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Hydromagnetic Natural Convection Heat Transfer in a Partially Heated Enclosure Filled with Porous Medium Saturated by Nanofluid

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Abstract

In this paper, a computational analysis has been performed for hydromagnetic natural convection in a partially heated porous square enclosure filled with Al2O3-water nanofluid. The bottom wall of the enclosure is partially heated at constant high temperature Th and the vertical walls are kept at constant temperature Tc which is lower than that of hot wall while the remaining walls are thermally insulated. A modified model for effective thermal conductivity of nanofluids is introduced by taking into account the random motion of nanoparticles. Finite element method is implemented to solve the governing partial differential equations which have been formulated based on Navier–Stokes and energy balance equations along with Brinkman equation. The numerical simulation has been carried out for a range of Rayleigh number (103–106), solid volume fraction of nanoparticles (0–5%), Hartmann number (0–100) and Darcy number (0.001–1.0) and detailed discussion has been presented based on results in terms of streamlines, isotherms, average Nusselt number and average velocity, respectively. Comparison of the present results with the previously published results has been performed and excellent agreements were found. The results show that the flow and temperature fields inside the enclosure are sensitive due to the variation of Rayleigh number, concentration of nanoparticles, Hartmann number and Darcy number. It is also found that optimum heat transfer take place in higher Rayleigh number, concentration of nanoparticles and Darcy number. In addition, nanofluid shows a greater heat transfer enhancement as compared to base fluid for all concentrations of nanoparticles considered.

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Abbreviations

\( c_{p} \) :

Specific heat at constant pressure \( ({\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} ) \)

\( g \) :

Gravitational acceleration \( ({\text{ms}}^{ - 2} ) \)

L:

Length of the enclosure \( ({\text{m}}) \)

\( k \) :

Thermal conductivity \( ({\text{Wm}}^{ - 1} \;{\text{K}}^{ - 1} ) \)

K:

Permeability of the porous medium \( ({\text{m}}^{2} ) \)

p:

Dimensional pressure \( ({\text{Nm}}^{ - 2} ) \)

P:

Dimensionless pressure

\( q_{w} \) :

Heat flux \( ({\text{Wm}}^{ - 2} ) \)

\( T \) :

Dimensional temperature \( ({\text{K}}) \)

\( u\,,\,v \) :

Dimensional velocity components \( ({\text{ms}}^{ - 1} ) \)

\( U\,,\,V \) :

Dimensionless velocity components

\( x\,,\,y \) :

Dimensional coordinates \( ({\text{m}}) \)

\( X\,,\,Y \) :

Dimensionless coordinates

\( \alpha \) :

Fluid thermal diffusivity \( ({\text{m}}^{2} \;{\text{s}}^{ - 1} ) \)

\( \beta \) :

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

\( \varphi \) :

Volume fraction of nanoparticles

\( \theta \) :

Dimensionless temperature \( \theta \, = \,(T - T_{c} )/(T_{h} - T_{c} ) \)

\( \mu \) :

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

\( \nu \) :

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

\( \rho \) :

density \( ({\text{kg}}\,{\text{m}}^{ - 3} ) \)

f :

Fluid

h :

Hot

c :

Cold

nf :

Nanofluid

References

  1. Basak, T., Roy, S., Thirumalesha, C.: Finite element analysis of natural convection in a triangular enclosure: effects of various thermal boundary conditions. Chem. Eng. Sci. 62, 2623–2640 (2007)

    Article  Google Scholar 

  2. Basak, T., Roy, S., Pop, I.: Heat flow analysis for natural convection within trapezoidal enclosures based on heat line concept. Int. J. Heat Mass Transf. 52, 2471–2483 (2009)

    Article  Google Scholar 

  3. Oztop, H.F., Abu-Nada, E., Varol, Y., Chamkha, A.: Natural convection in wavy enclosures with volumetric heat sources. Int. J. Therm. Sci. 50, 502–514 (2011)

    Article  Google Scholar 

  4. Yesiloz, G., Aydin, O.: Laminar natural convection in right-angled triangular enclosures heated and cooled on adjacent walls. Int. J. Heat Mass Transf. 60, 365–374 (2013)

    Article  Google Scholar 

  5. Khanafer, K., Vafai, K., Lightstone, M.: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int. J. Heat Mass Transf. 46, 3639–3653 (2003)

