A multi-grid embedded multi-scale approach is presented for conjugate heat transfer analysis of systems with a wide range of length scales of interest. The multi-scale analysis involves a sequential two-step “zoom-in” approach to resolve both the large length scales associated with the system enclosure, and the smaller length scales associated with fine spatial structures of discrete heat sources contained within. With this approach, computation time is shortened significantly, compared to conventional single-step computational fluid dynamics/computational heat transfer (CFD/CHT) modeling, with a very fine mesh. Performance of the two-step multi-scale approach is further enhanced by integrating the multi-grid technique in the CFD/CHT solver. Implementation of the enhanced approach is demonstrated for thermal analysis of an array of substrate mounted discrete heat sources cooled by mixed and forced convection, with accompanying experiments performed for validation and for the assessment of the importance of mixed convection. It is found that the multi-grid embedded multi-scale thermal analysis reduces simulation run time by 90% compared to the multi-grid integrated single step solution. The computed temperatures were in good agreement with measurements, with maximum deviation of 8%.

Keywords:
differential equations, thermal analysis, forced convection, natural convection, heat conduction, heat radiation, cooling, heating
Topics:
Flow (Dynamics), Forced convection, Heat, Heat conduction, Junctions, Mixed convection, Modeling, Printed circuit boards, Temperature, Thermal analysis, Radiation (Physics), Heating, Thermal conductivity
1.
Aung
,
W.
,
1972
, “
Fully Developed Laminar Free Convection Between Vertical Plates Heated Asymetrically
,”
Int. J. Heat Mass Transfer
,
15
, pp.
1577
–
1580
.
2.
Miyatake
,
O.
, and
Fujii
,
T.
,
1974
, “
Natural Convection Heat Transfer Between Vertical Parallel Plates With Unequal Heat Fluxes
,”
Heat Transfer-Jpn. Res.
,
3
(
3
), pp.
29
–
33
.
3.
Bar Cohen
,
A.
, and
Rohsenow
,
W. M.
,
1980
, “
Thermally Optimal Spacing of Vertical Natural Convection Cooled Parallel Plates
,”
ASME J. Heat Transfer
,
102
, pp.
221
–
227
.
4.
Bar Cohen
,
A.
, and
Rohsenow
,
W. M.
,
1984
, “
Thermal Spacing of Vertical, Natural Convection Cooled Parallel Plates
,”
ASME J. Heat Transfer
,
106
, pp.
116
–
123
.
5.
Wirtz
,
R. A.
, and
Stutzman
,
R. J.
,
1982
, “
Experiments on Free Convection Between Vertical Plates With Symmetric Heating
,”
ASME J. Heat Transfer
,
104
, pp.
501
–
507
.
6.
Wills
,
M.
,
1983
, “
Thermal Analysis of Air-Cooled PCBs
,”
Electronic Products
, May, pp.
11
–
18
.
7.
Landis, F., and Elenbaas, W., 1986, “Heat Dissipation of Parallel Plates by Free Convection,” ASME HTD-57, pp. 11–22.
8.
Moffat, R. J., and Ortega, A., 1988, “Direct Air Cooling of Electronic Components,” Advances in Thermal Modeling of Electronic Components and Systems, A Bar-Cohen and A. D. Kraus, eds., Hemisphere Publishing Corp., New York, NY.
9.
Braaten
,
M. E.
, and
Patankar
,
S. V.
,
1989
, “
A Block-Correction Subdomain Solution Procedure for Recirculating Flow Calculations
,”
Numer. Heat Transfer, Part B
,
15
, pp.
1
–
20
.
10.
Kelkar
,
K. M.
,
1990
, “
Iterative Method for the Numerical Prediction of Heat Transfer in Problems Involving Large Differences in Thermal Conductivities
,”
Numer. Heat Transfer, Part B
,
17
, pp.
113
–
128
.
11.
Brandt
,
A.
,
1977
, “
Multi-Level Adaptive Solutions to Boundary Valued Problems
,”
Math. Comput.
,
31
, pp.
333
–
390
.
12.
Vanka
,
S. P.
,
1986
, “
Block-Implicit Multi-Grid Solution of Navier-Stokes Equations in Primitive Variables
,”
J. Comput. Phys.
,
65
, pp.
