Investigation of free convection heat transfer and entropy generation of nanofluid flow inside a cavity affected by magnetic field and thermal radiation

  • Ahmad Hajatzadeh Pordanjani
  • Saeed Aghakhani
  • Arash Karimipour
  • Masoud Afrand
  • Marjan GoodarziEmail author


In this paper, the effect of the presence of radiation on the convection heat transfer rate and the nanofluid entropy generation within a diagonal rectangular chamber is investigated numerically in the presence of a magnetic field. The governing equations have been solved via finite volume method using the simple algorithm. In this paper, the effects of Rayleigh number, Hartmann number, magnetic field angle changes, chamber angle changes, entropy parameter, radiation parameter and volume percent of nanoparticles on the entropy generation and heat transfer have been investigated. The results show that with increasing Rayleigh number and decreasing the Hartmann number, the Nusselt number and entropy generation increase and the Bejan number decreases. By increasing the angle of the magnetic field, the heat transfer rate and the entropy generation are reduced and the Bejan number increases. By increasing the angle of the chamber at high Rayleigh numbers, the heat transfer rate increases, or by adding 6% of the nanoparticles to the base fluid, the heat transfer rate increases by 9.3% and the entropy generation increases by 15.5% in the absence of radiation. This increase in the Rd = 3 radiation parameter is 5.4% and 6.2%, respectively. It was also observed that the Nusselt number and the entropy generation increased, and with increasing the radiation parameter, the Bejan number decreased. Increasing the heat transfer rate is more significant at higher Rayleigh numbers by increasing the radiation parameter.


Radiation effects Entropy generation Magnetic field Nanofluid 

List of symbols


Magnetic field strength


Bejan number, \(Be = {\raise0.7ex\hbox{${S_{\text{gen,T}} }$} \!\mathord{\left/ {\vphantom {{S_{\text{gen,T}} } {S_{\text{Total}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${S_{\text{Total}} }$}}\)


Specific heat at constant pressure (J kg−1 K−1)


Gravity (m s−2)


Enclosure height


Hartmann number, \(Ha = B_{0} l\sqrt {\frac{{\sigma_{\text{f}} }}{{\rho_{\text{f}} \vartheta_{\text{f}} }}}\)


Thermal conductivity (W mK−1)


Thickness of enclosure


Nusselt number, hl/k


Pressure (Pa)


Prandtl number, \({\raise0.7ex\hbox{${\vartheta_{\text{f}} }$} \!\mathord{\left/ {\vphantom {{\vartheta_{\text{f}} } {\alpha_{\text{f}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\alpha_{\text{f}} }$}}\)


Rayleigh number, \(Ra = \frac{{g\beta_{\text{f}} l^{3} (T_{\text{h}} - T_{\text{c}} )}}{{\alpha_{\text{f}} \vartheta_{\text{f}} }}\)


Radiation parameter, \({\text{Rd}} = \frac{{\sigma_{\text{e}} }}{{\beta_{\text{R}} }}\left( {\left( {\rho C_{\text{P}} } \right)_{\text{f}} } \right)\)


Entropy (J K−1)


Temperature (K)


Velocity in x direction (m s−1)


Velocity in y direction (m s−1)


Cartesian coordinates (m)

Greek letters


The electrical conductivity [(Ω m)−1]


Solid volume fraction


Thermal diffusivity (m2 s−1), \({\raise0.7ex\hbox{$k$} \!\mathord{\left/ {\vphantom {k {\rho C_{\text{p}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\rho C_{\text{p}} }$}}\)


Density (kg m−2)


Dynamic viscosity (kg m−2)


Stream function value





Fluid (pure water)











