Poiseuille number, the product of friction factor and Reynolds number (fRe) for quasi-fully-developed gas microchannel flow in the slip flow regime, was obtained numerically based on the arbitrary-Lagrangian-Eulerian method. Two-dimensional compressible momentum and energy equations were solved for a wide range of Reynolds and Mach numbers for constant wall temperatures that are lower or higher than the inlet temperature. The channel height ranges from 2 μm to 10 μm and the channel aspect ratio is 200. The stagnation pressure $pstg$ is chosen such that the exit Mach number ranges from 0.1 to 1.0. The outlet pressure is fixed at atmospheric conditon. Mach and Knudsen numbers are systematically varied to determine their effects on $fRe$. The correlation for $fRe$ for the slip flow is obtained from that of $fRe$ of no-slip flow and incompressible theory as a function of Mach and Knudsen numbers. The results are in excellent agreement with the available experimental measurements. It was found that $fRe$ is a function of Mach and Knudsen numbers and is different from the values by 96/(1+12Kn) obtained from the incompressible flow theory.

1.
Wu
,
P.
, and
Little
,
W. A.
, 1983, “
Measurement of Friction Factors for the Flow of Gases in Very Fine Channels Used for Microminiature Joule-Thomson Refrigerator
,”
Cryogenics
0011-2275,
23
, pp.
273
277
.
2.
Wu
,
P.
, and
Little
,
W. A.
, 1984, “
Measurement of the Heat Transfer Characteristics of Gas Flow in Fine Channel Heat Exchangers Used for Microminiature Refrigerators
,”
Cryogenics
0011-2275,
24
, pp.
415
420
.
3.
Arkilic
,
E. B.
,
Schmidt
,
M. A.
, and
Breuer
,
K. S.
, 1994, “
Gaseous Flow in Microchannel
,”
ASME Symposium on Micro Machining and Fluid Mechanics
, pp.
1
10
.
4.
Beskok
,
A.
, and
,
G. E.
, 1994, “
Simulation of Heat and Momentum Transfer in Complex Microgeometries
,”
J. Thermophys. Heat Transfer
0887-8722,
8
(
4
), pp.
647
655
.
5.
Beskok
,
A.
,
,
G. E.
, and
Trimmer
,
W.
, 1996, “
Rarefaction and Compressibility Effects in Gas Microflows
,”
ASME J. Fluids Eng.
0098-2202,
118
, pp.
448
456
.
6.
Ameel
,
T. A.
,
Wang
,
X.
,
Barron
,
R. F.
, and
Warrington
,
R. O.
, 1997, “
Laminar Forced Convection in a Circular Tube With Constant Heat Flux and Slip Flow
,”
Microscale Thermophys. Eng.
1089-3954,
1
(
4
), pp.
303
320
.
7.
Ahmed
,
I.
, and
Beskok
,
A.
, 2002, “
Rarefaction, Compressibility, and Viscous Heating in Gas Microfilters
,”
J. Thermophys. Heat Transfer
0887-8722,
16
(
2
), pp.
161
170
.
8.
Renksizbulut
,
M.
,
Niazmand
,
H.
, and
Teracan
,
G.
, 2006, “
Slip-Flow and Heat Transfer in Rectangular Microchannels With Constant Wall Temperature
,”
Int. J. Therm. Sci.
1290-0729,
45
, pp.
870
881
.
9.
Prud"homme
,
R. K.
,
Chapman
,
T. W.
, and
Bowen
,
J. R.
, 1986, “
Laminar Compressible Flow in a Tube
,”
Appl. Sci. Res.
0003-6994,
43
, pp.
67
74
.
10.
Berg
,
H. R.
,
Seldam
,
C. A.
, and
Gulik
,
P. S.
, 1983, “
Compressible Laminar Flow in a Capillary
,”
J. Fluid Mech.
0022-1120,
246
, pp.
1
20
.
11.
Guo
,
Z. Y.
, 2000, “
Size Effect on Flow and Heat Transfer Characteristics in MEMS
,”
Proceedings of the International Conference on Heat Transfer and Transport Phenomena in Microscale
,
, pp.
24
31
.
12.
Sayegh
,
R.
