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# Exact Closed-Form Solution of the Nonlinear Fin Problem With Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient

[+] Author and Article Information
Mahdi Anbarloei

Department of Mathematics,
Imam Khomeini International University,
Qazvin 34149-16818, Iran
e-mail: m.anbarloei@sci.ikiu.ac.ir

Elyas Shivanian

Department of Mathematics,
Imam Khomeini International University,
Qazvin 34149-16818, Iran
e-mail: shivanian@sci.ikiu.ac.ir

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received November 12, 2015; final manuscript received May 26, 2016; published online June 28, 2016. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 138(11), 114501 (Jun 28, 2016) (6 pages) Paper No: HT-15-1726; doi: 10.1115/1.4033809 History: Received November 12, 2015; Revised May 26, 2016

## Abstract

In the current paper, the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient is revisited. In this problem, it has been assumed that the heat transfer coefficient is expressed in a power-law form and the thermal conductivity is a linear function of temperature. It is shown that its governing nonlinear differential equation is exactly solvable. A full discussion and exact analytical solution in the implicit form are given for further physical interpretation and it is proved that three possible cases may occur: there is no solution to the problem, the solution is unique, and the solutions are dual depending on the values of the parameters of the model. Furthermore, we give exact analytical expressions of fin efficiency as a function of thermogeometric fin parameter.

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## Figures

Fig. 1

Diagram of θ0 versus M for different n in four cases: β=−1,0,1,2 (left to right and up to down)

Fig. 2

Diagram of θ0 versus β for different n in four cases: M=0.2,0.4,0.6,0.8 (left to right and up to down)

Fig. 3

(Left) Unique temperature distribution with M = 1 and β = 0 for different n. (Right) Dual temperature distributions with β = 2 and different n for and M = 0.8.

Fig. 4

Fin efficiency variation for different n: (left) β = 0, (right) β=0.5

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