

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CHAPTER 4 FLOW CONFIGURATION AND TEST CASES As noted earlier in the previous investigations, the flow and heat transfer in microscale flow passages exhibit some unusual behavior and unique performance enhancement. There are also some questions surrounding these issues and some significant differences from the conventional situation that needs to be clarified. In the current investigation, an attempt is made to examine computationally the forced flow characteristics of water flowing through microchannels with dimensions as stated below, with and without surface roughness, to better understand the fundamental physical nature associated with this type of fluid flow. The 2-D parallel plate channels used for the simulations of the flows are as shown below. The simulations were performed on 3 sets of channels as stated below. The 2-D parallel plate channel is characterized by width of the channel (W) and the length of the channel (L). The surface roughness is characterized by the average height of the roughness height (h). The flow inside the channel is characterized by the initial velocity (U 0 ), viscosity (). The Reynolds number is based on the width of the channel (W). Case 1: Channel with no surface roughness. W = 0.1mm, L = 5 mm Case 2: Channels with sawtooth surface roughness. W = 0.1mm, L = 5mm Case 3: Channels with random surface roughness. W = 0.1mm, L = 5mm The ratio of the height of surface roughness elements (h) to the height of the channel (W) i.e. 2h/W ranges from 1% to 6%. Therefore the surface roughness may have a profound effect on the velocity field and the flow friction in microchannels. 40

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41 Uniform inlet velocity Outflow W L Fig 5.1 Parallel plates configuration used in the current study. For the channels with sawtooth surface roughness, the maximum height of the surface roughness goes from 1m to 6m. The profiles of the channels with sawtooth surface roughness are as shown the figures below. Fig 4.1. Section of a channel with sawtooth surface roughness. Maximum height of surface roughness is 1m.

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42 Fig 4.2. Section of a channel with sawtooth surface roughness. Maximum height of surface roughness is 2m. Fig 4.3. Section of a channel with sawtooth surface roughness. Maximum height of surface roughness is 4m.

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43 For the channels with random surface roughness, random roughness was generated using the random number generator in FORTRAN. The random numbers generated were then fitted to generate a curve using the sine function. The height of the surface roughness varies from 6m to 1m, but the average of height of the surface roughness element is maintained in accordance with the average height of the sawtooth surface. Following are the profiles of the channels with random surface roughness. Fig 4.4. Section of a channel with random surface roughness. Average height of surface roughness is 1m.

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44 Fig 4.5. Section of a channel with random surface roughness. Average height of surface roughness is 2m. Fig 4.6. Profile of random surface roughness. Average height of surface roughness is 6m.

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45 Table 4.1 Geometric parameters of microchannels ChannelL0) h/H Sawtoot0.1 50 00.01Sawtoot0.1 50 00.02Sawtoot0.1 50 00.03Sawtoot0.1 50 00.04Sawtoot0.1 50 00.05Sawtoot0.1 50 00.06Random0.1 50 00.01Random0.1 50 00.02Random 3 0.1 5 50 0.3 0.030.4 0.04Random 5 0.1 5 50 0.5 0.05 of channels for Reynolds number rangint accuracy is achieved using this grid. A comparison of pressure drop in no Heig ht (10-3 m) ength (10-3 m )L /Hh (1 -5 m h 1 5 .1 h 2 5 .2 h 3 5 .3 h 4 5 .4 h 5 5 .5 h 6 5 .6 1 5 .1 2 5 .2 Random 4 0.1 5 50 Random 6 0.1 5 50 0.6 0.06In Table 4.1 lists all the channels that are used for the simulations. Simulations were run of the above mentioned set g from 50 to 800. The flow was fully developed at the end of the runtime. Reynolds number was based on the entrance velocity. The grid used in the computation of flow in the channel is 50x500 i.e. 25000 grid points. Sufficien channels Sawtooth 1 and Sawtooth 2 with 25000 grid points and 50000 grid points are as shown below. Pressure drop Vs Re20000400006000080000100000120000Pressure drop 25000 grid points 50000 grid points Fig 4.7. A comparison of pressure drop in Sawtooth 1 channel with 25000 grid points and 50000 grid points 00200 4006008001000Re

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46 Pressure drop Vs Re02000040000600008000010000012000014000016000002004006008001000RePressure drop 25000 grid points 50000 grid points Fig 4.8. A comparison of pressure drop in Sawtooth 2 channel with 25000 grid points and 50000 grid points As we see that even if we increase in the grid points in X direction by 2 times (i.e. 1000 grid points) there is not a significant difference in pressure drop of the channels. So a sufficient accuracy is obtained even with 500 grid points in the X direction.

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CHAPTER 5 RESULTS AND DISCUSSIONS 5.1 Introduction This chapter presents the results of our simulations. First, the results of the pressure drop in microchannels are presented. The deviation of pressure drop from the conventional theory for microchannels due to the presence of surface roughness is studied. Next, a set of results are presented which shows the deviation of computed friction factor from conventional theory predictions. 5.2 Pressure Drop For the microchannels used in this study, the computed pressure gradients are plotted in Fig 5.1 to 5.6, for all the three sets of channels (smooth channel, sawtooth roughness channels, and random roughness channels). The pressure drop is defined as the difference between the inlet and the exit pressure values. For each calculated pressure gradient the Reynolds number (Re) is calculated using the uniform entrance velocity of the fluid. The physical properties involved in these calculations, such as density and dynamic viscosity, were determined from properties of water at STP conditions. Other useful parameters such as the average velocity u m flow rate Q, apparent friction factor f app and friction factor constant C f were determined from the velocity field. 47

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48 1m average surface roughness 02000040000600008000010000012000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.1. A comparison of pressure drop in channels with average surface roughness of 1m to that of theoretical pressure drop in a plain channel. 2m average surface roughness 02000040000600008000010000012000014000016000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.2 A comparison of pressure drop in channels with average surface roughness of 2m to that of theoretical pressure drop in a plain channel.

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49 3m average surface roughness 05000010000015000020000025000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.3 A comparison of pressure drop in channels with average surface roughness of 3m to that of theoretical pressure drop in a plain channel. 4m avera g e surface rou g hness 05000010000015000020000025000030000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.4 A comparison of pressure drop in channels with average surface roughness of 4m to that of theoretical pressure drop in a plain channel.

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50 5m average surface roughness 05000010000015000020000025000030000035000040000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.5 A comparison of pressure drop in channels with average surface roughness of 5m to that of theoretical pressure drop in a plain channel. 6m average surface roughness 05000010000015000020000025000030000035000040000045000050000050150250350450550650750Reynolds number RePressure drop Pa smooth channel sawtooth surfaceroughness random surface roughness Fig 5.6 A comparison of pressure drop in channels with average surface roughness of 6m to that of theoretical pressure drop in a plain channel.

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51 As seen in the Fig5.1-Fig5.6 we see that, the pressure gradient in the case of rough channels is more as compared to that of the plain channels. In the case of plain channels, the pressure gradient is linear as required by conventional laminar flow theory. As the pressure gradient in case of rough channels is more than that of the plain channels, for a given flow rate, a higher pressure gradient is required to force the liquid to flow through rough microchannels than that of the plain channels. For the microchannels with 1% of surface roughness, the pressure gradient is in close resemblance with the pressure gradient predicted by the conventional theory. As the surface roughness of the channels is increased we see more and more deviation from the theoretical values of the pressure gradient. Table 5.1 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 1m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 3868.23 4046.562 1.08393 1.133901 100 7506.738 8125.7 8311.249 1.082454 1.107172 150 11761.416 12868.6 13062.19 1.094137 1.110597 200 16420.464 17945.2 18147.4 1.092856 1.10517 250 21266.613 23471.5 23683.06 1.103678 1.113627 300 26622.414 29394.7 29616.31 1.104134 1.112458 350 31951.404 35754.3 35986.69 1.119021 1.126294 400 37862.244 42277.1 42520.55 1.116603 1.123033 450 44092.17 49317.6 49572.98 1.118512 1.124304 500 50219.325 56780.4 57048.43 1.130648 1.135986 550 56554.425 64340.4 64621.25 1.137672 1.142638 600 63526.419 72515.4 72810.11 1.1415 1.146139 650 70299.072 81191.7 81501.12 1.154947 1.159348 700 77916.357 89909.2 90233.4 1.153919 1.15808 750 85148.73 98995.2 99334.8 1.162615 1.166603 800 93328.551 108502.6 108858.3 1.162587 1.166399 In the above table we see that the pressure gradients for the channels with 1m average surface roughness, varies from 1.08 to 1.16 for sawtooth surface roughness and

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52 1.13 to 1.16 for random surface roughness. So the pressure gradients for these channels are as good as the theoretical analysis. Table 5.2 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 2m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 4355.28 4237.9 1.220408 1.187517 100 7506.738 9770.58 9186.3 1.301575 1.223741 150 11761.416 15325.38 14806.7 1.303022 1.258922 200 16420.464 21818.61 21106.4 1.328745 1.285372 250 21266.613 29601.54 28066.5 1.391925 1.319745 300 26622.414 36817.47 35628.4 1.38295 1.338286 350 31951.404 45377.82 43806.3 1.420214 1.371029 400 37862.244 54542.07 52717.5 1.44054 1.39235 450 44092.17 64402.02 62127.3 1.460623 1.409032 500 50219.325 74919.69 72165.8 1.49185 1.437013 550 56554.425 86112.18 82879 1.522643 1.465473 600 63526.419 97363.98 94145.2 1.532653 1.481985 650 70299.072 110440.6 106059.3 1.571011 1.508687 700 77916.357 123660.6 118619.8 1.587095 1.522399 750 85148.73 137653.8 131847.2 1.616628 1.548434 800 93328.551 151840.4 145341.5 1.626944 1.55731 Table 5.3 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 3m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 5147.1 4479.03 1.442287 1.255085 100 7506.738 11315.8 10458.54 1.507419 1.39322 150 11761.416 18475.4 17796.33 1.570848 1.513111 200 16420.464 26603.5 26359.65 1.620143 1.605293 250 21266.613 35703.2 36221.94 1.678838 1.70323 300 26622.414 45591.7 46944.18 1.712531 1.763333 350 31951.404 56750.5 59086.71 1.77615 1.849268 400 37862.244 68803.9 72390.06 1.817217 1.911933 450 44092.17 82397.2 86843.88 1.868749 1.969599 500 50219.325 95486.4 102549.2 1.901388 2.042027 550 56554.425 110267.5 118913.1 1.949759 2.102632 600 63526.419 125847.5 136679.9 1.981026 2.151545 650 70299.072 142486 155768.5 2.026855 2.215797 700 77916.357 160063 176022.7 2.054293 2.259124 750 85148.73 178320.3 197959.1 2.094221 2.324862 800 93328.551 197684.9 220924.2 2.118161 2.367166

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53 Table 5.4 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 4m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 5497.087 4563.82 1.540358 1.278844 100 7506.738 12922.67 11067.85 1.721476 1.474389 150 11761.416 20766.94 19359.94 1.765683 1.646055 200 16420.464 30292.39 29431.42 1.844795 1.792362 250 21266.613 42117.78 41188.88 1.980465 1.936786 300 26622.414 55096.92 54763.8 2.069569 2.057056 350 31951.404 69818.89 70057.26 2.185159 2.192619 400 37862.244 86065.4 86767.24 2.273119 2.291656 450 44092.17 100563.3 105437.2 2.280753 2.391291 500 50219.325 119049.5 125597.8 2.370592 2.500985 550 56554.425 138814 147563.1 2.454521 2.609224 600 63526.419 160217.3 170860.4 2.522058 2.689595 650 70299.072 181466.3 196452.7 2.581347 2.794527 700 77916.357 205386.2 224184.1 2.635984 2.877241 750 85148.73 228800.4 253703.2 2.687067 2.979531 800 93328.551 254713.6 284471.4 2.729215 3.048064 Table 5.5 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 5m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 5210.09 5633.3 1.459938 1.578527 100 7506.738 12921.36 13783.6 1.721301 1.836164 150 11761.416 22689.75 24308.3 1.929168 2.066783 200 16420.464 34789.06 37152 2.11864 2.262543 250 21266.613 49378.99 52153.8 2.321902 2.452379 300 26622.414 65549.24 69518.1 2.462182 2.611262 350 31951.404 85241.28 88914 2.667841 2.782789 400 37862.244 105809.9 110588.3 2.794602 2.920807 450 44092.17 128239.9 134099.8 2.90845 3.041352 500 50219.325 154202.6 160075.5 3.070582 3.187528 550 56554.425 181742.1 187787.8 3.213579 3.320479 600 63526.419 211934.1 219053.2 3.336158 3.448222 650 70299.072 243120.1 252465.6 3.458368 3.591308 700 77916.357 275690.3 288280.1 3.538285 3.699866 750 85148.73 310924.1 325894.9 3.651541 3.827361 800 93328.551 352944.1 365819.1 3.781738 3.919691

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54 Table 5.6 Comparison of P sawtooth /P theo and P sawtooth /P theo for channels with 6m average surface roughness. Re P theo P sawtooth P random P sawtooth /P theo P random /P theo 50 3568.707 5098.68 6015.7 1.428719 1.685681 100 7506.738 13198.14 15249 1.758172 2.031375 150 11761.416 24674.49 27451.1 2.097918 2.333996 200 16420.464 39037.68 42741.8 2.37738 2.602959 250 21266.613 56093.49 60892.9 2.637632 2.86331 300 26622.414 75839.49 82108.6 2.848708 3.084191 350 31951.404 100022.9 106301.9 3.130468 3.326987 400 37862.244 124617.8 133447.3 3.291347 3.524548 450 44092.17 153900.8 164732.6 3.490434 3.736096 500 50219.325 186491.7 196553.4 3.713545 3.9139 550 56554.425 220943.1 231795.4 3.906734 4.098625 600 63526.419 258915.2 270629.4 4.07571 4.260108 650 70299.072 298794.4 313164.5 4.250332 4.454746 700 77916.357 339122.1 357832.3 4.352386 4.592518 750 85148.73 384631 404674.1 4.517167 4.752556 800 93328.551 440443.5 455079.6 4.71928 4.876103 Pressure drop in microchannel. L = 5mm, width = 0.1mm (sawtooh surface roughness)05000010000015000020000025000030000035000040000045000050000050150250350450550650750RePressure Drop (Pa) smooth channel sawtooth 1 sawtooth 2 sawtooth 3 sawtooth 4 sawtooth 5 sawtooth 6 Fig 5.7 A comparison of pressure drop in channels with sawtooth surface roughness to that of theoretical pressure drop in a plain channel as the roughness is increased.