    Article  Google Scholar 

  6. Jou, R.-Y., Tzeng, S.-C.: Numerical research of nature convective heat transfer enhancement filled with nanofluids in rectangular enclosures. Int. Commun. Heat Mass Transf. 33, 727–736 (2006)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Rashidi, I., Mahian, O., Lorenzini, G., Biserni, C., Wongwises, S.: Natural convection of Al2O3/water nanofluid in a square cavity: effects of heterogeneous heating. Int. J. Heat Mass Transf. 74, 391–402 (2014)

    Article  Google Scholar 

  9. Pirmohammadi, M., Ghassemi, M., Sheikhzadeh, G.A.: Effect of a magnetic field on buoyancy-driven convection in differentially heated square cavity. IEEE Trans. Magn. 45, 407–411 (2009)

    Article  Google Scholar 

  10. Sheikholeslami, M., Bandpy, M.G., Ellahi, R., Hassan, M., Soleimani, S.: Effects of MHD on Cu–water nanofluid flow and heat transfer by means of CVFEM. J. Magn. Magn. Mater. 349, 188–200 (2014)

    Article  Google Scholar 

  11. Mhamed, B., Sidik, N.A.C., Yazid, M.N.A.M., Mamat, R., Najafi, G., Kefayati, G.H.R.: A review on why researchers apply external magnetic field on nanofluids. Int. Commun. Heat Mass Transf. 78, 60–67 (2016)

    Article  Google Scholar 

  12. Ali, M.M., Alim, M.A., Akhter, R., Ahmed, S.S.: MHD natural convection flow of CuO/water nanofluid in a differentially heated hexagonal enclosure with a tilted square block. Int. J. Appl. Comput. Math. 3, 1047–1069 (2017)

    Article  MathSciNet  Google Scholar 

  13. Ali, M.M., Alim, M.A., Ahmed, S.S.: Numerical simulation of hydromagnetic natural convection flow in a grooved enclosure filled with CuO–water nanofluid considering Brownian motion. Int. J. Appl. Comput. Math. 4, 125 (2018)

    Article  MathSciNet  Google Scholar 

  14. Sathiyamoorthy, M., Basak, T., Roy, S., Pop, I.: Steady natural convection flows in a square cavity with a porous medium for linearly heated side wall(s). Int. J. Heat Mass Transf. 50, 1892–1901 (2007)

    Article  Google Scholar 

  15. Basak, T., Roy, S., Singh, S.K., Pop, I.: Analysis of mixed convection in a lid-driven porous square cavity with linearly heated side wall(s). Int. J. Heat Mass Transf. 53, 1819–1840 (2010)

    Article  Google Scholar 

  16. Lam, P.A.K., Prakash, K.A.: A numerical study on natural convection and entropy generation in a porous enclosure with heat sources. Int. J. Heat Mass Transf. 69, 390–407 (2014)

    Article  Google Scholar 

  17. Khanafer, K., AlAmiri, A., Bull, J.: Laminar natural convection heat transfer in a differentially heated cavity with a thin porous fin attached to the hot wall. Int. J. Heat Mass Transf. 87, 59–70 (2015)

    Article  Google Scholar 

  18. Torabi, M., Peterson, G.P.: Effects of velocity slip and temperature jump on the heat transfer and entropy generation in micro porous channels under magnetic field. Int. J. Heat Mass Transf. 102, 585–595 (2016)

    Article  Google Scholar 

  19. Sheremet, M.A., Grosan, T., Pop, I.: Free convection in a square cavity filled with a porous medium saturated by nanofluid using Tiwari and Das’ nanofluid model. Transp. Porous Med. 106, 595–610 (2015)

    Article  Google Scholar 

  20. Solomon, A.B., Sharifpur, M., Ottermann, T., Grobler, C., Joubert, M., Meyer, J.P.: Natural convection enhancement in a porous cavity with Al2O3-Ethylene glycol/water nanofluids. Int. J. Heat Mass Transf. 108, 1324–1334 (2017)

    Article  Google Scholar 

  21. Sheremet, M.A., Pop, I.: Natural convection in a square porous cavity with sinusoidal temperature distributions on both side walls filled with a nanofluid: Buongiorno’s Mathematical Model. Transp. Porous Med. 105, 411–429 (2014)

