Advertisement

Transport in Porous Media

, Volume 28, Issue 1, pp 19–50 | Cite as

Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure

  • B. Goyeau
  • T. Benihaddadene
  • D. Gobin
  • M. Quintard
Article

Abstract

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.

momentum equation nonhomogeneous porous media evolving heterogeneities volume averaging closure problem. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Barrère, J.: 1990, Modélisation des ecoulements de Stokes et Navier- Stokes en milieu poreux, PhD thesis, Université de Bordeaux.Google Scholar
  2. Barrère, J., Gipouloux, O. and Whitaker, S.: 1992, On the closure problem for Darcy's law, Transport in Porous Media 7, 209–222.Google Scholar
  3. Bear, J.: 1972, Dynamics of Fluids in Porous Media, Elsevier, New York.Google Scholar
  4. Beckermann, C.: 1987, Melting and solidification of binary mixtures with double-diffusive convection in the melt, PhD thesis, Purdue University, West Lafayette.Google Scholar
  5. Beckermann, C and Viskanta, R.: 1988, Double-diffusive convection during dendritic solidification of a Binary mixture, Phys. Chem. Hydrodynamics 10, 195–213.Google Scholar
  6. Bennon, W. D. and Incropera, F. P.: 1987, A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems I. Model formulation, Int. J. Heat Mass Transfer 30(10), 2161–2170.Google Scholar
  7. Bensoussan, J., Lions, L. and Papanicolao, G.: 1987, Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam.Google Scholar
  8. Carbonell, R. G. and Whitaker, S.: 1984, Heat and Mass Transfer in Porous Media, Martinus Nijhoff, Dordrecht, pp. 121–198.Google Scholar
  9. Christenson, M. S., Bennon, W. D. and Incropera, F. P.: 1989, Solidification of an aqueous ammonium chloride solution in a sectangular cavity - II. Comparison of predicted and mesured results, Int. J. Heat Mass Transfer 32, 69–79.Google Scholar
  10. Cushman, J. H.: 1983, Multiphase transport equations: I General equation for macroscopic statistical, local, space-time, Transport Theory and Statistical Physic 12(1), 35–71.Google Scholar
  11. Cushman, H. H. and Ginn, T. R.: 1983 Non-local dispersion in media with continuous evolving scales of heterogeneity, Transport in Porous Media 13, 123–138.Google Scholar
  12. Ding, A. and Candela, D.: 1996, Probing nonlocal tracer dispersion in flows through random porous media, Phys. Rev. E 54(1), 656–660.Google Scholar
  13. Firdaous, M. and Guermond, J.: 1995, Sur l'homogénéisation des équations de Navier- Stokes à faible nombre de Reynolds, C. R. Acad. Sci. Paris Série I 320, 245–251.Google Scholar
  14. Flemings, M.: 1974, Solidification Processing, McGraw-Hill, New YorkGoogle Scholar
  15. Ganesan, S. and Poirier, D. R.: 1990, Conservation of mass and momentum for the flow of interdendritic liquid during solidification, Met. Trans. 21B, 173–181.Google Scholar
  16. Godrèche, C.: 1991, Solids Far from Equilibrium, Cambridge University Press.Google Scholar
  17. Gray, W. G.: 1975, A derivation of the equations for multi-phase transport, Chem. Eng. Sci. 3, 229–233.Google Scholar
  18. Heinrich, J. C., Felicelli, S., Nandapurkar, P. and Poirier, D. R.: 1989, Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection, Met. Trans. 20B, 883–891.Google Scholar
  19. Hills, R. N., Lopper, D. E. and Roberts, P. H.: 1983, A thermodymically consistent model of a mushy zone, Q. J. Mech. Appl. Math. 36, 505–539.Google Scholar
  20. Howes, F. W. and Whitaker, S.: 1985, The spatial averaging theorem revisited, Chem. Eng. Sci. 40, 1387–1392.Google Scholar
  21. Huppert, H. E.: 1990, The fluid mechanics of solidification, J. Fluid Mech. 212, 209–240.Google Scholar
  22. Koch, D. L. and Brady, J. L.: 1987, A non-local description of advection-diffusion with application to dispersion in porous media, J. Fluid Mech. 180, 387–403.Google Scholar
  23. Marle, C. M.: 1967, Ecoulements monophasiques en milieu poreux, Rev. Inst. Francais Pétrole, pp. 1471–1509.Google Scholar
  24. Marle, C.M.: 1982, On macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media, Int. J. Engng. Sci. 20, 643–662.Google Scholar
  25. Matheron, G.: 1982, Les variables régionalisées et leur estimation: une aplication de la théorie des fonctions aléatoires aux sciences de la nature, Masson, Paris.Google Scholar
  26. Mei, C. C. and Auriault, J. L.: 1991, The effect of weak inertia on flow through a porous media, J. Fluid Mech. 222, 647–663.Google Scholar
