The present investigation is devoted to the fully developed slip flow mixed convection in vertical microducts of two different cross sections, namely, polygon, with circle as a limiting case, and rectangle. The two axially constant heat flux boundary conditions of H1 and H2 are considered in the analysis. The velocity and temperature discontinuities at the boundary are incorporated into the solutions using the first-order slip boundary conditions. The method considered is mainly analytical in which the governing equations in cylindrical coordinates along with the symmetry conditions and finiteness of the flow parameter at the origin are exactly satisfied. The first-order slip boundary conditions are then applied to the solution using the point matching technique. The results show that both the Nusselt number and the pressure drop parameter are increasing functions of the Grashof to Reynolds ratio. It is also found that, with the exception of the H2 Nusselt number of the triangular duct, which shows an opposite trend, both the Nusselt number and the pressure drop are decreased by increasing the Knudsen number. Furthermore, the pressure drop of the H2 case is found to be higher than that obtained by assuming an H1 thermal boundary condition.

References

References
1.
Barber
,
R. W.
, and
Emerson
,
D. R.
,
2006
, “
Challenges in Modeling Gas-Phase Flow in Microchannels: From Slip to Transition
,”
Heat Transfer Eng.
,
27
(
4
), pp.
3
12
.10.1080/01457630500522271
2.
Harley
,
J. C.
,
Huang
,
Y.
,
Bau
,
H. H.
, and
Zemel
,
J. N.
,
1995
, “
Gas Flow in Micro-Channels
,”
J. Fluid Mech.
,
284
, pp.
257
274
.10.1017/S0022112095000358
3.
Araki
,
T.
,
Kim
,
M. S.
,
Iwai
,
H.
, and
Suzuki
,
K.
,
2002
, “
An Experimental Investigation of Gaseous Flow Characteristics in Microchannels
,”
Microscale Thermophys. Eng.
,
6
(
2
), pp.
117
130
.10.1080/10893950252901268
4.
Arkilic
,
E. B.
,
Schmidt
,
M. A.
, and
Breuer
,
K. S.
,
1997
, “
Gaseous Slip Flow in Long Microchannels
,”
J. Microelectromech. Syst.
,
6
(
2
), pp.
167
178
.10.1109/84.585795
5.
Arkilic
,
E. B.
,
Breuer
,
K. S.
, and
Schmidt
,
M. A.
,
2001
, “
Mass Flow and Tangential Momentum Accomodation in Silicon Micromachined Channels
,”
J. Fluid Mech.
,
437
, pp.
29
43
.10.1017/S0022112001004128
6.
Liu
,
J.
,
Tai
,
Y.-C.
, and
Ho
,
C.-M.
, “
MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels
,”
Proceedings of the IEEE Micro Electro Mechanical Systems
, pp.
209
215
.
7.
Choi
,
S. B.
,
Barron
,
R. F.
, and
Warrington
,
R. O.
,
1991
, “
Fluid Flow and Heat Transfer in Microtubes
,”
Proceedings of Micromechanical Sensors, Actuators, and Systems
, ASME, pp.
123
134
.
8.
Taheri
,
P.
,
Torrilhon
,
M.
, and
Struchtrup
,
H.
,
2009
, “
Couette and Poiseuille Microflows: Analytical Solutions for Regularized 13-Moment Equations
,”
Phys. Fluids
,
21
(
1
), p.
017102
.10.1063/1.3064123
9.
Beskok
,
A.
, and
Karniadakis
,
G. E.
,
1994
, “
Simulation of Heat and Momentum Transfer in Complex Microgeometries
,”
J. Thermophys. Heat Transfer
,
8
(
4
), pp.
647
655
.10.2514/3.594
10.
Hadjiconstantinou
,
N. G.
,
2006
, “
The Limits of Navier-Stokes Theory and Kinetic Extensions for Describing Small-Scale Gaseous Hydrodynamics
,”
Phys. Fluids
,
18
(
11
), p.
111301
.10.1063/1.2393436
11.
Aydin
,
O.
