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Journal of Thermal Analysis and Calorimetry

, Volume 136, Issue 4, pp 1489–1514 | Cite as

Effects of nonhomogeneous nanofluid model on convective heat transfer in partially heated square cavity with conducting solid block

  • A. I. AlsaberyEmail author
  • M. H. Yazdi
  • A. A. Altawallbeh
  • I. Hashim
Article
  • 53 Downloads

Abstract

In this study, the conjugate natural convection in a square cavity filled with \(\hbox {Al}_2\hbox {O}_3\)–water nanofluid with an inner conducting solid block is studied numerically using nonhomogeneous Buongiorno’s two-phase model. The left wall of the cavity is partially heated and the remaining parts of the wall are adiabatic, while the right wall is fully cooled. The top and bottom horizontal walls are adiabatic. The numerical simulations are based on the finite difference method. The results are simulated for various values of the nanoparticle volume fraction \((0\le \phi \le 0.04)\), Rayleigh number \((10^2\le Ra\le 10^6)\), thermal conductivity of the conjugate square \((k_{\mathrm{w}}=0.28, 0.76, 1.95, 7.0)\) and 16.0 (epoxy: 0.28, brickwork: 0.76, granite: 1.95, solid rock: 7, stainless steel: 16), the size of the inner solid \((0\le D\le 0.7)\), and the length of the heater (\(0.1\le H\le 1.0\)). The numerical results for the average and local Nusselt numbers, isotherms, distribution of nanoparticles, and streamlines are presented graphically. The findings indicate that increasing the average solid volume fraction and the size of the solid block as well as the thermal conductivity will enhance the rate of the heat transfer at low values of Rayleigh number \(Ra=10^3\). On the other hand, increasing these parameters at high values of Rayleigh number (\(Ra>10^5\)) decreases the average Nusselt number.

Keywords

Conjugate natural convection Square cavity Brownian motion Thermophoresis effect Nanoparticle distribution Partially heating 

List of symbols

\(C_{\mathrm{p}}\)

Specific heat capacity

d

Width and height of inner square

\(d_{\mathrm{f}}\)

Diameter of the base fluid molecule

\(d_{\mathrm{p}}\)

Diameter of the nanoparticle

D

Dimensionless length of the conductive solid block (\(D=d/L\))

\(D_{\mathrm{B}}\)

Brownian diffusion coefficient

\(D_{\mathrm{B}0}\)

Reference Brownian diffusion coefficient

\(D_{\mathrm{T}}\)

Thermophoretic diffusivity coefficient

\(D_{\mathrm{T}0}\)

Reference thermophoretic diffusion coefficient

\({\mathbf {g}}\)

Gravitational acceleration

H

Dimensionless length of the heat source (\(H=h/L\))

k

Thermal conductivity

\(K_{\mathrm{r}}\)

Square wall to base fluid thermal conductivity ratio (\(K_{\mathrm{r}}=k_{\mathrm{w}}/k_{\mathrm{nf}}\))

L

Width and height of cavity

Le

Lewis number

\(N_{\mathrm{BT}}\)

Ratio of Brownian to thermophoretic diffusivity

\(\overline{Nu}\)

Average Nusselt number

Pr

Prandtl number

Ra

Rayleigh number

\(Re_{\mathrm{B}}\)

Brownian motion Reynolds number

T

Temperature

\(T_0\)

Reference temperature (310 K)

\(T_{\mathrm{fr}}\)

Freezing point of the base fluid (273.15 K)

\({\mathbf{v}}\), \({\mathbf{V}}\)

Velocity and dimensionless velocity vector

\(u_{\mathrm{B}}\)

Brownian velocity of the nanoparticle

x, y and X, Y

Space coordinates and dimensionless space coordinates

Greek symbols

\(\alpha\)

Thermal diffusivity

\(\beta\)

Thermal expansion coefficient

\(\delta\)

Normalized temperature parameter

\(\theta\)

Dimensionless temperature

\(\mu\)

Dynamic viscosity

\(\nu\)

Kinematic viscosity

\(\rho\)

Density

\(\varphi\)

Solid volume fraction

\(\varphi ^*\)

Normalized solid volume fraction

\(\phi\)

Average solid volume fraction

\(\psi\) and \(\varPsi\)

Stream function and dimensionless stream function

\(\omega\) and \(\varOmega\)

Vorticity and dimensionless vorticity

Subscripts

c

Cold

f

Base fluid

h

Hot

nf

Nanofluid

p

Solid nanoparticles

w

Solid wall

Notes

Acknowledgements

The work was supported by the Universiti Kebangsaan Malaysia (UKM) research Grant DIP-2017-010. We thank the respected reviewers for their constructive comments which clearly enhanced the quality of the manuscript.

