Effect of instantaneous change of surface temperature and density on an unsteady liquid–vapour front in a porous medium

Abstract

This article presents a comprehensive analysis of time dependent condensation model embedded in a porous medium with variations in liquid–vapour densities. Both similarity and asymptotic solutions for the unsteady liquid–vapour phase change front are obtained with the manifestation of various pertinent parameters. The obtained results are compared which congregate well as depicted clearly in graphs. Results indicate that with different diffusivity and contrast ratios, the similarity front parameter is found to be gradually declining with variation in a density ratio. We have shown for the condensation process, the ratio of sensible to latent heat is independent of time and is equal to the half of the Stefan number of the liquid phase.

This is a preview of subscription content, access via your institution.

References

  1. Bear, J., Buchlin, J.-M. 1981. Modelling and Applications of Transport Phenomena in Porous Media. Kluwer Academic Publishers.

  2. Beckett, G., MacKenzie, J. A., Robertson, M. L. 2001. A moving mesh finite element method for the solution of two-dimensional Stefan problems. J Comput Phys, 168: 500–518.

    MathSciNet  Article  Google Scholar 

  3. Bernoff, A. J., Witelski, T. P. 2010. Stability and dynamics of self-similarity in evolution equations. J Eng Math, 66: 11–31.

    MathSciNet  Article  Google Scholar 

  4. Bodvarsson, G., Pruess, K., Lippmann, M. 1986. Modeling of geothermal systems. J Petrol Technol, 38: 1007–1021.

    Article  Google Scholar 

  5. Bonacina, C., Comini, G., Fasano, A., Primicerio, M. 1973. Numerical solution of phase-change problems. Int J Heat Mass Tran, 16: 1825–1832.

    Article  Google Scholar 

  6. Carey, V. P. 2007. Liquid-Vapor Phase-Change Phenomena. Taylor & Francis, Inc.

  7. Carslaw, H. S., Jaeger, J. C. 1959. Conduction of Heat in Solids. Clarendon Press.

  8. Chiareli, A. O. P., Huppert, H. E., Worster, M. G. 1994. Segregation and flow during the solidification of alloys. J Cryst Growth, 139: 134–146.

    Article  Google Scholar 

  9. Date, A. W. 1991. A strong enthalpy formulation for the Stefan problem. Int J Heat Mass Tran, 34: 2231–2235.

    Article  Google Scholar 

  10. Douglas, J. 1957. A uniqueness theorem for the solution of a Stefan problem. Proc Am Math Soc, 8: 402–408.

    MathSciNet  Article  Google Scholar 

  11. Dutil, Y., Rousse, D. R., Salah, N. B., Lassue, S., Zalewski, L. 2011. A review on phase-change materials: Mathematical modeling and simulations. Renew Sust Energ Rev, 15: 112–130.

    Article  Google Scholar 

  12. Evans, G. W. 1951. A note on the existence of a solution to a problem of Stefan. Q Appl Math, 9: 185–193.

    MathSciNet  Article  Google Scholar 

  13. Gupta, S. C. 2003. The Classical Stefan Problem: Basic Concepts, Modelling and Analysis. Elsevier.

  14. Hager, J., Whitaker, S. 2000. Vapor–liquid jump conditions within a porous medium: Results for mass and energy. Transport Porous Med, 40: 73–111.

    MathSciNet  Article  Google Scholar 

  15. Harris, K. T., Haji-Sheikh, A., Agwu Nnanna, A. G. 2001. Phase-change phenomena in porous media—a non-local thermal equilibrium model. Int J Heat Mass Tran, 44: 1619–1625.

    Article  Google Scholar 

  16. Khan, Z. H. 2014. Transition to instability of liquid–vapour front in a porous medium cooled from above. Int J Heat Mass Tran, 70: 610–620.

    Article  Google Scholar 

  17. Khan, Z. H., Pritchard, D. 2013. Liquid–vapour fronts in porous media: Multiplicity and stability of front positions. Int J Heat Mass Tran, 61: 1–17.

    Article  Google Scholar 

  18. Khan, Z. H., Pritchard, D. 2015. Anomaly of spontaneous transition to instability of liquid–vapour front in a porous medium. Int J Heat Mass Tran, 84: 448–455.

    Article  Google Scholar 

  19. Lunardini, V. J. 1981. Heat Transfer in Cold Climates. Van Nostrand Reinhold Company.

  20. Masur, L. J., Mortensen, A., Cornie, J. A., Flemings, M. C. 1989. Infiltration of fibrous preforms by a pure metal: Part II. Experiment. Metall Trans A, 20: 2549–2557.

    Article  Google Scholar 

  21. Mattheij, R. M. M., Rienstra, S. W., ten Thije Boonkkamp, J. H. M. 2005. Partial Differential Equations: Modeling, Analysis, Computation. SIAM.

  22. Mortensen, A., Masur, L. J., Cornie, J. A., Flemings, M. C. 1989. Infiltration of fibrous preforms by a pure metal: Part I. Theory. Metall Trans A, 20: 2535–2547.

    Article  Google Scholar 

  23. Ochoa-Tapia, J. A., Whitaker, S. 1997. Heat transfer at the boundary between a porous medium and a homogeneous fluid. Int J Heat Mass Tran, 40: 2691–2707.

    Article  Google Scholar 

  24. Rubin, A., Schweitzer, S. 1972. Heat transfer in porous media with phase change. Int J Heat Mass Tran, 15: 43–60.

    Article  Google Scholar 

  25. Solomon, A. 1981. A note on the Stefan number in slab melting and solidification. Lett Heat Mass Trans, 8: 229–235.

    Article  Google Scholar 

  26. Torranc, K. E. 1986. Phase-change heat transfer in porous media. Heat Transfer, 1: 181–188.

    Google Scholar 

Download references

Acknowledgements

The corresponding author is profoundly grateful to the financial support of the National Natural Science Foundation of China (Grant Nos. 51709191, 51706149, and 51606130), Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (Grant No. ARES-2018-10), and State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University (Grant No. Skhl1803).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zafar Hayat Khan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khan, Z.H., Ahmad, R. & Sun, L. Effect of instantaneous change of surface temperature and density on an unsteady liquid–vapour front in a porous medium. Exp. Comput. Multiph. Flow 2, 115–121 (2020). https://doi.org/10.1007/s42757-019-0027-9

Download citation

Keywords

  • unsteady liquid–vapour front
  • porous medium
  • Stefan number
  • similarity and asymptotic solutions
  • density variations