The Espresso Coffee Problem

  • Antonio Fasano
  • F. Talamucci
  • M. Petracco
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


We review the results of a long research project on the espresso coffee brewing process, carried out jointly by the industrial mathematics research group at the Department of Mathematics “U. Dini” of the University of Florence and the Italian company illycaffè s. p. a. (Trieste).

We describe the main experimental steps of the research and present the mathematical models developed in order to interpret the data correctly. The models are of increasing complexity, the first being confined to the mechanical phenomena (experiments performed with cold water), while the most comprehensive includes the influence of dissolution. Particular emphasis is put on the fact that the process deviates significantly from usual filtration in standard porous media, although the classical Darcy’s law is assumed as the fundamental flow mechanism.


Porous Medium Fine Particle Free Boundary Injection Pressure Removal Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baldini, G., Filtrazione non lineare di un fluido attraverso un mezzo poroso deformabile, Thesis, University of Florence, Italy, 1992.Google Scholar
  2. 2.
    Baldini, G., and Petracco, M., Models for water percolation during the preparation of espresso coffee, 7th ECMI Conference, edited by A. Fasano and M. Primicerio, Teubner, Stuttgart, 131–8 (1994).Google Scholar
  3. 3.
    Bandini, S., Illy, E., Simone, C., and Suggi Liverani, and F., A computational model based on the reaction-diffusion machine to simulate transportation phenomena: The case of coffee percolation, in Cellular Automata: Research towards Industry, Proc. ACRI’98, edited by S. Bandini, R. Serra, and F. Suggi Liverani, Springer Verlag (1998).CrossRefGoogle Scholar
  4. 4.
    Bear, J.Dynamics of Fluids in Porous Media, Elsevier (1972).Google Scholar
  5. 5.
    Bertaccini, D., Simulation of a filtration in a deformable porous medium. A numerical approach, Nonlinear Analysis, Theory e4 Applications, 30, 1, 663–8 (1997) (Proc. 2nd World Congress of Nonlinear Analysis).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Borsani, C., Cattaneo, G., De Mattei, V., Jocher, U., and Zampini, B., 2D and 3D lattice gas technique for fluid-dynamics simulation, in S. Bandini, R. Serra, and F. Suggi Liverani, Cellular Automata: Research Towards Industry, Proc. ACRI’98, edited by Springer-Verlag (1998).Google Scholar
  7. 7.
    Bullo, T., and Illy, E., Considérations sur le procédé d’extractionCafé, Cacao, 7, 4, 395–9 (1963).Google Scholar
  8. 8.
    Chadam, J., Chen, X., Comparini, E., and Ricci, R., Travelling wave solutions of a reaction-infiltration problem and a related free boundary problem, European J. Appl. Math., 5, 255–66 (1994).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chadam, J., Chen, X., Gianni, E., and Ricci, R., A reaction infiltration problem: Existence, uniqueness and regularity of solutions in two space dimensions, Math. Models Methods and Appl. Sci., 5, 599–618 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chadam, J., Hoff, D., Merino, E., Ortoleva, P., and Sem, A., Reactive infiltration instabilities, IMA J. Appl. Math., 36, 207–20 (1987).CrossRefGoogle Scholar
  11. 11.
    Chadam, J., Peirle, A., and Sem, A., Weakly nonlinear stability of reaction-infiltration interfaces, SIAM J. Appl. Math., 48, 1362–78 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Comparini, E., and Mannucci, P., Penetration of a wetting front in a porous medium interacting with the flow, Nonlinear Differential Equations and Applications, 4, 425–38 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Comparini, E., and Ughi, M., On the existence of shock propagation in a flow through deformable porous media. Preprint of the Mathematics Department of the University of Florence, 8, 1999. Submitted to Interfaces and Free Boundaries.Google Scholar
  14. 14.
