An extension of a recently proposed single domain formulation of conjugated conduction–convection heat transfer problems is presented, taking into account the axial diffusion effects at both the walls and fluid regions, which are often of relevance in microchannels flows. The single domain formulation simultaneously models the heat transfer phenomena at both the fluid stream and the channel walls, by making use of coefficients represented as space variable functions, with abrupt transitions occurring at the fluid-wall interface. The generalized integral transform technique (GITT) is then employed in the hybrid numerical–analytical solution of the resulting convection–diffusion problem with variable coefficients. With axial diffusion included in the formulation, a nonclassical eigenvalue problem may be preferred in the solution procedure, which is itself handled with the GITT. To allow for critical comparisons against the results obtained by means of this alternative solution path, we have also proposed a more direct solution involving a pseudotransient term, but with the aid of a classical Sturm-Liouville eigenvalue problem. The fully converged results confirm the adequacy of this single domain approach in handling conjugated heat transfer problems in microchannels, when axial diffusion effects must be accounted for.

References

References
1.
Tuckerman
,
D. B.
, and
Pease
,
R. F. W.
,
1981
, “
High Performance Heat Sinking in VLSI
,”
IEEE Electron. Device Lett.
,
2
(
5
), pp.
126
129
.10.1109/EDL.1981.25367
2.
Tabeling
,
P.
,
2003
,
Introduction a la Microfluidique
,
Belin, Collection Échelles
,
Paris
.
3.
Morini
,
G. L.
,
2004
, “
Single-Phase Convective Heat Transfer in Microchannels: A Review of Experimental Results
,”
Int. J. Therm. Sci.
,
43
, pp.
631
651
.10.1016/j.ijthermalsci.2004.01.003
4.
Yener
,
Y.
,
Kakaç
,
S.
,
Avelino
,
M.
, and
Okutucu
,
T.
,
2005
, “
Single-Phase Forced Convection in Microchannels—A State-of-the-Art Review
,”
Microscale Heat Transfer—Fundamentals and Applications
(NATO ASI Series),
S.
Kakaç
,
L. L.
Vasiliev
,
Y.
Bayazitoglu
, and
Y.
Yener
, eds.,
Kluwer Academic Publishers
,
The Netherlands
, pp.
1
24
.
5.
Yu
,
S.
, and
Ameel
,
T. A.
,
2001
, “
Slip Flow Heat Transfer in Rectangular Microchannels
,”
Int. J. Heat Mass Transfer
,
44
, pp.
4225
4234
.10.1016/S0017-9310(01)00075-8
6.
Tunc
,
G.
, and
Bayazitoglu
,
Y.
,
2001
, “
Heat Transfer in Microtubes With Viscous Dissipation
,”
Int. J. Heat Mass Transfer
,
44
, pp.
2395
2403
.10.1016/S0017-9310(00)00298-2
7.
Tunc
,
G.
, and
Bayazitoglu
,
Y.
,
2002
, “
Heat Transfer in Rectangular Microchannels
,”
Int. J. Heat Mass Transfer
,
45
, pp.
765
773
.10.1016/S0017-9310(01)00201-0
8.
Mikhailov
,
M. D.
, and
Cotta
,
R. M.
,
2005
, “
Mixed Symbolic-Numerical Computation of Convective Heat Transfer With Slip Flow in Microchannels
,”
Int. Commun. Heat Mass Transfer
,
32
, pp.
341
348
.10.1016/j.icheatmasstransfer.2004.04.030
9.
Cotta
,
R. M.
,
Kakaç
,
S.
,
Mikhailov
,
M. D.
,
Castellões
,
F. V.
, and
Cardoso
,
C. R.
,
2005
, “
Transient Flow and Thermal Analysis in Microfluidics
,”
Microscale Heat Transfer—Fundamentals and Applications
(NATO ASI Series),
S.
Kakaç
,
L. L.
Vasiliev
,
Y.
Bayazitoglu
, and
Y.
Yener
, eds.,
Kluwer Academic Publishers
,
The Netherlands
, pp.