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55 Pressure drop in microchannel. L = 5mm, width = 0.1mm (random surface roughness)05000010000015000020000025000030000035000040000045000050000050150250350450550650750RePressure Drop (Pa) smooth channel random1 random 2 random 3 random 4 random 5 random 6 Fig 5.8 A comparison of pressure drop in channels with random surface roughness to that of theoretical pressure drop in a plain channel as the roughness is increased. Fig 5.9 Experimentally measured pressure gradient (a) SS and (b) FS microtubes, and comparison with the classical theory. (Mala, Li, Int. J. heat and Fluid flow (1999) 142-148)

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56 Fig. 5.10 A comparison of the measured data of pressure gradient vs. Reynolds number with the predictions of conventional laminar flow theory for trapezoidal microchannels. (a) (1) dh = 51.3 mm; (2) dh= 62.3 mm;(3) dh = 64.9 mm (Qu et al. Int. J. of Heat and Mass Transfer 43 (2000) 353-364) In Fig 5.7 and 5.8 we make a comparison of the pressure gradients in microchannels with sawtooth surface roughness and random surface roughness to those of a plain channel for increasing surface roughness. It is observed as the roughness in the channels is increased, the pressure gradient increases accordingly. Fig 5.9 gives the experimental results for flow in SS and FS microtubes having diameters in range of 63.5m to 254m for SS microtubes and 50.0m to 250m for FS microtubes. It is seen in Fig 9 the experimental curves are above the theoretical curves. For small Re the pressure gradient is approximately equal to that predicted conventional theory. As Re increases the measured pressure gradient is significantly higher than that predicted by conventional theory. According to the authors reason for this deviation of

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57 pressure gradient from the conventional theory was due to the presence of the surface roughness in the tubes. Fig 5.10 gives the experimental results for flow in trapezoidal silicon microchannels having hydraulic diameters from 51.3m to 168.9m. It is seen in Fig 5.10 the theoretical curves are below the experimental curves. For small Re the pressure gradient is approximately equal to that predicted conventional theory. As Re increases the measured pressure gradient is significantly higher than that predicted by conventional theory. According to the authors reason for this deviation of pressure gradient from the conventional theory was due to the presence of the surface roughness in the tubes. During the current study, we also find similar results. In Fig 5.7 and Fig 5.8 we also see that the pressure gradient deviates more from that of smooth channel in almost similar manner. We see that as Reynolds number increases, the difference between the rough surfaces microchannels and smooth channels is very significant. Table 1 Table 6 tabulates the ratio P sawtooth /P smooth and P sawtooth /P smooth for all the surface roughness at different Reynolds number. We see that at high Reynolds number pressure gradients in rough microchannels are 3.2 to 4.6 times that of pressure gradients in smooth channels. Therefore the surface roughness does have a major effect on the pressure gradients in microchannels, which is observed in the computed results as well as the experimental results done by researchers. To investigate the reasons behind the nature of increase in pressure drop in rough microchannels, we further run simulations of smooth channels with reduced width of channel, which might be caused due to the roughness element present in the microchannels. For these cases we reduce the width of channel by the value of average

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58 surface roughness height in the channels. Results of the following simulations are as presented below. 1m average surface roughness02000040000600008000010000012000002004006008001000RePressure drop Sawtooth surface 0.1 Random surface 0.1 Reduced channel smooth channel Fig 5.11 A comparison of pressure drop in rough channels, reduced width channels and smooth channel. 2m average surface roughness02000040000600008000010000012000014000016000002004006008001000RePressure drop Sawtooth surface 0.2 Random surface 0.2 Reduced channel smooth channel Fig 5.12 A comparison of pressure drop in rough channels, reduced width channels and smooth channel.

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59 3m average surface roughness05000010000015000020000025000002004006008001000RePressure drop Sawtooth surface 0.3 Random surface 0.3 Reduced channel smooth channel Fig 5.13 A comparison of pressure drop rough channels, reduced width channels and smooth channel. In Fig 5.11 Fig 5.13 we see that as we reduce the width of the channel the pressure drop in the reduced channel increases as compared to that of a smooth channel. So the increase in pressure drop of the rough channels may be due to the decrease in the width of channel. So the decrease in width of channel is one of the factors in increase of pressure drop for the rough microchannels. 5.3 Flow Friction The flow behavior of water through microchannels can be further interpreted in terms of the flow friction. In the following study we use the fanning friction factor for the total pressure drop in the channel, which is defined as 212happmDfPLu where, P pressure drop between the inlet and outlet. D h Hydraulic diameter. L Length of channel

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60 u m average velocity of fluid. density of water In the figures below fanning friction factor is plotted as a function of the Reynolds number. For comparison the relationships between friction factors and Reynolds number predicted by the conventional theory are also plotted. 1m average surface roughness 0.0000.1000.2000.3000.4000.5000.600102104106108101010Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surface roughness Fig 5.14 A comparison of friction factor in channels with average surface roughness of 1m to that of theoretical friction factor in a plain channel.

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61 2m average surface roughness 0.0000.1000.2000.3000.4000.5000.60050150250350450550650750Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surface roughness Fig 5.15 A comparison of friction factor in channels with average surface roughness of 2m to that of theoretical friction factor in a plain channel. 3m average surface roughness 0.0000.1000.2000.3000.4000.5000.60050150250350450550650750Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surface roughness Fig 5.16 A comparison of friction factor in channels with average surface roughness of 3m to that of theoretical friction factor in a plain channel.

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62 4m average surface roughness 0.0000.1000.2000.3000.4000.5000.60050150250350450550650750Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surface roughness Fig 5.17 A comparison of friction factor in channels with average surface roughness of 4m to that of theoretical friction factor in a plain channel. 5m avera g e surface rou g hness 0.0000.1000.2000.3000.4000.5000.60050150250350450550650750Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surfaceroughness Fig 5.18 A comparison of friction factor in channels with average surface roughness of 5m to that of theoretical friction factor in a plain channel.

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63 6m average surface roughness 0.0000.1000.2000.3000.4000.5000.60050150250350450550650750Reynolds number ReFriction factor theoretical friction factor sawtooth surfaceroughness random surface roughness Fig 5.19 A comparison of friction factor in channels with average surface roughness of 6m to that of theoretical friction factor in a plain channel. As seen in the above figures Fig 5.14Fig 5.19 we found that, the friction factor in the case of rough channels is more as compared to that of the smooth channels. In case of the smooth channels, the friction factor behaves as predicted conventional laminar flow theory. For the microchannels with 1% of surface roughness, the friction factor is in close resemblance with the friction factor predicted by the conventional theory. As the surface roughness of the channels is increased we see more and more deviation from the theoretical values of the friction factor.

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64 Fanning Friction factor Vs Reynolds number(random surface roughness) 0.010.1102004006008001000Reynolds numberFanning friction factor smooth channel random 1 random 2 random 3 random 4 random 5 random 6 Fig 5.20 A comparison of friction factor in channels with random surface roughness to that of theoretical friction factor in a plain channel as the roughness is increased Fanning Friction factor Vs Reynolds number(sawtooth surface roughness) 0.010.1102004006008001000Reynolds numberFanning friction factor smooth channel sawtooth 1 sawtooth 2 sawtooth 3 sawtooth 4 sawtooth 5 sawtooth 6 Fig 5.21 A comparison of friction factor in channels with sawtooth surface roughness to that of theoretical friction factor in a plain channel as the roughness is increased

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65 Fig 5.22 Friction factor f exp vs Re for some SS and FS microtubes and comparison with the classical theory. ( Mala et al. (1999)) In the Fig 5.20 and Fig 5.21 a comparison is made between the Fanning friction factors for the channels with surface roughness having average height of the roughness elements from 1% to 6% with that of a smooth channel. We see that as the average height of the roughness element in the channel is increased, there is more deviation of the Fanning friction factor from that of a smooth microchannel. It can be seen that the higher Reynolds number range all the curves are above the curve for the plain channel, which means that for a given Reynolds number the flow friction in the microchannels is higher than that obtained by using the plain channel, which is due to the presence of the surface roughness elements present in the channels. Fig 5.22 gives the experimental results for variation of friction factor in some SS and FS microtubes having diameters in range of 63.5m to 254m for SS microtubes and

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66 50.0m to 250m for FS microtubes. It is seen in Fig 5.22 that for small Re the friction factor decreases linearly with increase in Re on semi-log plots. But as Re becomes large Re > 1500 the slope of the curve decreases and approaches to zero. Even in the laminar flow regime it is seen the friction factors deviate from the conventional theory. According to the authors reason for this kind of behavior of friction factors was due to the presence of the surface roughness in the tubes. In Fig 5.20 and Fig 5.21 we find similar results for the friction factors. When the Re is small friction factors are in agreement with the conventional theory for small roughness channels, but as Re increases the difference between the friction factors is significant. This is due to the presence of the surface roughness. As we increase the surface roughness the deviation goes on increasing. So surface roughness does indeed play a role in friction factors for microchannels. Another important parameter used to describe the flow friction in channels is the friction factor constant C f which is the product of the friction factor and Reynolds number. RefCf It is well known from the conventional laminar flow theory that the friction factor constant is dependent on geometry of channel cross-section (R.K Shah et al. 1978). Hence for the present case a constant value of C f should be expected. However according to the computed data we find that C f is no longer constant for the flow in a microchannel.

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67 Friction factor constant Vs Re0510152025303540455002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.23 A comparison of f.Re in channels with average surface roughness of 1m to that of theoretical f.Re in a plain channel. Friction factor constant Vs Re010203040506002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.24 A comparison of f.Re in channels with average surface roughness of 2m to that of theoretical f.Re in a plain channel.

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68 Friction factor constant Vs Re010203040506070809002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.25 A comparison of f.Re in channels with average surface roughness of 3m to that of theoretical f.Re in a plain channel. Friction factor constant Vs Re02040608010012002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.26 A comparison of f.Re in channels with average surface roughness of 4m to that of theoretical f.Re in a plain channel.

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69 Friction factor constant Vs Re02040608010012014002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.27 A comparison of f.Re in channels with average surface roughness of 5m to that of theoretical f.Re in a plain channel. Friction factor constant Vs Re02040608010012014016018002004006008001000Ref.Re smooth channel sawtooth surface roughness random surface roughness Fig 5.28 A comparison of f.Re in channels with average surface roughness of 6m to that of theoretical f.Re in a plain channel.

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70 Friction constant C f is plotted as a function of Reynolds number in Fig 5.23 to 5.28. As it can be seen in the above figures, C f for the rough channels is not constant as compared to that of a smooth channel. However for the same average surface roughness the friction constant is in close agreement. Also we find that the flow friction in the channel is 1-41% higher than theoretical prediction in the channels with average surface roughness height of 0.1m and it also goes on increasing as the roughness element is increased. Qu et al. (2000), Mala et al. (1999) Toh et al. (2002), Peng et al. (1994), Wu et al. (2003a, 2003b) also observed such dependence of friction factor on Reynolds number. All the above authors quoted this deviation of frictional factor from the conventional theory was due to the presence of the surface roughness present in the channels. In the above study we see the same behavior of the friction factor. Friction constant Vs Reynolds number(sawtooth surface roughness) 02040608010012014016018050100150200250300350400450500550600650700750800Reynolds numberFriction constant smooth channel sawtooth 1 sawtooth 2 sawtooth 3 sawtooth 4 sawtooth 5 sawtooth 6 Fig 5.29 A comparison of friction constant in channels with sawtooth surface roughness to that of theoretical friction constant in a plain channel as the roughness is increased

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71 Friction constant Vs Reynolds number(random surface roughness) 02040608010012014016018050150250350450550650750Reynolds numberFriction constant smooth channel random 1 random 2 random 3 random 4 random 5 random 6 Fig 5.30 A comparison of friction constant in channels with sawtooth surface roughness to that of theoretical friction constant in a plain channel as the roughness is increased Fig 5.31 A comparison of f.Re in channels with rough surfaces (Wu et al. 2003)

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72 In the Fig 5.29 and Fig 5.30 a comparison is made between the friction factor constant for the channels with random surface roughness having average height of the roughness elements from 1% to 6% with that of a smooth channel. We see that as the average height of the roughness element in the channel is increased, there is more deviation of the friction factor constant from that of a smooth microchannel. It can be seen that the higher Reynolds number range all the curves are above the curve for the plain channel, which means that for a given Reynolds number the flow friction in the microchannels is higher than that obtained by using the plain channel, which is due to the presence of the surface roughness elements present in the channels. In Fig 5.31 gives the experimental results to two pairs of the silicon microchannels with different surface roughness. Microchannels #7 and #9 are two trapezoidal channels having the same geometric parameters but with relative different relative surface roughness (3.26x10 -5 5.87x10 -3 ), while microchannels #8 and #10 have triangular cross-section with same geometric parameters but different relative surface roughness (3.62x10 -5 1.09x10 -2 ). As we see in Fig 5.28 friction constant of trapezoidal microchannel #9 are larger than those of the trapezoidal microchannel #7 which has much lower surface roughness than microchannel #9. Similar increase in friction constant is observed for triangular microchannel #10 (1.09x10 -2 ) which has higher surface roughness when compared to the triangular microchannel #8 (3.26x10 -5 ). From Fig 5.29 Fig 5.30 it can be observed that friction factor increases with increase in Reynolds number. Also as the surface roughness is increased the friction factor and friction constant also increases. Similar results are obtained for microchannels

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73 with trapezoidal and triangular cross sections by Wu et al. (2003) as surface roughness is increased.

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CHAPTER 6 CONCLUSION Numerical simulations have been used to investigate the effects of surface roughness on the microchannel flows. In order to study the effects two different types of rough surface profiles were used for computation, the sawtooth surface roughness profile and the random surface roughness profile with the maximum height of the surface roughness and the randomness present in the channel specified so as to maintain a constant average surface roughness height. The actual geometry is then converted into the computational geometry using the transformation on coordinates. The 2-D set of Navier-Stokes equations are then solved on the transformed grid, using SIMPLE method along with the momentum interpolation technique on a non-staggered grid. During the first part of the investigation we study the effects of surface roughness in the pressure drop in the flow between two parallel plates. We can see from the results plotted that as the surface roughness is increased there is more deviation of the pressure drop from that estimated by conventional laminar theory as well as that estimated by the flow in smooth channel without any surface roughness. Also we see that as Reynolds number is increased the deviation between the computed and the estimated value of the pressure drop increases. The results are in agreement with the results obtained by the other researchers using different channels for the test. In all the cases the pressure drop in the rough microchannels was more than that of the conventional results. As we decrease the width of the microchannels for their corresponding surface roughness we see that the 74

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75 pressure drop in the reduced width microchannels increases. So the decrease in the width of the microchannels is one of the reasons of the increase in pressure drop of rough microchannels. In laminar flow friction factor is independent of the surface roughness. But in the second part of the investigation we see that, as the surface roughness is increased on the walls of the channels, friction factor increases. Also we see that as the Reynolds number is increased the friction factor for the given channel increases. So infact for microchannel flows surface roughness plays a very important role in friction factor. The results presented above are in agreement with the experiments done my researchers in the same field. So during this study we conclude that surface roughness plays a very important role in microchannel flows. We see that the pressure drop in the channels is more than the estimated value due to the presence of the surface roughness. Similarly the friction factor is also dependent on the surface roughness.

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76 LIST OF REFERENCES Adams, T.M., Abdel Khalik, S.I., Jeter, S.M., Qureshi, Z.H., An Experimental Investigation of Single phase Forced Convection in Microchannels, Int ernational J ournal of Heat Mass Transfer 41 1988 pp. 851 857. Amato, I. Formenting a Revolution, in Miniature, Science 282(5388), October 16 th 1998, pp. 402 405. Ammar, E.S., and Rodgers, T.J., UMOS Transistrors on (110) Silicon, IEEE Trans actions Electron Devices, ED 27, 1980, pp. 907 914. Angell, J.B., Terry, S.C. and Barth, P.W. Silicon Micromechanical Devices, Faraday Trans act ions I 68, 1983, pp 744 748 Anderson, D.A., Tannehill, J.C., and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer Hemisphere, 1984, New York. Ashley, S. Getting a Microgrip in the Operating Room, Mech anical Eng ineernig 118, September, 1996, pp. 91 93. Baker, D.R. Capillary Electrophoresis, Techniques in Analytical Chemistry Series John Wiley & Sons, New York 1995. Brandner, J., Fitchner, M., Schygulla, U., and Schubert, K. Improving the Efficiency of Micro Heat Exchangers and Reactors, in Proc. 4 th Int ernational Conf erence Microreaction Technology, AIChE, March 5 th 9 th Atlanta, GA, 2000, pp. 244 -249. Chalmers, P. Relay Races, Mech. Eng. 123, January, 2001, pp.66 68. Choi, S.B., Barron, R.F., Warrington, R.O., Fluid Flow and H eat T ransfer in icrotubes, Micormechanical Sensors, Actuators, and Systems ASME 1991, pp. 123 134. Fletcher, C.A.J., Computational Methods for Fluid Dynamics 2 Volumes. Springer, 1991, Berling. Ferziger, J.H., Peric, M., Computational Methods for Fluid Dynamics 3 rd Ed. Springer, 2002, New York. Gabriel, K.J., Jarvis, J. and Trimmer, W. eds. Small Machines, Large Opportunities: A R eport on the Emerging Field of Microdynamics, National Science Foundation, AT&T Bell Laboratories, Murray Hill, NJ. 1 998.