  1. 1.
    Ostrach S. Natural convection in enclosures. J Heat Transfer. 1988;110:1175–90.CrossRefGoogle Scholar
  2. 2.
    Selimefendigil F, Öztop HF. Laminar convective nanofluid flow over a backward-facing step with an elastic bottom wall. J Therm Sci Eng Appl. 2018;10:041003.CrossRefGoogle Scholar
  3. 3.
    Selimefendigil F, Öztop HF. Mixed convection in a partially heated triangular cavity filled with nanofluid having a partially flexible wall and internal heat generation. J Taiwan Inst Chem Eng. 2017;70:168–78.CrossRefGoogle Scholar
  4. 4.
    Selimefendigil F, Öztop HF. Mixed convection of nanofluid filled cavity with oscillating lid under the influence of an inclined magnetic field. J Taiwan Inst Chem Eng. 2016;63:202–15.CrossRefGoogle Scholar
  5. 5.
    Aghakhani S, Pordanjani AH, Karimipour A, Abdollahi A, Afrand M. Numerical Investigation of heat transfer in a power-law non-Newtonian fluid in a C-shaped cavity with magnetic field effect using finite difference lattice Boltzmann method. Comput Fluids. 2018;176:51–67.CrossRefGoogle Scholar
  6. 6.
    Kefayati GR, Hosseinizadeh S, Gorji M, Sajjadi H. Lattice Boltzmann simulation of natural convection in tall enclosures using water/SiO2 nanofluid. Int Commun Heat Mass Transfer. 2011;38:798–805.CrossRefGoogle Scholar
  7. 7.
    Selimefendigil F, Ismael MA, Chamkha AJ. Mixed convection in superposed nanofluid and porous layers in square enclosure with inner rotating cylinder. Int J Mech Sci. 2017;124:95–108.CrossRefGoogle Scholar
  8. 8.
    Lai F-H, Yang Y-T. Lattice Boltzmann simulation of natural convection heat transfer of Al2O3/water nanofluids in a square enclosure. Int J Therm Sci. 2011;50:1930–41.CrossRefGoogle Scholar
  9. 9.
    Lin KC, Violi A. Natural convection heat transfer of nanofluids in a vertical cavity: effects of non-uniform particle diameter and temperature on thermal conductivity. Int J Heat Fluid Flow. 2010;31:236–45.CrossRefGoogle Scholar
  10. 10.
    Öğüt EB. Natural convection of water-based nanofluids in an inclined enclosure with a heat source. Int J Therm Sci. 2009;48:2063–73.CrossRefGoogle Scholar
  11. 11.
    Abu-Nada E, Oztop HF. Effects of inclination angle on natural convection in enclosures filled with Cu–water nanofluid. Int J Heat Fluid Flow. 2009;30:669–78.CrossRefGoogle Scholar
  12. 12.
    Sajjadi H, Gorji M, Kefayati GR, Ganji DD. Lattice Boltzmann simulation of natural convection in an inclined heated cavity partially using Cu/water nanofluid. Int J Fluid Mech Res. 2012. Scholar
  13. 13.
    Selimefendigil F, Öztop HF, Al-Salem K. Natural convection of ferrofluids in partially heated square enclosures. J Magn Magn Mater. 2014;372:122–33.CrossRefGoogle Scholar
  14. 14.
    Hosseini M, Mustafa M, Jafaryar M, Mohammadian E. Nanofluid in tilted cavity with partially heated walls. J Mol Liq. 2014;199:545–51.CrossRefGoogle Scholar
  15. 15.
    Selimefendigil F, Öztop HF. Numerical study and pod-based prediction of natural convection in a ferrofluids–filled triangular cavity with generalized neural networks. Numer Heat Transf Part A Appl. 2015;67:1136–61.CrossRefGoogle Scholar
  16. 16.
    Karimipour A, Esfe MH, Safaei MR, Semiromi DT, Jafari S, Kazi S. Mixed convection of copper–water nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method. Phys A. 2014;402:150–68.CrossRefGoogle Scholar
  17. 17.
    Sheikholeslami M, Ganji DD. Entropy generation of nanofluid in presence of magnetic field using lattice Boltzmann method. Phys A. 2015;417:273–86.CrossRefGoogle Scholar
  18. 18.
    Sheikholeslami M, Rashidi M, Hayat T, Ganji D. Free convection of magnetic nanofluid considering MFD viscosity effect. J Mol Liq. 2016;218:393–9.CrossRefGoogle Scholar
  19. 19.
