Natural Convection and Surface Radiation Heat Transfer in a Square Cavity with an Inner Wavy Body

Abstract

An in-house CFD code has been developed to address the coupling between natural convection and surface radiation in irregular geometries encountered in cooling electronic components. In the current numerical study, the discrete ordinate methods with the concept of blocked-off region are used in order to investigate the conjugate natural convection and surface radiation in an enclosure containing a wavy inner circular cylinder. The validity of the numerical code is ascertained by comparing our results with previously published results dealing with the limiting case of a smooth circular cylinder enclosed in a cavity. Extensive computations have been performed to study in detail the effect of the Rayleigh number (\(10^{3} \le {\text{Ra}} \le 10^{7}\)), the inner and outer surface emissivities (\(0 \le \upvarepsilon_{{{\text{E}}\,{\text{or I}}}} \le 1\)), the undulations number (\(4 \le {\text{N}} \le 1 2\), their amplitude (\(0 \le {\upalpha} \le 1\)) and the average radius of the wavy circular cylinder (\(0.05 \le {\text{R}}_{ 0} \le 0.25\)) on the flow structure as well as on the convective and radiative heat transfer characteristics. The obtained results showed that the amplitude and the average radius have more significant effect on the total heat transfer rate than the undulations number of the wavy wall.

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Abbreviations

g:

Gravitational acceleration, m·s−2

H:

Height of the cavity, m

I:

Dimensionless intensity

k:

Thermal conductivity, W·m−1K−1

L:

Width of the cavity, m

N:

Undulations number

Nu:

Nusselt number

P:

Pressure, Pa

Pl:

Planck number

Pr:

Prandtl number

Ra:

Rayleigh number

T:

Temperature, K

TR :

Temperature ratio

U, V:

Velocity components

x, y:

Cartesian coordinates, m

X, Y:

Dimensionless Cartesian coordinates

α:

Thermal diffisuvity, m2·s−1, undulations amplitude

β:

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

\({{\Delta {\mathbf{T}}}}\) :

Temperature difference, \({\text{K}}\)

δ:

Kronecker symbol

ε:

Emissivity of the radiative surface

θ:

Dimensionless temperature

\({\upsigma }\) :

Stefan-Boltzmann constant, W·K−4·m−2

\({{\upnu }}\) :

Kinematic viscosity, m2·s−1

ρ:

Density, kg·m−3

τ:

Time, s, optical thickness

\({{\upxi }}\) :

Direction cosines

ω:

Scattering albedo

B:

Black body

C:

Cold, convective

E:

External

H:

Hot

i, j:

Directions

I:

Internal

L:

Local

max:

Maximum

min:

Minimum

R:

Radiative

S:

Surface

References

  1. 1.

    C.C. Wang, W.L. Fu, C.T. Chang, Heat transfer and friction characteristics of typical wavy fin-and-tube heat exchangers. Exp. Thermal Fluid Sci. 14(2), 174–186 (1997). https://doi.org/10.1016/s0894-1777(96)00056-8

    Article  Google Scholar 

  2. 2.

    J.Y. Jang, L.K. Chen, Numerical analysis of heat transfer and fluid flow in a three-dimensional wavy-fin and tube heat exchanger. Int. J. Heat Mass Transf. 40(16), 3981–3990 (1997). https://doi.org/10.1016/s0017-9310(97)00047-1

    Article  Google Scholar 

  3. 3.

    M. Aounallah, Y. Addad, S. Benhamadouch, O. Imine, L. Adjlout, D. Laurence, Numerical investigation of turbulent natural convection in an inclined square cavity with a hot wavy wall. Int. J. Heat Mass Transf. 50(9–10), 1683–1693 (2007). https://doi.org/10.1016/j.ijheatmasstransfer.2006.10.015

    Article  MATH  Google Scholar 

  4. 4.

    S. Morsli, A. Sabeur, M. El Ganaoui, Influence of aspect ratio on the natural convection and entropy generation in rectangular cavities with wavy-wall. Energy Procedia 139, 29–36 (2017). https://doi.org/10.1016/j.egypro.2017.11.168

    Article  Google Scholar 

  5. 5.

