Numerical Simulations of High Knudsen Number Gas Flows and
Microchannel Electrokinetic Liquid Flows
A Thesis
Submitted to the Faculty
of
Drexel University
by
Fang Yan
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
May 2003
ii
ACKNOWLEDGMENTS
I express my gratitude to my advisor, Prof. B. Farouk at Drexel University for his valuable guidance, support, encouragement and patience during my studies at Drexel
University. I am also grateful to Prof. N. Cernansky, Prof. Y. Gogotsi, Prof. M. Vallieres, Prof. D.
Wootton and Prof. K. Choi from Drexel University for their Valuable advice on this thesis as committee members.
I acknowledge the financial support from Department of Mechanical Engineering and
Mechanics, Drexel University that provide me the opportunity to pursue my doctoral study. I appreciate the help from the fellow laboratory members Dr. K. Bera, Dr. J. Yi, and Mr. M. Aktas.
I wish to express my special gratitude to my parents and wife for their understanding and encouragement during my stay in the U. S. A.
iii
TABLE OF CONTENTS
LIST OF TABLES...... v
LIST OF FIGURES...... vi
ABSTRACT ...... x
1. INTRODUCTION ...... 1
1.1. Motivation ...... 1
1.2. Objectives...... 6
1.3. Outline of the thesis...... 7
2. BACKGROUND AND LITERATURE SURVEY ...... 10
2.1. High Kn number gas flows...... 10
2.2. Liquid flows in microchannels...... 13
3. NUMERICAL AND MATHEMATICAL MODELS...... 16
3.1. Overview of research work ...... 16
3.2. Direct simulation Monte Carlo model for high Knudsen number gas flows ...... 19
3.3. Mathematical model of microchannel liquid flows...... 27
4. LOW PRESSURE GAS FLOW AND HEAT TRANSFER IN DUCTS ...... 38
4.1. Introduction...... 38
4.2. Problem description...... 39
4.3. Model description ...... 40
4.4. Results and discussion...... 42
5. MIXING GAS FLOWS IN A MICROCHANNEL ...... 60
5.1. Introduction...... 60
5.2. Problem description...... 62
5.3. Model description ...... 62
5.4. Results and discussion...... 63
iv
6. THREE DIMENSIONAL LOW PRESSURE COUETTE FLOWS...... 77
6.1. Introduction...... 77
6.2. Problem description...... 79
6.3. Parallelization of DSMC...... 79
6.4. Results and discussion...... 82
7. NUMERICAL SIMULATION OF CAPILLARY ELECTROKINETIC SEPARATION FLOWS IN MICROCHANNEL ...... 93
7.1. Introduction...... 93
7.2. Problem geometry ...... 95
7.3. Mathematical modeling...... 96
7.4 Results and discussion...... 99
8. SUMMARY AND CONCLUSIONS ...... 118
8.1. Summary ...... 118
8.2. Recommendations for future work...... 120
LIST OF REFERENCES ...... 123
VITA ...... 134
v
LIST OF TABLES
4.1. Values of Nu vs Re, Kn and Pr for Pin = 6.7 Pa and Pout = 5.3 Pa ...... 49
4.2. Values of Nu vs Re, Kn and Pr for Pin = 0.67 Pa and Pout = 0.53 Pa ...... 49
4.3. Values of Nu vs Re, Kn and Pr for Pin = 0.067 Pa and Pout = 0.053 Pa...... 50
5.1a. Inlet and solid wall conditions for equal inlet pressure cases ...... 69
5.1b. Inlet and solid wall conditions for cases with variable inlet pressure of the H2 stream...... 69
5.1c. Inlet and solid wall conditions for cases with variable accommodation coefficients ...... 69
6.1. Inlet and solid wall conditions for cases studied for 3D Couette flows...... 85
6.2. Inlet and solid wall conditions for cases studied for ‘ultimate pressure’ calculations...... 85
7.1. Electric field distribution of cases considered for simulation...... 106
vi
LIST OF FIGURES
1.1. The Knudsen number limits on the mathematical models (Bird, 1994) ...... 3
3.1. Flow chart of DSMC procedure ...... 34
3.2. Schematic of single component gas flow in microchannel ...... 35
3.3. Comparison of axial pressure distribution for channel flow (with slip-walls) ...... 35
3.4. Comparison of numerical results with experiment data of pressure distribution along the channel...... 36
3.5. Cross-channel device for electrophoresis separation...... 37
3.6. Description of electric double layer (a) formation of electric double layer (b) potential ψ distribution within the electric double layer (c) velocity distribution within the electric double layer ...... 37
4.1. Schematic of the problem geometry...... 50
4.2a. The effect of cell size on temperature profile at x = 0 .0 m (50 particles / cell) ...... 51
4.2b. The effect of cell size on temperature profile at x = 0.4 m (50 particles / cell) ...... 51
4.3. The effect of particle number on temperature profile at x = 0 (350 × 80 cells) ...... 52
4.4. Temperature contours for the base case...... 52
4.5. Velocity vectors for the base case...... 53
4.6. Pressure distribution along centerline for the base case...... 53
4.7. Heat flux along the plate for the base case ...... 54
4.8. Mean Nu as a function of position along the plate for the base case ...... 54
4.9a. Kn as a function of position along the plate for the base case ...... 55
4.9b. Re as a function of position along the plate for the base case ...... 55
4.10. Temperature contours for transitional flow (case 8) ...... 56
4.11. Velocity vectors for transitional flow (case 8) ...... 56
4.12. Pressure distribution along the centerline for transitional flow (case 8)...... 57
vii
4.13. Heat flux along the plate for transition flow (case 8)...... 57
4.14. Local Nu as a function of position along the plate for transitional flow (case 8)...... 58
4.15a. Kn as the function of position along the axis for transitional flow (case 8)...... 58
4.15b. Re as a function of position along the plate for transitional flow (case 8)...... 59
4.16. Parity plot showing correlation of computed data...... 59
5.1. Schematic of the mixing problem geometry ...... 70
5.2. The effect of cell size on streamwise velocity profiles at the inlet interface (base case) ...... 70
5.3a. Mass density contours for the base case ...... 71
5.3b. Detail of the mass density contours near the inlet region for the base case (values in kg/m3) ...... 71
5.4a. Mass averaged velocity vectors for the base case ...... 72
5.4b. Details of the mass averaged velocity vectors near the inlet region for the base case...... 72
5.5a. Diffusion velocity vectors for H2 at the inlet for the base case...... 73
5.5b. Diffusion velocity vectors for O2 at the inlet for the base case...... 73
5.6. Temperature contours for the base case (values in K)...... 74
5.7. The variation of mixing length for cases shown in Table 5.1a (inlet pressure ratio = 1.0) ...... 74
5.8. The variation of mixing length for different pressure ratios (PH2/PO2) (cases shown in Table 5.1b) ...... 75
5.9a. Mass averaged velocity vectors for case 11, Table 5.1c (α = 0.01)...... 75
5.9b. Mass density contours for case 11, Table 5.1c (α = 0.01)...... 76
5.10. The variation of mixing length with accommodation coefficient (for cases shown in Table 5.1c) ...... 76
6.1. Schematic of channel flow problem...... 86
6.2. The effect of granularity on the parallel performance (a) speed up (b) efficiency...... 87
6.3. Parallel efficiency for different problem sizes ...... 88
viii
6.4. Velocity profiles in Couette flow for three cases (a) Kn = 0.01, (b) Kn = 0.1, and (c) Kn = 1.0 ...... 88
6.5. Pressure contours inside the duct for three cases (a) Kn = 1.0, (b) Kn = 0.1, and (c) Kn = 0.01 ...... 89
6.6. Comparison of velocity vectors for three cases for ultimate pressure calculation with U = 250 m/s (a) Kn = 5.0, (b) Kn = 0.5 and (c) Kn = 0.05...... 91
6.7. Variation of compression ratio as a function of Kn number ...... 92
7.1. Schematic of the cross channel geometry (the arrows indicate the flow direction along each channel during (a) loading step and (b) dispensing step) ...... 107
7.2. Comparison between experimental (Jacobson and Ramsey, 1997) (dashed line) and simulated (solid line) concentration profiles across the injection and waste channels (a) 3 µm upstream from the intersection chamber (b) 9 µm into the intersection chamber (c) at the exit of the intersection chamber (d) 60 µm downstream from the intersection chamber...... 108
7.3. Electric field schemes of the base case for the loading and dispensing steps. Arrows show the direction of the field, and numbers correspond to the relative electric field strength in the channels...... 110
7.4. Velocity vectors plot of species 1 at channel section (base case) (a) loading step (b) dispensing step for the base case in Table 7.1 ...... 111
7.5. Concentration contours of species 1 at different time levels for the dispensing step (a) t = 0.0 s, (b) t = 0.1 s, (c) t = 0.2 s and (d) t = 0.3 s for the base case...... 112
7.6. Representation of the resolution of species separation...... 114
7.7. Temporal evolution of species concentration at the end of the separation channel (base case) ...... 114
7.8. Temporal evolution of species 1 concentration at the end of the separation channel under different loading conditions
with |ε4D| = 0.3 ...... 115
7.9. Variation of retention time of different species as function of |ε4D| with |ε1L| = 0.3 ...... 115
7.10. Variation of band width ‘tw1’ for species 1 as function of |ε1L| and |ε4D| ...... 116
7.11. Variation of maximum concentration Cmax for species 1 as function of |ε1L| and |ε4D| ...... 116
ix
7.12. Separation resolution as a function of dispensing electric field schemes |ε4D| with |ε1L| = 0.8 ...... 117
x
ABSTRACT Numerical Simulations of High Knudsen Number Gas Flows and Microchannel Electrokinetic Liquid Flows Fang Yan Bakhtier Farouk, Ph. D.
Low pressure and microchannel gas flows are characterized by high Knudsen numbers.
Liquid flows in microchannels are characterized by non-conventional driving potentials like
electrokinetic forces. The main thrust of the dissertation is to investigate these two different
kinds of flows in gases and liquids respectively.
High Knudsen number (Kn) gas flows were characterized by ‘rarified’ or ‘microscale’
behavior. Because of significant non-continuum effect, traditional CFD techniques are often
inaccurate for analyzing high Kn number gas flows. The direct simulation Monte Carlo (DSMC)
method offers an alternative to traditional CFD which retains its validity in slip and transition flow
regimes. To validate the DSMC code, comparisons of simulation results with theoretical
analysis and experimental data are made. The DSMC method was first applied to compute low
pressure, high Kn flow fields in partially heated two dimensional channels. The effects of varying
pressure, inlet flow and gas transport properties (Kn, Reynolds number, Re and the Prandtl
number, Pr respectively) on the wall heat transfer (Nusselt number, Nu) were examined.
The DSMC method was employed to explore mixing gas flows in two dimensional
microchannels. Mixing of two gas streams (H2 and O2) was considered within a microchannel. The effect of the inlet-outlet pressure difference, the pressure ratio of the incoming streams and the accommodation coefficient of the solid wall on mixing length were all examined.
Parallelization of a three-dimensional DSMC code was implemented using OpenMP procedure on a shared memory multi-processor computer. The parallel code was used to simulate 3D high Kn number Couette flow and the flow characteristics are found to be very different from their continuum counterparts.
xi
A mathematical model describing electrokinetically driven mass transport phenomena in microfabricated chip devices will also be presented. The model accounts for the principal physical phenomena affecting sample mass transport in microchip channels. Numerical simulations were performed to study the capillary electrophoresis flow in microchannels. Flow fields and species distribution are simulated for both the loading and the dispensing steps in a two dimensional cross channel device for species separation. The goal of these simulations is to identify the parameters providing optimal separation performance.
1
1. INTRODUCTION
1.1. Motivation
When the device length scale is on the order of the mean free path, gas flow
experiences large Knudsen numbers and is characterized by so called high Knudsen number flows. There are two distinct classes of high Knudsen number non-continuum flow problems - one due to micro length scale and the other due to large mean free path (very low pressure) of the gas molecules. Both classes of problems offer unique challenges in obtaining their solutions. With a variety of novel applications such as micro-electromechanical systems
(MEMS), low-pressure chemical vapor deposition (CVD) reactors, etc., the transport problem in the non-continuum (high Kn number) gas flow regime has opened up a new territory for flow and heat transfer research. It is necessary to examine the unique features of high Kn number gas flows. At high Kn number, the Reynolds number is typically very small. Due to the
rarefaction effect caused by the high Knudsen number, the flow velocity at the surface starts to
slip (Knudsen, 1909); therefore, the viscous shear stress is much reduced.
For liquid flows in microchannels, the distance between molecules is on the order of
angstroms. Typical microscale devices have length scale much larger than the molecular
spacing of simple liquid and hence the flow is still in the continuum region. The non-slip
boundary condition holds in the absence of a moving contact line at the fluid/fluid/solid interface.
In some situations when immiscible fluids are considered in microchannel flows, surface tension
might have significant effect on the transport behavior.
Electrokinetically driven phenomenon has been utilized to induce liquid flows in
complex micro-geometries while most gas flows in micro-devices are driven by pressure
differences. Micro-fluidic systems with electrokinetically driven liquid flows inside can be utilized
for bio-medical drug delivery, DNA analysis as well as in the development of laboratory on a
microchip to detect biological and chemical warfare agents. The electrostatic charges on the
solid surfaces can attract ions in liquid flows to form an electric double layer (EDL) (Audubert 2
and Mende, 1959). This electrically charged layer can affect the bulk flow. Due to the presence
of a surface potential at the interface of a non-conducting surface and a dilute solution,
electroosmotic flow is formed. Under the action of an applied electric field, charged species also
move in a solution with different velocities. This phenomenon is called electrophoresis and can
be utilized in species separation.
In the present thesis, both high Knudsen number gas flows and electrokinetic liquid
flows in microchannels are addressed. Computation models are developed and validated for
both high Kn number gas flows and liquid flows in microchannels.
1.1.1. High Knudsen number gas flows
During the past decade (Robinson et al., 1995; Koester et al., 1996), a variety of novel applications viz. micro-electromechanical systems (MEMS), low-pressure chemical vapor deposition (CVD) reactors, etc., have become important in many disciplines because of their extraordinary advantage for practical use. There are two distinct classes of high Knudsen number non-continuum flow problems. In the field of materials processing, a variety of vapor-phase processing and plasma-etch applications are used to produce thin films. The densities of the
gases are quite low (a few Torr or less). In the relatively new development of microminiaturization of devices, such as microactuators, microrefrigerators, and microgenerators, gases flow in micron-sized channels at relatively high densities (atmospheric, perhaps higher). Each of these distinct applications of high-Knudsen-number flows has generated great scientific and engineering interests (Oh et al., 1997; Nanbu et al., 1991).
Both classes of problems offer unique challenges in obtaining their solutions. Traditional continuum computational fluid dynamic (CFD) techniques are often invalid for analyzing gas flows in MEMS or in low-pressure devices. This inaccuracy stems from their calculation of molecular transport effects such as viscous dissipation and thermal conduction from bulk flow quantities such as mean velocity and temperature. This approximation of continuum phenomena fails as the characteristic length of the flow gradients (L) approaches the average 3 distance traveled by molecules between collisions (mean free path, λ). The ratio of these
λ quantities is known as the Knudsen number ( Kn = ) and is used to indicate the degree of L flow rarefaction. The Navier-Stokes equations ignore rarefaction effects and are therefore only strictly accurate at a vanishingly small Kn. The Knudsen domain (0 < Kn < ∞) is often divided into four regimes. When Kn < 0.01, the flow is considered to be continuum; for 0.01 < Kn < 0.1, it is in the slip-flow regime, for 0.1 < Kn < 3, it is called the transition-flow regime. When Kn > 3, the flow is considered to be free molecular and is sufficiently rarefied to allow molecular collisions to be completely neglected in analysis. The collisionless Boltzmann equation is therefore applicable for such flows. These Knudsen number limits on the conventional mathematical formulations are shown schematically in Figure 1.1.
Discrete Collisionless Particle or Boltzmann Equation Boltzmann Molecular Equation Model
Navier-Stokes Continuum Conservation Equations Euler Equations Model Do Not Form a Closed Set Eqs. (with slip or non- slop boundary conditions)
0 0.01 0.1 1 10 100 ∞ Inviscid Free Li mit Molecular Li mit
Local Knudsen Number
Figure 1.1. The Knudsen number limits on the mathematical models (Bird, 1994) 4
The direct simulation Monte Carlo (DSMC) method is a well-established alternative to
traditional CFD that has been used widely and successfully to simulate high Kn number gas
flow problems (Bird, 1994). This technique makes no continuum assumption. Instead, it models
the flow as it physically exists: a collection of discrete particles, each with a position, a velocity,
an internal energy, a species identity, etc. These particles are moved and allowed to interact
with the domain boundaries in small time steps. Macroscopic quantities, such as flow speed and
temperature, are then obtained by sampling the microscopic state of all particles in the region of
interest.
Three-dimensional simulations of flows at near-continuum (low) values of Kn and low
Ma (Mach number Ma = u/a, a is the sound speed) are often prohibitive because of very high
computing requirements dictated by higher collision rates and larger sample sizes. The
significant reduction in computational time is realistic through the recent advances in parallel
computer hardware and the development of efficient parallel DSMC algorithm. Different
paradigms must be employed for parallel programming to take advantage of the memory
architecture. MPI (Gropp et al., 1999) is a natural paradigm for distributed memory machines
while OpenMP (Chandra 2001) is a natural paradigm for shared memory machines.
1.1.2. Electroosmotic and electrophoretic microchannel liquid flows
The area of microfabricated fluidic devices that perform chemical and biochemical analysis has developed into one of the most dynamic fields in analytical chemistry during the past decade (Chiem and Harrison 1997; Effenhauser et al. 1993; Effenhauser et al. 1995; Fan and Harrison 1994; Harrison et al. 1993; Harrison et al. 1992; Jacobson et al. 1998; Jacobson et al. 1994a; Waters et al. 1998). Microfabricated fluidic devices can be utilized for medical, pharmaceutical, defense, and environmental monitoring applications. Example of such applications are drug delivery (Santini et al., 2000), DNA analysis/sequencing systems (Chang et al., 2000), and biological/chemical agent detection sensors on microchips. Miniaturized analysis systems can provide several advantages over conventional instruments, including 5 typically 2 orders of magnitude shorter analysis time, smaller injection volumes in the nanoliter to picoliter range, lower cost, greater portability, and potentially greater separation resolution.
These microscale devices associated with the Lab-on-a-Chip concept where sample handling and reactions are integrated with chemical separations almost exclusively use electrokinetic manipulations for fluid handling and analysis as they have the advantage of only requiring application of electric potentials to the devices. These manipulations exploit two physical phenomena: electrophoresis and electroosmosis. Electroosmosis refers to the bulk movement of an aqueous solution past a stationary solid surface due to an externally applied electric filed.
This requires the existence of a charged double-layer at the solid-liquid interface.
Electrophoresis, on the other hand, moves charged species in a buffer solution under the action of an applied electric field. Velocity of species depends on the electrophoretic mobility and local electric field. Development of these microfluidic devices is possible with reliable micro-pumps maintaining a desired flow rate. In the past micro-pumps with moving components have been designed and analyzed (Smits, 1990). Moving components in MEMS are prone to mechanical failure and fatigue. Robust micro-pumps for liquids with non-moving components can be developed by utilization of the electrokinetic phenomenon. The possibility of using the electrokinetic pumping in micro-capillaries and micro-channels has been demonstrated by Paul et al. (1998).
Microchannels are one of the primary components of microfluidic systems. Extensive experimental studies of capillary eletroosmotic and electrophoretic flows in microchannel have been performed recently. Considering the difficulties associated with performing experiments in micro-scales, it is desirable to develop reliable numerical models that describe the electrokinetically driven flow in complex micro-geometries. These numerical models provide better understanding of coupled effects of pressure, surface and electrokinetic forces on fluid motion, and hence can be utilized for an optimized micro-fluidic systems design, prior to hardware fabrication and experimental verification. 6
Electroosmotic and electrophoretic liquid flows in microchannels rely on fluidic,
electrokinetic, thermal and chemical interactions for performing the intended function. Advances
have been made in understanding these areas individually but coupled effects of these
phenomena are not well understood. Numerical simulations of liquid flows in microchannels are
very challenging due to coupling of flow with electrokinetic and thermal effects, association-
dissociation reactions for different species during transport process, non-uniform properties of
liquids, and non-Newtonian behavior if biological fluids are considered. A mathematical model
which can predict liquid flows in microchannels in the presence of electric and thermal fields is
desirable.
1.2. Objectives
The aim of this thesis is to describe, by the use of numerical models, the behavior of
high Kn number gas flows and microchannel liquid flows. Due to very different physical
characteristics of gas and liquid flows, different numerical models are implemented to
investigate high Kn gas flows and electroosmotic liquid flows. A comprehensive direct
simulation Monte Carlo (DSMC) model is developed to describe the significant non-continuum
impacts, usually neglected in traditional analysis, on the behavior of gas flow and heat transfer
for high Ku number. Flows in ducts and microchannels are chosen for investigation because
they represent the basic geometry components of many MEMS and low pressure devices. A
mathematical model has also been developed to simulate electroosmotic and electrophoretic liquid flows in microchannels. The modeled phenomena include fluidic and electrokinetic effects, which occur during the liquid transport process. The interactions between the physical effects are also considered.
