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A new fractional model for convective straight fins with temperature-dependent thermal conductivity

The key aim of this work is to present a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative. The fractional energy balance equation is solved by using homotopy perturbation method coupled with Laplace transform method. The efficiency of straight fin has been derived in terms of thermo-geometric fin parameter. The numerical results derived by the application of suggested scheme are demonstrated graphically. The subsequent correlation equations are very helpful for thermal design scientists and engineers to design straight fins having temperature-dependent thermal conductivity.
PAPER REVISED: 2017-03-11
PAPER ACCEPTED: 2017-03-24
  1. Ganji, D.D., Rajabi, A., Assessment of homotopy perturbation methods in heat radiation equations, Int. Commun. Heat Mass Transfer, 33 (2006), pp. 391-400.
  2. Aziz, A., Nguyen, H., Two dimensional performance of convecting-radiating fins of different profile shapes, Waerme Stoffuebertrag, 28 (1993), pp. 481-487.
  3. Cuce, E., Cuce, P.M., Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fines with temperature-dependent thermal conductivity, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci., (2012), pp. 1-7.
  4. Kern, D., Kraus, A., Extended Surface heat transfer, McGraw Hill Book Comp., N.Y., USA, 1972.
  5. Domairry, G., Fazeli, M., Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), pp. 489-499.
  6. Chiu C.H., Chen, C.K., A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, Int. Journal Heat and Mass Transfer, 45(10) (2002), pp. 2067-2075.
  7. Chiu, C.H., Chen, C.K., Applications of adomian's decomposition procedure to the analysis of convective-radiative fins, Journal of Heat Transfer, 125 (2003), 2, pp. 312-316.
  8. Bartas, J.G., Sellers, W.H., Radiation fin effectiveness, Journal of Heat Transfer, Series C, 82 (1960), pp. 73-75.
  9. Coskun, S.B., Atay, M.D., Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis, Math. Prob. Eng., (2007),, Article ID: 42072, p. 15.
  10. Arslanturk, C., Optimum design of space radiators with temperature-dependent thermal conductivity, Applied Thermal Engineering, 26 (2006), 11-12, pp. 1149-1157.
  11. Aziz, A., Hug, S.M.E., Perturbation solution for convecting fin with variable thermal conductivity, J. Heat Transfer, 97 (1975), pp. 300-301.
  12. Patra, A., Ray, S.S., Analysis for fin efficiency with temperature dependent thermal conductivity of fractional order energy balance equation using HPST method, Alexandria Eng. J., 55 (2016), pp. 77-85.
  13. Hilfer, R. (Ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore-New Jersey-Hong Kong, 2000, pp. 87-130.
  14. Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.
  15. Miller, K.S., Ross, B., An Introduction to the fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
  16. Baleanu, D., Guvenc, Z.B., Machado, J.A.T. (Ed.), New Trends in Nanotechnology and Fractional Calculus Applications, Springer Dordrecht Heidelberg, London New York, 2010.
  17. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  18. Kumar, D., Singh, J., Baleanu, D, Numerical computation of a fractional model of differential-difference equation, Journal of Computational and Nonlinear Dynamics, 11 (2016), doi: 10.1115/1.4033899.
  19. Singh, J., Kumar, D., Nieto, J.J., A reliable algorithm for local fractional Tricomi equation arising in fractal transonic flow, Entropy, 18 (2016), 6, doi: 10.3390/e18060206.
  20. Bhrawy, A.H., Zaky, M.A., Baleanu, D., New numerical approximations for space-time fractional Burgers' equations via a Legendre spectral-collocation method, Romanian Reports in Physics, 67 (2015), 2, pp. 340-349.
  21. Area, I., Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Torres, A., On a fractional order Ebola epidemic model, Advances in Difference Equations, (2015), doi: 10.1186/s13662-015-0613-5.
  22. Carvalho, A., Pinto, C.M.A., A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J. Dynam. Control., (2016), doi:10.1007/s40435-016-0224-3.
  23. Srivastava, H.M., Kumar, D., Singh, J., An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45 (2017), pp. 192-204.
  24. Yang, X.J., Machado, J.A.T., Baleanu D., On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, 26, Chaos, (2016), 084312.
  25. Jafari, H., Kamil, J.H., Tchier, F., Baleanu D., On the Approximate Solutions of Local Fractional Differential Equations with Local Fractional Operators, Entropy, 18 (2016), 4, Article Number: 150.
  26. Yang, X.J., Zhang, Zhi-Zhen; Machado, J. A. T., On local fractional operators view of computational complexity diffusion and relaxation defined on cantor sets, Thermal Science, 20, (2016), pp. S755-S767.