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77 Gabriel, K.J., Tabata, O. Shimaoka, K., Sugiyama, S., and Fujita, H., Surface Normal Electrostatic/Pneumatic Acutator, in Proc. IEEE Micro Electro Mechanical Systems February 4 th 7 th 1992, pp. 128 131. Gad el Hak, M. The Fluid Mechanics of Microd evices The Freeman Scholar Lecture, Journal of Fluids. Eng. Vol. 121, 1999, pp. 257 274. Ghaddar, N.K., Korczak, K.Z., Mikic, B.B., Patera, A.T., Numerical I nvestigation of I ncompressible F low in G rooved C hannels. Part 1. Stability and S elf S ustained O scillations, Journal of Fluid Mechanics Vol. 163, 1986, pp. 99 127. Ghaddar, N.K., Korczak, K.Z., Mikic, B.B., Patera, A.T., Numerical I nvestigation of I ncompressible F low in G rooved C hannels. Part 2. Resonance and O scillatory Heat Transfer Enhancement, Journal of Fluid Mechanics Vol. 168, 1986, pp. 541567. Ghia, U., Ghia, N., Shin, C.T., High Re Solutions for Incompressible Flow Using the Navier Stokes Equations and a Multigrid Method, Journal of Computational Physics Vol. 48, 1982, pp. 387 411. G uo, Z.Y., Li, Z.X., Size E ffect on M icroscale S ingle P hase F low and H eat T ransfer, International Journal of Heat and Mass Transfer 46 2003, 149 159. Gravesen, P., Branebjerg, J., and Jensen, O.S., Microfluidics A Review, Journal of Micromechanics and Mircroengineering, Vol. 3, 1993, pp. 168 182. Harley, J., Bau, H., Fluid Flow in Micron and Submicron Size Channels, IEEE Trans. THO249 3 1989, 25 28. Hirsch, C., Numerical Computation for Internal and External Flows Vol. 2, Wiley, 1990, New York. K night, J. Dust Mites Dilemma, New Sci. 162(2180) 29 th May, 1999, pp. 40 43. Kwang Hua, C.Z., Flow in a M icrotube with C orrugated W all, Mechanics Research Communications, Vol. 26(4), pp. 457 462. Karniadakis, G., Beskok. A., Microflows: Fundamentals an d Applications 2 nd Ed. Springer, 2001, New York. Lim, S.K., Wong, T.N., Ooi, K.T., Toh, K.C., Suzuki, K. Analytical S tudy of l iquid flow in a M icrotube, Engineering A dvances at the D awn of 21 st century, Proceedings of the Seminar on Integrated Engineeri ng, 2000, pp. 472 478. Ma, H.B., and Peterson, G.P, Laminar Friction Factor in Microscale Ducts of Irregular Cross section, Microscale Thermophyscial Engineering 1 (3) 1997, pp. 253265. Madou, M.J., and Morrison, S.J., Chemical Sensing with Solid State Devices Academic Press, 1997 New York

PAGE 92

78 Mala, G.M., Li, D., Flow C haracteristics of W ater in M icrotubes, International Journal of Heat and Fluid Flow Vol. 20 1999142 148. McDonald, A.T., Fox, R.W., Introduction to Fluid Mechanics 2 nd Ed. John Wile y & Sons, 1995, New York. Nguyen, N.T., Wereley, S.T., Fundamentals and Applications of Microfluidics 1 st Ed. Artech House, 2002, Boston. Patankar, S.V., Numerical Heat Transfer and Fluid Flow Hemisphere, Washington DC. 1980. Peng, X.F., Peterson, G .P., Convective H eat T ransfer and F low F riction for W ater F low in M icrochannel S tructures, Int. J. Heat Mass Transfer 39 1996 pp. 2599 2608. Peter Gravensen, Jens Barenbjerg and Ole Sondergard Jensen, Microfluidics A Review, MME, 1993, p. 143 164 Pet ersen, K.E ., McMillan W A. Kovacs G T. A. Northrup M.A ., Christel L A ., and Pourahmadi F ., Toward Next Generation Clinical Diagnostic Instruments: Scaling and New Processing Paradigms , Journal of Biomedical Microdevices, Vol. 2, No. 1, 1999. pp. 71 79. Qu, W., Mala, G.M., Li, D., Pressure D riven W ater F lows in T rapezoidal S ilicon M icrochannels, Int ernational J ournal of Heat and Mass Transfer 43 2000 353 364. Qu, W., Mala, G.M., Li, D., Heat T ransfer for W ater F low in T rapezoidal S ilicon M icrochan nels, Int ernational J ournal of Heat and Mass Transfer 43 2000 3925 3936 Roache, P.J., Computational Fluid Dynamics Hermosa Publishers, 1972, Albuquerque, NM. Schaller, Th. Bolin, L., Mayer, J., and Schubbert, K. Microstructure Grooves with a Width Le ss that 50m Cut with Ground Hard Metal Micro End Mills, Precision Eng. 23, 1999, pp. 229 235. Shoji, S., and Esashi, M., Microflow Devices and Systems, Journal of Micromechanics and Microengineering, Vol. 4, No. 4, 1994, pp. 157 171. Shyy, W., Computa tional Modelling of Fluid Flow and Interfacial Dynamics Elsevier, Amsterdam, Netherlands. 1994. Shyy, W., Thakur, S., Wright, J., Second order Upwind Scheme in a Sequential Solver for Recirculating Flows, AIAA Journal, Vol 30(4), pp. 923 932. System Pla nning Corporation, Microelectromechanical Systems (MEMS): A SPC Market Study, January 1999, Arlington VA.

PAGE 93

79 Tabeling, P., Some Basic Problems of Microfluidics, 14 th Australian Fluid Mechanics Conference, December 10 th 14 th 2001. Toh, K.C., Chen, X.Y., Chai, J.C., Numerical Computation of Fluid Flow and Heat Transfer in Microchannels, International Journal of Heat and Mass Transfer 45 2002, pp. 5133-5141. Thakur, S.S., and Shyy, W., Development of High Accuracy Convection Schemes for Sequential Solvers, Numerical Heat and Transfer, Vol. 23, pp. 175-199. Warming, R.F., and Beam, R.M., Upwind Second-order Difference Schemes and Applications in Aerodynamic Flows, AIAA Journal, Vol. 14 (9), pp. 1241-1249. White, F.M., Viscous Fluid Flow, 2 nd Ed. McGraw-Hill, 1991, New York. Wu, H.Y., Cheng, P., An Experimental Study of Convective Heat Transfer in Silicon Microchannels with Different Surface Conditions, International Journal of Heat and Mass Transfer 46 2003, pp. 2547-2556. Wu, P.Y., Little, W.A., Measurement of Friction Factor for Flow of Gases in Very Fine Channels Used for Micro-miniature Joule Thompson Refrigerators, Cryogenics 24(8) 1983, pp. 273-277. Yu, D., Warrington, R., Barren, T., An Experimental and Theoretical Investigation of Fluid Flow and Heat Transfer in Micr otubes, Proceedings of the ASME/JSME Thermal Engineering Conference, ASME, 1, 1995, 523-530.

PAGE 94

BIOGRAPHICAL SKETCH Amit Kulkarni was born in Pune, India, in December 1979. He graduated from the University of Pune with a Bachelor of Engineering degree in mechanical engineering in July 2001. He joined University of Florida in Fall 2001 to pursue his masters in the Department of Mechanical and Aerospace Engineering. He will be conferred the Master of Science degree by the University of Florida in May 2004. 80


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EFFECTS OF SURFACE ROUGHNESS IN MICROCHANNEL FLOWS


By

AMIT S. KULKARNI















A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2004

































Copyright 2004

by

Amit S. Kulkarni


































This document is dedicated to my parents for their ever-extending support and
contribution. .















ACKNOWLEDGMENTS

This thesis would not have been possible without help of certain people and hence I

would like to use this opportunity to thank them. I would like to thank my parents in the

first place for their constant support, encouragement and love all along my way. I would

like to express big thanks for Dr. J. N. Chung for allowing me to work on this project

under his tutelage and showing confidence in me and giving me insightful talks that went

as building blocks in my thesis as well as my master's education. I would also like to

express my sincere gratitude towards Dr. William Lear, Dr. Darryl Butt and Dr. Wei

Shyy for their time and effort as my committee member. I would also like to thank Dr.

Siddarth Thakur for his assistance with regard to development of the code for the current

study. I would also like to thank my roommates for their support during my master's

studies. Lastly I would also like to acknowledge the financial support from Motorola.















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ....................................................... ............ .............. .. vii

L IST O F FIG U R E S .............. ............................ ............. ........... ... ........ viii

N O M E N C L A T U R E ......................................................................................................... x ii

A B S T R A C T .......................................... ..................................................x iii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 Introduction .................................................................................................. ....... 1
1.2 Liquid Flows in Microchannel............................................ 4
1.3 Unique Aspects of Liquids in Microchannels ............................................7
1.4 Commercial Aspects of M icrofluidics ............... ............................................ 8
1.5 Scientific Aspects of M icrofluidics ................................................................... 11
1.6 M ilestones of M icrofluidics............................................................................... 11
1.6.1 D evice D evelopm ent ............................................. ................ ............... 11
1.6.1.1 M iniaturization approach .................................... ....................... 11
1.6.1.2 Exploration of new effects .............. ................. ............ 12
1.6.1.3 Application developments ................ .......... ......... ................12
1.6.2 Technology D evelopm ent ........................................ ....... ............... 13

2 LITERA TU RE SU RVEY ................................................. .............................. 14

2.1 Introduction........................................................................ ....... ...... 14
2.2 Wet Bulk Micromachining ......... ........ .... ......... ............... 14
2.2.1 W et Isotropic and Anisotropic Etching ................................................ 15
2.2.2 Surface Roughness and Notching.................... .......... ............ 16

3 GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM...................... 22

3 .1 In tro d u ctio n ................................ ... .. .... .................................... .... ............... 2 2
3.2 Transformation to Body-Fitted Coordinates for 2D geometries...........................23
3.3 D iscretized Form of Equations .................................... .......................... ......... 26
3.4 SIMPLE Method.................. .. ..............................29


v









3.4.1 Staggered Grid: .... ... .. ...................... ........ 34
3.4.2 N on-staggered G rids:........................................................... .................34
3.5 M om entum Interpolation Technique ........................................ .....................35
3.6 Validation of Computational Model................. .............. ...... ..............36

4 FLOW CONFIGURATION AND TEST CASES ............................................. 40

5 RESULTS AND DISCU SSION S......................................... .......................... 47

5 .1 Introdu action ..................................................................................... 4 7
5 .2 P re ssu re D ro p .................................................................................................. 4 7
5 .3 F low F friction ................................................................59

6 C O N C L U SIO N ......... ......................................................................... ........ .. ..... .. 74

L IST O F R EFER EN CE S ........................................... ............................. ............... 76

B IO G R A PH IC A L SK E TCH ..................................................................... ..................80
















LIST OF TABLES


Table page

4.1 Geometric parameters of microchannels .............. ......................................45

5.1 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 1 [im
average surface roughness. ............................................ ............................ 51

5.2 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 2im
average surface roughness. ............................................ ............................ 52

5.3 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 3 im
average surface roughness. ............................................ ............................ 52

5.4 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 4rm
average surface roughness. ............................................ ............................ 53

5.5 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 5.im
average surface roughness. ............................................ ............................ 53

5.6 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 6rm
average surface roughness. ............................................ ............................ 54
















LIST OF FIGURES


Figure pge

1.1 Scaling of thing s ....................................................... 2

1.2 Micro heat exchanger constructed from rectangular channels machined in
m etal ....... ................................................................... . 5

1.3 Blood sample cartridge using microfluidic channels............. ... .................6

1.4 K nudsen num ber regim es. ........................................................................... 7

1.5 Estimated sales of microfluidic components compared to other MEMS
d ev ic e s................................. ......................................................... ............... 1 0

2.1 A wet bulk micromachining process is used to craft a membrane with
pierzoresistive elem ents. ..................................... ............... ......... 15

2.2 Roughness caused in microchannels during anisotropic etching process..............16

2.3 Experimentally measured pressure gradient........... ................................. 19

2.4 A comparison of the measured data of pressure gradient vs. Reynolds
number with the predictions of conventional laminar flow theory.....................20

2.5 A comparison of the experimental data of pressure gradient vs. Reynolds
number with the predictions of Roughness viscosity model. ............. ...............21

3.1 Collocated grid and notation for a 2-D grid on a physical plane.........................27

3.2 Collocated grid and notation for a 2-D grid on a Transformed
(C om putational) plane. ........................................ ............................................28

3.3 T he SIM P L E algorithm ............................................................... .....................33

3.4 Streamlines for the driven cavity problem at Re = 100 on a grid of 100x100.......37

3.5 U-component of velocity contours for the driven cavity problem at Re = 100
on a grid of 100x 00 .... ...................................................................... ......... 3 8

3.6 V-component of velocity contours for the driven cavity problem at Re = 100
on a grid of 100x 100 ...................................................................... ...... 3 8









3.7 U-component of velocity profile along the vertical centerline for the driven
cavity problem at Re = 100 on a grid of 100x100 ............. ............................39

5.1 Parallel plates configuration used in the current study. .....................................41

4.1 Section of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 1 m ............................................................ ............... 4 1

4.2 Selection of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 2 m ........................................................... ............... 42

4.3 Section of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 4 m ........................................................... ............... 42

4.4. Section of a channel with random surface roughness. Average height of
surface roughness is 1 m ............................................................. ............... 43

4.5. Section of a channel with random surface roughness. Average height of
surface roughness is 2 m ............................................................ ............... 44

4.6. Profile of random surface roughness. Average height of surface roughness
is 6 im ............................................................................4 4

4.7. A comparison of pressure drop in Sawtooth 1 channel with 25000 grid
points and 50000 grid points.......................... ............................. ............... 45

4.8. A comparison of pressure drop in Sawtooth 2 channel with 25000 grid
points and 50000 grid points.................... .... ............ .................. 46

5.1. A comparison of pressure drop in channels with average surface roughness
of 1 lm to that of theoretical pressure drop in a plain channel. ...........................48

5.2 A comparison of pressure drop in channels with average surface roughness
of 2.im to that of theoretical pressure drop in a plain channel. ............................48

5.3 A comparison of pressure drop in channels with average surface roughness
of 3 tm to that of theoretical pressure drop in a plain channel. ............................49

5.4 A comparison of pressure drop in channels with average surface roughness
of 4tm to that of theoretical pressure drop in a plain channel. ............................49

5.5 A comparison of pressure drop in channels with average surface roughness
of 5 m to that of theoretical pressure drop in a plain channel. ............................50

5.6 A comparison of pressure drop in channels with average surface roughness
of 6Lim to that of theoretical pressure drop in a plain channel. ...........................50









5.7 A comparison of pressure drop in channels with sawtooth surface roughness
to that of theoretical pressure drop in a plain channel as the roughness is
in creased ........................................................ ................ 54

5.8 A comparison of pressure drop in channels with random surface roughness
to that of theoretical pressure drop in a plain channel as the roughness is
in creased ........................................................ ................ 5 5

5.9 Experimentally measured pressure gradient (a) SS and (b) FS microtubes,
and comparison with the classical theory. .................................. .................55

5.10 A comparison of the measured data of pressure gradient vs. Reynolds
number with the predictions of conventional laminar flow theory for
trapezoidal m icrochannels. ............................................ ............................ 56

5.11 A comparison of pressure drop in rough channels, reduced width channels and
sm ooth channel. ................................................... ................. 58

5.12 A comparison of pressure drop in rough channels, reduced width channels
an d sm ooth ch ann el.......... .............................................................. .. .... .. .... .. 58

5.13 A comparison of pressure drop rough channels, reduced width channels and
sm ooth channel. ................................................... ................. 59

5.14 A comparison of friction factor in channels with average surface roughness
of 1 lm to that of theoretical friction factor in a plain channel ...........................60

5.15 A comparison of friction factor in channels with average surface roughness
of 2rm to that of theoretical friction factor in a plain channel ...........................61

5.16 A comparison of friction factor in channels with average surface roughness
of 3 Cm to that of theoretical friction factor in a plain channel...........................61

5.17 A comparison of friction factor in channels with average surface roughness
of 4rm to that of theoretical friction factor in a plain channel...........................62

5.18 A comparison of friction factor in channels with average surface roughness
of 5 m to that of theoretical friction factor in a plain channel...........................62

5.19 A comparison of friction factor in channels with average surface roughness
of 6rm to that of theoretical friction factor in a plain channel...........................63

