Journal of Computational Applied Mechanics

Analysis of Heat transfer in Porous Fin with Temperature-dependent Thermal Conductivity and Internal Heat Generation using Chebychev Spectral Collocation Method

Document Type : Research Paper

Author

Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria

Abstract

In this work, analysis of heat transfer in porous fin with temperature-dependent thermal conductivity and internal heat generation is carried out using Chebychev spectral collocation method. The numerical solutions are used to investigate the influence of various parameters on the thermal performance of the porous fin. The results show that increase in convective parameter, porosity parameter, Nusselt, Darcy and Rayleigh numbers and thickness-length ratio of the fin, the rate of heat transfer from the base of the fin increases and consequently improve the efficiency of the fin. However, the rate of heat transfer from the base of the fin increases with decrease in thermal conductivity material. Also, from the parametric studies, an optimum value is reached beyond which further increase in porosity, Nusselt, Darcy and Rayleigh numbers, thermal conductivity ratio and thickness-length ratio has no significant influence on the rate of heat transfer. It is established that the temperature predictions in the fin using the Chebychev spectral collocation method are in excellent agreement with the results of homotopy perturbation method and that of numerical methods using Runge-Kutta coupled with shooting method.

Keywords

Main Subjects

[1] S. Kiwan, A. Al-Nimr. Using Porous Fins for Heat Transfer Enhancement. ASME J. Heat Transfer 2001; 123:790–5.
[2] S. Kiwan, Effect of radiative losses on the heat transfer from porous fins. Int. J. Therm. Sci. 46(2007a)., 1046-1055
[3] S. Kiwan. Thermal analysis of natural convection porous fins. Tran. Porous Media 67(2007b), 17-29.
[4] S. Kiwan, O. Zeitoun, Natural convection in a horizontal cylindrical annulus using porous fins. Int. J. Numer. Heat Fluid Flow 18 (5)(2008), 618-634.
[5] R. S. Gorla, A. Y. Bakier. Thermal analysis of natural convection and radiation in porous fins. Int. Commun. Heat Mass Transfer 38(2011), 638-645.
[6] B. Kundu, D. Bhanji. An analytical prediction for performance and optimum design analysis of porous fins. Int. J. Refrigeration 34(2011), 337-352.
[7] B. Kundu, D. Bhanja, K. S. Lee. A model on the basis of analytics for computing maximum heat transfer in porous fins. Int. J. Heat Mass Transfer 55 (25-26)(2012) 7611-7622.
[8] Taklifi, C. Aghanajafi, H. Akrami. The effect of MHD on a porous fin attached to a vertical isothermal surface. Transp Porous Med. 85(2010) 215–31.
[9] D. Bhanja, B. Kundu. Thermal analysis of a constructal T-shaped porous fin with radiation effects. Int J Refrigerat 34(2011) 1483–96.
[10] B. Kundu, Performance and optimization analysis of SRC profile fins subject to simultaneous heat and mass transfer. Int. J. Heat Mass Transfer 50(2007) 1545-1558.
[11] S. Saedodin, S. Sadeghi, S. Temperature distribution in long porous fins in natural convection condition. Middle-east J. Sci. Res. 13 (6)(2013) 812-817.
[12] S. Saedodin, M. Olank, 2011. Temperature Distribution in Porous Fins in Natural Convection Condition, Journal of American Science 7(6)(2011) 476-481.
[13] M. T. Darvishi, R. Gorla, R.S., Khani, F., Aziz, A.-E. Thermal performance of a porus radial fin with natural convection and radiative heat losses. Thermal Science, 19(2) (2015)  669-678.
[14] M. Hatami , D. D. Ganji. Thermal performance of circular convective-radiative porous fins with different section shapes and materials. Energy Conversion and Management, 76(2013)185−193.
[15] M. Hatami , D. D. Ganji. Thermal behavior of longitudinal convective–radiative porous fins with different section shapes and ceramic materials (SiC and Si3N4). International of J. Ceramics International, 40(2014), 6765−6775.
[16] M. Hatami, A. Hasanpour, D. D. Ganji, Heat transfer study through porous fins (Si3N4 and AL) with temperature-dependent heat generation. Energ. Convers. Manage. 74(2013) 9-16.
[17] M. Hatami , D. D. Ganji. Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis. International Journal of Refrigeration, 40(2014) 140−151.
[18] M. Hatami, G. H. R. M. Ahangar, D. D. Ganji,, K. Boubaker. Refrigeration efficiency analysis for fully wet semi-spherical porous fins. Energy Conversion and Management, 84(2014) 533−540.
[19] R. Gorla, R.S., Darvishi, M. T. Khani, F.  Effects of variable Thermal conductivity on natural convection and radiation in porous fins. Int. Commun. Heat Mass Transfer 38(2013), 638-645.
