In this paper, we consider the heat transfer problems associated with a periodic array of triangular, longitudinal, axisymmetric, and pin fins. The problems are modeled as a wall where the flat side is isothermal and the other side, which has extended surfaces/fins, is subjected to convection with a uniform heat transfer coefficient. Hence, our analysis differs from the classical approach because (i) we consider multidimensional heat conduction and (ii) the wall on which the fins are attached is included in the analysis. The latter results in a nonisothermal temperature distribution along the base of the fin. The Biot number ($Bi=ht/k$) characterizing the heat transfer process is defined with respect to the thickness/diameter of the fins (t). Numerical results demonstrate that the fins would enhance the heat transfer rate only if the Biot number is less than a critical value, which, in general, depends on the geometrical parameters, i.e., the thickness of the wall, the length of the fins, and the period. For pin fins, similar to rectangular fins, the critical Biot number is independent of the geometry and is approximately equal to 3.1. The physical argument is that, under strong convection, a thick fin introduces an additional resistance to heat conduction.

## References

References
1.
Incropera
,
F. P.
, and
DeWitt
,
D. P.
,
1990
,
Fundamentals of Heat and Mass Transfer
,
Wiley
,
New York.
2.
Leontiou
,
T.
, and
Fyrillas
,
M. M.
,
2015
, “
Critical Thickness of an Optimum Extended Surface Characterized by Uniform Heat Transfer Coefficient
,” arXiv:1503.05148.
3.
Fyrillas
,
M. M.
, and
Leontiou
,
T.
,
2016
, “
Critical Biot Number of a Periodic Array of Rectangular Fins
,”
ASME J. Heat Transfer
,
138
(
2
), p.
024504
.
4.
Fyrillas
,
M. M.
, and
Stone
,
H. A.
,
2011
, “
Critical Insulation Thickness of a Slab Embedded With a Periodic Array of Isothermal Strips
,”
Int. J. Heat Mass Transfer
,
54
(
1–3
), pp. 180–185.
5.
Leontiou
,
T.
,
Ikram
,
M.
,
Beketayev
,
K.
, and
Fyrillas
,
M. M.
,
2016
, “
Heat Transfer Enhancement of a Periodic Array of Isothermal Pipes
,”
Int. J. Therm. Sci.
,
104
, pp.
480
488
.
6.
Sahin
,
A.
,
2012
, “
Critical Insulation Thickness for Maximum Entropy Generation
,”
Int. J. Energy
,
10
(
1
), pp. 34–43.
7.
Bau
,
H. B.
,
1984
, “
Convective Heat Losses From a Pipe Buried in a Semi-Infinite Porous Medium
,”
Int. J. Heat Mass Transfer
,
27
(
11
), pp.
2047
2056
.
8.
Fyrillas
,
M. M.
,
2017
, “
Critical Depth of Buried Isothermal Circular Pipes
,”
Heat Transfer Eng.
,
38
(
18
).
9.
Bobaru
,
F.
, and
Rachakonda
,
S.
,
2004
, “
Boundary Layer in Shape Optimization of Convective Fins Using a Meshfree Approach
,”
Int. J. Numer. Methods Eng.
,
60
(
7
), pp.
1215
1236
.
10.
Fyrillas
,
M. M.
, and
Pozrikidis
,
C.
,
2001
, “
Conductive Heat Transport Across Rough Surfaces and Interfaces Between Two Conforming Media
,”
Int. J. Heat Mass Transfer
,
44
(
9
), pp. 1789–1801.
11.
Fyrillas
,
M. M.
,
2009
, “
Shape Optimization for 2D Diffusive Scalar Transport
,”
Optim. Eng.
,
10
(
4
), p.
477
.
12.
Leontiou
,
T.
,
Kotsonis
,
M.
, and
Fyrillas
,
M. M.
,
2013
, “
Optimum Isothermal Surfaces That Maximize Heat Transfer
,”
Int. J. Heat Mass Transfer
,
63
, pp. 13–19.
13.
,
M.
, and
Pozrikidis
,
C.
,
1993
, “
Diffusive Transport Across Irregular and Fractal Walls
,”
Proc. R. Soc. London A
,
442
(
1916
), p.
571
.
14.
Rohsenow
,
W. M.
,
Hartnett
,
J. P.
, and
Cho
,
Y. I.
,
1998
,
Handbook of Heat Transfer
,
McGraw-Hill
,
New York
.
15.
Karagiozis
,
A.
,
Raithy
,
G. D.
, and
Hollands
,
K. G. T.
,
1994
, “
Natural Convection Heat Transfer From Arrays of Isothermal Triangular Fins to Air
,”
ASME J. Heat Transfer
,
116
(
1
), pp. 105–111.
16.
Bobaru
,
F.
, and
Rachakonda
,
S.
,
2004
, “
Optimal Shape Profiles for Cooling Fins of High and Low Conductivity
,”
Int. J. Heat Mass Transfer
,
47
(
23
), pp. 4953–4966.
17.
Fyrillas
,
M. M.
,
2008
, “
Heat Conduction in a Solid Slab Embedded With a Pipe of General Cross-Section: Shape Factor and Shape Optimization
,”
Int. J. Eng. Sci.
,
46
(
9
), pp. 907–916.
18.
Fyrillas
,
M. M.
,
2010
, “
Shape Factor and Shape Optimization for a Periodic Array of Isothermal Pipes
,”
Int. J. Heat Mass Transfer
,
53
(
5–6
), pp. 982–989.
19.
Leontiou
,
T.
, and
Fyrillas
,
M. M.
,
2014
, “
Shape Optimization With Isoperimetric Constraints for Isothermal Pipes Embedded in an Insulated Slab
,”
ASME J. Heat Transfer
,
136
(
9
), p.
094502
.
20.
Fyrillas
,
M. M.
, and
Leontiou
,
T.
,
2016
, “
Optimum Interfaces That Maximize the Heat Transfer Rate Between Two Conforming Conductive Media
,” Internal Report, Frederick University Cyprus, Report No. 13/2016.
21.
,
D.
, and
Alimbayev
,
T.
,
2014
, “
Conjugate Heat Transfer in a Developing Laminar Boundary Layer
,”
World Congress on Engineering
(
WCE
), London, July 2–4, Vol.
II
, pp. 1387–1392.
22.
Campo
,
A.
, and
Celentano
,
D. J.
,
2016
, “
Absolute Maximum Heat Transfer Rendered by Straight Fins With Quarter Circle Profile Using Finite Element Analysis
,”
Appl. Therm. Eng.
,
105
, pp.
85
92
.
23.
Shah
,
R. K.
,
2003
,
Fundamental of Heat Exchanger Design
,
Wiley
,
Hoboken, NJ
.
24.
Kraus
,
A. D.
,
Aziz
,
A.
, and
Welty
,
J.
,
2001
,
Extended Surface Heat Transfer
,
Wiley
,
New York
.
25.
COMSOL
,
2008
, “
Heat Transfer Module, COMSOL Multiphysics User's Guide, Version 3.4
,” COMSOL AB, Stockholm, Sweden.