This work utilizes the direct simulation Monte Carlo (DSMC) calculations and examines the influence of rarefication on the mixing length and effective diffusion coefficient in a two-species mixing problem. There have been efforts in past rarefied mixing flow studies to bridge between the mixing evolution rate and Knudsen number. A careful review of those efforts shows that the past derived relations did not determine the weights of Reynolds (or Peclet) number in the rarefaction influences. Although they indicated that an increase in Knudsen would decrease the mixing length, such reductions were primarily due to the Reynolds (or Peclet) reduction. Therefore, those studies could not explicitly appraise the contribution of rarefaction in the total mass diffusion magnitude. This work focuses specifically on the role of rarefaction in the total diffusive mass transfer magnitude in rarefied gas mixing problems. It excludes the contributions of momentum and heat to the mass diffusion via imposing suitable velocity, pressure, and temperature fields in the mixer domain. The results show that there will be some decreases in the diffusive mass fluxes and some increases in the mixing length as Knudsen increases. Using the Fick’s law, the effective diffusion coefficient is then calculated in the mixer zone. The results show that this coefficient may vary considerably throughout the mixer zone due to the local rarefaction level variation. The results of all investigated cases indicate that the trends of their effective diffusion coefficient variations approach to a limiting value as the rarefaction level decreases.

References

References
1.
Agarwal
,
R. K.
,
Yun
,
K.-Y.
, and
Balakrishnan
,
R.
,
2001
, “
Beyond Navier–Stokes: Burnett Equations for Flows in the Continuum–Transition Regime
,”
Phys. Fluids
,
13
(
10
), pp.
3061
3085
.
2.
Torrilhon
,
M.
,
2011
, “
Regularization of Grad’s 13-Moment-Equations in Kinetic Gas Theory
,” RWTH Aachen University, Aachen, Germany, Report No.
RTO-EN-AVT-194
http://www.mathcces.rwth-aachen.de/torrilhon/Torrilhon_VKI2011.pdf.
3.
Gu
,
X.-J.
, and
Emerson
,
D. R.
,
2009
, “
A High-Order Moment Approach for Capturing Non-Equilibrium Phenomena in the Transition Regime
,”
J. Fluid Mech.
,
636
, pp.
177
216
.
4.
Torrilhon
,
M.
,
2016
, “
Modeling Nonequilibrium Gas Flow Based on Moment Equations
,”
Annu. Rev. Fluid Mech.
,
48
, pp.
429
458
.
5.
Lockerby
,
D. A.
,
Reese
,
J. M.
, and
Gallis
,
M. A.
,
2005
, “
The Usefulness of Higher-Order Constitutive Relations for Describing the Knudsen Layer
,”
Phys. Fluids
,
17
(
10
), p.
100609
.
6.
O’Hare
,
L.
,
Lockerby
,
D. A.
,
Reese
,
J. M.
, and
Emerson
,
D. R.
,
2007
, “
Near-Wall Effects in Rarefied Gas Micro-Flows: Some Modern Hydrodynamic Approaches
,”
Int. J. Heat Fluid Flow
,
28
(
1
), pp.
37
43
.
7.
Watari
,
M.
,
2015
, “
Is the Lattice Boltzmann Method Applicable to Rarefied Gas Flows? Comprehensive Evaluation of the Higher-Order Models
,”
ASME J. Fluids Eng.
,
138
(
1
), p.
011202
.
8.
Karniadakis
,
G.
,
Beskok
,
A.
, and
Aluru
,
N. R.
,
2005
,
Microflows and Nanoflows: Fundamentals and Simulation
,
Springer
,
New York
, Chap. 4.
9.
Roohi
,
E.
, and
Darbandi
,
M.
,
2009
, “
Extending the Navier–Stokes Solutions to Transition Regime in Two-Dimensional Micro- and Nanochannel Flows Using Information Preservation Scheme
,”
Phys. Fluids
,
21
(
8
), p.
082001
.
10.
Michalis
,
V. K.
,
Kalarakis
,
A. N.
,
Skouras
,
E. D.
, and
Burganos
,
V. N.
,
2010
, “
Rarefaction Effects on Gas Viscosity in the Knudsen Transition Regime
,”
Microfluid. Nanofluid.
,
9
(
4–5
), pp.
847
853
.
11.
To
,
Q.
,
Leonard
,
C.
, and
Lauriat
,
G.
,
2015
, “
Free-Path Distribution and Knudsen-Layer Modeling for Gaseous Flows in the Transition Regime
,”
Phys. Rev. E
,
91
(
2
), p.
023015
.
12.
Tu
,
C.
,
Qian
,
L.
,
Bao
,
F.
, and
Yan
,
W.