5.20 A comparison of friction factor in channels with random surface roughness
to that of theoretical friction factor in a plain channel as the roughness is
in creased ........................................................................... 6 4









5.21 A comparison of friction factor in channels with sawtooth surface roughness
to that of theoretical friction factor in a plain channel as the roughness is
in creased ........................................................................... 6 4

5.22 Friction factor fexp vs Re for some SS and FS microtubes and comparison
w ith the classical theory............. .... ................................ ...... ........ ............... 65

5.23 A comparison of fRe in channels with average surface roughness of 1 lm to
that of theoretical fRe in a plain channel............ ........ ........ ....................67

5.24 A comparison of fRe in channels with average surface roughness of 2[m to
that of theoretical fRe in a plain channel............... .......... .... ............... 67

5.25 A comparison of fRe in channels with average surface roughness of 3 m to
that of theoretical fRe in a plain channel............ ........ ........ ....................68

5.26 A comparison of fRe in channels with average surface roughness of 4rm to
that of theoretical fRe in a plain channel......................... .... ............... 68

5.27 A comparison of fRe in channels with average surface roughness of 5[m to
that of theoretical fRe in a plain channel....................... ...................69

5.28 A comparison of fRe in channels with average surface roughness of 6[m to
that of theoretical fRe in a plain channel......................... .... ............... 69

5.29 A comparison of friction constant in channels with sawtooth surface
roughness to that of theoretical friction constant in a plain channel as the
roughness is increased........ ............................................................. .... ........ .. 70

5.30 A comparison of friction constant in channels with sawtooth surface
roughness to that of theoretical friction constant in a plain channel as the
roughness is increased........ ............................................................. .... ........ .. 7 1















NOMENCLATURE


W = Width of the channel
L = Length of the channel.
x,y = Coordinate directions
Re = Reynolds number
u = Velocity in x-direction
v = Velocity in y-direction
Um = Average velocity of fluid.
Uo = Initial velocity of fluid.
p = Pressure
AP = Pressure difference
p = Viscosity
p = Density
f = Friction factor
Cf = Coefficient of friction
Q = Flow rate
h = Average height of surface roughness
Dh = Hydraulic diameter
c, qr = Computational coordinate directions
J = Jacobian
U = Contravariant velocity in x-direction
V = Contravariant velocity in y-direction
W = Matrix of variables discretized in time
E = Matrix of Flux vectors in C direction
F = Matrix of Flux vectors in r direction
S = Matrix of Source terms
F = Mass flux at the control volume
H, = Compact notation for advective and convective flux
Rp = Stencil error
S, = Momentum source term
p* = Guessed pressure
p' = Correction pressure
u* = Guessed u-component of velocity
u' = Correction u-component of velocity
v* = Guessed v-component of velocity
v' = Correction v-component of velocity















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

EFFECTS OF SURFACE ROUGHNESS IN MICROCHANNEL FLOWS

By

Amit S. Kulkarni

May 2004

Chair: J.N.Chung
Major Department: Mechanical and Aerospace Engineering

An incompressible 2-D Navier Stokes solver is used to study the effects of surface

roughness in microchannel flows. A rough microchannel is generated by selecting the

maximum height of the roughness element in the microchannel, and the randomness

present in the length of the channel so as to maintain a constant average surface

roughness in the channel. Two different types of surface roughness, namely, random

surface roughness and sawtooth surface roughness profiles, are generated for the study.

The actual geometry is converted into the computational geometry using the

transformation of coordinates. The geometry is solved for the specified boundary

conditions using body fitted coordinate system. The governing equations are discretized

using the non-staggered manner. The governing equations are solved using a SIMPLE

method along with the momentum interpolation technique.

The study investigates the pressure drop and the friction factor in rough channels.

During the first part the pressure drop in the channels is found to be more than the









pressure drop estimated by the conventional laminar flow theory. The surface roughness

is the factor causing the change in pressure drop.

The second part of the study investigates the departure of the friction behavior from

the classical thermofluid correlations. Although the friction factor in the laminar flow is

supposed to be independent of surface roughness, this does not hold true in the case of

microchannels. As we increase the surface roughness in the channels, we see more

deviation of friction factor from the standard values.

These numerical studies are compared with the experimental results done by

researchers in this field. The study attempts to set caveats for innovative and inquisitive

minds that aspire to study actual flow behavior in microchannels.














CHAPTER 1
INTRODUCTION

1.1 Introduction

Tool making has always differentiated our species from all others on earth.

Aerodynamically correct spears were carved by Homo sapiens close to 40,000 years ago.

Man builds size consistent with his size, typically in the range of two orders of magnitude

larger or smaller than himself as shown in Fig 1.1. Humans have always striven to

explore, build and control the extremes of length and time scales. The Great Pyramid of

Khufu was originally 147m high when completed around 2600 B.C., while the Empire

State Building is 449m high, constructed during 1950. At the other end of the spectrum of

man-made artifacts, a dime is less than 2cm in diameter. Watchmakers have practiced the

art of miniaturization since the 13th century. The invention of the microscope in the 17th

century opened the way for direct observation of microbes and plant and animal cells.

Smaller things were man-made in the later half of 20th century. The transistor-invented in

1947, in today's integrated circuits has a size or 0.18lm in production and approaches

10nm in research laboratories using electron beams.

Manufacturing processes that can create extremely small machines have been

developed in recent years (Angell et al. 1983; Gabriel et al. 1988, 1992; Ashley, 1996;

Amato, 1998; Knight 1999; Chalmers, 2001). Electrostatic, magnetic, electromagnetic,

pneumatic and thermal actuators, motors, valves, gears etc less than size of 100lm in size











Diameter of Earth


Light Year


104 106 + 1 10 10 T 1012 10"


1018 1020

meter


Voyage to Brobdingnag

Voyage to Lilliput

meter
6 1014 1012 1010 10 10 10 10 108 110 00 102


Diameter of Proton H-Atom Diameter Human Hair Man


Nanodevices (IJ Typical Man-Madee
Devices


Fig 1.1 Scaling of things, in meters. Lower scale continues in the upper bar from left to
right.

have been fabricated. These have been used as sensors for pressure, temperature,

mass flow, velocity, sound and chemical composition; as actuators for linear and angular

motions and also part of complex systems such as robots, micro heat engines, and micro

heat pumps.

Integrated microfluidic systems with a complex network of fluidic channels are

routinely used for chemical and biological analysis and sensing. They have generated a

considerable activity, at economic and scientific levels, and their importance in our

everyday life, is expected to considerably increase over the next few years. The rapid

development of microelectronics and molecular biology has simulated an increasing

interest in miniaturization characterized by flow and heat transfer in confined tiny

geometries.


Astronomical Unit









Micorelectromechanical systems (MEMS) refer to devices that have a characteristic

length of less than 1mm but more than 1 lm, that combine electrical and mechanical

components that are fabricated using integrated circuit batch-processing technologies.

Current manufacturing techniques for MEMS include surface silicon micromachining;

bulk silicon micromachining; lithography, electrodeposition and plastic molding, and

electrodischarge machining (EDM). As shown in Figure 1.1, MEMS are more than four

order magnitudes larger than hydrogen atom, but about four order magnitude smaller than

the traditional man-made artifacts. Microdevices can have characteristic lengths smaller

than the diameter of a human hair.

Some of the MicroElectroMechanical Systems (MEMS) devices used for

momentum and energy transfer have characteristic lengths of microns. Microfluidics is

about flows of liquids and gases, single or multiphase, through microdevices fabricated

by MEMS. It is mainly fostered by the development of lab-on a chip devices, i.e. systems

able to perform in impressive number of tasks on a small chip, such as mixing,

separating, analyzing, detecting molecules. In the next few years, a variety of such

systems, designed to identify and analyze DNA from a drop of blood will be made

available. Also in context of propulsion of miniaturized rockets, for space applications,

microfluidics arose into a new discipline.

MEMS are finding increased applications in a variety of industrial and medical

fields, with a potential market in billions of dollars. Accelerometers for automobile

airbags, keyless entry systems, dense arrays of micromirrors for high definition optical

displays, scanning electron microscope tip to image single atoms, micro-heat exchangers

for cooling electronic circuits, reactors for separating biological cells, blood analyzers









and pressure sensors for catheter tips are but a few use. Microducts are used in infrared

detectors, diode lasers, miniature gas chromatographs and high frequency fluidic control

systems. Micropumps are used for ink-jet printing, environmental testing and electronic

cooling. Potential medical applications for small pumps include controlled delivery and

monitoring of minute amounts of medication, manufacturing of nanoliters of chemicals

and development of artificial pancreas.

1.2 Liquid Flows in Microchannel

Nominally, microchannels can be defined as channels whose dimensions are less

than 1mm and greater than 1lm. Above 1 m the flow exhibits behavior that is same as

macroscopic flows. Below 1 lm the flow is characterized as nanoscopic. Currently, most

microchannels fall into the range of 30 to 300[im. Microchannels can be fabricated in

many materials-glass, polymers, silicon metals using various processes including surface

micromachining, bulk micromachining, molding, embossing and conventional machining

with microcutters.

Microchannels offer advantages due to their high surface-to-volume ratio and their

small volumes. The large surface to volume ration leads to a high rate of heat and mass

transfer, making microdevices excellent tools for compact heat exchangers. For example,

the device in Fig 1.2 is a cross-flow heat exchanger constructed from a stack of 50

14mm-14mm foils, each containing 34 200[pm wide x 100lm deep channels machined

into the 200 [m thick stainless steel foils by the process of direct high precision

mechanical micromachining (Brander et al., 2000; Schaller et al., 1999). The direction of

the flow in adjacent foils is alternated 90, and the foils are attached by the means of

diffusion bonding to create a stack of cross-flow heat exchangers capable of transferring

10kW at a temperature difference of 80K using water flowing at 750kg/hr. The









impressive large of heat transfer is accomplished mainly by the large surface area

covered by the interior of the microchannel: approximately 3600mm2 packed into a

14mm cube.

A second example of the application of microchannels is in the area of

microelectromechanical systems (MEMS) devices for biological and chemical analyses.

The primary advantages of microscale devices in these applications are the good match

with the scale of biological structures and potential for placing multiple functions for

chemical analysis on a small are.

Microchannels are used to transport biological materials such as (in order of size)

proteins, DNA, cells and embryos or to transport chemical samples. Flow in biological

devices and chemical analysis microdevices are usually much slower than those in heat

transfer and chemical reactor microdevices.






















Fig 1.2 Micro heat exchanger constructed from rectangular channels machined in metal.
(The MEMS Handbook (2002))











6 -STAT


EG6+ j



L r

~iJ I


Fig 1.3 Blood sample cartridge using microfluidic channels (The MEMS Handbook
(2002))

Knudsen Number:
It is defined as the ratio of the mean free path over the characteristic geometry

length or a length over which very large variations of macroscopic quantity make take

place.

It is given by the following formula
A
Kn =
L
where

S= mean free path
L = characteristic length
According to the Knudsen number the flow regimes can be divided into various

regimes. These are: continuum, slip, transition and free-molecular flow regimes. Discrete

particle or molecular based model is the Boltzmann equation. The continuum based

models are the Navier-Stokes equations. Euler equations correspond to inviscid

continuum limit which shows a singular limit since the fluid is assumed to be inviscid









and non-conducting. Euler flow corresponds to Kn = 0.0. Navier-Stokes equations with

slip boundary conditions are used for Slip flow regime.

Kn = 0.0001 0.001 0.01 0.1 1 10 100







ontinum flow Transition regime

Slip-flow regime Free-molecule
flow
Fig 1.4 Knudsen number regimes. (The MEMS Handbook (2002))

Since the density of liquids is 1000 times the density of gases, Kn doesn't play an

important role. The flow is in the continuum regime. For example, water has lattice

spacing of 0.3 nm. In a l1m gap and 50 [m diameter channel Knudsen number is 3x10-4

and 6x10-4, respectively, which are well within the continuum flow regime.

1.3 Unique Aspects of Liquids in Microchannels

Flow in microscale devices differ from the macroscopic counterparts for two

reasons:

1. The small scale makes molecular effects such as wall slip more important.

2. Small scale amplifies the magnitudes of certain ordinary continuum effects to
extreme levels.

Consider, strain rate and shear rate which scale in proportion to the velocity scale,

Us, and inverse proportion to the length scale, Ls. Thus, 100 mm/s flow in a 10tm

channel experiences a shear rate of order of 104 s-1. Acceleration scales are also similarly

enhanced. The effect is even more dramatic if one tries to maintain the same volume flux

while scaling down. The flux scales as Q UsL2, so at constant flux Q L-2s, and both

shear and acceleration go as L-3s. Fluids that are Newtonian at ordinary rates of shear and









extension can become Non-Newtonian at very high rates. The pressure gradient becomes

especially large in small cross-section channels. For fixed volume flux, the pressure

gradient increases as L4s.

Electrokinetic effects occur at the interface between liquids and solids such as glass

due to chemical interaction. The result is electrically charged double layer that indicates a

charge distribution in a very thin layer of fluid close to the wall. Application of an

electric field to this layer creates a body force capable of moving the fluid as if it were

slipping over the wall.

Molecular effects in liquids are difficult to predict because the transport theory is

less well developed than the kinetic theory of gases. For this reason, studies of liquid

microflows in which molecular effects play a role are much more convincing if done

experimentally.

Liquids are essentially incompressible. Consequently, the density of a liquid in

microchannel flow remains very nearly constant as a function of distance along the

channel, despite the very large pressure gradients that characterize the microscale flow.

This behavior greatly simplifies the analysis of liquid flows relative to gas flows wherein

the large pressure drop in a channel leads to large expansion and large changes of thermal

heat capacity.

Liquids in contact with solids or gases have surface tension in the interface. At the

microscale, the surface tension force becomes one of the most important forces, far

exceeding body forces such as gravity and electrostatic fields.

1.4 Commercial Aspects of Microfluidics

With the recent achievement in the Human Genome Project and the huge potential

of biotechnology, microfluidic devices promise to be a big commercial success.









Microfluidic devices are tools that enable novel applications unrealizable with

conventional equipment. The apparent interest and participation of the industry in

microfluidics research and development show the commercial values of devices for

practical applications. With this commercial potential, microfluidics is poised to become

the most dynamic segment of the MEMS technology thrust. From its beginnings with the

now-traditional microfluidic devices, such as inkjet print heads and pressure sensors, a

much broader microfluidic market is now emerging.

Fig 1.4 shows the estimated sales of microfluidic devices in comparison to other

MEMS devices. The estimation assumes an exponential growth curve based on the

survey data of 1996[System Planning Corporation, 1999) The projection considers four

types of microfluidic devices: fluid control devices, gas and fluid measurement devices,

medical testing devices, and miscellaneous devices such as implantable drug pumps. The

curve in the fig shows that microfluidic devices sales exceed all other application areas,

even the emerging radio frequency MEMS devices (RF-MEMS). Commercial interests

are focused on plastic microfabrication for single-use disposable microfluidic devices.

The major application applications of microfluidics are medical diagnostics, genetic

sequencing, chemistry production, drug discovery and proteomics.

Microfluidics can have a revolutionizing impact on chemical analysis and

synthesis, similar to the impact of integrated circuits on computers and electronics.

Microfluidic devices could change the way instrument companies do business. Instead of

selling a few expensive systems, companies could have a mass market of cheap,

disposable drug dispensers available for everyone will secure a huge market similar to

that of computers today.









As computing power has improved from generation to generation by higher

operation frequency as well as parallel architecture, in the same way, microfluidics

revolutionizes chemical screening power. Furthermore, microfluidics will allow the

pharmaceutical industry to screen combinatorial libraries with high throughput-not

previously possible with manual, bench-top experiments. Fast analysis is enabled by the

smaller quantities of materials in assays. Massively parallel analysis on the same

microfluidic chip allows higher screening throughput. Microfluidic assay can have

several hundred to several hundred thousand parallel processes. The high performance is

extremely important for DNA-based diagnostics in pharmaceutical and health care

applications.



