PREDICTION OF MIXING OF TWO PARALLEL GAS STREAMS IN A MICROCHANNEL USING
THE DIRECT SIMULATION MONTE CARLO METHOD
F.YanandB.Farouk @
Mechanical Engineering and Mechanics Department
Drexel University
Philadelphia, PA 19104
Telephone: 215 895 2287; Fax: 215 895 1478; e-mail: bfarouk@coe.drexel.edu
PACS numbers: 47.60 Fluid flow in ducts, channels, and conduits, 47.45 rarefied gas dynamics
ABSTRACT
Most micro-structure-related gas flows are usually in the slip and transitional flow regimes. Such flows
are characterized by non-continuum behavior. For numerical analysis, the direct simulation Monte Carlo (DSMC)
method offers a well established method to investigate such gas flows at high Knudsen number (Kn). In this paper,
the DSMC method was employed to explore mixing flows in microchannels. Mixing of two parallel gas streams
(H2 and 02) was considered within a microchannel. The effects of inlet velocity, the inlet-outlet pressure difference
and the pressure ratio of the incoming streams f>H2,Met/Po2,Met)
on
mixing length were examined. The simulation
results indicate that mixing decreases with the increase of inlet-outlet pressure difference. When the two streams
enter the microchannel with different pressures, the mixing is also found to decrease with the increase of the
pressure ratio. When the inlet and outlet pressures are held fixed along with Q inlet velocity, higher inlet H>
velocity is also found to inhibit mixing.
INTRODUCTION
During the past decade, microelectromechanical systems (MEMS)1'
2
have become important in many
disciplines because of their extraordinary advantages for practical use. These devices, such as microactuators,
microsensors, microgenerators, etc. are very small (usually micron-sized) and are manufactured with the techniques
developed from those used for integrated circuits. The applications of MEMS have expanded rapidly into such fields as
instrumentation, microelectronics, bioengineering 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 tens to hundreds of microns is critical3. It is
absolutely necessary to understand these phenomena to develop this relatively new technology. Both experimental4 and
numerical efforts5'6 have been reported in the literature towards this objective.
Traditional 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. This approximation of microscale
phenomena fails as the characteristic length of the flow gradients (L) approaches the average distance traveled by
molecules between collisions (mean free path A). The ratio of these quantities is known as the Knudsen number (Kn =
@
Corresponding author
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
510
A/L ) and is used to indicate the degree of flow rarefaction. The Navier-Stokes equations ignore rarefaction effects and
are therefore only strictly accurate at small Kn. The Kn domain (0 < Kn < °o) is often divided into four regimes. When
Kn < 0.01, the flow is considered to be in the continuum regime; for 0.01 < Kn < 0.1, it is in the slip-flow regime, for 0.1
< Kn < 3, it is called the transitional flow regime. When Kn > 3, the fluid flow is considered to be a free molecular flow
and is sufficiently rarefied to allow molecular collisions to be completely neglected in analysis. The collisionless
Boltzmann equation can then be applied. In microchannel gas flows, the Kn is in the rarefied gas flow regime (0.01 <
Kn<3).
The direct simulation Monte Carlo method (DSMC) is a well-established7 alternative 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. Further, the DSMC
uncouples the analysis of the molecular motion from that of the molecular collisions by use of time steps smaller than
the real physical collision time. Continuous improvements in efficiency and accuracy of DSMC code have been
accomplished by Bird7 and other researchers over the last several decades. His method has been used widely and
successfully to simulate variety of high Kn gas flow problems. DSMC technique has also been implemented for the
analysis of both low-pressure flows8 and MEMS flows9'10.
Mixing of gas streams 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 mixing process is a necessary prerequisite to setting criteria for the control of
energy release in reaction chambers. Both supersonic and subsonic mixing layers have been studied intensely using
traditional CFD techniques. However, similar mixing 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 is now necessary and the knowledge can
provide tools to facilitate the design and optimization of such devices.
In this study, we investigate the gas flow and heat transfer for moderately high Kn ( > 0.01) flows in a
microchannel using DSMC. Parallel H2 and O^ streams with different inlet pressures 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 mass density field. Mixing problems with different inlet pressure ratios, PH2,inie/Po2,iniet (with the
same inlet velocities), different inlet pressures (with the same inlet pressure ratio and inlet velocities) and different inlet
velocities VH2,Met/Vo2,Met (with same inlet pressures) were calculated.
PROBLEM GEOMETRY
A slip-flow case with nitrogen gas (non-mixing case) is considered first. The two-dimensional channel we consider is
1.2 jim in width and 36 jim in length. The inlet pressure is 191 kPa while the outlet pressure is atmospheric. The inlet and the
wall temperatures are set to 300 K.
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The mixing in a microchannel was simulated next for parallel streams of hydrogen and oxygen entering the
computational domain with different inlet pressures and inlet streamwise velocities. Two streams of different gases formed the
mixing flow system we simulated, as shown in Figure 1.
