Skip to main content
SpringerLink
Log in

A two-dimensional model of the pressing section of a paper machine including dynamic capillary effects

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

Paper production is a problem with significant importance for society; it is also a challenging topic for scientific investigation. This study is concerned with the simulation of the pressing section of a paper machine. A two-dimensional model is developed to account for the water flow within the pressing zone. A Richards-type equation is used to describe the flow in the unsaturated zone. The dynamic capillary pressure–saturation relation is adopted for the paper production process. The mathematical model accounts for the coexistence of saturated and unsaturated zones in a multilayer computational domain. The discretization is performed by the MPFA-O method. Numerical experiments are carried out for parameters that are typical of the production process. The static and dynamic capillary pressure–saturation relations are tested to evaluate the influence of the dynamic capillary effect.

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
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

References

  1. Metso Corporation (2010) http://www.metso.com/pulpandpaper

  2. Paper Academy (2011) http://www.paperacademy.net/855/paper-papermaking-manufacturing/paper-machine-press-section/

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

    MATH  Google Scholar 

  4. Bezanovic D, van Duijn CJ, Kaasschieter EF (2007) Analysis of wet pressing of paper: the three-phase model. Part 1: Constant air density. Report CASA 05-16 of the Department of Mathematics and Computer Science, Eindhoven, University of Technology

  5. Bezanovic D, van Duijn CJ, Kaasschieter EF (2007) Analysis of wet pressing of paper: the three-phase model. Part 2: Compressible air case. Transp Porous Med 67:171–187

    Article  Google Scholar 

  6. Hiltunen K (1995) Mathematical and numerical modelling of consolidation processes in paper machines. PhD Thesis, University of Jyväskylä, Finland

  7. Fitt AD, Howell PD, King JR, Please CP, Schwendeman DW (2002) Multiphase flow in a roll press nip. Eur J Appl Math 13:225–259

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 25:1053–1063

    Article  ADS  Google Scholar 

  9. Bermond C (1997) Establishing the scientific base for energy efficiency in emerging pressing and drying technologies. Appl Therm Eng 17(8–10):901–910

    Article  Google Scholar 

  10. Rief S (2005) Nonlinear flow in porous media. Dissertation, University of Kaiserslautern, Germany

  11. Rief S (2007) Modeling and simulation of the pressing section of a paper machine. Berichte des Fraunhofer ITWM, Nr 113

  12. Velten K, Best W (2000) Rolling of unsaturated porous materials: evolution of a fully saturated zone. Phys Rev E 62:3891–3899

    Article  ADS  Google Scholar 

  13. Bear J (1972) Dynamics of fluids in porous media. American Elsevier, New York

    MATH  Google Scholar 

  14. Bear J, Bachmat Y (1990) Introduction to modeling of transport phenomena in porous media. Kluwer, Dordrecht

    Book  Google Scholar 

  15. Bear J, Verruijt A (1987) Modeling groundwater flow and pollution. Reidel, Dordrecht

  16. Helmig R (1997) Multiphase flow and transport processes in the subsurface. Springer (Environmental Engineering), Berlin

  17. Broocks RH, Corey AT (1964) Hydraulic properties of porous media. In: Hydrology paper, vol 3. Colorado State University, Fort Collins

  18. Leverett MC (1941) Capillary behavior in porous solids. Trans AIME 142:152–169

    Google Scholar 

  19. Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898

    Article  Google Scholar 

  20. Barenblatt GI, Gilman AA (1987) Nonequilibrium counterflow capillary impregnation. J Eng Phys 52:335–339

    Article  Google Scholar 

  21. Barenblatt GI, Patzek TW, Silin DB (2002) The mathematical model of non-equilibrium effects in water–oil displacement. In: SPE/DOE 13th symposium on improved oil recovery, vol SPE 75169, Tusla, USA

  22. Bourgeat A, Panfilov M (1998) Effective two-phase flow through highly heterogeneous porous media: capillary nonequilibrium effects. Comput Geosci 2:191–215

    Article  MathSciNet  MATH  Google Scholar 

  23. Kalaydjian F (1992) Dynamic capillary pressure curve for water/oil displacement in porous media: theory vs. experiment. Soc Petrol Eng SPE 24813:491–506