3,500
Microfluidics

3,000

o
0 2,500


5 2,000

03
1,500
Pressure sensors ..-- ,




1500 \ ^ 9- 9 1 2
6 1 997 1998 1999 2000 2001 2002 2003
Year

Fig 1.5 Estimated sales of microfluidic components compared to other MEMS devices.
(Fund. and App. of Microfluidics (2002))









1.5 Scientific Aspects of Microfluidics

Scientists from almost all traditional engineering and science disciplines have

begun pursuing mircrofluidics research, making it a truly multidisciplinary field

representative of the new economy of the 21st century.

Electrical and mechanical engineers contribute novel enabling technologies to

microfluidics. Fluid mechanics researchers are interested in the new fluid phenomena

possible at the microscale. In contrast to the continuum based hypotheses of conventional

macroscale flows, flows physics in microfluidic devices is governed by a transitional

regime between the continuum and molecular-dominated regimes. Besides new analytical

and computational models, microfluidics has enabled a new class of fluid measurements

of microscale flows using in situ microinstruments. Life scientists and chemists also find

in microfluidics novel, useful tools. Mircofluidic tools allow them to explore new effects

not possible in traditional devices. These new effects, new chemical reactions, and new

microinsturments lead to new applications in chemistry and bioengineering. These

reasons explain the enormous interest of research disciplines in microfluidics.

1.6 Milestones of Microfluidics

There are two major aspects considered as the milestones of microfluidics: the

applications-driven development of devices and development of fabrication technologies.

1.6.1 Device Development

1.6.1.1 Miniaturization approach

With silicon micromachining as the enabling technology, researchers have been

developing silicon microfluidic devices. The first approach of making miniaturized

devices was shirking down conventional principles. This approach is representative of the

research conducted in the 1980s through the mid-1990s. In this phase of microfluidics









development, a number of silicon microvalves, micropumps, and microflow sensors were

developed and investigated (Shoji S. et al. 1994).

Two general observations of scaling laws can be made in this development stage:

the power limit and the size limit of the devices. Assuming that the energy density of

actuators is independent of their size, scaling down the size will decrease the power of the

device by the length scale cubed. This means we cannot expect micropumps and

microvalves to deliver the same power level as conventional devices. The surface-to-

volume ratio varies as the inverse of the length scale. Large surface area means large

viscous forces, which in turn requires powerful actuators to be overcome. Often

integrated microactuators cannot deliver enough power, force, or displacement to drive a

microfluidic device, so an external actuator is the only option for microvalves and

micropumps. The use of external actuators limits the size of those microfluidic devices,

which range from several millimeters to several centimeters.

1.6.1.2 Exploration of new effects

Since the mid-1990s, development has shifted to the exploration of new actuating

schemes of microfluidics. Because of the power and size constraints research efforts have

concentrated on actuators with no moving parts and non-mechanical pumping principles.

Electrokinetic pumping, surface tension-driven flows, electromagnetic forces and

acoustic streaming are effects that usually have no impact at macroscopic length scales.

However, at the microscale they offer particular advantages over mechanical principles.

1.6.1.3 Application developments

Concurrent with the exploration of the new effects, microfluidics today is looking

for further application fields beyond conventional fields such as flow control, chemical

analysis, biomedical diagnostics and drug discovery. New applications utilizing









microfluidics for distributed energy supply, distributed thermal management and

chemical production are promising.

1.6.2 Technology Development

Similar to trends in device development, the technology of making microfluidic

devices has also seen a paradigm shift. Since mid-1990, with chemists joining the field,

microfabrication technology has been moving to plastic micromachining. With the

philosophy of functionality above miniaturization and simplicity above complexity,

microfluidic devices have been kept simple, sometimes only with a passive system of

microchannels. The actuating and sensing devices are not necessarily integrated into

microdevices. These microdevices are incorporated as replaceable elements in bench top

and handheld tools. Batch fabrication of plastic devices is possible with many replication

and forming techniques. The master for replication can be fabricated with traditional

silicon-based micromachining technologies. Complex based microfluidic devices based

on plastic microfabrication could be expected in the near future with further

achievements of plastic-based microelectronics.














CHAPTER 2
LITERATURE SURVEY

2.1 Introduction

Rough walls exist in all flow systems, where they may lead either to deterioration

or improvement of the desired functionality. Wall roughness can be increased to promote

mixing of the fluid, or reduced to eliminate flow disturbances. The related problem of the

laminar-turbulent transition over a rough wall is one of the classical problems in fluid

mechanics that has so far defied all analytical efforts. Recently, the effects of surface

roughness became of interest from the point of view of passive/active flow control

strategies, where once is interested in determining the smallest possible surface

modification that may induce the largest possible changes in flow field.

2.2 Wet Bulk Micromachining

In wet bulk micromachining, features are sculpted in the bulk of materials such as

silicon, quartz, SiC, GaAs, InP, Ge and glass by orientation-dependent (anisotropic)

and/or by orientation-dependent isotropicc) wet etchants. The technology employs pools

as tools instead of plasma. A vast majority of micromachining work is based on single

crystal silicon. These tools are used to fabricate microstructures either in parallel or serial

processes. The principle commercial Si micromachining tools used today are the well-

established wet bulk micromachining. A typical structure fashioned in a bulk

micromachining process is shown in Fig 2.1. Despite all the emerging technologies, Si

wet bulk micromachining, being the best characterized micromachining tool, remains the









most popular in industry. Two types of etching used for wet bulk micromachining are wet

isotropic etching and anisotropic etching which are discussed as below.


Deposit Photoresist Open Contacts
'm


Silicon Wafer
e11vlp JiUV L



Develop Resist


gight
Mask


Deposit Aluminum


Pattern Aluminum
[-^"SBS--s


I11 1 Implant
Boron Pattern Back Oxide


An neal and Oxidation Sjlicon Etch



Fig 2.1 A wet bulk micromachining process is used to craft a membrane with
pierzoresistive elements. (The MEMS Handbook (2002)).

2.2.1 Wet Isotropic and Anisotropic Etching

Wet etching of Si is used mainly for cleaning, shaping, polishing and characterizing

structural and compositional features. Wet etching is also faster as compared to typical

dry etching. Modification of wet etchant and/or temperature can alter the selectivity to

silicon dopant concentration and type especially when using alkaline etchants, to

crystallographic orientation.

Isotropic etchants, also polishing etchants, etch in all crystallographic directions at

the same rate; they usually are acidic, and lead to rounded isotropic features in single

crystalline Si. They are usually used at room temperatures or slightly above (< 50C).

Anisotropic etchants, etches away crystalline silicon at different rates depending on

the orientation of exposed crystal plane. Typically the pH stays above 12, while more









elevated temperatures are used for these slower type etchants (> 50C). These etchants

are reaction rate limited. When carried out properly, anisotropic etching results in

geometric shapes bounded by perfectly defined crystallographic planes.

2.2.2 Surface Roughness and Notching

Anisotropic etchants frequently leave too rough surface behind, and a use of

isotropic etch is required to 'touch-up'. Roughness also referred to as notching or

pillowing, results when centers of exposed areas etch with a seemingly lower average

speed compared with the borders of areas. This difference can be as 1 to 2[im, which is

quiet considerable if one is etching 10 to 20m thick structures. Fig 2.2shows these kinds

of roughness in the channels.




"Roughness" Effect













Fig 2.2 Roughness caused in microchannels during anisotropic etching process. (The
MEMS Handbook (2002)).


The liquid flow characteristics in microchannels are important in the design and the

process control of MEMS and micro-fluidic devices. It has been found that for fluid flow

in microchannels, the flow behavior often deviates significantly from the predictions of

conventional theories of fluid mechanics.









There has been a lot of research work done in the areas, of heat transfer and fluid

flow through microchannels. Wu and Little (1983) measured the friction factors for the

flow of gases in miniature channels. The test channels were etched in glass and silicon

with hydraulic diameters ranging from 55.81 to 83.08 im. The tests involved both

laminar and turbulent flow regimes. They found that the friction factors of both flow

regimes in these channels were larger than predictions form the correlations for the

macroscale pipes. The transition to turbulent flow was found to be as low as 350. They

attributed these deviations to relatively high surface roughness, asymmetric roughness

and uncertainty in the determination of channel dimensions.

Harley and Bau (1989) measured the friction factors in microchannels with

trapezoidal and rectangular cross sections. The trapezoidal microchannels had the

dimensions of 33 pm (depth), 111 pm (top width), and 63 pm (bottom width), and a

rectangular channel had the dimensions of 100pm in depth, and 50pm in width. They

found that the product off/Re ranged from 49 for the rectangular channel to 512 for the

trapezoidal channel in contrast to the classical value of 48.

Adams et al. (1998) conducted single phase flow studies in microchannels using

water as working fluid. Two diameters of circular microchannels, namely 0.76 and 1.09

mm were used for investigation. It was found that the Nusselt numbers are larger than

those in macrochannels.

Peng and Peterson (1996) investigated water flows in rectangular microchannels

with hydraulic diameters ranging from 0.133 to 0.367 mm and width to height ratios from

0.333 to 1. Their experimental results indicated that the flow transition occurs at

Reynolds number 200-700. This transitional Re decreases as the size of the microchannel









decreases. The flow friction behaviors of both the laminar and turbulent flows were found

to depart from classical correlations. The friction factors were either large or small than

the predictions of the classical theories. The geometrical parameters, such as hydraulic

diameter and aspect ratio, were found to have important effects on flow.

Yu et al. (1995) studied the fluid flow and heat transfer characteristics of dry

nitrogen gas and water in microtubes with diameters of 19, 52 and 102 im. The Reynolds

number in their study ranges from 250 to over 20,000. The average relative roughness for

53 gm microtube was measured and the value is approximately 0.0003. The flow friction

results indicate that for laminar flow, in microtubes, the value of the product fRe, is

between 49.35 and 51.56, instead of 64.

Lim et al. (2000) conducted experimental study on water flow in microtubes. The

diameters range from 49.3 to 701.9 im. It was found that when microtube diameters are

below 300 gm, thefRe decreases (departs from fully developedf/Re of 64) as the

diameter decreases.

However one of the very few works done in understanding the effect of surface

roughness on momentum and heat transfer was done by Mala and Li (1999) and Qu et al.

(2000). They investigated flow in trapezoidal silicon microchannels. A significant

difference between the experimental data and the theoretical predictions was found.

Experimental results indicated that pressure gradient and flow friction in microchannels

are higher that those given by conventional laminar flow theory. The measured higher

pressure gradient and flow friction was attributed to the surface roughness of

microchannels.







19


Mala and Li (1999) measured pressure gradients of water flow in microtubes with

inner diameter ranging from 50.0 to 254.0 [im. The Reynolds number was up to 2100.

They found for larger microtubes with inner diameter above 150 [im, the experimental

results were in rough agreement with conventional theory. For smaller microtubes, the

pressure gradients are up to 35% higher than these predicted by the conventional theory.

As the Reynolds number increased the difference between the experimental and the

conventional results also increased. They attributed these effects to the change in flow

mode from laminar to turbulent at low Reynolds number, or due to the effects of surface

roughness.

2.5 '63.5 H exp I i I
S101.6pm exp
2.0 130pm exp
152P mexp
1.5 203pm exp .
254pm exp
1.0 .--" Classical thry.

0.5


3.5 80m exp


2.5 1 50gm exp
S205 m exp
2.0 Classical thry.
S1_5 ..'0
1.0 ...." ____ .......
0.5 ;.
0.0 ". --
0 300 600 900 1200 1500 1800 2100
Reynolds number
Fig 2.3 Experimentally measured pressure gradient (a) SS and (b) FS microtubes, and
comparison with the classical theory. (Mala, Li, Int. J. heat and Fluid flow
(1999) 142-148)









6 I -1
(a)
(33)





R .nm r Re' '
4 : 4
















mm; (2) dh= 62.3 mm;( 3) dh= 64.9 mm (Qu et al. Int. J. of Heat and Mass
A ExpiDIUF)
I "^_ Theory'


0 too 200 300 400 500 600
Reynoids number, lRe
Fig. 2.4. A comparison of the measured data of pressure gradient vs. Reynolds number
with the predictions of conventional laminar flow theory. (a) (1) dh = 51.3
mm; (2) dh= 62.3 mm ;( 3) dh = 64.9 mm (Qu et al. Int. J. of Heat and Mass
Transfer 43 (2000) 353-364)

As seen in the above Fig 2.4 theoretical curves fall below the experimental curves,

which mean that at any given flow rate, a higher pressure gradient is required to force the

liquid to flow through those microchannels than the predictions of the conventional

laminar flow theory.

They proposed a Roughness-Viscosity model to interpret the experimental data

more correctly. According to this concept the value of roughness viscosity fR should have

a higher value near the wall and gradually diminish as the distance from the wall

increases. Roughness viscosity /R should also increase as Re increases.












0 r I



1(1












0 100 200 300 400 500 600
Reynolds number, Re
Fig. 2.5. A comparison of the experimental data of pressure gradient vs. Reynolds
number with the predictions of Roughness viscosity model. (a) (1) dh= 51.3

mm; (2) dh= 62.3 mm ;( 3) dh = 64.9 mm (Qu et al. Int. J. of Heat and Mass
Transfer 43 (2000) 353-364)

However, not a lot of work is done in computational modeling the microchannels

with surface roughness. In the following work, we try to model the surface roughness in

the microchannels and model the flow field in the microchannels to study the effect of the

surface roughness on the pressure drop and the friction factor in the microchannel.














CHAPTER 3
GOVERNING EQUATIONS AND OVERVIEW OF ALGORITHM

3.1 Introduction

Simulation of full Navier-Stokes equations for different fluid flows arises in

various engineering problems. Various different algorithms have been proposed and

developed by various researchers. But an approach that is fully robust from the point of

view of numerical and modeling accuracy as well as efficiency has yet to be developed.

Existing algorithms for Navier-Stokes equations can be generally classified as density-

based methods and pressure-based methods. For both of these methods, the velocity field

is normally obtained using the momentum equations.

Density based methods are employed for compressible flows where the continuity

equation is used to obtain the density of the fluid, while pressure information is obtained

using the equation of state. The system of equations is solved simultaneously. These

methods can be extended with modification to the low Mach number regime where the

flows are incompressible where the density has no role to play in determining, the

pressure field. (Fletcher 1988, Hirsch 1990).

Pressure based methods (Patankar 1980, Shyy 1994) are developed for the

incompressible flow regime. These obtain the pressure field via a pressure correction

equation which is formulated by manipulating the continuity and momentum equations.

The solution procedure is conventionally sequential in nature, and hence, can more easily

accommodate a varying number of equations depending on the physics of the problem

involved, without necessity of reformulating the entire algorithm. These methods can be









extended to compressible flows by taking in account the dependence of density on

pressure, via the equation of state.

The discretization schemes used in the present algorithm have been developed

primarily for incompressible flows and are differentiated by the geometric shape of the

interpolation function used to estimate the fluxes on a control volume interface. The first-

order upwind, central difference, second-order upwind (Warming & Beam 1976, Shyy et

al. 1992, Thakur & Shyy 1993) and QUICK (Quadratic Upwind Interpolation for

Convective Kinematics) are the examples of these kind of schemes.

The continuity and the momentum equation in the Cartesian coordinates can be

written as follows

Op + 8(pu) (pv) 0 3.1

8(pu) 8(puu) 8(puv) __p 8- ( + 3.
+ + + 8 +- a 3.2
8({pv) 8p) (v8(pvv) ^ 8p 8 vY S ( v3
at x y 8y 8x 8) y 8y y
3.2 Transformation to Body-Fitted Coordinates for 2D geometries

The above form of Navier-Stokes equations is for Cartesian coordinates (x, y). For

arbitrarily shaped geometries generalized body-fitted coordinates are employed, denoted

by (, fl) where = (x,y), fr = rl(x,y). The transformation of the physical domain (x, y) to

the computational domain (, fl) is achieved by transformation metrics which are related

to the physical coordinates as follows (Anderson et. al 1984).