,
2017
, “
Local Effective Viscosity of Gas in Nano-Scale Channels
,”
Eur. J. Mech. B/Fluids
,
64
, pp.
55
59
.
13.
Zhu
,
C.-Y.
,
Li
,
Z.-Y.
, and
Tao
,
W.-Q.
,
2017
, “
Theoretical and DSMC Studies on Heat Conduction of Gas Confined in a Cuboid Nanopore
,”
ASME J. Heat Transfer
,
139
(
5
), p.
052405
.
14.
Fichman
,
M.
, and
Hetsroni
,
G.
,
2005
, “
Viscosity and Slip Velocity in Gas Flow in Microchannels
,”
Phys. Fluids
,
17
(
12
), p.
123102
.
15.
Lockerby
,
D. A.
,
Reese
,
J. M.
, and
Gallis
,
M. A.
,
2005
, “
Capturing the Knudsen Layer in Continuum-Fluid Models of Non-Equilibrium Gas Flows
,”
AIAA J.
,
43
(
6
), pp.
1391
1393
.
16.
Guo
,
Z. L.
,
Shi
,
B. C.
, and
Zheng
,
C. G.
,
2007
, “
An Extended Navier–Stokes Formulation for Gas Flows in the Knudsen Layer Near a Wall
,”
EPL
,
80
(
2
), p.
24001
.
17.
Lilley
,
C. R.
, and
Sader
,
J. E.
,
2007
, “
Velocity Gradient Singularity and Structure of the Velocity Profile in the Knudsen Layer According to the Boltzmann Equation
,”
Phys. Rev. E
,
76
(
2
), p.
026315
.
18.
O’Hare
,
L.
,
Scanlon
,
T. J.
,
Emerson
,
D. R.
, and
Reese
,
J. M.
,
2008
, “
Evaluating Constitutive Scaling Models for Application to Compressible Microflows
,”
Int. J. Heat Mass Transfer
,
51
(
5
), pp.
1281
1292
.
19.
Lilley
,
C. R.
, and
Sader
,
J. E.
,
2008
, “
Velocity Profile in the Knudsen Layer According to the Boltzmann Equation
,”
Proc. R. Soc. A
,
464
(
2096
), pp.
2015
2035
.
20.
Lockerby
,
D. A.
, and
Reese
,
J. M.
,
2008
, “
On the Modelling of Isothermal Gas Flows at the Microscale
,”
J. Fluid Mech.
,
604
, pp.
235
261
.
21.
Dongari
,
N.
,
Zhang
,
Y.
, and
Reese
,
J. M.
,
2011
, “
Modeling of Knudsen Layer Effects in Micro/Nanoscale Gas Flows
,”
ASME J. Fluids Eng.
,
133
(
7
), p.
071101
.
22.
Zhang
,
Y.-H.
,
Gu
,
X.-J.
,
Barber
,
R. W.
, and
Emerson
,
D. R.
,
2006
, “
Capturing Knudsen Layer Phenomena Using a Lattice Boltzmann Model
,”
Phys. Rev. E
,
74
(
4
), p.
046704
.
23.
Norouzi
,
A.
, and
Esfahani
,
J. A.
,
2015
, “
Two Relaxation Time Lattice Boltzmann Equation for High Knudsen Number Flows Using Wall Function Approach
,”
Microfluid. Nanofluid.
,
18
(
2
), pp.
323
332
.
24.
Norouzi
,
A.
, and
Esfahani
,
J. A.
,
2016
, “
Capturing Non-Equilibrium Effects of Micro/Nano Scale Gaseous Flow Using a Novel Lattice Boltzmann Model
,”
J. Stat. Phys.
,
162
(
3
), pp.
712
726
.
25.
Wang
,
L.
,
Xu
,
Z.
, and
Guo
,
Z.
,
2016
, “
Lattice Boltzmann Simulation of Separation Phenomenon in a Binary Gaseous Flow Through a Microchannel
,”
J. Appl. Phys.
,
120
, p.
134306
.
26.
Yuan
,
Y.
, and
Rahman
,
S.
,
2016
, “
Extended Application of Lattice Boltzmann Method to Rarefied Gas Flow in Micro-Channels
,”
Phys. A
,
463
, pp.
25
36
.
27.
Mejia
,
J. M.
,
Sadiki
,
A.
,
Molina
,
A.
,
Chejne
,
F.
, and
Pantangi
,
P.
,
2015
, “
Large Eddy Simulation of the Mixing of a Passive Scalar in a High-Schmidt Turbulent Jet
,”
ASME J. Fluids Eng.
,
137
(
3
), p.
031301
.
28.
Lindner
,
G.
,
Schmelter
,
S.
,
Model
,
R.
