Technical Brief

Critical Biot Number of a Periodic Array of Rectangular Fins

[+] Author and Article Information
Marios M. Fyrillas

Department of Mechanical Engineering,
Nazarbayev University,
Astana 010000, Republic of Kazakhstan;
Department of Mechanical Engineering,
Frederick University,
Nicosia 1303, Cyprus
e-mail: m.fyrillas@gmail.com

Theodoros Leontiou

General Department,
Frederick University,
Nicosia 1303, Cyprus
e-mail: eng.lt@fit.ac.cy

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 28, 2015; final manuscript received August 23, 2015; published online October 13, 2015. Assoc. Editor: Oronzio Manca.

J. Heat Transfer 138(2), 024504 (Oct 13, 2015) (4 pages) Paper No: HT-15-1308; doi: 10.1115/1.4031640 History: Received April 28, 2015; Revised August 23, 2015

We consider the heat transfer problem associated with a periodic array of rectangular fins subjected to convection heat transfer with a uniform heat transfer coefficient. Our analysis differs from the classical approach as (i) we consider two-dimensional (2D) heat conduction and (ii) the wall, to which the fins are attached, is included in the analysis. The problem is modeled as a 2D channel whose upper surface is flat and isothermal, while the lower surface has a periodic array of rectangular extensions/fins which are subjected to heat convection. The Biot number (Bi=h t/k) characterizing the heat transfer process is defined with respect to the thickness of the fins (t). Numerical results suggest that the fins would enhance the heat transfer rate only if the Biot number is less than a critical value which is independent of the thickness of the wall, the length of the fins, and the period; the critical Biot number is approximately equal to 1.64. The optimum fins are infinitely thin and long, and densely packed, i.e., hairlike.

Copyright © 2016 by ASME
Topics: Fins , Heat transfer
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Lavrentev, M. A. , and Shabat, B. A. , 2001, Theoretical Methods of Functions of Complex Variables, Nauka, Moscow, p. 1789.
Incropera, F. P. , and DeWitt, D. P. , 1990, Fundamentals of Heat and Mass Transfer, Wiley, New York.
Kraus, A. D. , Aziz, A. , and Welty, J. , 2001, Extended Surface Heat Transfer, Wiley, Hoboken, NJ.
Holmes, M. H. , 2013, Introduction to Perturbation Methods, Springer-Verlag, New York.
Fyrillas, M. M. , and Szeri, A. J. , 1994, “ Dissolution or Growth of Soluble, Spherical, Oscillating Bubbles,” J. Fluid Mech., 277, pp. 381–407. [CrossRef]
Fyrillas, M. M. , and Szeri, A. J. , 1995, “ Dissolution or Growth of Soluble, Spherical, Oscillating Bubbles: The Effect of Surfactants,” J. Fluid Mech., 289, pp. 295–314. [CrossRef]
Fyrillas, M. M. , and Szeri, A. J. , 1995, “ Surfactant Dynamics and Rectified Diffusion of Microbubbles,” J. Fluid Mech., 311, pp. 361–378. [CrossRef]
Stone, H. A. , 1989, “ Heat/Mass Transfer From Surface Films to Shear Flows at Arbitrary Peclet Numbers,” Phys. Fluids A, 1(7), pp. 1112–1122. [CrossRef]
Fyrillas, M. M. , 2000, “ Advection-Dispersion Mass Transport Associated With a Non-Aqueous-Phase Liquid Pool,” J. Fluid Mech., 413, pp. 49–63. [CrossRef]
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. 94502. [CrossRef]
Fyrillas, M. M. , and Kontoghiorghes, E. J. , 2004, “ Numerical Calculation of Mass Transfer From Elliptical Pools in Uniform Flow Using the Boundary Element Method,” Transp. Porous Media, 55(1), pp. 91–102. [CrossRef]
Ioannou, Y. , Fyrillas, M. M. , and Doumanidis, H. , 2012, “ Approximate Solution to Fredholm Integral Equations Using Linear Regression and Applications to Heat and Mass Transfer,” Eng. Anal. Boundary Elem., 36(8), pp. 1278–1283. [CrossRef]
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. [CrossRef]
Sahin, A. , 2012, “ Critical Insulation Thickness for Maximum Entropy Generation,” Int. J. Exergy, 10(1), pp. 34–43. [CrossRef]
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. [CrossRef]
Brady, M. , and Pozrikidis, C. , 1993, “ Diffusive Transport Across Irregular and Fractal Walls,” Proc. R. Soc. London, Ser. A, 442(1916), pp. 571–583. [CrossRef]
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. [CrossRef]
Mazloomi, A. , Sharifi, F. , Salimpour, M. R. , and Moosavi, A. , 2012, “ Optimization of Highly Conductive Insert Architecture for Cooling a Rectangular Chip,” Int. Commun. Heat Mass Transfer, 39(8), pp. 1265–1271. [CrossRef]
Ganzelves, F. L. A. , and van der Geld, C. W. M. , 1997, “ The Shape Factor of Conduction in a Multiple Channel Slab and the Effect Non-Uniform Temperature,” Int. J. Heat Mass Transfer, 40(10), pp. 2493–2498. [CrossRef]
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. [CrossRef]
Moharana, M. K. , and Das, P. K. , 2008, “ Heat Conduction Through Heat Exchanger Tubes of Noncircular Cross Section,” ASME J. Heat Transfer, 130(1), p. 011301. [CrossRef]
Kundu, B. , and Das, P. K. , 2005, “ Optimum Profile of Thin Fins With Volumetric Heat Generation: A Unified Approach,” ASME J. Heat Transfer, 127(8), pp. 945–948. [CrossRef]
Aziz, A. , 1992, “ Optimum Dimensions of Extended Surfaces Operating in a Convective Environment,” ASME Appl. Mech. Rev., 45(5), pp. 155–173. [CrossRef]
COMSOL, 2008, Heat Transfer Module: COMSOL Multiphysics User’s Guide, Version 3.4, COMSOL AB, Stockholm, Sweden.
Fyrillas, M. M. , 2009, “ Shape Optimization for 2D Diffusive Scalar Transport,” Optim. Eng., 10(4), pp. 477–489. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Leontiou, T. , and Fyrillas, M. M. , 2015, “ Critical Thickness of an Optimum Extended Surface Characterized by Uniform Heat Transfer Coefficient,” e-print arXiv:1503.05148 [physics.class-ph].


Grahic Jump Location
Fig. 1

Schematic representation of the problem in the physical domain. The fins are of unit span and extend periodically in the horizontal direction. All variables are nondimensional; lengths have been nondimensionalized with the thickness of the fins t. The (dimensionless) thickness of the wall is Hb, the (dimensionless) length of the fin is H, and the (dimensionless) distance between fins is L (period). The nondimensional temperatures are T = 0 at the top boundary and T = 1 at the far field.

Grahic Jump Location
Fig. 2

Numerical domain and boundary conditions associated with a periodic array of rectangular fins. Note that the domain has been truncated taking into account periodicity/symmetry.

Grahic Jump Location
Fig. 3

Effectiveness of rectangular fins (εf) versus Biot number. a: H = 3,Hb = 0.5, and L = 2 ; b: H = 1.5,Hb = 0.1, and L = 2; and c: H = 0.5,Hb = 0.5, and L = 2. All curves cross the line εf = 1 at approximately Bi = 1.64. For Bi = 0, the effectiveness is equal to εf = 2H/L+1 (within numerical error).

Grahic Jump Location
Fig. 4

A density/contour plot of the temperature field obtained through finite element numerical simulations (Bi = 1.64). The domain and the boundary conditions are indicated in Fig. 2.



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