References

  1. 1.
    Nield DA, Bejan A. Convection in porous media, vol. 3. Berlin: Springer; 2006.Google Scholar
  2. 2.
    Bergman TL, Incropera FP. Introduction to heat transfer. 6th ed. New York: Wiley; 2011.Google Scholar
  3. 3.
    Corcione M. Heat transfer features of buoyancy-driven nanofluids inside rectangular cavities differentially heated at the sidewalls. Int J Therm Sci. 2010;49(9):1536.CrossRefGoogle Scholar
  4. 4.
    Ebrahimnia-Bajestan E, Niazmand H, Etminan-Farooji V, Ebrahimnia E. Numerical modeling of the freezing of a porous humid food inside a cavity due to natural convection. Numer Heat Transf Part A Appl. 2012;62(3):250.CrossRefGoogle Scholar
  5. 5.
    Altawallbeh AA, Saeid NH, Hashim I. Magnetic field effect on natural convection in a porous cavity heating from below and salting from side. Adv Mech Eng. 2013;5:183079.CrossRefGoogle Scholar
  6. 6.
    Umavathi JC, Sheremet MA, Ojjela O, Reddy GJ. The onset of double-diffusive convection in a nanofluid saturated porous layer: cross-diffusion effects. Eur J Mech B Fluids. 2017;65:70.CrossRefGoogle Scholar
  7. 7.
    Sheikholeslami M, Shehzad SA, Li Z. Water based nanofluid free convection heat transfer in a three dimensional porous cavity with hot sphere obstacle in existence of Lorenz forces. Int J Heat Mass Transf. 2018;125:375.CrossRefGoogle Scholar
  8. 8.
    Dogonchi AS, Sheremet MA, Ganji DD, Pop I. Free convection of copper–water nanofluid in a porous gap between hot rectangular cylinder and cold circular cylinder under the effect of inclined magnetic field. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7396-3 Google Scholar
  9. 9.
    Sheikholeslami M, Shehzad SA, Li Z, Shafee A. Numerical modeling for alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law. Int J Heat Mass Transf. 2018;127:614.CrossRefGoogle Scholar
  10. 10.
    Abedini A, Armaghani T, Chamkha AJ. MHD free convection heat transfer of a water–\(\text{Fe}_3\text{O}_4\) nanofluid in a baffled C-shaped enclosure. J Therm Anal Calorim. 2018.  https://doi.org/10.1007/s10973-018-7225-8 Google Scholar
  11. 11.
    Sheikholeslami M. Application of Darcy law for nanofluid flow in a porous cavity under the impact of Lorentz forces. J Mol Liq. 2018;266:495.CrossRefGoogle Scholar
  12. 12.
    Motlagh SY, Soltanipour H. Natural convection of Al2O3-water nanofluid in an inclined cavity using Buongiorno’s two-phase model. Int J Therm Sci. 2017;111:310.CrossRefGoogle Scholar
  13. 13.
    Alsabery AI, Sheremet MA, Chamkha AJ, Hashim I. Conjugate natural convection of Al2O3-water nanofluid in a square cavity with a concentric solid insert using Buongiornos two-phase model. Int J Mech Sci. 2018;136:200.CrossRefGoogle Scholar
  14. 14.
    Jayhooni SMH, Rahimpour MR. Effect of different types of nanofluids on free convection heat transfer around spherical mini-reactor. Superlattices Microstruct. 2013;58:205.CrossRefGoogle Scholar
  15. 15.
    Khanafer K, Vafai K, Lightstone M. Buoyancy-driven heat transfer enhancement in a two-dimensional cavity utilizing nanofluids. Int J Heat Mass Transf. 2003;46(19):3639.CrossRefGoogle Scholar
  16. 16.
    Hu Y, He Y, Wang S, Wang Q, Schlaberg HI. Experimental and numerical investigation on natural convection heat transfer of \(\text{TiO}_2\)–water nanofluids in a square enclosure. J Heat Transf. 2014;136(2):022502.CrossRefGoogle Scholar
  17. 17.
    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 Stat Mech Appl. 2014;402:150.CrossRefGoogle Scholar
  18. 18.