,
Nowak
,
A.
,
Ebert
,
V.
, and
Bär
,
M.
,
2015
, “
A Computational Fluid Dynamics Study on the Gas Mixing Capabilities of a Multiple Inlet System
,”
ASME J. Fluids Eng.
,
138
(
3
), p.
031302.
29.
Park
,
J.
,
Pagan-Vazquez
,
A.
,
Alvarado
,
J. L.
,
Chamorro
,
L. P.
,
Lux
,
S. M.
, and
Marsh
,
C. P.
,
2016
, “
Characterization of Tab-Induced Counter-Rotating Vortex Pair for Mixing Applications
,”
ASME J. Fluids Eng.
,
139
(
3
), p.
031102
.
30.
Zhou
,
T.
,
Xu
,
Y.
,
Liu
,
Z.
, and
Joo
,
S. W.
,
2015
, “
An Enhanced One-Layer Passive Microfluidic Mixer With an Optimized Lateral Structure With the Dean Effect
,”
ASME J. Fluids Eng.
,
137
(
9
), p.
091102
.
31.
Viktorov
,
V.
,
Visconte
,
C.
, and
Mahmud
,
M. R.
,
2016
, “
Analysis of a Novel Y-Y Micromixer for Mixing at a Wide Range of Reynolds Numbers
,”
ASME J. Fluids Eng.
,
138
(
9
), p.
091201
.
32.
Zhang
,
W.
,
Wang
,
X.
,
Feng
,
X.
,
Yang
,
C.
, and
Mao
,
Z.-S.
,
2016
, “
Investigation of Mixing Performance in Passive Micromixers
,”
Ind. Eng. Chem. Res.
,
55
(
38
), pp.
10036
10043
.
33.
Wang
,
M.
, and
Li
,
Z.
,
2006
, “
Gas Mixing in Microchannels Using the Direct Simulation Monte Carlo Method
,”
Int. J. Heat Mass Transfer
,
49
(
9–10
), pp.
1696
1702
.
34.
Le
,
M.
, and
Hassan
,
I.
,
2007
, “
DSMC Simulation of Gas Mixing in T-Shape Micromixer
,”
Appl. Therm. Eng.
,
27
(
14
), pp.
2370
2377
.
35.
Darbandi
,
M.
, and
Sabouri
,
M.
,
2015
, “
Detail Study on Improving Micro/Nano Gas Mixer Performances in Slip and Transitional Flow Regimes
,”
Sens. Actuators, B
,
218
, pp.
78
88
.
36.
Gad-el-hak
,
M.
,
1999
, “
The Fluid Mechanics of Microdevices—The Freeman Scholar Lecture
,”
ASME J. Fluids Eng.
,
121
(
1
), pp.
5
33
.
37.
Gavriilidis
,
A.
,
Angeli
,
P.
,
Cao
,
E.
,
Yeong
,
K. K.
, and
Wan
,
Y. S. S.
,
2002
, “
Technology and Applications of Microengineered Reactors
,”
Chem. Eng. Res. Des.
,
80
(
1
), pp.
3
30
.
38.
Stephani
,
K. A.
,
Goldstein
,
D. B.
, and
Varghese
,
P. L.
,
2012
, “
Consistent Treatment of Transport Properties for Five-Species Air Direct Simulation Monte Carlo/Navier–Stokes Applications
,”
Phys. Fluids
,
24
(
7
), p.
077101
.
39.
Bird
,
G. A.
,
1994
,
Molecular Gas Dynamics and Direct Simulation of Gas Flows
,
Clarendon Press
,
Oxford, UK
.
40.
Piekos
,
E. S.
, and
Breuer
,
K. S.
,
1996
, “
Numerical Modeling of Micromechanical Devices Using the Direct Simulation Monte Carlo Method
,”
ASME J. Fluids Eng.
,
118
(
3
), pp.
464
469
.
41.
Fang
,
Y.
, and
Liou
,
W. W.
,
2002
, “
Computations of the Flow and Heat Transfer in Microdevices Using DSMC With Implicit Boundary Conditions
,”
ASME J. Heat Transfer
,
124
(
2
), pp.
338
345
.
42.
Wang
,
M.
, and
Li
,
Z.
,
2004
, “
Simulations for Gas Flows in Microgeometries Using the Direct Simulation Monte Carlo Method
,”
Int. J. Heat Fluid Flow
,
25
(
6
), pp.
975
985
.
43.
Le
,
M.
,
Hassan
,
I.
, and
Esmail
,
N.
,
2006
, “
DSMC Simulation of Subsonic Flows in Parallel and Series Microchannels
,”
ASME J. Fluids Eng.
,
128
(
6
), pp.
1153
1163
.
44.
