Abstract-- This note presents an explicit formula for evaluating the total heat flux in an infinite cylindrical fin. The formula is valid even for nonlinear descriptions, especially when the thermal conductivity depends on the temperature and when thermal radiation can not be neglected.
Keywords-- Nonlinear ordinary equation, unbounded domain, exact result.
I. INTRODUCTION
The steady-state heat transfer process in a fin is usually described by a boundary value problem, involving a second-order ordinary differential equation.
Many times the descriptions are non-linear (especially when thermal conductivity depends on the temperature, when there exists radiation heat transfer, or when there exists natural convection in porous fins) and the calculation of the total heat flux requires a considerable effort, since, in general, the authors look for determining the temperature distribution for, in a second step, evaluating the heat flux.
Nevertheless, when the fin is cylindrical and may be considered infinite, the heat transferred from/to the fin can be calculated in a very simple and exact way, even when the heat transfer process is described by a nonlinear differential equation.
II. METHODS
Let us begin considering the problem below, which describes the heat transfer process in an infinite cylindrical fin (Incropera and Dewitt, 1996)
(1)
in which 
depends on 
(usually an absolute temperature) in such a way that
(2)
=constant, 
and the function 
(thermal conductivity) is a function such that (Arpaci, 1966; Woodcraft,
2005, Gama, 2017)
(3)
for any , 
, 
constants.
Taking into account that the heat flux along the fin is always nonzero, let us write, from Eq. (1), the following equation
(4)
that enables us to write the following separable differential equation (Wylie, 1975)
(5)
yielding
(6)
in which 
the (unique) root of 
. Since it is expected that, when 
, we reach the thermal equilibrium, the following
holds
(7)
Now,
let us take into account that the original problem represents a mathematical
description of a cooling fin (the case in which 
). In such case, the temperature must be a
non-increasing function of the position 
and, therefore
at 
. (8)
Equation (8) may be improved in order to describe heating fins too. Aiming to do this, we can rewrite Eq. (8) as follows
at . (9)
in which 
when 
and 
when 
.
The total heat transferred by the fin is easily, and without the need of knowing the temperature distribution, obtained from
(10)
in which 
is the sectional area of the fin.
III. SOME EXAMPLES
The most classical problem
involving cylindrical fins is the linear one in which is a constant and the governing equation may be written as
In this case, we have one of the most known results in heat transfer (Incropera and Dewitt, 1996). The temperature distribution is given by
(12)
The total heat flux (transferred by the fin) is given by the well-known result below
Using Eq.(10) we obtain
(note that, here, 
) one of the most known results in heat transfer
(Holman 1976)
(14)
The
above results refer to a situation in which the thermal conductivity does not
depend on the temperature. Nevertheless, Eq. (10) allows the calculation of the
total heat, directly, even when the thermal conductivity is
temperature-dependent. In other words, we can calculate, directly, the total
heat flux even if is the solution of
, 
(15)
at 
at .
For
instance, suppose that 
in which 
is a positive constant (representing the thermal conductivity for 
and 
is a constant different from zero. In
this case, Eq. (15) is a nonlinear problem whose solution seems to be very
difficult to be obtained. Nevertheless, the total heat flux is easily obtained
from Eq. (10) as follows
(16)
Despite
is a non-zero constant, it is
interesting to evaluate the limit of the above expression as 
. Aiming to this, we have
(17)
An
interesting description is found in the article of Gorla and Bakier (2011),
which considers a porous cylindrical fin. The governing equation employed in
that paper may be represented as (the authors considered only temperatures
above and constant thermal conductivity)
(18)
at
at .
in which and 
are positive constants.
The
cases considered by Gorla and Bakier (2011) involving infinite fins, were
approximated by a numerical scheme, using boundary conditions and choosing a
large 
.
With
the result presented here, no numerical scheme is necessary and the total heat
flux for an infinite fin is exactly reached, even if the thermal conductivity
was assumed temperature-dependent. The total heat flux for the problem (Gorla
and Bakier, 2011), when 
, is given (exactly) by
(19)
IV. NUMERICAL APPROXIMATION
Many times the integral that appears in Eq. (10) requires the use of a numerical approximation. Despite the original problem involves an unbounded domain, the integration is carried out between two finite temperatures. So, Eq. (10) can always be approximated by a numerical integration formula (Wylie ,1975).
This common procedure makes the calculation of the total heat flux even simpler to be carried out, even in situations involving complicated functions.
V. CONCLUSIONS
Many authors spent a lot of time and effort evaluating the total heat flux in infinite cylindrical fins. Even so, the obtained results, many times, lie far from the exact result, since the considered fins are very long, but finite and the solutions usually are obtained with the aid of numerical schemes involving second-order differential equations and two boundary conditions. These schemes were used in Kiwan (2007), Darvish et al. (2013), Waseem et al. (2020) and Li and Pang (2020).
The exact solutions obtained by Moitsheki et al. (2020) also involve only finite fins, but consider temperature-dependent thermal conductivity. The dependence of thermal conductivity on the temperature must be considered in all cases involving large temperature variations (Goldberg et al., 2001).
The extremely simple and special result presented here provides the most efficient tool for determining the total heat flux in an infinite fin with temperature dependent thermal conductivity.
It is to be noticed that infinite fins consist of a bound for finite fins and the knowledge of the total heat transferred by an infinite fin is an important information.
ACKNOWLEDGMENT
The author R. M. S. Gama, gratefully, acknowledge the support provided by Brazilian Agency CNPq (Grant 306364/2018-1). The authors acknowledge the partial support provided by the Brazilian Agency CAPES (Finance code 001).
REFERENCES
Arpaci, V.S. (1966) Conduction Heat Transfer. Addison-Wesley Publishing Company, Massachusetts.
Darvish, M.T., Gorla, R.S.R. and Khani, F. (2013) Natural convection and radiation in porous fins. International Journal of Numerical Methods for Heat & Fluid Flow, 23, 1406-1420.
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Waseem, W., Sulaiman, M., Islam, S., Kumam, P., Nawaz, R., Raja, M. A. Z., Farooq, M. and Shoaib, M. (2020) A Study of Changes in Temperature Profile of Porous Fin using Cuckoo search Algorithm, Alexandria Engineering Journal. 59, 11-24.
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Received: October 3, 2021
Sent to Subject Editor: November 20, 2021
Accepted: January 13, 2022
Recommended by Subject Editor Fabio Giannetti