, and
Avci
,
M.
,
2006
, “
Heat and Fluid Flow Characteristics of Gases in Micropipes
,”
Int. J. Heat Mass Transfer
,
49
(
9–10
), pp.
1723
1730
.10.1016/j.ijheatmasstransfer.2005.10.020
12.
Aydin
,
O.
, and
Avci
,
M.
,
2007
, “
Analysis of Laminar Heat Transfer in Micro-Poiseuille Flow
,”
Int. J. Therm. Sci.
,
46
(
1
), pp.
30
37
.10.1016/j.ijthermalsci.2006.04.003
13.
Jeong
,
H. E.
, and
Jeong
,
J. T.
,
2006
, “
Extended Graetz Problem Including Axial Conduction and Viscous Dissipation in Microtube
,”
J. Mech. Sci. Technol.
,
20
(
1
), pp.
158
166
.10.1007/BF02916209
14.
Jeong
,
H. E.
, and
Jeong
,
J. T.
,
2006
, “
Extended Graetz Problem Including Streamwise Conduction and Viscous Dissipation in Microchannel
,”
Int. J. Heat Mass Transfer
,
49
(
13–14
), pp.
2151
2157
.10.1016/j.ijheatmasstransfer.2005.11.026
15.
Sadeghi
,
A.
,
Asgarshamsi
,
A.
, and
Saidi
,
M. H.
,
2009
, “
Analysis of Laminar Flow in the Entrance Region of Parallel Plate Microchannels for Slip Flow
,”
ASME Conf. Proc.
,
2009
(
43499
), pp.
345
352
.
16.
Ghodoossi
,
L.
, and
Eğrican
,
N.
,
2005
, “
Prediction of Heat Transfer Characteristics in Rectangular Microchannels for Slip Flow Regime and H1 Boundary Condition
,”
Int. J. Therm. Sci.
,
44
(
6
), pp.
513
520
.10.1016/j.ijthermalsci.2005.01.006
17.
Kuddusi
,
L.
, and
Çetegen
,
E.
,
2007
, “
Prediction of Temperature Distribution and Nusselt Number in Rectangular Microchannels at Wall Slip Condition for All Versions of Constant Heat Flux
,”
Int. J. Heat Fluid Flow
,
28
(
4
), pp.
777
786
.10.1016/j.ijheatfluidflow.2006.09.002
18.
Yu
,
S.
, and
Ameel
,
T. A.
,
2001
, “
Slip-Flow Heat Transfer in Rectangular Microchannels
,”
Int. J. Heat Mass Transfer
,
44
(
22
), pp.
4225
4234
.10.1016/S0017-9310(01)00075-8
19.
Yu
,
S.
, and
Ameel
,
T. A.
,
2002
, “
Slip Flow Convection in Isoflux Rectangular Microchannels
,”
ASME J. Heat Transfer
,
124
(
2
), pp.
346
355
.10.1115/1.1447932
20.
Khan
,
W. A.
, and
Yovanovich
,
M. M.
,
2008
, “
Analytical Modeling of Fluid Flow and Heat Transfer in Microchannel/ Nanochannel Heat Sinks
,”
J. Thermophys. Heat Transfer
,
22
(
3
), pp.
352
359
.10.2514/1.35621
21.
Sadeghi
,
A.
, and
Saidi
,
M. H.
,
2010
, “
Viscous Dissipation and Rarefaction Effects on Laminar Forced Convection in Microchannels
,”
ASME J. Heat Transfer
,
132
(
7
), p.
072401
.10.1115/1.4001100
22.
Sadeghi
,
A.
,
Asgarshamsi
,
A.
, and
Saidi
,
M. H.
,
2010
, “
Thermodynamic Analysis of Slip Flow Forced Convection Through a Microannulus
,”
J. Thermophys. Heat Transfer
,
24
(
4
), pp.
785
795
.10.2514/1.48036
23.
Sadeghi
,
A.