The specific objectives of the dissertation are:
• to employ the direct simulation Monte Carlo (DSMC) method to predict high Kn gas flows. 7
• to investigate low pressure, high Kn gas flow fields and temperature fields in partially
heated channels.
• to investigate mixing flows of two gas streams (H2 and O2) within a microchannel.
• to develop a parallel 3D DSMC model from Bird’s original serial DSMC code on a
shared memory multi-processor machine using OpenMP.
• to develop a numerical scheme to model electrokinetically driven liquid mass transport
phenomena in microchannels.
• to study the capillary electrophoresis separation in microchannels using numerical
simulation.
1.3. Outline of the thesis
The present research addresses modeling of high Kn number gas flows and microchannel electrokinetic liquid flows. The objectives, methodology and achievements of the research work are presented as follows.
The motivation and objectives of the present research are presented in Chapter 1.
Chapter 2 describes the present status of the literature for high Kn number gas flows and microchannel liquid flows. The importance of simulation of high Kn number gas flows and microchannel liquid flows is also discussed.
A detailed description of the mathematical modeling development for present research work is discussed in Chapter 3. Direct simulation Monte Carlo model is applied to high Kn number gas flows. Appropriate boundary conditions are developed for pressure driven flows in ducts. The DSMC model is validated by comparing the simulation results with theoretical analysis and experimental data. A numerical model is developed for electrokinetic liquid flows in microchannels. The model accounts for both electroosmotic and electrophoretic effects. All of the equations which describe the physical effects are solved in the same source code structure.
In Chapter 4, DSMC model is implemented to investigate low pressure gas flow fields and temperature fields in partially heated channels. Heat transfer from the channel walls are 8 calculated and compared with the classical Graetz solution. The effects of varying pressure, inlet flow and gas transport properties (Kn, Reynolds number, Re and the Prandtl number, Pr respectively) on the wall heat transfer (Nusselt number, Nu) are examined. A simplified correlation is presented for predicting Nu as a function of the Peclet number, Pe and Kn. A simplified correlation is presented for predicting Nu as a function of the Peclet number, Pe and
Kn.
In Chapter 5, mixing flows of two gas streams (H2 and O2) within a microchannel are investigated using DSMC model. The effects of the inlet-outlet pressure difference, the pressure ratio of the incoming streams and the accommodation coefficient of the solid wall on mixing length are all examined.
In Chapter 6, a parallel 3D DSMC model is developed from Bird’s original serial DSMC code on a shared memory multi-processor machine using OpenMP. Numerical results are then presented for the subsonic rarefied gas flow in a three dimensional duct with a high speed moving wall from slip flow to free molecular flow regime.
In Chapter 7, the capillary electrophoresis liquid flows in microchannels are studied using numerical simulation. The sample stream is focused during the loading step and driven into a separation channel during the dispensing step. Flow fields and species distribution are simulated for both loading and dispensing steps in a two dimensional cross channel device. The evolution of each sample species concentration at the end of the separation channel is predicted. The separation resolution is defined from the sample species concentration band retention time and band width. Different separation performances can be obtained by manipulating electric field strength. A series of simulations for different electric field distributions and field magnitudes in the channel are presented. The goal of these simulations is to identify the parameters providing optimal separation performance. The effect of both loading and dispensing schemes on species concentration and separation resolution is presented. 9
Chapter 8 summarizes the research presented in this thesis and recommends future work that can be done to further advance the state of numerical simulations of high Kn number gas flows and microchannel liquid flows. 10
2. BACKGROUND AND LITERATURE SURVEY
2.1. High Kn number gas flows
Gas flows around spacecraft reentry into planetary atmospheres are important
problems in rarefied atmospheric gas dynamics. In the field of materials processing, a variety of
vapor-phase processing and plasma-etch applications are used to produce thin films. In both
atmospheric dynamics and materials processing, the densities of the gases are quite low (a few
torrs or less). However, in the relatively new development of microsystems, such as
microelectro-mechanical systems (MEMS), gases flow in micron-sized channels at relatively
high densities (atmospheric, perhaps higher). Each of these distinct applications of high-
Knudsen-number flows is now of practical scientific and engineering importance.
As computer speed and memory have increased and become more generally available,
the direct simulation Monte Carlo (DSMC) method has emerged as the primary method for solving realistic problems with high-Kn flows. The fundamental work on DSMC is described by
Bird (1976, 1994), its pioneer and inventor. Bird has also written a review of DSMC research
(Bird, 1978), giving the early background of DSMC and comparing it with molecular dynamics and the other methods. The DSMC method may be used (Bird, 1994) for dilute gases when the
ratio of the mean-free-path to the molecular diameter λ/d is at least ≥ 10. DSMC is, by far, the most popular particle methods for the analysis of collisional flow, i.e., flows for which intermolecular collisions significantly affect fluid behavior. It is called a simulation (rather than a solution) scheme because it was originally formulated to capture the important physical features of the flow, not to solve a particular set of equations.
In rarefied gas dynamics, at one extreme there are very basic, careful studies of the structure of shock waves or flows around specific configurations such as blunt bodies (Moss et
al., 1994; Dogra et al., 1995a, b). These calculations and their calibrations give important
physical insights into flow structure. The structure of normal shocks in high-Kn, nonequilibrium flow is a good example of one of the basic configurations. When modeled with DSMC, simple 11
problems provide a wealth of information useful for experiments and more complicated simulations. The shocks are relatively thick, and the properties of the gas change greatly over a few mean-free-paths. Thick shock waves in high-speed, dilute gases are important because of
the significant radiative and chemical reaction processes that occur in the shock wave (Bird,
1985, 1986a, 1987; Moss, 1987; Muntz, 1989). At the other extreme are computations of very
complex problems that describe entire vehicles maneuvering through rarefied, reacting, space
atmospheres. Computations of the nonequilibrium flows around spacecraft are perhaps the
most ambitious and complex DSMC calculations (see Moss, 1995; Haas et al., 1995). The
atmosphere surrounding the spacecraft is of low density and composed of atomic and molecular
species with a wide range of molecular weights and concentrations. The physics involves
processes such as surface desorption, reflection, absorption, and adsorption.
Perhaps one of the greatest challenges today in materials sciences is the design and
economical production of materials that have specific physical properties. Now there is an
increasing emphasis on low-pressure processes that could improve film uniformity and quality- control doping processes, grow larger films, and etch smaller features. In many of these cases,
the characteristic sizes are small enough (microns) and the densities are low enough
(atmospheric or less) that the flow is in the transition regime. Simulations of low-pressure CVD have been used in semiconductor growth to study the deposition over micron-sized trenches and flow through and around growth substrates. The vapor-phase process in CVD must be
designed to produce uniform deposition across each wafer and from wafer to wafer, and to
produce good step coverage over topographical features. For example, Ikegawa and Kobayashi
(1989), and Coronell and Jensen (1992, 1993) analyzed the flow in and around systems of wafers. It is important to create a uniform surface. This uniformity may be optimized by correctly choosing the total pressure, composition of the entering gas, and spacing of the wafers.
The etching and processing techniques described above are often used to make
components of microsystems, a general class of systems on the micron or submicron scale.
Such systems include microminiaturized devices, commonly called micro-electro-mechanical 12
systems (MEMS), such as microaccelerators, microactuators, microflaps, microsatellites,
microrobots, microengines, and the head-platter systems in computer disk drives (Gabriel, 1995;
Ho and Tai, 1997). MEMS have become important because of their current and potential use in a variety of medical and engineering problems. For example, combining MEMS devices with electronic sensors could result in highly controllable integrated systems. The success of designing and fabricating microsystems depends on the operation of a number of components,
the most fundamental of which are flows through narrow channels, diffusers, and nozzles. In
many systems using gases, the flow may be in the high-Kn regime.
In the past few years, an increasing effort has been made to understand and predict the properties of flows in microsystems, of which the microchannel is a basic element. In these
systems, the mean-free-path of the molecules may be the order of the system size, so that Kn is in the transition regime. Viscous momentum transport from the wall to the flow is an important aspect of the system. Near the walls, rough surfaces and high gradients in the fluid variables,
such as velocity and pressure, may affect transport in ways different than those in larger systems. There have been experiments on microchannels (Pong et al., 1994; Harley et al.,
1995). Helium is a common gas used in most experiments because it has a large mean free path (about 2 x 10-7 m under laboratory conditions). The Knudsen number based on a channel
height of 1 micron is 0.2 at atmospheric pressure. A micro channel with integrated micro pressure sensors was fabricated to study the flow field (Liu et al., 1993b; Pong et al., 1994). Slip flow is observed, and the measured mass flow rate (Pfahler et al., 1991; Pong et al., 1994;
Arkilic et al., 1995; Harley et al., 1995; Liu et al., 1995; Shih et al., 1995, 1996) is higher than that based on the non-slip boundary condition. For other gases (e.g. nitrogen, oxygen, and
nitrous oxide), the Knudsen number is about a factor of four smaller, but surface slip still exits. In the microchannel, high pressure drops are observed along the streamwise direction. This is because of the small transverse dimension, which causes high viscous dissipation. A drop of a few atmospheres in several mm is common (Pong et al., 1994; Shih et al., 1995). The density of
the gas can change so much that the pressure does not decrease linearly with streamwise 13
distance as in typical creeping flows. Rather, the compressibility effect causes the pressure to
decrease more slowly. On the other hand, the rarefaction effect caused by the high Knudsen
number works against the compressibility and keeps the pressure toward the linear distribution
(Arkilic and Breuer, 1993). The two effects are not equal, so the net result is a nonlinear
pressure distribution.
There are also a number of theoretical and computational efforts based on extensions and modification of the Navier-Stokes equations to accommodate relatively high-Kn effects
(Beskok and Karniadakis, 1994; Arkilic et al., 1995, 1997; Harley et al., 1995; Beskok et al.,
1996). The mass flow rate can be calculated from the Navier-Stokes equation with a slip boundary condition (Kennard, 1938; Beskok and Karniadakis, 1992, 1993; Arkilic and Breuer,
1993). An accommodation coefficient is introduced to represent the tangential momentum
transfer between the impinging molecules and the wall. The value of the coefficient should be ≤
1. However, the predicted mass flow rate is sensitive to the accommodation constant, which actually functions as a matching coefficient. In past few years, some attempts have been made
to use DSMC to simulate gas flows in microchannels (Oh et al., 1995, 1997; Ahn et al., 1996;
Piekos and Breuer, 1996; Beskok et al., 1996; Oran et al., 1998). When the mean streamwise
velocity in the microchannel is in very low subsonic range (<1 m/s), the molecular thermal
velocity of 1000 m/s (Oh et al., 1995) can be several orders of magnitude greater than the flow velocity. Computing the solution is a challenge for very low Mach-number flows.
2.2. Liquid flows in microchannels
Even though the non-slip boundary of simple liquids with molecules is established by experimental studies and by molecular dynamics simulation (Koplik, 1989), this does not make the study of liquid flow through microchannels a routine process. To the contrary, it seems to be
an even richer problem than gas flows in microchannels. For liquid flows through capillary tubes
(Migun and Prokhorenko, 1987) or micromachined channels (Pfahler et al., 1990, 1991), the
measured flow rates and pressure drops across the channel were compared with the Stokes 14
flow solution. Size effects are apparent in these results. The value of viscosity deviates from the conventional value for micron-size channels.
The molecular structure of the liquid also affects the flow. Molecules with no electrical
charge can have a dipole configuration (e.g. water). Pfahler et al (1991) found that the value of the viscosity of polar isopropanol decreases from the nominal value for a channel height smaller
than 40 microns and reaches an asymptotic value at a channel height of about 10 microns. It is reasonable that the size effect becomes more pronounced for a narrow channel.
Recently, electroosmotic pumping has been demonstrated as a suitable way to drive
liquid flows in microchannels. Electroosmosis describes the motion of fluids with respect to a
fixed, charged solid surface. If a surface is positively charged, the negative ions in the solution
will be attracted by the Coulomb forces toward the solid wall. There is thus a thin solvated layer
of negatively charged particles in contact with the solid surface. Electroosmotic flow results
when an electric field is applied to this electrical double layer that spontaneously forms at the
interface between a liquid and solid when they are brought into contact (Bockris and Reddy,
1998). The thickness of this double layer is determined by the opposing forces of electrostatic
attraction and thermal diffusion and is of the order of the Debye length of the solution. The
advent of microfluidic fabrication technologies has seen an application of electroosmosis as a
method of controlling fluids in microfluidic chips (Jacobson and Hergenroder, 1994). On-chip
electroosmotic pumping is readily incorporated into electrophoretic and chromatographic
separations, and these "laboratories on a chip" offer distinct advantages over the traditional,
free-standing capillary systems including the ability to network multiple channel streams and
reduce analysis time.
Considering the difficulties associated with performing experiments in micro-scales, it is
desirable to develop reliable numerical analysis tools that can model the electrokinetically driven
flows in complex geometries. These numerical models can provide better understanding of
coupled effects of electric field with fluid motion, and hence can be utilized for an optimized
micro-fluidic system design, prior to hardware fabrication and experimental verification. Since 15 the electrokinetic phenomena was first described by Reuss in 1809 (Reuss, 1809), there have been numerous theoretical investigations on electrokinetic microflows (Burgreen and Nakache,
1964; Keh and Liu, 1995; Ohshima and Kondo, 1990). Various studies dealing with electroosmotic and electrophoretic flow theory and modeling have analyzed the effects of flow inertia, pressure gradients, and nonuniform ζ potentials (the potentials due to the surface charge) (Santiago, 2001) in microfluidic channels. Anderson and Idol (1985) analyzed electroosmotic flow in channels with nonuniform surface charge. Anderson (1985) presented an analysis that shows electrical double layers reach a local equilibrium in the presence of electroosmosis when the channel length is long compared to the characteristic hydraulic diameter. Dose and Guiochon (1993) and Sodermann and Jonsson (1996) presented an analysis of the time scales of transient electroosmotic flows. Dose and Guiochon (1993) presented a numerical simulation of impulsively started electroosmotic flow, while Sodermann and Jonsson presented analytical models for start-up electroosmotic flows in a variety of flow geometries including flow over a single flat plate and flow between two flat plates. Most successful theoretical studies of sample transport in electrophoresis and electroosmosis have been conducted using computer modeling. Computers have been applied to the modeling of most common electrokinetic techniques and classes of samples (Mosher et al. 1992; Mosher et al. 1993). Computer modeling has also been extended to simulate capillary electrophoresis and electroosmosis in 2D and 3D geometries (Bianchi et al., 2000; Ermakov et al., 1998; Ermakov et al., 2000; Patankar and Hu, 1998; Yang et al., 1998). Both qualitative and quantitative agreement between simulation and experiment has been obtained (Ermakov et al., 1998;
Ermakov et al., 2000; Patankar and Hu, 1998). Recently, Ermakov et al. (2000) investigated electrokinetic injection techniques in microchips using computer simulation. A two dimensional cross channel was used for sample manipulation. Both pinched injection valve (Jacobson et al.,
1994a) and gated injection valve (Jacobson et al., 1994b) were investigated. They reported the operating parameters providing optimal valve performance for both injection techniques according to their simulation results. 16
3. NUMERICAL AND MATHEMATICAL MODELS
3.1. Overview of research work
The overall goal of the present research program is to develop computational models
for predicting behaviors of high Kn number gas flows and liquid flows in microchannels. The
issues for gas and liquid flows are fundamentally different.
3.1.1. High Knudsen number gas flows
Gas flows and heat transfer in channels and tubes have been intensely studied in the past (Kakac and Ozgu, 1969) due to their practical use as well as their theoretical importance.
However, most of the past studies for gas flows considered continuum formulations, which is only valid for low Knudsen numbers, Kn < 0.01. In the present research program, the numerical simulation is implemented to study the effect of Kn, Re and Pr on heat transfer from the heated walls under the rarefied flow conditions. Specifically, subsonic pressure-driven flows are considered. The predicted heat flux results are compared with the Graetz solution (Kakac and
Ozgu, 1969) with uniform flow inlet. A large number of cases are calculated with varying inlet
Kn, Re and Pr from the slip-flow regime to the transition-flow regime. Nitrogen, hydrogen, oxygen and noble gas mixtures are employed in the simulation. Based upon these simulations, a simple correlation connecting Nu with Pe and Kn is developed (Yan and Farouk, 2002a).
Mixing is another area of intense research in continuum fluid flow and heat transfer.
Adequate mixing is an essential requirement for propulsion devices designed to provide
chemical reactions and heat release in streams of gas and fuel. For this reason, the study of
mixing process is a necessary prerequisite to setting criteria for the control of energy release in
reaction chambers. Both supersonic and subsonic shear layers have been studied intensely
(Farouk et al., 1991). However, similar problems in microchannels have not been investigated.
With the development of MEMS-assisted fuel injectors and other propulsion devices being
examined for aerospace applications, an understanding of the behavior of mixing in the micro- 17
scale counterpart can provide tools to facilitate the design and optimization of such innovative
devices. DSMC is utilized to investigate the gas flow and heat transfer for high Kn number flow
in a microchannel. The aim is to show the basic characteristics of the mixing process in
microchannel (Yan and Farouk, 2002b). H2 and O2 with different pressure as well as different streamwise velocities were used for the computations. The length required for the two gases to be fully mixed was determined by examining the predicted density field. Mixing problems with different inlet pressure ratios, different inlet pressures and different solid wall accommodation coefficients were calculated. The effect of these parameters are analyzed according to the numerical predictions.
Three-dimensional simulations of flows at near-continuum (low) values of Kn and low
Ma are often prohibitive because of very high computing requirements dictated by higher collision rates and larger sample sizes. A significant reduction in computational time is realistic due to recent advances in parallel computer hardware and the development of efficient parallel
DSMC algorithm. A considerable amount of effort has already been put into the parallelization of the DSMC algorithm (LeBeau, 1999; Oh et al., 1996; Wilmoth, 1992). Most work has involved implementing parallelization of DSMC procedures on distributed memory computers (Aktas et al., 2001; Oh et al., 1996; Wilmoth, 1992). The parallel method is based on domain decomposition in which a portion of the physical space along with its associated cells and molecules is allocated to each processor. A great deal of inter-processor communication is required and message passing interface, MPI (Gropp et al., 1999) is a common communication paradigm that supports data sending and receiving. However, different paradigms must be employed for parallel programming on a shared memory machine to take advantage of the memory architecture. One of the objectives of this thesis is to develop a parallel DSMC model from Bird’s original serial DSMC code on a shared memory multi-processor machine using
OpenMP. Numerical results are then presented for the subsonic rarefied gas flow in a three dimensional duct with a high speed moving wall from slip flow to free molecular flow regime
(Yan and Farouk, 2002). 18
3.1.2. Electroosmotic and electrophoretic microchannel liquid flows
Electroosmotic liquid flow describes the motion of fluids with respect to a fixed, charged solid surface. This flow results when an electric field is applied to the electrical double layer that spontaneously forms at the interface between a liquid and solid when they are brought into contact (Audubert and Mende, 1959). The thickness of this double layer is determined by the opposing forces of electrostatic attraction and thermal diffusion and is on the order of the Debye length of the solution. The advent of microfluidic fabrication technologies has seen an application of electroosmosis as a method of controlling fluids in microfluidic chips (Harrison et al., 1992; Jacobson et al., 1994). On-chip electroosmotic pumping is readily incorporated into electrophoretic and chromatographic separations, and these "laboratories on a chip" offer distinct advantages over the traditional, free-standing capillary systems including the ability to network multiple channel streams and reduce analysis time.
Various studies dealing with theory and modeling have been performed recently for electroosmotic and electrophoretic flows in microfluidic channels. Most models are limited to 1-
D cases. There are few reports which solve 2-D and 3-D problems such as electroosmotic flow in a rectangular channel intersection studied by Patankar and Hu (1998) recently. It is desirable to develop a 3-D numerical model to investigate various physical effects on electroosmotic liquid flows in complicated microfluidic devices which are not well understood.
In the present research program, a numerical model for electroosmotic and electrophoretic flows is tested. The primary aim is to develop a robust numerical scheme that can be applied to various flow conditions. The continuity equation, momentum equation and the governing equation for potential distribution are presented. A numerical model is implemented to solve these coupled equations.
Recently, Ermakov et al. (2000) investigated electrokinetic injection techniques in microchips using computer simulation. A two dimensional cross channel was used for sample manipulation. Both pinched injection valve (Jacobson et al., 1998) and gated injection valve
(Jacobson et al., 1994) were investigated. They reported the operating parameters providing 19
optimal valve performance for both injection techniques according to their simulation results.
However, species separation from each other was not examined. To provide optimal electric field
strength manipulation for species separation, the electrophoresis species separation in
microchannels is studied using computer modeling. The numerical model accounts for the principal
physical phenomena affection sample mass transport in microchannels such as the electrophoretic
motion, electroosmotic motion and diffusion. Unlike Ermakov et al. (2000) in which electrokinetic
injection performance was the target for simulation, we concentrated on species separation under
various operating conditions. Variation of species concentration and separation resolution are
predicted for different loading and dispensing scheme combination. This work can help to improve
the understanding of the physical phenomena of electrophoresis separation and the design of
capillary-electrophoresis microchips.
