Skip to main content
SpringerLink
Log in

Influence of the Prandtl Number on Wall-to-Fluid Thermal Transfer Rate in a Cubic Cavity

  • Published:
Flow, Turbulence and Combustion Aims and scope Submit manuscript

Abstract

The present paper describes a detailed qualitative and quantitative study of the impacts of the overall Prandtl number on the wall-to-fluid thermal transfer mechanisms in two-phase flows using CFD. The physical model consisted of a bubble inside a cubic cavity subjected to the gravity acceleration and to a non-isothermal field. The variables studied in the present investigation were the size of the bubble radius and the overall Prandtl number of flow. The variations of the overall Prandtl number of the flow and the bubble size were evaluated in order to understand their impact on the spatial mean Nusselt number using a computational fluid dynamics model. The bubble size was controlled modifying the bubbles’ initial radius in the computational simulations and the overall Prandtl number was computed according to the Prandtl number of each phase and its volume fraction. Several computational simulations were performed and the numerical model employed in these simulations were validated with previous numerical results in the literature. The present paper reported the influence of the overall Prandtl number on regulating the thermal transfer rate in the classical problem of a differentially heated cavity. According to the numerical results, the introduction of a dispersed phase in single-phase problems may increase or reduce the thermal transfer rate, depending on the overall Prandtl number. The influence of the bubble size has also played an important role in the variations of the thermal transfer rate, since the bubble movement in the domain was modified according to to the size of this radius, changing the distance between the walls and the dispersed phase. Finally, the generally accepted hypothesis elaborated from the literature regarding the increase of the thermal transfer rate due to the introduction of bubbles in single-phase flows has failed in the cubic cavity configuration from the present research.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Deen, N., Kuipers, J.: Direct numerical simulation of wall-to liquid heat transfer in dispersed gas-liquid two-pahse flow using a volume of fluid approach. Chem. Eng. Sci. 102, 268–282 (2013). https://doi.org/10.1016/j.ces.2013.08.025

    Article  Google Scholar 

  2. Deckwer, W.D.: On the mechanism of heat transfer in bubble column reactor. Chem. Eng. Sci. 92, 1341–1346 (1980). https://doi.org/10.1016/0009-2509(80)85127-X

    Article  Google Scholar 

  3. Oresta, P., Verzicco, R., Lohse, D., Prosperetti, A.: Heat transfer mechanisms in bubbly Rayleigh-Bénard convection. Phys. Rev. 80, 259–293 (2007). https://doi.org/10.1103/PhysRevE.80.026304

    Google Scholar 

  4. Tamari, M., Nishikawa, K.: The stirring effect of bubbles upon the heat transfer to liquids. Heat Transfer – Japonese Res. 5, 31–44 (1976). https://doi.org/10.1299/kikai1938.33.87

    Google Scholar 

  5. Dabiri, S., Tryggvason, G.: Heat transfer in turbulent bubbly flow in vertical channels. Chem. Eng. Sci. 122, 106–113 (2015). https://doi.org/10.1016/j.ces.2014.09.006

    Article  Google Scholar 

  6. Bukhari, S., Siddiqui, M.: Characteristics of air and water velocity fields during natural convection. Heat Mass Transfer 43, 415–525 (2007). https://doi.org/10.1007/s00231-005-0072-8

    Article  Google Scholar 

  7. Chandra, A., Chhabra, R.P.: Effect of prandtl number on natural convection heat transfer from a heated semi-circular cylinder. Int. J. Mech. Aerosp. Ind. Mechatron. Manuf. Eng. 6, 106–113 (2012). https://doi.org/10.1016/j.ces.2014.09.006

    Google Scholar 

  8. Qiu, R., Wang, A., Jiang, T.: Lattice boltzmann method for natural convection with multicomponent and multiphase fluids in a two-dimensional square cavity. Canad. J. Chem. Eng. 92, 1121–1129 (2014). https://doi.org/10.1002/cjce.21950

    Article  Google Scholar 

  9. Wan, D., Patnaik, B., Wei, G.: A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numer. Heat Transfer 40, 199–228 (2001). https://doi.org/10.1080/104077901752379620

    Article  Google Scholar 

  10. Kizildag, D., Rodríquez, I., Oliva, A., Lehmkuhl, O.: Limits of the oberbeck-boussinesq approximation in a tall differentially heated cavity filled with water. Int. J. Heat Mass Transfer 68, 489–499 (2014). https://doi.org/10.1016/j.ijheatmasstransfer.2013.09.046