    Alsabery AI, Saleh H, Hashim I, Siddheshwar PG. Transient natural convection heat transfer in nanoliquid-saturated porous oblique cavity using thermal non-equilibrium model. Int J Mech Sci. 2016;114:233.CrossRefGoogle Scholar
  19. 19.
    Chen S, Yang B, Luo KH, Xiong X, Zheng C. Double diffusion natural convection in a square cavity filled with nanofluid. Int J Heat Mass Transf. 2016;95:1070.CrossRefGoogle Scholar
  20. 20.
    Gupta U, Ahuja J, Wanchoo R. Magneto convection in a nanofluid layer. Int J Heat Mass Transf. 2013;64:1163.CrossRefGoogle Scholar
  21. 21.
    Umavathi JC, Ojjela O, Vajravelu K. Numerical analysis of natural convective flow and heat transfer of nanofluids in a vertical rectangular duct using Darcy–Forchheimer–Brinkman model. Int J Therm Sci. 2017;111:511.CrossRefGoogle Scholar
  22. 22.
    Umavathi J, Sheremet MA. Influence of temperature dependent conductivity of a nanofluid in a vertical rectangular duct. Int J Non-Linear Mech. 2016;78:17.CrossRefGoogle Scholar
  23. 23.
    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(3):1765.CrossRefGoogle Scholar
  24. 24.
    Selimefendigil F, Öztop HF. Role of magnetic field and surface corrugation on natural convection in a nanofluid filled 3D trapezoidal cavity. Int Commun Heat Mass Transf. 2018;95:182.CrossRefGoogle Scholar
  25. 25.
    Sheikholeslami M. Solidification of NEPCM under the effect of magnetic field in a porous thermal energy storage cavity using CuO nanoparticles. J Mol Liq. 2018;263:303.CrossRefGoogle Scholar
  26. 26.
    Dogonchi AS, Ismael MA, Chamkha AJ, Ganji DD. Numerical analysis of natural convection of Cu–water nanofluid filling triangular cavity with semicircular bottom wall. J Therm Anal Calorim. 2018.  https://doi.org/10.1007/s10973-018-7520-4 Google Scholar
  27. 27.
    Corcione M. Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers Manag. 2011;52(1):789.CrossRefGoogle Scholar
  28. 28.
    Wen D, Ding Y. Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. Int J Heat Mass Transf. 2004;47(24):5181.CrossRefGoogle Scholar
  29. 29.
    He Y, Men Y, Zhao Y, Lu H, Ding Y. Numerical investigation into the convective heat transfer of \(\text{TiO}_2\) nanofluids flowing through a straight tube under the laminar flow conditions. Appl Therm Eng. 2009;29(10):1965.CrossRefGoogle Scholar
  30. 30.
    Selimefendigil F, Öztop HF. Forced convection and thermal predictions of pulsating nanofluid flow over a backward facing step with a corrugated bottom wall. Int J Heat Mass Transf. 2017;110:231.CrossRefGoogle Scholar
  31. 31.
    Selimefendigil F, Öztop HF. Jet impingement cooling and optimization study for a partly curved isothermal surface with CuO–water nanofluid. Int Commun Heat Mass Transf. 2017;89:211.CrossRefGoogle Scholar
  32. 32.
    Selimefendigil F, Öztop HF, Chamkha AJ. Fluid-structure-magnetic field interaction in a nanofluid filled lid-driven cavity with flexible side wall. Eur J Mech B Fluids. 2017;61:77.CrossRefGoogle Scholar
  33. 33.
    Sheikholeslami M, Darzi M, Sadoughi MK. Heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid; an experimental procedure. Int J Heat Mass Transf. 2018;122:643.CrossRefGoogle Scholar
  34. 34.
    Sheikholeslami M, Darzi M, Li Z. Experimental investigation for entropy generation and exergy loss of nano-refrigerant condensation process. Int J Heat Mass Transf. 2018;125:1087.CrossRefGoogle Scholar
  35. 35.
    Buongiorno J. Convective transport in nanofluids. J Heat Transf. 2006;128(3):240.CrossRefGoogle Scholar
