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Technical Brief

# Critical Biot Numbers of Periodic Arrays of 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

Sayat Ospanov

Department of Mechanical Engineering,
Nazarbayev University,
Astana 010000, Republic of Kazakhstan
e-mail: sayat.ospanov@nu.edu.kz

Ulmeken Kaibaldiyeva

Department of Chemical Engineering,
Nazarbayev University,
Astana 010000, Republic of Kazakhstan
e-mail: ulmeken.kaibaldiyeva@nu.edu.kz

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received August 25, 2016; final manuscript received January 11, 2017; published online April 19, 2017. Assoc. Editor: Amir Jokar.

J. Thermal Sci. Eng. Appl 9(4), 044502 (Apr 19, 2017) (6 pages) Paper No: TSEA-16-1238; doi: 10.1115/1.4035971 History: Received August 25, 2016; Revised January 11, 2017

## Abstract

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.

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## Figures

Fig. 1

Schematic representation of the four types of fins we considered. All the variables are nondimensional; lengths have been nondimensionalized with the thickness of the fins t, or their diameter in the case of pin fins. 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 inside surface of the wall, and T = 1 at the far field. The governing equation is the Laplace equation (∇2T = 0), i.e., conduction heat transfer, with a convection boundary condition on the outer surface: (a) triangular fins, (b) longitudinal fins, (c) axisymmetric fins, and (d) pin fins.

Fig. 2

Critical Biot number (Bicrit) versus the length H of triangular fins

Fig. 3

A density plot of the temperature field in a tube with four longitudinal fins at critical Biot number. Higher temperatures are shown darker, as shown on the legend on the right. We also showed a number of isothermal contours 0.1, 0.2,..., 0.8. The domain and the boundary conditions are indicated in Fig. 1(b). We have only considered one quarter of the domain due to symmetry.

Fig. 4

A density plot of the temperature field associated with a pin fin at the critical Biot number (Bicrit = 3.1). Higher temperatures are shown darker, as shown on the legend on the right. We also showed three isothermal contours 0.3, 0.6, and 0.9. The domain and the boundary conditions are indicated in Fig. 1(d). We have only considered a quarter of the domain in view of symmetries.

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