  27. Murakami, K., Shiraishi, A. and Okamoto, T.: 1983, Interdendritic fluid flow normal to primary dendrite-arms in cubic alloys, Acta Metal. 31, 1417–1424.Google Scholar
  28. Murakami, K., Shiraishi, A. and Okamoto, A. 1984, Fluid flow in interdendritic space in cubic alloys, Acta Metal. 32, 1423–1428.Google Scholar
  29. Nandapurkar, P., Poirier, D. R. and Heinrich, J. C.: 1991, Momentum equation for dendritic solidification, Numerical Heat Transfer 19A, 297–311.Google Scholar
  30. Nasser-Rafi, R., Deshmukh, R. and Poirier, D. R.: 1985, Flow of interdendritic liquid and permeability in Pb-20 Wt Pct Sn alloys, Met. Trans. 16A, 2263–2271.Google Scholar
  31. Ni, J. and Beckermann, C. 1991, A volume-averaged two-phase model for transport phenomena during solidification, Met. Trans. 22B, 349–361.Google Scholar
  32. Poirier, D. R.: 1987, Permeability for flow of interdendritic liquid in columnar-dendritic alloys, Met. Trans. 18B, 245–255.Google Scholar
  33. Prescott, P. J., Incropera, F. P. and Gaskell, D. R.: 1994, Convective transport phenomena and macrosegregation during solidification of binary metal alloy: II - Experiments and comparisons with numerical predictions, J. Heat Transfer 116, 742–749.Google Scholar
  34. Quintard, M. and Whitaker, S.: 1993, Transport in ordered and disordered porous media: Volume averaged equations, closure problems and comparison with experiment, Chem. Eng. Sci. 48, 2537–2564.Google Scholar
  35. Quintard, M. and Whitaker, S., 1994a, Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions, Transport in Porous Media 14, 163–177.Google Scholar
  36. Quintard, M. and Whitaker, S.: 1994b, Transport in ordered and disordered porous media II: Generalized volume averaging, Transport in Porous Media 14, 179–206.Google Scholar
  37. Quintard, M. and Whitaker, S.: 1994c, Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment, Transport in Porous Media 15, 31–49.Google Scholar
  38. Quintard, M. and Whitaker, S.: 1994d, Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems, Transport in Porous Media 15, 51–70.Google Scholar
  39. Quintard, M. and Whitaker, S.: 1994e Transport in ordered and disordered porous Media V: Geometrical results for two-dimensional systems, Transport in Porous Media 15, 183–196.Google Scholar
  40. Sanchez-Palencia, E. 1980, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Phys. 127, Springer, New York.Google Scholar
  41. Schwartz, L., 1987, Théorie des distributions, Hermann, Paris.Google Scholar
  42. Sinha, S.K., Sundararajan, T. and Garg, V. K.: 1992, Avariable property analysis of alloy solidification using the anisotropic porous medium approach, Int. J. Heat and Mass Transfer 35, 2865–2877.Google Scholar
  43. Slattery, J. C., 1967, Flow of viscoelastic fluids through porous media, AIChE J. 13, 1066–1071.Google Scholar
  44. Sternberg, S. P. K., Cushman, J. H. and Greenkorn, R. A., 1996, Laboratory observation of nonlocal dispersion, Transport in Porous Media 25(2), 35–151.Google Scholar
  45. Szekely, J. and Jassal, A. S.: 1978, An experimental and analytical study of the solidification of a binary dendritic system, Met. Trans. 9B, 389–398.Google Scholar
  46. Voller, V. R. and Prakash, C.: 1987, A fixed grid numerical modelling methodology for connectiondiffusion mushy region phase-change problems, Int. J. Heat Mass Transfer 30(8), 1709–1719.Google Scholar
  47. Whitaker, S.: 1969, Advances in theory of fluid motion in porous media, Ind. Eng. Chem. 12, 12–28.Google Scholar
  48. Whitaker, S.: 1986, Flow in porous media I: A theoretical derivation of Darcy's law, Transport in Porous Media, 1, 3–35.Google Scholar
  49. Whitaker, S.: 1996, The Forchheimer equation: A theoretical development, Transport in Porous Media 25, 27–61.Google Scholar
  50. Wodie, J. C. and Levy, T.: 1991, Correction non linéaire de la loi de Darcy, C.R. Acad. Sci. Paris Serie II 312, 157–161.Google Scholar
  51. Worster, M. G.: 1991, Natural convection in a mushy layer, J. Fluid Mech. 224, 335–359.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • B. Goyeau
    • 1
  • T. Benihaddadene
    • 1
  • D. Gobin
    • 1
  • M. Quintard
    • 2
  1. 1.Laboratoire FAST – URA CNRS 871Universités Paris VI et Pasis XIOrsay CedexFrance
  2. 2.Laboratoire Energétique et Phénomènes de Transfert, URA CNRS 873 ENSAM, Esplanade des Arts et MétiersTalence CedexFrance

Personalised recommendations