Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1337–1351 | Cite as

Investigation of the Local Thermal Nonequilibrium Conditions for a Convective Heat Transfer Flow in an Inclined Square Enclosure Filled with Cu-Water Nanofluid

  • K. S. Al Kalbani
  • M. M. RahmanEmail author
Research Article - Mechanical Engineering


In this paper, the local thermal nonequilibrium conditions between the base fluid and the nanoparticles inside an inclined square enclosure have been investigated and analyzed numerically. The effects of magnetic field intensity and the geometry inclination angle on the heat exchange between base fluid and nanoparticles are also taken into account. Two opposite walls of the enclosure are insulated, and the other two walls are kept at different temperatures. A PDE solver, Comsol Multiphysics which uses the Galerkin weighted residual finite element technique, has been employed to solve the governing nonlinear dimensionless equations. Comparisons with previously published works are performed, and excellent agreement is obtained. Numerical simulations are accomplished to calculate the dimensionless temperature profiles along lines \(X=0.05\) and \(Y=X\) inside the enclosure. A single-phase approach with two temperature equations is applied in this work for the first time. The results indicate that the local thermal nonequilibrium conditions are highly controlled by the Nield number and the nanoparticles volume fraction. The domains at which base fluid and nanoparticles are at local thermal equilibrium or local thermal nonequilibrium have been calculated. These findings may open a door for the researchers to choose the suitable model in analyzing the dynamics of nanofluids and will be helpful in investigating the heat transfer rate based on fluid/particle interface. It will also provide the basis for the future research on the entropy generation investigation during natural convection in order to improve the energy efficiency which may be applicable for different renewable energy systems.


Convection Thermal nonequilibrium Heat transfer Inclined square enclosure Nanofluids Finite element method 

List of Symbols


Magnetic field strength (\(\hbox {kg}\,\hbox {s}^{-2}\,\hbox {A}^{-1}\))


Gravitational acceleration (\(\hbox {m s}^{-2}\))


Volumetric heat transfer coefficient (\(\hbox {W}\,\hbox {m}^{-2}\hbox {K}^{-1}\))


Hartmann number (–)


Thermal conductivity (\(\hbox {W m}^{-1}\,\hbox {K}^{-1}\))


Enclosure length (m)


Nusselt number (–)


Dimensional fluid pressure (Pa)


Dimensionless fluid pressure (–)


Time (s)


Prandtl number (–)


Rayleigh number (–)


Temperature (K)


Dimensional velocity components (\(\hbox {m s}^{-1}\))


Dimensionless velocities (–)


Dimensional coordinates (m)


Dimensionless coordinates (–)

Greek Symbols

\(\delta \)

Geometry inclination angle (\(^\circ \))

\(\beta \)

Thermal expansion coefficient (\(\hbox {K}^{-1}\))

\(\alpha \)

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

\(\gamma \)

Magnetic field inclination angle (\(^\circ \))

\(\mu \)

Dynamic viscosity (Ns \(\hbox {m}^{-2}\))

\(\upsilon \)

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

\(\rho \)

Density (\(\hbox {kg m}^{-3}\))

\(\sigma \)

Electrical conductivity (\(\hbox {S m}^{-1}\))

\(\theta \)

Dimensionless temperature (–)

\(\phi \)

Nanoparticle volume fraction (–)

\(\tau \)

Dimensionless time (–)


\({\hbox {av}}\)


\({\hbox {C}}\)

Cold wall

\({\hbox {f}}\)

Base fluid

\({\hbox {H}}\)

Hot wall

\({\hbox {nf}}\)


\({\hbox {p}}\)

Solid particle


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the anonymous referees for their constructive comments for the further improvement of the paper. M. M. Rahman is thankful to The Research Council (TRC) of Oman for funding under the Open Research Grant Program ORG/SQU/CBS/14/007 and College of Science for the Grant No. IG/SCI/DOMS/16/15. K. S. Al Kalbani is grateful to TRC for a Doctoral Sponsorship.


