This paper addresses the fundamental problem of maximizing the thermal contact between an entire heat-generating volume and a pulsating stream of coolant that bathes the volume. The coolant flows through an array of round and equidistant tubes. Two laminar flow configurations are considered: stop-and-go flow, where the reservoir of coolant is on one side of the volume, and back-and-forth flow, where the volume is sandwiched between two reservoirs of coolant. The total heat transfer rate between the volume and the coolant is determined numerically for many geometric configurations in the pressure drop number range $102⩽B⩽106,$ and $Pr⩾1.$ The optimal tube radius and the maximum volumetric heat transfer rate are determined numerically. The numerical optimization results are later predicted based on scale analysis by matching the longitudinal and transversal time scales of the temperature field in each tube, for each pulsation stroke. The predicted scales lead to power-law formulas that correlate the results and summarize the optimal geometry. The optimal tube size is nearly the same in stop-and-go flow and back-and-forth flow, and is independent of the pulsation frequency.

1.
Bejan, A., 2000, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK.
1.
Schmidt
,
E.
,
1926
, “
Die Wa¨rmeu¨bertragung durch Rippen
,”
Z. Ver. Dt. Ing.
,
70
, pp.
885
889
;
2.
70
, pp.
947
951
.
1.
Snider, A. D., and Kraus, A. D., 1986, “The Quest for the Optimum Longitudinal Fin Profile,” ASME HTD-Vol. 64, pp. 43–48.
2.
Bejan, A., 1993, Heat Transfer, Wiley, New York.
3.
Bejan, A., 1984, Convection Heat Transfer, Wiley, New York, chap. 4, Problem 11, p. 157.
4.
Bar-Cohen
,
A.
, and
Rohsenow
,
W. M.
,
1984
, “
Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates
,”
ASME J. Heat Transfer
,
106
, pp.
116
123
.
5.
Anand
,
N. K.
,
Kim
,
S. H.
, and
Fletcher
,
L. S.
,
1992
, “
The Effect of Plate Spacing on Free Convection between Heated Parallel Plates
,”
ASME J. Heat Transfer
,
114
, pp.
515
518
.
6.
Bejan
,
A.
,
Fowler
,
A. J.
, and
Stanescu
,
G.
,
1995
, “
The Optimal Spacing Between Horizontal Cylinders in a Fixed Volume Cooled by Natural Convection
,”
Int. J. Heat Mass Transf.
,
38
, pp.
2047
2055
.
7.
Ledezma
,
G. A.
, and
Bejan
,
A.
,
1997
, “
Optimal Geometric Arrangement of Staggered Vertical Plates in Natural Convection
,”
ASME J. Heat Transfer
,
119
, pp.
700
708
.
8.
Nakayama, W., Matsushima, H., and Goel, P., 1988, “Forced Convective Heat Transfer from Arrays of Finned Packages,” in Cooling Technology for Electronic Equipment, W. Aung, ed., pp. 195–210, Hemisphere, New York.
9.
Nakayama, W., 1994, “Information Processing and Heat Transfer Engineering: Some Generic Views on Future Research Needs,” in Cooling of Electronic Systems, S. Kakac, H. Yu¨ncu¨ and K. Hijikata, eds., Kluwer Academic, Dordrecht, The Netherlands, pp. 911–943.
10.
Knight
,
R. W.
,
Goodling
,
J. S.
, and
Hall
,
D. J.
,
1991
, “
Optimal Thermal Design of Forced Convection Heat Sinks—Analytical
,”
ASME J. Electron. Packag.
,
113
, pp.
313
321
.
11.
Matsushima
,
H.
,
Yanagida
,
T.
, and
Kondo
,
Y.
,
1992
, “
Algorithm for Predicting the Thermal Resistance of Finned LSI Packages Mounted on a Circuit Board
,”
Heat Transfer Jap. Res.
,
21
, No.
5
pp.
504
517
.
12.
Bejan
,
A.
, and
Sciubba
,
E.
,
1992
, “
The Optimal Spacing for Parallel Plates Cooled by Forced Convection
,”
Int. J. Heat Mass Transf.
,
35
, pp.
3259
3264
.
13.
Bejan, A., 1995, Convection Heat Transfer, 2nd Ed., Wiley, New York.
14.
Fowler
,
A. J.
,
Ledezma
,
G. A.
, and
Bejan
,
A.
,
1997
, “
Optimal Geometric Arrangement of Staggered Plates in Forced Convection
,”
Int. J. Heat Mass Transf.
,
40
, pp.
1795
1805
.
15.
Bejan
,
A.
,
1995
, “
The Optimal Spacings for Cylinders in Crossflow Forced Convection
,”
ASME J. Heat Transfer
,
117
, pp.
767
770
.
16.
Stanescu
,
G.
,
Fowler
,
A. J.
, and
Bejan
,
A.
,
1996
, “
The Optimal Spacing of Cylinders in Free-Stream Cross-Flow Forced Convection
,”
Int. J. Heat Mass Transf.
,
39
, pp
311
317
.
17.
Petrescu
,
S.
,
1994
, “
Comments on the Optimal Spacing of Parallel Plates Cooled by Forced Convection
,”
Int. J. Heat Mass Transf.
,
37
, p.
1283
1283
.
18.
Aung, W., ed., 1988, Cooling Technology for Electronic Equipment, Hemisphere, New York.
19.
Moffat, R. J., and Ortega, A., 1988, “Direct Air Cooling of Electronic Components,” in Advances in Thermal Modeling of Electronic Components and Systems Vol. I, A. Bar-Cohen and A. D. Kraus, eds., Hemisphere, New York, pp. 129–282.
20.
Peterson
,
G. P.
, and
Ortega
,
A.
,
1990
, “
Thermal Control of Electronic Equipment and Devices
,”
,
20
, pp.
181
314
.
21.
Kakac, S., Yu¨ncu¨, H., and Hijikata, K., eds., 1994, Cooling of Electronic Systems, Kluwer, Dordrecht, The Netherlands.
22.
Kraus, A. D., and Bar-Cohen, A., 1995, Design and Analysis of Heat Sinks, Wiley, New York.
23.
Kim, S. J., and Lee, S. W., eds., 1996, Air Cooling Technology for Electronic Equipment, CRC Press, Boca Raton, FL.
24.
Shah, R. K., and London, A. L., 1978, Laminar Flow Forced Convection in Ducts, Supplement 1 to Advances in Heat Transfer, Academic Press, New York.
25.
O¨zisik, M. N., 1994, Finite Difference Methods in Heat Transfer, CRC Press, Boca Raton, FL.
26.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992, Numerical Recipes in FORTRAN, Cambridge University Press, Cambridge, New York.
27.
Chatwin
,
P. C.
,
1975
, “
On the Longitudinal Dispersion of Passive Contaminant in Oscillatory Flows in Tubes
,”
J. Fluid Mech.
,
71
, pp.
513
527
.
28.
Watson
,
E. J.
,
1983
, “
Diffusion in Oscillatory Pipe Flow
,”
J. Fluid Mech.
,
133
, pp.
233
244
.
29.
Kurzweg
,
U. H.
, and
Zhao
,
L. D.
,
1984
, “
Heat Transfer by High-Frequency Oscillations: a New Hydrodynamic Technique for Achieving Large Effective Thermal Conductivities
,”
Phys. Fluids
,
27
, pp.
2624
2627
.
30.
Kurzweg
,
U. H.
,
1985
, “
Enhanced Heat Conduction in Oscillating Viscous Flows Within Parallel-Plate Channels
,”
J. Fluid Mech.
,
156
, pp.
291
300
.