175
196
.
10.
Castellões
,
F. V.
, and
Cotta
,
R. M.
,
2006
, “
Analysis of Transient and Periodic Convection in Microchannels via Integral Transforms
,”
Prog. Comput. Fluid Dyn.
,
6
(
6
), pp.
321
326
.10.1504/PCFD.2006.010772
11.
Castellões
,
F. V.
,
Cardoso
,
C. R.
,
Couto
,
P.
, and
Cotta
,
R. M.
,
2007
, “
Transient Analysis of Slip Flow and Heat Transfer in Microchannels
,”
Heat Transfer Eng.
,
28
(
6
), pp.
549
558
.10.1080/01457630701193971
12.
Castellões
,
F. V.
, and
Cotta
,
R. M.
,
2008
, “
Heat Transfer Enhancement in Smooth and Corrugated Microchannels
,”
Proceedings of the 7th Minsk International Seminar on Heat Pipes, Heat Pumps, Refrigerators, Invited Lecture, Minsk, Belarus
, Sept. 8–11.
13.
Castellões
,
F. V.
,
Quaresma
,
J. N. N.
, and
Cotta
,
R. M.
,
2010
, “
Convective Heat Transfer Enhancement in Low Reynolds Number Flows With Wavy Walls
,”
Int. J. Heat Mass Transfer
,
53
, pp.
2022
2034
.10.1016/j.ijheatmasstransfer.2009.12.054
14.
Naveira-Cotta
,
C. P.
,
Orlande
,
H. R. B.
, and
Cotta
,
R. M.
,
2010
, “
Inverse Analysis of Forced Convection in Microchannels With Slip Flow via Integral Transforms and Bayesian Inference
,”
Int. J. Therm. Sci.
,
49
, pp.
879
888
.10.1016/j.ijthermalsci.2009.12.009
15.
Dharaiya
,
V. V.
, and
Kandlikar
,
S. G.
,
2012
, “
Numerical Investigation of Heat Transfer in Rectangular Microchannels Under H2 Boundary Condition During Developing and Fully Developed Laminar Flow
,”
ASME J. Heat Transfer
,
134
, p.
020911
.10.1115/1.4004934
16.
Lin
,
T.-Y.
, and
Kandlikar
,
S. G.
,
2012
, “
A Theoretical Model for Axial Heat Conduction Effects During Single-Phase Flow in Microchannels
,”
ASME J. Heat Transfer
,
134
, p.
020902
.10.1115/1.4004936
17.
Nunes
,
J. S.
,
Cotta
,
R. M.
,
Avelino
,
M.
, and
Kakaç
,
S.
,
2010
, “
Conjugated Heat Transfer in Microchannels
,”
Microfluidics Based Microsystems: Fundamentals and Applications
(NATO Science for Peace and Security Series A: Chemistry and Biology),
S.
Kakaç
,
B.
Kosoy
, and
A.
Pramuanjaroenkij
, eds., Vol.
1
, pp.
61
82
.
18.
Maranzana
,
G.
,
Perry
,
I.
, and
Maillet
,
D.
,
2004
, “
Mini and Microchannels: Influence of Axial Conduction in the Walls
,”
Int. J. Heat Mass Transfer
,
47
, pp.
3993
4004
.10.1016/j.ijheatmasstransfer.2004.04.016
19.
Guedes
,
R. O. C.
,
Cotta
,
R. M.
, and
Brum
,
N. C. L.
,
1991
, “
Heat Transfer in Laminar Tube Flow With Wall Axial Conduction Effects
,”
J. Thermophys. Heat Transfer
,
5
(
4
), pp.
508
513
.10.2514/3.294
20.
Guedes
,
R. O. C.
, and
Cotta
,
R. M.
,
1991
, “
Periodic Laminar Forced Convection Within Ducts Including Wall Heat Conduction Effects
,”
Int. J. Eng. Sci.
,
29
(
5
), pp.
535
547
.10.1016/0020-7225(91)90059-C
21.