02
Twan = 300 K
Figure 1. Schematic of the micro-mixing problem geometry
Two partially diffuse reflecting parallel walls bound the flow domain. The width of the channel was chosen as 1 jiim
while the length was 8 jiim. The wall temperatures were set to 300 K. The outlet pressure was kept at 50 kPa for all
cases. The length required for the two gases to be fully mixed (defined here as the 'mixing length') was determined
from the predicted mass density field. A 'mixing length' is defined as the length required for the mass density
contour to be transversely symmetric. Due to the transverse temperature gradients, the defining mass density contour
is taken as symmetric instead of being vertical. Conceptually, the mixing length can be also determined when the
diffusion velocity of any species becomes zero. However, because of statistical scatter, such an approach is not followed
and we calculated the mixing length from the computed mass density field.
The computations performed are divided into three categories:
(a) Inlet pressures (with PH2,iniet/Po2,iniet= 1) are varied while the inlet velocities are held fixed.
(b) Inlet pressure ratio is varied by varying PH2,Met> while Po2,Met and the inlet velocities are held fixed.
(c) Inlet velocity ratio is varied by varying Vn2,Met? while Vo2,Met and the inlet pressures are held fixed.
Table 1 below lists the mixing flow cases considered.
Table 1. Inlet conditions for the mixing flows
Case
PH2,Met (kPa)
VH2(m/s)
Po2,Met (kPa)
Vo2(m/s)
1 (base case)
200
150
200
60
2
60
150
60
60
3
150
150
150
60
4
300
150
300
60
5
100
150
200
60
6
400
150
200
60
7
200
200
200
60
8
200
100
200
60
512
COMPUTATIONAL METHOD
The direct simulation Monte Carlo method 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 a large number of statistically representative particles. The
positions, velocities and internal energies of these simulated particles are stored and modified in time in the process of
motion and interaction of the particles. DSMC calculations are all treated as unsteady, even though the boundary
condition may be given as steady state. In the DSMC method, each simulation consists of four primary processes: (a)
move the particles, (b) index and cross-reference the particles, (c) simulate collisions, and (d) sample the field. These
processes are uncoupled during each time step. The basic flow chart of the conventional DSMC can be found in Oran et
al9 and Bird7.
A 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 conditions. 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. After steady-state conditions were reached, computations are continued to increase the
sample sizes (per cell) for computing the macroscopic flow variables. The sample size used in the present study is of the
order of 107. The time step is chosen to be less than one-third of the typical particle mean collision time. The variables
hard sphere (VMS) model is used to model the collision because it has been widely used in both slip and transitional
flow simulations7. The gas properties used in calculations such as molecular diameter, molecular mass, viscosity-
temperature index and reference temperature were obtained from Bird7.
For single component flow, (following Piekos and Breuer11 ) the temperature, pressure and the transverse
velocity are specified at the inlet, while the streamwise velocity is determined according to the streamwise velocity inside
the channel. For mixing problems (Fig. 1), we specified the streamwise velocities, transverse velocities and pressures of
both streams, while the temperatures of two streams are calculated from the flow state inside the channel. At the exit
face, only pressure is specified and temperature, transverse velocity and streamwise velocity are all set to have zero
streamwise gradients. In the calculations, the solid walls are all considered as diffuse reflectors with thermal and
momentum accommodation coefficient a = 0.85. At the start of a calculation, all particles are initialized with the
temperature 300 K and a streamwise velocity equal to zero.
RESULTS AND DISCUSSION
MicroChannel flow
For the non-mixing case, non-linear pressure profile along the channel was observed. The agreement between the
simulation result and experimental data4 was excellent. Corresponding low-pressure gas flows with similar Kn in wide-gap
channels were also simulated. Bulk gas flow and heat transfer characteristics were found to be essentially same when the Kn is
same and the boundaries are geometrically similar. According to analytical results based on the slip-flow solution of Navier-Stokes
equations , 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 lengths when the flow is isothermal. Non-linear
pressure profile along the channel is observed. The agreement between the simulation results and experimental data13 is excellent.
513
Mixing flow simulation
The mixing fluid flows in a microchannel were simulated for a gas mixture of hydrogen and oxygen with
different inlet pressures and different inlet streamwise velocities. The inflowing lower stream is H2, at a higher velocity
than the inflowing O2 stream with velocity 60 m/s, held fixed for all cases (see Table 1). The outlet pressure is kept as 50
kPa for all cases.
For the base case, H2 flows into the microchannel with velocity 150 m/s . The inlet pressures of H2 and 0^ were
both 200 kPa. The calculations for the base case were repeated with different mesh sizes to establish the grid independent
results. A mesh size of 600 x 80 was found to be adequate. After the fluid flow reached steady state, there were about
2.5 x ICf particles contained in the flow domain. The computed mass density contours for the base case are shown in
Figure 2.