  36. 36.
    Nield DA, Kuznetsov AV. Thermal instability in a porous medium layer saturated by a nanofluid. Int J Heat Mass Transf. 2009;52(25–26):5796.CrossRefGoogle Scholar
  37. 37.
    Tzou DY. Thermal instability of nanofluids in natural convection. Int J Heat Mass Transf. 2008;51(11–12):2967.CrossRefGoogle Scholar
  38. 38.
    Corcione M, Cianfrini M, Quintino A. Two-phase mixture modeling of natural convection of nanofluids with temperature-dependent properties. Int J Therm Sci. 2013;71:182.CrossRefGoogle Scholar
  39. 39.
    Pakravan HA, Yaghoubi M. Analysis of nanoparticles migration on natural convective heat transfer of nanofluids. Int J Therm Sci. 2013;68:79.CrossRefGoogle Scholar
  40. 40.
    Sheremet MA, Groşan T, Pop I. Steady-state free convection in right-angle porous trapezoidal cavity filled by a nanofluid: Buongiornos mathematical model. Eur J Mech B Fluids. 2015;53:241.CrossRefGoogle Scholar
  41. 41.
    Sheikholeslami M, Gorji-Bandpy M, Ganji D, Soleimani S. Thermal management for free convection of nanofluid using two phase model. J Mol Liq. 2014;194:179.CrossRefGoogle Scholar
  42. 42.
    Sheremet MA, Pop I, Rahman MM. Three-dimensional natural convection in a porous cavity filled with a nanofluid using Buongiorno’s mathematical model. Int J Heat Mass Transf. 2015;82:396.CrossRefGoogle Scholar
  43. 43.
    Shahriari A, Javaran EJ, Rahnama M. Effect of nanoparticles Brownian motion and uniform sinusoidal roughness elements on natural convection in an enclosure. J Therm Anal Calorim. 2018;131(3):2865.CrossRefGoogle Scholar
  44. 44.
    Sheikholeslami M, Rokni HB. Magnetic nanofluid flow and convective heat transfer in a porous cavity considering Brownian motion effects. Phys Fluids. 2018;30(1):012003.CrossRefGoogle Scholar
  45. 45.
    Alsabery AI, Armaghani T, Chamkha AJ, Hashim I. Conjugate heat transfer of Al2O3–water nanofluid in a square cavity heated by a triangular thick wall using Buongiornos two-phase model. J Therm Anal Calorim. 2018.  https://doi.org/10.1007/s10973-018-7473-7 Google Scholar
  46. 46.
    Oztop HF, Varol Y, Koca A. Natural convection in a vertically divided square cavity by a solid partition into air and water regions. Int J Heat Mass Transf. 2009;52(25–26):5909.CrossRefGoogle Scholar
  47. 47.
    Das MK, Reddy KSK. Conjugate natural convection heat transfer in an inclined square cavity containing a conducting block. Int J Heat Mass Transf. 2006;49(25):4987.Google Scholar
  48. 48.
    Kim DM, Viskanta R. Effect of wall heat conduction on natural convection heat transfer in a square enclosure. J Heat Transf. 1985;107(1):139.CrossRefGoogle Scholar
  49. 49.
    Merrikh AA, Lage JL. Natural convection in an cavity with disconnected and conducting solid blocks. Int J Heat Mass Transf. 2005;48(7):1361.CrossRefGoogle Scholar
  50. 50.
    House JM, Beckermann C, Smith TF. Effect of a centered conducting body on natural convection heat transfer in an enclosure. Numer Heat Transf. 1990;18(2):213.CrossRefGoogle Scholar
  51. 51.
    Zhao FY, Liu D, Tang GF. Conjugate heat transfer in square enclosures. Heat Mass Transf. 2007;43(9):907.CrossRefGoogle Scholar
  52. 52.
    Saeid NH. Conjugate natural convection in a porous enclosure: effect of conduction in one of the vertical walls. Int J Therm Sci. 2007;46(6):531.CrossRefGoogle Scholar
  53. 53.
    Sheikholeslami M, Shehzad SA, Abbasi FM, Li Z. Nanofluid flow and forced convection heat transfer due to Lorentz forces in a porous lid driven cubic cavity with hot obstacle. Comput Methods Appl Mech Eng. 2018;338:491.CrossRefGoogle Scholar