Cotta
,
R. M.
,
1990
, “
Hybrid Numerical-Analytical Approach to Nonlinear Diffusion Problems
,”
Numer. Heat Transfer, Part B
,
127
, pp.
217
226
.10.1080/10407799008961740
22.
Cotta
,
R. M.
,
1993
,
Integral Transforms in Computational Heat and Fluid Flow
,
CRC Press
,
Boca Raton, FL
.
23.
Cotta
,
R. M.
,
1994
, “
Benchmark Results in Computational Heat and Fluid Flow: The Integral Transform Method
,”
Int. J. Heat Mass Transfer
(Invited Paper),
37
(
Suppl. 1
), pp.
381
394
.10.1016/0017-9310(94)90038-8
24.
Cotta
,
R. M.
, and
Mikhailov
,
M. D.
,
1997
,
Heat Conduction: Lumped Analysis, Integral Transforms, Symbolic Computation
,
Wiley-Interscience
,
Chichester, UK
.
25.
Cotta
,
R. M.
,
1998
,
The Integral Transform Method in Thermal and Fluids Sciences and Engineering
,
Begell House
,
New York
.
26.
Cotta
,
R. M.
, and
Mikhailov
,
M. D.
,
2006
, “
Hybrid Methods and Symbolic Computations
,”
Handbook of Numerical Heat Transfer
,
2nd ed.
,
W. J.
Minkowycz
,
E. M.
Sparrow
, and
J. Y.
Murthy
, eds.,
John Wiley
,
New York
, Chap. 16.
27.
Knupp
,
D. C.
,
Naveira-Cotta
,
C. P.
, and
Cotta
,
R. M.
,
2012
, “
Theoretical Analysis of Conjugated Heat Transfer With a Single Domain Formulation and Integral Transforms
,”
Int. Commun. Heat Mass Transfer
,
39
(
3
), pp.
355
362
.10.1016/j.icheatmasstransfer.2011.12.012
28.
Vick
,
B.
, and
Özisik
,
M. N.
,
1981
, “
Effects of Axial Conduction and Convective Boundary Conditions in Slug Flow Inside a Circular Tube
,”
ASME J. Heat Transfer
,
103
, pp.
436
440
.10.1115/1.3244482
29.
Oliveira
,
M. C.
,
Ramos
,
R.
, and
Cotta
,
R. M.
,
1995
, “
On the Eigenvalues Basic to the Analytical Solution of Convective Heat Transfer With Axial Diffusion Effects
,”
Commun. Numer. Methods Eng.
,
11
, pp.
287
296
.10.1002/cnm.1640110402
30.
Wolfram
,
S.
,
2005
,
The Mathematica Book
,
Wolfram Media
,
Cambridge
.
31.
Naveira-Cotta
,
C. P.
,
Cotta
,
R. M.
,
Orlande
,
H. R. B.
, and
Fudym
,
O.
,
2009
, “
Eigenfunction Expansions for Transient Diffusion in Heterogeneous Media
,”
Int. J. Heat Mass Transfer
,
52
, pp.
5029
5039
.10.1016/j.ijheatmasstransfer.2009.04.014
32.
Ayres
,
J. V. C.
,
Knupp
,
D. C.
,
Naveira-Cotta
,
C. P.
,
Ferreira
,
L. O. S.
, and
Cotta
,
R. M.
,
2011
, “
Evaluation of Infrared Thermography Experimental Analysis of a Single Microchannel Heat Spreader
,”
21st International Congress of Mechanical Engineering, COBEM-2011
,
ABCM, Natal, RN, Brazil
.
33.
Knupp
,
D. C.
,
Naveira-Cotta
,
C. P.
,
Ayres
,
J. V. C.
,
Cotta
,
R. M.
, and
Orlande
,
H. R. B.
,
2012
, “
Theoretical-Experimental Analysis of Heat Transfer in Non-Homogeneous Solids Via Improved Lumped Formulation, Integral Transforms and Infrared Thermography
,”
Int. J. Thermal Sciences
,
62
, pp.
71
84
.10.1016/j.ijthermalsci.2012.01.005
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