38
–
158
.
13.
Sathyamurthy
,
P. S.
, and
Patankar
,
S. V.
,
1994
, “
Block-Correction-Based Multi-Grid Method for Fluid Flow Problems
,”
Numer. Heat Transfer, Part B
,
25
, pp.
375
–
394
.
14.
Heindel
,
T. J.
,
Ramadhyani
,
S.
, and
Incropera
,
F. P.
,
1996
, “
Conjugate Natural Convection From an Array of Protruding Heat Sources
,”
Numer. Heat Transfer, Part A
,
29
, pp.
1
–
18
.
15.
Linton, R. L., and Agonafer, D., 1994, “Coarse and Detailed CFD Modeling of a Finned Heat Sink,” Proceedings of I-THERM IV, InterSociety Conference on Thermal Phenomena in Electronic Systems, pp. 156–161.
16.
Ewes, I., 1995, “Modeling of IC-Packages Based on Thermal Characteristics,” EUROTHERM Seminar No. 45, Thermal Management of Electronic Systems, Delft Conference Proceedings, Leuven, Belgium.
17.
Rosten, H. I., Parry, J. D., Addison, J. S., Viswanath, R., Davies, M., and Fitzgerald, E., 1995, “Development, Validation, and Application of a Thermal Model of a Plastic Quad Flat Pack,” Proceedings of 45th Electronic Components and Technology Conference, pp. 1140–1151.
18.
Adams, V. H., Blackburn D. L., Joshi, Y., and Berning, D. W., 1997, “Issues in Validating Package Compact Thermal Models for Natural Convection Cooled Electronic Systems,” Proceedings of the 13th IEEE SEMI-THERM Symposium, Austin, TX.
19.
Tang
,
L.
, and
Joshi
,
Y.
,
1999
, “
Integrated Thermal Analysis of Natural Convection Air Cooled Electronic Enclosure
,”
ASME J. Electron. Packag.
,
121
, pp.
108
–
115
.
20.
Azar
,
K.
, and
Graebner
,
J. E.
,
1997
, “
Thermal Conductivity Measurements in Printed Wiring Boards
,”
ASME J. Heat Transfer
,
119
(
3
), pp.
401
–
405
.
21.
Nelson, R., 2001, “Wiring Statistics and Printed Wiring Board Thermal Conductivity,” Proceedings of SEMI-THERHM 2001, p. 252.
22.
Tang, L., 1998, “A Multi-Scale Conjugate Thermal Analysis Methodology for Convectively Cooled Electronic Enclosures,” Ph.D. dissertation, Univ. of Maryland, College Park, MD.
23.
Sparrow
,
E. M.
,
Eichhorn
,
R.
, and
Gregg
,
J. L.
,
1959
, “
Combined Forced and Free Convection in a Boundary Layer
,”
Phys. Fluids
,
2
, pp.
319
–
329
.
24.
Choi
,
C. Y.
, and
Kim
,
S. J.
,
1996
, “
Conjugate Mixed Convection in a Channel: Modified Five Percent Deviation Rule
,”
Int. J. Heat Mass Transfer
,
39
(
6
), pp.
1223
–
1234
.
25.
Sikka
,
K. K.
,
Fisher
,
T. S.
,
Torrance
,
K. E.
, and
Lamb
,
C. R.
,
1997
, “
Effects of Package Orientation and Mixed Convection on Heat Transfer From a PQFP
,”
IEEE Trans. Compon., Packag. Manuf. Technol., Part A
,
20
(
2
), pp.
152
–
159
.
26.
Churchill
,
S. W.
, and
Usagi
,
R.
,
1972
, “
A General Expression for the Correlation of Rates of Transfer and Other Phenomena
,”
AIChE J.
,
18
(
6
), pp.
1121
–
1128
.
27.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York.
28.
Tang
,
L.
, and
Joshi
,
Y.
,
1999
, “
Application of Block-Implicit Multi-Grid Approach to Three-Dimensional Heat Transfer Problems Involving Discrete Heating
,”
Numer. Heat Transfer, Part A
,
35
, pp.
717
–
734
.
Copyright © 2005
 by ASME
You do not currently have access to this content.