Existence, Uniqueness and Numerical Modeling of Wine Fermentation Based on Integro-Differential Equations

TL;DR

This paper develops a detailed mathematical model for white wine fermentation using integro-differential equations, analyzing its theoretical properties and numerical solutions, and comparing it with simpler models to understand initial cell mass effects.

Contribution

Introduces a novel, highly nonlinear integro-differential model for wine fermentation, along with existence, uniqueness analysis, and a numerical scheme for its solution.

Findings

The detailed model captures initial cell mass distribution effects.

Numerical results show smaller differences between models than expected.

The model provides insights useful for mathematicians, winemakers, and engineers.

Abstract

Predictive modeling is the key factor for saving time and resources with respect to manufacturing processes such as fermentation processes arising e.g.\ in food and chemical manufacturing processes. According to Zhang et al. (2002), the open-loop dynamics of yeast are highly dependent on the initial cell mass distribution. This can be modeled via population balance models describing the single-cell behavior of the yeast cell. There have already been several population balance models for wine fermentation in the literature. However, the new model introduced in this paper is much more detailed than the ones studied previously. This new model for the white wine fermentation process is based on a combination of components previously introduced in literature. It turns it into a system of highly nonlinear weakly hyperbolic partial/ordinary integro-differential equations. This model becomes very challenging from a theoretical and numerical point of view. Existence and uniqueness of solutions to a simplified version of the introduced problem is studied based on semigroup theory. For its numerical solution a numerical methodology based on a finite volume scheme combined with a time implicit scheme is derived. The impact of the initial cell mass distribution on the solution is studied and underlined with numerical results. The detailed model is compared to a simpler model based on ordinary differential equations. The observed differences for different initial distributions and the different models turn out to be smaller than expected. The outcomes of this paper are very interesting and useful for applied mathematicians, winemakers and process engineers.