    M. Hatami, D. Jing, Optimization of Wavy Direct Absorber Solar Collector (WDASC) using Al2O3-water Nanofluid and RSM analysis. Appl. Therm. Eng. 121, 1040–1050 (2017). https://doi.org/10.1016/j.applthermaleng.2017.04.137

    Article  Google Scholar 

  6. 6.

    M.A. Mansour, M.A.Y. Bakier, “Free convection heat transfer in complex-wavy-wall enclosed cavity filled with nanofluid. Int. Commun. Heat Mass Transfer 44, 108–115 (2013). https://doi.org/10.1016/j.icheatmasstransfer.2013.02.015

    Article  Google Scholar 

  7. 7.

    M. Esmaeilpour, M. Abdollahzadeh, “Free convection and entropy generation of nanofluid inside an enclosure with different patterns of vertical wavy walls. Int. J. Therm. Sci. 52, 127–136 (2012). https://doi.org/10.1016/j.ijthermalsci.2011.08.019

    Article  Google Scholar 

  8. 8.

    C.C. Cho, C.L. Chen, C.K. Chen, Natural convection heat transfer performance in complex-wavy-wall enclosed cavity filled with nanofluid. Int. J. Therm. Sci. 60, 255–263 (2012). https://doi.org/10.1016/j.ijthermalsci.2012.05.001

    Article  Google Scholar 

  9. 9.

    C.C. Cho, C.H. Chiu, C.Y. Lai, Natural convection and entropy generation of Al2O3–water nanofluid in an inclined wavy-wall cavity. Int. J. Heat Mass Transf. 97, 511–520 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.01.078

    Article  Google Scholar 

  10. 10.

    C.C. Cho, Influence of magnetic field on natural convection and entropy generation in Cu–water nanofluid-filled cavity with wavy surfaces. Int. J. Heat Mass Transf. 101, 637–647 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.05.044

    Article  Google Scholar 

  11. 11.

    M. Nikfar, M. Mahmoodi, Meshless local Petrov-Galerkin analysis of free convection of nanofluid in a cavity with wavy side walls. Eng. Anal. Boundary Elem. 36(3), 433–445 (2012). https://doi.org/10.1016/j.enganabound.2011.09.017

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    K.M. Shirvan, R. Ellahi, M. Mamourian, M. Moghiman, Effects of wavy surface characteristics on natural convection heat transfer in a cosine corrugated square cavity filled with nanofluid. Int. J. Heat Mass Transf. 107, 1110–1118 (2017). https://doi.org/10.1016/j.ijheatmasstransfer.2016.11.022

    Article  Google Scholar 

  13. 13.

    H.R. Ashorynejad, A. Shahriari, MHD natural convection of hybrid nanofluid in an open wavy cavity. Results Phys. 9, 440–455 (2018). https://doi.org/10.1016/j.rinp.2018.02.045

    ADS  Article  Google Scholar 

  14. 14.

    N.S. Bondareva, M.A. Sheremet, H.F. Oztop, N. Abu-Hamdeh, Heatline visualization of MHD natural convection in an inclined wavy open porous cavity filled with a nanofluid with a local heater. Int. J. Heat Mass Transf. 99, 872–881 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.04.055

    Article  Google Scholar 

  15. 15.

    M.A. Sheremet, I. Pop, A. Shenoy, Unsteady free convection in a porous open wavy cavity filled with a nanofluid using Buongiorno’s mathematical model. Int. Commun. Heat Mass Transfer 67, 66–72 (2015). https://doi.org/10.1016/j.icheatmasstransfer.2015.07.007

    Article  Google Scholar 

  16. 16.

    C.C. Cho, C.L. Chen, C.K. Chen, Mixed convection heat transfer performance of water-based nanofluids in lid-driven cavity with wavy surfaces. Int. J. Therm. Sci. 68, 181–190 (2013). https://doi.org/10.1016/j.ijthermalsci.2013.01.013

    Article  Google Scholar 

  17. 17.

    M. Mamourian, K.M. Shirvan, R. Ellahi, A.B. Rahimi, Optimization of mixed convection heat transfer with entropy generation in a wavy surface square lid-driven cavity by means of Taguchi approach. Int. J. Heat Mass Transf. 102, 544–554 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.056

    Article  Google Scholar 

  18. 18.

