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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 everextending 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 BodyFitted 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 onstaggered 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 2D grid on a physical plane.........................27 3.2 Collocated grid and notation for a 2D 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 Ucomponent of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x 00 .... ...................................................................... ......... 3 8 3.6 Vcomponent of velocity contours for the driven cavity problem at Re = 100 on a grid of 100x 100 ...................................................................... ...... 3 8 3.7 Ucomponent 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 xdirection v = Velocity in ydirection 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 xdirection V = Contravariant velocity in ydirection 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 ucomponent of velocity u' = Correction ucomponent of velocity v* = Guessed vcomponent of velocity v' = Correction vcomponent 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 2D 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 nonstaggered 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 manmade 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 manmade in the later half of 20th century. The transistorinvented 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 HAtom Diameter Human Hair Man Nanodevices (IJ Typical ManMadee 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 batchprocessing 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 manmade 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 labon 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, microheat 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 inkjet 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 materialsglass, 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 surfacetovolume 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 crossflow heat exchanger constructed from a stack of 50 14mm14mm 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 crossflow 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 freemolecular flow regimes. Discrete particle or molecular based model is the Boltzmann equation. The continuum based models are the NavierStokes equations. Euler equations correspond to inviscid continuum limit which shows a singular limit since the fluid is assumed to be inviscid and nonconducting. Euler flow corresponds to Kn = 0.0. NavierStokes 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 Slipflow regime Freemolecule 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 3x104 and 6x104, 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 s1. 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 L2s, and both shear and acceleration go as L3s. Fluids that are Newtonian at ordinary rates of shear and extension can become NonNewtonian at very high rates. The pressure gradient becomes especially large in small crosssection 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 nowtraditional 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 (RFMEMS). Commercial interests are focused on plastic microfabrication for singleuse 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 throughputnot previously possible with manual, benchtop 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 DNAbased 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 moleculardominated 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 applicationsdriven 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 mid1990s. 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 surfaceto 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 mid1990s, 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 nonmechanical pumping principles. Electrokinetic pumping, surface tensiondriven 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 mid1990, 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 siliconbased micromachining technologies. Complex based microfluidic devices based on plastic microfabrication could be expected in the near future with further achievements of plasticbased 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 laminarturbulent 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 orientationdependent (anisotropic) and/or by orientationdependent 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 [^"SBSs 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 'touchup'. 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 microfluidic 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 200700. 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) 142148) 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) 353364) 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 RoughnessViscosity 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) 353364) 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 NavierStokes 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 NavierStokes equations can be generally classified as density based methods and pressurebased 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, secondorder 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 BodyFitted Coordinates for 2D geometries The above form of NavierStokes equations is for Cartesian coordinates (x, y). For arbitrarily shaped geometries generalized bodyfitted 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. 3.13.3) is transformed to the (5, rl) coordinate system. The resulting governing equations in generalized bodyfitted 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 ait) 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 nonstaggered 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 ucomponent 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 2D 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 2D 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 qe 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 pressurecorrection) equation to enforce mass conservation at each time step or otherwise known as outer iteration for steady solvers. The acronym SIMPLE stands for SemiImplicit Method for Pressure Linked Equations. The algorithm was originally put forward by Patankar and Splading and is essentially a guessandcorrect 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 pressurecorrection schemes on a regular grid. During the current investigation we use SIMPLE method on nonstaggered grids. The disadvantages mentioned above in using nonstaggered grid are taken care by using the RhieChow Momentum interpolation technique. Discretized Umomentum and Vmomentum equations are written in following form app P= 'anbu"b +(pW 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 ucontrol 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 Umomentum and Vmomentum 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. (3.213.23) 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 nonCartesian, nonorthogonal meshes (especially 3D meshes) 1. More memory is required for storage of variables. 2. Difficult and inefficient for multigrid solvers. 3.4.2 Nonstaggered Grids: In nonstaggered 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 Nonstaggered grid without certain modifications to the original staggered grid scheme is highlighted. The necessary remedies must be applied to the nonstaggered 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 RhieChow momentuminterpolation technique to satisfy the continuity and using the nonstaggered 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 'umomentum' 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 nonstaggered 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 liddriven 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 Ucomponent of velocity contours for the driven grid of 100x100 cavity problem at Re Fig 3.6 Vcomponent 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 'III 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 8722272 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 Ucomponent 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 2D 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 2D 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 5E05 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 8E05 4E05 2E05 0 5E05 0.0001 0.00015 X 0.0001 5E05 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 9E05 8E05 7E05 6E05 5E05 4E05  3E05 2E05 1E05 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 9E05 8E05 7E05 6E05 S5E05 4E05 3E05 2E05 1E05 0 0.0001 0.00015 0.0002 x 0.0001 8E05 6E05 4E05 '2t05 0 2E05 4E05 6E05 5E05 0.0001 0.00015 0.0002 XC Table 4.1 Geometric parameters of microchannels Channel no Height (103 m) Length (103 m) L/H h (105 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 Ssmooth 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 Arandom 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 usaw tooth surface a 150000 S 0 roughness S Arandom 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 Fig5.1Fig5.6 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 S4sawtooth 5 a. 150000 Isawtooth 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 Ssmooth channel 350000 random 300000 *. random 2 S250000  random 3 S200000 Arandom 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) 142148) 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) 353364) 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 Krandom 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 semilog 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 crosssection (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 141% 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 4sawtooth 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 Krandom 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 5.87x103), while microchannels #8 and #10 have triangular cross section with same geometric parameters but different relative surface roughness (3.62x105 1.09x102). 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 (1.09x102) which has higher surface roughness when compared to the triangular microchannel #8 (3.26x105). 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 2D set of NavierStokes equations are then solved on the transformed grid, using SIMPLE method along with the momentum interpolation technique on a nonstaggered 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. LIST OF REFERENCES Adams, T.M., AbdelKhalik, S.I., Jeter, S.M., Qureshi, Z.H., "An Experimental Investigation of Singlephase Forced Convection in Microchannels," International Journal of Heat Mass Transfer 41 1988, pp. 851857. 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White, F.M., Viscous Fluid Flow, 2nd Ed. McGrawHill, 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. 25472556. Wu, P.Y., Little, W.A., "Measurement of Friction Factor for Flow of Gases in Very Fine Channels Used for Microminiature Joule Thompson Refrigerators," Cryogenics 24(8) 1983, pp. 273277. Yu, D., Warrington, R., Barren, T., "An Experimental and Theoretical Investigation of Fluid Flow and Heat Transfer in Microtubes," Proceedings of the ASME/JSME Thermal Engineering Conference, ASME, 1, 1995, 523530. 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. 