    Article  MathSciNet  Google Scholar 

  22. Sheremet, M.A., Oztop, H.F., Pop, I., Al-Salem, K.: MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater. Int. J. Heat Mass Transf. 103, 955–964 (2016)

    Article  Google Scholar 

  23. Muthtamilselvan, M., Kandaswamy, P., Lee, J.: Hydromagnetic mixed convection in a lid-driven cavity filled with a fluid-saturated porous medium. Int. J. Appl. Math. Mech. 5(7), 28–44 (2009)

    Google Scholar 

  24. Muthtamilselvan, M., Doh, D.-H.: Magnetic field effect on mixed convection in a lid-driven square cavity filled with nanofluids. J. Mech. Sci. Technol. 28(1), 137–143 (2014)

    Article  Google Scholar 

  25. Muthtamilselvan, M., Sureshkumar, S.: Convective heat transfer in a nanofluid-saturated porous cavity with the effects of various aspect ratios and thermal radiation. Phys. Chem. Liq. 55(5), 617–636 (2017)

    Article  Google Scholar 

  26. Acharya, N., Das, K., Kundu, P.K.: Ramification of variable thickness on MHD TiO2 and Ag nanofluid flow over a slendering stretching sheet using NDM. Eur. Phys. J. Plus 131(9), 303 (2016)

    Article  Google Scholar 

  27. Kefayati, G.H.R.: Lattice Boltzmann simulation of natural convection in a nanouid filled inclined square cavity at presence of magnetic field. Sci. Iran. 20, 1517–1527 (2013)

    Google Scholar 

  28. Kefayati, G.H.R.: Lattice Boltzmann simulation of natural convection in partially heated cavities utilizing Kerosene/Cobalt ferrofluid. Iran. J. Sci. Technol. Trans. Mech. Eng. 37, 107–118 (2013)

    Google Scholar 

  29. Kefayati, G.H.R.: FDLBM simulation of magnetic field effect on natural convection of non-Newtonian power-law uids in a linearly heated cavity. Powder Technol. 256, 87–99 (2014)

    Article  Google Scholar 

  30. Kefayati, G.H.R.: Mixed convection of non-Newtonian nanouids ows in a lid-driven enclosure with sinusoidal temperature profileusing FDLBM. Powder Technol. 266, 268–281 (2014)

    Article  Google Scholar 

  31. Kefayati, G.H.R.: Simulation of double diffusive natural convection and entropy generation of power-law fluids in an inclined porous cavity with Soret and Dufour effects (Part I: Study of fluid ow, heat and mass transfer). Int. J. Heat Mass Transf. 94, 539–581 (2014)

    Article  Google Scholar 

  32. Kefayati, G.H.R.: Simulation of double diffusive natural convection and entropy generation of power-law fluids in an inclined porous cavity with Soret and Dufour effects (Part II: entropy generation). Int. J. Heat Mass Transf. 94, 582–624 (2016)

    Article  Google Scholar 

  33. Kefayati, G.H.R.: Nor Azwadi Che Sidik, Simulation of natural convection and entropy generation of non-Newtonian nanouid in an inclined cavity using Buongiorno’s mathematical model (Part II, entropy generation). Powder Technol. 305, 679–703 (2017)

    Article  Google Scholar 

  34. Kefayati, G.H.R.: Mixed convection of non-Newtonian nanofluid in an enclosure using Buongiornos mathematical model. Int. J. Heat Mass Transf. 108, 1481–1500 (2017)

    Article  Google Scholar 

  35. Kefayati, G.H.R.: Simulation of natural convection and entropy generation of non-Newtonian nanofluid in a porous cavity using Buongiorno’s mathematical model. Int. J. Heat Mass Transf. 112, 709–744 (2017)

    Article  Google Scholar 

  36. Kefayati, G.H.R., Tang, H., Chan, A., Wang, X.: A lattice Boltzmann model for thermal non-Newtonian fluid flows through porous media. Comput. Fluids 176, 226–244 (2018)

    Article  MathSciNet  Google Scholar 

  37. Armaghani, T., Kasaeipoor, A., Alavi, N., Rashidi, M.M.: Numerical investigation of water alumina nanofluid natural convection heat transfer and entropy generation in a baffled L shaped cavity. J. Mol. Liq. 223, 243–251 (2016)

    Article  Google Scholar 

  38. Hussain, S., Mehmood, K., Sagheer, M.: MHD mixed convection and entropy generation of water–alumina nanofluid flow in a double lid driven cavity with discrete heating. J. Magn. Magn. Mater. 419, 140–155 (2016)

