On the role of enclosure side walls thickness and heater geometry in heat transfer enhancement of water–Al2O3 nanofluid in presence of a magnetic field

Sensitivity analysis and optimization
  • Seyed Masoud VahediEmail author
  • Ahmad Hajatzadeh Pordanjani
  • Somchai Wongwises
  • Masoud Afrand


The natural heat convection within a square enclosure filled with water–Al2O3 nanofluid has been studied numerically in the presence of a magnetic field. The effect of heat source geometry attached to the bottom wall on the Nusselt number was investigated by changing its nondimensional width and height, and side walls thickness of the enclosure ranging from 0.1 to 0.5, 0.1 to 0.8 and 0.05 to 0.2, respectively. A regression model has been obtained along with conducting a sensitivity analysis seeking an optimal heat transfer. Results, reveal that Nusselt number increases by enlarging the fin, and reaching a peak point before it declines. Thus, interestingly, the ever-increasing heat transfer by means of fin size does not retain and there is an optimal point wherein the maximum heat transfer occurs. Moreover, the thermal performance of the system largely depends on the fin size rather than the relative side walls thickness. However, its effect intensifies as the fin width increases. Results of optimization show that the maximum heat transfer occurs at \(W = 0.4615\), \(H = 0.6467\) and \(L_{\text{b}} = 0.2\).


Water–Al2O3 nanofluid Square enclosure Brownian motion MHD flow Response surface methodology 

List of symbols


Magnetic intensity


Specific heat \(\left( {{\text{j}}{\kern 1pt} \,{\text{kg}}^{ - 1} .{\text{k}}^{ - 1} } \right)\)


Nanoparticles diameter \(({\text{m}})\)


Convection heat transfer coefficient \(({\text{wm}}^{ - 2} \,{\text{k}}^{ - 1} )\)


Heater nondimensional height (-)


Hartmann number


Thermal conductivity \(\left( {{\text{wm}}^{ - 1} {\text{k}}^{ - 1} } \right)\)


Conductivity ratio (\(k_{\text{s}} /k_{\text{f}}\))


Side walls nondimensional thickness (-)


Nusselt number (\({\text{hl}}/k_{\text{f}}\))


Local Nusselt number \(\left( {{\text{hl}}/k_{\text{f}} } \right)\)


Averaged Nusselt number \(\left( {{\text{hl}}/k_{\text{f}} } \right)\)


Modified pressure (\(p + \rho gy\))


Pressure \(\left( {\bar{P}l^{2} /\rho_{\text{nf}} \alpha_{\text{f}}^{2} } \right)\)


Prandtl number \((\vartheta_{\text{f}} /\alpha_{\text{f}} )\)


Rayleigh number \(\left( {g\beta_{\text{f}} l^{3} (T_{\text{h}} - T_{{{\text{c}})}} /\alpha_{\text{f}} \vartheta_{f} } \right)\)


Temperature (K)


Brownian motion velocity \(\left( {{\text{ms}}^{ - 1} } \right)\)


Interstitial velocity components \(\left( {U = ul/\alpha_{\text{f}} ,\;V = \nu l/\alpha_{\text{f}} } \right)\)


Heater nondimensional width (-)

X, Y

Coordinates (x/l, y/l) \(\left( {X = x/l, \,Y = y/l} \right)\)

Greek symbols


Magnetic field angle (°)


Thermal diffusivity \(({\text{m}}^{2} {\text{s}}^{ - 1} )\)


Nanofluid concentration (-)


Boltzmann constant


Nondimensional temperature(-)


Dynamic viscosity \(({\text{wm}}^{ - 1} \,{\text{k}}^{ - 1} )\)


Kinematic viscosity \(({\text{m}}^{2} \,{\text{s}}^{ - 1} )\)


Density (\({\text{kgm}}^{ - 3} )\)


Electrical conductivity \(\left( {\varOmega .{\text{m}}} \right)\)


Inclination angle (°)