Ewart
,
T.
,
Firpo
,
J. L.
,
Graur
,
I. A.
,
Perrier
,
P.
, and
Me´olans
,
J. G.
,
2009
, “
DSMC Simulation: Validation and Application to Low Speed Gas Flows in Microchannels
,”
ASME J. Fluids Eng.
,
131
(
1
), p.
014501
.
45.
Roohi
,
E.
,
Darbandi
,
M.
, and
Mirjalili
,
V.
,
2009
, “
Direct Simulation Monte Carlo Solution of Subsonic Flow Through Micro/Nanoscale Channels
,”
ASME J. Heat Transfer
,
131
(
9
), p.
092402
.
46.
Yang
,
J.
,
Ye
,
J. J.
,
Zheng
,
J. Y.
,
Wong
,
I.
,
Lam
,
C. K.
,
Xu
,
P.
,
Chen
,
R. X.
, and
Zhu
,
Z. H.
,
2010
, “
Using Direct Simulation Monte Carlo With Improved Boundary Conditions for Heat and Mass Transfer in Microchannels
,”
ASME J. Heat Transfer
,
132
(
4
), p.
041008
.
47.
Darbandi
,
M.
, and
Roohi
,
E.
,
2011
, “
Study of Subsonic–Supersonic Gas Flow Through Micro/Nanoscale Nozzles Using Unstructured DSMC Solver
,”
Microfluid. Nanofluid.
,
10
(
2
), pp.
321
335
.
48.
Darbandi
,
M.
, and
Roohi
,
E.
,
2011
, “
DSMC Simulation of Subsonic Flow Through Nanochannels and Micro/Nano Backward-Facing Steps
,”
Int. Commun. Heat Mass Transfer
,
38
(
10
), pp.
1443
1448
.
49.
Roohi
,
E.
, and
Darbandi
,
M.
,
2012
, “
Recommendations on Performance of Parallel DSMC Algorithm in Solving Subsonic Nanoflows
,”
Appl. Math. Model.
,
36
(
5
), pp.
2314
2321
.
50.
Hadj-Nacer
,
M.
,
Maharjan
,
D.
,
Ho
,
M.-T.
,
Stefanov
,
S. K.
,
Graur
,
I.
, and
Greiner
,
M.
,
2017
, “
Continuum and Kinetic Simulations of Heat Transfer Though Rarefied Gas in Annular and Planar Geometries in the Slip Regime
,”
ASME J. Heat Transfer
,
139
(
4
), p.
042002
.
51.
Roohi
,
E.
, and
Stefanov
,
S.
,
2016
, “
Collision Partner Selection Schemes in DSMC: From Micro/Nano Flows to Hypersonic Flows
,”
Phys. Rep.
,
656
, pp.
1
38
.
52.
Yan
,
F.
, and
Farouk
,
B.
,
2002
, “
Numerical Simulation of Gas Flow and Mixing in a Microchannel Using the Direct Simulation Monte Carlo Method
,”
Microscale Thermophys. Eng.
,
6
(
3
), pp.
235
251
.
53.
Qazi Zade
,
A.
,
Ahmadzadegan
,
A.
, and
Renksizbulut
,
M.
,
2012
, “
A Detailed Comparison Between Navier–Stokes and DSMC Simulations of Multicomponent Gaseous Flow in Microchannels
,”
Int. J. Heat Mass Transfer
,
55
(
17
), pp.
4673
4681
.
54.
Darbandi
,
M.
, and
Sabouri
,
M.
,
2015
, “
Numerical Study of Mixing Enhancement Through Nanomixers Using the Throttling Approach
,”
Sci. Iran. Trans. F: Nanotechnol.
,
22
(
3
), pp.
1306
1316
.http://scientiairanica.sharif.edu/article_3721.html
55.
Zhang
,
H.
, and
Xie
,
M.
,
2015
, “
DSMC Simulation of Non-Premixed Combustion of H2/O2 in a Y-Shaped Microchannel
,”
Nanoscale Microscale Thermophys. Eng.
,
19
(
1
), pp.
31
62
.
56.
Scanlon
,
T. J.
,
Roohi
,
E.
,
White
,
C.
,
Darbandi
,
M.
, and
Reese
,
J. M.
,
2010
, “
An Open Source, Parallel DSMC Code for Rarefied Gas Flows in Arbitrary Geometries
,”
Comput. Fluids
,
39
(
10
), pp.
2078
2089
.
57.
OpenFOAM, 2017, “
OpenFOAM User Guide Version 5.0
,” The OpenFOAM Foundation, London, accessed Oct. 13, 2017, http://foam.sourceforge.net/docs/Guides-a4/OpenFOAMUserGuide-A4.pdf
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