    Selimefendigil F, Öztop HF, Chamkha AJ. MHD mixed convection and entropy generation of nanofluid filled lid driven cavity under the influence of inclined magnetic fields imposed to its upper and lower diagonal triangular domains. J Magn Magn Mater. 2016;406:266–81.CrossRefGoogle Scholar
  20. 20.
    Rashidi S, Bovand M, Esfahani JA. Opposition of magnetohydrodynamic and Al2O3–water nanofluid flow around a vertex facing triangular obstacle. J Mol Liq. 2016;215:276–84.CrossRefGoogle Scholar
  21. 21.
    Rashidi S, Esfahani JA, Maskaniyan M. Applications of magnetohydrodynamics in biological systems—a review on the numerical studies. J Magn Magn Mater. 2017;439:358–72.CrossRefGoogle Scholar
  22. 22.
    Kefayati G. Natural convection of ferrofluid in a linearly heated cavity utilizing LBM. J Mol Liq. 2014;191:1–9.CrossRefGoogle Scholar
  23. 23.
    Pirmohammadi M, Ghassemi M. Effect of magnetic field on convection heat transfer inside a tilted square enclosure. Int Commun Heat Mass Transfer. 2009;36:776–80.CrossRefGoogle Scholar
  24. 24.
    Sheikholeslami M, Shehzad S. Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM. Int J Heat Mass Transf. 2017;113:796–805.CrossRefGoogle Scholar
  25. 25.
    Selimefendigil F, Öztop HF, Abu-Hamdeh N. Natural convection and entropy generation in nanofluid filled entrapped trapezoidal cavities under the influence of magnetic field. Entropy. 2016;18:43.CrossRefGoogle Scholar
  26. 26.
    Pordanjani AH, Jahanbakhshi A, Nadooshan AA, Afrand M. Effect of two isothermal obstacles on the natural convection of nanofluid in the presence of magnetic field inside an enclosure with sinusoidal wall temperature distribution. Int J Heat Mass Transf. 2018;121:565–78.CrossRefGoogle Scholar
  27. 27.
    Oztop HF, Al-Salem K. A review on entropy generation in natural and mixed convection heat transfer for energy systems. Renew Sustain Energy Rev. 2012;16:911–20.CrossRefGoogle Scholar
  28. 28.
    Bejan A. Second law analysis in heat transfer. Energy. 1980;5:720–32.CrossRefGoogle Scholar
  29. 29.
    Cengel YA, Boles MA. Thermodynamics: an engineering approach. Sea. 2002;1000:8862.Google Scholar
  30. 30.
    Maskaniyan M, Nazari M, Rashidi S, Mahian O. Natural convection and entropy generation analysis inside a channel with a porous plate mounted as a cooling system. Therm Sci Eng Progr. 2018;6:186–93.CrossRefGoogle Scholar
  31. 31.
    Mahmoudi AH, Shahi M, Talebi F. Entropy generation due to natural convection in a partially open cavity with a thin heat source subjected to a nanofluid. Numer Heat Transf Part A Appl. 2012;61:283–305.CrossRefGoogle Scholar
  32. 32.
    Shahi M, Mahmoudi AH, Talebi F. Entropy generation due to natural convection cooling of a horizontal heat source mounted inside a square cavity filled with nanofluid. Heat Transf Res. 2012;43:19–46.CrossRefGoogle Scholar
  33. 33.
    Rashidi S, Javadi P, Esfahani JA. Second law of thermodynamics analysis for nanofluid turbulent flow inside a solar heater with the ribbed absorber plate. J Therm Anal Calorim. 2018. Scholar
  34. 34.
    Khorasanizadeh H, Amani J, Nikfar M. Numerical investigation of Cu–water nanofluid natural convection and entropy generation within a cavity with an embedded conductive baffle. Sci Iran. 2012;19:1996–2003.CrossRefGoogle Scholar
  35. 35.
    Esmaeilpour M, Abdollahzadeh M. Free convection and entropy generation of nanofluid inside an enclosure with different patterns of vertical wavy walls. Int J Therm Sci. 2012;52:127–36.CrossRefGoogle Scholar
  36. 36.
    Mahmoudi AH, Pop I, Shahi M, Talebi F. MHD natural convection and entropy generation in a trapezoidal enclosure using Cu–water nanofluid. Comput Fluids. 2013;72:46–62.CrossRefGoogle Scholar
  37. 37.
    Mahmoudi AH, Hooman K. Effect of a discrete heat source location on entropy generation in mixed convective cooling of a nanofluid inside the ventilated cavity. Int J Exergy. 2013;13:299–319.CrossRefGoogle Scholar
