Thermal Science 2023 Volume 27, Issue 4 Part A, Pages: 2831-2837
https://doi.org/10.2298/TSCI220917207W
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A new fractal model of the convective-radiative fins with temperature-dependent thermal conductivity
Wang Kang-Jia (School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, China), sfhpu@outlook.com
Shi Feng (School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, China)
In this paper, the convective-radiative fins of rectangular profile with temperature-dependent thermal conductivity are considered. By studying the conventional heat transfer equation, its modified fractal form, which can describe the problem in the porous medium, is presented based on He’s fractal derivative for the first time. The fractal two-scale transform method together with the Taylor series are applied to deal with fractal model, and an analytical approximate solution is obtained. The impact of the different fractal orders on the thermal behavior of the fins is also elaborated in detail. In addition, a comparison between our solution and the existing one is given to prove the correctness of the proposed method, which shows that the proposed method is easy but effective, and are expected to shed a bright light on practical applications of fractal calculus.
Keywords: He’s fractal derivative, Porous medium, Taylor series, Fractal two-scale transform method
Show references
Kraus, A. D., et al ., Extended Surface Heat Transfer, John Wiley, New York, USA, 2002
Cui, M., Song, R., Comprehensive Performance Investigation and Optimization of a Plate Fin Heat Exchanger with Wavy Fins, Thermal Science, 26 (2021), 3A, pp. 2261-2273
Khani F, Aziz A., Thermal Analysis of a Longitudinal Trapezoidal Fin with Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient, Communications in Non-linear Science and Numerical Simulation, 15 (2010), 3, pp. 590-601
Hagen, K. D., Perturbation Analysis of Tapered Fins with Non-Linear Thermal Properties, Journal of Thermophysics and Heat Transfer, 2 (1988), 2, pp. 276-279
Aziz, A., Fang, T., Thermal Analysis of an Annular fin with (a) Simultaneously Imposed Base Temperature and Base Heat Flux and (b) Fixed Base and Tip Temperatures, Energy Conversion and Management, 52 (2011), 7, pp. 2467-2478
Aziz, A., Benzies, J. Y., Application of Perturbation Techniques to Heat Transfer Problems with Variable Thermal Properties, Int. J. Heat Mass Transf., 19 (1976), 3, pp. 271-276
Domairry, G., Fazeli, M., Homotopy Analysis Method to Determine the Fin Efficiency of Convective Straight Fins with Temperature-Dependent Thermal Conductivity, Commun. Non-lin. Sci. Numer. Simul., 14 (2009), 2, pp. 489-499
Ganji, D. D., et al., Determining the Fin Efficiency of Convective Straight Fins with Temperature Dependent Thermal Conductivity by Using Homotopy Perturbation Method, Int. J. Numer. Methods Heat Fluid-Flow., 22 (2012), 2, pp. 263-272
Chiu, C. H., Chen, C. K., Application of Adomian’s Decomposition Procedure to the Analysis of Convective-Radiative Fins, Journal Heat Transf., 125 (2003), 2, pp. 312-316
Kundu, B., Wongwises, S., A Decomposition Analysis on Convecting-Radiating Rectangular Plate Fins for Variable Thermal Conductivity and Heat Transfer Coefficient, Journal Frankl. Inst., 349 (2012), 3, pp. 966-984
Moradi, A., et al., Convection-Radiation Thermal Analysis of Triangular Porous Fins with Temperature-Dependent Thermal Conductivity by DTM, Energy Convers. Manag., 77 (2014), Jan., pp. 70-77
Torabi, M., Zhang, Q. B., Analytical Solution for Evaluating the Thermal Performance and Efficiency of Convective-Radiative Straight Fins with Various Profiles and Considering All Non-Linearities, Energy Convers. Manag., 66 (2013), Feb., pp. 199-210
Anbarloei, M., Shivanian, E., Exact Closed-Form Solution of the Non-Linear Fin Problem with Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient, Journal Heat Transf., 138 (2016), 11, pp. 114501-1145016