3.2. Direct simulation Monte Carlo model for high Knudsen number gas flows
3.2.1. Direct simulation Monte Carlo model development
3.2.1.a. Principles of the DSMC model
DSMC is a direct particle simulation method based on the kinetic theory of gases. The fundamental idea is to track a large number of statistically representative particles. The particles' motion and interactions are then used to modify their positions, velocities, or chemical reactions.
Conservation of mass, momentum, and energy is enforced to machine accuracy.
The primary approximation of DSMC is to uncouple the molecular motions and the intermolecular collisions over small time intervals. Particle motions are modeled deterministically, while the collisions are treated statistically. Symmetries in physical space can be used to reduce the number of grid dimensions and to reduce the storage requirements for molecular spatial information. However, modeling collisions is always a three-dimensional
calculation. The limitations of DSMC are the same as those of classical kinetic theory: the assumption of molecular chaos and the restriction to dilute gases. DSMC is equivalent to solving 20
the Boltzmann equation for a monatomic gas undergoing binary collisions (Nanbu, 1980;
Babovsky and Illner, 1989). Comparisons of DSMC with molecular dynamics are very
expensive, but excellent agreement has been shown in shock computations (Salomons and
Mareschal, 1992) and slip lengths (Morris et al., 1992). In addition, many computational studies
have shown that DSMC solutions approach Navier-Stokes solutions in the limit of very low Kn
(Muntz, 1989; Cheng, 1992; and many articles in the proceedings of the International Symposia on Rarefied Gas Dynamics).
Using each simulated molecule to represent a large number of actual molecules
reduces computational requirements to a manageable level but introduces statistical error. The statistical error of a DSMC solution is inversely proportional to the square root of the total number N of simulated particles (or sample size). The computational work is directly proportional to N. The simplicity of the algorithm allows for straightforward incorporation of higher-order physical models and for application to complex geometries. The primary drawback of DSMC is its cost. Whereas idealized calculations are possible on small computers, significant
computational resources are required to simulate realistic flows. However, this drawback has
been mitigated by the fact that the method is capable of giving quantitative results in the high-Kn flow regime and that large-scale computer resources are more generally available.
Figure 3.1 shows the procedures involved in applying DSMC to an unsteady or steady
flow problem. Execution of the method requires using a cell, or spatial element, network in
physical space. The cell network provides geometric areas and volumes required to evaluate macroscopic flow properties. It is also used by the collision process model, in which only particles located within the same cell, at a given time, are allowed to interact.
A DSMC simulation, like a continuum computational fluid dynamics (CFD) calculation,
proceeds from a set of prescribed initial conditions. The particle positions, velocities, collision cross sections, and boundary conditions determine the subsequent evolution of the system. As
in a continuum solution or an experiment, the solution that evolves may be very sensitive to the
particular choice of initial, boundary, and input conditions. 21
Initially, the number of simulated particles in the computational domain is assigned
arbitrarily. For example, consider a computational domain with volume 1 m3 and 50 cells. The initial gas has a number density 1020 m-3 at a temperature of 300 K. In order to obtain 1000 molecules with average 20 molecules in each cell, the number of real molecule that is represented by each simulated molecules is set to 1017.
One computational approximation associated with the DSMC method is the ratio of the
number of simulated molecules to the number of real molecules. This approximation leads to
the statistical scatter. This statistical scatter can generally be assumed to follow the Poisson
distribution, and the standard deviation is of the order of the inverse square root of the sample
size. With a cumulative sample, the statistical fluctuations declines with the square root of the
sample size and it should be possible to achieve any desired level of accuracy by continuing or
repeating the simulation to build up the size of the sample to the required magnitude. The
gradual approach of a sampled quantity to its correct value is sometimes referred to as
‘convergence’. It is possible to argue qualitatively that fluctuations may become unstable in
some flow configurations. This behavior has been found to be due to simulation procedures that
allow a random walk in one or more variables. One characteristic of a random walk is that
displacement from mean position or value increases with square root of time. Random walk can
arise whenever one of the molecular quantities is conserved only on the average, rather than
exactly, in any of the simulation procedures. For example, the application of species based
weighting factors leads to random walks. Such weighting factors have been used when dealing
with trace species, because these would be represented by a very small sample if the number
of real molecules represent by each simulated molecule was the same for each species. The
sample of the trace species is increased if each of the simulated molecules of this species
represents fewer real molecules than those of the dominant species, and this difference is
termed a weighting factor. In this thesis, no weighting factor is applied and all DSMC
simulations have shown that the fluctuations are stable and convergence is achieved. This is an
empirical result and it would be desirable to have an analytical proof of this stability. 22
3.2.1.b. The DSMC procedure
The core of the DSMC algorithm consists of four primary processes: move the particles, index and cross-reference the particles, simulate collisions, and sample the flow field. These procedures are uncoupled during each time-step. Of primary importance is the selection of a timestep that is less than the mean collision time. Simulation results are independent of the timestep increment as long as this requirement and the cell-size requirement on the gradient resolution are satisfied.
The first process, moving simulated molecules, enforces the boundary conditions and samples macroscopic properties along solid surfaces. Modeling molecule-surface interactions requires applying the conservation laws to individual molecules instead of using the velocity distribution function. Such treatment of the boundary conditions allows DSMC to be extended to include physical effects such as chemical reactions, catalytic walls, radiation effects, three-body
collisions, and ionized flows, without major modifications in the basic procedure.
The second DSMC process involves indexing and tracking particles. A scheme for
molecular referencing is the prerequisite for the next two steps: modeling collisions and sampling the flow field. Accurate and fast indexing and tracking are key to practical DSMC
applications for large-scale processing. This subject is discussed further in the section on
Lagrangian DSMC.
The next step, simulating collisions, is a probabilistic process that sets DSMC apart from deterministic simulation methods such as molecular dynamics. Several collisional modeling
techniques have been applied successfully within the framework of DSMC. All simulate an
appropriate number of representative collisions between randomly selected pairs of molecules within each cell but with various degrees of computational efficiency. The currently preferred model is the no-time-counter technique (Bird, 1994), used in conjunction with the sub-cell technique (Bird, 1986b). The sub-cell method calculates local collision rates based on the 23
individual cells but restricts possible collision pairs to sub-cells. This procedure improves accuracy by ensuring that collisions occur only between near neighbors.
The final process is sampling the macroscopic flow properties. The spatial coordinates
and velocity components of molecules in a particular cell are used to calculate macroscopic quantities at the geometric center of the cell. For example, the macroscopic density is equal to the sum of the individual particle mass per unit volume. In DSMC, it can be written as the product of number of simulated particles, individual particle mass and represented particle ratio
(number of real particles represented by one simulated particles) divided by the cell volume.
With different represented particle ratio, different number of simulated particles will present in the cell. However, the sampled macroscopic density will be the same. The other steps of the
DSMC procedure do not depend on the sampling process. Therefore, one way to reduce
computational time is to sample the flow properties every nth timestep.
The DSMC technique is explicit and time-marching, so that it always produces a flow
simulation that is unsteady. For an unsteady flow application, an ensemble of many
independent computations may be assembled and averaged to obtain final results with an
acceptable statistical accuracy. An ensemble average (the instantaneous average over area or volume elements of an arbitrarily large group of similar systems) is commonly used to present
unsteady DSMC results. To simulate a steady problem, each independent computation
proceeds until a steady flow is established at a sufficiently large time, and the desired steady
result is a time average of all values calculated after reaching the steady state.
3.2.1.c. Boundary conditions
Two dimensional subsonic flow simulations (considered in the present study) by the direct simulation Monte Carlo technique in the slip and transitional regimes are rare (Nanbu et al., 1991). One of the difficulties lies in specifying the outlet boundary conditions where particles may move upstream and reenter the flow domain. In hypersonic flows or free molecular flows, the particles re-entering from the outlet boundary can be neglected. In subsonic flows in the 24
transitional regime, we cannot neglect particles, which go out of the boundary and may reenter.
Nanbu et al. (1991) applied a porous wall at the inlet of a duct to prevent molecules from going
upstream. Another difficulty in two dimensional subsonic-flow particle simulations in transitional
regime is in controlling the flow velocity inside the domain. If particles do not re-enter from the
exit (assuming expansion to vacuum), they are accelerated to the exit and velocities inside the
domain do not stay uniform.
In the present calculations, the inlet and outlet boundary conditions were formulated
following the procedure outlined by Piekos and Breuer (1995,1996). The inlet and outlet
pressures are considered to be specified. This pressure drop drives the gas flow through the
channel. At the inlet boundary, temperature is kept constant (300 K), and the density is
specified (via ideal-gas law) according to the given inlet pressure. The transverse velocity of
flow is set to zero. Thermal velocities of each simulated particle are set randomly from a
Maxwell-Boltzmann velocity distribution. Velocities of each simulated particles are therefore the
summation of thermal velocities and mass average velocities. At the outlet, pressure is specified
and zero gradient boundary conditions are considered for temperature, transverse velocity and
streamwise velocity. Only the number density is calculated according to the given outlet
pressure. It has been shown (Cai et al., 2000) that using mean velocity of inflow and outflow is
not accurate enough in the DSMC simulation. However the velocities used to implement the
boundary conditions are the local velocities at each inlet and outlet cell.
3.2.1.d. Validation of the DSMC model
Two independent comparisons are carried out to validate our DSMC results. First, the
DSMC result is compared with an analytical solution (Arkilic et al., 1994) for a long channel without any heated section. Then, DSMC results obtained for the schematic described in Figure
3.2 are compared with the analytic Graetz solution (presented in the next chapter).
For the first comparison, a pressure driven flow is considered in which the channel
length, L, is long enough, i.e., L/d >> 1 and isothermal as well as steady-flow are assumed. The 25
Navier-Stokes equations can be simplified and solved analytically with a velocity-slip boundary
condition. The solution (Arkilic et al., 1994) gives a pressure profile, which is uniform across the
channel and changes along the channel according to
2 - α P * ( x *) = - 6 Kn o + α
2 - α 2 * 2 2 - α * 2 - α (6 Kn o ) + ( P + 12 Kn o P (1 - x *) + (1 + 12 Kn o ) x * α i α i α (3.1)
* * where P (x*) and Pi are the local pressure along the channel and inlet pressure, both
normalized by the outlet pressure, Kno is the outlet Knudsen number, x* is the streamwise coordinate normalized by L and α is the accommodation coefficient. The channel length is chosen to be 2.4 m and the width is 0.05 m to make the aspect ratio equal to 48. The flow domain is divided into 28000 cells (350 × 80) and initially 50 stationary particles per cell are set at random position. Every cell is subdivided into two sub-cells in each direction. The sub-cell is introduced to cause all collisions to occur between particles in neighboring cells. As the simulation starts, the gas particles are accelerated and eventually when the temporal variation of the outlet velocity profile is small enough, (the maximum changes of the outlet velocity at all exit locations less than 1%), the flow is assumed to have reached the steady state condition.
The time step used is less than one-third of the typical particle mean collision time (Bird, 1994).
The typical molecular displacement in each direction during a time step is thus less than one third of the cell size. The ‘variable hard sphere’ (VHS) model is used to account for the collisions
(Bird, 1994) since VHS model has been widely used in both slip flow and transition flow simulation (Oh et al., 1997; Mavriplis et al., 1997; Piekos and Breuer, 1995). The gas properties used in the calculation such as molecular diameter, molecular mass, viscosity-temperature index and reference temperature can be obtained according to Bird (1994). The inlet pressure is
6.7 Pa while the outlet pressure is 2.23 Pa. Temperature is 300 K everywhere and the outlet
Kno is about 0.048. The accommodation coefficient α is set to be 0.85. It can be seen from
Figure 3.3 that the agreement between the DSMC result and the analytical solution is excellent. 26
The most studied micro-flow problem is the pressure-driven channel flow with a micron
size channel cross section. The inlet and outlet pressure difference is specified which drives the
gas flow through the channel. Eventually, steady flow states are established inside the channel.
Experimental measurements on microchannel flow are difficult due to the very small size. Recent
development of MEMS technology allows micro-sized instrumentations such as pressure sensors
to be integrated with the microchannel flow system (Pong et al., 1994). The experimental results
used here for comparison are from Pong et al. (1994). In their experiment, an integrated
microchannel/pressure sensor system was used, featuring a micro-flow channel with 4500 µm length, 40 µm width and 1.2 µm height. Eleven sensors were arranged along the channel. The
pressure sensors are 250 × 250 µm2 in area. During the experiment, the outlet is open to air. The
inlet is connected to a pressure reservoir. The experiment ran at room temperature.
A standard test for slip flow case with nitrogen gas is considered first. The inlet pressure is
191 kPa and the outlet pressure is 101 kPa, i.e., atmospheric. The pressure ratio is then 1.88. The
environment temperature and the two wall temperatures are all set to be 300 K. According to
analytical results based on the slip-flow solution of Navier-Stokes equations (Arklic, 1993), the
absolute value of the channel length is not important for the pressure profile as long as the channel
is long enough L/H >> 1. This allows comparisons of pressure profile between channels with
different length when the flow is isothermal. For our simulation, a channel with 1.2 µm in width and
36 µm in length is used. The inlet and outlet Kn numbers are 0.028 and 0.053 respectively.
Corresponding to the upstream and down stream pressures in the experiment channel (Pong et al.,
1994), the pressure distribution is shown in Figure 3.4 for the simulation. The measurements (Pong
et al., 1994) are also plotted. The axial distance is normalized with the channel length in Figure 3.4.
Total 90000 (60 × 1500) cells were used in order to satisfy the conditions that the cell size must be
less than half the mean free path. Initially, 50 particles were set in each cell. Non-linear pressure
profile along the channel is observed. The agreement between the simulation results and
experimental data are excellent.
27
3.3. Mathematical model of microchannel liquid flows
3.3.1. Problem description
Capillary electrophoresis (CE) is a separation method that could be coupled with flow
injection analysis on a planar substrate to explore the µ-TAS concept (the miniaturization of a
total chemical analysis system onto a monolithic structure) which can be used to monitor
chemical concentrations continuously in industrial chemical and biochemical processes
(Harrison et al., 1992).
Extensive experimental studies of capillary electrophoresis using micromachined
channels have been performed recently. Harrison et al. (1992) integrated a capillary
electrophoresis and sample injection system on a planar glass chip. They observed that the
solvent flow could be directed along a specified capillary by the application of appropriate
voltages, so that valveless switching of fluid flow between the capillaries could be achieved.
Seiler et al. (1993) presented improvements in the instrumentation and experimental method for
the device described by Harrison et al. (1992). Harrison et al. (1993) performed electroosmotic
pumping and electrophoretic separation of samples using a device consisting of two intersecting
channels micromachined in silicon. Fan and Harrison (1994) used a similar cross-channel
device integrated on a glass chip and evaluated its performance. Seiler et al. (1994) used a
device in which the electroosmotic flow was driven within a network of intersecting capillaries
integrated on a glass chip.
In most of these experiments, the injection and the separation of the sample were achieved using the electroosmotic and electrophoretic flows driven by an applied potential along the channels. The injection channel and the separation channel were perpendicular to each other. The electroosmotic and electrophoretic flows themselves are influenced by various parameters. Here a numerical model is developed to do such investigations since one can better control various parameters involved.
The geometry of the device shown in Figure 3.5 is similar to that used by Fan and
Harrison (1994). There is an intersection of cross-channels with four reservoirs numbered 1-4. 28
The cross-channels are closed by walls from above and below. The injection of the sample
solution (containing various species) is effected by applying an electric potential between
reservoirs 1 and 2. This potential drives the sample solution from reservoir 1 across the
intersection (as shown in Figure 3.5), creating a sample plug in the path between reservoirs 3
and 4. Subsequent application of a voltage difference between reservoir 3 and 4 should then
drive a small plug of sample along the channel 4, effecting electrophoresis separation of the
sample plug due to different electrophoretic mobility of different species. Sample separation
does not occur during the injection when the reservoir 3 and 4 are kept at the same voltage
since the sample solution runs continuously in the injection channel. The separation of the
species in the sample requires the sample to be arranged initially in a zone so that the species
can move away from each other due to the difference in their electrophoretic mobilities (as the
zone spreads) (Saville and Palusinski, 1986). During the separation process, the electric
potential between reservoirs 3 and 4 drives the sample plug to move toward reservoir 4 and
causes the separation of various species.
3.3.2. Mathematical formulation
The diffusion of the sample solution from the injection channel into the separation
channel is not important, since the diffusion time is much longer than the relevant convective
time scale of the flow. The equation for the transport of the species i is (dilute solution
assumed)
∂ C r r i + ∇. ((V + z µ E )C ) = D ∇ 2 C (3.2) ∂ t i ep ,i i i i where Ci is the species concentration, Di is the diffusion coefficient, zi is the valence and µep is r r the electrophoretic mobility of the ith species. V is the velocity vector and E is the electric field
r vector. The term zi µep, i E accounts for the electrophoretic velocity of each species relative to
r r r the bulk flow. The term (V + zi µ ep ,i E ) denotes the species velocityVi . 29
Let U be the convective velocity scale and h be some length scale, e.g., the channel
width. If we assume there is no wall reaction and nondimensionalize the velocity by U, length by
h, and time by h/U, (3.2) becomes
∂C r 1 i + (V ⋅ ∇)C = D ∇2C (3.3) ∂t i i Sc Re i i where the nondimensional parameters Sc = ν/D is the Schmidt number and Re = Uh/ν is the
Reynolds number. ν is the kinematic viscosity of the sample solution. For the problem at hand,
D is of the order of 10-11 m2/s, and ν is of the order of 10-6 m2/s; thus Sc is of the order of 105
and Re is of order one. Therefore, the diffusion is negligible as compared to the convection,
since Sc·Re is large. This means that the sample species convect with the flow.
Since the sample solution is generally a dilute solution of certain species in the buffer, it
is assumed that the mechanical properties (density and viscosity) of the sample solution are the
same as that of the buffer. The Joule heating, which gives rise to temperature variations in the
channels, may also make the properties of the solution nonuniform. However, since we are
considering the capillary electrophoresis where temperature variations are usually small due to
the efficient heat dissipation, we can neglect the Joule heating and assume that the mechanical
properties of the solution in the cross-channel are uniform.
Given the above assumptions, the basic equations describing the flow are the continuity
equation, r ∇ ⋅V = 0 (3.4)
and the momentum equation, r ∂V r r r r ρ[ + (V ⋅ ∇)V ] = −∇P + µ∇ 2V + ρ E (3.5) ∂t e r where V is the bulk flow velocity, ρ is the density, p is the pressure, µ is the viscosity, ρe is the r electric charge density, and E is the electric field intensity. The electric field is related to the electric potential Φ by, 30
r E = −∇Φ (3.6)
while the electric potential is governed by,
2 ∇ Φ = −(ρe / ε ) (3.7) where ε is the electrical permittivity of the solution. Substituting equations 3.6 and 3.7 into equation 3.5, the momentum equation can be rewritten as, r ∂V r r r ρ[ + (V ⋅ ∇)V ] = −∇P + µ∇2V + ε[∇2Φ ⋅ ∇Φ] (3.8) ∂t
Equations 3.6 and 3.7 can be modified to obtain a more solvable form. The channel walls in contact with the solution have a certain potential due to the charge on them. This potential is called the ζ potential. The magnitude of the ζ potential depends on the character of the non-conducting surface and the solution. The electroosmotic flow is primarily caused by the migration of the charged species next to the channel walls subjected to an externally applied electric field. The distribution of the charged species in the domain is governed by the potential at the walls and by the externally applied electric field. When the Debye thickness is small and the charge at the walls is not large, this distribution is governed mainly by the ζ potential at the
wall and is affected very little by the external electric field (Saville, 1977). Thus, the charge
distribution near the walls can be determined independent of the external electric field. The
charge distribution may be affected by fluid motion since the charged species convect with the
flow. However, the effect of fluid motion on the charge redistribution can be neglected when the
fluid velocity is small, i.e., when the inertial terms in the momentum equation are not dominant
(Saville, 1977) or when the Debye thickness is small. In summary, the electric field equations
and the fluid flow equations can be decoupled, and the potential Φ can be decomposed into a
potential due to the external electric field, φ, and a potential due to the charge at the walls, ψ,
Φ = φ + ψ (3.9)
Consequently (3.7) can be written as two separate equations,
2 ∇ φ = 0 (3.10) 31
and
2 ∇ ψ = - (ρe / ε) (3.11)
The external electric field, which generates a force on the charged species causing the electroosmotic flow, is given by r E = −∇φ (3.12)
This decomposition is similar to that given by Grossman and Colburn (1992). Equation 3.11 can be further simplified following a treatment similar to that of Hiemenz (1986) where two important steps are involved. First is the use of an expression for the number density of ions in solution near a charged surface as proposed by Debye and Hückel, and in the second step, the Debye-
Hückel approximation was made to further simplify this expression. Details of the steps applied to the one-dimensional case can be found in Grossman and Colburn (1992). They obtained the following simplified one-dimensional form of (3.11),
d 2 ψ ψ = (3.13) d 2 x κ 2
where κ is called the Debye length and it corresponds to the thickness of the Debye layer. The
Debye length is generally a constant for a given solid-liquid interface. For the cross-channel
geometry, The form of (3.13) can be simplified as
∇2 ψ = ψ / κ2 (3.14)
From (3.11) and (3.14) we get the following expression for the charge density,
ρ = - εψ / κ2 (3.15) e
Using (3.12) and (3.15), we rewrite the momentum equation 3.5 as,
r ∂V r r r εψ∇Φ ρ[ + (V ⋅ ∇)V ] = −∇P + µ∇ 2V + ( 3.16) ∂t κ 2
Similar equations were obtained by Saville (1977) by using perturbation methods for the flow around a charged particle. 32
The Debye thickness κ-1 is usually of the order 10-9 m. Hence, for geometry being
studied (the channel width h is of the order 10-5 m), the non-dimensional Debye length is (κh)-1
is of the order 1.0-4. It is very computationally expensive to resolve the electric double layer with thickness of Debye length where most of the electric charge is concentrated because very fine mesh size is required near the solid wall. The features of electric double layer is described further in Figure 3.6. It can be seen from Figure 3.6(a) that ions from the sample liquid are attracted by oppositely charged ions from the solid side to the interface to form the electric double layer. Figure 3.6(b) displays the distribution of potential ψ due to electric charge. The
potential value at the solid wall is ψ0. Outside the electric double layer, the potential ψ decays very fast and approaches zero rapidly. Figure 3.6(c) exhibits the velocity distribution within the electric double layer. A uniform velocity profile is presented in the region outside the electric double layer. Due to this feature, an alternative formulation for momentum equation is expressed as r ∂ V r r r ρ [ + (V ⋅ ∇ ) V ] = -∇ P + µ∇ 2V (3.17) ∂ t
with the bulk fluid velocity along the wall due to electroosmotic mobility expressed as: r r Veo = µ eo E (3.18)
r where µeo is the electroosmotic mobility. The velocity,Veo is imposed as slip velocity boundary condition along walls of the cross channel. The electroosmotic mobility is related to ζ potential with the equation (Probstein, 1994)
ψ β µ = 0 (3.19) eo 4πµ
where ψ0 is the potential at the solid-liquid interface and β is the dielectric constant. Since the electric body force term is no longer presented in the momentum equation, it is not necessary to r r resolve the thin electric double layer. With slip electroosmotic velocity (Veo = µ eo E ) boundary
condition imposed, relatively coarse mesh size can be implemented in the simulation while the 33 main physical characteristic of electrokinetic flow can still be predicted. In the present thesis, the momentum equation 3.17 with boundary condition 3.18 is adopted to solve for the bulk flow.
