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Fractional Modeling of Fin on non-Fourier Heat Conduction via Modern Fractional Differential Operators

  • Research Article-Mechanical Engineering
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Abstract

The enhancement of heat transfer for electronic kits and automobiles has become highly dependent on the finned heat exchangers; this is because fin provides high heat transfer rate and superior performance with a significant temperature reduction. In this manuscript, a fractional modeling of non-Fourier heat conduction problem of a fin is proposed within the periodic temperature boundary condition. The mathematical modeling is performed via classical theory of heat conduction that is directly proportional to temperature gradient through which hyperbolic heat conduction equation for a fin is generated. The hyperbolic heat conduction equation for a fin is fractionalized via modern approaches of fractional differentiations, namely Atangana–Baleanu and Caputo–Fabrizio differential operators. In order to have analyticity of hyperbolic heat conduction equation for a fin, we invoked the mathematical techniques of Laplace transform. The exact solutions of temperature distribution have been obtained in terms of Fox-H and Mittag–Leffler functions with the product of convolution. The solutions of temperature distribution have been classified into integer verses non-integer theories by making fractional parameters \( \alpha = \beta = 1 \) and \( \alpha \ne \beta \ne 1 \), respectively. Our results suggest that due to variability of different rheological parameters on temperature distribution, the cooling process is faster via fractional models in comparison to non-fractional model. Additionally, it is also observed that thermal wave propagates at a specific time results the reciprocal trend in temperature distribution.

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Abbreviations

\( b \) :

Straight fin with uniform thickness

\( L \) :

Length

\( T_{0} \) :

Initial temperature

\( w \) :

Width

\( \omega \) :

Temperature oscillation frequency

\( A \) :

Input temperature amplitude

\( \alpha ,\beta \) :

Fractional parameters

\( T_{\text{b,m}} \) :

Mean base temperature

\( T_{\infty } \) :

Ambient temperature

\( T_{\text{b}} \left( {0,t} \right) \) :

Periodic base temperature

\( \tau \) :

Relaxation time

\( t \) :

Time

\( T\left( {y,t} \right) \) :

Temperature

\( k \) :

Thermal conductivity

\( \rho \) :

Density

\( C \) :

Specific heat capacity in a medium

\( \lambda_{1} \) :

Relaxation time

\( \lambda_{2} \) :

Relaxation time

\( \lambda_{3} \) :

Convective heat transfer

\( \theta \) :

Temperature distribution

\( \Omega \) :

Frequency of the temperature oscillation

\( A_{0} - A_{4} \) :

Letting parameters of Caputo–Fabrizio fractional operator

\( B_{0} - B_{4} \) :

Letting parameters of Atangana–Baleanu fractional operator

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Acknowledgements

The author Kashif Ali Abro is highly thankful and grateful to Mehran university of Engineering and Technology, Jamshoro, Pakistan, for generous support and facilities of this research work.

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Authors and Affiliations

  1. Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

    Kashif Ali Abro

  2. Departamento de Ingeniería Electrónica, CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

    Jose Francisco Gomez-Aguilar

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Correspondence to Kashif Ali Abro.

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Appendix

Appendix

$$ {\mathcal{L}}^{ - 1} \left( {\frac{1}{{\left( {s^{\alpha } - \beta } \right)}}} \right) = t^{\alpha - 1} {\mathbf{E}}_{\alpha ,\alpha } \left( { - \beta t^{\alpha } } \right), $$
(A1)
$$ {\mathcal{L}}^{ - 1} \left( {\frac{{s^{\alpha } }}{{s\left( {s^{\alpha } + \beta } \right)}}} \right) = {\mathbf{E}}_{\alpha } \left( { - \beta t^{\alpha } } \right), $$
(A2)

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Abro, K.A., Gomez-Aguilar, J.F. Fractional Modeling of Fin on non-Fourier Heat Conduction via Modern Fractional Differential Operators. Arab J Sci Eng 46, 2901–2910 (2021). https://doi.org/10.1007/s13369-020-05243-6

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  • DOI: https://doi.org/10.1007/s13369-020-05243-6

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