A heat equation for freezing processes with phase change: stability analysis and applications

ABSTRACT

In this work, the stability properties as well as possible applications of a partial differential equation (PDE) with state-dependent parameters are investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (potential) Burgers’ equation. We show that for certain forms of coefficient functions, the PDE converges to a stationary solution given by (fixed) boundary conditions that make physical sense. These boundary conditions are either symmetric or asymmetric of Dirichlet type. Furthermore, we present an observer design based on the PDE model for estimation of inner-domain temperatures in block-frozen fish and for monitoring freezing time. We illustrate the results with numerical simulations.

Keywords:

• Distributed parameter systems
• stability analysis
• observer design
• freezing process

Acknowledgements

The authors wish to thank the anonymous referees for their highly appreciated comments, which helped in improving the comprehensibility and readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The standard EN 60584 defines three accuracy classes, where the first allows deviations of ±1.5K

Additional information

Funding

J. Leth was supported by the Southern Danish Growth Forum and the European Regional Development Fund, under the project ‘Smart & Cool’. C.J. Backi was supported by the Norwegian Research Council as part of the project DANTEQ (Development and assessment of novel technologies improving the fishing operation and on board processing with respect to environmental impact and fish quality) – Project no. 199447/I10.