34
Start
Read data
Set constants
Initialize molecules and boundaries
Move molecules within ∆t : compute interactions with boundaries
Reset molecular indexing
Compute collisions
No Interval > ∆t ? Unsteady flow: repeat until required sample Sample flow properties obtained
No Time > t ?
Unsteady flow: average runs Steady flow: average samples after establishing steady flow
Print final results
Stop
Figure 3.1. Flow chart of DSMC procedure
35
N2 Twall = 300 K
d
L
Figure 3.2. Schematic of single component gas flow in microchannel
3 Analytical solution (α = 0.85) (Arkilic et al., 1994)
DSMC result(α =0.85) 2.75 DSMC result(α =1.0)
2.5
2.25
2 P*(X*)
1.75
1.5
1.25
1 0 0.25 0.5 0.75 1 X*
Figure 3.3. Comparison of axial pressure distribution for channel flow (with slip-walls) 36
1
DSMC result 0.9 Experimental data[19] 0.8 Linear pressure
0.7
0.6
0.5
0.4
0.3 (P(x)-Pout)/(Pin-Pout)
0.2
0.1
0.25 0.5 0.75 1 X/L
Figure 3.4. Comparison of numerical results with experiment data of pressure distribution along the channel 37
3 buffer
1 2 h Injection channel W sample h
y Separation channel x
4 buffer
H
Figure 3.5. Cross-channel device for electrophoresis separation
liquid y y
Electric double layer
ψ0 ψ solid 0 0 (a) (b) (c)
Figure 3.6. Description of electric double layer (a) formation of electric double layer (b) potential ψ distribution within the electric double layer (c) velocity distribution within the electric double layer 38
4. LOW PRESSURE GAS FLOW AND HEAT TRANSFER IN DUCTS1
4.1. Introduction
Heat transfer in developing flows in channels and tubes has been intensely studied in the past (Kakac and Ozgu, 1969) due to their practical use as well as their theoretical importance. In the combined entrance region, simultaneous development of the hydrodynamic and thermal boundary layers have to be considered. Most of the past studies considered continuum formulations, which is valid for low Knudsen numbers, Kn < 0.01. Due to a variety of novel applications such as micro-electromechanical systems (MEMS), low-pressure chemical vapor deposition (CVD) reactors, etc., the transport problem in the non-continuum (Kn > 0.01) regime has generated renewed interests in the above problem (Oh et al., 1997; Nanbu et al.,
1991). There are two distinct classes of high Knudsen number non-continuum flow problems - one due to micro length-scale and the other due to large mean free path (very low pressure) of the gas molecules. Both classes of problems offer unique challenges in obtaining their solutions. Though approximate techniques have been attempted, particle methods such as molecular dynamics (MD), particle-in-cell (PIC) and the direct simulation Monte Carlo (DSMC) are attractive tools for the study of such flows.
Traditional continuum computational fluid dynamic (CFD) techniques are often invalid for analyzing flows in MEMS or in low-pressure devices. This inaccuracy stems from their calculation of molecular transport effects such as viscous dissipation and thermal conduction from bulk flow quantities such as mean velocity and temperature. This approximation of continuum phenomena fails as the characteristic length of the flow gradients (L) approaches the average distance traveled by molecules between collisions (mean free path, λ). The direct simulation Monte Carlo (DSMC) method is a well-established approach that has been used widely and successfully to simulate high Kn number gas flow problems (Bird, 1994). Several numerical studies using the DSMC method (Beskok and Karniadakis, 1994; Piekos and Breuer,
1 The results presented in this chapter appeared in ASME Journal of Heat Transfer, Vol. 124, No. 4, pp. 609-616, Yan and Farouk, 2002 39
1995; Oh et al., 1997; Oran et al., 1998) have been reported. In this chapter, the fluid flow and
heat transfer for high Kn in a partially heated duct is investigated using DSMC. The objective of
the simulation is to study the effect of Kn, Re and Pr on heat transfer from the heated walls
under the rarefied flow conditions. Specifically, subsonic pressure-driven flows are considered.
The predicted heat flux results are compared with the Graetz solution with uniform flow inlet. A
large number of cases are calculated with varying inlet Kn, Re and Pr from the slip-flow regime
to the transition-flow regime. Based upon these simulations, a simple correlation connecting Nu
with Pe and Kn is developed.
4.2. Problem description
Rarefied subsonic gas flows between parallel plates (60 cm long and 10 cm apart) as shown in Figure 4.1 is considered. Molecules come in from the left side and go out through the right side. A portion of the plates (20 cm) at the entrance region is unheated, so that the flow can develop. In the next portion, the walls are heated and held at constant temperature. Also, this configuration resembles that of the classical Graetz problem. The flow fields are computed for specified values of pressure at the inlet and the outlet. The inlet velocity is determined according to the information from the calculated domain rather than being specified. The inlet temperature is 300 K for all cases. The temperature of the heated walls is held at 600 K.
Computations were carried out for nitrogen, argon, hydrogen, oxygen and noble gas mixture flows. Results were obtained for a relatively wide range of Re, Kn and Pr. It is interesting to note here that for the highly rarefied conditions and the non-isothermal flow fields considered, the parameters, Kn, and Re vary substantially along the flow directions. For reporting the results, axially averaged values, Re, and Kn are considered. The values of Pr (for the cases reported here) do not vary appreciably along the length of channels.
The thermal boundary conditions were chosen following the Graetz problem (1883).
Unlike the Graetz solution, only developing flows are considered here. However, due to the other similarities, the present heat transfer predictions are compared with those obtained from 40 the Graetz solution with either parabolic and uniform velocity profiles. It has been shown that molecular gas flow characteristics in the channel are essentially the same when inlet Kn is the same and the boundaries are geometrically similar (Marvriplis et al., 1997). This means that the characteristic properties of flow are independent of the size of channel if the same inlet Kn and
-1 boundary conditions are maintained. A fairly large cross section (O[10 ]m) channel is considered but at lower inlet pressures (about 7 Pa) instead of the atmospheric pressure and micron-sized cross sections found in microchannels.
4.3. Model description
The direct simulation Monte Carlo (DSMC) method described in Chapter III was used to obtain the density, pressure, velocity and the temperature fields in the channels. The DSMC method retains its validity at high Kn because no continuum assumptions are made. In the
DSMC method, a real gas is simulated by thousands or millions of simulated particles. The positions, velocities and initial states of these simulated particles are stored and modified in time in the process of particles moving, colliding among themselves, and interacting with boundaries in the simulated physical space. Each simulated particle represents a very large number of physical molecules. In this fashion, the number of molecular trajectories and molecular collisions that must be calculated is substantially reduced, while the physical velocities, molecular size and internal energies are preserved in the simulation. Further, the DSMC method uncouples the analysis of the molecular motion from that of the molecular collisions by use of a time step smaller than the real physical collision time. The DSMC method applied to low-pressure fluid flow simulation is intuitively attractive because it is valid for all flow regimes.
Generally, in the DSMC framework, the flows with high velocities are much easier to compute than flows with low velocity because of the statistical scatter (Bird, 1994). In DSMC, the computational time step must be less than the mean collision time. Another condition that must be satisfied during a DSMC procedure is that the smallest dimensions of the computational cells must not be greater than one-third of the mean-free path. In addition, the importance of 41
statistical scatter caused by small perturbation increases as the flow velocity becomes smaller.
For the present problem, the thermal speed is approximately 500 m/s. For the flow speeds
considered here (10 m/s ~ 100 m/s), the statistical noise is thus significant. However, as the
sample sizes from the ensemble or time average increase, the correct result will in principle
emerge. The statistical fluctuations decrease with the square root of the sample size. The
sample sizes used in the present study is of the order of 106. The accuracy of the calculation presented is thus acceptable (Bird, 1994) for the large sample size considered.
Two dimensional subsonic flow simulations (considered in the present study) by the direct simulation Monte Carlo technique in the slip and transitional regimes are rare (Nanbu et al., 1991). One of the difficulties lies in specifying the outlet boundary conditions where particles may move upstream and reenter the flow domain. In hypersonic flows or free molecular flows, we can neglect the particles re-entering from the outlet boundary. In subsonic flows in the transitional regime, we cannot neglect particles which go out of the boundary and may reenter.
Nanbu et al. (1991) applied a porous wall at the inlet of a duct to prevent molecules from going upstream. Another difficulty in two dimensional subsonic-flow particle simulations in transitional regime is in controlling the flow velocity inside the domain. If particles do not re-enter from the exit (assuming expansion to vacuum), they are accelerated to the exit and velocities inside the domain do not stay uniform.
In the present calculations, the inlet and outlet boundary conditions were formulated following the procedure outlined by Piekos and Breuer (1995,1996). The inlet and outlet pressures are considered to be specified. This pressure drop drives the gas flow through the channel. At the inlet boundary, temperature is kept constant (300 K), and the density is specified (via ideal-gas law) according to the given inlet pressure. The transverse velocity of flow is set to zero. At the outlet, pressure is specified and zero gradient boundary conditions are considered for temperature, transverse velocity and streamwise velocity. Only the number density is calculated according to the given outlet pressure.
42
4.4. Results and discussion
In this section, results are presented for the low-pressure subsonic flows in a partially
heated duct, as shown schematically in Figure 4.1. A range of Kn (from 0.01 to 3.0) was considered for the simulations. The variation of inlet Kn was implemented by the change of the inflow number density. In addition to nitrogen, argon, hydrogen and oxygen, different noble gas mixtures are used for the simulations. Since mixing helium with other noble gases can change the mixture Pr significantly, simulations are carried out with such gas mixtures. The mixtures considered are He-Xe (62%/38%), He-Kr (60%/40%), He-Ar (58%/42%) with Pr 0.18, 0.23 and
0.39 respectively (Wetzel and Herman, 1997). The plates are considered as diffuse reflectors with full thermal and momentum accommodation. Experiments with ‘engineering’ surfaces in contact with gases at normal temperatures indicate that the reflection process approximates diffuse reflection with complete thermal accommodation (Bird, 1994).
Viscosity and thermal conductivity of the gases at low pressure are calculated using the kinetic theory for pure gases (Vincenti and Kruger, 1965). Viscosity and thermal conductivity both increase as temperature increases while the specific heat changes negligibly with the variation of temperature for the cases considered. The Prandtl number Pr, thus is found to be almost independent of temperature for the cases studied here. For noble gas mixtures, the above properties are calculated following Giacobbe (1994).
Calculations were carried out for a large number of cases. The Re, Kn and Pr were systematically varied to obtain detailed information about the flow fields and to estimate the Nu .
Tables 4.1, 4.2 and 4.3 list the cases considered in the study. Table 4.1 lists the cases studied
with inlet pressure of 6.7 Pa and outlet pressure of 5.3 Pa. Tables 4.2 and 4.3 list cases studied
with inlet pressures of 0.67 and 0.067 Pa and outlet pressures of 0.53 and 0.053 Pa
respectively. In addition to the parameters of the cases, the tables also contain the computed
values of Re, Kn and Nu . In the problem formulation, the coordinate origin was set at the transition point between the unheated section and the heated section. That is, for x < 0, the wall temperature is 300 K, and for x ≥ 0, the wall temperature is 600 K. Hence Nu , Re and Kn are 43
the average values obtained from the heated section since the fluid flow and heat transfer in this
section are of most interest.
The Kn in the tables ranges from 0.01 to 3.0 to cover both slip-flow and transitional flow regime for all gases and gas-mixtures studied. The computed Re is found to decrease with the
increase of the Kn number. This is an expected result. Also under the conditions of same inlet and outlet pressure, the Kn decreases for molecules with larger diameters.
The simulations were done for nitrogen, argon, hydrogen, oxygen and noble gas mixtures, where the inlet Kn was varied systematically over two orders of magnitude. In case 1
(the base case), a slip flow regime with nitrogen is considered. The calculations in this study were carried out with a mesh size that met the general requirement that were discussed by Bird
(1994). As described earlier, every cell is subdivided into two sub-cells in each direction. When the temporal variation of the outlet velocity profile is small enough, (the maximum changes of the outlet velocity at all exit locations less than 1%), the flow is assumed to have reached the steady state condition. The time step used is less than one-third of the typical particle mean collision time (Bird, 1994). The ‘variable hard sphere’ (VHS) model is used to account for the collisions (Bird, 1994).
The calculations for the base case were repeated with different mesh size to establish the grid independency of the results presented. First, computations were made with 420 x 100 cells. Computations were then carried out with 350 × 80 and 200 × 60 mesh sizes. A comparison of the temperature profiles across the duct at point x = 0 (the transition point from the unheated section to the heated section) and at point x = 0.4 are shown in Figures 4.2a and
4.2b respectively for the three mesh sizes considered. There was no significant change in the results for the mesh sizes 420 × 100 and 350 × 80. The mesh size (350 × 80) was considered adequate for the simulations as case 1 gives the highest Re.
The effect of the initial ‘particle numbers per cell’ on the simulation results was also investigated. Figure 4.3 shows the temperature profiles obtained with different particle numbers for the base case with a mesh size of 350 × 80. Results were obtained with 60, 50 and 40 44 particles per cell. The variation in the predicted temperature profiles was small. 50 particles per cell are used for the cases studied. All computations were performed on an IBM-RISC-6000
(Model 360) workstation. Typical computation time ranged from 20 to 30 hours for a case.
Figure 4.4 shows the temperature contours in the flow domain for the base case. In some regions of the unheated section, temperature is a little higher than 300 K due to molecular diffusion. In the heated section, temperature gradually increases due to the heating and the exit gas temperature is almost equal to the wall temperature. Figure 4.5 shows the velocity vectors for the base case. Since the Kn is fairly low for this case, the velocity profiles are parabolic as expected. However, slight slip can be observed in the present results. The slip magnitude of velocity is about 7 m/s at the inlet and 23 m/s at the outlet. The Mach number at the outlet for this case is 0.2.
In addition to using the VHS model, the base case calculations were repeated with the
VSS (variable soft sphere) model. The variation of pressure along the axis (transversely averaged) is shown in Figure 4.6 for both the VHS and VSS models. The difference between the predictions is very small. It is noted that a significant nonlinear characteristic in the predicted pressure distribution. It is interesting to note that the slope of the pressure distribution increases at x = 0.0. This is caused by the introduction of the heated section in the channel. It is well known that pressure is proportional to temperature and number density. The number density decreases along the channel while temperature increases for x > 0.
Heat flux results from the DSMC simulation (for the base case) are next compared with the well-known Graetz solution (Graetz, 1883) with both ‘parabolic flow’ and ‘uniform flow’ conditions. Wall heat fluxes are calculated from the differences between the energies of the impinging particles and the reflected particles. Figure 4.7 shows the heat flux variation along the flow direction. The results from the DSMC simulation and Graetz solution agree qualitatively in the heated region. Negative heat fluxes (heat transfer from the gas to the wall) are predicted for some portion of the unheated walls. This shows heat conduction due to molecular diffusion is not negligible and the unheated sections of the walls cool the heated particles. Additional 45
differences between the DSMC results and the Graetz solution exist. These differences are
expected as the Graetz solution is valid for fully developed incompressible continuum flows
whereas rarefied developing flow conditions are considered in the present case. Although the
velocity at the heated section is more parabolic than uniform (see Figure 4.5), the Graetz
solution for heat flux with ‘uniform velocity’ agrees better with the DSMC results than the
Graetz solution with ‘parabolic velocity’. This is because of the presence of slip in the DSMC
results. It is well known that slip at the wall can enhance the heat transfer. The Graetz
solutions do not provide exact validation because the DSMC computations cannot be performed
under identical conditions to those assumed in deriving the analytic solution.
The heat transfer coefficient h(x) is calculated as
q wall hx()= (4.1) Txwb()− Tx ()
where Tw(x) and Tb(x) are the local wall and bulk-mean temperatures respectively. Figure 4.8 shows the variation of local Nu(x) [= h(x)d/κ(x)] along the axial direction for the heated portion of the walls. Nu varies significantly in the heated section of the flow. Since the temperature increases along the channel, thermal conductivity increases accordingly. Thus the variation of
Nu(x) is due to both the heat flux and thermal conductivity variation along the flow direction.
Significant variations are also observed in the variation of local (transversely averaged)
Kn(x) and Re(x). The axial variation of Kn(x) and Re(x) for the base case are shown in Figures
4.9a and 4.9b respectively. Due to the sharp drop of density in the axial direction, Kn increases rapidly in the heated section while Re decreases on the other hand. It is seen from Figure 4.9a that most of the flow is in the slip-flow regime. Although velocity and viscosity increase along the channel, the rapid decrease of density causes the Re to drop along the channel.
Now results of nitrogen flows in the transition flow regime are presented (case 8 in
Table 4.2). The inlet pressure here is 0.67 Pa, and the outlet pressure is 0.53 Pa to give a pressure ratio of 1.27. For this high Kn flow (inlet Kn = 0.08, outlet Kn = 0.22), diffusion is much more pronounced. This can be seen clearly from the temperature contours (Figure 4.10). 46
Figure 4.11 shows the velocity vectors as predicted by DSMC calculations for case 8. The
velocity gradient across the channel width is smaller. The amount of velocity slip at the wall is
about 13 m/s near the inlet and 30 m/s at the outlet, which are large, compared to those for the
base case. Because of the very low density, the Re is extremely low. In the transition area
between the unheated and the heated regions, a strong nonlinear pressure distribution is again
observed (Figure 4.12). It can be seen from Figure 4.12 that the pressure rises to a local
maximum near x = 0 due to the rapid increase of temperature. This is because the diffusion
effect is more pronounced here compared to the base case.
The heat flux along the length for case 8 is shown in Figure 4.13. The predictions are
also compared with the Graetz solution with ‘parabolic velocity’. The heat transfer is highest at
the channel transition region where the temperature difference is the largest. Compared to the
base case, the heat flux values along the wall are considerably smaller for case 8. The Graetz
solution is again found to qualitatively agree with the DSMC prediction of the heat flux. The
variation of the local Nu along the axis for case 8 is shown in Figure 4.14. Compared with the
former slip flow case (base case), the variation of Nu near the origin is relatively smaller in the
transitional flow case. It shows that molecular diffusion in transition flow is dominant.
Figure 4.15a shows the variation of Kn along the axial direction for case 8 for the heated section. Changes in both pressure and temperature contribute to the increase of velocity in the axial direction. Figure 4.15b shows the axial variation of Re along the axis. In this case, towards the end of the channel, the temperature of the fluid approaches the wall temperature.
The statistical scatter of temperature thus becomes more pronounced. The temperature fluctuation induces the fluctuation shown in Re in Figure 4.15b.
The effect of diffusion is found to be even stronger in case 15. Simulations for similar conditions (Cases 1, 8 and 15) were repeated for different gases such as argon, hydrogen, oxygen and noble gas mixtures, as shown in Tables 4.1, 4.2 and 4.3. Comparing the results for cases 1, 2, 3 and 4 (Table 4.1, it can be seen that for similar values of Kn, the diatomic gases 47
(nitrogen, oxygen and hydrogen) have lower values of Nu compared to that of Ar. Nu again
decreases with increasing Kn for all gases considered.
