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Analysis of wet pressing of paper: the three-phase model. Part I: constant air density

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Abstract

In this study, we consider a one-dimensional three-phase model describing wet pressing of paper. Part I is devoted to the simplified case in which air is assumed incompressible. In Part II we drop this assumption. The model is formulated in terms of water saturation and void ratio and it uses a material coordinate to describe spatial dependence. It also involves cross or matching conditions between the wet paper and the felt. In mathematical terms, we end up with a coupled system of equations: a nonlinear diffusion equation and a first order hyperbolic equation. We present some analytical observations to explain the essential behaviour of the model and we carry out numerical experiments using an upwind and a front tracking method.

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References

  1. Bear J. (1972) Dynamics of Fluids in Porous Media. Elsevier, New York

  2. Bežanović, D.: Mathematical modelling of wet paper pressing. PhD thesis, Eindhoven University of Technology (2005)

  3. Bežanović D., van Duijn C.J., Kaasschieter E.F. (2006) Analysis of paper pressing: the saturated one-dimensional case. J. Appl. Math. Mech. 86(1): 18–36

  4. Clague D.S., Kandhai B.D., Zhang R., Sloot P.M.A. (2000) Hydraulic conductivity of (un)bounded fibrous media using the lattice Boltzmann method. Phys. Rev. E. 61(1): 616–625

  5. van Duijn C.J., Mikelic A., de Pop I.S. (2002) Effective equations for two-phase flow with trapping on the micro scale. SIAM J. Appl. Math 62(5): 1531–1568

  6. van Duijn C.J., Molenaar J., de Neef M.J. (1995) The effect of capillary forces on immiscible two-phase flow in heterogeneous porous media. Transport Porous Media 21, 71–93

  7. Helmig R. (1997) Multiphase Flow and Transport Processes in the Subsurface. Springer-Verlag, Berlin

  8. Hiltunen K. Mathematical modelling for consolidation process in paper Machines. PhD thesis, University of Jyväskylä, Finland (1995)

  9. Kataja M., Hiltunen K., Timonen J. (1992) Flow of water and air in a compressible porous medium. A model of wet pressing of paper. J. Phys. D: Appl. Phys. 25, 1053–1063 (1992)

  10. Ladyzhenskaya O.A., Solonnikov V.A., Uraltceva N.N. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc. Transl. Math. Mono 23, Providence R.I. (1968)

  11. Leveque R.J. Numerical Methods for Conservation Laws. Birkhauser Verlag, (1999)

  12. Lewis R.W., Schrefler B.A. (1998) The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. John Wiley and Sons Ltd, Chichester

  13. Mulder B.M.P., Riepen M. Wet press modelling, a one-dimensional approach. Report to Netherlands Agency for Energy and Enivironment, project nr. 96.13.6.3201 (1994)

  14. Renardy M. (1996) On an equation describing the spreading of surfactants on thin films. Nonlinear Anal. 26, 1207–1219

  15. Renardy M.(1997) A degenerate parabolic-hyperbolic system modeling the spreading of surfactants. Siam J. Math. Anal. 28(5): 1048–1063

  16. Paulapuro H. (2001) Wet pressing-present understanding and future challenges. 12th Fundamental Research Symposium, Oxford

  17. Singh K.M. Mathematical analysis of the wet pressing of paper. PhD thesis, SUNY College of Environmental Science and Forestry, Syracuse, New York (1994)

  18. Smoller J. (1980) Shock Waves and Reaction-diffuion Equations. Springer-Verlag, New York

  19. Ta-tsien L., Wen-Tzu Y., We-shi S. (1981) Boundary value problems and free boundary problems for quasilinear hyperbolic-parabolic coupled systems. Technical Summary Report 2273, University of Wisconsin, September (1981)

  20. Velten K., Best W. (2002) Rolling of unsaturated porous materials: evolution of fully saturated zone. Phys. Rev. E 62 (3): 3891–3899

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Correspondence to D. Bežanović.

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Bežanović, D., van Duijn, C.J. & Kaasschieter, E.F. Analysis of wet pressing of paper: the three-phase model. Part I: constant air density. Transp Porous Med 67, 93–113 (2007). https://doi.org/10.1007/s11242-006-9002-6

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Keywords

  • Paper pressing
  • Parabolic-hyperbolic system
  • Cross conditions
  • Upwind and front tracking method