~'x ~'y 1 f i f1, 3.4
77x ly ] J If21 /22.
where,

fil =y,
fi2 = X
f21 =
f22 = X

and J is the Jacobian of the transformation given by,

J = x yq xq y

Each term in the equations (Eq. 3.1-3.3) is transformed to the (5, rl) coordinate

system. The resulting governing equations in generalized body-fitted coordinates are

presented in the complete form as follows:

Continuity:


S+ (pU)+ a (pV)= 0 3.5
at 0c 0r7
U momentum equation:

p +- (V =- (fp) +-_(f2 ) +- q q2
Sac a rJ a 1 r 3.6)
3.6

0, T127 afc 16 ]









V momentum equation:

6Q) a a a a f a-it)

3.7
[ ] J al a +q3 o

where,

2 2


q3 = X X + y2y7
and U, V are the components of the contravariant velocity (which is the scalar product of

the velocity vector and the area vector at a control volume interface).

U = uy7 vx7
V = vx uy,
The contravariant velocity components can be interpreted as the volume flux normal to

control volume interfaces; specifically, U is the local volume flux along the 5 coordinate,

V along the rI coordinate.

In a compact form, the governing equations can be written as follows

aW OE OF
+-+-= S 3.8
Ot O 0r
where,



W=pu 3.9


and the flux and source vectors are given by











pU

E= -pUu- fp+ q -q2 > U 3.10

Jp a8u au'
-PVv f2lp+- -q2 + q3



pV

F = -pVu f2+ -q2 + q 3.11
O u 0u

-pVv- f22+ -q2


S= {S" 3.12


3.3 Discretized Form of Equations

The governing equations presented in the previous section are discretized on a

structured grid. The velocity components and the scalar variables like pressure are

located on the grid in a non-staggered manner as shown in Fig 3.1. For the flow domain,

the positive direction (increasing index i) is denoted as the east direction and the

negative direction (decreasing index i) as the west. Similarly, the north and south

directions are along the positive f (increasing j) and negative f (decreasing j) directions,

respectively. The index notation for momentum control volumes is illustrated using a u-

component control volume. The u-component associated with a representative grid point

(i, j) is labeled E (index i + 1) and EE (index i + 2). Similarly, the first and second

neighbors along the west, north, and south directions are labeled, respectively, as W and

WW, N and NN, S and SS. The east face is also denoted as i + 12 interface indicating that









it lies between the u's located at (i, j) and) (i+1, j). Similarly, the west, north, south faces

are also denoted as i 12, j + 12, j /2, respectively.


Fig 3.1 Collocated grid and notation for a 2-D grid on a physical plane.

The governing equations written in the generalized body fitted coordinates are

integrated over the control volume whose dimensions in the computational domain are

given by A( x Aq. The discretized form of the governing equations is finally obtained by

choosing the dimensions of the control volume as A = 1, Aq = 1. The discretized form of

the continuity and momentum equations are as presented below.

The discretized form of the continuity equation can be written as:

o[u]e + [po] =0 3.13
where,









[*'] = (*), (). etc.
We denote the mass flux at each control volume face by F and rewrite the continuity

equation as

[F]e +[F]: =0 3.14


Fig 3.2 Collocated grid and notation for a 2-D grid on a Transformed (Computational)
plane.

The discretized form of the momentum equations can be obtained in a similar manner.

The details of the momentum equations are as follows.


n









U momentum equation


[Uuw + [pVu] = -[flp]-[f21p] f+ Fq q-e

F r( u a u\
2 w 3.15

+ -q2-+3


V momentum equation


L[uv] + lpvvl = 22p : [/ 1 2 -
S 3.16

+ q2 +q3 )

The two terms on the left hand side of the equation (write equation number), are the

convective fluxes at the control volume faces; the first two terms on the right hand side

are the pressure fluxes and the last two terms are the diffusion fluxes. The standard

central difference operator is employed for the pressure and the diffusive fluxes.

3.4 SIMPLE Method

Implicit methods are preferred for steady and slow transient flows, because they

have less stringent time steps restrictions as compared the explicit schemes. Many

solutions methods for steady incompressible flows use a pressure (or pressure-correction)

equation to enforce mass conservation at each time step or otherwise known as outer

iteration for steady solvers.

The acronym SIMPLE stands for Semi-Implicit Method for Pressure Linked

Equations. The algorithm was originally put forward by Patankar and Splading and is

essentially a guess-and-correct procedure for the calculation of pressure. Until the early

1980s the SIMPLE family of methods was generally only employed on staggered grids.









However later several investigators (Rhie et al. 1981) reported success in implementing

pressure-correction schemes on a regular grid.

During the current investigation we use SIMPLE method on non-staggered grids.

The disadvantages mentioned above in using non-staggered grid are taken care by using

the Rhie-Chow Momentum interpolation technique.

Discretized U-momentum and V-momentum equations are written in following

form

app P= 'anbu"b +(p-W pP)AP +SP 3.17
aPvp = a 'vb +(pS ) AP +SP 3.18
where,

S,:is the source momentum term
Ap : cell face area of the u-control volume.
To initiate the SIMPLE calculation process a pressure field* is guessed. Discretized

momentum equations Eqn. 3.17 and Eqn. 318 are solved using the guessed pressure field

to yield velocity components u* and v* as follows:

ap P = a b u +(p*w p )A +SP 3.19
apvp = anbvb +(p' Pp) Ap + Sp 3.20
Now we define the correction p' as the difference between the correct pressure field p

and the guessed pressure fieldp, so that

p = P +p' 3.21
Similarly we define velocity corrections u' and v' to relate the correct u and v velocities

to the guessed solutions u* and v*

u =* +u' 3.22
v = v' +v' 3.23
Substitution of correct pressure field p into momentum equations yields the correct

velocity field (u,v). Discretized U-momentum and V-momentum equations Eq. (3.17) and

Eq. (3.18) link the correct velocity fields with the correct pressure field.








Subtraction of Eq. (3.19) and Eq. (20) from Eq. (3.17) and Eq. (3.18) respectively gives

ap(U p)= anb (u U) + [(pW p) (pp p)] AP 3.24
aP(vP vP) = a (vb- vb)+ [(P -s (p -P P)]A 3.25
Using the correction formulae Eq. (3.21-3.23) the Eq. (3.24) and Eq. (3.25) may be

written as follows:

apU' = Z aC b + ( p' ) Ap 3.26
aPvP = alb +(p' p')A 3.27
At this point an approximation is introduced an u'b and anu are dropped to

simplify equations (3.26) and (3.27) for velocity corrections. Omission of these terms is

the main approximation of the SIMPLE algorithm. We obtain

u' = dp(p p') 3.28
v' = d, p p',) 3.29
where

dp Ap
ap
Equations (3.28) and (3.29) describe the corrections to be applied to velocities

through equations (3.22) and (3.23) which gives

u=u*+d (p -p) 3.30
S= + dp (p' p',) 3.31
The velocity field obtained above is subjected to the constraint that it should satisfy

continuity equation. Continuity equation is the discretized form is as shown below.

[(puA)E -(pu]A)P][(A)N -(pvA) ] = 0 3.32
Substitution of the corrected velocities Eq.(3.30) and Eq. (3.31) into discretized

continuity equation (3.32) gives the equation for pressure correction p'. The pressure

correction equation in the symbolic form is as shown below

appp = a + awp + aNp + asp + b 3.33









where,

a, = aE + a, + aN + a, + b' are the coefficients obtained from the continuity equation.

b' is the imbalance arising from the incorrect velocity field u*, v*.

By solving Eq. (3.33) the pressure correction' can be obtained at all points. Once

the pressure correction field is known the correct pressure field is obtained from Eq.

(3.21) and velocity components through correction formulas Eq.(3.30) and Eq.(3.31).

The omission of terms such as anb u' and an nu' does not affect the final

solution because the pressure correction and velocity corrections will be zero in a

converged solution giving p* = p, u* = u, v* = v.










S START



STEP 1: Solve the discretized momentum
equations
apu' = Z abu* +(p*- pp)Ap ++ S
av*p = anbvn +(p*s p*p)Ap + S




Set
p; STEP 2: Solve pressure correction equation
V* v app, = aEPE + aW + aNN + aspi +




STEP 3: Correct pressure and velocities
u=u*+ d(p -pp)
v=v* +d(p'-p'p)

p = Pp +Pp




No
Convergence



Yes


SSTOP

Fig 3.3 The SIMPLE algorithm

Above we present an algorithm of the SIMPLE method which is employed in this


code.









3.4.1 Staggered Grid:

The main idea of the staggered grid is to evaluate scalar variables, such as pressure,

density, temperature etc., at ordinary nodal points but to calculate velocity components

on staggered grids centered around the cell faces.

Advantages of the staggered grid:

1. Checkered board problem is solved.

2. Divergence condition is satisfied.

3. Compact stencil is used for the Pressure Poison equation

4. Strong coupling between, and u,v.

However, the disadvantages of staggered grid are:

There is more computational complexity especially for the non-Cartesian, non-orthogonal

meshes (especially 3D meshes)

1. More memory is required for storage of variables.

2. Difficult and inefficient for multi-grid solvers.

3.4.2 Non-staggered Grids:

In non-staggered grid we have all the variables p, u, v are solved at same point.

This retains a strong coupling between and u, v. The checkered board pattern of the

pressure field that results when using a Non-staggered grid without certain modifications

to the original staggered grid scheme is highlighted. The necessary remedies must be

applied to the non-staggered scheme to overcome the checkered board pressure field.

In the approach presented here all quantities are solved and stored at the element

centroid. The face values of the velocity components have to be calculated from these

element based values. This leads to the need to employ an alternative interpolation










method, which does not suffer from the checkerboard effect. The Rhie Chow

interpolation method offers one approach which satisfies these requirements

We need to use the Rhie-Chow momentum-interpolation technique to satisfy the

continuity and using the non-staggered grid.

3.5 Momentum Interpolation Technique

Consider a control volume as shown in the figure3.1,

We write the u momentum equation in the symbolic form as follows


Au,= Au +b AQK,
M=E,W k & p

Apup =H' -AQp KP
P
where,

H'= ZAMuM +b
M=E,W
H AQ, (3P^
u =H F,

where,

H
H p
P A
P
Now, we write 'u-momentum' equation at face 'e' in the same form.


u,= He A- P 7.1

where,


H= (HP +HE)

But P
&) Ax
But,

















1 1A, pPP-





term in [] is the pressure smoothening term. It is of the following form,


4A

which is 3rd order pressure distribution. When we use it in the momentum equation it is
4th order dissipation. This 4th order dissipation is added the continuity equation. The cell


face values of velocities computed as indicated above are then used to compute the

continuity terms.
Several investigators have compared the accuracy and computational efficiency of

the non-staggered and staggered grid versions of SIMPLE family of methods. Among

these studies are the works of Burns et al. (1986), Peric et al. (1988) and Malaaen (1992).
Generally the accuracy and convergence rate of both formulations have been found

comparable. The difference between the two results has been less than estimated
numerical error in the calculations of the either scheme.


3.6 Validation of Computational Model

The standard lid-driven cavity flow problem is presented as a test case, in order to

validate the code. The problem has been extensively used to asses various codes and

schemes by several researchers, and serves as a useful test case owing to substantial












skewness of the streamlines of the flow relative to the grid employed for numerical


simulation.


The streamline contours obtained for Reynolds number of 100 on a grid of 100x100


using the SIMPLE method described above are plotted in Fig 3.1 after the steady state


has reached. Fig 3.2 illustrates the u velocity profiles plotted along the center line of


the cavity for 100x100 grid at Reynolds number of 100. The well known results of Ghia


et al. (1982) have been used as a benchmark to assess the performance of the code.


0.5 1
XC

Fig 3.4 Streamlines for the driven cavity problem at Re = 100 on a grid of 100x100


Sbtamline plot for cavity flow


53

59

J;

.I I:

O
:I.j

JJ

:I i

5'









































Fig 3.5 U-component of velocity contours for the driven
grid of 100x100


cavity problem at Re


Fig 3.6 V-component of velocity contours for the driven cavity problem at Re
grid of 100x100


U component of velocity contours


0.9

0.8

0.7

0.6 -

>- 0.5 -

0.4 -

0.3 'I-II-

0.2

0.1


0.5 1
X


V component of velocity contour plots

0.9



08 "1
0.5 -
0.6 .. ..

0.5

0.4 2 'x

0.3
v 270626 II,, i
0.2 02220226

0.1 -D061 872-2272
O2 681:272

0.5 1
x


100 on a































100 on a












u comp of velocity


1 -

0.9

0.8

0.7

0.6

> 0.5

0.4

0.3

0.2

0.1

0
-0.4


--100x100
* Ghia


0.8 1 1.2


Fig 3.7 U-component of velocity profile along the vertical centerline for the driven cavity
problem at Re = 100 on a grid of 100x100


-0.2 0 0.2 0.4 0.6
u comp of velocity














CHAPTER 4
FLOW CONFIGURATION AND TEST CASES

As noted earlier in the previous investigations, the flow and heat transfer in

microscale flow passages exhibit some unusual behavior and unique performance

enhancement. There are also some questions surrounding these issues and some

significant differences from the conventional situation that needs to be clarified. In the

current investigation, an attempt is made to examine computationally the forced flow

characteristics of water flowing through microchannels with dimensions as stated below,

with and without surface roughness, to better understand the fundamental physical nature

associated with this type of fluid flow.

The 2-D parallel plate channels used for the simulations of the flows are as shown

below. The simulations were performed on 3 sets of channels as stated below. The 2-D

parallel plate channel is characterized by width of the channel (W) and the length of the

channel (L). The surface roughness is characterized by the average height of the

roughness height (h). The flow inside the channel is characterized by the initial velocity

(Uo), viscosity ([). The Reynolds number is based on the width of the channel (W).

Case 1: Channel with no surface roughness. W = 0.1mm, L = 5 mm
Case 2: Channels with sawtooth surface roughness. W = 0.1mm, L = 5mm
Case 3: Channels with random surface roughness. W = 0.1mm, L = 5mm

The ratio of the height of surface roughness elements (h) to the height of the

channel (W) i.e. 2h/W ranges from 1% to 6%. Therefore the surface roughness may have

a profound effect on the velocity field and the flow friction in microchannels.











Uniform inlet
velocity


Outflow


L

Fig 5.1 Parallel plates configuration used in the current study.

For the channels with sawtooth surface roughness, the maximum height of the

surface roughness goes from 1 m to 6gm. The profiles of the channels with sawtooth

surface roughness are as shown the figures below.


0.0001





5E-05







0.0001 0.00015
x

Fig 4.1. Section of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 1 m.







































Fig 4.2. Section of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 2gm.


Fig 4.3. Section of a channel with sawtooth surface roughness. Maximum height of
surface roughness is 4gm.


0.00012


0.0001


8E-05





4E-05


2E-05


0
5E-05 0.0001 0.00015
X


0.0001






5E-05






0

0.0001 0.00015 0.0002
X







43


For the channels with random surface roughness, random roughness was generated


using the random number generator in FORTRAN. The random numbers generated were


then fitted to generate a curve using the sine function. The height of the surface


roughness varies from 6gm to 1gm, but the average of height of the surface roughness


element is maintained in accordance with the average height of the sawtooth surface.


Following are the profiles of the channels with random surface roughness.


0.0001
9E-05
8E-05
7E-05
6E-05
5E-05
4E-05 -
3E-05
2E-05
1E-05


0.0001 0.00015 0.0002
x

Fig 4.4. Section of a channel with random surface roughness. Average height of surface
roughness is 1 m.








































Fig 4.5. Section of a channel with random surface roughness. Average height of surface
roughness is 2gm.


Fig 4.6. Profile of random surface roughness. Average height of surface roughness is
6gm.


0.0001

9E-05

8E-05

7E-05

6E-05

S5E-05
4E-05

3E-05

2E-05

1E-05

0

0.0001 0.00015 0.0002
x


0.0001

8E-05

6E-05

4E-05

'2t-05

0

-2E-05

-4E-05

-6E-05

5E-05 0.0001 0.00015 0.0002
XC










Table 4.1 Geometric parameters of microchannels
Channel no Height (10-3 m) Length (10-3 m) L/H h (10-5 m) h/H
Sawtooth 1 0.1 5 50 0.1 0.01
Sawtooth 2 0.1 5 50 0.2 0.02
Sawtooth 3 0.1 5 50 0.3 0.03
Sawtooth 4 0.1 5 50 0.4 0.04
Sawtooth 5 0.1 5 50 0.5 0.05
Sawtooth 6 0.1 5 50 0.6 0.06
Random 1 0.1 5 50 0.1 0.01
Random 2 0.1 5 50 0.2 0.02
Random 3 0.1 5 50 0.3 0.03
Random 4 0.1 5 50 0.4 0.04
Random 5 0.1 5 50 0.5 0.05
Random 6 0.1 5 50 0.6 0.06

In Table 4.1 lists all the channels that are used for the simulations.