Noble gas mixtures were used in order to study the effect of Pr on heat transfer for the channel flows in low pressure. The Pr is 0.67 for all noble gases (at the reference temperature of 293 K). However, certain properties of the gaseous mixtures are not necessarily functions that vary linearly with subcomponent mole fractions. For example, the viscosities of some binary mixtures vary oddly with mixture composition. A variety of intermolecular attractive and repulsive forces that exist between real gas molecules must be considered. The Pr of gas mixtures were calculated following Giacobbe (1994). For the noble gas mixtures considered, the
Pr varied from 0.18 to 0.39 (see Tables 4.1, 4.2 and 4.3). Compared with the values of Nu
predicted for Argon, the Nu values for noble gas mixtures are found to be smaller. Low Pr of the noble gas mixtures thus reduces the values of Nu .
A simplified expression for predicting Nu as a function of Pe and Kn was obtained
following the correlations given by Graetz where Peclet number, Pe = Re⋅Pr. Using a linear
regression technique, the following form is obtained from the data shown in Tables 4.1, 4.2 and
4.3:
072. Nu=+054.ln(..) Pe 778 05 Kn (4.2)
A parity plot showing the degree of correlation of the computed data is shown in Figure 4.16
with a confidence level of 85%. That is, the maximum relative error between the correlation and
the computed data is less than 15%.
The coefficient of Kn is positive which indicates that for a given Pe, Nu increases as the
Kn increases. However, it must be pointed out that for most gases, Pe usually decreases as Kn
is increased. Since the additive constant in the above correlation is large (7.78), the effect of
Kn is rather weak on Nu . Since the correlation is based on a data set that only contains low values of Re, the continuum correlation given by Graetz for uniform inlet flow is not recovered by setting Kn Æ 0. Comparing the present correlation with that for Graetz (Kakac, 1969), the 48
exponential coefficient (0.72) of Pe is found to be larger than its continuum counterpart (0.33).
Hence, for low-pressure flow the effect of Pe on convection heat transfer becomes more pronounced.
49
Table 4.1. Values of Nu , Re, Kn and Pr for Pin = 6.7 Pa and Pout = 5.3 Pa
Case Gas Molecular Pr Knin Re Kn Nu
D × 1010 m
1 N2 4.17 0.68 0.008 11.22 0.016 4.646
2 Ar 4.17 0.667 0.008 17.26 0.016 4.977
3 H2 2.92 0.70 0.0163 2.536 0.037 2.623
4 O2 4.07 0.65 0.0084 9.483 0.018 3.012
5 He-Xe(68%-- 3.42 0.18 0.0119 15.83 0.024 2.843
32%)
6 He-Kr(60%--40%) 3.30 0.23 0.0128 10.02 0.029 2.080
7 He-Ar(58%--42%) 3.10 0.39 0.0145 6.484 0.033 2.610
Table 4.2. Values of Nu , Re, Kn and Pr for Pin = 0.67 Pa and Pout = 0.53 Pa
Case Gas Molecular Pr Knin Re Kn Nu
D × 1010 m
8 N2 4.17 0.68 0.08 0.614 0.178 0.822
9 Ar 4.17 0.667 0.08 0.974 0.177 1.029
10 H2 2.92 0.70 0.163 0.191 0.358 0.575
11 O2 4.07 0.65 0.084 0.524 0.195 0.882
12 He-Xe(68%--32%) 3.42 0.18 0.119 0.947 0.241 0.521
13 He-Kr(60%--40%) 3.30 0.23 0.128 0.620 0.293 0.433
14 He-Ar(58%--42%) 3.10 0.39 0.145 0.426 0.332 0.432
50
Table 4.3. Values of Nu , Re, Kn and Pr for Pin = 0.067 Pa and Pout = 0.053 Pa
Case Gas Molecular Pr Knin Re Kn Nu
D × 1010 m
15 N2 4.17 0.68 0.8 0.067 1.619 0.112
16 Ar 4.17 0.667 0.8 0.105 1.606 0.139
17 H2 2.92 0.70 1.63 0.027 3.276 0.052
18 O2 4.07 0.65 0.84 0.061 1.789 0.091
19 He-Xe(68%--32%) 3.42 0.18 1.19 0.098 2.288 0.076
20 He-Kr(60%--40%) 3.30 0.23 1.28 0.069 2.729 0.065
21 He-Ar(58%--42%) 3.10 0.39 1.45 0.050 3.069 0.062
0.1m
0.2m 0.4m
Y
X
Figure 4.1. Schematic of the problem geometry
51
380 420 X 100 cells 350 X 80 cells 370 200 x 60 cells
360
350
T(K) 340
330
320
310
0 0.025 0.05 0.075 0.1 Y(m)
Figure 4.2a. The effect of cell size on temperature profile at x = 0 .0 m (50 particles / cell)
590 420X100cells 350 X 80 cells 580 200 X 60 cells
570
560 T(K)
550
540
530
520 0 0.025 0.05 0.075 0.1 Y(m)
Figure 4.2b. The effect of cell size on temperature profile at x = 0.4 m (50 particles / cell)
52
380 6 0 particles/cell 370 50 particles/cell 40 particles/cell
360
350
T(K) 340
330
320
310
300 0 0.025 0.05 0.075 0.1 Y(m)
Figure 4.3. The effect of particle number on temperature profile at x=0 (350 × 80 cells)
Level T (K) 9571.4 7510.4 5449.3 3388.2 1327.2
9 1 3 5 7
-0.1 0.0 0.1 0.2 0.3 X
Figure 4.4. Temperature contour for the base case 53
80 m/s
-0.1 0 0.1 0.2 0.3 X(m)
Figure 4.5. Velocity vectors for the base case
6.6 VHS model result
VSS model result 6.4
6.2
6
Pressure (Pa) 5.8
5.6
5.4
0.1 0.2 0.3 0.4 0.5 X(m)
Figure 4.6. Pressure distribution along centerline for the base case 54
700 DSMC result Graetz solution 600 (uniform inlet velocity) Graetz solution 500 (parabolic inlet velocity) ) 2 400
300
200
100 Heat Flux (W/m 0
-100
-200
-300 -0.1 0 0.1 0.2 0.3 X(m)
Figure 4.7. Heat flux along the plate for the base case
10
9
8
7
6
Nu 5
4
3
2
1
0 0.1 0.2 0.3 X(m)
Figure 4.8. Mean Nu as a function of position along the plate for the base case 55
0.02
0.018
0.016
Kn 0.014
0.012
0.01
0.008 0.1 0.2 0.3 X(m)
Figure 4.9a. Kn as a function of position along the plate for the base case
14
13
12 Re 11
10
9
8 0.1 0.2 0.3 X(m)
Figure 4.9b. Re as a function of position along the plate for the base case
56
Level T(K) 9 570.3 7 510.3 5 450.3 3 390.4 1 330.4
5 7 1 3 9
-0.1 0 0.1 0.2 0.3 X(m)
Figure 4.10. Temperature contours for transition flow (case 8)
40 m/s
-0.1 0 0.1 0.2 0.3 X(m)
Figure 4.11. Velocity vectors for transitional flow (case 8) 57
0.66
0.64
0.62
0.6
Pressure(Pa) 0.58
0.56
0.54
-0.1 0 0.1 0.2 0.3 X(m)
Figure 4.12. Pressure distribution along the centerline for transitional flow (case 8)
100
DSMC result 80 Graetz solution (parabolic inlet velocity) 60 ) 2 40
20
0
Heat Flux (W/m -20
-40
-60
-0.1 0 0.1 0.2 0.3 X(m)
Figure 4.13. Heat flux along the plate for transition flow (case 8) 58
1.6
1.4
1.2 Nu 1
0.8
0.6
0.4 0.1 0.2 0.3 X(m)
Figure 4.14. Local Nu as a function of position along the plate for transitional flow (case 8)
0.22
0.21
0.2
0.19
0.18
0.17 Kn
0.16
0.15
0.14
0.13
0.12
0.1 0.2 0.3 X(m)
Figure 4.15a. Kn as the function of position along the axis for transitional flow (case 8) 59
0.72
0.7
0.68
0.66
Re 0.64
0.62
0.6
0.58
0.56
0.1 0.2 0.3 X(m)
Figure 4.15b. Re as a function of position along the plate for transitional flow (case 8)
100 Nu
10-1
10-1 100 0.54Pe0.72ln(7.78+0.5Kn)
Figure 4.16. Parity plot showing correlation of computed data
60
5. MIXING GAS FLOWS IN A MICROCHANNEL2
5.1. Introduction
During the past decade, microelectromechanical systems (MEMS) have become important
in many disciplines (Robinson et al., 1995; Koester et al., 1996) because of their extraordinary
advantages for practical usage. These devices such as microactuators, microsensors,
microgenerators, etc. (Wise, 1996) are very small (usually micron-sized) and are manufactured with the techniques developed from those used for integrated circuits. The application of MEMS have expanded rapidly into such fields as instrumentation (Gass et al., 1993), microelectronics
(Koester et al., 1996), bioengineering (Lammerink et al., 1993), and resulted in remarkable
contributions to the development of advanced technology. In order to optimize the design of some
MEMS application, the analysis of fluid flow, heat transfer and mixing in narrow microchannels with
dimensions ranging from tenth to hundreds of microns is critical (Ho, 1998). It is absolutely
necessary to understand these phenomena to develop this relatively new technology. Both
experimental (Harley et al., 1995) and numerical efforts (Piekos and Breuer, 1996; Oh et al., 1997;
Beskok et al., 1994) to study flows in microchannels have been reported in the literature.
Microscale gas flow phenomena are quite different from their counterparts in meso- and
large-scale situations and pose unique challenges in their study. Unfortunately, traditional
continuum CFD (computational fluid dynamic) techniques are often invalid for analyzing
microchannel flows. This inaccuracy stems from the calculation of molecular transport effects such
as viscous dissipation and thermal conduction from bulk flow quantities such as mean velocity and
temperature. The approximations for continuum flow analysis fail for microscale flows as the
characteristic length of the flow gradients (L) approaches the average distance traveled by
molecules between collisions (mean free path λ). The ratio of these quantities is known as the
2 The results presented in this chapter appeared in Microscale Thermophysical Engineering, Vol. 6, No. 4, pp. 235-251, Yan and Farouk, 2002
61
λ Knudsen number ( Kn = ) and is used to indicate the degree of flow rarefaction or scale of L
the flow-problem. The Navier-Stokes equations ignore rarefaction or microscale effects and is
therefore only strictly accurate at a vanishingly small Kn.
Different methods (Barron et al., 1997; Oran et al., 1998) can be used to model the
flows for different regimes for rarefied or microscale gas flows. The direct simulation Monte
Carlo method (DSMC) is a well-established alternative (Bird, 1994) to traditional CFD technique
which retains its validity at high Knudsen number as no continuum assumption is invoked in
DSMC. In a conventional DSMC method, each simulated particle represents a very large
number of physical molecules. In this fashion, the number of molecular trajectories and
molecular collisions that must be calculated is substantially reduced, while the physical
velocities, molecular size and internal energies are preserved in the simulation.
Mixing is an area of intense research in continuum fluid flow and heat transfer.
Adequate mixing is an essential requirement for propulsion devices designed to provide
chemical reactions and heat release in streams of gas and fuel. For this reason, the study of the
mixing process is a necessary prerequisite to setting criteria for the control of energy release in
reaction chambers. Both supersonic and subsonic shear layers have been studied intensely
(Farouk et al., 1991). However, similar problems in microchannels have not been investigated.
With the development of MEMS-assisted fuel injectors and other propulsion devices being
examined for aerospace applications, an understanding of the behavior of mixing in the micro-
scale counterpart can provide tools to facilitate the design and optimization of such innovative
devices.
In this chapter, the gas flow and heat transfer are investigated for high Kn number flow in a
microchannel using the DSMC method. The basic characteristics of the mixing process are shown
for flow in a microchannel. H2 and O2 gases with different inlet pressures as well as with different streamwise velocities were used for our computations. The length required for the two gases to be fully mixed was determined by examining the predicted density field. Mixing problems with different 62
inlet pressure ratios, different inlet pressures and different solid wall accommodation coefficients
were calculated. The effects of these parameters on the mixing behavior are analyzed.
5.2. Problem description
The schematic for mixing of two different gases in a microchannel is illustrated in Figure
5.1. The wall temperatures are kept as 300 K for all cases. Hydrogen and oxygen streams with
different inlet pressures and different inlet streamwise velocities were considered. The inflowing
lower stream was H2 at a higher velocity than the inflowing O2 stream. Before the mixing chamber,
a splitter with length 7.5 µm is placed as indicated in Figure 5.1 to form two leading inlets for both
inflow streams respectively. The width of the channel was chosen as 1.0 µm and the length was
32.0 µm. Both the wall and environment temperatures were set to be 300 K. The outlet pressure
was kept as 50 kPa.
5.3. Model description
The DSMC method is based on the kinetic theory of gases. The DSMC method retains
its validity at high Kn number because no continuum assumptions are made.
The most typical microchannel flow is the pressure driven channel flow (Piekos and
Breuer, 1996). Pressure difference between inlet and outlet drives the gas flow through the
channel. Eventually steady flow states are established inside the channel. In the DSMC framework,
the computational time step must be less than the mean collision time. Another condition that must
be satisfied during the DSMC procedure is that the smallest dimensions of the computational cells
must not be greater than half of the local mean free path based on the localized flow gradients. In
addition, the importance of statistical scatter caused by small perturbation increases as the flow
velocity becomes smaller. However, as the sample sizes from either the ensemble or time average
increase, the noise decreases with the square root of the sample size. The sample sizes used in
the present study is of the order of 107. The accuracy of the calculation presented is acceptable
(Bird, 1994) for the large sample size. All computations were performed on an IBM RS/6000 63
enterprise server S80 (model 7017). The CPU time for one typical case is about 90 hours. In the
present calculations, every cell is subdivided into two subcells in each direction. The time step is
chosen to be less than one-third of the typical particles mean collision time. The variables hard
sphere (VHS) model is used to model the collision because it has been widely used in both slip
flow and transition flow simulations (Oh et al., 1997; Piekos and Breuer, 1996). The gas properties
used in calculations such as molecular diameter, molecular mass, viscosity-temperature index and
reference temperature can be obtained according to procedures outlined in Bird (1994).
According to Piekos and Breuer (1996) and Piekos (1995), at the inlet boundary, temperature is kept constant (in our case always at 300 K), the density is specified according to the given inlet pressure, and the transverse velocity of flow is set to zero while the streamwise velocity is determined according to the streamwise velocity inside the channel. For the mixing problem
(Figure 5.1), because the inlet streams are separated by the splitter, different pressures are set at the inlet interfaces. Also, the incoming streamwise velocities for each species are calculated individually. At the exit face, temperature, transverse velocity and streamwise velocity are all set to be the same values as those calculated in the last cell center inside the domain respectively. Only the number density is determined according to the given outlet pressure. For most calculations in this chapter, the solid walls are all considered as diffuse reflectors with thermal and momentum accommodation coefficient α = 0.85. This is the typical coefficient used in past microchannel flow studies (Piekos and Breuer, 1996). Finally, in order to investigate the effect of accommodation coefficients on the mixing flow α is varied from 0.01 (nearly specular reflection) to unity (completely diffuse reflection). At the start of a calculation, all particles are initialized with the temperature 300 K and zero mean velocity everywhere.
5.4. Results and discussion
The mixing flows in a microchannel were simulated for two parallel streams of hydrogen
and oxygen entering the microchannel with different inlet pressures and different inlet streamwise
velocities (see Figure 5.1). Due to much smaller molecular mass in the pressure driven flow, the 64
inflowing lower stream H2 is at a higher velocity than the inflowing O2 stream (as listed in Table
5.1). The streamwise velocity of H2 is more sensitive to the pressure variation owing to the same
reason. Both the wall temperature and environment temperatures are set to be 300 K. The outlet
pressure was held at 50 kPa.
The computations performed can be grouped into three categories:
(a) Equal inlet pressure cases: both streams are at the same inlet pressure, calculations were
performed by systematically varying the inlet pressure.
(b) Variable inlet pressure of H2 stream: inlet conditions of O2 stream are kept constant. The inlet
pressure of H2 stream was varied to investigate the effect of different pressure ratios on the
mixing process.
(c)Variable accommodation coefficient: the accommodation coefficient α of the solid walls is
changed from 0.01 to one to investigate the effect of wall reflection on mixing.
Tables 5.1a – 5.1c list the mixing flow cases calculated in the present study. For the base case, H2 flows into the microchannel with an average velocity 60.6 m/s at the bottom half part of the
channel and O2 flows at the upper half part with an average velocity of 19.2 m/s. The inlet
pressures of H2 and O2 are both 200 kPa and the exit pressure was held at 50 kPa. The channel length was long enough (determined via trial calculations) for H2 and O2 to mix completely before they left the channel. Thus the outlet Kn number for base case was equal to 0.157. Because of the existence of the splitter at the inlet interface, the inlet Knin number is defined as the Kn at the
entrance to the mixing chamber (x = L1) instead of inlet x = 0. The flow regime is changing from slip flow at the inlet to transition flow at the outlet. The calculations for the base case were repeated with different mesh sizes to establish grid independent results. The cell size is uniform in y direction but non-uniform in x direction. First, 600 × 80 cells were used. Computations were also carried out with 750 × 100 and 360 × 50 mesh sizes. A comparison of the streamwise velocity profiles across the channel at inlet for the base case is shown in Figure 5.2. The differences between the results for the mesh size 750 × 100 and 600 × 80 are small. This indicates that the original choice of mesh 65 size (600 × 80) is adequate. After the fluid flow reached steady state, there were 2,448,000 particles contained in the flow domain.
Figures 5.3a and 5.3b show the computed mass density contours for the base case. In the figures, the dash dot line indicates the position of splitter, the vertical coordinate has been expanded for better view by a factor of 4.0 due to the large aspect ratio of the channel. The density contours along the channel are plotted in Figure 5.3a from which the mixing process after entering the duct can be seen. Although the inflowing number densities of the two streams are the same, because of the large mass difference between H2 and O2, the mass density values at the inlet region show significant variation. Due to the molecular diffusion, the two different gases eventually mix completely with each other. The detail of the mass density contours near the inlet region is shown in Figure 5.3b. Here the ‘mixing length’ is defined as the length of the microchannel required for the two different gases to be fully mixed. Conceptually, the mixing length can also be characterized as the length where the diffusion velocity is nearly zero. The diffusion velocity is defined as the relative velocity of a particular species mean velocity to the mass average velocity
(Bird, 1994). However, due to the inherent statistical nature of the present calculations, it is not easy to determine the mixing length accurately from the information of diffusion velocity. Therefore, the ‘mixing length’ was determined from the computed mass density values in the flow domain. The mixing is assumed to be complete when the mass density contours across the channel are symmetric across the center line of the channel. From the mass density contours shown in Figures
5.3a and 5.3b, it can be determined that the mixing length is about 2.8 µm for the base case. The two dashed vertical lines in Figure 5.3b are used to determine if the density contours are symmetric about the centerline of the microchannel.
The mass averaged velocity vectors are presented in Figures 5.4a and 5.4b for the base case. In the Figure 5.4, the y coordinate is also stretched by a factor of 4.0 for better depict. Both streams enter the microchannel with the same pressure (200 kPa), but with different inlet velocities.
The O2 (upper) stream enters the microchannel with an average velocity of 19.2 m/s while the H2
(lower) stream enters with an average velocity of 60.6 m/s. The diffusion velocities of H2 and O2 at 66
the inlet are shown in Figures 5.5a and 5.5b. As shown O2 diffuses from the top part to the bottom
part and the H2 diffuses in the opposite direction. It is clear that the diffusion velocity is to the
reverse streamwise direction. When the mixing is complete, the diffusion velocities of both species
are nearly zero. It is noted that due to this diffusion process, premixing in the leading channels is
inevitable before the mixing chamber. Furthermore, because of larger incoming momentum, O2
diffusion velocity has more pronounced effect on the inflowing stream of H2. As seen from Figure
5.4, the streamwise velocity at the lower part is significantly slowed down near the entrance region
to the mixing chamber (from x = 6.0 µm to x = 7.5 µm). As long as the length of the splitter is long
enough (in our case L1 = 7.5 µm), the back flow due to diffusion is not important compared with the
main flow because the partial pressure of H2 at the top half part of inlet is negligible compared to
the incoming O2 pressure and vice versa at the bottom half part of inlet. The diffusion momentum of both H2 and O2 are less than 0.01 percent of the total flow momentum at the inlet section.