    S.K. Pal, S. Bhattacharyya, I. Pop, Effect of solid-to-fluid conductivity ratio on mixed convection and entropy generation of a nanofluid in a lid-driven enclosure with a thick wavy wall. Int. J. Heat Mass Transf. 127, 885–900 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.078

    Article  Google Scholar 

  19. 19.

    A. Al-Amiri, K. Khanafer, J. Bull, I. Pop, Effect of sinusoidal wavy bottom surface on mixed convection heat transfer in a lid-driven cavity. Int. J. Heat Mass Transf. 50, 1771–1780 (2007). https://doi.org/10.1016/j.ijheatmasstransfer.2006.10.008

    Article  MATH  Google Scholar 

  20. 20.

    C.C. Cho, Heat transfer and entropy generation of mixed convection flow in Cu-water nanofluid-filled lid-driven cavity with wavy surface. Int. J. Heat Mass Transf. 119, 163–174 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.090

    Article  Google Scholar 

  21. 21.

    C.C. Cho, Mixed convection heat transfer and entropy generation of Cu-water nanofluid in wavy-wall lid-driven cavity in presence of inclined magnetic field. Int. J. Mech. Sci. 151, 703–714 (2019). https://doi.org/10.1016/j.ijmecsci.2018.12.017

    Article  Google Scholar 

  22. 22.

    T. Kousksou, M. Mahdaoui, A. Ahmed, A. Ait Msaad, Melting over a wavy surface in a rectangular cavity heated from below. Energy 64, 212–219 (2014). https://doi.org/10.1016/j.energy.2013.11.033

    Article  Google Scholar 

  23. 23.

    M. Abdollahzadeh, M. Esmaeilpour, Enhancement of phase change material (PCM) based latent heat storage system with nano fluid and wavy surface. Int. J. Heat Mass Transf. 80, 376–385 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2014.09.007

    Article  Google Scholar 

  24. 24.

    F. Zemani, A. Sabeur-Bendehina, M. Boussoufi, Numerical investigation of natural convection in air filled cubical enclosure with hot wavy surface and partial partitions. Procedia Comput. Sci. 32, 622–630 (2014). https://doi.org/10.1016/j.procs.2014.05.469

    Article  Google Scholar 

  25. 25.

    B.V.R. Kumar, Shalini, Double diffusive natural convection in a doubly stratified wavy porous enclosure. Appl. Math. Comput. 171, 180–202 (2005). https://doi.org/10.1016/j.amc.2005.01.061

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    K. Khanafer, B. Al-Azmi, A. Marafie, I. Pop, Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure. Int. J. Heat Mass Transf. 52(7–8), 1887–1896 (2009). https://doi.org/10.1016/j.ijheatmasstransfer.2008.08.040

    Article  MATH  Google Scholar 

  27. 27.

    S. Bhardwaj, A. Dalal, S. Pati, Influence of wavy wall and non-uniform heating on natural convection heat transfer and entropy generation inside porous complex enclosure. Energy 79, 467–481 (2015). https://doi.org/10.1016/j.energy.2014.11.036

    Article  Google Scholar 

  28. 28.

    M.A. Sheremet, I. Pop, N. Bachok, Effect of thermal dispersion on transient natural convection in a wavy-walled porous cavity filled with a nanofluid: tiwari and Das’ nanofluid model. Int. J. Heat Mass Transf. 92, 1053–1060 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2015.09.071

    Article  Google Scholar 

  29. 29.

    C.C. Cho, Heat transfer and entropy generation of natural convection in nanofluid-filled square cavity with partially-heated wavy surface. Int. J. Heat Mass Transf. 77, 818–827 (2014). https://doi.org/10.1016/j.ijheatmasstransfer.2014.05.063

    Article  Google Scholar 

  30. 30.

    A.K. Hussein, S.H. Hussain, Heatline visualization of natural convection heat transfer in an inclined wavy cavities filled with nanofluids and subjected to a discrete isoflux heating from its left sidewall. Alexandria Engineering Journal 55(1), 169–186 (2016). https://doi.org/10.1016/j.aej.2015.12.014

    MathSciNet  Article  Google Scholar 

  31. 31.

    N.S. Gibanov, M.A. Sheremet, I. Pop, Natural convection of micropolar fluid in a wavy differentially heated cavity. J. Mol. Liq. 221, 518–525 (2016). https://doi.org/10.1016/j.molliq.2016.06.033

    Article  Google Scholar 

  32. 32.