  1. 1.
    Ting, H.H.; Hou, S.S.: Numerical study of laminar flow and convective heat transfer utilizing nanofluids in equilateral triangular ducts with constant heat flux. Materials 9(7), 1–17 (2016)CrossRefGoogle Scholar
  2. 2.
    Ganvir, R.P.; Walke, P.V.; Kriplai, V.M.: Heat transfer characteristics in nanofluid—a review. Renew. Sustain. Energy Rev. 75, 451–460 (2017)CrossRefGoogle Scholar
  3. 3.
    Abu-Nada,; Eiyad, : Simulation of heat transfer enhancement in nanofluids using dissipative particle dynamics. Int. Commun. Heat Mass Transf. 85, 1–11 (2017)CrossRefzbMATHGoogle Scholar
  4. 4.
    Balla, H.H.; Abdullah, S.; Faizal, W.M.; Zulkifli, R.; Sopian, K.: Enhancement of heat transfer coefficient multi-metallic nanofluid with ANFIS modeling for thermophysical properties. Therm. Sci. 19(5), 1613–1620 (2015)CrossRefGoogle Scholar
  5. 5.
    Uddin, M.J.; Rahman, M.M.; Alam, M.S.: Analysis of natural convective heat transport in a homocentric annuli containing nanofluids with an oriented magnetic field using nonhomogeneous dynamic model. Neural Comput. Appl. (2017). Google Scholar
  6. 6.
    Al Kalbani, K.S.; Rahman, M.M.; Alam, M.S.; Al-Salti, N.; Eltayeb, I.A.: Buoyancy induced heat transfer flow inside a tilted square enclosure filled with nanofluids in the presence of oriented magnetic field. Heat Transf. Eng. 39(6), 511–525 (2018)CrossRefGoogle Scholar
  7. 7.
    Uddin, M.J.; Alam, M.S.; Rahman, M.M.: Natural convective heat transfer flow of nanofluids inside a quarter-circular-shaped enclosure using nonhomogeneous dynamic model. Arabian J. Sci. Eng. 42(5), 1883–1901 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sheikholeslami, M.; Zeeshan, A.: Analysis of flow and heat transfer in water based nanofluid due to magnetic field in a porous enclosure with constant heat flux using CVFEM. Comput. Methods Appl. Mech. Eng. 320, 68–81 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mohebbi, R.; Rashidi, M.M.: Numerical simulation of natural convection heat transfer of a nanofluid in L-shaped enclosure with a heating obstacle. J. Taiwan Inst. of Chem. Eng. 72, 70–84 (2017)CrossRefGoogle Scholar
  10. 10.
    Hussein, A.K.; Bakier, M.A.Y.; Hamida, M.B.B.; Sivasankaran, S.: Magneto-hydrodynamic natural convection in an inclined T-shaped enclosure for different nanofluids and subjected to a uniform heat source. Alex. Eng. J. 55(3), 2157–2169 (2016)CrossRefGoogle Scholar
  11. 11.
    Hatami, M.; Safari, H.: Effect of inside heated cylinder on the natural convection heat transfer of nanofluids in a wavy-wall enclosure. Int. J. Heat Mass Transf. 103, 1053–1057 (2016)CrossRefGoogle Scholar
  12. 12.
    Al-Weheibi, S.M.; Rahman, M.M.; Alam, M.S.; Vajravelu, K.: Nanofluids and the enhancement of natural convection heat transfer. Int. J. Mech. Sci. 131–132, 599–612 (2017)CrossRefGoogle Scholar
  13. 13.
    Quintino, A.; Ricci, E.; Habib, E.; Corcione, M.: Buoyancy-driven convection of nanofluids in an inclined enclosures. Chem. Eng. Res. Des. 122, 63–76 (2017)CrossRefGoogle Scholar
  14. 14.
    Raisi, A.: Heat transfer in an enclosure filled with a nanofluid and containing a heat-generating conductive body. Appl. Therm. Eng. 110, 469–480 (2017)CrossRefGoogle Scholar
  15. 15.
    Kalidasan, K.; Velkennedy, R.; Kanna, P.R.: Natural convection heat transfer enhancement using nanofluid and time-variant temperature on the square enclosure with diagonally constructed twin adiabatic blocks. Appl. Therm. Eng. 92, 219–235 (2016)CrossRefGoogle Scholar
  16. 16.
    Corcione, M.; Cianfrini, M.; Quintino, A.: Enhanced natural convection heat transfer of nanofluids in enclosures with two adjacent walls heated and the two opposite walls cooled. Int. J. Heat Mass Transf. 88, 902–913 (2015)CrossRefGoogle Scholar
  17. 17.
    Fontes, D.H.; dos Santos, D.D.O.; Padilla, E.L.M.; Filho, E.P.B.: Two numerical modelings of free convection heat transfer using nanofluids inside a square enclosure. Mech. Res. Commun. 66, 34–43 (2015)CrossRefGoogle Scholar