Simulations were run of the above mentioned set of channels for Reynolds number

ranging from 50 to 800. The flow was fully developed at the end of the runtime.

Reynolds number was based on the entrance velocity.

The grid used in the computation of flow in the channel is 50x500 i.e. 25000 grid

points. Sufficient accuracy is achieved using this grid. A comparison of pressure drop in

channels Sawtooth 1 and Sawtooth 2 with 25000 grid points and 50000 grid points are as

shown below.


Pressure drop Vs Re

120000
100000
2 80000
-- 25000 grid points
2. 60000
50000 grid points
40000
20000
0
0 200 400 600 800 1000
Re

Fig 4.7. A comparison of pressure drop in Sawtooth 1 channel with 25000 grid points and
50000 grid points












Pressure drop Vs Re


160000
140000
120000
100000
80000
60000
40000
20000
0
0 200 400 600 800
Re


- 25000 grid points
50000 grid points


Fig 4.8. A comparison of pressure drop in Sawtooth 2 channel with 25000 grid points and
50000 grid points

As we see that even if we increase in the grid points in X direction by 2 times (i.e.


1000 grid points) there is not a significant difference in pressure drop of the channels. So


a sufficient accuracy is obtained even with 500 grid points in the X -direction.














CHAPTER 5
RESULTS AND DISCUSSIONS

5.1 Introduction

This chapter presents the results of our simulations. First, the results of the pressure

drop in microchannels are presented. The deviation of pressure drop from the

conventional theory for microchannels due to the presence of surface roughness is

studied. Next, a set of results are presented which shows the deviation of computed

friction factor from conventional theory predictions.

5.2 Pressure Drop

For the microchannels used in this study, the computed pressure gradients are

plotted in Fig 5.1 to 5.6, for all the three sets of channels (smooth channel, sawtooth

roughness channels, and random roughness channels). The pressure drop is defined as the

difference between the inlet and the exit pressure values. For each calculated pressure

gradient the Reynolds number (Re) is calculated using the uniform entrance velocity of

the fluid. The physical properties involved in these calculations, such as density and

dynamic viscosity, were determined from properties of water at STP conditions. Other

useful parameters such as the average velocity um, flow rate Q, apparent friction factor

fapp, and friction factor constant Cf, were determined from the velocity field.












1pm average surface roughness


120000


100000


80000


60000


40000


-*-smooth channel

---sawtooth surface
roughness
-A- random surface roughness


20000 -





Reynolds number Re


Fig 5.1. A comparison of pressure drop in channels with average surface roughness of
1 lm to that of theoretical pressure drop in a plain channel.


2pm average surface roughness


160000

140000

120000
S--smooth channel
100000
8- ---sawtooth surface
80000
roughness
60000- -Arandom surface roughness
e 60000

40000

20000

0


Reynolds number Re



Fig 5.2 A comparison of pressure drop in channels with average surface roughness of
2im to that of theoretical pressure drop in a plain channel.













3pm average surface roughness


250000



200000


150000
0


g 100000
0

50000


0 -J

4


--smooth channel

--saw tooth surface
roughness
-A-random surface roughness


Reynolds number Re


Fig 5.3 A comparison of pressure drop in channels with average surface roughness of
3 pm to that of theoretical pressure drop in a plain channel.



4pm average surface roughness


300000


250000


200000 --smooth channel
a-
2 -u-saw tooth surface
-a 150000
S 0 roughness

S- -A-random surface roughness
S100000


50000


0



Reynolds number Re



Fig 5.4 A comparison of pressure drop in channels with average surface roughness of
4pm to that of theoretical pressure drop in a plain channel.












5pm average surface roughness


-*-smooth channel

----sawtooth surface
roughness
-Arandom surface roughness


Reynolds number Re



Fig 5.5 A comparison of pressure drop in channels with average surface roughness of
5 gm to that of theoretical pressure drop in a plain channel.


6pm average surface roughness


500000

450000

400000

350000 -
L. -*-esmooth channel
300000
0000 ---sawtooth surface
250000
roughness
S200000 --random surface roughness
M. 150000

100000

50000

0


Reynolds number Re



Fig 5.6 A comparison of pressure drop in channels with average surface roughness of
6gm to that of theoretical pressure drop in a plain channel.


400000

350000

300000

C 250000
2
200000

S150000

100000

50000

0










As seen in the Fig5.1-Fig5.6 we see that, the pressure gradient in the case of rough

channels is more as compared to that of the plain channels. In the case of plain channels,

the pressure gradient is linear as required by conventional laminar flow theory. As the

pressure gradient in case of rough channels is more than that of the plain channels, for a

given flow rate, a higher pressure gradient is required to force the liquid to flow through

rough microchannels than that of the plain channels.

For the microchannels with 1% of surface roughness, the pressure gradient is in

close resemblance with the pressure gradient predicted by the conventional theory. As the

surface roughness of the channels is increased we see more and more deviation from the

theoretical values of the pressure gradient.

Table 5.1 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 1 lm
average surface roughness.
Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 3868.23 4046.562 1.08393 1.133901
100 7506.738 8125.7 8311.249 1.082454 1.107172
150 11761.416 12868.6 13062.19 1.094137 1.110597
200 16420.464 17945.2 18147.4 1.092856 1.10517
250 21266.613 23471.5 23683.06 1.103678 1.113627
300 26622.414 29394.7 29616.31 1.104134 1.112458
350 31951.404 35754.3 35986.69 1.119021 1.126294
400 37862.244 42277.1 42520.55 1.116603 1.123033
450 44092.17 49317.6 49572.98 1.118512 1.124304
500 50219.325 56780.4 57048.43 1.130648 1.135986
550 56554.425 64340.4 64621.25 1.137672 1.142638
600 63526.419 72515.4 72810.11 1.1415 1.146139
650 70299.072 81191.7 81501.12 1.154947 1.159348
700 77916.357 89909.2 90233.4 1.153919 1.15808
750 85148.73 98995.2 99334.8 1.162615 1.166603
800 93328.551 108502.6 108858.3 1.162587 1.166399

In the above table we see that the pressure gradients for the channels with 1 .im

average surface roughness, varies from 1.08 to 1.16 for sawtooth surface roughness and










1.13 to 1.16 for random surface roughness. So the pressure gradients for these channels

are as good as the theoretical analysis.

Table 5.2 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 2im
average surface roughness.
Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 4355.28 4237.9 1.220408 1.187517
100 7506.738 9770.58 9186.3 1.301575 1.223741
150 11761.416 15325.38 14806.7 1.303022 1.258922
200 16420.464 21818.61 21106.4 1.328745 1.285372
250 21266.613 29601.54 28066.5 1.391925 1.319745
300 26622.414 36817.47 35628.4 1.38295 1.338286
350 31951.404 45377.82 43806.3 1.420214 1.371029
400 37862.244 54542.07 52717.5 1.44054 1.39235
450 44092.17 64402.02 62127.3 1.460623 1.409032
500 50219.325 74919.69 72165.8 1.49185 1.437013
550 56554.425 86112.18 82879 1.522643 1.465473
600 63526.419 97363.98 94145.2 1.532653 1.481985
650 70299.072 110440.6 106059.3 1.571011 1.508687
700 77916.357 123660.6 118619.8 1.587095 1.522399
750 85148.73 137653.8 131847.2 1.616628 1.548434
800 93328.551 151840.4 145341.5 1.626944 1.55731


Table 5.3 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for
f^ i


channels with 3 im


average surface rougnness.
Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 5147.1 4479.03 1.442287 1.255085
100 7506.738 11315.8 10458.54 1.507419 1.39322
150 11761.416 18475.4 17796.33 1.570848 1.513111
200 16420.464 26603.5 26359.65 1.620143 1.605293
250 21266.613 35703.2 36221.94 1.678838 1.70323
300 26622.414 45591.7 46944.18 1.712531 1.763333
350 31951.404 56750.5 59086.71 1.77615 1.849268
400 37862.244 68803.9 72390.06 1.817217 1.911933
450 44092.17 82397.2 86843.88 1.868749 1.969599
500 50219.325 95486.4 102549.2 1.901388 2.042027
550 56554.425 110267.5 118913.1 1.949759 2.102632
600 63526.419 125847.5 136679.9 1.981026 2.151545
650 70299.072 142486 155768.5 2.026855 2.215797
700 77916.357 160063 176022.7 2.054293 2.259124
750 85148.73 178320.3 197959.1 2.094221 2.324862
800 93328.551 197684.9 220924.2 2.118161 2.367166










Table 5.4 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 4[m
average surface roughness.
Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 5497.087 4563.82 1.540358 1.278844
100 7506.738 12922.67 11067.85 1.721476 1.474389
150 11761.416 20766.94 19359.94 1.765683 1.646055
200 16420.464 30292.39 29431.42 1.844795 1.792362
250 21266.613 42117.78 41188.88 1.980465 1.936786
300 26622.414 55096.92 54763.8 2.069569 2.057056
350 31951.404 69818.89 70057.26 2.185159 2.192619
400 37862.244 86065.4 86767.24 2.273119 2.291656
450 44092.17 100563.3 105437.2 2.280753 2.391291
500 50219.325 119049.5 125597.8 2.370592 2.500985
550 56554.425 138814 147563.1 2.454521 2.609224
600 63526.419 160217.3 170860.4 2.522058 2.689595
650 70299.072 181466.3 196452.7 2.581347 2.794527
700 77916.357 205386.2 224184.1 2.635984 2.877241
750 85148.73 228800.4 253703.2 2.687067 2.979531
800 93328.551 254713.6 284471.4 2.729215 3.048064


Table 5.5


Comparison of APsawtooth /APtheo and APsawtooh th/APtheo for channels with 5 .m
-__- -* 1_ 1- -._ _


average surface roughness.
Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 5210.09 5633.3 1.459938 1.578527
100 7506.738 12921.36 13783.6 1.721301 1.836164
150 11761.416 22689.75 24308.3 1.929168 2.066783
200 16420.464 34789.06 37152 2.11864 2.262543
250 21266.613 49378.99 52153.8 2.321902 2.452379
300 26622.414 65549.24 69518.1 2.462182 2.611262
350 31951.404 85241.28 88914 2.667841 2.782789
400 37862.244 105809.9 110588.3 2.794602 2.920807
450 44092.17 128239.9 134099.8 2.90845 3.041352
500 50219.325 154202.6 160075.5 3.070582 3.187528
550 56554.425 181742.1 187787.8 3.213579 3.320479
600 63526.419 211934.1 219053.2 3.336158 3.448222
650 70299.072 243120.1 252465.6 3.458368 3.591308
700 77916.357 275690.3 288280.1 3.538285 3.699866
750 85148.73 310924.1 325894.9 3.651541 3.827361
800 93328.551 352944.1 365819.1 3.781738 3.919691











Table 5.6 Comparison of APsawtooth /APtheo and APsawtooth /APtheo for channels with 6im
average surface roughness.

Re APtheo APsawtooth APrandom APsawtooth /APtheo APrandom /APtheo
50 3568.707 5098.68 6015.7 1.428719 1.685681
100 7506.738 13198.14 15249 1.758172 2.031375
150 11761.416 24674.49 27451.1 2.097918 2.333996
200 16420.464 39037.68 42741.8 2.37738 2.602959
250 21266.613 56093.49 60892.9 2.637632 2.86331
300 26622.414 75839.49 82108.6 2.848708 3.084191
350 31951.404 100022.9 106301.9 3.130468 3.326987
400 37862.244 124617.8 133447.3 3.291347 3.524548
450 44092.17 153900.8 164732.6 3.490434 3.736096
500 50219.325 186491.7 196553.4 3.713545 3.9139
550 56554.425 220943.1 231795.4 3.906734 4.098625
600 63526.419 258915.2 270629.4 4.07571 4.260108
650 70299.072 298794.4 313164.5 4.250332 4.454746
700 77916.357 339122.1 357832.3 4.352386 4.592518
750 85148.73 384631 404674.1 4.517167 4.752556
800 93328.551 440443.5 455079.6 4.71928 4.876103


Pressure drop in microchannel. L = 5mm, width = 0.1mm
(sawtooh surface roughness)


500000
450000
400000
9 350000 smooth channel
Ssawtooth 1
,- 300000
300000 sawtooth 2
S250000 -- sawtooth 3
S200000 sawtooth 4
2 200000
S-4--sawtooth 5
a. 150000 I-sawtooth 6
100000
50000
0


Re



Fig 5.7 A comparison of pressure drop in channels with sawtooth surface roughness to
that of theoretical pressure drop in a plain channel as the roughness is
increased.












Pressure drop in microchannel. L = 5mm, width = 0.1mm
(random surface roughness)


500000

450000
400000
S--smooth channel
350000 random

300000 -*. random 2

S250000 -- random 3
S200000 -A-random 4
S--- random 5
150000
a- random 6
100000

50000

0


Re



Fig 5.8 A comparison of pressure drop in channels with random surface roughness to that
of theoretical pressure drop in a plain channel as the roughness is increased.

2.5 '63.5prt exp I
101.6jAm exp
2.0 130pm exp
S152pm exp
S1.5 203pm exp (a)
0 254pim exp
1.0 Classical thry.



.0 '

50Am exp
35 80jtm exp
S3.0 b) 101 pm exp
2.5 150m exp
S2054m exp
2.0 Classical thry
1.5 -

S10.0

0 300 600 900 1200 1500 1800 2100
Reynolds number


Fig 5.9 Experimentally measured pressure gradient (a) SS and (b) FS microtubes, and
comparison with the classical theory. (Mala, Li, Int. J. heat and Fluid flow
(1999) 142-148)










6 I I
(a)

(3)
(R~,ls, (3)











Fig. 5.10 A comparison of the measured data of pressure gradient vs. Reynolds number
S.. E.,DIUF


1 -Theory
a-


0 tOO 200 300 400 500 6O
Reynolds nunfbe,, Re

Fig. 5.10 A comparison of the measured data of pressure gradient vs. Reynolds number
with the predictions of conventional laminar flow theory for trapezoidal
microchannels. (a) (1) dh = 51.3 mm; (2) dh= 62.3 mm;(3) dh = 64.9 mm (Qu
et al. Int. J. of Heat and Mass Transfer 43 (2000) 353-364)

In Fig 5.7 and 5.8 we make a comparison of the pressure gradients in

microchannels with sawtooth surface roughness and random surface roughness to those

of a plain channel for increasing surface roughness. It is observed as the roughness in the

channels is increased, the pressure gradient increases accordingly.

Fig 5.9 gives the experimental results for flow in SS and FS microtubes having

diameters in range of 63.5km to 254im for SS microtubes and 50.0km to 250im for FS

microtubes. It is seen in Fig 9 the experimental curves are above the theoretical curves.

For small Re the pressure gradient is approximately equal to that predicted conventional

theory. As Re increases the measured pressure gradient is significantly higher than that

predicted by conventional theory. According to the authors reason for this deviation of









pressure gradient from the conventional theory was due to the presence of the surface

roughness in the tubes.

Fig 5.10 gives the experimental results for flow in trapezoidal silicon

microchannels having hydraulic diameters from 51.3 .m to 168.9[im. It is seen in Fig

5.10 the theoretical curves are below the experimental curves. For small Re the pressure

gradient is approximately equal to that predicted conventional theory. As Re increases the

measured pressure gradient is significantly higher than that predicted by conventional

theory. According to the authors reason for this deviation of pressure gradient from the

conventional theory was due to the presence of the surface roughness in the tubes.