, and
Saidi
,
M. H.
,
2010
, “
Second Law Analysis of Slip Flow Forced Convection Through a Parallel Plate Microchannel
,”
Nanoscale Microscale Thermophys. Eng.
,
14
(
4
), pp.
209
228
.10.1080/15567265.2010.502924
24.
Sadeghi
,
A.
,
Salarieh
,
H.
,
Saidi
,
M. H.
, and
Mozafari
,
A. A.
,
2011
, “
Effects of Corrugated Roughness on Gaseous Slip Flow Forced Convection in Microtubes
,”
J. Thermophys. Heat Transfer
,
25
(
2
), pp.
262
271
.10.2514/1.51797
25.
Chen
,
C. K.
, and
Weng
,
H. C.
,
2005
, “
Natural Convection in a Vertical Microchannel
,”
ASME J. Heat Transfer
,
127
(
9
), pp.
1053
1056
.10.1115/1.1999651
26.
Weng
,
H. C.
, and
Chen
,
C. K.
,
2008
, “
Variable Physical Properties in Natural Convective Gas Microflow
,”
ASME J. Heat Transfer
,
130
(
8
), p.
082401
.10.1115/1.2927400
27.
Weng
,
H. C.
, and
Chen
,
C. K.
,
2008
, “
On the Importance of Thermal Creep in Natural Convective Gas Microflow With Wall Heat Fluxes
,”
J. Phys. D: Appl. Phys.
,
41
(
11
), p.
115501
.10.1088/0022-3727/41/11/115501
28.
Chen
,
C. K.
, and
Weng
,
H. C.
,
2006
, “
Developing Natural Convection With Thermal Creep in a Vertical Microchannel
,”
J. Phys. D: Appl. Phys.
,
39
(
14
), pp.
3107
3118
.10.1088/0022-3727/39/14/034
29.
Weng
,
H. C.
, and
Chen
,
C. K.
,
2009
, “
Drag Reduction and Heat Transfer Enhancement Over a Heated Wall of a Vertical Annular Microchannel
,”
Int. J. Heat Mass Transfer
,
52
(
3–4
), pp.
1075
1079
.10.1016/j.ijheatmasstransfer.2008.06.022
30.
Biswal
,
L.
,
Som
,
S. K.
, and
Chakraborty
,
S.
,
2007
, “
Effects of Entrance Region Transport Processes on Free Convection Slip Flow in Vertical Microchannels With Isothermally Heated Walls
,”
Int. J. Heat Mass Transfer
,
50
(
7–8
), pp.
1248
1254
.10.1016/j.ijheatmasstransfer.2006.09.025
31.
Chakraborty
,
S.
,
Som
,
S. K.
, and
Rahul
,
2008
, “
A Boundary Layer Analysis for Entrance Region Heat Transfer in Vertical Microchannels Within the Slip Flow Regime
,”
Int. J. Heat Mass Transfer
,
51
(
11–12
), pp.
3245
3250
.10.1016/j.ijheatmasstransfer.2008.01.019
32.
Buonomo
,
B.
, and
Manca
,
O.
,
2010
, “
Natural Convection Slip Flow in a Vertical Microchannel Heated at Uniform Heat Flux
,”
Int. J. Therm. Sci.
,
49
(
8
), pp.
1333
1344
.10.1016/j.ijthermalsci.2010.03.005
33.
Buonomo
,
B.
, and
Manca
,
O.
,
2012
, “
Transient Natural Convection in a Vertical Microchannel Heated at Uniform Heat Flux
,”
Int. J. Therm. Sci.
,
56
, pp.
35
47
.10.1016/j.ijthermalsci.2012.01.013
34.
Avci
,
M.
, and
Aydin
,
O.
,
2007
, “
Mixed Convection in a Vertical Parallel Plate Microchannel
,”
ASME J. Heat Transfer
,
129
(
2
), pp.
162
166
.10.1115/1.2422741
35.
Avci
,
M.
, and
Aydin
,
O.
,
2007
, “
Mixed Convection in a Vertical Parallel Plate Microchannel With Asymmetric Wall Heat Fluxes
,”
ASME J. Heat Transfer
,
129
(
8
), pp.
