### Abstract

The packaging industry wants to produce a foil for food packaging purposes,
which is transparent and lets very little oxygen pass. To accomplish this they add
a scavenger material to the foil which reacts with the oxygen that diffuses through
the foil. We model this process by a system of partial differential equations: a
reaction-diffusion equation for the oxygen concentration and a reaction equation for
the scavenger concentration. A probabilistic background of this model is given and
different methods are used to get information from the model. Homogenization theory
is used to describe the influence of the shape of the scavenger droplets on the oxygen
flux, an argument using the Fourier number of the foil leads to insight into the
dependency on the position of the scavenger and a method via conformal mappings is
proposed to find out more about the role of the size of the droplet. Also simulations
with Mathematica were done, leading to comparisons between different placements
and shapes of the scavenger material in one- and two-dimensional foils.

Original language | English |
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Title of host publication | Mathematics in Industry (Proceedings 55th European Study Group Mathematics with Industry, ESGI55/SWI2006, Eindhoven, The Netherlands, January 30-February 3, 2006) |

Editors | E.R. Fledderus, R.W. Hofstad, van der, E. Jochemsz, J. Molenaar, T.J.J. Mussche, M.A. Peletier, G. Prokert |

Place of Publication | Eindhoven |

Publisher | Technische Universiteit Eindhoven |

Pages | 69-100 |

Publication status | Published - 2006 |

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## Cite this

Mourik, van, S., Gennip, van, Y., Peletier, M. A., Hlod, A. V., Shcherbakov, V., Panhuis, in 't, P. H. M. W., ... Berg, van den, J. B. (2006). Catching gas with droplets : modelling and simulation of a diffusion-reaction process. In E. R. Fledderus, R. W. Hofstad, van der, E. Jochemsz, J. Molenaar, T. J. J. Mussche, M. A. Peletier, & G. Prokert (Eds.),

*Mathematics in Industry (Proceedings 55th European Study Group Mathematics with Industry, ESGI55/SWI2006, Eindhoven, The Netherlands, January 30-February 3, 2006)*(pp. 69-100). Eindhoven: Technische Universiteit Eindhoven.