During the current study, we also find similar results. In Fig 5.7 and Fig 5.8 we also

see that the pressure gradient deviates more from that of smooth channel in almost similar

manner. We see that as Reynolds number increases, the difference between the rough

surfaces microchannels and smooth channels is very significant. Table 1 Table 6

tabulates the ratio APsawtooth /APsmooth and APsawtooth /APsmooth for all the surface roughness

at different Reynolds number. We see that at high Reynolds number pressure gradients in

rough microchannels are 3.2 to 4.6 times that of pressure gradients in smooth channels.

Therefore the surface roughness does have a major effect on the pressure gradients

in microchannels, which is observed in the computed results as well as the experimental

results done by researchers.

To investigate the reasons behind the nature of increase in pressure drop in rough

microchannels, we further run simulations of smooth channels with reduced width of

channel, which might be caused due to the roughness element present in the

microchannels. For these cases we reduce the width of channel by the value of average







58



surface roughness height in the channels. Results of the following simulations are as


presented below.


1pm average surface roughness


120000

100000

80000

60000

40000


- Sawtooth surface 0.1
-- Random surface 0.1
- Reduced channel
smooth channel


20000


0 200 400 600 800 1000
Re



Fig 5.11 A comparison of pressure drop in rough channels, reduced width channels and
smooth channel.


2pm average surface roughness


160000

140000

120000

100000

80000

60000

40000

20000

0


- Sawtooth surface 0.2
-W- Random surface 0.2
- Reduced channel
smooth channel


0 200 400 600 800 1000
Re



Fig 5.12 A comparison of pressure drop in rough channels, reduced width channels and
smooth channel.










3pm average surface roughness

250000

200000
a. -- Sawtooth surface 0.3
2 150000 -
0 -- Random surface 0.3

I 100000 --s- Reduced channel
g smooth channel
a- 50000


0 200 400 600 800 1000
Re



Fig 5.13 A comparison of pressure drop rough channels, reduced width channels and
smooth channel.

In Fig 5.11 Fig 5.13 we see that as we reduce the width of the channel the

pressure drop in the reduced channel increases as compared to that of a smooth channel.

So the increase in pressure drop of the rough channels may be due to the decrease in the

width of channel. So the decrease in width of channel is one of the factors in increase of

pressure drop for the rough microchannels.

5.3 Flow Friction

The flow behavior of water through microchannels can be further interpreted in

terms of the flow friction. In the following study we use the fanning friction factor for the

total pressure drop in the channel, which is defined as

f Dh 1
f/PP = AP 12
L 2pum

where,

AP pressure drop between the inlet and outlet.

Dh Hydraulic diameter.

LLength of channel







60


Um average velocity of fluid.

p density of water

In the figures below fanning friction factor is plotted as a function of the Reynolds

number. For comparison the relationships between friction factors and Reynolds number

predicted by the conventional theory are also plotted.


1pm average surface roughness


0.600


0.500


0.400 theoretical friction factor

03 sawtooth surface
C 0.300 -
o roughness
A random surface roughness
"- 0.200


0.100 L


0.000
10 210 410 610 810 1010
Reynolds number Re


Fig 5.14 A comparison of friction factor in channels with average surface roughness of
1 lm to that of theoretical friction factor in a plain channel.













2pm average surface roughness


0.600


0.500


0.400
o
-

-
" 0.300
o
.-
" 0.200


0.100


0.000


-- theoretical friction factor


sawtooth surface
roughness
A random surface roughness


Reynolds number Re


Fig 5.15 A comparison of friction factor in channels with average surface roughness of
2gm to that of theoretical friction factor in a plain channel.




3pm average surface roughness


0.600


0.500 -


0.400 -- theoretical friction factor
o
\ sawtooth surface
c 0.300 -
S0.300 roughness
.o A random surface roughness
t- A


u" 0.200


0.100


0.000


Reynolds number Re


Fig 5.16 A comparison of friction factor in channels with average surface roughness of
3 m to that of theoretical friction factor in a plain channel.


I
,_ K R K K K


S0- E f m m








62




4pm average surface roughness


-- theoretical friction factor


m sawtooth surface
roughness
A random surface roughness


Reynolds number Re


Fig 5.17 A comparison of friction factor in channels with average surface roughness of
4pm to that of theoretical friction factor in a plain channel.




5pm average surface roughness


- theoretical friction factor


sawtooth surface
roughness
A random surface
roughness


Reynolds number Re


Fig 5.18 A comparison of friction factor in channels with average surface roughness of
5 pm to that of theoretical friction factor in a plain channel.


0.600


0.500


0.400
o
-

" 0.300
o
.-
" 0.200


0.100


0.000


E~


0.600 -


0.500 -


0.400
-

S0.300 -
o

" 0.200 -


0.100 -


n n -


& Aa E K


SK


.~ u











6pm average surface roughness

0.600

0.500


S0.400 -- theoretical friction factor
S\A
0.300 sawtooth surface
0.00 roughness
2 A random surface roughness
0.200- E U... a

0.100-

0.000


Reynolds number Re

Fig 5.19 A comparison of friction factor in channels with average surface roughness of
6rm to that of theoretical friction factor in a plain channel.

As seen in the above figures Fig 5.14- Fig 5.19 we found that, the friction factor in

the case of rough channels is more as compared to that of the smooth channels. In case of

the smooth channels, the friction factor behaves as predicted conventional laminar flow

theory.

For the microchannels with 1% of surface roughness, the friction factor is in close

resemblance with the friction factor predicted by the conventional theory. As the surface

roughness of the channels is increased we see more and more deviation from the

theoretical values of the friction factor.






















--smooth channel
-- random 1
-- random 2
X random 3
-K-random 4
-*--random 5
--random 6


Reynolds number


Fig 5.20 A comparison of friction factor in channels with random surface roughness to
that of theoretical friction factor in a plain channel as the roughness is
increased






Fanning Friction factor Vs Reynolds number
sawtoothh surface roughness)


---smooth channel


-* sawtooth 1


-- sawtooth 2


X sawtooth 3


-- sawtooth 4


--- sawtooth 5


-- sawtooth 6


Fig 5.21 A comparison of friction factor in channels with sawtooth surface roughness to
that of theoretical friction factor in a plain channel as the roughness is
increased


64




Fanning Friction factor Vs Reynolds number
(random surface roughness)


600 800 1(


1






0
C
0
. 0.1

E

L-



0.01


400 600 800 1C


1






0
o 0.1

.E

0L


0.01


Reynolds number


200 400


200










0.8 50 m FST
63.5 pm SST
0.6* 101,6 Rm SST
130 pmSST
0.4 152pimSST:
150 pm FST
254 pm SST
c. classical thry
S- Blasius Eq.
In



u. 0.08
So.o6


0.04 ----


500 1000 1500 2000
Reynolds number

Fig 5.22 Friction factorfexp vs Re for some SS and FS microtubes and comparison with
the classical theory. (Mala et al. (1999))

In the Fig 5.20 and Fig 5.21 a comparison is made between the Fanning friction

factors for the channels with surface roughness having average height of the roughness

elements from 1% to 6% with that of a smooth channel. We see that as the average height

of the roughness element in the channel is increased, there is more deviation of the

Fanning friction factor from that of a smooth microchannel. It can be seen that the higher

Reynolds number range all the curves are above the curve for the plain channel, which

means that for a given Reynolds number the flow friction in the microchannels is higher

than that obtained by using the plain channel, which is due to the presence of the surface

roughness elements present in the channels.

Fig 5.22 gives the experimental results for variation of friction factor in some SS

and FS microtubes having diameters in range of 63.5pm to 254pm for SS microtubes and









50.0tm to 250tm for FS microtubes. It is seen in Fig 5.22 that for small Re the friction

factor decreases linearly with increase in Re on semi-log plots. But as Re becomes large

Re > 1500 the slope of the curve decreases and approaches to zero. Even in the laminar

flow regime it is seen the friction factors deviate from the conventional theory. According

to the authors reason for this kind of behavior of friction factors was due to the presence

of the surface roughness in the tubes.

In Fig 5.20 and Fig 5.21 we find similar results for the friction factors. When the

Re is small friction factors are in agreement with the conventional theory for small

roughness channels, but as Re increases the difference between the friction factors is

significant. This is due to the presence of the surface roughness. As we increase the

surface roughness the deviation goes on increasing. So surface roughness does indeed

play a role in friction factors for microchannels.

Another important parameter used to describe the flow friction in channels is the

friction factor constant Cf, which is the product of the friction factor and Reynolds

number.

C = f.Re

It is well known from the conventional laminar flow theory that the friction factor

constant is dependent on geometry of channel cross-section (R.K Shah et al. 1978).

Hence for the present case a constant value of Cf should be expected. However according

to the computed data we find that Cfis no longer constant for the flow in a microchannel.







67



Friction factor constant Vs Re


AlA

ii li i


---smooth channel

sawtooth surface roughness


A random surface roughness
15
10
5-
0
0 ----------------------
0 200 400 600 800 1000
Re


Fig 5.23 A comparison offRe in channels with average surface roughness of 1 lm to that
of theoretical fRe in a plain channel.


Friction factor constant Vs Re


50 I *

40 -n m m

1II
S30 -

20-


---smooth channel

sawtooth surface roughness

A random surface roughness


10

0
0 200 400 600 800 1000
Re


Fig 5.24 A comparison offRe in channels with average surface roughness of 2[m to that
of theoretical fRe in a plain channel.


45
40
35
30
S 25







68



Friction factor constant Vs Re


n IkI '


- m~mIl


Am


--smooth channel

sawtooth surface roughness

A random surface roughness


0 200 400 600


800 1000


Fig 5.25 A comparison offRe in channels with average surface roughness of 3 m to that
of theoretical fRe in a plain channel.





Friction factor constant Vs Re


A


mIi
ml. *m l
---
Im


---smooth channel

sawtooth surface roughness

A random surface roughness


0 200 400 600 800 1000
Re


Fig 5.26 A comparison offRe in channels with average surface roughness of 4pm to that
of theoretical fRe in a plain channel.


70
60
S50
S40
30
20


120

100

80

8 60

40

20







69



Friction factor constant Vs Re


140


100 -

80 m

60- A

40
-m


--smooth channel

sawtooth surface roughness

A random surface roughness


0
0 200 400 600 800 1000
Re


Fig 5.27 A comparison offRe in channels with average surface roughness of 5[m to that
of theoretical fRe in a plain channel.


Friction factor constant Vs Re


Em
U
U


---smooth channel


m


am ,A .*
a__


* sawtooth surface roughness

A random surface roughness


0 200 400 600 800 1000
Re


Fig 5.28 A comparison offRe in channels with average surface roughness of 6[m to that
of theoretical fRe in a plain channel.


100
S80
60
40
20










Friction constant C is plotted as a function of Reynolds number in Fig 5.23 to 5.28.

As it can be seen in the above figures, Cffor the rough channels is not constant as

compared to that of a smooth channel. However for the same average surface roughness

the friction constant is in close agreement. Also we find that the flow friction in the

channel is 1-41% higher than theoretical prediction in the channels with average surface

roughness height of 0.1 lm and it also goes on increasing as the roughness element is

increased.

Qu et al. (2000), Mala et al. (1999) Toh et al. (2002), Peng et al. (1994), Wu et al.

(2003a, 2003b) also observed such dependence of friction factor on Reynolds number.

All the above authors quoted this deviation of frictional factor from the conventional

theory was due to the presence of the surface roughness present in the channels. In the

above study we see the same behavior of the friction factor.


Friction constant Vs Reynolds number
sawtoothh surface roughness)


180 --smooth channel
160
140 -- sawtooth 1
S120
100 -- sawtooth 2
o100 -
C 80-
.2 80 X X sawtooth 3
& 60- XX
L-
40 X* sawtooth 4
20
0 4-sawtooth 5
o000000000000000
0 0 0 0 0 0 0 0 0 0 0" 0 0 0 0I 0
Reynolds number sawtooth 6




Fig 5.29 A comparison of friction constant in channels with sawtooth surface roughness
to that of theoretical friction constant in a plain channel as the roughness is
increased







71



Friction constant Vs Reynolds number
(random surface roughness)


Reynolds number


--smooth channel

-random 1

--random 2

X random 3

-K-random 4

--- random 5


-Irandom 6


Fig 5.30 A comparison of friction constant in channels with sawtooth surface roughness
to that of theoretical friction constant in a plain channel as the roughness is
increased


40-


g-


90 -





: 20-




10-




0-


I4
400


' I
8eO
Re


' I
120


1600


Fig 5.31 A comparison offRe in channels with rough surfaces (Wu et al. 2003)


Iet6~











In the Fig 5.29 and Fig 5.30 a comparison is made between the friction factor

constant for the channels with random surface roughness having average height of the

roughness elements from 1% to 6% with that of a smooth channel. We see that as the

average height of the roughness element in the channel is increased, there is more

deviation of the friction factor constant from that of a smooth microchannel. It can be

seen that the higher Reynolds number range all the curves are above the curve for the

plain channel, which means that for a given Reynolds number the flow friction in the

microchannels is higher than that obtained by using the plain channel, which is due to the

presence of the surface roughness elements present in the channels.

In Fig 5.31 gives the experimental results to two pairs of the silicon microchannels

with different surface roughness. Microchannels #7 and #9 are two trapezoidal channels

having the same geometric parameters but with relative different relative surface

roughness (3.26x10 5- 5.87x10-3), while microchannels #8 and #10 have triangular cross-

section with same geometric parameters but different relative surface roughness

(3.62x10-5- 1.09x10-2). As we see in Fig 5.28 friction constant of trapezoidal

microchannel #9 are larger than those of the trapezoidal microchannel #7 which has

much lower surface roughness than microchannel #9. Similar increase in friction constant

is observed for triangular microchannel #10 (1.09x10-2) which has higher surface

roughness when compared to the triangular microchannel #8 (3.26x10-5).

From Fig 5.29 Fig 5.30 it can be observed that friction factor increases with

increase in Reynolds number. Also as the surface roughness is increased the friction

factor and friction constant also increases. Similar results are obtained for microchannels






73


with trapezoidal and triangular cross sections by Wu et al. (2003) as surface roughness is

increased.














CHAPTER 6
CONCLUSION

Numerical simulations have been used to investigate the effects of surface

roughness on the microchannel flows. In order to study the effects two different types of

rough surface profiles were used for computation, the sawtooth surface roughness profile

and the random surface roughness profile with the maximum height of the surface

roughness and the randomness present in the channel specified so as to maintain a

constant average surface roughness height.

The actual geometry is then converted into the computational geometry using the

transformation on coordinates. The 2-D set of Navier-Stokes equations are then solved on

the transformed grid, using SIMPLE method along with the momentum interpolation

technique on a non-staggered grid.

During the first part of the investigation we study the effects of surface roughness

in the pressure drop in the flow between two parallel plates. We can see from the results

plotted that as the surface roughness is increased there is more deviation of the pressure

drop from that estimated by conventional laminar theory as well as that estimated by the

flow in smooth channel without any surface roughness. Also we see that as Reynolds

number is increased the deviation between the computed and the estimated value of the

pressure drop increases. The results are in agreement with the results obtained by the

other researchers using different channels for the test. In all the cases the pressure drop in

the rough microchannels was more than that of the conventional results. As we decrease

the width of the microchannels for their corresponding surface roughness we see that the









pressure drop in the reduced width microchannels increases. So the decrease in the width

of the microchannels is one of the reasons of the increase in pressure drop of rough

microchannels.

In laminar flow friction factor is independent of the surface roughness. But in the

second part of the investigation we see that, as the surface roughness is increased on the

walls of the channels, friction factor increases. Also we see that as the Reynolds number

is increased the friction factor for the given channel increases. So infact for microchannel

flows surface roughness plays a very important role in friction factor. The results

presented above are in agreement with the experiments done my researchers in the same

field.

So during this study we conclude that surface roughness plays a very important role

in microchannel flows. We see that the pressure drop in the channels is more than the

estimated value due to the presence of the surface roughness. Similarly the friction factor

is also dependent on the surface roughness.
















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BIOGRAPHICAL SKETCH

Amit Kulkarni was born in Pune, India, in December 1979. He graduated from the

University of Pune with a Bachelor of Engineering degree in mechanical engineering in

July 2001. He joined University of Florida in Fall 2001 to pursue his master's in the

Department of Mechanical and Aerospace Engineering. He will be conferred the Master

of Science degree by the University of Florida in May 2004.