  18. 18.
    Tiwari, R.K.; Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50, 2002–2018 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Buongiorno, J.: A non-homogeneous equilibrium model for convective transport in flowing nanofluid. J. Heat Transf. 2, 599–607 (2005)Google Scholar
  20. 20.
    Jou, R.Y.; Tzeng, S.C.: Numerical research on nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures. Int. Commun. Heat Mass Transf. 33, 727–736 (2006)CrossRefGoogle Scholar
  21. 21.
    Esfandiary, M.; Habibzadeh, A.; Sayehvand, H.; Mekanik, A.: Convective heat transfer and pressure drop study in a developing laminar flow using \(\text{ Al }_{{2}}\text{ O }_{{3}}\) nanofluid. Prof. J. Eng. Res. 1(1), 1–9 (2013)Google Scholar
  22. 22.
    Göktepe, S.; Atalık, K.; Ertürk, H.: Comparison of single and two-phase models for nanofluid convection at the entrance of a uniformly heated tube. Int. J. Therm. Sci. 80, 83–92 (2014)CrossRefGoogle Scholar
  23. 23.
    Hajmohammadi, M.R.: Cylindrical couette flow and heat transfer properties of nanofluids; single-phase and two-phase analyses. J. Mol. Liq. 240, 45–55 (2017)CrossRefGoogle Scholar
  24. 24.
    Kuznetsov, A.V.; Nield, D.A.: Effect of local thermal non-equilibrium on the onset of convection in a porous medium layer saturated by a nanofluid. Transp. Porous Med. 83, 425–436 (2010)CrossRefGoogle Scholar
  25. 25.
    Armaghani, T.; Maghrebi, M.J.; Chamkha, A.J.; Nazari, M.: Effects of particle migration on nanofluid forced convection heat transfer in a local thermal non-equilibrium porous channel. J. Nanofluids 3(1), 51–59 (2014)CrossRefGoogle Scholar
  26. 26.
    Nazari, M.; Maghrebi, M.J.; Armaghani, T.; Chamkha, A.J.: New models for heat flux splitting at the boundary of a porous medium: three energy equations for nanofluid flow under local thermal nonequilibrium conditions. J. Phys. 92(11), 1312–1319 (2014)Google Scholar
  27. 27.
    Armaghani, T.; Chamkha, A.J.; Maghrebi, M.J.; Nazari, M.: Numerical analysis of a nanofluid forced convection in a porous channel; a new heat flux model in LTNE condition. J. Porous Media 17(7), 637–646 (2014)CrossRefGoogle Scholar
  28. 28.
    Seetharamu, K.N.; Leela, V.; Kotloni, N.: Numerical investigation of heat transfer in a micro-porous-channel under variable wall heat flux and variable wall temperature boundary conditions using local thermal non-equilibrium model with internal heat generation. Int. J. Heat Mass Transf. 112, 201–215 (2017)CrossRefGoogle Scholar
  29. 29.
    Öztürk, A.; Kahveci, K.: Slip flow of nanofluids between parallel plates heated with a constant heat flux. Strojniški Vestnik-J. Mech. Eng. 62(9), 511–520 (2016)CrossRefGoogle Scholar
  30. 30.
    Mebrouk, R.; Kadja, M.; Lachi, M.; Fohanno, S.: Numerical study of natural turbulent convection of nanofluids in a tall cavity heated from below. Therm. Sci. 20(6), 2051–2064 (2016)CrossRefGoogle Scholar
  31. 31.
    Al Kalbani, K.S.; Alam, M.S.; Rahman, M.M.: Finite element analysis of unsteady natural convective heat transfer and fluid flow of nanofluids inside a tilted square enclosure in the presence of oriented magnetic field. Am. J. Heat Mass Transf. 3(3), 186–224 (2016)Google Scholar
  32. 32.
    Rahman, M.M.; Al-Hatmi, M.M.: Hydromagnetic boundary layer flow and heat transfer characteristics of a nanofluid over an inclined stretching surface in the presence of convective surface: a comprehensive study. SQU J. Sci. 19(2), 53–76 (2014)CrossRefGoogle Scholar
  33. 33.
    Oztop, H.F.; Abu-Nada, E.: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow 29, 1326–1336 (2008)CrossRefGoogle Scholar
  34. 34.
    Zienkiewicz, O.C.; Taylor, R.L.: The Finite Element Method, 4th edn. McGraw-Hill, New York (1991)Google Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of ScienceSultan Qaboos UniversityAl-Khod, MuscatSultanate of Oman

Personalised recommendations