Therefore, the inlet boundary conditions can not be modified by this diffusion flow. It can be seen from Figure 5.4b that after a short distance (less than 0.8 µm), the mass averaged velocities of the gas mixtures become parabolic across the channel and the diffusion velocity of both H2 and O2
were nearly zero (Figures 5.5a and 5.5b). The mixing process shown by the density contours
(Figure 5.3b) indicate that the mixing length is about 2.8 µm, which is considerably higher than 0.8
µm. It can be concluded that the ‘velocity mixing’ process is much faster than its ‘mass mixing’
counterpart in the flow studied. Comparing Figure 5.5a with Figure 5.5b, it can be seen that the
diffusion effect of H2 is much stronger compared to that of O2. The diffusion velocities of O2 are non- zero only in the region very near the bottom half part of the entrance. It is because the momentum of O2 is much larger compared to that of H2 for the present case. It is noted that the mixing flow at
high Kn number is very different from its continuum counterpart. In microchannel mixing, the
diffusion process is significant in the mixing process while in continuum mixing shear layers form
discrete large-scale structures (Brown and Roshko, 1974).
Figure 5.6 depicts the predicted temperature contours for the base case. The y
coordinate is also stretched by a factor of 4.0. Although the initial inlet temperature and wall 67
temperatures are all set as 300 K, the microchannel flow cannot be regarded as isothermal.
Due to the effect of compressibility, both streams attain temperatures that are slightly lower than
the wall temperature. It is also seen in Figure 5.6, that as the fluid reaches the exit, a low
temperature region is gradually formed due to the acceleration of the gas mixture. For the same
reason, there is a small cool zone near the O2 inlet where the expansion is rapid.
Calculations were repeated by changing the inlet pressures of H2 and O2 while the
pressure ratio (P /P ) was kept equal to unity and the outlet conditions were unchanged H2 O2 inlet
(Table 5.1a). The pressure values of 100 kPa, 150 kPa and 300 kPa were used. The mixing length for each case was determined from the density contours as 2.0 µm, 2.5 µm and 3.0 µm
respectively. The variation of mixing length versus the inlet pressure is plotted in Figure 5.7. The
mixing length increases with the increment of inlet pressure monotonically. When the inlet-outlet
pressure difference is increased, the convection effect becomes more pronounced compared with
the molecular diffusion. Therefore, the ‘mixing length’ is longer for higher pressure cases to be
mixed completely.
Next the effects of different inlet pressure ratios on mixing is investigated. Two additional
cases (Table 5.1b) are calculated by changing the inlet pressure of H2 stream (to 100 and 400 kPa
respectively) while keeping other conditions unchanged. With the base case, the pressure ratio
varied as 0.5, 1.0 and 2.0. The variation of the ‘mixing length’ versus the inlet pressure ratio is
plotted in Figure 5.8. The mixing length increases as the inlet pressure ratio becomes higher. For
all of the three cases considered in Figure 5.8, the inlet momentum of O2 was larger than those of the inlet momentum of H2. When the pressure of H2 is increased, the momentum of the H2 stream
also increases while the inlet momentum of O2 decreases. H2 particles traveled longer distance in streamwise direction before penetrating the O2 stream.
Finally, the effect of the wall reflection conditions on the mixing process is studied by
changing the accommodation coefficient α from 0.01 (for nearly specular reflection) to unity (for
completely diffuse reflection). When α is changed from 0.85 to 1.0, the flow fields are similar and
the variation for the mixing length is very small, (2.7 µm compared to 2.8 µm). However, when a 68
nearly specularly reflective wall is considered ( α = 0.01 ), the mixing process is significantly
different. Figure 5.9a shows the mass averaged velocity vectors this case (Table 1d). In the Figure
5.9, the y coordinate is also stretched by a factor of 4.0. The velocity vectors along the channel
are basically flat rather than parabolic. Because of very small surface friction, both streams have
very large average inlet velocities with H2 at 303.2 m/s and O2 at 53.6 m/s. The slip velocities near the solid walls are also very large. The surface condition has significant influence on the microchannel flow. The effects on the velocity and temperature fields (due to the reflection conditions at the wall) can spread quickly into the main stream through molecular collisions. For nearly specular reflection at the solid walls, both streams accelerate rapidly. The mass density contours for the case with α = 0.01 is shown in Figure 5.9b. The mixing length for this case is 8.30
µm, which is almost three times as that for the base case. Additional cases (Table 5.1c) are calculated with different surface conditions and the relationship between the mixing length and accommodation coefficients is plotted in Figure 5.10. The slope of the curve near zero accommodation coefficient is very sharp. It indicates that the mixing process is very sensitive to the surface condition when specular reflection is approached. The gradient near complete accommodation region is, however, flat. It also can be seen from Table 5.1c that the inlet Kn decreases as the accommodation coefficient decreases. This is because when the surface friction is small, the pressure drop in the leading channels decreases accordingly. 69
Table 5.1a. Inlet and solid wall conditions for equal inlet pressure cases.
Case P (kPa) V (m/s) P (kPa) V (m/s) Kn H2,inlet H2 O2,inlet O2 in α
1 (base case) 200 60.6 200 19.2 0.058 0.85
2 100 35.7 100 12.4 0.104 0.85
3 150 45.6 150 15.9 0.067 0.85
4 300 75.2 300 24.8 0.049 0.85
Table 5.1b. Inlet and solid wall conditions for cases with variable inlet pressure of the H2 stream
Case P (kPa) V (m/s) P (kPa) V (m/s) Kn H2,inlet H2 O2,inlet O2 in α
1 (base case) 200 60.6 200 19.2 0.058 0.85
5 100 28.6 200 22.5 0.071 0.85
6 400 84.7 200 16.4 0.045 0.85
Table 5.1c. Inlet and solid wall conditions for cases with variable accommodation coefficients
Case P (kPa) V (m/s) P (kPa) V (m/s) Kn H2,inlet H2 O2,inlet O2 in α
1 (base case) 200 60.6 200 19.2 0.058 0.85
7 200 57.6 200 18.5 0.059 1.00
8 200 76.3 200 21.7 0.053 0.5
9 200 96.3 200 26.8 0.051 0.28
10 200 155.8 200 35.1 0.047 0.1
11 200 303.2 200 53.6 0.043 0.01
70
Twall = 300 K O2
L1 d
H2
L
Figure 5.1. Schematic of the mixing problem geometry
80
750X100cells 70 600 X 80 cells 360 X 50 cells 60
50
40
30 uvelocity(m/s)
20
10
0 0 2E-07 4E-07 6E-07 8E-07 1E-06 y(m)
Figure 5.2. The effect of cell size on streamwise velocity profiles at the inlet interfaces (base case) 71
Level Density (kg/m3)
15 2.42 13 2.12 11 1.82 91.51 71.21 50.91 30.61 10.35
15 13 11 9 7 5 3
1
1
0 5E-06 1E-05 1.5E-05 2E-05 2.5E-05 3E-05 X(m)
Figure 5.3a. Mass density contours for the base case
1.21 0.61 0.91
0
.
5
3
0.54
0.51
0.35 0.50
0.53
6E-06 8E-06 1E-05 X(m)
Figure 5.3b. Detail of the mass density contours near the inlet region for the base case (values in kg/m3)
72
40 m/s
0 1E-05 2E-05 3E-05 X(m)
Figure 5.4a. Mass averaged velocity vectors for the base case
40 m/s
6E-06 7E-06 8E-06 9E-06 1E-05 X(m)
Figure 5.4b. Details of the mass averaged velocity vectors near the inlet region for the base case
73
50 m/s
5E-06 6E-06 7E-06 8E-06 9E-06 X(m)
Figure 5.5a. Diffusion velocity vectors for H2 at the inlet for the base case
50 m/s
5E-06 6E-06 7E-06 8E-06 9E-06 1E-05 X(m)
Figure 5.5b. Diffusion velocity vectors for O2 at the inlet for the base case 74
299.3 297.3 298.3 299.3
297.3 298.3 296.4
0 5E-06 1E-05 1.5E-05 X(m)
Figure 5.6. Temperature contours for the base case (values in K)
3 m) -6
2.5 Mixing length (10
2
50 100 150 200 250 300 350 Pressure (kPa)
Figure 5.7. The variation of mixing length for cases shown in Table 5.1a (inlet pressure ratio = 1.0)
75
3.5
3 m) -6
2.5 Mixing length (10
2
00.511.522.5 Pressure ratio
Figure 5.8. The variation of mixing length for different inlet pressure ratios (PH2/PO2) (cases shown in Table 5.1b)
300 m/s
5E-06 1E-05 1.5E-05 X(m)
Figure 5.9a. Mass averaged velocity vectors for case 11, Table 5.1c ( α = 0.01 )
76
Level Density (kg/m3) 13 2.37 11 1.88 91.39 70.90 50.40 30.32 10.29
7 13 5
3 1 1 3
5E-06 1E-05 1.5E-05 X(m)
Figure 5.9b. Mass density contours for case 11, Table 5.1c ( α = 0.01 )
8
7 m) -6 6
5 Mixing length (10 4
3
0 0.25 0.5 0.75 1 Accommodation Coefficient
Figure 5.10. The variation of mixing length with accommodation coefficient (for cases shown in Table 5.1c) 77
6. THREE DIMENSIONAL LOW PRESSURE COUETTE FLOWS3
6.1. Introduction
Molecular drag pumps have played a successful role in molecular flow pumping (Atta,
1965; Bhatti et al., 2001). According to the molecular drag principle, if a long rectangular duct has one of its walls moving in the direction of its length, any gas trapped in the duct is forced to travel along the direction of its motion and a pumping action thus results. Due to the growing needs for maintaining conditions in semiconductor fabrication technology, it is indispensable to improve the performance of currently available pumps. The use of numerical method to simulate the flow field of a vacuum pump makes it possible to have a better understanding of the flow phenomena in such pumps and to calculate the needed pumping characteristics before a new vacuum pump has been made.
Traditional continuum computational fluid dynamic (CFD) techniques are often invalid for analyzing gas flows in micro-electro-mechanical systems (MEMS) or in low-pressure devices. This inaccuracy stems from their calculation of molecular transport effects such as viscous dissipation and thermal conduction from bulk flow quantities such as mean velocity and temperature. This approximation of continuum phenomena fails as the characteristic length of the flow gradients (L) approaches the average distance traveled by molecules between collisions (mean free path, λ).
The direct simulation Monte Carlo method (DSMC) is a well-established alternative
(Bird, 1994) to traditional CFD technique, which retains its validity at high Knudsen number as no continuum assumption is invoked. In a conventional DSMC method, each simulated particle represents a very large number of physical molecules. In this fashion, the number of molecular trajectories and molecular collisions that must be calculated is substantially reduced, while the physical velocities, molecular size and internal energies are preserved in the simulation.
Continuous improvement in efficiency and accuracy of DSMC codes has been accomplished by
3 The results presented in this chapter appeared in ASME IMECE, Vol. 4, pp. 1-8, Yan, Farouk and Johnson, 2002 78
Bird and other researchers (Oran et al., 1998; Piekos, 1995) over the last several decades. His
method has been used widely and successfully to simulate high Kn number gas flow problems.
Some other advanced DSMC techniques (Oh et al., 1997; Piekos and Breuer, 1996) have also
been implemented for the practical analysis of both low-pressure flows and microchannel flows
(Mavriplis et al., 1997).
Three-dimensional simulations of flows at near-continuum (low) values of Kn and low
Ma are often prohibitive because of very high computing requirements dictated by higher
collision rates and larger sample sizes. The significant reduction in computational time is
realistic through the recent advances in parallel computer hardware and the development of
efficient parallel DSMC algorithm. A considerable amount of effort has already been put into the
parallelization of DSMC algorithm (LeBeau, 1999; Oh et al., 1996; Wilmoth, 1992). Most work
has involved implementing parallelization of DSMC procedures on distributed memory
computers (Aktas et al., 2001; Oh et al., 1996; Wilmoth, 1992). The parallel method is based on
domain decomposition in which a portion of the physical space along with its associated cells
and molecules is allocated to each processor. A great deal of inter-processor communication is
required and message passing interface, MPI (Gropp et al., 1999) is a common communication
paradigm that supports data sending and receiving. However, different paradigms must be
employed for parallel programming on a shared memory machine to take advantage of the
memory architecture.
One of the objectives of this work is to develop a parallel DSMC model from Bird’s
original serial DSMC code on a shared memory multi-processor machine. Numerical results are
then presented for the subsonic rarefied gas flow in a three dimensional duct with a high speed moving wall from slip flow to free molecular flow regime.
79
6.2. Problem description
The schematic for flow through a channel is illustrated in Figure 6.1. The wall temperatures are kept at 300 K for all cases. The channel has a width of 1 mm, a depth of 1 mm and a length of 12 mm. Inlet pressure P1 and outlet pressure P2 are specified as the boundary condition for the calculations. The geometric dimensions are kept constant in all simulations while the Kn number is changed with the variation of the inlet and outlet pressures. The top wall is moving at a high speed U driving the gas flows from the left hand side to the right hand side. This is also the fundamental operating principle of the molecular drag pump (Atta, 1965). Nitrogen is considered for all cases.
6.3. Parallelization of DSMC
The parallelization technique used in this study was developed for the DSMC algorithm and implemented on an IBM RS/6000 enterprise sever S80 (model 7017). The computer is a shared memory multi-processor (SMP) machine with twelve 450 MHz RS64 III processors.
Each processor has a 128 Kbytes / 128 Kbytes L1 cache and a total of 8 Gbytes shared RAM is available. Both OpenMP and MPI (Message-Passing Interface) can be utilized in parallel programming. For this application, it is most convenient to use OpenMP model. OpenMP is a natural way for parallel programming on shared memory computer (Chandra et al., 2001).
Providing OpenMP directive, the master thread creates additional slave threads when it encounters a parallel region. The computational tasks are then divided among these parallel threads. Unlike MPI, there is no explicit data sending and receiving between processors. Data communication is controlled by data scope clauses. By using OpenMP, the programming complexity incurred in adding parallelism to the existing DSMC sequential code to attain reasonable speed-up performance can be minimized. To test the performance of our parallel code, the channel flow displayed in Figure 6.1 is simulated. For the initial test case, both inlet and outlet pressures are set to be 5.36 Pa. The Kn number is 0.1. The speed of the moving wall is 250 m/s. Totally, 120 × 10 × 10 grid points are used. 80
As described above, DSMC simulation for each time step includes particle movement,
collision, indexing and flow field sampling. Since all the particles are moving independently
without considering the interaction with each other, we can divide the whole bunch of particles
into several chunks and feed them to each thread separately. In particle collisions, collision
partners are selected only from the same cell. The flow domain can thus be decomposed into
several small sub-domains and each thread operates on individual sub-domain simultaneously.
Flow field quantities can also be sampled in parallel for each cell. The particle indexing has to
be run sequentially if the data structure of the original serial code is kept unchanged. In the
serial DSMC code, for the test problem with 445,000 particles in the channel, indexing process
accounts for about 2.5 percent of the CPU time. Therefore, serial execution of the indexing
process does not have significant effect on the performance of the DSMC parallel processing on
a 12-processor computer. But this can be the severe bottleneck for the parallel running of
DSMC on much more processors since in theory the maximum speed up that can be reached
for this parallelization scheme is 40 (Chandra et al., 2001)
There are two major factors hampering the parallelization of DSMC code using
OpenMP. First, synchronization among parallel threads is necessary to ensure correct
execution and data access. The overhead paid for synchronization can be costly if the
executions on different threads are highly dependent on each other. Second, load balance
among the threads can be poor so that some processors can be idle while only just a few are
active. Therefore, an efficient way to manipulate these two issues is essential.
The parallel technique developed here addresses both questions. First, in the particle
moving process, all the particles are indexed and a pointer refers to the tail of the particle list.
When one particle is flowing out of the flow domain, it will be simply erased from the list. The
methodology to fulfill this is to replace the presently removed particle with the last particle in the
list. The number of particles and the pointer indicating the list tail need to be updated
accordingly. If several threads have particles in hand to be removed, they all try to grab the tail of the particle list. A critical section is necessary for mutual exclusion synchronization. It will 81 inevitably cause a lot of overhead, which decreases the efficiency. To relieve this data contention, a buffer recorder is set for each thread. Particles once found out of the flow region are not deleted immediately but recorded in the buffer. They are removed altogether after motions of all particles at each time step are completed. Second, collision frequency is proportional to the number of particles and the particle number density is usually non-uniform in flow field. In the collision calculation, an adaptive scheme is used to redistribute the cells among threads with each thread having approximately equal number of particles. In the particle moving simulation, to keep load balance among the processors, it is necessary to investigate the effect of granularity on efficiency. With N threads in use, the coarsest granularity is to divide the particles uniformly into N pieces and send each thread with one piece. However, the run times for different particles are quite different. If particles collide with a solid wall, much more CPU time has to be spent on calculating the post-reflection parameters. Therefore, uniform division of particles does not mean uniform computational tasks for each thread. To improve this, it needs to have many smaller pieces of tasks for processors to work on concurrently, that is, finer granularity. However, the benefit from a high finer granularity can be canceled out due to increased overhead. There exists an optimal granularity. Figure 6.2 exhibits the effect of the granularity on the parallel performance for the test problem with 445,000 particles considered.
The speed up of the computational time is illustrated in Figure 6.2(a) as a function of the number of processors compared with the runtime for the same calculation using the sequential code. It can be see that the granularity with 1000 particles per piece has relatively better performance than others. The efficiency, defined as the ratio of the speed up over the number of processors, is drawn in Figure 6.2(b). The variation of ideal performance curve along with the number of processors is also indicated in the figure. The ideal efficiency of this parallel code can not reach 100 percent as explained before. Here it is assumed the indexing part takes up 2.5 percent of the sequential code runtime.
The ratio of computational time to the synchronization overhead can be varied by scaling the size of the problem. With different numbers of particles in simulation, different 82
efficiency can be reached. The effect of the problem size on the parallel performance is shown
in Figure 6.3. Obviously, the efficiency increases to certain extent for larger problem.
6.4. Results and discussion
The solution for continuum laminar Couette flow is well known. Velocity profile across the channel is linear and the pressure field is uniform everywhere. However, rarefied gas
Couette flow has very different characteristics. A number of authors have worked in this field to understand Couette flow phenomena in transition regime with kinetic theory description and
DSMC simulation (Boulon et al., 1999; Liu and Lees, 1961; Nanbu, 1983). Liu and Lees (1961) obtained theoretical results for plane Couette flow in transition regime with the six moments solution (from Boltzmann equation). Nanbu (1983) performed DSMC computation on hypersonic flow (M = 3) and compared his DSMC results with analytical solution. He noted very good agreement between DSMC results and the theoretical results. All this work is limited to two dimensional plane. This Couette flow investigation is extended into three dimensions using the parallel DSMC model described above. Table 6.1 lists the cases studied for three different Kn numbers (0.01, 0.1 and 1.0). The top wall velocity is 250 m/s. Both inlet and outlet pressures are set to the same value.
The velocity profiles are presented in each case to see the evolution along the channel in Figure 6.4. Velocity vectors are displayed only in the plane y = h/2. For case 1 (Kn = 0.001), the velocity profile is close to the continuum Couette flow solution. A linear velocity variation from the static to the moving wall can be observed. The velocity slips near the walls are quite small. As the Kn number increases, the velocity profile becomes more uniform. The velocity profiles for different Kn numbers obtained from our computational results are similar to what
Nanbu (1983) and Liu and Lees (1960) obtained although the geometry is slightly different.
Figure 6.5 shows the pressure contours in the duct for each case. Although equal pressures are specified at the inlet and outlet interface, pressure gradients still can be observed inside the flow domain. Due to the rarefaction effects, it is not realistic to keep the pressure field 83
uniform everywhere. It can be seen from the figure that near the moving top wall the pressures
are higher than other region. The pressure gradient for the case Kn = 1.0 is pronounced. As the
Kn increases, the pressure field is becoming more uniform. As the Kn approaching zero, the
pressure gradient will disappear.
Now that we have a good understanding of the flow phenomena for different regimes, it
is important to link the results to more quantitative values such as the ‘ultimate pressure’ for
given geometry and wall velocity. Because the moving top wall keeps on driving gas in one
direction, a pressure difference is built up between the ends, as P1 < P2. A back-diffusive flow is thus induced by the pressure gradient. Keeping back pressure P2 constant, the ‘ultimate pressure’ P1 for a vacuum pump is reached under equilibrium condition when the flow driven by
the moving wall is balanced by the flow due to the pressure gradients. This is a mixed Couette-
Poiseuille flow with zero flow rate. The ultimate pressures are calculated from molecular to slip
flow regimes and for three wall speeds (50 m/s, 100 m/s and 250 m/s). It is noted that in
simulations the inlet pressure is the parameter needed to be solved. A scheme proposed by
Chang et al. (2001) is employed to determine the ultimate pressure P1. At the beginning, An
initial inlet pressure is guessed for P1. At each time step, the mass flow rate at the inlet (denoted
as m& in) is calculated by recording the incoming and outgoing molecules at the inlet interface. The inlet pressure is updated according to the zero flow rate condition until it converges.
1 m2 * & in if m > 0 P1 = P1 - β 2 & in 2 mnin Ain
2 * 1 m& in if m in < 0 (6.1) P1 = P1 + β 2 & 2 mnin A in
* Here P1 is the inlet pressure from the last iteration and β is the relaxation factor which is chosen
-4 as 1×10 .
Table 6.2 summarizes the cases we have studied for ultimate pressure prediction.
