Efficient Iterative Method for Investigation of Convective–Radiative Porous Fin with Internal Heat Generation Under a Uniform Magnetic Field

  • George OguntalaEmail author
  • Gbeminiyi Sobamowo
  • Raed Abd-Alhameed
  • Stephen Jones
Original Paper


This paper is aimed at presenting an efficient iterative approach using Daftardar-Gejiji and Jafari method (DJM) for the analysis of thermal behaviour of convective–radiative porous fin with internal heat generation under a uniform magnetic field. The developed heat transfer models are used to investigate the effects of convective, radiative, and magnetic parameters on the thermal performance of the porous fin. From the study, we establish that increase in porosity, convective, radiative and magnetic parameters increase the heat transferred by the fin, which subsequently improves the fin efficiency. In addition, there is significant increase in heat transfer at the base of the fin whenever the thermal conductivity of the fin decreases. The result of DJM is validated by an established result of Adomian decomposition method, and compared with the results of numerical method using first-order Runge–Kutta with shooting method and homotopy analysis method. The comparison shows that Daftardar-Gejiji and Jafari’s method exhibits higher accuracy than the established two results.


Daftardar-Gejiji and Jafari method Iterative method Thermal analysis Porous fin Convective–radiative fin Magnetic field 

List of Symbols

\( A \)

Cross-sectional area

\( x \)

Axial distance of fin

\( w \)

Width of fin

\( X \)

Dimensionless length of the fin

\( P \)

Perimeter of fin


Fin length


Velocity of fin

\( t \)

Fin thickness

\( Da \)

Darcy constant

\( g \)

Gravitational constant

\( P_{e} \)

Peclet constant

\( R_{a} \)

Modified Rayleigh number

\( R_{d} \)

Radiation–condition number


Convective dimensionless parameter


Dimensionless radiation number

\( N_{c} \)

Dimensionless convective parameter

\( N_{r} \)

Dimensionless radiative parameter

\( S_{h} \)

Porosity parameter


Total current density

\( J_{c} \)

Conduction current density

\( T \)

Fin temperature

\( T_{b} \)

Fin base temperature

\( q \)

Rate of heat transfer

\( q_{c} \)

Rate of heat transfer by convection

\( q_{r} \)

Rate of heat transfer by radiation

\( k \)

Thermal conductivity of fin material

\( k_{a} \)

Thermal conductivity of fin material at ambient temperature

\( Q \)

Dimensionless heat transfer rate per unit area

\( h \)

Coefficient of heat transfer over the fin surface

\( H \)

Coefficient of dimensionless heat transfer at the fin base

\( \rho \)

Density of saturates single-phase fluid

\( \dot{m} \)

Mass flowage of saturated single-phase fluid

\( v_{w} \)

Velocity of saturated single-phase fluid at any point



Greek Symbol

\( \alpha \)

Thermal diffusivity of the fin

\( \sigma \)

Stefan–Boltzmann constant

\( \sigma_{e} \)

Electric conductivity

\( \lambda \)

Variable thermal conductivity with temperature

\( \beta \)

Thermal conductivity parameter or nonlinear parameter

\( q \)

Heat transfer per unit area

\( \eta \)

Fin efficiency



This work is supported in part by the European Union’s Horizon 2020 research and innovation programme under Grant Agreement H2020-MSCA-ITN-2016 SECRET-722424. In addition, the authors wish to thank the reviewers for their constructive comments.