    Article  Google Scholar 

  11. White, F.M.: Viscous Fluid Flow. McGraw-Hill (1974)

  12. Wang, P., Zhang, Y., Guo, Z.: Numerical study of three-dimensional natural convection in a cubical cavity at high rayleigh numbers. Int. J. Heat Mass Transf. 113, 217–228 (2017). https://doi.org/10.1016/j.ijheatmasstransfer.2017.05.057

    Article  Google Scholar 

  13. Brackbill, J.U., Kothe, D.B., Zemach, C.: A continuum method for modeling surface-tension. J. Comput. Phys. 100, 335–354 (1992). https://doi.org/10.1016/0021-9991(92)90240-Y

    Article  MathSciNet  MATH  Google Scholar 

  14. Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968). https://doi.org/10.1090/s0025-5718-1968-0242392-2

    Article  MathSciNet  MATH  Google Scholar 

  15. Centrella, J.M., Wilson, J.R.: Planar numerical cosmology. II - the difference equations and numerical tests. Astrophys. J. Suppl. Series 54, 229–249 (1984). https://doi.org/10.1086/190927

    Article  Google Scholar 

  16. Germano, M., Piomelli, U., Moin, P., Cabot, W.H.: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A: Fluid Dyn. 3, 1760–1765 (1991)

    Article  MATH  Google Scholar 

  17. Lilly, D.: A proposed modification of the germano subgrid-scale closure method. Phys. Fluids A: Fluid Dyn. 4, 633 (1992)

    Article  Google Scholar 

  18. Akhtar, M., Kleis, S.: Boiling flow simulations on adaptive octree grids. Int. J. Multiphase Flow 53, 88–99 (2013). https://doi.org/10.1016/j.ijmultiphaseflow.2013.01.008

    Article  Google Scholar 

  19. Hirt, C.W., Nichols, B.D.: Volume of fluid (vof) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225 (1981). https://doi.org/10.1016/0021-9991(81)90145-5

    Article  MATH  Google Scholar 

  20. Wachem, B.G.M., Schouten, J.C.: Experimental validation of 3-d lagragian vof model: Bubble shape and rise velocity. AIChE J 48, 253–282 (2002). https://doi.org/10.1002/aic.690481205

    Google Scholar 

  21. Dennner, F., Wachem, B.G.M.: Fully-coupled balanced-force vof framework for arbitrary meshes with least-squares curvature evaluation from volume fractions. Numer. Heat Transfer Part B: Fund. 65, 218–255 (2014). https://doi.org/10.1080/10407790.2013.849996

    Article  Google Scholar 

  22. Barbi, F., Pivello, M.R., Villar, M.M., Serfaty, R., Roma, A.M., Silveira-Neto, A.: Numerical experiments of ascending bubbles for fluid dynamic force calculations. J. Braz. Soc. Mech. Sci. Eng. 40, 519–531 (2018). https://doi.org/10.1007/s40430-018-1435-7

    Article  Google Scholar 

  23. Krane, R.J., Jessee, J.: Some detailed field measurements for a natural convection flow in a vertical square enclosure. Proc. First ASME-JSME Thermal Eng. Joint Conf. 1, 323–329 (1983)

    Google Scholar 

  24. Padilla, E., Lourenço, M., Silveira-Neto, A.: Natural convection inside cubical cavities: Numerical solutions with two boundary conditions. J. Braz. Soc. Mech. Sci. Eng. 35, 275–283 (2013). https://doi.org/10.1007/s40430-013-0033-y

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support from Petrobras, CNPq, Fapemig and Capes. The authors are also grateful to the mechanical engineering graduate program from the Federal University of Uberlândia (UFU).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Federal University of Uberlândia, Av. João Naves de Ávila, 2121, Bloco 5P, Uberlândia, Minas Gerais, 38400-902, Brazil

    Bernardo Alan de Freitas Duarte & Aristeu da Silveira Neto

  2. Petrobras, Rio de Janeiro, 21941-915, Brazil

    Ricardo Serfaty

Authors
  1. Bernardo Alan de Freitas Duarte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo Serfaty
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aristeu da Silveira Neto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernardo Alan de Freitas Duarte.

Ethics declarations

Conflict of interests

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Freitas Duarte, B.A., Serfaty, R. & da Silveira Neto, A. Influence of the Prandtl Number on Wall-to-Fluid Thermal Transfer Rate in a Cubic Cavity. Flow Turbulence Combust 103, 345–367 (2019). https://doi.org/10.1007/s10494-019-00025-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10494-019-00025-z

Keywords

Search

Navigation