Dushman (1962) provided a relationship of pressure difference as the function of moving speed
U and geometry parameters for both continuum and molecular flows. If the back pressure P2 is 84 kept at atmospheric pressure, that is, the flow is in continuum flow, according to his analysis, for the geometry and moving speed considered here, the pressure difference P2 - P1 is very small compared to P2. The compression ratio P2 / P1 is thus approximately equal to one. It means, the molecular drag pump is not efficient at continuum flow condition. In free molecular flow regime,
Dushman derived a relation for compression ratio, which is independent of Kn number. The compression ratio is investigated from our computation for the three wall velocities. In Figure
6.6, the velocity fields are demonstrated for case 26, 29 and 32. The moving speed is 250 m/s.
The Kn numbers for three cases are 0.05, 0.5 and 5.0 respectively. For high Kn number flow, there are not enough collisions between particles. From Figure 6.6(a), it can be seen that only particles near the moving wall are driven out of the channel.
Figure 6.7 illustrates the comparison of the compression ratio for a wide range of Kn with three moving speed U = 50 m/s, 100 m/s and 250 m/s. The compression ratio is pretty low at low Kn number region. As Kn increases, the compression ratio also increases. The compression ratio reaches a maximum value in the region the between Kn = 0.1 and Kn = 1.0.
The compression ratio then drops approaching a certain value as the Kn keeps increasing.
Obviously, larger moving speed can significantly increases the pumping performance. It is also noted the maximum compression ratio point moves to lower Kn region as the moving wall speed increases. 85
Table 6.1. Inlet and solid wall conditions for cases studied for 3D Couette flows
Case P1 (Pa) P2 (Pa) U (m/s) Kn
1 53.6 53.6 250 0.01
2 5.36 5.36 250 0.1
3 0.536 0.536 250 1.0
Table 6.2. Inlet and solid wall conditions for cases studied for ‘ultimate pressure’ calculations
Case P1(Pa) (calculated) P2 (Pa) U (m/s) Kn
4 48.84 53.6 50 0.01
5 20.85 26.8 50 0.02
6 6.56 10.7 50 0.05
7 2.58 5.36 50 0.1
8 1.09 2.68 50 0.2
9 0.42 1.07 50 0.5
10 0.21 0.536 50 1.0
11 0.11 0.268 50 2.0
12 0.048 0.107 50 5.0
13 0.025 0.0536 50 10.0
14 44.82 53.6 100 0.01
15 16.88 26.8 100 0.02
16 3.50 10.7 100 0.05
17 1.08 5.36 100 0.1
18 0.44 2.68 100 0.2
19 0.17 1.07 100 0.5 86
Table 6.2. (continued)
20 0.096 0.536 100 1.0
21 0.053 0.268 100 2.0
22 0.024 0.107 100 5.0
23 0.013 0.0536 100 10.0
24 40.30 53.6 250 0.01
25 13.88 26.8 250 0.02
26 0.31 10.7 250 0.05
27 0.11 5.36 250 0.1
28 0.057 2.68 250 0.2
29 0.032 1.07 250 0.5
30 0.024 0.536 250 1.0
31 0.015 0.268 250 2.0
32 0.0078 0.107 250 5.0
33 0.0044 0.0536 250 10.0
U
h P1 d P2
L
Figure 6.1. Schematic of channel flow problem
87
9
Ideal 8 500 particles / piece 1000 particles / piece 7 2000 particles / piece 5000 particles / piece 10000 particles / piece 6
5 Speed up 4
3
2
1 510 Number of processors
(a)
100 Ideal 500 particles / piece 95 1000 particles / piece 2000 particles / piece 5000 particles / piece 90 10000 particles / piece
85
fiiny( %Efficiency ) 80
75
70 510 Number of processors
(b)
Figure 6.2. The effect of granularity on the parallel performance (a) speed up (b) efficiency
88
100
Ideal 95 950,000 particles
445,000 particles 90 250,000 particles
85
80 Efficiency ( % )
75
70
65 510 Number of processors
Figure 6.3. Parallel efficiency for different problem sizes
250 m/s
0.00 0.05 0.10 x(m)
(a)
250 m/s
0.00 0.05 0.10 x(m)
(b)
Figure 6.4 (continued) 89
250 m/s
0 0.05 0.1 x(m)
(c)
Figure 6.4. Velocity profiles in Couette flow for three cases (a) Kn = 0.01, (b) Kn = 0.1, and (c) Kn = 1.0
Y Z
X
2
5 .
0
2
5
.
0
0.53
0 0 . . 5 0.54 4 0 9
2
0 5 z(m) . . 0.004 0.008 4 0 9 3 5 0 . 0
0.02 8 0 0 .0 . 5 0 0.04 2 0 5
.
3
0
5
.
0.06 0 ) x 4 m (m 0 ( ) 0.08 .0 0 y 0.1
0.12
(a)
Figure 6.5 (continued) 90
Z Y
5.17 X
5
. 1
7 4.84
4.84
0. 00 5.50 z(m)
0. 5.17 05 x
5.17 0.000 0.004 0.008 (m 8 ) 00 0.
0.1 04 ) 0 .0 (m 0 y
(b)
Z
Y
X
53.56
53.56 z(m) 0. 000 53.56
0.0 50 0.000 0.005 0.010 08 x ( 53.56 .0 m) 0 4 ) 0. .00 (m 100 0 y 00 0.0
(c)
Figure 6.5. Pressure contours inside the duct for three cases (a) Kn = 1.0, (b) Kn = 0.1, and (c) Kn = 0.01
91
50 m/s
00.050.1 x(m)
(a)
100 m/s
0 0.02 0.04 0.06 0.08 0.1 0.12 x(m)
(b)
200 m/s
0 0.05 0.1 x(m)
(c)
Figure 6.6. Comparison of velocity vectors for three cases for ultimate pressure calculation with U = 250 m/s (a) Kn = 5.0, (b) Kn = 0.5 and (c) Kn = 0.05
92
50 U=50m/s 40 U = 100 m/s 30 U = 250 m/s
20
10 opeso ratio Compression
10-2 10-1 100 101 Kn
Figure 6.7. Variation of compression ratio as a function of Kn number
93
7. NUMERICAL SIMULATION OF CAPILLARY ELECTROKINETIC SEPARATION FLOWS IN MICROCHANNEL4
7.1. Introduction
The area of microfabricated fluidic devices that perform chemical and biochemical analysis has developed into one of the most dynamic fields in analytical chemistry during the past decade (Chiem and Harrison, 1997; Effenhauser et al., 1993, 1995; Fan and Harrison,
1994; Harrison et al., 1993; Harrison et al., 1992; Jacobson et al., 1998; Jacobson et al., 1994a;
Waters et al., 1998). Microfabricated fluidic devices can be utilized for medical, pharmaceutical, defense, and environmental monitoring applications. Example of such applications are drug delivery (Santini et al., 2000), DNA analysis/sequencing systems (Chang et al., 2000), and biological/chemical agent detection sensors on microchips. Miniaturized analysis systems can provide several advantages over conventional instruments, including typically two orders of magnitude shorter analysis time, smaller injection volumes in the nanoliter to picoliter range, lower cost, greater portability, and potentially greater separation resolution. These microscale devices associated with the Lab-on-a-Chip concept where sample handling and reactions are integrated with chemical separations almost exclusively use electrokinetic manipulations for fluid handling and analysis as they have the advantage of only requiring application of electric potentials to the devices. These manipulations exploit two physical phenomena: electrophoresis and electroosmosis. In electrophoresis, charged species move in a buffer solution under the action of an applied electric field. Velocity of species depends on the electrophoretic mobility and local electric field. Electroosmosis on the other hand, refers to the bulk movement of an aqueous solution past a stationary solid surface due to an externally applied electric filed. This requires the existence of a charged double-layer at the solid-liquid interface.
Considering the difficulties associated with performing experiments in micro-scales, it is desirable to develop reliable numerical analysis tools that can simulate the electrokinetically driven flows in complex geometries. These numerical models can provide better understanding
4 The results presented in this chapter are under review, Analytical Chemistry, Yan and Farouk, 2003 94
of coupled effects of electric field with fluid motion, and hence can be utilized for an optimized
micro-fluidic system design, prior to hardware fabrication and experimental verification. Since
the electrokinetic phenomena was first described by Reuss in 1809 (Reuss, 1809), there have
been numerous theoretical investigations on electrokinetic flows (Burgreen and Nakache, 1964;
Keh and Liu, 1995; Ohshima and Kondo, 1990). Computers have been applied to the modeling
of the most common electrokinetic techniques and classes of samples (Mosher et al., 1992;
Mosher et al., 1993). Numerical modeling has also been extended to simulate capillary
electrophoresis and electroosmosis in 2D and 3D geometries (Bianchi et al., 2000; Ermakov et
al., 1998, 2000; Patankar and Hu, 1998; Yang et al., 1998). Both qualitative and quantitative agreement between simulation and experiment has been obtained (Ermakov et al., 1998, 2000;
Patankar and Hu, 1998). Recently, Ermakov et al. (2000) investigated electrokinetic injection techniques in microchips using computer simulation. A two dimensional cross channel was used for sample manipulation. Both pinched injection valve (Jacobson et al., 1994a) and gated injection valve (Jacobson et al., 1994b) were investigated. They reported the operating parameters providing optimal valve performance for both injection techniques according to their simulation results. However, species separation from each other was not examined.
To provide optimal electric field strength manipulation for species separation, this paper reports on a study of the electrophoretic species separation in microchannels using numerical modeling and simulation. The numerical model accounts for the principal physical phenomena affecting ‘sample’ mass transport in microchannels viz. the electrophoretic motion, electroosmotic motion and diffusion. Unlike Ermakov et al. (2000) in which only electrokinetic injection performance was the target for simulation, we concentrated on the studies of the species separation under various operating conditions. Variation of species concentration and separation resolution is predicted for different loading and dispensing scheme combinations.
The models developed can help improve the understanding of the physical phenomena of electrophoretic separation and the design of capillary-electrophoresis microchips.
95
7.2. Problem geometry
The geometry of a two dimensional cross channel device studied here (see Figure 7.1) is similar to that used by Ermakov et al. (2000). This is a typical geometry used in species separation on microfluidic devices(Ermakov et al., 2000; Patankar and Hu, 1998). The injection channel is for the passage of sample solution containing various species to be separated. The other channel (orthogonal to the injection channel) is the separation channel where separation of the species occurs. The injection channel is connected to sample reservoir 1 at the left in which sample species are stored and the intersection chamber. The buffer inlet channel is connected to the buffer reservoir 3 at the top and the intersection chamber while the separation channel is connected to a collection reservoir 4 at the bottom. The injection of the sample solution is effected by applying an electric potential between reservoirs 1 and 2. This potential drives the sample solution from reservoir 1 across the intersection, creating a sample plug in the path between reservoirs 3 and 4. During the loading step, to prevent the sample species flow into the separation channel, certain voltages are also applied at reservoirs 3 and 4. The sample in the intersection is spatially confined by the incoming buffer streams from reservoirs 3 and 4.
This electrokinetic focusing during the loading process can generate short axial extent sample plug suitable for high performance separation (Jacobson et al., 1998). The extent of the sample focusing is regulated by the electric field strength in the vertical (separation) channel and in the horizontal (injection) channel. First, the electric field is applied to the injection channel. Once the sample flow has reached a steady state, the electric field is switched from the injection step to the separation step. Subsequent application of a voltage difference between reservoir 3 and 4 drives buffer flow vertically from reservoir 3 toward reservoir 4. This buffer flow thus pushes a small plug of sample to move toward reservoir 4 and causes the separation of various species due to different electrophoretic mobility along the separation channel. In the separation step, in order to prevent sample leakage into the separation channel and achieve a short axial extent sample plug, an electric field is also applied to reservoir 1 and 2 to draw sample from the intersection back to reservoirs 1 and 2. The separated species in the sample solution can be 96
detected either by an amperometric electrochemical detector or by a fluorescence detector as
described by Harrison et al. (1993) and Woolley and Mathies (1994).
The injection and separation processes described above are also termed as loading
and dispensing steps, and therefore, two electric field distributions are specified for each step.
Following the nomenclature for electric field used in Ermakov et al. (2000), Eij is the electric field
strength at the reservoir i for either the loading or dispensing step ( j = L or D). A relative electric
field strength in the channels, εij = Eij / Emax is used for convenience, where Emax is the maximum
field value. It is assumed that εij is positive if the vector Eij is directed toward the channel intersection and negative if directed away. Because we consider only channels of equal width and solutions with constant conductivity, according to the Kirchhoff’s law, the relative electric field at each reservoir εij satisfies the following equation
∑ε ij = 0 (7.1) i
where again i indicates the reservoir number (1 ~ 4) and j is either for loading (L) or dispensing
(D).
7.3. Mathematical modeling
As discussed in Chapter 3, a major assumption has been made to simplify the modeling of
electroosmotic flows. The Debye layer thickness is usually of the order 10-9 m. For the geometry
being studied, the channel width h is of the order 10-5 m. It is very computationally expensive to
resolve the electric double layer with thickness of Debye length where most of the electric charge is
concentrated because very fine mesh size is required near the solid wall. Inside the electric double
layer, very large velocity gradient is presented and outside the electric double layer, a uniform
velocity profile is exhibited. Due to this feature, it is reasonable to assume a slip electroosmotic
velocity boundary condition along solid walls without resolving the thin electric double layer. With
this assumption, relatively coarse mesh size can be implemented in the simulation while the main
physical characteristic of electrokinetic flow can still be predicted. 97
To model electrokinetic flow of charged species, the equations governing
electroosmotic, electrophoretic sample transport and diffusion in an electric field must be solved
(Saville and Palusinski, 1986). Both Patankar and Hu (1998) and Ermakov et al. presented a
mathematical models describing electrokinetically driven mass transport phenomena in
microfabricated chip devices. Here we briefly present the mathematical formulation for the
studies of microfluidic devices. The conservation equation for a given specie “i” is expressed as:
∂ C r r 2 i + ∇. ((V + z µ E )C ) = D ∇ C (7.2) ∂ t i ep ,i i i i
In the above equation Ci is the species concentration, Di is the diffusion coefficient, zi is the
th r r valence and µep is the electrophoretic mobility of the i species. V is the velocity vector and E
r is the electric field vector. The term zi µep, i E accounts for the electrophoretic velocity of each
r r r species relative to the bulk flow. The term (V + zi µ ep ,i E ) denotes the species velocityVi .
The momentum equation describing the bulk flow of the solution is given by: r ∂ V r r 1 2 r + (V∇ ) V = - ∇ P + ν∇ V (7.3) ∂ t ρ
In order to calculate the electromigration term, we need to calculate the electric field and the r electric potential. The electrical field E is related to the electric potential Φ by r E = -∇ Φ (7.4)
while the electric potential is governed by
2 ∇ Φ = 0 (7.5)
We need to specify the boundary conditions for the problem now. All the side, top and bottom walls are assumed to be insulated and non-flux boundary condition is applied to both species and electric potential equations. Voltages are specified at the inlet and exit of the channels. For the flow field, we set the pressures to be zero at all the four reservoirs. A slip velocity is implemented at side walls for momentum equation of bulk flow. Electroosmosis is 98
simulated via applying a slip velocity at the wall calculated using specified electroosmotic
mobility. The bulk fluid velocity along the wall due to electroosmotic mobility is expressed as: r r Veo = µ eo E (7.6)
r where µeo is the electroosmotic mobility (Ermakov et al., 1998). The velocity,Veo is imposed as
slip velocity boundary condition along walls of the cross channel.
For the numerical scheme, the diffusion terms are modeled using second order central
differences, and the third-order accurate QUICK differencing scheme (Freitas et al., 1985) is
used for the advection terms. The third-order scheme possesses the stability of the first-order
upwind formula and is free from substantial numerical diffusion experienced with the usual first-
order methods. The SIMPLE algorithm is implemented to achieve the pressure-correction. The
second order central differencing scheme is also used for potential equation (7.6) discretization.
Totally 12,800 grid points were placed in the computational domain. Dense grids were arranged
-4 in the cross section area with grid spacing as small as 4 × 10 mm near intersection channel
corners. For the loading step, steady state solutions were obtained. A second order Crank
Nicolson differencing scheme was chosen for the time derivative in the dispensing step
simulation. A time step as small as 0.02 s is used for these computations. With the electric field
distributions considered, time for sample plug to completely transport out of separation channel
was in the order of seconds. All computations were performed on an IBM-RISC-6000 (Model
7044) workstation. Typical computation time ranged from 200 to 300 minutes for a case. A self-
written Poisson equation solver is coupled with commercial software CFX 4.2 (AEA Technology,
1997) to implement simulations.
99
7.4. Results and discussion
7.4.1. Focusing process only
Electrokinetic focusing in microchannels was implemented by experiment previously
(Jacobson and Ramsey, 1997). Sample (single species) stream measured concentration
profiles at different sections also were reported. We consider the same geometry and conditions
for our simulations as in the above experimental work. The channel width was set to be 18 µm.
In the simulations L and H were chosen to be 600 µm long. For the loading step, electric field
strength values were taken from the reference (Jacobson and Ramsey, 1997) and were used as
boundary conditions at the entry of each reservoir, that is, E1 = 0.11 kV/cm, E2 = 2.43 kV/cm, E3
= E4 = 1.16 kV/cm. The sample (rhodamine 6G) with an electrophoretic mobility of µep = 1.4 × 10
–4 cm2/ (V⋅s) and a diffusion coefficient D = 3.0 × 10 –6 cm2/s was used as the material to be
spatially confined. Other physical properties of the sample are similar to those of water
(Jacobson and Ramsey, 1997). The buffer is also assumed to have the similar physical
properties as water (Patankar and Hu, 1998). The electroosmotic mobility has the value µeo =
4.0 × 10 –4 cm2/ (V⋅s).
The simulated concentration distribution is compared in Figure 7.2 with measurement data extracted from experimentally obtained CCD images. All concentration values were normalized for comparison purpose. Sample concentration profiles were plotted at four injection and sample waste channel cross sections. Figure 7.2(a) shows the sample concentration distribution at 3 µm upstream form the intersection. Profile 7.2(b) is 9 µm into the chamber, i.e., in the center of the focusing chamber. Profile 7.2(c) is for the exit of the chamber and profile
2(d) is at 60 µm downstream of the chamber. It can be seen from figures 7.2(a) – 7.2(d) that the agreement between the simulation results and experimental measurements is good.
100
7.4.2. Focusing process followed by separation process
Simulations were next carried out for different loading and dispensing conditions. For a
given loading condition, simulations were carried out for various dispensing conditions. Table
7.1 lists the loading and dispensing conditions studied in the paper. Following Ermakov et al.
(2000), in the loading step we keep the focusing electric fields from reservoir 3 and 4 (Figure
7.1) to be equal, that is, ε3L = ε4L, and in the dispensing step the pullback electric field on reservoir 1 and 2 are kept the same, ε1D = ε2D. Therefore, due to equation (7.1), in the loading step, the electric field distribution is characterized by ε1L and in the dispensing step the electric field is characterized by ε4D (Ermakov et al., 2000). The sample loading confinement is weak if
|ε1L| > |ε3L|, strong if |ε1L| < |ε3L|, and medium if |ε1L| ≈ |ε3L|. Similarly in the dispensing step, the
sample pull back is weak if |ε4D| > |ε2D|, strong if |ε4D| < |ε2D|, and medium if |ε4D| ≈ |ε2D|. In all simulations the electric field scale was set to Emax = 500 V/cm. Emax = 500 is a typical value in
electrically driven microchannel flows.
An example of one electric field distribution used in the loading step and dispensing step is depicted in Figure 7.3. This is the electric field configuration for the base case simulation listed in Table 7.1. It can be seen from Figure 7.3(a) that the focusing electric field is weak during the loading step with ε3L = ε4L = 0.2. Since ε3L and ε4L are both small, the focusing
strength from reservoir 3 and 4 are both weak. During the dispensing step, the sample pull
back force is also small with ε1D = ε2D = 0.3. The second subscript L and D denote the loading
and dispensing step respectively.
Electrophoretic separation studies in microchannels were then conducted using
simulations based on the mathematical model described above. For the cases described in this
section, the channel width was set to be 50 µm. In the simulations L and H were chosen to be
1000 µm long. Three hypothetical species with the same concentrations (33.3%) in sample reservoir were introduced into the cross microchannel for separation. All the species are positively charged and are assumed to have electrophoretic mobility 1.0 × 10 –4 cm2/V⋅s, 101
(species 1) 2.0 × 10 –4 cm2/V⋅s (species 2) and 3.0 × 10 –4 cm2/V⋅s (species 3) respectively. An
electroosmotic mobility 3.0 × 10 –4 cm2/V⋅s of the solution (equation (7.6)) is applied and a diffusion coefficient 3×10-6 cm2/s is used. Other physical properties of both sample solution and buffer are again assumed to be similar to those of water. r Figures 7.4(a) and 7.4(b) show the velocity vector (Vi ) plots near the intersection
channel at the focusing and separation stages respectively for species 1 (for the base case)
with |ε1L| = 0.6 in the loading step and |ε4D| = 0.3 in the dispensing step. We see that the velocity
vectors stay are parallel across the channels, away from the intersection. The variation in the
velocity distribution around the intersection of the channels is due to the interaction of two
mutually orthogonal flow streams. From Figure 7.4(a) it can be seen that during the loading
step, sample stream flows through the intersection from left side to right side and buffer streams
from top and bottom reservoirs are squeezing the sample flow. Figure 7.4(b) shows the buffer
stream flowing down toward the separation channel during the dispensing step and sample
streams along the injection channel are pulled away from the intersection chamber. This
distribution can be described as a weak pinch during loading combined with a medium sample
pullback during dispensing. Figures 7.5(a) – 7.5(d) depict a sequence of concentration contours
of species 1 at different time levels during the dispensing step. The concentration is again
normalized with its initial sample concentration in sample reservoir for convenience. Figure
7.5(a) is the concentration distribution in dispensing stage at t = 0 s which is also the end of
loading step. The sample is introduced from left of the channel in a continuous fashion during
the loading step. Transport from the two side channels, 3 and 4, confines the sample stream in
the intersection. Due to weak confinement and diffusion, small amount of sample stream
disperse into the adjacent focusing streams. The sample stream reaches a minimum width and
then broadens because of diffusion. The species is diluted in the waste channel (between the
intersection and reservoir 2) due to mixing with buffer flow from the side reservoirs. At times t =
0.1 s, 0.2 s and 0.3 s, as shown in Figures 7.5(b), 7.5(c) and 7.5(d) respectively, a short axial
extent species plug is driven into the separation channel. To prevent additional amounts of 102
sample to flow from the injection channel into the separation channel, the sample flow is
electrokinetically pulled back from intersection to reservoir 1 and 2.
