ANNALI DELL'UNIVERSITA' DI FERRARA

, Volume 53, Issue 2, pp 333–359 | Cite as

Rigorous upscaling of the infinite adsorption rate reactive flow under dominant Peclet number through a pore

Article

Abstract

In this paper we present a rigorous derivation of the effective model for enhanced diffusion through a narrow and long 2D pore. The analysis uses the anisotropic singular perturbation technique. Starting point is a local pore scale model describing the transport by convection and diffusion of a reactive solute. The solute particles undergo an adsorption process at the lateral tube boundary, with high adsorption rate. The transport and reaction parameters are such that we have large, dominant Peclet number with respect to the ratio of characteristic transversal and longitudinal lengths (the small parameter \(\varepsilon\)). We give a formal derivation of the model using the anisotropic multiscale expansion with respect to \(\varepsilon\) . Error estimates for the approximation of the physical solution, by the upscaled one, are presented in the energy norm as well as in L and L1 norms with respect to the space variable. They give the approximation error as a power of \(\varepsilon\) and guarantee the validity of the upscaled model through the rigorous mathematical justification of the effective behavior for small \(\varepsilon\).

Keywords

Taylor’s dispersion Large Peclet number Singular perturbation Surface chemical reaction Large Damkohler number 

Mathematics Subject Classification (2000)

35B25 92E20 76F25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aris R. (1956). On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Sect. A. 235: 67–77 CrossRefGoogle Scholar
  2. 2.
    Bourgeat A., Jurak M. and Piatnitski A.L. (2003). Averaging a transport equation with small diffusion and oscillating velocity. Math. Methods Appl. Sci. 26: 95–117 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Grenier E. and Rousset F. (2001). Stability of one-dimensional boundary layers by using Green’s functions. Commun. Pure Appl. Math. LIV: 1343–1385 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Knabner P., van Duijn C.J., Hengst S. (1995). An analysis of crystal dissolution fronts in flows through porous media. Part 1: compatible boundary conditions. Adv. Water Resour. 18: 171–185 CrossRefGoogle Scholar
  5. 5.
    Mercer G.N. and Roberts A.J. (1990). A centre manifold description of contaminant dispersion in channels with varying flow profiles. SIAM J. Appl. Math. 50: 1547–1565 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Mikelić A., Devigne V., van Duijn C.J. (2006). Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers. SIAM J. Math. Anal. 38(4): 1262–1287 CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Oleinik O.A. and Iosif’jan G.A. (1981). On the behavior at infinity of solutions of second order elliptic equations in domains with noncompact boundary. Math. USSR Sbornik 40: 527–548 CrossRefGoogle Scholar
  8. 8.
    Paine M.A., Carbonell R.G. and Whitaker S. (1983). Dispersion in pulsed systems—I, heterogeneous reaction and reversible adsorption in capillary tubes. Chem. Eng. Sci. 38: 1781–1793 CrossRefGoogle Scholar
  9. 9.
    Pyatniskii A.I. (1984). Averaging singularly perturbed equation with rapidly oscillating coefficients in a layer. Math. USSR Sbornik 49: 19–40 CrossRefGoogle Scholar
  10. 10.
    Pyatniskii, A.I.: Personal communication (2005)Google Scholar
  11. 11.
    Rosencrans S. (1997). Taylor dispersion in curved channels. SIAM J. Appl. Math. 57: 1216–1241 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rubinstein J. and Mauri R. (1986). Dispersion and convection in porous media. SIAM J. Appl. Math. 46: 1018–1023 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Taylor G.I. (1953). Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. A 219: 186–203 CrossRefGoogle Scholar
  14. 14.
    van Duijn, C.J., Knabner, P., Schotting, R.J. (1996). An analysis of crystal dissolution fronts in flows through porous media. Part 2: Incompatible boundary conditions. CWI, Amsterdam Google Scholar
  15. 15.
    Knabner P. and Duijn C.J. (1997). Travelling wave behavior of crystal dissolution in porous media flow. Eur. J. Appl. Math. 8: 49–72 MATHGoogle Scholar
  16. 16.
    Pop I.S. and Duijn C.J. (2004). Crystal dissolution and precipitation in porous media: pore scale analysis. J. Reine Angew. Math. 577: 171–211 MATHMathSciNetGoogle Scholar
  17. 17.
    van Duijn, C.J., Mikelić, A., Pop, I.S., Rosier, C.: Effective dispersion equations for reactive flows with dominant Peclet and Damkohler numbers. CASA Report 07-20, TU Eindhoven, The Netherlands (June 2007). Under revision in “Advances in Chemical Engineering”Google Scholar
  18. 18.
    van Genuchten M.Th., Cleary R.W. (1979). Movement of solutes in soil: computer-simulated and laboratory results. In: Bolt, G.H. (eds) “Soil Chemistry B. Physico-Chemical Models, Chap. 10”. Developements in Soil Sciences 5B, pp 349–386. Elsevier, AmsterdamGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2007

Authors and Affiliations

  1. 1.Université de LyonLyonFrance
  2. 2.Institut Camille Jordan, UFR MathématiquesUniversité Lyon 1Lyon Cedex 07France
  3. 3.Laboratoire de mathématiques pures et appliquées “Joseph Liouville”Université du LittoralCalais CedexFrance

Personalised recommendations