  38. 38.
    Khorasanizadeh H, Nikfar M, Amani J. Entropy generation of Cu–water nanofluid mixed convection in a cavity. Eur J Mech B Fluids. 2013;37:143–52.CrossRefGoogle Scholar
  39. 39.
    Chai JC, Lee HS, Patankar SV. Finite volume method for radiation heat transfer. J Thermophys Heat Transfer. 1994;8:419–25.CrossRefGoogle Scholar
  40. 40.
    Chang L, Yang K, Lloyd J. Radiation-natural convection interactions in two-dimensional complex enclosures. J Heat Transfer. 1983;105:89–95.CrossRefGoogle Scholar
  41. 41.
    Ridouane E, Hasnaoui M, Amahmid A, Raji A. Interaction between natural convection and radiation in a square cavity heated from below. Numer Heat Transf Part A Appl. 2004;45:289–311.CrossRefGoogle Scholar
  42. 42.
    Ridouane EH, Hasnaoui M, Campo A. Effects of surface radiation on natural convection in a Rayleigh–Benard square enclosure: steady and unsteady conditions. Heat Mass Transf. 2006;42:214.CrossRefGoogle Scholar
  43. 43.
    Sheikholeslami M. Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure. J Mol Liq. 2018;249:1212–21.CrossRefGoogle Scholar
  44. 44.
    Karimipour A. A novel case study for thermal radiation through a nanofluid as a semitransparent medium via discrete ordinates method to consider the absorption and scattering of nanoparticles along the radiation beams coupled with natural convection. Int Commun Heat Mass Transfer. 2017;87:256–69.CrossRefGoogle Scholar
  45. 45.
    Sheikholeslami M, Hayat T, Alsaedi A. MHD free convection of Al2O3–water nanofluid considering thermal radiation: a numerical study. Int J Heat Mass Transf. 2016;96:513–24.CrossRefGoogle Scholar
  46. 46.
    Ghalambaz M, Sabour M, Pop I. Free convection in a square cavity filled by a porous medium saturated by a nanofluid: viscous dissipation and radiation effects. Eng Sci Technol Int J. 2016;19:1244–53.CrossRefGoogle Scholar
  47. 47.
    Hayat T, Qayyum S, Imtiaz M, Alsaedi A. Comparative study of silver and copper water nanofluids with mixed convection and nonlinear thermal radiation. Int J Heat Mass Transf. 2016;102:723–32.CrossRefGoogle Scholar
  48. 48.
    Jaballah S, Sammouda H, Belghith A. Effect of surface radiation on the natural-convection stability in a two-dimensional enclosure with diffusely emitting boundary walls. Numer Heat Transf Part A Appl. 2007;51:495–516.CrossRefGoogle Scholar
  49. 49.
    Bianco N, Langellotto L, Manca O, Naso V. Numerical analysis of radiative effects on natural convection in vertical convergent and symmetrically heated channels. Numer Heat Transf Part A Appl. 2006;49:369–91.CrossRefGoogle Scholar
  50. 50.
    Kuznik F, Vareilles J, Rusaouen G, Krauss G. A double-population lattice Boltzmann method with non-uniform mesh for the simulation of natural convection in a square cavity. Int J Heat Fluid Flow. 2007;28:862–70.CrossRefGoogle Scholar
  51. 51.
    Jami M, Mezrhab A, Bouzidi MH, Lallemand P. Lattice Boltzmann method applied to the laminar natural convection in an enclosure with a heat-generating cylinder conducting body. Int J Therm Sci. 2007;46:38–47.CrossRefGoogle Scholar
  52. 52.
    Nkurikiyimfura I, Wang Y, Pan Z. Heat transfer enhancement by magnetic nanofluids—a review. Renew Sustain Energy Rev. 2013;21:548–61.CrossRefGoogle Scholar
  53. 53.
    Odenbach S. Colloidal magnetic fluids: basics, development and application of ferrofluids, vol. 763. Berlin: Springer; 2009.CrossRefGoogle Scholar
  54. 54.
    Bellos E, Tzivanidis C. A review of concentrating solar thermal collectors with and without nanofluids. J Therm Anal Calorim. 2018. Scholar
  55. 55.
    Rashidi S, Eskandarian M, Mahian O, Poncet S. Combination of nanofluid and inserts for heat transfer enhancement. J Therm Anal Calorim. 2018. Scholar
  56. 56.