For adequate separation of the species, we must have quantitative measure of the
achieved separation resolution of a given sample mixture. Figure 7.6 displays the desired
typical concentration variation of each species versus time at the end of the separation channel.
It can be seen from Figure 7.6 that each species concentration varies as a function of time and
has a form of a symmetrical, bell-shaped band. Each band emerges from the separation
channel end at a characteristic time that can be used to identify the species. This retention time
is measured from the time of starting dispensing to the time the concentration band maximum
leaves the separation channel as shown in Figure 7.6. Each band is also characterized by a
band width, tw, which can be seen in Figure 7.6. The band width tw is determined for each
species as demonstrated in Figure 7.6. Tangents are drawn to each side of the band and
extended to touch the baseline. Separation is better for narrower bands and smaller values of
tw. The resolution of two adjacent concentration bands species 1 and species 2 is defined as the
time between the occurrence of two concentration peaks divided by average band width
(Snyder and Kirkland, 1979) (see Figure 7.6)
(t2 − t1 ) R1,2 = (7.7) (1/ 2)(tw1 + tw2 )
In the above equation, times t1 and t2 are the retention times of species1 and 2. Larger values of resolution R1,2 means better separation of species 1 and 2. The larger the difference between t1
and t2, the easier is the separation of the two species. The separation resolution between
species 2 and 3 can be specified in the same way.
Returning to Figure 7.5, due to band broadening, the spatial extent of the sample plug
is enlarged as the sample flows along the separation channel. The concentration value of the
sample also drops due to dilution. The concentration contour evolutions of species 2 and 3
experience similar process as described for species 1 in Figure 7.5. Eventually various species
reach the end of the separation channel and carried off to the detector where their 103
concentrations are recorded as a function of separation time. To simulate the detector signal
obtained from a single-point detection scheme (Jacobson and Ramsey, 1996), the average
value of the sample concentration across channel at the end of the separation channel is
calculated for each species at different time steps.
Figure 7.7 displays the temporal variation of the species concentration at the end of the
separation channel for the base case. It is clear that, for a given length of channel, the traveling
time of a species is inversely proportional to it mobility. Species 3, which has the highest
electrophoretic mobility, will be detected by the detector first. The peak concentration of species
1, 2 and 3 reach the end of the separation channel at time t = 0.91 s, 0.73 s and 0.60 s
respectively after the beginning of the dispensing step. Due to band broadening, the peak
concentration value Cmax of each species is much smaller than the initial value. It can also be
seen that the peak concentration value Cmax at the end of the separation channel increases
slightly for species with larger electrophoretic mobility, with Cmax,1 = 0.15, Cmax,2 = 0.157 and
Cmax,3 = 0.165 respectively. This is because during the loading step, species with larger electrophoretic mobility has more mass confined in the focusing chamber. Consequently, more sample plug will flow into the separation channel during the dispensing step. Species with larger mobility also has a stronger convection effect during mass transport. It also reduces the band broadening. The retention times is calculated for species 1 and 2 to be 0.91 s and 0.73 s and the band widths tw1 and tw2 are be found to be 0.72 s and 0.61 s from Figure 7.7. The resolution
R1,2 (equation 7.7) can thus be calculated as 0.27. Similarly, the band width tw3 is found to be
0.50 s. The band width of each species decreases with larger electrophoretic mobility. The resolution R2,3 between species 2 and 3 can be calculated to be 0.24. The resolutions of both
R1,2 and R2,3 are not good mainly due to the relatively short separation channel considered here.
With longer separation channel, we can expect better separation performance.
To control separation, we can also investigate how the resolution varies with other
parameters such as electric field. Here we keep the geometry constant and attempt to find an
optimal electric field distribution for best separation performance. Besides the resolution, the 104
maximum value of concentration band is another parameter served to measure separation
operation. Larger value of peak sample concentration is desirable. However, the requirements
for maximum resolution and maximum sample concentration may result in opposite trends.
Narrower concentration bands resulting in larger resolution are obtained through stronger
sample confinement, which reduces the sample concentration Cmax on the other hand. The optimal electric field distribution is the one with the product of peak concentration value and resolution maximized.
Table 7.1 lists the cases we have studied for determining optimal electric field distribution for given geometry and species characteristics. The simulations cover wide range of loading confinement and pullback combination as detailed in Table 7.1. The effect of different parameters on separation behavior is investigated based on numerical results. Figure 7.8
displays the concentration bands for species 1 with |ε4D| = 0.3 and |ε1L| varying from 0.2 to 0.8
(i.e., cases 7, 12, 1 and 18). It is interesting to see that when |ε4D| is fixed, the retention times of
concentration bands for different |ε1L| is nearly the same, that is, about 0.91 s. This phenomena
is also observed with other |ε4D| and repeated for species 2 and 3. This indicates that the
retention time is almost independent of the loading scheme. For different |ε1L|, the retention time
reaches approximately the same value as long as |ε4D| is constant.
Figure 7.9 displays the variation of the retention time (t1, t2, etc. in Figure 7.6) for each
species with different |ε4D| values and |ε1L| = 0.3. The retention time for each species is mainly dependent on the dispensing scheme and species mobility. With stronger dispensing and larger mobility, the retention time decreases significantly.
The effect of different parameters on the band width ( tw1, tw2, etc. in Figure 7.6) is then
considered. Figure 7.10 shows the variation of band width tw1 for species 1 with different electric distributions |ε1L| and |ε4D|. The band width decreases significantly with |ε4D| increases because
larger |ε4D| provides stronger electric field along the separation channel and consequently more
pronounced convection effect. The band width also decreases slightly as |ε1L| drops since 105
smaller band width can be achieved with stronger focus (smaller |ε1L|). However, it is surprising to see that the band width changes only slightly with |ε1L| for constant |ε4D|. It means stronger
confinement does not decreases the band width significantly and therefore contributes little to
the improvement of separation resolution.
On the other hand, it can be seen from Figure 7.11 that the peak concentration values
for species 1 grow quickly as both |ε1L| and |ε4D| increase. Species 2 and 3 also exhibit the same behavior. This feature is very desirable since weak focus (|ε1L|>0.5) can be used in species separation with larger sample concentration albeit with somewhat less resolution. In our cases, for weak focus, sample tailing (sample migration to separation channel during the loading step) contributes little to enlarge the band width.
Finally, in Figure 7.12, the separation resolutions R1,2 and R2,3 are plotted as a function of dispensing |ε4D| electric field schemes. Due to insignificant effect of loading schemes on band
width and retention time as shown above, the separation resolution is also slightly dependent on
|ε1L|. Figure 7.12 only shows the results for |ε1L| equal to 0.8. It can be seen that the separation
resolution R1,2 increases with increases |ε4D| and reaches a maximum value near 0.7 then drops a little at 0.9. For separation between species 2 and 3, the curve has the same trend.
Based on the simulation results, a loading scheme with |ε1L| equal to 0.8 combined with a dispensing scheme with |ε4D| equal to 0.7 is suggested which gives maximum sample
concentration and maximum separation resolution amongst the cases considered. These values
are different from the optimal electric field distribution for injection found by Ermakov et al.
(2000). In Ermakov et al. (2000), optimal electric field distribution was investigated based on
loading step only without considering species separation resolution and the dispensing step.
They obtained the optimal electric field distribution with the value |ε1L| = 0.6 and |ε4D| = 0.3. In the present work, an optimal electric field is investigated for both injection and species separation resolution. The optimal injection from their study does not match the optimal species separation here. 106
Table 7.1. Electric field distribution of cases considered for simulation
Case ε1L ε4D
1 (base case) 0.6 0.3
2 0.6 0.1
3 0.6 0.5
4 0.6 0.7
5 0.6 0.9
6 0.2 0.1
7 0.2 0.3
8 0.2 0.5
9 0.2 0.7
10 0.2 0.9
11 0.4 0.1
12 0.4 0.3
13 0.4 0.5
14 0.4 0.7
15 0.4 0.9
16 0.8 0.1
17 0.8 0.3
18 0.8 0.5
19 0.8 0.7
20 0.8 0.9
107
Buffer reservoir 3
Buffer inlet channel
Injection channel Sample Sample waste H reservoir 1 d reservoir 2
Separation channel
d
Collection reservoir 4
L
(a)
Buffer reservoir 3
Buffer inlet channel
Injection channel Sample Sample waste H reservoir 1 d reservoir 2
Separation channel
d
Collection reservoir 4
L
(b)
Figure 7.1. Schematic of the cross channel geometry (the arrows indicate the flow direction along each channel during (a) loading step and (b) dispensing step) 108
1
0.8
0.6
0.4
Sample concentration 0.2
0
-20 -15 -10 -5 0 5 10 15 20 Channel width (µm)
(a)
1
0.8
0.6
0.4
Sample0.2 concentration
0
-20 -15 -10 -5 0 5 10 15 20 channel width (µm)
(b)
Figure 7.2 (continued) 109
1
0.8
0.6
0.4
Sample0.2 concentration
0
-20 -10 0 10 20 Channel width (µm)
(c)
1
0.8
0.6
0.4
Sample concentration 0.2
0
-20 -15 -10 -5 0 5 10 15 20 Channel width (µm)
(d)
Figure 7.2. Comparison between experimental (Jacobson and Ramsey, 1997) (dashed line) and simulated (solid line) concentration profiles across the injection and waste channels (a) 3 µm upstream from the intersection chamber (b) 9 µm into the intersection chamber (c) at the exit of the intersection chamber (d) 60 µm downstream from the intersection chamber
110
Loading step Dispensing step
0.2 1.0
0.6 -1.0 -0.35 -0.35
0.2 -0.3 (a) (b)
Figure 7.3. Electric field schemes of the base case for the loading and dispensing steps Arrows show the direction of the field, and numbers correspond to the relative electric field strength in the channels
111
2mm/s
(a)
2mm/s
(b)
Figure 7.4. Velocity vectors plot of species 1 at channel section (base case) (a) loading step (b) dispensing step for the base case in Table 7.1 112
Concentration
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
(a)
Concentration
0.92 0.83 0.75 0.67 0.58 0.50 0.42 0.33 0.25 0.17 0.08
(b)
Figure 7.5 (continued) 113
Concentration
0.91 0.83 0.74 0.66 0.58 0.50 0.41 0.33 0.25 0.17 0.08
(c)
Concentration
0.89 0.80 0.72 0.63 0.55 0.47 0.38 0.30 0.21 0.13 0.04
(d)
Figure 7.5. Concentration contours of species 1 at different time levels for the dispensing step (a) t = 0.0 s,(b) t = 0.1 s, (c) t = 0.2 s and (d) t = 0.3 s for the base case 114
t3
t2
t1 concentration
t tw3 w1 time tw2
Figure 7.6. Representation of the resolution of species separation
species 1 0.15 species 2
species 3
0.1 Concentration
0.05
0 00.511.52 Time (s)
Figure 7.7. Temporal evolution of species concentration at the end of the separation channel (base case)
115
0.2 | ε1L |=0.2
| ε1L |=0.4
| ε1L |=0.6
| ε1L |=0.8
0.1 Concentration
0 00.511.52 Time (s)
Figure 7.8. Temporal evolution of species 1 concentration at the end of the separation
channel under different loading conditions with |ε4D| = 0.3
3
species 1 species 2 2 species 3
Retention time (s) 1
0 00.20.40.60.81
|ε4D|
Figure 7.9. Variation of retention time of different species as function of |ε4D| with |ε1L| = 0.3 116
3
2
1 Band width (s)
|ε4D|=0.1
|ε4D|=0.3
|ε4D|=0.5
|ε4D|=0.7
|ε4D|=0.9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
|ε1L|
Figure 7.10. Variation of band width ‘tw1’ for species 1 as function of |ε1L| and |ε4D|
0.5
|ε4D|=0.1
|ε4D|=0.3
|ε4D|=0.5 0.4 |ε4D|=0.7
|ε4D|=0.9
0.3
0.2 Maximum concentration
0.1
0 0 0.2 0.4 0.6 0.8 1
|ε1L|
Figure 7.11. Variation of maximum concentration Cmax for species 1 as function of |ε1L| and |ε4D| 117
0.4
R1,2
R2,3
0.3
0.2 Separation resolution
0.1 0 0.2 0.4 0.6 0.8 1
|ε4D|
Figure 7.12. Separation resolution as a function of dispensing electric field schemes |ε4D| with |ε1L| = 0.8 118
8. SUMMARY AND CONCLUSIONS
8.1. Summary
8.1.1. High Knudsen number gas flows
First, the development and validation of DSMC code were carried out for simulating high Kn number flows. Numerical results were compared with analytical results and experimental data. The agreement obtained was excellent.
High Kn number low speed flows through partially heated parallel plates were studied
using the DSMC technique. Diatomic and monatomic gases as well as noble gas mixtures were
considered. Heat fluxes along the plates were calculated and compared with the Graetz
analytical solution (with uniform inlet flow). Heat fluxes predicted by the DSMC method agreed
qualitatively with the Graetz solution in the heated region. However, significant differences
existed because of the rarefied flows considered. Molecular diffusion dominates at large Kn.
Nonlinear pressure distribution was observed along the channel. As the Kn number increases, the heat flux decreases monotonically along the channel wall. This research work is published in Yan and Farouk (2002a).
DSMC calculations of mixing of parallel H2 and O2 streams in a microchannel were performed. The effects of inlet-outlet pressure ratio on the mixing length were investigated. The effect of varying the respective inlet pressures and inlet velocities of the two streams on the mixing behavior was also investigated. Temperatures varied both in streamwise and transverse directions for the gases. Diffusion velocities of both H2 and O2 were plotted to show the diffusion effect during the mixing process. A ‘mixing length’ was specified according to the variation of mass density contours. The effects of inlet-outlet pressure difference and inlet pressure ratio on mixing were also systematically investigated. The effect of the wall ‘accomodation coefficient’ on the mixing process was found to be significant for the cases studied. The results indicate that mixing behavior of parallel gas streams under rarefied and/or in microchannel geometries are 119
fundamentally different from the mixing behavior under continuum conditions. The results
presented can be of value in the design of microdevices intended for gas mixing. The
relationships of mixing length versus inlet-outlet pressure difference and inlet pressure ratio were
found for a specific geometry and physical conditions. This research work is published in Yan and
Farouk (2002b).
A method for implementing parallel processing of Bird’s DSMC algorithm for three-
dimensional flows using OpenMP was developed. The performance of the parallel method was
evaluated by simulating three-dimensional Couette flows. Application of the code to simulations
of subsonic, rarefied flow through a channel was carried out on an IBM shared memory parallel
computer.
The performance of the method demonstrates that parallel processing can clearly
provide significant reduction in computation times for DSMC calculations. Proper granularity
provides reasonable load balancing among threads. Avoiding critical region for mutual exclusion
can decrease the synchronization overhead. For larger problems, parallel processing can
provide more computational efficiency. The parallelized DSMC code was applied to investigate three-dimensional rarefied Couette flow problems. The Couette flow in high Kn number regime is found to be very different from its laminar continuum counterpart. Velocity profiles become more uniform as Kn number increases. Pressure gradient is presented in all three directions even when equal pressure boundary conditions are imposed at the inlet and outlet faces. The
‘ultimate pressures’ from slip flow to the molecular flow regime were also calculated for three different moving speeds. Higher speed moving wall can provide a better pumping effect. The variation of compression ratio with Kn numbers was also computed. The maximum compression ratio is reached in the transition flow regime. The model developed will find potential application in the characterization and design of molecular drag pumps. This research work is published in
Yan et al. (2002). 120
8.1.2. Electroosmotic and electrophoretic microchannel liquid flows
A mathematical model describing electrokinetically driven mass transport phenomena in microfabricated chip devices was presented. The model accounts for the principal physical phenomena affecting sample mass transport in microchip channels. The set of equations describing the electric field distribution and sample mass transport is formulated. A numerical scheme that implements the model was developed to simulate transport of materials during electrokinetic manipulation in channel structures. The computer code allows arbitrary channel geometries with various boundary conditions for electric field and the sample concentration.
Numerical simulations were carried out to investigate the species separation in a microfabricated device. Both electroosmosis and electrophoresis transport phenomena were considered in microchannel fluid flow. Fluid flow fields and concentration contour evolution were simulated for loading step and dispensing step. The model predicts that the shape of the sample plug and species separation are strongly dependent on the electric field distribution. The species separation resolution was calculated from retention time and sample concentration band width. A series of simulations were employed for a wide range of electric field distributions. The retention time of each species is highly dependent on the electric distribution in the dispensing step while it is almost independent of the loading schemes. The maximum concentration value increases with weak confinement and weak pullback. The band width of species decreases significantly with weak pullback but is lowly dependent on different loading steps. High species separation resolution is found for weak pullback.
8.2. Recommendations for future work
The present research provides insight on high Kn number gas flows and microchannel
electrokinetic liquid flows by numerical simulations. To further advance the knowledge in both
areas, the following tasks are recommended for future researchers.
121
8.2.1. Application of DSMC code for plasma-gas simulations
The present DSMC code can be combined with particle-in-cell (PIC) into a PIC-DSMC model for self-consistent simulations of thermal and non-thermal plasmas. In plasma simulation using PIC-DSMC model, chemical reactions such as dissociation and recombination need to be taken into consideration. This approach is based on the weighting collision simulation scheme allowing for disparate number densities and time scales of different species. This algorithm can be applied to plasma-gas simulation for such technologies as sputtering, dry etching, plasma enhanced vapor deposition, etc. The simulations can be beneficial to the semiconductor industries in that they will reduce the cost of expensive experiments.
8.2.1. Development of hybrid model of DSMC and Navier-Stokes solver
The hybrid model uses different models to describe the physics in different regimes of physical time or space, such as combining Navier-Stokes and DSMC for various regions of space, depending on the properties of the flow in the different regimes. This type of computations could reduce the computational time and cost, as well as increase the accuracy, of many types of practical simulations. In hybrid DSMC and Navier-Stokes solvers, it is necessary to specify certain criterion to decide where to use which method, and build up several different spatial interface conditions for connecting the two methods. Also, the difficult problem coupling the physical variables at the interface, where particles (density), momentum, and energy must be conserved needs to be addressed.
8.2.3. Parallelization of DSMC using MPI
MPI is a natural paradigm for parallel processing on distributed memory multi-processor computers while OpenMP is preferred on shared memory machines. Supercomputer with hundreds of processors is always built up with distributed memory architecture. Better performance can therefore be expected from parallelized DSMC code with MPI than using
OpenMP on computers with a large number of processors. However, more efforts are required 122 in programming. For example, the data structure in parallel DSMC code can no longer be inherited from its sequential counterpart. Moreover, because in MPI data communication must be implemented explicitly, more difficulty may be encountered in synchronization.
8.2.4. Extension of electrokinetically driven flow model to predict microfluidic temperature gradient focusing
A new technique is developed recently for the concentration and separation of ionic species in solution within microchannels or capillaries. Concentration is achieved by balancing the electrophoretic velocity of a species against the bulk flow of solution in the presence of a temperature gradient. With an appropriate buffer, the temperature gradient can generate a corresponding gradient in the electrophoretic velocity, so that the electrophoretic and bulk velocities sum to zero at a unique point, and the species will be focused at that point. This method potentially simplifies device construction and operation and allows for highly efficient focusing of any charged species. A numerical model is desirable to predict this process with
Joule heating considered. The energy equation must be coupled with the momentum equation and Poisson equation to simulate the physical phenomenon. 123
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VITA
Fang Yan was born on October 31, 1971 in Chengdu, China. He received his B.S. and
M.S. degrees in Fluid Engineering from Xi’an Jiaotong University, Xi’an, China in 1993 and
1996 respectively. He joined the doctoral program in the department of Mechanical Engineering and Mechanics at Drexel University, Philadelphia in 1998. His research interests include numerical simulations of high Kn number gas flows, electro-kinetically driven microchannel liquid flows, free surface flows, electrohydrodynamically driven capillary jets and nanofibers, turbulent flows, plasma discharges and material processing, and high performance computing.