    Comparini, E., and Ughi, M., Shock propagation in a one-dimensional flow through deformable porous media. Preprint of the Mathematics Department of the University of Florence, 7, 1999. Submitted to Interfaces and Free Boundaries.Google Scholar
  15. 15.
    Fasano, A., The penetration of a wetting front through a porous medium accompanied by the dissolution of a substance, in International Congress on Math Modelling of Flow in Porous Media, edited by A. Bourgeat et al., World Scientific, 183–95 (1995).Google Scholar
  16. 16.
    Fasano, A., Some nonstandard one-dimensional filtration problems, The Bulletin of the Faculty of Education, 44, Chiba University (1996).Google Scholar
  17. 17.
    Fasano, A., and Mikelic, A., On the filtration through porous media with partially soluble permeable grains, to appear. in NoDEA.Google Scholar
  18. 18.
    Fasano, A., Mikelic, A., and Primicerio, M., Homogenization of flows through porous media with permeable grains, Adv. Math. Sci. Appl., 8, 1–31 (1998).MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fasano, A., and Primicerio, M., Flows through saturated mass exchanging porous media under high pressure gradients, Proc. of Calculus of Variations, Applications and Computations, edited by C. Bandle et al., Pitman Res. Notes Math. Series, 326 (1994).Google Scholar
  20. 20.
    Fasano, A., and Primicerio, M., Mathematical models for filtration through porous media interacting with the flow, in Nonlinear Mathematical Problems in Industry, I. M. Kawarada, N. Kenmochi, and N. Yanagihara, Math. Sci. Appl., 1, 61–85, edited by Gakkotosho, Tokyo.Google Scholar
  21. 21.
    Fasano, A., Primicerio, M., and Watts, A., On a filtration problem with flow-induced displacement of fine particles, in Boundary Control and Boundary Variation, edited by J. P. Zolesio, M. Dekker, New York (1994), 205–32.Google Scholar
  22. 22.
    Fasano, A., and Talamucci, F., A comprehensive mathematical model for a multi-species flow through ground coffee, to appear in SIAM J. Math. Anal. Google Scholar
  23. 23.
    Fasano, A., and Tani, P., Penetration of a wetting front in a porous medium with time dependent hydraulic parameters, in Nonlinear Problems in Applied Mathematics, edited by K. Cooke et al., SIAM (1995).Google Scholar
  24. 24.
    Gianni, R., and Ricci, R., Existence and uniqueness for a reaction-diffusion problem in infiltration, Ann. Mat. Pura Appl., 168, 373–94 (1995).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Knabner, P., Van Duijn, C., and Nengst, S., An analysis of crystal dissolution fronts in flows through porous media. Part I: Compatible boundary conditions, Advances in Water Resources, 18, 3, 171–85 (1995).CrossRefGoogle Scholar
  26. 26.
    Pawell, A., and Krannich K-D, Dissolution effects in transport in porous media, arising in transport in porous media, SIAM J. Appl. Math., 56, 1, 89–118 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Petracco, M., Espresso coffee brewing dynamics: Development of mathematical and computational models, 15éme Colloque Scient. Internat. sur le Café, Associat. Scientif. Internat. du Café, Paris (1993).Google Scholar
  28. 28.
    Talamucci, F., Analysis of coupled heat-mass transport in freezing saturated soils, Surveys on Mathematics for Industry, 7, Springer-Verlag, 93–139, (1997).MathSciNetzbMATHGoogle Scholar
  29. 29.
    Talamucci, F., Flow through a porous medium with mass removal and diffusionNonlinear Differential Equations and Applications, 5, 427–44 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Van Duijn, C., and Knabner, P., Solute transport in porous media with equilibrium and non-equilibrium multiplicity adsorption in travelling waves, J. Reine Angew. Math., 1–49 (1991).Google Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Antonio Fasano
    • 1
  • F. Talamucci
    • 1
  • M. Petracco
    • 2
  1. 1.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly
  2. 2.illycaffè s.p.aTriesteItaly

Personalised recommendations