Pure fluid











  1. 1.
    Rahimi A, Rahjoo M, Hashemi SS, Sarlak MR, Malekshah MH, Malekshah EH. Combination of Dual-MRT lattice Boltzmann method with experimental observations during free convection in enclosure filled with MWCNT-MgO/Water hybrid nanofluid. Therm Sci Eng Prog. 2018;5:422–36. Scholar
  2. 2.
    Estellé P, Mahian O, Maré T, Öztop HF. Natural convection of CNT water-based nanofluids in a differentially heated square cavity. J Therm Anal Calorim. 2017;128:1765–70. Scholar
  3. 3.
    Afrand M, Rostami S, Akbari M, Wongwises S, Esfe MH, Karimipour A. Effect of induced electric field on magneto-natural convection in a vertical cylindrical annulus filled with liquid potassium. Int J Heat Mass Transf. 2015;90:418–26. Scholar
  4. 4.
    Öztop HF, Sakhrieh A, Abu-Nada E, Al-Salem K. Mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure. Int Commun Heat Mass Transf. 2017;86:42–51. Scholar
  5. 5.
    Chen CL, Chang SC, Chen CK, Chang CK. Lattice boltzmann simulation for mixed convection of nanofluids in a square enclosure. Appl Math Model. 2015;39:2436–51. Scholar
  6. 6.
    Arefmanesh A, Aghaei A, Ehteram H. Mixed convection heat transfer in a CuO-water filled trapezoidal enclosure, effects of various constant and variable properties of the nanofluid. Appl Math Model. 2016;40:815–31. Scholar
  7. 7.
    Rashidi S, Mahian O, Languri EM. Applications of nanofluids in condensing and evaporating systems: a review. J Therm Anal Calorim. 2018;131:2027–39. Scholar
  8. 8.
    Al Kalbani KS, Rahman MM, Alam S, Al-Salti N, Eltayeb IA. Buoyancy induced heat transfer flow inside a tilted square enclosure filled with nanofluids in the presence of oriented magnetic field. Heat Transf Eng. 2018;39:511–25. Scholar
  9. 9.
    Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int J Heat Mass Transf. 2003;46:3639–53. Scholar
  10. 10.
    Venkatachalappa M, Subbaraya CK. Natural convection in a rectangular enclosure in the presence of a magnetic field with uniform heat flux from the side walls. Acta Mech. 1993;96:13–26. Scholar
  11. 11.
    Teamah MAMA. Numerical simulation of double diffusive natural convection in rectangular enclosure in the presences of magnetic field and heat source. Int J Therm Sci. 2008;47:237–48. Scholar
  12. 12.
    Xu B, Li BQ, Stock DE. An experimental study of thermally induced convection of molten gallium in magnetic fields. Int J Heat Mass Transf. 2006;49:2009–19. Scholar
  13. 13.
    Rudraiah N, Barron RM, Venkatachalappa M, Subbaraya CK. Effect of a magnetic field on free convection in a rectangular enclosure. Int J Eng Sci. 1995;33:1075–84. Scholar
  14. 14.
    Mehryan SAM, Ghalambaz M, Ismael MA, Chamkha AJ. Analysis of fluid-solid interaction in MHD natural convection in a square cavity equally partitioned by a vertical flexible membrane. J Magn Magn Mater. 2017;424:161–73. Scholar
  15. 15.
    Ghalambaz M, Jamesahar E, Ismael MA, Chamkha AJ. Fluid-structure interaction study of natural convection heat transfer over a flexible oscillating fin in a square cavity. Int J Therm Sci. 2017;111:256–73. Scholar
  16. 16.
    Ismael MA, Jasim HF. Role of the fluid-structure interaction in mixed convection in a vented cavity. Int J Mech Sci. 2018;135:190–202. Scholar
  17. 17.
    Doostani A, Ghalambaz M, Chamkha AJ. MHD natural convection phase-change heat transfer in a cavity: analysis of the magnetic field effect. J Braz Soc Mech Sci Eng. 2017;39:2831–46. Scholar
  18. 18.
    Shahriari A, Jahanshahi Javaran E, Rahnama M. Effect of nanoparticles Brownian motion and uniform sinusoidal roughness elements on natural convection in an enclosure. J Therm Anal Calorim. 2018;131:2865–84. Scholar
