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Taylor–Galerkin–Legendre-wavelet approach to the analysis of a moving fin with size-dependent thermal conductivity and temperature-dependent internal heat generation

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Abstract

In heat transfer, fins are commonly used to enhance the heat transfer rate from surfaces and are widely applicable in heat exchangers and thermal energy storage systems. The material used for fins typically has a high heat conductivity. While the study of temperature-dependent heat conductivity for fins is already available, an insufficient mathematical description is observed in the case of size-dependent heat conductivity and convective heat transfer. In the current work, a heat transfer study is presented to estimate the temperature field in a moving fin that accounts for size-dependent heat conductivity and internal heat generation that depends on temperature under periodic boundary conditions. To observe the temperature field, we have developed a hybrid numerical method based on Taylor–Galerkin and Legendre wavelets. The stability analysis of the developed method is discussed in detail. Our numerical method shows excellent agreement with the analytical solution obtained in a special case. The impact of problem parameters is extensively discussed. This study shows that fin temperature decreases periodically with a space-dependent heat conductivity. In addition, for a problem which accounts for constant heat conductivity and movable fin, have greater temperature response, and standard problem which accounts for constant heat conductivity have weaker temperature response while it is between them for a problem that includes size-dependent heat conductivity and moving fin. It is shown that fin efficiency can be improved by lowering the value of the Knudsen number. Moreover, fin problem with fixed thermal conductivity offer greater efficiency in comparison with size-dependent thermal conductivity.

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Abbreviations

\(A_\textrm{c}\) :

Cross-sectional area (\(\mathrm m^2\))

c :

Specific heat (\(\mathrm{Jkg^{-1} K^{-1}}\))

h :

Coefficient of heat transfer (\(\mathrm{ Wm^{-2}K^{-1}}\))

P :

Perimeter (\(\mathrm m\))

\(q^*\) :

Heat generated inside fin (\(\mathrm {W m^{-3}}\))

\(q_{\infty }^*\) :

Heat generated inside fin at sink temperature (\(\mathrm {Wm^{-3}}\))

u :

Unidirectional speed of fin (\(\mathrm {m s^{-1}}\))

F :

Temperature (\(\mathrm K\))

\(F_\infty\) :

Sink temperature (\(\mathrm K\))

\(F_\textrm{b,m}\) :

Base temperature (\(\mathrm K\))

T :

Non-dimensional temperature

L :

Characteristic length of the system (\(\mathrm m\))

l :

Phonon mean free path (\(\mathrm m\))

X :

Spatial variable (\(\mathrm m\))

x :

Dimensionless spatial variable

J :

Non-dimensional fin parameter

G :

Non-dimensional generation number

\(A _{\text m}\) :

Non-dimensional amplitude

t :

Non-dimensional time

Pe :

Peclet number

\(K _{\text n}\) :

Knudsen number

\(\epsilon _{\text G}\) :

Non-dimensional internal heat generation

\(\omega\) :

Non-dimensional frequency

\(\epsilon\) :

Parameter of internal heat generation (\(\mathrm K^{-1}\))

\(\Omega\) :

Periodicity (\(\mathrm s^{-1}\))

\(\tau\) :

Time (\(\mathrm s\))

\(\rho\) :

Density (\(\mathrm {kgm^{-3}}\))

\(\kappa\) :

Heat conductivity (\(\mathrm {Wm^{-1} K^{-1}}\))

\(\kappa _0\) :

Bulk heat conductivity (\(\mathrm {Wm^{-1}K^{-1}}\))

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Acknowledgements

Vikas Chaurasiya, one of the authors, is grateful to DST (INSPIRE)-New Delhi (India) for the Senior Research Fellowship vide Ref. No. DST/INSPIRE/03/2017/000184 (i) Ref. no/Math/2017-18/March 18/347.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Vikas Chaurasiya & Jitendra Singh

  2. Department of Mathematics, Dehradun Institute of Technology University, Dehradun, India

    Vikas Chaurasiya

  3. Faculty of BS & H, Indian Naval Academy Ezhimala Kannur, Kerala, India

    Subrahamanyam Upadhyay

  4. Department of Mathematical Sciences, IIT (BHU), Varanasi, 221005, India

    K. N. Rai

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  1. Vikas Chaurasiya
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  2. Subrahamanyam Upadhyay
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  3. K. N. Rai
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  4. Jitendra Singh
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Contributions

VC: Formulation of model and numerical method, analytical solution of mathematical model, figure plotting, writing, analysis and data curation and original draft preparation. SU: Formulation of numerical method, and reviewing manuscript. KNR: Supervision, and reviewing manuscript. JS: Supervision and formulation of model.

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Correspondence to Jitendra Singh.

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Chaurasiya, V., Upadhyay, S., Rai, K.N. et al. Taylor–Galerkin–Legendre-wavelet approach to the analysis of a moving fin with size-dependent thermal conductivity and temperature-dependent internal heat generation. J Therm Anal Calorim 148, 12565–12581 (2023). https://doi.org/10.1007/s10973-023-12613-3

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