,
Faghri
,
M.
,
Asako
, and
Y.
,
Sunden
,
B.
, 1999, “
Direct Simulation Monte Carlo of Gaseous Flow in Micro-Channel
,”
Proceedings of the ASME National Heat Transfer Conference
,
Albuquerque, NM
, Paper No. HTD99-256.
13.
Sun
,
H.
, and
Faghri
,
M.
, 1999, “
Effect of Rarefaction and Compressibility of Gaseous Flow in Microchannel Using DSMC
,”
Numer. Heat Transfer, Part A
1040-7782,
38
, pp.
153
158
.
14.
Araki
,
T.
,
Kim
,
M. S.
,
Inaoka
,
K.
, and
Suzuki
,
K.
, 1999, “
An Experimental Investigation of Gaseous Flow Characteristics in Micro-Tubes
,”
Proceedings of Kyoto University-Tsinghua University Joint Conference on Energy and Environment
, pp.
7
12
.
15.
Araki
,
T.
,
Kim
,
M. S.
,
Inaoka
,
K.
, and
Suzuki
,
K.
, 2002, “
An Experimental Investigation of Gaseous Flow Characteristics in Microchannels
,”
Microscale Thermophys. Eng.
1089-3954,
6
, pp.
117
130
.
16.
Turner
,
S. E.
,
Sun
,
H.
,
Faghri
,
M.
, and
Gregory
,
O. J.
, 2000, “
Effect of Surface Roughness on Gaseous Flow Through Microchannels
,” IMECE, HTD-Vol
366
, pp.
291
298
.
17.
Turner
,
S. E.
,
Lam
,
L. C.
,
Faghri
,
M.
, and
Gregory
,
O. J.
, 2004, “
Experimental Investigation of Gas Flow in Microchannel
,”
ASME J. Heat Transfer
0022-1481,
127
, pp.
753
763
.
18.
Asako
,
Y.
,
Pi
,
T.
,
Turner
,
S. E.
, and
Faghri
,
M.
, 2003, “
Effect of Compressibility on Gaseous Flows in Micro-Channels
,”
Int. J. Heat Mass Transfer
0017-9310,
46
, pp.
3041
3050
.
19.
Asako
,
Y.
,
Nakayama
,
K.
, and
Shinozuka
,
T.
, 2005, “
Effect of Compressibility on Gaseous Flows in a Micro Tube
,”
Int. J. Heat Mass Transfer
0017-9310,
48
, pp.
4985
4994
.
20.
Ji
,
Y.
,
Yuan
,
K.
, and
Chung
,
J. N.
, 2006, “
Numerical Simulation of Wall Roughness on Gaseous Flow and Hea Transfer in a Microchannel
,”
Int. J. Heat Mass Transfer
0017-9310,
49
, pp.
1329
1339
.
21.
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
1070-6631,
17
(
10
), p.
100609
.
22.
Sparrow
,
E. M.
, and
Lin
,
S. H.
, 1962, “
Laminar Heat Transfer in Tubes Under Slip-Flow Conditions
,”
ASME J. Heat Transfer
0022-1481,
84
(
4
), pp.
363
369
.
23.
Karki
,
K. C.
, 1986, “
A Calculation Procedure for Viscous Flows at All Speeds in Complex Geometries
,” Ph.D. thesis, University of Minnesota, Minneapolis.
24.
Churchill
,
S. W.
, 1988,
Viscous Flows: The Practical Use of Theory
,
Butterworths
,
Boston
.
25.
Amsden
,
A. A.
,
Ruppel
,
H. M.
, and
Hire
,
C. W.
, 1980, “
SALE a Simplified ALE Computer Program for Fluid Flow at All Speeds
,” Los Alamos Scientific Laboratory Report No. LA-8095.
26.
Roche
,
P. J.
, 1994, “
Perspective: A Method for Uniform Reporting of Grid refinement Studies
,”
ASME J. Fluids Eng.
0098-2202,
116
, pp.
405
413
.
27.
Shah
,
R. K.
, and
London
,
A. L.
, 1978,
Laminar Flow Forced Convection in Ducts, Advances in Heat Transfer Supplement, 1
,