    Rashidi S, Mahian O, Languri EM. Applications of nanofluids in condensing and evaporating systems. J Therm Anal Calorim. 2018;131(3):2027–39.CrossRefGoogle Scholar
  57. 57.
    Rashidi S, Karimi N, Mahian O, Esfahani JA. A concise review on the role of nanoparticles upon the productivity of solar desalination systems. J Therm Anal Calorim. 2018. Scholar
  58. 58.
    Ghasemi B, Aminossadati S, Raisi A. Magnetic field effect on natural convection in a nanofluid-filled square enclosure. Int J Therm Sci. 2011;50:1748–56.CrossRefGoogle Scholar
  59. 59.
    Vajjha RS, Das DK. Experimental determination of thermal conductivity of three nanofluids and development of new correlations. Int J Heat Mass Transf. 2009;52:4675–82.CrossRefGoogle Scholar
  60. 60.
    Hassani S, Saidur R, Mekhilef S, Hepbasli A. A new correlation for predicting the thermal conductivity of nanofluids; using dimensional analysis. Int J Heat Mass Transf. 2015;90:121–30.CrossRefGoogle Scholar
  61. 61.
    Maxwell JC, Thompson JJ. A treatise on electricity and magnetism, vol. 2. Oxford: Clarendon; 1904.Google Scholar
  62. 62.
    Brinkman H. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20:571.CrossRefGoogle Scholar
  63. 63.
    Patankar S. Numerical heat transfer and fluid flow. Boca Raton: CRC Press; 1980.Google Scholar
  64. 64.
    Oliveski RDC, Macagnan MH, Copetti JB. Entropy generation and natural convection in rectangular cavities. Appl Therm Eng. 2009;29:1417–25.CrossRefGoogle Scholar
  65. 65.
    Krane RJ, Jessee J. Some detailed field measurements for a natural convection flow in a vertical square enclosure. In: Proceedings of the first ASME–JSME thermal engineering joint conference; 1983. Vol. 1, p. 323–329.Google Scholar
  66. 66.
    Dehghani Y, Abdollahi A, Karimipour A. Experimental investigation toward obtaining a new correlation for viscosity of WO3 and Al2O3 nanoparticles-loaded nanofluid within aqueous and non-aqueous basefluids. J Therm Anal Calorim. 2018. Scholar
  67. 67.
    Arabpour A, Karimipour A, Toghraie D, Akbari OA. Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel. J Therm Anal Calorim. 2018;131(3):2975–91.CrossRefGoogle Scholar
  68. 68.
    Arabpour A, Karimipour A, Toghraie D. The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (MWCNTs) nanofluid in the microchannel heat sink with slip boundary condition. J Therm Anal Calorim. 2018;131(2):1553–66.CrossRefGoogle Scholar
  69. 69.
    Zadkhast M, Toghraie D, Karimipour A. Developing a new correlation to estimate the thermal conductivity of MWCNT-CuO/water hybrid nanofluid via an experimental investigation. J Therm Anal Calorim. 2017;129(2):859–67.CrossRefGoogle Scholar
  70. 70.
    Alrashed AA, Karimipour A, Bagherzadeh SA, Safaei MR, Afrand M. Electro-and thermophysical properties of water-based nanofluids containing copper ferrite nanoparticles coated with silica: experimental data, modeling through enhanced ANN and curve fitting. Int J Heat Mass Transf. 2018;127:925–35.CrossRefGoogle Scholar
  71. 71.
    Goodarzi M, Safaei MR, Oztop HF, Karimipour A, Sadeghinezhad E, Dahari M, Kazi SN, Jomhari N. Numerical study of entropy generation due to coupled laminar and turbulent mixed convection and thermal radiation in an enclosure filled with a semitransparent medium. Sci World J. 2014;2014:761745.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  • Ahmad Hajatzadeh Pordanjani
    • 1
  • Saeed Aghakhani
    • 2
  • Arash Karimipour
    • 2
  • Masoud Afrand
    • 2
  • Marjan Goodarzi
    • 3
    Email author
  1. 1.Department of Mechanical EngineeringShahrekord UniversityShahrekordIran
  2. 2.Department of Mechanical Engineering, Najafabad BranchIslamic Azad UniversityNajafabadIran
  3. 3.Sustainable Management of Natural Resources and Environment Research Group, Faculty of Environment and Labour SafetyTon Duc Thang UniversityHo Chi Minh CityVietnam

Personalised recommendations