In this paper, a generalized methodology for the optimum design of thin fins with uniform volumetric heat generation is described. Using variational calculus, the optimum profiles of longitudinal, annular, and pin fins are determined from the basic fin equation under the constraint of specified fin volume. From a common optimality criteria, a generalized closed form expression for the fin thickness is obtained for the above three types of fins. Closed-form expressions are also obtained for the optimum profiles of longitudinal and pin fins. As a special case, both the temperature profile and the shape of optimum fins without heat generation are determined.

Issue Section:
Technical Briefs
Keywords:
variational techniques, heat conduction, convection, Annular Fin, Calculus of Variation, Heat Generation, Longitudinal Fin, Optimum Profile, Pin Fin
Topics:
Fins, Heat, Temperature profiles, Heat transfer, Variational techniques
1.
Gardener
,
K. A.
, 1945, “
Efficiency of Extended Surface
,”
ASME J. Heat Transfer
0022-1481,
67
, pp.
621
–
631
.
2.
Schmidt
,
E.
, 1926, “
Die Warmenbertranung durch Rippen
,”
Z. VDI
,
70
, pp.
885
–
951
.
3.
Duffin
,
R.
, 1959, “
A Variational Problem Relating to Cooling Fins
,”
J. Math. Mech.
0095-9057,
8
, pp.
47
–
56
.
4.
Minkler
,
W. S.
, and
Rouleau
,
W. T.
, 1960, “
The Effects of Internal Heat Generation on Heat Transfer in Thin Fins
,”
Nucl. Sci. Eng.
0029-5639,
7
, pp.
400
–
406
.
Crossref
Search ADS
 
5.
Wilkins
,
J. E.
, Jr., 1962, “
Minimum Mass Thin Fins With Internal Heat Generation
,”
Nucl. Sci. Eng.
0029-5639,
14
, pp.
203
–
204
.
Crossref
Search ADS
 
6.
Liu
,
C. Y.
, 1961, “
A Variational Problem With Applications to Cooling Fins With Heat Generation
,”
J. Soc. Ind. Appl. Math.
0368-4245,
9
, pp.
194
–
201
.
7.
Natarajan
,
U.
, and
Shenoy
,
U. V.
, 1990, “
Optimum Shapes Convective Pin Fins With Variable Heat Transfer Coefficient
,”
J. Franklin Inst.
0016-0032,
327
, pp.
965
–
982
.
Crossref
Search ADS
 
8.
Murray
,
W. M.
, 1938, “
Heat Transfer Through an Annular Disc or Fin of Uniform Thickness
,”
ASME J. Appl. Mech.
0021-8936,
60
, p.
A
-
78
.
9.
Kraus
,
A. D.
, 1982,
Analysis and Evaluation of Extended Surface Thermal Systems
,
Hemisphere
, Washington, DC.
10.
Elsgolts
,
L.
, 1977,
Differential Equation and the Calculus of Variations
,
Mir Publishers
, Moscow.
Copyright © 2005
 by American Society of Mechanical Engineers
You do not currently have access to this content.