  19. 19.
    Dogonchi AS, Chamkha AJ, Ganji DD. A numerical investigation of magneto-hydrodynamic natural convection of Cu–water nanofluid in a wavy cavity using CVFEM. J Therm Anal Calorim. 2018. Scholar
  20. 20.
    Teamah MA, El-Maghlany WM. Augmentation of natural convective heat transfer in square cavity by utilizing nanofluids in the presence of magnetic field and uniform heat generation/absorption. Int J Therm Sci. 2012;58:130–42.CrossRefGoogle Scholar
  21. 21.
    Ghasemi B, Aminossadati SMM, Raisi A. Magnetic field effect on natural convection in a nanofluid-filled square enclosure. Int J Therm Sci. 2011;50:1748–56. Scholar
  22. 22.
    Mahmoudi A, Mejri I, Omri A. Study of natural convection cooling of a nanofluid subjected to a magnetic field. Phys A Stat Mech Its Appl. 2016;451:333–48. Scholar
  23. 23.
    Sheikholeslami M, Ellahi R. Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid. Int J Heat Mass Transf. 2015;89:799–808. Scholar
  24. 24.
    Aminossadati SM, Ghasemi B. Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure. Eur J Mech B/Fluids. 2009;28:630–40. Scholar
  25. 25.
    Mahmoodia M, Esfeb MH, Akbari M, Karimipour A, Afrand M. Magneto-natural convection in square cavities with a source-sink pair on different walls. Int J Appl Electromagn Mech. 2015;47:21–32. Scholar
  26. 26.
    Karimipour A, Hossein Nezhad A, D’Orazio A, Hemmat Esfe M, Safaei MR, Shirani E. Simulation of copper-water nanofluid in a microchannel in slip flow regime using the lattice Boltzmann method. Eur J Mech B/Fluids. 2015;49:89–99. Scholar
  27. 27.
    Karimipour A, Taghipour A, Malvandi A. Developing the laminar MHD forced convection flow of water/FMWNT carbon nanotubes in a microchannel imposed the uniform heat flux. J Magn Magn Mater. 2016;419:420–8. Scholar
  28. 28.
    Karimipour A, D’Orazio A, Shadloo MS. The effects of different nano particles of Al2O3 and Ag on the MHD nano fluid flow and heat transfer in a microchannel including slip velocity and temperature jump. Phys E Low-Dimens Syst Nanostruct. 2017;86:146–53. Scholar
  29. 29.
    Esfe MH, Akbar A, Arani A. Numerical simulation of natural convection around an obstacle placed in an enclosure filled with different types of nanofluids. Heat Transf Res. 2014;45:279–92. Scholar
  30. 30.
    Shekar BC, Kishan N, Chamkha AJ. Soret and dufour effects on MHD natural convective heat and solute transfer in a fluid-saturated porous cavity. J Porous Media. 2016;19:669–86. Scholar
  31. 31.
    Yamaguchi H, Zhang XR, Niu XD, Yoshikawa K. Thermomagnetic natural convection of thermo-sensitive magnetic fluids in cubic cavity with heat generating object inside. J Magn Magn Mater. 2010;322:698–704. Scholar
  32. 32.
    Sheikholeslami M, Rashidi MM, Hayat T, Ganji DD. Free convection of magnetic nanofluid considering MFD viscosity effect. J Mol Liq. 2016;218:393–9. Scholar
  33. 33.
    Mahian O, Kianifar A, Heris SZ, Wongwises S. Natural convection of silica nanofluids in square and triangular enclosures: theoretical and experimental study. Int J Heat Mass Transf. 2016;99:792–804. Scholar
  34. 34.
    Esfandiary M, Mehmandoust B, Karimipour A, Pakravan HA. Natural convection of Al2O3-water nanofluid in an inclined enclosure with the effects of slip velocity mechanisms: Brownian motion and thermophoresis phenomenon. Int J Therm Sci. 2016;105:137–58. Scholar
  35. 35.
    Karimipour A, Hemmat Esfe M, Safaei MR, Toghraie Semiromi D, Jafari S, Kazi SN. Mixed convection of copper-water nanofluid in a shallow inclined lid driven cavity using the lattice Boltzmann method. Phys A Stat Mech Its Appl. 2014;402:150–68. Scholar
  36. 36.