    M. Song, D. Jing, “Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition. Int. J. Heat Mass Transf. 98, 758–767 (2016). https://doi.org/10.1016/j.ijheatmasstransfer.2016.03.063

    Article  Google Scholar 

  33. 33.

    I.V. Miroshnichenko, M.A. Sheremet, I. Pop, A. Ishak, Convective heat transfer of micropolar fluid in a horizontal wavy channel under the local heating. Int. J. Mech. Sci. 128–129, 541–549 (2017). https://doi.org/10.1016/j.ijmecsci.2017.05.013

    Article  Google Scholar 

  34. 34.

    F.B. Abdul Hasis, P.M. Mithun Krishna, G.P. Aravind, M. Deepu, S.R. Shine, Thermo hydraulic performance analysis of twisted sinusoidal wavy microchannels. Int. J. Therm. Sci. 128, 124–136 (2018). https://doi.org/10.1016/j.ijthermalsci.2018.02.018

    Article  Google Scholar 

  35. 35.

    R. Dormohammadi, M.F. Gord, A.E. Moghadam, M.H. Ahmadi, Heat transfer and entropy generation of the nanofluid flow inside sinusoidal wavy channels. J. Mol. Liq. 269, 229–240 (2018). https://doi.org/10.1016/j.molliq.2018.07.119

    Article  Google Scholar 

  36. 36.

    S. Harikrishnan, S. Tiwari, Effect of skewness on flow and heat transfer characteristics of a wavy channel. Int. J. Heat Mass Transf. 120, 956–969 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.120

    Article  Google Scholar 

  37. 37.

    C.Y. Cheng, Combined heat and mass transfer in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium with thermal and mass stratification. Int. Commun. Heat Mass Transfer 36, 351–356 (2009). https://doi.org/10.1016/j.icheatmasstransfer.2009.01.003

    Article  Google Scholar 

  38. 38.

    C.Y. Cheng, Natural convection heat transfer from an inclined wavy plate in a bidisperse porous medium. Int. Commun. Heat Mass Transfer 43, 69–74 (2013). https://doi.org/10.1016/j.icheatmasstransfer.2013.01.001

    Article  Google Scholar 

  39. 39.

    A.I. Alsabery, T. Tayebic, A.J. Chamkha, I. Hashim, Effect of rotating solid cylinder on entropy generation and convective heat transfer in a wavy porous cavity heated from below. Int. Commun. Heat Mass Transfer 95, 197–209 (2018). https://doi.org/10.1016/j.icheatmasstransfer.2018.05.003

    Article  Google Scholar 

  40. 40.

    A.I. Alsabery, M.A. Sheremet, A.J. Chamkha, I. Hashim, Impact of nonhomogeneous nanofluid model on transient mixed convection in a double lid-driven wavy cavity involving solid circular cylinder. Int. J. Mech. Sci. 150, 637–655 (2019). https://doi.org/10.1016/j.ijmecsci.2018.10.069

    Article  Google Scholar 

  41. 41.

    A.S. Dogonchi, I. Hashim, Heat transfer by natural convection of Fe3O4-water nanofluid in an annulus between a wavy circular cylinder and a rhombus. Int. J. Heat Mass Transf. 130, 320–332 (2019). https://doi.org/10.1016/j.ijheatmasstransfer.2018.10.086

    Article  Google Scholar 

  42. 42.

    I. Hashim, A.I. Alsabery, M.A. Sheremet, A.J. Chamkha, Numerical investigation of natural convection of Al2O3-water nanofluid in a wavy cavity with conductive inner block using Buongiorno’s two-phase model. Adv. Powder Technol. 30(2), 399–414 (2019). https://doi.org/10.1016/j.apt.2018.11.017

    Article  Google Scholar 

  43. 43.

    A. Mezrhab, H. Bouali, Modeling of combined radiative and convective heat transfer in an enclosure with a heat-generating conducting body. Int. J. Comput. Methods 2(3), 431–450 (2005). https://doi.org/10.1142/s0219876205000521

    Article  MATH  Google Scholar 

  44. 44.

    S. Saravanan, C. Sivaraj, Combined thermal radiation and natural convection in a cavity containing a discrete heater: effects of nature of heating and heater aspect ratio. Int. J. Heat Fluid Flow 66, 70–82 (2017). https://doi.org/10.1016/j.ijheatfluidflow.2017.05.004

    Article  Google Scholar 

  45. 45.