Sen, A. K., Trinh, S., An Exact Solution for the Rate of Heat Transfer from a Rectangular Fin Governed by Power Law-Type Temperature Dependence, Joornal Heat Transf., 108 (1986), 2, pp. 457-459
Huang, Y., Li, X. F., Exact and Approximate Solutions of Convective-Radiative Fins with Temperature-Dependent Thermal Conductivity Using Integral Equation Method, International Journal of Heat and Mass Transfer, 150 (2020), 119303
He, J. H., et al., Solitary Waves Travelling along an Unsmooth Boundary, Results in Physics, 24 (2021), 104104
Wang, K. J., Backlund Transformation and Diverse Exact Explicit Solutions of the Fractal Combined KdV-mKdV Equation, Fractals, 30 (2022), 9, 2250189
Wang, K. J., et al., Periodic Wave Structure of the Fractal Generalized Fourth Order Boussinesq Equation Travelling Along the Non-Smooth Boundary, Fractals, 30 (2022), 9, 2250168
He, J. H., Abd Elazem, N. Y., The Carbon Nanotube-Embedded Boundary-Layer Theory for Energy Harvesting, Facta Universitatis, Series: Mechanical Engineering, 20 (2022), 2, pp. 211-235
Wang, K. L., et al., New Properties of the Fractal Boussinesq-Kadomtsev-Petviashvili-Like Equation with Unsmooth Boundaries. Fractals, 30 (2022), 9, 2250175
Wang, Q., et al., Intelligent Nanomaterials for Solar Energy Harvesting: From Polar Bear Hairs to Unsmooth Nanofiber Fabrication, Frontiers in Bioengineering and Biotechnology, 10 (2022), 926253
He, J. H., Thermal Science for the Real World: Reality and Challenge, Thermal Science, 24 (2020), 4, pp. 2289-2294
Wang, K. L., A Study of the Fractal Foam Drainage Model in a Microgravity Space, Mathematical Methods in the Applied Sciences, 44 (2021), 13, pp. 10530-10540
El-Nabulsi, R. A., Anukool, W., Fractal Non-Local Thermoelasticity of Thin Elastic Nanobeam with Apparent Negative Thermal Conductivity, Journal of Thermal Stresses, 45 (2022), 4, pp. 303-318
El-Nabulsi, R. A., Thermal Transport Equations in Porous Media from Product-Like Fractal Measure, Journal of Thermal Stresses, 44 (2021), 7, pp. 899-918
Wang, K. J., Si, J., On the Non-Differentiable Exact Solutions of the (2+1)-Dimensional Local Fractional Breaking Soliton Equation on Cantor sets, Mathematical Methods in the Applied Sciences, 46 (2023), 2, pp. 1456-1465
Wang, K. J., Exact Traveling Wave Solutions to the Local Fractional (3+1)-Dimensional Jimbo-Miwa Equation on Cantor Sets, Fractals, 30 (2022), 6, 2250102
Wang, K. J., et al., Generalized Variational Structure of the Fractal Modified KdV-Zakharov-Kuznetsov Equation, Fractals, 31 (2023), 7, 2350084
He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
Wang, K. J., Variational Approach for the Fractional Exothermic Reactions Model with Constant heat Source in Porous Medium, Thermal Science, 27 (2023), 4A, pp. 2879-2885
He, J. H., Taylor Series Solution for a Third Order Boundary Value Problem Arising in Architectural Engineering, Ain Shams Engineering Journal, 11 (2020), 4, pp. 1411-1414
Liu, F., et al., Thermal Oscillation Arising in a Heat Shock of a Porous Hierarchy and Its Application, Facta Universitatis, Series: Mechanical Engineering, 20 (2022), 3, pp. 633-645
Liang, Y. H., Wang, K. J., Taylor Series Solution for the Non-Linear Emden-Fowler Equations, Thermal Science, 26 (2022), 3B, pp. 2693-2697
Wang, K. J., A Simple Approach for the Fractal Riccati Differential Equation, Journal of Applied and Computational Mechanics, 7 (2021), 1, pp. 177-181
Ain, Q. T., He, J. H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
He, J. H., Ain, Q. T., New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermodynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681
He, J. H., Qian, M. Y., A Fractal Approach to the Diffusion Process of Red Ink in a Saline Water, Thermal Science, 26 (2022), 3, pp. 2447-2451
He, J., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199