1091
1095
.10.1115/1.2737483
36.
Avci
,
M.
, and
Aydin
,
O.
,
2009
, “
Mixed Convection in a Vertical Microannulus Between Two Concentric Microtubes
,”
ASME J. Heat Transfer
,
131
(
1
), pp.
1
4
.10.1115/1.2977552
37.
Weng
,
H. C.
, and
Jian
,
S. J.
,
2012
, “
Developing Mixed Convection in a Vertical Microchannel
,”
Adv. Sci. Lett.
,
9
, pp.
908
913
.10.1166/asl.2012.2609
38.
Shah
,
R. K.
, and
London
,
A. L.
,
1978
,
Laminar Flow Forced Convection in Ducts
,
Academic Press
,
New York
.
39.
Sparrow
,
E. M.
, and
Loeffler
,
A. L.
,
1959
, “
Longitudinal Laminar Flow Between Cylinders Arranged in Regular Array
,”
AIChE J.
,
5
(
3
), pp.
325
330
.10.1002/aic.690050315
40.
Cheng
,
K. C.
,
1966
, “
Analog Solution of Laminar Heat Transfer in Noncircular Ducts by Moire Method and Point-Matching
,”
ASME J. Heat Transfer
,
88
(
2
), pp.
175
181
.10.1115/1.3691508
41.
Beskok
,
A.
,
Karniadakis
,
G. E.
, and
Trimmer
,
W.
,
1996
, “
Rarefaction and Compressibility Effects in Gas Microflows
,”
ASME J. Fluids Eng.
,
118
(
3
), pp.
448
456
.10.1115/1.2817779
42.
Schaaf
,
S. A.
,
1963
, “
Mechanics of Rarefied Gases
,”
Encyclopedia of Physics
,
Springer
,
Berlin
, pp.
591
624
.
43.
Barletta
,
A.
, and
Zanchini
,
E.
,
1999
, “
On the Choice of the Reference Temperature for Fully-Developed Mixed Convection in a Vertical Channel
,”
Int. J. Heat Mass Transfer
,
42
(
16
), pp.
3169
3181
.10.1016/S0017-9310(99)00011-3
44.
Barletta
,
A.
,
2002
, “
Fully Developed Mixed Convection and Flow Reversal in a Vertical Rectangular Duct With Uniform Wall Heat Flux
,”
Int. J. Heat Mass Transfer
,
45
(
3
), pp.
641
654
.10.1016/S0017-9310(01)00160-0
45.
McLachlan
,
N. W.
,
1961
,
Bessel Functions for Engineers
,
Clarendon Press
,
London
.
46.
Barletta
,
A.
,
Rossi di Schio
,
E.
, and
Zanchini
,
E.
,
2003
, “
Combined Forced and Free Flow in a Vertical Rectangular Duct With Prescribed Wall Heat Flux
,”
Int. J. Heat Fluid Flow
,
24
(
6
), pp.
874
887
.10.1016/S0142-727X(03)00090-0
47.
Ameel
,
T. A.
,
Wang
,
X.
,
Barron
,
R. F.
, and
Warrington
Jr.,
R. O.
,
1997
, “
Laminar Forced Convection in a Circular Tube With Constant Heat Flux and Slip Flow
,”
Microscale Thermophys. Eng.
,
1
(
4
), pp.
303
320
.10.1080/108939597200160
48.
Bejan
,
A.
,
2004
,
Convection Heat Transfer
,
Wiley
,
Hoboken, NJ
.
49.
Shen
,
C.
,
2005
,
Rarefied Gas Dynamics
,
Springer
,
Berlin
.
50.
Maurer
,
J.
,
Tabeling
,
P.
,
Joseph
,
P.
, and
Willaime
,
H.
,
2003
, “
Second-Order Slip Laws in Microchannels for Helium and Nitrogen
,”
Phys. Fluids
,
15
, pp.
2613
2621
.10.1063/1.1599355
You do not currently have access to this content.