  27. He, J.H., et al., A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University Science, 2015.
  28. Wang K.L., Liu S.Y., He's fractional derivative for non-linear fractional heat transfer equation, Therm. Sci., 20 (2016), 3, pp. 793-796.
  29. Liu F.J., et al., He's Fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy, Therm. Sci., 19 (2015), 4, pp. 1155-1159.
  30. Sayevand, K., Pichaghchi , K., Analysis of nonlinear fractional KdV equation based on He's fractional derivative, Nonlinear Sci. Lett. A, 7 (2016), 3, pp. 77-85.
  31. Hu, Y., et al. On fractal space-time and fractional calculus, Therm. Sci., 20 (2016), 3, pp. 773-777.
  32. Liu, F.J., et al., A fractional model for insulation clothings with cocoon-like porous structure, Therm. Sci., 20 (2016), (3), pp. 779-784.
  33. Caputo, M., Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969.
  34. Yang, X.J., Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  35. He, J.H., A new fractal derivation. Therm. Sci. 15(2011), pp. 145-147.
  36. He, J.H., A tutorial review on fractal space time and fractional calculus, Int. J. Theor. Phys. 53(2014), 11, pp.3698-3718.
  37. Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1(2015), pp. 73-85.
  38. Atangana, A., On the new fractional derivative and application to nonlinear Fisher's reaction- diffusion equation, Applied Mathematics and Computation, 273 (2016), pp. 948-956.
  39. Atangana, A., Koca, I., On the new fractional derivative and application to nonlinear Baggs and Freedman Model, J. Nonlinear Sci. Appl., 9 (2016), pp. 2467-2480.
  40. Hristov, J., Transient Heat Diffusion with aNon-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey's kernel to the Caputo-Fabrizio time-fractional derivative, Thermal Science, 20 (2016), 2, pp.765-770.
  41. Hristov, J., Steady-State Heat Conduction in a Medium with Spatial Non-Singular Fading Memory: Derivation of Caputo-Fabrizio space-fractional derivative with Jeffrey's kernel and analytical solutions, in press; Thermal Science, (2016) OnLine-First (00):115-115; DOI:10.2298/TSCI160229115H.
  42. Sun, H.G., Hao, X., Zhang, Y., Baleanu, D., Relaxation and diffusion models with non-singular kernels, Physica A, 468 (2017), pp. 590-596.
  43. Yang, X.J., Srivastava, H.M., Machado, J.A.T., A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Thermal Science, 20, (2016), pp. 753-756.
  44. Atangana A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel, Theory and application to heat transfer model, Thermal Science, 20, (2016), 2, pp. 763-769.
  45. Mirza, I.A., Vieru, D., Fundamental solutions to advection-diffusion equation with time-fractional Caputo-Fabrizio derivative, Comput. Math. Appl., 73 (2017), 1, pp. 1-10.
  46. Atangana, A., Baleanu, D., Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., (2016), doi: /10.1061/(ASCE) EM.1943-7889.0001091.
  47. Ali, F., Saqib, M., Khan, I., Sheikh, N.A., Application of Caputo-Fabrizio derivatives to MHD free convection flow of generalized Walters'-B fluid model, Eur. Phys. J. Plus, 131, (2016), Article no. 377.
  48. Baleanu, D., Agheli, B., Al Qurashi, M.M., Fractional advection differential equation within Caputo and Caputo-Fabrizio derivatives, Adv. Mech. Eng., 8 (2017), 12, doi: 10.1177/1687814016683305.
  49. Algahtani, O.J.J., Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos Solitons Fractals, 89 (2016), pp. 552-559.
  50. Khan, Y., Wu, Q., Homotopy perturbation transform method for nonlinear equations using He's polynomials, Computer and Mathematics with Applications, 61 (2011), 8, pp. 1963-1967.
  51. Goswami, A., Singh, J., Kumar, D., A reliable algorithm for KdV equations arising in warm plasma, Nonlinear Eng., 5 (2016), 1, pp. 7-16.
  52. Kumar, D., Singh, J., Kumar, S., Analytic and approximate solutions of space- and time-fractional telegraph equations via Laplace transform, Walailak Journal of Science and Technology, 11 (2014), (8, pp 711-728.
  53. He, J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), pp. 257-262.
  54. He, J.H., Homotopy perturbation method: a new nonlinear analytical technique, Appl. Math. Comput., 135 (2003), pp. 73-79.
  55. He, J.H., New interpretation of homotopy perturbation method, Int. J. Mod. Phys. B, 20 (2006), pp. 2561-2568.
  56. Ghorbani, A., Beyond adomian's polynomials: He polynomials, Chaos Solitons Fractals, 39 (2009), pp. 1486-1492.
  57. Losada, J., Nieto, J.J., Properties of the new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), pp. 87-92.