[20] Moradi, A., Hayat, T. and Alsaedi, A. Convective-radiative thermal analysis of triangular fins with temperature-dependent thermal conductivity by DTM. Energy Conversion and Management 77 (2014) 70–77
[21] S. Saedodin. M. Shahbabaei. Thermal Analysis of Natural Convection in Porous Fins with Homotopy Perturbation Method (HPM). Arab J Sci Eng (2013) 38:2227–2231.
[22] H. Ha, Ganji D. D and Abbasi M. Determination of Temperature Distribution for Porous Fin with Temperature-Dependent Heat Generation by Homotopy Analysis Method. J Appl Mech Eng., 4(1) (2005).
[23] H. A. Hoshyar,  I. Rahimipetroudi, D. D. Ganji, A. R. Majidian. Thermal performance of porous fins with temperature-dependent heat generation via Homotopy perturbation method and collocation method. Journal of Applied Mathematics and Computational Mechanics. 14(4) (2015), 53-65.
[24] Y. Rostamiyan,, D. D. Ganji , I. R. Petroudi, and M. K. Nejad. Analytical Investigation of Nonlinear Model Arising in Heat Transfer Through the Porous Fin.Thermal Science. 18(2)(2014), 409-417.
[25] S. E. Ghasemi, P. Valipour, M. Hatami, D. D. Ganji.. Heat transfer study on solid and porous convective fins with temperature-dependent heat -generation using efficient analytical method J. Cent. South Univ. 21(2014), 4592−4598.
[26] D.D. Ganji, A.S. Dogonchi, Analytical investigation of convective heat transfer of a longitudinal fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation, Int. J. Phys. Sci. 9 (21) (2014), 466–474.
[27] S. Dogonchi and D. D. Ganji Convection-radiation heat transfer study of moving fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation. Applied thermal engineering 103 (2016) 705-712.
[28] Aziz and M. N. Bouaziz. A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity. Energ Conver and Manage, 52(2011): 2876−2882.
[29] Fernandez. On some approximate methods for nonlinear models. Appl Math Comput., 215(2009). :168-74.
[30] D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: Theory and applications, in: Regional Conference Series in Applied Mathematics, vol. 28, SIAM, Philadelphia, 1977, pp. 1–168.
[31] Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag,  New York, 1988.
[32] R. Peyret, Spectral Methods for Incompressible Viscous Flow, SpringerVerlag, New York, 2002.
[33] F.B. Belgacem, M. Grundmann, Approximation of the wave and electromagnetic diffusion equations by spectral methods, SIAM Journal on Scientific Computing 20 (1), (1998), 13–32. 
[34] X.W. Shan, D. Montgomery, H.D. Chen, Nonlinear magnetohydrodynamics by Galerkin-method computation, Physical Review A 44 (10) (1991) 6800–6818.
[35] X.W. Shan, Magnetohydrodynamic stabilization through rotation, Physical Review Letters 73 (12) (1994) 1624–1627.
[36] J.P. Wang, Fundamental problems in spectral methods and finite spectral method, Sinica Acta Aerodynamica 19 (2) (2001) 161–171.
[37] E.M.E. Elbarbary, M. El-kady, Chebyshev finite difference approximation for the boundary value problems, Applied Mathematics and Computation 139 (2003) 513–523.
[38] Z.J. Huang, and Z.J. Zhu, Chebyshev spectral collocation method for solution of Burgers’ equation and laminar natural convection in two-dimensional cavities,Bachelor Thesis, University of Science and Technology of China, Hefei, 2009.
[39] N.T. Eldabe, M.E.M. Ouaf, Chebyshev finite difference method for heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching surface with Ohmic heating and viscous dissipation, Applied Mathematics and Computation 177 (2006) 561–571.
[40] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers'-type equations, Journal of Computational and Applied Mathematics 222 (2008) 333–350.
[41] Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.
[42] E.H. Doha, A.H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math. 58 (2008) 1224–1244.
 [43] E.H. Doha, A.H. Bhrawy, Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations, Numer. Methods Partial Differential Equations 25 (2009) 712–739.
[44] E.H. Doha, A.H. Bhrawy, R.M. Hafez, A Jacobi–Jacobi dual-Petrov–Galerkin method for third- and fifth-order differential equations, Math. Computer Modelling 53 (2011) 1820–1832.
[45] E.H. Doha, A.H. Bhrawy, S.S. Ezzeldeen, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model. (2011) doi:10.1016/j.apm.2011.05.011.
Journal of Computational Applied Mechanics
Volume 48, Issue 2
December 2017
Pages 271-284
  • Receive Date: 03 August 2017
  • Revise Date: 25 August 2017
  • Accept Date: 04 September 2017