  1. 1.
    Bhanja, D., Kundu, B., Aziz, A.: Enhancement of heat transfer from a continuously moving porous fin exposed in convective–radiative environment. Energy Convers. Manag. 88, 842–853 (2014)CrossRefGoogle Scholar
  2. 2.
    Kiwan, S., Al-Nimr, M.A.: Using porous fins for heat transfer enhancement. J. Heat Transf. 123, 790–795 (2000)CrossRefGoogle Scholar
  3. 3.
    Kiwan, S.: Thermal analysis of natural convection porous fins. Transp. Porous Media 67, 17 (2006)CrossRefGoogle Scholar
  4. 4.
    Kiwan, S.: Effect of radiative losses on the heat transfer from porous fins. Int. J. Therm. Sci. 46, 1046–1055 (2007)CrossRefGoogle Scholar
  5. 5.
    Kiwan, S., Zeitoun, O.: Natural convection in a horizontal cylindrical annulus using porous fins. Int. J. Numer. Methods Heat Fluid Flow 18, 618–634 (2008)CrossRefGoogle Scholar
  6. 6.
    Rahimi-Gorji, M., Pourmehran, O., Hatami, M., Ganji, D.D.: Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis. Eur. Phys. J. Plus 130, 22 (2015)CrossRefGoogle Scholar
  7. 7.
    Gorla, R.S.R., Bakier, A.Y.: Thermal analysis of natural convection and radiation in porous fins. Int. Commun. Heat Mass Transf. 38, 638–645 (2011)CrossRefGoogle Scholar
  8. 8.
    Kundu, B.: Performance and optimization analysis of SRC profile fins subject to simultaneous heat and mass transfer. Int. J. Heat Mass Transf. 50, 1545–1558 (2007)CrossRefGoogle Scholar
  9. 9.
    Kundu, B., Bhanja, D.: An analytical prediction for performance and optimum design analysis of porous fins. Int. J. Refrig 34, 337–352 (2011)CrossRefGoogle Scholar
  10. 10.
    Kundu, B., Bhanja, D., Lee, K.-S.: A model on the basis of analytics for computing maximum heat transfer in porous fins. Int. J. Heat Mass Transf. 55, 7611–7622 (2012)CrossRefGoogle Scholar
  11. 11.
    Bhanja, D., Kundu, B.: Thermal analysis of a constructal T-shaped porous fin with radiation effects. Int. J. Refrig 34, 1483–1496 (2011)CrossRefGoogle Scholar
  12. 12.
    Taklifi, A., Aghanajafi, C., Akrami, H.: The effect of MHD on a porous fin attached to a vertical isothermal surface. Transp. Porous Media 85, 215–231 (2010)CrossRefGoogle Scholar
  13. 13.
    Seyfolah Saedodin, M.O.: Temperature distribution in porous fins in natural convection condition. J. Am. Sci. 7, 476–481 (2011)Google Scholar
  14. 14.
    Saedodin, S., Sadeghi, S.: Temperature distribution in long porous fins in natural convection condition. Middle East J. Sci. Res. 13, 812 (2013)Google Scholar
  15. 15.
    Darvishi, M.T., Subba, Rama, Gorla, Rama, Aziz, Abdul: Thermal performance of a porous radial fin with natural convection and radiative heat losses. Therm. Sci. 19, 669–678 (2012)CrossRefGoogle Scholar
  16. 16.
    Hatami, M., Ganji, D.D.: Thermal performance of circular convective–radiative porous fins with different section shapes and materials. Energy Convers. Manag. 76, 185–193 (2013)CrossRefGoogle Scholar
  17. 17.
    Hatami, M., Ganji, D.D.: Thermal behaviour of longitudinal convective–radiative porous fins with different section shapes and ceramic materials (SiC and Si3N4). Ceram. Int. 40, 6765–6775 (2014)CrossRefGoogle Scholar
  18. 18.
    Hatami, M., Ganji, D.D.: Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis. Int. J. Refrig 40, 140–151 (2014)CrossRefGoogle Scholar
  19. 19.
    Oguntala, G., Abd-Alhameed, R., Sobamowo, G.: On the effect of magnetic field on thermal performance of convective–radiative fin with temperature-dependent thermal conductivity. Karbala Int. J. Mod. Sci. 4, 1–11 (2018)CrossRefGoogle Scholar
  20. 20.
    Mosayebidorcheh, S., Ganji, D.D., Farzinpoor, M.: Approximate solution of the nonlinear heat transfer equation of a fin with the power-law temperature-dependent thermal conductivity and heat transfer coefficient. Propuls. Power Res. 3, 41–47 (2014)CrossRefGoogle Scholar
  21. 21.
    Oguntala, G.A., Abd-Alhameed, R.A., Sobamowo, G.M., Eya, N.: Effects of particles deposition on thermal performance of a convective–radiative heat sink porous fin of an electronic component. Therm. Sci. Eng. Prog. 6, 177–185 (2018)CrossRefGoogle Scholar
  22. 22.
    Roy, P.K., Mondal, H., Mallick, A.: A decomposition method for convective–radiative fin with heat generation. Ain Shams Eng. J. 6, 307–313 (2015)CrossRefGoogle Scholar
  23. 23.
    Hoshyar, H., Ganji, D.D., Majidian, A.R.: Least square method for porous fin in the presence of uniform magnetic field. J. Appl. Fluid Mech. 9, 661–668 (2016)CrossRefGoogle Scholar
  24. 24.
    Saedodin, S., Shahbabaei, M.: Thermal analysis of natural convection in porous fins with homotopy perturbation method (HPM). Arab. J. Sci. Eng. 38, 2227–2231 (2013)CrossRefGoogle Scholar