    Google Scholar 

  24. Ross PJ, Smettem KRJ (2000) A simple treatment of physical nonequilibrium water flow in soils. Soil Sci Soc Am J 64:1926–1930

    Article  Google Scholar 

  25. Hassanizadeh SM, Celia MA, Dahle HK (2002) Dynamic effect in the capillary pressure–saturation relationship and its impacts on unsaturated flow. Vadose Zone J 1:38–57

    Google Scholar 

  26. Hassanizadeh SM, Gray WG (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv Water Resour 13:169–186

    Article  ADS  Google Scholar 

  27. Hassanizadeh SM, Gray WG (1993) Thermodynamic basis of capillary pressure in porous media. Water Resour Res 29:3389–3405

    Article  ADS  Google Scholar 

  28. Manthey S (2006) Two-phase flow processes with dynamic effects in porous media—parameter estimation and simulation. Dissertation, Institute of Hydraulic Engineering of Stuttgart, Germany

  29. Aavatsmark I (2002) An introduction to multipoint flux approximations for quadrilateral grids. Comput Geosci 6:405–432

    Article  MathSciNet  MATH  Google Scholar 

  30. Aavatsmark I (2007) Multipoint flux approximation methods for quadrilateral grids. In: 9th International forum on reservoir simulation, Abu Dhabi

  31. Edwards MG (2002) Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids. Comput Geosci 2:433–452

    Article  Google Scholar 

  32. Herbin R, Hubert F (2008) Benchmark on discretization schemes for anisotropic diffusion problems on general grids for anisotropic heterogeneous diffusion problems. In: Eymard R, Hrard J-M (eds) Finite volumes for complex applications V. Wiley, Hoboken, pp 659–692

  33. Eigestad GT, Klausen RA (2005) On the convergence of the multi-point flux approximation O-method: numerical experiments for discontinuous permeability. Wiley Interscience, New York

  34. Iliev O, Printsypar G, Rief S (2012) On mathematical modeling and simulation of the pressing section of a paper machine including dynamic capillary effects: one-dimensional model. Transp Porous Med 92(1):41–59. doi:10.1007/s11242-011-9890-y

    Article  MathSciNet  Google Scholar 

  35. Jewett K, Ceckler W, Busker L, Co A (1980) Computer model of a transversal flow nip. AIChE symposium series, New York, vol 76, pp 59–70 (200)

  36. Eymard R, Gallouet T, Herbin R (2006) Finite volume methods. An update of the preprint no. 97-19 du LATP, UMR 6632, Marseille, September 1997

  37. Deuflhard P (2004) Newton methods for nonlinear problems. Affine invariance and adaptive algorithms. Computational mathematics. Springer, New York

  38. Kelley CT (1995) Iterative methods for linear and nonlinear equations. Fundamental algorithms for numerical calculations. SIAM, Philadelphia

  39. Printsypar G (2012) Mathematical modeling and simulation of two-phase flow in porous media with application to the pressing section of a paper machine. Dissertation, University of Kaiserslautern, Germany

  40. Beck D (1983) Fluid pressure in a press nip: measurements and conclusions. In: Engineering conference proceedings, TAPPI, Atlanta, GA, pp 475–487

Download references

Acknowledgments

The authors would like to thank our industrial partner, Voith Paper Fabrics GmbH&Co. KG at Heidenheim for the interesting discussions and for the experimental data, which allowed us to perform the experiments with realistic sets of parameters.

Author information

Author notes

  1. Galina Printsypar

    Present address: Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

Authors and Affiliations

  1. Department of Flow and Material Simulation, Fraunhofer Institute for Industrial Mathematics (ITWM), Kaiserslautern, Germany

    Oleg Iliev, Galina Printsypar & Stefan Rief

  2. Technical University of Kaiserslautern, Kaiserslautern, Germany

    Oleg Iliev & Galina Printsypar

Authors
  1. Oleg Iliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Galina Printsypar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Rief
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Galina Printsypar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iliev, O., Printsypar, G. & Rief, S. A two-dimensional model of the pressing section of a paper machine including dynamic capillary effects. J Eng Math 83, 81–107 (2013). https://doi.org/10.1007/s10665-012-9619-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-012-9619-0

Keywords

Search

Navigation