0
5E-07
1E-06
1 .5 E -0 6
X
2 E - 0 6
2 . 5 E - 0 6
( m )
Figure 2. Mass density contours near the inlet region for the base case (values in kg/m3)
The channel length was long enough for H and Q to mix with each other completely before they left the
channel. The outlet Kn was found to be 0.157. Although the inflowing number densities of two streams are the same,
the mass density contour at the inlet region is quite asymmetric along the centerline of the channel, because of large mass
difference between H2 and C^,. The mixing is complete when the density contours across the channel are symmetric
according to the center line of the channel. From the density contours we can discern that the 'mixing length' is about 2.4
jiim for the base case.
Figure 3 shows the 'diffusion' velocity vectors for hydrogen in the microchannel. Because of molecular
diffusion, two different gases eventually mix completely with each other. The diffusion velocity of a particular species is
defined here as its mean velocity relative to the mass averaged velocity 7. Hydrogen is found to vigorously diffuse in to
514
the upper layer near the inlet. As the mixing progresses, the diffusion velocity of hydrogen becomes smaller.
;
;
^3A>V
-^^^"t^ \C* \£ *'s */• ',' *'*
—-..•-*•—•
••- ——-*•— *^*
- _.--^—
•*»• ...•—••^ —-«——
—•» —••"
.-» **•—.• -».** —•—
5
E
- 0
.
;
S
/i
7
1
. 5
E
- 0
6
Figure 3. The diffusion velocity of H2 at the inlet for the base case
Figure 4 depicts the temperature contours for the base case. Due to the effect of rarefaction, both streams attain
temperatures that are slightly lower than the wall temperature. It is also seen in Figure 4, 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 midplane of the inlet where the transverse velocities are rather high. A thermal boundary
layer type structure gradually forms along the walls due to the imposed thermal boundary condition.
4 E-06
X (m)
Figure 4. Temperature contours for the base case (values in K)
Calculations were repeated by changing the inlet pressures of IJ and Q while the pressure ratio
PH2,Met/Po2,inietwas kept equal to 1.0 (cases 1 to 4 in Table 1). The inlet pressure values 60 kPa, 150 kPa and 300
kPa were used in addition to the base case value of 200 kPa. The mixing lengths for the above cases were
determined from the mass density contours as 1.8 |nm, 2.0 |nm and 3.0 |im respectively. The variation of mixing
length versus the inlet pressure is plotted in Figure 5. The mixing length increases with the increase of inlet pressure
(with outlet pressure held at 50 kPa). When the inlet-outlet pressure difference is large, the convection effect
becomes more pronounced compared to molecular diffusion.
515
3.4 •
3.2 ;
3;
2.8
?
b
2.6
JC
B
2.4
1
2.2
I
2
1.8
2 00
1
300
15
Pressure Ratio
Pressure (kPa)
Figure 5. The variation of mixing length for
different inlet pressure (inlet pressure ratio = 1.0)
Figure 6. The variation of mixing length for
different inlet pressure ratio (Po Met = 200 kPa)
We then investigated how different inlet pressure ratios affect the mixing length. We calculated two additional
cases (5 and 6 in Table 1) by changing the inlet pressure of H2 stream (to 100 and 400 kPa respectively) while keeping
other conditions unchanged. The variation of mixing length versus the inlet pressure ratio is plotted in Figure 6. The
mixing length increases as the pressure ratio becomes higher. For all of the three cases considered in Figure 6, the inlet
momentum of Oz were held fixed and 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. H2 particles traveled longer distance in streamwise direction
before penetrating the Oz stream. It is interesting to note that we obtain a linear relationship between the mixing length
and the inlet pressure ratio.
C 2.5E-06
£
|
1 2.4E-06
Velocity ( m i s )
Figure 7. The variation of mixing length for different inlet velocity of H2
The effect of inlet velocity ratio on the mixing was also investigated. The inlet velocity, and pressure of (>> are
kept the same as in the base case. The inlet pressure of H2 is also set to the base case value, i.e., 200 kPa. We calculated
two cases (7 and 8 in Table 1) with inlet H2 velocity equal to 100 m/s and 200 m/s respectively. The variation of mixing
516
length versus the H2 inlet velocity is shown in Figure 7. The mixing length increases with the increase of the inlet
velocity ratio. The increase of inlet velocity of Ek means more momentum for the E^ stream. Longer distance is thus
necessary for mixing. The increase of inlet velocity-ratio thus has similar effect on mixing as the increase of inlet
pressure ratio.
CONCLUSIONS
DSMC calculations of mixing of parallel E| and 0^ 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. Along the microchannel, thermal boundary layer like
structures are formed. 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 results indicate that
mixing behavior of parallel gas streams under rarefied and/or in microchannel geometries are 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,
inlet pressure ratio and inlet velocity were found for a specific geometry and physical conditions..
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