    Article  Google Scholar 

  39. Hussain, S., Sheikholeslami, M., Bandpy, M.G., Ganji, D.: Numerical investigation of MHD effects on Al2O3-water nanofluid flow and heat transfer in a semi-annulusenclosure using LBM. Energy 60, 501–510 (2013)

    Article  Google Scholar 

  40. Selimefendigil, F., Oztop, H.F.: Numerical study of MHD mixed convection in a nanofluid filled lid driven square enclosure with a rotating cylinder. Int. J. Heat Mass Transf. 78, 741–754 (2014)

    Article  Google Scholar 

  41. Das, S., Jana, R.N., Makinde, O.D.: Mixed convective magnetohydrodynamic flow in a vertical channel filled with nanofluids. Eng. Sci. Technol. Int. J. 18, 244–255 (2015)

    Article  Google Scholar 

  42. Maxwell-Garnett, J.C.: Colours in metal glasses and in metallic films. Philos. Trans. R. Soc. A 203, 385–420 (1904)

    Article  Google Scholar 

  43. Pak, B.C., Cho, Y.I.: Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles. Exp. Heat Transf. 11, 151–170 (1998)

    Article  Google Scholar 

  44. Cui, W., Shen, Z., Yang, J., Shaohua, W.: A modified prediction model for thermal conductivity of spherical nanoparticle suspensions (nanofluids) by introducing static and dynamic mechanisms. Ind. Eng. Chem. Res. 53(46), 18071–18080 (2014)

    Article  Google Scholar 

  45. Nasrin, R., Alim, M.A., Chamkha, A.J.: Buoyancy-driven heat transfer of water-Al2O3 nanofluid in a closed chamber: effects of solid volume fraction, Prandtl number and aspect ratio. Int. J. Heat Mass Transf. 55, 7355–7365 (2012)

    Article  Google Scholar 

  46. Mehmood, K., Hussain, S., Sagheer, M.: Mixed convection in alumina-water nanofluid filled lid-driven square cavity with an isothermally heated square blockage inside with magnetic field effect: introduction. Int. J. Heat Mass Transf. 109, 397–409 (2017)

    Article  Google Scholar 

  47. Hussain, S., Ahmed, S.E., Akbar, T.: Entropy generation analysis in MHD mixed convection of hybrid nanofluid in an open cavity with a horizontal channel containing an adiabatic obstacle. Int. J. Heat Mass Transf. 114, 1054–1066 (2017)

    Article  Google Scholar 

  48. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, 4th edn. McGraw-Hill, New York (1991)

    Google Scholar 

  49. Reddy, J.N.: An Introduction to Finite Element Analysis. McGraw-Hill, New York (1993)

    Google Scholar 

  50. Zeinkiewicz, O.C., Taylor, R.L., Too, J.M.: Reduced integration technique in general analysis of plates and shells. Int. J. Numer. Meth. Eng. 3, 275–290 (1971)

    Article  Google Scholar 

  51. Taylor, C., Hood, P.: A numerical solution of the Navier–Stokes equations using the finite element technique. Comput. Fluids 1, 73–89 (1973)

    Article  MathSciNet  Google Scholar 

  52. Dechaumphai, P.: Finite Element Method in Engineering, 2nd edn. Chulalongkorn University Press, Bangkok (1999)

    Google Scholar 

  53. de Vahl Davis, G.: Natural convection of air in a square cavity a bench mark solution. Int. J. Numer. Methods Fluids 3, 249–264 (1983)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Engineering, International University of Scholars, Dhaka, 1212, Bangladesh

    Rowsanara Akhter

  2. Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail, 1902, Bangladesh

    Mohammad Mokaddes Ali

  3. Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh

    Md. Abdul Alim

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  1. Rowsanara Akhter
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  2. Mohammad Mokaddes Ali
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  3. Md. Abdul Alim
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Correspondence to Mohammad Mokaddes Ali.

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Akhter, R., Ali, M.M. & Alim, M.A. Hydromagnetic Natural Convection Heat Transfer in a Partially Heated Enclosure Filled with Porous Medium Saturated by Nanofluid. Int. J. Appl. Comput. Math 5, 52 (2019). https://doi.org/10.1007/s40819-019-0638-7

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