  25. 25.
    Darvishi, M.T., Subba, R., Gorla, R., Khani, F., Aziz, A.: Thermal performance of a porous radial fin with natural convection and radiative heat losses. Therm. Sci. 19, 669–678 (2015)CrossRefGoogle Scholar
  26. 26.
    Moradi, A., Hayat, T., Alsaedi, A.: Convection–radiation thermal analysis of triangular porous fins with temperature-dependent thermal conductivity by DTM. Energy Convers. Manag. 77, 70–77 (2014)CrossRefGoogle Scholar
  27. 27.
    Hoshyar, H., Ganji, D.D., Abbasi, M.: Determination of temperature distribution for porous fin with temperature-dependent heat generation by homotopy analysis method. J. Appl. Mech. Eng. 4, 153 (2015)zbMATHGoogle Scholar
  28. 28.
    Oguntala, G.A., Abd-Alhameed, R.A.: Haar wavelet collocation method for thermal analysis of porous fin with temperature-dependent thermal conductivity and internal heat generation. J. Appl. Comput. Mech. 3, 185–191 (2017)Google Scholar
  29. 29.
    Oguntala, G., Abd-Alhameed, R.: Thermal analysis of convective–radiative fin with temperature-dependent thermal conductivity using Chebychev spectral collocation method. J. Appl. Comput. Mech. 4(2), 87–94 (2018). CrossRefGoogle Scholar
  30. 30.
    Das, R.: Forward and inverse solutions of a conductive, convective and radiative cylindrical porous fin. Energy Convers. Manag. 87, 96–106 (2014)CrossRefGoogle Scholar
  31. 31.
    Rostamiyan, Y., Ganji, D.D., Petroudi, R.I., Nejad, K.M.: Analytical investigation of nonlinear model arising in heat transfer through the porous fin. Therm. Sci. 18, 409–417 (2014)CrossRefGoogle Scholar
  32. 32.
    Ghasemi, S.E., Valipour, P., Hatami, M., Ganji, D.D.: Heat transfer study on solid and porous convective fins with temperature-dependent heat generation using efficient analytical method. J. Cent. South Univ. 21, 4592–4598 (2014)CrossRefGoogle Scholar
  33. 33.
    Sobamowo, M.G., Kamiyo, O.M., Adeleye, O.A.: Thermal performance analysis of a natural convection porous fin with temperature-dependent thermal conductivity and internal heat generation. Therm. Sci. Eng. Prog. 1, 39–52 (2017)CrossRefGoogle Scholar
  34. 34.
    Hoshyar, H., Rahimipetroudi, I., Ganji, D.D., Majidian, A.R.: Thermal performance of porous fins with temperature-dependent heat generation via the homotopy perturbation method and collocation method. J. Appl. Math. Comput. Mech. 14, 53–65 (2015)CrossRefGoogle Scholar
  35. 35.
    Rezazadeh Amirkolaei, S., Ganji, D.D., Salarian, H.: Determination of temperature distribution for porous fin which is exposed to uniform magnetic field to a vertical isothermal surface by homotopy analysis method and collocation method. Indian J. Sci. Res. 1, 215–222 (2014)Google Scholar
  36. 36.
    Mosayebidorcheh, S., Sheikholeslami, M., Hatami, M., Ganji, D.D.: Analysis of turbulent MHD Couette nanofluid flow and heat transfer using hybrid DTM–FDM. Particuology 26, 95–101 (2016)CrossRefGoogle Scholar
  37. 37.
    Daftardar-Gejji, V., Jafari, H.: An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316, 753–763 (2006)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Bhalekar, S., Daftardar-Gejji, V.: New iterative method: application to partial differential equations. Appl. Math. Comput. 203, 778–783 (2008)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Daftardar-Gejji, V., Bhalekar, S.: Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method. Comput. Math Appl. 59, 1801–1809 (2010)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Daftardar-Gejji, V., Bhalekar, S.: An iterative method for solving fractional differential equations. PAMM 7, 2050017–2050018 (2007)CrossRefGoogle Scholar
  41. 41.
    Bhalekar, S., Daftardar-Gejji, V.: Solving evolution equations using a new iterative method. Numer. Methods Partial Differ. Equ. 26, 906–916 (2010)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Jafari, H., Seifi, S., Alipoor, A., Zabihi, M.: An iterative method for solving linear and nonlinear fractional diffusion-wave equation. Int. e-J. Numer. Anal. Relat. Top. 3, 20–32 (2009)Google Scholar
  43. 43.
    Yaseen, M., Samraiz, M.: The modified new iterative method for solving linear and nonlinear Klein-Gordon equations. Appl. Math. Sci. 6, 2979–2987 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.School of Electrical Engineering, Faculty of Engineering and InformaticsUniversity of BradfordBradfordUK
  2. 2.Department of Mechanical EngineeringUniversity of LagosAkokaNigeria

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