  54. 54.
    Oztop HF, Abu-Nada E. Numerical study of natural convection in partially heated rectangular cavities filled with nanofluids. Int J Heat Fluid Flow. 2008;29(5):1326.CrossRefGoogle Scholar
  55. 55.
    Altawallbeh A, Saeid N, Hashim I. Numerical solution of double-diffusive natural convection in a porous cavity partially heated from below and partially salted from the side. J Porous Media. 2013;16(10):903–19.CrossRefGoogle Scholar
  56. 56.
    Öztop HF, Estellé P, Yan WM, Al-Salem K, Orfi J, Mahian O. A brief review of natural convection in cavities under localized heating with and without nanofluids. Int Commun Heat Mass Transf. 2015;60:37.CrossRefGoogle Scholar
  57. 57.
    Jmai R, Ben-Beya B, Lili T. Numerical analysis of mixed convection at various walls speed ratios in two-sided lid-driven cavity partially heated and filled with nanofluid. J Mol Liq. 2016;221:691.CrossRefGoogle Scholar
  58. 58.
    Purusothaman A, Nithyadevi N, Oztop H, Divya V, Al-Salem K. Three dimensional numerical analysis of natural convection cooling with an array of discrete heaters embedded in nanofluid filled enclosure. Adv Powder Technol. 2016;27(1):268.CrossRefGoogle Scholar
  59. 59.
    Sheremet MA, Cimpean DS, Pop I. Free convection in a partially heated wavy porous cavity filled with a nanofluid under the effects of Brownian diffusion and thermophoresis. Appl Therm Eng. 2017;113:413.CrossRefGoogle Scholar
  60. 60.
    Selimefendigil F, Öztop HF. Analysis and predictive modeling of nanofluid-jet impingement cooling of an isothermal surface under the influence of a rotating cylinder. Int J Heat Mass Transf. 2018;121:233.CrossRefGoogle Scholar
  61. 61.
    Ho CJ, Liu WK, Chang YS, Lin CC. Natural convection heat transfer of alumina–water nanofluid in vertical square enclosures: An experimental study. Int J Therm Sci. 2010;49(8):1345.CrossRefGoogle Scholar
  62. 62.
    Sheikhzadeh GA, Dastmalchi M, Khorasanizadeh H. Effects of nanoparticles transport mechanisms on \(\text{Al}_2\text{O}_3\)–water nanofluid natural convection in a square enclosure. Int J Therm Sci. 2013;66:51.CrossRefGoogle Scholar
  63. 63.
    Cianfrini C, Corcione M, Habib E, Quintino A. Buoyancy-induced convection in/water nanofluids from an enclosed heater. Eur J Mech B Fluids. 2014;48:123.CrossRefGoogle Scholar
  64. 64.
    Chon CH, Kihm KD, Lee SP, Choi SU. Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement. Appl Phys Lett. 2005;87(15):3107.CrossRefGoogle Scholar
  65. 65.
    Heris SZ, Etemad SGh, Esfahany MN. Experimental investigation of oxide nanofluids laminar flow convective heat transfer. Int Commun Heat Mass Transf. 2006;33(4):529.CrossRefGoogle Scholar
  66. 66.
    Polidori G, Fohanno S, Nguyen CT. A note on heat transfer modelling of Newtonian nanofluids in laminar free convection. Int J Therm Sci. 2007;46(8):739.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Refrigeration & Air-Conditioning Technical Engineering Department, College of Technical EngineeringThe Islamic UniversityNajafIraq
  2. 2.School of Mathematical Sciences, Faculty of Science & TechnologyUniversiti Kebangsaan Malaysia (UKM)BangiMalaysia
  3. 3.Department of Mechanical Engineering, Neyshabur BranchIslamic Azad UniversityNeyshaburIran
  4. 4.Department of Basic Sciences and Mathematics, Faculty of SciencePhiladelphia UniversityAmmanJordan

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