    A. Mezrhab, M.A. Moussaoui, H. Naji, Lattice Boltzmann simulation of surface radiation and natural convection in a square cavity with an inner cylinder. J. Phys. D Appl. Phys. 41(11), 115502 (2008). https://doi.org/10.1088/0022-3727/41/11/115502

    ADS  Article  Google Scholar 

  46. 46.

    M. Yousaf, S. Usman, Natural convection heat transfer in a square cavity with sinusoidal roughness elements. Int. J. Heat Mass Transf. 90, 180–190 (2015). https://doi.org/10.1016/j.ijheatmasstransfer.2015.06.049

    Article  Google Scholar 

  47. 47.

    E.L.M. Padilla, A. Silveira-Neto, Large-eddy simulation of transition to turbulence in natural convection in a horizontal annular cavity. Int. J. Heat Mass Transf. 51(13–14), 3656–3668 (2008). https://doi.org/10.1016/j.ijheatmasstransfer.2007.07.025

    Article  MATH  Google Scholar 

  48. 48.

    Z.T. Yu, X. Xu, Y.C. Hu, L.W. Fan, K.F. Cen, Unsteady natural convection heat transfer from a heated horizontal circular cylinder to its air-filled coaxial triangular enclosure. Int. J. Heat Mass Transf. 54(7–8), 1563–1571 (2011). https://doi.org/10.1016/j.ijheatmasstransfer.2010.11.032

    Article  MATH  Google Scholar 

  49. 49.

    T. Fusegi, B. Farouk, Laminar and turbulent natural convection– radiation interactions in a square enclosure filled with a nongray gas. Numer. Heat Transfer, Part A 15(3), 303–322 (1989). https://doi.org/10.1080/10407788908944690

    ADS  Article  MATH  Google Scholar 

  50. 50.

    G. Desrayaud, G. Lauriat, Unsteady confined buoyant plumes. J. Fluid Mech. 252, 617 (1993). https://doi.org/10.1017/s002211209300391x

    ADS  Article  Google Scholar 

  51. 51.

    K. Lari, M. Baneshi, S.A. Gandjalikhan Nassab, A. Komiya, S. Maruyama, Combined heat transfer of radiation and natural convection in a square cavity containing participating gases. Int. J. Heat Mass Transf. 54(23–24), 5087–5099 (2011). https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.026

    Article  MATH  Google Scholar 

  52. 52.

    L. El Moutaouakil, M. Boukendil, Z. Zrikem, A. Abdelbaki, Numerical analysis of natural convection with thermal radiation in a partially heated 3D cavity. Int. J. Thermophys. 40(11), 1–22 (2019). https://doi.org/10.1007/s10765-019-2569-2

    Article  Google Scholar 

  53. 53.

    L. Eloutaouakil, Z. Zrikem, A. Abdelbaki, Interaction of surface radiation with laminar and turbulent natural convection in tall vertical cavities: analysis and heat transfer correlations. Heat Transfer Eng. 36(17), 1472–1484 (2015). https://doi.org/10.1080/01457632.2015.1010934

    ADS  Article  Google Scholar 

  54. 54.

    H.S. Choi, K. Suzuki, Large eddy simulation of turbulent flow and heat transfer in a channel with one wavy wall. Int. J. Heat Fluid Flow 26(5), 681–694 (2005).

    Article  Google Scholar 

  55. 55.

    W.S. Fu, K.R. Huang, C.X. Lo, W.H. Wang, An investigation of natural convection in a three dimensional square wavy channel. Int. Commun. Heat Mass Transfer 66, 122–132 (2015). https://doi.org/10.1016/j.icheatmasstransfer.2015.05.012

    Article  Google Scholar 

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El Moutaouakil, L., Boukendil, M., Zrikem, Z. et al. Natural Convection and Surface Radiation Heat Transfer in a Square Cavity with an Inner Wavy Body. Int J Thermophys 41, 109 (2020). https://doi.org/10.1007/s10765-020-02688-7

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Keywords

  • Amplitude
  • Average radius
  • Inner and outer surface emissivities
  • Surface radiation
  • Undulations
  • Wavy surface