    Mejri I, Mahmoudi A, Abbassi MA, Omri A. MHD natural convection in a nanofluid-filled enclosure with non-uniform heating on both side walls. Fluid Dyn Mater Process. 2014;10:83–114.Google Scholar
  37. 37.
    Pordanjani AH, Jahanbakhshi A, Ahmadi Nadooshan A, 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. Scholar
  38. 38.
    Mahmoudi A, Mejri I, Abbassi MA, Omri A. Lattice Boltzmann simulation of MHD natural convection in a nanofluid-filled cavity with linear temperature distribution. Powder Technol. 2014;256:257–71.CrossRefGoogle Scholar
  39. 39.
    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. Scholar
  40. 40.
    Maxwell JC. A treatise on electricity and magnetism, Vol. II. J Frankl Inst. 1954;258:534. Scholar
  41. 41.
    Brinkman HC. The viscosity of concentrated suspensions and solutions. J Chem Phys. 1952;20:571. Scholar
  42. 42.
    Patankar S. Numerical heat transfer and fluid flow: computational methods in mechanics and thermal science. New York: McGraw-Hill Education; 1980. Scholar
  43. 43.
    Krane R, Jessee J. Some detailed field measurements for a natural convection flow in a vertical square enclosure. Proc. First ASME-JSME Thermal Engineering Joint Conference, 1983, pp. 323–9.Google Scholar
  44. 44.
    Rashidi S, Bovand M, Esfahani JA. Heat transfer enhancement and pressure drop penalty in porous solar heat exchangers: a sensitivity analysis. Energy Convers Manag. 2015;103:726–38. Scholar
  45. 45.
    Pordanjani AH, Vahedi SM, Rikhtegar FWS. Optimization and sensitivity analysis of magneto-hydrodynamic natural convection nanofluid flow inside a square enclosure using response surface methodology. J Therm Anal Calorim. 2018. Scholar
  46. 46.
    Bovand M, Rashidi S, Esfahani JA. Optimum interaction between magnetohydrodynamics and nanofluid for thermal and drag management. J Thermophys Heat Transf. 2016;31:1–12. Scholar
  47. 47.
    Milani Shirvan K, Mirzakhanlari S, Kalogirou SA, Öztop HF, Mamourian M. Heat transfer and sensitivity analysis in a double pipe heat exchanger filled with porous medium. Int J Therm Sci. 2017;121:124–37. Scholar
  48. 48.
    Rashidi S, Bovand M, Esfahani JA. Sensitivity analysis for entropy generation in porous solar heat exchangers by RSM. J Thermophys Heat Transf. 2017;31:390–402. Scholar
  49. 49.
    Akbarzadeh M, Rashidi S, Bovand M, Ellahi R. A sensitivity analysis on thermal and pumping power for the flow of nanofluid inside a wavy channel. J Mol Liq. 2016;220:1–13. Scholar
  50. 50.
    Rashidi S, Bovand M, Esfahani JA. Optimization of partitioning inside a single slope solar still for performance improvement. Desalination. 2016;395:79–91. Scholar
  51. 51.
    Vahedi SM, Zare Ghadi A, Valipour MS. Application of response surface methodology in the optimization of magneto-hydrodynamic flow around and through a porous circular cylinder. J Mech. 2018;34:695–710. Scholar
  52. 52.
    Myer RH, Montgomery DC. Response surface methodology: process and product optimization using designed experiments. 2nd ed. Hoboken: Wiley; 2002. Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringSemnan UniversitySemnanIran
  2. 2.Gas Refining Technology Group, Gas Research DivisionResearch Institute of Petroleum Industry (RIPI)TehranIran
  3. 3.Department of Mechanical EngineeringShahrekord UniversityShahrekordIran
  4. 4.Fluid Mechanics, Thermal Engineering and Multiphase Flow Research Lab. (FUTURE), Department of Mechanical Engineering, Faculty of EngineeringKing Mongkut’s University of Technology ThonburiBangkokThailand
  5. 5.Department of Mechanical Engineering, Najafabad BranchIslamic Azad UniversityNajafabadIran

Personalised recommendations